From ed9ec629d9ef4e5e9d8e29b665b3202cffd9c827 Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 18:49:40 +0200 Subject: [PATCH 01/10] site: agda web --- site/docusaurus.config.ts | 17 + site/scripts/build-agda-docs.sh | 23 ++ site/scripts/dev-with-formal-spec.sh | 31 ++ site/scripts/process-agda-html.js | 334 +++++++++++++++++++ site/scripts/test-formal-spec-integration.sh | 56 ++++ site/src/pages/formal-spec/index.tsx | 84 +++++ site/src/pages/formal-spec/styles.module.css | 53 +++ 7 files changed, 598 insertions(+) create mode 100755 site/scripts/build-agda-docs.sh create mode 100755 site/scripts/dev-with-formal-spec.sh create mode 100755 site/scripts/process-agda-html.js create mode 100755 site/scripts/test-formal-spec-integration.sh create mode 100644 site/src/pages/formal-spec/index.tsx create mode 100644 site/src/pages/formal-spec/styles.module.css diff --git a/site/docusaurus.config.ts b/site/docusaurus.config.ts index a69bd2929..384befa6d 100644 --- a/site/docusaurus.config.ts +++ b/site/docusaurus.config.ts @@ -1,6 +1,7 @@ import type * as Preset from "@docusaurus/preset-classic"; import type { Config } from "@docusaurus/types"; import { themes as prismThemes } from "prism-react-renderer"; +import path from 'path'; const config: Config = { title: "Ouroboros Leios", @@ -37,6 +38,13 @@ const config: Config = { }, ], + // Configure static file serving + staticDirectories: ['static', 'public'], + + // Configure plugins + plugins: [ + ], + presets: [ [ "classic", @@ -87,6 +95,11 @@ const config: Config = { position: "left", label: "Development", }, + { + to: "/formal-spec/", + label: "Formal Specification", + position: "left", + }, { to: "/news", label: "Weekly updates", position: "right" }, { type: "dropdown", @@ -142,6 +155,10 @@ const config: Config = { label: "How it works", to: "/docs/how-it-works", }, + { + label: "Formal Specification", + to: "/formal-spec/", + }, { label: "FAQs", to: "/docs/faq", diff --git a/site/scripts/build-agda-docs.sh b/site/scripts/build-agda-docs.sh new file mode 100755 index 000000000..ffea0690c --- /dev/null +++ b/site/scripts/build-agda-docs.sh @@ -0,0 +1,23 @@ +#!/bin/bash + +# Exit on error +set -e + +# Get the directory of this script +SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)" +SITE_DIR="$(dirname "$SCRIPT_DIR")" +FORMAL_SPEC_DIR="$(cd "$SITE_DIR/../../../ouroboros-leios-formal-spec" && pwd)" + +echo "Building Agda documentation..." +cd "$FORMAL_SPEC_DIR" +nix build .#leiosDocs + +echo "Copying Agda HTML files..." +mkdir -p "$SITE_DIR/static/agda_html" +cp -r result/share/doc/agda/html/* "$SITE_DIR/static/agda_html/" + +echo "Processing Agda HTML files..." +cd "$SITE_DIR" +node scripts/process-agda-html.js + +echo "Done! The Agda specification is now available at /agda_html/" \ No newline at end of file diff --git a/site/scripts/dev-with-formal-spec.sh b/site/scripts/dev-with-formal-spec.sh new file mode 100755 index 000000000..26b7892fc --- /dev/null +++ b/site/scripts/dev-with-formal-spec.sh @@ -0,0 +1,31 @@ +#!/bin/bash + +# Exit on error +set -e + +# Get the directory of this script +SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)" +SITE_DIR="$(dirname "$SCRIPT_DIR")" +FORMAL_SPEC_DIR="$(cd "$SITE_DIR/../../ouroboros-leios-formal-spec" && pwd)" + +echo "Building Agda documentation..." +cd "$FORMAL_SPEC_DIR" + +# Add Nix configuration for trusted users +export NIX_CONFIG="trusted-users = root $USER +substituters = https://cache.nixos.org/ +trusted-public-keys = cache.nixos.org-1:6NCHdD59X431o0gWypbMrAURkbJ16ZPMQFGspcDShjY=" + +# Build the docs with --impure to handle dirty git tree +nix build .#leiosDocs --impure + +echo "Copying Agda HTML files..." +mkdir -p "$SITE_DIR/static/agda_html" +cp -r result/html/* "$SITE_DIR/static/agda_html/" + +echo "Processing Agda HTML files..." +cd "$SITE_DIR" +node scripts/process-agda-html.js + +echo "Starting development server..." +yarn start diff --git a/site/scripts/process-agda-html.js b/site/scripts/process-agda-html.js new file mode 100755 index 000000000..3b0cd8917 --- /dev/null +++ b/site/scripts/process-agda-html.js @@ -0,0 +1,334 @@ +const fs = require('fs'); +const path = require('path'); + +const AGDA_HTML_DIR = path.join(__dirname, '../static/agda_html'); + +// First, collect all module information and create search index +const modules = new Map(); +const moduleGroups = new Map(); +const searchIndex = []; + +fs.readdirSync(AGDA_HTML_DIR).forEach(file => { + if (file.endsWith('.html')) { + const filePath = path.join(AGDA_HTML_DIR, file); + const content = fs.readFileSync(filePath, 'utf8'); + // Look for both h1 and title tags to find module names + const h1Match = content.match(/]*>([^<]+)<\/h1>/); + const titleMatch = content.match(/]*>([^<]+)<\/title>/); + const moduleName = h1Match ? h1Match[1].trim() : (titleMatch ? titleMatch[1].trim() : file); + + // Only include Leios package modules + if (moduleName.startsWith('Leios.') || + moduleName.startsWith('Ouroboros.') || + moduleName.startsWith('Cardano.')) { + + // Get the top-level module name for grouping + const topLevel = moduleName.split('.')[0]; + if (!moduleGroups.has(topLevel)) { + moduleGroups.set(topLevel, []); + } + + const moduleInfo = { + name: moduleName, + path: file, + group: topLevel + }; + + modules.set(file, moduleInfo); + moduleGroups.get(topLevel).push(moduleInfo); + + // Add to search index + const lines = content.split('\n'); + lines.forEach((line, index) => { + if (line.trim()) { // Only index non-empty lines + // Try to extract the type/expression from the line + const typeMatch = line.match(/^([^:]+):/); + const expressionMatch = line.match(/^([^=]+)=/); + const title = typeMatch ? typeMatch[1].trim() : + expressionMatch ? expressionMatch[1].trim() : + line.trim(); + + searchIndex.push({ + moduleName: moduleName, + path: file, + group: topLevel, + lineNumber: index + 1, + title: title, + content: line.trim(), + searchableContent: line.toLowerCase() + }); + } + }); + } + } +}); + +// Create the script file +const scriptContent = ` +// Search functionality +const searchIndex = ${JSON.stringify(searchIndex)}; +const searchInput = document.querySelector('.search-input'); +const searchResults = document.querySelector('.search-results'); +const searchOverlay = document.querySelector('.search-overlay'); + +function toggleSearch() { + searchOverlay.classList.toggle('active'); + if (searchOverlay.classList.contains('active')) { + searchInput.focus(); + } +} + +if (searchInput && searchResults) { + // Update the search results HTML generation + function generateSearchResults(results) { + return results.map(result => { + const highlightedTitle = result.title.replace( + new RegExp(result.term, 'gi'), + match => \`\${match}\` + ); + + return \` + +

+ \${highlightedTitle} + \${result.moduleName} +

+ \${result.content} + + \`; + }).join(''); + } + + searchInput.addEventListener('input', (e) => { + const query = e.target.value.toLowerCase(); + if (query.length < 2) { + searchResults.innerHTML = ''; + return; + } + + const results = searchIndex + .filter(item => item.searchableContent.includes(query)) + .map(item => ({ + ...item, + term: query + })) + .slice(0, 10); + + if (results.length > 0) { + searchResults.innerHTML = generateSearchResults(results); + } else { + searchResults.innerHTML = '
No results found
'; + } + }); + + // Add scroll-to-line functionality + document.addEventListener('click', (e) => { + const result = e.target.closest('.search-result'); + if (result) { + const lineNumber = result.dataset.line; + const targetElement = document.querySelector(\`#L\${lineNumber}\`); + if (targetElement) { + targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + targetElement.classList.add('highlight-line'); + setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); + } + searchOverlay.classList.remove('active'); + } + }); + + // Close search when clicking outside or pressing Escape + document.addEventListener('click', (e) => { + if (e.target === searchOverlay) { + searchOverlay.classList.remove('active'); + } + }); + + document.addEventListener('keydown', (e) => { + if (e.key === 'Escape' && searchOverlay.classList.contains('active')) { + searchOverlay.classList.remove('active'); + } + }); +} + +// Theme toggle functionality +function toggleTheme() { + const html = document.documentElement; + const currentTheme = html.getAttribute('data-theme'); + const newTheme = currentTheme === 'dark' ? 'light' : 'dark'; + html.setAttribute('data-theme', newTheme); + localStorage.setItem('theme', newTheme); +} + +// Initialize theme from localStorage or system preference +const savedTheme = localStorage.getItem('theme'); +if (savedTheme) { + document.documentElement.setAttribute('data-theme', savedTheme); +} else if (window.matchMedia('(prefers-color-scheme: dark)').matches) { + document.documentElement.setAttribute('data-theme', 'dark'); +} +`; + +fs.writeFileSync(path.join(AGDA_HTML_DIR, 'agda.js'), scriptContent); + +// Process all HTML files in the agda_html directory +fs.readdirSync(AGDA_HTML_DIR).forEach(file => { + if (file.endsWith('.html')) { + const filePath = path.join(AGDA_HTML_DIR, file); + let content = fs.readFileSync(filePath, 'utf8'); + + // Add our custom CSS and theme toggle + content = content.replace( + //, + ` + + ` + ); + + // Add header, sidebar, and theme toggle + content = content.replace( + /]*>/, + ` +
+
+ Back + +
+
+ + + + + + +
+
+
+
+
+ +
+
+
+
+
+ +
+ ` + ); + + // Close the main content div + content = content.replace( + /<\/body>/, + `
` + ); + + // Fix internal links + content = content.replace( + /href="([^"]*\.html)(#[^"]*)?"/g, + (match, file, anchor) => { + if (file.startsWith('http')) { + return match; // Keep external links as is + } + // Remove any existing /agda_html/ prefix to prevent duplication + const cleanFile = file.replace(/^\/?agda_html\//, ''); + return `href="/agda_html/${cleanFile}${anchor || ''}"`; + } + ); + + // Update the theme toggle button HTML + content = content.replace( + /` + ); + + // Write the processed file + fs.writeFileSync(filePath, content); + console.log(`Processed ${file}`); + } +}); + +console.log('Finished processing Agda HTML files'); + +// Update the module list to ensure all modules are properly linked +const AGDA_MODULES = [ + { + name: 'Leios.Base', + path: 'Leios.Base.html', + description: 'Core definitions and types' + }, + { + name: 'Leios.Voting', + path: 'Leios.Voting.html', + description: 'Voting mechanism and rules' + }, + { + name: 'Leios.Protocol', + path: 'Leios.Protocol.html', + description: 'Protocol state and transitions' + }, + { + name: 'Leios.Blocks', + path: 'Leios.Blocks.html', + description: 'Block structure and validation' + }, + { + name: 'Leios.Network', + path: 'Leios.Network.html', + description: 'Network communication and messages' + }, + { + name: 'Leios.Chain', + path: 'Leios.Chain.html', + description: 'Chain state and updates' + }, + { + name: 'Leios.Validation', + path: 'Leios.Validation.html', + description: 'Transaction and block validation' + }, + { + name: 'Leios.Properties', + path: 'Leios.Properties.html', + description: 'Protocol properties and proofs' + } +]; \ No newline at end of file diff --git a/site/scripts/test-formal-spec-integration.sh b/site/scripts/test-formal-spec-integration.sh new file mode 100755 index 000000000..62665bcb9 --- /dev/null +++ b/site/scripts/test-formal-spec-integration.sh @@ -0,0 +1,56 @@ +#!/bin/bash + +# Exit on error +set -e + +# Get the directory of this script +SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)" +SITE_DIR="$(dirname "$SCRIPT_DIR")" +FORMAL_SPEC_DIR="$(cd "$SITE_DIR/../../../ouroboros-leios-formal-spec" && pwd)" + +echo "Testing formal spec integration..." + +# Step 1: Build formal spec docs (simulating formal-spec.yaml) +echo "Building formal spec documentation..." +cd "$FORMAL_SPEC_DIR" + +# Clean up any previous build artifacts +rm -rf result + +# Add Nix configuration for trusted users +export NIX_CONFIG="trusted-users = root $USER +substituters = https://cache.nixos.org/ +trusted-public-keys = cache.nixos.org-1:6NCHdD59X431o0gWypbMrAURkbJ16ZPMQFGspcDShjY=" + +# Build the docs with --impure to handle dirty git tree +echo "Running nix build..." +nix build .#leiosDocs --impure + +# Step 2: Copy docs to site (simulating formal-spec-integration.yaml) +echo "Copying docs to site..." +mkdir -p "$SITE_DIR/static/agda_html" + +# The HTML files are directly in result/html +HTML_DIR="result/html" +if [ ! -d "$HTML_DIR" ]; then + echo "Error: Could not find HTML directory at $HTML_DIR" + echo "Contents of result directory:" + ls -R result + exit 1 +fi + +# Copy the files +echo "Copying from $HTML_DIR to $SITE_DIR/static/agda_html/" +cp -r "$HTML_DIR"/* "$SITE_DIR/static/agda_html/" + +# Step 3: Process the HTML files +echo "Processing HTML files..." +cd "$SITE_DIR" +node scripts/process-agda-html.js + +# Step 4: Build the site +echo "Building site..." +yarn build + +echo "Done! You can test the site with:" +echo "cd $SITE_DIR && yarn serve" \ No newline at end of file diff --git a/site/src/pages/formal-spec/index.tsx b/site/src/pages/formal-spec/index.tsx new file mode 100644 index 000000000..65eeb0fc6 --- /dev/null +++ b/site/src/pages/formal-spec/index.tsx @@ -0,0 +1,84 @@ +import React from 'react'; +import Layout from '@theme/Layout'; +import styles from './styles.module.css'; + +// List of available Agda modules with their actual file paths +const AGDA_MODULES = [ + { + name: 'Leios.Base', + path: 'Leios.Base.html', + description: 'Core definitions and types' + }, + { + name: 'Leios.Voting', + path: 'Leios.Voting.html', + description: 'Voting mechanism and rules' + }, + { + name: 'Leios.Protocol', + path: 'Leios.Protocol.html', + description: 'Protocol state and transitions' + }, + { + name: 'Leios.Blocks', + path: 'Leios.Blocks.html', + description: 'Block structure and validation' + }, + { + name: 'Leios.Network', + path: 'Leios.Network.html', + description: 'Network communication and messages' + } +]; + +export default function FormalSpecPage(): React.ReactElement { + return ( + +
+
+

Ouroboros Leios Formal Specification

+

+ This section contains the formal specification of the Ouroboros Leios protocol, + written in Agda. The specification provides a mathematical foundation for the + protocol's properties and guarantees. +

+ +

Available Modules

+
+ {AGDA_MODULES.map(module => ( + +
{module.name}
+
{module.description}
+ + ))} +
+ +

Getting Started

+

+ The formal specification is organized into modules, each focusing on different + aspects of the protocol. Start with these key modules: +

+
    + {AGDA_MODULES.slice(0, 3).map(module => ( +
  • + + {module.name} + + {' - '}{module.description} +
    • + ))} +
+
+
+ + ); +} \ No newline at end of file diff --git a/site/src/pages/formal-spec/styles.module.css b/site/src/pages/formal-spec/styles.module.css new file mode 100644 index 000000000..228c45e7f --- /dev/null +++ b/site/src/pages/formal-spec/styles.module.css @@ -0,0 +1,53 @@ +.main { + padding: 2rem 0; + background-color: var(--ifm-background-color); +} + +.container { + max-width: 1200px; + margin: 0 auto; + padding: 0 1rem; +} + +.fileGrid { + display: grid; + grid-template-columns: repeat(auto-fill, minmax(300px, 1fr)); + gap: 1.5rem; + margin: 2rem 0; +} + +.fileLink { + display: block; + padding: 1.5rem; + background-color: var(--ifm-card-background-color); + border: 1px solid var(--ifm-color-emphasis-200); + border-radius: var(--ifm-card-border-radius); + color: var(--ifm-font-color-base); + text-decoration: none; + transition: all 0.2s ease; + height: 100%; +} + +.fileLink:hover { + background-color: var(--ifm-color-emphasis-100); + text-decoration: none; + transform: translateY(-2px); + box-shadow: 0 4px 8px rgba(0, 0, 0, 0.1); +} + +.fileLink:active { + transform: translateY(0); +} + +.moduleName { + font-size: 1.2rem; + font-weight: bold; + margin-bottom: 0.5rem; + color: var(--ifm-color-primary); +} + +.moduleDescription { + font-size: 0.9rem; + color: var(--ifm-color-emphasis-700); + line-height: 1.4; +} \ No newline at end of file From fe9eb8b85bba61f88ed4ffb6809fb9dc85af74e8 Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 18:49:54 +0200 Subject: [PATCH 02/10] scripts: remove test script --- site/scripts/test-formal-spec-integration.sh | 56 -------------------- 1 file changed, 56 deletions(-) delete mode 100755 site/scripts/test-formal-spec-integration.sh diff --git a/site/scripts/test-formal-spec-integration.sh b/site/scripts/test-formal-spec-integration.sh deleted file mode 100755 index 62665bcb9..000000000 --- a/site/scripts/test-formal-spec-integration.sh +++ /dev/null @@ -1,56 +0,0 @@ -#!/bin/bash - -# Exit on error -set -e - -# Get the directory of this script -SCRIPT_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)" -SITE_DIR="$(dirname "$SCRIPT_DIR")" -FORMAL_SPEC_DIR="$(cd "$SITE_DIR/../../../ouroboros-leios-formal-spec" && pwd)" - -echo "Testing formal spec integration..." - -# Step 1: Build formal spec docs (simulating formal-spec.yaml) -echo "Building formal spec documentation..." -cd "$FORMAL_SPEC_DIR" - -# Clean up any previous build artifacts -rm -rf result - -# Add Nix configuration for trusted users -export NIX_CONFIG="trusted-users = root $USER -substituters = https://cache.nixos.org/ -trusted-public-keys = cache.nixos.org-1:6NCHdD59X431o0gWypbMrAURkbJ16ZPMQFGspcDShjY=" - -# Build the docs with --impure to handle dirty git tree -echo "Running nix build..." -nix build .#leiosDocs --impure - -# Step 2: Copy docs to site (simulating formal-spec-integration.yaml) -echo "Copying docs to site..." -mkdir -p "$SITE_DIR/static/agda_html" - -# The HTML files are directly in result/html -HTML_DIR="result/html" -if [ ! -d "$HTML_DIR" ]; then - echo "Error: Could not find HTML directory at $HTML_DIR" - echo "Contents of result directory:" - ls -R result - exit 1 -fi - -# Copy the files -echo "Copying from $HTML_DIR to $SITE_DIR/static/agda_html/" -cp -r "$HTML_DIR"/* "$SITE_DIR/static/agda_html/" - -# Step 3: Process the HTML files -echo "Processing HTML files..." -cd "$SITE_DIR" -node scripts/process-agda-html.js - -# Step 4: Build the site -echo "Building site..." -yarn build - -echo "Done! You can test the site with:" -echo "cd $SITE_DIR && yarn serve" \ No newline at end of file From 550bf25e71b2a6d6814c9baa0f0a5de368760c06 Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 18:57:14 +0200 Subject: [PATCH 03/10] scripts: add transform script --- site/src/pages/formal-spec/index.tsx | 18 +- site/static/agda_html/Agda.css | 546 +++++++++++++++++++++++++++ site/static/agda_html/agda.js | 102 +++++ 3 files changed, 649 insertions(+), 17 deletions(-) create mode 100644 site/static/agda_html/Agda.css create mode 100644 site/static/agda_html/agda.js diff --git a/site/src/pages/formal-spec/index.tsx b/site/src/pages/formal-spec/index.tsx index 65eeb0fc6..231da62d3 100644 --- a/site/src/pages/formal-spec/index.tsx +++ b/site/src/pages/formal-spec/index.tsx @@ -46,7 +46,7 @@ export default function FormalSpecPage(): React.ReactElement { protocol's properties and guarantees.

-

Available Modules

+

Modules

{AGDA_MODULES.map(module => ( ))}
- -

Getting Started

-

- The formal specification is organized into modules, each focusing on different - aspects of the protocol. Start with these key modules: -

- diff --git a/site/static/agda_html/Agda.css b/site/static/agda_html/Agda.css new file mode 100644 index 000000000..fc8cb347d --- /dev/null +++ b/site/static/agda_html/Agda.css @@ -0,0 +1,546 @@ +:root { + --bg-color: #ffffff; + --text-color: #24292e; + --border-color: #e1e4e8; + --code-bg: #f6f8fa; + --link-color: #0366d6; + --header-bg: transparent; + --sidebar-width: 280px; + --content-width: 800px; + --search-width: 300px; + --hover-color: #f6f8fa; + --highlight-color: #ffeb3b; + --muted-color: #6a737d; + --header-height: 60px; + --font-sans: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; + --font-mono: "Fira Code", "SF Mono", "Roboto Mono", Menlo, Consolas, monospace; +} + +[data-theme="dark"] { + --bg-color: #0d1117; + --text-color: #c9d1d9; + --border-color: #30363d; + --code-bg: #161b22; + --link-color: #58a6ff; + --header-bg: transparent; + --hover-color: #161b22; + --highlight-color: #ffeb3b; + --muted-color: #8b949e; +} + +/* Layout */ +body { + margin: 0; + padding: 0; + min-height: 100vh; + display: flex; + flex-direction: column; + background-color: var(--bg-color); + color: var(--text-color); + font-family: var(--font-sans); + line-height: 1.6; +} + +.agda-header { + position: fixed; + top: 0; + left: 0; + right: 0; + height: var(--header-height); + background-color: var(--bg-color); + display: flex; + justify-content: space-between; + align-items: center; + padding: 0 1.5rem; + z-index: 100; + border-bottom: none; +} + +.agda-header-left { + display: flex; + align-items: center; + gap: 1.5rem; +} + +.agda-container { + display: flex; + margin-top: var(--header-height); + min-height: calc(100vh - var(--header-height)); +} + +.agda-sidebar { + position: fixed; + top: var(--header-height); + left: 0; + bottom: 0; + width: var(--sidebar-width); + background-color: var(--bg-color); + overflow-y: auto; + padding: 1.5rem; + border-right: none; +} + +.agda-sidebar h3 { + margin: 0 0 1.5rem 0; + font-size: 1.1rem; + font-weight: 600; + color: var(--text-color); + letter-spacing: -0.01em; +} + +.agda-sidebar h4 { + margin: 1.5rem 0 0.75rem 0; + font-size: 0.9rem; + font-weight: 600; + color: var(--muted-color); + text-transform: uppercase; + letter-spacing: 0.05em; +} + +.agda-sidebar ul { + list-style: none; + padding: 0; + margin: 0; +} + +.agda-sidebar li { + margin: 0.25rem 0; +} + +.agda-sidebar a { + color: var(--text-color); + text-decoration: none; + font-size: 0.9rem; + display: block; + padding: 0.35rem 0.5rem; + border-radius: 4px; + transition: all 0.2s; +} + +.agda-sidebar a:hover { + background-color: var(--hover-color); + color: var(--link-color); +} + +.agda-sidebar a.active { + background-color: var(--hover-color); + color: var(--link-color); + font-weight: 500; +} + +.agda-content { + flex: 1; + margin-left: var(--sidebar-width); + padding: 2rem; + max-width: var(--content-width); +} + +/* Agda Syntax Highlighting */ +.Agda { + font-family: var(--font-mono); + font-size: 0.95rem; + line-height: 1.6; + background-color: var(--bg-color); + color: var(--text-color); +} + +.Agda code { + font-family: var(--font-mono); + font-size: 0.95rem; + background-color: var(--code-bg); + padding: 0.2em 0.4em; + border-radius: 3px; +} + +.Agda a { + color: var(--link-color); + text-decoration: none; +} + +.Agda a:hover { + text-decoration: underline; +} + +/* Aspects. */ +.Agda .Comment { color: #B22222 } +.Agda .Background {} +.Agda .Markup { color: #000000 } +.Agda .Keyword { color: #CD6600 } +.Agda .String { color: #B22222 } +.Agda .Number { color: #A020F0 } +.Agda .Symbol { color: #404040 } +.Agda .PrimitiveType { color: #0000CD } +.Agda .Pragma { color: black } +.Agda .Operator {} +.Agda .Hole { background: #B4EEB4 } + +/* NameKinds. */ +.Agda .Bound { color: black } +.Agda .Generalizable { color: black } +.Agda .InductiveConstructor { color: #008B00 } +.Agda .CoinductiveConstructor { color: #8B7500 } +.Agda .Datatype { color: #0000CD } +.Agda .Field { color: #EE1289 } +.Agda .Function { color: #0000CD } +.Agda .Macro { color: #0000CD } +.Agda .Module { color: #A020F0 } +.Agda .Postulate { color: #0000CD } +.Agda .Primitive { color: #0000CD } +.Agda .Record { color: #0000CD } + +/* OtherAspects. */ +.Agda .DottedPattern {} +.Agda .UnsolvedMeta { color: black; background: yellow } +.Agda .UnsolvedConstraint { color: black; background: yellow } +.Agda .TerminationProblem { color: black; background: #FFA07A } +.Agda .IncompletePattern { color: black; background: #F5DEB3 } +.Agda .Error { color: red; text-decoration: underline } +.Agda .TypeChecks { color: black; background: #ADD8E6 } +.Agda .Deadcode { color: black; background: #808080 } +.Agda .ShadowingInTelescope { color: black; background: #808080 } + +/* Standard attributes. */ +.Agda a { text-decoration: none } +.Agda a[href]:hover { background-color: #B4EEB4 } +.Agda [href].hover-highlight { background-color: #B4EEB4; } + +/* Back link styling */ +.agda-header-left a { + color: var(--muted-color); + text-decoration: none; + font-size: 0.9rem; + display: flex; + align-items: center; + gap: 0.5rem; + transition: all 0.2s; +} + +.agda-header-left a:hover { + color: var(--link-color); +} + +.agda-header-left a::before { + content: "←"; + font-size: 1.1em; +} + +/* Dark mode syntax highlighting improvements */ +[data-theme="dark"] .Agda .Comment { + color: #8b949e; +} + +[data-theme="dark"] .Agda .Keyword { + color: #ff7b72; +} + +[data-theme="dark"] .Agda .Number { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .String { + color: #a5d6ff; +} + +[data-theme="dark"] .Agda .Symbol { + color: #c9d1d9; +} + +[data-theme="dark"] .Agda .Function { + color: #d2a8ff; +} + +[data-theme="dark"] .Agda .Module { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .Postulate { + color: #ffa657; +} + +[data-theme="dark"] .Agda .Type { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .Record { + color: #d2a8ff; +} + +[data-theme="dark"] .Agda .Constructor { + color: #7ee787; +} + +[data-theme="dark"] .Agda .Field { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .Bound { + color: #c9d1d9; +} + +[data-theme="dark"] .Agda .Generalizable { + color: #c9d1d9; +} + +[data-theme="dark"] .Agda .InductiveConstructor { + color: #7ee787; +} + +[data-theme="dark"] .Agda .CoinductiveConstructor { + color: #ffa657; +} + +[data-theme="dark"] .Agda .Datatype { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .Macro { + color: #d2a8ff; +} + +[data-theme="dark"] .Agda .Primitive { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .PrimitiveType { + color: #79c0ff; +} + +[data-theme="dark"] .Agda .Operator { + color: #c9d1d9; +} + +[data-theme="dark"] .Agda .Hole { + background: #1b4b1b; +} + +[data-theme="dark"] .Agda .UnsolvedMeta { + color: #c9d1d9; + background: #5a1a1a; +} + +[data-theme="dark"] .Agda .UnsolvedConstraint { + color: #c9d1d9; + background: #5a1a1a; +} + +[data-theme="dark"] .Agda .TerminationProblem { + color: #c9d1d9; + background: #5a1a1a; +} + +[data-theme="dark"] .Agda .IncompletePattern { + color: #c9d1d9; + background: #5a1a1a; +} + +[data-theme="dark"] .Agda .Error { + color: #ff7b72; + text-decoration: underline; +} + +[data-theme="dark"] .Agda .TypeChecks { + color: #c9d1d9; + background: #1b4b1b; +} + +[data-theme="dark"] .Agda .Deadcode { + color: #8b949e; + background: #30363d; +} + +[data-theme="dark"] .Agda .ShadowingInTelescope { + color: #8b949e; + background: #30363d; +} + +/* Search and Theme Toggle */ +.search-button { + display: flex; + align-items: center; + gap: 0.5rem; + padding: 0.5rem 1rem; + border: none; + background: none; + color: var(--text-color); + cursor: pointer; + font-size: 0.9rem; + transition: all 0.2s; + opacity: 0.8; + font-family: var(--font-sans); +} + +.search-button:hover { + opacity: 1; + color: var(--link-color); +} + +.theme-toggle { + background: none; + border: none; + cursor: pointer; + padding: 0.5rem; + width: 24px; + height: 24px; + opacity: 0.8; + transition: opacity 0.2s; + color: var(--text-color); + display: flex; + align-items: center; + justify-content: center; +} + +.theme-toggle:hover { + opacity: 1; + color: var(--link-color); +} + +.search-button svg { + width: 16px; + height: 16px; +} + +.search-overlay { + position: fixed; + top: 0; + left: 0; + right: 0; + bottom: 0; + background-color: rgba(0, 0, 0, 0.5); + display: none; + z-index: 1000; +} + +.search-overlay.active { + display: flex; + align-items: flex-start; + justify-content: center; + padding-top: 10vh; +} + +.search-modal { + background-color: var(--bg-color); + border-radius: 8px; + box-shadow: 0 4px 12px rgba(0, 0, 0, 0.15); + width: 90%; + max-width: 800px; + max-height: 80vh; + overflow: hidden; +} + +.search-container { + padding: 1rem; +} + +.search-input { + width: 100%; + padding: 0.75rem 1rem; + border: 1px solid var(--border-color); + border-radius: 4px; + background-color: var(--bg-color); + color: var(--text-color); + font-size: 1rem; + transition: border-color 0.2s; + margin-bottom: 1rem; +} + +.search-input:focus { + outline: none; + border-color: var(--link-color); +} + +.search-results { + max-height: calc(80vh - 5rem); + overflow-y: auto; +} + +.search-result { + display: block; + padding: 1rem; + border-bottom: 1px solid var(--border-color); + text-decoration: none; + color: var(--text-color); + transition: background-color 0.2s; +} + +.search-result:hover { + background-color: var(--hover-color); +} + +.search-header { + margin: 0 0 0.5rem 0; + font-size: 1.1em; + display: flex; + justify-content: space-between; + align-items: center; +} + +.search-identifier { + font-weight: 500; +} + +.search-module { + font-size: 0.9em; + color: var(--muted-color); + margin-left: 1rem; +} + +.search-type { + font-family: var(--font-mono); + font-size: 0.9em; + color: var(--muted-color); + display: block; + white-space: nowrap; + overflow: hidden; + text-overflow: ellipsis; +} + +.search-match { + background-color: var(--highlight-color); + color: var(--text-color); + padding: 0 2px; + border-radius: 2px; +} + +.search-no-results { + padding: 1rem; + color: var(--muted-color); + text-align: center; +} + +.theme-toggle svg { + width: 24px; + height: 24px; + fill: currentColor; +} + +[data-theme="dark"] .light-icon, +[data-theme="light"] .dark-icon { + display: none; +} + +.agda-header-right { + display: flex; + align-items: center; + gap: 1rem; +} + +.github-link { + display: flex; + align-items: center; + justify-content: center; + width: 24px; + height: 24px; + color: var(--text-color); + opacity: 0.8; + transition: opacity 0.2s; +} + +.github-link:hover { + opacity: 1; +} + +.github-link svg { + width: 24px; + height: 24px; + fill: currentColor; +} diff --git a/site/static/agda_html/agda.js b/site/static/agda_html/agda.js new file mode 100644 index 000000000..111ae3fe4 --- /dev/null +++ b/site/static/agda_html/agda.js @@ -0,0 +1,102 @@ + +// Search functionality +const searchIndex = [{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":2,"title":"Leios.Abstract
{-# OPTIONS --safe #-}","searchableContent":"leios.abstract
{-# options --safe #-}"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":4,"title":"module Leios.Abstract where","searchableContent":"module leios.abstract where"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":6,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":8,"title":"record LeiosAbstract ","content":"record LeiosAbstract : Type₁ where","searchableContent":"record leiosabstract : type₁ where"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":9,"title":"field Tx ","content":"field Tx : Type","searchableContent":"  field tx : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":10,"title":"⦃ DecEq-Tx ⦄ ","content":"⦃ DecEq-Tx ⦄ : DecEq Tx","searchableContent":"        ⦃ deceq-tx ⦄ : deceq tx"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":11,"title":"PoolID ","content":"PoolID : Type","searchableContent":"        poolid : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":12,"title":"⦃ DecEq-PoolID ⦄ ","content":"⦃ DecEq-PoolID ⦄ : DecEq PoolID","searchableContent":"        ⦃ deceq-poolid ⦄ : deceq poolid"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":13,"title":"BodyHash VrfPf PrivKey Sig Hash ","content":"BodyHash VrfPf PrivKey Sig Hash : Type -- these could have been byte strings, but this is safer","searchableContent":"        bodyhash vrfpf privkey sig hash : type -- these could have been byte strings, but this is safer"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":14,"title":"⦃ DecEq-Hash ⦄ ","content":"⦃ DecEq-Hash ⦄ : DecEq Hash","searchableContent":"        ⦃ deceq-hash ⦄ : deceq hash"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":15,"title":"⦃ DecEq-VrfPf ⦄ ","content":"⦃ DecEq-VrfPf ⦄ : DecEq VrfPf","searchableContent":"        ⦃ deceq-vrfpf ⦄ : deceq vrfpf"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":16,"title":"⦃ DecEq-Sig ⦄ ","content":"⦃ DecEq-Sig ⦄ : DecEq Sig","searchableContent":"        ⦃ deceq-sig ⦄ : deceq sig"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":17,"title":"Vote ","content":"Vote : Type","searchableContent":"        vote : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":18,"title":"⦃ DecEq-Vote ⦄ ","content":"⦃ DecEq-Vote ⦄ : DecEq Vote","searchableContent":"        ⦃ deceq-vote ⦄ : deceq vote"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":19,"title":"vote ","content":"vote : PrivKey → Hash → Vote","searchableContent":"        vote : privkey → hash → vote"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":20,"title":"sign ","content":"sign : PrivKey → Hash → Sig","searchableContent":"        sign : privkey → hash → sig"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":21,"title":"⦃ Hashable-Txs ⦄ ","content":"⦃ Hashable-Txs ⦄ : Hashable (List Tx) Hash","searchableContent":"        ⦃ hashable-txs ⦄ : hashable (list tx) hash"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":22,"title":"L ","content":"L : ℕ","searchableContent":"        l : ℕ"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":23,"title":"⦃ NonZero-L ⦄ ","content":"⦃ NonZero-L ⦄ : NonZero L","searchableContent":"        ⦃ nonzero-l ⦄ : nonzero l"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":24,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":2,"title":"Leios.Base
{-# OPTIONS --safe #-}","searchableContent":"leios.base
{-# options --safe #-}"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":6,"title":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":8,"title":"module Leios.Base (a ","content":"module Leios.Base (a : LeiosAbstract) (open LeiosAbstract a) (vrf' : LeiosVRF a)","searchableContent":"module leios.base (a : leiosabstract) (open leiosabstract a) (vrf' : leiosvrf a)"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":9,"title":"(let open LeiosVRF vrf') where","searchableContent":"  (let open leiosvrf vrf') where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":11,"title":"open import Leios.Blocks a using (EndorserBlock)","searchableContent":"open import leios.blocks a using (endorserblock)"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":13,"title":"StakeDistr ","content":"StakeDistr : Type","searchableContent":"stakedistr : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":14,"title":"StakeDistr = TotalMap PoolID ℕ","searchableContent":"stakedistr = totalmap poolid ℕ"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":16,"title":"RankingBlock ","content":"RankingBlock : Type","searchableContent":"rankingblock : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":17,"title":"RankingBlock = These EndorserBlock (List Tx)","searchableContent":"rankingblock = these endorserblock (list tx)"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":19,"title":"record BaseAbstract ","content":"record BaseAbstract : Type₁ where","searchableContent":"record baseabstract : type₁ where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":20,"title":"field Cert ","content":"field Cert : Type","searchableContent":"  field cert : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":21,"title":"VTy ","content":"VTy : Type","searchableContent":"        vty : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":22,"title":"initSlot ","content":"initSlot : VTy → ℕ","searchableContent":"        initslot : vty → ℕ"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":23,"title":"V-chkCerts ","content":"V-chkCerts : List PubKey → EndorserBlock × Cert → Bool","searchableContent":"        v-chkcerts : list pubkey → endorserblock × cert → bool"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":25,"title":"data Input ","content":"data Input : Type where","searchableContent":"  data input : type where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":26,"title":"INIT   ","content":"INIT   : (EndorserBlock × Cert → Bool) → Input","searchableContent":"    init   : (endorserblock × cert → bool) → input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":27,"title":"SUBMIT ","content":"SUBMIT : RankingBlock → Input","searchableContent":"    submit : rankingblock → input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":28,"title":"FTCH-LDG ","content":"FTCH-LDG : Input","searchableContent":"    ftch-ldg : input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":30,"title":"data Output ","content":"data Output : Type where","searchableContent":"  data output : type where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":31,"title":"STAKE ","content":"STAKE : StakeDistr → Output","searchableContent":"    stake : stakedistr → output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":32,"title":"EMPTY ","content":"EMPTY : Output","searchableContent":"    empty : output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":33,"title":"BASE-LDG ","content":"BASE-LDG : List RankingBlock → Output","searchableContent":"    base-ldg : list rankingblock → output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":35,"title":"record Functionality ","content":"record Functionality : Type₁ where","searchableContent":"  record functionality : type₁ where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":36,"title":"field State ","content":"field State : Type","searchableContent":"    field state : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":37,"title":"_-⟦_/_⟧⇀_ ","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"          _-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":38,"title":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ ","content":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : {s : State} → {i : Input} → {o : Output} → {s' : State} → (s -⟦ i / o ⟧⇀ s') ⁇","searchableContent":"          ⦃ dec-_-⟦_/_⟧⇀_ ⦄ : {s : state} → {i : input} → {o : output} → {s' : state} → (s -⟦ i / o ⟧⇀ s') ⁇"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":39,"title":"SUBMIT-total ","content":"SUBMIT-total : ∀ {s b} → ∃[ s' ] s -⟦ SUBMIT b / EMPTY ⟧⇀ s'","searchableContent":"          submit-total : ∀ {s b} → ∃[ s' ] s -⟦ submit b / empty ⟧⇀ s'"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":41,"title":"open Input public","searchableContent":"    open input public"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":42,"title":"open Output public","searchableContent":"    open output public"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":43,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":2,"title":"Leios.Blocks
{-# OPTIONS --safe #-}","searchableContent":"leios.blocks
{-# options --safe #-}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":6,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":7,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":8,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":10,"title":"module Leios.Blocks (a ","content":"module Leios.Blocks (a : LeiosAbstract) (let open LeiosAbstract a) where","searchableContent":"module leios.blocks (a : leiosabstract) (let open leiosabstract a) where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":12,"title":"-- IsBlock typeclass (could do a closed-world approach instead)","searchableContent":"-- isblock typeclass (could do a closed-world approach instead)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":13,"title":"-- Q","content":"-- Q: should votes have an instance of this class?","searchableContent":"-- q: should votes have an instance of this class?"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":14,"title":"record IsBlock (B ","content":"record IsBlock (B : Type) : Type where","searchableContent":"record isblock (b : type) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":15,"title":"field slotNumber ","content":"field slotNumber : B → ℕ","searchableContent":"  field slotnumber : b → ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":16,"title":"producerID ","content":"producerID : B → PoolID","searchableContent":"        producerid : b → poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":17,"title":"lotteryPf  ","content":"lotteryPf  : B → VrfPf","searchableContent":"        lotterypf  : b → vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":18,"title":"signature  ","content":"signature  : B → Sig","searchableContent":"        signature  : b → sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":20,"title":"infix 4 _∈ᴮ_","searchableContent":"  infix 4 _∈ᴮ_"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":22,"title":"_∈ᴮ_ ","content":"_∈ᴮ_ : B → ℙ ℕ → Type","searchableContent":"  _∈ᴮ_ : b → ℙ ℕ → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":23,"title":"b ∈ᴮ X = slotNumber b ∈ X","searchableContent":"  b ∈ᴮ x = slotnumber b ∈ x"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":25,"title":"open IsBlock ⦃...⦄ public","searchableContent":"open isblock ⦃...⦄ public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":27,"title":"IBRef = Hash","searchableContent":"ibref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":28,"title":"EBRef = Hash","searchableContent":"ebref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":30,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":31,"title":"-- Input Blocks","searchableContent":"-- input blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":32,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":34,"title":"record IBHeaderOSig (sig ","content":"record IBHeaderOSig (sig : Type) : Type where","searchableContent":"record ibheaderosig (sig : type) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":35,"title":"field slotNumber ","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":36,"title":"producerID ","content":"producerID : PoolID","searchableContent":"        producerid : poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":37,"title":"lotteryPf  ","content":"lotteryPf  : VrfPf","searchableContent":"        lotterypf  : vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":38,"title":"bodyHash   ","content":"bodyHash   : Hash","searchableContent":"        bodyhash   : hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":39,"title":"signature  ","content":"signature  : sig","searchableContent":"        signature  : sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":41,"title":"IBHeader    = IBHeaderOSig Sig","searchableContent":"ibheader    = ibheaderosig sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":42,"title":"PreIBHeader = IBHeaderOSig ⊤","searchableContent":"preibheader = ibheaderosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":44,"title":"record IBBody ","content":"record IBBody : Type where","searchableContent":"record ibbody : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":45,"title":"field txs ","content":"field txs : List Tx","searchableContent":"  field txs : list tx"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":47,"title":"record InputBlock ","content":"record InputBlock : Type where","searchableContent":"record inputblock : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":48,"title":"field header ","content":"field header : IBHeader","searchableContent":"  field header : ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":49,"title":"body   ","content":"body   : IBBody","searchableContent":"        body   : ibbody"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":51,"title":"open IBHeaderOSig header public","searchableContent":"  open ibheaderosig header public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":53,"title":"unquoteDecl DecEq-IBBody DecEq-IBHeaderOSig DecEq-InputBlock =","searchableContent":"unquotedecl deceq-ibbody deceq-ibheaderosig deceq-inputblock ="},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":54,"title":"derive-DecEq (","searchableContent":"  derive-deceq ("},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":55,"title":"(quote IBBody , DecEq-IBBody)","searchableContent":"      (quote ibbody , deceq-ibbody)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":56,"title":"∷ (quote IBHeaderOSig , DecEq-IBHeaderOSig)","searchableContent":"    ∷ (quote ibheaderosig , deceq-ibheaderosig)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":57,"title":"∷ (quote InputBlock , DecEq-InputBlock) ∷ [])","searchableContent":"    ∷ (quote inputblock , deceq-inputblock) ∷ [])"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":59,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":60,"title":"IsBlock-IBHeader ","content":"IsBlock-IBHeader : IsBlock IBHeader","searchableContent":"  isblock-ibheader : isblock ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":61,"title":"IsBlock-IBHeader = record { IBHeaderOSig }","searchableContent":"  isblock-ibheader = record { ibheaderosig }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":63,"title":"Hashable-IBBody ","content":"Hashable-IBBody : Hashable IBBody Hash","searchableContent":"  hashable-ibbody : hashable ibbody hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":64,"title":"Hashable-IBBody .hash b = hash (b .IBBody.txs)","searchableContent":"  hashable-ibbody .hash b = hash (b .ibbody.txs)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":66,"title":"Hashable-IBHeader ","content":"Hashable-IBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → Hashable IBHeader Hash","searchableContent":"  hashable-ibheader : ⦃ hashable preibheader hash ⦄ → hashable ibheader hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":67,"title":"Hashable-IBHeader .hash b = hash {T = PreIBHeader}","searchableContent":"  hashable-ibheader .hash b = hash {t = preibheader}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":68,"title":"record { IBHeaderOSig b hiding (signature) ; signature = _ }","searchableContent":"    record { ibheaderosig b hiding (signature) ; signature = _ }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":70,"title":"IsBlock-InputBlock ","content":"IsBlock-InputBlock : IsBlock InputBlock","searchableContent":"  isblock-inputblock : isblock inputblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":71,"title":"IsBlock-InputBlock = record { InputBlock }","searchableContent":"  isblock-inputblock = record { inputblock }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":73,"title":"mkIBHeader ","content":"mkIBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List Tx → IBHeader","searchableContent":"mkibheader : ⦃ hashable preibheader hash ⦄ → ℕ → poolid → vrfpf → privkey → list tx → ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":74,"title":"mkIBHeader slot id π pKey txs = record { signature = sign pKey (hash h) ; IBHeaderOSig h }","searchableContent":"mkibheader slot id π pkey txs = record { signature = sign pkey (hash h) ; ibheaderosig h }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":75,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":76,"title":"h ","content":"h : IBHeaderOSig ⊤","searchableContent":"    h : ibheaderosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":77,"title":"h = record { slotNumber = slot","searchableContent":"    h = record { slotnumber = slot"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":78,"title":"; producerID = id","searchableContent":"               ; producerid = id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":79,"title":"; lotteryPf  = π","searchableContent":"               ; lotterypf  = π"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":80,"title":"; bodyHash   = hash txs","searchableContent":"               ; bodyhash   = hash txs"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":81,"title":"; signature  = _","searchableContent":"               ; signature  = _"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":82,"title":"}","searchableContent":"               }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":84,"title":"getIBRef ","content":"getIBRef : ⦃ Hashable PreIBHeader Hash ⦄ → InputBlock → IBRef","searchableContent":"getibref : ⦃ hashable preibheader hash ⦄ → inputblock → ibref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":85,"title":"getIBRef = hash ∘ InputBlock.header","searchableContent":"getibref = hash ∘ inputblock.header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":87,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":88,"title":"-- Endorser Blocks","searchableContent":"-- endorser blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":89,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":91,"title":"record EndorserBlockOSig (sig ","content":"record EndorserBlockOSig (sig : Type) : Type where","searchableContent":"record endorserblockosig (sig : type) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":92,"title":"field slotNumber ","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":93,"title":"producerID ","content":"producerID : PoolID","searchableContent":"        producerid : poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":94,"title":"lotteryPf  ","content":"lotteryPf  : VrfPf","searchableContent":"        lotterypf  : vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":95,"title":"ibRefs     ","content":"ibRefs     : List IBRef","searchableContent":"        ibrefs     : list ibref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":96,"title":"ebRefs     ","content":"ebRefs     : List EBRef","searchableContent":"        ebrefs     : list ebref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":97,"title":"signature  ","content":"signature  : sig","searchableContent":"        signature  : sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":99,"title":"EndorserBlock    = EndorserBlockOSig Sig","searchableContent":"endorserblock    = endorserblockosig sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":100,"title":"PreEndorserBlock = EndorserBlockOSig ⊤","searchableContent":"preendorserblock = endorserblockosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":102,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":103,"title":"Hashable-EndorserBlock ","content":"Hashable-EndorserBlock : ⦃ Hashable PreEndorserBlock Hash ⦄ → Hashable EndorserBlock Hash","searchableContent":"  hashable-endorserblock : ⦃ hashable preendorserblock hash ⦄ → hashable endorserblock hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":104,"title":"Hashable-EndorserBlock .hash b = hash {T = PreEndorserBlock}","searchableContent":"  hashable-endorserblock .hash b = hash {t = preendorserblock}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":105,"title":"record { EndorserBlockOSig b hiding (signature) ; signature = _ }","searchableContent":"    record { endorserblockosig b hiding (signature) ; signature = _ }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":107,"title":"IsBlock-EndorserBlock ","content":"IsBlock-EndorserBlock : IsBlock EndorserBlock","searchableContent":"  isblock-endorserblock : isblock endorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":108,"title":"IsBlock-EndorserBlock = record { EndorserBlockOSig }","searchableContent":"  isblock-endorserblock = record { endorserblockosig }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":110,"title":"unquoteDecl DecEq-EndorserBlockOSig = derive-DecEq ((quote EndorserBlockOSig , DecEq-EndorserBlockOSig) ∷ [])","searchableContent":"unquotedecl deceq-endorserblockosig = derive-deceq ((quote endorserblockosig , deceq-endorserblockosig) ∷ [])"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":112,"title":"mkEB ","content":"mkEB : ⦃ Hashable PreEndorserBlock Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List IBRef → List EBRef → EndorserBlock","searchableContent":"mkeb : ⦃ hashable preendorserblock hash ⦄ → ℕ → poolid → vrfpf → privkey → list ibref → list ebref → endorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":113,"title":"mkEB slot id π pKey LI LE = record { signature = sign pKey (hash b) ; EndorserBlockOSig b }","searchableContent":"mkeb slot id π pkey li le = record { signature = sign pkey (hash b) ; endorserblockosig b }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":114,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":115,"title":"b ","content":"b : PreEndorserBlock","searchableContent":"    b : preendorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":116,"title":"b = record { slotNumber = slot","searchableContent":"    b = record { slotnumber = slot"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":117,"title":"; producerID = id","searchableContent":"               ; producerid = id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":118,"title":"; lotteryPf  = π","searchableContent":"               ; lotterypf  = π"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":119,"title":"; ibRefs     = LI","searchableContent":"               ; ibrefs     = li"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":120,"title":"; ebRefs     = LE","searchableContent":"               ; ebrefs     = le"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":121,"title":"; signature  = _","searchableContent":"               ; signature  = _"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":122,"title":"}","searchableContent":"               }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":124,"title":"getEBRef ","content":"getEBRef : ⦃ Hashable PreEndorserBlock Hash ⦄ → EndorserBlock → EBRef","searchableContent":"getebref : ⦃ hashable preendorserblock hash ⦄ → endorserblock → ebref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":125,"title":"getEBRef = hash","searchableContent":"getebref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":127,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":128,"title":"-- Votes","searchableContent":"-- votes"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":129,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":133,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":134,"title":"-- FFD for Leios Blocks","searchableContent":"-- ffd for leios blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":135,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":137,"title":"module GenFFD ⦃ _ ","content":"module GenFFD ⦃ _ : IsBlock (List Vote) ⦄ where","searchableContent":"module genffd ⦃ _ : isblock (list vote) ⦄ where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":138,"title":"data Header ","content":"data Header : Type where","searchableContent":"  data header : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":139,"title":"ibHeader ","content":"ibHeader : IBHeader → Header","searchableContent":"    ibheader : ibheader → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":140,"title":"ebHeader ","content":"ebHeader : EndorserBlock → Header","searchableContent":"    ebheader : endorserblock → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":141,"title":"vtHeader ","content":"vtHeader : List Vote → Header","searchableContent":"    vtheader : list vote → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":143,"title":"data Body ","content":"data Body : Type where","searchableContent":"  data body : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":144,"title":"ibBody ","content":"ibBody : IBBody → Body","searchableContent":"    ibbody : ibbody → body"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":146,"title":"unquoteDecl DecEq-Header DecEq-Body =","searchableContent":"  unquotedecl deceq-header deceq-body ="},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":147,"title":"derive-DecEq ((quote Header , DecEq-Header) ∷ (quote Body , DecEq-Body) ∷ [])","searchableContent":"    derive-deceq ((quote header , deceq-header) ∷ (quote body , deceq-body) ∷ [])"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":149,"title":"ID ","content":"ID : Type","searchableContent":"  id : type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":150,"title":"ID = ℕ × PoolID","searchableContent":"  id = ℕ × poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":152,"title":"matchIB ","content":"matchIB : IBHeader → IBBody → Type","searchableContent":"  matchib : ibheader → ibbody → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":153,"title":"matchIB h b = bodyHash ≡ hash b","searchableContent":"  matchib h b = bodyhash ≡ hash b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":154,"title":"where open IBHeaderOSig h; open IBBody b","searchableContent":"    where open ibheaderosig h; open ibbody b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":156,"title":"matchIB? ","content":"matchIB? :  ∀ (h : IBHeader) → (b : IBBody) → Dec (matchIB h b)","searchableContent":"  matchib? :  ∀ (h : ibheader) → (b : ibbody) → dec (matchib h b)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":157,"title":"matchIB? h b = bodyHash ≟ hash b","searchableContent":"  matchib? h b = bodyhash ≟ hash b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":158,"title":"where open IBHeaderOSig h; open IBBody b","searchableContent":"    where open ibheaderosig h; open ibbody b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":160,"title":"match ","content":"match : Header → Body → Type","searchableContent":"  match : header → body → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":161,"title":"match (ibHeader h) (ibBody b) = matchIB h b","searchableContent":"  match (ibheader h) (ibbody b) = matchib h b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":162,"title":"match _ _ = ⊥","searchableContent":"  match _ _ = ⊥"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":164,"title":"-- can we express uniqueness wrt pipelines as a property?","searchableContent":"  -- can we express uniqueness wrt pipelines as a property?"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":165,"title":"msgID ","content":"msgID : Header → ID","searchableContent":"  msgid : header → id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":166,"title":"msgID (ibHeader h) = (slotNumber h , producerID h)","searchableContent":"  msgid (ibheader h) = (slotnumber h , producerid h)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":167,"title":"msgID (ebHeader h) = (slotNumber h , producerID h) -- NOTE","content":"msgID (ebHeader h) = (slotNumber h , producerID h) -- NOTE: this isn't in the paper","searchableContent":"  msgid (ebheader h) = (slotnumber h , producerid h) -- note: this isn't in the paper"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":168,"title":"msgID (vtHeader  h) = (slotNumber h , producerID h) -- NOTE","content":"msgID (vtHeader  h) = (slotNumber h , producerID h) -- NOTE: this isn't in the paper","searchableContent":"  msgid (vtheader  h) = (slotnumber h , producerid h) -- note: this isn't in the paper"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":170,"title":"ffdAbstract ","content":"ffdAbstract : ⦃ _ : IsBlock (List Vote) ⦄ → FFDAbstract","searchableContent":"ffdabstract : ⦃ _ : isblock (list vote) ⦄ → ffdabstract"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":171,"title":"ffdAbstract = record { GenFFD }","searchableContent":"ffdabstract = record { genffd }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":172,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":2,"title":"Leios.Config
open import Leios.Prelude","searchableContent":"leios.config
open import leios.prelude"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":3,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":4,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":6,"title":"module Leios.Config where","searchableContent":"module leios.config where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":8,"title":"data BlockType ","content":"data BlockType : Type where","searchableContent":"data blocktype : type where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":9,"title":"IB EB VT ","content":"IB EB VT : BlockType","searchableContent":"  ib eb vt : blocktype"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":11,"title":"unquoteDecl DecEq-BlockType = derive-DecEq ((quote BlockType , DecEq-BlockType) ∷ [])","searchableContent":"unquotedecl deceq-blocktype = derive-deceq ((quote blocktype , deceq-blocktype) ∷ [])"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":13,"title":"record Params ","content":"record Params : Type where","searchableContent":"record params : type where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":14,"title":"field numberOfParties ","content":"field numberOfParties : ℕ","searchableContent":"  field numberofparties : ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":15,"title":"sutId ","content":"sutId : Fin numberOfParties","searchableContent":"        sutid : fin numberofparties"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":16,"title":"stakeDistribution ","content":"stakeDistribution : TotalMap (Fin numberOfParties) ℕ","searchableContent":"        stakedistribution : totalmap (fin numberofparties) ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":17,"title":"stageLength ","content":"stageLength : ℕ","searchableContent":"        stagelength : ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":18,"title":"⦃ NonZero-stageLength ⦄ ","content":"⦃ NonZero-stageLength ⦄ : NonZero stageLength","searchableContent":"        ⦃ nonzero-stagelength ⦄ : nonzero stagelength"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":19,"title":"winning-slots ","content":"winning-slots : ℙ (BlockType × ℕ)","searchableContent":"        winning-slots : ℙ (blocktype × ℕ)"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":20,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":2,"title":"Leios.Defaults
open import Leios.Prelude","searchableContent":"leios.defaults
open import leios.prelude"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":3,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":4,"title":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":5,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":7,"title":"open import Axiom.Set.Properties th","searchableContent":"open import axiom.set.properties th"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":8,"title":"open import Data.Nat.Show as N","searchableContent":"open import data.nat.show as n"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":9,"title":"open import Data.Integer hiding (_≟_)","searchableContent":"open import data.integer hiding (_≟_)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":10,"title":"open import Data.String as S using (intersperse)","searchableContent":"open import data.string as s using (intersperse)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":11,"title":"open import Function.Related.TypeIsomorphisms","searchableContent":"open import function.related.typeisomorphisms"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":12,"title":"open import Relation.Binary.Structures","searchableContent":"open import relation.binary.structures"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":14,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":15,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":17,"title":"open Equivalence","searchableContent":"open equivalence"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":19,"title":"-- The module contains very simple implementations for the functionalities","searchableContent":"-- the module contains very simple implementations for the functionalities"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":20,"title":"-- that allow to build examples for traces for the different Leios variants","searchableContent":"-- that allow to build examples for traces for the different leios variants"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":21,"title":"module Leios.Defaults (params ","content":"module Leios.Defaults (params : Params) (let open Params params) where","searchableContent":"module leios.defaults (params : params) (let open params params) where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":23,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":24,"title":"htx ","content":"htx : Hashable (List ℕ) (List ℕ)","searchableContent":"  htx : hashable (list ℕ) (list ℕ)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":25,"title":"htx = record { hash = id }","searchableContent":"  htx = record { hash = id }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":27,"title":"d-Abstract ","content":"d-Abstract : LeiosAbstract","searchableContent":"d-abstract : leiosabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":28,"title":"d-Abstract =","searchableContent":"d-abstract ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":29,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":30,"title":"{ Tx       = ℕ","searchableContent":"    { tx       = ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":31,"title":"; PoolID   = Fin numberOfParties","searchableContent":"    ; poolid   = fin numberofparties"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":32,"title":"; BodyHash = List ℕ","searchableContent":"    ; bodyhash = list ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":33,"title":"; VrfPf    = ⊤","searchableContent":"    ; vrfpf    = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":34,"title":"; PrivKey  = BlockType × ⊤","searchableContent":"    ; privkey  = blocktype × ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":35,"title":"; Sig      = ⊤","searchableContent":"    ; sig      = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":36,"title":"; Hash     = List ℕ","searchableContent":"    ; hash     = list ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":37,"title":"; Vote     = ⊤","searchableContent":"    ; vote     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":38,"title":"; vote     = λ _ _ → tt","searchableContent":"    ; vote     = λ _ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":39,"title":"; sign     = λ _ _ → tt","searchableContent":"    ; sign     = λ _ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":40,"title":"; L        = stageLength","searchableContent":"    ; l        = stagelength"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":41,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":43,"title":"open LeiosAbstract d-Abstract public","searchableContent":"open leiosabstract d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":45,"title":"open import Leios.VRF d-Abstract public","searchableContent":"open import leios.vrf d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":47,"title":"sutStake ","content":"sutStake : ℕ","searchableContent":"sutstake : ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":48,"title":"sutStake = TotalMap.lookup stakeDistribution sutId","searchableContent":"sutstake = totalmap.lookup stakedistribution sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":50,"title":"sortition ","content":"sortition : BlockType → ℕ → ℕ","searchableContent":"sortition : blocktype → ℕ → ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":51,"title":"sortition b n with (b , n) ∈? winning-slots","searchableContent":"sortition b n with (b , n) ∈? winning-slots"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":52,"title":"... | yes _ = 0","searchableContent":"... | yes _ = 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":53,"title":"... | no _ = sutStake","searchableContent":"... | no _ = sutstake"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":55,"title":"d-VRF ","content":"d-VRF : LeiosVRF","searchableContent":"d-vrf : leiosvrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":56,"title":"d-VRF =","searchableContent":"d-vrf ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":57,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":58,"title":"{ PubKey     = Fin numberOfParties × ⊤","searchableContent":"    { pubkey     = fin numberofparties × ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":59,"title":"; vrf        =","searchableContent":"    ; vrf        ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":60,"title":"record","searchableContent":"        record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":61,"title":"{ isKeyPair = λ _ _ → ⊤","searchableContent":"          { iskeypair = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":62,"title":"; eval      = λ (b , _) y → sortition b y , tt","searchableContent":"          ; eval      = λ (b , _) y → sortition b y , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":63,"title":"; verify    = λ _ _ _ _ → ⊤","searchableContent":"          ; verify    = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":64,"title":"; verify?   = λ _ _ _ _ → yes tt","searchableContent":"          ; verify?   = λ _ _ _ _ → yes tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":65,"title":"}","searchableContent":"          }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":66,"title":"; genIBInput = id","searchableContent":"    ; genibinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":67,"title":"; genEBInput = id","searchableContent":"    ; genebinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":68,"title":"; genVInput  = id","searchableContent":"    ; genvinput  = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":69,"title":"; genV1Input = id","searchableContent":"    ; genv1input = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":70,"title":"; genV2Input = id","searchableContent":"    ; genv2input = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":71,"title":"; poolID     = proj₁","searchableContent":"    ; poolid     = proj₁"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":72,"title":"; verifySig  = λ _ _ → ⊤","searchableContent":"    ; verifysig  = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":73,"title":"; verifySig? = λ _ _ → yes tt","searchableContent":"    ; verifysig? = λ _ _ → yes tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":74,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":76,"title":"open LeiosVRF d-VRF public","searchableContent":"open leiosvrf d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":78,"title":"open import Leios.Blocks d-Abstract public","searchableContent":"open import leios.blocks d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":79,"title":"open import Leios.KeyRegistration d-Abstract d-VRF public","searchableContent":"open import leios.keyregistration d-abstract d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":81,"title":"d-KeyRegistration ","content":"d-KeyRegistration : KeyRegistrationAbstract","searchableContent":"d-keyregistration : keyregistrationabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":82,"title":"d-KeyRegistration = _","searchableContent":"d-keyregistration = _"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":84,"title":"d-KeyRegistrationFunctionality ","content":"d-KeyRegistrationFunctionality : KeyRegistrationAbstract.Functionality d-KeyRegistration","searchableContent":"d-keyregistrationfunctionality : keyregistrationabstract.functionality d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":85,"title":"d-KeyRegistrationFunctionality =","searchableContent":"d-keyregistrationfunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":86,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":87,"title":"{ State     = ⊤","searchableContent":"    { state     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":88,"title":"; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤","searchableContent":"    ; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":89,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":91,"title":"open import Leios.Base d-Abstract d-VRF public","searchableContent":"open import leios.base d-abstract d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":93,"title":"d-Base ","content":"d-Base : BaseAbstract","searchableContent":"d-base : baseabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":94,"title":"d-Base =","searchableContent":"d-base ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":95,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":96,"title":"{ Cert       = ⊤","searchableContent":"    { cert       = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":97,"title":"; VTy        = ⊤","searchableContent":"    ; vty        = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":98,"title":"; initSlot   = λ _ → 0","searchableContent":"    ; initslot   = λ _ → 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":99,"title":"; V-chkCerts = λ _ _ → true","searchableContent":"    ; v-chkcerts = λ _ _ → true"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":100,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":102,"title":"d-BaseFunctionality ","content":"d-BaseFunctionality : BaseAbstract.Functionality d-Base","searchableContent":"d-basefunctionality : baseabstract.functionality d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":103,"title":"d-BaseFunctionality =","searchableContent":"d-basefunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":104,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":105,"title":"{ State         = ⊤","searchableContent":"    { state         = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":106,"title":"; _-⟦_/_⟧⇀_     = λ _ _ _ _ → ⊤","searchableContent":"    ; _-⟦_/_⟧⇀_     = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":107,"title":"; Dec-_-⟦_/_⟧⇀_ = ⁇ (yes tt)","searchableContent":"    ; dec-_-⟦_/_⟧⇀_ = ⁇ (yes tt)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":108,"title":"; SUBMIT-total  = tt , tt","searchableContent":"    ; submit-total  = tt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":109,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":111,"title":"open import Leios.FFD public","searchableContent":"open import leios.ffd public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":113,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":114,"title":"isb ","content":"isb : IsBlock (List ⊤)","searchableContent":"  isb : isblock (list ⊤)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":115,"title":"isb =","searchableContent":"  isb ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":116,"title":"record","searchableContent":"    record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":117,"title":"{ slotNumber = λ _ → 0","searchableContent":"      { slotnumber = λ _ → 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":118,"title":"; producerID = λ _ → sutId","searchableContent":"      ; producerid = λ _ → sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":119,"title":"; lotteryPf  = λ _ → tt","searchableContent":"      ; lotterypf  = λ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":120,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":122,"title":"hhs ","content":"hhs : Hashable PreIBHeader (List ℕ)","searchableContent":"  hhs : hashable preibheader (list ℕ)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":123,"title":"hhs .hash = IBHeaderOSig.bodyHash","searchableContent":"  hhs .hash = ibheaderosig.bodyhash"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":125,"title":"hpe ","content":"hpe : Hashable PreEndorserBlock (List ℕ)","searchableContent":"  hpe : hashable preendorserblock (list ℕ)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":126,"title":"hpe .hash = L.concat ∘ EndorserBlockOSig.ibRefs","searchableContent":"  hpe .hash = l.concat ∘ endorserblockosig.ibrefs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":128,"title":"record FFDBuffers ","content":"record FFDBuffers : Type where","searchableContent":"record ffdbuffers : type where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":129,"title":"field inIBs ","content":"field inIBs : List InputBlock","searchableContent":"  field inibs : list inputblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":130,"title":"inEBs ","content":"inEBs : List EndorserBlock","searchableContent":"        inebs : list endorserblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":131,"title":"inVTs ","content":"inVTs : List (List Vote)","searchableContent":"        invts : list (list vote)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":133,"title":"outIBs ","content":"outIBs : List InputBlock","searchableContent":"        outibs : list inputblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":134,"title":"outEBs ","content":"outEBs : List EndorserBlock","searchableContent":"        outebs : list endorserblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":135,"title":"outVTs ","content":"outVTs : List (List Vote)","searchableContent":"        outvts : list (list vote)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":137,"title":"unquoteDecl DecEq-FFDBuffers = derive-DecEq ((quote FFDBuffers , DecEq-FFDBuffers) ∷ [])","searchableContent":"unquotedecl deceq-ffdbuffers = derive-deceq ((quote ffdbuffers , deceq-ffdbuffers) ∷ [])"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":139,"title":"open GenFFD.Header","searchableContent":"open genffd.header"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":140,"title":"open GenFFD.Body","searchableContent":"open genffd.body"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":141,"title":"open FFDBuffers","searchableContent":"open ffdbuffers"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":143,"title":"flushIns ","content":"flushIns : FFDBuffers → List (GenFFD.Header ⊎ GenFFD.Body)","searchableContent":"flushins : ffdbuffers → list (genffd.header ⊎ genffd.body)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":144,"title":"flushIns record { inIBs = ibs ; inEBs = ebs ; inVTs = vts } =","searchableContent":"flushins record { inibs = ibs ; inebs = ebs ; invts = vts } ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":145,"title":"flushIBs ibs ++ L.map (inj₁ ∘ ebHeader) ebs ++ L.map (inj₁ ∘ vtHeader) vts","searchableContent":"  flushibs ibs ++ l.map (inj₁ ∘ ebheader) ebs ++ l.map (inj₁ ∘ vtheader) vts"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":146,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":147,"title":"flushIBs ","content":"flushIBs : List InputBlock → List (GenFFD.Header ⊎ GenFFD.Body)","searchableContent":"    flushibs : list inputblock → list (genffd.header ⊎ genffd.body)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":148,"title":"flushIBs [] = []","searchableContent":"    flushibs [] = []"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":149,"title":"flushIBs (record {header = h; body = b} ∷ ibs) = inj₁ (ibHeader h) ∷ inj₂ (ibBody b) ∷ flushIBs ibs","searchableContent":"    flushibs (record {header = h; body = b} ∷ ibs) = inj₁ (ibheader h) ∷ inj₂ (ibbody b) ∷ flushibs ibs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":151,"title":"data SimpleFFD ","content":"data SimpleFFD : FFDBuffers → FFDAbstract.Input ffdAbstract → FFDAbstract.Output ffdAbstract → FFDBuffers → Type where","searchableContent":"data simpleffd : ffdbuffers → ffdabstract.input ffdabstract → ffdabstract.output ffdabstract → ffdbuffers → type where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":152,"title":"SendIB ","content":"SendIB : ∀ {s h b}    → SimpleFFD s (FFDAbstract.Send (ibHeader h) (just (ibBody b))) FFDAbstract.SendRes (record s { outIBs = record {header = h; body = b} ∷ outIBs s})","searchableContent":"  sendib : ∀ {s h b}    → simpleffd s (ffdabstract.send (ibheader h) (just (ibbody b))) ffdabstract.sendres (record s { outibs = record {header = h; body = b} ∷ outibs s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":153,"title":"SendEB ","content":"SendEB : ∀ {s eb}     → SimpleFFD s (FFDAbstract.Send (ebHeader eb) nothing) FFDAbstract.SendRes (record s { outEBs = eb ∷ outEBs s})","searchableContent":"  sendeb : ∀ {s eb}     → simpleffd s (ffdabstract.send (ebheader eb) nothing) ffdabstract.sendres (record s { outebs = eb ∷ outebs s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":154,"title":"SendVS ","content":"SendVS : ∀ {s vs}     → SimpleFFD s (FFDAbstract.Send (vtHeader vs) nothing) FFDAbstract.SendRes (record s { outVTs = vs ∷ outVTs s})","searchableContent":"  sendvs : ∀ {s vs}     → simpleffd s (ffdabstract.send (vtheader vs) nothing) ffdabstract.sendres (record s { outvts = vs ∷ outvts s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":156,"title":"BadSendIB ","content":"BadSendIB : ∀ {s h}   → SimpleFFD s (FFDAbstract.Send (ibHeader h) nothing) FFDAbstract.SendRes s","searchableContent":"  badsendib : ∀ {s h}   → simpleffd s (ffdabstract.send (ibheader h) nothing) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":157,"title":"BadSendEB ","content":"BadSendEB : ∀ {s h b} → SimpleFFD s (FFDAbstract.Send (ebHeader h) (just b)) FFDAbstract.SendRes s","searchableContent":"  badsendeb : ∀ {s h b} → simpleffd s (ffdabstract.send (ebheader h) (just b)) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":158,"title":"BadSendVS ","content":"BadSendVS : ∀ {s h b} → SimpleFFD s (FFDAbstract.Send (vtHeader h) (just b)) FFDAbstract.SendRes s","searchableContent":"  badsendvs : ∀ {s h b} → simpleffd s (ffdabstract.send (vtheader h) (just b)) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":160,"title":"Fetch ","content":"Fetch : ∀ {s}         → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes (flushIns s)) (record s { inIBs = [] ; inEBs = [] ; inVTs = [] })","searchableContent":"  fetch : ∀ {s}         → simpleffd s ffdabstract.fetch (ffdabstract.fetchres (flushins s)) (record s { inibs = [] ; inebs = [] ; invts = [] })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":162,"title":"send-total ","content":"send-total : ∀ {s h b} → ∃[ s' ] (SimpleFFD s (FFDAbstract.Send h b) FFDAbstract.SendRes s')","searchableContent":"send-total : ∀ {s h b} → ∃[ s' ] (simpleffd s (ffdabstract.send h b) ffdabstract.sendres s')"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":163,"title":"send-total {s} {ibHeader h} {just (ibBody b)} = record s { outIBs = record {header = h; body = b} ∷ outIBs s} , SendIB","searchableContent":"send-total {s} {ibheader h} {just (ibbody b)} = record s { outibs = record {header = h; body = b} ∷ outibs s} , sendib"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":164,"title":"send-total {s} {ebHeader eb} {nothing}        = record s { outEBs = eb ∷ outEBs s} , SendEB","searchableContent":"send-total {s} {ebheader eb} {nothing}        = record s { outebs = eb ∷ outebs s} , sendeb"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":165,"title":"send-total {s} {vtHeader vs} {nothing}        = record s { outVTs = vs ∷ outVTs s} , SendVS","searchableContent":"send-total {s} {vtheader vs} {nothing}        = record s { outvts = vs ∷ outvts s} , sendvs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":167,"title":"send-total {s} {ibHeader h} {nothing} = s , BadSendIB","searchableContent":"send-total {s} {ibheader h} {nothing} = s , badsendib"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":168,"title":"send-total {s} {ebHeader eb} {just _} = s , BadSendEB","searchableContent":"send-total {s} {ebheader eb} {just _} = s , badsendeb"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":169,"title":"send-total {s} {vtHeader vs} {just _} = s , BadSendVS","searchableContent":"send-total {s} {vtheader vs} {just _} = s , badsendvs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":171,"title":"fetch-total ","content":"fetch-total : ∀ {s} → ∃[ x ] (∃[ s' ] (SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes x) s'))","searchableContent":"fetch-total : ∀ {s} → ∃[ x ] (∃[ s' ] (simpleffd s ffdabstract.fetch (ffdabstract.fetchres x) s'))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":172,"title":"fetch-total {s} = flushIns s , (record s { inIBs = [] ; inEBs = [] ; inVTs = [] } , Fetch)","searchableContent":"fetch-total {s} = flushins s , (record s { inibs = [] ; inebs = [] ; invts = [] } , fetch)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":174,"title":"send-complete ","content":"send-complete : ∀ {s h b s'} → SimpleFFD s (FFDAbstract.Send h b) FFDAbstract.SendRes s' → s' ≡ proj₁ (send-total {s} {h} {b})","searchableContent":"send-complete : ∀ {s h b s'} → simpleffd s (ffdabstract.send h b) ffdabstract.sendres s' → s' ≡ proj₁ (send-total {s} {h} {b})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":175,"title":"send-complete SendIB    = refl","searchableContent":"send-complete sendib    = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":176,"title":"send-complete SendEB    = refl","searchableContent":"send-complete sendeb    = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":177,"title":"send-complete SendVS    = refl","searchableContent":"send-complete sendvs    = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":178,"title":"send-complete BadSendIB = refl","searchableContent":"send-complete badsendib = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":179,"title":"send-complete BadSendEB = refl","searchableContent":"send-complete badsendeb = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":180,"title":"send-complete BadSendVS = refl","searchableContent":"send-complete badsendvs = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":182,"title":"fetch-complete₁ ","content":"fetch-complete₁ : ∀ {s r s'} → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes r) s' → s' ≡ proj₁ (proj₂ (fetch-total {s}))","searchableContent":"fetch-complete₁ : ∀ {s r s'} → simpleffd s ffdabstract.fetch (ffdabstract.fetchres r) s' → s' ≡ proj₁ (proj₂ (fetch-total {s}))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":183,"title":"fetch-complete₁ Fetch = refl","searchableContent":"fetch-complete₁ fetch = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":185,"title":"fetch-complete₂ ","content":"fetch-complete₂ : ∀ {s r s'} → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes r) s' → r ≡ proj₁ (fetch-total {s})","searchableContent":"fetch-complete₂ : ∀ {s r s'} → simpleffd s ffdabstract.fetch (ffdabstract.fetchres r) s' → r ≡ proj₁ (fetch-total {s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":186,"title":"fetch-complete₂ Fetch = refl","searchableContent":"fetch-complete₂ fetch = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":188,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":189,"title":"Dec-SimpleFFD ","content":"Dec-SimpleFFD : ∀ {s i o s'} → SimpleFFD s i o s' ⁇","searchableContent":"  dec-simpleffd : ∀ {s i o s'} → simpleffd s i o s' ⁇"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":190,"title":"Dec-SimpleFFD {s} {FFDAbstract.Send h b} {FFDAbstract.SendRes} {s'} with s' ≟ proj₁ (send-total {s} {h} {b})","searchableContent":"  dec-simpleffd {s} {ffdabstract.send h b} {ffdabstract.sendres} {s'} with s' ≟ proj₁ (send-total {s} {h} {b})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":191,"title":"... | yes p rewrite p = ⁇ yes (proj₂ send-total)","searchableContent":"  ... | yes p rewrite p = ⁇ yes (proj₂ send-total)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":192,"title":"... | no ¬p = ⁇ no λ x → ⊥-elim (¬p (send-complete x))","searchableContent":"  ... | no ¬p = ⁇ no λ x → ⊥-elim (¬p (send-complete x))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":193,"title":"Dec-SimpleFFD {_} {FFDAbstract.Send _ _} {FFDAbstract.FetchRes _} {_} = ⁇ no λ ()","searchableContent":"  dec-simpleffd {_} {ffdabstract.send _ _} {ffdabstract.fetchres _} {_} = ⁇ no λ ()"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":194,"title":"Dec-SimpleFFD {s} {FFDAbstract.Fetch} {FFDAbstract.FetchRes r} {s'}","searchableContent":"  dec-simpleffd {s} {ffdabstract.fetch} {ffdabstract.fetchres r} {s'}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":195,"title":"with s' ≟ proj₁ (proj₂ (fetch-total {s}))","searchableContent":"    with s' ≟ proj₁ (proj₂ (fetch-total {s}))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":196,"title":"| r ≟ proj₁ (fetch-total {s})","searchableContent":"      | r ≟ proj₁ (fetch-total {s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":197,"title":"... | yes p | yes q rewrite p rewrite q = ⁇ yes (proj₂ (proj₂ (fetch-total {s})))","searchableContent":"  ... | yes p | yes q rewrite p rewrite q = ⁇ yes (proj₂ (proj₂ (fetch-total {s})))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":198,"title":"... | yes p | no ¬q = ⁇ no λ x → ⊥-elim (¬q (fetch-complete₂ x))","searchableContent":"  ... | yes p | no ¬q = ⁇ no λ x → ⊥-elim (¬q (fetch-complete₂ x))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":199,"title":"... | no ¬p | _ = ⁇ no λ x → ⊥-elim (¬p (fetch-complete₁ x))","searchableContent":"  ... | no ¬p | _ = ⁇ no λ x → ⊥-elim (¬p (fetch-complete₁ x))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":200,"title":"Dec-SimpleFFD {_} {FFDAbstract.Fetch} {FFDAbstract.SendRes} {_} = ⁇ no λ ()","searchableContent":"  dec-simpleffd {_} {ffdabstract.fetch} {ffdabstract.sendres} {_} = ⁇ no λ ()"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":202,"title":"d-FFDFunctionality ","content":"d-FFDFunctionality : FFDAbstract.Functionality ffdAbstract","searchableContent":"d-ffdfunctionality : ffdabstract.functionality ffdabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":203,"title":"d-FFDFunctionality =","searchableContent":"d-ffdfunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":204,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":205,"title":"{ State         = FFDBuffers","searchableContent":"    { state         = ffdbuffers"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":206,"title":"; initFFDState  = record { inIBs = []; inEBs = []; inVTs = []; outIBs = []; outEBs = []; outVTs = [] }","searchableContent":"    ; initffdstate  = record { inibs = []; inebs = []; invts = []; outibs = []; outebs = []; outvts = [] }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":207,"title":"; _-⟦_/_⟧⇀_     = SimpleFFD","searchableContent":"    ; _-⟦_/_⟧⇀_     = simpleffd"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":208,"title":"; Dec-_-⟦_/_⟧⇀_ = Dec-SimpleFFD","searchableContent":"    ; dec-_-⟦_/_⟧⇀_ = dec-simpleffd"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":209,"title":"; Send-total    = send-total","searchableContent":"    ; send-total    = send-total"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":210,"title":"; Fetch-total   = fetch-total","searchableContent":"    ; fetch-total   = fetch-total"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":211,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":213,"title":"open import Leios.Voting public","searchableContent":"open import leios.voting public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":215,"title":"d-VotingAbstract ","content":"d-VotingAbstract : VotingAbstract (Fin 1 × EndorserBlock)","searchableContent":"d-votingabstract : votingabstract (fin 1 × endorserblock)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":216,"title":"d-VotingAbstract =","searchableContent":"d-votingabstract ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":217,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":218,"title":"{ VotingState     = ⊤","searchableContent":"    { votingstate     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":219,"title":"; initVotingState = tt","searchableContent":"    ; initvotingstate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":220,"title":"; isVoteCertified = λ _ _ → ⊤","searchableContent":"    ; isvotecertified = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":221,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":223,"title":"d-VotingAbstract-2 ","content":"d-VotingAbstract-2 : VotingAbstract (Fin 2 × EndorserBlock)","searchableContent":"d-votingabstract-2 : votingabstract (fin 2 × endorserblock)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":224,"title":"d-VotingAbstract-2 =","searchableContent":"d-votingabstract-2 ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":225,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":226,"title":"{ VotingState     = ⊤","searchableContent":"    { votingstate     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":227,"title":"; initVotingState = tt","searchableContent":"    ; initvotingstate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":228,"title":"; isVoteCertified = λ _ _ → ⊤","searchableContent":"    ; isvotecertified = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":229,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":231,"title":"d-SpecStructure ","content":"d-SpecStructure : SpecStructure 1","searchableContent":"d-specstructure : specstructure 1"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":232,"title":"d-SpecStructure = record","searchableContent":"d-specstructure = record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":233,"title":"{ a                         = d-Abstract","searchableContent":"      { a                         = d-abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":234,"title":"; Hashable-PreIBHeader      = hhs","searchableContent":"      ; hashable-preibheader      = hhs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":235,"title":"; Hashable-PreEndorserBlock = hpe","searchableContent":"      ; hashable-preendorserblock = hpe"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":236,"title":"; id                        = sutId","searchableContent":"      ; id                        = sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":237,"title":"; FFD'                      = d-FFDFunctionality","searchableContent":"      ; ffd'                      = d-ffdfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":238,"title":"; vrf'                      = d-VRF","searchableContent":"      ; vrf'                      = d-vrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":239,"title":"; sk-IB                     = IB , tt","searchableContent":"      ; sk-ib                     = ib , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":240,"title":"; sk-EB                     = EB , tt","searchableContent":"      ; sk-eb                     = eb , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":241,"title":"; sk-VT                     = VT , tt","searchableContent":"      ; sk-vt                     = vt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":242,"title":"; pk-IB                     = sutId , tt","searchableContent":"      ; pk-ib                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":243,"title":"; pk-EB                     = sutId , tt","searchableContent":"      ; pk-eb                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":244,"title":"; pk-VT                     = sutId , tt","searchableContent":"      ; pk-vt                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":245,"title":"; B'                        = d-Base","searchableContent":"      ; b'                        = d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":246,"title":"; BF                        = d-BaseFunctionality","searchableContent":"      ; bf                        = d-basefunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":247,"title":"; initBaseState             = tt","searchableContent":"      ; initbasestate             = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":248,"title":"; K'                        = d-KeyRegistration","searchableContent":"      ; k'                        = d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":249,"title":"; KF                        = d-KeyRegistrationFunctionality","searchableContent":"      ; kf                        = d-keyregistrationfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":250,"title":"; va                        = d-VotingAbstract","searchableContent":"      ; va                        = d-votingabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":251,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":253,"title":"d-SpecStructure-2 ","content":"d-SpecStructure-2 : SpecStructure 2","searchableContent":"d-specstructure-2 : specstructure 2"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":254,"title":"d-SpecStructure-2 = record","searchableContent":"d-specstructure-2 = record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":255,"title":"{ a                         = d-Abstract","searchableContent":"      { a                         = d-abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":256,"title":"; Hashable-PreIBHeader      = hhs","searchableContent":"      ; hashable-preibheader      = hhs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":257,"title":"; Hashable-PreEndorserBlock = hpe","searchableContent":"      ; hashable-preendorserblock = hpe"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":258,"title":"; id                        = sutId","searchableContent":"      ; id                        = sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":259,"title":"; FFD'                      = d-FFDFunctionality","searchableContent":"      ; ffd'                      = d-ffdfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":260,"title":"; vrf'                      = d-VRF","searchableContent":"      ; vrf'                      = d-vrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":261,"title":"; sk-IB                     = IB , tt","searchableContent":"      ; sk-ib                     = ib , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":262,"title":"; sk-EB                     = EB , tt","searchableContent":"      ; sk-eb                     = eb , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":263,"title":"; sk-VT                     = VT , tt","searchableContent":"      ; sk-vt                     = vt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":264,"title":"; pk-IB                     = sutId , tt","searchableContent":"      ; pk-ib                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":265,"title":"; pk-EB                     = sutId , tt","searchableContent":"      ; pk-eb                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":266,"title":"; pk-VT                     = sutId , tt","searchableContent":"      ; pk-vt                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":267,"title":"; B'                        = d-Base","searchableContent":"      ; b'                        = d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":268,"title":"; BF                        = d-BaseFunctionality","searchableContent":"      ; bf                        = d-basefunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":269,"title":"; initBaseState             = tt","searchableContent":"      ; initbasestate             = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":270,"title":"; K'                        = d-KeyRegistration","searchableContent":"      ; k'                        = d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":271,"title":"; KF                        = d-KeyRegistrationFunctionality","searchableContent":"      ; kf                        = d-keyregistrationfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":272,"title":"; va                        = d-VotingAbstract-2","searchableContent":"      ; va                        = d-votingabstract-2"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":273,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":275,"title":"open import Leios.Short d-SpecStructure public","searchableContent":"open import leios.short d-specstructure public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":276,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":2,"title":"Leios.FFD
{-# OPTIONS --safe #-}","searchableContent":"leios.ffd
{-# options --safe #-}"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":4,"title":"module Leios.FFD where","searchableContent":"module leios.ffd where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":6,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":8,"title":"record FFDAbstract ","content":"record FFDAbstract : Type₁ where","searchableContent":"record ffdabstract : type₁ where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":9,"title":"field Header Body ID ","content":"field Header Body ID : Type","searchableContent":"  field header body id : type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":10,"title":"⦃ DecEq-Header ⦄ ","content":"⦃ DecEq-Header ⦄ : DecEq Header","searchableContent":"        ⦃ deceq-header ⦄ : deceq header"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":11,"title":"⦃ DecEq-Body ⦄ ","content":"⦃ DecEq-Body ⦄ : DecEq Body","searchableContent":"        ⦃ deceq-body ⦄ : deceq body"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":12,"title":"match ","content":"match : Header → Body → Type","searchableContent":"        match : header → body → type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":13,"title":"msgID ","content":"msgID : Header → ID","searchableContent":"        msgid : header → id"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":15,"title":"data Input ","content":"data Input : Type where","searchableContent":"  data input : type where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":16,"title":"Send  ","content":"Send  : Header → Maybe Body → Input","searchableContent":"    send  : header → maybe body → input"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":17,"title":"Fetch ","content":"Fetch : Input","searchableContent":"    fetch : input"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":19,"title":"data Output ","content":"data Output : Type where","searchableContent":"  data output : type where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":20,"title":"SendRes  ","content":"SendRes  : Output","searchableContent":"    sendres  : output"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":21,"title":"FetchRes ","content":"FetchRes : List (Header ⊎ Body) → Output","searchableContent":"    fetchres : list (header ⊎ body) → output"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":23,"title":"record Functionality ","content":"record Functionality : Type₁ where","searchableContent":"  record functionality : type₁ where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":24,"title":"field State ","content":"field State : Type","searchableContent":"    field state : type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":25,"title":"initFFDState ","content":"initFFDState : State","searchableContent":"          initffdstate : state"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":26,"title":"_-⟦_/_⟧⇀_ ","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"          _-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":27,"title":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ ","content":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : {s : State} → {i : Input} → {o : Output} → {s' : State} → (s -⟦ i / o ⟧⇀ s') ⁇","searchableContent":"          ⦃ dec-_-⟦_/_⟧⇀_ ⦄ : {s : state} → {i : input} → {o : output} → {s' : state} → (s -⟦ i / o ⟧⇀ s') ⁇"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":28,"title":"Send-total ","content":"Send-total : ∀ {ffds h b} → ∃[ ffds' ] ffds -⟦ Send h b / SendRes ⟧⇀ ffds'","searchableContent":"          send-total : ∀ {ffds h b} → ∃[ ffds' ] ffds -⟦ send h b / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":29,"title":"Fetch-total ","content":"Fetch-total : ∀ {ffds} → ∃[ r ] (∃[ ffds' ] (ffds -⟦ Fetch / FetchRes r ⟧⇀ ffds'))","searchableContent":"          fetch-total : ∀ {ffds} → ∃[ r ] (∃[ ffds' ] (ffds -⟦ fetch / fetchres r ⟧⇀ ffds'))"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":31,"title":"open Input public","searchableContent":"    open input public"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":32,"title":"open Output public","searchableContent":"    open output public"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":33,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.BaseTypes
module Leios.Foreign.BaseTypes where","searchableContent":"leios.foreign.basetypes
module leios.foreign.basetypes where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":4,"title":"-- TODO","content":"-- TODO: copied from the formal-ledger project for now","searchableContent":"-- todo: copied from the formal-ledger project for now"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":5,"title":"-- Added","content":"-- Added: * TotalMap","searchableContent":"-- added: * totalmap"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":7,"title":"open import Data.Rational","searchableContent":"open import data.rational"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":11,"title":"open import Data.Fin","searchableContent":"open import data.fin"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":12,"title":"open import Function.Related.TypeIsomorphisms","searchableContent":"open import function.related.typeisomorphisms"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":13,"title":"open import Relation.Binary.Structures","searchableContent":"open import relation.binary.structures"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":15,"title":"open import Tactic.Derive.Convertible","searchableContent":"open import tactic.derive.convertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":16,"title":"open import Tactic.Derive.HsType","searchableContent":"open import tactic.derive.hstype"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":18,"title":"open import Class.Convertible","searchableContent":"open import class.convertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":19,"title":"open import Class.Decidable.Instances","searchableContent":"open import class.decidable.instances"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":20,"title":"open import Class.HasHsType","searchableContent":"open import class.hashstype"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":22,"title":"open import Leios.Foreign.HsTypes as F","searchableContent":"open import leios.foreign.hstypes as f"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":23,"title":"open import Leios.Foreign.Util","searchableContent":"open import leios.foreign.util"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":24,"title":"open import Foreign.Haskell","searchableContent":"open import foreign.haskell"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":26,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":27,"title":"iConvTop    = Convertible-Refl {⊤}","searchableContent":"  iconvtop    = convertible-refl {⊤}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":28,"title":"iConvNat    = Convertible-Refl {ℕ}","searchableContent":"  iconvnat    = convertible-refl {ℕ}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":29,"title":"iConvString = Convertible-Refl {String}","searchableContent":"  iconvstring = convertible-refl {string}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":30,"title":"iConvBool   = Convertible-Refl {Bool}","searchableContent":"  iconvbool   = convertible-refl {bool}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":32,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":34,"title":"-- * Unit and empty","searchableContent":"  -- * unit and empty"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":36,"title":"HsTy-⊥ = MkHsType ⊥ F.Empty","searchableContent":"  hsty-⊥ = mkhstype ⊥ f.empty"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":37,"title":"Conv-⊥ = autoConvert ⊥","searchableContent":"  conv-⊥ = autoconvert ⊥"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":39,"title":"HsTy-⊤ = MkHsType ⊤ ⊤","searchableContent":"  hsty-⊤ = mkhstype ⊤ ⊤"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":41,"title":"-- * Rational numbers","searchableContent":"  -- * rational numbers"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":43,"title":"HsTy-Rational = MkHsType ℚ F.Rational","searchableContent":"  hsty-rational = mkhstype ℚ f.rational"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":44,"title":"Conv-Rational ","content":"Conv-Rational : HsConvertible ℚ","searchableContent":"  conv-rational : hsconvertible ℚ"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":45,"title":"Conv-Rational = λ where","searchableContent":"  conv-rational = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":46,"title":".to (mkℚ n d _)       → n F., suc d","searchableContent":"    .to (mkℚ n d _)       → n f., suc d"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":47,"title":".from (n F., zero)    → 0ℚ -- TODO is there a safer way to do this?","searchableContent":"    .from (n f., zero)    → 0ℚ -- todo is there a safer way to do this?"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":48,"title":".from (n F., (suc d)) → n Data.Rational./ suc d","searchableContent":"    .from (n f., (suc d)) → n data.rational./ suc d"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":50,"title":"-- * Maps and Sets","searchableContent":"  -- * maps and sets"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":52,"title":"HsTy-HSSet ","content":"HsTy-HSSet : ∀ {A} → ⦃ HasHsType A ⦄ → HasHsType (ℙ A)","searchableContent":"  hsty-hsset : ∀ {a} → ⦃ hashstype a ⦄ → hashstype (ℙ a)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":53,"title":"HsTy-HSSet {A} = MkHsType _ (F.HSSet (HsType A))","searchableContent":"  hsty-hsset {a} = mkhstype _ (f.hsset (hstype a))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":55,"title":"Conv-HSSet ","content":"Conv-HSSet : ∀ {A} ⦃ _ : HasHsType A ⦄","searchableContent":"  conv-hsset : ∀ {a} ⦃ _ : hashstype a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":56,"title":"→ ⦃ HsConvertible A ⦄","searchableContent":"             → ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":57,"title":"→ HsConvertible (ℙ A)","searchableContent":"             → hsconvertible (ℙ a)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":58,"title":"Conv-HSSet = λ where","searchableContent":"  conv-hsset = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":59,"title":".to   → F.MkHSSet ∘ to ∘ setToList","searchableContent":"    .to   → f.mkhsset ∘ to ∘ settolist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":60,"title":".from → fromList ∘ from ∘ F.HSSet.elems","searchableContent":"    .from → fromlist ∘ from ∘ f.hsset.elems"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":62,"title":"Convertible-FinSet ","content":"Convertible-FinSet : Convertible₁ ℙ_ List","searchableContent":"  convertible-finset : convertible₁ ℙ_ list"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":63,"title":"Convertible-FinSet = λ where","searchableContent":"  convertible-finset = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":64,"title":".to   → map to   ∘ setToList","searchableContent":"    .to   → map to   ∘ settolist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":65,"title":".from → fromList ∘ map from","searchableContent":"    .from → fromlist ∘ map from"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":67,"title":"Convertible-Map ","content":"Convertible-Map : ∀ {K K' V V'} → ⦃ DecEq K ⦄","searchableContent":"  convertible-map : ∀ {k k' v v'} → ⦃ deceq k ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":68,"title":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","searchableContent":"    → ⦃ convertible k k' ⦄ → ⦃ convertible v v' ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":69,"title":"→ Convertible (K ⇀ V) (List $ Pair K' V')","searchableContent":"    → convertible (k ⇀ v) (list $ pair k' v')"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":70,"title":"Convertible-Map = λ where","searchableContent":"  convertible-map = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":71,"title":".to   → to        ∘ proj₁","searchableContent":"    .to   → to        ∘ proj₁"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":72,"title":".from → fromListᵐ ∘ map from","searchableContent":"    .from → fromlistᵐ ∘ map from"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":74,"title":"HsTy-Map ","content":"HsTy-Map : ∀ {A B} → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (A ⇀ B)","searchableContent":"  hsty-map : ∀ {a b} → ⦃ hashstype a ⦄ → ⦃ hashstype b ⦄ → hashstype (a ⇀ b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":75,"title":"HsTy-Map {A} {B} = MkHsType _ (F.HSMap (HsType A) (HsType B))","searchableContent":"  hsty-map {a} {b} = mkhstype _ (f.hsmap (hstype a) (hstype b))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":77,"title":"Conv-HSMap ","content":"Conv-HSMap : ∀ {A B} ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄","searchableContent":"  conv-hsmap : ∀ {a b} ⦃ _ : hashstype a ⦄ ⦃ _ : hashstype b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":78,"title":"→ ⦃ DecEq A ⦄","searchableContent":"    → ⦃ deceq a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":79,"title":"→ ⦃ HsConvertible A ⦄","searchableContent":"    → ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":80,"title":"→ ⦃ HsConvertible B ⦄","searchableContent":"    → ⦃ hsconvertible b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":81,"title":"→ HsConvertible (A ⇀ B)","searchableContent":"    → hsconvertible (a ⇀ b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":82,"title":"Conv-HSMap = λ where","searchableContent":"  conv-hsmap = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":83,"title":".to   → F.MkHSMap ∘ to","searchableContent":"    .to   → f.mkhsmap ∘ to"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":84,"title":".from → from ∘ F.HSMap.assocList","searchableContent":"    .from → from ∘ f.hsmap.assoclist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":86,"title":"Convertible-TotalMap ","content":"Convertible-TotalMap : ∀ {K K' V V'} → ⦃ DecEq K ⦄ → ⦃ Listable K ⦄","searchableContent":"  convertible-totalmap : ∀ {k k' v v'} → ⦃ deceq k ⦄ → ⦃ listable k ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":87,"title":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","searchableContent":"    → ⦃ convertible k k' ⦄ → ⦃ convertible v v' ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":88,"title":"→ Convertible (TotalMap K V) (List $ Pair K' V')","searchableContent":"    → convertible (totalmap k v) (list $ pair k' v')"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":89,"title":"Convertible-TotalMap {K} = λ where","searchableContent":"  convertible-totalmap {k} = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":90,"title":".to   → to ∘ TotalMap.rel","searchableContent":"    .to   → to ∘ totalmap.rel"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":91,"title":".from → λ x →","searchableContent":"    .from → λ x →"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":92,"title":"let (r , l) = fromListᵐ (map from x)","searchableContent":"      let (r , l) = fromlistᵐ (map from x)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":93,"title":"in case (¿ total r ¿) of λ where","searchableContent":"      in case (¿ total r ¿) of λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":94,"title":"(yes p) → record { rel = r ; left-unique-rel = l ; total-rel = p }","searchableContent":"           (yes p) → record { rel = r ; left-unique-rel = l ; total-rel = p }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":95,"title":"(no p) → error "Expected total map"","searchableContent":"           (no p) → error "expected total map""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":97,"title":"HsTy-TotalMap ","content":"HsTy-TotalMap : ∀ {A B} → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (TotalMap A B)","searchableContent":"  hsty-totalmap : ∀ {a b} → ⦃ hashstype a ⦄ → ⦃ hashstype b ⦄ → hashstype (totalmap a b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":98,"title":"HsTy-TotalMap {A} {B} = MkHsType _ (F.HSMap (HsType A) (HsType B))","searchableContent":"  hsty-totalmap {a} {b} = mkhstype _ (f.hsmap (hstype a) (hstype b))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":100,"title":"Conv-HSTotalMap ","content":"Conv-HSTotalMap : ∀ {A B} ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄","searchableContent":"  conv-hstotalmap : ∀ {a b} ⦃ _ : hashstype a ⦄ ⦃ _ : hashstype b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":101,"title":"→ ⦃ DecEq A ⦄","searchableContent":"    → ⦃ deceq a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":102,"title":"→ ⦃ Listable A ⦄","searchableContent":"    → ⦃ listable a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":103,"title":"→ ⦃ HsConvertible A ⦄","searchableContent":"    → ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":104,"title":"→ ⦃ HsConvertible B ⦄","searchableContent":"    → ⦃ hsconvertible b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":105,"title":"→ HsConvertible (TotalMap A B)","searchableContent":"    → hsconvertible (totalmap a b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":106,"title":"Conv-HSTotalMap = λ where","searchableContent":"  conv-hstotalmap = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":107,"title":".to   → MkHSMap ∘ to","searchableContent":"    .to   → mkhsmap ∘ to"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":108,"title":".from → from ∘ F.HSMap.assocList","searchableContent":"    .from → from ∘ f.hsmap.assoclist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":110,"title":"-- * ComputationResult","searchableContent":"  -- * computationresult"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":112,"title":"open import Class.Computational as C","searchableContent":"  open import class.computational as c"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":114,"title":"HsTy-ComputationResult ","content":"HsTy-ComputationResult : ∀ {l} {Err} {A : Type l}","searchableContent":"  hsty-computationresult : ∀ {l} {err} {a : type l}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":115,"title":"→ ⦃ HasHsType Err ⦄ → ⦃ HasHsType A ⦄","searchableContent":"                           → ⦃ hashstype err ⦄ → ⦃ hashstype a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":116,"title":"→ HasHsType (C.ComputationResult Err A)","searchableContent":"                           → hashstype (c.computationresult err a)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":117,"title":"HsTy-ComputationResult {Err = Err} {A} = MkHsType _ (F.ComputationResult (HsType Err) (HsType A))","searchableContent":"  hsty-computationresult {err = err} {a} = mkhstype _ (f.computationresult (hstype err) (hstype a))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":119,"title":"Conv-ComputationResult ","content":"Conv-ComputationResult : ConvertibleType C.ComputationResult F.ComputationResult","searchableContent":"  conv-computationresult : convertibletype c.computationresult f.computationresult"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":120,"title":"Conv-ComputationResult = autoConvertible","searchableContent":"  conv-computationresult = autoconvertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":121,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.HsTypes
module Leios.Foreign.HsTypes where","searchableContent":"leios.foreign.hstypes
module leios.foreign.hstypes where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":4,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":5,"title":"{-# LANGUAGE DeriveGeneric #-}","searchableContent":"  {-# language derivegeneric #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":6,"title":"{-# LANGUAGE DeriveFunctor #-}","searchableContent":"  {-# language derivefunctor #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":7,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":11,"title":"open import Foreign.Haskell","searchableContent":"open import foreign.haskell"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":12,"title":"open import Foreign.Haskell.Coerce","searchableContent":"open import foreign.haskell.coerce"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":13,"title":"open import Foreign.Haskell.Either","searchableContent":"open import foreign.haskell.either"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":14,"title":"open import Data.Rational.Base","searchableContent":"open import data.rational.base"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":16,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":17,"title":"import GHC.Generics (Generic)","searchableContent":"  import ghc.generics (generic)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":18,"title":"import Data.Void (Void)","searchableContent":"  import data.void (void)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":19,"title":"import Prelude hiding (Rational)","searchableContent":"  import prelude hiding (rational)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":20,"title":"import GHC.Real (Ratio(..))","searchableContent":"  import ghc.real (ratio(..))"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":21,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":23,"title":"-- * The empty type","searchableContent":"-- * the empty type"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":25,"title":"data Empty ","content":"data Empty : Type where","searchableContent":"data empty : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":26,"title":"{-# COMPILE GHC Empty = data Void () #-}","searchableContent":"{-# compile ghc empty = data void () #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":28,"title":"-- * Rational","searchableContent":"-- * rational"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":30,"title":"data Rational ","content":"data Rational : Type where","searchableContent":"data rational : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":31,"title":"_,_ ","content":"_,_ : ℤ → ℕ → Rational","searchableContent":"  _,_ : ℤ → ℕ → rational"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":32,"title":"{-# COMPILE GHC Rational = data Rational ((","content":"{-# COMPILE GHC Rational = data Rational ((:%)) #-}","searchableContent":"{-# compile ghc rational = data rational ((:%)) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":34,"title":"-- We'll generate code with qualified references to Rational in this","searchableContent":"-- we'll generate code with qualified references to rational in this"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":35,"title":"-- module, so make sure to define it.","searchableContent":"-- module, so make sure to define it."},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":36,"title":"{-# FOREIGN GHC type Rational = Ratio Integer #-}","searchableContent":"{-# foreign ghc type rational = ratio integer #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":38,"title":"-- * Maps and Sets","searchableContent":"-- * maps and sets"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":40,"title":"record HSMap K V ","content":"record HSMap K V : Type where","searchableContent":"record hsmap k v : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":41,"title":"constructor MkHSMap","searchableContent":"  constructor mkhsmap"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":42,"title":"field assocList ","content":"field assocList : List (Pair K V)","searchableContent":"  field assoclist : list (pair k v)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":44,"title":"record HSSet A ","content":"record HSSet A : Type where","searchableContent":"record hsset a : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":45,"title":"constructor MkHSSet","searchableContent":"  constructor mkhsset"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":46,"title":"field elems ","content":"field elems : List A","searchableContent":"  field elems : list a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":48,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":49,"title":"newtype HSMap k v = MkHSMap [(k, v)]","searchableContent":"  newtype hsmap k v = mkhsmap [(k, v)]"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":50,"title":"deriving (Generic, Show, Eq, Ord)","searchableContent":"    deriving (generic, show, eq, ord)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":51,"title":"newtype HSSet a = MkHSSet [a]","searchableContent":"  newtype hsset a = mkhsset [a]"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":52,"title":"deriving (Generic, Show, Eq, Ord)","searchableContent":"    deriving (generic, show, eq, ord)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":53,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":54,"title":"{-# COMPILE GHC HSMap = data HSMap (MkHSMap) #-}","searchableContent":"{-# compile ghc hsmap = data hsmap (mkhsmap) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":55,"title":"{-# COMPILE GHC HSSet = data HSSet (MkHSSet) #-}","searchableContent":"{-# compile ghc hsset = data hsset (mkhsset) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":57,"title":"-- * ComputationResult","searchableContent":"-- * computationresult"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":59,"title":"data ComputationResult E A ","content":"data ComputationResult E A : Type where","searchableContent":"data computationresult e a : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":60,"title":"Success ","content":"Success : A → ComputationResult E A","searchableContent":"  success : a → computationresult e a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":61,"title":"Failure ","content":"Failure : E → ComputationResult E A","searchableContent":"  failure : e → computationresult e a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":63,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":64,"title":"data ComputationResult e a = Success a | Failure e","searchableContent":"  data computationresult e a = success a | failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":65,"title":"deriving (Functor, Eq, Show, Generic)","searchableContent":"    deriving (functor, eq, show, generic)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":67,"title":"instance Applicative (ComputationResult e) where","searchableContent":"  instance applicative (computationresult e) where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":68,"title":"pure = Success","searchableContent":"    pure = success"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":69,"title":"(Success f) <*> x = f <$> x","searchableContent":"    (success f) <*> x = f <$> x"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":70,"title":"(Failure e) <*> _ = Failure e","searchableContent":"    (failure e) <*> _ = failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":72,"title":"instance Monad (ComputationResult e) where","searchableContent":"  instance monad (computationresult e) where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":73,"title":"return = pure","searchableContent":"    return = pure"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":74,"title":"(Success a) >>= m = m a","searchableContent":"    (success a) >>= m = m a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":75,"title":"(Failure e) >>= _ = Failure e","searchableContent":"    (failure e) >>= _ = failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":76,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":77,"title":"{-# COMPILE GHC ComputationResult = data ComputationResult (Success | Failure) #-}","searchableContent":"{-# compile ghc computationresult = data computationresult (success | failure) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":78,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.Types
open import Data.Char.Base as C using (Char)","searchableContent":"leios.foreign.types
open import data.char.base as c using (char)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":3,"title":"import Data.String as S","searchableContent":"import data.string as s"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":4,"title":"open import Data.Integer using (+_; ∣_∣)","searchableContent":"open import data.integer using (+_; ∣_∣)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":6,"title":"open import Class.Convertible","searchableContent":"open import class.convertible"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":7,"title":"open import Class.HasHsType","searchableContent":"open import class.hashstype"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":8,"title":"open import Tactic.Derive.Convertible","searchableContent":"open import tactic.derive.convertible"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":9,"title":"open import Tactic.Derive.HsType","searchableContent":"open import tactic.derive.hstype"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":11,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":13,"title":"open import Leios.Foreign.BaseTypes","searchableContent":"open import leios.foreign.basetypes"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":14,"title":"open import Leios.Foreign.HsTypes","searchableContent":"open import leios.foreign.hstypes"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":15,"title":"open import Leios.Foreign.Util","searchableContent":"open import leios.foreign.util"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":17,"title":"module Leios.Foreign.Types where","searchableContent":"module leios.foreign.types where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":19,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":20,"title":"{-# LANGUAGE DuplicateRecordFields #-}","searchableContent":"  {-# language duplicaterecordfields #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":21,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":23,"title":"-- TODO","content":"-- TODO: Get rid of hardcoded parameters in this module","searchableContent":"-- todo: get rid of hardcoded parameters in this module"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":25,"title":"{-","searchableContent":"{-"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":26,"title":"numberOfParties","content":"numberOfParties : ℕ","searchableContent":"numberofparties : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":27,"title":"numberOfParties","content":"numberOfParties = 2","searchableContent":"numberofparties = 2"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":29,"title":"open import Leios.Defaults numberOfParties fzero","content":"open import Leios.Defaults numberOfParties fzero","searchableContent":"open import leios.defaults numberofparties fzero"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":30,"title":"renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)","content":"renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)","searchableContent":"  renaming (endorserblock to endorserblockagda; ibheader to ibheaderagda)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":32,"title":"dropDash","content":"dropDash : S.String → S.String","searchableContent":"dropdash : s.string → s.string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":33,"title":"dropDash","content":"dropDash = S.concat ∘ S.wordsByᵇ ('-' C.≈ᵇ_)","searchableContent":"dropdash = s.concat ∘ s.wordsbyᵇ ('-' c.≈ᵇ_)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":35,"title":"prefix","content":"prefix : S.String → S.String → S.String","searchableContent":"prefix : s.string → s.string → s.string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":36,"title":"prefix","content":"prefix = S._++_","searchableContent":"prefix = s._++_"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":38,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":39,"title":"HsTy-SlotUpkeep","content":"HsTy-SlotUpkeep = autoHsType SlotUpkeep ⊣ onConstructors dropDash","searchableContent":"  hsty-slotupkeep = autohstype slotupkeep ⊣ onconstructors dropdash"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":40,"title":"Conv-SlotUpkeep","content":"Conv-SlotUpkeep = autoConvert SlotUpkeep","searchableContent":"  conv-slotupkeep = autoconvert slotupkeep"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":42,"title":"record IBHeader","content":"record IBHeader : Type where","searchableContent":"record ibheader : type where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":43,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":44,"title":"producerID","content":"producerID : ℕ","searchableContent":"        producerid : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":45,"title":"bodyHash","content":"bodyHash : List ℕ","searchableContent":"        bodyhash : list ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":47,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":48,"title":"data IBHeader = IBHeader {slotNumber","content":"data IBHeader = IBHeader {slotNumber :: Integer, producerID :: Integer, bodyHash :: Data.Text.Text }","searchableContent":"data ibheader = ibheader {slotnumber :: integer, producerid :: integer, bodyhash :: data.text.text }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":49,"title":"deriving (Show, Eq, Generic)","content":"deriving (Show, Eq, Generic)","searchableContent":"  deriving (show, eq, generic)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":50,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":52,"title":"{-# COMPILE GHC IBHeader","content":"{-# COMPILE GHC IBHeader = data IBHeader (IBHeader) #-}","searchableContent":"{-# compile ghc ibheader = data ibheader (ibheader) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":54,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":55,"title":"HsTy-IBHeader","content":"HsTy-IBHeader = MkHsType IBHeaderAgda IBHeader","searchableContent":"  hsty-ibheader = mkhstype ibheaderagda ibheader"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":57,"title":"Conv-IBHeader","content":"Conv-IBHeader : Convertible IBHeaderAgda IBHeader","searchableContent":"  conv-ibheader : convertible ibheaderagda ibheader"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":58,"title":"Conv-IBHeader","content":"Conv-IBHeader = record","searchableContent":"  conv-ibheader = record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":59,"title":"{ to","content":"{ to = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →","searchableContent":"    { to = λ (record { slotnumber = s ; producerid = p ; bodyhash = h }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":60,"title":"record { slotNumber","content":"record { slotNumber = s ; producerID = toℕ p ; bodyHash = h}","searchableContent":"        record { slotnumber = s ; producerid = toℕ p ; bodyhash = h}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":61,"title":"; from","content":"; from = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →","searchableContent":"    ; from = λ (record { slotnumber = s ; producerid = p ; bodyhash = h }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":62,"title":"case p <? numberOfParties of λ where","content":"case p <? numberOfParties of λ where","searchableContent":"        case p <? numberofparties of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":63,"title":"(yes q) → record { slotNumber","content":"(yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; bodyHash = h ; signature = tt }","searchableContent":"          (yes q) → record { slotnumber = s ; producerid = #_ p {numberofparties} {fromwitness q} ; lotterypf = tt ; bodyhash = h ; signature = tt }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":64,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"          (no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":65,"title":"}","content":"}","searchableContent":"    }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":67,"title":"HsTy-IBBody","content":"HsTy-IBBody = autoHsType IBBody","searchableContent":"  hsty-ibbody = autohstype ibbody"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":68,"title":"Conv-IBBody","content":"Conv-IBBody = autoConvert IBBody","searchableContent":"  conv-ibbody = autoconvert ibbody"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":70,"title":"HsTy-InputBlock","content":"HsTy-InputBlock = autoHsType InputBlock","searchableContent":"  hsty-inputblock = autohstype inputblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":71,"title":"Conv-InputBlock","content":"Conv-InputBlock = autoConvert InputBlock","searchableContent":"  conv-inputblock = autoconvert inputblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":73,"title":"Conv-ℕ","content":"Conv-ℕ : HsConvertible ℕ","searchableContent":"  conv-ℕ : hsconvertible ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":74,"title":"Conv-ℕ","content":"Conv-ℕ = Convertible-Refl","searchableContent":"  conv-ℕ = convertible-refl"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":76,"title":"record EndorserBlock","content":"record EndorserBlock : Type where","searchableContent":"record endorserblock : type where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":77,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":78,"title":"producerID","content":"producerID : ℕ","searchableContent":"        producerid : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":79,"title":"ibRefs","content":"ibRefs     : List (List IBRef)","searchableContent":"        ibrefs     : list (list ibref)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":81,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":82,"title":"data EndorserBlock = EndorserBlock { slotNumber","content":"data EndorserBlock = EndorserBlock { slotNumber :: Integer, producerID :: Integer, ibRefs :: [Data.Text.Text] }","searchableContent":"data endorserblock = endorserblock { slotnumber :: integer, producerid :: integer, ibrefs :: [data.text.text] }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":83,"title":"deriving (Show, Eq, Generic)","content":"deriving (Show, Eq, Generic)","searchableContent":"  deriving (show, eq, generic)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":84,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":86,"title":"{-# COMPILE GHC EndorserBlock","content":"{-# COMPILE GHC EndorserBlock = data EndorserBlock (EndorserBlock) #-}","searchableContent":"{-# compile ghc endorserblock = data endorserblock (endorserblock) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":88,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":89,"title":"HsTy-EndorserBlock","content":"HsTy-EndorserBlock = MkHsType EndorserBlockAgda EndorserBlock","searchableContent":"  hsty-endorserblock = mkhstype endorserblockagda endorserblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":91,"title":"Conv-EndorserBlock","content":"Conv-EndorserBlock : Convertible EndorserBlockAgda EndorserBlock","searchableContent":"  conv-endorserblock : convertible endorserblockagda endorserblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":92,"title":"Conv-EndorserBlock","content":"Conv-EndorserBlock =","searchableContent":"  conv-endorserblock ="},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":93,"title":"record","content":"record","searchableContent":"    record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":94,"title":"{ to","content":"{ to = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →","searchableContent":"      { to = λ (record { slotnumber = s ; producerid = p ; ibrefs = refs }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":95,"title":"record { slotNumber","content":"record { slotNumber = s ; producerID = toℕ p ; ibRefs = refs }","searchableContent":"          record { slotnumber = s ; producerid = toℕ p ; ibrefs = refs }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":96,"title":"; from","content":"; from = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →","searchableContent":"      ; from = λ (record { slotnumber = s ; producerid = p ; ibrefs = refs }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":97,"title":"case p <? numberOfParties of λ where","content":"case p <? numberOfParties of λ where","searchableContent":"        case p <? numberofparties of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":98,"title":"(yes q) → record { slotNumber","content":"(yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; signature = tt ; ibRefs = refs ; ebRefs = [] }","searchableContent":"          (yes q) → record { slotnumber = s ; producerid = #_ p {numberofparties} {fromwitness q} ; lotterypf = tt ; signature = tt ; ibrefs = refs ; ebrefs = [] }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":99,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"          (no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":100,"title":"}","content":"}","searchableContent":"      }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":102,"title":"HsTy-FFDState","content":"HsTy-FFDState = autoHsType FFDState","searchableContent":"  hsty-ffdstate = autohstype ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":103,"title":"Conv-FFDState","content":"Conv-FFDState = autoConvert FFDState","searchableContent":"  conv-ffdstate = autoconvert ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":105,"title":"HsTy-Fin","content":"HsTy-Fin : ∀ {n} → HasHsType (Fin n)","searchableContent":"  hsty-fin : ∀ {n} → hashstype (fin n)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":106,"title":"HsTy-Fin .HasHsType.HsType","content":"HsTy-Fin .HasHsType.HsType = ℕ","searchableContent":"  hsty-fin .hashstype.hstype = ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":108,"title":"Conv-Fin","content":"Conv-Fin : ∀ {n} → HsConvertible (Fin n)","searchableContent":"  conv-fin : ∀ {n} → hsconvertible (fin n)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":109,"title":"Conv-Fin {n}","content":"Conv-Fin {n} =","searchableContent":"  conv-fin {n} ="},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":110,"title":"record","content":"record","searchableContent":"    record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":111,"title":"{ to","content":"{ to = toℕ","searchableContent":"      { to = toℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":112,"title":"; from","content":"; from = λ m →","searchableContent":"      ; from = λ m →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":113,"title":"case m <? n of λ where","content":"case m <? n of λ where","searchableContent":"          case m <? n of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":114,"title":"(yes p) → #_ m {n} {fromWitness p}","content":"(yes p) → #_ m {n} {fromWitness p}","searchableContent":"            (yes p) → #_ m {n} {fromwitness p}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":115,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"            (no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":116,"title":"}","content":"}","searchableContent":"      }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":118,"title":"HsTy-LeiosState","content":"HsTy-LeiosState = autoHsType LeiosState","searchableContent":"  hsty-leiosstate = autohstype leiosstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":119,"title":"Conv-LeiosState","content":"Conv-LeiosState = autoConvert LeiosState","searchableContent":"  conv-leiosstate = autoconvert leiosstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":121,"title":"HsTy-LeiosInput","content":"HsTy-LeiosInput = autoHsType LeiosInput ⊣ onConstructors (prefix "I_" ∘ dropDash)","searchableContent":"  hsty-leiosinput = autohstype leiosinput ⊣ onconstructors (prefix "i_" ∘ dropdash)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":122,"title":"Conv-LeiosInput","content":"Conv-LeiosInput = autoConvert LeiosInput","searchableContent":"  conv-leiosinput = autoconvert leiosinput"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":124,"title":"HsTy-LeiosOutput","content":"HsTy-LeiosOutput = autoHsType LeiosOutput ⊣ onConstructors (prefix "O_" ∘ dropDash)","searchableContent":"  hsty-leiosoutput = autohstype leiosoutput ⊣ onconstructors (prefix "o_" ∘ dropdash)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":125,"title":"Conv-LeiosOutput","content":"Conv-LeiosOutput = autoConvert LeiosOutput","searchableContent":"  conv-leiosoutput = autoconvert leiosoutput"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":127,"title":"open import Class.Computational as C","content":"open import Class.Computational as C","searchableContent":"open import class.computational as c"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":128,"title":"open import Class.Computational22","content":"open import Class.Computational22","searchableContent":"open import class.computational22"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":130,"title":"open Computational22","content":"open Computational22","searchableContent":"open computational22"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":131,"title":"open BaseAbstract","content":"open BaseAbstract","searchableContent":"open baseabstract"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":132,"title":"open FFDAbstract","content":"open FFDAbstract","searchableContent":"open ffdabstract"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":134,"title":"open GenFFD.Header using (ibHeader; ebHeader; vtHeader)","content":"open GenFFD.Header using (ibHeader; ebHeader; vtHeader)","searchableContent":"open genffd.header using (ibheader; ebheader; vtheader)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":135,"title":"open GenFFD.Body using (ibBody)","content":"open GenFFD.Body using (ibBody)","searchableContent":"open genffd.body using (ibbody)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":136,"title":"open FFDState","content":"open FFDState","searchableContent":"open ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":138,"title":"open import Leios.Short.Deterministic d-SpecStructure public","content":"open import Leios.Short.Deterministic d-SpecStructure public","searchableContent":"open import leios.short.deterministic d-specstructure public"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":140,"title":"open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)","content":"open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)","searchableContent":"open funtot (completefin numberofparties) (maximalfin numberofparties)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":142,"title":"d-StakeDistribution","content":"d-StakeDistribution : TotalMap (Fin numberOfParties) ℕ","searchableContent":"d-stakedistribution : totalmap (fin numberofparties) ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":143,"title":"d-StakeDistribution","content":"d-StakeDistribution = Fun⇒TotalMap (const 100000000)","searchableContent":"d-stakedistribution = fun⇒totalmap (const 100000000)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":145,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":146,"title":"Computational-B","content":"Computational-B : Computational22 (BaseAbstract.Functionality._-⟦_/_⟧⇀_ d-BaseFunctionality) String","searchableContent":"  computational-b : computational22 (baseabstract.functionality._-⟦_/_⟧⇀_ d-basefunctionality) string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":147,"title":"Computational-B .computeProof s (INIT x)","content":"Computational-B .computeProof s (INIT x) = success ((STAKE d-StakeDistribution , tt) , tt)","searchableContent":"  computational-b .computeproof s (init x) = success ((stake d-stakedistribution , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":148,"title":"Computational-B .computeProof s (SUBMIT x)","content":"Computational-B .computeProof s (SUBMIT x) = success ((EMPTY , tt) , tt)","searchableContent":"  computational-b .computeproof s (submit x) = success ((empty , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":149,"title":"Computational-B .computeProof s FTCH-LDG","content":"Computational-B .computeProof s FTCH-LDG = success (((BASE-LDG []) , tt) , tt)","searchableContent":"  computational-b .computeproof s ftch-ldg = success (((base-ldg []) , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":150,"title":"Computational-B .completeness _ _ _ _ _ = {!!} -- TODO","content":"Computational-B .completeness _ _ _ _ _ = {!!} -- TODO: Completeness proof","searchableContent":"  computational-b .completeness _ _ _ _ _ = {!!} -- todo: completeness proof"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":152,"title":"Computational-FFD","content":"Computational-FFD : Computational22 (FFDAbstract.Functionality._-⟦_/_⟧⇀_ d-FFDFunctionality) String","searchableContent":"  computational-ffd : computational22 (ffdabstract.functionality._-⟦_/_⟧⇀_ d-ffdfunctionality) string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":153,"title":"Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b)))","content":"Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b))) = success ((SendRes , record s {outIBs = record {header = h; body = b} ∷ outIBs s}) , SendIB)","searchableContent":"  computational-ffd .computeproof s (send (ibheader h) (just (ibbody b))) = success ((sendres , record s {outibs = record {header = h; body = b} ∷ outibs s}) , sendib)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":154,"title":"Computational-FFD .computeProof s (Send (ebHeader h) nothing)","content":"Computational-FFD .computeProof s (Send (ebHeader h) nothing) = success ((SendRes , record s {outEBs = h ∷ outEBs s}) , SendEB)","searchableContent":"  computational-ffd .computeproof s (send (ebheader h) nothing) = success ((sendres , record s {outebs = h ∷ outebs s}) , sendeb)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":155,"title":"Computational-FFD .computeProof s (Send (vtHeader h) nothing)","content":"Computational-FFD .computeProof s (Send (vtHeader h) nothing) = success ((SendRes , record s {outVTs = h ∷ outVTs s}) , SendVS)","searchableContent":"  computational-ffd .computeproof s (send (vtheader h) nothing) = success ((sendres , record s {outvts = h ∷ outvts s}) , sendvs)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":156,"title":"Computational-FFD .computeProof s Fetch","content":"Computational-FFD .computeProof s Fetch = success ((FetchRes (flushIns s) , record s {inIBs = []; inEBs = []; inVTs = []}) , Fetch)","searchableContent":"  computational-ffd .computeproof s fetch = success ((fetchres (flushins s) , record s {inibs = []; inebs = []; invts = []}) , fetch)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":158,"title":"Computational-FFD .computeProof _ _","content":"Computational-FFD .computeProof _ _ = failure "FFD error"","searchableContent":"  computational-ffd .computeproof _ _ = failure "ffd error""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":159,"title":"Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO","content":"Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO:Completeness proof","searchableContent":"  computational-ffd .completeness _ _ _ _ _ = {!!} -- todo:completeness proof"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":161,"title":"stepHs","content":"stepHs : HsType (LeiosState → LeiosInput → C.ComputationResult String (LeiosOutput × LeiosState))","searchableContent":"stephs : hstype (leiosstate → leiosinput → c.computationresult string (leiosoutput × leiosstate))"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":162,"title":"stepHs","content":"stepHs = to (compute Computational--⟦/⟧⇀)","searchableContent":"stephs = to (compute computational--⟦/⟧⇀)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":164,"title":"{-# COMPILE GHC stepHs as step #-}","content":"{-# COMPILE GHC stepHs as step #-}","searchableContent":"{-# compile ghc stephs as step #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":165,"title":"-}","content":"-}","searchableContent":"-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":166,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.Util
module Leios.Foreign.Util where","searchableContent":"leios.foreign.util
module leios.foreign.util where"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":6,"title":"postulate","searchableContent":"postulate"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":7,"title":"error ","content":"error : {A : Set} → String → A","searchableContent":"  error : {a : set} → string → a"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":8,"title":"{-# FOREIGN GHC import Data.Text #-}","searchableContent":"{-# foreign ghc import data.text #-}"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":9,"title":"{-# COMPILE GHC error = \\ _ s -> error (unpack s) #-}","searchableContent":"{-# compile ghc error = \\ _ s -> error (unpack s) #-}"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":10,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":2,"title":"Leios.KeyRegistration
{-# OPTIONS --safe #-}","searchableContent":"leios.keyregistration
{-# options --safe #-}"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":6,"title":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":8,"title":"module Leios.KeyRegistration (a ","content":"module Leios.KeyRegistration (a : LeiosAbstract) (open LeiosAbstract a)","searchableContent":"module leios.keyregistration (a : leiosabstract) (open leiosabstract a)"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":9,"title":"(vrf ","content":"(vrf : LeiosVRF a) (let open LeiosVRF vrf) where","searchableContent":"  (vrf : leiosvrf a) (let open leiosvrf vrf) where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":11,"title":"record KeyRegistrationAbstract ","content":"record KeyRegistrationAbstract : Type₁ where","searchableContent":"record keyregistrationabstract : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":13,"title":"data Input ","content":"data Input : Type₁ where","searchableContent":"  data input : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":14,"title":"INIT ","content":"INIT : PubKey → PubKey → PubKey → Input","searchableContent":"    init : pubkey → pubkey → pubkey → input"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":16,"title":"data Output ","content":"data Output : Type where","searchableContent":"  data output : type where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":17,"title":"PUBKEYS ","content":"PUBKEYS : List PubKey → Output","searchableContent":"    pubkeys : list pubkey → output"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":19,"title":"record Functionality ","content":"record Functionality : Type₁ where","searchableContent":"  record functionality : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":20,"title":"field State ","content":"field State : Type","searchableContent":"    field state : type"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":21,"title":"_-⟦_/_⟧⇀_ ","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"          _-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":23,"title":"open Input public","searchableContent":"    open input public"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":24,"title":"open Output public","searchableContent":"    open output public"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":25,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":2,"title":"Leios.Network
module Leios.Network where","searchableContent":"leios.network
module leios.network where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":4,"title":"open import abstract-set-theory.Prelude hiding (_∘_; _⊗_)","searchableContent":"open import abstract-set-theory.prelude hiding (_∘_; _⊗_)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":5,"title":"open import abstract-set-theory.FiniteSetTheory using (ℙ_; _∈_; _∪_; ❴_❵; _∉_)","searchableContent":"open import abstract-set-theory.finitesettheory using (ℙ_; _∈_; _∪_; ❴_❵; _∉_)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":7,"title":"open import CategoricalCrypto","searchableContent":"open import categoricalcrypto"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":9,"title":"record Abstract ","content":"record Abstract : Set₁ where","searchableContent":"record abstract : set₁ where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":10,"title":"field Header Body ID ","content":"field Header Body ID : Set","searchableContent":"  field header body id : set"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":11,"title":"match ","content":"match : Header → Body → Set","searchableContent":"        match : header → body → set"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":12,"title":"msgID ","content":"msgID : Header → ID","searchableContent":"        msgid : header → id"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":14,"title":"module Broadcast (M Peer ","content":"module Broadcast (M Peer : Set) where","searchableContent":"module broadcast (m peer : set) where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":15,"title":"open Channel","searchableContent":"  open channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":17,"title":"C ","content":"C : Channel","searchableContent":"  c : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":18,"title":"C .P = Peer","searchableContent":"  c .p = peer"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":19,"title":"C .rcvType _ = Peer × M","searchableContent":"  c .rcvtype _ = peer × m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":20,"title":"C .sndType _ = M","searchableContent":"  c .sndtype _ = m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":22,"title":"postulate Functionality ","content":"postulate Functionality : Machine I C","searchableContent":"  postulate functionality : machine i c"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":24,"title":"Single ","content":"Single : Channel","searchableContent":"  single : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":25,"title":"Single .P = ⊤","searchableContent":"  single .p = ⊤"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":26,"title":"Single .rcvType _ = Peer × M","searchableContent":"  single .rcvtype _ = peer × m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":27,"title":"Single .sndType _ = M","searchableContent":"  single .sndtype _ = m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":29,"title":"postulate SingleFunctionality ","content":"postulate SingleFunctionality : Machine I Single","searchableContent":"  postulate singlefunctionality : machine i single"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":31,"title":"-- connectWithBroadcast","content":"-- connectWithBroadcast : ∀ {A} → (Peer → Machine Single A) → Machine I A","searchableContent":"  -- connectwithbroadcast : ∀ {a} → (peer → machine single a) → machine i a"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":32,"title":"-- connectWithBroadcast = {!!}","searchableContent":"  -- connectwithbroadcast = {!!}"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":34,"title":"module HeaderDiffusion (a ","content":"module HeaderDiffusion (a : Abstract) (Peer : Set) (self : Peer) where","searchableContent":"module headerdiffusion (a : abstract) (peer : set) (self : peer) where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":35,"title":"open Channel","searchableContent":"  open channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":36,"title":"open Abstract a","searchableContent":"  open abstract a"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":37,"title":"module B = Broadcast Header Peer","searchableContent":"  module b = broadcast header peer"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":39,"title":"data Port ","content":"data Port : Set where","searchableContent":"  data port : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":40,"title":"Send    ","content":"Send    : Port -- we want to send a header","searchableContent":"    send    : port -- we want to send a header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":41,"title":"Forward ","content":"Forward : Port -- we want to forward a header","searchableContent":"    forward : port -- we want to forward a header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":43,"title":"C ","content":"C : Channel","searchableContent":"  c : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":44,"title":"C .P = Port","searchableContent":"  c .p = port"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":45,"title":"C .sndType _ = Header","searchableContent":"  c .sndtype _ = header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":46,"title":"C .rcvType Forward = Header","searchableContent":"  c .rcvtype forward = header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":47,"title":"C .rcvType Send = ⊥","searchableContent":"  c .rcvtype send = ⊥"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":49,"title":"data Input ","content":"data Input : Set where","searchableContent":"  data input : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":50,"title":"S ","content":"S : Header → Input","searchableContent":"    s : header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":51,"title":"F ","content":"F : Header → Input","searchableContent":"    f : header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":52,"title":"R ","content":"R : Peer → Header → Input","searchableContent":"    r : peer → header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":54,"title":"data Output ","content":"data Output : Set where","searchableContent":"  data output : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":55,"title":"Verify ","content":"Verify : Header → Output","searchableContent":"    verify : header → output"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":57,"title":"private variable","searchableContent":"  private variable"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":58,"title":"h ","content":"h : Header","searchableContent":"    h : header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":59,"title":"s ","content":"s : ℙ ID","searchableContent":"    s : ℙ id"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":61,"title":"data Step ","content":"data Step : ∃ (rcvType (B.Single ⊗ C ᵀ)) → ℙ ID → ℙ ID × Maybe (∃ (sndType (B.Single ⊗ C ᵀ))) → Set where","searchableContent":"  data step : ∃ (rcvtype (b.single ⊗ c ᵀ)) → ℙ id → ℙ id × maybe (∃ (sndtype (b.single ⊗ c ᵀ))) → set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":62,"title":"Init ","content":"Init : Step (inj₂ Send , h) s (s ∪ ❴ msgID h ❵ , just (inj₁ _ , h))","searchableContent":"    init : step (inj₂ send , h) s (s ∪ ❴ msgid h ❵ , just (inj₁ _ , h))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":63,"title":"Receive1  ","content":"Receive1  : ∀ {p} → Step (inj₁ _ , p , h) s (s , just (inj₂ Forward , h))","searchableContent":"    receive1  : ∀ {p} → step (inj₁ _ , p , h) s (s , just (inj₂ forward , h))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":64,"title":"Receive2  ","content":"Receive2  : msgID h ∉ s → Step (inj₂ Forward , h) s (s ∪ ❴ msgID h ❵ , just (inj₁ _ , h))","searchableContent":"    receive2  : msgid h ∉ s → step (inj₂ forward , h) s (s ∪ ❴ msgid h ❵ , just (inj₁ _ , h))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":65,"title":"Receive2' ","content":"Receive2' : msgID h ∈ s → Step (inj₂ Forward , h) s (s , nothing)","searchableContent":"    receive2' : msgid h ∈ s → step (inj₂ forward , h) s (s , nothing)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":67,"title":"step ","content":"step : ∃ (rcvType (B.Single ⊗ C ᵀ)) → ∃ (sndType (B.Single ⊗ C ᵀ))","searchableContent":"  step : ∃ (rcvtype (b.single ⊗ c ᵀ)) → ∃ (sndtype (b.single ⊗ c ᵀ))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":68,"title":"step (inj₁ _   , _ , h) = (inj₂ Forward , h)","searchableContent":"  step (inj₁ _   , _ , h) = (inj₂ forward , h)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":69,"title":"step (inj₂ Forward , h) = (inj₁ _ , h)","searchableContent":"  step (inj₂ forward , h) = (inj₁ _ , h)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":70,"title":"step (inj₂ Send    , h) = (inj₁ _ , h)","searchableContent":"  step (inj₂ send    , h) = (inj₁ _ , h)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":72,"title":"Functionality ","content":"Functionality : Machine I C","searchableContent":"  functionality : machine i c"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":73,"title":"Functionality = MkMachine' Step ∘ B.SingleFunctionality","searchableContent":"  functionality = mkmachine' step ∘ b.singlefunctionality"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":74,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":2,"title":"Leios.Prelude
{-# OPTIONS --safe #-}","searchableContent":"leios.prelude
{-# options --safe #-}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":4,"title":"module Leios.Prelude where","searchableContent":"module leios.prelude where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":6,"title":"open import abstract-set-theory.FiniteSetTheory public","searchableContent":"open import abstract-set-theory.finitesettheory public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":7,"title":"open import abstract-set-theory.Prelude public","searchableContent":"open import abstract-set-theory.prelude public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":8,"title":"open import Data.List using (upTo)","searchableContent":"open import data.list using (upto)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":10,"title":"open import Class.HasAdd public","searchableContent":"open import class.hasadd public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":11,"title":"open import Class.HasOrder public","searchableContent":"open import class.hasorder public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":12,"title":"open import Class.Hashable public","searchableContent":"open import class.hashable public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":13,"title":"open import Prelude.InferenceRules public","searchableContent":"open import prelude.inferencerules public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":15,"title":"module T where","searchableContent":"module t where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":16,"title":"open import Data.These public","searchableContent":"  open import data.these public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":17,"title":"open T public using (These; this; that)","searchableContent":"open t public using (these; this; that)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":19,"title":"module L where","searchableContent":"module l where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":20,"title":"open import Data.List public","searchableContent":"  open import data.list public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":21,"title":"open L public using (List; []; _∷_; _++_; catMaybes; head; length; sum; and; or; any)","searchableContent":"open l public using (list; []; _∷_; _++_; catmaybes; head; length; sum; and; or; any)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":23,"title":"module A where","searchableContent":"module a where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":24,"title":"open import Data.List.Relation.Unary.Any public","searchableContent":"  open import data.list.relation.unary.any public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":25,"title":"open A public using (here; there)","searchableContent":"open a public using (here; there)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":27,"title":"module N where","searchableContent":"module n where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":28,"title":"open import Data.Nat public","searchableContent":"  open import data.nat public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":29,"title":"open import Data.Nat.Properties public","searchableContent":"  open import data.nat.properties public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":30,"title":"open N public using (ℕ; zero; suc)","searchableContent":"open n public using (ℕ; zero; suc)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":32,"title":"module F where","searchableContent":"module f where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":33,"title":"open import Data.Fin public","searchableContent":"  open import data.fin public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":34,"title":"open import Data.Fin.Properties public","searchableContent":"  open import data.fin.properties public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":35,"title":"open F public using (Fin; toℕ; #_) renaming (zero to fzero; suc to fsuc)","searchableContent":"open f public using (fin; toℕ; #_) renaming (zero to fzero; suc to fsuc)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":37,"title":"fromTo ","content":"fromTo : ℕ → ℕ → List ℕ","searchableContent":"fromto : ℕ → ℕ → list ℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":38,"title":"fromTo m n = map (_+ m) (upTo (n ∸ m))","searchableContent":"fromto m n = map (_+ m) (upto (n ∸ m))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":40,"title":"slice ","content":"slice : (L : ℕ) → ⦃ NonZero L ⦄ → ℕ → ℕ → ℙ ℕ","searchableContent":"slice : (l : ℕ) → ⦃ nonzero l ⦄ → ℕ → ℕ → ℙ ℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":41,"title":"slice L s x = fromList (fromTo s' (s' + (L ∸ 1)))","searchableContent":"slice l s x = fromlist (fromto s' (s' + (l ∸ 1)))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":42,"title":"where s' = ((s / L) ∸ x) * L -- equivalent to the formula in the paper","searchableContent":"  where s' = ((s / l) ∸ x) * l -- equivalent to the formula in the paper"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":44,"title":"filter ","content":"filter : {A : Set} → (P : A → Type) ⦃ _ : P ⁇¹ ⦄ → List A → List A","searchableContent":"filter : {a : set} → (p : a → type) ⦃ _ : p ⁇¹ ⦄ → list a → list a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":45,"title":"filter P = L.filter ¿ P ¿¹","searchableContent":"filter p = l.filter ¿ p ¿¹"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":47,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":48,"title":"IsSet-List ","content":"IsSet-List : {A : Set} → IsSet (List A) A","searchableContent":"  isset-list : {a : set} → isset (list a) a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":49,"title":"IsSet-List .toSet A = fromList A","searchableContent":"  isset-list .toset a = fromlist a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":51,"title":"open import Data.List.Relation.Unary.Any","searchableContent":"open import data.list.relation.unary.any"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":52,"title":"open Properties","searchableContent":"open properties"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":54,"title":"finite⇒A≡∅⊎∃a∈A ","content":"finite⇒A≡∅⊎∃a∈A : {X : Type} → {A : ℙ X} → finite A → (A ≡ᵉ ∅) ⊎ Σ[ a ∈ X ] a ∈ A","searchableContent":"finite⇒a≡∅⊎∃a∈a : {x : type} → {a : ℙ x} → finite a → (a ≡ᵉ ∅) ⊎ σ[ a ∈ x ] a ∈ a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":55,"title":"finite⇒A≡∅⊎∃a∈A ([]    , h) = inj₁ (∅-least (λ a∈A → ⊥-elim (case Equivalence.to h a∈A of λ ())))","searchableContent":"finite⇒a≡∅⊎∃a∈a ([]    , h) = inj₁ (∅-least (λ a∈a → ⊥-elim (case equivalence.to h a∈a of λ ())))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":56,"title":"finite⇒A≡∅⊎∃a∈A (x ∷ _ , h) = inj₂ (x , Equivalence.from h (here refl))","searchableContent":"finite⇒a≡∅⊎∃a∈a (x ∷ _ , h) = inj₂ (x , equivalence.from h (here refl))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":58,"title":"completeFin ","content":"completeFin : ∀ (n : ℕ) → ℙ (Fin n)","searchableContent":"completefin : ∀ (n : ℕ) → ℙ (fin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":59,"title":"completeFin zero = ∅","searchableContent":"completefin zero = ∅"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":60,"title":"completeFin (ℕ.suc n) = singleton (F.fromℕ n) ∪ mapˢ F.inject₁ (completeFin n)","searchableContent":"completefin (ℕ.suc n) = singleton (f.fromℕ n) ∪ mapˢ f.inject₁ (completefin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":62,"title":"m≤n∧n≤m⇒m≡n ","content":"m≤n∧n≤m⇒m≡n : ∀ {n m : ℕ} → n N.≤ m → m N.≤ n → m ≡ n","searchableContent":"m≤n∧n≤m⇒m≡n : ∀ {n m : ℕ} → n n.≤ m → m n.≤ n → m ≡ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":63,"title":"m≤n∧n≤m⇒m≡n z≤n z≤n = refl","searchableContent":"m≤n∧n≤m⇒m≡n z≤n z≤n = refl"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":64,"title":"m≤n∧n≤m⇒m≡n (s≤s n≤m) (s≤s m≤n) = cong N.suc (m≤n∧n≤m⇒m≡n n≤m m≤n)","searchableContent":"m≤n∧n≤m⇒m≡n (s≤s n≤m) (s≤s m≤n) = cong n.suc (m≤n∧n≤m⇒m≡n n≤m m≤n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":66,"title":"toℕ-fromℕ ","content":"toℕ-fromℕ : ∀ {n} {a : Fin (N.suc n)} → toℕ a ≡ n → a ≡ F.fromℕ n","searchableContent":"toℕ-fromℕ : ∀ {n} {a : fin (n.suc n)} → toℕ a ≡ n → a ≡ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":67,"title":"toℕ-fromℕ {zero} {fzero} x = refl","searchableContent":"toℕ-fromℕ {zero} {fzero} x = refl"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":68,"title":"toℕ-fromℕ {N.suc n} {fsuc a} x = cong fsuc (toℕ-fromℕ {n} {a} (N.suc-injective x))","searchableContent":"toℕ-fromℕ {n.suc n} {fsuc a} x = cong fsuc (toℕ-fromℕ {n} {a} (n.suc-injective x))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":70,"title":"open Equivalence","searchableContent":"open equivalence"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":72,"title":"maximalFin ","content":"maximalFin : ∀ (n : ℕ) → isMaximal (completeFin n)","searchableContent":"maximalfin : ∀ (n : ℕ) → ismaximal (completefin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":73,"title":"maximalFin (ℕ.suc n) {a} with toℕ a N.<? n","searchableContent":"maximalfin (ℕ.suc n) {a} with toℕ a n.<? n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":74,"title":"... | yes p =","searchableContent":"... | yes p ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":75,"title":"let n≢toℕ = ≢-sym (N.<⇒≢ p)","searchableContent":"  let n≢toℕ = ≢-sym (n.<⇒≢ p)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":76,"title":"fn = F.lower₁ a n≢toℕ","searchableContent":"      fn = f.lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":77,"title":"fn≡a = F.inject₁-lower₁ a n≢toℕ","searchableContent":"      fn≡a = f.inject₁-lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":78,"title":"in (to ∈-∪) (inj₂ ((to ∈-map) (fn , (sym fn≡a , maximalFin n))))","searchableContent":"  in (to ∈-∪) (inj₂ ((to ∈-map) (fn , (sym fn≡a , maximalfin n))))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":79,"title":"... | no ¬p with a F.≟ F.fromℕ n","searchableContent":"... | no ¬p with a f.≟ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":80,"title":"... | yes q = (to ∈-∪) (inj₁ ((to ∈-singleton) q))","searchableContent":"... | yes q = (to ∈-∪) (inj₁ ((to ∈-singleton) q))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":81,"title":"... | no ¬q =","searchableContent":"... | no ¬q ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":82,"title":"let n≢toℕ = N.≰⇒> ¬p","searchableContent":"  let n≢toℕ = n.≰⇒> ¬p"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":83,"title":"a<sucn = F.toℕ<n a","searchableContent":"      a<sucn = f.toℕ<n a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":84,"title":"in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (N.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))","searchableContent":"  in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (n.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":86,"title":"record Listable (A ","content":"record Listable (A : Type) : Type where","searchableContent":"record listable (a : type) : type where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":87,"title":"field","searchableContent":"  field"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":88,"title":"listing  ","content":"listing  : ℙ A","searchableContent":"    listing  : ℙ a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":89,"title":"complete ","content":"complete : ∀ {a : A} → a ∈ listing","searchableContent":"    complete : ∀ {a : a} → a ∈ listing"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":91,"title":"totalDec ","content":"totalDec : ∀ {A B : Type} → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → {R : Rel A B} → Dec (total R)","searchableContent":"totaldec : ∀ {a b : type} → ⦃ deceq a ⦄ → ⦃ listable a ⦄ → {r : rel a b} → dec (total r)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":92,"title":"totalDec {A} {B} {R} with all? (_∈? dom R)","searchableContent":"totaldec {a} {b} {r} with all? (_∈? dom r)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":93,"title":"... | yes p = yes λ {a} → p {a} ((Listable.complete it) {a})","searchableContent":"... | yes p = yes λ {a} → p {a} ((listable.complete it) {a})"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":94,"title":"... | no ¬p = no λ x → ¬p λ {a} _ → x {a}","searchableContent":"... | no ¬p = no λ x → ¬p λ {a} _ → x {a}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":96,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":97,"title":"total? ","content":"total? : ∀ {A B : Type} → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → {R : Rel A B} → ({a : A} → a ∈ dom R) ⁇","searchableContent":"  total? : ∀ {a b : type} → ⦃ deceq a ⦄ → ⦃ listable a ⦄ → {r : rel a b} → ({a : a} → a ∈ dom r) ⁇"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":98,"title":"total? = ⁇ totalDec","searchableContent":"  total? = ⁇ totaldec"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":100,"title":"Listable-Fin ","content":"Listable-Fin : ∀ {n} → Listable (Fin n)","searchableContent":"  listable-fin : ∀ {n} → listable (fin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":101,"title":"Listable-Fin {zero} = record { listing = ∅ ; complete = λ {a} → ⊥-elim $ (Inverse.to F.0↔⊥) a }","searchableContent":"  listable-fin {zero} = record { listing = ∅ ; complete = λ {a} → ⊥-elim $ (inverse.to f.0↔⊥) a }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":102,"title":"Listable-Fin {suc n} =","searchableContent":"  listable-fin {suc n} ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":103,"title":"let record { listing = l ; complete = c } = Listable-Fin {n}","searchableContent":"    let record { listing = l ; complete = c } = listable-fin {n}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":104,"title":"in record","searchableContent":"    in record"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":105,"title":"{ listing = singleton (F.fromℕ n) ∪ mapˢ F.inject₁ l","searchableContent":"         { listing = singleton (f.fromℕ n) ∪ mapˢ f.inject₁ l"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":106,"title":"; complete = complete","searchableContent":"         ; complete = complete"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":107,"title":"}","searchableContent":"         }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":108,"title":"where","searchableContent":"       where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":109,"title":"complete ","content":"complete : ∀ {a} → a ∈ singleton (F.fromℕ n) ∪ mapˢ F.inject₁ (let record { listing = l } = Listable-Fin {n} in l)","searchableContent":"         complete : ∀ {a} → a ∈ singleton (f.fromℕ n) ∪ mapˢ f.inject₁ (let record { listing = l } = listable-fin {n} in l)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":110,"title":"complete {a} with F.toℕ a N.<? n","searchableContent":"         complete {a} with f.toℕ a n.<? n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":111,"title":"... | yes p =","searchableContent":"         ... | yes p ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":112,"title":"let record { listing = l ; complete = c } = Listable-Fin {n}","searchableContent":"           let record { listing = l ; complete = c } = listable-fin {n}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":113,"title":"n≢toℕ = ≢-sym (N.<⇒≢ p)","searchableContent":"               n≢toℕ = ≢-sym (n.<⇒≢ p)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":114,"title":"fn = F.lower₁ a n≢toℕ","searchableContent":"               fn = f.lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":115,"title":"fn≡a = F.inject₁-lower₁ a n≢toℕ","searchableContent":"               fn≡a = f.inject₁-lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":116,"title":"in (Equivalence.to ∈-∪) (inj₂ ((Equivalence.to ∈-map) (fn , (sym fn≡a , c))))","searchableContent":"           in (equivalence.to ∈-∪) (inj₂ ((equivalence.to ∈-map) (fn , (sym fn≡a , c))))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":117,"title":"... | no ¬p with a F.≟ F.fromℕ n","searchableContent":"         ... | no ¬p with a f.≟ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":118,"title":"... | yes q = (Equivalence.to ∈-∪) (inj₁ ((Equivalence.to ∈-singleton) q))","searchableContent":"         ... | yes q = (equivalence.to ∈-∪) (inj₁ ((equivalence.to ∈-singleton) q))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":119,"title":"... | no ¬q =","searchableContent":"         ... | no ¬q ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":120,"title":"let n≢toℕ = N.≰⇒> ¬p","searchableContent":"           let n≢toℕ = n.≰⇒> ¬p"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":121,"title":"a<sucn = F.toℕ<n a","searchableContent":"               a<sucn = f.toℕ<n a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":122,"title":"in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (N.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))","searchableContent":"           in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (n.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":124,"title":"completeFinL ","content":"completeFinL : ∀ (n : ℕ) → List (Fin n)","searchableContent":"completefinl : ∀ (n : ℕ) → list (fin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":125,"title":"completeFinL zero = []","searchableContent":"completefinl zero = []"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":126,"title":"completeFinL (ℕ.suc n) = F.fromℕ n ∷ L.map F.inject₁ (completeFinL n)","searchableContent":"completefinl (ℕ.suc n) = f.fromℕ n ∷ l.map f.inject₁ (completefinl n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":127,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":2,"title":"Leios.Protocol
{-# OPTIONS --safe #-}","searchableContent":"leios.protocol
{-# options --safe #-}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":5,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":6,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":8,"title":"module Leios.Protocol {n} (⋯ ","content":"module Leios.Protocol {n} (⋯ : SpecStructure n) (let open SpecStructure ⋯) (SlotUpkeep : Type) where","searchableContent":"module leios.protocol {n} (⋯ : specstructure n) (let open specstructure ⋯) (slotupkeep : type) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":10,"title":"open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)","searchableContent":"open baseabstract b' using (cert; v-chkcerts; vty; initslot)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":11,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":13,"title":"-- High level structure","content":"-- High level structure:","searchableContent":"-- high level structure:"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":16,"title":"--                                      (simple) Leios","searchableContent":"--                                      (simple) leios"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":17,"title":"--                                        /         |","searchableContent":"--                                        /         |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":18,"title":"-- +-------------------------------------+          |","searchableContent":"-- +-------------------------------------+          |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":19,"title":"-- | Header Diffusion     Body Diffusion |          |","searchableContent":"-- | header diffusion     body diffusion |          |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":20,"title":"-- +-------------------------------------+       Base Protocol","searchableContent":"-- +-------------------------------------+       base protocol"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":21,"title":"--                                        \\      /","searchableContent":"--                                        \\      /"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":22,"title":"--                                        Network","searchableContent":"--                                        network"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":24,"title":"data LeiosInput ","content":"data LeiosInput : Type where","searchableContent":"data leiosinput : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":25,"title":"INIT     ","content":"INIT     : VTy → LeiosInput","searchableContent":"  init     : vty → leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":26,"title":"SUBMIT   ","content":"SUBMIT   : EndorserBlock ⊎ List Tx → LeiosInput","searchableContent":"  submit   : endorserblock ⊎ list tx → leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":27,"title":"SLOT     ","content":"SLOT     : LeiosInput","searchableContent":"  slot     : leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":28,"title":"FTCH-LDG ","content":"FTCH-LDG : LeiosInput","searchableContent":"  ftch-ldg : leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":30,"title":"data LeiosOutput ","content":"data LeiosOutput : Type where","searchableContent":"data leiosoutput : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":31,"title":"FTCH-LDG ","content":"FTCH-LDG : List Tx → LeiosOutput","searchableContent":"  ftch-ldg : list tx → leiosoutput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":32,"title":"EMPTY    ","content":"EMPTY    : LeiosOutput","searchableContent":"  empty    : leiosoutput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":34,"title":"record LeiosState ","content":"record LeiosState : Type where","searchableContent":"record leiosstate : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":35,"title":"field V           ","content":"field V           : VTy","searchableContent":"  field v           : vty"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":36,"title":"SD          ","content":"SD          : StakeDistr","searchableContent":"        sd          : stakedistr"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":37,"title":"FFDState    ","content":"FFDState    : FFD.State","searchableContent":"        ffdstate    : ffd.state"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":38,"title":"Ledger      ","content":"Ledger      : List Tx","searchableContent":"        ledger      : list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":39,"title":"ToPropose   ","content":"ToPropose   : List Tx","searchableContent":"        topropose   : list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":40,"title":"IBs         ","content":"IBs         : List InputBlock","searchableContent":"        ibs         : list inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":41,"title":"EBs         ","content":"EBs         : List EndorserBlock","searchableContent":"        ebs         : list endorserblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":42,"title":"Vs          ","content":"Vs          : List (List Vote)","searchableContent":"        vs          : list (list vote)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":43,"title":"slot        ","content":"slot        : ℕ","searchableContent":"        slot        : ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":44,"title":"IBHeaders   ","content":"IBHeaders   : List IBHeader","searchableContent":"        ibheaders   : list ibheader"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":45,"title":"IBBodies    ","content":"IBBodies    : List IBBody","searchableContent":"        ibbodies    : list ibbody"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":46,"title":"Upkeep      ","content":"Upkeep      : ℙ SlotUpkeep","searchableContent":"        upkeep      : ℙ slotupkeep"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":47,"title":"BaseState   ","content":"BaseState   : B.State","searchableContent":"        basestate   : b.state"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":48,"title":"votingState ","content":"votingState : VotingState","searchableContent":"        votingstate : votingstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":49,"title":"PubKeys     ","content":"PubKeys     : List PubKey","searchableContent":"        pubkeys     : list pubkey"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":51,"title":"lookupEB ","content":"lookupEB : EBRef → Maybe EndorserBlock","searchableContent":"  lookupeb : ebref → maybe endorserblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":52,"title":"lookupEB r = find (λ b → getEBRef b ≟ r) EBs","searchableContent":"  lookupeb r = find (λ b → getebref b ≟ r) ebs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":54,"title":"lookupIB ","content":"lookupIB : IBRef → Maybe InputBlock","searchableContent":"  lookupib : ibref → maybe inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":55,"title":"lookupIB r = find (λ b → getIBRef b ≟ r) IBs","searchableContent":"  lookupib r = find (λ b → getibref b ≟ r) ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":57,"title":"lookupTxs ","content":"lookupTxs : EndorserBlock → List Tx","searchableContent":"  lookuptxs : endorserblock → list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":58,"title":"lookupTxs eb = do","searchableContent":"  lookuptxs eb = do"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":59,"title":"eb′ ← mapMaybe lookupEB $ ebRefs eb","searchableContent":"    eb′ ← mapmaybe lookupeb $ ebrefs eb"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":60,"title":"ib  ← mapMaybe lookupIB $ ibRefs eb′","searchableContent":"    ib  ← mapmaybe lookupib $ ibrefs eb′"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":61,"title":"txs $ body ib","searchableContent":"    txs $ body ib"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":62,"title":"where open EndorserBlockOSig","searchableContent":"    where open endorserblockosig"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":63,"title":"open IBBody","searchableContent":"          open ibbody"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":64,"title":"open InputBlock","searchableContent":"          open inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":66,"title":"constructLedger ","content":"constructLedger : List RankingBlock → List Tx","searchableContent":"  constructledger : list rankingblock → list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":67,"title":"constructLedger = L.concat ∘ L.map (T.mergeThese L._++_ ∘ T.map₁ lookupTxs)","searchableContent":"  constructledger = l.concat ∘ l.map (t.mergethese l._++_ ∘ t.map₁ lookuptxs)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":69,"title":"needsUpkeep ","content":"needsUpkeep : SlotUpkeep → Set","searchableContent":"  needsupkeep : slotupkeep → set"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":70,"title":"needsUpkeep = _∉ Upkeep","searchableContent":"  needsupkeep = _∉ upkeep"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":72,"title":"Dec-needsUpkeep ","content":"Dec-needsUpkeep : ∀ {u : SlotUpkeep} → ⦃ DecEq SlotUpkeep ⦄ → needsUpkeep u ⁇","searchableContent":"  dec-needsupkeep : ∀ {u : slotupkeep} → ⦃ deceq slotupkeep ⦄ → needsupkeep u ⁇"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":73,"title":"Dec-needsUpkeep {u} .dec = ¬? (u ∈? Upkeep)","searchableContent":"  dec-needsupkeep {u} .dec = ¬? (u ∈? upkeep)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":75,"title":"addUpkeep ","content":"addUpkeep : LeiosState → SlotUpkeep → LeiosState","searchableContent":"addupkeep : leiosstate → slotupkeep → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":76,"title":"addUpkeep s u = let open LeiosState s in record s { Upkeep = Upkeep ∪ ❴ u ❵ }","searchableContent":"addupkeep s u = let open leiosstate s in record s { upkeep = upkeep ∪ ❴ u ❵ }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":77,"title":"{-# INJECTIVE_FOR_INFERENCE addUpkeep #-}","searchableContent":"{-# injective_for_inference addupkeep #-}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":79,"title":"initLeiosState ","content":"initLeiosState : VTy → StakeDistr → B.State → List PubKey → LeiosState","searchableContent":"initleiosstate : vty → stakedistr → b.state → list pubkey → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":80,"title":"initLeiosState V SD bs pks = record","searchableContent":"initleiosstate v sd bs pks = record"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":81,"title":"{ V           = V","searchableContent":"  { v           = v"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":82,"title":"; SD          = SD","searchableContent":"  ; sd          = sd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":83,"title":"; FFDState    = FFD.initFFDState","searchableContent":"  ; ffdstate    = ffd.initffdstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":84,"title":"; Ledger      = []","searchableContent":"  ; ledger      = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":85,"title":"; ToPropose   = []","searchableContent":"  ; topropose   = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":86,"title":"; IBs         = []","searchableContent":"  ; ibs         = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":87,"title":"; EBs         = []","searchableContent":"  ; ebs         = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":88,"title":"; Vs          = []","searchableContent":"  ; vs          = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":89,"title":"; slot        = initSlot V","searchableContent":"  ; slot        = initslot v"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":90,"title":"; IBHeaders   = []","searchableContent":"  ; ibheaders   = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":91,"title":"; IBBodies    = []","searchableContent":"  ; ibbodies    = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":92,"title":"; Upkeep      = ∅","searchableContent":"  ; upkeep      = ∅"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":93,"title":"; BaseState   = bs","searchableContent":"  ; basestate   = bs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":94,"title":"; votingState = initVotingState","searchableContent":"  ; votingstate = initvotingstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":95,"title":"; PubKeys     = pks","searchableContent":"  ; pubkeys     = pks"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":96,"title":"}","searchableContent":"  }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":98,"title":"stake' ","content":"stake' : PoolID → LeiosState → ℕ","searchableContent":"stake' : poolid → leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":99,"title":"stake' pid record { SD = SD } = TotalMap.lookup SD pid","searchableContent":"stake' pid record { sd = sd } = totalmap.lookup sd pid"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":101,"title":"stake'' ","content":"stake'' : PubKey → LeiosState → ℕ","searchableContent":"stake'' : pubkey → leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":102,"title":"stake'' pk = stake' (poolID pk)","searchableContent":"stake'' pk = stake' (poolid pk)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":104,"title":"stake ","content":"stake : LeiosState → ℕ","searchableContent":"stake : leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":105,"title":"stake = stake' id","searchableContent":"stake = stake' id"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":107,"title":"lookupPubKeyAndStake ","content":"lookupPubKeyAndStake : ∀ {B} → ⦃ _ : IsBlock B ⦄ → LeiosState → B → Maybe (PubKey × ℕ)","searchableContent":"lookuppubkeyandstake : ∀ {b} → ⦃ _ : isblock b ⦄ → leiosstate → b → maybe (pubkey × ℕ)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":108,"title":"lookupPubKeyAndStake s b =","searchableContent":"lookuppubkeyandstake s b ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":109,"title":"L.head $","searchableContent":"  l.head $"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":110,"title":"L.map (λ pk → (pk , stake'' pk s)) $","searchableContent":"    l.map (λ pk → (pk , stake'' pk s)) $"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":111,"title":"L.filter (λ pk → producerID b ≟ poolID pk) (LeiosState.PubKeys s)","searchableContent":"      l.filter (λ pk → producerid b ≟ poolid pk) (leiosstate.pubkeys s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":113,"title":"module _ (s ","content":"module _ (s : LeiosState)  where","searchableContent":"module _ (s : leiosstate)  where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":115,"title":"record ibHeaderValid (h ","content":"record ibHeaderValid (h : IBHeader) (pk : PubKey) (st : ℕ) : Type where","searchableContent":"  record ibheadervalid (h : ibheader) (pk : pubkey) (st : ℕ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":116,"title":"field lotteryPfValid ","content":"field lotteryPfValid : verify pk (slotNumber h) st (lotteryPf h)","searchableContent":"    field lotterypfvalid : verify pk (slotnumber h) st (lotterypf h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":117,"title":"signatureValid ","content":"signatureValid : verifySig pk (signature h)","searchableContent":"          signaturevalid : verifysig pk (signature h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":119,"title":"record ibBodyValid (b ","content":"record ibBodyValid (b : IBBody) : Type where","searchableContent":"  record ibbodyvalid (b : ibbody) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":121,"title":"ibHeaderValid? ","content":"ibHeaderValid? : (h : IBHeader) (pk : PubKey) (st : ℕ) → Dec (ibHeaderValid h pk st)","searchableContent":"  ibheadervalid? : (h : ibheader) (pk : pubkey) (st : ℕ) → dec (ibheadervalid h pk st)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":122,"title":"ibHeaderValid? h pk st","searchableContent":"  ibheadervalid? h pk st"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":123,"title":"with verify? pk (slotNumber h) st (lotteryPf h)","searchableContent":"    with verify? pk (slotnumber h) st (lotterypf h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":124,"title":"... | no ¬p = no (¬p ∘ ibHeaderValid.lotteryPfValid)","searchableContent":"  ... | no ¬p = no (¬p ∘ ibheadervalid.lotterypfvalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":125,"title":"... | yes p","searchableContent":"  ... | yes p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":126,"title":"with verifySig? pk (signature h)","searchableContent":"    with verifysig? pk (signature h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":127,"title":"... | yes q = yes (record { lotteryPfValid = p ; signatureValid = q })","searchableContent":"  ... | yes q = yes (record { lotterypfvalid = p ; signaturevalid = q })"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":128,"title":"... | no ¬q = no (¬q ∘ ibHeaderValid.signatureValid)","searchableContent":"  ... | no ¬q = no (¬q ∘ ibheadervalid.signaturevalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":130,"title":"ibBodyValid? ","content":"ibBodyValid? : (b : IBBody) → Dec (ibBodyValid b)","searchableContent":"  ibbodyvalid? : (b : ibbody) → dec (ibbodyvalid b)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":131,"title":"ibBodyValid? _ = yes record {}","searchableContent":"  ibbodyvalid? _ = yes record {}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":133,"title":"ibValid ","content":"ibValid : InputBlock → Type","searchableContent":"  ibvalid : inputblock → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":134,"title":"ibValid record { header = h ; body = b }","searchableContent":"  ibvalid record { header = h ; body = b }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":135,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":136,"title":"... | just (pk , pid) = ibHeaderValid h pk (stake'' pk s) × ibBodyValid b","searchableContent":"  ... | just (pk , pid) = ibheadervalid h pk (stake'' pk s) × ibbodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":137,"title":"... | nothing = ⊥","searchableContent":"  ... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":139,"title":"ibValid? ","content":"ibValid? : (ib : InputBlock) → Dec (ibValid ib)","searchableContent":"  ibvalid? : (ib : inputblock) → dec (ibvalid ib)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":140,"title":"ibValid? record { header = h ; body = b }","searchableContent":"  ibvalid? record { header = h ; body = b }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":141,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":142,"title":"... | just (pk , pid) = ibHeaderValid? h pk (stake'' pk s) ×-dec ibBodyValid? b","searchableContent":"  ... | just (pk , pid) = ibheadervalid? h pk (stake'' pk s) ×-dec ibbodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":143,"title":"... | nothing = no λ x → x","searchableContent":"  ... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":145,"title":"record ebValid (eb ","content":"record ebValid (eb : EndorserBlock) (pk : PubKey) (st : ℕ) : Type where","searchableContent":"  record ebvalid (eb : endorserblock) (pk : pubkey) (st : ℕ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":146,"title":"field lotteryPfValid ","content":"field lotteryPfValid : verify pk (slotNumber eb) st (lotteryPf eb)","searchableContent":"    field lotterypfvalid : verify pk (slotnumber eb) st (lotterypf eb)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":147,"title":"signatureValid ","content":"signatureValid : verifySig pk (signature eb)","searchableContent":"          signaturevalid : verifysig pk (signature eb)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":148,"title":"-- TODO","searchableContent":"    -- todo"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":149,"title":"-- ibRefsValid","content":"-- ibRefsValid : ?","searchableContent":"    -- ibrefsvalid : ?"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":150,"title":"-- ebRefsValid","content":"-- ebRefsValid : ?","searchableContent":"    -- ebrefsvalid : ?"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":152,"title":"ebValid? ","content":"ebValid? : (eb : EndorserBlock) (pk : PubKey) (st : ℕ) → Dec (ebValid eb pk st)","searchableContent":"  ebvalid? : (eb : endorserblock) (pk : pubkey) (st : ℕ) → dec (ebvalid eb pk st)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":153,"title":"ebValid? h pk st","searchableContent":"  ebvalid? h pk st"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":154,"title":"with verify? pk (slotNumber h) st (lotteryPf h)","searchableContent":"    with verify? pk (slotnumber h) st (lotterypf h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":155,"title":"... | no ¬p = no (¬p ∘ ebValid.lotteryPfValid)","searchableContent":"  ... | no ¬p = no (¬p ∘ ebvalid.lotterypfvalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":156,"title":"... | yes p","searchableContent":"  ... | yes p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":157,"title":"with verifySig? pk (signature h)","searchableContent":"    with verifysig? pk (signature h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":158,"title":"... | yes q = yes (record { lotteryPfValid = p ; signatureValid = q })","searchableContent":"  ... | yes q = yes (record { lotterypfvalid = p ; signaturevalid = q })"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":159,"title":"... | no ¬q = no (¬q ∘ ebValid.signatureValid)","searchableContent":"  ... | no ¬q = no (¬q ∘ ebvalid.signaturevalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":161,"title":"-- TODO","searchableContent":"  -- todo"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":162,"title":"record vsValid (vs ","content":"record vsValid (vs : List Vote) : Type where","searchableContent":"  record vsvalid (vs : list vote) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":164,"title":"vsValid? ","content":"vsValid? : (vs : List Vote) → Dec (vsValid vs)","searchableContent":"  vsvalid? : (vs : list vote) → dec (vsvalid vs)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":165,"title":"vsValid? _ = yes record {}","searchableContent":"  vsvalid? _ = yes record {}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":167,"title":"headerValid ","content":"headerValid : Header → Type","searchableContent":"  headervalid : header → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":168,"title":"headerValid (ibHeader h)","searchableContent":"  headervalid (ibheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":169,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":170,"title":"... | just (pk , pid) = ibHeaderValid h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ibheadervalid h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":171,"title":"... | nothing = ⊥","searchableContent":"  ... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":172,"title":"headerValid (ebHeader h)","searchableContent":"  headervalid (ebheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":173,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":174,"title":"... | just (pk , pid) = ebValid h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ebvalid h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":175,"title":"... | nothing = ⊥","searchableContent":"  ... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":176,"title":"headerValid (vtHeader h) = vsValid h","searchableContent":"  headervalid (vtheader h) = vsvalid h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":178,"title":"headerValid? ","content":"headerValid? : (h : Header) → Dec (headerValid h)","searchableContent":"  headervalid? : (h : header) → dec (headervalid h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":179,"title":"headerValid? (ibHeader h)","searchableContent":"  headervalid? (ibheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":180,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":181,"title":"... | just (pk , pid) = ibHeaderValid? h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ibheadervalid? h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":182,"title":"... | nothing = no λ x → x","searchableContent":"  ... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":183,"title":"headerValid? (ebHeader h)","searchableContent":"  headervalid? (ebheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":184,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":185,"title":"... | just (pk , pid) = ebValid? h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ebvalid? h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":186,"title":"... | nothing = no λ x → x","searchableContent":"  ... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":187,"title":"headerValid? (vtHeader h) = vsValid? h","searchableContent":"  headervalid? (vtheader h) = vsvalid? h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":189,"title":"bodyValid ","content":"bodyValid : Body → Type","searchableContent":"  bodyvalid : body → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":190,"title":"bodyValid (ibBody b) = ibBodyValid b","searchableContent":"  bodyvalid (ibbody b) = ibbodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":192,"title":"bodyValid? ","content":"bodyValid? : (b : Body) → Dec (bodyValid b)","searchableContent":"  bodyvalid? : (b : body) → dec (bodyvalid b)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":193,"title":"bodyValid? (ibBody b) = ibBodyValid? b","searchableContent":"  bodyvalid? (ibbody b) = ibbodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":195,"title":"isValid ","content":"isValid : Header ⊎ Body → Type","searchableContent":"  isvalid : header ⊎ body → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":196,"title":"isValid (inj₁ h) = headerValid h","searchableContent":"  isvalid (inj₁ h) = headervalid h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":197,"title":"isValid (inj₂ b) = bodyValid b","searchableContent":"  isvalid (inj₂ b) = bodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":199,"title":"isValid? ","content":"isValid? : ∀ (x : Header ⊎ Body) → Dec (isValid x)","searchableContent":"  isvalid? : ∀ (x : header ⊎ body) → dec (isvalid x)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":200,"title":"isValid? (inj₁ h) = headerValid? h","searchableContent":"  isvalid? (inj₁ h) = headervalid? h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":201,"title":"isValid? (inj₂ b) = bodyValid? b","searchableContent":"  isvalid? (inj₂ b) = bodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":203,"title":"-- some predicates about EBs","searchableContent":"-- some predicates about ebs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":204,"title":"module _ (s ","content":"module _ (s : LeiosState) (eb : EndorserBlock) where","searchableContent":"module _ (s : leiosstate) (eb : endorserblock) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":205,"title":"open EndorserBlockOSig eb","searchableContent":"  open endorserblockosig eb"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":206,"title":"open LeiosState s","searchableContent":"  open leiosstate s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":208,"title":"allIBRefsKnown ","content":"allIBRefsKnown : Type","searchableContent":"  allibrefsknown : type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":209,"title":"allIBRefsKnown = ∀[ ref ∈ fromList ibRefs ] ref ∈ˡ map getIBRef IBs","searchableContent":"  allibrefsknown = ∀[ ref ∈ fromlist ibrefs ] ref ∈ˡ map getibref ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":211,"title":"module _ (s ","content":"module _ (s : LeiosState) where","searchableContent":"module _ (s : leiosstate) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":213,"title":"open LeiosState s","searchableContent":"  open leiosstate s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":215,"title":"upd ","content":"upd : Header ⊎ Body → LeiosState","searchableContent":"  upd : header ⊎ body → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":216,"title":"upd (inj₁ (ebHeader eb)) = record s { EBs = eb ∷ EBs }","searchableContent":"  upd (inj₁ (ebheader eb)) = record s { ebs = eb ∷ ebs }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":217,"title":"upd (inj₁ (vtHeader vs)) = record s { Vs = vs ∷ Vs }","searchableContent":"  upd (inj₁ (vtheader vs)) = record s { vs = vs ∷ vs }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":218,"title":"upd (inj₁ (ibHeader h)) with A.any? (matchIB? h) IBBodies","searchableContent":"  upd (inj₁ (ibheader h)) with a.any? (matchib? h) ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":219,"title":"... | yes p =","searchableContent":"  ... | yes p ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":220,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":221,"title":"{ IBs = record { header = h ; body = A.lookup p } ∷ IBs","searchableContent":"      { ibs = record { header = h ; body = a.lookup p } ∷ ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":222,"title":"; IBBodies = IBBodies A.─ p","searchableContent":"      ; ibbodies = ibbodies a.─ p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":223,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":224,"title":"... | no _ =","searchableContent":"  ... | no _ ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":225,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":226,"title":"{ IBHeaders = h ∷ IBHeaders","searchableContent":"      { ibheaders = h ∷ ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":227,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":228,"title":"upd (inj₂ (ibBody b)) with A.any? (flip matchIB? b) IBHeaders","searchableContent":"  upd (inj₂ (ibbody b)) with a.any? (flip matchib? b) ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":229,"title":"... | yes p =","searchableContent":"  ... | yes p ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":230,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":231,"title":"{ IBs = record { header = A.lookup p ; body = b } ∷ IBs","searchableContent":"      { ibs = record { header = a.lookup p ; body = b } ∷ ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":232,"title":"; IBHeaders = IBHeaders A.─ p","searchableContent":"      ; ibheaders = ibheaders a.─ p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":233,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":234,"title":"... | no _ =","searchableContent":"  ... | no _ ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":235,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":236,"title":"{ IBBodies = b ∷ IBBodies","searchableContent":"      { ibbodies = b ∷ ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":237,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":239,"title":"module _ {s s'} where","searchableContent":"module _ {s s'} where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":240,"title":"open LeiosState s'","searchableContent":"  open leiosstate s'"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":242,"title":"upd-preserves-Upkeep ","content":"upd-preserves-Upkeep : ∀ {x} → LeiosState.Upkeep s ≡ LeiosState.Upkeep s'","searchableContent":"  upd-preserves-upkeep : ∀ {x} → leiosstate.upkeep s ≡ leiosstate.upkeep s'"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":243,"title":"→ LeiosState.Upkeep s ≡ LeiosState.Upkeep (upd s' x)","searchableContent":"                               → leiosstate.upkeep s ≡ leiosstate.upkeep (upd s' x)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":244,"title":"upd-preserves-Upkeep {inj₁ (ibHeader x)} refl with A.any? (matchIB? x) IBBodies","searchableContent":"  upd-preserves-upkeep {inj₁ (ibheader x)} refl with a.any? (matchib? x) ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":245,"title":"... | yes p = refl","searchableContent":"  ... | yes p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":246,"title":"... | no ¬p = refl","searchableContent":"  ... | no ¬p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":247,"title":"upd-preserves-Upkeep {inj₁ (ebHeader x)} refl = refl","searchableContent":"  upd-preserves-upkeep {inj₁ (ebheader x)} refl = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":248,"title":"upd-preserves-Upkeep {inj₁ (vtHeader x)} refl = refl","searchableContent":"  upd-preserves-upkeep {inj₁ (vtheader x)} refl = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":249,"title":"upd-preserves-Upkeep {inj₂ (ibBody x)} refl with A.any? (flip matchIB? x) IBHeaders","searchableContent":"  upd-preserves-upkeep {inj₂ (ibbody x)} refl with a.any? (flip matchib? x) ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":250,"title":"... | yes p = refl","searchableContent":"  ... | yes p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":251,"title":"... | no ¬p = refl","searchableContent":"  ... | no ¬p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":253,"title":"infix 25 _↑_","searchableContent":"infix 25 _↑_"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":254,"title":"_↑_ ","content":"_↑_ : LeiosState → List (Header ⊎ Body) → LeiosState","searchableContent":"_↑_ : leiosstate → list (header ⊎ body) → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":255,"title":"_↑_ = foldr (flip upd)","searchableContent":"_↑_ = foldr (flip upd)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":257,"title":"↑-preserves-Upkeep ","content":"↑-preserves-Upkeep : ∀ {s x} → LeiosState.Upkeep s ≡ LeiosState.Upkeep (s ↑ x)","searchableContent":"↑-preserves-upkeep : ∀ {s x} → leiosstate.upkeep s ≡ leiosstate.upkeep (s ↑ x)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":258,"title":"↑-preserves-Upkeep {x = []} = refl","searchableContent":"↑-preserves-upkeep {x = []} = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":259,"title":"↑-preserves-Upkeep {s = s} {x = x ∷ x₁} =","searchableContent":"↑-preserves-upkeep {s = s} {x = x ∷ x₁} ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":260,"title":"upd-preserves-Upkeep {s = s} {x = x} (↑-preserves-Upkeep {x = x₁})","searchableContent":"  upd-preserves-upkeep {s = s} {x = x} (↑-preserves-upkeep {x = x₁})"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":261,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Decidable
open import Leios.Prelude hiding (id)","searchableContent":"leios.short.decidable
open import leios.prelude hiding (id)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":3,"title":"open import Leios.SpecStructure using (SpecStructure)","searchableContent":"open import leios.specstructure using (specstructure)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":5,"title":"module Leios.Short.Decidable (⋯ ","content":"module Leios.Short.Decidable (⋯ : SpecStructure 1) (let open SpecStructure ⋯) where","searchableContent":"module leios.short.decidable (⋯ : specstructure 1) (let open specstructure ⋯) where"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":7,"title":"open import Leios.Short ⋯ renaming (isVoteCertified to isVoteCertified')","searchableContent":"open import leios.short ⋯ renaming (isvotecertified to isvotecertified')"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":8,"title":"open B hiding (_-⟦_/_⟧⇀_)","searchableContent":"open b hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":9,"title":"open FFD hiding (_-⟦_/_⟧⇀_)","searchableContent":"open ffd hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":11,"title":"module _ {s ","content":"module _ {s : LeiosState} (let open LeiosState s renaming (FFDState to ffds; BaseState to bs)) where","searchableContent":"module _ {s : leiosstate} (let open leiosstate s renaming (ffdstate to ffds; basestate to bs)) where"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":13,"title":"IB-Role? ","content":"IB-Role? : ∀ {π ffds'} →","searchableContent":"  ib-role? : ∀ {π ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":14,"title":"let b = GenFFD.ibBody (record { txs = ToPropose })","searchableContent":"           let b = genffd.ibbody (record { txs = topropose })"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":15,"title":"h = GenFFD.ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"               h = genffd.ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":16,"title":"in","searchableContent":"           in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":17,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep IB-Role }","searchableContent":"           { _ : auto∶ needsupkeep ib-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":18,"title":"{ _ ","content":"{ _ : auto∶ canProduceIB slot sk-IB (stake s) π }","searchableContent":"           { _ : auto∶ canproduceib slot sk-ib (stake s) π }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":19,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send h (just b) / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"           { _ : auto∶ ffds ffd.-⟦ ffd.send h (just b) / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":20,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"           ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":21,"title":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"           s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":22,"title":"IB-Role? {_} {_} {p} {q} {r} = IB-Role (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  ib-role? {_} {_} {p} {q} {r} = ib-role (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":24,"title":"{-","searchableContent":"  {-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":25,"title":"No-IB-Role?","content":"No-IB-Role? :","searchableContent":"  no-ib-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":26,"title":"{ _","content":"{ _ : auto∶ needsUpkeep IB-Role }","searchableContent":"              { _ : auto∶ needsupkeep ib-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":27,"title":"{ _","content":"{ _ : auto∶ ∀ π → ¬ canProduceIB slot sk-IB (stake s) π } →","searchableContent":"              { _ : auto∶ ∀ π → ¬ canproduceib slot sk-ib (stake s) π } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":28,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"              ─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":29,"title":"s ↝ addUpkeep s IB-Role","content":"s ↝ addUpkeep s IB-Role","searchableContent":"              s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":30,"title":"No-IB-Role? {p} {q}","content":"No-IB-Role? {p} {q} = No-IB-Role (toWitness p) (toWitness q)","searchableContent":"  no-ib-role? {p} {q} = no-ib-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":31,"title":"-}","content":"-}","searchableContent":"  -}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":33,"title":"EB-Role? ","content":"EB-Role? : ∀ {π ffds'} →","searchableContent":"  eb-role? : ∀ {π ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":34,"title":"let LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"           let li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":35,"title":"h = mkEB slot id π sk-EB LI []","searchableContent":"               h = mkeb slot id π sk-eb li []"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":36,"title":"in","searchableContent":"           in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":37,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep EB-Role }","searchableContent":"           { _ : auto∶ needsupkeep eb-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":38,"title":"{ _ ","content":"{ _ : auto∶ canProduceEB slot sk-EB (stake s) π }","searchableContent":"           { _ : auto∶ canproduceeb slot sk-eb (stake s) π }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":39,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send (GenFFD.ebHeader h) nothing / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"           { _ : auto∶ ffds ffd.-⟦ ffd.send (genffd.ebheader h) nothing / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":40,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"           ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":41,"title":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"           s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":42,"title":"EB-Role? {_} {_} {p} {q} {r} = EB-Role (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  eb-role? {_} {_} {p} {q} {r} = eb-role (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":44,"title":"{-","searchableContent":"  {-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":45,"title":"No-EB-Role?","content":"No-EB-Role? :","searchableContent":"  no-eb-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":46,"title":"{ _","content":"{ _ : auto∶ needsUpkeep EB-Role }","searchableContent":"              { _ : auto∶ needsupkeep eb-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":47,"title":"{ _","content":"{ _ : auto∶ ∀ π → ¬ canProduceEB slot sk-EB (stake s) π } →","searchableContent":"              { _ : auto∶ ∀ π → ¬ canproduceeb slot sk-eb (stake s) π } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":48,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"              ─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":49,"title":"s ↝ addUpkeep s EB-Role","content":"s ↝ addUpkeep s EB-Role","searchableContent":"              s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":50,"title":"No-EB-Role? {_} {p} {q}","content":"No-EB-Role? {_} {p} {q} = No-EB-Role (toWitness p) (toWitness q)","searchableContent":"  no-eb-role? {_} {p} {q} = no-eb-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":51,"title":"-}","content":"-}","searchableContent":"  -}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":53,"title":"V-Role? ","content":"V-Role? : ∀ {ffds'} →","searchableContent":"  v-role? : ∀ {ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":54,"title":"let EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"          let ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":55,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"              votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":56,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":57,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep VT-Role }","searchableContent":"          { _ : auto∶ needsupkeep vt-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":58,"title":"{ _ ","content":"{ _ : auto∶ canProduceV slot sk-VT (stake s) }","searchableContent":"          { _ : auto∶ canproducev slot sk-vt (stake s) }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":59,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send (GenFFD.vtHeader votes) nothing / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"          { _ : auto∶ ffds ffd.-⟦ ffd.send (genffd.vtheader votes) nothing / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":60,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":61,"title":"s ↝ addUpkeep record s { FFDState = ffds' } VT-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } vt-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":62,"title":"V-Role? {_} {p} {q} {r} = VT-Role (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  v-role? {_} {p} {q} {r} = vt-role (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":64,"title":"No-V-Role? ","content":"No-V-Role? :","searchableContent":"  no-v-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":65,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep VT-Role }","searchableContent":"             { _ : auto∶ needsupkeep vt-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":66,"title":"{ _ ","content":"{ _ : auto∶ ¬ canProduceV slot sk-VT (stake s) } →","searchableContent":"             { _ : auto∶ ¬ canproducev slot sk-vt (stake s) } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":67,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":68,"title":"s ↝ addUpkeep s VT-Role","searchableContent":"             s ↝ addupkeep s vt-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":69,"title":"No-V-Role? {p} {q} = No-VT-Role (toWitness p) (toWitness q)","searchableContent":"  no-v-role? {p} {q} = no-vt-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":71,"title":"{-","searchableContent":"  {-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":72,"title":"Init?","content":"Init? : ∀ {ks pks ks' SD bs' V} →","searchableContent":"  init? : ∀ {ks pks ks' sd bs' v} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":73,"title":"{ _","content":"{ _ : auto∶ ks K.-⟦ K.INIT pk-IB pk-EB pk-V / K.PUBKEYS pks ⟧⇀ ks' }","searchableContent":"        { _ : auto∶ ks k.-⟦ k.init pk-ib pk-eb pk-v / k.pubkeys pks ⟧⇀ ks' }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":74,"title":"{ _","content":"{ _ : initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs' } →","searchableContent":"        { _ : initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":75,"title":"────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────","searchableContent":"        ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":76,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'","content":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'","searchableContent":"        nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":77,"title":"Init?","content":"Init? = ?","searchableContent":"  init? = ?"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":78,"title":"-}","content":"-}","searchableContent":"  -}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":80,"title":"Base₂a? ","content":"Base₂a? : ∀ {eb bs'} →","searchableContent":"  base₂a? : ∀ {eb bs'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":81,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep Base }","searchableContent":"          { _ : auto∶ needsupkeep base }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":82,"title":"{ _ ","content":"{ _ : auto∶ eb ∈ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2) EBs }","searchableContent":"          { _ : auto∶ eb ∈ filter (λ eb → isvotecertified' s eb × eb ∈ᴮ slice l slot 2) ebs }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":83,"title":"{ _ ","content":"{ _ : auto∶ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs' } →","searchableContent":"          { _ : auto∶ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":84,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":85,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":86,"title":"Base₂a? {_} {_} {p} {q} {r} = Base₂a (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  base₂a? {_} {_} {p} {q} {r} = base₂a (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":88,"title":"Base₂b? ","content":"Base₂b? : ∀ {bs'} →","searchableContent":"  base₂b? : ∀ {bs'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":89,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep Base }","searchableContent":"          { _ : auto∶ needsupkeep base }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":90,"title":"{ _ ","content":"{ _ : auto∶ [] ≡ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2) EBs }","searchableContent":"          { _ : auto∶ [] ≡ filter (λ eb → isvotecertified' s eb × eb ∈ᴮ slice l slot 2) ebs }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":91,"title":"{ _ ","content":"{ _ : auto∶ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs' } →","searchableContent":"          { _ : auto∶ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":92,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":93,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":94,"title":"Base₂b? {_} {p} {q} {r} = Base₂b (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  base₂b? {_} {p} {q} {r} = base₂b (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":96,"title":"Slot? ","content":"Slot? : ∀ {rbs bs' msgs ffds'} →","searchableContent":"  slot? : ∀ {rbs bs' msgs ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":97,"title":"{ _ ","content":"{ _ : auto∶ allDone s }","searchableContent":"         { _ : auto∶ alldone s }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":98,"title":"{ _ ","content":"{ _ : auto∶ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs' }","searchableContent":"         { _ : auto∶ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs' }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":99,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds' } →","searchableContent":"         { _ : auto∶ ffds ffd.-⟦ ffd.fetch / ffd.fetchres msgs ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":100,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"         ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":101,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"         just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":102,"title":"{ FFDState  = ffds'","searchableContent":"             { ffdstate  = ffds'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":103,"title":"; BaseState = bs'","searchableContent":"             ; basestate = bs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":104,"title":"; Ledger    = constructLedger rbs","searchableContent":"             ; ledger    = constructledger rbs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":105,"title":"; slot      = suc slot","searchableContent":"             ; slot      = suc slot"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":106,"title":"; Upkeep    = ∅","searchableContent":"             ; upkeep    = ∅"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":107,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"             } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":108,"title":"Slot? {_} {_} {_} {_} {p} {q} {r} = Slot (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  slot? {_} {_} {_} {_} {p} {q} {r} = slot (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":109,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Trace.Verifier.Test
open import Leios.Prelude hiding (id)","searchableContent":"leios.short.trace.verifier.test
open import leios.prelude hiding (id)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":3,"title":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":5,"title":"module Leios.Short.Trace.Verifier.Test where","searchableContent":"module leios.short.trace.verifier.test where"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":7,"title":"params ","content":"params : Params","searchableContent":"params : params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":8,"title":"params =","searchableContent":"params ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":9,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":10,"title":"{ numberOfParties = 2","searchableContent":"    { numberofparties = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":11,"title":"; sutId = fzero","searchableContent":"    ; sutid = fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":12,"title":"; stakeDistribution =","searchableContent":"    ; stakedistribution ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":13,"title":"let open FunTot (completeFin 2) (maximalFin 2)","searchableContent":"        let open funtot (completefin 2) (maximalfin 2)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":14,"title":"in Fun⇒TotalMap (const 100000000)","searchableContent":"        in fun⇒totalmap (const 100000000)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":15,"title":"; stageLength = 2","searchableContent":"    ; stagelength = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":16,"title":"; winning-slots = fromList $","searchableContent":"    ; winning-slots = fromlist $"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":17,"title":"(IB , 0) ∷ (EB , 0) ∷ (VT , 0) ∷","searchableContent":"        (ib , 0) ∷ (eb , 0) ∷ (vt , 0) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":18,"title":"(IB , 1) ∷ (EB , 1) ∷ (VT , 1) ∷","searchableContent":"        (ib , 1) ∷ (eb , 1) ∷ (vt , 1) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":19,"title":"(IB , 2) ∷ (EB , 2) ∷ (VT , 2) ∷","searchableContent":"        (ib , 2) ∷ (eb , 2) ∷ (vt , 2) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":20,"title":"(IB , 3) ∷ (EB , 3) ∷ (VT , 3) ∷","searchableContent":"        (ib , 3) ∷ (eb , 3) ∷ (vt , 3) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":21,"title":"(IB , 4) ∷ (EB , 4) ∷ (VT , 4) ∷","searchableContent":"        (ib , 4) ∷ (eb , 4) ∷ (vt , 4) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":22,"title":"(IB , 5) ∷ (EB , 5) ∷ (VT , 5) ∷","searchableContent":"        (ib , 5) ∷ (eb , 5) ∷ (vt , 5) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":23,"title":"(VT , 6) ∷","searchableContent":"                              (vt , 6) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":24,"title":"(IB , 7) ∷ (EB , 7) ∷ (VT , 7) ∷","searchableContent":"        (ib , 7) ∷ (eb , 7) ∷ (vt , 7) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":25,"title":"(IB , 8) ∷ (EB , 8) ∷ (VT , 8) ∷","searchableContent":"        (ib , 8) ∷ (eb , 8) ∷ (vt , 8) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":26,"title":"[]","searchableContent":"        []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":27,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":29,"title":"open Params params","searchableContent":"open params params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":30,"title":"open import Leios.Short.Trace.Verifier params","searchableContent":"open import leios.short.trace.verifier params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":32,"title":"private","searchableContent":"private"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":33,"title":"opaque","searchableContent":"  opaque"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":34,"title":"unfolding List-Model","searchableContent":"    unfolding list-model"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":36,"title":"test₁ ","content":"test₁ : Bool","searchableContent":"    test₁ : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":37,"title":"test₁ = ¿ ValidTrace (inj₁ (IB-Role-Action 0 , SLOT) ∷ []) ¿ᵇ","searchableContent":"    test₁ = ¿ validtrace (inj₁ (ib-role-action 0 , slot) ∷ []) ¿ᵇ"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":39,"title":"_ ","content":"_ : test₁ ≡ true","searchableContent":"    _ : test₁ ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":40,"title":"_ = refl","searchableContent":"    _ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":42,"title":"test-valid-ib ","content":"test-valid-ib : Bool","searchableContent":"    test-valid-ib : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":43,"title":"test-valid-ib =","searchableContent":"    test-valid-ib ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":44,"title":"let h = record { slotNumber = 1","searchableContent":"      let h = record { slotnumber = 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":45,"title":"; producerID = fsuc fzero","searchableContent":"                     ; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":46,"title":"; lotteryPf = tt","searchableContent":"                     ; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":47,"title":"; bodyHash = 0 ∷ 1 ∷ 2 ∷ []","searchableContent":"                     ; bodyhash = 0 ∷ 1 ∷ 2 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":48,"title":"; signature = tt","searchableContent":"                     ; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":49,"title":"}","searchableContent":"                     }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":50,"title":"b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }","searchableContent":"          b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":51,"title":"ib = record { header = h ; body = b }","searchableContent":"          ib = record { header = h ; body = b }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":52,"title":"pks = L.zip (completeFinL numberOfParties) (L.replicate numberOfParties tt)","searchableContent":"          pks = l.zip (completefinl numberofparties) (l.replicate numberofparties tt)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":53,"title":"s = initLeiosState tt stakeDistribution tt pks","searchableContent":"          s = initleiosstate tt stakedistribution tt pks"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":54,"title":"in isYes (ibValid? s ib)","searchableContent":"      in isyes (ibvalid? s ib)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":56,"title":"_ ","content":"_ : test-valid-ib ≡ true","searchableContent":"    _ : test-valid-ib ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":57,"title":"_ = refl","searchableContent":"    _ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":59,"title":"test₂ ","content":"test₂ : Bool","searchableContent":"    test₂ : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":60,"title":"test₂ =","searchableContent":"    test₂ ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":61,"title":"let t = L.reverse $","searchableContent":"      let t = l.reverse $"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":62,"title":"-- slot 0","searchableContent":"            -- slot 0"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":63,"title":"inj₁ (IB-Role-Action 0    , SLOT)","searchableContent":"              inj₁ (ib-role-action 0    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":64,"title":"∷ inj₁ (EB-Role-Action 0 [] , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 0 [] , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":65,"title":"∷ inj₁ (VT-Role-Action 0    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 0    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":66,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":67,"title":"∷ inj₁ (Slot-Action    0    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    0    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":68,"title":"-- slot 1","searchableContent":"            -- slot 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":69,"title":"∷ inj₁ (IB-Role-Action 1    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 1    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":70,"title":"∷ inj₁ (VT-Role-Action 1    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 1    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":71,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":72,"title":"∷ inj₁ (Slot-Action    1    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    1    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":73,"title":"-- slot 2","searchableContent":"            -- slot 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":74,"title":"∷ inj₂ (IB-Recv-Update","searchableContent":"            ∷ inj₂ (ib-recv-update"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":75,"title":"(record { header =","searchableContent":"                (record { header ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":76,"title":"record { slotNumber = 1","searchableContent":"                  record { slotnumber = 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":77,"title":"; producerID = fsuc fzero","searchableContent":"                         ; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":78,"title":"; lotteryPf = tt","searchableContent":"                         ; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":79,"title":"; bodyHash = 0 ∷ 1 ∷ 2 ∷ []","searchableContent":"                         ; bodyhash = 0 ∷ 1 ∷ 2 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":80,"title":"; signature = tt","searchableContent":"                         ; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":81,"title":"}","searchableContent":"                         }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":82,"title":"; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))","searchableContent":"                        ; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":83,"title":"∷ inj₁ (IB-Role-Action 2    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 2    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":84,"title":"∷ inj₁ (EB-Role-Action 2 [] , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 2 [] , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":85,"title":"∷ inj₁ (VT-Role-Action 2    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 2    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":86,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":87,"title":"∷ inj₁ (Slot-Action    2    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    2    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":88,"title":"-- slot 3","searchableContent":"            -- slot 3"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":89,"title":"∷ inj₂ (IB-Recv-Update","searchableContent":"            ∷ inj₂ (ib-recv-update"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":90,"title":"(record { header =","searchableContent":"                (record { header ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":91,"title":"record { slotNumber = 2","searchableContent":"                  record { slotnumber = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":92,"title":"; producerID = fsuc fzero","searchableContent":"                         ; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":93,"title":"; lotteryPf = tt","searchableContent":"                         ; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":94,"title":"; bodyHash = 3 ∷ 4 ∷ 5 ∷ []","searchableContent":"                         ; bodyhash = 3 ∷ 4 ∷ 5 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":95,"title":"; signature = tt","searchableContent":"                         ; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":96,"title":"}","searchableContent":"                         }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":97,"title":"; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))","searchableContent":"                        ; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":98,"title":"∷ inj₁ (IB-Role-Action 3    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 3    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":99,"title":"∷ inj₁ (VT-Role-Action 3    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 3    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":100,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":101,"title":"∷ inj₁ (Slot-Action    3    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    3    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":102,"title":"-- slot 4","searchableContent":"            -- slot 4"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":103,"title":"∷ inj₁ (IB-Role-Action 4    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 4    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":104,"title":"∷ inj₁ (EB-Role-Action 4 [] , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 4 [] , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":105,"title":"∷ inj₁ (VT-Role-Action 4    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 4    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":106,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":107,"title":"∷ inj₁ (Slot-Action    4    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    4    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":108,"title":"-- slot 5","searchableContent":"            -- slot 5"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":109,"title":"∷ inj₁ (IB-Role-Action 5    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 5    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":110,"title":"∷ inj₁ (VT-Role-Action 5    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 5    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":111,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":112,"title":"∷ inj₁ (Slot-Action    5    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    5    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":113,"title":"-- slot 6","searchableContent":"            -- slot 6"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":114,"title":"∷ inj₁ (No-IB-Role-Action   , SLOT)","searchableContent":"            ∷ inj₁ (no-ib-role-action   , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":115,"title":"∷ inj₁ (No-EB-Role-Action   , SLOT)","searchableContent":"            ∷ inj₁ (no-eb-role-action   , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":116,"title":"∷ inj₁ (VT-Role-Action 6    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 6    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":117,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":118,"title":"∷ inj₁ (Slot-Action    6    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    6    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":119,"title":"-- slot 7","searchableContent":"            -- slot 7"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":120,"title":"∷ inj₁ (IB-Role-Action 7    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 7    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":121,"title":"∷ inj₁ (VT-Role-Action 7    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 7    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":122,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":123,"title":"∷ inj₁ (Slot-Action    7    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    7    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":124,"title":"-- slot 8","searchableContent":"            -- slot 8"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":125,"title":"∷ inj₁ (IB-Role-Action 8    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 8    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":126,"title":"∷ inj₁ (EB-Role-Action 8 ((3 ∷ 4 ∷ 5 ∷ []) ∷ []) , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 8 ((3 ∷ 4 ∷ 5 ∷ []) ∷ []) , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":127,"title":"∷ inj₁ (VT-Role-Action 8    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 8    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":128,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":129,"title":"∷ inj₁ (Slot-Action    8    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    8    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":130,"title":"∷ []","searchableContent":"            ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":131,"title":"in ¿ ValidTrace t ¿ᵇ","searchableContent":"      in ¿ validtrace t ¿ᵇ"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":133,"title":"_ ","content":"_ : test₂ ≡ true","searchableContent":"    _ : test₂ ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":134,"title":"_ = refl","searchableContent":"    _ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":135,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Trace.Verifier
open import Leios.Prelude hiding (id)","searchableContent":"leios.short.trace.verifier
open import leios.prelude hiding (id)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":3,"title":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":5,"title":"-- TODO","content":"-- TODO: SpecStructure as parameter","searchableContent":"-- todo: specstructure as parameter"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":6,"title":"module Leios.Short.Trace.Verifier (params ","content":"module Leios.Short.Trace.Verifier (params : Params) (let open Params params) where","searchableContent":"module leios.short.trace.verifier (params : params) (let open params params) where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":8,"title":"open import Leios.Defaults params","searchableContent":"open import leios.defaults params"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":9,"title":"using (LeiosState; initLeiosState; isb; hpe; hhs; htx; SendIB; FFDBuffers; Dec-SimpleFFD)","searchableContent":"  using (leiosstate; initleiosstate; isb; hpe; hhs; htx; sendib; ffdbuffers; dec-simpleffd)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":10,"title":"renaming (d-SpecStructure to traceSpecStructure) public","searchableContent":"  renaming (d-specstructure to tracespecstructure) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":12,"title":"open import Leios.SpecStructure using (SpecStructure)","searchableContent":"open import leios.specstructure using (specstructure)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":13,"title":"open SpecStructure traceSpecStructure hiding (Hashable-IBHeader; Hashable-EndorserBlock; isVoteCertified) public","searchableContent":"open specstructure tracespecstructure hiding (hashable-ibheader; hashable-endorserblock; isvotecertified) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":15,"title":"open import Leios.Short traceSpecStructure hiding (LeiosState; initLeiosState) public","searchableContent":"open import leios.short tracespecstructure hiding (leiosstate; initleiosstate) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":16,"title":"open import Prelude.Closures _↝_","searchableContent":"open import prelude.closures _↝_"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":17,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":19,"title":"data FFDUpdate ","content":"data FFDUpdate : Type where","searchableContent":"data ffdupdate : type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":20,"title":"IB-Recv-Update ","content":"IB-Recv-Update : InputBlock → FFDUpdate","searchableContent":"  ib-recv-update : inputblock → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":21,"title":"EB-Recv-Update ","content":"EB-Recv-Update : EndorserBlock → FFDUpdate","searchableContent":"  eb-recv-update : endorserblock → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":22,"title":"VT-Recv-Update ","content":"VT-Recv-Update : List Vote → FFDUpdate","searchableContent":"  vt-recv-update : list vote → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":24,"title":"data Action ","content":"data Action : Type where","searchableContent":"data action : type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":25,"title":"IB-Role-Action ","content":"IB-Role-Action : ℕ → Action","searchableContent":"  ib-role-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":26,"title":"EB-Role-Action ","content":"EB-Role-Action : ℕ → List IBRef → Action","searchableContent":"  eb-role-action : ℕ → list ibref → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":27,"title":"VT-Role-Action ","content":"VT-Role-Action : ℕ → Action","searchableContent":"  vt-role-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":28,"title":"No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action ","content":"No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action : Action","searchableContent":"  no-ib-role-action no-eb-role-action no-vt-role-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":29,"title":"Ftch-Action ","content":"Ftch-Action : Action","searchableContent":"  ftch-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":30,"title":"Slot-Action ","content":"Slot-Action : ℕ → Action","searchableContent":"  slot-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":31,"title":"Base₁-Action ","content":"Base₁-Action : Action","searchableContent":"  base₁-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":32,"title":"Base₂a-Action ","content":"Base₂a-Action : EndorserBlock → Action","searchableContent":"  base₂a-action : endorserblock → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":33,"title":"Base₂b-Action ","content":"Base₂b-Action : Action","searchableContent":"  base₂b-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":35,"title":"Actions = List (Action × LeiosInput)","searchableContent":"actions = list (action × leiosinput)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":37,"title":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":38,"title":"s s′ ","content":"s s′ : LeiosState","searchableContent":"  s s′ : leiosstate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":39,"title":"α ","content":"α : Action","searchableContent":"  α : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":41,"title":"data ValidUpdate ","content":"data ValidUpdate : FFDUpdate → LeiosState → Type where","searchableContent":"data validupdate : ffdupdate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":43,"title":"IB-Recv ","content":"IB-Recv : ∀ {ib} →","searchableContent":"  ib-recv : ∀ {ib} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":44,"title":"ValidUpdate (IB-Recv-Update ib) s","searchableContent":"    validupdate (ib-recv-update ib) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":46,"title":"EB-Recv ","content":"EB-Recv : ∀ {eb} →","searchableContent":"  eb-recv : ∀ {eb} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":47,"title":"ValidUpdate (EB-Recv-Update eb) s","searchableContent":"    validupdate (eb-recv-update eb) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":49,"title":"VT-Recv ","content":"VT-Recv : ∀ {vt} →","searchableContent":"  vt-recv : ∀ {vt} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":50,"title":"ValidUpdate (VT-Recv-Update vt) s","searchableContent":"    validupdate (vt-recv-update vt) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":52,"title":"data ValidAction ","content":"data ValidAction : Action → LeiosState → LeiosInput → Type where","searchableContent":"data validaction : action → leiosstate → leiosinput → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":54,"title":"IB-Role ","content":"IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":55,"title":"b = record { txs = ToPropose }","searchableContent":"                b = record { txs = topropose }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":56,"title":"h = mkIBHeader slot id tt sk-IB ToPropose","searchableContent":"                h = mkibheader slot id tt sk-ib topropose"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":57,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ibHeader h} {just (ibBody b)})","searchableContent":"                ffds' = proj₁ (ffd.send-total {ffds} {ibheader h} {just (ibbody b)})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":58,"title":"in .(needsUpkeep IB-Role) →","searchableContent":"            in .(needsupkeep ib-role) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":59,"title":".(canProduceIB slot sk-IB (stake s) tt) →","searchableContent":"               .(canproduceib slot sk-ib (stake s) tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":60,"title":".(ffds FFD.-⟦ FFD.Send (ibHeader h) (just (ibBody b)) / FFD.SendRes ⟧⇀ ffds') →","searchableContent":"               .(ffds ffd.-⟦ ffd.send (ibheader h) (just (ibbody b)) / ffd.sendres ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":61,"title":"ValidAction (IB-Role-Action slot) s SLOT","searchableContent":"               validaction (ib-role-action slot) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":63,"title":"EB-Role ","content":"EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":64,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":65,"title":"h = mkEB slot id tt sk-EB LI []","searchableContent":"                h = mkeb slot id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":66,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ebHeader h} {nothing})","searchableContent":"                ffds' = proj₁ (ffd.send-total {ffds} {ebheader h} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":67,"title":"in .(needsUpkeep EB-Role) →","searchableContent":"            in .(needsupkeep eb-role) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":68,"title":".(canProduceEB slot sk-EB (stake s) tt) →","searchableContent":"               .(canproduceeb slot sk-eb (stake s) tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":69,"title":".(ffds FFD.-⟦ FFD.Send (ebHeader h) nothing / FFD.SendRes ⟧⇀ ffds') →","searchableContent":"               .(ffds ffd.-⟦ ffd.send (ebheader h) nothing / ffd.sendres ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":70,"title":"ValidAction (EB-Role-Action slot LI) s SLOT","searchableContent":"               validaction (eb-role-action slot li) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":72,"title":"VT-Role ","content":"VT-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  vt-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":73,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":74,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":75,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {vtHeader votes} {nothing})","searchableContent":"                ffds' = proj₁ (ffd.send-total {ffds} {vtheader votes} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":76,"title":"in .(needsUpkeep VT-Role) →","searchableContent":"            in .(needsupkeep vt-role) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":77,"title":".(canProduceV slot sk-VT (stake s)) →","searchableContent":"               .(canproducev slot sk-vt (stake s)) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":78,"title":".(ffds FFD.-⟦ FFD.Send (vtHeader votes) nothing / FFD.SendRes ⟧⇀ ffds') →","searchableContent":"               .(ffds ffd.-⟦ ffd.send (vtheader votes) nothing / ffd.sendres ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":79,"title":"ValidAction (VT-Role-Action slot) s SLOT","searchableContent":"               validaction (vt-role-action slot) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":81,"title":"No-IB-Role ","content":"No-IB-Role : let open LeiosState s","searchableContent":"  no-ib-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":82,"title":"in needsUpkeep IB-Role →","searchableContent":"               in needsupkeep ib-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":83,"title":"(∀ π → ¬ canProduceIB slot sk-IB (stake s) π) →","searchableContent":"                  (∀ π → ¬ canproduceib slot sk-ib (stake s) π) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":84,"title":"ValidAction No-IB-Role-Action s SLOT","searchableContent":"                  validaction no-ib-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":86,"title":"No-EB-Role ","content":"No-EB-Role : let open LeiosState s","searchableContent":"  no-eb-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":87,"title":"in needsUpkeep EB-Role →","searchableContent":"               in needsupkeep eb-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":88,"title":"(∀ π → ¬ canProduceEB slot sk-EB (stake s) π) →","searchableContent":"                  (∀ π → ¬ canproduceeb slot sk-eb (stake s) π) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":89,"title":"ValidAction No-EB-Role-Action s SLOT","searchableContent":"                  validaction no-eb-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":91,"title":"No-VT-Role ","content":"No-VT-Role : let open LeiosState s","searchableContent":"  no-vt-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":92,"title":"in needsUpkeep VT-Role →","searchableContent":"               in needsupkeep vt-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":93,"title":"(¬ canProduceV slot sk-VT (stake s)) →","searchableContent":"                  (¬ canproducev slot sk-vt (stake s)) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":94,"title":"ValidAction No-VT-Role-Action s SLOT","searchableContent":"                  validaction no-vt-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":96,"title":"Slot ","content":"Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs)","searchableContent":"  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":97,"title":"(msgs , (ffds' , _)) = FFD.Fetch-total {ffds}","searchableContent":"             (msgs , (ffds' , _)) = ffd.fetch-total {ffds}"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":98,"title":"in .(allDone s) →","searchableContent":"         in .(alldone s) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":99,"title":".(bs B.-⟦ B.FTCH-LDG / B.BASE-LDG [] ⟧⇀ tt) →","searchableContent":"            .(bs b.-⟦ b.ftch-ldg / b.base-ldg [] ⟧⇀ tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":100,"title":".(ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds') →","searchableContent":"            .(ffds ffd.-⟦ ffd.fetch / ffd.fetchres msgs ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":101,"title":"ValidAction (Slot-Action slot) s SLOT","searchableContent":"            validaction (slot-action slot) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":103,"title":"Ftch ","content":"Ftch : ValidAction Ftch-Action s FTCH-LDG","searchableContent":"  ftch : validaction ftch-action s ftch-ldg"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":105,"title":"Base₁ ","content":"Base₁ : ∀ {txs} → ValidAction Base₁-Action s (SUBMIT (inj₂ txs))","searchableContent":"  base₁ : ∀ {txs} → validaction base₁-action s (submit (inj₂ txs))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":107,"title":"Base₂a ","content":"Base₂a : ∀ {eb} → let open LeiosState s renaming (BaseState to bs)","searchableContent":"  base₂a : ∀ {eb} → let open leiosstate s renaming (basestate to bs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":108,"title":"in .(needsUpkeep Base) →","searchableContent":"           in .(needsupkeep base) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":109,"title":".(eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs) →","searchableContent":"              .(eb ∈ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":110,"title":".(bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ tt) →","searchableContent":"              .(bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":111,"title":"ValidAction (Base₂a-Action eb) s SLOT","searchableContent":"              validaction (base₂a-action eb) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":113,"title":"Base₂b ","content":"Base₂b : let open LeiosState s renaming (BaseState to bs)","searchableContent":"  base₂b : let open leiosstate s renaming (basestate to bs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":114,"title":"in .(needsUpkeep Base) →","searchableContent":"           in .(needsupkeep base) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":115,"title":".([] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs) →","searchableContent":"              .([] ≡ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":116,"title":".(bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ tt) →","searchableContent":"              .(bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":117,"title":"ValidAction Base₂b-Action s SLOT","searchableContent":"              validaction base₂b-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":119,"title":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":120,"title":"i ","content":"i : LeiosInput","searchableContent":"  i : leiosinput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":121,"title":"o ","content":"o : LeiosOutput","searchableContent":"  o : leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":123,"title":"⟦_⟧ ","content":"⟦_⟧ : ValidAction α s i → LeiosState × LeiosOutput","searchableContent":"⟦_⟧ : validaction α s i → leiosstate × leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":124,"title":"⟦ IB-Role {s} _ _ _ ⟧ =","searchableContent":"⟦ ib-role {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":125,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":126,"title":"b = record { txs = ToPropose }","searchableContent":"      b = record { txs = topropose }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":127,"title":"h = mkIBHeader slot id tt sk-IB ToPropose","searchableContent":"      h = mkibheader slot id tt sk-ib topropose"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":128,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ibHeader h} {just (ibBody b)})","searchableContent":"      ffds' = proj₁ (ffd.send-total {ffds} {ibheader h} {just (ibbody b)})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":129,"title":"in addUpkeep record s { FFDState = ffds' } IB-Role , EMPTY","searchableContent":"  in addupkeep record s { ffdstate = ffds' } ib-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":130,"title":"⟦ EB-Role {s} _ _ _ ⟧ =","searchableContent":"⟦ eb-role {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":131,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":132,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"      li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":133,"title":"h = mkEB slot id tt sk-EB LI []","searchableContent":"      h = mkeb slot id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":134,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ebHeader h} {nothing})","searchableContent":"      ffds' = proj₁ (ffd.send-total {ffds} {ebheader h} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":135,"title":"in addUpkeep record s { FFDState = ffds' } EB-Role , EMPTY","searchableContent":"  in addupkeep record s { ffdstate = ffds' } eb-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":136,"title":"⟦ VT-Role {s} _ _ _ ⟧ =","searchableContent":"⟦ vt-role {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":137,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":138,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"      ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":139,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"      votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":140,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {vtHeader votes} {nothing})","searchableContent":"      ffds' = proj₁ (ffd.send-total {ffds} {vtheader votes} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":141,"title":"in addUpkeep record s { FFDState = ffds' } VT-Role , EMPTY","searchableContent":"  in addupkeep record s { ffdstate = ffds' } vt-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":142,"title":"⟦ No-IB-Role {s} _ _ ⟧ = addUpkeep s IB-Role , EMPTY","searchableContent":"⟦ no-ib-role {s} _ _ ⟧ = addupkeep s ib-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":143,"title":"⟦ No-EB-Role {s} _ _ ⟧ = addUpkeep s EB-Role , EMPTY","searchableContent":"⟦ no-eb-role {s} _ _ ⟧ = addupkeep s eb-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":144,"title":"⟦ No-VT-Role {s} _ _ ⟧ = addUpkeep s VT-Role , EMPTY","searchableContent":"⟦ no-vt-role {s} _ _ ⟧ = addupkeep s vt-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":145,"title":"⟦ Slot {s} _ _ _ ⟧ =","searchableContent":"⟦ slot {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":146,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":147,"title":"(msgs , (ffds' , _)) = FFD.Fetch-total {ffds}","searchableContent":"      (msgs , (ffds' , _)) = ffd.fetch-total {ffds}"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":148,"title":"in","searchableContent":"  in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":149,"title":"(record s","searchableContent":"  (record s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":150,"title":"{ FFDState  = ffds'","searchableContent":"     { ffdstate  = ffds'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":151,"title":"; BaseState = tt","searchableContent":"     ; basestate = tt"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":152,"title":"; Ledger    = constructLedger []","searchableContent":"     ; ledger    = constructledger []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":153,"title":"; slot      = suc slot","searchableContent":"     ; slot      = suc slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":154,"title":"; Upkeep    = ∅","searchableContent":"     ; upkeep    = ∅"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":155,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"     } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":156,"title":", EMPTY)","searchableContent":"  , empty)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":157,"title":"⟦ Ftch {s} ⟧ = s , FTCH-LDG (LeiosState.Ledger s)","searchableContent":"⟦ ftch {s} ⟧ = s , ftch-ldg (leiosstate.ledger s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":158,"title":"⟦ Base₁ {s} {txs} ⟧ = record s { ToPropose = txs } , EMPTY","searchableContent":"⟦ base₁ {s} {txs} ⟧ = record s { topropose = txs } , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":159,"title":"⟦ Base₂a {s} _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY","searchableContent":"⟦ base₂a {s} _ _ _ ⟧ = addupkeep record s { basestate = tt } base , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":160,"title":"⟦ Base₂b {s} _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY","searchableContent":"⟦ base₂b {s} _ _ _ ⟧ = addupkeep record s { basestate = tt } base , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":162,"title":"open LeiosState","searchableContent":"open leiosstate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":163,"title":"open FFDBuffers","searchableContent":"open ffdbuffers"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":165,"title":"ValidAction→Eq-Slot ","content":"ValidAction→Eq-Slot : ∀ {s sl} → ValidAction (Slot-Action sl) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-slot : ∀ {s sl} → validaction (slot-action sl) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":166,"title":"ValidAction→Eq-Slot (Slot _ _ _) = refl","searchableContent":"validaction→eq-slot (slot _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":168,"title":"ValidAction→Eq-IB ","content":"ValidAction→Eq-IB : ∀ {s sl} → ValidAction (IB-Role-Action sl) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-ib : ∀ {s sl} → validaction (ib-role-action sl) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":169,"title":"ValidAction→Eq-IB (IB-Role _ _ _) = refl","searchableContent":"validaction→eq-ib (ib-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":171,"title":"ValidAction→Eq-EB ","content":"ValidAction→Eq-EB : ∀ {s sl ibs} → ValidAction (EB-Role-Action sl ibs) s SLOT → sl ≡ slot s × ibs ≡ (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))","searchableContent":"validaction→eq-eb : ∀ {s sl ibs} → validaction (eb-role-action sl ibs) s slot → sl ≡ slot s × ibs ≡ (map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":172,"title":"ValidAction→Eq-EB (EB-Role _ _ _) = refl , refl","searchableContent":"validaction→eq-eb (eb-role _ _ _) = refl , refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":174,"title":"ValidAction→Eq-VT ","content":"ValidAction→Eq-VT : ∀ {s sl} → ValidAction (VT-Role-Action sl) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-vt : ∀ {s sl} → validaction (vt-role-action sl) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":175,"title":"ValidAction→Eq-VT (VT-Role _ _ _) = refl","searchableContent":"validaction→eq-vt (vt-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":177,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":178,"title":"Dec-ValidAction ","content":"Dec-ValidAction : ValidAction ⁇³","searchableContent":"  dec-validaction : validaction ⁇³"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":179,"title":"Dec-ValidAction {IB-Role-Action sl} {s} {SLOT} .dec","searchableContent":"  dec-validaction {ib-role-action sl} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":180,"title":"with sl ≟ slot s","searchableContent":"    with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":181,"title":"... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-IB x))","searchableContent":"  ... | no ¬p = no λ x → ⊥-elim (¬p (validaction→eq-ib x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":182,"title":"... | yes p rewrite p","searchableContent":"  ... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":183,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":184,"title":"... | yes x | yes y | yes z = yes (IB-Role x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (ib-role x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":185,"title":"... | no ¬p | _ | _ = no λ where (IB-Role p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (ib-role p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":186,"title":"... | _ | no ¬p | _ = no λ where (IB-Role _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (ib-role _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":187,"title":"... | _ | _ | no ¬p = no λ where (IB-Role _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (ib-role _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":188,"title":"Dec-ValidAction {IB-Role-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ib-role-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":189,"title":"Dec-ValidAction {IB-Role-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ib-role-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":190,"title":"Dec-ValidAction {IB-Role-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {ib-role-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":191,"title":"Dec-ValidAction {EB-Role-Action sl ibs} {s} {SLOT} .dec","searchableContent":"  dec-validaction {eb-role-action sl ibs} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":192,"title":"with sl ≟ slot s | ibs ≟ (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))","searchableContent":"    with sl ≟ slot s | ibs ≟ (map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":193,"title":"... | no ¬p | _ = no λ x → ⊥-elim (¬p (proj₁ $ ValidAction→Eq-EB x))","searchableContent":"  ... | no ¬p | _ = no λ x → ⊥-elim (¬p (proj₁ $ validaction→eq-eb x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":194,"title":"... | _ | no ¬q = no λ x → ⊥-elim (¬q (proj₂ $ ValidAction→Eq-EB x))","searchableContent":"  ... | _ | no ¬q = no λ x → ⊥-elim (¬q (proj₂ $ validaction→eq-eb x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":195,"title":"... | yes p | yes q rewrite p rewrite q","searchableContent":"  ... | yes p | yes q rewrite p rewrite q"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":196,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":197,"title":"... | yes x | yes y | yes z = yes (EB-Role x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (eb-role x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":198,"title":"... | no ¬p | _ | _ = no λ where (EB-Role p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (eb-role p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":199,"title":"... | _ | no ¬p | _ = no λ where (EB-Role _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (eb-role _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":200,"title":"... | _ | _ | no ¬p = no λ where (EB-Role _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (eb-role _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":201,"title":"Dec-ValidAction {EB-Role-Action _ _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {eb-role-action _ _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":202,"title":"Dec-ValidAction {EB-Role-Action _ _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {eb-role-action _ _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":203,"title":"Dec-ValidAction {EB-Role-Action _ _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {eb-role-action _ _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":204,"title":"Dec-ValidAction {VT-Role-Action sl} {s} {SLOT} .dec","searchableContent":"  dec-validaction {vt-role-action sl} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":205,"title":"with sl ≟ slot s","searchableContent":"    with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":206,"title":"... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-VT x))","searchableContent":"  ... | no ¬p = no λ x → ⊥-elim (¬p (validaction→eq-vt x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":207,"title":"... | yes p rewrite p","searchableContent":"  ... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":208,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":209,"title":"... | yes x | yes y | yes z = yes (VT-Role x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (vt-role x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":210,"title":"... | no ¬p | _ | _ = no λ where (VT-Role p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (vt-role p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":211,"title":"... | _ | no ¬p | _ = no λ where (VT-Role _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (vt-role _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":212,"title":"... | _ | _ | no ¬p = no λ where (VT-Role _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (vt-role _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":213,"title":"Dec-ValidAction {VT-Role-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {vt-role-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":214,"title":"Dec-ValidAction {VT-Role-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {vt-role-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":215,"title":"Dec-ValidAction {VT-Role-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {vt-role-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":216,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {no-ib-role-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":217,"title":"with dec | dec","searchableContent":"    with dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":218,"title":"... | yes p | yes q = yes (No-IB-Role p q)","searchableContent":"  ... | yes p | yes q = yes (no-ib-role p q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":219,"title":"... | no ¬p | _ = no λ where (No-IB-Role p _) → ⊥-elim (¬p p)","searchableContent":"  ... | no ¬p | _ = no λ where (no-ib-role p _) → ⊥-elim (¬p p)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":220,"title":"... | _ | no ¬q = no λ where (No-IB-Role _ q) → ⊥-elim (¬q q)","searchableContent":"  ... | _ | no ¬q = no λ where (no-ib-role _ q) → ⊥-elim (¬q q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":221,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-ib-role-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":222,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-ib-role-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":223,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {no-ib-role-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":224,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {no-eb-role-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":225,"title":"with dec | dec","searchableContent":"    with dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":226,"title":"... | yes p | yes q = yes (No-EB-Role p q)","searchableContent":"  ... | yes p | yes q = yes (no-eb-role p q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":227,"title":"... | no ¬p | _ = no λ where (No-EB-Role p _) → ⊥-elim (¬p p)","searchableContent":"  ... | no ¬p | _ = no λ where (no-eb-role p _) → ⊥-elim (¬p p)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":228,"title":"... | _ | no ¬q = no λ where (No-EB-Role _ q) → ⊥-elim (¬q q)","searchableContent":"  ... | _ | no ¬q = no λ where (no-eb-role _ q) → ⊥-elim (¬q q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":229,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-eb-role-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":230,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-eb-role-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":231,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {no-eb-role-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":232,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {no-vt-role-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":233,"title":"with dec | dec","searchableContent":"    with dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":234,"title":"... | yes p | yes q = yes (No-VT-Role p q)","searchableContent":"  ... | yes p | yes q = yes (no-vt-role p q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":235,"title":"... | no ¬p | _ = no λ where (No-VT-Role p _) → ⊥-elim (¬p p)","searchableContent":"  ... | no ¬p | _ = no λ where (no-vt-role p _) → ⊥-elim (¬p p)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":236,"title":"... | _ | no ¬q = no λ where (No-VT-Role _ q) → ⊥-elim (¬q q)","searchableContent":"  ... | _ | no ¬q = no λ where (no-vt-role _ q) → ⊥-elim (¬q q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":237,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-vt-role-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":238,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-vt-role-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":239,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {no-vt-role-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":240,"title":"Dec-ValidAction {Slot-Action sl} {s} {SLOT} .dec","searchableContent":"  dec-validaction {slot-action sl} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":241,"title":"with sl ≟ slot s","searchableContent":"    with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":242,"title":"... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-Slot x))","searchableContent":"  ... | no ¬p = no λ x → ⊥-elim (¬p (validaction→eq-slot x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":243,"title":"... | yes p rewrite p","searchableContent":"  ... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":244,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":245,"title":"... | yes x | yes y | yes z = yes (Slot x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (slot x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":246,"title":"... | no ¬p | _ | _ = no λ where (Slot p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (slot p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":247,"title":"... | _ | no ¬p | _ = no λ where (Slot _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (slot _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":248,"title":"... | _ | _ | no ¬p = no λ where (Slot _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (slot _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":249,"title":"Dec-ValidAction {Slot-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {slot-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":250,"title":"Dec-ValidAction {Slot-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {slot-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":251,"title":"Dec-ValidAction {Slot-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {slot-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":252,"title":"Dec-ValidAction {Ftch-Action} {s} {FTCH-LDG} .dec = yes Ftch","searchableContent":"  dec-validaction {ftch-action} {s} {ftch-ldg} .dec = yes ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":253,"title":"Dec-ValidAction {Ftch-Action} {s} {SLOT} .dec = no λ ()","searchableContent":"  dec-validaction {ftch-action} {s} {slot} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":254,"title":"Dec-ValidAction {Ftch-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ftch-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":255,"title":"Dec-ValidAction {Ftch-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ftch-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":256,"title":"Dec-ValidAction {Base₁-Action} {s} {SUBMIT (inj₁ ebs)} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {submit (inj₁ ebs)} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":257,"title":"Dec-ValidAction {Base₁-Action} {s} {SUBMIT (inj₂ txs)} .dec = yes (Base₁ {s} {txs})","searchableContent":"  dec-validaction {base₁-action} {s} {submit (inj₂ txs)} .dec = yes (base₁ {s} {txs})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":258,"title":"Dec-ValidAction {Base₁-Action} {s} {SLOT} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {slot} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":259,"title":"Dec-ValidAction {Base₁-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":260,"title":"Dec-ValidAction {Base₁-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":261,"title":"Dec-ValidAction {Base₂a-Action eb} {s} {SLOT} .dec","searchableContent":"  dec-validaction {base₂a-action eb} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":262,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":263,"title":"... | yes x | yes y | yes z = yes (Base₂a x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (base₂a x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":264,"title":"... | no ¬p | _ | _ = no λ where (Base₂a p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (base₂a p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":265,"title":"... | _ | no ¬p | _ = no λ where (Base₂a {s} {eb} _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (base₂a {s} {eb} _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":266,"title":"... | _ | _ | no ¬p = no λ where (Base₂a _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (base₂a _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":267,"title":"Dec-ValidAction {Base₂a-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂a-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":268,"title":"Dec-ValidAction {Base₂a-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {base₂a-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":269,"title":"Dec-ValidAction {Base₂a-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂a-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":270,"title":"Dec-ValidAction {Base₂b-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {base₂b-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":271,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":272,"title":"... | yes x | yes y | yes z = yes (Base₂b x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (base₂b x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":273,"title":"... | no ¬p | _ | _ = no λ where (Base₂b p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (base₂b p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":274,"title":"... | _ | no ¬p | _ = no λ where (Base₂b _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (base₂b _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":275,"title":"... | _ | _ | no ¬p = no λ where (Base₂b _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (base₂b _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":276,"title":"Dec-ValidAction {Base₂b-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂b-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":277,"title":"Dec-ValidAction {Base₂b-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {base₂b-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":278,"title":"Dec-ValidAction {Base₂b-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂b-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":280,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":281,"title":"Dec-ValidUpdate ","content":"Dec-ValidUpdate : ValidUpdate ⁇²","searchableContent":"  dec-validupdate : validupdate ⁇²"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":282,"title":"Dec-ValidUpdate {IB-Recv-Update _} .dec = yes IB-Recv","searchableContent":"  dec-validupdate {ib-recv-update _} .dec = yes ib-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":283,"title":"Dec-ValidUpdate {EB-Recv-Update _} .dec = yes EB-Recv","searchableContent":"  dec-validupdate {eb-recv-update _} .dec = yes eb-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":284,"title":"Dec-ValidUpdate {VT-Recv-Update _} .dec = yes VT-Recv","searchableContent":"  dec-validupdate {vt-recv-update _} .dec = yes vt-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":286,"title":"mutual","searchableContent":"mutual"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":287,"title":"data ValidTrace ","content":"data ValidTrace : List ((Action × LeiosInput) ⊎ FFDUpdate) → Type where","searchableContent":"  data validtrace : list ((action × leiosinput) ⊎ ffdupdate) → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":288,"title":"[] ","content":"[] :","searchableContent":"    [] :"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":289,"title":"─────────────","searchableContent":"      ─────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":290,"title":"ValidTrace []","searchableContent":"      validtrace []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":292,"title":"_/_∷_⊣_ ","content":"_/_∷_⊣_ : ∀ α i {αs} →","searchableContent":"    _/_∷_⊣_ : ∀ α i {αs} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":293,"title":"∀ (tr ","content":"∀ (tr : ValidTrace αs) →","searchableContent":"      ∀ (tr : validtrace αs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":294,"title":"∙ ValidAction α (proj₁ ⟦ tr ⟧∗) i","searchableContent":"      ∙ validaction α (proj₁ ⟦ tr ⟧∗) i"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":295,"title":"───────────────────","searchableContent":"        ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":296,"title":"ValidTrace (inj₁ (α , i) ∷ αs)","searchableContent":"        validtrace (inj₁ (α , i) ∷ αs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":298,"title":"_↥_ ","content":"_↥_ : ∀ {f αs} →","searchableContent":"    _↥_ : ∀ {f αs} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":299,"title":"∀ (tr ","content":"∀ (tr : ValidTrace αs) →","searchableContent":"      ∀ (tr : validtrace αs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":300,"title":"(vu ","content":"(vu : ValidUpdate f (proj₁ ⟦ tr ⟧∗)) →","searchableContent":"        (vu : validupdate f (proj₁ ⟦ tr ⟧∗)) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":301,"title":"───────────────────","searchableContent":"        ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":302,"title":"ValidTrace (inj₂ f ∷ αs)","searchableContent":"        validtrace (inj₂ f ∷ αs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":305,"title":"⟦_⟧∗ ","content":"⟦_⟧∗ : ∀ {αs : List ((Action × LeiosInput) ⊎ FFDUpdate)} → ValidTrace αs → LeiosState × LeiosOutput","searchableContent":"  ⟦_⟧∗ : ∀ {αs : list ((action × leiosinput) ⊎ ffdupdate)} → validtrace αs → leiosstate × leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":306,"title":"⟦ [] ⟧∗ = initLeiosState tt stakeDistribution tt pks , EMPTY","searchableContent":"  ⟦ [] ⟧∗ = initleiosstate tt stakedistribution tt pks , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":307,"title":"where pks = L.zip (completeFinL numberOfParties) (L.replicate numberOfParties tt)","searchableContent":"    where pks = l.zip (completefinl numberofparties) (l.replicate numberofparties tt)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":308,"title":"⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧","searchableContent":"  ⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":309,"title":"⟦ _↥_ {IB-Recv-Update ib} tr vu ⟧∗ =","searchableContent":"  ⟦ _↥_ {ib-recv-update ib} tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":310,"title":"let (s , o) = ⟦ tr ⟧∗","searchableContent":"    let (s , o) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":311,"title":"in record s { FFDState = record (FFDState s) { inIBs = ib ∷ inIBs (FFDState s)}} , o","searchableContent":"    in record s { ffdstate = record (ffdstate s) { inibs = ib ∷ inibs (ffdstate s)}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":312,"title":"⟦ _↥_ {EB-Recv-Update eb} tr vu ⟧∗ =","searchableContent":"  ⟦ _↥_ {eb-recv-update eb} tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":313,"title":"let (s , o) = ⟦ tr ⟧∗","searchableContent":"    let (s , o) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":314,"title":"in record s { FFDState = record (FFDState s) { inEBs = eb ∷ inEBs (FFDState s)}} , o","searchableContent":"    in record s { ffdstate = record (ffdstate s) { inebs = eb ∷ inebs (ffdstate s)}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":315,"title":"⟦ _↥_ {VT-Recv-Update vt} tr vu ⟧∗ =","searchableContent":"  ⟦ _↥_ {vt-recv-update vt} tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":316,"title":"let (s , o) = ⟦ tr ⟧∗","searchableContent":"    let (s , o) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":317,"title":"in record s { FFDState = record (FFDState s) { inVTs = vt ∷ inVTs (FFDState s)}} , o","searchableContent":"    in record s { ffdstate = record (ffdstate s) { invts = vt ∷ invts (ffdstate s)}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":319,"title":"Irr-ValidAction ","content":"Irr-ValidAction : Irrelevant (ValidAction α s i)","searchableContent":"irr-validaction : irrelevant (validaction α s i)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":320,"title":"Irr-ValidAction (IB-Role _ _ _) (IB-Role _ _ _)   = refl","searchableContent":"irr-validaction (ib-role _ _ _) (ib-role _ _ _)   = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":321,"title":"Irr-ValidAction (EB-Role _ _ _) (EB-Role _ _ _)   = refl","searchableContent":"irr-validaction (eb-role _ _ _) (eb-role _ _ _)   = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":322,"title":"Irr-ValidAction (VT-Role _ _ _) (VT-Role _ _ _)   = refl","searchableContent":"irr-validaction (vt-role _ _ _) (vt-role _ _ _)   = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":323,"title":"Irr-ValidAction (No-IB-Role _ _) (No-IB-Role _ _) = refl","searchableContent":"irr-validaction (no-ib-role _ _) (no-ib-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":324,"title":"Irr-ValidAction (No-EB-Role _ _) (No-EB-Role _ _) = refl","searchableContent":"irr-validaction (no-eb-role _ _) (no-eb-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":325,"title":"Irr-ValidAction (No-VT-Role _ _) (No-VT-Role _ _) = refl","searchableContent":"irr-validaction (no-vt-role _ _) (no-vt-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":326,"title":"Irr-ValidAction (Slot _ _ _) (Slot _ _ _)         = refl","searchableContent":"irr-validaction (slot _ _ _) (slot _ _ _)         = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":327,"title":"Irr-ValidAction Ftch Ftch                         = refl","searchableContent":"irr-validaction ftch ftch                         = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":328,"title":"Irr-ValidAction Base₁ Base₁                       = refl","searchableContent":"irr-validaction base₁ base₁                       = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":329,"title":"Irr-ValidAction (Base₂a _ _ _) (Base₂a _ _ _)     = refl","searchableContent":"irr-validaction (base₂a _ _ _) (base₂a _ _ _)     = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":330,"title":"Irr-ValidAction (Base₂b _ _ _) (Base₂b _ _ _)     = refl","searchableContent":"irr-validaction (base₂b _ _ _) (base₂b _ _ _)     = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":332,"title":"Irr-ValidUpdate ","content":"Irr-ValidUpdate : ∀ {f} → Irrelevant (ValidUpdate f s)","searchableContent":"irr-validupdate : ∀ {f} → irrelevant (validupdate f s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":333,"title":"Irr-ValidUpdate IB-Recv IB-Recv = refl","searchableContent":"irr-validupdate ib-recv ib-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":334,"title":"Irr-ValidUpdate EB-Recv EB-Recv = refl","searchableContent":"irr-validupdate eb-recv eb-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":335,"title":"Irr-ValidUpdate VT-Recv VT-Recv = refl","searchableContent":"irr-validupdate vt-recv vt-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":337,"title":"Irr-ValidTrace ","content":"Irr-ValidTrace : ∀ {αs} → Irrelevant (ValidTrace αs)","searchableContent":"irr-validtrace : ∀ {αs} → irrelevant (validtrace αs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":338,"title":"Irr-ValidTrace [] [] = refl","searchableContent":"irr-validtrace [] [] = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":339,"title":"Irr-ValidTrace (α / i ∷ vαs ⊣ vα) (.α / .i ∷ vαs′ ⊣ vα′)","searchableContent":"irr-validtrace (α / i ∷ vαs ⊣ vα) (.α / .i ∷ vαs′ ⊣ vα′)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":340,"title":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidAction vα vα′","searchableContent":"  rewrite irr-validtrace vαs vαs′ | irr-validaction vα vα′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":341,"title":"= refl","searchableContent":"  = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":342,"title":"Irr-ValidTrace (vαs ↥ u) (vαs′ ↥ u′)","searchableContent":"irr-validtrace (vαs ↥ u) (vαs′ ↥ u′)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":343,"title":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidUpdate u u′","searchableContent":"  rewrite irr-validtrace vαs vαs′ | irr-validupdate u u′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":344,"title":"= refl","searchableContent":"  = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":346,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":347,"title":"Dec-ValidTrace ","content":"Dec-ValidTrace : ValidTrace ⁇¹","searchableContent":"  dec-validtrace : validtrace ⁇¹"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":348,"title":"Dec-ValidTrace {tr} .dec with tr","searchableContent":"  dec-validtrace {tr} .dec with tr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":349,"title":"... | [] = yes []","searchableContent":"  ... | [] = yes []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":350,"title":"... | inj₁ (α , i) ∷ αs","searchableContent":"  ... | inj₁ (α , i) ∷ αs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":351,"title":"with ¿ ValidTrace αs ¿","searchableContent":"    with ¿ validtrace αs ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":352,"title":"... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs","searchableContent":"  ... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":353,"title":"... | yes vαs","searchableContent":"  ... | yes vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":354,"title":"with ¿ ValidAction α (proj₁ ⟦ vαs ⟧∗) i ¿","searchableContent":"    with ¿ validaction α (proj₁ ⟦ vαs ⟧∗) i ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":355,"title":"... | no ¬vα = no λ where","searchableContent":"  ... | no ¬vα = no λ where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":356,"title":"(_ / _ ∷ tr ⊣ vα) → ¬vα","searchableContent":"    (_ / _ ∷ tr ⊣ vα) → ¬vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":357,"title":"$ subst (λ x → ValidAction α x i) (cong (proj₁ ∘ ⟦_⟧∗) $ Irr-ValidTrace tr vαs) vα","searchableContent":"                  $ subst (λ x → validaction α x i) (cong (proj₁ ∘ ⟦_⟧∗) $ irr-validtrace tr vαs) vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":358,"title":"... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα","searchableContent":"  ... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":359,"title":"Dec-ValidTrace {tr} .dec | inj₂ u ∷ αs","searchableContent":"  dec-validtrace {tr} .dec | inj₂ u ∷ αs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":360,"title":"with ¿ ValidTrace αs ¿","searchableContent":"    with ¿ validtrace αs ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":361,"title":"... | no ¬vαs = no λ where (vαs ↥ _) → ¬vαs vαs","searchableContent":"  ... | no ¬vαs = no λ where (vαs ↥ _) → ¬vαs vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":362,"title":"... | yes vαs","searchableContent":"  ... | yes vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":363,"title":"with ¿ ValidUpdate u (proj₁ ⟦ vαs ⟧∗) ¿","searchableContent":"    with ¿ validupdate u (proj₁ ⟦ vαs ⟧∗) ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":364,"title":"... | yes vu = yes (vαs ↥ vu)","searchableContent":"  ... | yes vu = yes (vαs ↥ vu)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":365,"title":"... | no ¬vu = no λ where","searchableContent":"  ... | no ¬vu = no λ where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":366,"title":"(tr ↥ vu) → ¬vu $ subst (λ x → ValidUpdate u x) (cong (proj₁ ∘ ⟦_⟧∗) $ Irr-ValidTrace tr vαs) vu","searchableContent":"    (tr ↥ vu) → ¬vu $ subst (λ x → validupdate u x) (cong (proj₁ ∘ ⟦_⟧∗) $ irr-validtrace tr vαs) vu"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":368,"title":"data _⇑_ ","content":"data _⇑_ : LeiosState → LeiosState → Type where","searchableContent":"data _⇑_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":370,"title":"UpdateIB ","content":"UpdateIB : ∀ {s ib} → let open LeiosState s renaming (FFDState to ffds) in","searchableContent":"  updateib : ∀ {s ib} → let open leiosstate s renaming (ffdstate to ffds) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":371,"title":"s ⇑ record s { FFDState = record ffds { inIBs = ib ∷ inIBs ffds } }","searchableContent":"    s ⇑ record s { ffdstate = record ffds { inibs = ib ∷ inibs ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":373,"title":"UpdateEB ","content":"UpdateEB : ∀ {s eb} → let open LeiosState s renaming (FFDState to ffds) in","searchableContent":"  updateeb : ∀ {s eb} → let open leiosstate s renaming (ffdstate to ffds) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":374,"title":"s ⇑ record s { FFDState = record ffds { inEBs = eb ∷ inEBs ffds } }","searchableContent":"    s ⇑ record s { ffdstate = record ffds { inebs = eb ∷ inebs ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":376,"title":"UpdateVT ","content":"UpdateVT : ∀ {s vt} → let open LeiosState s renaming (FFDState to ffds) in","searchableContent":"  updatevt : ∀ {s vt} → let open leiosstate s renaming (ffdstate to ffds) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":377,"title":"s ⇑ record s { FFDState = record ffds { inVTs = vt ∷ inVTs ffds } }","searchableContent":"    s ⇑ record s { ffdstate = record ffds { invts = vt ∷ invts ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":379,"title":"data LocalStep ","content":"data LocalStep : LeiosState → LeiosState → Type where","searchableContent":"data localstep : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":381,"title":"StateStep ","content":"StateStep : ∀ {s i o s′} →","searchableContent":"  statestep : ∀ {s i o s′} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":382,"title":"∙ just s -⟦ i / o ⟧⇀ s′","searchableContent":"    ∙ just s -⟦ i / o ⟧⇀ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":383,"title":"───────────────────","searchableContent":"      ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":384,"title":"LocalStep s s′","searchableContent":"      localstep s s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":386,"title":"UpdateState ","content":"UpdateState : ∀ {s s′} →","searchableContent":"  updatestate : ∀ {s s′} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":387,"title":"∙ s ⇑ s′","searchableContent":"    ∙ s ⇑ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":388,"title":"───────────────────","searchableContent":"      ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":389,"title":"LocalStep s s′","searchableContent":"      localstep s s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":391,"title":"-- TODO","content":"-- TODO: add base layer update","searchableContent":"  -- todo: add base layer update"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":393,"title":"getLabel ","content":"getLabel : just s -⟦ i / o ⟧⇀ s′ → Action","searchableContent":"getlabel : just s -⟦ i / o ⟧⇀ s′ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":394,"title":"getLabel (Slot {s} _ _ _)            = Slot-Action (slot s)","searchableContent":"getlabel (slot {s} _ _ _)            = slot-action (slot s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":395,"title":"getLabel Ftch                        = Ftch-Action","searchableContent":"getlabel ftch                        = ftch-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":396,"title":"getLabel Base₁                       = Base₁-Action","searchableContent":"getlabel base₁                       = base₁-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":397,"title":"getLabel (Base₂a {s} {eb} _ _ _)     = Base₂a-Action eb","searchableContent":"getlabel (base₂a {s} {eb} _ _ _)     = base₂a-action eb"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":398,"title":"getLabel (Base₂b _ _ _)              = Base₂b-Action","searchableContent":"getlabel (base₂b _ _ _)              = base₂b-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":399,"title":"getLabel (Roles (IB-Role {s} _ _ _)) = IB-Role-Action (slot s)","searchableContent":"getlabel (roles (ib-role {s} _ _ _)) = ib-role-action (slot s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":400,"title":"getLabel (Roles (EB-Role {s} _ _ _)) = EB-Role-Action (slot s) (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))","searchableContent":"getlabel (roles (eb-role {s} _ _ _)) = eb-role-action (slot s) (map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":401,"title":"getLabel (Roles (VT-Role {s} _ _ _)) = VT-Role-Action (slot s)","searchableContent":"getlabel (roles (vt-role {s} _ _ _)) = vt-role-action (slot s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":402,"title":"getLabel (Roles (No-IB-Role _ _))    = No-IB-Role-Action","searchableContent":"getlabel (roles (no-ib-role _ _))    = no-ib-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":403,"title":"getLabel (Roles (No-EB-Role _ _))    = No-EB-Role-Action","searchableContent":"getlabel (roles (no-eb-role _ _))    = no-eb-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":404,"title":"getLabel (Roles (No-VT-Role _ _))    = No-VT-Role-Action","searchableContent":"getlabel (roles (no-vt-role _ _))    = no-vt-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":406,"title":"ValidAction-sound ","content":"ValidAction-sound : (vα : ValidAction α s i) → let (s′ , o) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′","searchableContent":"validaction-sound : (vα : validaction α s i) → let (s′ , o) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":407,"title":"ValidAction-sound (Slot x x₁ x₂)    = Slot {rbs = []} (recompute dec x) (recompute dec x₁) (recompute dec x₂)","searchableContent":"validaction-sound (slot x x₁ x₂)    = slot {rbs = []} (recompute dec x) (recompute dec x₁) (recompute dec x₂)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":408,"title":"ValidAction-sound Ftch              = Ftch","searchableContent":"validaction-sound ftch              = ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":409,"title":"ValidAction-sound Base₁             = Base₁","searchableContent":"validaction-sound base₁             = base₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":410,"title":"ValidAction-sound (Base₂a x x₁ x₂)  = Base₂a (recompute dec x) (recompute dec x₁) (recompute dec x₂)","searchableContent":"validaction-sound (base₂a x x₁ x₂)  = base₂a (recompute dec x) (recompute dec x₁) (recompute dec x₂)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":411,"title":"ValidAction-sound (Base₂b x x₁ x₂)  = Base₂b (recompute dec x) (recompute dec x₁) (recompute dec x₂)","searchableContent":"validaction-sound (base₂b x x₁ x₂)  = base₂b (recompute dec x) (recompute dec x₁) (recompute dec x₂)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":412,"title":"ValidAction-sound (IB-Role x x₁ x₂) = Roles (IB-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))","searchableContent":"validaction-sound (ib-role x x₁ x₂) = roles (ib-role (recompute dec x) (recompute dec x₁) (recompute dec x₂))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":413,"title":"ValidAction-sound (EB-Role x x₁ x₂) = Roles (EB-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))","searchableContent":"validaction-sound (eb-role x x₁ x₂) = roles (eb-role (recompute dec x) (recompute dec x₁) (recompute dec x₂))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":414,"title":"ValidAction-sound (VT-Role x x₁ x₂) = Roles (VT-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))","searchableContent":"validaction-sound (vt-role x x₁ x₂) = roles (vt-role (recompute dec x) (recompute dec x₁) (recompute dec x₂))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":415,"title":"ValidAction-sound (No-IB-Role x x₁) = Roles (No-IB-Role x x₁)","searchableContent":"validaction-sound (no-ib-role x x₁) = roles (no-ib-role x x₁)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":416,"title":"ValidAction-sound (No-EB-Role x x₁) = Roles (No-EB-Role x x₁)","searchableContent":"validaction-sound (no-eb-role x x₁) = roles (no-eb-role x x₁)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":417,"title":"ValidAction-sound (No-VT-Role x x₁) = Roles (No-VT-Role x x₁)","searchableContent":"validaction-sound (no-vt-role x x₁) = roles (no-vt-role x x₁)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":419,"title":"ValidAction-complete ","content":"ValidAction-complete : (st : just s -⟦ i / o ⟧⇀ s′) → ValidAction (getLabel st) s i","searchableContent":"validaction-complete : (st : just s -⟦ i / o ⟧⇀ s′) → validaction (getlabel st) s i"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":420,"title":"ValidAction-complete {s} {s′} (Roles (IB-Role {s} {π} {ffds'} x x₁ _)) =","searchableContent":"validaction-complete {s} {s′} (roles (ib-role {s} {π} {ffds'} x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":421,"title":"let b  = record { txs = ToPropose s }","searchableContent":"  let b  = record { txs = topropose s }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":422,"title":"h  = mkIBHeader (slot s) id tt sk-IB (ToPropose s)","searchableContent":"      h  = mkibheader (slot s) id tt sk-ib (topropose s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":423,"title":"pr = proj₂ (FFD.Send-total {FFDState s} {ibHeader h} {just (ibBody b)})","searchableContent":"      pr = proj₂ (ffd.send-total {ffdstate s} {ibheader h} {just (ibbody b)})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":424,"title":"in IB-Role {s} x x₁ pr","searchableContent":"  in ib-role {s} x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":425,"title":"ValidAction-complete {s} (Roles (EB-Role x x₁ _)) =","searchableContent":"validaction-complete {s} (roles (eb-role x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":426,"title":"let LI = map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s)","searchableContent":"  let li = map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":427,"title":"h  = mkEB (slot s) id tt sk-EB LI []","searchableContent":"      h  = mkeb (slot s) id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":428,"title":"pr = proj₂ (FFD.Send-total {FFDState s} {ebHeader h} {nothing})","searchableContent":"      pr = proj₂ (ffd.send-total {ffdstate s} {ebheader h} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":429,"title":"in EB-Role {s} x x₁ pr","searchableContent":"  in eb-role {s} x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":430,"title":"ValidAction-complete {s} (Roles (VT-Role x x₁ _))  =","searchableContent":"validaction-complete {s} (roles (vt-role x x₁ _))  ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":431,"title":"let EBs'  = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L (slot s) 1) (EBs s)","searchableContent":"  let ebs'  = filter (allibrefsknown s) $ filter (_∈ᴮ slice l (slot s) 1) (ebs s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":432,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"      votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":433,"title":"pr    = proj₂ (FFD.Send-total {FFDState s} {vtHeader votes} {nothing})","searchableContent":"      pr    = proj₂ (ffd.send-total {ffdstate s} {vtheader votes} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":434,"title":"in VT-Role {s} x x₁ pr","searchableContent":"  in vt-role {s} x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":435,"title":"ValidAction-complete (Roles (No-IB-Role x x₁)) = No-IB-Role x x₁","searchableContent":"validaction-complete (roles (no-ib-role x x₁)) = no-ib-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":436,"title":"ValidAction-complete (Roles (No-EB-Role x x₁)) = No-EB-Role x x₁","searchableContent":"validaction-complete (roles (no-eb-role x x₁)) = no-eb-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":437,"title":"ValidAction-complete (Roles (No-VT-Role x x₁)) = No-VT-Role x x₁","searchableContent":"validaction-complete (roles (no-vt-role x x₁)) = no-vt-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":438,"title":"ValidAction-complete Ftch                      = Ftch","searchableContent":"validaction-complete ftch                      = ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":439,"title":"ValidAction-complete Base₁                     = Base₁","searchableContent":"validaction-complete base₁                     = base₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":440,"title":"ValidAction-complete (Base₂a x x₁ x₂)          = Base₂a x x₁ x₂","searchableContent":"validaction-complete (base₂a x x₁ x₂)          = base₂a x x₁ x₂"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":441,"title":"ValidAction-complete (Base₂b x x₁ x₂)          = Base₂b x x₁ x₂","searchableContent":"validaction-complete (base₂b x x₁ x₂)          = base₂b x x₁ x₂"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":442,"title":"ValidAction-complete {s} (Slot x x₁ _)         = Slot x x₁ (proj₂ (proj₂ (FFD.Fetch-total {FFDState s})))","searchableContent":"validaction-complete {s} (slot x x₁ _)         = slot x x₁ (proj₂ (proj₂ (ffd.fetch-total {ffdstate s})))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":443,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":2,"title":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":3,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":4,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":5,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":6,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":7,"title":"Leios.Short","content":"Leios.Short","searchableContent":" leios.short"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":8,"title":"","content":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":25,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":26,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":27,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":28,"title":"

Short-Pipeline Leios

","searchableContent":"

short-pipeline leios

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":29,"title":"","content":"-->","searchableContent":"-->"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":43,"title":"

This document is a specification of Short-Pipeline Leios, usually","content":"

This document is a specification of Short-Pipeline Leios, usually","searchableContent":"

this document is a specification of short-pipeline leios, usually"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":44,"title":"abbreviated as Short Leios. On a high level, the pipeline looks like","content":"abbreviated as Short Leios. On a high level, the pipeline looks like","searchableContent":"abbreviated as short leios. on a high level, the pipeline looks like"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":45,"title":"this","content":"this:

","searchableContent":"this:

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":46,"title":"
    ","searchableContent":"
      "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":47,"title":"
    1. If elected, propose IB
      2. ","content":"
    2. If elected, propose IB
      3. ","searchableContent":"
    3. if elected, propose ib
      4. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":48,"title":"
    4. Wait
      5. ","content":"
    5. Wait
      6. ","searchableContent":"
    6. wait
      7. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":49,"title":"
    7. Wait
      8. ","content":"
    8. Wait
      9. ","searchableContent":"
    9. wait
      10. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":50,"title":"
    10. If elected, propose EB
      11. ","content":"
    11. If elected, propose EB
      12. ","searchableContent":"
    12. if elected, propose eb
      13. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":51,"title":"
    13. If elected, vote If elected, propose RB
      14. ","content":"
    14. If elected, vote If elected, propose RB
      15. ","searchableContent":"
    15. if elected, vote if elected, propose rb
      16. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":52,"title":"
    ","content":"
","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":53,"title":"

Upkeep

","searchableContent":"

upkeep

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":54,"title":"

A node that never produces a block even though it could is not","content":"

A node that never produces a block even though it could is not","searchableContent":"

a node that never produces a block even though it could is not"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":55,"title":"supposed to be an honest node, and we prevent that by tracking whether a","content":"supposed to be an honest node, and we prevent that by tracking whether a","searchableContent":"supposed to be an honest node, and we prevent that by tracking whether a"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":56,"title":"node has checked if it can make a block in a particular slot.","content":"node has checked if it can make a block in a particular slot.","searchableContent":"node has checked if it can make a block in a particular slot."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":57,"title":"LeiosState contains a set of SlotUpkeep and we","content":"LeiosState contains a set of SlotUpkeep and we","searchableContent":"leiosstate contains a set of slotupkeep and we"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":58,"title":"ensure that this set contains all elements before we can advance to the","content":"ensure that this set contains all elements before we can advance to the","searchableContent":"ensure that this set contains all elements before we can advance to the"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":59,"title":"next slot, resetting this field to the empty set.

","content":"next slot, resetting this field to the empty set.

","searchableContent":"next slot, resetting this field to the empty set.

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":60,"title":"
data SlotUpkeep ","content":"
data SlotUpkeep : Type where","searchableContent":"
data slotupkeep : type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":61,"title":"Base IB-Role EB-Role VT-Role ","content":"Base IB-Role EB-Role VT-Role : SlotUpkeep","searchableContent":"  base ib-role eb-role vt-role : slotupkeep"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":62,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":63,"title":"","content":"-->","searchableContent":"-->"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":85,"title":"

Block/Vote production rules

","searchableContent":"

block/vote production rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":86,"title":"

We now define the rules for block production given by the relation","content":"

We now define the rules for block production given by the relation","searchableContent":"

we now define the rules for block production given by the relation"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":87,"title":"_↝_. These are split in two","content":"_↝_. These are split in two:

","searchableContent":"_↝_. these are split in two:

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":88,"title":"
    ","searchableContent":"
      "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":89,"title":"
    1. Positive rules, when we do need to create a block.
      2. ","content":"
    2. Positive rules, when we do need to create a block.
      3. ","searchableContent":"
    3. positive rules, when we do need to create a block.
      4. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":90,"title":"
    4. Negative rules, when we cannot create a block.
      5. ","content":"
    5. Negative rules, when we cannot create a block.
      6. ","searchableContent":"
    6. negative rules, when we cannot create a block.
      7. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":91,"title":"
    ","content":"
","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":92,"title":"

The purpose of the negative rules is to properly adjust the upkeep if","content":"

The purpose of the negative rules is to properly adjust the upkeep if","searchableContent":"

the purpose of the negative rules is to properly adjust the upkeep if"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":93,"title":"we cannot make a block.

","content":"we cannot make a block.

","searchableContent":"we cannot make a block.

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":94,"title":"

Note that _↝_, starting with an empty upkeep can always","content":"

Note that _↝_, starting with an empty upkeep can always","searchableContent":"

note that _↝_, starting with an empty upkeep can always"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":95,"title":"make exactly three steps corresponding to the three types of Leios","content":"make exactly three steps corresponding to the three types of Leios","searchableContent":"make exactly three steps corresponding to the three types of leios"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":96,"title":"specific blocks.

","content":"specific blocks.

","searchableContent":"specific blocks.

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":97,"title":"
data _↝_ ","content":"
data _↝_ : LeiosState → LeiosState → Type where","searchableContent":"
data _↝_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":98,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":99,"title":"

Positive rules

","searchableContent":"

positive rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":100,"title":"
  IB-Role ","content":"
  IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"
  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":101,"title":"b = ibBody (record { txs = ToPropose })","searchableContent":"                b = ibbody (record { txs = topropose })"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":102,"title":"h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"                h = ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":103,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":104,"title":"∙ needsUpkeep IB-Role","searchableContent":"          ∙ needsupkeep ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":105,"title":"∙ canProduceIB slot sk-IB (stake s) π","searchableContent":"          ∙ canproduceib slot sk-ib (stake s) π"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":106,"title":"∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send h (just b) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":107,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":108,"title":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":109,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":110,"title":"
  EB-Role ","content":"
  EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"
  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":111,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":112,"title":"h = mkEB slot id π sk-EB LI []","searchableContent":"                h = mkeb slot id π sk-eb li []"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":113,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":114,"title":"∙ needsUpkeep EB-Role","searchableContent":"          ∙ needsupkeep eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":115,"title":"∙ canProduceEB slot sk-EB (stake s) π","searchableContent":"          ∙ canproduceeb slot sk-eb (stake s) π"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":116,"title":"∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (ebheader h) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":117,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":118,"title":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":119,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":120,"title":"
  VT-Role ","content":"
  VT-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"
  vt-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":121,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":122,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":123,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":124,"title":"∙ needsUpkeep VT-Role","searchableContent":"          ∙ needsupkeep vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":125,"title":"∙ canProduceV slot sk-VT (stake s)","searchableContent":"          ∙ canproducev slot sk-vt (stake s)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":126,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":127,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":128,"title":"s ↝ addUpkeep record s { FFDState = ffds' } VT-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":129,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":130,"title":"

Negative rules

","searchableContent":"

negative rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":131,"title":"
  No-IB-Role ","content":"
  No-IB-Role : let open LeiosState s in","searchableContent":"
  no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":132,"title":"∙ needsUpkeep IB-Role","searchableContent":"             ∙ needsupkeep ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":133,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)","searchableContent":"             ∙ (∀ π → ¬ canproduceib slot sk-ib (stake s) π)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":134,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":135,"title":"s ↝ addUpkeep s IB-Role","searchableContent":"             s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":136,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":137,"title":"
  No-EB-Role ","content":"
  No-EB-Role : let open LeiosState s in","searchableContent":"
  no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":138,"title":"∙ needsUpkeep EB-Role","searchableContent":"             ∙ needsupkeep eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":139,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)","searchableContent":"             ∙ (∀ π → ¬ canproduceeb slot sk-eb (stake s) π)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":140,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":141,"title":"s ↝ addUpkeep s EB-Role","searchableContent":"             s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":142,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":143,"title":"
  No-VT-Role ","content":"
  No-VT-Role : let open LeiosState s in","searchableContent":"
  no-vt-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":144,"title":"∙ needsUpkeep VT-Role","searchableContent":"             ∙ needsupkeep vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":145,"title":"∙ ¬ canProduceV slot sk-VT (stake s)","searchableContent":"             ∙ ¬ canproducev slot sk-vt (stake s)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":146,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":147,"title":"s ↝ addUpkeep s VT-Role","searchableContent":"             s ↝ addupkeep s vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":148,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":149,"title":"

Uniform short-pipeline

","searchableContent":"

uniform short-pipeline

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":150,"title":"
stage ","content":"
stage : ℕ → ⦃ _ : NonZero L ⦄ → ℕ","searchableContent":"
stage : ℕ → ⦃ _ : nonzero l ⦄ → ℕ"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":151,"title":"stage s = s / L","searchableContent":"stage s = s / l"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":153,"title":"beginningOfStage ","content":"beginningOfStage : ℕ → Type","searchableContent":"beginningofstage : ℕ → type"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":154,"title":"beginningOfStage s = stage s * L ≡ s","searchableContent":"beginningofstage s = stage s * l ≡ s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":156,"title":"allDone ","content":"allDone : LeiosState → Type","searchableContent":"alldone : leiosstate → type"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":157,"title":"allDone s =","searchableContent":"alldone s ="},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":158,"title":"let open LeiosState s","searchableContent":"  let open leiosstate s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":159,"title":"in   (beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ EB-Role ∷ VT-Role ∷ Base ∷ []))","searchableContent":"  in   (beginningofstage slot × upkeep ≡ᵉ fromlist (ib-role ∷ eb-role ∷ vt-role ∷ base ∷ []))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":160,"title":"⊎ (¬ beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ VT-Role ∷ Base ∷ []))","searchableContent":"   ⊎ (¬ beginningofstage slot × upkeep ≡ᵉ fromlist (ib-role ∷ vt-role ∷ base ∷ []))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":162,"title":"data _-⟦_/_⟧⇀_ ","content":"data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":163,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":164,"title":"

Initialization

","searchableContent":"

initialization

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":165,"title":"
  Init ","content":"
  Init :","searchableContent":"
  init :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":166,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"       ∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":167,"title":"∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'","searchableContent":"       ∙ initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":168,"title":"────────────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":169,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","searchableContent":"       nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":170,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":171,"title":"

Network and Ledger

","searchableContent":"

network and ledger

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":172,"title":"
  Slot ","content":"
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in","searchableContent":"
  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":173,"title":"∙ allDone s","searchableContent":"       ∙ alldone s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":174,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"       ∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":175,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"       ∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":176,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":177,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"       just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":178,"title":"{ FFDState  = ffds'","searchableContent":"           { ffdstate  = ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":179,"title":"; BaseState = bs'","searchableContent":"           ; basestate = bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":180,"title":"; Ledger    = constructLedger rbs","searchableContent":"           ; ledger    = constructledger rbs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":181,"title":"; slot      = suc slot","searchableContent":"           ; slot      = suc slot"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":182,"title":"; Upkeep    = ∅","searchableContent":"           ; upkeep    = ∅"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":183,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"           } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":184,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":185,"title":"
  Ftch ","content":"
  Ftch :","searchableContent":"
  ftch :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":186,"title":"────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":187,"title":"just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s","searchableContent":"       just s -⟦ ftch-ldg / ftch-ldg (leiosstate.ledger s) ⟧⇀ s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":188,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":189,"title":"

Base chain

","searchableContent":"

base chain

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":190,"title":"Note","content":"Note: Submitted data to the base chain is only taken into account if the","searchableContent":"note: submitted data to the base chain is only taken into account if the"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":191,"title":"party submitting is the block producer on the base chain for the given","content":"party submitting is the block producer on the base chain for the given","searchableContent":"party submitting is the block producer on the base chain for the given"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":192,"title":"slot","content":"slot","searchableContent":"slot"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":193,"title":"
  Base₁   ","content":"
  Base₁   :","searchableContent":"
  base₁   :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":194,"title":"───────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":195,"title":"just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"          just s -⟦ submit (inj₂ txs) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":196,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":197,"title":"
  Base₂a  ","content":"
  Base₂a  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"
  base₂a  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":198,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":199,"title":"∙ eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ eb ∈ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":200,"title":"∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":201,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":202,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":204,"title":"Base₂b  ","content":"Base₂b  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂b  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":205,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":206,"title":"∙ [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ [] ≡ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":207,"title":"∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":208,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":209,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":210,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":211,"title":"

Protocol rules

","searchableContent":"

protocol rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":212,"title":"
  Roles ","content":"
  Roles :","searchableContent":"
  roles :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":213,"title":"∙ s ↝ s'","searchableContent":"        ∙ s ↝ s'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":214,"title":"─────────────────────────────","searchableContent":"        ─────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":215,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"        just s -⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":216,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":217,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":218,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":2,"title":"Leios.Simplified.Deterministic
--{-# OPTIONS --safe #-}","searchableContent":"leios.simplified.deterministic
--{-# options --safe #-}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":3,"title":"{-# OPTIONS --allow-unsolved-metas #-}","searchableContent":"{-# options --allow-unsolved-metas #-}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":5,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":6,"title":"-- Deterministic variant of simple Leios","searchableContent":"-- deterministic variant of simple leios"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":7,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":10,"title":"open import Prelude.Init using (∃₂-syntax)","searchableContent":"open import prelude.init using (∃₂-syntax)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":11,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":13,"title":"import Data.List as L","searchableContent":"import data.list as l"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":14,"title":"open import Data.List.Relation.Unary.Any using (here)","searchableContent":"open import data.list.relation.unary.any using (here)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":16,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":18,"title":"module Leios.Simplified.Deterministic (⋯ ","content":"module Leios.Simplified.Deterministic (⋯ : SpecStructure 2) (let open SpecStructure ⋯) (Λ μ : ℕ) where","searchableContent":"module leios.simplified.deterministic (⋯ : specstructure 2) (let open specstructure ⋯) (λ μ : ℕ) where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":20,"title":"import Leios.Simplified","searchableContent":"import leios.simplified"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":21,"title":"open import Leios.Simplified ⋯ Λ μ hiding (_-⟦_/_⟧⇀_)","searchableContent":"open import leios.simplified ⋯ λ μ hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":22,"title":"module ND = Leios.Simplified ⋯ Λ μ","searchableContent":"module nd = leios.simplified ⋯ λ μ"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":24,"title":"open import Class.Computational","searchableContent":"open import class.computational"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":25,"title":"open import Class.Computational22","searchableContent":"open import class.computational22"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":26,"title":"open import StateMachine","searchableContent":"open import statemachine"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":28,"title":"open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)","searchableContent":"open baseabstract b' using (cert; v-chkcerts; vty; initslot)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":29,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":31,"title":"open FFD hiding (_-⟦_/_⟧⇀_)","searchableContent":"open ffd hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":33,"title":"private variable s s' s0 s1 s2 s3 s4 s5 ","content":"private variable s s' s0 s1 s2 s3 s4 s5 : LeiosState","searchableContent":"private variable s s' s0 s1 s2 s3 s4 s5 : leiosstate"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":34,"title":"i      ","content":"i      : LeiosInput","searchableContent":"                 i      : leiosinput"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":35,"title":"o      ","content":"o      : LeiosOutput","searchableContent":"                 o      : leiosoutput"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":36,"title":"ffds'  ","content":"ffds'  : FFD.State","searchableContent":"                 ffds'  : ffd.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":37,"title":"π      ","content":"π      : VrfPf","searchableContent":"                 π      : vrfpf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":38,"title":"bs'    ","content":"bs'    : B.State","searchableContent":"                 bs'    : b.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":39,"title":"ks ks' ","content":"ks ks' : K.State","searchableContent":"                 ks ks' : k.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":40,"title":"msgs   ","content":"msgs   : List (FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract)","searchableContent":"                 msgs   : list (ffdabstract.header ffdabstract ⊎ ffdabstract.body ffdabstract)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":41,"title":"eb     ","content":"eb     : EndorserBlock","searchableContent":"                 eb     : endorserblock"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":42,"title":"rbs    ","content":"rbs    : List RankingBlock","searchableContent":"                 rbs    : list rankingblock"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":43,"title":"txs    ","content":"txs    : List Tx","searchableContent":"                 txs    : list tx"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":44,"title":"V      ","content":"V      : VTy","searchableContent":"                 v      : vty"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":45,"title":"SD     ","content":"SD     : StakeDistr","searchableContent":"                 sd     : stakedistr"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":46,"title":"pks    ","content":"pks    : List PubKey","searchableContent":"                 pks    : list pubkey"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":48,"title":"lemma ","content":"lemma : ∀ {u} → u ∈ LeiosState.Upkeep (addUpkeep s u)","searchableContent":"lemma : ∀ {u} → u ∈ leiosstate.upkeep (addupkeep s u)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":49,"title":"lemma = to ∈-∪ (inj₂ (to ∈-singleton refl))","searchableContent":"lemma = to ∈-∪ (inj₂ (to ∈-singleton refl))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":50,"title":"where open Equivalence","searchableContent":"  where open equivalence"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":52,"title":"addUpkeep⇒¬needsUpkeep ","content":"addUpkeep⇒¬needsUpkeep : ∀ {u} → ¬ LeiosState.needsUpkeep (addUpkeep s u) u","searchableContent":"addupkeep⇒¬needsupkeep : ∀ {u} → ¬ leiosstate.needsupkeep (addupkeep s u) u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":53,"title":"addUpkeep⇒¬needsUpkeep {s = s} = λ x → x (lemma {s = s})","searchableContent":"addupkeep⇒¬needsupkeep {s = s} = λ x → x (lemma {s = s})"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":55,"title":"data _⊢_ ","content":"data _⊢_ : LeiosInput → LeiosState → Type where","searchableContent":"data _⊢_ : leiosinput → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":56,"title":"Init ","content":"Init :","searchableContent":"  init :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":57,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"       ∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":58,"title":"∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'","searchableContent":"       ∙ initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":59,"title":"────────────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":60,"title":"INIT V ⊢ initLeiosState V SD bs' pks","searchableContent":"       init v ⊢ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":62,"title":"data _-⟦Base⟧⇀_ ","content":"data _-⟦Base⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦base⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":64,"title":"Base₂a  ","content":"Base₂a  : ∀ {ebs} → let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂a  : ∀ {ebs} → let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":65,"title":"∙ eb ∷ ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ eb ∷ ebs ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":66,"title":"∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":67,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":68,"title":"s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          s -⟦base⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":70,"title":"Base₂b  ","content":"Base₂b  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂b  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":71,"title":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ [] ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":72,"title":"∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":73,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":74,"title":"s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          s -⟦base⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":76,"title":"Base⇒ND ","content":"Base⇒ND : LeiosState.needsUpkeep s Base → s -⟦Base⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"base⇒nd : leiosstate.needsupkeep s base → s -⟦base⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":77,"title":"Base⇒ND u (Base₂a x₁ x₂) = Base₂a u (subst (_ ∈_) x₁ (Equivalence.to ∈-fromList (here refl))) x₂","searchableContent":"base⇒nd u (base₂a x₁ x₂) = base₂a u (subst (_ ∈_) x₁ (equivalence.to ∈-fromlist (here refl))) x₂"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":78,"title":"Base⇒ND u (Base₂b x₁ x₂) = Base₂b u x₁ x₂","searchableContent":"base⇒nd u (base₂b x₁ x₂) = base₂b u x₁ x₂"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":80,"title":"Base-Upkeep ","content":"Base-Upkeep : ∀ {u} → u ≢ Base → LeiosState.needsUpkeep s u → s -⟦Base⟧⇀ s'","searchableContent":"base-upkeep : ∀ {u} → u ≢ base → leiosstate.needsupkeep s u → s -⟦base⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":81,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":82,"title":"Base-Upkeep u≢Base h (Base₂a _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"base-upkeep u≢base h (base₂a _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":83,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":84,"title":"(inj₂ y) → u≢Base (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢base (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":85,"title":"Base-Upkeep u≢Base h (Base₂b _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"base-upkeep u≢base h (base₂b _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":86,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":87,"title":"(inj₂ y) → u≢Base (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢base (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":89,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":90,"title":"Base-total ","content":"Base-total : ∃[ s' ] s -⟦Base⟧⇀ s'","searchableContent":"  base-total : ∃[ s' ] s -⟦base⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":91,"title":"Base-total {s = s} with","searchableContent":"  base-total {s = s} with"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":92,"title":"(let open LeiosState s in filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs)","searchableContent":"    (let open leiosstate s in filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":93,"title":"in eq","searchableContent":"    in eq"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":94,"title":"... | []    = -, Base₂b (sym eq) (proj₂ B.SUBMIT-total)","searchableContent":"  ... | []    = -, base₂b (sym eq) (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":95,"title":"... | x ∷ l = -, Base₂a (sym eq) (proj₂ B.SUBMIT-total)","searchableContent":"  ... | x ∷ l = -, base₂a (sym eq) (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":97,"title":"Base-total' ","content":"Base-total' : ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄","searchableContent":"  base-total' : ⦃ computational-b : computational22 b._-⟦_/_⟧⇀_ string ⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":98,"title":"→ ∃[ bs ] s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs } Base","searchableContent":"              → ∃[ bs ] s -⟦base⟧⇀ addupkeep record s { basestate = bs } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":99,"title":"Base-total' {s = s} = let open LeiosState s in","searchableContent":"  base-total' {s = s} = let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":100,"title":"case ∃[ ebs ] ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs ∋ -, refl","searchableContent":"    case ∃[ ebs ] ebs ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs ∋ -, refl"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":101,"title":"of λ where","searchableContent":"      of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":102,"title":"(eb ∷ _ , eq) → -, Base₂a eq (proj₂ B.SUBMIT-total)","searchableContent":"        (eb ∷ _ , eq) → -, base₂a eq (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":103,"title":"([]     , eq) → -, Base₂b eq (proj₂ B.SUBMIT-total)","searchableContent":"        ([]     , eq) → -, base₂b eq (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":105,"title":"data _-⟦IB-Role⟧⇀_ ","content":"data _-⟦IB-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦ib-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":107,"title":"IB-Role ","content":"IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":108,"title":"b = ibBody (record { txs = ToPropose })","searchableContent":"                b = ibbody (record { txs = topropose })"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":109,"title":"h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"                h = ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":110,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":111,"title":"∙ canProduceIB slot sk-IB (stake s) π","searchableContent":"          ∙ canproduceib slot sk-ib (stake s) π"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":112,"title":"∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send h (just b) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":113,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":114,"title":"s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"          s -⟦ib-role⟧⇀ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":116,"title":"No-IB-Role ","content":"No-IB-Role : let open LeiosState s in","searchableContent":"  no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":117,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceib slot sk-ib (stake s) π)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":118,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":119,"title":"s -⟦IB-Role⟧⇀ addUpkeep s IB-Role","searchableContent":"          s -⟦ib-role⟧⇀ addupkeep s ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":121,"title":"IB-Role⇒ND ","content":"IB-Role⇒ND : LeiosState.needsUpkeep s IB-Role → s -⟦IB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"ib-role⇒nd : leiosstate.needsupkeep s ib-role → s -⟦ib-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":122,"title":"IB-Role⇒ND u (IB-Role x₁ x₂) = Roles (IB-Role u x₁ x₂)","searchableContent":"ib-role⇒nd u (ib-role x₁ x₂) = roles (ib-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":123,"title":"IB-Role⇒ND u (No-IB-Role x₁) = Roles (No-IB-Role u x₁)","searchableContent":"ib-role⇒nd u (no-ib-role x₁) = roles (no-ib-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":125,"title":"IB-Role-Upkeep ","content":"IB-Role-Upkeep : ∀ {u} → u ≢ IB-Role → LeiosState.needsUpkeep s u → s -⟦IB-Role⟧⇀ s'","searchableContent":"ib-role-upkeep : ∀ {u} → u ≢ ib-role → leiosstate.needsupkeep s u → s -⟦ib-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":126,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":127,"title":"IB-Role-Upkeep u≢IB-Role h (IB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"ib-role-upkeep u≢ib-role h (ib-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":128,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":129,"title":"(inj₂ y) → u≢IB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢ib-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":130,"title":"IB-Role-Upkeep u≢IB-Role h (No-IB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"ib-role-upkeep u≢ib-role h (no-ib-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":131,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":132,"title":"(inj₂ y) → u≢IB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢ib-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":134,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":135,"title":"IB-Role-total ","content":"IB-Role-total : ∃[ s' ] s -⟦IB-Role⟧⇀ s'","searchableContent":"  ib-role-total : ∃[ s' ] s -⟦ib-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":136,"title":"IB-Role-total {s = s} = let open LeiosState s in case Dec-canProduceIB of λ where","searchableContent":"  ib-role-total {s = s} = let open leiosstate s in case dec-canproduceib of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":137,"title":"(inj₁ (π , pf)) → -, IB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, ib-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":138,"title":"(inj₂ pf)       → -, No-IB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-ib-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":140,"title":"IB-Role-total' ","content":"IB-Role-total' : ∃[ ffds ] s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds } IB-Role","searchableContent":"  ib-role-total' : ∃[ ffds ] s -⟦ib-role⟧⇀ addupkeep record s { ffdstate = ffds } ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":141,"title":"IB-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceIB of λ where","searchableContent":"  ib-role-total' {s = s} = let open leiosstate s in case dec-canproduceib of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":142,"title":"(inj₁ (π , pf)) → -, IB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, ib-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":143,"title":"(inj₂ pf)       → -, No-IB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-ib-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":145,"title":"data _-⟦EB-Role⟧⇀_ ","content":"data _-⟦EB-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦eb-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":147,"title":"EB-Role ","content":"EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":148,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot (Λ + 1)) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot (λ + 1)) ibs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":149,"title":"LE = map getEBRef $ filter (isVote1Certified s) $","searchableContent":"                le = map getebref $ filter (isvote1certified s) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":150,"title":"filter (_∈ᴮ slice L slot (μ + 2)) EBs","searchableContent":"                           filter (_∈ᴮ slice l slot (μ + 2)) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":151,"title":"h = mkEB slot id π sk-EB LI LE","searchableContent":"                h = mkeb slot id π sk-eb li le"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":152,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":153,"title":"∙ canProduceEB slot sk-EB (stake s) π","searchableContent":"          ∙ canproduceeb slot sk-eb (stake s) π"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":154,"title":"∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (ebheader h) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":155,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":156,"title":"s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"          s -⟦eb-role⟧⇀ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":158,"title":"No-EB-Role ","content":"No-EB-Role : let open LeiosState s in","searchableContent":"  no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":159,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceeb slot sk-eb (stake s) π)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":160,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":161,"title":"s -⟦EB-Role⟧⇀ addUpkeep s EB-Role","searchableContent":"          s -⟦eb-role⟧⇀ addupkeep s eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":163,"title":"EB-Role⇒ND ","content":"EB-Role⇒ND : LeiosState.needsUpkeep s EB-Role → s -⟦EB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"eb-role⇒nd : leiosstate.needsupkeep s eb-role → s -⟦eb-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":164,"title":"EB-Role⇒ND u (EB-Role x₁ x₂) = Roles (EB-Role u x₁ x₂)","searchableContent":"eb-role⇒nd u (eb-role x₁ x₂) = roles (eb-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":165,"title":"EB-Role⇒ND u (No-EB-Role x₁) = Roles (No-EB-Role u x₁)","searchableContent":"eb-role⇒nd u (no-eb-role x₁) = roles (no-eb-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":167,"title":"EB-Role-Upkeep ","content":"EB-Role-Upkeep : ∀ {u} → u ≢ EB-Role → LeiosState.needsUpkeep s u → s -⟦EB-Role⟧⇀ s'","searchableContent":"eb-role-upkeep : ∀ {u} → u ≢ eb-role → leiosstate.needsupkeep s u → s -⟦eb-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":168,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":169,"title":"EB-Role-Upkeep u≢EB-Role h (EB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"eb-role-upkeep u≢eb-role h (eb-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":170,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":171,"title":"(inj₂ y) → u≢EB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢eb-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":172,"title":"EB-Role-Upkeep u≢EB-Role h (No-EB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"eb-role-upkeep u≢eb-role h (no-eb-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":173,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":174,"title":"(inj₂ y) → u≢EB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢eb-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":176,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":177,"title":"EB-Role-total ","content":"EB-Role-total : ∃[ s' ] s -⟦EB-Role⟧⇀ s'","searchableContent":"  eb-role-total : ∃[ s' ] s -⟦eb-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":178,"title":"EB-Role-total {s = s} = let open LeiosState s in case Dec-canProduceEB of λ where","searchableContent":"  eb-role-total {s = s} = let open leiosstate s in case dec-canproduceeb of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":179,"title":"(inj₁ (π , pf)) → -, EB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, eb-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":180,"title":"(inj₂ pf)       → -, No-EB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-eb-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":182,"title":"EB-Role-total' ","content":"EB-Role-total' : ∃[ ffds ] s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds } EB-Role","searchableContent":"  eb-role-total' : ∃[ ffds ] s -⟦eb-role⟧⇀ addupkeep record s { ffdstate = ffds } eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":183,"title":"EB-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceEB of λ where","searchableContent":"  eb-role-total' {s = s} = let open leiosstate s in case dec-canproduceeb of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":184,"title":"(inj₁ (π , pf)) → -, EB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, eb-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":185,"title":"(inj₂ pf)       → -, No-EB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-eb-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":187,"title":"data _-⟦V1-Role⟧⇀_ ","content":"data _-⟦V1-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦v1-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":189,"title":"V1-Role ","content":"V1-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v1-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":190,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot (μ + 1)) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot (μ + 1)) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":191,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":192,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":193,"title":"∙ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":194,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":195,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":196,"title":"s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V1-Role","searchableContent":"          s -⟦v1-role⟧⇀ addupkeep record s { ffdstate = ffds' } v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":198,"title":"No-V1-Role ","content":"No-V1-Role : let open LeiosState s in","searchableContent":"  no-v1-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":199,"title":"∙ ¬ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":200,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":201,"title":"s -⟦V1-Role⟧⇀ addUpkeep s V1-Role","searchableContent":"          s -⟦v1-role⟧⇀ addupkeep s v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":203,"title":"V1-Role⇒ND ","content":"V1-Role⇒ND : LeiosState.needsUpkeep s V1-Role → s -⟦V1-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"v1-role⇒nd : leiosstate.needsupkeep s v1-role → s -⟦v1-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":204,"title":"V1-Role⇒ND u (V1-Role x₁ x₂) = Roles (V1-Role u x₁ x₂)","searchableContent":"v1-role⇒nd u (v1-role x₁ x₂) = roles (v1-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":205,"title":"V1-Role⇒ND u (No-V1-Role x₁) = Roles (No-V1-Role u x₁)","searchableContent":"v1-role⇒nd u (no-v1-role x₁) = roles (no-v1-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":207,"title":"V1-Role-Upkeep ","content":"V1-Role-Upkeep : ∀ {u} → u ≢ V1-Role → LeiosState.needsUpkeep s u → s -⟦V1-Role⟧⇀ s'","searchableContent":"v1-role-upkeep : ∀ {u} → u ≢ v1-role → leiosstate.needsupkeep s u → s -⟦v1-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":208,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":209,"title":"V1-Role-Upkeep u≢V1-Role h (V1-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v1-role-upkeep u≢v1-role h (v1-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":210,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":211,"title":"(inj₂ y) → u≢V1-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v1-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":212,"title":"V1-Role-Upkeep u≢V1-Role h (No-V1-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v1-role-upkeep u≢v1-role h (no-v1-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":213,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":214,"title":"(inj₂ y) → u≢V1-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v1-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":216,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":217,"title":"V1-Role-total ","content":"V1-Role-total : ∃[ s' ] s -⟦V1-Role⟧⇀ s'","searchableContent":"  v1-role-total : ∃[ s' ] s -⟦v1-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":218,"title":"V1-Role-total {s = s} = let open LeiosState s in case Dec-canProduceV1 of λ where","searchableContent":"  v1-role-total {s = s} = let open leiosstate s in case dec-canproducev1 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":219,"title":"(yes p) → -, V1-Role p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v1-role p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":220,"title":"(no ¬p) → -, No-V1-Role ¬p","searchableContent":"    (no ¬p) → -, no-v1-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":222,"title":"V1-Role-total' ","content":"V1-Role-total' : ∃[ ffds ] s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds } V1-Role","searchableContent":"  v1-role-total' : ∃[ ffds ] s -⟦v1-role⟧⇀ addupkeep record s { ffdstate = ffds } v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":223,"title":"V1-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceV1 of λ where","searchableContent":"  v1-role-total' {s = s} = let open leiosstate s in case dec-canproducev1 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":224,"title":"(yes p) → -, V1-Role    p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v1-role    p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":225,"title":"(no ¬p) → -, No-V1-Role ¬p","searchableContent":"    (no ¬p) → -, no-v1-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":227,"title":"data _-⟦V2-Role⟧⇀_ ","content":"data _-⟦V2-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦v2-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":229,"title":"V2-Role ","content":"V2-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v2-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":230,"title":"EBs' = filter (vote2Eligible s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (vote2eligible s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":231,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":232,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":233,"title":"∙ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":234,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":235,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":236,"title":"s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V2-Role","searchableContent":"          s -⟦v2-role⟧⇀ addupkeep record s { ffdstate = ffds' } v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":238,"title":"No-V2-Role ","content":"No-V2-Role : let open LeiosState s in","searchableContent":"  no-v2-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":239,"title":"∙ ¬ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":240,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":241,"title":"s -⟦V2-Role⟧⇀ addUpkeep s V2-Role","searchableContent":"          s -⟦v2-role⟧⇀ addupkeep s v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":243,"title":"V2-Role⇒ND ","content":"V2-Role⇒ND : LeiosState.needsUpkeep s V2-Role → s -⟦V2-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"v2-role⇒nd : leiosstate.needsupkeep s v2-role → s -⟦v2-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":244,"title":"V2-Role⇒ND u (V2-Role x₁ x₂) = Roles (V2-Role u x₁ x₂)","searchableContent":"v2-role⇒nd u (v2-role x₁ x₂) = roles (v2-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":245,"title":"V2-Role⇒ND u (No-V2-Role x₁) = Roles (No-V2-Role u x₁)","searchableContent":"v2-role⇒nd u (no-v2-role x₁) = roles (no-v2-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":247,"title":"V2-Role-Upkeep ","content":"V2-Role-Upkeep : ∀ {u} → u ≢ V2-Role → LeiosState.needsUpkeep s u → s -⟦V2-Role⟧⇀ s'","searchableContent":"v2-role-upkeep : ∀ {u} → u ≢ v2-role → leiosstate.needsupkeep s u → s -⟦v2-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":248,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":249,"title":"V2-Role-Upkeep u≢V2-Role h (V2-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v2-role-upkeep u≢v2-role h (v2-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":250,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":251,"title":"(inj₂ y) → u≢V2-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v2-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":252,"title":"V2-Role-Upkeep u≢V2-Role h (No-V2-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v2-role-upkeep u≢v2-role h (no-v2-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":253,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":254,"title":"(inj₂ y) → u≢V2-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v2-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":256,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":257,"title":"V2-Role-total ","content":"V2-Role-total : ∃[ s' ] s -⟦V2-Role⟧⇀ s'","searchableContent":"  v2-role-total : ∃[ s' ] s -⟦v2-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":258,"title":"V2-Role-total {s = s} = let open LeiosState s in case Dec-canProduceV2 of λ where","searchableContent":"  v2-role-total {s = s} = let open leiosstate s in case dec-canproducev2 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":259,"title":"(yes p) → -, V2-Role p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v2-role p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":260,"title":"(no ¬p) → -, No-V2-Role ¬p","searchableContent":"    (no ¬p) → -, no-v2-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":262,"title":"V2-Role-total' ","content":"V2-Role-total' : ∃[ ffds ] s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds } V2-Role","searchableContent":"  v2-role-total' : ∃[ ffds ] s -⟦v2-role⟧⇀ addupkeep record s { ffdstate = ffds } v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":263,"title":"V2-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceV2 of λ where","searchableContent":"  v2-role-total' {s = s} = let open leiosstate s in case dec-canproducev2 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":264,"title":"(yes p) → -, V2-Role    p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v2-role    p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":265,"title":"(no ¬p) → -, No-V2-Role ¬p","searchableContent":"    (no ¬p) → -, no-v2-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":267,"title":"data _-⟦_/_⟧⇀_ ","content":"data _-⟦_/_⟧⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":269,"title":"-- Network and Ledger","searchableContent":"  -- network and ledger"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":271,"title":"Slot ","content":"Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs)","searchableContent":"  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":272,"title":"s0 = record s","searchableContent":"             s0 = record s"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":273,"title":"{ FFDState = ffds'","searchableContent":"                    { ffdstate = ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":274,"title":"; Ledger   = constructLedger rbs","searchableContent":"                    ; ledger   = constructledger rbs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":275,"title":"; slot     = suc slot","searchableContent":"                    ; slot     = suc slot"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":276,"title":"; Upkeep   = ∅","searchableContent":"                    ; upkeep   = ∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":277,"title":"; BaseState = bs'","searchableContent":"                    ; basestate = bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":278,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"                    } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":279,"title":"in","searchableContent":"       in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":280,"title":"∙ Upkeep ≡ᵉ allUpkeep","searchableContent":"       ∙ upkeep ≡ᵉ allupkeep"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":281,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"       ∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":282,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"       ∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":283,"title":"∙ s0 -⟦Base⟧⇀    s1","searchableContent":"       ∙ s0 -⟦base⟧⇀    s1"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":284,"title":"∙ s1 -⟦IB-Role⟧⇀ s2","searchableContent":"       ∙ s1 -⟦ib-role⟧⇀ s2"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":285,"title":"∙ s2 -⟦EB-Role⟧⇀ s3","searchableContent":"       ∙ s2 -⟦eb-role⟧⇀ s3"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":286,"title":"∙ s3 -⟦V1-Role⟧⇀ s4","searchableContent":"       ∙ s3 -⟦v1-role⟧⇀ s4"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":287,"title":"∙ s4 -⟦V2-Role⟧⇀ s5","searchableContent":"       ∙ s4 -⟦v2-role⟧⇀ s5"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":288,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":289,"title":"s -⟦ SLOT / EMPTY ⟧⇀ s5","searchableContent":"       s -⟦ slot / empty ⟧⇀ s5"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":291,"title":"Ftch ","content":"Ftch :","searchableContent":"  ftch :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":292,"title":"───────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":293,"title":"s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s","searchableContent":"       s -⟦ ftch-ldg / ftch-ldg (leiosstate.ledger s) ⟧⇀ s"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":295,"title":"-- Base chain","searchableContent":"  -- base chain"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":296,"title":"--","searchableContent":"  --"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":297,"title":"-- Note","content":"-- Note: Submitted data to the base chain is only taken into account","searchableContent":"  -- note: submitted data to the base chain is only taken into account"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":298,"title":"--       if the party submitting is the block producer on the base chain","searchableContent":"  --       if the party submitting is the block producer on the base chain"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":299,"title":"--       for the given slot","searchableContent":"  --       for the given slot"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":301,"title":"Base₁   ","content":"Base₁   :","searchableContent":"  base₁   :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":302,"title":"──────────────────────────────────────────────────────────────","searchableContent":"          ──────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":303,"title":"s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"          s -⟦ submit (inj₂ txs) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":305,"title":"-- Protocol rules","searchableContent":"  -- protocol rules"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":307,"title":"_-⟦_/_⟧ⁿᵈ⇀_ ","content":"_-⟦_/_⟧ⁿᵈ⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type","searchableContent":"_-⟦_/_⟧ⁿᵈ⇀_ : leiosstate → leiosinput → leiosoutput → leiosstate → type"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":308,"title":"s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s ND.-⟦ i / o ⟧⇀ s'","searchableContent":"s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s nd.-⟦ i / o ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":310,"title":"_-⟦_/_⟧ⁿᵈ*⇀_ ","content":"_-⟦_/_⟧ⁿᵈ*⇀_ : LeiosState → List LeiosInput → List LeiosOutput → LeiosState → Type","searchableContent":"_-⟦_/_⟧ⁿᵈ*⇀_ : leiosstate → list leiosinput → list leiosoutput → leiosstate → type"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":311,"title":"_-⟦_/_⟧ⁿᵈ*⇀_ = ReflexiveTransitiveClosure _-⟦_/_⟧ⁿᵈ⇀_","searchableContent":"_-⟦_/_⟧ⁿᵈ*⇀_ = reflexivetransitiveclosure _-⟦_/_⟧ⁿᵈ⇀_"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":313,"title":"-- Key fact","content":"-- Key fact: stepping with the deterministic relation means we can","searchableContent":"-- key fact: stepping with the deterministic relation means we can"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":314,"title":"-- also step with the non-deterministic one","searchableContent":"-- also step with the non-deterministic one"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":315,"title":"-- TODO","content":"-- TODO: this is a lot like a weak simulation, can we do something prettier?","searchableContent":"-- todo: this is a lot like a weak simulation, can we do something prettier?"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":316,"title":"-⟦/⟧⇀⇒ND ","content":"-⟦/⟧⇀⇒ND : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] (s -⟦ i / o ⟧ⁿᵈ*⇀ s')","searchableContent":"-⟦/⟧⇀⇒nd : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] (s -⟦ i / o ⟧ⁿᵈ*⇀ s')"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":317,"title":"-⟦/⟧⇀⇒ND (Slot {s = s} {msgs = msgs} {s1 = s1} {s2 = s2} {s3 = s3} {s4 = s4} x x₁ x₂ hB hIB hEB hV1 hV2) = replicate 6 SLOT , replicate 6 EMPTY ,","searchableContent":"-⟦/⟧⇀⇒nd (slot {s = s} {msgs = msgs} {s1 = s1} {s2 = s2} {s3 = s3} {s4 = s4} x x₁ x₂ hb hib heb hv1 hv2) = replicate 6 slot , replicate 6 empty ,"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":318,"title":"let","searchableContent":"  let"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":319,"title":"s0 = _","searchableContent":"    s0 = _"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":320,"title":"upkeep≡∅ ","content":"upkeep≡∅ : LeiosState.Upkeep s0 ≡ ∅","searchableContent":"    upkeep≡∅ : leiosstate.upkeep s0 ≡ ∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":321,"title":"upkeep≡∅ = sym (↑-preserves-Upkeep {x = L.filter (isValid? s) msgs})","searchableContent":"    upkeep≡∅ = sym (↑-preserves-upkeep {x = l.filter (isvalid? s) msgs})"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":322,"title":"needsAllUpkeep ","content":"needsAllUpkeep : ∀ {u} → LeiosState.needsUpkeep s0 u","searchableContent":"    needsallupkeep : ∀ {u} → leiosstate.needsupkeep s0 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":323,"title":"needsAllUpkeep {u} = subst (u ∉_) (sym upkeep≡∅) Properties.∉-∅","searchableContent":"    needsallupkeep {u} = subst (u ∉_) (sym upkeep≡∅) properties.∉-∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":324,"title":"needsUpkeep1 ","content":"needsUpkeep1 : ∀ {u} → u ≢ Base → LeiosState.needsUpkeep s1 u","searchableContent":"    needsupkeep1 : ∀ {u} → u ≢ base → leiosstate.needsupkeep s1 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":325,"title":"needsUpkeep1 h1 = Base-Upkeep h1 needsAllUpkeep hB","searchableContent":"    needsupkeep1 h1 = base-upkeep h1 needsallupkeep hb"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":326,"title":"needsUpkeep2 ","content":"needsUpkeep2 : ∀ {u} → u ≢ Base → u ≢ IB-Role → LeiosState.needsUpkeep s2 u","searchableContent":"    needsupkeep2 : ∀ {u} → u ≢ base → u ≢ ib-role → leiosstate.needsupkeep s2 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":327,"title":"needsUpkeep2 h1 h2 = IB-Role-Upkeep h2 (needsUpkeep1 h1) hIB","searchableContent":"    needsupkeep2 h1 h2 = ib-role-upkeep h2 (needsupkeep1 h1) hib"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":328,"title":"needsUpkeep3 ","content":"needsUpkeep3 : ∀ {u} → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → LeiosState.needsUpkeep s3 u","searchableContent":"    needsupkeep3 : ∀ {u} → u ≢ base → u ≢ ib-role → u ≢ eb-role → leiosstate.needsupkeep s3 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":329,"title":"needsUpkeep3 h1 h2 h3 = EB-Role-Upkeep h3 (needsUpkeep2 h1 h2) hEB","searchableContent":"    needsupkeep3 h1 h2 h3 = eb-role-upkeep h3 (needsupkeep2 h1 h2) heb"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":330,"title":"needsUpkeep4 ","content":"needsUpkeep4 : ∀ {u} → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → u ≢ V1-Role → LeiosState.needsUpkeep s4 u","searchableContent":"    needsupkeep4 : ∀ {u} → u ≢ base → u ≢ ib-role → u ≢ eb-role → u ≢ v1-role → leiosstate.needsupkeep s4 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":331,"title":"needsUpkeep4 h1 h2 h3 h4 = V1-Role-Upkeep h4 (needsUpkeep3 h1 h2 h3) hV1","searchableContent":"    needsupkeep4 h1 h2 h3 h4 = v1-role-upkeep h4 (needsupkeep3 h1 h2 h3) hv1"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":332,"title":"in (BS-ind (ND.Slot x x₁ x₂) $","searchableContent":"  in (bs-ind (nd.slot x x₁ x₂) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":333,"title":"BS-ind (Base⇒ND {s = s0} needsAllUpkeep hB) $","searchableContent":"      bs-ind (base⇒nd {s = s0} needsallupkeep hb) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":334,"title":"BS-ind (IB-Role⇒ND (needsUpkeep1 (λ ())) hIB) $","searchableContent":"      bs-ind (ib-role⇒nd (needsupkeep1 (λ ())) hib) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":335,"title":"BS-ind (EB-Role⇒ND (needsUpkeep2 (λ ()) (λ ())) hEB) $","searchableContent":"      bs-ind (eb-role⇒nd (needsupkeep2 (λ ()) (λ ())) heb) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":336,"title":"BS-ind (V1-Role⇒ND (needsUpkeep3 (λ ()) (λ ()) (λ ())) hV1) $","searchableContent":"      bs-ind (v1-role⇒nd (needsupkeep3 (λ ()) (λ ()) (λ ())) hv1) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":337,"title":"STS⇒RTC (V2-Role⇒ND (needsUpkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hV2))","searchableContent":"      sts⇒rtc (v2-role⇒nd (needsupkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hv2))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":338,"title":"-⟦/⟧⇀⇒ND Ftch = _ , _ , STS⇒RTC Ftch","searchableContent":"-⟦/⟧⇀⇒nd ftch = _ , _ , sts⇒rtc ftch"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":339,"title":"-⟦/⟧⇀⇒ND Base₁ = _ , _ , STS⇒RTC Base₁","searchableContent":"-⟦/⟧⇀⇒nd base₁ = _ , _ , sts⇒rtc base₁"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":341,"title":"open Computational22 ⦃...⦄","searchableContent":"open computational22 ⦃...⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":343,"title":"module _ ⦃ Computational-B ","content":"module _ ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄","searchableContent":"module _ ⦃ computational-b : computational22 b._-⟦_/_⟧⇀_ string ⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":344,"title":"⦃ Computational-FFD ","content":"⦃ Computational-FFD : Computational22 FFD._-⟦_/_⟧⇀_ String ⦄ where","searchableContent":"         ⦃ computational-ffd : computational22 ffd._-⟦_/_⟧⇀_ string ⦄ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":346,"title":"instance","searchableContent":"  instance"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":347,"title":"Computational--⟦/⟧⇀ ","content":"Computational--⟦/⟧⇀ : Computational22 _-⟦_/_⟧⇀_ String","searchableContent":"    computational--⟦/⟧⇀ : computational22 _-⟦_/_⟧⇀_ string"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":348,"title":"Computational--⟦/⟧⇀ .computeProof s (INIT x) = failure "No handling of INIT here"","searchableContent":"    computational--⟦/⟧⇀ .computeproof s (init x) = failure "no handling of init here""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":349,"title":"Computational--⟦/⟧⇀ .computeProof s (SUBMIT (inj₁ eb)) = failure "Cannot submit EB here"","searchableContent":"    computational--⟦/⟧⇀ .computeproof s (submit (inj₁ eb)) = failure "cannot submit eb here""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":350,"title":"Computational--⟦/⟧⇀ .computeProof s (SUBMIT (inj₂ txs)) = success (-, Base₁)","searchableContent":"    computational--⟦/⟧⇀ .computeproof s (submit (inj₂ txs)) = success (-, base₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":351,"title":"Computational--⟦/⟧⇀ .computeProof s* SLOT = let open LeiosState s* in","searchableContent":"    computational--⟦/⟧⇀ .computeproof s* slot = let open leiosstate s* in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":352,"title":"case (¿ Upkeep ≡ᵉ allUpkeep ¿ ,′ computeProof BaseState B.FTCH-LDG ,′ computeProof FFDState FFD.Fetch) of λ where","searchableContent":"      case (¿ upkeep ≡ᵉ allupkeep ¿ ,′ computeproof basestate b.ftch-ldg ,′ computeproof ffdstate ffd.fetch) of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":353,"title":"(yes p , success ((B.BASE-LDG l , bs) , p₁) , success ((FFD.FetchRes msgs , ffds) , p₂)) →","searchableContent":"        (yes p , success ((b.base-ldg l , bs) , p₁) , success ((ffd.fetchres msgs , ffds) , p₂)) →"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":354,"title":"success ((_ , (Slot p p₁ p₂ (proj₂ Base-total) (proj₂ IB-Role-total) (proj₂ EB-Role-total) (proj₂ V1-Role-total) (proj₂ V2-Role-total))))","searchableContent":"          success ((_ , (slot p p₁ p₂ (proj₂ base-total) (proj₂ ib-role-total) (proj₂ eb-role-total) (proj₂ v1-role-total) (proj₂ v2-role-total))))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":355,"title":"(yes p , _ , _) → failure "Subsystem failed"","searchableContent":"        (yes p , _ , _) → failure "subsystem failed""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":356,"title":"(no ¬p , _) → failure "Upkeep incorrect"","searchableContent":"        (no ¬p , _) → failure "upkeep incorrect""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":357,"title":"Computational--⟦/⟧⇀ .computeProof s FTCH-LDG = success (-, Ftch)","searchableContent":"    computational--⟦/⟧⇀ .computeproof s ftch-ldg = success (-, ftch)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":358,"title":"Computational--⟦/⟧⇀ .completeness = {!!}","searchableContent":"    computational--⟦/⟧⇀ .completeness = {!!}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":359,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":2,"title":"Leios.Simplified
{-# OPTIONS --safe #-}","searchableContent":"leios.simplified
{-# options --safe #-}"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":5,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":6,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":7,"title":"open import Data.Fin.Patterns","searchableContent":"open import data.fin.patterns"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":9,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":10,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":12,"title":"module Leios.Simplified (⋯ ","content":"module Leios.Simplified (⋯ : SpecStructure 2) (let open SpecStructure ⋯) (Λ μ : ℕ) where","searchableContent":"module leios.simplified (⋯ : specstructure 2) (let open specstructure ⋯) (λ μ : ℕ) where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":14,"title":"data SlotUpkeep ","content":"data SlotUpkeep : Type where","searchableContent":"data slotupkeep : type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":15,"title":"Base IB-Role EB-Role V1-Role V2-Role ","content":"Base IB-Role EB-Role V1-Role V2-Role : SlotUpkeep","searchableContent":"  base ib-role eb-role v1-role v2-role : slotupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":17,"title":"unquoteDecl DecEq-SlotUpkeep = derive-DecEq ((quote SlotUpkeep , DecEq-SlotUpkeep) ∷ [])","searchableContent":"unquotedecl deceq-slotupkeep = derive-deceq ((quote slotupkeep , deceq-slotupkeep) ∷ [])"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":19,"title":"allUpkeep ","content":"allUpkeep : ℙ SlotUpkeep","searchableContent":"allupkeep : ℙ slotupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":20,"title":"allUpkeep = fromList (Base ∷ IB-Role ∷ EB-Role ∷ V1-Role ∷ V2-Role ∷ [])","searchableContent":"allupkeep = fromlist (base ∷ ib-role ∷ eb-role ∷ v1-role ∷ v2-role ∷ [])"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":22,"title":"open import Leios.Protocol (⋯) SlotUpkeep public","searchableContent":"open import leios.protocol (⋯) slotupkeep public"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":24,"title":"open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)","searchableContent":"open baseabstract b' using (cert; v-chkcerts; vty; initslot)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":25,"title":"open FFD hiding (_-⟦_/_⟧⇀_)","searchableContent":"open ffd hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":26,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":28,"title":"isVote1Certified ","content":"isVote1Certified : LeiosState → EndorserBlock → Type","searchableContent":"isvote1certified : leiosstate → endorserblock → type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":29,"title":"isVote1Certified s eb = isVoteCertified (LeiosState.votingState s) (0F , eb)","searchableContent":"isvote1certified s eb = isvotecertified (leiosstate.votingstate s) (0f , eb)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":31,"title":"isVote2Certified ","content":"isVote2Certified : LeiosState → EndorserBlock → Type","searchableContent":"isvote2certified : leiosstate → endorserblock → type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":32,"title":"isVote2Certified s eb = isVoteCertified (LeiosState.votingState s) (1F , eb)","searchableContent":"isvote2certified s eb = isvotecertified (leiosstate.votingstate s) (1f , eb)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":34,"title":"-- Predicates about EBs","searchableContent":"-- predicates about ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":35,"title":"module _ (s ","content":"module _ (s : LeiosState) (eb : EndorserBlock) where","searchableContent":"module _ (s : leiosstate) (eb : endorserblock) where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":36,"title":"open EndorserBlockOSig eb","searchableContent":"  open endorserblockosig eb"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":37,"title":"open LeiosState s","searchableContent":"  open leiosstate s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":39,"title":"vote2Eligible ","content":"vote2Eligible : Type","searchableContent":"  vote2eligible : type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":40,"title":"vote2Eligible = length ebRefs ≥ lengthˢ candidateEBs / 2 -- should this be `>`?","searchableContent":"  vote2eligible = length ebrefs ≥ lengthˢ candidateebs / 2 -- should this be `>`?"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":41,"title":"× fromList ebRefs ⊆ candidateEBs","searchableContent":"                × fromlist ebrefs ⊆ candidateebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":42,"title":"where candidateEBs ","content":"where candidateEBs : ℙ Hash","searchableContent":"    where candidateebs : ℙ hash"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":43,"title":"candidateEBs = mapˢ getEBRef $ filterˢ (_∈ᴮ slice L slot (μ + 3)) (fromList EBs)","searchableContent":"          candidateebs = mapˢ getebref $ filterˢ (_∈ᴮ slice l slot (μ + 3)) (fromlist ebs)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":45,"title":"private variable s s'   ","content":"private variable s s'   : LeiosState","searchableContent":"private variable s s'   : leiosstate"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":46,"title":"ffds'  ","content":"ffds'  : FFD.State","searchableContent":"                 ffds'  : ffd.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":47,"title":"π      ","content":"π      : VrfPf","searchableContent":"                 π      : vrfpf"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":48,"title":"bs'    ","content":"bs'    : B.State","searchableContent":"                 bs'    : b.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":49,"title":"ks ks' ","content":"ks ks' : K.State","searchableContent":"                 ks ks' : k.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":50,"title":"msgs   ","content":"msgs   : List (FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract)","searchableContent":"                 msgs   : list (ffdabstract.header ffdabstract ⊎ ffdabstract.body ffdabstract)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":51,"title":"eb     ","content":"eb     : EndorserBlock","searchableContent":"                 eb     : endorserblock"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":52,"title":"rbs    ","content":"rbs    : List RankingBlock","searchableContent":"                 rbs    : list rankingblock"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":53,"title":"txs    ","content":"txs    : List Tx","searchableContent":"                 txs    : list tx"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":54,"title":"V      ","content":"V      : VTy","searchableContent":"                 v      : vty"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":55,"title":"SD     ","content":"SD     : StakeDistr","searchableContent":"                 sd     : stakedistr"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":56,"title":"pks    ","content":"pks    : List PubKey","searchableContent":"                 pks    : list pubkey"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":58,"title":"data _↝_ ","content":"data _↝_ : LeiosState → LeiosState → Type where","searchableContent":"data _↝_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":60,"title":"IB-Role ","content":"IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":61,"title":"b = ibBody (record { txs = ToPropose })","searchableContent":"                b = ibbody (record { txs = topropose })"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":62,"title":"h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"                h = ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":63,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":64,"title":"∙ needsUpkeep IB-Role","searchableContent":"          ∙ needsupkeep ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":65,"title":"∙ canProduceIB slot sk-IB (stake s) π","searchableContent":"          ∙ canproduceib slot sk-ib (stake s) π"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":66,"title":"∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send h (just b) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":67,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":68,"title":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":70,"title":"EB-Role ","content":"EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":71,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot (Λ + 1)) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot (λ + 1)) ibs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":72,"title":"LE = map getEBRef $ filter (isVote1Certified s) $","searchableContent":"                le = map getebref $ filter (isvote1certified s) $"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":73,"title":"filter (_∈ᴮ slice L slot (μ + 2)) EBs","searchableContent":"                           filter (_∈ᴮ slice l slot (μ + 2)) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":74,"title":"h = mkEB slot id π sk-EB LI LE","searchableContent":"                h = mkeb slot id π sk-eb li le"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":75,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":76,"title":"∙ needsUpkeep EB-Role","searchableContent":"          ∙ needsupkeep eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":77,"title":"∙ canProduceEB slot sk-EB (stake s) π","searchableContent":"          ∙ canproduceeb slot sk-eb (stake s) π"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":78,"title":"∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (ebheader h) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":79,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":80,"title":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":82,"title":"V1-Role ","content":"V1-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v1-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":83,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot (μ + 1)) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot (μ + 1)) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":84,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":85,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":86,"title":"∙ needsUpkeep V1-Role","searchableContent":"          ∙ needsupkeep v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":87,"title":"∙ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":88,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":89,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":90,"title":"s ↝ addUpkeep record s { FFDState = ffds' } V1-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":92,"title":"V2-Role ","content":"V2-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v2-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":93,"title":"EBs' = filter (vote2Eligible s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (vote2eligible s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":94,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":95,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":96,"title":"∙ needsUpkeep V2-Role","searchableContent":"          ∙ needsupkeep v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":97,"title":"∙ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":98,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":99,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":100,"title":"s ↝ addUpkeep record s { FFDState = ffds' } V2-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":102,"title":"-- Note","content":"-- Note: Base doesn't need a negative rule, since it can always be invoked","searchableContent":"  -- note: base doesn't need a negative rule, since it can always be invoked"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":104,"title":"No-IB-Role ","content":"No-IB-Role : let open LeiosState s in","searchableContent":"  no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":105,"title":"∙ needsUpkeep IB-Role","searchableContent":"          ∙ needsupkeep ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":106,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceib slot sk-ib (stake s) π)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":107,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":108,"title":"s ↝ addUpkeep s IB-Role","searchableContent":"          s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":110,"title":"No-EB-Role ","content":"No-EB-Role : let open LeiosState s in","searchableContent":"  no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":111,"title":"∙ needsUpkeep EB-Role","searchableContent":"          ∙ needsupkeep eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":112,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceeb slot sk-eb (stake s) π)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":113,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":114,"title":"s ↝ addUpkeep s EB-Role","searchableContent":"          s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":116,"title":"No-V1-Role ","content":"No-V1-Role : let open LeiosState s in","searchableContent":"  no-v1-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":117,"title":"∙ needsUpkeep V1-Role","searchableContent":"          ∙ needsupkeep v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":118,"title":"∙ ¬ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":119,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":120,"title":"s ↝ addUpkeep s V1-Role","searchableContent":"          s ↝ addupkeep s v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":122,"title":"No-V2-Role ","content":"No-V2-Role : let open LeiosState s in","searchableContent":"  no-v2-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":123,"title":"∙ needsUpkeep V2-Role","searchableContent":"          ∙ needsupkeep v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":124,"title":"∙ ¬ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":125,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":126,"title":"s ↝ addUpkeep s V2-Role","searchableContent":"          s ↝ addupkeep s v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":128,"title":"data _-⟦_/_⟧⇀_ ","content":"data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":130,"title":"-- Initialization","searchableContent":"  -- initialization"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":132,"title":"Init ","content":"Init :","searchableContent":"  init :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":133,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"       ∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":134,"title":"∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'","searchableContent":"       ∙ initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":135,"title":"────────────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":136,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","searchableContent":"       nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":138,"title":"-- Network and Ledger","searchableContent":"  -- network and ledger"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":140,"title":"Slot ","content":"Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in","searchableContent":"  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":141,"title":"∙ Upkeep ≡ᵉ allUpkeep","searchableContent":"       ∙ upkeep ≡ᵉ allupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":142,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"       ∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":143,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"       ∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":144,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":145,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"       just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":146,"title":"{ FFDState  = ffds'","searchableContent":"           { ffdstate  = ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":147,"title":"; Ledger    = constructLedger rbs","searchableContent":"           ; ledger    = constructledger rbs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":148,"title":"; slot      = suc slot","searchableContent":"           ; slot      = suc slot"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":149,"title":"; Upkeep    = ∅","searchableContent":"           ; upkeep    = ∅"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":150,"title":"; BaseState = bs'","searchableContent":"           ; basestate = bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":151,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"           } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":153,"title":"Ftch ","content":"Ftch :","searchableContent":"  ftch :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":154,"title":"────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":155,"title":"just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s","searchableContent":"       just s -⟦ ftch-ldg / ftch-ldg (leiosstate.ledger s) ⟧⇀ s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":157,"title":"-- Base chain","searchableContent":"  -- base chain"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":158,"title":"--","searchableContent":"  --"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":159,"title":"-- Note","content":"-- Note: Submitted data to the base chain is only taken into account","searchableContent":"  -- note: submitted data to the base chain is only taken into account"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":160,"title":"--       if the party submitting is the block producer on the base chain","searchableContent":"  --       if the party submitting is the block producer on the base chain"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":161,"title":"--       for the given slot","searchableContent":"  --       for the given slot"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":163,"title":"Base₁   ","content":"Base₁   :","searchableContent":"  base₁   :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":164,"title":"───────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":165,"title":"just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"          just s -⟦ submit (inj₂ txs) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":167,"title":"Base₂a  ","content":"Base₂a  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂a  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":168,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":169,"title":"∙ eb ∈ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ eb ∈ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":170,"title":"∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":171,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":172,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":174,"title":"Base₂b  ","content":"Base₂b  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂b  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":175,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":176,"title":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ [] ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":177,"title":"∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":178,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":179,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":181,"title":"-- Protocol rules","searchableContent":"  -- protocol rules"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":183,"title":"Roles ","content":"Roles : ∙ s ↝ s'","searchableContent":"  roles : ∙ s ↝ s'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":184,"title":"─────────────────────────────","searchableContent":"          ─────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":185,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"          just s -⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":186,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":2,"title":"Leios.SpecStructure
{-# OPTIONS --safe #-}","searchableContent":"leios.specstructure
{-# options --safe #-}"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":6,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":7,"title":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":9,"title":"import Leios.Base","searchableContent":"import leios.base"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":10,"title":"import Leios.Blocks","searchableContent":"import leios.blocks"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":11,"title":"import Leios.KeyRegistration","searchableContent":"import leios.keyregistration"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":12,"title":"import Leios.Voting","searchableContent":"import leios.voting"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":14,"title":"open import Data.Fin","searchableContent":"open import data.fin"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":16,"title":"module Leios.SpecStructure where","searchableContent":"module leios.specstructure where"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":18,"title":"record SpecStructure (rounds ","content":"record SpecStructure (rounds : ℕ) : Type₁ where","searchableContent":"record specstructure (rounds : ℕ) : type₁ where"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":19,"title":"field a ","content":"field a : LeiosAbstract","searchableContent":"  field a : leiosabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":21,"title":"open LeiosAbstract a public","searchableContent":"  open leiosabstract a public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":22,"title":"open Leios.Blocks a public","searchableContent":"  open leios.blocks a public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":24,"title":"field ⦃ IsBlock-Vote ⦄ ","content":"field ⦃ IsBlock-Vote ⦄ : IsBlock (List Vote)","searchableContent":"  field ⦃ isblock-vote ⦄ : isblock (list vote)"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":25,"title":"⦃ Hashable-PreIBHeader ⦄ ","content":"⦃ Hashable-PreIBHeader ⦄ : Hashable PreIBHeader Hash","searchableContent":"        ⦃ hashable-preibheader ⦄ : hashable preibheader hash"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":26,"title":"⦃ Hashable-PreEndorserBlock ⦄ ","content":"⦃ Hashable-PreEndorserBlock ⦄ : Hashable PreEndorserBlock Hash","searchableContent":"        ⦃ hashable-preendorserblock ⦄ : hashable preendorserblock hash"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":27,"title":"id ","content":"id : PoolID","searchableContent":"        id : poolid"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":28,"title":"FFD' ","content":"FFD' : FFDAbstract.Functionality ffdAbstract","searchableContent":"        ffd' : ffdabstract.functionality ffdabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":29,"title":"vrf' ","content":"vrf' : LeiosVRF a","searchableContent":"        vrf' : leiosvrf a"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":31,"title":"open LeiosVRF vrf' public","searchableContent":"  open leiosvrf vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":33,"title":"field sk-IB sk-EB sk-VT ","content":"field sk-IB sk-EB sk-VT : PrivKey","searchableContent":"  field sk-ib sk-eb sk-vt : privkey"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":34,"title":"pk-IB pk-EB pk-VT ","content":"pk-IB pk-EB pk-VT : PubKey","searchableContent":"        pk-ib pk-eb pk-vt : pubkey"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":36,"title":"open Leios.Base a vrf' public","searchableContent":"  open leios.base a vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":38,"title":"field B' ","content":"field B' : BaseAbstract","searchableContent":"  field b' : baseabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":39,"title":"BF ","content":"BF : BaseAbstract.Functionality B'","searchableContent":"        bf : baseabstract.functionality b'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":40,"title":"initBaseState ","content":"initBaseState : BaseAbstract.Functionality.State BF","searchableContent":"        initbasestate : baseabstract.functionality.state bf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":42,"title":"open Leios.KeyRegistration a vrf' public","searchableContent":"  open leios.keyregistration a vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":44,"title":"field K' ","content":"field K' : KeyRegistrationAbstract","searchableContent":"  field k' : keyregistrationabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":45,"title":"KF ","content":"KF : KeyRegistrationAbstract.Functionality K'","searchableContent":"        kf : keyregistrationabstract.functionality k'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":47,"title":"module B   = BaseAbstract.Functionality BF","searchableContent":"  module b   = baseabstract.functionality bf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":48,"title":"module K   = KeyRegistrationAbstract.Functionality KF","searchableContent":"  module k   = keyregistrationabstract.functionality kf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":49,"title":"module FFD = FFDAbstract.Functionality FFD'","searchableContent":"  module ffd = ffdabstract.functionality ffd'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":51,"title":"open Leios.Voting public","searchableContent":"  open leios.voting public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":53,"title":"field va ","content":"field va : VotingAbstract (Fin rounds × EndorserBlock)","searchableContent":"  field va : votingabstract (fin rounds × endorserblock)"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":54,"title":"open VotingAbstract va public","searchableContent":"  open votingabstract va public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":55,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":2,"title":"Leios.Traces
{-# OPTIONS --safe #-}","searchableContent":"leios.traces
{-# options --safe #-}"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":5,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":7,"title":"import Leios.Protocol","searchableContent":"import leios.protocol"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":9,"title":"module Leios.Traces {n} (⋯ ","content":"module Leios.Traces {n} (⋯ : SpecStructure n) {u : Type} (let open Leios.Protocol ⋯ u)","searchableContent":"module leios.traces {n} (⋯ : specstructure n) {u : type} (let open leios.protocol ⋯ u)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":10,"title":"(_-⟦_/_⟧⇀_ ","content":"(_-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type)","searchableContent":"  (_-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":11,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":13,"title":"_⇉_ ","content":"_⇉_ : LeiosState → LeiosState → Type","searchableContent":"_⇉_ : leiosstate → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":14,"title":"s₁ ⇉ s₂ = Σ[ (i , o) ∈ LeiosInput × LeiosOutput ] (just s₁ -⟦ i / o ⟧⇀ s₂)","searchableContent":"s₁ ⇉ s₂ = σ[ (i , o) ∈ leiosinput × leiosoutput ] (just s₁ -⟦ i / o ⟧⇀ s₂)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":16,"title":"_⇉[_]_ ","content":"_⇉[_]_ : LeiosState → ℕ → LeiosState → Type","searchableContent":"_⇉[_]_ : leiosstate → ℕ → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":17,"title":"s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂","searchableContent":"s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":18,"title":"s₁ ⇉[ suc m ] sₙ = Σ[ s₂ ∈ LeiosState ] (s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ)","searchableContent":"s₁ ⇉[ suc m ] sₙ = σ[ s₂ ∈ leiosstate ] (s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":20,"title":"_⇉⋆_ ","content":"_⇉⋆_ : LeiosState → LeiosState → Type","searchableContent":"_⇉⋆_ : leiosstate → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":21,"title":"s₁ ⇉⋆ sₙ = Σ[ n ∈  ℕ ] (s₁ ⇉[ n ] sₙ)","searchableContent":"s₁ ⇉⋆ sₙ = σ[ n ∈  ℕ ] (s₁ ⇉[ n ] sₙ)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":22,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":2,"title":"Leios.VRF
{-# OPTIONS --safe #-}","searchableContent":"leios.vrf
{-# options --safe #-}"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":7,"title":"module Leios.VRF (a ","content":"module Leios.VRF (a : LeiosAbstract) (let open LeiosAbstract a) where","searchableContent":"module leios.vrf (a : leiosabstract) (let open leiosabstract a) where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":9,"title":"record VRF (Dom Range PubKey ","content":"record VRF (Dom Range PubKey : Type) : Type₁ where","searchableContent":"record vrf (dom range pubkey : type) : type₁ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":10,"title":"field isKeyPair ","content":"field isKeyPair : PubKey → PrivKey → Type","searchableContent":"  field iskeypair : pubkey → privkey → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":11,"title":"eval ","content":"eval : PrivKey → Dom → Range × VrfPf","searchableContent":"        eval : privkey → dom → range × vrfpf"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":12,"title":"verify ","content":"verify : PubKey → Dom → Range → VrfPf → Type","searchableContent":"        verify : pubkey → dom → range → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":13,"title":"verify? ","content":"verify? : (pk : PubKey) → (d : Dom) → (r : Range) → (pf : VrfPf) → Dec (verify pk d r pf)","searchableContent":"        verify? : (pk : pubkey) → (d : dom) → (r : range) → (pf : vrfpf) → dec (verify pk d r pf)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":15,"title":"record LeiosVRF ","content":"record LeiosVRF : Type₁ where","searchableContent":"record leiosvrf : type₁ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":16,"title":"field PubKey ","content":"field PubKey : Type","searchableContent":"  field pubkey : type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":17,"title":"poolID ","content":"poolID : PubKey → PoolID","searchableContent":"        poolid : pubkey → poolid"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":18,"title":"verifySig ","content":"verifySig : PubKey → Sig → Type","searchableContent":"        verifysig : pubkey → sig → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":19,"title":"verifySig? ","content":"verifySig? : (pk : PubKey) → (sig : Sig) → Dec (verifySig pk sig)","searchableContent":"        verifysig? : (pk : pubkey) → (sig : sig) → dec (verifysig pk sig)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":21,"title":"vrf ","content":"vrf : VRF ℕ ℕ PubKey","searchableContent":"        vrf : vrf ℕ ℕ pubkey"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":23,"title":"open VRF vrf public","searchableContent":"  open vrf vrf public"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":25,"title":"-- transforming slot numbers into VRF seeds","searchableContent":"  -- transforming slot numbers into vrf seeds"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":26,"title":"field genIBInput genEBInput genVInput genV1Input genV2Input ","content":"field genIBInput genEBInput genVInput genV1Input genV2Input : ℕ → ℕ","searchableContent":"  field genibinput genebinput genvinput genv1input genv2input : ℕ → ℕ"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":28,"title":"canProduceIB ","content":"canProduceIB : ℕ → PrivKey → ℕ → VrfPf → Type","searchableContent":"  canproduceib : ℕ → privkey → ℕ → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":29,"title":"canProduceIB slot k stake π = let (val , pf) = eval k (genIBInput slot) in val < stake × pf ≡ π","searchableContent":"  canproduceib slot k stake π = let (val , pf) = eval k (genibinput slot) in val < stake × pf ≡ π"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":31,"title":"Dec-canProduceIB ","content":"Dec-canProduceIB : ∀ {slot k stake} → (∃[ π ] canProduceIB slot k stake π) ⊎ (∀ π → ¬ canProduceIB slot k stake π)","searchableContent":"  dec-canproduceib : ∀ {slot k stake} → (∃[ π ] canproduceib slot k stake π) ⊎ (∀ π → ¬ canproduceib slot k stake π)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":32,"title":"Dec-canProduceIB {slot} {k} {stake} with eval k (genIBInput slot)","searchableContent":"  dec-canproduceib {slot} {k} {stake} with eval k (genibinput slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":33,"title":"... | (val , pf) = case ¿ val < stake ¿ of λ where","searchableContent":"  ... | (val , pf) = case ¿ val < stake ¿ of λ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":34,"title":"(yes p) → inj₁ (pf , p , refl)","searchableContent":"    (yes p) → inj₁ (pf , p , refl)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":35,"title":"(no ¬p) → inj₂ (λ π (h , _) → ¬p h)","searchableContent":"    (no ¬p) → inj₂ (λ π (h , _) → ¬p h)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":37,"title":"canProduceIBPub ","content":"canProduceIBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type","searchableContent":"  canproduceibpub : ℕ → ℕ → pubkey → vrfpf → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":38,"title":"canProduceIBPub slot val k pf stake = verify k (genIBInput slot) val pf × val < stake","searchableContent":"  canproduceibpub slot val k pf stake = verify k (genibinput slot) val pf × val < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":40,"title":"canProduceEB ","content":"canProduceEB : ℕ → PrivKey → ℕ → VrfPf → Type","searchableContent":"  canproduceeb : ℕ → privkey → ℕ → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":41,"title":"canProduceEB slot k stake π = let (val , pf) = eval k (genEBInput slot) in val < stake × pf ≡ π","searchableContent":"  canproduceeb slot k stake π = let (val , pf) = eval k (genebinput slot) in val < stake × pf ≡ π"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":43,"title":"Dec-canProduceEB ","content":"Dec-canProduceEB : ∀ {slot k stake} → (∃[ π ] canProduceEB slot k stake π) ⊎ (∀ π → ¬ canProduceEB slot k stake π)","searchableContent":"  dec-canproduceeb : ∀ {slot k stake} → (∃[ π ] canproduceeb slot k stake π) ⊎ (∀ π → ¬ canproduceeb slot k stake π)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":44,"title":"Dec-canProduceEB {slot} {k} {stake} with eval k (genEBInput slot)","searchableContent":"  dec-canproduceeb {slot} {k} {stake} with eval k (genebinput slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":45,"title":"... | (val , pf) = case ¿ val < stake ¿ of λ where","searchableContent":"  ... | (val , pf) = case ¿ val < stake ¿ of λ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":46,"title":"(yes p) → inj₁ (pf , p , refl)","searchableContent":"    (yes p) → inj₁ (pf , p , refl)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":47,"title":"(no ¬p) → inj₂ (λ π (h , _) → ¬p h)","searchableContent":"    (no ¬p) → inj₂ (λ π (h , _) → ¬p h)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":49,"title":"canProduceEBPub ","content":"canProduceEBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type","searchableContent":"  canproduceebpub : ℕ → ℕ → pubkey → vrfpf → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":50,"title":"canProduceEBPub slot val k pf stake = verify k (genEBInput slot) val pf × val < stake","searchableContent":"  canproduceebpub slot val k pf stake = verify k (genebinput slot) val pf × val < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":52,"title":"canProduceV ","content":"canProduceV : ℕ → PrivKey → ℕ → Type","searchableContent":"  canproducev : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":53,"title":"canProduceV slot k stake = proj₁ (eval k (genVInput slot)) < stake","searchableContent":"  canproducev slot k stake = proj₁ (eval k (genvinput slot)) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":55,"title":"Dec-canProduceV ","content":"Dec-canProduceV : ∀ {slot k stake} → Dec (canProduceV slot k stake)","searchableContent":"  dec-canproducev : ∀ {slot k stake} → dec (canproducev slot k stake)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":56,"title":"Dec-canProduceV {slot} {k} {stake} with eval k (genVInput slot)","searchableContent":"  dec-canproducev {slot} {k} {stake} with eval k (genvinput slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":57,"title":"... | (val , pf) = ¿ val < stake ¿","searchableContent":"  ... | (val , pf) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":59,"title":"canProduceV1 ","content":"canProduceV1 : ℕ → PrivKey → ℕ → Type","searchableContent":"  canproducev1 : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":60,"title":"canProduceV1 slot k stake = proj₁ (eval k (genV1Input slot)) < stake","searchableContent":"  canproducev1 slot k stake = proj₁ (eval k (genv1input slot)) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":62,"title":"Dec-canProduceV1 ","content":"Dec-canProduceV1 : ∀ {slot k stake} → Dec (canProduceV1 slot k stake)","searchableContent":"  dec-canproducev1 : ∀ {slot k stake} → dec (canproducev1 slot k stake)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":63,"title":"Dec-canProduceV1 {slot} {k} {stake} with eval k (genV1Input slot)","searchableContent":"  dec-canproducev1 {slot} {k} {stake} with eval k (genv1input slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":64,"title":"... | (val , pf) = ¿ val < stake ¿","searchableContent":"  ... | (val , pf) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":66,"title":"canProduceV2 ","content":"canProduceV2 : ℕ → PrivKey → ℕ → Type","searchableContent":"  canproducev2 : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":67,"title":"canProduceV2 slot k stake = proj₁ (eval k (genV2Input slot)) < stake","searchableContent":"  canproducev2 slot k stake = proj₁ (eval k (genv2input slot)) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":69,"title":"Dec-canProduceV2 ","content":"Dec-canProduceV2 : ∀ {slot k stake} → Dec (canProduceV2 slot k stake)","searchableContent":"  dec-canproducev2 : ∀ {slot k stake} → dec (canproducev2 slot k stake)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":70,"title":"Dec-canProduceV2 {slot} {k} {stake} with eval k (genV2Input slot)","searchableContent":"  dec-canproducev2 {slot} {k} {stake} with eval k (genv2input slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":71,"title":"... | (val , pf) = ¿ val < stake ¿","searchableContent":"  ... | (val , pf) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":72,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":2,"title":"Leios.Voting
{-# OPTIONS --safe #-}","searchableContent":"leios.voting
{-# options --safe #-}"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":6,"title":"module Leios.Voting where","searchableContent":"module leios.voting where"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":8,"title":"record VotingAbstract (X ","content":"record VotingAbstract (X : Type) : Type₁ where","searchableContent":"record votingabstract (x : type) : type₁ where"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":9,"title":"field VotingState     ","content":"field VotingState     : Type","searchableContent":"  field votingstate     : type"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":10,"title":"initVotingState ","content":"initVotingState : VotingState","searchableContent":"        initvotingstate : votingstate"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":11,"title":"isVoteCertified ","content":"isVoteCertified : VotingState → X → Type","searchableContent":"        isvotecertified : votingstate → x → type"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":13,"title":"⦃ isVoteCertified⁇ ⦄ ","content":"⦃ isVoteCertified⁇ ⦄ : ∀ {vs x} → isVoteCertified vs x ⁇","searchableContent":"        ⦃ isvotecertified⁇ ⦄ : ∀ {vs x} → isvotecertified vs x ⁇"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":14,"title":"
","content":"
","searchableContent":""}]; +const searchInput = document.querySelector('.search-input'); +const searchResults = document.querySelector('.search-results'); +const searchOverlay = document.querySelector('.search-overlay'); + +function toggleSearch() { + searchOverlay.classList.toggle('active'); + if (searchOverlay.classList.contains('active')) { + searchInput.focus(); + } +} + +if (searchInput && searchResults) { + // Update the search results HTML generation + function generateSearchResults(results) { + return results.map(result => { + const highlightedTitle = result.title.replace( + new RegExp(result.term, 'gi'), + match => `${match}` + ); + + return ` + +

+ ${highlightedTitle} + ${result.moduleName} +

+ ${result.content} + + `; + }).join(''); + } + + searchInput.addEventListener('input', (e) => { + const query = e.target.value.toLowerCase(); + if (query.length < 2) { + searchResults.innerHTML = ''; + return; + } + + const results = searchIndex + .filter(item => item.searchableContent.includes(query)) + .map(item => ({ + ...item, + term: query + })) + .slice(0, 10); + + if (results.length > 0) { + searchResults.innerHTML = generateSearchResults(results); + } else { + searchResults.innerHTML = '
No results found
'; + } + }); + + // Add scroll-to-line functionality + document.addEventListener('click', (e) => { + const result = e.target.closest('.search-result'); + if (result) { + const lineNumber = result.dataset.line; + const targetElement = document.querySelector(`#L${lineNumber}`); + if (targetElement) { + targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + targetElement.classList.add('highlight-line'); + setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); + } + searchOverlay.classList.remove('active'); + } + }); + + // Close search when clicking outside or pressing Escape + document.addEventListener('click', (e) => { + if (e.target === searchOverlay) { + searchOverlay.classList.remove('active'); + } + }); + + document.addEventListener('keydown', (e) => { + if (e.key === 'Escape' && searchOverlay.classList.contains('active')) { + searchOverlay.classList.remove('active'); + } + }); +} + +// Theme toggle functionality +function toggleTheme() { + const html = document.documentElement; + const currentTheme = html.getAttribute('data-theme'); + const newTheme = currentTheme === 'dark' ? 'light' : 'dark'; + html.setAttribute('data-theme', newTheme); + localStorage.setItem('theme', newTheme); +} + +// Initialize theme from localStorage or system preference +const savedTheme = localStorage.getItem('theme'); +if (savedTheme) { + document.documentElement.setAttribute('data-theme', savedTheme); +} else if (window.matchMedia('(prefers-color-scheme: dark)').matches) { + document.documentElement.setAttribute('data-theme', 'dark'); +} From 0bf798aa7fbd90aae06a8da500ffe9bdb30702ea Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 19:40:05 +0200 Subject: [PATCH 04/10] fix: css restore; add search --- site/scripts/dev-with-formal-spec.sh | 12 +++ site/scripts/process-agda-html.js | 92 ++++++++++-------- site/static/agda_html/Agda.css | 133 ++++++++++++++++++--------- site/static/agda_html/agda.js | 42 +++++---- 4 files changed, 182 insertions(+), 97 deletions(-) diff --git a/site/scripts/dev-with-formal-spec.sh b/site/scripts/dev-with-formal-spec.sh index 26b7892fc..95b249812 100755 --- a/site/scripts/dev-with-formal-spec.sh +++ b/site/scripts/dev-with-formal-spec.sh @@ -21,8 +21,20 @@ nix build .#leiosDocs --impure echo "Copying Agda HTML files..." mkdir -p "$SITE_DIR/static/agda_html" + +# Backup our custom CSS if it exists +if [ -f "$SITE_DIR/static/agda_html/agda.css" ]; then + cp "$SITE_DIR/static/agda_html/agda.css" "$SITE_DIR/static/agda_html/agda.css.bak" +fi + +# Copy all files except Agda.css cp -r result/html/* "$SITE_DIR/static/agda_html/" +# Restore our custom CSS +if [ -f "$SITE_DIR/static/agda_html/agda.css.bak" ]; then + mv "$SITE_DIR/static/agda_html/agda.css.bak" "$SITE_DIR/static/agda_html/agda.css" +fi + echo "Processing Agda HTML files..." cd "$SITE_DIR" node scripts/process-agda-html.js diff --git a/site/scripts/process-agda-html.js b/site/scripts/process-agda-html.js index 3b0cd8917..7ac65d0ff 100755 --- a/site/scripts/process-agda-html.js +++ b/site/scripts/process-agda-html.js @@ -41,22 +41,26 @@ fs.readdirSync(AGDA_HTML_DIR).forEach(file => { const lines = content.split('\n'); lines.forEach((line, index) => { if (line.trim()) { // Only index non-empty lines - // Try to extract the type/expression from the line - const typeMatch = line.match(/^([^:]+):/); - const expressionMatch = line.match(/^([^=]+)=/); - const title = typeMatch ? typeMatch[1].trim() : - expressionMatch ? expressionMatch[1].trim() : - line.trim(); - - searchIndex.push({ - moduleName: moduleName, - path: file, - group: topLevel, - lineNumber: index + 1, - title: title, - content: line.trim(), - searchableContent: line.toLowerCase() - }); + // Strip HTML tags from the line + const cleanLine = line.replace(/<[^>]*>/g, ' ').replace(/\s+/g, ' ').trim(); + if (cleanLine) { + // Try to extract the type/expression from the line + const typeMatch = cleanLine.match(/^([^:]+):/); + const expressionMatch = cleanLine.match(/^([^=]+)=/); + const title = typeMatch ? typeMatch[1].trim() : + expressionMatch ? expressionMatch[1].trim() : + cleanLine.trim(); + + searchIndex.push({ + moduleName: moduleName, + path: file, + group: topLevel, + lineNumber: index + 1, + title: title, + content: cleanLine, + searchableContent: cleanLine.toLowerCase() + }); + } } }); } @@ -82,20 +86,19 @@ if (searchInput && searchResults) { // Update the search results HTML generation function generateSearchResults(results) { return results.map(result => { - const highlightedTitle = result.title.replace( + // The content is already clean (no HTML tags) + const highlightedContent = result.content.replace( new RegExp(result.term, 'gi'), - match => \`\${match}\` + match => '' + match + '' ); - return \` - -

- \${highlightedTitle} - \${result.moduleName} -

- \${result.content} - - \`; + // Ensure the path is properly encoded + const encodedPath = encodeURIComponent(result.path); + + return '' + + '' + highlightedContent + '' + + '' + result.moduleName + '' + + ''; }).join(''); } @@ -111,8 +114,7 @@ if (searchInput && searchResults) { .map(item => ({ ...item, term: query - })) - .slice(0, 10); + })); if (results.length > 0) { searchResults.innerHTML = generateSearchResults(results); @@ -126,13 +128,25 @@ if (searchInput && searchResults) { const result = e.target.closest('.search-result'); if (result) { const lineNumber = result.dataset.line; - const targetElement = document.querySelector(\`#L\${lineNumber}\`); + const targetElement = document.querySelector('#L' + lineNumber); if (targetElement) { - targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + // Calculate the offset to account for the header height and add some padding + const headerHeight = 60; // matches --header-height in CSS + const offset = headerHeight + 40; // add 20px padding + const elementPosition = targetElement.getBoundingClientRect().top; + const offsetPosition = elementPosition + window.pageYOffset - offset; + + window.scrollTo({ + top: offsetPosition, + behavior: 'smooth' + }); + targetElement.classList.add('highlight-line'); setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); } searchOverlay.classList.remove('active'); + searchInput.value = ''; // Clear the search input + searchResults.innerHTML = ''; // Clear the results } }); @@ -271,14 +285,14 @@ fs.readdirSync(AGDA_HTML_DIR).forEach(file => { - - - - - - - - ` + + + + + + + + ` ); // Write the processed file diff --git a/site/static/agda_html/Agda.css b/site/static/agda_html/Agda.css index fc8cb347d..d9d162328 100644 --- a/site/static/agda_html/Agda.css +++ b/site/static/agda_html/Agda.css @@ -6,7 +6,7 @@ --link-color: #0366d6; --header-bg: transparent; --sidebar-width: 280px; - --content-width: 800px; + --content-width: calc(100vw - var(--sidebar-width) - 4rem); --search-width: 300px; --hover-color: #f6f8fa; --highlight-color: #ffeb3b; @@ -132,7 +132,16 @@ body { flex: 1; margin-left: var(--sidebar-width); padding: 2rem; - max-width: var(--content-width); + width: var(--content-width); + overflow-x: hidden; +} + +/* Add padding to Agda content to prevent overlap */ +.agda-content .Agda { + padding-left: 1rem; + width: 100%; + overflow-x: visible; + max-width: none; } /* Agda Syntax Highlighting */ @@ -353,6 +362,10 @@ body { background: #30363d; } +[data-theme="dark"] .Agda .Pragma { + color: #c9d1d9; +} + /* Search and Theme Toggle */ .search-button { display: flex; @@ -408,103 +421,134 @@ body { background-color: rgba(0, 0, 0, 0.5); display: none; z-index: 1000; + backdrop-filter: blur(4px); + overflow: hidden; } .search-overlay.active { display: flex; align-items: flex-start; justify-content: center; - padding-top: 10vh; + padding-top: 15vh; } .search-modal { background-color: var(--bg-color); - border-radius: 8px; - box-shadow: 0 4px 12px rgba(0, 0, 0, 0.15); + border-radius: 12px; + box-shadow: 0 8px 24px rgba(0, 0, 0, 0.15); width: 90%; - max-width: 800px; - max-height: 80vh; + max-width: 700px; + max-height: 70vh; overflow: hidden; + border: 1px solid var(--border-color); + display: flex; + flex-direction: column; } .search-container { - padding: 1rem; + padding: 1.5rem; + display: flex; + flex-direction: column; + height: 100%; + overflow: hidden; } .search-input { width: 100%; - padding: 0.75rem 1rem; + padding: 0.875rem 1rem; border: 1px solid var(--border-color); - border-radius: 4px; + border-radius: 8px; background-color: var(--bg-color); color: var(--text-color); font-size: 1rem; - transition: border-color 0.2s; + transition: all 0.2s; margin-bottom: 1rem; + box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05); + box-sizing: border-box; + flex-shrink: 0; } .search-input:focus { outline: none; border-color: var(--link-color); + box-shadow: 0 0 0 3px rgba(3, 102, 214, 0.1); } .search-results { - max-height: calc(80vh - 5rem); overflow-y: auto; + margin: 0; + padding: 0; + width: 100%; + flex: 1; + min-height: 0; + border: 1px solid var(--border-color); + border-radius: 8px; + background-color: var(--bg-color); } .search-result { - display: block; - padding: 1rem; - border-bottom: 1px solid var(--border-color); - text-decoration: none; - color: var(--text-color); - transition: background-color 0.2s; -} - -.search-result:hover { - background-color: var(--hover-color); -} - -.search-header { - margin: 0 0 0.5rem 0; - font-size: 1.1em; display: flex; justify-content: space-between; align-items: center; + padding: 0.75rem 1rem; + text-decoration: none; + color: var(--text-color); + transition: background-color 0.15s ease; + border-bottom: 1px solid var(--border-color); + font-family: var(--font-sans); + font-size: 0.9rem; + line-height: 1.4; + cursor: pointer; + width: 100%; + box-sizing: border-box; } -.search-identifier { - font-weight: 500; +.search-result:last-child { + border-bottom: none; } -.search-module { - font-size: 0.9em; - color: var(--muted-color); - margin-left: 1rem; +.search-result:hover { + background-color: var(--hover-color); + text-decoration: none; } -.search-type { - font-family: var(--font-mono); - font-size: 0.9em; - color: var(--muted-color); - display: block; - white-space: nowrap; +.result-match { + flex: 1; + margin-right: 1rem; overflow: hidden; text-overflow: ellipsis; + white-space: nowrap; + color: var(--text-color); +} + +.result-file { + color: var(--muted-color); + font-size: 0.85em; + flex-shrink: 0; + font-family: var(--font-sans); } .search-match { background-color: var(--highlight-color); - color: var(--text-color); - padding: 0 2px; + padding: 0.1em 0.2em; border-radius: 2px; + font-weight: 500; + color: var(--text-color); +} + +/* Remove unused search result styles */ +.search-header, +.search-identifier, +.search-module, +.search-type { + display: none; } .search-no-results { - padding: 1rem; + padding: 2rem; color: var(--muted-color); text-align: center; + font-size: 0.95rem; } .theme-toggle svg { @@ -544,3 +588,8 @@ body { height: 24px; fill: currentColor; } + +/* Remove the search-type class as we're simplifying the results */ +.search-type { + display: none; +} diff --git a/site/static/agda_html/agda.js b/site/static/agda_html/agda.js index 111ae3fe4..3b0203d8e 100644 --- a/site/static/agda_html/agda.js +++ b/site/static/agda_html/agda.js @@ -1,6 +1,6 @@ // Search functionality -const searchIndex = [{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":2,"title":"Leios.Abstract
{-# OPTIONS --safe #-}","searchableContent":"leios.abstract
{-# options --safe #-}"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":4,"title":"module Leios.Abstract where","searchableContent":"module leios.abstract where"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":6,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":8,"title":"record LeiosAbstract ","content":"record LeiosAbstract : Type₁ where","searchableContent":"record leiosabstract : type₁ where"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":9,"title":"field Tx ","content":"field Tx : Type","searchableContent":"  field tx : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":10,"title":"⦃ DecEq-Tx ⦄ ","content":"⦃ DecEq-Tx ⦄ : DecEq Tx","searchableContent":"        ⦃ deceq-tx ⦄ : deceq tx"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":11,"title":"PoolID ","content":"PoolID : Type","searchableContent":"        poolid : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":12,"title":"⦃ DecEq-PoolID ⦄ ","content":"⦃ DecEq-PoolID ⦄ : DecEq PoolID","searchableContent":"        ⦃ deceq-poolid ⦄ : deceq poolid"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":13,"title":"BodyHash VrfPf PrivKey Sig Hash ","content":"BodyHash VrfPf PrivKey Sig Hash : Type -- these could have been byte strings, but this is safer","searchableContent":"        bodyhash vrfpf privkey sig hash : type -- these could have been byte strings, but this is safer"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":14,"title":"⦃ DecEq-Hash ⦄ ","content":"⦃ DecEq-Hash ⦄ : DecEq Hash","searchableContent":"        ⦃ deceq-hash ⦄ : deceq hash"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":15,"title":"⦃ DecEq-VrfPf ⦄ ","content":"⦃ DecEq-VrfPf ⦄ : DecEq VrfPf","searchableContent":"        ⦃ deceq-vrfpf ⦄ : deceq vrfpf"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":16,"title":"⦃ DecEq-Sig ⦄ ","content":"⦃ DecEq-Sig ⦄ : DecEq Sig","searchableContent":"        ⦃ deceq-sig ⦄ : deceq sig"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":17,"title":"Vote ","content":"Vote : Type","searchableContent":"        vote : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":18,"title":"⦃ DecEq-Vote ⦄ ","content":"⦃ DecEq-Vote ⦄ : DecEq Vote","searchableContent":"        ⦃ deceq-vote ⦄ : deceq vote"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":19,"title":"vote ","content":"vote : PrivKey → Hash → Vote","searchableContent":"        vote : privkey → hash → vote"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":20,"title":"sign ","content":"sign : PrivKey → Hash → Sig","searchableContent":"        sign : privkey → hash → sig"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":21,"title":"⦃ Hashable-Txs ⦄ ","content":"⦃ Hashable-Txs ⦄ : Hashable (List Tx) Hash","searchableContent":"        ⦃ hashable-txs ⦄ : hashable (list tx) hash"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":22,"title":"L ","content":"L : ℕ","searchableContent":"        l : ℕ"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":23,"title":"⦃ NonZero-L ⦄ ","content":"⦃ NonZero-L ⦄ : NonZero L","searchableContent":"        ⦃ nonzero-l ⦄ : nonzero l"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":24,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":2,"title":"Leios.Base
{-# OPTIONS --safe #-}","searchableContent":"leios.base
{-# options --safe #-}"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":6,"title":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":8,"title":"module Leios.Base (a ","content":"module Leios.Base (a : LeiosAbstract) (open LeiosAbstract a) (vrf' : LeiosVRF a)","searchableContent":"module leios.base (a : leiosabstract) (open leiosabstract a) (vrf' : leiosvrf a)"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":9,"title":"(let open LeiosVRF vrf') where","searchableContent":"  (let open leiosvrf vrf') where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":11,"title":"open import Leios.Blocks a using (EndorserBlock)","searchableContent":"open import leios.blocks a using (endorserblock)"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":13,"title":"StakeDistr ","content":"StakeDistr : Type","searchableContent":"stakedistr : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":14,"title":"StakeDistr = TotalMap PoolID ℕ","searchableContent":"stakedistr = totalmap poolid ℕ"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":16,"title":"RankingBlock ","content":"RankingBlock : Type","searchableContent":"rankingblock : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":17,"title":"RankingBlock = These EndorserBlock (List Tx)","searchableContent":"rankingblock = these endorserblock (list tx)"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":19,"title":"record BaseAbstract ","content":"record BaseAbstract : Type₁ where","searchableContent":"record baseabstract : type₁ where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":20,"title":"field Cert ","content":"field Cert : Type","searchableContent":"  field cert : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":21,"title":"VTy ","content":"VTy : Type","searchableContent":"        vty : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":22,"title":"initSlot ","content":"initSlot : VTy → ℕ","searchableContent":"        initslot : vty → ℕ"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":23,"title":"V-chkCerts ","content":"V-chkCerts : List PubKey → EndorserBlock × Cert → Bool","searchableContent":"        v-chkcerts : list pubkey → endorserblock × cert → bool"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":25,"title":"data Input ","content":"data Input : Type where","searchableContent":"  data input : type where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":26,"title":"INIT   ","content":"INIT   : (EndorserBlock × Cert → Bool) → Input","searchableContent":"    init   : (endorserblock × cert → bool) → input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":27,"title":"SUBMIT ","content":"SUBMIT : RankingBlock → Input","searchableContent":"    submit : rankingblock → input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":28,"title":"FTCH-LDG ","content":"FTCH-LDG : Input","searchableContent":"    ftch-ldg : input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":30,"title":"data Output ","content":"data Output : Type where","searchableContent":"  data output : type where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":31,"title":"STAKE ","content":"STAKE : StakeDistr → Output","searchableContent":"    stake : stakedistr → output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":32,"title":"EMPTY ","content":"EMPTY : Output","searchableContent":"    empty : output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":33,"title":"BASE-LDG ","content":"BASE-LDG : List RankingBlock → Output","searchableContent":"    base-ldg : list rankingblock → output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":35,"title":"record Functionality ","content":"record Functionality : Type₁ where","searchableContent":"  record functionality : type₁ where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":36,"title":"field State ","content":"field State : Type","searchableContent":"    field state : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":37,"title":"_-⟦_/_⟧⇀_ ","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"          _-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":38,"title":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ ","content":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : {s : State} → {i : Input} → {o : Output} → {s' : State} → (s -⟦ i / o ⟧⇀ s') ⁇","searchableContent":"          ⦃ dec-_-⟦_/_⟧⇀_ ⦄ : {s : state} → {i : input} → {o : output} → {s' : state} → (s -⟦ i / o ⟧⇀ s') ⁇"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":39,"title":"SUBMIT-total ","content":"SUBMIT-total : ∀ {s b} → ∃[ s' ] s -⟦ SUBMIT b / EMPTY ⟧⇀ s'","searchableContent":"          submit-total : ∀ {s b} → ∃[ s' ] s -⟦ submit b / empty ⟧⇀ s'"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":41,"title":"open Input public","searchableContent":"    open input public"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":42,"title":"open Output public","searchableContent":"    open output public"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":43,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":2,"title":"Leios.Blocks
{-# OPTIONS --safe #-}","searchableContent":"leios.blocks
{-# options --safe #-}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":6,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":7,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":8,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":10,"title":"module Leios.Blocks (a ","content":"module Leios.Blocks (a : LeiosAbstract) (let open LeiosAbstract a) where","searchableContent":"module leios.blocks (a : leiosabstract) (let open leiosabstract a) where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":12,"title":"-- IsBlock typeclass (could do a closed-world approach instead)","searchableContent":"-- isblock typeclass (could do a closed-world approach instead)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":13,"title":"-- Q","content":"-- Q: should votes have an instance of this class?","searchableContent":"-- q: should votes have an instance of this class?"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":14,"title":"record IsBlock (B ","content":"record IsBlock (B : Type) : Type where","searchableContent":"record isblock (b : type) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":15,"title":"field slotNumber ","content":"field slotNumber : B → ℕ","searchableContent":"  field slotnumber : b → ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":16,"title":"producerID ","content":"producerID : B → PoolID","searchableContent":"        producerid : b → poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":17,"title":"lotteryPf  ","content":"lotteryPf  : B → VrfPf","searchableContent":"        lotterypf  : b → vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":18,"title":"signature  ","content":"signature  : B → Sig","searchableContent":"        signature  : b → sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":20,"title":"infix 4 _∈ᴮ_","searchableContent":"  infix 4 _∈ᴮ_"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":22,"title":"_∈ᴮ_ ","content":"_∈ᴮ_ : B → ℙ ℕ → Type","searchableContent":"  _∈ᴮ_ : b → ℙ ℕ → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":23,"title":"b ∈ᴮ X = slotNumber b ∈ X","searchableContent":"  b ∈ᴮ x = slotnumber b ∈ x"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":25,"title":"open IsBlock ⦃...⦄ public","searchableContent":"open isblock ⦃...⦄ public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":27,"title":"IBRef = Hash","searchableContent":"ibref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":28,"title":"EBRef = Hash","searchableContent":"ebref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":30,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":31,"title":"-- Input Blocks","searchableContent":"-- input blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":32,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":34,"title":"record IBHeaderOSig (sig ","content":"record IBHeaderOSig (sig : Type) : Type where","searchableContent":"record ibheaderosig (sig : type) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":35,"title":"field slotNumber ","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":36,"title":"producerID ","content":"producerID : PoolID","searchableContent":"        producerid : poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":37,"title":"lotteryPf  ","content":"lotteryPf  : VrfPf","searchableContent":"        lotterypf  : vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":38,"title":"bodyHash   ","content":"bodyHash   : Hash","searchableContent":"        bodyhash   : hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":39,"title":"signature  ","content":"signature  : sig","searchableContent":"        signature  : sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":41,"title":"IBHeader    = IBHeaderOSig Sig","searchableContent":"ibheader    = ibheaderosig sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":42,"title":"PreIBHeader = IBHeaderOSig ⊤","searchableContent":"preibheader = ibheaderosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":44,"title":"record IBBody ","content":"record IBBody : Type where","searchableContent":"record ibbody : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":45,"title":"field txs ","content":"field txs : List Tx","searchableContent":"  field txs : list tx"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":47,"title":"record InputBlock ","content":"record InputBlock : Type where","searchableContent":"record inputblock : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":48,"title":"field header ","content":"field header : IBHeader","searchableContent":"  field header : ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":49,"title":"body   ","content":"body   : IBBody","searchableContent":"        body   : ibbody"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":51,"title":"open IBHeaderOSig header public","searchableContent":"  open ibheaderosig header public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":53,"title":"unquoteDecl DecEq-IBBody DecEq-IBHeaderOSig DecEq-InputBlock =","searchableContent":"unquotedecl deceq-ibbody deceq-ibheaderosig deceq-inputblock ="},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":54,"title":"derive-DecEq (","searchableContent":"  derive-deceq ("},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":55,"title":"(quote IBBody , DecEq-IBBody)","searchableContent":"      (quote ibbody , deceq-ibbody)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":56,"title":"∷ (quote IBHeaderOSig , DecEq-IBHeaderOSig)","searchableContent":"    ∷ (quote ibheaderosig , deceq-ibheaderosig)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":57,"title":"∷ (quote InputBlock , DecEq-InputBlock) ∷ [])","searchableContent":"    ∷ (quote inputblock , deceq-inputblock) ∷ [])"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":59,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":60,"title":"IsBlock-IBHeader ","content":"IsBlock-IBHeader : IsBlock IBHeader","searchableContent":"  isblock-ibheader : isblock ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":61,"title":"IsBlock-IBHeader = record { IBHeaderOSig }","searchableContent":"  isblock-ibheader = record { ibheaderosig }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":63,"title":"Hashable-IBBody ","content":"Hashable-IBBody : Hashable IBBody Hash","searchableContent":"  hashable-ibbody : hashable ibbody hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":64,"title":"Hashable-IBBody .hash b = hash (b .IBBody.txs)","searchableContent":"  hashable-ibbody .hash b = hash (b .ibbody.txs)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":66,"title":"Hashable-IBHeader ","content":"Hashable-IBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → Hashable IBHeader Hash","searchableContent":"  hashable-ibheader : ⦃ hashable preibheader hash ⦄ → hashable ibheader hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":67,"title":"Hashable-IBHeader .hash b = hash {T = PreIBHeader}","searchableContent":"  hashable-ibheader .hash b = hash {t = preibheader}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":68,"title":"record { IBHeaderOSig b hiding (signature) ; signature = _ }","searchableContent":"    record { ibheaderosig b hiding (signature) ; signature = _ }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":70,"title":"IsBlock-InputBlock ","content":"IsBlock-InputBlock : IsBlock InputBlock","searchableContent":"  isblock-inputblock : isblock inputblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":71,"title":"IsBlock-InputBlock = record { InputBlock }","searchableContent":"  isblock-inputblock = record { inputblock }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":73,"title":"mkIBHeader ","content":"mkIBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List Tx → IBHeader","searchableContent":"mkibheader : ⦃ hashable preibheader hash ⦄ → ℕ → poolid → vrfpf → privkey → list tx → ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":74,"title":"mkIBHeader slot id π pKey txs = record { signature = sign pKey (hash h) ; IBHeaderOSig h }","searchableContent":"mkibheader slot id π pkey txs = record { signature = sign pkey (hash h) ; ibheaderosig h }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":75,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":76,"title":"h ","content":"h : IBHeaderOSig ⊤","searchableContent":"    h : ibheaderosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":77,"title":"h = record { slotNumber = slot","searchableContent":"    h = record { slotnumber = slot"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":78,"title":"; producerID = id","searchableContent":"               ; producerid = id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":79,"title":"; lotteryPf  = π","searchableContent":"               ; lotterypf  = π"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":80,"title":"; bodyHash   = hash txs","searchableContent":"               ; bodyhash   = hash txs"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":81,"title":"; signature  = _","searchableContent":"               ; signature  = _"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":82,"title":"}","searchableContent":"               }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":84,"title":"getIBRef ","content":"getIBRef : ⦃ Hashable PreIBHeader Hash ⦄ → InputBlock → IBRef","searchableContent":"getibref : ⦃ hashable preibheader hash ⦄ → inputblock → ibref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":85,"title":"getIBRef = hash ∘ InputBlock.header","searchableContent":"getibref = hash ∘ inputblock.header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":87,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":88,"title":"-- Endorser Blocks","searchableContent":"-- endorser blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":89,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":91,"title":"record EndorserBlockOSig (sig ","content":"record EndorserBlockOSig (sig : Type) : Type where","searchableContent":"record endorserblockosig (sig : type) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":92,"title":"field slotNumber ","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":93,"title":"producerID ","content":"producerID : PoolID","searchableContent":"        producerid : poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":94,"title":"lotteryPf  ","content":"lotteryPf  : VrfPf","searchableContent":"        lotterypf  : vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":95,"title":"ibRefs     ","content":"ibRefs     : List IBRef","searchableContent":"        ibrefs     : list ibref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":96,"title":"ebRefs     ","content":"ebRefs     : List EBRef","searchableContent":"        ebrefs     : list ebref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":97,"title":"signature  ","content":"signature  : sig","searchableContent":"        signature  : sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":99,"title":"EndorserBlock    = EndorserBlockOSig Sig","searchableContent":"endorserblock    = endorserblockosig sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":100,"title":"PreEndorserBlock = EndorserBlockOSig ⊤","searchableContent":"preendorserblock = endorserblockosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":102,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":103,"title":"Hashable-EndorserBlock ","content":"Hashable-EndorserBlock : ⦃ Hashable PreEndorserBlock Hash ⦄ → Hashable EndorserBlock Hash","searchableContent":"  hashable-endorserblock : ⦃ hashable preendorserblock hash ⦄ → hashable endorserblock hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":104,"title":"Hashable-EndorserBlock .hash b = hash {T = PreEndorserBlock}","searchableContent":"  hashable-endorserblock .hash b = hash {t = preendorserblock}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":105,"title":"record { EndorserBlockOSig b hiding (signature) ; signature = _ }","searchableContent":"    record { endorserblockosig b hiding (signature) ; signature = _ }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":107,"title":"IsBlock-EndorserBlock ","content":"IsBlock-EndorserBlock : IsBlock EndorserBlock","searchableContent":"  isblock-endorserblock : isblock endorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":108,"title":"IsBlock-EndorserBlock = record { EndorserBlockOSig }","searchableContent":"  isblock-endorserblock = record { endorserblockosig }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":110,"title":"unquoteDecl DecEq-EndorserBlockOSig = derive-DecEq ((quote EndorserBlockOSig , DecEq-EndorserBlockOSig) ∷ [])","searchableContent":"unquotedecl deceq-endorserblockosig = derive-deceq ((quote endorserblockosig , deceq-endorserblockosig) ∷ [])"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":112,"title":"mkEB ","content":"mkEB : ⦃ Hashable PreEndorserBlock Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List IBRef → List EBRef → EndorserBlock","searchableContent":"mkeb : ⦃ hashable preendorserblock hash ⦄ → ℕ → poolid → vrfpf → privkey → list ibref → list ebref → endorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":113,"title":"mkEB slot id π pKey LI LE = record { signature = sign pKey (hash b) ; EndorserBlockOSig b }","searchableContent":"mkeb slot id π pkey li le = record { signature = sign pkey (hash b) ; endorserblockosig b }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":114,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":115,"title":"b ","content":"b : PreEndorserBlock","searchableContent":"    b : preendorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":116,"title":"b = record { slotNumber = slot","searchableContent":"    b = record { slotnumber = slot"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":117,"title":"; producerID = id","searchableContent":"               ; producerid = id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":118,"title":"; lotteryPf  = π","searchableContent":"               ; lotterypf  = π"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":119,"title":"; ibRefs     = LI","searchableContent":"               ; ibrefs     = li"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":120,"title":"; ebRefs     = LE","searchableContent":"               ; ebrefs     = le"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":121,"title":"; signature  = _","searchableContent":"               ; signature  = _"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":122,"title":"}","searchableContent":"               }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":124,"title":"getEBRef ","content":"getEBRef : ⦃ Hashable PreEndorserBlock Hash ⦄ → EndorserBlock → EBRef","searchableContent":"getebref : ⦃ hashable preendorserblock hash ⦄ → endorserblock → ebref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":125,"title":"getEBRef = hash","searchableContent":"getebref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":127,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":128,"title":"-- Votes","searchableContent":"-- votes"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":129,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":133,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":134,"title":"-- FFD for Leios Blocks","searchableContent":"-- ffd for leios blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":135,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":137,"title":"module GenFFD ⦃ _ ","content":"module GenFFD ⦃ _ : IsBlock (List Vote) ⦄ where","searchableContent":"module genffd ⦃ _ : isblock (list vote) ⦄ where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":138,"title":"data Header ","content":"data Header : Type where","searchableContent":"  data header : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":139,"title":"ibHeader ","content":"ibHeader : IBHeader → Header","searchableContent":"    ibheader : ibheader → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":140,"title":"ebHeader ","content":"ebHeader : EndorserBlock → Header","searchableContent":"    ebheader : endorserblock → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":141,"title":"vtHeader ","content":"vtHeader : List Vote → Header","searchableContent":"    vtheader : list vote → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":143,"title":"data Body ","content":"data Body : Type where","searchableContent":"  data body : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":144,"title":"ibBody ","content":"ibBody : IBBody → Body","searchableContent":"    ibbody : ibbody → body"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":146,"title":"unquoteDecl DecEq-Header DecEq-Body =","searchableContent":"  unquotedecl deceq-header deceq-body ="},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":147,"title":"derive-DecEq ((quote Header , DecEq-Header) ∷ (quote Body , DecEq-Body) ∷ [])","searchableContent":"    derive-deceq ((quote header , deceq-header) ∷ (quote body , deceq-body) ∷ [])"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":149,"title":"ID ","content":"ID : Type","searchableContent":"  id : type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":150,"title":"ID = ℕ × PoolID","searchableContent":"  id = ℕ × poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":152,"title":"matchIB ","content":"matchIB : IBHeader → IBBody → Type","searchableContent":"  matchib : ibheader → ibbody → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":153,"title":"matchIB h b = bodyHash ≡ hash b","searchableContent":"  matchib h b = bodyhash ≡ hash b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":154,"title":"where open IBHeaderOSig h; open IBBody b","searchableContent":"    where open ibheaderosig h; open ibbody b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":156,"title":"matchIB? ","content":"matchIB? :  ∀ (h : IBHeader) → (b : IBBody) → Dec (matchIB h b)","searchableContent":"  matchib? :  ∀ (h : ibheader) → (b : ibbody) → dec (matchib h b)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":157,"title":"matchIB? h b = bodyHash ≟ hash b","searchableContent":"  matchib? h b = bodyhash ≟ hash b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":158,"title":"where open IBHeaderOSig h; open IBBody b","searchableContent":"    where open ibheaderosig h; open ibbody b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":160,"title":"match ","content":"match : Header → Body → Type","searchableContent":"  match : header → body → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":161,"title":"match (ibHeader h) (ibBody b) = matchIB h b","searchableContent":"  match (ibheader h) (ibbody b) = matchib h b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":162,"title":"match _ _ = ⊥","searchableContent":"  match _ _ = ⊥"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":164,"title":"-- can we express uniqueness wrt pipelines as a property?","searchableContent":"  -- can we express uniqueness wrt pipelines as a property?"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":165,"title":"msgID ","content":"msgID : Header → ID","searchableContent":"  msgid : header → id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":166,"title":"msgID (ibHeader h) = (slotNumber h , producerID h)","searchableContent":"  msgid (ibheader h) = (slotnumber h , producerid h)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":167,"title":"msgID (ebHeader h) = (slotNumber h , producerID h) -- NOTE","content":"msgID (ebHeader h) = (slotNumber h , producerID h) -- NOTE: this isn't in the paper","searchableContent":"  msgid (ebheader h) = (slotnumber h , producerid h) -- note: this isn't in the paper"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":168,"title":"msgID (vtHeader  h) = (slotNumber h , producerID h) -- NOTE","content":"msgID (vtHeader  h) = (slotNumber h , producerID h) -- NOTE: this isn't in the paper","searchableContent":"  msgid (vtheader  h) = (slotnumber h , producerid h) -- note: this isn't in the paper"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":170,"title":"ffdAbstract ","content":"ffdAbstract : ⦃ _ : IsBlock (List Vote) ⦄ → FFDAbstract","searchableContent":"ffdabstract : ⦃ _ : isblock (list vote) ⦄ → ffdabstract"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":171,"title":"ffdAbstract = record { GenFFD }","searchableContent":"ffdabstract = record { genffd }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":172,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":2,"title":"Leios.Config
open import Leios.Prelude","searchableContent":"leios.config
open import leios.prelude"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":3,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":4,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":6,"title":"module Leios.Config where","searchableContent":"module leios.config where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":8,"title":"data BlockType ","content":"data BlockType : Type where","searchableContent":"data blocktype : type where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":9,"title":"IB EB VT ","content":"IB EB VT : BlockType","searchableContent":"  ib eb vt : blocktype"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":11,"title":"unquoteDecl DecEq-BlockType = derive-DecEq ((quote BlockType , DecEq-BlockType) ∷ [])","searchableContent":"unquotedecl deceq-blocktype = derive-deceq ((quote blocktype , deceq-blocktype) ∷ [])"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":13,"title":"record Params ","content":"record Params : Type where","searchableContent":"record params : type where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":14,"title":"field numberOfParties ","content":"field numberOfParties : ℕ","searchableContent":"  field numberofparties : ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":15,"title":"sutId ","content":"sutId : Fin numberOfParties","searchableContent":"        sutid : fin numberofparties"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":16,"title":"stakeDistribution ","content":"stakeDistribution : TotalMap (Fin numberOfParties) ℕ","searchableContent":"        stakedistribution : totalmap (fin numberofparties) ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":17,"title":"stageLength ","content":"stageLength : ℕ","searchableContent":"        stagelength : ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":18,"title":"⦃ NonZero-stageLength ⦄ ","content":"⦃ NonZero-stageLength ⦄ : NonZero stageLength","searchableContent":"        ⦃ nonzero-stagelength ⦄ : nonzero stagelength"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":19,"title":"winning-slots ","content":"winning-slots : ℙ (BlockType × ℕ)","searchableContent":"        winning-slots : ℙ (blocktype × ℕ)"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":20,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":2,"title":"Leios.Defaults
open import Leios.Prelude","searchableContent":"leios.defaults
open import leios.prelude"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":3,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":4,"title":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":5,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":7,"title":"open import Axiom.Set.Properties th","searchableContent":"open import axiom.set.properties th"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":8,"title":"open import Data.Nat.Show as N","searchableContent":"open import data.nat.show as n"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":9,"title":"open import Data.Integer hiding (_≟_)","searchableContent":"open import data.integer hiding (_≟_)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":10,"title":"open import Data.String as S using (intersperse)","searchableContent":"open import data.string as s using (intersperse)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":11,"title":"open import Function.Related.TypeIsomorphisms","searchableContent":"open import function.related.typeisomorphisms"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":12,"title":"open import Relation.Binary.Structures","searchableContent":"open import relation.binary.structures"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":14,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":15,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":17,"title":"open Equivalence","searchableContent":"open equivalence"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":19,"title":"-- The module contains very simple implementations for the functionalities","searchableContent":"-- the module contains very simple implementations for the functionalities"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":20,"title":"-- that allow to build examples for traces for the different Leios variants","searchableContent":"-- that allow to build examples for traces for the different leios variants"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":21,"title":"module Leios.Defaults (params ","content":"module Leios.Defaults (params : Params) (let open Params params) where","searchableContent":"module leios.defaults (params : params) (let open params params) where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":23,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":24,"title":"htx ","content":"htx : Hashable (List ℕ) (List ℕ)","searchableContent":"  htx : hashable (list ℕ) (list ℕ)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":25,"title":"htx = record { hash = id }","searchableContent":"  htx = record { hash = id }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":27,"title":"d-Abstract ","content":"d-Abstract : LeiosAbstract","searchableContent":"d-abstract : leiosabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":28,"title":"d-Abstract =","searchableContent":"d-abstract ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":29,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":30,"title":"{ Tx       = ℕ","searchableContent":"    { tx       = ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":31,"title":"; PoolID   = Fin numberOfParties","searchableContent":"    ; poolid   = fin numberofparties"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":32,"title":"; BodyHash = List ℕ","searchableContent":"    ; bodyhash = list ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":33,"title":"; VrfPf    = ⊤","searchableContent":"    ; vrfpf    = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":34,"title":"; PrivKey  = BlockType × ⊤","searchableContent":"    ; privkey  = blocktype × ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":35,"title":"; Sig      = ⊤","searchableContent":"    ; sig      = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":36,"title":"; Hash     = List ℕ","searchableContent":"    ; hash     = list ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":37,"title":"; Vote     = ⊤","searchableContent":"    ; vote     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":38,"title":"; vote     = λ _ _ → tt","searchableContent":"    ; vote     = λ _ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":39,"title":"; sign     = λ _ _ → tt","searchableContent":"    ; sign     = λ _ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":40,"title":"; L        = stageLength","searchableContent":"    ; l        = stagelength"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":41,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":43,"title":"open LeiosAbstract d-Abstract public","searchableContent":"open leiosabstract d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":45,"title":"open import Leios.VRF d-Abstract public","searchableContent":"open import leios.vrf d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":47,"title":"sutStake ","content":"sutStake : ℕ","searchableContent":"sutstake : ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":48,"title":"sutStake = TotalMap.lookup stakeDistribution sutId","searchableContent":"sutstake = totalmap.lookup stakedistribution sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":50,"title":"sortition ","content":"sortition : BlockType → ℕ → ℕ","searchableContent":"sortition : blocktype → ℕ → ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":51,"title":"sortition b n with (b , n) ∈? winning-slots","searchableContent":"sortition b n with (b , n) ∈? winning-slots"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":52,"title":"... | yes _ = 0","searchableContent":"... | yes _ = 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":53,"title":"... | no _ = sutStake","searchableContent":"... | no _ = sutstake"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":55,"title":"d-VRF ","content":"d-VRF : LeiosVRF","searchableContent":"d-vrf : leiosvrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":56,"title":"d-VRF =","searchableContent":"d-vrf ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":57,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":58,"title":"{ PubKey     = Fin numberOfParties × ⊤","searchableContent":"    { pubkey     = fin numberofparties × ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":59,"title":"; vrf        =","searchableContent":"    ; vrf        ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":60,"title":"record","searchableContent":"        record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":61,"title":"{ isKeyPair = λ _ _ → ⊤","searchableContent":"          { iskeypair = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":62,"title":"; eval      = λ (b , _) y → sortition b y , tt","searchableContent":"          ; eval      = λ (b , _) y → sortition b y , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":63,"title":"; verify    = λ _ _ _ _ → ⊤","searchableContent":"          ; verify    = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":64,"title":"; verify?   = λ _ _ _ _ → yes tt","searchableContent":"          ; verify?   = λ _ _ _ _ → yes tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":65,"title":"}","searchableContent":"          }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":66,"title":"; genIBInput = id","searchableContent":"    ; genibinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":67,"title":"; genEBInput = id","searchableContent":"    ; genebinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":68,"title":"; genVInput  = id","searchableContent":"    ; genvinput  = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":69,"title":"; genV1Input = id","searchableContent":"    ; genv1input = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":70,"title":"; genV2Input = id","searchableContent":"    ; genv2input = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":71,"title":"; poolID     = proj₁","searchableContent":"    ; poolid     = proj₁"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":72,"title":"; verifySig  = λ _ _ → ⊤","searchableContent":"    ; verifysig  = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":73,"title":"; verifySig? = λ _ _ → yes tt","searchableContent":"    ; verifysig? = λ _ _ → yes tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":74,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":76,"title":"open LeiosVRF d-VRF public","searchableContent":"open leiosvrf d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":78,"title":"open import Leios.Blocks d-Abstract public","searchableContent":"open import leios.blocks d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":79,"title":"open import Leios.KeyRegistration d-Abstract d-VRF public","searchableContent":"open import leios.keyregistration d-abstract d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":81,"title":"d-KeyRegistration ","content":"d-KeyRegistration : KeyRegistrationAbstract","searchableContent":"d-keyregistration : keyregistrationabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":82,"title":"d-KeyRegistration = _","searchableContent":"d-keyregistration = _"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":84,"title":"d-KeyRegistrationFunctionality ","content":"d-KeyRegistrationFunctionality : KeyRegistrationAbstract.Functionality d-KeyRegistration","searchableContent":"d-keyregistrationfunctionality : keyregistrationabstract.functionality d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":85,"title":"d-KeyRegistrationFunctionality =","searchableContent":"d-keyregistrationfunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":86,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":87,"title":"{ State     = ⊤","searchableContent":"    { state     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":88,"title":"; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤","searchableContent":"    ; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":89,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":91,"title":"open import Leios.Base d-Abstract d-VRF public","searchableContent":"open import leios.base d-abstract d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":93,"title":"d-Base ","content":"d-Base : BaseAbstract","searchableContent":"d-base : baseabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":94,"title":"d-Base =","searchableContent":"d-base ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":95,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":96,"title":"{ Cert       = ⊤","searchableContent":"    { cert       = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":97,"title":"; VTy        = ⊤","searchableContent":"    ; vty        = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":98,"title":"; initSlot   = λ _ → 0","searchableContent":"    ; initslot   = λ _ → 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":99,"title":"; V-chkCerts = λ _ _ → true","searchableContent":"    ; v-chkcerts = λ _ _ → true"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":100,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":102,"title":"d-BaseFunctionality ","content":"d-BaseFunctionality : BaseAbstract.Functionality d-Base","searchableContent":"d-basefunctionality : baseabstract.functionality d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":103,"title":"d-BaseFunctionality =","searchableContent":"d-basefunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":104,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":105,"title":"{ State         = ⊤","searchableContent":"    { state         = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":106,"title":"; _-⟦_/_⟧⇀_     = λ _ _ _ _ → ⊤","searchableContent":"    ; _-⟦_/_⟧⇀_     = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":107,"title":"; Dec-_-⟦_/_⟧⇀_ = ⁇ (yes tt)","searchableContent":"    ; dec-_-⟦_/_⟧⇀_ = ⁇ (yes tt)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":108,"title":"; SUBMIT-total  = tt , tt","searchableContent":"    ; submit-total  = tt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":109,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":111,"title":"open import Leios.FFD public","searchableContent":"open import leios.ffd public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":113,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":114,"title":"isb ","content":"isb : IsBlock (List ⊤)","searchableContent":"  isb : isblock (list ⊤)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":115,"title":"isb =","searchableContent":"  isb ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":116,"title":"record","searchableContent":"    record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":117,"title":"{ slotNumber = λ _ → 0","searchableContent":"      { slotnumber = λ _ → 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":118,"title":"; producerID = λ _ → sutId","searchableContent":"      ; producerid = λ _ → sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":119,"title":"; lotteryPf  = λ _ → tt","searchableContent":"      ; lotterypf  = λ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":120,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":122,"title":"hhs ","content":"hhs : Hashable PreIBHeader (List ℕ)","searchableContent":"  hhs : hashable preibheader (list ℕ)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":123,"title":"hhs .hash = IBHeaderOSig.bodyHash","searchableContent":"  hhs .hash = ibheaderosig.bodyhash"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":125,"title":"hpe ","content":"hpe : Hashable PreEndorserBlock (List ℕ)","searchableContent":"  hpe : hashable preendorserblock (list ℕ)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":126,"title":"hpe .hash = L.concat ∘ EndorserBlockOSig.ibRefs","searchableContent":"  hpe .hash = l.concat ∘ endorserblockosig.ibrefs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":128,"title":"record FFDBuffers ","content":"record FFDBuffers : Type where","searchableContent":"record ffdbuffers : type where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":129,"title":"field inIBs ","content":"field inIBs : List InputBlock","searchableContent":"  field inibs : list inputblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":130,"title":"inEBs ","content":"inEBs : List EndorserBlock","searchableContent":"        inebs : list endorserblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":131,"title":"inVTs ","content":"inVTs : List (List Vote)","searchableContent":"        invts : list (list vote)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":133,"title":"outIBs ","content":"outIBs : List InputBlock","searchableContent":"        outibs : list inputblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":134,"title":"outEBs ","content":"outEBs : List EndorserBlock","searchableContent":"        outebs : list endorserblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":135,"title":"outVTs ","content":"outVTs : List (List Vote)","searchableContent":"        outvts : list (list vote)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":137,"title":"unquoteDecl DecEq-FFDBuffers = derive-DecEq ((quote FFDBuffers , DecEq-FFDBuffers) ∷ [])","searchableContent":"unquotedecl deceq-ffdbuffers = derive-deceq ((quote ffdbuffers , deceq-ffdbuffers) ∷ [])"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":139,"title":"open GenFFD.Header","searchableContent":"open genffd.header"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":140,"title":"open GenFFD.Body","searchableContent":"open genffd.body"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":141,"title":"open FFDBuffers","searchableContent":"open ffdbuffers"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":143,"title":"flushIns ","content":"flushIns : FFDBuffers → List (GenFFD.Header ⊎ GenFFD.Body)","searchableContent":"flushins : ffdbuffers → list (genffd.header ⊎ genffd.body)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":144,"title":"flushIns record { inIBs = ibs ; inEBs = ebs ; inVTs = vts } =","searchableContent":"flushins record { inibs = ibs ; inebs = ebs ; invts = vts } ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":145,"title":"flushIBs ibs ++ L.map (inj₁ ∘ ebHeader) ebs ++ L.map (inj₁ ∘ vtHeader) vts","searchableContent":"  flushibs ibs ++ l.map (inj₁ ∘ ebheader) ebs ++ l.map (inj₁ ∘ vtheader) vts"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":146,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":147,"title":"flushIBs ","content":"flushIBs : List InputBlock → List (GenFFD.Header ⊎ GenFFD.Body)","searchableContent":"    flushibs : list inputblock → list (genffd.header ⊎ genffd.body)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":148,"title":"flushIBs [] = []","searchableContent":"    flushibs [] = []"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":149,"title":"flushIBs (record {header = h; body = b} ∷ ibs) = inj₁ (ibHeader h) ∷ inj₂ (ibBody b) ∷ flushIBs ibs","searchableContent":"    flushibs (record {header = h; body = b} ∷ ibs) = inj₁ (ibheader h) ∷ inj₂ (ibbody b) ∷ flushibs ibs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":151,"title":"data SimpleFFD ","content":"data SimpleFFD : FFDBuffers → FFDAbstract.Input ffdAbstract → FFDAbstract.Output ffdAbstract → FFDBuffers → Type where","searchableContent":"data simpleffd : ffdbuffers → ffdabstract.input ffdabstract → ffdabstract.output ffdabstract → ffdbuffers → type where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":152,"title":"SendIB ","content":"SendIB : ∀ {s h b}    → SimpleFFD s (FFDAbstract.Send (ibHeader h) (just (ibBody b))) FFDAbstract.SendRes (record s { outIBs = record {header = h; body = b} ∷ outIBs s})","searchableContent":"  sendib : ∀ {s h b}    → simpleffd s (ffdabstract.send (ibheader h) (just (ibbody b))) ffdabstract.sendres (record s { outibs = record {header = h; body = b} ∷ outibs s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":153,"title":"SendEB ","content":"SendEB : ∀ {s eb}     → SimpleFFD s (FFDAbstract.Send (ebHeader eb) nothing) FFDAbstract.SendRes (record s { outEBs = eb ∷ outEBs s})","searchableContent":"  sendeb : ∀ {s eb}     → simpleffd s (ffdabstract.send (ebheader eb) nothing) ffdabstract.sendres (record s { outebs = eb ∷ outebs s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":154,"title":"SendVS ","content":"SendVS : ∀ {s vs}     → SimpleFFD s (FFDAbstract.Send (vtHeader vs) nothing) FFDAbstract.SendRes (record s { outVTs = vs ∷ outVTs s})","searchableContent":"  sendvs : ∀ {s vs}     → simpleffd s (ffdabstract.send (vtheader vs) nothing) ffdabstract.sendres (record s { outvts = vs ∷ outvts s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":156,"title":"BadSendIB ","content":"BadSendIB : ∀ {s h}   → SimpleFFD s (FFDAbstract.Send (ibHeader h) nothing) FFDAbstract.SendRes s","searchableContent":"  badsendib : ∀ {s h}   → simpleffd s (ffdabstract.send (ibheader h) nothing) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":157,"title":"BadSendEB ","content":"BadSendEB : ∀ {s h b} → SimpleFFD s (FFDAbstract.Send (ebHeader h) (just b)) FFDAbstract.SendRes s","searchableContent":"  badsendeb : ∀ {s h b} → simpleffd s (ffdabstract.send (ebheader h) (just b)) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":158,"title":"BadSendVS ","content":"BadSendVS : ∀ {s h b} → SimpleFFD s (FFDAbstract.Send (vtHeader h) (just b)) FFDAbstract.SendRes s","searchableContent":"  badsendvs : ∀ {s h b} → simpleffd s (ffdabstract.send (vtheader h) (just b)) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":160,"title":"Fetch ","content":"Fetch : ∀ {s}         → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes (flushIns s)) (record s { inIBs = [] ; inEBs = [] ; inVTs = [] })","searchableContent":"  fetch : ∀ {s}         → simpleffd s ffdabstract.fetch (ffdabstract.fetchres (flushins s)) (record s { inibs = [] ; inebs = [] ; invts = [] })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":162,"title":"send-total ","content":"send-total : ∀ {s h b} → ∃[ s' ] (SimpleFFD s (FFDAbstract.Send h b) FFDAbstract.SendRes s')","searchableContent":"send-total : ∀ {s h b} → ∃[ s' ] (simpleffd s (ffdabstract.send h b) ffdabstract.sendres s')"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":163,"title":"send-total {s} {ibHeader h} {just (ibBody b)} = record s { outIBs = record {header = h; body = b} ∷ outIBs s} , SendIB","searchableContent":"send-total {s} {ibheader h} {just (ibbody b)} = record s { outibs = record {header = h; body = b} ∷ outibs s} , sendib"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":164,"title":"send-total {s} {ebHeader eb} {nothing}        = record s { outEBs = eb ∷ outEBs s} , SendEB","searchableContent":"send-total {s} {ebheader eb} {nothing}        = record s { outebs = eb ∷ outebs s} , sendeb"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":165,"title":"send-total {s} {vtHeader vs} {nothing}        = record s { outVTs = vs ∷ outVTs s} , SendVS","searchableContent":"send-total {s} {vtheader vs} {nothing}        = record s { outvts = vs ∷ outvts s} , sendvs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":167,"title":"send-total {s} {ibHeader h} {nothing} = s , BadSendIB","searchableContent":"send-total {s} {ibheader h} {nothing} = s , badsendib"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":168,"title":"send-total {s} {ebHeader eb} {just _} = s , BadSendEB","searchableContent":"send-total {s} {ebheader eb} {just _} = s , badsendeb"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":169,"title":"send-total {s} {vtHeader vs} {just _} = s , BadSendVS","searchableContent":"send-total {s} {vtheader vs} {just _} = s , badsendvs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":171,"title":"fetch-total ","content":"fetch-total : ∀ {s} → ∃[ x ] (∃[ s' ] (SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes x) s'))","searchableContent":"fetch-total : ∀ {s} → ∃[ x ] (∃[ s' ] (simpleffd s ffdabstract.fetch (ffdabstract.fetchres x) s'))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":172,"title":"fetch-total {s} = flushIns s , (record s { inIBs = [] ; inEBs = [] ; inVTs = [] } , Fetch)","searchableContent":"fetch-total {s} = flushins s , (record s { inibs = [] ; inebs = [] ; invts = [] } , fetch)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":174,"title":"send-complete ","content":"send-complete : ∀ {s h b s'} → SimpleFFD s (FFDAbstract.Send h b) FFDAbstract.SendRes s' → s' ≡ proj₁ (send-total {s} {h} {b})","searchableContent":"send-complete : ∀ {s h b s'} → simpleffd s (ffdabstract.send h b) ffdabstract.sendres s' → s' ≡ proj₁ (send-total {s} {h} {b})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":175,"title":"send-complete SendIB    = refl","searchableContent":"send-complete sendib    = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":176,"title":"send-complete SendEB    = refl","searchableContent":"send-complete sendeb    = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":177,"title":"send-complete SendVS    = refl","searchableContent":"send-complete sendvs    = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":178,"title":"send-complete BadSendIB = refl","searchableContent":"send-complete badsendib = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":179,"title":"send-complete BadSendEB = refl","searchableContent":"send-complete badsendeb = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":180,"title":"send-complete BadSendVS = refl","searchableContent":"send-complete badsendvs = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":182,"title":"fetch-complete₁ ","content":"fetch-complete₁ : ∀ {s r s'} → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes r) s' → s' ≡ proj₁ (proj₂ (fetch-total {s}))","searchableContent":"fetch-complete₁ : ∀ {s r s'} → simpleffd s ffdabstract.fetch (ffdabstract.fetchres r) s' → s' ≡ proj₁ (proj₂ (fetch-total {s}))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":183,"title":"fetch-complete₁ Fetch = refl","searchableContent":"fetch-complete₁ fetch = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":185,"title":"fetch-complete₂ ","content":"fetch-complete₂ : ∀ {s r s'} → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes r) s' → r ≡ proj₁ (fetch-total {s})","searchableContent":"fetch-complete₂ : ∀ {s r s'} → simpleffd s ffdabstract.fetch (ffdabstract.fetchres r) s' → r ≡ proj₁ (fetch-total {s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":186,"title":"fetch-complete₂ Fetch = refl","searchableContent":"fetch-complete₂ fetch = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":188,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":189,"title":"Dec-SimpleFFD ","content":"Dec-SimpleFFD : ∀ {s i o s'} → SimpleFFD s i o s' ⁇","searchableContent":"  dec-simpleffd : ∀ {s i o s'} → simpleffd s i o s' ⁇"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":190,"title":"Dec-SimpleFFD {s} {FFDAbstract.Send h b} {FFDAbstract.SendRes} {s'} with s' ≟ proj₁ (send-total {s} {h} {b})","searchableContent":"  dec-simpleffd {s} {ffdabstract.send h b} {ffdabstract.sendres} {s'} with s' ≟ proj₁ (send-total {s} {h} {b})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":191,"title":"... | yes p rewrite p = ⁇ yes (proj₂ send-total)","searchableContent":"  ... | yes p rewrite p = ⁇ yes (proj₂ send-total)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":192,"title":"... | no ¬p = ⁇ no λ x → ⊥-elim (¬p (send-complete x))","searchableContent":"  ... | no ¬p = ⁇ no λ x → ⊥-elim (¬p (send-complete x))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":193,"title":"Dec-SimpleFFD {_} {FFDAbstract.Send _ _} {FFDAbstract.FetchRes _} {_} = ⁇ no λ ()","searchableContent":"  dec-simpleffd {_} {ffdabstract.send _ _} {ffdabstract.fetchres _} {_} = ⁇ no λ ()"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":194,"title":"Dec-SimpleFFD {s} {FFDAbstract.Fetch} {FFDAbstract.FetchRes r} {s'}","searchableContent":"  dec-simpleffd {s} {ffdabstract.fetch} {ffdabstract.fetchres r} {s'}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":195,"title":"with s' ≟ proj₁ (proj₂ (fetch-total {s}))","searchableContent":"    with s' ≟ proj₁ (proj₂ (fetch-total {s}))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":196,"title":"| r ≟ proj₁ (fetch-total {s})","searchableContent":"      | r ≟ proj₁ (fetch-total {s})"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":197,"title":"... | yes p | yes q rewrite p rewrite q = ⁇ yes (proj₂ (proj₂ (fetch-total {s})))","searchableContent":"  ... | yes p | yes q rewrite p rewrite q = ⁇ yes (proj₂ (proj₂ (fetch-total {s})))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":198,"title":"... | yes p | no ¬q = ⁇ no λ x → ⊥-elim (¬q (fetch-complete₂ x))","searchableContent":"  ... | yes p | no ¬q = ⁇ no λ x → ⊥-elim (¬q (fetch-complete₂ x))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":199,"title":"... | no ¬p | _ = ⁇ no λ x → ⊥-elim (¬p (fetch-complete₁ x))","searchableContent":"  ... | no ¬p | _ = ⁇ no λ x → ⊥-elim (¬p (fetch-complete₁ x))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":200,"title":"Dec-SimpleFFD {_} {FFDAbstract.Fetch} {FFDAbstract.SendRes} {_} = ⁇ no λ ()","searchableContent":"  dec-simpleffd {_} {ffdabstract.fetch} {ffdabstract.sendres} {_} = ⁇ no λ ()"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":202,"title":"d-FFDFunctionality ","content":"d-FFDFunctionality : FFDAbstract.Functionality ffdAbstract","searchableContent":"d-ffdfunctionality : ffdabstract.functionality ffdabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":203,"title":"d-FFDFunctionality =","searchableContent":"d-ffdfunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":204,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":205,"title":"{ State         = FFDBuffers","searchableContent":"    { state         = ffdbuffers"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":206,"title":"; initFFDState  = record { inIBs = []; inEBs = []; inVTs = []; outIBs = []; outEBs = []; outVTs = [] }","searchableContent":"    ; initffdstate  = record { inibs = []; inebs = []; invts = []; outibs = []; outebs = []; outvts = [] }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":207,"title":"; _-⟦_/_⟧⇀_     = SimpleFFD","searchableContent":"    ; _-⟦_/_⟧⇀_     = simpleffd"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":208,"title":"; Dec-_-⟦_/_⟧⇀_ = Dec-SimpleFFD","searchableContent":"    ; dec-_-⟦_/_⟧⇀_ = dec-simpleffd"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":209,"title":"; Send-total    = send-total","searchableContent":"    ; send-total    = send-total"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":210,"title":"; Fetch-total   = fetch-total","searchableContent":"    ; fetch-total   = fetch-total"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":211,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":213,"title":"open import Leios.Voting public","searchableContent":"open import leios.voting public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":215,"title":"d-VotingAbstract ","content":"d-VotingAbstract : VotingAbstract (Fin 1 × EndorserBlock)","searchableContent":"d-votingabstract : votingabstract (fin 1 × endorserblock)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":216,"title":"d-VotingAbstract =","searchableContent":"d-votingabstract ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":217,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":218,"title":"{ VotingState     = ⊤","searchableContent":"    { votingstate     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":219,"title":"; initVotingState = tt","searchableContent":"    ; initvotingstate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":220,"title":"; isVoteCertified = λ _ _ → ⊤","searchableContent":"    ; isvotecertified = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":221,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":223,"title":"d-VotingAbstract-2 ","content":"d-VotingAbstract-2 : VotingAbstract (Fin 2 × EndorserBlock)","searchableContent":"d-votingabstract-2 : votingabstract (fin 2 × endorserblock)"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":224,"title":"d-VotingAbstract-2 =","searchableContent":"d-votingabstract-2 ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":225,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":226,"title":"{ VotingState     = ⊤","searchableContent":"    { votingstate     = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":227,"title":"; initVotingState = tt","searchableContent":"    ; initvotingstate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":228,"title":"; isVoteCertified = λ _ _ → ⊤","searchableContent":"    ; isvotecertified = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":229,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":231,"title":"d-SpecStructure ","content":"d-SpecStructure : SpecStructure 1","searchableContent":"d-specstructure : specstructure 1"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":232,"title":"d-SpecStructure = record","searchableContent":"d-specstructure = record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":233,"title":"{ a                         = d-Abstract","searchableContent":"      { a                         = d-abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":234,"title":"; Hashable-PreIBHeader      = hhs","searchableContent":"      ; hashable-preibheader      = hhs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":235,"title":"; Hashable-PreEndorserBlock = hpe","searchableContent":"      ; hashable-preendorserblock = hpe"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":236,"title":"; id                        = sutId","searchableContent":"      ; id                        = sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":237,"title":"; FFD'                      = d-FFDFunctionality","searchableContent":"      ; ffd'                      = d-ffdfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":238,"title":"; vrf'                      = d-VRF","searchableContent":"      ; vrf'                      = d-vrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":239,"title":"; sk-IB                     = IB , tt","searchableContent":"      ; sk-ib                     = ib , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":240,"title":"; sk-EB                     = EB , tt","searchableContent":"      ; sk-eb                     = eb , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":241,"title":"; sk-VT                     = VT , tt","searchableContent":"      ; sk-vt                     = vt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":242,"title":"; pk-IB                     = sutId , tt","searchableContent":"      ; pk-ib                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":243,"title":"; pk-EB                     = sutId , tt","searchableContent":"      ; pk-eb                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":244,"title":"; pk-VT                     = sutId , tt","searchableContent":"      ; pk-vt                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":245,"title":"; B'                        = d-Base","searchableContent":"      ; b'                        = d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":246,"title":"; BF                        = d-BaseFunctionality","searchableContent":"      ; bf                        = d-basefunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":247,"title":"; initBaseState             = tt","searchableContent":"      ; initbasestate             = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":248,"title":"; K'                        = d-KeyRegistration","searchableContent":"      ; k'                        = d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":249,"title":"; KF                        = d-KeyRegistrationFunctionality","searchableContent":"      ; kf                        = d-keyregistrationfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":250,"title":"; va                        = d-VotingAbstract","searchableContent":"      ; va                        = d-votingabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":251,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":253,"title":"d-SpecStructure-2 ","content":"d-SpecStructure-2 : SpecStructure 2","searchableContent":"d-specstructure-2 : specstructure 2"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":254,"title":"d-SpecStructure-2 = record","searchableContent":"d-specstructure-2 = record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":255,"title":"{ a                         = d-Abstract","searchableContent":"      { a                         = d-abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":256,"title":"; Hashable-PreIBHeader      = hhs","searchableContent":"      ; hashable-preibheader      = hhs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":257,"title":"; Hashable-PreEndorserBlock = hpe","searchableContent":"      ; hashable-preendorserblock = hpe"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":258,"title":"; id                        = sutId","searchableContent":"      ; id                        = sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":259,"title":"; FFD'                      = d-FFDFunctionality","searchableContent":"      ; ffd'                      = d-ffdfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":260,"title":"; vrf'                      = d-VRF","searchableContent":"      ; vrf'                      = d-vrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":261,"title":"; sk-IB                     = IB , tt","searchableContent":"      ; sk-ib                     = ib , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":262,"title":"; sk-EB                     = EB , tt","searchableContent":"      ; sk-eb                     = eb , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":263,"title":"; sk-VT                     = VT , tt","searchableContent":"      ; sk-vt                     = vt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":264,"title":"; pk-IB                     = sutId , tt","searchableContent":"      ; pk-ib                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":265,"title":"; pk-EB                     = sutId , tt","searchableContent":"      ; pk-eb                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":266,"title":"; pk-VT                     = sutId , tt","searchableContent":"      ; pk-vt                     = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":267,"title":"; B'                        = d-Base","searchableContent":"      ; b'                        = d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":268,"title":"; BF                        = d-BaseFunctionality","searchableContent":"      ; bf                        = d-basefunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":269,"title":"; initBaseState             = tt","searchableContent":"      ; initbasestate             = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":270,"title":"; K'                        = d-KeyRegistration","searchableContent":"      ; k'                        = d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":271,"title":"; KF                        = d-KeyRegistrationFunctionality","searchableContent":"      ; kf                        = d-keyregistrationfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":272,"title":"; va                        = d-VotingAbstract-2","searchableContent":"      ; va                        = d-votingabstract-2"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":273,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":275,"title":"open import Leios.Short d-SpecStructure public","searchableContent":"open import leios.short d-specstructure public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":276,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":2,"title":"Leios.FFD
{-# OPTIONS --safe #-}","searchableContent":"leios.ffd
{-# options --safe #-}"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":4,"title":"module Leios.FFD where","searchableContent":"module leios.ffd where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":6,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":8,"title":"record FFDAbstract ","content":"record FFDAbstract : Type₁ where","searchableContent":"record ffdabstract : type₁ where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":9,"title":"field Header Body ID ","content":"field Header Body ID : Type","searchableContent":"  field header body id : type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":10,"title":"⦃ DecEq-Header ⦄ ","content":"⦃ DecEq-Header ⦄ : DecEq Header","searchableContent":"        ⦃ deceq-header ⦄ : deceq header"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":11,"title":"⦃ DecEq-Body ⦄ ","content":"⦃ DecEq-Body ⦄ : DecEq Body","searchableContent":"        ⦃ deceq-body ⦄ : deceq body"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":12,"title":"match ","content":"match : Header → Body → Type","searchableContent":"        match : header → body → type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":13,"title":"msgID ","content":"msgID : Header → ID","searchableContent":"        msgid : header → id"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":15,"title":"data Input ","content":"data Input : Type where","searchableContent":"  data input : type where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":16,"title":"Send  ","content":"Send  : Header → Maybe Body → Input","searchableContent":"    send  : header → maybe body → input"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":17,"title":"Fetch ","content":"Fetch : Input","searchableContent":"    fetch : input"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":19,"title":"data Output ","content":"data Output : Type where","searchableContent":"  data output : type where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":20,"title":"SendRes  ","content":"SendRes  : Output","searchableContent":"    sendres  : output"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":21,"title":"FetchRes ","content":"FetchRes : List (Header ⊎ Body) → Output","searchableContent":"    fetchres : list (header ⊎ body) → output"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":23,"title":"record Functionality ","content":"record Functionality : Type₁ where","searchableContent":"  record functionality : type₁ where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":24,"title":"field State ","content":"field State : Type","searchableContent":"    field state : type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":25,"title":"initFFDState ","content":"initFFDState : State","searchableContent":"          initffdstate : state"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":26,"title":"_-⟦_/_⟧⇀_ ","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"          _-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":27,"title":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ ","content":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : {s : State} → {i : Input} → {o : Output} → {s' : State} → (s -⟦ i / o ⟧⇀ s') ⁇","searchableContent":"          ⦃ dec-_-⟦_/_⟧⇀_ ⦄ : {s : state} → {i : input} → {o : output} → {s' : state} → (s -⟦ i / o ⟧⇀ s') ⁇"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":28,"title":"Send-total ","content":"Send-total : ∀ {ffds h b} → ∃[ ffds' ] ffds -⟦ Send h b / SendRes ⟧⇀ ffds'","searchableContent":"          send-total : ∀ {ffds h b} → ∃[ ffds' ] ffds -⟦ send h b / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":29,"title":"Fetch-total ","content":"Fetch-total : ∀ {ffds} → ∃[ r ] (∃[ ffds' ] (ffds -⟦ Fetch / FetchRes r ⟧⇀ ffds'))","searchableContent":"          fetch-total : ∀ {ffds} → ∃[ r ] (∃[ ffds' ] (ffds -⟦ fetch / fetchres r ⟧⇀ ffds'))"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":31,"title":"open Input public","searchableContent":"    open input public"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":32,"title":"open Output public","searchableContent":"    open output public"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":33,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.BaseTypes
module Leios.Foreign.BaseTypes where","searchableContent":"leios.foreign.basetypes
module leios.foreign.basetypes where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":4,"title":"-- TODO","content":"-- TODO: copied from the formal-ledger project for now","searchableContent":"-- todo: copied from the formal-ledger project for now"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":5,"title":"-- Added","content":"-- Added: * TotalMap","searchableContent":"-- added: * totalmap"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":7,"title":"open import Data.Rational","searchableContent":"open import data.rational"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":11,"title":"open import Data.Fin","searchableContent":"open import data.fin"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":12,"title":"open import Function.Related.TypeIsomorphisms","searchableContent":"open import function.related.typeisomorphisms"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":13,"title":"open import Relation.Binary.Structures","searchableContent":"open import relation.binary.structures"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":15,"title":"open import Tactic.Derive.Convertible","searchableContent":"open import tactic.derive.convertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":16,"title":"open import Tactic.Derive.HsType","searchableContent":"open import tactic.derive.hstype"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":18,"title":"open import Class.Convertible","searchableContent":"open import class.convertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":19,"title":"open import Class.Decidable.Instances","searchableContent":"open import class.decidable.instances"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":20,"title":"open import Class.HasHsType","searchableContent":"open import class.hashstype"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":22,"title":"open import Leios.Foreign.HsTypes as F","searchableContent":"open import leios.foreign.hstypes as f"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":23,"title":"open import Leios.Foreign.Util","searchableContent":"open import leios.foreign.util"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":24,"title":"open import Foreign.Haskell","searchableContent":"open import foreign.haskell"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":26,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":27,"title":"iConvTop    = Convertible-Refl {⊤}","searchableContent":"  iconvtop    = convertible-refl {⊤}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":28,"title":"iConvNat    = Convertible-Refl {ℕ}","searchableContent":"  iconvnat    = convertible-refl {ℕ}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":29,"title":"iConvString = Convertible-Refl {String}","searchableContent":"  iconvstring = convertible-refl {string}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":30,"title":"iConvBool   = Convertible-Refl {Bool}","searchableContent":"  iconvbool   = convertible-refl {bool}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":32,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":34,"title":"-- * Unit and empty","searchableContent":"  -- * unit and empty"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":36,"title":"HsTy-⊥ = MkHsType ⊥ F.Empty","searchableContent":"  hsty-⊥ = mkhstype ⊥ f.empty"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":37,"title":"Conv-⊥ = autoConvert ⊥","searchableContent":"  conv-⊥ = autoconvert ⊥"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":39,"title":"HsTy-⊤ = MkHsType ⊤ ⊤","searchableContent":"  hsty-⊤ = mkhstype ⊤ ⊤"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":41,"title":"-- * Rational numbers","searchableContent":"  -- * rational numbers"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":43,"title":"HsTy-Rational = MkHsType ℚ F.Rational","searchableContent":"  hsty-rational = mkhstype ℚ f.rational"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":44,"title":"Conv-Rational ","content":"Conv-Rational : HsConvertible ℚ","searchableContent":"  conv-rational : hsconvertible ℚ"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":45,"title":"Conv-Rational = λ where","searchableContent":"  conv-rational = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":46,"title":".to (mkℚ n d _)       → n F., suc d","searchableContent":"    .to (mkℚ n d _)       → n f., suc d"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":47,"title":".from (n F., zero)    → 0ℚ -- TODO is there a safer way to do this?","searchableContent":"    .from (n f., zero)    → 0ℚ -- todo is there a safer way to do this?"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":48,"title":".from (n F., (suc d)) → n Data.Rational./ suc d","searchableContent":"    .from (n f., (suc d)) → n data.rational./ suc d"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":50,"title":"-- * Maps and Sets","searchableContent":"  -- * maps and sets"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":52,"title":"HsTy-HSSet ","content":"HsTy-HSSet : ∀ {A} → ⦃ HasHsType A ⦄ → HasHsType (ℙ A)","searchableContent":"  hsty-hsset : ∀ {a} → ⦃ hashstype a ⦄ → hashstype (ℙ a)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":53,"title":"HsTy-HSSet {A} = MkHsType _ (F.HSSet (HsType A))","searchableContent":"  hsty-hsset {a} = mkhstype _ (f.hsset (hstype a))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":55,"title":"Conv-HSSet ","content":"Conv-HSSet : ∀ {A} ⦃ _ : HasHsType A ⦄","searchableContent":"  conv-hsset : ∀ {a} ⦃ _ : hashstype a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":56,"title":"→ ⦃ HsConvertible A ⦄","searchableContent":"             → ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":57,"title":"→ HsConvertible (ℙ A)","searchableContent":"             → hsconvertible (ℙ a)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":58,"title":"Conv-HSSet = λ where","searchableContent":"  conv-hsset = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":59,"title":".to   → F.MkHSSet ∘ to ∘ setToList","searchableContent":"    .to   → f.mkhsset ∘ to ∘ settolist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":60,"title":".from → fromList ∘ from ∘ F.HSSet.elems","searchableContent":"    .from → fromlist ∘ from ∘ f.hsset.elems"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":62,"title":"Convertible-FinSet ","content":"Convertible-FinSet : Convertible₁ ℙ_ List","searchableContent":"  convertible-finset : convertible₁ ℙ_ list"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":63,"title":"Convertible-FinSet = λ where","searchableContent":"  convertible-finset = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":64,"title":".to   → map to   ∘ setToList","searchableContent":"    .to   → map to   ∘ settolist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":65,"title":".from → fromList ∘ map from","searchableContent":"    .from → fromlist ∘ map from"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":67,"title":"Convertible-Map ","content":"Convertible-Map : ∀ {K K' V V'} → ⦃ DecEq K ⦄","searchableContent":"  convertible-map : ∀ {k k' v v'} → ⦃ deceq k ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":68,"title":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","searchableContent":"    → ⦃ convertible k k' ⦄ → ⦃ convertible v v' ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":69,"title":"→ Convertible (K ⇀ V) (List $ Pair K' V')","searchableContent":"    → convertible (k ⇀ v) (list $ pair k' v')"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":70,"title":"Convertible-Map = λ where","searchableContent":"  convertible-map = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":71,"title":".to   → to        ∘ proj₁","searchableContent":"    .to   → to        ∘ proj₁"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":72,"title":".from → fromListᵐ ∘ map from","searchableContent":"    .from → fromlistᵐ ∘ map from"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":74,"title":"HsTy-Map ","content":"HsTy-Map : ∀ {A B} → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (A ⇀ B)","searchableContent":"  hsty-map : ∀ {a b} → ⦃ hashstype a ⦄ → ⦃ hashstype b ⦄ → hashstype (a ⇀ b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":75,"title":"HsTy-Map {A} {B} = MkHsType _ (F.HSMap (HsType A) (HsType B))","searchableContent":"  hsty-map {a} {b} = mkhstype _ (f.hsmap (hstype a) (hstype b))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":77,"title":"Conv-HSMap ","content":"Conv-HSMap : ∀ {A B} ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄","searchableContent":"  conv-hsmap : ∀ {a b} ⦃ _ : hashstype a ⦄ ⦃ _ : hashstype b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":78,"title":"→ ⦃ DecEq A ⦄","searchableContent":"    → ⦃ deceq a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":79,"title":"→ ⦃ HsConvertible A ⦄","searchableContent":"    → ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":80,"title":"→ ⦃ HsConvertible B ⦄","searchableContent":"    → ⦃ hsconvertible b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":81,"title":"→ HsConvertible (A ⇀ B)","searchableContent":"    → hsconvertible (a ⇀ b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":82,"title":"Conv-HSMap = λ where","searchableContent":"  conv-hsmap = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":83,"title":".to   → F.MkHSMap ∘ to","searchableContent":"    .to   → f.mkhsmap ∘ to"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":84,"title":".from → from ∘ F.HSMap.assocList","searchableContent":"    .from → from ∘ f.hsmap.assoclist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":86,"title":"Convertible-TotalMap ","content":"Convertible-TotalMap : ∀ {K K' V V'} → ⦃ DecEq K ⦄ → ⦃ Listable K ⦄","searchableContent":"  convertible-totalmap : ∀ {k k' v v'} → ⦃ deceq k ⦄ → ⦃ listable k ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":87,"title":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","searchableContent":"    → ⦃ convertible k k' ⦄ → ⦃ convertible v v' ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":88,"title":"→ Convertible (TotalMap K V) (List $ Pair K' V')","searchableContent":"    → convertible (totalmap k v) (list $ pair k' v')"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":89,"title":"Convertible-TotalMap {K} = λ where","searchableContent":"  convertible-totalmap {k} = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":90,"title":".to   → to ∘ TotalMap.rel","searchableContent":"    .to   → to ∘ totalmap.rel"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":91,"title":".from → λ x →","searchableContent":"    .from → λ x →"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":92,"title":"let (r , l) = fromListᵐ (map from x)","searchableContent":"      let (r , l) = fromlistᵐ (map from x)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":93,"title":"in case (¿ total r ¿) of λ where","searchableContent":"      in case (¿ total r ¿) of λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":94,"title":"(yes p) → record { rel = r ; left-unique-rel = l ; total-rel = p }","searchableContent":"           (yes p) → record { rel = r ; left-unique-rel = l ; total-rel = p }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":95,"title":"(no p) → error "Expected total map"","searchableContent":"           (no p) → error "expected total map""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":97,"title":"HsTy-TotalMap ","content":"HsTy-TotalMap : ∀ {A B} → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (TotalMap A B)","searchableContent":"  hsty-totalmap : ∀ {a b} → ⦃ hashstype a ⦄ → ⦃ hashstype b ⦄ → hashstype (totalmap a b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":98,"title":"HsTy-TotalMap {A} {B} = MkHsType _ (F.HSMap (HsType A) (HsType B))","searchableContent":"  hsty-totalmap {a} {b} = mkhstype _ (f.hsmap (hstype a) (hstype b))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":100,"title":"Conv-HSTotalMap ","content":"Conv-HSTotalMap : ∀ {A B} ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄","searchableContent":"  conv-hstotalmap : ∀ {a b} ⦃ _ : hashstype a ⦄ ⦃ _ : hashstype b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":101,"title":"→ ⦃ DecEq A ⦄","searchableContent":"    → ⦃ deceq a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":102,"title":"→ ⦃ Listable A ⦄","searchableContent":"    → ⦃ listable a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":103,"title":"→ ⦃ HsConvertible A ⦄","searchableContent":"    → ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":104,"title":"→ ⦃ HsConvertible B ⦄","searchableContent":"    → ⦃ hsconvertible b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":105,"title":"→ HsConvertible (TotalMap A B)","searchableContent":"    → hsconvertible (totalmap a b)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":106,"title":"Conv-HSTotalMap = λ where","searchableContent":"  conv-hstotalmap = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":107,"title":".to   → MkHSMap ∘ to","searchableContent":"    .to   → mkhsmap ∘ to"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":108,"title":".from → from ∘ F.HSMap.assocList","searchableContent":"    .from → from ∘ f.hsmap.assoclist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":110,"title":"-- * ComputationResult","searchableContent":"  -- * computationresult"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":112,"title":"open import Class.Computational as C","searchableContent":"  open import class.computational as c"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":114,"title":"HsTy-ComputationResult ","content":"HsTy-ComputationResult : ∀ {l} {Err} {A : Type l}","searchableContent":"  hsty-computationresult : ∀ {l} {err} {a : type l}"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":115,"title":"→ ⦃ HasHsType Err ⦄ → ⦃ HasHsType A ⦄","searchableContent":"                           → ⦃ hashstype err ⦄ → ⦃ hashstype a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":116,"title":"→ HasHsType (C.ComputationResult Err A)","searchableContent":"                           → hashstype (c.computationresult err a)"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":117,"title":"HsTy-ComputationResult {Err = Err} {A} = MkHsType _ (F.ComputationResult (HsType Err) (HsType A))","searchableContent":"  hsty-computationresult {err = err} {a} = mkhstype _ (f.computationresult (hstype err) (hstype a))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":119,"title":"Conv-ComputationResult ","content":"Conv-ComputationResult : ConvertibleType C.ComputationResult F.ComputationResult","searchableContent":"  conv-computationresult : convertibletype c.computationresult f.computationresult"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":120,"title":"Conv-ComputationResult = autoConvertible","searchableContent":"  conv-computationresult = autoconvertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":121,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.HsTypes
module Leios.Foreign.HsTypes where","searchableContent":"leios.foreign.hstypes
module leios.foreign.hstypes where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":4,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":5,"title":"{-# LANGUAGE DeriveGeneric #-}","searchableContent":"  {-# language derivegeneric #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":6,"title":"{-# LANGUAGE DeriveFunctor #-}","searchableContent":"  {-# language derivefunctor #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":7,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":11,"title":"open import Foreign.Haskell","searchableContent":"open import foreign.haskell"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":12,"title":"open import Foreign.Haskell.Coerce","searchableContent":"open import foreign.haskell.coerce"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":13,"title":"open import Foreign.Haskell.Either","searchableContent":"open import foreign.haskell.either"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":14,"title":"open import Data.Rational.Base","searchableContent":"open import data.rational.base"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":16,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":17,"title":"import GHC.Generics (Generic)","searchableContent":"  import ghc.generics (generic)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":18,"title":"import Data.Void (Void)","searchableContent":"  import data.void (void)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":19,"title":"import Prelude hiding (Rational)","searchableContent":"  import prelude hiding (rational)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":20,"title":"import GHC.Real (Ratio(..))","searchableContent":"  import ghc.real (ratio(..))"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":21,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":23,"title":"-- * The empty type","searchableContent":"-- * the empty type"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":25,"title":"data Empty ","content":"data Empty : Type where","searchableContent":"data empty : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":26,"title":"{-# COMPILE GHC Empty = data Void () #-}","searchableContent":"{-# compile ghc empty = data void () #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":28,"title":"-- * Rational","searchableContent":"-- * rational"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":30,"title":"data Rational ","content":"data Rational : Type where","searchableContent":"data rational : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":31,"title":"_,_ ","content":"_,_ : ℤ → ℕ → Rational","searchableContent":"  _,_ : ℤ → ℕ → rational"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":32,"title":"{-# COMPILE GHC Rational = data Rational ((","content":"{-# COMPILE GHC Rational = data Rational ((:%)) #-}","searchableContent":"{-# compile ghc rational = data rational ((:%)) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":34,"title":"-- We'll generate code with qualified references to Rational in this","searchableContent":"-- we'll generate code with qualified references to rational in this"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":35,"title":"-- module, so make sure to define it.","searchableContent":"-- module, so make sure to define it."},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":36,"title":"{-# FOREIGN GHC type Rational = Ratio Integer #-}","searchableContent":"{-# foreign ghc type rational = ratio integer #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":38,"title":"-- * Maps and Sets","searchableContent":"-- * maps and sets"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":40,"title":"record HSMap K V ","content":"record HSMap K V : Type where","searchableContent":"record hsmap k v : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":41,"title":"constructor MkHSMap","searchableContent":"  constructor mkhsmap"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":42,"title":"field assocList ","content":"field assocList : List (Pair K V)","searchableContent":"  field assoclist : list (pair k v)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":44,"title":"record HSSet A ","content":"record HSSet A : Type where","searchableContent":"record hsset a : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":45,"title":"constructor MkHSSet","searchableContent":"  constructor mkhsset"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":46,"title":"field elems ","content":"field elems : List A","searchableContent":"  field elems : list a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":48,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":49,"title":"newtype HSMap k v = MkHSMap [(k, v)]","searchableContent":"  newtype hsmap k v = mkhsmap [(k, v)]"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":50,"title":"deriving (Generic, Show, Eq, Ord)","searchableContent":"    deriving (generic, show, eq, ord)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":51,"title":"newtype HSSet a = MkHSSet [a]","searchableContent":"  newtype hsset a = mkhsset [a]"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":52,"title":"deriving (Generic, Show, Eq, Ord)","searchableContent":"    deriving (generic, show, eq, ord)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":53,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":54,"title":"{-# COMPILE GHC HSMap = data HSMap (MkHSMap) #-}","searchableContent":"{-# compile ghc hsmap = data hsmap (mkhsmap) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":55,"title":"{-# COMPILE GHC HSSet = data HSSet (MkHSSet) #-}","searchableContent":"{-# compile ghc hsset = data hsset (mkhsset) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":57,"title":"-- * ComputationResult","searchableContent":"-- * computationresult"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":59,"title":"data ComputationResult E A ","content":"data ComputationResult E A : Type where","searchableContent":"data computationresult e a : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":60,"title":"Success ","content":"Success : A → ComputationResult E A","searchableContent":"  success : a → computationresult e a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":61,"title":"Failure ","content":"Failure : E → ComputationResult E A","searchableContent":"  failure : e → computationresult e a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":63,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":64,"title":"data ComputationResult e a = Success a | Failure e","searchableContent":"  data computationresult e a = success a | failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":65,"title":"deriving (Functor, Eq, Show, Generic)","searchableContent":"    deriving (functor, eq, show, generic)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":67,"title":"instance Applicative (ComputationResult e) where","searchableContent":"  instance applicative (computationresult e) where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":68,"title":"pure = Success","searchableContent":"    pure = success"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":69,"title":"(Success f) <*> x = f <$> x","searchableContent":"    (success f) <*> x = f <$> x"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":70,"title":"(Failure e) <*> _ = Failure e","searchableContent":"    (failure e) <*> _ = failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":72,"title":"instance Monad (ComputationResult e) where","searchableContent":"  instance monad (computationresult e) where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":73,"title":"return = pure","searchableContent":"    return = pure"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":74,"title":"(Success a) >>= m = m a","searchableContent":"    (success a) >>= m = m a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":75,"title":"(Failure e) >>= _ = Failure e","searchableContent":"    (failure e) >>= _ = failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":76,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":77,"title":"{-# COMPILE GHC ComputationResult = data ComputationResult (Success | Failure) #-}","searchableContent":"{-# compile ghc computationresult = data computationresult (success | failure) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":78,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.Types
open import Data.Char.Base as C using (Char)","searchableContent":"leios.foreign.types
open import data.char.base as c using (char)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":3,"title":"import Data.String as S","searchableContent":"import data.string as s"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":4,"title":"open import Data.Integer using (+_; ∣_∣)","searchableContent":"open import data.integer using (+_; ∣_∣)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":6,"title":"open import Class.Convertible","searchableContent":"open import class.convertible"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":7,"title":"open import Class.HasHsType","searchableContent":"open import class.hashstype"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":8,"title":"open import Tactic.Derive.Convertible","searchableContent":"open import tactic.derive.convertible"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":9,"title":"open import Tactic.Derive.HsType","searchableContent":"open import tactic.derive.hstype"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":11,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":13,"title":"open import Leios.Foreign.BaseTypes","searchableContent":"open import leios.foreign.basetypes"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":14,"title":"open import Leios.Foreign.HsTypes","searchableContent":"open import leios.foreign.hstypes"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":15,"title":"open import Leios.Foreign.Util","searchableContent":"open import leios.foreign.util"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":17,"title":"module Leios.Foreign.Types where","searchableContent":"module leios.foreign.types where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":19,"title":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":20,"title":"{-# LANGUAGE DuplicateRecordFields #-}","searchableContent":"  {-# language duplicaterecordfields #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":21,"title":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":23,"title":"-- TODO","content":"-- TODO: Get rid of hardcoded parameters in this module","searchableContent":"-- todo: get rid of hardcoded parameters in this module"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":25,"title":"{-","searchableContent":"{-"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":26,"title":"numberOfParties","content":"numberOfParties : ℕ","searchableContent":"numberofparties : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":27,"title":"numberOfParties","content":"numberOfParties = 2","searchableContent":"numberofparties = 2"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":29,"title":"open import Leios.Defaults numberOfParties fzero","content":"open import Leios.Defaults numberOfParties fzero","searchableContent":"open import leios.defaults numberofparties fzero"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":30,"title":"renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)","content":"renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)","searchableContent":"  renaming (endorserblock to endorserblockagda; ibheader to ibheaderagda)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":32,"title":"dropDash","content":"dropDash : S.String → S.String","searchableContent":"dropdash : s.string → s.string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":33,"title":"dropDash","content":"dropDash = S.concat ∘ S.wordsByᵇ ('-' C.≈ᵇ_)","searchableContent":"dropdash = s.concat ∘ s.wordsbyᵇ ('-' c.≈ᵇ_)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":35,"title":"prefix","content":"prefix : S.String → S.String → S.String","searchableContent":"prefix : s.string → s.string → s.string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":36,"title":"prefix","content":"prefix = S._++_","searchableContent":"prefix = s._++_"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":38,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":39,"title":"HsTy-SlotUpkeep","content":"HsTy-SlotUpkeep = autoHsType SlotUpkeep ⊣ onConstructors dropDash","searchableContent":"  hsty-slotupkeep = autohstype slotupkeep ⊣ onconstructors dropdash"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":40,"title":"Conv-SlotUpkeep","content":"Conv-SlotUpkeep = autoConvert SlotUpkeep","searchableContent":"  conv-slotupkeep = autoconvert slotupkeep"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":42,"title":"record IBHeader","content":"record IBHeader : Type where","searchableContent":"record ibheader : type where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":43,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":44,"title":"producerID","content":"producerID : ℕ","searchableContent":"        producerid : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":45,"title":"bodyHash","content":"bodyHash : List ℕ","searchableContent":"        bodyhash : list ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":47,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":48,"title":"data IBHeader = IBHeader {slotNumber","content":"data IBHeader = IBHeader {slotNumber :: Integer, producerID :: Integer, bodyHash :: Data.Text.Text }","searchableContent":"data ibheader = ibheader {slotnumber :: integer, producerid :: integer, bodyhash :: data.text.text }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":49,"title":"deriving (Show, Eq, Generic)","content":"deriving (Show, Eq, Generic)","searchableContent":"  deriving (show, eq, generic)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":50,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":52,"title":"{-# COMPILE GHC IBHeader","content":"{-# COMPILE GHC IBHeader = data IBHeader (IBHeader) #-}","searchableContent":"{-# compile ghc ibheader = data ibheader (ibheader) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":54,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":55,"title":"HsTy-IBHeader","content":"HsTy-IBHeader = MkHsType IBHeaderAgda IBHeader","searchableContent":"  hsty-ibheader = mkhstype ibheaderagda ibheader"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":57,"title":"Conv-IBHeader","content":"Conv-IBHeader : Convertible IBHeaderAgda IBHeader","searchableContent":"  conv-ibheader : convertible ibheaderagda ibheader"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":58,"title":"Conv-IBHeader","content":"Conv-IBHeader = record","searchableContent":"  conv-ibheader = record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":59,"title":"{ to","content":"{ to = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →","searchableContent":"    { to = λ (record { slotnumber = s ; producerid = p ; bodyhash = h }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":60,"title":"record { slotNumber","content":"record { slotNumber = s ; producerID = toℕ p ; bodyHash = h}","searchableContent":"        record { slotnumber = s ; producerid = toℕ p ; bodyhash = h}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":61,"title":"; from","content":"; from = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →","searchableContent":"    ; from = λ (record { slotnumber = s ; producerid = p ; bodyhash = h }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":62,"title":"case p <? numberOfParties of λ where","content":"case p <? numberOfParties of λ where","searchableContent":"        case p <? numberofparties of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":63,"title":"(yes q) → record { slotNumber","content":"(yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; bodyHash = h ; signature = tt }","searchableContent":"          (yes q) → record { slotnumber = s ; producerid = #_ p {numberofparties} {fromwitness q} ; lotterypf = tt ; bodyhash = h ; signature = tt }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":64,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"          (no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":65,"title":"}","content":"}","searchableContent":"    }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":67,"title":"HsTy-IBBody","content":"HsTy-IBBody = autoHsType IBBody","searchableContent":"  hsty-ibbody = autohstype ibbody"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":68,"title":"Conv-IBBody","content":"Conv-IBBody = autoConvert IBBody","searchableContent":"  conv-ibbody = autoconvert ibbody"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":70,"title":"HsTy-InputBlock","content":"HsTy-InputBlock = autoHsType InputBlock","searchableContent":"  hsty-inputblock = autohstype inputblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":71,"title":"Conv-InputBlock","content":"Conv-InputBlock = autoConvert InputBlock","searchableContent":"  conv-inputblock = autoconvert inputblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":73,"title":"Conv-ℕ","content":"Conv-ℕ : HsConvertible ℕ","searchableContent":"  conv-ℕ : hsconvertible ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":74,"title":"Conv-ℕ","content":"Conv-ℕ = Convertible-Refl","searchableContent":"  conv-ℕ = convertible-refl"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":76,"title":"record EndorserBlock","content":"record EndorserBlock : Type where","searchableContent":"record endorserblock : type where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":77,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"  field slotnumber : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":78,"title":"producerID","content":"producerID : ℕ","searchableContent":"        producerid : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":79,"title":"ibRefs","content":"ibRefs     : List (List IBRef)","searchableContent":"        ibrefs     : list (list ibref)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":81,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":82,"title":"data EndorserBlock = EndorserBlock { slotNumber","content":"data EndorserBlock = EndorserBlock { slotNumber :: Integer, producerID :: Integer, ibRefs :: [Data.Text.Text] }","searchableContent":"data endorserblock = endorserblock { slotnumber :: integer, producerid :: integer, ibrefs :: [data.text.text] }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":83,"title":"deriving (Show, Eq, Generic)","content":"deriving (Show, Eq, Generic)","searchableContent":"  deriving (show, eq, generic)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":84,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":86,"title":"{-# COMPILE GHC EndorserBlock","content":"{-# COMPILE GHC EndorserBlock = data EndorserBlock (EndorserBlock) #-}","searchableContent":"{-# compile ghc endorserblock = data endorserblock (endorserblock) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":88,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":89,"title":"HsTy-EndorserBlock","content":"HsTy-EndorserBlock = MkHsType EndorserBlockAgda EndorserBlock","searchableContent":"  hsty-endorserblock = mkhstype endorserblockagda endorserblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":91,"title":"Conv-EndorserBlock","content":"Conv-EndorserBlock : Convertible EndorserBlockAgda EndorserBlock","searchableContent":"  conv-endorserblock : convertible endorserblockagda endorserblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":92,"title":"Conv-EndorserBlock","content":"Conv-EndorserBlock =","searchableContent":"  conv-endorserblock ="},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":93,"title":"record","content":"record","searchableContent":"    record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":94,"title":"{ to","content":"{ to = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →","searchableContent":"      { to = λ (record { slotnumber = s ; producerid = p ; ibrefs = refs }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":95,"title":"record { slotNumber","content":"record { slotNumber = s ; producerID = toℕ p ; ibRefs = refs }","searchableContent":"          record { slotnumber = s ; producerid = toℕ p ; ibrefs = refs }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":96,"title":"; from","content":"; from = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →","searchableContent":"      ; from = λ (record { slotnumber = s ; producerid = p ; ibrefs = refs }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":97,"title":"case p <? numberOfParties of λ where","content":"case p <? numberOfParties of λ where","searchableContent":"        case p <? numberofparties of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":98,"title":"(yes q) → record { slotNumber","content":"(yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; signature = tt ; ibRefs = refs ; ebRefs = [] }","searchableContent":"          (yes q) → record { slotnumber = s ; producerid = #_ p {numberofparties} {fromwitness q} ; lotterypf = tt ; signature = tt ; ibrefs = refs ; ebrefs = [] }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":99,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"          (no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":100,"title":"}","content":"}","searchableContent":"      }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":102,"title":"HsTy-FFDState","content":"HsTy-FFDState = autoHsType FFDState","searchableContent":"  hsty-ffdstate = autohstype ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":103,"title":"Conv-FFDState","content":"Conv-FFDState = autoConvert FFDState","searchableContent":"  conv-ffdstate = autoconvert ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":105,"title":"HsTy-Fin","content":"HsTy-Fin : ∀ {n} → HasHsType (Fin n)","searchableContent":"  hsty-fin : ∀ {n} → hashstype (fin n)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":106,"title":"HsTy-Fin .HasHsType.HsType","content":"HsTy-Fin .HasHsType.HsType = ℕ","searchableContent":"  hsty-fin .hashstype.hstype = ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":108,"title":"Conv-Fin","content":"Conv-Fin : ∀ {n} → HsConvertible (Fin n)","searchableContent":"  conv-fin : ∀ {n} → hsconvertible (fin n)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":109,"title":"Conv-Fin {n}","content":"Conv-Fin {n} =","searchableContent":"  conv-fin {n} ="},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":110,"title":"record","content":"record","searchableContent":"    record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":111,"title":"{ to","content":"{ to = toℕ","searchableContent":"      { to = toℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":112,"title":"; from","content":"; from = λ m →","searchableContent":"      ; from = λ m →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":113,"title":"case m <? n of λ where","content":"case m <? n of λ where","searchableContent":"          case m <? n of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":114,"title":"(yes p) → #_ m {n} {fromWitness p}","content":"(yes p) → #_ m {n} {fromWitness p}","searchableContent":"            (yes p) → #_ m {n} {fromwitness p}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":115,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"            (no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":116,"title":"}","content":"}","searchableContent":"      }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":118,"title":"HsTy-LeiosState","content":"HsTy-LeiosState = autoHsType LeiosState","searchableContent":"  hsty-leiosstate = autohstype leiosstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":119,"title":"Conv-LeiosState","content":"Conv-LeiosState = autoConvert LeiosState","searchableContent":"  conv-leiosstate = autoconvert leiosstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":121,"title":"HsTy-LeiosInput","content":"HsTy-LeiosInput = autoHsType LeiosInput ⊣ onConstructors (prefix "I_" ∘ dropDash)","searchableContent":"  hsty-leiosinput = autohstype leiosinput ⊣ onconstructors (prefix "i_" ∘ dropdash)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":122,"title":"Conv-LeiosInput","content":"Conv-LeiosInput = autoConvert LeiosInput","searchableContent":"  conv-leiosinput = autoconvert leiosinput"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":124,"title":"HsTy-LeiosOutput","content":"HsTy-LeiosOutput = autoHsType LeiosOutput ⊣ onConstructors (prefix "O_" ∘ dropDash)","searchableContent":"  hsty-leiosoutput = autohstype leiosoutput ⊣ onconstructors (prefix "o_" ∘ dropdash)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":125,"title":"Conv-LeiosOutput","content":"Conv-LeiosOutput = autoConvert LeiosOutput","searchableContent":"  conv-leiosoutput = autoconvert leiosoutput"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":127,"title":"open import Class.Computational as C","content":"open import Class.Computational as C","searchableContent":"open import class.computational as c"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":128,"title":"open import Class.Computational22","content":"open import Class.Computational22","searchableContent":"open import class.computational22"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":130,"title":"open Computational22","content":"open Computational22","searchableContent":"open computational22"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":131,"title":"open BaseAbstract","content":"open BaseAbstract","searchableContent":"open baseabstract"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":132,"title":"open FFDAbstract","content":"open FFDAbstract","searchableContent":"open ffdabstract"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":134,"title":"open GenFFD.Header using (ibHeader; ebHeader; vtHeader)","content":"open GenFFD.Header using (ibHeader; ebHeader; vtHeader)","searchableContent":"open genffd.header using (ibheader; ebheader; vtheader)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":135,"title":"open GenFFD.Body using (ibBody)","content":"open GenFFD.Body using (ibBody)","searchableContent":"open genffd.body using (ibbody)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":136,"title":"open FFDState","content":"open FFDState","searchableContent":"open ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":138,"title":"open import Leios.Short.Deterministic d-SpecStructure public","content":"open import Leios.Short.Deterministic d-SpecStructure public","searchableContent":"open import leios.short.deterministic d-specstructure public"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":140,"title":"open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)","content":"open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)","searchableContent":"open funtot (completefin numberofparties) (maximalfin numberofparties)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":142,"title":"d-StakeDistribution","content":"d-StakeDistribution : TotalMap (Fin numberOfParties) ℕ","searchableContent":"d-stakedistribution : totalmap (fin numberofparties) ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":143,"title":"d-StakeDistribution","content":"d-StakeDistribution = Fun⇒TotalMap (const 100000000)","searchableContent":"d-stakedistribution = fun⇒totalmap (const 100000000)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":145,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":146,"title":"Computational-B","content":"Computational-B : Computational22 (BaseAbstract.Functionality._-⟦_/_⟧⇀_ d-BaseFunctionality) String","searchableContent":"  computational-b : computational22 (baseabstract.functionality._-⟦_/_⟧⇀_ d-basefunctionality) string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":147,"title":"Computational-B .computeProof s (INIT x)","content":"Computational-B .computeProof s (INIT x) = success ((STAKE d-StakeDistribution , tt) , tt)","searchableContent":"  computational-b .computeproof s (init x) = success ((stake d-stakedistribution , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":148,"title":"Computational-B .computeProof s (SUBMIT x)","content":"Computational-B .computeProof s (SUBMIT x) = success ((EMPTY , tt) , tt)","searchableContent":"  computational-b .computeproof s (submit x) = success ((empty , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":149,"title":"Computational-B .computeProof s FTCH-LDG","content":"Computational-B .computeProof s FTCH-LDG = success (((BASE-LDG []) , tt) , tt)","searchableContent":"  computational-b .computeproof s ftch-ldg = success (((base-ldg []) , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":150,"title":"Computational-B .completeness _ _ _ _ _ = {!!} -- TODO","content":"Computational-B .completeness _ _ _ _ _ = {!!} -- TODO: Completeness proof","searchableContent":"  computational-b .completeness _ _ _ _ _ = {!!} -- todo: completeness proof"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":152,"title":"Computational-FFD","content":"Computational-FFD : Computational22 (FFDAbstract.Functionality._-⟦_/_⟧⇀_ d-FFDFunctionality) String","searchableContent":"  computational-ffd : computational22 (ffdabstract.functionality._-⟦_/_⟧⇀_ d-ffdfunctionality) string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":153,"title":"Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b)))","content":"Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b))) = success ((SendRes , record s {outIBs = record {header = h; body = b} ∷ outIBs s}) , SendIB)","searchableContent":"  computational-ffd .computeproof s (send (ibheader h) (just (ibbody b))) = success ((sendres , record s {outibs = record {header = h; body = b} ∷ outibs s}) , sendib)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":154,"title":"Computational-FFD .computeProof s (Send (ebHeader h) nothing)","content":"Computational-FFD .computeProof s (Send (ebHeader h) nothing) = success ((SendRes , record s {outEBs = h ∷ outEBs s}) , SendEB)","searchableContent":"  computational-ffd .computeproof s (send (ebheader h) nothing) = success ((sendres , record s {outebs = h ∷ outebs s}) , sendeb)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":155,"title":"Computational-FFD .computeProof s (Send (vtHeader h) nothing)","content":"Computational-FFD .computeProof s (Send (vtHeader h) nothing) = success ((SendRes , record s {outVTs = h ∷ outVTs s}) , SendVS)","searchableContent":"  computational-ffd .computeproof s (send (vtheader h) nothing) = success ((sendres , record s {outvts = h ∷ outvts s}) , sendvs)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":156,"title":"Computational-FFD .computeProof s Fetch","content":"Computational-FFD .computeProof s Fetch = success ((FetchRes (flushIns s) , record s {inIBs = []; inEBs = []; inVTs = []}) , Fetch)","searchableContent":"  computational-ffd .computeproof s fetch = success ((fetchres (flushins s) , record s {inibs = []; inebs = []; invts = []}) , fetch)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":158,"title":"Computational-FFD .computeProof _ _","content":"Computational-FFD .computeProof _ _ = failure "FFD error"","searchableContent":"  computational-ffd .computeproof _ _ = failure "ffd error""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":159,"title":"Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO","content":"Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO:Completeness proof","searchableContent":"  computational-ffd .completeness _ _ _ _ _ = {!!} -- todo:completeness proof"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":161,"title":"stepHs","content":"stepHs : HsType (LeiosState → LeiosInput → C.ComputationResult String (LeiosOutput × LeiosState))","searchableContent":"stephs : hstype (leiosstate → leiosinput → c.computationresult string (leiosoutput × leiosstate))"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":162,"title":"stepHs","content":"stepHs = to (compute Computational--⟦/⟧⇀)","searchableContent":"stephs = to (compute computational--⟦/⟧⇀)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":164,"title":"{-# COMPILE GHC stepHs as step #-}","content":"{-# COMPILE GHC stepHs as step #-}","searchableContent":"{-# compile ghc stephs as step #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":165,"title":"-}","content":"-}","searchableContent":"-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":166,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.Util
module Leios.Foreign.Util where","searchableContent":"leios.foreign.util
module leios.foreign.util where"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":6,"title":"postulate","searchableContent":"postulate"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":7,"title":"error ","content":"error : {A : Set} → String → A","searchableContent":"  error : {a : set} → string → a"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":8,"title":"{-# FOREIGN GHC import Data.Text #-}","searchableContent":"{-# foreign ghc import data.text #-}"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":9,"title":"{-# COMPILE GHC error = \\ _ s -> error (unpack s) #-}","searchableContent":"{-# compile ghc error = \\ _ s -> error (unpack s) #-}"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":10,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":2,"title":"Leios.KeyRegistration
{-# OPTIONS --safe #-}","searchableContent":"leios.keyregistration
{-# options --safe #-}"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":6,"title":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":8,"title":"module Leios.KeyRegistration (a ","content":"module Leios.KeyRegistration (a : LeiosAbstract) (open LeiosAbstract a)","searchableContent":"module leios.keyregistration (a : leiosabstract) (open leiosabstract a)"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":9,"title":"(vrf ","content":"(vrf : LeiosVRF a) (let open LeiosVRF vrf) where","searchableContent":"  (vrf : leiosvrf a) (let open leiosvrf vrf) where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":11,"title":"record KeyRegistrationAbstract ","content":"record KeyRegistrationAbstract : Type₁ where","searchableContent":"record keyregistrationabstract : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":13,"title":"data Input ","content":"data Input : Type₁ where","searchableContent":"  data input : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":14,"title":"INIT ","content":"INIT : PubKey → PubKey → PubKey → Input","searchableContent":"    init : pubkey → pubkey → pubkey → input"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":16,"title":"data Output ","content":"data Output : Type where","searchableContent":"  data output : type where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":17,"title":"PUBKEYS ","content":"PUBKEYS : List PubKey → Output","searchableContent":"    pubkeys : list pubkey → output"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":19,"title":"record Functionality ","content":"record Functionality : Type₁ where","searchableContent":"  record functionality : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":20,"title":"field State ","content":"field State : Type","searchableContent":"    field state : type"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":21,"title":"_-⟦_/_⟧⇀_ ","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"          _-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":23,"title":"open Input public","searchableContent":"    open input public"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":24,"title":"open Output public","searchableContent":"    open output public"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":25,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":2,"title":"Leios.Network
module Leios.Network where","searchableContent":"leios.network
module leios.network where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":4,"title":"open import abstract-set-theory.Prelude hiding (_∘_; _⊗_)","searchableContent":"open import abstract-set-theory.prelude hiding (_∘_; _⊗_)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":5,"title":"open import abstract-set-theory.FiniteSetTheory using (ℙ_; _∈_; _∪_; ❴_❵; _∉_)","searchableContent":"open import abstract-set-theory.finitesettheory using (ℙ_; _∈_; _∪_; ❴_❵; _∉_)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":7,"title":"open import CategoricalCrypto","searchableContent":"open import categoricalcrypto"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":9,"title":"record Abstract ","content":"record Abstract : Set₁ where","searchableContent":"record abstract : set₁ where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":10,"title":"field Header Body ID ","content":"field Header Body ID : Set","searchableContent":"  field header body id : set"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":11,"title":"match ","content":"match : Header → Body → Set","searchableContent":"        match : header → body → set"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":12,"title":"msgID ","content":"msgID : Header → ID","searchableContent":"        msgid : header → id"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":14,"title":"module Broadcast (M Peer ","content":"module Broadcast (M Peer : Set) where","searchableContent":"module broadcast (m peer : set) where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":15,"title":"open Channel","searchableContent":"  open channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":17,"title":"C ","content":"C : Channel","searchableContent":"  c : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":18,"title":"C .P = Peer","searchableContent":"  c .p = peer"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":19,"title":"C .rcvType _ = Peer × M","searchableContent":"  c .rcvtype _ = peer × m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":20,"title":"C .sndType _ = M","searchableContent":"  c .sndtype _ = m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":22,"title":"postulate Functionality ","content":"postulate Functionality : Machine I C","searchableContent":"  postulate functionality : machine i c"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":24,"title":"Single ","content":"Single : Channel","searchableContent":"  single : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":25,"title":"Single .P = ⊤","searchableContent":"  single .p = ⊤"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":26,"title":"Single .rcvType _ = Peer × M","searchableContent":"  single .rcvtype _ = peer × m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":27,"title":"Single .sndType _ = M","searchableContent":"  single .sndtype _ = m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":29,"title":"postulate SingleFunctionality ","content":"postulate SingleFunctionality : Machine I Single","searchableContent":"  postulate singlefunctionality : machine i single"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":31,"title":"-- connectWithBroadcast","content":"-- connectWithBroadcast : ∀ {A} → (Peer → Machine Single A) → Machine I A","searchableContent":"  -- connectwithbroadcast : ∀ {a} → (peer → machine single a) → machine i a"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":32,"title":"-- connectWithBroadcast = {!!}","searchableContent":"  -- connectwithbroadcast = {!!}"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":34,"title":"module HeaderDiffusion (a ","content":"module HeaderDiffusion (a : Abstract) (Peer : Set) (self : Peer) where","searchableContent":"module headerdiffusion (a : abstract) (peer : set) (self : peer) where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":35,"title":"open Channel","searchableContent":"  open channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":36,"title":"open Abstract a","searchableContent":"  open abstract a"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":37,"title":"module B = Broadcast Header Peer","searchableContent":"  module b = broadcast header peer"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":39,"title":"data Port ","content":"data Port : Set where","searchableContent":"  data port : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":40,"title":"Send    ","content":"Send    : Port -- we want to send a header","searchableContent":"    send    : port -- we want to send a header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":41,"title":"Forward ","content":"Forward : Port -- we want to forward a header","searchableContent":"    forward : port -- we want to forward a header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":43,"title":"C ","content":"C : Channel","searchableContent":"  c : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":44,"title":"C .P = Port","searchableContent":"  c .p = port"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":45,"title":"C .sndType _ = Header","searchableContent":"  c .sndtype _ = header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":46,"title":"C .rcvType Forward = Header","searchableContent":"  c .rcvtype forward = header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":47,"title":"C .rcvType Send = ⊥","searchableContent":"  c .rcvtype send = ⊥"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":49,"title":"data Input ","content":"data Input : Set where","searchableContent":"  data input : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":50,"title":"S ","content":"S : Header → Input","searchableContent":"    s : header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":51,"title":"F ","content":"F : Header → Input","searchableContent":"    f : header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":52,"title":"R ","content":"R : Peer → Header → Input","searchableContent":"    r : peer → header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":54,"title":"data Output ","content":"data Output : Set where","searchableContent":"  data output : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":55,"title":"Verify ","content":"Verify : Header → Output","searchableContent":"    verify : header → output"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":57,"title":"private variable","searchableContent":"  private variable"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":58,"title":"h ","content":"h : Header","searchableContent":"    h : header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":59,"title":"s ","content":"s : ℙ ID","searchableContent":"    s : ℙ id"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":61,"title":"data Step ","content":"data Step : ∃ (rcvType (B.Single ⊗ C ᵀ)) → ℙ ID → ℙ ID × Maybe (∃ (sndType (B.Single ⊗ C ᵀ))) → Set where","searchableContent":"  data step : ∃ (rcvtype (b.single ⊗ c ᵀ)) → ℙ id → ℙ id × maybe (∃ (sndtype (b.single ⊗ c ᵀ))) → set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":62,"title":"Init ","content":"Init : Step (inj₂ Send , h) s (s ∪ ❴ msgID h ❵ , just (inj₁ _ , h))","searchableContent":"    init : step (inj₂ send , h) s (s ∪ ❴ msgid h ❵ , just (inj₁ _ , h))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":63,"title":"Receive1  ","content":"Receive1  : ∀ {p} → Step (inj₁ _ , p , h) s (s , just (inj₂ Forward , h))","searchableContent":"    receive1  : ∀ {p} → step (inj₁ _ , p , h) s (s , just (inj₂ forward , h))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":64,"title":"Receive2  ","content":"Receive2  : msgID h ∉ s → Step (inj₂ Forward , h) s (s ∪ ❴ msgID h ❵ , just (inj₁ _ , h))","searchableContent":"    receive2  : msgid h ∉ s → step (inj₂ forward , h) s (s ∪ ❴ msgid h ❵ , just (inj₁ _ , h))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":65,"title":"Receive2' ","content":"Receive2' : msgID h ∈ s → Step (inj₂ Forward , h) s (s , nothing)","searchableContent":"    receive2' : msgid h ∈ s → step (inj₂ forward , h) s (s , nothing)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":67,"title":"step ","content":"step : ∃ (rcvType (B.Single ⊗ C ᵀ)) → ∃ (sndType (B.Single ⊗ C ᵀ))","searchableContent":"  step : ∃ (rcvtype (b.single ⊗ c ᵀ)) → ∃ (sndtype (b.single ⊗ c ᵀ))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":68,"title":"step (inj₁ _   , _ , h) = (inj₂ Forward , h)","searchableContent":"  step (inj₁ _   , _ , h) = (inj₂ forward , h)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":69,"title":"step (inj₂ Forward , h) = (inj₁ _ , h)","searchableContent":"  step (inj₂ forward , h) = (inj₁ _ , h)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":70,"title":"step (inj₂ Send    , h) = (inj₁ _ , h)","searchableContent":"  step (inj₂ send    , h) = (inj₁ _ , h)"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":72,"title":"Functionality ","content":"Functionality : Machine I C","searchableContent":"  functionality : machine i c"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":73,"title":"Functionality = MkMachine' Step ∘ B.SingleFunctionality","searchableContent":"  functionality = mkmachine' step ∘ b.singlefunctionality"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":74,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":2,"title":"Leios.Prelude
{-# OPTIONS --safe #-}","searchableContent":"leios.prelude
{-# options --safe #-}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":4,"title":"module Leios.Prelude where","searchableContent":"module leios.prelude where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":6,"title":"open import abstract-set-theory.FiniteSetTheory public","searchableContent":"open import abstract-set-theory.finitesettheory public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":7,"title":"open import abstract-set-theory.Prelude public","searchableContent":"open import abstract-set-theory.prelude public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":8,"title":"open import Data.List using (upTo)","searchableContent":"open import data.list using (upto)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":10,"title":"open import Class.HasAdd public","searchableContent":"open import class.hasadd public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":11,"title":"open import Class.HasOrder public","searchableContent":"open import class.hasorder public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":12,"title":"open import Class.Hashable public","searchableContent":"open import class.hashable public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":13,"title":"open import Prelude.InferenceRules public","searchableContent":"open import prelude.inferencerules public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":15,"title":"module T where","searchableContent":"module t where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":16,"title":"open import Data.These public","searchableContent":"  open import data.these public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":17,"title":"open T public using (These; this; that)","searchableContent":"open t public using (these; this; that)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":19,"title":"module L where","searchableContent":"module l where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":20,"title":"open import Data.List public","searchableContent":"  open import data.list public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":21,"title":"open L public using (List; []; _∷_; _++_; catMaybes; head; length; sum; and; or; any)","searchableContent":"open l public using (list; []; _∷_; _++_; catmaybes; head; length; sum; and; or; any)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":23,"title":"module A where","searchableContent":"module a where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":24,"title":"open import Data.List.Relation.Unary.Any public","searchableContent":"  open import data.list.relation.unary.any public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":25,"title":"open A public using (here; there)","searchableContent":"open a public using (here; there)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":27,"title":"module N where","searchableContent":"module n where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":28,"title":"open import Data.Nat public","searchableContent":"  open import data.nat public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":29,"title":"open import Data.Nat.Properties public","searchableContent":"  open import data.nat.properties public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":30,"title":"open N public using (ℕ; zero; suc)","searchableContent":"open n public using (ℕ; zero; suc)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":32,"title":"module F where","searchableContent":"module f where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":33,"title":"open import Data.Fin public","searchableContent":"  open import data.fin public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":34,"title":"open import Data.Fin.Properties public","searchableContent":"  open import data.fin.properties public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":35,"title":"open F public using (Fin; toℕ; #_) renaming (zero to fzero; suc to fsuc)","searchableContent":"open f public using (fin; toℕ; #_) renaming (zero to fzero; suc to fsuc)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":37,"title":"fromTo ","content":"fromTo : ℕ → ℕ → List ℕ","searchableContent":"fromto : ℕ → ℕ → list ℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":38,"title":"fromTo m n = map (_+ m) (upTo (n ∸ m))","searchableContent":"fromto m n = map (_+ m) (upto (n ∸ m))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":40,"title":"slice ","content":"slice : (L : ℕ) → ⦃ NonZero L ⦄ → ℕ → ℕ → ℙ ℕ","searchableContent":"slice : (l : ℕ) → ⦃ nonzero l ⦄ → ℕ → ℕ → ℙ ℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":41,"title":"slice L s x = fromList (fromTo s' (s' + (L ∸ 1)))","searchableContent":"slice l s x = fromlist (fromto s' (s' + (l ∸ 1)))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":42,"title":"where s' = ((s / L) ∸ x) * L -- equivalent to the formula in the paper","searchableContent":"  where s' = ((s / l) ∸ x) * l -- equivalent to the formula in the paper"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":44,"title":"filter ","content":"filter : {A : Set} → (P : A → Type) ⦃ _ : P ⁇¹ ⦄ → List A → List A","searchableContent":"filter : {a : set} → (p : a → type) ⦃ _ : p ⁇¹ ⦄ → list a → list a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":45,"title":"filter P = L.filter ¿ P ¿¹","searchableContent":"filter p = l.filter ¿ p ¿¹"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":47,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":48,"title":"IsSet-List ","content":"IsSet-List : {A : Set} → IsSet (List A) A","searchableContent":"  isset-list : {a : set} → isset (list a) a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":49,"title":"IsSet-List .toSet A = fromList A","searchableContent":"  isset-list .toset a = fromlist a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":51,"title":"open import Data.List.Relation.Unary.Any","searchableContent":"open import data.list.relation.unary.any"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":52,"title":"open Properties","searchableContent":"open properties"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":54,"title":"finite⇒A≡∅⊎∃a∈A ","content":"finite⇒A≡∅⊎∃a∈A : {X : Type} → {A : ℙ X} → finite A → (A ≡ᵉ ∅) ⊎ Σ[ a ∈ X ] a ∈ A","searchableContent":"finite⇒a≡∅⊎∃a∈a : {x : type} → {a : ℙ x} → finite a → (a ≡ᵉ ∅) ⊎ σ[ a ∈ x ] a ∈ a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":55,"title":"finite⇒A≡∅⊎∃a∈A ([]    , h) = inj₁ (∅-least (λ a∈A → ⊥-elim (case Equivalence.to h a∈A of λ ())))","searchableContent":"finite⇒a≡∅⊎∃a∈a ([]    , h) = inj₁ (∅-least (λ a∈a → ⊥-elim (case equivalence.to h a∈a of λ ())))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":56,"title":"finite⇒A≡∅⊎∃a∈A (x ∷ _ , h) = inj₂ (x , Equivalence.from h (here refl))","searchableContent":"finite⇒a≡∅⊎∃a∈a (x ∷ _ , h) = inj₂ (x , equivalence.from h (here refl))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":58,"title":"completeFin ","content":"completeFin : ∀ (n : ℕ) → ℙ (Fin n)","searchableContent":"completefin : ∀ (n : ℕ) → ℙ (fin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":59,"title":"completeFin zero = ∅","searchableContent":"completefin zero = ∅"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":60,"title":"completeFin (ℕ.suc n) = singleton (F.fromℕ n) ∪ mapˢ F.inject₁ (completeFin n)","searchableContent":"completefin (ℕ.suc n) = singleton (f.fromℕ n) ∪ mapˢ f.inject₁ (completefin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":62,"title":"m≤n∧n≤m⇒m≡n ","content":"m≤n∧n≤m⇒m≡n : ∀ {n m : ℕ} → n N.≤ m → m N.≤ n → m ≡ n","searchableContent":"m≤n∧n≤m⇒m≡n : ∀ {n m : ℕ} → n n.≤ m → m n.≤ n → m ≡ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":63,"title":"m≤n∧n≤m⇒m≡n z≤n z≤n = refl","searchableContent":"m≤n∧n≤m⇒m≡n z≤n z≤n = refl"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":64,"title":"m≤n∧n≤m⇒m≡n (s≤s n≤m) (s≤s m≤n) = cong N.suc (m≤n∧n≤m⇒m≡n n≤m m≤n)","searchableContent":"m≤n∧n≤m⇒m≡n (s≤s n≤m) (s≤s m≤n) = cong n.suc (m≤n∧n≤m⇒m≡n n≤m m≤n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":66,"title":"toℕ-fromℕ ","content":"toℕ-fromℕ : ∀ {n} {a : Fin (N.suc n)} → toℕ a ≡ n → a ≡ F.fromℕ n","searchableContent":"toℕ-fromℕ : ∀ {n} {a : fin (n.suc n)} → toℕ a ≡ n → a ≡ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":67,"title":"toℕ-fromℕ {zero} {fzero} x = refl","searchableContent":"toℕ-fromℕ {zero} {fzero} x = refl"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":68,"title":"toℕ-fromℕ {N.suc n} {fsuc a} x = cong fsuc (toℕ-fromℕ {n} {a} (N.suc-injective x))","searchableContent":"toℕ-fromℕ {n.suc n} {fsuc a} x = cong fsuc (toℕ-fromℕ {n} {a} (n.suc-injective x))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":70,"title":"open Equivalence","searchableContent":"open equivalence"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":72,"title":"maximalFin ","content":"maximalFin : ∀ (n : ℕ) → isMaximal (completeFin n)","searchableContent":"maximalfin : ∀ (n : ℕ) → ismaximal (completefin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":73,"title":"maximalFin (ℕ.suc n) {a} with toℕ a N.<? n","searchableContent":"maximalfin (ℕ.suc n) {a} with toℕ a n.<? n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":74,"title":"... | yes p =","searchableContent":"... | yes p ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":75,"title":"let n≢toℕ = ≢-sym (N.<⇒≢ p)","searchableContent":"  let n≢toℕ = ≢-sym (n.<⇒≢ p)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":76,"title":"fn = F.lower₁ a n≢toℕ","searchableContent":"      fn = f.lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":77,"title":"fn≡a = F.inject₁-lower₁ a n≢toℕ","searchableContent":"      fn≡a = f.inject₁-lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":78,"title":"in (to ∈-∪) (inj₂ ((to ∈-map) (fn , (sym fn≡a , maximalFin n))))","searchableContent":"  in (to ∈-∪) (inj₂ ((to ∈-map) (fn , (sym fn≡a , maximalfin n))))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":79,"title":"... | no ¬p with a F.≟ F.fromℕ n","searchableContent":"... | no ¬p with a f.≟ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":80,"title":"... | yes q = (to ∈-∪) (inj₁ ((to ∈-singleton) q))","searchableContent":"... | yes q = (to ∈-∪) (inj₁ ((to ∈-singleton) q))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":81,"title":"... | no ¬q =","searchableContent":"... | no ¬q ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":82,"title":"let n≢toℕ = N.≰⇒> ¬p","searchableContent":"  let n≢toℕ = n.≰⇒> ¬p"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":83,"title":"a<sucn = F.toℕ<n a","searchableContent":"      a<sucn = f.toℕ<n a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":84,"title":"in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (N.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))","searchableContent":"  in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (n.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":86,"title":"record Listable (A ","content":"record Listable (A : Type) : Type where","searchableContent":"record listable (a : type) : type where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":87,"title":"field","searchableContent":"  field"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":88,"title":"listing  ","content":"listing  : ℙ A","searchableContent":"    listing  : ℙ a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":89,"title":"complete ","content":"complete : ∀ {a : A} → a ∈ listing","searchableContent":"    complete : ∀ {a : a} → a ∈ listing"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":91,"title":"totalDec ","content":"totalDec : ∀ {A B : Type} → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → {R : Rel A B} → Dec (total R)","searchableContent":"totaldec : ∀ {a b : type} → ⦃ deceq a ⦄ → ⦃ listable a ⦄ → {r : rel a b} → dec (total r)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":92,"title":"totalDec {A} {B} {R} with all? (_∈? dom R)","searchableContent":"totaldec {a} {b} {r} with all? (_∈? dom r)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":93,"title":"... | yes p = yes λ {a} → p {a} ((Listable.complete it) {a})","searchableContent":"... | yes p = yes λ {a} → p {a} ((listable.complete it) {a})"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":94,"title":"... | no ¬p = no λ x → ¬p λ {a} _ → x {a}","searchableContent":"... | no ¬p = no λ x → ¬p λ {a} _ → x {a}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":96,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":97,"title":"total? ","content":"total? : ∀ {A B : Type} → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → {R : Rel A B} → ({a : A} → a ∈ dom R) ⁇","searchableContent":"  total? : ∀ {a b : type} → ⦃ deceq a ⦄ → ⦃ listable a ⦄ → {r : rel a b} → ({a : a} → a ∈ dom r) ⁇"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":98,"title":"total? = ⁇ totalDec","searchableContent":"  total? = ⁇ totaldec"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":100,"title":"Listable-Fin ","content":"Listable-Fin : ∀ {n} → Listable (Fin n)","searchableContent":"  listable-fin : ∀ {n} → listable (fin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":101,"title":"Listable-Fin {zero} = record { listing = ∅ ; complete = λ {a} → ⊥-elim $ (Inverse.to F.0↔⊥) a }","searchableContent":"  listable-fin {zero} = record { listing = ∅ ; complete = λ {a} → ⊥-elim $ (inverse.to f.0↔⊥) a }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":102,"title":"Listable-Fin {suc n} =","searchableContent":"  listable-fin {suc n} ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":103,"title":"let record { listing = l ; complete = c } = Listable-Fin {n}","searchableContent":"    let record { listing = l ; complete = c } = listable-fin {n}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":104,"title":"in record","searchableContent":"    in record"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":105,"title":"{ listing = singleton (F.fromℕ n) ∪ mapˢ F.inject₁ l","searchableContent":"         { listing = singleton (f.fromℕ n) ∪ mapˢ f.inject₁ l"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":106,"title":"; complete = complete","searchableContent":"         ; complete = complete"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":107,"title":"}","searchableContent":"         }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":108,"title":"where","searchableContent":"       where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":109,"title":"complete ","content":"complete : ∀ {a} → a ∈ singleton (F.fromℕ n) ∪ mapˢ F.inject₁ (let record { listing = l } = Listable-Fin {n} in l)","searchableContent":"         complete : ∀ {a} → a ∈ singleton (f.fromℕ n) ∪ mapˢ f.inject₁ (let record { listing = l } = listable-fin {n} in l)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":110,"title":"complete {a} with F.toℕ a N.<? n","searchableContent":"         complete {a} with f.toℕ a n.<? n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":111,"title":"... | yes p =","searchableContent":"         ... | yes p ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":112,"title":"let record { listing = l ; complete = c } = Listable-Fin {n}","searchableContent":"           let record { listing = l ; complete = c } = listable-fin {n}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":113,"title":"n≢toℕ = ≢-sym (N.<⇒≢ p)","searchableContent":"               n≢toℕ = ≢-sym (n.<⇒≢ p)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":114,"title":"fn = F.lower₁ a n≢toℕ","searchableContent":"               fn = f.lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":115,"title":"fn≡a = F.inject₁-lower₁ a n≢toℕ","searchableContent":"               fn≡a = f.inject₁-lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":116,"title":"in (Equivalence.to ∈-∪) (inj₂ ((Equivalence.to ∈-map) (fn , (sym fn≡a , c))))","searchableContent":"           in (equivalence.to ∈-∪) (inj₂ ((equivalence.to ∈-map) (fn , (sym fn≡a , c))))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":117,"title":"... | no ¬p with a F.≟ F.fromℕ n","searchableContent":"         ... | no ¬p with a f.≟ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":118,"title":"... | yes q = (Equivalence.to ∈-∪) (inj₁ ((Equivalence.to ∈-singleton) q))","searchableContent":"         ... | yes q = (equivalence.to ∈-∪) (inj₁ ((equivalence.to ∈-singleton) q))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":119,"title":"... | no ¬q =","searchableContent":"         ... | no ¬q ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":120,"title":"let n≢toℕ = N.≰⇒> ¬p","searchableContent":"           let n≢toℕ = n.≰⇒> ¬p"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":121,"title":"a<sucn = F.toℕ<n a","searchableContent":"               a<sucn = f.toℕ<n a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":122,"title":"in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (N.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))","searchableContent":"           in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (n.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":124,"title":"completeFinL ","content":"completeFinL : ∀ (n : ℕ) → List (Fin n)","searchableContent":"completefinl : ∀ (n : ℕ) → list (fin n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":125,"title":"completeFinL zero = []","searchableContent":"completefinl zero = []"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":126,"title":"completeFinL (ℕ.suc n) = F.fromℕ n ∷ L.map F.inject₁ (completeFinL n)","searchableContent":"completefinl (ℕ.suc n) = f.fromℕ n ∷ l.map f.inject₁ (completefinl n)"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":127,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":2,"title":"Leios.Protocol
{-# OPTIONS --safe #-}","searchableContent":"leios.protocol
{-# options --safe #-}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":5,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":6,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":8,"title":"module Leios.Protocol {n} (⋯ ","content":"module Leios.Protocol {n} (⋯ : SpecStructure n) (let open SpecStructure ⋯) (SlotUpkeep : Type) where","searchableContent":"module leios.protocol {n} (⋯ : specstructure n) (let open specstructure ⋯) (slotupkeep : type) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":10,"title":"open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)","searchableContent":"open baseabstract b' using (cert; v-chkcerts; vty; initslot)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":11,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":13,"title":"-- High level structure","content":"-- High level structure:","searchableContent":"-- high level structure:"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":16,"title":"--                                      (simple) Leios","searchableContent":"--                                      (simple) leios"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":17,"title":"--                                        /         |","searchableContent":"--                                        /         |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":18,"title":"-- +-------------------------------------+          |","searchableContent":"-- +-------------------------------------+          |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":19,"title":"-- | Header Diffusion     Body Diffusion |          |","searchableContent":"-- | header diffusion     body diffusion |          |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":20,"title":"-- +-------------------------------------+       Base Protocol","searchableContent":"-- +-------------------------------------+       base protocol"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":21,"title":"--                                        \\      /","searchableContent":"--                                        \\      /"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":22,"title":"--                                        Network","searchableContent":"--                                        network"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":24,"title":"data LeiosInput ","content":"data LeiosInput : Type where","searchableContent":"data leiosinput : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":25,"title":"INIT     ","content":"INIT     : VTy → LeiosInput","searchableContent":"  init     : vty → leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":26,"title":"SUBMIT   ","content":"SUBMIT   : EndorserBlock ⊎ List Tx → LeiosInput","searchableContent":"  submit   : endorserblock ⊎ list tx → leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":27,"title":"SLOT     ","content":"SLOT     : LeiosInput","searchableContent":"  slot     : leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":28,"title":"FTCH-LDG ","content":"FTCH-LDG : LeiosInput","searchableContent":"  ftch-ldg : leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":30,"title":"data LeiosOutput ","content":"data LeiosOutput : Type where","searchableContent":"data leiosoutput : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":31,"title":"FTCH-LDG ","content":"FTCH-LDG : List Tx → LeiosOutput","searchableContent":"  ftch-ldg : list tx → leiosoutput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":32,"title":"EMPTY    ","content":"EMPTY    : LeiosOutput","searchableContent":"  empty    : leiosoutput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":34,"title":"record LeiosState ","content":"record LeiosState : Type where","searchableContent":"record leiosstate : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":35,"title":"field V           ","content":"field V           : VTy","searchableContent":"  field v           : vty"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":36,"title":"SD          ","content":"SD          : StakeDistr","searchableContent":"        sd          : stakedistr"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":37,"title":"FFDState    ","content":"FFDState    : FFD.State","searchableContent":"        ffdstate    : ffd.state"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":38,"title":"Ledger      ","content":"Ledger      : List Tx","searchableContent":"        ledger      : list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":39,"title":"ToPropose   ","content":"ToPropose   : List Tx","searchableContent":"        topropose   : list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":40,"title":"IBs         ","content":"IBs         : List InputBlock","searchableContent":"        ibs         : list inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":41,"title":"EBs         ","content":"EBs         : List EndorserBlock","searchableContent":"        ebs         : list endorserblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":42,"title":"Vs          ","content":"Vs          : List (List Vote)","searchableContent":"        vs          : list (list vote)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":43,"title":"slot        ","content":"slot        : ℕ","searchableContent":"        slot        : ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":44,"title":"IBHeaders   ","content":"IBHeaders   : List IBHeader","searchableContent":"        ibheaders   : list ibheader"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":45,"title":"IBBodies    ","content":"IBBodies    : List IBBody","searchableContent":"        ibbodies    : list ibbody"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":46,"title":"Upkeep      ","content":"Upkeep      : ℙ SlotUpkeep","searchableContent":"        upkeep      : ℙ slotupkeep"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":47,"title":"BaseState   ","content":"BaseState   : B.State","searchableContent":"        basestate   : b.state"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":48,"title":"votingState ","content":"votingState : VotingState","searchableContent":"        votingstate : votingstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":49,"title":"PubKeys     ","content":"PubKeys     : List PubKey","searchableContent":"        pubkeys     : list pubkey"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":51,"title":"lookupEB ","content":"lookupEB : EBRef → Maybe EndorserBlock","searchableContent":"  lookupeb : ebref → maybe endorserblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":52,"title":"lookupEB r = find (λ b → getEBRef b ≟ r) EBs","searchableContent":"  lookupeb r = find (λ b → getebref b ≟ r) ebs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":54,"title":"lookupIB ","content":"lookupIB : IBRef → Maybe InputBlock","searchableContent":"  lookupib : ibref → maybe inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":55,"title":"lookupIB r = find (λ b → getIBRef b ≟ r) IBs","searchableContent":"  lookupib r = find (λ b → getibref b ≟ r) ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":57,"title":"lookupTxs ","content":"lookupTxs : EndorserBlock → List Tx","searchableContent":"  lookuptxs : endorserblock → list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":58,"title":"lookupTxs eb = do","searchableContent":"  lookuptxs eb = do"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":59,"title":"eb′ ← mapMaybe lookupEB $ ebRefs eb","searchableContent":"    eb′ ← mapmaybe lookupeb $ ebrefs eb"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":60,"title":"ib  ← mapMaybe lookupIB $ ibRefs eb′","searchableContent":"    ib  ← mapmaybe lookupib $ ibrefs eb′"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":61,"title":"txs $ body ib","searchableContent":"    txs $ body ib"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":62,"title":"where open EndorserBlockOSig","searchableContent":"    where open endorserblockosig"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":63,"title":"open IBBody","searchableContent":"          open ibbody"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":64,"title":"open InputBlock","searchableContent":"          open inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":66,"title":"constructLedger ","content":"constructLedger : List RankingBlock → List Tx","searchableContent":"  constructledger : list rankingblock → list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":67,"title":"constructLedger = L.concat ∘ L.map (T.mergeThese L._++_ ∘ T.map₁ lookupTxs)","searchableContent":"  constructledger = l.concat ∘ l.map (t.mergethese l._++_ ∘ t.map₁ lookuptxs)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":69,"title":"needsUpkeep ","content":"needsUpkeep : SlotUpkeep → Set","searchableContent":"  needsupkeep : slotupkeep → set"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":70,"title":"needsUpkeep = _∉ Upkeep","searchableContent":"  needsupkeep = _∉ upkeep"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":72,"title":"Dec-needsUpkeep ","content":"Dec-needsUpkeep : ∀ {u : SlotUpkeep} → ⦃ DecEq SlotUpkeep ⦄ → needsUpkeep u ⁇","searchableContent":"  dec-needsupkeep : ∀ {u : slotupkeep} → ⦃ deceq slotupkeep ⦄ → needsupkeep u ⁇"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":73,"title":"Dec-needsUpkeep {u} .dec = ¬? (u ∈? Upkeep)","searchableContent":"  dec-needsupkeep {u} .dec = ¬? (u ∈? upkeep)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":75,"title":"addUpkeep ","content":"addUpkeep : LeiosState → SlotUpkeep → LeiosState","searchableContent":"addupkeep : leiosstate → slotupkeep → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":76,"title":"addUpkeep s u = let open LeiosState s in record s { Upkeep = Upkeep ∪ ❴ u ❵ }","searchableContent":"addupkeep s u = let open leiosstate s in record s { upkeep = upkeep ∪ ❴ u ❵ }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":77,"title":"{-# INJECTIVE_FOR_INFERENCE addUpkeep #-}","searchableContent":"{-# injective_for_inference addupkeep #-}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":79,"title":"initLeiosState ","content":"initLeiosState : VTy → StakeDistr → B.State → List PubKey → LeiosState","searchableContent":"initleiosstate : vty → stakedistr → b.state → list pubkey → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":80,"title":"initLeiosState V SD bs pks = record","searchableContent":"initleiosstate v sd bs pks = record"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":81,"title":"{ V           = V","searchableContent":"  { v           = v"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":82,"title":"; SD          = SD","searchableContent":"  ; sd          = sd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":83,"title":"; FFDState    = FFD.initFFDState","searchableContent":"  ; ffdstate    = ffd.initffdstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":84,"title":"; Ledger      = []","searchableContent":"  ; ledger      = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":85,"title":"; ToPropose   = []","searchableContent":"  ; topropose   = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":86,"title":"; IBs         = []","searchableContent":"  ; ibs         = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":87,"title":"; EBs         = []","searchableContent":"  ; ebs         = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":88,"title":"; Vs          = []","searchableContent":"  ; vs          = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":89,"title":"; slot        = initSlot V","searchableContent":"  ; slot        = initslot v"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":90,"title":"; IBHeaders   = []","searchableContent":"  ; ibheaders   = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":91,"title":"; IBBodies    = []","searchableContent":"  ; ibbodies    = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":92,"title":"; Upkeep      = ∅","searchableContent":"  ; upkeep      = ∅"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":93,"title":"; BaseState   = bs","searchableContent":"  ; basestate   = bs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":94,"title":"; votingState = initVotingState","searchableContent":"  ; votingstate = initvotingstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":95,"title":"; PubKeys     = pks","searchableContent":"  ; pubkeys     = pks"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":96,"title":"}","searchableContent":"  }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":98,"title":"stake' ","content":"stake' : PoolID → LeiosState → ℕ","searchableContent":"stake' : poolid → leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":99,"title":"stake' pid record { SD = SD } = TotalMap.lookup SD pid","searchableContent":"stake' pid record { sd = sd } = totalmap.lookup sd pid"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":101,"title":"stake'' ","content":"stake'' : PubKey → LeiosState → ℕ","searchableContent":"stake'' : pubkey → leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":102,"title":"stake'' pk = stake' (poolID pk)","searchableContent":"stake'' pk = stake' (poolid pk)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":104,"title":"stake ","content":"stake : LeiosState → ℕ","searchableContent":"stake : leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":105,"title":"stake = stake' id","searchableContent":"stake = stake' id"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":107,"title":"lookupPubKeyAndStake ","content":"lookupPubKeyAndStake : ∀ {B} → ⦃ _ : IsBlock B ⦄ → LeiosState → B → Maybe (PubKey × ℕ)","searchableContent":"lookuppubkeyandstake : ∀ {b} → ⦃ _ : isblock b ⦄ → leiosstate → b → maybe (pubkey × ℕ)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":108,"title":"lookupPubKeyAndStake s b =","searchableContent":"lookuppubkeyandstake s b ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":109,"title":"L.head $","searchableContent":"  l.head $"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":110,"title":"L.map (λ pk → (pk , stake'' pk s)) $","searchableContent":"    l.map (λ pk → (pk , stake'' pk s)) $"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":111,"title":"L.filter (λ pk → producerID b ≟ poolID pk) (LeiosState.PubKeys s)","searchableContent":"      l.filter (λ pk → producerid b ≟ poolid pk) (leiosstate.pubkeys s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":113,"title":"module _ (s ","content":"module _ (s : LeiosState)  where","searchableContent":"module _ (s : leiosstate)  where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":115,"title":"record ibHeaderValid (h ","content":"record ibHeaderValid (h : IBHeader) (pk : PubKey) (st : ℕ) : Type where","searchableContent":"  record ibheadervalid (h : ibheader) (pk : pubkey) (st : ℕ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":116,"title":"field lotteryPfValid ","content":"field lotteryPfValid : verify pk (slotNumber h) st (lotteryPf h)","searchableContent":"    field lotterypfvalid : verify pk (slotnumber h) st (lotterypf h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":117,"title":"signatureValid ","content":"signatureValid : verifySig pk (signature h)","searchableContent":"          signaturevalid : verifysig pk (signature h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":119,"title":"record ibBodyValid (b ","content":"record ibBodyValid (b : IBBody) : Type where","searchableContent":"  record ibbodyvalid (b : ibbody) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":121,"title":"ibHeaderValid? ","content":"ibHeaderValid? : (h : IBHeader) (pk : PubKey) (st : ℕ) → Dec (ibHeaderValid h pk st)","searchableContent":"  ibheadervalid? : (h : ibheader) (pk : pubkey) (st : ℕ) → dec (ibheadervalid h pk st)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":122,"title":"ibHeaderValid? h pk st","searchableContent":"  ibheadervalid? h pk st"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":123,"title":"with verify? pk (slotNumber h) st (lotteryPf h)","searchableContent":"    with verify? pk (slotnumber h) st (lotterypf h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":124,"title":"... | no ¬p = no (¬p ∘ ibHeaderValid.lotteryPfValid)","searchableContent":"  ... | no ¬p = no (¬p ∘ ibheadervalid.lotterypfvalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":125,"title":"... | yes p","searchableContent":"  ... | yes p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":126,"title":"with verifySig? pk (signature h)","searchableContent":"    with verifysig? pk (signature h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":127,"title":"... | yes q = yes (record { lotteryPfValid = p ; signatureValid = q })","searchableContent":"  ... | yes q = yes (record { lotterypfvalid = p ; signaturevalid = q })"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":128,"title":"... | no ¬q = no (¬q ∘ ibHeaderValid.signatureValid)","searchableContent":"  ... | no ¬q = no (¬q ∘ ibheadervalid.signaturevalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":130,"title":"ibBodyValid? ","content":"ibBodyValid? : (b : IBBody) → Dec (ibBodyValid b)","searchableContent":"  ibbodyvalid? : (b : ibbody) → dec (ibbodyvalid b)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":131,"title":"ibBodyValid? _ = yes record {}","searchableContent":"  ibbodyvalid? _ = yes record {}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":133,"title":"ibValid ","content":"ibValid : InputBlock → Type","searchableContent":"  ibvalid : inputblock → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":134,"title":"ibValid record { header = h ; body = b }","searchableContent":"  ibvalid record { header = h ; body = b }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":135,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":136,"title":"... | just (pk , pid) = ibHeaderValid h pk (stake'' pk s) × ibBodyValid b","searchableContent":"  ... | just (pk , pid) = ibheadervalid h pk (stake'' pk s) × ibbodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":137,"title":"... | nothing = ⊥","searchableContent":"  ... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":139,"title":"ibValid? ","content":"ibValid? : (ib : InputBlock) → Dec (ibValid ib)","searchableContent":"  ibvalid? : (ib : inputblock) → dec (ibvalid ib)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":140,"title":"ibValid? record { header = h ; body = b }","searchableContent":"  ibvalid? record { header = h ; body = b }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":141,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":142,"title":"... | just (pk , pid) = ibHeaderValid? h pk (stake'' pk s) ×-dec ibBodyValid? b","searchableContent":"  ... | just (pk , pid) = ibheadervalid? h pk (stake'' pk s) ×-dec ibbodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":143,"title":"... | nothing = no λ x → x","searchableContent":"  ... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":145,"title":"record ebValid (eb ","content":"record ebValid (eb : EndorserBlock) (pk : PubKey) (st : ℕ) : Type where","searchableContent":"  record ebvalid (eb : endorserblock) (pk : pubkey) (st : ℕ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":146,"title":"field lotteryPfValid ","content":"field lotteryPfValid : verify pk (slotNumber eb) st (lotteryPf eb)","searchableContent":"    field lotterypfvalid : verify pk (slotnumber eb) st (lotterypf eb)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":147,"title":"signatureValid ","content":"signatureValid : verifySig pk (signature eb)","searchableContent":"          signaturevalid : verifysig pk (signature eb)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":148,"title":"-- TODO","searchableContent":"    -- todo"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":149,"title":"-- ibRefsValid","content":"-- ibRefsValid : ?","searchableContent":"    -- ibrefsvalid : ?"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":150,"title":"-- ebRefsValid","content":"-- ebRefsValid : ?","searchableContent":"    -- ebrefsvalid : ?"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":152,"title":"ebValid? ","content":"ebValid? : (eb : EndorserBlock) (pk : PubKey) (st : ℕ) → Dec (ebValid eb pk st)","searchableContent":"  ebvalid? : (eb : endorserblock) (pk : pubkey) (st : ℕ) → dec (ebvalid eb pk st)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":153,"title":"ebValid? h pk st","searchableContent":"  ebvalid? h pk st"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":154,"title":"with verify? pk (slotNumber h) st (lotteryPf h)","searchableContent":"    with verify? pk (slotnumber h) st (lotterypf h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":155,"title":"... | no ¬p = no (¬p ∘ ebValid.lotteryPfValid)","searchableContent":"  ... | no ¬p = no (¬p ∘ ebvalid.lotterypfvalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":156,"title":"... | yes p","searchableContent":"  ... | yes p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":157,"title":"with verifySig? pk (signature h)","searchableContent":"    with verifysig? pk (signature h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":158,"title":"... | yes q = yes (record { lotteryPfValid = p ; signatureValid = q })","searchableContent":"  ... | yes q = yes (record { lotterypfvalid = p ; signaturevalid = q })"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":159,"title":"... | no ¬q = no (¬q ∘ ebValid.signatureValid)","searchableContent":"  ... | no ¬q = no (¬q ∘ ebvalid.signaturevalid)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":161,"title":"-- TODO","searchableContent":"  -- todo"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":162,"title":"record vsValid (vs ","content":"record vsValid (vs : List Vote) : Type where","searchableContent":"  record vsvalid (vs : list vote) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":164,"title":"vsValid? ","content":"vsValid? : (vs : List Vote) → Dec (vsValid vs)","searchableContent":"  vsvalid? : (vs : list vote) → dec (vsvalid vs)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":165,"title":"vsValid? _ = yes record {}","searchableContent":"  vsvalid? _ = yes record {}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":167,"title":"headerValid ","content":"headerValid : Header → Type","searchableContent":"  headervalid : header → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":168,"title":"headerValid (ibHeader h)","searchableContent":"  headervalid (ibheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":169,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":170,"title":"... | just (pk , pid) = ibHeaderValid h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ibheadervalid h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":171,"title":"... | nothing = ⊥","searchableContent":"  ... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":172,"title":"headerValid (ebHeader h)","searchableContent":"  headervalid (ebheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":173,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":174,"title":"... | just (pk , pid) = ebValid h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ebvalid h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":175,"title":"... | nothing = ⊥","searchableContent":"  ... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":176,"title":"headerValid (vtHeader h) = vsValid h","searchableContent":"  headervalid (vtheader h) = vsvalid h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":178,"title":"headerValid? ","content":"headerValid? : (h : Header) → Dec (headerValid h)","searchableContent":"  headervalid? : (h : header) → dec (headervalid h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":179,"title":"headerValid? (ibHeader h)","searchableContent":"  headervalid? (ibheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":180,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":181,"title":"... | just (pk , pid) = ibHeaderValid? h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ibheadervalid? h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":182,"title":"... | nothing = no λ x → x","searchableContent":"  ... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":183,"title":"headerValid? (ebHeader h)","searchableContent":"  headervalid? (ebheader h)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":184,"title":"with lookupPubKeyAndStake s h","searchableContent":"    with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":185,"title":"... | just (pk , pid) = ebValid? h pk (stake'' pk s)","searchableContent":"  ... | just (pk , pid) = ebvalid? h pk (stake'' pk s)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":186,"title":"... | nothing = no λ x → x","searchableContent":"  ... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":187,"title":"headerValid? (vtHeader h) = vsValid? h","searchableContent":"  headervalid? (vtheader h) = vsvalid? h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":189,"title":"bodyValid ","content":"bodyValid : Body → Type","searchableContent":"  bodyvalid : body → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":190,"title":"bodyValid (ibBody b) = ibBodyValid b","searchableContent":"  bodyvalid (ibbody b) = ibbodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":192,"title":"bodyValid? ","content":"bodyValid? : (b : Body) → Dec (bodyValid b)","searchableContent":"  bodyvalid? : (b : body) → dec (bodyvalid b)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":193,"title":"bodyValid? (ibBody b) = ibBodyValid? b","searchableContent":"  bodyvalid? (ibbody b) = ibbodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":195,"title":"isValid ","content":"isValid : Header ⊎ Body → Type","searchableContent":"  isvalid : header ⊎ body → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":196,"title":"isValid (inj₁ h) = headerValid h","searchableContent":"  isvalid (inj₁ h) = headervalid h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":197,"title":"isValid (inj₂ b) = bodyValid b","searchableContent":"  isvalid (inj₂ b) = bodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":199,"title":"isValid? ","content":"isValid? : ∀ (x : Header ⊎ Body) → Dec (isValid x)","searchableContent":"  isvalid? : ∀ (x : header ⊎ body) → dec (isvalid x)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":200,"title":"isValid? (inj₁ h) = headerValid? h","searchableContent":"  isvalid? (inj₁ h) = headervalid? h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":201,"title":"isValid? (inj₂ b) = bodyValid? b","searchableContent":"  isvalid? (inj₂ b) = bodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":203,"title":"-- some predicates about EBs","searchableContent":"-- some predicates about ebs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":204,"title":"module _ (s ","content":"module _ (s : LeiosState) (eb : EndorserBlock) where","searchableContent":"module _ (s : leiosstate) (eb : endorserblock) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":205,"title":"open EndorserBlockOSig eb","searchableContent":"  open endorserblockosig eb"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":206,"title":"open LeiosState s","searchableContent":"  open leiosstate s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":208,"title":"allIBRefsKnown ","content":"allIBRefsKnown : Type","searchableContent":"  allibrefsknown : type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":209,"title":"allIBRefsKnown = ∀[ ref ∈ fromList ibRefs ] ref ∈ˡ map getIBRef IBs","searchableContent":"  allibrefsknown = ∀[ ref ∈ fromlist ibrefs ] ref ∈ˡ map getibref ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":211,"title":"module _ (s ","content":"module _ (s : LeiosState) where","searchableContent":"module _ (s : leiosstate) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":213,"title":"open LeiosState s","searchableContent":"  open leiosstate s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":215,"title":"upd ","content":"upd : Header ⊎ Body → LeiosState","searchableContent":"  upd : header ⊎ body → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":216,"title":"upd (inj₁ (ebHeader eb)) = record s { EBs = eb ∷ EBs }","searchableContent":"  upd (inj₁ (ebheader eb)) = record s { ebs = eb ∷ ebs }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":217,"title":"upd (inj₁ (vtHeader vs)) = record s { Vs = vs ∷ Vs }","searchableContent":"  upd (inj₁ (vtheader vs)) = record s { vs = vs ∷ vs }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":218,"title":"upd (inj₁ (ibHeader h)) with A.any? (matchIB? h) IBBodies","searchableContent":"  upd (inj₁ (ibheader h)) with a.any? (matchib? h) ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":219,"title":"... | yes p =","searchableContent":"  ... | yes p ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":220,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":221,"title":"{ IBs = record { header = h ; body = A.lookup p } ∷ IBs","searchableContent":"      { ibs = record { header = h ; body = a.lookup p } ∷ ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":222,"title":"; IBBodies = IBBodies A.─ p","searchableContent":"      ; ibbodies = ibbodies a.─ p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":223,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":224,"title":"... | no _ =","searchableContent":"  ... | no _ ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":225,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":226,"title":"{ IBHeaders = h ∷ IBHeaders","searchableContent":"      { ibheaders = h ∷ ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":227,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":228,"title":"upd (inj₂ (ibBody b)) with A.any? (flip matchIB? b) IBHeaders","searchableContent":"  upd (inj₂ (ibbody b)) with a.any? (flip matchib? b) ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":229,"title":"... | yes p =","searchableContent":"  ... | yes p ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":230,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":231,"title":"{ IBs = record { header = A.lookup p ; body = b } ∷ IBs","searchableContent":"      { ibs = record { header = a.lookup p ; body = b } ∷ ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":232,"title":"; IBHeaders = IBHeaders A.─ p","searchableContent":"      ; ibheaders = ibheaders a.─ p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":233,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":234,"title":"... | no _ =","searchableContent":"  ... | no _ ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":235,"title":"record s","searchableContent":"    record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":236,"title":"{ IBBodies = b ∷ IBBodies","searchableContent":"      { ibbodies = b ∷ ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":237,"title":"}","searchableContent":"      }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":239,"title":"module _ {s s'} where","searchableContent":"module _ {s s'} where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":240,"title":"open LeiosState s'","searchableContent":"  open leiosstate s'"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":242,"title":"upd-preserves-Upkeep ","content":"upd-preserves-Upkeep : ∀ {x} → LeiosState.Upkeep s ≡ LeiosState.Upkeep s'","searchableContent":"  upd-preserves-upkeep : ∀ {x} → leiosstate.upkeep s ≡ leiosstate.upkeep s'"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":243,"title":"→ LeiosState.Upkeep s ≡ LeiosState.Upkeep (upd s' x)","searchableContent":"                               → leiosstate.upkeep s ≡ leiosstate.upkeep (upd s' x)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":244,"title":"upd-preserves-Upkeep {inj₁ (ibHeader x)} refl with A.any? (matchIB? x) IBBodies","searchableContent":"  upd-preserves-upkeep {inj₁ (ibheader x)} refl with a.any? (matchib? x) ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":245,"title":"... | yes p = refl","searchableContent":"  ... | yes p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":246,"title":"... | no ¬p = refl","searchableContent":"  ... | no ¬p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":247,"title":"upd-preserves-Upkeep {inj₁ (ebHeader x)} refl = refl","searchableContent":"  upd-preserves-upkeep {inj₁ (ebheader x)} refl = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":248,"title":"upd-preserves-Upkeep {inj₁ (vtHeader x)} refl = refl","searchableContent":"  upd-preserves-upkeep {inj₁ (vtheader x)} refl = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":249,"title":"upd-preserves-Upkeep {inj₂ (ibBody x)} refl with A.any? (flip matchIB? x) IBHeaders","searchableContent":"  upd-preserves-upkeep {inj₂ (ibbody x)} refl with a.any? (flip matchib? x) ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":250,"title":"... | yes p = refl","searchableContent":"  ... | yes p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":251,"title":"... | no ¬p = refl","searchableContent":"  ... | no ¬p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":253,"title":"infix 25 _↑_","searchableContent":"infix 25 _↑_"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":254,"title":"_↑_ ","content":"_↑_ : LeiosState → List (Header ⊎ Body) → LeiosState","searchableContent":"_↑_ : leiosstate → list (header ⊎ body) → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":255,"title":"_↑_ = foldr (flip upd)","searchableContent":"_↑_ = foldr (flip upd)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":257,"title":"↑-preserves-Upkeep ","content":"↑-preserves-Upkeep : ∀ {s x} → LeiosState.Upkeep s ≡ LeiosState.Upkeep (s ↑ x)","searchableContent":"↑-preserves-upkeep : ∀ {s x} → leiosstate.upkeep s ≡ leiosstate.upkeep (s ↑ x)"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":258,"title":"↑-preserves-Upkeep {x = []} = refl","searchableContent":"↑-preserves-upkeep {x = []} = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":259,"title":"↑-preserves-Upkeep {s = s} {x = x ∷ x₁} =","searchableContent":"↑-preserves-upkeep {s = s} {x = x ∷ x₁} ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":260,"title":"upd-preserves-Upkeep {s = s} {x = x} (↑-preserves-Upkeep {x = x₁})","searchableContent":"  upd-preserves-upkeep {s = s} {x = x} (↑-preserves-upkeep {x = x₁})"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":261,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Decidable
open import Leios.Prelude hiding (id)","searchableContent":"leios.short.decidable
open import leios.prelude hiding (id)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":3,"title":"open import Leios.SpecStructure using (SpecStructure)","searchableContent":"open import leios.specstructure using (specstructure)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":5,"title":"module Leios.Short.Decidable (⋯ ","content":"module Leios.Short.Decidable (⋯ : SpecStructure 1) (let open SpecStructure ⋯) where","searchableContent":"module leios.short.decidable (⋯ : specstructure 1) (let open specstructure ⋯) where"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":7,"title":"open import Leios.Short ⋯ renaming (isVoteCertified to isVoteCertified')","searchableContent":"open import leios.short ⋯ renaming (isvotecertified to isvotecertified')"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":8,"title":"open B hiding (_-⟦_/_⟧⇀_)","searchableContent":"open b hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":9,"title":"open FFD hiding (_-⟦_/_⟧⇀_)","searchableContent":"open ffd hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":11,"title":"module _ {s ","content":"module _ {s : LeiosState} (let open LeiosState s renaming (FFDState to ffds; BaseState to bs)) where","searchableContent":"module _ {s : leiosstate} (let open leiosstate s renaming (ffdstate to ffds; basestate to bs)) where"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":13,"title":"IB-Role? ","content":"IB-Role? : ∀ {π ffds'} →","searchableContent":"  ib-role? : ∀ {π ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":14,"title":"let b = GenFFD.ibBody (record { txs = ToPropose })","searchableContent":"           let b = genffd.ibbody (record { txs = topropose })"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":15,"title":"h = GenFFD.ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"               h = genffd.ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":16,"title":"in","searchableContent":"           in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":17,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep IB-Role }","searchableContent":"           { _ : auto∶ needsupkeep ib-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":18,"title":"{ _ ","content":"{ _ : auto∶ canProduceIB slot sk-IB (stake s) π }","searchableContent":"           { _ : auto∶ canproduceib slot sk-ib (stake s) π }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":19,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send h (just b) / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"           { _ : auto∶ ffds ffd.-⟦ ffd.send h (just b) / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":20,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"           ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":21,"title":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"           s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":22,"title":"IB-Role? {_} {_} {p} {q} {r} = IB-Role (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  ib-role? {_} {_} {p} {q} {r} = ib-role (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":24,"title":"{-","searchableContent":"  {-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":25,"title":"No-IB-Role?","content":"No-IB-Role? :","searchableContent":"  no-ib-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":26,"title":"{ _","content":"{ _ : auto∶ needsUpkeep IB-Role }","searchableContent":"              { _ : auto∶ needsupkeep ib-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":27,"title":"{ _","content":"{ _ : auto∶ ∀ π → ¬ canProduceIB slot sk-IB (stake s) π } →","searchableContent":"              { _ : auto∶ ∀ π → ¬ canproduceib slot sk-ib (stake s) π } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":28,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"              ─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":29,"title":"s ↝ addUpkeep s IB-Role","content":"s ↝ addUpkeep s IB-Role","searchableContent":"              s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":30,"title":"No-IB-Role? {p} {q}","content":"No-IB-Role? {p} {q} = No-IB-Role (toWitness p) (toWitness q)","searchableContent":"  no-ib-role? {p} {q} = no-ib-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":31,"title":"-}","content":"-}","searchableContent":"  -}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":33,"title":"EB-Role? ","content":"EB-Role? : ∀ {π ffds'} →","searchableContent":"  eb-role? : ∀ {π ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":34,"title":"let LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"           let li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":35,"title":"h = mkEB slot id π sk-EB LI []","searchableContent":"               h = mkeb slot id π sk-eb li []"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":36,"title":"in","searchableContent":"           in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":37,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep EB-Role }","searchableContent":"           { _ : auto∶ needsupkeep eb-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":38,"title":"{ _ ","content":"{ _ : auto∶ canProduceEB slot sk-EB (stake s) π }","searchableContent":"           { _ : auto∶ canproduceeb slot sk-eb (stake s) π }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":39,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send (GenFFD.ebHeader h) nothing / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"           { _ : auto∶ ffds ffd.-⟦ ffd.send (genffd.ebheader h) nothing / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":40,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"           ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":41,"title":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"           s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":42,"title":"EB-Role? {_} {_} {p} {q} {r} = EB-Role (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  eb-role? {_} {_} {p} {q} {r} = eb-role (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":44,"title":"{-","searchableContent":"  {-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":45,"title":"No-EB-Role?","content":"No-EB-Role? :","searchableContent":"  no-eb-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":46,"title":"{ _","content":"{ _ : auto∶ needsUpkeep EB-Role }","searchableContent":"              { _ : auto∶ needsupkeep eb-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":47,"title":"{ _","content":"{ _ : auto∶ ∀ π → ¬ canProduceEB slot sk-EB (stake s) π } →","searchableContent":"              { _ : auto∶ ∀ π → ¬ canproduceeb slot sk-eb (stake s) π } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":48,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"              ─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":49,"title":"s ↝ addUpkeep s EB-Role","content":"s ↝ addUpkeep s EB-Role","searchableContent":"              s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":50,"title":"No-EB-Role? {_} {p} {q}","content":"No-EB-Role? {_} {p} {q} = No-EB-Role (toWitness p) (toWitness q)","searchableContent":"  no-eb-role? {_} {p} {q} = no-eb-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":51,"title":"-}","content":"-}","searchableContent":"  -}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":53,"title":"V-Role? ","content":"V-Role? : ∀ {ffds'} →","searchableContent":"  v-role? : ∀ {ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":54,"title":"let EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"          let ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":55,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"              votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":56,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":57,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep VT-Role }","searchableContent":"          { _ : auto∶ needsupkeep vt-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":58,"title":"{ _ ","content":"{ _ : auto∶ canProduceV slot sk-VT (stake s) }","searchableContent":"          { _ : auto∶ canproducev slot sk-vt (stake s) }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":59,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send (GenFFD.vtHeader votes) nothing / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"          { _ : auto∶ ffds ffd.-⟦ ffd.send (genffd.vtheader votes) nothing / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":60,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":61,"title":"s ↝ addUpkeep record s { FFDState = ffds' } VT-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } vt-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":62,"title":"V-Role? {_} {p} {q} {r} = VT-Role (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  v-role? {_} {p} {q} {r} = vt-role (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":64,"title":"No-V-Role? ","content":"No-V-Role? :","searchableContent":"  no-v-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":65,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep VT-Role }","searchableContent":"             { _ : auto∶ needsupkeep vt-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":66,"title":"{ _ ","content":"{ _ : auto∶ ¬ canProduceV slot sk-VT (stake s) } →","searchableContent":"             { _ : auto∶ ¬ canproducev slot sk-vt (stake s) } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":67,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":68,"title":"s ↝ addUpkeep s VT-Role","searchableContent":"             s ↝ addupkeep s vt-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":69,"title":"No-V-Role? {p} {q} = No-VT-Role (toWitness p) (toWitness q)","searchableContent":"  no-v-role? {p} {q} = no-vt-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":71,"title":"{-","searchableContent":"  {-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":72,"title":"Init?","content":"Init? : ∀ {ks pks ks' SD bs' V} →","searchableContent":"  init? : ∀ {ks pks ks' sd bs' v} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":73,"title":"{ _","content":"{ _ : auto∶ ks K.-⟦ K.INIT pk-IB pk-EB pk-V / K.PUBKEYS pks ⟧⇀ ks' }","searchableContent":"        { _ : auto∶ ks k.-⟦ k.init pk-ib pk-eb pk-v / k.pubkeys pks ⟧⇀ ks' }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":74,"title":"{ _","content":"{ _ : initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs' } →","searchableContent":"        { _ : initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":75,"title":"────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────","searchableContent":"        ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":76,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'","content":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'","searchableContent":"        nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":77,"title":"Init?","content":"Init? = ?","searchableContent":"  init? = ?"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":78,"title":"-}","content":"-}","searchableContent":"  -}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":80,"title":"Base₂a? ","content":"Base₂a? : ∀ {eb bs'} →","searchableContent":"  base₂a? : ∀ {eb bs'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":81,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep Base }","searchableContent":"          { _ : auto∶ needsupkeep base }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":82,"title":"{ _ ","content":"{ _ : auto∶ eb ∈ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2) EBs }","searchableContent":"          { _ : auto∶ eb ∈ filter (λ eb → isvotecertified' s eb × eb ∈ᴮ slice l slot 2) ebs }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":83,"title":"{ _ ","content":"{ _ : auto∶ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs' } →","searchableContent":"          { _ : auto∶ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":84,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":85,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":86,"title":"Base₂a? {_} {_} {p} {q} {r} = Base₂a (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  base₂a? {_} {_} {p} {q} {r} = base₂a (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":88,"title":"Base₂b? ","content":"Base₂b? : ∀ {bs'} →","searchableContent":"  base₂b? : ∀ {bs'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":89,"title":"{ _ ","content":"{ _ : auto∶ needsUpkeep Base }","searchableContent":"          { _ : auto∶ needsupkeep base }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":90,"title":"{ _ ","content":"{ _ : auto∶ [] ≡ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2) EBs }","searchableContent":"          { _ : auto∶ [] ≡ filter (λ eb → isvotecertified' s eb × eb ∈ᴮ slice l slot 2) ebs }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":91,"title":"{ _ ","content":"{ _ : auto∶ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs' } →","searchableContent":"          { _ : auto∶ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":92,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":93,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":94,"title":"Base₂b? {_} {p} {q} {r} = Base₂b (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  base₂b? {_} {p} {q} {r} = base₂b (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":96,"title":"Slot? ","content":"Slot? : ∀ {rbs bs' msgs ffds'} →","searchableContent":"  slot? : ∀ {rbs bs' msgs ffds'} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":97,"title":"{ _ ","content":"{ _ : auto∶ allDone s }","searchableContent":"         { _ : auto∶ alldone s }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":98,"title":"{ _ ","content":"{ _ : auto∶ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs' }","searchableContent":"         { _ : auto∶ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs' }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":99,"title":"{ _ ","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds' } →","searchableContent":"         { _ : auto∶ ffds ffd.-⟦ ffd.fetch / ffd.fetchres msgs ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":100,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"         ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":101,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"         just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":102,"title":"{ FFDState  = ffds'","searchableContent":"             { ffdstate  = ffds'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":103,"title":"; BaseState = bs'","searchableContent":"             ; basestate = bs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":104,"title":"; Ledger    = constructLedger rbs","searchableContent":"             ; ledger    = constructledger rbs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":105,"title":"; slot      = suc slot","searchableContent":"             ; slot      = suc slot"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":106,"title":"; Upkeep    = ∅","searchableContent":"             ; upkeep    = ∅"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":107,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"             } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":108,"title":"Slot? {_} {_} {_} {_} {p} {q} {r} = Slot (toWitness p) (toWitness q) (toWitness r)","searchableContent":"  slot? {_} {_} {_} {_} {p} {q} {r} = slot (towitness p) (towitness q) (towitness r)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":109,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Trace.Verifier.Test
open import Leios.Prelude hiding (id)","searchableContent":"leios.short.trace.verifier.test
open import leios.prelude hiding (id)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":3,"title":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":5,"title":"module Leios.Short.Trace.Verifier.Test where","searchableContent":"module leios.short.trace.verifier.test where"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":7,"title":"params ","content":"params : Params","searchableContent":"params : params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":8,"title":"params =","searchableContent":"params ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":9,"title":"record","searchableContent":"  record"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":10,"title":"{ numberOfParties = 2","searchableContent":"    { numberofparties = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":11,"title":"; sutId = fzero","searchableContent":"    ; sutid = fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":12,"title":"; stakeDistribution =","searchableContent":"    ; stakedistribution ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":13,"title":"let open FunTot (completeFin 2) (maximalFin 2)","searchableContent":"        let open funtot (completefin 2) (maximalfin 2)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":14,"title":"in Fun⇒TotalMap (const 100000000)","searchableContent":"        in fun⇒totalmap (const 100000000)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":15,"title":"; stageLength = 2","searchableContent":"    ; stagelength = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":16,"title":"; winning-slots = fromList $","searchableContent":"    ; winning-slots = fromlist $"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":17,"title":"(IB , 0) ∷ (EB , 0) ∷ (VT , 0) ∷","searchableContent":"        (ib , 0) ∷ (eb , 0) ∷ (vt , 0) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":18,"title":"(IB , 1) ∷ (EB , 1) ∷ (VT , 1) ∷","searchableContent":"        (ib , 1) ∷ (eb , 1) ∷ (vt , 1) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":19,"title":"(IB , 2) ∷ (EB , 2) ∷ (VT , 2) ∷","searchableContent":"        (ib , 2) ∷ (eb , 2) ∷ (vt , 2) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":20,"title":"(IB , 3) ∷ (EB , 3) ∷ (VT , 3) ∷","searchableContent":"        (ib , 3) ∷ (eb , 3) ∷ (vt , 3) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":21,"title":"(IB , 4) ∷ (EB , 4) ∷ (VT , 4) ∷","searchableContent":"        (ib , 4) ∷ (eb , 4) ∷ (vt , 4) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":22,"title":"(IB , 5) ∷ (EB , 5) ∷ (VT , 5) ∷","searchableContent":"        (ib , 5) ∷ (eb , 5) ∷ (vt , 5) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":23,"title":"(VT , 6) ∷","searchableContent":"                              (vt , 6) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":24,"title":"(IB , 7) ∷ (EB , 7) ∷ (VT , 7) ∷","searchableContent":"        (ib , 7) ∷ (eb , 7) ∷ (vt , 7) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":25,"title":"(IB , 8) ∷ (EB , 8) ∷ (VT , 8) ∷","searchableContent":"        (ib , 8) ∷ (eb , 8) ∷ (vt , 8) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":26,"title":"[]","searchableContent":"        []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":27,"title":"}","searchableContent":"    }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":29,"title":"open Params params","searchableContent":"open params params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":30,"title":"open import Leios.Short.Trace.Verifier params","searchableContent":"open import leios.short.trace.verifier params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":32,"title":"private","searchableContent":"private"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":33,"title":"opaque","searchableContent":"  opaque"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":34,"title":"unfolding List-Model","searchableContent":"    unfolding list-model"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":36,"title":"test₁ ","content":"test₁ : Bool","searchableContent":"    test₁ : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":37,"title":"test₁ = ¿ ValidTrace (inj₁ (IB-Role-Action 0 , SLOT) ∷ []) ¿ᵇ","searchableContent":"    test₁ = ¿ validtrace (inj₁ (ib-role-action 0 , slot) ∷ []) ¿ᵇ"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":39,"title":"_ ","content":"_ : test₁ ≡ true","searchableContent":"    _ : test₁ ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":40,"title":"_ = refl","searchableContent":"    _ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":42,"title":"test-valid-ib ","content":"test-valid-ib : Bool","searchableContent":"    test-valid-ib : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":43,"title":"test-valid-ib =","searchableContent":"    test-valid-ib ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":44,"title":"let h = record { slotNumber = 1","searchableContent":"      let h = record { slotnumber = 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":45,"title":"; producerID = fsuc fzero","searchableContent":"                     ; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":46,"title":"; lotteryPf = tt","searchableContent":"                     ; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":47,"title":"; bodyHash = 0 ∷ 1 ∷ 2 ∷ []","searchableContent":"                     ; bodyhash = 0 ∷ 1 ∷ 2 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":48,"title":"; signature = tt","searchableContent":"                     ; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":49,"title":"}","searchableContent":"                     }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":50,"title":"b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }","searchableContent":"          b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":51,"title":"ib = record { header = h ; body = b }","searchableContent":"          ib = record { header = h ; body = b }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":52,"title":"pks = L.zip (completeFinL numberOfParties) (L.replicate numberOfParties tt)","searchableContent":"          pks = l.zip (completefinl numberofparties) (l.replicate numberofparties tt)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":53,"title":"s = initLeiosState tt stakeDistribution tt pks","searchableContent":"          s = initleiosstate tt stakedistribution tt pks"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":54,"title":"in isYes (ibValid? s ib)","searchableContent":"      in isyes (ibvalid? s ib)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":56,"title":"_ ","content":"_ : test-valid-ib ≡ true","searchableContent":"    _ : test-valid-ib ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":57,"title":"_ = refl","searchableContent":"    _ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":59,"title":"test₂ ","content":"test₂ : Bool","searchableContent":"    test₂ : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":60,"title":"test₂ =","searchableContent":"    test₂ ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":61,"title":"let t = L.reverse $","searchableContent":"      let t = l.reverse $"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":62,"title":"-- slot 0","searchableContent":"            -- slot 0"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":63,"title":"inj₁ (IB-Role-Action 0    , SLOT)","searchableContent":"              inj₁ (ib-role-action 0    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":64,"title":"∷ inj₁ (EB-Role-Action 0 [] , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 0 [] , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":65,"title":"∷ inj₁ (VT-Role-Action 0    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 0    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":66,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":67,"title":"∷ inj₁ (Slot-Action    0    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    0    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":68,"title":"-- slot 1","searchableContent":"            -- slot 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":69,"title":"∷ inj₁ (IB-Role-Action 1    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 1    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":70,"title":"∷ inj₁ (VT-Role-Action 1    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 1    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":71,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":72,"title":"∷ inj₁ (Slot-Action    1    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    1    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":73,"title":"-- slot 2","searchableContent":"            -- slot 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":74,"title":"∷ inj₂ (IB-Recv-Update","searchableContent":"            ∷ inj₂ (ib-recv-update"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":75,"title":"(record { header =","searchableContent":"                (record { header ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":76,"title":"record { slotNumber = 1","searchableContent":"                  record { slotnumber = 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":77,"title":"; producerID = fsuc fzero","searchableContent":"                         ; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":78,"title":"; lotteryPf = tt","searchableContent":"                         ; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":79,"title":"; bodyHash = 0 ∷ 1 ∷ 2 ∷ []","searchableContent":"                         ; bodyhash = 0 ∷ 1 ∷ 2 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":80,"title":"; signature = tt","searchableContent":"                         ; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":81,"title":"}","searchableContent":"                         }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":82,"title":"; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))","searchableContent":"                        ; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":83,"title":"∷ inj₁ (IB-Role-Action 2    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 2    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":84,"title":"∷ inj₁ (EB-Role-Action 2 [] , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 2 [] , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":85,"title":"∷ inj₁ (VT-Role-Action 2    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 2    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":86,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":87,"title":"∷ inj₁ (Slot-Action    2    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    2    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":88,"title":"-- slot 3","searchableContent":"            -- slot 3"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":89,"title":"∷ inj₂ (IB-Recv-Update","searchableContent":"            ∷ inj₂ (ib-recv-update"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":90,"title":"(record { header =","searchableContent":"                (record { header ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":91,"title":"record { slotNumber = 2","searchableContent":"                  record { slotnumber = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":92,"title":"; producerID = fsuc fzero","searchableContent":"                         ; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":93,"title":"; lotteryPf = tt","searchableContent":"                         ; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":94,"title":"; bodyHash = 3 ∷ 4 ∷ 5 ∷ []","searchableContent":"                         ; bodyhash = 3 ∷ 4 ∷ 5 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":95,"title":"; signature = tt","searchableContent":"                         ; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":96,"title":"}","searchableContent":"                         }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":97,"title":"; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))","searchableContent":"                        ; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":98,"title":"∷ inj₁ (IB-Role-Action 3    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 3    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":99,"title":"∷ inj₁ (VT-Role-Action 3    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 3    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":100,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":101,"title":"∷ inj₁ (Slot-Action    3    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    3    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":102,"title":"-- slot 4","searchableContent":"            -- slot 4"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":103,"title":"∷ inj₁ (IB-Role-Action 4    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 4    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":104,"title":"∷ inj₁ (EB-Role-Action 4 [] , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 4 [] , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":105,"title":"∷ inj₁ (VT-Role-Action 4    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 4    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":106,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":107,"title":"∷ inj₁ (Slot-Action    4    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    4    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":108,"title":"-- slot 5","searchableContent":"            -- slot 5"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":109,"title":"∷ inj₁ (IB-Role-Action 5    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 5    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":110,"title":"∷ inj₁ (VT-Role-Action 5    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 5    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":111,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":112,"title":"∷ inj₁ (Slot-Action    5    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    5    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":113,"title":"-- slot 6","searchableContent":"            -- slot 6"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":114,"title":"∷ inj₁ (No-IB-Role-Action   , SLOT)","searchableContent":"            ∷ inj₁ (no-ib-role-action   , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":115,"title":"∷ inj₁ (No-EB-Role-Action   , SLOT)","searchableContent":"            ∷ inj₁ (no-eb-role-action   , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":116,"title":"∷ inj₁ (VT-Role-Action 6    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 6    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":117,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":118,"title":"∷ inj₁ (Slot-Action    6    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    6    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":119,"title":"-- slot 7","searchableContent":"            -- slot 7"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":120,"title":"∷ inj₁ (IB-Role-Action 7    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 7    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":121,"title":"∷ inj₁ (VT-Role-Action 7    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 7    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":122,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":123,"title":"∷ inj₁ (Slot-Action    7    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    7    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":124,"title":"-- slot 8","searchableContent":"            -- slot 8"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":125,"title":"∷ inj₁ (IB-Role-Action 8    , SLOT)","searchableContent":"            ∷ inj₁ (ib-role-action 8    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":126,"title":"∷ inj₁ (EB-Role-Action 8 ((3 ∷ 4 ∷ 5 ∷ []) ∷ []) , SLOT)","searchableContent":"            ∷ inj₁ (eb-role-action 8 ((3 ∷ 4 ∷ 5 ∷ []) ∷ []) , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":127,"title":"∷ inj₁ (VT-Role-Action 8    , SLOT)","searchableContent":"            ∷ inj₁ (vt-role-action 8    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":128,"title":"∷ inj₁ (Base₂b-Action       , SLOT)","searchableContent":"            ∷ inj₁ (base₂b-action       , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":129,"title":"∷ inj₁ (Slot-Action    8    , SLOT)","searchableContent":"            ∷ inj₁ (slot-action    8    , slot)"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":130,"title":"∷ []","searchableContent":"            ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":131,"title":"in ¿ ValidTrace t ¿ᵇ","searchableContent":"      in ¿ validtrace t ¿ᵇ"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":133,"title":"_ ","content":"_ : test₂ ≡ true","searchableContent":"    _ : test₂ ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":134,"title":"_ = refl","searchableContent":"    _ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":135,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Trace.Verifier
open import Leios.Prelude hiding (id)","searchableContent":"leios.short.trace.verifier
open import leios.prelude hiding (id)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":3,"title":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":5,"title":"-- TODO","content":"-- TODO: SpecStructure as parameter","searchableContent":"-- todo: specstructure as parameter"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":6,"title":"module Leios.Short.Trace.Verifier (params ","content":"module Leios.Short.Trace.Verifier (params : Params) (let open Params params) where","searchableContent":"module leios.short.trace.verifier (params : params) (let open params params) where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":8,"title":"open import Leios.Defaults params","searchableContent":"open import leios.defaults params"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":9,"title":"using (LeiosState; initLeiosState; isb; hpe; hhs; htx; SendIB; FFDBuffers; Dec-SimpleFFD)","searchableContent":"  using (leiosstate; initleiosstate; isb; hpe; hhs; htx; sendib; ffdbuffers; dec-simpleffd)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":10,"title":"renaming (d-SpecStructure to traceSpecStructure) public","searchableContent":"  renaming (d-specstructure to tracespecstructure) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":12,"title":"open import Leios.SpecStructure using (SpecStructure)","searchableContent":"open import leios.specstructure using (specstructure)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":13,"title":"open SpecStructure traceSpecStructure hiding (Hashable-IBHeader; Hashable-EndorserBlock; isVoteCertified) public","searchableContent":"open specstructure tracespecstructure hiding (hashable-ibheader; hashable-endorserblock; isvotecertified) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":15,"title":"open import Leios.Short traceSpecStructure hiding (LeiosState; initLeiosState) public","searchableContent":"open import leios.short tracespecstructure hiding (leiosstate; initleiosstate) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":16,"title":"open import Prelude.Closures _↝_","searchableContent":"open import prelude.closures _↝_"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":17,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":19,"title":"data FFDUpdate ","content":"data FFDUpdate : Type where","searchableContent":"data ffdupdate : type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":20,"title":"IB-Recv-Update ","content":"IB-Recv-Update : InputBlock → FFDUpdate","searchableContent":"  ib-recv-update : inputblock → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":21,"title":"EB-Recv-Update ","content":"EB-Recv-Update : EndorserBlock → FFDUpdate","searchableContent":"  eb-recv-update : endorserblock → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":22,"title":"VT-Recv-Update ","content":"VT-Recv-Update : List Vote → FFDUpdate","searchableContent":"  vt-recv-update : list vote → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":24,"title":"data Action ","content":"data Action : Type where","searchableContent":"data action : type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":25,"title":"IB-Role-Action ","content":"IB-Role-Action : ℕ → Action","searchableContent":"  ib-role-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":26,"title":"EB-Role-Action ","content":"EB-Role-Action : ℕ → List IBRef → Action","searchableContent":"  eb-role-action : ℕ → list ibref → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":27,"title":"VT-Role-Action ","content":"VT-Role-Action : ℕ → Action","searchableContent":"  vt-role-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":28,"title":"No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action ","content":"No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action : Action","searchableContent":"  no-ib-role-action no-eb-role-action no-vt-role-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":29,"title":"Ftch-Action ","content":"Ftch-Action : Action","searchableContent":"  ftch-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":30,"title":"Slot-Action ","content":"Slot-Action : ℕ → Action","searchableContent":"  slot-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":31,"title":"Base₁-Action ","content":"Base₁-Action : Action","searchableContent":"  base₁-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":32,"title":"Base₂a-Action ","content":"Base₂a-Action : EndorserBlock → Action","searchableContent":"  base₂a-action : endorserblock → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":33,"title":"Base₂b-Action ","content":"Base₂b-Action : Action","searchableContent":"  base₂b-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":35,"title":"Actions = List (Action × LeiosInput)","searchableContent":"actions = list (action × leiosinput)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":37,"title":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":38,"title":"s s′ ","content":"s s′ : LeiosState","searchableContent":"  s s′ : leiosstate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":39,"title":"α ","content":"α : Action","searchableContent":"  α : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":41,"title":"data ValidUpdate ","content":"data ValidUpdate : FFDUpdate → LeiosState → Type where","searchableContent":"data validupdate : ffdupdate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":43,"title":"IB-Recv ","content":"IB-Recv : ∀ {ib} →","searchableContent":"  ib-recv : ∀ {ib} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":44,"title":"ValidUpdate (IB-Recv-Update ib) s","searchableContent":"    validupdate (ib-recv-update ib) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":46,"title":"EB-Recv ","content":"EB-Recv : ∀ {eb} →","searchableContent":"  eb-recv : ∀ {eb} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":47,"title":"ValidUpdate (EB-Recv-Update eb) s","searchableContent":"    validupdate (eb-recv-update eb) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":49,"title":"VT-Recv ","content":"VT-Recv : ∀ {vt} →","searchableContent":"  vt-recv : ∀ {vt} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":50,"title":"ValidUpdate (VT-Recv-Update vt) s","searchableContent":"    validupdate (vt-recv-update vt) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":52,"title":"data ValidAction ","content":"data ValidAction : Action → LeiosState → LeiosInput → Type where","searchableContent":"data validaction : action → leiosstate → leiosinput → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":54,"title":"IB-Role ","content":"IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":55,"title":"b = record { txs = ToPropose }","searchableContent":"                b = record { txs = topropose }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":56,"title":"h = mkIBHeader slot id tt sk-IB ToPropose","searchableContent":"                h = mkibheader slot id tt sk-ib topropose"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":57,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ibHeader h} {just (ibBody b)})","searchableContent":"                ffds' = proj₁ (ffd.send-total {ffds} {ibheader h} {just (ibbody b)})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":58,"title":"in .(needsUpkeep IB-Role) →","searchableContent":"            in .(needsupkeep ib-role) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":59,"title":".(canProduceIB slot sk-IB (stake s) tt) →","searchableContent":"               .(canproduceib slot sk-ib (stake s) tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":60,"title":".(ffds FFD.-⟦ FFD.Send (ibHeader h) (just (ibBody b)) / FFD.SendRes ⟧⇀ ffds') →","searchableContent":"               .(ffds ffd.-⟦ ffd.send (ibheader h) (just (ibbody b)) / ffd.sendres ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":61,"title":"ValidAction (IB-Role-Action slot) s SLOT","searchableContent":"               validaction (ib-role-action slot) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":63,"title":"EB-Role ","content":"EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":64,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":65,"title":"h = mkEB slot id tt sk-EB LI []","searchableContent":"                h = mkeb slot id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":66,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ebHeader h} {nothing})","searchableContent":"                ffds' = proj₁ (ffd.send-total {ffds} {ebheader h} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":67,"title":"in .(needsUpkeep EB-Role) →","searchableContent":"            in .(needsupkeep eb-role) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":68,"title":".(canProduceEB slot sk-EB (stake s) tt) →","searchableContent":"               .(canproduceeb slot sk-eb (stake s) tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":69,"title":".(ffds FFD.-⟦ FFD.Send (ebHeader h) nothing / FFD.SendRes ⟧⇀ ffds') →","searchableContent":"               .(ffds ffd.-⟦ ffd.send (ebheader h) nothing / ffd.sendres ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":70,"title":"ValidAction (EB-Role-Action slot LI) s SLOT","searchableContent":"               validaction (eb-role-action slot li) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":72,"title":"VT-Role ","content":"VT-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  vt-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":73,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":74,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":75,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {vtHeader votes} {nothing})","searchableContent":"                ffds' = proj₁ (ffd.send-total {ffds} {vtheader votes} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":76,"title":"in .(needsUpkeep VT-Role) →","searchableContent":"            in .(needsupkeep vt-role) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":77,"title":".(canProduceV slot sk-VT (stake s)) →","searchableContent":"               .(canproducev slot sk-vt (stake s)) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":78,"title":".(ffds FFD.-⟦ FFD.Send (vtHeader votes) nothing / FFD.SendRes ⟧⇀ ffds') →","searchableContent":"               .(ffds ffd.-⟦ ffd.send (vtheader votes) nothing / ffd.sendres ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":79,"title":"ValidAction (VT-Role-Action slot) s SLOT","searchableContent":"               validaction (vt-role-action slot) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":81,"title":"No-IB-Role ","content":"No-IB-Role : let open LeiosState s","searchableContent":"  no-ib-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":82,"title":"in needsUpkeep IB-Role →","searchableContent":"               in needsupkeep ib-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":83,"title":"(∀ π → ¬ canProduceIB slot sk-IB (stake s) π) →","searchableContent":"                  (∀ π → ¬ canproduceib slot sk-ib (stake s) π) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":84,"title":"ValidAction No-IB-Role-Action s SLOT","searchableContent":"                  validaction no-ib-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":86,"title":"No-EB-Role ","content":"No-EB-Role : let open LeiosState s","searchableContent":"  no-eb-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":87,"title":"in needsUpkeep EB-Role →","searchableContent":"               in needsupkeep eb-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":88,"title":"(∀ π → ¬ canProduceEB slot sk-EB (stake s) π) →","searchableContent":"                  (∀ π → ¬ canproduceeb slot sk-eb (stake s) π) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":89,"title":"ValidAction No-EB-Role-Action s SLOT","searchableContent":"                  validaction no-eb-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":91,"title":"No-VT-Role ","content":"No-VT-Role : let open LeiosState s","searchableContent":"  no-vt-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":92,"title":"in needsUpkeep VT-Role →","searchableContent":"               in needsupkeep vt-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":93,"title":"(¬ canProduceV slot sk-VT (stake s)) →","searchableContent":"                  (¬ canproducev slot sk-vt (stake s)) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":94,"title":"ValidAction No-VT-Role-Action s SLOT","searchableContent":"                  validaction no-vt-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":96,"title":"Slot ","content":"Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs)","searchableContent":"  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":97,"title":"(msgs , (ffds' , _)) = FFD.Fetch-total {ffds}","searchableContent":"             (msgs , (ffds' , _)) = ffd.fetch-total {ffds}"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":98,"title":"in .(allDone s) →","searchableContent":"         in .(alldone s) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":99,"title":".(bs B.-⟦ B.FTCH-LDG / B.BASE-LDG [] ⟧⇀ tt) →","searchableContent":"            .(bs b.-⟦ b.ftch-ldg / b.base-ldg [] ⟧⇀ tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":100,"title":".(ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds') →","searchableContent":"            .(ffds ffd.-⟦ ffd.fetch / ffd.fetchres msgs ⟧⇀ ffds') →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":101,"title":"ValidAction (Slot-Action slot) s SLOT","searchableContent":"            validaction (slot-action slot) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":103,"title":"Ftch ","content":"Ftch : ValidAction Ftch-Action s FTCH-LDG","searchableContent":"  ftch : validaction ftch-action s ftch-ldg"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":105,"title":"Base₁ ","content":"Base₁ : ∀ {txs} → ValidAction Base₁-Action s (SUBMIT (inj₂ txs))","searchableContent":"  base₁ : ∀ {txs} → validaction base₁-action s (submit (inj₂ txs))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":107,"title":"Base₂a ","content":"Base₂a : ∀ {eb} → let open LeiosState s renaming (BaseState to bs)","searchableContent":"  base₂a : ∀ {eb} → let open leiosstate s renaming (basestate to bs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":108,"title":"in .(needsUpkeep Base) →","searchableContent":"           in .(needsupkeep base) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":109,"title":".(eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs) →","searchableContent":"              .(eb ∈ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":110,"title":".(bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ tt) →","searchableContent":"              .(bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":111,"title":"ValidAction (Base₂a-Action eb) s SLOT","searchableContent":"              validaction (base₂a-action eb) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":113,"title":"Base₂b ","content":"Base₂b : let open LeiosState s renaming (BaseState to bs)","searchableContent":"  base₂b : let open leiosstate s renaming (basestate to bs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":114,"title":"in .(needsUpkeep Base) →","searchableContent":"           in .(needsupkeep base) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":115,"title":".([] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs) →","searchableContent":"              .([] ≡ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":116,"title":".(bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ tt) →","searchableContent":"              .(bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ tt) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":117,"title":"ValidAction Base₂b-Action s SLOT","searchableContent":"              validaction base₂b-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":119,"title":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":120,"title":"i ","content":"i : LeiosInput","searchableContent":"  i : leiosinput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":121,"title":"o ","content":"o : LeiosOutput","searchableContent":"  o : leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":123,"title":"⟦_⟧ ","content":"⟦_⟧ : ValidAction α s i → LeiosState × LeiosOutput","searchableContent":"⟦_⟧ : validaction α s i → leiosstate × leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":124,"title":"⟦ IB-Role {s} _ _ _ ⟧ =","searchableContent":"⟦ ib-role {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":125,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":126,"title":"b = record { txs = ToPropose }","searchableContent":"      b = record { txs = topropose }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":127,"title":"h = mkIBHeader slot id tt sk-IB ToPropose","searchableContent":"      h = mkibheader slot id tt sk-ib topropose"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":128,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ibHeader h} {just (ibBody b)})","searchableContent":"      ffds' = proj₁ (ffd.send-total {ffds} {ibheader h} {just (ibbody b)})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":129,"title":"in addUpkeep record s { FFDState = ffds' } IB-Role , EMPTY","searchableContent":"  in addupkeep record s { ffdstate = ffds' } ib-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":130,"title":"⟦ EB-Role {s} _ _ _ ⟧ =","searchableContent":"⟦ eb-role {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":131,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":132,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"      li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":133,"title":"h = mkEB slot id tt sk-EB LI []","searchableContent":"      h = mkeb slot id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":134,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {ebHeader h} {nothing})","searchableContent":"      ffds' = proj₁ (ffd.send-total {ffds} {ebheader h} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":135,"title":"in addUpkeep record s { FFDState = ffds' } EB-Role , EMPTY","searchableContent":"  in addupkeep record s { ffdstate = ffds' } eb-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":136,"title":"⟦ VT-Role {s} _ _ _ ⟧ =","searchableContent":"⟦ vt-role {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":137,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":138,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"      ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":139,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"      votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":140,"title":"ffds' = proj₁ (FFD.Send-total {ffds} {vtHeader votes} {nothing})","searchableContent":"      ffds' = proj₁ (ffd.send-total {ffds} {vtheader votes} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":141,"title":"in addUpkeep record s { FFDState = ffds' } VT-Role , EMPTY","searchableContent":"  in addupkeep record s { ffdstate = ffds' } vt-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":142,"title":"⟦ No-IB-Role {s} _ _ ⟧ = addUpkeep s IB-Role , EMPTY","searchableContent":"⟦ no-ib-role {s} _ _ ⟧ = addupkeep s ib-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":143,"title":"⟦ No-EB-Role {s} _ _ ⟧ = addUpkeep s EB-Role , EMPTY","searchableContent":"⟦ no-eb-role {s} _ _ ⟧ = addupkeep s eb-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":144,"title":"⟦ No-VT-Role {s} _ _ ⟧ = addUpkeep s VT-Role , EMPTY","searchableContent":"⟦ no-vt-role {s} _ _ ⟧ = addupkeep s vt-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":145,"title":"⟦ Slot {s} _ _ _ ⟧ =","searchableContent":"⟦ slot {s} _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":146,"title":"let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":147,"title":"(msgs , (ffds' , _)) = FFD.Fetch-total {ffds}","searchableContent":"      (msgs , (ffds' , _)) = ffd.fetch-total {ffds}"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":148,"title":"in","searchableContent":"  in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":149,"title":"(record s","searchableContent":"  (record s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":150,"title":"{ FFDState  = ffds'","searchableContent":"     { ffdstate  = ffds'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":151,"title":"; BaseState = tt","searchableContent":"     ; basestate = tt"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":152,"title":"; Ledger    = constructLedger []","searchableContent":"     ; ledger    = constructledger []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":153,"title":"; slot      = suc slot","searchableContent":"     ; slot      = suc slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":154,"title":"; Upkeep    = ∅","searchableContent":"     ; upkeep    = ∅"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":155,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"     } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":156,"title":", EMPTY)","searchableContent":"  , empty)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":157,"title":"⟦ Ftch {s} ⟧ = s , FTCH-LDG (LeiosState.Ledger s)","searchableContent":"⟦ ftch {s} ⟧ = s , ftch-ldg (leiosstate.ledger s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":158,"title":"⟦ Base₁ {s} {txs} ⟧ = record s { ToPropose = txs } , EMPTY","searchableContent":"⟦ base₁ {s} {txs} ⟧ = record s { topropose = txs } , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":159,"title":"⟦ Base₂a {s} _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY","searchableContent":"⟦ base₂a {s} _ _ _ ⟧ = addupkeep record s { basestate = tt } base , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":160,"title":"⟦ Base₂b {s} _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY","searchableContent":"⟦ base₂b {s} _ _ _ ⟧ = addupkeep record s { basestate = tt } base , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":162,"title":"open LeiosState","searchableContent":"open leiosstate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":163,"title":"open FFDBuffers","searchableContent":"open ffdbuffers"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":165,"title":"ValidAction→Eq-Slot ","content":"ValidAction→Eq-Slot : ∀ {s sl} → ValidAction (Slot-Action sl) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-slot : ∀ {s sl} → validaction (slot-action sl) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":166,"title":"ValidAction→Eq-Slot (Slot _ _ _) = refl","searchableContent":"validaction→eq-slot (slot _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":168,"title":"ValidAction→Eq-IB ","content":"ValidAction→Eq-IB : ∀ {s sl} → ValidAction (IB-Role-Action sl) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-ib : ∀ {s sl} → validaction (ib-role-action sl) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":169,"title":"ValidAction→Eq-IB (IB-Role _ _ _) = refl","searchableContent":"validaction→eq-ib (ib-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":171,"title":"ValidAction→Eq-EB ","content":"ValidAction→Eq-EB : ∀ {s sl ibs} → ValidAction (EB-Role-Action sl ibs) s SLOT → sl ≡ slot s × ibs ≡ (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))","searchableContent":"validaction→eq-eb : ∀ {s sl ibs} → validaction (eb-role-action sl ibs) s slot → sl ≡ slot s × ibs ≡ (map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":172,"title":"ValidAction→Eq-EB (EB-Role _ _ _) = refl , refl","searchableContent":"validaction→eq-eb (eb-role _ _ _) = refl , refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":174,"title":"ValidAction→Eq-VT ","content":"ValidAction→Eq-VT : ∀ {s sl} → ValidAction (VT-Role-Action sl) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-vt : ∀ {s sl} → validaction (vt-role-action sl) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":175,"title":"ValidAction→Eq-VT (VT-Role _ _ _) = refl","searchableContent":"validaction→eq-vt (vt-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":177,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":178,"title":"Dec-ValidAction ","content":"Dec-ValidAction : ValidAction ⁇³","searchableContent":"  dec-validaction : validaction ⁇³"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":179,"title":"Dec-ValidAction {IB-Role-Action sl} {s} {SLOT} .dec","searchableContent":"  dec-validaction {ib-role-action sl} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":180,"title":"with sl ≟ slot s","searchableContent":"    with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":181,"title":"... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-IB x))","searchableContent":"  ... | no ¬p = no λ x → ⊥-elim (¬p (validaction→eq-ib x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":182,"title":"... | yes p rewrite p","searchableContent":"  ... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":183,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":184,"title":"... | yes x | yes y | yes z = yes (IB-Role x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (ib-role x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":185,"title":"... | no ¬p | _ | _ = no λ where (IB-Role p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (ib-role p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":186,"title":"... | _ | no ¬p | _ = no λ where (IB-Role _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (ib-role _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":187,"title":"... | _ | _ | no ¬p = no λ where (IB-Role _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (ib-role _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":188,"title":"Dec-ValidAction {IB-Role-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ib-role-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":189,"title":"Dec-ValidAction {IB-Role-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ib-role-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":190,"title":"Dec-ValidAction {IB-Role-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {ib-role-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":191,"title":"Dec-ValidAction {EB-Role-Action sl ibs} {s} {SLOT} .dec","searchableContent":"  dec-validaction {eb-role-action sl ibs} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":192,"title":"with sl ≟ slot s | ibs ≟ (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))","searchableContent":"    with sl ≟ slot s | ibs ≟ (map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":193,"title":"... | no ¬p | _ = no λ x → ⊥-elim (¬p (proj₁ $ ValidAction→Eq-EB x))","searchableContent":"  ... | no ¬p | _ = no λ x → ⊥-elim (¬p (proj₁ $ validaction→eq-eb x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":194,"title":"... | _ | no ¬q = no λ x → ⊥-elim (¬q (proj₂ $ ValidAction→Eq-EB x))","searchableContent":"  ... | _ | no ¬q = no λ x → ⊥-elim (¬q (proj₂ $ validaction→eq-eb x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":195,"title":"... | yes p | yes q rewrite p rewrite q","searchableContent":"  ... | yes p | yes q rewrite p rewrite q"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":196,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":197,"title":"... | yes x | yes y | yes z = yes (EB-Role x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (eb-role x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":198,"title":"... | no ¬p | _ | _ = no λ where (EB-Role p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (eb-role p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":199,"title":"... | _ | no ¬p | _ = no λ where (EB-Role _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (eb-role _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":200,"title":"... | _ | _ | no ¬p = no λ where (EB-Role _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (eb-role _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":201,"title":"Dec-ValidAction {EB-Role-Action _ _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {eb-role-action _ _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":202,"title":"Dec-ValidAction {EB-Role-Action _ _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {eb-role-action _ _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":203,"title":"Dec-ValidAction {EB-Role-Action _ _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {eb-role-action _ _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":204,"title":"Dec-ValidAction {VT-Role-Action sl} {s} {SLOT} .dec","searchableContent":"  dec-validaction {vt-role-action sl} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":205,"title":"with sl ≟ slot s","searchableContent":"    with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":206,"title":"... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-VT x))","searchableContent":"  ... | no ¬p = no λ x → ⊥-elim (¬p (validaction→eq-vt x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":207,"title":"... | yes p rewrite p","searchableContent":"  ... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":208,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":209,"title":"... | yes x | yes y | yes z = yes (VT-Role x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (vt-role x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":210,"title":"... | no ¬p | _ | _ = no λ where (VT-Role p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (vt-role p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":211,"title":"... | _ | no ¬p | _ = no λ where (VT-Role _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (vt-role _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":212,"title":"... | _ | _ | no ¬p = no λ where (VT-Role _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (vt-role _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":213,"title":"Dec-ValidAction {VT-Role-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {vt-role-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":214,"title":"Dec-ValidAction {VT-Role-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {vt-role-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":215,"title":"Dec-ValidAction {VT-Role-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {vt-role-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":216,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {no-ib-role-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":217,"title":"with dec | dec","searchableContent":"    with dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":218,"title":"... | yes p | yes q = yes (No-IB-Role p q)","searchableContent":"  ... | yes p | yes q = yes (no-ib-role p q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":219,"title":"... | no ¬p | _ = no λ where (No-IB-Role p _) → ⊥-elim (¬p p)","searchableContent":"  ... | no ¬p | _ = no λ where (no-ib-role p _) → ⊥-elim (¬p p)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":220,"title":"... | _ | no ¬q = no λ where (No-IB-Role _ q) → ⊥-elim (¬q q)","searchableContent":"  ... | _ | no ¬q = no λ where (no-ib-role _ q) → ⊥-elim (¬q q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":221,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-ib-role-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":222,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-ib-role-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":223,"title":"Dec-ValidAction {No-IB-Role-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {no-ib-role-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":224,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {no-eb-role-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":225,"title":"with dec | dec","searchableContent":"    with dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":226,"title":"... | yes p | yes q = yes (No-EB-Role p q)","searchableContent":"  ... | yes p | yes q = yes (no-eb-role p q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":227,"title":"... | no ¬p | _ = no λ where (No-EB-Role p _) → ⊥-elim (¬p p)","searchableContent":"  ... | no ¬p | _ = no λ where (no-eb-role p _) → ⊥-elim (¬p p)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":228,"title":"... | _ | no ¬q = no λ where (No-EB-Role _ q) → ⊥-elim (¬q q)","searchableContent":"  ... | _ | no ¬q = no λ where (no-eb-role _ q) → ⊥-elim (¬q q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":229,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-eb-role-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":230,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-eb-role-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":231,"title":"Dec-ValidAction {No-EB-Role-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {no-eb-role-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":232,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {no-vt-role-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":233,"title":"with dec | dec","searchableContent":"    with dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":234,"title":"... | yes p | yes q = yes (No-VT-Role p q)","searchableContent":"  ... | yes p | yes q = yes (no-vt-role p q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":235,"title":"... | no ¬p | _ = no λ where (No-VT-Role p _) → ⊥-elim (¬p p)","searchableContent":"  ... | no ¬p | _ = no λ where (no-vt-role p _) → ⊥-elim (¬p p)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":236,"title":"... | _ | no ¬q = no λ where (No-VT-Role _ q) → ⊥-elim (¬q q)","searchableContent":"  ... | _ | no ¬q = no λ where (no-vt-role _ q) → ⊥-elim (¬q q)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":237,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-vt-role-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":238,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {no-vt-role-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":239,"title":"Dec-ValidAction {No-VT-Role-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {no-vt-role-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":240,"title":"Dec-ValidAction {Slot-Action sl} {s} {SLOT} .dec","searchableContent":"  dec-validaction {slot-action sl} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":241,"title":"with sl ≟ slot s","searchableContent":"    with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":242,"title":"... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-Slot x))","searchableContent":"  ... | no ¬p = no λ x → ⊥-elim (¬p (validaction→eq-slot x))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":243,"title":"... | yes p rewrite p","searchableContent":"  ... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":244,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":245,"title":"... | yes x | yes y | yes z = yes (Slot x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (slot x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":246,"title":"... | no ¬p | _ | _ = no λ where (Slot p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (slot p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":247,"title":"... | _ | no ¬p | _ = no λ where (Slot _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (slot _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":248,"title":"... | _ | _ | no ¬p = no λ where (Slot _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (slot _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":249,"title":"Dec-ValidAction {Slot-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {slot-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":250,"title":"Dec-ValidAction {Slot-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {slot-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":251,"title":"Dec-ValidAction {Slot-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {slot-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":252,"title":"Dec-ValidAction {Ftch-Action} {s} {FTCH-LDG} .dec = yes Ftch","searchableContent":"  dec-validaction {ftch-action} {s} {ftch-ldg} .dec = yes ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":253,"title":"Dec-ValidAction {Ftch-Action} {s} {SLOT} .dec = no λ ()","searchableContent":"  dec-validaction {ftch-action} {s} {slot} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":254,"title":"Dec-ValidAction {Ftch-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ftch-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":255,"title":"Dec-ValidAction {Ftch-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {ftch-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":256,"title":"Dec-ValidAction {Base₁-Action} {s} {SUBMIT (inj₁ ebs)} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {submit (inj₁ ebs)} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":257,"title":"Dec-ValidAction {Base₁-Action} {s} {SUBMIT (inj₂ txs)} .dec = yes (Base₁ {s} {txs})","searchableContent":"  dec-validaction {base₁-action} {s} {submit (inj₂ txs)} .dec = yes (base₁ {s} {txs})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":258,"title":"Dec-ValidAction {Base₁-Action} {s} {SLOT} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {slot} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":259,"title":"Dec-ValidAction {Base₁-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":260,"title":"Dec-ValidAction {Base₁-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₁-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":261,"title":"Dec-ValidAction {Base₂a-Action eb} {s} {SLOT} .dec","searchableContent":"  dec-validaction {base₂a-action eb} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":262,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":263,"title":"... | yes x | yes y | yes z = yes (Base₂a x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (base₂a x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":264,"title":"... | no ¬p | _ | _ = no λ where (Base₂a p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (base₂a p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":265,"title":"... | _ | no ¬p | _ = no λ where (Base₂a {s} {eb} _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (base₂a {s} {eb} _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":266,"title":"... | _ | _ | no ¬p = no λ where (Base₂a _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (base₂a _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":267,"title":"Dec-ValidAction {Base₂a-Action _} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂a-action _} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":268,"title":"Dec-ValidAction {Base₂a-Action _} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {base₂a-action _} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":269,"title":"Dec-ValidAction {Base₂a-Action _} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂a-action _} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":270,"title":"Dec-ValidAction {Base₂b-Action} {s} {SLOT} .dec","searchableContent":"  dec-validaction {base₂b-action} {s} {slot} .dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":271,"title":"with dec | dec | dec","searchableContent":"    with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":272,"title":"... | yes x | yes y | yes z = yes (Base₂b x y z)","searchableContent":"  ... | yes x | yes y | yes z = yes (base₂b x y z)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":273,"title":"... | no ¬p | _ | _ = no λ where (Base₂b p _ _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | no ¬p | _ | _ = no λ where (base₂b p _ _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":274,"title":"... | _ | no ¬p | _ = no λ where (Base₂b _ p _) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | no ¬p | _ = no λ where (base₂b _ p _) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":275,"title":"... | _ | _ | no ¬p = no λ where (Base₂b _ _ p) → ⊥-elim (¬p (recompute dec p))","searchableContent":"  ... | _ | _ | no ¬p = no λ where (base₂b _ _ p) → ⊥-elim (¬p (recompute dec p))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":276,"title":"Dec-ValidAction {Base₂b-Action} {s} {SUBMIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂b-action} {s} {submit _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":277,"title":"Dec-ValidAction {Base₂b-Action} {s} {FTCH-LDG} .dec = no λ ()","searchableContent":"  dec-validaction {base₂b-action} {s} {ftch-ldg} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":278,"title":"Dec-ValidAction {Base₂b-Action} {s} {INIT _} .dec = no λ ()","searchableContent":"  dec-validaction {base₂b-action} {s} {init _} .dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":280,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":281,"title":"Dec-ValidUpdate ","content":"Dec-ValidUpdate : ValidUpdate ⁇²","searchableContent":"  dec-validupdate : validupdate ⁇²"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":282,"title":"Dec-ValidUpdate {IB-Recv-Update _} .dec = yes IB-Recv","searchableContent":"  dec-validupdate {ib-recv-update _} .dec = yes ib-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":283,"title":"Dec-ValidUpdate {EB-Recv-Update _} .dec = yes EB-Recv","searchableContent":"  dec-validupdate {eb-recv-update _} .dec = yes eb-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":284,"title":"Dec-ValidUpdate {VT-Recv-Update _} .dec = yes VT-Recv","searchableContent":"  dec-validupdate {vt-recv-update _} .dec = yes vt-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":286,"title":"mutual","searchableContent":"mutual"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":287,"title":"data ValidTrace ","content":"data ValidTrace : List ((Action × LeiosInput) ⊎ FFDUpdate) → Type where","searchableContent":"  data validtrace : list ((action × leiosinput) ⊎ ffdupdate) → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":288,"title":"[] ","content":"[] :","searchableContent":"    [] :"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":289,"title":"─────────────","searchableContent":"      ─────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":290,"title":"ValidTrace []","searchableContent":"      validtrace []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":292,"title":"_/_∷_⊣_ ","content":"_/_∷_⊣_ : ∀ α i {αs} →","searchableContent":"    _/_∷_⊣_ : ∀ α i {αs} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":293,"title":"∀ (tr ","content":"∀ (tr : ValidTrace αs) →","searchableContent":"      ∀ (tr : validtrace αs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":294,"title":"∙ ValidAction α (proj₁ ⟦ tr ⟧∗) i","searchableContent":"      ∙ validaction α (proj₁ ⟦ tr ⟧∗) i"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":295,"title":"───────────────────","searchableContent":"        ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":296,"title":"ValidTrace (inj₁ (α , i) ∷ αs)","searchableContent":"        validtrace (inj₁ (α , i) ∷ αs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":298,"title":"_↥_ ","content":"_↥_ : ∀ {f αs} →","searchableContent":"    _↥_ : ∀ {f αs} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":299,"title":"∀ (tr ","content":"∀ (tr : ValidTrace αs) →","searchableContent":"      ∀ (tr : validtrace αs) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":300,"title":"(vu ","content":"(vu : ValidUpdate f (proj₁ ⟦ tr ⟧∗)) →","searchableContent":"        (vu : validupdate f (proj₁ ⟦ tr ⟧∗)) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":301,"title":"───────────────────","searchableContent":"        ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":302,"title":"ValidTrace (inj₂ f ∷ αs)","searchableContent":"        validtrace (inj₂ f ∷ αs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":305,"title":"⟦_⟧∗ ","content":"⟦_⟧∗ : ∀ {αs : List ((Action × LeiosInput) ⊎ FFDUpdate)} → ValidTrace αs → LeiosState × LeiosOutput","searchableContent":"  ⟦_⟧∗ : ∀ {αs : list ((action × leiosinput) ⊎ ffdupdate)} → validtrace αs → leiosstate × leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":306,"title":"⟦ [] ⟧∗ = initLeiosState tt stakeDistribution tt pks , EMPTY","searchableContent":"  ⟦ [] ⟧∗ = initleiosstate tt stakedistribution tt pks , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":307,"title":"where pks = L.zip (completeFinL numberOfParties) (L.replicate numberOfParties tt)","searchableContent":"    where pks = l.zip (completefinl numberofparties) (l.replicate numberofparties tt)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":308,"title":"⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧","searchableContent":"  ⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":309,"title":"⟦ _↥_ {IB-Recv-Update ib} tr vu ⟧∗ =","searchableContent":"  ⟦ _↥_ {ib-recv-update ib} tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":310,"title":"let (s , o) = ⟦ tr ⟧∗","searchableContent":"    let (s , o) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":311,"title":"in record s { FFDState = record (FFDState s) { inIBs = ib ∷ inIBs (FFDState s)}} , o","searchableContent":"    in record s { ffdstate = record (ffdstate s) { inibs = ib ∷ inibs (ffdstate s)}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":312,"title":"⟦ _↥_ {EB-Recv-Update eb} tr vu ⟧∗ =","searchableContent":"  ⟦ _↥_ {eb-recv-update eb} tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":313,"title":"let (s , o) = ⟦ tr ⟧∗","searchableContent":"    let (s , o) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":314,"title":"in record s { FFDState = record (FFDState s) { inEBs = eb ∷ inEBs (FFDState s)}} , o","searchableContent":"    in record s { ffdstate = record (ffdstate s) { inebs = eb ∷ inebs (ffdstate s)}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":315,"title":"⟦ _↥_ {VT-Recv-Update vt} tr vu ⟧∗ =","searchableContent":"  ⟦ _↥_ {vt-recv-update vt} tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":316,"title":"let (s , o) = ⟦ tr ⟧∗","searchableContent":"    let (s , o) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":317,"title":"in record s { FFDState = record (FFDState s) { inVTs = vt ∷ inVTs (FFDState s)}} , o","searchableContent":"    in record s { ffdstate = record (ffdstate s) { invts = vt ∷ invts (ffdstate s)}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":319,"title":"Irr-ValidAction ","content":"Irr-ValidAction : Irrelevant (ValidAction α s i)","searchableContent":"irr-validaction : irrelevant (validaction α s i)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":320,"title":"Irr-ValidAction (IB-Role _ _ _) (IB-Role _ _ _)   = refl","searchableContent":"irr-validaction (ib-role _ _ _) (ib-role _ _ _)   = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":321,"title":"Irr-ValidAction (EB-Role _ _ _) (EB-Role _ _ _)   = refl","searchableContent":"irr-validaction (eb-role _ _ _) (eb-role _ _ _)   = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":322,"title":"Irr-ValidAction (VT-Role _ _ _) (VT-Role _ _ _)   = refl","searchableContent":"irr-validaction (vt-role _ _ _) (vt-role _ _ _)   = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":323,"title":"Irr-ValidAction (No-IB-Role _ _) (No-IB-Role _ _) = refl","searchableContent":"irr-validaction (no-ib-role _ _) (no-ib-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":324,"title":"Irr-ValidAction (No-EB-Role _ _) (No-EB-Role _ _) = refl","searchableContent":"irr-validaction (no-eb-role _ _) (no-eb-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":325,"title":"Irr-ValidAction (No-VT-Role _ _) (No-VT-Role _ _) = refl","searchableContent":"irr-validaction (no-vt-role _ _) (no-vt-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":326,"title":"Irr-ValidAction (Slot _ _ _) (Slot _ _ _)         = refl","searchableContent":"irr-validaction (slot _ _ _) (slot _ _ _)         = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":327,"title":"Irr-ValidAction Ftch Ftch                         = refl","searchableContent":"irr-validaction ftch ftch                         = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":328,"title":"Irr-ValidAction Base₁ Base₁                       = refl","searchableContent":"irr-validaction base₁ base₁                       = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":329,"title":"Irr-ValidAction (Base₂a _ _ _) (Base₂a _ _ _)     = refl","searchableContent":"irr-validaction (base₂a _ _ _) (base₂a _ _ _)     = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":330,"title":"Irr-ValidAction (Base₂b _ _ _) (Base₂b _ _ _)     = refl","searchableContent":"irr-validaction (base₂b _ _ _) (base₂b _ _ _)     = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":332,"title":"Irr-ValidUpdate ","content":"Irr-ValidUpdate : ∀ {f} → Irrelevant (ValidUpdate f s)","searchableContent":"irr-validupdate : ∀ {f} → irrelevant (validupdate f s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":333,"title":"Irr-ValidUpdate IB-Recv IB-Recv = refl","searchableContent":"irr-validupdate ib-recv ib-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":334,"title":"Irr-ValidUpdate EB-Recv EB-Recv = refl","searchableContent":"irr-validupdate eb-recv eb-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":335,"title":"Irr-ValidUpdate VT-Recv VT-Recv = refl","searchableContent":"irr-validupdate vt-recv vt-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":337,"title":"Irr-ValidTrace ","content":"Irr-ValidTrace : ∀ {αs} → Irrelevant (ValidTrace αs)","searchableContent":"irr-validtrace : ∀ {αs} → irrelevant (validtrace αs)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":338,"title":"Irr-ValidTrace [] [] = refl","searchableContent":"irr-validtrace [] [] = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":339,"title":"Irr-ValidTrace (α / i ∷ vαs ⊣ vα) (.α / .i ∷ vαs′ ⊣ vα′)","searchableContent":"irr-validtrace (α / i ∷ vαs ⊣ vα) (.α / .i ∷ vαs′ ⊣ vα′)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":340,"title":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidAction vα vα′","searchableContent":"  rewrite irr-validtrace vαs vαs′ | irr-validaction vα vα′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":341,"title":"= refl","searchableContent":"  = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":342,"title":"Irr-ValidTrace (vαs ↥ u) (vαs′ ↥ u′)","searchableContent":"irr-validtrace (vαs ↥ u) (vαs′ ↥ u′)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":343,"title":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidUpdate u u′","searchableContent":"  rewrite irr-validtrace vαs vαs′ | irr-validupdate u u′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":344,"title":"= refl","searchableContent":"  = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":346,"title":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":347,"title":"Dec-ValidTrace ","content":"Dec-ValidTrace : ValidTrace ⁇¹","searchableContent":"  dec-validtrace : validtrace ⁇¹"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":348,"title":"Dec-ValidTrace {tr} .dec with tr","searchableContent":"  dec-validtrace {tr} .dec with tr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":349,"title":"... | [] = yes []","searchableContent":"  ... | [] = yes []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":350,"title":"... | inj₁ (α , i) ∷ αs","searchableContent":"  ... | inj₁ (α , i) ∷ αs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":351,"title":"with ¿ ValidTrace αs ¿","searchableContent":"    with ¿ validtrace αs ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":352,"title":"... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs","searchableContent":"  ... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":353,"title":"... | yes vαs","searchableContent":"  ... | yes vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":354,"title":"with ¿ ValidAction α (proj₁ ⟦ vαs ⟧∗) i ¿","searchableContent":"    with ¿ validaction α (proj₁ ⟦ vαs ⟧∗) i ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":355,"title":"... | no ¬vα = no λ where","searchableContent":"  ... | no ¬vα = no λ where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":356,"title":"(_ / _ ∷ tr ⊣ vα) → ¬vα","searchableContent":"    (_ / _ ∷ tr ⊣ vα) → ¬vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":357,"title":"$ subst (λ x → ValidAction α x i) (cong (proj₁ ∘ ⟦_⟧∗) $ Irr-ValidTrace tr vαs) vα","searchableContent":"                  $ subst (λ x → validaction α x i) (cong (proj₁ ∘ ⟦_⟧∗) $ irr-validtrace tr vαs) vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":358,"title":"... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα","searchableContent":"  ... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":359,"title":"Dec-ValidTrace {tr} .dec | inj₂ u ∷ αs","searchableContent":"  dec-validtrace {tr} .dec | inj₂ u ∷ αs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":360,"title":"with ¿ ValidTrace αs ¿","searchableContent":"    with ¿ validtrace αs ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":361,"title":"... | no ¬vαs = no λ where (vαs ↥ _) → ¬vαs vαs","searchableContent":"  ... | no ¬vαs = no λ where (vαs ↥ _) → ¬vαs vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":362,"title":"... | yes vαs","searchableContent":"  ... | yes vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":363,"title":"with ¿ ValidUpdate u (proj₁ ⟦ vαs ⟧∗) ¿","searchableContent":"    with ¿ validupdate u (proj₁ ⟦ vαs ⟧∗) ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":364,"title":"... | yes vu = yes (vαs ↥ vu)","searchableContent":"  ... | yes vu = yes (vαs ↥ vu)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":365,"title":"... | no ¬vu = no λ where","searchableContent":"  ... | no ¬vu = no λ where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":366,"title":"(tr ↥ vu) → ¬vu $ subst (λ x → ValidUpdate u x) (cong (proj₁ ∘ ⟦_⟧∗) $ Irr-ValidTrace tr vαs) vu","searchableContent":"    (tr ↥ vu) → ¬vu $ subst (λ x → validupdate u x) (cong (proj₁ ∘ ⟦_⟧∗) $ irr-validtrace tr vαs) vu"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":368,"title":"data _⇑_ ","content":"data _⇑_ : LeiosState → LeiosState → Type where","searchableContent":"data _⇑_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":370,"title":"UpdateIB ","content":"UpdateIB : ∀ {s ib} → let open LeiosState s renaming (FFDState to ffds) in","searchableContent":"  updateib : ∀ {s ib} → let open leiosstate s renaming (ffdstate to ffds) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":371,"title":"s ⇑ record s { FFDState = record ffds { inIBs = ib ∷ inIBs ffds } }","searchableContent":"    s ⇑ record s { ffdstate = record ffds { inibs = ib ∷ inibs ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":373,"title":"UpdateEB ","content":"UpdateEB : ∀ {s eb} → let open LeiosState s renaming (FFDState to ffds) in","searchableContent":"  updateeb : ∀ {s eb} → let open leiosstate s renaming (ffdstate to ffds) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":374,"title":"s ⇑ record s { FFDState = record ffds { inEBs = eb ∷ inEBs ffds } }","searchableContent":"    s ⇑ record s { ffdstate = record ffds { inebs = eb ∷ inebs ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":376,"title":"UpdateVT ","content":"UpdateVT : ∀ {s vt} → let open LeiosState s renaming (FFDState to ffds) in","searchableContent":"  updatevt : ∀ {s vt} → let open leiosstate s renaming (ffdstate to ffds) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":377,"title":"s ⇑ record s { FFDState = record ffds { inVTs = vt ∷ inVTs ffds } }","searchableContent":"    s ⇑ record s { ffdstate = record ffds { invts = vt ∷ invts ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":379,"title":"data LocalStep ","content":"data LocalStep : LeiosState → LeiosState → Type where","searchableContent":"data localstep : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":381,"title":"StateStep ","content":"StateStep : ∀ {s i o s′} →","searchableContent":"  statestep : ∀ {s i o s′} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":382,"title":"∙ just s -⟦ i / o ⟧⇀ s′","searchableContent":"    ∙ just s -⟦ i / o ⟧⇀ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":383,"title":"───────────────────","searchableContent":"      ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":384,"title":"LocalStep s s′","searchableContent":"      localstep s s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":386,"title":"UpdateState ","content":"UpdateState : ∀ {s s′} →","searchableContent":"  updatestate : ∀ {s s′} →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":387,"title":"∙ s ⇑ s′","searchableContent":"    ∙ s ⇑ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":388,"title":"───────────────────","searchableContent":"      ───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":389,"title":"LocalStep s s′","searchableContent":"      localstep s s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":391,"title":"-- TODO","content":"-- TODO: add base layer update","searchableContent":"  -- todo: add base layer update"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":393,"title":"getLabel ","content":"getLabel : just s -⟦ i / o ⟧⇀ s′ → Action","searchableContent":"getlabel : just s -⟦ i / o ⟧⇀ s′ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":394,"title":"getLabel (Slot {s} _ _ _)            = Slot-Action (slot s)","searchableContent":"getlabel (slot {s} _ _ _)            = slot-action (slot s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":395,"title":"getLabel Ftch                        = Ftch-Action","searchableContent":"getlabel ftch                        = ftch-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":396,"title":"getLabel Base₁                       = Base₁-Action","searchableContent":"getlabel base₁                       = base₁-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":397,"title":"getLabel (Base₂a {s} {eb} _ _ _)     = Base₂a-Action eb","searchableContent":"getlabel (base₂a {s} {eb} _ _ _)     = base₂a-action eb"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":398,"title":"getLabel (Base₂b _ _ _)              = Base₂b-Action","searchableContent":"getlabel (base₂b _ _ _)              = base₂b-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":399,"title":"getLabel (Roles (IB-Role {s} _ _ _)) = IB-Role-Action (slot s)","searchableContent":"getlabel (roles (ib-role {s} _ _ _)) = ib-role-action (slot s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":400,"title":"getLabel (Roles (EB-Role {s} _ _ _)) = EB-Role-Action (slot s) (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))","searchableContent":"getlabel (roles (eb-role {s} _ _ _)) = eb-role-action (slot s) (map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":401,"title":"getLabel (Roles (VT-Role {s} _ _ _)) = VT-Role-Action (slot s)","searchableContent":"getlabel (roles (vt-role {s} _ _ _)) = vt-role-action (slot s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":402,"title":"getLabel (Roles (No-IB-Role _ _))    = No-IB-Role-Action","searchableContent":"getlabel (roles (no-ib-role _ _))    = no-ib-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":403,"title":"getLabel (Roles (No-EB-Role _ _))    = No-EB-Role-Action","searchableContent":"getlabel (roles (no-eb-role _ _))    = no-eb-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":404,"title":"getLabel (Roles (No-VT-Role _ _))    = No-VT-Role-Action","searchableContent":"getlabel (roles (no-vt-role _ _))    = no-vt-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":406,"title":"ValidAction-sound ","content":"ValidAction-sound : (vα : ValidAction α s i) → let (s′ , o) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′","searchableContent":"validaction-sound : (vα : validaction α s i) → let (s′ , o) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":407,"title":"ValidAction-sound (Slot x x₁ x₂)    = Slot {rbs = []} (recompute dec x) (recompute dec x₁) (recompute dec x₂)","searchableContent":"validaction-sound (slot x x₁ x₂)    = slot {rbs = []} (recompute dec x) (recompute dec x₁) (recompute dec x₂)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":408,"title":"ValidAction-sound Ftch              = Ftch","searchableContent":"validaction-sound ftch              = ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":409,"title":"ValidAction-sound Base₁             = Base₁","searchableContent":"validaction-sound base₁             = base₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":410,"title":"ValidAction-sound (Base₂a x x₁ x₂)  = Base₂a (recompute dec x) (recompute dec x₁) (recompute dec x₂)","searchableContent":"validaction-sound (base₂a x x₁ x₂)  = base₂a (recompute dec x) (recompute dec x₁) (recompute dec x₂)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":411,"title":"ValidAction-sound (Base₂b x x₁ x₂)  = Base₂b (recompute dec x) (recompute dec x₁) (recompute dec x₂)","searchableContent":"validaction-sound (base₂b x x₁ x₂)  = base₂b (recompute dec x) (recompute dec x₁) (recompute dec x₂)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":412,"title":"ValidAction-sound (IB-Role x x₁ x₂) = Roles (IB-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))","searchableContent":"validaction-sound (ib-role x x₁ x₂) = roles (ib-role (recompute dec x) (recompute dec x₁) (recompute dec x₂))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":413,"title":"ValidAction-sound (EB-Role x x₁ x₂) = Roles (EB-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))","searchableContent":"validaction-sound (eb-role x x₁ x₂) = roles (eb-role (recompute dec x) (recompute dec x₁) (recompute dec x₂))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":414,"title":"ValidAction-sound (VT-Role x x₁ x₂) = Roles (VT-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))","searchableContent":"validaction-sound (vt-role x x₁ x₂) = roles (vt-role (recompute dec x) (recompute dec x₁) (recompute dec x₂))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":415,"title":"ValidAction-sound (No-IB-Role x x₁) = Roles (No-IB-Role x x₁)","searchableContent":"validaction-sound (no-ib-role x x₁) = roles (no-ib-role x x₁)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":416,"title":"ValidAction-sound (No-EB-Role x x₁) = Roles (No-EB-Role x x₁)","searchableContent":"validaction-sound (no-eb-role x x₁) = roles (no-eb-role x x₁)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":417,"title":"ValidAction-sound (No-VT-Role x x₁) = Roles (No-VT-Role x x₁)","searchableContent":"validaction-sound (no-vt-role x x₁) = roles (no-vt-role x x₁)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":419,"title":"ValidAction-complete ","content":"ValidAction-complete : (st : just s -⟦ i / o ⟧⇀ s′) → ValidAction (getLabel st) s i","searchableContent":"validaction-complete : (st : just s -⟦ i / o ⟧⇀ s′) → validaction (getlabel st) s i"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":420,"title":"ValidAction-complete {s} {s′} (Roles (IB-Role {s} {π} {ffds'} x x₁ _)) =","searchableContent":"validaction-complete {s} {s′} (roles (ib-role {s} {π} {ffds'} x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":421,"title":"let b  = record { txs = ToPropose s }","searchableContent":"  let b  = record { txs = topropose s }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":422,"title":"h  = mkIBHeader (slot s) id tt sk-IB (ToPropose s)","searchableContent":"      h  = mkibheader (slot s) id tt sk-ib (topropose s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":423,"title":"pr = proj₂ (FFD.Send-total {FFDState s} {ibHeader h} {just (ibBody b)})","searchableContent":"      pr = proj₂ (ffd.send-total {ffdstate s} {ibheader h} {just (ibbody b)})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":424,"title":"in IB-Role {s} x x₁ pr","searchableContent":"  in ib-role {s} x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":425,"title":"ValidAction-complete {s} (Roles (EB-Role x x₁ _)) =","searchableContent":"validaction-complete {s} (roles (eb-role x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":426,"title":"let LI = map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s)","searchableContent":"  let li = map getibref $ filter (_∈ᴮ slice l (slot s) 3) (ibs s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":427,"title":"h  = mkEB (slot s) id tt sk-EB LI []","searchableContent":"      h  = mkeb (slot s) id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":428,"title":"pr = proj₂ (FFD.Send-total {FFDState s} {ebHeader h} {nothing})","searchableContent":"      pr = proj₂ (ffd.send-total {ffdstate s} {ebheader h} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":429,"title":"in EB-Role {s} x x₁ pr","searchableContent":"  in eb-role {s} x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":430,"title":"ValidAction-complete {s} (Roles (VT-Role x x₁ _))  =","searchableContent":"validaction-complete {s} (roles (vt-role x x₁ _))  ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":431,"title":"let EBs'  = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L (slot s) 1) (EBs s)","searchableContent":"  let ebs'  = filter (allibrefsknown s) $ filter (_∈ᴮ slice l (slot s) 1) (ebs s)"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":432,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"      votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":433,"title":"pr    = proj₂ (FFD.Send-total {FFDState s} {vtHeader votes} {nothing})","searchableContent":"      pr    = proj₂ (ffd.send-total {ffdstate s} {vtheader votes} {nothing})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":434,"title":"in VT-Role {s} x x₁ pr","searchableContent":"  in vt-role {s} x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":435,"title":"ValidAction-complete (Roles (No-IB-Role x x₁)) = No-IB-Role x x₁","searchableContent":"validaction-complete (roles (no-ib-role x x₁)) = no-ib-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":436,"title":"ValidAction-complete (Roles (No-EB-Role x x₁)) = No-EB-Role x x₁","searchableContent":"validaction-complete (roles (no-eb-role x x₁)) = no-eb-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":437,"title":"ValidAction-complete (Roles (No-VT-Role x x₁)) = No-VT-Role x x₁","searchableContent":"validaction-complete (roles (no-vt-role x x₁)) = no-vt-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":438,"title":"ValidAction-complete Ftch                      = Ftch","searchableContent":"validaction-complete ftch                      = ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":439,"title":"ValidAction-complete Base₁                     = Base₁","searchableContent":"validaction-complete base₁                     = base₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":440,"title":"ValidAction-complete (Base₂a x x₁ x₂)          = Base₂a x x₁ x₂","searchableContent":"validaction-complete (base₂a x x₁ x₂)          = base₂a x x₁ x₂"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":441,"title":"ValidAction-complete (Base₂b x x₁ x₂)          = Base₂b x x₁ x₂","searchableContent":"validaction-complete (base₂b x x₁ x₂)          = base₂b x x₁ x₂"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":442,"title":"ValidAction-complete {s} (Slot x x₁ _)         = Slot x x₁ (proj₂ (proj₂ (FFD.Fetch-total {FFDState s})))","searchableContent":"validaction-complete {s} (slot x x₁ _)         = slot x x₁ (proj₂ (proj₂ (ffd.fetch-total {ffdstate s})))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":443,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":2,"title":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":3,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":4,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":5,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":6,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":7,"title":"Leios.Short","content":"Leios.Short","searchableContent":" leios.short"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":8,"title":"","content":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":25,"title":"","searchableContent":" "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":26,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":27,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":28,"title":"

Short-Pipeline Leios

","searchableContent":"

short-pipeline leios

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":29,"title":"","content":"-->","searchableContent":"-->"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":43,"title":"

This document is a specification of Short-Pipeline Leios, usually","content":"

This document is a specification of Short-Pipeline Leios, usually","searchableContent":"

this document is a specification of short-pipeline leios, usually"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":44,"title":"abbreviated as Short Leios. On a high level, the pipeline looks like","content":"abbreviated as Short Leios. On a high level, the pipeline looks like","searchableContent":"abbreviated as short leios. on a high level, the pipeline looks like"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":45,"title":"this","content":"this:

","searchableContent":"this:

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":46,"title":"
    ","searchableContent":"
      "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":47,"title":"
    1. If elected, propose IB
      2. ","content":"
    2. If elected, propose IB
      3. ","searchableContent":"
    3. if elected, propose ib
      4. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":48,"title":"
    4. Wait
      5. ","content":"
    5. Wait
      6. ","searchableContent":"
    6. wait
      7. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":49,"title":"
    7. Wait
      8. ","content":"
    8. Wait
      9. ","searchableContent":"
    9. wait
      10. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":50,"title":"
    10. If elected, propose EB
      11. ","content":"
    11. If elected, propose EB
      12. ","searchableContent":"
    12. if elected, propose eb
      13. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":51,"title":"
    13. If elected, vote If elected, propose RB
      14. ","content":"
    14. If elected, vote If elected, propose RB
      15. ","searchableContent":"
    15. if elected, vote if elected, propose rb
      16. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":52,"title":"
    ","content":"
","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":53,"title":"

Upkeep

","searchableContent":"

upkeep

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":54,"title":"

A node that never produces a block even though it could is not","content":"

A node that never produces a block even though it could is not","searchableContent":"

a node that never produces a block even though it could is not"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":55,"title":"supposed to be an honest node, and we prevent that by tracking whether a","content":"supposed to be an honest node, and we prevent that by tracking whether a","searchableContent":"supposed to be an honest node, and we prevent that by tracking whether a"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":56,"title":"node has checked if it can make a block in a particular slot.","content":"node has checked if it can make a block in a particular slot.","searchableContent":"node has checked if it can make a block in a particular slot."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":57,"title":"LeiosState contains a set of SlotUpkeep and we","content":"LeiosState contains a set of SlotUpkeep and we","searchableContent":"leiosstate contains a set of slotupkeep and we"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":58,"title":"ensure that this set contains all elements before we can advance to the","content":"ensure that this set contains all elements before we can advance to the","searchableContent":"ensure that this set contains all elements before we can advance to the"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":59,"title":"next slot, resetting this field to the empty set.

","content":"next slot, resetting this field to the empty set.

","searchableContent":"next slot, resetting this field to the empty set.

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":60,"title":"
data SlotUpkeep ","content":"
data SlotUpkeep : Type where","searchableContent":"
data slotupkeep : type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":61,"title":"Base IB-Role EB-Role VT-Role ","content":"Base IB-Role EB-Role VT-Role : SlotUpkeep","searchableContent":"  base ib-role eb-role vt-role : slotupkeep"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":62,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":63,"title":"","content":"-->","searchableContent":"-->"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":85,"title":"

Block/Vote production rules

","searchableContent":"

block/vote production rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":86,"title":"

We now define the rules for block production given by the relation","content":"

We now define the rules for block production given by the relation","searchableContent":"

we now define the rules for block production given by the relation"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":87,"title":"_↝_. These are split in two","content":"_↝_. These are split in two:

","searchableContent":"_↝_. these are split in two:

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":88,"title":"
    ","searchableContent":"
      "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":89,"title":"
    1. Positive rules, when we do need to create a block.
      2. ","content":"
    2. Positive rules, when we do need to create a block.
      3. ","searchableContent":"
    3. positive rules, when we do need to create a block.
      4. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":90,"title":"
    4. Negative rules, when we cannot create a block.
      5. ","content":"
    5. Negative rules, when we cannot create a block.
      6. ","searchableContent":"
    6. negative rules, when we cannot create a block.
      7. "},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":91,"title":"
    ","content":"
","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":92,"title":"

The purpose of the negative rules is to properly adjust the upkeep if","content":"

The purpose of the negative rules is to properly adjust the upkeep if","searchableContent":"

the purpose of the negative rules is to properly adjust the upkeep if"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":93,"title":"we cannot make a block.

","content":"we cannot make a block.

","searchableContent":"we cannot make a block.

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":94,"title":"

Note that _↝_, starting with an empty upkeep can always","content":"

Note that _↝_, starting with an empty upkeep can always","searchableContent":"

note that _↝_, starting with an empty upkeep can always"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":95,"title":"make exactly three steps corresponding to the three types of Leios","content":"make exactly three steps corresponding to the three types of Leios","searchableContent":"make exactly three steps corresponding to the three types of leios"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":96,"title":"specific blocks.

","content":"specific blocks.

","searchableContent":"specific blocks.

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":97,"title":"
data _↝_ ","content":"
data _↝_ : LeiosState → LeiosState → Type where","searchableContent":"
data _↝_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":98,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":99,"title":"

Positive rules

","searchableContent":"

positive rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":100,"title":"
  IB-Role ","content":"
  IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"
  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":101,"title":"b = ibBody (record { txs = ToPropose })","searchableContent":"                b = ibbody (record { txs = topropose })"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":102,"title":"h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"                h = ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":103,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":104,"title":"∙ needsUpkeep IB-Role","searchableContent":"          ∙ needsupkeep ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":105,"title":"∙ canProduceIB slot sk-IB (stake s) π","searchableContent":"          ∙ canproduceib slot sk-ib (stake s) π"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":106,"title":"∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send h (just b) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":107,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":108,"title":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":109,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":110,"title":"
  EB-Role ","content":"
  EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"
  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":111,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot 3) ibs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":112,"title":"h = mkEB slot id π sk-EB LI []","searchableContent":"                h = mkeb slot id π sk-eb li []"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":113,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":114,"title":"∙ needsUpkeep EB-Role","searchableContent":"          ∙ needsupkeep eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":115,"title":"∙ canProduceEB slot sk-EB (stake s) π","searchableContent":"          ∙ canproduceeb slot sk-eb (stake s) π"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":116,"title":"∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (ebheader h) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":117,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":118,"title":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":119,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":120,"title":"
  VT-Role ","content":"
  VT-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"
  vt-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":121,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":122,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":123,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":124,"title":"∙ needsUpkeep VT-Role","searchableContent":"          ∙ needsupkeep vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":125,"title":"∙ canProduceV slot sk-VT (stake s)","searchableContent":"          ∙ canproducev slot sk-vt (stake s)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":126,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":127,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":128,"title":"s ↝ addUpkeep record s { FFDState = ffds' } VT-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":129,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":130,"title":"

Negative rules

","searchableContent":"

negative rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":131,"title":"
  No-IB-Role ","content":"
  No-IB-Role : let open LeiosState s in","searchableContent":"
  no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":132,"title":"∙ needsUpkeep IB-Role","searchableContent":"             ∙ needsupkeep ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":133,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)","searchableContent":"             ∙ (∀ π → ¬ canproduceib slot sk-ib (stake s) π)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":134,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":135,"title":"s ↝ addUpkeep s IB-Role","searchableContent":"             s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":136,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":137,"title":"
  No-EB-Role ","content":"
  No-EB-Role : let open LeiosState s in","searchableContent":"
  no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":138,"title":"∙ needsUpkeep EB-Role","searchableContent":"             ∙ needsupkeep eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":139,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)","searchableContent":"             ∙ (∀ π → ¬ canproduceeb slot sk-eb (stake s) π)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":140,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":141,"title":"s ↝ addUpkeep s EB-Role","searchableContent":"             s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":142,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":143,"title":"
  No-VT-Role ","content":"
  No-VT-Role : let open LeiosState s in","searchableContent":"
  no-vt-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":144,"title":"∙ needsUpkeep VT-Role","searchableContent":"             ∙ needsupkeep vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":145,"title":"∙ ¬ canProduceV slot sk-VT (stake s)","searchableContent":"             ∙ ¬ canproducev slot sk-vt (stake s)"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":146,"title":"─────────────────────────────────────────────","searchableContent":"             ─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":147,"title":"s ↝ addUpkeep s VT-Role","searchableContent":"             s ↝ addupkeep s vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":148,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":149,"title":"

Uniform short-pipeline

","searchableContent":"

uniform short-pipeline

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":150,"title":"
stage ","content":"
stage : ℕ → ⦃ _ : NonZero L ⦄ → ℕ","searchableContent":"
stage : ℕ → ⦃ _ : nonzero l ⦄ → ℕ"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":151,"title":"stage s = s / L","searchableContent":"stage s = s / l"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":153,"title":"beginningOfStage ","content":"beginningOfStage : ℕ → Type","searchableContent":"beginningofstage : ℕ → type"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":154,"title":"beginningOfStage s = stage s * L ≡ s","searchableContent":"beginningofstage s = stage s * l ≡ s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":156,"title":"allDone ","content":"allDone : LeiosState → Type","searchableContent":"alldone : leiosstate → type"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":157,"title":"allDone s =","searchableContent":"alldone s ="},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":158,"title":"let open LeiosState s","searchableContent":"  let open leiosstate s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":159,"title":"in   (beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ EB-Role ∷ VT-Role ∷ Base ∷ []))","searchableContent":"  in   (beginningofstage slot × upkeep ≡ᵉ fromlist (ib-role ∷ eb-role ∷ vt-role ∷ base ∷ []))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":160,"title":"⊎ (¬ beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ VT-Role ∷ Base ∷ []))","searchableContent":"   ⊎ (¬ beginningofstage slot × upkeep ≡ᵉ fromlist (ib-role ∷ vt-role ∷ base ∷ []))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":162,"title":"data _-⟦_/_⟧⇀_ ","content":"data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":163,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":164,"title":"

Initialization

","searchableContent":"

initialization

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":165,"title":"
  Init ","content":"
  Init :","searchableContent":"
  init :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":166,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"       ∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":167,"title":"∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'","searchableContent":"       ∙ initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":168,"title":"────────────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":169,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","searchableContent":"       nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":170,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":171,"title":"

Network and Ledger

","searchableContent":"

network and ledger

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":172,"title":"
  Slot ","content":"
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in","searchableContent":"
  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":173,"title":"∙ allDone s","searchableContent":"       ∙ alldone s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":174,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"       ∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":175,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"       ∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":176,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":177,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"       just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":178,"title":"{ FFDState  = ffds'","searchableContent":"           { ffdstate  = ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":179,"title":"; BaseState = bs'","searchableContent":"           ; basestate = bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":180,"title":"; Ledger    = constructLedger rbs","searchableContent":"           ; ledger    = constructledger rbs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":181,"title":"; slot      = suc slot","searchableContent":"           ; slot      = suc slot"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":182,"title":"; Upkeep    = ∅","searchableContent":"           ; upkeep    = ∅"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":183,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"           } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":184,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":185,"title":"
  Ftch ","content":"
  Ftch :","searchableContent":"
  ftch :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":186,"title":"────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":187,"title":"just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s","searchableContent":"       just s -⟦ ftch-ldg / ftch-ldg (leiosstate.ledger s) ⟧⇀ s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":188,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":189,"title":"

Base chain

","searchableContent":"

base chain

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":190,"title":"Note","content":"Note: Submitted data to the base chain is only taken into account if the","searchableContent":"note: submitted data to the base chain is only taken into account if the"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":191,"title":"party submitting is the block producer on the base chain for the given","content":"party submitting is the block producer on the base chain for the given","searchableContent":"party submitting is the block producer on the base chain for the given"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":192,"title":"slot","content":"slot","searchableContent":"slot"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":193,"title":"
  Base₁   ","content":"
  Base₁   :","searchableContent":"
  base₁   :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":194,"title":"───────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":195,"title":"just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"          just s -⟦ submit (inj₂ txs) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":196,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":197,"title":"
  Base₂a  ","content":"
  Base₂a  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"
  base₂a  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":198,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":199,"title":"∙ eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ eb ∈ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":200,"title":"∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":201,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":202,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":204,"title":"Base₂b  ","content":"Base₂b  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂b  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":205,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":206,"title":"∙ [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ [] ≡ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":207,"title":"∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":208,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":209,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":210,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":211,"title":"

Protocol rules

","searchableContent":"

protocol rules

"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":212,"title":"
  Roles ","content":"
  Roles :","searchableContent":"
  roles :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":213,"title":"∙ s ↝ s'","searchableContent":"        ∙ s ↝ s'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":214,"title":"─────────────────────────────","searchableContent":"        ─────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":215,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"        just s -⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":216,"title":"
","content":"
","searchableContent":"
"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":217,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":218,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":2,"title":"Leios.Simplified.Deterministic
--{-# OPTIONS --safe #-}","searchableContent":"leios.simplified.deterministic
--{-# options --safe #-}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":3,"title":"{-# OPTIONS --allow-unsolved-metas #-}","searchableContent":"{-# options --allow-unsolved-metas #-}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":5,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":6,"title":"-- Deterministic variant of simple Leios","searchableContent":"-- deterministic variant of simple leios"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":7,"title":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":10,"title":"open import Prelude.Init using (∃₂-syntax)","searchableContent":"open import prelude.init using (∃₂-syntax)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":11,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":13,"title":"import Data.List as L","searchableContent":"import data.list as l"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":14,"title":"open import Data.List.Relation.Unary.Any using (here)","searchableContent":"open import data.list.relation.unary.any using (here)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":16,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":18,"title":"module Leios.Simplified.Deterministic (⋯ ","content":"module Leios.Simplified.Deterministic (⋯ : SpecStructure 2) (let open SpecStructure ⋯) (Λ μ : ℕ) where","searchableContent":"module leios.simplified.deterministic (⋯ : specstructure 2) (let open specstructure ⋯) (λ μ : ℕ) where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":20,"title":"import Leios.Simplified","searchableContent":"import leios.simplified"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":21,"title":"open import Leios.Simplified ⋯ Λ μ hiding (_-⟦_/_⟧⇀_)","searchableContent":"open import leios.simplified ⋯ λ μ hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":22,"title":"module ND = Leios.Simplified ⋯ Λ μ","searchableContent":"module nd = leios.simplified ⋯ λ μ"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":24,"title":"open import Class.Computational","searchableContent":"open import class.computational"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":25,"title":"open import Class.Computational22","searchableContent":"open import class.computational22"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":26,"title":"open import StateMachine","searchableContent":"open import statemachine"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":28,"title":"open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)","searchableContent":"open baseabstract b' using (cert; v-chkcerts; vty; initslot)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":29,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":31,"title":"open FFD hiding (_-⟦_/_⟧⇀_)","searchableContent":"open ffd hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":33,"title":"private variable s s' s0 s1 s2 s3 s4 s5 ","content":"private variable s s' s0 s1 s2 s3 s4 s5 : LeiosState","searchableContent":"private variable s s' s0 s1 s2 s3 s4 s5 : leiosstate"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":34,"title":"i      ","content":"i      : LeiosInput","searchableContent":"                 i      : leiosinput"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":35,"title":"o      ","content":"o      : LeiosOutput","searchableContent":"                 o      : leiosoutput"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":36,"title":"ffds'  ","content":"ffds'  : FFD.State","searchableContent":"                 ffds'  : ffd.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":37,"title":"π      ","content":"π      : VrfPf","searchableContent":"                 π      : vrfpf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":38,"title":"bs'    ","content":"bs'    : B.State","searchableContent":"                 bs'    : b.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":39,"title":"ks ks' ","content":"ks ks' : K.State","searchableContent":"                 ks ks' : k.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":40,"title":"msgs   ","content":"msgs   : List (FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract)","searchableContent":"                 msgs   : list (ffdabstract.header ffdabstract ⊎ ffdabstract.body ffdabstract)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":41,"title":"eb     ","content":"eb     : EndorserBlock","searchableContent":"                 eb     : endorserblock"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":42,"title":"rbs    ","content":"rbs    : List RankingBlock","searchableContent":"                 rbs    : list rankingblock"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":43,"title":"txs    ","content":"txs    : List Tx","searchableContent":"                 txs    : list tx"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":44,"title":"V      ","content":"V      : VTy","searchableContent":"                 v      : vty"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":45,"title":"SD     ","content":"SD     : StakeDistr","searchableContent":"                 sd     : stakedistr"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":46,"title":"pks    ","content":"pks    : List PubKey","searchableContent":"                 pks    : list pubkey"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":48,"title":"lemma ","content":"lemma : ∀ {u} → u ∈ LeiosState.Upkeep (addUpkeep s u)","searchableContent":"lemma : ∀ {u} → u ∈ leiosstate.upkeep (addupkeep s u)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":49,"title":"lemma = to ∈-∪ (inj₂ (to ∈-singleton refl))","searchableContent":"lemma = to ∈-∪ (inj₂ (to ∈-singleton refl))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":50,"title":"where open Equivalence","searchableContent":"  where open equivalence"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":52,"title":"addUpkeep⇒¬needsUpkeep ","content":"addUpkeep⇒¬needsUpkeep : ∀ {u} → ¬ LeiosState.needsUpkeep (addUpkeep s u) u","searchableContent":"addupkeep⇒¬needsupkeep : ∀ {u} → ¬ leiosstate.needsupkeep (addupkeep s u) u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":53,"title":"addUpkeep⇒¬needsUpkeep {s = s} = λ x → x (lemma {s = s})","searchableContent":"addupkeep⇒¬needsupkeep {s = s} = λ x → x (lemma {s = s})"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":55,"title":"data _⊢_ ","content":"data _⊢_ : LeiosInput → LeiosState → Type where","searchableContent":"data _⊢_ : leiosinput → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":56,"title":"Init ","content":"Init :","searchableContent":"  init :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":57,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"       ∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":58,"title":"∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'","searchableContent":"       ∙ initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":59,"title":"────────────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":60,"title":"INIT V ⊢ initLeiosState V SD bs' pks","searchableContent":"       init v ⊢ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":62,"title":"data _-⟦Base⟧⇀_ ","content":"data _-⟦Base⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦base⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":64,"title":"Base₂a  ","content":"Base₂a  : ∀ {ebs} → let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂a  : ∀ {ebs} → let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":65,"title":"∙ eb ∷ ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ eb ∷ ebs ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":66,"title":"∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":67,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":68,"title":"s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          s -⟦base⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":70,"title":"Base₂b  ","content":"Base₂b  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂b  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":71,"title":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ [] ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":72,"title":"∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":73,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":74,"title":"s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          s -⟦base⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":76,"title":"Base⇒ND ","content":"Base⇒ND : LeiosState.needsUpkeep s Base → s -⟦Base⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"base⇒nd : leiosstate.needsupkeep s base → s -⟦base⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":77,"title":"Base⇒ND u (Base₂a x₁ x₂) = Base₂a u (subst (_ ∈_) x₁ (Equivalence.to ∈-fromList (here refl))) x₂","searchableContent":"base⇒nd u (base₂a x₁ x₂) = base₂a u (subst (_ ∈_) x₁ (equivalence.to ∈-fromlist (here refl))) x₂"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":78,"title":"Base⇒ND u (Base₂b x₁ x₂) = Base₂b u x₁ x₂","searchableContent":"base⇒nd u (base₂b x₁ x₂) = base₂b u x₁ x₂"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":80,"title":"Base-Upkeep ","content":"Base-Upkeep : ∀ {u} → u ≢ Base → LeiosState.needsUpkeep s u → s -⟦Base⟧⇀ s'","searchableContent":"base-upkeep : ∀ {u} → u ≢ base → leiosstate.needsupkeep s u → s -⟦base⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":81,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":82,"title":"Base-Upkeep u≢Base h (Base₂a _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"base-upkeep u≢base h (base₂a _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":83,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":84,"title":"(inj₂ y) → u≢Base (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢base (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":85,"title":"Base-Upkeep u≢Base h (Base₂b _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"base-upkeep u≢base h (base₂b _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":86,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":87,"title":"(inj₂ y) → u≢Base (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢base (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":89,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":90,"title":"Base-total ","content":"Base-total : ∃[ s' ] s -⟦Base⟧⇀ s'","searchableContent":"  base-total : ∃[ s' ] s -⟦base⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":91,"title":"Base-total {s = s} with","searchableContent":"  base-total {s = s} with"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":92,"title":"(let open LeiosState s in filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs)","searchableContent":"    (let open leiosstate s in filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":93,"title":"in eq","searchableContent":"    in eq"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":94,"title":"... | []    = -, Base₂b (sym eq) (proj₂ B.SUBMIT-total)","searchableContent":"  ... | []    = -, base₂b (sym eq) (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":95,"title":"... | x ∷ l = -, Base₂a (sym eq) (proj₂ B.SUBMIT-total)","searchableContent":"  ... | x ∷ l = -, base₂a (sym eq) (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":97,"title":"Base-total' ","content":"Base-total' : ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄","searchableContent":"  base-total' : ⦃ computational-b : computational22 b._-⟦_/_⟧⇀_ string ⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":98,"title":"→ ∃[ bs ] s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs } Base","searchableContent":"              → ∃[ bs ] s -⟦base⟧⇀ addupkeep record s { basestate = bs } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":99,"title":"Base-total' {s = s} = let open LeiosState s in","searchableContent":"  base-total' {s = s} = let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":100,"title":"case ∃[ ebs ] ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs ∋ -, refl","searchableContent":"    case ∃[ ebs ] ebs ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs ∋ -, refl"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":101,"title":"of λ where","searchableContent":"      of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":102,"title":"(eb ∷ _ , eq) → -, Base₂a eq (proj₂ B.SUBMIT-total)","searchableContent":"        (eb ∷ _ , eq) → -, base₂a eq (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":103,"title":"([]     , eq) → -, Base₂b eq (proj₂ B.SUBMIT-total)","searchableContent":"        ([]     , eq) → -, base₂b eq (proj₂ b.submit-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":105,"title":"data _-⟦IB-Role⟧⇀_ ","content":"data _-⟦IB-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦ib-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":107,"title":"IB-Role ","content":"IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":108,"title":"b = ibBody (record { txs = ToPropose })","searchableContent":"                b = ibbody (record { txs = topropose })"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":109,"title":"h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"                h = ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":110,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":111,"title":"∙ canProduceIB slot sk-IB (stake s) π","searchableContent":"          ∙ canproduceib slot sk-ib (stake s) π"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":112,"title":"∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send h (just b) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":113,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":114,"title":"s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"          s -⟦ib-role⟧⇀ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":116,"title":"No-IB-Role ","content":"No-IB-Role : let open LeiosState s in","searchableContent":"  no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":117,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceib slot sk-ib (stake s) π)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":118,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":119,"title":"s -⟦IB-Role⟧⇀ addUpkeep s IB-Role","searchableContent":"          s -⟦ib-role⟧⇀ addupkeep s ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":121,"title":"IB-Role⇒ND ","content":"IB-Role⇒ND : LeiosState.needsUpkeep s IB-Role → s -⟦IB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"ib-role⇒nd : leiosstate.needsupkeep s ib-role → s -⟦ib-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":122,"title":"IB-Role⇒ND u (IB-Role x₁ x₂) = Roles (IB-Role u x₁ x₂)","searchableContent":"ib-role⇒nd u (ib-role x₁ x₂) = roles (ib-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":123,"title":"IB-Role⇒ND u (No-IB-Role x₁) = Roles (No-IB-Role u x₁)","searchableContent":"ib-role⇒nd u (no-ib-role x₁) = roles (no-ib-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":125,"title":"IB-Role-Upkeep ","content":"IB-Role-Upkeep : ∀ {u} → u ≢ IB-Role → LeiosState.needsUpkeep s u → s -⟦IB-Role⟧⇀ s'","searchableContent":"ib-role-upkeep : ∀ {u} → u ≢ ib-role → leiosstate.needsupkeep s u → s -⟦ib-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":126,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":127,"title":"IB-Role-Upkeep u≢IB-Role h (IB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"ib-role-upkeep u≢ib-role h (ib-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":128,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":129,"title":"(inj₂ y) → u≢IB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢ib-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":130,"title":"IB-Role-Upkeep u≢IB-Role h (No-IB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"ib-role-upkeep u≢ib-role h (no-ib-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":131,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":132,"title":"(inj₂ y) → u≢IB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢ib-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":134,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":135,"title":"IB-Role-total ","content":"IB-Role-total : ∃[ s' ] s -⟦IB-Role⟧⇀ s'","searchableContent":"  ib-role-total : ∃[ s' ] s -⟦ib-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":136,"title":"IB-Role-total {s = s} = let open LeiosState s in case Dec-canProduceIB of λ where","searchableContent":"  ib-role-total {s = s} = let open leiosstate s in case dec-canproduceib of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":137,"title":"(inj₁ (π , pf)) → -, IB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, ib-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":138,"title":"(inj₂ pf)       → -, No-IB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-ib-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":140,"title":"IB-Role-total' ","content":"IB-Role-total' : ∃[ ffds ] s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds } IB-Role","searchableContent":"  ib-role-total' : ∃[ ffds ] s -⟦ib-role⟧⇀ addupkeep record s { ffdstate = ffds } ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":141,"title":"IB-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceIB of λ where","searchableContent":"  ib-role-total' {s = s} = let open leiosstate s in case dec-canproduceib of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":142,"title":"(inj₁ (π , pf)) → -, IB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, ib-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":143,"title":"(inj₂ pf)       → -, No-IB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-ib-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":145,"title":"data _-⟦EB-Role⟧⇀_ ","content":"data _-⟦EB-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦eb-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":147,"title":"EB-Role ","content":"EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":148,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot (Λ + 1)) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot (λ + 1)) ibs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":149,"title":"LE = map getEBRef $ filter (isVote1Certified s) $","searchableContent":"                le = map getebref $ filter (isvote1certified s) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":150,"title":"filter (_∈ᴮ slice L slot (μ + 2)) EBs","searchableContent":"                           filter (_∈ᴮ slice l slot (μ + 2)) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":151,"title":"h = mkEB slot id π sk-EB LI LE","searchableContent":"                h = mkeb slot id π sk-eb li le"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":152,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":153,"title":"∙ canProduceEB slot sk-EB (stake s) π","searchableContent":"          ∙ canproduceeb slot sk-eb (stake s) π"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":154,"title":"∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (ebheader h) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":155,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":156,"title":"s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"          s -⟦eb-role⟧⇀ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":158,"title":"No-EB-Role ","content":"No-EB-Role : let open LeiosState s in","searchableContent":"  no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":159,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceeb slot sk-eb (stake s) π)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":160,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":161,"title":"s -⟦EB-Role⟧⇀ addUpkeep s EB-Role","searchableContent":"          s -⟦eb-role⟧⇀ addupkeep s eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":163,"title":"EB-Role⇒ND ","content":"EB-Role⇒ND : LeiosState.needsUpkeep s EB-Role → s -⟦EB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"eb-role⇒nd : leiosstate.needsupkeep s eb-role → s -⟦eb-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":164,"title":"EB-Role⇒ND u (EB-Role x₁ x₂) = Roles (EB-Role u x₁ x₂)","searchableContent":"eb-role⇒nd u (eb-role x₁ x₂) = roles (eb-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":165,"title":"EB-Role⇒ND u (No-EB-Role x₁) = Roles (No-EB-Role u x₁)","searchableContent":"eb-role⇒nd u (no-eb-role x₁) = roles (no-eb-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":167,"title":"EB-Role-Upkeep ","content":"EB-Role-Upkeep : ∀ {u} → u ≢ EB-Role → LeiosState.needsUpkeep s u → s -⟦EB-Role⟧⇀ s'","searchableContent":"eb-role-upkeep : ∀ {u} → u ≢ eb-role → leiosstate.needsupkeep s u → s -⟦eb-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":168,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":169,"title":"EB-Role-Upkeep u≢EB-Role h (EB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"eb-role-upkeep u≢eb-role h (eb-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":170,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":171,"title":"(inj₂ y) → u≢EB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢eb-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":172,"title":"EB-Role-Upkeep u≢EB-Role h (No-EB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"eb-role-upkeep u≢eb-role h (no-eb-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":173,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":174,"title":"(inj₂ y) → u≢EB-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢eb-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":176,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":177,"title":"EB-Role-total ","content":"EB-Role-total : ∃[ s' ] s -⟦EB-Role⟧⇀ s'","searchableContent":"  eb-role-total : ∃[ s' ] s -⟦eb-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":178,"title":"EB-Role-total {s = s} = let open LeiosState s in case Dec-canProduceEB of λ where","searchableContent":"  eb-role-total {s = s} = let open leiosstate s in case dec-canproduceeb of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":179,"title":"(inj₁ (π , pf)) → -, EB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, eb-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":180,"title":"(inj₂ pf)       → -, No-EB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-eb-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":182,"title":"EB-Role-total' ","content":"EB-Role-total' : ∃[ ffds ] s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds } EB-Role","searchableContent":"  eb-role-total' : ∃[ ffds ] s -⟦eb-role⟧⇀ addupkeep record s { ffdstate = ffds } eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":183,"title":"EB-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceEB of λ where","searchableContent":"  eb-role-total' {s = s} = let open leiosstate s in case dec-canproduceeb of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":184,"title":"(inj₁ (π , pf)) → -, EB-Role    pf (proj₂ FFD.Send-total)","searchableContent":"    (inj₁ (π , pf)) → -, eb-role    pf (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":185,"title":"(inj₂ pf)       → -, No-EB-Role pf","searchableContent":"    (inj₂ pf)       → -, no-eb-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":187,"title":"data _-⟦V1-Role⟧⇀_ ","content":"data _-⟦V1-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦v1-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":189,"title":"V1-Role ","content":"V1-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v1-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":190,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot (μ + 1)) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot (μ + 1)) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":191,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":192,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":193,"title":"∙ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":194,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":195,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":196,"title":"s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V1-Role","searchableContent":"          s -⟦v1-role⟧⇀ addupkeep record s { ffdstate = ffds' } v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":198,"title":"No-V1-Role ","content":"No-V1-Role : let open LeiosState s in","searchableContent":"  no-v1-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":199,"title":"∙ ¬ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":200,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":201,"title":"s -⟦V1-Role⟧⇀ addUpkeep s V1-Role","searchableContent":"          s -⟦v1-role⟧⇀ addupkeep s v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":203,"title":"V1-Role⇒ND ","content":"V1-Role⇒ND : LeiosState.needsUpkeep s V1-Role → s -⟦V1-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"v1-role⇒nd : leiosstate.needsupkeep s v1-role → s -⟦v1-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":204,"title":"V1-Role⇒ND u (V1-Role x₁ x₂) = Roles (V1-Role u x₁ x₂)","searchableContent":"v1-role⇒nd u (v1-role x₁ x₂) = roles (v1-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":205,"title":"V1-Role⇒ND u (No-V1-Role x₁) = Roles (No-V1-Role u x₁)","searchableContent":"v1-role⇒nd u (no-v1-role x₁) = roles (no-v1-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":207,"title":"V1-Role-Upkeep ","content":"V1-Role-Upkeep : ∀ {u} → u ≢ V1-Role → LeiosState.needsUpkeep s u → s -⟦V1-Role⟧⇀ s'","searchableContent":"v1-role-upkeep : ∀ {u} → u ≢ v1-role → leiosstate.needsupkeep s u → s -⟦v1-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":208,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":209,"title":"V1-Role-Upkeep u≢V1-Role h (V1-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v1-role-upkeep u≢v1-role h (v1-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":210,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":211,"title":"(inj₂ y) → u≢V1-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v1-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":212,"title":"V1-Role-Upkeep u≢V1-Role h (No-V1-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v1-role-upkeep u≢v1-role h (no-v1-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":213,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":214,"title":"(inj₂ y) → u≢V1-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v1-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":216,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":217,"title":"V1-Role-total ","content":"V1-Role-total : ∃[ s' ] s -⟦V1-Role⟧⇀ s'","searchableContent":"  v1-role-total : ∃[ s' ] s -⟦v1-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":218,"title":"V1-Role-total {s = s} = let open LeiosState s in case Dec-canProduceV1 of λ where","searchableContent":"  v1-role-total {s = s} = let open leiosstate s in case dec-canproducev1 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":219,"title":"(yes p) → -, V1-Role p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v1-role p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":220,"title":"(no ¬p) → -, No-V1-Role ¬p","searchableContent":"    (no ¬p) → -, no-v1-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":222,"title":"V1-Role-total' ","content":"V1-Role-total' : ∃[ ffds ] s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds } V1-Role","searchableContent":"  v1-role-total' : ∃[ ffds ] s -⟦v1-role⟧⇀ addupkeep record s { ffdstate = ffds } v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":223,"title":"V1-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceV1 of λ where","searchableContent":"  v1-role-total' {s = s} = let open leiosstate s in case dec-canproducev1 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":224,"title":"(yes p) → -, V1-Role    p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v1-role    p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":225,"title":"(no ¬p) → -, No-V1-Role ¬p","searchableContent":"    (no ¬p) → -, no-v1-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":227,"title":"data _-⟦V2-Role⟧⇀_ ","content":"data _-⟦V2-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦v2-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":229,"title":"V2-Role ","content":"V2-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v2-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":230,"title":"EBs' = filter (vote2Eligible s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (vote2eligible s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":231,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":232,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":233,"title":"∙ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":234,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":235,"title":"────────────────────────────────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":236,"title":"s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V2-Role","searchableContent":"          s -⟦v2-role⟧⇀ addupkeep record s { ffdstate = ffds' } v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":238,"title":"No-V2-Role ","content":"No-V2-Role : let open LeiosState s in","searchableContent":"  no-v2-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":239,"title":"∙ ¬ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":240,"title":"────────────────────────────────────────","searchableContent":"          ────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":241,"title":"s -⟦V2-Role⟧⇀ addUpkeep s V2-Role","searchableContent":"          s -⟦v2-role⟧⇀ addupkeep s v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":243,"title":"V2-Role⇒ND ","content":"V2-Role⇒ND : LeiosState.needsUpkeep s V2-Role → s -⟦V2-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"v2-role⇒nd : leiosstate.needsupkeep s v2-role → s -⟦v2-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":244,"title":"V2-Role⇒ND u (V2-Role x₁ x₂) = Roles (V2-Role u x₁ x₂)","searchableContent":"v2-role⇒nd u (v2-role x₁ x₂) = roles (v2-role u x₁ x₂)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":245,"title":"V2-Role⇒ND u (No-V2-Role x₁) = Roles (No-V2-Role u x₁)","searchableContent":"v2-role⇒nd u (no-v2-role x₁) = roles (no-v2-role u x₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":247,"title":"V2-Role-Upkeep ","content":"V2-Role-Upkeep : ∀ {u} → u ≢ V2-Role → LeiosState.needsUpkeep s u → s -⟦V2-Role⟧⇀ s'","searchableContent":"v2-role-upkeep : ∀ {u} → u ≢ v2-role → leiosstate.needsupkeep s u → s -⟦v2-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":248,"title":"→ LeiosState.needsUpkeep s' u","searchableContent":"                  → leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":249,"title":"V2-Role-Upkeep u≢V2-Role h (V2-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v2-role-upkeep u≢v2-role h (v2-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":250,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":251,"title":"(inj₂ y) → u≢V2-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v2-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":252,"title":"V2-Role-Upkeep u≢V2-Role h (No-V2-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v2-role-upkeep u≢v2-role h (no-v2-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":253,"title":"(inj₁ x) → h x","searchableContent":"  (inj₁ x) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":254,"title":"(inj₂ y) → u≢V2-Role (Equivalence.from ∈-singleton y)","searchableContent":"  (inj₂ y) → u≢v2-role (equivalence.from ∈-singleton y)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":256,"title":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":257,"title":"V2-Role-total ","content":"V2-Role-total : ∃[ s' ] s -⟦V2-Role⟧⇀ s'","searchableContent":"  v2-role-total : ∃[ s' ] s -⟦v2-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":258,"title":"V2-Role-total {s = s} = let open LeiosState s in case Dec-canProduceV2 of λ where","searchableContent":"  v2-role-total {s = s} = let open leiosstate s in case dec-canproducev2 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":259,"title":"(yes p) → -, V2-Role p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v2-role p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":260,"title":"(no ¬p) → -, No-V2-Role ¬p","searchableContent":"    (no ¬p) → -, no-v2-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":262,"title":"V2-Role-total' ","content":"V2-Role-total' : ∃[ ffds ] s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds } V2-Role","searchableContent":"  v2-role-total' : ∃[ ffds ] s -⟦v2-role⟧⇀ addupkeep record s { ffdstate = ffds } v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":263,"title":"V2-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceV2 of λ where","searchableContent":"  v2-role-total' {s = s} = let open leiosstate s in case dec-canproducev2 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":264,"title":"(yes p) → -, V2-Role    p (proj₂ FFD.Send-total)","searchableContent":"    (yes p) → -, v2-role    p (proj₂ ffd.send-total)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":265,"title":"(no ¬p) → -, No-V2-Role ¬p","searchableContent":"    (no ¬p) → -, no-v2-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":267,"title":"data _-⟦_/_⟧⇀_ ","content":"data _-⟦_/_⟧⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":269,"title":"-- Network and Ledger","searchableContent":"  -- network and ledger"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":271,"title":"Slot ","content":"Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs)","searchableContent":"  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":272,"title":"s0 = record s","searchableContent":"             s0 = record s"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":273,"title":"{ FFDState = ffds'","searchableContent":"                    { ffdstate = ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":274,"title":"; Ledger   = constructLedger rbs","searchableContent":"                    ; ledger   = constructledger rbs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":275,"title":"; slot     = suc slot","searchableContent":"                    ; slot     = suc slot"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":276,"title":"; Upkeep   = ∅","searchableContent":"                    ; upkeep   = ∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":277,"title":"; BaseState = bs'","searchableContent":"                    ; basestate = bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":278,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"                    } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":279,"title":"in","searchableContent":"       in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":280,"title":"∙ Upkeep ≡ᵉ allUpkeep","searchableContent":"       ∙ upkeep ≡ᵉ allupkeep"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":281,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"       ∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":282,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"       ∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":283,"title":"∙ s0 -⟦Base⟧⇀    s1","searchableContent":"       ∙ s0 -⟦base⟧⇀    s1"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":284,"title":"∙ s1 -⟦IB-Role⟧⇀ s2","searchableContent":"       ∙ s1 -⟦ib-role⟧⇀ s2"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":285,"title":"∙ s2 -⟦EB-Role⟧⇀ s3","searchableContent":"       ∙ s2 -⟦eb-role⟧⇀ s3"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":286,"title":"∙ s3 -⟦V1-Role⟧⇀ s4","searchableContent":"       ∙ s3 -⟦v1-role⟧⇀ s4"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":287,"title":"∙ s4 -⟦V2-Role⟧⇀ s5","searchableContent":"       ∙ s4 -⟦v2-role⟧⇀ s5"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":288,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":289,"title":"s -⟦ SLOT / EMPTY ⟧⇀ s5","searchableContent":"       s -⟦ slot / empty ⟧⇀ s5"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":291,"title":"Ftch ","content":"Ftch :","searchableContent":"  ftch :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":292,"title":"───────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":293,"title":"s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s","searchableContent":"       s -⟦ ftch-ldg / ftch-ldg (leiosstate.ledger s) ⟧⇀ s"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":295,"title":"-- Base chain","searchableContent":"  -- base chain"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":296,"title":"--","searchableContent":"  --"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":297,"title":"-- Note","content":"-- Note: Submitted data to the base chain is only taken into account","searchableContent":"  -- note: submitted data to the base chain is only taken into account"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":298,"title":"--       if the party submitting is the block producer on the base chain","searchableContent":"  --       if the party submitting is the block producer on the base chain"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":299,"title":"--       for the given slot","searchableContent":"  --       for the given slot"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":301,"title":"Base₁   ","content":"Base₁   :","searchableContent":"  base₁   :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":302,"title":"──────────────────────────────────────────────────────────────","searchableContent":"          ──────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":303,"title":"s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"          s -⟦ submit (inj₂ txs) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":305,"title":"-- Protocol rules","searchableContent":"  -- protocol rules"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":307,"title":"_-⟦_/_⟧ⁿᵈ⇀_ ","content":"_-⟦_/_⟧ⁿᵈ⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type","searchableContent":"_-⟦_/_⟧ⁿᵈ⇀_ : leiosstate → leiosinput → leiosoutput → leiosstate → type"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":308,"title":"s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s ND.-⟦ i / o ⟧⇀ s'","searchableContent":"s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s nd.-⟦ i / o ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":310,"title":"_-⟦_/_⟧ⁿᵈ*⇀_ ","content":"_-⟦_/_⟧ⁿᵈ*⇀_ : LeiosState → List LeiosInput → List LeiosOutput → LeiosState → Type","searchableContent":"_-⟦_/_⟧ⁿᵈ*⇀_ : leiosstate → list leiosinput → list leiosoutput → leiosstate → type"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":311,"title":"_-⟦_/_⟧ⁿᵈ*⇀_ = ReflexiveTransitiveClosure _-⟦_/_⟧ⁿᵈ⇀_","searchableContent":"_-⟦_/_⟧ⁿᵈ*⇀_ = reflexivetransitiveclosure _-⟦_/_⟧ⁿᵈ⇀_"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":313,"title":"-- Key fact","content":"-- Key fact: stepping with the deterministic relation means we can","searchableContent":"-- key fact: stepping with the deterministic relation means we can"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":314,"title":"-- also step with the non-deterministic one","searchableContent":"-- also step with the non-deterministic one"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":315,"title":"-- TODO","content":"-- TODO: this is a lot like a weak simulation, can we do something prettier?","searchableContent":"-- todo: this is a lot like a weak simulation, can we do something prettier?"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":316,"title":"-⟦/⟧⇀⇒ND ","content":"-⟦/⟧⇀⇒ND : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] (s -⟦ i / o ⟧ⁿᵈ*⇀ s')","searchableContent":"-⟦/⟧⇀⇒nd : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] (s -⟦ i / o ⟧ⁿᵈ*⇀ s')"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":317,"title":"-⟦/⟧⇀⇒ND (Slot {s = s} {msgs = msgs} {s1 = s1} {s2 = s2} {s3 = s3} {s4 = s4} x x₁ x₂ hB hIB hEB hV1 hV2) = replicate 6 SLOT , replicate 6 EMPTY ,","searchableContent":"-⟦/⟧⇀⇒nd (slot {s = s} {msgs = msgs} {s1 = s1} {s2 = s2} {s3 = s3} {s4 = s4} x x₁ x₂ hb hib heb hv1 hv2) = replicate 6 slot , replicate 6 empty ,"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":318,"title":"let","searchableContent":"  let"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":319,"title":"s0 = _","searchableContent":"    s0 = _"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":320,"title":"upkeep≡∅ ","content":"upkeep≡∅ : LeiosState.Upkeep s0 ≡ ∅","searchableContent":"    upkeep≡∅ : leiosstate.upkeep s0 ≡ ∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":321,"title":"upkeep≡∅ = sym (↑-preserves-Upkeep {x = L.filter (isValid? s) msgs})","searchableContent":"    upkeep≡∅ = sym (↑-preserves-upkeep {x = l.filter (isvalid? s) msgs})"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":322,"title":"needsAllUpkeep ","content":"needsAllUpkeep : ∀ {u} → LeiosState.needsUpkeep s0 u","searchableContent":"    needsallupkeep : ∀ {u} → leiosstate.needsupkeep s0 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":323,"title":"needsAllUpkeep {u} = subst (u ∉_) (sym upkeep≡∅) Properties.∉-∅","searchableContent":"    needsallupkeep {u} = subst (u ∉_) (sym upkeep≡∅) properties.∉-∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":324,"title":"needsUpkeep1 ","content":"needsUpkeep1 : ∀ {u} → u ≢ Base → LeiosState.needsUpkeep s1 u","searchableContent":"    needsupkeep1 : ∀ {u} → u ≢ base → leiosstate.needsupkeep s1 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":325,"title":"needsUpkeep1 h1 = Base-Upkeep h1 needsAllUpkeep hB","searchableContent":"    needsupkeep1 h1 = base-upkeep h1 needsallupkeep hb"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":326,"title":"needsUpkeep2 ","content":"needsUpkeep2 : ∀ {u} → u ≢ Base → u ≢ IB-Role → LeiosState.needsUpkeep s2 u","searchableContent":"    needsupkeep2 : ∀ {u} → u ≢ base → u ≢ ib-role → leiosstate.needsupkeep s2 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":327,"title":"needsUpkeep2 h1 h2 = IB-Role-Upkeep h2 (needsUpkeep1 h1) hIB","searchableContent":"    needsupkeep2 h1 h2 = ib-role-upkeep h2 (needsupkeep1 h1) hib"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":328,"title":"needsUpkeep3 ","content":"needsUpkeep3 : ∀ {u} → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → LeiosState.needsUpkeep s3 u","searchableContent":"    needsupkeep3 : ∀ {u} → u ≢ base → u ≢ ib-role → u ≢ eb-role → leiosstate.needsupkeep s3 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":329,"title":"needsUpkeep3 h1 h2 h3 = EB-Role-Upkeep h3 (needsUpkeep2 h1 h2) hEB","searchableContent":"    needsupkeep3 h1 h2 h3 = eb-role-upkeep h3 (needsupkeep2 h1 h2) heb"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":330,"title":"needsUpkeep4 ","content":"needsUpkeep4 : ∀ {u} → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → u ≢ V1-Role → LeiosState.needsUpkeep s4 u","searchableContent":"    needsupkeep4 : ∀ {u} → u ≢ base → u ≢ ib-role → u ≢ eb-role → u ≢ v1-role → leiosstate.needsupkeep s4 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":331,"title":"needsUpkeep4 h1 h2 h3 h4 = V1-Role-Upkeep h4 (needsUpkeep3 h1 h2 h3) hV1","searchableContent":"    needsupkeep4 h1 h2 h3 h4 = v1-role-upkeep h4 (needsupkeep3 h1 h2 h3) hv1"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":332,"title":"in (BS-ind (ND.Slot x x₁ x₂) $","searchableContent":"  in (bs-ind (nd.slot x x₁ x₂) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":333,"title":"BS-ind (Base⇒ND {s = s0} needsAllUpkeep hB) $","searchableContent":"      bs-ind (base⇒nd {s = s0} needsallupkeep hb) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":334,"title":"BS-ind (IB-Role⇒ND (needsUpkeep1 (λ ())) hIB) $","searchableContent":"      bs-ind (ib-role⇒nd (needsupkeep1 (λ ())) hib) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":335,"title":"BS-ind (EB-Role⇒ND (needsUpkeep2 (λ ()) (λ ())) hEB) $","searchableContent":"      bs-ind (eb-role⇒nd (needsupkeep2 (λ ()) (λ ())) heb) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":336,"title":"BS-ind (V1-Role⇒ND (needsUpkeep3 (λ ()) (λ ()) (λ ())) hV1) $","searchableContent":"      bs-ind (v1-role⇒nd (needsupkeep3 (λ ()) (λ ()) (λ ())) hv1) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":337,"title":"STS⇒RTC (V2-Role⇒ND (needsUpkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hV2))","searchableContent":"      sts⇒rtc (v2-role⇒nd (needsupkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hv2))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":338,"title":"-⟦/⟧⇀⇒ND Ftch = _ , _ , STS⇒RTC Ftch","searchableContent":"-⟦/⟧⇀⇒nd ftch = _ , _ , sts⇒rtc ftch"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":339,"title":"-⟦/⟧⇀⇒ND Base₁ = _ , _ , STS⇒RTC Base₁","searchableContent":"-⟦/⟧⇀⇒nd base₁ = _ , _ , sts⇒rtc base₁"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":341,"title":"open Computational22 ⦃...⦄","searchableContent":"open computational22 ⦃...⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":343,"title":"module _ ⦃ Computational-B ","content":"module _ ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄","searchableContent":"module _ ⦃ computational-b : computational22 b._-⟦_/_⟧⇀_ string ⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":344,"title":"⦃ Computational-FFD ","content":"⦃ Computational-FFD : Computational22 FFD._-⟦_/_⟧⇀_ String ⦄ where","searchableContent":"         ⦃ computational-ffd : computational22 ffd._-⟦_/_⟧⇀_ string ⦄ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":346,"title":"instance","searchableContent":"  instance"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":347,"title":"Computational--⟦/⟧⇀ ","content":"Computational--⟦/⟧⇀ : Computational22 _-⟦_/_⟧⇀_ String","searchableContent":"    computational--⟦/⟧⇀ : computational22 _-⟦_/_⟧⇀_ string"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":348,"title":"Computational--⟦/⟧⇀ .computeProof s (INIT x) = failure "No handling of INIT here"","searchableContent":"    computational--⟦/⟧⇀ .computeproof s (init x) = failure "no handling of init here""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":349,"title":"Computational--⟦/⟧⇀ .computeProof s (SUBMIT (inj₁ eb)) = failure "Cannot submit EB here"","searchableContent":"    computational--⟦/⟧⇀ .computeproof s (submit (inj₁ eb)) = failure "cannot submit eb here""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":350,"title":"Computational--⟦/⟧⇀ .computeProof s (SUBMIT (inj₂ txs)) = success (-, Base₁)","searchableContent":"    computational--⟦/⟧⇀ .computeproof s (submit (inj₂ txs)) = success (-, base₁)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":351,"title":"Computational--⟦/⟧⇀ .computeProof s* SLOT = let open LeiosState s* in","searchableContent":"    computational--⟦/⟧⇀ .computeproof s* slot = let open leiosstate s* in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":352,"title":"case (¿ Upkeep ≡ᵉ allUpkeep ¿ ,′ computeProof BaseState B.FTCH-LDG ,′ computeProof FFDState FFD.Fetch) of λ where","searchableContent":"      case (¿ upkeep ≡ᵉ allupkeep ¿ ,′ computeproof basestate b.ftch-ldg ,′ computeproof ffdstate ffd.fetch) of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":353,"title":"(yes p , success ((B.BASE-LDG l , bs) , p₁) , success ((FFD.FetchRes msgs , ffds) , p₂)) →","searchableContent":"        (yes p , success ((b.base-ldg l , bs) , p₁) , success ((ffd.fetchres msgs , ffds) , p₂)) →"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":354,"title":"success ((_ , (Slot p p₁ p₂ (proj₂ Base-total) (proj₂ IB-Role-total) (proj₂ EB-Role-total) (proj₂ V1-Role-total) (proj₂ V2-Role-total))))","searchableContent":"          success ((_ , (slot p p₁ p₂ (proj₂ base-total) (proj₂ ib-role-total) (proj₂ eb-role-total) (proj₂ v1-role-total) (proj₂ v2-role-total))))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":355,"title":"(yes p , _ , _) → failure "Subsystem failed"","searchableContent":"        (yes p , _ , _) → failure "subsystem failed""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":356,"title":"(no ¬p , _) → failure "Upkeep incorrect"","searchableContent":"        (no ¬p , _) → failure "upkeep incorrect""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":357,"title":"Computational--⟦/⟧⇀ .computeProof s FTCH-LDG = success (-, Ftch)","searchableContent":"    computational--⟦/⟧⇀ .computeproof s ftch-ldg = success (-, ftch)"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":358,"title":"Computational--⟦/⟧⇀ .completeness = {!!}","searchableContent":"    computational--⟦/⟧⇀ .completeness = {!!}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":359,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":2,"title":"Leios.Simplified
{-# OPTIONS --safe #-}","searchableContent":"leios.simplified
{-# options --safe #-}"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":5,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":6,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":7,"title":"open import Data.Fin.Patterns","searchableContent":"open import data.fin.patterns"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":9,"title":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":10,"title":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":12,"title":"module Leios.Simplified (⋯ ","content":"module Leios.Simplified (⋯ : SpecStructure 2) (let open SpecStructure ⋯) (Λ μ : ℕ) where","searchableContent":"module leios.simplified (⋯ : specstructure 2) (let open specstructure ⋯) (λ μ : ℕ) where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":14,"title":"data SlotUpkeep ","content":"data SlotUpkeep : Type where","searchableContent":"data slotupkeep : type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":15,"title":"Base IB-Role EB-Role V1-Role V2-Role ","content":"Base IB-Role EB-Role V1-Role V2-Role : SlotUpkeep","searchableContent":"  base ib-role eb-role v1-role v2-role : slotupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":17,"title":"unquoteDecl DecEq-SlotUpkeep = derive-DecEq ((quote SlotUpkeep , DecEq-SlotUpkeep) ∷ [])","searchableContent":"unquotedecl deceq-slotupkeep = derive-deceq ((quote slotupkeep , deceq-slotupkeep) ∷ [])"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":19,"title":"allUpkeep ","content":"allUpkeep : ℙ SlotUpkeep","searchableContent":"allupkeep : ℙ slotupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":20,"title":"allUpkeep = fromList (Base ∷ IB-Role ∷ EB-Role ∷ V1-Role ∷ V2-Role ∷ [])","searchableContent":"allupkeep = fromlist (base ∷ ib-role ∷ eb-role ∷ v1-role ∷ v2-role ∷ [])"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":22,"title":"open import Leios.Protocol (⋯) SlotUpkeep public","searchableContent":"open import leios.protocol (⋯) slotupkeep public"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":24,"title":"open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)","searchableContent":"open baseabstract b' using (cert; v-chkcerts; vty; initslot)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":25,"title":"open FFD hiding (_-⟦_/_⟧⇀_)","searchableContent":"open ffd hiding (_-⟦_/_⟧⇀_)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":26,"title":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":28,"title":"isVote1Certified ","content":"isVote1Certified : LeiosState → EndorserBlock → Type","searchableContent":"isvote1certified : leiosstate → endorserblock → type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":29,"title":"isVote1Certified s eb = isVoteCertified (LeiosState.votingState s) (0F , eb)","searchableContent":"isvote1certified s eb = isvotecertified (leiosstate.votingstate s) (0f , eb)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":31,"title":"isVote2Certified ","content":"isVote2Certified : LeiosState → EndorserBlock → Type","searchableContent":"isvote2certified : leiosstate → endorserblock → type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":32,"title":"isVote2Certified s eb = isVoteCertified (LeiosState.votingState s) (1F , eb)","searchableContent":"isvote2certified s eb = isvotecertified (leiosstate.votingstate s) (1f , eb)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":34,"title":"-- Predicates about EBs","searchableContent":"-- predicates about ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":35,"title":"module _ (s ","content":"module _ (s : LeiosState) (eb : EndorserBlock) where","searchableContent":"module _ (s : leiosstate) (eb : endorserblock) where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":36,"title":"open EndorserBlockOSig eb","searchableContent":"  open endorserblockosig eb"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":37,"title":"open LeiosState s","searchableContent":"  open leiosstate s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":39,"title":"vote2Eligible ","content":"vote2Eligible : Type","searchableContent":"  vote2eligible : type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":40,"title":"vote2Eligible = length ebRefs ≥ lengthˢ candidateEBs / 2 -- should this be `>`?","searchableContent":"  vote2eligible = length ebrefs ≥ lengthˢ candidateebs / 2 -- should this be `>`?"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":41,"title":"× fromList ebRefs ⊆ candidateEBs","searchableContent":"                × fromlist ebrefs ⊆ candidateebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":42,"title":"where candidateEBs ","content":"where candidateEBs : ℙ Hash","searchableContent":"    where candidateebs : ℙ hash"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":43,"title":"candidateEBs = mapˢ getEBRef $ filterˢ (_∈ᴮ slice L slot (μ + 3)) (fromList EBs)","searchableContent":"          candidateebs = mapˢ getebref $ filterˢ (_∈ᴮ slice l slot (μ + 3)) (fromlist ebs)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":45,"title":"private variable s s'   ","content":"private variable s s'   : LeiosState","searchableContent":"private variable s s'   : leiosstate"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":46,"title":"ffds'  ","content":"ffds'  : FFD.State","searchableContent":"                 ffds'  : ffd.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":47,"title":"π      ","content":"π      : VrfPf","searchableContent":"                 π      : vrfpf"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":48,"title":"bs'    ","content":"bs'    : B.State","searchableContent":"                 bs'    : b.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":49,"title":"ks ks' ","content":"ks ks' : K.State","searchableContent":"                 ks ks' : k.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":50,"title":"msgs   ","content":"msgs   : List (FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract)","searchableContent":"                 msgs   : list (ffdabstract.header ffdabstract ⊎ ffdabstract.body ffdabstract)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":51,"title":"eb     ","content":"eb     : EndorserBlock","searchableContent":"                 eb     : endorserblock"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":52,"title":"rbs    ","content":"rbs    : List RankingBlock","searchableContent":"                 rbs    : list rankingblock"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":53,"title":"txs    ","content":"txs    : List Tx","searchableContent":"                 txs    : list tx"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":54,"title":"V      ","content":"V      : VTy","searchableContent":"                 v      : vty"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":55,"title":"SD     ","content":"SD     : StakeDistr","searchableContent":"                 sd     : stakedistr"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":56,"title":"pks    ","content":"pks    : List PubKey","searchableContent":"                 pks    : list pubkey"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":58,"title":"data _↝_ ","content":"data _↝_ : LeiosState → LeiosState → Type where","searchableContent":"data _↝_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":60,"title":"IB-Role ","content":"IB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  ib-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":61,"title":"b = ibBody (record { txs = ToPropose })","searchableContent":"                b = ibbody (record { txs = topropose })"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":62,"title":"h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)","searchableContent":"                h = ibheader (mkibheader slot id π sk-ib topropose)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":63,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":64,"title":"∙ needsUpkeep IB-Role","searchableContent":"          ∙ needsupkeep ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":65,"title":"∙ canProduceIB slot sk-IB (stake s) π","searchableContent":"          ∙ canproduceib slot sk-ib (stake s) π"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":66,"title":"∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send h (just b) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":67,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":68,"title":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":70,"title":"EB-Role ","content":"EB-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  eb-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":71,"title":"LI = map getIBRef $ filter (_∈ᴮ slice L slot (Λ + 1)) IBs","searchableContent":"                li = map getibref $ filter (_∈ᴮ slice l slot (λ + 1)) ibs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":72,"title":"LE = map getEBRef $ filter (isVote1Certified s) $","searchableContent":"                le = map getebref $ filter (isvote1certified s) $"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":73,"title":"filter (_∈ᴮ slice L slot (μ + 2)) EBs","searchableContent":"                           filter (_∈ᴮ slice l slot (μ + 2)) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":74,"title":"h = mkEB slot id π sk-EB LI LE","searchableContent":"                h = mkeb slot id π sk-eb li le"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":75,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":76,"title":"∙ needsUpkeep EB-Role","searchableContent":"          ∙ needsupkeep eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":77,"title":"∙ canProduceEB slot sk-EB (stake s) π","searchableContent":"          ∙ canproduceeb slot sk-eb (stake s) π"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":78,"title":"∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (ebheader h) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":79,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":80,"title":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":82,"title":"V1-Role ","content":"V1-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v1-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":83,"title":"EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot (μ + 1)) EBs","searchableContent":"                ebs' = filter (allibrefsknown s) $ filter (_∈ᴮ slice l slot (μ + 1)) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":84,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":85,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":86,"title":"∙ needsUpkeep V1-Role","searchableContent":"          ∙ needsupkeep v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":87,"title":"∙ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":88,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":89,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":90,"title":"s ↝ addUpkeep record s { FFDState = ffds' } V1-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":92,"title":"V2-Role ","content":"V2-Role : let open LeiosState s renaming (FFDState to ffds)","searchableContent":"  v2-role : let open leiosstate s renaming (ffdstate to ffds)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":93,"title":"EBs' = filter (vote2Eligible s) $ filter (_∈ᴮ slice L slot 1) EBs","searchableContent":"                ebs' = filter (vote2eligible s) $ filter (_∈ᴮ slice l slot 1) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":94,"title":"votes = map (vote sk-VT ∘ hash) EBs'","searchableContent":"                votes = map (vote sk-vt ∘ hash) ebs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":95,"title":"in","searchableContent":"          in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":96,"title":"∙ needsUpkeep V2-Role","searchableContent":"          ∙ needsupkeep v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":97,"title":"∙ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":98,"title":"∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'","searchableContent":"          ∙ ffds ffd.-⟦ send (vtheader votes) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":99,"title":"─────────────────────────────────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":100,"title":"s ↝ addUpkeep record s { FFDState = ffds' } V2-Role","searchableContent":"          s ↝ addupkeep record s { ffdstate = ffds' } v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":102,"title":"-- Note","content":"-- Note: Base doesn't need a negative rule, since it can always be invoked","searchableContent":"  -- note: base doesn't need a negative rule, since it can always be invoked"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":104,"title":"No-IB-Role ","content":"No-IB-Role : let open LeiosState s in","searchableContent":"  no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":105,"title":"∙ needsUpkeep IB-Role","searchableContent":"          ∙ needsupkeep ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":106,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceib slot sk-ib (stake s) π)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":107,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":108,"title":"s ↝ addUpkeep s IB-Role","searchableContent":"          s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":110,"title":"No-EB-Role ","content":"No-EB-Role : let open LeiosState s in","searchableContent":"  no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":111,"title":"∙ needsUpkeep EB-Role","searchableContent":"          ∙ needsupkeep eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":112,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)","searchableContent":"          ∙ (∀ π → ¬ canproduceeb slot sk-eb (stake s) π)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":113,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":114,"title":"s ↝ addUpkeep s EB-Role","searchableContent":"          s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":116,"title":"No-V1-Role ","content":"No-V1-Role : let open LeiosState s in","searchableContent":"  no-v1-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":117,"title":"∙ needsUpkeep V1-Role","searchableContent":"          ∙ needsupkeep v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":118,"title":"∙ ¬ canProduceV1 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev1 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":119,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":120,"title":"s ↝ addUpkeep s V1-Role","searchableContent":"          s ↝ addupkeep s v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":122,"title":"No-V2-Role ","content":"No-V2-Role : let open LeiosState s in","searchableContent":"  no-v2-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":123,"title":"∙ needsUpkeep V2-Role","searchableContent":"          ∙ needsupkeep v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":124,"title":"∙ ¬ canProduceV2 slot sk-VT (stake s)","searchableContent":"          ∙ ¬ canproducev2 slot sk-vt (stake s)"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":125,"title":"─────────────────────────────────────────────","searchableContent":"          ─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":126,"title":"s ↝ addUpkeep s V2-Role","searchableContent":"          s ↝ addupkeep s v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":128,"title":"data _-⟦_/_⟧⇀_ ","content":"data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":130,"title":"-- Initialization","searchableContent":"  -- initialization"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":132,"title":"Init ","content":"Init :","searchableContent":"  init :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":133,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"       ∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":134,"title":"∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'","searchableContent":"       ∙ initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":135,"title":"────────────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":136,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","searchableContent":"       nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":138,"title":"-- Network and Ledger","searchableContent":"  -- network and ledger"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":140,"title":"Slot ","content":"Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in","searchableContent":"  slot : let open leiosstate s renaming (ffdstate to ffds; basestate to bs) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":141,"title":"∙ Upkeep ≡ᵉ allUpkeep","searchableContent":"       ∙ upkeep ≡ᵉ allupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":142,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"       ∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":143,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"       ∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":144,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"       ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":145,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"       just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":146,"title":"{ FFDState  = ffds'","searchableContent":"           { ffdstate  = ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":147,"title":"; Ledger    = constructLedger rbs","searchableContent":"           ; ledger    = constructledger rbs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":148,"title":"; slot      = suc slot","searchableContent":"           ; slot      = suc slot"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":149,"title":"; Upkeep    = ∅","searchableContent":"           ; upkeep    = ∅"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":150,"title":"; BaseState = bs'","searchableContent":"           ; basestate = bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":151,"title":"} ↑ L.filter (isValid? s) msgs","searchableContent":"           } ↑ l.filter (isvalid? s) msgs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":153,"title":"Ftch ","content":"Ftch :","searchableContent":"  ftch :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":154,"title":"────────────────────────────────────────────────────────","searchableContent":"       ────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":155,"title":"just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s","searchableContent":"       just s -⟦ ftch-ldg / ftch-ldg (leiosstate.ledger s) ⟧⇀ s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":157,"title":"-- Base chain","searchableContent":"  -- base chain"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":158,"title":"--","searchableContent":"  --"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":159,"title":"-- Note","content":"-- Note: Submitted data to the base chain is only taken into account","searchableContent":"  -- note: submitted data to the base chain is only taken into account"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":160,"title":"--       if the party submitting is the block producer on the base chain","searchableContent":"  --       if the party submitting is the block producer on the base chain"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":161,"title":"--       for the given slot","searchableContent":"  --       for the given slot"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":163,"title":"Base₁   ","content":"Base₁   :","searchableContent":"  base₁   :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":164,"title":"───────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":165,"title":"just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"          just s -⟦ submit (inj₂ txs) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":167,"title":"Base₂a  ","content":"Base₂a  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂a  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":168,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":169,"title":"∙ eb ∈ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ eb ∈ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":170,"title":"∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (this eb) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":171,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":172,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":174,"title":"Base₂b  ","content":"Base₂b  : let open LeiosState s renaming (BaseState to bs) in","searchableContent":"  base₂b  : let open leiosstate s renaming (basestate to bs) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":175,"title":"∙ needsUpkeep Base","searchableContent":"          ∙ needsupkeep base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":176,"title":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs","searchableContent":"          ∙ [] ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":177,"title":"∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'","searchableContent":"          ∙ bs b.-⟦ b.submit (that topropose) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":178,"title":"───────────────────────────────────────────────────────────────────────","searchableContent":"          ───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":179,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"          just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":181,"title":"-- Protocol rules","searchableContent":"  -- protocol rules"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":183,"title":"Roles ","content":"Roles : ∙ s ↝ s'","searchableContent":"  roles : ∙ s ↝ s'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":184,"title":"─────────────────────────────","searchableContent":"          ─────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":185,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"          just s -⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":186,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":2,"title":"Leios.SpecStructure
{-# OPTIONS --safe #-}","searchableContent":"leios.specstructure
{-# options --safe #-}"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding (id)","searchableContent":"open import leios.prelude hiding (id)"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":6,"title":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":7,"title":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":9,"title":"import Leios.Base","searchableContent":"import leios.base"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":10,"title":"import Leios.Blocks","searchableContent":"import leios.blocks"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":11,"title":"import Leios.KeyRegistration","searchableContent":"import leios.keyregistration"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":12,"title":"import Leios.Voting","searchableContent":"import leios.voting"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":14,"title":"open import Data.Fin","searchableContent":"open import data.fin"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":16,"title":"module Leios.SpecStructure where","searchableContent":"module leios.specstructure where"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":18,"title":"record SpecStructure (rounds ","content":"record SpecStructure (rounds : ℕ) : Type₁ where","searchableContent":"record specstructure (rounds : ℕ) : type₁ where"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":19,"title":"field a ","content":"field a : LeiosAbstract","searchableContent":"  field a : leiosabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":21,"title":"open LeiosAbstract a public","searchableContent":"  open leiosabstract a public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":22,"title":"open Leios.Blocks a public","searchableContent":"  open leios.blocks a public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":24,"title":"field ⦃ IsBlock-Vote ⦄ ","content":"field ⦃ IsBlock-Vote ⦄ : IsBlock (List Vote)","searchableContent":"  field ⦃ isblock-vote ⦄ : isblock (list vote)"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":25,"title":"⦃ Hashable-PreIBHeader ⦄ ","content":"⦃ Hashable-PreIBHeader ⦄ : Hashable PreIBHeader Hash","searchableContent":"        ⦃ hashable-preibheader ⦄ : hashable preibheader hash"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":26,"title":"⦃ Hashable-PreEndorserBlock ⦄ ","content":"⦃ Hashable-PreEndorserBlock ⦄ : Hashable PreEndorserBlock Hash","searchableContent":"        ⦃ hashable-preendorserblock ⦄ : hashable preendorserblock hash"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":27,"title":"id ","content":"id : PoolID","searchableContent":"        id : poolid"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":28,"title":"FFD' ","content":"FFD' : FFDAbstract.Functionality ffdAbstract","searchableContent":"        ffd' : ffdabstract.functionality ffdabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":29,"title":"vrf' ","content":"vrf' : LeiosVRF a","searchableContent":"        vrf' : leiosvrf a"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":31,"title":"open LeiosVRF vrf' public","searchableContent":"  open leiosvrf vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":33,"title":"field sk-IB sk-EB sk-VT ","content":"field sk-IB sk-EB sk-VT : PrivKey","searchableContent":"  field sk-ib sk-eb sk-vt : privkey"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":34,"title":"pk-IB pk-EB pk-VT ","content":"pk-IB pk-EB pk-VT : PubKey","searchableContent":"        pk-ib pk-eb pk-vt : pubkey"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":36,"title":"open Leios.Base a vrf' public","searchableContent":"  open leios.base a vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":38,"title":"field B' ","content":"field B' : BaseAbstract","searchableContent":"  field b' : baseabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":39,"title":"BF ","content":"BF : BaseAbstract.Functionality B'","searchableContent":"        bf : baseabstract.functionality b'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":40,"title":"initBaseState ","content":"initBaseState : BaseAbstract.Functionality.State BF","searchableContent":"        initbasestate : baseabstract.functionality.state bf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":42,"title":"open Leios.KeyRegistration a vrf' public","searchableContent":"  open leios.keyregistration a vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":44,"title":"field K' ","content":"field K' : KeyRegistrationAbstract","searchableContent":"  field k' : keyregistrationabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":45,"title":"KF ","content":"KF : KeyRegistrationAbstract.Functionality K'","searchableContent":"        kf : keyregistrationabstract.functionality k'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":47,"title":"module B   = BaseAbstract.Functionality BF","searchableContent":"  module b   = baseabstract.functionality bf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":48,"title":"module K   = KeyRegistrationAbstract.Functionality KF","searchableContent":"  module k   = keyregistrationabstract.functionality kf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":49,"title":"module FFD = FFDAbstract.Functionality FFD'","searchableContent":"  module ffd = ffdabstract.functionality ffd'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":51,"title":"open Leios.Voting public","searchableContent":"  open leios.voting public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":53,"title":"field va ","content":"field va : VotingAbstract (Fin rounds × EndorserBlock)","searchableContent":"  field va : votingabstract (fin rounds × endorserblock)"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":54,"title":"open VotingAbstract va public","searchableContent":"  open votingabstract va public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":55,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":2,"title":"Leios.Traces
{-# OPTIONS --safe #-}","searchableContent":"leios.traces
{-# options --safe #-}"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":5,"title":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":7,"title":"import Leios.Protocol","searchableContent":"import leios.protocol"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":9,"title":"module Leios.Traces {n} (⋯ ","content":"module Leios.Traces {n} (⋯ : SpecStructure n) {u : Type} (let open Leios.Protocol ⋯ u)","searchableContent":"module leios.traces {n} (⋯ : specstructure n) {u : type} (let open leios.protocol ⋯ u)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":10,"title":"(_-⟦_/_⟧⇀_ ","content":"(_-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type)","searchableContent":"  (_-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":11,"title":"where","searchableContent":"  where"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":13,"title":"_⇉_ ","content":"_⇉_ : LeiosState → LeiosState → Type","searchableContent":"_⇉_ : leiosstate → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":14,"title":"s₁ ⇉ s₂ = Σ[ (i , o) ∈ LeiosInput × LeiosOutput ] (just s₁ -⟦ i / o ⟧⇀ s₂)","searchableContent":"s₁ ⇉ s₂ = σ[ (i , o) ∈ leiosinput × leiosoutput ] (just s₁ -⟦ i / o ⟧⇀ s₂)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":16,"title":"_⇉[_]_ ","content":"_⇉[_]_ : LeiosState → ℕ → LeiosState → Type","searchableContent":"_⇉[_]_ : leiosstate → ℕ → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":17,"title":"s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂","searchableContent":"s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":18,"title":"s₁ ⇉[ suc m ] sₙ = Σ[ s₂ ∈ LeiosState ] (s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ)","searchableContent":"s₁ ⇉[ suc m ] sₙ = σ[ s₂ ∈ leiosstate ] (s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":20,"title":"_⇉⋆_ ","content":"_⇉⋆_ : LeiosState → LeiosState → Type","searchableContent":"_⇉⋆_ : leiosstate → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":21,"title":"s₁ ⇉⋆ sₙ = Σ[ n ∈  ℕ ] (s₁ ⇉[ n ] sₙ)","searchableContent":"s₁ ⇉⋆ sₙ = σ[ n ∈  ℕ ] (s₁ ⇉[ n ] sₙ)"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":22,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":2,"title":"Leios.VRF
{-# OPTIONS --safe #-}","searchableContent":"leios.vrf
{-# options --safe #-}"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":7,"title":"module Leios.VRF (a ","content":"module Leios.VRF (a : LeiosAbstract) (let open LeiosAbstract a) where","searchableContent":"module leios.vrf (a : leiosabstract) (let open leiosabstract a) where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":9,"title":"record VRF (Dom Range PubKey ","content":"record VRF (Dom Range PubKey : Type) : Type₁ where","searchableContent":"record vrf (dom range pubkey : type) : type₁ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":10,"title":"field isKeyPair ","content":"field isKeyPair : PubKey → PrivKey → Type","searchableContent":"  field iskeypair : pubkey → privkey → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":11,"title":"eval ","content":"eval : PrivKey → Dom → Range × VrfPf","searchableContent":"        eval : privkey → dom → range × vrfpf"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":12,"title":"verify ","content":"verify : PubKey → Dom → Range → VrfPf → Type","searchableContent":"        verify : pubkey → dom → range → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":13,"title":"verify? ","content":"verify? : (pk : PubKey) → (d : Dom) → (r : Range) → (pf : VrfPf) → Dec (verify pk d r pf)","searchableContent":"        verify? : (pk : pubkey) → (d : dom) → (r : range) → (pf : vrfpf) → dec (verify pk d r pf)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":15,"title":"record LeiosVRF ","content":"record LeiosVRF : Type₁ where","searchableContent":"record leiosvrf : type₁ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":16,"title":"field PubKey ","content":"field PubKey : Type","searchableContent":"  field pubkey : type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":17,"title":"poolID ","content":"poolID : PubKey → PoolID","searchableContent":"        poolid : pubkey → poolid"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":18,"title":"verifySig ","content":"verifySig : PubKey → Sig → Type","searchableContent":"        verifysig : pubkey → sig → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":19,"title":"verifySig? ","content":"verifySig? : (pk : PubKey) → (sig : Sig) → Dec (verifySig pk sig)","searchableContent":"        verifysig? : (pk : pubkey) → (sig : sig) → dec (verifysig pk sig)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":21,"title":"vrf ","content":"vrf : VRF ℕ ℕ PubKey","searchableContent":"        vrf : vrf ℕ ℕ pubkey"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":23,"title":"open VRF vrf public","searchableContent":"  open vrf vrf public"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":25,"title":"-- transforming slot numbers into VRF seeds","searchableContent":"  -- transforming slot numbers into vrf seeds"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":26,"title":"field genIBInput genEBInput genVInput genV1Input genV2Input ","content":"field genIBInput genEBInput genVInput genV1Input genV2Input : ℕ → ℕ","searchableContent":"  field genibinput genebinput genvinput genv1input genv2input : ℕ → ℕ"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":28,"title":"canProduceIB ","content":"canProduceIB : ℕ → PrivKey → ℕ → VrfPf → Type","searchableContent":"  canproduceib : ℕ → privkey → ℕ → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":29,"title":"canProduceIB slot k stake π = let (val , pf) = eval k (genIBInput slot) in val < stake × pf ≡ π","searchableContent":"  canproduceib slot k stake π = let (val , pf) = eval k (genibinput slot) in val < stake × pf ≡ π"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":31,"title":"Dec-canProduceIB ","content":"Dec-canProduceIB : ∀ {slot k stake} → (∃[ π ] canProduceIB slot k stake π) ⊎ (∀ π → ¬ canProduceIB slot k stake π)","searchableContent":"  dec-canproduceib : ∀ {slot k stake} → (∃[ π ] canproduceib slot k stake π) ⊎ (∀ π → ¬ canproduceib slot k stake π)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":32,"title":"Dec-canProduceIB {slot} {k} {stake} with eval k (genIBInput slot)","searchableContent":"  dec-canproduceib {slot} {k} {stake} with eval k (genibinput slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":33,"title":"... | (val , pf) = case ¿ val < stake ¿ of λ where","searchableContent":"  ... | (val , pf) = case ¿ val < stake ¿ of λ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":34,"title":"(yes p) → inj₁ (pf , p , refl)","searchableContent":"    (yes p) → inj₁ (pf , p , refl)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":35,"title":"(no ¬p) → inj₂ (λ π (h , _) → ¬p h)","searchableContent":"    (no ¬p) → inj₂ (λ π (h , _) → ¬p h)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":37,"title":"canProduceIBPub ","content":"canProduceIBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type","searchableContent":"  canproduceibpub : ℕ → ℕ → pubkey → vrfpf → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":38,"title":"canProduceIBPub slot val k pf stake = verify k (genIBInput slot) val pf × val < stake","searchableContent":"  canproduceibpub slot val k pf stake = verify k (genibinput slot) val pf × val < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":40,"title":"canProduceEB ","content":"canProduceEB : ℕ → PrivKey → ℕ → VrfPf → Type","searchableContent":"  canproduceeb : ℕ → privkey → ℕ → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":41,"title":"canProduceEB slot k stake π = let (val , pf) = eval k (genEBInput slot) in val < stake × pf ≡ π","searchableContent":"  canproduceeb slot k stake π = let (val , pf) = eval k (genebinput slot) in val < stake × pf ≡ π"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":43,"title":"Dec-canProduceEB ","content":"Dec-canProduceEB : ∀ {slot k stake} → (∃[ π ] canProduceEB slot k stake π) ⊎ (∀ π → ¬ canProduceEB slot k stake π)","searchableContent":"  dec-canproduceeb : ∀ {slot k stake} → (∃[ π ] canproduceeb slot k stake π) ⊎ (∀ π → ¬ canproduceeb slot k stake π)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":44,"title":"Dec-canProduceEB {slot} {k} {stake} with eval k (genEBInput slot)","searchableContent":"  dec-canproduceeb {slot} {k} {stake} with eval k (genebinput slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":45,"title":"... | (val , pf) = case ¿ val < stake ¿ of λ where","searchableContent":"  ... | (val , pf) = case ¿ val < stake ¿ of λ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":46,"title":"(yes p) → inj₁ (pf , p , refl)","searchableContent":"    (yes p) → inj₁ (pf , p , refl)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":47,"title":"(no ¬p) → inj₂ (λ π (h , _) → ¬p h)","searchableContent":"    (no ¬p) → inj₂ (λ π (h , _) → ¬p h)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":49,"title":"canProduceEBPub ","content":"canProduceEBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type","searchableContent":"  canproduceebpub : ℕ → ℕ → pubkey → vrfpf → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":50,"title":"canProduceEBPub slot val k pf stake = verify k (genEBInput slot) val pf × val < stake","searchableContent":"  canproduceebpub slot val k pf stake = verify k (genebinput slot) val pf × val < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":52,"title":"canProduceV ","content":"canProduceV : ℕ → PrivKey → ℕ → Type","searchableContent":"  canproducev : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":53,"title":"canProduceV slot k stake = proj₁ (eval k (genVInput slot)) < stake","searchableContent":"  canproducev slot k stake = proj₁ (eval k (genvinput slot)) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":55,"title":"Dec-canProduceV ","content":"Dec-canProduceV : ∀ {slot k stake} → Dec (canProduceV slot k stake)","searchableContent":"  dec-canproducev : ∀ {slot k stake} → dec (canproducev slot k stake)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":56,"title":"Dec-canProduceV {slot} {k} {stake} with eval k (genVInput slot)","searchableContent":"  dec-canproducev {slot} {k} {stake} with eval k (genvinput slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":57,"title":"... | (val , pf) = ¿ val < stake ¿","searchableContent":"  ... | (val , pf) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":59,"title":"canProduceV1 ","content":"canProduceV1 : ℕ → PrivKey → ℕ → Type","searchableContent":"  canproducev1 : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":60,"title":"canProduceV1 slot k stake = proj₁ (eval k (genV1Input slot)) < stake","searchableContent":"  canproducev1 slot k stake = proj₁ (eval k (genv1input slot)) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":62,"title":"Dec-canProduceV1 ","content":"Dec-canProduceV1 : ∀ {slot k stake} → Dec (canProduceV1 slot k stake)","searchableContent":"  dec-canproducev1 : ∀ {slot k stake} → dec (canproducev1 slot k stake)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":63,"title":"Dec-canProduceV1 {slot} {k} {stake} with eval k (genV1Input slot)","searchableContent":"  dec-canproducev1 {slot} {k} {stake} with eval k (genv1input slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":64,"title":"... | (val , pf) = ¿ val < stake ¿","searchableContent":"  ... | (val , pf) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":66,"title":"canProduceV2 ","content":"canProduceV2 : ℕ → PrivKey → ℕ → Type","searchableContent":"  canproducev2 : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":67,"title":"canProduceV2 slot k stake = proj₁ (eval k (genV2Input slot)) < stake","searchableContent":"  canproducev2 slot k stake = proj₁ (eval k (genv2input slot)) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":69,"title":"Dec-canProduceV2 ","content":"Dec-canProduceV2 : ∀ {slot k stake} → Dec (canProduceV2 slot k stake)","searchableContent":"  dec-canproducev2 : ∀ {slot k stake} → dec (canproducev2 slot k stake)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":70,"title":"Dec-canProduceV2 {slot} {k} {stake} with eval k (genV2Input slot)","searchableContent":"  dec-canproducev2 {slot} {k} {stake} with eval k (genv2input slot)"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":71,"title":"... | (val , pf) = ¿ val < stake ¿","searchableContent":"  ... | (val , pf) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":72,"title":"
","content":"
","searchableContent":""},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":1,"title":"","content":"","searchableContent":""},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":2,"title":"Leios.Voting
{-# OPTIONS --safe #-}","searchableContent":"leios.voting
{-# options --safe #-}"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":6,"title":"module Leios.Voting where","searchableContent":"module leios.voting where"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":8,"title":"record VotingAbstract (X ","content":"record VotingAbstract (X : Type) : Type₁ where","searchableContent":"record votingabstract (x : type) : type₁ where"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":9,"title":"field VotingState     ","content":"field VotingState     : Type","searchableContent":"  field votingstate     : type"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":10,"title":"initVotingState ","content":"initVotingState : VotingState","searchableContent":"        initvotingstate : votingstate"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":11,"title":"isVoteCertified ","content":"isVoteCertified : VotingState → X → Type","searchableContent":"        isvotecertified : votingstate → x → type"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":13,"title":"⦃ isVoteCertified⁇ ⦄ ","content":"⦃ isVoteCertified⁇ ⦄ : ∀ {vs x} → isVoteCertified vs x ⁇","searchableContent":"        ⦃ isvotecertified⁇ ⦄ : ∀ {vs x} → isvotecertified vs x ⁇"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":14,"title":"
","content":"
","searchableContent":""}]; +const searchIndex = [{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":2,"title":"Leios.Abstract {-# OPTIONS --safe #-}","content":"Leios.Abstract {-# OPTIONS --safe #-}","searchableContent":"leios.abstract {-# options --safe #-}"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":4,"title":"module Leios.Abstract where","content":"module Leios.Abstract where","searchableContent":"module leios.abstract where"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":6,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":8,"title":"record LeiosAbstract","content":"record LeiosAbstract : Type₁ where","searchableContent":"record leiosabstract : type₁ where"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":9,"title":"field Tx","content":"field Tx : Type","searchableContent":"field tx : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":10,"title":"⦃ DecEq-Tx ⦄","content":"⦃ DecEq-Tx ⦄ : DecEq Tx","searchableContent":"⦃ deceq-tx ⦄ : deceq tx"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":11,"title":"PoolID","content":"PoolID : Type","searchableContent":"poolid : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":12,"title":"⦃ DecEq-PoolID ⦄","content":"⦃ DecEq-PoolID ⦄ : DecEq PoolID","searchableContent":"⦃ deceq-poolid ⦄ : deceq poolid"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":13,"title":"BodyHash VrfPf PrivKey Sig Hash","content":"BodyHash VrfPf PrivKey Sig Hash : Type -- these could have been byte strings, but this is safer","searchableContent":"bodyhash vrfpf privkey sig hash : type -- these could have been byte strings, but this is safer"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":14,"title":"⦃ DecEq-Hash ⦄","content":"⦃ DecEq-Hash ⦄ : DecEq Hash","searchableContent":"⦃ deceq-hash ⦄ : deceq hash"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":15,"title":"⦃ DecEq-VrfPf ⦄","content":"⦃ DecEq-VrfPf ⦄ : DecEq VrfPf","searchableContent":"⦃ deceq-vrfpf ⦄ : deceq vrfpf"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":16,"title":"⦃ DecEq-Sig ⦄","content":"⦃ DecEq-Sig ⦄ : DecEq Sig","searchableContent":"⦃ deceq-sig ⦄ : deceq sig"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":17,"title":"Vote","content":"Vote : Type","searchableContent":"vote : type"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":18,"title":"⦃ DecEq-Vote ⦄","content":"⦃ DecEq-Vote ⦄ : DecEq Vote","searchableContent":"⦃ deceq-vote ⦄ : deceq vote"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":19,"title":"vote","content":"vote : PrivKey → Hash → Vote","searchableContent":"vote : privkey → hash → vote"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":20,"title":"sign","content":"sign : PrivKey → Hash → Sig","searchableContent":"sign : privkey → hash → sig"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":21,"title":"⦃ Hashable-Txs ⦄","content":"⦃ Hashable-Txs ⦄ : Hashable ( List Tx ) Hash","searchableContent":"⦃ hashable-txs ⦄ : hashable ( list tx ) hash"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":22,"title":"L","content":"L : ℕ","searchableContent":"l : ℕ"},{"moduleName":"Leios.Abstract","path":"Leios.Abstract.html","group":"Leios","lineNumber":23,"title":"⦃ NonZero-L ⦄","content":"⦃ NonZero-L ⦄ : NonZero L","searchableContent":"⦃ nonzero-l ⦄ : nonzero l"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":2,"title":"Leios.Base {-# OPTIONS --safe #-}","content":"Leios.Base {-# OPTIONS --safe #-}","searchableContent":"leios.base {-# options --safe #-}"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","content":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":6,"title":"open import Leios.VRF","content":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":8,"title":"module Leios.Base ( a","content":"module Leios.Base ( a : LeiosAbstract ) ( open LeiosAbstract a ) ( vrf' : LeiosVRF a )","searchableContent":"module leios.base ( a : leiosabstract ) ( open leiosabstract a ) ( vrf' : leiosvrf a )"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":9,"title":"( let open LeiosVRF vrf' ) where","content":"( let open LeiosVRF vrf' ) where","searchableContent":"( let open leiosvrf vrf' ) where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":11,"title":"open import Leios.Blocks a using ( EndorserBlock )","content":"open import Leios.Blocks a using ( EndorserBlock )","searchableContent":"open import leios.blocks a using ( endorserblock )"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":13,"title":"StakeDistr","content":"StakeDistr : Type","searchableContent":"stakedistr : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":14,"title":"StakeDistr","content":"StakeDistr = TotalMap PoolID ℕ","searchableContent":"stakedistr = totalmap poolid ℕ"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":16,"title":"RankingBlock","content":"RankingBlock : Type","searchableContent":"rankingblock : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":17,"title":"RankingBlock","content":"RankingBlock = These EndorserBlock ( List Tx )","searchableContent":"rankingblock = these endorserblock ( list tx )"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":19,"title":"record BaseAbstract","content":"record BaseAbstract : Type₁ where","searchableContent":"record baseabstract : type₁ where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":20,"title":"field Cert","content":"field Cert : Type","searchableContent":"field cert : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":21,"title":"VTy","content":"VTy : Type","searchableContent":"vty : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":22,"title":"initSlot","content":"initSlot : VTy → ℕ","searchableContent":"initslot : vty → ℕ"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":23,"title":"V-chkCerts","content":"V-chkCerts : List PubKey → EndorserBlock × Cert → Bool","searchableContent":"v-chkcerts : list pubkey → endorserblock × cert → bool"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":25,"title":"data Input","content":"data Input : Type where","searchableContent":"data input : type where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":26,"title":"INIT","content":"INIT : ( EndorserBlock × Cert → Bool ) → Input","searchableContent":"init : ( endorserblock × cert → bool ) → input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":27,"title":"SUBMIT","content":"SUBMIT : RankingBlock → Input","searchableContent":"submit : rankingblock → input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":28,"title":"FTCH-LDG","content":"FTCH-LDG : Input","searchableContent":"ftch-ldg : input"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":30,"title":"data Output","content":"data Output : Type where","searchableContent":"data output : type where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":31,"title":"STAKE","content":"STAKE : StakeDistr → Output","searchableContent":"stake : stakedistr → output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":32,"title":"EMPTY","content":"EMPTY : Output","searchableContent":"empty : output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":33,"title":"BASE-LDG","content":"BASE-LDG : List RankingBlock → Output","searchableContent":"base-ldg : list rankingblock → output"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":35,"title":"record Functionality","content":"record Functionality : Type₁ where","searchableContent":"record functionality : type₁ where"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":36,"title":"field State","content":"field State : Type","searchableContent":"field state : type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":37,"title":"_-⟦_/_⟧⇀_","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"_-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":38,"title":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄","content":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : { s : State } → { i : Input } → { o : Output } → { s' : State } → ( s -⟦ i / o ⟧⇀ s' ) ⁇","searchableContent":"⦃ dec-_-⟦_/_⟧⇀_ ⦄ : { s : state } → { i : input } → { o : output } → { s' : state } → ( s -⟦ i / o ⟧⇀ s' ) ⁇"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":39,"title":"SUBMIT-total","content":"SUBMIT-total : ∀ { s b } → ∃[ s' ] s -⟦ SUBMIT b / EMPTY ⟧⇀ s'","searchableContent":"submit-total : ∀ { s b } → ∃[ s' ] s -⟦ submit b / empty ⟧⇀ s'"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":41,"title":"open Input public","content":"open Input public","searchableContent":"open input public"},{"moduleName":"Leios.Base","path":"Leios.Base.html","group":"Leios","lineNumber":42,"title":"open Output public","content":"open Output public","searchableContent":"open output public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":2,"title":"Leios.Blocks {-# OPTIONS --safe #-}","content":"Leios.Blocks {-# OPTIONS --safe #-}","searchableContent":"leios.blocks {-# options --safe #-}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","content":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":6,"title":"open import Leios.FFD","content":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":7,"title":"open import Tactic.Defaults","content":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":8,"title":"open import Tactic.Derive.DecEq","content":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":10,"title":"module Leios.Blocks ( a","content":"module Leios.Blocks ( a : LeiosAbstract ) ( let open LeiosAbstract a ) where","searchableContent":"module leios.blocks ( a : leiosabstract ) ( let open leiosabstract a ) where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":12,"title":"-- IsBlock typeclass (could do a closed-world approach instead)","content":"-- IsBlock typeclass (could do a closed-world approach instead)","searchableContent":"-- isblock typeclass (could do a closed-world approach instead)"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":13,"title":"-- Q","content":"-- Q: should votes have an instance of this class?","searchableContent":"-- q: should votes have an instance of this class?"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":14,"title":"record IsBlock ( B","content":"record IsBlock ( B : Type ) : Type where","searchableContent":"record isblock ( b : type ) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":15,"title":"field slotNumber","content":"field slotNumber : B → ℕ","searchableContent":"field slotnumber : b → ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":16,"title":"producerID","content":"producerID : B → PoolID","searchableContent":"producerid : b → poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":17,"title":"lotteryPf","content":"lotteryPf : B → VrfPf","searchableContent":"lotterypf : b → vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":18,"title":"signature","content":"signature : B → Sig","searchableContent":"signature : b → sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":20,"title":"infix 4 _∈ᴮ_","content":"infix 4 _∈ᴮ_","searchableContent":"infix 4 _∈ᴮ_"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":22,"title":"_∈ᴮ_","content":"_∈ᴮ_ : B → ℙ ℕ → Type","searchableContent":"_∈ᴮ_ : b → ℙ ℕ → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":23,"title":"b ∈ᴮ X","content":"b ∈ᴮ X = slotNumber b ∈ X","searchableContent":"b ∈ᴮ x = slotnumber b ∈ x"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":25,"title":"open IsBlock ⦃...⦄ public","content":"open IsBlock ⦃...⦄ public","searchableContent":"open isblock ⦃...⦄ public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":27,"title":"IBRef","content":"IBRef = Hash","searchableContent":"ibref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":28,"title":"EBRef","content":"EBRef = Hash","searchableContent":"ebref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":30,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":31,"title":"-- Input Blocks","content":"-- Input Blocks","searchableContent":"-- input blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":32,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":34,"title":"record IBHeaderOSig ( sig","content":"record IBHeaderOSig ( sig : Type ) : Type where","searchableContent":"record ibheaderosig ( sig : type ) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":35,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"field slotnumber : ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":36,"title":"producerID","content":"producerID : PoolID","searchableContent":"producerid : poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":37,"title":"lotteryPf","content":"lotteryPf : VrfPf","searchableContent":"lotterypf : vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":38,"title":"bodyHash","content":"bodyHash : Hash","searchableContent":"bodyhash : hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":39,"title":"signature","content":"signature : sig","searchableContent":"signature : sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":41,"title":"IBHeader","content":"IBHeader = IBHeaderOSig Sig","searchableContent":"ibheader = ibheaderosig sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":42,"title":"PreIBHeader","content":"PreIBHeader = IBHeaderOSig ⊤","searchableContent":"preibheader = ibheaderosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":44,"title":"record IBBody","content":"record IBBody : Type where","searchableContent":"record ibbody : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":45,"title":"field txs","content":"field txs : List Tx","searchableContent":"field txs : list tx"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":47,"title":"record InputBlock","content":"record InputBlock : Type where","searchableContent":"record inputblock : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":48,"title":"field header","content":"field header : IBHeader","searchableContent":"field header : ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":49,"title":"body","content":"body : IBBody","searchableContent":"body : ibbody"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":51,"title":"open IBHeaderOSig header public","content":"open IBHeaderOSig header public","searchableContent":"open ibheaderosig header public"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":53,"title":"unquoteDecl DecEq-IBBody DecEq-IBHeaderOSig DecEq-InputBlock","content":"unquoteDecl DecEq-IBBody DecEq-IBHeaderOSig DecEq-InputBlock =","searchableContent":"unquotedecl deceq-ibbody deceq-ibheaderosig deceq-inputblock ="},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":54,"title":"derive-DecEq (","content":"derive-DecEq (","searchableContent":"derive-deceq ("},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":55,"title":"( quote IBBody , DecEq-IBBody )","content":"( quote IBBody , DecEq-IBBody )","searchableContent":"( quote ibbody , deceq-ibbody )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":56,"title":"∷ ( quote IBHeaderOSig , DecEq-IBHeaderOSig )","content":"∷ ( quote IBHeaderOSig , DecEq-IBHeaderOSig )","searchableContent":"∷ ( quote ibheaderosig , deceq-ibheaderosig )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":57,"title":"∷ ( quote InputBlock , DecEq-InputBlock ) ∷ [] )","content":"∷ ( quote InputBlock , DecEq-InputBlock ) ∷ [] )","searchableContent":"∷ ( quote inputblock , deceq-inputblock ) ∷ [] )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":59,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":60,"title":"IsBlock-IBHeader","content":"IsBlock-IBHeader : IsBlock IBHeader","searchableContent":"isblock-ibheader : isblock ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":61,"title":"IsBlock-IBHeader","content":"IsBlock-IBHeader = record { IBHeaderOSig }","searchableContent":"isblock-ibheader = record { ibheaderosig }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":63,"title":"Hashable-IBBody","content":"Hashable-IBBody : Hashable IBBody Hash","searchableContent":"hashable-ibbody : hashable ibbody hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":64,"title":"Hashable-IBBody . hash b","content":"Hashable-IBBody . hash b = hash ( b . IBBody.txs )","searchableContent":"hashable-ibbody . hash b = hash ( b . ibbody.txs )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":66,"title":"Hashable-IBHeader","content":"Hashable-IBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → Hashable IBHeader Hash","searchableContent":"hashable-ibheader : ⦃ hashable preibheader hash ⦄ → hashable ibheader hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":67,"title":"Hashable-IBHeader . hash b","content":"Hashable-IBHeader . hash b = hash { T = PreIBHeader }","searchableContent":"hashable-ibheader . hash b = hash { t = preibheader }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":68,"title":"record { IBHeaderOSig b hiding ( signature ) ; signature","content":"record { IBHeaderOSig b hiding ( signature ) ; signature = _ }","searchableContent":"record { ibheaderosig b hiding ( signature ) ; signature = _ }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":70,"title":"IsBlock-InputBlock","content":"IsBlock-InputBlock : IsBlock InputBlock","searchableContent":"isblock-inputblock : isblock inputblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":71,"title":"IsBlock-InputBlock","content":"IsBlock-InputBlock = record { InputBlock }","searchableContent":"isblock-inputblock = record { inputblock }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":73,"title":"mkIBHeader","content":"mkIBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List Tx → IBHeader","searchableContent":"mkibheader : ⦃ hashable preibheader hash ⦄ → ℕ → poolid → vrfpf → privkey → list tx → ibheader"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":74,"title":"mkIBHeader slot id π pKey txs","content":"mkIBHeader slot id π pKey txs = record { signature = sign pKey ( hash h ) ; IBHeaderOSig h }","searchableContent":"mkibheader slot id π pkey txs = record { signature = sign pkey ( hash h ) ; ibheaderosig h }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":75,"title":"where","content":"where","searchableContent":"where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":76,"title":"h","content":"h : IBHeaderOSig ⊤","searchableContent":"h : ibheaderosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":77,"title":"h","content":"h = record { slotNumber = slot","searchableContent":"h = record { slotnumber = slot"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":78,"title":"; producerID","content":"; producerID = id","searchableContent":"; producerid = id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":79,"title":"; lotteryPf","content":"; lotteryPf = π","searchableContent":"; lotterypf = π"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":80,"title":"; bodyHash","content":"; bodyHash = hash txs","searchableContent":"; bodyhash = hash txs"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":81,"title":"; signature","content":"; signature = _","searchableContent":"; signature = _"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":82,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":84,"title":"getIBRef","content":"getIBRef : ⦃ Hashable PreIBHeader Hash ⦄ → InputBlock → IBRef","searchableContent":"getibref : ⦃ hashable preibheader hash ⦄ → inputblock → ibref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":85,"title":"getIBRef","content":"getIBRef = hash ∘ InputBlock.header","searchableContent":"getibref = hash ∘ inputblock.header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":87,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":88,"title":"-- Endorser Blocks","content":"-- Endorser Blocks","searchableContent":"-- endorser blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":89,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":91,"title":"record EndorserBlockOSig ( sig","content":"record EndorserBlockOSig ( sig : Type ) : Type where","searchableContent":"record endorserblockosig ( sig : type ) : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":92,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"field slotnumber : ℕ"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":93,"title":"producerID","content":"producerID : PoolID","searchableContent":"producerid : poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":94,"title":"lotteryPf","content":"lotteryPf : VrfPf","searchableContent":"lotterypf : vrfpf"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":95,"title":"ibRefs","content":"ibRefs : List IBRef","searchableContent":"ibrefs : list ibref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":96,"title":"ebRefs","content":"ebRefs : List EBRef","searchableContent":"ebrefs : list ebref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":97,"title":"signature","content":"signature : sig","searchableContent":"signature : sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":99,"title":"EndorserBlock","content":"EndorserBlock = EndorserBlockOSig Sig","searchableContent":"endorserblock = endorserblockosig sig"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":100,"title":"PreEndorserBlock","content":"PreEndorserBlock = EndorserBlockOSig ⊤","searchableContent":"preendorserblock = endorserblockosig ⊤"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":102,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":103,"title":"Hashable-EndorserBlock","content":"Hashable-EndorserBlock : ⦃ Hashable PreEndorserBlock Hash ⦄ → Hashable EndorserBlock Hash","searchableContent":"hashable-endorserblock : ⦃ hashable preendorserblock hash ⦄ → hashable endorserblock hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":104,"title":"Hashable-EndorserBlock . hash b","content":"Hashable-EndorserBlock . hash b = hash { T = PreEndorserBlock }","searchableContent":"hashable-endorserblock . hash b = hash { t = preendorserblock }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":105,"title":"record { EndorserBlockOSig b hiding ( signature ) ; signature","content":"record { EndorserBlockOSig b hiding ( signature ) ; signature = _ }","searchableContent":"record { endorserblockosig b hiding ( signature ) ; signature = _ }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":107,"title":"IsBlock-EndorserBlock","content":"IsBlock-EndorserBlock : IsBlock EndorserBlock","searchableContent":"isblock-endorserblock : isblock endorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":108,"title":"IsBlock-EndorserBlock","content":"IsBlock-EndorserBlock = record { EndorserBlockOSig }","searchableContent":"isblock-endorserblock = record { endorserblockosig }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":110,"title":"unquoteDecl DecEq-EndorserBlockOSig","content":"unquoteDecl DecEq-EndorserBlockOSig = derive-DecEq (( quote EndorserBlockOSig , DecEq-EndorserBlockOSig ) ∷ [] )","searchableContent":"unquotedecl deceq-endorserblockosig = derive-deceq (( quote endorserblockosig , deceq-endorserblockosig ) ∷ [] )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":112,"title":"mkEB","content":"mkEB : ⦃ Hashable PreEndorserBlock Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List IBRef → List EBRef → EndorserBlock","searchableContent":"mkeb : ⦃ hashable preendorserblock hash ⦄ → ℕ → poolid → vrfpf → privkey → list ibref → list ebref → endorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":113,"title":"mkEB slot id π pKey LI LE","content":"mkEB slot id π pKey LI LE = record { signature = sign pKey ( hash b ) ; EndorserBlockOSig b }","searchableContent":"mkeb slot id π pkey li le = record { signature = sign pkey ( hash b ) ; endorserblockosig b }"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":114,"title":"where","content":"where","searchableContent":"where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":115,"title":"b","content":"b : PreEndorserBlock","searchableContent":"b : preendorserblock"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":116,"title":"b","content":"b = record { slotNumber = slot","searchableContent":"b = record { slotnumber = slot"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":117,"title":"; producerID","content":"; producerID = id","searchableContent":"; producerid = id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":118,"title":"; lotteryPf","content":"; lotteryPf = π","searchableContent":"; lotterypf = π"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":119,"title":"; ibRefs","content":"; ibRefs = LI","searchableContent":"; ibrefs = li"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":120,"title":"; ebRefs","content":"; ebRefs = LE","searchableContent":"; ebrefs = le"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":121,"title":"; signature","content":"; signature = _","searchableContent":"; signature = _"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":122,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":124,"title":"getEBRef","content":"getEBRef : ⦃ Hashable PreEndorserBlock Hash ⦄ → EndorserBlock → EBRef","searchableContent":"getebref : ⦃ hashable preendorserblock hash ⦄ → endorserblock → ebref"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":125,"title":"getEBRef","content":"getEBRef = hash","searchableContent":"getebref = hash"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":127,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":128,"title":"-- Votes","content":"-- Votes","searchableContent":"-- votes"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":129,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":133,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":134,"title":"-- FFD for Leios Blocks","content":"-- FFD for Leios Blocks","searchableContent":"-- ffd for leios blocks"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":135,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":137,"title":"module GenFFD ⦃ _","content":"module GenFFD ⦃ _ : IsBlock ( List Vote ) ⦄ where","searchableContent":"module genffd ⦃ _ : isblock ( list vote ) ⦄ where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":138,"title":"data Header","content":"data Header : Type where","searchableContent":"data header : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":139,"title":"ibHeader","content":"ibHeader : IBHeader → Header","searchableContent":"ibheader : ibheader → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":140,"title":"ebHeader","content":"ebHeader : EndorserBlock → Header","searchableContent":"ebheader : endorserblock → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":141,"title":"vtHeader","content":"vtHeader : List Vote → Header","searchableContent":"vtheader : list vote → header"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":143,"title":"data Body","content":"data Body : Type where","searchableContent":"data body : type where"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":144,"title":"ibBody","content":"ibBody : IBBody → Body","searchableContent":"ibbody : ibbody → body"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":146,"title":"unquoteDecl DecEq-Header DecEq-Body","content":"unquoteDecl DecEq-Header DecEq-Body =","searchableContent":"unquotedecl deceq-header deceq-body ="},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":147,"title":"derive-DecEq (( quote Header , DecEq-Header ) ∷ ( quote Body , DecEq-Body ) ∷ [] )","content":"derive-DecEq (( quote Header , DecEq-Header ) ∷ ( quote Body , DecEq-Body ) ∷ [] )","searchableContent":"derive-deceq (( quote header , deceq-header ) ∷ ( quote body , deceq-body ) ∷ [] )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":149,"title":"ID","content":"ID : Type","searchableContent":"id : type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":150,"title":"ID","content":"ID = ℕ × PoolID","searchableContent":"id = ℕ × poolid"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":152,"title":"matchIB","content":"matchIB : IBHeader → IBBody → Type","searchableContent":"matchib : ibheader → ibbody → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":153,"title":"matchIB h b","content":"matchIB h b = bodyHash ≡ hash b","searchableContent":"matchib h b = bodyhash ≡ hash b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":154,"title":"where open IBHeaderOSig h ; open IBBody b","content":"where open IBHeaderOSig h ; open IBBody b","searchableContent":"where open ibheaderosig h ; open ibbody b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":156,"title":"matchIB?","content":"matchIB? : ∀ ( h : IBHeader ) → ( b : IBBody ) → Dec ( matchIB h b )","searchableContent":"matchib? : ∀ ( h : ibheader ) → ( b : ibbody ) → dec ( matchib h b )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":157,"title":"matchIB? h b","content":"matchIB? h b = bodyHash ≟ hash b","searchableContent":"matchib? h b = bodyhash ≟ hash b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":158,"title":"where open IBHeaderOSig h ; open IBBody b","content":"where open IBHeaderOSig h ; open IBBody b","searchableContent":"where open ibheaderosig h ; open ibbody b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":160,"title":"match","content":"match : Header → Body → Type","searchableContent":"match : header → body → type"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":161,"title":"match ( ibHeader h ) ( ibBody b )","content":"match ( ibHeader h ) ( ibBody b ) = matchIB h b","searchableContent":"match ( ibheader h ) ( ibbody b ) = matchib h b"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":162,"title":"match _ _","content":"match _ _ = ⊥","searchableContent":"match _ _ = ⊥"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":164,"title":"-- can we express uniqueness wrt pipelines as a property?","content":"-- can we express uniqueness wrt pipelines as a property?","searchableContent":"-- can we express uniqueness wrt pipelines as a property?"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":165,"title":"msgID","content":"msgID : Header → ID","searchableContent":"msgid : header → id"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":166,"title":"msgID ( ibHeader h )","content":"msgID ( ibHeader h ) = ( slotNumber h , producerID h )","searchableContent":"msgid ( ibheader h ) = ( slotnumber h , producerid h )"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":167,"title":"msgID ( ebHeader h ) = ( slotNumber h , producerID h ) -- NOTE","content":"msgID ( ebHeader h ) = ( slotNumber h , producerID h ) -- NOTE: this isn't in the paper","searchableContent":"msgid ( ebheader h ) = ( slotnumber h , producerid h ) -- note: this isn't in the paper"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":168,"title":"msgID ( vtHeader h ) = ( slotNumber h , producerID h ) -- NOTE","content":"msgID ( vtHeader h ) = ( slotNumber h , producerID h ) -- NOTE: this isn't in the paper","searchableContent":"msgid ( vtheader h ) = ( slotnumber h , producerid h ) -- note: this isn't in the paper"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":170,"title":"ffdAbstract","content":"ffdAbstract : ⦃ _ : IsBlock ( List Vote ) ⦄ → FFDAbstract","searchableContent":"ffdabstract : ⦃ _ : isblock ( list vote ) ⦄ → ffdabstract"},{"moduleName":"Leios.Blocks","path":"Leios.Blocks.html","group":"Leios","lineNumber":171,"title":"ffdAbstract","content":"ffdAbstract = record { GenFFD }","searchableContent":"ffdabstract = record { genffd }"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":2,"title":"Leios.Config open import Leios.Prelude","content":"Leios.Config open import Leios.Prelude","searchableContent":"leios.config open import leios.prelude"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":3,"title":"open import Tactic.Defaults","content":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":4,"title":"open import Tactic.Derive.DecEq","content":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":6,"title":"module Leios.Config where","content":"module Leios.Config where","searchableContent":"module leios.config where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":8,"title":"data BlockType","content":"data BlockType : Type where","searchableContent":"data blocktype : type where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":9,"title":"IB EB VT","content":"IB EB VT : BlockType","searchableContent":"ib eb vt : blocktype"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":11,"title":"unquoteDecl DecEq-BlockType","content":"unquoteDecl DecEq-BlockType = derive-DecEq (( quote BlockType , DecEq-BlockType ) ∷ [] )","searchableContent":"unquotedecl deceq-blocktype = derive-deceq (( quote blocktype , deceq-blocktype ) ∷ [] )"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":13,"title":"record Params","content":"record Params : Type where","searchableContent":"record params : type where"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":14,"title":"field numberOfParties","content":"field numberOfParties : ℕ","searchableContent":"field numberofparties : ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":15,"title":"sutId","content":"sutId : Fin numberOfParties","searchableContent":"sutid : fin numberofparties"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":16,"title":"stakeDistribution","content":"stakeDistribution : TotalMap ( Fin numberOfParties ) ℕ","searchableContent":"stakedistribution : totalmap ( fin numberofparties ) ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":17,"title":"stageLength","content":"stageLength : ℕ","searchableContent":"stagelength : ℕ"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":18,"title":"⦃ NonZero-stageLength ⦄","content":"⦃ NonZero-stageLength ⦄ : NonZero stageLength","searchableContent":"⦃ nonzero-stagelength ⦄ : nonzero stagelength"},{"moduleName":"Leios.Config","path":"Leios.Config.html","group":"Leios","lineNumber":19,"title":"winning-slots","content":"winning-slots : ℙ ( BlockType × ℕ )","searchableContent":"winning-slots : ℙ ( blocktype × ℕ )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":2,"title":"Leios.Defaults open import Leios.Prelude","content":"Leios.Defaults open import Leios.Prelude","searchableContent":"leios.defaults open import leios.prelude"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":3,"title":"open import Leios.Abstract","content":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":4,"title":"open import Leios.Config","content":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":5,"title":"open import Leios.SpecStructure","content":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":7,"title":"open import Axiom.Set.Properties th","content":"open import Axiom.Set.Properties th","searchableContent":"open import axiom.set.properties th"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":8,"title":"open import Data.Nat.Show as N","content":"open import Data.Nat.Show as N","searchableContent":"open import data.nat.show as n"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":9,"title":"open import Data.Integer hiding ( _≟_ )","content":"open import Data.Integer hiding ( _≟_ )","searchableContent":"open import data.integer hiding ( _≟_ )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":10,"title":"open import Data.String as S using ( intersperse )","content":"open import Data.String as S using ( intersperse )","searchableContent":"open import data.string as s using ( intersperse )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":11,"title":"open import Function.Related.TypeIsomorphisms","content":"open import Function.Related.TypeIsomorphisms","searchableContent":"open import function.related.typeisomorphisms"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":12,"title":"open import Relation.Binary.Structures","content":"open import Relation.Binary.Structures","searchableContent":"open import relation.binary.structures"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":14,"title":"open import Tactic.Defaults","content":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":15,"title":"open import Tactic.Derive.DecEq","content":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":17,"title":"open Equivalence","content":"open Equivalence","searchableContent":"open equivalence"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":19,"title":"-- The module contains very simple implementations for the functionalities","content":"-- The module contains very simple implementations for the functionalities","searchableContent":"-- the module contains very simple implementations for the functionalities"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":20,"title":"-- that allow to build examples for traces for the different Leios variants","content":"-- that allow to build examples for traces for the different Leios variants","searchableContent":"-- that allow to build examples for traces for the different leios variants"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":21,"title":"module Leios.Defaults ( params","content":"module Leios.Defaults ( params : Params ) ( let open Params params ) where","searchableContent":"module leios.defaults ( params : params ) ( let open params params ) where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":23,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":24,"title":"htx","content":"htx : Hashable ( List ℕ ) ( List ℕ )","searchableContent":"htx : hashable ( list ℕ ) ( list ℕ )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":25,"title":"htx","content":"htx = record { hash = id }","searchableContent":"htx = record { hash = id }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":27,"title":"d-Abstract","content":"d-Abstract : LeiosAbstract","searchableContent":"d-abstract : leiosabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":28,"title":"d-Abstract","content":"d-Abstract =","searchableContent":"d-abstract ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":29,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":30,"title":"{ Tx","content":"{ Tx = ℕ","searchableContent":"{ tx = ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":31,"title":"; PoolID","content":"; PoolID = Fin numberOfParties","searchableContent":"; poolid = fin numberofparties"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":32,"title":"; BodyHash","content":"; BodyHash = List ℕ","searchableContent":"; bodyhash = list ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":33,"title":"; VrfPf","content":"; VrfPf = ⊤","searchableContent":"; vrfpf = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":34,"title":"; PrivKey","content":"; PrivKey = BlockType × ⊤","searchableContent":"; privkey = blocktype × ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":35,"title":"; Sig","content":"; Sig = ⊤","searchableContent":"; sig = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":36,"title":"; Hash","content":"; Hash = List ℕ","searchableContent":"; hash = list ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":37,"title":"; Vote","content":"; Vote = ⊤","searchableContent":"; vote = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":38,"title":"; vote","content":"; vote = λ _ _ → tt","searchableContent":"; vote = λ _ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":39,"title":"; sign","content":"; sign = λ _ _ → tt","searchableContent":"; sign = λ _ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":40,"title":"; L","content":"; L = stageLength","searchableContent":"; l = stagelength"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":41,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":43,"title":"open LeiosAbstract d-Abstract public","content":"open LeiosAbstract d-Abstract public","searchableContent":"open leiosabstract d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":45,"title":"open import Leios.VRF d-Abstract public","content":"open import Leios.VRF d-Abstract public","searchableContent":"open import leios.vrf d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":47,"title":"sutStake","content":"sutStake : ℕ","searchableContent":"sutstake : ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":48,"title":"sutStake","content":"sutStake = TotalMap.lookup stakeDistribution sutId","searchableContent":"sutstake = totalmap.lookup stakedistribution sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":50,"title":"sortition","content":"sortition : BlockType → ℕ → ℕ","searchableContent":"sortition : blocktype → ℕ → ℕ"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":51,"title":"sortition b n with ( b , n ) ∈? winning-slots","content":"sortition b n with ( b , n ) ∈? winning-slots","searchableContent":"sortition b n with ( b , n ) ∈? winning-slots"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":52,"title":"... | yes _","content":"... | yes _ = 0","searchableContent":"... | yes _ = 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":53,"title":"... | no _","content":"... | no _ = sutStake","searchableContent":"... | no _ = sutstake"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":55,"title":"d-VRF","content":"d-VRF : LeiosVRF","searchableContent":"d-vrf : leiosvrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":56,"title":"d-VRF","content":"d-VRF =","searchableContent":"d-vrf ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":57,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":58,"title":"{ PubKey","content":"{ PubKey = Fin numberOfParties × ⊤","searchableContent":"{ pubkey = fin numberofparties × ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":59,"title":"; vrf","content":"; vrf =","searchableContent":"; vrf ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":60,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":61,"title":"{ isKeyPair","content":"{ isKeyPair = λ _ _ → ⊤","searchableContent":"{ iskeypair = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":62,"title":"; eval","content":"; eval = λ ( b , _) y → sortition b y , tt","searchableContent":"; eval = λ ( b , _) y → sortition b y , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":63,"title":"; verify","content":"; verify = λ _ _ _ _ → ⊤","searchableContent":"; verify = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":64,"title":"; verify?","content":"; verify? = λ _ _ _ _ → yes tt","searchableContent":"; verify? = λ _ _ _ _ → yes tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":65,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":66,"title":"; genIBInput","content":"; genIBInput = id","searchableContent":"; genibinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":67,"title":"; genEBInput","content":"; genEBInput = id","searchableContent":"; genebinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":68,"title":"; genVInput","content":"; genVInput = id","searchableContent":"; genvinput = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":69,"title":"; genV1Input","content":"; genV1Input = id","searchableContent":"; genv1input = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":70,"title":"; genV2Input","content":"; genV2Input = id","searchableContent":"; genv2input = id"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":71,"title":"; poolID","content":"; poolID = proj₁","searchableContent":"; poolid = proj₁"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":72,"title":"; verifySig","content":"; verifySig = λ _ _ → ⊤","searchableContent":"; verifysig = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":73,"title":"; verifySig?","content":"; verifySig? = λ _ _ → yes tt","searchableContent":"; verifysig? = λ _ _ → yes tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":74,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":76,"title":"open LeiosVRF d-VRF public","content":"open LeiosVRF d-VRF public","searchableContent":"open leiosvrf d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":78,"title":"open import Leios.Blocks d-Abstract public","content":"open import Leios.Blocks d-Abstract public","searchableContent":"open import leios.blocks d-abstract public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":79,"title":"open import Leios.KeyRegistration d-Abstract d-VRF public","content":"open import Leios.KeyRegistration d-Abstract d-VRF public","searchableContent":"open import leios.keyregistration d-abstract d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":81,"title":"d-KeyRegistration","content":"d-KeyRegistration : KeyRegistrationAbstract","searchableContent":"d-keyregistration : keyregistrationabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":82,"title":"d-KeyRegistration","content":"d-KeyRegistration = _","searchableContent":"d-keyregistration = _"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":84,"title":"d-KeyRegistrationFunctionality","content":"d-KeyRegistrationFunctionality : KeyRegistrationAbstract.Functionality d-KeyRegistration","searchableContent":"d-keyregistrationfunctionality : keyregistrationabstract.functionality d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":85,"title":"d-KeyRegistrationFunctionality","content":"d-KeyRegistrationFunctionality =","searchableContent":"d-keyregistrationfunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":86,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":87,"title":"{ State","content":"{ State = ⊤","searchableContent":"{ state = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":88,"title":"; _-⟦_/_⟧⇀_","content":"; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤","searchableContent":"; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":89,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":91,"title":"open import Leios.Base d-Abstract d-VRF public","content":"open import Leios.Base d-Abstract d-VRF public","searchableContent":"open import leios.base d-abstract d-vrf public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":93,"title":"d-Base","content":"d-Base : BaseAbstract","searchableContent":"d-base : baseabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":94,"title":"d-Base","content":"d-Base =","searchableContent":"d-base ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":95,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":96,"title":"{ Cert","content":"{ Cert = ⊤","searchableContent":"{ cert = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":97,"title":"; VTy","content":"; VTy = ⊤","searchableContent":"; vty = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":98,"title":"; initSlot","content":"; initSlot = λ _ → 0","searchableContent":"; initslot = λ _ → 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":99,"title":"; V-chkCerts","content":"; V-chkCerts = λ _ _ → true","searchableContent":"; v-chkcerts = λ _ _ → true"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":100,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":102,"title":"d-BaseFunctionality","content":"d-BaseFunctionality : BaseAbstract.Functionality d-Base","searchableContent":"d-basefunctionality : baseabstract.functionality d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":103,"title":"d-BaseFunctionality","content":"d-BaseFunctionality =","searchableContent":"d-basefunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":104,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":105,"title":"{ State","content":"{ State = ⊤","searchableContent":"{ state = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":106,"title":"; _-⟦_/_⟧⇀_","content":"; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤","searchableContent":"; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":107,"title":"; Dec-_-⟦_/_⟧⇀_","content":"; Dec-_-⟦_/_⟧⇀_ = ⁇ ( yes tt )","searchableContent":"; dec-_-⟦_/_⟧⇀_ = ⁇ ( yes tt )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":108,"title":"; SUBMIT-total","content":"; SUBMIT-total = tt , tt","searchableContent":"; submit-total = tt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":109,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":111,"title":"open import Leios.FFD public","content":"open import Leios.FFD public","searchableContent":"open import leios.ffd public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":113,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":114,"title":"isb","content":"isb : IsBlock ( List ⊤ )","searchableContent":"isb : isblock ( list ⊤ )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":115,"title":"isb","content":"isb =","searchableContent":"isb ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":116,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":117,"title":"{ slotNumber","content":"{ slotNumber = λ _ → 0","searchableContent":"{ slotnumber = λ _ → 0"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":118,"title":"; producerID","content":"; producerID = λ _ → sutId","searchableContent":"; producerid = λ _ → sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":119,"title":"; lotteryPf","content":"; lotteryPf = λ _ → tt","searchableContent":"; lotterypf = λ _ → tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":120,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":122,"title":"hhs","content":"hhs : Hashable PreIBHeader ( List ℕ )","searchableContent":"hhs : hashable preibheader ( list ℕ )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":123,"title":"hhs . hash","content":"hhs . hash = IBHeaderOSig.bodyHash","searchableContent":"hhs . hash = ibheaderosig.bodyhash"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":125,"title":"hpe","content":"hpe : Hashable PreEndorserBlock ( List ℕ )","searchableContent":"hpe : hashable preendorserblock ( list ℕ )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":126,"title":"hpe . hash","content":"hpe . hash = L.concat ∘ EndorserBlockOSig.ibRefs","searchableContent":"hpe . hash = l.concat ∘ endorserblockosig.ibrefs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":128,"title":"record FFDBuffers","content":"record FFDBuffers : Type where","searchableContent":"record ffdbuffers : type where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":129,"title":"field inIBs","content":"field inIBs : List InputBlock","searchableContent":"field inibs : list inputblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":130,"title":"inEBs","content":"inEBs : List EndorserBlock","searchableContent":"inebs : list endorserblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":131,"title":"inVTs","content":"inVTs : List ( List Vote )","searchableContent":"invts : list ( list vote )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":133,"title":"outIBs","content":"outIBs : List InputBlock","searchableContent":"outibs : list inputblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":134,"title":"outEBs","content":"outEBs : List EndorserBlock","searchableContent":"outebs : list endorserblock"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":135,"title":"outVTs","content":"outVTs : List ( List Vote )","searchableContent":"outvts : list ( list vote )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":137,"title":"unquoteDecl DecEq-FFDBuffers","content":"unquoteDecl DecEq-FFDBuffers = derive-DecEq (( quote FFDBuffers , DecEq-FFDBuffers ) ∷ [] )","searchableContent":"unquotedecl deceq-ffdbuffers = derive-deceq (( quote ffdbuffers , deceq-ffdbuffers ) ∷ [] )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":139,"title":"open GenFFD.Header","content":"open GenFFD.Header","searchableContent":"open genffd.header"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":140,"title":"open GenFFD.Body","content":"open GenFFD.Body","searchableContent":"open genffd.body"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":141,"title":"open FFDBuffers","content":"open FFDBuffers","searchableContent":"open ffdbuffers"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":143,"title":"flushIns","content":"flushIns : FFDBuffers → List ( GenFFD.Header ⊎ GenFFD.Body )","searchableContent":"flushins : ffdbuffers → list ( genffd.header ⊎ genffd.body )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":144,"title":"flushIns record { inIBs","content":"flushIns record { inIBs = ibs ; inEBs = ebs ; inVTs = vts } =","searchableContent":"flushins record { inibs = ibs ; inebs = ebs ; invts = vts } ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":145,"title":"flushIBs ibs ++ L.map ( inj₁ ∘ ebHeader ) ebs ++ L.map ( inj₁ ∘ vtHeader ) vts","content":"flushIBs ibs ++ L.map ( inj₁ ∘ ebHeader ) ebs ++ L.map ( inj₁ ∘ vtHeader ) vts","searchableContent":"flushibs ibs ++ l.map ( inj₁ ∘ ebheader ) ebs ++ l.map ( inj₁ ∘ vtheader ) vts"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":146,"title":"where","content":"where","searchableContent":"where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":147,"title":"flushIBs","content":"flushIBs : List InputBlock → List ( GenFFD.Header ⊎ GenFFD.Body )","searchableContent":"flushibs : list inputblock → list ( genffd.header ⊎ genffd.body )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":148,"title":"flushIBs []","content":"flushIBs [] = []","searchableContent":"flushibs [] = []"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":149,"title":"flushIBs ( record { header","content":"flushIBs ( record { header = h ; body = b } ∷ ibs ) = inj₁ ( ibHeader h ) ∷ inj₂ ( ibBody b ) ∷ flushIBs ibs","searchableContent":"flushibs ( record { header = h ; body = b } ∷ ibs ) = inj₁ ( ibheader h ) ∷ inj₂ ( ibbody b ) ∷ flushibs ibs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":151,"title":"data SimpleFFD","content":"data SimpleFFD : FFDBuffers → FFDAbstract.Input ffdAbstract → FFDAbstract.Output ffdAbstract → FFDBuffers → Type where","searchableContent":"data simpleffd : ffdbuffers → ffdabstract.input ffdabstract → ffdabstract.output ffdabstract → ffdbuffers → type where"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":152,"title":"SendIB","content":"SendIB : ∀ { s h b } → SimpleFFD s ( FFDAbstract.Send ( ibHeader h ) ( just ( ibBody b ))) FFDAbstract.SendRes ( record s { outIBs = record { header = h ; body = b } ∷ outIBs s })","searchableContent":"sendib : ∀ { s h b } → simpleffd s ( ffdabstract.send ( ibheader h ) ( just ( ibbody b ))) ffdabstract.sendres ( record s { outibs = record { header = h ; body = b } ∷ outibs s })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":153,"title":"SendEB","content":"SendEB : ∀ { s eb } → SimpleFFD s ( FFDAbstract.Send ( ebHeader eb ) nothing ) FFDAbstract.SendRes ( record s { outEBs = eb ∷ outEBs s })","searchableContent":"sendeb : ∀ { s eb } → simpleffd s ( ffdabstract.send ( ebheader eb ) nothing ) ffdabstract.sendres ( record s { outebs = eb ∷ outebs s })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":154,"title":"SendVS","content":"SendVS : ∀ { s vs } → SimpleFFD s ( FFDAbstract.Send ( vtHeader vs ) nothing ) FFDAbstract.SendRes ( record s { outVTs = vs ∷ outVTs s })","searchableContent":"sendvs : ∀ { s vs } → simpleffd s ( ffdabstract.send ( vtheader vs ) nothing ) ffdabstract.sendres ( record s { outvts = vs ∷ outvts s })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":156,"title":"BadSendIB","content":"BadSendIB : ∀ { s h } → SimpleFFD s ( FFDAbstract.Send ( ibHeader h ) nothing ) FFDAbstract.SendRes s","searchableContent":"badsendib : ∀ { s h } → simpleffd s ( ffdabstract.send ( ibheader h ) nothing ) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":157,"title":"BadSendEB","content":"BadSendEB : ∀ { s h b } → SimpleFFD s ( FFDAbstract.Send ( ebHeader h ) ( just b )) FFDAbstract.SendRes s","searchableContent":"badsendeb : ∀ { s h b } → simpleffd s ( ffdabstract.send ( ebheader h ) ( just b )) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":158,"title":"BadSendVS","content":"BadSendVS : ∀ { s h b } → SimpleFFD s ( FFDAbstract.Send ( vtHeader h ) ( just b )) FFDAbstract.SendRes s","searchableContent":"badsendvs : ∀ { s h b } → simpleffd s ( ffdabstract.send ( vtheader h ) ( just b )) ffdabstract.sendres s"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":160,"title":"Fetch","content":"Fetch : ∀ { s } → SimpleFFD s FFDAbstract.Fetch ( FFDAbstract.FetchRes ( flushIns s )) ( record s { inIBs = [] ; inEBs = [] ; inVTs = [] })","searchableContent":"fetch : ∀ { s } → simpleffd s ffdabstract.fetch ( ffdabstract.fetchres ( flushins s )) ( record s { inibs = [] ; inebs = [] ; invts = [] })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":162,"title":"send-total","content":"send-total : ∀ { s h b } → ∃[ s' ] ( SimpleFFD s ( FFDAbstract.Send h b ) FFDAbstract.SendRes s' )","searchableContent":"send-total : ∀ { s h b } → ∃[ s' ] ( simpleffd s ( ffdabstract.send h b ) ffdabstract.sendres s' )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":163,"title":"send-total { s } { ibHeader h } { just ( ibBody b )}","content":"send-total { s } { ibHeader h } { just ( ibBody b )} = record s { outIBs = record { header = h ; body = b } ∷ outIBs s } , SendIB","searchableContent":"send-total { s } { ibheader h } { just ( ibbody b )} = record s { outibs = record { header = h ; body = b } ∷ outibs s } , sendib"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":164,"title":"send-total { s } { ebHeader eb } { nothing }","content":"send-total { s } { ebHeader eb } { nothing } = record s { outEBs = eb ∷ outEBs s } , SendEB","searchableContent":"send-total { s } { ebheader eb } { nothing } = record s { outebs = eb ∷ outebs s } , sendeb"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":165,"title":"send-total { s } { vtHeader vs } { nothing }","content":"send-total { s } { vtHeader vs } { nothing } = record s { outVTs = vs ∷ outVTs s } , SendVS","searchableContent":"send-total { s } { vtheader vs } { nothing } = record s { outvts = vs ∷ outvts s } , sendvs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":167,"title":"send-total { s } { ibHeader h } { nothing }","content":"send-total { s } { ibHeader h } { nothing } = s , BadSendIB","searchableContent":"send-total { s } { ibheader h } { nothing } = s , badsendib"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":168,"title":"send-total { s } { ebHeader eb } { just _}","content":"send-total { s } { ebHeader eb } { just _} = s , BadSendEB","searchableContent":"send-total { s } { ebheader eb } { just _} = s , badsendeb"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":169,"title":"send-total { s } { vtHeader vs } { just _}","content":"send-total { s } { vtHeader vs } { just _} = s , BadSendVS","searchableContent":"send-total { s } { vtheader vs } { just _} = s , badsendvs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":171,"title":"fetch-total","content":"fetch-total : ∀ { s } → ∃[ x ] ( ∃[ s' ] ( SimpleFFD s FFDAbstract.Fetch ( FFDAbstract.FetchRes x ) s' ))","searchableContent":"fetch-total : ∀ { s } → ∃[ x ] ( ∃[ s' ] ( simpleffd s ffdabstract.fetch ( ffdabstract.fetchres x ) s' ))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":172,"title":"fetch-total { s }","content":"fetch-total { s } = flushIns s , ( record s { inIBs = [] ; inEBs = [] ; inVTs = [] } , Fetch )","searchableContent":"fetch-total { s } = flushins s , ( record s { inibs = [] ; inebs = [] ; invts = [] } , fetch )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":174,"title":"send-complete","content":"send-complete : ∀ { s h b s' } → SimpleFFD s ( FFDAbstract.Send h b ) FFDAbstract.SendRes s' → s' ≡ proj₁ ( send-total { s } { h } { b })","searchableContent":"send-complete : ∀ { s h b s' } → simpleffd s ( ffdabstract.send h b ) ffdabstract.sendres s' → s' ≡ proj₁ ( send-total { s } { h } { b })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":175,"title":"send-complete SendIB","content":"send-complete SendIB = refl","searchableContent":"send-complete sendib = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":176,"title":"send-complete SendEB","content":"send-complete SendEB = refl","searchableContent":"send-complete sendeb = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":177,"title":"send-complete SendVS","content":"send-complete SendVS = refl","searchableContent":"send-complete sendvs = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":178,"title":"send-complete BadSendIB","content":"send-complete BadSendIB = refl","searchableContent":"send-complete badsendib = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":179,"title":"send-complete BadSendEB","content":"send-complete BadSendEB = refl","searchableContent":"send-complete badsendeb = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":180,"title":"send-complete BadSendVS","content":"send-complete BadSendVS = refl","searchableContent":"send-complete badsendvs = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":182,"title":"fetch-complete₁","content":"fetch-complete₁ : ∀ { s r s' } → SimpleFFD s FFDAbstract.Fetch ( FFDAbstract.FetchRes r ) s' → s' ≡ proj₁ ( proj₂ ( fetch-total { s }))","searchableContent":"fetch-complete₁ : ∀ { s r s' } → simpleffd s ffdabstract.fetch ( ffdabstract.fetchres r ) s' → s' ≡ proj₁ ( proj₂ ( fetch-total { s }))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":183,"title":"fetch-complete₁ Fetch","content":"fetch-complete₁ Fetch = refl","searchableContent":"fetch-complete₁ fetch = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":185,"title":"fetch-complete₂","content":"fetch-complete₂ : ∀ { s r s' } → SimpleFFD s FFDAbstract.Fetch ( FFDAbstract.FetchRes r ) s' → r ≡ proj₁ ( fetch-total { s })","searchableContent":"fetch-complete₂ : ∀ { s r s' } → simpleffd s ffdabstract.fetch ( ffdabstract.fetchres r ) s' → r ≡ proj₁ ( fetch-total { s })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":186,"title":"fetch-complete₂ Fetch","content":"fetch-complete₂ Fetch = refl","searchableContent":"fetch-complete₂ fetch = refl"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":188,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":189,"title":"Dec-SimpleFFD","content":"Dec-SimpleFFD : ∀ { s i o s' } → SimpleFFD s i o s' ⁇","searchableContent":"dec-simpleffd : ∀ { s i o s' } → simpleffd s i o s' ⁇"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":190,"title":"Dec-SimpleFFD { s } { FFDAbstract.Send h b } { FFDAbstract.SendRes } { s' } with s' ≟ proj₁ ( send-total { s } { h } { b })","content":"Dec-SimpleFFD { s } { FFDAbstract.Send h b } { FFDAbstract.SendRes } { s' } with s' ≟ proj₁ ( send-total { s } { h } { b })","searchableContent":"dec-simpleffd { s } { ffdabstract.send h b } { ffdabstract.sendres } { s' } with s' ≟ proj₁ ( send-total { s } { h } { b })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":191,"title":"... | yes p rewrite p","content":"... | yes p rewrite p = ⁇ yes ( proj₂ send-total )","searchableContent":"... | yes p rewrite p = ⁇ yes ( proj₂ send-total )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":192,"title":"... | no ¬p","content":"... | no ¬p = ⁇ no λ x → ⊥-elim ( ¬p ( send-complete x ))","searchableContent":"... | no ¬p = ⁇ no λ x → ⊥-elim ( ¬p ( send-complete x ))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":193,"title":"Dec-SimpleFFD {_} { FFDAbstract.Send _ _} { FFDAbstract.FetchRes _} {_}","content":"Dec-SimpleFFD {_} { FFDAbstract.Send _ _} { FFDAbstract.FetchRes _} {_} = ⁇ no λ ()","searchableContent":"dec-simpleffd {_} { ffdabstract.send _ _} { ffdabstract.fetchres _} {_} = ⁇ no λ ()"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":194,"title":"Dec-SimpleFFD { s } { FFDAbstract.Fetch } { FFDAbstract.FetchRes r } { s' }","content":"Dec-SimpleFFD { s } { FFDAbstract.Fetch } { FFDAbstract.FetchRes r } { s' }","searchableContent":"dec-simpleffd { s } { ffdabstract.fetch } { ffdabstract.fetchres r } { s' }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":195,"title":"with s' ≟ proj₁ ( proj₂ ( fetch-total { s }))","content":"with s' ≟ proj₁ ( proj₂ ( fetch-total { s }))","searchableContent":"with s' ≟ proj₁ ( proj₂ ( fetch-total { s }))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":196,"title":"| r ≟ proj₁ ( fetch-total { s })","content":"| r ≟ proj₁ ( fetch-total { s })","searchableContent":"| r ≟ proj₁ ( fetch-total { s })"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":197,"title":"... | yes p | yes q rewrite p rewrite q","content":"... | yes p | yes q rewrite p rewrite q = ⁇ yes ( proj₂ ( proj₂ ( fetch-total { s })))","searchableContent":"... | yes p | yes q rewrite p rewrite q = ⁇ yes ( proj₂ ( proj₂ ( fetch-total { s })))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":198,"title":"... | yes p | no ¬q","content":"... | yes p | no ¬q = ⁇ no λ x → ⊥-elim ( ¬q ( fetch-complete₂ x ))","searchableContent":"... | yes p | no ¬q = ⁇ no λ x → ⊥-elim ( ¬q ( fetch-complete₂ x ))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":199,"title":"... | no ¬p | _","content":"... | no ¬p | _ = ⁇ no λ x → ⊥-elim ( ¬p ( fetch-complete₁ x ))","searchableContent":"... | no ¬p | _ = ⁇ no λ x → ⊥-elim ( ¬p ( fetch-complete₁ x ))"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":200,"title":"Dec-SimpleFFD {_} { FFDAbstract.Fetch } { FFDAbstract.SendRes } {_}","content":"Dec-SimpleFFD {_} { FFDAbstract.Fetch } { FFDAbstract.SendRes } {_} = ⁇ no λ ()","searchableContent":"dec-simpleffd {_} { ffdabstract.fetch } { ffdabstract.sendres } {_} = ⁇ no λ ()"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":202,"title":"d-FFDFunctionality","content":"d-FFDFunctionality : FFDAbstract.Functionality ffdAbstract","searchableContent":"d-ffdfunctionality : ffdabstract.functionality ffdabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":203,"title":"d-FFDFunctionality","content":"d-FFDFunctionality =","searchableContent":"d-ffdfunctionality ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":204,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":205,"title":"{ State","content":"{ State = FFDBuffers","searchableContent":"{ state = ffdbuffers"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":206,"title":"; initFFDState","content":"; initFFDState = record { inIBs = [] ; inEBs = [] ; inVTs = [] ; outIBs = [] ; outEBs = [] ; outVTs = [] }","searchableContent":"; initffdstate = record { inibs = [] ; inebs = [] ; invts = [] ; outibs = [] ; outebs = [] ; outvts = [] }"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":207,"title":"; _-⟦_/_⟧⇀_","content":"; _-⟦_/_⟧⇀_ = SimpleFFD","searchableContent":"; _-⟦_/_⟧⇀_ = simpleffd"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":208,"title":"; Dec-_-⟦_/_⟧⇀_","content":"; Dec-_-⟦_/_⟧⇀_ = Dec-SimpleFFD","searchableContent":"; dec-_-⟦_/_⟧⇀_ = dec-simpleffd"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":209,"title":"; Send-total","content":"; Send-total = send-total","searchableContent":"; send-total = send-total"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":210,"title":"; Fetch-total","content":"; Fetch-total = fetch-total","searchableContent":"; fetch-total = fetch-total"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":211,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":213,"title":"open import Leios.Voting public","content":"open import Leios.Voting public","searchableContent":"open import leios.voting public"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":215,"title":"d-VotingAbstract","content":"d-VotingAbstract : VotingAbstract ( Fin 1 × EndorserBlock )","searchableContent":"d-votingabstract : votingabstract ( fin 1 × endorserblock )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":216,"title":"d-VotingAbstract","content":"d-VotingAbstract =","searchableContent":"d-votingabstract ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":217,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":218,"title":"{ VotingState","content":"{ VotingState = ⊤","searchableContent":"{ votingstate = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":219,"title":"; initVotingState","content":"; initVotingState = tt","searchableContent":"; initvotingstate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":220,"title":"; isVoteCertified","content":"; isVoteCertified = λ _ _ → ⊤","searchableContent":"; isvotecertified = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":221,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":223,"title":"d-VotingAbstract-2","content":"d-VotingAbstract-2 : VotingAbstract ( Fin 2 × EndorserBlock )","searchableContent":"d-votingabstract-2 : votingabstract ( fin 2 × endorserblock )"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":224,"title":"d-VotingAbstract-2","content":"d-VotingAbstract-2 =","searchableContent":"d-votingabstract-2 ="},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":225,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":226,"title":"{ VotingState","content":"{ VotingState = ⊤","searchableContent":"{ votingstate = ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":227,"title":"; initVotingState","content":"; initVotingState = tt","searchableContent":"; initvotingstate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":228,"title":"; isVoteCertified","content":"; isVoteCertified = λ _ _ → ⊤","searchableContent":"; isvotecertified = λ _ _ → ⊤"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":229,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":231,"title":"d-SpecStructure","content":"d-SpecStructure : SpecStructure 1","searchableContent":"d-specstructure : specstructure 1"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":232,"title":"d-SpecStructure","content":"d-SpecStructure = record","searchableContent":"d-specstructure = record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":233,"title":"{ a","content":"{ a = d-Abstract","searchableContent":"{ a = d-abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":234,"title":"; Hashable-PreIBHeader","content":"; Hashable-PreIBHeader = hhs","searchableContent":"; hashable-preibheader = hhs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":235,"title":"; Hashable-PreEndorserBlock","content":"; Hashable-PreEndorserBlock = hpe","searchableContent":"; hashable-preendorserblock = hpe"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":236,"title":"; id","content":"; id = sutId","searchableContent":"; id = sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":237,"title":"; FFD'","content":"; FFD' = d-FFDFunctionality","searchableContent":"; ffd' = d-ffdfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":238,"title":"; vrf'","content":"; vrf' = d-VRF","searchableContent":"; vrf' = d-vrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":239,"title":"; sk-IB","content":"; sk-IB = IB , tt","searchableContent":"; sk-ib = ib , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":240,"title":"; sk-EB","content":"; sk-EB = EB , tt","searchableContent":"; sk-eb = eb , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":241,"title":"; sk-VT","content":"; sk-VT = VT , tt","searchableContent":"; sk-vt = vt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":242,"title":"; pk-IB","content":"; pk-IB = sutId , tt","searchableContent":"; pk-ib = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":243,"title":"; pk-EB","content":"; pk-EB = sutId , tt","searchableContent":"; pk-eb = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":244,"title":"; pk-VT","content":"; pk-VT = sutId , tt","searchableContent":"; pk-vt = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":245,"title":"; B'","content":"; B' = d-Base","searchableContent":"; b' = d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":246,"title":"; BF","content":"; BF = d-BaseFunctionality","searchableContent":"; bf = d-basefunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":247,"title":"; initBaseState","content":"; initBaseState = tt","searchableContent":"; initbasestate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":248,"title":"; K'","content":"; K' = d-KeyRegistration","searchableContent":"; k' = d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":249,"title":"; KF","content":"; KF = d-KeyRegistrationFunctionality","searchableContent":"; kf = d-keyregistrationfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":250,"title":"; va","content":"; va = d-VotingAbstract","searchableContent":"; va = d-votingabstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":251,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":253,"title":"d-SpecStructure-2","content":"d-SpecStructure-2 : SpecStructure 2","searchableContent":"d-specstructure-2 : specstructure 2"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":254,"title":"d-SpecStructure-2","content":"d-SpecStructure-2 = record","searchableContent":"d-specstructure-2 = record"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":255,"title":"{ a","content":"{ a = d-Abstract","searchableContent":"{ a = d-abstract"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":256,"title":"; Hashable-PreIBHeader","content":"; Hashable-PreIBHeader = hhs","searchableContent":"; hashable-preibheader = hhs"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":257,"title":"; Hashable-PreEndorserBlock","content":"; Hashable-PreEndorserBlock = hpe","searchableContent":"; hashable-preendorserblock = hpe"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":258,"title":"; id","content":"; id = sutId","searchableContent":"; id = sutid"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":259,"title":"; FFD'","content":"; FFD' = d-FFDFunctionality","searchableContent":"; ffd' = d-ffdfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":260,"title":"; vrf'","content":"; vrf' = d-VRF","searchableContent":"; vrf' = d-vrf"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":261,"title":"; sk-IB","content":"; sk-IB = IB , tt","searchableContent":"; sk-ib = ib , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":262,"title":"; sk-EB","content":"; sk-EB = EB , tt","searchableContent":"; sk-eb = eb , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":263,"title":"; sk-VT","content":"; sk-VT = VT , tt","searchableContent":"; sk-vt = vt , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":264,"title":"; pk-IB","content":"; pk-IB = sutId , tt","searchableContent":"; pk-ib = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":265,"title":"; pk-EB","content":"; pk-EB = sutId , tt","searchableContent":"; pk-eb = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":266,"title":"; pk-VT","content":"; pk-VT = sutId , tt","searchableContent":"; pk-vt = sutid , tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":267,"title":"; B'","content":"; B' = d-Base","searchableContent":"; b' = d-base"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":268,"title":"; BF","content":"; BF = d-BaseFunctionality","searchableContent":"; bf = d-basefunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":269,"title":"; initBaseState","content":"; initBaseState = tt","searchableContent":"; initbasestate = tt"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":270,"title":"; K'","content":"; K' = d-KeyRegistration","searchableContent":"; k' = d-keyregistration"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":271,"title":"; KF","content":"; KF = d-KeyRegistrationFunctionality","searchableContent":"; kf = d-keyregistrationfunctionality"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":272,"title":"; va","content":"; va = d-VotingAbstract-2","searchableContent":"; va = d-votingabstract-2"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":273,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Defaults","path":"Leios.Defaults.html","group":"Leios","lineNumber":275,"title":"open import Leios.Short d-SpecStructure public","content":"open import Leios.Short d-SpecStructure public","searchableContent":"open import leios.short d-specstructure public"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":2,"title":"Leios.FFD {-# OPTIONS --safe #-}","content":"Leios.FFD {-# OPTIONS --safe #-}","searchableContent":"leios.ffd {-# options --safe #-}"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":4,"title":"module Leios.FFD where","content":"module Leios.FFD where","searchableContent":"module leios.ffd where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":6,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":8,"title":"record FFDAbstract","content":"record FFDAbstract : Type₁ where","searchableContent":"record ffdabstract : type₁ where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":9,"title":"field Header Body ID","content":"field Header Body ID : Type","searchableContent":"field header body id : type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":10,"title":"⦃ DecEq-Header ⦄","content":"⦃ DecEq-Header ⦄ : DecEq Header","searchableContent":"⦃ deceq-header ⦄ : deceq header"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":11,"title":"⦃ DecEq-Body ⦄","content":"⦃ DecEq-Body ⦄ : DecEq Body","searchableContent":"⦃ deceq-body ⦄ : deceq body"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":12,"title":"match","content":"match : Header → Body → Type","searchableContent":"match : header → body → type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":13,"title":"msgID","content":"msgID : Header → ID","searchableContent":"msgid : header → id"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":15,"title":"data Input","content":"data Input : Type where","searchableContent":"data input : type where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":16,"title":"Send","content":"Send : Header → Maybe Body → Input","searchableContent":"send : header → maybe body → input"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":17,"title":"Fetch","content":"Fetch : Input","searchableContent":"fetch : input"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":19,"title":"data Output","content":"data Output : Type where","searchableContent":"data output : type where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":20,"title":"SendRes","content":"SendRes : Output","searchableContent":"sendres : output"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":21,"title":"FetchRes","content":"FetchRes : List ( Header ⊎ Body ) → Output","searchableContent":"fetchres : list ( header ⊎ body ) → output"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":23,"title":"record Functionality","content":"record Functionality : Type₁ where","searchableContent":"record functionality : type₁ where"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":24,"title":"field State","content":"field State : Type","searchableContent":"field state : type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":25,"title":"initFFDState","content":"initFFDState : State","searchableContent":"initffdstate : state"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":26,"title":"_-⟦_/_⟧⇀_","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"_-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":27,"title":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄","content":"⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : { s : State } → { i : Input } → { o : Output } → { s' : State } → ( s -⟦ i / o ⟧⇀ s' ) ⁇","searchableContent":"⦃ dec-_-⟦_/_⟧⇀_ ⦄ : { s : state } → { i : input } → { o : output } → { s' : state } → ( s -⟦ i / o ⟧⇀ s' ) ⁇"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":28,"title":"Send-total","content":"Send-total : ∀ { ffds h b } → ∃[ ffds' ] ffds -⟦ Send h b / SendRes ⟧⇀ ffds'","searchableContent":"send-total : ∀ { ffds h b } → ∃[ ffds' ] ffds -⟦ send h b / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":29,"title":"Fetch-total","content":"Fetch-total : ∀ { ffds } → ∃[ r ] ( ∃[ ffds' ] ( ffds -⟦ Fetch / FetchRes r ⟧⇀ ffds' ))","searchableContent":"fetch-total : ∀ { ffds } → ∃[ r ] ( ∃[ ffds' ] ( ffds -⟦ fetch / fetchres r ⟧⇀ ffds' ))"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":31,"title":"open Input public","content":"open Input public","searchableContent":"open input public"},{"moduleName":"Leios.FFD","path":"Leios.FFD.html","group":"Leios","lineNumber":32,"title":"open Output public","content":"open Output public","searchableContent":"open output public"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.BaseTypes module Leios.Foreign.BaseTypes where","content":"Leios.Foreign.BaseTypes module Leios.Foreign.BaseTypes where","searchableContent":"leios.foreign.basetypes module leios.foreign.basetypes where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":4,"title":"-- TODO","content":"-- TODO: copied from the formal-ledger project for now","searchableContent":"-- todo: copied from the formal-ledger project for now"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":5,"title":"-- Added","content":"-- Added: * TotalMap","searchableContent":"-- added: * totalmap"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":7,"title":"open import Data.Rational","content":"open import Data.Rational","searchableContent":"open import data.rational"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":11,"title":"open import Data.Fin","content":"open import Data.Fin","searchableContent":"open import data.fin"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":12,"title":"open import Function.Related.TypeIsomorphisms","content":"open import Function.Related.TypeIsomorphisms","searchableContent":"open import function.related.typeisomorphisms"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":13,"title":"open import Relation.Binary.Structures","content":"open import Relation.Binary.Structures","searchableContent":"open import relation.binary.structures"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":15,"title":"open import Tactic.Derive.Convertible","content":"open import Tactic.Derive.Convertible","searchableContent":"open import tactic.derive.convertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":16,"title":"open import Tactic.Derive.HsType","content":"open import Tactic.Derive.HsType","searchableContent":"open import tactic.derive.hstype"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":18,"title":"open import Class.Convertible","content":"open import Class.Convertible","searchableContent":"open import class.convertible"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":19,"title":"open import Class.Decidable.Instances","content":"open import Class.Decidable.Instances","searchableContent":"open import class.decidable.instances"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":20,"title":"open import Class.HasHsType","content":"open import Class.HasHsType","searchableContent":"open import class.hashstype"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":22,"title":"open import Leios.Foreign.HsTypes as F","content":"open import Leios.Foreign.HsTypes as F","searchableContent":"open import leios.foreign.hstypes as f"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":23,"title":"open import Leios.Foreign.Util","content":"open import Leios.Foreign.Util","searchableContent":"open import leios.foreign.util"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":24,"title":"open import Foreign.Haskell","content":"open import Foreign.Haskell","searchableContent":"open import foreign.haskell"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":26,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":27,"title":"iConvTop","content":"iConvTop = Convertible-Refl { ⊤ }","searchableContent":"iconvtop = convertible-refl { ⊤ }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":28,"title":"iConvNat","content":"iConvNat = Convertible-Refl { ℕ }","searchableContent":"iconvnat = convertible-refl { ℕ }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":29,"title":"iConvString","content":"iConvString = Convertible-Refl { String }","searchableContent":"iconvstring = convertible-refl { string }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":30,"title":"iConvBool","content":"iConvBool = Convertible-Refl { Bool }","searchableContent":"iconvbool = convertible-refl { bool }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":32,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":34,"title":"-- * Unit and empty","content":"-- * Unit and empty","searchableContent":"-- * unit and empty"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":36,"title":"HsTy-⊥","content":"HsTy-⊥ = MkHsType ⊥ F.Empty","searchableContent":"hsty-⊥ = mkhstype ⊥ f.empty"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":37,"title":"Conv-⊥","content":"Conv-⊥ = autoConvert ⊥","searchableContent":"conv-⊥ = autoconvert ⊥"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":39,"title":"HsTy-⊤","content":"HsTy-⊤ = MkHsType ⊤ ⊤","searchableContent":"hsty-⊤ = mkhstype ⊤ ⊤"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":41,"title":"-- * Rational numbers","content":"-- * Rational numbers","searchableContent":"-- * rational numbers"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":43,"title":"HsTy-Rational","content":"HsTy-Rational = MkHsType ℚ F.Rational","searchableContent":"hsty-rational = mkhstype ℚ f.rational"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":44,"title":"Conv-Rational","content":"Conv-Rational : HsConvertible ℚ","searchableContent":"conv-rational : hsconvertible ℚ"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":45,"title":"Conv-Rational","content":"Conv-Rational = λ where","searchableContent":"conv-rational = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":46,"title":". to ( mkℚ n d _) → n F., suc d","content":". to ( mkℚ n d _) → n F., suc d","searchableContent":". to ( mkℚ n d _) → n f., suc d"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":47,"title":". from ( n F., zero ) → 0ℚ -- TODO is there a safer way to do this?","content":". from ( n F., zero ) → 0ℚ -- TODO is there a safer way to do this?","searchableContent":". from ( n f., zero ) → 0ℚ -- todo is there a safer way to do this?"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":48,"title":". from ( n F., ( suc d )) → n Data.Rational./ suc d","content":". from ( n F., ( suc d )) → n Data.Rational./ suc d","searchableContent":". from ( n f., ( suc d )) → n data.rational./ suc d"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":50,"title":"-- * Maps and Sets","content":"-- * Maps and Sets","searchableContent":"-- * maps and sets"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":52,"title":"HsTy-HSSet","content":"HsTy-HSSet : ∀ { A } → ⦃ HasHsType A ⦄ → HasHsType ( ℙ A )","searchableContent":"hsty-hsset : ∀ { a } → ⦃ hashstype a ⦄ → hashstype ( ℙ a )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":53,"title":"HsTy-HSSet { A }","content":"HsTy-HSSet { A } = MkHsType _ ( F.HSSet ( HsType A ))","searchableContent":"hsty-hsset { a } = mkhstype _ ( f.hsset ( hstype a ))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":55,"title":"Conv-HSSet","content":"Conv-HSSet : ∀ { A } ⦃ _ : HasHsType A ⦄","searchableContent":"conv-hsset : ∀ { a } ⦃ _ : hashstype a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":56,"title":"→ ⦃ HsConvertible A ⦄","content":"→ ⦃ HsConvertible A ⦄","searchableContent":"→ ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":57,"title":"→ HsConvertible ( ℙ A )","content":"→ HsConvertible ( ℙ A )","searchableContent":"→ hsconvertible ( ℙ a )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":58,"title":"Conv-HSSet","content":"Conv-HSSet = λ where","searchableContent":"conv-hsset = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":59,"title":". to → F.MkHSSet ∘ to ∘ setToList","content":". to → F.MkHSSet ∘ to ∘ setToList","searchableContent":". to → f.mkhsset ∘ to ∘ settolist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":60,"title":". from → fromList ∘ from ∘ F.HSSet.elems","content":". from → fromList ∘ from ∘ F.HSSet.elems","searchableContent":". from → fromlist ∘ from ∘ f.hsset.elems"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":62,"title":"Convertible-FinSet","content":"Convertible-FinSet : Convertible₁ ℙ_ List","searchableContent":"convertible-finset : convertible₁ ℙ_ list"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":63,"title":"Convertible-FinSet","content":"Convertible-FinSet = λ where","searchableContent":"convertible-finset = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":64,"title":". to → map to ∘ setToList","content":". to → map to ∘ setToList","searchableContent":". to → map to ∘ settolist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":65,"title":". from → fromList ∘ map from","content":". from → fromList ∘ map from","searchableContent":". from → fromlist ∘ map from"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":67,"title":"Convertible-Map","content":"Convertible-Map : ∀ { K K' V V' } → ⦃ DecEq K ⦄","searchableContent":"convertible-map : ∀ { k k' v v' } → ⦃ deceq k ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":68,"title":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","content":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","searchableContent":"→ ⦃ convertible k k' ⦄ → ⦃ convertible v v' ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":69,"title":"→ Convertible ( K ⇀ V ) ( List $ Pair K' V' )","content":"→ Convertible ( K ⇀ V ) ( List $ Pair K' V' )","searchableContent":"→ convertible ( k ⇀ v ) ( list $ pair k' v' )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":70,"title":"Convertible-Map","content":"Convertible-Map = λ where","searchableContent":"convertible-map = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":71,"title":". to → to ∘ proj₁","content":". to → to ∘ proj₁","searchableContent":". to → to ∘ proj₁"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":72,"title":". from → fromListᵐ ∘ map from","content":". from → fromListᵐ ∘ map from","searchableContent":". from → fromlistᵐ ∘ map from"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":74,"title":"HsTy-Map","content":"HsTy-Map : ∀ { A B } → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType ( A ⇀ B )","searchableContent":"hsty-map : ∀ { a b } → ⦃ hashstype a ⦄ → ⦃ hashstype b ⦄ → hashstype ( a ⇀ b )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":75,"title":"HsTy-Map { A } { B }","content":"HsTy-Map { A } { B } = MkHsType _ ( F.HSMap ( HsType A ) ( HsType B ))","searchableContent":"hsty-map { a } { b } = mkhstype _ ( f.hsmap ( hstype a ) ( hstype b ))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":77,"title":"Conv-HSMap","content":"Conv-HSMap : ∀ { A B } ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄","searchableContent":"conv-hsmap : ∀ { a b } ⦃ _ : hashstype a ⦄ ⦃ _ : hashstype b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":78,"title":"→ ⦃ DecEq A ⦄","content":"→ ⦃ DecEq A ⦄","searchableContent":"→ ⦃ deceq a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":79,"title":"→ ⦃ HsConvertible A ⦄","content":"→ ⦃ HsConvertible A ⦄","searchableContent":"→ ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":80,"title":"→ ⦃ HsConvertible B ⦄","content":"→ ⦃ HsConvertible B ⦄","searchableContent":"→ ⦃ hsconvertible b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":81,"title":"→ HsConvertible ( A ⇀ B )","content":"→ HsConvertible ( A ⇀ B )","searchableContent":"→ hsconvertible ( a ⇀ b )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":82,"title":"Conv-HSMap","content":"Conv-HSMap = λ where","searchableContent":"conv-hsmap = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":83,"title":". to → F.MkHSMap ∘ to","content":". to → F.MkHSMap ∘ to","searchableContent":". to → f.mkhsmap ∘ to"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":84,"title":". from → from ∘ F.HSMap.assocList","content":". from → from ∘ F.HSMap.assocList","searchableContent":". from → from ∘ f.hsmap.assoclist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":86,"title":"Convertible-TotalMap","content":"Convertible-TotalMap : ∀ { K K' V V' } → ⦃ DecEq K ⦄ → ⦃ Listable K ⦄","searchableContent":"convertible-totalmap : ∀ { k k' v v' } → ⦃ deceq k ⦄ → ⦃ listable k ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":87,"title":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","content":"→ ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄","searchableContent":"→ ⦃ convertible k k' ⦄ → ⦃ convertible v v' ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":88,"title":"→ Convertible ( TotalMap K V ) ( List $ Pair K' V' )","content":"→ Convertible ( TotalMap K V ) ( List $ Pair K' V' )","searchableContent":"→ convertible ( totalmap k v ) ( list $ pair k' v' )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":89,"title":"Convertible-TotalMap { K }","content":"Convertible-TotalMap { K } = λ where","searchableContent":"convertible-totalmap { k } = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":90,"title":". to → to ∘ TotalMap.rel","content":". to → to ∘ TotalMap.rel","searchableContent":". to → to ∘ totalmap.rel"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":91,"title":". from → λ x →","content":". from → λ x →","searchableContent":". from → λ x →"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":92,"title":"let ( r , l )","content":"let ( r , l ) = fromListᵐ ( map from x )","searchableContent":"let ( r , l ) = fromlistᵐ ( map from x )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":93,"title":"in case ( ¿ total r ¿ ) of λ where","content":"in case ( ¿ total r ¿ ) of λ where","searchableContent":"in case ( ¿ total r ¿ ) of λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":94,"title":"( yes p ) → record { rel","content":"( yes p ) → record { rel = r ; left-unique-rel = l ; total-rel = p }","searchableContent":"( yes p ) → record { rel = r ; left-unique-rel = l ; total-rel = p }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":95,"title":"( no p ) → error "Expected total map"","content":"( no p ) → error "Expected total map"","searchableContent":"( no p ) → error "expected total map""},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":97,"title":"HsTy-TotalMap","content":"HsTy-TotalMap : ∀ { A B } → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType ( TotalMap A B )","searchableContent":"hsty-totalmap : ∀ { a b } → ⦃ hashstype a ⦄ → ⦃ hashstype b ⦄ → hashstype ( totalmap a b )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":98,"title":"HsTy-TotalMap { A } { B }","content":"HsTy-TotalMap { A } { B } = MkHsType _ ( F.HSMap ( HsType A ) ( HsType B ))","searchableContent":"hsty-totalmap { a } { b } = mkhstype _ ( f.hsmap ( hstype a ) ( hstype b ))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":100,"title":"Conv-HSTotalMap","content":"Conv-HSTotalMap : ∀ { A B } ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄","searchableContent":"conv-hstotalmap : ∀ { a b } ⦃ _ : hashstype a ⦄ ⦃ _ : hashstype b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":101,"title":"→ ⦃ DecEq A ⦄","content":"→ ⦃ DecEq A ⦄","searchableContent":"→ ⦃ deceq a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":102,"title":"→ ⦃ Listable A ⦄","content":"→ ⦃ Listable A ⦄","searchableContent":"→ ⦃ listable a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":103,"title":"→ ⦃ HsConvertible A ⦄","content":"→ ⦃ HsConvertible A ⦄","searchableContent":"→ ⦃ hsconvertible a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":104,"title":"→ ⦃ HsConvertible B ⦄","content":"→ ⦃ HsConvertible B ⦄","searchableContent":"→ ⦃ hsconvertible b ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":105,"title":"→ HsConvertible ( TotalMap A B )","content":"→ HsConvertible ( TotalMap A B )","searchableContent":"→ hsconvertible ( totalmap a b )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":106,"title":"Conv-HSTotalMap","content":"Conv-HSTotalMap = λ where","searchableContent":"conv-hstotalmap = λ where"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":107,"title":". to → MkHSMap ∘ to","content":". to → MkHSMap ∘ to","searchableContent":". to → mkhsmap ∘ to"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":108,"title":". from → from ∘ F.HSMap.assocList","content":". from → from ∘ F.HSMap.assocList","searchableContent":". from → from ∘ f.hsmap.assoclist"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":110,"title":"-- * ComputationResult","content":"-- * ComputationResult","searchableContent":"-- * computationresult"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":112,"title":"open import Class.Computational as C","content":"open import Class.Computational as C","searchableContent":"open import class.computational as c"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":114,"title":"HsTy-ComputationResult","content":"HsTy-ComputationResult : ∀ { l } { Err } { A : Type l }","searchableContent":"hsty-computationresult : ∀ { l } { err } { a : type l }"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":115,"title":"→ ⦃ HasHsType Err ⦄ → ⦃ HasHsType A ⦄","content":"→ ⦃ HasHsType Err ⦄ → ⦃ HasHsType A ⦄","searchableContent":"→ ⦃ hashstype err ⦄ → ⦃ hashstype a ⦄"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":116,"title":"→ HasHsType ( C.ComputationResult Err A )","content":"→ HasHsType ( C.ComputationResult Err A )","searchableContent":"→ hashstype ( c.computationresult err a )"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":117,"title":"HsTy-ComputationResult { Err","content":"HsTy-ComputationResult { Err = Err } { A } = MkHsType _ ( F.ComputationResult ( HsType Err ) ( HsType A ))","searchableContent":"hsty-computationresult { err = err } { a } = mkhstype _ ( f.computationresult ( hstype err ) ( hstype a ))"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":119,"title":"Conv-ComputationResult","content":"Conv-ComputationResult : ConvertibleType C.ComputationResult F.ComputationResult","searchableContent":"conv-computationresult : convertibletype c.computationresult f.computationresult"},{"moduleName":"Leios.Foreign.BaseTypes","path":"Leios.Foreign.BaseTypes.html","group":"Leios","lineNumber":120,"title":"Conv-ComputationResult","content":"Conv-ComputationResult = autoConvertible","searchableContent":"conv-computationresult = autoconvertible"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.HsTypes module Leios.Foreign.HsTypes where","content":"Leios.Foreign.HsTypes module Leios.Foreign.HsTypes where","searchableContent":"leios.foreign.hstypes module leios.foreign.hstypes where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":4,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":5,"title":"{-# LANGUAGE DeriveGeneric #-}","content":"{-# LANGUAGE DeriveGeneric #-}","searchableContent":"{-# language derivegeneric #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":6,"title":"{-# LANGUAGE DeriveFunctor #-}","content":"{-# LANGUAGE DeriveFunctor #-}","searchableContent":"{-# language derivefunctor #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":7,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":11,"title":"open import Foreign.Haskell","content":"open import Foreign.Haskell","searchableContent":"open import foreign.haskell"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":12,"title":"open import Foreign.Haskell.Coerce","content":"open import Foreign.Haskell.Coerce","searchableContent":"open import foreign.haskell.coerce"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":13,"title":"open import Foreign.Haskell.Either","content":"open import Foreign.Haskell.Either","searchableContent":"open import foreign.haskell.either"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":14,"title":"open import Data.Rational.Base","content":"open import Data.Rational.Base","searchableContent":"open import data.rational.base"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":16,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":17,"title":"import GHC.Generics (Generic)","content":"import GHC.Generics (Generic)","searchableContent":"import ghc.generics (generic)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":18,"title":"import Data.Void (Void)","content":"import Data.Void (Void)","searchableContent":"import data.void (void)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":19,"title":"import Prelude hiding (Rational)","content":"import Prelude hiding (Rational)","searchableContent":"import prelude hiding (rational)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":20,"title":"import GHC.Real (Ratio(..))","content":"import GHC.Real (Ratio(..))","searchableContent":"import ghc.real (ratio(..))"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":21,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":23,"title":"-- * The empty type","content":"-- * The empty type","searchableContent":"-- * the empty type"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":25,"title":"data Empty","content":"data Empty : Type where","searchableContent":"data empty : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":26,"title":"{-# COMPILE GHC Empty","content":"{-# COMPILE GHC Empty = data Void () #-}","searchableContent":"{-# compile ghc empty = data void () #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":28,"title":"-- * Rational","content":"-- * Rational","searchableContent":"-- * rational"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":30,"title":"data Rational","content":"data Rational : Type where","searchableContent":"data rational : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":31,"title":"_,_","content":"_,_ : ℤ → ℕ → Rational","searchableContent":"_,_ : ℤ → ℕ → rational"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":32,"title":"{-# COMPILE GHC Rational = data Rational ((","content":"{-# COMPILE GHC Rational = data Rational ((:%)) #-}","searchableContent":"{-# compile ghc rational = data rational ((:%)) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":34,"title":"-- We'll generate code with qualified references to Rational in this","content":"-- We'll generate code with qualified references to Rational in this","searchableContent":"-- we'll generate code with qualified references to rational in this"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":35,"title":"-- module, so make sure to define it.","content":"-- module, so make sure to define it.","searchableContent":"-- module, so make sure to define it."},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":36,"title":"{-# FOREIGN GHC type Rational","content":"{-# FOREIGN GHC type Rational = Ratio Integer #-}","searchableContent":"{-# foreign ghc type rational = ratio integer #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":38,"title":"-- * Maps and Sets","content":"-- * Maps and Sets","searchableContent":"-- * maps and sets"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":40,"title":"record HSMap K V","content":"record HSMap K V : Type where","searchableContent":"record hsmap k v : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":41,"title":"constructor MkHSMap","content":"constructor MkHSMap","searchableContent":"constructor mkhsmap"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":42,"title":"field assocList","content":"field assocList : List ( Pair K V )","searchableContent":"field assoclist : list ( pair k v )"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":44,"title":"record HSSet A","content":"record HSSet A : Type where","searchableContent":"record hsset a : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":45,"title":"constructor MkHSSet","content":"constructor MkHSSet","searchableContent":"constructor mkhsset"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":46,"title":"field elems","content":"field elems : List A","searchableContent":"field elems : list a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":48,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":49,"title":"newtype HSMap k v","content":"newtype HSMap k v = MkHSMap [(k, v)]","searchableContent":"newtype hsmap k v = mkhsmap [(k, v)]"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":50,"title":"deriving (Generic, Show, Eq, Ord)","content":"deriving (Generic, Show, Eq, Ord)","searchableContent":"deriving (generic, show, eq, ord)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":51,"title":"newtype HSSet a","content":"newtype HSSet a = MkHSSet [a]","searchableContent":"newtype hsset a = mkhsset [a]"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":52,"title":"deriving (Generic, Show, Eq, Ord)","content":"deriving (Generic, Show, Eq, Ord)","searchableContent":"deriving (generic, show, eq, ord)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":53,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":54,"title":"{-# COMPILE GHC HSMap","content":"{-# COMPILE GHC HSMap = data HSMap (MkHSMap) #-}","searchableContent":"{-# compile ghc hsmap = data hsmap (mkhsmap) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":55,"title":"{-# COMPILE GHC HSSet","content":"{-# COMPILE GHC HSSet = data HSSet (MkHSSet) #-}","searchableContent":"{-# compile ghc hsset = data hsset (mkhsset) #-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":57,"title":"-- * ComputationResult","content":"-- * ComputationResult","searchableContent":"-- * computationresult"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":59,"title":"data ComputationResult E A","content":"data ComputationResult E A : Type where","searchableContent":"data computationresult e a : type where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":60,"title":"Success","content":"Success : A → ComputationResult E A","searchableContent":"success : a → computationresult e a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":61,"title":"Failure","content":"Failure : E → ComputationResult E A","searchableContent":"failure : e → computationresult e a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":63,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":64,"title":"data ComputationResult e a","content":"data ComputationResult e a = Success a | Failure e","searchableContent":"data computationresult e a = success a | failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":65,"title":"deriving (Functor, Eq, Show, Generic)","content":"deriving (Functor, Eq, Show, Generic)","searchableContent":"deriving (functor, eq, show, generic)"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":67,"title":"instance Applicative (ComputationResult e) where","content":"instance Applicative (ComputationResult e) where","searchableContent":"instance applicative (computationresult e) where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":68,"title":"pure","content":"pure = Success","searchableContent":"pure = success"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":69,"title":"(Success f) <*> x","content":"(Success f) <*> x = f <$> x","searchableContent":"(success f) <*> x = f <$> x"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":70,"title":"(Failure e) <*> _","content":"(Failure e) <*> _ = Failure e","searchableContent":"(failure e) <*> _ = failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":72,"title":"instance Monad (ComputationResult e) where","content":"instance Monad (ComputationResult e) where","searchableContent":"instance monad (computationresult e) where"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":73,"title":"return","content":"return = pure","searchableContent":"return = pure"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":74,"title":"(Success a) >>","content":"(Success a) >>= m = m a","searchableContent":"(success a) >>= m = m a"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":75,"title":"(Failure e) >>","content":"(Failure e) >>= _ = Failure e","searchableContent":"(failure e) >>= _ = failure e"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":76,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.HsTypes","path":"Leios.Foreign.HsTypes.html","group":"Leios","lineNumber":77,"title":"{-# COMPILE GHC ComputationResult","content":"{-# COMPILE GHC ComputationResult = data ComputationResult (Success | Failure) #-}","searchableContent":"{-# compile ghc computationresult = data computationresult (success | failure) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.Types open import Data.Char.Base as C using ( Char )","content":"Leios.Foreign.Types open import Data.Char.Base as C using ( Char )","searchableContent":"leios.foreign.types open import data.char.base as c using ( char )"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":3,"title":"import Data.String as S","content":"import Data.String as S","searchableContent":"import data.string as s"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":4,"title":"open import Data.Integer using ( +_ ; ∣_∣ )","content":"open import Data.Integer using ( +_ ; ∣_∣ )","searchableContent":"open import data.integer using ( +_ ; ∣_∣ )"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":6,"title":"open import Class.Convertible","content":"open import Class.Convertible","searchableContent":"open import class.convertible"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":7,"title":"open import Class.HasHsType","content":"open import Class.HasHsType","searchableContent":"open import class.hashstype"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":8,"title":"open import Tactic.Derive.Convertible","content":"open import Tactic.Derive.Convertible","searchableContent":"open import tactic.derive.convertible"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":9,"title":"open import Tactic.Derive.HsType","content":"open import Tactic.Derive.HsType","searchableContent":"open import tactic.derive.hstype"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":11,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":13,"title":"open import Leios.Foreign.BaseTypes","content":"open import Leios.Foreign.BaseTypes","searchableContent":"open import leios.foreign.basetypes"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":14,"title":"open import Leios.Foreign.HsTypes","content":"open import Leios.Foreign.HsTypes","searchableContent":"open import leios.foreign.hstypes"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":15,"title":"open import Leios.Foreign.Util","content":"open import Leios.Foreign.Util","searchableContent":"open import leios.foreign.util"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":17,"title":"module Leios.Foreign.Types where","content":"module Leios.Foreign.Types where","searchableContent":"module leios.foreign.types where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":19,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":20,"title":"{-# LANGUAGE DuplicateRecordFields #-}","content":"{-# LANGUAGE DuplicateRecordFields #-}","searchableContent":"{-# language duplicaterecordfields #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":21,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":23,"title":"-- TODO","content":"-- TODO: Get rid of hardcoded parameters in this module","searchableContent":"-- todo: get rid of hardcoded parameters in this module"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":25,"title":"{-","content":"{-","searchableContent":"{-"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":26,"title":"numberOfParties","content":"numberOfParties : ℕ","searchableContent":"numberofparties : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":27,"title":"numberOfParties","content":"numberOfParties = 2","searchableContent":"numberofparties = 2"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":29,"title":"open import Leios.Defaults numberOfParties fzero","content":"open import Leios.Defaults numberOfParties fzero","searchableContent":"open import leios.defaults numberofparties fzero"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":30,"title":"renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)","content":"renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)","searchableContent":"renaming (endorserblock to endorserblockagda; ibheader to ibheaderagda)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":32,"title":"dropDash","content":"dropDash : S.String → S.String","searchableContent":"dropdash : s.string → s.string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":33,"title":"dropDash","content":"dropDash = S.concat ∘ S.wordsByᵇ ('-' C.≈ᵇ_)","searchableContent":"dropdash = s.concat ∘ s.wordsbyᵇ ('-' c.≈ᵇ_)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":35,"title":"prefix","content":"prefix : S.String → S.String → S.String","searchableContent":"prefix : s.string → s.string → s.string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":36,"title":"prefix","content":"prefix = S._++_","searchableContent":"prefix = s._++_"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":38,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":39,"title":"HsTy-SlotUpkeep","content":"HsTy-SlotUpkeep = autoHsType SlotUpkeep ⊣ onConstructors dropDash","searchableContent":"hsty-slotupkeep = autohstype slotupkeep ⊣ onconstructors dropdash"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":40,"title":"Conv-SlotUpkeep","content":"Conv-SlotUpkeep = autoConvert SlotUpkeep","searchableContent":"conv-slotupkeep = autoconvert slotupkeep"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":42,"title":"record IBHeader","content":"record IBHeader : Type where","searchableContent":"record ibheader : type where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":43,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"field slotnumber : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":44,"title":"producerID","content":"producerID : ℕ","searchableContent":"producerid : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":45,"title":"bodyHash","content":"bodyHash : List ℕ","searchableContent":"bodyhash : list ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":47,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":48,"title":"data IBHeader = IBHeader {slotNumber","content":"data IBHeader = IBHeader {slotNumber :: Integer, producerID :: Integer, bodyHash :: Data.Text.Text }","searchableContent":"data ibheader = ibheader {slotnumber :: integer, producerid :: integer, bodyhash :: data.text.text }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":49,"title":"deriving (Show, Eq, Generic)","content":"deriving (Show, Eq, Generic)","searchableContent":"deriving (show, eq, generic)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":50,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":52,"title":"{-# COMPILE GHC IBHeader","content":"{-# COMPILE GHC IBHeader = data IBHeader (IBHeader) #-}","searchableContent":"{-# compile ghc ibheader = data ibheader (ibheader) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":54,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":55,"title":"HsTy-IBHeader","content":"HsTy-IBHeader = MkHsType IBHeaderAgda IBHeader","searchableContent":"hsty-ibheader = mkhstype ibheaderagda ibheader"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":57,"title":"Conv-IBHeader","content":"Conv-IBHeader : Convertible IBHeaderAgda IBHeader","searchableContent":"conv-ibheader : convertible ibheaderagda ibheader"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":58,"title":"Conv-IBHeader","content":"Conv-IBHeader = record","searchableContent":"conv-ibheader = record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":59,"title":"{ to","content":"{ to = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →","searchableContent":"{ to = λ (record { slotnumber = s ; producerid = p ; bodyhash = h }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":60,"title":"record { slotNumber","content":"record { slotNumber = s ; producerID = toℕ p ; bodyHash = h}","searchableContent":"record { slotnumber = s ; producerid = toℕ p ; bodyhash = h}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":61,"title":"; from","content":"; from = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →","searchableContent":"; from = λ (record { slotnumber = s ; producerid = p ; bodyhash = h }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":62,"title":"case p <? numberOfParties of λ where","content":"case p <? numberOfParties of λ where","searchableContent":"case p <? numberofparties of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":63,"title":"(yes q) → record { slotNumber","content":"(yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; bodyHash = h ; signature = tt }","searchableContent":"(yes q) → record { slotnumber = s ; producerid = #_ p {numberofparties} {fromwitness q} ; lotterypf = tt ; bodyhash = h ; signature = tt }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":64,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"(no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":65,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":67,"title":"HsTy-IBBody","content":"HsTy-IBBody = autoHsType IBBody","searchableContent":"hsty-ibbody = autohstype ibbody"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":68,"title":"Conv-IBBody","content":"Conv-IBBody = autoConvert IBBody","searchableContent":"conv-ibbody = autoconvert ibbody"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":70,"title":"HsTy-InputBlock","content":"HsTy-InputBlock = autoHsType InputBlock","searchableContent":"hsty-inputblock = autohstype inputblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":71,"title":"Conv-InputBlock","content":"Conv-InputBlock = autoConvert InputBlock","searchableContent":"conv-inputblock = autoconvert inputblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":73,"title":"Conv-ℕ","content":"Conv-ℕ : HsConvertible ℕ","searchableContent":"conv-ℕ : hsconvertible ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":74,"title":"Conv-ℕ","content":"Conv-ℕ = Convertible-Refl","searchableContent":"conv-ℕ = convertible-refl"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":76,"title":"record EndorserBlock","content":"record EndorserBlock : Type where","searchableContent":"record endorserblock : type where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":77,"title":"field slotNumber","content":"field slotNumber : ℕ","searchableContent":"field slotnumber : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":78,"title":"producerID","content":"producerID : ℕ","searchableContent":"producerid : ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":79,"title":"ibRefs","content":"ibRefs : List (List IBRef)","searchableContent":"ibrefs : list (list ibref)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":81,"title":"{-# FOREIGN GHC","content":"{-# FOREIGN GHC","searchableContent":"{-# foreign ghc"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":82,"title":"data EndorserBlock = EndorserBlock { slotNumber","content":"data EndorserBlock = EndorserBlock { slotNumber :: Integer, producerID :: Integer, ibRefs :: [Data.Text.Text] }","searchableContent":"data endorserblock = endorserblock { slotnumber :: integer, producerid :: integer, ibrefs :: [data.text.text] }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":83,"title":"deriving (Show, Eq, Generic)","content":"deriving (Show, Eq, Generic)","searchableContent":"deriving (show, eq, generic)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":84,"title":"#-}","content":"#-}","searchableContent":"#-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":86,"title":"{-# COMPILE GHC EndorserBlock","content":"{-# COMPILE GHC EndorserBlock = data EndorserBlock (EndorserBlock) #-}","searchableContent":"{-# compile ghc endorserblock = data endorserblock (endorserblock) #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":88,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":89,"title":"HsTy-EndorserBlock","content":"HsTy-EndorserBlock = MkHsType EndorserBlockAgda EndorserBlock","searchableContent":"hsty-endorserblock = mkhstype endorserblockagda endorserblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":91,"title":"Conv-EndorserBlock","content":"Conv-EndorserBlock : Convertible EndorserBlockAgda EndorserBlock","searchableContent":"conv-endorserblock : convertible endorserblockagda endorserblock"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":92,"title":"Conv-EndorserBlock","content":"Conv-EndorserBlock =","searchableContent":"conv-endorserblock ="},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":93,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":94,"title":"{ to","content":"{ to = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →","searchableContent":"{ to = λ (record { slotnumber = s ; producerid = p ; ibrefs = refs }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":95,"title":"record { slotNumber","content":"record { slotNumber = s ; producerID = toℕ p ; ibRefs = refs }","searchableContent":"record { slotnumber = s ; producerid = toℕ p ; ibrefs = refs }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":96,"title":"; from","content":"; from = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →","searchableContent":"; from = λ (record { slotnumber = s ; producerid = p ; ibrefs = refs }) →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":97,"title":"case p <? numberOfParties of λ where","content":"case p <? numberOfParties of λ where","searchableContent":"case p <? numberofparties of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":98,"title":"(yes q) → record { slotNumber","content":"(yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; signature = tt ; ibRefs = refs ; ebRefs = [] }","searchableContent":"(yes q) → record { slotnumber = s ; producerid = #_ p {numberofparties} {fromwitness q} ; lotterypf = tt ; signature = tt ; ibrefs = refs ; ebrefs = [] }"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":99,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"(no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":100,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":102,"title":"HsTy-FFDState","content":"HsTy-FFDState = autoHsType FFDState","searchableContent":"hsty-ffdstate = autohstype ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":103,"title":"Conv-FFDState","content":"Conv-FFDState = autoConvert FFDState","searchableContent":"conv-ffdstate = autoconvert ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":105,"title":"HsTy-Fin","content":"HsTy-Fin : ∀ {n} → HasHsType (Fin n)","searchableContent":"hsty-fin : ∀ {n} → hashstype (fin n)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":106,"title":"HsTy-Fin .HasHsType.HsType","content":"HsTy-Fin .HasHsType.HsType = ℕ","searchableContent":"hsty-fin .hashstype.hstype = ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":108,"title":"Conv-Fin","content":"Conv-Fin : ∀ {n} → HsConvertible (Fin n)","searchableContent":"conv-fin : ∀ {n} → hsconvertible (fin n)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":109,"title":"Conv-Fin {n}","content":"Conv-Fin {n} =","searchableContent":"conv-fin {n} ="},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":110,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":111,"title":"{ to","content":"{ to = toℕ","searchableContent":"{ to = toℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":112,"title":"; from","content":"; from = λ m →","searchableContent":"; from = λ m →"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":113,"title":"case m <? n of λ where","content":"case m <? n of λ where","searchableContent":"case m <? n of λ where"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":114,"title":"(yes p) → #_ m {n} {fromWitness p}","content":"(yes p) → #_ m {n} {fromWitness p}","searchableContent":"(yes p) → #_ m {n} {fromwitness p}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":115,"title":"(no _) → error "Conversion to Fin not possible!"","content":"(no _) → error "Conversion to Fin not possible!"","searchableContent":"(no _) → error "conversion to fin not possible!""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":116,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":118,"title":"HsTy-LeiosState","content":"HsTy-LeiosState = autoHsType LeiosState","searchableContent":"hsty-leiosstate = autohstype leiosstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":119,"title":"Conv-LeiosState","content":"Conv-LeiosState = autoConvert LeiosState","searchableContent":"conv-leiosstate = autoconvert leiosstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":121,"title":"HsTy-LeiosInput","content":"HsTy-LeiosInput = autoHsType LeiosInput ⊣ onConstructors (prefix "I_" ∘ dropDash)","searchableContent":"hsty-leiosinput = autohstype leiosinput ⊣ onconstructors (prefix "i_" ∘ dropdash)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":122,"title":"Conv-LeiosInput","content":"Conv-LeiosInput = autoConvert LeiosInput","searchableContent":"conv-leiosinput = autoconvert leiosinput"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":124,"title":"HsTy-LeiosOutput","content":"HsTy-LeiosOutput = autoHsType LeiosOutput ⊣ onConstructors (prefix "O_" ∘ dropDash)","searchableContent":"hsty-leiosoutput = autohstype leiosoutput ⊣ onconstructors (prefix "o_" ∘ dropdash)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":125,"title":"Conv-LeiosOutput","content":"Conv-LeiosOutput = autoConvert LeiosOutput","searchableContent":"conv-leiosoutput = autoconvert leiosoutput"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":127,"title":"open import Class.Computational as C","content":"open import Class.Computational as C","searchableContent":"open import class.computational as c"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":128,"title":"open import Class.Computational22","content":"open import Class.Computational22","searchableContent":"open import class.computational22"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":130,"title":"open Computational22","content":"open Computational22","searchableContent":"open computational22"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":131,"title":"open BaseAbstract","content":"open BaseAbstract","searchableContent":"open baseabstract"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":132,"title":"open FFDAbstract","content":"open FFDAbstract","searchableContent":"open ffdabstract"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":134,"title":"open GenFFD.Header using (ibHeader; ebHeader; vtHeader)","content":"open GenFFD.Header using (ibHeader; ebHeader; vtHeader)","searchableContent":"open genffd.header using (ibheader; ebheader; vtheader)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":135,"title":"open GenFFD.Body using (ibBody)","content":"open GenFFD.Body using (ibBody)","searchableContent":"open genffd.body using (ibbody)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":136,"title":"open FFDState","content":"open FFDState","searchableContent":"open ffdstate"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":138,"title":"open import Leios.Short.Deterministic d-SpecStructure public","content":"open import Leios.Short.Deterministic d-SpecStructure public","searchableContent":"open import leios.short.deterministic d-specstructure public"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":140,"title":"open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)","content":"open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)","searchableContent":"open funtot (completefin numberofparties) (maximalfin numberofparties)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":142,"title":"d-StakeDistribution","content":"d-StakeDistribution : TotalMap (Fin numberOfParties) ℕ","searchableContent":"d-stakedistribution : totalmap (fin numberofparties) ℕ"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":143,"title":"d-StakeDistribution","content":"d-StakeDistribution = Fun⇒TotalMap (const 100000000)","searchableContent":"d-stakedistribution = fun⇒totalmap (const 100000000)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":145,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":146,"title":"Computational-B","content":"Computational-B : Computational22 (BaseAbstract.Functionality._-⟦_/_⟧⇀_ d-BaseFunctionality) String","searchableContent":"computational-b : computational22 (baseabstract.functionality._-⟦_/_⟧⇀_ d-basefunctionality) string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":147,"title":"Computational-B .computeProof s (INIT x)","content":"Computational-B .computeProof s (INIT x) = success ((STAKE d-StakeDistribution , tt) , tt)","searchableContent":"computational-b .computeproof s (init x) = success ((stake d-stakedistribution , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":148,"title":"Computational-B .computeProof s (SUBMIT x)","content":"Computational-B .computeProof s (SUBMIT x) = success ((EMPTY , tt) , tt)","searchableContent":"computational-b .computeproof s (submit x) = success ((empty , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":149,"title":"Computational-B .computeProof s FTCH-LDG","content":"Computational-B .computeProof s FTCH-LDG = success (((BASE-LDG []) , tt) , tt)","searchableContent":"computational-b .computeproof s ftch-ldg = success (((base-ldg []) , tt) , tt)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":150,"title":"Computational-B .completeness _ _ _ _ _ = {!!} -- TODO","content":"Computational-B .completeness _ _ _ _ _ = {!!} -- TODO: Completeness proof","searchableContent":"computational-b .completeness _ _ _ _ _ = {!!} -- todo: completeness proof"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":152,"title":"Computational-FFD","content":"Computational-FFD : Computational22 (FFDAbstract.Functionality._-⟦_/_⟧⇀_ d-FFDFunctionality) String","searchableContent":"computational-ffd : computational22 (ffdabstract.functionality._-⟦_/_⟧⇀_ d-ffdfunctionality) string"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":153,"title":"Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b)))","content":"Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b))) = success ((SendRes , record s {outIBs = record {header = h; body = b} ∷ outIBs s}) , SendIB)","searchableContent":"computational-ffd .computeproof s (send (ibheader h) (just (ibbody b))) = success ((sendres , record s {outibs = record {header = h; body = b} ∷ outibs s}) , sendib)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":154,"title":"Computational-FFD .computeProof s (Send (ebHeader h) nothing)","content":"Computational-FFD .computeProof s (Send (ebHeader h) nothing) = success ((SendRes , record s {outEBs = h ∷ outEBs s}) , SendEB)","searchableContent":"computational-ffd .computeproof s (send (ebheader h) nothing) = success ((sendres , record s {outebs = h ∷ outebs s}) , sendeb)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":155,"title":"Computational-FFD .computeProof s (Send (vtHeader h) nothing)","content":"Computational-FFD .computeProof s (Send (vtHeader h) nothing) = success ((SendRes , record s {outVTs = h ∷ outVTs s}) , SendVS)","searchableContent":"computational-ffd .computeproof s (send (vtheader h) nothing) = success ((sendres , record s {outvts = h ∷ outvts s}) , sendvs)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":156,"title":"Computational-FFD .computeProof s Fetch","content":"Computational-FFD .computeProof s Fetch = success ((FetchRes (flushIns s) , record s {inIBs = []; inEBs = []; inVTs = []}) , Fetch)","searchableContent":"computational-ffd .computeproof s fetch = success ((fetchres (flushins s) , record s {inibs = []; inebs = []; invts = []}) , fetch)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":158,"title":"Computational-FFD .computeProof _ _","content":"Computational-FFD .computeProof _ _ = failure "FFD error"","searchableContent":"computational-ffd .computeproof _ _ = failure "ffd error""},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":159,"title":"Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO","content":"Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO:Completeness proof","searchableContent":"computational-ffd .completeness _ _ _ _ _ = {!!} -- todo:completeness proof"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":161,"title":"stepHs","content":"stepHs : HsType (LeiosState → LeiosInput → C.ComputationResult String (LeiosOutput × LeiosState))","searchableContent":"stephs : hstype (leiosstate → leiosinput → c.computationresult string (leiosoutput × leiosstate))"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":162,"title":"stepHs","content":"stepHs = to (compute Computational--⟦/⟧⇀)","searchableContent":"stephs = to (compute computational--⟦/⟧⇀)"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":164,"title":"{-# COMPILE GHC stepHs as step #-}","content":"{-# COMPILE GHC stepHs as step #-}","searchableContent":"{-# compile ghc stephs as step #-}"},{"moduleName":"Leios.Foreign.Types","path":"Leios.Foreign.Types.html","group":"Leios","lineNumber":165,"title":"-}","content":"-}","searchableContent":"-}"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":2,"title":"Leios.Foreign.Util module Leios.Foreign.Util where","content":"Leios.Foreign.Util module Leios.Foreign.Util where","searchableContent":"leios.foreign.util module leios.foreign.util where"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":6,"title":"postulate","content":"postulate","searchableContent":"postulate"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":7,"title":"error","content":"error : { A : Set } → String → A","searchableContent":"error : { a : set } → string → a"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":8,"title":"{-# FOREIGN GHC import Data.Text #-}","content":"{-# FOREIGN GHC import Data.Text #-}","searchableContent":"{-# foreign ghc import data.text #-}"},{"moduleName":"Leios.Foreign.Util","path":"Leios.Foreign.Util.html","group":"Leios","lineNumber":9,"title":"{-# COMPILE GHC error","content":"{-# COMPILE GHC error = \\ _ s -> error (unpack s) #-}","searchableContent":"{-# compile ghc error = \\ _ s -> error (unpack s) #-}"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":2,"title":"Leios.KeyRegistration {-# OPTIONS --safe #-}","content":"Leios.KeyRegistration {-# OPTIONS --safe #-}","searchableContent":"leios.keyregistration {-# options --safe #-}"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","content":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":6,"title":"open import Leios.VRF","content":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":8,"title":"module Leios.KeyRegistration ( a","content":"module Leios.KeyRegistration ( a : LeiosAbstract ) ( open LeiosAbstract a )","searchableContent":"module leios.keyregistration ( a : leiosabstract ) ( open leiosabstract a )"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":9,"title":"( vrf","content":"( vrf : LeiosVRF a ) ( let open LeiosVRF vrf ) where","searchableContent":"( vrf : leiosvrf a ) ( let open leiosvrf vrf ) where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":11,"title":"record KeyRegistrationAbstract","content":"record KeyRegistrationAbstract : Type₁ where","searchableContent":"record keyregistrationabstract : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":13,"title":"data Input","content":"data Input : Type₁ where","searchableContent":"data input : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":14,"title":"INIT","content":"INIT : PubKey → PubKey → PubKey → Input","searchableContent":"init : pubkey → pubkey → pubkey → input"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":16,"title":"data Output","content":"data Output : Type where","searchableContent":"data output : type where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":17,"title":"PUBKEYS","content":"PUBKEYS : List PubKey → Output","searchableContent":"pubkeys : list pubkey → output"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":19,"title":"record Functionality","content":"record Functionality : Type₁ where","searchableContent":"record functionality : type₁ where"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":20,"title":"field State","content":"field State : Type","searchableContent":"field state : type"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":21,"title":"_-⟦_/_⟧⇀_","content":"_-⟦_/_⟧⇀_ : State → Input → Output → State → Type","searchableContent":"_-⟦_/_⟧⇀_ : state → input → output → state → type"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":23,"title":"open Input public","content":"open Input public","searchableContent":"open input public"},{"moduleName":"Leios.KeyRegistration","path":"Leios.KeyRegistration.html","group":"Leios","lineNumber":24,"title":"open Output public","content":"open Output public","searchableContent":"open output public"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":2,"title":"Leios.Network module Leios.Network where","content":"Leios.Network module Leios.Network where","searchableContent":"leios.network module leios.network where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":4,"title":"open import abstract-set-theory.Prelude hiding ( _∘_ ; _⊗_ )","content":"open import abstract-set-theory.Prelude hiding ( _∘_ ; _⊗_ )","searchableContent":"open import abstract-set-theory.prelude hiding ( _∘_ ; _⊗_ )"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":5,"title":"open import abstract-set-theory.FiniteSetTheory using ( ℙ_ ; _∈_ ; _∪_ ; ❴_❵ ; _∉_ )","content":"open import abstract-set-theory.FiniteSetTheory using ( ℙ_ ; _∈_ ; _∪_ ; ❴_❵ ; _∉_ )","searchableContent":"open import abstract-set-theory.finitesettheory using ( ℙ_ ; _∈_ ; _∪_ ; ❴_❵ ; _∉_ )"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":7,"title":"open import CategoricalCrypto","content":"open import CategoricalCrypto","searchableContent":"open import categoricalcrypto"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":9,"title":"record Abstract","content":"record Abstract : Set₁ where","searchableContent":"record abstract : set₁ where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":10,"title":"field Header Body ID","content":"field Header Body ID : Set","searchableContent":"field header body id : set"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":11,"title":"match","content":"match : Header → Body → Set","searchableContent":"match : header → body → set"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":12,"title":"msgID","content":"msgID : Header → ID","searchableContent":"msgid : header → id"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":14,"title":"module Broadcast ( M Peer","content":"module Broadcast ( M Peer : Set ) where","searchableContent":"module broadcast ( m peer : set ) where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":15,"title":"open Channel","content":"open Channel","searchableContent":"open channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":17,"title":"C","content":"C : Channel","searchableContent":"c : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":18,"title":"C . P","content":"C . P = Peer","searchableContent":"c . p = peer"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":19,"title":"C . rcvType _","content":"C . rcvType _ = Peer × M","searchableContent":"c . rcvtype _ = peer × m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":20,"title":"C . sndType _","content":"C . sndType _ = M","searchableContent":"c . sndtype _ = m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":22,"title":"postulate Functionality","content":"postulate Functionality : Machine I C","searchableContent":"postulate functionality : machine i c"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":24,"title":"Single","content":"Single : Channel","searchableContent":"single : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":25,"title":"Single . P","content":"Single . P = ⊤","searchableContent":"single . p = ⊤"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":26,"title":"Single . rcvType _","content":"Single . rcvType _ = Peer × M","searchableContent":"single . rcvtype _ = peer × m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":27,"title":"Single . sndType _","content":"Single . sndType _ = M","searchableContent":"single . sndtype _ = m"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":29,"title":"postulate SingleFunctionality","content":"postulate SingleFunctionality : Machine I Single","searchableContent":"postulate singlefunctionality : machine i single"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":31,"title":"-- connectWithBroadcast","content":"-- connectWithBroadcast : ∀ {A} → (Peer → Machine Single A) → Machine I A","searchableContent":"-- connectwithbroadcast : ∀ {a} → (peer → machine single a) → machine i a"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":32,"title":"-- connectWithBroadcast","content":"-- connectWithBroadcast = {!!}","searchableContent":"-- connectwithbroadcast = {!!}"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":34,"title":"module HeaderDiffusion ( a","content":"module HeaderDiffusion ( a : Abstract ) ( Peer : Set ) ( self : Peer ) where","searchableContent":"module headerdiffusion ( a : abstract ) ( peer : set ) ( self : peer ) where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":35,"title":"open Channel","content":"open Channel","searchableContent":"open channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":36,"title":"open Abstract a","content":"open Abstract a","searchableContent":"open abstract a"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":37,"title":"module B","content":"module B = Broadcast Header Peer","searchableContent":"module b = broadcast header peer"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":39,"title":"data Port","content":"data Port : Set where","searchableContent":"data port : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":40,"title":"Send","content":"Send : Port -- we want to send a header","searchableContent":"send : port -- we want to send a header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":41,"title":"Forward","content":"Forward : Port -- we want to forward a header","searchableContent":"forward : port -- we want to forward a header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":43,"title":"C","content":"C : Channel","searchableContent":"c : channel"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":44,"title":"C . P","content":"C . P = Port","searchableContent":"c . p = port"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":45,"title":"C . sndType _","content":"C . sndType _ = Header","searchableContent":"c . sndtype _ = header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":46,"title":"C . rcvType Forward","content":"C . rcvType Forward = Header","searchableContent":"c . rcvtype forward = header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":47,"title":"C . rcvType Send","content":"C . rcvType Send = ⊥","searchableContent":"c . rcvtype send = ⊥"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":49,"title":"data Input","content":"data Input : Set where","searchableContent":"data input : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":50,"title":"S","content":"S : Header → Input","searchableContent":"s : header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":51,"title":"F","content":"F : Header → Input","searchableContent":"f : header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":52,"title":"R","content":"R : Peer → Header → Input","searchableContent":"r : peer → header → input"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":54,"title":"data Output","content":"data Output : Set where","searchableContent":"data output : set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":55,"title":"Verify","content":"Verify : Header → Output","searchableContent":"verify : header → output"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":57,"title":"private variable","content":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":58,"title":"h","content":"h : Header","searchableContent":"h : header"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":59,"title":"s","content":"s : ℙ ID","searchableContent":"s : ℙ id"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":61,"title":"data Step","content":"data Step : ∃ ( rcvType ( B.Single ⊗ C ᵀ )) → ℙ ID → ℙ ID × Maybe ( ∃ ( sndType ( B.Single ⊗ C ᵀ ))) → Set where","searchableContent":"data step : ∃ ( rcvtype ( b.single ⊗ c ᵀ )) → ℙ id → ℙ id × maybe ( ∃ ( sndtype ( b.single ⊗ c ᵀ ))) → set where"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":62,"title":"Init","content":"Init : Step ( inj₂ Send , h ) s ( s ∪ ❴ msgID h ❵ , just ( inj₁ _ , h ))","searchableContent":"init : step ( inj₂ send , h ) s ( s ∪ ❴ msgid h ❵ , just ( inj₁ _ , h ))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":63,"title":"Receive1","content":"Receive1 : ∀ { p } → Step ( inj₁ _ , p , h ) s ( s , just ( inj₂ Forward , h ))","searchableContent":"receive1 : ∀ { p } → step ( inj₁ _ , p , h ) s ( s , just ( inj₂ forward , h ))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":64,"title":"Receive2","content":"Receive2 : msgID h ∉ s → Step ( inj₂ Forward , h ) s ( s ∪ ❴ msgID h ❵ , just ( inj₁ _ , h ))","searchableContent":"receive2 : msgid h ∉ s → step ( inj₂ forward , h ) s ( s ∪ ❴ msgid h ❵ , just ( inj₁ _ , h ))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":65,"title":"Receive2'","content":"Receive2' : msgID h ∈ s → Step ( inj₂ Forward , h ) s ( s , nothing )","searchableContent":"receive2' : msgid h ∈ s → step ( inj₂ forward , h ) s ( s , nothing )"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":67,"title":"step","content":"step : ∃ ( rcvType ( B.Single ⊗ C ᵀ )) → ∃ ( sndType ( B.Single ⊗ C ᵀ ))","searchableContent":"step : ∃ ( rcvtype ( b.single ⊗ c ᵀ )) → ∃ ( sndtype ( b.single ⊗ c ᵀ ))"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":68,"title":"step ( inj₁ _ , _ , h )","content":"step ( inj₁ _ , _ , h ) = ( inj₂ Forward , h )","searchableContent":"step ( inj₁ _ , _ , h ) = ( inj₂ forward , h )"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":69,"title":"step ( inj₂ Forward , h )","content":"step ( inj₂ Forward , h ) = ( inj₁ _ , h )","searchableContent":"step ( inj₂ forward , h ) = ( inj₁ _ , h )"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":70,"title":"step ( inj₂ Send , h )","content":"step ( inj₂ Send , h ) = ( inj₁ _ , h )","searchableContent":"step ( inj₂ send , h ) = ( inj₁ _ , h )"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":72,"title":"Functionality","content":"Functionality : Machine I C","searchableContent":"functionality : machine i c"},{"moduleName":"Leios.Network","path":"Leios.Network.html","group":"Leios","lineNumber":73,"title":"Functionality","content":"Functionality = MkMachine' Step ∘ B.SingleFunctionality","searchableContent":"functionality = mkmachine' step ∘ b.singlefunctionality"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":2,"title":"Leios.Prelude {-# OPTIONS --safe #-}","content":"Leios.Prelude {-# OPTIONS --safe #-}","searchableContent":"leios.prelude {-# options --safe #-}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":4,"title":"module Leios.Prelude where","content":"module Leios.Prelude where","searchableContent":"module leios.prelude where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":6,"title":"open import abstract-set-theory.FiniteSetTheory public","content":"open import abstract-set-theory.FiniteSetTheory public","searchableContent":"open import abstract-set-theory.finitesettheory public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":7,"title":"open import abstract-set-theory.Prelude public","content":"open import abstract-set-theory.Prelude public","searchableContent":"open import abstract-set-theory.prelude public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":8,"title":"open import Data.List using ( upTo )","content":"open import Data.List using ( upTo )","searchableContent":"open import data.list using ( upto )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":10,"title":"open import Class.HasAdd public","content":"open import Class.HasAdd public","searchableContent":"open import class.hasadd public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":11,"title":"open import Class.HasOrder public","content":"open import Class.HasOrder public","searchableContent":"open import class.hasorder public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":12,"title":"open import Class.Hashable public","content":"open import Class.Hashable public","searchableContent":"open import class.hashable public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":13,"title":"open import Prelude.InferenceRules public","content":"open import Prelude.InferenceRules public","searchableContent":"open import prelude.inferencerules public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":15,"title":"module T where","content":"module T where","searchableContent":"module t where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":16,"title":"open import Data.These public","content":"open import Data.These public","searchableContent":"open import data.these public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":17,"title":"open T public using ( These ; this ; that )","content":"open T public using ( These ; this ; that )","searchableContent":"open t public using ( these ; this ; that )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":19,"title":"module L where","content":"module L where","searchableContent":"module l where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":20,"title":"open import Data.List public","content":"open import Data.List public","searchableContent":"open import data.list public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":21,"title":"open L public using ( List ; [] ; _∷_ ; _++_ ; catMaybes ; head ; length ; sum ; and ; or ; any )","content":"open L public using ( List ; [] ; _∷_ ; _++_ ; catMaybes ; head ; length ; sum ; and ; or ; any )","searchableContent":"open l public using ( list ; [] ; _∷_ ; _++_ ; catmaybes ; head ; length ; sum ; and ; or ; any )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":23,"title":"module A where","content":"module A where","searchableContent":"module a where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":24,"title":"open import Data.List.Relation.Unary.Any public","content":"open import Data.List.Relation.Unary.Any public","searchableContent":"open import data.list.relation.unary.any public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":25,"title":"open A public using ( here ; there )","content":"open A public using ( here ; there )","searchableContent":"open a public using ( here ; there )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":27,"title":"module N where","content":"module N where","searchableContent":"module n where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":28,"title":"open import Data.Nat public","content":"open import Data.Nat public","searchableContent":"open import data.nat public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":29,"title":"open import Data.Nat.Properties public","content":"open import Data.Nat.Properties public","searchableContent":"open import data.nat.properties public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":30,"title":"open N public using ( ℕ ; zero ; suc )","content":"open N public using ( ℕ ; zero ; suc )","searchableContent":"open n public using ( ℕ ; zero ; suc )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":32,"title":"module F where","content":"module F where","searchableContent":"module f where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":33,"title":"open import Data.Fin public","content":"open import Data.Fin public","searchableContent":"open import data.fin public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":34,"title":"open import Data.Fin.Properties public","content":"open import Data.Fin.Properties public","searchableContent":"open import data.fin.properties public"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":35,"title":"open F public using ( Fin ; toℕ ; #_ ) renaming ( zero to fzero ; suc to fsuc )","content":"open F public using ( Fin ; toℕ ; #_ ) renaming ( zero to fzero ; suc to fsuc )","searchableContent":"open f public using ( fin ; toℕ ; #_ ) renaming ( zero to fzero ; suc to fsuc )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":37,"title":"fromTo","content":"fromTo : ℕ → ℕ → List ℕ","searchableContent":"fromto : ℕ → ℕ → list ℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":38,"title":"fromTo m n","content":"fromTo m n = map ( _+ m ) ( upTo ( n ∸ m ))","searchableContent":"fromto m n = map ( _+ m ) ( upto ( n ∸ m ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":40,"title":"slice","content":"slice : ( L : ℕ ) → ⦃ NonZero L ⦄ → ℕ → ℕ → ℙ ℕ","searchableContent":"slice : ( l : ℕ ) → ⦃ nonzero l ⦄ → ℕ → ℕ → ℙ ℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":41,"title":"slice L s x","content":"slice L s x = fromList ( fromTo s' ( s' + ( L ∸ 1 )))","searchableContent":"slice l s x = fromlist ( fromto s' ( s' + ( l ∸ 1 )))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":42,"title":"where s'","content":"where s' = (( s / L ) ∸ x ) * L -- equivalent to the formula in the paper","searchableContent":"where s' = (( s / l ) ∸ x ) * l -- equivalent to the formula in the paper"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":44,"title":"filter","content":"filter : { A : Set } → ( P : A → Type ) ⦃ _ : P ⁇¹ ⦄ → List A → List A","searchableContent":"filter : { a : set } → ( p : a → type ) ⦃ _ : p ⁇¹ ⦄ → list a → list a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":45,"title":"filter P","content":"filter P = L.filter ¿ P ¿¹","searchableContent":"filter p = l.filter ¿ p ¿¹"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":47,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":48,"title":"IsSet-List","content":"IsSet-List : { A : Set } → IsSet ( List A ) A","searchableContent":"isset-list : { a : set } → isset ( list a ) a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":49,"title":"IsSet-List . toSet A","content":"IsSet-List . toSet A = fromList A","searchableContent":"isset-list . toset a = fromlist a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":51,"title":"open import Data.List.Relation.Unary.Any","content":"open import Data.List.Relation.Unary.Any","searchableContent":"open import data.list.relation.unary.any"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":52,"title":"open Properties","content":"open Properties","searchableContent":"open properties"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":54,"title":"finite⇒A≡∅⊎∃a∈A","content":"finite⇒A≡∅⊎∃a∈A : { X : Type } → { A : ℙ X } → finite A → ( A ≡ᵉ ∅ ) ⊎ Σ[ a ∈ X ] a ∈ A","searchableContent":"finite⇒a≡∅⊎∃a∈a : { x : type } → { a : ℙ x } → finite a → ( a ≡ᵉ ∅ ) ⊎ σ[ a ∈ x ] a ∈ a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":55,"title":"finite⇒A≡∅⊎∃a∈A ( [] , h )","content":"finite⇒A≡∅⊎∃a∈A ( [] , h ) = inj₁ ( ∅-least (λ a∈A → ⊥-elim ( case Equivalence.to h a∈A of λ ())))","searchableContent":"finite⇒a≡∅⊎∃a∈a ( [] , h ) = inj₁ ( ∅-least (λ a∈a → ⊥-elim ( case equivalence.to h a∈a of λ ())))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":56,"title":"finite⇒A≡∅⊎∃a∈A ( x ∷ _ , h )","content":"finite⇒A≡∅⊎∃a∈A ( x ∷ _ , h ) = inj₂ ( x , Equivalence.from h ( here refl ))","searchableContent":"finite⇒a≡∅⊎∃a∈a ( x ∷ _ , h ) = inj₂ ( x , equivalence.from h ( here refl ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":58,"title":"completeFin","content":"completeFin : ∀ ( n : ℕ ) → ℙ ( Fin n )","searchableContent":"completefin : ∀ ( n : ℕ ) → ℙ ( fin n )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":59,"title":"completeFin zero","content":"completeFin zero = ∅","searchableContent":"completefin zero = ∅"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":60,"title":"completeFin ( ℕ.suc n )","content":"completeFin ( ℕ.suc n ) = singleton ( F.fromℕ n ) ∪ mapˢ F.inject₁ ( completeFin n )","searchableContent":"completefin ( ℕ.suc n ) = singleton ( f.fromℕ n ) ∪ mapˢ f.inject₁ ( completefin n )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":62,"title":"m≤n∧n≤m⇒m≡n","content":"m≤n∧n≤m⇒m≡n : ∀ { n m : ℕ } → n N.≤ m → m N.≤ n → m ≡ n","searchableContent":"m≤n∧n≤m⇒m≡n : ∀ { n m : ℕ } → n n.≤ m → m n.≤ n → m ≡ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":63,"title":"m≤n∧n≤m⇒m≡n z≤n z≤n","content":"m≤n∧n≤m⇒m≡n z≤n z≤n = refl","searchableContent":"m≤n∧n≤m⇒m≡n z≤n z≤n = refl"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":64,"title":"m≤n∧n≤m⇒m≡n ( s≤s n≤m ) ( s≤s m≤n )","content":"m≤n∧n≤m⇒m≡n ( s≤s n≤m ) ( s≤s m≤n ) = cong N.suc ( m≤n∧n≤m⇒m≡n n≤m m≤n )","searchableContent":"m≤n∧n≤m⇒m≡n ( s≤s n≤m ) ( s≤s m≤n ) = cong n.suc ( m≤n∧n≤m⇒m≡n n≤m m≤n )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":66,"title":"toℕ-fromℕ","content":"toℕ-fromℕ : ∀ { n } { a : Fin ( N.suc n )} → toℕ a ≡ n → a ≡ F.fromℕ n","searchableContent":"toℕ-fromℕ : ∀ { n } { a : fin ( n.suc n )} → toℕ a ≡ n → a ≡ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":67,"title":"toℕ-fromℕ { zero } { fzero } x","content":"toℕ-fromℕ { zero } { fzero } x = refl","searchableContent":"toℕ-fromℕ { zero } { fzero } x = refl"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":68,"title":"toℕ-fromℕ { N.suc n } { fsuc a } x","content":"toℕ-fromℕ { N.suc n } { fsuc a } x = cong fsuc ( toℕ-fromℕ { n } { a } ( N.suc-injective x ))","searchableContent":"toℕ-fromℕ { n.suc n } { fsuc a } x = cong fsuc ( toℕ-fromℕ { n } { a } ( n.suc-injective x ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":70,"title":"open Equivalence","content":"open Equivalence","searchableContent":"open equivalence"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":72,"title":"maximalFin","content":"maximalFin : ∀ ( n : ℕ ) → isMaximal ( completeFin n )","searchableContent":"maximalfin : ∀ ( n : ℕ ) → ismaximal ( completefin n )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":73,"title":"maximalFin ( ℕ.suc n ) { a } with toℕ a N.<? n","content":"maximalFin ( ℕ.suc n ) { a } with toℕ a N.<? n","searchableContent":"maximalfin ( ℕ.suc n ) { a } with toℕ a n.<? n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":74,"title":"... | yes p","content":"... | yes p =","searchableContent":"... | yes p ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":75,"title":"let n≢toℕ","content":"let n≢toℕ = ≢-sym ( N.<⇒≢ p )","searchableContent":"let n≢toℕ = ≢-sym ( n.<⇒≢ p )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":76,"title":"fn","content":"fn = F.lower₁ a n≢toℕ","searchableContent":"fn = f.lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":77,"title":"fn≡a","content":"fn≡a = F.inject₁-lower₁ a n≢toℕ","searchableContent":"fn≡a = f.inject₁-lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":78,"title":"in ( to ∈-∪ ) ( inj₂ (( to ∈-map ) ( fn , ( sym fn≡a , maximalFin n ))))","content":"in ( to ∈-∪ ) ( inj₂ (( to ∈-map ) ( fn , ( sym fn≡a , maximalFin n ))))","searchableContent":"in ( to ∈-∪ ) ( inj₂ (( to ∈-map ) ( fn , ( sym fn≡a , maximalfin n ))))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":79,"title":"... | no ¬p with a F.≟ F.fromℕ n","content":"... | no ¬p with a F.≟ F.fromℕ n","searchableContent":"... | no ¬p with a f.≟ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":80,"title":"... | yes q","content":"... | yes q = ( to ∈-∪ ) ( inj₁ (( to ∈-singleton ) q ))","searchableContent":"... | yes q = ( to ∈-∪ ) ( inj₁ (( to ∈-singleton ) q ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":81,"title":"... | no ¬q","content":"... | no ¬q =","searchableContent":"... | no ¬q ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":82,"title":"let n≢toℕ","content":"let n≢toℕ = N.≰⇒> ¬p","searchableContent":"let n≢toℕ = n.≰⇒> ¬p"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":83,"title":"a<sucn","content":"a<sucn = F.toℕ<n a","searchableContent":"a<sucn = f.toℕ<n a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":84,"title":"in ⊥-elim $ ( ¬q ∘ toℕ-fromℕ ) ( N.suc-injective ( m≤n∧n≤m⇒m≡n n≢toℕ a<sucn ))","content":"in ⊥-elim $ ( ¬q ∘ toℕ-fromℕ ) ( N.suc-injective ( m≤n∧n≤m⇒m≡n n≢toℕ a<sucn ))","searchableContent":"in ⊥-elim $ ( ¬q ∘ toℕ-fromℕ ) ( n.suc-injective ( m≤n∧n≤m⇒m≡n n≢toℕ a<sucn ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":86,"title":"record Listable ( A","content":"record Listable ( A : Type ) : Type where","searchableContent":"record listable ( a : type ) : type where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":87,"title":"field","content":"field","searchableContent":"field"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":88,"title":"listing","content":"listing : ℙ A","searchableContent":"listing : ℙ a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":89,"title":"complete","content":"complete : ∀ { a : A } → a ∈ listing","searchableContent":"complete : ∀ { a : a } → a ∈ listing"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":91,"title":"totalDec","content":"totalDec : ∀ { A B : Type } → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → { R : Rel A B } → Dec ( total R )","searchableContent":"totaldec : ∀ { a b : type } → ⦃ deceq a ⦄ → ⦃ listable a ⦄ → { r : rel a b } → dec ( total r )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":92,"title":"totalDec { A } { B } { R } with all? ( _∈? dom R )","content":"totalDec { A } { B } { R } with all? ( _∈? dom R )","searchableContent":"totaldec { a } { b } { r } with all? ( _∈? dom r )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":93,"title":"... | yes p","content":"... | yes p = yes λ { a } → p { a } (( Listable.complete it ) { a })","searchableContent":"... | yes p = yes λ { a } → p { a } (( listable.complete it ) { a })"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":94,"title":"... | no ¬p","content":"... | no ¬p = no λ x → ¬p λ { a } _ → x { a }","searchableContent":"... | no ¬p = no λ x → ¬p λ { a } _ → x { a }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":96,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":97,"title":"total?","content":"total? : ∀ { A B : Type } → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → { R : Rel A B } → ({ a : A } → a ∈ dom R ) ⁇","searchableContent":"total? : ∀ { a b : type } → ⦃ deceq a ⦄ → ⦃ listable a ⦄ → { r : rel a b } → ({ a : a } → a ∈ dom r ) ⁇"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":98,"title":"total?","content":"total? = ⁇ totalDec","searchableContent":"total? = ⁇ totaldec"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":100,"title":"Listable-Fin","content":"Listable-Fin : ∀ { n } → Listable ( Fin n )","searchableContent":"listable-fin : ∀ { n } → listable ( fin n )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":101,"title":"Listable-Fin { zero }","content":"Listable-Fin { zero } = record { listing = ∅ ; complete = λ { a } → ⊥-elim $ ( Inverse.to F.0↔⊥ ) a }","searchableContent":"listable-fin { zero } = record { listing = ∅ ; complete = λ { a } → ⊥-elim $ ( inverse.to f.0↔⊥ ) a }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":102,"title":"Listable-Fin { suc n }","content":"Listable-Fin { suc n } =","searchableContent":"listable-fin { suc n } ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":103,"title":"let record { listing","content":"let record { listing = l ; complete = c } = Listable-Fin { n }","searchableContent":"let record { listing = l ; complete = c } = listable-fin { n }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":104,"title":"in record","content":"in record","searchableContent":"in record"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":105,"title":"{ listing","content":"{ listing = singleton ( F.fromℕ n ) ∪ mapˢ F.inject₁ l","searchableContent":"{ listing = singleton ( f.fromℕ n ) ∪ mapˢ f.inject₁ l"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":106,"title":"; complete","content":"; complete = complete","searchableContent":"; complete = complete"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":107,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":108,"title":"where","content":"where","searchableContent":"where"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":109,"title":"complete","content":"complete : ∀ { a } → a ∈ singleton ( F.fromℕ n ) ∪ mapˢ F.inject₁ ( let record { listing = l } = Listable-Fin { n } in l )","searchableContent":"complete : ∀ { a } → a ∈ singleton ( f.fromℕ n ) ∪ mapˢ f.inject₁ ( let record { listing = l } = listable-fin { n } in l )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":110,"title":"complete { a } with F.toℕ a N.<? n","content":"complete { a } with F.toℕ a N.<? n","searchableContent":"complete { a } with f.toℕ a n.<? n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":111,"title":"... | yes p","content":"... | yes p =","searchableContent":"... | yes p ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":112,"title":"let record { listing","content":"let record { listing = l ; complete = c } = Listable-Fin { n }","searchableContent":"let record { listing = l ; complete = c } = listable-fin { n }"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":113,"title":"n≢toℕ","content":"n≢toℕ = ≢-sym ( N.<⇒≢ p )","searchableContent":"n≢toℕ = ≢-sym ( n.<⇒≢ p )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":114,"title":"fn","content":"fn = F.lower₁ a n≢toℕ","searchableContent":"fn = f.lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":115,"title":"fn≡a","content":"fn≡a = F.inject₁-lower₁ a n≢toℕ","searchableContent":"fn≡a = f.inject₁-lower₁ a n≢toℕ"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":116,"title":"in ( Equivalence.to ∈-∪ ) ( inj₂ (( Equivalence.to ∈-map ) ( fn , ( sym fn≡a , c ))))","content":"in ( Equivalence.to ∈-∪ ) ( inj₂ (( Equivalence.to ∈-map ) ( fn , ( sym fn≡a , c ))))","searchableContent":"in ( equivalence.to ∈-∪ ) ( inj₂ (( equivalence.to ∈-map ) ( fn , ( sym fn≡a , c ))))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":117,"title":"... | no ¬p with a F.≟ F.fromℕ n","content":"... | no ¬p with a F.≟ F.fromℕ n","searchableContent":"... | no ¬p with a f.≟ f.fromℕ n"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":118,"title":"... | yes q","content":"... | yes q = ( Equivalence.to ∈-∪ ) ( inj₁ (( Equivalence.to ∈-singleton ) q ))","searchableContent":"... | yes q = ( equivalence.to ∈-∪ ) ( inj₁ (( equivalence.to ∈-singleton ) q ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":119,"title":"... | no ¬q","content":"... | no ¬q =","searchableContent":"... | no ¬q ="},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":120,"title":"let n≢toℕ","content":"let n≢toℕ = N.≰⇒> ¬p","searchableContent":"let n≢toℕ = n.≰⇒> ¬p"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":121,"title":"a<sucn","content":"a<sucn = F.toℕ<n a","searchableContent":"a<sucn = f.toℕ<n a"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":122,"title":"in ⊥-elim $ ( ¬q ∘ toℕ-fromℕ ) ( N.suc-injective ( m≤n∧n≤m⇒m≡n n≢toℕ a<sucn ))","content":"in ⊥-elim $ ( ¬q ∘ toℕ-fromℕ ) ( N.suc-injective ( m≤n∧n≤m⇒m≡n n≢toℕ a<sucn ))","searchableContent":"in ⊥-elim $ ( ¬q ∘ toℕ-fromℕ ) ( n.suc-injective ( m≤n∧n≤m⇒m≡n n≢toℕ a<sucn ))"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":124,"title":"completeFinL","content":"completeFinL : ∀ ( n : ℕ ) → List ( Fin n )","searchableContent":"completefinl : ∀ ( n : ℕ ) → list ( fin n )"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":125,"title":"completeFinL zero","content":"completeFinL zero = []","searchableContent":"completefinl zero = []"},{"moduleName":"Leios.Prelude","path":"Leios.Prelude.html","group":"Leios","lineNumber":126,"title":"completeFinL ( ℕ.suc n )","content":"completeFinL ( ℕ.suc n ) = F.fromℕ n ∷ L.map F.inject₁ ( completeFinL n )","searchableContent":"completefinl ( ℕ.suc n ) = f.fromℕ n ∷ l.map f.inject₁ ( completefinl n )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":2,"title":"Leios.Protocol {-# OPTIONS --safe #-}","content":"Leios.Protocol {-# OPTIONS --safe #-}","searchableContent":"leios.protocol {-# options --safe #-}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding ( id )","content":"open import Leios.Prelude hiding ( id )","searchableContent":"open import leios.prelude hiding ( id )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":5,"title":"open import Leios.FFD","content":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":6,"title":"open import Leios.SpecStructure","content":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":8,"title":"module Leios.Protocol { n } ( ⋯","content":"module Leios.Protocol { n } ( ⋯ : SpecStructure n ) ( let open SpecStructure ⋯ ) ( SlotUpkeep : Type ) where","searchableContent":"module leios.protocol { n } ( ⋯ : specstructure n ) ( let open specstructure ⋯ ) ( slotupkeep : type ) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":10,"title":"open BaseAbstract B' using ( Cert ; V-chkCerts ; VTy ; initSlot )","content":"open BaseAbstract B' using ( Cert ; V-chkCerts ; VTy ; initSlot )","searchableContent":"open baseabstract b' using ( cert ; v-chkcerts ; vty ; initslot )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":11,"title":"open GenFFD","content":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":13,"title":"-- High level structure","content":"-- High level structure:","searchableContent":"-- high level structure:"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":16,"title":"-- (simple) Leios","content":"-- (simple) Leios","searchableContent":"-- (simple) leios"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":17,"title":"-- / |","content":"-- / |","searchableContent":"-- / |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":18,"title":"-- +-------------------------------------+ |","content":"-- +-------------------------------------+ |","searchableContent":"-- +-------------------------------------+ |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":19,"title":"-- | Header Diffusion Body Diffusion | |","content":"-- | Header Diffusion Body Diffusion | |","searchableContent":"-- | header diffusion body diffusion | |"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":20,"title":"-- +-------------------------------------+ Base Protocol","content":"-- +-------------------------------------+ Base Protocol","searchableContent":"-- +-------------------------------------+ base protocol"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":21,"title":"-- \\ /","content":"-- \\ /","searchableContent":"-- \\ /"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":22,"title":"-- Network","content":"-- Network","searchableContent":"-- network"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":24,"title":"data LeiosInput","content":"data LeiosInput : Type where","searchableContent":"data leiosinput : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":25,"title":"INIT","content":"INIT : VTy → LeiosInput","searchableContent":"init : vty → leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":26,"title":"SUBMIT","content":"SUBMIT : EndorserBlock ⊎ List Tx → LeiosInput","searchableContent":"submit : endorserblock ⊎ list tx → leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":27,"title":"SLOT","content":"SLOT : LeiosInput","searchableContent":"slot : leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":28,"title":"FTCH-LDG","content":"FTCH-LDG : LeiosInput","searchableContent":"ftch-ldg : leiosinput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":30,"title":"data LeiosOutput","content":"data LeiosOutput : Type where","searchableContent":"data leiosoutput : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":31,"title":"FTCH-LDG","content":"FTCH-LDG : List Tx → LeiosOutput","searchableContent":"ftch-ldg : list tx → leiosoutput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":32,"title":"EMPTY","content":"EMPTY : LeiosOutput","searchableContent":"empty : leiosoutput"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":34,"title":"record LeiosState","content":"record LeiosState : Type where","searchableContent":"record leiosstate : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":35,"title":"field V","content":"field V : VTy","searchableContent":"field v : vty"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":36,"title":"SD","content":"SD : StakeDistr","searchableContent":"sd : stakedistr"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":37,"title":"FFDState","content":"FFDState : FFD.State","searchableContent":"ffdstate : ffd.state"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":38,"title":"Ledger","content":"Ledger : List Tx","searchableContent":"ledger : list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":39,"title":"ToPropose","content":"ToPropose : List Tx","searchableContent":"topropose : list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":40,"title":"IBs","content":"IBs : List InputBlock","searchableContent":"ibs : list inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":41,"title":"EBs","content":"EBs : List EndorserBlock","searchableContent":"ebs : list endorserblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":42,"title":"Vs","content":"Vs : List ( List Vote )","searchableContent":"vs : list ( list vote )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":43,"title":"slot","content":"slot : ℕ","searchableContent":"slot : ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":44,"title":"IBHeaders","content":"IBHeaders : List IBHeader","searchableContent":"ibheaders : list ibheader"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":45,"title":"IBBodies","content":"IBBodies : List IBBody","searchableContent":"ibbodies : list ibbody"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":46,"title":"Upkeep","content":"Upkeep : ℙ SlotUpkeep","searchableContent":"upkeep : ℙ slotupkeep"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":47,"title":"BaseState","content":"BaseState : B.State","searchableContent":"basestate : b.state"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":48,"title":"votingState","content":"votingState : VotingState","searchableContent":"votingstate : votingstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":49,"title":"PubKeys","content":"PubKeys : List PubKey","searchableContent":"pubkeys : list pubkey"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":51,"title":"lookupEB","content":"lookupEB : EBRef → Maybe EndorserBlock","searchableContent":"lookupeb : ebref → maybe endorserblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":52,"title":"lookupEB r","content":"lookupEB r = find (λ b → getEBRef b ≟ r ) EBs","searchableContent":"lookupeb r = find (λ b → getebref b ≟ r ) ebs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":54,"title":"lookupIB","content":"lookupIB : IBRef → Maybe InputBlock","searchableContent":"lookupib : ibref → maybe inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":55,"title":"lookupIB r","content":"lookupIB r = find (λ b → getIBRef b ≟ r ) IBs","searchableContent":"lookupib r = find (λ b → getibref b ≟ r ) ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":57,"title":"lookupTxs","content":"lookupTxs : EndorserBlock → List Tx","searchableContent":"lookuptxs : endorserblock → list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":58,"title":"lookupTxs eb","content":"lookupTxs eb = do","searchableContent":"lookuptxs eb = do"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":59,"title":"eb′ ← mapMaybe lookupEB $ ebRefs eb","content":"eb′ ← mapMaybe lookupEB $ ebRefs eb","searchableContent":"eb′ ← mapmaybe lookupeb $ ebrefs eb"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":60,"title":"ib ← mapMaybe lookupIB $ ibRefs eb′","content":"ib ← mapMaybe lookupIB $ ibRefs eb′","searchableContent":"ib ← mapmaybe lookupib $ ibrefs eb′"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":61,"title":"txs $ body ib","content":"txs $ body ib","searchableContent":"txs $ body ib"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":62,"title":"where open EndorserBlockOSig","content":"where open EndorserBlockOSig","searchableContent":"where open endorserblockosig"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":63,"title":"open IBBody","content":"open IBBody","searchableContent":"open ibbody"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":64,"title":"open InputBlock","content":"open InputBlock","searchableContent":"open inputblock"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":66,"title":"constructLedger","content":"constructLedger : List RankingBlock → List Tx","searchableContent":"constructledger : list rankingblock → list tx"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":67,"title":"constructLedger","content":"constructLedger = L.concat ∘ L.map ( T.mergeThese L._++_ ∘ T.map₁ lookupTxs )","searchableContent":"constructledger = l.concat ∘ l.map ( t.mergethese l._++_ ∘ t.map₁ lookuptxs )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":69,"title":"needsUpkeep","content":"needsUpkeep : SlotUpkeep → Set","searchableContent":"needsupkeep : slotupkeep → set"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":70,"title":"needsUpkeep","content":"needsUpkeep = _∉ Upkeep","searchableContent":"needsupkeep = _∉ upkeep"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":72,"title":"Dec-needsUpkeep","content":"Dec-needsUpkeep : ∀ { u : SlotUpkeep } → ⦃ DecEq SlotUpkeep ⦄ → needsUpkeep u ⁇","searchableContent":"dec-needsupkeep : ∀ { u : slotupkeep } → ⦃ deceq slotupkeep ⦄ → needsupkeep u ⁇"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":73,"title":"Dec-needsUpkeep { u } . dec","content":"Dec-needsUpkeep { u } . dec = ¬? ( u ∈? Upkeep )","searchableContent":"dec-needsupkeep { u } . dec = ¬? ( u ∈? upkeep )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":75,"title":"addUpkeep","content":"addUpkeep : LeiosState → SlotUpkeep → LeiosState","searchableContent":"addupkeep : leiosstate → slotupkeep → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":76,"title":"addUpkeep s u","content":"addUpkeep s u = let open LeiosState s in record s { Upkeep = Upkeep ∪ ❴ u ❵ }","searchableContent":"addupkeep s u = let open leiosstate s in record s { upkeep = upkeep ∪ ❴ u ❵ }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":77,"title":"{-# INJECTIVE_FOR_INFERENCE addUpkeep #-}","content":"{-# INJECTIVE_FOR_INFERENCE addUpkeep #-}","searchableContent":"{-# injective_for_inference addupkeep #-}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":79,"title":"initLeiosState","content":"initLeiosState : VTy → StakeDistr → B.State → List PubKey → LeiosState","searchableContent":"initleiosstate : vty → stakedistr → b.state → list pubkey → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":80,"title":"initLeiosState V SD bs pks","content":"initLeiosState V SD bs pks = record","searchableContent":"initleiosstate v sd bs pks = record"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":81,"title":"{ V","content":"{ V = V","searchableContent":"{ v = v"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":82,"title":"; SD","content":"; SD = SD","searchableContent":"; sd = sd"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":83,"title":"; FFDState","content":"; FFDState = FFD.initFFDState","searchableContent":"; ffdstate = ffd.initffdstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":84,"title":"; Ledger","content":"; Ledger = []","searchableContent":"; ledger = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":85,"title":"; ToPropose","content":"; ToPropose = []","searchableContent":"; topropose = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":86,"title":"; IBs","content":"; IBs = []","searchableContent":"; ibs = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":87,"title":"; EBs","content":"; EBs = []","searchableContent":"; ebs = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":88,"title":"; Vs","content":"; Vs = []","searchableContent":"; vs = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":89,"title":"; slot","content":"; slot = initSlot V","searchableContent":"; slot = initslot v"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":90,"title":"; IBHeaders","content":"; IBHeaders = []","searchableContent":"; ibheaders = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":91,"title":"; IBBodies","content":"; IBBodies = []","searchableContent":"; ibbodies = []"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":92,"title":"; Upkeep","content":"; Upkeep = ∅","searchableContent":"; upkeep = ∅"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":93,"title":"; BaseState","content":"; BaseState = bs","searchableContent":"; basestate = bs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":94,"title":"; votingState","content":"; votingState = initVotingState","searchableContent":"; votingstate = initvotingstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":95,"title":"; PubKeys","content":"; PubKeys = pks","searchableContent":"; pubkeys = pks"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":96,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":98,"title":"stake'","content":"stake' : PoolID → LeiosState → ℕ","searchableContent":"stake' : poolid → leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":99,"title":"stake' pid record { SD","content":"stake' pid record { SD = SD } = TotalMap.lookup SD pid","searchableContent":"stake' pid record { sd = sd } = totalmap.lookup sd pid"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":101,"title":"stake''","content":"stake'' : PubKey → LeiosState → ℕ","searchableContent":"stake'' : pubkey → leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":102,"title":"stake'' pk","content":"stake'' pk = stake' ( poolID pk )","searchableContent":"stake'' pk = stake' ( poolid pk )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":104,"title":"stake","content":"stake : LeiosState → ℕ","searchableContent":"stake : leiosstate → ℕ"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":105,"title":"stake","content":"stake = stake' id","searchableContent":"stake = stake' id"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":107,"title":"lookupPubKeyAndStake","content":"lookupPubKeyAndStake : ∀ { B } → ⦃ _ : IsBlock B ⦄ → LeiosState → B → Maybe ( PubKey × ℕ )","searchableContent":"lookuppubkeyandstake : ∀ { b } → ⦃ _ : isblock b ⦄ → leiosstate → b → maybe ( pubkey × ℕ )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":108,"title":"lookupPubKeyAndStake s b","content":"lookupPubKeyAndStake s b =","searchableContent":"lookuppubkeyandstake s b ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":109,"title":"L.head $","content":"L.head $","searchableContent":"l.head $"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":110,"title":"L.map (λ pk → ( pk , stake'' pk s )) $","content":"L.map (λ pk → ( pk , stake'' pk s )) $","searchableContent":"l.map (λ pk → ( pk , stake'' pk s )) $"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":111,"title":"L.filter (λ pk → producerID b ≟ poolID pk ) ( LeiosState.PubKeys s )","content":"L.filter (λ pk → producerID b ≟ poolID pk ) ( LeiosState.PubKeys s )","searchableContent":"l.filter (λ pk → producerid b ≟ poolid pk ) ( leiosstate.pubkeys s )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":113,"title":"module _ ( s","content":"module _ ( s : LeiosState ) where","searchableContent":"module _ ( s : leiosstate ) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":115,"title":"record ibHeaderValid ( h","content":"record ibHeaderValid ( h : IBHeader ) ( pk : PubKey ) ( st : ℕ ) : Type where","searchableContent":"record ibheadervalid ( h : ibheader ) ( pk : pubkey ) ( st : ℕ ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":116,"title":"field lotteryPfValid","content":"field lotteryPfValid : verify pk ( slotNumber h ) st ( lotteryPf h )","searchableContent":"field lotterypfvalid : verify pk ( slotnumber h ) st ( lotterypf h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":117,"title":"signatureValid","content":"signatureValid : verifySig pk ( signature h )","searchableContent":"signaturevalid : verifysig pk ( signature h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":119,"title":"record ibBodyValid ( b","content":"record ibBodyValid ( b : IBBody ) : Type where","searchableContent":"record ibbodyvalid ( b : ibbody ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":121,"title":"ibHeaderValid?","content":"ibHeaderValid? : ( h : IBHeader ) ( pk : PubKey ) ( st : ℕ ) → Dec ( ibHeaderValid h pk st )","searchableContent":"ibheadervalid? : ( h : ibheader ) ( pk : pubkey ) ( st : ℕ ) → dec ( ibheadervalid h pk st )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":122,"title":"ibHeaderValid? h pk st","content":"ibHeaderValid? h pk st","searchableContent":"ibheadervalid? h pk st"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":123,"title":"with verify? pk ( slotNumber h ) st ( lotteryPf h )","content":"with verify? pk ( slotNumber h ) st ( lotteryPf h )","searchableContent":"with verify? pk ( slotnumber h ) st ( lotterypf h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":124,"title":"... | no ¬p","content":"... | no ¬p = no ( ¬p ∘ ibHeaderValid.lotteryPfValid )","searchableContent":"... | no ¬p = no ( ¬p ∘ ibheadervalid.lotterypfvalid )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":125,"title":"... | yes p","content":"... | yes p","searchableContent":"... | yes p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":126,"title":"with verifySig? pk ( signature h )","content":"with verifySig? pk ( signature h )","searchableContent":"with verifysig? pk ( signature h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":127,"title":"... | yes q","content":"... | yes q = yes ( record { lotteryPfValid = p ; signatureValid = q })","searchableContent":"... | yes q = yes ( record { lotterypfvalid = p ; signaturevalid = q })"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":128,"title":"... | no ¬q","content":"... | no ¬q = no ( ¬q ∘ ibHeaderValid.signatureValid )","searchableContent":"... | no ¬q = no ( ¬q ∘ ibheadervalid.signaturevalid )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":130,"title":"ibBodyValid?","content":"ibBodyValid? : ( b : IBBody ) → Dec ( ibBodyValid b )","searchableContent":"ibbodyvalid? : ( b : ibbody ) → dec ( ibbodyvalid b )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":131,"title":"ibBodyValid? _","content":"ibBodyValid? _ = yes record {}","searchableContent":"ibbodyvalid? _ = yes record {}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":133,"title":"ibValid","content":"ibValid : InputBlock → Type","searchableContent":"ibvalid : inputblock → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":134,"title":"ibValid record { header","content":"ibValid record { header = h ; body = b }","searchableContent":"ibvalid record { header = h ; body = b }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":135,"title":"with lookupPubKeyAndStake s h","content":"with lookupPubKeyAndStake s h","searchableContent":"with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":136,"title":"... | just ( pk , pid )","content":"... | just ( pk , pid ) = ibHeaderValid h pk ( stake'' pk s ) × ibBodyValid b","searchableContent":"... | just ( pk , pid ) = ibheadervalid h pk ( stake'' pk s ) × ibbodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":137,"title":"... | nothing","content":"... | nothing = ⊥","searchableContent":"... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":139,"title":"ibValid?","content":"ibValid? : ( ib : InputBlock ) → Dec ( ibValid ib )","searchableContent":"ibvalid? : ( ib : inputblock ) → dec ( ibvalid ib )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":140,"title":"ibValid? record { header","content":"ibValid? record { header = h ; body = b }","searchableContent":"ibvalid? record { header = h ; body = b }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":141,"title":"with lookupPubKeyAndStake s h","content":"with lookupPubKeyAndStake s h","searchableContent":"with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":142,"title":"... | just ( pk , pid )","content":"... | just ( pk , pid ) = ibHeaderValid? h pk ( stake'' pk s ) ×-dec ibBodyValid? b","searchableContent":"... | just ( pk , pid ) = ibheadervalid? h pk ( stake'' pk s ) ×-dec ibbodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":143,"title":"... | nothing","content":"... | nothing = no λ x → x","searchableContent":"... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":145,"title":"record ebValid ( eb","content":"record ebValid ( eb : EndorserBlock ) ( pk : PubKey ) ( st : ℕ ) : Type where","searchableContent":"record ebvalid ( eb : endorserblock ) ( pk : pubkey ) ( st : ℕ ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":146,"title":"field lotteryPfValid","content":"field lotteryPfValid : verify pk ( slotNumber eb ) st ( lotteryPf eb )","searchableContent":"field lotterypfvalid : verify pk ( slotnumber eb ) st ( lotterypf eb )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":147,"title":"signatureValid","content":"signatureValid : verifySig pk ( signature eb )","searchableContent":"signaturevalid : verifysig pk ( signature eb )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":148,"title":"-- TODO","content":"-- TODO","searchableContent":"-- todo"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":149,"title":"-- ibRefsValid","content":"-- ibRefsValid : ?","searchableContent":"-- ibrefsvalid : ?"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":150,"title":"-- ebRefsValid","content":"-- ebRefsValid : ?","searchableContent":"-- ebrefsvalid : ?"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":152,"title":"ebValid?","content":"ebValid? : ( eb : EndorserBlock ) ( pk : PubKey ) ( st : ℕ ) → Dec ( ebValid eb pk st )","searchableContent":"ebvalid? : ( eb : endorserblock ) ( pk : pubkey ) ( st : ℕ ) → dec ( ebvalid eb pk st )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":153,"title":"ebValid? h pk st","content":"ebValid? h pk st","searchableContent":"ebvalid? h pk st"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":154,"title":"with verify? pk ( slotNumber h ) st ( lotteryPf h )","content":"with verify? pk ( slotNumber h ) st ( lotteryPf h )","searchableContent":"with verify? pk ( slotnumber h ) st ( lotterypf h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":155,"title":"... | no ¬p","content":"... | no ¬p = no ( ¬p ∘ ebValid.lotteryPfValid )","searchableContent":"... | no ¬p = no ( ¬p ∘ ebvalid.lotterypfvalid )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":156,"title":"... | yes p","content":"... | yes p","searchableContent":"... | yes p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":157,"title":"with verifySig? pk ( signature h )","content":"with verifySig? pk ( signature h )","searchableContent":"with verifysig? pk ( signature h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":158,"title":"... | yes q","content":"... | yes q = yes ( record { lotteryPfValid = p ; signatureValid = q })","searchableContent":"... | yes q = yes ( record { lotterypfvalid = p ; signaturevalid = q })"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":159,"title":"... | no ¬q","content":"... | no ¬q = no ( ¬q ∘ ebValid.signatureValid )","searchableContent":"... | no ¬q = no ( ¬q ∘ ebvalid.signaturevalid )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":161,"title":"-- TODO","content":"-- TODO","searchableContent":"-- todo"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":162,"title":"record vsValid ( vs","content":"record vsValid ( vs : List Vote ) : Type where","searchableContent":"record vsvalid ( vs : list vote ) : type where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":164,"title":"vsValid?","content":"vsValid? : ( vs : List Vote ) → Dec ( vsValid vs )","searchableContent":"vsvalid? : ( vs : list vote ) → dec ( vsvalid vs )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":165,"title":"vsValid? _","content":"vsValid? _ = yes record {}","searchableContent":"vsvalid? _ = yes record {}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":167,"title":"headerValid","content":"headerValid : Header → Type","searchableContent":"headervalid : header → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":168,"title":"headerValid ( ibHeader h )","content":"headerValid ( ibHeader h )","searchableContent":"headervalid ( ibheader h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":169,"title":"with lookupPubKeyAndStake s h","content":"with lookupPubKeyAndStake s h","searchableContent":"with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":170,"title":"... | just ( pk , pid )","content":"... | just ( pk , pid ) = ibHeaderValid h pk ( stake'' pk s )","searchableContent":"... | just ( pk , pid ) = ibheadervalid h pk ( stake'' pk s )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":171,"title":"... | nothing","content":"... | nothing = ⊥","searchableContent":"... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":172,"title":"headerValid ( ebHeader h )","content":"headerValid ( ebHeader h )","searchableContent":"headervalid ( ebheader h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":173,"title":"with lookupPubKeyAndStake s h","content":"with lookupPubKeyAndStake s h","searchableContent":"with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":174,"title":"... | just ( pk , pid )","content":"... | just ( pk , pid ) = ebValid h pk ( stake'' pk s )","searchableContent":"... | just ( pk , pid ) = ebvalid h pk ( stake'' pk s )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":175,"title":"... | nothing","content":"... | nothing = ⊥","searchableContent":"... | nothing = ⊥"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":176,"title":"headerValid ( vtHeader h )","content":"headerValid ( vtHeader h ) = vsValid h","searchableContent":"headervalid ( vtheader h ) = vsvalid h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":178,"title":"headerValid?","content":"headerValid? : ( h : Header ) → Dec ( headerValid h )","searchableContent":"headervalid? : ( h : header ) → dec ( headervalid h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":179,"title":"headerValid? ( ibHeader h )","content":"headerValid? ( ibHeader h )","searchableContent":"headervalid? ( ibheader h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":180,"title":"with lookupPubKeyAndStake s h","content":"with lookupPubKeyAndStake s h","searchableContent":"with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":181,"title":"... | just ( pk , pid )","content":"... | just ( pk , pid ) = ibHeaderValid? h pk ( stake'' pk s )","searchableContent":"... | just ( pk , pid ) = ibheadervalid? h pk ( stake'' pk s )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":182,"title":"... | nothing","content":"... | nothing = no λ x → x","searchableContent":"... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":183,"title":"headerValid? ( ebHeader h )","content":"headerValid? ( ebHeader h )","searchableContent":"headervalid? ( ebheader h )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":184,"title":"with lookupPubKeyAndStake s h","content":"with lookupPubKeyAndStake s h","searchableContent":"with lookuppubkeyandstake s h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":185,"title":"... | just ( pk , pid )","content":"... | just ( pk , pid ) = ebValid? h pk ( stake'' pk s )","searchableContent":"... | just ( pk , pid ) = ebvalid? h pk ( stake'' pk s )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":186,"title":"... | nothing","content":"... | nothing = no λ x → x","searchableContent":"... | nothing = no λ x → x"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":187,"title":"headerValid? ( vtHeader h )","content":"headerValid? ( vtHeader h ) = vsValid? h","searchableContent":"headervalid? ( vtheader h ) = vsvalid? h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":189,"title":"bodyValid","content":"bodyValid : Body → Type","searchableContent":"bodyvalid : body → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":190,"title":"bodyValid ( ibBody b )","content":"bodyValid ( ibBody b ) = ibBodyValid b","searchableContent":"bodyvalid ( ibbody b ) = ibbodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":192,"title":"bodyValid?","content":"bodyValid? : ( b : Body ) → Dec ( bodyValid b )","searchableContent":"bodyvalid? : ( b : body ) → dec ( bodyvalid b )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":193,"title":"bodyValid? ( ibBody b )","content":"bodyValid? ( ibBody b ) = ibBodyValid? b","searchableContent":"bodyvalid? ( ibbody b ) = ibbodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":195,"title":"isValid","content":"isValid : Header ⊎ Body → Type","searchableContent":"isvalid : header ⊎ body → type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":196,"title":"isValid ( inj₁ h )","content":"isValid ( inj₁ h ) = headerValid h","searchableContent":"isvalid ( inj₁ h ) = headervalid h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":197,"title":"isValid ( inj₂ b )","content":"isValid ( inj₂ b ) = bodyValid b","searchableContent":"isvalid ( inj₂ b ) = bodyvalid b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":199,"title":"isValid?","content":"isValid? : ∀ ( x : Header ⊎ Body ) → Dec ( isValid x )","searchableContent":"isvalid? : ∀ ( x : header ⊎ body ) → dec ( isvalid x )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":200,"title":"isValid? ( inj₁ h )","content":"isValid? ( inj₁ h ) = headerValid? h","searchableContent":"isvalid? ( inj₁ h ) = headervalid? h"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":201,"title":"isValid? ( inj₂ b )","content":"isValid? ( inj₂ b ) = bodyValid? b","searchableContent":"isvalid? ( inj₂ b ) = bodyvalid? b"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":203,"title":"-- some predicates about EBs","content":"-- some predicates about EBs","searchableContent":"-- some predicates about ebs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":204,"title":"module _ ( s","content":"module _ ( s : LeiosState ) ( eb : EndorserBlock ) where","searchableContent":"module _ ( s : leiosstate ) ( eb : endorserblock ) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":205,"title":"open EndorserBlockOSig eb","content":"open EndorserBlockOSig eb","searchableContent":"open endorserblockosig eb"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":206,"title":"open LeiosState s","content":"open LeiosState s","searchableContent":"open leiosstate s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":208,"title":"allIBRefsKnown","content":"allIBRefsKnown : Type","searchableContent":"allibrefsknown : type"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":209,"title":"allIBRefsKnown","content":"allIBRefsKnown = ∀[ ref ∈ fromList ibRefs ] ref ∈ˡ map getIBRef IBs","searchableContent":"allibrefsknown = ∀[ ref ∈ fromlist ibrefs ] ref ∈ˡ map getibref ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":211,"title":"module _ ( s","content":"module _ ( s : LeiosState ) where","searchableContent":"module _ ( s : leiosstate ) where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":213,"title":"open LeiosState s","content":"open LeiosState s","searchableContent":"open leiosstate s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":215,"title":"upd","content":"upd : Header ⊎ Body → LeiosState","searchableContent":"upd : header ⊎ body → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":216,"title":"upd ( inj₁ ( ebHeader eb ))","content":"upd ( inj₁ ( ebHeader eb )) = record s { EBs = eb ∷ EBs }","searchableContent":"upd ( inj₁ ( ebheader eb )) = record s { ebs = eb ∷ ebs }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":217,"title":"upd ( inj₁ ( vtHeader vs ))","content":"upd ( inj₁ ( vtHeader vs )) = record s { Vs = vs ∷ Vs }","searchableContent":"upd ( inj₁ ( vtheader vs )) = record s { vs = vs ∷ vs }"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":218,"title":"upd ( inj₁ ( ibHeader h )) with A.any? ( matchIB? h ) IBBodies","content":"upd ( inj₁ ( ibHeader h )) with A.any? ( matchIB? h ) IBBodies","searchableContent":"upd ( inj₁ ( ibheader h )) with a.any? ( matchib? h ) ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":219,"title":"... | yes p","content":"... | yes p =","searchableContent":"... | yes p ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":220,"title":"record s","content":"record s","searchableContent":"record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":221,"title":"{ IBs","content":"{ IBs = record { header = h ; body = A.lookup p } ∷ IBs","searchableContent":"{ ibs = record { header = h ; body = a.lookup p } ∷ ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":222,"title":"; IBBodies","content":"; IBBodies = IBBodies A.─ p","searchableContent":"; ibbodies = ibbodies a.─ p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":223,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":224,"title":"... | no _","content":"... | no _ =","searchableContent":"... | no _ ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":225,"title":"record s","content":"record s","searchableContent":"record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":226,"title":"{ IBHeaders","content":"{ IBHeaders = h ∷ IBHeaders","searchableContent":"{ ibheaders = h ∷ ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":227,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":228,"title":"upd ( inj₂ ( ibBody b )) with A.any? ( flip matchIB? b ) IBHeaders","content":"upd ( inj₂ ( ibBody b )) with A.any? ( flip matchIB? b ) IBHeaders","searchableContent":"upd ( inj₂ ( ibbody b )) with a.any? ( flip matchib? b ) ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":229,"title":"... | yes p","content":"... | yes p =","searchableContent":"... | yes p ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":230,"title":"record s","content":"record s","searchableContent":"record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":231,"title":"{ IBs","content":"{ IBs = record { header = A.lookup p ; body = b } ∷ IBs","searchableContent":"{ ibs = record { header = a.lookup p ; body = b } ∷ ibs"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":232,"title":"; IBHeaders","content":"; IBHeaders = IBHeaders A.─ p","searchableContent":"; ibheaders = ibheaders a.─ p"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":233,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":234,"title":"... | no _","content":"... | no _ =","searchableContent":"... | no _ ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":235,"title":"record s","content":"record s","searchableContent":"record s"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":236,"title":"{ IBBodies","content":"{ IBBodies = b ∷ IBBodies","searchableContent":"{ ibbodies = b ∷ ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":237,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":239,"title":"module _ { s s' } where","content":"module _ { s s' } where","searchableContent":"module _ { s s' } where"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":240,"title":"open LeiosState s'","content":"open LeiosState s'","searchableContent":"open leiosstate s'"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":242,"title":"upd-preserves-Upkeep","content":"upd-preserves-Upkeep : ∀ { x } → LeiosState.Upkeep s ≡ LeiosState.Upkeep s'","searchableContent":"upd-preserves-upkeep : ∀ { x } → leiosstate.upkeep s ≡ leiosstate.upkeep s'"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":243,"title":"→ LeiosState.Upkeep s ≡ LeiosState.Upkeep ( upd s' x )","content":"→ LeiosState.Upkeep s ≡ LeiosState.Upkeep ( upd s' x )","searchableContent":"→ leiosstate.upkeep s ≡ leiosstate.upkeep ( upd s' x )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":244,"title":"upd-preserves-Upkeep { inj₁ ( ibHeader x )} refl with A.any? ( matchIB? x ) IBBodies","content":"upd-preserves-Upkeep { inj₁ ( ibHeader x )} refl with A.any? ( matchIB? x ) IBBodies","searchableContent":"upd-preserves-upkeep { inj₁ ( ibheader x )} refl with a.any? ( matchib? x ) ibbodies"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":245,"title":"... | yes p","content":"... | yes p = refl","searchableContent":"... | yes p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":246,"title":"... | no ¬p","content":"... | no ¬p = refl","searchableContent":"... | no ¬p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":247,"title":"upd-preserves-Upkeep { inj₁ ( ebHeader x )} refl","content":"upd-preserves-Upkeep { inj₁ ( ebHeader x )} refl = refl","searchableContent":"upd-preserves-upkeep { inj₁ ( ebheader x )} refl = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":248,"title":"upd-preserves-Upkeep { inj₁ ( vtHeader x )} refl","content":"upd-preserves-Upkeep { inj₁ ( vtHeader x )} refl = refl","searchableContent":"upd-preserves-upkeep { inj₁ ( vtheader x )} refl = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":249,"title":"upd-preserves-Upkeep { inj₂ ( ibBody x )} refl with A.any? ( flip matchIB? x ) IBHeaders","content":"upd-preserves-Upkeep { inj₂ ( ibBody x )} refl with A.any? ( flip matchIB? x ) IBHeaders","searchableContent":"upd-preserves-upkeep { inj₂ ( ibbody x )} refl with a.any? ( flip matchib? x ) ibheaders"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":250,"title":"... | yes p","content":"... | yes p = refl","searchableContent":"... | yes p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":251,"title":"... | no ¬p","content":"... | no ¬p = refl","searchableContent":"... | no ¬p = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":253,"title":"infix 25 _↑_","content":"infix 25 _↑_","searchableContent":"infix 25 _↑_"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":254,"title":"_↑_","content":"_↑_ : LeiosState → List ( Header ⊎ Body ) → LeiosState","searchableContent":"_↑_ : leiosstate → list ( header ⊎ body ) → leiosstate"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":255,"title":"_↑_","content":"_↑_ = foldr ( flip upd )","searchableContent":"_↑_ = foldr ( flip upd )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":257,"title":"↑-preserves-Upkeep","content":"↑-preserves-Upkeep : ∀ { s x } → LeiosState.Upkeep s ≡ LeiosState.Upkeep ( s ↑ x )","searchableContent":"↑-preserves-upkeep : ∀ { s x } → leiosstate.upkeep s ≡ leiosstate.upkeep ( s ↑ x )"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":258,"title":"↑-preserves-Upkeep { x","content":"↑-preserves-Upkeep { x = [] } = refl","searchableContent":"↑-preserves-upkeep { x = [] } = refl"},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":259,"title":"↑-preserves-Upkeep { s","content":"↑-preserves-Upkeep { s = s } { x = x ∷ x₁ } =","searchableContent":"↑-preserves-upkeep { s = s } { x = x ∷ x₁ } ="},{"moduleName":"Leios.Protocol","path":"Leios.Protocol.html","group":"Leios","lineNumber":260,"title":"upd-preserves-Upkeep { s","content":"upd-preserves-Upkeep { s = s } { x = x } ( ↑-preserves-Upkeep { x = x₁ })","searchableContent":"upd-preserves-upkeep { s = s } { x = x } ( ↑-preserves-upkeep { x = x₁ })"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Decidable open import Leios.Prelude hiding ( id )","content":"Leios.Short.Decidable open import Leios.Prelude hiding ( id )","searchableContent":"leios.short.decidable open import leios.prelude hiding ( id )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":3,"title":"open import Leios.SpecStructure using ( SpecStructure )","content":"open import Leios.SpecStructure using ( SpecStructure )","searchableContent":"open import leios.specstructure using ( specstructure )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":5,"title":"module Leios.Short.Decidable ( ⋯","content":"module Leios.Short.Decidable ( ⋯ : SpecStructure 1 ) ( let open SpecStructure ⋯ ) where","searchableContent":"module leios.short.decidable ( ⋯ : specstructure 1 ) ( let open specstructure ⋯ ) where"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":7,"title":"open import Leios.Short ⋯ renaming ( isVoteCertified to isVoteCertified' )","content":"open import Leios.Short ⋯ renaming ( isVoteCertified to isVoteCertified' )","searchableContent":"open import leios.short ⋯ renaming ( isvotecertified to isvotecertified' )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":8,"title":"open B hiding ( _-⟦_/_⟧⇀_ )","content":"open B hiding ( _-⟦_/_⟧⇀_ )","searchableContent":"open b hiding ( _-⟦_/_⟧⇀_ )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":9,"title":"open FFD hiding ( _-⟦_/_⟧⇀_ )","content":"open FFD hiding ( _-⟦_/_⟧⇀_ )","searchableContent":"open ffd hiding ( _-⟦_/_⟧⇀_ )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":11,"title":"module _ { s","content":"module _ { s : LeiosState } ( let open LeiosState s renaming ( FFDState to ffds ; BaseState to bs )) where","searchableContent":"module _ { s : leiosstate } ( let open leiosstate s renaming ( ffdstate to ffds ; basestate to bs )) where"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":13,"title":"IB-Role?","content":"IB-Role? : ∀ { π ffds' } →","searchableContent":"ib-role? : ∀ { π ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":14,"title":"let b","content":"let b = GenFFD.ibBody ( record { txs = ToPropose })","searchableContent":"let b = genffd.ibbody ( record { txs = topropose })"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":15,"title":"h","content":"h = GenFFD.ibHeader ( mkIBHeader slot id π sk-IB ToPropose )","searchableContent":"h = genffd.ibheader ( mkibheader slot id π sk-ib topropose )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":16,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":17,"title":"{ _","content":"{ _ : auto∶ needsUpkeep IB-Role }","searchableContent":"{ _ : auto∶ needsupkeep ib-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":18,"title":"{ _","content":"{ _ : auto∶ canProduceIB slot sk-IB ( stake s ) π }","searchableContent":"{ _ : auto∶ canproduceib slot sk-ib ( stake s ) π }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":19,"title":"{ _","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send h ( just b ) / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"{ _ : auto∶ ffds ffd.-⟦ ffd.send h ( just b ) / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":20,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":21,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":22,"title":"IB-Role? {_} {_} { p } { q } { r }","content":"IB-Role? {_} {_} { p } { q } { r } = IB-Role ( toWitness p ) ( toWitness q ) ( toWitness r )","searchableContent":"ib-role? {_} {_} { p } { q } { r } = ib-role ( towitness p ) ( towitness q ) ( towitness r )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":24,"title":"{-","content":"{-","searchableContent":"{-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":25,"title":"No-IB-Role?","content":"No-IB-Role? :","searchableContent":"no-ib-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":26,"title":"{ _","content":"{ _ : auto∶ needsUpkeep IB-Role }","searchableContent":"{ _ : auto∶ needsupkeep ib-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":27,"title":"{ _","content":"{ _ : auto∶ ∀ π → ¬ canProduceIB slot sk-IB (stake s) π } →","searchableContent":"{ _ : auto∶ ∀ π → ¬ canproduceib slot sk-ib (stake s) π } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":28,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":29,"title":"s ↝ addUpkeep s IB-Role","content":"s ↝ addUpkeep s IB-Role","searchableContent":"s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":30,"title":"No-IB-Role? {p} {q}","content":"No-IB-Role? {p} {q} = No-IB-Role (toWitness p) (toWitness q)","searchableContent":"no-ib-role? {p} {q} = no-ib-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":31,"title":"-}","content":"-}","searchableContent":"-}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":33,"title":"EB-Role?","content":"EB-Role? : ∀ { π ffds' } →","searchableContent":"eb-role? : ∀ { π ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":34,"title":"let LI","content":"let LI = map getIBRef $ filter ( _∈ᴮ slice L slot 3 ) IBs","searchableContent":"let li = map getibref $ filter ( _∈ᴮ slice l slot 3 ) ibs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":35,"title":"h","content":"h = mkEB slot id π sk-EB LI []","searchableContent":"h = mkeb slot id π sk-eb li []"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":36,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":37,"title":"{ _","content":"{ _ : auto∶ needsUpkeep EB-Role }","searchableContent":"{ _ : auto∶ needsupkeep eb-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":38,"title":"{ _","content":"{ _ : auto∶ canProduceEB slot sk-EB ( stake s ) π }","searchableContent":"{ _ : auto∶ canproduceeb slot sk-eb ( stake s ) π }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":39,"title":"{ _","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send ( GenFFD.ebHeader h ) nothing / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"{ _ : auto∶ ffds ffd.-⟦ ffd.send ( genffd.ebheader h ) nothing / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":40,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":41,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":42,"title":"EB-Role? {_} {_} { p } { q } { r }","content":"EB-Role? {_} {_} { p } { q } { r } = EB-Role ( toWitness p ) ( toWitness q ) ( toWitness r )","searchableContent":"eb-role? {_} {_} { p } { q } { r } = eb-role ( towitness p ) ( towitness q ) ( towitness r )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":44,"title":"{-","content":"{-","searchableContent":"{-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":45,"title":"No-EB-Role?","content":"No-EB-Role? :","searchableContent":"no-eb-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":46,"title":"{ _","content":"{ _ : auto∶ needsUpkeep EB-Role }","searchableContent":"{ _ : auto∶ needsupkeep eb-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":47,"title":"{ _","content":"{ _ : auto∶ ∀ π → ¬ canProduceEB slot sk-EB (stake s) π } →","searchableContent":"{ _ : auto∶ ∀ π → ¬ canproduceeb slot sk-eb (stake s) π } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":48,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":49,"title":"s ↝ addUpkeep s EB-Role","content":"s ↝ addUpkeep s EB-Role","searchableContent":"s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":50,"title":"No-EB-Role? {_} {p} {q}","content":"No-EB-Role? {_} {p} {q} = No-EB-Role (toWitness p) (toWitness q)","searchableContent":"no-eb-role? {_} {p} {q} = no-eb-role (towitness p) (towitness q)"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":51,"title":"-}","content":"-}","searchableContent":"-}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":53,"title":"V-Role?","content":"V-Role? : ∀ { ffds' } →","searchableContent":"v-role? : ∀ { ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":54,"title":"let EBs'","content":"let EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L slot 1 ) EBs","searchableContent":"let ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l slot 1 ) ebs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":55,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":56,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":57,"title":"{ _","content":"{ _ : auto∶ needsUpkeep VT-Role }","searchableContent":"{ _ : auto∶ needsupkeep vt-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":58,"title":"{ _","content":"{ _ : auto∶ canProduceV slot sk-VT ( stake s ) }","searchableContent":"{ _ : auto∶ canproducev slot sk-vt ( stake s ) }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":59,"title":"{ _","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Send ( GenFFD.vtHeader votes ) nothing / FFD.SendRes ⟧⇀ ffds' } →","searchableContent":"{ _ : auto∶ ffds ffd.-⟦ ffd.send ( genffd.vtheader votes ) nothing / ffd.sendres ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":60,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":61,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } VT-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } vt-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":62,"title":"V-Role? {_} { p } { q } { r }","content":"V-Role? {_} { p } { q } { r } = VT-Role ( toWitness p ) ( toWitness q ) ( toWitness r )","searchableContent":"v-role? {_} { p } { q } { r } = vt-role ( towitness p ) ( towitness q ) ( towitness r )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":64,"title":"No-V-Role?","content":"No-V-Role? :","searchableContent":"no-v-role? :"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":65,"title":"{ _","content":"{ _ : auto∶ needsUpkeep VT-Role }","searchableContent":"{ _ : auto∶ needsupkeep vt-role }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":66,"title":"{ _","content":"{ _ : auto∶ ¬ canProduceV slot sk-VT ( stake s ) } →","searchableContent":"{ _ : auto∶ ¬ canproducev slot sk-vt ( stake s ) } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":67,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":68,"title":"s ↝ addUpkeep s VT-Role","content":"s ↝ addUpkeep s VT-Role","searchableContent":"s ↝ addupkeep s vt-role"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":69,"title":"No-V-Role? { p } { q }","content":"No-V-Role? { p } { q } = No-VT-Role ( toWitness p ) ( toWitness q )","searchableContent":"no-v-role? { p } { q } = no-vt-role ( towitness p ) ( towitness q )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":71,"title":"{-","content":"{-","searchableContent":"{-"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":72,"title":"Init?","content":"Init? : ∀ {ks pks ks' SD bs' V} →","searchableContent":"init? : ∀ {ks pks ks' sd bs' v} →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":73,"title":"{ _","content":"{ _ : auto∶ ks K.-⟦ K.INIT pk-IB pk-EB pk-V / K.PUBKEYS pks ⟧⇀ ks' }","searchableContent":"{ _ : auto∶ ks k.-⟦ k.init pk-ib pk-eb pk-v / k.pubkeys pks ⟧⇀ ks' }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":74,"title":"{ _","content":"{ _ : initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs' } →","searchableContent":"{ _ : initbasestate b.-⟦ b.init (v-chkcerts pks) / b.stake sd ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":75,"title":"────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":76,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'","content":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'","searchableContent":"nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":77,"title":"Init?","content":"Init? = ?","searchableContent":"init? = ?"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":78,"title":"-}","content":"-}","searchableContent":"-}"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":80,"title":"Base₂a?","content":"Base₂a? : ∀ { eb bs' } →","searchableContent":"base₂a? : ∀ { eb bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":81,"title":"{ _","content":"{ _ : auto∶ needsUpkeep Base }","searchableContent":"{ _ : auto∶ needsupkeep base }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":82,"title":"{ _","content":"{ _ : auto∶ eb ∈ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2 ) EBs }","searchableContent":"{ _ : auto∶ eb ∈ filter (λ eb → isvotecertified' s eb × eb ∈ᴮ slice l slot 2 ) ebs }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":83,"title":"{ _","content":"{ _ : auto∶ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs' } →","searchableContent":"{ _ : auto∶ bs b.-⟦ b.submit ( this eb ) / b.empty ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":84,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":85,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState","content":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":86,"title":"Base₂a? {_} {_} { p } { q } { r }","content":"Base₂a? {_} {_} { p } { q } { r } = Base₂a ( toWitness p ) ( toWitness q ) ( toWitness r )","searchableContent":"base₂a? {_} {_} { p } { q } { r } = base₂a ( towitness p ) ( towitness q ) ( towitness r )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":88,"title":"Base₂b?","content":"Base₂b? : ∀ { bs' } →","searchableContent":"base₂b? : ∀ { bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":89,"title":"{ _","content":"{ _ : auto∶ needsUpkeep Base }","searchableContent":"{ _ : auto∶ needsupkeep base }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":90,"title":"{ _","content":"{ _ : auto∶ [] ≡ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2 ) EBs }","searchableContent":"{ _ : auto∶ [] ≡ filter (λ eb → isvotecertified' s eb × eb ∈ᴮ slice l slot 2 ) ebs }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":91,"title":"{ _","content":"{ _ : auto∶ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs' } →","searchableContent":"{ _ : auto∶ bs b.-⟦ b.submit ( that topropose ) / b.empty ⟧⇀ bs' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":92,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":93,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState","content":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":94,"title":"Base₂b? {_} { p } { q } { r }","content":"Base₂b? {_} { p } { q } { r } = Base₂b ( toWitness p ) ( toWitness q ) ( toWitness r )","searchableContent":"base₂b? {_} { p } { q } { r } = base₂b ( towitness p ) ( towitness q ) ( towitness r )"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":96,"title":"Slot?","content":"Slot? : ∀ { rbs bs' msgs ffds' } →","searchableContent":"slot? : ∀ { rbs bs' msgs ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":97,"title":"{ _","content":"{ _ : auto∶ allDone s }","searchableContent":"{ _ : auto∶ alldone s }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":98,"title":"{ _","content":"{ _ : auto∶ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs' }","searchableContent":"{ _ : auto∶ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs' }"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":99,"title":"{ _","content":"{ _ : auto∶ ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds' } →","searchableContent":"{ _ : auto∶ ffds ffd.-⟦ ffd.fetch / ffd.fetchres msgs ⟧⇀ ffds' } →"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":100,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":101,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","content":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":102,"title":"{ FFDState","content":"{ FFDState = ffds'","searchableContent":"{ ffdstate = ffds'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":103,"title":"; BaseState","content":"; BaseState = bs'","searchableContent":"; basestate = bs'"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":104,"title":"; Ledger","content":"; Ledger = constructLedger rbs","searchableContent":"; ledger = constructledger rbs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":105,"title":"; slot","content":"; slot = suc slot","searchableContent":"; slot = suc slot"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":106,"title":"; Upkeep","content":"; Upkeep = ∅","searchableContent":"; upkeep = ∅"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":107,"title":"} ↑ L.filter ( isValid? s ) msgs","content":"} ↑ L.filter ( isValid? s ) msgs","searchableContent":"} ↑ l.filter ( isvalid? s ) msgs"},{"moduleName":"Leios.Short.Decidable","path":"Leios.Short.Decidable.html","group":"Leios","lineNumber":108,"title":"Slot? {_} {_} {_} {_} { p } { q } { r }","content":"Slot? {_} {_} {_} {_} { p } { q } { r } = Slot ( toWitness p ) ( toWitness q ) ( toWitness r )","searchableContent":"slot? {_} {_} {_} {_} { p } { q } { r } = slot ( towitness p ) ( towitness q ) ( towitness r )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Trace.Verifier.Test open import Leios.Prelude hiding ( id )","content":"Leios.Short.Trace.Verifier.Test open import Leios.Prelude hiding ( id )","searchableContent":"leios.short.trace.verifier.test open import leios.prelude hiding ( id )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":3,"title":"open import Leios.Config","content":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":5,"title":"module Leios.Short.Trace.Verifier.Test where","content":"module Leios.Short.Trace.Verifier.Test where","searchableContent":"module leios.short.trace.verifier.test where"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":7,"title":"params","content":"params : Params","searchableContent":"params : params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":8,"title":"params","content":"params =","searchableContent":"params ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":9,"title":"record","content":"record","searchableContent":"record"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":10,"title":"{ numberOfParties","content":"{ numberOfParties = 2","searchableContent":"{ numberofparties = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":11,"title":"; sutId","content":"; sutId = fzero","searchableContent":"; sutid = fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":12,"title":"; stakeDistribution","content":"; stakeDistribution =","searchableContent":"; stakedistribution ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":13,"title":"let open FunTot ( completeFin 2 ) ( maximalFin 2 )","content":"let open FunTot ( completeFin 2 ) ( maximalFin 2 )","searchableContent":"let open funtot ( completefin 2 ) ( maximalfin 2 )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":14,"title":"in Fun⇒TotalMap ( const 100000000 )","content":"in Fun⇒TotalMap ( const 100000000 )","searchableContent":"in fun⇒totalmap ( const 100000000 )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":15,"title":"; stageLength","content":"; stageLength = 2","searchableContent":"; stagelength = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":16,"title":"; winning-slots","content":"; winning-slots = fromList $","searchableContent":"; winning-slots = fromlist $"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":17,"title":"( IB , 0 ) ∷ ( EB , 0 ) ∷ ( VT , 0 ) ∷","content":"( IB , 0 ) ∷ ( EB , 0 ) ∷ ( VT , 0 ) ∷","searchableContent":"( ib , 0 ) ∷ ( eb , 0 ) ∷ ( vt , 0 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":18,"title":"( IB , 1 ) ∷ ( EB , 1 ) ∷ ( VT , 1 ) ∷","content":"( IB , 1 ) ∷ ( EB , 1 ) ∷ ( VT , 1 ) ∷","searchableContent":"( ib , 1 ) ∷ ( eb , 1 ) ∷ ( vt , 1 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":19,"title":"( IB , 2 ) ∷ ( EB , 2 ) ∷ ( VT , 2 ) ∷","content":"( IB , 2 ) ∷ ( EB , 2 ) ∷ ( VT , 2 ) ∷","searchableContent":"( ib , 2 ) ∷ ( eb , 2 ) ∷ ( vt , 2 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":20,"title":"( IB , 3 ) ∷ ( EB , 3 ) ∷ ( VT , 3 ) ∷","content":"( IB , 3 ) ∷ ( EB , 3 ) ∷ ( VT , 3 ) ∷","searchableContent":"( ib , 3 ) ∷ ( eb , 3 ) ∷ ( vt , 3 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":21,"title":"( IB , 4 ) ∷ ( EB , 4 ) ∷ ( VT , 4 ) ∷","content":"( IB , 4 ) ∷ ( EB , 4 ) ∷ ( VT , 4 ) ∷","searchableContent":"( ib , 4 ) ∷ ( eb , 4 ) ∷ ( vt , 4 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":22,"title":"( IB , 5 ) ∷ ( EB , 5 ) ∷ ( VT , 5 ) ∷","content":"( IB , 5 ) ∷ ( EB , 5 ) ∷ ( VT , 5 ) ∷","searchableContent":"( ib , 5 ) ∷ ( eb , 5 ) ∷ ( vt , 5 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":23,"title":"( VT , 6 ) ∷","content":"( VT , 6 ) ∷","searchableContent":"( vt , 6 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":24,"title":"( IB , 7 ) ∷ ( EB , 7 ) ∷ ( VT , 7 ) ∷","content":"( IB , 7 ) ∷ ( EB , 7 ) ∷ ( VT , 7 ) ∷","searchableContent":"( ib , 7 ) ∷ ( eb , 7 ) ∷ ( vt , 7 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":25,"title":"( IB , 8 ) ∷ ( EB , 8 ) ∷ ( VT , 8 ) ∷","content":"( IB , 8 ) ∷ ( EB , 8 ) ∷ ( VT , 8 ) ∷","searchableContent":"( ib , 8 ) ∷ ( eb , 8 ) ∷ ( vt , 8 ) ∷"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":26,"title":"[]","content":"[]","searchableContent":"[]"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":27,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":29,"title":"open Params params","content":"open Params params","searchableContent":"open params params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":30,"title":"open import Leios.Short.Trace.Verifier params","content":"open import Leios.Short.Trace.Verifier params","searchableContent":"open import leios.short.trace.verifier params"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":32,"title":"private","content":"private","searchableContent":"private"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":33,"title":"opaque","content":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":34,"title":"unfolding List-Model","content":"unfolding List-Model","searchableContent":"unfolding list-model"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":36,"title":"test₁","content":"test₁ : Bool","searchableContent":"test₁ : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":37,"title":"test₁","content":"test₁ = ¿ ValidTrace ( inj₁ ( IB-Role-Action 0 , SLOT ) ∷ [] ) ¿ᵇ","searchableContent":"test₁ = ¿ validtrace ( inj₁ ( ib-role-action 0 , slot ) ∷ [] ) ¿ᵇ"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":39,"title":"_","content":"_ : test₁ ≡ true","searchableContent":"_ : test₁ ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":40,"title":"_","content":"_ = refl","searchableContent":"_ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":42,"title":"test-valid-ib","content":"test-valid-ib : Bool","searchableContent":"test-valid-ib : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":43,"title":"test-valid-ib","content":"test-valid-ib =","searchableContent":"test-valid-ib ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":44,"title":"let h","content":"let h = record { slotNumber = 1","searchableContent":"let h = record { slotnumber = 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":45,"title":"; producerID","content":"; producerID = fsuc fzero","searchableContent":"; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":46,"title":"; lotteryPf","content":"; lotteryPf = tt","searchableContent":"; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":47,"title":"; bodyHash","content":"; bodyHash = 0 ∷ 1 ∷ 2 ∷ []","searchableContent":"; bodyhash = 0 ∷ 1 ∷ 2 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":48,"title":"; signature","content":"; signature = tt","searchableContent":"; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":49,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":50,"title":"b","content":"b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }","searchableContent":"b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":51,"title":"ib","content":"ib = record { header = h ; body = b }","searchableContent":"ib = record { header = h ; body = b }"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":52,"title":"pks","content":"pks = L.zip ( completeFinL numberOfParties ) ( L.replicate numberOfParties tt )","searchableContent":"pks = l.zip ( completefinl numberofparties ) ( l.replicate numberofparties tt )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":53,"title":"s","content":"s = initLeiosState tt stakeDistribution tt pks","searchableContent":"s = initleiosstate tt stakedistribution tt pks"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":54,"title":"in isYes ( ibValid? s ib )","content":"in isYes ( ibValid? s ib )","searchableContent":"in isyes ( ibvalid? s ib )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":56,"title":"_","content":"_ : test-valid-ib ≡ true","searchableContent":"_ : test-valid-ib ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":57,"title":"_","content":"_ = refl","searchableContent":"_ = refl"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":59,"title":"test₂","content":"test₂ : Bool","searchableContent":"test₂ : bool"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":60,"title":"test₂","content":"test₂ =","searchableContent":"test₂ ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":61,"title":"let t","content":"let t = L.reverse $","searchableContent":"let t = l.reverse $"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":62,"title":"-- slot 0","content":"-- slot 0","searchableContent":"-- slot 0"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":63,"title":"inj₁ ( IB-Role-Action 0 , SLOT )","content":"inj₁ ( IB-Role-Action 0 , SLOT )","searchableContent":"inj₁ ( ib-role-action 0 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":64,"title":"∷ inj₁ ( EB-Role-Action 0 [] , SLOT )","content":"∷ inj₁ ( EB-Role-Action 0 [] , SLOT )","searchableContent":"∷ inj₁ ( eb-role-action 0 [] , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":65,"title":"∷ inj₁ ( VT-Role-Action 0 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 0 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 0 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":66,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":67,"title":"∷ inj₁ ( Slot-Action 0 , SLOT )","content":"∷ inj₁ ( Slot-Action 0 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 0 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":68,"title":"-- slot 1","content":"-- slot 1","searchableContent":"-- slot 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":69,"title":"∷ inj₁ ( IB-Role-Action 1 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 1 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 1 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":70,"title":"∷ inj₁ ( VT-Role-Action 1 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 1 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 1 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":71,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":72,"title":"∷ inj₁ ( Slot-Action 1 , SLOT )","content":"∷ inj₁ ( Slot-Action 1 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 1 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":73,"title":"-- slot 2","content":"-- slot 2","searchableContent":"-- slot 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":74,"title":"∷ inj₂ ( IB-Recv-Update","content":"∷ inj₂ ( IB-Recv-Update","searchableContent":"∷ inj₂ ( ib-recv-update"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":75,"title":"( record { header","content":"( record { header =","searchableContent":"( record { header ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":76,"title":"record { slotNumber","content":"record { slotNumber = 1","searchableContent":"record { slotnumber = 1"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":77,"title":"; producerID","content":"; producerID = fsuc fzero","searchableContent":"; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":78,"title":"; lotteryPf","content":"; lotteryPf = tt","searchableContent":"; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":79,"title":"; bodyHash","content":"; bodyHash = 0 ∷ 1 ∷ 2 ∷ []","searchableContent":"; bodyhash = 0 ∷ 1 ∷ 2 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":80,"title":"; signature","content":"; signature = tt","searchableContent":"; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":81,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":82,"title":"; body","content":"; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))","searchableContent":"; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":83,"title":"∷ inj₁ ( IB-Role-Action 2 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 2 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 2 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":84,"title":"∷ inj₁ ( EB-Role-Action 2 [] , SLOT )","content":"∷ inj₁ ( EB-Role-Action 2 [] , SLOT )","searchableContent":"∷ inj₁ ( eb-role-action 2 [] , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":85,"title":"∷ inj₁ ( VT-Role-Action 2 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 2 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 2 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":86,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":87,"title":"∷ inj₁ ( Slot-Action 2 , SLOT )","content":"∷ inj₁ ( Slot-Action 2 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 2 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":88,"title":"-- slot 3","content":"-- slot 3","searchableContent":"-- slot 3"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":89,"title":"∷ inj₂ ( IB-Recv-Update","content":"∷ inj₂ ( IB-Recv-Update","searchableContent":"∷ inj₂ ( ib-recv-update"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":90,"title":"( record { header","content":"( record { header =","searchableContent":"( record { header ="},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":91,"title":"record { slotNumber","content":"record { slotNumber = 2","searchableContent":"record { slotnumber = 2"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":92,"title":"; producerID","content":"; producerID = fsuc fzero","searchableContent":"; producerid = fsuc fzero"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":93,"title":"; lotteryPf","content":"; lotteryPf = tt","searchableContent":"; lotterypf = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":94,"title":"; bodyHash","content":"; bodyHash = 3 ∷ 4 ∷ 5 ∷ []","searchableContent":"; bodyhash = 3 ∷ 4 ∷ 5 ∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":95,"title":"; signature","content":"; signature = tt","searchableContent":"; signature = tt"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":96,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":97,"title":"; body","content":"; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))","searchableContent":"; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":98,"title":"∷ inj₁ ( IB-Role-Action 3 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 3 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 3 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":99,"title":"∷ inj₁ ( VT-Role-Action 3 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 3 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 3 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":100,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":101,"title":"∷ inj₁ ( Slot-Action 3 , SLOT )","content":"∷ inj₁ ( Slot-Action 3 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 3 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":102,"title":"-- slot 4","content":"-- slot 4","searchableContent":"-- slot 4"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":103,"title":"∷ inj₁ ( IB-Role-Action 4 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 4 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 4 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":104,"title":"∷ inj₁ ( EB-Role-Action 4 [] , SLOT )","content":"∷ inj₁ ( EB-Role-Action 4 [] , SLOT )","searchableContent":"∷ inj₁ ( eb-role-action 4 [] , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":105,"title":"∷ inj₁ ( VT-Role-Action 4 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 4 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 4 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":106,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":107,"title":"∷ inj₁ ( Slot-Action 4 , SLOT )","content":"∷ inj₁ ( Slot-Action 4 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 4 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":108,"title":"-- slot 5","content":"-- slot 5","searchableContent":"-- slot 5"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":109,"title":"∷ inj₁ ( IB-Role-Action 5 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 5 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 5 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":110,"title":"∷ inj₁ ( VT-Role-Action 5 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 5 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 5 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":111,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":112,"title":"∷ inj₁ ( Slot-Action 5 , SLOT )","content":"∷ inj₁ ( Slot-Action 5 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 5 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":113,"title":"-- slot 6","content":"-- slot 6","searchableContent":"-- slot 6"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":114,"title":"∷ inj₁ ( No-IB-Role-Action , SLOT )","content":"∷ inj₁ ( No-IB-Role-Action , SLOT )","searchableContent":"∷ inj₁ ( no-ib-role-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":115,"title":"∷ inj₁ ( No-EB-Role-Action , SLOT )","content":"∷ inj₁ ( No-EB-Role-Action , SLOT )","searchableContent":"∷ inj₁ ( no-eb-role-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":116,"title":"∷ inj₁ ( VT-Role-Action 6 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 6 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 6 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":117,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":118,"title":"∷ inj₁ ( Slot-Action 6 , SLOT )","content":"∷ inj₁ ( Slot-Action 6 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 6 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":119,"title":"-- slot 7","content":"-- slot 7","searchableContent":"-- slot 7"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":120,"title":"∷ inj₁ ( IB-Role-Action 7 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 7 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 7 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":121,"title":"∷ inj₁ ( VT-Role-Action 7 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 7 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 7 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":122,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":123,"title":"∷ inj₁ ( Slot-Action 7 , SLOT )","content":"∷ inj₁ ( Slot-Action 7 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 7 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":124,"title":"-- slot 8","content":"-- slot 8","searchableContent":"-- slot 8"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":125,"title":"∷ inj₁ ( IB-Role-Action 8 , SLOT )","content":"∷ inj₁ ( IB-Role-Action 8 , SLOT )","searchableContent":"∷ inj₁ ( ib-role-action 8 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":126,"title":"∷ inj₁ ( EB-Role-Action 8 (( 3 ∷ 4 ∷ 5 ∷ [] ) ∷ [] ) , SLOT )","content":"∷ inj₁ ( EB-Role-Action 8 (( 3 ∷ 4 ∷ 5 ∷ [] ) ∷ [] ) , SLOT )","searchableContent":"∷ inj₁ ( eb-role-action 8 (( 3 ∷ 4 ∷ 5 ∷ [] ) ∷ [] ) , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":127,"title":"∷ inj₁ ( VT-Role-Action 8 , SLOT )","content":"∷ inj₁ ( VT-Role-Action 8 , SLOT )","searchableContent":"∷ inj₁ ( vt-role-action 8 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":128,"title":"∷ inj₁ ( Base₂b-Action , SLOT )","content":"∷ inj₁ ( Base₂b-Action , SLOT )","searchableContent":"∷ inj₁ ( base₂b-action , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":129,"title":"∷ inj₁ ( Slot-Action 8 , SLOT )","content":"∷ inj₁ ( Slot-Action 8 , SLOT )","searchableContent":"∷ inj₁ ( slot-action 8 , slot )"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":130,"title":"∷ []","content":"∷ []","searchableContent":"∷ []"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":131,"title":"in ¿ ValidTrace t ¿ᵇ","content":"in ¿ ValidTrace t ¿ᵇ","searchableContent":"in ¿ validtrace t ¿ᵇ"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":133,"title":"_","content":"_ : test₂ ≡ true","searchableContent":"_ : test₂ ≡ true"},{"moduleName":"Leios.Short.Trace.Verifier.Test","path":"Leios.Short.Trace.Verifier.Test.html","group":"Leios","lineNumber":134,"title":"_","content":"_ = refl","searchableContent":"_ = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":2,"title":"Leios.Short.Trace.Verifier open import Leios.Prelude hiding ( id )","content":"Leios.Short.Trace.Verifier open import Leios.Prelude hiding ( id )","searchableContent":"leios.short.trace.verifier open import leios.prelude hiding ( id )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":3,"title":"open import Leios.Config","content":"open import Leios.Config","searchableContent":"open import leios.config"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":5,"title":"-- TODO","content":"-- TODO: SpecStructure as parameter","searchableContent":"-- todo: specstructure as parameter"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":6,"title":"module Leios.Short.Trace.Verifier ( params","content":"module Leios.Short.Trace.Verifier ( params : Params ) ( let open Params params ) where","searchableContent":"module leios.short.trace.verifier ( params : params ) ( let open params params ) where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":8,"title":"open import Leios.Defaults params","content":"open import Leios.Defaults params","searchableContent":"open import leios.defaults params"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":9,"title":"using ( LeiosState ; initLeiosState ; isb ; hpe ; hhs ; htx ; SendIB ; FFDBuffers ; Dec-SimpleFFD )","content":"using ( LeiosState ; initLeiosState ; isb ; hpe ; hhs ; htx ; SendIB ; FFDBuffers ; Dec-SimpleFFD )","searchableContent":"using ( leiosstate ; initleiosstate ; isb ; hpe ; hhs ; htx ; sendib ; ffdbuffers ; dec-simpleffd )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":10,"title":"renaming ( d-SpecStructure to traceSpecStructure ) public","content":"renaming ( d-SpecStructure to traceSpecStructure ) public","searchableContent":"renaming ( d-specstructure to tracespecstructure ) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":12,"title":"open import Leios.SpecStructure using ( SpecStructure )","content":"open import Leios.SpecStructure using ( SpecStructure )","searchableContent":"open import leios.specstructure using ( specstructure )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":13,"title":"open SpecStructure traceSpecStructure hiding ( Hashable-IBHeader ; Hashable-EndorserBlock ; isVoteCertified ) public","content":"open SpecStructure traceSpecStructure hiding ( Hashable-IBHeader ; Hashable-EndorserBlock ; isVoteCertified ) public","searchableContent":"open specstructure tracespecstructure hiding ( hashable-ibheader ; hashable-endorserblock ; isvotecertified ) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":15,"title":"open import Leios.Short traceSpecStructure hiding ( LeiosState ; initLeiosState ) public","content":"open import Leios.Short traceSpecStructure hiding ( LeiosState ; initLeiosState ) public","searchableContent":"open import leios.short tracespecstructure hiding ( leiosstate ; initleiosstate ) public"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":16,"title":"open import Prelude.Closures _↝_","content":"open import Prelude.Closures _↝_","searchableContent":"open import prelude.closures _↝_"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":17,"title":"open GenFFD","content":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":19,"title":"data FFDUpdate","content":"data FFDUpdate : Type where","searchableContent":"data ffdupdate : type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":20,"title":"IB-Recv-Update","content":"IB-Recv-Update : InputBlock → FFDUpdate","searchableContent":"ib-recv-update : inputblock → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":21,"title":"EB-Recv-Update","content":"EB-Recv-Update : EndorserBlock → FFDUpdate","searchableContent":"eb-recv-update : endorserblock → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":22,"title":"VT-Recv-Update","content":"VT-Recv-Update : List Vote → FFDUpdate","searchableContent":"vt-recv-update : list vote → ffdupdate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":24,"title":"data Action","content":"data Action : Type where","searchableContent":"data action : type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":25,"title":"IB-Role-Action","content":"IB-Role-Action : ℕ → Action","searchableContent":"ib-role-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":26,"title":"EB-Role-Action","content":"EB-Role-Action : ℕ → List IBRef → Action","searchableContent":"eb-role-action : ℕ → list ibref → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":27,"title":"VT-Role-Action","content":"VT-Role-Action : ℕ → Action","searchableContent":"vt-role-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":28,"title":"No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action","content":"No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action : Action","searchableContent":"no-ib-role-action no-eb-role-action no-vt-role-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":29,"title":"Ftch-Action","content":"Ftch-Action : Action","searchableContent":"ftch-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":30,"title":"Slot-Action","content":"Slot-Action : ℕ → Action","searchableContent":"slot-action : ℕ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":31,"title":"Base₁-Action","content":"Base₁-Action : Action","searchableContent":"base₁-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":32,"title":"Base₂a-Action","content":"Base₂a-Action : EndorserBlock → Action","searchableContent":"base₂a-action : endorserblock → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":33,"title":"Base₂b-Action","content":"Base₂b-Action : Action","searchableContent":"base₂b-action : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":35,"title":"Actions","content":"Actions = List ( Action × LeiosInput )","searchableContent":"actions = list ( action × leiosinput )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":37,"title":"private variable","content":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":38,"title":"s s′","content":"s s′ : LeiosState","searchableContent":"s s′ : leiosstate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":39,"title":"α","content":"α : Action","searchableContent":"α : action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":41,"title":"data ValidUpdate","content":"data ValidUpdate : FFDUpdate → LeiosState → Type where","searchableContent":"data validupdate : ffdupdate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":43,"title":"IB-Recv","content":"IB-Recv : ∀ { ib } →","searchableContent":"ib-recv : ∀ { ib } →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":44,"title":"ValidUpdate ( IB-Recv-Update ib ) s","content":"ValidUpdate ( IB-Recv-Update ib ) s","searchableContent":"validupdate ( ib-recv-update ib ) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":46,"title":"EB-Recv","content":"EB-Recv : ∀ { eb } →","searchableContent":"eb-recv : ∀ { eb } →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":47,"title":"ValidUpdate ( EB-Recv-Update eb ) s","content":"ValidUpdate ( EB-Recv-Update eb ) s","searchableContent":"validupdate ( eb-recv-update eb ) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":49,"title":"VT-Recv","content":"VT-Recv : ∀ { vt } →","searchableContent":"vt-recv : ∀ { vt } →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":50,"title":"ValidUpdate ( VT-Recv-Update vt ) s","content":"ValidUpdate ( VT-Recv-Update vt ) s","searchableContent":"validupdate ( vt-recv-update vt ) s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":52,"title":"data ValidAction","content":"data ValidAction : Action → LeiosState → LeiosInput → Type where","searchableContent":"data validaction : action → leiosstate → leiosinput → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":54,"title":"IB-Role","content":"IB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"ib-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":55,"title":"b","content":"b = record { txs = ToPropose }","searchableContent":"b = record { txs = topropose }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":56,"title":"h","content":"h = mkIBHeader slot id tt sk-IB ToPropose","searchableContent":"h = mkibheader slot id tt sk-ib topropose"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":57,"title":"ffds'","content":"ffds' = proj₁ ( FFD.Send-total { ffds } { ibHeader h } { just ( ibBody b )})","searchableContent":"ffds' = proj₁ ( ffd.send-total { ffds } { ibheader h } { just ( ibbody b )})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":58,"title":"in .( needsUpkeep IB-Role ) →","content":"in .( needsUpkeep IB-Role ) →","searchableContent":"in .( needsupkeep ib-role ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":59,"title":".( canProduceIB slot sk-IB ( stake s ) tt ) →","content":".( canProduceIB slot sk-IB ( stake s ) tt ) →","searchableContent":".( canproduceib slot sk-ib ( stake s ) tt ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":60,"title":".( ffds FFD.-⟦ FFD.Send ( ibHeader h ) ( just ( ibBody b )) / FFD.SendRes ⟧⇀ ffds' ) →","content":".( ffds FFD.-⟦ FFD.Send ( ibHeader h ) ( just ( ibBody b )) / FFD.SendRes ⟧⇀ ffds' ) →","searchableContent":".( ffds ffd.-⟦ ffd.send ( ibheader h ) ( just ( ibbody b )) / ffd.sendres ⟧⇀ ffds' ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":61,"title":"ValidAction ( IB-Role-Action slot ) s SLOT","content":"ValidAction ( IB-Role-Action slot ) s SLOT","searchableContent":"validaction ( ib-role-action slot ) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":63,"title":"EB-Role","content":"EB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"eb-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":64,"title":"LI","content":"LI = map getIBRef $ filter ( _∈ᴮ slice L slot 3 ) IBs","searchableContent":"li = map getibref $ filter ( _∈ᴮ slice l slot 3 ) ibs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":65,"title":"h","content":"h = mkEB slot id tt sk-EB LI []","searchableContent":"h = mkeb slot id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":66,"title":"ffds'","content":"ffds' = proj₁ ( FFD.Send-total { ffds } { ebHeader h } { nothing })","searchableContent":"ffds' = proj₁ ( ffd.send-total { ffds } { ebheader h } { nothing })"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":67,"title":"in .( needsUpkeep EB-Role ) →","content":"in .( needsUpkeep EB-Role ) →","searchableContent":"in .( needsupkeep eb-role ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":68,"title":".( canProduceEB slot sk-EB ( stake s ) tt ) →","content":".( canProduceEB slot sk-EB ( stake s ) tt ) →","searchableContent":".( canproduceeb slot sk-eb ( stake s ) tt ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":69,"title":".( ffds FFD.-⟦ FFD.Send ( ebHeader h ) nothing / FFD.SendRes ⟧⇀ ffds' ) →","content":".( ffds FFD.-⟦ FFD.Send ( ebHeader h ) nothing / FFD.SendRes ⟧⇀ ffds' ) →","searchableContent":".( ffds ffd.-⟦ ffd.send ( ebheader h ) nothing / ffd.sendres ⟧⇀ ffds' ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":70,"title":"ValidAction ( EB-Role-Action slot LI ) s SLOT","content":"ValidAction ( EB-Role-Action slot LI ) s SLOT","searchableContent":"validaction ( eb-role-action slot li ) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":72,"title":"VT-Role","content":"VT-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"vt-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":73,"title":"EBs'","content":"EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L slot 1 ) EBs","searchableContent":"ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l slot 1 ) ebs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":74,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":75,"title":"ffds'","content":"ffds' = proj₁ ( FFD.Send-total { ffds } { vtHeader votes } { nothing })","searchableContent":"ffds' = proj₁ ( ffd.send-total { ffds } { vtheader votes } { nothing })"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":76,"title":"in .( needsUpkeep VT-Role ) →","content":"in .( needsUpkeep VT-Role ) →","searchableContent":"in .( needsupkeep vt-role ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":77,"title":".( canProduceV slot sk-VT ( stake s )) →","content":".( canProduceV slot sk-VT ( stake s )) →","searchableContent":".( canproducev slot sk-vt ( stake s )) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":78,"title":".( ffds FFD.-⟦ FFD.Send ( vtHeader votes ) nothing / FFD.SendRes ⟧⇀ ffds' ) →","content":".( ffds FFD.-⟦ FFD.Send ( vtHeader votes ) nothing / FFD.SendRes ⟧⇀ ffds' ) →","searchableContent":".( ffds ffd.-⟦ ffd.send ( vtheader votes ) nothing / ffd.sendres ⟧⇀ ffds' ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":79,"title":"ValidAction ( VT-Role-Action slot ) s SLOT","content":"ValidAction ( VT-Role-Action slot ) s SLOT","searchableContent":"validaction ( vt-role-action slot ) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":81,"title":"No-IB-Role","content":"No-IB-Role : let open LeiosState s","searchableContent":"no-ib-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":82,"title":"in needsUpkeep IB-Role →","content":"in needsUpkeep IB-Role →","searchableContent":"in needsupkeep ib-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":83,"title":"(∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π ) →","content":"(∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π ) →","searchableContent":"(∀ π → ¬ canproduceib slot sk-ib ( stake s ) π ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":84,"title":"ValidAction No-IB-Role-Action s SLOT","content":"ValidAction No-IB-Role-Action s SLOT","searchableContent":"validaction no-ib-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":86,"title":"No-EB-Role","content":"No-EB-Role : let open LeiosState s","searchableContent":"no-eb-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":87,"title":"in needsUpkeep EB-Role →","content":"in needsUpkeep EB-Role →","searchableContent":"in needsupkeep eb-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":88,"title":"(∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π ) →","content":"(∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π ) →","searchableContent":"(∀ π → ¬ canproduceeb slot sk-eb ( stake s ) π ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":89,"title":"ValidAction No-EB-Role-Action s SLOT","content":"ValidAction No-EB-Role-Action s SLOT","searchableContent":"validaction no-eb-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":91,"title":"No-VT-Role","content":"No-VT-Role : let open LeiosState s","searchableContent":"no-vt-role : let open leiosstate s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":92,"title":"in needsUpkeep VT-Role →","content":"in needsUpkeep VT-Role →","searchableContent":"in needsupkeep vt-role →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":93,"title":"( ¬ canProduceV slot sk-VT ( stake s )) →","content":"( ¬ canProduceV slot sk-VT ( stake s )) →","searchableContent":"( ¬ canproducev slot sk-vt ( stake s )) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":94,"title":"ValidAction No-VT-Role-Action s SLOT","content":"ValidAction No-VT-Role-Action s SLOT","searchableContent":"validaction no-vt-role-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":96,"title":"Slot","content":"Slot : let open LeiosState s renaming ( FFDState to ffds ; BaseState to bs )","searchableContent":"slot : let open leiosstate s renaming ( ffdstate to ffds ; basestate to bs )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":97,"title":"( msgs , ( ffds' , _))","content":"( msgs , ( ffds' , _)) = FFD.Fetch-total { ffds }","searchableContent":"( msgs , ( ffds' , _)) = ffd.fetch-total { ffds }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":98,"title":"in .( allDone s ) →","content":"in .( allDone s ) →","searchableContent":"in .( alldone s ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":99,"title":".( bs B.-⟦ B.FTCH-LDG / B.BASE-LDG [] ⟧⇀ tt ) →","content":".( bs B.-⟦ B.FTCH-LDG / B.BASE-LDG [] ⟧⇀ tt ) →","searchableContent":".( bs b.-⟦ b.ftch-ldg / b.base-ldg [] ⟧⇀ tt ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":100,"title":".( ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds' ) →","content":".( ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds' ) →","searchableContent":".( ffds ffd.-⟦ ffd.fetch / ffd.fetchres msgs ⟧⇀ ffds' ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":101,"title":"ValidAction ( Slot-Action slot ) s SLOT","content":"ValidAction ( Slot-Action slot ) s SLOT","searchableContent":"validaction ( slot-action slot ) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":103,"title":"Ftch","content":"Ftch : ValidAction Ftch-Action s FTCH-LDG","searchableContent":"ftch : validaction ftch-action s ftch-ldg"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":105,"title":"Base₁","content":"Base₁ : ∀ { txs } → ValidAction Base₁-Action s ( SUBMIT ( inj₂ txs ))","searchableContent":"base₁ : ∀ { txs } → validaction base₁-action s ( submit ( inj₂ txs ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":107,"title":"Base₂a","content":"Base₂a : ∀ { eb } → let open LeiosState s renaming ( BaseState to bs )","searchableContent":"base₂a : ∀ { eb } → let open leiosstate s renaming ( basestate to bs )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":108,"title":"in .( needsUpkeep Base ) →","content":"in .( needsUpkeep Base ) →","searchableContent":"in .( needsupkeep base ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":109,"title":".( eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs ) →","content":".( eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs ) →","searchableContent":".( eb ∈ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2 ) ebs ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":110,"title":".( bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ tt ) →","content":".( bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ tt ) →","searchableContent":".( bs b.-⟦ b.submit ( this eb ) / b.empty ⟧⇀ tt ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":111,"title":"ValidAction ( Base₂a-Action eb ) s SLOT","content":"ValidAction ( Base₂a-Action eb ) s SLOT","searchableContent":"validaction ( base₂a-action eb ) s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":113,"title":"Base₂b","content":"Base₂b : let open LeiosState s renaming ( BaseState to bs )","searchableContent":"base₂b : let open leiosstate s renaming ( basestate to bs )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":114,"title":"in .( needsUpkeep Base ) →","content":"in .( needsUpkeep Base ) →","searchableContent":"in .( needsupkeep base ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":115,"title":".( [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs ) →","content":".( [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs ) →","searchableContent":".( [] ≡ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2 ) ebs ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":116,"title":".( bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ tt ) →","content":".( bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ tt ) →","searchableContent":".( bs b.-⟦ b.submit ( that topropose ) / b.empty ⟧⇀ tt ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":117,"title":"ValidAction Base₂b-Action s SLOT","content":"ValidAction Base₂b-Action s SLOT","searchableContent":"validaction base₂b-action s slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":119,"title":"private variable","content":"private variable","searchableContent":"private variable"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":120,"title":"i","content":"i : LeiosInput","searchableContent":"i : leiosinput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":121,"title":"o","content":"o : LeiosOutput","searchableContent":"o : leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":123,"title":"⟦_⟧","content":"⟦_⟧ : ValidAction α s i → LeiosState × LeiosOutput","searchableContent":"⟦_⟧ : validaction α s i → leiosstate × leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":124,"title":"⟦ IB-Role { s } _ _ _ ⟧","content":"⟦ IB-Role { s } _ _ _ ⟧ =","searchableContent":"⟦ ib-role { s } _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":125,"title":"let open LeiosState s renaming ( FFDState to ffds )","content":"let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":126,"title":"b","content":"b = record { txs = ToPropose }","searchableContent":"b = record { txs = topropose }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":127,"title":"h","content":"h = mkIBHeader slot id tt sk-IB ToPropose","searchableContent":"h = mkibheader slot id tt sk-ib topropose"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":128,"title":"ffds'","content":"ffds' = proj₁ ( FFD.Send-total { ffds } { ibHeader h } { just ( ibBody b )})","searchableContent":"ffds' = proj₁ ( ffd.send-total { ffds } { ibheader h } { just ( ibbody b )})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":129,"title":"in addUpkeep record s { FFDState","content":"in addUpkeep record s { FFDState = ffds' } IB-Role , EMPTY","searchableContent":"in addupkeep record s { ffdstate = ffds' } ib-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":130,"title":"⟦ EB-Role { s } _ _ _ ⟧","content":"⟦ EB-Role { s } _ _ _ ⟧ =","searchableContent":"⟦ eb-role { s } _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":131,"title":"let open LeiosState s renaming ( FFDState to ffds )","content":"let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":132,"title":"LI","content":"LI = map getIBRef $ filter ( _∈ᴮ slice L slot 3 ) IBs","searchableContent":"li = map getibref $ filter ( _∈ᴮ slice l slot 3 ) ibs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":133,"title":"h","content":"h = mkEB slot id tt sk-EB LI []","searchableContent":"h = mkeb slot id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":134,"title":"ffds'","content":"ffds' = proj₁ ( FFD.Send-total { ffds } { ebHeader h } { nothing })","searchableContent":"ffds' = proj₁ ( ffd.send-total { ffds } { ebheader h } { nothing })"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":135,"title":"in addUpkeep record s { FFDState","content":"in addUpkeep record s { FFDState = ffds' } EB-Role , EMPTY","searchableContent":"in addupkeep record s { ffdstate = ffds' } eb-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":136,"title":"⟦ VT-Role { s } _ _ _ ⟧","content":"⟦ VT-Role { s } _ _ _ ⟧ =","searchableContent":"⟦ vt-role { s } _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":137,"title":"let open LeiosState s renaming ( FFDState to ffds )","content":"let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":138,"title":"EBs'","content":"EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L slot 1 ) EBs","searchableContent":"ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l slot 1 ) ebs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":139,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":140,"title":"ffds'","content":"ffds' = proj₁ ( FFD.Send-total { ffds } { vtHeader votes } { nothing })","searchableContent":"ffds' = proj₁ ( ffd.send-total { ffds } { vtheader votes } { nothing })"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":141,"title":"in addUpkeep record s { FFDState","content":"in addUpkeep record s { FFDState = ffds' } VT-Role , EMPTY","searchableContent":"in addupkeep record s { ffdstate = ffds' } vt-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":142,"title":"⟦ No-IB-Role { s } _ _ ⟧","content":"⟦ No-IB-Role { s } _ _ ⟧ = addUpkeep s IB-Role , EMPTY","searchableContent":"⟦ no-ib-role { s } _ _ ⟧ = addupkeep s ib-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":143,"title":"⟦ No-EB-Role { s } _ _ ⟧","content":"⟦ No-EB-Role { s } _ _ ⟧ = addUpkeep s EB-Role , EMPTY","searchableContent":"⟦ no-eb-role { s } _ _ ⟧ = addupkeep s eb-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":144,"title":"⟦ No-VT-Role { s } _ _ ⟧","content":"⟦ No-VT-Role { s } _ _ ⟧ = addUpkeep s VT-Role , EMPTY","searchableContent":"⟦ no-vt-role { s } _ _ ⟧ = addupkeep s vt-role , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":145,"title":"⟦ Slot { s } _ _ _ ⟧","content":"⟦ Slot { s } _ _ _ ⟧ =","searchableContent":"⟦ slot { s } _ _ _ ⟧ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":146,"title":"let open LeiosState s renaming ( FFDState to ffds )","content":"let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":147,"title":"( msgs , ( ffds' , _))","content":"( msgs , ( ffds' , _)) = FFD.Fetch-total { ffds }","searchableContent":"( msgs , ( ffds' , _)) = ffd.fetch-total { ffds }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":148,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":149,"title":"( record s","content":"( record s","searchableContent":"( record s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":150,"title":"{ FFDState","content":"{ FFDState = ffds'","searchableContent":"{ ffdstate = ffds'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":151,"title":"; BaseState","content":"; BaseState = tt","searchableContent":"; basestate = tt"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":152,"title":"; Ledger","content":"; Ledger = constructLedger []","searchableContent":"; ledger = constructledger []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":153,"title":"; slot","content":"; slot = suc slot","searchableContent":"; slot = suc slot"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":154,"title":"; Upkeep","content":"; Upkeep = ∅","searchableContent":"; upkeep = ∅"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":155,"title":"} ↑ L.filter ( isValid? s ) msgs","content":"} ↑ L.filter ( isValid? s ) msgs","searchableContent":"} ↑ l.filter ( isvalid? s ) msgs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":156,"title":", EMPTY )","content":", EMPTY )","searchableContent":", empty )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":157,"title":"⟦ Ftch { s } ⟧","content":"⟦ Ftch { s } ⟧ = s , FTCH-LDG ( LeiosState.Ledger s )","searchableContent":"⟦ ftch { s } ⟧ = s , ftch-ldg ( leiosstate.ledger s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":158,"title":"⟦ Base₁ { s } { txs } ⟧","content":"⟦ Base₁ { s } { txs } ⟧ = record s { ToPropose = txs } , EMPTY","searchableContent":"⟦ base₁ { s } { txs } ⟧ = record s { topropose = txs } , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":159,"title":"⟦ Base₂a { s } _ _ _ ⟧","content":"⟦ Base₂a { s } _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY","searchableContent":"⟦ base₂a { s } _ _ _ ⟧ = addupkeep record s { basestate = tt } base , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":160,"title":"⟦ Base₂b { s } _ _ _ ⟧","content":"⟦ Base₂b { s } _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY","searchableContent":"⟦ base₂b { s } _ _ _ ⟧ = addupkeep record s { basestate = tt } base , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":162,"title":"open LeiosState","content":"open LeiosState","searchableContent":"open leiosstate"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":163,"title":"open FFDBuffers","content":"open FFDBuffers","searchableContent":"open ffdbuffers"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":165,"title":"ValidAction→Eq-Slot","content":"ValidAction→Eq-Slot : ∀ { s sl } → ValidAction ( Slot-Action sl ) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-slot : ∀ { s sl } → validaction ( slot-action sl ) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":166,"title":"ValidAction→Eq-Slot ( Slot _ _ _)","content":"ValidAction→Eq-Slot ( Slot _ _ _) = refl","searchableContent":"validaction→eq-slot ( slot _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":168,"title":"ValidAction→Eq-IB","content":"ValidAction→Eq-IB : ∀ { s sl } → ValidAction ( IB-Role-Action sl ) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-ib : ∀ { s sl } → validaction ( ib-role-action sl ) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":169,"title":"ValidAction→Eq-IB ( IB-Role _ _ _)","content":"ValidAction→Eq-IB ( IB-Role _ _ _) = refl","searchableContent":"validaction→eq-ib ( ib-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":171,"title":"ValidAction→Eq-EB","content":"ValidAction→Eq-EB : ∀ { s sl ibs } → ValidAction ( EB-Role-Action sl ibs ) s SLOT → sl ≡ slot s × ibs ≡ ( map getIBRef $ filter ( _∈ᴮ slice L ( slot s ) 3 ) ( IBs s ))","searchableContent":"validaction→eq-eb : ∀ { s sl ibs } → validaction ( eb-role-action sl ibs ) s slot → sl ≡ slot s × ibs ≡ ( map getibref $ filter ( _∈ᴮ slice l ( slot s ) 3 ) ( ibs s ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":172,"title":"ValidAction→Eq-EB ( EB-Role _ _ _)","content":"ValidAction→Eq-EB ( EB-Role _ _ _) = refl , refl","searchableContent":"validaction→eq-eb ( eb-role _ _ _) = refl , refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":174,"title":"ValidAction→Eq-VT","content":"ValidAction→Eq-VT : ∀ { s sl } → ValidAction ( VT-Role-Action sl ) s SLOT → sl ≡ slot s","searchableContent":"validaction→eq-vt : ∀ { s sl } → validaction ( vt-role-action sl ) s slot → sl ≡ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":175,"title":"ValidAction→Eq-VT ( VT-Role _ _ _)","content":"ValidAction→Eq-VT ( VT-Role _ _ _) = refl","searchableContent":"validaction→eq-vt ( vt-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":177,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":178,"title":"Dec-ValidAction","content":"Dec-ValidAction : ValidAction ⁇³","searchableContent":"dec-validaction : validaction ⁇³"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":179,"title":"Dec-ValidAction { IB-Role-Action sl } { s } { SLOT } . dec","content":"Dec-ValidAction { IB-Role-Action sl } { s } { SLOT } . dec","searchableContent":"dec-validaction { ib-role-action sl } { s } { slot } . dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":180,"title":"with sl ≟ slot s","content":"with sl ≟ slot s","searchableContent":"with sl ≟ slot s"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":181,"title":"... | no ¬p","content":"... | no ¬p = no λ x → ⊥-elim ( ¬p ( ValidAction→Eq-IB x ))","searchableContent":"... | no ¬p = no λ x → ⊥-elim ( ¬p ( validaction→eq-ib x ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":182,"title":"... | yes p rewrite p","content":"... | yes p rewrite p","searchableContent":"... | yes p rewrite p"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":183,"title":"with dec | dec | dec","content":"with dec | dec | dec","searchableContent":"with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":184,"title":"... | yes x | yes y | yes z","content":"... | yes x | yes y | yes z = yes ( IB-Role x y z )","searchableContent":"... | yes 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( ¬p ( recompute dec p ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":248,"title":"... | _ | _ | no ¬p","content":"... | _ | _ | no ¬p = no λ where ( Slot _ _ p ) → ⊥-elim ( ¬p ( recompute dec p ))","searchableContent":"... | _ | _ | no ¬p = no λ where ( slot _ _ p ) → ⊥-elim ( ¬p ( recompute dec p ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":249,"title":"Dec-ValidAction { Slot-Action _} { s } { INIT _} . dec","content":"Dec-ValidAction { Slot-Action _} { s } { INIT _} . dec = no λ ()","searchableContent":"dec-validaction { slot-action _} { s } { init _} . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":250,"title":"Dec-ValidAction { Slot-Action _} { s } { SUBMIT _} . dec","content":"Dec-ValidAction { Slot-Action _} { s } { SUBMIT _} . dec = no λ ()","searchableContent":"dec-validaction { slot-action _} { s } { submit _} . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":251,"title":"Dec-ValidAction { Slot-Action _} { s } { FTCH-LDG } . dec","content":"Dec-ValidAction { Slot-Action _} { s } { FTCH-LDG } . dec = no λ ()","searchableContent":"dec-validaction { slot-action _} { s } { ftch-ldg } . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":252,"title":"Dec-ValidAction { Ftch-Action } { s } { FTCH-LDG } . dec","content":"Dec-ValidAction { Ftch-Action } { s } { FTCH-LDG } . dec = yes Ftch","searchableContent":"dec-validaction { ftch-action } { s } { ftch-ldg } . dec = yes ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":253,"title":"Dec-ValidAction { Ftch-Action } { s } { SLOT } . 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dec","content":"Dec-ValidAction { Base₁-Action } { s } { SLOT } . dec = no λ ()","searchableContent":"dec-validaction { base₁-action } { s } { slot } . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":259,"title":"Dec-ValidAction { Base₁-Action } { s } { FTCH-LDG } . dec","content":"Dec-ValidAction { Base₁-Action } { s } { FTCH-LDG } . dec = no λ ()","searchableContent":"dec-validaction { base₁-action } { s } { ftch-ldg } . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":260,"title":"Dec-ValidAction { Base₁-Action } { s } { INIT _} . dec","content":"Dec-ValidAction { Base₁-Action } { s } { INIT _} . dec = no λ ()","searchableContent":"dec-validaction { base₁-action } { s } { init _} . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":261,"title":"Dec-ValidAction { Base₂a-Action eb } { s } { SLOT } . dec","content":"Dec-ValidAction { Base₂a-Action eb } { s } { SLOT } . dec","searchableContent":"dec-validaction { base₂a-action eb } { s } { slot } . dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":262,"title":"with dec | dec | dec","content":"with dec | dec | dec","searchableContent":"with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":263,"title":"... | yes x | yes y | yes z","content":"... | yes x | yes y | yes z = yes ( Base₂a x y z )","searchableContent":"... | yes x | yes y | yes z = yes ( base₂a x y z )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":264,"title":"... | no ¬p | _ | _","content":"... | no ¬p | _ | _ = no λ where ( Base₂a p _ _) → ⊥-elim ( ¬p ( recompute dec p ))","searchableContent":"... | no ¬p | _ | _ = no λ where ( base₂a p _ _) → ⊥-elim ( ¬p ( recompute dec p ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":265,"title":"... | _ | no ¬p | _","content":"... | _ | no ¬p | _ = no λ where ( Base₂a { s } { eb } _ p _) → ⊥-elim ( ¬p ( recompute dec p ))","searchableContent":"... | _ | no ¬p | _ = no λ where ( base₂a { s } { eb } _ p _) → ⊥-elim ( ¬p ( recompute dec p ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":266,"title":"... | _ | _ | no ¬p","content":"... | _ | _ | no ¬p = no λ where ( Base₂a _ _ p ) → ⊥-elim ( ¬p ( recompute dec p ))","searchableContent":"... | _ | _ | no ¬p = no λ where ( base₂a _ _ p ) → ⊥-elim ( ¬p ( recompute dec p 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()","searchableContent":"dec-validaction { base₂a-action _} { s } { init _} . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":270,"title":"Dec-ValidAction { Base₂b-Action } { s } { SLOT } . dec","content":"Dec-ValidAction { Base₂b-Action } { s } { SLOT } . dec","searchableContent":"dec-validaction { base₂b-action } { s } { slot } . dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":271,"title":"with dec | dec | dec","content":"with dec | dec | dec","searchableContent":"with dec | dec | dec"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":272,"title":"... | yes x | yes y | yes z","content":"... | yes x | yes y | yes z = yes ( Base₂b x y z )","searchableContent":"... | yes x | yes y | yes z = yes ( base₂b x y z 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} { s } { INIT _} . dec = no λ ()","searchableContent":"dec-validaction { base₂b-action } { s } { init _} . dec = no λ ()"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":280,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":281,"title":"Dec-ValidUpdate","content":"Dec-ValidUpdate : ValidUpdate ⁇²","searchableContent":"dec-validupdate : validupdate ⁇²"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":282,"title":"Dec-ValidUpdate { IB-Recv-Update _} . dec","content":"Dec-ValidUpdate { IB-Recv-Update _} . dec = yes IB-Recv","searchableContent":"dec-validupdate { ib-recv-update _} . dec = yes ib-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":283,"title":"Dec-ValidUpdate { EB-Recv-Update _} . dec","content":"Dec-ValidUpdate { EB-Recv-Update _} . dec = yes EB-Recv","searchableContent":"dec-validupdate { eb-recv-update _} . dec = yes eb-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":284,"title":"Dec-ValidUpdate { VT-Recv-Update _} . dec","content":"Dec-ValidUpdate { VT-Recv-Update _} . dec = yes VT-Recv","searchableContent":"dec-validupdate { vt-recv-update _} . dec = yes vt-recv"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":286,"title":"mutual","content":"mutual","searchableContent":"mutual"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":287,"title":"data ValidTrace","content":"data ValidTrace : List (( Action × LeiosInput ) ⊎ FFDUpdate ) → Type where","searchableContent":"data validtrace : list (( action × leiosinput ) ⊎ ffdupdate ) → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":288,"title":"[]","content":"[] :","searchableContent":"[] :"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":289,"title":"─────────────","content":"─────────────","searchableContent":"─────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":290,"title":"ValidTrace []","content":"ValidTrace []","searchableContent":"validtrace []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":292,"title":"_/_∷_⊣_","content":"_/_∷_⊣_ : ∀ α i { αs } →","searchableContent":"_/_∷_⊣_ : ∀ α i { αs } →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":293,"title":"∀ ( tr","content":"∀ ( tr : ValidTrace αs ) →","searchableContent":"∀ ( tr : validtrace αs ) →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":294,"title":"∙ ValidAction α ( proj₁ ⟦ tr ⟧∗ ) i","content":"∙ ValidAction α ( proj₁ ⟦ tr ⟧∗ ) i","searchableContent":"∙ validaction α ( proj₁ ⟦ tr ⟧∗ ) i"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":295,"title":"───────────────────","content":"───────────────────","searchableContent":"───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":296,"title":"ValidTrace ( inj₁ ( α , i ) ∷ αs )","content":"ValidTrace ( inj₁ ( α , i ) ∷ αs )","searchableContent":"validtrace ( inj₁ ( α , i ) ∷ αs 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→"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":301,"title":"───────────────────","content":"───────────────────","searchableContent":"───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":302,"title":"ValidTrace ( inj₂ f ∷ αs )","content":"ValidTrace ( inj₂ f ∷ αs )","searchableContent":"validtrace ( inj₂ f ∷ αs )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":305,"title":"⟦_⟧∗","content":"⟦_⟧∗ : ∀ { αs : List (( Action × LeiosInput ) ⊎ FFDUpdate )} → ValidTrace αs → LeiosState × LeiosOutput","searchableContent":"⟦_⟧∗ : ∀ { αs : list (( action × leiosinput ) ⊎ ffdupdate )} → validtrace αs → leiosstate × leiosoutput"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":306,"title":"⟦ [] ⟧∗","content":"⟦ [] ⟧∗ = initLeiosState tt stakeDistribution tt pks , EMPTY","searchableContent":"⟦ [] ⟧∗ = initleiosstate tt stakedistribution tt pks , empty"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":307,"title":"where pks","content":"where pks = L.zip ( completeFinL numberOfParties ) ( L.replicate numberOfParties tt )","searchableContent":"where pks = l.zip ( completefinl numberofparties ) ( l.replicate numberofparties tt )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":308,"title":"⟦ _ / _ ∷ _ ⊣ vα ⟧∗","content":"⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧","searchableContent":"⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":309,"title":"⟦ _↥_ { IB-Recv-Update ib } tr vu ⟧∗","content":"⟦ _↥_ { IB-Recv-Update ib } tr vu ⟧∗ =","searchableContent":"⟦ _↥_ { ib-recv-update ib } tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":310,"title":"let ( s , o )","content":"let ( s , o ) = ⟦ tr ⟧∗","searchableContent":"let ( s , o ) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":311,"title":"in record s { FFDState","content":"in record s { FFDState = record ( FFDState s ) { inIBs = ib ∷ inIBs ( FFDState s )}} , o","searchableContent":"in record s { ffdstate = record ( ffdstate s ) { inibs = ib ∷ inibs ( ffdstate s )}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":312,"title":"⟦ _↥_ { EB-Recv-Update eb } tr vu ⟧∗","content":"⟦ _↥_ { EB-Recv-Update eb } tr vu ⟧∗ =","searchableContent":"⟦ _↥_ { eb-recv-update eb } tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":313,"title":"let ( s , o )","content":"let ( s , o ) = ⟦ tr ⟧∗","searchableContent":"let ( s , o ) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":314,"title":"in record s { FFDState","content":"in record s { FFDState = record ( FFDState s ) { inEBs = eb ∷ inEBs ( FFDState s )}} , o","searchableContent":"in record s { ffdstate = record ( ffdstate s ) { inebs = eb ∷ inebs ( ffdstate s )}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":315,"title":"⟦ _↥_ { VT-Recv-Update vt } tr vu ⟧∗","content":"⟦ _↥_ { VT-Recv-Update vt } tr vu ⟧∗ =","searchableContent":"⟦ _↥_ { vt-recv-update vt } tr vu ⟧∗ ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":316,"title":"let ( s , o )","content":"let ( s , o ) = ⟦ tr ⟧∗","searchableContent":"let ( s , o ) = ⟦ tr ⟧∗"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":317,"title":"in record s { FFDState","content":"in record s { FFDState = record ( FFDState s ) { inVTs = vt ∷ inVTs ( FFDState s )}} , o","searchableContent":"in record s { ffdstate = record ( ffdstate s ) { invts = vt ∷ invts ( ffdstate s )}} , o"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":319,"title":"Irr-ValidAction","content":"Irr-ValidAction : Irrelevant ( ValidAction α s i )","searchableContent":"irr-validaction : irrelevant ( validaction α s i )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":320,"title":"Irr-ValidAction ( IB-Role _ _ _) ( IB-Role _ _ _)","content":"Irr-ValidAction ( IB-Role _ _ _) ( IB-Role _ _ _) = refl","searchableContent":"irr-validaction ( ib-role _ _ _) ( ib-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":321,"title":"Irr-ValidAction ( EB-Role _ _ _) ( EB-Role _ _ _)","content":"Irr-ValidAction ( EB-Role _ _ _) ( EB-Role _ _ _) = refl","searchableContent":"irr-validaction ( eb-role _ _ _) ( eb-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":322,"title":"Irr-ValidAction ( VT-Role _ _ _) ( VT-Role _ _ _)","content":"Irr-ValidAction ( VT-Role _ _ _) ( VT-Role _ _ _) = refl","searchableContent":"irr-validaction ( vt-role _ _ _) ( vt-role _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":323,"title":"Irr-ValidAction ( No-IB-Role _ _) ( No-IB-Role _ _)","content":"Irr-ValidAction ( No-IB-Role _ _) ( No-IB-Role _ _) = refl","searchableContent":"irr-validaction ( no-ib-role _ _) ( no-ib-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":324,"title":"Irr-ValidAction ( No-EB-Role _ _) ( No-EB-Role _ _)","content":"Irr-ValidAction ( No-EB-Role _ _) ( No-EB-Role _ _) = refl","searchableContent":"irr-validaction ( no-eb-role _ _) ( no-eb-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":325,"title":"Irr-ValidAction ( No-VT-Role _ _) ( No-VT-Role _ _)","content":"Irr-ValidAction ( No-VT-Role _ _) ( No-VT-Role _ _) = refl","searchableContent":"irr-validaction ( no-vt-role _ _) ( no-vt-role _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":326,"title":"Irr-ValidAction ( Slot _ _ _) ( Slot _ _ _)","content":"Irr-ValidAction ( Slot _ _ _) ( Slot _ _ _) = refl","searchableContent":"irr-validaction ( slot _ _ _) ( slot _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":327,"title":"Irr-ValidAction Ftch Ftch","content":"Irr-ValidAction Ftch Ftch = refl","searchableContent":"irr-validaction ftch ftch = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":328,"title":"Irr-ValidAction Base₁ Base₁","content":"Irr-ValidAction Base₁ Base₁ = refl","searchableContent":"irr-validaction base₁ base₁ = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":329,"title":"Irr-ValidAction ( Base₂a _ _ _) ( Base₂a _ _ _)","content":"Irr-ValidAction ( Base₂a _ _ _) ( Base₂a _ _ _) = refl","searchableContent":"irr-validaction ( base₂a _ _ _) ( base₂a _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":330,"title":"Irr-ValidAction ( Base₂b _ _ _) ( Base₂b _ _ _)","content":"Irr-ValidAction ( Base₂b _ _ _) ( Base₂b _ _ _) = refl","searchableContent":"irr-validaction ( base₂b _ _ _) ( base₂b _ _ _) = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":332,"title":"Irr-ValidUpdate","content":"Irr-ValidUpdate : ∀ { f } → Irrelevant ( ValidUpdate f s )","searchableContent":"irr-validupdate : ∀ { f } → irrelevant ( validupdate f s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":333,"title":"Irr-ValidUpdate IB-Recv IB-Recv","content":"Irr-ValidUpdate IB-Recv IB-Recv = refl","searchableContent":"irr-validupdate ib-recv ib-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":334,"title":"Irr-ValidUpdate EB-Recv EB-Recv","content":"Irr-ValidUpdate EB-Recv EB-Recv = refl","searchableContent":"irr-validupdate eb-recv eb-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":335,"title":"Irr-ValidUpdate VT-Recv VT-Recv","content":"Irr-ValidUpdate VT-Recv VT-Recv = refl","searchableContent":"irr-validupdate vt-recv vt-recv = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":337,"title":"Irr-ValidTrace","content":"Irr-ValidTrace : ∀ { αs } → Irrelevant ( ValidTrace αs )","searchableContent":"irr-validtrace : ∀ { αs } → irrelevant ( validtrace αs )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":338,"title":"Irr-ValidTrace [] []","content":"Irr-ValidTrace [] [] = refl","searchableContent":"irr-validtrace [] [] = refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":339,"title":"Irr-ValidTrace ( α / i ∷ vαs ⊣ vα ) ( . α / . i ∷ vαs′ ⊣ vα′ )","content":"Irr-ValidTrace ( α / i ∷ vαs ⊣ vα ) ( . α / . i ∷ vαs′ ⊣ vα′ )","searchableContent":"irr-validtrace ( α / i ∷ vαs ⊣ vα ) ( . α / . i ∷ vαs′ ⊣ vα′ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":340,"title":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidAction vα vα′","content":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidAction vα vα′","searchableContent":"rewrite irr-validtrace vαs vαs′ | irr-validaction vα vα′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":341,"title":"= refl","content":"= refl","searchableContent":"= refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":342,"title":"Irr-ValidTrace ( vαs ↥ u ) ( vαs′ ↥ u′ )","content":"Irr-ValidTrace ( vαs ↥ u ) ( vαs′ ↥ u′ )","searchableContent":"irr-validtrace ( vαs ↥ u ) ( vαs′ ↥ u′ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":343,"title":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidUpdate u u′","content":"rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidUpdate u u′","searchableContent":"rewrite irr-validtrace vαs vαs′ | irr-validupdate u u′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":344,"title":"= refl","content":"= refl","searchableContent":"= refl"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":346,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":347,"title":"Dec-ValidTrace","content":"Dec-ValidTrace : ValidTrace ⁇¹","searchableContent":"dec-validtrace : validtrace ⁇¹"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":348,"title":"Dec-ValidTrace { tr } . dec with tr","content":"Dec-ValidTrace { tr } . dec with tr","searchableContent":"dec-validtrace { tr } . dec with tr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":349,"title":"... | []","content":"... | [] = yes []","searchableContent":"... | [] = yes []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":350,"title":"... | inj₁ ( α , i ) ∷ αs","content":"... | inj₁ ( α , i ) ∷ αs","searchableContent":"... | inj₁ ( α , i ) ∷ αs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":351,"title":"with ¿ ValidTrace αs ¿","content":"with ¿ ValidTrace αs ¿","searchableContent":"with ¿ validtrace αs ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":352,"title":"... | no ¬vαs","content":"... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs","searchableContent":"... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":353,"title":"... | yes vαs","content":"... | yes vαs","searchableContent":"... | yes vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":354,"title":"with ¿ ValidAction α ( proj₁ ⟦ vαs ⟧∗ ) i ¿","content":"with ¿ ValidAction α ( proj₁ ⟦ vαs ⟧∗ ) i ¿","searchableContent":"with ¿ validaction α ( proj₁ ⟦ vαs ⟧∗ ) i ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":355,"title":"... | no ¬vα","content":"... | no ¬vα = no λ where","searchableContent":"... | no ¬vα = no λ where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":356,"title":"(_ / _ ∷ tr ⊣ vα ) → ¬vα","content":"(_ / _ ∷ tr ⊣ vα ) → ¬vα","searchableContent":"(_ / _ ∷ tr ⊣ vα ) → ¬vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":357,"title":"$ subst (λ x → ValidAction α x i ) ( cong ( proj₁ ∘ ⟦_⟧∗ ) $ Irr-ValidTrace tr vαs ) vα","content":"$ subst (λ x → ValidAction α x i ) ( cong ( proj₁ ∘ ⟦_⟧∗ ) $ Irr-ValidTrace tr vαs ) vα","searchableContent":"$ subst (λ x → validaction α x i ) ( cong ( proj₁ ∘ ⟦_⟧∗ ) $ irr-validtrace tr vαs ) vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":358,"title":"... | yes vα","content":"... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα","searchableContent":"... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":359,"title":"Dec-ValidTrace { tr } . dec | inj₂ u ∷ αs","content":"Dec-ValidTrace { tr } . dec | inj₂ u ∷ αs","searchableContent":"dec-validtrace { tr } . dec | inj₂ u ∷ αs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":360,"title":"with ¿ ValidTrace αs ¿","content":"with ¿ ValidTrace αs ¿","searchableContent":"with ¿ validtrace αs ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":361,"title":"... | no ¬vαs","content":"... | no ¬vαs = no λ where ( vαs ↥ _) → ¬vαs vαs","searchableContent":"... | no ¬vαs = no λ where ( vαs ↥ _) → ¬vαs vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":362,"title":"... | yes vαs","content":"... | yes vαs","searchableContent":"... | yes vαs"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":363,"title":"with ¿ ValidUpdate u ( proj₁ ⟦ vαs ⟧∗ ) ¿","content":"with ¿ ValidUpdate u ( proj₁ ⟦ vαs ⟧∗ ) ¿","searchableContent":"with ¿ validupdate u ( proj₁ ⟦ vαs ⟧∗ ) ¿"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":364,"title":"... | yes vu","content":"... | yes vu = yes ( vαs ↥ vu )","searchableContent":"... | yes vu = yes ( vαs ↥ vu )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":365,"title":"... | no ¬vu","content":"... | no ¬vu = no λ where","searchableContent":"... | no ¬vu = no λ where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":366,"title":"( tr ↥ vu ) → ¬vu $ subst (λ x → ValidUpdate u x ) ( cong ( proj₁ ∘ ⟦_⟧∗ ) $ Irr-ValidTrace tr vαs ) vu","content":"( tr ↥ vu ) → ¬vu $ subst (λ x → ValidUpdate u x ) ( cong ( proj₁ ∘ ⟦_⟧∗ ) $ Irr-ValidTrace tr vαs ) vu","searchableContent":"( tr ↥ vu ) → ¬vu $ subst (λ x → validupdate u x ) ( cong ( proj₁ ∘ ⟦_⟧∗ ) $ irr-validtrace tr vαs ) vu"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":368,"title":"data _⇑_","content":"data _⇑_ : LeiosState → LeiosState → Type where","searchableContent":"data _⇑_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":370,"title":"UpdateIB","content":"UpdateIB : ∀ { s ib } → let open LeiosState s renaming ( FFDState to ffds ) in","searchableContent":"updateib : ∀ { s ib } → let open leiosstate s renaming ( ffdstate to ffds ) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":371,"title":"s ⇑ record s { FFDState","content":"s ⇑ record s { FFDState = record ffds { inIBs = ib ∷ inIBs ffds } }","searchableContent":"s ⇑ record s { ffdstate = record ffds { inibs = ib ∷ inibs ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":373,"title":"UpdateEB","content":"UpdateEB : ∀ { s eb } → let open LeiosState s renaming ( FFDState to ffds ) in","searchableContent":"updateeb : ∀ { s eb } → let open leiosstate s renaming ( ffdstate to ffds ) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":374,"title":"s ⇑ record s { FFDState","content":"s ⇑ record s { FFDState = record ffds { inEBs = eb ∷ inEBs ffds } }","searchableContent":"s ⇑ record s { ffdstate = record ffds { inebs = eb ∷ inebs ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":376,"title":"UpdateVT","content":"UpdateVT : ∀ { s vt } → let open LeiosState s renaming ( FFDState to ffds ) in","searchableContent":"updatevt : ∀ { s vt } → let open leiosstate s renaming ( ffdstate to ffds ) in"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":377,"title":"s ⇑ record s { FFDState","content":"s ⇑ record s { FFDState = record ffds { inVTs = vt ∷ inVTs ffds } }","searchableContent":"s ⇑ record s { ffdstate = record ffds { invts = vt ∷ invts ffds } }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":379,"title":"data LocalStep","content":"data LocalStep : LeiosState → LeiosState → Type where","searchableContent":"data localstep : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":381,"title":"StateStep","content":"StateStep : ∀ { s i o s′ } →","searchableContent":"statestep : ∀ { s i o s′ } →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":382,"title":"∙ just s -⟦ i / o ⟧⇀ s′","content":"∙ just s -⟦ i / o ⟧⇀ s′","searchableContent":"∙ just s -⟦ i / o ⟧⇀ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":383,"title":"───────────────────","content":"───────────────────","searchableContent":"───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":384,"title":"LocalStep s s′","content":"LocalStep s s′","searchableContent":"localstep s s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":386,"title":"UpdateState","content":"UpdateState : ∀ { s s′ } →","searchableContent":"updatestate : ∀ { s s′ } →"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":387,"title":"∙ s ⇑ s′","content":"∙ s ⇑ s′","searchableContent":"∙ s ⇑ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":388,"title":"───────────────────","content":"───────────────────","searchableContent":"───────────────────"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":389,"title":"LocalStep s s′","content":"LocalStep s s′","searchableContent":"localstep s s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":391,"title":"-- TODO","content":"-- TODO: add base layer update","searchableContent":"-- todo: add base layer update"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":393,"title":"getLabel","content":"getLabel : just s -⟦ i / o ⟧⇀ s′ → Action","searchableContent":"getlabel : just s -⟦ i / o ⟧⇀ s′ → action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":394,"title":"getLabel ( Slot { s } _ _ _)","content":"getLabel ( Slot { s } _ _ _) = Slot-Action ( slot s )","searchableContent":"getlabel ( slot { s } _ _ _) = slot-action ( slot s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":395,"title":"getLabel Ftch","content":"getLabel Ftch = Ftch-Action","searchableContent":"getlabel ftch = ftch-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":396,"title":"getLabel Base₁","content":"getLabel Base₁ = Base₁-Action","searchableContent":"getlabel base₁ = base₁-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":397,"title":"getLabel ( Base₂a { s } { eb } _ _ _)","content":"getLabel ( Base₂a { s } { eb } _ _ _) = Base₂a-Action eb","searchableContent":"getlabel ( base₂a { s } { eb } _ _ _) = base₂a-action eb"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":398,"title":"getLabel ( Base₂b _ _ _)","content":"getLabel ( Base₂b _ _ _) = Base₂b-Action","searchableContent":"getlabel ( base₂b _ _ _) = base₂b-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":399,"title":"getLabel ( Roles ( IB-Role { s } _ _ _))","content":"getLabel ( Roles ( IB-Role { s } _ _ _)) = IB-Role-Action ( slot s )","searchableContent":"getlabel ( roles ( ib-role { s } _ _ _)) = ib-role-action ( slot s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":400,"title":"getLabel ( Roles ( EB-Role { s } _ _ _))","content":"getLabel ( Roles ( EB-Role { s } _ _ _)) = EB-Role-Action ( slot s ) ( map getIBRef $ filter ( _∈ᴮ slice L ( slot s ) 3 ) ( IBs s ))","searchableContent":"getlabel ( roles ( eb-role { s } _ _ _)) = eb-role-action ( slot s ) ( map getibref $ filter ( _∈ᴮ slice l ( slot s ) 3 ) ( ibs s ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":401,"title":"getLabel ( Roles ( VT-Role { s } _ _ _))","content":"getLabel ( Roles ( VT-Role { s } _ _ _)) = VT-Role-Action ( slot s )","searchableContent":"getlabel ( roles ( vt-role { s } _ _ _)) = vt-role-action ( slot s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":402,"title":"getLabel ( Roles ( No-IB-Role _ _))","content":"getLabel ( Roles ( No-IB-Role _ _)) = No-IB-Role-Action","searchableContent":"getlabel ( roles ( no-ib-role _ _)) = no-ib-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":403,"title":"getLabel ( Roles ( No-EB-Role _ _))","content":"getLabel ( Roles ( No-EB-Role _ _)) = No-EB-Role-Action","searchableContent":"getlabel ( roles ( no-eb-role _ _)) = no-eb-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":404,"title":"getLabel ( Roles ( No-VT-Role _ _))","content":"getLabel ( Roles ( No-VT-Role _ _)) = No-VT-Role-Action","searchableContent":"getlabel ( roles ( no-vt-role _ _)) = no-vt-role-action"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":406,"title":"ValidAction-sound","content":"ValidAction-sound : ( vα : ValidAction α s i ) → let ( s′ , o ) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′","searchableContent":"validaction-sound : ( vα : validaction α s i ) → let ( s′ , o ) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":407,"title":"ValidAction-sound ( Slot x x₁ x₂ )","content":"ValidAction-sound ( Slot x x₁ x₂ ) = Slot { rbs = [] } ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ )","searchableContent":"validaction-sound ( slot x x₁ x₂ ) = slot { rbs = [] } ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":408,"title":"ValidAction-sound Ftch","content":"ValidAction-sound Ftch = Ftch","searchableContent":"validaction-sound ftch = ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":409,"title":"ValidAction-sound Base₁","content":"ValidAction-sound Base₁ = Base₁","searchableContent":"validaction-sound base₁ = base₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":410,"title":"ValidAction-sound ( Base₂a x x₁ x₂ )","content":"ValidAction-sound ( Base₂a x x₁ x₂ ) = Base₂a ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ )","searchableContent":"validaction-sound ( base₂a x x₁ x₂ ) = base₂a ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":411,"title":"ValidAction-sound ( Base₂b x x₁ x₂ )","content":"ValidAction-sound ( Base₂b x x₁ x₂ ) = Base₂b ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ )","searchableContent":"validaction-sound ( base₂b x x₁ x₂ ) = base₂b ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":412,"title":"ValidAction-sound ( IB-Role x x₁ x₂ )","content":"ValidAction-sound ( IB-Role x x₁ x₂ ) = Roles ( IB-Role ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ ))","searchableContent":"validaction-sound ( ib-role x x₁ x₂ ) = roles ( ib-role ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":413,"title":"ValidAction-sound ( EB-Role x x₁ x₂ )","content":"ValidAction-sound ( EB-Role x x₁ x₂ ) = Roles ( EB-Role ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ ))","searchableContent":"validaction-sound ( eb-role x x₁ x₂ ) = roles ( eb-role ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":414,"title":"ValidAction-sound ( VT-Role x x₁ x₂ )","content":"ValidAction-sound ( VT-Role x x₁ x₂ ) = Roles ( VT-Role ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ ))","searchableContent":"validaction-sound ( vt-role x x₁ x₂ ) = roles ( vt-role ( recompute dec x ) ( recompute dec x₁ ) ( recompute dec x₂ ))"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":415,"title":"ValidAction-sound ( No-IB-Role x x₁ )","content":"ValidAction-sound ( No-IB-Role x x₁ ) = Roles ( No-IB-Role x x₁ )","searchableContent":"validaction-sound ( no-ib-role x x₁ ) = roles ( no-ib-role x x₁ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":416,"title":"ValidAction-sound ( No-EB-Role x x₁ )","content":"ValidAction-sound ( No-EB-Role x x₁ ) = Roles ( No-EB-Role x x₁ )","searchableContent":"validaction-sound ( no-eb-role x x₁ ) = roles ( no-eb-role x x₁ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":417,"title":"ValidAction-sound ( No-VT-Role x x₁ )","content":"ValidAction-sound ( No-VT-Role x x₁ ) = Roles ( No-VT-Role x x₁ )","searchableContent":"validaction-sound ( no-vt-role x x₁ ) = roles ( no-vt-role x x₁ )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":419,"title":"ValidAction-complete","content":"ValidAction-complete : ( st : just s -⟦ i / o ⟧⇀ s′ ) → ValidAction ( getLabel st ) s i","searchableContent":"validaction-complete : ( st : just s -⟦ i / o ⟧⇀ s′ ) → validaction ( getlabel st ) s i"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":420,"title":"ValidAction-complete { s } { s′ } ( Roles ( IB-Role { s } { π } { ffds' } x x₁ _))","content":"ValidAction-complete { s } { s′ } ( Roles ( IB-Role { s } { π } { ffds' } x x₁ _)) =","searchableContent":"validaction-complete { s } { s′ } ( roles ( ib-role { s } { π } { ffds' } x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":421,"title":"let b","content":"let b = record { txs = ToPropose s }","searchableContent":"let b = record { txs = topropose s }"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":422,"title":"h","content":"h = mkIBHeader ( slot s ) id tt sk-IB ( ToPropose s )","searchableContent":"h = mkibheader ( slot s ) id tt sk-ib ( topropose s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":423,"title":"pr","content":"pr = proj₂ ( FFD.Send-total { FFDState s } { ibHeader h } { just ( ibBody b )})","searchableContent":"pr = proj₂ ( ffd.send-total { ffdstate s } { ibheader h } { just ( ibbody b )})"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":424,"title":"in IB-Role { s } x x₁ pr","content":"in IB-Role { s } x x₁ pr","searchableContent":"in ib-role { s } x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":425,"title":"ValidAction-complete { s } ( Roles ( EB-Role x x₁ _))","content":"ValidAction-complete { s } ( Roles ( EB-Role x x₁ _)) =","searchableContent":"validaction-complete { s } ( roles ( eb-role x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":426,"title":"let LI","content":"let LI = map getIBRef $ filter ( _∈ᴮ slice L ( slot s ) 3 ) ( IBs s )","searchableContent":"let li = map getibref $ filter ( _∈ᴮ slice l ( slot s ) 3 ) ( ibs s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":427,"title":"h","content":"h = mkEB ( slot s ) id tt sk-EB LI []","searchableContent":"h = mkeb ( slot s ) id tt sk-eb li []"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":428,"title":"pr","content":"pr = proj₂ ( FFD.Send-total { FFDState s } { ebHeader h } { nothing })","searchableContent":"pr = proj₂ ( ffd.send-total { ffdstate s } { ebheader h } { nothing })"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":429,"title":"in EB-Role { s } x x₁ pr","content":"in EB-Role { s } x x₁ pr","searchableContent":"in eb-role { s } x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":430,"title":"ValidAction-complete { s } ( Roles ( VT-Role x x₁ _))","content":"ValidAction-complete { s } ( Roles ( VT-Role x x₁ _)) =","searchableContent":"validaction-complete { s } ( roles ( vt-role x x₁ _)) ="},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":431,"title":"let EBs'","content":"let EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L ( slot s ) 1 ) ( EBs s )","searchableContent":"let ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l ( slot s ) 1 ) ( ebs s )"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":432,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":433,"title":"pr","content":"pr = proj₂ ( FFD.Send-total { FFDState s } { vtHeader votes } { nothing })","searchableContent":"pr = proj₂ ( ffd.send-total { ffdstate s } { vtheader votes } { nothing })"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":434,"title":"in VT-Role { s } x x₁ pr","content":"in VT-Role { s } x x₁ pr","searchableContent":"in vt-role { s } x x₁ pr"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":435,"title":"ValidAction-complete ( Roles ( No-IB-Role x x₁ ))","content":"ValidAction-complete ( Roles ( No-IB-Role x x₁ )) = No-IB-Role x x₁","searchableContent":"validaction-complete ( roles ( no-ib-role x x₁ )) = no-ib-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":436,"title":"ValidAction-complete ( Roles ( No-EB-Role x x₁ ))","content":"ValidAction-complete ( Roles ( No-EB-Role x x₁ )) = No-EB-Role x x₁","searchableContent":"validaction-complete ( roles ( no-eb-role x x₁ )) = no-eb-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":437,"title":"ValidAction-complete ( Roles ( No-VT-Role x x₁ ))","content":"ValidAction-complete ( Roles ( No-VT-Role x x₁ )) = No-VT-Role x x₁","searchableContent":"validaction-complete ( roles ( no-vt-role x x₁ )) = no-vt-role x x₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":438,"title":"ValidAction-complete Ftch","content":"ValidAction-complete Ftch = Ftch","searchableContent":"validaction-complete ftch = ftch"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":439,"title":"ValidAction-complete Base₁","content":"ValidAction-complete Base₁ = Base₁","searchableContent":"validaction-complete base₁ = base₁"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":440,"title":"ValidAction-complete ( Base₂a x x₁ x₂ )","content":"ValidAction-complete ( Base₂a x x₁ x₂ ) = Base₂a x x₁ x₂","searchableContent":"validaction-complete ( base₂a x x₁ x₂ ) = base₂a x x₁ x₂"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":441,"title":"ValidAction-complete ( Base₂b x x₁ x₂ )","content":"ValidAction-complete ( Base₂b x x₁ x₂ ) = Base₂b x x₁ x₂","searchableContent":"validaction-complete ( base₂b x x₁ x₂ ) = base₂b x x₁ x₂"},{"moduleName":"Leios.Short.Trace.Verifier","path":"Leios.Short.Trace.Verifier.html","group":"Leios","lineNumber":442,"title":"ValidAction-complete { s } ( Slot x x₁ _)","content":"ValidAction-complete { s } ( Slot x x₁ _) = Slot x x₁ ( proj₂ ( proj₂ ( FFD.Fetch-total { FFDState s })))","searchableContent":"validaction-complete { s } ( slot x x₁ _) = slot x x₁ ( proj₂ ( proj₂ ( ffd.fetch-total { ffdstate s })))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":7,"title":"Leios.Short","content":"Leios.Short","searchableContent":"leios.short"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":9,"title":"code{white-space","content":"code{white-space: pre-wrap;}","searchableContent":"code{white-space: pre-wrap;}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":10,"title":"span.smallcaps{font-variant","content":"span.smallcaps{font-variant: small-caps;}","searchableContent":"span.smallcaps{font-variant: small-caps;}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":11,"title":"div.columns{display","content":"div.columns{display: flex; gap: min(4vw, 1.5em);}","searchableContent":"div.columns{display: flex; gap: min(4vw, 1.5em);}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":12,"title":"div.column{flex","content":"div.column{flex: auto; overflow-x: auto;}","searchableContent":"div.column{flex: auto; overflow-x: auto;}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":13,"title":"div.hanging-indent{margin-left","content":"div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}","searchableContent":"div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":14,"title":"/* The extra [class] is a hack that increases specificity enough to","content":"/* The extra [class] is a hack that increases specificity enough to","searchableContent":"/* the extra [class] is a hack that increases specificity enough to"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":15,"title":"override a similar rule in reveal.js */","content":"override a similar rule in reveal.js */","searchableContent":"override a similar rule in reveal.js */"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":16,"title":"ul.task-list[class]{list-style","content":"ul.task-list[class]{list-style: none;}","searchableContent":"ul.task-list[class]{list-style: none;}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":17,"title":"ul.task-list li input[type","content":"ul.task-list li input[type=\"checkbox\"] {","searchableContent":"ul.task-list li input[type=\"checkbox\"] {"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":18,"title":"font-size","content":"font-size: inherit;","searchableContent":"font-size: inherit;"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":19,"title":"width","content":"width: 0.8em;","searchableContent":"width: 0.8em;"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":20,"title":"margin","content":"margin: 0 0.8em 0.2em -1.6em;","searchableContent":"margin: 0 0.8em 0.2em -1.6em;"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":21,"title":"vertical-align","content":"vertical-align: middle;","searchableContent":"vertical-align: middle;"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":22,"title":"}","content":"}","searchableContent":"}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":23,"title":".display.math{display","content":".display.math{display: block; text-align: center; margin: 0.5rem auto;}","searchableContent":".display.math{display: block; text-align: center; margin: 0.5rem auto;}"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":28,"title":"Short-Pipeline Leios","content":"Short-Pipeline Leios","searchableContent":"short-pipeline leios"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":29,"title":"","content":"-->","searchableContent":"-->"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":43,"title":"This document is a specification of Short-Pipeline Leios, usually","content":"This document is a specification of Short-Pipeline Leios, usually","searchableContent":"this document is a specification of short-pipeline leios, usually"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":44,"title":"abbreviated as Short Leios. On a high level, the pipeline looks like","content":"abbreviated as Short Leios. On a high level, the pipeline looks like","searchableContent":"abbreviated as short leios. on a high level, the pipeline looks like"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":45,"title":"this","content":"this:","searchableContent":"this:"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":47,"title":"If elected, propose IB","content":"If elected, propose IB","searchableContent":"if elected, propose ib"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":48,"title":"Wait","content":"Wait","searchableContent":"wait"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":49,"title":"Wait","content":"Wait","searchableContent":"wait"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":50,"title":"If elected, propose EB","content":"If elected, propose EB","searchableContent":"if elected, propose eb"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":51,"title":"If elected, vote If elected, propose RB","content":"If elected, vote If elected, propose RB","searchableContent":"if elected, vote if elected, propose rb"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":53,"title":"Upkeep","content":"Upkeep","searchableContent":"upkeep"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":54,"title":"A node that never produces a block even though it could is not","content":"A node that never produces a block even though it could is not","searchableContent":"a node that never produces a block even though it could is not"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":55,"title":"supposed to be an honest node, and we prevent that by tracking whether a","content":"supposed to be an honest node, and we prevent that by tracking whether a","searchableContent":"supposed to be an honest node, and we prevent that by tracking whether a"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":56,"title":"node has checked if it can make a block in a particular slot.","content":"node has checked if it can make a block in a particular slot.","searchableContent":"node has checked if it can make a block in a particular slot."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":57,"title":"LeiosState contains a set of SlotUpkeep and we","content":"LeiosState contains a set of SlotUpkeep and we","searchableContent":"leiosstate contains a set of slotupkeep and we"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":58,"title":"ensure that this set contains all elements before we can advance to the","content":"ensure that this set contains all elements before we can advance to the","searchableContent":"ensure that this set contains all elements before we can advance to the"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":59,"title":"next slot, resetting this field to the empty set.","content":"next slot, resetting this field to the empty set.","searchableContent":"next slot, resetting this field to the empty set."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":60,"title":"data SlotUpkeep","content":"data SlotUpkeep : Type where","searchableContent":"data slotupkeep : type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":61,"title":"Base IB-Role EB-Role VT-Role","content":"Base IB-Role EB-Role VT-Role : SlotUpkeep","searchableContent":"base ib-role eb-role vt-role : slotupkeep"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":63,"title":"","content":"-->","searchableContent":"-->"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":85,"title":"Block/Vote production rules","content":"Block/Vote production rules","searchableContent":"block/vote production rules"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":86,"title":"We now define the rules for block production given by the relation","content":"We now define the rules for block production given by the relation","searchableContent":"we now define the rules for block production given by the relation"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":87,"title":"_↝_ . These are split in two","content":"_↝_ . These are split in two:","searchableContent":"_↝_ . these are split in two:"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":89,"title":"Positive rules, when we do need to create a block.","content":"Positive rules, when we do need to create a block.","searchableContent":"positive rules, when we do need to create a block."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":90,"title":"Negative rules, when we cannot create a block.","content":"Negative rules, when we cannot create a block.","searchableContent":"negative rules, when we cannot create a block."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":92,"title":"The purpose of the negative rules is to properly adjust the upkeep if","content":"The purpose of the negative rules is to properly adjust the upkeep if","searchableContent":"the purpose of the negative rules is to properly adjust the upkeep if"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":93,"title":"we cannot make a block.","content":"we cannot make a block.","searchableContent":"we cannot make a block."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":94,"title":"Note that _↝_ , starting with an empty upkeep can always","content":"Note that _↝_ , starting with an empty upkeep can always","searchableContent":"note that _↝_ , starting with an empty upkeep can always"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":95,"title":"make exactly three steps corresponding to the three types of Leios","content":"make exactly three steps corresponding to the three types of Leios","searchableContent":"make exactly three steps corresponding to the three types of leios"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":96,"title":"specific blocks.","content":"specific blocks.","searchableContent":"specific blocks."},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":97,"title":"data _↝_","content":"data _↝_ : LeiosState → LeiosState → Type where","searchableContent":"data _↝_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":99,"title":"Positive rules","content":"Positive rules","searchableContent":"positive rules"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":100,"title":"IB-Role","content":"IB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"ib-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":101,"title":"b","content":"b = ibBody ( record { txs = ToPropose })","searchableContent":"b = ibbody ( record { txs = topropose })"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":102,"title":"h","content":"h = ibHeader ( mkIBHeader slot id π sk-IB ToPropose )","searchableContent":"h = ibheader ( mkibheader slot id π sk-ib topropose )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":103,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":104,"title":"∙ needsUpkeep IB-Role","content":"∙ needsUpkeep IB-Role","searchableContent":"∙ needsupkeep ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":105,"title":"∙ canProduceIB slot sk-IB ( stake s ) π","content":"∙ canProduceIB slot sk-IB ( stake s ) π","searchableContent":"∙ canproduceib slot sk-ib ( stake s ) π"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":106,"title":"∙ ffds FFD.-⟦ Send h ( just b ) / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send h ( just b ) / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send h ( just b ) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":107,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":108,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":110,"title":"EB-Role","content":"EB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"eb-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":111,"title":"LI","content":"LI = map getIBRef $ filter ( _∈ᴮ slice L slot 3 ) IBs","searchableContent":"li = map getibref $ filter ( _∈ᴮ slice l slot 3 ) ibs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":112,"title":"h","content":"h = mkEB slot id π sk-EB LI []","searchableContent":"h = mkeb slot id π sk-eb li []"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":113,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":114,"title":"∙ needsUpkeep EB-Role","content":"∙ needsUpkeep EB-Role","searchableContent":"∙ needsupkeep eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":115,"title":"∙ canProduceEB slot sk-EB ( stake s ) π","content":"∙ canProduceEB slot sk-EB ( stake s ) π","searchableContent":"∙ canproduceeb slot sk-eb ( stake s ) π"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":116,"title":"∙ ffds FFD.-⟦ Send ( ebHeader h ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( ebHeader h ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( ebheader h ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":117,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":118,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":120,"title":"VT-Role","content":"VT-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"vt-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":121,"title":"EBs'","content":"EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L slot 1 ) EBs","searchableContent":"ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l slot 1 ) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":122,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":123,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":124,"title":"∙ needsUpkeep VT-Role","content":"∙ needsUpkeep VT-Role","searchableContent":"∙ needsupkeep vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":125,"title":"∙ canProduceV slot sk-VT ( stake s )","content":"∙ canProduceV slot sk-VT ( stake s )","searchableContent":"∙ canproducev slot sk-vt ( stake s )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":126,"title":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( vtheader votes ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":127,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":128,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } VT-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":130,"title":"Negative rules","content":"Negative rules","searchableContent":"negative rules"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":131,"title":"No-IB-Role","content":"No-IB-Role : let open LeiosState s in","searchableContent":"no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":132,"title":"∙ needsUpkeep IB-Role","content":"∙ needsUpkeep IB-Role","searchableContent":"∙ needsupkeep ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":133,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π )","content":"∙ (∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π )","searchableContent":"∙ (∀ π → ¬ canproduceib slot sk-ib ( stake s ) π )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":134,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":135,"title":"s ↝ addUpkeep s IB-Role","content":"s ↝ addUpkeep s IB-Role","searchableContent":"s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":137,"title":"No-EB-Role","content":"No-EB-Role : let open LeiosState s in","searchableContent":"no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":138,"title":"∙ needsUpkeep EB-Role","content":"∙ needsUpkeep EB-Role","searchableContent":"∙ needsupkeep eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":139,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π )","content":"∙ (∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π )","searchableContent":"∙ (∀ π → ¬ canproduceeb slot sk-eb ( stake s ) π )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":140,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":141,"title":"s ↝ addUpkeep s EB-Role","content":"s ↝ addUpkeep s EB-Role","searchableContent":"s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":143,"title":"No-VT-Role","content":"No-VT-Role : let open LeiosState s in","searchableContent":"no-vt-role : let open leiosstate s in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":144,"title":"∙ needsUpkeep VT-Role","content":"∙ needsUpkeep VT-Role","searchableContent":"∙ needsupkeep vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":145,"title":"∙ ¬ canProduceV slot sk-VT ( stake s )","content":"∙ ¬ canProduceV slot sk-VT ( stake s )","searchableContent":"∙ ¬ canproducev slot sk-vt ( stake s )"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":146,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":147,"title":"s ↝ addUpkeep s VT-Role","content":"s ↝ addUpkeep s VT-Role","searchableContent":"s ↝ addupkeep s vt-role"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":149,"title":"Uniform short-pipeline","content":"Uniform short-pipeline","searchableContent":"uniform short-pipeline"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":150,"title":"stage","content":"stage : ℕ → ⦃ _ : NonZero L ⦄ → ℕ","searchableContent":"stage : ℕ → ⦃ _ : nonzero l ⦄ → ℕ"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":151,"title":"stage s","content":"stage s = s / L","searchableContent":"stage s = s / l"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":153,"title":"beginningOfStage","content":"beginningOfStage : ℕ → Type","searchableContent":"beginningofstage : ℕ → type"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":154,"title":"beginningOfStage s","content":"beginningOfStage s = stage s * L ≡ s","searchableContent":"beginningofstage s = stage s * l ≡ s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":156,"title":"allDone","content":"allDone : LeiosState → Type","searchableContent":"alldone : leiosstate → type"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":157,"title":"allDone s","content":"allDone s =","searchableContent":"alldone s ="},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":158,"title":"let open LeiosState s","content":"let open LeiosState s","searchableContent":"let open leiosstate s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":159,"title":"in ( beginningOfStage slot × Upkeep ≡ᵉ fromList ( IB-Role ∷ EB-Role ∷ VT-Role ∷ Base ∷ [] ))","content":"in ( beginningOfStage slot × Upkeep ≡ᵉ fromList ( IB-Role ∷ EB-Role ∷ VT-Role ∷ Base ∷ [] ))","searchableContent":"in ( beginningofstage slot × upkeep ≡ᵉ fromlist ( ib-role ∷ eb-role ∷ vt-role ∷ base ∷ [] ))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":160,"title":"⊎ ( ¬ beginningOfStage slot × Upkeep ≡ᵉ fromList ( IB-Role ∷ VT-Role ∷ Base ∷ [] ))","content":"⊎ ( ¬ beginningOfStage slot × Upkeep ≡ᵉ fromList ( IB-Role ∷ VT-Role ∷ Base ∷ [] ))","searchableContent":"⊎ ( ¬ beginningofstage slot × upkeep ≡ᵉ fromlist ( ib-role ∷ vt-role ∷ base ∷ [] ))"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":162,"title":"data _-⟦_/_⟧⇀_","content":"data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":164,"title":"Initialization","content":"Initialization","searchableContent":"initialization"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":165,"title":"Init","content":"Init :","searchableContent":"init :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":166,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","content":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":167,"title":"∙ initBaseState B.-⟦ B.INIT ( V-chkCerts pks ) / B.STAKE SD ⟧⇀ bs'","content":"∙ initBaseState B.-⟦ B.INIT ( V-chkCerts pks ) / B.STAKE SD ⟧⇀ bs'","searchableContent":"∙ initbasestate b.-⟦ b.init ( v-chkcerts pks ) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":168,"title":"────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":169,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","content":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","searchableContent":"nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":171,"title":"Network and Ledger","content":"Network and Ledger","searchableContent":"network and ledger"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":172,"title":"Slot","content":"Slot : let open LeiosState s renaming ( FFDState to ffds ; BaseState to bs ) in","searchableContent":"slot : let open leiosstate s renaming ( ffdstate to ffds ; basestate to bs ) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":173,"title":"∙ allDone s","content":"∙ allDone s","searchableContent":"∙ alldone s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":174,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","content":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":175,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":176,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":177,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","content":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":178,"title":"{ FFDState","content":"{ FFDState = ffds'","searchableContent":"{ ffdstate = ffds'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":179,"title":"; BaseState","content":"; BaseState = bs'","searchableContent":"; basestate = bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":180,"title":"; Ledger","content":"; Ledger = constructLedger rbs","searchableContent":"; ledger = constructledger rbs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":181,"title":"; slot","content":"; slot = suc slot","searchableContent":"; slot = suc slot"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":182,"title":"; Upkeep","content":"; Upkeep = ∅","searchableContent":"; upkeep = ∅"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":183,"title":"} ↑ L.filter ( isValid? s ) msgs","content":"} ↑ L.filter ( isValid? s ) msgs","searchableContent":"} ↑ l.filter ( isvalid? s ) msgs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":185,"title":"Ftch","content":"Ftch :","searchableContent":"ftch :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":186,"title":"────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":187,"title":"just s -⟦ FTCH-LDG / FTCH-LDG ( LeiosState.Ledger s ) ⟧⇀ s","content":"just s -⟦ FTCH-LDG / FTCH-LDG ( LeiosState.Ledger s ) ⟧⇀ s","searchableContent":"just s -⟦ ftch-ldg / ftch-ldg ( leiosstate.ledger s ) ⟧⇀ s"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":189,"title":"Base chain","content":"Base chain","searchableContent":"base chain"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":190,"title":"Note","content":"Note: Submitted data to the base chain is only taken into account if the","searchableContent":"note: submitted data to the base chain is only taken into account if the"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":191,"title":"party submitting is the block producer on the base chain for the given","content":"party submitting is the block producer on the base chain for the given","searchableContent":"party submitting is the block producer on the base chain for the given"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":192,"title":"slot","content":"slot","searchableContent":"slot"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":193,"title":"Base₁","content":"Base₁ :","searchableContent":"base₁ :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":194,"title":"───────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":195,"title":"just s -⟦ SUBMIT ( inj₂ txs ) / EMPTY ⟧⇀ record s { ToPropose","content":"just s -⟦ SUBMIT ( inj₂ txs ) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"just s -⟦ submit ( inj₂ txs ) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":197,"title":"Base₂a","content":"Base₂a : let open LeiosState s renaming ( BaseState to bs ) in","searchableContent":"base₂a : let open leiosstate s renaming ( basestate to bs ) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":198,"title":"∙ needsUpkeep Base","content":"∙ needsUpkeep Base","searchableContent":"∙ needsupkeep base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":199,"title":"∙ eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs","content":"∙ eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs","searchableContent":"∙ eb ∈ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2 ) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":200,"title":"∙ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs'","content":"∙ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.submit ( this eb ) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":201,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":202,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState","content":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":204,"title":"Base₂b","content":"Base₂b : let open LeiosState s renaming ( BaseState to bs ) in","searchableContent":"base₂b : let open leiosstate s renaming ( basestate to bs ) in"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":205,"title":"∙ needsUpkeep Base","content":"∙ needsUpkeep Base","searchableContent":"∙ needsupkeep base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":206,"title":"∙ [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs","content":"∙ [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2 ) EBs","searchableContent":"∙ [] ≡ filter (λ eb → isvotecertified s eb × eb ∈ᴮ slice l slot 2 ) ebs"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":207,"title":"∙ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs'","content":"∙ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.submit ( that topropose ) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":208,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":209,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState","content":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":211,"title":"Protocol rules","content":"Protocol rules","searchableContent":"protocol rules"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":212,"title":"Roles","content":"Roles :","searchableContent":"roles :"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":213,"title":"∙ s ↝ s'","content":"∙ s ↝ s'","searchableContent":"∙ s ↝ s'"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":214,"title":"─────────────────────────────","content":"─────────────────────────────","searchableContent":"─────────────────────────────"},{"moduleName":"Leios.Short","path":"Leios.Short.html","group":"Leios","lineNumber":215,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","content":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"just s -⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":2,"title":"Leios.Simplified.Deterministic --{-# OPTIONS --safe #-}","content":"Leios.Simplified.Deterministic --{-# OPTIONS --safe #-}","searchableContent":"leios.simplified.deterministic --{-# options --safe #-}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":3,"title":"{-# OPTIONS --allow-unsolved-metas #-}","content":"{-# OPTIONS --allow-unsolved-metas #-}","searchableContent":"{-# options --allow-unsolved-metas #-}"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":5,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":6,"title":"-- Deterministic variant of simple Leios","content":"-- Deterministic variant of simple Leios","searchableContent":"-- deterministic variant of simple leios"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":7,"title":"--------------------------------------------------------------------------------","content":"--------------------------------------------------------------------------------","searchableContent":"--------------------------------------------------------------------------------"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":9,"title":"open import Leios.Prelude hiding ( id )","content":"open import Leios.Prelude hiding ( id )","searchableContent":"open import leios.prelude hiding ( id )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":10,"title":"open import Prelude.Init using ( ∃₂-syntax )","content":"open import Prelude.Init using ( ∃₂-syntax )","searchableContent":"open import prelude.init using ( ∃₂-syntax )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":11,"title":"open import Leios.FFD","content":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":13,"title":"import Data.List as L","content":"import Data.List as L","searchableContent":"import data.list as l"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":14,"title":"open import Data.List.Relation.Unary.Any using ( here )","content":"open import Data.List.Relation.Unary.Any using ( here )","searchableContent":"open import data.list.relation.unary.any using ( here )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":16,"title":"open import Leios.SpecStructure","content":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":18,"title":"module Leios.Simplified.Deterministic ( ⋯","content":"module Leios.Simplified.Deterministic ( ⋯ : SpecStructure 2 ) ( let open SpecStructure ⋯ ) ( Λ μ : ℕ ) where","searchableContent":"module leios.simplified.deterministic ( ⋯ : specstructure 2 ) ( let open specstructure ⋯ ) ( λ μ : ℕ ) where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":20,"title":"import Leios.Simplified","content":"import Leios.Simplified","searchableContent":"import leios.simplified"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":21,"title":"open import Leios.Simplified ⋯ Λ μ hiding ( _-⟦_/_⟧⇀_ )","content":"open import Leios.Simplified ⋯ Λ μ hiding ( _-⟦_/_⟧⇀_ )","searchableContent":"open import leios.simplified ⋯ λ μ hiding ( _-⟦_/_⟧⇀_ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":22,"title":"module ND","content":"module ND = Leios.Simplified ⋯ Λ μ","searchableContent":"module nd = leios.simplified ⋯ λ μ"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":24,"title":"open import Class.Computational","content":"open import Class.Computational","searchableContent":"open import class.computational"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":25,"title":"open import Class.Computational22","content":"open import Class.Computational22","searchableContent":"open import class.computational22"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":26,"title":"open import StateMachine","content":"open import StateMachine","searchableContent":"open import statemachine"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":28,"title":"open BaseAbstract B' using ( Cert ; V-chkCerts ; VTy ; initSlot )","content":"open BaseAbstract B' using ( Cert ; V-chkCerts ; VTy ; initSlot )","searchableContent":"open baseabstract b' using ( cert ; v-chkcerts ; vty ; initslot )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":29,"title":"open GenFFD","content":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":31,"title":"open FFD hiding ( _-⟦_/_⟧⇀_ )","content":"open FFD hiding ( _-⟦_/_⟧⇀_ )","searchableContent":"open ffd hiding ( _-⟦_/_⟧⇀_ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":33,"title":"private variable s s' s0 s1 s2 s3 s4 s5","content":"private variable s s' s0 s1 s2 s3 s4 s5 : LeiosState","searchableContent":"private variable s s' s0 s1 s2 s3 s4 s5 : leiosstate"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":34,"title":"i","content":"i : LeiosInput","searchableContent":"i : leiosinput"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":35,"title":"o","content":"o : LeiosOutput","searchableContent":"o : leiosoutput"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":36,"title":"ffds'","content":"ffds' : FFD.State","searchableContent":"ffds' : ffd.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":37,"title":"π","content":"π : VrfPf","searchableContent":"π : vrfpf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":38,"title":"bs'","content":"bs' : B.State","searchableContent":"bs' : b.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":39,"title":"ks ks'","content":"ks ks' : K.State","searchableContent":"ks ks' : k.state"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":40,"title":"msgs","content":"msgs : List ( FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract )","searchableContent":"msgs : list ( ffdabstract.header ffdabstract ⊎ ffdabstract.body ffdabstract )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":41,"title":"eb","content":"eb : EndorserBlock","searchableContent":"eb : endorserblock"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":42,"title":"rbs","content":"rbs : List RankingBlock","searchableContent":"rbs : list rankingblock"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":43,"title":"txs","content":"txs : List Tx","searchableContent":"txs : list tx"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":44,"title":"V","content":"V : VTy","searchableContent":"v : vty"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":45,"title":"SD","content":"SD : StakeDistr","searchableContent":"sd : stakedistr"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":46,"title":"pks","content":"pks : List PubKey","searchableContent":"pks : list pubkey"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":48,"title":"lemma","content":"lemma : ∀ { u } → u ∈ LeiosState.Upkeep ( addUpkeep s u )","searchableContent":"lemma : ∀ { u } → u ∈ leiosstate.upkeep ( addupkeep s u )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":49,"title":"lemma","content":"lemma = to ∈-∪ ( inj₂ ( to ∈-singleton refl ))","searchableContent":"lemma = to ∈-∪ ( inj₂ ( to ∈-singleton refl ))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":50,"title":"where open Equivalence","content":"where open Equivalence","searchableContent":"where open equivalence"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":52,"title":"addUpkeep⇒¬needsUpkeep","content":"addUpkeep⇒¬needsUpkeep : ∀ { u } → ¬ LeiosState.needsUpkeep ( addUpkeep s u ) u","searchableContent":"addupkeep⇒¬needsupkeep : ∀ { u } → ¬ leiosstate.needsupkeep ( addupkeep s u ) u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":53,"title":"addUpkeep⇒¬needsUpkeep { s","content":"addUpkeep⇒¬needsUpkeep { s = s } = λ x → x ( lemma { s = s })","searchableContent":"addupkeep⇒¬needsupkeep { s = s } = λ x → x ( lemma { s = s })"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":55,"title":"data _⊢_","content":"data _⊢_ : LeiosInput → LeiosState → Type where","searchableContent":"data _⊢_ : leiosinput → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":56,"title":"Init","content":"Init :","searchableContent":"init :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":57,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","content":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":58,"title":"∙ initBaseState B.-⟦ B.INIT ( V-chkCerts pks ) / B.STAKE SD ⟧⇀ bs'","content":"∙ initBaseState B.-⟦ B.INIT ( V-chkCerts pks ) / B.STAKE SD ⟧⇀ bs'","searchableContent":"∙ initbasestate b.-⟦ b.init ( v-chkcerts pks ) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":59,"title":"────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":60,"title":"INIT V ⊢ initLeiosState V SD bs' pks","content":"INIT V ⊢ initLeiosState V SD bs' pks","searchableContent":"init v ⊢ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":62,"title":"data _-⟦Base⟧⇀_","content":"data _-⟦Base⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦base⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":64,"title":"Base₂a","content":"Base₂a : ∀ { ebs } → let open LeiosState s renaming ( BaseState to bs ) in","searchableContent":"base₂a : ∀ { ebs } → let open leiosstate s renaming ( basestate to bs ) in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":65,"title":"∙ eb ∷ ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","content":"∙ eb ∷ ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","searchableContent":"∙ eb ∷ ebs ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2 ) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":66,"title":"∙ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs'","content":"∙ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.submit ( this eb ) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":67,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":68,"title":"s -⟦Base⟧⇀ addUpkeep record s { BaseState","content":"s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"s -⟦base⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":70,"title":"Base₂b","content":"Base₂b : let open LeiosState s renaming ( BaseState to bs ) in","searchableContent":"base₂b : let open leiosstate s renaming ( basestate to bs ) in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":71,"title":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","content":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","searchableContent":"∙ [] ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2 ) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":72,"title":"∙ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs'","content":"∙ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.submit ( that topropose ) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":73,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":74,"title":"s -⟦Base⟧⇀ addUpkeep record s { BaseState","content":"s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"s -⟦base⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":76,"title":"Base⇒ND","content":"Base⇒ND : LeiosState.needsUpkeep s Base → s -⟦Base⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"base⇒nd : leiosstate.needsupkeep s base → s -⟦base⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":77,"title":"Base⇒ND u ( Base₂a x₁ x₂ )","content":"Base⇒ND u ( Base₂a x₁ x₂ ) = Base₂a u ( subst (_ ∈_ ) x₁ ( Equivalence.to ∈-fromList ( here refl ))) x₂","searchableContent":"base⇒nd u ( base₂a x₁ x₂ ) = base₂a u ( subst (_ ∈_ ) x₁ ( equivalence.to ∈-fromlist ( here refl ))) x₂"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":78,"title":"Base⇒ND u ( Base₂b x₁ x₂ )","content":"Base⇒ND u ( Base₂b x₁ x₂ ) = Base₂b u x₁ x₂","searchableContent":"base⇒nd u ( base₂b x₁ x₂ ) = base₂b u x₁ x₂"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":80,"title":"Base-Upkeep","content":"Base-Upkeep : ∀ { u } → u ≢ Base → LeiosState.needsUpkeep s u → s -⟦Base⟧⇀ s'","searchableContent":"base-upkeep : ∀ { u } → u ≢ base → leiosstate.needsupkeep s u → s -⟦base⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":81,"title":"→ LeiosState.needsUpkeep s' u","content":"→ LeiosState.needsUpkeep s' u","searchableContent":"→ leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":82,"title":"Base-Upkeep u≢Base h ( Base₂a _ _) u∈su","content":"Base-Upkeep u≢Base h ( Base₂a _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"base-upkeep u≢base h ( base₂a _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":83,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":84,"title":"( inj₂ y ) → u≢Base ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢Base ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢base ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":85,"title":"Base-Upkeep u≢Base h ( Base₂b _ _) u∈su","content":"Base-Upkeep u≢Base h ( Base₂b _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"base-upkeep u≢base h ( base₂b _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":86,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":87,"title":"( inj₂ y ) → u≢Base ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢Base ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢base ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":89,"title":"opaque","content":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":90,"title":"Base-total","content":"Base-total : ∃[ s' ] s -⟦Base⟧⇀ s'","searchableContent":"base-total : ∃[ s' ] s -⟦base⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":91,"title":"Base-total { s","content":"Base-total { s = s } with","searchableContent":"base-total { s = s } with"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":92,"title":"( let open LeiosState s in filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs )","content":"( let open LeiosState s in filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs )","searchableContent":"( let open leiosstate s in filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2 ) ebs )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":93,"title":"in eq","content":"in eq","searchableContent":"in eq"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":94,"title":"... | []","content":"... | [] = -, Base₂b ( sym eq ) ( proj₂ B.SUBMIT-total )","searchableContent":"... | [] = -, base₂b ( sym eq ) ( proj₂ b.submit-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":95,"title":"... | x ∷ l","content":"... | x ∷ l = -, Base₂a ( sym eq ) ( proj₂ B.SUBMIT-total )","searchableContent":"... | x ∷ l = -, base₂a ( sym eq ) ( proj₂ b.submit-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":97,"title":"Base-total'","content":"Base-total' : ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄","searchableContent":"base-total' : ⦃ computational-b : computational22 b._-⟦_/_⟧⇀_ string ⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":98,"title":"→ ∃[ bs ] s -⟦Base⟧⇀ addUpkeep record s { BaseState","content":"→ ∃[ bs ] s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs } Base","searchableContent":"→ ∃[ bs ] s -⟦base⟧⇀ addupkeep record s { basestate = bs } base"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":99,"title":"Base-total' { s","content":"Base-total' { s = s } = let open LeiosState s in","searchableContent":"base-total' { s = s } = let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":100,"title":"case ∃[ ebs ] ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs ∋ -, refl","content":"case ∃[ ebs ] ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs ∋ -, refl","searchableContent":"case ∃[ ebs ] ebs ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2 ) ebs ∋ -, refl"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":101,"title":"of λ where","content":"of λ where","searchableContent":"of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":102,"title":"( eb ∷ _ , eq ) → -, Base₂a eq ( proj₂ B.SUBMIT-total )","content":"( eb ∷ _ , eq ) → -, Base₂a eq ( proj₂ B.SUBMIT-total )","searchableContent":"( eb ∷ _ , eq ) → -, base₂a eq ( proj₂ b.submit-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":103,"title":"( [] , eq ) → -, Base₂b eq ( proj₂ B.SUBMIT-total )","content":"( [] , eq ) → -, Base₂b eq ( proj₂ B.SUBMIT-total )","searchableContent":"( [] , eq ) → -, base₂b eq ( proj₂ b.submit-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":105,"title":"data _-⟦IB-Role⟧⇀_","content":"data _-⟦IB-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦ib-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":107,"title":"IB-Role","content":"IB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"ib-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":108,"title":"b","content":"b = ibBody ( record { txs = ToPropose })","searchableContent":"b = ibbody ( record { txs = topropose })"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":109,"title":"h","content":"h = ibHeader ( mkIBHeader slot id π sk-IB ToPropose )","searchableContent":"h = ibheader ( mkibheader slot id π sk-ib topropose )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":110,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":111,"title":"∙ canProduceIB slot sk-IB ( stake s ) π","content":"∙ canProduceIB slot sk-IB ( stake s ) π","searchableContent":"∙ canproduceib slot sk-ib ( stake s ) π"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":112,"title":"∙ ffds FFD.-⟦ Send h ( just b ) / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send h ( just b ) / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send h ( just b ) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":113,"title":"────────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":114,"title":"s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState","content":"s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"s -⟦ib-role⟧⇀ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":116,"title":"No-IB-Role","content":"No-IB-Role : let open LeiosState s in","searchableContent":"no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":117,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π )","content":"∙ (∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π )","searchableContent":"∙ (∀ π → ¬ canproduceib slot sk-ib ( stake s ) π )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":118,"title":"────────────────────────────────────────","content":"────────────────────────────────────────","searchableContent":"────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":119,"title":"s -⟦IB-Role⟧⇀ addUpkeep s IB-Role","content":"s -⟦IB-Role⟧⇀ addUpkeep s IB-Role","searchableContent":"s -⟦ib-role⟧⇀ addupkeep s ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":121,"title":"IB-Role⇒ND","content":"IB-Role⇒ND : LeiosState.needsUpkeep s IB-Role → s -⟦IB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"ib-role⇒nd : leiosstate.needsupkeep s ib-role → s -⟦ib-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":122,"title":"IB-Role⇒ND u ( IB-Role x₁ x₂ )","content":"IB-Role⇒ND u ( IB-Role x₁ x₂ ) = Roles ( IB-Role u x₁ x₂ )","searchableContent":"ib-role⇒nd u ( ib-role x₁ x₂ ) = roles ( ib-role u x₁ x₂ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":123,"title":"IB-Role⇒ND u ( No-IB-Role x₁ )","content":"IB-Role⇒ND u ( No-IB-Role x₁ ) = Roles ( No-IB-Role u x₁ )","searchableContent":"ib-role⇒nd u ( no-ib-role x₁ ) = roles ( no-ib-role u x₁ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":125,"title":"IB-Role-Upkeep","content":"IB-Role-Upkeep : ∀ { u } → u ≢ IB-Role → LeiosState.needsUpkeep s u → s -⟦IB-Role⟧⇀ s'","searchableContent":"ib-role-upkeep : ∀ { u } → u ≢ ib-role → leiosstate.needsupkeep s u → s -⟦ib-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":126,"title":"→ LeiosState.needsUpkeep s' u","content":"→ LeiosState.needsUpkeep s' u","searchableContent":"→ leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":127,"title":"IB-Role-Upkeep u≢IB-Role h ( IB-Role _ _) u∈su","content":"IB-Role-Upkeep u≢IB-Role h ( IB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"ib-role-upkeep u≢ib-role h ( ib-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":128,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":129,"title":"( inj₂ y ) → u≢IB-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢IB-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢ib-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":130,"title":"IB-Role-Upkeep u≢IB-Role h ( No-IB-Role _) u∈su","content":"IB-Role-Upkeep u≢IB-Role h ( No-IB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"ib-role-upkeep u≢ib-role h ( no-ib-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":131,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":132,"title":"( inj₂ y ) → u≢IB-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢IB-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢ib-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":134,"title":"opaque","content":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":135,"title":"IB-Role-total","content":"IB-Role-total : ∃[ s' ] s -⟦IB-Role⟧⇀ s'","searchableContent":"ib-role-total : ∃[ s' ] s -⟦ib-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":136,"title":"IB-Role-total { s","content":"IB-Role-total { s = s } = let open LeiosState s in case Dec-canProduceIB of λ where","searchableContent":"ib-role-total { s = s } = let open leiosstate s in case dec-canproduceib of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":137,"title":"( inj₁ ( π , pf )) → -, IB-Role pf ( proj₂ FFD.Send-total )","content":"( inj₁ ( π , pf )) → -, IB-Role pf ( proj₂ FFD.Send-total )","searchableContent":"( inj₁ ( π , pf )) → -, ib-role pf ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":138,"title":"( inj₂ pf ) → -, No-IB-Role pf","content":"( inj₂ pf ) → -, No-IB-Role pf","searchableContent":"( inj₂ pf ) → -, no-ib-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":140,"title":"IB-Role-total'","content":"IB-Role-total' : ∃[ ffds ] s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds } IB-Role","searchableContent":"ib-role-total' : ∃[ ffds ] s -⟦ib-role⟧⇀ addupkeep record s { ffdstate = ffds } ib-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":141,"title":"IB-Role-total' { s","content":"IB-Role-total' { s = s } = let open LeiosState s in case Dec-canProduceIB of λ where","searchableContent":"ib-role-total' { s = s } = let open leiosstate s in case dec-canproduceib of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":142,"title":"( inj₁ ( π , pf )) → -, IB-Role pf ( proj₂ FFD.Send-total )","content":"( inj₁ ( π , pf )) → -, IB-Role pf ( proj₂ FFD.Send-total )","searchableContent":"( inj₁ ( π , pf )) → -, ib-role pf ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":143,"title":"( inj₂ pf ) → -, No-IB-Role pf","content":"( inj₂ pf ) → -, No-IB-Role pf","searchableContent":"( inj₂ pf ) → -, no-ib-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":145,"title":"data _-⟦EB-Role⟧⇀_","content":"data _-⟦EB-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦eb-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":147,"title":"EB-Role","content":"EB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"eb-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":148,"title":"LI","content":"LI = map getIBRef $ filter ( _∈ᴮ slice L slot ( Λ + 1 )) IBs","searchableContent":"li = map getibref $ filter ( _∈ᴮ slice l slot ( λ + 1 )) ibs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":149,"title":"LE","content":"LE = map getEBRef $ filter ( isVote1Certified s ) $","searchableContent":"le = map getebref $ filter ( isvote1certified s ) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":150,"title":"filter ( _∈ᴮ slice L slot ( μ + 2 )) EBs","content":"filter ( _∈ᴮ slice L slot ( μ + 2 )) EBs","searchableContent":"filter ( _∈ᴮ slice l slot ( μ + 2 )) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":151,"title":"h","content":"h = mkEB slot id π sk-EB LI LE","searchableContent":"h = mkeb slot id π sk-eb li le"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":152,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":153,"title":"∙ canProduceEB slot sk-EB ( stake s ) π","content":"∙ canProduceEB slot sk-EB ( stake s ) π","searchableContent":"∙ canproduceeb slot sk-eb ( stake s ) π"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":154,"title":"∙ ffds FFD.-⟦ Send ( ebHeader h ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( ebHeader h ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( ebheader h ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":155,"title":"────────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":156,"title":"s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState","content":"s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"s -⟦eb-role⟧⇀ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":158,"title":"No-EB-Role","content":"No-EB-Role : let open LeiosState s in","searchableContent":"no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":159,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π )","content":"∙ (∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π )","searchableContent":"∙ (∀ π → ¬ canproduceeb slot sk-eb ( stake s ) π )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":160,"title":"────────────────────────────────────────","content":"────────────────────────────────────────","searchableContent":"────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":161,"title":"s -⟦EB-Role⟧⇀ addUpkeep s EB-Role","content":"s -⟦EB-Role⟧⇀ addUpkeep s EB-Role","searchableContent":"s -⟦eb-role⟧⇀ addupkeep s eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":163,"title":"EB-Role⇒ND","content":"EB-Role⇒ND : LeiosState.needsUpkeep s EB-Role → s -⟦EB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"eb-role⇒nd : leiosstate.needsupkeep s eb-role → s -⟦eb-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":164,"title":"EB-Role⇒ND u ( EB-Role x₁ x₂ )","content":"EB-Role⇒ND u ( EB-Role x₁ x₂ ) = Roles ( EB-Role u x₁ x₂ )","searchableContent":"eb-role⇒nd u ( eb-role x₁ x₂ ) = roles ( eb-role u x₁ x₂ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":165,"title":"EB-Role⇒ND u ( No-EB-Role x₁ )","content":"EB-Role⇒ND u ( No-EB-Role x₁ ) = Roles ( No-EB-Role u x₁ )","searchableContent":"eb-role⇒nd u ( no-eb-role x₁ ) = roles ( no-eb-role u x₁ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":167,"title":"EB-Role-Upkeep","content":"EB-Role-Upkeep : ∀ { u } → u ≢ EB-Role → LeiosState.needsUpkeep s u → s -⟦EB-Role⟧⇀ s'","searchableContent":"eb-role-upkeep : ∀ { u } → u ≢ eb-role → leiosstate.needsupkeep s u → s -⟦eb-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":168,"title":"→ LeiosState.needsUpkeep s' u","content":"→ LeiosState.needsUpkeep s' u","searchableContent":"→ leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":169,"title":"EB-Role-Upkeep u≢EB-Role h ( EB-Role _ _) u∈su","content":"EB-Role-Upkeep u≢EB-Role h ( EB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"eb-role-upkeep u≢eb-role h ( eb-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":170,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":171,"title":"( inj₂ y ) → u≢EB-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢EB-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢eb-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":172,"title":"EB-Role-Upkeep u≢EB-Role h ( No-EB-Role _) u∈su","content":"EB-Role-Upkeep u≢EB-Role h ( No-EB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"eb-role-upkeep u≢eb-role h ( no-eb-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":173,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":174,"title":"( inj₂ y ) → u≢EB-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢EB-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢eb-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":176,"title":"opaque","content":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":177,"title":"EB-Role-total","content":"EB-Role-total : ∃[ s' ] s -⟦EB-Role⟧⇀ s'","searchableContent":"eb-role-total : ∃[ s' ] s -⟦eb-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":178,"title":"EB-Role-total { s","content":"EB-Role-total { s = s } = let open LeiosState s in case Dec-canProduceEB of λ where","searchableContent":"eb-role-total { s = s } = let open leiosstate s in case dec-canproduceeb of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":179,"title":"( inj₁ ( π , pf )) → -, EB-Role pf ( proj₂ FFD.Send-total )","content":"( inj₁ ( π , pf )) → -, EB-Role pf ( proj₂ FFD.Send-total )","searchableContent":"( inj₁ ( π , pf )) → -, eb-role pf ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":180,"title":"( inj₂ pf ) → -, No-EB-Role pf","content":"( inj₂ pf ) → -, No-EB-Role pf","searchableContent":"( inj₂ pf ) → -, no-eb-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":182,"title":"EB-Role-total'","content":"EB-Role-total' : ∃[ ffds ] s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds } EB-Role","searchableContent":"eb-role-total' : ∃[ ffds ] s -⟦eb-role⟧⇀ addupkeep record s { ffdstate = ffds } eb-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":183,"title":"EB-Role-total' { s","content":"EB-Role-total' { s = s } = let open LeiosState s in case Dec-canProduceEB of λ where","searchableContent":"eb-role-total' { s = s } = let open leiosstate s in case dec-canproduceeb of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":184,"title":"( inj₁ ( π , pf )) → -, EB-Role pf ( proj₂ FFD.Send-total )","content":"( inj₁ ( π , pf )) → -, EB-Role pf ( proj₂ FFD.Send-total )","searchableContent":"( inj₁ ( π , pf )) → -, eb-role pf ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":185,"title":"( inj₂ pf ) → -, No-EB-Role pf","content":"( inj₂ pf ) → -, No-EB-Role pf","searchableContent":"( inj₂ pf ) → -, no-eb-role pf"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":187,"title":"data _-⟦V1-Role⟧⇀_","content":"data _-⟦V1-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦v1-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":189,"title":"V1-Role","content":"V1-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"v1-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":190,"title":"EBs'","content":"EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L slot ( μ + 1 )) EBs","searchableContent":"ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l slot ( μ + 1 )) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":191,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":192,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":193,"title":"∙ canProduceV1 slot sk-VT ( stake s )","content":"∙ canProduceV1 slot sk-VT ( stake s )","searchableContent":"∙ canproducev1 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":194,"title":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( vtheader votes ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":195,"title":"────────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":196,"title":"s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState","content":"s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V1-Role","searchableContent":"s -⟦v1-role⟧⇀ addupkeep record s { ffdstate = ffds' } v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":198,"title":"No-V1-Role","content":"No-V1-Role : let open LeiosState s in","searchableContent":"no-v1-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":199,"title":"∙ ¬ canProduceV1 slot sk-VT ( stake s )","content":"∙ ¬ canProduceV1 slot sk-VT ( stake s )","searchableContent":"∙ ¬ canproducev1 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":200,"title":"────────────────────────────────────────","content":"────────────────────────────────────────","searchableContent":"────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":201,"title":"s -⟦V1-Role⟧⇀ addUpkeep s V1-Role","content":"s -⟦V1-Role⟧⇀ addUpkeep s V1-Role","searchableContent":"s -⟦v1-role⟧⇀ addupkeep s v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":203,"title":"V1-Role⇒ND","content":"V1-Role⇒ND : LeiosState.needsUpkeep s V1-Role → s -⟦V1-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"v1-role⇒nd : leiosstate.needsupkeep s v1-role → s -⟦v1-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":204,"title":"V1-Role⇒ND u ( V1-Role x₁ x₂ )","content":"V1-Role⇒ND u ( V1-Role x₁ x₂ ) = Roles ( V1-Role u x₁ x₂ )","searchableContent":"v1-role⇒nd u ( v1-role x₁ x₂ ) = roles ( v1-role u x₁ x₂ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":205,"title":"V1-Role⇒ND u ( No-V1-Role x₁ )","content":"V1-Role⇒ND u ( No-V1-Role x₁ ) = Roles ( No-V1-Role u x₁ )","searchableContent":"v1-role⇒nd u ( no-v1-role x₁ ) = roles ( no-v1-role u x₁ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":207,"title":"V1-Role-Upkeep","content":"V1-Role-Upkeep : ∀ { u } → u ≢ V1-Role → LeiosState.needsUpkeep s u → s -⟦V1-Role⟧⇀ s'","searchableContent":"v1-role-upkeep : ∀ { u } → u ≢ v1-role → leiosstate.needsupkeep s u → s -⟦v1-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":208,"title":"→ LeiosState.needsUpkeep s' u","content":"→ LeiosState.needsUpkeep s' u","searchableContent":"→ leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":209,"title":"V1-Role-Upkeep u≢V1-Role h ( V1-Role _ _) u∈su","content":"V1-Role-Upkeep u≢V1-Role h ( V1-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v1-role-upkeep u≢v1-role h ( v1-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":210,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":211,"title":"( inj₂ y ) → u≢V1-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢V1-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢v1-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":212,"title":"V1-Role-Upkeep u≢V1-Role h ( No-V1-Role _) u∈su","content":"V1-Role-Upkeep u≢V1-Role h ( No-V1-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v1-role-upkeep u≢v1-role h ( no-v1-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":213,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":214,"title":"( inj₂ y ) → u≢V1-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢V1-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢v1-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":216,"title":"opaque","content":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":217,"title":"V1-Role-total","content":"V1-Role-total : ∃[ s' ] s -⟦V1-Role⟧⇀ s'","searchableContent":"v1-role-total : ∃[ s' ] s -⟦v1-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":218,"title":"V1-Role-total { s","content":"V1-Role-total { s = s } = let open LeiosState s in case Dec-canProduceV1 of λ where","searchableContent":"v1-role-total { s = s } = let open leiosstate s in case dec-canproducev1 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":219,"title":"( yes p ) → -, V1-Role p ( proj₂ FFD.Send-total )","content":"( yes p ) → -, V1-Role p ( proj₂ FFD.Send-total )","searchableContent":"( yes p ) → -, v1-role p ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":220,"title":"( no ¬p ) → -, No-V1-Role ¬p","content":"( no ¬p ) → -, No-V1-Role ¬p","searchableContent":"( no ¬p ) → -, no-v1-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":222,"title":"V1-Role-total'","content":"V1-Role-total' : ∃[ ffds ] s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds } V1-Role","searchableContent":"v1-role-total' : ∃[ ffds ] s -⟦v1-role⟧⇀ addupkeep record s { ffdstate = ffds } v1-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":223,"title":"V1-Role-total' { s","content":"V1-Role-total' { s = s } = let open LeiosState s in case Dec-canProduceV1 of λ where","searchableContent":"v1-role-total' { s = s } = let open leiosstate s in case dec-canproducev1 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":224,"title":"( yes p ) → -, V1-Role p ( proj₂ FFD.Send-total )","content":"( yes p ) → -, V1-Role p ( proj₂ FFD.Send-total )","searchableContent":"( yes p ) → -, v1-role p ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":225,"title":"( no ¬p ) → -, No-V1-Role ¬p","content":"( no ¬p ) → -, No-V1-Role ¬p","searchableContent":"( no ¬p ) → -, no-v1-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":227,"title":"data _-⟦V2-Role⟧⇀_","content":"data _-⟦V2-Role⟧⇀_ : LeiosState → LeiosState → Type where","searchableContent":"data _-⟦v2-role⟧⇀_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":229,"title":"V2-Role","content":"V2-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"v2-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":230,"title":"EBs'","content":"EBs' = filter ( vote2Eligible s ) $ filter ( _∈ᴮ slice L slot 1 ) EBs","searchableContent":"ebs' = filter ( vote2eligible s ) $ filter ( _∈ᴮ slice l slot 1 ) ebs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":231,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":232,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":233,"title":"∙ canProduceV2 slot sk-VT ( stake s )","content":"∙ canProduceV2 slot sk-VT ( stake s )","searchableContent":"∙ canproducev2 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":234,"title":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( vtheader votes ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":235,"title":"────────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":236,"title":"s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState","content":"s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V2-Role","searchableContent":"s -⟦v2-role⟧⇀ addupkeep record s { ffdstate = ffds' } v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":238,"title":"No-V2-Role","content":"No-V2-Role : let open LeiosState s in","searchableContent":"no-v2-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":239,"title":"∙ ¬ canProduceV2 slot sk-VT ( stake s )","content":"∙ ¬ canProduceV2 slot sk-VT ( stake s )","searchableContent":"∙ ¬ canproducev2 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":240,"title":"────────────────────────────────────────","content":"────────────────────────────────────────","searchableContent":"────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":241,"title":"s -⟦V2-Role⟧⇀ addUpkeep s V2-Role","content":"s -⟦V2-Role⟧⇀ addUpkeep s V2-Role","searchableContent":"s -⟦v2-role⟧⇀ addupkeep s v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":243,"title":"V2-Role⇒ND","content":"V2-Role⇒ND : LeiosState.needsUpkeep s V2-Role → s -⟦V2-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"v2-role⇒nd : leiosstate.needsupkeep s v2-role → s -⟦v2-role⟧⇀ s' → just s nd.-⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":244,"title":"V2-Role⇒ND u ( V2-Role x₁ x₂ )","content":"V2-Role⇒ND u ( V2-Role x₁ x₂ ) = Roles ( V2-Role u x₁ x₂ )","searchableContent":"v2-role⇒nd u ( v2-role x₁ x₂ ) = roles ( v2-role u x₁ x₂ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":245,"title":"V2-Role⇒ND u ( No-V2-Role x₁ )","content":"V2-Role⇒ND u ( No-V2-Role x₁ ) = Roles ( No-V2-Role u x₁ )","searchableContent":"v2-role⇒nd u ( no-v2-role x₁ ) = roles ( no-v2-role u x₁ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":247,"title":"V2-Role-Upkeep","content":"V2-Role-Upkeep : ∀ { u } → u ≢ V2-Role → LeiosState.needsUpkeep s u → s -⟦V2-Role⟧⇀ s'","searchableContent":"v2-role-upkeep : ∀ { u } → u ≢ v2-role → leiosstate.needsupkeep s u → s -⟦v2-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":248,"title":"→ LeiosState.needsUpkeep s' u","content":"→ LeiosState.needsUpkeep s' u","searchableContent":"→ leiosstate.needsupkeep s' u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":249,"title":"V2-Role-Upkeep u≢V2-Role h ( V2-Role _ _) u∈su","content":"V2-Role-Upkeep u≢V2-Role h ( V2-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v2-role-upkeep u≢v2-role h ( v2-role _ _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":250,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":251,"title":"( inj₂ y ) → u≢V2-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢V2-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢v2-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":252,"title":"V2-Role-Upkeep u≢V2-Role h ( No-V2-Role _) u∈su","content":"V2-Role-Upkeep u≢V2-Role h ( No-V2-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where","searchableContent":"v2-role-upkeep u≢v2-role h ( no-v2-role _) u∈su = case equivalence.from ∈-∪ u∈su of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":253,"title":"( inj₁ x ) → h x","content":"( inj₁ x ) → h x","searchableContent":"( inj₁ x ) → h x"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":254,"title":"( inj₂ y ) → u≢V2-Role ( Equivalence.from ∈-singleton y )","content":"( inj₂ y ) → u≢V2-Role ( Equivalence.from ∈-singleton y )","searchableContent":"( inj₂ y ) → u≢v2-role ( equivalence.from ∈-singleton y )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":256,"title":"opaque","content":"opaque","searchableContent":"opaque"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":257,"title":"V2-Role-total","content":"V2-Role-total : ∃[ s' ] s -⟦V2-Role⟧⇀ s'","searchableContent":"v2-role-total : ∃[ s' ] s -⟦v2-role⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":258,"title":"V2-Role-total { s","content":"V2-Role-total { s = s } = let open LeiosState s in case Dec-canProduceV2 of λ where","searchableContent":"v2-role-total { s = s } = let open leiosstate s in case dec-canproducev2 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":259,"title":"( yes p ) → -, V2-Role p ( proj₂ FFD.Send-total )","content":"( yes p ) → -, V2-Role p ( proj₂ FFD.Send-total )","searchableContent":"( yes p ) → -, v2-role p ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":260,"title":"( no ¬p ) → -, No-V2-Role ¬p","content":"( no ¬p ) → -, No-V2-Role ¬p","searchableContent":"( no ¬p ) → -, no-v2-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":262,"title":"V2-Role-total'","content":"V2-Role-total' : ∃[ ffds ] s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds } V2-Role","searchableContent":"v2-role-total' : ∃[ ffds ] s -⟦v2-role⟧⇀ addupkeep record s { ffdstate = ffds } v2-role"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":263,"title":"V2-Role-total' { s","content":"V2-Role-total' { s = s } = let open LeiosState s in case Dec-canProduceV2 of λ where","searchableContent":"v2-role-total' { s = s } = let open leiosstate s in case dec-canproducev2 of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":264,"title":"( yes p ) → -, V2-Role p ( proj₂ FFD.Send-total )","content":"( yes p ) → -, V2-Role p ( proj₂ FFD.Send-total )","searchableContent":"( yes p ) → -, v2-role p ( proj₂ ffd.send-total )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":265,"title":"( no ¬p ) → -, No-V2-Role ¬p","content":"( no ¬p ) → -, No-V2-Role ¬p","searchableContent":"( no ¬p ) → -, no-v2-role ¬p"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":267,"title":"data _-⟦_/_⟧⇀_","content":"data _-⟦_/_⟧⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":269,"title":"-- Network and Ledger","content":"-- Network and Ledger","searchableContent":"-- network and ledger"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":271,"title":"Slot","content":"Slot : let open LeiosState s renaming ( FFDState to ffds ; BaseState to bs )","searchableContent":"slot : let open leiosstate s renaming ( ffdstate to ffds ; basestate to bs )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":272,"title":"s0","content":"s0 = record s","searchableContent":"s0 = record s"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":273,"title":"{ FFDState","content":"{ FFDState = ffds'","searchableContent":"{ ffdstate = ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":274,"title":"; Ledger","content":"; Ledger = constructLedger rbs","searchableContent":"; ledger = constructledger rbs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":275,"title":"; slot","content":"; slot = suc slot","searchableContent":"; slot = suc slot"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":276,"title":"; Upkeep","content":"; Upkeep = ∅","searchableContent":"; upkeep = ∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":277,"title":"; BaseState","content":"; BaseState = bs'","searchableContent":"; basestate = bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":278,"title":"} ↑ L.filter ( isValid? s ) msgs","content":"} ↑ L.filter ( isValid? s ) msgs","searchableContent":"} ↑ l.filter ( isvalid? s ) msgs"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":279,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":280,"title":"∙ Upkeep ≡ᵉ allUpkeep","content":"∙ Upkeep ≡ᵉ allUpkeep","searchableContent":"∙ upkeep ≡ᵉ allupkeep"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":281,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","content":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":282,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":283,"title":"∙ s0 -⟦Base⟧⇀ s1","content":"∙ s0 -⟦Base⟧⇀ s1","searchableContent":"∙ s0 -⟦base⟧⇀ s1"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":284,"title":"∙ s1 -⟦IB-Role⟧⇀ s2","content":"∙ s1 -⟦IB-Role⟧⇀ s2","searchableContent":"∙ s1 -⟦ib-role⟧⇀ s2"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":285,"title":"∙ s2 -⟦EB-Role⟧⇀ s3","content":"∙ s2 -⟦EB-Role⟧⇀ s3","searchableContent":"∙ s2 -⟦eb-role⟧⇀ s3"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":286,"title":"∙ s3 -⟦V1-Role⟧⇀ s4","content":"∙ s3 -⟦V1-Role⟧⇀ s4","searchableContent":"∙ s3 -⟦v1-role⟧⇀ s4"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":287,"title":"∙ s4 -⟦V2-Role⟧⇀ s5","content":"∙ s4 -⟦V2-Role⟧⇀ s5","searchableContent":"∙ s4 -⟦v2-role⟧⇀ s5"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":288,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":289,"title":"s -⟦ SLOT / EMPTY ⟧⇀ s5","content":"s -⟦ SLOT / EMPTY ⟧⇀ s5","searchableContent":"s -⟦ slot / empty ⟧⇀ s5"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":291,"title":"Ftch","content":"Ftch :","searchableContent":"ftch :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":292,"title":"───────────────────────────────────────────────────","content":"───────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":293,"title":"s -⟦ FTCH-LDG / FTCH-LDG ( LeiosState.Ledger s ) ⟧⇀ s","content":"s -⟦ FTCH-LDG / FTCH-LDG ( LeiosState.Ledger s ) ⟧⇀ s","searchableContent":"s -⟦ ftch-ldg / ftch-ldg ( leiosstate.ledger s ) ⟧⇀ s"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":295,"title":"-- Base chain","content":"-- Base chain","searchableContent":"-- base chain"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":296,"title":"--","content":"--","searchableContent":"--"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":297,"title":"-- Note","content":"-- Note: Submitted data to the base chain is only taken into account","searchableContent":"-- note: submitted data to the base chain is only taken into account"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":298,"title":"-- if the party submitting is the block producer on the base chain","content":"-- if the party submitting is the block producer on the base chain","searchableContent":"-- if the party submitting is the block producer on the base chain"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":299,"title":"-- for the given slot","content":"-- for the given slot","searchableContent":"-- for the given slot"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":301,"title":"Base₁","content":"Base₁ :","searchableContent":"base₁ :"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":302,"title":"──────────────────────────────────────────────────────────────","content":"──────────────────────────────────────────────────────────────","searchableContent":"──────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":303,"title":"s -⟦ SUBMIT ( inj₂ txs ) / EMPTY ⟧⇀ record s { ToPropose","content":"s -⟦ SUBMIT ( inj₂ txs ) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"s -⟦ submit ( inj₂ txs ) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":305,"title":"-- Protocol rules","content":"-- Protocol rules","searchableContent":"-- protocol rules"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":307,"title":"_-⟦_/_⟧ⁿᵈ⇀_","content":"_-⟦_/_⟧ⁿᵈ⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type","searchableContent":"_-⟦_/_⟧ⁿᵈ⇀_ : leiosstate → leiosinput → leiosoutput → leiosstate → type"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":308,"title":"s -⟦ i / o ⟧ⁿᵈ⇀ s'","content":"s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s ND.-⟦ i / o ⟧⇀ s'","searchableContent":"s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s nd.-⟦ i / o ⟧⇀ s'"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":310,"title":"_-⟦_/_⟧ⁿᵈ*⇀_","content":"_-⟦_/_⟧ⁿᵈ*⇀_ : LeiosState → List LeiosInput → List LeiosOutput → LeiosState → Type","searchableContent":"_-⟦_/_⟧ⁿᵈ*⇀_ : leiosstate → list leiosinput → list leiosoutput → leiosstate → type"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":311,"title":"_-⟦_/_⟧ⁿᵈ*⇀_","content":"_-⟦_/_⟧ⁿᵈ*⇀_ = ReflexiveTransitiveClosure _-⟦_/_⟧ⁿᵈ⇀_","searchableContent":"_-⟦_/_⟧ⁿᵈ*⇀_ = reflexivetransitiveclosure _-⟦_/_⟧ⁿᵈ⇀_"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":313,"title":"-- Key fact","content":"-- Key fact: stepping with the deterministic relation means we can","searchableContent":"-- key fact: stepping with the deterministic relation means we can"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":314,"title":"-- also step with the non-deterministic one","content":"-- also step with the non-deterministic one","searchableContent":"-- also step with the non-deterministic one"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":315,"title":"-- TODO","content":"-- TODO: this is a lot like a weak simulation, can we do something prettier?","searchableContent":"-- todo: this is a lot like a weak simulation, can we do something prettier?"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":316,"title":"-⟦/⟧⇀⇒ND","content":"-⟦/⟧⇀⇒ND : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] ( s -⟦ i / o ⟧ⁿᵈ*⇀ s' )","searchableContent":"-⟦/⟧⇀⇒nd : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] ( s -⟦ i / o ⟧ⁿᵈ*⇀ s' )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":317,"title":"-⟦/⟧⇀⇒ND ( Slot { s","content":"-⟦/⟧⇀⇒ND ( Slot { s = s } { msgs = msgs } { s1 = s1 } { s2 = s2 } { s3 = s3 } { s4 = s4 } x x₁ x₂ hB hIB hEB hV1 hV2 ) = replicate 6 SLOT , replicate 6 EMPTY ,","searchableContent":"-⟦/⟧⇀⇒nd ( slot { s = s } { msgs = msgs } { s1 = s1 } { s2 = s2 } { s3 = s3 } { s4 = s4 } x x₁ x₂ hb hib heb hv1 hv2 ) = replicate 6 slot , replicate 6 empty ,"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":318,"title":"let","content":"let","searchableContent":"let"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":319,"title":"s0","content":"s0 = _","searchableContent":"s0 = _"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":320,"title":"upkeep≡∅","content":"upkeep≡∅ : LeiosState.Upkeep s0 ≡ ∅","searchableContent":"upkeep≡∅ : leiosstate.upkeep s0 ≡ ∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":321,"title":"upkeep≡∅","content":"upkeep≡∅ = sym ( ↑-preserves-Upkeep { x = L.filter ( isValid? s ) msgs })","searchableContent":"upkeep≡∅ = sym ( ↑-preserves-upkeep { x = l.filter ( isvalid? s ) msgs })"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":322,"title":"needsAllUpkeep","content":"needsAllUpkeep : ∀ { u } → LeiosState.needsUpkeep s0 u","searchableContent":"needsallupkeep : ∀ { u } → leiosstate.needsupkeep s0 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":323,"title":"needsAllUpkeep { u }","content":"needsAllUpkeep { u } = subst ( u ∉_ ) ( sym upkeep≡∅ ) Properties.∉-∅","searchableContent":"needsallupkeep { u } = subst ( u ∉_ ) ( sym upkeep≡∅ ) properties.∉-∅"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":324,"title":"needsUpkeep1","content":"needsUpkeep1 : ∀ { u } → u ≢ Base → LeiosState.needsUpkeep s1 u","searchableContent":"needsupkeep1 : ∀ { u } → u ≢ base → leiosstate.needsupkeep s1 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":325,"title":"needsUpkeep1 h1","content":"needsUpkeep1 h1 = Base-Upkeep h1 needsAllUpkeep hB","searchableContent":"needsupkeep1 h1 = base-upkeep h1 needsallupkeep hb"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":326,"title":"needsUpkeep2","content":"needsUpkeep2 : ∀ { u } → u ≢ Base → u ≢ IB-Role → LeiosState.needsUpkeep s2 u","searchableContent":"needsupkeep2 : ∀ { u } → u ≢ base → u ≢ ib-role → leiosstate.needsupkeep s2 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":327,"title":"needsUpkeep2 h1 h2","content":"needsUpkeep2 h1 h2 = IB-Role-Upkeep h2 ( needsUpkeep1 h1 ) hIB","searchableContent":"needsupkeep2 h1 h2 = ib-role-upkeep h2 ( needsupkeep1 h1 ) hib"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":328,"title":"needsUpkeep3","content":"needsUpkeep3 : ∀ { u } → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → LeiosState.needsUpkeep s3 u","searchableContent":"needsupkeep3 : ∀ { u } → u ≢ base → u ≢ ib-role → u ≢ eb-role → leiosstate.needsupkeep s3 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":329,"title":"needsUpkeep3 h1 h2 h3","content":"needsUpkeep3 h1 h2 h3 = EB-Role-Upkeep h3 ( needsUpkeep2 h1 h2 ) hEB","searchableContent":"needsupkeep3 h1 h2 h3 = eb-role-upkeep h3 ( needsupkeep2 h1 h2 ) heb"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":330,"title":"needsUpkeep4","content":"needsUpkeep4 : ∀ { u } → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → u ≢ V1-Role → LeiosState.needsUpkeep s4 u","searchableContent":"needsupkeep4 : ∀ { u } → u ≢ base → u ≢ ib-role → u ≢ eb-role → u ≢ v1-role → leiosstate.needsupkeep s4 u"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":331,"title":"needsUpkeep4 h1 h2 h3 h4","content":"needsUpkeep4 h1 h2 h3 h4 = V1-Role-Upkeep h4 ( needsUpkeep3 h1 h2 h3 ) hV1","searchableContent":"needsupkeep4 h1 h2 h3 h4 = v1-role-upkeep h4 ( needsupkeep3 h1 h2 h3 ) hv1"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":332,"title":"in ( BS-ind ( ND.Slot x x₁ x₂ ) $","content":"in ( BS-ind ( ND.Slot x x₁ x₂ ) $","searchableContent":"in ( bs-ind ( nd.slot x x₁ x₂ ) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":333,"title":"BS-ind ( Base⇒ND { s","content":"BS-ind ( Base⇒ND { s = s0 } needsAllUpkeep hB ) $","searchableContent":"bs-ind ( base⇒nd { s = s0 } needsallupkeep hb ) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":334,"title":"BS-ind ( IB-Role⇒ND ( needsUpkeep1 (λ ())) hIB ) $","content":"BS-ind ( IB-Role⇒ND ( needsUpkeep1 (λ ())) hIB ) $","searchableContent":"bs-ind ( ib-role⇒nd ( needsupkeep1 (λ ())) hib ) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":335,"title":"BS-ind ( EB-Role⇒ND ( needsUpkeep2 (λ ()) (λ ())) hEB ) $","content":"BS-ind ( EB-Role⇒ND ( needsUpkeep2 (λ ()) (λ ())) hEB ) $","searchableContent":"bs-ind ( eb-role⇒nd ( needsupkeep2 (λ ()) (λ ())) heb ) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":336,"title":"BS-ind ( V1-Role⇒ND ( needsUpkeep3 (λ ()) (λ ()) (λ ())) hV1 ) $","content":"BS-ind ( V1-Role⇒ND ( needsUpkeep3 (λ ()) (λ ()) (λ ())) hV1 ) $","searchableContent":"bs-ind ( v1-role⇒nd ( needsupkeep3 (λ ()) (λ ()) (λ ())) hv1 ) $"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":337,"title":"STS⇒RTC ( V2-Role⇒ND ( needsUpkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hV2 ))","content":"STS⇒RTC ( V2-Role⇒ND ( needsUpkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hV2 ))","searchableContent":"sts⇒rtc ( v2-role⇒nd ( needsupkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hv2 ))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":338,"title":"-⟦/⟧⇀⇒ND Ftch","content":"-⟦/⟧⇀⇒ND Ftch = _ , _ , STS⇒RTC Ftch","searchableContent":"-⟦/⟧⇀⇒nd ftch = _ , _ , sts⇒rtc ftch"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":339,"title":"-⟦/⟧⇀⇒ND Base₁","content":"-⟦/⟧⇀⇒ND Base₁ = _ , _ , STS⇒RTC Base₁","searchableContent":"-⟦/⟧⇀⇒nd base₁ = _ , _ , sts⇒rtc base₁"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":341,"title":"open Computational22 ⦃...⦄","content":"open Computational22 ⦃...⦄","searchableContent":"open computational22 ⦃...⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":343,"title":"module _ ⦃ Computational-B","content":"module _ ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄","searchableContent":"module _ ⦃ computational-b : computational22 b._-⟦_/_⟧⇀_ string ⦄"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":344,"title":"⦃ Computational-FFD","content":"⦃ Computational-FFD : Computational22 FFD._-⟦_/_⟧⇀_ String ⦄ where","searchableContent":"⦃ computational-ffd : computational22 ffd._-⟦_/_⟧⇀_ string ⦄ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":346,"title":"instance","content":"instance","searchableContent":"instance"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":347,"title":"Computational--⟦/⟧⇀","content":"Computational--⟦/⟧⇀ : Computational22 _-⟦_/_⟧⇀_ String","searchableContent":"computational--⟦/⟧⇀ : computational22 _-⟦_/_⟧⇀_ string"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":348,"title":"Computational--⟦/⟧⇀ . computeProof s ( INIT x )","content":"Computational--⟦/⟧⇀ . computeProof s ( INIT x ) = failure "No handling of INIT here"","searchableContent":"computational--⟦/⟧⇀ . computeproof s ( init x ) = failure "no handling of init here""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":349,"title":"Computational--⟦/⟧⇀ . computeProof s ( SUBMIT ( inj₁ eb ))","content":"Computational--⟦/⟧⇀ . computeProof s ( SUBMIT ( inj₁ eb )) = failure "Cannot submit EB here"","searchableContent":"computational--⟦/⟧⇀ . computeproof s ( submit ( inj₁ eb )) = failure "cannot submit eb here""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":350,"title":"Computational--⟦/⟧⇀ . computeProof s ( SUBMIT ( inj₂ txs ))","content":"Computational--⟦/⟧⇀ . computeProof s ( SUBMIT ( inj₂ txs )) = success ( -, Base₁ )","searchableContent":"computational--⟦/⟧⇀ . computeproof s ( submit ( inj₂ txs )) = success ( -, base₁ )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":351,"title":"Computational--⟦/⟧⇀ . computeProof s* SLOT","content":"Computational--⟦/⟧⇀ . computeProof s* SLOT = let open LeiosState s* in","searchableContent":"computational--⟦/⟧⇀ . computeproof s* slot = let open leiosstate s* in"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":352,"title":"case ( ¿ Upkeep ≡ᵉ allUpkeep ¿ ,′ computeProof BaseState B.FTCH-LDG ,′ computeProof FFDState FFD.Fetch ) of λ where","content":"case ( ¿ Upkeep ≡ᵉ allUpkeep ¿ ,′ computeProof BaseState B.FTCH-LDG ,′ computeProof FFDState FFD.Fetch ) of λ where","searchableContent":"case ( ¿ upkeep ≡ᵉ allupkeep ¿ ,′ computeproof basestate b.ftch-ldg ,′ computeproof ffdstate ffd.fetch ) of λ where"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":353,"title":"( yes p , success (( B.BASE-LDG l , bs ) , p₁ ) , success (( FFD.FetchRes msgs , ffds ) , p₂ )) →","content":"( yes p , success (( B.BASE-LDG l , bs ) , p₁ ) , success (( FFD.FetchRes msgs , ffds ) , p₂ )) →","searchableContent":"( yes p , success (( b.base-ldg l , bs ) , p₁ ) , success (( ffd.fetchres msgs , ffds ) , p₂ )) →"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":354,"title":"success ((_ , ( Slot p p₁ p₂ ( proj₂ Base-total ) ( proj₂ IB-Role-total ) ( proj₂ EB-Role-total ) ( proj₂ V1-Role-total ) ( proj₂ V2-Role-total ))))","content":"success ((_ , ( Slot p p₁ p₂ ( proj₂ Base-total ) ( proj₂ IB-Role-total ) ( proj₂ EB-Role-total ) ( proj₂ V1-Role-total ) ( proj₂ V2-Role-total ))))","searchableContent":"success ((_ , ( slot p p₁ p₂ ( proj₂ base-total ) ( proj₂ ib-role-total ) ( proj₂ eb-role-total ) ( proj₂ v1-role-total ) ( proj₂ v2-role-total ))))"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":355,"title":"( yes p , _ , _) → failure "Subsystem failed"","content":"( yes p , _ , _) → failure "Subsystem failed"","searchableContent":"( yes p , _ , _) → failure "subsystem failed""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":356,"title":"( no ¬p , _) → failure "Upkeep incorrect"","content":"( no ¬p , _) → failure "Upkeep incorrect"","searchableContent":"( no ¬p , _) → failure "upkeep incorrect""},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":357,"title":"Computational--⟦/⟧⇀ . computeProof s FTCH-LDG","content":"Computational--⟦/⟧⇀ . computeProof s FTCH-LDG = success ( -, Ftch )","searchableContent":"computational--⟦/⟧⇀ . computeproof s ftch-ldg = success ( -, ftch )"},{"moduleName":"Leios.Simplified.Deterministic","path":"Leios.Simplified.Deterministic.html","group":"Leios","lineNumber":358,"title":"Computational--⟦/⟧⇀ . completeness","content":"Computational--⟦/⟧⇀ . completeness = {!!}","searchableContent":"computational--⟦/⟧⇀ . completeness = {!!}"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":2,"title":"Leios.Simplified {-# OPTIONS --safe #-}","content":"Leios.Simplified {-# OPTIONS --safe #-}","searchableContent":"leios.simplified {-# options --safe #-}"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding ( id )","content":"open import Leios.Prelude hiding ( id )","searchableContent":"open import leios.prelude hiding ( id )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":5,"title":"open import Leios.FFD","content":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":6,"title":"open import Leios.SpecStructure","content":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":7,"title":"open import Data.Fin.Patterns","content":"open import Data.Fin.Patterns","searchableContent":"open import data.fin.patterns"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":9,"title":"open import Tactic.Defaults","content":"open import Tactic.Defaults","searchableContent":"open import tactic.defaults"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":10,"title":"open import Tactic.Derive.DecEq","content":"open import Tactic.Derive.DecEq","searchableContent":"open import tactic.derive.deceq"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":12,"title":"module Leios.Simplified ( ⋯","content":"module Leios.Simplified ( ⋯ : SpecStructure 2 ) ( let open SpecStructure ⋯ ) ( Λ μ : ℕ ) where","searchableContent":"module leios.simplified ( ⋯ : specstructure 2 ) ( let open specstructure ⋯ ) ( λ μ : ℕ ) where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":14,"title":"data SlotUpkeep","content":"data SlotUpkeep : Type where","searchableContent":"data slotupkeep : type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":15,"title":"Base IB-Role EB-Role V1-Role V2-Role","content":"Base IB-Role EB-Role V1-Role V2-Role : SlotUpkeep","searchableContent":"base ib-role eb-role v1-role v2-role : slotupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":17,"title":"unquoteDecl DecEq-SlotUpkeep","content":"unquoteDecl DecEq-SlotUpkeep = derive-DecEq (( quote SlotUpkeep , DecEq-SlotUpkeep ) ∷ [] )","searchableContent":"unquotedecl deceq-slotupkeep = derive-deceq (( quote slotupkeep , deceq-slotupkeep ) ∷ [] )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":19,"title":"allUpkeep","content":"allUpkeep : ℙ SlotUpkeep","searchableContent":"allupkeep : ℙ slotupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":20,"title":"allUpkeep","content":"allUpkeep = fromList ( Base ∷ IB-Role ∷ EB-Role ∷ V1-Role ∷ V2-Role ∷ [] )","searchableContent":"allupkeep = fromlist ( base ∷ ib-role ∷ eb-role ∷ v1-role ∷ v2-role ∷ [] )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":22,"title":"open import Leios.Protocol ( ⋯ ) SlotUpkeep public","content":"open import Leios.Protocol ( ⋯ ) SlotUpkeep public","searchableContent":"open import leios.protocol ( ⋯ ) slotupkeep public"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":24,"title":"open BaseAbstract B' using ( Cert ; V-chkCerts ; VTy ; initSlot )","content":"open BaseAbstract B' using ( Cert ; V-chkCerts ; VTy ; initSlot )","searchableContent":"open baseabstract b' using ( cert ; v-chkcerts ; vty ; initslot )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":25,"title":"open FFD hiding ( _-⟦_/_⟧⇀_ )","content":"open FFD hiding ( _-⟦_/_⟧⇀_ )","searchableContent":"open ffd hiding ( _-⟦_/_⟧⇀_ )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":26,"title":"open GenFFD","content":"open GenFFD","searchableContent":"open genffd"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":28,"title":"isVote1Certified","content":"isVote1Certified : LeiosState → EndorserBlock → Type","searchableContent":"isvote1certified : leiosstate → endorserblock → type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":29,"title":"isVote1Certified s eb","content":"isVote1Certified s eb = isVoteCertified ( LeiosState.votingState s ) ( 0F , eb )","searchableContent":"isvote1certified s eb = isvotecertified ( leiosstate.votingstate s ) ( 0f , eb )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":31,"title":"isVote2Certified","content":"isVote2Certified : LeiosState → EndorserBlock → Type","searchableContent":"isvote2certified : leiosstate → endorserblock → type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":32,"title":"isVote2Certified s eb","content":"isVote2Certified s eb = isVoteCertified ( LeiosState.votingState s ) ( 1F , eb )","searchableContent":"isvote2certified s eb = isvotecertified ( leiosstate.votingstate s ) ( 1f , eb )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":34,"title":"-- Predicates about EBs","content":"-- Predicates about EBs","searchableContent":"-- predicates about ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":35,"title":"module _ ( s","content":"module _ ( s : LeiosState ) ( eb : EndorserBlock ) where","searchableContent":"module _ ( s : leiosstate ) ( eb : endorserblock ) where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":36,"title":"open EndorserBlockOSig eb","content":"open EndorserBlockOSig eb","searchableContent":"open endorserblockosig eb"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":37,"title":"open LeiosState s","content":"open LeiosState s","searchableContent":"open leiosstate s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":39,"title":"vote2Eligible","content":"vote2Eligible : Type","searchableContent":"vote2eligible : type"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":40,"title":"vote2Eligible","content":"vote2Eligible = length ebRefs ≥ lengthˢ candidateEBs / 2 -- should this be `>`?","searchableContent":"vote2eligible = length ebrefs ≥ lengthˢ candidateebs / 2 -- should this be `>`?"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":41,"title":"× fromList ebRefs ⊆ candidateEBs","content":"× fromList ebRefs ⊆ candidateEBs","searchableContent":"× fromlist ebrefs ⊆ candidateebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":42,"title":"where candidateEBs","content":"where candidateEBs : ℙ Hash","searchableContent":"where candidateebs : ℙ hash"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":43,"title":"candidateEBs","content":"candidateEBs = mapˢ getEBRef $ filterˢ ( _∈ᴮ slice L slot ( μ + 3 )) ( fromList EBs )","searchableContent":"candidateebs = mapˢ getebref $ filterˢ ( _∈ᴮ slice l slot ( μ + 3 )) ( fromlist ebs )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":45,"title":"private variable s s'","content":"private variable s s' : LeiosState","searchableContent":"private variable s s' : leiosstate"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":46,"title":"ffds'","content":"ffds' : FFD.State","searchableContent":"ffds' : ffd.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":47,"title":"π","content":"π : VrfPf","searchableContent":"π : vrfpf"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":48,"title":"bs'","content":"bs' : B.State","searchableContent":"bs' : b.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":49,"title":"ks ks'","content":"ks ks' : K.State","searchableContent":"ks ks' : k.state"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":50,"title":"msgs","content":"msgs : List ( FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract )","searchableContent":"msgs : list ( ffdabstract.header ffdabstract ⊎ ffdabstract.body ffdabstract )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":51,"title":"eb","content":"eb : EndorserBlock","searchableContent":"eb : endorserblock"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":52,"title":"rbs","content":"rbs : List RankingBlock","searchableContent":"rbs : list rankingblock"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":53,"title":"txs","content":"txs : List Tx","searchableContent":"txs : list tx"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":54,"title":"V","content":"V : VTy","searchableContent":"v : vty"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":55,"title":"SD","content":"SD : StakeDistr","searchableContent":"sd : stakedistr"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":56,"title":"pks","content":"pks : List PubKey","searchableContent":"pks : list pubkey"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":58,"title":"data _↝_","content":"data _↝_ : LeiosState → LeiosState → Type where","searchableContent":"data _↝_ : leiosstate → leiosstate → type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":60,"title":"IB-Role","content":"IB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"ib-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":61,"title":"b","content":"b = ibBody ( record { txs = ToPropose })","searchableContent":"b = ibbody ( record { txs = topropose })"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":62,"title":"h","content":"h = ibHeader ( mkIBHeader slot id π sk-IB ToPropose )","searchableContent":"h = ibheader ( mkibheader slot id π sk-ib topropose )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":63,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":64,"title":"∙ needsUpkeep IB-Role","content":"∙ needsUpkeep IB-Role","searchableContent":"∙ needsupkeep ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":65,"title":"∙ canProduceIB slot sk-IB ( stake s ) π","content":"∙ canProduceIB slot sk-IB ( stake s ) π","searchableContent":"∙ canproduceib slot sk-ib ( stake s ) π"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":66,"title":"∙ ffds FFD.-⟦ Send h ( just b ) / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send h ( just b ) / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send h ( just b ) / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":67,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":68,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } IB-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":70,"title":"EB-Role","content":"EB-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"eb-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":71,"title":"LI","content":"LI = map getIBRef $ filter ( _∈ᴮ slice L slot ( Λ + 1 )) IBs","searchableContent":"li = map getibref $ filter ( _∈ᴮ slice l slot ( λ + 1 )) ibs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":72,"title":"LE","content":"LE = map getEBRef $ filter ( isVote1Certified s ) $","searchableContent":"le = map getebref $ filter ( isvote1certified s ) $"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":73,"title":"filter ( _∈ᴮ slice L slot ( μ + 2 )) EBs","content":"filter ( _∈ᴮ slice L slot ( μ + 2 )) EBs","searchableContent":"filter ( _∈ᴮ slice l slot ( μ + 2 )) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":74,"title":"h","content":"h = mkEB slot id π sk-EB LI LE","searchableContent":"h = mkeb slot id π sk-eb li le"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":75,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":76,"title":"∙ needsUpkeep EB-Role","content":"∙ needsUpkeep EB-Role","searchableContent":"∙ needsupkeep eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":77,"title":"∙ canProduceEB slot sk-EB ( stake s ) π","content":"∙ canProduceEB slot sk-EB ( stake s ) π","searchableContent":"∙ canproduceeb slot sk-eb ( stake s ) π"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":78,"title":"∙ ffds FFD.-⟦ Send ( ebHeader h ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( ebHeader h ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( ebheader h ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":79,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":80,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } EB-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":82,"title":"V1-Role","content":"V1-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"v1-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":83,"title":"EBs'","content":"EBs' = filter ( allIBRefsKnown s ) $ filter ( _∈ᴮ slice L slot ( μ + 1 )) EBs","searchableContent":"ebs' = filter ( allibrefsknown s ) $ filter ( _∈ᴮ slice l slot ( μ + 1 )) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":84,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":85,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":86,"title":"∙ needsUpkeep V1-Role","content":"∙ needsUpkeep V1-Role","searchableContent":"∙ needsupkeep v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":87,"title":"∙ canProduceV1 slot sk-VT ( stake s )","content":"∙ canProduceV1 slot sk-VT ( stake s )","searchableContent":"∙ canproducev1 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":88,"title":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( vtheader votes ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":89,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":90,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } V1-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":92,"title":"V2-Role","content":"V2-Role : let open LeiosState s renaming ( FFDState to ffds )","searchableContent":"v2-role : let open leiosstate s renaming ( ffdstate to ffds )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":93,"title":"EBs'","content":"EBs' = filter ( vote2Eligible s ) $ filter ( _∈ᴮ slice L slot 1 ) EBs","searchableContent":"ebs' = filter ( vote2eligible s ) $ filter ( _∈ᴮ slice l slot 1 ) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":94,"title":"votes","content":"votes = map ( vote sk-VT ∘ hash ) EBs'","searchableContent":"votes = map ( vote sk-vt ∘ hash ) ebs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":95,"title":"in","content":"in","searchableContent":"in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":96,"title":"∙ needsUpkeep V2-Role","content":"∙ needsUpkeep V2-Role","searchableContent":"∙ needsupkeep v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":97,"title":"∙ canProduceV2 slot sk-VT ( stake s )","content":"∙ canProduceV2 slot sk-VT ( stake s )","searchableContent":"∙ canproducev2 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":98,"title":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Send ( vtHeader votes ) nothing / SendRes ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ send ( vtheader votes ) nothing / sendres ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":99,"title":"─────────────────────────────────────────────────────────────────────────","content":"─────────────────────────────────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":100,"title":"s ↝ addUpkeep record s { FFDState","content":"s ↝ addUpkeep record s { FFDState = ffds' } V2-Role","searchableContent":"s ↝ addupkeep record s { ffdstate = ffds' } v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":102,"title":"-- Note","content":"-- Note: Base doesn't need a negative rule, since it can always be invoked","searchableContent":"-- note: base doesn't need a negative rule, since it can always be invoked"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":104,"title":"No-IB-Role","content":"No-IB-Role : let open LeiosState s in","searchableContent":"no-ib-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":105,"title":"∙ needsUpkeep IB-Role","content":"∙ needsUpkeep IB-Role","searchableContent":"∙ needsupkeep ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":106,"title":"∙ (∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π )","content":"∙ (∀ π → ¬ canProduceIB slot sk-IB ( stake s ) π )","searchableContent":"∙ (∀ π → ¬ canproduceib slot sk-ib ( stake s ) π )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":107,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":108,"title":"s ↝ addUpkeep s IB-Role","content":"s ↝ addUpkeep s IB-Role","searchableContent":"s ↝ addupkeep s ib-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":110,"title":"No-EB-Role","content":"No-EB-Role : let open LeiosState s in","searchableContent":"no-eb-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":111,"title":"∙ needsUpkeep EB-Role","content":"∙ needsUpkeep EB-Role","searchableContent":"∙ needsupkeep eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":112,"title":"∙ (∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π )","content":"∙ (∀ π → ¬ canProduceEB slot sk-EB ( stake s ) π )","searchableContent":"∙ (∀ π → ¬ canproduceeb slot sk-eb ( stake s ) π )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":113,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":114,"title":"s ↝ addUpkeep s EB-Role","content":"s ↝ addUpkeep s EB-Role","searchableContent":"s ↝ addupkeep s eb-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":116,"title":"No-V1-Role","content":"No-V1-Role : let open LeiosState s in","searchableContent":"no-v1-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":117,"title":"∙ needsUpkeep V1-Role","content":"∙ needsUpkeep V1-Role","searchableContent":"∙ needsupkeep v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":118,"title":"∙ ¬ canProduceV1 slot sk-VT ( stake s )","content":"∙ ¬ canProduceV1 slot sk-VT ( stake s )","searchableContent":"∙ ¬ canproducev1 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":119,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":120,"title":"s ↝ addUpkeep s V1-Role","content":"s ↝ addUpkeep s V1-Role","searchableContent":"s ↝ addupkeep s v1-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":122,"title":"No-V2-Role","content":"No-V2-Role : let open LeiosState s in","searchableContent":"no-v2-role : let open leiosstate s in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":123,"title":"∙ needsUpkeep V2-Role","content":"∙ needsUpkeep V2-Role","searchableContent":"∙ needsupkeep v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":124,"title":"∙ ¬ canProduceV2 slot sk-VT ( stake s )","content":"∙ ¬ canProduceV2 slot sk-VT ( stake s )","searchableContent":"∙ ¬ canproducev2 slot sk-vt ( stake s )"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":125,"title":"─────────────────────────────────────────────","content":"─────────────────────────────────────────────","searchableContent":"─────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":126,"title":"s ↝ addUpkeep s V2-Role","content":"s ↝ addUpkeep s V2-Role","searchableContent":"s ↝ addupkeep s v2-role"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":128,"title":"data _-⟦_/_⟧⇀_","content":"data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where","searchableContent":"data _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type where"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":130,"title":"-- Initialization","content":"-- Initialization","searchableContent":"-- initialization"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":132,"title":"Init","content":"Init :","searchableContent":"init :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":133,"title":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","content":"∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'","searchableContent":"∙ ks k.-⟦ k.init pk-ib pk-eb pk-vt / k.pubkeys pks ⟧⇀ ks'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":134,"title":"∙ initBaseState B.-⟦ B.INIT ( V-chkCerts pks ) / B.STAKE SD ⟧⇀ bs'","content":"∙ initBaseState B.-⟦ B.INIT ( V-chkCerts pks ) / B.STAKE SD ⟧⇀ bs'","searchableContent":"∙ initbasestate b.-⟦ b.init ( v-chkcerts pks ) / b.stake sd ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":135,"title":"────────────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":136,"title":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","content":"nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks","searchableContent":"nothing -⟦ init v / empty ⟧⇀ initleiosstate v sd bs' pks"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":138,"title":"-- Network and Ledger","content":"-- Network and Ledger","searchableContent":"-- network and ledger"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":140,"title":"Slot","content":"Slot : let open LeiosState s renaming ( FFDState to ffds ; BaseState to bs ) in","searchableContent":"slot : let open leiosstate s renaming ( ffdstate to ffds ; basestate to bs ) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":141,"title":"∙ Upkeep ≡ᵉ allUpkeep","content":"∙ Upkeep ≡ᵉ allUpkeep","searchableContent":"∙ upkeep ≡ᵉ allupkeep"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":142,"title":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","content":"∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.ftch-ldg / b.base-ldg rbs ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":143,"title":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","content":"∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'","searchableContent":"∙ ffds ffd.-⟦ fetch / fetchres msgs ⟧⇀ ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":144,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":145,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","content":"just s -⟦ SLOT / EMPTY ⟧⇀ record s","searchableContent":"just s -⟦ slot / empty ⟧⇀ record s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":146,"title":"{ FFDState","content":"{ FFDState = ffds'","searchableContent":"{ ffdstate = ffds'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":147,"title":"; Ledger","content":"; Ledger = constructLedger rbs","searchableContent":"; ledger = constructledger rbs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":148,"title":"; slot","content":"; slot = suc slot","searchableContent":"; slot = suc slot"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":149,"title":"; Upkeep","content":"; Upkeep = ∅","searchableContent":"; upkeep = ∅"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":150,"title":"; BaseState","content":"; BaseState = bs'","searchableContent":"; basestate = bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":151,"title":"} ↑ L.filter ( isValid? s ) msgs","content":"} ↑ L.filter ( isValid? s ) msgs","searchableContent":"} ↑ l.filter ( isvalid? s ) msgs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":153,"title":"Ftch","content":"Ftch :","searchableContent":"ftch :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":154,"title":"────────────────────────────────────────────────────────","content":"────────────────────────────────────────────────────────","searchableContent":"────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":155,"title":"just s -⟦ FTCH-LDG / FTCH-LDG ( LeiosState.Ledger s ) ⟧⇀ s","content":"just s -⟦ FTCH-LDG / FTCH-LDG ( LeiosState.Ledger s ) ⟧⇀ s","searchableContent":"just s -⟦ ftch-ldg / ftch-ldg ( leiosstate.ledger s ) ⟧⇀ s"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":157,"title":"-- Base chain","content":"-- Base chain","searchableContent":"-- base chain"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":158,"title":"--","content":"--","searchableContent":"--"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":159,"title":"-- Note","content":"-- Note: Submitted data to the base chain is only taken into account","searchableContent":"-- note: submitted data to the base chain is only taken into account"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":160,"title":"-- if the party submitting is the block producer on the base chain","content":"-- if the party submitting is the block producer on the base chain","searchableContent":"-- if the party submitting is the block producer on the base chain"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":161,"title":"-- for the given slot","content":"-- for the given slot","searchableContent":"-- for the given slot"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":163,"title":"Base₁","content":"Base₁ :","searchableContent":"base₁ :"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":164,"title":"───────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":165,"title":"just s -⟦ SUBMIT ( inj₂ txs ) / EMPTY ⟧⇀ record s { ToPropose","content":"just s -⟦ SUBMIT ( inj₂ txs ) / EMPTY ⟧⇀ record s { ToPropose = txs }","searchableContent":"just s -⟦ submit ( inj₂ txs ) / empty ⟧⇀ record s { topropose = txs }"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":167,"title":"Base₂a","content":"Base₂a : let open LeiosState s renaming ( BaseState to bs ) in","searchableContent":"base₂a : let open leiosstate s renaming ( basestate to bs ) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":168,"title":"∙ needsUpkeep Base","content":"∙ needsUpkeep Base","searchableContent":"∙ needsupkeep base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":169,"title":"∙ eb ∈ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","content":"∙ eb ∈ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","searchableContent":"∙ eb ∈ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2 ) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":170,"title":"∙ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs'","content":"∙ bs B.-⟦ B.SUBMIT ( this eb ) / B.EMPTY ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.submit ( this eb ) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":171,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":172,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState","content":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":174,"title":"Base₂b","content":"Base₂b : let open LeiosState s renaming ( BaseState to bs ) in","searchableContent":"base₂b : let open leiosstate s renaming ( basestate to bs ) in"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":175,"title":"∙ needsUpkeep Base","content":"∙ needsUpkeep Base","searchableContent":"∙ needsupkeep base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":176,"title":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","content":"∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2 ) EBs","searchableContent":"∙ [] ≡ filter (λ eb → isvote2certified s eb × eb ∈ᴮ slice l slot 2 ) ebs"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":177,"title":"∙ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs'","content":"∙ bs B.-⟦ B.SUBMIT ( that ToPropose ) / B.EMPTY ⟧⇀ bs'","searchableContent":"∙ bs b.-⟦ b.submit ( that topropose ) / b.empty ⟧⇀ bs'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":178,"title":"───────────────────────────────────────────────────────────────────────","content":"───────────────────────────────────────────────────────────────────────","searchableContent":"───────────────────────────────────────────────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":179,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState","content":"just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base","searchableContent":"just s -⟦ slot / empty ⟧⇀ addupkeep record s { basestate = bs' } base"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":181,"title":"-- Protocol rules","content":"-- Protocol rules","searchableContent":"-- protocol rules"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":183,"title":"Roles","content":"Roles : ∙ s ↝ s'","searchableContent":"roles : ∙ s ↝ s'"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":184,"title":"─────────────────────────────","content":"─────────────────────────────","searchableContent":"─────────────────────────────"},{"moduleName":"Leios.Simplified","path":"Leios.Simplified.html","group":"Leios","lineNumber":185,"title":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","content":"just s -⟦ SLOT / EMPTY ⟧⇀ s'","searchableContent":"just s -⟦ slot / empty ⟧⇀ s'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":2,"title":"Leios.SpecStructure {-# OPTIONS --safe #-}","content":"Leios.SpecStructure {-# OPTIONS --safe #-}","searchableContent":"leios.specstructure {-# options --safe #-}"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude hiding ( id )","content":"open import Leios.Prelude hiding ( id )","searchableContent":"open import leios.prelude hiding ( id )"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","content":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":6,"title":"open import Leios.FFD","content":"open import Leios.FFD","searchableContent":"open import leios.ffd"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":7,"title":"open import Leios.VRF","content":"open import Leios.VRF","searchableContent":"open import leios.vrf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":9,"title":"import Leios.Base","content":"import Leios.Base","searchableContent":"import leios.base"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":10,"title":"import Leios.Blocks","content":"import Leios.Blocks","searchableContent":"import leios.blocks"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":11,"title":"import Leios.KeyRegistration","content":"import Leios.KeyRegistration","searchableContent":"import leios.keyregistration"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":12,"title":"import Leios.Voting","content":"import Leios.Voting","searchableContent":"import leios.voting"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":14,"title":"open import Data.Fin","content":"open import Data.Fin","searchableContent":"open import data.fin"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":16,"title":"module Leios.SpecStructure where","content":"module Leios.SpecStructure where","searchableContent":"module leios.specstructure where"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":18,"title":"record SpecStructure ( rounds","content":"record SpecStructure ( rounds : ℕ ) : Type₁ where","searchableContent":"record specstructure ( rounds : ℕ ) : type₁ where"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":19,"title":"field a","content":"field a : LeiosAbstract","searchableContent":"field a : leiosabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":21,"title":"open LeiosAbstract a public","content":"open LeiosAbstract a public","searchableContent":"open leiosabstract a public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":22,"title":"open Leios.Blocks a public","content":"open Leios.Blocks a public","searchableContent":"open leios.blocks a public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":24,"title":"field ⦃ IsBlock-Vote ⦄","content":"field ⦃ IsBlock-Vote ⦄ : IsBlock ( List Vote )","searchableContent":"field ⦃ isblock-vote ⦄ : isblock ( list vote )"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":25,"title":"⦃ Hashable-PreIBHeader ⦄","content":"⦃ Hashable-PreIBHeader ⦄ : Hashable PreIBHeader Hash","searchableContent":"⦃ hashable-preibheader ⦄ : hashable preibheader hash"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":26,"title":"⦃ Hashable-PreEndorserBlock ⦄","content":"⦃ Hashable-PreEndorserBlock ⦄ : Hashable PreEndorserBlock Hash","searchableContent":"⦃ hashable-preendorserblock ⦄ : hashable preendorserblock hash"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":27,"title":"id","content":"id : PoolID","searchableContent":"id : poolid"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":28,"title":"FFD'","content":"FFD' : FFDAbstract.Functionality ffdAbstract","searchableContent":"ffd' : ffdabstract.functionality ffdabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":29,"title":"vrf'","content":"vrf' : LeiosVRF a","searchableContent":"vrf' : leiosvrf a"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":31,"title":"open LeiosVRF vrf' public","content":"open LeiosVRF vrf' public","searchableContent":"open leiosvrf vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":33,"title":"field sk-IB sk-EB sk-VT","content":"field sk-IB sk-EB sk-VT : PrivKey","searchableContent":"field sk-ib sk-eb sk-vt : privkey"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":34,"title":"pk-IB pk-EB pk-VT","content":"pk-IB pk-EB pk-VT : PubKey","searchableContent":"pk-ib pk-eb pk-vt : pubkey"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":36,"title":"open Leios.Base a vrf' public","content":"open Leios.Base a vrf' public","searchableContent":"open leios.base a vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":38,"title":"field B'","content":"field B' : BaseAbstract","searchableContent":"field b' : baseabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":39,"title":"BF","content":"BF : BaseAbstract.Functionality B'","searchableContent":"bf : baseabstract.functionality b'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":40,"title":"initBaseState","content":"initBaseState : BaseAbstract.Functionality.State BF","searchableContent":"initbasestate : baseabstract.functionality.state bf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":42,"title":"open Leios.KeyRegistration a vrf' public","content":"open Leios.KeyRegistration a vrf' public","searchableContent":"open leios.keyregistration a vrf' public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":44,"title":"field K'","content":"field K' : KeyRegistrationAbstract","searchableContent":"field k' : keyregistrationabstract"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":45,"title":"KF","content":"KF : KeyRegistrationAbstract.Functionality K'","searchableContent":"kf : keyregistrationabstract.functionality k'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":47,"title":"module B","content":"module B = BaseAbstract.Functionality BF","searchableContent":"module b = baseabstract.functionality bf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":48,"title":"module K","content":"module K = KeyRegistrationAbstract.Functionality KF","searchableContent":"module k = keyregistrationabstract.functionality kf"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":49,"title":"module FFD","content":"module FFD = FFDAbstract.Functionality FFD'","searchableContent":"module ffd = ffdabstract.functionality ffd'"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":51,"title":"open Leios.Voting public","content":"open Leios.Voting public","searchableContent":"open leios.voting public"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":53,"title":"field va","content":"field va : VotingAbstract ( Fin rounds × EndorserBlock )","searchableContent":"field va : votingabstract ( fin rounds × endorserblock )"},{"moduleName":"Leios.SpecStructure","path":"Leios.SpecStructure.html","group":"Leios","lineNumber":54,"title":"open VotingAbstract va public","content":"open VotingAbstract va public","searchableContent":"open votingabstract va public"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":2,"title":"Leios.Traces {-# OPTIONS --safe #-}","content":"Leios.Traces {-# OPTIONS --safe #-}","searchableContent":"leios.traces {-# options --safe #-}"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":5,"title":"open import Leios.SpecStructure","content":"open import Leios.SpecStructure","searchableContent":"open import leios.specstructure"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":7,"title":"import Leios.Protocol","content":"import Leios.Protocol","searchableContent":"import leios.protocol"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":9,"title":"module Leios.Traces { n } ( ⋯","content":"module Leios.Traces { n } ( ⋯ : SpecStructure n ) { u : Type } ( let open Leios.Protocol ⋯ u )","searchableContent":"module leios.traces { n } ( ⋯ : specstructure n ) { u : type } ( let open leios.protocol ⋯ u )"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":10,"title":"( _-⟦_/_⟧⇀_","content":"( _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type )","searchableContent":"( _-⟦_/_⟧⇀_ : maybe leiosstate → leiosinput → leiosoutput → leiosstate → type )"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":11,"title":"where","content":"where","searchableContent":"where"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":13,"title":"_⇉_","content":"_⇉_ : LeiosState → LeiosState → Type","searchableContent":"_⇉_ : leiosstate → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":14,"title":"s₁ ⇉ s₂","content":"s₁ ⇉ s₂ = Σ[ ( i , o ) ∈ LeiosInput × LeiosOutput ] ( just s₁ -⟦ i / o ⟧⇀ s₂ )","searchableContent":"s₁ ⇉ s₂ = σ[ ( i , o ) ∈ leiosinput × leiosoutput ] ( just s₁ -⟦ i / o ⟧⇀ s₂ )"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":16,"title":"_⇉[_]_","content":"_⇉[_]_ : LeiosState → ℕ → LeiosState → Type","searchableContent":"_⇉[_]_ : leiosstate → ℕ → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":17,"title":"s₁ ⇉[ zero ] s₂","content":"s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂","searchableContent":"s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":18,"title":"s₁ ⇉[ suc m ] sₙ","content":"s₁ ⇉[ suc m ] sₙ = Σ[ s₂ ∈ LeiosState ] ( s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ )","searchableContent":"s₁ ⇉[ suc m ] sₙ = σ[ s₂ ∈ leiosstate ] ( s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ )"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":20,"title":"_⇉⋆_","content":"_⇉⋆_ : LeiosState → LeiosState → Type","searchableContent":"_⇉⋆_ : leiosstate → leiosstate → type"},{"moduleName":"Leios.Traces","path":"Leios.Traces.html","group":"Leios","lineNumber":21,"title":"s₁ ⇉⋆ sₙ","content":"s₁ ⇉⋆ sₙ = Σ[ n ∈ ℕ ] ( s₁ ⇉[ n ] sₙ )","searchableContent":"s₁ ⇉⋆ sₙ = σ[ n ∈ ℕ ] ( s₁ ⇉[ n ] sₙ )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":2,"title":"Leios.VRF {-# OPTIONS --safe #-}","content":"Leios.VRF {-# OPTIONS --safe #-}","searchableContent":"leios.vrf {-# options --safe #-}"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":5,"title":"open import Leios.Abstract","content":"open import Leios.Abstract","searchableContent":"open import leios.abstract"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":7,"title":"module Leios.VRF ( a","content":"module Leios.VRF ( a : LeiosAbstract ) ( let open LeiosAbstract a ) where","searchableContent":"module leios.vrf ( a : leiosabstract ) ( let open leiosabstract a ) where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":9,"title":"record VRF ( Dom Range PubKey","content":"record VRF ( Dom Range PubKey : Type ) : Type₁ where","searchableContent":"record vrf ( dom range pubkey : type ) : type₁ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":10,"title":"field isKeyPair","content":"field isKeyPair : PubKey → PrivKey → Type","searchableContent":"field iskeypair : pubkey → privkey → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":11,"title":"eval","content":"eval : PrivKey → Dom → Range × VrfPf","searchableContent":"eval : privkey → dom → range × vrfpf"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":12,"title":"verify","content":"verify : PubKey → Dom → Range → VrfPf → Type","searchableContent":"verify : pubkey → dom → range → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":13,"title":"verify?","content":"verify? : ( pk : PubKey ) → ( d : Dom ) → ( r : Range ) → ( pf : VrfPf ) → Dec ( verify pk d r pf )","searchableContent":"verify? : ( pk : pubkey ) → ( d : dom ) → ( r : range ) → ( pf : vrfpf ) → dec ( verify pk d r pf )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":15,"title":"record LeiosVRF","content":"record LeiosVRF : Type₁ where","searchableContent":"record leiosvrf : type₁ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":16,"title":"field PubKey","content":"field PubKey : Type","searchableContent":"field pubkey : type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":17,"title":"poolID","content":"poolID : PubKey → PoolID","searchableContent":"poolid : pubkey → poolid"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":18,"title":"verifySig","content":"verifySig : PubKey → Sig → Type","searchableContent":"verifysig : pubkey → sig → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":19,"title":"verifySig?","content":"verifySig? : ( pk : PubKey ) → ( sig : Sig ) → Dec ( verifySig pk sig )","searchableContent":"verifysig? : ( pk : pubkey ) → ( sig : sig ) → dec ( verifysig pk sig )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":21,"title":"vrf","content":"vrf : VRF ℕ ℕ PubKey","searchableContent":"vrf : vrf ℕ ℕ pubkey"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":23,"title":"open VRF vrf public","content":"open VRF vrf public","searchableContent":"open vrf vrf public"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":25,"title":"-- transforming slot numbers into VRF seeds","content":"-- transforming slot numbers into VRF seeds","searchableContent":"-- transforming slot numbers into vrf seeds"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":26,"title":"field genIBInput genEBInput genVInput genV1Input genV2Input","content":"field genIBInput genEBInput genVInput genV1Input genV2Input : ℕ → ℕ","searchableContent":"field genibinput genebinput genvinput genv1input genv2input : ℕ → ℕ"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":28,"title":"canProduceIB","content":"canProduceIB : ℕ → PrivKey → ℕ → VrfPf → Type","searchableContent":"canproduceib : ℕ → privkey → ℕ → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":29,"title":"canProduceIB slot k stake π","content":"canProduceIB slot k stake π = let ( val , pf ) = eval k ( genIBInput slot ) in val < stake × pf ≡ π","searchableContent":"canproduceib slot k stake π = let ( val , pf ) = eval k ( genibinput slot ) in val < stake × pf ≡ π"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":31,"title":"Dec-canProduceIB","content":"Dec-canProduceIB : ∀ { slot k stake } → ( ∃[ π ] canProduceIB slot k stake π ) ⊎ (∀ π → ¬ canProduceIB slot k stake π )","searchableContent":"dec-canproduceib : ∀ { slot k stake } → ( ∃[ π ] canproduceib slot k stake π ) ⊎ (∀ π → ¬ canproduceib slot k stake π )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":32,"title":"Dec-canProduceIB { slot } { k } { stake } with eval k ( genIBInput slot )","content":"Dec-canProduceIB { slot } { k } { stake } with eval k ( genIBInput slot )","searchableContent":"dec-canproduceib { slot } { k } { stake } with eval k ( genibinput slot )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":33,"title":"... | ( val , pf )","content":"... | ( val , pf ) = case ¿ val < stake ¿ of λ where","searchableContent":"... | ( val , pf ) = case ¿ val < stake ¿ of λ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":34,"title":"( yes p ) → inj₁ ( pf , p , refl )","content":"( yes p ) → inj₁ ( pf , p , refl )","searchableContent":"( yes p ) → inj₁ ( pf , p , refl )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":35,"title":"( no ¬p ) → inj₂ (λ π ( h , _) → ¬p h )","content":"( no ¬p ) → inj₂ (λ π ( h , _) → ¬p h )","searchableContent":"( no ¬p ) → inj₂ (λ π ( h , _) → ¬p h )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":37,"title":"canProduceIBPub","content":"canProduceIBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type","searchableContent":"canproduceibpub : ℕ → ℕ → pubkey → vrfpf → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":38,"title":"canProduceIBPub slot val k pf stake","content":"canProduceIBPub slot val k pf stake = verify k ( genIBInput slot ) val pf × val < stake","searchableContent":"canproduceibpub slot val k pf stake = verify k ( genibinput slot ) val pf × val < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":40,"title":"canProduceEB","content":"canProduceEB : ℕ → PrivKey → ℕ → VrfPf → Type","searchableContent":"canproduceeb : ℕ → privkey → ℕ → vrfpf → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":41,"title":"canProduceEB slot k stake π","content":"canProduceEB slot k stake π = let ( val , pf ) = eval k ( genEBInput slot ) in val < stake × pf ≡ π","searchableContent":"canproduceeb slot k stake π = let ( val , pf ) = eval k ( genebinput slot ) in val < stake × pf ≡ π"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":43,"title":"Dec-canProduceEB","content":"Dec-canProduceEB : ∀ { slot k stake } → ( ∃[ π ] canProduceEB slot k stake π ) ⊎ (∀ π → ¬ canProduceEB slot k stake π )","searchableContent":"dec-canproduceeb : ∀ { slot k stake } → ( ∃[ π ] canproduceeb slot k stake π ) ⊎ (∀ π → ¬ canproduceeb slot k stake π )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":44,"title":"Dec-canProduceEB { slot } { k } { stake } with eval k ( genEBInput slot )","content":"Dec-canProduceEB { slot } { k } { stake } with eval k ( genEBInput slot )","searchableContent":"dec-canproduceeb { slot } { k } { stake } with eval k ( genebinput slot )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":45,"title":"... | ( val , pf )","content":"... | ( val , pf ) = case ¿ val < stake ¿ of λ where","searchableContent":"... | ( val , pf ) = case ¿ val < stake ¿ of λ where"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":46,"title":"( yes p ) → inj₁ ( pf , p , refl )","content":"( yes p ) → inj₁ ( pf , p , refl )","searchableContent":"( yes p ) → inj₁ ( pf , p , refl )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":47,"title":"( no ¬p ) → inj₂ (λ π ( h , _) → ¬p h )","content":"( no ¬p ) → inj₂ (λ π ( h , _) → ¬p h )","searchableContent":"( no ¬p ) → inj₂ (λ π ( h , _) → ¬p h )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":49,"title":"canProduceEBPub","content":"canProduceEBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type","searchableContent":"canproduceebpub : ℕ → ℕ → pubkey → vrfpf → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":50,"title":"canProduceEBPub slot val k pf stake","content":"canProduceEBPub slot val k pf stake = verify k ( genEBInput slot ) val pf × val < stake","searchableContent":"canproduceebpub slot val k pf stake = verify k ( genebinput slot ) val pf × val < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":52,"title":"canProduceV","content":"canProduceV : ℕ → PrivKey → ℕ → Type","searchableContent":"canproducev : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":53,"title":"canProduceV slot k stake","content":"canProduceV slot k stake = proj₁ ( eval k ( genVInput slot )) < stake","searchableContent":"canproducev slot k stake = proj₁ ( eval k ( genvinput slot )) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":55,"title":"Dec-canProduceV","content":"Dec-canProduceV : ∀ { slot k stake } → Dec ( canProduceV slot k stake )","searchableContent":"dec-canproducev : ∀ { slot k stake } → dec ( canproducev slot k stake )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":56,"title":"Dec-canProduceV { slot } { k } { stake } with eval k ( genVInput slot )","content":"Dec-canProduceV { slot } { k } { stake } with eval k ( genVInput slot )","searchableContent":"dec-canproducev { slot } { k } { stake } with eval k ( genvinput slot )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":57,"title":"... | ( val , pf )","content":"... | ( val , pf ) = ¿ val < stake ¿","searchableContent":"... | ( val , pf ) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":59,"title":"canProduceV1","content":"canProduceV1 : ℕ → PrivKey → ℕ → Type","searchableContent":"canproducev1 : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":60,"title":"canProduceV1 slot k stake","content":"canProduceV1 slot k stake = proj₁ ( eval k ( genV1Input slot )) < stake","searchableContent":"canproducev1 slot k stake = proj₁ ( eval k ( genv1input slot )) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":62,"title":"Dec-canProduceV1","content":"Dec-canProduceV1 : ∀ { slot k stake } → Dec ( canProduceV1 slot k stake )","searchableContent":"dec-canproducev1 : ∀ { slot k stake } → dec ( canproducev1 slot k stake )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":63,"title":"Dec-canProduceV1 { slot } { k } { stake } with eval k ( genV1Input slot )","content":"Dec-canProduceV1 { slot } { k } { stake } with eval k ( genV1Input slot )","searchableContent":"dec-canproducev1 { slot } { k } { stake } with eval k ( genv1input slot )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":64,"title":"... | ( val , pf )","content":"... | ( val , pf ) = ¿ val < stake ¿","searchableContent":"... | ( val , pf ) = ¿ val < stake ¿"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":66,"title":"canProduceV2","content":"canProduceV2 : ℕ → PrivKey → ℕ → Type","searchableContent":"canproducev2 : ℕ → privkey → ℕ → type"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":67,"title":"canProduceV2 slot k stake","content":"canProduceV2 slot k stake = proj₁ ( eval k ( genV2Input slot )) < stake","searchableContent":"canproducev2 slot k stake = proj₁ ( eval k ( genv2input slot )) < stake"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":69,"title":"Dec-canProduceV2","content":"Dec-canProduceV2 : ∀ { slot k stake } → Dec ( canProduceV2 slot k stake )","searchableContent":"dec-canproducev2 : ∀ { slot k stake } → dec ( canproducev2 slot k stake )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":70,"title":"Dec-canProduceV2 { slot } { k } { stake } with eval k ( genV2Input slot )","content":"Dec-canProduceV2 { slot } { k } { stake } with eval k ( genV2Input slot )","searchableContent":"dec-canproducev2 { slot } { k } { stake } with eval k ( genv2input slot )"},{"moduleName":"Leios.VRF","path":"Leios.VRF.html","group":"Leios","lineNumber":71,"title":"... | ( val , pf )","content":"... | ( val , pf ) = ¿ val < stake ¿","searchableContent":"... | ( val , pf ) = ¿ val < stake ¿"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":2,"title":"Leios.Voting {-# OPTIONS --safe #-}","content":"Leios.Voting {-# OPTIONS --safe #-}","searchableContent":"leios.voting {-# options --safe #-}"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":4,"title":"open import Leios.Prelude","content":"open import Leios.Prelude","searchableContent":"open import leios.prelude"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":6,"title":"module Leios.Voting where","content":"module Leios.Voting where","searchableContent":"module leios.voting where"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":8,"title":"record VotingAbstract ( X","content":"record VotingAbstract ( X : Type ) : Type₁ where","searchableContent":"record votingabstract ( x : type ) : type₁ where"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":9,"title":"field VotingState","content":"field VotingState : Type","searchableContent":"field votingstate : type"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":10,"title":"initVotingState","content":"initVotingState : VotingState","searchableContent":"initvotingstate : votingstate"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":11,"title":"isVoteCertified","content":"isVoteCertified : VotingState → X → Type","searchableContent":"isvotecertified : votingstate → x → type"},{"moduleName":"Leios.Voting","path":"Leios.Voting.html","group":"Leios","lineNumber":13,"title":"⦃ isVoteCertified⁇ ⦄","content":"⦃ isVoteCertified⁇ ⦄ : ∀ { vs x } → isVoteCertified vs x ⁇","searchableContent":"⦃ isvotecertified⁇ ⦄ : ∀ { vs x } → isvotecertified vs x ⁇"}]; const searchInput = document.querySelector('.search-input'); const searchResults = document.querySelector('.search-results'); const searchOverlay = document.querySelector('.search-overlay'); @@ -16,20 +16,19 @@ if (searchInput && searchResults) { // Update the search results HTML generation function generateSearchResults(results) { return results.map(result => { - const highlightedTitle = result.title.replace( + // The content is already clean (no HTML tags) + const highlightedContent = result.content.replace( new RegExp(result.term, 'gi'), - match => `${match}` + match => '' + match + '' ); - return ` - -

- ${highlightedTitle} - ${result.moduleName} -

- ${result.content} - - `; + // Ensure the path is properly encoded + const encodedPath = encodeURIComponent(result.path); + + return '' + + '' + highlightedContent + '' + + '' + result.moduleName + '' + + ''; }).join(''); } @@ -45,8 +44,7 @@ if (searchInput && searchResults) { .map(item => ({ ...item, term: query - })) - .slice(0, 10); + })); if (results.length > 0) { searchResults.innerHTML = generateSearchResults(results); @@ -60,13 +58,25 @@ if (searchInput && searchResults) { const result = e.target.closest('.search-result'); if (result) { const lineNumber = result.dataset.line; - const targetElement = document.querySelector(`#L${lineNumber}`); + const targetElement = document.querySelector('#L' + lineNumber); if (targetElement) { - targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + // Calculate the offset to account for the header height and add some padding + const headerHeight = 60; // matches --header-height in CSS + const offset = headerHeight + 40; // add 20px padding + const elementPosition = targetElement.getBoundingClientRect().top; + const offsetPosition = elementPosition + window.pageYOffset - offset; + + window.scrollTo({ + top: offsetPosition, + behavior: 'smooth' + }); + targetElement.classList.add('highlight-line'); setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); } searchOverlay.classList.remove('active'); + searchInput.value = ''; // Clear the search input + searchResults.innerHTML = ''; // Clear the results } }); From a1ddae984a1cad5b810f3edf4b7c701b22f8bdd6 Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 19:51:22 +0200 Subject: [PATCH 05/10] fix: offset and link selection --- site/scripts/process-agda-html.js | 12 +----------- site/static/agda_html/Agda.css | 30 ++++++++++++++++++++++++++---- site/static/agda_html/agda.js | 12 +----------- 3 files changed, 28 insertions(+), 26 deletions(-) diff --git a/site/scripts/process-agda-html.js b/site/scripts/process-agda-html.js index 7ac65d0ff..52acad5d5 100755 --- a/site/scripts/process-agda-html.js +++ b/site/scripts/process-agda-html.js @@ -130,17 +130,7 @@ if (searchInput && searchResults) { const lineNumber = result.dataset.line; const targetElement = document.querySelector('#L' + lineNumber); if (targetElement) { - // Calculate the offset to account for the header height and add some padding - const headerHeight = 60; // matches --header-height in CSS - const offset = headerHeight + 40; // add 20px padding - const elementPosition = targetElement.getBoundingClientRect().top; - const offsetPosition = elementPosition + window.pageYOffset - offset; - - window.scrollTo({ - top: offsetPosition, - behavior: 'smooth' - }); - + targetElement.scrollIntoView({ behavior: 'smooth' }); targetElement.classList.add('highlight-line'); setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); } diff --git a/site/static/agda_html/Agda.css b/site/static/agda_html/Agda.css index d9d162328..c4915f556 100644 --- a/site/static/agda_html/Agda.css +++ b/site/static/agda_html/Agda.css @@ -8,7 +8,7 @@ --sidebar-width: 280px; --content-width: calc(100vw - var(--sidebar-width) - 4rem); --search-width: 300px; - --hover-color: #f6f8fa; + --hover-color: rgba(0, 0, 0, 0.05); --highlight-color: #ffeb3b; --muted-color: #6a737d; --header-height: 60px; @@ -23,7 +23,7 @@ --code-bg: #161b22; --link-color: #58a6ff; --header-bg: transparent; - --hover-color: #161b22; + --hover-color: rgba(255, 255, 255, 0.05); --highlight-color: #ffeb3b; --muted-color: #8b949e; } @@ -64,8 +64,13 @@ body { .agda-container { display: flex; - margin-top: var(--header-height); - min-height: calc(100vh - var(--header-height)); + position: fixed; + top: var(--header-height); + left: 0; + right: 0; + bottom: 0; + background-color: var(--bg-color); + overflow: hidden; } .agda-sidebar { @@ -78,6 +83,7 @@ body { overflow-y: auto; padding: 1.5rem; border-right: none; + z-index: 10; } .agda-sidebar h3 { @@ -133,7 +139,11 @@ body { margin-left: var(--sidebar-width); padding: 2rem; width: var(--content-width); + height: 100%; + overflow-y: auto; overflow-x: hidden; + position: relative; + box-sizing: border-box; } /* Add padding to Agda content to prevent overlap */ @@ -142,6 +152,18 @@ body { width: 100%; overflow-x: visible; max-width: none; + padding-bottom: 4rem; /* Add some padding at the bottom */ +} + +/* Add styles for highlighted line */ +.highlight-line { + background-color: var(--highlight-color); + transition: background-color 0.3s ease; +} + +/* Add styles for line anchors */ +[id^="L"] { + scroll-margin-top: calc(var(--header-height) + 2rem); } /* Agda Syntax Highlighting */ diff --git a/site/static/agda_html/agda.js b/site/static/agda_html/agda.js index 3b0203d8e..792eb5340 100644 --- a/site/static/agda_html/agda.js +++ b/site/static/agda_html/agda.js @@ -60,17 +60,7 @@ if (searchInput && searchResults) { const lineNumber = result.dataset.line; const targetElement = document.querySelector('#L' + lineNumber); if (targetElement) { - // Calculate the offset to account for the header height and add some padding - const headerHeight = 60; // matches --header-height in CSS - const offset = headerHeight + 40; // add 20px padding - const elementPosition = targetElement.getBoundingClientRect().top; - const offsetPosition = elementPosition + window.pageYOffset - offset; - - window.scrollTo({ - top: offsetPosition, - behavior: 'smooth' - }); - + targetElement.scrollIntoView({ behavior: 'smooth' }); targetElement.classList.add('highlight-line'); setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); } From e6427af0682023fa8a8632ae439bae9458b4d330 Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 20:01:36 +0200 Subject: [PATCH 06/10] style: content --- site/static/agda_html/Agda.css | 46 ++++++++++++++++++++++++---------- 1 file changed, 33 insertions(+), 13 deletions(-) diff --git a/site/static/agda_html/Agda.css b/site/static/agda_html/Agda.css index c4915f556..5840de398 100644 --- a/site/static/agda_html/Agda.css +++ b/site/static/agda_html/Agda.css @@ -39,6 +39,7 @@ body { color: var(--text-color); font-family: var(--font-sans); line-height: 1.6; + padding-top: var(--header-height); /* Added to offset fixed header */ } .agda-header { @@ -64,26 +65,19 @@ body { .agda-container { display: flex; - position: fixed; - top: var(--header-height); - left: 0; - right: 0; - bottom: 0; + flex-direction: row; background-color: var(--bg-color); - overflow: hidden; + min-height: calc(100vh - var(--header-height)); + position: relative; + top: 0; } .agda-sidebar { - position: fixed; - top: var(--header-height); - left: 0; - bottom: 0; width: var(--sidebar-width); background-color: var(--bg-color); overflow-y: auto; padding: 1.5rem; border-right: none; - z-index: 10; } .agda-sidebar h3 { @@ -136,10 +130,9 @@ body { .agda-content { flex: 1; - margin-left: var(--sidebar-width); padding: 2rem; width: var(--content-width); - height: 100%; + min-width: 0; overflow-y: auto; overflow-x: hidden; position: relative; @@ -615,3 +608,30 @@ body { .search-type { display: none; } + +/* Search term highlighting */ +.search-term-highlight mark { + background-color: var(--highlight-color); + color: var(--text-color); + padding: 0.1em 0.2em; + border-radius: 2px; + font-weight: 500; + animation: fadeHighlight 5s ease-out; +} + +@keyframes fadeHighlight { + 0% { + background-color: var(--highlight-color); + } + 80% { + background-color: var(--highlight-color); + } + 100% { + background-color: transparent; + } +} + +[data-theme="dark"] .search-term-highlight mark { + background-color: var(--highlight-color); + color: var(--text-color); +} From 4a432beab689dad93d4fc37558c52828c6aa5f30 Mon Sep 17 00:00:00 2001 From: William Wolff Date: Wed, 14 May 2025 20:05:10 +0200 Subject: [PATCH 07/10] fix: ux --- site/scripts/process-agda-html.js | 84 ++++++++++++++++++++++++++++--- site/static/agda_html/Agda.css | 6 ++- site/static/agda_html/agda.js | 84 ++++++++++++++++++++++++++++--- 3 files changed, 161 insertions(+), 13 deletions(-) diff --git a/site/scripts/process-agda-html.js b/site/scripts/process-agda-html.js index 52acad5d5..78bce8865 100755 --- a/site/scripts/process-agda-html.js +++ b/site/scripts/process-agda-html.js @@ -95,7 +95,7 @@ if (searchInput && searchResults) { // Ensure the path is properly encoded const encodedPath = encodeURIComponent(result.path); - return '' + + return '' + '' + highlightedContent + '' + '' + result.moduleName + '' + ''; @@ -127,19 +127,91 @@ if (searchInput && searchResults) { document.addEventListener('click', (e) => { const result = e.target.closest('.search-result'); if (result) { + e.preventDefault(); // Prevent default link behavior const lineNumber = result.dataset.line; - const targetElement = document.querySelector('#L' + lineNumber); - if (targetElement) { - targetElement.scrollIntoView({ behavior: 'smooth' }); - targetElement.classList.add('highlight-line'); - setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); + const searchTerm = result.dataset.term; + const href = result.getAttribute('href'); + + // If we're already on the target page + if (window.location.pathname.endsWith(href.split('#')[0])) { + scrollToLine(lineNumber, searchTerm); + } else { + // If we're not on the target page, navigate to it + // Store the line number and search term in sessionStorage + sessionStorage.setItem('scrollToLine', lineNumber); + sessionStorage.setItem('searchTerm', searchTerm); + window.location.href = href; } + searchOverlay.classList.remove('active'); searchInput.value = ''; // Clear the search input searchResults.innerHTML = ''; // Clear the results } }); + // Function to scroll to a specific line and highlight terms + function scrollToLine(lineNumber, searchTerm) { + // Wait for the next frame to ensure content is rendered + requestAnimationFrame(() => { + const targetElement = document.querySelector('#L' + lineNumber); + if (targetElement) { + // Add a small delay to ensure content is fully rendered + setTimeout(() => { + targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + targetElement.classList.add('highlight-line'); + setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); + + // Highlight all matching terms on the page + const content = document.querySelector('.agda-content'); + if (content && searchTerm) { + const regex = new RegExp(searchTerm, 'gi'); + const walker = document.createTreeWalker(content, NodeFilter.SHOW_TEXT, null, false); + const nodesToHighlight = []; + + while (walker.nextNode()) { + const node = walker.currentNode; + if (node.textContent.match(regex)) { + nodesToHighlight.push(node); + } + } + + nodesToHighlight.forEach(node => { + const span = document.createElement('span'); + span.className = 'search-term-highlight'; + span.innerHTML = node.textContent.replace(regex, match => '' + match + ''); + node.parentNode.replaceChild(span, node); + }); + + // Remove highlights after 5 seconds + setTimeout(() => { + const highlights = document.querySelectorAll('.search-term-highlight'); + highlights.forEach(highlight => { + const text = highlight.textContent; + const textNode = document.createTextNode(text); + highlight.parentNode.replaceChild(textNode, highlight); + }); + }, 5000); + } + }, 100); // Small delay to ensure content is rendered + } + }); + } + + // Check for stored line number and search term on page load + window.addEventListener('load', () => { + // Wait for a short time to ensure all content is loaded and rendered + setTimeout(() => { + const storedLine = sessionStorage.getItem('scrollToLine'); + const storedTerm = sessionStorage.getItem('searchTerm'); + if (storedLine) { + scrollToLine(storedLine, storedTerm); + // Clear the stored values + sessionStorage.removeItem('scrollToLine'); + sessionStorage.removeItem('searchTerm'); + } + }, 100); + }); + // Close search when clicking outside or pressing Escape document.addEventListener('click', (e) => { if (e.target === searchOverlay) { diff --git a/site/static/agda_html/Agda.css b/site/static/agda_html/Agda.css index 5840de398..4f69b4acd 100644 --- a/site/static/agda_html/Agda.css +++ b/site/static/agda_html/Agda.css @@ -134,7 +134,7 @@ body { width: var(--content-width); min-width: 0; overflow-y: auto; - overflow-x: hidden; + overflow-x: auto; position: relative; box-sizing: border-box; } @@ -146,6 +146,7 @@ body { overflow-x: visible; max-width: none; padding-bottom: 4rem; /* Add some padding at the bottom */ + white-space: pre; /* Preserve whitespace and line breaks */ } /* Add styles for highlighted line */ @@ -174,6 +175,9 @@ body { background-color: var(--code-bg); padding: 0.2em 0.4em; border-radius: 3px; + white-space: pre; /* Preserve whitespace and line breaks */ + overflow-x: auto; /* Allow horizontal scrolling */ + display: inline-block; /* Ensure proper overflow behavior */ } .Agda a { diff --git a/site/static/agda_html/agda.js b/site/static/agda_html/agda.js index 792eb5340..81e777d34 100644 --- a/site/static/agda_html/agda.js +++ b/site/static/agda_html/agda.js @@ -25,7 +25,7 @@ if (searchInput && searchResults) { // Ensure the path is properly encoded const encodedPath = encodeURIComponent(result.path); - return '' + + return '' + '' + highlightedContent + '' + '' + result.moduleName + '' + ''; @@ -57,19 +57,91 @@ if (searchInput && searchResults) { document.addEventListener('click', (e) => { const result = e.target.closest('.search-result'); if (result) { + e.preventDefault(); // Prevent default link behavior const lineNumber = result.dataset.line; - const targetElement = document.querySelector('#L' + lineNumber); - if (targetElement) { - targetElement.scrollIntoView({ behavior: 'smooth' }); - targetElement.classList.add('highlight-line'); - setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); + const searchTerm = result.dataset.term; + const href = result.getAttribute('href'); + + // If we're already on the target page + if (window.location.pathname.endsWith(href.split('#')[0])) { + scrollToLine(lineNumber, searchTerm); + } else { + // If we're not on the target page, navigate to it + // Store the line number and search term in sessionStorage + sessionStorage.setItem('scrollToLine', lineNumber); + sessionStorage.setItem('searchTerm', searchTerm); + window.location.href = href; } + searchOverlay.classList.remove('active'); searchInput.value = ''; // Clear the search input searchResults.innerHTML = ''; // Clear the results } }); + // Function to scroll to a specific line and highlight terms + function scrollToLine(lineNumber, searchTerm) { + // Wait for the next frame to ensure content is rendered + requestAnimationFrame(() => { + const targetElement = document.querySelector('#L' + lineNumber); + if (targetElement) { + // Add a small delay to ensure content is fully rendered + setTimeout(() => { + targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + targetElement.classList.add('highlight-line'); + setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); + + // Highlight all matching terms on the page + const content = document.querySelector('.agda-content'); + if (content && searchTerm) { + const regex = new RegExp(searchTerm, 'gi'); + const walker = document.createTreeWalker(content, NodeFilter.SHOW_TEXT, null, false); + const nodesToHighlight = []; + + while (walker.nextNode()) { + const node = walker.currentNode; + if (node.textContent.match(regex)) { + nodesToHighlight.push(node); + } + } + + nodesToHighlight.forEach(node => { + const span = document.createElement('span'); + span.className = 'search-term-highlight'; + span.innerHTML = node.textContent.replace(regex, match => '' + match + ''); + node.parentNode.replaceChild(span, node); + }); + + // Remove highlights after 5 seconds + setTimeout(() => { + const highlights = document.querySelectorAll('.search-term-highlight'); + highlights.forEach(highlight => { + const text = highlight.textContent; + const textNode = document.createTextNode(text); + highlight.parentNode.replaceChild(textNode, highlight); + }); + }, 5000); + } + }, 100); // Small delay to ensure content is rendered + } + }); + } + + // Check for stored line number and search term on page load + window.addEventListener('load', () => { + // Wait for a short time to ensure all content is loaded and rendered + setTimeout(() => { + const storedLine = sessionStorage.getItem('scrollToLine'); + const storedTerm = sessionStorage.getItem('searchTerm'); + if (storedLine) { + scrollToLine(storedLine, storedTerm); + // Clear the stored values + sessionStorage.removeItem('scrollToLine'); + sessionStorage.removeItem('searchTerm'); + } + }, 100); + }); + // Close search when clicking outside or pressing Escape document.addEventListener('click', (e) => { if (e.target === searchOverlay) { From d9e5e9629bb3f5181de123bcdd0d4fd08f1528ba Mon Sep 17 00:00:00 2001 From: William Wolff Date: Thu, 15 May 2025 12:28:36 +0200 Subject: [PATCH 08/10] fix: scripts for search --- site/scripts/dev-with-formal-spec.sh | 2 +- site/scripts/process-agda-html.js | 116 +++++++++++++++++---------- 2 files changed, 74 insertions(+), 44 deletions(-) diff --git a/site/scripts/dev-with-formal-spec.sh b/site/scripts/dev-with-formal-spec.sh index 95b249812..dee99f140 100755 --- a/site/scripts/dev-with-formal-spec.sh +++ b/site/scripts/dev-with-formal-spec.sh @@ -28,7 +28,7 @@ if [ -f "$SITE_DIR/static/agda_html/agda.css" ]; then fi # Copy all files except Agda.css -cp -r result/html/* "$SITE_DIR/static/agda_html/" +cp -r result/share/doc/agda/html/* "$SITE_DIR/static/agda_html/" # Restore our custom CSS if [ -f "$SITE_DIR/static/agda_html/agda.css.bak" ]; then diff --git a/site/scripts/process-agda-html.js b/site/scripts/process-agda-html.js index 78bce8865..7bed5cafa 100755 --- a/site/scripts/process-agda-html.js +++ b/site/scripts/process-agda-html.js @@ -151,50 +151,54 @@ if (searchInput && searchResults) { // Function to scroll to a specific line and highlight terms function scrollToLine(lineNumber, searchTerm) { - // Wait for the next frame to ensure content is rendered - requestAnimationFrame(() => { - const targetElement = document.querySelector('#L' + lineNumber); - if (targetElement) { - // Add a small delay to ensure content is fully rendered - setTimeout(() => { - targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); - targetElement.classList.add('highlight-line'); - setTimeout(() => targetElement.classList.remove('highlight-line'), 2000); - - // Highlight all matching terms on the page - const content = document.querySelector('.agda-content'); - if (content && searchTerm) { - const regex = new RegExp(searchTerm, 'gi'); - const walker = document.createTreeWalker(content, NodeFilter.SHOW_TEXT, null, false); - const nodesToHighlight = []; - - while (walker.nextNode()) { - const node = walker.currentNode; - if (node.textContent.match(regex)) { - nodesToHighlight.push(node); - } + const targetElement = document.querySelector('.Agda .line:nth-child(' + (lineNumber - 1) + ')'); + if (targetElement) { + // Remove any existing highlights + document.querySelectorAll('.line.highlight-line').forEach(line => { + line.classList.remove('highlight-line'); + }); + + // Add highlight to target line + targetElement.classList.add('highlight-line'); + + // Scroll to the line + targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' }); + + // Highlight matching terms + if (searchTerm) { + const content = document.querySelector('.agda-content'); + if (content) { + const regex = new RegExp(searchTerm, 'gi'); + const walker = document.createTreeWalker(content, NodeFilter.SHOW_TEXT, null, false); + const nodesToHighlight = []; + + while (walker.nextNode()) { + const node = walker.currentNode; + if (node.textContent.match(regex)) { + nodesToHighlight.push(node); + } + } + + nodesToHighlight.forEach(node => { + const span = document.createElement('span'); + span.className = 'search-term-highlight'; + span.innerHTML = node.textContent.replace(regex, match => '' + match + ''); + node.parentNode.replaceChild(span, node); + }); + + // Remove highlights after 5 seconds + setTimeout(() => { + const highlights = document.querySelectorAll('.search-term-highlight'); + highlights.forEach(highlight => { + const text = highlight.textContent; + const textNode = document.createTextNode(text); + highlight.parentNode.replaceChild(textNode, highlight); + }); + targetElement.classList.remove('highlight-line'); + }, 5000); } - - nodesToHighlight.forEach(node => { - const span = document.createElement('span'); - span.className = 'search-term-highlight'; - span.innerHTML = node.textContent.replace(regex, match => '' + match + ''); - node.parentNode.replaceChild(span, node); - }); - - // Remove highlights after 5 seconds - setTimeout(() => { - const highlights = document.querySelectorAll('.search-term-highlight'); - highlights.forEach(highlight => { - const text = highlight.textContent; - const textNode = document.createTextNode(text); - highlight.parentNode.replaceChild(textNode, highlight); - }); - }, 5000); - } - }, 100); // Small delay to ensure content is rendered - } - }); + } + } } // Check for stored line number and search term on page load @@ -212,6 +216,17 @@ if (searchInput && searchResults) { }, 100); }); + // Check for line number in URL hash on page load + window.addEventListener('load', () => { + const hash = window.location.hash; + if (hash.startsWith('#L')) { + const lineNumber = parseInt(hash.substring(2)); + if (!isNaN(lineNumber)) { + scrollToLine(lineNumber); + } + } + }); + // Close search when clicking outside or pressing Escape document.addEventListener('click', (e) => { if (e.target === searchOverlay) { @@ -260,6 +275,21 @@ fs.readdirSync(AGDA_HTML_DIR).forEach(file => { ` ); + // Wrap each line in the Agda content with a numbered div + content = content.replace( + /
([\s\S]*?)<\/pre>/g,
+            (match, content) => {
+                const lines = content.split('\n');
+                const numberedLines = lines.map(line => {
+                    if (line.trim() === '') {
+                        return '
';
+                    }
+                    return `
${line}
`;
+                }).join('\n');
+                return `
${numberedLines}
`;
+            }
+        );
+
         // Add header, sidebar, and theme toggle
         content = content.replace(
             /]*>/,

From e7c6518fe2231154ad6a0dd2b9faabb8187245bb Mon Sep 17 00:00:00 2001
From: William Wolff 
Date: Thu, 15 May 2025 12:29:08 +0200
Subject: [PATCH 09/10] site: add static formal spec

---
 site/static/agda_html/Agda.Builtin.Bool.html  |   92 +
 .../Agda.Builtin.Char.Properties.html         |   87 +
 site/static/agda_html/Agda.Builtin.Char.html  |   95 +
 .../agda_html/Agda.Builtin.Equality.html      |   86 +
 .../Agda.Builtin.Float.Properties.html        |   87 +
 site/static/agda_html/Agda.Builtin.Float.html |  286 ++
 site/static/agda_html/Agda.Builtin.IO.html    |   87 +
 site/static/agda_html/Agda.Builtin.Int.html   |   95 +
 site/static/agda_html/Agda.Builtin.List.html  |   93 +
 site/static/agda_html/Agda.Builtin.Maybe.html |   86 +
 site/static/agda_html/Agda.Builtin.Nat.html   |  211 ++
 .../Agda.Builtin.Reflection.Properties.html   |   88 +
 .../agda_html/Agda.Builtin.Reflection.html    |  558 ++++
 site/static/agda_html/Agda.Builtin.Sigma.html |   94 +
 .../static/agda_html/Agda.Builtin.Strict.html |   86 +
 .../Agda.Builtin.String.Properties.html       |   88 +
 .../static/agda_html/Agda.Builtin.String.html |  113 +
 site/static/agda_html/Agda.Builtin.Unit.html  |   87 +
 .../Agda.Builtin.Word.Properties.html         |   87 +
 site/static/agda_html/Agda.Builtin.Word.html  |   90 +
 site/static/agda_html/Agda.Primitive.html     |  118 +
 site/static/agda_html/Agda.css                |   55 +-
 .../agda_html/Algebra.Apartness.Bundles.html  |  142 +
 .../Algebra.Apartness.Structures.html         |  129 +
 site/static/agda_html/Algebra.Apartness.html  |   89 +
 .../static/agda_html/Algebra.Bundles.Raw.html |  412 +++
 site/static/agda_html/Algebra.Bundles.html    | 1335 +++++++++
 .../agda_html/Algebra.Consequences.Base.html  |  120 +
 .../Algebra.Consequences.Propositional.html   |  281 ++
 .../Algebra.Consequences.Setoid.html          |  522 ++++
 .../Algebra.Construct.LiftedChoice.html       |  264 ++
 .../Algebra.Construct.NaturalChoice.Base.html |  138 +
 .../Algebra.Construct.NaturalChoice.Max.html  |  124 +
 ...Algebra.Construct.NaturalChoice.MaxOp.html |  158 ++
 .../Algebra.Construct.NaturalChoice.Min.html  |  130 +
 ...ebra.Construct.NaturalChoice.MinMaxOp.html |  222 ++
 ...Algebra.Construct.NaturalChoice.MinOp.html |  334 +++
 site/static/agda_html/Algebra.Core.html       |   99 +
 .../Algebra.Definitions.RawMagma.html         |  163 ++
 .../Algebra.Definitions.RawMonoid.html        |  144 +
 .../Algebra.Definitions.RawSemiring.html      |  162 ++
 .../static/agda_html/Algebra.Definitions.html |  311 ++
 site/static/agda_html/Algebra.Function.html   |  159 ++
 .../Algebra.Lattice.Bundles.Raw.html          |  110 +
 .../agda_html/Algebra.Lattice.Bundles.html    |  306 ++
 ...Lattice.Construct.NaturalChoice.MaxOp.html |  103 +
 ...tice.Construct.NaturalChoice.MinMaxOp.html |  172 ++
 ...Lattice.Construct.NaturalChoice.MinOp.html |  116 +
 ....Lattice.Morphism.LatticeMonomorphism.html |  200 ++
 .../Algebra.Lattice.Morphism.Structures.html  |  195 ++
 ...bra.Lattice.Properties.BooleanAlgebra.html |  615 ++++
 ...attice.Properties.DistributiveLattice.html |  116 +
 .../Algebra.Lattice.Properties.Lattice.html   |  257 ++
 ...lgebra.Lattice.Properties.Semilattice.html |  134 +
 .../Algebra.Lattice.Structures.Biased.html    |  200 ++
 .../agda_html/Algebra.Lattice.Structures.html |  269 ++
 site/static/agda_html/Algebra.Lattice.html    |   95 +
 .../Algebra.Morphism.Definitions.html         |  123 +
 .../Algebra.Morphism.GroupMonomorphism.html   |  174 ++
 .../Algebra.Morphism.MagmaMonomorphism.html   |  200 ++
 .../Algebra.Morphism.MonoidMonomorphism.html  |  171 ++
 .../Algebra.Morphism.RingMonomorphism.html    |  237 ++
 .../Algebra.Morphism.Structures.html          |  866 ++++++
 site/static/agda_html/Algebra.Morphism.html   |  286 ++
 .../Algebra.Properties.AbelianGroup.html      |  118 +
 .../Algebra.Properties.CommutativeMonoid.html |  119 +
 ...gebra.Properties.CommutativeSemigroup.html |  251 ++
 .../agda_html/Algebra.Properties.Group.html   |  247 ++
 .../agda_html/Algebra.Properties.Loop.html    |  134 +
 .../Algebra.Properties.Monoid.Mult.html       |  157 ++
 .../Algebra.Properties.Quasigroup.html        |  122 +
 .../agda_html/Algebra.Properties.Ring.html    |  109 +
 .../Algebra.Properties.RingWithoutOne.html    |  151 +
 .../Algebra.Properties.Semigroup.html         |  107 +
 .../Algebra.Properties.Semiring.Exp.html      |  149 +
 ...bra.Solver.Ring.AlmostCommutativeRing.html |  228 ++
 .../agda_html/Algebra.Solver.Ring.Lemmas.html |  191 ++
 .../agda_html/Algebra.Solver.Ring.Simple.html |   98 +
 .../static/agda_html/Algebra.Solver.Ring.html |  629 +++++
 .../agda_html/Algebra.Structures.Biased.html  |  355 +++
 site/static/agda_html/Algebra.Structures.html | 1112 ++++++++
 site/static/agda_html/Algebra.html            |   93 +
 .../Axiom.Extensionality.Propositional.html   |  139 +
 site/static/agda_html/Axiom.Set.Factor.html   |  149 +
 site/static/agda_html/Axiom.Set.List.html     |  140 +
 site/static/agda_html/Axiom.Set.Map.Dec.html  |  159 ++
 site/static/agda_html/Axiom.Set.Map.html      |  483 ++++
 .../agda_html/Axiom.Set.Properties.html       |  517 ++++
 site/static/agda_html/Axiom.Set.Rel.html      |  364 +++
 site/static/agda_html/Axiom.Set.Sum.html      |  241 ++
 site/static/agda_html/Axiom.Set.TotalMap.html |  174 ++
 .../agda_html/Axiom.Set.TotalMapOn.html       |  136 +
 site/static/agda_html/Axiom.Set.html          |  458 +++
 .../Axiom.UniquenessOfIdentityProofs.html     |  153 +
 site/static/agda_html/CategoricalCrypto.html  |  409 +++
 .../agda_html/Class.Applicative.Core.html     |  131 +
 .../Class.Applicative.Instances.html          |  139 +
 site/static/agda_html/Class.Applicative.html  |   82 +
 site/static/agda_html/Class.Bifunctor.html    |  142 +
 .../Class.CommutativeMonoid.Core.html         |  103 +
 .../Class.CommutativeMonoid.Instances.html    |   88 +
 .../agda_html/Class.CommutativeMonoid.html    |   83 +
 .../static/agda_html/Class.Computational.html |  269 ++
 .../agda_html/Class.Computational22.html      |  123 +
 site/static/agda_html/Class.Convertible.html  |  161 ++
 site/static/agda_html/Class.Core.html         |  100 +
 site/static/agda_html/Class.DecEq.Core.html   |   99 +
 .../agda_html/Class.DecEq.Instances.html      |  155 +
 site/static/agda_html/Class.DecEq.html        |   83 +
 .../agda_html/Class.Decidable.Core.html       |  159 ++
 .../agda_html/Class.Decidable.Instances.html  |  185 ++
 site/static/agda_html/Class.Decidable.html    |   82 +
 site/static/agda_html/Class.Functor.Core.html |  109 +
 .../agda_html/Class.Functor.Instances.html    |  132 +
 site/static/agda_html/Class.Functor.html      |   82 +
 site/static/agda_html/Class.HasAdd.Core.html  |   85 +
 .../agda_html/Class.HasAdd.Instance.html      |   98 +
 site/static/agda_html/Class.HasAdd.html       |   82 +
 site/static/agda_html/Class.HasEmptySet.html  |  100 +
 site/static/agda_html/Class.HasHsType.html    |  128 +
 .../static/agda_html/Class.HasOrder.Core.html |  237 ++
 .../agda_html/Class.HasOrder.Instance.html    |  112 +
 site/static/agda_html/Class.HasOrder.html     |   83 +
 site/static/agda_html/Class.HasSingleton.html |  100 +
 site/static/agda_html/Class.Hashable.html     |   92 +
 site/static/agda_html/Class.IsSet.html        |  130 +
 site/static/agda_html/Class.Monad.Core.html   |  142 +
 .../agda_html/Class.Monad.Instances.html      |   98 +
 site/static/agda_html/Class.Monad.html        |   82 +
 .../agda_html/Class.MonadError.Instances.html |   82 +
 site/static/agda_html/Class.MonadError.html   |  120 +
 .../Class.MonadReader.Instances.html          |   82 +
 site/static/agda_html/Class.MonadReader.html  |  118 +
 .../agda_html/Class.MonadTC.Instances.html    |   82 +
 site/static/agda_html/Class.MonadTC.html      |  444 +++
 site/static/agda_html/Class.Monoid.Core.html  |  101 +
 .../agda_html/Class.Monoid.Instances.html     |  159 ++
 site/static/agda_html/Class.Monoid.html       |   82 +
 site/static/agda_html/Class.Prelude.html      |  151 +
 .../agda_html/Class.Semigroup.Core.html       |  101 +
 .../agda_html/Class.Semigroup.Instances.html  |  159 ++
 site/static/agda_html/Class.Semigroup.html    |   82 +
 site/static/agda_html/Class.Show.Core.html    |   91 +
 .../agda_html/Class.Show.Instances.html       |  146 +
 site/static/agda_html/Class.Show.html         |   82 +
 site/static/agda_html/Class.ToBool.html       |  129 +
 .../agda_html/Class.Traversable.Core.html     |   91 +
 .../Class.Traversable.Instances.html          |   99 +
 site/static/agda_html/Class.Traversable.html  |   82 +
 site/static/agda_html/Data.Bool.Base.html     |  152 +
 .../agda_html/Data.Bool.Properties.html       |  865 ++++++
 site/static/agda_html/Data.Bool.Show.html     |   98 +
 site/static/agda_html/Data.Bool.html          |   97 +
 site/static/agda_html/Data.Char.Base.html     |  138 +
 .../agda_html/Data.Char.Properties.html       |  382 +++
 site/static/agda_html/Data.Char.html          |   93 +
 .../static/agda_html/Data.Container.Core.html |  148 +
 .../agda_html/Data.Container.Membership.html  |   99 +
 .../Data.Container.Morphism.Properties.html   |  150 +
 .../agda_html/Data.Container.Morphism.html    |  104 +
 .../agda_html/Data.Container.Properties.html  |  102 +
 .../agda_html/Data.Container.Related.html     |  132 +
 ...ainer.Relation.Binary.Equality.Setoid.html |  134 +
 ....Relation.Binary.Pointwise.Properties.html |  119 +
 ...a.Container.Relation.Binary.Pointwise.html |  112 +
 .../Data.Container.Relation.Unary.All.html    |  120 +
 .../Data.Container.Relation.Unary.Any.html    |  122 +
 site/static/agda_html/Data.Container.html     |  133 +
 site/static/agda_html/Data.Digit.html         |  217 ++
 .../agda_html/Data.Empty.Polymorphic.html     |   96 +
 site/static/agda_html/Data.Empty.html         |  117 +
 site/static/agda_html/Data.Fin.Base.html      |  420 +++
 site/static/agda_html/Data.Fin.Patterns.html  |  102 +
 .../static/agda_html/Data.Fin.Properties.html | 1266 +++++++++
 site/static/agda_html/Data.Fin.html           |  107 +
 site/static/agda_html/Data.Float.Base.html    |  164 ++
 .../agda_html/Data.Float.Properties.html      |  184 ++
 site/static/agda_html/Data.Float.html         |   92 +
 site/static/agda_html/Data.Integer.Base.html  |  425 +++
 .../agda_html/Data.Integer.Coprimality.html   |  116 +
 .../agda_html/Data.Integer.Divisibility.html  |  132 +
 site/static/agda_html/Data.Integer.GCD.html   |  140 +
 .../Data.Integer.Properties.NatLemmas.html    |  146 +
 .../agda_html/Data.Integer.Properties.html    | 2459 ++++++++++++++++
 site/static/agda_html/Data.Integer.Show.html  |  101 +
 .../static/agda_html/Data.Integer.Solver.html |   99 +
 site/static/agda_html/Data.Integer.html       |  124 +
 site/static/agda_html/Data.Irrelevant.html    |  131 +
 site/static/agda_html/Data.List.Base.html     |  659 +++++
 .../static/agda_html/Data.List.Effectful.html |  392 +++
 .../agda_html/Data.List.Ext.Properties.html   |  140 +
 .../agda_html/Data.List.Extrema.Core.html     |  198 ++
 site/static/agda_html/Data.List.Extrema.html  |  324 +++
 ...Data.List.Membership.DecPropositional.html |  100 +
 .../Data.List.Membership.DecSetoid.html       |  110 +
 ...bership.Propositional.Properties.Core.html |  162 ++
 ...t.Membership.Propositional.Properties.html |  480 ++++
 .../Data.List.Membership.Propositional.html   |  112 +
 ...ata.List.Membership.Setoid.Properties.html |  525 ++++
 .../Data.List.Membership.Setoid.html          |  133 +
 .../agda_html/Data.List.NonEmpty.Base.html    |  415 +++
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 ...Data.List.NonEmpty.Relation.Unary.All.html |  116 +
 site/static/agda_html/Data.List.NonEmpty.html |  117 +
 .../agda_html/Data.List.Properties.html       | 1698 +++++++++++
 ...ist.Relation.Binary.BagAndSetEquality.html |  681 +++++
 ...elation.Binary.Disjoint.Propositional.html |   95 +
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 ....List.Relation.Binary.Disjoint.Setoid.html |  114 +
 ...elation.Binary.Equality.Propositional.html |  112 +
 ....List.Relation.Binary.Equality.Setoid.html |  236 ++
 .../Data.List.Relation.Binary.Lex.Core.html   |  124 +
 .../Data.List.Relation.Binary.Lex.Strict.html |  326 +++
 .../Data.List.Relation.Binary.Lex.html        |  195 ++
 ...lation.Binary.Permutation.Homogeneous.html |  138 +
 ....Permutation.Propositional.Properties.html |  473 ++++
 ...tion.Binary.Permutation.Propositional.html |  177 ++
 ....Binary.Permutation.Setoid.Properties.html |  582 ++++
 ...st.Relation.Binary.Permutation.Setoid.html |  192 ++
 ...a.List.Relation.Binary.Pointwise.Base.html |  142 +
 ....Relation.Binary.Pointwise.Properties.html |  154 +
 .../Data.List.Relation.Binary.Pointwise.html  |  354 +++
 ...ion.Binary.Sublist.Heterogeneous.Core.html |  107 +
 ...nary.Sublist.Heterogeneous.Properties.html |  797 ++++++
 ...Relation.Binary.Sublist.Heterogeneous.html |  187 ++
 ...nary.Sublist.Propositional.Properties.html |  311 ++
 ...Relation.Binary.Sublist.Propositional.html |  234 ++
 ...tion.Binary.Sublist.Setoid.Properties.html |  428 +++
 ...a.List.Relation.Binary.Sublist.Setoid.html |  368 +++
 ...inary.Subset.Propositional.Properties.html |  404 +++
 ....Relation.Binary.Subset.Propositional.html |   95 +
 ...ation.Binary.Subset.Setoid.Properties.html |  393 +++
 ...ta.List.Relation.Binary.Subset.Setoid.html |  115 +
 ...ta.List.Relation.Unary.All.Properties.html |  881 ++++++
 .../Data.List.Relation.Unary.All.html         |  324 +++
 ...ata.List.Relation.Unary.AllPairs.Core.html |  111 +
 ...st.Relation.Unary.AllPairs.Properties.html |  228 ++
 .../Data.List.Relation.Unary.AllPairs.html    |  157 ++
 ...ta.List.Relation.Unary.Any.Properties.html |  826 ++++++
 .../Data.List.Relation.Unary.Any.html         |  186 ++
 ....List.Relation.Unary.First.Properties.html |  194 ++
 .../Data.List.Relation.Unary.First.html       |  185 ++
 ...List.Relation.Unary.Linked.Properties.html |  228 ++
 .../Data.List.Relation.Unary.Linked.html      |  190 ++
 ...on.Unary.Sorted.TotalOrder.Properties.html |  229 ++
 ...List.Relation.Unary.Sorted.TotalOrder.html |  126 +
 ...ry.Unique.DecPropositional.Properties.html |  107 +
 ...elation.Unary.Unique.DecPropositional.html |   96 +
 ...ion.Unary.Unique.DecSetoid.Properties.html |  109 +
 ....List.Relation.Unary.Unique.DecSetoid.html |  105 +
 ...Unique.Propositional.Properties.WithK.html |  125 +
 ...Unary.Unique.Propositional.Properties.html |  233 ++
 ...t.Relation.Unary.Unique.Propositional.html |   94 +
 ...lation.Unary.Unique.Setoid.Properties.html |  235 ++
 ...ata.List.Relation.Unary.Unique.Setoid.html |  105 +
 .../agda_html/Data.List.Scans.Base.html       |  126 +
 .../static/agda_html/Data.List.Sort.Base.html |  109 +
 .../agda_html/Data.List.Sort.MergeSort.html   |  194 ++
 site/static/agda_html/Data.List.Sort.html     |  118 +
 site/static/agda_html/Data.List.html          |   98 +
 site/static/agda_html/Data.Maybe.Base.html    |  222 ++
 .../agda_html/Data.Maybe.Effectful.html       |  185 ++
 .../agda_html/Data.Maybe.Properties.html      |  265 ++
 .../Data.Maybe.Relation.Binary.Connected.html |  144 +
 .../Data.Maybe.Relation.Binary.Pointwise.html |  188 ++
 ...a.Maybe.Relation.Unary.All.Properties.html |  147 +
 .../Data.Maybe.Relation.Unary.All.html        |  197 ++
 .../Data.Maybe.Relation.Unary.Any.html        |  153 +
 site/static/agda_html/Data.Maybe.html         |  118 +
 site/static/agda_html/Data.Nat.Base.html      |  521 ++++
 .../agda_html/Data.Nat.Binary.Base.html       |  271 ++
 .../agda_html/Data.Nat.Binary.Properties.html | 1617 +++++++++++
 site/static/agda_html/Data.Nat.Binary.html    |   89 +
 .../agda_html/Data.Nat.Coprimality.html       |  213 ++
 .../agda_html/Data.Nat.DivMod.Core.html       |  341 +++
 site/static/agda_html/Data.Nat.DivMod.html    |  572 ++++
 .../agda_html/Data.Nat.Divisibility.Core.html |  158 ++
 .../agda_html/Data.Nat.Divisibility.html      |  437 +++
 .../static/agda_html/Data.Nat.GCD.Lemmas.html |  342 +++
 site/static/agda_html/Data.Nat.GCD.html       |  481 ++++
 .../Data.Nat.GeneralisedArithmetic.html       |  185 ++
 site/static/agda_html/Data.Nat.Induction.html |  187 ++
 site/static/agda_html/Data.Nat.Primality.html |  502 ++++
 .../static/agda_html/Data.Nat.Properties.html | 2491 +++++++++++++++++
 site/static/agda_html/Data.Nat.Show.html      |  162 ++
 site/static/agda_html/Data.Nat.Solver.html    |   99 +
 site/static/agda_html/Data.Nat.html           |  126 +
 site/static/agda_html/Data.Parity.Base.html   |  194 ++
 .../agda_html/Data.Product.Algebra.html       |  256 ++
 site/static/agda_html/Data.Product.Base.html  |  270 ++
 ...duct.Function.Dependent.Propositional.html |  395 +++
 ...t.Function.NonDependent.Propositional.html |  164 ++
 ....Product.Function.NonDependent.Setoid.html |  213 ++
 .../Data.Product.Nary.NonDependent.html       |  324 +++
 .../Data.Product.Properties.Ext.html          |  122 +
 .../agda_html/Data.Product.Properties.html    |  190 ++
 ...elation.Binary.Pointwise.NonDependent.html |  283 ++
 .../Data.Product.Relation.Unary.All.html      |   98 +
 site/static/agda_html/Data.Product.html       |  140 +
 site/static/agda_html/Data.Rational.Base.html |  446 +++
 .../agda_html/Data.Rational.Properties.html   | 1865 ++++++++++++
 .../Data.Rational.Unnormalised.Base.html      |  464 +++
 ...Data.Rational.Unnormalised.Properties.html | 2067 ++++++++++++++
 site/static/agda_html/Data.Rational.html      |  114 +
 site/static/agda_html/Data.Record.html        |  255 ++
 site/static/agda_html/Data.Refinement.html    |  123 +
 site/static/agda_html/Data.Sign.Base.html     |  139 +
 .../agda_html/Data.Sign.Properties.html       |  286 ++
 site/static/agda_html/Data.Sign.html          |   93 +
 site/static/agda_html/Data.String.Base.html   |  273 ++
 .../agda_html/Data.String.Properties.html     |  254 ++
 site/static/agda_html/Data.String.html        |  147 +
 site/static/agda_html/Data.Sum.Algebra.html   |  197 ++
 site/static/agda_html/Data.Sum.Base.html      |  151 +
 .../agda_html/Data.Sum.Effectful.Left.html    |  171 ++
 .../Data.Sum.Function.Propositional.html      |  163 ++
 .../agda_html/Data.Sum.Function.Setoid.html   |  251 ++
 .../static/agda_html/Data.Sum.Properties.html |  229 ++
 .../Data.Sum.Relation.Binary.Pointwise.html   |  295 ++
 .../Data.Sum.Relation.Unary.All.html          |  116 +
 site/static/agda_html/Data.Sum.html           |  141 +
 site/static/agda_html/Data.These.Base.html    |  157 ++
 .../agda_html/Data.These.Properties.html      |  128 +
 site/static/agda_html/Data.These.html         |  131 +
 site/static/agda_html/Data.Unit.Base.html     |   98 +
 .../agda_html/Data.Unit.Polymorphic.Base.html |   98 +
 .../Data.Unit.Polymorphic.Properties.html     |  189 ++
 .../agda_html/Data.Unit.Polymorphic.html      |   96 +
 .../agda_html/Data.Unit.Properties.html       |  171 ++
 site/static/agda_html/Data.Unit.html          |   97 +
 site/static/agda_html/Data.Universe.html      |   96 +
 site/static/agda_html/Data.Vec.Base.html      |  450 +++
 .../agda_html/Data.Vec.Bounded.Base.html      |  240 ++
 .../static/agda_html/Data.Vec.Functional.html |  253 ++
 .../Data.Vec.Membership.Propositional.html    |  105 +
 .../agda_html/Data.Vec.Membership.Setoid.html |  135 +
 site/static/agda_html/Data.Vec.N-ary.html     |  265 ++
 .../static/agda_html/Data.Vec.Properties.html | 1530 ++++++++++
 ...ata.Vec.Relation.Binary.Equality.Cast.html |  205 ++
 ...c.Relation.Binary.Pointwise.Inductive.html |  348 +++
 ...ata.Vec.Relation.Unary.All.Properties.html |  258 ++
 .../Data.Vec.Relation.Unary.All.html          |  217 ++
 ...ata.Vec.Relation.Unary.Any.Properties.html |  500 ++++
 .../Data.Vec.Relation.Unary.Any.html          |  165 ++
 site/static/agda_html/Data.Vec.html           |  132 +
 site/static/agda_html/Data.W.html             |  143 +
 site/static/agda_html/Data.Word64.Base.html   |  118 +
 .../agda_html/Data.Word64.Properties.html     |  185 ++
 site/static/agda_html/Data.Word64.html        |   92 +
 site/static/agda_html/Effect.Applicative.html |  210 ++
 site/static/agda_html/Effect.Choice.html      |  103 +
 site/static/agda_html/Effect.Comonad.html     |  125 +
 site/static/agda_html/Effect.Empty.html       |  101 +
 site/static/agda_html/Effect.Functor.html     |  126 +
 site/static/agda_html/Effect.Monad.html       |  211 ++
 site/static/agda_html/Everything.html         |  108 +
 .../agda_html/Foreign.Haskell.Coerce.html     |  248 ++
 .../agda_html/Foreign.Haskell.Either.html     |  124 +
 .../Foreign.Haskell.List.NonEmpty.html        |  108 +
 .../agda_html/Foreign.Haskell.Pair.html       |  125 +
 site/static/agda_html/Foreign.Haskell.html    |  106 +
 site/static/agda_html/Function.Base.html      |  347 +++
 site/static/agda_html/Function.Bundles.html   |  628 +++++
 .../Function.Consequences.Propositional.html  |  130 +
 .../Function.Consequences.Setoid.html         |  169 ++
 .../agda_html/Function.Consequences.html      |  191 ++
 .../Function.Construct.Composition.html       |  364 +++
 .../Function.Construct.Identity.html          |  340 +++
 .../Function.Construct.Symmetry.html          |  322 +++
 site/static/agda_html/Function.Core.html      |  105 +
 .../agda_html/Function.Definitions.html       |  141 +
 .../agda_html/Function.Dependent.Bundles.html |  127 +
 .../Function.Identity.Effectful.html          |  124 +
 ...tion.Indexed.Relation.Binary.Equality.html |  104 +
 .../agda_html/Function.Metric.Bundles.html    |  226 ++
 .../agda_html/Function.Metric.Core.html       |   97 +
 .../Function.Metric.Definitions.html          |  140 +
 .../Function.Metric.Nat.Bundles.html          |  217 ++
 .../agda_html/Function.Metric.Nat.Core.html   |   95 +
 .../Function.Metric.Nat.Definitions.html      |  145 +
 .../Function.Metric.Nat.Structures.html       |  153 +
 .../static/agda_html/Function.Metric.Nat.html |   91 +
 .../agda_html/Function.Metric.Structures.html |  174 ++
 .../Function.Nary.NonDependent.Base.html      |  220 ++
 .../agda_html/Function.Nary.NonDependent.html |  177 ++
 .../Function.Properties.Bijection.html        |  155 +
 ...erties.Inverse.HalfAdjointEquivalence.html |  200 ++
 .../Function.Properties.Inverse.html          |  236 ++
 .../Function.Properties.RightInverse.html     |  158 ++
 .../Function.Properties.Surjection.html       |  158 ++
 .../Function.Related.Propositional.html       |  468 ++++
 .../Function.Related.TypeIsomorphisms.html    |  411 +++
 site/static/agda_html/Function.Strict.html    |  152 +
 .../agda_html/Function.Structures.Biased.html |  204 ++
 .../static/agda_html/Function.Structures.html |  266 ++
 site/static/agda_html/Function.html           |   94 +
 site/static/agda_html/IO.Primitive.Core.html  |  118 +
 .../agda_html/Induction.Lexicographic.html    |  160 ++
 .../agda_html/Induction.WellFounded.html      |  342 +++
 site/static/agda_html/Induction.html          |  141 +
 site/static/agda_html/Leios.Abstract.html     |   99 +
 site/static/agda_html/Leios.Base.html         |  118 +
 site/static/agda_html/Leios.Blocks.html       |  247 ++
 site/static/agda_html/Leios.Config.html       |   95 +
 site/static/agda_html/Leios.Defaults.html     |  351 +++
 site/static/agda_html/Leios.FFD.html          |  108 +
 .../agda_html/Leios.Foreign.BaseTypes.html    |  196 ++
 .../agda_html/Leios.Foreign.HsTypes.html      |  153 +
 .../static/agda_html/Leios.Foreign.Types.html |  241 ++
 site/static/agda_html/Leios.Foreign.Util.html |   85 +
 .../agda_html/Leios.KeyRegistration.html      |  100 +
 site/static/agda_html/Leios.Network.html      |  149 +
 site/static/agda_html/Leios.Prelude.html      |  202 ++
 site/static/agda_html/Leios.Protocol.html     |  336 +++
 .../agda_html/Leios.Short.Decidable.html      |  184 ++
 .../Leios.Short.Trace.Verifier.Test.html      |  210 ++
 .../agda_html/Leios.Short.Trace.Verifier.html |  518 ++++
 site/static/agda_html/Leios.Short.html        |  291 ++
 site/static/agda_html/Leios.Short.md          |  182 ++
 .../Leios.Simplified.Deterministic.html       |  434 +++
 site/static/agda_html/Leios.Simplified.html   |  261 ++
 .../static/agda_html/Leios.SpecStructure.html |  130 +
 site/static/agda_html/Leios.Traces.html       |   97 +
 site/static/agda_html/Leios.VRF.html          |  147 +
 site/static/agda_html/Leios.Voting.html       |   89 +
 site/static/agda_html/Level.html              |  111 +
 site/static/agda_html/Meta.Init.html          |   92 +
 site/static/agda_html/Meta.Prelude.html       |  108 +
 site/static/agda_html/Prelude.Closures.html   |   99 +
 .../agda_html/Prelude.InferenceRules.html     | 1161 ++++++++
 site/static/agda_html/Prelude.Init.html       |  253 ++
 .../agda_html/Reflection.AST.Abstraction.html |  142 +
 .../Reflection.AST.Argument.Information.html  |  135 +
 .../Reflection.AST.Argument.Modality.html     |  135 +
 .../Reflection.AST.Argument.Quantity.html     |  107 +
 .../Reflection.AST.Argument.Relevance.html    |  107 +
 .../Reflection.AST.Argument.Visibility.html   |  112 +
 .../agda_html/Reflection.AST.Argument.html    |  170 ++
 .../agda_html/Reflection.AST.DeBruijn.html    |  211 ++
 .../agda_html/Reflection.AST.Definition.html  |  193 ++
 .../agda_html/Reflection.AST.Literal.html     |  186 ++
 .../static/agda_html/Reflection.AST.Meta.html |  113 +
 .../static/agda_html/Reflection.AST.Name.html |  127 +
 .../agda_html/Reflection.AST.Pattern.html     |  106 +
 .../static/agda_html/Reflection.AST.Show.html |  231 ++
 .../static/agda_html/Reflection.AST.Term.html |  560 ++++
 .../agda_html/Reflection.AST.Traversal.html   |  210 ++
 site/static/agda_html/Reflection.AST.html     |  136 +
 site/static/agda_html/Reflection.Debug.html   |  197 ++
 site/static/agda_html/Reflection.Syntax.html  |  188 ++
 site/static/agda_html/Reflection.TCI.html     |  212 ++
 .../agda_html/Reflection.TCM.Format.html      |  162 ++
 .../agda_html/Reflection.TCM.Syntax.html      |  129 +
 site/static/agda_html/Reflection.TCM.html     |  122 +
 site/static/agda_html/Reflection.Tactic.html  |  126 +
 .../Reflection.Utils.Substitute.html          |  123 +
 .../agda_html/Reflection.Utils.TCI.html       |  377 +++
 site/static/agda_html/Reflection.Utils.html   |  330 +++
 site/static/agda_html/Reflection.html         |  262 ++
 .../agda_html/Relation.Binary.Bundles.html    |  469 ++++
 .../Relation.Binary.Consequences.html         |  392 +++
 ...onstruct.Closure.Reflexive.Properties.html |  224 ++
 ...on.Binary.Construct.Closure.Reflexive.html |  159 ++
 ...Relation.Binary.Construct.Composition.html |  164 ++
 ...lation.Binary.Construct.Constant.Core.html |  101 +
 ...lation.Binary.Construct.Flip.EqAndOrd.html |  277 ++
 ...elation.Binary.Construct.Intersection.html |  221 ++
 ...on.Binary.Construct.NaturalOrder.Left.html |  266 ++
 ...on.Binary.Construct.NonStrictToStrict.html |  225 ++
 .../Relation.Binary.Construct.On.html         |  301 ++
 ...on.Binary.Construct.StrictToNonStrict.html |  241 ++
 ...ation.Binary.Construct.Subst.Equality.html |  121 +
 .../agda_html/Relation.Binary.Core.html       |  143 +
 .../Relation.Binary.Definitions.html          |  336 +++
 ....Binary.Indexed.Heterogeneous.Bundles.html |  134 +
 ...dexed.Heterogeneous.Construct.Trivial.html |  136 +
 ...ion.Binary.Indexed.Heterogeneous.Core.html |  115 +
 ...ary.Indexed.Heterogeneous.Definitions.html |  110 +
 ...nary.Indexed.Heterogeneous.Structures.html |  123 +
 ...Relation.Binary.Indexed.Heterogeneous.html |   94 +
 .../Relation.Binary.Lattice.Bundles.html      |  305 ++
 .../Relation.Binary.Lattice.Definitions.html  |  114 +
 ...ice.Properties.BoundedJoinSemilattice.html |  131 +
 ...ry.Lattice.Properties.JoinSemilattice.html |  203 ++
 .../Relation.Binary.Lattice.Structures.html   |  263 ++
 .../agda_html/Relation.Binary.Lattice.html    |   93 +
 .../Relation.Binary.Morphism.Bundles.html     |  181 ++
 .../Relation.Binary.Morphism.Definitions.html |  109 +
 ...ion.Binary.Morphism.OrderMonomorphism.html |  187 ++
 ...ation.Binary.Morphism.RelMonomorphism.html |  143 +
 .../Relation.Binary.Morphism.Structures.html  |  194 ++
 .../agda_html/Relation.Binary.Morphism.html   |   95 +
 ...n.Binary.Properties.ApartnessRelation.html |  106 +
 .../Relation.Binary.Properties.DecSetoid.html |  120 +
 ...ation.Binary.Properties.DecTotalOrder.html |  173 ++
 .../Relation.Binary.Properties.Poset.html     |  203 ++
 .../Relation.Binary.Properties.Preorder.html  |  140 +
 .../Relation.Binary.Properties.Setoid.html    |  179 ++
 ...Relation.Binary.Properties.TotalOrder.html |  177 ++
 ....Binary.PropositionalEquality.Algebra.html |  112 +
 ...ion.Binary.PropositionalEquality.Core.html |  179 ++
 ...nary.PropositionalEquality.Properties.html |  282 ++
 ...Relation.Binary.PropositionalEquality.html |  209 ++
 ...Relation.Binary.Reasoning.Base.Double.html |  178 ++
 ...Relation.Binary.Reasoning.Base.Single.html |  127 +
 ...Relation.Binary.Reasoning.Base.Triple.html |  205 ++
 ...elation.Binary.Reasoning.PartialOrder.html |  140 +
 .../Relation.Binary.Reasoning.Preorder.html   |  113 +
 .../Relation.Binary.Reasoning.Setoid.html     |  120 +
 .../Relation.Binary.Reasoning.Syntax.html     |  522 ++++
 .../agda_html/Relation.Binary.Reflection.html |  180 ++
 .../Relation.Binary.Structures.Biased.html    |  126 +
 .../agda_html/Relation.Binary.Structures.html |  395 +++
 site/static/agda_html/Relation.Binary.html    |   95 +
 .../Relation.Nullary.Decidable.Core.html      |  322 +++
 .../agda_html/Relation.Nullary.Decidable.html |  159 ++
 .../agda_html/Relation.Nullary.Indexed.html   |   96 +
 .../Relation.Nullary.Negation.Core.html       |  155 +
 .../agda_html/Relation.Nullary.Negation.html  |  185 ++
 .../Relation.Nullary.Recomputable.html        |  140 +
 .../agda_html/Relation.Nullary.Reflects.html  |  194 ++
 site/static/agda_html/Relation.Nullary.html   |  119 +
 .../Relation.Unary.PredicateTransformer.html  |  204 ++
 .../agda_html/Relation.Unary.Properties.html  |  320 +++
 site/static/agda_html/Relation.Unary.html     |  398 +++
 site/static/agda_html/StateMachine.html       |  133 +
 site/static/agda_html/Tactic.AnyOf.html       |  113 +
 site/static/agda_html/Tactic.Assumption.html  |  129 +
 site/static/agda_html/Tactic.ByEq.html        |  135 +
 .../agda_html/Tactic.ClauseBuilder.html       |  419 +++
 site/static/agda_html/Tactic.Defaults.html    |   95 +
 .../agda_html/Tactic.Derive.Convertible.html  |  262 ++
 .../static/agda_html/Tactic.Derive.DecEq.html |  203 ++
 .../agda_html/Tactic.Derive.HsType.html       |  438 +++
 site/static/agda_html/Tactic.Derive.Show.html |  169 ++
 .../agda_html/Tactic.Derive.TestTypes.html    |  161 ++
 site/static/agda_html/Tactic.Derive.html      |  198 ++
 .../static/agda_html/Text.Format.Generic.html |  204 ++
 site/static/agda_html/Text.Format.html        |  134 +
 .../static/agda_html/Text.Printf.Generic.html |  159 ++
 site/static/agda_html/Text.Printf.html        |  111 +
 .../abstract-set-theory.FiniteSetTheory.html  |  205 ++
 .../abstract-set-theory.Prelude.html          |  139 +
 site/static/agda_html/agda.js                 |  101 +-
 545 files changed, 122965 insertions(+), 57 deletions(-)
 create mode 100644 site/static/agda_html/Agda.Builtin.Bool.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Char.Properties.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Char.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Equality.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Float.Properties.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Float.html
 create mode 100644 site/static/agda_html/Agda.Builtin.IO.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Int.html
 create mode 100644 site/static/agda_html/Agda.Builtin.List.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Maybe.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Nat.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Reflection.Properties.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Reflection.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Sigma.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Strict.html
 create mode 100644 site/static/agda_html/Agda.Builtin.String.Properties.html
 create mode 100644 site/static/agda_html/Agda.Builtin.String.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Unit.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Word.Properties.html
 create mode 100644 site/static/agda_html/Agda.Builtin.Word.html
 create mode 100644 site/static/agda_html/Agda.Primitive.html
 create mode 100644 site/static/agda_html/Algebra.Apartness.Bundles.html
 create mode 100644 site/static/agda_html/Algebra.Apartness.Structures.html
 create mode 100644 site/static/agda_html/Algebra.Apartness.html
 create mode 100644 site/static/agda_html/Algebra.Bundles.Raw.html
 create mode 100644 site/static/agda_html/Algebra.Bundles.html
 create mode 100644 site/static/agda_html/Algebra.Consequences.Base.html
 create mode 100644 site/static/agda_html/Algebra.Consequences.Propositional.html
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diff --git a/site/static/agda_html/Agda.Builtin.Bool.html b/site/static/agda_html/Agda.Builtin.Bool.html
new file mode 100644
index 000000000..79db56a94
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Bool.html
@@ -0,0 +1,92 @@
+
+Agda.Builtin.Bool
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-universe-polymorphism

+
            --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Bool where

+

+
data Bool : Set where

+
  false true : Bool

+

+
{-# BUILTIN BOOL  Bool  #-}

+
{-# BUILTIN FALSE false #-}

+
{-# BUILTIN TRUE  true  #-}

+

+
{-# COMPILE JS Bool  = function (x,v) { return ((x)? v["true"]() : v["false"]()); } #-}

+
{-# COMPILE JS false = false #-}

+
{-# COMPILE JS true  = true  #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Char.Properties.html b/site/static/agda_html/Agda.Builtin.Char.Properties.html
new file mode 100644
index 000000000..4bdb529af
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Char.Properties.html
@@ -0,0 +1,87 @@
+
+Agda.Builtin.Char.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Char.Properties where

+

+
open import Agda.Builtin.Char

+
open import Agda.Builtin.Equality

+

+
primitive

+

+
  primCharToNatInjective : ∀ a b → primCharToNat a ≡ primCharToNat b → a ≡ b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Char.html b/site/static/agda_html/Agda.Builtin.Char.html
new file mode 100644
index 000000000..08e29618a
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Char.html
@@ -0,0 +1,95 @@
+
+Agda.Builtin.Char
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-universe-polymorphism

+
            --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Char where

+

+
open import Agda.Builtin.Nat

+
open import Agda.Builtin.Bool

+

+
postulate Char : Set

+
{-# BUILTIN CHAR Char #-}

+

+
primitive

+
  primIsLower primIsDigit primIsAlpha primIsSpace primIsAscii

+
    primIsLatin1 primIsPrint primIsHexDigit : Char → Bool

+
  primToUpper primToLower : Char → Char

+
  primCharToNat : Char → Nat

+
  primNatToChar : Nat → Char

+
  primCharEquality : Char → Char → Bool

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Equality.html b/site/static/agda_html/Agda.Builtin.Equality.html
new file mode 100644
index 000000000..695266334
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Equality.html
@@ -0,0 +1,86 @@
+
+Agda.Builtin.Equality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

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+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Equality where

+

+
infix 4 _≡_

+
data _≡_ {a} {A : Set a} (x : A) : A → Set a where

+
  instance refl : x ≡ x

+

+
{-# BUILTIN EQUALITY _≡_ #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Float.Properties.html b/site/static/agda_html/Agda.Builtin.Float.Properties.html
new file mode 100644
index 000000000..9cd3da96e
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Float.Properties.html
@@ -0,0 +1,87 @@
+
+Agda.Builtin.Float.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

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+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Float.Properties where

+

+
open import Agda.Builtin.Float

+
open import Agda.Builtin.Equality

+

+
primitive

+

+
  primFloatToWord64Injective : ∀ a b → primFloatToWord64 a ≡ primFloatToWord64 b → a ≡ b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Float.html b/site/static/agda_html/Agda.Builtin.Float.html
new file mode 100644
index 000000000..9b96e641f
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Float.html
@@ -0,0 +1,286 @@
+
+Agda.Builtin.Float
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

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+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Float where

+

+
open import Agda.Builtin.Bool

+
open import Agda.Builtin.Int

+
open import Agda.Builtin.Maybe

+
open import Agda.Builtin.Nat

+
open import Agda.Builtin.Sigma

+
open import Agda.Builtin.String

+
open import Agda.Builtin.Word

+

+
postulate Float : Set

+
{-# BUILTIN FLOAT Float #-}

+

+
primitive

+
  -- Relations

+
  primFloatInequality        : Float → Float → Bool

+
  primFloatEquality          : Float → Float → Bool

+
  primFloatLess              : Float → Float → Bool

+
  primFloatIsInfinite        : Float → Bool

+
  primFloatIsNaN             : Float → Bool

+
  primFloatIsNegativeZero    : Float → Bool

+
  primFloatIsSafeInteger     : Float → Bool

+
  -- Conversions

+
  primFloatToWord64          : Float → Maybe Word64

+
  primNatToFloat             : Nat → Float

+
  primIntToFloat             : Int → Float

+
  primFloatRound             : Float → Maybe Int

+
  primFloatFloor             : Float → Maybe Int

+
  primFloatCeiling           : Float → Maybe Int

+
  primFloatToRatio           : Float → (Σ Int λ _ → Int)

+
  primRatioToFloat           : Int → Int → Float

+
  primFloatDecode            : Float → Maybe (Σ Int λ _ → Int)

+
  primFloatEncode            : Int → Int → Maybe Float

+
  primShowFloat              : Float → String

+
  -- Operations

+
  primFloatPlus              : Float → Float → Float

+
  primFloatMinus             : Float → Float → Float

+
  primFloatTimes             : Float → Float → Float

+
  primFloatDiv               : Float → Float → Float

+
  primFloatPow               : Float → Float → Float

+
  primFloatNegate            : Float → Float

+
  primFloatSqrt              : Float → Float

+
  primFloatExp               : Float → Float

+
  primFloatLog               : Float → Float

+
  primFloatSin               : Float → Float

+
  primFloatCos               : Float → Float

+
  primFloatTan               : Float → Float

+
  primFloatASin              : Float → Float

+
  primFloatACos              : Float → Float

+
  primFloatATan              : Float → Float

+
  primFloatATan2             : Float → Float → Float

+
  primFloatSinh              : Float → Float

+
  primFloatCosh              : Float → Float

+
  primFloatTanh              : Float → Float

+
  primFloatASinh             : Float → Float

+
  primFloatACosh             : Float → Float

+
  primFloatATanh             : Float → Float

+

+
{-# COMPILE JS

+
    primFloatRound = function(x) {

+
        x = agdaRTS._primFloatRound(x);

+
        if (x === null) {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["nothing"];

+
        }

+
        else {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["just"](x);

+
        }

+
    };

+
#-}

+
{-# COMPILE JS

+
    primFloatFloor = function(x) {

+
        x = agdaRTS._primFloatFloor(x);

+
        if (x === null) {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["nothing"];

+
        }

+
        else {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["just"](x);

+
        }

+
    };

+
#-}

+
{-# COMPILE JS

+
    primFloatCeiling = function(x) {

+
        x = agdaRTS._primFloatCeiling(x);

+
        if (x === null) {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["nothing"];

+
        }

+
        else {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["just"](x);

+
        }

+
    };

+
#-}

+
{-# COMPILE JS

+
    primFloatToRatio = function(x) {

+
        x = agdaRTS._primFloatToRatio(x);

+
        return z_jAgda_Agda_Builtin_Sigma["_,_"](x.numerator)(x.denominator);

+
    };

+
#-}

+
{-# COMPILE JS

+
    primFloatDecode = function(x) {

+
        x = agdaRTS._primFloatDecode(x);

+
        if (x === null) {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["nothing"];

+
        }

+
        else {

+
            return z_jAgda_Agda_Builtin_Maybe["Maybe"]["just"](

+
                z_jAgda_Agda_Builtin_Sigma["_,_"](x.mantissa)(x.exponent));

+
        }

+
    };

+
#-}

+
{-# COMPILE JS

+
    primFloatEncode = function(x) {

+
        return function (y) {

+
            x = agdaRTS.uprimFloatEncode(x, y);

+
            if (x === null) {

+
                return z_jAgda_Agda_Builtin_Maybe["Maybe"]["nothing"];

+
            }

+
            else {

+
                return z_jAgda_Agda_Builtin_Maybe["Maybe"]["just"](x);

+
            }

+
        }

+
    };

+
#-}

+

+
primFloatNumericalEquality = primFloatEquality

+
{-# WARNING_ON_USAGE primFloatNumericalEquality

+
"Warning: primFloatNumericalEquality was deprecated in Agda v2.6.2.

+
Please use primFloatEquality instead."

+
#-}

+

+
primFloatNumericalLess = primFloatLess

+
{-# WARNING_ON_USAGE primFloatNumericalLess

+
"Warning: primFloatNumericalLess was deprecated in Agda v2.6.2.

+
Please use primFloatLess instead."

+
#-}

+

+
primRound = primFloatRound

+
{-# WARNING_ON_USAGE primRound

+
"Warning: primRound was deprecated in Agda v2.6.2.

+
Please use primFloatRound instead."

+
#-}

+

+
primFloor = primFloatFloor

+
{-# WARNING_ON_USAGE primFloor

+
"Warning: primFloor was deprecated in Agda v2.6.2.

+
Please use primFloatFloor instead."

+
#-}

+

+
primCeiling = primFloatCeiling

+
{-# WARNING_ON_USAGE primCeiling

+
"Warning: primCeiling was deprecated in Agda v2.6.2.

+
Please use primFloatCeiling instead."

+
#-}

+

+
primExp = primFloatExp

+
{-# WARNING_ON_USAGE primExp

+
"Warning: primExp was deprecated in Agda v2.6.2.

+
Please use primFloatExp instead."

+
#-}

+

+
primLog = primFloatLog

+
{-# WARNING_ON_USAGE primLog

+
"Warning: primLog was deprecated in Agda v2.6.2.

+
Please use primFloatLog instead."

+
#-}

+

+
primSin = primFloatSin

+
{-# WARNING_ON_USAGE primSin

+
"Warning: primSin was deprecated in Agda v2.6.2.

+
Please use primFloatSin instead."

+
#-}

+

+
primCos = primFloatCos

+
{-# WARNING_ON_USAGE primCos

+
"Warning: primCos was deprecated in Agda v2.6.2.

+
Please use primFloatCos instead."

+
#-}

+

+
primTan = primFloatTan

+
{-# WARNING_ON_USAGE primTan

+
"Warning: primTan was deprecated in Agda v2.6.2.

+
Please use primFloatTan instead."

+
#-}

+

+
primASin = primFloatASin

+
{-# WARNING_ON_USAGE primASin

+
"Warning: primASin was deprecated in Agda v2.6.2.

+
Please use primFloatASin instead."

+
#-}

+

+

+
primACos = primFloatACos

+
{-# WARNING_ON_USAGE primACos

+
"Warning: primACos was deprecated in Agda v2.6.2.

+
Please use primFloatACos instead."

+
#-}

+

+
primATan = primFloatATan

+
{-# WARNING_ON_USAGE primATan

+
"Warning: primATan was deprecated in Agda v2.6.2.

+
Please use primFloatATan instead."

+
#-}

+

+
primATan2 = primFloatATan2

+
{-# WARNING_ON_USAGE primATan2

+
"Warning: primATan2 was deprecated in Agda v2.6.2.

+
Please use primFloatATan2 instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.IO.html b/site/static/agda_html/Agda.Builtin.IO.html
new file mode 100644
index 000000000..9455d8463
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.IO.html
@@ -0,0 +1,87 @@
+
+Agda.Builtin.IO
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.IO where

+

+
postulate IO : ∀ {a} → Set a → Set a

+
{-# POLARITY IO ++ ++ #-}

+
{-# BUILTIN IO IO #-}

+

+
{-# FOREIGN GHC type AgdaIO a b = IO b #-}

+
{-# COMPILE GHC IO = type AgdaIO #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Int.html b/site/static/agda_html/Agda.Builtin.Int.html
new file mode 100644
index 000000000..362f35515
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Int.html
@@ -0,0 +1,95 @@
+
+Agda.Builtin.Int
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Int where

+

+
open import Agda.Builtin.Nat

+
open import Agda.Builtin.String

+

+
infix 8 pos  -- Standard library uses this as +_

+

+
data Int : Set where

+
  pos    : (n : Nat) → Int

+
  negsuc : (n : Nat) → Int

+

+
{-# BUILTIN INTEGER       Int    #-}

+
{-# BUILTIN INTEGERPOS    pos    #-}

+
{-# BUILTIN INTEGERNEGSUC negsuc #-}

+

+
primitive primShowInteger : Int → String

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.List.html b/site/static/agda_html/Agda.Builtin.List.html
new file mode 100644
index 000000000..cacbfb622
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.List.html
@@ -0,0 +1,93 @@
+
+Agda.Builtin.List
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.List where

+

+
infixr 5 _∷_

+
data List {a} (A : Set a) : Set a where

+
  []  : List A

+
  _∷_ : (x : A) (xs : List A) → List A

+

+
{-# BUILTIN LIST List #-}

+

+
{-# COMPILE JS  List = function(x,v) {

+
  if (x.length < 1) { return v["[]"](); } else { return v["_∷_"](x[0], x.slice(1)); }

+
} #-}

+
{-# COMPILE JS [] = Array() #-}

+
{-# COMPILE JS _∷_ = function (x) { return function(y) { return Array(x).concat(y); }; } #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Maybe.html b/site/static/agda_html/Agda.Builtin.Maybe.html
new file mode 100644
index 000000000..87afc420b
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Maybe.html
@@ -0,0 +1,86 @@
+
+Agda.Builtin.Maybe
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Maybe where

+

+
data Maybe {a} (A : Set a) : Set a where

+
  just : A → Maybe A

+
  nothing : Maybe A

+

+
{-# BUILTIN MAYBE Maybe #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Nat.html b/site/static/agda_html/Agda.Builtin.Nat.html
new file mode 100644
index 000000000..09585917b
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Nat.html
@@ -0,0 +1,211 @@
+
+Agda.Builtin.Nat
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-universe-polymorphism

+
            --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Nat where

+

+
open import Agda.Builtin.Bool

+

+
data Nat : Set where

+
  zero : Nat

+
  suc  : (n : Nat) → Nat

+

+
{-# BUILTIN NATURAL Nat #-}

+

+
infix  4 _==_ _<_

+
infixl 6 _+_ _-_

+
infixl 7 _*_

+

+
_+_ : Nat → Nat → Nat

+
zero  + m = m

+
suc n + m = suc (n + m)

+

+
{-# BUILTIN NATPLUS _+_ #-}

+

+
_-_ : Nat → Nat → Nat

+
n     - zero = n

+
zero  - suc m = zero

+
suc n - suc m = n - m

+

+
{-# BUILTIN NATMINUS _-_ #-}

+

+
_*_ : Nat → Nat → Nat

+
zero  * m = zero

+
suc n * m = m + n * m

+

+
{-# BUILTIN NATTIMES _*_ #-}

+

+
_==_ : Nat → Nat → Bool

+
zero  == zero  = true

+
suc n == suc m = n == m

+
_     == _     = false

+

+
{-# BUILTIN NATEQUALS _==_ #-}

+

+
_<_ : Nat → Nat → Bool

+
_     < zero  = false

+
zero  < suc _ = true

+
suc n < suc m = n < m

+

+
{-# BUILTIN NATLESS _<_ #-}

+

+
-- Helper function  div-helper  for Euclidean division.

+
---------------------------------------------------------------------------

+
--

+
-- div-helper computes n / 1+m via iteration on n.

+
--

+
--   n div (suc m) = div-helper 0 m n m

+
--

+
-- The state of the iterator has two accumulator variables:

+
--

+
--   k: The quotient, returned once n=0.  Initialized to 0.

+
--

+
--   j: A counter, initialized to the divisor m, decreased on each iteration step.

+
--      Once it reaches 0, the quotient k is increased and j reset to m,

+
--      starting the next countdown.

+
--

+
-- Under the precondition j ≤ m, the invariant is

+
--

+
--   div-helper k m n j = k + (n + m - j) div (1 + m)

+

+
div-helper : (k m n j : Nat) → Nat

+
div-helper k m  zero    j      = k

+
div-helper k m (suc n)  zero   = div-helper (suc k) m n m

+
div-helper k m (suc n) (suc j) = div-helper k       m n j

+

+
{-# BUILTIN NATDIVSUCAUX div-helper #-}

+

+
-- Proof of the invariant by induction on n.

+
--

+
--   clause 1: div-helper k m 0 j

+
--           = k                                        by definition

+
--           = k + (0 + m - j) div (1 + m)              since m - j < 1 + m

+
--

+
--   clause 2: div-helper k m (1 + n) 0

+
--           = div-helper (1 + k) m n m                 by definition

+
--           = 1 + k + (n + m - m) div (1 + m)          by induction hypothesis

+
--           = 1 + k +          n  div (1 + m)          by simplification

+
--           = k +   (n + (1 + m)) div (1 + m)          by expansion

+
--           = k + (1 + n + m - 0) div (1 + m)          by expansion

+
--

+
--   clause 3: div-helper k m (1 + n) (1 + j)

+
--           = div-helper k m n j                       by definition

+
--           = k + (n + m - j) div (1 + m)              by induction hypothesis

+
--           = k + ((1 + n) + m - (1 + j)) div (1 + m)  by expansion

+
--

+
-- Q.e.d.

+

+
-- Helper function  mod-helper  for the remainder computation.

+
---------------------------------------------------------------------------

+
--

+
-- (Analogous to div-helper.)

+
--

+
-- mod-helper computes n % 1+m via iteration on n.

+
--

+
--   n mod (suc m) = mod-helper 0 m n m

+
--

+
-- The invariant is:

+
--

+
--   m = k + j  ==>  mod-helper k m n j = (n + k) mod (1 + m).

+

+
mod-helper : (k m n j : Nat) → Nat

+
mod-helper k m  zero    j      = k

+
mod-helper k m (suc n)  zero   = mod-helper 0       m n m

+
mod-helper k m (suc n) (suc j) = mod-helper (suc k) m n j

+

+
{-# BUILTIN NATMODSUCAUX mod-helper #-}

+

+
-- Proof of the invariant by induction on n.

+
--

+
--   clause 1: mod-helper k m 0 j

+
--           = k                               by definition

+
--           = (0 + k) mod (1 + m)             since m = k + j, thus k < m

+
--

+
--   clause 2: mod-helper k m (1 + n) 0

+
--           = mod-helper 0 m n m              by definition

+
--           = (n + 0)       mod (1 + m)       by induction hypothesis

+
--           = (n + (1 + m)) mod (1 + m)       by expansion

+
--           = (1 + n) + k)  mod (1 + m)       since k = m (as l = 0)

+
--

+
--   clause 3: mod-helper k m (1 + n) (1 + j)

+
--           = mod-helper (1 + k) m n j        by definition

+
--           = (n + (1 + k)) mod (1 + m)       by induction hypothesis

+
--           = ((1 + n) + k) mod (1 + m)       by commutativity

+
--

+
-- Q.e.d.

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Reflection.Properties.html b/site/static/agda_html/Agda.Builtin.Reflection.Properties.html
new file mode 100644
index 000000000..fe41fd6cf
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Reflection.Properties.html
@@ -0,0 +1,88 @@
+
+Agda.Builtin.Reflection.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Reflection.Properties where

+

+
open import Agda.Builtin.Reflection

+
open import Agda.Builtin.Equality

+

+
primitive

+

+
  primMetaToNatInjective : ∀ a b → primMetaToNat a ≡ primMetaToNat b → a ≡ b

+
  primQNameToWord64sInjective : ∀ a b → primQNameToWord64s a ≡ primQNameToWord64s b → a ≡ b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Reflection.html b/site/static/agda_html/Agda.Builtin.Reflection.html
new file mode 100644
index 000000000..58336da99
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Reflection.html
@@ -0,0 +1,558 @@
+
+Agda.Builtin.Reflection
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Reflection where

+

+
open import Agda.Builtin.Unit

+
open import Agda.Builtin.Bool

+
open import Agda.Builtin.Nat

+
open import Agda.Builtin.Word

+
open import Agda.Builtin.List

+
open import Agda.Builtin.String

+
open import Agda.Builtin.Char

+
open import Agda.Builtin.Float

+
open import Agda.Builtin.Int

+
open import Agda.Builtin.Sigma

+
open import Agda.Primitive

+

+
-- Names --

+

+
postulate Name : Set

+
{-# BUILTIN QNAME Name #-}

+

+
primitive

+
  primQNameEquality : Name → Name → Bool

+
  primQNameLess     : Name → Name → Bool

+
  primShowQName     : Name → String

+

+
-- Fixity --

+

+
data Associativity : Set where

+
  left-assoc  : Associativity

+
  right-assoc : Associativity

+
  non-assoc   : Associativity

+

+
data Precedence : Set where

+
  related   : Float → Precedence

+
  unrelated : Precedence

+

+
data Fixity : Set where

+
  fixity : Associativity → Precedence → Fixity

+

+
{-# BUILTIN ASSOC      Associativity #-}

+
{-# BUILTIN ASSOCLEFT  left-assoc    #-}

+
{-# BUILTIN ASSOCRIGHT right-assoc   #-}

+
{-# BUILTIN ASSOCNON   non-assoc     #-}

+

+
{-# BUILTIN PRECEDENCE    Precedence #-}

+
{-# BUILTIN PRECRELATED   related    #-}

+
{-# BUILTIN PRECUNRELATED unrelated  #-}

+

+
{-# BUILTIN FIXITY       Fixity #-}

+
{-# BUILTIN FIXITYFIXITY fixity #-}

+

+
{-# COMPILE GHC Associativity = data MAlonzo.RTE.Assoc (MAlonzo.RTE.LeftAssoc | MAlonzo.RTE.RightAssoc | MAlonzo.RTE.NonAssoc) #-}

+
{-# COMPILE GHC Precedence    = data MAlonzo.RTE.Precedence (MAlonzo.RTE.Related | MAlonzo.RTE.Unrelated) #-}

+
{-# COMPILE GHC Fixity        = data MAlonzo.RTE.Fixity (MAlonzo.RTE.Fixity) #-}

+

+
{-# COMPILE JS Associativity  = function (x,v) { return v[x](); } #-}

+
{-# COMPILE JS left-assoc     = "left-assoc"  #-}

+
{-# COMPILE JS right-assoc    = "right-assoc" #-}

+
{-# COMPILE JS non-assoc      = "non-assoc"   #-}

+

+
{-# COMPILE JS Precedence     =

+
  function (x,v) {

+
    if (x === "unrelated") { return v[x](); } else { return v["related"](x); }} #-}

+
{-# COMPILE JS related        = function(x) { return x; } #-}

+
{-# COMPILE JS unrelated      = "unrelated"               #-}

+

+
{-# COMPILE JS Fixity         = function (x,v) { return v["fixity"](x["assoc"], x["prec"]); } #-}

+
{-# COMPILE JS fixity         = function (x) { return function (y) { return { "assoc": x, "prec": y}; }; } #-}

+

+
primitive

+
  primQNameFixity : Name → Fixity

+
  primQNameToWord64s : Name → Σ Word64 (λ _ → Word64)

+

+
-- Metavariables --

+

+
postulate Meta : Set

+
{-# BUILTIN AGDAMETA Meta #-}

+

+
primitive

+
  primMetaEquality : Meta → Meta → Bool

+
  primMetaLess     : Meta → Meta → Bool

+
  primShowMeta     : Meta → String

+
  primMetaToNat    : Meta → Nat

+

+
-- Arguments --

+

+
-- Arguments can be (visible), {hidden}, or {{instance}}.

+
data Visibility : Set where

+
  visible hidden instance′ : Visibility

+

+
{-# BUILTIN HIDING   Visibility #-}

+
{-# BUILTIN VISIBLE  visible    #-}

+
{-# BUILTIN HIDDEN   hidden     #-}

+
{-# BUILTIN INSTANCE instance′  #-}

+

+
-- Arguments can be relevant or irrelevant.

+
data Relevance : Set where

+
  relevant irrelevant : Relevance

+

+
{-# BUILTIN RELEVANCE  Relevance  #-}

+
{-# BUILTIN RELEVANT   relevant   #-}

+
{-# BUILTIN IRRELEVANT irrelevant #-}

+

+
-- Arguments also have a quantity.

+
data Quantity : Set where

+
  quantity-0 quantity-ω : Quantity

+

+
{-# BUILTIN QUANTITY   Quantity   #-}

+
{-# BUILTIN QUANTITY-0 quantity-0 #-}

+
{-# BUILTIN QUANTITY-ω quantity-ω #-}

+

+
-- Relevance and quantity are combined into a modality.

+
data Modality : Set where

+
  modality : (r : Relevance) (q : Quantity) → Modality

+

+
{-# BUILTIN MODALITY             Modality #-}

+
{-# BUILTIN MODALITY-CONSTRUCTOR modality #-}

+

+
data ArgInfo : Set where

+
  arg-info : (v : Visibility) (m : Modality) → ArgInfo

+

+
data Arg {a} (A : Set a) : Set a where

+
  arg : (i : ArgInfo) (x : A) → Arg A

+

+
{-# BUILTIN ARGINFO    ArgInfo  #-}

+
{-# BUILTIN ARGARGINFO arg-info #-}

+
{-# BUILTIN ARG        Arg      #-}

+
{-# BUILTIN ARGARG     arg      #-}

+

+
data Blocker : Set where

+
  blockerAny  : List Blocker → Blocker

+
  blockerAll  : List Blocker → Blocker

+
  blockerMeta : Meta → Blocker

+

+
{-# BUILTIN AGDABLOCKER     Blocker #-}

+
{-# BUILTIN AGDABLOCKERANY  blockerAny #-}

+
{-# BUILTIN AGDABLOCKERALL  blockerAll #-}

+
{-# BUILTIN AGDABLOCKERMETA blockerMeta #-}

+

+
-- Name abstraction --

+

+
data Abs {a} (A : Set a) : Set a where

+
  abs : (s : String) (x : A) → Abs A

+

+
{-# BUILTIN ABS    Abs #-}

+
{-# BUILTIN ABSABS abs #-}

+

+
-- Literals --

+

+
data Literal : Set where

+
  nat    : (n : Nat)    → Literal

+
  word64 : (n : Word64) → Literal

+
  float  : (x : Float)  → Literal

+
  char   : (c : Char)   → Literal

+
  string : (s : String) → Literal

+
  name   : (x : Name)   → Literal

+
  meta   : (x : Meta)   → Literal

+

+
{-# BUILTIN AGDALITERAL   Literal #-}

+
{-# BUILTIN AGDALITNAT    nat     #-}

+
{-# BUILTIN AGDALITWORD64 word64  #-}

+
{-# BUILTIN AGDALITFLOAT  float   #-}

+
{-# BUILTIN AGDALITCHAR   char    #-}

+
{-# BUILTIN AGDALITSTRING string  #-}

+
{-# BUILTIN AGDALITQNAME  name    #-}

+
{-# BUILTIN AGDALITMETA   meta    #-}

+

+

+
-- Terms and patterns --

+

+
data Term    : Set

+
data Sort    : Set

+
data Pattern : Set

+
data Clause  : Set

+
Type = Term

+
Telescope = List (Σ String λ _ → Arg Type)

+

+
data Term where

+
  var       : (x : Nat) (args : List (Arg Term)) → Term

+
  con       : (c : Name) (args : List (Arg Term)) → Term

+
  def       : (f : Name) (args : List (Arg Term)) → Term

+
  lam       : (v : Visibility) (t : Abs Term) → Term

+
  pat-lam   : (cs : List Clause) (args : List (Arg Term)) → Term

+
  pi        : (a : Arg Type) (b : Abs Type) → Term

+
  agda-sort : (s : Sort) → Term

+
  lit       : (l : Literal) → Term

+
  meta      : (x : Meta) → List (Arg Term) → Term

+
  unknown   : Term

+

+
data Sort where

+
  set     : (t : Term) → Sort

+
  lit     : (n : Nat) → Sort

+
  prop    : (t : Term) → Sort

+
  propLit : (n : Nat) → Sort

+
  inf     : (n : Nat) → Sort

+
  unknown : Sort

+

+
data Pattern where

+
  con    : (c : Name) (ps : List (Arg Pattern)) → Pattern

+
  dot    : (t : Term)    → Pattern

+
  var    : (x : Nat)     → Pattern

+
  lit    : (l : Literal) → Pattern

+
  proj   : (f : Name)    → Pattern

+
  absurd : (x : Nat)     → Pattern  -- absurd patterns counts as variables

+

+
data Clause where

+
  clause        : (tel : Telescope) (ps : List (Arg Pattern)) (t : Term) → Clause

+
  absurd-clause : (tel : Telescope) (ps : List (Arg Pattern)) → Clause

+

+
{-# BUILTIN AGDATERM      Term    #-}

+
{-# BUILTIN AGDASORT      Sort    #-}

+
{-# BUILTIN AGDAPATTERN   Pattern #-}

+
{-# BUILTIN AGDACLAUSE    Clause  #-}

+

+
{-# BUILTIN AGDATERMVAR         var       #-}

+
{-# BUILTIN AGDATERMCON         con       #-}

+
{-# BUILTIN AGDATERMDEF         def       #-}

+
{-# BUILTIN AGDATERMMETA        meta      #-}

+
{-# BUILTIN AGDATERMLAM         lam       #-}

+
{-# BUILTIN AGDATERMEXTLAM      pat-lam   #-}

+
{-# BUILTIN AGDATERMPI          pi        #-}

+
{-# BUILTIN AGDATERMSORT        agda-sort #-}

+
{-# BUILTIN AGDATERMLIT         lit       #-}

+
{-# BUILTIN AGDATERMUNSUPPORTED unknown   #-}

+

+
{-# BUILTIN AGDASORTSET         set     #-}

+
{-# BUILTIN AGDASORTLIT         lit     #-}

+
{-# BUILTIN AGDASORTPROP        prop    #-}

+
{-# BUILTIN AGDASORTPROPLIT     propLit #-}

+
{-# BUILTIN AGDASORTINF         inf     #-}

+
{-# BUILTIN AGDASORTUNSUPPORTED unknown #-}

+

+
{-# BUILTIN AGDAPATCON    con     #-}

+
{-# BUILTIN AGDAPATDOT    dot     #-}

+
{-# BUILTIN AGDAPATVAR    var     #-}

+
{-# BUILTIN AGDAPATLIT    lit     #-}

+
{-# BUILTIN AGDAPATPROJ   proj    #-}

+
{-# BUILTIN AGDAPATABSURD absurd  #-}

+

+
{-# BUILTIN AGDACLAUSECLAUSE clause        #-}

+
{-# BUILTIN AGDACLAUSEABSURD absurd-clause #-}

+

+
-- Definitions --

+

+
data Definition : Set where

+
  function    : (cs : List Clause) → Definition

+
  data-type   : (pars : Nat) (cs : List Name) → Definition

+
  record-type : (c : Name) (fs : List (Arg Name)) → Definition

+
  data-cons   : (d : Name) (q : Quantity) → Definition

+
  axiom       : Definition

+
  prim-fun    : Definition

+

+
{-# BUILTIN AGDADEFINITION                Definition  #-}

+
{-# BUILTIN AGDADEFINITIONFUNDEF          function    #-}

+
{-# BUILTIN AGDADEFINITIONDATADEF         data-type   #-}

+
{-# BUILTIN AGDADEFINITIONRECORDDEF       record-type #-}

+
{-# BUILTIN AGDADEFINITIONDATACONSTRUCTOR data-cons   #-}

+
{-# BUILTIN AGDADEFINITIONPOSTULATE       axiom       #-}

+
{-# BUILTIN AGDADEFINITIONPRIMITIVE       prim-fun    #-}

+

+
-- Errors --

+

+
data ErrorPart : Set where

+
  strErr  : String → ErrorPart

+
  termErr : Term → ErrorPart

+
  pattErr : Pattern → ErrorPart

+
  nameErr : Name → ErrorPart

+

+
{-# BUILTIN AGDAERRORPART       ErrorPart #-}

+
{-# BUILTIN AGDAERRORPARTSTRING strErr    #-}

+
{-# BUILTIN AGDAERRORPARTTERM   termErr   #-}

+
{-# BUILTIN AGDAERRORPARTPATT   pattErr   #-}

+
{-# BUILTIN AGDAERRORPARTNAME   nameErr   #-}

+

+
-- TC monad --

+

+
postulate

+
  TC               : ∀ {a} → Set a → Set a

+
  returnTC         : ∀ {a} {A : Set a} → A → TC A

+
  bindTC           : ∀ {a b} {A : Set a} {B : Set b} → TC A → (A → TC B) → TC B

+
  unify            : Term → Term → TC ⊤

+
  typeError        : ∀ {a} {A : Set a} → List ErrorPart → TC A

+
  inferType        : Term → TC Type

+
  checkType        : Term → Type → TC Term

+
  normalise        : Term → TC Term

+
  reduce           : Term → TC Term

+
  catchTC          : ∀ {a} {A : Set a} → TC A → TC A → TC A

+
  quoteTC          : ∀ {a} {A : Set a} → A → TC Term

+
  unquoteTC        : ∀ {a} {A : Set a} → Term → TC A

+
  quoteωTC         : ∀ {A : Setω} → A → TC Term

+
  getContext       : TC Telescope

+
  extendContext    : ∀ {a} {A : Set a} → String → Arg Type → TC A → TC A

+
  inContext        : ∀ {a} {A : Set a} → Telescope → TC A → TC A

+
  freshName        : String → TC Name

+
  declareDef       : Arg Name → Type → TC ⊤

+
  declarePostulate : Arg Name → Type → TC ⊤

+
  declareData      : Name → Nat → Type → TC ⊤

+
  defineData       : Name → List (Σ Name (λ _ → Σ Quantity (λ _ → Type))) → TC ⊤

+
  defineFun        : Name → List Clause → TC ⊤

+
  getType          : Name → TC Type

+
  getDefinition    : Name → TC Definition

+
  blockTC          : ∀ {a} {A : Set a} → Blocker → TC A

+
  commitTC         : TC ⊤

+
  isMacro          : Name → TC Bool

+
  pragmaForeign    : String → String → TC ⊤

+
  pragmaCompile    : String → Name → String → TC ⊤

+

+
  -- If 'true', makes the following primitives also normalise

+
  -- their results: inferType, checkType, quoteTC, getType, and getContext

+
  withNormalisation : ∀ {a} {A : Set a} → Bool → TC A → TC A

+
  askNormalisation  : TC Bool

+

+
  -- If 'true', makes the following primitives to reconstruct hidden arguments:

+
  -- getDefinition, normalise, reduce, inferType, checkType and getContext

+
  withReconstructed : ∀ {a} {A : Set a} → Bool → TC A → TC A

+
  askReconstructed  : TC Bool

+

+
  -- Whether implicit arguments at the end should be turned into metavariables

+
  withExpandLast : ∀ {a} {A : Set a} → Bool → TC A → TC A

+
  askExpandLast  : TC Bool

+

+
  -- White/blacklist specific definitions for reduction while executing the TC computation

+
  -- 'true' for whitelist, 'false' for blacklist

+
  withReduceDefs : ∀ {a} {A : Set a} → (Σ Bool λ _ → List Name) → TC A → TC A

+
  askReduceDefs  : TC (Σ Bool λ _ → List Name)

+

+
  formatErrorParts : List ErrorPart → TC String

+
  -- Prints the third argument if the corresponding verbosity level is turned

+
  -- on (with the -v flag to Agda).

+
  debugPrint : String → Nat → List ErrorPart → TC ⊤

+

+
  -- Fail if the given computation gives rise to new, unsolved

+
  -- "blocking" constraints.

+
  noConstraints : ∀ {a} {A : Set a} → TC A → TC A

+

+
  -- Run the given computation at the type level, allowing use of erased things.

+
  workOnTypes : ∀ {a} {A : Set a} → TC A → TC A

+

+
  -- Run the given TC action and return the first component. Resets to

+
  -- the old TC state if the second component is 'false', or keep the

+
  -- new TC state if it is 'true'.

+
  runSpeculative : ∀ {a} {A : Set a} → TC (Σ A λ _ → Bool) → TC A

+

+
  -- Get a list of all possible instance candidates for the given meta

+
  -- variable (it does not have to be an instance meta).

+
  getInstances : Meta → TC (List Term)

+

+
  -- Try to solve open instance constraints. When wrapped in `noConstraints`,

+
  -- fails if there are unsolved instance constraints left over that originate

+
  -- from the current macro invokation. Outside constraints are still attempted,

+
  -- but failure to solve them are ignored by `noConstraints`.

+
  solveInstanceConstraints : TC ⊤

+

+
{-# BUILTIN AGDATCM                           TC                         #-}

+
{-# BUILTIN AGDATCMRETURN                     returnTC                   #-}

+
{-# BUILTIN AGDATCMBIND                       bindTC                     #-}

+
{-# BUILTIN AGDATCMUNIFY                      unify                      #-}

+
{-# BUILTIN AGDATCMTYPEERROR                  typeError                  #-}

+
{-# BUILTIN AGDATCMINFERTYPE                  inferType                  #-}

+
{-# BUILTIN AGDATCMCHECKTYPE                  checkType                  #-}

+
{-# BUILTIN AGDATCMNORMALISE                  normalise                  #-}

+
{-# BUILTIN AGDATCMREDUCE                     reduce                     #-}

+
{-# BUILTIN AGDATCMCATCHERROR                 catchTC                    #-}

+
{-# BUILTIN AGDATCMQUOTETERM                  quoteTC                    #-}

+
{-# BUILTIN AGDATCMUNQUOTETERM                unquoteTC                  #-}

+
{-# BUILTIN AGDATCMQUOTEOMEGATERM             quoteωTC                   #-}

+
{-# BUILTIN AGDATCMGETCONTEXT                 getContext                 #-}

+
{-# BUILTIN AGDATCMEXTENDCONTEXT              extendContext              #-}

+
{-# BUILTIN AGDATCMINCONTEXT                  inContext                  #-}

+
{-# BUILTIN AGDATCMFRESHNAME                  freshName                  #-}

+
{-# BUILTIN AGDATCMDECLAREDEF                 declareDef                 #-}

+
{-# BUILTIN AGDATCMDECLAREPOSTULATE           declarePostulate           #-}

+
{-# BUILTIN AGDATCMDECLAREDATA                declareData                #-}

+
{-# BUILTIN AGDATCMDEFINEDATA                 defineData                 #-}

+
{-# BUILTIN AGDATCMDEFINEFUN                  defineFun                  #-}

+
{-# BUILTIN AGDATCMGETTYPE                    getType                    #-}

+
{-# BUILTIN AGDATCMGETDEFINITION              getDefinition              #-}

+
{-# BUILTIN AGDATCMBLOCK                      blockTC                    #-}

+
{-# BUILTIN AGDATCMCOMMIT                     commitTC                   #-}

+
{-# BUILTIN AGDATCMISMACRO                    isMacro                    #-}

+
{-# BUILTIN AGDATCMPRAGMAFOREIGN              pragmaForeign              #-}

+
{-# BUILTIN AGDATCMPRAGMACOMPILE              pragmaCompile              #-}

+
{-# BUILTIN AGDATCMWITHNORMALISATION          withNormalisation          #-}

+
{-# BUILTIN AGDATCMWITHRECONSTRUCTED          withReconstructed          #-}

+
{-# BUILTIN AGDATCMWITHEXPANDLAST             withExpandLast             #-}

+
{-# BUILTIN AGDATCMWITHREDUCEDEFS             withReduceDefs             #-}

+
{-# BUILTIN AGDATCMASKNORMALISATION           askNormalisation           #-}

+
{-# BUILTIN AGDATCMASKRECONSTRUCTED           askReconstructed           #-}

+
{-# BUILTIN AGDATCMASKEXPANDLAST              askExpandLast              #-}

+
{-# BUILTIN AGDATCMASKREDUCEDEFS              askReduceDefs              #-}

+
{-# BUILTIN AGDATCMFORMATERRORPARTS           formatErrorParts           #-}

+
{-# BUILTIN AGDATCMDEBUGPRINT                 debugPrint                 #-}

+
{-# BUILTIN AGDATCMNOCONSTRAINTS              noConstraints              #-}

+
{-# BUILTIN AGDATCMWORKONTYPES                workOnTypes                #-}

+
{-# BUILTIN AGDATCMRUNSPECULATIVE             runSpeculative             #-}

+
{-# BUILTIN AGDATCMGETINSTANCES               getInstances               #-}

+
{-# BUILTIN AGDATCMSOLVEINSTANCES             solveInstanceConstraints   #-}

+

+
-- All the TC primitives are compiled to functions that return

+
-- undefined, rather than just undefined, in an attempt to make sure

+
-- that code will run properly.

+
{-# COMPILE JS returnTC          = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS bindTC            = _ => _ => _ => _ =>

+
                                   _ => _ =>           undefined #-}

+
{-# COMPILE JS unify             = _ => _ =>           undefined #-}

+
{-# COMPILE JS typeError         = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS inferType         = _ =>                undefined #-}

+
{-# COMPILE JS checkType         = _ => _ =>           undefined #-}

+
{-# COMPILE JS normalise         = _ =>                undefined #-}

+
{-# COMPILE JS reduce            = _ =>                undefined #-}

+
{-# COMPILE JS catchTC           = _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS quoteTC           = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS unquoteTC         = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS quoteωTC          = _ => _ =>           undefined #-}

+
{-# COMPILE JS getContext        =                     undefined #-}

+
{-# COMPILE JS extendContext     = _ => _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS inContext         = _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS freshName         = _ =>                undefined #-}

+
{-# COMPILE JS declareDef        = _ => _ =>           undefined #-}

+
{-# COMPILE JS declarePostulate  = _ => _ =>           undefined #-}

+
{-# COMPILE JS declareData       = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS defineData        = _ => _ =>           undefined #-}

+
{-# COMPILE JS defineFun         = _ => _ =>           undefined #-}

+
{-# COMPILE JS getType           = _ =>                undefined #-}

+
{-# COMPILE JS getDefinition     = _ =>                undefined #-}

+
{-# COMPILE JS blockTC           = _ => _ =>           undefined #-}

+
{-# COMPILE JS commitTC          =                     undefined #-}

+
{-# COMPILE JS isMacro           = _ =>                undefined #-}

+
{-# COMPILE JS pragmaForeign     = _ => _ =>           undefined #-}

+
{-# COMPILE JS pragmaCompile     = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS withNormalisation = _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS withReconstructed = _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS withExpandLast    = _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS withReduceDefs    = _ => _ => _ => _ => undefined #-}

+
{-# COMPILE JS askNormalisation  =                     undefined #-}

+
{-# COMPILE JS askReconstructed  =                     undefined #-}

+
{-# COMPILE JS askExpandLast     =                     undefined #-}

+
{-# COMPILE JS askReduceDefs     =                     undefined #-}

+
{-# COMPILE JS debugPrint        = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS noConstraints     = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS runSpeculative    = _ => _ => _ =>      undefined #-}

+
{-# COMPILE JS getInstances      = _ =>                undefined #-}

+

+
private

+
  filter : (Name → Bool) → List Name → List Name

+
  filter p [] = []

+
  filter p (x ∷ xs) with p x

+
  ... | true  = x ∷ filter p xs

+
  ... | false = filter p xs

+

+
  _∈_ : Name → List Name → Bool

+
  n ∈ []      = false

+
  n ∈ (n' ∷ l) with primQNameEquality n n'

+
  ... | true  = true

+
  ... | false = n ∈ l

+

+
  _∉_ : Name → List Name → Bool

+
  n ∉ l with n ∈ l

+
  ... | true  = false

+
  ... | false = true

+

+
  _++_ : List Name → List Name → List Name

+
  [] ++ l       = l

+
  (x ∷ xs) ++ l = x ∷ (xs ++ l)

+

+
  combineReduceDefs : (Σ Bool λ _ → List Name) → (Σ Bool λ _ → List Name) → (Σ Bool λ _ → List Name)

+
  combineReduceDefs (true  , defs₁) (true  , defs₂) = (true  , filter (_∈ defs₁) defs₂)

+
  combineReduceDefs (false , defs₁) (true  , defs₂) = (true  , filter (_∉ defs₁) defs₂)

+
  combineReduceDefs (true  , defs₁) (false , defs₂) = (true  , filter (_∉ defs₂) defs₁)

+
  combineReduceDefs (false , defs₁) (false , defs₂) = (false , defs₁ ++ defs₂)

+

+
onlyReduceDefs dontReduceDefs : ∀ {a} {A : Set a} → List Name → TC A → TC A

+
onlyReduceDefs defs x = bindTC askReduceDefs (λ exDefs → withReduceDefs (combineReduceDefs (true  , defs) exDefs) x)

+
dontReduceDefs defs x = bindTC askReduceDefs (λ exDefs → withReduceDefs (combineReduceDefs (false , defs) exDefs) x)

+

+
blockOnMeta   : ∀ {a} {A : Set a} → Meta → TC A

+
blockOnMeta m = blockTC (blockerMeta m)

+

+
{-# WARNING_ON_USAGE onlyReduceDefs "DEPRECATED: Use `withReduceDefs` instead of `onlyReduceDefs`" #-}

+
{-# WARNING_ON_USAGE dontReduceDefs "DEPRECATED: Use `withReduceDefs` instead of `dontReduceDefs`" #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Sigma.html b/site/static/agda_html/Agda.Builtin.Sigma.html
new file mode 100644
index 000000000..b3cf2dc56
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Sigma.html
@@ -0,0 +1,94 @@
+
+Agda.Builtin.Sigma
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Sigma where

+

+
open import Agda.Primitive

+

+
record Σ {a b} (A : Set a) (B : A → Set b) : Set (a ⊔ b) where

+
  constructor _,_

+
  field

+
    fst : A

+
    snd : B fst

+

+
open Σ public

+

+
infixr 4 _,_

+

+
{-# BUILTIN SIGMA Σ #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Strict.html b/site/static/agda_html/Agda.Builtin.Strict.html
new file mode 100644
index 000000000..73f783d8c
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Strict.html
@@ -0,0 +1,86 @@
+
+Agda.Builtin.Strict
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Strict where

+

+
open import Agda.Builtin.Equality

+

+
primitive

+
  primForce      : ∀ {a b} {A : Set a} {B : A → Set b} (x : A) → (∀ x → B x) → B x

+
  primForceLemma : ∀ {a b} {A : Set a} {B : A → Set b} (x : A) (f : ∀ x → B x) → primForce x f ≡ f x

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.String.Properties.html b/site/static/agda_html/Agda.Builtin.String.Properties.html
new file mode 100644
index 000000000..51d14effc
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.String.Properties.html
@@ -0,0 +1,88 @@
+
+Agda.Builtin.String.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.String.Properties where

+

+
open import Agda.Builtin.String

+
open import Agda.Builtin.Equality

+

+
primitive

+

+
  primStringToListInjective : ∀ a b → primStringToList a ≡ primStringToList b → a ≡ b

+
  primStringFromListInjective : ∀ a b → primStringFromList a ≡ primStringFromList b → a ≡ b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.String.html b/site/static/agda_html/Agda.Builtin.String.html
new file mode 100644
index 000000000..865fe6223
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.String.html
@@ -0,0 +1,113 @@
+
+Agda.Builtin.String
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.String where

+

+
open import Agda.Builtin.Bool

+
open import Agda.Builtin.Char

+
open import Agda.Builtin.List

+
open import Agda.Builtin.Maybe

+
open import Agda.Builtin.Nat using (Nat)

+
open import Agda.Builtin.Sigma

+

+
postulate String : Set

+
{-# BUILTIN STRING String #-}

+

+
primitive

+
  primStringUncons   : String → Maybe (Σ Char (λ _ → String))

+
  primStringToList   : String → List Char

+
  primStringFromList : List Char → String

+
  primStringAppend   : String → String → String

+
  primStringEquality : String → String → Bool

+
  primShowChar       : Char → String

+
  primShowString     : String → String

+
  primShowNat        : Nat → String

+

+
{-# COMPILE JS primStringUncons = function(x) {

+
   if (x === "") { return z_jAgda_Agda_Builtin_Maybe["Maybe"]["nothing"]; };

+
   return z_jAgda_Agda_Builtin_Maybe["Maybe"]["just"](z_jAgda_Agda_Builtin_Sigma["_,_"](x.charAt(0))(x.slice(1)));

+
   }

+
 #-}

+
{-# COMPILE JS primStringToList = function(x) { return x.split(""); } #-}

+
{-# COMPILE JS primStringFromList = function(x) { return x.join(""); } #-}

+
{-# COMPILE JS primStringAppend = function(x) { return function(y) { return x+y; }; } #-}

+
{-# COMPILE JS primStringEquality = function(x) { return function(y) { return x===y; }; } #-}

+
{-# COMPILE JS primShowChar = function(x) { return JSON.stringify(x); } #-}

+
{-# COMPILE JS primShowString = function(x) { return JSON.stringify(x); } #-}

+
{-# COMPILE JS primShowNat = function(x) { return JSON.stringify(x); } #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Unit.html b/site/static/agda_html/Agda.Builtin.Unit.html
new file mode 100644
index 000000000..497b943ae
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Unit.html
@@ -0,0 +1,87 @@
+
+Agda.Builtin.Unit
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-universe-polymorphism

+
            --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Unit where

+

+
record ⊤ : Set where

+
  instance constructor tt

+

+
{-# BUILTIN UNIT ⊤ #-}

+
{-# COMPILE GHC ⊤ = data () (()) #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Word.Properties.html b/site/static/agda_html/Agda.Builtin.Word.Properties.html
new file mode 100644
index 000000000..38f1fe004
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Word.Properties.html
@@ -0,0 +1,87 @@
+
+Agda.Builtin.Word.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Word.Properties where

+

+
open import Agda.Builtin.Word

+
open import Agda.Builtin.Equality

+

+
primitive

+

+
  primWord64ToNatInjective : ∀ a b → primWord64ToNat a ≡ primWord64ToNat b → a ≡ b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Builtin.Word.html b/site/static/agda_html/Agda.Builtin.Word.html
new file mode 100644
index 000000000..44888182b
--- /dev/null
+++ b/site/static/agda_html/Agda.Builtin.Word.html
@@ -0,0 +1,90 @@
+
+Agda.Builtin.Word
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --cubical-compatible --safe --no-universe-polymorphism

+
            --no-sized-types --no-guardedness --level-universe #-}

+

+
module Agda.Builtin.Word where

+

+
open import Agda.Builtin.Nat

+

+
postulate Word64 : Set

+
{-# BUILTIN WORD64 Word64 #-}

+

+
primitive

+
  primWord64ToNat   : Word64 → Nat

+
  primWord64FromNat : Nat → Word64

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.Primitive.html b/site/static/agda_html/Agda.Primitive.html
new file mode 100644
index 000000000..8926f949a
--- /dev/null
+++ b/site/static/agda_html/Agda.Primitive.html
@@ -0,0 +1,118 @@
+
+Agda.Primitive
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
-- The Agda primitives (preloaded).

+

+
{-# OPTIONS --cubical-compatible --no-import-sorts --level-universe #-}

+

+
module Agda.Primitive where

+

+
------------------------------------------------------------------------

+
-- Universe levels

+
------------------------------------------------------------------------

+

+
infixl 6 _⊔_

+

+
{-# BUILTIN PROP           Prop      #-}

+
{-# BUILTIN TYPE           Set       #-}

+
{-# BUILTIN STRICTSET      SSet      #-}

+

+
{-# BUILTIN PROPOMEGA      Propω     #-}

+
{-# BUILTIN SETOMEGA       Setω      #-}

+
{-# BUILTIN STRICTSETOMEGA SSetω     #-}

+

+
{-# BUILTIN LEVELUNIV      LevelUniv #-}

+

+
-- Level is the first thing we need to define.

+
-- The other postulates can only be checked if built-in Level is known.

+

+
postulate

+
  Level : LevelUniv

+

+
-- MAlonzo compiles Level to (). This should be safe, because it is

+
-- not possible to pattern match on levels.

+

+
{-# BUILTIN LEVEL Level #-}

+

+
postulate

+
  lzero : Level

+
  lsuc  : (ℓ : Level) → Level

+
  _⊔_   : (ℓ₁ ℓ₂ : Level) → Level

+

+
{-# BUILTIN LEVELZERO lzero #-}

+
{-# BUILTIN LEVELSUC  lsuc  #-}

+
{-# BUILTIN LEVELMAX  _⊔_   #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Agda.css b/site/static/agda_html/Agda.css
index 4f69b4acd..87a6be959 100644
--- a/site/static/agda_html/Agda.css
+++ b/site/static/agda_html/Agda.css
@@ -9,7 +9,7 @@
   --content-width: calc(100vw - var(--sidebar-width) - 4rem);
   --search-width: 300px;
   --hover-color: rgba(0, 0, 0, 0.05);
-  --highlight-color: #ffeb3b;
+  --highlight-color: rgba(255, 235, 59, 0.2);
   --muted-color: #6a737d;
   --header-height: 60px;
   --font-sans: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
@@ -24,7 +24,7 @@
   --link-color: #58a6ff;
   --header-bg: transparent;
   --hover-color: rgba(255, 255, 255, 0.05);
-  --highlight-color: #ffeb3b;
+  --highlight-color: rgba(255, 235, 59, 0.15);
   --muted-color: #8b949e;
 }
 
@@ -39,14 +39,11 @@ body {
   color: var(--text-color);
   font-family: var(--font-sans);
   line-height: 1.6;
-  padding-top: var(--header-height); /* Added to offset fixed header */
 }
 
 .agda-header {
-  position: fixed;
-  top: 0;
-  left: 0;
-  right: 0;
+  position: relative;
+  flex: 1;
   height: var(--header-height);
   background-color: var(--bg-color);
   display: flex;
@@ -65,11 +62,10 @@ body {
 
 .agda-container {
   display: flex;
-  flex-direction: row;
+  flex: 1;
   background-color: var(--bg-color);
-  min-height: calc(100vh - var(--header-height));
+  min-height: 0; /* Important for proper flex behavior */
   position: relative;
-  top: 0;
 }
 
 .agda-sidebar {
@@ -78,6 +74,7 @@ body {
   overflow-y: auto;
   padding: 1.5rem;
   border-right: none;
+  flex-shrink: 0;
 }
 
 .agda-sidebar h3 {
@@ -131,7 +128,6 @@ body {
 .agda-content {
   flex: 1;
   padding: 2rem;
-  width: var(--content-width);
   min-width: 0;
   overflow-y: auto;
   overflow-x: auto;
@@ -145,8 +141,9 @@ body {
   width: 100%;
   overflow-x: visible;
   max-width: none;
-  padding-bottom: 4rem; /* Add some padding at the bottom */
-  white-space: pre; /* Preserve whitespace and line breaks */
+  padding-bottom: 4rem;
+  white-space: pre;
+  margin: 0;
 }
 
 /* Add styles for highlighted line */
@@ -164,9 +161,39 @@ body {
 .Agda {
   font-family: var(--font-mono);
   font-size: 0.95rem;
-  line-height: 1.6;
+  line-height: 1;
   background-color: var(--bg-color);
   color: var(--text-color);
+  counter-reset: line;
+  position: relative;
+}
+
+.Agda .line {
+  position: relative;
+  display: block;
+  padding-left: 5em;
+  min-height: 1em;
+}
+
+.Agda .line::before {
+  counter-increment: line;
+  content: counter(line);
+  position: absolute;
+  left: 0;
+  width: 3em;
+  text-align: right;
+  color: var(--muted-color);
+  padding-right: 1em;
+  user-select: none;
+}
+
+.Agda .line.highlight-line {
+  background-color: var(--highlight-color);
+}
+
+.Agda .line.highlight-line::before {
+  color: var(--text-color);
+  font-weight: bold;
 }
 
 .Agda code {
diff --git a/site/static/agda_html/Algebra.Apartness.Bundles.html b/site/static/agda_html/Algebra.Apartness.Bundles.html
new file mode 100644
index 000000000..2de64996e
--- /dev/null
+++ b/site/static/agda_html/Algebra.Apartness.Bundles.html
@@ -0,0 +1,142 @@
+
+Algebra.Apartness.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for local algebraic structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Apartness.Bundles where

+

+
open import Level using (_⊔_; suc)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (ApartnessRelation)

+
open import Algebra.Core using (Op₁; Op₂)

+
open import Algebra.Bundles using (CommutativeRing)

+
open import Algebra.Apartness.Structures

+

+
record HeytingCommutativeRing c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_ _#_

+
  field

+
    Carrier                  : Set c

+
    _≈_                      : Rel Carrier ℓ₁

+
    _#_                      : Rel Carrier ℓ₂

+
    _+_                      : Op₂ Carrier

+
    _*_                      : Op₂ Carrier

+
    -_                       : Op₁ Carrier

+
    0#                       : Carrier

+
    1#                       : Carrier

+
    isHeytingCommutativeRing : IsHeytingCommutativeRing _≈_ _#_ _+_ _*_ -_ 0# 1#

+

+
  open IsHeytingCommutativeRing isHeytingCommutativeRing public

+

+
  commutativeRing : CommutativeRing c ℓ₁

+
  commutativeRing = record { isCommutativeRing = isCommutativeRing }

+

+
  apartnessRelation : ApartnessRelation c ℓ₁ ℓ₂

+
  apartnessRelation = record { isApartnessRelation = isApartnessRelation }

+

+

+
record HeytingField c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_ _#_

+
  field

+
    Carrier        : Set c

+
    _≈_            : Rel Carrier ℓ₁

+
    _#_            : Rel Carrier ℓ₂

+
    _+_            : Op₂ Carrier

+
    _*_            : Op₂ Carrier

+
    -_             : Op₁ Carrier

+
    0#             : Carrier

+
    1#             : Carrier

+
    isHeytingField : IsHeytingField _≈_ _#_ _+_ _*_ -_ 0# 1#

+

+
  open IsHeytingField isHeytingField public

+

+
  heytingCommutativeRing : HeytingCommutativeRing c ℓ₁ ℓ₂

+
  heytingCommutativeRing = record { isHeytingCommutativeRing = isHeytingCommutativeRing }

+

+
  apartnessRelation : ApartnessRelation c ℓ₁ ℓ₂

+
  apartnessRelation = record { isApartnessRelation = isApartnessRelation }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Apartness.Structures.html b/site/static/agda_html/Algebra.Apartness.Structures.html
new file mode 100644
index 000000000..a433d7819
--- /dev/null
+++ b/site/static/agda_html/Algebra.Apartness.Structures.html
@@ -0,0 +1,129 @@
+
+Algebra.Apartness.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Algebraic structures with an apartness relation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core using (Op₁; Op₂)

+
open import Relation.Binary.Core using (Rel)

+

+
module Algebra.Apartness.Structures

+
  {c ℓ₁ ℓ₂} {Carrier : Set c}

+
  (_≈_ : Rel Carrier ℓ₁)

+
  (_#_ : Rel Carrier ℓ₂)

+
  (_+_ _*_ : Op₂ Carrier) (-_ : Op₁ Carrier) (0# 1# : Carrier)

+
  where

+

+
open import Level using (_⊔_; suc)

+
open import Data.Product.Base using (∃-syntax; _×_; _,_; proj₂)

+
open import Algebra.Definitions _≈_ using (Invertible)

+
open import Algebra.Structures _≈_ using (IsCommutativeRing)

+
open import Relation.Binary.Structures using (IsEquivalence; IsApartnessRelation)

+
open import Relation.Binary.Definitions using (Tight)

+
open import Relation.Nullary.Negation using (¬_)

+
import Relation.Binary.Properties.ApartnessRelation as AR

+

+

+
record IsHeytingCommutativeRing : Set (c ⊔ ℓ₁ ⊔ ℓ₂) where

+

+
  field

+
    isCommutativeRing   : IsCommutativeRing _+_ _*_ -_ 0# 1#

+
    isApartnessRelation : IsApartnessRelation _≈_ _#_

+

+
  open IsCommutativeRing isCommutativeRing public

+
  open IsApartnessRelation isApartnessRelation public

+

+
  field

+
    #⇒invertible : ∀ {x y} → x # y → Invertible 1# _*_ (x - y)

+
    invertible⇒# : ∀ {x y} → Invertible 1# _*_ (x - y) → x # y

+

+
  ¬#-isEquivalence : IsEquivalence _¬#_

+
  ¬#-isEquivalence = AR.¬#-isEquivalence refl isApartnessRelation

+

+

+
record IsHeytingField : Set (c ⊔ ℓ₁ ⊔ ℓ₂) where

+

+
  field

+
    isHeytingCommutativeRing : IsHeytingCommutativeRing

+
    tight                    : Tight _≈_ _#_

+

+
  open IsHeytingCommutativeRing isHeytingCommutativeRing public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Apartness.html b/site/static/agda_html/Algebra.Apartness.html
new file mode 100644
index 000000000..96fe9c0ac
--- /dev/null
+++ b/site/static/agda_html/Algebra.Apartness.html
@@ -0,0 +1,89 @@
+
+Algebra.Apartness
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Algebraic objects with an apartness relation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Apartness where

+

+
open import Algebra.Apartness.Structures public

+
open import Algebra.Apartness.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Bundles.Raw.html b/site/static/agda_html/Algebra.Bundles.Raw.html
new file mode 100644
index 000000000..6e878ba3e
--- /dev/null
+++ b/site/static/agda_html/Algebra.Bundles.Raw.html
@@ -0,0 +1,412 @@
+
+Algebra.Bundles.Raw
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of 'raw' bundles

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Bundles.Raw where

+

+
open import Algebra.Core

+
open import Relation.Binary.Core using (Rel)

+
open import Level using (suc; _⊔_)

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
-- A raw SuccessorSet is a SuccessorSet without any laws.

+

+
record RawSuccessorSet c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    suc#    : Op₁ Carrier

+
    zero#   : Carrier

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 1 binary operation

+
------------------------------------------------------------------------

+

+
record RawMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+

+
  infix 4 _≉_

+
  _≉_ : Rel Carrier _

+
  x ≉ y = ¬ (x ≈ y)

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 1 binary operation & 1 element

+
------------------------------------------------------------------------

+

+
-- A raw monoid is a monoid without any laws.

+

+
record RawMonoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    ε       : Carrier

+

+
  rawMagma : RawMagma c ℓ

+
  rawMagma = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _∙_

+
    }

+

+
  open RawMagma rawMagma public

+
    using (_≉_)

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 1 binary operation, 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record RawGroup c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⁻¹

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    ε       : Carrier

+
    _⁻¹     : Op₁ Carrier

+

+
  rawMonoid : RawMonoid c ℓ

+
  rawMonoid = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _∙_

+
    ; ε   = ε

+
    }

+

+
  open RawMonoid rawMonoid public

+
    using (_≉_; rawMagma)

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 2 binary operations & 1 element

+
------------------------------------------------------------------------

+

+
record RawNearSemiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _+_     : Op₂ Carrier

+
    _*_     : Op₂ Carrier

+
    0#      : Carrier

+

+
  +-rawMonoid : RawMonoid c ℓ

+
  +-rawMonoid = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _+_

+
    ;  ε  = 0#

+
    }

+

+
  open RawMonoid +-rawMonoid public

+
    using (_≉_) renaming (rawMagma to +-rawMagma)

+

+
  *-rawMagma : RawMagma c ℓ

+
  *-rawMagma = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _*_

+
    }

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 2 binary operations & 2 elements

+
------------------------------------------------------------------------

+

+
record RawSemiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _+_     : Op₂ Carrier

+
    _*_     : Op₂ Carrier

+
    0#      : Carrier

+
    1#      : Carrier

+

+
  rawNearSemiring : RawNearSemiring c ℓ

+
  rawNearSemiring = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; 0#  = 0#

+
    }

+

+
  open RawNearSemiring rawNearSemiring public

+
    using (_≉_; +-rawMonoid; +-rawMagma; *-rawMagma)

+

+
  *-rawMonoid : RawMonoid c ℓ

+
  *-rawMonoid = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _*_

+
    ; ε   = 1#

+
    }

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 2 binary operations, 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record RawRingWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _+_     : Op₂ Carrier

+
    _*_     : Op₂ Carrier

+
    -_      : Op₁ Carrier

+
    0#      : Carrier

+

+
  +-rawGroup : RawGroup c ℓ

+
  +-rawGroup = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _+_

+
    ; ε   = 0#

+
    ; _⁻¹ = -_

+
    }

+

+
  open RawGroup +-rawGroup public

+
    using (_≉_) renaming (rawMagma to +-rawMagma; rawMonoid to +-rawMonoid)

+

+
  *-rawMagma : RawMagma c ℓ

+
  *-rawMagma = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _*_

+
    }

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 2 binary operations, 1 unary operation & 2 elements

+
------------------------------------------------------------------------

+

+
-- A raw ring is a ring without any laws.

+

+
record RawRing c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _+_     : Op₂ Carrier

+
    _*_     : Op₂ Carrier

+
    -_      : Op₁ Carrier

+
    0#      : Carrier

+
    1#      : Carrier

+

+
  rawSemiring : RawSemiring c ℓ

+
  rawSemiring = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; 0#  = 0#

+
    ; 1#  = 1#

+
    }

+

+
  open RawSemiring rawSemiring public

+
    using

+
    ( _≉_

+
    ; +-rawMagma; +-rawMonoid

+
    ; *-rawMagma; *-rawMonoid

+
    )

+

+
  rawRingWithoutOne : RawRingWithoutOne c ℓ

+
  rawRingWithoutOne = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; -_ = -_

+
    ; 0# = 0#

+
    }

+

+
  open RawRingWithoutOne rawRingWithoutOne public

+
    using (+-rawGroup)

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 3 binary operations

+
------------------------------------------------------------------------

+

+
record RawQuasigroup c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    _\\_    : Op₂ Carrier

+
    _//_    : Op₂ Carrier

+

+
  ∙-rawMagma : RawMagma c ℓ

+
  ∙-rawMagma = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _∙_

+
    }

+

+
  \\-rawMagma : RawMagma c ℓ

+
  \\-rawMagma = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _\\_

+
    }

+

+
  //-rawMagma : RawMagma c ℓ

+
  //-rawMagma = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _//_

+
    }

+

+
  open RawMagma \\-rawMagma public

+
    using (_≉_)

+

+
------------------------------------------------------------------------

+
-- Raw bundles with 3 binary operations & 1 element

+
------------------------------------------------------------------------

+

+
record RawLoop  c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    _\\_    : Op₂ Carrier

+
    _//_    : Op₂ Carrier

+
    ε       : Carrier

+

+
  rawQuasigroup : RawQuasigroup c ℓ

+
  rawQuasigroup = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _∙_

+
    ; _\\_ = _\\_

+
    ; _//_ = _//_

+
    }

+

+
  open RawQuasigroup rawQuasigroup public

+
    using (_≉_ ; ∙-rawMagma; \\-rawMagma; //-rawMagma)

+

+
record RawKleeneAlgebra c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⋆

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _+_     : Op₂ Carrier

+
    _*_     : Op₂ Carrier

+
    _⋆      : Op₁ Carrier

+
    0#      : Carrier

+
    1#      : Carrier

+

+
  rawSemiring : RawSemiring c ℓ

+
  rawSemiring = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; 0#  = 0#

+
    ; 1#  = 1#

+
    }

+

+
  open RawSemiring rawSemiring public

+
    using

+
    ( _≉_

+
    ; +-rawMagma; +-rawMonoid

+
    ; *-rawMagma; *-rawMonoid

+
    )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Bundles.html b/site/static/agda_html/Algebra.Bundles.html
new file mode 100644
index 000000000..0119a3f62
--- /dev/null
+++ b/site/static/agda_html/Algebra.Bundles.html
@@ -0,0 +1,1335 @@
+
+Algebra.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of algebraic structures like monoids and rings

+
-- (packed in records together with sets, operations, etc.)

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Bundles where

+

+
import Algebra.Bundles.Raw as Raw

+
open import Algebra.Core

+
open import Algebra.Structures

+
open import Relation.Binary.Core using (Rel)

+
open import Level

+

+
------------------------------------------------------------------------

+
-- Re-export definitions of 'raw' bundles

+

+
open Raw public

+
  using ( RawSuccessorSet; RawMagma; RawMonoid; RawGroup

+
        ; RawNearSemiring; RawSemiring

+
        ; RawRingWithoutOne; RawRing

+
        ; RawQuasigroup; RawLoop; RawKleeneAlgebra)

+

+
------------------------------------------------------------------------

+
-- Bundles with 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record SuccessorSet c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  4 _≈_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ

+
    suc#             : Op₁ Carrier

+
    zero#            : Carrier

+
    isSuccessorSet   : IsSuccessorSet _≈_ suc# zero#

+

+
  open IsSuccessorSet isSuccessorSet public

+

+
  rawSuccessorSet : RawSuccessorSet _ _

+
  rawSuccessorSet = record { _≈_ = _≈_; suc# = suc#; zero# = zero# }

+

+
  open RawSuccessorSet rawSuccessorSet public

+

+
------------------------------------------------------------------------

+
-- Bundles with 1 binary operation

+
------------------------------------------------------------------------

+

+
record Magma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isMagma : IsMagma _≈_ _∙_

+

+
  open IsMagma isMagma public

+

+
  rawMagma : RawMagma _ _

+
  rawMagma = record { _≈_ = _≈_; _∙_ = _∙_ }

+

+
  open RawMagma rawMagma public

+
    using (_≉_)

+

+

+
record SelectiveMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ

+
    _∙_              : Op₂ Carrier

+
    isSelectiveMagma : IsSelectiveMagma _≈_ _∙_

+

+
  open IsSelectiveMagma isSelectiveMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public using (rawMagma)

+

+

+
record CommutativeMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier            : Set c

+
    _≈_                : Rel Carrier ℓ

+
    _∙_                : Op₂ Carrier

+
    isCommutativeMagma : IsCommutativeMagma _≈_ _∙_

+

+
  open IsCommutativeMagma isCommutativeMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public using (rawMagma)

+

+
record IdempotentMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isIdempotentMagma  : IsIdempotentMagma _≈_ _∙_

+

+
  open IsIdempotentMagma isIdempotentMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (rawMagma)

+

+
record AlternativeMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isAlternativeMagma  : IsAlternativeMagma _≈_ _∙_

+

+
  open IsAlternativeMagma isAlternativeMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (rawMagma)

+

+
record FlexibleMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isFlexibleMagma  : IsFlexibleMagma _≈_ _∙_

+

+
  open IsFlexibleMagma isFlexibleMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (rawMagma)

+

+
record MedialMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isMedialMagma  : IsMedialMagma _≈_ _∙_

+

+
  open IsMedialMagma isMedialMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (rawMagma)

+

+
record SemimedialMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isSemimedialMagma  : IsSemimedialMagma _≈_ _∙_

+

+
  open IsSemimedialMagma isSemimedialMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (rawMagma)

+

+

+
record Semigroup c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier     : Set c

+
    _≈_         : Rel Carrier ℓ

+
    _∙_         : Op₂ Carrier

+
    isSemigroup : IsSemigroup _≈_ _∙_

+

+
  open IsSemigroup isSemigroup public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (_≉_; rawMagma)

+

+

+
record Band c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    isBand  : IsBand _≈_ _∙_

+

+
  open IsBand isBand public

+

+
  semigroup : Semigroup c ℓ

+
  semigroup = record { isSemigroup = isSemigroup }

+

+
  open Semigroup semigroup public

+
    using (_≉_; magma; rawMagma)

+

+

+
record CommutativeSemigroup c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier                 : Set c

+
    _≈_                     : Rel Carrier ℓ

+
    _∙_                     : Op₂ Carrier

+
    isCommutativeSemigroup  : IsCommutativeSemigroup _≈_ _∙_

+

+
  open IsCommutativeSemigroup isCommutativeSemigroup public

+

+
  semigroup : Semigroup c ℓ

+
  semigroup = record { isSemigroup = isSemigroup }

+

+
  open Semigroup semigroup public

+
    using (_≉_; magma; rawMagma)

+

+
  commutativeMagma : CommutativeMagma c ℓ

+
  commutativeMagma = record { isCommutativeMagma = isCommutativeMagma }

+

+
record CommutativeBand c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier            : Set c

+
    _≈_                : Rel Carrier ℓ

+
    _∙_                : Op₂ Carrier

+
    isCommutativeBand  : IsCommutativeBand _≈_ _∙_

+

+
  open IsCommutativeBand isCommutativeBand public

+

+
  band : Band _ _

+
  band = record { isBand = isBand }

+

+
  open Band band public

+
    using (_≉_; magma; rawMagma; semigroup)

+

+
  commutativeSemigroup : CommutativeSemigroup c ℓ

+
  commutativeSemigroup = record { isCommutativeSemigroup = isCommutativeSemigroup }

+

+
  open CommutativeSemigroup commutativeSemigroup public

+
    using (commutativeMagma)

+

+

+
------------------------------------------------------------------------

+
-- Bundles with 1 binary operation & 1 element

+
------------------------------------------------------------------------

+

+
record UnitalMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier  : Set c

+
    _≈_      : Rel Carrier ℓ

+
    _∙_      : Op₂ Carrier

+
    ε        : Carrier

+
    isUnitalMagma : IsUnitalMagma _≈_ _∙_ ε

+

+
  open IsUnitalMagma isUnitalMagma public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (_≉_; rawMagma)

+

+

+
record Monoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier  : Set c

+
    _≈_      : Rel Carrier ℓ

+
    _∙_      : Op₂ Carrier

+
    ε        : Carrier

+
    isMonoid : IsMonoid _≈_ _∙_ ε

+

+
  open IsMonoid isMonoid public

+

+
  semigroup : Semigroup _ _

+
  semigroup = record { isSemigroup = isSemigroup }

+

+
  open Semigroup semigroup public

+
    using (_≉_; rawMagma; magma)

+

+
  rawMonoid : RawMonoid _ _

+
  rawMonoid = record { _≈_ = _≈_; _∙_ = _∙_; ε = ε}

+

+
  unitalMagma : UnitalMagma _ _

+
  unitalMagma = record { isUnitalMagma = isUnitalMagma  }

+

+

+
record CommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier             : Set c

+
    _≈_                 : Rel Carrier ℓ

+
    _∙_                 : Op₂ Carrier

+
    ε                   : Carrier

+
    isCommutativeMonoid : IsCommutativeMonoid _≈_ _∙_ ε

+

+
  open IsCommutativeMonoid isCommutativeMonoid public

+

+
  monoid : Monoid _ _

+
  monoid = record { isMonoid = isMonoid }

+

+
  open Monoid monoid public

+
    using (_≉_; rawMagma; magma; semigroup; unitalMagma; rawMonoid)

+

+
  commutativeSemigroup : CommutativeSemigroup _ _

+
  commutativeSemigroup = record { isCommutativeSemigroup = isCommutativeSemigroup }

+

+
  open CommutativeSemigroup commutativeSemigroup public

+
    using (commutativeMagma)

+

+
record IdempotentMonoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier            : Set c

+
    _≈_                : Rel Carrier ℓ

+
    _∙_                : Op₂ Carrier

+
    ε                  : Carrier

+
    isIdempotentMonoid : IsIdempotentMonoid _≈_ _∙_ ε

+

+
  open IsIdempotentMonoid isIdempotentMonoid public

+

+
  monoid : Monoid _ _

+
  monoid = record { isMonoid = isMonoid }

+

+
  open Monoid monoid public

+
    using (_≉_; rawMagma; magma; semigroup; unitalMagma; rawMonoid)

+

+
  band : Band _ _

+
  band = record { isBand = isBand }

+

+

+
record IdempotentCommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier                       : Set c

+
    _≈_                           : Rel Carrier ℓ

+
    _∙_                           : Op₂ Carrier

+
    ε                             : Carrier

+
    isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid _≈_ _∙_ ε

+

+
  open IsIdempotentCommutativeMonoid isIdempotentCommutativeMonoid public

+

+
  commutativeMonoid : CommutativeMonoid _ _

+
  commutativeMonoid = record { isCommutativeMonoid = isCommutativeMonoid }

+

+
  idempotentMonoid : IdempotentMonoid _ _

+
  idempotentMonoid = record { isIdempotentMonoid = isIdempotentMonoid }

+

+
  commutativeBand : CommutativeBand _ _

+
  commutativeBand = record { isCommutativeBand = isCommutativeBand }

+

+
  open CommutativeMonoid commutativeMonoid public

+
    using

+
    ( _≉_; rawMagma; magma; unitalMagma; commutativeMagma

+
    ; semigroup; commutativeSemigroup

+
    ; rawMonoid; monoid

+
    )

+

+
  open CommutativeBand commutativeBand public

+
    using (band)

+

+
-- Idempotent commutative monoids are also known as bounded lattices.

+
-- Note that the BoundedLattice necessarily uses the notation inherited

+
-- from monoids rather than lattices.

+

+
BoundedLattice = IdempotentCommutativeMonoid

+

+
module BoundedLattice {c ℓ} (idemCommMonoid : IdempotentCommutativeMonoid c ℓ) =

+
       IdempotentCommutativeMonoid idemCommMonoid

+

+

+
------------------------------------------------------------------------

+
-- Bundles with 1 binary operation, 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record InvertibleMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⁻¹

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    ε       : Carrier

+
    _⁻¹     : Op₁ Carrier

+
    isInvertibleMagma : IsInvertibleMagma _≈_ _∙_ ε _⁻¹

+

+
  open IsInvertibleMagma isInvertibleMagma public

+

+
  magma : Magma _ _

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (_≉_; rawMagma)

+

+

+
record InvertibleUnitalMagma c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⁻¹

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier                  : Set c

+
    _≈_                      : Rel Carrier ℓ

+
    _∙_                      : Op₂ Carrier

+
    ε                        : Carrier

+
    _⁻¹                      : Op₁ Carrier

+
    isInvertibleUnitalMagma  : IsInvertibleUnitalMagma _≈_ _∙_ ε _⁻¹

+

+
  open IsInvertibleUnitalMagma isInvertibleUnitalMagma public

+

+
  invertibleMagma : InvertibleMagma _ _

+
  invertibleMagma = record { isInvertibleMagma = isInvertibleMagma }

+

+
  open InvertibleMagma invertibleMagma public

+
    using (_≉_; rawMagma; magma)

+

+
record Group c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⁻¹

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    ε       : Carrier

+
    _⁻¹     : Op₁ Carrier

+
    isGroup : IsGroup _≈_ _∙_ ε _⁻¹

+

+
  open IsGroup isGroup public

+

+
  rawGroup : RawGroup _ _

+
  rawGroup = record { _≈_ = _≈_; _∙_ = _∙_; ε = ε; _⁻¹ = _⁻¹}

+

+
  monoid : Monoid _ _

+
  monoid = record { isMonoid = isMonoid }

+

+
  open Monoid monoid public

+
    using (_≉_; rawMagma; magma; semigroup; unitalMagma; rawMonoid)

+

+
  invertibleMagma : InvertibleMagma c ℓ

+
  invertibleMagma = record

+
    { isInvertibleMagma = isInvertibleMagma

+
    }

+

+
  invertibleUnitalMagma : InvertibleUnitalMagma c ℓ

+
  invertibleUnitalMagma = record

+
    { isInvertibleUnitalMagma = isInvertibleUnitalMagma

+
    }

+

+
record AbelianGroup c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⁻¹

+
  infixl 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier        : Set c

+
    _≈_            : Rel Carrier ℓ

+
    _∙_            : Op₂ Carrier

+
    ε              : Carrier

+
    _⁻¹            : Op₁ Carrier

+
    isAbelianGroup : IsAbelianGroup _≈_ _∙_ ε _⁻¹

+

+
  open IsAbelianGroup isAbelianGroup public

+

+
  group : Group _ _

+
  group = record { isGroup = isGroup }

+

+
  open Group group public using

+
    (_≉_; rawMagma; magma; semigroup

+
    ; rawMonoid; monoid; rawGroup; invertibleMagma; invertibleUnitalMagma

+
    )

+

+
  commutativeMonoid : CommutativeMonoid _ _

+
  commutativeMonoid = record { isCommutativeMonoid = isCommutativeMonoid }

+

+
  open CommutativeMonoid commutativeMonoid public

+
    using (commutativeMagma; commutativeSemigroup)

+

+
------------------------------------------------------------------------

+
-- Bundles with 2 binary operations & 1 element

+
------------------------------------------------------------------------

+

+
record NearSemiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier        : Set c

+
    _≈_            : Rel Carrier ℓ

+
    _+_            : Op₂ Carrier

+
    _*_            : Op₂ Carrier

+
    0#             : Carrier

+
    isNearSemiring : IsNearSemiring _≈_ _+_ _*_ 0#

+

+
  open IsNearSemiring isNearSemiring public

+

+
  rawNearSemiring : RawNearSemiring _ _

+
  rawNearSemiring = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; 0#  = 0#

+
    }

+

+
  +-monoid : Monoid _ _

+
  +-monoid = record { isMonoid = +-isMonoid }

+

+
  open Monoid +-monoid public

+
    using (_≉_) renaming

+
    ( rawMagma    to  +-rawMagma

+
    ; magma       to  +-magma

+
    ; semigroup   to  +-semigroup

+
    ; unitalMagma to  +-unitalMagma

+
    ; rawMonoid   to  +-rawMonoid

+
    )

+

+
  *-semigroup : Semigroup _ _

+
  *-semigroup = record { isSemigroup = *-isSemigroup }

+

+
  open Semigroup *-semigroup public

+
    using () renaming

+
    ( rawMagma to *-rawMagma

+
    ; magma    to *-magma

+
    )

+

+

+
record SemiringWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier              : Set c

+
    _≈_                  : Rel Carrier ℓ

+
    _+_                  : Op₂ Carrier

+
    _*_                  : Op₂ Carrier

+
    0#                   : Carrier

+
    isSemiringWithoutOne : IsSemiringWithoutOne _≈_ _+_ _*_ 0#

+

+
  open IsSemiringWithoutOne isSemiringWithoutOne public

+

+
  nearSemiring : NearSemiring _ _

+
  nearSemiring = record { isNearSemiring = isNearSemiring }

+

+
  open NearSemiring nearSemiring public

+
    using

+
    ( +-rawMagma; +-magma; +-unitalMagma; +-semigroup

+
    ; +-rawMonoid; +-monoid

+
    ; *-rawMagma; *-magma; *-semigroup

+
    ; rawNearSemiring

+
    )

+

+
  +-commutativeMonoid : CommutativeMonoid _ _

+
  +-commutativeMonoid = record { isCommutativeMonoid = +-isCommutativeMonoid }

+

+
  open CommutativeMonoid +-commutativeMonoid public

+
    using () renaming

+
    ( commutativeMagma     to +-commutativeMagma

+
    ; commutativeSemigroup to +-commutativeSemigroup

+
    )

+

+

+
record CommutativeSemiringWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier                         : Set c

+
    _≈_                             : Rel Carrier ℓ

+
    _+_                             : Op₂ Carrier

+
    _*_                             : Op₂ Carrier

+
    0#                              : Carrier

+
    isCommutativeSemiringWithoutOne :

+
      IsCommutativeSemiringWithoutOne _≈_ _+_ _*_ 0#

+

+
  open IsCommutativeSemiringWithoutOne

+
         isCommutativeSemiringWithoutOne public

+

+
  semiringWithoutOne : SemiringWithoutOne _ _

+
  semiringWithoutOne =

+
    record { isSemiringWithoutOne = isSemiringWithoutOne }

+

+
  open SemiringWithoutOne semiringWithoutOne public

+
    using

+
    ( +-rawMagma; +-magma; +-unitalMagma; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-semigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; nearSemiring; rawNearSemiring

+
    )

+

+
------------------------------------------------------------------------

+
-- Bundles with 2 binary operations & 2 elements

+
------------------------------------------------------------------------

+

+
record SemiringWithoutAnnihilatingZero c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier                           : Set c

+
    _≈_                               : Rel Carrier ℓ

+
    _+_                               : Op₂ Carrier

+
    _*_                               : Op₂ Carrier

+
    0#                                : Carrier

+
    1#                                : Carrier

+
    isSemiringWithoutAnnihilatingZero :

+
      IsSemiringWithoutAnnihilatingZero _≈_ _+_ _*_ 0# 1#

+

+
  open IsSemiringWithoutAnnihilatingZero

+
         isSemiringWithoutAnnihilatingZero public

+

+
  rawSemiring : RawSemiring c ℓ

+
  rawSemiring = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; 0#  = 0#

+
    ; 1#  = 1#

+
    }

+

+
  open RawSemiring rawSemiring public

+
    using (rawNearSemiring)

+

+
  +-commutativeMonoid : CommutativeMonoid _ _

+
  +-commutativeMonoid =

+
    record { isCommutativeMonoid = +-isCommutativeMonoid }

+

+
  open CommutativeMonoid +-commutativeMonoid public

+
    using (_≉_) renaming

+
    ( rawMagma             to +-rawMagma

+
    ; magma                to +-magma

+
    ; unitalMagma          to +-unitalMagma

+
    ; commutativeMagma     to +-commutativeMagma

+
    ; semigroup            to +-semigroup

+
    ; commutativeSemigroup to +-commutativeSemigroup

+
    ; rawMonoid            to +-rawMonoid

+
    ; monoid               to +-monoid

+
    )

+

+
  *-monoid : Monoid _ _

+
  *-monoid = record { isMonoid = *-isMonoid }

+

+
  open Monoid *-monoid public

+
    using () renaming

+
    ( rawMagma  to *-rawMagma

+
    ; magma     to *-magma

+
    ; semigroup to *-semigroup

+
    ; rawMonoid to *-rawMonoid

+
    )

+

+

+
record Semiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier    : Set c

+
    _≈_        : Rel Carrier ℓ

+
    _+_        : Op₂ Carrier

+
    _*_        : Op₂ Carrier

+
    0#         : Carrier

+
    1#         : Carrier

+
    isSemiring : IsSemiring _≈_ _+_ _*_ 0# 1#

+

+
  open IsSemiring isSemiring public

+

+
  semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero _ _

+
  semiringWithoutAnnihilatingZero = record

+
    { isSemiringWithoutAnnihilatingZero =

+
        isSemiringWithoutAnnihilatingZero

+
    }

+

+
  open SemiringWithoutAnnihilatingZero

+
         semiringWithoutAnnihilatingZero public

+
    using

+
    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma;  *-magma;  *-semigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid

+
    ; rawNearSemiring ; rawSemiring

+
    )

+

+
  semiringWithoutOne : SemiringWithoutOne _ _

+
  semiringWithoutOne =

+
    record { isSemiringWithoutOne = isSemiringWithoutOne }

+

+
  open SemiringWithoutOne semiringWithoutOne public

+
    using (nearSemiring)

+

+

+
record CommutativeSemiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier               : Set c

+
    _≈_                   : Rel Carrier ℓ

+
    _+_                   : Op₂ Carrier

+
    _*_                   : Op₂ Carrier

+
    0#                    : Carrier

+
    1#                    : Carrier

+
    isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1#

+

+
  open IsCommutativeSemiring isCommutativeSemiring public

+

+
  semiring : Semiring _ _

+
  semiring = record { isSemiring = isSemiring }

+

+
  open Semiring semiring public

+
    using

+
    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-semigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid

+
    ; nearSemiring; semiringWithoutOne

+
    ; semiringWithoutAnnihilatingZero

+
    ; rawSemiring

+
    )

+

+
  *-commutativeMonoid : CommutativeMonoid _ _

+
  *-commutativeMonoid = record

+
    { isCommutativeMonoid = *-isCommutativeMonoid

+
    }

+

+
  open CommutativeMonoid *-commutativeMonoid public

+
    using () renaming

+
    ( commutativeMagma     to *-commutativeMagma

+
    ; commutativeSemigroup to *-commutativeSemigroup

+
    )

+

+
  commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne _ _

+
  commutativeSemiringWithoutOne = record

+
    { isCommutativeSemiringWithoutOne = isCommutativeSemiringWithoutOne

+
    }

+

+

+
record CancellativeCommutativeSemiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier                           : Set c

+
    _≈_                               : Rel Carrier ℓ

+
    _+_                               : Op₂ Carrier

+
    _*_                               : Op₂ Carrier

+
    0#                                : Carrier

+
    1#                                : Carrier

+
    isCancellativeCommutativeSemiring : IsCancellativeCommutativeSemiring _≈_ _+_ _*_ 0# 1#

+

+
  open IsCancellativeCommutativeSemiring isCancellativeCommutativeSemiring public

+

+
  commutativeSemiring : CommutativeSemiring c ℓ

+
  commutativeSemiring = record

+
    { isCommutativeSemiring = isCommutativeSemiring

+
    }

+

+
  open CommutativeSemiring commutativeSemiring public

+
    using

+
    ( +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-commutativeMagma; *-semigroup; *-commutativeSemigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid; *-commutativeMonoid

+
    ; nearSemiring; semiringWithoutOne

+
    ; semiringWithoutAnnihilatingZero

+
    ; rawSemiring

+
    ; semiring

+
    ; _≉_

+
    )

+

+
record IdempotentSemiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier                : Set c

+
    _≈_                    : Rel Carrier ℓ

+
    _+_                    : Op₂ Carrier

+
    _*_                    : Op₂ Carrier

+
    0#                     : Carrier

+
    1#                     : Carrier

+
    isIdempotentSemiring   : IsIdempotentSemiring _≈_ _+_ _*_ 0# 1#

+

+
  open IsIdempotentSemiring isIdempotentSemiring public

+

+
  semiring : Semiring _ _

+
  semiring = record { isSemiring = isSemiring }

+

+
  open Semiring semiring public

+
    using

+
    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-semigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid

+
    ; nearSemiring; semiringWithoutOne

+
    ; semiringWithoutAnnihilatingZero

+
    ; rawSemiring

+
    )

+

+
  +-idempotentCommutativeMonoid : IdempotentCommutativeMonoid _ _

+
  +-idempotentCommutativeMonoid = record { isIdempotentCommutativeMonoid = +-isIdempotentCommutativeMonoid }

+

+
  open IdempotentCommutativeMonoid +-idempotentCommutativeMonoid public

+
    using ()

+
    renaming ( band to +-band

+
             ; commutativeBand to +-commutativeBand

+
             ; idempotentMonoid to +-idempotentMonoid

+
             )

+

+
record KleeneAlgebra c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 _⋆

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier               : Set c

+
    _≈_                   : Rel Carrier ℓ

+
    _+_                   : Op₂ Carrier

+
    _*_                   : Op₂ Carrier

+
    _⋆                    : Op₁ Carrier

+
    0#                    : Carrier

+
    1#                    : Carrier

+
    isKleeneAlgebra       : IsKleeneAlgebra _≈_ _+_ _*_ _⋆ 0# 1#

+

+
  open IsKleeneAlgebra isKleeneAlgebra public

+

+
  idempotentSemiring : IdempotentSemiring _ _

+
  idempotentSemiring = record { isIdempotentSemiring = isIdempotentSemiring }

+

+
  open IdempotentSemiring idempotentSemiring public

+
    using

+
    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-semigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid

+
    ; nearSemiring; semiringWithoutOne

+
    ; semiringWithoutAnnihilatingZero

+
    ; rawSemiring; semiring

+
    )

+

+
record Quasiring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier       : Set c

+
    _≈_           : Rel Carrier ℓ

+
    _+_           : Op₂ Carrier

+
    _*_           : Op₂ Carrier

+
    0#            : Carrier

+
    1#            : Carrier

+
    isQuasiring   : IsQuasiring _≈_ _+_ _*_ 0# 1#

+

+
  open IsQuasiring isQuasiring public

+

+
  +-monoid : Monoid _ _

+
  +-monoid = record { isMonoid = +-isMonoid }

+

+
  open Monoid +-monoid public

+
    using (_≉_) renaming

+
    ( rawMagma    to  +-rawMagma

+
    ; magma       to  +-magma

+
    ; semigroup   to  +-semigroup

+
    ; unitalMagma to  +-unitalMagma

+
    ; rawMonoid   to  +-rawMonoid

+
    )

+

+
  *-monoid : Monoid _ _

+
  *-monoid = record { isMonoid = *-isMonoid }

+

+
  open Monoid *-monoid public

+
    using () renaming

+
    ( rawMagma  to *-rawMagma

+
    ; magma     to *-magma

+
    ; semigroup to *-semigroup

+
    ; rawMonoid to *-rawMonoid

+
    )

+

+
------------------------------------------------------------------------

+
-- Bundles with 2 binary operations, 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record RingWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ

+
    _+_               : Op₂ Carrier

+
    _*_               : Op₂ Carrier

+
    -_                : Op₁ Carrier

+
    0#                : Carrier

+
    isRingWithoutOne  : IsRingWithoutOne _≈_ _+_ _*_ -_ 0#

+

+
  open IsRingWithoutOne isRingWithoutOne public

+

+
  +-abelianGroup : AbelianGroup _ _

+
  +-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }

+

+
  *-semigroup : Semigroup _ _

+
  *-semigroup = record { isSemigroup = *-isSemigroup }

+

+
  open AbelianGroup +-abelianGroup public

+
    using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)

+

+
  open Semigroup *-semigroup public

+
    using () renaming

+
    ( rawMagma to *-rawMagma

+
    ; magma    to *-magma

+
    )

+

+
------------------------------------------------------------------------

+
-- Bundles with 2 binary operations, 1 unary operation & 2 elements

+
------------------------------------------------------------------------

+

+
record NonAssociativeRing c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier               : Set c

+
    _≈_                   : Rel Carrier ℓ

+
    _+_                   : Op₂ Carrier

+
    _*_                   : Op₂ Carrier

+
    -_                    : Op₁ Carrier

+
    0#                    : Carrier

+
    1#                    : Carrier

+
    isNonAssociativeRing  : IsNonAssociativeRing _≈_ _+_ _*_ -_ 0# 1#

+

+
  open IsNonAssociativeRing isNonAssociativeRing public

+

+
  +-abelianGroup : AbelianGroup _ _

+
  +-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }

+

+
  open AbelianGroup +-abelianGroup public

+
    using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)

+

+
  *-unitalMagma : UnitalMagma _ _

+
  *-unitalMagma = record { isUnitalMagma = *-isUnitalMagma}

+

+
  open UnitalMagma *-unitalMagma public

+
    using () renaming (magma to *-magma; identity to *-identity)

+

+
record Nearring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier       : Set c

+
    _≈_           : Rel Carrier ℓ

+
    _+_           : Op₂ Carrier

+
    _*_           : Op₂ Carrier

+
    -_            : Op₁ Carrier

+
    0#            : Carrier

+
    1#            : Carrier

+
    isNearring    : IsNearring _≈_ _+_ _*_ 0# 1# -_

+

+
  open IsNearring isNearring public

+

+
  quasiring : Quasiring _ _

+
  quasiring = record { isQuasiring = isQuasiring }

+

+
  open Quasiring quasiring public

+
    using

+
    (_≉_; +-rawMagma; +-magma; +-unitalMagma; +-semigroup; +-monoid; +-rawMonoid

+
    ;*-rawMagma; *-magma; *-semigroup; *-monoid

+
    )

+

+

+
record Ring c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _+_     : Op₂ Carrier

+
    _*_     : Op₂ Carrier

+
    -_      : Op₁ Carrier

+
    0#      : Carrier

+
    1#      : Carrier

+
    isRing  : IsRing _≈_ _+_ _*_ -_ 0# 1#

+

+
  open IsRing isRing public

+

+
  +-abelianGroup : AbelianGroup _ _

+
  +-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }

+

+
  ringWithoutOne : RingWithoutOne _ _

+
  ringWithoutOne = record { isRingWithoutOne = isRingWithoutOne }

+

+
  semiring : Semiring _ _

+
  semiring = record { isSemiring = isSemiring }

+

+
  open Semiring semiring public

+
    using

+
    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-semigroup

+
    ; +-rawMonoid; +-monoid ; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid

+
    ; nearSemiring; semiringWithoutOne

+
    ; semiringWithoutAnnihilatingZero

+
    )

+

+
  open NearSemiring nearSemiring public

+
    using (rawNearSemiring)

+

+
  open AbelianGroup +-abelianGroup public

+
    using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)

+

+
  rawRing : RawRing _ _

+
  rawRing = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; -_  = -_

+
    ; 0#  = 0#

+
    ; 1#  = 1#

+
    }

+

+
  open RawRing rawRing public

+
    using (rawRingWithoutOne; +-rawGroup)

+

+

+
record CommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ

+
    _+_               : Op₂ Carrier

+
    _*_               : Op₂ Carrier

+
    -_                : Op₁ Carrier

+
    0#                : Carrier

+
    1#                : Carrier

+
    isCommutativeRing : IsCommutativeRing _≈_ _+_ _*_ -_ 0# 1#

+

+
  open IsCommutativeRing isCommutativeRing public

+

+
  ring : Ring _ _

+
  ring = record { isRing = isRing }

+

+
  open Ring ring public using (_≉_; rawRing; +-invertibleMagma; +-invertibleUnitalMagma; +-group; +-abelianGroup)

+

+
  commutativeSemiring : CommutativeSemiring _ _

+
  commutativeSemiring =

+
    record { isCommutativeSemiring = isCommutativeSemiring }

+

+
  open CommutativeSemiring commutativeSemiring public

+
    using

+
    ( +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma

+
    ; +-semigroup; +-commutativeSemigroup

+
    ; *-rawMagma; *-magma; *-commutativeMagma; *-semigroup; *-commutativeSemigroup

+
    ; +-rawMonoid; +-monoid; +-commutativeMonoid

+
    ; *-rawMonoid; *-monoid; *-commutativeMonoid

+
    ; nearSemiring; semiringWithoutOne

+
    ; semiringWithoutAnnihilatingZero; semiring

+
    ; commutativeSemiringWithoutOne

+
    )

+

+
------------------------------------------------------------------------

+
-- Bundles with 3 binary operations

+
------------------------------------------------------------------------

+

+
record Quasigroup c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier      : Set c

+
    _≈_          : Rel Carrier ℓ

+
    _∙_          : Op₂ Carrier

+
    _\\_         : Op₂ Carrier

+
    _//_         : Op₂ Carrier

+
    isQuasigroup : IsQuasigroup  _≈_ _∙_ _\\_ _//_

+

+
  open IsQuasigroup isQuasigroup public

+

+
  magma : Magma c ℓ

+
  magma = record { isMagma = isMagma }

+

+
  open Magma magma public

+
    using (_≉_; rawMagma)

+

+
  rawQuasigroup : RawQuasigroup c ℓ

+
  rawQuasigroup = record

+
    { _≈_  = _≈_

+
    ; _∙_  = _∙_

+
    ; _\\_  = _\\_

+
    ; _//_  = _//_

+
    }

+

+
  open RawQuasigroup rawQuasigroup public

+
    using (//-rawMagma; \\-rawMagma; ∙-rawMagma)

+

+
record Loop  c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    _\\_    : Op₂ Carrier

+
    _//_    : Op₂ Carrier

+
    ε       : Carrier

+
    isLoop  : IsLoop  _≈_ _∙_ _\\_ _//_ ε

+

+
  open IsLoop isLoop public

+

+
  rawLoop : RawLoop c ℓ

+
  rawLoop = record

+
    { _≈_ = _≈_

+
    ; _∙_ = _∙_

+
    ; _\\_ = _\\_

+
    ; _//_ = _//_

+
    ; ε = ε

+
    }

+

+
  quasigroup : Quasigroup _ _

+
  quasigroup = record { isQuasigroup = isQuasigroup }

+

+
  open Quasigroup quasigroup public

+
    using (_≉_; ∙-rawMagma; \\-rawMagma; //-rawMagma)

+

+
record LeftBolLoop c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    _\\_    : Op₂ Carrier

+
    _//_    : Op₂ Carrier

+
    ε       : Carrier

+
    isLeftBolLoop : IsLeftBolLoop  _≈_ _∙_ _\\_ _//_ ε

+

+
  open IsLeftBolLoop isLeftBolLoop public

+

+
  loop : Loop _ _

+
  loop = record { isLoop = isLoop }

+

+
  open Loop loop public

+
    using (quasigroup)

+

+
record RightBolLoop c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    _\\_    : Op₂ Carrier

+
    _//_    : Op₂ Carrier

+
    ε       : Carrier

+
    isRightBolLoop : IsRightBolLoop  _≈_ _∙_ _\\_ _//_ ε

+

+
  open IsRightBolLoop isRightBolLoop public

+

+
  loop : Loop _ _

+
  loop = record { isLoop = isLoop }

+

+
  open Loop loop public

+
    using (quasigroup)

+

+
record MoufangLoop c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∙_     : Op₂ Carrier

+
    _\\_    : Op₂ Carrier

+
    _//_    : Op₂ Carrier

+
    ε       : Carrier

+
    isMoufangLoop : IsMoufangLoop  _≈_ _∙_ _\\_ _//_ ε

+

+
  open IsMoufangLoop isMoufangLoop public

+

+
  leftBolLoop : LeftBolLoop _ _

+
  leftBolLoop = record { isLeftBolLoop = isLeftBolLoop }

+

+
  open LeftBolLoop leftBolLoop public

+
    using (loop)

+

+
record MiddleBolLoop c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixl 7 _∙_

+
  infixl 7 _\\_

+
  infixl 7 _//_

+
  infix  4 _≈_

+
  field

+
    Carrier         : Set c

+
    _≈_             : Rel Carrier ℓ

+
    _∙_             : Op₂ Carrier

+
    _\\_            : Op₂ Carrier

+
    _//_            : Op₂ Carrier

+
    ε               : Carrier

+
    isMiddleBolLoop : IsMiddleBolLoop  _≈_ _∙_ _\\_ _//_ ε

+

+
  open IsMiddleBolLoop isMiddleBolLoop public

+

+
  loop : Loop _ _

+
  loop = record { isLoop = isLoop }

+

+
  open Loop loop public

+
    using (quasigroup)

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Consequences.Base.html b/site/static/agda_html/Algebra.Consequences.Base.html
new file mode 100644
index 000000000..ada04aeb4
--- /dev/null
+++ b/site/static/agda_html/Algebra.Consequences.Base.html
@@ -0,0 +1,120 @@
+
+Algebra.Consequences.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lemmas relating algebraic definitions (such as associativity and

+
-- commutativity) that don't require the equality relation to be a setoid.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Consequences.Base

+
  {a} {A : Set a} where

+

+
open import Algebra.Core

+
open import Algebra.Definitions

+
open import Data.Sum.Base

+
open import Relation.Binary.Core

+
open import Relation.Binary.Definitions using (Reflexive)

+

+
module _ {ℓ} {_•_ : Op₂ A} (_≈_ : Rel A ℓ) where

+

+
  sel⇒idem : Selective _≈_ _•_ → Idempotent _≈_ _•_

+
  sel⇒idem sel x = reduce (sel x x)

+

+
module _ {ℓ} {f : Op₁ A} (_≈_ : Rel A ℓ) where

+

+
  reflexive∧selfInverse⇒involutive : Reflexive _≈_ →

+
                                     SelfInverse _≈_ f →

+
                                     Involutive _≈_ f

+
  reflexive∧selfInverse⇒involutive refl inv _ = inv refl

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
reflexive+selfInverse⇒involutive = reflexive∧selfInverse⇒involutive

+
{-# WARNING_ON_USAGE reflexive+selfInverse⇒involutive

+
"Warning: reflexive+selfInverse⇒involutive was deprecated in v2.0.

+
Please use reflexive∧selfInverse⇒involutive instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Consequences.Propositional.html b/site/static/agda_html/Algebra.Consequences.Propositional.html
new file mode 100644
index 000000000..9fdab02cb
--- /dev/null
+++ b/site/static/agda_html/Algebra.Consequences.Propositional.html
@@ -0,0 +1,281 @@
+
+Algebra.Consequences.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Relations between properties of functions, such as associativity and

+
-- commutativity (specialised to propositional equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Consequences.Propositional

+
  {a} {A : Set a} where

+

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (Symmetric; Total)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; cong₂; subst)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid)

+
open import Relation.Unary using (Pred)

+

+
open import Algebra.Core

+
open import Algebra.Definitions {A = A} _≡_

+
import Algebra.Consequences.Setoid (setoid A) as Base

+

+
------------------------------------------------------------------------

+
-- Re-export all proofs that don't require congruence or substitutivity

+

+
open Base public

+
  hiding

+
  ( comm∧assoc⇒middleFour

+
  ; identity∧middleFour⇒assoc

+
  ; identity∧middleFour⇒comm

+
  ; assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ

+
  ; assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ

+
  ; assoc∧id∧invʳ⇒invˡ-unique

+
  ; assoc∧id∧invˡ⇒invʳ-unique

+
  ; comm∧distrˡ⇒distrʳ

+
  ; comm∧distrʳ⇒distrˡ

+
  ; comm⇒sym[distribˡ]

+
  ; subst∧comm⇒sym

+
  ; wlog

+
  ; sel⇒idem

+
-- plus all the deprecated versions of the above

+
  ; comm+assoc⇒middleFour

+
  ; identity+middleFour⇒assoc

+
  ; identity+middleFour⇒comm

+
  ; assoc+distribʳ+idʳ+invʳ⇒zeˡ

+
  ; assoc+distribˡ+idʳ+invʳ⇒zeʳ

+
  ; assoc+id+invʳ⇒invˡ-unique

+
  ; assoc+id+invˡ⇒invʳ-unique

+
  ; comm+distrˡ⇒distrʳ

+
  ; comm+distrʳ⇒distrˡ

+
  ; subst+comm⇒sym

+
  )

+

+
------------------------------------------------------------------------

+
-- Group-like structures

+

+
module _ {_∙_ _⁻¹ ε} where

+

+
  assoc∧id∧invʳ⇒invˡ-unique : Associative _∙_ → Identity ε _∙_ →

+
                              RightInverse ε _⁻¹ _∙_ →

+
                              ∀ x y → (x ∙ y) ≡ ε → x ≡ (y ⁻¹)

+
  assoc∧id∧invʳ⇒invˡ-unique = Base.assoc∧id∧invʳ⇒invˡ-unique (cong₂ _)

+

+
  assoc∧id∧invˡ⇒invʳ-unique : Associative _∙_ → Identity ε _∙_ →

+
                              LeftInverse ε _⁻¹ _∙_ →

+
                              ∀ x y → (x ∙ y) ≡ ε → y ≡ (x ⁻¹)

+
  assoc∧id∧invˡ⇒invʳ-unique = Base.assoc∧id∧invˡ⇒invʳ-unique (cong₂ _)

+

+
------------------------------------------------------------------------

+
-- Ring-like structures

+

+
module _ {_+_ _*_ -_ 0#} where

+

+
  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →

+
                                RightIdentity 0# _+_ → RightInverse 0# -_ _+_ →

+
                                LeftZero 0# _*_

+
  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ =

+
    Base.assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ (cong₂ _+_) (cong₂ _*_)

+

+
  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →

+
                                RightIdentity 0# _+_ → RightInverse 0# -_ _+_ →

+
                                RightZero 0# _*_

+
  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ =

+
    Base.assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ (cong₂ _+_) (cong₂ _*_)

+

+
------------------------------------------------------------------------

+
-- Bisemigroup-like structures

+

+
module _ {_∙_ _◦_ : Op₂ A} (∙-comm : Commutative _∙_) where

+

+
  comm∧distrˡ⇒distrʳ : _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_

+
  comm∧distrˡ⇒distrʳ = Base.comm+distrˡ⇒distrʳ (cong₂ _) ∙-comm

+

+
  comm∧distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_

+
  comm∧distrʳ⇒distrˡ = Base.comm∧distrʳ⇒distrˡ (cong₂ _) ∙-comm

+

+
  comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≡ ((x ◦ y) ∙ (x ◦ z)))

+
  comm⇒sym[distribˡ] = Base.comm⇒sym[distribˡ] (cong₂ _◦_) ∙-comm

+

+
------------------------------------------------------------------------

+
-- Selectivity

+

+
module _ {_∙_ : Op₂ A} where

+

+
  sel⇒idem : Selective _∙_ → Idempotent _∙_

+
  sel⇒idem = Base.sel⇒idem _≡_

+

+
------------------------------------------------------------------------

+
-- Middle-Four Exchange

+

+
module _ {_∙_ : Op₂ A} where

+

+
  comm∧assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →

+
                          _∙_ MiddleFourExchange _∙_

+
  comm∧assoc⇒middleFour = Base.comm∧assoc⇒middleFour (cong₂ _∙_)

+

+
  identity∧middleFour⇒assoc : {e : A} → Identity e _∙_ →

+
                              _∙_ MiddleFourExchange _∙_ →

+
                              Associative _∙_

+
  identity∧middleFour⇒assoc = Base.identity∧middleFour⇒assoc (cong₂ _∙_)

+

+
  identity∧middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →

+
                             _∙_ MiddleFourExchange _+_ →

+
                             Commutative _∙_

+
  identity∧middleFour⇒comm = Base.identity∧middleFour⇒comm (cong₂ _∙_)

+

+
------------------------------------------------------------------------

+
-- Without Loss of Generality

+

+
module _ {p} {P : Pred A p} where

+

+
  subst∧comm⇒sym : ∀ {f} (f-comm : Commutative f) →

+
                   Symmetric (λ a b → P (f a b))

+
  subst∧comm⇒sym = Base.subst∧comm⇒sym {P = P} subst

+

+
  wlog : ∀ {f} (f-comm : Commutative f) →

+
         ∀ {r} {_R_ : Rel _ r} → Total _R_ →

+
         (∀ a b → a R b → P (f a b)) →

+
         ∀ a b → P (f a b)

+
  wlog = Base.wlog {P = P} subst

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
comm+assoc⇒middleFour = comm∧assoc⇒middleFour

+
{-# WARNING_ON_USAGE comm+assoc⇒middleFour

+
"Warning: comm+assoc⇒middleFour was deprecated in v2.0.

+
Please use comm∧assoc⇒middleFour instead."

+
#-}

+
identity+middleFour⇒assoc = identity∧middleFour⇒assoc

+
{-# WARNING_ON_USAGE identity+middleFour⇒assoc

+
"Warning: identity+middleFour⇒assoc was deprecated in v2.0.

+
Please use identity∧middleFour⇒assoc instead."

+
#-}

+
identity+middleFour⇒comm = identity∧middleFour⇒comm

+
{-# WARNING_ON_USAGE identity+middleFour⇒comm

+
"Warning: identity+middleFour⇒comm was deprecated in v2.0.

+
Please use identity∧middleFour⇒comm instead."

+
#-}

+
comm+distrˡ⇒distrʳ = comm∧distrˡ⇒distrʳ

+
{-# WARNING_ON_USAGE comm+distrˡ⇒distrʳ

+
"Warning: comm+distrˡ⇒distrʳ was deprecated in v2.0.

+
Please use comm∧distrˡ⇒distrʳ instead."

+
#-}

+
comm+distrʳ⇒distrˡ = comm∧distrʳ⇒distrˡ

+
{-# WARNING_ON_USAGE comm+distrʳ⇒distrˡ

+
"Warning: comm+distrʳ⇒distrˡ was deprecated in v2.0.

+
Please use comm∧distrʳ⇒distrˡ instead."

+
#-}

+
assoc+distribʳ+idʳ+invʳ⇒zeˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ

+
{-# WARNING_ON_USAGE assoc+distribʳ+idʳ+invʳ⇒zeˡ

+
"Warning: assoc+distribʳ+idʳ+invʳ⇒zeˡ was deprecated in v2.0.

+
Please use assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ instead."

+
#-}

+
assoc+distribˡ+idʳ+invʳ⇒zeʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ

+
{-# WARNING_ON_USAGE assoc+distribˡ+idʳ+invʳ⇒zeʳ

+
"Warning: assoc+distribˡ+idʳ+invʳ⇒zeʳ was deprecated in v2.0.

+
Please use assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ instead."

+
#-}

+
assoc+id+invʳ⇒invˡ-unique = assoc∧id∧invʳ⇒invˡ-unique

+
{-# WARNING_ON_USAGE assoc+id+invʳ⇒invˡ-unique

+
"Warning: assoc+id+invʳ⇒invˡ-unique was deprecated in v2.0.

+
Please use assoc∧id∧invʳ⇒invˡ-unique instead."

+
#-}

+
assoc+id+invˡ⇒invʳ-unique = assoc∧id∧invˡ⇒invʳ-unique

+
{-# WARNING_ON_USAGE assoc+id+invˡ⇒invʳ-unique

+
"Warning: assoc+id+invˡ⇒invʳ-unique was deprecated in v2.0.

+
Please use assoc∧id∧invˡ⇒invʳ-unique instead."

+
#-}

+
subst+comm⇒sym = subst∧comm⇒sym

+
{-# WARNING_ON_USAGE subst+comm⇒sym

+
"Warning: subst+comm⇒sym was deprecated in v2.0.

+
Please use subst∧comm⇒sym instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Consequences.Setoid.html b/site/static/agda_html/Algebra.Consequences.Setoid.html
new file mode 100644
index 000000000..0af361c49
--- /dev/null
+++ b/site/static/agda_html/Algebra.Consequences.Setoid.html
@@ -0,0 +1,522 @@
+
+Algebra.Consequences.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Relations between properties of functions, such as associativity and

+
-- commutativity, when the underlying relation is a setoid

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions

+
  using (Substitutive; Symmetric; Total)

+

+
module Algebra.Consequences.Setoid {a ℓ} (S : Setoid a ℓ) where

+

+
open Setoid S renaming (Carrier to A)

+
open import Algebra.Core

+
open import Algebra.Definitions _≈_

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_$_; id; _∘_)

+
open import Function.Definitions

+
import Relation.Binary.Consequences as Bin

+
open import Relation.Binary.Reasoning.Setoid S

+
open import Relation.Unary using (Pred)

+

+
------------------------------------------------------------------------

+
-- Re-exports

+

+
-- Export base lemmas that don't require the setoid

+

+
open import Algebra.Consequences.Base public

+

+
------------------------------------------------------------------------

+
-- MiddleFourExchange

+

+
module _ {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where

+

+
  comm∧assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →

+
                          _∙_ MiddleFourExchange _∙_

+
  comm∧assoc⇒middleFour comm assoc w x y z = begin

+
    (w ∙ x) ∙ (y ∙ z) ≈⟨ assoc w x (y ∙ z) ⟩

+
    w ∙ (x ∙ (y ∙ z)) ≈⟨ cong refl (sym (assoc x y z)) ⟩

+
    w ∙ ((x ∙ y) ∙ z) ≈⟨ cong refl (cong (comm x y) refl) ⟩

+
    w ∙ ((y ∙ x) ∙ z) ≈⟨ cong refl (assoc y x z) ⟩

+
    w ∙ (y ∙ (x ∙ z)) ≈⟨ sym (assoc w y (x ∙ z)) ⟩

+
    (w ∙ y) ∙ (x ∙ z) ∎

+

+
  identity∧middleFour⇒assoc : {e : A} → Identity e _∙_ →

+
                              _∙_ MiddleFourExchange _∙_ →

+
                              Associative _∙_

+
  identity∧middleFour⇒assoc {e} (identityˡ , identityʳ) middleFour x y z = begin

+
    (x ∙ y) ∙ z       ≈⟨ cong refl (sym (identityˡ z)) ⟩

+
    (x ∙ y) ∙ (e ∙ z) ≈⟨ middleFour x y e z ⟩

+
    (x ∙ e) ∙ (y ∙ z) ≈⟨ cong (identityʳ x) refl ⟩

+
    x ∙ (y ∙ z)       ∎

+

+
  identity∧middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →

+
                             _∙_ MiddleFourExchange _+_ →

+
                             Commutative _∙_

+
  identity∧middleFour⇒comm {_+_} {e} (identityˡ , identityʳ) middleFour x y

+
    = begin

+
    x ∙ y             ≈⟨ sym (cong (identityˡ x) (identityʳ y)) ⟩

+
    (e + x) ∙ (y + e) ≈⟨ middleFour e x y e ⟩

+
    (e + y) ∙ (x + e) ≈⟨ cong (identityˡ y) (identityʳ x) ⟩

+
    y ∙ x             ∎

+

+
------------------------------------------------------------------------

+
-- SelfInverse

+

+
module _ {f : Op₁ A} (self : SelfInverse f) where

+

+
  selfInverse⇒involutive : Involutive f

+
  selfInverse⇒involutive = reflexive∧selfInverse⇒involutive _≈_ refl self

+

+
  selfInverse⇒congruent : Congruent _≈_ _≈_ f

+
  selfInverse⇒congruent {x} {y} x≈y = sym (self (begin

+
    f (f x) ≈⟨ selfInverse⇒involutive x ⟩

+
    x       ≈⟨ x≈y ⟩

+
    y       ∎))

+

+
  selfInverse⇒inverseᵇ : Inverseᵇ _≈_ _≈_ f f

+
  selfInverse⇒inverseᵇ = self ∘ sym , self ∘ sym

+

+
  selfInverse⇒surjective : Surjective _≈_ _≈_ f

+
  selfInverse⇒surjective y = f y , self ∘ sym

+

+
  selfInverse⇒injective : Injective _≈_ _≈_ f

+
  selfInverse⇒injective {x} {y} x≈y = begin

+
    x       ≈⟨ self x≈y ⟨

+
    f (f y) ≈⟨ selfInverse⇒involutive y ⟩

+
    y       ∎

+

+
  selfInverse⇒bijective : Bijective _≈_ _≈_ f

+
  selfInverse⇒bijective = selfInverse⇒injective , selfInverse⇒surjective

+

+
------------------------------------------------------------------------

+
-- Magma-like structures

+

+
module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) where

+

+
  comm∧cancelˡ⇒cancelʳ : LeftCancellative _∙_ → RightCancellative _∙_

+
  comm∧cancelˡ⇒cancelʳ cancelˡ x y z eq = cancelˡ x y z $ begin

+
    x ∙ y ≈⟨ comm x y ⟩

+
    y ∙ x ≈⟨ eq ⟩

+
    z ∙ x ≈⟨ comm z x ⟩

+
    x ∙ z ∎

+

+
  comm∧cancelʳ⇒cancelˡ : RightCancellative _∙_ → LeftCancellative _∙_

+
  comm∧cancelʳ⇒cancelˡ cancelʳ x y z eq = cancelʳ x y z $ begin

+
    y ∙ x ≈⟨ comm y x ⟩

+
    x ∙ y ≈⟨ eq ⟩

+
    x ∙ z ≈⟨ comm x z ⟩

+
    z ∙ x ∎

+

+
------------------------------------------------------------------------

+
-- Monoid-like structures

+

+
module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) {e : A} where

+

+
  comm∧idˡ⇒idʳ : LeftIdentity e _∙_ → RightIdentity e _∙_

+
  comm∧idˡ⇒idʳ idˡ x = begin

+
    x ∙ e ≈⟨ comm x e ⟩

+
    e ∙ x ≈⟨ idˡ x ⟩

+
    x     ∎

+

+
  comm∧idʳ⇒idˡ : RightIdentity e _∙_ → LeftIdentity e _∙_

+
  comm∧idʳ⇒idˡ idʳ x = begin

+
    e ∙ x ≈⟨ comm e x ⟩

+
    x ∙ e ≈⟨ idʳ x ⟩

+
    x     ∎

+

+
  comm∧idˡ⇒id : LeftIdentity e _∙_ → Identity e _∙_

+
  comm∧idˡ⇒id idˡ = idˡ , comm∧idˡ⇒idʳ idˡ

+

+
  comm∧idʳ⇒id : RightIdentity e _∙_ → Identity e _∙_

+
  comm∧idʳ⇒id idʳ = comm∧idʳ⇒idˡ idʳ , idʳ

+

+
  comm∧zeˡ⇒zeʳ : LeftZero e _∙_ → RightZero e _∙_

+
  comm∧zeˡ⇒zeʳ zeˡ x = begin

+
    x ∙ e ≈⟨ comm x e ⟩

+
    e ∙ x ≈⟨ zeˡ x ⟩

+
    e     ∎

+

+
  comm∧zeʳ⇒zeˡ : RightZero e _∙_ → LeftZero e _∙_

+
  comm∧zeʳ⇒zeˡ zeʳ x = begin

+
    e ∙ x ≈⟨ comm e x ⟩

+
    x ∙ e ≈⟨ zeʳ x ⟩

+
    e     ∎

+

+
  comm∧zeˡ⇒ze : LeftZero e _∙_ → Zero e _∙_

+
  comm∧zeˡ⇒ze zeˡ = zeˡ , comm∧zeˡ⇒zeʳ zeˡ

+

+
  comm∧zeʳ⇒ze : RightZero e _∙_ → Zero e _∙_

+
  comm∧zeʳ⇒ze zeʳ = comm∧zeʳ⇒zeˡ zeʳ , zeʳ

+

+
  comm∧almostCancelˡ⇒almostCancelʳ : AlmostLeftCancellative e _∙_ →

+
                                     AlmostRightCancellative e _∙_

+
  comm∧almostCancelˡ⇒almostCancelʳ cancelˡ-nonZero x y z x≉e yx≈zx =

+
    cancelˡ-nonZero x y z x≉e $ begin

+
      x ∙ y ≈⟨ comm x y ⟩

+
      y ∙ x ≈⟨ yx≈zx ⟩

+
      z ∙ x ≈⟨ comm z x ⟩

+
      x ∙ z ∎

+

+
  comm∧almostCancelʳ⇒almostCancelˡ : AlmostRightCancellative e _∙_ →

+
                                     AlmostLeftCancellative e _∙_

+
  comm∧almostCancelʳ⇒almostCancelˡ cancelʳ-nonZero x y z x≉e xy≈xz =

+
    cancelʳ-nonZero x y z x≉e $ begin

+
      y ∙ x ≈⟨ comm y x ⟩

+
      x ∙ y ≈⟨ xy≈xz ⟩

+
      x ∙ z ≈⟨ comm x z ⟩

+
      z ∙ x ∎

+

+
------------------------------------------------------------------------

+
-- Group-like structures

+

+
module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (comm : Commutative _∙_) where

+

+
  comm∧invˡ⇒invʳ : LeftInverse e _⁻¹ _∙_ → RightInverse e _⁻¹ _∙_

+
  comm∧invˡ⇒invʳ invˡ x = begin

+
    x ∙ (x ⁻¹) ≈⟨ comm x (x ⁻¹) ⟩

+
    (x ⁻¹) ∙ x ≈⟨ invˡ x ⟩

+
    e          ∎

+

+
  comm∧invˡ⇒inv : LeftInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_

+
  comm∧invˡ⇒inv invˡ = invˡ , comm∧invˡ⇒invʳ invˡ

+

+
  comm∧invʳ⇒invˡ : RightInverse e _⁻¹ _∙_ → LeftInverse e _⁻¹ _∙_

+
  comm∧invʳ⇒invˡ invʳ x = begin

+
    (x ⁻¹) ∙ x ≈⟨ comm (x ⁻¹) x ⟩

+
    x ∙ (x ⁻¹) ≈⟨ invʳ x ⟩

+
    e          ∎

+

+
  comm∧invʳ⇒inv : RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_

+
  comm∧invʳ⇒inv invʳ = comm∧invʳ⇒invˡ invʳ , invʳ

+

+
module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _∙_) where

+

+
  assoc∧id∧invʳ⇒invˡ-unique : Associative _∙_ →

+
                              Identity e _∙_ → RightInverse e _⁻¹ _∙_ →

+
                              ∀ x y → (x ∙ y) ≈ e → x ≈ (y ⁻¹)

+
  assoc∧id∧invʳ⇒invˡ-unique assoc (idˡ , idʳ) invʳ x y eq = begin

+
    x                ≈⟨ sym (idʳ x) ⟩

+
    x ∙ e            ≈⟨ cong refl (sym (invʳ y)) ⟩

+
    x ∙ (y ∙ (y ⁻¹)) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩

+
    (x ∙ y) ∙ (y ⁻¹) ≈⟨ cong eq refl ⟩

+
    e ∙ (y ⁻¹)       ≈⟨ idˡ (y ⁻¹) ⟩

+
    y ⁻¹             ∎

+

+
  assoc∧id∧invˡ⇒invʳ-unique : Associative _∙_ →

+
                              Identity e _∙_ → LeftInverse e _⁻¹ _∙_ →

+
                              ∀ x y → (x ∙ y) ≈ e → y ≈ (x ⁻¹)

+
  assoc∧id∧invˡ⇒invʳ-unique assoc (idˡ , idʳ) invˡ x y eq = begin

+
    y                ≈⟨ sym (idˡ y) ⟩

+
    e ∙ y            ≈⟨ cong (sym (invˡ x)) refl ⟩

+
    ((x ⁻¹) ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩

+
    (x ⁻¹) ∙ (x ∙ y) ≈⟨ cong refl eq ⟩

+
    (x ⁻¹) ∙ e       ≈⟨ idʳ (x ⁻¹) ⟩

+
    x ⁻¹             ∎

+

+
------------------------------------------------------------------------

+
-- Bisemigroup-like structures

+

+
module _ {_∙_ _◦_ : Op₂ A}

+
         (◦-cong : Congruent₂ _◦_)

+
         (∙-comm : Commutative _∙_)

+
         where

+

+
  comm∧distrˡ⇒distrʳ :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_

+
  comm∧distrˡ⇒distrʳ distrˡ x y z = begin

+
    (y ◦ z) ∙ x       ≈⟨ ∙-comm (y ◦ z) x ⟩

+
    x ∙ (y ◦ z)       ≈⟨ distrˡ x y z ⟩

+
    (x ∙ y) ◦ (x ∙ z) ≈⟨ ◦-cong (∙-comm x y) (∙-comm x z) ⟩

+
    (y ∙ x) ◦ (z ∙ x) ∎

+

+
  comm∧distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_

+
  comm∧distrʳ⇒distrˡ distrˡ x y z = begin

+
    x ∙ (y ◦ z)       ≈⟨ ∙-comm x (y ◦ z) ⟩

+
    (y ◦ z) ∙ x       ≈⟨ distrˡ x y z ⟩

+
    (y ∙ x) ◦ (z ∙ x) ≈⟨ ◦-cong (∙-comm y x) (∙-comm z x) ⟩

+
    (x ∙ y) ◦ (x ∙ z) ∎

+

+
  comm∧distrˡ⇒distr :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_

+
  comm∧distrˡ⇒distr distrˡ = distrˡ , comm∧distrˡ⇒distrʳ distrˡ

+

+
  comm∧distrʳ⇒distr :  _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_

+
  comm∧distrʳ⇒distr distrʳ = comm∧distrʳ⇒distrˡ distrʳ , distrʳ

+

+
  comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≈ ((x ◦ y) ∙ (x ◦ z)))

+
  comm⇒sym[distribˡ] x {y} {z} prf = begin

+
    x ◦ (z ∙ y)       ≈⟨ ◦-cong refl (∙-comm z y) ⟩

+
    x ◦ (y ∙ z)       ≈⟨ prf ⟩

+
    (x ◦ y) ∙ (x ◦ z) ≈⟨ ∙-comm (x ◦ y) (x ◦ z) ⟩

+
    (x ◦ z) ∙ (x ◦ y) ∎

+

+

+
module _ {_∙_ _◦_ : Op₂ A}

+
         (∙-cong  : Congruent₂ _∙_)

+
         (∙-assoc : Associative _∙_)

+
         (◦-comm  : Commutative _◦_)

+
         where

+

+
  distrib∧absorbs⇒distribˡ : _∙_ Absorbs _◦_ →

+
                             _◦_ Absorbs _∙_ →

+
                             _◦_ DistributesOver _∙_ →

+
                             _∙_ DistributesOverˡ _◦_

+
  distrib∧absorbs⇒distribˡ ∙-absorbs-◦ ◦-absorbs-∙ (◦-distribˡ-∙ , ◦-distribʳ-∙) x y z = begin

+
    x ∙ (y ◦ z)                    ≈⟨ ∙-cong (∙-absorbs-◦ _ _) refl ⟨

+
    (x ∙ (x ◦ y)) ∙ (y ◦ z)        ≈⟨  ∙-cong (∙-cong refl (◦-comm _ _)) refl ⟩

+
    (x ∙ (y ◦ x)) ∙ (y ◦ z)        ≈⟨  ∙-assoc _ _ _ ⟩

+
    x ∙ ((y ◦ x) ∙ (y ◦ z))        ≈⟨ ∙-cong refl (◦-distribˡ-∙ _ _ _) ⟨

+
    x ∙ (y ◦ (x ∙ z))              ≈⟨ ∙-cong (◦-absorbs-∙ _ _) refl ⟨

+
    (x ◦ (x ∙ z)) ∙ (y ◦ (x ∙ z))  ≈⟨ ◦-distribʳ-∙ _ _ _ ⟨

+
    (x ∙ y) ◦ (x ∙ z)              ∎

+

+
------------------------------------------------------------------------

+
-- Ring-like structures

+

+
module _ {_+_ _*_ : Op₂ A}

+
         {_⁻¹ : Op₁ A} {0# : A}

+
         (+-cong : Congruent₂ _+_)

+
         (*-cong : Congruent₂ _*_)

+
         where

+

+
  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →

+
                                RightIdentity 0# _+_ → RightInverse 0# _⁻¹ _+_ →

+
                                LeftZero 0# _*_

+
  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ +-assoc distribʳ idʳ invʳ  x = begin

+
    0# * x                                 ≈⟨ sym (idʳ _) ⟩

+
    (0# * x) + 0#                          ≈⟨ +-cong refl (sym (invʳ _)) ⟩

+
    (0# * x) + ((0# * x)  + ((0# * x)⁻¹))  ≈⟨ sym (+-assoc _ _ _) ⟩

+
    ((0# * x) +  (0# * x)) + ((0# * x)⁻¹)  ≈⟨ +-cong (sym (distribʳ _ _ _)) refl ⟩

+
    ((0# + 0#) * x) + ((0# * x)⁻¹)         ≈⟨ +-cong (*-cong (idʳ _) refl) refl ⟩

+
    (0# * x) + ((0# * x)⁻¹)                ≈⟨ invʳ _ ⟩

+
    0#                                     ∎

+

+
  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →

+
                                RightIdentity 0# _+_ → RightInverse 0# _⁻¹ _+_ →

+
                                RightZero 0# _*_

+
  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ +-assoc distribˡ idʳ invʳ  x = begin

+
     x * 0#                                ≈⟨ sym (idʳ _) ⟩

+
     (x * 0#) + 0#                         ≈⟨ +-cong refl (sym (invʳ _)) ⟩

+
     (x * 0#) + ((x * 0#) + ((x * 0#)⁻¹))  ≈⟨ sym (+-assoc _ _ _) ⟩

+
     ((x * 0#) + (x * 0#)) + ((x * 0#)⁻¹)  ≈⟨ +-cong (sym (distribˡ _ _ _)) refl ⟩

+
     (x * (0# + 0#)) + ((x * 0#)⁻¹)        ≈⟨ +-cong (*-cong refl (idʳ _)) refl ⟩

+
     ((x * 0#) + ((x * 0#)⁻¹))             ≈⟨ invʳ _ ⟩

+
     0#                                    ∎

+

+
------------------------------------------------------------------------

+
-- Without Loss of Generality

+

+
module _ {p} {f : Op₂ A} {P : Pred A p}

+
         (≈-subst : Substitutive _≈_ p)

+
         (comm : Commutative f)

+
         where

+

+
  subst∧comm⇒sym : Symmetric (λ a b → P (f a b))

+
  subst∧comm⇒sym = ≈-subst P (comm _ _)

+

+
  wlog : ∀ {r} {_R_ : Rel _ r} → Total _R_ →

+
         (∀ a b → a R b → P (f a b)) →

+
         ∀ a b → P (f a b)

+
  wlog r-total = Bin.wlog r-total subst∧comm⇒sym

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
comm+assoc⇒middleFour = comm∧assoc⇒middleFour

+
{-# WARNING_ON_USAGE comm+assoc⇒middleFour

+
"Warning: comm+assoc⇒middleFour was deprecated in v2.0.

+
Please use comm∧assoc⇒middleFour instead."

+
#-}

+
identity+middleFour⇒assoc = identity∧middleFour⇒assoc

+
{-# WARNING_ON_USAGE identity+middleFour⇒assoc

+
"Warning: identity+middleFour⇒assoc was deprecated in v2.0.

+
Please use identity∧middleFour⇒assoc instead."

+
#-}

+
identity+middleFour⇒comm = identity∧middleFour⇒comm

+
{-# WARNING_ON_USAGE identity+middleFour⇒comm

+
"Warning: identity+middleFour⇒comm was deprecated in v2.0.

+
Please use identity∧middleFour⇒comm instead."

+
#-}

+
comm+cancelˡ⇒cancelʳ = comm∧cancelˡ⇒cancelʳ

+
{-# WARNING_ON_USAGE comm+cancelˡ⇒cancelʳ

+
"Warning: comm+cancelˡ⇒cancelʳ was deprecated in v2.0.

+
Please use comm∧cancelˡ⇒cancelʳ instead."

+
#-}

+
comm+cancelʳ⇒cancelˡ = comm∧cancelʳ⇒cancelˡ

+
{-# WARNING_ON_USAGE comm+cancelʳ⇒cancelˡ

+
"Warning: comm+cancelʳ⇒cancelˡ was deprecated in v2.0.

+
Please use comm∧cancelʳ⇒cancelˡ instead."

+
#-}

+
comm+idˡ⇒idʳ = comm∧idˡ⇒idʳ

+
{-# WARNING_ON_USAGE comm+idˡ⇒idʳ

+
"Warning: comm+idˡ⇒idʳ was deprecated in v2.0.

+
Please use comm∧idˡ⇒idʳ instead."

+
#-}

+
comm+idʳ⇒idˡ = comm∧idʳ⇒idˡ

+
{-# WARNING_ON_USAGE comm+idʳ⇒idˡ

+
"Warning: comm+idʳ⇒idˡ was deprecated in v2.0.

+
Please use comm∧idʳ⇒idˡ instead."

+
#-}

+
comm+zeˡ⇒zeʳ = comm∧zeˡ⇒zeʳ

+
{-# WARNING_ON_USAGE comm+zeˡ⇒zeʳ

+
"Warning: comm+zeˡ⇒zeʳ was deprecated in v2.0.

+
Please use comm∧zeˡ⇒zeʳ instead."

+
#-}

+
comm+zeʳ⇒zeˡ = comm∧zeʳ⇒zeˡ

+
{-# WARNING_ON_USAGE comm+zeʳ⇒zeˡ

+
"Warning: comm+zeʳ⇒zeˡ was deprecated in v2.0.

+
Please use comm∧zeʳ⇒zeˡ instead."

+
#-}

+
comm+almostCancelˡ⇒almostCancelʳ = comm∧almostCancelˡ⇒almostCancelʳ

+
{-# WARNING_ON_USAGE comm+almostCancelˡ⇒almostCancelʳ

+
"Warning: comm+almostCancelˡ⇒almostCancelʳ was deprecated in v2.0.

+
Please use comm∧almostCancelˡ⇒almostCancelʳ instead."

+
#-}

+
comm+almostCancelʳ⇒almostCancelˡ = comm∧almostCancelʳ⇒almostCancelˡ

+
{-# WARNING_ON_USAGE comm+almostCancelʳ⇒almostCancelˡ

+
"Warning: comm+almostCancelʳ⇒almostCancelˡ was deprecated in v2.0.

+
Please use comm∧almostCancelʳ⇒almostCancelˡ instead."

+
#-}

+
comm+invˡ⇒invʳ = comm∧invˡ⇒invʳ

+
{-# WARNING_ON_USAGE comm+invˡ⇒invʳ

+
"Warning: comm+invˡ⇒invʳ was deprecated in v2.0.

+
Please use comm∧invˡ⇒invʳ instead."

+
#-}

+
comm+invʳ⇒invˡ = comm∧invʳ⇒invˡ

+
{-# WARNING_ON_USAGE comm+invʳ⇒invˡ

+
"Warning: comm+invʳ⇒invˡ was deprecated in v2.0.

+
Please use comm∧invʳ⇒invˡ instead."

+
#-}

+
comm+invˡ⇒inv = comm∧invˡ⇒inv

+
{-# WARNING_ON_USAGE comm+invˡ⇒inv

+
"Warning: comm+invˡ⇒inv was deprecated in v2.0.

+
Please use comm∧invˡ⇒inv instead."

+
#-}

+
comm+invʳ⇒inv = comm∧invʳ⇒inv

+
{-# WARNING_ON_USAGE comm+invʳ⇒inv

+
"Warning: comm+invʳ⇒inv was deprecated in v2.0.

+
Please use comm∧invʳ⇒inv instead."

+
#-}

+
comm+distrˡ⇒distrʳ = comm∧distrˡ⇒distrʳ

+
{-# WARNING_ON_USAGE comm+distrˡ⇒distrʳ

+
"Warning: comm+distrˡ⇒distrʳ was deprecated in v2.0.

+
Please use comm∧distrˡ⇒distrʳ instead."

+
#-}

+
comm+distrʳ⇒distrˡ = comm∧distrʳ⇒distrˡ

+
{-# WARNING_ON_USAGE comm+distrʳ⇒distrˡ

+
"Warning: comm+distrʳ⇒distrˡ was deprecated in v2.0.

+
Please use comm∧distrʳ⇒distrˡ instead."

+
#-}

+
assoc+distribʳ+idʳ+invʳ⇒zeˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ

+
{-# WARNING_ON_USAGE assoc+distribʳ+idʳ+invʳ⇒zeˡ

+
"Warning: assoc+distribʳ+idʳ+invʳ⇒zeˡ was deprecated in v2.0.

+
Please use assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ instead."

+
#-}

+
assoc+distribˡ+idʳ+invʳ⇒zeʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ

+
{-# WARNING_ON_USAGE assoc+distribˡ+idʳ+invʳ⇒zeʳ

+
"Warning: assoc+distribˡ+idʳ+invʳ⇒zeʳ was deprecated in v2.0.

+
Please use assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ instead."

+
#-}

+
assoc+id+invʳ⇒invˡ-unique = assoc∧id∧invʳ⇒invˡ-unique

+
{-# WARNING_ON_USAGE assoc+id+invʳ⇒invˡ-unique

+
"Warning: assoc+id+invʳ⇒invˡ-unique was deprecated in v2.0.

+
Please use assoc∧id∧invʳ⇒invˡ-unique instead."

+
#-}

+
assoc+id+invˡ⇒invʳ-unique = assoc∧id∧invˡ⇒invʳ-unique

+
{-# WARNING_ON_USAGE assoc+id+invˡ⇒invʳ-unique

+
"Warning: assoc+id+invˡ⇒invʳ-unique was deprecated in v2.0.

+
Please use assoc∧id∧invˡ⇒invʳ-unique instead."

+
#-}

+
subst+comm⇒sym = subst∧comm⇒sym

+
{-# WARNING_ON_USAGE subst+comm⇒sym

+
"Warning: subst+comm⇒sym was deprecated in v2.0.

+
Please use subst∧comm⇒sym instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.LiftedChoice.html b/site/static/agda_html/Algebra.Construct.LiftedChoice.html
new file mode 100644
index 000000000..af86ea954
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.LiftedChoice.html
@@ -0,0 +1,264 @@
+
+Algebra.Construct.LiftedChoice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Choosing between elements based on the result of applying a function

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra

+

+
module Algebra.Construct.LiftedChoice where

+

+
open import Algebra.Consequences.Base

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Function.Base using (const; _$_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (Rel; _⇒_; _Preserves_⟶_)

+
open import Relation.Nullary using (¬_; yes; no)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Unary using (Pred)

+

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
private

+
  variable

+
    a b p ℓ : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
module _ (_≈_ : Rel B ℓ) (_•_ : Op₂ B) where

+

+
  Lift : Selective _≈_ _•_ → (A → B) → Op₂ A

+
  Lift ∙-sel f x y = [ const x , const y ]′ $ ∙-sel (f x) (f y)

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
module _ {_≈_ : Rel B ℓ} {_∙_ : Op₂ B}

+
         (∙-isSelectiveMagma : IsSelectiveMagma _≈_ _∙_) where

+

+
  private module M = IsSelectiveMagma ∙-isSelectiveMagma

+
  open M hiding (sel; isMagma)

+
  open ≈-Reasoning setoid

+

+
  module _ (f : A → B) where

+

+
    private

+
      _◦_ = Lift _≈_ _∙_ M.sel f

+

+
    sel-≡ : Selective _≡_ _◦_

+
    sel-≡ x y with M.sel (f x) (f y)

+
    ... | inj₁ _ = inj₁ ≡.refl

+
    ... | inj₂ _ = inj₂ ≡.refl

+

+
    distrib : ∀ x y → ((f x) ∙ (f y)) ≈ f (x ◦ y)

+
    distrib x y with M.sel (f x) (f y)

+
    ... | inj₁ fx∙fy≈fx = fx∙fy≈fx

+
    ... | inj₂ fx∙fy≈fy = fx∙fy≈fy

+

+
  module _ (f : A → B) {_≈′_ : Rel A ℓ}

+
           (≈-reflexive : _≡_ ⇒ _≈′_) where

+

+
    private

+
      _◦_ = Lift _≈_ _∙_ M.sel f

+

+
    sel : Selective _≈′_ _◦_

+
    sel x y = Sum.map ≈-reflexive ≈-reflexive (sel-≡ f x y)

+

+
    idem : Idempotent _≈′_ _◦_

+
    idem = sel⇒idem _≈′_ sel

+

+
  module _ {f : A → B} {_≈′_ : Rel A ℓ}

+
           (f-injective : ∀ {x y} → f x ≈ f y → x ≈′ y)

+
           where

+

+
    private

+
      _◦_ = Lift _≈_ _∙_ M.sel f

+

+
    cong : f Preserves _≈′_ ⟶ _≈_ → Congruent₂ _≈′_ _◦_

+
    cong f-cong {x} {y} {u} {v} x≈y u≈v

+
      with M.sel (f x) (f u) | M.sel (f y) (f v)

+
    ... | inj₁ fx∙fu≈fx | inj₁ fy∙fv≈fy = x≈y

+
    ... | inj₂ fx∙fu≈fu | inj₂ fy∙fv≈fv = u≈v

+
    ... | inj₁ fx∙fu≈fx | inj₂ fy∙fv≈fv = f-injective (begin

+
      f x       ≈⟨ sym fx∙fu≈fx ⟩

+
      f x ∙ f u ≈⟨ ∙-cong (f-cong x≈y) (f-cong u≈v) ⟩

+
      f y ∙ f v ≈⟨ fy∙fv≈fv ⟩

+
      f v       ∎)

+
    ... | inj₂ fx∙fu≈fu | inj₁ fy∙fv≈fy = f-injective (begin

+
      f u       ≈⟨ sym fx∙fu≈fu ⟩

+
      f x ∙ f u ≈⟨ ∙-cong (f-cong x≈y) (f-cong u≈v) ⟩

+
      f y ∙ f v ≈⟨ fy∙fv≈fy ⟩

+
      f y       ∎)

+

+
    assoc : Associative _≈_ _∙_ → Associative _≈′_ _◦_

+
    assoc ∙-assoc x y z = f-injective (begin

+
      f ((x ◦ y) ◦ z)   ≈⟨ distrib f (x ◦ y) z ⟨

+
      f (x ◦ y) ∙ f z   ≈⟨ ∙-congʳ (distrib f x y) ⟨

+
      (f x ∙ f y) ∙ f z ≈⟨  ∙-assoc (f x) (f y) (f z) ⟩

+
      f x ∙ (f y ∙ f z) ≈⟨  ∙-congˡ (distrib f y z) ⟩

+
      f x ∙ f (y ◦ z)   ≈⟨  distrib f x (y ◦ z) ⟩

+
      f (x ◦ (y ◦ z))   ∎)

+

+
    comm : Commutative _≈_ _∙_ → Commutative _≈′_ _◦_

+
    comm ∙-comm x y = f-injective (begin

+
      f (x ◦ y) ≈⟨ distrib f x y ⟨

+
      f x ∙ f y ≈⟨  ∙-comm (f x) (f y) ⟩

+
      f y ∙ f x ≈⟨  distrib f y x ⟩

+
      f (y ◦ x) ∎)

+

+
------------------------------------------------------------------------

+
-- Algebraic structures

+

+
  module _ {_≈′_ : Rel A ℓ} {f : A → B}

+
           (f-injective : ∀ {x y} → f x ≈ f y → x ≈′ y)

+
           (f-cong : f Preserves _≈′_ ⟶ _≈_)

+
           (≈′-isEquivalence : IsEquivalence _≈′_)

+
           where

+

+
    private

+
      module E = IsEquivalence ≈′-isEquivalence

+
      _◦_ = Lift _≈_ _∙_ M.sel f

+

+
    isMagma : IsMagma _≈′_ _◦_

+
    isMagma = record

+
      { isEquivalence = ≈′-isEquivalence

+
      ; ∙-cong        = cong (λ {x} → f-injective {x}) f-cong

+
      }

+

+
    isSemigroup : Associative _≈_ _∙_ → IsSemigroup _≈′_ _◦_

+
    isSemigroup ∙-assoc = record

+
      { isMagma = isMagma

+
      ; assoc   = assoc (λ {x} → f-injective {x}) ∙-assoc

+
      }

+

+
    isBand : Associative _≈_ _∙_ → IsBand _≈′_ _◦_

+
    isBand ∙-assoc = record

+
      { isSemigroup = isSemigroup ∙-assoc

+
      ; idem        = idem f E.reflexive

+
      }

+

+
    isSelectiveMagma : IsSelectiveMagma _≈′_ _◦_

+
    isSelectiveMagma = record

+
      { isMagma = isMagma

+
      ; sel     = sel f E.reflexive

+
      }

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
  module _ {P : Pred A p} (f : A → B) where

+

+
    private

+
      _◦_ = Lift _≈_ _∙_ M.sel f

+

+
    preservesᵒ : (∀ {x y} → P x → (f x ∙ f y) ≈ f y → P y) →

+
                 (∀ {x y} → P y → (f x ∙ f y) ≈ f x → P x) →

+
                 ∀ x y → P x ⊎ P y → P (x ◦ y)

+
    preservesᵒ left right x y (inj₁ px) with M.sel (f x) (f y)

+
    ... | inj₁ _        = px

+
    ... | inj₂ fx∙fy≈fx = left px fx∙fy≈fx

+
    preservesᵒ left right x y (inj₂ py) with M.sel (f x) (f y)

+
    ... | inj₁ fx∙fy≈fy = right py fx∙fy≈fy

+
    ... | inj₂ _        = py

+

+
    preservesʳ : (∀ {x y} → P y → (f x ∙ f y) ≈ f x → P x) →

+
                 ∀ x {y} → P y → P (x ◦ y)

+
    preservesʳ right x {y} Py with M.sel (f x) (f y)

+
    ... | inj₁ fx∙fy≈fx = right Py fx∙fy≈fx

+
    ... | inj₂ fx∙fy≈fy = Py

+

+
    preservesᵇ : ∀ {x y} → P x → P y → P (x ◦ y)

+
    preservesᵇ {x} {y} Px Py with M.sel (f x) (f y)

+
    ... | inj₁ _ = Px

+
    ... | inj₂ _ = Py

+

+
    forcesᵇ : (∀ {x y} → P x → (f x ∙ f y) ≈ f x → P y) →

+
              (∀ {x y} → P y → (f x ∙ f y) ≈ f y → P x) →

+
              ∀ x y → P (x ◦ y) → P x × P y

+
    forcesᵇ presˡ presʳ x y P[x∙y] with M.sel (f x) (f y)

+
    ... | inj₁ fx∙fy≈fx = P[x∙y] , presˡ P[x∙y] fx∙fy≈fx

+
    ... | inj₂ fx∙fy≈fy = presʳ P[x∙y] fx∙fy≈fy , P[x∙y]

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.NaturalChoice.Base.html b/site/static/agda_html/Algebra.Construct.NaturalChoice.Base.html
new file mode 100644
index 000000000..40b2f5eb6
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.NaturalChoice.Base.html
@@ -0,0 +1,138 @@
+
+Algebra.Construct.NaturalChoice.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic definition of an operator that computes the min/max value

+
-- with respect to a total preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Level as L hiding (_⊔_)

+
open import Function.Base using (flip)

+
open import Relation.Binary.Bundles using (TotalPreorder)

+
open import Relation.Binary.Construct.Flip.EqAndOrd using ()

+
  renaming (totalPreorder to flipOrder)

+
import Relation.Binary.Properties.TotalOrder as TotalOrderProperties

+

+
module Algebra.Construct.NaturalChoice.Base where

+

+
private

+
  variable

+
    a ℓ₁ ℓ₂ : Level

+
    O : TotalPreorder a ℓ₁ ℓ₂

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
module _ (O : TotalPreorder a ℓ₁ ℓ₂) where

+
  open TotalPreorder O renaming (_≲_ to _≤_)

+
  private _≥_ = flip _≤_

+

+
  record MinOperator : Set (a L.⊔ ℓ₁ L.⊔ ℓ₂) where

+
    infixl 7 _⊓_

+
    field

+
      _⊓_       : Op₂ Carrier

+
      x≤y⇒x⊓y≈x : ∀ {x y} → x ≤ y → x ⊓ y ≈ x

+
      x≥y⇒x⊓y≈y : ∀ {x y} → x ≥ y → x ⊓ y ≈ y

+

+
  record MaxOperator : Set (a L.⊔ ℓ₁ L.⊔ ℓ₂) where

+
    infixl 6 _⊔_

+
    field

+
      _⊔_       : Op₂ Carrier

+
      x≤y⇒x⊔y≈y : ∀ {x y} → x ≤ y → x ⊔ y ≈ y

+
      x≥y⇒x⊔y≈x : ∀ {x y} → x ≥ y → x ⊔ y ≈ x

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
MinOp⇒MaxOp : MinOperator O → MaxOperator (flipOrder O)

+
MinOp⇒MaxOp minOp = record

+
  { _⊔_       = _⊓_

+
  ; x≤y⇒x⊔y≈y = x≥y⇒x⊓y≈y

+
  ; x≥y⇒x⊔y≈x = x≤y⇒x⊓y≈x

+
  } where open MinOperator minOp

+

+
MaxOp⇒MinOp : MaxOperator O → MinOperator (flipOrder O)

+
MaxOp⇒MinOp maxOp = record

+
  { _⊓_       = _⊔_

+
  ; x≤y⇒x⊓y≈x = x≥y⇒x⊔y≈x

+
  ; x≥y⇒x⊓y≈y = x≤y⇒x⊔y≈y

+
  } where open MaxOperator maxOp

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.NaturalChoice.Max.html b/site/static/agda_html/Algebra.Construct.NaturalChoice.Max.html
new file mode 100644
index 000000000..b41a2ce99
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.NaturalChoice.Max.html
@@ -0,0 +1,124 @@
+
+Algebra.Construct.NaturalChoice.Max
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The max operator derived from an arbitrary total preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (TotalOrder)

+

+
module Algebra.Construct.NaturalChoice.Max

+
  {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂) where

+

+
open import Algebra.Core

+
open import Algebra.Definitions

+
open import Algebra.Construct.NaturalChoice.Base

+
open import Relation.Binary.Construct.Flip.EqAndOrd using ()

+
  renaming (totalOrder to flip)

+

+
open TotalOrder totalOrder renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Max is just min with a flipped order

+

+
import Algebra.Construct.NaturalChoice.Min (flip totalOrder) as Min

+

+
infixl 6 _⊔_

+

+
_⊔_ : Op₂ A

+
_⊔_ = Min._⊓_

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
open Min public using ()

+
  renaming

+
  ( x≤y⇒x⊓y≈x to x≤y⇒y⊔x≈y

+
  ; x≤y⇒y⊓x≈x to x≤y⇒x⊔y≈y

+
  )

+

+
maxOperator : MaxOperator totalPreorder

+
maxOperator = record

+
  { x≤y⇒x⊔y≈y = x≤y⇒x⊔y≈y

+
  ; x≥y⇒x⊔y≈x = x≤y⇒y⊔x≈y

+
  }

+

+
open import Algebra.Construct.NaturalChoice.MaxOp maxOperator public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.NaturalChoice.MaxOp.html b/site/static/agda_html/Algebra.Construct.NaturalChoice.MaxOp.html
new file mode 100644
index 000000000..32dde8fa0
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.NaturalChoice.MaxOp.html
@@ -0,0 +1,158 @@
+
+Algebra.Construct.NaturalChoice.MaxOp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of a max operator derived from a spec over a total

+
-- preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Construct.NaturalChoice.Base

+
import Algebra.Construct.NaturalChoice.MinOp as MinOp

+
open import Function.Base using (flip)

+
open import Relation.Binary.Core using (_Preserves_⟶_)

+
open import Relation.Binary.Bundles using (TotalPreorder)

+
open import Relation.Binary.Construct.Flip.EqAndOrd using ()

+
  renaming (totalPreorder to flipOrder)

+

+
module Algebra.Construct.NaturalChoice.MaxOp

+
  {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂} (maxOp : MaxOperator O)

+
  where

+

+
open TotalPreorder O renaming (Carrier to A; _≲_ to _≤_)

+
open MaxOperator maxOp

+

+
-- Max is just min with a flipped order

+

+
private

+
  module Min = MinOp (MaxOp⇒MinOp maxOp)

+

+
open Min public

+
  using ()

+
  renaming

+
  ( ⊓-cong       to  ⊔-cong

+
  ; ⊓-congʳ      to  ⊔-congʳ

+
  ; ⊓-congˡ      to  ⊔-congˡ

+
  ; ⊓-idem       to  ⊔-idem

+
  ; ⊓-sel        to  ⊔-sel

+
  ; ⊓-assoc      to  ⊔-assoc

+
  ; ⊓-comm       to  ⊔-comm

+

+
  ; ⊓-identityˡ  to  ⊔-identityˡ

+
  ; ⊓-identityʳ  to  ⊔-identityʳ

+
  ; ⊓-identity   to  ⊔-identity

+
  ; ⊓-zeroˡ      to  ⊔-zeroˡ

+
  ; ⊓-zeroʳ      to  ⊔-zeroʳ

+
  ; ⊓-zero       to  ⊔-zero

+

+
  ; ⊓-isMagma                 to  ⊔-isMagma

+
  ; ⊓-isSemigroup             to  ⊔-isSemigroup

+
  ; ⊓-isCommutativeSemigroup  to  ⊔-isCommutativeSemigroup

+
  ; ⊓-isBand                  to  ⊔-isBand

+
  ; ⊓-isMonoid                to  ⊔-isMonoid

+
  ; ⊓-isSelectiveMagma        to  ⊔-isSelectiveMagma

+

+
  ; ⊓-magma                   to  ⊔-magma

+
  ; ⊓-semigroup               to  ⊔-semigroup

+
  ; ⊓-commutativeSemigroup    to  ⊔-commutativeSemigroup

+
  ; ⊓-band                    to  ⊔-band

+
  ; ⊓-monoid                  to  ⊔-monoid

+
  ; ⊓-selectiveMagma          to  ⊔-selectiveMagma

+

+
  ; x⊓y≈y⇒y≤x  to x⊔y≈y⇒x≤y

+
  ; x⊓y≈x⇒x≤y  to x⊔y≈x⇒y≤x

+
  ; x⊓y≤x      to x≤x⊔y

+
  ; x⊓y≤y      to x≤y⊔x

+
  ; x≤y⇒x⊓z≤y  to x≤y⇒x≤y⊔z

+
  ; x≤y⇒z⊓x≤y  to x≤y⇒x≤z⊔y

+
  ; x≤y⊓z⇒x≤y  to x⊔y≤z⇒x≤z

+
  ; x≤y⊓z⇒x≤z  to x⊔y≤z⇒y≤z

+

+
  ; ⊓-glb              to  ⊔-lub

+
  ; ⊓-triangulate      to  ⊔-triangulate

+
  ; ⊓-mono-≤           to  ⊔-mono-≤

+
  ; ⊓-monoˡ-≤          to  ⊔-monoˡ-≤

+
  ; ⊓-monoʳ-≤          to  ⊔-monoʳ-≤

+
  )

+

+
mono-≤-distrib-⊔ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ x y → f (x ⊔ y) ≈ f x ⊔ f y

+
mono-≤-distrib-⊔ cong pres = Min.mono-≤-distrib-⊓ cong pres

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.NaturalChoice.Min.html b/site/static/agda_html/Algebra.Construct.NaturalChoice.Min.html
new file mode 100644
index 000000000..f4a1c6ef0
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.NaturalChoice.Min.html
@@ -0,0 +1,130 @@
+
+Algebra.Construct.NaturalChoice.Min
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The min operator derived from an arbitrary total preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
open import Algebra.Construct.NaturalChoice.Base

+
open import Data.Sum.Base using (inj₁; inj₂; [_,_])

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (id)

+
open import Relation.Binary.Bundles using (TotalOrder)

+
import Algebra.Construct.NaturalChoice.MinOp as MinOp

+

+
module Algebra.Construct.NaturalChoice.Min

+
  {a ℓ₁ ℓ₂} (O : TotalOrder a ℓ₁ ℓ₂)

+
  where

+

+
open TotalOrder O renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infixl 7 _⊓_

+

+
_⊓_ : Op₂ A

+
x ⊓ y with total x y

+
... | inj₁ x≤y = x

+
... | inj₂ y≤x = y

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
x≤y⇒x⊓y≈x : ∀ {x y} → x ≤ y → x ⊓ y ≈ x

+
x≤y⇒x⊓y≈x {x} {y} x≤y with total x y

+
... | inj₁ _   = Eq.refl

+
... | inj₂ y≤x = antisym y≤x x≤y

+

+
x≤y⇒y⊓x≈x : ∀ {x y} → x ≤ y → y ⊓ x ≈ x

+
x≤y⇒y⊓x≈x {x} {y} x≤y with total y x

+
... | inj₁ y≤x = antisym y≤x x≤y

+
... | inj₂ _   = Eq.refl

+

+
minOperator : MinOperator totalPreorder

+
minOperator = record

+
  { x≤y⇒x⊓y≈x = x≤y⇒x⊓y≈x

+
  ; x≥y⇒x⊓y≈y = x≤y⇒y⊓x≈x

+
  }

+

+
open MinOp minOperator public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.NaturalChoice.MinMaxOp.html b/site/static/agda_html/Algebra.Construct.NaturalChoice.MinMaxOp.html
new file mode 100644
index 000000000..3149de481
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.NaturalChoice.MinMaxOp.html
@@ -0,0 +1,222 @@
+
+Algebra.Construct.NaturalChoice.MinMaxOp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of min and max operators specified over a total

+
-- preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
open import Algebra.Construct.NaturalChoice.Base

+
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (id; _∘_; flip)

+
open import Relation.Binary.Core using (_Preserves_⟶_)

+
open import Relation.Binary.Bundles using (TotalPreorder)

+
open import Relation.Binary.Consequences

+

+
module Algebra.Construct.NaturalChoice.MinMaxOp

+
  {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂}

+
  (minOp : MinOperator O)

+
  (maxOp : MaxOperator O)

+
  where

+

+
open TotalPreorder O renaming

+
  ( Carrier   to A

+
  ; _≲_       to _≤_

+
  ; ≲-resp-≈  to ≤-resp-≈

+
  ; ≲-respʳ-≈ to ≤-respʳ-≈

+
  ; ≲-respˡ-≈ to ≤-respˡ-≈

+
  )

+
open MinOperator minOp

+
open MaxOperator maxOp

+

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures _≈_

+
open import Algebra.Consequences.Setoid Eq.setoid

+
open import Relation.Binary.Reasoning.Preorder preorder

+

+
------------------------------------------------------------------------

+
-- Re-export properties of individual operators

+

+
open import Algebra.Construct.NaturalChoice.MinOp minOp public

+
open import Algebra.Construct.NaturalChoice.MaxOp maxOp public

+

+
------------------------------------------------------------------------

+
-- Joint algebraic structures

+

+
⊓-distribˡ-⊔ : _⊓_ DistributesOverˡ _⊔_

+
⊓-distribˡ-⊔ x y z with total y z

+
... | inj₁ y≤z = begin-equality

+
  x ⊓ (y ⊔ z)       ≈⟨  ⊓-congˡ x (x≤y⇒x⊔y≈y y≤z) ⟩

+
  x ⊓ z             ≈⟨ x≤y⇒x⊔y≈y (⊓-monoʳ-≤ x y≤z) ⟨

+
  (x ⊓ y) ⊔ (x ⊓ z) ∎

+
... | inj₂ y≥z = begin-equality

+
  x ⊓ (y ⊔ z)       ≈⟨  ⊓-congˡ x (x≥y⇒x⊔y≈x y≥z) ⟩

+
  x ⊓ y             ≈⟨ x≥y⇒x⊔y≈x (⊓-monoʳ-≤ x y≥z) ⟨

+
  (x ⊓ y) ⊔ (x ⊓ z) ∎

+

+
⊓-distribʳ-⊔ : _⊓_ DistributesOverʳ _⊔_

+
⊓-distribʳ-⊔ = comm+distrˡ⇒distrʳ ⊔-cong ⊓-comm ⊓-distribˡ-⊔

+

+
⊓-distrib-⊔ : _⊓_ DistributesOver _⊔_

+
⊓-distrib-⊔ = ⊓-distribˡ-⊔ , ⊓-distribʳ-⊔

+

+
⊔-distribˡ-⊓ : _⊔_ DistributesOverˡ _⊓_

+
⊔-distribˡ-⊓ x y z with total y z

+
... | inj₁ y≤z = begin-equality

+
  x ⊔ (y ⊓ z)       ≈⟨  ⊔-congˡ x (x≤y⇒x⊓y≈x y≤z) ⟩

+
  x ⊔ y             ≈⟨ x≤y⇒x⊓y≈x (⊔-monoʳ-≤ x y≤z) ⟨

+
  (x ⊔ y) ⊓ (x ⊔ z) ∎

+
... | inj₂ y≥z = begin-equality

+
  x ⊔ (y ⊓ z)       ≈⟨  ⊔-congˡ x (x≥y⇒x⊓y≈y y≥z) ⟩

+
  x ⊔ z             ≈⟨ x≥y⇒x⊓y≈y (⊔-monoʳ-≤ x y≥z) ⟨

+
  (x ⊔ y) ⊓ (x ⊔ z) ∎

+

+
⊔-distribʳ-⊓ : _⊔_ DistributesOverʳ _⊓_

+
⊔-distribʳ-⊓ = comm+distrˡ⇒distrʳ ⊓-cong ⊔-comm ⊔-distribˡ-⊓

+

+
⊔-distrib-⊓ : _⊔_ DistributesOver _⊓_

+
⊔-distrib-⊓ = ⊔-distribˡ-⊓ , ⊔-distribʳ-⊓

+

+
⊓-absorbs-⊔ : _⊓_ Absorbs _⊔_

+
⊓-absorbs-⊔ x y with total x y

+
... | inj₁ x≤y = begin-equality

+
  x ⊓ (x ⊔ y)  ≈⟨ ⊓-congˡ x (x≤y⇒x⊔y≈y x≤y) ⟩

+
  x ⊓ y        ≈⟨ x≤y⇒x⊓y≈x x≤y ⟩

+
  x            ∎

+
... | inj₂ y≤x = begin-equality

+
  x ⊓ (x ⊔ y)  ≈⟨ ⊓-congˡ x (x≥y⇒x⊔y≈x y≤x) ⟩

+
  x ⊓ x        ≈⟨ ⊓-idem x ⟩

+
  x            ∎

+

+
⊔-absorbs-⊓ : _⊔_ Absorbs _⊓_

+
⊔-absorbs-⊓ x y with total x y

+
... | inj₁ x≤y = begin-equality

+
  x ⊔ (x ⊓ y)  ≈⟨ ⊔-congˡ x (x≤y⇒x⊓y≈x x≤y) ⟩

+
  x ⊔ x        ≈⟨ ⊔-idem x ⟩

+
  x            ∎

+
... | inj₂ y≤x = begin-equality

+
  x ⊔ (x ⊓ y)  ≈⟨ ⊔-congˡ x (x≥y⇒x⊓y≈y y≤x) ⟩

+
  x ⊔ y        ≈⟨ x≥y⇒x⊔y≈x y≤x ⟩

+
  x            ∎

+

+
⊔-⊓-absorptive : Absorptive _⊔_ _⊓_

+
⊔-⊓-absorptive = ⊔-absorbs-⊓ , ⊓-absorbs-⊔

+

+
⊓-⊔-absorptive : Absorptive _⊓_ _⊔_

+
⊓-⊔-absorptive = ⊓-absorbs-⊔ , ⊔-absorbs-⊓

+

+
------------------------------------------------------------------------

+
-- Other joint properties

+

+
private _≥_ = flip _≤_

+

+
antimono-≤-distrib-⊓ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ x y → f (x ⊓ y) ≈ f x ⊔ f y

+
antimono-≤-distrib-⊓ {f} cong antimono x y with total x y

+
... | inj₁ x≤y = begin-equality

+
  f (x ⊓ y)  ≈⟨ cong (x≤y⇒x⊓y≈x x≤y) ⟩

+
  f x        ≈⟨ x≥y⇒x⊔y≈x (antimono x≤y) ⟨

+
  f x ⊔ f y  ∎

+
... | inj₂ y≤x = begin-equality

+
  f (x ⊓ y)  ≈⟨ cong (x≥y⇒x⊓y≈y y≤x) ⟩

+
  f y        ≈⟨ x≤y⇒x⊔y≈y (antimono y≤x) ⟨

+
  f x ⊔ f y  ∎

+

+
antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ x y → f (x ⊔ y) ≈ f x ⊓ f y

+
antimono-≤-distrib-⊔ {f} cong antimono x y with total x y

+
... | inj₁ x≤y = begin-equality

+
  f (x ⊔ y)  ≈⟨ cong (x≤y⇒x⊔y≈y x≤y) ⟩

+
  f y        ≈⟨ x≥y⇒x⊓y≈y (antimono x≤y) ⟨

+
  f x ⊓ f y  ∎

+
... | inj₂ y≤x = begin-equality

+
  f (x ⊔ y)  ≈⟨ cong (x≥y⇒x⊔y≈x y≤x) ⟩

+
  f x        ≈⟨ x≤y⇒x⊓y≈x (antimono y≤x) ⟨

+
  f x ⊓ f y  ∎

+

+
x⊓y≤x⊔y : ∀ x y → x ⊓ y ≤ x ⊔ y

+
x⊓y≤x⊔y x y = begin

+
  x ⊓ y ∼⟨ x⊓y≤x x y ⟩

+
  x     ∼⟨ x≤x⊔y x y ⟩

+
  x ⊔ y ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Construct.NaturalChoice.MinOp.html b/site/static/agda_html/Algebra.Construct.NaturalChoice.MinOp.html
new file mode 100644
index 000000000..ca8e9f0e5
--- /dev/null
+++ b/site/static/agda_html/Algebra.Construct.NaturalChoice.MinOp.html
@@ -0,0 +1,334 @@
+
+Algebra.Construct.NaturalChoice.MinOp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of a min operator derived from a spec over a total

+
-- preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
open import Algebra.Construct.NaturalChoice.Base

+
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (id; _∘_)

+
open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.Bundles using (TotalPreorder)

+
open import Relation.Binary.Definitions using (Maximum; Minimum)

+
open import Relation.Binary.Consequences

+

+
module Algebra.Construct.NaturalChoice.MinOp

+
  {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂} (minOp : MinOperator O) where

+

+
open TotalPreorder O renaming

+
  ( Carrier   to A

+
  ; _≲_       to _≤_

+
  ; ≲-resp-≈  to ≤-resp-≈

+
  ; ≲-respʳ-≈ to ≤-respʳ-≈

+
  ; ≲-respˡ-≈ to ≤-respˡ-≈

+
  )

+
open MinOperator minOp

+

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures _≈_

+
open import Relation.Binary.Reasoning.Preorder preorder

+

+
------------------------------------------------------------------------

+
-- Helpful properties

+

+
x⊓y≤x : ∀ x y → x ⊓ y ≤ x

+
x⊓y≤x x y with total x y

+
... | inj₁ x≤y = reflexive (x≤y⇒x⊓y≈x x≤y)

+
... | inj₂ y≤x = ≤-respˡ-≈ (Eq.sym (x≥y⇒x⊓y≈y y≤x)) y≤x

+

+
x⊓y≤y : ∀ x y → x ⊓ y ≤ y

+
x⊓y≤y x y with total x y

+
... | inj₁ x≤y = ≤-respˡ-≈ (Eq.sym (x≤y⇒x⊓y≈x x≤y)) x≤y

+
... | inj₂ y≤x = reflexive (x≥y⇒x⊓y≈y y≤x)

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
⊓-comm : Commutative _⊓_

+
⊓-comm x y with total x y

+
... | inj₁ x≤y = Eq.trans (x≤y⇒x⊓y≈x x≤y) (Eq.sym (x≥y⇒x⊓y≈y x≤y))

+
... | inj₂ y≤x = Eq.trans (x≥y⇒x⊓y≈y y≤x) (Eq.sym (x≤y⇒x⊓y≈x y≤x))

+

+
⊓-congˡ : ∀ x → Congruent₁ (x ⊓_)

+
⊓-congˡ x {y} {r} y≈r with total x y

+
... | inj₁ x≤y = begin-equality

+
  x ⊓ y  ≈⟨  x≤y⇒x⊓y≈x x≤y ⟩

+
  x      ≈⟨ x≤y⇒x⊓y≈x (≤-respʳ-≈ y≈r x≤y) ⟨

+
  x ⊓ r  ∎

+
... | inj₂ y≤x = begin-equality

+
  x ⊓ y  ≈⟨  x≥y⇒x⊓y≈y y≤x ⟩

+
  y      ≈⟨  y≈r ⟩

+
  r      ≈⟨ x≥y⇒x⊓y≈y (≤-respˡ-≈ y≈r y≤x) ⟨

+
  x ⊓ r  ∎

+

+
⊓-congʳ : ∀ x → Congruent₁ (_⊓ x)

+
⊓-congʳ x {y₁} {y₂} y₁≈y₂ = begin-equality

+
  y₁ ⊓ x  ≈⟨ ⊓-comm x y₁ ⟨

+
  x  ⊓ y₁ ≈⟨  ⊓-congˡ x y₁≈y₂ ⟩

+
  x  ⊓ y₂ ≈⟨  ⊓-comm x y₂ ⟩

+
  y₂ ⊓ x  ∎

+

+
⊓-cong : Congruent₂ _⊓_

+
⊓-cong {x₁} {x₂} {y₁} {y₂} x₁≈x₂ y₁≈y₂ = Eq.trans (⊓-congˡ x₁ y₁≈y₂) (⊓-congʳ y₂ x₁≈x₂)

+

+
⊓-assoc : Associative _⊓_

+
⊓-assoc x y r with total x y | total y r

+
⊓-assoc x y r | inj₁ x≤y | inj₁ y≤r = begin-equality

+
  (x ⊓ y) ⊓ r  ≈⟨ ⊓-congʳ r (x≤y⇒x⊓y≈x x≤y) ⟩

+
  x ⊓ r        ≈⟨ x≤y⇒x⊓y≈x (trans x≤y y≤r) ⟩

+
  x            ≈⟨ x≤y⇒x⊓y≈x x≤y ⟨

+
  x ⊓ y        ≈⟨ ⊓-congˡ x (x≤y⇒x⊓y≈x y≤r) ⟨

+
  x ⊓ (y ⊓ r)  ∎

+
⊓-assoc x y r | inj₁ x≤y | inj₂ r≤y = begin-equality

+
  (x ⊓ y) ⊓ r  ≈⟨ ⊓-congʳ r (x≤y⇒x⊓y≈x x≤y) ⟩

+
  x ⊓ r        ≈⟨ ⊓-congˡ x (x≥y⇒x⊓y≈y r≤y) ⟨

+
  x ⊓ (y ⊓ r)  ∎

+
⊓-assoc x y r | inj₂ y≤x | _ = begin-equality

+
  (x ⊓ y) ⊓ r  ≈⟨ ⊓-congʳ r (x≥y⇒x⊓y≈y y≤x) ⟩

+
  y ⊓ r        ≈⟨ x≥y⇒x⊓y≈y (trans (x⊓y≤x y r) y≤x) ⟨

+
  x ⊓ (y ⊓ r)  ∎

+

+
⊓-idem : Idempotent _⊓_

+
⊓-idem x = x≤y⇒x⊓y≈x (refl {x})

+

+
⊓-sel : Selective _⊓_

+
⊓-sel x y = Sum.map x≤y⇒x⊓y≈x x≥y⇒x⊓y≈y (total x y)

+

+
⊓-identityˡ : ∀ {⊤} → Maximum _≤_ ⊤ → LeftIdentity ⊤ _⊓_

+
⊓-identityˡ max = x≥y⇒x⊓y≈y ∘ max

+

+
⊓-identityʳ : ∀ {⊤} → Maximum _≤_ ⊤ → RightIdentity ⊤ _⊓_

+
⊓-identityʳ max = x≤y⇒x⊓y≈x ∘ max

+

+
⊓-identity : ∀ {⊤} → Maximum _≤_ ⊤ → Identity ⊤ _⊓_

+
⊓-identity max = ⊓-identityˡ max , ⊓-identityʳ max

+

+
⊓-zeroˡ : ∀ {⊥} → Minimum _≤_ ⊥ → LeftZero ⊥ _⊓_

+
⊓-zeroˡ min = x≤y⇒x⊓y≈x ∘ min

+

+
⊓-zeroʳ : ∀ {⊥} → Minimum _≤_ ⊥ → RightZero ⊥ _⊓_

+
⊓-zeroʳ min = x≥y⇒x⊓y≈y ∘ min

+

+
⊓-zero : ∀ {⊥} → Minimum _≤_ ⊥ → Zero ⊥ _⊓_

+
⊓-zero min = ⊓-zeroˡ min , ⊓-zeroʳ min

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
⊓-isMagma : IsMagma _⊓_

+
⊓-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = ⊓-cong

+
  }

+

+
⊓-isSemigroup : IsSemigroup _⊓_

+
⊓-isSemigroup = record

+
  { isMagma = ⊓-isMagma

+
  ; assoc   = ⊓-assoc

+
  }

+

+
⊓-isBand : IsBand _⊓_

+
⊓-isBand = record

+
  { isSemigroup = ⊓-isSemigroup

+
  ; idem        = ⊓-idem

+
  }

+

+
⊓-isCommutativeSemigroup : IsCommutativeSemigroup _⊓_

+
⊓-isCommutativeSemigroup = record

+
  { isSemigroup = ⊓-isSemigroup

+
  ; comm        = ⊓-comm

+
  }

+

+
⊓-isSelectiveMagma : IsSelectiveMagma _⊓_

+
⊓-isSelectiveMagma = record

+
  { isMagma = ⊓-isMagma

+
  ; sel     = ⊓-sel

+
  }

+

+
⊓-isMonoid : ∀ {⊤} → Maximum _≤_ ⊤ → IsMonoid _⊓_ ⊤

+
⊓-isMonoid max = record

+
  { isSemigroup = ⊓-isSemigroup

+
  ; identity    = ⊓-identity max

+
  }

+

+
------------------------------------------------------------------------

+
-- Raw bundles

+

+
⊓-rawMagma : RawMagma _ _

+
⊓-rawMagma = record { _≈_ = _≈_ ; _∙_ = _⊓_ }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
⊓-magma : Magma _ _

+
⊓-magma = record

+
  { isMagma = ⊓-isMagma

+
  }

+

+
⊓-semigroup : Semigroup _ _

+
⊓-semigroup = record

+
  { isSemigroup = ⊓-isSemigroup

+
  }

+

+
⊓-band : Band _ _

+
⊓-band = record

+
  { isBand = ⊓-isBand

+
  }

+

+
⊓-commutativeSemigroup : CommutativeSemigroup _ _

+
⊓-commutativeSemigroup = record

+
  { isCommutativeSemigroup = ⊓-isCommutativeSemigroup

+
  }

+

+
⊓-selectiveMagma : SelectiveMagma _ _

+
⊓-selectiveMagma = record

+
  { isSelectiveMagma = ⊓-isSelectiveMagma

+
  }

+

+
⊓-monoid : ∀ {⊤} → Maximum _≤_ ⊤ → Monoid a ℓ₁

+
⊓-monoid max = record

+
  { isMonoid = ⊓-isMonoid max

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
x⊓y≈x⇒x≤y : ∀ {x y} → x ⊓ y ≈ x → x ≤ y

+
x⊓y≈x⇒x≤y {x} {y} x⊓y≈x with total x y

+
... | inj₁ x≤y = x≤y

+
... | inj₂ y≤x = reflexive (Eq.trans (Eq.sym x⊓y≈x) (x≥y⇒x⊓y≈y y≤x))

+

+
x⊓y≈y⇒y≤x : ∀ {x y} → x ⊓ y ≈ y → y ≤ x

+
x⊓y≈y⇒y≤x {x} {y} x⊓y≈y = x⊓y≈x⇒x≤y (begin-equality

+
  y ⊓ x  ≈⟨ ⊓-comm y x ⟩

+
  x ⊓ y  ≈⟨ x⊓y≈y ⟩

+
  y      ∎)

+

+
mono-≤-distrib-⊓ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ x y → f (x ⊓ y) ≈ f x ⊓ f y

+
mono-≤-distrib-⊓ {f} cong mono x y with total x y

+
... | inj₁ x≤y = begin-equality

+
  f (x ⊓ y)  ≈⟨ cong (x≤y⇒x⊓y≈x x≤y) ⟩

+
  f x        ≈⟨ x≤y⇒x⊓y≈x (mono x≤y) ⟨

+
  f x ⊓ f y  ∎

+
... | inj₂ y≤x = begin-equality

+
  f (x ⊓ y)  ≈⟨ cong (x≥y⇒x⊓y≈y y≤x) ⟩

+
  f y        ≈⟨ x≥y⇒x⊓y≈y (mono y≤x) ⟨

+
  f x ⊓ f y  ∎

+

+
x≤y⇒x⊓z≤y : ∀ {x y} z → x ≤ y → x ⊓ z ≤ y

+
x≤y⇒x⊓z≤y z x≤y = trans (x⊓y≤x _ z) x≤y

+

+
x≤y⇒z⊓x≤y : ∀ {x y} z → x ≤ y → z ⊓ x ≤ y

+
x≤y⇒z⊓x≤y y x≤y = trans (x⊓y≤y y _) x≤y

+

+
x≤y⊓z⇒x≤y : ∀ {x} y z → x ≤ y ⊓ z → x ≤ y

+
x≤y⊓z⇒x≤y y z x≤y⊓z = trans x≤y⊓z (x⊓y≤x y z)

+

+
x≤y⊓z⇒x≤z : ∀ {x} y z → x ≤ y ⊓ z → x ≤ z

+
x≤y⊓z⇒x≤z y z x≤y⊓z = trans x≤y⊓z (x⊓y≤y y z)

+

+
⊓-mono-≤ : _⊓_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
⊓-mono-≤ {x} {y} {u} {v} x≤y u≤v with ⊓-sel y v

+
... | inj₁ y⊓v≈y = ≤-respʳ-≈ (Eq.sym y⊓v≈y) (trans (x⊓y≤x x u) x≤y)

+
... | inj₂ y⊓v≈v = ≤-respʳ-≈ (Eq.sym y⊓v≈v) (trans (x⊓y≤y x u) u≤v)

+

+
⊓-monoˡ-≤ : ∀ x → (_⊓ x) Preserves _≤_ ⟶ _≤_

+
⊓-monoˡ-≤ x y≤z = ⊓-mono-≤ y≤z (refl {x})

+

+
⊓-monoʳ-≤ : ∀ x → (x ⊓_) Preserves _≤_ ⟶ _≤_

+
⊓-monoʳ-≤ x y≤z = ⊓-mono-≤ (refl {x}) y≤z

+

+
⊓-glb : ∀ {x y z} → x ≤ y → x ≤ z → x ≤ y ⊓ z

+
⊓-glb {x} x≤y x≤z = ≤-respˡ-≈ (⊓-idem x) (⊓-mono-≤ x≤y x≤z)

+

+
⊓-triangulate : ∀ x y z → x ⊓ y ⊓ z ≈ (x ⊓ y) ⊓ (y ⊓ z)

+
⊓-triangulate x y z = begin-equality

+
  x ⊓ y ⊓ z           ≈⟨ ⊓-congʳ z (⊓-congˡ x (⊓-idem y)) ⟨

+
  x ⊓ (y ⊓ y) ⊓ z     ≈⟨  ⊓-assoc x _ _ ⟩

+
  x ⊓ ((y ⊓ y) ⊓ z)   ≈⟨  ⊓-congˡ x (⊓-assoc y y z) ⟩

+
  x ⊓ (y ⊓ (y ⊓ z))   ≈⟨ ⊓-assoc x y (y ⊓ z) ⟨

+
  (x ⊓ y) ⊓ (y ⊓ z)   ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Core.html b/site/static/agda_html/Algebra.Core.html
new file mode 100644
index 000000000..d844d5e78
--- /dev/null
+++ b/site/static/agda_html/Algebra.Core.html
@@ -0,0 +1,99 @@
+
+Algebra.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core algebraic definitions

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Core where

+

+
open import Level using (_⊔_)

+

+
------------------------------------------------------------------------

+
-- Unary and binary operations

+

+
Op₁ : ∀ {ℓ} → Set ℓ → Set ℓ

+
Op₁ A = A → A

+

+
Op₂ : ∀ {ℓ} → Set ℓ → Set ℓ

+
Op₂ A = A → A → A

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Definitions.RawMagma.html b/site/static/agda_html/Algebra.Definitions.RawMagma.html
new file mode 100644
index 000000000..bdb876070
--- /dev/null
+++ b/site/static/agda_html/Algebra.Definitions.RawMagma.html
@@ -0,0 +1,163 @@
+
+Algebra.Definitions.RawMagma
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic auxiliary definitions for magma-like structures

+
------------------------------------------------------------------------

+

+
-- You're unlikely to want to use this module directly. Instead you

+
-- probably want to be importing the appropriate module from

+
-- `Algebra.Properties.(Magma/Semigroup/...).Divisibility`

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles.Raw using (RawMagma)

+
open import Data.Product.Base using (_×_; ∃)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
module Algebra.Definitions.RawMagma

+
  {a ℓ} (M : RawMagma a ℓ)

+
  where

+

+
open RawMagma M renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Divisibility

+

+
infix 5 _∣ˡ_ _∤ˡ_ _∣ʳ_ _∤ʳ_ _∣_ _∤_ _∣∣_ _∤∤_

+

+
-- Divisibility from the left.

+
--

+
-- This and, the definition of right divisibility below, are defined as

+
-- records rather than in terms of the base product type in order to

+
-- make the use of pattern synonyms more ergonomic (see #2216 for

+
-- further details). The record field names are not designed to be

+
-- used explicitly and indeed aren't re-exported publicly by

+
-- `Algebra.X.Properties.Divisibility` modules.

+

+
record _∣ˡ_ (x y : A) : Set (a ⊔ ℓ) where

+
  constructor _,_

+
  field

+
    quotient : A

+
    equality : x ∙ quotient ≈ y

+

+
_∤ˡ_ : Rel A (a ⊔ ℓ)

+
x ∤ˡ y = ¬ x ∣ˡ y

+

+
-- Divisibility from the right

+

+
record _∣ʳ_ (x y : A) : Set (a ⊔ ℓ) where

+
  constructor _,_

+
  field

+
    quotient : A

+
    equality : quotient ∙ x ≈ y

+

+
_∤ʳ_ : Rel A (a ⊔ ℓ)

+
x ∤ʳ y = ¬ x ∣ʳ y

+

+
-- General divisibility

+

+
-- The relations _∣ˡ_ and _∣ʳ_ are only equivalent when _∙_ is

+
-- commutative. When that is not the case we take `_∣ʳ_` to be the

+
-- primary one.

+

+
_∣_ : Rel A (a ⊔ ℓ)

+
_∣_ = _∣ʳ_

+

+
_∤_ : Rel A (a ⊔ ℓ)

+
x ∤ y = ¬ x ∣ y

+

+
------------------------------------------------------------------------

+
-- Mutual divisibility.

+

+
-- In a  monoid, this is an equivalence relation extending _≈_.

+
-- When in a cancellative monoid,  elements related by _∣∣_ are called

+
-- associated, and `x ∣∣ y` means that `x` and `y` differ by some

+
-- invertible factor.

+

+
-- Example: for ℕ  this is equivalent to x ≡ y,

+
--          for ℤ  this is equivalent to (x ≡ y or x ≡ - y).

+

+
_∣∣_ : Rel A (a ⊔ ℓ)

+
x ∣∣ y = x ∣ y × y ∣ x

+

+
_∤∤_ : Rel A (a ⊔ ℓ)

+
x ∤∤ y = ¬ x ∣∣ y

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Definitions.RawMonoid.html b/site/static/agda_html/Algebra.Definitions.RawMonoid.html
new file mode 100644
index 000000000..c3f5fc27d
--- /dev/null
+++ b/site/static/agda_html/Algebra.Definitions.RawMonoid.html
@@ -0,0 +1,144 @@
+
+Algebra.Definitions.RawMonoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic auxiliary definitions for monoid-like structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles using (RawMonoid)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)

+
open import Data.Vec.Functional as Vector using (Vector)

+

+
module Algebra.Definitions.RawMonoid {a ℓ} (M : RawMonoid a ℓ) where

+

+
open RawMonoid M renaming ( _∙_ to _+_ ; ε to 0# )

+

+
------------------------------------------------------------------------

+
-- Re-export definitions over a magma

+
------------------------------------------------------------------------

+

+
open import Algebra.Definitions.RawMagma rawMagma public

+

+
------------------------------------------------------------------------

+
-- Multiplication by natural number

+
------------------------------------------------------------------------

+
-- Standard definition

+

+
-- A simple definition, easy to use and prove properties about.

+

+
infixr 8 _×_

+

+
_×_ : ℕ → Carrier → Carrier

+
0     × x = 0#

+
suc n × x = x + (n × x)

+

+
------------------------------------------------------------------------

+
-- Type-checking optimised definition

+

+
-- For use in code where high performance at type-checking time is

+
-- important, e.g. solvers and tactics. Firstly it avoids unnecessarily

+
-- multiplying by the unit if possible, speeding up type-checking and

+
-- makes for much more readable proofs:

+
--

+
--     Standard definition:  x * 2 = x + x + 0#

+
--     Optimised definition: x * 2 = x + x

+
--

+
-- Secondly, associates to the left which, counterintuitive as it may

+
-- seem, also speeds up typechecking.

+
--

+
--     Standard definition: x * 3 = x + (x + (x + 0#))

+
--     Our definition:      x * 3 = (x + x) + x

+

+
infixl 8 _×′_

+

+
_×′_ : ℕ → Carrier → Carrier

+
0     ×′ x = 0#

+
1     ×′ x = x

+
suc n ×′ x = n ×′ x + x

+

+
{-# INLINE _×′_ #-}

+

+
------------------------------------------------------------------------

+
-- Summation

+
------------------------------------------------------------------------

+

+
sum : ∀ {n} → Vector Carrier n → Carrier

+
sum = Vector.foldr _+_ 0#

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Definitions.RawSemiring.html b/site/static/agda_html/Algebra.Definitions.RawSemiring.html
new file mode 100644
index 000000000..d1acbd2a2
--- /dev/null
+++ b/site/static/agda_html/Algebra.Definitions.RawSemiring.html
@@ -0,0 +1,162 @@
+
+Algebra.Definitions.RawSemiring
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic auxiliary definitions for semiring-like structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles using (RawSemiring)

+
open import Data.Sum.Base using (_⊎_)

+
open import Data.Nat.Base using (ℕ; zero; suc)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (Rel)

+

+
module Algebra.Definitions.RawSemiring {a ℓ} (M : RawSemiring a ℓ) where

+

+
open RawSemiring M renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definitions over _+_

+

+
open import Algebra.Definitions.RawMonoid +-rawMonoid public

+
  using

+
  ( _×_   -- : ℕ → A → A

+
  ; _×′_  -- : ℕ → A → A

+
  ; sum   -- : Vector A n → A

+
  )

+

+
------------------------------------------------------------------------

+
-- Definitions over _*_

+

+
open import Algebra.Definitions.RawMonoid *-rawMonoid as Mult public

+
  using

+
  ( _∣_

+
  ; _∤_

+
  )

+
  renaming

+
  ( sum to product

+
  )

+

+
-- Unlike `sum` to `product`, can't simply rename multiplication to

+
-- exponentation as the argument order is reversed.

+

+
-- Standard exponentiation

+

+
infixr 8 _^_

+
_^_ : A → ℕ → A

+
x ^ n  = n Mult.× x

+

+
-- Exponentiation optimised for type-checking

+

+
infixr 8 _^′_

+
_^′_ : A → ℕ → A

+
x ^′ n  = n Mult.×′ x

+
{-# INLINE _^′_ #-}

+

+
-- Exponentiation optimised for tail-recursion

+

+
infixr 8 _^[_]*_ _^ᵗ_

+

+
_^[_]*_ : A → ℕ → A → A

+
x ^[ zero ]*  y = y

+
x ^[ suc n ]* y = x ^[ n ]* (x * y)

+

+
_^ᵗ_ : A → ℕ → A

+
x ^ᵗ n = x ^[ n ]* 1#

+

+
------------------------------------------------------------------------

+
-- Primality

+

+
Coprime : Rel A (a ⊔ ℓ)

+
Coprime x y = ∀ {z} → z ∣ x → z ∣ y → z ∣ 1#

+

+
record Irreducible (p : A) : Set (a ⊔ ℓ) where

+
  constructor mkIrred

+
  field

+
    p∤1       : p ∤ 1#

+
    split-∣1 : ∀ {x y} → p ≈ (x * y) → x ∣ 1# ⊎ y ∣ 1#

+

+
record Prime (p : A) : Set (a ⊔ ℓ) where

+
  constructor mkPrime

+
  field

+
    p≉0     : p ≉ 0#

+
    p∤1     : p ∤ 1#

+
    split-∣ : ∀ {x y} → p ∣ x * y → p ∣ x ⊎ p ∣ y

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Definitions.html b/site/static/agda_html/Algebra.Definitions.html
new file mode 100644
index 000000000..7e84723f9
--- /dev/null
+++ b/site/static/agda_html/Algebra.Definitions.html
@@ -0,0 +1,311 @@
+
+Algebra.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of functions, such as associativity and commutativity

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra`, unless

+
-- you want to parameterise it via the equality relation.

+

+
-- Note that very few of the element arguments are made implicit here,

+
-- as we do not assume that the Agda can infer either the right or left

+
-- argument of the binary operators. This is despite the fact that the

+
-- library defines most of its concrete operators (e.g. in

+
-- `Data.Nat.Base`) as being left-biased.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
module Algebra.Definitions

+
  {a ℓ} {A : Set a}   -- The underlying set

+
  (_≈_ : Rel A ℓ)     -- The underlying equality

+
  where

+

+
open import Algebra.Core using (Op₁; Op₂)

+
open import Data.Product.Base using (_×_; ∃-syntax)

+
open import Data.Sum.Base using (_⊎_)

+

+
------------------------------------------------------------------------

+
-- Properties of operations

+

+
Congruent₁ : Op₁ A → Set _

+
Congruent₁ f = f Preserves _≈_ ⟶ _≈_

+

+
Congruent₂ : Op₂ A → Set _

+
Congruent₂ ∙ = ∙ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_

+

+
LeftCongruent : Op₂ A → Set _

+
LeftCongruent _∙_ = ∀ {x} → (x ∙_) Preserves _≈_ ⟶ _≈_

+

+
RightCongruent : Op₂ A → Set _

+
RightCongruent _∙_ = ∀ {x} → (_∙ x) Preserves _≈_ ⟶ _≈_

+

+
Associative : Op₂ A → Set _

+
Associative _∙_ = ∀ x y z → ((x ∙ y) ∙ z) ≈ (x ∙ (y ∙ z))

+

+
Commutative : Op₂ A → Set _

+
Commutative _∙_ = ∀ x y → (x ∙ y) ≈ (y ∙ x)

+

+
LeftIdentity : A → Op₂ A → Set _

+
LeftIdentity e _∙_ = ∀ x → (e ∙ x) ≈ x

+

+
RightIdentity : A → Op₂ A → Set _

+
RightIdentity e _∙_ = ∀ x → (x ∙ e) ≈ x

+

+
Identity : A → Op₂ A → Set _

+
Identity e ∙ = (LeftIdentity e ∙) × (RightIdentity e ∙)

+

+
LeftZero : A → Op₂ A → Set _

+
LeftZero z _∙_ = ∀ x → (z ∙ x) ≈ z

+

+
RightZero : A → Op₂ A → Set _

+
RightZero z _∙_ = ∀ x → (x ∙ z) ≈ z

+

+
Zero : A → Op₂ A → Set _

+
Zero z ∙ = (LeftZero z ∙) × (RightZero z ∙)

+

+
LeftInverse : A → Op₁ A → Op₂ A → Set _

+
LeftInverse e _⁻¹ _∙_ = ∀ x → ((x ⁻¹) ∙ x) ≈ e

+

+
RightInverse : A → Op₁ A → Op₂ A → Set _

+
RightInverse e _⁻¹ _∙_ = ∀ x → (x ∙ (x ⁻¹)) ≈ e

+

+
Inverse : A → Op₁ A → Op₂ A → Set _

+
Inverse e ⁻¹ ∙ = (LeftInverse e ⁻¹) ∙ × (RightInverse e ⁻¹ ∙)

+

+
-- For structures in which not every element has an inverse (e.g. Fields)

+
LeftInvertible : A → Op₂ A → A → Set _

+
LeftInvertible e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e

+

+
RightInvertible : A → Op₂ A → A → Set _

+
RightInvertible e _∙_ x = ∃[ x⁻¹ ] (x ∙ x⁻¹) ≈ e

+

+
-- NB: this is not quite the same as

+
-- LeftInvertible e ∙ x × RightInvertible e ∙ x

+
-- since the left and right inverses have to coincide.

+
Invertible : A → Op₂ A → A → Set _

+
Invertible e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e × (x ∙ x⁻¹) ≈ e

+

+
LeftConical : A → Op₂ A → Set _

+
LeftConical e _∙_ = ∀ x y → (x ∙ y) ≈ e → x ≈ e

+

+
RightConical : A → Op₂ A → Set _

+
RightConical e _∙_ = ∀ x y → (x ∙ y) ≈ e → y ≈ e

+

+
Conical : A → Op₂ A → Set _

+
Conical e ∙ = (LeftConical e ∙) × (RightConical e ∙)

+

+
infix 4 _DistributesOverˡ_ _DistributesOverʳ_ _DistributesOver_

+

+
_DistributesOverˡ_ : Op₂ A → Op₂ A → Set _

+
_*_ DistributesOverˡ _+_ =

+
  ∀ x y z → (x * (y + z)) ≈ ((x * y) + (x * z))

+

+
_DistributesOverʳ_ : Op₂ A → Op₂ A → Set _

+
_*_ DistributesOverʳ _+_ =

+
  ∀ x y z → ((y + z) * x) ≈ ((y * x) + (z * x))

+

+
_DistributesOver_ : Op₂ A → Op₂ A → Set _

+
* DistributesOver + = (* DistributesOverˡ +) × (* DistributesOverʳ +)

+

+
infix 4 _MiddleFourExchange_ _IdempotentOn_ _Absorbs_

+

+
_MiddleFourExchange_ : Op₂ A → Op₂ A → Set _

+
_*_ MiddleFourExchange _+_ =

+
  ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))

+

+
_IdempotentOn_ : Op₂ A → A → Set _

+
_∙_ IdempotentOn x = (x ∙ x) ≈ x

+

+
Idempotent : Op₂ A → Set _

+
Idempotent ∙ = ∀ x → ∙ IdempotentOn x

+

+
IdempotentFun : Op₁ A → Set _

+
IdempotentFun f = ∀ x → f (f x) ≈ f x

+

+
Selective : Op₂ A → Set _

+
Selective _∙_ = ∀ x y → (x ∙ y) ≈ x ⊎ (x ∙ y) ≈ y

+

+
_Absorbs_ : Op₂ A → Op₂ A → Set _

+
_∙_ Absorbs _∘_ = ∀ x y → (x ∙ (x ∘ y)) ≈ x

+

+
Absorptive : Op₂ A → Op₂ A → Set _

+
Absorptive ∙ ∘ = (∙ Absorbs ∘) × (∘ Absorbs ∙)

+

+
SelfInverse : Op₁ A → Set _

+
SelfInverse f = ∀ {x y} → f x ≈ y → f y ≈ x

+

+
Involutive : Op₁ A → Set _

+
Involutive f = ∀ x → f (f x) ≈ x

+

+
LeftCancellative : Op₂ A → Set _

+
LeftCancellative _•_ = ∀ x y z → (x • y) ≈ (x • z) → y ≈ z

+

+
RightCancellative : Op₂ A → Set _

+
RightCancellative _•_ = ∀ x y z → (y • x) ≈ (z • x) → y ≈ z

+

+
Cancellative : Op₂ A → Set _

+
Cancellative _•_ = (LeftCancellative _•_) × (RightCancellative _•_)

+

+
AlmostLeftCancellative : A → Op₂ A → Set _

+
AlmostLeftCancellative e _•_ = ∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z

+

+
AlmostRightCancellative : A → Op₂ A → Set _

+
AlmostRightCancellative e _•_ = ∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z

+

+
AlmostCancellative : A → Op₂ A → Set _

+
AlmostCancellative e _•_ = AlmostLeftCancellative e _•_ × AlmostRightCancellative e _•_

+

+
Interchangable : Op₂ A → Op₂ A → Set _

+
Interchangable _∘_ _∙_ = ∀ w x y z → ((w ∙ x) ∘ (y ∙ z)) ≈ ((w ∘ y) ∙ (x ∘ z))

+

+
LeftDividesˡ : Op₂ A → Op₂ A → Set _

+
LeftDividesˡ _∙_  _\\_ = ∀ x y → (x ∙ (x \\ y)) ≈ y

+

+
LeftDividesʳ : Op₂ A → Op₂ A → Set _

+
LeftDividesʳ _∙_ _\\_ = ∀ x y → (x \\ (x ∙ y)) ≈ y

+

+
RightDividesˡ : Op₂ A → Op₂ A → Set _

+
RightDividesˡ _∙_ _//_ = ∀ x y → ((y // x) ∙ x) ≈ y

+

+
RightDividesʳ : Op₂ A → Op₂ A → Set _

+
RightDividesʳ _∙_ _//_ = ∀ x y → ((y ∙ x) // x) ≈ y

+

+
LeftDivides : Op₂ A → Op₂ A → Set _

+
LeftDivides ∙ \\ = (LeftDividesˡ ∙ \\) × (LeftDividesʳ ∙ \\)

+

+
RightDivides : Op₂ A → Op₂ A → Set _

+
RightDivides ∙ // = (RightDividesˡ ∙ //) × (RightDividesʳ ∙ //)

+

+
StarRightExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _

+
StarRightExpansive e _+_ _∙_ _* = ∀ x → (e + (x ∙ (x *))) ≈ (x *)

+

+
StarLeftExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _

+
StarLeftExpansive e _+_ _∙_ _* = ∀ x →  (e + ((x *) ∙ x)) ≈ (x *)

+

+
StarExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _

+
StarExpansive e _+_ _∙_ _* = (StarLeftExpansive e _+_ _∙_ _*) × (StarRightExpansive e _+_ _∙_ _*)

+

+
StarLeftDestructive : Op₂ A → Op₂ A → Op₁ A → Set _

+
StarLeftDestructive _+_ _∙_ _* = ∀ a b x → (b + (a ∙ x)) ≈ x → ((a *) ∙ b) ≈ x

+

+
StarRightDestructive : Op₂ A → Op₂ A → Op₁ A → Set _

+
StarRightDestructive _+_ _∙_ _* = ∀ a b x → (b + (x ∙ a)) ≈ x → (b ∙ (a *)) ≈ x

+

+
StarDestructive : Op₂ A → Op₂ A → Op₁ A → Set _

+
StarDestructive _+_ _∙_ _* = (StarLeftDestructive _+_ _∙_ _*) × (StarRightDestructive _+_ _∙_ _*)

+

+
LeftAlternative : Op₂ A → Set _

+
LeftAlternative _∙_ = ∀ x y  →  ((x ∙ x) ∙ y) ≈ (x ∙ (x ∙ y))

+

+
RightAlternative : Op₂ A → Set _

+
RightAlternative _∙_ = ∀ x y → (x ∙ (y ∙ y)) ≈ ((x ∙ y) ∙ y)

+

+
Alternative : Op₂ A → Set _

+
Alternative _∙_ = (LeftAlternative _∙_ ) × (RightAlternative _∙_)

+

+
Flexible : Op₂ A → Set _

+
Flexible _∙_ = ∀ x y → ((x ∙ y) ∙ x) ≈ (x ∙ (y ∙ x))

+

+
Medial : Op₂ A → Set _

+
Medial _∙_ = ∀ x y u z → ((x ∙ y) ∙ (u ∙ z)) ≈ ((x ∙ u) ∙ (y ∙ z))

+

+
LeftSemimedial : Op₂ A → Set _

+
LeftSemimedial _∙_ = ∀ x y z → ((x ∙ x) ∙ (y ∙ z)) ≈ ((x ∙ y) ∙ (x ∙ z))

+

+
RightSemimedial : Op₂ A → Set _

+
RightSemimedial _∙_ = ∀ x y z → ((y ∙ z) ∙ (x ∙ x)) ≈ ((y ∙ x) ∙ (z ∙ x))

+

+
Semimedial : Op₂ A → Set _

+
Semimedial _∙_ = (LeftSemimedial _∙_) × (RightSemimedial _∙_)

+

+
LeftBol : Op₂ A → Set _

+
LeftBol _∙_ = ∀ x y z → (x ∙ (y ∙ (x ∙ z))) ≈ ((x ∙ (y ∙ x)) ∙ z )

+

+
RightBol : Op₂ A → Set _

+
RightBol _∙_ = ∀ x y z → (((z ∙ x) ∙ y) ∙ x) ≈ (z ∙ ((x ∙ y) ∙ x))

+

+
MiddleBol : Op₂ A → Op₂ A  → Op₂ A  → Set _

+
MiddleBol _∙_ _\\_ _//_ = ∀ x y z → (x ∙ ((y ∙ z) \\ x)) ≈ ((x // z) ∙ (y \\ x))

+

+
Identical : Op₂ A → Set _

+
Identical _∙_ = ∀ x y z → ((z ∙ x) ∙ (y ∙ z)) ≈ (z ∙ ((x ∙ y) ∙ z))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Function.html b/site/static/agda_html/Algebra.Function.html
new file mode 100644
index 000000000..538cd3239
--- /dev/null
+++ b/site/static/agda_html/Algebra.Function.html
@@ -0,0 +1,159 @@
+
+Algebra.Function
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
open import Level

+
open import Algebra.Lattice

+
open import Data.Product

+

+
module Algebra.Function {a b c} (A : Set a) (B : BooleanAlgebra b c) where

+
  module B = BooleanAlgebra B

+

+
  Carrier : Set (a ⊔ b)

+
  Carrier = A → B.Carrier

+

+
  _≈_ : Carrier → Carrier → Set (a ⊔ c)

+
  _≈_ f g = ∀ a → f a B.≈ g a

+

+
  _∨_ : Carrier → Carrier → Carrier

+
  _∨_ f g a = f a B.∨ g a

+

+
  _∧_ : Carrier → Carrier → Carrier

+
  _∧_ f g a = f a B.∧ g a

+

+
  ¬_ : Carrier → Carrier

+
  ¬_ f a = B.¬ f a

+

+
  ⊤ : Carrier

+
  ⊤ a = B.⊤

+

+
  ⊥ : Carrier

+
  ⊥ a = B.⊥

+

+
  open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)

+
  open import Relation.Binary using (Setoid)

+
  import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial

+
    as Trivial

+

+
  open IsLattice

+
  isLattice' : IsLattice _≈_ _∨_ _∧_

+
  isLattice' = record

+
    { isEquivalence = Setoid.isEquivalence (≡-setoid A (Trivial.indexedSetoid B.setoid))

+
    ; ∨-comm = λ f g a → B.isLattice .∨-comm (f a) (g a)

+
    ; ∨-assoc = λ x y z a → B.isLattice .∨-assoc (x a) (y a) (z a)

+
    ; ∨-cong = λ eq₁ eq₂ a → B.isLattice .∨-cong (eq₁ a) (eq₂ a)

+
    ; ∧-comm = λ f g a → B.isLattice .∧-comm (f a) (g a)

+
    ; ∧-assoc = λ x y z a → B.isLattice .∧-assoc (x a) (y a) (z a)

+
    ; ∧-cong = λ eq₁ eq₂ a → B.isLattice .∧-cong (eq₁ a) (eq₂ a)

+
    ; absorptive =

+
      (λ x y a → B.isLattice .absorptive .proj₁ (x a) (y a)) ,

+
      (λ x y a → B.isLattice .absorptive .proj₂ (x a) (y a))

+
    }

+

+
  open IsDistributiveLattice

+
  isDistributiveLattice' : IsDistributiveLattice _≈_ _∨_ _∧_

+
  isDistributiveLattice' .isLattice = isLattice'

+
  isDistributiveLattice' .∨-distrib-∧ =

+
    (λ x y z a → B.isDistributiveLattice .∨-distrib-∧ .proj₁ (x a) (y a) (z a)) ,

+
    (λ x y z a → B.isDistributiveLattice .∨-distrib-∧ .proj₂ (x a) (y a) (z a))

+
  isDistributiveLattice' .∧-distrib-∨ =

+
    (λ x y z a → B.isDistributiveLattice .∧-distrib-∨ .proj₁ (x a) (y a) (z a)) ,

+
    (λ x y z a → B.isDistributiveLattice .∧-distrib-∨ .proj₂ (x a) (y a) (z a))

+

+
  open IsBooleanAlgebra

+
  isBooleanAlgebra : IsBooleanAlgebra _≈_ _∨_ _∧_ ¬_ ⊤ ⊥

+
  isBooleanAlgebra .isDistributiveLattice = isDistributiveLattice'

+
  isBooleanAlgebra .∨-complement =

+
    (λ f a → B.isBooleanAlgebra .∨-complement .proj₁ (f a)) ,

+
    (λ f a → B.isBooleanAlgebra .∨-complement .proj₂ (f a))

+
  isBooleanAlgebra .∧-complement =

+
    (λ f a → B.isBooleanAlgebra .∧-complement .proj₁ (f a)) ,

+
    (λ f a → B.isBooleanAlgebra .∧-complement .proj₂ (f a))

+
  isBooleanAlgebra .¬-cong = λ f a → B.isBooleanAlgebra .¬-cong (f a)

+

+
  hom : BooleanAlgebra _ _

+
  hom = record

+
          { Carrier = Carrier

+
          ; _≈_ = _≈_

+
          ; _∨_ = _∨_

+
          ; _∧_ = _∧_

+
          ; ¬_ = ¬_

+
          ; ⊤ = ⊤

+
          ; ⊥ = ⊥

+
          ; isBooleanAlgebra = isBooleanAlgebra

+
          }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Bundles.Raw.html b/site/static/agda_html/Algebra.Lattice.Bundles.Raw.html
new file mode 100644
index 000000000..26768cde1
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Bundles.Raw.html
@@ -0,0 +1,110 @@
+
+Algebra.Lattice.Bundles.Raw
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of 'raw' bundles

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Lattice.Bundles.Raw where

+

+
open import Algebra.Core

+
open import Algebra.Bundles.Raw using (RawMagma)

+
open import Level using (suc; _⊔_)

+
open import Relation.Binary.Core using (Rel)

+

+
record RawLattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∧_

+
  infixr 6 _∨_

+
  infix  4 _≈_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ

+
    _∧_     : Op₂ Carrier

+
    _∨_     : Op₂ Carrier

+

+
  ∨-rawMagma : RawMagma c ℓ

+
  ∨-rawMagma = record { _≈_ = _≈_; _∙_ = _∨_ }

+

+
  ∧-rawMagma : RawMagma c ℓ

+
  ∧-rawMagma = record { _≈_ = _≈_; _∙_ = _∧_ }

+

+
  open RawMagma ∨-rawMagma public

+
    using (_≉_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Bundles.html b/site/static/agda_html/Algebra.Lattice.Bundles.html
new file mode 100644
index 000000000..cf40a7ef2
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Bundles.html
@@ -0,0 +1,306 @@
+
+Algebra.Lattice.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of algebraic structures like semilattices and lattices

+
-- (packed in records together with sets, operations, etc.), defined via

+
-- meet/join operations and their properties

+
--

+
-- For lattices defined via an order relation, see

+
-- Relation.Binary.Lattice.

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra.Lattice`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Lattice.Bundles where

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
open import Algebra.Structures

+
import Algebra.Lattice.Bundles.Raw as Raw

+
open import Algebra.Lattice.Structures

+
open import Level using (suc; _⊔_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Core using (Rel)

+

+
------------------------------------------------------------------------

+
-- Re-export definitions of 'raw' bundles

+

+
open Raw public

+
  using (RawLattice)

+

+
------------------------------------------------------------------------

+
-- Bundles

+
------------------------------------------------------------------------

+

+
record Semilattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier       : Set c

+
    _≈_           : Rel Carrier ℓ

+
    _∙_           : Op₂ Carrier

+
    isSemilattice : IsSemilattice _≈_ _∙_

+

+
  open IsSemilattice _≈_ isSemilattice public

+

+
  band : Band c ℓ

+
  band = record { isBand = isBand }

+

+
  open Band band public

+
    using (_≉_; rawMagma; magma; isMagma; semigroup; isSemigroup; isBand)

+

+

+
record MeetSemilattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∧_

+
  infix  4 _≈_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ

+
    _∧_               : Op₂ Carrier

+
    isMeetSemilattice : IsSemilattice _≈_ _∧_

+

+
  open IsMeetSemilattice _≈_ isMeetSemilattice public

+

+
  semilattice : Semilattice c ℓ

+
  semilattice = record { isSemilattice = isMeetSemilattice }

+

+
  open Semilattice semilattice public

+
    using (rawMagma; magma; semigroup; band)

+

+

+
record JoinSemilattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∨_

+
  infix  4 _≈_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ

+
    _∨_               : Op₂ Carrier

+
    isJoinSemilattice : IsSemilattice _≈_ _∨_

+

+
  open IsJoinSemilattice _≈_ isJoinSemilattice public

+

+
  semilattice : Semilattice c ℓ

+
  semilattice = record { isSemilattice = isJoinSemilattice }

+

+
  open Semilattice semilattice public

+
    using (rawMagma; magma; semigroup; band)

+

+

+
record BoundedSemilattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∙_

+
  infix  4 _≈_

+
  field

+
    Carrier              : Set c

+
    _≈_                  : Rel Carrier ℓ

+
    _∙_                  : Op₂ Carrier

+
    ε                    : Carrier

+
    isBoundedSemilattice : IsBoundedSemilattice _≈_ _∙_ ε

+

+
  open IsBoundedSemilattice _≈_ isBoundedSemilattice public

+

+
  semilattice : Semilattice c ℓ

+
  semilattice = record { isSemilattice = isSemilattice }

+

+
  open Semilattice semilattice public using (rawMagma; magma; semigroup; band)

+

+

+
record BoundedMeetSemilattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∧_

+
  infix  4 _≈_

+
  field

+
    Carrier                  : Set c

+
    _≈_                      : Rel Carrier ℓ

+
    _∧_                      : Op₂ Carrier

+
    ⊤                        : Carrier

+
    isBoundedMeetSemilattice : IsBoundedSemilattice _≈_ _∧_ ⊤

+

+
  open IsBoundedMeetSemilattice _≈_ isBoundedMeetSemilattice public

+

+
  boundedSemilattice : BoundedSemilattice c ℓ

+
  boundedSemilattice = record

+
    { isBoundedSemilattice = isBoundedMeetSemilattice }

+

+
  open BoundedSemilattice boundedSemilattice public

+
    using (rawMagma; magma; semigroup; band; semilattice)

+

+

+
record BoundedJoinSemilattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∨_

+
  infix  4 _≈_

+
  field

+
    Carrier                  : Set c

+
    _≈_                      : Rel Carrier ℓ

+
    _∨_                      : Op₂ Carrier

+
    ⊥                        : Carrier

+
    isBoundedJoinSemilattice : IsBoundedSemilattice _≈_ _∨_ ⊥

+

+
  open IsBoundedJoinSemilattice _≈_ isBoundedJoinSemilattice public

+

+
  boundedSemilattice : BoundedSemilattice c ℓ

+
  boundedSemilattice = record

+
    { isBoundedSemilattice = isBoundedJoinSemilattice }

+

+
  open BoundedSemilattice boundedSemilattice public

+
    using (rawMagma; magma; semigroup; band; semilattice)

+

+

+
record Lattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∧_

+
  infixr 6 _∨_

+
  infix  4 _≈_

+
  field

+
    Carrier   : Set c

+
    _≈_       : Rel Carrier ℓ

+
    _∨_       : Op₂ Carrier

+
    _∧_       : Op₂ Carrier

+
    isLattice : IsLattice _≈_ _∨_ _∧_

+

+
  open IsLattice isLattice public

+

+
  rawLattice : RawLattice c ℓ

+
  rawLattice = record

+
    { _≈_  = _≈_

+
    ; _∧_  = _∧_

+
    ; _∨_  = _∨_

+
    }

+

+
  open RawLattice rawLattice public

+
    using (∨-rawMagma; ∧-rawMagma)

+

+
  setoid : Setoid c ℓ

+
  setoid = record { isEquivalence = isEquivalence }

+

+
  open Setoid setoid public

+
    using (_≉_)

+

+

+
record DistributiveLattice c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infixr 7 _∧_

+
  infixr 6 _∨_

+
  infix  4 _≈_

+
  field

+
    Carrier               : Set c

+
    _≈_                   : Rel Carrier ℓ

+
    _∨_                   : Op₂ Carrier

+
    _∧_                   : Op₂ Carrier

+
    isDistributiveLattice : IsDistributiveLattice _≈_ _∨_ _∧_

+

+
  open IsDistributiveLattice isDistributiveLattice public

+

+
  lattice : Lattice _ _

+
  lattice = record { isLattice = isLattice }

+

+
  open Lattice lattice public

+
    using

+
    ( _≉_; setoid; rawLattice

+
    ; ∨-rawMagma; ∧-rawMagma

+
    )

+

+

+
record BooleanAlgebra c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 ¬_

+
  infixr 7 _∧_

+
  infixr 6 _∨_

+
  infix  4 _≈_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ

+
    _∨_              : Op₂ Carrier

+
    _∧_              : Op₂ Carrier

+
    ¬_               : Op₁ Carrier

+
    ⊤                : Carrier

+
    ⊥                : Carrier

+
    isBooleanAlgebra : IsBooleanAlgebra _≈_ _∨_ _∧_ ¬_ ⊤ ⊥

+

+
  open IsBooleanAlgebra isBooleanAlgebra public

+

+
  distributiveLattice : DistributiveLattice _ _

+
  distributiveLattice = record

+
    { isDistributiveLattice = isDistributiveLattice

+
    }

+

+
  open DistributiveLattice distributiveLattice public

+
    using

+
    ( _≉_; setoid; rawLattice

+
    ; ∨-rawMagma; ∧-rawMagma

+
    ; lattice

+
    )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MaxOp.html b/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MaxOp.html
new file mode 100644
index 000000000..594b2a7da
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MaxOp.html
@@ -0,0 +1,103 @@
+
+Algebra.Lattice.Construct.NaturalChoice.MaxOp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of a max operator derived from a spec over a total

+
-- preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Construct.NaturalChoice.Base

+
import Algebra.Lattice.Construct.NaturalChoice.MinOp as MinOp

+
open import Relation.Binary.Bundles using (TotalPreorder)

+

+
module Algebra.Lattice.Construct.NaturalChoice.MaxOp

+
  {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂} (maxOp : MaxOperator O)

+
  where

+

+
private

+
  module Min = MinOp (MaxOp⇒MinOp maxOp)

+

+
open Min public

+
  using ()

+
  renaming

+
  ( ⊓-isSemilattice           to  ⊔-isSemilattice

+
  ; ⊓-semilattice             to  ⊔-semilattice

+
  )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.html b/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.html
new file mode 100644
index 000000000..3878ce472
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MinMaxOp.html
@@ -0,0 +1,172 @@
+
+Algebra.Lattice.Construct.NaturalChoice.MinMaxOp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of min and max operators specified over a total preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Lattice.Bundles

+
open import Algebra.Construct.NaturalChoice.Base

+
open import Relation.Binary.Bundles using (TotalPreorder)

+

+
module Algebra.Lattice.Construct.NaturalChoice.MinMaxOp

+
  {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂}

+
  (minOp : MinOperator O)

+
  (maxOp : MaxOperator O)

+
  where

+

+
open TotalPreorder O

+
open MinOperator minOp

+
open MaxOperator maxOp

+

+
open import Algebra.Lattice.Structures _≈_

+
open import Algebra.Construct.NaturalChoice.MinMaxOp minOp maxOp

+
open import Relation.Binary.Reasoning.Preorder preorder

+

+
------------------------------------------------------------------------

+
-- Re-export properties of individual operators

+

+
open import Algebra.Lattice.Construct.NaturalChoice.MinOp minOp public

+
open import Algebra.Lattice.Construct.NaturalChoice.MaxOp maxOp public

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
⊔-⊓-isLattice : IsLattice _⊔_ _⊓_

+
⊔-⊓-isLattice = record

+
  { isEquivalence = isEquivalence

+
  ; ∨-comm        = ⊔-comm

+
  ; ∨-assoc       = ⊔-assoc

+
  ; ∨-cong        = ⊔-cong

+
  ; ∧-comm        = ⊓-comm

+
  ; ∧-assoc       = ⊓-assoc

+
  ; ∧-cong        = ⊓-cong

+
  ; absorptive    = ⊔-⊓-absorptive

+
  }

+

+
⊓-⊔-isLattice : IsLattice _⊓_ _⊔_

+
⊓-⊔-isLattice = record

+
  { isEquivalence = isEquivalence

+
  ; ∨-comm        = ⊓-comm

+
  ; ∨-assoc       = ⊓-assoc

+
  ; ∨-cong        = ⊓-cong

+
  ; ∧-comm        = ⊔-comm

+
  ; ∧-assoc       = ⊔-assoc

+
  ; ∧-cong        = ⊔-cong

+
  ; absorptive    = ⊓-⊔-absorptive

+
  }

+

+
⊓-⊔-isDistributiveLattice : IsDistributiveLattice _⊓_ _⊔_

+
⊓-⊔-isDistributiveLattice = record

+
  { isLattice    = ⊓-⊔-isLattice

+
  ; ∨-distrib-∧  = ⊓-distrib-⊔

+
  ; ∧-distrib-∨  = ⊔-distrib-⊓

+
  }

+

+
⊔-⊓-isDistributiveLattice : IsDistributiveLattice _⊔_ _⊓_

+
⊔-⊓-isDistributiveLattice = record

+
  { isLattice    = ⊔-⊓-isLattice

+
  ; ∨-distrib-∧  = ⊔-distrib-⊓

+
  ; ∧-distrib-∨  = ⊓-distrib-⊔

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
⊔-⊓-lattice : Lattice _ _

+
⊔-⊓-lattice = record

+
  { isLattice = ⊔-⊓-isLattice

+
  }

+

+
⊓-⊔-lattice : Lattice _ _

+
⊓-⊔-lattice = record

+
  { isLattice = ⊓-⊔-isLattice

+
  }

+

+
⊔-⊓-distributiveLattice : DistributiveLattice _ _

+
⊔-⊓-distributiveLattice = record

+
  { isDistributiveLattice = ⊔-⊓-isDistributiveLattice

+
  }

+

+
⊓-⊔-distributiveLattice : DistributiveLattice _ _

+
⊓-⊔-distributiveLattice = record

+
  { isDistributiveLattice = ⊓-⊔-isDistributiveLattice

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MinOp.html b/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MinOp.html
new file mode 100644
index 000000000..36805f698
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Construct.NaturalChoice.MinOp.html
@@ -0,0 +1,116 @@
+
+Algebra.Lattice.Construct.NaturalChoice.MinOp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of a min operator derived from a spec over a total

+
-- preorder.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles

+
open import Algebra.Lattice.Bundles

+
open import Algebra.Construct.NaturalChoice.Base

+
open import Relation.Binary.Bundles using (TotalPreorder)

+

+
module Algebra.Lattice.Construct.NaturalChoice.MinOp

+
  {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂} (minOp : MinOperator O) where

+

+
open TotalPreorder O

+
open MinOperator minOp

+

+
open import Algebra.Lattice.Structures _≈_

+
open import Algebra.Construct.NaturalChoice.MinOp minOp

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
⊓-isSemilattice : IsSemilattice _⊓_

+
⊓-isSemilattice = record

+
  { isBand = ⊓-isBand

+
  ; comm   = ⊓-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
⊓-semilattice : Semilattice _ _

+
⊓-semilattice = record

+
  { isSemilattice = ⊓-isSemilattice

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Morphism.LatticeMonomorphism.html b/site/static/agda_html/Algebra.Lattice.Morphism.LatticeMonomorphism.html
new file mode 100644
index 000000000..678c2549e
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Morphism.LatticeMonomorphism.html
@@ -0,0 +1,200 @@
+
+Algebra.Lattice.Morphism.LatticeMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between lattice-like structures

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra

+
open import Algebra.Lattice

+
open import Algebra.Lattice.Morphism.Structures

+
import Algebra.Consequences.Setoid as Consequences

+
import Algebra.Morphism.MagmaMonomorphism as MagmaMonomorphisms

+
import Algebra.Lattice.Properties.Lattice as LatticeProperties

+
open import Data.Product.Base using (_,_; map)

+
open import Relation.Binary.Bundles using (Setoid)

+
import Relation.Binary.Morphism.RelMonomorphism as RelMonomorphisms

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
module Algebra.Lattice.Morphism.LatticeMonomorphism

+
  {a b ℓ₁ ℓ₂} {L₁ : RawLattice a ℓ₁} {L₂ : RawLattice b ℓ₂} {⟦_⟧}

+
  (isLatticeMonomorphism : IsLatticeMonomorphism L₁ L₂ ⟦_⟧)

+
  where

+

+
open IsLatticeMonomorphism isLatticeMonomorphism

+
open RawLattice L₁ renaming (_≈_ to _≈₁_; _∨_ to _∨_; _∧_ to _∧_)

+
open RawLattice L₂ renaming (_≈_ to _≈₂_; _∨_ to _⊔_; _∧_ to _⊓_)

+

+
------------------------------------------------------------------------

+
-- Re-export all properties of magma monomorphisms

+

+
open MagmaMonomorphisms ∨-isMagmaMonomorphism public

+
  using () renaming

+
  ( cong    to ∨-cong

+
  ; assoc   to ∨-assoc

+
  ; comm    to ∨-comm

+
  ; idem    to ∨-idem

+
  ; sel     to ∨-sel

+
  ; cancelˡ to ∨-cancelˡ

+
  ; cancelʳ to ∨-cancelʳ

+
  ; cancel  to ∨-cancel

+
  )

+

+
open MagmaMonomorphisms ∧-isMagmaMonomorphism public

+
  using () renaming

+
  ( cong    to ∧-cong

+
  ; assoc   to ∧-assoc

+
  ; comm    to ∧-comm

+
  ; idem    to ∧-idem

+
  ; sel     to ∧-sel

+
  ; cancelˡ to ∧-cancelˡ

+
  ; cancelʳ to ∧-cancelʳ

+
  ; cancel  to ∧-cancel

+
  )

+

+
------------------------------------------------------------------------

+
-- Lattice-specific properties

+

+
module _ (⊔-⊓-isLattice : IsLattice _≈₂_ _⊔_ _⊓_) where

+

+
  open IsLattice ⊔-⊓-isLattice using (isEquivalence) renaming

+
    ( ∨-congˡ     to ⊔-congˡ

+
    ; ∨-congʳ     to ⊔-congʳ

+
    ; ∧-cong      to ⊓-cong

+
    ; ∧-congˡ     to ⊓-congˡ

+
    ; ∨-absorbs-∧ to ⊔-absorbs-⊓

+
    ; ∧-absorbs-∨ to ⊓-absorbs-⊔

+
    )

+

+
  setoid : Setoid b ℓ₂

+
  setoid = record { isEquivalence = isEquivalence }

+

+
  open ≈-Reasoning setoid

+

+
  ∨-absorbs-∧ : _Absorbs_ _≈₁_ _∨_ _∧_

+
  ∨-absorbs-∧ x y = injective (begin

+
    ⟦ x ∨ x ∧ y ⟧                        ≈⟨ ∨-homo x (x ∧ y) ⟩

+
    ⟦ x ⟧ ⊔ ⟦ x ∧ y ⟧                    ≈⟨ ⊔-congˡ (∧-homo x y) ⟩

+
    ⟦ x ⟧ ⊔ ⟦ x ⟧ ⊓ ⟦ y ⟧                ≈⟨ ⊔-absorbs-⊓ ⟦ x ⟧ ⟦ y ⟧ ⟩

+
    ⟦ x ⟧                                ∎)

+

+
  ∧-absorbs-∨ : _Absorbs_ _≈₁_ _∧_ _∨_

+
  ∧-absorbs-∨ x y = injective (begin

+
    ⟦ x ∧ (x ∨ y) ⟧                      ≈⟨ ∧-homo x (x ∨ y) ⟩

+
    ⟦ x ⟧ ⊓ ⟦ x ∨ y ⟧                    ≈⟨ ⊓-congˡ (∨-homo x y) ⟩

+
    ⟦ x ⟧ ⊓ (⟦ x ⟧ ⊔ ⟦ y ⟧)              ≈⟨ ⊓-absorbs-⊔ ⟦ x ⟧ ⟦ y ⟧ ⟩

+
    ⟦ x ⟧                                ∎)

+

+
  absorptive : Absorptive _≈₁_ _∨_ _∧_

+
  absorptive = ∨-absorbs-∧ , ∧-absorbs-∨

+

+
  distribʳ : _DistributesOverʳ_ _≈₂_ _⊔_ _⊓_ → _DistributesOverʳ_ _≈₁_ _∨_ _∧_

+
  distribʳ distribʳ x y z = injective (begin

+
    ⟦ y ∧ z ∨ x ⟧                        ≈⟨ ∨-homo (y ∧ z) x ⟩

+
    ⟦ y ∧ z ⟧ ⊔ ⟦ x ⟧                    ≈⟨ ⊔-congʳ (∧-homo y z) ⟩

+
    ⟦ y ⟧ ⊓ ⟦ z ⟧ ⊔ ⟦ x ⟧                ≈⟨ distribʳ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩

+
    (⟦ y ⟧ ⊔ ⟦ x ⟧) ⊓ (⟦ z ⟧ ⊔ ⟦ x ⟧)    ≈⟨ ⊓-cong (∨-homo y x) (∨-homo z x) ⟨

+
    ⟦ y ∨ x ⟧ ⊓ ⟦ z ∨ x ⟧                ≈⟨ ∧-homo (y ∨ x) (z ∨ x) ⟨

+
    ⟦ (y ∨ x) ∧ (z ∨ x) ⟧                ∎)

+

+
isLattice : IsLattice _≈₂_ _⊔_ _⊓_ → IsLattice _≈₁_ _∨_ _∧_

+
isLattice isLattice = record

+
  { isEquivalence = RelMonomorphisms.isEquivalence isRelMonomorphism L.isEquivalence

+
  ; ∨-comm        = ∨-comm  LP.∨-isMagma L.∨-comm

+
  ; ∨-assoc       = ∨-assoc LP.∨-isMagma L.∨-assoc

+
  ; ∨-cong        = ∨-cong  LP.∨-isMagma

+
  ; ∧-comm        = ∧-comm  LP.∧-isMagma L.∧-comm

+
  ; ∧-assoc       = ∧-assoc LP.∧-isMagma L.∧-assoc

+
  ; ∧-cong        = ∧-cong  LP.∧-isMagma

+
  ; absorptive    = absorptive isLattice

+
  } where

+
    module L  = IsLattice isLattice

+
    module LP = LatticeProperties (record { isLattice = isLattice })

+

+
isDistributiveLattice : IsDistributiveLattice _≈₂_ _⊔_ _⊓_ →

+
                        IsDistributiveLattice _≈₁_ _∨_ _∧_

+
isDistributiveLattice isDL = isDistributiveLatticeʳʲᵐ (record

+
  { isLattice     = isLattice L.isLattice

+
  ; ∨-distribʳ-∧  = distribʳ  L.isLattice L.∨-distribʳ-∧

+
  }) where module L = IsDistributiveLattice isDL

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Morphism.Structures.html b/site/static/agda_html/Algebra.Lattice.Morphism.Structures.html
new file mode 100644
index 000000000..e2b545559
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Morphism.Structures.html
@@ -0,0 +1,195 @@
+
+Algebra.Lattice.Morphism.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Morphisms between algebraic lattice structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
open import Algebra.Morphism

+
open import Algebra.Lattice.Bundles

+
import Algebra.Morphism.Definitions as MorphismDefinitions

+
open import Level using (Level; _⊔_)

+
open import Function.Definitions

+
open import Relation.Binary.Morphism.Structures

+
open import Relation.Binary.Core

+

+
module Algebra.Lattice.Morphism.Structures where

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+

+
------------------------------------------------------------------------

+
-- Morphisms over lattice-like structures

+
------------------------------------------------------------------------

+

+
module LatticeMorphisms (L₁ : RawLattice a ℓ₁) (L₂ : RawLattice b ℓ₂) where

+

+
  open RawLattice L₁ renaming

+
    ( Carrier to A; _≈_ to _≈₁_

+
    ; _∧_ to _∧₁_; _∨_ to _∨₁_

+
    ; ∧-rawMagma to ∧-rawMagma₁

+
    ; ∨-rawMagma to ∨-rawMagma₁)

+

+
  open RawLattice L₂ renaming

+
    ( Carrier to B; _≈_ to _≈₂_

+
    ; _∧_ to _∧₂_; _∨_ to _∨₂_

+
    ; ∧-rawMagma to ∧-rawMagma₂

+
    ; ∨-rawMagma to ∨-rawMagma₂)

+

+
  module ∨ = MagmaMorphisms ∨-rawMagma₁ ∨-rawMagma₂

+
  module ∧ = MagmaMorphisms ∧-rawMagma₁ ∧-rawMagma₂

+

+
  open MorphismDefinitions A B _≈₂_

+

+

+
  record IsLatticeHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧

+
      ∧-homo            : Homomorphic₂ ⟦_⟧ _∧₁_ _∧₂_

+
      ∨-homo            : Homomorphic₂ ⟦_⟧ _∨₁_ _∨₂_

+

+
    open IsRelHomomorphism isRelHomomorphism public

+
      renaming (cong to ⟦⟧-cong)

+

+
    ∧-isMagmaHomomorphism : ∧.IsMagmaHomomorphism ⟦_⟧

+
    ∧-isMagmaHomomorphism = record

+
      { isRelHomomorphism = isRelHomomorphism

+
      ; homo = ∧-homo

+
      }

+

+
    ∨-isMagmaHomomorphism : ∨.IsMagmaHomomorphism ⟦_⟧

+
    ∨-isMagmaHomomorphism = record

+
      { isRelHomomorphism = isRelHomomorphism

+
      ; homo = ∨-homo

+
      }

+

+
  record IsLatticeMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isLatticeHomomorphism : IsLatticeHomomorphism ⟦_⟧

+
      injective             : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsLatticeHomomorphism isLatticeHomomorphism public

+

+
    ∨-isMagmaMonomorphism : ∨.IsMagmaMonomorphism ⟦_⟧

+
    ∨-isMagmaMonomorphism = record

+
      { isMagmaHomomorphism = ∨-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
    ∧-isMagmaMonomorphism : ∧.IsMagmaMonomorphism ⟦_⟧

+
    ∧-isMagmaMonomorphism = record

+
      { isMagmaHomomorphism = ∧-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
    open ∧.IsMagmaMonomorphism ∧-isMagmaMonomorphism public

+
      using (isRelMonomorphism)

+

+

+
  record IsLatticeIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isLatticeMonomorphism : IsLatticeMonomorphism ⟦_⟧

+
      surjective            : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsLatticeMonomorphism isLatticeMonomorphism public

+

+
    ∨-isMagmaIsomorphism : ∨.IsMagmaIsomorphism ⟦_⟧

+
    ∨-isMagmaIsomorphism = record

+
      { isMagmaMonomorphism = ∨-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    ∧-isMagmaIsomorphism : ∧.IsMagmaIsomorphism ⟦_⟧

+
    ∧-isMagmaIsomorphism = record

+
      { isMagmaMonomorphism = ∧-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    open ∧.IsMagmaIsomorphism ∧-isMagmaIsomorphism public

+
      using (isRelIsomorphism)

+

+
------------------------------------------------------------------------

+
-- Re-export contents of modules publicly

+

+
open LatticeMorphisms public

+

\ No newline at end of file
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+
+Algebra.Lattice.Properties.BooleanAlgebra
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties of Boolean algebras

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Lattice.Bundles

+

+
module Algebra.Lattice.Properties.BooleanAlgebra

+
  {b₁ b₂} (B : BooleanAlgebra b₁ b₂)

+
  where

+

+
open BooleanAlgebra B

+

+
import Algebra.Lattice.Properties.DistributiveLattice as DistribLatticeProperties

+
open import Algebra.Core

+
open import Algebra.Structures _≈_

+
open import Algebra.Definitions _≈_

+
open import Algebra.Consequences.Setoid setoid

+
open import Algebra.Bundles

+
open import Algebra.Lattice.Structures _≈_

+
open import Relation.Binary.Reasoning.Setoid setoid

+
open import Function.Base using (id; _$_; _⟨_⟩_)

+
open import Function.Bundles using (_⇔_; module Equivalence)

+
open import Data.Product.Base using (_,_)

+

+
------------------------------------------------------------------------

+
-- Export properties from distributive lattices

+

+
open DistribLatticeProperties distributiveLattice public

+

+
------------------------------------------------------------------------

+
-- The dual construction is also a boolean algebra

+

+
∧-∨-isBooleanAlgebra : IsBooleanAlgebra _∧_ _∨_ ¬_ ⊥ ⊤

+
∧-∨-isBooleanAlgebra = record

+
  { isDistributiveLattice = ∧-∨-isDistributiveLattice

+
  ; ∨-complement          = ∧-complement

+
  ; ∧-complement          = ∨-complement

+
  ; ¬-cong                = ¬-cong

+
  }

+

+
∧-∨-booleanAlgebra : BooleanAlgebra _ _

+
∧-∨-booleanAlgebra = record

+
  { isBooleanAlgebra = ∧-∨-isBooleanAlgebra

+
  }

+

+
------------------------------------------------------------------------

+
-- (∨, ∧, ⊥, ⊤) and (∧, ∨, ⊤, ⊥) are commutative semirings

+

+
∧-identityʳ : RightIdentity ⊤ _∧_

+
∧-identityʳ x = begin

+
  x ∧ ⊤          ≈⟨ ∧-congˡ (sym (∨-complementʳ _)) ⟩

+
  x ∧ (x ∨ ¬ x)  ≈⟨ ∧-absorbs-∨ _ _ ⟩

+
  x              ∎

+

+
∧-identityˡ : LeftIdentity ⊤ _∧_

+
∧-identityˡ = comm∧idʳ⇒idˡ ∧-comm ∧-identityʳ

+

+
∧-identity : Identity ⊤ _∧_

+
∧-identity = ∧-identityˡ , ∧-identityʳ

+

+
∨-identityʳ : RightIdentity ⊥ _∨_

+
∨-identityʳ x = begin

+
  x ∨ ⊥          ≈⟨ ∨-congˡ $ sym (∧-complementʳ _) ⟩

+
  x ∨ x ∧ ¬ x    ≈⟨ ∨-absorbs-∧ _ _ ⟩

+
  x              ∎

+

+
∨-identityˡ : LeftIdentity ⊥ _∨_

+
∨-identityˡ = comm∧idʳ⇒idˡ ∨-comm ∨-identityʳ

+

+
∨-identity : Identity ⊥ _∨_

+
∨-identity = ∨-identityˡ , ∨-identityʳ

+

+
∧-zeroʳ : RightZero ⊥ _∧_

+
∧-zeroʳ x = begin

+
  x ∧ ⊥          ≈⟨ ∧-congˡ (∧-complementʳ x) ⟨

+
  x ∧  x  ∧ ¬ x  ≈⟨ ∧-assoc x x (¬ x) ⟨

+
  (x ∧ x) ∧ ¬ x  ≈⟨  ∧-congʳ (∧-idem x) ⟩

+
  x       ∧ ¬ x  ≈⟨  ∧-complementʳ x ⟩

+
  ⊥              ∎

+

+
∧-zeroˡ : LeftZero ⊥ _∧_

+
∧-zeroˡ = comm∧zeʳ⇒zeˡ ∧-comm ∧-zeroʳ

+

+
∧-zero : Zero ⊥ _∧_

+
∧-zero = ∧-zeroˡ , ∧-zeroʳ

+

+
∨-zeroʳ : ∀ x → x ∨ ⊤ ≈ ⊤

+
∨-zeroʳ x = begin

+
  x ∨ ⊤          ≈⟨ ∨-congˡ (∨-complementʳ x) ⟨

+
  x ∨  x  ∨ ¬ x  ≈⟨ ∨-assoc x x (¬ x) ⟨

+
  (x ∨ x) ∨ ¬ x  ≈⟨ ∨-congʳ (∨-idem x) ⟩

+
  x       ∨ ¬ x  ≈⟨ ∨-complementʳ x ⟩

+
  ⊤              ∎

+

+
∨-zeroˡ : LeftZero ⊤ _∨_

+
∨-zeroˡ = comm∧zeʳ⇒zeˡ ∨-comm ∨-zeroʳ

+

+
∨-zero : Zero ⊤ _∨_

+
∨-zero = ∨-zeroˡ , ∨-zeroʳ

+

+
∨-⊥-isMonoid : IsMonoid _∨_ ⊥

+
∨-⊥-isMonoid = record

+
  { isSemigroup = ∨-isSemigroup

+
  ; identity    = ∨-identity

+
  }

+

+
∧-⊤-isMonoid : IsMonoid _∧_ ⊤

+
∧-⊤-isMonoid = record

+
  { isSemigroup = ∧-isSemigroup

+
  ; identity    = ∧-identity

+
  }

+

+
∨-⊥-isCommutativeMonoid : IsCommutativeMonoid _∨_ ⊥

+
∨-⊥-isCommutativeMonoid = record

+
  { isMonoid = ∨-⊥-isMonoid

+
  ; comm     = ∨-comm

+
  }

+

+
∧-⊤-isCommutativeMonoid : IsCommutativeMonoid _∧_ ⊤

+
∧-⊤-isCommutativeMonoid = record

+
  { isMonoid = ∧-⊤-isMonoid

+
  ; comm     = ∧-comm

+
  }

+

+
∨-∧-isSemiring : IsSemiring _∨_ _∧_ ⊥ ⊤

+
∨-∧-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = ∨-⊥-isCommutativeMonoid

+
    ; *-cong = ∧-cong

+
    ; *-assoc = ∧-assoc

+
    ; *-identity = ∧-identity

+
    ; distrib = ∧-distrib-∨

+
    }

+
  ; zero = ∧-zero

+
  }

+

+
∧-∨-isSemiring : IsSemiring _∧_ _∨_ ⊤ ⊥

+
∧-∨-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = ∧-⊤-isCommutativeMonoid

+
    ; *-cong = ∨-cong

+
    ; *-assoc = ∨-assoc

+
    ; *-identity = ∨-identity

+
    ; distrib = ∨-distrib-∧

+
    }

+
  ; zero = ∨-zero

+
  }

+

+
∨-∧-isCommutativeSemiring : IsCommutativeSemiring _∨_ _∧_ ⊥ ⊤

+
∨-∧-isCommutativeSemiring = record

+
  { isSemiring = ∨-∧-isSemiring

+
  ; *-comm = ∧-comm

+
  }

+

+
∧-∨-isCommutativeSemiring : IsCommutativeSemiring _∧_ _∨_ ⊤ ⊥

+
∧-∨-isCommutativeSemiring = record

+
  { isSemiring = ∧-∨-isSemiring

+
  ; *-comm = ∨-comm

+
  }

+

+
∨-∧-commutativeSemiring : CommutativeSemiring _ _

+
∨-∧-commutativeSemiring = record

+
  { isCommutativeSemiring = ∨-∧-isCommutativeSemiring

+
  }

+

+
∧-∨-commutativeSemiring : CommutativeSemiring _ _

+
∧-∨-commutativeSemiring = record

+
  { isCommutativeSemiring = ∧-∨-isCommutativeSemiring

+
  }

+

+
------------------------------------------------------------------------

+
-- Some other properties

+

+
-- I took the statement of this lemma (called Uniqueness of

+
-- Complements) from some course notes, "Boolean Algebra", written

+
-- by Gert Smolka.

+

+
private

+
  lemma : ∀ x y → x ∧ y ≈ ⊥ → x ∨ y ≈ ⊤ → ¬ x ≈ y

+
  lemma x y x∧y=⊥ x∨y=⊤ = begin

+
    ¬ x                ≈⟨ ∧-identityʳ _ ⟨

+
    ¬ x ∧ ⊤            ≈⟨ ∧-congˡ x∨y=⊤ ⟨

+
    ¬ x ∧ (x ∨ y)      ≈⟨  ∧-distribˡ-∨ _ _ _ ⟩

+
    ¬ x ∧ x ∨ ¬ x ∧ y  ≈⟨  ∨-congʳ $ ∧-complementˡ _ ⟩

+
    ⊥ ∨ ¬ x ∧ y        ≈⟨ ∨-congʳ x∧y=⊥ ⟨

+
    x ∧ y ∨ ¬ x ∧ y    ≈⟨ ∧-distribʳ-∨ _ _ _ ⟨

+
    (x ∨ ¬ x) ∧ y      ≈⟨  ∧-congʳ $ ∨-complementʳ _ ⟩

+
    ⊤ ∧ y              ≈⟨  ∧-identityˡ _ ⟩

+
    y                  ∎

+

+
⊥≉⊤ : ¬ ⊥ ≈ ⊤

+
⊥≉⊤ = lemma ⊥ ⊤ (∧-identityʳ _) (∨-zeroʳ _)

+

+
⊤≉⊥ : ¬ ⊤ ≈ ⊥

+
⊤≉⊥ = lemma ⊤ ⊥ (∧-zeroʳ _) (∨-identityʳ _)

+

+
¬-involutive : Involutive ¬_

+
¬-involutive x = lemma (¬ x) x (∧-complementˡ _) (∨-complementˡ _)

+

+
deMorgan₁ : ∀ x y → ¬ (x ∧ y) ≈ ¬ x ∨ ¬ y

+
deMorgan₁ x y = lemma (x ∧ y) (¬ x ∨ ¬ y) lem₁ lem₂

+
  where

+
  lem₁ = begin

+
    (x ∧ y) ∧ (¬ x ∨ ¬ y)          ≈⟨ ∧-distribˡ-∨ _ _ _ ⟩

+
    (x ∧ y) ∧ ¬ x ∨ (x ∧ y) ∧ ¬ y  ≈⟨ ∨-congʳ $ ∧-congʳ $ ∧-comm _ _ ⟩

+
    (y ∧ x) ∧ ¬ x ∨ (x ∧ y) ∧ ¬ y  ≈⟨ ∧-assoc _ _ _ ⟨ ∨-cong ⟩ ∧-assoc _ _ _ ⟩

+
    y ∧ (x ∧ ¬ x) ∨ x ∧ (y ∧ ¬ y)  ≈⟨ (∧-congˡ $ ∧-complementʳ _) ⟨ ∨-cong ⟩

+
                                      (∧-congˡ $ ∧-complementʳ _) ⟩

+
    (y ∧ ⊥) ∨ (x ∧ ⊥)              ≈⟨ ∧-zeroʳ _ ⟨ ∨-cong ⟩ ∧-zeroʳ _ ⟩

+
    ⊥ ∨ ⊥                          ≈⟨ ∨-identityʳ _ ⟩

+
    ⊥                              ∎

+

+
  lem₃ = begin

+
    (x ∧ y) ∨ ¬ x          ≈⟨ ∨-distribʳ-∧ _ _ _ ⟩

+
    (x ∨ ¬ x) ∧ (y ∨ ¬ x)  ≈⟨ ∧-congʳ $ ∨-complementʳ _ ⟩

+
    ⊤ ∧ (y ∨ ¬ x)          ≈⟨ ∧-identityˡ _ ⟩

+
    y ∨ ¬ x                ≈⟨ ∨-comm _ _ ⟩

+
    ¬ x ∨ y                ∎

+

+
  lem₂ = begin

+
    (x ∧ y) ∨ (¬ x ∨ ¬ y)  ≈⟨ ∨-assoc _ _ _ ⟨

+
    ((x ∧ y) ∨ ¬ x) ∨ ¬ y  ≈⟨ ∨-congʳ lem₃ ⟩

+
    (¬ x ∨ y) ∨ ¬ y        ≈⟨ ∨-assoc _ _ _ ⟩

+
    ¬ x ∨ (y ∨ ¬ y)        ≈⟨ ∨-congˡ $ ∨-complementʳ _ ⟩

+
    ¬ x ∨ ⊤                ≈⟨ ∨-zeroʳ _ ⟩

+
    ⊤                      ∎

+

+
deMorgan₂ : ∀ x y → ¬ (x ∨ y) ≈ ¬ x ∧ ¬ y

+
deMorgan₂ x y = begin

+
  ¬ (x ∨ y)          ≈⟨ ¬-cong $ ((¬-involutive _) ⟨ ∨-cong ⟩ (¬-involutive _)) ⟨

+
  ¬ (¬ ¬ x ∨ ¬ ¬ y)  ≈⟨ ¬-cong $ deMorgan₁ _ _ ⟨

+
  ¬ ¬ (¬ x ∧ ¬ y)    ≈⟨ ¬-involutive _ ⟩

+
  ¬ x ∧ ¬ y          ∎

+

+
------------------------------------------------------------------------

+
-- (⊕, ∧, id, ⊥, ⊤) is a commutative ring

+

+
-- This construction is parameterised over the definition of xor.

+

+
module XorRing

+
  (xor : Op₂ Carrier)

+
  (⊕-def : ∀ x y → xor x y ≈ (x ∨ y) ∧ ¬ (x ∧ y))

+
  where

+

+
  private

+
    infixl 6 _⊕_

+

+
    _⊕_ : Op₂ Carrier

+
    _⊕_ = xor

+

+
    helper : ∀ {x y u v} → x ≈ y → u ≈ v → x ∧ ¬ u ≈ y ∧ ¬ v

+
    helper x≈y u≈v = x≈y ⟨ ∧-cong ⟩ ¬-cong u≈v

+

+
  ⊕-cong : Congruent₂ _⊕_

+
  ⊕-cong {x} {y} {u} {v} x≈y u≈v = begin

+
    x ⊕ u                ≈⟨  ⊕-def _ _ ⟩

+
    (x ∨ u) ∧ ¬ (x ∧ u)  ≈⟨  helper (x≈y ⟨ ∨-cong ⟩ u≈v)

+
                                    (x≈y ⟨ ∧-cong ⟩ u≈v) ⟩

+
    (y ∨ v) ∧ ¬ (y ∧ v)  ≈⟨ ⊕-def _ _ ⟨

+
    y ⊕ v                ∎

+

+
  ⊕-comm : Commutative _⊕_

+
  ⊕-comm x y = begin

+
    x ⊕ y                ≈⟨  ⊕-def _ _ ⟩

+
    (x ∨ y) ∧ ¬ (x ∧ y)  ≈⟨  helper (∨-comm _ _) (∧-comm _ _) ⟩

+
    (y ∨ x) ∧ ¬ (y ∧ x)  ≈⟨ ⊕-def _ _ ⟨

+
    y ⊕ x                ∎

+

+
  ¬-distribˡ-⊕ : ∀ x y → ¬ (x ⊕ y) ≈ ¬ x ⊕ y

+
  ¬-distribˡ-⊕ x y = begin

+
    ¬ (x ⊕ y)                              ≈⟨ ¬-cong $ ⊕-def _ _ ⟩

+
    ¬ ((x ∨ y) ∧ (¬ (x ∧ y)))              ≈⟨ ¬-cong (∧-distribʳ-∨ _ _ _) ⟩

+
    ¬ ((x ∧ ¬ (x ∧ y)) ∨ (y ∧ ¬ (x ∧ y)))  ≈⟨ ¬-cong $ ∨-congˡ $ ∧-congˡ $ ¬-cong (∧-comm _ _) ⟩

+
    ¬ ((x ∧ ¬ (x ∧ y)) ∨ (y ∧ ¬ (y ∧ x)))  ≈⟨ ¬-cong $ lem _ _ ⟨ ∨-cong ⟩ lem _ _ ⟩

+
    ¬ ((x ∧ ¬ y) ∨ (y ∧ ¬ x))              ≈⟨ deMorgan₂ _ _ ⟩

+
    ¬ (x ∧ ¬ y) ∧ ¬ (y ∧ ¬ x)              ≈⟨ ∧-congʳ $ deMorgan₁ _ _ ⟩

+
    (¬ x ∨ (¬ ¬ y)) ∧ ¬ (y ∧ ¬ x)          ≈⟨ helper (∨-congˡ $ ¬-involutive _) (∧-comm _ _) ⟩

+
    (¬ x ∨ y) ∧ ¬ (¬ x ∧ y)                ≈⟨ ⊕-def _ _ ⟨

+
    ¬ x ⊕ y                                ∎

+
    where

+
    lem : ∀ x y → x ∧ ¬ (x ∧ y) ≈ x ∧ ¬ y

+
    lem x y = begin

+
      x ∧ ¬ (x ∧ y)          ≈⟨ ∧-congˡ $ deMorgan₁ _ _ ⟩

+
      x ∧ (¬ x ∨ ¬ y)        ≈⟨ ∧-distribˡ-∨ _ _ _ ⟩

+
      (x ∧ ¬ x) ∨ (x ∧ ¬ y)  ≈⟨ ∨-congʳ $ ∧-complementʳ _ ⟩

+
      ⊥ ∨ (x ∧ ¬ y)          ≈⟨ ∨-identityˡ _ ⟩

+
      x ∧ ¬ y                ∎

+

+
  ¬-distribʳ-⊕ : ∀ x y → ¬ (x ⊕ y) ≈ x ⊕ ¬ y

+
  ¬-distribʳ-⊕ x y = begin

+
    ¬ (x ⊕ y)  ≈⟨ ¬-cong $ ⊕-comm _ _ ⟩

+
    ¬ (y ⊕ x)  ≈⟨ ¬-distribˡ-⊕ _ _ ⟩

+
    ¬ y ⊕ x    ≈⟨ ⊕-comm _ _ ⟩

+
    x ⊕ ¬ y    ∎

+

+
  ⊕-annihilates-¬ : ∀ x y → x ⊕ y ≈ ¬ x ⊕ ¬ y

+
  ⊕-annihilates-¬ x y = begin

+
    x ⊕ y        ≈⟨ ¬-involutive _ ⟨

+
    ¬ ¬ (x ⊕ y)  ≈⟨  ¬-cong $ ¬-distribˡ-⊕ _ _ ⟩

+
    ¬ (¬ x ⊕ y)  ≈⟨  ¬-distribʳ-⊕ _ _ ⟩

+
    ¬ x ⊕ ¬ y    ∎

+

+
  ⊕-identityˡ : LeftIdentity ⊥ _⊕_

+
  ⊕-identityˡ x = begin

+
    ⊥ ⊕ x                ≈⟨ ⊕-def _ _ ⟩

+
    (⊥ ∨ x) ∧ ¬ (⊥ ∧ x)  ≈⟨ helper (∨-identityˡ _) (∧-zeroˡ _) ⟩

+
    x ∧ ¬ ⊥              ≈⟨ ∧-congˡ ⊥≉⊤ ⟩

+
    x ∧ ⊤                ≈⟨ ∧-identityʳ _ ⟩

+
    x                    ∎

+

+
  ⊕-identityʳ : RightIdentity ⊥ _⊕_

+
  ⊕-identityʳ _ = ⊕-comm _ _ ⟨ trans ⟩ ⊕-identityˡ _

+

+
  ⊕-identity : Identity ⊥ _⊕_

+
  ⊕-identity = ⊕-identityˡ , ⊕-identityʳ

+

+
  ⊕-inverseˡ : LeftInverse ⊥ id _⊕_

+
  ⊕-inverseˡ x = begin

+
    x ⊕ x               ≈⟨ ⊕-def _ _ ⟩

+
    (x ∨ x) ∧ ¬ (x ∧ x) ≈⟨ helper (∨-idem _) (∧-idem _) ⟩

+
    x ∧ ¬ x             ≈⟨ ∧-complementʳ _ ⟩

+
    ⊥                   ∎

+

+
  ⊕-inverseʳ : RightInverse ⊥ id _⊕_

+
  ⊕-inverseʳ _ = ⊕-comm _ _ ⟨ trans ⟩ ⊕-inverseˡ _

+

+
  ⊕-inverse : Inverse ⊥ id _⊕_

+
  ⊕-inverse = ⊕-inverseˡ , ⊕-inverseʳ

+

+
  ∧-distribˡ-⊕ : _∧_ DistributesOverˡ _⊕_

+
  ∧-distribˡ-⊕ x y z = begin

+
    x ∧ (y ⊕ z)                ≈⟨ ∧-congˡ $ ⊕-def _ _ ⟩

+
    x ∧ ((y ∨ z) ∧ ¬ (y ∧ z))  ≈⟨ ∧-assoc _ _ _ ⟨

+
    (x ∧ (y ∨ z)) ∧ ¬ (y ∧ z)  ≈⟨ ∧-congˡ $ deMorgan₁ _ _ ⟩

+
    (x ∧ (y ∨ z)) ∧

+
    (¬ y ∨ ¬ z)                ≈⟨ ∨-identityˡ _ ⟨

+
    ⊥ ∨

+
    ((x ∧ (y ∨ z)) ∧

+
    (¬ y ∨ ¬ z))               ≈⟨ ∨-congʳ lem₃ ⟩

+
    ((x ∧ (y ∨ z)) ∧ ¬ x) ∨

+
    ((x ∧ (y ∨ z)) ∧

+
    (¬ y ∨ ¬ z))               ≈⟨ ∧-distribˡ-∨ _ _ _ ⟨

+
    (x ∧ (y ∨ z)) ∧

+
    (¬ x ∨ (¬ y ∨ ¬ z))        ≈⟨ ∧-congˡ $ ∨-congˡ (deMorgan₁ _ _) ⟨

+
    (x ∧ (y ∨ z)) ∧

+
    (¬ x ∨ ¬ (y ∧ z))          ≈⟨ ∧-congˡ (deMorgan₁ _ _) ⟨

+
    (x ∧ (y ∨ z)) ∧

+
    ¬ (x ∧ (y ∧ z))            ≈⟨ helper refl lem₁ ⟩

+
    (x ∧ (y ∨ z)) ∧

+
    ¬ ((x ∧ y) ∧ (x ∧ z))      ≈⟨ ∧-congʳ $ ∧-distribˡ-∨ _ _ _ ⟩

+
    ((x ∧ y) ∨ (x ∧ z)) ∧

+
    ¬ ((x ∧ y) ∧ (x ∧ z))      ≈⟨ ⊕-def _ _ ⟨

+
    (x ∧ y) ⊕ (x ∧ z)          ∎

+
      where

+
      lem₂ = begin

+
        x ∧ (y ∧ z)  ≈⟨ ∧-assoc _ _ _ ⟨

+
        (x ∧ y) ∧ z  ≈⟨ ∧-congʳ $ ∧-comm _ _ ⟩

+
        (y ∧ x) ∧ z  ≈⟨ ∧-assoc _ _ _ ⟩

+
        y ∧ (x ∧ z)  ∎

+

+
      lem₁ = begin

+
        x ∧ (y ∧ z)        ≈⟨ ∧-congʳ (∧-idem _) ⟨

+
        (x ∧ x) ∧ (y ∧ z)  ≈⟨ ∧-assoc _ _ _ ⟩

+
        x ∧ (x ∧ (y ∧ z))  ≈⟨ ∧-congˡ lem₂ ⟩

+
        x ∧ (y ∧ (x ∧ z))  ≈⟨ ∧-assoc _ _ _ ⟨

+
        (x ∧ y) ∧ (x ∧ z)  ∎

+

+
      lem₃ = begin

+
        ⊥                      ≈⟨ ∧-zeroʳ _ ⟨

+
        (y ∨ z) ∧ ⊥            ≈⟨ ∧-congˡ (∧-complementʳ _) ⟨

+
        (y ∨ z) ∧ (x ∧ ¬ x)    ≈⟨ ∧-assoc _ _ _ ⟨

+
        ((y ∨ z) ∧ x) ∧ ¬ x    ≈⟨ ∧-congʳ (∧-comm _ _) ⟩

+
        (x ∧ (y ∨ z)) ∧ ¬ x    ∎

+

+
  ∧-distribʳ-⊕ : _∧_ DistributesOverʳ _⊕_

+
  ∧-distribʳ-⊕ = comm∧distrˡ⇒distrʳ ⊕-cong ∧-comm ∧-distribˡ-⊕

+

+
  ∧-distrib-⊕ : _∧_ DistributesOver _⊕_

+
  ∧-distrib-⊕ = ∧-distribˡ-⊕ , ∧-distribʳ-⊕

+

+
  private

+

+
    lemma₂ : ∀ x y u v →

+
             (x ∧ y) ∨ (u ∧ v) ≈

+
             ((x ∨ u) ∧ (y ∨ u)) ∧

+
             ((x ∨ v) ∧ (y ∨ v))

+
    lemma₂ x y u v = begin

+
        (x ∧ y) ∨ (u ∧ v)              ≈⟨ ∨-distribˡ-∧ _ _ _ ⟩

+
        ((x ∧ y) ∨ u) ∧ ((x ∧ y) ∨ v)  ≈⟨ ∨-distribʳ-∧ _ _ _

+
                                            ⟨ ∧-cong ⟩

+
                                          ∨-distribʳ-∧ _ _ _ ⟩

+
        ((x ∨ u) ∧ (y ∨ u)) ∧

+
        ((x ∨ v) ∧ (y ∨ v))            ∎

+

+
  ⊕-assoc : Associative _⊕_

+
  ⊕-assoc x y z = sym $ begin

+
    x ⊕ (y ⊕ z)                                ≈⟨ ⊕-cong refl (⊕-def _ _) ⟩

+
    x ⊕ ((y ∨ z) ∧ ¬ (y ∧ z))                  ≈⟨ ⊕-def _ _ ⟩

+
      (x ∨ ((y ∨ z) ∧ ¬ (y ∧ z))) ∧

+
    ¬ (x ∧ ((y ∨ z) ∧ ¬ (y ∧ z)))              ≈⟨ ∧-cong lem₃ lem₄ ⟩

+
    (((x ∨ y) ∨ z) ∧ ((x ∨ ¬ y) ∨ ¬ z)) ∧

+
    (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ ∧-assoc _ _ _ ⟩

+
    ((x ∨ y) ∨ z) ∧

+
    (((x ∨ ¬ y) ∨ ¬ z) ∧

+
     (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z)))  ≈⟨ ∧-congˡ lem₅ ⟩

+
    ((x ∨ y) ∨ z) ∧

+
    (((¬ x ∨ ¬ y) ∨ z) ∧

+
     (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z)))  ≈⟨ ∧-assoc _ _ _ ⟨

+
    (((x ∨ y) ∨ z) ∧ ((¬ x ∨ ¬ y) ∨ z)) ∧

+
    (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ ∧-cong lem₁ lem₂ ⟩

+
      (((x ∨ y) ∧ ¬ (x ∧ y)) ∨ z) ∧

+
    ¬ (((x ∨ y) ∧ ¬ (x ∧ y)) ∧ z)              ≈⟨ ⊕-def _ _ ⟨

+
    ((x ∨ y) ∧ ¬ (x ∧ y)) ⊕ z                  ≈⟨ ⊕-cong (⊕-def _ _) refl ⟨

+
    (x ⊕ y) ⊕ z                                ∎

+
    where

+
    lem₁ = begin

+
      ((x ∨ y) ∨ z) ∧ ((¬ x ∨ ¬ y) ∨ z)  ≈⟨ ∨-distribʳ-∧ _ _ _ ⟨

+
      ((x ∨ y) ∧ (¬ x ∨ ¬ y)) ∨ z        ≈⟨ ∨-congʳ $ ∧-congˡ (deMorgan₁ _ _) ⟨

+
      ((x ∨ y) ∧ ¬ (x ∧ y)) ∨ z          ∎

+

+
    lem₂′ = begin

+
      (x ∨ ¬ y) ∧ (¬ x ∨ y)              ≈⟨ ∧-cong (∧-identityˡ _) (∧-identityʳ _) ⟨

+
      (⊤ ∧ (x ∨ ¬ y)) ∧ ((¬ x ∨ y) ∧ ⊤)  ≈⟨ ∧-cong

+
                                              (∧-cong (∨-complementˡ _) (∨-comm _ _))

+
                                              (∧-congˡ $ ∨-complementˡ _) ⟨

+
      ((¬ x ∨ x) ∧ (¬ y ∨ x)) ∧

+
      ((¬ x ∨ y) ∧ (¬ y ∨ y))            ≈⟨ lemma₂ _ _ _ _ ⟨

+
      (¬ x ∧ ¬ y) ∨ (x ∧ y)              ≈⟨ ∨-cong (deMorgan₂ _ _) (¬-involutive _) ⟨

+
      ¬ (x ∨ y) ∨ ¬ ¬ (x ∧ y)            ≈⟨ deMorgan₁ _ _ ⟨

+
      ¬ ((x ∨ y) ∧ ¬ (x ∧ y))            ∎

+

+
    lem₂ = begin

+
      ((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z)  ≈⟨ ∨-distribʳ-∧ _ _ _ ⟨

+
      ((x ∨ ¬ y) ∧ (¬ x ∨ y)) ∨ ¬ z          ≈⟨ ∨-congʳ lem₂′ ⟩

+
      ¬ ((x ∨ y) ∧ ¬ (x ∧ y)) ∨ ¬ z          ≈⟨ deMorgan₁ _ _ ⟨

+
      ¬ (((x ∨ y) ∧ ¬ (x ∧ y)) ∧ z)          ∎

+

+
    lem₃ = begin

+
      x ∨ ((y ∨ z) ∧ ¬ (y ∧ z))          ≈⟨ ∨-congˡ $ ∧-congˡ $ deMorgan₁ _ _ ⟩

+
      x ∨ ((y ∨ z) ∧ (¬ y ∨ ¬ z))        ≈⟨ ∨-distribˡ-∧ _ _ _ ⟩

+
      (x ∨ (y ∨ z)) ∧ (x ∨ (¬ y ∨ ¬ z))  ≈⟨ ∨-assoc _ _ _ ⟨ ∧-cong ⟩ ∨-assoc _ _ _ ⟨

+
      ((x ∨ y) ∨ z) ∧ ((x ∨ ¬ y) ∨ ¬ z)  ∎

+

+
    lem₄′ = begin

+
      ¬ ((y ∨ z) ∧ ¬ (y ∧ z))    ≈⟨ deMorgan₁ _ _ ⟩

+
      ¬ (y ∨ z) ∨ ¬ ¬ (y ∧ z)    ≈⟨ deMorgan₂ _ _ ⟨ ∨-cong ⟩ ¬-involutive _ ⟩

+
      (¬ y ∧ ¬ z) ∨ (y ∧ z)      ≈⟨ lemma₂ _ _ _ _ ⟩

+
      ((¬ y ∨ y) ∧ (¬ z ∨ y)) ∧

+
      ((¬ y ∨ z) ∧ (¬ z ∨ z))    ≈⟨ (∨-complementˡ _ ⟨ ∧-cong ⟩ ∨-comm _ _)

+
                                      ⟨ ∧-cong ⟩

+
                                   (∧-congˡ $ ∨-complementˡ _) ⟩

+
      (⊤ ∧ (y ∨ ¬ z)) ∧

+
      ((¬ y ∨ z) ∧ ⊤)            ≈⟨ ∧-identityˡ _ ⟨ ∧-cong ⟩ ∧-identityʳ _ ⟩

+
      (y ∨ ¬ z) ∧ (¬ y ∨ z)      ∎

+

+
    lem₄ = begin

+
      ¬ (x ∧ ((y ∨ z) ∧ ¬ (y ∧ z)))  ≈⟨ deMorgan₁ _ _ ⟩

+
      ¬ x ∨ ¬ ((y ∨ z) ∧ ¬ (y ∧ z))  ≈⟨ ∨-congˡ lem₄′ ⟩

+
      ¬ x ∨ ((y ∨ ¬ z) ∧ (¬ y ∨ z))  ≈⟨ ∨-distribˡ-∧ _ _ _ ⟩

+
      (¬ x ∨ (y     ∨ ¬ z)) ∧

+
      (¬ x ∨ (¬ y ∨ z))              ≈⟨ ∨-assoc _ _ _ ⟨ ∧-cong ⟩ ∨-assoc _ _ _ ⟨

+
      ((¬ x ∨ y)     ∨ ¬ z) ∧

+
      ((¬ x ∨ ¬ y) ∨ z)              ≈⟨ ∧-comm _ _ ⟩

+
      ((¬ x ∨ ¬ y) ∨ z) ∧

+
      ((¬ x ∨ y)     ∨ ¬ z)          ∎

+

+
    lem₅ = begin

+
      ((x ∨ ¬ y) ∨ ¬ z) ∧

+
      (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ ∧-assoc _ _ _ ⟨

+
      (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ ¬ y) ∨ z)) ∧

+
      ((¬ x ∨ y) ∨ ¬ z)                          ≈⟨ ∧-congʳ $ ∧-comm _ _ ⟩

+
      (((¬ x ∨ ¬ y) ∨ z) ∧ ((x ∨ ¬ y) ∨ ¬ z)) ∧

+
      ((¬ x ∨ y) ∨ ¬ z)                          ≈⟨ ∧-assoc _ _ _ ⟩

+
      ((¬ x ∨ ¬ y) ∨ z) ∧

+
      (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ∎

+

+
  ⊕-isMagma : IsMagma _⊕_

+
  ⊕-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = ⊕-cong

+
    }

+

+
  ⊕-isSemigroup : IsSemigroup _⊕_

+
  ⊕-isSemigroup = record

+
    { isMagma = ⊕-isMagma

+
    ; assoc   = ⊕-assoc

+
    }

+

+
  ⊕-⊥-isMonoid : IsMonoid _⊕_ ⊥

+
  ⊕-⊥-isMonoid = record

+
    { isSemigroup = ⊕-isSemigroup

+
    ; identity    = ⊕-identity

+
    }

+

+
  ⊕-⊥-isGroup : IsGroup _⊕_ ⊥ id

+
  ⊕-⊥-isGroup = record

+
    { isMonoid = ⊕-⊥-isMonoid

+
    ; inverse  = ⊕-inverse

+
    ; ⁻¹-cong  = id

+
    }

+

+
  ⊕-⊥-isAbelianGroup : IsAbelianGroup _⊕_ ⊥ id

+
  ⊕-⊥-isAbelianGroup = record

+
    { isGroup = ⊕-⊥-isGroup

+
    ; comm    = ⊕-comm

+
    }

+

+
  ⊕-∧-isRing : IsRing _⊕_ _∧_ id ⊥ ⊤

+
  ⊕-∧-isRing = record

+
    { +-isAbelianGroup = ⊕-⊥-isAbelianGroup

+
    ; *-cong = ∧-cong

+
    ; *-assoc = ∧-assoc

+
    ; *-identity = ∧-identity

+
    ; distrib = ∧-distrib-⊕

+
    }

+

+
  ⊕-∧-isCommutativeRing : IsCommutativeRing _⊕_ _∧_ id ⊥ ⊤

+
  ⊕-∧-isCommutativeRing = record

+
    { isRing = ⊕-∧-isRing

+
    ; *-comm = ∧-comm

+
    }

+

+
  ⊕-∧-commutativeRing : CommutativeRing _ _

+
  ⊕-∧-commutativeRing = record

+
    { isCommutativeRing = ⊕-∧-isCommutativeRing

+
    }

+

+

+
infixl 6 _⊕_

+

+
_⊕_ : Op₂ Carrier

+
x ⊕ y = (x ∨ y) ∧ ¬ (x ∧ y)

+

+
module DefaultXorRing = XorRing _⊕_ (λ _ _ → refl)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Properties.DistributiveLattice.html b/site/static/agda_html/Algebra.Lattice.Properties.DistributiveLattice.html
new file mode 100644
index 000000000..2018a42ea
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Properties.DistributiveLattice.html
@@ -0,0 +1,116 @@
+
+Algebra.Lattice.Properties.DistributiveLattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Lattice.Bundles

+
import Algebra.Lattice.Properties.Lattice as LatticeProperties

+

+
module Algebra.Lattice.Properties.DistributiveLattice

+
  {dl₁ dl₂} (DL : DistributiveLattice dl₁ dl₂)

+
  where

+

+
open DistributiveLattice DL

+
open import Algebra.Definitions _≈_

+
open import Algebra.Lattice.Structures _≈_

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
------------------------------------------------------------------------

+
-- Export properties of lattices

+

+
open LatticeProperties lattice public

+

+
------------------------------------------------------------------------

+
-- The dual construction is also a distributive lattice.

+

+
∧-∨-isDistributiveLattice : IsDistributiveLattice _∧_ _∨_

+
∧-∨-isDistributiveLattice = record

+
  { isLattice   = ∧-∨-isLattice

+
  ; ∨-distrib-∧ = ∧-distrib-∨

+
  ; ∧-distrib-∨ = ∨-distrib-∧

+
  }

+

+
∧-∨-distributiveLattice : DistributiveLattice _ _

+
∧-∨-distributiveLattice = record

+
  { isDistributiveLattice = ∧-∨-isDistributiveLattice

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Properties.Lattice.html b/site/static/agda_html/Algebra.Lattice.Properties.Lattice.html
new file mode 100644
index 000000000..655e3f4d6
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Properties.Lattice.html
@@ -0,0 +1,257 @@
+
+Algebra.Lattice.Properties.Lattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties of lattices

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Lattice.Bundles

+
import Algebra.Lattice.Properties.Semilattice as SemilatticeProperties

+
open import Relation.Binary.Bundles using (Poset)

+
import Relation.Binary.Lattice as R

+
open import Function.Base

+
open import Data.Product.Base using (_,_; swap)

+

+
module Algebra.Lattice.Properties.Lattice

+
  {l₁ l₂} (L : Lattice l₁ l₂) where

+

+
open Lattice L

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures _≈_

+
open import Algebra.Lattice.Structures _≈_

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
------------------------------------------------------------------------

+
-- _∧_ is a semilattice

+

+
∧-idem : Idempotent _∧_

+
∧-idem x = begin

+
  x ∧ x            ≈⟨ ∧-congˡ (∨-absorbs-∧ _ _) ⟨

+
  x ∧ (x ∨ x ∧ x)  ≈⟨  ∧-absorbs-∨ _ _ ⟩

+
  x                ∎

+

+
∧-isMagma : IsMagma _∧_

+
∧-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = ∧-cong

+
  }

+

+
∧-isSemigroup : IsSemigroup _∧_

+
∧-isSemigroup = record

+
  { isMagma = ∧-isMagma

+
  ; assoc   = ∧-assoc

+
  }

+

+
∧-isBand : IsBand _∧_

+
∧-isBand = record

+
  { isSemigroup = ∧-isSemigroup

+
  ; idem        = ∧-idem

+
  }

+

+
∧-isSemilattice : IsSemilattice _∧_

+
∧-isSemilattice = record

+
  { isBand = ∧-isBand

+
  ; comm   = ∧-comm

+
  }

+

+
∧-semilattice : Semilattice l₁ l₂

+
∧-semilattice = record

+
  { isSemilattice = ∧-isSemilattice

+
  }

+

+
open SemilatticeProperties ∧-semilattice public

+
  using

+
  ( ∧-isOrderTheoreticMeetSemilattice

+
  ; ∧-isOrderTheoreticJoinSemilattice

+
  ; ∧-orderTheoreticMeetSemilattice

+
  ; ∧-orderTheoreticJoinSemilattice

+
  )

+

+
------------------------------------------------------------------------

+
-- _∨_ is a semilattice

+

+
∨-idem : Idempotent _∨_

+
∨-idem x = begin

+
  x ∨ x      ≈⟨ ∨-congˡ (∧-idem _) ⟨

+
  x ∨ x ∧ x  ≈⟨  ∨-absorbs-∧ _ _ ⟩

+
  x          ∎

+

+
∨-isMagma : IsMagma _∨_

+
∨-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = ∨-cong

+
  }

+

+
∨-isSemigroup : IsSemigroup _∨_

+
∨-isSemigroup = record

+
  { isMagma = ∨-isMagma

+
  ; assoc   = ∨-assoc

+
  }

+

+
∨-isBand : IsBand _∨_

+
∨-isBand = record

+
  { isSemigroup = ∨-isSemigroup

+
  ; idem        = ∨-idem

+
  }

+

+
∨-isSemilattice : IsSemilattice _∨_

+
∨-isSemilattice = record

+
  { isBand = ∨-isBand

+
  ; comm   = ∨-comm

+
  }

+

+
∨-semilattice : Semilattice l₁ l₂

+
∨-semilattice = record

+
  { isSemilattice = ∨-isSemilattice

+
  }

+

+
open SemilatticeProperties ∨-semilattice public

+
  using ()

+
  renaming

+
  ( ∧-isOrderTheoreticMeetSemilattice to ∨-isOrderTheoreticMeetSemilattice

+
  ; ∧-isOrderTheoreticJoinSemilattice to ∨-isOrderTheoreticJoinSemilattice

+
  ; ∧-orderTheoreticMeetSemilattice   to ∨-orderTheoreticMeetSemilattice

+
  ; ∧-orderTheoreticJoinSemilattice   to ∨-orderTheoreticJoinSemilattice

+
  )

+

+
------------------------------------------------------------------------

+
-- The dual construction is also a lattice.

+

+
∧-∨-isLattice : IsLattice _∧_ _∨_

+
∧-∨-isLattice = record

+
  { isEquivalence = isEquivalence

+
  ; ∨-comm        = ∧-comm

+
  ; ∨-assoc       = ∧-assoc

+
  ; ∨-cong        = ∧-cong

+
  ; ∧-comm        = ∨-comm

+
  ; ∧-assoc       = ∨-assoc

+
  ; ∧-cong        = ∨-cong

+
  ; absorptive    = swap absorptive

+
  }

+

+
∧-∨-lattice : Lattice _ _

+
∧-∨-lattice = record

+
  { isLattice = ∧-∨-isLattice

+
  }

+

+
------------------------------------------------------------------------

+
-- Every algebraic lattice can be turned into an order-theoretic one.

+

+
open SemilatticeProperties ∧-semilattice public using (poset)

+
open Poset poset using (_≤_; isPartialOrder)

+

+
∨-∧-isOrderTheoreticLattice : R.IsLattice _≈_ _≤_ _∨_ _∧_

+
∨-∧-isOrderTheoreticLattice = record

+
  { isPartialOrder = isPartialOrder

+
  ; supremum       = supremum

+
  ; infimum        = infimum

+
  }

+
  where

+
  open R.MeetSemilattice ∧-orderTheoreticMeetSemilattice using (infimum)

+
  open R.JoinSemilattice ∨-orderTheoreticJoinSemilattice using (x≤x∨y; y≤x∨y; ∨-least)

+
    renaming (_≤_ to _≤′_)

+

+
  -- An alternative but equivalent interpretation of the order _≤_.

+

+
  sound : ∀ {x y} → x ≤′ y → x ≤ y

+
  sound {x} {y} y≈y∨x = sym $ begin

+
    x ∧ y        ≈⟨ ∧-congˡ y≈y∨x ⟩

+
    x ∧ (y ∨ x)  ≈⟨ ∧-congˡ (∨-comm y x) ⟩

+
    x ∧ (x ∨ y)  ≈⟨ ∧-absorbs-∨ x y ⟩

+
    x            ∎

+

+
  complete : ∀ {x y} → x ≤ y → x ≤′ y

+
  complete {x} {y} x≈x∧y = sym $ begin

+
    y ∨ x        ≈⟨ ∨-congˡ x≈x∧y ⟩

+
    y ∨ (x ∧ y)  ≈⟨ ∨-congˡ (∧-comm x y) ⟩

+
    y ∨ (y ∧ x)  ≈⟨ ∨-absorbs-∧ y x ⟩

+
    y            ∎

+

+
  supremum : R.Supremum _≤_ _∨_

+
  supremum x y =

+
     sound (x≤x∨y x y) ,

+
     sound (y≤x∨y x y) ,

+
     λ z x≤z y≤z → sound (∨-least (complete x≤z) (complete y≤z))

+

+
∨-∧-orderTheoreticLattice : R.Lattice _ _ _

+
∨-∧-orderTheoreticLattice = record

+
  { isLattice = ∨-∧-isOrderTheoreticLattice

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Properties.Semilattice.html b/site/static/agda_html/Algebra.Lattice.Properties.Semilattice.html
new file mode 100644
index 000000000..4bf8a06d9
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Properties.Semilattice.html
@@ -0,0 +1,134 @@
+
+Algebra.Lattice.Properties.Semilattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties of semilattices

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Lattice.Bundles using (Semilattice)

+
open import Relation.Binary.Bundles using (Poset)

+
import Relation.Binary.Lattice as B

+
import Relation.Binary.Properties.Poset as PosetProperties

+

+
module Algebra.Lattice.Properties.Semilattice

+
  {c ℓ} (L : Semilattice c ℓ) where

+

+
open Semilattice L renaming (_∙_ to _∧_)

+

+
open import Relation.Binary.Reasoning.Setoid setoid

+
import Relation.Binary.Construct.NaturalOrder.Left _≈_ _∧_

+
  as LeftNaturalOrder

+

+
------------------------------------------------------------------------

+
-- Every semilattice can be turned into a poset via the left natural

+
-- order.

+

+
poset : Poset c ℓ ℓ

+
poset = LeftNaturalOrder.poset isSemilattice

+

+
open Poset poset using (_≤_; _≥_; isPartialOrder)

+
open PosetProperties poset using (≥-isPartialOrder)

+

+
------------------------------------------------------------------------

+
-- Every algebraic semilattice can be turned into an order-theoretic one.

+

+
∧-isOrderTheoreticMeetSemilattice : B.IsMeetSemilattice _≈_ _≤_ _∧_

+
∧-isOrderTheoreticMeetSemilattice = record

+
  { isPartialOrder = isPartialOrder

+
  ; infimum        = LeftNaturalOrder.infimum isSemilattice

+
  }

+

+
∧-isOrderTheoreticJoinSemilattice : B.IsJoinSemilattice _≈_ _≥_ _∧_

+
∧-isOrderTheoreticJoinSemilattice = record

+
  { isPartialOrder = ≥-isPartialOrder

+
  ; supremum       = B.IsMeetSemilattice.infimum

+
                       ∧-isOrderTheoreticMeetSemilattice

+
  }

+

+
∧-orderTheoreticMeetSemilattice : B.MeetSemilattice c ℓ ℓ

+
∧-orderTheoreticMeetSemilattice = record

+
  { isMeetSemilattice = ∧-isOrderTheoreticMeetSemilattice

+
  }

+

+
∧-orderTheoreticJoinSemilattice : B.JoinSemilattice c ℓ ℓ

+
∧-orderTheoreticJoinSemilattice = record

+
  { isJoinSemilattice = ∧-isOrderTheoreticJoinSemilattice

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Structures.Biased.html b/site/static/agda_html/Algebra.Lattice.Structures.Biased.html
new file mode 100644
index 000000000..f600f6614
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Structures.Biased.html
@@ -0,0 +1,200 @@
+
+Algebra.Lattice.Structures.Biased
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some biased records for lattice-like structures. Such records are

+
-- often easier to construct, but are suboptimal to use more widely and

+
-- should be converted to the standard record definitions immediately

+
-- using the provided conversion functions.

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra.Lattice`,

+
-- unless you want to parameterise it via the equality relation.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Consequences.Setoid

+
open import Data.Product.Base using (proj₁; proj₂)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
module Algebra.Lattice.Structures.Biased

+
  {a ℓ} {A : Set a}  -- The underlying set

+
  (_≈_ : Rel A ℓ)    -- The underlying equality relation

+
  where

+

+
open import Algebra.Definitions _≈_

+
open import Algebra.Lattice.Structures _≈_

+

+
private

+
  variable

+
    ∧ ∨ : Op₂ A

+
    ¬   : Op₁ A

+
    ⊤ ⊥ : A

+

+
------------------------------------------------------------------------

+
-- Lattice

+

+
-- An alternative form of `IsLattice` defined in terms of

+
-- `IsJoinSemilattice` and `IsMeetLattice`. This form may be desirable

+
-- to use when constructing a lattice object as it requires fewer

+
-- arguments, but is often a mistake to use as an argument as it

+
-- contains two, *potentially different*, proofs that the equality

+
-- relation _≈_ is an equivalence.

+

+
record IsLattice₂ (∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isJoinSemilattice : IsJoinSemilattice ∨

+
    isMeetSemilattice : IsMeetSemilattice ∧

+
    absorptive        : Absorptive ∨ ∧

+

+
  module ML = IsMeetSemilattice isMeetSemilattice

+
  module JL = IsJoinSemilattice isJoinSemilattice

+

+
  isLattice₂ : IsLattice ∨ ∧

+
  isLattice₂ = record

+
    { isEquivalence = ML.isEquivalence

+
    ; ∨-comm        = JL.comm

+
    ; ∨-assoc       = JL.assoc

+
    ; ∨-cong        = JL.∨-cong

+
    ; ∧-comm        = ML.comm

+
    ; ∧-assoc       = ML.assoc

+
    ; ∧-cong        = ML.∧-cong

+
    ; absorptive    = absorptive

+
    }

+

+
open IsLattice₂ public using (isLattice₂)

+

+
------------------------------------------------------------------------

+
-- DistributiveLattice

+

+
-- A version of distributive lattice that is biased towards the (r)ight

+
-- distributivity law for (j)oin and (m)eet.

+
record IsDistributiveLatticeʳʲᵐ (∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isLattice    : IsLattice ∨ ∧

+
    ∨-distribʳ-∧ : ∨ DistributesOverʳ ∧

+

+
  open IsLattice isLattice public

+

+
  setoid : Setoid a ℓ

+
  setoid = record { isEquivalence = isEquivalence }

+

+
  ∨-distrib-∧  = comm∧distrʳ⇒distr setoid ∧-cong ∨-comm ∨-distribʳ-∧

+
  ∧-distribˡ-∨ = distrib∧absorbs⇒distribˡ setoid ∧-cong ∧-assoc ∨-comm ∧-absorbs-∨ ∨-absorbs-∧ ∨-distrib-∧

+
  ∧-distrib-∨  = comm∧distrˡ⇒distr setoid ∨-cong ∧-comm ∧-distribˡ-∨

+

+
  isDistributiveLatticeʳʲᵐ : IsDistributiveLattice ∨ ∧

+
  isDistributiveLatticeʳʲᵐ = record

+
    { isLattice   = isLattice

+
    ; ∨-distrib-∧ = ∨-distrib-∧

+
    ; ∧-distrib-∨ = ∧-distrib-∨

+
    }

+

+
open IsDistributiveLatticeʳʲᵐ public using (isDistributiveLatticeʳʲᵐ)

+

+
------------------------------------------------------------------------

+
-- BooleanAlgebra

+

+
-- A (r)ight biased version of a boolean algebra.

+
record IsBooleanAlgebraʳ

+
         (∨ ∧ : Op₂ A) (¬ : Op₁ A) (⊤ ⊥ : A) : Set (a ⊔ ℓ) where

+
  field

+
    isDistributiveLattice : IsDistributiveLattice ∨ ∧

+
    ∨-complementʳ         : RightInverse ⊤ ¬ ∨

+
    ∧-complementʳ         : RightInverse ⊥ ¬ ∧

+
    ¬-cong                : Congruent₁ ¬

+

+
  open IsDistributiveLattice isDistributiveLattice public

+

+
  setoid : Setoid a ℓ

+
  setoid = record { isEquivalence = isEquivalence }

+

+
  isBooleanAlgebraʳ : IsBooleanAlgebra  ∨ ∧ ¬ ⊤ ⊥

+
  isBooleanAlgebraʳ = record

+
    { isDistributiveLattice = isDistributiveLattice

+
    ; ∨-complement          = comm∧invʳ⇒inv setoid ∨-comm ∨-complementʳ

+
    ; ∧-complement          = comm∧invʳ⇒inv setoid ∧-comm ∧-complementʳ

+
    ; ¬-cong                = ¬-cong

+
    }

+

+
open IsBooleanAlgebraʳ public using (isBooleanAlgebraʳ)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.Structures.html b/site/static/agda_html/Algebra.Lattice.Structures.html
new file mode 100644
index 000000000..0f9af2c95
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.Structures.html
@@ -0,0 +1,269 @@
+
+Algebra.Lattice.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some lattice-like structures defined by properties of _∧_ and _∨_

+
-- (not packed up with sets, operations, etc.)

+
--

+
-- For lattices defined via an order relation, see

+
-- Relation.Binary.Lattice.

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra.Lattice`,

+
-- unless you want to parameterise it via the equality relation.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Data.Product.Base using (proj₁; proj₂)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
module Algebra.Lattice.Structures

+
  {a ℓ} {A : Set a}  -- The underlying set

+
  (_≈_ : Rel A ℓ)    -- The underlying equality relation

+
  where

+

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures _≈_

+

+
------------------------------------------------------------------------

+
-- Structures with 1 binary operation

+

+
IsSemilattice = IsCommutativeBand

+
module IsSemilattice {∙} (L : IsSemilattice ∙) where

+
  open IsCommutativeBand L public

+
    using (isBand; comm)

+
  open IsBand isBand public

+

+
-- Used to bring names appropriate for a meet semilattice into scope.

+
IsMeetSemilattice = IsSemilattice

+
module IsMeetSemilattice {∧} (L : IsMeetSemilattice ∧) where

+
  open IsSemilattice L public

+
    renaming

+
    ( ∙-cong  to ∧-cong

+
    ; ∙-congˡ to ∧-congˡ

+
    ; ∙-congʳ to ∧-congʳ

+
    )

+

+
-- Used to bring names appropriate for a join semilattice into scope.

+
IsJoinSemilattice = IsSemilattice

+
module IsJoinSemilattice {∨} (L : IsJoinSemilattice ∨) where

+
  open IsSemilattice L public

+
    renaming

+
    ( ∙-cong  to ∨-cong

+
    ; ∙-congˡ to ∨-congˡ

+
    ; ∙-congʳ to ∨-congʳ

+
    )

+

+
------------------------------------------------------------------------

+
-- Structures with 1 binary operation & 1 element

+

+
-- A bounded semi-lattice is the same thing as an idempotent commutative

+
-- monoid.

+
IsBoundedSemilattice = IsIdempotentCommutativeMonoid

+
module IsBoundedSemilattice {∙ ε} (L : IsBoundedSemilattice ∙ ε) where

+

+
  open IsIdempotentCommutativeMonoid L public

+
    renaming (isCommutativeBand to isSemilattice)

+

+

+
-- Used to bring names appropriate for a bounded meet semilattice

+
-- into scope.

+
IsBoundedMeetSemilattice = IsBoundedSemilattice

+
module IsBoundedMeetSemilattice {∧ ⊤} (L : IsBoundedMeetSemilattice ∧ ⊤)

+
  where

+

+
  open IsBoundedSemilattice L public

+
    using (identity; identityˡ; identityʳ)

+
    renaming (isSemilattice to isMeetSemilattice)

+

+
  open IsMeetSemilattice isMeetSemilattice public

+

+

+
-- Used to bring names appropriate for a bounded join semilattice

+
-- into scope.

+
IsBoundedJoinSemilattice = IsBoundedSemilattice

+
module IsBoundedJoinSemilattice {∨ ⊥} (L : IsBoundedJoinSemilattice ∨ ⊥)

+
  where

+

+
  open IsBoundedSemilattice L public

+
    using (identity; identityˡ; identityʳ)

+
    renaming (isSemilattice to isJoinSemilattice)

+

+
  open IsJoinSemilattice isJoinSemilattice public

+

+
------------------------------------------------------------------------

+
-- Structures with 2 binary operations

+

+
-- Note that `IsLattice` is not defined in terms of `IsMeetSemilattice`

+
-- and `IsJoinSemilattice` for two reasons:

+
--   1) it would result in a structure with two *different* proofs that

+
--   the equality relation `≈` is an equivalence relation.

+
--   2) the idempotence laws of ∨ and ∧ can be derived from the

+
--   absorption laws, which makes the corresponding "idem" fields

+
--   redundant.

+
--

+
-- It is possible to construct the `IsLattice` record from

+
-- `IsMeetSemilattice` and `IsJoinSemilattice` via the `IsLattice₂`

+
-- record found in `Algebra.Lattice.Structures.Biased`.

+
--

+
-- The derived idempotence laws are stated and proved in

+
-- `Algebra.Lattice.Properties.Lattice` along with the fact that every

+
-- lattice consists of two semilattices.

+

+
record IsLattice (∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isEquivalence : IsEquivalence _≈_

+
    ∨-comm        : Commutative ∨

+
    ∨-assoc       : Associative ∨

+
    ∨-cong        : Congruent₂ ∨

+
    ∧-comm        : Commutative ∧

+
    ∧-assoc       : Associative ∧

+
    ∧-cong        : Congruent₂ ∧

+
    absorptive    : Absorptive ∨ ∧

+

+
  open IsEquivalence isEquivalence public

+

+
  ∨-absorbs-∧ : ∨ Absorbs ∧

+
  ∨-absorbs-∧ = proj₁ absorptive

+

+
  ∧-absorbs-∨ : ∧ Absorbs ∨

+
  ∧-absorbs-∨ = proj₂ absorptive

+

+
  ∧-congˡ : LeftCongruent ∧

+
  ∧-congˡ y≈z = ∧-cong refl y≈z

+

+
  ∧-congʳ : RightCongruent ∧

+
  ∧-congʳ y≈z = ∧-cong y≈z refl

+

+
  ∨-congˡ : LeftCongruent ∨

+
  ∨-congˡ y≈z = ∨-cong refl y≈z

+

+
  ∨-congʳ : RightCongruent ∨

+
  ∨-congʳ y≈z = ∨-cong y≈z refl

+

+

+
record IsDistributiveLattice (∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isLattice   : IsLattice ∨ ∧

+
    ∨-distrib-∧ : ∨ DistributesOver ∧

+
    ∧-distrib-∨ : ∧ DistributesOver ∨

+

+
  open IsLattice isLattice public

+

+
  ∨-distribˡ-∧ : ∨ DistributesOverˡ ∧

+
  ∨-distribˡ-∧ = proj₁ ∨-distrib-∧

+

+
  ∨-distribʳ-∧ : ∨ DistributesOverʳ ∧

+
  ∨-distribʳ-∧ = proj₂ ∨-distrib-∧

+

+
  ∧-distribˡ-∨ : ∧ DistributesOverˡ ∨

+
  ∧-distribˡ-∨ = proj₁ ∧-distrib-∨

+

+
  ∧-distribʳ-∨ : ∧ DistributesOverʳ ∨

+
  ∧-distribʳ-∨ = proj₂ ∧-distrib-∨

+

+
------------------------------------------------------------------------

+
-- Structures with 2 binary ops, 1 unary op and 2 elements.

+

+
record IsBooleanAlgebra (∨ ∧ : Op₂ A) (¬ : Op₁ A) (⊤ ⊥ : A) : Set (a ⊔ ℓ)

+
  where

+

+
  field

+
    isDistributiveLattice : IsDistributiveLattice ∨ ∧

+
    ∨-complement          : Inverse ⊤ ¬ ∨

+
    ∧-complement          : Inverse ⊥ ¬ ∧

+
    ¬-cong                : Congruent₁ ¬

+

+
  open IsDistributiveLattice isDistributiveLattice public

+

+
  ∨-complementˡ : LeftInverse ⊤ ¬ ∨

+
  ∨-complementˡ = proj₁ ∨-complement

+

+
  ∨-complementʳ : RightInverse ⊤ ¬ ∨

+
  ∨-complementʳ = proj₂ ∨-complement

+

+
  ∧-complementˡ : LeftInverse ⊥ ¬ ∧

+
  ∧-complementˡ = proj₁ ∧-complement

+

+
  ∧-complementʳ : RightInverse ⊥ ¬ ∧

+
  ∧-complementʳ = proj₂ ∧-complement

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Lattice.html b/site/static/agda_html/Algebra.Lattice.html
new file mode 100644
index 000000000..eb7c3982d
--- /dev/null
+++ b/site/static/agda_html/Algebra.Lattice.html
@@ -0,0 +1,95 @@
+
+Algebra.Lattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of algebraic structures like semilattices and lattices

+
-- (packed in records together with sets, operations, etc.), defined via

+
-- meet/join operations and their properties

+
--

+
-- For lattices defined via an order relation, see

+
-- Relation.Binary.Lattice.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Lattice where

+

+
open import Algebra.Lattice.Structures public

+
open import Algebra.Lattice.Structures.Biased public

+
open import Algebra.Lattice.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.Definitions.html b/site/static/agda_html/Algebra.Morphism.Definitions.html
new file mode 100644
index 000000000..a84ffe4dd
--- /dev/null
+++ b/site/static/agda_html/Algebra.Morphism.Definitions.html
@@ -0,0 +1,123 @@
+
+Algebra.Morphism.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic definitions for morphisms between algebraic structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core

+

+
module Algebra.Morphism.Definitions

+
  {a} (A : Set a)     -- The domain of the morphism

+
  {b} (B : Set b)     -- The codomain of the morphism

+
  {ℓ} (_≈_ : Rel B ℓ)  -- The equality relation over the codomain

+
  where

+

+
open import Algebra.Core

+
  using (Op₁; Op₂)

+

+
------------------------------------------------------------------------

+
-- Basic definitions

+

+
Homomorphic₀ : (A → B) → A → B → Set _

+
Homomorphic₀ ⟦_⟧ ∙ ∘ = ⟦ ∙ ⟧ ≈ ∘

+

+
Homomorphic₁ : (A → B) → Op₁ A → Op₁ B → Set _

+
Homomorphic₁ ⟦_⟧ ∙_ ∘_ = ∀ x → ⟦ ∙ x ⟧ ≈ (∘ ⟦ x ⟧)

+

+
Homomorphic₂ : (A → B) → Op₂ A → Op₂ B → Set _

+
Homomorphic₂ ⟦_⟧ _∙_ _∘_ = ∀ x y → ⟦ x ∙ y ⟧ ≈ (⟦ x ⟧ ∘ ⟦ y ⟧)

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.3

+

+
Morphism : Set _

+
Morphism = A → B

+

+
{-# WARNING_ON_USAGE Morphism

+
"Warning: Morphism was deprecated in v1.3.

+
Please use the standard function notation (e.g. A → B) instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.GroupMonomorphism.html b/site/static/agda_html/Algebra.Morphism.GroupMonomorphism.html
new file mode 100644
index 000000000..f7f0ca157
--- /dev/null
+++ b/site/static/agda_html/Algebra.Morphism.GroupMonomorphism.html
@@ -0,0 +1,174 @@
+
+Algebra.Morphism.GroupMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between group-like structures

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles

+
open import Algebra.Morphism.Structures

+
open import Relation.Binary.Core

+

+
module Algebra.Morphism.GroupMonomorphism

+
  {a b ℓ₁ ℓ₂} {G₁ : RawGroup a ℓ₁} {G₂ : RawGroup b ℓ₂} {⟦_⟧}

+
  (isGroupMonomorphism : IsGroupMonomorphism G₁ G₂ ⟦_⟧)

+
  where

+

+
open IsGroupMonomorphism isGroupMonomorphism

+
open RawGroup G₁ renaming

+
  (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; _⁻¹ to _⁻¹₁; ε to ε₁)

+
open RawGroup G₂ renaming

+
  (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; _⁻¹ to _⁻¹₂; ε to ε₂)

+

+
open import Algebra.Definitions

+
open import Algebra.Structures

+
open import Data.Product.Base using (_,_)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
------------------------------------------------------------------------

+
-- Re-export all properties of monoid monomorphisms

+

+
open import Algebra.Morphism.MonoidMonomorphism

+
  isMonoidMonomorphism public

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where

+

+
  open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)

+
  open ≈-Reasoning setoid

+

+
  inverseˡ : LeftInverse _≈₂_ ε₂ _⁻¹₂ _◦_ → LeftInverse _≈₁_ ε₁ _⁻¹₁ _∙_

+
  inverseˡ invˡ x = injective (begin

+
    ⟦ x ⁻¹₁ ∙ x ⟧     ≈⟨ ∙-homo (x ⁻¹₁ ) x ⟩

+
    ⟦ x ⁻¹₁ ⟧ ◦ ⟦ x ⟧ ≈⟨ ◦-cong (⁻¹-homo x) refl ⟩

+
    ⟦ x ⟧ ⁻¹₂ ◦ ⟦ x ⟧ ≈⟨ invˡ ⟦ x ⟧ ⟩

+
    ε₂                ≈⟨ ε-homo ⟨

+
    ⟦ ε₁ ⟧ ∎)

+

+
  inverseʳ : RightInverse _≈₂_ ε₂ _⁻¹₂ _◦_ → RightInverse _≈₁_ ε₁ _⁻¹₁ _∙_

+
  inverseʳ invʳ x = injective (begin

+
    ⟦ x ∙ x ⁻¹₁ ⟧     ≈⟨ ∙-homo x (x ⁻¹₁) ⟩

+
    ⟦ x ⟧ ◦ ⟦ x ⁻¹₁ ⟧ ≈⟨ ◦-cong refl (⁻¹-homo x) ⟩

+
    ⟦ x ⟧ ◦ ⟦ x ⟧ ⁻¹₂ ≈⟨ invʳ ⟦ x ⟧ ⟩

+
    ε₂                ≈⟨ ε-homo ⟨

+
    ⟦ ε₁ ⟧ ∎)

+

+
  inverse : Inverse _≈₂_ ε₂ _⁻¹₂ _◦_ → Inverse _≈₁_ ε₁ _⁻¹₁ _∙_

+
  inverse (invˡ , invʳ) = inverseˡ invˡ , inverseʳ invʳ

+

+
  ⁻¹-cong : Congruent₁ _≈₂_ _⁻¹₂ → Congruent₁ _≈₁_ _⁻¹₁

+
  ⁻¹-cong ⁻¹-cong {x} {y} x≈y = injective (begin

+
    ⟦ x ⁻¹₁ ⟧ ≈⟨ ⁻¹-homo x ⟩

+
    ⟦ x ⟧ ⁻¹₂ ≈⟨ ⁻¹-cong (⟦⟧-cong x≈y) ⟩

+
    ⟦ y ⟧ ⁻¹₂ ≈⟨ ⁻¹-homo y ⟨

+
    ⟦ y ⁻¹₁ ⟧ ∎)

+

+
module _ (◦-isAbelianGroup : IsAbelianGroup _≈₂_ _◦_ ε₂ _⁻¹₂) where

+

+
  open IsAbelianGroup ◦-isAbelianGroup renaming (∙-cong to ◦-cong; ⁻¹-cong to ⁻¹₂-cong)

+
  open ≈-Reasoning setoid

+

+
  ⁻¹-distrib-∙ : (∀ x y → (x ◦ y) ⁻¹₂ ≈₂ (x ⁻¹₂) ◦ (y ⁻¹₂)) → (∀ x y → (x ∙ y) ⁻¹₁ ≈₁ (x ⁻¹₁) ∙ (y ⁻¹₁))

+
  ⁻¹-distrib-∙ ⁻¹-distrib-∙ x y = injective (begin

+
    ⟦ (x ∙ y) ⁻¹₁ ⟧       ≈⟨ ⁻¹-homo (x ∙ y) ⟩

+
    ⟦ x ∙ y ⟧ ⁻¹₂         ≈⟨ ⁻¹₂-cong (∙-homo x y) ⟩

+
    (⟦ x ⟧ ◦ ⟦ y ⟧) ⁻¹₂   ≈⟨ ⁻¹-distrib-∙ ⟦ x ⟧ ⟦ y ⟧ ⟩

+
    ⟦ x ⟧ ⁻¹₂ ◦ ⟦ y ⟧ ⁻¹₂ ≈⟨ sym (◦-cong (⁻¹-homo x) (⁻¹-homo y)) ⟩

+
    ⟦ x ⁻¹₁ ⟧ ◦ ⟦ y ⁻¹₁ ⟧ ≈⟨ sym (∙-homo (x ⁻¹₁) (y ⁻¹₁)) ⟩

+
    ⟦ (x ⁻¹₁) ∙ (y ⁻¹₁) ⟧ ∎)

+

+
isGroup : IsGroup _≈₂_ _◦_ ε₂ _⁻¹₂ → IsGroup _≈₁_ _∙_ ε₁ _⁻¹₁

+
isGroup isGroup = record

+
  { isMonoid = isMonoid G.isMonoid

+
  ; inverse  = inverse  G.isMagma G.inverse

+
  ; ⁻¹-cong  = ⁻¹-cong  G.isMagma G.⁻¹-cong

+
  } where module G = IsGroup isGroup

+

+
isAbelianGroup : IsAbelianGroup _≈₂_ _◦_ ε₂ _⁻¹₂ → IsAbelianGroup _≈₁_ _∙_ ε₁ _⁻¹₁

+
isAbelianGroup isAbelianGroup = record

+
  { isGroup = isGroup G.isGroup

+
  ; comm    = comm G.isMagma G.comm

+
  } where module G = IsAbelianGroup isAbelianGroup

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.MagmaMonomorphism.html b/site/static/agda_html/Algebra.Morphism.MagmaMonomorphism.html
new file mode 100644
index 000000000..9880859e2
--- /dev/null
+++ b/site/static/agda_html/Algebra.Morphism.MagmaMonomorphism.html
@@ -0,0 +1,200 @@
+
+Algebra.Morphism.MagmaMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between magma-like structures

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
open import Algebra.Morphism.Structures

+
open import Relation.Binary.Core

+

+
module Algebra.Morphism.MagmaMonomorphism

+
  {a b ℓ₁ ℓ₂} {M₁ : RawMagma a ℓ₁} {M₂ : RawMagma b ℓ₂} {⟦_⟧}

+
  (isMagmaMonomorphism : IsMagmaMonomorphism M₁ M₂ ⟦_⟧)

+
  where

+

+
open IsMagmaMonomorphism isMagmaMonomorphism

+
open RawMagma M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_)

+
open RawMagma M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_)

+

+
open import Algebra.Structures

+
open import Algebra.Definitions

+
open import Data.Product.Base using (map)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
import Relation.Binary.Morphism.RelMonomorphism isRelMonomorphism as RelMorphism

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where

+

+
  open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)

+
  open ≈-Reasoning setoid

+

+
  cong : Congruent₂ _≈₁_ _∙_

+
  cong {x} {y} {u} {v} x≈y u≈v = injective (begin

+
    ⟦ x ∙ u ⟧      ≈⟨  homo x u ⟩

+
    ⟦ x ⟧ ◦ ⟦ u ⟧  ≈⟨  ◦-cong (⟦⟧-cong x≈y) (⟦⟧-cong u≈v) ⟩

+
    ⟦ y ⟧ ◦ ⟦ v ⟧  ≈⟨ homo y v ⟨

+
    ⟦ y ∙ v ⟧      ∎)

+

+
  assoc : Associative _≈₂_ _◦_ → Associative _≈₁_ _∙_

+
  assoc assoc x y z = injective (begin

+
    ⟦ (x ∙ y) ∙ z ⟧          ≈⟨  homo (x ∙ y) z ⟩

+
    ⟦ x ∙ y ⟧ ◦ ⟦ z ⟧        ≈⟨  ◦-cong (homo x y) refl ⟩

+
    (⟦ x ⟧ ◦ ⟦ y ⟧) ◦ ⟦ z ⟧  ≈⟨  assoc ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩

+
    ⟦ x ⟧ ◦ (⟦ y ⟧ ◦ ⟦ z ⟧)  ≈⟨ ◦-cong refl (homo y z) ⟨

+
    ⟦ x ⟧ ◦ ⟦ y ∙ z ⟧        ≈⟨ homo x (y ∙ z) ⟨

+
    ⟦ x ∙ (y ∙ z) ⟧          ∎)

+

+
  comm : Commutative _≈₂_ _◦_ → Commutative _≈₁_ _∙_

+
  comm comm x y = injective (begin

+
    ⟦ x ∙ y ⟧      ≈⟨  homo x y ⟩

+
    ⟦ x ⟧ ◦ ⟦ y ⟧  ≈⟨  comm ⟦ x ⟧ ⟦ y ⟧ ⟩

+
    ⟦ y ⟧ ◦ ⟦ x ⟧  ≈⟨ homo y x ⟨

+
    ⟦ y ∙ x ⟧      ∎)

+

+
  idem : Idempotent _≈₂_ _◦_ → Idempotent _≈₁_ _∙_

+
  idem idem x = injective (begin

+
    ⟦ x ∙ x ⟧     ≈⟨ homo x x ⟩

+
    ⟦ x ⟧ ◦ ⟦ x ⟧ ≈⟨ idem ⟦ x ⟧ ⟩

+
    ⟦ x ⟧         ∎)

+

+
  sel : Selective _≈₂_ _◦_ → Selective _≈₁_ _∙_

+
  sel sel x y with sel ⟦ x ⟧ ⟦ y ⟧

+
  ... | inj₁ x◦y≈x = inj₁ (injective (begin

+
    ⟦ x ∙ y ⟧      ≈⟨ homo x y ⟩

+
    ⟦ x ⟧ ◦ ⟦ y ⟧  ≈⟨ x◦y≈x ⟩

+
    ⟦ x ⟧          ∎))

+
  ... | inj₂ x◦y≈y = inj₂ (injective (begin

+
    ⟦ x ∙ y ⟧      ≈⟨ homo x y ⟩

+
    ⟦ x ⟧ ◦ ⟦ y ⟧  ≈⟨ x◦y≈y ⟩

+
    ⟦ y ⟧          ∎))

+

+
  cancelˡ : LeftCancellative _≈₂_ _◦_ → LeftCancellative _≈₁_ _∙_

+
  cancelˡ cancelˡ x y z x∙y≈x∙z = injective (cancelˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ (begin

+
    ⟦ x ⟧ ◦ ⟦ y ⟧  ≈⟨ homo x y ⟨

+
    ⟦ x ∙ y ⟧      ≈⟨  ⟦⟧-cong x∙y≈x∙z ⟩

+
    ⟦ x ∙ z ⟧      ≈⟨  homo x z ⟩

+
    ⟦ x ⟧ ◦ ⟦ z ⟧  ∎))

+

+
  cancelʳ : RightCancellative _≈₂_ _◦_ → RightCancellative _≈₁_ _∙_

+
  cancelʳ cancelʳ x y z y∙x≈z∙x = injective (cancelʳ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ (begin

+
    ⟦ y ⟧ ◦ ⟦ x ⟧  ≈⟨ homo y x ⟨

+
    ⟦ y ∙ x ⟧      ≈⟨  ⟦⟧-cong y∙x≈z∙x ⟩

+
    ⟦ z ∙ x ⟧      ≈⟨  homo z x ⟩

+
    ⟦ z ⟧ ◦ ⟦ x ⟧  ∎))

+

+
  cancel : Cancellative _≈₂_ _◦_ → Cancellative _≈₁_ _∙_

+
  cancel = map cancelˡ cancelʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isMagma : IsMagma _≈₂_ _◦_ → IsMagma _≈₁_ _∙_

+
isMagma isMagma = record

+
  { isEquivalence = RelMorphism.isEquivalence M.isEquivalence

+
  ; ∙-cong        = cong isMagma

+
  } where module M = IsMagma isMagma

+

+
isSemigroup : IsSemigroup _≈₂_ _◦_ → IsSemigroup _≈₁_ _∙_

+
isSemigroup isSemigroup = record

+
  { isMagma = isMagma S.isMagma

+
  ; assoc   = assoc   S.isMagma S.assoc

+
  } where module S = IsSemigroup isSemigroup

+

+
isBand : IsBand _≈₂_ _◦_ → IsBand _≈₁_ _∙_

+
isBand isBand = record

+
  { isSemigroup = isSemigroup B.isSemigroup

+
  ; idem        = idem        B.isMagma B.idem

+
  } where module B = IsBand isBand

+

+
isSelectiveMagma : IsSelectiveMagma _≈₂_ _◦_ → IsSelectiveMagma _≈₁_ _∙_

+
isSelectiveMagma isSelMagma = record

+
  { isMagma = isMagma S.isMagma

+
  ; sel     = sel     S.isMagma S.sel

+
  } where module S = IsSelectiveMagma isSelMagma

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.MonoidMonomorphism.html b/site/static/agda_html/Algebra.Morphism.MonoidMonomorphism.html
new file mode 100644
index 000000000..e3bde70e2
--- /dev/null
+++ b/site/static/agda_html/Algebra.Morphism.MonoidMonomorphism.html
@@ -0,0 +1,171 @@
+
+Algebra.Morphism.MonoidMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between monoid-like structures

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles

+
open import Algebra.Morphism.Structures

+
open import Relation.Binary.Core

+

+
module Algebra.Morphism.MonoidMonomorphism

+
  {a b ℓ₁ ℓ₂} {M₁ : RawMonoid a ℓ₁} {M₂ : RawMonoid b ℓ₂} {⟦_⟧}

+
  (isMonoidMonomorphism : IsMonoidMonomorphism M₁ M₂ ⟦_⟧)

+
  where

+

+
open IsMonoidMonomorphism isMonoidMonomorphism

+
open RawMonoid M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; ε to ε₁)

+
open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; ε to ε₂)

+

+
open import Algebra.Definitions

+
open import Algebra.Structures

+
open import Data.Product.Base using (map)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
------------------------------------------------------------------------

+
-- Re-export all properties of magma monomorphisms

+

+
open import Algebra.Morphism.MagmaMonomorphism

+
  isMagmaMonomorphism public

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where

+

+
  open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)

+
  open ≈-Reasoning setoid

+

+
  identityˡ : LeftIdentity _≈₂_ ε₂ _◦_ → LeftIdentity _≈₁_ ε₁ _∙_

+
  identityˡ idˡ x = injective (begin

+
    ⟦ ε₁ ∙ x ⟧      ≈⟨ homo ε₁ x ⟩

+
    ⟦ ε₁ ⟧ ◦ ⟦ x ⟧  ≈⟨ ◦-cong ε-homo refl ⟩

+
    ε₂ ◦ ⟦ x ⟧      ≈⟨ idˡ ⟦ x ⟧ ⟩

+
    ⟦ x ⟧           ∎)

+

+
  identityʳ : RightIdentity _≈₂_ ε₂ _◦_ → RightIdentity _≈₁_ ε₁ _∙_

+
  identityʳ idʳ x = injective (begin

+
    ⟦ x ∙ ε₁ ⟧      ≈⟨ homo x ε₁ ⟩

+
    ⟦ x ⟧ ◦ ⟦ ε₁ ⟧  ≈⟨ ◦-cong refl ε-homo ⟩

+
    ⟦ x ⟧ ◦ ε₂      ≈⟨ idʳ ⟦ x ⟧ ⟩

+
    ⟦ x ⟧           ∎)

+

+
  identity : Identity _≈₂_ ε₂ _◦_ → Identity _≈₁_ ε₁ _∙_

+
  identity = map identityˡ identityʳ

+

+
  zeroˡ : LeftZero _≈₂_ ε₂ _◦_ → LeftZero _≈₁_ ε₁ _∙_

+
  zeroˡ zeˡ x = injective (begin

+
    ⟦ ε₁ ∙ x ⟧     ≈⟨  homo ε₁ x ⟩

+
    ⟦ ε₁ ⟧ ◦ ⟦ x ⟧ ≈⟨  ◦-cong ε-homo refl ⟩

+
    ε₂   ◦ ⟦ x ⟧   ≈⟨  zeˡ ⟦ x ⟧ ⟩

+
    ε₂             ≈⟨ ε-homo ⟨

+
    ⟦ ε₁ ⟧         ∎)

+

+
  zeroʳ : RightZero _≈₂_ ε₂ _◦_ → RightZero _≈₁_ ε₁ _∙_

+
  zeroʳ zeʳ x = injective (begin

+
    ⟦ x ∙ ε₁ ⟧     ≈⟨  homo x ε₁ ⟩

+
    ⟦ x ⟧ ◦ ⟦ ε₁ ⟧ ≈⟨  ◦-cong refl ε-homo ⟩

+
    ⟦ x ⟧ ◦ ε₂     ≈⟨  zeʳ ⟦ x ⟧ ⟩

+
    ε₂             ≈⟨ ε-homo ⟨

+
    ⟦ ε₁ ⟧         ∎)

+

+
  zero : Zero _≈₂_ ε₂ _◦_ → Zero _≈₁_ ε₁ _∙_

+
  zero = map zeroˡ zeroʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isMonoid : IsMonoid _≈₂_ _◦_ ε₂ → IsMonoid _≈₁_ _∙_ ε₁

+
isMonoid isMonoid = record

+
  { isSemigroup = isSemigroup M.isSemigroup

+
  ; identity    = identity    M.isMagma M.identity

+
  } where module M = IsMonoid isMonoid

+

+
isCommutativeMonoid : IsCommutativeMonoid _≈₂_ _◦_ ε₂ →

+
                      IsCommutativeMonoid _≈₁_ _∙_ ε₁

+
isCommutativeMonoid isCommMonoid = record

+
  { isMonoid = isMonoid C.isMonoid

+
  ; comm     = comm     C.isMagma C.comm

+
  } where module C = IsCommutativeMonoid isCommMonoid

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.RingMonomorphism.html b/site/static/agda_html/Algebra.Morphism.RingMonomorphism.html
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@@ -0,0 +1,237 @@
+
+Algebra.Morphism.RingMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between ring-like structures

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles

+
open import Algebra.Morphism.Structures

+
import Algebra.Morphism.GroupMonomorphism  as GroupMonomorphism

+
import Algebra.Morphism.MonoidMonomorphism as MonoidMonomorphism

+
open import Relation.Binary.Core

+

+
module Algebra.Morphism.RingMonomorphism

+
  {a b ℓ₁ ℓ₂} {R₁ : RawRing a ℓ₁} {R₂ : RawRing b ℓ₂} {⟦_⟧}

+
  (isRingMonomorphism : IsRingMonomorphism R₁ R₂ ⟦_⟧)

+
  where

+

+
open IsRingMonomorphism isRingMonomorphism

+
open RawRing R₁ renaming (Carrier to A; _≈_ to _≈₁_)

+
open RawRing R₂ renaming

+
  ( Carrier to B; _≈_ to _≈₂_; _+_ to _⊕_

+
  ; _*_ to _⊛_; 1# to 1#₂; 0# to 0#₂; -_ to ⊝_)

+

+
open import Algebra.Definitions

+
open import Algebra.Structures

+
open import Data.Product.Base using (proj₁; proj₂; _,_)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
------------------------------------------------------------------------

+
-- Re-export all properties of group and monoid monomorphisms

+

+
open GroupMonomorphism +-isGroupMonomorphism public

+
  renaming

+
  ( assoc   to +-assoc

+
  ; comm    to +-comm

+
  ; cong    to +-cong

+
  ; idem    to +-idem

+
  ; sel     to +-sel

+
  ; ⁻¹-cong to neg-cong

+

+
  ; identity to +-identity; identityˡ to +-identityˡ; identityʳ to +-identityʳ

+
  ; cancel   to +-cancel;   cancelˡ   to +-cancelˡ;   cancelʳ   to +-cancelʳ

+
  ; zero     to +-zero;     zeroˡ     to +-zeroˡ;     zeroʳ     to +-zeroʳ

+

+
  ; isMagma             to +-isMagma

+
  ; isSemigroup         to +-isSemigroup

+
  ; isMonoid            to +-isMonoid

+
  ; isSelectiveMagma    to +-isSelectiveMagma

+
  ; isBand              to +-isBand

+
  ; isCommutativeMonoid to +-isCommutativeMonoid

+
  )

+

+
open MonoidMonomorphism *-isMonoidMonomorphism public

+
  renaming

+
  ( assoc to *-assoc

+
  ; comm  to *-comm

+
  ; cong  to *-cong

+
  ; idem  to *-idem

+
  ; sel   to *-sel

+

+
  ; identity to *-identity; identityˡ to *-identityˡ; identityʳ to *-identityʳ

+
  ; cancel   to *-cancel;   cancelˡ   to *-cancelˡ;   cancelʳ   to *-cancelʳ

+
  ; zero     to *-zero;     zeroˡ     to *-zeroˡ;     zeroʳ     to *-zeroʳ

+

+
  ; isMagma             to *-isMagma

+
  ; isSemigroup         to *-isSemigroup

+
  ; isMonoid            to *-isMonoid

+
  ; isSelectiveMagma    to *-isSelectiveMagma

+
  ; isBand              to *-isBand

+
  ; isCommutativeMonoid to *-isCommutativeMonoid

+
  )

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)

+
         (*-isMagma : IsMagma _≈₂_ _⊛_) where

+

+
  open IsGroup +-isGroup hiding (setoid; refl; sym)

+
  open IsMagma *-isMagma renaming (∙-cong to ◦-cong)

+
  open ≈-Reasoning setoid

+

+
  distribˡ : _DistributesOverˡ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverˡ_ _≈₁_ _*_ _+_

+
  distribˡ distribˡ x y z = injective (begin

+
    ⟦ x * (y + z) ⟧               ≈⟨ *-homo x (y + z) ⟩

+
    ⟦ x ⟧ ⊛ ⟦ y + z ⟧             ≈⟨ ◦-cong refl (+-homo y z) ⟩

+
    ⟦ x ⟧ ⊛ (⟦ y ⟧ ⊕ ⟦ z ⟧)       ≈⟨ distribˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩

+
    ⟦ x ⟧ ⊛ ⟦ y ⟧ ⊕ ⟦ x ⟧ ⊛ ⟦ z ⟧ ≈⟨ ∙-cong (*-homo x y) (*-homo x z) ⟨

+
    ⟦ x * y ⟧ ⊕ ⟦ x * z ⟧         ≈⟨ +-homo (x * y) (x * z) ⟨

+
    ⟦ x * y + x * z ⟧ ∎)

+

+
  distribʳ : _DistributesOverʳ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverʳ_ _≈₁_ _*_ _+_

+
  distribʳ distribˡ x y z = injective (begin

+
    ⟦ (y + z) * x ⟧               ≈⟨ *-homo (y + z) x ⟩

+
    ⟦ y + z ⟧ ⊛ ⟦ x ⟧             ≈⟨ ◦-cong (+-homo y z) refl ⟩

+
    (⟦ y ⟧ ⊕ ⟦ z ⟧) ⊛ ⟦ x ⟧       ≈⟨ distribˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩

+
    ⟦ y ⟧ ⊛ ⟦ x ⟧ ⊕ ⟦ z ⟧ ⊛ ⟦ x ⟧ ≈⟨ ∙-cong (*-homo y x) (*-homo z x) ⟨

+
    ⟦ y * x ⟧ ⊕ ⟦ z * x ⟧         ≈⟨ +-homo (y * x) (z * x) ⟨

+
    ⟦ y * x + z * x ⟧ ∎)

+

+
  distrib : _DistributesOver_ _≈₂_ _⊛_ _⊕_ → _DistributesOver_ _≈₁_ _*_ _+_

+
  distrib distrib = distribˡ (proj₁ distrib) , distribʳ (proj₂ distrib)

+

+
  zeroˡ : LeftZero _≈₂_ 0#₂ _⊛_ → LeftZero _≈₁_ 0# _*_

+
  zeroˡ zeroˡ x = injective (begin

+
    ⟦ 0# * x ⟧     ≈⟨ *-homo 0# x ⟩

+
    ⟦ 0# ⟧ ⊛ ⟦ x ⟧ ≈⟨ ◦-cong 0#-homo refl ⟩

+
    0#₂ ⊛ ⟦ x ⟧    ≈⟨ zeroˡ ⟦ x ⟧ ⟩

+
    0#₂            ≈⟨ 0#-homo ⟨

+
    ⟦ 0# ⟧         ∎)

+

+
  zeroʳ : RightZero _≈₂_ 0#₂ _⊛_ → RightZero _≈₁_ 0# _*_

+
  zeroʳ zeroʳ x = injective (begin

+
    ⟦ x * 0# ⟧     ≈⟨ *-homo x 0# ⟩

+
    ⟦ x ⟧ ⊛ ⟦ 0# ⟧ ≈⟨ ◦-cong refl 0#-homo ⟩

+
    ⟦ x ⟧ ⊛ 0#₂    ≈⟨ zeroʳ ⟦ x ⟧ ⟩

+
    0#₂            ≈⟨ 0#-homo ⟨

+
    ⟦ 0# ⟧         ∎)

+

+
  zero : Zero _≈₂_ 0#₂ _⊛_ → Zero _≈₁_ 0# _*_

+
  zero zero = zeroˡ (proj₁ zero) , zeroʳ (proj₂ zero)

+

+
  neg-distribˡ-* : (∀ x y → (⊝ (x ⊛ y)) ≈₂ ((⊝ x) ⊛ y)) → (∀ x y → (- (x * y)) ≈₁ ((- x) * y))

+
  neg-distribˡ-* neg-distribˡ-* x y = injective (begin

+
    ⟦ - (x * y) ⟧     ≈⟨ -‿homo (x * y) ⟩

+
    ⊝ ⟦ x * y ⟧       ≈⟨ ⁻¹-cong (*-homo x y) ⟩

+
    ⊝ (⟦ x ⟧ ⊛ ⟦ y ⟧) ≈⟨ neg-distribˡ-* ⟦ x ⟧ ⟦ y ⟧ ⟩

+
    ⊝ ⟦ x ⟧ ⊛ ⟦ y ⟧   ≈⟨ ◦-cong (sym (-‿homo x)) refl ⟩

+
    ⟦ - x ⟧ ⊛ ⟦ y ⟧   ≈⟨ sym (*-homo (- x) y) ⟩

+
    ⟦ - x * y ⟧       ∎)

+

+
  neg-distribʳ-* : (∀ x y → (⊝ (x ⊛ y)) ≈₂ (x ⊛ (⊝ y))) → (∀ x y → (- (x * y)) ≈₁ (x * (- y)))

+
  neg-distribʳ-* neg-distribʳ-* x y = injective (begin

+
    ⟦ - (x * y) ⟧     ≈⟨ -‿homo (x * y) ⟩

+
    ⊝ ⟦ x * y ⟧       ≈⟨ ⁻¹-cong (*-homo x y) ⟩

+
    ⊝ (⟦ x ⟧ ⊛ ⟦ y ⟧) ≈⟨ neg-distribʳ-* ⟦ x ⟧ ⟦ y ⟧ ⟩

+
    ⟦ x ⟧ ⊛ ⊝ ⟦ y ⟧   ≈⟨ ◦-cong refl (sym (-‿homo y)) ⟩

+
    ⟦ x ⟧ ⊛ ⟦ - y ⟧   ≈⟨ sym (*-homo x (- y)) ⟩

+
    ⟦ x * - y ⟧       ∎)

+

+
isRing : IsRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ → IsRing _≈₁_ _+_ _*_ -_ 0# 1#

+
isRing isRing = record

+
  { +-isAbelianGroup = isAbelianGroup R.+-isAbelianGroup

+
  ; *-cong           = *-cong R.*-isMagma

+
  ; *-assoc          = *-assoc R.*-isMagma R.*-assoc

+
  ; *-identity       = *-identity R.*-isMagma R.*-identity

+
  ; distrib          = distrib R.+-isGroup R.*-isMagma R.distrib

+
  } where module R = IsRing isRing

+

+
isCommutativeRing : IsCommutativeRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ →

+
                    IsCommutativeRing _≈₁_ _+_ _*_ -_ 0# 1#

+
isCommutativeRing isCommRing = record

+
  { isRing = isRing C.isRing

+
  ; *-comm = *-comm C.*-isMagma C.*-comm

+
  } where module C = IsCommutativeRing isCommRing

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.Structures.html b/site/static/agda_html/Algebra.Morphism.Structures.html
new file mode 100644
index 000000000..494522a62
--- /dev/null
+++ b/site/static/agda_html/Algebra.Morphism.Structures.html
@@ -0,0 +1,866 @@
+
+Algebra.Morphism.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Morphisms between algebraic structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Morphism.Structures where

+

+
open import Algebra.Core

+
open import Algebra.Bundles

+
import Algebra.Morphism.Definitions as MorphismDefinitions

+
open import Level using (Level; _⊔_)

+
open import Function.Definitions

+
open import Relation.Binary.Core

+
open import Relation.Binary.Morphism.Structures

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+

+
------------------------------------------------------------------------

+
-- Morphisms over SuccessorSet-like structures

+
------------------------------------------------------------------------

+

+
module SuccessorSetMorphisms

+
  (N₁ : RawSuccessorSet a ℓ₁) (N₂ : RawSuccessorSet b ℓ₂)

+
  where

+

+
  open RawSuccessorSet N₁

+
    renaming (Carrier to A; _≈_ to _≈₁_; suc# to suc#₁; zero# to zero#₁)

+
  open RawSuccessorSet N₂

+
    renaming (Carrier to B; _≈_ to _≈₂_; suc# to suc#₂; zero# to zero#₂)

+
  open MorphismDefinitions A B _≈₂_

+

+

+
  record IsSuccessorSetHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧

+
      suc#-homo         : Homomorphic₁ ⟦_⟧ suc#₁ suc#₂

+
      zero#-homo        : Homomorphic₀ ⟦_⟧ zero#₁ zero#₂

+

+
  record IsSuccessorSetMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isSuccessorSetHomomorphism : IsSuccessorSetHomomorphism ⟦_⟧

+
      injective                  : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsSuccessorSetHomomorphism isSuccessorSetHomomorphism public

+

+
    isRelMonomorphism : IsRelMonomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelMonomorphism = record

+
      { isHomomorphism = isRelHomomorphism

+
      ; injective      = injective

+
      }

+

+

+
  record IsSuccessorSetIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isSuccessorSetMonomorphism : IsSuccessorSetMonomorphism ⟦_⟧

+
      surjective                 : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsSuccessorSetMonomorphism isSuccessorSetMonomorphism public

+

+
    isRelIsomorphism : IsRelIsomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelIsomorphism = record

+
      { isMonomorphism = isRelMonomorphism

+
      ; surjective     = surjective

+
      }

+

+

+
------------------------------------------------------------------------

+
-- Morphisms over magma-like structures

+
------------------------------------------------------------------------

+

+
module MagmaMorphisms (M₁ : RawMagma a ℓ₁) (M₂ : RawMagma b ℓ₂) where

+

+
  open RawMagma M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_)

+
  open RawMagma M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_)

+
  open MorphismDefinitions A B _≈₂_

+

+

+
  record IsMagmaHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧

+
      homo              : Homomorphic₂ ⟦_⟧ _∙_ _◦_

+

+
    open IsRelHomomorphism isRelHomomorphism public

+
      renaming (cong to ⟦⟧-cong)

+

+

+
  record IsMagmaMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isMagmaHomomorphism : IsMagmaHomomorphism ⟦_⟧

+
      injective           : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsMagmaHomomorphism isMagmaHomomorphism public

+

+
    isRelMonomorphism : IsRelMonomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelMonomorphism = record

+
      { isHomomorphism = isRelHomomorphism

+
      ; injective      = injective

+
      }

+

+

+
  record IsMagmaIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isMagmaMonomorphism : IsMagmaMonomorphism ⟦_⟧

+
      surjective          : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsMagmaMonomorphism isMagmaMonomorphism public

+

+
    isRelIsomorphism : IsRelIsomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelIsomorphism = record

+
      { isMonomorphism = isRelMonomorphism

+
      ; surjective     = surjective

+
      }

+

+

+
------------------------------------------------------------------------

+
-- Morphisms over monoid-like structures

+
------------------------------------------------------------------------

+

+
module MonoidMorphisms (M₁ : RawMonoid a ℓ₁) (M₂ : RawMonoid b ℓ₂) where

+

+
  open RawMonoid M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; ε to ε₁)

+
  open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; ε to ε₂)

+
  open MorphismDefinitions A B _≈₂_

+
  open MagmaMorphisms (RawMonoid.rawMagma M₁) (RawMonoid.rawMagma M₂)

+

+

+
  record IsMonoidHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isMagmaHomomorphism : IsMagmaHomomorphism ⟦_⟧

+
      ε-homo              : Homomorphic₀ ⟦_⟧ ε₁ ε₂

+

+
    open IsMagmaHomomorphism isMagmaHomomorphism public

+

+

+
  record IsMonoidMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isMonoidHomomorphism : IsMonoidHomomorphism ⟦_⟧

+
      injective            : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsMonoidHomomorphism isMonoidHomomorphism public

+

+
    isMagmaMonomorphism : IsMagmaMonomorphism ⟦_⟧

+
    isMagmaMonomorphism = record

+
      { isMagmaHomomorphism = isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
    open IsMagmaMonomorphism isMagmaMonomorphism public

+
      using (isRelMonomorphism)

+

+

+
  record IsMonoidIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isMonoidMonomorphism : IsMonoidMonomorphism ⟦_⟧

+
      surjective           : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsMonoidMonomorphism isMonoidMonomorphism public

+

+
    isMagmaIsomorphism : IsMagmaIsomorphism ⟦_⟧

+
    isMagmaIsomorphism = record

+
      { isMagmaMonomorphism = isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    open IsMagmaIsomorphism isMagmaIsomorphism public

+
      using (isRelIsomorphism)

+

+

+
------------------------------------------------------------------------

+
-- Morphisms over group-like structures

+
------------------------------------------------------------------------

+

+
module GroupMorphisms (G₁ : RawGroup a ℓ₁) (G₂ : RawGroup b ℓ₂) where

+

+
  open RawGroup G₁ renaming

+
    (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; _⁻¹ to _⁻¹₁; ε to ε₁)

+
  open RawGroup G₂ renaming

+
    (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; _⁻¹ to _⁻¹₂; ε to ε₂)

+
  open MorphismDefinitions A B _≈₂_

+
  open MagmaMorphisms  (RawGroup.rawMagma  G₁) (RawGroup.rawMagma  G₂)

+
  open MonoidMorphisms (RawGroup.rawMonoid G₁) (RawGroup.rawMonoid G₂)

+

+

+
  record IsGroupHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isMonoidHomomorphism : IsMonoidHomomorphism ⟦_⟧

+
      ⁻¹-homo              : Homomorphic₁ ⟦_⟧ _⁻¹₁ _⁻¹₂

+

+
    open IsMonoidHomomorphism isMonoidHomomorphism public

+

+

+
  record IsGroupMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isGroupHomomorphism : IsGroupHomomorphism ⟦_⟧

+
      injective           : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsGroupHomomorphism isGroupHomomorphism public

+
      renaming (homo to ∙-homo)

+

+
    isMonoidMonomorphism : IsMonoidMonomorphism ⟦_⟧

+
    isMonoidMonomorphism = record

+
      { isMonoidHomomorphism = isMonoidHomomorphism

+
      ; injective            = injective

+
      }

+

+
    open IsMonoidMonomorphism isMonoidMonomorphism public

+
      using (isRelMonomorphism; isMagmaMonomorphism)

+

+

+
  record IsGroupIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isGroupMonomorphism : IsGroupMonomorphism ⟦_⟧

+
      surjective          : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsGroupMonomorphism isGroupMonomorphism public

+

+
    isMonoidIsomorphism : IsMonoidIsomorphism ⟦_⟧

+
    isMonoidIsomorphism = record

+
      { isMonoidMonomorphism = isMonoidMonomorphism

+
      ; surjective           = surjective

+
      }

+

+
    open IsMonoidIsomorphism isMonoidIsomorphism public

+
      using (isRelIsomorphism; isMagmaIsomorphism)

+

+

+
------------------------------------------------------------------------

+
-- Morphisms over near-semiring-like structures

+
------------------------------------------------------------------------

+

+
module NearSemiringMorphisms (R₁ : RawNearSemiring a ℓ₁) (R₂ : RawNearSemiring b ℓ₂) where

+

+
  open RawNearSemiring R₁ renaming

+
    ( Carrier to A; _≈_ to _≈₁_

+
    ; +-rawMonoid to +-rawMonoid₁

+
    ; _*_ to _*₁_

+
    ; *-rawMagma to *-rawMagma₁)

+

+
  open RawNearSemiring R₂ renaming

+
    ( Carrier to B; _≈_ to _≈₂_

+
    ; +-rawMonoid to +-rawMonoid₂

+
    ; _*_ to _*₂_

+
    ; *-rawMagma to *-rawMagma₂)

+

+
  private

+
    module + = MonoidMorphisms +-rawMonoid₁ +-rawMonoid₂

+
    module * = MagmaMorphisms *-rawMagma₁ *-rawMagma₂

+

+
  open MorphismDefinitions A B _≈₂_

+

+

+
  record IsNearSemiringHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      +-isMonoidHomomorphism : +.IsMonoidHomomorphism ⟦_⟧

+
      *-homo : Homomorphic₂ ⟦_⟧ _*₁_ _*₂_

+

+
    open +.IsMonoidHomomorphism +-isMonoidHomomorphism public

+
      renaming (homo to +-homo; ε-homo to 0#-homo; isMagmaHomomorphism to +-isMagmaHomomorphism)

+

+
    *-isMagmaHomomorphism : *.IsMagmaHomomorphism ⟦_⟧

+
    *-isMagmaHomomorphism = record

+
      { isRelHomomorphism = isRelHomomorphism

+
      ; homo = *-homo

+
      }

+

+
  record IsNearSemiringMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isNearSemiringHomomorphism : IsNearSemiringHomomorphism ⟦_⟧

+
      injective          : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsNearSemiringHomomorphism isNearSemiringHomomorphism public

+

+
    +-isMonoidMonomorphism : +.IsMonoidMonomorphism ⟦_⟧

+
    +-isMonoidMonomorphism = record

+
      { isMonoidHomomorphism = +-isMonoidHomomorphism

+
      ; injective            = injective

+
      }

+

+
    open +.IsMonoidMonomorphism +-isMonoidMonomorphism public

+
      using (isRelMonomorphism)

+
      renaming (isMagmaMonomorphism to +-isMagmaMonomorphsm)

+

+
    *-isMagmaMonomorphism : *.IsMagmaMonomorphism ⟦_⟧

+
    *-isMagmaMonomorphism = record

+
      { isMagmaHomomorphism = *-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
  record IsNearSemiringIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isNearSemiringMonomorphism : IsNearSemiringMonomorphism ⟦_⟧

+
      surjective                 : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsNearSemiringMonomorphism isNearSemiringMonomorphism public

+

+
    +-isMonoidIsomorphism : +.IsMonoidIsomorphism ⟦_⟧

+
    +-isMonoidIsomorphism = record

+
      { isMonoidMonomorphism = +-isMonoidMonomorphism

+
      ; surjective           = surjective

+
      }

+

+
    open +.IsMonoidIsomorphism +-isMonoidIsomorphism public

+
      using (isRelIsomorphism)

+
      renaming (isMagmaIsomorphism to +-isMagmaIsomorphism)

+

+
    *-isMagmaIsomorphism : *.IsMagmaIsomorphism ⟦_⟧

+
    *-isMagmaIsomorphism = record

+
      { isMagmaMonomorphism = *-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
------------------------------------------------------------------------

+
-- Morphisms over semiring-like structures

+
------------------------------------------------------------------------

+

+
module SemiringMorphisms (R₁ : RawSemiring a ℓ₁) (R₂ : RawSemiring b ℓ₂) where

+

+
  open RawSemiring R₁ renaming

+
    ( Carrier to A; _≈_ to _≈₁_

+
    ; 1# to 1#₁

+
    ; rawNearSemiring to rawNearSemiring₁

+
    ; *-rawMonoid to *-rawMonoid₁)

+

+
  open RawSemiring R₂ renaming

+
    ( Carrier to B; _≈_ to _≈₂_

+
    ; 1# to 1#₂

+
    ; rawNearSemiring to rawNearSemiring₂

+
    ; *-rawMonoid to *-rawMonoid₂)

+

+
  private

+
    module * = MonoidMorphisms *-rawMonoid₁ *-rawMonoid₂

+

+
  open MorphismDefinitions A B _≈₂_

+
  open NearSemiringMorphisms rawNearSemiring₁ rawNearSemiring₂

+

+
  record IsSemiringHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isNearSemiringHomomorphism : IsNearSemiringHomomorphism ⟦_⟧

+
      1#-homo : Homomorphic₀ ⟦_⟧ 1#₁ 1#₂

+

+
    open IsNearSemiringHomomorphism isNearSemiringHomomorphism public

+

+
    *-isMonoidHomomorphism : *.IsMonoidHomomorphism ⟦_⟧

+
    *-isMonoidHomomorphism = record

+
      { isMagmaHomomorphism = *-isMagmaHomomorphism

+
      ; ε-homo = 1#-homo

+
      }

+

+
  record IsSemiringMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isSemiringHomomorphism : IsSemiringHomomorphism ⟦_⟧

+
      injective              : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsSemiringHomomorphism isSemiringHomomorphism public

+

+
    isNearSemiringMonomorphism : IsNearSemiringMonomorphism ⟦_⟧

+
    isNearSemiringMonomorphism = record

+
      { isNearSemiringHomomorphism = isNearSemiringHomomorphism

+
      ; injective = injective

+
      }

+

+
    open IsNearSemiringMonomorphism isNearSemiringMonomorphism public

+
      using (+-isMonoidMonomorphism; *-isMagmaMonomorphism)

+

+
    *-isMonoidMonomorphism : *.IsMonoidMonomorphism ⟦_⟧

+
    *-isMonoidMonomorphism = record

+
      { isMonoidHomomorphism = *-isMonoidHomomorphism

+
      ; injective            = injective

+
      }

+

+
  record IsSemiringIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isSemiringMonomorphism : IsSemiringMonomorphism ⟦_⟧

+
      surjective             : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsSemiringMonomorphism isSemiringMonomorphism public

+

+
    isNearSemiringIsomorphism : IsNearSemiringIsomorphism ⟦_⟧

+
    isNearSemiringIsomorphism = record

+
      { isNearSemiringMonomorphism = isNearSemiringMonomorphism

+
      ; surjective = surjective

+
      }

+

+
    open IsNearSemiringIsomorphism isNearSemiringIsomorphism public

+
      using (+-isMonoidIsomorphism; *-isMagmaIsomorphism)

+

+
    *-isMonoidIsomorphism : *.IsMonoidIsomorphism ⟦_⟧

+
    *-isMonoidIsomorphism = record

+
      { isMonoidMonomorphism = *-isMonoidMonomorphism

+
      ; surjective           = surjective

+
      }

+

+
------------------------------------------------------------------------

+
-- Morphisms over ringWithoutOne-like structures

+
------------------------------------------------------------------------

+

+
module RingWithoutOneMorphisms (R₁ : RawRingWithoutOne a ℓ₁) (R₂ : RawRingWithoutOne b ℓ₂) where

+

+
  open RawRingWithoutOne R₁ renaming

+
    ( Carrier to A; _≈_ to _≈₁_

+
    ; _*_ to _*₁_

+
    ; *-rawMagma to *-rawMagma₁

+
    ; +-rawGroup to +-rawGroup₁)

+

+
  open RawRingWithoutOne R₂ renaming

+
    ( Carrier to B; _≈_ to _≈₂_

+
    ; _*_ to _*₂_

+
    ; *-rawMagma to *-rawMagma₂

+
    ; +-rawGroup to +-rawGroup₂)

+

+
  private

+
    module + = GroupMorphisms  +-rawGroup₁  +-rawGroup₂

+
    module * = MagmaMorphisms *-rawMagma₁ *-rawMagma₂

+

+
  open MorphismDefinitions A B _≈₂_

+

+
  record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      +-isGroupHomomorphism : +.IsGroupHomomorphism ⟦_⟧

+
      *-homo : Homomorphic₂ ⟦_⟧ _*₁_ _*₂_

+

+
    open +.IsGroupHomomorphism +-isGroupHomomorphism public

+
      renaming (homo to +-homo; ε-homo to 0#-homo; isMagmaHomomorphism to +-isMagmaHomomorphism)

+

+
    *-isMagmaHomomorphism : *.IsMagmaHomomorphism ⟦_⟧

+
    *-isMagmaHomomorphism = record

+
      { isRelHomomorphism = isRelHomomorphism

+
      ; homo = *-homo

+
      }

+

+
  record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRingWithoutOneHomomorphism : IsRingWithoutOneHomomorphism ⟦_⟧

+
      injective                    : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsRingWithoutOneHomomorphism isRingWithoutOneHomomorphism public

+

+
    +-isGroupMonomorphism : +.IsGroupMonomorphism ⟦_⟧

+
    +-isGroupMonomorphism = record

+
      { isGroupHomomorphism = +-isGroupHomomorphism

+
      ; injective            = injective

+
      }

+

+
    open +.IsGroupMonomorphism +-isGroupMonomorphism public

+
      using (isRelMonomorphism)

+
      renaming (isMagmaMonomorphism to +-isMagmaMonomorphsm; isMonoidMonomorphism to +-isMonoidMonomorphism)

+

+
    *-isMagmaMonomorphism : *.IsMagmaMonomorphism ⟦_⟧

+
    *-isMagmaMonomorphism = record

+
      { isMagmaHomomorphism = *-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
  record IsRingWithoutOneIsoMorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRingWithoutOneMonomorphism : IsRingWithoutOneMonomorphism ⟦_⟧

+
      surjective                   : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsRingWithoutOneMonomorphism isRingWithoutOneMonomorphism public

+

+
    +-isGroupIsomorphism   : +.IsGroupIsomorphism ⟦_⟧

+
    +-isGroupIsomorphism  = record

+
      { isGroupMonomorphism = +-isGroupMonomorphism

+
      ; surjective           = surjective

+
      }

+

+
    open +.IsGroupIsomorphism +-isGroupIsomorphism public

+
      using (isRelIsomorphism)

+
      renaming (isMagmaIsomorphism to +-isMagmaIsomorphism; isMonoidIsomorphism to +-isMonoidIsomorphism)

+

+
    *-isMagmaIsomorphism : *.IsMagmaIsomorphism ⟦_⟧

+
    *-isMagmaIsomorphism = record

+
      { isMagmaMonomorphism = *-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+

+
------------------------------------------------------------------------

+
-- Morphisms over ring-like structures

+
------------------------------------------------------------------------

+

+
module RingMorphisms (R₁ : RawRing a ℓ₁) (R₂ : RawRing b ℓ₂) where

+

+
  open RawRing R₁ renaming

+
    ( Carrier to A; _≈_ to _≈₁_

+
    ; -_ to -₁_

+
    ; rawSemiring to rawSemiring₁

+
    ; *-rawMonoid to *-rawMonoid₁

+
    ; +-rawGroup to +-rawGroup₁)

+

+
  open RawRing R₂ renaming

+
    ( Carrier to B; _≈_ to _≈₂_

+
    ; -_ to -₂_

+
    ; rawSemiring to rawSemiring₂

+
    ; *-rawMonoid to *-rawMonoid₂

+
    ; +-rawGroup to +-rawGroup₂)

+

+
  module + = GroupMorphisms  +-rawGroup₁  +-rawGroup₂

+
  module * = MonoidMorphisms *-rawMonoid₁ *-rawMonoid₂

+

+
  open MorphismDefinitions A B _≈₂_

+
  open SemiringMorphisms rawSemiring₁ rawSemiring₂

+

+

+
  record IsRingHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isSemiringHomomorphism : IsSemiringHomomorphism ⟦_⟧

+
      -‿homo : Homomorphic₁ ⟦_⟧ -₁_ -₂_

+

+
    open IsSemiringHomomorphism isSemiringHomomorphism public

+

+
    +-isGroupHomomorphism : +.IsGroupHomomorphism ⟦_⟧

+
    +-isGroupHomomorphism = record

+
      { isMonoidHomomorphism = +-isMonoidHomomorphism

+
      ; ⁻¹-homo = -‿homo

+
      }

+

+
  record IsRingMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRingHomomorphism : IsRingHomomorphism ⟦_⟧

+
      injective          : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsRingHomomorphism isRingHomomorphism public

+

+
    isSemiringMonomorphism : IsSemiringMonomorphism ⟦_⟧

+
    isSemiringMonomorphism = record

+
      { isSemiringHomomorphism = isSemiringHomomorphism

+
      ; injective = injective

+
      }

+

+
    +-isGroupMonomorphism : +.IsGroupMonomorphism ⟦_⟧

+
    +-isGroupMonomorphism = record

+
      { isGroupHomomorphism = +-isGroupHomomorphism

+
      ; injective           = injective

+
      }

+

+
    open +.IsGroupMonomorphism +-isGroupMonomorphism

+
      using (isRelMonomorphism)

+
      renaming ( isMagmaMonomorphism to +-isMagmaMonomorphism

+
               ; isMonoidMonomorphism to +-isMonoidMonomorphism

+
               )

+

+
    *-isMonoidMonomorphism : *.IsMonoidMonomorphism ⟦_⟧

+
    *-isMonoidMonomorphism = record

+
      { isMonoidHomomorphism = *-isMonoidHomomorphism

+
      ; injective            = injective

+
      }

+

+
    open *.IsMonoidMonomorphism *-isMonoidMonomorphism public

+
      using ()

+
      renaming (isMagmaMonomorphism to *-isMagmaMonomorphism)

+

+

+
  record IsRingIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRingMonomorphism : IsRingMonomorphism ⟦_⟧

+
      surjective         : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsRingMonomorphism isRingMonomorphism public

+

+
    isSemiringIsomorphism : IsSemiringIsomorphism ⟦_⟧

+
    isSemiringIsomorphism = record

+
      { isSemiringMonomorphism = isSemiringMonomorphism

+
      ; surjective = surjective

+
      }

+

+
    +-isGroupIsomorphism : +.IsGroupIsomorphism ⟦_⟧

+
    +-isGroupIsomorphism = record

+
      { isGroupMonomorphism = +-isGroupMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    open +.IsGroupIsomorphism +-isGroupIsomorphism

+
      using (isRelIsomorphism)

+
      renaming ( isMagmaIsomorphism to +-isMagmaIsomorphism

+
               ; isMonoidIsomorphism to +-isMonoidIsomorphisn

+
               )

+

+
    *-isMonoidIsomorphism : *.IsMonoidIsomorphism ⟦_⟧

+
    *-isMonoidIsomorphism = record

+
      { isMonoidMonomorphism = *-isMonoidMonomorphism

+
      ; surjective           = surjective

+
      }

+

+
    open *.IsMonoidIsomorphism *-isMonoidIsomorphism public

+
      using ()

+
      renaming (isMagmaIsomorphism to *-isMagmaIsomorphisn)

+

+
------------------------------------------------------------------------

+
-- Morphisms over quasigroup-like structures

+
------------------------------------------------------------------------

+

+
module QuasigroupMorphisms (Q₁ : RawQuasigroup a ℓ₁) (Q₂ : RawQuasigroup b ℓ₂) where

+

+
  open RawQuasigroup Q₁ renaming (Carrier to A; ∙-rawMagma to ∙-rawMagma₁;

+
                                  \\-rawMagma to \\-rawMagma₁; //-rawMagma to //-rawMagma₁;

+
                                  _≈_ to _≈₁_; _∙_ to _∙₁_; _\\_ to _\\₁_; _//_ to _//₁_)

+
  open RawQuasigroup Q₂ renaming (Carrier to B; ∙-rawMagma to ∙-rawMagma₂;

+
                                  \\-rawMagma to \\-rawMagma₂; //-rawMagma to //-rawMagma₂;

+
                                  _≈_ to _≈₂_; _∙_ to _∙₂_; _\\_ to _\\₂_; _//_ to _//₂_)

+

+
  module ∙  = MagmaMorphisms ∙-rawMagma₁ ∙-rawMagma₂

+
  module \\ = MagmaMorphisms \\-rawMagma₁ \\-rawMagma₂

+
  module // = MagmaMorphisms //-rawMagma₁ //-rawMagma₂

+

+
  open MorphismDefinitions A B _≈₂_

+

+
  record IsQuasigroupHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧

+
      ∙-homo            : Homomorphic₂ ⟦_⟧ _∙₁_ _∙₂_

+
      \\-homo           : Homomorphic₂ ⟦_⟧ _\\₁_ _\\₂_

+
      //-homo           : Homomorphic₂ ⟦_⟧ _//₁_ _//₂_

+

+
    open IsRelHomomorphism isRelHomomorphism public

+
      renaming (cong to ⟦⟧-cong)

+

+
    ∙-isMagmaHomomorphism : ∙.IsMagmaHomomorphism ⟦_⟧

+
    ∙-isMagmaHomomorphism = record

+
      { isRelHomomorphism = isRelHomomorphism

+
      ; homo = ∙-homo

+
      }

+

+
    \\-isMagmaHomomorphism : \\.IsMagmaHomomorphism ⟦_⟧

+
    \\-isMagmaHomomorphism = record

+
      { isRelHomomorphism  = isRelHomomorphism

+
      ; homo = \\-homo

+
      }

+

+
    //-isMagmaHomomorphism : //.IsMagmaHomomorphism ⟦_⟧

+
    //-isMagmaHomomorphism = record

+
      { isRelHomomorphism  = isRelHomomorphism

+
      ; homo = //-homo

+
      }

+

+
  record IsQuasigroupMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isQuasigroupHomomorphism : IsQuasigroupHomomorphism ⟦_⟧

+
      injective                : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsQuasigroupHomomorphism isQuasigroupHomomorphism public

+

+

+
    ∙-isMagmaMonomorphism   : ∙.IsMagmaMonomorphism ⟦_⟧

+
    ∙-isMagmaMonomorphism   = record

+
      { isMagmaHomomorphism = ∙-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
    \\-isMagmaMonomorphism  : \\.IsMagmaMonomorphism ⟦_⟧

+
    \\-isMagmaMonomorphism  = record

+
      { isMagmaHomomorphism = \\-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
    //-isMagmaMonomorphism  : //.IsMagmaMonomorphism ⟦_⟧

+
    //-isMagmaMonomorphism  = record

+
      { isMagmaHomomorphism = //-isMagmaHomomorphism

+
      ; injective           = injective

+
      }

+

+
    open //.IsMagmaMonomorphism //-isMagmaMonomorphism public

+
      using (isRelMonomorphism)

+

+

+
  record IsQuasigroupIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isQuasigroupMonomorphism : IsQuasigroupMonomorphism ⟦_⟧

+
      surjective               : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsQuasigroupMonomorphism isQuasigroupMonomorphism public

+

+
    ∙-isMagmaIsomorphism    : ∙.IsMagmaIsomorphism ⟦_⟧

+
    ∙-isMagmaIsomorphism    = record

+
      { isMagmaMonomorphism = ∙-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    \\-isMagmaIsomorphism   : \\.IsMagmaIsomorphism ⟦_⟧

+
    \\-isMagmaIsomorphism   = record

+
      { isMagmaMonomorphism = \\-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    //-isMagmaIsomorphism   : //.IsMagmaIsomorphism ⟦_⟧

+
    //-isMagmaIsomorphism   = record

+
      { isMagmaMonomorphism = //-isMagmaMonomorphism

+
      ; surjective          = surjective

+
      }

+

+
    open //.IsMagmaIsomorphism //-isMagmaIsomorphism public

+
      using (isRelIsomorphism)

+

+
------------------------------------------------------------------------

+
-- Morphisms over loop-like structures

+
------------------------------------------------------------------------

+

+
module LoopMorphisms (L₁ : RawLoop a ℓ₁) (L₂ : RawLoop b ℓ₂) where

+

+
  open RawLoop L₁ renaming (Carrier to A; ∙-rawMagma to ∙-rawMagma₁;

+
                            \\-rawMagma to \\-rawMagma₁; //-rawMagma to //-rawMagma₁;

+
                             _≈_ to _≈₁_; _∙_ to _∙₁_; _\\_ to _\\₁_; _//_ to _//₁_; ε to ε₁)

+
  open RawLoop L₂ renaming (Carrier to B; ∙-rawMagma to ∙-rawMagma₂;

+
                            \\-rawMagma to \\-rawMagma₂; //-rawMagma to //-rawMagma₂;

+
                            _≈_ to _≈₂_; _∙_ to _∙₂_; _\\_ to _\\₂_; _//_ to _//₂_ ; ε to ε₂)

+
  open MorphismDefinitions A B _≈₂_

+

+
  open QuasigroupMorphisms (RawLoop.rawQuasigroup L₁) (RawLoop.rawQuasigroup L₂)

+

+
  record IsLoopHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isQuasigroupHomomorphism : IsQuasigroupHomomorphism ⟦_⟧

+
      ε-homo                   : Homomorphic₀ ⟦_⟧ ε₁ ε₂

+

+
    open IsQuasigroupHomomorphism isQuasigroupHomomorphism public

+

+
  record IsLoopMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isLoopHomomorphism   : IsLoopHomomorphism ⟦_⟧

+
      injective            : Injective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsLoopHomomorphism isLoopHomomorphism public

+

+
  record IsLoopIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isLoopMonomorphism   : IsLoopMonomorphism ⟦_⟧

+
      surjective           : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsLoopMonomorphism isLoopMonomorphism public

+

+
------------------------------------------------------------------------

+
-- Morphisms over Kleene algebra structures

+
------------------------------------------------------------------------

+
module KleeneAlgebraMorphisms (R₁ : RawKleeneAlgebra a ℓ₁) (R₂ : RawKleeneAlgebra b ℓ₂) where

+

+
  open RawKleeneAlgebra R₁ renaming

+
    ( Carrier to A; _≈_ to _≈₁_

+
    ; _⋆ to _⋆₁

+
    ; rawSemiring to rawSemiring₁

+
    )

+

+
  open RawKleeneAlgebra R₂ renaming

+
    ( Carrier to B; _≈_ to _≈₂_

+
    ; _⋆ to _⋆₂

+
    ; rawSemiring to rawSemiring₂

+
    )

+

+
  open MorphismDefinitions A B _≈₂_

+
  open SemiringMorphisms rawSemiring₁ rawSemiring₂

+

+
  record IsKleeneAlgebraHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isSemiringHomomorphism : IsSemiringHomomorphism ⟦_⟧

+
      ⋆-homo :  Homomorphic₁ ⟦_⟧ _⋆₁ _⋆₂

+

+
    open IsSemiringHomomorphism isSemiringHomomorphism public

+

+
  record IsKleeneAlgebraMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isKleeneAlgebraHomomorphism   : IsKleeneAlgebraHomomorphism ⟦_⟧

+
      injective                     : Injective  _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsKleeneAlgebraHomomorphism isKleeneAlgebraHomomorphism public

+

+
  record IsKleeneAlgebraIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      isKleeneAlgebraMonomorphism   : IsKleeneAlgebraMonomorphism ⟦_⟧

+
      surjective                    : Surjective  _≈₁_ _≈₂_ ⟦_⟧

+

+
    open IsKleeneAlgebraMonomorphism isKleeneAlgebraMonomorphism public

+

+
------------------------------------------------------------------------

+
-- Re-export contents of modules publicly

+

+
open MagmaMorphisms public

+
open MonoidMorphisms public

+
open GroupMorphisms public

+
open NearSemiringMorphisms public

+
open SemiringMorphisms public

+
open RingWithoutOneMorphisms public

+
open RingMorphisms public

+
open QuasigroupMorphisms public

+
open LoopMorphisms public

+
open KleeneAlgebraMorphisms public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Morphism.html b/site/static/agda_html/Algebra.Morphism.html
new file mode 100644
index 000000000..b3713227d
--- /dev/null
+++ b/site/static/agda_html/Algebra.Morphism.html
@@ -0,0 +1,286 @@
+
+Algebra.Morphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Morphisms between algebraic structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Morphism where

+

+
import Algebra.Morphism.Definitions as MorphismDefinitions

+
open import Algebra

+
import Algebra.Properties.Group as GroupP

+
open import Function.Base

+
open import Level

+
open import Relation.Binary.Core using (Rel; _Preserves_⟶_)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Re-export

+

+
module Definitions {a b ℓ₁} (A : Set a) (B : Set b) (_≈_ : Rel B ℓ₁) where

+
  open MorphismDefinitions A B _≈_ public

+

+
open import Algebra.Morphism.Structures public

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new definitions re-exported from

+
-- `Algebra.Morphism.Structures` as continuing support for the below is

+
-- no guaranteed.

+

+
-- Version 1.5

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : Semigroup c₁ ℓ₁)

+
         (To   : Semigroup c₂ ℓ₂) where

+

+
  private

+
    module F = Semigroup From

+
    module T = Semigroup To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsSemigroupMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      ⟦⟧-cong : ⟦_⟧ Preserves F._≈_ ⟶ T._≈_

+
      ∙-homo  : Homomorphic₂ ⟦_⟧ F._∙_ T._∙_

+

+
  IsSemigroupMorphism-syntax = IsSemigroupMorphism

+
  syntax IsSemigroupMorphism-syntax From To F = F Is From -Semigroup⟶ To

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : Monoid c₁ ℓ₁)

+
         (To   : Monoid c₂ ℓ₂) where

+

+
  private

+
    module F = Monoid From

+
    module T = Monoid To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsMonoidMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      sm-homo : IsSemigroupMorphism F.semigroup T.semigroup ⟦_⟧

+
      ε-homo  : Homomorphic₀ ⟦_⟧ F.ε T.ε

+

+
    open IsSemigroupMorphism sm-homo public

+

+
  IsMonoidMorphism-syntax = IsMonoidMorphism

+
  syntax IsMonoidMorphism-syntax From To F = F Is From -Monoid⟶ To

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : CommutativeMonoid c₁ ℓ₁)

+
         (To   : CommutativeMonoid c₂ ℓ₂) where

+

+
  private

+
    module F = CommutativeMonoid From

+
    module T = CommutativeMonoid To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsCommutativeMonoidMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      mn-homo : IsMonoidMorphism F.monoid T.monoid ⟦_⟧

+

+
    open IsMonoidMorphism mn-homo public

+

+
  IsCommutativeMonoidMorphism-syntax = IsCommutativeMonoidMorphism

+
  syntax IsCommutativeMonoidMorphism-syntax From To F = F Is From -CommutativeMonoid⟶ To

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : IdempotentCommutativeMonoid c₁ ℓ₁)

+
         (To   : IdempotentCommutativeMonoid c₂ ℓ₂) where

+

+
  private

+
    module F = IdempotentCommutativeMonoid From

+
    module T = IdempotentCommutativeMonoid To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsIdempotentCommutativeMonoidMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      mn-homo : IsMonoidMorphism F.monoid T.monoid ⟦_⟧

+

+
    open IsMonoidMorphism mn-homo public

+

+
    isCommutativeMonoidMorphism :

+
      IsCommutativeMonoidMorphism F.commutativeMonoid T.commutativeMonoid ⟦_⟧

+
    isCommutativeMonoidMorphism = record { mn-homo = mn-homo }

+

+
  IsIdempotentCommutativeMonoidMorphism-syntax = IsIdempotentCommutativeMonoidMorphism

+
  syntax IsIdempotentCommutativeMonoidMorphism-syntax From To F = F Is From -IdempotentCommutativeMonoid⟶ To

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : Group c₁ ℓ₁)

+
         (To   : Group c₂ ℓ₂) where

+

+
  private

+
    module F = Group From

+
    module T = Group To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsGroupMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      mn-homo : IsMonoidMorphism F.monoid T.monoid ⟦_⟧

+

+
    open IsMonoidMorphism mn-homo public

+

+
    ⁻¹-homo : Homomorphic₁ ⟦_⟧ F._⁻¹ T._⁻¹

+
    ⁻¹-homo x = let open ≈-Reasoning T.setoid in T.uniqueˡ-⁻¹ ⟦ x F.⁻¹ ⟧ ⟦ x ⟧ $ begin

+
      ⟦ x F.⁻¹ ⟧ T.∙ ⟦ x ⟧ ≈⟨ T.sym (∙-homo (x F.⁻¹) x) ⟩

+
      ⟦ x F.⁻¹ F.∙ x ⟧     ≈⟨ ⟦⟧-cong (F.inverseˡ x) ⟩

+
      ⟦ F.ε ⟧              ≈⟨ ε-homo ⟩

+
      T.ε ∎

+

+
  IsGroupMorphism-syntax = IsGroupMorphism

+
  syntax IsGroupMorphism-syntax From To F = F Is From -Group⟶ To

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : AbelianGroup c₁ ℓ₁)

+
         (To   : AbelianGroup c₂ ℓ₂) where

+

+
  private

+
    module F = AbelianGroup From

+
    module T = AbelianGroup To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsAbelianGroupMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      gp-homo : IsGroupMorphism F.group T.group ⟦_⟧

+

+
    open IsGroupMorphism gp-homo public

+

+
  IsAbelianGroupMorphism-syntax = IsAbelianGroupMorphism

+
  syntax IsAbelianGroupMorphism-syntax From To F = F Is From -AbelianGroup⟶ To

+

+
module _ {c₁ ℓ₁ c₂ ℓ₂}

+
         (From : Ring c₁ ℓ₁)

+
         (To   : Ring c₂ ℓ₂) where

+

+
  private

+
    module F = Ring From

+
    module T = Ring To

+
  open Definitions F.Carrier T.Carrier T._≈_

+

+
  record IsRingMorphism (⟦_⟧ : Morphism) :

+
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where

+
    field

+
      +-abgp-homo : ⟦_⟧ Is F.+-abelianGroup -AbelianGroup⟶ T.+-abelianGroup

+
      *-mn-homo   : ⟦_⟧ Is F.*-monoid -Monoid⟶ T.*-monoid

+

+
  IsRingMorphism-syntax = IsRingMorphism

+
  syntax IsRingMorphism-syntax From To F = F Is From -Ring⟶ To

+

+
{-# WARNING_ON_USAGE IsSemigroupMorphism

+
"Warning: IsSemigroupMorphism was deprecated in v1.5.

+
Please use IsMagmaHomomorphism instead."

+
#-}

+
{-# WARNING_ON_USAGE IsMonoidMorphism

+
"Warning: IsMonoidMorphism was deprecated in v1.5.

+
Please use IsMonoidHomomorphism instead."

+
#-}

+
{-# WARNING_ON_USAGE IsCommutativeMonoidMorphism

+
"Warning: IsCommutativeMonoidMorphism was deprecated in v1.5.

+
Please use IsMonoidHomomorphism instead."

+
#-}

+
{-# WARNING_ON_USAGE IsIdempotentCommutativeMonoidMorphism

+
"Warning: IsIdempotentCommutativeMonoidMorphism was deprecated in v1.5.

+
Please use IsMonoidHomomorphism instead."

+
#-}

+
{-# WARNING_ON_USAGE IsGroupMorphism

+
"Warning: IsGroupMorphism was deprecated in v1.5.

+
Please use IsGroupHomomorphism instead."

+
#-}

+
{-# WARNING_ON_USAGE IsAbelianGroupMorphism

+
"Warning: IsAbelianGroupMorphism was deprecated in v1.5.

+
Please use IsGroupHomomorphism instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.AbelianGroup.html b/site/static/agda_html/Algebra.Properties.AbelianGroup.html
new file mode 100644
index 000000000..62be00731
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.AbelianGroup.html
@@ -0,0 +1,118 @@
+
+Algebra.Properties.AbelianGroup
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra

+

+
module Algebra.Properties.AbelianGroup

+
  {a ℓ} (G : AbelianGroup a ℓ) where

+

+
open AbelianGroup G

+
open import Function.Base using (_$_)

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
------------------------------------------------------------------------

+
-- Publicly re-export group properties

+

+
open import Algebra.Properties.Group group public

+

+
------------------------------------------------------------------------

+
-- Properties of abelian groups

+

+
⁻¹-anti-homo‿- : ∀ x y → (x - y) ⁻¹ ≈ y - x

+
⁻¹-anti-homo‿- = ⁻¹-anti-homo-//

+

+
xyx⁻¹≈y : ∀ x y → x ∙ y ∙ x ⁻¹ ≈ y

+
xyx⁻¹≈y x y = begin

+
  x ∙ y ∙ x ⁻¹    ≈⟨ ∙-congʳ $ comm _ _ ⟩

+
  y ∙ x ∙ x ⁻¹    ≈⟨ assoc _ _ _ ⟩

+
  y ∙ (x ∙ x ⁻¹)  ≈⟨ ∙-congˡ $ inverseʳ _ ⟩

+
  y ∙ ε           ≈⟨ identityʳ _ ⟩

+
  y               ∎

+

+
⁻¹-∙-comm : ∀ x y → x ⁻¹ ∙ y ⁻¹ ≈ (x ∙ y) ⁻¹

+
⁻¹-∙-comm x y = begin

+
  x ⁻¹ ∙ y ⁻¹ ≈⟨ ⁻¹-anti-homo-∙ y x ⟨

+
  (y ∙ x) ⁻¹  ≈⟨ ⁻¹-cong $ comm y x ⟩

+
  (x ∙ y) ⁻¹  ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.CommutativeMonoid.html b/site/static/agda_html/Algebra.Properties.CommutativeMonoid.html
new file mode 100644
index 000000000..2725cfd47
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.CommutativeMonoid.html
@@ -0,0 +1,119 @@
+
+Algebra.Properties.CommutativeMonoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles using (CommutativeMonoid)

+
open import Algebra.Definitions using (LeftInvertible; RightInvertible; Invertible)

+
open import Data.Product.Base using (_,_; proj₂)

+

+
module Algebra.Properties.CommutativeMonoid

+
  {g₁ g₂} (M : CommutativeMonoid g₁ g₂) where

+

+
open CommutativeMonoid M

+
  renaming

+
  ( ε         to 0#

+
  ; _∙_       to _+_

+
  ; ∙-cong    to +-cong

+
  ; ∙-congˡ   to +-congˡ

+
  ; ∙-congʳ   to +-congʳ

+
  ; identityˡ to +-identityˡ

+
  ; identityʳ to +-identityʳ

+
  ; assoc     to +-assoc

+
  ; comm      to +-comm

+
  )

+

+
private variable

+
  x : Carrier

+

+
invertibleˡ⇒invertibleʳ : LeftInvertible _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x

+
invertibleˡ⇒invertibleʳ {x} (-x , -x+x≈1) = -x , trans (+-comm x -x) -x+x≈1

+

+
invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible _≈_ 0# _+_ x

+
invertibleʳ⇒invertibleˡ {x} (-x , x+-x≈1) = -x , trans (+-comm -x x) x+-x≈1

+

+
invertibleˡ⇒invertible : LeftInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x

+
invertibleˡ⇒invertible left@(-x , -x+x≈1) = -x , -x+x≈1 , invertibleˡ⇒invertibleʳ left .proj₂

+

+
invertibleʳ⇒invertible : RightInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x

+
invertibleʳ⇒invertible right@(-x , x+-x≈1) = -x , invertibleʳ⇒invertibleˡ right .proj₂ , x+-x≈1

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.CommutativeSemigroup.html b/site/static/agda_html/Algebra.Properties.CommutativeSemigroup.html
new file mode 100644
index 000000000..459c1739b
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.CommutativeSemigroup.html
@@ -0,0 +1,251 @@
+
+Algebra.Properties.CommutativeSemigroup
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some theory for commutative semigroup

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra using (CommutativeSemigroup)

+

+
module Algebra.Properties.CommutativeSemigroup

+
  {a ℓ} (CS : CommutativeSemigroup a ℓ)

+
  where

+

+
open CommutativeSemigroup CS

+

+
open import Algebra.Definitions _≈_

+
open import Relation.Binary.Reasoning.Setoid setoid

+
open import Data.Product.Base using (_,_)

+

+
------------------------------------------------------------------------

+
-- Re-export the contents of semigroup

+

+
open import Algebra.Properties.Semigroup semigroup public

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
interchange : Interchangable _∙_ _∙_

+
interchange a b c d = begin

+
  (a ∙ b) ∙ (c ∙ d)  ≈⟨  assoc a b (c ∙ d) ⟩

+
  a ∙ (b ∙ (c ∙ d))  ≈⟨ ∙-congˡ (assoc b c d) ⟨

+
  a ∙ ((b ∙ c) ∙ d)  ≈⟨  ∙-congˡ (∙-congʳ (comm b c)) ⟩

+
  a ∙ ((c ∙ b) ∙ d)  ≈⟨  ∙-congˡ (assoc c b d) ⟩

+
  a ∙ (c ∙ (b ∙ d))  ≈⟨ assoc a c (b ∙ d) ⟨

+
  (a ∙ c) ∙ (b ∙ d)  ∎

+

+
------------------------------------------------------------------------

+
-- Permutation laws for _∙_ for three factors.

+

+
-- There are five nontrivial permutations.

+

+
------------------------------------------------------------------------

+
-- Partitions (1,1).

+

+
x∙yz≈y∙xz :  ∀ x y z → x ∙ (y ∙ z) ≈ y ∙ (x ∙ z)

+
x∙yz≈y∙xz x y z = begin

+
  x ∙ (y ∙ z)    ≈⟨ sym (assoc x y z) ⟩

+
  (x ∙ y) ∙ z    ≈⟨ ∙-congʳ (comm x y) ⟩

+
  (y ∙ x) ∙ z    ≈⟨ assoc y x z ⟩

+
  y ∙ (x ∙ z)    ∎

+

+
x∙yz≈z∙yx :  ∀ x y z → x ∙ (y ∙ z) ≈ z ∙ (y ∙ x)

+
x∙yz≈z∙yx x y z = begin

+
  x ∙ (y ∙ z)    ≈⟨ ∙-congˡ (comm y z) ⟩

+
  x ∙ (z ∙ y)    ≈⟨ x∙yz≈y∙xz x z y ⟩

+
  z ∙ (x ∙ y)    ≈⟨ ∙-congˡ (comm x y) ⟩

+
  z ∙ (y ∙ x)    ∎

+

+
x∙yz≈x∙zy :  ∀ x y z → x ∙ (y ∙ z) ≈ x ∙ (z ∙ y)

+
x∙yz≈x∙zy _ y z =  ∙-congˡ (comm y z)

+

+
x∙yz≈y∙zx :  ∀ x y z → x ∙ (y ∙ z) ≈ y ∙ (z ∙ x)

+
x∙yz≈y∙zx x y z = begin

+
  x ∙ (y ∙ z)   ≈⟨ comm x _ ⟩

+
  (y ∙ z) ∙ x   ≈⟨ assoc y z x ⟩

+
  y ∙ (z ∙ x)   ∎

+

+
x∙yz≈z∙xy :  ∀ x y z → x ∙ (y ∙ z) ≈ z ∙ (x ∙ y)

+
x∙yz≈z∙xy x y z = begin

+
  x ∙ (y ∙ z)   ≈⟨ sym (assoc x y z) ⟩

+
  (x ∙ y) ∙ z   ≈⟨ comm _ z ⟩

+
  z ∙ (x ∙ y)   ∎

+

+
------------------------------------------------------------------------

+
-- Partitions (1,2).

+

+
-- These permutation laws are proved by composing the proofs for

+
-- partitions (1,1) with  \p → trans p (sym (assoc _ _ _)).

+

+
x∙yz≈yx∙z :  ∀ x y z → x ∙ (y ∙ z) ≈ (y ∙ x) ∙ z

+
x∙yz≈yx∙z x y z =  trans (x∙yz≈y∙xz x y z) (sym (assoc y x z))

+

+
x∙yz≈zy∙x :  ∀ x y z → x ∙ (y ∙ z) ≈ (z ∙ y) ∙ x

+
x∙yz≈zy∙x x y z =  trans (x∙yz≈z∙yx x y z) (sym (assoc z y x))

+

+
x∙yz≈xz∙y :  ∀ x y z → x ∙ (y ∙ z) ≈ (x ∙ z) ∙ y

+
x∙yz≈xz∙y x y z =  trans (x∙yz≈x∙zy x y z) (sym (assoc x z y))

+

+
x∙yz≈yz∙x :  ∀ x y z → x ∙ (y ∙ z) ≈ (y ∙ z) ∙ x

+
x∙yz≈yz∙x x y z =  trans (x∙yz≈y∙zx _ _ _) (sym (assoc y z x))

+

+
x∙yz≈zx∙y :  ∀ x y z → x ∙ (y ∙ z) ≈ (z ∙ x) ∙ y

+
x∙yz≈zx∙y x y z =  trans (x∙yz≈z∙xy x y z) (sym (assoc z x y))

+

+
------------------------------------------------------------------------

+
-- Partitions (2,1).

+

+
-- Their laws are proved by composing proofs for partitions (1,1) with

+
-- trans (assoc x y z).

+

+
xy∙z≈y∙xz :  ∀ x y z → (x ∙ y) ∙ z ≈ y ∙ (x ∙ z)

+
xy∙z≈y∙xz x y z =  trans (assoc x y z) (x∙yz≈y∙xz x y z)

+

+
xy∙z≈z∙yx :  ∀ x y z → (x ∙ y) ∙ z ≈ z ∙ (y ∙ x)

+
xy∙z≈z∙yx x y z =  trans (assoc x y z) (x∙yz≈z∙yx x y z)

+

+
xy∙z≈x∙zy :  ∀ x y z → (x ∙ y) ∙ z ≈ x ∙ (z ∙ y)

+
xy∙z≈x∙zy x y z =  trans (assoc x y z) (x∙yz≈x∙zy x y z)

+

+
xy∙z≈y∙zx :  ∀ x y z → (x ∙ y) ∙ z ≈ y ∙ (z ∙ x)

+
xy∙z≈y∙zx x y z =  trans (assoc x y z) (x∙yz≈y∙zx x y z)

+

+
xy∙z≈z∙xy :  ∀ x y z → (x ∙ y) ∙ z ≈ z ∙ (x ∙ y)

+
xy∙z≈z∙xy x y z =  trans (assoc x y z) (x∙yz≈z∙xy x y z)

+

+
------------------------------------------------------------------------

+
-- Partitions (2,2).

+

+
-- These proofs are by composing with the proofs for (2,1).

+

+
xy∙z≈yx∙z :  ∀ x y z → (x ∙ y) ∙ z ≈ (y ∙ x) ∙ z

+
xy∙z≈yx∙z x y z =  trans (xy∙z≈y∙xz _ _ _) (sym (assoc y x z))

+

+
xy∙z≈zy∙x :  ∀ x y z → (x ∙ y) ∙ z ≈ (z ∙ y) ∙ x

+
xy∙z≈zy∙x x y z =  trans (xy∙z≈z∙yx x y z) (sym (assoc z y x))

+

+
xy∙z≈xz∙y :  ∀ x y z → (x ∙ y) ∙ z ≈ (x ∙ z) ∙ y

+
xy∙z≈xz∙y x y z =  trans (xy∙z≈x∙zy x y z) (sym (assoc x z y))

+

+
xy∙z≈yz∙x :  ∀ x y z → (x ∙ y) ∙ z ≈ (y ∙ z) ∙ x

+
xy∙z≈yz∙x x y z =  trans (xy∙z≈y∙zx x y z) (sym (assoc y z x))

+

+
xy∙z≈zx∙y :  ∀ x y z → (x ∙ y) ∙ z ≈ (z ∙ x) ∙ y

+
xy∙z≈zx∙y x y z =  trans (xy∙z≈z∙xy x y z) (sym (assoc z x y))

+

+
------------------------------------------------------------------------

+
-- commutative semigroup has Jordan identity

+

+
xy∙xx≈x∙yxx : ∀ x y → (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))

+
xy∙xx≈x∙yxx x y = assoc x y ((x ∙ x))

+

+
------------------------------------------------------------------------

+
-- commutative semigroup is left/right/middle semiMedial

+

+
semimedialˡ : LeftSemimedial _∙_

+
semimedialˡ x y z = begin

+
  (x ∙ x) ∙ (y ∙ z) ≈⟨ assoc x x (y ∙ z) ⟩

+
  x ∙ (x ∙ (y ∙ z)) ≈⟨ ∙-congˡ (sym (assoc x y z)) ⟩

+
  x ∙ ((x ∙ y) ∙ z) ≈⟨ ∙-congˡ (∙-congʳ (comm x y)) ⟩

+
  x ∙ ((y ∙ x) ∙ z) ≈⟨ ∙-congˡ (assoc y x z) ⟩

+
  x ∙ (y ∙ (x ∙ z)) ≈⟨ sym (assoc x y ((x ∙ z))) ⟩

+
  (x ∙ y) ∙ (x ∙ z) ∎

+

+
semimedialʳ : RightSemimedial _∙_

+
semimedialʳ x y z = begin

+
  (y ∙ z) ∙ (x ∙ x) ≈⟨ assoc y z (x ∙ x) ⟩

+
  y ∙ (z ∙ (x ∙ x)) ≈⟨ ∙-congˡ (sym (assoc z x x)) ⟩

+
  y ∙ ((z ∙ x) ∙ x) ≈⟨ ∙-congˡ (∙-congʳ (comm z x)) ⟩

+
  y ∙ ((x ∙ z) ∙ x) ≈⟨ ∙-congˡ (assoc x z x) ⟩

+
  y ∙ (x ∙ (z ∙ x)) ≈⟨ sym (assoc y x ((z ∙ x))) ⟩

+
  (y ∙ x) ∙ (z ∙ x) ∎

+

+
middleSemimedial : ∀ x y z → (x ∙ y) ∙ (z ∙ x) ≈ (x ∙ z) ∙ (y ∙ x)

+
middleSemimedial x y z = begin

+
  (x ∙ y) ∙ (z ∙ x) ≈⟨ assoc x y ((z ∙ x)) ⟩

+
  x ∙ (y ∙ (z ∙ x)) ≈⟨ ∙-congˡ (sym (assoc y z x)) ⟩

+
  x ∙ ((y ∙ z) ∙ x) ≈⟨ ∙-congˡ (∙-congʳ (comm y z)) ⟩

+
  x ∙ ((z ∙ y) ∙ x) ≈⟨ ∙-congˡ ( assoc z y x) ⟩

+
  x ∙ (z ∙ (y ∙ x)) ≈⟨ sym (assoc x z ((y ∙ x))) ⟩

+
  (x ∙ z) ∙ (y ∙ x) ∎

+

+
semimedial : Semimedial _∙_

+
semimedial = semimedialˡ , semimedialʳ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Group.html b/site/static/agda_html/Algebra.Properties.Group.html
new file mode 100644
index 000000000..7d7e07adf
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Group.html
@@ -0,0 +1,247 @@
+
+Algebra.Properties.Group
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some derivable properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles

+

+
module Algebra.Properties.Group {g₁ g₂} (G : Group g₁ g₂) where

+

+
import Algebra.Properties.Loop as LoopProperties

+
import Algebra.Properties.Quasigroup as QuasigroupProperties

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_$_)

+
open import Function.Definitions

+

+
open Group G

+
open import Algebra.Consequences.Setoid setoid

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures _≈_ using (IsLoop; IsQuasigroup)

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
\\-cong₂ : Congruent₂ _\\_

+
\\-cong₂ x≈y u≈v = ∙-cong (⁻¹-cong x≈y) u≈v

+

+
//-cong₂ : Congruent₂ _//_

+
//-cong₂ x≈y u≈v = ∙-cong x≈y (⁻¹-cong u≈v)

+

+
------------------------------------------------------------------------

+
-- Groups are quasi-groups

+

+
\\-leftDividesˡ : LeftDividesˡ _∙_ _\\_

+
\\-leftDividesˡ x y = begin

+
  x  ∙ (x \\ y)  ≈⟨ assoc x (x ⁻¹) y ⟨

+
  x ∙ x ⁻¹ ∙ y   ≈⟨ ∙-congʳ (inverseʳ x) ⟩

+
  ε ∙ y          ≈⟨ identityˡ y ⟩

+
  y              ∎

+

+
\\-leftDividesʳ : LeftDividesʳ _∙_ _\\_

+
\\-leftDividesʳ x y = begin

+
  x \\ x ∙ y     ≈⟨ assoc (x ⁻¹) x y ⟨

+
  x ⁻¹ ∙ x ∙ y   ≈⟨ ∙-congʳ (inverseˡ x) ⟩

+
  ε ∙ y          ≈⟨ identityˡ y ⟩

+
  y              ∎

+

+

+
\\-leftDivides : LeftDivides _∙_ _\\_

+
\\-leftDivides = \\-leftDividesˡ , \\-leftDividesʳ

+

+
//-rightDividesˡ : RightDividesˡ _∙_ _//_

+
//-rightDividesˡ x y = begin

+
  (y // x) ∙ x    ≈⟨ assoc y (x ⁻¹) x ⟩

+
  y ∙ (x ⁻¹ ∙ x)  ≈⟨ ∙-congˡ (inverseˡ x) ⟩

+
  y ∙ ε           ≈⟨ identityʳ y ⟩

+
  y               ∎

+

+
//-rightDividesʳ : RightDividesʳ _∙_ _//_

+
//-rightDividesʳ x y = begin

+
  y ∙ x // x    ≈⟨ assoc y x (x ⁻¹) ⟩

+
  y ∙ (x // x)  ≈⟨ ∙-congˡ (inverseʳ x) ⟩

+
  y ∙ ε         ≈⟨ identityʳ y ⟩

+
  y             ∎

+

+
//-rightDivides : RightDivides _∙_ _//_

+
//-rightDivides = //-rightDividesˡ , //-rightDividesʳ

+

+
isQuasigroup : IsQuasigroup _∙_ _\\_ _//_

+
isQuasigroup = record

+
  { isMagma = isMagma

+
  ; \\-cong = \\-cong₂

+
  ; //-cong = //-cong₂

+
  ; leftDivides = \\-leftDivides

+
  ; rightDivides = //-rightDivides

+
  }

+

+
quasigroup : Quasigroup _ _

+
quasigroup = record { isQuasigroup = isQuasigroup }

+

+
open QuasigroupProperties quasigroup public

+
  using (x≈z//y; y≈x\\z)

+
  renaming (cancelˡ to ∙-cancelˡ; cancelʳ to ∙-cancelʳ; cancel to ∙-cancel)

+

+
------------------------------------------------------------------------

+
-- Groups are loops

+

+
isLoop : IsLoop _∙_ _\\_ _//_ ε

+
isLoop = record { isQuasigroup = isQuasigroup ; identity = identity }

+

+
loop : Loop _ _

+
loop = record { isLoop = isLoop }

+

+
open LoopProperties loop public

+
  using (identityˡ-unique; identityʳ-unique; identity-unique)

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
inverseˡ-unique : ∀ x y → x ∙ y ≈ ε → x ≈ y ⁻¹

+
inverseˡ-unique x y eq = trans (x≈z//y x y ε eq) (identityˡ _)

+

+
inverseʳ-unique : ∀ x y → x ∙ y ≈ ε → y ≈ x ⁻¹

+
inverseʳ-unique x y eq = trans (y≈x\\z x y ε eq) (identityʳ _)

+

+
ε⁻¹≈ε : ε ⁻¹ ≈ ε

+
ε⁻¹≈ε = sym $ inverseˡ-unique _ _ (identityˡ ε)

+

+
⁻¹-selfInverse : SelfInverse _⁻¹

+
⁻¹-selfInverse {x} {y} eq = sym $ inverseˡ-unique x y $ begin

+
  x ∙ y    ≈⟨ ∙-congˡ eq ⟨

+
  x ∙ x ⁻¹ ≈⟨ inverseʳ x  ⟩

+
  ε        ∎

+

+
⁻¹-involutive : Involutive _⁻¹

+
⁻¹-involutive = selfInverse⇒involutive ⁻¹-selfInverse

+

+
x∙y⁻¹≈ε⇒x≈y : ∀ x y → (x ∙ y ⁻¹) ≈ ε → x ≈ y

+
x∙y⁻¹≈ε⇒x≈y x y x∙y⁻¹≈ε = begin

+
  x         ≈⟨ inverseˡ-unique x (y ⁻¹) x∙y⁻¹≈ε ⟩

+
  y ⁻¹ ⁻¹   ≈⟨ ⁻¹-involutive y ⟩

+
  y         ∎

+

+
x≈y⇒x∙y⁻¹≈ε : ∀ {x y} → x ≈ y → (x ∙ y ⁻¹) ≈ ε

+
x≈y⇒x∙y⁻¹≈ε {x} {y} x≈y = begin

+
  x ∙ y ⁻¹ ≈⟨ ∙-congʳ x≈y ⟩

+
  y ∙ y ⁻¹ ≈⟨ inverseʳ y ⟩

+
  ε        ∎

+

+
⁻¹-injective : Injective _≈_ _≈_ _⁻¹

+
⁻¹-injective = selfInverse⇒injective ⁻¹-selfInverse

+

+
⁻¹-anti-homo-∙ : ∀ x y → (x ∙ y) ⁻¹ ≈ y ⁻¹ ∙ x ⁻¹

+
⁻¹-anti-homo-∙ x y = ∙-cancelˡ _ _ _ $ begin

+
  x ∙ y ∙ (x ∙ y) ⁻¹    ≈⟨ inverseʳ _ ⟩

+
  ε                     ≈⟨ inverseʳ _ ⟨

+
  x ∙ x ⁻¹              ≈⟨ ∙-congʳ (//-rightDividesʳ y x) ⟨

+
  (x ∙ y) ∙ y ⁻¹ ∙ x ⁻¹ ≈⟨ assoc (x ∙ y) (y ⁻¹) (x ⁻¹) ⟩

+
  x ∙ y ∙ (y ⁻¹ ∙ x ⁻¹) ∎

+

+
⁻¹-anti-homo-// : ∀ x y → (x // y) ⁻¹ ≈ y // x

+
⁻¹-anti-homo-// x y = begin

+
  (x // y) ⁻¹      ≡⟨⟩

+
  (x ∙ y ⁻¹) ⁻¹    ≈⟨ ⁻¹-anti-homo-∙ x (y ⁻¹) ⟩

+
  (y ⁻¹) ⁻¹ ∙ x ⁻¹ ≈⟨ ∙-congʳ (⁻¹-involutive y) ⟩

+
  y ∙ x ⁻¹         ≡⟨⟩

+
  y // x ∎

+

+
⁻¹-anti-homo-\\ : ∀ x y → (x \\ y) ⁻¹ ≈ y \\ x

+
⁻¹-anti-homo-\\ x y = begin

+
  (x \\ y) ⁻¹      ≡⟨⟩

+
  (x ⁻¹ ∙ y) ⁻¹    ≈⟨ ⁻¹-anti-homo-∙ (x ⁻¹) y ⟩

+
  y ⁻¹ ∙ (x ⁻¹) ⁻¹ ≈⟨ ∙-congˡ (⁻¹-involutive x) ⟩

+
  y ⁻¹ ∙ x         ≡⟨⟩

+
  y \\ x ∎

+

+
\\≗flip-//⇒comm : (∀ x y → x \\ y ≈ y // x) → Commutative _∙_

+
\\≗flip-//⇒comm \\≗//ᵒ x y = begin

+
  x ∙ y                ≈⟨ ∙-congˡ (//-rightDividesˡ x y) ⟨

+
  x ∙ ((y // x) ∙ x)   ≈⟨ ∙-congˡ (∙-congʳ (\\≗//ᵒ x y)) ⟨

+
  x ∙ ((x \\ y) ∙ x)   ≈⟨ assoc x (x \\ y) x ⟨

+
  x ∙ (x \\ y) ∙ x     ≈⟨ ∙-congʳ (\\-leftDividesˡ x y) ⟩

+
  y ∙ x                ∎

+

+
comm⇒\\≗flip-// : Commutative _∙_ → ∀ x y → x \\ y ≈ y // x

+
comm⇒\\≗flip-// comm x y = begin

+
  x \\ y    ≡⟨⟩

+
  x ⁻¹ ∙ y  ≈⟨ comm _ _ ⟩

+
  y ∙ x ⁻¹  ≡⟨⟩

+
  y // x    ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Loop.html b/site/static/agda_html/Algebra.Properties.Loop.html
new file mode 100644
index 000000000..f52924ee6
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Loop.html
@@ -0,0 +1,134 @@
+
+Algebra.Properties.Loop
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some basic properties of Loop

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles using (Loop)

+

+
module Algebra.Properties.Loop {l₁ l₂} (L : Loop l₁ l₂) where

+

+
open Loop L

+
open import Algebra.Definitions _≈_

+
open import Algebra.Properties.Quasigroup quasigroup

+
open import Data.Product.Base using (proj₂)

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
x//x≈ε : ∀ x → x // x ≈ ε

+
x//x≈ε x = begin

+
  x // x       ≈⟨ //-congʳ (identityˡ x) ⟨

+
  (ε ∙ x) // x ≈⟨ rightDividesʳ x ε ⟩

+
  ε            ∎

+

+
x\\x≈ε : ∀ x → x \\ x ≈ ε

+
x\\x≈ε x = begin

+
  x \\ x       ≈⟨ \\-congˡ (identityʳ x ) ⟨

+
  x \\ (x ∙ ε) ≈⟨ leftDividesʳ x ε ⟩

+
  ε            ∎

+

+
ε\\x≈x : ∀ x → ε \\ x ≈ x

+
ε\\x≈x x = begin

+
  ε \\ x       ≈⟨ identityˡ (ε \\ x) ⟨

+
  ε ∙ (ε \\ x) ≈⟨ leftDividesˡ ε x ⟩

+
  x            ∎

+

+
x//ε≈x : ∀ x → x // ε ≈ x

+
x//ε≈x x = begin

+
 x // ε       ≈⟨ identityʳ (x // ε) ⟨

+
 (x // ε) ∙ ε ≈⟨ rightDividesˡ ε x ⟩

+
 x            ∎

+

+
identityˡ-unique : ∀ x y → x ∙ y ≈ y → x ≈ ε

+
identityˡ-unique x y eq = begin

+
  x      ≈⟨ x≈z//y x y y eq ⟩

+
  y // y ≈⟨ x//x≈ε y ⟩

+
  ε      ∎

+

+
identityʳ-unique : ∀ x y → x ∙ y ≈ x → y ≈ ε

+
identityʳ-unique x y eq = begin

+
  y       ≈⟨ y≈x\\z x y x eq ⟩

+
  x \\ x  ≈⟨ x\\x≈ε x ⟩

+
  ε       ∎

+

+
identity-unique : ∀ {x} → Identity x _∙_ → x ≈ ε

+
identity-unique {x} id = identityˡ-unique x x (proj₂ id x)

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Monoid.Mult.html b/site/static/agda_html/Algebra.Properties.Monoid.Mult.html
new file mode 100644
index 000000000..c6b1d0987
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Monoid.Mult.html
@@ -0,0 +1,157 @@
+
+Algebra.Properties.Monoid.Mult
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Multiplication over a monoid (i.e. repeated addition)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles using (Monoid)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero)

+
open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+

+
module Algebra.Properties.Monoid.Mult {a ℓ} (M : Monoid a ℓ) where

+

+
-- View of the monoid operator as addition

+
open Monoid M

+
  renaming

+
  ( _∙_       to _+_

+
  ; ∙-cong    to +-cong

+
  ; ∙-congʳ   to +-congʳ

+
  ; ∙-congˡ   to +-congˡ

+
  ; identityˡ to +-identityˡ

+
  ; identityʳ to +-identityʳ

+
  ; assoc     to +-assoc

+
  ; ε         to 0#

+
  )

+

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
open import Algebra.Definitions _≈_

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open import Algebra.Definitions.RawMonoid rawMonoid public

+
  using (_×_)

+

+
------------------------------------------------------------------------

+
-- Properties of _×_

+

+
×-congʳ : ∀ n → (n ×_) Preserves _≈_ ⟶ _≈_

+
×-congʳ 0       x≈x′ = refl

+
×-congʳ (suc n) x≈x′ = +-cong x≈x′ (×-congʳ n x≈x′)

+

+
×-cong : _×_ Preserves₂ _≡_ ⟶ _≈_ ⟶ _≈_

+
×-cong {n} ≡.refl x≈x′ = ×-congʳ n x≈x′

+

+
×-congˡ : ∀ {x} → (_× x) Preserves _≡_ ⟶ _≈_

+
×-congˡ m≡n = ×-cong m≡n refl

+

+
-- _×_ is homomorphic with respect to _ℕ+_/_+_.

+

+
×-homo-0 : ∀ x → 0 × x ≈ 0#

+
×-homo-0 x = refl

+

+
×-homo-1 : ∀ x → 1 × x ≈ x

+
×-homo-1 = +-identityʳ

+

+
×-homo-+ : ∀ x m n → (m ℕ.+ n) × x ≈ m × x + n × x

+
×-homo-+ x 0       n = sym (+-identityˡ (n × x))

+
×-homo-+ x (suc m) n = begin

+
  x + (m ℕ.+ n) × x    ≈⟨ +-cong refl (×-homo-+ x m n) ⟩

+
  x + (m × x + n × x)  ≈⟨ sym (+-assoc x (m × x) (n × x)) ⟩

+
  x + m × x + n × x    ∎

+

+
×-idem : ∀ {c} → _+_ IdempotentOn c →

+
         ∀ n → .{{_ : NonZero n}} → n × c ≈ c

+
×-idem {c} idem (suc zero)    = +-identityʳ c

+
×-idem {c} idem (suc n@(suc _)) = begin

+
  c + (n × c) ≈⟨ +-congˡ (×-idem idem n ) ⟩

+
  c + c       ≈⟨ idem ⟩

+
  c           ∎

+

+
×-assocˡ : ∀ x m n → m × (n × x) ≈ (m ℕ.* n) × x

+
×-assocˡ x zero    n = refl

+
×-assocˡ x (suc m) n = begin

+
  n × x + m × n × x     ≈⟨ +-congˡ (×-assocˡ x m n) ⟩

+
  n × x + (m ℕ.* n) × x ≈⟨ ×-homo-+ x n (m ℕ.* n) ⟨

+
  (suc m ℕ.* n) × x     ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Quasigroup.html b/site/static/agda_html/Algebra.Properties.Quasigroup.html
new file mode 100644
index 000000000..fa0cb16a8
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Quasigroup.html
@@ -0,0 +1,122 @@
+
+Algebra.Properties.Quasigroup
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some basic properties of Quasigroup

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles using (Quasigroup)

+

+
module Algebra.Properties.Quasigroup {q₁ q₂} (Q : Quasigroup q₁ q₂) where

+

+
open Quasigroup Q

+
open import Algebra.Definitions _≈_

+
open import Relation.Binary.Reasoning.Setoid setoid

+
open import Data.Product.Base using (_,_)

+

+
cancelˡ : LeftCancellative _∙_

+
cancelˡ x y z eq = begin

+
  y             ≈⟨ leftDividesʳ x y ⟨

+
  x \\ (x ∙ y)  ≈⟨ \\-congˡ eq ⟩

+
  x \\ (x ∙ z)  ≈⟨ leftDividesʳ x z ⟩

+
  z             ∎

+

+
cancelʳ : RightCancellative _∙_

+
cancelʳ x y z eq = begin

+
  y             ≈⟨ rightDividesʳ x y ⟨

+
  (y ∙ x) // x  ≈⟨ //-congʳ eq ⟩

+
  (z ∙ x) // x  ≈⟨ rightDividesʳ x z ⟩

+
  z             ∎

+

+
cancel : Cancellative _∙_

+
cancel = cancelˡ , cancelʳ

+

+
y≈x\\z : ∀ x y z → x ∙ y ≈ z → y ≈ x \\ z

+
y≈x\\z x y z eq = begin

+
  y            ≈⟨ leftDividesʳ x y ⟨

+
  x \\ (x ∙ y) ≈⟨ \\-congˡ eq ⟩

+
  x \\ z       ∎

+

+
x≈z//y : ∀ x y z → x ∙ y ≈ z → x ≈ z // y

+
x≈z//y x y z eq = begin

+
  x            ≈⟨ rightDividesʳ y x ⟨

+
  (x ∙ y) // y ≈⟨ //-congʳ eq ⟩

+
  z // y       ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Ring.html b/site/static/agda_html/Algebra.Properties.Ring.html
new file mode 100644
index 000000000..c45088d1c
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Ring.html
@@ -0,0 +1,109 @@
+
+Algebra.Properties.Ring
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some basic properties of Rings

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra using (Ring)

+

+
module Algebra.Properties.Ring {r₁ r₂} (R : Ring r₁ r₂) where

+

+
open Ring R

+

+
import Algebra.Properties.RingWithoutOne as RingWithoutOneProperties

+
open import Function.Base using (_$_)

+
open import Relation.Binary.Reasoning.Setoid setoid

+
open import Algebra.Definitions _≈_

+

+
------------------------------------------------------------------------

+
-- Export properties of rings without a 1#.

+

+
open RingWithoutOneProperties ringWithoutOne public

+

+
------------------------------------------------------------------------

+
-- Extra properties of 1#

+

+
-1*x≈-x : ∀ x → - 1# * x ≈ - x

+
-1*x≈-x x = begin

+
  - 1# * x    ≈⟨ -‿distribˡ-* 1# x ⟨

+
  - (1# * x)  ≈⟨ -‿cong ( *-identityˡ x ) ⟩

+
  - x         ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.RingWithoutOne.html b/site/static/agda_html/Algebra.Properties.RingWithoutOne.html
new file mode 100644
index 000000000..5e7fbe8e5
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.RingWithoutOne.html
@@ -0,0 +1,151 @@
+
+Algebra.Properties.RingWithoutOne
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some basic properties of RingWithoutOne

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra

+

+
module Algebra.Properties.RingWithoutOne {r₁ r₂} (R : RingWithoutOne r₁ r₂) where

+

+
open RingWithoutOne R

+

+
import Algebra.Properties.AbelianGroup as AbelianGroupProperties

+
open import Function.Base using (_$_)

+
open import Relation.Binary.Reasoning.Setoid setoid

+

+
------------------------------------------------------------------------

+
-- Export properties of abelian groups

+

+
open AbelianGroupProperties +-abelianGroup public

+
  renaming

+
  ( ε⁻¹≈ε            to -0#≈0#

+
  ; ∙-cancelˡ        to +-cancelˡ

+
  ; ∙-cancelʳ        to +-cancelʳ

+
  ; ∙-cancel         to +-cancel

+
  ; ⁻¹-involutive    to -‿involutive

+
  ; ⁻¹-injective     to -‿injective

+
  ; ⁻¹-anti-homo-∙   to -‿anti-homo-+

+
  ; identityˡ-unique to +-identityˡ-unique

+
  ; identityʳ-unique to +-identityʳ-unique

+
  ; identity-unique  to +-identity-unique

+
  ; inverseˡ-unique  to +-inverseˡ-unique

+
  ; inverseʳ-unique  to +-inverseʳ-unique

+
  ; ⁻¹-∙-comm        to -‿+-comm

+
  )

+

+
-‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y

+
-‿distribˡ-* x y = sym $ begin

+
  - x * y                        ≈⟨ +-identityʳ (- x * y) ⟨

+
  - x * y + 0#                   ≈⟨ +-congˡ $ -‿inverseʳ (x * y) ⟨

+
  - x * y + (x * y + - (x * y))  ≈⟨ +-assoc (- x * y) (x * y) (- (x * y)) ⟨

+
  - x * y + x * y + - (x * y)    ≈⟨ +-congʳ $ distribʳ y (- x) x ⟨

+
  (- x + x) * y + - (x * y)      ≈⟨ +-congʳ $ *-congʳ $ -‿inverseˡ x ⟩

+
  0# * y + - (x * y)             ≈⟨ +-congʳ $ zeroˡ y ⟩

+
  0# + - (x * y)                 ≈⟨ +-identityˡ (- (x * y)) ⟩

+
  - (x * y)                      ∎

+

+
-‿distribʳ-* : ∀ x y → - (x * y) ≈ x * - y

+
-‿distribʳ-* x y = sym $ begin

+
  x * - y                        ≈⟨ +-identityˡ (x * - y) ⟨

+
  0# + x * - y                   ≈⟨ +-congʳ $ -‿inverseˡ (x * y) ⟨

+
  - (x * y) + x * y + x * - y    ≈⟨ +-assoc (- (x * y)) (x * y) (x * - y) ⟩

+
  - (x * y) + (x * y + x * - y)  ≈⟨ +-congˡ $ distribˡ x y (- y) ⟨

+
  - (x * y) + x * (y + - y)      ≈⟨ +-congˡ $ *-congˡ $ -‿inverseʳ y ⟩

+
  - (x * y) + x * 0#             ≈⟨ +-congˡ $ zeroʳ x ⟩

+
  - (x * y) + 0#                 ≈⟨ +-identityʳ (- (x * y)) ⟩

+
  - (x * y)                      ∎

+

+
x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#

+
x+x≈x⇒x≈0 x eq = +-identityˡ-unique x x eq

+

+
x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z

+
x[y-z]≈xy-xz x y z = begin

+
  x * (y - z)      ≈⟨ distribˡ x y (- z) ⟩

+
  x * y + x * - z  ≈⟨ +-congˡ (sym (-‿distribʳ-* x z)) ⟩

+
  x * y - x * z    ∎

+

+
[y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)

+
[y-z]x≈yx-zx x y z = begin

+
  (y - z) * x      ≈⟨ distribʳ x y (- z) ⟩

+
  y * x + - z * x  ≈⟨ +-congˡ (sym (-‿distribˡ-* z x)) ⟩

+
  y * x - z * x    ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Semigroup.html b/site/static/agda_html/Algebra.Properties.Semigroup.html
new file mode 100644
index 000000000..1e117a636
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Semigroup.html
@@ -0,0 +1,107 @@
+
+Algebra.Properties.Semigroup
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some theory for Semigroup

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra using (Semigroup)

+

+
module Algebra.Properties.Semigroup {a ℓ} (S : Semigroup a ℓ) where

+

+
open Semigroup S

+
open import Algebra.Definitions _≈_

+
open import Data.Product.Base using (_,_)

+

+
x∙yz≈xy∙z : ∀ x y z → x ∙ (y ∙ z) ≈ (x ∙ y) ∙ z

+
x∙yz≈xy∙z x y z = sym (assoc x y z)

+

+
alternativeˡ : LeftAlternative _∙_

+
alternativeˡ x y = assoc x x y

+

+
alternativeʳ : RightAlternative _∙_

+
alternativeʳ x y = sym (assoc x y y)

+

+
alternative : Alternative _∙_

+
alternative = alternativeˡ , alternativeʳ

+

+
flexible : Flexible _∙_

+
flexible x y = assoc x y x

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Properties.Semiring.Exp.html b/site/static/agda_html/Algebra.Properties.Semiring.Exp.html
new file mode 100644
index 000000000..d1b045973
--- /dev/null
+++ b/site/static/agda_html/Algebra.Properties.Semiring.Exp.html
@@ -0,0 +1,149 @@
+
+Algebra.Properties.Semiring.Exp
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Exponentiation defined over a semiring as repeated multiplication

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)

+
open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
import Data.Nat.Properties as ℕ

+

+
module Algebra.Properties.Semiring.Exp

+
  {a ℓ} (S : Semiring a ℓ) where

+

+
open Semiring S

+
open import Relation.Binary.Reasoning.Setoid setoid

+
import Algebra.Properties.Monoid.Mult *-monoid as Mult

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open import Algebra.Definitions.RawSemiring rawSemiring public

+
  using (_^_)

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
^-congˡ : ∀ n → (_^ n) Preserves _≈_ ⟶ _≈_

+
^-congˡ = Mult.×-congʳ

+

+
^-cong : _^_ Preserves₂ _≈_ ⟶ _≡_ ⟶ _≈_

+
^-cong x≈y u≡v = Mult.×-cong u≡v x≈y

+

+
^-congʳ : ∀ x → (x ^_) Preserves _≡_ ⟶ _≈_

+
^-congʳ x = Mult.×-congˡ

+

+
-- xᵐ⁺ⁿ ≈ xᵐxⁿ

+
^-homo-* : ∀ x m n → x ^ (m ℕ.+ n) ≈ (x ^ m) * (x ^ n)

+
^-homo-* = Mult.×-homo-+

+

+
-- (xᵐ)ⁿ≈xᵐ*ⁿ

+
^-assocʳ : ∀ x m n → (x ^ m) ^ n ≈ x ^ (m ℕ.* n)

+
^-assocʳ x m n rewrite ℕ.*-comm m n = Mult.×-assocˡ x n m

+

+
------------------------------------------------------------------------

+
-- A lemma using commutativity, needed for the Binomial Theorem

+

+
y*x^m*y^n≈x^m*y^[n+1] : ∀ {x} {y} (x*y≈y*x : x * y ≈ y * x) →

+
                        ∀ m n → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n

+
y*x^m*y^n≈x^m*y^[n+1] {x} {y} x*y≈y*x = helper

+
  where

+
    helper : ∀ m n → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n

+
    helper zero    n = begin

+
      y * (x ^ ℕ.zero * y ^ n)  ≡⟨⟩

+
      y * (1# * y ^ n)          ≈⟨ *-congˡ (*-identityˡ (y ^ n)) ⟩

+
      y * (y ^ n)               ≡⟨⟩

+
      y ^ (suc n)               ≈⟨ *-identityˡ (y ^ suc n) ⟨

+
      1# * y ^ (suc n)          ≡⟨⟩

+
      x ^ ℕ.zero * y ^ (suc n)  ∎

+
    helper (suc m) n = begin

+
      y * (x ^ suc m * y ^ n)    ≡⟨⟩

+
      y * ((x * x ^ m) * y ^ n)  ≈⟨ *-congˡ (*-assoc x (x ^ m) (y ^ n)) ⟩

+
      y * (x * (x ^ m * y ^ n))  ≈⟨ *-assoc y x (x ^ m * y ^ n) ⟨

+
      y * x * (x ^ m * y ^ n)    ≈⟨ *-congʳ x*y≈y*x ⟨

+
      x * y * (x ^ m * y ^ n)    ≈⟨ *-assoc x y _ ⟩

+
      x * (y * (x ^ m * y ^ n))  ≈⟨ *-congˡ (helper m n) ⟩

+
      x * (x ^ m * y ^ suc n)    ≈⟨ *-assoc x (x ^ m) (y ^ suc n) ⟨

+
      (x * x ^ m) * y ^ suc n    ≡⟨⟩

+
      x ^ suc m * y ^ suc n      ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Solver.Ring.AlmostCommutativeRing.html b/site/static/agda_html/Algebra.Solver.Ring.AlmostCommutativeRing.html
new file mode 100644
index 000000000..cf52572e7
--- /dev/null
+++ b/site/static/agda_html/Algebra.Solver.Ring.AlmostCommutativeRing.html
@@ -0,0 +1,228 @@
+
+Algebra.Solver.Ring.AlmostCommutativeRing
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Commutative semirings with some additional structure ("almost"

+
-- commutative rings), used by the ring solver

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra.Solver.Ring.AlmostCommutativeRing where

+

+
open import Algebra

+
open import Algebra.Structures

+
open import Algebra.Definitions

+
import Algebra.Morphism as Morphism

+
import Algebra.Morphism.Definitions as MorphismDefinitions

+
open import Function.Base using (id)

+
open import Level

+
open import Relation.Binary.Core using (Rel)

+

+

+
record IsAlmostCommutativeRing {a ℓ} {A : Set a} (_≈_ : Rel A ℓ)

+
                               (_+_ _*_ : Op₂ A) (-_ : Op₁ A)

+
                               (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1#

+
    -‿cong                : Congruent₁ _≈_ -_

+
    -‿*-distribˡ          : ∀ x y → ((- x) * y)     ≈ (- (x * y))

+
    -‿+-comm              : ∀ x y → ((- x) + (- y)) ≈ (- (x + y))

+

+
  open IsCommutativeSemiring isCommutativeSemiring public

+

+

+
record AlmostCommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix  8 -_

+
  infixl 7 _*_

+
  infixl 6 _+_

+
  infix  4 _≈_

+
  field

+
    Carrier                 : Set c

+
    _≈_                     : Rel Carrier ℓ

+
    _+_                     : Op₂ Carrier

+
    _*_                     : Op₂ Carrier

+
    -_                      : Op₁ Carrier

+
    0#                      : Carrier

+
    1#                      : Carrier

+
    isAlmostCommutativeRing : IsAlmostCommutativeRing _≈_ _+_ _*_ -_ 0# 1#

+

+
  open IsAlmostCommutativeRing isAlmostCommutativeRing public

+

+
  commutativeSemiring : CommutativeSemiring _ _

+
  commutativeSemiring = record

+
    { isCommutativeSemiring = isCommutativeSemiring

+
    }

+

+
  open CommutativeSemiring commutativeSemiring public

+
    using

+
    ( +-magma; +-semigroup

+
    ; *-magma; *-semigroup; *-commutativeSemigroup

+
    ; +-monoid; +-commutativeMonoid

+
    ; *-monoid; *-commutativeMonoid

+
    ; semiring

+
    )

+

+
  rawRing : RawRing _ _

+
  rawRing = record

+
    { _≈_ = _≈_

+
    ; _+_ = _+_

+
    ; _*_ = _*_

+
    ; -_  = -_

+
    ; 0#  = 0#

+
    ; 1#  = 1#

+
    }

+

+

+
------------------------------------------------------------------------

+
-- Homomorphisms

+

+
infix 4 _-Raw-AlmostCommutative⟶_

+

+
record _-Raw-AlmostCommutative⟶_

+
  {r₁ r₂ r₃ r₄}

+
  (From : RawRing r₁ r₄)

+
  (To : AlmostCommutativeRing r₂ r₃) : Set (r₁ ⊔ r₂ ⊔ r₃) where

+
  private

+
    module F = RawRing From

+
    module T = AlmostCommutativeRing To

+
  open MorphismDefinitions F.Carrier T.Carrier T._≈_

+
  field

+
    ⟦_⟧    : Morphism

+
    +-homo : Homomorphic₂ ⟦_⟧ F._+_ T._+_

+
    *-homo : Homomorphic₂ ⟦_⟧ F._*_ T._*_

+
    -‿homo : Homomorphic₁ ⟦_⟧ F.-_  T.-_

+
    0-homo : Homomorphic₀ ⟦_⟧ F.0#  T.0#

+
    1-homo : Homomorphic₀ ⟦_⟧ F.1#  T.1#

+

+
-raw-almostCommutative⟶ :

+
  ∀ {r₁ r₂} (R : AlmostCommutativeRing r₁ r₂) →

+
  AlmostCommutativeRing.rawRing R -Raw-AlmostCommutative⟶ R

+
-raw-almostCommutative⟶ R = record

+
  { ⟦_⟧    = id

+
  ; +-homo = λ _ _ → refl

+
  ; *-homo = λ _ _ → refl

+
  ; -‿homo = λ _ → refl

+
  ; 0-homo = refl

+
  ; 1-homo = refl

+
  }

+
  where open AlmostCommutativeRing R

+

+
Induced-equivalence : ∀ {c₁ c₂ ℓ₁ ℓ₂} {Coeff : RawRing c₁ ℓ₁}

+
                      {R : AlmostCommutativeRing c₂ ℓ₂} →

+
                      Coeff -Raw-AlmostCommutative⟶ R →

+
                      Rel (RawRing.Carrier Coeff) ℓ₂

+
Induced-equivalence {R = R} morphism a b = ⟦ a ⟧ ≈ ⟦ b ⟧

+
  where

+
  open AlmostCommutativeRing R

+
  open _-Raw-AlmostCommutative⟶_ morphism

+

+
------------------------------------------------------------------------

+
-- Conversions

+

+
-- Commutative rings are almost commutative rings.

+

+
fromCommutativeRing : ∀ {r₁ r₂} → CommutativeRing r₁ r₂ → AlmostCommutativeRing r₁ r₂

+
fromCommutativeRing CR = record

+
  { isAlmostCommutativeRing = record

+
      { isCommutativeSemiring = isCommutativeSemiring

+
      ; -‿cong                = -‿cong

+
      ; -‿*-distribˡ          = λ x y → sym (-‿distribˡ-* x y)

+
      ; -‿+-comm              = ⁻¹-∙-comm

+
      }

+
  }

+
  where

+
  open CommutativeRing CR

+
  open import Algebra.Properties.Ring ring

+
  open import Algebra.Properties.AbelianGroup +-abelianGroup

+

+
-- Commutative semirings can be viewed as almost commutative rings by

+
-- using identity as the "almost negation".

+

+
fromCommutativeSemiring : ∀ {r₁ r₂} → CommutativeSemiring r₁ r₂ → AlmostCommutativeRing _ _

+
fromCommutativeSemiring CS = record

+
  { -_                      = id

+
  ; isAlmostCommutativeRing = record

+
      { isCommutativeSemiring = isCommutativeSemiring

+
      ; -‿cong                = id

+
      ; -‿*-distribˡ          = λ _ _ → refl

+
      ; -‿+-comm              = λ _ _ → refl

+
      }

+
  }

+
  where open CommutativeSemiring CS

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Solver.Ring.Lemmas.html b/site/static/agda_html/Algebra.Solver.Ring.Lemmas.html
new file mode 100644
index 000000000..310d8dcf3
--- /dev/null
+++ b/site/static/agda_html/Algebra.Solver.Ring.Lemmas.html
@@ -0,0 +1,191 @@
+
+Algebra.Solver.Ring.Lemmas
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some boring lemmas used by the ring solver

+
------------------------------------------------------------------------

+

+
-- Note that these proofs use all "almost commutative ring" properties.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra

+
open import Algebra.Solver.Ring.AlmostCommutativeRing

+

+
module Algebra.Solver.Ring.Lemmas

+
  {r₁ r₂ r₃ r₄}

+
  (coeff : RawRing r₁ r₄)

+
  (r : AlmostCommutativeRing r₂ r₃)

+
  (morphism : coeff -Raw-AlmostCommutative⟶ r)

+
  where

+

+
private

+
  module C = RawRing coeff

+
open AlmostCommutativeRing r

+
open import Algebra.Morphism

+
open _-Raw-AlmostCommutative⟶_ morphism

+
open import Relation.Binary.Reasoning.Setoid setoid

+
open import Function.Base using (_⟨_⟩_; _$_)

+

+
lemma₀ : ∀ a b c x →

+
         (a + b) * x + c ≈ a * x + (b * x + c)

+
lemma₀ a b c x = begin

+
  (a + b) * x + c      ≈⟨ distribʳ _ _ _ ⟨ +-cong ⟩ refl ⟩

+
  (a * x + b * x) + c  ≈⟨ +-assoc _ _ _ ⟩

+
  a * x + (b * x + c)  ∎

+

+
lemma₁ : ∀ a b c d x →

+
         (a + b) * x + (c + d) ≈ (a * x + c) + (b * x + d)

+
lemma₁ a b c d x = begin

+
  (a + b) * x + (c + d)      ≈⟨ lemma₀ _ _ _ _ ⟩

+
  a * x + (b * x + (c + d))  ≈⟨ refl ⟨ +-cong ⟩ sym (+-assoc _ _ _) ⟩

+
  a * x + ((b * x + c) + d)  ≈⟨ refl ⟨ +-cong ⟩ (+-comm _ _ ⟨ +-cong ⟩ refl) ⟩

+
  a * x + ((c + b * x) + d)  ≈⟨ refl ⟨ +-cong ⟩ +-assoc _ _ _ ⟩

+
  a * x + (c + (b * x + d))  ≈⟨ sym $ +-assoc _ _ _ ⟩

+
  (a * x + c) + (b * x + d)  ∎

+

+
lemma₂ : ∀ a b c x → a * c * x + b * c ≈ (a * x + b) * c

+
lemma₂ a b c x = begin

+
  a * c * x + b * c  ≈⟨ lem ⟨ +-cong ⟩ refl ⟩

+
  a * x * c + b * c  ≈⟨ sym $ distribʳ _ _ _ ⟩

+
  (a * x + b) * c    ∎

+
  where

+
  lem = begin

+
    a * c * x    ≈⟨ *-assoc _ _ _ ⟩

+
    a * (c * x)  ≈⟨ refl ⟨ *-cong ⟩ *-comm _ _ ⟩

+
    a * (x * c)  ≈⟨ sym $ *-assoc _ _ _ ⟩

+
    a * x * c    ∎

+

+
lemma₃ : ∀ a b c x → a * b * x + a * c ≈ a * (b * x + c)

+
lemma₃ a b c x = begin

+
  a * b * x + a * c    ≈⟨ *-assoc _ _ _ ⟨ +-cong ⟩ refl ⟩

+
  a * (b * x) + a * c  ≈⟨ sym $ distribˡ _ _ _ ⟩

+
  a * (b * x + c)      ∎

+

+
lemma₄ : ∀ a b c d x →

+
         (a * c * x + (a * d + b * c)) * x + b * d ≈

+
         (a * x + b) * (c * x + d)

+
lemma₄ a b c d x = begin

+
  (a * c * x + (a * d + b * c)) * x + b * d              ≈⟨ distribʳ _ _ _ ⟨ +-cong ⟩ refl ⟩

+
  (a * c * x * x + (a * d + b * c) * x) + b * d          ≈⟨ refl ⟨ +-cong ⟩ ((refl ⟨ +-cong ⟩ refl) ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ refl ⟩

+
  (a * c * x * x + (a * d + b * c) * x) + b * d          ≈⟨ +-assoc _ _ _  ⟩

+
  a * c * x * x + ((a * d + b * c) * x + b * d)          ≈⟨ lem₁ ⟨ +-cong ⟩ (lem₂ ⟨ +-cong ⟩ refl) ⟩

+
  a * x * (c * x) + (a * x * d + b * (c * x) + b * d)    ≈⟨ refl ⟨ +-cong ⟩ +-assoc _ _ _ ⟩

+
  a * x * (c * x) + (a * x * d + (b * (c * x) + b * d))  ≈⟨ sym $ +-assoc _ _ _ ⟩

+
  a * x * (c * x) + a * x * d + (b * (c * x) + b * d)    ≈⟨ sym $ distribˡ _ _ _ ⟨ +-cong ⟩ distribˡ _ _ _ ⟩

+
  a * x * (c * x + d) + b * (c * x + d)                  ≈⟨ sym $ distribʳ _ _ _ ⟩

+
  (a * x + b) * (c * x + d)                              ∎

+
  where

+
  lem₁′ = begin

+
    a * c * x    ≈⟨ *-assoc _ _ _ ⟩

+
    a * (c * x)  ≈⟨ refl ⟨ *-cong ⟩ *-comm _ _ ⟩

+
    a * (x * c)  ≈⟨ sym $ *-assoc _ _ _ ⟩

+
    a * x * c    ∎

+

+
  lem₁ = begin

+
    a * c * x * x    ≈⟨ lem₁′ ⟨ *-cong ⟩ refl ⟩

+
    a * x * c * x    ≈⟨ *-assoc _ _ _ ⟩

+
    a * x * (c * x)  ∎

+

+
  lem₂ = begin

+
    (a * d + b * c) * x        ≈⟨ distribʳ _ _ _ ⟩

+
    a * d * x + b * c * x      ≈⟨ *-assoc _ _ _ ⟨ +-cong ⟩ *-assoc _ _ _ ⟩

+
    a * (d * x) + b * (c * x)  ≈⟨ (refl ⟨ *-cong ⟩ *-comm _ _) ⟨ +-cong ⟩ refl ⟩

+
    a * (x * d) + b * (c * x)  ≈⟨ sym $ *-assoc _ _ _ ⟨ +-cong ⟩ refl ⟩

+
    a * x * d + b * (c * x)    ∎

+

+
lemma₅ : ∀ x → (0# * x + 1#) * x + 0# ≈ x

+
lemma₅ x = begin

+
  (0# * x + 1#) * x + 0#   ≈⟨ ((zeroˡ _ ⟨ +-cong ⟩ refl) ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ refl ⟩

+
  (0# + 1#) * x + 0#       ≈⟨ (+-identityˡ _ ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ refl ⟩

+
  1# * x + 0#              ≈⟨ +-identityʳ _ ⟩

+
  1# * x                   ≈⟨ *-identityˡ _ ⟩

+
  x                        ∎

+

+
lemma₆ : ∀ a x → 0# * x + a ≈ a

+
lemma₆ a x = begin

+
  0# * x + a    ≈⟨ zeroˡ _ ⟨ +-cong ⟩ refl ⟩

+
  0# + a        ≈⟨ +-identityˡ _ ⟩

+
  a             ∎

+

+
lemma₇ : ∀ x → - 1# * x ≈ - x

+
lemma₇ x = begin

+
  - 1# * x      ≈⟨ -‿*-distribˡ _ _ ⟩

+
  - (1# * x)    ≈⟨ -‿cong (*-identityˡ _) ⟩

+
  - x           ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Solver.Ring.Simple.html b/site/static/agda_html/Algebra.Solver.Ring.Simple.html
new file mode 100644
index 000000000..44cf97203
--- /dev/null
+++ b/site/static/agda_html/Algebra.Solver.Ring.Simple.html
@@ -0,0 +1,98 @@
+
+Algebra.Solver.Ring.Simple
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Instantiates the ring solver with two copies of the same ring with

+
-- decidable equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Solver.Ring.AlmostCommutativeRing

+
open import Relation.Binary.Definitions using (Decidable)

+
open import Relation.Binary.Consequences using (dec⇒weaklyDec)

+

+
module Algebra.Solver.Ring.Simple

+
  {r₁ r₂} (R : AlmostCommutativeRing r₁ r₂)

+
  (_≟_ : Decidable (AlmostCommutativeRing._≈_ R))

+
  where

+

+
open AlmostCommutativeRing R

+
import Algebra.Solver.Ring as RS

+
open RS rawRing R (-raw-almostCommutative⟶ R) (dec⇒weaklyDec _≟_) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Solver.Ring.html b/site/static/agda_html/Algebra.Solver.Ring.html
new file mode 100644
index 000000000..7c78de202
--- /dev/null
+++ b/site/static/agda_html/Algebra.Solver.Ring.html
@@ -0,0 +1,629 @@
+
+Algebra.Solver.Ring
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Old solver for commutative ring or semiring equalities

+
------------------------------------------------------------------------

+

+
-- Uses ideas from the Coq ring tactic. See "Proving Equalities in a

+
-- Commutative Ring Done Right in Coq" by Grégoire and Mahboubi. The

+
-- code below is not optimised like theirs, though (in particular, our

+
-- Horner normal forms are not sparse).

+
--

+
-- At first the `WeaklyDecidable` type may at first glance look useless

+
-- as there is no guarantee that it doesn't always return `nothing`.

+
-- However the implementation of it affects the power of the solver. The

+
-- more equalities it returns, the more expressions the ring solver can

+
-- solve.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Bundles

+
open import Algebra.Solver.Ring.AlmostCommutativeRing

+
open import Relation.Binary.Definitions using (WeaklyDecidable)

+

+
module Algebra.Solver.Ring

+
  {r₁ r₂ r₃ r₄}

+
  (Coeff : RawRing r₁ r₄)            -- Coefficient "ring".

+
  (R : AlmostCommutativeRing r₂ r₃)  -- Main "ring".

+
  (morphism : Coeff -Raw-AlmostCommutative⟶ R)

+
  (_coeff≟_ : WeaklyDecidable (Induced-equivalence morphism))

+
  where

+

+
open import Algebra.Core

+
open import Algebra.Solver.Ring.Lemmas Coeff R morphism

+
private module C = RawRing Coeff

+
open AlmostCommutativeRing R

+
  renaming (zero to *-zero; zeroˡ to *-zeroˡ; zeroʳ to *-zeroʳ)

+
open import Algebra.Definitions _≈_

+
open import Algebra.Morphism

+
open _-Raw-AlmostCommutative⟶_ morphism renaming (⟦_⟧ to ⟦_⟧′)

+
open import Algebra.Properties.Semiring.Exp semiring

+

+
open import Relation.Nullary.Decidable using (yes; no)

+
open import Relation.Binary.Reasoning.Setoid setoid

+
import Relation.Binary.PropositionalEquality.Core as ≡

+
import Relation.Binary.Reflection as Reflection

+

+
open import Data.Nat.Base using (ℕ; suc; zero)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Vec.Base using (Vec; []; _∷_; lookup)

+
open import Data.Maybe.Base using (just; nothing)

+
open import Function.Base using (_⟨_⟩_; _$_)

+
open import Level using (_⊔_)

+

+
infix  9 :-_ -H_ -N_

+
infixr 9 _:×_ _:^_ _^N_

+
infix  8 _*x+_ _*x+HN_ _*x+H_

+
infixl 8 _:*_ _*N_ _*H_ _*NH_ _*HN_

+
infixl 7 _:+_ _:-_ _+H_ _+N_

+
infix  4 _≈H_ _≈N_

+

+
------------------------------------------------------------------------

+
-- Polynomials

+

+
data Op : Set where

+
  [+] : Op

+
  [*] : Op

+

+
-- The polynomials are indexed by the number of variables.

+

+
data Polynomial (m : ℕ) : Set r₁ where

+
  op   : (o : Op) (p₁ : Polynomial m) (p₂ : Polynomial m) → Polynomial m

+
  con  : (c : C.Carrier) → Polynomial m

+
  var  : (x : Fin m) → Polynomial m

+
  _:^_ : (p : Polynomial m) (n : ℕ) → Polynomial m

+
  :-_  : (p : Polynomial m) → Polynomial m

+

+
-- Short-hand notation.

+

+
_:+_ : ∀ {n} → Polynomial n → Polynomial n → Polynomial n

+
_:+_ = op [+]

+

+
_:*_ : ∀ {n} → Polynomial n → Polynomial n → Polynomial n

+
_:*_ = op [*]

+

+
_:-_ : ∀ {n} → Polynomial n → Polynomial n → Polynomial n

+
x :- y = x :+ :- y

+

+
_:×_ : ∀ {n} → ℕ → Polynomial n → Polynomial n

+
zero :× p = con C.0#

+
suc m :× p = p :+ m :× p

+

+
-- Semantics.

+

+
sem : Op → Op₂ Carrier

+
sem [+] = _+_

+
sem [*] = _*_

+

+
⟦_⟧ : ∀ {n} → Polynomial n → Vec Carrier n → Carrier

+
⟦ op o p₁ p₂ ⟧ ρ = ⟦ p₁ ⟧ ρ ⟨ sem o ⟩ ⟦ p₂ ⟧ ρ

+
⟦ con c      ⟧ ρ = ⟦ c ⟧′

+
⟦ var x      ⟧ ρ = lookup ρ x

+
⟦ p :^ n     ⟧ ρ = ⟦ p ⟧ ρ ^ n

+
⟦ :- p       ⟧ ρ = - ⟦ p ⟧ ρ

+

+
------------------------------------------------------------------------

+
-- Normal forms of polynomials

+

+
-- A univariate polynomial of degree d,

+
--

+
--     p = a_d x^d + a_{d-1}x^{d-1} + … + a_0,

+
--

+
-- is represented in Horner normal form by

+
--

+
--     p = ((a_d x + a_{d-1})x + …)x + a_0.

+
--

+
-- Note that Horner normal forms can be represented as lists, with the

+
-- empty list standing for the zero polynomial of degree "-1".

+
--

+
-- Given this representation of univariate polynomials over an

+
-- arbitrary ring, polynomials in any number of variables over the

+
-- ring C can be represented via the isomorphisms

+
--

+
--     C[] ≅ C

+
--

+
-- and

+
--

+
--     C[X_0,...X_{n+1}] ≅ C[X_0,...,X_n][X_{n+1}].

+

+
mutual

+

+
  -- The polynomial representations are indexed by the polynomial's

+
  -- degree.

+

+
  data HNF : ℕ → Set r₁ where

+
    ∅     : ∀ {n} → HNF (suc n)

+
    _*x+_ : ∀ {n} → HNF (suc n) → Normal n → HNF (suc n)

+

+
  data Normal : ℕ → Set r₁ where

+
    con  : C.Carrier → Normal zero

+
    poly : ∀ {n} → HNF (suc n) → Normal (suc n)

+

+
  -- Note that the data types above do /not/ ensure uniqueness of

+
  -- normal forms: the zero polynomial of degree one can be

+
  -- represented using both ∅ and ∅ *x+ con C.0#.

+

+
mutual

+

+
  -- Semantics.

+

+
  ⟦_⟧H : ∀ {n} → HNF (suc n) → Vec Carrier (suc n) → Carrier

+
  ⟦ ∅       ⟧H _       = 0#

+
  ⟦ p *x+ c ⟧H (x ∷ ρ) = ⟦ p ⟧H (x ∷ ρ) * x + ⟦ c ⟧N ρ

+

+
  ⟦_⟧N : ∀ {n} → Normal n → Vec Carrier n → Carrier

+
  ⟦ con c  ⟧N _ = ⟦ c ⟧′

+
  ⟦ poly p ⟧N ρ = ⟦ p ⟧H ρ

+

+
------------------------------------------------------------------------

+
-- Equality and decidability

+

+
mutual

+

+
  -- Equality.

+

+
  data _≈H_ : ∀ {n} → HNF n → HNF n → Set (r₁ ⊔ r₃) where

+
    ∅     : ∀ {n} → _≈H_ {suc n} ∅ ∅

+
    _*x+_ : ∀ {n} {p₁ p₂ : HNF (suc n)} {c₁ c₂ : Normal n} →

+
            p₁ ≈H p₂ → c₁ ≈N c₂ → (p₁ *x+ c₁) ≈H (p₂ *x+ c₂)

+

+
  data _≈N_ : ∀ {n} → Normal n → Normal n → Set (r₁ ⊔ r₃) where

+
    con  : ∀ {c₁ c₂} → ⟦ c₁ ⟧′ ≈ ⟦ c₂ ⟧′ → con c₁ ≈N con c₂

+
    poly : ∀ {n} {p₁ p₂ : HNF (suc n)} → p₁ ≈H p₂ → poly p₁ ≈N poly p₂

+

+
mutual

+

+
  -- Equality is weakly decidable.

+

+
  infix 4 _≟H_ _≟N_

+

+
  _≟H_ : ∀ {n} → WeaklyDecidable (_≈H_ {n = n})

+
  ∅           ≟H ∅           = just ∅

+
  ∅           ≟H (_ *x+ _)   = nothing

+
  (_ *x+ _)   ≟H ∅           = nothing

+
  (p₁ *x+ c₁) ≟H (p₂ *x+ c₂) with p₁ ≟H p₂ | c₁ ≟N c₂

+
  ... | just p₁≈p₂ | just c₁≈c₂ = just (p₁≈p₂ *x+ c₁≈c₂)

+
  ... | _          | nothing    = nothing

+
  ... | nothing    | _          = nothing

+

+
  _≟N_ : ∀ {n} → WeaklyDecidable (_≈N_ {n = n})

+
  con c₁ ≟N con c₂ with c₁ coeff≟ c₂

+
  ... | just c₁≈c₂ = just (con c₁≈c₂)

+
  ... | nothing    = nothing

+
  poly p₁ ≟N poly p₂ with p₁ ≟H p₂

+
  ... | just p₁≈p₂ = just (poly p₁≈p₂)

+
  ... | nothing    = nothing

+

+
mutual

+

+
  -- The semantics respect the equality relations defined above.

+

+
  ⟦_⟧H-cong : ∀ {n} {p₁ p₂ : HNF (suc n)} →

+
              p₁ ≈H p₂ → ∀ ρ → ⟦ p₁ ⟧H ρ ≈ ⟦ p₂ ⟧H ρ

+
  ⟦ ∅               ⟧H-cong _       = refl

+
  ⟦ p₁≈p₂ *x+ c₁≈c₂ ⟧H-cong (x ∷ ρ) =

+
    (⟦ p₁≈p₂ ⟧H-cong (x ∷ ρ) ⟨ *-cong ⟩ refl)

+
      ⟨ +-cong ⟩

+
    ⟦ c₁≈c₂ ⟧N-cong ρ

+

+
  ⟦_⟧N-cong :

+
    ∀ {n} {p₁ p₂ : Normal n} →

+
    p₁ ≈N p₂ → ∀ ρ → ⟦ p₁ ⟧N ρ ≈ ⟦ p₂ ⟧N ρ

+
  ⟦ con c₁≈c₂  ⟧N-cong _ = c₁≈c₂

+
  ⟦ poly p₁≈p₂ ⟧N-cong ρ = ⟦ p₁≈p₂ ⟧H-cong ρ

+

+
------------------------------------------------------------------------

+
-- Ring operations on Horner normal forms

+

+
-- Zero.

+

+
0H : ∀ {n} → HNF (suc n)

+
0H = ∅

+

+
0N : ∀ {n} → Normal n

+
0N {zero}  = con C.0#

+
0N {suc n} = poly 0H

+

+
mutual

+

+
  -- One.

+

+
  1H : ∀ {n} → HNF (suc n)

+
  1H {n} = ∅ *x+ 1N {n}

+

+
  1N : ∀ {n} → Normal n

+
  1N {zero}  = con C.1#

+
  1N {suc n} = poly 1H

+

+
-- A simplifying variant of _*x+_.

+

+
_*x+HN_ : ∀ {n} → HNF (suc n) → Normal n → HNF (suc n)

+
(p *x+ c′) *x+HN c = (p *x+ c′) *x+ c

+
∅          *x+HN c with c ≟N 0N

+
... | just c≈0 = ∅

+
... | nothing  = ∅ *x+ c

+

+
mutual

+

+
  -- Addition.

+

+
  _+H_ : ∀ {n} → HNF (suc n) → HNF (suc n) → HNF (suc n)

+
  ∅           +H p           = p

+
  p           +H ∅           = p

+
  (p₁ *x+ c₁) +H (p₂ *x+ c₂) = (p₁ +H p₂) *x+HN (c₁ +N c₂)

+

+
  _+N_ : ∀ {n} → Normal n → Normal n → Normal n

+
  con c₁  +N con c₂  = con (c₁ C.+ c₂)

+
  poly p₁ +N poly p₂ = poly (p₁ +H p₂)

+

+
-- Multiplication.

+

+
_*x+H_ : ∀ {n} → HNF (suc n) → HNF (suc n) → HNF (suc n)

+
p₁         *x+H (p₂ *x+ c) = (p₁ +H p₂) *x+HN c

+
∅          *x+H ∅          = ∅

+
(p₁ *x+ c) *x+H ∅          = (p₁ *x+ c) *x+ 0N

+

+
mutual

+

+
  _*NH_ : ∀ {n} → Normal n → HNF (suc n) → HNF (suc n)

+
  c *NH ∅          = ∅

+
  c *NH (p *x+ c′) with c ≟N 0N

+
  ... | just c≈0 = ∅

+
  ... | nothing  = (c *NH p) *x+ (c *N c′)

+

+
  _*HN_ : ∀ {n} → HNF (suc n) → Normal n → HNF (suc n)

+
  ∅          *HN c = ∅

+
  (p *x+ c′) *HN c with c ≟N 0N

+
  ... | just c≈0 = ∅

+
  ... | nothing  = (p *HN c) *x+ (c′ *N c)

+

+
  _*H_ : ∀ {n} → HNF (suc n) → HNF (suc n) → HNF (suc n)

+
  ∅           *H _           = ∅

+
  (_ *x+ _)   *H ∅           = ∅

+
  (p₁ *x+ c₁) *H (p₂ *x+ c₂) =

+
    ((p₁ *H p₂) *x+H (p₁ *HN c₂ +H c₁ *NH p₂)) *x+HN (c₁ *N c₂)

+

+
  _*N_ : ∀ {n} → Normal n → Normal n → Normal n

+
  con c₁  *N con c₂  = con (c₁ C.* c₂)

+
  poly p₁ *N poly p₂ = poly (p₁ *H p₂)

+

+
-- Exponentiation.

+

+
_^N_ : ∀ {n} → Normal n → ℕ → Normal n

+
p ^N zero  = 1N

+
p ^N suc n = p *N (p ^N n)

+

+
mutual

+

+
  -- Negation.

+

+
  -H_ : ∀ {n} → HNF (suc n) → HNF (suc n)

+
  -H p = (-N 1N) *NH p

+

+
  -N_ : ∀ {n} → Normal n → Normal n

+
  -N con c  = con (C.- c)

+
  -N poly p = poly (-H p)

+

+
------------------------------------------------------------------------

+
-- Normalisation

+

+
normalise-con : ∀ {n} → C.Carrier → Normal n

+
normalise-con {zero}  c = con c

+
normalise-con {suc n} c = poly (∅ *x+HN normalise-con c)

+

+
normalise-var : ∀ {n} → Fin n → Normal n

+
normalise-var zero    = poly ((∅ *x+ 1N) *x+ 0N)

+
normalise-var (suc i) = poly (∅ *x+HN normalise-var i)

+

+
normalise : ∀ {n} → Polynomial n → Normal n

+
normalise (op [+] t₁ t₂) = normalise t₁ +N normalise t₂

+
normalise (op [*] t₁ t₂) = normalise t₁ *N normalise t₂

+
normalise (con c)        = normalise-con c

+
normalise (var i)        = normalise-var i

+
normalise (t :^ k)       = normalise t ^N k

+
normalise (:- t)         = -N normalise t

+

+
-- Evaluation after normalisation.

+

+
⟦_⟧↓ : ∀ {n} → Polynomial n → Vec Carrier n → Carrier

+
⟦ p ⟧↓ ρ = ⟦ normalise p ⟧N ρ

+

+
------------------------------------------------------------------------

+
-- Homomorphism lemmas

+

+
0N-homo : ∀ {n} ρ → ⟦ 0N {n} ⟧N ρ ≈ 0#

+
0N-homo []      = 0-homo

+
0N-homo (x ∷ ρ) = refl

+

+
-- If c is equal to 0N, then c is semantically equal to 0#.

+

+
0≈⟦0⟧ : ∀ {n} {c : Normal n} → c ≈N 0N → ∀ ρ → 0# ≈ ⟦ c ⟧N ρ

+
0≈⟦0⟧ {c = c} c≈0 ρ = sym (begin

+
  ⟦ c  ⟧N ρ  ≈⟨ ⟦ c≈0 ⟧N-cong ρ ⟩

+
  ⟦ 0N ⟧N ρ  ≈⟨ 0N-homo ρ ⟩

+
  0#         ∎)

+

+
1N-homo : ∀ {n} ρ → ⟦ 1N {n} ⟧N ρ ≈ 1#

+
1N-homo []      = 1-homo

+
1N-homo (x ∷ ρ) = begin

+
  0# * x + ⟦ 1N ⟧N ρ  ≈⟨ refl ⟨ +-cong ⟩ 1N-homo ρ ⟩

+
  0# * x + 1#         ≈⟨ lemma₆ _ _ ⟩

+
  1#                  ∎

+

+
-- _*x+HN_ is equal to _*x+_.

+

+
*x+HN≈*x+ : ∀ {n} (p : HNF (suc n)) (c : Normal n) →

+
            ∀ ρ → ⟦ p *x+HN c ⟧H ρ ≈ ⟦ p *x+ c ⟧H ρ

+
*x+HN≈*x+ (p *x+ c′) c ρ       = refl

+
*x+HN≈*x+ ∅          c (x ∷ ρ) with c ≟N 0N

+
... | just c≈0 = begin

+
  0#                 ≈⟨ 0≈⟦0⟧ c≈0 ρ ⟩

+
  ⟦ c ⟧N ρ           ≈⟨ sym $ lemma₆ _ _ ⟩

+
  0# * x + ⟦ c ⟧N ρ  ∎

+
... | nothing = refl

+

+
∅*x+HN-homo : ∀ {n} (c : Normal n) x ρ →

+
              ⟦ ∅ *x+HN c ⟧H (x ∷ ρ) ≈ ⟦ c ⟧N ρ

+
∅*x+HN-homo c x ρ with c ≟N 0N

+
... | just c≈0 = 0≈⟦0⟧ c≈0 ρ

+
... | nothing = lemma₆ _ _

+

+
mutual

+

+
  +H-homo : ∀ {n} (p₁ p₂ : HNF (suc n)) →

+
            ∀ ρ → ⟦ p₁ +H p₂ ⟧H ρ ≈ ⟦ p₁ ⟧H ρ + ⟦ p₂ ⟧H ρ

+
  +H-homo ∅           p₂          ρ       = sym (+-identityˡ _)

+
  +H-homo (p₁ *x+ x₁) ∅           ρ       = sym (+-identityʳ _)

+
  +H-homo (p₁ *x+ c₁) (p₂ *x+ c₂) (x ∷ ρ) = begin

+
    ⟦ (p₁ +H p₂) *x+HN (c₁ +N c₂) ⟧H (x ∷ ρ)                           ≈⟨ *x+HN≈*x+ (p₁ +H p₂) (c₁ +N c₂) (x ∷ ρ) ⟩

+

+
    ⟦ p₁ +H p₂ ⟧H (x ∷ ρ) * x + ⟦ c₁ +N c₂ ⟧N ρ                        ≈⟨ (+H-homo p₁ p₂ (x ∷ ρ) ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ +N-homo c₁ c₂ ρ ⟩

+

+
    (⟦ p₁ ⟧H (x ∷ ρ) + ⟦ p₂ ⟧H (x ∷ ρ)) * x + (⟦ c₁ ⟧N ρ + ⟦ c₂ ⟧N ρ)  ≈⟨ lemma₁ _ _ _ _ _ ⟩

+

+
    (⟦ p₁ ⟧H (x ∷ ρ) * x + ⟦ c₁ ⟧N ρ) +

+
    (⟦ p₂ ⟧H (x ∷ ρ) * x + ⟦ c₂ ⟧N ρ)                                  ∎

+

+
  +N-homo : ∀ {n} (p₁ p₂ : Normal n) →

+
            ∀ ρ → ⟦ p₁ +N p₂ ⟧N ρ ≈ ⟦ p₁ ⟧N ρ + ⟦ p₂ ⟧N ρ

+
  +N-homo (con c₁)  (con c₂)  _ = +-homo _ _

+
  +N-homo (poly p₁) (poly p₂) ρ = +H-homo p₁ p₂ ρ

+

+
*x+H-homo :

+
  ∀ {n} (p₁ p₂ : HNF (suc n)) x ρ →

+
  ⟦ p₁ *x+H p₂ ⟧H (x ∷ ρ) ≈

+
  ⟦ p₁ ⟧H (x ∷ ρ) * x + ⟦ p₂ ⟧H (x ∷ ρ)

+
*x+H-homo ∅         ∅ _ _ = sym $ lemma₆ _ _

+
*x+H-homo (p *x+ c) ∅ x ρ = begin

+
  ⟦ p *x+ c ⟧H (x ∷ ρ) * x + ⟦ 0N ⟧N ρ  ≈⟨ refl ⟨ +-cong ⟩ 0N-homo ρ ⟩

+
  ⟦ p *x+ c ⟧H (x ∷ ρ) * x + 0#         ∎

+
*x+H-homo p₁         (p₂ *x+ c₂) x ρ = begin

+
  ⟦ (p₁ +H p₂) *x+HN c₂ ⟧H (x ∷ ρ)                         ≈⟨ *x+HN≈*x+ (p₁ +H p₂) c₂ (x ∷ ρ) ⟩

+
  ⟦ p₁ +H p₂ ⟧H (x ∷ ρ) * x + ⟦ c₂ ⟧N ρ                    ≈⟨ (+H-homo p₁ p₂ (x ∷ ρ) ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ refl ⟩

+
  (⟦ p₁ ⟧H (x ∷ ρ) + ⟦ p₂ ⟧H (x ∷ ρ)) * x + ⟦ c₂ ⟧N ρ      ≈⟨ lemma₀ _ _ _ _ ⟩

+
  ⟦ p₁ ⟧H (x ∷ ρ) * x + (⟦ p₂ ⟧H (x ∷ ρ) * x + ⟦ c₂ ⟧N ρ)  ∎

+

+
mutual

+

+
  *NH-homo :

+
    ∀ {n} (c : Normal n) (p : HNF (suc n)) x ρ →

+
    ⟦ c *NH p ⟧H (x ∷ ρ) ≈ ⟦ c ⟧N ρ * ⟦ p ⟧H (x ∷ ρ)

+
  *NH-homo c ∅          x ρ = sym (*-zeroʳ _)

+
  *NH-homo c (p *x+ c′) x ρ with c ≟N 0N

+
  ... | just c≈0 = begin

+
    0#                                            ≈⟨ sym (*-zeroˡ _) ⟩

+
    0# * (⟦ p ⟧H (x ∷ ρ) * x + ⟦ c′ ⟧N ρ)         ≈⟨ 0≈⟦0⟧ c≈0 ρ ⟨ *-cong ⟩ refl ⟩

+
    ⟦ c ⟧N ρ  * (⟦ p ⟧H (x ∷ ρ) * x + ⟦ c′ ⟧N ρ)  ∎

+
  ... | nothing = begin

+
    ⟦ c *NH p ⟧H (x ∷ ρ) * x + ⟦ c *N c′ ⟧N ρ                 ≈⟨ (*NH-homo c p x ρ ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ *N-homo c c′ ρ ⟩

+
    (⟦ c ⟧N ρ * ⟦ p ⟧H (x ∷ ρ)) * x + (⟦ c ⟧N ρ * ⟦ c′ ⟧N ρ)  ≈⟨ lemma₃ _ _ _ _ ⟩

+
    ⟦ c ⟧N ρ * (⟦ p ⟧H (x ∷ ρ) * x + ⟦ c′ ⟧N ρ)               ∎

+

+
  *HN-homo :

+
    ∀ {n} (p : HNF (suc n)) (c : Normal n) x ρ →

+
    ⟦ p *HN c ⟧H (x ∷ ρ) ≈ ⟦ p ⟧H (x ∷ ρ) * ⟦ c ⟧N ρ

+
  *HN-homo ∅          c x ρ = sym (*-zeroˡ _)

+
  *HN-homo (p *x+ c′) c x ρ with c ≟N 0N

+
  ... | just c≈0 = begin

+
    0#                                           ≈⟨ sym (*-zeroʳ _) ⟩

+
    (⟦ p ⟧H (x ∷ ρ) * x + ⟦ c′ ⟧N ρ) * 0#        ≈⟨ refl ⟨ *-cong ⟩ 0≈⟦0⟧ c≈0 ρ ⟩

+
    (⟦ p ⟧H (x ∷ ρ) * x + ⟦ c′ ⟧N ρ) * ⟦ c ⟧N ρ  ∎

+
  ... | nothing = begin

+
    ⟦ p *HN c ⟧H (x ∷ ρ) * x + ⟦ c′ *N c ⟧N ρ                 ≈⟨ (*HN-homo p c x ρ ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ *N-homo c′ c ρ ⟩

+
    (⟦ p ⟧H (x ∷ ρ) * ⟦ c ⟧N ρ) * x + (⟦ c′ ⟧N ρ * ⟦ c ⟧N ρ)  ≈⟨ lemma₂ _ _ _ _ ⟩

+
    (⟦ p ⟧H (x ∷ ρ) * x + ⟦ c′ ⟧N ρ) * ⟦ c ⟧N ρ               ∎

+

+
  *H-homo : ∀ {n} (p₁ p₂ : HNF (suc n)) →

+
            ∀ ρ → ⟦ p₁ *H p₂ ⟧H ρ ≈ ⟦ p₁ ⟧H ρ * ⟦ p₂ ⟧H ρ

+
  *H-homo ∅           p₂          ρ       = sym $ *-zeroˡ _

+
  *H-homo (p₁ *x+ c₁) ∅           ρ       = sym $ *-zeroʳ _

+
  *H-homo (p₁ *x+ c₁) (p₂ *x+ c₂) (x ∷ ρ) = begin

+
    ⟦ ((p₁ *H p₂) *x+H ((p₁ *HN c₂) +H (c₁ *NH p₂))) *x+HN

+
      (c₁ *N c₂) ⟧H (x ∷ ρ)                                              ≈⟨ *x+HN≈*x+ ((p₁ *H p₂) *x+H ((p₁ *HN c₂) +H (c₁ *NH p₂)))

+
                                                                                      (c₁ *N c₂) (x ∷ ρ) ⟩

+
    ⟦ (p₁ *H p₂) *x+H

+
      ((p₁ *HN c₂) +H (c₁ *NH p₂)) ⟧H (x ∷ ρ) * x +

+
    ⟦ c₁ *N c₂ ⟧N ρ                                                      ≈⟨ (*x+H-homo (p₁ *H p₂) ((p₁ *HN c₂) +H (c₁ *NH p₂)) x ρ

+
                                                                               ⟨ *-cong ⟩

+
                                                                             refl)

+
                                                                              ⟨ +-cong ⟩

+
                                                                            *N-homo c₁ c₂ ρ ⟩

+
    (⟦ p₁ *H p₂ ⟧H (x ∷ ρ) * x +

+
     ⟦ (p₁ *HN c₂) +H (c₁ *NH p₂) ⟧H (x ∷ ρ)) * x +

+
    ⟦ c₁ ⟧N ρ * ⟦ c₂ ⟧N ρ                                                ≈⟨ (((*H-homo p₁ p₂ (x ∷ ρ) ⟨ *-cong ⟩ refl)

+
                                                                                ⟨ +-cong ⟩

+
                                                                              (+H-homo (p₁ *HN c₂) (c₁ *NH p₂) (x ∷ ρ)))

+
                                                                               ⟨ *-cong ⟩

+
                                                                             refl)

+
                                                                              ⟨ +-cong ⟩

+
                                                                            refl ⟩

+
    (⟦ p₁ ⟧H (x ∷ ρ) * ⟦ p₂ ⟧H (x ∷ ρ) * x +

+
     (⟦ p₁ *HN c₂ ⟧H (x ∷ ρ) + ⟦ c₁ *NH p₂ ⟧H (x ∷ ρ))) * x +

+
    ⟦ c₁ ⟧N ρ * ⟦ c₂ ⟧N ρ                                                ≈⟨ ((refl ⟨ +-cong ⟩ (*HN-homo p₁ c₂ x ρ ⟨ +-cong ⟩ *NH-homo c₁ p₂ x ρ))

+
                                                                               ⟨ *-cong ⟩

+
                                                                             refl)

+
                                                                              ⟨ +-cong ⟩

+
                                                                            refl ⟩

+
    (⟦ p₁ ⟧H (x ∷ ρ) * ⟦ p₂ ⟧H (x ∷ ρ) * x +

+
     (⟦ p₁ ⟧H (x ∷ ρ) * ⟦ c₂ ⟧N ρ + ⟦ c₁ ⟧N ρ * ⟦ p₂ ⟧H (x ∷ ρ))) * x +

+
    (⟦ c₁ ⟧N ρ * ⟦ c₂ ⟧N ρ)                                              ≈⟨ lemma₄ _ _ _ _ _ ⟩

+

+
    (⟦ p₁ ⟧H (x ∷ ρ) * x + ⟦ c₁ ⟧N ρ) *

+
    (⟦ p₂ ⟧H (x ∷ ρ) * x + ⟦ c₂ ⟧N ρ)                                    ∎

+

+
  *N-homo : ∀ {n} (p₁ p₂ : Normal n) →

+
            ∀ ρ → ⟦ p₁ *N p₂ ⟧N ρ ≈ ⟦ p₁ ⟧N ρ * ⟦ p₂ ⟧N ρ

+
  *N-homo (con c₁)  (con c₂)  _ = *-homo _ _

+
  *N-homo (poly p₁) (poly p₂) ρ = *H-homo p₁ p₂ ρ

+

+
^N-homo : ∀ {n} (p : Normal n) (k : ℕ) →

+
          ∀ ρ → ⟦ p ^N k ⟧N ρ ≈ ⟦ p ⟧N ρ ^ k

+
^N-homo p zero    ρ = 1N-homo ρ

+
^N-homo p (suc k) ρ = begin

+
  ⟦ p *N (p ^N k) ⟧N ρ       ≈⟨ *N-homo p (p ^N k) ρ ⟩

+
  ⟦ p ⟧N ρ * ⟦ p ^N k ⟧N ρ   ≈⟨ refl ⟨ *-cong ⟩ ^N-homo p k ρ ⟩

+
  ⟦ p ⟧N ρ * (⟦ p ⟧N ρ ^ k)  ∎

+

+
mutual

+

+
  -H‿-homo : ∀ {n} (p : HNF (suc n)) →

+
             ∀ ρ → ⟦ -H p ⟧H ρ ≈ - ⟦ p ⟧H ρ

+
  -H‿-homo p (x ∷ ρ) = begin

+
    ⟦ (-N 1N) *NH p ⟧H (x ∷ ρ)     ≈⟨ *NH-homo (-N 1N) p x ρ ⟩

+
    ⟦ -N 1N ⟧N ρ * ⟦ p ⟧H (x ∷ ρ)  ≈⟨ trans (-N‿-homo 1N ρ) (-‿cong (1N-homo ρ)) ⟨ *-cong ⟩ refl ⟩

+
    - 1# * ⟦ p ⟧H (x ∷ ρ)          ≈⟨ lemma₇ _ ⟩

+
    - ⟦ p ⟧H (x ∷ ρ)               ∎

+

+
  -N‿-homo : ∀ {n} (p : Normal n) →

+
             ∀ ρ → ⟦ -N p ⟧N ρ ≈ - ⟦ p ⟧N ρ

+
  -N‿-homo (con c)  _ = -‿homo _

+
  -N‿-homo (poly p) ρ = -H‿-homo p ρ

+

+
------------------------------------------------------------------------

+
-- Correctness

+

+
correct-con : ∀ {n} (c : C.Carrier) (ρ : Vec Carrier n) →

+
              ⟦ normalise-con c ⟧N ρ ≈ ⟦ c ⟧′

+
correct-con c []      = refl

+
correct-con c (x ∷ ρ) = begin

+
  ⟦ ∅ *x+HN normalise-con c ⟧H (x ∷ ρ)  ≈⟨ ∅*x+HN-homo (normalise-con c) x ρ ⟩

+
  ⟦ normalise-con c ⟧N ρ            ≈⟨ correct-con c ρ ⟩

+
  ⟦ c ⟧′                                   ∎

+

+
correct-var : ∀ {n} (i : Fin n) →

+
              ∀ ρ → ⟦ normalise-var i ⟧N ρ ≈ lookup ρ i

+
correct-var (suc i) (x ∷ ρ) = begin

+
  ⟦ ∅ *x+HN normalise-var i ⟧H (x ∷ ρ)  ≈⟨ ∅*x+HN-homo (normalise-var i) x ρ ⟩

+
  ⟦ normalise-var i ⟧N ρ                ≈⟨ correct-var i ρ ⟩

+
  lookup ρ i                            ∎

+
correct-var zero (x ∷ ρ) = begin

+
  (0# * x + ⟦ 1N ⟧N ρ) * x + ⟦ 0N ⟧N ρ  ≈⟨ ((refl ⟨ +-cong ⟩ 1N-homo ρ) ⟨ *-cong ⟩ refl) ⟨ +-cong ⟩ 0N-homo ρ ⟩

+
  (0# * x + 1#) * x + 0#                ≈⟨ lemma₅ _ ⟩

+
  x                                     ∎

+

+
correct : ∀ {n} (p : Polynomial n) → ∀ ρ → ⟦ p ⟧↓ ρ ≈ ⟦ p ⟧ ρ

+
correct (op [+] p₁ p₂) ρ = begin

+
  ⟦ normalise p₁ +N normalise p₂ ⟧N ρ  ≈⟨ +N-homo (normalise p₁) (normalise p₂) ρ ⟩

+
  ⟦ p₁ ⟧↓ ρ + ⟦ p₂ ⟧↓ ρ                ≈⟨ correct p₁ ρ ⟨ +-cong ⟩ correct p₂ ρ ⟩

+
  ⟦ p₁ ⟧ ρ + ⟦ p₂ ⟧ ρ                  ∎

+
correct (op [*] p₁ p₂) ρ = begin

+
  ⟦ normalise p₁ *N normalise p₂ ⟧N ρ  ≈⟨ *N-homo (normalise p₁) (normalise p₂) ρ ⟩

+
  ⟦ p₁ ⟧↓ ρ * ⟦ p₂ ⟧↓ ρ                ≈⟨ correct p₁ ρ ⟨ *-cong ⟩ correct p₂ ρ ⟩

+
  ⟦ p₁ ⟧ ρ * ⟦ p₂ ⟧ ρ                  ∎

+
correct (con c)  ρ = correct-con c ρ

+
correct (var i)  ρ = correct-var i ρ

+
correct (p :^ k) ρ = begin

+
  ⟦ normalise p ^N k ⟧N ρ  ≈⟨ ^N-homo (normalise p) k ρ ⟩

+
  ⟦ p ⟧↓ ρ ^ k             ≈⟨ correct p ρ ⟨ ^-cong ⟩ ≡.refl {x = k} ⟩

+
  ⟦ p ⟧ ρ ^ k              ∎

+
correct (:- p) ρ = begin

+
  ⟦ -N normalise p ⟧N ρ  ≈⟨ -N‿-homo (normalise p) ρ ⟩

+
  - ⟦ p ⟧↓ ρ             ≈⟨ -‿cong (correct p ρ) ⟩

+
  - ⟦ p ⟧ ρ              ∎

+

+
------------------------------------------------------------------------

+
-- "Tactic.

+

+
open Reflection setoid var ⟦_⟧ ⟦_⟧↓ correct public

+
  using (prove; solve) renaming (_⊜_ to _:=_)

+

+
-- For examples of how solve and _:=_ can be used to

+
-- semi-automatically prove ring equalities, see, for instance,

+
-- Data.Digit or Data.Nat.DivMod.

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Structures.Biased.html b/site/static/agda_html/Algebra.Structures.Biased.html
new file mode 100644
index 000000000..a8a566970
--- /dev/null
+++ b/site/static/agda_html/Algebra.Structures.Biased.html
@@ -0,0 +1,355 @@
+
+Algebra.Structures.Biased
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Ways to give instances of certain structures where some fields can

+
-- be given in terms of others. Re-exported via `Algebra`.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Algebra.Consequences.Setoid

+
open import Data.Product.Base using (_,_; proj₁; proj₂)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
module Algebra.Structures.Biased

+
  {a ℓ} {A : Set a}  -- The underlying set

+
  (_≈_ : Rel A ℓ)    -- The underlying equality relation

+
  where

+

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures  _≈_

+

+
------------------------------------------------------------------------

+
-- IsCommutativeMonoid

+

+
record IsCommutativeMonoidˡ (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemigroup : IsSemigroup ∙

+
    identityˡ   : LeftIdentity ε ∙

+
    comm        : Commutative ∙

+

+
  isCommutativeMonoid : IsCommutativeMonoid ∙ ε

+
  isCommutativeMonoid = record

+
    { isMonoid = record

+
      { isSemigroup = isSemigroup

+
      ; identity    = comm∧idˡ⇒id setoid comm identityˡ

+
      }

+
    ; comm = comm

+
    } where open IsSemigroup isSemigroup

+

+
open IsCommutativeMonoidˡ public

+
  using () renaming (isCommutativeMonoid to isCommutativeMonoidˡ)

+

+

+
record IsCommutativeMonoidʳ (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemigroup : IsSemigroup ∙

+
    identityʳ   : RightIdentity ε ∙

+
    comm        : Commutative ∙

+

+
  isCommutativeMonoid : IsCommutativeMonoid ∙ ε

+
  isCommutativeMonoid = record

+
    { isMonoid = record

+
      { isSemigroup = isSemigroup

+
      ; identity    = comm∧idʳ⇒id setoid comm identityʳ

+
      }

+
    ; comm = comm

+
    } where open IsSemigroup isSemigroup

+

+
open IsCommutativeMonoidʳ public

+
  using () renaming (isCommutativeMonoid to isCommutativeMonoidʳ)

+

+
------------------------------------------------------------------------

+
-- IsSemiringWithoutOne

+

+
record IsSemiringWithoutOne* (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isCommutativeMonoid : IsCommutativeMonoid + 0#

+
    *-isSemigroup         : IsSemigroup *

+
    distrib               : * DistributesOver +

+
    zero                  : Zero 0# *

+

+
  isSemiringWithoutOne : IsSemiringWithoutOne + * 0#

+
  isSemiringWithoutOne = record

+
    { +-isCommutativeMonoid = +-isCommutativeMonoid

+
    ; *-cong = ∙-cong

+
    ; *-assoc = assoc

+
    ; distrib = distrib

+
    ; zero = zero

+
    } where open IsSemigroup *-isSemigroup

+

+
open IsSemiringWithoutOne* public

+
  using () renaming (isSemiringWithoutOne to isSemiringWithoutOne*)

+

+
------------------------------------------------------------------------

+
-- IsNearSemiring

+

+
record IsNearSemiring* (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isMonoid    : IsMonoid + 0#

+
    *-isSemigroup : IsSemigroup *

+
    distribʳ      : * DistributesOverʳ +

+
    zeroˡ         : LeftZero 0# *

+

+
  isNearSemiring : IsNearSemiring + * 0#

+
  isNearSemiring = record

+
    { +-isMonoid = +-isMonoid

+
    ; *-cong = ∙-cong

+
    ; *-assoc = assoc

+
    ; distribʳ = distribʳ

+
    ; zeroˡ = zeroˡ

+
    } where open IsSemigroup *-isSemigroup

+

+
open IsNearSemiring* public

+
  using () renaming (isNearSemiring to isNearSemiring*)

+

+
------------------------------------------------------------------------

+
-- IsSemiringWithoutAnnihilatingZero

+

+
record IsSemiringWithoutAnnihilatingZero* (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isCommutativeMonoid : IsCommutativeMonoid + 0#

+
    *-isMonoid            : IsMonoid * 1#

+
    distrib               : * DistributesOver +

+

+
  isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero + * 0# 1#

+
  isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = +-isCommutativeMonoid

+
    ; *-cong = ∙-cong

+
    ; *-assoc = assoc

+
    ; *-identity = identity

+
    ; distrib = distrib

+
    } where open IsMonoid *-isMonoid

+

+
open IsSemiringWithoutAnnihilatingZero* public

+
  using () renaming (isSemiringWithoutAnnihilatingZero to isSemiringWithoutAnnihilatingZero*)

+

+
------------------------------------------------------------------------

+
-- IsCommutativeSemiring

+

+
record IsCommutativeSemiringˡ (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isCommutativeMonoid : IsCommutativeMonoid + 0#

+
    *-isCommutativeMonoid : IsCommutativeMonoid * 1#

+
    distribʳ              : * DistributesOverʳ +

+
    zeroˡ                 : LeftZero 0# *

+

+
  isCommutativeSemiring : IsCommutativeSemiring + * 0# 1#

+
  isCommutativeSemiring = record

+
    { isSemiring = record

+
      { isSemiringWithoutAnnihilatingZero = record

+
        { +-isCommutativeMonoid = +-isCommutativeMonoid

+
        ; *-cong                = *.∙-cong

+
        ; *-assoc               = *.assoc

+
        ; *-identity            = *.identity

+
        ; distrib               = comm∧distrʳ⇒distr +.setoid +.∙-cong *.comm distribʳ

+
        }

+
      ; zero = comm∧zeˡ⇒ze +.setoid *.comm zeroˡ

+
      }

+
    ; *-comm = *.comm

+
    }

+
    where

+
    module + = IsCommutativeMonoid +-isCommutativeMonoid

+
    module * = IsCommutativeMonoid *-isCommutativeMonoid

+

+
open IsCommutativeSemiringˡ public

+
  using () renaming (isCommutativeSemiring to isCommutativeSemiringˡ)

+

+

+
record IsCommutativeSemiringʳ (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isCommutativeMonoid : IsCommutativeMonoid + 0#

+
    *-isCommutativeMonoid : IsCommutativeMonoid * 1#

+
    distribˡ              : * DistributesOverˡ +

+
    zeroʳ                 : RightZero 0# *

+

+
  isCommutativeSemiring : IsCommutativeSemiring + * 0# 1#

+
  isCommutativeSemiring = record

+
    { isSemiring = record

+
      { isSemiringWithoutAnnihilatingZero = record

+
        { +-isCommutativeMonoid = +-isCommutativeMonoid

+
        ; *-cong                = *.∙-cong

+
        ; *-assoc               = *.assoc

+
        ; *-identity            = *.identity

+
        ; distrib               = comm∧distrˡ⇒distr +.setoid +.∙-cong *.comm distribˡ

+
        }

+
      ; zero = comm∧zeʳ⇒ze +.setoid *.comm zeroʳ

+
      }

+
    ; *-comm = *.comm

+
    }

+
    where

+
    module + = IsCommutativeMonoid +-isCommutativeMonoid

+
    module * = IsCommutativeMonoid *-isCommutativeMonoid

+

+
open IsCommutativeSemiringʳ public

+
  using () renaming (isCommutativeSemiring to isCommutativeSemiringʳ)

+

+

+
------------------------------------------------------------------------

+
-- IsRing

+

+
record IsRing* (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isAbelianGroup : IsAbelianGroup + 0# -_

+
    *-isMonoid       : IsMonoid * 1#

+
    distrib          : * DistributesOver +

+
    zero             : Zero 0# *

+

+
  isRing : IsRing + * -_ 0# 1#

+
  isRing = record

+
    { +-isAbelianGroup = +-isAbelianGroup

+
    ; *-cong = ∙-cong

+
    ; *-assoc = assoc

+
    ; *-identity = identity

+
    ; distrib = distrib

+
    } where open IsMonoid *-isMonoid

+

+
open IsRing* public

+
  using () renaming (isRing to isRing*)

+

+

+

+
------------------------------------------------------------------------

+
-- Deprecated

+
------------------------------------------------------------------------

+

+
-- Version 2.0

+

+
-- We can recover a ring without proving that 0# annihilates *.

+
record IsRingWithoutAnnihilatingZero (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A)

+
                                     : Set (a ⊔ ℓ) where

+
  field

+
    +-isAbelianGroup : IsAbelianGroup + 0# -_

+
    *-isMonoid       : IsMonoid * 1#

+
    distrib          : * DistributesOver +

+

+
  module + = IsAbelianGroup +-isAbelianGroup

+
  module * = IsMonoid *-isMonoid

+

+
  open + using (setoid) renaming (∙-cong to +-cong)

+
  open * using ()       renaming (∙-cong to *-cong)

+

+
  zeroˡ : LeftZero 0# *

+
  zeroˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ setoid

+
            +-cong *-cong +.assoc (proj₂ distrib) +.identityʳ +.inverseʳ

+

+
  zeroʳ : RightZero 0# *

+
  zeroʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ setoid

+
            +-cong *-cong +.assoc (proj₁ distrib) +.identityʳ +.inverseʳ

+

+
  zero : Zero 0# *

+
  zero = (zeroˡ , zeroʳ)

+

+
  isRing : IsRing + * -_ 0# 1#

+
  isRing = record

+
    { +-isAbelianGroup = +-isAbelianGroup

+
    ; *-cong           = *.∙-cong

+
    ; *-assoc          = *.assoc

+
    ; *-identity       = *.identity

+
    ; distrib          = distrib

+
    }

+

+
open IsRingWithoutAnnihilatingZero public

+
  using () renaming (isRing to isRingWithoutAnnihilatingZero)

+

+
{-# WARNING_ON_USAGE IsRingWithoutAnnihilatingZero

+
"Warning: IsRingWithoutAnnihilatingZero was deprecated in v2.0.

+
Please use the standard `IsRing` instead."

+
#-}

+
{-# WARNING_ON_USAGE isRingWithoutAnnihilatingZero

+
"Warning: isRingWithoutAnnihilatingZero was deprecated in v2.0.

+
Please use the standard `isRing` instead."

+
#-}

+

+
-- Version 2.1

+

+
-- issue #2253

+
{-# WARNING_ON_USAGE IsRing*

+
"Warning: IsRing* was deprecated in v2.1.

+
Please use the standard `IsRing` instead."

+
#-}

+
{-# WARNING_ON_USAGE isRing*

+
"Warning: isRing* was deprecated in v2.1.

+
Please use the standard `isRing` instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.Structures.html b/site/static/agda_html/Algebra.Structures.html
new file mode 100644
index 000000000..034d19fe5
--- /dev/null
+++ b/site/static/agda_html/Algebra.Structures.html
@@ -0,0 +1,1112 @@
+
+Algebra.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some algebraic structures (not packed up with sets, operations, etc.)

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Algebra`, unless

+
-- you want to parameterise it via the equality relation.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
module Algebra.Structures

+
  {a ℓ} {A : Set a}  -- The underlying set

+
  (_≈_ : Rel A ℓ)    -- The underlying equality relation

+
  where

+

+
-- The file is divided into sections depending on the arities of the

+
-- components of the algebraic structure.

+

+
open import Algebra.Core using (Op₁; Op₂)

+
open import Algebra.Definitions _≈_

+
import Algebra.Consequences.Setoid as Consequences

+
open import Data.Product.Base using (_,_; proj₁; proj₂)

+
open import Level using (_⊔_)

+

+
------------------------------------------------------------------------

+
-- Structures with 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record IsSuccessorSet (suc# : Op₁ A) (zero# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isEquivalence : IsEquivalence _≈_

+
    suc#-cong     : Congruent₁ suc#

+

+
  open IsEquivalence isEquivalence public

+

+
  setoid : Setoid a ℓ

+
  setoid = record { isEquivalence = isEquivalence }

+

+
------------------------------------------------------------------------

+
-- Structures with 1 binary operation

+
------------------------------------------------------------------------

+

+
record IsMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isEquivalence : IsEquivalence _≈_

+
    ∙-cong        : Congruent₂ ∙

+

+
  open IsEquivalence isEquivalence public

+

+
  setoid : Setoid a ℓ

+
  setoid = record { isEquivalence = isEquivalence }

+

+
  ∙-congˡ : LeftCongruent ∙

+
  ∙-congˡ y≈z = ∙-cong refl y≈z

+

+
  ∙-congʳ : RightCongruent ∙

+
  ∙-congʳ y≈z = ∙-cong y≈z refl

+

+

+
record IsCommutativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma : IsMagma ∙

+
    comm    : Commutative ∙

+

+
  open IsMagma isMagma public

+

+
record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma : IsMagma ∙

+
    idem    : Idempotent ∙

+

+
  open IsMagma isMagma public

+

+
record IsAlternativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma  : IsMagma ∙

+
    alter    : Alternative ∙

+

+
  open IsMagma isMagma public

+

+
  alternativeˡ : LeftAlternative ∙

+
  alternativeˡ = proj₁ alter

+

+
  alternativeʳ : RightAlternative ∙

+
  alternativeʳ = proj₂ alter

+

+
record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma  : IsMagma ∙

+
    flex     : Flexible ∙

+

+
  open IsMagma isMagma public

+

+
record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma : IsMagma ∙

+
    medial  : Medial ∙

+

+
  open IsMagma isMagma public

+

+
record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma    : IsMagma ∙

+
    semiMedial : Semimedial ∙

+

+
  open IsMagma isMagma public

+

+
  semimedialˡ : LeftSemimedial ∙

+
  semimedialˡ = proj₁ semiMedial

+

+
  semimedialʳ : RightSemimedial ∙

+
  semimedialʳ = proj₂ semiMedial

+

+
record IsSelectiveMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma : IsMagma ∙

+
    sel     : Selective ∙

+

+
  open IsMagma isMagma public

+

+

+
record IsSemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma : IsMagma ∙

+
    assoc   : Associative ∙

+

+
  open IsMagma isMagma public

+

+

+
record IsBand (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemigroup : IsSemigroup ∙

+
    idem        : Idempotent ∙

+

+
  open IsSemigroup isSemigroup public

+

+

+
record IsCommutativeSemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemigroup : IsSemigroup ∙

+
    comm        : Commutative ∙

+

+
  open IsSemigroup isSemigroup public

+

+
  isCommutativeMagma : IsCommutativeMagma ∙

+
  isCommutativeMagma = record

+
    { isMagma = isMagma

+
    ; comm    = comm

+
    }

+

+

+
record IsCommutativeBand (∙ : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isBand : IsBand ∙

+
    comm   : Commutative ∙

+

+
  open IsBand isBand public

+

+
  isCommutativeSemigroup : IsCommutativeSemigroup ∙

+
  isCommutativeSemigroup = record { isSemigroup = isSemigroup ; comm = comm }

+

+
  open IsCommutativeSemigroup isCommutativeSemigroup public

+
    using (isCommutativeMagma)

+

+
------------------------------------------------------------------------

+
-- Structures with 1 binary operation & 1 element

+
------------------------------------------------------------------------

+

+
record IsUnitalMagma (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma  : IsMagma ∙

+
    identity : Identity ε ∙

+

+
  open IsMagma isMagma public

+

+
  identityˡ : LeftIdentity ε ∙

+
  identityˡ = proj₁ identity

+

+
  identityʳ : RightIdentity ε ∙

+
  identityʳ = proj₂ identity

+

+

+
record IsMonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemigroup : IsSemigroup ∙

+
    identity    : Identity ε ∙

+

+
  open IsSemigroup isSemigroup public

+

+
  identityˡ : LeftIdentity ε ∙

+
  identityˡ = proj₁ identity

+

+
  identityʳ : RightIdentity ε ∙

+
  identityʳ = proj₂ identity

+

+
  isUnitalMagma : IsUnitalMagma ∙ ε

+
  isUnitalMagma = record

+
    { isMagma  = isMagma

+
    ; identity = identity

+
    }

+

+

+
record IsCommutativeMonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isMonoid : IsMonoid ∙ ε

+
    comm     : Commutative ∙

+

+
  open IsMonoid isMonoid public

+

+
  isCommutativeSemigroup : IsCommutativeSemigroup ∙

+
  isCommutativeSemigroup = record

+
    { isSemigroup = isSemigroup

+
    ; comm        = comm

+
    }

+

+
  open IsCommutativeSemigroup isCommutativeSemigroup public

+
    using (isCommutativeMagma)

+

+

+
record IsIdempotentMonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isMonoid : IsMonoid ∙ ε

+
    idem     : Idempotent ∙

+

+
  open IsMonoid isMonoid public

+

+
  isBand : IsBand ∙

+
  isBand = record { isSemigroup = isSemigroup ; idem = idem }

+

+

+
record IsIdempotentCommutativeMonoid (∙ : Op₂ A)

+
                                     (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isCommutativeMonoid : IsCommutativeMonoid ∙ ε

+
    idem                : Idempotent ∙

+

+
  open IsCommutativeMonoid isCommutativeMonoid public

+

+
  isIdempotentMonoid : IsIdempotentMonoid ∙ ε

+
  isIdempotentMonoid = record { isMonoid = isMonoid ; idem = idem }

+

+
  open IsIdempotentMonoid isIdempotentMonoid public

+
    using (isBand)

+

+
  isCommutativeBand : IsCommutativeBand ∙

+
  isCommutativeBand = record { isBand = isBand ; comm = comm }

+

+
------------------------------------------------------------------------

+
-- Structures with 1 binary operation, 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record IsInvertibleMagma (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma  : IsMagma _∙_

+
    inverse   : Inverse ε _⁻¹ _∙_

+
    ⁻¹-cong   : Congruent₁ _⁻¹

+

+
  open IsMagma isMagma public

+

+
  inverseˡ : LeftInverse ε _⁻¹ _∙_

+
  inverseˡ = proj₁ inverse

+

+
  inverseʳ : RightInverse ε _⁻¹ _∙_

+
  inverseʳ = proj₂ inverse

+

+

+
record IsInvertibleUnitalMagma (_∙_ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where

+
  field

+
    isInvertibleMagma : IsInvertibleMagma _∙_  ε ⁻¹

+
    identity          : Identity ε _∙_

+

+
  open IsInvertibleMagma isInvertibleMagma public

+

+
  identityˡ : LeftIdentity ε _∙_

+
  identityˡ = proj₁ identity

+

+
  identityʳ : RightIdentity ε _∙_

+
  identityʳ = proj₂ identity

+

+
  isUnitalMagma : IsUnitalMagma _∙_  ε

+
  isUnitalMagma = record

+
    { isMagma  = isMagma

+
    ; identity = identity

+
    }

+

+

+
record IsGroup (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMonoid  : IsMonoid _∙_ ε

+
    inverse   : Inverse ε _⁻¹ _∙_

+
    ⁻¹-cong   : Congruent₁ _⁻¹

+

+
  open IsMonoid isMonoid public

+

+
  infixr 6 _\\_

+
  _\\_ : Op₂ A

+
  x \\ y = (x ⁻¹) ∙ y

+

+
  infixl 6 _//_

+
  _//_ : Op₂ A

+
  x // y = x ∙ (y ⁻¹)

+

+
  -- Deprecated.

+
  infixl 6 _-_

+
  _-_ : Op₂ A

+
  _-_ = _//_

+
  {-# WARNING_ON_USAGE _-_

+
  "Warning: _-_ was deprecated in v2.1.

+
  Please use _//_ instead. "

+
  #-}

+

+
  inverseˡ : LeftInverse ε _⁻¹ _∙_

+
  inverseˡ = proj₁ inverse

+

+
  inverseʳ : RightInverse ε _⁻¹ _∙_

+
  inverseʳ = proj₂ inverse

+

+
  uniqueˡ-⁻¹ : ∀ x y → (x ∙ y) ≈ ε → x ≈ (y ⁻¹)

+
  uniqueˡ-⁻¹ = Consequences.assoc∧id∧invʳ⇒invˡ-unique

+
                setoid ∙-cong assoc identity inverseʳ

+

+
  uniqueʳ-⁻¹ : ∀ x y → (x ∙ y) ≈ ε → y ≈ (x ⁻¹)

+
  uniqueʳ-⁻¹ = Consequences.assoc∧id∧invˡ⇒invʳ-unique

+
                setoid ∙-cong assoc identity inverseˡ

+

+
  isInvertibleMagma : IsInvertibleMagma _∙_ ε _⁻¹

+
  isInvertibleMagma = record

+
    { isMagma = isMagma

+
    ; inverse = inverse

+
    ; ⁻¹-cong = ⁻¹-cong

+
    }

+

+
  isInvertibleUnitalMagma : IsInvertibleUnitalMagma _∙_ ε _⁻¹

+
  isInvertibleUnitalMagma = record

+
    { isInvertibleMagma = isInvertibleMagma

+
    ; identity = identity

+
    }

+

+

+
record IsAbelianGroup (∙ : Op₂ A)

+
                      (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where

+
  field

+
    isGroup : IsGroup ∙ ε ⁻¹

+
    comm    : Commutative ∙

+

+
  open IsGroup isGroup public renaming (_//_ to _-_) hiding (_\\_; _-_)

+

+
  isCommutativeMonoid : IsCommutativeMonoid ∙ ε

+
  isCommutativeMonoid = record

+
    { isMonoid = isMonoid

+
    ; comm     = comm

+
    }

+

+
  open IsCommutativeMonoid isCommutativeMonoid public

+
    using (isCommutativeMagma; isCommutativeSemigroup)

+

+

+
------------------------------------------------------------------------

+
-- Structures with 2 binary operations & 1 element

+
------------------------------------------------------------------------

+

+
record IsNearSemiring (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isMonoid    : IsMonoid + 0#

+
    *-cong        : Congruent₂ *

+
    *-assoc       : Associative *

+
    distribʳ      : * DistributesOverʳ +

+
    zeroˡ         : LeftZero 0# *

+

+
  open IsMonoid +-isMonoid public

+
    renaming

+
    ( assoc         to +-assoc

+
    ; ∙-cong        to +-cong

+
    ; ∙-congˡ       to +-congˡ

+
    ; ∙-congʳ       to +-congʳ

+
    ; identity      to +-identity

+
    ; identityˡ     to +-identityˡ

+
    ; identityʳ     to +-identityʳ

+
    ; isMagma       to +-isMagma

+
    ; isUnitalMagma to +-isUnitalMagma

+
    ; isSemigroup   to +-isSemigroup

+
    )

+

+
  *-isMagma : IsMagma *

+
  *-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = *-cong

+
    }

+

+
  *-isSemigroup : IsSemigroup *

+
  *-isSemigroup = record

+
    { isMagma = *-isMagma

+
    ; assoc   = *-assoc

+
    }

+

+
  open IsMagma *-isMagma public

+
    using ()

+
    renaming

+
    ( ∙-congˡ  to *-congˡ

+
    ; ∙-congʳ  to *-congʳ

+
    )

+

+

+
record IsSemiringWithoutOne (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isCommutativeMonoid : IsCommutativeMonoid + 0#

+
    *-cong                : Congruent₂ *

+
    *-assoc               : Associative *

+
    distrib               : * DistributesOver +

+
    zero                  : Zero 0# *

+

+
  open IsCommutativeMonoid +-isCommutativeMonoid public

+
    using (setoid)

+
    renaming

+
    ( comm                   to +-comm

+
    ; isMonoid               to +-isMonoid

+
    ; isCommutativeMagma     to +-isCommutativeMagma

+
    ; isCommutativeSemigroup to +-isCommutativeSemigroup

+
    )

+

+
  open Setoid setoid public

+

+
  *-isMagma : IsMagma *

+
  *-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = *-cong

+
    }

+

+
  *-isSemigroup : IsSemigroup *

+
  *-isSemigroup = record

+
    { isMagma = *-isMagma

+
    ; assoc   = *-assoc

+
    }

+

+
  open IsMagma *-isMagma public

+
    using ()

+
    renaming

+
    ( ∙-congˡ to *-congˡ

+
    ; ∙-congʳ to *-congʳ

+
    )

+

+
  zeroˡ : LeftZero 0# *

+
  zeroˡ = proj₁ zero

+

+
  zeroʳ : RightZero 0# *

+
  zeroʳ = proj₂ zero

+

+
  isNearSemiring : IsNearSemiring + * 0#

+
  isNearSemiring = record

+
    { +-isMonoid    = +-isMonoid

+
    ; *-cong        = *-cong

+
    ; *-assoc       = *-assoc

+
    ; distribʳ      = proj₂ distrib

+
    ; zeroˡ         = zeroˡ

+
    }

+

+
record IsCommutativeSemiringWithoutOne

+
         (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemiringWithoutOne : IsSemiringWithoutOne + * 0#

+
    *-comm               : Commutative *

+

+
  open IsSemiringWithoutOne isSemiringWithoutOne public

+

+
  *-isCommutativeSemigroup : IsCommutativeSemigroup *

+
  *-isCommutativeSemigroup = record

+
    { isSemigroup = *-isSemigroup

+
    ; comm        = *-comm

+
    }

+

+
  open IsCommutativeSemigroup *-isCommutativeSemigroup public

+
    using () renaming (isCommutativeMagma to *-isCommutativeMagma)

+

+
------------------------------------------------------------------------

+
-- Structures with 2 binary operations & 2 elements

+
------------------------------------------------------------------------

+

+
record IsSemiringWithoutAnnihilatingZero (+ * : Op₂ A)

+
                                         (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    -- Note that these structures do have an additive unit, but this

+
    -- unit does not necessarily annihilate multiplication.

+
    +-isCommutativeMonoid : IsCommutativeMonoid + 0#

+
    *-cong                : Congruent₂ *

+
    *-assoc               : Associative *

+
    *-identity            : Identity 1# *

+
    distrib               : * DistributesOver +

+

+
  distribˡ : * DistributesOverˡ +

+
  distribˡ = proj₁ distrib

+

+
  distribʳ : * DistributesOverʳ +

+
  distribʳ = proj₂ distrib

+

+
  open IsCommutativeMonoid +-isCommutativeMonoid public

+
    renaming

+
    ( assoc                  to +-assoc

+
    ; ∙-cong                 to +-cong

+
    ; ∙-congˡ                to +-congˡ

+
    ; ∙-congʳ                to +-congʳ

+
    ; identity               to +-identity

+
    ; identityˡ              to +-identityˡ

+
    ; identityʳ              to +-identityʳ

+
    ; comm                   to +-comm

+
    ; isMagma                to +-isMagma

+
    ; isSemigroup            to +-isSemigroup

+
    ; isMonoid               to +-isMonoid

+
    ; isUnitalMagma          to +-isUnitalMagma

+
    ; isCommutativeMagma     to +-isCommutativeMagma

+
    ; isCommutativeSemigroup to +-isCommutativeSemigroup

+
    )

+

+
  *-isMagma : IsMagma *

+
  *-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = *-cong

+
    }

+

+
  *-isSemigroup : IsSemigroup *

+
  *-isSemigroup = record

+
    { isMagma = *-isMagma

+
    ; assoc   = *-assoc

+
    }

+

+
  *-isMonoid : IsMonoid * 1#

+
  *-isMonoid = record

+
    { isSemigroup = *-isSemigroup

+
    ; identity    = *-identity

+
    }

+

+
  open IsMonoid *-isMonoid public

+
    using ()

+
    renaming

+
    ( ∙-congˡ     to *-congˡ

+
    ; ∙-congʳ     to *-congʳ

+
    ; identityˡ   to *-identityˡ

+
    ; identityʳ   to *-identityʳ

+
    )

+

+

+
record IsSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemiringWithoutAnnihilatingZero :

+
      IsSemiringWithoutAnnihilatingZero + * 0# 1#

+
    zero : Zero 0# *

+

+
  open IsSemiringWithoutAnnihilatingZero

+
         isSemiringWithoutAnnihilatingZero public

+

+
  isSemiringWithoutOne : IsSemiringWithoutOne + * 0#

+
  isSemiringWithoutOne = record

+
    { +-isCommutativeMonoid = +-isCommutativeMonoid

+
    ; *-cong                = *-cong

+
    ; *-assoc               = *-assoc

+
    ; distrib               = distrib

+
    ; zero                  = zero

+
    }

+

+
  open IsSemiringWithoutOne isSemiringWithoutOne public

+
    using

+
    ( isNearSemiring

+
    ; zeroˡ

+
    ; zeroʳ

+
    )

+

+

+
record IsCommutativeSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemiring : IsSemiring + * 0# 1#

+
    *-comm     : Commutative *

+

+
  open IsSemiring isSemiring public

+

+
  isCommutativeSemiringWithoutOne :

+
    IsCommutativeSemiringWithoutOne + * 0#

+
  isCommutativeSemiringWithoutOne = record

+
    { isSemiringWithoutOne = isSemiringWithoutOne

+
    ; *-comm = *-comm

+
    }

+

+
  open IsCommutativeSemiringWithoutOne isCommutativeSemiringWithoutOne public

+
    using

+
    ( *-isCommutativeMagma

+
    ; *-isCommutativeSemigroup

+
    )

+

+
  *-isCommutativeMonoid : IsCommutativeMonoid * 1#

+
  *-isCommutativeMonoid = record

+
    { isMonoid = *-isMonoid

+
    ; comm     = *-comm

+
    }

+

+

+
record IsCancellativeCommutativeSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isCommutativeSemiring : IsCommutativeSemiring + * 0# 1#

+
    *-cancelˡ-nonZero     : AlmostLeftCancellative 0# *

+

+
  open IsCommutativeSemiring isCommutativeSemiring public

+

+
  *-cancelʳ-nonZero : AlmostRightCancellative 0# *

+
  *-cancelʳ-nonZero = Consequences.comm∧almostCancelˡ⇒almostCancelʳ setoid

+
      *-comm *-cancelˡ-nonZero

+

+
record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isSemiring     : IsSemiring + * 0# 1#

+
    +-idem         : Idempotent +

+

+
  open IsSemiring isSemiring public

+

+
  +-isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid + 0#

+
  +-isIdempotentCommutativeMonoid = record

+
    { isCommutativeMonoid = +-isCommutativeMonoid

+
    ; idem = +-idem

+
    }

+

+
  open IsIdempotentCommutativeMonoid +-isIdempotentCommutativeMonoid public

+
    using ()

+
    renaming ( isCommutativeBand to +-isCommutativeBand

+
             ; isBand to +-isBand

+
             ; isIdempotentMonoid to +-isIdempotentMonoid

+
             )

+

+

+
record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isIdempotentSemiring  : IsIdempotentSemiring + * 0# 1#

+
    starExpansive         : StarExpansive 1# + * ⋆

+
    starDestructive       : StarDestructive + * ⋆

+

+
  open IsIdempotentSemiring isIdempotentSemiring public

+

+
  starExpansiveˡ : StarLeftExpansive 1# + * ⋆

+
  starExpansiveˡ = proj₁ starExpansive

+

+
  starExpansiveʳ : StarRightExpansive 1# + * ⋆

+
  starExpansiveʳ = proj₂ starExpansive

+

+
  starDestructiveˡ : StarLeftDestructive + * ⋆

+
  starDestructiveˡ = proj₁ starDestructive

+

+
  starDestructiveʳ : StarRightDestructive + * ⋆

+
  starDestructiveʳ = proj₂ starDestructive

+

+
record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isMonoid    : IsMonoid + 0#

+
    *-cong        : Congruent₂ *

+
    *-assoc       : Associative *

+
    *-identity    : Identity 1# *

+
    distrib       : * DistributesOver +

+
    zero          : Zero 0# *

+

+
  open IsMonoid +-isMonoid public

+
    renaming

+
    ( assoc         to +-assoc

+
    ; ∙-cong        to +-cong

+
    ; ∙-congˡ       to +-congˡ

+
    ; ∙-congʳ       to +-congʳ

+
    ; identity      to +-identity

+
    ; identityˡ     to +-identityˡ

+
    ; identityʳ     to +-identityʳ

+
    ; isMagma       to +-isMagma

+
    ; isUnitalMagma to +-isUnitalMagma

+
    ; isSemigroup   to +-isSemigroup

+
    )

+

+
  distribˡ : * DistributesOverˡ +

+
  distribˡ = proj₁ distrib

+

+
  distribʳ : * DistributesOverʳ +

+
  distribʳ = proj₂ distrib

+

+
  zeroˡ : LeftZero 0# *

+
  zeroˡ = proj₁ zero

+

+
  zeroʳ : RightZero 0# *

+
  zeroʳ = proj₂ zero

+

+
  identityˡ : LeftIdentity 1# *

+
  identityˡ = proj₁ *-identity

+

+
  identityʳ : RightIdentity 1# *

+
  identityʳ = proj₂ *-identity

+

+
  *-isMagma : IsMagma *

+
  *-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = *-cong

+
    }

+

+
  *-isSemigroup : IsSemigroup *

+
  *-isSemigroup = record

+
    { isMagma = *-isMagma

+
    ; assoc   = *-assoc

+
    }

+

+
  *-isMonoid : IsMonoid * 1#

+
  *-isMonoid = record

+
    { isSemigroup = *-isSemigroup

+
    ; identity    = *-identity

+
    }

+

+
  open IsMonoid *-isMonoid public

+
    using ()

+
    renaming

+
    ( ∙-congˡ     to *-congˡ

+
    ; ∙-congʳ     to *-congʳ

+
    ; identityˡ   to *-identityˡ

+
    ; identityʳ   to *-identityʳ

+
    )

+

+
------------------------------------------------------------------------

+
-- Structures with 2 binary operations, 1 unary operation & 1 element

+
------------------------------------------------------------------------

+

+
record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isAbelianGroup : IsAbelianGroup + 0# -_

+
    *-cong           : Congruent₂ *

+
    *-assoc          : Associative *

+
    distrib          : * DistributesOver +

+

+
  open IsAbelianGroup +-isAbelianGroup public

+
    renaming

+
    ( assoc                   to +-assoc

+
    ; ∙-cong                  to +-cong

+
    ; ∙-congˡ                 to +-congˡ

+
    ; ∙-congʳ                 to +-congʳ

+
    ; identity                to +-identity

+
    ; identityˡ               to +-identityˡ

+
    ; identityʳ               to +-identityʳ

+
    ; inverse                 to -‿inverse

+
    ; inverseˡ                to -‿inverseˡ

+
    ; inverseʳ                to -‿inverseʳ

+
    ; ⁻¹-cong                 to -‿cong

+
    ; comm                    to +-comm

+
    ; isMagma                 to +-isMagma

+
    ; isSemigroup             to +-isSemigroup

+
    ; isMonoid                to +-isMonoid

+
    ; isUnitalMagma           to +-isUnitalMagma

+
    ; isCommutativeMagma      to +-isCommutativeMagma

+
    ; isCommutativeMonoid     to +-isCommutativeMonoid

+
    ; isCommutativeSemigroup  to +-isCommutativeSemigroup

+
    ; isInvertibleMagma       to +-isInvertibleMagma

+
    ; isInvertibleUnitalMagma to +-isInvertibleUnitalMagma

+
    ; isGroup                 to +-isGroup

+
    )

+

+
  distribˡ : * DistributesOverˡ +

+
  distribˡ = proj₁ distrib

+

+
  distribʳ : * DistributesOverʳ +

+
  distribʳ = proj₂ distrib

+

+
  zeroˡ : LeftZero 0# *

+
  zeroˡ = Consequences.assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ setoid

+
    +-cong *-cong +-assoc distribʳ +-identityʳ -‿inverseʳ

+

+
  zeroʳ : RightZero 0# *

+
  zeroʳ = Consequences.assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ setoid

+
    +-cong *-cong +-assoc distribˡ +-identityʳ -‿inverseʳ

+

+
  zero : Zero 0# *

+
  zero = zeroˡ , zeroʳ

+

+
  *-isMagma : IsMagma *

+
  *-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = *-cong

+
    }

+

+
  *-isSemigroup : IsSemigroup *

+
  *-isSemigroup = record

+
    { isMagma = *-isMagma

+
    ; assoc   = *-assoc

+
    }

+

+
  open IsSemigroup *-isSemigroup public

+
    using ()

+
    renaming

+
    ( ∙-congˡ   to *-congˡ

+
    ; ∙-congʳ   to *-congʳ

+
    )

+

+
------------------------------------------------------------------------

+
-- Structures with 2 binary operations, 1 unary operation & 2 elements

+
------------------------------------------------------------------------

+

+
record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isAbelianGroup : IsAbelianGroup + 0# -_

+
    *-cong           : Congruent₂ *

+
    *-identity       : Identity 1# *

+
    distrib          : * DistributesOver +

+
    zero             : Zero 0# *

+

+
  open IsAbelianGroup +-isAbelianGroup public

+
    renaming

+
    ( assoc                   to +-assoc

+
    ; ∙-cong                  to +-cong

+
    ; ∙-congˡ                 to +-congˡ

+
    ; ∙-congʳ                 to +-congʳ

+
    ; identity                to +-identity

+
    ; identityˡ               to +-identityˡ

+
    ; identityʳ               to +-identityʳ

+
    ; inverse                 to -‿inverse

+
    ; inverseˡ                to -‿inverseˡ

+
    ; inverseʳ                to -‿inverseʳ

+
    ; ⁻¹-cong                 to -‿cong

+
    ; comm                    to +-comm

+
    ; isMagma                 to +-isMagma

+
    ; isSemigroup             to +-isSemigroup

+
    ; isMonoid                to +-isMonoid

+
    ; isUnitalMagma           to +-isUnitalMagma

+
    ; isCommutativeMagma      to +-isCommutativeMagma

+
    ; isCommutativeMonoid     to +-isCommutativeMonoid

+
    ; isCommutativeSemigroup  to +-isCommutativeSemigroup

+
    ; isInvertibleMagma       to +-isInvertibleMagma

+
    ; isInvertibleUnitalMagma to +-isInvertibleUnitalMagma

+
    ; isGroup                 to +-isGroup

+
    )

+

+
  zeroˡ : LeftZero 0# *

+
  zeroˡ = proj₁ zero

+

+
  zeroʳ : RightZero 0# *

+
  zeroʳ = proj₂ zero

+

+
  distribˡ : * DistributesOverˡ +

+
  distribˡ = proj₁ distrib

+

+
  distribʳ : * DistributesOverʳ +

+
  distribʳ = proj₂ distrib

+

+
  *-isMagma : IsMagma *

+
  *-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = *-cong

+
    }

+

+
  *-identityˡ : LeftIdentity 1# *

+
  *-identityˡ = proj₁ *-identity

+

+
  *-identityʳ : RightIdentity 1# *

+
  *-identityʳ = proj₂ *-identity

+

+
  *-isUnitalMagma : IsUnitalMagma * 1#

+
  *-isUnitalMagma = record

+
    { isMagma = *-isMagma

+
    ; identity = *-identity

+
    }

+

+
  open IsUnitalMagma *-isUnitalMagma public

+
    using ()

+
    renaming

+
    ( ∙-congˡ   to *-congˡ

+
    ; ∙-congʳ   to *-congʳ

+
    )

+

+
record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where

+
  field

+
    isQuasiring : IsQuasiring + * 0# 1#

+
    +-inverse   : Inverse 0# _⁻¹ +

+
    ⁻¹-cong     : Congruent₁ _⁻¹

+

+
  open IsQuasiring isQuasiring public

+

+
  +-inverseˡ : LeftInverse 0# _⁻¹ +

+
  +-inverseˡ = proj₁ +-inverse

+

+
  +-inverseʳ : RightInverse 0# _⁻¹ +

+
  +-inverseʳ = proj₂ +-inverse

+

+
record IsRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    +-isAbelianGroup : IsAbelianGroup + 0# -_

+
    *-cong           : Congruent₂ *

+
    *-assoc          : Associative *

+
    *-identity       : Identity 1# *

+
    distrib          : * DistributesOver +

+

+
  isRingWithoutOne : IsRingWithoutOne + * -_ 0#

+
  isRingWithoutOne = record

+
    { +-isAbelianGroup = +-isAbelianGroup

+
    ; *-cong  = *-cong

+
    ; *-assoc = *-assoc

+
    ; distrib = distrib

+
    }

+

+
  open IsRingWithoutOne isRingWithoutOne public

+
    hiding (+-isAbelianGroup; *-cong; *-assoc; distrib)

+

+
  *-isMonoid : IsMonoid * 1#

+
  *-isMonoid = record

+
    { isSemigroup = *-isSemigroup

+
    ; identity    = *-identity

+
    }

+

+
  open IsMonoid *-isMonoid public

+
    using ()

+
    renaming

+
    ( identityˡ   to *-identityˡ

+
    ; identityʳ   to *-identityʳ

+
    )

+

+
  isSemiringWithoutAnnihilatingZero

+
    : IsSemiringWithoutAnnihilatingZero + * 0# 1#

+
  isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = +-isCommutativeMonoid

+
    ; *-cong                = *-cong

+
    ; *-assoc               = *-assoc

+
    ; *-identity            = *-identity

+
    ; distrib               = distrib

+
    }

+

+
  isSemiring : IsSemiring + * 0# 1#

+
  isSemiring = record

+
    { isSemiringWithoutAnnihilatingZero =

+
        isSemiringWithoutAnnihilatingZero

+
    ; zero = zero

+
    }

+

+
  open IsSemiring isSemiring public

+
    using (isNearSemiring; isSemiringWithoutOne)

+

+

+
record IsCommutativeRing

+
         (+ * : Op₂ A) (- : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where

+
  field

+
    isRing : IsRing + * - 0# 1#

+
    *-comm : Commutative *

+

+
  open IsRing isRing public

+

+
  isCommutativeSemiring : IsCommutativeSemiring + * 0# 1#

+
  isCommutativeSemiring = record

+
    { isSemiring = isSemiring

+
    ; *-comm = *-comm

+
    }

+

+
  open IsCommutativeSemiring isCommutativeSemiring public

+
    using

+
    ( isCommutativeSemiringWithoutOne

+
    ; *-isCommutativeMagma

+
    ; *-isCommutativeSemigroup

+
    ; *-isCommutativeMonoid

+
    )

+

+
------------------------------------------------------------------------

+
-- Structures with 3 binary operations

+
------------------------------------------------------------------------

+

+
record IsQuasigroup (∙ \\ // : Op₂ A) : Set (a ⊔ ℓ) where

+
  field

+
    isMagma       : IsMagma ∙

+
    \\-cong       : Congruent₂ \\

+
    //-cong       : Congruent₂ //

+
    leftDivides   : LeftDivides ∙ \\

+
    rightDivides  : RightDivides ∙ //

+

+
  open IsMagma isMagma public

+

+
  \\-congˡ : LeftCongruent \\

+
  \\-congˡ y≈z = \\-cong refl y≈z

+

+
  \\-congʳ : RightCongruent \\

+
  \\-congʳ y≈z = \\-cong y≈z refl

+

+
  //-congˡ : LeftCongruent //

+
  //-congˡ y≈z = //-cong refl y≈z

+

+
  //-congʳ : RightCongruent //

+
  //-congʳ y≈z = //-cong y≈z refl

+

+
  leftDividesˡ : LeftDividesˡ ∙ \\

+
  leftDividesˡ = proj₁ leftDivides

+

+
  leftDividesʳ : LeftDividesʳ ∙ \\

+
  leftDividesʳ = proj₂ leftDivides

+

+
  rightDividesˡ : RightDividesˡ ∙ //

+
  rightDividesˡ = proj₁ rightDivides

+

+
  rightDividesʳ : RightDividesʳ ∙ //

+
  rightDividesʳ = proj₂ rightDivides

+

+

+
record IsLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isQuasigroup : IsQuasigroup ∙ \\ //

+
    identity     : Identity ε ∙

+

+
  open IsQuasigroup isQuasigroup public

+

+
  identityˡ : LeftIdentity ε ∙

+
  identityˡ = proj₁ identity

+

+
  identityʳ : RightIdentity ε ∙

+
  identityʳ = proj₂ identity

+

+
record IsLeftBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isLoop  : IsLoop ∙ \\ //  ε

+
    leftBol : LeftBol ∙

+

+
  open IsLoop isLoop public

+

+
record IsRightBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isLoop   : IsLoop ∙ \\ //  ε

+
    rightBol : RightBol ∙

+

+
  open IsLoop isLoop public

+

+
record IsMoufangLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isLeftBolLoop  : IsLeftBolLoop ∙ \\ //  ε

+
    rightBol       : RightBol ∙

+
    identical      : Identical ∙

+

+
  open IsLeftBolLoop isLeftBolLoop public

+

+
record IsMiddleBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where

+
  field

+
    isLoop    : IsLoop ∙ \\ //  ε

+
    middleBol : MiddleBol ∙ \\ //

+

+
  open IsLoop isLoop public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Algebra.html b/site/static/agda_html/Algebra.html
new file mode 100644
index 000000000..55c0e0816
--- /dev/null
+++ b/site/static/agda_html/Algebra.html
@@ -0,0 +1,93 @@
+
+Algebra
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of algebraic structures like monoids and rings

+
-- (packed in records together with sets, operations, etc.)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Algebra where

+

+
open import Algebra.Core public

+
open import Algebra.Definitions public

+
open import Algebra.Structures public

+
open import Algebra.Structures.Biased public

+
open import Algebra.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Extensionality.Propositional.html b/site/static/agda_html/Axiom.Extensionality.Propositional.html
new file mode 100644
index 000000000..65c119e77
--- /dev/null
+++ b/site/static/agda_html/Axiom.Extensionality.Propositional.html
@@ -0,0 +1,139 @@
+
+Axiom.Extensionality.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Results concerning function extensionality for propositional equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Axiom.Extensionality.Propositional where

+

+
open import Function.Base

+
open import Level using (Level; _⊔_; suc; lift)

+
open import Relation.Binary.Core

+
open import Relation.Binary.PropositionalEquality.Core

+

+
------------------------------------------------------------------------

+
-- Function extensionality states that if two functions are

+
-- propositionally equal for every input, then the functions themselves

+
-- must be propositionally equal.

+

+
Extensionality : (a b : Level) → Set _

+
Extensionality a b =

+
  {A : Set a} {B : A → Set b} {f g : (x : A) → B x} →

+
  (∀ x → f x ≡ g x) → f ≡ g

+

+
-- A variant for implicit function spaces.

+

+
ExtensionalityImplicit : (a b : Level) → Set _

+
ExtensionalityImplicit a b =

+
  {A : Set a} {B : A → Set b} {f g : {x : A} → B x} →

+
  (∀ {x} → f {x} ≡ g {x}) → (λ {x} → f {x}) ≡ (λ {x} → g {x})

+

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
-- If extensionality holds for a given universe level, then it also

+
-- holds for lower ones.

+

+
lower-extensionality : ∀ {a₁ b₁} a₂ b₂ →

+
                       Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →

+
                       Extensionality a₁ b₁

+
lower-extensionality a₂ b₂ ext f≡g = cong (λ h → Level.lower ∘ h ∘ lift) $

+
    ext (cong (lift {ℓ = b₂}) ∘ f≡g ∘ Level.lower {ℓ = a₂})

+

+
-- Functional extensionality implies a form of extensionality for

+
-- Π-types.

+

+
∀-extensionality : ∀ {a b} → Extensionality a (suc b) →

+
                   {A : Set a} (B₁ B₂ : A → Set b) →

+
                   (∀ x → B₁ x ≡ B₂ x) →

+
                   (∀ x → B₁ x) ≡ (∀ x → B₂ x)

+
∀-extensionality ext B₁ B₂ B₁≡B₂ with ext B₁≡B₂

+
... | refl = refl

+

+
-- Extensionality for explicit function spaces implies extensionality

+
-- for implicit function spaces.

+

+
implicit-extensionality : ∀ {a b} →

+
                          Extensionality a b →

+
                          ExtensionalityImplicit a b

+
implicit-extensionality ext f≡g = cong _$- (ext (λ x → f≡g))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.Factor.html b/site/static/agda_html/Axiom.Set.Factor.html
new file mode 100644
index 000000000..bfc46a0e6
--- /dev/null
+++ b/site/static/agda_html/Axiom.Set.Factor.html
@@ -0,0 +1,149 @@
+
+Axiom.Set.Factor
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import abstract-set-theory.Prelude

+
open import Axiom.Set

+

+
module Axiom.Set.Factor (th : Theory) where

+

+
open Theory th

+
open import Axiom.Set.Properties th

+

+
import Function.Related.Propositional as R

+
open import Data.List.Ext.Properties

+
open import Data.List.Relation.Binary.BagAndSetEquality

+
open import Data.List.Relation.Binary.Disjoint.Propositional

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
open import Data.List.Relation.Unary.Unique.DecPropositional.Properties

+
open import Data.List.Relation.Unary.Unique.Propositional

+
open import Data.List.Relation.Unary.Unique.Propositional.Properties.WithK

+
open import Relation.Binary

+

+
open Equivalence

+

+
private variable A B : Type

+

+
_ᶠ : (X : Set A) → ⦃ finite X ⦄ → FinSet A

+
_ᶠ X ⦃ Xᶠ ⦄ = X , Xᶠ

+

+
instance

+
  ∪-preserves-finite' : {X Y : Set A} → ⦃ finite X ⦄ → ⦃ finite Y ⦄ → finite (X ∪ Y)

+
  ∪-preserves-finite' ⦃ Xᶠ ⦄ ⦃ Yᶠ ⦄ = ∪-preserves-finite Xᶠ Yᶠ

+

+
module Factor (_≈_ : B → B → Type) (f : List A → B) (f-cong : ∀ {l l'} → l ∼[ set ] l' → f l ≈ f l') where

+

+
  factor : FinSet A → B

+
  factor (_ , l , _) = f l

+

+
  factor-cong : factor Preserves (_≡ᵉ_ on proj₁) ⟶ _≈_

+
  factor-cong {X , Xˡ , hX} {Y , Yˡ , hY} X≡ᵉY = f-cong λ {a} →

+
    a ∈ˡ Xˡ ∼⟨ R.SK-sym hX ⟩ a ∈  X  ∼⟨ to ≡ᵉ⇔≡ᵉ' X≡ᵉY _ ⟩

+
    a ∈  Y  ∼⟨ hY ⟩          a ∈ˡ Yˡ ∎

+
    where open R.EquationalReasoning

+

+
  factor-∪ : ∀ {R : B → B → B → Type} {X Y : Set A} ⦃ Xᶠ : finite X ⦄ ⦃ Yᶠ : finite Y ⦄

+
           → (∀ {l l'} → R (f l) (f l') (f (l ++ l')))

+
           → R (factor (X ᶠ)) (factor (Y ᶠ)) (factor ((X ∪ Y) ᶠ))

+
  factor-∪ hR = hR

+

+
module FactorUnique ⦃ _ : DecEq A ⦄ (_≈_ : B → B → Type) (f : (Σ (List A) Unique) → B)

+
                    (f-cong : ∀ {l l'} → proj₁ l ↭ proj₁ l' → f l ≈ f l') where

+

+
  f-cong' : ∀ {l l'} → (∀ {a} → a ∈ˡ proj₁ l ⇔ a ∈ˡ proj₁ l') → f l ≈ f l'

+
  f-cong' {l} {l'} h = f-cong (∼bag⇒↭ (unique∧set⇒bag (proj₂ l) (proj₂ l') h))

+

+
  deduplicate-Σ : List A → Σ (List A) Unique

+
  deduplicate-Σ l = (deduplicate _≟_ l , deduplicate-! _≟_ _)

+

+
  ext : List A → B

+
  ext = f ∘ deduplicate-Σ

+

+
  ext-cong : ∀ {l l'} → l ∼[ set ] l' → ext l ≈ ext l'

+
  ext-cong {l} {l'} h = f-cong' λ {a} →

+
    a ∈ˡ deduplicate _≟_ l  ∼⟨ R.SK-sym ∈-dedup ⟩ a ∈ˡ l                  ∼⟨ h ⟩

+
    a ∈ˡ l'                 ∼⟨ ∈-dedup ⟩          a ∈ˡ deduplicate _≟_ l' ∎

+
    where open R.EquationalReasoning

+

+
  open Factor _≈_ ext ext-cong public

+

+
  factor-∪' : ∀ {R : B → B → B → Type} {X Y : Set A} ⦃ Xᶠ : finite X ⦄ ⦃ Yᶠ : finite Y ⦄

+
            → disjoint X Y

+
            → (∀ {l l'} → Disjoint l l' → R (ext l) (ext l') (ext (l ++ l')))

+
            → R (factor (X ᶠ)) (factor (Y ᶠ)) (factor ((X ∪ Y) ᶠ))

+
  factor-∪' ⦃ _ , Xᶠ ⦄ ⦃ _ , Yᶠ ⦄ disj hR = hR λ where (a∈X , a∈Y) → ⊥-elim $ disj (from Xᶠ a∈X) (from Yᶠ a∈Y)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.List.html b/site/static/agda_html/Axiom.Set.List.html
new file mode 100644
index 000000000..aa1f748de
--- /dev/null
+++ b/site/static/agda_html/Axiom.Set.List.html
@@ -0,0 +1,140 @@
+
+Axiom.Set.List
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
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+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+

+
module Axiom.Set.List where

+

+
open import abstract-set-theory.Prelude hiding (find)

+

+
open import Axiom.Set

+

+
open import Data.List as L using (filter)

+
import Data.List.Relation.Unary.All as All

+
import Data.List.Relation.Unary.Any as Any

+
import Function.Properties.Inverse as I

+
import Function.Related.Propositional as R

+
import Relation.Nullary.Decidable as Dec

+
open import Data.List.Membership.Propositional using (find; lose) renaming (_∈_ to _∈ˡ_)

+
open import Data.List.Membership.Propositional.Properties

+
open import Data.Product using (swap)

+
open import Data.Product.Algebra

+
open import Data.Product.Properties.Ext

+
open import Relation.Binary using () renaming (Decidable to Dec₂)

+
open import Relation.Nullary.Decidable

+

+
List-Model : Theory

+
List-Model = λ where

+
  .Set           → List

+
  ._∈_           → _∈ˡ_

+
  .sp            → Dec-SpecProperty

+
  .specification → λ X P? → filter P? X , mk⇔

+
    (λ where (Pa , a∈X) → ∈-filter⁺ P? a∈X Pa)

+
    (λ a∈f → swap (∈-filter⁻ P? a∈f))

+
  .unions        → λ X → concat X , mk⇔

+
    (λ where (T , T∈X , a∈T) → ∈-concat⁺′ a∈T T∈X)

+
    (λ a∈cX → case ∈-concat⁻′ _ a∈cX of λ where (T , a∈T , T∈X) → (T , T∈X , a∈T))

+
  .replacement   → λ f X → L.map f X , λ {b} →

+
    (∃[ a ] b ≡ f a × a ∈ˡ X) ∼⟨ ∃-cong′ (I.↔⇒⇔ (×-comm _ _)) ⟩

+
    (∃[ a ] a ∈ˡ X × b ≡ f a) ⤖⟨ I.↔⇒⤖ (map-∈↔ f) ⟩

+
    b ∈ˡ L.map f X    ∎

+
  .listing → λ l → l , mk⇔ id id

+
    where open Theory hiding (filter)

+
          open R.EquationalReasoning

+

+
List-Modelᶠ : Theoryᶠ

+
List-Modelᶠ = λ where

+
  .theory     → List-Model

+
  .finiteness → λ X → X , mk⇔ id id

+
    where open Theoryᶠ

+

+
module Decˡ {A : Type} ⦃ _ : DecEq A ⦄ where

+
  open Theory List-Model

+

+
  _∈?_ : Dec₂ (_∈ˡ_ {A = A})

+
  _∈?_ a = Any.any? (a ≟_)

+

+
  DecEq-Set : DecEq (Set A)

+
  DecEq-Set = DecEq-List

+

+
List-Modelᵈ : Theoryᵈ

+
List-Modelᵈ = record

+
  { th = List-Model

+
  ; ∈-sp = Decˡ._∈? _

+
  ; _∈?_ = Decˡ._∈?_

+
  ; all? = λ P? {X} → Dec.map (mk⇔ All.lookup All.tabulate) (All.all? P? X)

+
  ; any? = λ P? X → Dec.map (mk⇔ find (uncurry lose ∘ proj₂)) (Any.any? P? X) }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.Map.Dec.html b/site/static/agda_html/Axiom.Set.Map.Dec.html
new file mode 100644
index 000000000..efb9d189c
--- /dev/null
+++ b/site/static/agda_html/Axiom.Set.Map.Dec.html
@@ -0,0 +1,159 @@
+
+Axiom.Set.Map.Dec
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
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+               

+             

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+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+
open import Axiom.Set using (Theoryᵈ; Theory)

+

+
module Axiom.Set.Map.Dec (thᵈ : Theoryᵈ) where

+

+
open import abstract-set-theory.Prelude hiding (map; Monoid)

+

+
import Data.Sum as Sum

+
open import Data.These hiding (map)

+

+
open Theoryᵈ thᵈ using (_∈?_; th; incl-set'; incl-set; incl-set-proj₁⊇)

+
open Theory th

+
open import Axiom.Set.Rel th using (dom; dom∈)

+
open import Axiom.Set.Map th

+
open import Data.Product.Properties using (×-≡,≡→≡; ×-≡,≡←≡)

+

+
open Equivalence

+

+
private variable A B C D : Type

+

+
module Lookupᵐᵈ (sp-∈ : spec-∈ A) where

+
  open Lookupᵐ sp-∈

+

+
  unionThese : ⦃ DecEq A ⦄ → (m : Map A B) (m' : Map A C) (x : A)

+
    → x ∈ dom (m ˢ) ∪ dom (m' ˢ) → These B C

+
  unionThese m m' x dp with x ∈? dom (m ˢ) | x ∈? dom (m' ˢ)

+
  ... | yes mr | yes mr' = these (lookupᵐ m x) (lookupᵐ m' x)

+
  ... | yes mr | no  mr' = this  (lookupᵐ m x)

+
  ... | no  mr | yes mr' = that  (lookupᵐ m' x)

+
  ... | no  mr | no  mr' = Sum.[ flip contradiction mr , flip contradiction mr' ]

+
                               (from ∈-∪ dp)

+

+
  unionWith : ⦃ DecEq A ⦄ → (These B C → D) → Map A B → Map A C → Map A D

+
  unionWith f m@(r , p) m'@(r' , p') = m'' , helper

+
     where

+
       d = dom r ∪ dom r'

+
       m'' = map (λ (x , p) → x , f (unionThese m m' x p)) (incl-set d)

+

+
       helper : left-unique m''

+
       helper q q'

+
         with _ , refl , t  ← from ∈-map q

+
         with _ , refl , t' ← from ∈-map q'

+
         with from (∈-mapPartial {f = incl-set' _}) t

+
            | from (∈-mapPartial {f = incl-set' _}) t'

+
       ... | z , _ | z' , _

+
         with z ∈? d in eq | z' ∈? d in eq'

+
       helper _ _ | _ , _ , refl | _ , _ , refl | yes _ | yes _

+
         with refl ← trans (sym eq) eq' = refl

+

+
  module _ {V : Type} ⦃ mon : CommutativeMonoid 0ℓ 0ℓ V ⦄ ⦃ _ : DecEq A ⦄ where

+
    infixr 6 _∪⁺_

+
    open CommutativeMonoid mon

+

+
    _∪⁺_ : Map A V → Map A V → Map A V

+
    _∪⁺_ = unionWith (fold id id _◇_)

+

+
    aggregate₊ : FinSet (A × V) → Map A V

+
    aggregate₊ (_ , l , _) = foldl (λ m x → m ∪⁺ ❴ x ❵ᵐ) ∅ᵐ l

+

+
    module _ {m m' : Map A V} where

+
      ∪dom-lookup : ∃[ a ] a ∈ dom (m ˢ) ∪ dom (m' ˢ) → A × V

+
      ∪dom-lookup (a , a∈) = a , (fold id id _◇_)(unionThese m m' a a∈)

+

+
      dom∪⁺⊆∪dom : dom ((m ∪⁺ m') ˢ) ⊆ dom (m ˢ) ∪ dom (m' ˢ)

+
      dom∪⁺⊆∪dom {a} a∈ = subst (_∈ dom (m ˢ) ∪ dom (m' ˢ))

+
                                  (sym $ proj₁ (×-≡,≡←≡ $ proj₁ (proj₂ ∈-dom∪⁺)))

+
                                  (proj₂ $ proj₁ ∈-dom∪⁺)

+
        where

+
        ∈-dom∪⁺ : ∃[ c ] (a , proj₁ (from dom∈ a∈)) ≡ ∪dom-lookup c

+
                          × c ∈ incl-set (dom (m ˢ) ∪ dom (m' ˢ))

+
        ∈-dom∪⁺ = from ∈-map $ proj₂ $ from dom∈ a∈

+

+
      ∪dom⊆dom∪⁺ : dom (m ˢ) ∪ dom (m' ˢ) ⊆ dom ((m ∪⁺ m') ˢ)

+
      ∪dom⊆dom∪⁺ {a} a∈ with from ∈-map (incl-set-proj₁⊇ a∈)

+
      ... | c' , a≡c₁' , c'∈ =

+
        to dom∈ (proj₂ (∪dom-lookup c') , to ∈-map (c' , ×-≡,≡→≡ (a≡c₁' , refl) , c'∈))

+

+
      dom∪⁺⇔∪dom : ∀ {a} → a ∈ dom ((m ∪⁺ m')ˢ) ⇔ a ∈ dom (m ˢ) ∪ dom (m' ˢ)

+
      dom∪⁺⇔∪dom {a} = mk⇔ dom∪⁺⊆∪dom ∪dom⊆dom∪⁺

+

+
      dom∪⁺≡∪dom : dom ((m ∪⁺ m')ˢ) ≡ᵉ dom (m ˢ) ∪ dom (m' ˢ)

+
      dom∪⁺≡∪dom = to dom∪⁺⇔∪dom , from dom∪⁺⇔∪dom

+

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+
+Axiom.Set.Map
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
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+                   
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+               

+             

+             

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{-# OPTIONS --safe --no-import-sorts #-}

+
{-# OPTIONS -v allTactics:100 #-}

+

+
open import abstract-set-theory.Prelude

+
open import Axiom.Set using (Theory)

+

+
module Axiom.Set.Map (th : Theory) where

+

+
open Theory th renaming (map to mapˢ)

+

+
open import Axiom.Set.Rel th hiding (_∣'_; _∣^'_)

+
open import Axiom.Set.Properties th

+

+
import Data.Sum as ⊎

+
open import Data.List.Ext.Properties using (AllPairs⇒≡∨R∨Rᵒᵖ)

+
open import Data.Product using (swap)

+
open import Data.Product.Properties using (×-≡,≡→≡; ×-≡,≡←≡)

+
open import Data.Maybe.Properties using (just-injective)

+
open import Relation.Unary using (Decidable)

+
import Relation.Binary.Reasoning.Setoid as SetoidReasoning

+

+
open import Reflection.Tactic using (initTac)

+
open import Tactic.AnyOf

+
open import Tactic.Assumption

+
open import Tactic.Defaults

+
open import Tactic.ByEq

+

+
open Equivalence

+

+
-- Because of missing macro hygiene, we have to copy&paste this.

+
-- c.f. https://github.com/agda/agda/issues/3819

+
private

+
  macro

+
    ∈⇒P = anyOfⁿᵗ

+
      (quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻ ∷ [])

+
    P⇒∈ = anyOfⁿᵗ

+
      (quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+
    ∈⇔P = anyOfⁿᵗ

+
      ( quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻

+
      ∷ quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+

+
  variable

+
    A A' B B' C D : Type

+
    R R' : Rel A B

+
    X Y : Set A

+
    a : A

+
    a' : A'

+
    b : B

+
    b' : B'

+

+
  ×-dup : ∀ {ℓ} {A : Type ℓ} → A → A × A

+
  ×-dup x = x , x

+

+
left-unique : Rel A B → Type

+
left-unique R = ∀ {a b b'} → (a , b) ∈ R → (a , b') ∈ R → b ≡ b'

+

+
record IsLeftUnique (R : Rel A B) : Type where

+
  field isLeftUnique : left-unique R

+

+
instance

+
  ∅-left-unique : IsLeftUnique {A = A} {B = B} ∅

+
  ∅-left-unique .IsLeftUnique.isLeftUnique h h' = ⊥-elim $ ∉-∅ h

+

+
⊆-left-unique : R ⊆ R' → left-unique R' → left-unique R

+
⊆-left-unique R⊆R' h = R⊆R' -⟨ h ⟩- R⊆R' -- on isn't dependent enough

+

+
left-unique-mapˢ : {f : A → B} (X : Set A) → left-unique (mapˢ (λ y → (y , f y)) X)

+
left-unique-mapˢ _ p q with from ∈-map p | from ∈-map q

+
... | _ , refl , _ | _ , refl , _ = refl

+

+
Map : Type → Type → Type

+
Map A B = Σ (Rel A B) left-unique

+

+
infix 4 _≡ᵐ_

+

+
_≡ᵐ_ : Map A B → Map A B → Type

+
(x , _) ≡ᵐ (y , _) = x ≡ᵉ y

+

+
private variable m m' : Map A B

+

+
_ˢ : Map A B → Rel A B

+
_ˢ = proj₁

+

+
_ᵐ : (R : Rel A B) → ⦃ IsLeftUnique R ⦄ → Map A B

+
_ᵐ R ⦃ record { isLeftUnique = h } ⦄ = R , h

+

+
⊆-map : (f : Rel A B → Rel A B) → (∀ {R} → f R ⊆ R) → Map A B → Map A B

+
⊆-map f H m = f (m ˢ) , ⊆-left-unique H (proj₂ m)

+

+
ˢ-left-unique : IsLeftUnique (m ˢ)

+
ˢ-left-unique {m = m} = record { isLeftUnique = proj₂ m }

+

+
instance

+
  _ = ˢ-left-unique

+

+
∅ᵐ : Map A B

+
∅ᵐ = _ᵐ ∅ ⦃ ∅-left-unique ⦄

+

+
fromListᵐ : ⦃ _ : DecEq A ⦄ → List (A × B) → Map A B

+
fromListᵐ l = fromList (deduplicate (λ x y → proj₁ x ≟ proj₁ y) l) ,

+
  (λ where (inj₁ refl)     → refl

+
           (inj₂ (inj₁ x)) → ⊥-elim (x refl)

+
           (inj₂ (inj₂ x)) → ⊥-elim (x refl))

+
  ∘₂ (∈⇒P -⟨ AllPairs⇒≡∨R∨Rᵒᵖ

+
           $ deduplicate-! (On.decSetoid (decSetoid _≟_) proj₁) l ⟩- ∈⇒P)

+
  where open import Data.List.Relation.Unary.Unique.DecSetoid.Properties

+
        import Relation.Binary.Construct.On as On

+

+
FinMap : Type → Type → Type

+
FinMap A B = Σ (Rel A B) (λ R → left-unique R × finite R)

+

+
toFinMap : (m : Map A B) → finite (m ˢ) → FinMap A B

+
toFinMap (m , h) fin = m , h , fin

+

+
toMap : FinMap A B → Map A B

+
toMap (m , l , _) = m , l

+

+
toRel : FinMap A B → Rel A B

+
toRel (m , l , _) = m

+

+
module Intersectionᵐ (sp-∈ : spec-∈ (A × B)) where

+
  open Intersection sp-∈

+
  open Intersectionᵖ sp-∈

+

+
  _∩ᵐ_ : Map A B → Map A B → Map A B

+
  m ∩ᵐ m' = (m ˢ ∩ m' ˢ , ⊆-left-unique ∩-⊆ˡ (proj₂ m))

+

+
disj-∪ : (m m' : Map A B) → disjoint (dom (m ˢ)) (dom (m' ˢ)) → Map A B

+
disj-∪ m m' disj = m ˢ ∪ m' ˢ , λ h h' → case ∈⇔P h , ∈⇔P h' of λ where

+
  (inj₁ hm  , inj₁ h'm)  → proj₂ m hm h'm

+
  (inj₂ hm' , inj₁ h'm)  → ⊥-elim $ disj (∈-map⁺'' h'm) (∈-map⁺'' hm')

+
  (inj₁ hm  , inj₂ h'm') → ⊥-elim $ disj (∈-map⁺'' hm)  (∈-map⁺'' h'm')

+
  (inj₂ hm' , inj₂ h'm') → proj₂ m' hm' h'm'

+

+
filterᵐ : {P : Pred (A × B) 0ℓ} → specProperty P → Map A B → Map A B

+
filterᵐ sp-P m = filter sp-P (m ˢ) , ⊆-left-unique filter-⊆ (proj₂ m)

+

+
filterᵐ-finite : {P : A × B → Type} → (sp : specProperty P) → Decidable P

+
  → finite (m ˢ) → finite (filterᵐ sp m ˢ)

+
filterᵐ-finite = filter-finite

+

+
filterKeys : {P : A → Type} → specProperty P → Map A B → Map A B

+
filterKeys sp-P = filterᵐ (sp-∘ sp-P proj₁)

+

+
singletonᵐ : A → B → Map A B

+
singletonᵐ a b = ❴ (a , b) ❵

+
               , (from ∈-singleton -⟨ (λ where refl refl → refl) ⟩- from ∈-singleton)

+

+
❴_❵ᵐ : A × B → Map A B

+
❴ k , v ❵ᵐ = singletonᵐ k v

+

+
disj-dom : ∀ {m m₁ m₂ : Map A B}

+
  → (m ˢ) ≡ (m₁ ˢ) ⨿ (m₂ ˢ)

+
  → disjoint (dom (m₁ ˢ)) (dom (m₂ ˢ))

+
disj-dom {m = m@(_ , uniq)} {m₁} {m₂} (m≡m₁∪m₂ , disj) a∈domm₁ a∈domm₂

+
  with (a , b₁) , (refl , h₁) ← ∈⇔P a∈domm₁

+
  with (_ , b₂) , (refl , h₂) ← ∈⇔P a∈domm₂

+
  = disj (subst _ (uniq (∈mᵢ⇒∈m (inj₁ h₁)) (∈mᵢ⇒∈m (inj₂ h₂))) h₁) h₂

+
  where

+
    ∈mᵢ⇒∈m : ∀ {a} → a ∈ (m₁ ˢ) ⊎ a ∈ (m₂ ˢ) → a ∈ (m ˢ)

+
    ∈mᵢ⇒∈m = proj₂ m≡m₁∪m₂ ∘ to ∈-∪

+

+
InjectiveOn : Set A → (A → B) → Type

+
InjectiveOn X f = ∀ {x y} → x ∈ X → y ∈ X → f x ≡ f y → x ≡ y

+

+
weaken-Injective : ∀ {X : Set A} {f : A → B} → Injective _≡_ _≡_ f → InjectiveOn X f

+
weaken-Injective p _ _ = p

+

+
mapˡ-uniq : {f : A → A'}

+
  → {@(tactic by-eq) inj : InjectiveOn (dom R) f}

+
  → left-unique R

+
  → left-unique (mapˡ f R)

+
mapˡ-uniq {inj = inj} uniq = λ h h' → case ∈⇔P h ,′ ∈⇔P h' of λ where

+
  (((_ , b) , refl , Ha) , ((_ , b') , eqb , Hb)) → uniq Ha

+
    $ subst _ ( sym

+
              $ ×-≡,≡→≡

+
              $ map₁ (inj (to dom∈ (b , Ha)) (to dom∈ (b' , Hb)))

+
                                  (×-≡,≡←≡ eqb))

+
              Hb

+

+
mapʳ-uniq : {f : B → B'} → left-unique R → left-unique (mapʳ f R)

+
mapʳ-uniq uniq = λ h h' → case ∈⇔P h ,′ ∈⇔P h' of λ where

+
  ((_ , refl , Ha) , (_ , refl , Hb)) → cong _ $ uniq Ha Hb

+

+
mapKeys : (f : A → A') → (m : Map A B)

+
  → {@(tactic by-eq) _ : InjectiveOn (dom (m ˢ)) f}

+
  → Map A' B

+
mapKeys f (R , uniq) {inj} = mapˡ f R , mapˡ-uniq {inj = inj} uniq

+

+
mapValues : (B → B') → Map A B → Map A B'

+
mapValues f (R , uniq) = mapʳ f R , mapʳ-uniq uniq

+

+
module Unionᵐ (sp-∈ : spec-∈ A) where

+
  infixr 6 _∪ˡ_

+

+
  _∪ˡ'_ : Rel A B → Rel A B → Rel A B

+
  m ∪ˡ' m' = m ∪ filter (sp-∘ (sp-¬ (sp-∈ {dom m})) proj₁) m'

+

+
  _∪ˡ_ : Map A B → Map A B → Map A B

+
  m ∪ˡ m' = disj-∪ m (filterᵐ (sp-∘ (sp-¬ sp-∈) proj₁) m')

+
      (∈⇔P -⟨ (λ where x (_ , refl , hy) → proj₁ (∈⇔P hy) (∈⇔P x)) ⟩- ∈⇔P)

+

+
  disjoint-∪ˡ-∪ : (H : disjoint (dom R) (dom R')) → R ∪ˡ' R' ≡ᵉ R ∪ R'

+
  disjoint-∪ˡ-∪ disj = from ≡ᵉ⇔≡ᵉ' λ _ → mk⇔

+
    (∈-∪⁺ ∘′ ⊎.map₂ (proj₂ ∘′ ∈⇔P) ∘′ ∈⇔P)

+
    (∈⇔P ∘′ ⊎.map₂ (to ∈-filter ∘′ (λ h → (flip disj (∈-map⁺'' h)) , h)) ∘ ∈⇔P)

+

+
  singleton-∈-∪ˡ : a ∈ dom (m ˢ) → m ∪ˡ ❴ (a , b) ❵ᵐ ≡ᵐ m

+
  singleton-∈-∪ˡ {m = m} a∈domm .proj₁ x with (from ∈-∪ x)

+
  ... | inj₁ h = h

+
  ... | inj₂ h = case from ∈-filter h of λ where

+
    (h₁ , h₂) → ⊥-elim $ h₁ $ subst (λ u → proj₁ u ∈ mapˢ proj₁ (proj₁ m))

+
                                    (sym (from ∈-singleton h₂)) a∈domm

+
  singleton-∈-∪ˡ _ .proj₂ = ∪-⊆ˡ

+

+
  insert : Map A B → A → B → Map A B

+
  insert m a b = ❴ a , b ❵ᵐ ∪ˡ m

+

+
  insertIfJust : ⦃ DecEq A ⦄ → A → Maybe B → Map A B → Map A B

+
  insertIfJust x nothing  m  = m

+
  insertIfJust x (just y) m  = insert m x y

+

+
  disjoint-∪ˡ-mapValues : {M M' : Map A B}

+
                          (f : B → C)

+
                          → (H : disjoint (dom (M ˢ)) (dom (M' ˢ)))

+
                          → (mapValues f (M ∪ˡ M')) ˢ ≡ᵉ (mapValues f M ∪ˡ mapValues f M') ˢ

+
  disjoint-∪ˡ-mapValues {M = M} {M'} f disj = begin

+
    proj₁ (mapValues f (M ∪ˡ M'))

+
    ≈⟨ map-≡ᵉ (disjoint-∪ˡ-∪ disj) ⟩

+
    (mapʳ f ((proj₁ M) ∪ (proj₁ M')))

+
    ≈⟨ map-∪ _ ⟩

+
    (mapʳ f (proj₁ M) ∪ mapʳ f (proj₁ M'))

+
    ≈˘⟨ disjoint-∪ˡ-∪ (λ x₁ x₂ → disj (dom-mapʳ⊆ x₁) (dom-mapʳ⊆ x₂)) ⟩

+
    proj₁ (mapValues f M ∪ˡ mapValues f M')

+
    ∎

+
   where open SetoidReasoning ≡ᵉ-Setoid

+

+
map⦅×-dup⦆-uniq : ∀ {x : Set A} → left-unique (mapˢ ×-dup x)

+
map⦅×-dup⦆-uniq x y with ∈-map⁻' x | ∈-map⁻' y

+
... | fst , refl , _ | .fst , refl , _ = refl

+

+
mapˡ∘map⦅×-dup⦆-uniq : ∀ {S : Set A} {f : A → B}

+
  → {@(tactic by-eq) inj : Injective _≡_ _≡_ f}

+
  → left-unique $ mapˡ f (mapˢ ×-dup S)

+
mapˡ∘map⦅×-dup⦆-uniq {inj = inj} = mapˡ-uniq {inj = λ _ _ → inj} map⦅×-dup⦆-uniq

+

+
idMap : Set A → Map A A

+
idMap s = -, map⦅×-dup⦆-uniq {x = s}

+

+
mapFromFun : (A → B) → Set A → Map A B

+
mapFromFun f s = mapValues f (idMap s)

+

+
mapWithKey-uniq : {f : A → B → B'}

+
  → left-unique R

+
  → left-unique (mapˢ (λ { (x , y) → x , f x y }) R)

+
mapWithKey-uniq {f = f} uniq p q with from ∈-map p | from ∈-map q

+
... | (x , y) , refl , xy∈r | (x' , y') , refl , xy'∈r = cong (f x) (uniq xy∈r xy'∈r)

+

+
mapWithKey : (A → B → B') → Map A B → Map A B'

+
mapWithKey f m@(r , p) = mapˢ (λ { (x , y) → x , f x y}) r , mapWithKey-uniq p

+

+
mapValues-dom : {f : B → C} → dom (m ˢ) ≡ᵉ dom (mapValues f m ˢ)

+
mapValues-dom {m = _ , _} = mapʳ-dom

+

+
_∣'_ : {P : A → Type} → Map A B → specProperty P → Map A B

+
m ∣' P? = filterᵐ (sp-∘ P? proj₁) m

+

+
_∣^'_ : {P : B → Type} → Map A B → specProperty P → Map A B

+
m ∣^' P? = filterᵐ (sp-∘ P? proj₂) m

+

+
constMap : Set A → B → Map A B

+
constMap X b = mapˢ (_, b) X , λ x x₁ →

+
  trans (proj₂ $ ×-≡,≡←≡ $ proj₁ $ proj₂ (∈⇔P x))

+
        (sym $ proj₂ $ ×-≡,≡←≡ $ proj₁ $ proj₂ (∈⇔P x₁))

+

+
constMap-dom : dom (constMap X b ˢ) ≡ᵉ X

+
constMap-dom .proj₁ a∈dom with from dom∈ a∈dom

+
... | b' , ab'∈map with from ∈-map ab'∈map

+
... | a' , ab'≡a'b , a'∈X = subst (_∈ _) (sym (proj₁ (×-≡,≡←≡ ab'≡a'b))) a'∈X

+
constMap-dom .proj₂ a∈X = to dom∈ (_ , to ∈-map (_ , refl , a∈X))

+

+
mapPartialLiftKey-just-uniq : ∀ {f : A → B → Maybe B'}

+
  → left-unique R

+
  → just (a , b)  ∈ mapˢ (mapPartialLiftKey f) R

+
  → just (a , b') ∈ mapˢ (mapPartialLiftKey f) R

+
  → b ≡ b'

+
mapPartialLiftKey-just-uniq {f = f} prop a∈ a'∈ =

+
  let _ , eq  , ax∈r  = mapPartialLiftKey-map {f = f} a∈

+
      _ , eq' , ax'∈r = mapPartialLiftKey-map {f = f} a'∈

+
  in

+
    just-injective $ trans eq (trans (cong (f _) (prop ax∈r ax'∈r)) (sym eq'))

+

+
mapPartial-uniq : ∀ {r : Rel A B} {f : A → B → Maybe B' }

+
  → left-unique r

+
  → left-unique (mapPartial (mapPartialLiftKey f) r)

+
mapPartial-uniq {f = f} prop {a} {b} {b'} p q =

+
  let p = ∈-map′ p

+
      q = ∈-map′ q

+
  in mapPartialLiftKey-just-uniq {f = f} prop (⊆-mapPartial p) (⊆-mapPartial q)

+

+
mapMaybeWithKeyᵐ : (A → B → Maybe B') → Map A B → Map A B'

+
mapMaybeWithKeyᵐ f (rel , prop) = mapMaybeWithKey f rel , mapPartial-uniq {f = f} prop

+

+
mapFromPartialFun : (A → Maybe B) → Set A → Map A B

+
mapFromPartialFun f s = mapMaybeWithKeyᵐ (λ _ → f) (idMap s)

+

+
module Restrictionᵐ (sp-∈ : spec-∈ A) where

+
  private module R = Restriction sp-∈

+
  open Unionᵐ sp-∈

+

+
  _∣_ : Map A B → Set A → Map A B

+
  m ∣ X = ⊆-map (R._∣ X) R.res-⊆ m

+

+
  _∣_ᶜ : Map A B → Set A → Map A B

+
  m ∣ X ᶜ = ⊆-map (R._∣ X ᶜ) R.ex-⊆ m

+

+
  resᵐ-∅ᶜ : {M : Map A B} → (M ∣ ∅ ᶜ) ˢ ≡ᵉ M ˢ

+
  resᵐ-∅ᶜ = R.res-∅ᶜ

+

+
  -- map only values in X

+
  mapValueRestricted : (B → B) → Map A B → Set A → Map A B

+
  mapValueRestricted f m X = mapValues f (m ∣ X) ∪ˡ m

+

+
  -- map only value at a

+
  mapSingleValue : (B → B) → Map A B → A → Map A B

+
  mapSingleValue f m a = mapValueRestricted f m ❴ a ❵

+

+
  curryᵐ : Map (A × B) C → A → Map B C

+
  curryᵐ m a = R.curryʳ (m ˢ) a , λ h h' → proj₂ m (R.∈-curryʳ h) (R.∈-curryʳ h')

+

+
  res-dom∈ : {R : Rel A B} → (a , b) ∈ (m ∣ dom R) ˢ ⇔ ((a , b) ∈ m ˢ × a ∈ dom R)

+
  res-dom∈ = mk⇔ (λ ab∈ → (R.res-⊆ ab∈ , R.res-dom (to dom∈ (_ , ab∈)))) (to ∈-filter ∘ swap)

+

+
  res-subset : R ⊆ m ˢ → (m ∣ dom R) ˢ ≡ᵉ R

+
  res-subset {m = _ , uniq} R⊆m .proj₁ {a , b} ab∈ with from dom∈ (R.res-dom $ to dom∈ (b , ab∈))

+
  ... | b' , ab'∈ = subst (λ u → (a , u) ∈ _) (uniq (R⊆m ab'∈) $ R.res-⊆ ab∈) ab'∈

+
  res-subset {m = m} R⊆m .proj₂ ab∈ = from (res-dom∈ {m = m}) (R⊆m ab∈ , to dom∈ (_ , ab∈))

+

+
  res-singleton : ∀ {k} → k ∈ dom (m ˢ) → ∃[ v ] m ∣ ❴ k ❵ ≡ᵐ ❴ k , v ❵ᵐ

+
  res-singleton {m = m@(_ , uniq)} k∈domm

+
    with (k , v) , (refl , h) ← ∈⇔P k∈domm

+
    = v

+
    , (λ a∈m∣k → to ∈-singleton $ case ∈⇔P a∈m∣k of λ (mem₁ , mem₂) →

+
         let eq = from ∈-singleton mem₁

+
         in  ×-≡,≡→≡ (eq , (uniq mem₂ (subst _ (sym eq) h))))

+
    , λ a∈❴k,v❵ → subst (_∈ ((m ∣ ❴ k ❵) ˢ))

+
                        (sym $ from ∈-singleton a∈❴k,v❵)

+
                        (∈⇔P (to ∈-singleton refl , h))

+

+
  res-singleton' : ∀ {k v} → (k , v) ∈ m ˢ → m ∣ ❴ k ❵ ≡ᵐ ❴ k , v ❵ᵐ

+
  res-singleton' {m = m} kv∈m

+
    with _ , h ← res-singleton {m = m} (∈⇔P (-, (refl , kv∈m)))

+
    = subst _ (sym $ proj₂ m kv∈m (R.res-⊆ $ proj₂ h $ to ∈-singleton refl)) h

+

+
  res-singleton⁺ : {k : A}{v : B} → (k , v) ∈ m ˢ → (k , v) ∈ (m ∣ ❴ k ❵)ˢ

+
  res-singleton⁺ kv∈m = to ∈-filter ((to ∈-singleton refl) , kv∈m)

+

+
  res-singleton-inhabited : ∀ {k a} → a ∈ (m ∣ ❴ k ❵) ˢ → k ∈ dom (m ˢ)

+
  res-singleton-inhabited {m = m} {k} {a} a∈ =

+
    to dom∈ ( proj₂ a , subst (λ x → (x , proj₂ a) ∈ (m ˢ))

+
                                (from ∈-singleton (R.res-dom (∈-dom a∈)))

+
                                (R.res-⊆ a∈) )

+

+
  res-singleton'' : ∀ {k a} → a ∈ (m ∣ ❴ k ❵)ˢ → ∃[ v ] a ≡ (k , v)

+
  res-singleton'' {m = m}{k}{a} a∈m =

+
    let (v , m|≡) = res-singleton {m = m} (res-singleton-inhabited{m = m} a∈m) in

+
    v , from ∈-singleton (proj₁ m|≡ a∈m)

+

+
  -- f(x,-)

+
  infix 30 _⦅_,-⦆

+
  _⦅_,-⦆ = curryᵐ

+

+
  update : A → Maybe B → Map A B → Map A B

+
  update x (just y) m = insert m x y

+
  update x nothing  m = m ∣ ❴ x ❵ ᶜ

+

+
module Lookupᵐ (sp-∈ : spec-∈ A) where

+
  open import Relation.Nullary.Decidable

+
  private module R = Restriction sp-∈

+
  open Unionᵐ sp-∈

+
  open Restriction sp-∈

+

+
  module _ (m : Map A B) (x : A) where

+
    lookupᵐ : {@(tactic initTac assumption') _ : x ∈ dom (m ˢ)} → B

+
    lookupᵐ {h} = proj₁ (from dom∈ h)

+

+
    lookupᵐ? : ⦃ (x ∈ dom (m ˢ)) ⁇ ⦄ → Maybe B

+
    lookupᵐ? ⦃ ⁇ no  _ ⦄ = nothing

+
    lookupᵐ? ⦃ ⁇ yes _ ⦄ = just lookupᵐ

+

+
  pullbackMap : (m : Map A B) → ⦃ ∀ {x} → (x ∈ dom (m ˢ)) ⁇ ⦄ → (A' → A) → Set A' → Map A' B

+
  pullbackMap m f s = mapMaybeWithKeyᵐ (λ a _ → lookupᵐ? m (f a)) (idMap s)

+

+
module Corestrictionᵐ (sp-∈ : spec-∈ B) where

+
  private module R = Corestriction sp-∈

+

+
  _∣^_ : Map A B → Set B → Map A B

+
  m ∣^ X = ⊆-map (R._∣^ X) R.cores-⊆ m

+

+
  _∣^_ᶜ : Map A B → Set B → Map A B

+
  m ∣^ X ᶜ = ⊆-map (R._∣^ X ᶜ) R.coex-⊆ m

+

+
  -- f⁻¹(x)

+
  infix 25 _⁻¹_

+
  _⁻¹_ : Map A B → B → Set A

+
  m ⁻¹ a = dom ((m ∣^ ❴ a ❵) ˢ)

+

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+
+Axiom.Set.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
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{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import Axiom.Set using (Theory)

+

+
module Axiom.Set.Properties {ℓ} (th : Theory {ℓ}) where

+

+
open import abstract-set-theory.Prelude hiding (isEquivalence; trans; map; map₂)

+
open Theory th

+

+
import Data.List as L

+
import Data.List.Relation.Unary.All as All

+
import Data.List.Relation.Unary.AllPairs as AllPairs

+
import Data.Sum as ⊎

+
import Function.Related.Propositional as R

+
import Relation.Binary.Lattice.Properties.BoundedJoinSemilattice as Bounded∨Semilattice

+
import Relation.Binary.Lattice.Properties.JoinSemilattice as ∨Semilattice

+
import Relation.Nullary.Decidable

+
open import Data.List.Ext.Properties using (_×-cong_; _⊎-cong_)

+
open import Data.List.Membership.DecPropositional using () renaming (_∈?_ to _∈ˡ?_)

+
open import Data.List.Membership.Propositional.Properties using (∈-filter⁺; ∈-filter⁻; ∈-++⁺ˡ; ∈-++⁺ʳ; ∈-++⁻)

+
open import Data.List.Relation.Binary.BagAndSetEquality using (∼bag⇒↭)

+
open import Data.List.Relation.Binary.Permutation.Propositional.Properties using (↭-length)

+
open import Data.List.Relation.Binary.Subset.Propositional using () renaming (_⊆_ to _⊆ˡ_)

+
open import Data.List.Relation.Unary.Any using (here; there)

+
open import Data.List.Relation.Unary.Unique.Propositional.Properties.WithK using (unique∧set⇒bag)

+
open import Data.Product using (map₂; swap)

+
open import Data.Product.Properties.Ext

+
open import Relation.Binary hiding (_⇔_)

+
open import Relation.Binary.Lattice

+
open import Relation.Binary.Morphism using (IsOrderHomomorphism)

+

+
open Equivalence

+

+
private variable

+
  A B C : Type ℓ

+
  X Y Z : Set A

+

+
module _ {f : A → B} {X} {b} where

+
  ∈-map⁻' : b ∈ map f X → (∃[ a ] b ≡ f a × a ∈ X)

+
  ∈-map⁻' = from ∈-map

+

+
  ∈-map⁺' : (∃[ a ] b ≡ f a × a ∈ X) → b ∈ map f X

+
  ∈-map⁺' = to ∈-map

+

+
∈-map⁺'' : ∀ {f : A → B} {X} {a} → a ∈ X → f a ∈ map f X

+
∈-map⁺'' h = to ∈-map (-, refl , h)

+

+
module _ {X : Set A} {P : A → Type} {sp-P : specProperty P} {a} where

+
  ∈-filter⁻' : a ∈ filter sp-P X → (P a × a ∈ X)

+
  ∈-filter⁻' = from ∈-filter

+

+
  ∈-filter⁺' : (P a × a ∈ X) → a ∈ filter sp-P X

+
  ∈-filter⁺' = to ∈-filter

+

+
module _ {X Y : Set A} {a} where

+
  ∈-∪⁻ : a ∈ X ∪ Y → a ∈ X ⊎ a ∈ Y

+
  ∈-∪⁻ = from ∈-∪

+

+
  ∈-∪⁺ : a ∈ X ⊎ a ∈ Y → a ∈ X ∪ Y

+
  ∈-∪⁺ = to ∈-∪

+

+
module _ {l : List A} {a} where

+
  ∈-fromList⁻ : a ∈ fromList l → a ∈ˡ l

+
  ∈-fromList⁻ = from ∈-fromList

+

+
  ∈-fromList⁺ : a ∈ˡ l → a ∈ fromList l

+
  ∈-fromList⁺ = to ∈-fromList

+

+
open import Tactic.AnyOf

+
open import Tactic.Defaults

+

+
-- Because of missing macro hygiene, we have to copy&paste this.

+
-- c.f. https://github.com/agda/agda/issues/3819

+
private macro

+
  ∈⇒P = anyOfⁿᵗ

+
    (quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻ ∷ [])

+
  P⇒∈ = anyOfⁿᵗ

+
    (quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+
  ∈⇔P = anyOfⁿᵗ

+
    ( quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻

+
    ∷ quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+

+
_≡_⨿_ : Set A → Set A → Set A → Type ℓ

+
X ≡ Y ⨿ Z = X ≡ᵉ Y ∪ Z × disjoint Y Z

+

+
-- FIXME: proving this has some weird issues when making a implicit in

+
-- in the definiton of _≡ᵉ'_

+
≡ᵉ⇔≡ᵉ' : X ≡ᵉ Y ⇔ X ≡ᵉ' Y

+
≡ᵉ⇔≡ᵉ' = mk⇔

+
  (λ where (X⊆Y , Y⊆X) _ → mk⇔ X⊆Y Y⊆X)

+
  (λ a∈X⇔a∈Y → (λ {_} → to (a∈X⇔a∈Y _)) , λ {_} → from (a∈X⇔a∈Y _))

+

+
cong-⊆⇒cong : {f : Set A → Set B} → f Preserves _⊆_ ⟶ _⊆_ → f Preserves _≡ᵉ_ ⟶ _≡ᵉ_

+
cong-⊆⇒cong h X≡ᵉX' = h (proj₁ X≡ᵉX') , h (proj₂ X≡ᵉX')

+

+
cong-⊆⇒cong₂ : {f : Set A → Set B → Set C}

+
             → f Preserves₂ _⊆_ ⟶ _⊆_ ⟶ _⊆_ → f Preserves₂ _≡ᵉ_ ⟶ _≡ᵉ_ ⟶ _≡ᵉ_

+
cong-⊆⇒cong₂ h X≡ᵉX' Y≡ᵉY' = h (proj₁ X≡ᵉX') (proj₁ Y≡ᵉY')

+
                           , h (proj₂ X≡ᵉX') (proj₂ Y≡ᵉY')

+

+
⊆-Transitive : Transitive (_⊆_ {A})

+
⊆-Transitive X⊆Y Y⊆Z = Y⊆Z ∘ X⊆Y

+

+
≡ᵉ-isEquivalence : IsEquivalence (_≡ᵉ_ {A})

+
≡ᵉ-isEquivalence = record

+
  { refl  = id , id

+
  ; sym   = λ where (h , h') → (h' , h)

+
  ; trans = λ eq₁ eq₂ → ⊆-Transitive (proj₁ eq₁) (proj₁ eq₂)

+
                      , ⊆-Transitive (proj₂ eq₂) (proj₂ eq₁)

+
  }

+

+
≡ᵉ-Setoid : ∀ {A} → Setoid ℓ ℓ

+
≡ᵉ-Setoid {A} = record

+
  { Carrier       = Set A

+
  ; _≈_           = _≡ᵉ_

+
  ; isEquivalence = ≡ᵉ-isEquivalence

+
  }

+

+
⊆-isPreorder : IsPreorder (_≡ᵉ_ {A}) _⊆_

+
⊆-isPreorder = λ where

+
  .isEquivalence → ≡ᵉ-isEquivalence

+
  .reflexive     → proj₁

+
  .trans         → ⊆-Transitive

+
    where open IsPreorder

+

+
⊆-Preorder : {A} → Preorder _ _ _

+
⊆-Preorder {A} = record

+
  { Carrier = Set A ; _≈_ = _≡ᵉ_ ; _≲_ = _⊆_ ; isPreorder = ⊆-isPreorder }

+

+
⊆-PartialOrder : IsPartialOrder (_≡ᵉ_ {A}) _⊆_

+
⊆-PartialOrder = record

+
  { isPreorder = ⊆-isPreorder

+
  ; antisym    = _,_ }

+

+
∈-× : {a : A} {b : B} → (a , b) ∈ X → (a ∈ map proj₁ X × b ∈ map proj₂ X)

+
∈-× {a = a} {b} x = to ∈-map ((a , b) , refl , x) , to ∈-map ((a , b) , refl , x)

+

+
module _ {f : A → B} {g : B → C} where

+
  map-⊆∘ : map g (map f X) ⊆ map (g ∘ f) X

+
  map-⊆∘ a∘∈

+
    with b , a≡gb , b∈prfX ← from ∈-map a∘∈

+
    with a , refl , a∈X    ← from ∈-map b∈prfX

+
    = to ∈-map (a , a≡gb , a∈X)

+

+
  map-∘⊆ : map (g ∘ f) X ⊆ map g (map f X)

+
  map-∘⊆ a∈∘ with from ∈-map a∈∘

+
  ... | a₁ , a₁≡gfa , a₁∈X = to ∈-map (f a₁ , a₁≡gfa , to ∈-map (a₁ , refl , a₁∈X))

+

+
  map-∘ : map g (map f X) ≡ᵉ map (g ∘ f) X

+
  map-∘ = map-⊆∘ , map-∘⊆

+

+
  ∈-map⁺-∘ : ∀ {b} → b ∈ map f X → g b ∈ map (g ∘ f) X

+
  ∈-map⁺-∘ = map-⊆∘ ∘ ∈-map⁺''

+

+
map-⊆ : {X Y : Set A} {f : A → B} → X ⊆ Y → map f X ⊆ map f Y

+
map-⊆ x⊆y a∈map with from ∈-map a∈map

+
... | a₁ , a≡fa₁ , a₁∈x = to ∈-map (a₁ , a≡fa₁ , x⊆y a₁∈x)

+

+
map-≡ᵉ : {X Y : Set A} {f : A → B} → X ≡ᵉ Y → map f X ≡ᵉ map f Y

+
map-≡ᵉ (x⊆y , y⊆x) = map-⊆ x⊆y , map-⊆ y⊆x

+

+
∉-∅ : {a : A} → a ∉ ∅

+
∉-∅ h = case ∈⇔P h of λ ()

+

+
∅-minimum : Minimum (_⊆_ {A}) ∅

+
∅-minimum = λ _ → ⊥-elim ∘ ∉-∅

+

+
∅-least : X ⊆ ∅ → X ≡ᵉ ∅

+
∅-least X⊆∅ = (X⊆∅ , ∅-minimum _)

+

+
∅-weakly-finite : weakly-finite {A = A} ∅

+
∅-weakly-finite = [] , ⊥-elim ∘ ∉-∅

+

+
∅-finite : finite {A = A} ∅

+
∅-finite = [] , mk⇔ (⊥-elim ∘ ∉-∅) λ ()

+

+
map-∅ : {X : Set A} {f : A → B} → map f ∅ ≡ᵉ ∅

+
map-∅ = ∅-least λ x∈map → case ∈-map⁻' x∈map of λ where (_ , _ , h) → ⊥-elim (∉-∅ h)

+

+
map-∪ : {X Y : Set A} → (f : A → B) → map f (X ∪ Y) ≡ᵉ map f X ∪ map f Y

+
map-∪ {X = X} {Y} f = from ≡ᵉ⇔≡ᵉ' λ b →

+
    b ∈ map f (X ∪ Y)

+
      ∼⟨ R.SK-sym ∈-map ⟩

+
    (∃[ a ] b ≡ f a × a ∈ X ∪ Y)

+
      ∼⟨ ∃-cong′ (R.K-refl ×-cong R.SK-sym ∈-∪) ⟩

+
    (∃[ a ] b ≡ f a × (a ∈ X ⊎ a ∈ Y))

+
      ↔⟨ ∃-cong′ ×-distribˡ-⊎' ⟩

+
    (∃[ a ] (b ≡ f a × a ∈ X ⊎ b ≡ f a × a ∈ Y))

+
      ↔⟨ ∃-distrib-⊎ ⟩

+
    (∃[ a ] b ≡ f a × a ∈ X ⊎ ∃[ a ] b ≡ f a × a ∈ Y)

+
      ∼⟨ ∈-map ⊎-cong ∈-map ⟩

+
    (b ∈ map f X ⊎ b ∈ map f Y)

+
      ∼⟨ ∈-∪ ⟩

+
    b ∈ map f X ∪ map f Y ∎

+
  where open R.EquationalReasoning

+

+
mapPartial-∅ : {f : A → Maybe B} → mapPartial f ∅ ≡ᵉ ∅

+
mapPartial-∅ {f = f} = ∅-least λ x∈map → case from (∈-mapPartial {f = f}) x∈map of λ where

+
  (_ , h , _) → ⊥-elim (∉-∅ h)

+

+
singleton-strongly-finite : {a : A} → strongly-finite (singleton a)

+
singleton-strongly-finite = _ , All.[] AllPairs.∷ AllPairs.[] , R.SK-sym ∈-fromList

+

+
card-singleton : {a : A} → card (singleton a , singleton-strongly-finite) ≡ 1

+
card-singleton = refl

+

+
card-≡ᵉ : (X Y : Σ (Set A) strongly-finite) → proj₁ X ≡ᵉ proj₁ Y → card X ≡ card Y

+
card-≡ᵉ (X , lX , lXᵘ , eqX) (Y , lY , lYᵘ , eqY) X≡Y =

+
  ↭-length $ ∼bag⇒↭ $ unique∧set⇒bag lXᵘ lYᵘ λ {a} →

+
    a ∈ˡ lX  ∼⟨ R.SK-sym eqX ⟩

+
    a ∈ X    ∼⟨ to ≡ᵉ⇔≡ᵉ' X≡Y a ⟩

+
    a ∈ Y    ∼⟨ eqY ⟩

+
    a ∈ˡ lY  ∎

+
  where open R.EquationalReasoning

+

+
singleton-≢-∅ : {a : A} → singleton a ≢ ∅

+
singleton-≢-∅ {A = A} {a = a} X≡∅ =

+
  case card-≡ᵉ (singleton a , singleton-strongly-finite)

+
               (∅ , ∅-strongly-finite)

+
               (subst (singleton a ≡ᵉ_) X≡∅  ≡ᵉ-refl)

+
    of λ ()

+
  where open IsEquivalence (≡ᵉ-isEquivalence {A}) renaming (refl to ≡ᵉ-refl)

+

+
module _ {P : A → Type} {sp-P : specProperty P} where

+

+
  filter-∅ : (∀ a → a ∈ X → ¬ P a) → filter sp-P X ≡ᵉ ∅

+
  proj₁ (filter-∅ h) {a} h' with from ∈-filter h'

+
  ... | (Pa , a∈) = ⊥-elim (h a a∈ Pa)

+
  proj₂ (filter-∅ h) {a} h' = ⊥-elim (∉-∅ h')

+

+
  filter-⊆ : filter sp-P X ⊆ X

+
  filter-⊆ = proj₂ ∘′ ∈⇔P

+

+
  filter-pres-⊆ : X ⊆ Y → filter sp-P X ⊆ filter sp-P Y

+
  filter-pres-⊆ xy a∈ = let Pa∈ = from ∈-filter a∈ in

+
    to ∈-filter (map₂ xy Pa∈)

+

+
  filter-pres-≡ᵉ : X ≡ᵉ Y → filter sp-P X ≡ᵉ filter sp-P Y

+
  filter-pres-≡ᵉ (X⊆Y , Y⊆X) = filter-pres-⊆ X⊆Y , filter-pres-⊆ Y⊆X

+

+
  filter-split-∪ : ∀ {a} → a ∈ filter sp-P (X ∪ Y) → (P a × a ∈ X) ⊎ (P a × a ∈ Y)

+
  filter-split-∪ a∈ = case (proj₁ (from ∈-filter a∈) , from ∈-∪ (proj₂ (from ∈-filter a∈))) of

+
    λ where

+
      (Pa , inj₁ a∈X) → inj₁ (Pa , a∈X)

+
      (Pa , inj₂ a∈Y) → inj₂ (Pa , a∈Y)

+

+
  filter-hom-⊆ : filter sp-P (X ∪ Y) ⊆ filter sp-P X ∪ filter sp-P Y

+
  filter-hom-⊆ {a = a} a∈ = to ∈-∪ (case filter-split-∪ a∈ of λ where

+
    (inj₁ v) → inj₁ (to ∈-filter v)

+
    (inj₂ v) → inj₂ (to ∈-filter v))

+

+
  filter-hom-⊇ : filter sp-P X ∪ filter sp-P Y ⊆ filter sp-P (X ∪ Y)

+
  filter-hom-⊇ a∈ = to ∈-filter (case (from ∈-∪ a∈) of λ where

+
       (inj₁ v) → proj₁ (from ∈-filter v) , to ∈-∪ (inj₁ (proj₂ (from ∈-filter v)))

+
       (inj₂ v) → proj₁ (from ∈-filter v) , to ∈-∪ (inj₂ (proj₂ (from ∈-filter v))) )

+

+
  filter-hom-∪ : filter sp-P (X ∪ Y) ≡ᵉ (filter sp-P X) ∪ (filter sp-P Y)

+
  filter-hom-∪ = filter-hom-⊆ , filter-hom-⊇

+

+
Dec-∈-fromList : ∀ {a : A} → ⦃ DecEq A ⦄ → (l : List A) → Decidable¹ (_∈ fromList l)

+
Dec-∈-fromList _ _ = Relation.Nullary.Decidable.map ∈-fromList (_∈ˡ?_ _≟_ _ _)

+

+
Dec-∈-singleton : ∀ {a : A} → ⦃ DecEq A ⦄ → Decidable¹ (_∈ ❴ a ❵)

+
Dec-∈-singleton _ = Relation.Nullary.Decidable.map ∈-singleton (_ ≟ _)

+

+
singleton-finite : ∀ {a : A} → finite ❴ a ❵

+
singleton-finite {a = a} = [ a ] , λ {x} →

+
  x ∈ ❴ a ❵  ∼⟨ R.SK-sym ∈-fromList ⟩

+
  x ∈ˡ [ a ] ∎

+
  where open R.EquationalReasoning

+

+
filter-finite : ∀ {P : A → Type}

+
              → (sp : specProperty P) → Decidable¹ P → finite X → finite (filter sp X)

+
filter-finite {X = X} {P} sp P? (l , hl) = L.filter P? l , λ {a} →

+
  a ∈ filter sp X            ∼⟨ R.SK-sym ∈-filter ⟩

+
  (P a × a ∈ X)              ∼⟨ R.K-refl ×-cong hl ⟩

+
  (P a × a ∈ˡ l)             ∼⟨ mk⇔ (uncurry $ flip $ ∈-filter⁺ P?)

+
                                    (swap ∘ ∈-filter⁻ P?) ⟩

+
  a ∈ˡ L.filter P? l ∎

+
  where open R.EquationalReasoning

+

+
∪-⊆ˡ : X ⊆ X ∪ Y

+
∪-⊆ˡ = ∈⇔P ∘′ inj₁

+

+
∪-⊆ʳ : Y ⊆ X ∪ Y

+
∪-⊆ʳ = ∈⇔P ∘′ inj₂

+

+
∪-⊆ : X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z

+
∪-⊆ X⊆Z Y⊆Z = λ a∈X∪Y → [ X⊆Z , Y⊆Z ]′ (∈⇔P a∈X∪Y)

+

+
⊆→∪ : X ⊆ Y → X ∪ Y ≡ᵉ Y

+
⊆→∪ X⊆Y = (λ {a} x → case from ∈-∪ x of λ where

+
            (inj₁ v) → X⊆Y v

+
            (inj₂ v) → v) , ∪-⊆ʳ

+

+
∪-Supremum : Supremum (_⊆_ {A}) _∪_

+
∪-Supremum _ _ = ∪-⊆ˡ , ∪-⊆ʳ , λ _ → ∪-⊆

+

+
∪-cong-⊆ : _∪_ {A} Preserves₂ _⊆_ ⟶ _⊆_ ⟶ _⊆_

+
∪-cong-⊆ X⊆X' Y⊆Y' = ∈⇔P ∘′ (⊎.map X⊆X' Y⊆Y') ∘′ ∈⇔P

+

+
∪-cong : _∪_ {A} Preserves₂ _≡ᵉ_ ⟶ _≡ᵉ_ ⟶ _≡ᵉ_

+
∪-cong = cong-⊆⇒cong₂ ∪-cong-⊆

+

+
∪-preserves-finite : _∪_ {A} Preservesˢ₂ finite

+
∪-preserves-finite {a = X} {Y} (l , hX) (l' , hY) = (l ++ l') , λ {a} →

+
  a ∈ X ∪ Y          ∼⟨ R.SK-sym ∈-∪ ⟩

+
  (a ∈ X ⊎ a ∈ Y)    ∼⟨ hX ⊎-cong hY ⟩

+
  (a ∈ˡ l ⊎ a ∈ˡ l') ∼⟨ mk⇔ ⊎.[ ∈-++⁺ˡ , ∈-++⁺ʳ _ ] (∈-++⁻ _) ⟩

+
  a ∈ˡ l ++ l'       ∎

+
  where open R.EquationalReasoning

+

+
∪-sym : X ∪ Y ≡ᵉ Y ∪ X

+
∪-sym = ∪-⊆ ∪-⊆ʳ ∪-⊆ˡ , ∪-⊆ ∪-⊆ʳ ∪-⊆ˡ

+

+
Set-JoinSemilattice : IsJoinSemilattice (_≡ᵉ_ {A}) _⊆_ _∪_

+
Set-JoinSemilattice = record

+
  { isPartialOrder = ⊆-PartialOrder ; supremum = ∪-Supremum }

+

+
Set-BoundedJoinSemilattice : IsBoundedJoinSemilattice (_≡ᵉ_ {A}) _⊆_ _∪_ ∅

+
Set-BoundedJoinSemilattice = record

+
  { isJoinSemilattice = Set-JoinSemilattice ; minimum = ∅-minimum }

+

+
Set-BddSemilattice : {A : Type ℓ} → BoundedJoinSemilattice _ _ _

+
Set-BddSemilattice {A} = record

+
                      { Carrier = Set A

+
                      ; _≈_ = _≡ᵉ_ {A}

+
                      ; _≤_ = _⊆_

+
                      ; _∨_ = _∪_

+
                      ; ⊥ = ∅

+
                      ; isBoundedJoinSemilattice = Set-BoundedJoinSemilattice

+
                      }

+

+
module _ {A : Type ℓ} where

+
  open Bounded∨Semilattice (Set-BddSemilattice {A})

+
  open ∨Semilattice (BoundedJoinSemilattice.joinSemilattice (Set-BddSemilattice {A}))

+
    using (∨-comm; ∨-assoc)

+

+
  ∪-identityˡ : (X : Set A) → ∅ ∪ X ≡ᵉ X

+
  ∪-identityˡ = identityˡ

+

+
  ∪-identityʳ : (X : Set A) → X ∪ ∅ ≡ᵉ X

+
  ∪-identityʳ = identityʳ

+

+
  ∪-comm : (X Y : Set A) → X ∪ Y ≡ᵉ Y ∪ X

+
  ∪-comm = ∨-comm

+

+
  ∪-assoc : (X Y Z : Set A) → (X ∪ Y) ∪ Z ≡ᵉ X ∪ (Y ∪ Z)

+
  ∪-assoc = ∨-assoc

+

+
fromList-∪-singleton : {A : Type ℓ} {x : A} {l : List A} → fromList (x ∷ l) ≡ᵉ ❴ x ❵ ∪ fromList l

+
fromList-∪-singleton .proj₁ h with from ∈-fromList h

+
... | here refl = ∈-∪⁺ (inj₁ (to ∈-fromList (here refl)))

+
... | there q = ∈-∪⁺ (inj₂ (to ∈-fromList q))

+
fromList-∪-singleton .proj₂ h with ∈-∪⁻ h

+
... | (inj₁ a∈) = to ∈-fromList (here (from ∈-singleton a∈))

+
... | (inj₂ a∈) = to ∈-fromList (there (from ∈-fromList a∈))

+

+
∪-fromList-++ : (ll lr : List A) → fromList ll ∪ fromList lr ≡ᵉ fromList (ll ++ lr)

+
∪-fromList-++ [] lr = ∪-identityˡ (fromList lr)

+
∪-fromList-++ (x ∷ l) lr =

+
  begin

+
  fromList (x ∷ l) ∪ fromList lr      ≈⟨ ∪-cong fromList-∪-singleton ≡ᵉ.refl ⟩

+
  (❴ x ❵ ∪ fromList l) ∪ fromList lr  ≈⟨ ∪-assoc ❴ x ❵ (fromList l) (fromList lr) ⟩

+
  ❴ x ❵ ∪ (fromList l ∪ fromList lr)  ≈⟨ ∪-cong ≡ᵉ.refl (∪-fromList-++ l lr) ⟩

+
  ❴ x ❵ ∪ fromList (l ++ lr)          ≈˘⟨ fromList-∪-singleton ⟩

+
  fromList (x ∷ (l ++ lr))            ∎

+
  where

+
  module ≡ᵉ = IsEquivalence (≡ᵉ-isEquivalence)

+
  open import Relation.Binary.Reasoning.Setoid ≡ᵉ-Setoid

+

+
disjoint-sym : disjoint X Y → disjoint Y X

+
disjoint-sym disj = flip disj

+

+
disjoint-subst : disjoint X Y → X ≡ᵉ Z → disjoint Z Y

+
disjoint-subst disj X≡Z = disj ∘ (proj₂ X≡Z)

+

+
module Intersectionᵖ (sp-∈ : spec-∈ A) where

+
  open Intersection sp-∈

+

+
  disjoint⇒disjoint' : disjoint X Y → disjoint' X Y

+
  disjoint⇒disjoint' h = ∅-least (⊥-elim ∘ uncurry h ∘ from ∈-∩)

+

+
  disjoint'⇒disjoint : disjoint' X Y → disjoint X Y

+
  disjoint'⇒disjoint h a∈X a∈Y = ∉-∅ (to (to ≡ᵉ⇔≡ᵉ' h _) (to ∈-∩ (a∈X , a∈Y)))

+

+
  ∩-⊆ˡ : X ∩ Y ⊆ X

+
  ∩-⊆ˡ = proj₁ ∘ from ∈-∩

+

+
  ∩-⊆ʳ : X ∩ Y ⊆ Y

+
  ∩-⊆ʳ = proj₂ ∘ from ∈-∩

+

+
  ∩-⊆ : Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y

+
  ∩-⊆ Z⊆X Z⊆Y = λ x∈Z → to ∈-∩ (< Z⊆X , Z⊆Y > x∈Z)

+

+
  ∩-Infimum : Infimum _⊆_ _∩_

+
  ∩-Infimum X Y = ∩-⊆ˡ , ∩-⊆ʳ , λ _ → ∩-⊆

+

+
  ∩-preserves-finite : _∩_ Preservesˢ₂ weakly-finite

+
  ∩-preserves-finite _ = ⊆-weakly-finite ∩-⊆ʳ

+

+
  ∩-cong-⊆ : _∩_ Preserves₂ _⊆_ ⟶ _⊆_ ⟶ _⊆_

+
  ∩-cong-⊆ X⊆X' Y⊆Y' a∈X∩Y = to ∈-∩ (Data.Product.map X⊆X' Y⊆Y' (from ∈-∩ a∈X∩Y))

+

+
  ∩-cong : _∩_ Preserves₂ _≡ᵉ_ ⟶ _≡ᵉ_ ⟶ _≡ᵉ_

+
  ∩-cong = cong-⊆⇒cong₂ ∩-cong-⊆

+

+
  ∩-OrderHomomorphismʳ : ∀ {X} → IsOrderHomomorphism _≡ᵉ_ _≡ᵉ_ _⊆_ _⊆_ (X ∩_)

+
  ∩-OrderHomomorphismʳ = record { cong = ∩-cong (id , id) ; mono = ∩-cong-⊆ id }

+

+
  ∩-OrderHomomorphismˡ : ∀ {X} → IsOrderHomomorphism _≡ᵉ_ _≡ᵉ_ _⊆_ _⊆_ (_∩ X)

+
  ∩-OrderHomomorphismˡ = record

+
    { cong = flip ∩-cong (id , id) ; mono = flip ∩-cong-⊆ id }

+

+
  Set-Lattice : IsLattice _≡ᵉ_ _⊆_ _∪_ _∩_

+
  Set-Lattice = record

+
    { isPartialOrder = ⊆-PartialOrder ; supremum = ∪-Supremum ; infimum = ∩-Infimum }

+

+
  ∩-sym⊆ : X ∩ Y ⊆ Y ∩ X

+
  ∩-sym⊆ a∈X∩Y with from ∈-∩ a∈X∩Y

+
  ... | a∈X , a∈Y = to ∈-∩ (a∈Y , a∈X)

+

+
  ∩-sym : X ∩ Y ≡ᵉ Y ∩ X

+
  ∩-sym = ∩-sym⊆ , ∩-sym⊆

+

+
-- Additional properties of lists and sets.

+
module _ {L : List A} where

+
  open Equivalence

+

+
  sublist-⇔ : {l : List A} → fromList l ⊆ fromList L ⇔ l ⊆ˡ L

+
  sublist-⇔ {[]} = mk⇔ (λ x ()) (λ _ {_} → ⊥-elim ∘ ∉-∅)

+
  sublist-⇔ {x ∷ xs} = mk⇔ onlyif (λ u → to ∈-fromList ∘ u ∘ from ∈-fromList)

+
    where

+
    onlyif : ({a : A} → a ∈ fromList (x ∷ xs) → a ∈ fromList L) → x ∷ xs ⊆ˡ L

+
    onlyif h (here refl) = from ∈-fromList (h (to ∈-fromList (here refl)))

+
    onlyif h (there x'∈) = from ∈-fromList (h (to ∈-fromList (there x'∈)))

+

+
  ∃-sublist-⇔ : {P : Pred (List A) ℓ} → (∃[ l ] fromList l ⊆ fromList L × P l) ⇔ (∃[ l ] l ⊆ˡ L × P l)

+
  ∃-sublist-⇔ = mk⇔ (λ (l , l⊆L , Pl) → l , to sublist-⇔ l⊆L , Pl)

+
                    (λ (l , l⊆L , Pl) → l , from sublist-⇔ l⊆L , Pl)

+

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+
+Axiom.Set.Rel
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

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+             
{-# OPTIONS --safe --no-import-sorts #-}

+
{-# OPTIONS -v allTactics:100 #-}

+

+
open import abstract-set-theory.Prelude hiding (map)

+
open import Axiom.Set using (Theory)

+

+
module Axiom.Set.Rel (th : Theory) where

+

+
import Relation.Binary.Reasoning.Setoid as SetoidReasoning

+
import Function.Related.Propositional as R

+

+
open Theory th

+
open import Axiom.Set.Properties th

+

+
import Data.Product as ×

+
open import Data.List.Ext.Properties using (_⊎-cong_)

+
open import Data.Maybe.Base using () renaming (map to map?)

+
open import Data.Product.Properties using (,-injectiveˡ; ×-≡,≡→≡; ×-≡,≡←≡)

+
open import Data.Product.Properties.Ext using (∃-cong′; ∃-distrib-⊎)

+
open import Relation.Unary using (Decidable)

+
open import Relation.Binary using (_Preserves_⟶_)

+

+
open import Tactic.AnyOf

+
open import Tactic.Defaults

+

+
open Equivalence

+

+
-- Because of missing macro hygiene, we have to copy&paste this.

+
-- c.f. https://github.com/agda/agda/issues/3819

+
private macro

+
  ∈⇒P = anyOfⁿᵗ

+
    (quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻ ∷ [])

+
  P⇒∈ = anyOfⁿᵗ

+
    (quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+
  ∈⇔P = anyOfⁿᵗ

+
    ( quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻

+
    ∷ quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+

+
Rel : Type → Type → Type

+
Rel A B = Set (A × B)

+

+
private variable A A' B B' C : Type

+
                 R R' : Rel A B

+
                 X : Set A

+

+
relatedˡ : Rel A B → Set A

+
relatedˡ = map proj₁

+

+
∅ʳ : Rel A B

+
∅ʳ = ∅

+

+
dom : Rel A B → Set A

+
dom = map proj₁

+

+
range : Rel A B → Set B

+
range = map proj₂

+

+
disjoint-dom⇒disjoint : disjoint (dom R) (dom R') → disjoint R R'

+
disjoint-dom⇒disjoint disj = ∈-map⁺'' -⟨ disj ⟩- ∈-map⁺''

+

+
_∣'_ : {P : A → Type} → Rel A B → specProperty P → Rel A B

+
m ∣' P? = filter (sp-∘ P? proj₁) m

+

+
_∣^'_ : {P : B → Type} → Rel A B → specProperty P → Rel A B

+
m ∣^' P? = filter (sp-∘ P? proj₂) m

+

+
impl⇒res⊆ : ∀ {X : Rel A B} {P P'} (sp-P : specProperty P) (sp-P' : specProperty P')

+
          → (∀ {a} → P a → P' a) → X ∣' sp-P ⊆ X ∣' sp-P'

+
impl⇒res⊆ sp-P sp-P' P⇒P' a∈X∣'P = ∈⇔P (×.map₁ P⇒P' (∈⇔P a∈X∣'P))

+

+
impl⇒cores⊆ : ∀ {X : Rel A B} {P P'} (sp-P : specProperty P) (sp-P' : specProperty P')

+
            → (∀ {b} → P b → P' b) → X ∣^' sp-P ⊆ X ∣^' sp-P'

+
impl⇒cores⊆ sp-P sp-P' P⇒P' a∈X∣^'P = ∈⇔P (×.map₁ P⇒P' (∈⇔P a∈X∣^'P))

+

+
mapˡ : (A → A') → Rel A B → Rel A' B

+
mapˡ f R = map (×.map₁ f) R

+

+
mapʳ : (B → B') → Rel A B → Rel A B'

+
mapʳ f R = map (×.map₂ f) R

+

+
dom∈ : ∀ {a} → (∃[ b ] (a , b) ∈ R) ⇔ a ∈ dom R

+
dom∈ {R = R} {a} =

+
  (∃[ b ] (a , b) ∈ R)            ∼⟨ R.SK-sym (mk⇔ (λ { ((_ , y) , refl , ay∈R) → y , ay∈R })

+
                                              (λ (x , ax∈R) → (a , x) , refl , ax∈R)) ⟩

+
  (∃[ a₁ ] a ≡ proj₁ a₁ × a₁ ∈ R) ∼⟨ ∈-map ⟩

+

+
  a ∈ dom R                       ∎

+
  where open R.EquationalReasoning

+

+
module _ {x : A} {y : B} where

+
  module _ {a : A} where

+
    ∈-dom-singleton-pair : a ≡ x ⇔ a ∈ dom ❴ x , y ❵

+
    ∈-dom-singleton-pair = mk⇔ (λ a≡x → to dom∈ (y , to ∈-singleton (×-≡,≡→≡ (a≡x , refl))))

+
                               (,-injectiveˡ ∘ from ∈-singleton ∘ proj₂ ∘ from dom∈)

+

+
    dom-single→single : a ∈ dom ❴ x , y ❵ → a ∈ ❴ x ❵

+
    dom-single→single = to ∈-singleton ∘ from ∈-dom-singleton-pair

+

+
    single→dom-single : a ∈ ❴ x ❵ → a ∈ dom ❴ x , y ❵

+
    single→dom-single = to ∈-dom-singleton-pair ∘ from ∈-singleton

+

+
  dom-single≡single : dom ❴ x , y ❵ ≡ᵉ ❴ x ❵

+
  dom-single≡single = dom-single→single , single→dom-single

+

+
∈-dom : {a : A × B} → a ∈ R → proj₁ a ∈ dom R

+
∈-dom {a = a} a∈ = to ∈-map (a , (refl , a∈))

+

+
∉-dom∅ : {a : A} → a ∉ dom{A}{B} ∅

+
∉-dom∅ {a} a∈dom∅ = ⊥-elim $ ∉-∅ $ proj₂ $ (from dom∈) a∈dom∅

+

+
dom∅ : dom{A}{B} ∅ ≡ᵉ ∅

+
dom∅ = ⊥-elim ∘ ∉-dom∅ , ∅-minimum (dom ∅)

+

+
dom∪ : dom (R ∪ R') ≡ᵉ dom R ∪ dom R'

+
dom∪ {R = R} {R'} = from ≡ᵉ⇔≡ᵉ' λ a →

+
  a ∈ dom (R ∪ R')                           ∼⟨ R.SK-sym dom∈ ⟩

+
  (∃[ b ] (a , b) ∈ R ∪ R')                  ∼⟨ ∃-cong′ (R.SK-sym ∈-∪) ⟩

+
  (∃[ b ] ((a , b) ∈ R ⊎ (a , b) ∈ R'))      ↔⟨ ∃-distrib-⊎ ⟩

+
  (∃[ b ] (a , b) ∈ R ⊎ ∃[ b ] (a , b) ∈ R') ∼⟨ dom∈ ⊎-cong dom∈ ⟩

+
  (a ∈ dom R ⊎ a ∈ dom R')                   ∼⟨ ∈-∪ ⟩

+
  a ∈ dom R ∪ dom R'                         ∎

+
  where open R.EquationalReasoning

+

+
dom⊆ : dom{A}{B} Preserves _⊆_ ⟶ _⊆_

+
dom⊆ R⊆R' a∈ = to dom∈ $ proj₁ (from dom∈ a∈) , R⊆R' (proj₂ (from dom∈ a∈))

+

+
dom-cong : R ≡ᵉ R' → dom R ≡ᵉ dom R'

+
dom-cong RR' = (dom⊆ (proj₁ RR')) , (dom⊆ (proj₂ RR'))

+

+
dom-⊆mapʳ : {f : B → B'} → dom R ⊆ dom (mapʳ f R)

+
dom-⊆mapʳ {f = f} {a} a∈domR with from dom∈ a∈domR

+
... | b , ab∈R = to dom∈ (f b , to ∈-map ((a , b) , refl , ab∈R))

+

+
dom-mapʳ⊆ : {f : B → B'} → dom (mapʳ f R) ⊆ dom R

+
dom-mapʳ⊆ a∈dmR with from dom∈ a∈dmR

+
... | _ , p∈map with from ∈-map p∈map

+
... | (_ , b) , refl , ab∈R = to dom∈ (b , ab∈R)

+

+
mapʳ-dom : {f : B → B'} → dom R ≡ᵉ dom (mapʳ f R)

+
mapʳ-dom = dom-⊆mapʳ , dom-mapʳ⊆

+

+
dom-mapˡ≡map-dom : {f : A → A'} → dom (mapˡ f R) ≡ᵉ map f (dom R)

+
dom-mapˡ≡map-dom .proj₁ a'∈dom with from ∈-map (proj₂ (from dom∈ a'∈dom))

+
... | (a , b) , a'b≡fab , ab∈R = to ∈-map (a , proj₁ (×-≡,≡←≡ a'b≡fab) , to dom∈ (b , ab∈R))

+
dom-mapˡ≡map-dom .proj₂ a'∈map with from ∈-map a'∈map

+
... | a , a'≡fa , a∈domR with from dom∈ a∈domR

+
... | b , ab∈R = to dom∈ (b , to ∈-map ((a , b) , ×-≡,≡→≡ (a'≡fa , refl) , ab∈R))

+

+
dom-∅ : dom R ⊆ ∅ → R ≡ᵉ ∅

+
dom-∅ dom⊆∅ = ∅-least (λ {x} x∈R → ⊥-elim $ ∉-∅ $ dom⊆∅ $ to dom∈ (-, x∈R))

+

+
mapPartialLiftKey : (A → B → Maybe B') → A × B → Maybe (A × B')

+
mapPartialLiftKey f (k , v) = map? (k ,_) (f k v)

+

+
mapPartialLiftKey-map : ∀ {a : A} {b' : B'} {f : A → B → Maybe B'} {r : Rel A B}

+
  → just (a , b') ∈ map (mapPartialLiftKey f) r

+
  → ∃[ b ] just b' ≡ f a b × (a , b) ∈ r

+
mapPartialLiftKey-map {f = f} ab∈m

+
  with from ∈-map ab∈m

+
... | (a' , b') , ≡ , a'b'∈r

+
  with f a' b' in eq

+
mapPartialLiftKey-map {f = f} ab∈m | (a' , b') , refl , a'b'∈r | just x

+
  = b' , sym eq , a'b'∈r

+

+
mapMaybeWithKey : (A → B → Maybe B') → Rel A B → Rel A B'

+
mapMaybeWithKey f r = mapPartial (mapPartialLiftKey f) r

+

+
∈-mapMaybeWithKey : ∀ {a : A} {b' : B'} {f : A → B → Maybe B'} {r : Rel A B}

+
  → (a , b') ∈ mapMaybeWithKey f r

+
  → ∃[ b ] (just b' ≡ f a b × (a , b) ∈ r)

+
∈-mapMaybeWithKey {a = a} {b'} {f} ab'∈

+
  = mapPartialLiftKey-map {f = f}

+
  $ ⊆-mapPartial

+
  $ to (∈-map {f = just}) ((a , b') , refl , ab'∈)

+

+
module Restriction (sp-∈ : spec-∈ A) where

+

+
  _∣_ : Rel A B → Set A → Rel A B

+
  m ∣ X = m ∣' sp-∈ {X}

+

+
  _∣_ᶜ : Rel A B → Set A → Rel A B

+
  m ∣ X ᶜ = m ∣' sp-¬ (sp-∈ {X})

+

+
  _⟪$⟫_ : Rel A B → Set A → Set B

+
  m ⟪$⟫ X = range (m ∣ X)

+

+
  res-cong : (R ∣_) Preserves _≡ᵉ_ ⟶ _≡ᵉ_

+
  res-cong (X⊆Y , Y⊆X) = (λ ∈R∣X → ∈⇔P (×.map₁ X⊆Y (∈⇔P ∈R∣X)))

+
                       , (λ ∈R∣Y → ∈⇔P (×.map₁ Y⊆X (∈⇔P ∈R∣Y)))

+

+
  res-dom : dom (R ∣ X) ⊆ X

+
  res-dom a∈dom with ∈⇔P a∈dom

+
  ... | _ , refl , h = proj₁ $ ∈⇔P h

+

+
  res-domᵐ : dom (R ∣ X) ⊆ dom R

+
  res-domᵐ a∈dom with ∈⇔P a∈dom

+
  ... | _ , refl , h = ∈-map⁺'' $ proj₂ (∈⇔P h)

+

+
  res-comp-cong : (R ∣_ᶜ) Preserves _≡ᵉ_ ⟶ _≡ᵉ_

+
  res-comp-cong (X⊆Y , Y⊆X) = (λ ∈R∣X → ∈⇔P (×.map₁ (_∘ Y⊆X) (∈⇔P ∈R∣X)))

+
                            , (λ ∈R∣Y → ∈⇔P (×.map₁ (_∘ X⊆Y) (∈⇔P ∈R∣Y)))

+

+
  res-comp-dom : ∀ {a} → a ∈ dom (R ∣ X ᶜ) → a ∉ X

+
  res-comp-dom a∈dom with ∈⇔P a∈dom

+
  ... | _ , refl , h = proj₁ $ ∈⇔P h

+

+
  res-comp-domᵐ : dom (R ∣ X ᶜ) ⊆ dom R

+
  res-comp-domᵐ a∈dom with ∈⇔P a∈dom

+
  ... | _ , refl , h = ∈-map⁺'' (proj₂ (∈⇔P h))

+

+
  res-⊆ : (R ∣ X) ⊆ R

+
  res-⊆ = proj₂ ∘′ ∈⇔P

+

+
  ex-⊆ : (R ∣ X ᶜ) ⊆ R

+
  ex-⊆ = proj₂ ∘′ ∈⇔P

+

+
  res-∅ : R ∣ ∅ ≡ᵉ ∅

+
  res-∅ = dom-∅ res-dom

+

+
  res-∅ᶜ : R ∣ ∅ ᶜ ≡ᵉ R

+
  res-∅ᶜ = ex-⊆ , λ a∈R → ∈⇔P (∉-∅ , a∈R)

+

+
  ∈-resᶜ-dom⁻ : ∀ {a} → a ∈ dom (R ∣ X ᶜ) → a ∉ X × ∃[ b ] (a , b) ∈ R

+
  ∈-resᶜ-dom⁻ a∈ = res-comp-dom a∈ , from dom∈ (dom⊆ ex-⊆ a∈)

+

+
  ∈-resᶜ-dom⁺ : ∀ {a} → a ∉ X × ∃[ b ] (a , b) ∈ R → a ∈ dom (R ∣ X ᶜ)

+
  ∈-resᶜ-dom⁺ (a∉X , (b , ab∈R)) = to dom∈ (b , (∈⇔P (a∉X , ab∈R)))

+

+
  ∈-resᶜ-dom : ∀ {a} → a ∈ dom (R ∣ X ᶜ) ⇔ (a ∉ X × ∃[ b ] (a , b) ∈ R)

+
  ∈-resᶜ-dom = mk⇔ ∈-resᶜ-dom⁻ ∈-resᶜ-dom⁺

+

+
  res-ex-∪ : Decidable (_∈ X) → (R ∣ X) ∪ (R ∣ X ᶜ) ≡ᵉ R

+
  res-ex-∪ ∈X? = ∪-⊆ res-⊆ ex-⊆ , λ {a} h → case ∈X? (proj₁ a) of λ where

+
    (yes p) → ∈⇔P (inj₁ (∈⇔P (p , h)))

+
    (no ¬p) → ∈⇔P (inj₂ (∈⇔P (¬p , h)))

+

+
  res-ex-disjoint : disjoint (dom (R ∣ X)) (dom (R ∣ X ᶜ))

+
  res-ex-disjoint h h' = res-comp-dom h' (res-dom h)

+

+
  res-ex-disj-∪ : Decidable (_∈ X) → R ≡ (R ∣ X) ⨿ (R ∣ X ᶜ)

+
  res-ex-disj-∪ ∈X? = IsEquivalence.sym ≡ᵉ-isEquivalence (res-ex-∪ ∈X?)

+
                    , disjoint-dom⇒disjoint res-ex-disjoint

+
    where open import Relation.Binary using (IsEquivalence)

+

+
  curryʳ : Rel (A × B) C → A → Rel B C

+
  curryʳ R a = mapˡ proj₂ (R ∣' (sp-∘ (sp-∈ {X = ❴ a ❵}) proj₁))

+

+
  ∈-curryʳ : ∀ {a} {b : B} {c : C} → (b , c) ∈ curryʳ R a → ((a , b) , c) ∈ R

+
  ∈-curryʳ h = case ∈⇔P h of λ where

+
    (((a , b) , c) , refl , h'') → case ∈⇔P h'' of λ where

+
      (p , p') → case from ∈-singleton p of λ where refl → p'

+

+
  open Intersection sp-∈

+
  open Intersectionᵖ sp-∈

+

+
  res-dom-comm⊆∩ : {m : Rel A B} {m' : Rel A C} → dom (m ∣ dom m') ⊆ dom m ∩ dom m'

+
  res-dom-comm⊆∩ x = to ∈-∩ (res-domᵐ x , res-dom x)

+

+
  res-dom-comm∩⊆ : {m : Rel A B} {m' : Rel A C} → dom m ∩ dom m' ⊆ dom (m ∣ dom m')

+
  res-dom-comm∩⊆ {m = m} {m' = m'} x with from ∈-∩ x

+
  ... | a∈dm , a∈dm' with from dom∈ a∈dm | from dom∈ a∈dm'

+
  ... | b , ab∈m | c , ac∈m = to dom∈ (b , to ∈-filter (a∈dm' , ab∈m))

+

+
  res-dom-comm' : {m : Rel A B} {m' : Rel A C} → dom (m ∣ dom m') ≡ᵉ dom m ∩ dom m'

+
  res-dom-comm' = res-dom-comm⊆∩ , res-dom-comm∩⊆

+

+
  res-dom-comm : {m : Rel A B} {m' : Rel A C} → dom (m ∣ dom m') ≡ᵉ dom (m' ∣ dom m)

+
  res-dom-comm {m = m} {m'} = begin

+
    dom (m ∣ dom m') ≈⟨ res-dom-comm' ⟩

+
    dom m ∩ dom m'   ≈˘⟨ ∩-sym ⟩

+
    dom m' ∩ dom m   ≈˘⟨ res-dom-comm' ⟩

+
    dom (m' ∣ dom m) ∎

+
    where open SetoidReasoning ≡ᵉ-Setoid

+

+
module Corestriction (sp-∈ : spec-∈ B) where

+

+
  _∣^_ : Rel A B → Set B → Rel A B

+
  m ∣^ X = m ∣^' sp-∈ {X}

+

+
  _∣^_ᶜ : Rel A B → Set B → Rel A B

+
  m ∣^ X ᶜ = m ∣^' sp-¬ (sp-∈ {X})

+

+
  cores-⊆ : (R ∣^ X) ⊆ R

+
  cores-⊆ = proj₂ ∘′ ∈⇔P

+

+
  coex-⊆ : (R ∣^ X ᶜ) ⊆ R

+
  coex-⊆ = proj₂ ∘′ ∈⇔P

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.Sum.html b/site/static/agda_html/Axiom.Set.Sum.html
new file mode 100644
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--- /dev/null
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+
+Axiom.Set.Sum
+             
+             
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+                 Back
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{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import Axiom.Set using (Theory)

+

+
open import abstract-set-theory.Prelude hiding (ε)

+

+
module Axiom.Set.Sum (th : Theory) where

+

+
open Theory th

+
open import Axiom.Set.Factor th

+
open import Axiom.Set.Properties th

+
open import Axiom.Set.Rel th

+
open import Axiom.Set.Map th

+

+
import Algebra

+
open import Algebra.Properties.CommutativeSemigroup using (x∙yz≈y∙xz)

+
open import Data.List.Ext.Properties using (dedup-++-↭)

+
open import Data.List.Properties using (foldr-++)

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
open import Data.List.Relation.Unary.Unique.Propositional using (Unique)

+
open import Relation.Binary using (_Preserves_⟶_; IsEquivalence)

+

+
open import Tactic.AnyOf

+
open import Tactic.Defaults

+

+
private module M-Reasoning {M : Type} ⦃ M-commMonoid : CommutativeMonoid 0ℓ 0ℓ M ⦄ where

+
  open Algebra.CommutativeMonoid (Conversion.toBundle M-commMonoid) renaming (trans to ≈-trans) public

+
  import Relation.Binary.Reasoning.Setoid as SetoidReasoning

+
  open SetoidReasoning (Algebra.CommutativeMonoid.setoid (Conversion.toBundle M-commMonoid)) public

+

+
-- Because of missing macro hygiene, we have to copy&paste this.

+
-- c.f. https://github.com/agda/agda/issues/3819

+
private macro

+
  ∈⇒P = anyOfⁿᵗ

+
    (quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻ ∷ [])

+
  P⇒∈ = anyOfⁿᵗ

+
    (quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+
  ∈⇔P = anyOfⁿᵗ

+
    ( quote ∈-filter⁻' ∷ quote ∈-∪⁻ ∷ quote ∈-map⁻' ∷ quote ∈-fromList⁻

+
    ∷ quote ∈-filter⁺' ∷ quote ∈-∪⁺ ∷ quote ∈-map⁺' ∷ quote ∈-fromList⁺ ∷ [])

+

+
private variable

+
  A B M : Type

+
  X Y : Set A

+
  f : A → M

+

+
module _ ⦃ M-commMonoid : CommutativeMonoid 0ℓ 0ℓ M ⦄ where

+

+
  open M-Reasoning ⦃ M-commMonoid ⦄

+

+
  indexedSumL : (A → M) → List A → M

+
  indexedSumL f = foldr (λ x → f x ∙_) ε

+

+
  syntax indexedSumL (λ a → x) m = ∑ˡ[ a ← m ] x

+

+
  indexedSumL' : (A → M) → Σ (List A) Unique → M

+
  indexedSumL' f = indexedSumL f ∘ proj₁

+

+
  fold-cong↭ : ∀ {l l' : List A}

+
    → l ↭ l'

+
    → foldr (λ x → f x ∙_) ε l ≈ foldr (λ x → f x ∙_) ε l'

+
  fold-cong↭ refl = begin _ ∎

+
  fold-cong↭ (prep _ h) = ∙-congˡ (fold-cong↭ h)

+
  fold-cong↭ {f = f} (swap {xs} {ys} x y h) = begin

+
    f x ∙ (f y ∙ indexedSumL f xs) ≈⟨ x∙yz≈y∙xz commutativeSemigroup _ _ _ ⟩

+
    f y ∙ (f x ∙ indexedSumL f xs) ≈⟨ ∙-congˡ (∙-congˡ (fold-cong↭ h)) ⟩

+
    f y ∙ (f x ∙ indexedSumL f ys) ∎

+
  fold-cong↭ (trans h h₁) = ≈-trans (fold-cong↭ h) (fold-cong↭ h₁)

+

+
  indexedSum : ⦃ _ : DecEq A ⦄ → (A → M) → FinSet A → M

+
  indexedSum f = let open FactorUnique _≈_ (indexedSumL' f) fold-cong↭ in factor

+

+
  indexedSumL-++ : {l l' : List A}

+
    → indexedSumL f (l ++ l') ≈ indexedSumL f l ∙ indexedSumL f l'

+
  indexedSumL-++ {f = f} {l = l} {l'} = begin

+
    indexedSumL f (l ++ l')                   ≡⟨ foldr-++ (λ x → f x ∙_) ε l l' ⟩

+
    foldr (λ x → f x ∙_) (indexedSumL f l') l ≈⟨ helper (indexedSumL f l') l f ⟩

+
    indexedSumL f l ∙ indexedSumL f l'        ∎

+
    where

+
      helper : ∀ m (l : List A) f → foldr (λ x → f x ∙_) m l ≈ indexedSumL f l ∙ m

+
      helper m [] f = begin m ≈˘⟨ identityˡ m ⟩ ε ∙ m ∎

+
      helper m (x ∷ l) f = begin

+
        f x ∙ foldr (λ y → f y ∙_) m l ≈⟨ ∙-congˡ (helper m l f) ⟩

+
        f x ∙ (indexedSumL f l ∙ m)    ≈˘⟨ assoc _ _ _ ⟩

+
        f x ∙ indexedSumL f l ∙ m      ∎

+

+

+
  module _ ⦃ _ : DecEq A ⦄ {f : A → M} where

+
    open FactorUnique _≈_ (indexedSumL' f) fold-cong↭

+

+
    indexedSum-cong : indexedSum f Preserves (_≡ᵉ_ on proj₁) ⟶ _≈_

+
    indexedSum-cong {x} {y} = factor-cong {x = x} {y}

+

+
    indexedSum-∅ : indexedSum f (∅ , ∅-finite) ≈ ε

+
    indexedSum-∅ = begin _ ∎

+

+
    indexedSum-∪ : ⦃ Xᶠ : finite X ⦄ ⦃ Yᶠ : finite Y ⦄ → disjoint X Y

+
      → indexedSum f ((X ∪ Y) ᶠ) ≈ indexedSum f (X ᶠ) ∙ indexedSum f (Y ᶠ)

+
    indexedSum-∪ disj = factor-∪' {λ x y z → z ≈ x ∙ y} disj

+
        λ {l} disj' → ≈-trans (fold-cong↭ (dedup-++-↭ disj'))

+
                              (indexedSumL-++ {l = deduplicate _≟_ l})

+

+
    indexedSum-singleton : ∀ {x} → indexedSum f (❴ x ❵ , singleton-finite) ≈ f x

+
    indexedSum-singleton = identityʳ _

+

+
    indexedSum-singleton' : ∀ {x} → (pf : finite ❴ x ❵)

+
      → indexedSum f (❴ x ❵ , pf) ≈ f x

+
    indexedSum-singleton' {x = x} pf =

+
      ≈-trans (indexedSum-cong {x = -, pf} {y = -, singleton-finite} ≡ᵉ.refl)

+
              indexedSum-singleton

+
      where module ≡ᵉ = IsEquivalence ≡ᵉ-isEquivalence

+

+
module _ ⦃ _ : DecEq A ⦄ ⦃ _ : DecEq B ⦄ ⦃ _ : CommutativeMonoid 0ℓ 0ℓ M ⦄ where

+

+
  indexedSumᵐ : (A × B → M) → FinMap A B → M

+
  indexedSumᵐ f (m , _ , h) = indexedSum f (m , h)

+

+
  indexedSumᵛ : (B → M) → FinMap A B → M

+
  indexedSumᵛ f = indexedSumᵐ (f ∘ proj₂)

+

+
  open M-Reasoning

+

+
  indexedSumᵐ-cong : {f : A × B → M}

+
    → indexedSumᵐ f Preserves (_≡ᵉ_ on proj₁) ⟶ _≈_

+
  indexedSumᵐ-cong {x = x , _ , h} {y , _ , h'} = indexedSum-cong {x = x , h} {y , h'}

+

+
module IndexedSumUnionᵐ ⦃ _ : DecEq A ⦄ ⦃ _ : DecEq B ⦄

+
  (sp-∈ : spec-∈ A) (∈-A-dec : {X : Set A} → Decidable¹ (_∈ X)) where

+

+
  open Unionᵐ sp-∈

+

+
  ∪ˡ-finite : {R R' : Rel A B} → finite R → finite R' → finite (R ∪ˡ' R')

+
  ∪ˡ-finite Rᶠ R'ᶠ = ∪-preserves-finite Rᶠ

+
                    $ filter-finite (sp-∘ (sp-¬ sp-∈) _) (¬? ∘ ∈-A-dec ∘ _) R'ᶠ

+

+
  _∪ˡᶠ_ : FinMap A B → FinMap A B → FinMap A B

+
  (_ , hX , Xᶠ) ∪ˡᶠ (_ , hY , Yᶠ) =

+
    toFinMap ((_ , hX) ∪ˡ (_ , hY)) (∪ˡ-finite Xᶠ Yᶠ)

+

+
  open M-Reasoning

+

+
  indexedSumᵐ-∪ : ∀ {X Y : FinMap A B} {f} ⦃ _ : CommutativeMonoid 0ℓ 0ℓ M ⦄

+
    → disjoint (dom (toRel X)) (dom (toRel Y))

+
    → indexedSumᵐ f (X ∪ˡᶠ Y) ≈ indexedSumᵐ f X ∙ indexedSumᵐ f Y

+
  indexedSumᵐ-∪ {X = X'@(X , _ , Xᶠ)} {Y'@(Y , _ , Yᶠ)} {f} disj = begin

+
    indexedSumᵐ f (X' ∪ˡᶠ Y')    ≈⟨ indexedSum-cong {x = -, ∪ˡ-finite Xᶠ Yᶠ} {(X ∪ Y) ᶠ}

+
                                       $ disjoint-∪ˡ-∪ disj ⟩

+
    indexedSum f ((X ∪ Y) ᶠ)      ≈⟨ indexedSum-∪ (disjoint-dom⇒disjoint disj) ⟩

+
    indexedSumᵐ f X' ∙ indexedSumᵐ f Y' ∎

+
    where instance _ = Xᶠ

+
                   _ = Yᶠ

+

+
  indexedSumᵐ-partition : ∀ {m m₁ m₂ : FinMap A B} {f} ⦃ _ : CommutativeMonoid 0ℓ 0ℓ M ⦄

+
                        → toRel m ≡ toRel m₁ ⨿ toRel m₂

+
                        → indexedSumᵐ f m ≈ indexedSumᵐ f m₁ ∙ indexedSumᵐ f m₂

+
  indexedSumᵐ-partition {m = m} {m₁} {m₂} {f} m≡m₁∪m₂ = begin

+
    indexedSumᵐ f m                     ≈⟨ indexedSumᵐ-cong {x = m} {m₁ ∪ˡᶠ m₂} helper ⟩

+
    indexedSumᵐ f (m₁ ∪ˡᶠ m₂)           ≈⟨ indexedSumᵐ-∪ {X = m₁} {Y = m₂} disj-dom' ⟩

+
    indexedSumᵐ f m₁ ∙ indexedSumᵐ f m₂  ∎

+
    where module ≡ᵉ = IsEquivalence ≡ᵉ-isEquivalence

+
          disj-dom' = disj-dom {m = toMap m} {toMap m₁} {toMap m₂} m≡m₁∪m₂

+

+
          helper : toRel m ≡ᵉ toRel (m₁ ∪ˡᶠ m₂)

+
          helper = ≡ᵉ.trans (proj₁ m≡m₁∪m₂) (≡ᵉ.sym $ disjoint-∪ˡ-∪ disj-dom')

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.TotalMap.html b/site/static/agda_html/Axiom.Set.TotalMap.html
new file mode 100644
index 000000000..433c1ee7f
--- /dev/null
+++ b/site/static/agda_html/Axiom.Set.TotalMap.html
@@ -0,0 +1,174 @@
+
+Axiom.Set.TotalMap
+             
+             
+             

+               

+                 Back
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{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import Axiom.Set using (Theory)

+

+
module Axiom.Set.TotalMap (th : Theory) where

+

+
open import abstract-set-theory.Prelude hiding (lookup; map)

+

+
open import Data.Product.Properties     using (Σ-≡,≡→≡)

+
open import Axiom.Set.Map th            using (left-unique; Map ; mapWithKey-uniq ; left-unique-mapˢ)

+
open import Axiom.Set.Rel th            using (Rel ; dom ; dom∈)

+

+
open Theory th

+
open Equivalence

+

+
private variable A B : Type

+

+
-- defines a total map for a given set

+
total : Rel A B → Type

+
total R = ∀ {a} → a ∈ dom R

+

+
record TotalMap (A B : Type) : Type where

+
  field rel             : Set (A × B)

+
        left-unique-rel : left-unique rel

+
        total-rel       : total rel

+

+
  toMap : Map A B

+
  toMap = rel , left-unique-rel

+

+
  lookup : A → B

+
  lookup _ = proj₁ (from dom∈ total-rel)

+

+
  -- verify that lookup is what we expect

+
  lookup∈rel : {a : A} → (a , lookup a) ∈ rel

+
  lookup∈rel = proj₂ (from dom∈ total-rel)

+

+
  -- this is useful for proving equalities involving lookup

+
  ∈-rel⇒lookup-≡ : {a : A}{b : B} → (a , b) ∈ rel → lookup a ≡ b

+
  ∈-rel⇒lookup-≡ ab∈rel = sym (left-unique-rel ab∈rel (proj₂ (from dom∈ total-rel)))

+

+
open TotalMap

+

+
module Update ⦃ _ : DecEq A ⦄ where

+

+
  private

+
    updateFn : A × B → A → B → B

+
    updateFn (a , b) x y with (x ≟ a)

+
    ... | yes  _ = b

+
    ... | no   _ = y

+

+
  updateFn-id : {a : A} {b b' : B} → b ≡ updateFn (a , b) a b'

+
  updateFn-id {a = a} with (a ≟ a)

+
  ... | yes _ = refl

+
  ... | no ¬p = ⊥-elim (¬p refl)

+

+
  mapWithKey : {B' : Type} → (A → B → B') → TotalMap A B → TotalMap A B'

+
  mapWithKey f tm .rel              = map (λ{(x , y) → x , f x y}) (rel tm)

+
  mapWithKey _ tm .left-unique-rel  = mapWithKey-uniq (left-unique-rel tm)

+
  mapWithKey _ tm .total-rel        = ∈-map′ (∈-map′ (proj₂ (from dom∈ (total-rel tm))))

+

+
  update : A → B → TotalMap A B → TotalMap A B

+
  update a b = mapWithKey (updateFn (a , b))

+

+

+
module LookupUpdate {X : Set A} {a : A} {b : B} {a∈X : a ∈ X} ⦃ _ : DecEq A ⦄ where

+

+
  open Update

+

+
  ∈-rel-update : (tm : TotalMap A B) → (a , b) ∈ rel (update a b tm)

+
  ∈-rel-update tm = to ∈-map ((a , lookup tm a) , Σ-≡,≡→≡ (refl , updateFn-id {A = A}) , lookup∈rel tm)

+

+
  lookup-update-id : (tm : TotalMap A B) → lookup (update a b tm) a ≡ b

+
  lookup-update-id tm = ∈-rel⇒lookup-≡ (update _ _ tm) (∈-rel-update tm)

+

+

+

+
------------------------------------------------------

+
-- Correspondences between total maps and functions --

+

+
module FunTot (X : Set A) (⋁A≡X : isMaximal X) where

+

+
  Fun⇒Map : ∀ {f : A → B} (X : Set A) → Map A B

+
  Fun⇒Map {f = f} X = map (λ x → (x , f x)) X , left-unique-mapˢ X

+

+

+
  Fun⇒TotalMap : (f : A → B) → TotalMap A B

+
  Fun⇒TotalMap f .rel              = map (λ x → (x , f x)) X

+
  Fun⇒TotalMap _ .left-unique-rel  = left-unique-mapˢ X

+
  Fun⇒TotalMap _ .total-rel        = ∈-map′ (∈-map′ ⋁A≡X)

+

+
  Fun∈TotalMap : {f : A → B} {a : A}

+
               → a ∈ X → (a , f a) ∈ rel (Fun⇒TotalMap f)

+
  Fun∈TotalMap a∈X = ∈-map′ a∈X

+

+
  lookup∘Fun⇒TotalMap-id : {f : A → B}{a : A}

+
                         → lookup (Fun⇒TotalMap f) a ≡ f a

+
  lookup∘Fun⇒TotalMap-id {f = f} = ∈-rel⇒lookup-≡ ((Fun⇒TotalMap f)) (Fun∈TotalMap ⋁A≡X)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.TotalMapOn.html b/site/static/agda_html/Axiom.Set.TotalMapOn.html
new file mode 100644
index 000000000..f243901e7
--- /dev/null
+++ b/site/static/agda_html/Axiom.Set.TotalMapOn.html
@@ -0,0 +1,136 @@
+
+Axiom.Set.TotalMapOn
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import Axiom.Set using (Theory)

+

+
module Axiom.Set.TotalMapOn (th : Theory) where

+

+
open import abstract-set-theory.Prelude hiding (lookup; map)

+

+
open import Axiom.Set.Map th using (left-unique ; Map ; mapWithKey-uniq)

+
open import Axiom.Set.Rel th using (Rel ; dom ; dom∈)

+
open import Class.DecEq      using (DecEq ; _≟_)

+

+
open Theory th    using ( Set ; _⊆_ ; _∈_ ; map ; ∈-map′ )

+
open Equivalence  using (from)

+

+
private variable A B : Type

+

+
_TotalOn_ : Rel A B → Set A → Type

+
R TotalOn X = X ⊆ dom R

+

+

+
record TotalMapOn {A : Type}(X : Set A)(B : Type) : Type where

+
  field  rel              : Set (A × B)

+
         left-unique-rel  : left-unique rel

+
         total-rel        : rel TotalOn X

+

+
  toMap : Map A B

+
  toMap = rel , left-unique-rel

+

+
  lookup : Σ A (_∈ X) → B

+
  lookup (_ , a∈X) = proj₁ (from dom∈ (total-rel a∈X))

+

+
  -- verify that lookup is what we expect

+
  lookup∈rel : {a : A} (a∈X : a ∈ X) → (a , lookup (a , a∈X)) ∈ rel

+
  lookup∈rel a∈X = proj₂ (from dom∈ (total-rel a∈X))

+

+
  -- this is useful for proving equalities involving lookup

+
  rel⇒lookup : {a : A} {a∈dom : a ∈ X} {b : B} → (a , b) ∈ rel → lookup (a , a∈dom) ≡ b

+
  rel⇒lookup {a} {a∈dom} ab∈rel = sym (left-unique-rel ab∈rel (proj₂ (from dom∈ (total-rel a∈dom))))

+

+

+
module UpdateOn {B : Type} ⦃ _ : DecEq A ⦄ where

+

+
  private

+
    updateFn : A × B → A → B → B

+
    updateFn (a , b) x y with (x ≟ a)

+
    ... | yes  _ = b

+
    ... | no   _ = y

+

+
  open TotalMapOn

+

+
  mapWithKeyOn : {X : Set A}{B' : Type} → (A → B → B') → TotalMapOn X B → TotalMapOn X B'

+
  mapWithKeyOn f tm .rel              = map (λ{(x , y) → x , f x y}) (rel tm)

+
  mapWithKeyOn _ tm .left-unique-rel  = mapWithKey-uniq (left-unique-rel tm)

+
  mapWithKeyOn _ tm .total-rel a∈X    = ∈-map′ (∈-map′ (proj₂ (from dom∈ ((total-rel tm) a∈X))))

+

+
  -- Return a new total map which is the same as the given total map except at a.

+
  update : {X : Set A} → A → B → TotalMapOn X B → TotalMapOn X B

+
  update a b = mapWithKeyOn (updateFn (a , b))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.Set.html b/site/static/agda_html/Axiom.Set.html
new file mode 100644
index 000000000..d187e3c87
--- /dev/null
+++ b/site/static/agda_html/Axiom.Set.html
@@ -0,0 +1,458 @@
+
+Axiom.Set
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+

+
module Axiom.Set where

+

+
open import abstract-set-theory.Prelude hiding (map)

+

+
import Function.Related.Propositional as R

+
open import Data.List.Ext.Properties using (∈-dedup; _×-cong_)

+
open import Data.List.Relation.Unary.Any using (here; there)

+
open import Data.List.Relation.Unary.Unique.DecPropositional.Properties using (deduplicate-!)

+
open import Data.List.Relation.Unary.Unique.Propositional using (Unique; [])

+
open import Data.Product.Algebra using (×-comm)

+
open import Data.Product.Properties using (∃∃↔∃∃)

+
open import Data.Product.Properties.Ext using (∃-cong′; ∃-≡)

+
open import Class.DecEq using (DecEq; _≟_)

+
open import Relation.Binary using () renaming (Decidable to Dec₂)

+

+
private variable

+
  ℓ : Level

+
  A B C : Type ℓ

+
  P : A → Type

+
  l : List A

+

+
_Preserves₁_⟶_ : {A : Type ℓ} → (A → B) → Pred A 0ℓ → Pred B 0ℓ → Type ℓ

+
f Preserves₁ P ⟶ Q = ∀ {a} → P a → Q (f a)

+

+
_Preserves₁₂_⟶_⟶_ : {A B : Type ℓ} → (A → B → C) → Pred A ℓ → Pred B ℓ → Pred C ℓ → Type ℓ

+
f Preserves₁₂ P ⟶ P' ⟶ Q = ∀ {a b} → P a → P' b → Q (f a b)

+

+
record SpecProperty {ℓ} : Type (sucˡ ℓ) where

+
  field specProperty : {A : Type ℓ} → (A → Type) → Type

+
        sp-∘ : specProperty P → (f : B → A) → specProperty (P ∘ f)

+
        sp-¬ : specProperty P → specProperty (¬_ ∘ P)

+

+
⊤-SpecProperty : ∀ {a} → SpecProperty {a}

+
⊤-SpecProperty = record

+
  { specProperty = λ _ → ⊤

+
  ; sp-∘         = λ _ _ → _

+
  ; sp-¬         = λ _ → _

+
  }

+

+
Dec-SpecProperty : SpecProperty

+
Dec-SpecProperty = record

+
  { specProperty = Decidable¹

+
  ; sp-∘         = λ P? → P? ∘_

+
  ; sp-¬         = λ P? → ¬? ∘ P?

+
  }

+

+
record Theory {ℓ} : Type (sucˡ ℓ) where

+
  infix 4 _⊆_ _≡ᵉ_ _∈_ _∉_

+
  infixr 6 _∪_

+

+
  field Set : Type ℓ → Type ℓ

+
        _∈_ : A → Set A → Type

+
        sp  : SpecProperty

+
  open SpecProperty sp public

+

+
  _⊆_ : Set A → Set A → Type ℓ

+
  X ⊆ Y = ∀ {a} → a ∈ X → a ∈ Y

+

+
  -- we might want to either have all properties or

+
  -- decidable properties allowed for specification

+
  field specification : (X : Set A)

+
                      → specProperty P → ∃[ Y ] ∀ {a} → (P a × a ∈ X) ⇔ a ∈ Y

+
        unions        : (X : Set (Set A))

+
                      → ∃[ Y ] ∀ {a} → (∃[ T ] (T ∈ X × a ∈ T)) ⇔ a ∈ Y

+
        replacement   : (f : A → B) (X : Set A)

+
                      → ∃[ Y ] ∀ {b} → (∃[ a ] b ≡ f a × a ∈ X) ⇔ b ∈ Y

+
        listing       : (l : List A)

+
                      → ∃[ X ] ∀ {a} → a ∈ˡ l ⇔ a ∈ X

+
                      -- ^ equivalent to pairing + empty set

+
        -- power-set     : (X : Set A) → ∃[ Y ] ∀ {T} → T ⊆ X → T ∈ Y

+

+
  private variable X X' Y : Set A

+

+
  _≡ᵉ_ : Set A → Set A → Type ℓ

+
  X ≡ᵉ Y = X ⊆ Y × Y ⊆ X

+

+
  _≡ᵉ'_ : Set A → Set A → Type ℓ

+
  X ≡ᵉ' Y = ∀ a → a ∈ X ⇔ a ∈ Y

+

+
  _∉_ : A → Set A → Type

+
  _∉_ = ¬_ ∘₂ _∈_

+

+
  ≡→∈ : {X : Set A} {a a' : A} → a ∈ X → a ≡ a' → a' ∈ X

+
  ≡→∈ a∈X refl = a∈X

+

+
  -- The following is useful in case we have `(a , p)` and `(a , q)`, where `p`

+
  -- and `q` are proofs of `a ∈ X`, and we want to prove `(a , p) ≡ (a , q)`.

+
  ∈-irrelevant : Set A → Type ℓ

+
  ∈-irrelevant X = ∀ {a} (p q : a ∈ X) → p ≡ q

+

+
  open Equivalence

+

+
  _Preservesˢ_ : (Set A → Set B) → (∀ {A} → Set A → Type) → Type ℓ

+
  f Preservesˢ P = f Preserves₁ P ⟶ P

+

+
  _Preservesˢ₂_ : (Set A → Set B → Set C) → (∀ {A : Type ℓ} → Set A → Type ℓ) → Type ℓ

+
  f Preservesˢ₂ P = f Preserves₁₂ P ⟶ P ⟶ P

+

+
  disjoint : Set A → Set A → Type ℓ

+
  disjoint X Y = ∀ {a} → a ∈ X → a ∈ Y → ⊥

+

+
  finite : Set A → Type ℓ

+
  finite X = ∃[ l ] ∀ {a} → a ∈ X ⇔ a ∈ˡ l

+

+
  Show-finite : ⦃ Show A ⦄ → Show (Σ (Set A) finite)

+
  Show.show Show-finite (X , (l , _)) = Show-List .show l

+

+
  weakly-finite : Set A → Type ℓ

+
  weakly-finite X = ∃[ l ] ∀ {a} → a ∈ X → a ∈ˡ l

+

+
  -- there exists a list without duplicates that has exactly the members of the set

+
  strongly-finite : Set A → Type ℓ

+
  strongly-finite X = ∃[ l ] Unique l × ∀ {a} → a ∈ X ⇔ a ∈ˡ l

+

+
  DecEq∧finite⇒strongly-finite : ⦃ _ : DecEq A ⦄ (X : Set A)

+
    → finite X → strongly-finite X

+
  DecEq∧finite⇒strongly-finite ⦃ eq? ⦄ X (l , h) = let _≟_ = eq? ._≟_ in

+
    deduplicate _≟_ l , deduplicate-! _≟_ l , λ {a} →

+
      a ∈ X                  ∼⟨ h ⟩

+
      a ∈ˡ l                 ∼⟨ ∈-dedup ⟩

+
      a ∈ˡ deduplicate _≟_ l ∎

+
    where open R.EquationalReasoning

+

+
  card : Σ (Set A) strongly-finite → ℕ

+
  card (_ , l , _) = length l

+

+
  ⊆-weakly-finite : X ⊆ Y → weakly-finite Y → weakly-finite X

+
  ⊆-weakly-finite X⊆Y (l , hl) = l , hl ∘ X⊆Y

+

+
  isMaximal : Set A → Type ℓ

+
  isMaximal {A} X = {a : A} → a ∈ X

+

+
  maximal-⊆ : isMaximal Y → X ⊆ Y

+
  maximal-⊆ maxY _ = maxY

+

+
  maximal-unique : isMaximal X → isMaximal Y → X ≡ᵉ Y

+
  maximal-unique maxX maxY = maximal-⊆ maxY , maximal-⊆ maxX

+

+
  FinSet : Type ℓ → Type ℓ

+
  FinSet A = Σ (Set A) finite

+

+
  -- if you can construct a set that contains all elements satisfying

+
  -- P, you can construct a set containing exactly the elements satisfying P

+
  strictify : specProperty P → (∃[ Y ] ∀ {a} → P a → a ∈ Y) → ∃[ Y ] ∀ {a} → P a ⇔ a ∈ Y

+
  strictify sp p with specification (proj₁ p) sp

+
  ... | (Y , p') = Y , (mk⇔ (λ a∈l → to p' (a∈l , proj₂ p a∈l)) (proj₁ ∘ from p'))

+

+
  map : (A → B) → Set A → Set B

+
  map = proj₁ ∘₂ replacement

+

+
  ∈-map : ∀ {f : A → B} {b} → (∃[ a ] b ≡ f a × a ∈ X) ⇔ b ∈ map f X

+
  ∈-map = proj₂ $ replacement _ _

+

+
  ∈-map′ : ∀ {f : A → B} {a} → a ∈ X → f a ∈ map f X

+
  ∈-map′ {a = a} a∈X = to ∈-map (a , refl , a∈X)

+

+
  -- don't know that there's a set containing all members of a type, which this is equivalent to

+
  -- _⁻¹_ : (A → B) → Set B → Set A

+
  -- f ⁻¹ X = {!!}

+

+
  filter : {P : A → Type} → specProperty P → Set A → Set A

+
  filter = proj₁ ∘₂ flip specification

+

+
  ∈-filter : ∀ {sp-P : specProperty P} {a} → (P a × a ∈ X) ⇔ a ∈ filter sp-P X

+
  ∈-filter = proj₂ $ specification _ _

+

+
  fromList : List A → Set A

+
  fromList = proj₁ ∘ listing

+

+
  ∈-fromList : ∀ {a} → a ∈ˡ l ⇔ a ∈ fromList l

+
  ∈-fromList = proj₂ $ listing _

+

+
  ∈-unions : {a : A} {U : Set (Set A)} → (∃[ T ] T ∈ U × a ∈ T) ⇔ a ∈ proj₁ (unions U)

+
  ∈-unions = proj₂ $ unions _

+

+
  ∅ : Set A

+
  ∅ = fromList []

+

+
  ∅-strongly-finite : strongly-finite {A} ∅

+
  ∅-strongly-finite = [] , [] , R.SK-sym ∈-fromList

+

+
  card-∅ : card (∅ {A} , ∅-strongly-finite) ≡ 0

+
  card-∅ = refl

+

+
  singleton : A → Set A

+
  singleton a = fromList [ a ]

+

+
  ❴_❵ = singleton

+

+
  ∈-singleton : {a b : A} → a ≡ b ⇔ a ∈ singleton b

+
  ∈-singleton {_} {a} {b} =

+
    a ≡ b           ∼⟨ mk⇔ (λ where refl → here refl) (λ where (here refl) → refl) ⟩

+
    a ∈ˡ [ b ]      ∼⟨ ∈-fromList ⟩

+
    a ∈ singleton b ∎

+
    where open R.EquationalReasoning

+

+
  partialToSet : (A → Maybe B) → A → Set B

+
  partialToSet f a = maybe (fromList ∘ [_]) ∅ (f a)

+

+
  ∈-partialToSet : ∀ {a : A} {b : B} {f} → f a ≡ just b ⇔ b ∈ partialToSet f a

+
  ∈-partialToSet {a = a} {b} {f} = mk⇔

+
    (λ h → subst (λ x → b ∈ maybe (fromList ∘ [_]) ∅ x) (sym h) (to ∈-singleton refl))

+
    (case f a returning (λ y → b ∈ maybe (λ x → fromList [ x ]) ∅ y → y ≡ just b) of

+
      λ where (just x) → λ h → cong just (sym $ from ∈-singleton h)

+
              nothing  → λ h → case from ∈-fromList h of λ ())

+

+
  concatMapˢ : (A → Set B) → Set A → Set B

+
  concatMapˢ f a = proj₁ $ unions (map f a)

+

+
  ∈-concatMapˢ : {y : B} {f : A → Set B}

+
    → (∃[ x ] x ∈ X × y ∈ f x) ⇔ y ∈ concatMapˢ f X

+
  ∈-concatMapˢ {X = X} {y} {f} =

+
    (∃[ x ] x ∈ X × y ∈ f x)

+
      ∼⟨ ∃-cong′ (λ {x} → ∃-≡ (λ T → x ∈ X × y ∈ T)) ⟩

+
    (∃[ x ] ∃[ T ] T ≡ f x × x ∈ X × y ∈ T)

+
      ↔⟨ ∃∃↔∃∃ (λ x T → T ≡ f x × x ∈ X × y ∈ T) ⟩

+
    (∃[ T ] ∃[ x ] T ≡ f x × x ∈ X × y ∈ T)

+
      ∼⟨ ∃-cong′ $ mk⇔

+
        (λ where (x , p₁ , p₂ , p₃) → (x , p₁ , p₂) , p₃)

+
        (λ where ((x , p₁ , p₂) , p₃) → x , p₁ , p₂ , p₃) ⟩

+
    (∃[ T ] (∃[ x ] T ≡ f x × x ∈ X) × y ∈ T)

+
      ∼⟨ ∃-cong′ (∈-map ×-cong R.K-refl) ⟩

+
    (∃[ T ] T ∈ map f X × y ∈ T)

+
      ∼⟨ ∈-unions ⟩

+
    y ∈ concatMapˢ f X ∎

+
    where open R.EquationalReasoning

+

+
  mapPartial : (A → Maybe B) → Set A → Set B

+
  mapPartial f = concatMapˢ (partialToSet f)

+

+
  ∈-mapPartial : {y : B} {f : A → Maybe B}

+
    → (∃[ x ] x ∈ X × f x ≡ just y) ⇔ y ∈ mapPartial f X

+
  ∈-mapPartial {X = X} {y} {f} =

+
    (∃[ x ] x ∈ X × f x ≡ just y)

+
      ∼⟨ ∃-cong′ (R.K-refl ×-cong (∈-partialToSet {f = f})) ⟩

+
    (∃[ x ] x ∈ X × y ∈ partialToSet f x)

+
      ∼⟨ ∈-concatMapˢ ⟩

+
    y ∈ mapPartial f X ∎

+
    where open R.EquationalReasoning

+

+
  ⊆-mapPartial : ∀ {f : A → Maybe B} → map just (mapPartial f X) ⊆ map f X

+
  ⊆-mapPartial {f = f} a∈m with from ∈-map a∈m

+
  ... | x , refl , a∈mp with from (∈-mapPartial {f = f}) a∈mp

+
  ... | x' , x'∈X , jx≡fx = to ∈-map (x' , sym jx≡fx , x'∈X)

+

+
  binary-unions : ∃[ Y ] ∀ {a} → (a ∈ X ⊎ a ∈ X') ⇔ a ∈ Y

+
  binary-unions {X = X} {X'} with unions (fromList (X ∷ [ X' ]))

+
  ... | (Y , h) = Y , mk⇔ (λ where

+
    (inj₁ a∈X)  → to h (X  , to ∈-fromList (here refl)         , a∈X)

+
    (inj₂ a∈X') → to h (X' , to ∈-fromList (there (here refl)) , a∈X'))

+
    (λ a∈Y → case from h a∈Y of λ (T , H , a∈T) → case from ∈-fromList H of λ where

+
      (here refl) → inj₁ a∈T

+
      (there (here refl)) → inj₂ a∈T)

+

+
  _∪_ : Set A → Set A → Set A

+
  X ∪ Y = proj₁ binary-unions

+

+
  ∈-∪ : ∀ {a} → (a ∈ X ⊎ a ∈ Y) ⇔ a ∈ X ∪ Y

+
  ∈-∪ = proj₂ binary-unions

+

+
  spec-∈ : Type ℓ → Type ℓ

+
  spec-∈ A = {X : Set A} → specProperty (_∈ X)

+

+
  -- membership needs to be a specProperty to have intersections

+
  module Intersection (sp-∈ : spec-∈ A) where

+

+
    infixr 7 _∩_

+
    _∩_ : Set A → Set A → Set A

+
    X ∩ Y = filter sp-∈ X

+

+
    ∈-∩ : ∀ {a} → (a ∈ X × a ∈ Y) ⇔ a ∈ X ∩ Y

+
    ∈-∩ {X} {Y} {a} = (a ∈ X × a ∈ Y) ↔⟨ ×-comm _ _ ⟩

+
                      (a ∈ Y × a ∈ X) ∼⟨ ∈-filter ⟩

+
                      a ∈ X ∩ Y       ∎

+
      where open R.EquationalReasoning

+

+
    disjoint' : Set A → Set A → Type ℓ

+
    disjoint' X Y = X ∩ Y ≡ᵉ ∅

+

+
    _＼_ : Set A → Set A → Set A

+
    X ＼ Y = filter (sp-¬ (sp-∈ {Y})) X

+

+
  All : (A → Type) → Set A → Type ℓ

+
  All P X = ∀ {a} → a ∈ X → P a

+

+
  Any : (A → Type) → Set A → Type ℓ

+
  Any P X = ∃[ a ] a ∈ X × P a

+

+
-- finite set theories

+
record Theoryᶠ : Type₁ where

+
  field theory : Theory

+
  open Theory theory public

+

+
  field finiteness : (X : Set A) → finite X

+

+
  DecEq⇒strongly-finite : ⦃ DecEq A ⦄ → (X : Set A) → strongly-finite X

+
  DecEq⇒strongly-finite X = DecEq∧finite⇒strongly-finite X (finiteness X)

+

+
  lengthˢ : ⦃ DecEq A ⦄ → Set A → ℕ

+
  lengthˢ X = card (X , DecEq⇒strongly-finite X)

+

+
  module _ {A : Type} ⦃ _ : Show A ⦄ where

+
    instance

+
      Show-Set : Show (Set A)

+
      Show-Set .show = λ x → Show-finite .show (x , (finiteness x))

+

+
-- set theories with an infinite set (containing all natural numbers)

+
record Theoryⁱ : Type₁ where

+
  field theory : Theory

+
  open Theory theory public

+

+
  field infinity : ∃[ Y ] ((n : ℕ) → n ∈ Y)

+

+
-- theories with decidable properties

+
record Theoryᵈ : Type₁ where

+
  field th : Theory

+
  open Theory th public

+
  open Equivalence

+

+
  field

+
    ∈-sp : ⦃ DecEq A ⦄ → spec-∈ A

+
    _∈?_ : ⦃ DecEq A ⦄ → Decidable² (_∈_ {A = A})

+
    all? : {P : A → Type} (P? : Decidable¹ P) {X : Set A} → Dec (All P X)

+
    any? : {P : A → Type} (P? : Decidable¹ P) (X : Set A) → Dec (Any P X)

+

+

+
  module _ {A : Type} {P : A → Type} where

+
    module _ ⦃ _ : P ⁇¹ ⦄ where instance

+
      Dec-Allˢ : All P ⁇¹

+
      Dec-Allˢ = ⁇¹ λ x → all? dec¹ {x}

+

+
      Dec-Anyˢ : Any P ⁇¹

+
      Dec-Anyˢ = ⁇¹ any? dec¹

+

+
    module _ (P? : Decidable¹ P) where

+
      allᵇ anyᵇ : (X : Set A) → Bool

+
      allᵇ X = ⌊ all? P? {X} ⌋

+
      anyᵇ X = ⌊ any? P? X   ⌋

+

+
  module _ {A : Type} ⦃ _ : DecEq A ⦄ where

+

+
    _∈ᵇ_ : A → Set A → Bool

+
    a ∈ᵇ X = ⌊ a ∈? X ⌋

+

+
    instance

+
      Dec-∈ : _∈_ {A = A} ⁇²

+
      Dec-∈ = ⁇² _∈?_

+

+
    _ = _∈_  {A = A} ⁇² ∋ it

+
    _ = _⊆_  {A = A} ⁇² ∋ it

+
    _ = _≡ᵉ_ {A = A} ⁇² ∋ it

+

+
    incl-set' : (X : Set A) → A → Maybe (∃[ a ] a ∈ X)

+
    incl-set' X x with x ∈? X

+
    ... | yes p = just (x , p)

+
    ... | no  p = nothing

+

+
    incl-set : (X : Set A) → Set (∃[ a ] a ∈ X)

+
    incl-set X = mapPartial (incl-set' X) X

+

+
    module _ {X : Set A} where

+
      incl-set-proj₁⊆ : map proj₁ (incl-set X) ⊆ X

+
      incl-set-proj₁⊆ x with from ∈-map x

+
      ... | (_ , pf) , refl , _ = pf

+

+
      incl-set-proj₁⊇ : X ⊆ map proj₁ (incl-set X)

+
      incl-set-proj₁⊇ {x} x∈X with x ∈? X in eq

+
      ... | no ¬p = contradiction x∈X ¬p

+
      ... | yes p = to ∈-map

+
        ( (x , p)

+
        , refl

+
        , to (∈-mapPartial {f = incl-set' X}) (x , x∈X , helper eq)

+
        )

+
        where helper : x ∈? X ≡ yes p → incl-set' X x ≡ just (x , p)

+
              helper h with x ∈? X | h

+
              ... | _ | refl = refl

+

+
      incl-set-proj₁ : map proj₁ (incl-set X) ≡ᵉ X

+
      incl-set-proj₁ = incl-set-proj₁⊆ , incl-set-proj₁⊇

+

\ No newline at end of file
diff --git a/site/static/agda_html/Axiom.UniquenessOfIdentityProofs.html b/site/static/agda_html/Axiom.UniquenessOfIdentityProofs.html
new file mode 100644
index 000000000..c64bcbcc4
--- /dev/null
+++ b/site/static/agda_html/Axiom.UniquenessOfIdentityProofs.html
@@ -0,0 +1,153 @@
+
+Axiom.UniquenessOfIdentityProofs
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Results concerning uniqueness of identity proofs

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Axiom.UniquenessOfIdentityProofs where

+

+
open import Level using (Level)

+
open import Relation.Nullary.Decidable.Core using (recompute; recompute-constant)

+
open import Relation.Binary.Core

+
open import Relation.Binary.Definitions

+
open import Relation.Binary.PropositionalEquality.Core

+
open import Relation.Binary.PropositionalEquality.Properties

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+
    x y : A

+

+
------------------------------------------------------------------------

+
-- Definition

+
--

+
-- Uniqueness of Identity Proofs (UIP) states that all proofs of

+
-- equality are themselves equal. In other words, the equality relation

+
-- is irrelevant. Here we define UIP relative to a given type.

+

+
UIP : (A : Set a) → Set a

+
UIP A = Irrelevant {A = A} _≡_

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
-- UIP always holds when using axiom K

+
-- (see `Axiom.UniquenessOfIdentityProofs.WithK`).

+

+
-- The existence of a constant function over proofs of equality for

+
-- elements in A is enough to prove UIP for A. Indeed, we can relate any

+
-- proof to its image via this function which we then know is equal to

+
-- the image of any other proof.

+

+
module Constant⇒UIP

+
  (f : _≡_ {A = A} ⇒ _≡_)

+
  (f-constant : ∀ {x y} (p q : x ≡ y) → f p ≡ f q)

+
  where

+

+
  ≡-canonical : (p : x ≡ y) → trans (sym (f refl)) (f p) ≡ p

+
  ≡-canonical refl = trans-symˡ (f refl)

+

+
  ≡-irrelevant : UIP A

+
  ≡-irrelevant p q = begin

+
    p                          ≡⟨ sym (≡-canonical p) ⟩

+
    trans (sym (f refl)) (f p) ≡⟨ cong (trans _) (f-constant p q) ⟩

+
    trans (sym (f refl)) (f q) ≡⟨ ≡-canonical q ⟩

+
    q                          ∎

+
    where open ≡-Reasoning

+

+
-- If equality is decidable for a given type, then we can prove UIP for

+
-- that type. Indeed, the decision procedure allows us to define a

+
-- function over proofs of equality which is constant: it returns the

+
-- proof produced by the decision procedure.

+

+
module Decidable⇒UIP (_≟_ : DecidableEquality A)

+
  where

+

+
  ≡-normalise : _≡_ {A = A} ⇒ _≡_

+
  ≡-normalise {x} {y} x≡y = recompute (x ≟ y) x≡y

+

+
  ≡-normalise-constant : (p q : x ≡ y) → ≡-normalise p ≡ ≡-normalise q

+
  ≡-normalise-constant {x = x} {y = y} = recompute-constant (x ≟ y)

+

+
  ≡-irrelevant : UIP A

+
  ≡-irrelevant = Constant⇒UIP.≡-irrelevant ≡-normalise ≡-normalise-constant

+

\ No newline at end of file
diff --git a/site/static/agda_html/CategoricalCrypto.html b/site/static/agda_html/CategoricalCrypto.html
new file mode 100644
index 000000000..912eb9318
--- /dev/null
+++ b/site/static/agda_html/CategoricalCrypto.html
@@ -0,0 +1,409 @@
+
+CategoricalCrypto
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module CategoricalCrypto where

+

+
open import abstract-set-theory.Prelude hiding (id; _∘_; _⊗_; lookup; Dec)

+

+
--------------------------------------------------------------------------------

+
-- Channels, which form the objects

+

+
record Channel : Set₁ where

+
  field P : Set

+
        rcvType sndType : P → Set

+

+
infixr 9 _⊗_

+

+
I : Channel

+
I .Channel.P = ⊥

+

+
_ᵀ : Channel → Channel

+
_ᵀ c = let open Channel c in λ where .P → P ; .rcvType → sndType ; .sndType → rcvType

+

+
_⊗_ : Channel → Channel → Channel

+
c₁ ⊗ c₂ = let open Channel c₁ renaming (P to P₁; rcvType to rcvType₁; sndType to sndType₁)

+
              open Channel c₂ renaming (P to P₂; rcvType to rcvType₂; sndType to sndType₂)

+
              open Channel

+
  in λ where

+
    .P → P₁ ⊎ P₂

+
    .rcvType (inj₁ a) → rcvType₁ a

+
    .rcvType (inj₂ b) → rcvType₂ b

+
    .sndType (inj₁ a) → sndType₁ a

+
    .sndType (inj₂ b) → sndType₂ b

+

+
snd₁ : ∀ {A B} → ∃ (Channel.sndType A) → ∃ (Channel.sndType (A ⊗ B))

+
snd₁ (p , m) = inj₁ p , m

+

+
snd₂ : ∀ {A B} → ∃ (Channel.sndType B) → ∃ (Channel.sndType (A ⊗ B))

+
snd₂ (p , m) = inj₂ p , m

+

+
-- being lazy here, this should be an iso instead

+
postulate

+
  ⊗-assoc : ∀ {A B C} → (A ⊗ B) ⊗ C ≡ A ⊗ (B ⊗ C)

+

+
--------------------------------------------------------------------------------

+
-- Machines, which form the morphisms

+

+
record Machine (A B : Channel) : Set₁ where

+
  constructor MkMachine

+
  open Channel (A ⊗ B ᵀ) public

+

+
  field State : Set

+
        stepRel : ∃ rcvType → State → State × Maybe (∃ sndType) → Set

+

+
module _ {A B} (let open Channel (A ⊗ B ᵀ)) where

+
  MkMachine' : ∀ {State} → (∃ rcvType → State → State × Maybe (∃ sndType) → Set) → Machine A B

+
  MkMachine' = MkMachine _

+

+
  StatelessMachine : (∃ rcvType → Maybe (∃ sndType) → Set) → Machine A B

+
  StatelessMachine R = MkMachine ⊤ (λ i _ (_ , o) → R i o)

+

+
  FunctionMachine : (∃ rcvType → Maybe (∃ sndType)) → Machine A B

+
  FunctionMachine f = StatelessMachine (λ i o → f i ≡ o)

+

+
id : ∀ {A} → Machine A A

+
id .Machine.State   = ⊤

+
id .Machine.stepRel (inj₁ a , m) _ (_ , m') = m' ≡ just (inj₂ a , m)

+
id .Machine.stepRel (inj₂ a , m) _ (_ , m') = m' ≡ just (inj₁ a , m)

+

+
module Tensor {A B C D} (M₁ : Machine A B) (M₂ : Machine C D) where

+
  open Machine M₁ renaming (State to State₁; stepRel to stepRel₁)

+
  open Machine M₂ renaming (State to State₂; stepRel to stepRel₂)

+

+
  State = State₁ × State₂

+

+
  AllCs = (A ⊗ B ᵀ) ⊗ (C ⊗ D ᵀ)

+

+
  data CompRel : ∃ (Channel.rcvType AllCs) → State → State × Maybe (∃ (Channel.sndType AllCs)) → Set where

+
    Step₁ : ∀ {p m m' s s' s₂} → stepRel₁ (p , m) s (s' , m')

+
          → CompRel (inj₁ p , m) (s , s₂) ((s' , s₂) , (snd₁ <$> m'))

+

+
    Step₂ : ∀ {p m m' s s' s₁} → stepRel₂ (p , m) s (s' , m')

+
          → CompRel (inj₂ p , m) (s₁ , s) ((s₁ , s') , (snd₂ <$> m'))

+

+

+
  _⊗'_ : Machine (A ⊗ C) (B ⊗ D)

+
  _⊗'_ = λ where

+
    .Machine.State → State

+
    .Machine.stepRel m s (s' , m') → CompRel

+
      (case m of λ where

+
        (inj₁ (inj₁ p) , m) → (inj₁ (inj₁ p) , m)

+
        (inj₁ (inj₂ p) , m) → (inj₂ (inj₁ p) , m)

+
        (inj₂ (inj₁ p) , m) → (inj₁ (inj₂ p) , m)

+
        (inj₂ (inj₂ p) , m) → (inj₂ (inj₂ p) , m))

+
      s

+
      (s' , ((λ where

+
        (inj₁ (inj₁ p) , m) → (inj₁ (inj₁ p) , m)

+
        (inj₁ (inj₂ p) , m) → (inj₂ (inj₁ p) , m)

+
        (inj₂ (inj₁ p) , m) → (inj₁ (inj₂ p) , m)

+
        (inj₂ (inj₂ p) , m) → (inj₂ (inj₂ p) , m)) <$> m'))

+

+
open Tensor using (_⊗'_) public

+

+
_⊗ˡ_ : ∀ {A B} → (C : Channel) → Machine A B → Machine (C ⊗ A) (C ⊗ B)

+
C ⊗ˡ M = id ⊗' M

+

+
_⊗ʳ_ : ∀ {A B} → Machine A B → (C : Channel) → Machine (A ⊗ C) (B ⊗ C)

+
M ⊗ʳ C = M ⊗' id

+

+
module Comp {A B C} (M₁ : Machine B C) (M₂ : Machine A B) where

+
  open Machine M₁ renaming (State to State₁; stepRel to stepRel₁)

+
  open Machine M₂ renaming (State to State₂; stepRel to stepRel₂)

+

+
  State = State₁ × State₂

+

+
  AllCs = (A ⊗ B ᵀ) ⊗ (B ⊗ C ᵀ)

+

+
  data CompRel : ∃ (Channel.rcvType AllCs) → State → State × Maybe (∃ (Channel.sndType AllCs)) → Set where

+
    Step₁ : ∀ {p m m' s s' s₂} → stepRel₁ (p , m) s (s' , m')

+
          → CompRel (inj₂ p , m) (s , s₂) ((s' , s₂) , (snd₂ <$> m'))

+

+
    Step₂ : ∀ {p m m' s s' s₁} → stepRel₂ (p , m) s (s' , m')

+
          → CompRel (inj₁ p , m) (s₁ , s) ((s₁ , s') , (snd₁ <$> m'))

+

+
    Multi₁ : ∀ {p m m' m'' s s' s''}

+
           → CompRel m s (s' , just ((inj₂ (inj₁ p) , m')))

+
           → CompRel (inj₁ (inj₂ p) , m') s' (s'' , m'')

+
           → CompRel m s (s'' , m'')

+

+
    Multi₂ : ∀ {p m m' m'' s s' s''}

+
           → CompRel m s (s' , just (inj₁ (inj₂ p) , m'))

+
           → CompRel (inj₂ (inj₁ p) , m') s' (s'' , m'')

+
           → CompRel m s (s'' , m'')

+

+
  infixr 9 _∘_

+

+
  _∘_ : Machine A C

+
  _∘_ = λ where

+
    .Machine.State → State

+
    .Machine.stepRel m s (s' , m') →

+
      CompRel

+
        (case m of λ where (inj₁ x , m) → inj₁ (inj₁ x) , m ; (inj₂ y , m) → inj₂ (inj₂ y) , m)

+
        s (s' ,  ((λ where (inj₁ x , m) → inj₁ (inj₁ x) , m ; (inj₂ y , m) → inj₂ (inj₂ y) , m) <$> m'))

+

+
open Comp using (_∘_) public

+

+
--------------------------------------------------------------------------------

+
-- Environment model

+

+
ℰ-Out : Channel

+
ℰ-Out .Channel.P = ⊤

+
ℰ-Out .Channel.sndType _ = Bool

+
ℰ-Out .Channel.rcvType _ = ⊥

+

+
-- Presheaf on the category of channels & machines

+
-- we just take machines that output a boolean

+
-- for now, not on the Kleisli construction

+
ℰ : Channel → Set₁

+
ℰ C = Machine C ℰ-Out

+

+
map-ℰ : ∀ {A B} → Machine A B → ℰ B → ℰ A

+
map-ℰ M E = E ∘ M

+

+
--------------------------------------------------------------------------------

+
-- UC relations

+

+
-- perfect equivalence

+
_≈ℰ_ : ∀ {A B} → Machine A B → Machine A B → Set₁

+
_≈ℰ_ {B = B} M M' = (E : ℰ B) → map-ℰ M E ≡ map-ℰ M' E

+

+
_≤UC_ : ∀ {A B E E''} → Machine A (B ⊗ E) → Machine A (B ⊗ E'') → Set₁

+
_≤UC_ {B = B} {E} R I = ∀ E' (A : Machine E E') → ∃[ S ] ((B ⊗ˡ A) ∘ R) ≈ℰ ((B ⊗ˡ S) ∘ I)

+

+
-- equivalent to _≤UC_ by "completeness of the dummy adversary"

+
_≤'UC_ : ∀ {A B E} → Machine A (B ⊗ E) → Machine A (B ⊗ E) → Set₁

+
_≤'UC_ {B = B} R I = ∃[ S ] R ≈ℰ (B ⊗ˡ S ∘ I)

+

+

+
--------------------------------------------------------------------------------

+
-- Example functionalities

+

+
module LeakyChannel (M : Set) where

+
  -- authenticated, non-lossy, leaks all messages

+

+
  open Channel

+

+
  A B E : Channel

+

+
  -- can receive messages from Alice (in reverse)

+
  A .P = ⊤

+
  A .sndType _ = M

+
  A .rcvType _ = ⊥

+

+
  -- can send messages to Bob (in reverse)

+
  B .P = ⊤

+
  B .sndType _ = ⊥

+
  B .rcvType _ = M

+

+
  -- upon request, can send next message to Eve (in reverse)

+
  E .P = ⊤

+
  E .sndType _ = ⊤

+
  E .rcvType _ = M

+

+
  open Machine hiding (rcvType; sndType)

+

+
  data R : ∃ (rcvType (I ⊗ (A ⊗ B ⊗ E) ᵀ))

+
         → List M

+
         → List M × Maybe (∃ (sndType (I ⊗ (A ⊗ B ⊗ E) ᵀ))) → Set where

+

+
    Send : ∀ {m s} → R (inj₂ (inj₁ _) , m) s (s ∷ʳ m , just (inj₂ (inj₂ (inj₁ _)) , m))

+
    Req  : ∀ {m s} → R (inj₂ (inj₂ (inj₂ _)) , _) (m ∷ s) (s , just (inj₂ (inj₂ (inj₂ _)) , m))

+

+
  Functionality : Machine I (A ⊗ B ⊗ E)

+
  Functionality .State = List M -- queue of messages

+
  Functionality .stepRel = R

+

+

+

+
module SecureChannel (M : Set) where

+
  -- authenticated, non-lossy, leaks only message length

+

+
  open Channel

+

+
  A B E : Channel

+

+
  -- can receive messages from Alice (in reverse)

+
  A .P = ⊤

+
  A .sndType _ = M

+
  A .rcvType _ = ⊥

+

+
  -- can send messages to Bob (in reverse)

+
  B .P = ⊤

+
  B .sndType _ = ⊥

+
  B .rcvType _ = M

+

+
  -- upon request, can send length of the next message to Eve (in reverse)

+
  E .P = ⊤

+
  E .sndType _ = ⊤

+
  E .rcvType _ = ℕ

+

+
  module _ (msgLen : M → ℕ) where

+

+
    open Machine hiding (rcvType; sndType)

+

+
    data R : ∃ (rcvType (I ⊗ (A ⊗ B ⊗ E) ᵀ))

+
           → List M

+
           → List M × Maybe (∃ (sndType (I ⊗ (A ⊗ B ⊗ E) ᵀ))) → Set where

+
      Send : ∀ {m s} → R (inj₂ (inj₁ _) , m) s (s ∷ʳ m , just (inj₂ (inj₂ (inj₁ _)) , m))

+
      Req  : ∀ {m s} → R (inj₂ (inj₂ (inj₂ _)) , _) (m ∷ s) (s , just (inj₂ (inj₂ (inj₂ _)) , msgLen m))

+

+
    Functionality : Machine I (A ⊗ B ⊗ E)

+
    Functionality .State = List M -- queue of messages

+
    Functionality .stepRel = R

+

+
    Functionality' : Machine I ((A ⊗ B) ⊗ E)

+
    Functionality' = subst (Machine I) (sym ⊗-assoc) Functionality

+

+

+

+
module Encryption (PlainText CipherText PubKey PrivKey : Set)

+
                  ⦃ _ : DecEq CipherText ⦄ ⦃ _ : DecEq PubKey ⦄

+
                  (genCT : ℕ → CipherText) (getPubKey : PrivKey → PubKey) where

+
  open Channel

+
  open Machine hiding (rcvType; sndType)

+

+
  C : Channel

+
  C .P = ⊤

+
  C .sndType _ = PlainText × PubKey ⊎ CipherText × PrivKey

+
  C .rcvType _ = CipherText ⊎ Maybe PlainText

+

+
  S : Set

+
  S = List (PubKey × PlainText × CipherText)

+

+
  lookup : {A : Set} → List A → (A → Bool) → Maybe A

+
  lookup [] f = nothing

+
  lookup (x ∷ l) f with f x

+
  ... | true  = just x

+
  ... | false = lookup l f

+

+
  lookupPlainText : S → CipherText × PubKey → Maybe PlainText

+
  lookupPlainText s (c , k) = proj₁ <$> (proj₂ <$> lookup s λ where (k' , _ , c') → ¿ k ≡ k' × c ≡ c' ¿ᵇ)

+

+
  data R : ∃ (rcvType (I ⊗ (C ᵀ))) → S → S × Maybe (∃ (sndType (I ⊗ (C ᵀ)))) → Set where

+
    Enc : ∀ {p k s} → let c = genCT (length s)

+
       in R (inj₂ tt , inj₁ (p , k)) s ((k , p , c) ∷ s , just (inj₂ tt , inj₁ c))

+
    Dec : ∀ {c k s} → let p = lookupPlainText s (c , getPubKey k)

+
       in R (inj₂ tt , inj₂ (c , k)) s (s , just (inj₂ tt , inj₂ p))

+

+
  Functionality : Machine I C

+
  Functionality .State   = S

+
  Functionality .stepRel = R

+

+
module EncryptionShim (PlainText CipherText PubKey PrivKey : Set)

+
                      ⦃ _ : DecEq CipherText ⦄ ⦃ _ : DecEq PubKey ⦄

+
                      (genCT : ℕ → CipherText) (getPubKey : PrivKey → PubKey)

+
                      (pubKey : PubKey) (privKey : PrivKey) where

+
  open Channel

+
  open Machine hiding (rcvType; sndType)

+

+
  module L = LeakyChannel CipherText

+
  module S = SecureChannel PlainText

+
  module E = Encryption PlainText CipherText PubKey PrivKey genCT getPubKey

+

+
  data R : ∃ (rcvType ((L.A ⊗ L.B ⊗ L.E) ⊗ ((S.A ⊗ S.B ⊗ S.E) ᵀ)))

+
         → E.Functionality .State

+
         → E.Functionality .State × Maybe (∃ (sndType ((L.A ⊗ L.B ⊗ L.E) ⊗ ((S.A ⊗ S.B ⊗ S.E) ᵀ))))

+
         → Set where

+
    EncSend : ∀ {m m' s s'}

+
            → E.R (inj₂ _ , inj₁ (m , pubKey)) s (s' , just (inj₂ _ , inj₁ m'))

+
            → R (inj₂ (inj₁ _) , m) s (s' , just (inj₁ (inj₁ _) , m'))

+
    DecRcv  : ∀ {m m' s s'}

+
            → E.R (inj₂ _ , inj₂ (m , privKey)) s (s' , just (inj₂ _ , inj₂ (just m')))

+
            → R (inj₁ (inj₂ (inj₁ _)) , m) s (s' , just (inj₂ (inj₂ (inj₁ _)) , m'))

+

+
  Functionality : Machine (L.A ⊗ L.B ⊗ L.E) (S.A ⊗ S.B ⊗ S.E)

+
  Functionality .State   = E.Functionality .State

+
  Functionality .stepRel = R

+

+
module SecureFromAuthenticated (PlainText CipherText PubKey PrivKey : Set)

+
                               ⦃ _ : DecEq CipherText ⦄ ⦃ _ : DecEq PubKey ⦄

+
                               (genCT : ℕ → CipherText) (getPubKey : PrivKey → PubKey)

+
                               (pubKey : PubKey) (privKey : PrivKey)

+
                               (msgLength : PlainText → ℕ) where

+

+
  module L  = LeakyChannel CipherText

+
  module S  = SecureChannel PlainText

+
  module SH = EncryptionShim PlainText CipherText PubKey PrivKey genCT getPubKey pubKey privKey

+

+
  Functionality : Machine I (S.A ⊗ S.B ⊗ S.E)

+
  Functionality = SH.Functionality ∘ L.Functionality

+

+
  Functionality' : Machine I ((S.A ⊗ S.B) ⊗ S.E)

+
  Functionality' = subst (Machine I) (sym ⊗-assoc) Functionality

+

+
  -- F≤Secure : Functionality' ≤'UC S.Functionality' msgLength

+
  -- F≤Secure = {!!}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Applicative.Core.html b/site/static/agda_html/Class.Applicative.Core.html
new file mode 100644
index 000000000..7102a971d
--- /dev/null
+++ b/site/static/agda_html/Class.Applicative.Core.html
@@ -0,0 +1,131 @@
+
+Class.Applicative.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
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+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Applicative.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+
open import Class.Functor.Core

+

+
record Applicative (F : Type↑) : Typeω where

+
  infixl 4 _<*>_ _⊛_ _<*_ _<⊛_ _*>_ _⊛>_

+
  infix  4 _⊗_

+

+
  field

+
    overlap ⦃ super ⦄ : Functor F

+
    pure : A → F A

+
    _<*>_  : F (A → B) → F A → F B

+
  _⊛_ = _<*>_

+

+
  _<*_ : F A → F B → F A

+
  x <* y = const <$> x ⊛ y

+

+
  _*>_ : F A → F B → F B

+
  x *> y = constᵣ <$> x ⊛ y

+

+
  _<⊛_ = _<*_; _⊛>_ = _*>_

+

+
  _⊗_ : F A → F B → F (A × B)

+
  x ⊗ y = (_,_) <$> x ⊛ y

+

+
  zipWithA : (A → B → C) → F A → F B → F C

+
  zipWithA f x y = f <$> x ⊛ y

+

+
  zipA : F A → F B → F (A × B)

+
  zipA = zipWithA _,_

+

+
open Applicative ⦃...⦄ public

+

+
record Applicative₀ (F : Type↑) : Typeω where

+
  field

+
    overlap ⦃ super ⦄ : Applicative F

+
    ε₀ : F A

+
open Applicative₀ ⦃...⦄ public

+

+
record Alternative (F : Type↑) : Typeω where

+
  infixr 3 _<|>_

+
  field _<|>_ : F A → F A → F A

+
open Alternative ⦃...⦄ public

+

+
infix -1 ⋃⁺_ ⋃_

+

+
⋃⁺_ : ⦃ Alternative F ⦄ → List⁺ (F A) → F A

+
⋃⁺_ = foldr₁ _<|>_

+

+
⋃_ : ⦃ Applicative₀ F ⦄ → ⦃ Alternative F ⦄ → List (F A) → F A

+
⋃_ = foldr _<|>_ ε₀

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Applicative.Instances.html b/site/static/agda_html/Class.Applicative.Instances.html
new file mode 100644
index 000000000..babe6d9d1
--- /dev/null
+++ b/site/static/agda_html/Class.Applicative.Instances.html
@@ -0,0 +1,139 @@
+
+Class.Applicative.Instances
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Applicative.Instances where

+

+
open import Class.Prelude

+
open import Class.Functor.Core

+
open import Class.Functor.Instances

+
open import Class.Applicative.Core

+

+
instance

+
  Applicative-Maybe : Applicative Maybe

+
  Applicative-Maybe = λ where

+
    .pure → just

+
    ._<*>_ → maybe fmap (const nothing)

+

+
  Applicative₀-Maybe : Applicative₀ Maybe

+
  Applicative₀-Maybe .ε₀ = nothing

+

+
  Alternative-Maybe : Alternative Maybe

+
  Alternative-Maybe ._<|>_ = May._<∣>_

+
    where import Data.Maybe as May

+

+
  Applicative-List : Applicative List

+
  Applicative-List = λ where

+
    .pure → [_]

+
    ._<*>_ → flip $ concatMap ∘ _<&>_

+

+
  Applicative₀-List : Applicative₀ List

+
  Applicative₀-List .ε₀ = []

+

+
  Alternative-List : Alternative List

+
  Alternative-List ._<|>_ = _++_

+

+
  Applicative-List⁺ : Applicative List⁺

+
  Applicative-List⁺ = λ where

+
    .pure → L⁺.[_]

+
    ._<*>_ → flip $ L⁺.concatMap ∘ _<&>_

+
   where import Data.List.NonEmpty as L⁺

+

+
  Applicative-Vec : ∀ {n} → Applicative (flip Vec n)

+
  Applicative-Vec = λ where

+
    .pure → V.replicate _

+
    ._<*>_ → V._⊛_

+
   where import Data.Vec as V

+

+

+
  Applicative₀-Vec : Applicative₀ (flip Vec 0)

+
  Applicative₀-Vec .ε₀ = []

+

+
  -- Applicative-∃Vec : Applicative (∃ ∘ Vec)

+
  -- Applicative-∃Vec = λ where

+
  --   .pure x → 1 , pure x

+
  --   ._<*>_ (n , xs) (m , ys) → {! (n ⊔ m) , zipWith _$_ xs ys  -- (+ zipWith-⊔ lemma) !}

+

+
  private module M where

+
    open import Reflection.TCM.Syntax public

+
    open import Reflection.TCM public

+

+
  Alternative-TC : Alternative TC

+
  Alternative-TC = record {M}

+

+
  Applicative-TC : Applicative TC

+
  Applicative-TC = record {M}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Applicative.html b/site/static/agda_html/Class.Applicative.html
new file mode 100644
index 000000000..547012b14
--- /dev/null
+++ b/site/static/agda_html/Class.Applicative.html
@@ -0,0 +1,82 @@
+
+Class.Applicative
+             
+             
+             

+               

+                 Back
+                 
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+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Applicative where

+

+
open import Class.Applicative.Core public

+
open import Class.Applicative.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Bifunctor.html b/site/static/agda_html/Class.Bifunctor.html
new file mode 100644
index 000000000..c76345ccf
--- /dev/null
+++ b/site/static/agda_html/Class.Bifunctor.html
@@ -0,0 +1,142 @@
+
+Class.Bifunctor
+             
+             
+             

+               

+                 Back
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+               

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+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Bifunctor where

+

+
open import Class.Prelude hiding (A; B; C)

+
open import Class.Core

+
import Data.Product as ×

+
import Data.Sum     as ⊎

+

+
private variable

+
  a b : Level

+
  A : Type a; A′ : Type a; A″ : Type a

+
  B : A → Type b; B′ : A → Type b; C : A′ → Type b

+

+
-- ** indexed/dependent version

+
record BifunctorI

+
  (F : (A : Type a) (B : A → Type b) → Type (a ⊔ b)) : Type (lsuc (a ⊔ b)) where

+
  field

+
    bimap′ : (f : A → A′) → (∀ {x} → B x → C (f x)) → F A B → F A′ C

+

+
  map₁′ : (A → A′) → F A (const A″) → F A′ (const A″)

+
  map₁′ f = bimap′ f id

+
  _<$>₁′_ = map₁′

+

+
  map₂′ : (∀ {x} → B x → B′ x) → F A B → F A B′

+
  map₂′ g = bimap′ id g

+
  _<$>₂′_ = map₂′

+

+
  infixl 4 _<$>₁′_ _<$>₂′_

+

+
open BifunctorI ⦃...⦄ public

+

+
instance

+
  Bifunctor-Σ : BifunctorI {a}{b} Σ

+
  Bifunctor-Σ .bimap′ = ×.map

+

+
-- ** non-dependent version

+
record Bifunctor (F : Type a → Type b → Type (a ⊔ b)) : Type (lsuc (a ⊔ b)) where

+
  field

+
    bimap : ∀ {A A′ : Type a} {B B′ : Type b} → (A → A′) → (B → B′) → F A B → F A′ B′

+

+
  map₁ : ∀ {A A′ : Type a} {B : Type b} → (A → A′) → F A B → F A′ B

+
  map₁ f = bimap f id

+
  _<$>₁_ = map₁

+

+
  map₂ : ∀ {A : Type a} {B B′ : Type b} → (B → B′) → F A B → F A B′

+
  map₂ g = bimap id g

+
  _<$>₂_ = map₂

+

+
  infixl 4 _<$>₁_ _<$>₂_

+

+
open Bifunctor ⦃...⦄ public

+

+
map₁₂ : ∀ {F : Type a → Type a → Type a} {A B : Type a}

+
  → ⦃ Bifunctor F ⦄

+
  → (A → B) → F A A → F B B

+
map₁₂ f = bimap f f

+
_<$>₁₂_ = map₁₂

+
infixl 4 _<$>₁₂_

+

+
instance

+
  Bifunctor-× : Bifunctor {a}{b} _×_

+
  Bifunctor-× .bimap f g = ×.map f g

+

+
  Bifunctor-⊎ : Bifunctor {a}{b} _⊎_

+
  Bifunctor-⊎ .bimap = ⊎.map

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.CommutativeMonoid.Core.html b/site/static/agda_html/Class.CommutativeMonoid.Core.html
new file mode 100644
index 000000000..51d93e6d0
--- /dev/null
+++ b/site/static/agda_html/Class.CommutativeMonoid.Core.html
@@ -0,0 +1,103 @@
+
+Class.CommutativeMonoid.Core
+             
+             
+             

+               

+                 Back
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+               

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+             
{-# OPTIONS --safe --without-K #-}

+

+
module Class.CommutativeMonoid.Core where

+

+
import Algebra as Alg

+
open import Class.Prelude

+
open import Class.Semigroup

+
open import Class.Monoid

+

+
record CommutativeMonoid c ℓ Carrier : Type (lsuc (c ⊔ ℓ)) where

+
  infix  4 _≈_

+
  field

+
    _≈_                 : Carrier → Carrier → Type ℓ

+
    ⦃ semigroup ⦄       : Semigroup Carrier

+
    ⦃ monoid ⦄          : Monoid Carrier

+
    isCommutativeMonoid : Alg.IsCommutativeMonoid {c} _≈_ _◇_ ε

+

+
module Conversion {c ℓ} where

+
  toBundle : ∀ {Carrier} → CommutativeMonoid c ℓ Carrier → Alg.CommutativeMonoid c ℓ

+
  toBundle c = record { CommutativeMonoid c }

+

+
  fromBundle : (m : Alg.CommutativeMonoid c ℓ) → CommutativeMonoid c ℓ (Alg.CommutativeMonoid.Carrier m)

+
  fromBundle c = record

+
    { Alg.CommutativeMonoid c using (_≈_; isCommutativeMonoid)

+
    ; semigroup = λ where ._◇_ → Alg.CommutativeMonoid._∙_ c

+
    ; monoid    = λ where .ε   → Alg.CommutativeMonoid.ε c }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.CommutativeMonoid.Instances.html b/site/static/agda_html/Class.CommutativeMonoid.Instances.html
new file mode 100644
index 000000000..c0cc4e11e
--- /dev/null
+++ b/site/static/agda_html/Class.CommutativeMonoid.Instances.html
@@ -0,0 +1,88 @@
+
+Class.CommutativeMonoid.Instances
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
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+                     
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+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module Class.CommutativeMonoid.Instances where

+

+
open import Class.CommutativeMonoid.Core

+

+
open import Data.Nat.Properties

+

+
module NonUniqueInstances where

+
  CommMonoid-ℕ-+ = Conversion.fromBundle +-0-commutativeMonoid

+
  CommMonoid-ℕ-* = Conversion.fromBundle *-1-commutativeMonoid

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.CommutativeMonoid.html b/site/static/agda_html/Class.CommutativeMonoid.html
new file mode 100644
index 000000000..9b4db86b7
--- /dev/null
+++ b/site/static/agda_html/Class.CommutativeMonoid.html
@@ -0,0 +1,83 @@
+
+Class.CommutativeMonoid
+             
+             
+             

+               

+                 Back
+                 
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+               

+                 
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+             
{-# OPTIONS --safe --without-K #-}

+

+
module Class.CommutativeMonoid where

+

+
open import Class.CommutativeMonoid.Core public

+
open import Class.CommutativeMonoid.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Computational.html b/site/static/agda_html/Class.Computational.html
new file mode 100644
index 000000000..25b4c3056
--- /dev/null
+++ b/site/static/agda_html/Class.Computational.html
@@ -0,0 +1,269 @@
+
+Class.Computational
+             
+             
+             

+               

+                 Back
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+               

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+                     
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+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Class.Computational where

+

+
open import Agda.Primitive using () renaming (Set to Type)

+
open import Level

+

+
open import Prelude.Init hiding (map)

+

+
open import Class.Applicative

+
open import Class.Bifunctor

+
open import Class.DecEq

+
open import Class.Functor renaming (fmap to map)

+
open import Class.Monad

+
open import Function

+
open import Relation.Binary.PropositionalEquality

+

+
private variable

+
  a : Level

+
  I O : Type

+
  i : I

+
  o o' : O

+
  Err Err₁ Err₂ : Type

+

+
data ComputationResult {a : Level} (Err : Type) (R : Type a) : Type a where

+
  success : R → ComputationResult Err R

+
  failure : Err → ComputationResult Err R

+

+
isFailure : ∀ {A : Type a} → ComputationResult Err A → Type a

+
isFailure x = ∃[ e ] x ≡ failure e

+

+
module _ {a b} {E : Type} {A : Type a} {B : Type b} where

+
  caseCR_∣_∣_ : (ma : ComputationResult E A) → (∀ {a} → ma ≡ success a → B) → (isFailure ma → B) → B

+
  caseCR ma ∣ f ∣ g with ma

+
  ... | success _ = f refl

+
  ... | failure e = g (e , refl)

+

+
  caseCR-success : ∀ {a} {ma : ComputationResult E A} {f : ∀ {a} → ma ≡ success a → B} {g : isFailure ma → B}

+
    → (eq : ma ≡ success a)

+
    → caseCR ma ∣ f ∣ g ≡ f eq

+
  caseCR-success refl = refl

+

+
  caseCR-failure : {ma : ComputationResult E A} {f : ∀ {a} → ma ≡ success a → B} {g : isFailure ma → B}

+
    → (eq : isFailure ma)

+
    → caseCR ma ∣ f ∣ g ≡ g eq

+
  caseCR-failure (_ , refl) = refl

+

+
instance

+
  Bifunctor-ComputationResult : ∀ {a : Level} → Bifunctor {_} {a} ComputationResult

+
  Bifunctor-ComputationResult .bimap _ f (success x) = success $ f x

+
  Bifunctor-ComputationResult .bimap f _ (failure x) = failure $ f x

+

+
  Functor-ComputationResult : ∀ {E : Type} → Functor (ComputationResult E)

+
  Functor-ComputationResult ._<$>_ f (success x) = success $ f x

+
  Functor-ComputationResult ._<$>_ _ (failure x) = failure x

+

+
  Applicative-ComputationResult : ∀ {E : Type} → Applicative (ComputationResult E)

+
  Applicative-ComputationResult .pure = success

+
  Applicative-ComputationResult ._<*>_ (success f) x = f <$> x

+
  Applicative-ComputationResult ._<*>_ (failure e) _ = failure e

+

+
  Monad-ComputationResult : ∀ {E : Type} → Monad (ComputationResult E)

+
  Monad-ComputationResult .return = success

+
  Monad-ComputationResult ._>>=_ (success a) m = m a

+
  Monad-ComputationResult ._>>=_ (failure e) _ = failure e

+

+
map-failure : ∀ {A B C : Type} {f : A → B} {x : C} {ma} → ma ≡ failure x → map f ma ≡ failure x

+
map-failure refl = refl

+

+
success-maybe : ∀ {R : Type} → ComputationResult Err R → Maybe R

+
success-maybe (success x) = just x

+
success-maybe (failure _) = nothing

+

+
failure-maybe : ∀ {R : Type} → ComputationResult Err R → Maybe Err

+
failure-maybe (success _) = nothing

+
failure-maybe (failure x) = just x

+

+
_≈ᶜʳ_ : ∀ {A} → ComputationResult Err A → ComputationResult Err A → Type

+
x ≈ᶜʳ y = success-maybe x ≡ success-maybe y

+

+
module _ (STS : I → O → Type) where

+

+
  ExtendedRel : I → ComputationResult Err O → Type

+
  ExtendedRel i (success o) = STS i o

+
  ExtendedRel i (failure _) = ∀ o → ¬ STS i o

+

+
  record Computational Err : Type₁ where

+
    constructor MkComputational

+
    field

+
      computeProof : (i : I) → ComputationResult Err (∃[ o ] STS i o)

+

+
    compute : I → ComputationResult Err O

+
    compute i = map proj₁ $ computeProof i

+

+
    field

+
      completeness : (i : I) → (o : O) → STS i o → compute i ≡ success o

+

+
    open ≡-Reasoning

+

+
    computeFail : I → Type

+
    computeFail i = isFailure $ compute i

+

+
    ≡-success⇔STS : compute i ≡ success o ⇔ STS i o

+
    ≡-success⇔STS {i} {o} with computeProof i in eq

+
    ... | success (o' , h) = mk⇔ (λ where refl → h) λ h' →

+
      begin success o' ≡˘⟨ completeness _ _ h ⟩

+
            compute i  ≡⟨ completeness _ _ h' ⟩

+
            success o  ∎

+
    ... | failure y = mk⇔ (λ ()) λ h →

+
      begin failure _ ≡˘⟨ map-failure eq ⟩

+
            compute i ≡⟨ completeness _ _ h ⟩

+
            success o ∎

+

+
    failure⇒∀¬STS : computeFail i → ∀ o → ¬ STS i o

+
    failure⇒∀¬STS comp≡nothing o h rewrite ≡-success⇔STS .Equivalence.from h =

+
      case comp≡nothing of λ ()

+

+
    ∀¬STS⇒failure : (∀ o → ¬ STS i o) → computeFail i

+
    ∀¬STS⇒failure {i = i} ¬sts with computeProof i

+
    ... | success (x , y) = ⊥-elim (¬sts x y)

+
    ... | failure y = (y , refl)

+

+
    failure⇔∀¬STS : computeFail i ⇔ ∀ o → ¬ STS i o

+
    failure⇔∀¬STS = mk⇔ failure⇒∀¬STS ∀¬STS⇒failure

+

+
    recomputeProof : STS i o → ComputationResult Err (∃[ o' ] STS i o')

+
    recomputeProof _ = computeProof _

+

+
module _ (STS : I → O → Type) ⦃ comp : Computational STS Err ⦄ where

+

+
  open Computational comp

+

+
  ExtendedRelSTS = ExtendedRel STS

+

+
  ExtendedRel-compute : ExtendedRelSTS i (compute i)

+
  ExtendedRel-compute {i} with compute i in eq

+
  ... | success o = Equivalence.to ≡-success⇔STS eq

+
  ... | failure _ = λ o h → case trans (sym $ Equivalence.from ≡-success⇔STS h) eq of λ ()

+

+
  open ≡-Reasoning

+

+
  ExtendedRel-rightUnique : ExtendedRelSTS i o → ExtendedRelSTS i o' → _≈ᶜʳ_ {Err = Err} o o'

+
  ExtendedRel-rightUnique {o = success x} {success x'} h h'

+
    with trans (sym $ Equivalence.from ≡-success⇔STS h) (Equivalence.from ≡-success⇔STS h')

+
  ... | refl = refl

+

+
  ExtendedRel-rightUnique {o = success x} {failure _}  h h' = ⊥-elim $ h' x h

+
  ExtendedRel-rightUnique {o = failure _} {success x'} h h' = ⊥-elim $ h x' h'

+
  ExtendedRel-rightUnique {o = failure a} {failure b}  h h' = refl

+

+
  computational⇒rightUnique : STS i o → STS i o' → o ≡ o'

+
  computational⇒rightUnique h h' with ExtendedRel-rightUnique h h'

+
  ... | refl = refl

+

+
  Computational⇒Dec : ⦃ _ : DecEq O ⦄ → Dec (STS i o)

+
  Computational⇒Dec {i} {o}

+
    with compute i | ExtendedRel-compute {i}

+
  ... | failure _ | ExSTS = no (ExSTS o)

+
  ... | success x | ExSTS with x ≟ o

+
  ... | no ¬p     = no  λ h → ¬p $ sym $ computational⇒rightUnique h ExSTS

+
  ... | yes refl  = yes ExSTS

+

+
module _ {STS : I → O → Type} (comp comp' : Computational STS Err) where

+

+
  open Computational comp  renaming (compute to compute₁)

+
  open Computational comp' renaming (compute to compute₂)

+

+
  compute-ext≡ : compute₁ i ≈ᶜʳ compute₂ i

+
  compute-ext≡ = ExtendedRel-rightUnique _ ⦃ comp ⦄

+
    (ExtendedRel-compute _ ⦃ comp ⦄) (ExtendedRel-compute _ ⦃ comp' ⦄)

+

+
open Computational ⦃...⦄

+

+
Computational⇒Dec' :

+
  {STS : I → O → Type} ⦃ comp : Computational STS Err ⦄ → Dec (∃[ o ] STS i o)

+
Computational⇒Dec' with computeProof _ in eq

+
... | success x = yes x

+
... | failure x = no λ (_ , h) → failure⇒∀¬STS (-, cong (map proj₁) eq) _ h

+

+
record InjectError Err₁ Err₂ : Type where

+
  field injectError : Err₁ → Err₂

+

+
open InjectError

+

+
instance

+
  InjectError-⊥ : InjectError ⊥ Err

+
  InjectError-⊥ = λ where

+
    .injectError ()

+

+
  InjectError-Id : InjectError Err Err

+
  InjectError-Id = λ where

+
    .injectError → id

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Computational22.html b/site/static/agda_html/Class.Computational22.html
new file mode 100644
index 000000000..f75a9bc1c
--- /dev/null
+++ b/site/static/agda_html/Class.Computational22.html
@@ -0,0 +1,123 @@
+
+Class.Computational22
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
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+               

+             

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+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
--------------------------------------------------------------------------------

+
-- Variant of `Computational` for relations with two inputs and outputs

+
--------------------------------------------------------------------------------

+

+
module Class.Computational22 where

+

+
open import Leios.Prelude

+
open import Class.Computational

+

+
private variable

+
  S I O Err : Type

+

+
module _ (STS : S → I → O → S → Type) where

+
  record Computational22 Err : Type₁ where

+
    constructor MkComputational

+
    field

+
      computeProof : (s : S) (i : I) → ComputationResult Err (∃[ (o , s') ] STS s i o s')

+

+
    compute : S → I → ComputationResult Err (O × S)

+
    compute s i = map proj₁ $ computeProof s i

+

+
    field

+
      completeness : ∀ s i o s' → STS s i o s' → compute s i ≡ success (o , s')

+

+
  STS' : S × I → O × S → Type

+
  STS' (s , i) (o , s') = STS s i o s'

+

+
  module _ ⦃ _ : Computational22 Err ⦄ where

+
    open Computational22 it

+
    instance

+
      comp22⇒comp : Computational STS' Err

+
      comp22⇒comp .Computational.computeProof (s , i)          = computeProof s i

+
      comp22⇒comp .Computational.completeness (s , i) (o , s') = completeness s i o s'

+

+
module _ {STS : S → I → O → S → Type} ⦃ _ : Computational22 STS Err ⦄ where

+
  open Computational22 it

+

+
  -- basically `traverse`

+
  computeTrace : S → List I → ComputationResult Err (List O × S)

+
  computeTrace s [] = success ([] , s)

+
  computeTrace s (x ∷ is) = do

+
    (o , s')   ← compute s x

+
    (os , s'') ← computeTrace s' is

+
    return (o ∷ os , s'')

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Convertible.html b/site/static/agda_html/Class.Convertible.html
new file mode 100644
index 000000000..20ef878fc
--- /dev/null
+++ b/site/static/agda_html/Class.Convertible.html
@@ -0,0 +1,161 @@
+
+Class.Convertible
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

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+               

+             

+             

+               
+               

+             
module Class.Convertible where

+

+
open import Function

+
open import Class.HasHsType

+
open import Class.Core

+
open import Class.Functor

+
open import Class.Bifunctor

+
open import Data.Maybe

+
open import Data.Product

+
open import Data.Sum

+
open import Data.List

+

+
record Convertible (A B : Set) : Set where

+
  field to   : A → B

+
        from : B → A

+
open Convertible ⦃...⦄ public

+

+
HsConvertible : (A : Set) → ⦃ HasHsType A ⦄ → Set

+
HsConvertible A = Convertible A (HsType A)

+

+
Convertible-Refl : ∀ {A} → Convertible A A

+
Convertible-Refl = λ where .to → id; .from → id

+

+
Convertible₁ : (Set → Set) → (Set → Set) → Set₁

+
Convertible₁ T U = ∀ {A B} → ⦃ Convertible A B ⦄ → Convertible (T A) (U B)

+

+
Convertible₂ : (Set → Set → Set) → (Set → Set → Set) → Set₁

+
Convertible₂ T U = ∀ {A B} → ⦃ Convertible A B ⦄ → Convertible₁ (T A) (U B)

+

+
Functor⇒Convertible : ∀ {F : Type↑} → ⦃ Functor F ⦄ → Convertible₁ F F

+
Functor⇒Convertible = λ where

+
  .to   → fmap to

+
  .from → fmap from

+

+
Bifunctor⇒Convertible : ∀ {F} → ⦃ Bifunctor F ⦄ → Convertible₂ F F

+
Bifunctor⇒Convertible = λ where

+
  .to   → bimap to to

+
  .from → bimap from from

+

+
_⨾_ : ∀ {A B C} → Convertible A B → Convertible B C → Convertible A C

+
(c ⨾ c') .to   = c' .to   ∘ c  .to

+
(c ⨾ c') .from = c  .from ∘ c' .from

+

+
-- ** instances

+

+
open import Foreign.Haskell

+
open import Foreign.Haskell.Coerce using (coerce)

+

+
open import Data.Nat

+

+
instance

+
  Convertible-ℕ : Convertible ℕ ℕ

+
  Convertible-ℕ = λ where

+
    .to   → id

+
    .from → id

+

+
  Convertible-Maybe : Convertible₁ Maybe Maybe

+
  Convertible-Maybe = Functor⇒Convertible

+

+
  Convertible-× : Convertible₂ _×_ _×_

+
  Convertible-× = Bifunctor⇒Convertible

+

+
  Convertible-Pair : Convertible₂ _×_ Pair

+
  Convertible-Pair = λ where

+
    .to   → coerce ∘ bimap to to

+
    .from → bimap from from ∘ coerce

+

+
  Convertible-⊎ : Convertible₂ _⊎_ _⊎_

+
  Convertible-⊎ = Bifunctor⇒Convertible

+

+
  Convertible-Either : Convertible₂ _⊎_ Either

+
  Convertible-Either = λ where

+
    .to   → coerce ∘ bimap to to

+
    .from → bimap from from ∘ coerce

+

+
  Convertible-List : Convertible₁ List List

+
  Convertible-List = λ where

+
    .to   → fmap to

+
    .from → fmap from

+

+
  Convertible-Fun : ∀ {A A' B B'} → ⦃ Convertible A A' ⦄ → ⦃ Convertible B B' ⦄ → Convertible (A → B) (A' → B')

+
  Convertible-Fun = λ where

+
    .to   → λ f → to   ∘ f ∘ from

+
    .from → λ f → from ∘ f ∘ to

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Core.html b/site/static/agda_html/Class.Core.html
new file mode 100644
index 000000000..36fb5b3d2
--- /dev/null
+++ b/site/static/agda_html/Class.Core.html
@@ -0,0 +1,100 @@
+
+Class.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
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+               

+             

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+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Core where

+

+
open import Class.Prelude

+

+
Type[_↝_] : ∀ ℓ ℓ′ → Type (lsuc ℓ ⊔ lsuc ℓ′)

+
Type[ ℓ ↝ ℓ′ ] = Type ℓ → Type ℓ′

+

+
Type↑ : Typeω

+
Type↑ = ∀ {ℓ} → Type[ ℓ ↝ ℓ ]

+

+
module _ (M : Type↑) where

+
  _¹ : (A → Type ℓ) → Type _

+
  _¹ P = ∀ {x} → M (P x)

+

+
  _² : (A → B → Type ℓ) → Type _

+
  _² _~_ = ∀ {x y} → M (x ~ y)

+

+
  _³ : (A → B → C → Type ℓ) → Type _

+
  _³ _~_~_ = ∀ {x y z} → M (x ~ y ~ z)

+

+
variable

+
  M F : Type↑

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.DecEq.Core.html b/site/static/agda_html/Class.DecEq.Core.html
new file mode 100644
index 000000000..fe235dd7f
--- /dev/null
+++ b/site/static/agda_html/Class.DecEq.Core.html
@@ -0,0 +1,99 @@
+
+Class.DecEq.Core
+             
+             
+             

+               

+                 Back
+                 
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+               

+                 
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+             
{-# OPTIONS --without-K #-}

+
module Class.DecEq.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+

+
record DecEq (A : Type ℓ) : Type ℓ where

+
  field _≟_ : DecidableEquality A

+

+
  _==_ _≡ᵇ_ : A → A → Bool

+
  x == y = ⌊ x ≟ y ⌋

+
  _≡ᵇ_ = _==_

+

+
  _≠_ : A → A → Bool

+
  x ≠ y = not (x == y)

+

+
  infix 4 _≟_ _≡ᵇ_ _==_ _≠_

+
open DecEq ⦃...⦄ public

+

+
DecEq¹ = DecEq ¹

+
DecEq² = DecEq ²

+
DecEq³ = DecEq ³

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.DecEq.Instances.html b/site/static/agda_html/Class.DecEq.Instances.html
new file mode 100644
index 000000000..04819a131
--- /dev/null
+++ b/site/static/agda_html/Class.DecEq.Instances.html
@@ -0,0 +1,155 @@
+
+Class.DecEq.Instances
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
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+             

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+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.DecEq.Instances where

+

+
open import Class.Prelude

+
open import Class.DecEq.Core

+

+
-- ** basic types

+
instance

+
  DecEq-⊥      = DecEq ⊥ ∋ λ where ._≟_ ()

+
  DecEq-⊤      = DecEq _ ∋ record {M} where import Data.Unit as M

+
  DecEq-Bool   = DecEq _ ∋ record {M} where import Data.Bool as M

+
  DecEq-ℕ      = DecEq _ ∋ record {M} where import Data.Nat as M

+
  DecEq-ℤ      = DecEq _ ∋ record {M} where import Data.Integer as M

+
  DecEq-ℚ      = DecEq _ ∋ record {M} where import Data.Rational.Properties as M

+
  DecEq-Char   = DecEq _ ∋ record {M} where import Data.Char as M

+
  DecEq-String = DecEq _ ∋ record {M} where import Data.String as M

+

+
  DecEq-Fin : DecEq¹ Fin

+
  DecEq-Fin = record {M} where import Data.Fin as M

+

+
  DecEq-List : ⦃ DecEq A ⦄ → DecEq (List A)

+
  DecEq-List ._≟_ = M.≡-dec _≟_ where import Data.List.Properties as M

+

+
-- ** containers of decidably equal elements

+
private

+
  ∷-injective : ∀ {x y xs ys} →

+
    (List⁺ A ∋ x ∷ xs) ≡ y ∷ ys → x ≡ y × xs ≡ ys

+
  ∷-injective refl = (refl , refl)

+
module _ ⦃ _ : DecEq A ⦄ where instance

+
  DecEq-List⁺ : DecEq (List⁺ A)

+
  DecEq-List⁺ ._≟_ (x ∷ xs) (y ∷ ys)

+
    with x ≟ y

+
  ... | no x≢y = no $ x≢y ∘ proj₁ ∘ ∷-injective

+
  ... | yes refl

+
    with xs ≟ ys

+
  ... | no xs≢ys = no $ xs≢ys ∘ proj₂ ∘ ∷-injective

+
  ... | yes refl = yes refl

+

+
  DecEq-Vec : DecEq¹ (Vec A)

+
  DecEq-Vec ._≟_ = M.≡-dec _≟_

+
    where import Data.Vec.Properties as M

+

+
  DecEq-Maybe : DecEq (Maybe A)

+
  DecEq-Maybe ._≟_ = M.≡-dec _≟_

+
    where import Data.Maybe.Properties as M

+

+
module _ ⦃ _ : DecEq A ⦄ ⦃ _ : DecEq B ⦄ where

+

+
  -- Not exported as instance so that users can also choose `Class.DecEq.WithK.DecEq-Σ`

+
  DecEq-× : DecEq (A × B)

+
  DecEq-× ._≟_ = ×.≡-dec _≟_ _≟_

+
    where import Data.Product.Properties as ×

+

+
  instance

+
    DecEq-⊎ : DecEq (A ⊎ B)

+
    DecEq-⊎ ._≟_ = ⊎.≡-dec _≟_ _≟_

+
      where import Data.Sum.Properties as ⊎

+

+
    DecEq-These : DecEq (These A B)

+
    DecEq-These ._≟_ = M.≡-dec _≟_ _≟_

+
      where import Data.These.Properties as M

+

+
-- ** reflection

+
instance

+
  DecEq-Name = DecEq _ ∋ record {M} where import Reflection.AST.Name as M

+
  DecEq-Lit  = DecEq _ ∋ record {M} where import Reflection.AST.Literal as M

+
  DecEq-Meta = DecEq _ ∋ record {M} where import Reflection.AST.Meta as M

+
  DecEq-Term = DecEq _ ∋ record {M} where import Reflection.AST.Term as M

+
  DecEq-Mod  = DecEq _ ∋ record {M}

+
    where import Reflection.AST.Argument.Modality as M

+
  DecEq-Vis  = DecEq _ ∋ record {M}

+
    where import Reflection.AST.Argument.Visibility as M

+
  DecEq-ArgI = DecEq _ ∋ record {M}

+
    where import Reflection.AST.Argument.Information as M

+

+
  -- put this last to work around Agda issue #6976

+
  DecEq-Arg : ⦃ DecEq A ⦄ → DecEq (Arg A)

+
  DecEq-Arg ._≟_ = M.≡-dec _≟_ where import Reflection.AST.Argument as M

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.DecEq.html b/site/static/agda_html/Class.DecEq.html
new file mode 100644
index 000000000..680d64148
--- /dev/null
+++ b/site/static/agda_html/Class.DecEq.html
@@ -0,0 +1,83 @@
+
+Class.DecEq
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
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+                 
+               

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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.DecEq where

+

+
open import Class.DecEq.Core public

+
open import Class.DecEq.Instances public

+
-- open import Class.DecEq.WithK public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Decidable.Core.html b/site/static/agda_html/Class.Decidable.Core.html
new file mode 100644
index 000000000..aad4464d4
--- /dev/null
+++ b/site/static/agda_html/Class.Decidable.Core.html
@@ -0,0 +1,159 @@
+
+Class.Decidable.Core
+             
+             
+             

+               

+                 Back
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+               

+                 
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+                     
+                   
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+               
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+             
{-# OPTIONS --without-K #-}

+
module Class.Decidable.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+

+
open import Relation.Nullary.Decidable using (True; False; toWitness; toWitnessFalse)

+

+
record _⁇ (P : Type ℓ) : Type ℓ where

+
  constructor ⁇_

+
  field dec : Dec P

+

+
  auto : ⦃ True dec ⦄ → P

+
  auto ⦃ pr ⦄ = toWitness pr

+

+
  contradict : ∀ {X : Type} {pr : False dec} → P → X

+
  contradict {pr = pr} = ⊥-elim ∘ toWitnessFalse pr

+

+
open _⁇ ⦃...⦄ public

+

+
¿_¿ : ∀ (X : Type ℓ) → ⦃ X ⁇ ⦄ → Dec X

+
¿ _ ¿ = dec

+

+
¿_¿ᵇ : (P : Type ℓ) → ⦃ P ⁇ ⦄ → Bool

+
¿ P ¿ᵇ = ⌊ ¿ P ¿ ⌋

+

+
infix 0 ifᵈ_then_else_

+
ifᵈ_then_else_ : ∀ {X : Type ℓ} (P : Type ℓ′)

+
  → ⦃ P ⁇ ⦄ → ({_ : P} → X) → ({_ : ¬ P} → X) → X

+
ifᵈ P then t else f with ¿ P ¿

+
... | yes p = t {p}

+
... | no ¬p = f {¬p}

+

+
_⁇¹ = _⁇ ¹

+
_⁇² = _⁇ ²

+
_⁇³ = _⁇ ³

+

+
module _ {A : Type ℓ} where

+

+
  module _ {P : Pred A ℓ′} where

+

+
    dec¹ : ⦃ P ⁇¹ ⦄ → Decidable¹ P

+
    dec¹ _ = dec

+

+
    ⁇¹_ : Decidable¹ P → P ⁇¹

+
    ⁇¹ p? = ⁇ (p? _)

+

+
  ¿_¿¹ : (P : Pred A ℓ′) → ⦃ P ⁇¹ ⦄ → Decidable¹ P

+
  ¿ _ ¿¹ = dec¹

+

+
module _ {A B : Type ℓ} where

+

+
  module _ {R : REL A B ℓ′} where

+

+
    dec² : ⦃ R ⁇² ⦄ → Decidable² R

+
    dec² _ _ = dec

+

+
    ⁇²_ : Decidable² R → R ⁇²

+
    ⁇² p? = ⁇ (p? _ _)

+

+
  ¿_¿² : (R : REL A B ℓ′) → ⦃ R ⁇² ⦄ → Decidable² R

+
  ¿ _ ¿² = dec²

+

+
module _ {A B C : Type ℓ} where

+

+
  module _ {R : 3REL A B C ℓ′} where

+

+
    dec³ : ⦃ R ⁇³ ⦄ → Decidable³ R

+
    dec³ _ _ _ = dec

+

+
    ⁇³_ : Decidable³ R → R ⁇³

+
    ⁇³_ p? = ⁇ (p? _ _ _)

+

+
  ¿_¿³ : (R : 3REL A B C ℓ′) → ⦃ R ⁇³ ⦄ → Decidable³ R

+
  ¿ _ ¿³ = dec³

+

+
infix -100 auto∶_

+
auto∶_ : ∀ (X : Type ℓ) → ⦃ X ⁇ ⦄ → Type

+
auto∶_ A = True ¿ A ¿

+

+
dec-✓ : ∀ {P : Type ℓ} ⦃ _ : P ⁇ ⦄ (p : P) → ∃[ p′ ] ¿ P ¿ ≡ yes p′

+
dec-✓ = dec-yes ¿ _ ¿

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Decidable.Instances.html b/site/static/agda_html/Class.Decidable.Instances.html
new file mode 100644
index 000000000..23e4c64a8
--- /dev/null
+++ b/site/static/agda_html/Class.Decidable.Instances.html
@@ -0,0 +1,185 @@
+
+Class.Decidable.Instances
+             
+             
+             

+               

+                 Back
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+               

+               

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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Decidable.Instances where

+

+
open import Class.Prelude

+
open import Class.Decidable.Core

+
open import Class.DecEq.Core

+

+
private variable

+
  n : ℕ

+
  x : A

+
  P Q : Pred A ℓ

+
  R : Rel A ℓ

+

+
instance

+

+
  -- ** deriving from DecEq

+
  DecEq⇒Dec : ⦃ DecEq A ⦄ → _≡_ {A = A} ⁇²

+
  DecEq⇒Dec = ⁇² _≟_

+

+
  -- ** basic type formers

+

+
  Dec-⊥ : ⊥ ⁇

+
  Dec-⊥ .dec = no λ()

+

+
  Dec-⊤ : ⊤ ⁇

+
  Dec-⊤ .dec = yes tt

+

+
  open import Relation.Nullary.Decidable as D

+

+
  Dec-→ : ⦃ A ⁇ ⦄ → ⦃ B ⁇ ⦄ → (A → B) ⁇

+
  Dec-→ .dec = dec D.→-dec dec

+

+
  -- ** products

+

+
  Dec-× : ⦃ A ⁇ ⦄ → ⦃ B ⁇ ⦄ → (A × B) ⁇

+
  Dec-× .dec = dec D.×-dec dec

+

+
  -- ** sums

+

+
  Dec-⊎ : ⦃ A ⁇ ⦄ → ⦃ B ⁇ ⦄ → (A ⊎ B) ⁇

+
  Dec-⊎ .dec = dec D.⊎-dec dec

+

+
  import Data.Sum.Relation.Unary.All as ⊎; open ⊎ using (inj₁; inj₂)

+
  open import Relation.Nullary.Decidable using () renaming (map′ to mapDec)

+

+
  Dec-⊎All : ⦃ P ⁇¹ ⦄ → ⦃ Q ⁇¹ ⦄ → ⊎.All P Q ⁇¹

+
  Dec-⊎All {P = P} {Q = Q} {x = inj₁ x} .dec = mapDec inj₁ inj₁˘ ¿ P x ¿

+
    where inj₁˘ : ⊎.All P Q (inj₁ x) → P x

+
          inj₁˘ (inj₁ x) = x

+
  Dec-⊎All {P = P} {Q = Q} {x = inj₂ y} .dec = mapDec inj₂ inj₂˘ ¿ Q y ¿

+
    where inj₂˘ : ⊎.All P Q (inj₂ x) → Q x

+
          inj₂˘ (inj₂ x) = x

+

+
  import Data.Bool as 𝔹

+

+
  Dec-T : T ⁇¹

+
  Dec-T = ⁇¹ 𝔹.T?

+

+
  import Data.List.Relation.Unary.All as L

+
  import Data.List.Relation.Unary.Any as L

+

+
  Dec-All : ⦃ P ⁇¹ ⦄ → L.All P ⁇¹

+
  Dec-All = ⁇¹ L.all? dec¹

+

+
  Dec-Any : ⦃ P ⁇¹ ⦄ → L.Any P ⁇¹

+
  Dec-Any = ⁇¹ L.any? dec¹

+

+
  import Data.List.Relation.Unary.AllPairs as AP

+

+
  Dec-AllPairs : ⦃ R ⁇² ⦄ → AP.AllPairs R ⁇¹

+
  Dec-AllPairs = ⁇¹ AP.allPairs? dec²

+

+
  open import Data.Vec as V

+
  open import Data.Vec.Relation.Unary.All as V

+
  open import Data.Vec.Relation.Unary.Any as V

+

+
  Dec-VAll : ⦃ P ⁇¹ ⦄ → V.All P {n} ⁇¹

+
  Dec-VAll = ⁇¹ V.all? dec¹

+

+
  Dec-VAny : ⦃ P ⁇¹ ⦄ → V.Any P {n} ⁇¹

+
  Dec-VAny = ⁇¹ V.any? dec¹

+

+
  import Data.Maybe as M

+
  import Data.Maybe.Relation.Unary.All as M renaming (dec to all?)

+
  import Data.Maybe.Relation.Unary.Any as M renaming (dec to any?)

+

+
  Dec-MAll : ⦃ P ⁇¹ ⦄ → M.All P ⁇¹

+
  Dec-MAll = ⁇¹ M.all? dec¹

+

+
  Dec-MAny : ⦃ P ⁇¹ ⦄ → M.Any P ⁇¹

+
  Dec-MAny = ⁇¹ M.any? dec¹

+

+
  -- ** inequalities

+

+
  import Data.Nat.Properties as ℕ

+

+
  ℕ-Dec-≤ = ⁇² ℕ._≤?_

+
  ℕ-Dec-< = ⁇² ℕ._<?_

+

+
  import Data.Integer.Properties as ℤ

+

+
  ℤ-Dec-≤ = ⁇² ℤ._≤?_

+
  ℤ-Dec-< = ⁇² ℤ._<?_

+

+
  import Data.Rational.Properties as Rat

+

+
  ℚ-Dec-≤ = ⁇² Rat._≤?_

+
  ℚ-Dec-< = ⁇² Rat._<?_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Decidable.html b/site/static/agda_html/Class.Decidable.html
new file mode 100644
index 000000000..de78d72b6
--- /dev/null
+++ b/site/static/agda_html/Class.Decidable.html
@@ -0,0 +1,82 @@
+
+Class.Decidable
+             
+             
+             

+               

+                 Back
+                 
+               

+               

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+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Decidable where

+

+
open import Class.Decidable.Core public

+
open import Class.Decidable.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Functor.Core.html b/site/static/agda_html/Class.Functor.Core.html
new file mode 100644
index 000000000..497a7b534
--- /dev/null
+++ b/site/static/agda_html/Class.Functor.Core.html
@@ -0,0 +1,109 @@
+
+Class.Functor.Core
+             
+             
+             

+               

+                 Back
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+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Functor.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+

+
private variable a b c : Level

+

+
record Functor (F : Type↑) : Typeω where

+
  infixl 4 _<$>_ _<$_

+
  infixl 1 _<&>_

+

+
  field _<$>_ : (A → B) → F A → F B

+
  fmap = _<$>_

+

+
  _<$_ : A → F B → F A

+
  x <$ y = const x <$> y

+

+
  _<&>_ : F A → (A → B) → F B

+
  _<&>_ = flip _<$>_

+
open Functor ⦃...⦄ public

+

+
record FunctorLaws (F : Type↑) ⦃ _ : Functor F ⦄ : Typeω where

+
  field

+
    -- preserves identity morphisms

+
    fmap-id : ∀ {A : Type a} (x : F A) →

+
      fmap id x ≡ x

+
    -- preserves composition of morphisms

+
    fmap-∘  : ∀ {A : Type a} {B : Type b} {C : Type c}

+
                {f : B → C} {g : A → B} (x : F A) →

+
      fmap (f ∘ g) x ≡ (fmap f ∘ fmap g) x

+
open FunctorLaws ⦃...⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Functor.Instances.html b/site/static/agda_html/Class.Functor.Instances.html
new file mode 100644
index 000000000..0281097c5
--- /dev/null
+++ b/site/static/agda_html/Class.Functor.Instances.html
@@ -0,0 +1,132 @@
+
+Class.Functor.Instances
+             
+             
+             

+               

+                 Back
+                 
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+               

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+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Functor.Instances where

+

+
open import Class.Prelude

+
open import Class.Functor.Core

+

+
instance

+
  Functor-Maybe : Functor Maybe

+
  Functor-Maybe = record {M}

+
    where import Data.Maybe as M renaming (map to _<$>_)

+

+
  FunctorLaws-Maybe : FunctorLaws Maybe

+
  FunctorLaws-Maybe = λ where

+
    .fmap-id → λ where (just _) → refl; nothing → refl

+
    .fmap-∘  → λ where (just _) → refl; nothing → refl

+

+
  Functor-List : Functor List

+
  Functor-List ._<$>_ = map

+

+
  FunctorLaws-List : FunctorLaws List

+
  FunctorLaws-List = record {fmap-id = p; fmap-∘ = q}

+
    where

+
      p : ∀ {A : Type ℓ} (x : List A) → fmap id x ≡ x

+
      p = λ where

+
        [] → refl

+
        (x ∷ xs) → cong (x ∷_) (p xs)

+

+
      q : ∀ {A : Type ℓ} {B : Type ℓ′} {C : Type ℓ″} {f : B → C} {g : A → B} (x : List A) →

+
        fmap (f ∘ g) x ≡ (fmap f ∘ fmap g) x

+
      q {f = f}{g} = λ where

+
        [] → refl

+
        (x ∷ xs) → cong (f (g x) ∷_) (q xs)

+

+
  Functor-List⁺ : Functor List⁺

+
  Functor-List⁺ = record {L}

+
    where import Data.List.NonEmpty as L renaming (map to _<$>_)

+

+
  Functor-Vec : ∀ {n} → Functor (flip Vec n)

+
  Functor-Vec = record {V}

+
    where import Data.Vec as V renaming (map to _<$>_)

+

+
  Functor-TC : Functor TC

+
  Functor-TC = record {R}

+
    where import Reflection.TCM.Syntax as R

+

+
  Functor-Abs : Functor Abs

+
  Functor-Abs = record {R}

+
    where import Reflection.AST.Abstraction as R renaming (map to _<$>_)

+

+
  Functor-Arg : Functor Arg

+
  Functor-Arg = record {R}

+
    where import Reflection.AST.Argument as R renaming (map to _<$>_)

+

+
  Functor-∃Vec : Functor (∃ ∘ Vec)

+
  Functor-∃Vec ._<$>_ f (_ , xs) = -, (f <$> xs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Functor.html b/site/static/agda_html/Class.Functor.html
new file mode 100644
index 000000000..0d5ad62a8
--- /dev/null
+++ b/site/static/agda_html/Class.Functor.html
@@ -0,0 +1,82 @@
+
+Class.Functor
+             
+             
+             

+               

+                 Back
+                 
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+               

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+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Functor where

+

+
open import Class.Functor.Core public

+
open import Class.Functor.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasAdd.Core.html b/site/static/agda_html/Class.HasAdd.Core.html
new file mode 100644
index 000000000..ac3f2de9d
--- /dev/null
+++ b/site/static/agda_html/Class.HasAdd.Core.html
@@ -0,0 +1,85 @@
+
+Class.HasAdd.Core
+             
+             
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+             
{-# OPTIONS --safe --cubical-compatible #-}

+
module Class.HasAdd.Core where

+

+
record HasAdd (A : Set) : Set where

+
  infixl 6 _+_

+
  field _+_ : A → A → A

+

+
open HasAdd ⦃ ... ⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasAdd.Instance.html b/site/static/agda_html/Class.HasAdd.Instance.html
new file mode 100644
index 000000000..9c5bf2b61
--- /dev/null
+++ b/site/static/agda_html/Class.HasAdd.Instance.html
@@ -0,0 +1,98 @@
+
+Class.HasAdd.Instance
+             
+             
+             

+               

+                 Back
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+               

+               

+                 
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+                     
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+             

+             

+               
+               

+             
{-# OPTIONS --safe --cubical-compatible #-}

+
module Class.HasAdd.Instance where

+

+
open import Class.HasAdd.Core

+
open import Data.Integer   as ℤ using (ℤ)

+
open import Data.Nat       as ℕ using (ℕ)

+
open import Data.Rational  as ℚ using (ℚ)

+
open import Data.String    as Str

+

+
instance

+
  addInt : HasAdd ℤ

+
  addInt ._+_ = ℤ._+_

+

+
  addNat : HasAdd ℕ

+
  addNat ._+_ = ℕ._+_

+

+
  addRat : HasAdd ℚ

+
  addRat ._+_ = ℚ._+_

+

+
  addString : HasAdd String

+
  addString ._+_ = Str._++_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasAdd.html b/site/static/agda_html/Class.HasAdd.html
new file mode 100644
index 000000000..150590386
--- /dev/null
+++ b/site/static/agda_html/Class.HasAdd.html
@@ -0,0 +1,82 @@
+
+Class.HasAdd
+             
+             
+             

+               

+                 Back
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+               

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+             

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+               
+               

+             
{-# OPTIONS --safe --cubical-compatible #-}

+
module Class.HasAdd where

+

+
open import Class.HasAdd.Core public

+
open import Class.HasAdd.Instance public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasEmptySet.html b/site/static/agda_html/Class.HasEmptySet.html
new file mode 100644
index 000000000..355ff62a5
--- /dev/null
+++ b/site/static/agda_html/Class.HasEmptySet.html
@@ -0,0 +1,100 @@
+
+Class.HasEmptySet
+             
+             
+             

+               

+                 Back
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+               

+               

+                 
+                   
+                     
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+             

+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import Axiom.Set using (Theory)

+

+
module Class.HasEmptySet (th : Theory) where

+

+
open import abstract-set-theory.Prelude

+

+
open Theory th renaming (∅ to ∅ˢ)

+
open import Axiom.Set.Map th

+

+
record HasEmptySet (A : Type) : Type where

+
  field

+
    ∅ : A

+

+
open HasEmptySet ⦃...⦄ public

+

+
instance

+
  HasEmptySet-Set : {A : Type} → HasEmptySet (Set A)

+
  HasEmptySet-Set = record { ∅ = ∅ˢ }

+

+
  HasEmptySet-Map : {A B : Type} → HasEmptySet (Map A B)

+
  HasEmptySet-Map = record { ∅ = ∅ᵐ }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasHsType.html b/site/static/agda_html/Class.HasHsType.html
new file mode 100644
index 000000000..48a78776c
--- /dev/null
+++ b/site/static/agda_html/Class.HasHsType.html
@@ -0,0 +1,128 @@
+
+Class.HasHsType
+             
+             
+             

+               

+                 Back
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+               

+               

+                 
+                   
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+             

+             

+               
+               

+             
module Class.HasHsType where

+

+
open import Level using (Level)

+
open import Data.Bool.Base using (Bool)

+
open import Data.Nat.Base using (ℕ)

+
open import Data.String.Base using (String)

+
open import Data.List.Base using (List)

+
open import Data.Maybe.Base using (Maybe)

+
open import Data.Sum.Base using (_⊎_)

+
open import Data.Product.Base using (_×_)

+
open import Data.Unit using (⊤)

+

+
open import Foreign.Haskell.Pair using (Pair)

+
open import Foreign.Haskell.Either using (Either)

+

+
private variable

+
  l : Level

+
  A B : Set l

+

+
record HasHsType (A : Set l) : Set₁ where

+
  field

+
    HsType : Set

+

+
HsType : (A : Set l) → ⦃ HasHsType A ⦄ → Set

+
HsType _ ⦃ i ⦄ = i .HasHsType.HsType

+

+
MkHsType : (A : Set l) (Hs : Set) → HasHsType A

+
MkHsType A Hs .HasHsType.HsType = Hs

+

+
instance

+

+
  iHsTy-ℕ      = MkHsType ℕ ℕ

+
  iHsTy-Bool   = MkHsType Bool Bool

+
  iHsTy-⊤      = MkHsType ⊤ ⊤

+
  iHsTy-String = MkHsType String String

+

+
  -- Could make a macro for these kind of congruence instances.

+
  iHsTy-List : ⦃ HasHsType A ⦄ → HasHsType (List A)

+
  iHsTy-List {A = A} .HasHsType.HsType = List (HsType A)

+

+
  iHsTy-Maybe : ⦃ HasHsType A ⦄ → HasHsType (Maybe A)

+
  iHsTy-Maybe {A = A} .HasHsType.HsType = Maybe (HsType A)

+

+
  iHsTy-Fun : ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (A → B)

+
  iHsTy-Fun {A = A} {B = B} .HasHsType.HsType = HsType A → HsType B

+

+
  iHsTy-Sum : ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (A ⊎ B)

+
  iHsTy-Sum {A = A} {B = B} .HasHsType.HsType = Either (HsType A) (HsType B)

+

+
  iHsTy-Pair : ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (A × B)

+
  iHsTy-Pair {A = A} {B = B} .HasHsType.HsType = Pair (HsType A) (HsType B)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasOrder.Core.html b/site/static/agda_html/Class.HasOrder.Core.html
new file mode 100644
index 000000000..f583310f3
--- /dev/null
+++ b/site/static/agda_html/Class.HasOrder.Core.html
@@ -0,0 +1,237 @@
+
+Class.HasOrder.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

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+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Class.HasOrder.Core where

+

+
open import Class.Decidable

+
open import Data.Empty

+
open import Data.Product

+
open import Data.Sum

+
open import Function

+
open import Level

+
open import Relation.Binary

+
open import Relation.Binary using () renaming (Decidable to Decidable²)

+
open import Relation.Binary.PropositionalEquality

+
open import Relation.Nullary

+

+
open Equivalence

+

+
module _ {a} {A : Set a} where

+
  module _ {_≈_ : Rel A a} where

+

+
    record HasPreorder : Set (suc a) where

+
      infix 4 _≤_ _<_ _≥_ _>_

+
      field

+
        _≤_ _<_       : Rel A a

+
        ≤-isPreorder  : IsPreorder _≈_ _≤_

+
        <-irrefl      : Irreflexive _≈_ _<_

+
        ≤⇔<∨≈         : ∀ {x y} → x ≤ y ⇔ (x < y ⊎ x ≈ y)

+

+
      _≥_ = flip _≤_

+
      _>_ = flip _<_

+

+
      open IsPreorder ≤-isPreorder public

+
        using ()

+
        renaming (isEquivalence to ≈-isEquivalence; refl to ≤-refl; trans to ≤-trans)

+

+
      _≤?_ : ⦃ _≤_ ⁇² ⦄ → Decidable _≤_

+
      _≤?_ = dec²

+

+
      _<?_ : ⦃ _<_ ⁇² ⦄ → Decidable _<_

+
      _<?_ = dec²

+

+
      infix 4 _<?_ _≤?_

+

+
      <⇒≤∧≉ : ∀ {x y} → x < y → x ≤ y × ¬ (x ≈ y)

+
      <⇒≤∧≉ x<y = from ≤⇔<∨≈ (inj₁ x<y) , λ x≈y → <-irrefl x≈y x<y

+

+
      ≤∧≉⇒< : ∀ {x y} → x ≤ y × ¬ (x ≈ y) → x < y

+
      ≤∧≉⇒< {x} {y} (x≤y , ¬x≈y) with to ≤⇔<∨≈ x≤y

+
      ... | inj₁ x<y = x<y

+
      ... | inj₂ x≈y = ⊥-elim (¬x≈y x≈y)

+

+
      ≤-antisym⇒<-asym : Antisymmetric _≈_ _≤_ → Asymmetric _<_

+
      ≤-antisym⇒<-asym antisym x<y y<x =

+
        proj₂ (<⇒≤∧≉ x<y) $ antisym (proj₁ $ <⇒≤∧≉ x<y) (proj₁ $ <⇒≤∧≉ y<x)

+

+
    open HasPreorder ⦃...⦄

+

+
    record HasDecPreorder : Set (suc a) where

+
      field ⦃ hasPreorder ⦄ : HasPreorder

+
            ⦃ dec-≤ ⦄ : _≤_ ⁇²

+
            ⦃ dec-< ⦄ : _<_ ⁇²

+

+
    record HasPartialOrder : Set (suc a) where

+
      field

+
        ⦃ hasPreorder ⦄ : HasPreorder

+
        ≤-antisym : Antisymmetric _≈_ _≤_

+

+
      ≤-isPartialOrder : IsPartialOrder _≈_ _≤_

+
      ≤-isPartialOrder = record { isPreorder = ≤-isPreorder ; antisym = ≤-antisym }

+

+
      <-asymmetric : Asymmetric _<_

+
      <-asymmetric = ≤-antisym⇒<-asym ≤-antisym

+

+
      open IsEquivalence ≈-isEquivalence renaming (sym to ≈-sym)

+

+
      <-trans : Transitive _<_

+
      <-trans i<j j<k =

+
        let

+
          j≤k = proj₁ $ <⇒≤∧≉ j<k; i≤j = proj₁ $ <⇒≤∧≉ i<j

+
          i≈k⇒j≈k = λ i≈k → ≤-antisym j≤k $ ≤-trans (from ≤⇔<∨≈ $ inj₂ (≈-sym i≈k)) i≤j

+
        in

+
          ≤∧≉⇒< (≤-trans i≤j j≤k , (proj₂ $ <⇒≤∧≉ j<k) ∘ i≈k⇒j≈k)

+

+
      <⇒¬>⊎≈ : ∀ {x y} → x < y → ¬ (y < x ⊎ y ≈ x)

+
      <⇒¬>⊎≈ x<y (inj₁ y<x) = <-asymmetric x<y y<x

+
      <⇒¬>⊎≈ x<y (inj₂ x≈y) = <-irrefl (≈-sym x≈y) x<y

+

+
    record HasDecPartialOrder : Set (suc a) where

+
      field

+
        ⦃ hasPartialOrder ⦄ : HasPartialOrder

+
        ⦃ dec-≤ ⦄ : _≤_ ⁇²

+
        ⦃ dec-< ⦄ : _<_ ⁇²

+

+
  HasPreorder≡        = HasPreorder {_≈_ = _≡_}

+
  HasDecPreorder≡     = HasDecPreorder {_≈_ = _≡_}

+
  HasPartialOrder≡    = HasPartialOrder {_≈_ = _≡_}

+
  HasDecPartialOrder≡ = HasDecPartialOrder {_≈_ = _≡_}

+

+
open HasPreorder ⦃...⦄ public

+
open HasPartialOrder ⦃...⦄ public hiding (hasPreorder)

+
open HasDecPartialOrder ⦃...⦄ public hiding (hasPartialOrder)

+

+
module _ {a} {A : Set a} {_≈_ : Rel A a} where

+

+
  module _ {_≤_ : Rel A a} where

+
    import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as SNS

+

+
    module _ (≤-isPreorder : IsPreorder _≈_ _≤_)

+
             (_≈?_ : Decidable² _≈_) where

+

+
      hasPreorderFromNonStrict : HasPreorder

+
      hasPreorderFromNonStrict = record

+
        { _≤_           = _≤_

+
        ; _<_           = SNS._<_

+
        ; ≤-isPreorder  = ≤-isPreorder

+
        ; <-irrefl      = SNS.<-irrefl

+
        ; ≤⇔<∨≈         = λ {a b} → mk⇔

+
          (λ a≤b → case (a ≈? b) of λ where (yes p) → inj₂ p ; (no ¬p) → inj₁ (a≤b , ¬p))

+
          λ where (inj₁ a<b) → proj₁ a<b ; (inj₂ a≈b) → IsPreorder.reflexive ≤-isPreorder a≈b

+
        }

+

+
      hasPartialOrderFromNonStrict : Antisymmetric _≈_ _≤_ → HasPartialOrder

+
      hasPartialOrderFromNonStrict antsym  = record

+
        { hasPreorder = hasPreorderFromNonStrict

+
        ; ≤-antisym = antsym

+
        }

+

+
  module _ {_<_ : Rel A a} where

+

+
    import Relation.Binary.Construct.StrictToNonStrict _≈_ _<_ as SNS

+

+
    module _ (<-isStrictPartialOrder : IsStrictPartialOrder _≈_ _<_) where

+

+
        hasPreorderFromStrictPartialOrder : HasPreorder

+
        hasPreorderFromStrictPartialOrder = record

+
          { _≤_ = SNS._≤_

+
          ; _<_ = _<_

+
          ; ≤-isPreorder = SNS.isPreorder₂ <-isStrictPartialOrder

+
          ; <-irrefl = <-isStrictPartialOrder .IsStrictPartialOrder.irrefl

+
          ; ≤⇔<∨≈ = mk⇔ id id

+
          }

+

+
        instance _ = hasPreorderFromStrictPartialOrder

+

+
        hasPartialOrderFromStrictPartialOrder : HasPartialOrder

+
        hasPartialOrderFromStrictPartialOrder = record

+
          { hasPreorder = hasPreorderFromStrictPartialOrder

+
          ; ≤-antisym = SNS.isPartialOrder <-isStrictPartialOrder .IsPartialOrder.antisym

+
          }

+

+

+
    module _ (<-isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_) where

+

+
      private spo = IsStrictTotalOrder.isStrictPartialOrder <-isStrictTotalOrder

+

+
      hasPreorderFromStrictTotalOrder : HasPreorder

+
      hasPreorderFromStrictTotalOrder = hasPreorderFromStrictPartialOrder spo

+

+
      hasPartialOrderFromStrictTotalOrder : HasPartialOrder

+
      hasPartialOrderFromStrictTotalOrder = hasPartialOrderFromStrictPartialOrder spo

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasOrder.Instance.html b/site/static/agda_html/Class.HasOrder.Instance.html
new file mode 100644
index 000000000..d357e948b
--- /dev/null
+++ b/site/static/agda_html/Class.HasOrder.Instance.html
@@ -0,0 +1,112 @@
+
+Class.HasOrder.Instance
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
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+                 
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+             

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+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Class.HasOrder.Instance where

+

+
open import Class.DecEq

+
open import Class.Decidable

+
open import Class.HasOrder.Core

+
open import Data.Integer using (ℤ)

+
open import Data.Nat using (ℕ)

+
open import Data.Rational using (ℚ)

+
open import Data.Sum

+
open import Function

+
open import Relation.Nullary

+

+
instance

+
  import Data.Nat.Properties as Nat hiding (_≟_)

+
  ℕ-hasPreorder = HasPreorder ∋ record {Nat; ≤⇔<∨≈ = let open Nat in mk⇔

+
    (λ a≤b → case _ ≟ _ of λ where (yes p) → inj₂ p ; (no ¬p) → inj₁ (≤∧≢⇒< a≤b ¬p))

+
    [ <⇒≤ , ≤-reflexive ] }

+
  ℕ-hasPartialOrder = HasPartialOrder ∋ record

+
    { ≤-antisym = Nat.≤-antisym }

+
  ℕ-hasDecPartialOrder = HasDecPartialOrder {A = ℕ} ∋ record {}

+

+
  import Data.Integer.Properties as Int hiding (_≟_)

+
  ℤ-hasPreorder = HasPreorder ∋ record {Int; ≤⇔<∨≈ = let open Int in mk⇔

+
    (λ a≤b → case _ ≟ _ of λ where (yes p) → inj₂ p ; (no ¬p) → inj₁ (≤∧≢⇒< a≤b ¬p))

+
    [ <⇒≤ , ≤-reflexive ] }

+
  ℤ-hasPartialOrder = HasPartialOrder ∋ record { ≤-antisym = Int.≤-antisym }

+
  ℤ-hasDecPartialOrder = HasDecPartialOrder {A = ℤ} ∋ record {}

+

+
  import Data.Rational.Properties as Rat hiding (_≟_)

+

+
  ℚ-hasPreorder = hasPreorderFromNonStrict Rat.≤-isPreorder _≟_

+
  ℚ-hasPartialOrder = HasPartialOrder ∋ record { ≤-antisym = Rat.≤-antisym }

+
  ℚ-hasDecPartialOrder = HasDecPartialOrder {A = ℚ} ∋ record {}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasOrder.html b/site/static/agda_html/Class.HasOrder.html
new file mode 100644
index 000000000..d8a7426b8
--- /dev/null
+++ b/site/static/agda_html/Class.HasOrder.html
@@ -0,0 +1,83 @@
+
+Class.HasOrder
+             
+             
+             

+               

+                 Back
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+               

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+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Class.HasOrder where

+

+
open import Class.HasOrder.Core public

+
open import Class.HasOrder.Instance public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.HasSingleton.html b/site/static/agda_html/Class.HasSingleton.html
new file mode 100644
index 000000000..947e9f853
--- /dev/null
+++ b/site/static/agda_html/Class.HasSingleton.html
@@ -0,0 +1,100 @@
+
+Class.HasSingleton
+             
+             
+             

+               

+                 Back
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+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Axiom.Set using (Theory)

+

+
module Class.HasSingleton (th : Theory) where

+

+
open Theory th renaming (Set to ℙ_; ❴_❵ to ❴_❵ˢ)

+

+
open import Axiom.Set.Map th

+
open import abstract-set-theory.Prelude

+

+
record HasSingleton (A B : Type) : Type where

+
  field

+
    ❴_❵ : A → B

+

+
open HasSingleton ⦃...⦄ public

+

+
instance

+
  HasSingletonSet-Set : {A : Type} → HasSingleton A (ℙ A)

+
  HasSingletonSet-Set = record { ❴_❵ = ❴_❵ˢ }

+

+
  HasSingletonSet-Map : {A B : Type} → HasSingleton (A × B) (Map A B)

+
  HasSingletonSet-Map = record { ❴_❵ = ❴_❵ᵐ }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Hashable.html b/site/static/agda_html/Class.Hashable.html
new file mode 100644
index 000000000..0943fc221
--- /dev/null
+++ b/site/static/agda_html/Class.Hashable.html
@@ -0,0 +1,92 @@
+
+Class.Hashable
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+
module Class.Hashable where

+

+
open import Agda.Primitive using () renaming (Set to Type)

+

+
record Hashable (T THash : Type) : Type where

+
  field hash : T → THash

+

+
open Hashable ⦃...⦄ public

+

+
Hashable₁ : (Type → Type) → Type → Type₁

+
Hashable₁ F THash = {A : Type} → ⦃ Hashable A THash ⦄ → Hashable (F A) THash

+

+
Hashable₂ : (Type → Type → Type) → Type → Type₁

+
Hashable₂ F THash = {A B : Type} → ⦃ Hashable A THash ⦄ → ⦃ Hashable B THash ⦄ → Hashable (F A B) THash

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.IsSet.html b/site/static/agda_html/Class.IsSet.html
new file mode 100644
index 000000000..08b303bdf
--- /dev/null
+++ b/site/static/agda_html/Class.IsSet.html
@@ -0,0 +1,130 @@
+
+Class.IsSet
+             
+             
+             

+               

+                 Back
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+               
+               

+             
{-# OPTIONS --safe --no-import-sorts #-}

+

+
open import Axiom.Set

+

+
module Class.IsSet (th : Theory) where

+

+
open Theory th renaming (_∈_ to _∈ᵗ_; _∉_ to _∉ᵗ_)

+

+
import Axiom.Set.Rel th as Rel

+
open import Axiom.Set.Map th as Map

+
open import Axiom.Set.TotalMap th as TotalMap

+
open import Data.Product

+
open import abstract-set-theory.Prelude

+

+
private variable A B X : Type

+

+
record IsSet (A B : Type) : Type where

+
  field

+
    toSet : A → Set B

+

+
  infix 4 _∈_ _∉_

+
  _∈_ _∉_ : B → A → Type

+
  a ∈ X = a ∈ᵗ (toSet X)

+
  a ∉ X = a ∉ᵗ (toSet X)

+

+
open IsSet ⦃...⦄ public

+

+
infix 2 All-syntax

+
All-syntax : ∀ {A X} ⦃ _ : IsSet X A ⦄ → (A → Type) → X → Type

+
All-syntax P X = All P (toSet X)

+
syntax All-syntax (λ x → P) l = ∀[ x ∈ l ] P

+

+
infix 2 Ex-syntax

+
Ex-syntax : ∀ {A X} ⦃ _ : IsSet X A ⦄ → (A → Type) → X → Type

+
Ex-syntax P X = Any P (toSet X)

+
syntax Ex-syntax (λ x → P) l = ∃[ x ∈ l ] P

+

+
module _ ⦃ _ : IsSet X (A × B) ⦄ where

+
  dom : X → Set A

+
  dom = Rel.dom ∘ toSet

+

+
  range : X → Set B

+
  range = Rel.range ∘ toSet

+

+
instance

+
  IsSet-Set : IsSet (Set A) A

+
  IsSet-Set .toSet A = A

+

+
  IsSet-Map : IsSet (Map A B) (A × B)

+
  IsSet-Map .toSet = _ˢ

+

+
  IsSet-TotalMap : IsSet (TotalMap A B) (A × B)

+
  IsSet-TotalMap .toSet = TotalMap.rel

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Monad.Core.html b/site/static/agda_html/Class.Monad.Core.html
new file mode 100644
index 000000000..050b210e1
--- /dev/null
+++ b/site/static/agda_html/Class.Monad.Core.html
@@ -0,0 +1,142 @@
+
+Class.Monad.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

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+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Monad.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+
open import Class.Functor

+
open import Class.Applicative

+

+
record Monad (M : Type↑) : Typeω where

+
  infixl 1 _>>=_ _>>_ _>=>_

+
  infixr 1 _=<<_ _<=<_

+

+
  field

+
    return : A → M A

+
    _>>=_  : M A → (A → M B) → M B

+

+
  _>>_ : M A → M B → M B

+
  m₁ >> m₂ = m₁ >>= λ _ → m₂

+

+
  _=<<_ : (A → M B) → M A → M B

+
  f =<< c = c >>= f

+

+
  _>=>_ : (A → M B) → (B → M C) → (A → M C)

+
  f >=> g = _=<<_ g ∘ f

+

+
  _<=<_ : (B → M C) → (A → M B) → (A → M C)

+
  g <=< f = f >=> g

+

+
  join : M (M A) → M A

+
  join m = m >>= id

+

+
  Functor-M : Functor M

+
  Functor-M = λ where ._<$>_ f x → return ∘ f =<< x

+

+
  instance _ = Functor-M

+

+
  mapM : (A → M B) → List A → M (List B)

+
  mapM f []       = return []

+
  mapM f (x ∷ xs) = do y ← f x; y ∷_ <$> mapM f xs

+

+
  concatMapM : (A → M (List B)) → List A → M (List B)

+
  concatMapM f xs = concat <$> mapM f xs

+

+
  forM : List A → (A → M B) → M (List B)

+
  forM []       _ = return []

+
  forM (x ∷ xs) f = do y ← f x; y ∷_ <$> forM xs f

+

+
  concatForM : List A → (A → M (List B)) → M (List B)

+
  concatForM xs f = concat <$> forM xs f

+

+
  return⊤ void : M A → M ⊤

+
  return⊤ k = k >> return tt

+
  void = return⊤

+

+
  filterM : (A → M Bool) → List A → M (List A)

+
  filterM _ [] = return []

+
  filterM p (x ∷ xs) = do

+
    b ← p x

+
    ((if b then [ x ] else []) ++_) <$> filterM p xs

+

+
  traverseM : ⦃ Applicative M ⦄ → ⦃ Monad M ⦄ → (A → M B) → List A → M (List B)

+
  traverseM f = λ where

+
    [] → return []

+
    (x ∷ xs) → ⦇ f x ∷ traverseM f xs ⦈

+
open Monad ⦃...⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Monad.Instances.html b/site/static/agda_html/Class.Monad.Instances.html
new file mode 100644
index 000000000..cbaec22c0
--- /dev/null
+++ b/site/static/agda_html/Class.Monad.Instances.html
@@ -0,0 +1,98 @@
+
+Class.Monad.Instances
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

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+                 

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+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Monad.Instances where

+

+
open import Class.Prelude

+
open import Class.Monad.Core

+

+
instance

+
  Monad-TC : Monad TC

+
  Monad-TC = record {R}

+
    where import Reflection as R renaming (pure to return)

+

+
  Monad-List : Monad List

+
  Monad-List = λ where

+
    .return → _∷ []

+
    ._>>=_  → flip concatMap

+

+
  Monad-Maybe : Monad Maybe

+
  Monad-Maybe = λ where

+
    .return → just

+
    ._>>=_  → Maybe._>>=_

+
   where import Data.Maybe as Maybe

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Monad.html b/site/static/agda_html/Class.Monad.html
new file mode 100644
index 000000000..350a30cea
--- /dev/null
+++ b/site/static/agda_html/Class.Monad.html
@@ -0,0 +1,82 @@
+
+Class.Monad
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
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+                 
+                 
+               

+             

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+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Monad where

+

+
open import Class.Monad.Core public

+
open import Class.Monad.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.MonadError.Instances.html b/site/static/agda_html/Class.MonadError.Instances.html
new file mode 100644
index 000000000..f3e23d5d4
--- /dev/null
+++ b/site/static/agda_html/Class.MonadError.Instances.html
@@ -0,0 +1,82 @@
+
+Class.MonadError.Instances
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
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+             

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+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
module Class.MonadError.Instances where

+

+
open import Class.MonadError public

+
open MonadError ⦃...⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.MonadError.html b/site/static/agda_html/Class.MonadError.html
new file mode 100644
index 000000000..289ba9900
--- /dev/null
+++ b/site/static/agda_html/Class.MonadError.html
@@ -0,0 +1,120 @@
+
+Class.MonadError
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
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+                   
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+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
open import Level

+

+
open import Meta.Prelude

+

+
open import Class.Monad

+
open import Reflection using (TC; ErrorPart; typeError; catchTC; strErr)

+

+
module Class.MonadError where

+

+
private

+
  variable

+
    e f : Level

+
    A : Set f

+

+
record MonadError (E : Set e) (M : ∀ {f} → Set f → Set f) : Setω where

+
  field

+
    error : E → M A

+
    catch : M A → (E → M A) → M A

+

+
open MonadError

+

+
MonadError-TC : MonadError (List ErrorPart) TC

+
MonadError-TC .error = typeError

+
MonadError-TC .catch x h = catchTC x (h [ strErr "TC doesn't provide which error to catch" ])

+

+
ErrorT : (E : Set) → (M : ∀ {f} → Set f → Set f) → ∀ {f} → Set f → Set f

+
ErrorT E M A = M (E ⊎ A)

+

+
module _ {E : Set} {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ where

+

+
  Monad-ErrorT : Monad (ErrorT E M)

+
  Monad-ErrorT .return a = return (inj₂ a)

+
  Monad-ErrorT ._>>=_ x f = x >>= λ where

+
    (inj₁ e) → return (inj₁ e)

+
    (inj₂ a) → f a

+

+
  MonadError-ErrorT : MonadError E (ErrorT E M)

+
  MonadError-ErrorT .error e = return (inj₁ e)

+
  MonadError-ErrorT .catch x h = x >>= λ where

+
    (inj₁ e) → h e

+
    (inj₂ a) → return (inj₂ a)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.MonadReader.Instances.html b/site/static/agda_html/Class.MonadReader.Instances.html
new file mode 100644
index 000000000..a6aa6bdfc
--- /dev/null
+++ b/site/static/agda_html/Class.MonadReader.Instances.html
@@ -0,0 +1,82 @@
+
+Class.MonadReader.Instances
+             
+             
+             

+               

+                 Back
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+               

+                 
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+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
module Class.MonadReader.Instances where

+

+
open import Class.MonadReader public

+
open MonadReader ⦃...⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.MonadReader.html b/site/static/agda_html/Class.MonadReader.html
new file mode 100644
index 000000000..5129a8f1e
--- /dev/null
+++ b/site/static/agda_html/Class.MonadReader.html
@@ -0,0 +1,118 @@
+
+Class.MonadReader
+             
+             
+             

+               

+                 Back
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+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module Class.MonadReader where

+

+
open import Meta.Prelude

+

+
open import Class.Monad

+
open import Class.Functor

+
open import Class.MonadError

+

+
private variable ℓ : Level

+

+
open MonadError ⦃...⦄

+

+
record MonadReader (R : Set ℓ) (M : ∀ {a} → Set a → Set a) ⦃ _ : Monad M ⦄ : Setω where

+
  field

+
    ask : M R

+
    local : ∀ {a} {A : Set a} → (R → R) → M A → M A

+

+
  reader : ∀ {a} {A : Set a} → (R → A) → M A

+
  reader f = f <$> ask

+
    where instance _ = Functor-M

+

+
open MonadReader ⦃...⦄

+

+
ReaderT : (R : Set) (M : ∀ {a} → Set a → Set a) → ∀ {a} → Set a → Set a

+
ReaderT R M A = R → M A

+

+
module _ {R : Set} {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ where

+

+
  Monad-ReaderT : Monad (ReaderT R M)

+
  Monad-ReaderT .return a = λ r → return a

+
  Monad-ReaderT ._>>=_ x f = λ r → x r >>= λ a → f a r

+

+
  MonadReader-ReaderT : MonadReader R (ReaderT R M) ⦃ Monad-ReaderT ⦄

+
  MonadReader-ReaderT .ask = λ r → return r

+
  MonadReader-ReaderT .local f x = x ∘ f

+

+
  MonadError-ReaderT : ∀ {e} {E : Set e} → ⦃ MonadError E M ⦄ → MonadError E (ReaderT R M)

+
  MonadError-ReaderT .error e = λ r → error e

+
  MonadError-ReaderT .catch x h = λ r → catch (x r) (λ e → h e r)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.MonadTC.Instances.html b/site/static/agda_html/Class.MonadTC.Instances.html
new file mode 100644
index 000000000..b4d2b21cd
--- /dev/null
+++ b/site/static/agda_html/Class.MonadTC.Instances.html
@@ -0,0 +1,82 @@
+
+Class.MonadTC.Instances
+             
+             
+             

+               

+                 Back
+                 
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+             
{-# OPTIONS --safe --without-K #-}

+
module Class.MonadTC.Instances where

+

+
open import Class.MonadTC public

+
open MonadTC ⦃...⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.MonadTC.html b/site/static/agda_html/Class.MonadTC.html
new file mode 100644
index 000000000..db242d632
--- /dev/null
+++ b/site/static/agda_html/Class.MonadTC.html
@@ -0,0 +1,444 @@
+
+Class.MonadTC
+             
+             
+             

+               

+                 Back
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+             
{-# OPTIONS --safe --without-K #-}

+

+
module Class.MonadTC where

+

+
open import Meta.Prelude

+

+
open import Data.List using (map)

+

+
import Agda.Builtin.Reflection as R'

+
import Reflection as R

+
open import Reflection.Syntax

+

+
open import Reflection.Debug

+

+
open import Class.MonadReader

+
open import Class.MonadError

+
open import Class.Functor

+
open import Class.Monad

+
open import Class.Traversable

+

+
private variable

+
  a f : Level

+
  A B : Set f

+

+
data ReductionOptions : Set where

+
  onlyReduce : List Name → ReductionOptions

+
  dontReduce : List Name → ReductionOptions

+

+
reduceAll : ReductionOptions

+
reduceAll = dontReduce []

+

+
record TCOptions : Set where

+
  field

+
    debug : DebugOptions

+
    fuel  : List (String × ℕ)

+

+
defaultTCOptions : TCOptions

+
defaultTCOptions = record

+
  { debug = defaultDebugOptions

+
  ; fuel  = [] }

+

+
record TCEnv : Set where

+
  field

+
    normalisation  : Bool

+
    reconstruction : Bool

+
    noConstraints  : Bool

+
    reduction      : ReductionOptions

+
    globalContext  : Telescope

+
    localContext   : Telescope

+
    goal           : Term ⊎ Type

+
    options        : TCOptions

+

+
  open TCOptions options public

+

+
initTCEnvWithGoal : Term → R.TC TCEnv

+
initTCEnvWithGoal goal = R.getContext <&> λ ctx → record

+
  { normalisation  = false

+
  ; reconstruction = false

+
  ; noConstraints  = false

+
  ; reduction      = reduceAll

+
  ; globalContext  = ctx

+
  ; localContext   = []

+
  ; goal           = inj₁ goal

+
  ; options        = defaultTCOptions

+
  }

+

+
initTCEnv : R.TC TCEnv

+
initTCEnv = initTCEnvWithGoal unknown

+

+
record MonadTC (M : ∀ {f} → Set f → Set f)

+
               ⦃ m : Monad M ⦄ ⦃ me : MonadError (List ErrorPart) M ⦄ : Setω₁ where

+
  field

+
    unify            : Term → Term → M ⊤

+
    typeError        : ∀ {A : Set f} → List ErrorPart → M A

+
    inferType        : Term → M Type

+
    checkType        : Term → Type → M Term

+
    normalise        : Term → M Term

+
    reduce           : Term → M Term

+
    quoteTC          : A → M Term

+
    unquoteTC        : Term → M A

+
    quoteωTC         : ∀ {A : Setω} → A → M Term

+
    freshName        : String → M Name

+
    declareDef       : Arg Name → Type → M ⊤

+
    declarePostulate : Arg Name → Type → M ⊤

+
    defineFun        : Name → List Clause → M ⊤

+
    getType          : Name → M Type

+
    getDefinition    : Name → M Definition

+
    blockOnMeta      : Meta → M A

+
    commitTC         : M ⊤

+
    isMacro          : Name → M Bool

+
    debugPrint       : String → ℕ → List ErrorPart → M ⊤

+
    runSpeculative   : M (A × Bool) → M A

+
    getInstances     : Meta → M (List Term)

+

+
  instance _ = Functor-M {M = M}

+
  open MonadError me

+

+
  runAndReset : M A → M A

+
  runAndReset x = runSpeculative ((_, false) <$> x)

+

+
  -- dry-run and return true iff no error occured

+
  -- does not change any state

+
  isSuccessful : M A → M Bool

+
  isSuccessful x = runAndReset (catch (x >> return true) (λ _ → return false))

+

+
  -- TODO: return true on function, axiom and prim-fun constructors?

+
  isDef : Name → M Bool

+
  isDef n = do

+
    constructor′ _ _ ← getDefinition n

+
      where _ → return true

+
    return false

+

+
  isCon : Name → M Bool

+
  isCon n = do

+
    constructor′ _ _ ← getDefinition n

+
      where _ → return false

+
    return true

+

+
  nameConstr : Name → List (Arg Term) → M Term

+
  nameConstr n args = do

+
    isD ← isDef n

+
    isC ← isCon n

+
    case (isD , isC) of λ where

+
      (true , _)      → return $ def n args

+
      (false , true)  → return $ con n args

+
      (false , false) → error ((R'.primShowQName n <+> "is neither a definition nor a constructor!") ∷ᵈ [])

+

+
  termFromName : Name → M Term

+
  termFromName n = nameConstr n []

+

+
  -- apply the unique constructor of the record to the arguments

+
  mkRecord : Name → List (Arg Term) → M Term

+
  mkRecord n args = do

+
    (record′ c _) ← getDefinition n

+
      where _ → error ("Not a record!" ∷ᵈ [])

+
    return $ con c args

+

+
  -- this allows mutual recursion

+
  declareAndDefineFuns : List (Arg Name × Type × List Clause) → M ⊤

+
  declareAndDefineFuns ds = do

+
    traverse (λ (n , t , _) → declareDef n t) ds

+
    traverse (λ where (arg _ n , _ , cs) → defineFun n cs) ds

+
    return _

+

+
  declareAndDefineFun : Arg Name → Type → List Clause → M ⊤

+
  declareAndDefineFun n ty cls = declareAndDefineFuns [ (n , ty , cls) ]

+

+
  newMeta : Term → M Term

+
  newMeta = checkType unknown

+

+
module _ {M : ∀ {f} → Set f → Set f}

+
  ⦃ m : Monad M ⦄ ⦃ me : MonadError (List ErrorPart) M ⦄ ⦃ mtc : MonadTC M ⦄ ⦃ mre : MonadReader TCEnv M ⦄ where

+

+
  instance _ = Functor-M {M = M}

+
  open MonadError me

+
  open MonadTC mtc

+
  open MonadReader mre

+

+
  record IsMErrorPart (A : Set a) : Setω where

+
    field toMErrorPart : A → M (List ErrorPart)

+

+
  open IsMErrorPart ⦃...⦄ public

+

+
  data MErrorPartWrap : Set where

+
    wrap : M (List ErrorPart) → MErrorPartWrap

+

+
  IsMErrorPart-IsErrorPart : ⦃ _ : IsErrorPart A ⦄ → IsMErrorPart A

+
  IsMErrorPart-IsErrorPart .toMErrorPart a = return [ toErrorPart a ]

+

+
  instance

+
    IsMErrorPart-String : IsMErrorPart String

+
    IsMErrorPart-String = IsMErrorPart-IsErrorPart

+

+
    IsMErrorPart-Term : IsMErrorPart Term

+
    IsMErrorPart-Term = IsMErrorPart-IsErrorPart

+

+
    IsMErrorPart-Name : IsMErrorPart Name

+
    IsMErrorPart-Name = IsMErrorPart-IsErrorPart

+

+
    IsMErrorPart-MErrorPartWrap : IsMErrorPart MErrorPartWrap

+
    IsMErrorPart-MErrorPartWrap .toMErrorPart (wrap a) = a

+

+
  []ᵐ : M (List ErrorPart)

+
  []ᵐ = return []

+

+
  infixr 5 _∷ᵈᵐ_

+
  _∷ᵈᵐ_ : A → ⦃ _ : IsMErrorPart A ⦄ → M (List ErrorPart) → M (List ErrorPart)

+
  e ∷ᵈᵐ es = do e ← toMErrorPart e; es ← es; return (e ++ es)

+

+
  _ᵛ : A → MErrorPartWrap

+
  a ᵛ = wrap do a ← quoteTC a; return (a ∷ᵈ [])

+

+
  _ᵛⁿ : A → MErrorPartWrap

+
  a ᵛⁿ = wrap do a ← local (λ env → record env { normalisation = true }) $ quoteTC a; return (a ∷ᵈ [])

+

+
  _ᵗ : Term → MErrorPartWrap

+
  t ᵗ = wrap do T ← inferType t; return (t ∷ᵈ " : " ∷ᵈ T ∷ᵈ [])

+

+
  debugLog : List ErrorPart → M ⊤

+
  debugLog es = do

+
    record { options = record { debug = debug } } ← ask

+
    if debug .DebugOptions.filter (debug .DebugOptions.path)

+
      then debugPrint (debugOptionsPath debug) (debug .DebugOptions.level)

+
             (debugPrintPrefix debug ∷ es)

+
      else return tt

+

+
  debugLogᵐ : M (List ErrorPart) → M ⊤

+
  debugLogᵐ x = do x ← x; debugLog x

+

+
  debugLog1ᵐ : A → ⦃ _ : IsMErrorPart A ⦄ → M ⊤

+
  debugLog1ᵐ a = debugLogᵐ (a ∷ᵈᵐ []ᵐ)

+

+
  -- withDebugOptions : DebugOptions → M A → M A

+
  -- withDebugOptions opts x = local (λ where

+
  --   env@record { debug = opts' } → record env { debug = specializeDebugOptions opts' opts }) x

+

+
  withAppendDebugPath : String → M A → M A

+
  withAppendDebugPath path x = local (λ where

+
    env@record { options = o@record { debug = opts } } → record env {

+
      options = record o { debug = record opts { path = opts .DebugOptions.path ∷ʳ path } } }) x

+

+
  noConstraints : M A → M A

+
  noConstraints = local λ e → record e { noConstraints = true }

+

+
  hasType : Term → Type → M Bool

+
  hasType t ty = isSuccessful (noConstraints $ checkType t ty)

+

+
  hasType' : Name → Type → M Bool

+
  hasType' n ty = isSuccessful $ noConstraints $ do

+
    t ← termFromName n

+
    t' ← checkType t ty

+
    if isAppliedToUnknownsAndMetas t'

+
      then return tt

+
      else error ("This makes the function return false" ∷ᵈ [])

+
    where

+
      isUnknownsAndMetas : List (Arg Term) → Bool

+
      isUnknownsAndMetas [] = true

+
      isUnknownsAndMetas (arg i (meta x x₁) ∷ l) = isUnknownsAndMetas l

+
      isUnknownsAndMetas (arg i unknown ∷ l) = isUnknownsAndMetas l

+
      isUnknownsAndMetas (arg i _ ∷ l) = false

+

+
      isAppliedToUnknownsAndMetas : Term → Bool

+
      isAppliedToUnknownsAndMetas (var x args)      = isUnknownsAndMetas args

+
      isAppliedToUnknownsAndMetas (con c args)      = isUnknownsAndMetas args

+
      isAppliedToUnknownsAndMetas (def f args)      = isUnknownsAndMetas args

+
      isAppliedToUnknownsAndMetas (pat-lam cs args) = isUnknownsAndMetas args

+
      isAppliedToUnknownsAndMetas (meta x args)     = isUnknownsAndMetas args

+
      isAppliedToUnknownsAndMetas _                 = true

+

+
  extendContext : String × Arg Type → M A → M A

+
  extendContext ty = local λ where e@record { localContext = Γ } → record e { localContext = ty ∷ Γ }

+

+
  getContext : M Telescope

+
  getContext = reader λ where

+
    e@record { localContext = Γ ; globalContext = Γ' } → Γ Data.List.++ Γ'

+

+
  getLocalContext : M Telescope

+
  getLocalContext = reader λ where e@record { localContext = Γ } → Γ

+

+
  inContext : Telescope → M A → M A

+
  inContext ctx = local λ e → record e { localContext = ctx }

+

+
  -- extend context by multiple variables

+
  extendContext' : Telescope → M A → M A

+
  extendContext' [] x = x

+
  extendContext' (c ∷ cs) x = extendContext c (extendContext' cs x)

+

+
  dropContext : ℕ → M A → M A

+
  dropContext n x = do

+
    ctx ← getContext

+
    inContext (drop n ctx) x

+

+
  logAndError : List ErrorPart → M A

+
  logAndError e = debugLog e >> error e

+

+
  error1 : ⦃ IsErrorPart A ⦄ → A → M B

+
  error1 a = error (a ∷ᵈ [])

+

+
  debugLog1 : ⦃ IsErrorPart A ⦄ → A → M ⊤

+
  debugLog1 a = debugLog (a ∷ᵈ [])

+

+
  logAndError1 : ⦃ IsErrorPart A ⦄ → A → M B

+
  logAndError1 a = debugLog1 a >> error1 a

+

+
  markDontFail : String → M A → M A

+
  markDontFail s x = catch x λ e → logAndError (s ∷ᵈ " should never fail! This is a bug!\nError:\n" ∷ᵈ e)

+

+
  goalTy : M Type

+
  goalTy = do

+
    inj₁ t ← reader TCEnv.goal

+
      where inj₂ T → return T

+
    inferType t

+

+
  -- take the first result if it's a just, otherwise reset the state and

+
  -- run the second computation

+
  runSpeculativeMaybe : M (Maybe A) → M A → M A

+
  runSpeculativeMaybe x y = do

+
    nothing ← runSpeculative (< id , is-just > <$> x)

+
      where just a → return a

+
    y

+

+
  try : List (M ⊤) → M ⊤ → M ⊤

+
  try [] e = e

+
  try (x ∷ cs) e = MonadError.catch me x (λ _ → try cs e)

+

+
  getConstrs : Name → M (List (Name × Type))

+
  getConstrs n = do

+
    d ← getDefinition n

+
    cs ← case d of λ where

+
      (record′ c fs)  → return [ c ]

+
      (data-type pars cs) → return cs

+
      _                   → error (n ∷ᵈ "is not a data or record definition!" ∷ᵈ [])

+
    traverse (λ n → (n ,_) <$> (normalise =<< getType n)) (List Name ∋ cs)

+

+
  getConstrsForType : Term → M (List (Name × Type))

+
  getConstrsForType ty = do

+
    (def n _) ← normalise ty

+
      where _ → error (ty ∷ᵈ "does not reduce to a definition!" ∷ᵈ [])

+
    getConstrs n

+

+
  getConstrsForTerm : Term → M (List (Name × Type))

+
  getConstrsForTerm t = getConstrsForType =<< inferType t

+

+
  -- run a TC computation to arrive at the term under the binder for the pattern

+
  withPattern : Telescope → List (Arg Pattern) → M Term → M Clause

+
  withPattern tel ps t = Clause.clause tel ps <$> extendContext' tel t

+

+
  unifyWithGoal : Term → M ⊤

+
  unifyWithGoal t = do

+
    inj₁ t' ← reader TCEnv.goal

+
      where _ → error1 "unifyWithGoal: Goal is not a term!"

+
    unify t' t

+

+
  runWithHole : Term → M A → M A

+
  runWithHole t = local (λ env → record env { goal = inj₁ t })

+

+
  runWithGoalTy : Term → M A → M A

+
  runWithGoalTy t = local (λ env → record env { goal = inj₂ t })

+

+
  goalHole : M Term

+
  goalHole = do

+
    inj₂ T ← reader TCEnv.goal

+
      where inj₁ hole → return hole

+
    newMeta T

+

+
  withGoalHole : M ⊤ → M Term

+
  withGoalHole tac = do

+
    hole ← goalHole

+
    runWithHole hole tac

+
    return hole

+

+
  -- Finding all instance candidates of a given type.

+
  findInstances : Type → M (List Term)

+
  findInstances ty = runAndReset $ do

+
    just m ← findMeta <$> newMeta ty

+
      where _ → error1 "[findInstances] newMeta did not produce meta-variable!"

+
    getInstances m

+
    where

+
      open import Data.Maybe using (_<∣>_)

+
      findMeta = λ where

+
        (pi (arg _ ty) (abs _ t)) → findMeta ty <∣> findMeta t

+
        (lam _ (abs _ t))         → findMeta t

+
        (meta m _)                → just m

+
        _                         → nothing

+

+
MonadTC-TC : MonadTC R.TC ⦃ Monad-TC ⦄ ⦃ MonadError-TC ⦄

+
MonadTC-TC = record { R; R' using (quoteωTC; withReconstructed; onlyReduceDefs; dontReduceDefs; getInstances) }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Monoid.Core.html b/site/static/agda_html/Class.Monoid.Core.html
new file mode 100644
index 000000000..fa8761d89
--- /dev/null
+++ b/site/static/agda_html/Class.Monoid.Core.html
@@ -0,0 +1,101 @@
+
+Class.Monoid.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Monoid.Core where

+

+
open import Class.Prelude

+
open import Class.Semigroup

+

+
record Monoid (A : Type ℓ) ⦃ _ : Semigroup A ⦄ : Type ℓ where

+
  field ε : A

+
open Monoid ⦃...⦄ public

+

+
module _ (A : Type ℓ) ⦃ _ : Semigroup A ⦄ ⦃ _ : Monoid A ⦄ where

+
  record MonoidLaws (_≈_ : Rel A ℓ′) : Type (ℓ ⊔ ℓ′) where

+
    open Alg _≈_

+
    field ε-identity : Identity ε _◇_

+
    ε-identityˡ = LeftIdentity  ε _◇_ ∋ ε-identity .proj₁

+
    ε-identityʳ = RightIdentity ε _◇_ ∋ ε-identity .proj₂

+
  open MonoidLaws ⦃...⦄ public

+

+
  MonoidLaws≡ : Type ℓ

+
  MonoidLaws≡ = MonoidLaws _≡_

+

+
open MonoidLaws ⦃...⦄ public

+
  renaming ( ε-identity to ε-identity≡

+
           ; ε-identityˡ to ε-identityˡ≡; ε-identityʳ to ε-identityʳ≡ )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Monoid.Instances.html b/site/static/agda_html/Class.Monoid.Instances.html
new file mode 100644
index 000000000..87ad63f06
--- /dev/null
+++ b/site/static/agda_html/Class.Monoid.Instances.html
@@ -0,0 +1,159 @@
+
+Class.Monoid.Instances
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
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+                     
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+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Monoid.Instances where

+

+
open import Class.Prelude

+
open import Class.Semigroup

+
open import Class.Monoid.Core

+

+
instance

+
  Monoid-List : Monoid (List A)

+
  Monoid-List .ε = []

+

+
  MonoidLaws-List : MonoidLaws≡ (List A)

+
  MonoidLaws-List .ε-identity = L.++-identityˡ , L.++-identityʳ

+
    where import Data.List.Properties as L

+

+
  Monoid-Vec : Monoid (∃ (Vec A))

+
  Monoid-Vec .ε = 0 , []

+

+
  Monoid-String : Monoid String

+
  Monoid-String .ε = ""

+

+
module _ ⦃ _ : Semigroup A ⦄ ⦃ _ : Monoid A ⦄ where instance

+
  Monoid-Maybe : Monoid (Maybe A)

+
  Monoid-Maybe .ε = nothing

+

+
  MonoidLaws-Maybe : ⦃ MonoidLaws≡ A ⦄ → MonoidLaws≡ (Maybe A)

+
  MonoidLaws-Maybe .ε-identity = p , q

+
    where open Alg≡

+
          p = LeftIdentity ε  _◇_ ∋ λ _ → refl

+
          q = RightIdentity ε _◇_ ∋ λ where (just _) → refl; nothing → refl

+

+
-- ** natural numbers

+
module _ where

+
  open import Data.Nat.Properties

+
  module _ where

+
    instance _ = Semigroup-ℕ-+

+
    Monoid-ℕ-+ = Monoid ℕ ∋ λ where .ε → 0

+
    instance _ = Monoid-ℕ-+

+
    MonoidLaws-ℕ-+ = MonoidLaws≡ ℕ ∋ record {ε-identity = +-identityˡ , +-identityʳ}

+
  module _ where

+
    instance _ = Semigroup-ℕ-*

+
    Monoid-ℕ-* = Monoid ℕ ∋ λ where .ε → 1

+
    instance _ = Monoid-ℕ-*

+
    MonoidLaws-ℕ-* = MonoidLaws≡ ℕ ∋ record {ε-identity = *-identityˡ , *-identityʳ}

+

+
-- ** natural numbers

+
module _ where

+
  open import Data.Nat.Binary

+
  open import Data.Nat.Binary.Properties

+

+
  instance _ = Semigroup-ℕᵇ-+

+
  Monoid-ℕᵇ-+ = Monoid ℕᵇ ∋ λ where .ε → zero

+
  instance _ = Monoid-ℕᵇ-+

+
  MonoidLaws-ℕᵇ-+ = MonoidLaws≡ ℕᵇ ∋ record {ε-identity = +-identityˡ , +-identityʳ}

+

+
-- ** integers

+
module _ where

+
  open import Data.Integer.Properties

+
  module _ where

+
    instance _ = Semigroup-ℤ-+

+
    Monoid-ℤ-+ = Monoid ℤ ∋ λ where .ε → 0ℤ

+
    instance _ = Monoid-ℤ-+

+
    MonoidLaws-ℤ-+ = MonoidLaws≡ ℤ ∋ record {ε-identity = +-identityˡ , +-identityʳ}

+
  module _ where

+
    instance _ = Semigroup-ℤ-*

+
    Monoid-ℤ-* = Monoid ℤ ∋ λ where .ε → 1ℤ

+
    instance _ = Monoid-ℤ-*

+
    MonoidLaws-ℤ-* = MonoidLaws≡ ℤ ∋ record {ε-identity = *-identityˡ , *-identityʳ}

+

+
-- ** maybes

+
module _ ⦃ _ : Semigroup A ⦄ ⦃ _ : Monoid A ⦄ ⦃ _ : MonoidLaws≡ A ⦄ (x : A) where

+
  just-◇ˡ : ∀ (mx : Maybe A) →

+
    just x ◇ mx ≡ just (x ◇ fromMaybe ε mx)

+
  just-◇ˡ = λ where

+
    (just _) → refl

+
    nothing  → cong just $ sym $ ε-identityʳ≡ x

+

+
  just-◇ʳ : ∀ (mx : Maybe A) →

+
    mx ◇ just x ≡ just (fromMaybe ε mx ◇ x)

+
  just-◇ʳ = λ where

+
    (just _) → refl

+
    nothing  → cong just $ sym $ ε-identityˡ≡ x

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Monoid.html b/site/static/agda_html/Class.Monoid.html
new file mode 100644
index 000000000..dbb2c1c39
--- /dev/null
+++ b/site/static/agda_html/Class.Monoid.html
@@ -0,0 +1,82 @@
+
+Class.Monoid
+             
+             
+             

+               

+                 Back
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+               

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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Monoid where

+

+
open import Class.Monoid.Core public

+
open import Class.Monoid.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Prelude.html b/site/static/agda_html/Class.Prelude.html
new file mode 100644
index 000000000..9a870f9f4
--- /dev/null
+++ b/site/static/agda_html/Class.Prelude.html
@@ -0,0 +1,151 @@
+
+Class.Prelude
+             
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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Prelude where

+

+
open import Agda.Primitive public

+
  using () renaming (Set to Type; Setω to Typeω)

+
open import Level public

+
  using (Level; _⊔_) renaming (suc to lsuc)

+
open import Function public

+
  using (id; _∘_; _∋_; _$_; const; constᵣ; flip; it)

+

+
open import Data.Empty public

+
  using (⊥; ⊥-elim)

+
open import Data.Unit public

+
  using (⊤; tt)

+
open import Data.Product public

+
  using (_×_; _,_; proj₁; proj₂; Σ; ∃; ∃-syntax; -,_)

+
open import Data.Sum public

+
  using (_⊎_; inj₁; inj₂)

+
open import Data.Bool public

+
  using (Bool; true; false; not; if_then_else_; T)

+
open import Data.Nat public

+
  using (ℕ; zero; suc)

+
open import Data.Fin as Fin public

+
  using (Fin; zero; suc)

+
open import Data.Integer public

+
  using (ℤ; 0ℤ; 1ℤ)

+
open import Data.Rational public

+
  using (ℚ)

+
open import Data.Float public

+
  using (Float)

+
open import Data.Char public

+
  using (Char)

+
open import Data.String public

+
  using (String)

+
open import Data.Maybe public

+
  using (Maybe; just; nothing; maybe; fromMaybe)

+
open import Data.List public

+
  using (List; []; _∷_; [_]; map; _++_; foldr; concat; concatMap)

+
open import Data.List.NonEmpty public

+
  using (List⁺; _∷_; _⁺++⁺_; foldr₁)

+
open import Data.Vec public

+
  using (Vec; []; _∷_)

+
open import Data.These public

+
  using (These; this; that; these)

+

+
open import Relation.Nullary public

+
  using (¬_; Dec; yes; no)

+
open import Relation.Nullary.Decidable public

+
  using (⌊_⌋; dec-yes; isYes)

+
open import Relation.Unary public

+
  using (Pred)

+
  renaming (Decidable to Decidable¹)

+
open import Relation.Binary public

+
  using (REL; Rel; DecidableEquality)

+
  renaming (Decidable to Decidable²)

+
module _ {a b c} where

+
  3REL : (A : Set a) (B : Set b) (C : Set c) (ℓ : Level) → Type _

+
  3REL A B C ℓ = A → B → C → Type ℓ

+

+
  Decidable³ : ∀ {A B C ℓ} → 3REL A B C ℓ → Type _

+
  Decidable³ _~_~_ = ∀ x y z → Dec (x ~ y ~ z)

+
open import Relation.Binary.PropositionalEquality public

+
  using (_≡_; refl; sym; cong; subst)

+

+
open import Reflection public

+
  using (TC; Arg; Abs)

+

+
variable

+
  ℓ ℓ′ ℓ″ : Level

+
  A B C : Type ℓ

+

+
module Alg (_~_ : Rel A ℓ) where

+
  open import Algebra.Definitions _~_ public

+
module Alg≡ {ℓ} {A : Type ℓ} = Alg (_≡_ {A = A})

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Semigroup.Core.html b/site/static/agda_html/Class.Semigroup.Core.html
new file mode 100644
index 000000000..ed240ad82
--- /dev/null
+++ b/site/static/agda_html/Class.Semigroup.Core.html
@@ -0,0 +1,101 @@
+
+Class.Semigroup.Core
+             
+             
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+               

+                 Back
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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Semigroup.Core where

+

+
open import Class.Prelude

+

+
record Semigroup (A : Type ℓ) : Type ℓ where

+
  infixr 5 _◇_ _<>_

+
  field _◇_ : A → A → A

+
  _<>_ = _◇_

+
open Semigroup ⦃...⦄ public

+

+
module _ (A : Type ℓ) ⦃ _ : Semigroup A ⦄ where

+
  record SemigroupLaws (_≈_ : Rel A ℓ′) : Type (ℓ ⊔ ℓ′) where

+
    open Alg _≈_

+
    field ◇-comm   : Commutative _◇_

+
          ◇-assocʳ : Associative _◇_

+
  open SemigroupLaws ⦃...⦄ public

+

+
  SemigroupLaws≡ : Type ℓ

+
  SemigroupLaws≡ = SemigroupLaws _≡_

+

+
module _ {A : Type ℓ} ⦃ _ : Semigroup A ⦄ ⦃ _ : SemigroupLaws≡ A ⦄ where

+
  ◇-assocˡ : ∀ (x y z : A) → (x ◇ (y ◇ z)) ≡ ((x ◇ y) ◇ z)

+
  ◇-assocˡ x y z = sym (◇-assocʳ x y z)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Semigroup.Instances.html b/site/static/agda_html/Class.Semigroup.Instances.html
new file mode 100644
index 000000000..3702f9b4b
--- /dev/null
+++ b/site/static/agda_html/Class.Semigroup.Instances.html
@@ -0,0 +1,159 @@
+
+Class.Semigroup.Instances
+             
+             
+             

+               

+                 Back
+                 
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+               

+                 
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+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Semigroup.Instances where

+

+
open import Class.Prelude

+
open import Class.Semigroup.Core

+

+
instance

+
  Semigroup-List : Semigroup (List A)

+
  Semigroup-List ._◇_ = _++_

+

+
  Semigroup-List⁺ : Semigroup (List⁺ A)

+
  Semigroup-List⁺ ._◇_ = _⁺++⁺_

+

+
  Semigroup-∃Vec : Semigroup (∃ (Vec A))

+
  Semigroup-∃Vec ._◇_ (_ , v) (_ , v′) = _ , (v V.++ v′)

+
    where import Data.Vec as V

+

+
  Semigroup-String : Semigroup String

+
  Semigroup-String ._◇_ = Str._++_

+
    where import Data.String as Str

+

+
  Semigroup-Maybe : ⦃ Semigroup A ⦄ → Semigroup (Maybe A)

+
  Semigroup-Maybe ._◇_ = λ where

+
    (just x) (just y) → just (x ◇ y)

+
    (just x) nothing  → just x

+
    nothing  m        → m

+

+
  SemigroupLaws-Maybe : ⦃ _ : Semigroup A ⦄ → ⦃ SemigroupLaws≡ A ⦄

+
    → SemigroupLaws≡ (Maybe A)

+
  SemigroupLaws-Maybe {A = A} = λ where

+
    .◇-assocʳ → λ where

+
      (just _) (just _) (just _) → cong just (◇-assocʳ _ _ _)

+
      (just _) (just _) nothing  → refl

+
      (just _) nothing  _        → refl

+
      nothing  (just _) _        → refl

+
      nothing  nothing  _        → refl

+
    .◇-comm → λ where

+
      (just x) (just y) → cong just (◇-comm x y)

+
      (just _) nothing  → refl

+
      nothing  (just _) → refl

+
      nothing  nothing  → refl

+
   where open Alg≡

+

+
-- ** natural numbers

+
module _ where

+
  open import Data.Nat

+
  open import Data.Nat.Properties

+

+
  Semigroup-ℕ-+ = Semigroup ℕ ∋ λ where ._◇_ → _+_

+
  SemigroupLaws-ℕ-+ = SemigroupLaws ℕ _≡_ ∋

+
    record {◇-assocʳ = +-assoc; ◇-comm = +-comm}

+
    where instance _ = Semigroup-ℕ-+

+

+
  Semigroup-ℕ-* = Semigroup ℕ ∋ λ where ._◇_ → _*_

+
  SemigroupLaws-ℕ-* = SemigroupLaws ℕ _≡_ ∋

+
    record {◇-assocʳ = *-assoc; ◇-comm = *-comm}

+
    where instance _ = Semigroup-ℕ-*

+

+
-- ** binary natural numbers

+
module _ where

+
  open import Data.Nat.Binary

+
  open import Data.Nat.Binary.Properties

+

+
  Semigroup-ℕᵇ-+ = Semigroup ℕᵇ ∋ λ where ._◇_ → _+_

+
  SemigroupLaws-ℕᵇ-+ = SemigroupLaws ℕᵇ _≡_ ∋

+
    record {◇-assocʳ = +-assoc; ◇-comm = +-comm}

+
    where instance _ = Semigroup-ℕᵇ-+

+

+
-- ** integers

+
module _ where

+
  open import Data.Integer

+
  open import Data.Integer.Properties

+

+
  Semigroup-ℤ-+ = Semigroup ℤ ∋ λ where ._◇_ → _+_

+
  SemigroupLaws-ℤ-+ = SemigroupLaws ℤ _≡_ ∋

+
    record {◇-assocʳ = +-assoc; ◇-comm = +-comm}

+
    where instance _ = Semigroup-ℤ-+

+

+
  Semigroup-ℤ-* = Semigroup ℤ ∋ λ where ._◇_ → _*_

+
  SemigroupLaws-ℤ-* = SemigroupLaws ℤ _≡_ ∋

+
    record {◇-assocʳ = *-assoc; ◇-comm = *-comm}

+
    where instance _ = Semigroup-ℤ-*

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Semigroup.html b/site/static/agda_html/Class.Semigroup.html
new file mode 100644
index 000000000..6c2b80373
--- /dev/null
+++ b/site/static/agda_html/Class.Semigroup.html
@@ -0,0 +1,82 @@
+
+Class.Semigroup
+             
+             
+             

+               

+                 Back
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+               

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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Semigroup where

+

+
open import Class.Semigroup.Core public

+
open import Class.Semigroup.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Show.Core.html b/site/static/agda_html/Class.Show.Core.html
new file mode 100644
index 000000000..afeb73178
--- /dev/null
+++ b/site/static/agda_html/Class.Show.Core.html
@@ -0,0 +1,91 @@
+
+Class.Show.Core
+             
+             
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+             

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+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Show.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+

+
record Show (A : Type ℓ) : Type ℓ where

+
  constructor mkShow

+
  field show : A → String

+
open Show ⦃...⦄ public

+

+
Show¹ = Show ¹

+
Show² = Show ²

+
Show³ = Show ³

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Show.Instances.html b/site/static/agda_html/Class.Show.Instances.html
new file mode 100644
index 000000000..77df9d060
--- /dev/null
+++ b/site/static/agda_html/Class.Show.Instances.html
@@ -0,0 +1,146 @@
+
+Class.Show.Instances
+             
+             
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+               

+                 Back
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+             
{-# OPTIONS --without-K #-}

+
module Class.Show.Instances where

+

+
open import Class.Prelude hiding (Type)

+
open import Data.String using (parens; braces; intersperse)

+
open import Class.Semigroup

+
open import Class.Show.Core

+

+
Show-× : ⦃ Show A ⦄ → ⦃ Show B ⦄ → Show (A × B)

+
Show-× .show (x , y) = parens $ show x ◇ " , " ◇ show y

+

+
Show-List : ⦃ Show A ⦄ → Show (List A)

+
Show-List .show = braces ∘ intersperse ", " ∘ map show

+

+
instance

+
  Show-String = mkShow id

+

+
  Show-⊤ = Show ⊤ ∋ λ where .show tt → "tt"

+

+
  Show-Char = Show _ ∋ record {M}

+
    where import Data.Char as M

+

+
  Show-Bool = Show _ ∋ record {M}

+
    where import Data.Bool.Show as M

+

+
  Show-ℕ = Show _ ∋ record {M}

+
    where import Data.Nat.Show as M

+

+
  Show-ℤ = Show _ ∋ record {M}

+
    where import Data.Integer.Show as M

+

+
  Show-Fin : Show¹ Fin

+
  Show-Fin .show = ("# " ◇_) ∘ show ∘ toℕ

+
    where open import Data.Fin using (toℕ)

+

+
  Show-Float = Show _ ∋ record {M}

+
    where import Data.Float as M

+

+
open import Reflection

+
open import Reflection.AST.Term

+
open import Reflection.AST.Meta

+

+
instance

+
  Show-Name = mkShow showName

+
  Show-Meta = mkShow showMeta

+
  Show-Relevance = mkShow showRel -- showRelevance

+
  Show-Vis = mkShow showVisibility

+
  Show-Literal = mkShow showLiteral

+

+
Show-Arg : ⦃ Show A ⦄ → Show (Arg A)

+
Show-Arg .show (arg (arg-info v _) x) = show v ◇ show x

+

+
Show-Abs : ⦃ Show A ⦄ → Show (Abs A)

+
Show-Abs .show (abs s x) = "abs " ◇ show s ◇ " " ◇ show x

+

+
instance

+
  Show-Names = Show (Args Term) ∋ mkShow showTerms

+
  Show-Term = mkShow showTerm

+
  Show-Sort = mkShow showSort

+
  Show-Patterns = Show (Args Pattern) ∋ mkShow showPatterns

+
  Show-Pattern = mkShow showPattern

+
  Show-Clause = mkShow showClause

+
  Show-Clauses = Show (List Clause) ∋ mkShow showClauses

+
  Show-Tel = Show Telescope ∋ mkShow showTel

+
  Show-Definition = mkShow showDefinition

+

+
  Show-AName = Show (Arg Name) ∋ Show-Arg

+
  Show-AType = Show (Arg Type) ∋ Show-Arg

+
  Show-ATerms = Show (Args Name) ∋ Show-List

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Show.html b/site/static/agda_html/Class.Show.html
new file mode 100644
index 000000000..53288b46b
--- /dev/null
+++ b/site/static/agda_html/Class.Show.html
@@ -0,0 +1,82 @@
+
+Class.Show
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Show where

+

+
open import Class.Show.Core public

+
open import Class.Show.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.ToBool.html b/site/static/agda_html/Class.ToBool.html
new file mode 100644
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--- /dev/null
+++ b/site/static/agda_html/Class.ToBool.html
@@ -0,0 +1,129 @@
+
+Class.ToBool
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+
module Class.ToBool where

+

+
open import Level

+
open import Data.Bool using (Bool; true; false)

+
open import Data.Sum

+
open import Function

+
open import Data.Unit.Polymorphic

+
open import Relation.Binary.PropositionalEquality

+
open import Relation.Nullary

+
open import Data.Maybe

+
open import Class.Decidable.Core

+

+
private variable

+
  ℓ ℓ′ : Level

+
  X : Set ℓ; P : X → Set ℓ

+

+
record ToBool′ (A : Set ℓ) (P 𝕋 𝔽 : A → Set ℓ′) : Set (ℓ ⊔ ℓ′) where

+
  field decide : (a : A) → ⦃ P a ⦄ → 𝕋 a ⊎ 𝔽 a

+

+
  infix -10 if_then_else_

+
  if_then_else_ : (a : A) ⦃ _ : P a ⦄ → ({𝕋 a} → X) → ({𝔽 a} → X) → X

+
  if a then t else f =

+
    case decide a of λ where

+
      (inj₁ 𝕥) → t {𝕥}

+
      (inj₂ 𝕗) → f {𝕗}

+

+
  toBool : (a : A) ⦃ _ : P a ⦄ → Bool

+
  toBool a = if a then true else false

+
open ToBool′ ⦃...⦄ public

+

+
ToBool : (A : Set ℓ) (𝕋 𝔽 : A → Set ℓ′) → Set (ℓ ⊔ ℓ′)

+
ToBool {ℓ} A = ToBool′ A (λ _ → ⊤)

+

+
instance

+
  ToBool-Bool : ToBool Bool (_≡ true) (_≡ false)

+
  ToBool-Bool .decide = λ where

+
    true  → inj₁ refl

+
    false → inj₂ refl

+

+
  ToBool-Dec : ToBool (Dec X) (const X) (const $ ¬ X)

+
  ToBool-Dec .decide = λ where

+
    (yes x) → inj₁ x

+
    (no ¬x) → inj₂ ¬x

+

+
  ToBool-Maybe : ToBool (Maybe X) (const X) (const ⊤)

+
  ToBool-Maybe .decide = λ where

+
    (just x) → inj₁ x

+
    nothing  → inj₂ tt

+

+
  ToBool-⁇ : ToBool′ (Set ℓ) _⁇ id ¬_

+
  ToBool-⁇ .decide _ = decide dec ⦃ _ ⦄

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Traversable.Core.html b/site/static/agda_html/Class.Traversable.Core.html
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,91 @@
+
+Class.Traversable.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

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+               

+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Traversable.Core where

+

+
open import Class.Prelude

+
open import Class.Core

+
open import Class.Functor.Core

+
open import Class.Monad

+

+
record Traversable (F : Type↑) ⦃ _ : Functor F ⦄ : Typeω where

+
  field sequence : ⦃ Monad M ⦄ → F (M A) → M (F A)

+

+
  traverse : ⦃ Monad M ⦄ → (A → M B) → F A → M (F B)

+
  traverse f = sequence ∘ fmap f

+
open Traversable ⦃...⦄ public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Traversable.Instances.html b/site/static/agda_html/Class.Traversable.Instances.html
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,99 @@
+
+Class.Traversable.Instances
+             
+             
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+               

+                 Back
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+               

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+             

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+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Traversable.Instances where

+

+
open import Class.Prelude

+
open import Class.Functor

+
open import Class.Monad

+
open import Class.Traversable.Core

+

+
instance

+
  Traversable-Maybe : Traversable Maybe

+
  Traversable-Maybe .sequence = λ where

+
    nothing  → return nothing

+
    (just x) → x >>= return ∘ just

+

+
  Traversable-List : Traversable List

+
  Traversable-List .sequence = go

+
    where go = λ where

+
      []       → return []

+
      (m ∷ ms) → do x ← m; xs ← go ms; return (x ∷ xs)

+

+
  Traversable-List⁺ : Traversable List⁺

+
  Traversable-List⁺ .sequence (m ∷ ms) = do x ← m; xs ← sequence ms; return (x ∷ xs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Class.Traversable.html b/site/static/agda_html/Class.Traversable.html
new file mode 100644
index 000000000..fc88e0d9c
--- /dev/null
+++ b/site/static/agda_html/Class.Traversable.html
@@ -0,0 +1,82 @@
+
+Class.Traversable
+             
+             
+             

+               

+                 Back
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+             

+             

+               
+               

+             
{-# OPTIONS --without-K #-}

+
module Class.Traversable where

+

+
open import Class.Traversable.Core public

+
open import Class.Traversable.Instances public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Bool.Base.html b/site/static/agda_html/Data.Bool.Base.html
new file mode 100644
index 000000000..57422a0b0
--- /dev/null
+++ b/site/static/agda_html/Data.Bool.Base.html
@@ -0,0 +1,152 @@
+
+Data.Bool.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

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+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The type for booleans and some operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Bool.Base where

+

+
open import Data.Unit.Base using (⊤)

+
open import Data.Empty

+
open import Level using (Level)

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- The boolean type

+

+
open import Agda.Builtin.Bool public

+

+
------------------------------------------------------------------------

+
-- Relations

+

+
infix 4 _≤_ _<_

+

+
data _≤_ : Bool → Bool → Set where

+
  f≤t : false ≤ true

+
  b≤b : ∀ {b} → b ≤ b

+

+
data _<_ : Bool → Bool → Set where

+
  f<t : false < true

+

+
------------------------------------------------------------------------

+
-- Boolean operations

+

+
infixr 6 _∧_

+
infixr 5 _∨_ _xor_

+

+
not : Bool → Bool

+
not true  = false

+
not false = true

+

+
_∧_ : Bool → Bool → Bool

+
true  ∧ b = b

+
false ∧ b = false

+

+
_∨_ : Bool → Bool → Bool

+
true  ∨ b = true

+
false ∨ b = b

+

+
_xor_ : Bool → Bool → Bool

+
true  xor b = not b

+
false xor b = b

+

+
------------------------------------------------------------------------

+
-- Conversion to Set

+

+
-- A function mapping true to an inhabited type and false to an empty

+
-- type.

+
T : Bool → Set

+
T true  = ⊤

+
T false = ⊥

+

+
------------------------------------------------------------------------

+
-- Other operations

+

+
infix 0 if_then_else_

+

+
if_then_else_ : Bool → A → A → A

+
if true  then t else f = t

+
if false then t else f = f

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Bool.Properties.html b/site/static/agda_html/Data.Bool.Properties.html
new file mode 100644
index 000000000..4992d43c0
--- /dev/null
+++ b/site/static/agda_html/Data.Bool.Properties.html
@@ -0,0 +1,865 @@
+
+Data.Bool.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- A bunch of properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Bool.Properties where

+

+
open import Algebra.Bundles

+
open import Algebra.Lattice.Bundles

+
import Algebra.Lattice.Properties.BooleanAlgebra as BooleanAlgebraProperties

+
open import Data.Bool.Base

+
open import Data.Empty

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])

+
open import Function.Base using (_⟨_⟩_; const; id)

+
open import Function.Bundles hiding (LeftInverse; RightInverse; Inverse)

+
open import Induction.WellFounded using (WellFounded; Acc; acc)

+
open import Level using (Level; 0ℓ)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Poset; Preorder; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Decidable; DecidableEquality; Reflexive; Transitive; Antisymmetric; Minimum; Maximum; Total; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>; _Respects₂_)

+
open import Relation.Binary.PropositionalEquality.Core

+
open import Relation.Binary.PropositionalEquality.Properties

+
open import Relation.Nullary.Decidable.Core using (True; yes; no; fromWitness)

+
import Relation.Unary as U

+

+
open import Algebra.Definitions {A = Bool} _≡_

+
open import Algebra.Structures {A = Bool} _≡_

+
open import Algebra.Lattice.Structures {A = Bool} _≡_

+

+
open ≡-Reasoning

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Bool

+
true  ≟ true  = yes refl

+
false ≟ false = yes refl

+
true  ≟ false = no λ()

+
false ≟ true  = no λ()

+

+
≡-setoid : Setoid 0ℓ 0ℓ

+
≡-setoid = setoid Bool

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+

+
-- Relational properties

+

+
≤-reflexive : _≡_ ⇒ _≤_

+
≤-reflexive refl = b≤b

+

+
≤-refl : Reflexive _≤_

+
≤-refl = ≤-reflexive refl

+

+
≤-trans : Transitive _≤_

+
≤-trans b≤b p   = p

+
≤-trans f≤t b≤b = f≤t

+

+
≤-antisym : Antisymmetric _≡_ _≤_

+
≤-antisym b≤b _ = refl

+

+
≤-minimum : Minimum _≤_ false

+
≤-minimum false = b≤b

+
≤-minimum true  = f≤t

+

+
≤-maximum : Maximum _≤_ true

+
≤-maximum false = f≤t

+
≤-maximum true  = b≤b

+

+
≤-total : Total _≤_

+
≤-total false b = inj₁ (≤-minimum b)

+
≤-total true  b = inj₂ (≤-maximum b)

+

+
infix 4 _≤?_

+

+
_≤?_ : Decidable _≤_

+
false ≤? b     = yes (≤-minimum b)

+
true  ≤? false = no λ ()

+
true  ≤? true  = yes b≤b

+

+
≤-irrelevant : Irrelevant _≤_

+
≤-irrelevant {_}     f≤t f≤t = refl

+
≤-irrelevant {false} b≤b b≤b = refl

+
≤-irrelevant {true}  b≤b b≤b = refl

+

+
-- Structures

+

+
≤-isPreorder : IsPreorder _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder _≡_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≤?_

+
  }

+

+
-- Bundles

+

+
≤-poset : Poset 0ℓ 0ℓ 0ℓ

+
≤-poset = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  }

+

+
≤-preorder : Preorder 0ℓ 0ℓ 0ℓ

+
≤-preorder = record

+
  { isPreorder = ≤-isPreorder

+
  }

+

+
≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ

+
≤-totalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  }

+

+
≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ

+
≤-decTotalOrder = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+

+
-- Relational properties

+

+
<-irrefl : Irreflexive _≡_ _<_

+
<-irrefl refl ()

+

+
<-asym : Asymmetric _<_

+
<-asym f<t ()

+

+
<-trans : Transitive _<_

+
<-trans f<t ()

+

+
<-transʳ : Trans _≤_ _<_ _<_

+
<-transʳ b≤b f<t = f<t

+

+
<-transˡ : Trans _<_ _≤_ _<_

+
<-transˡ f<t b≤b = f<t

+

+
<-cmp : Trichotomous _≡_ _<_

+
<-cmp false false = tri≈ (λ()) refl  (λ())

+
<-cmp false true  = tri< f<t   (λ()) (λ())

+
<-cmp true  false = tri> (λ()) (λ()) f<t

+
<-cmp true  true  = tri≈ (λ()) refl  (λ())

+

+
infix 4 _<?_

+

+
_<?_ : Decidable _<_

+
false <? false = no  (λ())

+
false <? true  = yes f<t

+
true  <? _     = no  (λ())

+

+
<-resp₂-≡ : _<_ Respects₂ _≡_

+
<-resp₂-≡ = subst (_ <_) , subst (_< _)

+

+
<-irrelevant : Irrelevant _<_

+
<-irrelevant f<t f<t = refl

+

+
<-wellFounded : WellFounded _<_

+
<-wellFounded _ = acc <-acc

+
  where

+
    <-acc : ∀ {x y} → y < x → Acc _<_ y

+
    <-acc f<t = acc λ ()

+

+
-- Structures

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = <-resp₂-≡

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
-- Bundles

+

+
<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _∨_

+

+
∨-assoc : Associative _∨_

+
∨-assoc true  y z = refl

+
∨-assoc false y z = refl

+

+
∨-comm : Commutative _∨_

+
∨-comm true  true  = refl

+
∨-comm true  false = refl

+
∨-comm false true  = refl

+
∨-comm false false = refl

+

+
∨-identityˡ : LeftIdentity false _∨_

+
∨-identityˡ _ = refl

+

+
∨-identityʳ : RightIdentity false _∨_

+
∨-identityʳ false = refl

+
∨-identityʳ true  = refl

+

+
∨-identity : Identity false _∨_

+
∨-identity = ∨-identityˡ , ∨-identityʳ

+

+
∨-zeroˡ : LeftZero true _∨_

+
∨-zeroˡ _ = refl

+

+
∨-zeroʳ : RightZero true _∨_

+
∨-zeroʳ false = refl

+
∨-zeroʳ true  = refl

+

+
∨-zero : Zero true _∨_

+
∨-zero = ∨-zeroˡ , ∨-zeroʳ

+

+
∨-inverseˡ : LeftInverse true not _∨_

+
∨-inverseˡ false = refl

+
∨-inverseˡ true  = refl

+

+
∨-inverseʳ : RightInverse true not _∨_

+
∨-inverseʳ x = ∨-comm x (not x) ⟨ trans ⟩ ∨-inverseˡ x

+

+
∨-inverse : Inverse true not _∨_

+
∨-inverse = ∨-inverseˡ , ∨-inverseʳ

+

+
∨-idem : Idempotent _∨_

+
∨-idem false = refl

+
∨-idem true  = refl

+

+
∨-sel : Selective _∨_

+
∨-sel false y = inj₂ refl

+
∨-sel true y  = inj₁ refl

+

+
∨-conicalˡ : LeftConical false _∨_

+
∨-conicalˡ false false _ = refl

+

+
∨-conicalʳ : RightConical false _∨_

+
∨-conicalʳ false false _ = refl

+

+
∨-conical : Conical false _∨_

+
∨-conical = ∨-conicalˡ , ∨-conicalʳ

+

+
∨-isMagma : IsMagma _∨_

+
∨-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _∨_

+
  }

+

+
∨-magma : Magma 0ℓ 0ℓ

+
∨-magma = record

+
  { isMagma = ∨-isMagma

+
  }

+

+
∨-isSemigroup : IsSemigroup _∨_

+
∨-isSemigroup = record

+
  { isMagma = ∨-isMagma

+
  ; assoc   = ∨-assoc

+
  }

+

+
∨-semigroup : Semigroup 0ℓ 0ℓ

+
∨-semigroup = record

+
  { isSemigroup = ∨-isSemigroup

+
  }

+

+
∨-isBand : IsBand _∨_

+
∨-isBand = record

+
  { isSemigroup = ∨-isSemigroup

+
  ; idem        = ∨-idem

+
  }

+

+
∨-band : Band 0ℓ 0ℓ

+
∨-band = record

+
  { isBand = ∨-isBand

+
  }

+

+
∨-isSemilattice : IsSemilattice _∨_

+
∨-isSemilattice = record

+
  { isBand = ∨-isBand

+
  ; comm   = ∨-comm

+
  }

+

+
∨-semilattice : Semilattice 0ℓ 0ℓ

+
∨-semilattice = record

+
  { isSemilattice = ∨-isSemilattice

+
  }

+

+
∨-isMonoid : IsMonoid _∨_ false

+
∨-isMonoid = record

+
  { isSemigroup = ∨-isSemigroup

+
  ; identity = ∨-identity

+
  }

+

+
∨-isCommutativeMonoid : IsCommutativeMonoid _∨_ false

+
∨-isCommutativeMonoid = record

+
  { isMonoid = ∨-isMonoid

+
  ; comm = ∨-comm

+
  }

+

+
∨-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
∨-commutativeMonoid = record

+
  { isCommutativeMonoid = ∨-isCommutativeMonoid

+
  }

+

+
∨-isIdempotentCommutativeMonoid :

+
  IsIdempotentCommutativeMonoid _∨_ false

+
∨-isIdempotentCommutativeMonoid = record

+
  { isCommutativeMonoid = ∨-isCommutativeMonoid

+
  ; idem                = ∨-idem

+
  }

+

+
∨-idempotentCommutativeMonoid : IdempotentCommutativeMonoid 0ℓ 0ℓ

+
∨-idempotentCommutativeMonoid = record

+
  { isIdempotentCommutativeMonoid = ∨-isIdempotentCommutativeMonoid

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _∧_

+

+
∧-assoc : Associative _∧_

+
∧-assoc true  y z = refl

+
∧-assoc false y z = refl

+

+
∧-comm : Commutative _∧_

+
∧-comm true  true  = refl

+
∧-comm true  false = refl

+
∧-comm false true  = refl

+
∧-comm false false = refl

+

+
∧-identityˡ : LeftIdentity true _∧_

+
∧-identityˡ _ = refl

+

+
∧-identityʳ : RightIdentity true _∧_

+
∧-identityʳ false = refl

+
∧-identityʳ true  = refl

+

+
∧-identity : Identity true _∧_

+
∧-identity = ∧-identityˡ , ∧-identityʳ

+

+
∧-zeroˡ : LeftZero false _∧_

+
∧-zeroˡ _ = refl

+

+
∧-zeroʳ : RightZero false _∧_

+
∧-zeroʳ false = refl

+
∧-zeroʳ true  = refl

+

+
∧-zero : Zero false _∧_

+
∧-zero = ∧-zeroˡ , ∧-zeroʳ

+

+
∧-inverseˡ : LeftInverse false not _∧_

+
∧-inverseˡ false = refl

+
∧-inverseˡ true = refl

+

+
∧-inverseʳ : RightInverse false not _∧_

+
∧-inverseʳ x = ∧-comm x (not x) ⟨ trans ⟩ ∧-inverseˡ x

+

+
∧-inverse : Inverse false not _∧_

+
∧-inverse = ∧-inverseˡ , ∧-inverseʳ

+

+
∧-idem : Idempotent _∧_

+
∧-idem false = refl

+
∧-idem true  = refl

+

+
∧-sel : Selective _∧_

+
∧-sel false y = inj₁ refl

+
∧-sel true y  = inj₂ refl

+

+
∧-conicalˡ : LeftConical true _∧_

+
∧-conicalˡ true true _ = refl

+

+
∧-conicalʳ : RightConical true _∧_

+
∧-conicalʳ true true _ = refl

+

+
∧-conical : Conical true _∧_

+
∧-conical = ∧-conicalˡ , ∧-conicalʳ

+

+
∧-distribˡ-∨ : _∧_ DistributesOverˡ _∨_

+
∧-distribˡ-∨ true  y z = refl

+
∧-distribˡ-∨ false y z = refl

+

+
∧-distribʳ-∨ : _∧_ DistributesOverʳ _∨_

+
∧-distribʳ-∨ x y z = begin

+
  (y ∨ z) ∧ x     ≡⟨ ∧-comm (y ∨ z) x ⟩

+
  x ∧ (y ∨ z)     ≡⟨ ∧-distribˡ-∨ x y z ⟩

+
  x ∧ y ∨ x ∧ z   ≡⟨ cong₂ _∨_ (∧-comm x y) (∧-comm x z) ⟩

+
  y ∧ x ∨ z ∧ x   ∎

+

+
∧-distrib-∨ : _∧_ DistributesOver _∨_

+
∧-distrib-∨ = ∧-distribˡ-∨ , ∧-distribʳ-∨

+

+
∨-distribˡ-∧ : _∨_ DistributesOverˡ _∧_

+
∨-distribˡ-∧ true  y z = refl

+
∨-distribˡ-∧ false y z = refl

+

+
∨-distribʳ-∧ : _∨_ DistributesOverʳ _∧_

+
∨-distribʳ-∧ x y z = begin

+
  (y ∧ z) ∨ x        ≡⟨ ∨-comm (y ∧ z) x ⟩

+
  x ∨ (y ∧ z)        ≡⟨ ∨-distribˡ-∧ x y z ⟩

+
  (x ∨ y) ∧ (x ∨ z)  ≡⟨ cong₂ _∧_ (∨-comm x y) (∨-comm x z) ⟩

+
  (y ∨ x) ∧ (z ∨ x)  ∎

+

+
∨-distrib-∧ : _∨_ DistributesOver _∧_

+
∨-distrib-∧ = ∨-distribˡ-∧ , ∨-distribʳ-∧

+

+
∧-abs-∨ : _∧_ Absorbs _∨_

+
∧-abs-∨ true  y = refl

+
∧-abs-∨ false y = refl

+

+
∨-abs-∧ : _∨_ Absorbs _∧_

+
∨-abs-∧ true  y = refl

+
∨-abs-∧ false y = refl

+

+
∨-∧-absorptive : Absorptive _∨_ _∧_

+
∨-∧-absorptive = ∨-abs-∧ , ∧-abs-∨

+

+
∧-isMagma : IsMagma _∧_

+
∧-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _∧_

+
  }

+

+
∧-magma : Magma 0ℓ 0ℓ

+
∧-magma = record

+
  { isMagma = ∧-isMagma

+
  }

+

+
∧-isSemigroup : IsSemigroup _∧_

+
∧-isSemigroup = record

+
  { isMagma = ∧-isMagma

+
  ; assoc   = ∧-assoc

+
  }

+

+
∧-semigroup : Semigroup 0ℓ 0ℓ

+
∧-semigroup = record

+
  { isSemigroup = ∧-isSemigroup

+
  }

+

+
∧-isBand : IsBand _∧_

+
∧-isBand = record

+
  { isSemigroup = ∧-isSemigroup

+
  ; idem        = ∧-idem

+
  }

+

+
∧-band : Band 0ℓ 0ℓ

+
∧-band = record

+
  { isBand = ∧-isBand

+
  }

+

+
∧-isSemilattice : IsSemilattice _∧_

+
∧-isSemilattice = record

+
  { isBand = ∧-isBand

+
  ; comm   = ∧-comm

+
  }

+

+
∧-semilattice : Semilattice 0ℓ 0ℓ

+
∧-semilattice = record

+
  { isSemilattice = ∧-isSemilattice

+
  }

+

+
∧-isMonoid : IsMonoid _∧_ true

+
∧-isMonoid = record

+
  { isSemigroup = ∧-isSemigroup

+
  ; identity = ∧-identity

+
  }

+

+
∧-isCommutativeMonoid : IsCommutativeMonoid _∧_ true

+
∧-isCommutativeMonoid = record

+
  { isMonoid = ∧-isMonoid

+
  ; comm = ∧-comm

+
  }

+

+
∧-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
∧-commutativeMonoid = record

+
  { isCommutativeMonoid = ∧-isCommutativeMonoid

+
  }

+

+
∧-isIdempotentCommutativeMonoid :

+
  IsIdempotentCommutativeMonoid _∧_ true

+
∧-isIdempotentCommutativeMonoid = record

+
  { isCommutativeMonoid = ∧-isCommutativeMonoid

+
  ; idem = ∧-idem

+
  }

+

+
∧-idempotentCommutativeMonoid : IdempotentCommutativeMonoid 0ℓ 0ℓ

+
∧-idempotentCommutativeMonoid = record

+
  { isIdempotentCommutativeMonoid = ∧-isIdempotentCommutativeMonoid

+
  }

+

+
∨-∧-isSemiring : IsSemiring _∨_ _∧_ false true

+
∨-∧-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = ∨-isCommutativeMonoid

+
    ; *-cong = cong₂ _∧_

+
    ; *-assoc = ∧-assoc

+
    ; *-identity = ∧-identity

+
    ; distrib = ∧-distrib-∨

+
    }

+
  ; zero = ∧-zero

+
  }

+

+
∨-∧-isCommutativeSemiring

+
  : IsCommutativeSemiring _∨_ _∧_ false true

+
∨-∧-isCommutativeSemiring = record

+
  { isSemiring = ∨-∧-isSemiring

+
  ; *-comm = ∧-comm

+
  }

+

+
∨-∧-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ

+
∨-∧-commutativeSemiring = record

+
  { _+_                   = _∨_

+
  ; _*_                   = _∧_

+
  ; 0#                    = false

+
  ; 1#                    = true

+
  ; isCommutativeSemiring = ∨-∧-isCommutativeSemiring

+
  }

+

+
∧-∨-isSemiring : IsSemiring _∧_ _∨_ true false

+
∧-∨-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = ∧-isCommutativeMonoid

+
    ; *-cong = cong₂ _∨_

+
    ; *-assoc = ∨-assoc

+
    ; *-identity = ∨-identity

+
    ; distrib = ∨-distrib-∧

+
    }

+
  ; zero = ∨-zero

+
  }

+

+
∧-∨-isCommutativeSemiring

+
  : IsCommutativeSemiring _∧_ _∨_ true false

+
∧-∨-isCommutativeSemiring = record

+
  { isSemiring = ∧-∨-isSemiring

+
  ; *-comm = ∨-comm

+
  }

+

+
∧-∨-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ

+
∧-∨-commutativeSemiring = record

+
  { _+_                   = _∧_

+
  ; _*_                   = _∨_

+
  ; 0#                    = true

+
  ; 1#                    = false

+
  ; isCommutativeSemiring = ∧-∨-isCommutativeSemiring

+
  }

+

+
∨-∧-isLattice : IsLattice _∨_ _∧_

+
∨-∧-isLattice = record

+
  { isEquivalence = isEquivalence

+
  ; ∨-comm        = ∨-comm

+
  ; ∨-assoc       = ∨-assoc

+
  ; ∨-cong        = cong₂ _∨_

+
  ; ∧-comm        = ∧-comm

+
  ; ∧-assoc       = ∧-assoc

+
  ; ∧-cong        = cong₂ _∧_

+
  ; absorptive    = ∨-∧-absorptive

+
  }

+

+
∨-∧-lattice : Lattice 0ℓ 0ℓ

+
∨-∧-lattice = record

+
  { isLattice = ∨-∧-isLattice

+
  }

+

+
∨-∧-isDistributiveLattice : IsDistributiveLattice _∨_ _∧_

+
∨-∧-isDistributiveLattice = record

+
  { isLattice   = ∨-∧-isLattice

+
  ; ∨-distrib-∧ = ∨-distrib-∧

+
  ; ∧-distrib-∨ = ∧-distrib-∨

+
  }

+

+
∨-∧-distributiveLattice : DistributiveLattice 0ℓ 0ℓ

+
∨-∧-distributiveLattice = record

+
  { isDistributiveLattice = ∨-∧-isDistributiveLattice

+
  }

+

+
∨-∧-isBooleanAlgebra : IsBooleanAlgebra _∨_ _∧_ not true false

+
∨-∧-isBooleanAlgebra = record

+
  { isDistributiveLattice = ∨-∧-isDistributiveLattice

+
  ; ∨-complement          = ∨-inverse

+
  ; ∧-complement          = ∧-inverse

+
  ; ¬-cong                = cong not

+
  }

+

+
∨-∧-booleanAlgebra : BooleanAlgebra 0ℓ 0ℓ

+
∨-∧-booleanAlgebra = record

+
  { isBooleanAlgebra = ∨-∧-isBooleanAlgebra

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of not

+

+
not-involutive : Involutive not

+
not-involutive true  = refl

+
not-involutive false = refl

+

+
not-injective : ∀ {x y} → not x ≡ not y → x ≡ y

+
not-injective {false} {false} nx≢ny = refl

+
not-injective {true}  {true}  nx≢ny = refl

+

+
not-¬ : ∀ {x y} → x ≡ y → x ≢ not y

+
not-¬ {true}  refl ()

+
not-¬ {false} refl ()

+

+
¬-not : ∀ {x y} → x ≢ y → x ≡ not y

+
¬-not {true}  {true}  x≢y = ⊥-elim (x≢y refl)

+
¬-not {true}  {false} _   = refl

+
¬-not {false} {true}  _   = refl

+
¬-not {false} {false} x≢y = ⊥-elim (x≢y refl)

+

+
------------------------------------------------------------------------

+
-- Properties of _xor_

+

+
xor-is-ok : ∀ x y → x xor y ≡ (x ∨ y) ∧ not (x ∧ y)

+
xor-is-ok true  y = refl

+
xor-is-ok false y = sym (∧-identityʳ _)

+

+
true-xor : ∀ x → true xor x ≡ not x

+
true-xor false = refl

+
true-xor true  = refl

+

+
xor-same : ∀ x → x xor x ≡ false

+
xor-same false = refl

+
xor-same true  = refl

+

+
not-distribˡ-xor : ∀ x y → not (x xor y) ≡ (not x) xor y

+
not-distribˡ-xor false y = refl

+
not-distribˡ-xor true  y = not-involutive _

+

+
not-distribʳ-xor : ∀ x y → not (x xor y) ≡ x xor (not y)

+
not-distribʳ-xor false y = refl

+
not-distribʳ-xor true  y = refl

+

+
xor-assoc : Associative _xor_

+
xor-assoc true  y z = sym (not-distribˡ-xor y z)

+
xor-assoc false y z = refl

+

+
xor-comm : Commutative _xor_

+
xor-comm false false = refl

+
xor-comm false true  = refl

+
xor-comm true  false = refl

+
xor-comm true  true  = refl

+

+
xor-identityˡ : LeftIdentity false _xor_

+
xor-identityˡ _ = refl

+

+
xor-identityʳ : RightIdentity false _xor_

+
xor-identityʳ false = refl

+
xor-identityʳ true  = refl

+

+
xor-identity : Identity false _xor_

+
xor-identity = xor-identityˡ , xor-identityʳ

+

+
xor-inverseˡ : LeftInverse true not _xor_

+
xor-inverseˡ false = refl

+
xor-inverseˡ true = refl

+

+
xor-inverseʳ : RightInverse true not _xor_

+
xor-inverseʳ x = xor-comm x (not x) ⟨ trans ⟩ xor-inverseˡ x

+

+
xor-inverse : Inverse true not _xor_

+
xor-inverse = xor-inverseˡ , xor-inverseʳ

+

+
∧-distribˡ-xor : _∧_ DistributesOverˡ _xor_

+
∧-distribˡ-xor false y z = refl

+
∧-distribˡ-xor true  y z = refl

+

+
∧-distribʳ-xor : _∧_ DistributesOverʳ _xor_

+
∧-distribʳ-xor x false z    = refl

+
∧-distribʳ-xor x true false = sym (xor-identityʳ x)

+
∧-distribʳ-xor x true true  = sym (xor-same x)

+

+
∧-distrib-xor : _∧_ DistributesOver _xor_

+
∧-distrib-xor = ∧-distribˡ-xor , ∧-distribʳ-xor

+

+
xor-annihilates-not : ∀ x y → (not x) xor (not y) ≡ x xor y

+
xor-annihilates-not false y = not-involutive _

+
xor-annihilates-not true  y = refl

+

+
xor-∧-commutativeRing : CommutativeRing 0ℓ 0ℓ

+
xor-∧-commutativeRing = ⊕-∧-commutativeRing

+
  where

+
  open BooleanAlgebraProperties ∨-∧-booleanAlgebra

+
  open XorRing _xor_ xor-is-ok

+

+
------------------------------------------------------------------------

+
-- Properties of if_then_else_

+

+
if-float : ∀ (f : A → B) b {x y} →

+
           f (if b then x else y) ≡ (if b then f x else f y)

+
if-float _ true  = refl

+
if-float _ false = refl

+

+
------------------------------------------------------------------------

+
-- Properties of T

+

+
open Relation.Nullary.Decidable.Core public using (T?)

+

+
T-≡ : ∀ {x} → T x ⇔ x ≡ true

+
T-≡ {false} = mk⇔ (λ ())       (λ ())

+
T-≡ {true}  = mk⇔ (const refl) (const _)

+

+
T-not-≡ : ∀ {x} → T (not x) ⇔ x ≡ false

+
T-not-≡ {false} = mk⇔ (const refl) (const _)

+
T-not-≡ {true}  = mk⇔ (λ ())       (λ ())

+

+
T-∧ : ∀ {x y} → T (x ∧ y) ⇔ (T x × T y)

+
T-∧ {true}  {true}  = mk⇔ (const (_ , _)) (const _)

+
T-∧ {true}  {false} = mk⇔ (λ ())          proj₂

+
T-∧ {false} {_}     = mk⇔ (λ ())          proj₁

+

+
T-∨ : ∀ {x y} → T (x ∨ y) ⇔ (T x ⊎ T y)

+
T-∨ {true}  {_}     = mk⇔ inj₁ (const _)

+
T-∨ {false} {true}  = mk⇔ inj₂ (const _)

+
T-∨ {false} {false} = mk⇔ inj₁ [ id , id ]

+

+
T-irrelevant : U.Irrelevant T

+
T-irrelevant {true}  _  _  = refl

+

+
T?-diag : ∀ b → T b → True (T? b)

+
T?-diag b = fromWitness

+

+
------------------------------------------------------------------------

+
-- Miscellaneous other properties

+

+
⇔→≡ : {x y z : Bool} → x ≡ z ⇔ y ≡ z → x ≡ y

+
⇔→≡ {true } {true }         hyp = refl

+
⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp refl)

+
⇔→≡ {true } {false} {false} hyp = Equivalence.from hyp refl

+
⇔→≡ {false} {true } {true } hyp = Equivalence.from hyp refl

+
⇔→≡ {false} {true } {false} hyp = sym (Equivalence.to hyp refl)

+
⇔→≡ {false} {false}         hyp = refl

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
push-function-into-if = if-float

+
{-# WARNING_ON_USAGE push-function-into-if

+
"Warning: push-function-into-if was deprecated in v2.0.

+
Please use if-float instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Bool.Show.html b/site/static/agda_html/Data.Bool.Show.html
new file mode 100644
index 000000000..b61c63df7
--- /dev/null
+++ b/site/static/agda_html/Data.Bool.Show.html
@@ -0,0 +1,98 @@
+
+Data.Bool.Show
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Showing booleans

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Bool.Show where

+

+
open import Data.Bool.Base using (Bool; false; true)

+
open import Data.Char.Base using (Char)

+
open import Data.String.Base using (String)

+

+
show : Bool → String

+
show true  = "true"

+
show false = "false"

+

+
showBit : Bool → Char

+
showBit true = '1'

+
showBit false = '0'

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Bool.html b/site/static/agda_html/Data.Bool.html
new file mode 100644
index 000000000..6dd1dc012
--- /dev/null
+++ b/site/static/agda_html/Data.Bool.html
@@ -0,0 +1,97 @@
+
+Data.Bool
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Booleans

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Bool where

+

+
------------------------------------------------------------------------

+
-- The boolean type and some operations

+

+
open import Data.Bool.Base public

+

+
------------------------------------------------------------------------

+
-- Publicly re-export queries

+

+
open import Data.Bool.Properties public

+
  using (T?; _≟_; _≤?_; _<?_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Char.Base.html b/site/static/agda_html/Data.Char.Base.html
new file mode 100644
index 000000000..3b1430ed8
--- /dev/null
+++ b/site/static/agda_html/Data.Char.Base.html
@@ -0,0 +1,138 @@
+
+Data.Char.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic definitions for Characters

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Char.Base where

+

+
open import Level using (zero)

+
import Data.Nat.Base as ℕ

+
open import Data.Bool.Base using (Bool)

+
open import Function.Base using (_on_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
open import Relation.Binary.Construct.Closure.Reflexive

+

+
------------------------------------------------------------------------

+
-- Re-export the type, and renamed primitives

+

+
open import Agda.Builtin.Char public using ( Char )

+
  renaming

+
  -- testing

+
  ( primIsLower    to isLower

+
  ; primIsDigit    to isDigit

+
  ; primIsAlpha    to isAlpha

+
  ; primIsSpace    to isSpace

+
  ; primIsAscii    to isAscii

+
  ; primIsLatin1   to isLatin1

+
  ; primIsPrint    to isPrint

+
  ; primIsHexDigit to isHexDigit

+
  -- transforming

+
  ; primToUpper to toUpper

+
  ; primToLower to toLower

+
  -- converting

+
  ; primCharToNat to toℕ

+
  ; primNatToChar to fromℕ

+
  )

+

+
open import Agda.Builtin.String public using ()

+
  renaming ( primShowChar to show )

+

+
infix 4 _≈_ _≉_

+
_≈_ : Rel Char zero

+
_≈_ = _≡_ on toℕ

+

+
_≉_ : Rel Char zero

+
_≉_ = _≢_ on toℕ

+

+
infix 4 _≈ᵇ_

+
_≈ᵇ_ : (c d : Char) → Bool

+
c ≈ᵇ d = toℕ c ℕ.≡ᵇ toℕ d

+

+
infix 4 _<_

+
_<_ : Rel Char zero

+
_<_ = ℕ._<_ on toℕ

+

+
infix 4 _≤_

+
_≤_ : Rel Char zero

+
_≤_ = ReflClosure _<_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Char.Properties.html b/site/static/agda_html/Data.Char.Properties.html
new file mode 100644
index 000000000..93555bcaf
--- /dev/null
+++ b/site/static/agda_html/Data.Char.Properties.html
@@ -0,0 +1,382 @@
+
+Data.Char.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of operations on characters

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Char.Properties where

+

+
open import Data.Bool.Base using (Bool)

+
open import Data.Char.Base

+
import Data.Nat.Base as ℕ

+
import Data.Nat.Properties as ℕ

+
open import Data.Product.Base using (_,_)

+

+
open import Function.Base

+
open import Relation.Nullary using (¬_; yes; no)

+
open import Relation.Nullary.Decidable using (map′; isYes)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; StrictPartialOrder; StrictTotalOrder; Preorder; Poset; DecPoset)

+
open import Relation.Binary.Structures

+
  using (IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Decidable; DecidableEquality; Trichotomous; Irreflexive; Transitive; Asymmetric; Antisymmetric; Symmetric; Substitutive; Reflexive; tri<; tri≈; tri>)

+
import Relation.Binary.Construct.On as On

+
import Relation.Binary.Construct.Subst.Equality as Subst

+
import Relation.Binary.Construct.Closure.Reflexive as Refl

+
import Relation.Binary.Construct.Closure.Reflexive.Properties as Refl

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; refl; cong; sym; trans; subst)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
------------------------------------------------------------------------

+
-- Primitive properties

+

+
open import Agda.Builtin.Char.Properties

+
  renaming ( primCharToNatInjective to toℕ-injective)

+
  public

+

+
------------------------------------------------------------------------

+
-- Properties of _≈_

+

+
≈⇒≡ : _≈_ ⇒ _≡_

+
≈⇒≡ = toℕ-injective _ _

+

+
≉⇒≢ : _≉_ ⇒ _≢_

+
≉⇒≢ p refl = p refl

+

+
≈-reflexive : _≡_ ⇒ _≈_

+
≈-reflexive = cong toℕ

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+

+
infix 4 _≟_

+
_≟_ : DecidableEquality Char

+
x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≟ toℕ y)

+

+
setoid : Setoid _ _

+
setoid = ≡.setoid Char

+

+
decSetoid : DecSetoid _ _

+
decSetoid = ≡.decSetoid _≟_

+

+
isDecEquivalence : IsDecEquivalence _≡_

+
isDecEquivalence = ≡.isDecEquivalence _≟_

+

+
------------------------------------------------------------------------

+
-- Boolean equality test.

+
--

+
-- Why is the definition _==_ = primCharEquality not used? One reason

+
-- is that the present definition can sometimes improve type

+
-- inference, at least with the version of Agda that is current at the

+
-- time of writing: see unit-test below.

+

+
infix 4 _==_

+
_==_ : Char → Char → Bool

+
c₁ == c₂ = isYes (c₁ ≟ c₂)

+

+
private

+

+
  -- The following unit test does not type-check (at the time of

+
  -- writing) if _==_ is replaced by primCharEquality.

+

+
  data P : (Char → Bool) → Set where

+
    MkP : (c : Char) → P (c ==_)

+

+
  unit-test : P ('x' ==_)

+
  unit-test = MkP _

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+

+
infix 4 _<?_

+
_<?_ : Decidable _<_

+
_<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_

+

+
<-cmp : Trichotomous _≡_ _<_

+
<-cmp c d with ℕ.<-cmp (toℕ c) (toℕ d)

+
... | tri< lt ¬eq ¬gt = tri< lt (≉⇒≢ ¬eq) ¬gt

+
... | tri≈ ¬lt eq ¬gt = tri≈ ¬lt (≈⇒≡ eq) ¬gt

+
... | tri> ¬lt ¬eq gt = tri> ¬lt (≉⇒≢ ¬eq) gt

+

+
<-irrefl : Irreflexive _≡_ _<_

+
<-irrefl = ℕ.<-irrefl ∘′ cong toℕ

+

+
<-trans : Transitive _<_

+
<-trans {c} {d} {e} = On.transitive toℕ ℕ._<_ ℕ.<-trans {c} {d} {e}

+

+
<-asym : Asymmetric _<_

+
<-asym {c} {d} = On.asymmetric toℕ ℕ._<_ ℕ.<-asym {c} {d}

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = ≡.isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = λ {a} {b} {c} → <-trans {a} {b} {c}

+
  ; <-resp-≈      = (λ {c} → ≡.subst (c <_))

+
                  , (λ {c} → ≡.subst (_< c))

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
<-strictPartialOrder : StrictPartialOrder _ _ _

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder _ _ _

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+

+
infix 4 _≤?_

+
_≤?_ : Decidable _≤_

+
_≤?_ = Refl.decidable <-cmp

+

+
≤-reflexive : _≡_ ⇒ _≤_

+
≤-reflexive = Refl.reflexive

+

+
≤-trans : Transitive _≤_

+
≤-trans = Refl.trans (λ {a} {b} {c} → <-trans {a} {b} {c})

+

+
≤-antisym : Antisymmetric _≡_ _≤_

+
≤-antisym = Refl.antisym _≡_ refl ℕ.<-asym

+

+
≤-isPreorder : IsPreorder _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = ≡.isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isDecPartialOrder : IsDecPartialOrder _≡_ _≤_

+
≤-isDecPartialOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; _≟_            = _≟_

+
  ; _≤?_           = _≤?_

+
  }

+

+
≤-preorder : Preorder _ _ _

+
≤-preorder = record { isPreorder = ≤-isPreorder }

+

+
≤-poset : Poset _ _ _

+
≤-poset = record { isPartialOrder = ≤-isPartialOrder }

+

+
≤-decPoset : DecPoset _ _ _

+
≤-decPoset = record { isDecPartialOrder = ≤-isDecPartialOrder }

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.5

+

+
≈-refl : Reflexive _≈_

+
≈-refl = refl

+
{-# WARNING_ON_USAGE ≈-refl

+
"Warning: ≈-refl was deprecated in v1.5.

+
Please use Propositional Equality's refl instead."

+
#-}

+

+
≈-sym : Symmetric _≈_

+
≈-sym = sym

+
{-# WARNING_ON_USAGE ≈-sym

+
"Warning: ≈-sym was deprecated in v1.5.

+
Please use Propositional Equality's sym instead."

+
#-}

+

+
≈-trans : Transitive _≈_

+
≈-trans = trans

+
{-# WARNING_ON_USAGE ≈-trans

+
"Warning: ≈-trans was deprecated in v1.5.

+
Please use Propositional Equality's trans instead."

+
#-}

+

+
≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ

+
≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p

+
{-# WARNING_ON_USAGE ≈-subst

+
"Warning: ≈-subst was deprecated in v1.5.

+
Please use Propositional Equality's subst instead."

+
#-}

+

+
infix 4 _≈?_

+
_≈?_ : Decidable _≈_

+
x ≈? y = toℕ x ℕ.≟ toℕ y

+

+
≈-isEquivalence : IsEquivalence _≈_

+
≈-isEquivalence = record

+
  { refl  = refl

+
  ; sym   = sym

+
  ; trans = trans

+
  }

+
≈-setoid : Setoid _ _

+
≈-setoid = record

+
  { isEquivalence = ≈-isEquivalence

+
  }

+
≈-isDecEquivalence : IsDecEquivalence _≈_

+
≈-isDecEquivalence = record

+
  { isEquivalence = ≈-isEquivalence

+
  ; _≟_           = _≈?_

+
  }

+
≈-decSetoid : DecSetoid _ _

+
≈-decSetoid = record

+
  { isDecEquivalence = ≈-isDecEquivalence

+
  }

+
{-# WARNING_ON_USAGE _≈?_

+
"Warning: _≈?_ was deprecated in v1.5.

+
Please use _≟_ instead."

+
#-}

+
{-# WARNING_ON_USAGE ≈-isEquivalence

+
"Warning: ≈-isEquivalence was deprecated in v1.5.

+
Please use Propositional Equality's isEquivalence instead."

+
#-}

+
{-# WARNING_ON_USAGE ≈-setoid

+
"Warning: ≈-setoid was deprecated in v1.5.

+
Please use Propositional Equality's setoid instead."

+
#-}

+
{-# WARNING_ON_USAGE ≈-isDecEquivalence

+
"Warning: ≈-isDecEquivalence was deprecated in v1.5.

+
Please use Propositional Equality's isDecEquivalence instead."

+
#-}

+
{-# WARNING_ON_USAGE ≈-decSetoid

+
"Warning: ≈-decSetoid was deprecated in v1.5.

+
Please use Propositional Equality's decSetoid instead."

+
#-}

+

+
≡-setoid : Setoid _ _

+
≡-setoid = setoid

+
{-# WARNING_ON_USAGE ≡-setoid

+
"Warning: ≡-setoid was deprecated in v1.5.

+
Please use setoid instead."

+
#-}

+

+
≡-decSetoid : DecSetoid _ _

+
≡-decSetoid = decSetoid

+
{-# WARNING_ON_USAGE ≡-decSetoid

+
"Warning: ≡-decSetoid was deprecated in v1.5.

+
Please use decSetoid instead."

+
#-}

+

+
<-isStrictPartialOrder-≈ : IsStrictPartialOrder _≈_ _<_

+
<-isStrictPartialOrder-≈ = On.isStrictPartialOrder toℕ ℕ.<-isStrictPartialOrder

+
{-# WARNING_ON_USAGE <-isStrictPartialOrder-≈

+
"Warning: <-isStrictPartialOrder-≈ was deprecated in v1.5.

+
Please use <-isStrictPartialOrder instead."

+
#-}

+

+
<-isStrictTotalOrder-≈ : IsStrictTotalOrder _≈_ _<_

+
<-isStrictTotalOrder-≈ = On.isStrictTotalOrder toℕ ℕ.<-isStrictTotalOrder

+
{-# WARNING_ON_USAGE <-isStrictTotalOrder-≈

+
"Warning: <-isStrictTotalOrder-≈ was deprecated in v1.5.

+
Please use <-isStrictTotalOrder instead."

+
#-}

+

+
<-strictPartialOrder-≈ : StrictPartialOrder _ _ _

+
<-strictPartialOrder-≈ = On.strictPartialOrder ℕ.<-strictPartialOrder toℕ

+
{-# WARNING_ON_USAGE <-strictPartialOrder-≈

+
"Warning: <-strictPartialOrder-≈ was deprecated in v1.5.

+
Please use <-strictPartialOrder instead."

+
#-}

+

+
<-strictTotalOrder-≈ : StrictTotalOrder _ _ _

+
<-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ

+
{-# WARNING_ON_USAGE <-strictTotalOrder-≈

+
"Warning: <-strictTotalOrder-≈ was deprecated in v1.5.

+
Please use <-strictTotalOrder instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Char.html b/site/static/agda_html/Data.Char.html
new file mode 100644
index 000000000..54f926c17
--- /dev/null
+++ b/site/static/agda_html/Data.Char.html
@@ -0,0 +1,93 @@
+
+Data.Char
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Characters

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Char where

+

+
------------------------------------------------------------------------

+
-- Re-export base definitions and decidability of equality

+

+
open import Data.Char.Base public

+
open import Data.Char.Properties

+
  using (_≈?_; _≟_; _<?_; _≤?_; _==_) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Core.html b/site/static/agda_html/Data.Container.Core.html
new file mode 100644
index 000000000..17ec22701
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Core.html
@@ -0,0 +1,148 @@
+
+Data.Container.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Containers core

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Core where

+

+
open import Level using (Level; _⊔_; suc)

+
open import Data.Product.Base as Product using (Σ-syntax)

+
open import Function.Base using (_∘_; _∘′_)

+
open import Function.Bundles using (Inverse; _↔_)

+
open import Relation.Unary using (Pred; _⊆_)

+

+
-- Definition of Containers

+

+
infix 5 _▷_

+
record Container (s p : Level) : Set (suc (s ⊔ p)) where

+
  constructor _▷_

+
  field

+
    Shape    : Set s

+
    Position : Shape → Set p

+
open Container public

+

+
-- The semantics ("extension") of a container.

+

+
⟦_⟧ : ∀ {s p ℓ} → Container s p → Set ℓ → Set (s ⊔ p ⊔ ℓ)

+
⟦ S ▷ P ⟧ X = Σ[ s ∈ S ] (P s → X)

+

+
-- The extension is a functor

+

+
map : ∀ {s p x y} {C : Container s p} {X : Set x} {Y : Set y} →

+
      (X → Y) → ⟦ C ⟧ X → ⟦ C ⟧ Y

+
map f = Product.map₂ (f ∘_)

+

+
-- Representation of container morphisms.

+

+
infixr 8 _⇒_ _⊸_

+

+
record _⇒_ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂)

+
           : Set (s₁ ⊔ s₂ ⊔ p₁ ⊔ p₂) where

+
  constructor _▷_

+
  field

+
    shape    : Shape C₁ → Shape C₂

+
    position : ∀ {s} → Position C₂ (shape s) → Position C₁ s

+

+
  ⟪_⟫ : ∀ {x} {X : Set x} → ⟦ C₁ ⟧ X → ⟦ C₂ ⟧ X

+
  ⟪_⟫ = Product.map shape (_∘′ position)

+

+
open _⇒_ public

+

+
-- Linear container morphisms

+

+
record _⊸_ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂)

+
  : Set (s₁ ⊔ s₂ ⊔ p₁ ⊔ p₂) where

+
  field

+
    shape⊸    : Shape C₁ → Shape C₂

+
    position⊸ : ∀ {s} → Position C₂ (shape⊸ s) ↔ Position C₁ s

+

+
  morphism : C₁ ⇒ C₂

+
  morphism = record

+
    { shape    = shape⊸

+
    ; position = Inverse.to position⊸

+
    }

+

+
  ⟪_⟫⊸ : ∀ {x} {X : Set x} → ⟦ C₁ ⟧ X → ⟦ C₂ ⟧ X

+
  ⟪_⟫⊸ = ⟪ morphism ⟫

+

+
open _⊸_ public using (shape⊸; position⊸; ⟪_⟫⊸)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Membership.html b/site/static/agda_html/Data.Container.Membership.html
new file mode 100644
index 000000000..e4458a873
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Membership.html
@@ -0,0 +1,99 @@
+
+Data.Container.Membership
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Membership for containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Membership where

+

+
open import Level using (_⊔_)

+
open import Relation.Unary using (Pred)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
open import Data.Container.Core using (Container; ⟦_⟧)

+
open import Data.Container.Relation.Unary.Any using (◇)

+

+
module _ {s p} {C : Container s p} {x} {X : Set x} where

+

+
  infix 4 _∈_

+
  _∈_ : X → Pred (⟦ C ⟧ X) (p ⊔ x)

+
  x ∈ xs = ◇ C (_≡_ x) xs

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Morphism.Properties.html b/site/static/agda_html/Data.Container.Morphism.Properties.html
new file mode 100644
index 000000000..c0beb4da2
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Morphism.Properties.html
@@ -0,0 +1,150 @@
+
+Data.Container.Morphism.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Propertiers of any for containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Morphism.Properties where

+

+
open import Level using (_⊔_; suc)

+
open import Function.Base as F using (_$_)

+
open import Data.Product.Base using (∃; proj₁; proj₂; _,_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; _≗_)

+

+
open import Data.Container.Core

+
open import Data.Container.Morphism

+
open import Data.Container.Relation.Binary.Equality.Setoid

+

+
-- Identity

+

+
module _ {s p} (C : Container s p) where

+

+
  id-correct : ∀ {x} {X : Set x} → ⟪ id C ⟫ {X = X} ≗ F.id

+
  id-correct x = ≡.refl

+

+
-- Composition.

+

+
module _ {s₁ s₂ s₃ p₁ p₂ p₃}

+
         {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂} {C₃ : Container s₃ p₃}

+
         where

+

+
  ∘-correct : (f : C₂ ⇒ C₃) (g : C₁ ⇒ C₂) → ∀ {x} {X : Set x} →

+
              ⟪ f ∘ g ⟫ {X = X} ≗ (⟪ f ⟫ F.∘ ⟪ g ⟫)

+
  ∘-correct f g xs = ≡.refl

+

+
module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂} where

+

+
  -- Naturality.

+

+
  Natural : ∀ x e → (∀ {X : Set x} → ⟦ C₁ ⟧ X → ⟦ C₂ ⟧ X) →

+
            Set (s₁ ⊔ s₂ ⊔ p₁ ⊔ p₂ ⊔ suc (x ⊔ e))

+
  Natural x e m =

+
    ∀ {X : Set x} (Y : Setoid x e) → let module Y = Setoid Y in

+
    (f : X → Y.Carrier) (xs : ⟦ C₁ ⟧ X) →

+
    Eq Y C₂ (m $ map f xs) (map f $ m xs)

+

+
  -- Container morphisms are natural.

+

+
  natural : ∀ (m : C₁ ⇒ C₂) x e → Natural x e ⟪ m ⟫

+
  natural m x e Y f xs = refl Y C₂

+

+
module _ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) where

+

+
  -- Natural transformations.

+

+
  NT : ∀ x e → Set (s₁ ⊔ s₂ ⊔ p₁ ⊔ p₂ ⊔ suc (x ⊔ e))

+
  NT x e = ∃ λ (m : ∀ {X : Set x} → ⟦ C₁ ⟧ X → ⟦ C₂ ⟧ X) → Natural x e m

+

+
module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂} where

+

+
  -- All natural functions of the right type are container morphisms.

+

+
  complete : ∀ {e} → (nt : NT C₁ C₂ p₁ e) →

+
      ∃ λ m → (X : Setoid p₁ e) → let module X = Setoid X in

+
      ∀ xs → Eq X C₂ (proj₁ nt xs) (⟪ m ⟫ xs)

+
  complete (nt , nat) =

+
    (m , λ X xs → nat X (proj₂ xs) (proj₁ xs , F.id)) where

+

+
    m : C₁ ⇒ C₂

+
    m .shape    = λ s → proj₁ (nt (s , F.id))

+
    m .position = proj₂ (nt (_ , F.id))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Morphism.html b/site/static/agda_html/Data.Container.Morphism.html
new file mode 100644
index 000000000..423aaf72c
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Morphism.html
@@ -0,0 +1,104 @@
+
+Data.Container.Morphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Container Morphisms

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Morphism where

+

+
open import Data.Container.Core

+
import Function.Base as F

+

+
module _ {s p} (C : Container s p) where

+

+
  id : C ⇒ C

+
  id .shape    = F.id

+
  id .position = F.id

+

+
module _ {s₁ s₂ s₃ p₁ p₂ p₃}

+
         {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂} {C₃ : Container s₃ p₃}

+
         where

+

+
  infixr 9 _∘_

+
  _∘_ : C₂ ⇒ C₃ → C₁ ⇒ C₂ → C₁ ⇒ C₃

+
  (f ∘ g) .shape    = shape    f F.∘′ shape    g

+
  (f ∘ g) .position = position g F.∘′ position f

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Properties.html b/site/static/agda_html/Data.Container.Properties.html
new file mode 100644
index 000000000..fc85f1ce1
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Properties.html
@@ -0,0 +1,102 @@
+
+Data.Container.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of operations on containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Properties where

+

+
import Function.Base as F

+
open import Relation.Binary.Bundles using (Setoid)

+

+
open import Data.Container.Core

+
open import Data.Container.Relation.Binary.Equality.Setoid

+

+
module _ {s p} {C : Container s p} where

+

+
  map-identity : ∀ {x e} (X : Setoid x e) xs →

+
                 Eq X C (map F.id xs) xs

+
  map-identity X xs = refl X C

+

+
  map-compose : ∀ {x y z e} {X : Set x} {Y : Set y} (Z : Setoid z e) g (f : X → Y) xs →

+
                Eq Z C (map g (map f xs)) (map (g F.∘′ f) xs)

+
  map-compose Z g f xs = refl Z C

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Related.html b/site/static/agda_html/Data.Container.Related.html
new file mode 100644
index 000000000..355155305
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Related.html
@@ -0,0 +1,132 @@
+
+Data.Container.Related
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Several kinds of "relatedness" for containers such as equivalences,

+
-- surjections and bijections

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Related where

+

+
open import Level using (_⊔_)

+
open import Data.Container.Core

+
import Function.Related.Propositional as Related

+
open import Relation.Binary.Bundles using (Preorder; Setoid)

+
open import Data.Container.Membership

+

+
open Related public

+
  using (Kind; SymmetricKind)

+
  renaming ( implication         to subset

+
           ; reverseImplication  to superset

+
           ; equivalence         to set

+
           ; injection           to subbag

+
           ; reverseInjection    to superbag

+
           ; bijection           to bag

+
           )

+

+
[_]-Order : ∀ {s p ℓ} → Kind → Container s p → Set ℓ →

+
            Preorder (s ⊔ p ⊔ ℓ) (s ⊔ p ⊔ ℓ) (p ⊔ ℓ)

+
[ k ]-Order C X = Related.InducedPreorder₂ k (_∈_ {C = C} {X = X})

+

+
[_]-Equality : ∀ {s p ℓ} → SymmetricKind → Container s p → Set ℓ →

+
               Setoid (s ⊔ p ⊔ ℓ) (p ⊔ ℓ)

+
[ k ]-Equality C X = Related.InducedEquivalence₂ k (_∈_ {C = C} {X = X})

+

+
infix 4 _≲[_]_

+
_≲[_]_ : ∀ {s p x} {C : Container s p} {X : Set x} →

+
         ⟦ C ⟧ X → Kind → ⟦ C ⟧ X → Set (p ⊔ x)

+
_≲[_]_ {C = C} {X} xs k ys = Preorder._≲_ ([ k ]-Order C X) xs ys

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
infix 4 _∼[_]_

+
_∼[_]_ = _≲[_]_

+
{-# WARNING_ON_USAGE _∼[_]_

+
"Warning: _∼[_]_ was deprecated in v2.0. Please use _≲[_]_ instead. "

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Relation.Binary.Equality.Setoid.html b/site/static/agda_html/Data.Container.Relation.Binary.Equality.Setoid.html
new file mode 100644
index 000000000..f81e24af0
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Relation.Binary.Equality.Setoid.html
@@ -0,0 +1,134 @@
+
+Data.Container.Relation.Binary.Equality.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Equality over container extensions parametrised by some setoid

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.Container.Relation.Binary.Equality.Setoid {c e} (S : Setoid c e) where

+

+
open import Level using (_⊔_; suc)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive)

+

+
open import Data.Container.Core

+
open import Data.Container.Relation.Binary.Pointwise

+
import Data.Container.Relation.Binary.Pointwise.Properties as Pw

+

+
private

+
  module S = Setoid S

+
open S using (_≈_) renaming (Carrier to X)

+

+
------------------------------------------------------------------------

+
-- Definition of equality

+

+
module _ {s p} (C : Container s p) where

+

+
  Eq : Rel (⟦ C ⟧ X) (e ⊔ s ⊔ p)

+
  Eq = Pointwise C _≈_

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
  refl : Reflexive Eq

+
  refl = Pw.refl C _ S.refl

+

+
  sym : Symmetric Eq

+
  sym = Pw.sym C _ S.sym

+

+
  trans : Transitive Eq

+
  trans = Pw.trans C _ S.trans

+

+
  isEquivalence : IsEquivalence Eq

+
  isEquivalence = record

+
    { refl  = refl

+
    ; sym   = sym

+
    ; trans = trans

+
    }

+

+
  setoid : Setoid (s ⊔ p ⊔ c) (s ⊔ p ⊔ e)

+
  setoid = record

+
    { isEquivalence = isEquivalence

+
    }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Relation.Binary.Pointwise.Properties.html b/site/static/agda_html/Data.Container.Relation.Binary.Pointwise.Properties.html
new file mode 100644
index 000000000..a54ad6bbb
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Relation.Binary.Pointwise.Properties.html
@@ -0,0 +1,119 @@
+
+Data.Container.Relation.Binary.Pointwise.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of pointwise equality for containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Relation.Binary.Pointwise.Properties where

+

+
open import Axiom.Extensionality.Propositional using (Extensionality)

+
open import Data.Container.Core using (Container; ⟦_⟧)

+
open import Data.Container.Relation.Binary.Pointwise

+
  using (Pointwise; _,_)

+
open import Data.Product.Base using (_,_; Σ-syntax; -,_)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; subst; cong)

+

+
module _ {s p x r} {X : Set x} (C : Container s p) (R : Rel X r) where

+

+
  refl : Reflexive R → Reflexive (Pointwise C R)

+
  refl R-refl = ≡.refl , λ p → R-refl

+

+
  sym : Symmetric R → Symmetric (Pointwise C R)

+
  sym R-sym (≡.refl , f) = ≡.refl , λ p → R-sym (f p)

+

+
  trans : Transitive R → Transitive (Pointwise C R)

+
  trans R-trans (≡.refl , f) (≡.refl , g) = ≡.refl , λ p → R-trans (f p) (g p)

+

+
private

+

+
  -- Note that, if propositional equality were extensional, then

+
  -- Eq _≡_ and _≡_ would coincide.

+

+
  Eq⇒≡ : ∀ {s p x} {C : Container s p} {X : Set x} {xs ys : ⟦ C ⟧ X} →

+
         Extensionality p x → Pointwise C _≡_ xs ys → xs ≡ ys

+
  Eq⇒≡ ext (≡.refl , f≈f′) = cong -,_ (ext f≈f′)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Relation.Binary.Pointwise.html b/site/static/agda_html/Data.Container.Relation.Binary.Pointwise.html
new file mode 100644
index 000000000..08e6b3f1a
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Relation.Binary.Pointwise.html
@@ -0,0 +1,112 @@
+
+Data.Container.Relation.Binary.Pointwise
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise equality for containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Relation.Binary.Pointwise where

+

+
open import Data.Product.Base using (_,_; Σ-syntax; -,_; proj₁; proj₂)

+
open import Function.Base using (_∘_)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (REL; _⇒_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; subst)

+

+
open import Data.Container.Core using (Container; ⟦_⟧)

+

+
-- Equality, parametrised on an underlying relation.

+

+
module _ {s p} (C : Container s p) where

+

+
  record Pointwise {x y e} {X : Set x} {Y : Set y} (R : REL X Y e)

+
                   (cx : ⟦ C ⟧ X) (cy : ⟦ C ⟧ Y) : Set (s ⊔ p ⊔ e) where

+
    constructor _,_

+
    field shape    : proj₁ cx ≡ proj₁ cy

+
          position : ∀ p → R (proj₂ cx p) (proj₂ cy (subst _ shape p))

+
  infixr 4 _,_

+

+
module _ {s p} {C : Container s p} {x y} {X : Set x} {Y : Set y}

+
         {ℓ ℓ′} {R : REL X Y ℓ} {R′ : REL X Y ℓ′}

+
         where

+

+
  map : R ⇒ R′ → Pointwise C R ⇒ Pointwise C R′

+
  map R⇒R′ (s , f) = s , R⇒R′ ∘ f

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Relation.Unary.All.html b/site/static/agda_html/Data.Container.Relation.Unary.All.html
new file mode 100644
index 000000000..6eb3aeca3
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Relation.Unary.All.html
@@ -0,0 +1,120 @@
+
+Data.Container.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- All (□) for containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Relation.Unary.All where

+

+
open import Level using (_⊔_)

+
open import Relation.Unary using (Pred; _⊆_)

+
open import Data.Product.Base using (_,_; proj₁; proj₂; ∃)

+
open import Function.Base using (_∘′_; id)

+

+
open import Data.Container.Core hiding (map)

+
import Data.Container.Morphism as M

+

+
record □ {s p} (C : Container s p) {x ℓ} {X : Set x}

+
         (P : Pred X ℓ) (cx : ⟦ C ⟧ X) : Set (p ⊔ ℓ) where

+
  constructor all

+
  field proof : ∀ p → P (proj₂ cx p)

+

+
module _ {s₁ p₁ s₂ p₂} {C : Container s₁ p₁} {D : Container s₂ p₂}

+
         {x ℓ ℓ′} {X : Set x} {P : Pred X ℓ} {Q : Pred X ℓ′}

+
         where

+

+
  map : (f : C ⇒ D) → P ⊆ Q → □ C P ⊆ □ D Q ∘′ ⟪ f ⟫

+
  map f P⊆Q (all prf) .□.proof p = P⊆Q (prf (f .position p))

+

+
module _ {s₁ p₁ s₂ p₂} {C : Container s₁ p₁} {D : Container s₂ p₂}

+
         {x ℓ} {X : Set x} {P : Pred X ℓ}

+
         where

+

+
  map₁ : (f : C ⇒ D) → □ C P ⊆ □ D P ∘′ ⟪ f ⟫

+
  map₁ f = map f id

+

+
module _ {s p} {C : Container s p}

+
         {x ℓ ℓ′} {X : Set x} {P : Pred X ℓ} {Q : Pred X ℓ′}

+
         where

+

+
  map₂ : P ⊆ Q → □ C P ⊆ □ C Q

+
  map₂ = map (M.id C)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.Relation.Unary.Any.html b/site/static/agda_html/Data.Container.Relation.Unary.Any.html
new file mode 100644
index 000000000..59b7f2c5d
--- /dev/null
+++ b/site/static/agda_html/Data.Container.Relation.Unary.Any.html
@@ -0,0 +1,122 @@
+
+Data.Container.Relation.Unary.Any
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Any (◇) for containers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Container.Relation.Unary.Any where

+

+
open import Level using (_⊔_)

+
open import Relation.Unary using (Pred; _⊆_)

+
open import Data.Product.Base using (_,_; proj₂; ∃)

+
open import Function.Base using (_∘′_; id)

+

+
open import Data.Container.Core hiding (map)

+
import Data.Container.Morphism as M

+

+
record ◇ {s p} (C : Container s p) {x ℓ} {X : Set x}

+
         (P : Pred X ℓ) (cx : ⟦ C ⟧ X) : Set (p ⊔ ℓ) where

+
  constructor any

+
  field proof : ∃ λ p → P (proj₂ cx p)

+

+
module _ {s₁ p₁ s₂ p₂} {C : Container s₁ p₁} {D : Container s₂ p₂}

+
         {x ℓ ℓ′} {X : Set x} {P : Pred X ℓ} {Q : Pred X ℓ′}

+
         where

+

+
  map : (f : C ⇒ D) → P ⊆ Q → ◇ D P ∘′ ⟪ f ⟫ ⊆ ◇ C Q

+
  map f P⊆Q (any (p , P)) .◇.proof = f .position p , P⊆Q P

+

+

+
module _ {s₁ p₁ s₂ p₂} {C : Container s₁ p₁} {D : Container s₂ p₂}

+
         {x ℓ} {X : Set x} {P : Pred X ℓ}

+
         where

+

+
  map₁ : (f : C ⇒ D) → ◇ D P ∘′ ⟪ f ⟫ ⊆ ◇ C P

+
  map₁ f = map f id

+

+

+
module _ {s p} {C : Container s p}

+
         {x ℓ ℓ′} {X : Set x} {P : Pred X ℓ} {Q : Pred X ℓ′}

+
         where

+

+
  map₂ : P ⊆ Q → ◇ C P ⊆ ◇ C Q

+
  map₂ = map (M.id C)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Container.html b/site/static/agda_html/Data.Container.html
new file mode 100644
index 000000000..4a53f2494
--- /dev/null
+++ b/site/static/agda_html/Data.Container.html
@@ -0,0 +1,133 @@
+
+Data.Container
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Containers, based on the work of Abbott and others

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Level using (Level; _⊔_)

+
open import Data.W using (W)

+

+
module Data.Container where

+

+
------------------------------------------------------------------------

+
-- Re-exporting content to maintain backwards compatibility

+

+
open import Data.Container.Core public

+
open import Data.Container.Relation.Unary.Any

+
  using (◇) renaming (map to ◇-map) public

+
open import Data.Container.Relation.Unary.All

+
  using (□) renaming (map to □-map) public

+
open import Data.Container.Membership

+
  using (_∈_) public

+
open import Data.Container.Relation.Binary.Pointwise

+
  using () renaming (Pointwise to Eq) public

+
open import Data.Container.Relation.Binary.Pointwise.Properties

+
  using (refl; sym; trans) public

+
open import Data.Container.Relation.Binary.Equality.Setoid

+
  using (isEquivalence; setoid) public

+
open import Data.Container.Properties

+
  using () renaming (map-identity to identity; map-compose to composition) public

+
open import Data.Container.Related public

+

+
module Morphism where

+

+
  open import Data.Container.Morphism.Properties

+
    using (Natural; NT; natural; complete; id-correct; ∘-correct) public

+

+
  open import Data.Container.Morphism

+
    using (id; _∘_) public

+

+
private

+
  variable

+
    s p : Level

+

+
-- The least fixpoint of a container.

+

+
μ : Container s p → Set (s ⊔ p)

+
μ = W

+

+
-- The greatest fixpoint of a container can be found

+
-- in `Data.Container.Fixpoints.Guarded` as it relies

+
-- on the `guardedness` flag.

+

+
-- You can find sized alternatives in `Data.Container.Fixpoints.Sized`

+
-- as they rely on the unsafe flag `--sized-types`.

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Digit.html b/site/static/agda_html/Data.Digit.html
new file mode 100644
index 000000000..0a1d3727c
--- /dev/null
+++ b/site/static/agda_html/Data.Digit.html
@@ -0,0 +1,217 @@
+
+Data.Digit
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Digits and digit expansions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Digit where

+

+
open import Data.Nat.Base

+
  using (ℕ; zero; suc; _<_; _/_; _%_; sz<ss; _+_; _*_; 2+; _≤′_)

+
open import Data.Nat.Properties

+
  using (_≤?_; _<?_; ≤⇒≤′; module ≤-Reasoning; m≤m+n; +-comm; +-assoc;

+
    *-distribˡ-+; *-identityʳ)

+
open import Data.Fin.Base as Fin using (Fin; zero; suc; toℕ)

+
open import Data.Bool.Base using (Bool; true; false)

+
open import Data.Char.Base using (Char)

+
open import Data.List.Base using (List; replicate; [_]; _∷_; [])

+
open import Data.Product.Base using (∃; _,_)

+
open import Data.Vec.Base as Vec using (Vec; _∷_; [])

+
open import Data.Nat.DivMod using (m/n<m; _divMod_; result)

+
open import Data.Nat.Induction

+
  using (Acc; acc; <-wellFounded-fast; <′-Rec; <′-rec)

+
open import Function.Base using (_$_)

+
open import Relation.Nullary.Decidable using (True; does; toWitness)

+
open import Relation.Binary.Definitions using (Decidable)

+
open import Relation.Binary.PropositionalEquality

+
  using (_≡_; refl; sym; cong; cong₂; module ≡-Reasoning)

+

+
------------------------------------------------------------------------

+
-- Digits

+

+
-- Digit b is the type of digits in base b.

+

+
Digit : ℕ → Set

+
Digit b = Fin b

+

+
-- Some specific digit kinds.

+

+
Decimal = Digit 10

+
Bit     = Digit 2

+

+
-- Some named digits.

+

+
0b : Bit

+
0b = zero

+

+
1b : Bit

+
1b = suc zero

+

+
------------------------------------------------------------------------

+
-- Converting between `ℕ` and `expansions of ℕ`

+

+
toNatDigits : (base : ℕ) {base≤16 : True (1 ≤? base)} → ℕ → List ℕ

+
toNatDigits base@(suc zero)    n = replicate n 1

+
toNatDigits base@(suc (suc _)) n = aux (<-wellFounded-fast n) []

+
  where

+
  aux : {n : ℕ} → Acc _<_ n → List ℕ → List ℕ

+
  aux {zero}        _      xs =  (0 ∷ xs)

+
  aux {n@(suc _)} (acc wf) xs with does (0 <? n / base)

+
  ... | false = (n % base) ∷ xs -- Could this more simply be n ∷ xs here?

+
  ... | true  = aux (wf (m/n<m n base sz<ss)) ((n % base) ∷ xs)

+

+
------------------------------------------------------------------------

+
-- Converting between `ℕ` and expansions of `Digit base`

+

+
Expansion : ℕ → Set

+
Expansion base = List (Digit base)

+

+
-- fromDigits takes a digit expansion of a natural number, starting

+
-- with the _least_ significant digit, and returns the corresponding

+
-- natural number.

+

+
fromDigits : ∀ {base} → Expansion base → ℕ

+
fromDigits        []       = 0

+
fromDigits {base} (d ∷ ds) = toℕ d + fromDigits ds * base

+

+
-- toDigits b n yields the digits of n, in base b, starting with the

+
-- _least_ significant digit.

+
--

+
-- Note that the list of digits is always non-empty.

+

+
toDigits : (base : ℕ) {base≥2 : True (2 ≤? base)} (n : ℕ) →

+
           ∃ λ (ds : Expansion base) → fromDigits ds ≡ n

+
toDigits base@(suc (suc k)) n = <′-rec Pred helper n

+
  where

+

+
  Pred = λ n → ∃ λ ds → fromDigits ds ≡ n

+

+
  cons : ∀ {m} (r : Digit base) → Pred m → Pred (toℕ r + m * base)

+
  cons r (ds , eq) = (r ∷ ds , cong (λ i → toℕ r + i * base) eq)

+

+
  lem′ : ∀ x k → x + x + (k + x * k) ≡ k + x * 2+ k

+
  lem′ x k = begin

+
    x + x + (k + x * k)   ≡⟨ +-assoc (x + x) k _ ⟨

+
    x + x + k + x * k     ≡⟨ cong (_+ x * k) (+-comm _ k) ⟩

+
    k + (x + x) + x * k   ≡⟨ +-assoc k (x + x) _ ⟩

+
    k + ((x + x) + x * k) ≡⟨ cong (k +_) (begin

+
      (x + x) + x * k       ≡⟨ +-assoc x x _ ⟩

+
      x + (x + x * k)       ≡⟨ cong (x +_) (cong (_+ x * k) (*-identityʳ x)) ⟨

+
      x + (x * 1 + x * k)   ≡⟨ cong₂ _+_ (*-identityʳ x) (*-distribˡ-+ x 1 k ) ⟨

+
      x * 1 + (x * suc k)   ≡⟨ *-distribˡ-+ x 1 (1 + k) ⟨

+
      x * 2+ k              ∎) ⟩

+
    k + x * 2+ k          ∎

+
    where open ≡-Reasoning

+

+
  lem : ∀ x k r → 2 + x ≤′ r + (1 + x) * (2 + k)

+
  lem x k r = ≤⇒≤′ $ begin

+
    2 + x                             ≤⟨ m≤m+n _ _ ⟩

+
    2 + x + (x + (1 + x) * k + r)     ≡⟨ cong ((2 + x) +_) (+-comm _ r) ⟩

+
    2 + x + (r + (x + (1 + x) * k))   ≡⟨ +-assoc (2 + x) r _ ⟨

+
    2 + x + r + (x + (1 + x) * k)     ≡⟨ cong (_+ (x + (1 + x) * k)) (+-comm (2 + x) r) ⟩

+
    r + (2 + x) + (x + (1 + x) * k)   ≡⟨ +-assoc r (2 + x) _ ⟩

+
    r + ((2 + x) + (x + (1 + x) * k)) ≡⟨ cong (r +_) (cong 2+ (+-assoc x x _)) ⟨

+
    r + (2 + (x + x + (1 + x) * k))   ≡⟨ cong (λ z → r + (2+ z)) (lem′ x k) ⟩

+
    r + (2 + (k + x * (2 + k)))       ≡⟨⟩

+
    r + (1 + x) * (2 + k)             ∎

+
    where open ≤-Reasoning

+

+
  helper : ∀ n → <′-Rec Pred n → Pred n

+
  helper n                       rec with n divMod base

+
  ... | result zero    r eq = ([ r ] , sym eq)

+
  ... | result (suc x) r refl = cons r (rec (lem x k (toℕ r)))

+

+
------------------------------------------------------------------------

+
-- Showing digits

+

+
-- The characters used to show the first 16 digits.

+

+
digitChars : Vec Char 16

+
digitChars =

+
  '0' ∷ '1' ∷ '2' ∷ '3' ∷ '4' ∷ '5' ∷ '6' ∷ '7' ∷ '8' ∷ '9' ∷

+
  'a' ∷ 'b' ∷ 'c' ∷ 'd' ∷ 'e' ∷ 'f' ∷ []

+

+
-- showDigit shows digits in base ≤ 16.

+

+
showDigit : ∀ {base} {base≤16 : True (base ≤? 16)} → Digit base → Char

+
showDigit {base≤16 = base≤16} d =

+
  Vec.lookup digitChars (Fin.inject≤ d (toWitness base≤16))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Empty.Polymorphic.html b/site/static/agda_html/Data.Empty.Polymorphic.html
new file mode 100644
index 000000000..127d0a196
--- /dev/null
+++ b/site/static/agda_html/Data.Empty.Polymorphic.html
@@ -0,0 +1,96 @@
+
+Data.Empty.Polymorphic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Level polymorphic Empty type

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Empty.Polymorphic where

+

+
import Data.Empty as Empty

+
open import Level

+

+
⊥ : {ℓ : Level} → Set ℓ

+
⊥ {ℓ} = Lift ℓ Empty.⊥

+

+
-- make ⊥-elim dependent too, as it does seem useful

+
⊥-elim : ∀ {w ℓ} {Whatever : ⊥ {ℓ} → Set w} → (witness : ⊥ {ℓ}) → Whatever witness

+
⊥-elim ()

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Empty.html b/site/static/agda_html/Data.Empty.html
new file mode 100644
index 000000000..db5da7a63
--- /dev/null
+++ b/site/static/agda_html/Data.Empty.html
@@ -0,0 +1,117 @@
+
+Data.Empty
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Empty type, judgementally proof irrelevant, Level-monomorphic

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Empty where

+

+
open import Data.Irrelevant using (Irrelevant)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Note that by default the empty type is not universe polymorphic as it

+
-- often results in unsolved metas. See `Data.Empty.Polymorphic` for a

+
-- universe polymorphic variant.

+

+
private

+
  data Empty : Set where

+

+
-- ⊥ is defined via Data.Irrelevant (a record with a single irrelevant

+
-- field) so that Agda can judgementally declare that all proofs of ⊥

+
-- are equal to each other. In particular this means that all functions

+
-- returning a proof of ⊥ are equal.

+

+
⊥ : Set

+
⊥ = Irrelevant Empty

+

+
{-# DISPLAY Irrelevant Empty = ⊥ #-}

+

+
------------------------------------------------------------------------

+
-- Functions

+

+
⊥-elim : ∀ {w} {Whatever : Set w} → ⊥ → Whatever

+
⊥-elim ()

+

+
⊥-elim-irr : ∀ {w} {Whatever : Set w} → .⊥ → Whatever

+
⊥-elim-irr ()

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Fin.Base.html b/site/static/agda_html/Data.Fin.Base.html
new file mode 100644
index 000000000..4d8b53afc
--- /dev/null
+++ b/site/static/agda_html/Data.Fin.Base.html
@@ -0,0 +1,420 @@
+
+Data.Fin.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Finite sets

+
------------------------------------------------------------------------

+

+
-- Note that elements of Fin n can be seen as natural numbers in the

+
-- set {m | m < n}. The notation "m" in comments below refers to this

+
-- natural number view.

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Fin.Base where

+

+
open import Data.Bool.Base using (Bool; T)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)

+
open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Function.Base using (id; _∘_; _on_; flip; _$_)

+
open import Level using (0ℓ)

+
open import Relation.Binary.Core

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; refl; cong)

+
open import Relation.Binary.Indexed.Heterogeneous.Core using (IRel)

+
open import Relation.Nullary.Negation.Core using (contradiction)

+

+
private

+
  variable

+
    m n : ℕ

+

+
------------------------------------------------------------------------

+
-- Types

+

+
-- Fin n is a type with n elements.

+

+
data Fin : ℕ → Set where

+
  zero : Fin (suc n)

+
  suc  : (i : Fin n) → Fin (suc n)

+

+
-- A conversion: toℕ "i" = i.

+

+
toℕ : Fin n → ℕ

+
toℕ zero    = 0

+
toℕ (suc i) = suc (toℕ i)

+

+
-- A Fin-indexed variant of Fin.

+

+
Fin′ : Fin n → Set

+
Fin′ i = Fin (toℕ i)

+

+
------------------------------------------------------------------------

+
-- A cast that actually computes on constructors (as opposed to subst)

+

+
cast : .(m ≡ n) → Fin m → Fin n

+
cast {zero}  {zero}  eq k       = k

+
cast {suc m} {suc n} eq zero    = zero

+
cast {suc m} {suc n} eq (suc k) = suc (cast (cong ℕ.pred eq) k)

+

+
------------------------------------------------------------------------

+
-- Conversions

+

+
-- toℕ is defined above.

+

+
-- fromℕ n = "n".

+

+
fromℕ : (n : ℕ) → Fin (suc n)

+
fromℕ zero    = zero

+
fromℕ (suc n) = suc (fromℕ n)

+

+
-- fromℕ< {m} _ = "m".

+

+
fromℕ< : .(m ℕ.< n) → Fin n

+
fromℕ< {zero}  {suc _} _   = zero

+
fromℕ< {suc m} {suc _} m<n = suc (fromℕ< (ℕ.s<s⁻¹ m<n))

+

+
-- fromℕ<″ m _ = "m".

+

+
fromℕ<″ : ∀ m {n} → .(m ℕ.<″ n) → Fin n

+
fromℕ<″ zero    {suc _} _    = zero

+
fromℕ<″ (suc m) {suc _} m<″n = suc (fromℕ<″ m (ℕ.s<″s⁻¹ m<″n))

+

+
-- canonical liftings of i:Fin m to larger index

+

+
-- injection on the left: "i" ↑ˡ n = "i" in Fin (m + n)

+
infixl 5 _↑ˡ_

+
_↑ˡ_ : ∀ {m} → Fin m → ∀ n → Fin (m ℕ.+ n)

+
zero    ↑ˡ n = zero

+
(suc i) ↑ˡ n = suc (i ↑ˡ n)

+

+
-- injection on the right: n ↑ʳ "i" = "n + i" in Fin (n + m)

+
infixr 5 _↑ʳ_

+
_↑ʳ_ : ∀ {m} n → Fin m → Fin (n ℕ.+ m)

+
zero    ↑ʳ i = i

+
(suc n) ↑ʳ i = suc (n ↑ʳ i)

+

+
-- reduce≥ "m + i" _ = "i".

+

+
reduce≥ : ∀ (i : Fin (m ℕ.+ n)) → .(m ℕ.≤ toℕ i) → Fin n

+
reduce≥ {zero}  i       _   = i

+
reduce≥ {suc _} (suc i) m≤i = reduce≥ i (ℕ.s≤s⁻¹ m≤i)

+

+
-- inject⋆ m "i" = "i".

+

+
inject : ∀ {i : Fin n} → Fin′ i → Fin n

+
inject {i = suc i} zero    = zero

+
inject {i = suc i} (suc j) = suc (inject j)

+

+
inject! : ∀ {i : Fin (suc n)} → Fin′ i → Fin n

+
inject! {n = suc _} {i = suc _} zero    = zero

+
inject! {n = suc _} {i = suc _} (suc j) = suc (inject! j)

+

+
inject₁ : Fin n → Fin (suc n)

+
inject₁ zero    = zero

+
inject₁ (suc i) = suc (inject₁ i)

+

+
inject≤ : Fin m → .(m ℕ.≤ n) → Fin n

+
inject≤ {n = suc _} zero    _   = zero

+
inject≤ {n = suc _} (suc i) m≤n = suc (inject≤ i (ℕ.s≤s⁻¹ m≤n))

+

+
-- lower₁ "i" _ = "i".

+

+
lower₁ : ∀ (i : Fin (suc n)) → n ≢ toℕ i → Fin n

+
lower₁ {zero}  zero    ne = contradiction refl ne

+
lower₁ {suc n} zero    _  = zero

+
lower₁ {suc n} (suc i) ne = suc (lower₁ i (ne ∘ cong suc))

+

+
-- A strengthening injection into the minimal Fin fibre.

+
strengthen : ∀ (i : Fin n) → Fin′ (suc i)

+
strengthen zero    = zero

+
strengthen (suc i) = suc (strengthen i)

+

+
-- splitAt m "i" = inj₁ "i"      if i < m

+
--                 inj₂ "i - m"  if i ≥ m

+
-- This is dual to splitAt from Data.Vec.

+

+
splitAt : ∀ m {n} → Fin (m ℕ.+ n) → Fin m ⊎ Fin n

+
splitAt zero    i       = inj₂ i

+
splitAt (suc m) zero    = inj₁ zero

+
splitAt (suc m) (suc i) = Sum.map₁ suc (splitAt m i)

+

+
-- inverse of above function

+
join : ∀ m n → Fin m ⊎ Fin n → Fin (m ℕ.+ n)

+
join m n = [ _↑ˡ n , m ↑ʳ_ ]′

+

+
-- quotRem k "i" = "i % k" , "i / k"

+
-- This is dual to group from Data.Vec.

+

+
quotRem : ∀ n → Fin (m ℕ.* n) → Fin n × Fin m

+
quotRem {suc m} n i =

+
  [ (_, zero)

+
  , Product.map₂ suc ∘ quotRem {m} n

+
  ]′ $ splitAt n i

+

+
-- a variant of quotRem the type of whose result matches the order of multiplication

+
remQuot : ∀ n → Fin (m ℕ.* n) → Fin m × Fin n

+
remQuot i = Product.swap ∘ quotRem i

+

+
quotient : ∀ n → Fin (m ℕ.* n) → Fin m

+
quotient n = proj₁ ∘ remQuot n

+

+
remainder : ∀ n → Fin (m ℕ.* n) → Fin n

+
remainder {m} n = proj₂ ∘ remQuot {m} n

+

+
-- inverse of remQuot

+
combine : Fin m → Fin n → Fin (m ℕ.* n)

+
combine {suc m} {n} zero    j = j ↑ˡ (m ℕ.* n)

+
combine {suc m} {n} (suc i) j = n ↑ʳ (combine i j)

+

+
-- Next in progression after splitAt and remQuot

+
finToFun : Fin (m ℕ.^ n) → (Fin n → Fin m)

+
finToFun {m} {suc n} i zero    = quotient (m ℕ.^ n) i

+
finToFun {m} {suc n} i (suc j) = finToFun (remainder {m} (m ℕ.^ n) i) j

+

+
-- inverse of above function

+
funToFin : (Fin m → Fin n) → Fin (n ℕ.^ m)

+
funToFin {zero}  f = zero

+
funToFin {suc m} f = combine (f zero) (funToFin (f ∘ suc))

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
-- Folds.

+

+
fold : ∀ {t} (T : ℕ → Set t) {m} →

+
       (∀ {n} → T n → T (suc n)) →

+
       (∀ {n} → T (suc n)) →

+
       Fin m → T m

+
fold T f x zero    = x

+
fold T f x (suc i) = f (fold T f x i)

+

+
fold′ : ∀ {n t} (T : Fin (suc n) → Set t) →

+
        (∀ i → T (inject₁ i) → T (suc i)) →

+
        T zero →

+
        ∀ i → T i

+
fold′             T f x zero     = x

+
fold′ {n = suc n} T f x (suc i)  =

+
  f i (fold′ (T ∘ inject₁) (f ∘ inject₁) x i)

+

+
-- Lifts functions.

+

+
lift : ∀ k → (Fin m → Fin n) → Fin (k ℕ.+ m) → Fin (k ℕ.+ n)

+
lift zero    f i       = f i

+
lift (suc k) f zero    = zero

+
lift (suc k) f (suc i) = suc (lift k f i)

+

+
-- "i" + "j" = "i + j".

+

+
infixl 6 _+_

+

+
_+_ : ∀ (i : Fin m) (j : Fin n) → Fin (toℕ i ℕ.+ n)

+
zero  + j = j

+
suc i + j = suc (i + j)

+

+
-- "i" - "j" = "i ∸ j".

+

+
infixl 6 _-_

+

+
_-_ : ∀ (i : Fin n) (j : Fin′ (suc i)) → Fin (n ℕ.∸ toℕ j)

+
i     - zero   = i

+
suc i - suc j  = i - j

+

+
-- m ℕ- "i" = "m ∸ i".

+

+
infixl 6 _ℕ-_

+

+
_ℕ-_ : (n : ℕ) (j : Fin (suc n)) → Fin (suc n ℕ.∸ toℕ j)

+
n     ℕ- zero   = fromℕ n

+
suc n ℕ- suc i  = n ℕ- i

+

+
-- m ℕ-ℕ "i" = m ∸ i.

+

+
infixl 6 _ℕ-ℕ_

+

+
_ℕ-ℕ_ : (n : ℕ) → Fin (suc n) → ℕ

+
n     ℕ-ℕ zero   = n

+
suc n ℕ-ℕ suc i  = n ℕ-ℕ i

+

+
-- pred "i" = "pred i".

+

+
pred : Fin n → Fin n

+
pred zero    = zero

+
pred (suc i) = inject₁ i

+

+
-- opposite "i" = "n - i" (i.e. the additive inverse).

+

+
opposite : Fin n → Fin n

+
opposite {suc n} zero    = fromℕ n

+
opposite {suc n} (suc i) = inject₁ (opposite i)

+

+
-- The function f(i,j) = if j>i then j-1 else j

+
-- This is a variant of the thick function from Conor

+
-- McBride's "First-order unification by structural recursion".

+

+
punchOut : ∀ {i j : Fin (suc n)} → i ≢ j → Fin n

+
punchOut {_}     {zero}   {zero}  i≢j = contradiction refl i≢j

+
punchOut {_}     {zero}   {suc j} _   = j

+
punchOut {suc _} {suc i}  {zero}  _   = zero

+
punchOut {suc _} {suc i}  {suc j} i≢j = suc (punchOut (i≢j ∘ cong suc))

+

+
-- The function f(i,j) = if j≥i then j+1 else j

+

+
punchIn : Fin (suc n) → Fin n → Fin (suc n)

+
punchIn zero    j       = suc j

+
punchIn (suc i) zero    = zero

+
punchIn (suc i) (suc j) = suc (punchIn i j)

+

+
-- The function f(i,j) such that f(i,j) = if j≤i then j else j-1

+

+
pinch : Fin n → Fin (suc n) → Fin n

+
pinch {suc n} _       zero    = zero

+
pinch {suc n} zero    (suc j) = j

+
pinch {suc n} (suc i) (suc j) = suc (pinch i j)

+

+
------------------------------------------------------------------------

+
-- Order relations

+

+
infix 4 _≤_ _≥_ _<_ _>_

+

+
_≤_ : IRel Fin 0ℓ

+
i ≤ j = toℕ i ℕ.≤ toℕ j

+

+
_≥_ : IRel Fin 0ℓ

+
i ≥ j = toℕ i ℕ.≥ toℕ j

+

+
_<_ : IRel Fin 0ℓ

+
i < j = toℕ i ℕ.< toℕ j

+

+
_>_ : IRel Fin 0ℓ

+
i > j = toℕ i ℕ.> toℕ j

+

+

+
------------------------------------------------------------------------

+
-- An ordering view.

+

+
data Ordering {n : ℕ} : Fin n → Fin n → Set where

+
  less    : ∀ greatest (least : Fin′ greatest) →

+
            Ordering (inject least) greatest

+
  equal   : ∀ i → Ordering i i

+
  greater : ∀ greatest (least : Fin′ greatest) →

+
            Ordering greatest (inject least)

+

+
compare : ∀ (i j : Fin n) → Ordering i j

+
compare zero    zero    = equal   zero

+
compare zero    (suc j) = less    (suc j) zero

+
compare (suc i) zero    = greater (suc i) zero

+
compare (suc i) (suc j) with compare i j

+
... | less    greatest least = less    (suc greatest) (suc least)

+
... | greater greatest least = greater (suc greatest) (suc least)

+
... | equal   i              = equal   (suc i)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
raise = _↑ʳ_

+
{-# WARNING_ON_USAGE raise

+
"Warning: raise was deprecated in v2.0.

+
Please use _↑ʳ_ instead."

+
#-}

+
inject+ : ∀ {m} n → Fin m → Fin (m ℕ.+ n)

+
inject+ n i = i ↑ˡ n

+
{-# WARNING_ON_USAGE inject+

+
"Warning: inject+ was deprecated in v2.0.

+
Please use _↑ˡ_ instead.

+
NB argument order has been flipped:

+
the left-hand argument is the Fin m

+
the right-hand is the Nat index increment."

+
#-}

+

+
data _≺_ : ℕ → ℕ → Set where

+
  _≻toℕ_ : ∀ n (i : Fin n) → toℕ i ≺ n

+

+
{-# WARNING_ON_USAGE _≺_

+
"Warning: _≺_ was deprecated in v2.0.

+
Please use equivalent relation _<_ instead."

+
#-}

+
{-# WARNING_ON_USAGE _≻toℕ_

+
"Warning: _≻toℕ_ was deprecated in v2.0.

+
Please use toℕ<n from Data.Fin.Properties instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Fin.Patterns.html b/site/static/agda_html/Data.Fin.Patterns.html
new file mode 100644
index 000000000..653cbc362
--- /dev/null
+++ b/site/static/agda_html/Data.Fin.Patterns.html
@@ -0,0 +1,102 @@
+
+Data.Fin.Patterns
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Patterns for Fin

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Fin.Patterns where

+

+
open import Data.Fin.Base

+

+
------------------------------------------------------------------------

+
-- Constants

+

+
pattern 0F = zero

+
pattern 1F = suc 0F

+
pattern 2F = suc 1F

+
pattern 3F = suc 2F

+
pattern 4F = suc 3F

+
pattern 5F = suc 4F

+
pattern 6F = suc 5F

+
pattern 7F = suc 6F

+
pattern 8F = suc 7F

+
pattern 9F = suc 8F

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Fin.Properties.html b/site/static/agda_html/Data.Fin.Properties.html
new file mode 100644
index 000000000..66ae720b0
--- /dev/null
+++ b/site/static/agda_html/Data.Fin.Properties.html
@@ -0,0 +1,1266 @@
+
+Data.Fin.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to Fin, and operations making use of these

+
-- properties (or other properties not available in Data.Fin)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+
{-# OPTIONS --warn=noUserWarning #-} -- for deprecated _≺_ and _≻toℕ_ (issue #1726)

+

+
module Data.Fin.Properties where

+

+
open import Axiom.Extensionality.Propositional

+
open import Algebra.Definitions using (Involutive)

+
open import Effect.Applicative using (RawApplicative)

+
open import Effect.Functor using (RawFunctor)

+
open import Data.Bool.Base using (Bool; true; false; not; _∧_; _∨_)

+
open import Data.Empty using (⊥; ⊥-elim)

+
open import Data.Fin.Base

+
open import Data.Fin.Patterns

+
open import Data.Nat.Base as ℕ

+
  using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; s<s⁻¹; _∸_; _^_)

+
import Data.Nat.Properties as ℕ

+
open import Data.Unit.Base using (⊤; tt)

+
open import Data.Product.Base as Product

+
  using (∃; ∃₂; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)

+
open import Data.Product.Properties using (,-injective)

+
open import Data.Product.Algebra using (×-cong)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)

+
open import Data.Sum.Properties using ([,]-map; [,]-∘)

+
open import Function.Base using (_∘_; id; _$_; flip)

+
open import Function.Bundles using (Injection; _↣_; _⇔_; _↔_; mk⇔; mk↔ₛ′)

+
open import Function.Definitions using (Injective; Surjective)

+
open import Function.Consequences.Propositional using (contraInjective)

+
open import Function.Construct.Composition as Comp hiding (injective)

+
open import Level using (Level)

+
open import Relation.Binary.Definitions as B hiding (Decidable)

+
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)

+
open import Relation.Binary.Bundles

+
  using (Preorder; Setoid; DecSetoid; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_)

+
open import Relation.Binary.PropositionalEquality.Properties as ≡

+
  using (module ≡-Reasoning)

+
open import Relation.Nullary.Decidable as Dec

+
  using (Dec; _because_; yes; no; _×-dec_; _⊎-dec_; map′)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (Reflects; invert)

+
open import Relation.Unary as U

+
  using (U; Pred; Decidable; _⊆_; Satisfiable; Universal)

+
open import Relation.Unary.Properties using (U?)

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+
    m n o : ℕ

+
    i j : Fin n

+

+
------------------------------------------------------------------------

+
-- Fin

+
------------------------------------------------------------------------

+

+
¬Fin0 : ¬ Fin 0

+
¬Fin0 ()

+

+
nonZeroIndex : Fin n → ℕ.NonZero n

+
nonZeroIndex {n = suc _} _ = _

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
0↔⊥ : Fin 0 ↔ ⊥

+
0↔⊥ = mk↔ₛ′ ¬Fin0 (λ ()) (λ ()) (λ ())

+

+
1↔⊤ : Fin 1 ↔ ⊤

+
1↔⊤ = mk↔ₛ′ (λ { 0F → tt }) (λ { tt → 0F }) (λ { tt → refl }) λ { 0F → refl }

+

+
2↔Bool : Fin 2 ↔ Bool

+
2↔Bool = mk↔ₛ′ (λ { 0F → false; 1F → true }) (λ { false → 0F ; true → 1F })

+
  (λ { false → refl ; true → refl }) (λ { 0F → refl ; 1F → refl })

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+
------------------------------------------------------------------------

+

+
0≢1+n : zero ≢ Fin.suc i

+
0≢1+n ()

+

+
suc-injective : Fin.suc i ≡ suc j → i ≡ j

+
suc-injective refl = refl

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality (Fin n)

+
zero  ≟ zero  = yes refl

+
zero  ≟ suc y = no λ()

+
suc x ≟ zero  = no λ()

+
suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≡-isDecEquivalence : IsDecEquivalence {A = Fin n} _≡_

+
≡-isDecEquivalence = record

+
  { isEquivalence = ≡.isEquivalence

+
  ; _≟_           = _≟_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≡-preorder : ℕ → Preorder _ _ _

+
≡-preorder n = ≡.preorder (Fin n)

+

+
≡-setoid : ℕ → Setoid _ _

+
≡-setoid n = ≡.setoid (Fin n)

+

+
≡-decSetoid : ℕ → DecSetoid _ _

+
≡-decSetoid n = record

+
  { isDecEquivalence = ≡-isDecEquivalence {n}

+
  }

+

+
------------------------------------------------------------------------

+
-- toℕ

+
------------------------------------------------------------------------

+

+
toℕ-injective : toℕ i ≡ toℕ j → i ≡ j

+
toℕ-injective {zero}  {}      {}      _

+
toℕ-injective {suc n} {zero}  {zero}  eq = refl

+
toℕ-injective {suc n} {suc i} {suc j} eq =

+
  cong suc (toℕ-injective (cong ℕ.pred eq))

+

+
toℕ-strengthen : ∀ (i : Fin n) → toℕ (strengthen i) ≡ toℕ i

+
toℕ-strengthen zero    = refl

+
toℕ-strengthen (suc i) = cong suc (toℕ-strengthen i)

+

+
------------------------------------------------------------------------

+
-- toℕ-↑ˡ: "i" ↑ˡ n = "i" in Fin (m + n)

+
------------------------------------------------------------------------

+

+
toℕ-↑ˡ : ∀ (i : Fin m) n → toℕ (i ↑ˡ n) ≡ toℕ i

+
toℕ-↑ˡ zero    n = refl

+
toℕ-↑ˡ (suc i) n = cong suc (toℕ-↑ˡ i n)

+

+
↑ˡ-injective : ∀ n (i j : Fin m) → i ↑ˡ n ≡ j ↑ˡ n → i ≡ j

+
↑ˡ-injective n zero zero refl = refl

+
↑ˡ-injective n (suc i) (suc j) eq =

+
  cong suc (↑ˡ-injective n i j (suc-injective eq))

+

+
------------------------------------------------------------------------

+
-- toℕ-↑ʳ: n ↑ʳ "i" = "n + i" in Fin (n + m)

+
------------------------------------------------------------------------

+

+
toℕ-↑ʳ : ∀ n (i : Fin m) → toℕ (n ↑ʳ i) ≡ n ℕ.+ toℕ i

+
toℕ-↑ʳ zero    i = refl

+
toℕ-↑ʳ (suc n) i = cong suc (toℕ-↑ʳ n i)

+

+
↑ʳ-injective : ∀ n (i j : Fin m) → n ↑ʳ i ≡ n ↑ʳ j → i ≡ j

+
↑ʳ-injective zero i i refl = refl

+
↑ʳ-injective (suc n) i j eq = ↑ʳ-injective n i j (suc-injective eq)

+

+
------------------------------------------------------------------------

+
-- toℕ and the ordering relations

+
------------------------------------------------------------------------

+

+
toℕ<n : ∀ (i : Fin n) → toℕ i ℕ.< n

+
toℕ<n {n = suc _} zero    = z<s

+
toℕ<n {n = suc _} (suc i) = s<s (toℕ<n i)

+

+
toℕ≤pred[n] : ∀ (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n

+
toℕ≤pred[n] zero                 = z≤n

+
toℕ≤pred[n] (suc {n = suc n} i)  = s≤s (toℕ≤pred[n] i)

+

+
toℕ≤n : ∀ (i : Fin n) → toℕ i ℕ.≤ n

+
toℕ≤n {suc n} i = ℕ.m≤n⇒m≤1+n (toℕ≤pred[n] i)

+

+
-- A simpler implementation of toℕ≤pred[n],

+
-- however, with a different reduction behavior.

+
-- If no one needs the reduction behavior of toℕ≤pred[n],

+
-- it can be removed in favor of toℕ≤pred[n]′.

+
toℕ≤pred[n]′ : ∀ (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n

+
toℕ≤pred[n]′ i = ℕ.<⇒≤pred (toℕ<n i)

+

+
toℕ-mono-< : i < j → toℕ i ℕ.< toℕ j

+
toℕ-mono-< i<j = i<j

+

+
toℕ-mono-≤ : i ≤ j → toℕ i ℕ.≤ toℕ j

+
toℕ-mono-≤ i≤j = i≤j

+

+
toℕ-cancel-≤ : toℕ i ℕ.≤ toℕ j → i ≤ j

+
toℕ-cancel-≤ i≤j = i≤j

+

+
toℕ-cancel-< : toℕ i ℕ.< toℕ j → i < j

+
toℕ-cancel-< i<j = i<j

+

+
------------------------------------------------------------------------

+
-- fromℕ

+
------------------------------------------------------------------------

+

+
toℕ-fromℕ : ∀ n → toℕ (fromℕ n) ≡ n

+
toℕ-fromℕ zero    = refl

+
toℕ-fromℕ (suc n) = cong suc (toℕ-fromℕ n)

+

+
fromℕ-toℕ : ∀ (i : Fin n) → fromℕ (toℕ i) ≡ strengthen i

+
fromℕ-toℕ zero    = refl

+
fromℕ-toℕ (suc i) = cong suc (fromℕ-toℕ i)

+

+
≤fromℕ : ∀ (i : Fin (suc n)) → i ≤ fromℕ n

+
≤fromℕ {n = n} i rewrite toℕ-fromℕ n = ℕ.s≤s⁻¹ (toℕ<n i)

+

+
------------------------------------------------------------------------

+
-- fromℕ<

+
------------------------------------------------------------------------

+

+
fromℕ<-toℕ : ∀ (i : Fin n) .(i<n : toℕ i ℕ.< n) → fromℕ< i<n ≡ i

+
fromℕ<-toℕ zero    _   = refl

+
fromℕ<-toℕ (suc i) i<n = cong suc (fromℕ<-toℕ i (ℕ.s<s⁻¹ i<n))

+

+
toℕ-fromℕ< : ∀ .(m<n : m ℕ.< n) → toℕ (fromℕ< m<n) ≡ m

+
toℕ-fromℕ< {m = zero}  {n = suc _} _   = refl

+
toℕ-fromℕ< {m = suc m} {n = suc _} m<n = cong suc (toℕ-fromℕ< (ℕ.s<s⁻¹ m<n))

+

+
-- fromℕ is a special case of fromℕ<.

+
fromℕ-def : ∀ n → fromℕ n ≡ fromℕ< ℕ.≤-refl

+
fromℕ-def zero    = refl

+
fromℕ-def (suc n) = cong suc (fromℕ-def n)

+

+
fromℕ<-cong : ∀ m n {o} → m ≡ n → .(m<o : m ℕ.< o) .(n<o : n ℕ.< o) →

+
              fromℕ< m<o ≡ fromℕ< n<o

+
fromℕ<-cong 0       0                   _ _   _   = refl

+
fromℕ<-cong (suc _) (suc _) {o = suc _} r m<n n<o

+
  = cong suc (fromℕ<-cong _ _ (ℕ.suc-injective r) (ℕ.s<s⁻¹ m<n) (ℕ.s<s⁻¹ n<o))

+

+
fromℕ<-injective : ∀ m n {o} → .(m<o : m ℕ.< o) .(n<o : n ℕ.< o) →

+
                   fromℕ< m<o ≡ fromℕ< n<o → m ≡ n

+
fromℕ<-injective 0       0                   _   _   _  = refl

+
fromℕ<-injective 0       (suc _) {o = suc _} _ _ ()

+
fromℕ<-injective (suc _) (suc _) {o = suc _} m<n n<o r

+
  = cong suc (fromℕ<-injective _ _ (ℕ.s<s⁻¹ m<n) (ℕ.s<s⁻¹ n<o) (suc-injective r))

+

+
------------------------------------------------------------------------

+
-- fromℕ<″

+
------------------------------------------------------------------------

+

+
fromℕ<≡fromℕ<″ : ∀ (m<n : m ℕ.< n) (m<″n : m ℕ.<″ n) →

+
                 fromℕ< m<n ≡ fromℕ<″ m m<″n

+
fromℕ<≡fromℕ<″ {m = zero}  {n = suc _} _ _ = refl

+
fromℕ<≡fromℕ<″ {m = suc m} {n = suc _} m<n m<″n

+
  = cong suc (fromℕ<≡fromℕ<″ (ℕ.s<s⁻¹ m<n) (ℕ.s<″s⁻¹ m<″n))

+

+
toℕ-fromℕ<″ : ∀ (m<n : m ℕ.<″ n) → toℕ (fromℕ<″ m m<n) ≡ m

+
toℕ-fromℕ<″ {m} {n} m<n = begin

+
  toℕ (fromℕ<″ m m<n)  ≡⟨ cong toℕ (sym (fromℕ<≡fromℕ<″ (ℕ.≤″⇒≤ m<n) m<n)) ⟩

+
  toℕ (fromℕ< _)       ≡⟨ toℕ-fromℕ< (ℕ.≤″⇒≤ m<n) ⟩

+
  m                    ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of cast

+
------------------------------------------------------------------------

+

+
toℕ-cast : ∀ .(eq : m ≡ n) (k : Fin m) → toℕ (cast eq k) ≡ toℕ k

+
toℕ-cast {n = suc n} eq zero    = refl

+
toℕ-cast {n = suc n} eq (suc k) = cong suc (toℕ-cast (cong ℕ.pred eq) k)

+

+
cast-is-id : .(eq : m ≡ m) (k : Fin m) → cast eq k ≡ k

+
cast-is-id eq zero    = refl

+
cast-is-id eq (suc k) = cong suc (cast-is-id (ℕ.suc-injective eq) k)

+

+
subst-is-cast : (eq : m ≡ n) (k : Fin m) → subst Fin eq k ≡ cast eq k

+
subst-is-cast refl k = sym (cast-is-id refl k)

+

+
cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (k : Fin m) →

+
             cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k

+
cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ zero = refl

+
cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (suc k) =

+
  cong suc (cast-trans (ℕ.suc-injective eq₁) (ℕ.suc-injective eq₂) k)

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
≤-reflexive : _≡_ ⇒ (_≤_ {n})

+
≤-reflexive refl = ℕ.≤-refl

+

+
≤-refl : Reflexive (_≤_ {n})

+
≤-refl = ≤-reflexive refl

+

+
≤-trans : Transitive (_≤_ {n})

+
≤-trans = ℕ.≤-trans

+

+
≤-antisym : Antisymmetric _≡_ (_≤_ {n})

+
≤-antisym x≤y y≤x = toℕ-injective (ℕ.≤-antisym x≤y y≤x)

+

+
≤-total : Total (_≤_ {n})

+
≤-total x y = ℕ.≤-total (toℕ x) (toℕ y)

+

+
≤-irrelevant : Irrelevant (_≤_ {m} {n})

+
≤-irrelevant = ℕ.≤-irrelevant

+

+
infix 4 _≤?_ _<?_

+

+
_≤?_ : B.Decidable (_≤_ {m} {n})

+
a ≤? b = toℕ a ℕ.≤? toℕ b

+

+
_<?_ : B.Decidable (_<_ {m} {n})

+
m <? n = suc (toℕ m) ℕ.≤? toℕ n

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≤-isPreorder : IsPreorder {A = Fin n} _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = ≡.isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isPartialOrder : IsPartialOrder {A = Fin n} _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder {A = Fin n} _≡_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder {A = Fin n} _≡_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≤-preorder : ℕ → Preorder _ _ _

+
≤-preorder n = record

+
  { isPreorder = ≤-isPreorder {n}

+
  }

+

+
≤-poset : ℕ → Poset _ _ _

+
≤-poset n = record

+
  { isPartialOrder = ≤-isPartialOrder {n}

+
  }

+

+
≤-totalOrder : ℕ → TotalOrder _ _ _

+
≤-totalOrder n = record

+
  { isTotalOrder = ≤-isTotalOrder {n}

+
  }

+

+
≤-decTotalOrder : ℕ → DecTotalOrder _ _ _

+
≤-decTotalOrder n = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder {n}

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
<-irrefl : Irreflexive _≡_ (_<_ {n})

+
<-irrefl refl = ℕ.<-irrefl refl

+

+
<-asym : Asymmetric (_<_ {n})

+
<-asym = ℕ.<-asym

+

+
<-trans : Transitive (_<_ {n})

+
<-trans = ℕ.<-trans

+

+
<-cmp : Trichotomous _≡_ (_<_ {n})

+
<-cmp zero    zero    = tri≈ (λ()) refl  (λ())

+
<-cmp zero    (suc j) = tri< z<s   (λ()) (λ())

+
<-cmp (suc i) zero    = tri> (λ()) (λ()) z<s

+
<-cmp (suc i) (suc j) with <-cmp i j

+
... | tri< i<j i≢j j≮i = tri< (s<s i<j)     (i≢j ∘ suc-injective) (j≮i ∘ s<s⁻¹)

+
... | tri> i≮j i≢j j<i = tri> (i≮j ∘ s<s⁻¹) (i≢j ∘ suc-injective) (s<s j<i)

+
... | tri≈ i≮j i≡j j≮i = tri≈ (i≮j ∘ s<s⁻¹) (cong suc i≡j)        (j≮i ∘ s<s⁻¹)

+

+
<-respˡ-≡ : (_<_ {m} {n}) Respectsˡ _≡_

+
<-respˡ-≡ refl x≤y = x≤y

+

+
<-respʳ-≡ : (_<_ {m} {n}) Respectsʳ _≡_

+
<-respʳ-≡ refl x≤y = x≤y

+

+
<-resp₂-≡ : (_<_ {n}) Respects₂ _≡_

+
<-resp₂-≡ = <-respʳ-≡ , <-respˡ-≡

+

+
<-irrelevant : Irrelevant (_<_ {m} {n})

+
<-irrelevant = ℕ.<-irrelevant

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
<-isStrictPartialOrder : IsStrictPartialOrder {A = Fin n} _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = ≡.isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = <-resp₂-≡

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder {A = Fin n} _≡_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
<-strictPartialOrder : ℕ → StrictPartialOrder _ _ _

+
<-strictPartialOrder n = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder {n}

+
  }

+

+
<-strictTotalOrder : ℕ → StrictTotalOrder _ _ _

+
<-strictTotalOrder n = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder {n}

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
i<1+i : ∀ (i : Fin n) → i < suc i

+
i<1+i = ℕ.n<1+n ∘ toℕ

+

+
<⇒≢ : i < j → i ≢ j

+
<⇒≢ i<i refl = ℕ.n≮n _ i<i

+

+
≤∧≢⇒< : i ≤ j → i ≢ j → i < j

+
≤∧≢⇒< {i = zero}  {zero}  _         0≢0   = contradiction refl 0≢0

+
≤∧≢⇒< {i = zero}  {suc j} _         _     = z<s

+
≤∧≢⇒< {i = suc i} {suc j} 1+i≤1+j 1+i≢1+j =

+
  s<s (≤∧≢⇒< (ℕ.s≤s⁻¹ 1+i≤1+j) (1+i≢1+j ∘ (cong suc)))

+

+
------------------------------------------------------------------------

+
-- inject

+
------------------------------------------------------------------------

+

+
toℕ-inject : ∀ {i : Fin n} (j : Fin′ i) → toℕ (inject j) ≡ toℕ j

+
toℕ-inject {i = suc i} zero    = refl

+
toℕ-inject {i = suc i} (suc j) = cong suc (toℕ-inject j)

+

+
------------------------------------------------------------------------

+
-- inject₁

+
------------------------------------------------------------------------

+

+
fromℕ≢inject₁ : fromℕ n ≢ inject₁ i

+
fromℕ≢inject₁ {i = suc i} eq = fromℕ≢inject₁ {i = i} (suc-injective eq)

+

+
inject₁-injective : inject₁ i ≡ inject₁ j → i ≡ j

+
inject₁-injective {i = zero}  {zero}  i≡j = refl

+
inject₁-injective {i = suc i} {suc j} i≡j =

+
  cong suc (inject₁-injective (suc-injective i≡j))

+

+
toℕ-inject₁ : ∀ (i : Fin n) → toℕ (inject₁ i) ≡ toℕ i

+
toℕ-inject₁ zero    = refl

+
toℕ-inject₁ (suc i) = cong suc (toℕ-inject₁ i)

+

+
toℕ-inject₁-≢ : ∀ (i : Fin n) → n ≢ toℕ (inject₁ i)

+
toℕ-inject₁-≢ (suc i) = toℕ-inject₁-≢ i ∘ ℕ.suc-injective

+

+
inject₁ℕ< : ∀ (i : Fin n) → toℕ (inject₁ i) ℕ.< n

+
inject₁ℕ< i rewrite toℕ-inject₁ i = toℕ<n i

+

+
inject₁ℕ≤ : ∀ (i : Fin n) → toℕ (inject₁ i) ℕ.≤ n

+
inject₁ℕ≤ = ℕ.<⇒≤ ∘ inject₁ℕ<

+

+
≤̄⇒inject₁< : i ≤ j → inject₁ i < suc j

+
≤̄⇒inject₁< {i = i} i≤j rewrite sym (toℕ-inject₁ i) = s<s i≤j

+

+
ℕ<⇒inject₁< : ∀ {i : Fin (ℕ.suc n)} {j : Fin n} → j < i → inject₁ j < i

+
ℕ<⇒inject₁< {i = suc i} j≤i = ≤̄⇒inject₁< (ℕ.s≤s⁻¹ j≤i)

+

+
i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j

+
i≤inject₁[j]⇒i≤1+j {i = zero}              _   = z≤n

+
i≤inject₁[j]⇒i≤1+j {i = suc i} {j = suc j} i≤j = s≤s (ℕ.m≤n⇒m≤1+n (subst (toℕ i ℕ.≤_) (toℕ-inject₁ j) (ℕ.s≤s⁻¹ i≤j)))

+

+
------------------------------------------------------------------------

+
-- lower₁

+
------------------------------------------------------------------------

+

+
toℕ-lower₁ : ∀ i (p : n ≢ toℕ i) → toℕ (lower₁ i p) ≡ toℕ i

+
toℕ-lower₁ {ℕ.zero}  zero    p = contradiction refl p

+
toℕ-lower₁ {ℕ.suc m} zero    p = refl

+
toℕ-lower₁ {ℕ.suc m} (suc i) p = cong ℕ.suc (toℕ-lower₁ i (p ∘ cong ℕ.suc))

+

+
lower₁-injective : ∀ {n≢i : n ≢ toℕ i} {n≢j : n ≢ toℕ j} →

+
                   lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j

+
lower₁-injective {zero}  {zero}  {_}     {n≢i} {_}   _    = contradiction refl n≢i

+
lower₁-injective {zero}  {_}     {zero}  {_}   {n≢j} _    = contradiction refl n≢j

+
lower₁-injective {suc n} {zero}  {zero}  {_}   {_}   refl = refl

+
lower₁-injective {suc n} {suc i} {suc j} {n≢i} {n≢j} eq   =

+
  cong suc (lower₁-injective (suc-injective eq))

+

+
------------------------------------------------------------------------

+
-- inject₁ and lower₁

+

+
inject₁-lower₁ : ∀ (i : Fin (suc n)) (n≢i : n ≢ toℕ i) →

+
                 inject₁ (lower₁ i n≢i) ≡ i

+
inject₁-lower₁ {zero}  zero     0≢0     = contradiction refl 0≢0

+
inject₁-lower₁ {suc n} zero     _       = refl

+
inject₁-lower₁ {suc n} (suc i)  n+1≢i+1 =

+
  cong suc (inject₁-lower₁ i  (n+1≢i+1 ∘ cong suc))

+

+
lower₁-inject₁′ : ∀ (i : Fin n) (n≢i : n ≢ toℕ (inject₁ i)) →

+
                  lower₁ (inject₁ i) n≢i ≡ i

+
lower₁-inject₁′ zero    _       = refl

+
lower₁-inject₁′ (suc i) n+1≢i+1 =

+
  cong suc (lower₁-inject₁′ i (n+1≢i+1 ∘ cong suc))

+

+
lower₁-inject₁ : ∀ (i : Fin n) →

+
                 lower₁ (inject₁ i) (toℕ-inject₁-≢ i) ≡ i

+
lower₁-inject₁ i = lower₁-inject₁′ i (toℕ-inject₁-≢ i)

+

+
lower₁-irrelevant : ∀ (i : Fin (suc n)) (n≢i₁ n≢i₂ : n ≢ toℕ i) →

+
                    lower₁ i n≢i₁ ≡ lower₁ i n≢i₂

+
lower₁-irrelevant {zero}  zero     0≢0 _ = contradiction refl 0≢0

+
lower₁-irrelevant {suc n} zero     _   _ = refl

+
lower₁-irrelevant {suc n} (suc i)  _   _ =

+
  cong suc (lower₁-irrelevant i _ _)

+

+
inject₁≡⇒lower₁≡ : ∀ {i : Fin n} {j : Fin (ℕ.suc n)} →

+
                  (n≢j : n ≢ toℕ j) → inject₁ i ≡ j → lower₁ j n≢j ≡ i

+
inject₁≡⇒lower₁≡ n≢j i≡j = inject₁-injective (trans (inject₁-lower₁ _ n≢j) (sym i≡j))

+

+
------------------------------------------------------------------------

+
-- inject≤

+
------------------------------------------------------------------------

+

+
toℕ-inject≤ : ∀ i .(m≤n : m ℕ.≤ n) → toℕ (inject≤ i m≤n) ≡ toℕ i

+
toℕ-inject≤ {_} {suc n} zero    _ = refl

+
toℕ-inject≤ {_} {suc n} (suc i) _ = cong suc (toℕ-inject≤ i _)

+

+
inject≤-refl : ∀ i .(n≤n : n ℕ.≤ n) → inject≤ i n≤n ≡ i

+
inject≤-refl {suc n} zero    _   = refl

+
inject≤-refl {suc n} (suc i) _ = cong suc (inject≤-refl i _)

+

+
inject≤-idempotent : ∀ (i : Fin m)

+
                     .(m≤n : m ℕ.≤ n) .(n≤o : n ℕ.≤ o) .(m≤o : m ℕ.≤ o) →

+
                     inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i m≤o

+
inject≤-idempotent {_} {suc n} {suc o} zero    _ _ _ = refl

+
inject≤-idempotent {_} {suc n} {suc o} (suc i) _ _ _ =

+
  cong suc (inject≤-idempotent i _ _ _)

+

+
inject≤-trans : ∀ (i : Fin m) .(m≤n : m ℕ.≤ n) .(n≤o : n ℕ.≤ o) →

+
                inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (ℕ.≤-trans m≤n n≤o)

+
inject≤-trans i _ _ = inject≤-idempotent i _ _ _

+

+
inject≤-injective : ∀ .(m≤n m≤n′ : m ℕ.≤ n) i j →

+
                    inject≤ i m≤n ≡ inject≤ j m≤n′ → i ≡ j

+
inject≤-injective {n = suc _} _ _ zero    zero    eq = refl

+
inject≤-injective {n = suc _} _ _ (suc i) (suc j) eq =

+
  cong suc (inject≤-injective _ _ i j (suc-injective eq))

+

+
inject≤-irrelevant : ∀ .(m≤n m≤n′ : m ℕ.≤ n) i →

+
                    inject≤ i m≤n ≡ inject≤ i m≤n′

+
inject≤-irrelevant _ _ i = refl

+

+
------------------------------------------------------------------------

+
-- pred

+
------------------------------------------------------------------------

+

+
pred< : ∀ (i : Fin (suc n)) → i ≢ zero → pred i < i

+
pred< zero    i≢0 = contradiction refl i≢0

+
pred< (suc i) _   = ≤̄⇒inject₁< ℕ.≤-refl

+

+
------------------------------------------------------------------------

+
-- splitAt

+
------------------------------------------------------------------------

+

+
-- Fin (m + n) ↔ Fin m ⊎ Fin n

+

+
splitAt-↑ˡ : ∀ m i n → splitAt m (i ↑ˡ n) ≡ inj₁ i

+
splitAt-↑ˡ (suc m) zero    n = refl

+
splitAt-↑ˡ (suc m) (suc i) n rewrite splitAt-↑ˡ m i n = refl

+

+
splitAt⁻¹-↑ˡ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i

+
splitAt⁻¹-↑ˡ {suc m} {n} {0F} {.0F} refl = refl

+
splitAt⁻¹-↑ˡ {suc m} {n} {suc i} {j} eq

+
  with inj₁ k ← splitAt m i in splitAt[m][i]≡inj₁[j]

+
  with refl ← eq

+
  = cong suc (splitAt⁻¹-↑ˡ {i = i} {j = k} splitAt[m][i]≡inj₁[j])

+

+
splitAt-↑ʳ : ∀ m n i → splitAt m (m ↑ʳ i) ≡ inj₂ {B = Fin n} i

+
splitAt-↑ʳ zero    n i = refl

+
splitAt-↑ʳ (suc m) n i rewrite splitAt-↑ʳ m n i = refl

+

+
splitAt⁻¹-↑ʳ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i

+
splitAt⁻¹-↑ʳ {zero}  {n} {i} {j} refl = refl

+
splitAt⁻¹-↑ʳ {suc m} {n} {suc i} {j} eq

+
  with inj₂ k ← splitAt m i in splitAt[m][i]≡inj₂[k]

+
  with refl ← eq

+
  = cong suc (splitAt⁻¹-↑ʳ {i = i} {j = k} splitAt[m][i]≡inj₂[k])

+

+
splitAt-join : ∀ m n i → splitAt m (join m n i) ≡ i

+
splitAt-join m n (inj₁ x) = splitAt-↑ˡ m x n

+
splitAt-join m n (inj₂ y) = splitAt-↑ʳ m n y

+

+
join-splitAt : ∀ m n i → join m n (splitAt m i) ≡ i

+
join-splitAt zero    n i       = refl

+
join-splitAt (suc m) n zero    = refl

+
join-splitAt (suc m) n (suc i) = begin

+
  [ _↑ˡ n , (suc m) ↑ʳ_ ]′ (splitAt (suc m) (suc i)) ≡⟨ [,]-map (splitAt m i) ⟩

+
  [ suc ∘ (_↑ˡ n) , suc ∘ (m ↑ʳ_) ]′ (splitAt m i)   ≡⟨ [,]-∘ suc (splitAt m i) ⟨

+
  suc ([ _↑ˡ n , m ↑ʳ_ ]′ (splitAt m i))             ≡⟨ cong suc (join-splitAt m n i) ⟩

+
  suc i                                                         ∎

+
  where open ≡-Reasoning

+

+
-- splitAt "m" "i" ≡ inj₁ "i" if i < m

+

+
splitAt-< : ∀ m {n} (i : Fin (m ℕ.+ n)) .(i<m : toℕ i ℕ.< m) →

+
            splitAt m i ≡ inj₁ (fromℕ< i<m)

+
splitAt-< (suc m) zero    _   = refl

+
splitAt-< (suc m) (suc i) i<m = cong (Sum.map suc id) (splitAt-< m i (ℕ.s<s⁻¹ i<m))

+

+
-- splitAt "m" "i" ≡ inj₂ "i - m" if i ≥ m

+

+
splitAt-≥ : ∀ m {n} (i : Fin (m ℕ.+ n)) .(i≥m : toℕ i ℕ.≥ m) →

+
            splitAt m i ≡ inj₂ (reduce≥ i i≥m)

+
splitAt-≥ zero    i       _   = refl

+
splitAt-≥ (suc m) (suc i) i≥m = cong (Sum.map suc id) (splitAt-≥ m i (ℕ.s≤s⁻¹ i≥m))

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
+↔⊎ : Fin (m ℕ.+ n) ↔ (Fin m ⊎ Fin n)

+
+↔⊎ {m} {n} = mk↔ₛ′ (splitAt m {n}) (join m n) (splitAt-join m n) (join-splitAt m n)

+

+
------------------------------------------------------------------------

+
-- remQuot

+
------------------------------------------------------------------------

+

+
-- Fin (m * n) ↔ Fin m × Fin n

+

+
remQuot-combine : ∀ {n k} (i : Fin n) j → remQuot k (combine i j) ≡ (i , j)

+
remQuot-combine {suc n} {k} zero    j rewrite splitAt-↑ˡ k j (n ℕ.* k) = refl

+
remQuot-combine {suc n} {k} (suc i) j rewrite splitAt-↑ʳ k   (n ℕ.* k) (combine i j) =

+
  cong (Product.map₁ suc) (remQuot-combine i j)

+

+
combine-remQuot : ∀ {n} k (i : Fin (n ℕ.* k)) → uncurry combine (remQuot {n} k i) ≡ i

+
combine-remQuot {suc n} k i with splitAt k i in eq

+
... | inj₁ j = begin

+
  join k (n ℕ.* k) (inj₁ j)      ≡⟨ cong (join k (n ℕ.* k)) eq ⟨

+
  join k (n ℕ.* k) (splitAt k i) ≡⟨ join-splitAt k (n ℕ.* k) i ⟩

+
  i                              ∎

+
  where open ≡-Reasoning

+
... | inj₂ j = begin

+
  k ↑ʳ (uncurry combine (remQuot {n} k j)) ≡⟨ cong (k ↑ʳ_) (combine-remQuot {n} k j) ⟩

+
  join k (n ℕ.* k) (inj₂ j)                ≡⟨ cong (join k (n ℕ.* k)) eq ⟨

+
  join k (n ℕ.* k) (splitAt k i)           ≡⟨ join-splitAt k (n ℕ.* k) i ⟩

+
  i                                        ∎

+
  where open ≡-Reasoning

+

+
toℕ-combine : ∀ (i : Fin m) (j : Fin n) → toℕ (combine i j) ≡ n ℕ.* toℕ i ℕ.+ toℕ j

+
toℕ-combine {suc m} {n} i@0F j = begin

+
  toℕ (combine i j)          ≡⟨⟩

+
  toℕ (j ↑ˡ (m ℕ.* n))       ≡⟨ toℕ-↑ˡ j (m ℕ.* n) ⟩

+
  toℕ j                      ≡⟨⟩

+
  0 ℕ.+ toℕ j                ≡⟨ cong (ℕ._+ toℕ j) (ℕ.*-zeroʳ n) ⟨

+
  n ℕ.* toℕ i ℕ.+ toℕ j      ∎

+
  where open ≡-Reasoning

+
toℕ-combine {suc m} {n} (suc i) j = begin

+
  toℕ (combine (suc i) j)            ≡⟨⟩

+
  toℕ (n ↑ʳ combine i j)             ≡⟨ toℕ-↑ʳ n (combine i j) ⟩

+
  n ℕ.+ toℕ (combine i j)            ≡⟨ cong (n ℕ.+_) (toℕ-combine i j) ⟩

+
  n ℕ.+ (n ℕ.* toℕ i ℕ.+ toℕ j)     ≡⟨ ℕ.+-assoc n _ (toℕ j) ⟨

+
  n ℕ.+ n ℕ.* toℕ i ℕ.+ toℕ j       ≡⟨ cong (λ z → z ℕ.+ n ℕ.* toℕ i ℕ.+ toℕ j) (ℕ.*-identityʳ n) ⟨

+
  n ℕ.* 1 ℕ.+ n ℕ.* toℕ i ℕ.+ toℕ j ≡⟨ cong (ℕ._+ toℕ j) (ℕ.*-distribˡ-+ n 1 (toℕ i) ) ⟨

+
  n ℕ.* toℕ (suc i) ℕ.+ toℕ j       ∎

+
  where open ≡-Reasoning

+

+
combine-monoˡ-< : ∀ {i j : Fin m} (k l : Fin n) →

+
                  i < j → combine i k < combine j l

+
combine-monoˡ-< {m} {n} {i} {j} k l i<j = begin-strict

+
  toℕ (combine i k)      ≡⟨ toℕ-combine i k ⟩

+
  n ℕ.* toℕ i ℕ.+ toℕ k  <⟨ ℕ.+-monoʳ-< (n ℕ.* toℕ i) (toℕ<n k) ⟩

+
  n ℕ.* toℕ i ℕ.+ n      ≡⟨ ℕ.+-comm _ n ⟩

+
  n ℕ.+ n ℕ.* toℕ i      ≡⟨ cong (n ℕ.+_) (ℕ.*-comm n _) ⟩

+
  n ℕ.+ toℕ i ℕ.* n      ≡⟨ ℕ.*-comm (suc (toℕ i)) n ⟩

+
  n ℕ.* suc (toℕ i)      ≤⟨ ℕ.*-monoʳ-≤ n (toℕ-mono-< i<j) ⟩

+
  n ℕ.* toℕ j            ≤⟨ ℕ.m≤m+n (n ℕ.* toℕ j) (toℕ l) ⟩

+
  n ℕ.* toℕ j ℕ.+ toℕ l  ≡⟨ toℕ-combine j l ⟨

+
  toℕ (combine j l)      ∎

+
  where open ℕ.≤-Reasoning

+

+
combine-injectiveˡ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →

+
                     combine i j ≡ combine k l → i ≡ k

+
combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ with <-cmp i k

+
... | tri< i<k _ _ = contradiction cᵢⱼ≡cₖₗ (<⇒≢ (combine-monoˡ-< j l i<k))

+
... | tri≈ _ i≡k _ = i≡k

+
... | tri> _ _ i>k = contradiction (sym cᵢⱼ≡cₖₗ) (<⇒≢ (combine-monoˡ-< l j i>k))

+

+
combine-injectiveʳ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →

+
                     combine i j ≡ combine k l → j ≡ l

+
combine-injectiveʳ {m} {n} i j k l cᵢⱼ≡cₖₗ

+
  with refl ← combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ

+
  = toℕ-injective (ℕ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin

+
  n ℕ.* toℕ i ℕ.+ toℕ j ≡⟨ toℕ-combine i j ⟨

+
  toℕ (combine i j)     ≡⟨ cong toℕ cᵢⱼ≡cₖₗ ⟩

+
  toℕ (combine i l)     ≡⟨ toℕ-combine i l ⟩

+
  n ℕ.* toℕ i ℕ.+ toℕ l ∎))

+
  where open ≡-Reasoning

+

+
combine-injective : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →

+
                    combine i j ≡ combine k l → i ≡ k × j ≡ l

+
combine-injective i j k l cᵢⱼ≡cₖₗ =

+
  combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ ,

+
  combine-injectiveʳ i j k l cᵢⱼ≡cₖₗ

+

+
combine-surjective : ∀ (i : Fin (m ℕ.* n)) → ∃₂ λ j k → combine j k ≡ i

+
combine-surjective {m} {n} i with j , k ← remQuot {m} n i in eq

+
  = j , k , (begin

+
  combine j k                       ≡⟨ uncurry (cong₂ combine) (,-injective eq) ⟨

+
  uncurry combine (remQuot {m} n i) ≡⟨ combine-remQuot {m} n i ⟩

+
  i                                 ∎)

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
*↔× : Fin (m ℕ.* n) ↔ (Fin m × Fin n)

+
*↔× {m} {n} = mk↔ₛ′ (remQuot {m} n) (uncurry combine)

+
  (uncurry remQuot-combine)

+
  (combine-remQuot {m} n)

+

+
------------------------------------------------------------------------

+
-- fin→fun

+
------------------------------------------------------------------------

+

+
funToFin-finToFin : funToFin {m} {n} ∘ finToFun ≗ id

+
funToFin-finToFin {zero}  {n} zero = refl

+
funToFin-finToFin {suc m} {n} k    =

+
  begin

+
    combine (finToFun {n} {suc m} k zero) (funToFin (finToFun {n} {suc m} k ∘ suc))

+
  ≡⟨⟩

+
    combine (quotient {n} (n ^ m) k)

+
      (funToFin (finToFun {n} {m} (remainder {n} (n ^ m) k)))

+
  ≡⟨ cong (combine (quotient {n} (n ^ m) k))

+
       (funToFin-finToFin {m} (remainder {n} (n ^ m) k)) ⟩

+
    combine (quotient {n} (n ^ m) k) (remainder {n} (n ^ m) k)

+
  ≡⟨⟩

+
    uncurry combine (remQuot {n} (n ^ m) k)

+
  ≡⟨ combine-remQuot {n = n} (n ^ m) k ⟩

+
    k

+
  ∎ where open ≡-Reasoning

+

+
finToFun-funToFin : (f : Fin m → Fin n) → finToFun (funToFin f) ≗ f

+
finToFun-funToFin {suc m} {n} f  zero   =

+
  begin

+
    quotient (n ^ m) (combine (f zero) (funToFin (f ∘ suc)))

+
  ≡⟨ cong proj₁ (remQuot-combine _ _) ⟩

+
    proj₁ (f zero , funToFin (f ∘ suc))

+
  ≡⟨⟩

+
    f zero

+
  ∎ where open ≡-Reasoning

+
finToFun-funToFin {suc m} {n} f (suc i) =

+
  begin

+
    finToFun (remainder {n} (n ^ m) (combine (f zero) (funToFin (f ∘ suc)))) i

+
  ≡⟨ cong (λ rq → finToFun (proj₂ rq) i) (remQuot-combine {n} _ _) ⟩

+
    finToFun (proj₂ (f zero , funToFin (f ∘ suc))) i

+
  ≡⟨⟩

+
    finToFun (funToFin (f ∘ suc)) i

+
  ≡⟨ finToFun-funToFin (f ∘ suc) i ⟩

+
    (f ∘ suc) i

+
  ≡⟨⟩

+
    f (suc i)

+
  ∎ where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
^↔→ : Extensionality _ _ → Fin (m ^ n) ↔ (Fin n → Fin m)

+
^↔→ {m} {n} ext = mk↔ₛ′ finToFun funToFin

+
  (ext ∘ finToFun-funToFin)

+
  (funToFin-finToFin {n} {m})

+

+
------------------------------------------------------------------------

+
-- lift

+
------------------------------------------------------------------------

+

+
lift-injective : ∀ (f : Fin m → Fin n) → Injective _≡_ _≡_ f →

+
                 ∀ k → Injective _≡_ _≡_ (lift k f)

+
lift-injective f inj zero    {_}     {_}     eq = inj eq

+
lift-injective f inj (suc k) {zero}  {zero}  eq = refl

+
lift-injective f inj (suc k) {suc _} {suc _} eq =

+
  cong suc (lift-injective f inj k (suc-injective eq))

+

+
------------------------------------------------------------------------

+
-- pred

+
------------------------------------------------------------------------

+

+
<⇒≤pred : i < j → i ≤ pred j

+
<⇒≤pred {i = zero}  {j = suc j} z<s       = z≤n

+
<⇒≤pred {i = suc i} {j = suc j} (s<s i<j) rewrite toℕ-inject₁ j = i<j

+

+
------------------------------------------------------------------------

+
-- _ℕ-_

+
------------------------------------------------------------------------

+

+
toℕ‿ℕ- : ∀ n i → toℕ (n ℕ- i) ≡ n ∸ toℕ i

+
toℕ‿ℕ- n       zero     = toℕ-fromℕ n

+
toℕ‿ℕ- (suc n) (suc i)  = toℕ‿ℕ- n i

+

+
------------------------------------------------------------------------

+
-- _ℕ-ℕ_

+
------------------------------------------------------------------------

+

+
ℕ-ℕ≡toℕ‿ℕ- : ∀ n i → n ℕ-ℕ i ≡ toℕ (n ℕ- i)

+
ℕ-ℕ≡toℕ‿ℕ- n       zero    = sym (toℕ-fromℕ n)

+
ℕ-ℕ≡toℕ‿ℕ- (suc n) (suc i) = ℕ-ℕ≡toℕ‿ℕ- n i

+

+
nℕ-ℕi≤n : ∀ n i → n ℕ-ℕ i ℕ.≤ n

+
nℕ-ℕi≤n n       zero     = ℕ.≤-refl

+
nℕ-ℕi≤n (suc n) (suc i)  = begin

+
  n ℕ-ℕ i  ≤⟨ nℕ-ℕi≤n n i ⟩

+
  n        ≤⟨ ℕ.n≤1+n n ⟩

+
  suc n    ∎

+
  where open ℕ.≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- punchIn

+
------------------------------------------------------------------------

+

+
punchIn-injective : ∀ i (j k : Fin n) →

+
                    punchIn i j ≡ punchIn i k → j ≡ k

+
punchIn-injective zero    _       _       refl      = refl

+
punchIn-injective (suc i) zero    zero    _         = refl

+
punchIn-injective (suc i) (suc j) (suc k) ↑j+1≡↑k+1 =

+
  cong suc (punchIn-injective i j k (suc-injective ↑j+1≡↑k+1))

+

+
punchInᵢ≢i : ∀ i (j : Fin n) → punchIn i j ≢ i

+
punchInᵢ≢i (suc i) (suc j) = punchInᵢ≢i i j ∘ suc-injective

+

+
------------------------------------------------------------------------

+
-- punchOut

+
------------------------------------------------------------------------

+

+
-- A version of 'cong' for 'punchOut' in which the inequality argument

+
-- can be changed out arbitrarily (reflecting the proof-irrelevance of

+
-- that argument).

+

+
punchOut-cong : ∀ (i : Fin (suc n)) {j k} {i≢j : i ≢ j} {i≢k : i ≢ k} →

+
                j ≡ k → punchOut i≢j ≡ punchOut i≢k

+
punchOut-cong {_}     zero    {zero}         {i≢j = 0≢0} = contradiction refl 0≢0

+
punchOut-cong {_}     zero    {suc j} {zero} {i≢k = 0≢0} = contradiction refl 0≢0

+
punchOut-cong {_}     zero    {suc j} {suc k}            = suc-injective

+
punchOut-cong {suc n} (suc i) {zero}  {zero}   _ = refl

+
punchOut-cong {suc n} (suc i) {suc j} {suc k}    = cong suc ∘ punchOut-cong i ∘ suc-injective

+

+
-- An alternative to 'punchOut-cong' in the which the new inequality

+
-- argument is specific. Useful for enabling the omission of that

+
-- argument during equational reasoning.

+

+
punchOut-cong′ : ∀ (i : Fin (suc n)) {j k} {p : i ≢ j} (q : j ≡ k) →

+
                 punchOut p ≡ punchOut (p ∘ sym ∘ trans q ∘ sym)

+
punchOut-cong′ i q = punchOut-cong i q

+

+
punchOut-injective : ∀ {i j k : Fin (suc n)}

+
                     (i≢j : i ≢ j) (i≢k : i ≢ k) →

+
                     punchOut i≢j ≡ punchOut i≢k → j ≡ k

+
punchOut-injective {_}     {zero}   {zero}  {_}     0≢0 _   _     = contradiction refl 0≢0

+
punchOut-injective {_}     {zero}   {_}     {zero}  _   0≢0 _     = contradiction refl 0≢0

+
punchOut-injective {_}     {zero}   {suc j} {suc k} _   _   pⱼ≡pₖ = cong suc pⱼ≡pₖ

+
punchOut-injective {suc n} {suc i}  {zero}  {zero}  _   _    _    = refl

+
punchOut-injective {suc n} {suc i}  {suc j} {suc k} i≢j i≢k pⱼ≡pₖ =

+
  cong suc (punchOut-injective (i≢j ∘ cong suc) (i≢k ∘ cong suc) (suc-injective pⱼ≡pₖ))

+

+
punchIn-punchOut : ∀ {i j : Fin (suc n)} (i≢j : i ≢ j) →

+
                   punchIn i (punchOut i≢j) ≡ j

+
punchIn-punchOut {_}     {zero}   {zero}  0≢0 = contradiction refl 0≢0

+
punchIn-punchOut {_}     {zero}   {suc j} _   = refl

+
punchIn-punchOut {suc m} {suc i}  {zero}  i≢j = refl

+
punchIn-punchOut {suc m} {suc i}  {suc j} i≢j =

+
  cong suc (punchIn-punchOut (i≢j ∘ cong suc))

+

+
punchOut-punchIn : ∀ i {j : Fin n} → punchOut {i = i} {j = punchIn i j} (punchInᵢ≢i i j ∘ sym) ≡ j

+
punchOut-punchIn zero    {j}     = refl

+
punchOut-punchIn (suc i) {zero}  = refl

+
punchOut-punchIn (suc i) {suc j} = cong suc (begin

+
  punchOut (punchInᵢ≢i i j ∘ suc-injective ∘ sym ∘ cong suc)  ≡⟨ punchOut-cong i refl ⟩

+
  punchOut (punchInᵢ≢i i j ∘ sym)                             ≡⟨ punchOut-punchIn i ⟩

+
  j                                                           ∎)

+
  where open ≡-Reasoning

+

+

+
------------------------------------------------------------------------

+
-- pinch

+
------------------------------------------------------------------------

+

+
pinch-surjective : ∀ (i : Fin n) → Surjective _≡_ _≡_ (pinch i)

+
pinch-surjective _       zero    = zero , λ { refl → refl }

+
pinch-surjective zero    (suc j) = suc (suc j) , λ { refl → refl }

+
pinch-surjective (suc i) (suc j) = map suc (λ {f refl → cong suc (f refl)}) (pinch-surjective i j)

+

+
pinch-mono-≤ : ∀ (i : Fin n) → (pinch i) Preserves _≤_ ⟶ _≤_

+
pinch-mono-≤ 0F      {0F}    {k}     0≤n = z≤n

+
pinch-mono-≤ 0F      {suc j} {suc k} j≤k = ℕ.s≤s⁻¹ j≤k

+
pinch-mono-≤ (suc i) {0F}    {k}     0≤n = z≤n

+
pinch-mono-≤ (suc i) {suc j} {suc k} j≤k = s≤s (pinch-mono-≤ i (ℕ.s≤s⁻¹ j≤k))

+

+
pinch-injective : ∀ {i : Fin n} {j k : Fin (ℕ.suc n)} →

+
                  suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k

+
pinch-injective {i = i}     {zero}  {zero}  _     _     _  = refl

+
pinch-injective {i = zero}  {zero}  {suc k} _     1+i≢k eq =

+
  contradiction (cong suc eq) 1+i≢k

+
pinch-injective {i = zero}  {suc j} {zero}  1+i≢j _     eq =

+
  contradiction (cong suc (sym eq)) 1+i≢j

+
pinch-injective {i = zero}  {suc j} {suc k} _     _     eq =

+
  cong suc eq

+
pinch-injective {i = suc i} {suc j} {suc k} 1+i≢j 1+i≢k eq =

+
  cong suc

+
    (pinch-injective (1+i≢j ∘ cong suc) (1+i≢k ∘ cong suc)

+
      (suc-injective eq))

+

+
------------------------------------------------------------------------

+
-- Quantification

+
------------------------------------------------------------------------

+

+
module _ {p} {P : Pred (Fin (suc n)) p} where

+

+
  ∀-cons : P zero → Π[ P ∘ suc ] → Π[ P ]

+
  ∀-cons z s zero    = z

+
  ∀-cons z s (suc i) = s i

+

+
  ∀-cons-⇔ : (P zero × Π[ P ∘ suc ]) ⇔ Π[ P ]

+
  ∀-cons-⇔ = mk⇔ (uncurry ∀-cons) < _$ zero , _∘ suc >

+

+
  ∃-here : P zero → ∃⟨ P ⟩

+
  ∃-here = zero ,_

+

+
  ∃-there : ∃⟨ P ∘ suc ⟩ → ∃⟨ P ⟩

+
  ∃-there = map suc id

+

+
  ∃-toSum : ∃⟨ P ⟩ → P zero ⊎ ∃⟨ P ∘ suc ⟩

+
  ∃-toSum ( zero , P₀ ) = inj₁ P₀

+
  ∃-toSum (suc f , P₁₊) = inj₂ (f , P₁₊)

+

+
  ⊎⇔∃ : (P zero ⊎ ∃⟨ P ∘ suc ⟩) ⇔ ∃⟨ P ⟩

+
  ⊎⇔∃ = mk⇔ [ ∃-here , ∃-there ] ∃-toSum

+

+
decFinSubset : ∀ {p q} {P : Pred (Fin n) p} {Q : Pred (Fin n) q} →

+
               Decidable Q → (∀ {i} → Q i → Dec (P i)) → Dec (Q ⊆ P)

+
decFinSubset {zero}  {_}     {_} Q? P? = yes λ {}

+
decFinSubset {suc n} {P = P} {Q} Q? P?

+
  with Q? zero | ∀-cons {P = λ x → Q x → P x}

+
... | false because [¬Q0] | cons =

+
  map′ (λ f {x} → cons (⊥-elim ∘ invert [¬Q0]) (λ x → f {x}) x)

+
       (λ f {x} → f {suc x})

+
       (decFinSubset (Q? ∘ suc) P?)

+
... | true  because  [Q0] | cons =

+
  map′ (uncurry λ P0 rec {x} → cons (λ _ → P0) (λ x → rec {x}) x)

+
       < _$ invert [Q0] , (λ f {x} → f {suc x}) >

+
       (P? (invert [Q0]) ×-dec decFinSubset (Q? ∘ suc) P?)

+

+
any? : ∀ {p} {P : Pred (Fin n) p} → Decidable P → Dec (∃ P)

+
any? {zero}  {P = _} P? = no λ { (() , _) }

+
any? {suc n} {P = P} P? = Dec.map ⊎⇔∃ (P? zero ⊎-dec any? (P? ∘ suc))

+

+
all? : ∀ {p} {P : Pred (Fin n) p} → Decidable P → Dec (∀ f → P f)

+
all? P? = map′ (λ ∀p f → ∀p tt) (λ ∀p {x} _ → ∀p x)

+
               (decFinSubset U? (λ {f} _ → P? f))

+

+
private

+
  -- A nice computational property of `all?`:

+
  -- The boolean component of the result is exactly the

+
  -- obvious fold of boolean tests (`foldr _∧_ true`).

+
  note : ∀ {p} {P : Pred (Fin 3) p} (P? : Decidable P) →

+
         ∃ λ z → Dec.does (all? P?) ≡ z

+
  note P? = Dec.does (P? 0F) ∧ Dec.does (P? 1F) ∧ Dec.does (P? 2F) ∧ true

+
          , refl

+

+
-- If a decidable predicate P over a finite set is sometimes false,

+
-- then we can find the smallest value for which this is the case.

+

+
¬∀⟶∃¬-smallest : ∀ n {p} (P : Pred (Fin n) p) → Decidable P →

+
                 ¬ (∀ i → P i) → ∃ λ i → ¬ P i × ((j : Fin′ i) → P (inject j))

+
¬∀⟶∃¬-smallest zero    P P? ¬∀P = contradiction (λ()) ¬∀P

+
¬∀⟶∃¬-smallest (suc n) P P? ¬∀P with P? zero

+
... | false because [¬P₀] = (zero , invert [¬P₀] , λ ())

+
... | true  because  [P₀] = map suc (map id (∀-cons (invert [P₀])))

+
  (¬∀⟶∃¬-smallest n (P ∘ suc) (P? ∘ suc) (¬∀P ∘ (∀-cons (invert [P₀]))))

+

+
-- When P is a decidable predicate over a finite set the following

+
-- lemma can be proved.

+

+
¬∀⟶∃¬ : ∀ n {p} (P : Pred (Fin n) p) → Decidable P →

+
          ¬ (∀ i → P i) → (∃ λ i → ¬ P i)

+
¬∀⟶∃¬ n P P? ¬P = map id proj₁ (¬∀⟶∃¬-smallest n P P? ¬P)

+

+
------------------------------------------------------------------------

+
-- Properties of functions to and from Fin

+
------------------------------------------------------------------------

+

+
-- The pigeonhole principle.

+

+
pigeonhole : m ℕ.< n → (f : Fin n → Fin m) → ∃₂ λ i j → i < j × f i ≡ f j

+
pigeonhole z<s               f = contradiction (f zero) λ()

+
pigeonhole (s<s m<n@(s≤s _)) f with any? (λ k → f zero ≟ f (suc k))

+
... | yes (j , f₀≡fⱼ) = zero , suc j , z<s , f₀≡fⱼ

+
... | no  f₀≢fₖ

+
  with i , j , i<j , fᵢ≡fⱼ ← pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))

+
  = suc i , suc j , s<s i<j , punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ

+

+
injective⇒≤ : ∀ {f : Fin m → Fin n} → Injective _≡_ _≡_ f → m ℕ.≤ n

+
injective⇒≤ {zero}  {_}     {f} _   = z≤n

+
injective⇒≤ {suc _} {zero}  {f} _   = contradiction (f zero) ¬Fin0

+
injective⇒≤ {suc _} {suc _} {f} inj = s≤s (injective⇒≤ (λ eq →

+
  suc-injective (inj (punchOut-injective

+
    (contraInjective inj 0≢1+n)

+
    (contraInjective inj 0≢1+n) eq))))

+

+
<⇒notInjective : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective _≡_ _≡_ f)

+
<⇒notInjective n<m inj = ℕ.≤⇒≯ (injective⇒≤ inj) n<m

+

+
ℕ→Fin-notInjective : ∀ (f : ℕ → Fin n) → ¬ (Injective _≡_ _≡_ f)

+
ℕ→Fin-notInjective f inj = ℕ.<-irrefl refl

+
  (injective⇒≤ (Comp.injective _≡_ _≡_ _≡_ toℕ-injective inj))

+

+
-- Cantor-Schröder-Bernstein for finite sets

+

+
cantor-schröder-bernstein : ∀ {f : Fin m → Fin n} {g : Fin n → Fin m} →

+
                            Injective _≡_ _≡_ f → Injective _≡_ _≡_ g →

+
                            m ≡ n

+
cantor-schröder-bernstein f-inj g-inj = ℕ.≤-antisym

+
  (injective⇒≤ f-inj) (injective⇒≤ g-inj)

+

+
------------------------------------------------------------------------

+
-- Effectful

+
------------------------------------------------------------------------

+

+
module _ {f} {F : Set f → Set f} (RA : RawApplicative F) where

+

+
  open RawApplicative RA

+

+
  sequence : ∀ {n} {P : Pred (Fin n) f} →

+
             (∀ i → F (P i)) → F (∀ i → P i)

+
  sequence {zero}  ∀iPi = pure λ()

+
  sequence {suc n} ∀iPi = ∀-cons <$> ∀iPi zero <*> sequence (∀iPi ∘ suc)

+

+
module _ {f} {F : Set f → Set f} (RF : RawFunctor F) where

+

+
  open RawFunctor RF

+

+
  sequence⁻¹ : ∀ {A : Set f} {P : Pred A f} →

+
               F (∀ i → P i) → (∀ i → F (P i))

+
  sequence⁻¹ F∀iPi i = (λ f → f i) <$> F∀iPi

+

+
------------------------------------------------------------------------

+
-- If there is an injection from a type A to a finite set, then the type

+
-- has decidable equality.

+

+
module _ {ℓ} {S : Setoid a ℓ} (inj : Injection S (≡-setoid n)) where

+
  open Setoid S

+

+
  inj⇒≟ : B.Decidable _≈_

+
  inj⇒≟ = Dec.via-injection inj _≟_

+

+
  inj⇒decSetoid : DecSetoid a ℓ

+
  inj⇒decSetoid = record

+
    { isDecEquivalence = record

+
      { isEquivalence = isEquivalence

+
      ; _≟_           = inj⇒≟

+
      }

+
    }

+

+
------------------------------------------------------------------------

+
-- Opposite

+
------------------------------------------------------------------------

+

+
opposite-prop : ∀ (i : Fin n) → toℕ (opposite i) ≡ n ∸ suc (toℕ i)

+
opposite-prop {suc n} zero    = toℕ-fromℕ n

+
opposite-prop {suc n} (suc i) = begin

+
  toℕ (inject₁ (opposite i)) ≡⟨ toℕ-inject₁ (opposite i) ⟩

+
  toℕ (opposite i)           ≡⟨ opposite-prop i ⟩

+
  n ∸ suc (toℕ i)            ∎

+
  where open ≡-Reasoning

+

+
opposite-involutive : Involutive {A = Fin n} _≡_ opposite

+
opposite-involutive {suc n} i = toℕ-injective (begin

+
  toℕ (opposite (opposite i)) ≡⟨ opposite-prop (opposite i) ⟩

+
  n ∸ (toℕ (opposite i))      ≡⟨ cong (n ∸_) (opposite-prop i) ⟩

+
  n ∸ (n ∸ (toℕ i))           ≡⟨ ℕ.m∸[m∸n]≡n (toℕ≤pred[n] i) ⟩

+
  toℕ i                       ∎)

+
  where open ≡-Reasoning

+

+
opposite-suc : ∀ (i : Fin n) → toℕ (opposite (suc i)) ≡ toℕ (opposite i)

+
opposite-suc {n} i = begin

+
  toℕ (opposite (suc i))     ≡⟨ opposite-prop (suc i) ⟩

+
  suc n ∸ suc (toℕ (suc i))  ≡⟨⟩

+
  n ∸ toℕ (suc i)            ≡⟨⟩

+
  n ∸ suc (toℕ i)            ≡⟨ opposite-prop i ⟨

+
  toℕ (opposite i)           ∎

+
  where open ≡-Reasoning

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.5

+

+
inject+-raise-splitAt = join-splitAt

+
{-# WARNING_ON_USAGE inject+-raise-splitAt

+
"Warning: inject+-raise-splitAt was deprecated in v1.5.

+
Please use join-splitAt instead."

+
#-}

+

+
-- Version 2.0

+

+
toℕ-raise = toℕ-↑ʳ

+
{-# WARNING_ON_USAGE toℕ-raise

+
"Warning: toℕ-raise was deprecated in v2.0.

+
Please use toℕ-↑ʳ instead."

+
#-}

+
toℕ-inject+ : ∀ {m} n (i : Fin m) → toℕ i ≡ toℕ (i ↑ˡ n)

+
toℕ-inject+ n i = sym (toℕ-↑ˡ i n)

+
{-# WARNING_ON_USAGE toℕ-inject+

+
"Warning: toℕ-inject+ was deprecated in v2.0.

+
Please use toℕ-↑ˡ instead.

+
NB argument order has been flipped:

+
the left-hand argument is the Fin m

+
the right-hand is the Nat index increment."

+
#-}

+
splitAt-inject+ : ∀ m n i → splitAt m (i ↑ˡ n) ≡ inj₁ i

+
splitAt-inject+ m n i = splitAt-↑ˡ m i n

+
{-# WARNING_ON_USAGE splitAt-inject+

+
"Warning: splitAt-inject+ was deprecated in v2.0.

+
Please use splitAt-↑ˡ instead.

+
NB argument order has been flipped."

+
#-}

+
splitAt-raise : ∀ m n i → splitAt m (m ↑ʳ i) ≡ inj₂ {B = Fin n} i

+
splitAt-raise = splitAt-↑ʳ

+
{-# WARNING_ON_USAGE splitAt-raise

+
"Warning: splitAt-raise was deprecated in v2.0.

+
Please use splitAt-↑ʳ instead."

+
#-}

+
Fin0↔⊥ : Fin 0 ↔ ⊥

+
Fin0↔⊥ = 0↔⊥

+
{-# WARNING_ON_USAGE Fin0↔⊥

+
"Warning: Fin0↔⊥ was deprecated in v2.0.

+
Please use 0↔⊥ instead."

+
#-}

+
eq? : A ↣ Fin n → DecidableEquality A

+
eq? = inj⇒≟

+
{-# WARNING_ON_USAGE eq?

+
"Warning: eq? was deprecated in v2.0.

+
Please use inj⇒≟ instead."

+
#-}

+

+
private

+

+
  z≺s : ∀ {n} → zero ≺ suc n

+
  z≺s = _ ≻toℕ zero

+

+
  s≺s : ∀ {m n} → m ≺ n → suc m ≺ suc n

+
  s≺s (n ≻toℕ i) = (suc n) ≻toℕ (suc i)

+

+
  <⇒≺ : ℕ._<_ ⇒ _≺_

+
  <⇒≺ {zero}  z<s      = z≺s

+
  <⇒≺ {suc m} (s<s lt) = s≺s (<⇒≺ lt)

+

+
  ≺⇒< : _≺_ ⇒ ℕ._<_

+
  ≺⇒< (n ≻toℕ i) = toℕ<n i

+

+
≺⇒<′ : _≺_ ⇒ ℕ._<′_

+
≺⇒<′ lt = ℕ.<⇒<′ (≺⇒< lt)

+
{-# WARNING_ON_USAGE ≺⇒<′

+
"Warning: ≺⇒<′ was deprecated in v2.0.

+
Please use <⇒<′ instead."

+
#-}

+

+
<′⇒≺ : ℕ._<′_ ⇒ _≺_

+
<′⇒≺ lt = <⇒≺ (ℕ.<′⇒< lt)

+
{-# WARNING_ON_USAGE <′⇒≺

+
"Warning: <′⇒≺ was deprecated in v2.0.

+
Please use <′⇒< instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Fin.html b/site/static/agda_html/Data.Fin.html
new file mode 100644
index 000000000..8803b351d
--- /dev/null
+++ b/site/static/agda_html/Data.Fin.html
@@ -0,0 +1,107 @@
+
+Data.Fin
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Finite sets

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Fin where

+

+
open import Relation.Nullary.Decidable.Core

+
import Data.Nat.Properties as ℕ

+

+
------------------------------------------------------------------------

+
-- Publicly re-export the contents of the base module

+

+
open import Data.Fin.Base public

+

+
------------------------------------------------------------------------

+
-- Publicly re-export queries

+

+
open import Data.Fin.Properties public

+
  using (_≟_; _≤?_; _<?_)

+

+
-- # m = "m".

+

+
infix 10 #_

+

+
#_ : ∀ m {n} {m<n : True (m ℕ.<? n)} → Fin n

+
#_ _ {m<n = m<n} = fromℕ< (toWitness m<n)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Float.Base.html b/site/static/agda_html/Data.Float.Base.html
new file mode 100644
index 000000000..1e3fb1dc4
--- /dev/null
+++ b/site/static/agda_html/Data.Float.Base.html
@@ -0,0 +1,164 @@
+
+Data.Float.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Floats: basic types and operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Float.Base where

+

+
open import Data.Bool.Base using (T)

+
import Data.Word64.Base as Word64

+
import Data.Maybe.Base as Maybe

+
open import Function.Base using (_on_; _∘_)

+
open import Agda.Builtin.Equality

+
open import Relation.Binary.Core using (Rel)

+

+
------------------------------------------------------------------------

+
-- Re-export built-ins publically

+

+
open import Agda.Builtin.Float public

+
  using (Float)

+
  renaming

+
  -- Relations

+
  ( primFloatInequality        to infix 4 _≤ᵇ_

+
  ; primFloatEquality          to infix 4 _≡ᵇ_

+
  ; primFloatLess              to infix 4 _<ᵇ_

+
  ; primFloatIsInfinite        to isInfinite

+
  ; primFloatIsNaN             to isNaN

+
  ; primFloatIsNegativeZero    to isNegativeZero

+
  ; primFloatIsSafeInteger     to isSafeInteger

+
  -- Conversions

+
  ; primFloatToWord64          to toWord64

+
  ; primNatToFloat             to fromℕ

+
  ; primIntToFloat             to fromℤ

+
  ; primFloatRound             to round

+
  ; primFloatFloor             to ⌊_⌋

+
  ; primFloatCeiling           to ⌈_⌉

+
  ; primFloatToRatio           to toRatio

+
  ; primRatioToFloat           to fromRatio

+
  ; primFloatDecode            to decode

+
  ; primFloatEncode            to encode

+
  ; primShowFloat              to show

+
  -- Operations

+
  ; primFloatPlus              to infixl 6 _+_

+
  ; primFloatMinus             to infixl 6 _-_

+
  ; primFloatTimes             to infixl 7 _*_

+
  ; primFloatDiv               to infixl 7 _÷_

+
  ; primFloatPow               to infixl 8 _**_

+
  ; primFloatNegate            to infixr 9 -_

+
  ; primFloatSqrt              to sqrt

+
  ; primFloatExp               to infixr 9 e^_

+
  ; primFloatLog               to log

+
  ; primFloatSin               to sin

+
  ; primFloatCos               to cos

+
  ; primFloatTan               to tan

+
  ; primFloatASin              to asin

+
  ; primFloatACos              to acos

+
  ; primFloatATan              to atan

+
  ; primFloatATan2             to atan2

+
  ; primFloatSinh              to sinh

+
  ; primFloatCosh              to cosh

+
  ; primFloatTanh              to tanh

+
  ; primFloatASinh             to asinh

+
  ; primFloatACosh             to acosh

+
  ; primFloatATanh             to atanh

+
  )

+

+
infix 4 _≈_

+

+
_≈_ : Rel Float _

+
_≈_ = _≡_ on Maybe.map Word64.toℕ ∘ toWord64

+

+

+
infix 4 _≤_

+
_≤_ : Rel Float _

+
x ≤ y = T (x ≤ᵇ y)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATIONS

+

+
toWord = toWord64

+
{-# WARNING_ON_USAGE toWord

+
"Warning: toWord was deprecated in v2.1.

+
Please use toWord64 instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Float.Properties.html b/site/static/agda_html/Data.Float.Properties.html
new file mode 100644
index 000000000..22feffd64
--- /dev/null
+++ b/site/static/agda_html/Data.Float.Properties.html
@@ -0,0 +1,184 @@
+
+Data.Float.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of operations on floats

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Float.Properties where

+

+
open import Data.Bool.Base as Bool using (Bool)

+
open import Data.Float.Base

+
import Data.Maybe.Base as Maybe

+
import Data.Maybe.Properties as Maybe

+
import Data.Nat.Properties as ℕ

+
import Data.Word64.Base as Word64

+
import Data.Word64.Properties as Word64

+
open import Function.Base using (_∘_)

+
open import Relation.Nullary.Decidable as RN using (map′)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles using (Setoid; DecSetoid)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive; Substitutive; Decidable; DecidableEquality)

+
import Relation.Binary.Construct.On as On

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; sym; trans; subst)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; decSetoid)

+

+
------------------------------------------------------------------------

+
-- Primitive properties

+

+
open import Agda.Builtin.Float.Properties

+
  renaming (primFloatToWord64Injective to toWord64-injective)

+
  public

+

+
------------------------------------------------------------------------

+
-- Properties of _≈_

+

+
≈⇒≡ : _≈_ ⇒ _≡_

+
≈⇒≡ eq = toWord64-injective _ _ (Maybe.map-injective Word64.≈⇒≡ eq)

+

+
≈-reflexive : _≡_ ⇒ _≈_

+
≈-reflexive eq = cong (Maybe.map Word64.toℕ ∘ toWord64) eq

+

+
≈-refl : Reflexive _≈_

+
≈-refl = refl

+

+
≈-sym : Symmetric _≈_

+
≈-sym = sym

+

+
≈-trans : Transitive _≈_

+
≈-trans = trans

+

+
≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ

+
≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p

+

+
infix 4 _≈?_

+
_≈?_ : Decidable _≈_

+
_≈?_ = On.decidable (Maybe.map Word64.toℕ ∘ toWord64) _≡_ (Maybe.≡-dec ℕ._≟_)

+

+
≈-isEquivalence : IsEquivalence _≈_

+
≈-isEquivalence = record

+
  { refl  = λ {i} → ≈-refl {i}

+
  ; sym   = λ {i j} → ≈-sym {i} {j}

+
  ; trans = λ {i j k} → ≈-trans {i} {j} {k}

+
  }

+

+
≈-setoid : Setoid _ _

+
≈-setoid = record

+
  { isEquivalence = ≈-isEquivalence

+
  }

+

+
≈-isDecEquivalence : IsDecEquivalence _≈_

+
≈-isDecEquivalence = record

+
  { isEquivalence = ≈-isEquivalence

+
  ; _≟_           = _≈?_

+
  }

+

+
≈-decSetoid : DecSetoid _ _

+
≈-decSetoid = record

+
  { isDecEquivalence = ≈-isDecEquivalence

+
  }

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+

+
infix 4 _≟_

+
_≟_ : DecidableEquality Float

+
x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)

+

+
≡-setoid : Setoid _ _

+
≡-setoid = setoid Float

+

+
≡-decSetoid : DecSetoid _ _

+
≡-decSetoid = decSetoid _≟_

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATIONS

+

+
toWord-injective = toWord64-injective

+
{-# WARNING_ON_USAGE toWord-injective

+
"Warning: toWord-injective was deprecated in v2.1.

+
Please use toWord64-injective instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Float.html b/site/static/agda_html/Data.Float.html
new file mode 100644
index 000000000..4919cec4a
--- /dev/null
+++ b/site/static/agda_html/Data.Float.html
@@ -0,0 +1,92 @@
+
+Data.Float
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Floating point numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Float where

+

+
------------------------------------------------------------------------

+
-- Re-export base definitions and decidability of equality

+

+
open import Data.Float.Base public

+
open import Data.Float.Properties using (_≈?_; _≟_) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Base.html b/site/static/agda_html/Data.Integer.Base.html
new file mode 100644
index 000000000..aadee2d13
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Base.html
@@ -0,0 +1,425 @@
+
+Data.Integer.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Integers, basic types and operations

+
------------------------------------------------------------------------

+

+
-- See README.Data.Integer for examples of how to use and reason about

+
-- integers.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Base where

+

+
open import Algebra.Bundles.Raw

+
  using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)

+
open import Data.Bool.Base using (Bool; T; true; false)

+
open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s) hiding (module ℕ)

+
open import Data.Sign.Base as Sign using (Sign)

+
open import Level using (0ℓ)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Unary using (Pred)

+

+
infix  8 -_

+
infixr 8 _^_

+
infixl 7 _*_ _⊓_ _/ℕ_ _/_ _%ℕ_ _%_

+
infixl 6 _+_ _-_ _⊖_ _⊔_

+
infix  4 _≤_ _≥_ _<_ _>_ _≰_ _≱_ _≮_ _≯_

+
infix  4 _≤ᵇ_

+

+
------------------------------------------------------------------------

+
-- Types

+

+
open import Agda.Builtin.Int public

+
  using ()

+
  renaming

+
  ( Int    to ℤ

+
  ; pos    to +_      -- "+ n"      stands for "n"

+
  ; negsuc to -[1+_]  -- "-[1+ n ]" stands for "- (1 + n)"

+
  )

+

+
-- Some additional patterns that provide symmetry around 0

+

+
pattern +0       = + 0

+
pattern +[1+_] n = + (ℕ.suc n)

+

+
------------------------------------------------------------------------

+
-- Constants

+

+
0ℤ : ℤ

+
0ℤ = +0

+

+
-1ℤ : ℤ

+
-1ℤ = -[1+ 0 ]

+

+
1ℤ : ℤ

+
1ℤ = +[1+ 0 ]

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
-- Absolute value.

+

+
∣_∣ : ℤ → ℕ

+
∣ + n      ∣ = n

+
∣ -[1+ n ] ∣ = ℕ.suc n

+

+
-- Gives the sign. For zero the sign is arbitrarily chosen to be +.

+

+
sign : ℤ → Sign

+
sign (+ _)    = Sign.+

+
sign -[1+ _ ] = Sign.-

+

+
------------------------------------------------------------------------

+
-- Ordering

+

+
data _≤_ : ℤ → ℤ → Set where

+
  -≤- : ∀ {m n} → (n≤m : n ℕ.≤ m) → -[1+ m ] ≤ -[1+ n ]

+
  -≤+ : ∀ {m n} → -[1+ m ] ≤ + n

+
  +≤+ : ∀ {m n} → (m≤n : m ℕ.≤ n) → + m ≤ + n

+

+
data _<_ : ℤ → ℤ → Set where

+
  -<- : ∀ {m n} → (n<m : n ℕ.< m) → -[1+ m ] < -[1+ n ]

+
  -<+ : ∀ {m n} → -[1+ m ] < + n

+
  +<+ : ∀ {m n} → (m<n : m ℕ.< n) → + m < + n

+

+
_≥_ : Rel ℤ 0ℓ

+
x ≥ y = y ≤ x

+

+
_>_ : Rel ℤ 0ℓ

+
x > y = y < x

+

+
_≰_ : Rel ℤ 0ℓ

+
x ≰ y = ¬ (x ≤ y)

+

+
_≱_ : Rel ℤ 0ℓ

+
x ≱ y = ¬ (x ≥ y)

+

+
_≮_ : Rel ℤ 0ℓ

+
x ≮ y = ¬ (x < y)

+

+
_≯_ : Rel ℤ 0ℓ

+
x ≯ y = ¬ (x > y)

+

+
------------------------------------------------------------------------

+
-- Boolean ordering

+

+
-- A boolean version.

+
_≤ᵇ_ : ℤ → ℤ → Bool

+
-[1+ m ] ≤ᵇ -[1+ n ] = n ℕ.≤ᵇ m

+
(+ m)    ≤ᵇ -[1+ n ] = false

+
-[1+ m ] ≤ᵇ (+ n)    = true

+
(+ m)    ≤ᵇ (+ n)    = m ℕ.≤ᵇ n

+

+
------------------------------------------------------------------------

+
-- Simple predicates

+

+
-- See `Data.Nat.Base` for a discussion on the design of these.

+

+
NonZero : Pred ℤ 0ℓ

+
NonZero i = ℕ.NonZero ∣ i ∣

+

+
record Positive (i : ℤ) : Set where

+
  field

+
    pos : T (1ℤ ≤ᵇ i)

+

+
record NonNegative (i : ℤ) : Set where

+
  field

+
    nonNeg : T (0ℤ ≤ᵇ i)

+

+
record NonPositive (i : ℤ) : Set where

+
  field

+
    nonPos : T (i ≤ᵇ 0ℤ)

+

+
record Negative (i : ℤ) : Set where

+
  field

+
    neg : T (i ≤ᵇ -1ℤ)

+

+
-- Instances

+

+
open ℕ public

+
  using (nonZero)

+

+
instance

+
  pos : ∀ {n} → Positive +[1+ n ]

+
  pos = _

+

+
  nonNeg : ∀ {n} → NonNegative (+ n)

+
  nonNeg = _

+

+
  nonPos0 : NonPositive 0ℤ

+
  nonPos0 = _

+

+
  nonPos : ∀ {n} → NonPositive -[1+ n ]

+
  nonPos = _

+

+
  neg : ∀ {n} → Negative -[1+ n ]

+
  neg = _

+

+
-- Constructors

+

+
≢-nonZero : ∀ {i} → i ≢ 0ℤ → NonZero i

+
≢-nonZero { +[1+ n ]} _   = _

+
≢-nonZero { +0}       0≢0 = contradiction refl 0≢0

+
≢-nonZero { -[1+ n ]} _   = _

+

+
>-nonZero : ∀ {i} → i > 0ℤ → NonZero i

+
>-nonZero (+<+ (s≤s m<n)) = _

+

+
<-nonZero : ∀ {i} → i < 0ℤ → NonZero i

+
<-nonZero -<+ = _

+

+
positive : ∀ {i} → i > 0ℤ → Positive i

+
positive (+<+ (s≤s m<n)) = _

+

+
negative : ∀ {i} → i < 0ℤ → Negative i

+
negative -<+ = _

+

+
nonPositive : ∀ {i} → i ≤ 0ℤ → NonPositive i

+
nonPositive -≤+       = _

+
nonPositive (+≤+ z≤n) = _

+

+
nonNegative : ∀ {i} → i ≥ 0ℤ → NonNegative i

+
nonNegative {+0}       _ = _

+
nonNegative {+[1+ n ]} _ = _

+

+
------------------------------------------------------------------------

+
-- A view of integers as sign + absolute value

+

+
infix 5 _◂_ _◃_

+

+
_◃_ : Sign → ℕ → ℤ

+
_      ◃ ℕ.zero  = +0

+
Sign.+ ◃ n       = + n

+
Sign.- ◃ ℕ.suc n = -[1+ n ]

+

+
data SignAbs : ℤ → Set where

+
  _◂_ : (s : Sign) (n : ℕ) → SignAbs (s ◃ n)

+

+
signAbs : ∀ i → SignAbs i

+
signAbs -[1+ n ] = Sign.- ◂ ℕ.suc n

+
signAbs +0       = Sign.+ ◂ ℕ.zero

+
signAbs +[1+ n ] = Sign.+ ◂ ℕ.suc n

+

+
------------------------------------------------------------------------

+
-- Arithmetic

+

+
-- Negation.

+

+
-_ : ℤ → ℤ

+
- -[1+ n ] = +[1+ n ]

+
- +0       = +0

+
- +[1+ n ] = -[1+ n ]

+

+
-- Subtraction of natural numbers.

+
-- We define it using _<ᵇ_ and _∸_ rather than inductively so that it

+
-- is backed by builtin operations. This makes it much faster.

+
_⊖_ : ℕ → ℕ → ℤ

+
m ⊖ n with m ℕ.<ᵇ n

+
... | true  = - + (n ℕ.∸ m)

+
... | false = + (m ℕ.∸ n)

+

+
-- Addition.

+

+
_+_ : ℤ → ℤ → ℤ

+
-[1+ m ] + -[1+ n ] = -[1+ ℕ.suc (m ℕ.+ n) ]

+
-[1+ m ] + +    n   = n ⊖ ℕ.suc m

+
+    m   + -[1+ n ] = m ⊖ ℕ.suc n

+
+    m   + +    n   = + (m ℕ.+ n)

+

+
-- Subtraction.

+

+
_-_ : ℤ → ℤ → ℤ

+
i - j = i + (- j)

+

+
-- Successor.

+

+
suc : ℤ → ℤ

+
suc i = 1ℤ + i

+

+
-- Predecessor.

+

+
pred : ℤ → ℤ

+
pred i = -1ℤ + i

+

+
-- Multiplication.

+

+
_*_ : ℤ → ℤ → ℤ

+
i * j = sign i Sign.* sign j ◃ ∣ i ∣ ℕ.* ∣ j ∣

+

+
-- Naïve exponentiation.

+

+
_^_ : ℤ → ℕ → ℤ

+
i ^ ℕ.zero    = 1ℤ

+
i ^ (ℕ.suc m) = i * i ^ m

+

+
-- Maximum.

+

+
_⊔_ : ℤ → ℤ → ℤ

+
-[1+ m ] ⊔ -[1+ n ] = -[1+ ℕ._⊓_ m n ]

+
-[1+ m ] ⊔ +    n   = + n

+
+    m   ⊔ -[1+ n ] = + m

+
+    m   ⊔ +    n   = + (ℕ._⊔_ m n)

+

+
-- Minimum.

+

+
_⊓_ : ℤ → ℤ → ℤ

+
-[1+ m ] ⊓ -[1+ n ] = -[1+ m ℕ.⊔ n ]

+
-[1+ m ] ⊓ +    n   = -[1+ m ]

+
+    m   ⊓ -[1+ n ] = -[1+ n ]

+
+    m   ⊓ +    n   = + (m ℕ.⊓ n)

+

+
-- Division by a natural

+

+
_/ℕ_ : (dividend : ℤ) (divisor : ℕ) .{{_ : ℕ.NonZero divisor}} → ℤ

+
(+ n      /ℕ d) = + (n ℕ./ d)

+
(-[1+ n ] /ℕ d) with ℕ.suc n ℕ.% d

+
... | ℕ.zero  = - (+ (ℕ.suc n ℕ./ d))

+
... | ℕ.suc r = -[1+ (ℕ.suc n ℕ./ d) ]

+

+
-- Division

+

+
_/_ : (dividend divisor : ℤ) .{{_ : NonZero divisor}} → ℤ

+
i / j = (sign j ◃ 1) * (i /ℕ ∣ j ∣)

+

+
-- Modulus by a natural

+

+
_%ℕ_ : (dividend : ℤ) (divisor : ℕ) .{{_ : ℕ.NonZero divisor}} → ℕ

+
(+ n      %ℕ d) = n ℕ.% d

+
(-[1+ n ] %ℕ d) with ℕ.suc n ℕ.% d

+
... | ℕ.zero      = 0

+
... | r@(ℕ.suc _) = d ℕ.∸ r

+

+
-- Modulus

+

+
_%_ : (dividend divisor : ℤ) .{{_ : NonZero divisor}} → ℕ

+
i % j = i %ℕ ∣ j ∣

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
+-rawMagma : RawMagma 0ℓ 0ℓ

+
+-rawMagma = record { _≈_ = _≡_ ; _∙_ = _+_ }

+

+
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ

+
+-0-rawMonoid = record { _≈_ = _≡_ ; _∙_ = _+_ ; ε = 0ℤ }

+

+
+-0-rawGroup : RawGroup 0ℓ 0ℓ

+
+-0-rawGroup = record { _≈_ = _≡_ ; _∙_ = _+_ ; _⁻¹ = -_; ε = 0ℤ }

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record { _≈_ = _≡_ ; _∙_ = _*_ }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record { _≈_ = _≡_ ; _∙_ = _*_ ; ε = 1ℤ }

+

+
+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ

+
+-*-rawNearSemiring = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0ℤ

+
  }

+

+
+-*-rawSemiring : RawSemiring 0ℓ 0ℓ

+
+-*-rawSemiring = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0ℤ

+
  ; 1# = 1ℤ

+
  }

+

+
+-*-rawRing : RawRing 0ℓ 0ℓ

+
+-*-rawRing = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; -_ = -_

+
  ; 0# = 0ℤ

+
  ; 1# = 1ℤ

+
  }

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Coprimality.html b/site/static/agda_html/Data.Integer.Coprimality.html
new file mode 100644
index 000000000..522c5a2f2
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Coprimality.html
@@ -0,0 +1,116 @@
+
+Data.Integer.Coprimality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Coprimality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Coprimality where

+

+
open import Data.Integer.Base

+
open import Data.Integer.Divisibility

+
open import Data.Integer.Properties

+
import Data.Nat.Coprimality as ℕ

+
import Data.Nat.Divisibility as ℕ

+
open import Function.Base using (_on_)

+
open import Level using (0ℓ)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions using (Decidable; Symmetric)

+
open import Relation.Binary.PropositionalEquality.Core using (subst)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
Coprime : Rel ℤ 0ℓ

+
Coprime = ℕ.Coprime on ∣_∣

+

+
------------------------------------------------------------------------

+
-- Properties of coprimality

+

+
sym : Symmetric Coprime

+
sym = ℕ.sym

+

+
coprime? : Decidable Coprime

+
coprime? x y = ℕ.coprime? ∣ x ∣ ∣ y ∣

+

+
coprime-divisor : ∀ i j k → Coprime i j → i ∣ j * k → i ∣ k

+
coprime-divisor i j k c eq =

+
  ℕ.coprime-divisor c (subst (∣ i ∣ ℕ.∣_ ) (abs-* j k) eq)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Divisibility.html b/site/static/agda_html/Data.Integer.Divisibility.html
new file mode 100644
index 000000000..f36b6bb1c
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Divisibility.html
@@ -0,0 +1,132 @@
+
+Data.Integer.Divisibility
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
-----------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Unsigned divisibility

+
------------------------------------------------------------------------

+

+
-- For signed divisibility see `Data.Integer.Divisibility.Signed`

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Divisibility where

+

+
open import Function.Base using (_on_; _$_)

+
open import Data.Integer.Base

+
open import Data.Integer.Properties

+
import Data.Nat.Base as ℕ

+
import Data.Nat.Divisibility as ℕ

+
open import Level

+
open import Relation.Binary.Core using (Rel; _Preserves_⟶_)

+

+

+
------------------------------------------------------------------------

+
-- Divisibility

+

+
infix 4 _∣_

+

+
_∣_ : Rel ℤ 0ℓ

+
_∣_ = ℕ._∣_ on ∣_∣

+

+
pattern divides k eq = ℕ.divides k eq

+

+
------------------------------------------------------------------------

+
-- Properties of divisibility

+

+
*-monoʳ-∣ : ∀ k → (k *_) Preserves _∣_ ⟶ _∣_

+
*-monoʳ-∣ k {i} {j} i∣j = begin

+
  ∣ k * i ∣       ≡⟨ abs-* k i ⟩

+
  ∣ k ∣ ℕ.* ∣ i ∣ ∣⟨ ℕ.*-monoʳ-∣ ∣ k ∣ i∣j ⟩

+
  ∣ k ∣ ℕ.* ∣ j ∣ ≡⟨ abs-* k j ⟨

+
  ∣ k * j ∣       ∎

+
  where open ℕ.∣-Reasoning

+

+
*-monoˡ-∣ : ∀ k → (_* k) Preserves _∣_ ⟶ _∣_

+
*-monoˡ-∣ k {i} {j} rewrite *-comm i k | *-comm j k = *-monoʳ-∣ k

+

+
*-cancelˡ-∣ : ∀ k {i j} .{{_ : NonZero k}} → k * i ∣ k * j → i ∣ j

+
*-cancelˡ-∣ k {i} {j} k*i∣k*j = ℕ.*-cancelˡ-∣ ∣ k ∣ $ begin

+
  ∣ k ∣ ℕ.* ∣ i ∣  ≡⟨ abs-* k i ⟨

+
  ∣ k * i ∣        ∣⟨ k*i∣k*j ⟩

+
  ∣ k * j ∣        ≡⟨ abs-* k j ⟩

+
  ∣ k ∣ ℕ.* ∣ j ∣  ∎

+
  where open ℕ.∣-Reasoning

+

+
*-cancelʳ-∣ : ∀ k {i j} .{{_ : NonZero k}} → i * k ∣ j * k → i ∣ j

+
*-cancelʳ-∣ k {i} {j} rewrite *-comm i k | *-comm j k = *-cancelˡ-∣ k

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.GCD.html b/site/static/agda_html/Data.Integer.GCD.html
new file mode 100644
index 000000000..985016521
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.GCD.html
@@ -0,0 +1,140 @@
+
+Data.Integer.GCD
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Greatest Common Divisor for integers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.GCD where

+

+
open import Data.Integer.Base

+
open import Data.Integer.Divisibility

+
open import Data.Integer.Properties

+
import Data.Nat.GCD as ℕ

+
open import Data.Product.Base using (_,_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)

+

+
open import Algebra.Definitions {A = ℤ} _≡_ as Algebra

+
  using (Associative; Commutative; LeftIdentity; RightIdentity; LeftZero; RightZero; Zero)

+

+
------------------------------------------------------------------------

+
-- Definition

+
------------------------------------------------------------------------

+

+
gcd : ℤ → ℤ → ℤ

+
gcd i j = + ℕ.gcd ∣ i ∣ ∣ j ∣

+

+
------------------------------------------------------------------------

+
-- Properties

+
------------------------------------------------------------------------

+

+
gcd[i,j]∣i : ∀ i j → gcd i j ∣ i

+
gcd[i,j]∣i i j = ℕ.gcd[m,n]∣m ∣ i ∣ ∣ j ∣

+

+
gcd[i,j]∣j : ∀ i j → gcd i j ∣ j

+
gcd[i,j]∣j i j = ℕ.gcd[m,n]∣n ∣ i ∣ ∣ j ∣

+

+
gcd-greatest : ∀ {i j c} → c ∣ i → c ∣ j → c ∣ gcd i j

+
gcd-greatest c∣i c∣j = ℕ.gcd-greatest c∣i c∣j

+

+
gcd[0,0]≡0 : gcd 0ℤ 0ℤ ≡ 0ℤ

+
gcd[0,0]≡0 = cong (+_) ℕ.gcd[0,0]≡0

+

+
gcd[i,j]≡0⇒i≡0 : ∀ i j → gcd i j ≡ 0ℤ → i ≡ 0ℤ

+
gcd[i,j]≡0⇒i≡0 i j eq = ∣i∣≡0⇒i≡0 (ℕ.gcd[m,n]≡0⇒m≡0 (+-injective eq))

+

+
gcd[i,j]≡0⇒j≡0 : ∀ {i j} → gcd i j ≡ 0ℤ → j ≡ 0ℤ

+
gcd[i,j]≡0⇒j≡0 {i} eq = ∣i∣≡0⇒i≡0 (ℕ.gcd[m,n]≡0⇒n≡0 ∣ i ∣ (+-injective eq))

+

+
gcd-comm : Commutative gcd

+
gcd-comm i j = cong (+_) (ℕ.gcd-comm ∣ i ∣ ∣ j ∣)

+

+
gcd-assoc : Associative gcd

+
gcd-assoc i j k = cong (+_) (ℕ.gcd-assoc ∣ i ∣ ∣ j ∣ (∣ k ∣))

+

+
gcd-zeroˡ : LeftZero 1ℤ gcd

+
gcd-zeroˡ i = cong (+_) (ℕ.gcd-zeroˡ ∣ i ∣)

+

+
gcd-zeroʳ : RightZero 1ℤ gcd

+
gcd-zeroʳ i = cong (+_) (ℕ.gcd-zeroʳ ∣ i ∣)

+

+
gcd-zero : Zero 1ℤ gcd

+
gcd-zero = gcd-zeroˡ , gcd-zeroʳ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Properties.NatLemmas.html b/site/static/agda_html/Data.Integer.Properties.NatLemmas.html
new file mode 100644
index 000000000..fdc820752
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Properties.NatLemmas.html
@@ -0,0 +1,146 @@
+
+Data.Integer.Properties.NatLemmas
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some extra lemmas about natural numbers only needed for

+
-- Data.Integer.Properties (for distributivity)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Properties.NatLemmas where

+

+
open import Data.Nat.Base using (ℕ; _+_; _*_; suc)

+
open import Data.Nat.Properties

+
  using (*-distribʳ-+; *-assoc; +-assoc; +-comm; +-suc)

+
open import Function.Base using (_∘_)

+
open import Relation.Binary.PropositionalEquality

+
  using (_≡_; cong; cong₂; sym; module ≡-Reasoning)

+

+
open ≡-Reasoning

+

+
inner-assoc : ∀ m n o → o + (n + m * suc n) * suc o

+
                ≡ o + n * suc o + m * suc (o + n * suc o)

+
inner-assoc m n o = begin

+
  o + (n + m * suc n) * suc o                ≡⟨ cong (o +_) (begin

+
    (n + m * suc n) * suc o             ≡⟨ *-distribʳ-+ (suc o) n (m * suc n) ⟩

+
    n * suc o + m * suc n * suc o       ≡⟨ cong (n * suc o +_) (*-assoc m (suc n) (suc o)) ⟩

+
    n * suc o + m * suc (o + n * suc o) ∎) ⟩

+
  o + (n * suc o + m * suc (o + n * suc o))  ≡⟨ +-assoc o _ _ ⟨

+
  o + n * suc o + m * suc (o + n * suc o)    ∎

+

+
private

+
  assoc-comm : ∀ a b c d → a + b + c + d ≡ (a + c) + (b + d)

+
  assoc-comm a b c d = begin

+
    a + b + c + d     ≡⟨ cong (_+ d) (+-assoc a b c) ⟩

+
    a + (b + c) + d   ≡⟨ cong (λ z → a + z + d) (+-comm b c) ⟩

+
    a + (c + b) + d   ≡⟨ cong (_+ d) (+-assoc a c b) ⟨

+
    (a + c) + b + d   ≡⟨ +-assoc (a + c) b d ⟩

+
    (a + c) + (b + d) ∎

+

+
  assoc-comm′ : ∀ a b c d → a + (b + (c + d)) ≡ a + c + (b + d)

+
  assoc-comm′ a b c d = begin

+
    a + (b + (c + d)) ≡⟨ cong (a +_) (+-assoc b c d) ⟨

+
    a + (b + c + d)   ≡⟨ cong (λ z → a + (z + d)) (+-comm b c) ⟩

+
    a + (c + b + d)   ≡⟨ cong (a +_) (+-assoc c b d) ⟩

+
    a + (c + (b + d)) ≡⟨ +-assoc a c _ ⟨

+
    a + c + (b + d)   ∎

+

+
assoc₁ : ∀ m n o → (2 + n + o) * (1 + m) ≡ (1 + n) * (1 + m) + (1 + o) * (1 + m)

+
assoc₁ m n o = begin

+
  (2 + n + o) * (1 + m)                  ≡⟨ cong (_* (1 + m)) (assoc-comm 1 1 n o) ⟩

+
  ((1 + n) + (1 + o)) * (1 + m)          ≡⟨ *-distribʳ-+ (1 + m) (1 + n) (1 + o) ⟩

+
  (1 + n) * (1 + m) + (1 + o) * (1 + m)   ∎

+

+
assoc₂ : ∀ m n o → (1 + n + (1 + o)) * (1 + m) ≡ (1 + n) * (1 + m) + (1 + o) * (1 + m)

+
assoc₂ m n o = *-distribʳ-+ (1 + m) (1 + n) (1 + o)

+

+
assoc₃ : ∀ m n o → m + (n + (1 + o)) * (1 + m) ≡ (1 + n) * (1 + m) + (m + o * (1 + m))

+
assoc₃ m n o = begin

+
  m + (n + (1 + o)) * (1 + m)           ≡⟨ cong (m +_) (*-distribʳ-+ (1 + m) n (1 + o)) ⟩

+
  m + (n * (1 + m) + (1 + o) * (1 + m)) ≡⟨ +-assoc m _ _ ⟨

+
  (m + n * (1 + m)) + (1 + o) * (1 + m) ≡⟨ +-suc _ _ ⟩

+
  (1 + n) * (1 + m) + (m + o * (1 + m)) ∎

+

+
assoc₄ : ∀ m n o → m + (1 + m + (n + o) * (1 + m)) ≡ (1 + n) * (1 + m) + (m + o * (1 + m))

+
assoc₄ m n o = begin

+
  m + (1 + m + (n + o) * (1 + m))           ≡⟨ +-suc _ _ ⟩

+
  1 + m + (m + (n + o) * (1 + m))           ≡⟨ cong (λ z → suc (m + (m + z))) (*-distribʳ-+ (suc m) n o) ⟩

+
  1 + m + (m + (n * (1 + m) + o * (1 + m))) ≡⟨ cong suc (assoc-comm′ m m _ _) ⟩

+
  (1 + n) * (1 + m) + (m + o * (1 + m))     ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Properties.html b/site/static/agda_html/Data.Integer.Properties.html
new file mode 100644
index 000000000..7babe26c4
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Properties.html
@@ -0,0 +1,2459 @@
+
+Data.Integer.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some properties about integers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Properties where

+

+
open import Algebra.Bundles

+
import Algebra.Morphism as Morphism

+
open import Algebra.Construct.NaturalChoice.Base

+
import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp

+
import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp

+
import Algebra.Properties.AbelianGroup

+
open import Data.Bool.Base using (T; true; false)

+
open import Data.Integer.Base renaming (suc to sucℤ)

+
open import Data.Integer.Properties.NatLemmas

+
open import Data.Nat.Base as ℕ

+
  using (ℕ; suc; zero; _∸_; s≤s; z≤n; s<s; z<s; s≤s⁻¹; s<s⁻¹)

+
  hiding (module ℕ)

+
import Data.Nat.Properties as ℕ

+
open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Data.Sign.Base as Sign using (Sign)

+
import Data.Sign.Properties as Sign

+
open import Function.Base using (_∘_; _$_; id)

+
open import Level using (0ℓ)

+
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.Bundles using

+
  (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; LeftTrans; RightTrans; Trichotomous; tri≈; tri<; tri>)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂; sym; _≢_; subst; resp₂; trans)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning; setoid; decSetoid; isEquivalence)

+
open import Relation.Nullary.Decidable.Core using (yes; no)

+
import Relation.Nullary.Reflects as Reflects

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
import Relation.Nullary.Decidable as Dec

+

+
open import Algebra.Definitions {A = ℤ} _≡_

+
open import Algebra.Consequences.Propositional

+
open import Algebra.Structures {A = ℤ} _≡_

+
module ℤtoℕ = Morphism.Definitions ℤ ℕ _≡_

+
module ℕtoℤ = Morphism.Definitions ℕ ℤ _≡_

+

+
private

+
  variable

+
    m n o : ℕ

+
    i j k : ℤ

+
    s t   : Sign

+

+
------------------------------------------------------------------------

+
-- Equality

+
------------------------------------------------------------------------

+

+
+-injective : + m ≡ + n → m ≡ n

+
+-injective refl = refl

+

+
-[1+-injective : -[1+ m ] ≡ -[1+ n ] → m ≡ n

+
-[1+-injective refl = refl

+

+
+[1+-injective : +[1+ m ] ≡ +[1+ n ] → m ≡ n

+
+[1+-injective refl = refl

+

+
infix 4 _≟_

+
_≟_ : DecidableEquality ℤ

+
+ m      ≟ + n      = Dec.map′ (cong (+_)) +-injective (m ℕ.≟ n)

+
+ m      ≟ -[1+ n ] = no λ()

+
-[1+ m ] ≟ + n      = no λ()

+
-[1+ m ] ≟ -[1+ n ] = Dec.map′ (cong -[1+_]) -[1+-injective (m ℕ.≟ n)

+

+
≡-setoid : Setoid 0ℓ 0ℓ

+
≡-setoid = setoid ℤ

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+
------------------------------------------------------------------------

+

+
drop‿+≤+ : + m ≤ + n → m ℕ.≤ n

+
drop‿+≤+ (+≤+ m≤n) = m≤n

+

+
drop‿-≤- : -[1+ m ] ≤ -[1+ n ] → n ℕ.≤ m

+
drop‿-≤- (-≤- n≤m) = n≤m

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
≤-reflexive : _≡_ ⇒ _≤_

+
≤-reflexive { -[1+ n ]} refl = -≤- ℕ.≤-refl

+
≤-reflexive {+ n}       refl = +≤+ ℕ.≤-refl

+

+
≤-refl : Reflexive _≤_

+
≤-refl = ≤-reflexive refl

+

+
≤-trans : Transitive _≤_

+
≤-trans -≤+       (+≤+ n≤m) = -≤+

+
≤-trans (-≤- n≤m) -≤+       = -≤+

+
≤-trans (-≤- n≤m) (-≤- k≤n) = -≤- (ℕ.≤-trans k≤n n≤m)

+
≤-trans (+≤+ m≤n) (+≤+ n≤k) = +≤+ (ℕ.≤-trans m≤n n≤k)

+

+
≤-antisym : Antisymmetric _≡_ _≤_

+
≤-antisym (-≤- n≤m) (-≤- m≤n) = cong -[1+_] $ ℕ.≤-antisym m≤n n≤m

+
≤-antisym (+≤+ m≤n) (+≤+ n≤m) = cong (+_)   $ ℕ.≤-antisym m≤n n≤m

+

+
≤-total : Total _≤_

+
≤-total (-[1+ m ]) (-[1+ n ]) = Sum.map -≤- -≤- (ℕ.≤-total n m)

+
≤-total (-[1+ m ]) (+    n  ) = inj₁ -≤+

+
≤-total (+    m  ) (-[1+ n ]) = inj₂ -≤+

+
≤-total (+    m  ) (+    n  ) = Sum.map +≤+ +≤+ (ℕ.≤-total m n)

+

+
infix  4 _≤?_

+
_≤?_ : Decidable _≤_

+
-[1+ m ] ≤? -[1+ n ] = Dec.map′ -≤- drop‿-≤- (n ℕ.≤? m)

+
-[1+ m ] ≤? +    n   = yes -≤+

+
+    m   ≤? -[1+ n ] = no λ ()

+
+    m   ≤? +    n   = Dec.map′ +≤+ drop‿+≤+ (m ℕ.≤? n)

+

+
≤-irrelevant : Irrelevant _≤_

+
≤-irrelevant -≤+       -≤+         = refl

+
≤-irrelevant (-≤- n≤m₁) (-≤- n≤m₂) = cong -≤- (ℕ.≤-irrelevant n≤m₁ n≤m₂)

+
≤-irrelevant (+≤+ n≤m₁) (+≤+ n≤m₂) = cong +≤+ (ℕ.≤-irrelevant n≤m₁ n≤m₂)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≤-isPreorder : IsPreorder _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isTotalPreorder : IsTotalPreorder _≡_ _≤_

+
≤-isTotalPreorder = record

+
  { isPreorder = ≤-isPreorder

+
  ; total      = ≤-total

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder _≡_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≤-preorder : Preorder 0ℓ 0ℓ 0ℓ

+
≤-preorder = record

+
  { isPreorder = ≤-isPreorder

+
  }

+

+
≤-totalPreorder : TotalPreorder 0ℓ 0ℓ 0ℓ

+
≤-totalPreorder = record

+
  { isTotalPreorder = ≤-isTotalPreorder

+
  }

+

+
≤-poset : Poset 0ℓ 0ℓ 0ℓ

+
≤-poset = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  }

+

+
≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ

+
≤-totalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  }

+

+
≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ

+
≤-decTotalOrder = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _≤ᵇ_

+
------------------------------------------------------------------------

+

+
≤ᵇ⇒≤ : T (i ≤ᵇ j) → i ≤ j

+
≤ᵇ⇒≤ {+ _}       {+ _}       i≤j = +≤+ (ℕ.≤ᵇ⇒≤ _ _ i≤j)

+
≤ᵇ⇒≤ { -[1+ _ ]} {+ _}       i≤j = -≤+

+
≤ᵇ⇒≤ { -[1+ _ ]} { -[1+ _ ]} i≤j = -≤- (ℕ.≤ᵇ⇒≤ _ _ i≤j)

+

+
≤⇒≤ᵇ : i ≤ j → T (i ≤ᵇ j)

+
≤⇒≤ᵇ (-≤- n≤m) = ℕ.≤⇒≤ᵇ n≤m

+
≤⇒≤ᵇ -≤+ = _

+
≤⇒≤ᵇ (+≤+ m≤n) = ℕ.≤⇒≤ᵇ m≤n

+

+
------------------------------------------------------------------------

+
-- Properties _<_

+
------------------------------------------------------------------------

+

+
drop‿+<+ : + m < + n → m ℕ.< n

+
drop‿+<+ (+<+ m<n) = m<n

+

+
drop‿-<- : -[1+ m ] < -[1+ n ] → n ℕ.< m

+
drop‿-<- (-<- n<m) = n<m

+

+
+≮0 : + n ≮ +0

+
+≮0 (+<+ ())

+

+
+≮- : + m ≮ -[1+ n ]

+
+≮- ()

+

+
------------------------------------------------------------------------

+
-- Relationship between other operators

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ (-<- i<j) = -≤- (ℕ.<⇒≤ i<j)

+
<⇒≤ -<+       = -≤+

+
<⇒≤ (+<+ i<j) = +≤+ (ℕ.<⇒≤ i<j)

+

+
<⇒≢ : _<_ ⇒ _≢_

+
<⇒≢ (-<- n<m) refl = ℕ.<⇒≢ n<m refl

+
<⇒≢ (+<+ m<n) refl = ℕ.<⇒≢ m<n refl

+

+
<⇒≱ : _<_ ⇒ _≱_

+
<⇒≱ (-<- n<m) = ℕ.<⇒≱ n<m ∘ drop‿-≤-

+
<⇒≱ (+<+ m<n) = ℕ.<⇒≱ m<n ∘ drop‿+≤+

+

+
≤⇒≯ : _≤_ ⇒ _≯_

+
≤⇒≯ (-≤- n≤m) (-<- n<m) = ℕ.≤⇒≯ n≤m n<m

+
≤⇒≯ -≤+ = +≮-

+
≤⇒≯ (+≤+ m≤n) (+<+ m<n) = ℕ.≤⇒≯ m≤n m<n

+

+
≰⇒> : _≰_ ⇒ _>_

+
≰⇒> {+ n}       {+_ n₁}      i≰j = +<+ (ℕ.≰⇒> (i≰j ∘ +≤+))

+
≰⇒> {+ n}       { -[1+_] n₁} i≰j = -<+

+
≰⇒> { -[1+_] n} {+_ n₁}      i≰j = contradiction -≤+ i≰j

+
≰⇒> { -[1+_] n} { -[1+_] n₁} i≰j = -<- (ℕ.≰⇒> (i≰j ∘ -≤-))

+

+
≮⇒≥ : _≮_ ⇒ _≥_

+
≮⇒≥ {+ i}       {+ j}       i≮j = +≤+ (ℕ.≮⇒≥ (i≮j ∘ +<+))

+
≮⇒≥ {+ i}       { -[1+_] j} i≮j = -≤+

+
≮⇒≥ { -[1+_] i} {+ j}       i≮j = contradiction -<+ i≮j

+
≮⇒≥ { -[1+_] i} { -[1+_] j} i≮j = -≤- (ℕ.≮⇒≥ (i≮j ∘ -<-))

+

+
>⇒≰ : _>_ ⇒ _≰_

+
>⇒≰ = <⇒≱

+

+
≤∧≢⇒< : i ≤ j → i ≢ j → i < j

+
≤∧≢⇒< (-≤- m≤n) i≢j = -<- (ℕ.≤∧≢⇒< m≤n (i≢j ∘ cong -[1+_] ∘ sym))

+
≤∧≢⇒< -≤+  i≢j      = -<+

+
≤∧≢⇒< (+≤+ n≤m) i≢j = +<+ (ℕ.≤∧≢⇒< n≤m (i≢j ∘ cong (+_)))

+

+
≤∧≮⇒≡ : i ≤ j → i ≮ j → i ≡ j

+
≤∧≮⇒≡ i≤j i≮j = ≤-antisym i≤j (≮⇒≥ i≮j)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
<-irrefl : Irreflexive _≡_ _<_

+
<-irrefl { -[1+ n ]} refl = ℕ.<-irrefl refl ∘ drop‿-<-

+
<-irrefl { +0}       refl (+<+ ())

+
<-irrefl { +[1+ n ]} refl = ℕ.<-irrefl refl ∘ drop‿+<+

+

+
<-asym : Asymmetric _<_

+
<-asym (-<- n<m) = ℕ.<-asym n<m ∘ drop‿-<-

+
<-asym (+<+ m<n) = ℕ.<-asym m<n ∘ drop‿+<+

+

+
≤-<-trans : LeftTrans _≤_ _<_

+
≤-<-trans (-≤- n≤m) (-<- o<n) = -<- (ℕ.<-≤-trans o<n n≤m)

+
≤-<-trans (-≤- n≤m) -<+       = -<+

+
≤-<-trans -≤+       (+<+ m<o) = -<+

+
≤-<-trans (+≤+ m≤n) (+<+ n<o) = +<+ (ℕ.≤-<-trans m≤n n<o)

+

+
<-≤-trans : RightTrans _<_ _≤_

+
<-≤-trans (-<- n<m) (-≤- o≤n) = -<- (ℕ.≤-<-trans o≤n n<m)

+
<-≤-trans (-<- n<m) -≤+       = -<+

+
<-≤-trans -<+       (+≤+ m≤n) = -<+

+
<-≤-trans (+<+ m<n) (+≤+ n≤o) = +<+ (ℕ.<-≤-trans m<n n≤o)

+

+
<-trans : Transitive _<_

+
<-trans m<n n<p = ≤-<-trans (<⇒≤ m<n) n<p

+

+
<-cmp : Trichotomous _≡_ _<_

+
<-cmp +0       +0       = tri≈ +≮0 refl +≮0

+
<-cmp +0       +[1+ n ] = tri< (+<+ z<s) (λ()) +≮0

+
<-cmp +[1+ n ] +0       = tri> +≮0 (λ()) (+<+ z<s)

+
<-cmp (+ m)    -[1+ n ] = tri> +≮- (λ()) -<+

+
<-cmp -[1+ m ] (+ n)    = tri< -<+ (λ()) +≮-

+
<-cmp -[1+ m ] -[1+ n ] with ℕ.<-cmp m n

+
... | tri< m<n m≢n n≯m = tri> (n≯m ∘ drop‿-<-) (m≢n ∘ -[1+-injective) (-<- m<n)

+
... | tri≈ m≮n m≡n n≯m = tri≈ (n≯m ∘ drop‿-<-) (cong -[1+_] m≡n) (m≮n ∘ drop‿-<-)

+
... | tri> m≮n m≢n n>m = tri< (-<- n>m) (m≢n ∘ -[1+-injective) (m≮n ∘ drop‿-<-)

+
<-cmp +[1+ m ] +[1+ n ] with ℕ.<-cmp m n

+
... | tri< m<n m≢n n≯m = tri< (+<+ (s<s m<n))              (m≢n ∘ +[1+-injective) (n≯m ∘ s<s⁻¹ ∘ drop‿+<+)

+
... | tri≈ m≮n m≡n n≯m = tri≈ (m≮n ∘ s<s⁻¹ ∘ drop‿+<+) (cong (+_ ∘ suc) m≡n)  (n≯m ∘ s<s⁻¹ ∘ drop‿+<+)

+
... | tri> m≮n m≢n n>m = tri> (m≮n ∘ s<s⁻¹ ∘ drop‿+<+) (m≢n ∘ +[1+-injective) (+<+ (s<s n>m))

+

+
infix 4 _<?_

+
_<?_ : Decidable _<_

+
-[1+ m ] <? -[1+ n ] = Dec.map′ -<- drop‿-<- (n ℕ.<? m)

+
-[1+ m ] <? + n      = yes -<+

+
+ m      <? -[1+ n ] = no λ()

+
+ m      <? + n      = Dec.map′ +<+ drop‿+<+ (m ℕ.<? n)

+

+
<-irrelevant : Irrelevant _<_

+
<-irrelevant (-<- n<m₁) (-<- n<m₂) = cong -<- (ℕ.<-irrelevant n<m₁ n<m₂)

+
<-irrelevant -<+       -<+         = refl

+
<-irrelevant (+<+ m<n₁) (+<+ m<n₂) = cong +<+ (ℕ.<-irrelevant m<n₁ m<n₂)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = subst (_ <_) , subst (_< _)

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _<_

+

+
i≮i : i ≮ i

+
i≮i = <-irrefl refl

+

+
>-irrefl : Irreflexive _≡_ _>_

+
>-irrefl = <-irrefl ∘ sym

+

+
------------------------------------------------------------------------

+
-- A specialised module for reasoning about the _≤_ and _<_ relations

+
------------------------------------------------------------------------

+

+
module ≤-Reasoning where

+
  open import Relation.Binary.Reasoning.Base.Triple

+
    ≤-isPreorder

+
    <-asym

+
    <-trans

+
    (resp₂ _<_)

+
    <⇒≤

+
    <-≤-trans

+
    ≤-<-trans

+
    public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+

+
------------------------------------------------------------------------

+
-- Properties of Positive/NonPositive/Negative/NonNegative and _≤_/_<_

+

+
positive⁻¹ : ∀ i → .{{Positive i}} → i > 0ℤ

+
positive⁻¹ +[1+ n ] = +<+ z<s

+

+
negative⁻¹ : ∀ i → .{{Negative i}} → i < 0ℤ

+
negative⁻¹ -[1+ n ] = -<+

+

+
nonPositive⁻¹ : ∀ i → .{{NonPositive i}} → i ≤ 0ℤ

+
nonPositive⁻¹ +0       = +≤+ z≤n

+
nonPositive⁻¹ -[1+ n ] = -≤+

+

+
nonNegative⁻¹ : ∀ i → .{{NonNegative i}} → i ≥ 0ℤ

+
nonNegative⁻¹ (+ n) = +≤+ z≤n

+

+
negative<positive : ∀ i j → .{{Negative i}} → .{{Positive j}} → i < j

+
negative<positive i j = <-trans (negative⁻¹ i) (positive⁻¹ j)

+

+
------------------------------------------------------------------------

+
-- Properties of -_

+
------------------------------------------------------------------------

+

+
neg-involutive : ∀ i → - - i ≡ i

+
neg-involutive -[1+ n ] = refl

+
neg-involutive +0       = refl

+
neg-involutive +[1+ n ] = refl

+

+
neg-injective : - i ≡ - j → i ≡ j

+
neg-injective {i} {j} -i≡-j = begin

+
  i     ≡⟨ neg-involutive i ⟨

+
  - - i ≡⟨  cong -_ -i≡-j ⟩

+
  - - j ≡⟨  neg-involutive j ⟩

+
  j     ∎ where open ≡-Reasoning

+

+
neg-≤-pos : ∀ {m n} → - (+ m) ≤ + n

+
neg-≤-pos {zero}  = +≤+ z≤n

+
neg-≤-pos {suc m} = -≤+

+

+
neg-mono-≤ : -_ Preserves _≤_ ⟶ _≥_

+
neg-mono-≤ -≤+             = neg-≤-pos

+
neg-mono-≤ (-≤- n≤m)       = +≤+ (s≤s n≤m)

+
neg-mono-≤ (+≤+ z≤n)       = neg-≤-pos

+
neg-mono-≤ (+≤+ (s≤s m≤n)) = -≤- m≤n

+

+
neg-cancel-≤ : - i ≤ - j → i ≥ j

+
neg-cancel-≤ { +[1+ m ]} { +[1+ n ]} (-≤- n≤m)        = +≤+ (s≤s n≤m)

+
neg-cancel-≤ { +[1+ m ]} { +0}        -≤+             = +≤+ z≤n

+
neg-cancel-≤ { +[1+ m ]} { -[1+ n ]}  -≤+             = -≤+

+
neg-cancel-≤ { +0}       { +0}        _               = +≤+ z≤n

+
neg-cancel-≤ { +0}       { -[1+ n ]}  _               = -≤+

+
neg-cancel-≤ { -[1+ m ]} { +0}        (+≤+ ())

+
neg-cancel-≤ { -[1+ m ]} { -[1+ n ]}  (+≤+ (s≤s m≤n)) = -≤- m≤n

+

+
neg-mono-< : -_ Preserves _<_ ⟶ _>_

+
neg-mono-< { -[1+ _ ]} { -[1+ _ ]} (-<- n<m) = +<+ (s<s n<m)

+
neg-mono-< { -[1+ _ ]} { +0}       -<+       = +<+ z<s

+
neg-mono-< { -[1+ _ ]} { +[1+ n ]} -<+       = -<+

+
neg-mono-< { +0}       { +[1+ n ]} (+<+ _)   = -<+

+
neg-mono-< { +[1+ m ]} { +[1+ n ]} (+<+ m<n) = -<- (s<s⁻¹ m<n)

+

+
neg-cancel-< : - i < - j → i > j

+
neg-cancel-< { +[1+ m ]} { +[1+ n ]} (-<- n<m)       = +<+ (s<s n<m)

+
neg-cancel-< { +[1+ m ]} { +0}        -<+            = +<+ z<s

+
neg-cancel-< { +[1+ m ]} { -[1+ n ]}  -<+            = -<+

+
neg-cancel-< { +0}       { +0}       (+<+ ())

+
neg-cancel-< { +0}       { -[1+ n ]} _               = -<+

+
neg-cancel-< { -[1+ m ]} { +0}       (+<+ ())

+
neg-cancel-< { -[1+ m ]} { -[1+ n ]} (+<+ m<n) = -<- (s<s⁻¹ m<n)

+

+
------------------------------------------------------------------------

+
-- Properties of ∣_∣

+
------------------------------------------------------------------------

+

+
∣i∣≡0⇒i≡0 : ∣ i ∣ ≡ 0 → i ≡ + 0

+
∣i∣≡0⇒i≡0 {+0} refl = refl

+

+
∣-i∣≡∣i∣ : ∀ i → ∣ - i ∣ ≡ ∣ i ∣

+
∣-i∣≡∣i∣ -[1+ n ] = refl

+
∣-i∣≡∣i∣ +0       = refl

+
∣-i∣≡∣i∣ +[1+ n ] = refl

+

+
0≤i⇒+∣i∣≡i : 0ℤ ≤ i → + ∣ i ∣ ≡ i

+
0≤i⇒+∣i∣≡i (+≤+ _) = refl

+

+
+∣i∣≡i⇒0≤i : + ∣ i ∣ ≡ i → 0ℤ ≤ i

+
+∣i∣≡i⇒0≤i {+ n} _ = +≤+ z≤n

+

+
+∣i∣≡i⊎+∣i∣≡-i : ∀ i → + ∣ i ∣ ≡ i ⊎ + ∣ i ∣ ≡ - i

+
+∣i∣≡i⊎+∣i∣≡-i (+ n)      = inj₁ refl

+
+∣i∣≡i⊎+∣i∣≡-i (-[1+ n ]) = inj₂ refl

+

+
∣m⊝n∣≤m⊔n : ∀ m n → ∣ m ⊖ n ∣ ℕ.≤ m ℕ.⊔ n

+
∣m⊝n∣≤m⊔n m n with m ℕ.<ᵇ n

+
... | true = begin

+
  ∣ - + (n ℕ.∸ m) ∣                ≡⟨ ∣-i∣≡∣i∣ (+ (n ℕ.∸ m)) ⟩

+
  ∣ + (n ℕ.∸ m) ∣                  ≡⟨⟩

+
  n ℕ.∸ m                          ≤⟨ ℕ.m∸n≤m n m ⟩

+
  n                                ≤⟨ ℕ.m≤n⊔m m n ⟩

+
  m ℕ.⊔ n                          ∎

+
  where open ℕ.≤-Reasoning

+
... | false = begin

+
  ∣ + (m ℕ.∸ n) ∣                  ≡⟨⟩

+
  m ℕ.∸ n                          ≤⟨ ℕ.m∸n≤m m n ⟩

+
  m                                ≤⟨ ℕ.m≤m⊔n m n ⟩

+
  m ℕ.⊔ n                          ∎

+
  where open ℕ.≤-Reasoning

+

+
∣i+j∣≤∣i∣+∣j∣ : ∀ i j → ∣ i + j ∣ ℕ.≤ ∣ i ∣ ℕ.+ ∣ j ∣

+
∣i+j∣≤∣i∣+∣j∣ +[1+ m ] (+ n)    = ℕ.≤-refl

+
∣i+j∣≤∣i∣+∣j∣ +0       (+ n)    = ℕ.≤-refl

+
∣i+j∣≤∣i∣+∣j∣ +0       -[1+ n ] = ℕ.≤-refl

+
∣i+j∣≤∣i∣+∣j∣ -[1+ m ] -[1+ n ] rewrite ℕ.+-suc (suc m) n = ℕ.≤-refl

+
∣i+j∣≤∣i∣+∣j∣ +[1+ m ] -[1+ n ] = begin

+
  ∣ suc m ⊖ suc n ∣  ≤⟨ ∣m⊝n∣≤m⊔n (suc m) (suc n) ⟩

+
  suc m ℕ.⊔ suc n    ≤⟨ ℕ.m⊔n≤m+n (suc m) (suc n) ⟩

+
  suc m ℕ.+ suc n    ∎

+
  where open ℕ.≤-Reasoning

+
∣i+j∣≤∣i∣+∣j∣ -[1+ m ] (+ n)    = begin

+
  ∣ n ⊖ suc m ∣  ≤⟨ ∣m⊝n∣≤m⊔n  n (suc m) ⟩

+
  n ℕ.⊔ suc m    ≤⟨ ℕ.m⊔n≤m+n n (suc m) ⟩

+
  n ℕ.+ suc m    ≡⟨ ℕ.+-comm  n (suc m) ⟩

+
  suc m ℕ.+ n    ∎

+
  where open ℕ.≤-Reasoning

+

+
∣i-j∣≤∣i∣+∣j∣ : ∀ i j → ∣ i - j ∣ ℕ.≤ ∣ i ∣ ℕ.+ ∣ j ∣

+
∣i-j∣≤∣i∣+∣j∣ i j = begin

+
  ∣ i   -       j ∣        ≤⟨ ∣i+j∣≤∣i∣+∣j∣ i (- j) ⟩

+
  ∣ i ∣ ℕ.+ ∣ - j ∣        ≡⟨ cong (∣ i ∣ ℕ.+_) (∣-i∣≡∣i∣ j) ⟩

+
  ∣ i ∣ ℕ.+ ∣   j ∣        ∎

+
  where open ℕ.≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of sign and _◃_

+

+
◃-nonZero : ∀ s n .{{_ : ℕ.NonZero n}} → NonZero (s ◃ n)

+
◃-nonZero Sign.- (ℕ.suc _) = _

+
◃-nonZero Sign.+ (ℕ.suc _) = _

+

+
◃-inverse : ∀ i → sign i ◃ ∣ i ∣ ≡ i

+
◃-inverse -[1+ n ] = refl

+
◃-inverse +0       = refl

+
◃-inverse +[1+ n ] = refl

+

+
◃-cong : sign i ≡ sign j → ∣ i ∣ ≡ ∣ j ∣ → i ≡ j

+
◃-cong {+ m}       {+ n }      ≡-sign refl = refl

+
◃-cong { -[1+ m ]} { -[1+ n ]} ≡-sign refl = refl

+

+
+◃n≡+n : ∀ n → Sign.+ ◃ n ≡ + n

+
+◃n≡+n zero    = refl

+
+◃n≡+n (suc _) = refl

+

+
-◃n≡-n : ∀ n → Sign.- ◃ n ≡ - + n

+
-◃n≡-n zero    = refl

+
-◃n≡-n (suc _) = refl

+

+
sign-◃ : ∀ s n .{{_ : ℕ.NonZero n}} → sign (s ◃ n) ≡ s

+
sign-◃ Sign.- (suc _) = refl

+
sign-◃ Sign.+ (suc _) = refl

+

+
abs-◃ : ∀ s n → ∣ s ◃ n ∣ ≡ n

+
abs-◃ _      zero    = refl

+
abs-◃ Sign.- (suc n) = refl

+
abs-◃ Sign.+ (suc n) = refl

+

+
signᵢ◃∣i∣≡i : ∀ i → sign i ◃ ∣ i ∣ ≡ i

+
signᵢ◃∣i∣≡i (+ n)    = +◃n≡+n n

+
signᵢ◃∣i∣≡i -[1+ n ] = refl

+

+
sign-cong : .{{_ : ℕ.NonZero m}} .{{_ : ℕ.NonZero n}} →

+
            s ◃ m ≡ t ◃ n → s ≡ t

+
sign-cong {n@(suc _)} {m@(suc _)} {s} {t} eq = begin

+
  s             ≡⟨ sign-◃ s n ⟨

+
  sign (s ◃ n)  ≡⟨  cong sign eq ⟩

+
  sign (t ◃ m)  ≡⟨  sign-◃ t m ⟩

+
  t             ∎ where open ≡-Reasoning

+

+
sign-cong′ : s ◃ m ≡ t ◃ n → s ≡ t ⊎ (m ≡ 0 × n ≡ 0)

+
sign-cong′ {s}       {zero}  {t}       {zero}  eq = inj₂ (refl , refl)

+
sign-cong′ {s}       {zero}  {Sign.- } {suc n} ()

+
sign-cong′ {s}       {zero}  {Sign.+ } {suc n} ()

+
sign-cong′ {Sign.- } {suc m} {t}       {zero}  ()

+
sign-cong′ {Sign.+ } {suc m} {t}       {zero}  ()

+
sign-cong′ {s}       {suc m} {t}       {suc n} eq = inj₁ (sign-cong eq)

+

+
abs-cong : s ◃ m ≡ t ◃ n → m ≡ n

+
abs-cong {s} {m} {t} {n} eq = begin

+
  m          ≡⟨ abs-◃ s m ⟨

+
  ∣ s ◃ m ∣  ≡⟨  cong ∣_∣ eq ⟩

+
  ∣ t ◃ n ∣  ≡⟨  abs-◃ t n ⟩

+
  n          ∎ where open ≡-Reasoning

+

+
∣s◃m∣*∣t◃n∣≡m*n : ∀ s t m n → ∣ s ◃ m ∣ ℕ.* ∣ t ◃ n ∣ ≡ m ℕ.* n

+
∣s◃m∣*∣t◃n∣≡m*n s t m n = cong₂ ℕ._*_ (abs-◃ s m) (abs-◃ t n)

+

+
+◃-mono-< : m ℕ.< n → Sign.+ ◃ m < Sign.+ ◃ n

+
+◃-mono-< {zero}  {suc n} m<n = +<+ m<n

+
+◃-mono-< {suc m} {suc n} m<n = +<+ m<n

+

+
+◃-cancel-< : Sign.+ ◃ m < Sign.+ ◃ n → m ℕ.< n

+
+◃-cancel-< {zero}  {zero}  (+<+ ())

+
+◃-cancel-< {suc m} {zero}  (+<+ ())

+
+◃-cancel-< {zero}  {suc n} (+<+ m<n) = m<n

+
+◃-cancel-< {suc m} {suc n} (+<+ m<n) = m<n

+

+
neg◃-cancel-< : Sign.- ◃ m < Sign.- ◃ n → n ℕ.< m

+
neg◃-cancel-< {zero}  {zero}  (+<+ ())

+
neg◃-cancel-< {suc m} {zero}  -<+       = z<s

+
neg◃-cancel-< {suc m} {suc n} (-<- n<m) = s<s n<m

+

+
-◃<+◃ : ∀ m n .{{_ : ℕ.NonZero m}} → Sign.- ◃ m < Sign.+ ◃ n

+
-◃<+◃ (suc _) zero    = -<+

+
-◃<+◃ (suc _) (suc _) = -<+

+

+
+◃≮-◃ : Sign.+ ◃ m ≮ Sign.- ◃ n

+
+◃≮-◃ {zero}  {zero} (+<+ ())

+
+◃≮-◃ {suc m} {zero} (+<+ ())

+

+
------------------------------------------------------------------------

+
-- Properties of _⊖_

+
------------------------------------------------------------------------

+

+
n⊖n≡0 : ∀ n → n ⊖ n ≡ 0ℤ

+
n⊖n≡0 n with n ℕ.<ᵇ n in leq

+
... | true  = cong (-_ ∘ +_) (ℕ.n∸n≡0 n) -- this is actually impossible!

+
... | false = cong +_ (ℕ.n∸n≡0 n)

+

+
[1+m]⊖[1+n]≡m⊖n : ∀ m n → suc m ⊖ suc n ≡ m ⊖ n

+
[1+m]⊖[1+n]≡m⊖n m n with m ℕ.<ᵇ n

+
... | true  = refl

+
... | false = refl

+

+
⊖-swap : ∀ m n → m ⊖ n ≡ - (n ⊖ m)

+
⊖-swap zero    zero    = refl

+
⊖-swap zero    (suc m) = refl

+
⊖-swap (suc m) zero    = refl

+
⊖-swap (suc m) (suc n) = begin

+
  suc m ⊖ suc n     ≡⟨ [1+m]⊖[1+n]≡m⊖n m n ⟩

+
  m ⊖ n             ≡⟨ ⊖-swap m n ⟩

+
  - (n ⊖ m)         ≡⟨ cong -_ ([1+m]⊖[1+n]≡m⊖n n m) ⟨

+
  - (suc n ⊖ suc m) ∎ where open ≡-Reasoning

+

+
⊖-≥ : m ℕ.≥ n → m ⊖ n ≡ + (m ∸ n)

+
⊖-≥ {m} {n} p with m ℕ.<ᵇ n | Reflects.invert (ℕ.<ᵇ-reflects-< m n)

+
... | true  | q = contradiction (ℕ.≤-<-trans p q) (ℕ.<-irrefl refl)

+
... | false | q = refl

+

+
≤-⊖ : m ℕ.≤ n → n ⊖ m ≡ + (n ∸ m)

+
≤-⊖ (z≤n {n})       = refl

+
≤-⊖ (s≤s {m} {n} p) = begin

+
  suc n ⊖ suc m     ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟩

+
  n ⊖ m             ≡⟨ ≤-⊖ p ⟩

+
  + (n ∸ m)         ≡⟨⟩

+
  + (suc n ∸ suc m) ∎ where open ≡-Reasoning

+

+
⊖-≤ : m ℕ.≤ n → m ⊖ n ≡ - + (n ∸ m)

+
⊖-≤ {m} {n} p with m ℕ.<ᵇ n | Reflects.invert (ℕ.<ᵇ-reflects-< m n)

+
... | true  | q = refl

+
... | false | q rewrite ℕ.≤-antisym p (ℕ.≮⇒≥ q) | ℕ.n∸n≡0 n = refl

+

+
⊖-< : m ℕ.< n → m ⊖ n ≡ - + (n ∸ m)

+
⊖-< = ⊖-≤ ∘ ℕ.<⇒≤

+

+
⊖-≰ : n ℕ.≰ m → m ⊖ n ≡ - + (n ∸ m)

+
⊖-≰ = ⊖-< ∘ ℕ.≰⇒>

+

+
∣⊖∣-≤ : m ℕ.≤ n → ∣ m ⊖ n ∣ ≡ n ∸ m

+
∣⊖∣-≤ {m} {n} p = begin

+
  ∣ m ⊖ n ∣         ≡⟨ cong ∣_∣ (⊖-≤ p) ⟩

+
  ∣ - (+ (n ∸ m)) ∣ ≡⟨ ∣-i∣≡∣i∣ (+ (n ∸ m)) ⟩

+
  ∣ + (n ∸ m) ∣     ≡⟨⟩

+
  n ∸ m             ∎ where open ≡-Reasoning

+

+
∣⊖∣-< : m ℕ.< n → ∣ m ⊖ n ∣ ≡ n ∸ m

+
∣⊖∣-< {m} {n} p = begin

+
  ∣ m ⊖ n ∣         ≡⟨ cong ∣_∣ (⊖-< p) ⟩

+
  ∣ - (+ (n ∸ m)) ∣ ≡⟨ ∣-i∣≡∣i∣ (+ (n ∸ m)) ⟩

+
  ∣ + (n ∸ m) ∣     ≡⟨⟩

+
  n ∸ m             ∎ where open ≡-Reasoning

+

+
∣⊖∣-≰ : n ℕ.≰ m → ∣ m ⊖ n ∣ ≡ n ∸ m

+
∣⊖∣-≰ = ∣⊖∣-< ∘ ℕ.≰⇒>

+

+
-m+n≡n⊖m : ∀ m n → - (+ m) + + n ≡ n ⊖ m

+
-m+n≡n⊖m zero    n = refl

+
-m+n≡n⊖m (suc m) n = refl

+

+
m-n≡m⊖n : ∀ m n → + m + (- + n) ≡ m ⊖ n

+
m-n≡m⊖n zero    zero    = refl

+
m-n≡m⊖n zero    (suc n) = refl

+
m-n≡m⊖n (suc m) zero    = cong +[1+_] (ℕ.+-identityʳ m)

+
m-n≡m⊖n (suc m) (suc n) = refl

+

+
-[n⊖m]≡-m+n : ∀ m n → - (m ⊖ n) ≡ (- (+ m)) + (+ n)

+
-[n⊖m]≡-m+n m n with m ℕ.<ᵇ n | Reflects.invert (ℕ.<ᵇ-reflects-< m n)

+
... | true  | p = begin

+
  - (- (+ (n ∸ m))) ≡⟨ neg-involutive (+ (n ∸ m)) ⟩

+
  + (n ∸ m)         ≡⟨ ⊖-≥ (ℕ.≤-trans (ℕ.m≤n+m m 1) p) ⟨

+
  n ⊖ m             ≡⟨ -m+n≡n⊖m m n ⟨

+
  - (+ m) + + n     ∎ where open ≡-Reasoning

+
... | false | p = begin

+
  - (+ (m ∸ n))     ≡⟨ ⊖-≤ (ℕ.≮⇒≥ p) ⟨

+
  n ⊖ m             ≡⟨ -m+n≡n⊖m m n ⟨

+
  - (+ m) + + n     ∎ where open ≡-Reasoning

+

+
∣m⊖n∣≡∣n⊖m∣ : ∀ m n → ∣ m ⊖ n ∣ ≡ ∣ n ⊖ m ∣

+
∣m⊖n∣≡∣n⊖m∣ m n = begin

+
  ∣ m ⊖ n ∣     ≡⟨ cong ∣_∣ (⊖-swap m n) ⟩

+
  ∣ - (n ⊖ m) ∣ ≡⟨ ∣-i∣≡∣i∣ (n ⊖ m) ⟩

+
  ∣ n ⊖ m ∣     ∎ where open ≡-Reasoning

+

+
+-cancelˡ-⊖ : ∀ m n o → (m ℕ.+ n) ⊖ (m ℕ.+ o) ≡ n ⊖ o

+
+-cancelˡ-⊖ zero    n o = refl

+
+-cancelˡ-⊖ (suc m) n o = begin

+
  suc (m ℕ.+ n) ⊖ suc (m ℕ.+ o) ≡⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ n) (m ℕ.+ o) ⟩

+
  m ℕ.+ n ⊖ (m ℕ.+ o)           ≡⟨ +-cancelˡ-⊖ m n o ⟩

+
  n ⊖ o                         ∎ where open ≡-Reasoning

+

+
m⊖n≤m : ∀ m n → m ⊖ n ≤ + m

+
m⊖n≤m m       zero    = ≤-refl

+
m⊖n≤m zero    (suc n) = -≤+

+
m⊖n≤m (suc m) (suc n) = begin

+
  suc m ⊖ suc n ≡⟨ [1+m]⊖[1+n]≡m⊖n m n ⟩

+
  m ⊖ n         ≤⟨ m⊖n≤m m n ⟩

+
  + m           ≤⟨ +≤+ (ℕ.n≤1+n m) ⟩

+
  +[1+ m ]      ∎ where open ≤-Reasoning

+

+
m⊖n<1+m : ∀ m n → m ⊖ n < +[1+ m ]

+
m⊖n<1+m m n = ≤-<-trans (m⊖n≤m m n) (+<+ (ℕ.m<n+m m z<s))

+

+
m⊖1+n<m : ∀ m n .{{_ : ℕ.NonZero n}} → m ⊖ n < + m

+
m⊖1+n<m zero    (suc n) = -<+

+
m⊖1+n<m (suc m) (suc n) = begin-strict

+
  suc m ⊖ suc n ≡⟨ [1+m]⊖[1+n]≡m⊖n m n ⟩

+
  m ⊖ n         <⟨ m⊖n<1+m m n ⟩

+
  +[1+ m ]      ∎ where open ≤-Reasoning

+

+
-1+m<n⊖m : ∀ m n → -[1+ m ] < n ⊖ m

+
-1+m<n⊖m zero    n       = -<+

+
-1+m<n⊖m (suc m) zero    = -<- ℕ.≤-refl

+
-1+m<n⊖m (suc m) (suc n) = begin-strict

+
  -[1+ suc m ]  <⟨ -<- ℕ.≤-refl ⟩

+
  -[1+ m ]      <⟨ -1+m<n⊖m m n ⟩

+
  n ⊖ m         ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟨

+
  suc n ⊖ suc m ∎ where open ≤-Reasoning

+

+
-[1+m]≤n⊖m+1 : ∀ m n → -[1+ m ] ≤ n ⊖ suc m

+
-[1+m]≤n⊖m+1 m zero    = ≤-refl

+
-[1+m]≤n⊖m+1 m (suc n) = begin

+
  -[1+ m ]      ≤⟨ <⇒≤ (-1+m<n⊖m m n) ⟩

+
  n ⊖ m         ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟨

+
  suc n ⊖ suc m ∎ where open ≤-Reasoning

+

+
-1+m≤n⊖m : ∀ m n → -[1+ m ] ≤ n ⊖ m

+
-1+m≤n⊖m m n = <⇒≤ (-1+m<n⊖m m n)

+

+
0⊖m≤+ : ∀ m {n} → 0 ⊖ m ≤ + n

+
0⊖m≤+ zero    = +≤+ z≤n

+
0⊖m≤+ (suc m) = -≤+

+

+
sign-⊖-< : m ℕ.< n → sign (m ⊖ n) ≡ Sign.-

+
sign-⊖-< {zero}          (ℕ.z<s) = refl

+
sign-⊖-< {suc m} {suc n} (ℕ.s<s m<n) = begin

+
  sign (suc m ⊖ suc n) ≡⟨ cong sign ([1+m]⊖[1+n]≡m⊖n m n) ⟩

+
  sign (m ⊖ n)         ≡⟨ sign-⊖-< m<n ⟩

+
  Sign.-               ∎ where open ≡-Reasoning

+

+
sign-⊖-≰ : n ℕ.≰ m → sign (m ⊖ n) ≡ Sign.-

+
sign-⊖-≰ = sign-⊖-< ∘ ℕ.≰⇒>

+

+
⊖-monoʳ-≥-≤ : ∀ n → (n ⊖_) Preserves ℕ._≥_ ⟶ _≤_

+
⊖-monoʳ-≥-≤ zero    {m}     z≤n      = 0⊖m≤+ m

+
⊖-monoʳ-≥-≤ zero    {_}    (s≤s m≤n) = -≤- m≤n

+
⊖-monoʳ-≥-≤ (suc n) {zero}  z≤n      = ≤-refl

+
⊖-monoʳ-≥-≤ (suc n) {suc m} z≤n      = begin

+
  suc n ⊖ suc m ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟩

+
  n ⊖ m         <⟨ m⊖n<1+m n m ⟩

+
  +[1+ n ]      ∎ where open ≤-Reasoning

+
⊖-monoʳ-≥-≤ (suc n) {suc m} {suc o} (s≤s m≤o) = begin

+
  suc n ⊖ suc m ≡⟨  [1+m]⊖[1+n]≡m⊖n n m ⟩

+
  n ⊖ m         ≤⟨  ⊖-monoʳ-≥-≤ n m≤o ⟩

+
  n ⊖ o         ≡⟨ [1+m]⊖[1+n]≡m⊖n n o ⟨

+
  suc n ⊖ suc o ∎ where open ≤-Reasoning

+

+
⊖-monoˡ-≤ : ∀ n → (_⊖ n) Preserves ℕ._≤_ ⟶ _≤_

+
⊖-monoˡ-≤ zero    {_} {_}     m≤o = +≤+ m≤o

+
⊖-monoˡ-≤ (suc n) {_} {0}     z≤n = ≤-refl

+
⊖-monoˡ-≤ (suc n) {_} {suc o} z≤n = begin

+
  zero ⊖ suc n  ≤⟨  ⊖-monoʳ-≥-≤ 0 (ℕ.n≤1+n n) ⟩

+
  zero ⊖ n      ≤⟨  ⊖-monoˡ-≤ n z≤n ⟩

+
  o ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n o n ⟨

+
  suc o ⊖ suc n ∎ where open ≤-Reasoning

+
⊖-monoˡ-≤ (suc n) {suc m} {suc o} (s≤s m≤o) = begin

+
  suc m ⊖ suc n ≡⟨  [1+m]⊖[1+n]≡m⊖n m n ⟩

+
  m ⊖ n         ≤⟨  ⊖-monoˡ-≤ n m≤o ⟩

+
  o ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n o n ⟨

+
  suc o ⊖ suc n ∎ where open ≤-Reasoning

+

+
⊖-monoʳ->-< : ∀ p → (p ⊖_) Preserves ℕ._>_ ⟶ _<_

+
⊖-monoʳ->-< zero    {_}     z<s       = -<+

+
⊖-monoʳ->-< zero    {_}     (s<s m<n@(s≤s _)) = -<- m<n

+
⊖-monoʳ->-< (suc p) {suc m} z<s       = begin-strict

+
  suc p ⊖ suc m ≡⟨ [1+m]⊖[1+n]≡m⊖n p m ⟩

+
  p ⊖ m         <⟨ m⊖n<1+m p m ⟩

+
  +[1+ p ]      ∎ where open ≤-Reasoning

+
⊖-monoʳ->-< (suc p) {suc m} {suc n} (s<s m<n@(s≤s _)) = begin-strict

+
  suc p ⊖ suc m ≡⟨  [1+m]⊖[1+n]≡m⊖n p m ⟩

+
  p ⊖ m         <⟨  ⊖-monoʳ->-< p m<n ⟩

+
  p ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n p n ⟨

+
  suc p ⊖ suc n ∎ where open ≤-Reasoning

+

+
⊖-monoˡ-< : ∀ n → (_⊖ n) Preserves ℕ._<_ ⟶ _<_

+
⊖-monoˡ-< zero    m<o             = +<+ m<o

+
⊖-monoˡ-< (suc n) {_} {suc o} z<s = begin-strict

+
  -[1+ n ]      <⟨  -1+m<n⊖m n _ ⟩

+
  o ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n o n ⟨

+
  suc o ⊖ suc n ∎ where open ≤-Reasoning

+
⊖-monoˡ-< (suc n) {suc m} {suc (suc o)} (s<s m<o@(s≤s _)) = begin-strict

+
  suc m ⊖ suc n       ≡⟨  [1+m]⊖[1+n]≡m⊖n m n ⟩

+
  m ⊖ n               <⟨  ⊖-monoˡ-< n m<o ⟩

+
  suc o ⊖ n           ≡⟨ [1+m]⊖[1+n]≡m⊖n (suc o) n ⟨

+
  suc (suc o) ⊖ suc n ∎ where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _+_

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Algebraic properties of _+_

+

+
+-comm : Commutative _+_

+
+-comm -[1+ m ] -[1+ n ] = cong (-[1+_] ∘ suc) (ℕ.+-comm m n)

+
+-comm (+ m)    (+ n)    = cong +_ (ℕ.+-comm m n)

+
+-comm -[1+ _ ] (+ _)    = refl

+
+-comm (+ _)    -[1+ _ ] = refl

+

+
+-identityˡ : LeftIdentity +0 _+_

+
+-identityˡ -[1+ _ ] = refl

+
+-identityˡ (+ _)    = refl

+

+
+-identityʳ : RightIdentity +0 _+_

+
+-identityʳ = comm∧idˡ⇒idʳ +-comm +-identityˡ

+

+
+-identity : Identity +0 _+_

+
+-identity = +-identityˡ , +-identityʳ

+

+
distribˡ-⊖-+-pos : ∀ m n o → n ⊖ o + + m ≡ n ℕ.+ m ⊖ o

+
distribˡ-⊖-+-pos _ zero    zero    = refl

+
distribˡ-⊖-+-pos _ zero    (suc _) = refl

+
distribˡ-⊖-+-pos _ (suc _) zero    = refl

+
distribˡ-⊖-+-pos m (suc n) (suc o) = begin

+
  suc n ⊖ suc o + + m   ≡⟨ cong (_+ + m) ([1+m]⊖[1+n]≡m⊖n n o) ⟩

+
  n ⊖ o + + m           ≡⟨ distribˡ-⊖-+-pos m n o ⟩

+
  n ℕ.+ m ⊖ o           ≡⟨ [1+m]⊖[1+n]≡m⊖n (n ℕ.+ m) o ⟨

+
  suc (n ℕ.+ m) ⊖ suc o ∎ where open ≡-Reasoning

+

+
distribˡ-⊖-+-neg : ∀ m n o → n ⊖ o + -[1+ m ] ≡ n ⊖ (suc o ℕ.+ m)

+
distribˡ-⊖-+-neg _ zero    zero    = refl

+
distribˡ-⊖-+-neg _ zero    (suc _) = refl

+
distribˡ-⊖-+-neg _ (suc _) zero    = refl

+
distribˡ-⊖-+-neg m (suc n) (suc o) = begin

+
  suc n ⊖ suc o + -[1+ m ]    ≡⟨ cong (_+ -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n n o) ⟩

+
  n ⊖ o + -[1+ m ]            ≡⟨ distribˡ-⊖-+-neg m n o ⟩

+
  n ⊖ (suc o ℕ.+ m)           ≡⟨ [1+m]⊖[1+n]≡m⊖n n (suc o ℕ.+ m) ⟨

+
  suc n ⊖ (suc (suc o) ℕ.+ m) ∎ where open ≡-Reasoning

+

+
distribʳ-⊖-+-pos : ∀ m n o → + m + (n ⊖ o) ≡ m ℕ.+ n ⊖ o

+
distribʳ-⊖-+-pos m n o = begin

+
  + m + (n ⊖ o) ≡⟨ +-comm (+ m) (n ⊖ o) ⟩

+
  (n ⊖ o) + + m ≡⟨ distribˡ-⊖-+-pos m n o ⟩

+
  n ℕ.+ m ⊖ o   ≡⟨ cong (_⊖ o) (ℕ.+-comm n m) ⟩

+
  m ℕ.+ n ⊖ o   ∎ where open ≡-Reasoning

+

+
distribʳ-⊖-+-neg : ∀ m n o → -[1+ m ] + (n ⊖ o) ≡ n ⊖ (suc m ℕ.+ o)

+
distribʳ-⊖-+-neg m n o = begin

+
  -[1+ m ] + (n ⊖ o) ≡⟨ +-comm -[1+ m ] (n ⊖ o) ⟩

+
  (n ⊖ o) + -[1+ m ] ≡⟨ distribˡ-⊖-+-neg m n o ⟩

+
  n ⊖ suc (o ℕ.+ m)  ≡⟨ cong (λ x → n ⊖ suc x) (ℕ.+-comm o m) ⟩

+
  n ⊖ suc (m ℕ.+ o)  ∎ where open ≡-Reasoning

+

+
+-assoc : Associative _+_

+
+-assoc +0 j k rewrite +-identityˡ      j  | +-identityˡ (j + k) = refl

+
+-assoc i +0 k rewrite +-identityʳ  i      | +-identityˡ      k  = refl

+
+-assoc i j +0 rewrite +-identityʳ (i + j) | +-identityʳ  j      = refl

+
+-assoc -[1+ m ] -[1+ n ] +[1+ o ] = begin

+
  suc o ⊖ suc (suc (m ℕ.+ n)) ≡⟨ [1+m]⊖[1+n]≡m⊖n o (suc m ℕ.+ n) ⟩

+
  o ⊖ (suc m ℕ.+ n)           ≡⟨ distribʳ-⊖-+-neg m o n ⟨

+
  -[1+ m ] + (o ⊖ n)          ≡⟨ cong (λ z → -[1+ m ] + z) ([1+m]⊖[1+n]≡m⊖n o n) ⟨

+
  -[1+ m ] + (suc o ⊖ suc n)  ∎ where open ≡-Reasoning

+
+-assoc -[1+ m ] +[1+ n ] +[1+ o ] = begin

+
  suc n ⊖ suc m + +[1+ o ]  ≡⟨ cong (_+ +[1+ o ]) ([1+m]⊖[1+n]≡m⊖n n m) ⟩

+
  (n ⊖ m) + +[1+ o ]        ≡⟨ distribˡ-⊖-+-pos (suc o) n m ⟩

+
  n ℕ.+ suc o ⊖ m           ≡⟨ [1+m]⊖[1+n]≡m⊖n (n ℕ.+ suc o) m ⟨

+
  suc (n ℕ.+ suc o) ⊖ suc m ∎ where open ≡-Reasoning

+
+-assoc +[1+ m ] -[1+ n ] -[1+ o ] = begin

+
  (suc m ⊖ suc n) + -[1+ o ]  ≡⟨ cong (_+ -[1+ o ]) ([1+m]⊖[1+n]≡m⊖n m n) ⟩

+
  (m ⊖ n) + -[1+ o ]          ≡⟨ distribˡ-⊖-+-neg o m n ⟩

+
  m ⊖ suc (n ℕ.+ o)           ≡⟨ [1+m]⊖[1+n]≡m⊖n m (suc n ℕ.+ o) ⟨

+
  suc m ⊖ suc (suc (n ℕ.+ o)) ∎ where open ≡-Reasoning

+
+-assoc +[1+ m ] -[1+ n ] +[1+ o ]

+
  rewrite [1+m]⊖[1+n]≡m⊖n m n

+
        | [1+m]⊖[1+n]≡m⊖n o n

+
        | distribˡ-⊖-+-pos (suc o) m n

+
        | distribʳ-⊖-+-pos (suc m) o n

+
        | sym (ℕ.+-assoc m 1 o)

+
        | ℕ.+-comm m 1

+
        = refl

+
+-assoc +[1+ m ] +[1+ n ] -[1+ o ]

+
  rewrite [1+m]⊖[1+n]≡m⊖n n o

+
        | [1+m]⊖[1+n]≡m⊖n (m ℕ.+ suc n) o

+
        | distribʳ-⊖-+-pos (suc m) n o

+
        | sym (ℕ.+-assoc m 1 n)

+
        | ℕ.+-comm m 1

+
        = refl

+
+-assoc -[1+ m ] -[1+ n ] -[1+ o ]

+
  rewrite sym (ℕ.+-assoc m 1 (n ℕ.+ o))

+
        | ℕ.+-comm m 1

+
        | ℕ.+-assoc m n o

+
        = refl

+
+-assoc -[1+ m ] +[1+ n ] -[1+ o ]

+
  rewrite [1+m]⊖[1+n]≡m⊖n n m

+
        | [1+m]⊖[1+n]≡m⊖n n o

+
        | distribʳ-⊖-+-neg m n o

+
        | distribˡ-⊖-+-neg o n m

+
        = refl

+
+-assoc +[1+ m ] +[1+ n ] +[1+ o ]

+
  rewrite ℕ.+-assoc (suc m) (suc n) (suc o)

+
        = refl

+

+
+-inverseˡ : LeftInverse +0 -_ _+_

+
+-inverseˡ -[1+ n ] = n⊖n≡0 (suc n)

+
+-inverseˡ +0       = refl

+
+-inverseˡ +[1+ n ] = n⊖n≡0 (suc n)

+

+
+-inverseʳ : RightInverse +0 -_ _+_

+
+-inverseʳ = comm∧invˡ⇒invʳ +-comm +-inverseˡ

+

+
+-inverse : Inverse +0 -_ _+_

+
+-inverse = +-inverseˡ , +-inverseʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
+-isMagma : IsMagma _+_

+
+-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _+_

+
  }

+

+
+-isSemigroup : IsSemigroup _+_

+
+-isSemigroup = record

+
  { isMagma = +-isMagma

+
  ; assoc   = +-assoc

+
  }

+

+
+-isCommutativeSemigroup : IsCommutativeSemigroup _+_

+
+-isCommutativeSemigroup = record

+
  { isSemigroup = +-isSemigroup

+
  ; comm        = +-comm

+
  }

+

+
+-0-isMonoid : IsMonoid _+_ +0

+
+-0-isMonoid = record

+
  { isSemigroup = +-isSemigroup

+
  ; identity    = +-identity

+
  }

+

+
+-0-isCommutativeMonoid : IsCommutativeMonoid _+_ +0

+
+-0-isCommutativeMonoid = record

+
  { isMonoid = +-0-isMonoid

+
  ; comm     = +-comm

+
  }

+

+
+-0-isGroup : IsGroup _+_ +0 (-_)

+
+-0-isGroup = record

+
  { isMonoid = +-0-isMonoid

+
  ; inverse  = +-inverse

+
  ; ⁻¹-cong  = cong (-_)

+
  }

+

+
+-0-isAbelianGroup : IsAbelianGroup _+_ +0 (-_)

+
+-0-isAbelianGroup = record

+
  { isGroup = +-0-isGroup

+
  ; comm    = +-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
+-magma : Magma 0ℓ 0ℓ

+
+-magma = record

+
  { isMagma = +-isMagma

+
  }

+

+
+-semigroup : Semigroup 0ℓ 0ℓ

+
+-semigroup = record

+
  { isSemigroup = +-isSemigroup

+
  }

+

+
+-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ

+
+-commutativeSemigroup = record

+
  { isCommutativeSemigroup = +-isCommutativeSemigroup

+
  }

+

+
+-0-monoid : Monoid 0ℓ 0ℓ

+
+-0-monoid = record

+
  { isMonoid = +-0-isMonoid

+
  }

+

+
+-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
+-0-commutativeMonoid = record

+
  { isCommutativeMonoid = +-0-isCommutativeMonoid

+
  }

+

+
+-0-abelianGroup : AbelianGroup 0ℓ 0ℓ

+
+-0-abelianGroup = record

+
  { isAbelianGroup = +-0-isAbelianGroup

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and +_/-_.

+

+
pos-+ : ℕtoℤ.Homomorphic₂ +_ ℕ._+_ _+_

+
pos-+ zero    n = refl

+
pos-+ (suc m) n = cong sucℤ (pos-+ m n)

+

+
neg-distrib-+ : ∀ i j → - (i + j) ≡ (- i) + (- j)

+
neg-distrib-+ +0        +0        = refl

+
neg-distrib-+ +0        +[1+ n ]  = refl

+
neg-distrib-+ +[1+ m ]  +0        = cong -[1+_] (ℕ.+-identityʳ m)

+
neg-distrib-+ +[1+ m ]  +[1+ n ]  = cong -[1+_] (ℕ.+-suc m n)

+
neg-distrib-+ -[1+ m ]  -[1+ n ]  = cong +[1+_] (sym (ℕ.+-suc m n))

+
neg-distrib-+ (+   m)   -[1+ n ]  = -[n⊖m]≡-m+n m (suc n)

+
neg-distrib-+ -[1+ m ]  (+   n)   =

+
  trans (-[n⊖m]≡-m+n n (suc m)) (+-comm (- + n) (+ suc m))

+

+
◃-distrib-+ : ∀ s m n → s ◃ (m ℕ.+ n) ≡ (s ◃ m) + (s ◃ n)

+
◃-distrib-+ Sign.- m n = begin

+
  Sign.- ◃ (m ℕ.+ n)          ≡⟨ -◃n≡-n (m ℕ.+ n) ⟩

+
  - (+ (m ℕ.+ n))             ≡⟨⟩

+
  - ((+ m) + (+ n))           ≡⟨ neg-distrib-+ (+ m) (+ n) ⟩

+
  (- (+ m)) + (- (+ n))       ≡⟨ sym (cong₂ _+_ (-◃n≡-n m) (-◃n≡-n n)) ⟩

+
  (Sign.- ◃ m) + (Sign.- ◃ n) ∎ where open ≡-Reasoning

+
◃-distrib-+ Sign.+ m n = begin

+
  Sign.+ ◃ (m ℕ.+ n)          ≡⟨ +◃n≡+n (m ℕ.+ n) ⟩

+
  + (m ℕ.+ n)                 ≡⟨⟩

+
  (+ m) + (+ n)               ≡⟨ sym (cong₂ _+_ (+◃n≡+n m) (+◃n≡+n n)) ⟩

+
  (Sign.+ ◃ m) + (Sign.+ ◃ n) ∎ where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _≤_

+

+
+-monoʳ-≤ : ∀ n → (_+_ n) Preserves _≤_ ⟶ _≤_

+
+-monoʳ-≤ (+ n)    {_}         (-≤- o≤m) = ⊖-monoʳ-≥-≤ n (s≤s o≤m)

+
+-monoʳ-≤ (+ n)    { -[1+ m ]} -≤+       = ≤-trans (m⊖n≤m n (suc m)) (+≤+ (ℕ.m≤m+n n _))

+
+-monoʳ-≤ (+ n)    {_}         (+≤+ m≤o) = +≤+ (ℕ.+-monoʳ-≤ n m≤o)

+
+-monoʳ-≤ -[1+ n ] {_} {_}     (-≤- n≤m) = -≤- (ℕ.+-monoʳ-≤ (suc n) n≤m)

+
+-monoʳ-≤ -[1+ n ] {_} {+ m}   -≤+       = ≤-trans (-≤- (ℕ.m≤m+n (suc n) _)) (-1+m≤n⊖m (suc n) m)

+
+-monoʳ-≤ -[1+ n ] {_} {_}     (+≤+ m≤n) = ⊖-monoˡ-≤ (suc n) m≤n

+

+
+-monoˡ-≤ : ∀ n → (_+ n) Preserves _≤_ ⟶ _≤_

+
+-monoˡ-≤ n {i} {j} rewrite +-comm i n | +-comm j n = +-monoʳ-≤ n

+

+
+-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
+-mono-≤ {m} {n} {i} {j} m≤n i≤j = begin

+
  m + i ≤⟨ +-monoˡ-≤ i m≤n ⟩

+
  n + i ≤⟨ +-monoʳ-≤ n i≤j ⟩

+
  n + j ∎

+
  where open ≤-Reasoning

+

+
i≤j⇒i≤k+j : ∀ k .{{_ : NonNegative k}} → i ≤ j → i ≤ k + j

+
i≤j⇒i≤k+j (+ n) i≤j = subst (_≤ _) (+-identityˡ _) (+-mono-≤ (+≤+ z≤n) i≤j)

+

+
i≤j+i : ∀ i j .{{_ : NonNegative j}} → i ≤ j + i

+
i≤j+i i j = i≤j⇒i≤k+j j ≤-refl

+

+
i≤i+j : ∀ i j .{{_ : NonNegative j}} → i ≤ i + j

+
i≤i+j i j rewrite +-comm i j = i≤j+i i j

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _<_

+

+
+-monoʳ-< : ∀ i → (_+_ i) Preserves _<_ ⟶ _<_

+
+-monoʳ-< (+ n)    {_} {_}   (-<- o<m) = ⊖-monoʳ->-< n (s<s o<m)

+
+-monoʳ-< (+ n)    {_} {_}   -<+       = <-≤-trans (m⊖1+n<m n _) (+≤+ (ℕ.m≤m+n n _))

+
+-monoʳ-< (+ n)    {_} {_}   (+<+ m<o) = +<+ (ℕ.+-monoʳ-< n m<o)

+
+-monoʳ-< -[1+ n ] {_} {_}   (-<- o<m) = -<- (ℕ.+-monoʳ-< (suc n) o<m)

+
+-monoʳ-< -[1+ n ] {_} {+ o} -<+       = <-≤-trans (-<- (ℕ.m≤m+n (suc n) _)) (-[1+m]≤n⊖m+1 n o)

+
+-monoʳ-< -[1+ n ] {_} {_}   (+<+ m<o) = ⊖-monoˡ-< (suc n) m<o

+

+
+-monoˡ-< : ∀ i → (_+ i) Preserves _<_ ⟶ _<_

+
+-monoˡ-< i {j} {k} rewrite +-comm j i | +-comm k i = +-monoʳ-< i

+

+
+-mono-< : _+_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
+-mono-< {i} {j} {k} {l} i<j k<l = begin-strict

+
  i + k <⟨ +-monoˡ-< k i<j ⟩

+
  j + k <⟨ +-monoʳ-< j k<l ⟩

+
  j + l ∎

+
  where open ≤-Reasoning

+

+
+-mono-≤-< : _+_ Preserves₂ _≤_ ⟶ _<_ ⟶ _<_

+
+-mono-≤-< {i} {j} {k} i≤j j<k = ≤-<-trans (+-monoˡ-≤ k i≤j) (+-monoʳ-< j j<k)

+

+
+-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_

+
+-mono-<-≤ {i} {j} {k} i<j j≤k = <-≤-trans (+-monoˡ-< k i<j) (+-monoʳ-≤ j j≤k)

+

+
------------------------------------------------------------------------

+
-- Properties of _-_

+
------------------------------------------------------------------------

+

+
neg-minus-pos : ∀ m n → -[1+ m ] - (+ n) ≡ -[1+ (n ℕ.+ m) ]

+
neg-minus-pos m       zero    = refl

+
neg-minus-pos zero    (suc n) = cong (-[1+_] ∘ suc) (sym (ℕ.+-identityʳ n))

+
neg-minus-pos (suc m) (suc n) = cong (-[1+_] ∘ suc) (ℕ.+-comm (suc m) n)

+

+
+-minus-telescope : ∀ i j k → (i - j) + (j - k) ≡ i - k

+
+-minus-telescope i j k = begin

+
  (i - j) + (j - k)   ≡⟨  +-assoc i (- j) (j - k) ⟩

+
  i + (- j + (j - k)) ≡⟨ cong (λ v → i + v) (+-assoc (- j) j _) ⟨

+
  i + ((- j + j) - k) ≡⟨ +-assoc i (- j + j) (- k) ⟨

+
  i + (- j + j) - k   ≡⟨  cong (λ a → i + a - k) (+-inverseˡ j) ⟩

+
  i + 0ℤ - k          ≡⟨  cong (_- k) (+-identityʳ i) ⟩

+
  i - k               ∎ where open ≡-Reasoning

+

+
[+m]-[+n]≡m⊖n : ∀ m n → (+ m) - (+ n) ≡ m ⊖ n

+
[+m]-[+n]≡m⊖n zero    zero    = refl

+
[+m]-[+n]≡m⊖n zero    (suc n) = refl

+
[+m]-[+n]≡m⊖n (suc m) zero    = cong +[1+_] (ℕ.+-identityʳ m)

+
[+m]-[+n]≡m⊖n (suc m) (suc n) = refl

+

+
∣i-j∣≡∣j-i∣ : ∀ i j → ∣ i - j ∣ ≡ ∣ j - i ∣

+
∣i-j∣≡∣j-i∣ -[1+ m ] -[1+ n ] = ∣m⊖n∣≡∣n⊖m∣ (suc n) (suc m)

+
∣i-j∣≡∣j-i∣ -[1+ m ] (+ n)    = begin

+
  ∣ -[1+ m ] - (+ n) ∣  ≡⟨  cong ∣_∣ (neg-minus-pos m n) ⟩

+
  suc (n ℕ.+ m)         ≡⟨ ℕ.+-suc n m ⟨

+
  n ℕ.+ suc m           ∎ where open ≡-Reasoning

+
∣i-j∣≡∣j-i∣ (+ m) -[1+ n ] = begin

+
  m ℕ.+ suc n          ≡⟨  ℕ.+-suc m n ⟩

+
  suc (m ℕ.+ n)        ≡⟨ cong ∣_∣ (neg-minus-pos n m) ⟨

+
  ∣ -[1+ n ] + - + m ∣ ∎ where open ≡-Reasoning

+
∣i-j∣≡∣j-i∣ (+ m) (+ n) = begin

+
  ∣ + m - + n ∣  ≡⟨  cong ∣_∣ ([+m]-[+n]≡m⊖n m n) ⟩

+
  ∣ m ⊖ n ∣      ≡⟨  ∣m⊖n∣≡∣n⊖m∣ m n ⟩

+
  ∣ n ⊖ m ∣      ≡⟨ cong ∣_∣ ([+m]-[+n]≡m⊖n n m) ⟨

+
  ∣ + n - + m ∣  ∎ where open ≡-Reasoning

+

+
∣-∣-≤ : i ≤ j → + ∣ i - j ∣ ≡ j - i

+
∣-∣-≤ (-≤- {m} {n} n≤m) = begin

+
  + ∣ -[1+ m ] + +[1+ n ] ∣ ≡⟨ cong (λ j → + ∣ j ∣) ([1+m]⊖[1+n]≡m⊖n n m) ⟩

+
  + ∣ n ⊖ m ∣               ≡⟨ cong +_ (∣⊖∣-≤ n≤m) ⟩

+
  + ( m ∸ n )              ≡⟨ sym (≤-⊖ n≤m) ⟩

+
  m ⊖ n                    ≡⟨ sym ([1+m]⊖[1+n]≡m⊖n m n) ⟩

+
  suc m ⊖ suc n            ∎ where open ≡-Reasoning

+
∣-∣-≤ (-≤+ {m} {zero}) = refl

+
∣-∣-≤ (-≤+ {m} {suc n}) = begin

+
  + ∣ -[1+ m ] - + suc n ∣ ≡⟨⟩

+
  + suc (suc m ℕ.+ n)    ≡⟨ cong (λ n → + suc n) (ℕ.+-comm (suc m) n) ⟩

+
  + (suc n ℕ.+ suc m)    ≡⟨⟩

+
  + suc n - -[1+ m ]      ∎ where open ≡-Reasoning

+
∣-∣-≤ (+≤+ {m} {n} m≤n) = begin

+
  + ∣ + m - + n ∣ ≡⟨ cong (λ j → + ∣ j ∣) (m-n≡m⊖n m n) ⟩

+
  + ∣ m ⊖ n ∣     ≡⟨ cong +_ ( ∣⊖∣-≤ m≤n ) ⟩

+
  + (n ∸ m)      ≡⟨ sym (≤-⊖  m≤n) ⟩

+
  n ⊖ m          ≡⟨ sym (m-n≡m⊖n n m) ⟩

+
  + n - + m      ∎ where open ≡-Reasoning

+

+
i≡j⇒i-j≡0 : i ≡ j → i - j ≡ 0ℤ

+
i≡j⇒i-j≡0 {i} refl = +-inverseʳ i

+

+
i-j≡0⇒i≡j : ∀ i j → i - j ≡ 0ℤ → i ≡ j

+
i-j≡0⇒i≡j i j i-j≡0 = begin

+
  i             ≡⟨ +-identityʳ i ⟨

+
  i + 0ℤ        ≡⟨ cong (_+_ i) (+-inverseˡ j) ⟨

+
  i + (- j + j) ≡⟨ +-assoc i (- j) j ⟨

+
  (i - j) + j   ≡⟨  cong (_+ j) i-j≡0 ⟩

+
  0ℤ + j        ≡⟨  +-identityˡ j ⟩

+
  j             ∎ where open ≡-Reasoning

+

+
i≤j⇒i-k≤j : ∀ k .{{_ : NonNegative k}} → i ≤ j → i - k ≤ j

+
i≤j⇒i-k≤j {i}         +0       i≤j rewrite +-identityʳ i = i≤j

+
i≤j⇒i-k≤j {+ m}       +[1+ n ] i≤j = ≤-trans (m⊖n≤m m (suc n)) i≤j

+
i≤j⇒i-k≤j { -[1+ m ]} +[1+ n ] i≤j = ≤-trans (-≤- (ℕ.≤-trans (ℕ.m≤m+n m n) (ℕ.n≤1+n _))) i≤j

+

+
i-j≤i : ∀ i j .{{_ : NonNegative j}} → i - j ≤ i

+
i-j≤i i j = i≤j⇒i-k≤j j ≤-refl

+

+
i≤j⇒i-j≤0 : i ≤ j → i - j ≤ 0ℤ

+
i≤j⇒i-j≤0 {_}         {j}         -≤+       = i≤j⇒i-k≤j j -≤+

+
i≤j⇒i-j≤0 { -[1+ m ]} { -[1+ n ]} (-≤- n≤m) = begin

+
  suc n ⊖ suc m ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟩

+
  n ⊖ m         ≤⟨ ⊖-monoʳ-≥-≤ n n≤m ⟩

+
  n ⊖ n         ≡⟨ n⊖n≡0 n ⟩

+
  0ℤ            ∎ where open ≤-Reasoning

+
i≤j⇒i-j≤0 {_}        {+0}       (+≤+ z≤n) = +≤+ z≤n

+
i≤j⇒i-j≤0 {_}        {+[1+ n ]} (+≤+ z≤n) = -≤+

+
i≤j⇒i-j≤0 {+[1+ m ]} {+[1+ n ]} (+≤+ (s≤s m≤n)) = begin

+
  suc m ⊖ suc n ≡⟨ [1+m]⊖[1+n]≡m⊖n m n ⟩

+
  m ⊖ n         ≤⟨ ⊖-monoʳ-≥-≤ m m≤n ⟩

+
  m ⊖ m         ≡⟨ n⊖n≡0 m ⟩

+
  0ℤ            ∎ where open ≤-Reasoning

+

+
i-j≤0⇒i≤j : i - j ≤ 0ℤ → i ≤ j

+
i-j≤0⇒i≤j {i} {j} i-j≤0 = begin

+
  i             ≡⟨ +-identityʳ i ⟨

+
  i + 0ℤ        ≡⟨ cong (_+_ i) (+-inverseˡ j) ⟨

+
  i + (- j + j) ≡⟨ +-assoc i (- j) j ⟨

+
  (i - j) + j   ≤⟨  +-monoˡ-≤ j i-j≤0 ⟩

+
  0ℤ + j        ≡⟨  +-identityˡ j ⟩

+
  j             ∎

+
  where open ≤-Reasoning

+

+
i≤j⇒0≤j-i : i ≤ j → 0ℤ ≤ j - i

+
i≤j⇒0≤j-i {i} {j} i≤j = begin

+
  0ℤ    ≡⟨ +-inverseʳ i ⟨

+
  i - i ≤⟨  +-monoˡ-≤ (- i) i≤j ⟩

+
  j - i ∎

+
  where open ≤-Reasoning

+

+
0≤i-j⇒j≤i : 0ℤ ≤ i - j → j ≤ i

+
0≤i-j⇒j≤i {i} {j} 0≤i-j = begin

+
  j             ≡⟨ +-identityˡ j ⟨

+
  0ℤ + j        ≤⟨  +-monoˡ-≤ j 0≤i-j ⟩

+
  i - j + j     ≡⟨  +-assoc i (- j) j ⟩

+
  i + (- j + j) ≡⟨  cong (_+_ i) (+-inverseˡ j) ⟩

+
  i + 0ℤ        ≡⟨  +-identityʳ i ⟩

+
  i             ∎

+
  where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of suc

+
------------------------------------------------------------------------

+

+
i≤j⇒i≤1+j : i ≤ j → i ≤ sucℤ j

+
i≤j⇒i≤1+j = i≤j⇒i≤k+j (+ 1)

+

+
i≤suc[i] : ∀ i → i ≤ sucℤ i

+
i≤suc[i] i = i≤j+i i (+ 1)

+

+
suc-+ : ∀ m n → +[1+ m ] + n ≡ sucℤ (+ m + n)

+
suc-+ m (+ n)      = refl

+
suc-+ m (-[1+ n ]) = sym (distribʳ-⊖-+-pos 1 m (suc n))

+

+
i≢suc[i] : i ≢ sucℤ i

+
i≢suc[i] {+ _}           ()

+
i≢suc[i] { -[1+ 0 ]}     ()

+
i≢suc[i] { -[1+ suc n ]} ()

+

+
1-[1+n]≡-n : ∀ n → sucℤ -[1+ n ] ≡ - (+ n)

+
1-[1+n]≡-n zero    = refl

+
1-[1+n]≡-n (suc n) = refl

+

+
suc-mono : sucℤ Preserves _≤_ ⟶ _≤_

+
suc-mono (-≤+ {m} {n}) = begin

+
  1 ⊖ suc m  ≡⟨ [1+m]⊖[1+n]≡m⊖n 0 m ⟩

+
  0 ⊖ m      ≤⟨ 0⊖m≤+ m ⟩

+
  sucℤ (+ n) ∎ where open ≤-Reasoning

+
suc-mono (-≤- n≤m) = ⊖-monoʳ-≥-≤ 1 (s≤s n≤m)

+
suc-mono (+≤+ m≤n) = +≤+ (s≤s m≤n)

+

+
suc[i]≤j⇒i<j : sucℤ i ≤ j → i < j

+
suc[i]≤j⇒i<j {+ i}           {+ _}       (+≤+ i≤j) = +<+ i≤j

+
suc[i]≤j⇒i<j { -[1+ 0 ]}     {+ j}       p         = -<+

+
suc[i]≤j⇒i<j { -[1+ suc i ]} {+ j}       -≤+       = -<+

+
suc[i]≤j⇒i<j { -[1+ suc i ]} { -[1+ j ]} (-≤- j≤i) = -<- (s≤s j≤i)

+

+
i<j⇒suc[i]≤j : i < j → sucℤ i ≤ j

+
i<j⇒suc[i]≤j {+ _}           {+ _}       (+<+ i<j) = +≤+ i<j

+
i<j⇒suc[i]≤j { -[1+ 0 ]}     {+ _}       -<+       = +≤+ z≤n

+
i<j⇒suc[i]≤j { -[1+ suc i ]} { -[1+ _ ]} (-<- j<i) = -≤- (s≤s⁻¹ j<i)

+
i<j⇒suc[i]≤j { -[1+ suc i ]} {+ _}       -<+       = -≤+

+

+
------------------------------------------------------------------------

+
-- Properties of pred

+
------------------------------------------------------------------------

+

+
suc-pred : ∀ i → sucℤ (pred i) ≡ i

+
suc-pred i = begin

+
  sucℤ (pred i) ≡⟨ +-assoc 1ℤ -1ℤ i ⟨

+
  0ℤ + i        ≡⟨  +-identityˡ i ⟩

+
  i             ∎ where open ≡-Reasoning

+

+
pred-suc : ∀ i → pred (sucℤ i) ≡ i

+
pred-suc i = begin

+
  pred (sucℤ i) ≡⟨ +-assoc -1ℤ 1ℤ i ⟨

+
  0ℤ + i        ≡⟨  +-identityˡ i ⟩

+
  i             ∎ where open ≡-Reasoning

+

+
+-pred : ∀ i j → i + pred j ≡ pred (i + j)

+
+-pred i j = begin

+
  i + (-1ℤ + j) ≡⟨ +-assoc i -1ℤ j ⟨

+
  i + -1ℤ + j   ≡⟨  cong (_+ j) (+-comm i -1ℤ) ⟩

+
  -1ℤ + i + j   ≡⟨  +-assoc -1ℤ i j ⟩

+
  -1ℤ + (i + j) ∎ where open ≡-Reasoning

+

+
pred-+ : ∀ i j → pred i + j ≡ pred (i + j)

+
pred-+ i j = begin

+
  pred i + j   ≡⟨ +-comm (pred i) j ⟩

+
  j + pred i   ≡⟨ +-pred j i ⟩

+
  pred (j + i) ≡⟨ cong pred (+-comm j i) ⟩

+
  pred (i + j) ∎ where open ≡-Reasoning

+

+
neg-suc : ∀ m → -[1+ m ] ≡ pred (- + m)

+
neg-suc zero    = refl

+
neg-suc (suc m) = refl

+

+
minus-suc : ∀ m n → m - +[1+ n ] ≡ pred (m - + n)

+
minus-suc m n = begin

+
  m + - +[1+ n ]     ≡⟨ cong (_+_ m) (neg-suc n) ⟩

+
  m + pred (- (+ n)) ≡⟨ +-pred m (- + n) ⟩

+
  pred (m - + n)     ∎ where open ≡-Reasoning

+

+
i≤pred[j]⇒i<j : i ≤ pred j → i < j

+
i≤pred[j]⇒i<j {_} { + n}      leq = ≤-<-trans leq (m⊖1+n<m n 1)

+
i≤pred[j]⇒i<j {_} { -[1+ n ]} leq = ≤-<-trans leq (-<- ℕ.≤-refl)

+

+
i<j⇒i≤pred[j] : i < j → i ≤ pred j

+
i<j⇒i≤pred[j] {_} { +0}       -<+       = -≤- z≤n

+
i<j⇒i≤pred[j] {_} { +[1+ n ]} -<+       = -≤+

+
i<j⇒i≤pred[j] {_} { +[1+ n ]} (+<+ m<n) = +≤+ (s≤s⁻¹ m<n)

+
i<j⇒i≤pred[j] {_} { -[1+ n ]} (-<- n<m) = -≤- n<m

+

+
i≤j⇒pred[i]≤j : i ≤ j → pred i ≤ j

+
i≤j⇒pred[i]≤j -≤+               = -≤+

+
i≤j⇒pred[i]≤j (-≤- n≤m)         = -≤- (ℕ.m≤n⇒m≤1+n n≤m)

+
i≤j⇒pred[i]≤j (+≤+ z≤n)         = -≤+

+
i≤j⇒pred[i]≤j (+≤+ (s≤s m≤n)) = +≤+ (ℕ.m≤n⇒m≤1+n m≤n)

+

+
pred-mono : pred Preserves _≤_ ⟶ _≤_

+
pred-mono (-≤+ {n = 0})     = -≤- z≤n

+
pred-mono (-≤+ {n = suc n}) = -≤+

+
pred-mono (-≤- n≤m)         = -≤- (s≤s n≤m)

+
pred-mono (+≤+ m≤n)         = ⊖-monoˡ-≤ 1 m≤n

+

+
------------------------------------------------------------------------

+
-- Properties of _*_

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
*-comm : Commutative _*_

+
*-comm -[1+ m ] -[1+ n ] rewrite ℕ.*-comm (suc m) (suc n) = refl

+
*-comm -[1+ m ] (+ n)    rewrite ℕ.*-comm (suc m) n       = refl

+
*-comm (+ m)    -[1+ n ] rewrite ℕ.*-comm m       (suc n) = refl

+
*-comm (+ m)    (+ n)    rewrite ℕ.*-comm m       n       = refl

+

+
*-identityˡ : LeftIdentity 1ℤ _*_

+
*-identityˡ -[1+ n ] rewrite ℕ.+-identityʳ n = refl

+
*-identityˡ +0       = refl

+
*-identityˡ +[1+ n ] rewrite ℕ.+-identityʳ n = refl

+

+
*-identityʳ : RightIdentity 1ℤ _*_

+
*-identityʳ = comm∧idˡ⇒idʳ *-comm *-identityˡ

+

+
*-identity : Identity 1ℤ _*_

+
*-identity = *-identityˡ , *-identityʳ

+

+
*-zeroˡ : LeftZero 0ℤ _*_

+
*-zeroˡ _ = refl

+

+
*-zeroʳ : RightZero 0ℤ _*_

+
*-zeroʳ = comm∧zeˡ⇒zeʳ *-comm *-zeroˡ

+

+
*-zero : Zero 0ℤ _*_

+
*-zero = *-zeroˡ , *-zeroʳ

+

+
*-assoc : Associative _*_

+
*-assoc +0 _ _ = refl

+
*-assoc i +0 _ rewrite ℕ.*-zeroʳ ∣ i ∣ = refl

+
*-assoc i j +0 rewrite

+
    ℕ.*-zeroʳ ∣ j ∣

+
  | ℕ.*-zeroʳ ∣ i ∣

+
  | ℕ.*-zeroʳ ∣ sign i Sign.* sign j ◃ ∣ i ∣ ℕ.* ∣ j ∣ ∣

+
  = refl

+
*-assoc -[1+ m ] -[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)

+
*-assoc -[1+ m ] +[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)

+
*-assoc +[1+ m ] +[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)

+
*-assoc +[1+ m ] -[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)

+
*-assoc -[1+ m ] -[1+ n ] -[1+ o ] = cong -[1+_] (inner-assoc m n o)

+
*-assoc -[1+ m ] +[1+ n ] +[1+ o ] = cong -[1+_] (inner-assoc m n o)

+
*-assoc +[1+ m ] -[1+ n ] +[1+ o ] = cong -[1+_] (inner-assoc m n o)

+
*-assoc +[1+ m ] +[1+ n ] -[1+ o ] = cong -[1+_] (inner-assoc m n o)

+

+
private

+

+
  -- lemma used to prove distributivity.

+
  distrib-lemma : ∀ m n o → (o ⊖ n) * -[1+ m ] ≡ m ℕ.+ n ℕ.* suc m ⊖ (m ℕ.+ o ℕ.* suc m)

+
  distrib-lemma m n o

+
    rewrite +-cancelˡ-⊖ m (n ℕ.* suc m) (o ℕ.* suc m)

+
          | ⊖-swap (n ℕ.* suc m) (o ℕ.* suc m)

+
    with n ℕ.≤? o

+
  ... | yes n≤o

+
    rewrite ⊖-≥ n≤o

+
          | ⊖-≥ (ℕ.*-mono-≤ n≤o (ℕ.≤-refl {x = suc m}))

+
          | -◃n≡-n ((o ∸ n) ℕ.* suc m)

+
          | ℕ.*-distribʳ-∸ (suc m) o n

+
          = refl

+
  ... | no n≰o

+
    rewrite sign-⊖-≰ n≰o

+
          | ∣⊖∣-≰ n≰o

+
          | +◃n≡+n ((n ∸ o) ℕ.* suc m)

+
          | ⊖-≰ (n≰o ∘ ℕ.*-cancelʳ-≤ n o (suc m))

+
          | neg-involutive (+ (n ℕ.* suc m ∸ o ℕ.* suc m))

+
          | ℕ.*-distribʳ-∸ (suc m) n o

+
          = refl

+

+
*-distribʳ-+ : _*_ DistributesOverʳ _+_

+
*-distribʳ-+ +0 y z

+
  rewrite ℕ.*-zeroʳ ∣ y ∣

+
        | ℕ.*-zeroʳ ∣ z ∣

+
        | ℕ.*-zeroʳ ∣ y + z ∣

+
        = refl

+
*-distribʳ-+ x +0 z

+
  rewrite +-identityˡ z

+
        | +-identityˡ (sign z Sign.* sign x ◃ ∣ z ∣ ℕ.* ∣ x ∣)

+
        = refl

+
*-distribʳ-+ x y +0

+
  rewrite +-identityʳ y

+
        | +-identityʳ (sign y Sign.* sign x ◃ ∣ y ∣ ℕ.* ∣ x ∣)

+
        = refl

+
*-distribʳ-+ -[1+ m ] -[1+ n ] -[1+ o ] = cong (+_) $ assoc₁ m n o

+
*-distribʳ-+ +[1+ m ] +[1+ n ] +[1+ o ] = cong +[1+_] $ ℕ.suc-injective (assoc₂ m n o)

+
*-distribʳ-+ -[1+ m ] +[1+ n ] +[1+ o ] = cong -[1+_] $ assoc₃ m n o

+
*-distribʳ-+ +[1+ m ] -[1+ n ] -[1+ o ] = cong -[1+_] $ assoc₄ m n o

+
*-distribʳ-+ -[1+ m ] -[1+ n ] +[1+ o ] = begin

+
  (suc o ⊖ suc n) * -[1+ m ]                ≡⟨ cong (_* -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n o n) ⟩

+
  (o ⊖ n) * -[1+ m ]                        ≡⟨ distrib-lemma m n o ⟩

+
  m ℕ.+ n ℕ.* suc m ⊖ (m ℕ.+ o ℕ.* suc m)   ≡⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ n ℕ.* suc m) (m ℕ.+ o ℕ.* suc m) ⟨

+
  -[1+ n ] * -[1+ m ] + +[1+ o ] * -[1+ m ] ∎ where open ≡-Reasoning

+
*-distribʳ-+ -[1+ m ] +[1+ n ] -[1+ o ] = begin

+
  (+[1+ n ] + -[1+ o ]) * -[1+ m ]          ≡⟨ cong (_* -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n n o) ⟩

+
  (n ⊖ o) * -[1+ m ]                        ≡⟨ distrib-lemma m o n ⟩

+
  m ℕ.+ o ℕ.* suc m ⊖ (m ℕ.+ n ℕ.* suc m)   ≡⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ o ℕ.* suc m) (m ℕ.+ n ℕ.* suc m) ⟨

+
  +[1+ n ] * -[1+ m ] + -[1+ o ] * -[1+ m ] ∎ where open ≡-Reasoning

+
*-distribʳ-+ +[1+ m ] -[1+ n ] +[1+ o ] with n ℕ.≤? o

+
... | yes n≤o

+
  rewrite [1+m]⊖[1+n]≡m⊖n o n

+
        | [1+m]⊖[1+n]≡m⊖n (m ℕ.+ o ℕ.* suc m) (m ℕ.+ n ℕ.* suc m)

+
        | +-cancelˡ-⊖ m (o ℕ.* suc m) (n ℕ.* suc m)

+
        | ⊖-≥ n≤o

+
        | +-comm (- (+ (m ℕ.+ n ℕ.* suc m))) (+ (m ℕ.+ o ℕ.* suc m))

+
        | ⊖-≥ (ℕ.*-mono-≤ n≤o (ℕ.≤-refl {x = suc m}))

+
        | ℕ.*-distribʳ-∸ (suc m) o n

+
        | +◃n≡+n (o ℕ.* suc m ∸ n ℕ.* suc m)

+
        = refl

+
... | no n≰o

+
  rewrite [1+m]⊖[1+n]≡m⊖n o n

+
        | [1+m]⊖[1+n]≡m⊖n (m ℕ.+ o ℕ.* suc m) (m ℕ.+ n ℕ.* suc m)

+
        | +-cancelˡ-⊖ m (o ℕ.* suc m) (n ℕ.* suc m)

+
        | sign-⊖-≰ n≰o

+
        | ∣⊖∣-≰ n≰o

+
        | -◃n≡-n ((n ∸ o) ℕ.* suc m)

+
        | ⊖-≰ (n≰o ∘ ℕ.*-cancelʳ-≤ n o (suc m))

+
        | ℕ.*-distribʳ-∸ (suc m) n o

+
        = refl

+
*-distribʳ-+ +[1+ o ] +[1+ m ] -[1+ n ] with n ℕ.≤? m

+
... | yes n≤m

+
  rewrite [1+m]⊖[1+n]≡m⊖n m n

+
        | [1+m]⊖[1+n]≡m⊖n (o ℕ.+ m ℕ.* suc o) (o ℕ.+ n ℕ.* suc o)

+
        | +-cancelˡ-⊖ o (m ℕ.* suc o) (n ℕ.* suc o)

+
        | ⊖-≥ n≤m

+
        | ⊖-≥ (ℕ.*-mono-≤ n≤m (ℕ.≤-refl {x = suc o}))

+
        | +◃n≡+n ((m ∸ n) ℕ.* suc o)

+
        | ℕ.*-distribʳ-∸ (suc o) m n

+
        = refl

+
... | no n≰m

+
  rewrite [1+m]⊖[1+n]≡m⊖n m n

+
        | [1+m]⊖[1+n]≡m⊖n (o ℕ.+ m ℕ.* suc o) (o ℕ.+ n ℕ.* suc o)

+
        | +-cancelˡ-⊖ o (m ℕ.* suc o) (n ℕ.* suc o)

+
        | sign-⊖-≰ n≰m

+
        | ∣⊖∣-≰ n≰m

+
        | ⊖-≰ (n≰m ∘ ℕ.*-cancelʳ-≤ n m (suc o))

+
        | -◃n≡-n ((n ∸ m) ℕ.* suc o)

+
        | ℕ.*-distribʳ-∸ (suc o) n m

+
        = refl

+

+
*-distribˡ-+ : _*_ DistributesOverˡ _+_

+
*-distribˡ-+ = comm∧distrʳ⇒distrˡ *-comm *-distribʳ-+

+

+
*-distrib-+ : _*_ DistributesOver _+_

+
*-distrib-+ = *-distribˡ-+ , *-distribʳ-+

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
*-isMagma : IsMagma _*_

+
*-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _*_

+
  }

+

+
*-isSemigroup : IsSemigroup _*_

+
*-isSemigroup = record

+
  { isMagma = *-isMagma

+
  ; assoc   = *-assoc

+
  }

+

+
*-isCommutativeSemigroup : IsCommutativeSemigroup _*_

+
*-isCommutativeSemigroup = record

+
  { isSemigroup = *-isSemigroup

+
  ; comm        = *-comm

+
  }

+

+
*-1-isMonoid : IsMonoid _*_ 1ℤ

+
*-1-isMonoid = record

+
  { isSemigroup = *-isSemigroup

+
  ; identity    = *-identity

+
  }

+

+
*-1-isCommutativeMonoid : IsCommutativeMonoid _*_ 1ℤ

+
*-1-isCommutativeMonoid = record

+
  { isMonoid = *-1-isMonoid

+
  ; comm     = *-comm

+
  }

+

+
+-*-isSemiring : IsSemiring _+_ _*_ 0ℤ 1ℤ

+
+-*-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = +-0-isCommutativeMonoid

+
    ; *-cong = cong₂ _*_

+
    ; *-assoc = *-assoc

+
    ; *-identity = *-identity

+
    ; distrib = *-distrib-+

+
    }

+
  ; zero = *-zero

+
  }

+

+
+-*-isCommutativeSemiring : IsCommutativeSemiring _+_ _*_ 0ℤ 1ℤ

+
+-*-isCommutativeSemiring = record

+
  { isSemiring = +-*-isSemiring

+
  ; *-comm = *-comm

+
  }

+

+
+-*-isRing : IsRing _+_ _*_ -_ 0ℤ 1ℤ

+
+-*-isRing = record

+
  { +-isAbelianGroup = +-0-isAbelianGroup

+
  ; *-cong           = cong₂ _*_

+
  ; *-assoc          = *-assoc

+
  ; *-identity       = *-identity

+
  ; distrib          = *-distrib-+

+
  }

+

+
+-*-isCommutativeRing : IsCommutativeRing _+_ _*_ -_ 0ℤ 1ℤ

+
+-*-isCommutativeRing = record

+
  { isRing = +-*-isRing

+
  ; *-comm = *-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
*-magma : Magma 0ℓ 0ℓ

+
*-magma = record

+
  { isMagma = *-isMagma

+
  }

+

+
*-semigroup : Semigroup 0ℓ 0ℓ

+
*-semigroup = record

+
  { isSemigroup = *-isSemigroup

+
  }

+

+
*-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ

+
*-commutativeSemigroup = record

+
  { isCommutativeSemigroup = *-isCommutativeSemigroup

+
  }

+

+
*-1-monoid : Monoid 0ℓ 0ℓ

+
*-1-monoid = record

+
  { isMonoid = *-1-isMonoid

+
  }

+

+
*-1-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
*-1-commutativeMonoid = record

+
  { isCommutativeMonoid = *-1-isCommutativeMonoid

+
  }

+

+
+-*-semiring : Semiring 0ℓ 0ℓ

+
+-*-semiring = record

+
  { isSemiring = +-*-isSemiring

+
  }

+

+
+-*-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ

+
+-*-commutativeSemiring = record

+
  { isCommutativeSemiring = +-*-isCommutativeSemiring

+
  }

+

+
+-*-ring : Ring 0ℓ 0ℓ

+
+-*-ring = record

+
  { isRing = +-*-isRing

+
  }

+

+
+-*-commutativeRing : CommutativeRing 0ℓ 0ℓ

+
+-*-commutativeRing = record

+
  { isCommutativeRing = +-*-isCommutativeRing

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _*_ and _≡_

+

+
abs-* : ℤtoℕ.Homomorphic₂ ∣_∣ _*_ ℕ._*_

+
abs-* i j = abs-◃ _ _

+

+
sign-* : ∀ i j → .{{NonZero (i * j)}} → sign (i * j) ≡ sign i Sign.* sign j

+
sign-* i j rewrite abs-* i j = sign-◃ (sign i Sign.* sign j) (∣ i ∣ ℕ.* ∣ j ∣)

+

+
*-cancelʳ-≡ : ∀ i j k .{{_ : NonZero k}} → i * k ≡ j * k → i ≡ j

+
*-cancelʳ-≡ i j k eq with sign-cong′ eq

+
... | inj₁ s[ik]≡s[jk] = ◃-cong

+
  (Sign.*-cancelʳ-≡ (sign k) (sign i) (sign j) s[ik]≡s[jk])

+
  (ℕ.*-cancelʳ-≡ ∣ i ∣ ∣ j ∣ _ (abs-cong eq))

+
... | inj₂ (∣ik∣≡0 , ∣jk∣≡0) = trans

+
  (∣i∣≡0⇒i≡0 (ℕ.m*n≡0⇒m≡0 _ _ ∣ik∣≡0))

+
  (sym (∣i∣≡0⇒i≡0 (ℕ.m*n≡0⇒m≡0 _ _ ∣jk∣≡0)))

+

+
*-cancelˡ-≡ : ∀ i j k .{{_ : NonZero i}} → i * j ≡ i * k → j ≡ k

+
*-cancelˡ-≡ i j k rewrite *-comm i j | *-comm i k = *-cancelʳ-≡ j k i

+

+
suc-* : ∀ i j → sucℤ i * j ≡ j + i * j

+
suc-* i j = begin

+
  sucℤ i * j      ≡⟨ *-distribʳ-+ j (+ 1) i ⟩

+
  + 1 * j + i * j ≡⟨ cong (_+ i * j) (*-identityˡ j) ⟩

+
  j + i * j       ∎

+
  where open ≡-Reasoning

+

+
*-suc : ∀ i j → i * sucℤ j ≡ i + i * j

+
*-suc i j = begin

+
  i * sucℤ j ≡⟨ *-comm i _ ⟩

+
  sucℤ j * i ≡⟨ suc-* j i ⟩

+
  i + j * i  ≡⟨ cong (λ v → i + v) (*-comm j i) ⟩

+
  i + i * j  ∎

+
  where open ≡-Reasoning

+

+
-1*i≡-i : ∀ i → -1ℤ * i ≡ - i

+
-1*i≡-i -[1+ n ] = cong +[1+_] (ℕ.+-identityʳ n)

+
-1*i≡-i +0       = refl

+
-1*i≡-i +[1+ n ] = cong -[1+_] (ℕ.+-identityʳ n)

+

+
i*j≡0⇒i≡0∨j≡0 : ∀ i {j} → i * j ≡ 0ℤ → i ≡ 0ℤ ⊎ j ≡ 0ℤ

+
i*j≡0⇒i≡0∨j≡0 i p with ℕ.m*n≡0⇒m≡0∨n≡0 ∣ i ∣ (abs-cong {t = Sign.+} p)

+
... | inj₁ ∣i∣≡0 = inj₁ (∣i∣≡0⇒i≡0 ∣i∣≡0)

+
... | inj₂ ∣j∣≡0 = inj₂ (∣i∣≡0⇒i≡0 ∣j∣≡0)

+

+
i*j≢0 : ∀ i j .{{_ : NonZero i}} .{{_ : NonZero j}} → NonZero (i * j)

+
i*j≢0 i j rewrite abs-* i j = ℕ.m*n≢0 ∣ i ∣ ∣ j ∣

+

+
------------------------------------------------------------------------

+
-- Properties of _^_

+
------------------------------------------------------------------------

+

+
^-identityʳ : ∀ i → i ^ 1 ≡ i

+
^-identityʳ =  *-identityʳ

+

+
^-zeroˡ : ∀ n → 1ℤ ^ n ≡ 1ℤ

+
^-zeroˡ zero  = refl

+
^-zeroˡ (suc n) = begin

+
  1ℤ ^ suc n    ≡⟨⟩

+
  1ℤ * (1ℤ ^ n) ≡⟨ *-identityˡ (1ℤ ^ n) ⟩

+
  1ℤ ^ n        ≡⟨ ^-zeroˡ n ⟩

+
  1ℤ            ∎

+
  where open ≡-Reasoning

+

+
^-distribˡ-+-* : ∀ i m n → i ^ (m ℕ.+ n) ≡ i ^ m * i ^ n

+
^-distribˡ-+-* i zero    n = sym (*-identityˡ (i ^ n))

+
^-distribˡ-+-* i (suc m) n = begin

+
  i * (i ^ (m ℕ.+ n))     ≡⟨ cong (i *_) (^-distribˡ-+-* i m n) ⟩

+
  i * ((i ^ m) * (i ^ n)) ≡⟨ sym (*-assoc i _ _) ⟩

+
  (i * (i ^ m)) * (i ^ n) ∎

+
  where open ≡-Reasoning

+

+
^-isMagmaHomomorphism : ∀ i → Morphism.IsMagmaHomomorphism ℕ.+-rawMagma *-rawMagma (i ^_)

+
^-isMagmaHomomorphism i = record

+
  { isRelHomomorphism = record { cong = cong (i ^_) }

+
  ; homo              = ^-distribˡ-+-* i

+
  }

+

+
^-isMonoidHomomorphism : ∀ i → Morphism.IsMonoidHomomorphism ℕ.+-0-rawMonoid *-1-rawMonoid (i ^_)

+
^-isMonoidHomomorphism i = record

+
  { isMagmaHomomorphism = ^-isMagmaHomomorphism i

+
  ; ε-homo              = refl

+
  }

+

+
^-*-assoc : ∀ i m n → (i ^ m) ^ n ≡ i ^ (m ℕ.* n)

+
^-*-assoc i m zero    = cong (i ^_) (sym $ ℕ.*-zeroʳ m)

+
^-*-assoc i m (suc n) = begin

+
  (i ^ m) * ((i ^ m) ^ n)   ≡⟨ cong ((i ^ m) *_) (^-*-assoc i m n) ⟩

+
  (i ^ m) * (i ^ (m ℕ.* n)) ≡⟨ sym (^-distribˡ-+-* i m (m ℕ.* n)) ⟩

+
  i ^ (m ℕ.+ m ℕ.* n)       ≡⟨ cong (i ^_) (sym (ℕ.*-suc m n)) ⟩

+
  i ^ (m ℕ.* suc n)         ∎

+
  where open ≡-Reasoning

+

+
i^n≡0⇒i≡0 : ∀ i n → i ^ n ≡ 0ℤ → i ≡ 0ℤ

+
i^n≡0⇒i≡0 i (suc n) eq = [ id , i^n≡0⇒i≡0 i n ]′ (i*j≡0⇒i≡0∨j≡0 i eq)

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and +_/-_

+

+
pos-* : ℕtoℤ.Homomorphic₂ +_ ℕ._*_ _*_

+
pos-* zero    n       = refl

+
pos-* (suc m) zero    = pos-* m zero

+
pos-* (suc m) (suc n) = refl

+

+
neg-distribˡ-* : ∀ i j → - (i * j) ≡ (- i) * j

+
neg-distribˡ-* i j = begin

+
  - (i * j)      ≡⟨ -1*i≡-i (i * j) ⟨

+
  -1ℤ * (i * j)  ≡⟨ *-assoc -1ℤ i j ⟨

+
  -1ℤ * i * j    ≡⟨ cong (_* j) (-1*i≡-i i) ⟩

+
  - i * j        ∎ where open ≡-Reasoning

+

+
neg-distribʳ-* : ∀ i j → - (i * j) ≡ i * (- j)

+
neg-distribʳ-* i j = begin

+
  - (i * j) ≡⟨ cong -_ (*-comm i j) ⟩

+
  - (j * i) ≡⟨ neg-distribˡ-* j i ⟩

+
  - j * i   ≡⟨ *-comm (- j) i ⟩

+
  i * (- j) ∎ where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _◃_

+

+
◃-distrib-* :  ∀ s t m n → (s Sign.* t) ◃ (m ℕ.* n) ≡ (s ◃ m) * (t ◃ n)

+
◃-distrib-* s t zero    zero    = refl

+
◃-distrib-* s t zero    (suc n) = refl

+
◃-distrib-* s t (suc m) zero    =

+
  trans

+
    (cong₂ _◃_ (Sign.*-comm s t) (ℕ.*-comm m 0))

+
    (*-comm (t ◃ zero) (s ◃ suc m))

+
◃-distrib-* s t (suc m) (suc n) =

+
  sym (cong₂ _◃_

+
    (cong₂ Sign._*_ (sign-◃ s (suc m)) (sign-◃ t (suc n)))

+
    (∣s◃m∣*∣t◃n∣≡m*n s t (suc m) (suc n)))

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _≤_

+

+
*-cancelʳ-≤-pos : ∀ i j k .{{_ : Positive k}} → i * k ≤ j * k → i ≤ j

+
*-cancelʳ-≤-pos -[1+ m ] -[1+ n ] +[1+ o ] (-≤- n≤m) =

+
  -≤- (s≤s⁻¹ (ℕ.*-cancelʳ-≤ (suc n) (suc m) (suc o) (s≤s n≤m)))

+
*-cancelʳ-≤-pos -[1+ _ ] (+ _)    +[1+ o ] _         = -≤+

+
*-cancelʳ-≤-pos +0       +0       +[1+ o ] _         = +≤+ z≤n

+
*-cancelʳ-≤-pos +0       +[1+ _ ] +[1+ o ] _         = +≤+ z≤n

+
*-cancelʳ-≤-pos +[1+ _ ] +0       +[1+ o ] (+≤+ ())

+
*-cancelʳ-≤-pos +[1+ m ] +[1+ n ] +[1+ o ] (+≤+ m≤n) =

+
  +≤+ (ℕ.*-cancelʳ-≤ (suc m) (suc n) (suc o) m≤n)

+

+
*-cancelˡ-≤-pos : ∀ i j k .{{_ : Positive k}} → k * i ≤ k * j → i ≤ j

+
*-cancelˡ-≤-pos i j k rewrite *-comm k i | *-comm k j = *-cancelʳ-≤-pos i j k

+

+
*-monoʳ-≤-nonNeg : ∀ i .{{_ : NonNegative i}} → (_* i) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤-nonNeg +0 {i} {j} i≤j rewrite *-zeroʳ i | *-zeroʳ j = +≤+ z≤n

+
*-monoʳ-≤-nonNeg +[1+ n ] (-≤+ {n = 0})         = -≤+

+
*-monoʳ-≤-nonNeg +[1+ n ] (-≤+ {n = suc _})     = -≤+

+
*-monoʳ-≤-nonNeg +[1+ n ] (-≤- n≤m) = -≤- (s≤s⁻¹ (ℕ.*-mono-≤ (s≤s n≤m) (ℕ.≤-refl {x = suc n})))

+
*-monoʳ-≤-nonNeg +[1+ n ] {+0}       {+0}       (+≤+ m≤n) = +≤+ m≤n

+
*-monoʳ-≤-nonNeg +[1+ n ] {+0}       {+[1+ _ ]} (+≤+ m≤n) = +≤+ z≤n

+
*-monoʳ-≤-nonNeg +[1+ n ] {+[1+ _ ]} {+[1+ _ ]} (+≤+ m≤n) = +≤+ (ℕ.*-monoˡ-≤ (suc n) m≤n)

+

+
*-monoˡ-≤-nonNeg : ∀ i .{{_ : NonNegative i}} → (i *_) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤-nonNeg i {j} {k} rewrite *-comm i j | *-comm i k = *-monoʳ-≤-nonNeg i

+

+
*-cancelˡ-≤-neg : ∀ i j k .{{_ : Negative i}} → i * j ≤ i * k → j ≥ k

+
*-cancelˡ-≤-neg i@(-[1+ _ ]) j k ij≤ik = neg-cancel-≤ (*-cancelˡ-≤-pos (- j) (- k) (- i) (begin

+
  - i * - j   ≡⟨ neg-distribʳ-* (- i) j ⟨

+
  -(- i * j)  ≡⟨  neg-distribˡ-* (- i) j ⟩

+
  i * j       ≤⟨  ij≤ik ⟩

+
  i * k       ≡⟨ neg-distribˡ-* (- i) k ⟨

+
  -(- i * k)  ≡⟨  neg-distribʳ-* (- i) k ⟩

+
  - i * - k   ∎))

+
  where open ≤-Reasoning

+

+
*-cancelʳ-≤-neg : ∀ i j k .{{_ : Negative k}} → i * k ≤ j * k → i ≥ j

+
*-cancelʳ-≤-neg i j k rewrite *-comm i k | *-comm j k = *-cancelˡ-≤-neg k i j

+

+
*-monoˡ-≤-nonPos : ∀ i .{{_ : NonPositive i}} → (i *_) Preserves _≤_ ⟶ _≥_

+
*-monoˡ-≤-nonPos +0           {j} {k} j≤k = +≤+ z≤n

+
*-monoˡ-≤-nonPos i@(-[1+ m ]) {j} {k} j≤k = begin

+
  i * k        ≡⟨ neg-distribˡ-* (- i) k ⟨

+
  -(- i * k)   ≡⟨  neg-distribʳ-* (- i) k ⟩

+
  - i * - k    ≤⟨  *-monoˡ-≤-nonNeg (- i) (neg-mono-≤ j≤k) ⟩

+
  - i * - j    ≡⟨ neg-distribʳ-* (- i) j ⟨

+
  -(- i * j)   ≡⟨  neg-distribˡ-* (- i) j ⟩

+
  i * j        ∎

+
  where open ≤-Reasoning

+

+
*-monoʳ-≤-nonPos : ∀ i .{{_ : NonPositive i}} → (_* i) Preserves _≤_ ⟶ _≥_

+
*-monoʳ-≤-nonPos i {j} {k} rewrite *-comm k i | *-comm j i = *-monoˡ-≤-nonPos i

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _<_

+

+
*-monoˡ-<-pos : ∀ i .{{_ : Positive i}} → (i *_) Preserves _<_ ⟶ _<_

+
*-monoˡ-<-pos +[1+ n ] {+ m}       {+ o}       (+<+ m<o) = +◃-mono-< (ℕ.+-mono-<-≤ m<o (ℕ.*-monoʳ-≤ n (ℕ.<⇒≤ m<o)))

+
*-monoˡ-<-pos +[1+ n ] { -[1+ m ]} {+ o}       leq       = -◃<+◃ _ (suc n ℕ.* o)

+
*-monoˡ-<-pos +[1+ n ] { -[1+ m ]} { -[1+ o ]} (-<- o<m) = -<- (ℕ.+-mono-<-≤ o<m (ℕ.*-monoʳ-≤ n (ℕ.<⇒≤ (s≤s o<m))))

+

+
*-monoʳ-<-pos : ∀ i .{{_ : Positive i}} → (_* i) Preserves _<_ ⟶ _<_

+
*-monoʳ-<-pos i {j} {k} rewrite *-comm j i | *-comm k i = *-monoˡ-<-pos i

+

+
*-cancelˡ-<-nonNeg : ∀ k .{{_ : NonNegative k}} → k * i < k * j → i < j

+
*-cancelˡ-<-nonNeg {+ i}       {+ j}       (+ n) leq = +<+ (ℕ.*-cancelˡ-< n _ _ (+◃-cancel-< leq))

+
*-cancelˡ-<-nonNeg {+ i}       { -[1+ j ]} (+ n) leq = contradiction leq +◃≮-◃

+
*-cancelˡ-<-nonNeg { -[1+ i ]} {+ j}       (+ n)leq = -<+

+
*-cancelˡ-<-nonNeg { -[1+ i ]} { -[1+ j ]} (+ n) leq = -<- (s<s⁻¹ (ℕ.*-cancelˡ-< n _ _ (neg◃-cancel-< leq)))

+

+
*-cancelʳ-<-nonNeg : ∀ k .{{_ : NonNegative k}} → i * k < j * k → i < j

+
*-cancelʳ-<-nonNeg {i} {j} k rewrite *-comm i k | *-comm j k = *-cancelˡ-<-nonNeg k

+

+
*-monoˡ-<-neg : ∀ i .{{_ : Negative i}} → (i *_) Preserves _<_ ⟶ _>_

+
*-monoˡ-<-neg i@(-[1+ _ ]) {j} {k} j<k = begin-strict

+
  i * k        ≡⟨ neg-distribˡ-* (- i) k ⟨

+
  -(- i * k)   ≡⟨  neg-distribʳ-* (- i) k ⟩

+
  - i * - k    <⟨  *-monoˡ-<-pos (- i) (neg-mono-< j<k) ⟩

+
  - i * - j    ≡⟨ neg-distribʳ-* (- i) j ⟨

+
  - (- i * j)  ≡⟨  neg-distribˡ-* (- i) j ⟩

+
  i * j        ∎

+
  where open ≤-Reasoning

+

+
*-monoʳ-<-neg : ∀ i .{{_ : Negative i}} → (_* i) Preserves _<_ ⟶ _>_

+
*-monoʳ-<-neg i {j} {k} rewrite *-comm k i | *-comm j i = *-monoˡ-<-neg i

+

+
*-cancelˡ-<-nonPos : ∀ k .{{_ : NonPositive k}} → k * i < k * j → i > j

+
*-cancelˡ-<-nonPos {i} {j} +0           (+<+ ())

+
*-cancelˡ-<-nonPos {i} {j} k@(-[1+ _ ]) ki<kj = neg-cancel-< (*-cancelˡ-<-nonNeg (- k) (begin-strict

+
  - k * - i   ≡⟨ neg-distribʳ-* (- k) i ⟨

+
  -(- k * i)  ≡⟨  neg-distribˡ-* (- k) i ⟩

+
  k * i       <⟨  ki<kj ⟩

+
  k * j       ≡⟨ neg-distribˡ-* (- k) j ⟨

+
  -(- k * j)  ≡⟨  neg-distribʳ-* (- k) j ⟩

+
  - k * - j   ∎))

+
  where open ≤-Reasoning

+

+
*-cancelʳ-<-nonPos : ∀ k .{{_ : NonPositive k}} → i * k < j * k → i > j

+
*-cancelʳ-<-nonPos {i} {j} k rewrite *-comm i k | *-comm j k = *-cancelˡ-<-nonPos k

+

+
*-cancelˡ-<-neg : ∀ n → -[1+ n ] * i < -[1+ n ] * j → i > j

+
*-cancelˡ-<-neg {i} {j} n = *-cancelˡ-<-nonPos -[1+ n ]

+

+
*-cancelʳ-<-neg : ∀ n → i * -[1+ n ] < j * -[1+ n ] → i > j

+
*-cancelʳ-<-neg {i} {j} n = *-cancelʳ-<-nonPos -[1+ n ]

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and ∣_∣

+

+
∣i*j∣≡∣i∣*∣j∣ : ∀ i j → ∣ i * j ∣ ≡ ∣ i ∣ ℕ.* ∣ j ∣

+
∣i*j∣≡∣i∣*∣j∣ = abs-*

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_ and _⊔_

+
------------------------------------------------------------------------

+
-- Basic specification in terms of _≤_

+

+
i≤j⇒i⊓j≡i : i ≤ j → i ⊓ j ≡ i

+
i≤j⇒i⊓j≡i (-≤- i≥j) = cong -[1+_] (ℕ.m≥n⇒m⊔n≡m i≥j)

+
i≤j⇒i⊓j≡i -≤+       = refl

+
i≤j⇒i⊓j≡i (+≤+ i≤j) = cong +_ (ℕ.m≤n⇒m⊓n≡m i≤j)

+

+
i≥j⇒i⊓j≡j : i ≥ j → i ⊓ j ≡ j

+
i≥j⇒i⊓j≡j (-≤- i≥j) = cong -[1+_] (ℕ.m≤n⇒m⊔n≡n i≥j)

+
i≥j⇒i⊓j≡j -≤+       = refl

+
i≥j⇒i⊓j≡j (+≤+ i≤j) = cong +_ (ℕ.m≥n⇒m⊓n≡n i≤j)

+

+
i≤j⇒i⊔j≡j : i ≤ j → i ⊔ j ≡ j

+
i≤j⇒i⊔j≡j (-≤- i≥j) = cong -[1+_] (ℕ.m≥n⇒m⊓n≡n i≥j)

+
i≤j⇒i⊔j≡j -≤+       = refl

+
i≤j⇒i⊔j≡j (+≤+ i≤j) = cong +_ (ℕ.m≤n⇒m⊔n≡n i≤j)

+

+
i≥j⇒i⊔j≡i : i ≥ j → i ⊔ j ≡ i

+
i≥j⇒i⊔j≡i (-≤- i≥j) = cong -[1+_] (ℕ.m≤n⇒m⊓n≡m i≥j)

+
i≥j⇒i⊔j≡i -≤+       = refl

+
i≥j⇒i⊔j≡i (+≤+ i≤j) = cong +_ (ℕ.m≥n⇒m⊔n≡m i≤j)

+

+
⊓-operator : MinOperator ≤-totalPreorder

+
⊓-operator = record

+
  { x≤y⇒x⊓y≈x = i≤j⇒i⊓j≡i

+
  ; x≥y⇒x⊓y≈y = i≥j⇒i⊓j≡j

+
  }

+

+
⊔-operator : MaxOperator ≤-totalPreorder

+
⊔-operator = record

+
  { x≤y⇒x⊔y≈y = i≤j⇒i⊔j≡j

+
  ; x≥y⇒x⊔y≈x = i≥j⇒i⊔j≡i

+
  }

+

+
------------------------------------------------------------------------

+
-- Automatically derived properties of _⊓_ and _⊔_

+

+
private

+
  module ⊓-⊔-properties        = MinMaxOp        ⊓-operator ⊔-operator

+
  module ⊓-⊔-latticeProperties = LatticeMinMaxOp ⊓-operator ⊔-operator

+

+
open ⊓-⊔-properties public

+
  using

+
  ( ⊓-idem                    -- : Idempotent _⊓_

+
  ; ⊓-sel                     -- : Selective _⊓_

+
  ; ⊓-assoc                   -- : Associative _⊓_

+
  ; ⊓-comm                    -- : Commutative _⊓_

+

+
  ; ⊔-idem                    -- : Idempotent _⊔_

+
  ; ⊔-sel                     -- : Selective _⊔_

+
  ; ⊔-assoc                   -- : Associative _⊔_

+
  ; ⊔-comm                    -- : Commutative _⊔_

+

+
  ; ⊓-distribˡ-⊔              -- : _⊓_ DistributesOverˡ _⊔_

+
  ; ⊓-distribʳ-⊔              -- : _⊓_ DistributesOverʳ _⊔_

+
  ; ⊓-distrib-⊔               -- : _⊓_ DistributesOver  _⊔_

+
  ; ⊔-distribˡ-⊓              -- : _⊔_ DistributesOverˡ _⊓_

+
  ; ⊔-distribʳ-⊓              -- : _⊔_ DistributesOverʳ _⊓_

+
  ; ⊔-distrib-⊓               -- : _⊔_ DistributesOver  _⊓_

+
  ; ⊓-absorbs-⊔               -- : _⊓_ Absorbs _⊔_

+
  ; ⊔-absorbs-⊓               -- : _⊔_ Absorbs _⊓_

+
  ; ⊔-⊓-absorptive            -- : Absorptive _⊔_ _⊓_

+
  ; ⊓-⊔-absorptive            -- : Absorptive _⊓_ _⊔_

+

+
  ; ⊓-isMagma                 -- : IsMagma _⊓_

+
  ; ⊓-isSemigroup             -- : IsSemigroup _⊓_

+
  ; ⊓-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊓_

+
  ; ⊓-isBand                  -- : IsBand _⊓_

+
  ; ⊓-isSelectiveMagma        -- : IsSelectiveMagma _⊓_

+

+
  ; ⊔-isMagma                 -- : IsMagma _⊔_

+
  ; ⊔-isSemigroup             -- : IsSemigroup _⊔_

+
  ; ⊔-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊔_

+
  ; ⊔-isBand                  -- : IsBand _⊔_

+
  ; ⊔-isSelectiveMagma        -- : IsSelectiveMagma _⊔_

+

+
  ; ⊓-magma                   -- : Magma _ _

+
  ; ⊓-semigroup               -- : Semigroup _ _

+
  ; ⊓-band                    -- : Band _ _

+
  ; ⊓-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊓-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊔-magma                   -- : Magma _ _

+
  ; ⊔-semigroup               -- : Semigroup _ _

+
  ; ⊔-band                    -- : Band _ _

+
  ; ⊔-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊔-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊓-glb                     -- : ∀ {m n o} → m ≥ o → n ≥ o → m ⊓ n ≥ o

+
  ; ⊓-triangulate             -- : ∀ m n o → m ⊓ n ⊓ o ≡ (m ⊓ n) ⊓ (n ⊓ o)

+
  ; ⊓-mono-≤                  -- : _⊓_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊓-monoˡ-≤                 -- : ∀ n → (_⊓ n) Preserves _≤_ ⟶ _≤_

+
  ; ⊓-monoʳ-≤                 -- : ∀ n → (n ⊓_) Preserves _≤_ ⟶ _≤_

+

+
  ; ⊔-lub                     -- : ∀ {m n o} → m ≤ o → n ≤ o → m ⊔ n ≤ o

+
  ; ⊔-triangulate             -- : ∀ m n o → m ⊔ n ⊔ o ≡ (m ⊔ n) ⊔ (n ⊔ o)

+
  ; ⊔-mono-≤                  -- : _⊔_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊔-monoˡ-≤                 -- : ∀ n → (_⊔ n) Preserves _≤_ ⟶ _≤_

+
  ; ⊔-monoʳ-≤                 -- : ∀ n → (n ⊔_) Preserves _≤_ ⟶ _≤_

+
  )

+
  renaming

+
  ( x⊓y≈y⇒y≤x to i⊓j≡j⇒j≤i    -- : ∀ {i j} → i ⊓ j ≡ j → j ≤ i

+
  ; x⊓y≈x⇒x≤y to i⊓j≡i⇒i≤j    -- : ∀ {i j} → i ⊓ j ≡ i → i ≤ j

+
  ; x⊓y≤x     to i⊓j≤i        -- : ∀ i j → i ⊓ j ≤ i

+
  ; x⊓y≤y     to i⊓j≤j        -- : ∀ i j → i ⊓ j ≤ j

+
  ; x≤y⇒x⊓z≤y to i≤j⇒i⊓k≤j    -- : ∀ {i j} k → i ≤ j → i ⊓ k ≤ j

+
  ; x≤y⇒z⊓x≤y to i≤j⇒k⊓i≤j    -- : ∀ {i j} k → i ≤ j → k ⊓ i ≤ j

+
  ; x≤y⊓z⇒x≤y to i≤j⊓k⇒i≤j    -- : ∀ {i} j k → i ≤ j ⊓ k → i ≤ j

+
  ; x≤y⊓z⇒x≤z to i≤j⊓k⇒i≤k    -- : ∀ {i} j k → i ≤ j ⊓ k → i ≤ k

+

+
  ; x⊔y≈y⇒x≤y to i⊔j≡j⇒i≤j    -- : ∀ {i j} → i ⊔ j ≡ j → i ≤ j

+
  ; x⊔y≈x⇒y≤x to i⊔j≡i⇒j≤i    -- : ∀ {i j} → i ⊔ j ≡ i → j ≤ i

+
  ; x≤x⊔y     to i≤i⊔j        -- : ∀ i j → i ≤ i ⊔ j

+
  ; x≤y⊔x     to i≤j⊔i        -- : ∀ i j → i ≤ j ⊔ i

+
  ; x≤y⇒x≤y⊔z to i≤j⇒i≤j⊔k    -- : ∀ {i j} k → i ≤ j → i ≤ j ⊔ k

+
  ; x≤y⇒x≤z⊔y to i≤j⇒i≤k⊔j    -- : ∀ {i j} k → i ≤ j → i ≤ k ⊔ j

+
  ; x⊔y≤z⇒x≤z to i⊔j≤k⇒i≤k    -- : ∀ i j {k} → i ⊔ j ≤ k → i ≤ k

+
  ; x⊔y≤z⇒y≤z to i⊔j≤k⇒j≤k    -- : ∀ i j {k} → i ⊔ j ≤ k → j ≤ k

+

+
  ; x⊓y≤x⊔y   to i⊓j≤i⊔j      -- : ∀ i j → i ⊓ j ≤ i ⊔ j

+
  )

+

+
open ⊓-⊔-latticeProperties public

+
  using

+
  ( ⊓-isSemilattice           -- : IsSemilattice _⊓_

+
  ; ⊔-isSemilattice           -- : IsSemilattice _⊔_

+
  ; ⊔-⊓-isLattice             -- : IsLattice _⊔_ _⊓_

+
  ; ⊓-⊔-isLattice             -- : IsLattice _⊓_ _⊔_

+
  ; ⊔-⊓-isDistributiveLattice -- : IsDistributiveLattice _⊔_ _⊓_

+
  ; ⊓-⊔-isDistributiveLattice -- : IsDistributiveLattice _⊓_ _⊔_

+

+
  ; ⊓-semilattice             -- : Semilattice _ _

+
  ; ⊔-semilattice             -- : Semilattice _ _

+
  ; ⊔-⊓-lattice               -- : Lattice _ _

+
  ; ⊓-⊔-lattice               -- : Lattice _ _

+
  ; ⊔-⊓-distributiveLattice   -- : DistributiveLattice _ _

+
  ; ⊓-⊔-distributiveLattice   -- : DistributiveLattice _ _

+
  )

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_ and _⊔_

+

+
mono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ i j → f (i ⊔ j) ≡ f i ⊔ f j

+
mono-≤-distrib-⊔ {f} = ⊓-⊔-properties.mono-≤-distrib-⊔ (cong f)

+

+
mono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ i j → f (i ⊓ j) ≡ f i ⊓ f j

+
mono-≤-distrib-⊓ {f} = ⊓-⊔-properties.mono-≤-distrib-⊓ (cong f)

+

+
antimono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ i j → f (i ⊓ j) ≡ f i ⊔ f j

+
antimono-≤-distrib-⊓ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊓ (cong f)

+

+
antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ i j → f (i ⊔ j) ≡ f i ⊓ f j

+
antimono-≤-distrib-⊔ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊔ (cong f)

+

+
mono-<-distrib-⊓ : ∀ f → f Preserves _<_ ⟶ _<_ → ∀ i j → f (i ⊓ j) ≡ f i ⊓ f j

+
mono-<-distrib-⊓ f f-mono-< i j with <-cmp i j

+
... | tri< i<j _    _   = trans (cong f (i≤j⇒i⊓j≡i (<⇒≤ i<j))) (sym (i≤j⇒i⊓j≡i (<⇒≤ (f-mono-< i<j))))

+
... | tri≈ _   refl _   = trans (cong f (i≤j⇒i⊓j≡i ≤-refl))    (sym (i≤j⇒i⊓j≡i ≤-refl))

+
... | tri> _   _    i>j = trans (cong f (i≥j⇒i⊓j≡j (<⇒≤ i>j))) (sym (i≥j⇒i⊓j≡j (<⇒≤ (f-mono-< i>j))))

+

+
mono-<-distrib-⊔ : ∀ f → f Preserves _<_ ⟶ _<_ → ∀ i j → f (i ⊔ j) ≡ f i ⊔ f j

+
mono-<-distrib-⊔ f f-mono-< i j with <-cmp i j

+
... | tri< i<j _    _   = trans (cong f (i≤j⇒i⊔j≡j (<⇒≤ i<j))) (sym (i≤j⇒i⊔j≡j (<⇒≤ (f-mono-< i<j))))

+
... | tri≈ _   refl _   = trans (cong f (i≤j⇒i⊔j≡j ≤-refl))    (sym (i≤j⇒i⊔j≡j ≤-refl))

+
... | tri> _   _    i>j = trans (cong f (i≥j⇒i⊔j≡i (<⇒≤ i>j))) (sym (i≥j⇒i⊔j≡i (<⇒≤ (f-mono-< i>j))))

+

+
antimono-<-distrib-⊔ : ∀ f  → f Preserves _<_ ⟶ _>_ → ∀ i j → f (i ⊔ j) ≡ f i ⊓ f j

+
antimono-<-distrib-⊔ f f-mono-< i j with <-cmp i j

+
... | tri< i<j _    _   = trans (cong f (i≤j⇒i⊔j≡j (<⇒≤ i<j))) (sym (i≥j⇒i⊓j≡j (<⇒≤ (f-mono-< i<j))))

+
... | tri≈ _   refl _   = trans (cong f (i≤j⇒i⊔j≡j ≤-refl))    (sym (i≥j⇒i⊓j≡j ≤-refl))

+
... | tri> _   _    i>j = trans (cong f (i≥j⇒i⊔j≡i (<⇒≤ i>j))) (sym (i≤j⇒i⊓j≡i (<⇒≤ (f-mono-< i>j))))

+

+
antimono-<-distrib-⊓ : ∀ f → f Preserves _<_ ⟶ _>_ → ∀ i j → f (i ⊓ j) ≡ f i ⊔ f j

+
antimono-<-distrib-⊓ f f-mono-< i j with <-cmp i j

+
... | tri< i<j _    _   = trans (cong f (i≤j⇒i⊓j≡i (<⇒≤ i<j))) (sym (i≥j⇒i⊔j≡i (<⇒≤ (f-mono-< i<j))))

+
... | tri≈ _   refl _   = trans (cong f (i≤j⇒i⊓j≡i ≤-refl))    (sym (i≥j⇒i⊔j≡i ≤-refl))

+
... | tri> _   _    i>j = trans (cong f (i≥j⇒i⊓j≡j (<⇒≤ i>j))) (sym (i≤j⇒i⊔j≡j (<⇒≤ (f-mono-< i>j))))

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_, _⊔_ and -_

+

+
neg-distrib-⊔-⊓ : ∀ i j → - (i ⊔ j) ≡ - i ⊓ - j

+
neg-distrib-⊔-⊓ = antimono-<-distrib-⊔ -_ neg-mono-<

+

+
neg-distrib-⊓-⊔ : ∀ i j → - (i ⊓ j) ≡ - i ⊔ - j

+
neg-distrib-⊓-⊔ = antimono-<-distrib-⊓ -_ neg-mono-<

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_, _⊔_ and _*_

+

+
*-distribˡ-⊓-nonNeg : ∀ i j k .{{_ : NonNegative i}} →

+
                      i * (j ⊓ k) ≡ (i * j) ⊓ (i * k)

+
*-distribˡ-⊓-nonNeg i j k = mono-≤-distrib-⊓ (*-monoˡ-≤-nonNeg i) j k

+

+
*-distribʳ-⊓-nonNeg : ∀ i j k .{{_ : NonNegative i}} →

+
                      (j ⊓ k) * i ≡ (j * i) ⊓ (k * i)

+
*-distribʳ-⊓-nonNeg i j k = mono-≤-distrib-⊓ (*-monoʳ-≤-nonNeg i) j k

+

+
*-distribˡ-⊓-nonPos : ∀ i j k .{{_ : NonPositive i}} →

+
                      i * (j ⊓ k) ≡ (i * j) ⊔ (i * k)

+
*-distribˡ-⊓-nonPos i j k = antimono-≤-distrib-⊓ (*-monoˡ-≤-nonPos i) j k

+

+
*-distribʳ-⊓-nonPos : ∀ i j k .{{_ : NonPositive i}} →

+
                      (j ⊓ k) * i ≡ (j * i) ⊔ (k * i)

+
*-distribʳ-⊓-nonPos i j k = antimono-≤-distrib-⊓ (*-monoʳ-≤-nonPos i) j k

+

+
*-distribˡ-⊔-nonNeg : ∀ i j k .{{_ : NonNegative i}} →

+
                      i * (j ⊔ k) ≡ (i * j) ⊔ (i * k)

+
*-distribˡ-⊔-nonNeg i j k = mono-≤-distrib-⊔ (*-monoˡ-≤-nonNeg i) j k

+

+
*-distribʳ-⊔-nonNeg : ∀ i j k .{{_ : NonNegative i}} →

+
                      (j ⊔ k) * i ≡ (j * i) ⊔ (k * i)

+
*-distribʳ-⊔-nonNeg i j k = mono-≤-distrib-⊔ (*-monoʳ-≤-nonNeg i) j k

+

+
*-distribˡ-⊔-nonPos : ∀ i j k .{{_ : NonPositive i}} →

+
                      i * (j ⊔ k) ≡ (i * j) ⊓ (i * k)

+
*-distribˡ-⊔-nonPos i j k = antimono-≤-distrib-⊔ (*-monoˡ-≤-nonPos i) j k

+

+
*-distribʳ-⊔-nonPos : ∀ i j k .{{_ : NonPositive i}} →

+
                      (j ⊔ k) * i ≡ (j * i) ⊓ (k * i)

+
*-distribʳ-⊔-nonPos i j k = antimono-≤-distrib-⊔ (*-monoʳ-≤-nonPos i) j k

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.5

+

+
neg-mono-<-> = neg-mono-<

+
{-# WARNING_ON_USAGE neg-mono-<->

+
"Warning: neg-mono-<-> was deprecated in v1.5.

+
Please use neg-mono-< instead."

+
#-}

+

+
neg-mono-≤-≥ = neg-mono-≤

+
{-# WARNING_ON_USAGE neg-mono-≤-≥

+
"Warning: neg-mono-≤-≥ was deprecated in v1.5.

+
Please use neg-mono-≤ instead."

+
#-}

+

+
*-monoʳ-≤-non-neg = *-monoʳ-≤-nonNeg

+
{-# WARNING_ON_USAGE *-monoʳ-≤-non-neg

+
"Warning: *-monoʳ-≤-non-neg was deprecated in v1.5.

+
Please use *-monoʳ-≤-nonNeg instead."

+
#-}

+

+
*-monoˡ-≤-non-neg = *-monoˡ-≤-nonNeg

+
{-# WARNING_ON_USAGE *-monoˡ-≤-non-neg

+
"Warning: *-monoˡ-≤-non-neg deprecated in v1.5.

+
Please use *-monoˡ-≤-nonNeg instead."

+
#-}

+

+
*-cancelˡ-<-non-neg = *-cancelˡ-<-nonNeg

+
{-# WARNING_ON_USAGE *-cancelˡ-<-non-neg

+
"Warning: *-cancelˡ-<-non-neg was deprecated in v1.5.

+
Please use *-cancelˡ-<-nonNeg instead."

+
#-}

+

+
*-cancelʳ-<-non-neg = *-cancelʳ-<-nonNeg

+
{-# WARNING_ON_USAGE *-cancelʳ-<-non-neg

+
"Warning: *-cancelʳ-<-non-neg was deprecated in v1.5.

+
Please use *-cancelʳ-<-nonNeg instead."

+
#-}

+

+
-- Version 1.6

+

+
m≤n⇒m⊓n≡m = i≤j⇒i⊓j≡i

+
{-# WARNING_ON_USAGE m≤n⇒m⊓n≡m

+
"Warning: m≤n⇒m⊓n≡m was deprecated in v1.6

+
Please use i≤j⇒i⊓j≡i instead."

+
#-}

+
m⊓n≡m⇒m≤n = i⊓j≡i⇒i≤j

+
{-# WARNING_ON_USAGE m⊓n≡m⇒m≤n

+
"Warning: m≤n⇒m⊓n≡m was deprecated in v1.6

+
Please use i⊓j≡i⇒i≤j instead."

+
#-}

+
m≥n⇒m⊓n≡n = i≥j⇒i⊓j≡j

+
{-# WARNING_ON_USAGE m≥n⇒m⊓n≡n

+
"Warning: m≥n⇒m⊓n≡n was deprecated in v1.6

+
Please use i≥j⇒i⊓j≡j instead."

+
#-}

+
m⊓n≡n⇒m≥n = i⊓j≡j⇒j≤i

+
{-# WARNING_ON_USAGE m⊓n≡n⇒m≥n

+
"Warning: m⊓n≡n⇒m≥n was deprecated in v1.6

+
Please use i⊓j≡j⇒j≤i instead."

+
#-}

+
m⊓n≤n = i⊓j≤j

+
{-# WARNING_ON_USAGE m⊓n≤n

+
"Warning: m⊓n≤n was deprecated in v1.6

+
Please use i⊓j≤j instead."

+
#-}

+
m⊓n≤m = i⊓j≤i

+
{-# WARNING_ON_USAGE m⊓n≤m

+
"Warning: m⊓n≤m was deprecated in v1.6

+
Please use i⊓j≤i instead."

+
#-}

+
m≤n⇒m⊔n≡n = i≤j⇒i⊔j≡j

+
{-# WARNING_ON_USAGE m≤n⇒m⊔n≡n

+
"Warning: m≤n⇒m⊔n≡n was deprecated in v1.6

+
Please use i≤j⇒i⊔j≡j instead."

+
#-}

+
m⊔n≡n⇒m≤n = i⊔j≡j⇒i≤j

+
{-# WARNING_ON_USAGE m⊔n≡n⇒m≤n

+
"Warning: m⊔n≡n⇒m≤n was deprecated in v1.6

+
Please use i⊔j≡j⇒i≤j instead."

+
#-}

+
m≥n⇒m⊔n≡m = i≥j⇒i⊔j≡i

+
{-# WARNING_ON_USAGE m≥n⇒m⊔n≡m

+
"Warning: m≥n⇒m⊔n≡m was deprecated in v1.6

+
Please use i≥j⇒i⊔j≡i instead."

+
#-}

+
m⊔n≡m⇒m≥n = i⊔j≡i⇒j≤i

+
{-# WARNING_ON_USAGE m⊔n≡m⇒m≥n

+
"Warning: m⊔n≡m⇒m≥n was deprecated in v1.6

+
Please use i⊔j≡i⇒j≤i instead."

+
#-}

+
m≤m⊔n = i≤i⊔j

+
{-# WARNING_ON_USAGE m≤m⊔n

+
"Warning: m≤m⊔n was deprecated in v1.6

+
Please use i≤i⊔j instead."

+
#-}

+
n≤m⊔n = i≤j⊔i

+
{-# WARNING_ON_USAGE n≤m⊔n

+
"Warning: n≤m⊔n was deprecated in v1.6

+
Please use i≤j⊔i instead."

+
#-}

+

+
-- Version 2.0

+

+
+-pos-monoʳ-≤ : ∀ n → (_+_ (+ n)) Preserves _≤_ ⟶ _≤_

+
+-pos-monoʳ-≤ n {_}         (-≤- o≤m) = ⊖-monoʳ-≥-≤ n (s≤s o≤m)

+
+-pos-monoʳ-≤ n { -[1+ m ]} -≤+       = ≤-trans (m⊖n≤m n (suc m)) (+≤+ (ℕ.m≤m+n n _))

+
+-pos-monoʳ-≤ n {_}         (+≤+ m≤o) = +≤+ (ℕ.+-monoʳ-≤ n m≤o)

+
{-# WARNING_ON_USAGE +-pos-monoʳ-≤

+
"Warning: +-pos-monoʳ-≤ was deprecated in v2.0

+
Please use +-monoʳ-≤ instead."

+
#-}

+
+-neg-monoʳ-≤ : ∀ n → (_+_ (-[1+ n ])) Preserves _≤_ ⟶ _≤_

+
+-neg-monoʳ-≤ n {_} {_}   (-≤- n≤m) = -≤- (ℕ.+-monoʳ-≤ (suc n) n≤m)

+
+-neg-monoʳ-≤ n {_} {+ m} -≤+       = ≤-trans (-≤- (ℕ.m≤m+n (suc n) _)) (-1+m≤n⊖m (suc n) m)

+
+-neg-monoʳ-≤ n {_} {_}   (+≤+ m≤n) = ⊖-monoˡ-≤ (suc n) m≤n

+
{-# WARNING_ON_USAGE +-neg-monoʳ-≤

+
"Warning: +-neg-monoʳ-≤ was deprecated in v2.0

+
Please use +-monoʳ-≤ instead."

+
#-}

+
n≮n = i≮i

+
{-# WARNING_ON_USAGE n≮n

+
"Warning: n≮n was deprecated in v2.0

+
Please use i≮i instead."

+
#-}

+
∣n∣≡0⇒n≡0 = ∣i∣≡0⇒i≡0

+
{-# WARNING_ON_USAGE ∣n∣≡0⇒n≡0

+
"Warning: ∣n∣≡0⇒n≡0 was deprecated in v2.0

+
Please use ∣i∣≡0⇒i≡0 instead."

+
#-}

+
∣-n∣≡∣n∣ = ∣-i∣≡∣i∣

+
{-# WARNING_ON_USAGE ∣-n∣≡∣n∣

+
"Warning: ∣-n∣≡∣n∣ was deprecated in v2.0

+
Please use ∣-i∣≡∣i∣ instead."

+
#-}

+
0≤n⇒+∣n∣≡n = 0≤i⇒+∣i∣≡i

+
{-# WARNING_ON_USAGE 0≤n⇒+∣n∣≡n

+
"Warning: 0≤n⇒+∣n∣≡n was deprecated in v2.0

+
Please use 0≤i⇒+∣i∣≡i instead."

+
#-}

+
+∣n∣≡n⇒0≤n = +∣i∣≡i⇒0≤i

+
{-# WARNING_ON_USAGE +∣n∣≡n⇒0≤n

+
"Warning: +∣n∣≡n⇒0≤n was deprecated in v2.0

+
Please use +∣i∣≡i⇒0≤i instead."

+
#-}

+
+∣n∣≡n⊎+∣n∣≡-n = +∣i∣≡i⊎+∣i∣≡-i

+
{-# WARNING_ON_USAGE +∣n∣≡n⊎+∣n∣≡-n

+
"Warning: +∣n∣≡n⊎+∣n∣≡-n was deprecated in v2.0

+
Please use +∣i∣≡i⊎+∣i∣≡-i instead."

+
#-}

+
∣m+n∣≤∣m∣+∣n∣ = ∣i+j∣≤∣i∣+∣j∣

+
{-# WARNING_ON_USAGE ∣m+n∣≤∣m∣+∣n∣

+
"Warning: ∣m+n∣≤∣m∣+∣n∣ was deprecated in v2.0

+
Please use ∣i+j∣≤∣i∣+∣j∣ instead."

+
#-}

+
∣m-n∣≤∣m∣+∣n∣ = ∣i-j∣≤∣i∣+∣j∣

+
{-# WARNING_ON_USAGE ∣m-n∣≤∣m∣+∣n∣

+
"Warning: ∣m-n∣≤∣m∣+∣n∣ was deprecated in v2.0

+
Please use ∣i-j∣≤∣i∣+∣j∣ instead."

+
#-}

+
signₙ◃∣n∣≡n = signᵢ◃∣i∣≡i

+
{-# WARNING_ON_USAGE signₙ◃∣n∣≡n

+
"Warning: signₙ◃∣n∣≡n was deprecated in v2.0

+
Please use signᵢ◃∣i∣≡i instead."

+
#-}

+
◃-≡ = ◃-cong

+
{-# WARNING_ON_USAGE ◃-≡

+
"Warning: ◃-≡ was deprecated in v2.0

+
Please use ◃-cong instead."

+
#-}

+
∣m-n∣≡∣n-m∣ = ∣i-j∣≡∣j-i∣

+
{-# WARNING_ON_USAGE ∣m-n∣≡∣n-m∣

+
"Warning: ∣m-n∣≡∣n-m∣ was deprecated in v2.0

+
Please use ∣i-j∣≡∣j-i∣ instead."

+
#-}

+
m≡n⇒m-n≡0 = i≡j⇒i-j≡0

+
{-# WARNING_ON_USAGE m≡n⇒m-n≡0

+
"Warning: m≡n⇒m-n≡0 was deprecated in v2.0

+
Please use i≡j⇒i-j≡0 instead."

+
#-}

+
m-n≡0⇒m≡n = i-j≡0⇒i≡j

+
{-# WARNING_ON_USAGE m-n≡0⇒m≡n

+
"Warning: m-n≡0⇒m≡n was deprecated in v2.0

+
Please use i-j≡0⇒i≡j instead."

+
#-}

+
≤-steps = i≤j⇒i≤k+j

+
{-# WARNING_ON_USAGE ≤-steps

+
"Warning: ≤-steps was deprecated in v2.0

+
Please use i≤j⇒i≤k+j instead."

+
#-}

+
≤-steps-neg = i≤j⇒i-k≤j

+
{-# WARNING_ON_USAGE ≤-steps-neg

+
"Warning: ≤-steps-neg was deprecated in v2.0

+
Please use i≤j⇒i-k≤j instead."

+
#-}

+
≤-step = i≤j⇒i≤1+j

+
{-# WARNING_ON_USAGE ≤-step

+
"Warning: ≤-step was deprecated in v2.0

+
Please use i≤j⇒i≤1+j instead."

+
#-}

+
≤-step-neg = i≤j⇒pred[i]≤j

+
{-# WARNING_ON_USAGE ≤-step-neg

+
"Warning: ≤-step-neg was deprecated in v2.0

+
Please use i≤j⇒pred[i]≤j instead."

+
#-}

+
m≤n⇒m-n≤0 = i≤j⇒i-j≤0

+
{-# WARNING_ON_USAGE m≤n⇒m-n≤0

+
"Warning: m≤n⇒m-n≤0 was deprecated in v2.0

+
Please use i≤j⇒i-j≤0 instead."

+
#-}

+
m-n≤0⇒m≤n = i-j≤0⇒i≤j

+
{-# WARNING_ON_USAGE m-n≤0⇒m≤n

+
"Warning: m-n≤0⇒m≤n was deprecated in v2.0

+
Please use i-j≤0⇒i≤j instead."

+
#-}

+
m≤n⇒0≤n-m = i≤j⇒0≤j-i

+
{-# WARNING_ON_USAGE m≤n⇒0≤n-m

+
"Warning: m≤n⇒0≤n-m was deprecated in v2.0

+
Please use i≤j⇒0≤j-i instead."

+
#-}

+
0≤n-m⇒m≤n = 0≤i-j⇒j≤i

+
{-# WARNING_ON_USAGE 0≤n-m⇒m≤n

+
"Warning: 0≤n-m⇒m≤n was deprecated in v2.0

+
Please use 0≤i-j⇒j≤i instead."

+
#-}

+
n≤1+n = i≤suc[i]

+
{-# WARNING_ON_USAGE n≤1+n

+
"Warning: n≤1+n was deprecated in v2.0

+
Please use i≤suc[i] instead."

+
#-}

+
n≢1+n = i≢suc[i]

+
{-# WARNING_ON_USAGE n≢1+n

+
"Warning: n≢1+n was deprecated in v2.0

+
Please use i≢suc[i] instead."

+
#-}

+
m≤pred[n]⇒m<n = i≤pred[j]⇒i<j

+
{-# WARNING_ON_USAGE m≤pred[n]⇒m<n

+
"Warning: m≤pred[n]⇒m<n was deprecated in v2.0

+
Please use i≤pred[j]⇒i<j instead."

+
#-}

+
m<n⇒m≤pred[n] = i<j⇒i≤pred[j]

+
{-# WARNING_ON_USAGE m<n⇒m≤pred[n]

+
"Warning: m<n⇒m≤pred[n] was deprecated in v2.0

+
Please use i<j⇒i≤pred[j] instead."

+
#-}

+
-1*n≡-n = -1*i≡-i

+
{-# WARNING_ON_USAGE -1*n≡-n

+
"Warning: -1*n≡-n was deprecated in v2.0

+
Please use -1*i≡-i instead."

+
#-}

+
m*n≡0⇒m≡0∨n≡0 = i*j≡0⇒i≡0∨j≡0

+
{-# WARNING_ON_USAGE m*n≡0⇒m≡0∨n≡0

+
"Warning: m*n≡0⇒m≡0∨n≡0 was deprecated in v2.0

+
Please use i*j≡0⇒i≡0∨j≡0 instead."

+
#-}

+
∣m*n∣≡∣m∣*∣n∣ = ∣i*j∣≡∣i∣*∣j∣

+
{-# WARNING_ON_USAGE ∣m*n∣≡∣m∣*∣n∣

+
"Warning: ∣m*n∣≡∣m∣*∣n∣ was deprecated in v2.0

+
Please use ∣i*j∣≡∣i∣*∣j∣ instead."

+
#-}

+
n≤m+n : ∀ n → i ≤ + n + i

+
n≤m+n {i} n = i≤j+i i (+ n)

+
{-# WARNING_ON_USAGE n≤m+n

+
"Warning: n≤m+n was deprecated in v2.0

+
Please use i≤j+i instead. Note the change of form of the explicit arguments."

+
#-}

+
m≤m+n : ∀ n → i ≤ i + + n

+
m≤m+n {i} n = i≤i+j i (+ n)

+
{-# WARNING_ON_USAGE m≤m+n

+
"Warning: m≤m+n was deprecated in v2.0

+
Please use i≤i+j instead. Note the change of form of the explicit arguments."

+
#-}

+
m-n≤m : ∀ i n → i - + n ≤ i

+
m-n≤m i n = i-j≤i i (+ n)

+
{-# WARNING_ON_USAGE m-n≤m

+
"Warning: m-n≤m was deprecated in v2.0

+
Please use i-j≤i instead. Note the change of form of the explicit arguments."

+
#-}

+
*-monoʳ-≤-pos : ∀ n → (_* + suc n) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤-pos n = *-monoʳ-≤-nonNeg +[1+ n ]

+
{-# WARNING_ON_USAGE *-monoʳ-≤-pos

+
"Warning: *-monoʳ-≤-pos was deprecated in v2.0

+
Please use *-monoʳ-≤-nonNeg instead."

+
#-}

+
*-monoˡ-≤-pos : ∀ n → (+ suc n *_) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤-pos n = *-monoˡ-≤-nonNeg +[1+ n ]

+
{-# WARNING_ON_USAGE *-monoˡ-≤-pos

+
"Warning: *-monoˡ-≤-pos was deprecated in v2.0

+
Please use *-monoˡ-≤-nonNeg instead."

+
#-}

+
*-monoˡ-≤-neg : ∀ m → (-[1+ m ] *_) Preserves _≤_ ⟶ _≥_

+
*-monoˡ-≤-neg m = *-monoˡ-≤-nonPos -[1+ m ]

+
{-# WARNING_ON_USAGE *-monoˡ-≤-neg

+
"Warning: *-monoˡ-≤-neg was deprecated in v2.0

+
Please use *-monoˡ-≤-nonPos instead."

+
#-}

+
*-monoʳ-≤-neg : ∀ m → (_* -[1+ m ]) Preserves _≤_ ⟶ _≥_

+
*-monoʳ-≤-neg m = *-monoʳ-≤-nonPos -[1+ m ]

+
{-# WARNING_ON_USAGE *-monoʳ-≤-neg

+
"Warning: *-monoʳ-≤-neg was deprecated in v2.0

+
Please use *-monoʳ-≤-nonPos instead."

+
#-}

+
pos-+-commute : ℕtoℤ.Homomorphic₂ +_ ℕ._+_ _+_

+
pos-+-commute = pos-+

+
{-# WARNING_ON_USAGE pos-+-commute

+
"Warning: pos-+-commute was deprecated in v2.0

+
Please use pos-+ instead."

+
#-}

+
abs-*-commute : ℤtoℕ.Homomorphic₂ ∣_∣ _*_ ℕ._*_

+
abs-*-commute = abs-*

+
{-# WARNING_ON_USAGE abs-*-commute

+
"Warning: abs-*-commute was deprecated in v2.0

+
Please use abs-* instead."

+
#-}

+
pos-distrib-* : ∀ m n → (+ m) * (+ n) ≡ + (m ℕ.* n)

+
pos-distrib-* m n = sym (pos-* m n)

+
{-# WARNING_ON_USAGE pos-distrib-*

+
"Warning: pos-distrib-* was deprecated in v2.0

+
Please use pos-* instead."

+
#-}

+
+-isAbelianGroup = +-0-isAbelianGroup

+
{-# WARNING_ON_USAGE +-isAbelianGroup

+
"Warning: +-isAbelianGroup was deprecated in v2.0

+
Please use +-0-isAbelianGroup instead."

+
#-}

+
{- issue1844/issue1755: raw bundles have moved to `Data.X.Base` -}

+
open Data.Integer.Base public

+
  using (*-rawMagma; *-1-rawMonoid)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Show.html b/site/static/agda_html/Data.Integer.Show.html
new file mode 100644
index 000000000..3dc45a480
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Show.html
@@ -0,0 +1,101 @@
+
+Data.Integer.Show
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Showing integers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Show where

+

+
open import Data.Integer.Base using (ℤ; +_; -[1+_])

+
open import Data.Nat.Base using (suc)

+
open import Data.Nat.Show using () renaming (show to showℕ)

+
open import Data.String.Base using (String; _++_)

+

+
------------------------------------------------------------------------

+
-- Show

+

+
-- Decimal notation

+
-- Time complexity is O(log₁₀(n))

+

+
show : ℤ → String

+
show (+ n)    = showℕ n

+
show -[1+ n ] = "-" ++ showℕ (suc n)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.Solver.html b/site/static/agda_html/Data.Integer.Solver.html
new file mode 100644
index 000000000..82dfac1c2
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.Solver.html
@@ -0,0 +1,99 @@
+
+Data.Integer.Solver
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Automatic solvers for equations over integers

+
------------------------------------------------------------------------

+

+
-- See README.Data.Integer for examples of how to use this solver

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer.Solver where

+

+
import Algebra.Solver.Ring.Simple as Solver

+
import Algebra.Solver.Ring.AlmostCommutativeRing as ACR

+
open import Data.Integer.Properties using (_≟_; +-*-commutativeRing)

+

+
------------------------------------------------------------------------

+
-- A module for automatically solving propositional equivalences

+
-- containing _+_ and _*_

+

+
module +-*-Solver =

+
  Solver (ACR.fromCommutativeRing +-*-commutativeRing) _≟_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Integer.html b/site/static/agda_html/Data.Integer.html
new file mode 100644
index 000000000..bb06807e2
--- /dev/null
+++ b/site/static/agda_html/Data.Integer.html
@@ -0,0 +1,124 @@
+
+Data.Integer
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Integers

+
------------------------------------------------------------------------

+

+
-- See README.Data.Integer for examples of how to use and reason about

+
-- integers.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Integer where

+

+
------------------------------------------------------------------------

+
-- Re-export basic definition, operations and queries

+

+
open import Data.Integer.Base public

+
open import Data.Integer.Properties public

+
  using (_≟_; _≤?_; _<?_)

+

+
------------------------------------------------------------------------

+
-- Deprecated

+

+
-- Version 0.17

+

+
open import Data.Integer.Properties public

+
  using (◃-cong; drop‿+≤+; drop‿-≤-)

+
  renaming (◃-inverse to ◃-left-inverse)

+

+
-- Version 1.5

+
-- Show

+

+
import Data.Nat.Show as ℕ using (show)

+
open import Data.Sign.Base as Sign using (Sign)

+
open import Data.String.Base using (String; _++_)

+

+
show : ℤ → String

+
show i = showSign (sign i) ++ ℕ.show ∣ i ∣

+
  where

+
  showSign : Sign → String

+
  showSign Sign.- = "-"

+
  showSign Sign.+ = ""

+

+
{-# WARNING_ON_USAGE show

+
"Warning: show was deprecated in v1.5.

+
Please use Data.Integer.Show's show instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Irrelevant.html b/site/static/agda_html/Data.Irrelevant.html
new file mode 100644
index 000000000..bcf42d972
--- /dev/null
+++ b/site/static/agda_html/Data.Irrelevant.html
@@ -0,0 +1,131 @@
+
+Data.Irrelevant
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Wrapper for the proof irrelevance modality

+
--

+
-- This allows us to store proof irrelevant witnesses in a record and

+
-- use projections to manipulate them without having to turn on the

+
-- unsafe option --irrelevant-projections.

+
-- Cf. Data.Refinement for a use case

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Irrelevant where

+

+
open import Level using (Level)

+

+
private

+
  variable

+
    a b c : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Type

+

+
record Irrelevant (A : Set a) : Set a where

+
  constructor [_]

+
  field .irrelevant : A

+
open Irrelevant public

+

+
------------------------------------------------------------------------

+
-- Algebraic structure: Functor, Appplicative and Monad-like

+

+
map : (A → B) → Irrelevant A → Irrelevant B

+
map f [ a ] = [ f a ]

+

+
pure : A → Irrelevant A

+
pure x = [ x ]

+

+
infixl 4 _<*>_

+
_<*>_ : Irrelevant (A → B) → Irrelevant A → Irrelevant B

+
[ f ] <*> [ a ] = [ f a ]

+

+
infixl 1 _>>=_

+
_>>=_ : Irrelevant A → (.A → Irrelevant B) → Irrelevant B

+
[ a ] >>= f = f a

+

+
------------------------------------------------------------------------

+
-- Other functions

+

+
zipWith : (A → B → C) → Irrelevant A → Irrelevant B → Irrelevant C

+
zipWith f a b = ⦇ f a b ⦈

+

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+
+Data.List.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists, basic types and operations

+
------------------------------------------------------------------------

+

+
-- See README.Data.List for examples of how to use and reason about

+
-- lists.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Base where

+

+
open import Algebra.Bundles.Raw using (RawMagma; RawMonoid)

+
open import Data.Bool.Base as Bool

+
  using (Bool; false; true; not; _∧_; _∨_; if_then_else_)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _*_ ; _≤_ ; s≤s)

+
open import Data.Product.Base as Product using (_×_; _,_; map₁; map₂′)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Data.These.Base as These using (These; this; that; these)

+
open import Function.Base

+
  using (id; _∘_ ; _∘′_; _∘₂_; _$_; const; flip)

+
open import Level using (Level)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Binary.Core using (Rel)

+
import Relation.Binary.Definitions as B

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+
open import Relation.Nullary.Decidable.Core using (T?; does; ¬?)

+

+
private

+
  variable

+
    a b c p ℓ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Types

+

+
open import Agda.Builtin.List public

+
  using (List; []; _∷_)

+

+
------------------------------------------------------------------------

+
-- Operations for transforming lists

+

+
map : (A → B) → List A → List B

+
map f []       = []

+
map f (x ∷ xs) = f x ∷ map f xs

+

+
infixr 5 _++_

+

+
_++_ : List A → List A → List A

+
[]       ++ ys = ys

+
(x ∷ xs) ++ ys = x ∷ (xs ++ ys)

+

+
intersperse : A → List A → List A

+
intersperse x []       = []

+
intersperse x (y ∷ []) = y ∷ []

+
intersperse x (y ∷ ys) = y ∷ x ∷ intersperse x ys

+

+
intercalate : List A → List (List A) → List A

+
intercalate xs []         = []

+
intercalate xs (ys ∷ [])  = ys

+
intercalate xs (ys ∷ yss) = ys ++ xs ++ intercalate xs yss

+

+
cartesianProductWith : (A → B → C) → List A → List B → List C

+
cartesianProductWith f []       _  = []

+
cartesianProductWith f (x ∷ xs) ys = map (f x) ys ++ cartesianProductWith f xs ys

+

+
cartesianProduct : List A → List B → List (A × B)

+
cartesianProduct = cartesianProductWith _,_

+

+
------------------------------------------------------------------------

+
-- Aligning and zipping

+

+
alignWith : (These A B → C) → List A → List B → List C

+
alignWith f []       bs       = map (f ∘′ that) bs

+
alignWith f as       []       = map (f ∘′ this) as

+
alignWith f (a ∷ as) (b ∷ bs) = f (these a b) ∷ alignWith f as bs

+

+
zipWith : (A → B → C) → List A → List B → List C

+
zipWith f (x ∷ xs) (y ∷ ys) = f x y ∷ zipWith f xs ys

+
zipWith f _        _        = []

+

+
unalignWith : (A → These B C) → List A → List B × List C

+
unalignWith f []       = [] , []

+
unalignWith f (a ∷ as) with f a

+
... | this b    = Product.map₁ (b ∷_) (unalignWith f as)

+
... | that c    = Product.map₂ (c ∷_) (unalignWith f as)

+
... | these b c = Product.map (b ∷_) (c ∷_) (unalignWith f as)

+

+
unzipWith : (A → B × C) → List A → List B × List C

+
unzipWith f []         = [] , []

+
unzipWith f (xy ∷ xys) = Product.zip _∷_ _∷_ (f xy) (unzipWith f xys)

+

+
partitionSumsWith : (A → B ⊎ C) → List A → List B × List C

+
partitionSumsWith f = unalignWith (These.fromSum ∘′ f)

+

+
align : List A → List B → List (These A B)

+
align = alignWith id

+

+
zip : List A → List B → List (A × B)

+
zip = zipWith (_,_)

+

+
unalign : List (These A B) → List A × List B

+
unalign = unalignWith id

+

+
unzip : List (A × B) → List A × List B

+
unzip = unzipWith id

+

+
partitionSums : List (A ⊎ B) → List A × List B

+
partitionSums = partitionSumsWith id

+

+
merge : {R : Rel A ℓ} → B.Decidable R → List A → List A → List A

+
merge R? []           ys           = ys

+
merge R? xs           []           = xs

+
merge R? x∷xs@(x ∷ xs) y∷ys@(y ∷ ys) = if does (R? x y)

+
  then x ∷ merge R? xs   y∷ys

+
  else y ∷ merge R? x∷xs ys

+

+
------------------------------------------------------------------------

+
-- Operations for reducing lists

+

+
foldr : (A → B → B) → B → List A → B

+
foldr c n []       = n

+
foldr c n (x ∷ xs) = c x (foldr c n xs)

+

+
foldl : (A → B → A) → A → List B → A

+
foldl c n []       = n

+
foldl c n (x ∷ xs) = foldl c (c n x) xs

+

+
concat : List (List A) → List A

+
concat = foldr _++_ []

+

+
concatMap : (A → List B) → List A → List B

+
concatMap f = concat ∘ map f

+

+
ap : List (A → B) → List A → List B

+
ap fs as = concatMap (flip map as) fs

+

+
catMaybes : List (Maybe A) → List A

+
catMaybes = foldr (maybe′ _∷_ id) []

+

+
mapMaybe : (A → Maybe B) → List A → List B

+
mapMaybe p = catMaybes ∘ map p

+

+
null : List A → Bool

+
null []       = true

+
null (x ∷ xs) = false

+

+
and : List Bool → Bool

+
and = foldr _∧_ true

+

+
or : List Bool → Bool

+
or = foldr _∨_ false

+

+
any : (A → Bool) → List A → Bool

+
any p = or ∘ map p

+

+
all : (A → Bool) → List A → Bool

+
all p = and ∘ map p

+

+
sum : List ℕ → ℕ

+
sum = foldr _+_ 0

+

+
product : List ℕ → ℕ

+
product = foldr _*_ 1

+

+
length : List A → ℕ

+
length = foldr (const suc) 0

+

+
------------------------------------------------------------------------

+
-- Operations for constructing lists

+

+
[_] : A → List A

+
[ x ] = x ∷ []

+

+
fromMaybe : Maybe A → List A

+
fromMaybe (just x) = [ x ]

+
fromMaybe nothing  = []

+

+
replicate : ℕ → A → List A

+
replicate zero    x = []

+
replicate (suc n) x = x ∷ replicate n x

+

+
iterate : (A → A) → A → ℕ → List A

+
iterate f e zero    = []

+
iterate f e (suc n) = e ∷ iterate f (f e) n

+

+
inits : List A → List (List A)

+
inits {A = A} = λ xs → [] ∷ tail xs

+
  module Inits where

+
    tail : List A → List (List A)

+
    tail []       = []

+
    tail (x ∷ xs) = [ x ] ∷ map (x ∷_) (tail xs)

+

+
tails : List A → List (List A)

+
tails {A = A} = λ xs → xs ∷ tail xs

+
  module Tails where

+
    tail : List A → List (List A)

+
    tail []       = []

+
    tail (_ ∷ xs) = xs ∷ tail xs

+

+
insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A

+
insertAt xs       zero    v = v ∷ xs

+
insertAt (x ∷ xs) (suc i) v = x ∷ insertAt xs i v

+

+
updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A

+
updateAt (x ∷ xs) zero    f = f x ∷ xs

+
updateAt (x ∷ xs) (suc i) f = x ∷ updateAt xs i f

+

+
-- Tabulation

+

+
applyUpTo : (ℕ → A) → ℕ → List A

+
applyUpTo f zero    = []

+
applyUpTo f (suc n) = f zero ∷ applyUpTo (f ∘ suc) n

+

+
applyDownFrom : (ℕ → A) → ℕ → List A

+
applyDownFrom f zero    = []

+
applyDownFrom f (suc n) = f n ∷ applyDownFrom f n

+

+
tabulate : ∀ {n} (f : Fin n → A) → List A

+
tabulate {n = zero}  f = []

+
tabulate {n = suc n} f = f zero ∷ tabulate (f ∘ suc)

+

+
lookup : ∀ (xs : List A) → Fin (length xs) → A

+
lookup (x ∷ xs) zero    = x

+
lookup (x ∷ xs) (suc i) = lookup xs i

+

+
-- Numerical

+

+
upTo : ℕ → List ℕ

+
upTo = applyUpTo id

+

+
downFrom : ℕ → List ℕ

+
downFrom = applyDownFrom id

+

+
allFin : ∀ n → List (Fin n)

+
allFin n = tabulate id

+

+
unfold : ∀ (P : ℕ → Set b)

+
         (f : ∀ {n} → P (suc n) → Maybe (A × P n)) →

+
         ∀ {n} → P n → List A

+
unfold P f {n = zero}  s = []

+
unfold P f {n = suc n} s = maybe′ (λ (x , s′) → x ∷ unfold P f s′) [] (f s)

+

+
------------------------------------------------------------------------

+
-- Operations for reversing lists

+

+
reverseAcc : List A → List A → List A

+
reverseAcc = foldl (flip _∷_)

+

+
reverse : List A → List A

+
reverse = reverseAcc []

+

+
-- "Reverse append" xs ʳ++ ys = reverse xs ++ ys

+

+
infixr 5 _ʳ++_

+

+
_ʳ++_ : List A → List A → List A

+
_ʳ++_ = flip reverseAcc

+

+
-- Snoc: Cons, but from the right.

+

+
infixl 6 _∷ʳ_

+

+
_∷ʳ_ : List A → A → List A

+
xs ∷ʳ x = xs ++ [ x ]

+

+

+

+
-- Backwards initialisation

+

+
infixl 5 _∷ʳ′_

+

+
data InitLast {A : Set a} : List A → Set a where

+
  []    : InitLast []

+
  _∷ʳ′_ : (xs : List A) (x : A) → InitLast (xs ∷ʳ x)

+

+
initLast : (xs : List A) → InitLast xs

+
initLast []               = []

+
initLast (x ∷ xs)         with initLast xs

+
... | []       = [] ∷ʳ′ x

+
... | ys ∷ʳ′ y = (x ∷ ys) ∷ʳ′ y

+

+
-- uncons, but from the right

+
unsnoc : List A → Maybe (List A × A)

+
unsnoc as with initLast as

+
... | []       = nothing

+
... | xs ∷ʳ′ x = just (xs , x)

+

+
------------------------------------------------------------------------

+
-- Operations for deconstructing lists

+

+
-- Note that although the following three combinators can be useful for

+
-- programming, when proving it is often a better idea to manually

+
-- destruct a list argument as each branch of the pattern-matching will

+
-- have a refined type.

+

+
uncons : List A → Maybe (A × List A)

+
uncons []       = nothing

+
uncons (x ∷ xs) = just (x , xs)

+

+
head : List A → Maybe A

+
head []      = nothing

+
head (x ∷ _) = just x

+

+
tail : List A → Maybe (List A)

+
tail []       = nothing

+
tail (_ ∷ xs) = just xs

+

+
last : List A → Maybe A

+
last []       = nothing

+
last (x ∷ []) = just x

+
last (_ ∷ xs) = last xs

+

+
take : ℕ → List A → List A

+
take zero    xs       = []

+
take (suc n) []       = []

+
take (suc n) (x ∷ xs) = x ∷ take n xs

+

+
drop : ℕ → List A → List A

+
drop zero    xs       = xs

+
drop (suc n) []       = []

+
drop (suc n) (x ∷ xs) = drop n xs

+

+
splitAt : ℕ → List A → List A × List A

+
splitAt zero    xs       = ([] , xs)

+
splitAt (suc n) []       = ([] , [])

+
splitAt (suc n) (x ∷ xs) = Product.map₁ (x ∷_) (splitAt n xs)

+

+
removeAt : (xs : List A) → Fin (length xs) → List A

+
removeAt (x ∷ xs) zero     = xs

+
removeAt (x ∷ xs) (suc i)  = x ∷ removeAt xs i

+

+
------------------------------------------------------------------------

+
-- Operations for filtering lists

+

+
-- The following are a variety of functions that can be used to

+
-- construct sublists using a predicate.

+
--

+
-- Each function has two forms. The first main variant uses a

+
-- proof-relevant decidable predicate, while the second variant uses

+
-- a irrelevant boolean predicate and are suffixed with a `ᵇ` character,

+
-- typed as \^b.

+
--

+
-- The decidable versions have several advantages: 1) easier to prove

+
-- properties, 2) better meta-variable inference and 3) most of the rest

+
-- of the library is set-up to work with decidable predicates. However,

+
-- in rare cases the boolean versions can be useful, mainly when one

+
-- wants to minimise dependencies.

+
--

+
-- In summary, in most cases you probably want to use the decidable

+
-- versions over the boolean versions, e.g. use `takeWhile (_≤? 10) xs`

+
-- rather than `takeWhileᵇ (_≤ᵇ 10) xs`.

+

+
takeWhile : ∀ {P : Pred A p} → Decidable P → List A → List A

+
takeWhile P? []       = []

+
takeWhile P? (x ∷ xs) with does (P? x)

+
... | true  = x ∷ takeWhile P? xs

+
... | false = []

+

+
takeWhileᵇ : (A → Bool) → List A → List A

+
takeWhileᵇ p = takeWhile (T? ∘ p)

+

+
dropWhile : ∀ {P : Pred A p} → Decidable P → List A → List A

+
dropWhile P? []       = []

+
dropWhile P? (x ∷ xs) with does (P? x)

+
... | true  = dropWhile P? xs

+
... | false = x ∷ xs

+

+
dropWhileᵇ : (A → Bool) → List A → List A

+
dropWhileᵇ p = dropWhile (T? ∘ p)

+

+
filter : ∀ {P : Pred A p} → Decidable P → List A → List A

+
filter P? [] = []

+
filter P? (x ∷ xs) with does (P? x)

+
... | false = filter P? xs

+
... | true  = x ∷ filter P? xs

+

+
filterᵇ : (A → Bool) → List A → List A

+
filterᵇ p = filter (T? ∘ p)

+

+
partition : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)

+
partition P? []       = ([] , [])

+
partition P? (x ∷ xs) with does (P? x) | partition P? xs

+
... | true  | (ys , zs) = (x ∷ ys , zs)

+
... | false | (ys , zs) = (ys , x ∷ zs)

+

+
partitionᵇ : (A → Bool) → List A → List A × List A

+
partitionᵇ p = partition (T? ∘ p)

+

+
span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)

+
span P? []       = ([] , [])

+
span P? ys@(x ∷ xs) with does (P? x)

+
... | true  = Product.map (x ∷_) id (span P? xs)

+
... | false = ([] , ys)

+

+

+
spanᵇ : (A → Bool) → List A → List A × List A

+
spanᵇ p = span (T? ∘ p)

+

+
break : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)

+
break P? = span (¬? ∘ P?)

+

+
breakᵇ : (A → Bool) → List A → List A × List A

+
breakᵇ p = break (T? ∘ p)

+

+
-- The predicate `P` represents the notion of newline character for the

+
-- type `A`. It is used to split the input list into a list of lines.

+
-- Some lines may be empty if the input contains at least two

+
-- consecutive newline characters.

+
linesBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A)

+
linesBy {A = A} P? = go nothing where

+

+
  go : Maybe (List A) → List A → List (List A)

+
  go acc []       = maybe′ ([_] ∘′ reverse) [] acc

+
  go acc (c ∷ cs) = if does (P? c)

+
    then reverse acc′ ∷ go nothing cs

+
    else go (just (c ∷ acc′)) cs

+
    where acc′ = Maybe.fromMaybe [] acc

+

+
linesByᵇ : (A → Bool) → List A → List (List A)

+
linesByᵇ p = linesBy (T? ∘ p)

+

+
-- The predicate `P` represents the notion of space character for the

+
-- type `A`. It is used to split the input list into a list of words.

+
-- All the words are non empty and the output does not contain any space

+
-- characters.

+
wordsBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A)

+
wordsBy {A = A} P? = go [] where

+

+
  cons : List A → List (List A) → List (List A)

+
  cons [] ass = ass

+
  cons as ass = reverse as ∷ ass

+

+
  go : List A → List A → List (List A)

+
  go acc []       = cons acc []

+
  go acc (c ∷ cs) = if does (P? c)

+
    then cons acc (go [] cs)

+
    else go (c ∷ acc) cs

+

+
wordsByᵇ : (A → Bool) → List A → List (List A)

+
wordsByᵇ p = wordsBy (T? ∘ p)

+

+
derun : ∀ {R : Rel A p} → B.Decidable R → List A → List A

+
derun R? [] = []

+
derun R? (x ∷ []) = x ∷ []

+
derun R? (x ∷ xs@(y ∷ _)) with does (R? x y) | derun R? xs

+
... | true  | ys = ys

+
... | false | ys = x ∷ ys

+

+
derunᵇ : (A → A → Bool) → List A → List A

+
derunᵇ r = derun (T? ∘₂ r)

+

+
deduplicate : ∀ {R : Rel A p} → B.Decidable R → List A → List A

+
deduplicate R? [] = []

+
deduplicate R? (x ∷ xs) = x ∷ filter (¬? ∘ R? x) (deduplicate R? xs)

+

+
deduplicateᵇ : (A → A → Bool) → List A → List A

+
deduplicateᵇ r = deduplicate (T? ∘₂ r)

+

+
-- Finds the first element satisfying the boolean predicate

+
find : ∀ {P : Pred A p} → Decidable P → List A → Maybe A

+
find P? []       = nothing

+
find P? (x ∷ xs) = if does (P? x) then just x else find P? xs

+

+
findᵇ : (A → Bool) → List A → Maybe A

+
findᵇ p = find (T? ∘ p)

+

+
-- Finds the index of the first element satisfying the boolean predicate

+
findIndex : ∀ {P : Pred A p} → Decidable P → (xs : List A) → Maybe $ Fin (length xs)

+
findIndex P? [] = nothing

+
findIndex P? (x ∷ xs) = if does (P? x)

+
  then just zero

+
  else Maybe.map suc (findIndex P? xs)

+

+
findIndexᵇ : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)

+
findIndexᵇ p = findIndex (T? ∘ p)

+

+
-- Finds indices of all the elements satisfying the boolean predicate

+
findIndices : ∀ {P : Pred A p} → Decidable P → (xs : List A) → List $ Fin (length xs)

+
findIndices P? []       = []

+
findIndices P? (x ∷ xs) = if does (P? x)

+
  then zero ∷ indices

+
  else indices

+
    where indices = map suc (findIndices P? xs)

+

+
findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)

+
findIndicesᵇ p = findIndices (T? ∘ p)

+

+
------------------------------------------------------------------------

+
-- Actions on single elements

+

+
infixl 5 _[_]%=_ _[_]∷=_

+

+
-- xs [ i ]%= f  modifies the i-th element of xs according to f

+

+
_[_]%=_ : (xs : List A) → Fin (length xs) → (A → A) → List A

+
xs [ i ]%= f = updateAt xs i f

+

+
-- xs [ i ]≔ y  overwrites the i-th element of xs with y

+

+
_[_]∷=_ : (xs : List A) → Fin (length xs) → A → List A

+
xs [ k ]∷= v = xs [ k ]%= const v

+

+
------------------------------------------------------------------------

+
-- Conditional versions of cons and snoc

+

+
infixr 5 _?∷_

+
_?∷_ : Maybe A → List A → List A

+
_?∷_ = maybe′ _∷_ id

+

+
infixl 6 _∷ʳ?_

+
_∷ʳ?_ : List A → Maybe A → List A

+
xs ∷ʳ? x = maybe′ (xs ∷ʳ_) xs x

+

+
------------------------------------------------------------------------

+
-- Raw algebraic bundles

+

+
module _ (A : Set a) where

+
  ++-rawMagma : RawMagma a _

+
  ++-rawMagma = record

+
    { Carrier = List A

+
    ; _≈_ = _≡_

+
    ; _∙_ = _++_

+
    }

+

+
  ++-[]-rawMonoid : RawMonoid a _

+
  ++-[]-rawMonoid = record

+
    { Carrier = List A

+
    ; _≈_ = _≡_

+
    ; _∙_ = _++_

+
    ; ε = []

+
    }

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.4

+

+
infixl 5 _∷ʳ'_

+
_∷ʳ'_ : (xs : List A) (x : A) → InitLast (xs ∷ʳ x)

+
_∷ʳ'_ = InitLast._∷ʳ′_

+
{-# WARNING_ON_USAGE _∷ʳ'_

+
"Warning: _∷ʳ'_ (ending in an apostrophe) was deprecated in v1.4.

+
Please use _∷ʳ′_ (ending in a prime) instead."

+
#-}

+

+
-- Version 2.0

+

+
infixl 5 _─_

+
_─_ = removeAt

+
{-# WARNING_ON_USAGE _─_

+
"Warning: _─_ was deprecated in v2.0.

+
Please use removeAt instead."

+
#-}

+

+
-- Version 2.1

+

+
scanr : (A → B → B) → B → List A → List B

+
scanr f e []       = e ∷ []

+
scanr f e (x ∷ xs) with scanr f e xs

+
... | []         = []                -- dead branch

+
... | ys@(y ∷ _) = f x y ∷ ys

+
{-# WARNING_ON_USAGE scanr

+
"Warning: scanr was deprecated in v2.1.

+
Please use Data.List.Scans.Base.scanr instead."

+
#-}

+

+
scanl : (A → B → A) → A → List B → List A

+
scanl f e []       = e ∷ []

+
scanl f e (x ∷ xs) = e ∷ scanl f (f e x) xs

+
{-# WARNING_ON_USAGE scanl

+
"Warning: scanl was deprecated in v2.1.

+
Please use Data.List.Scans.Base.scanl instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Effectful.html b/site/static/agda_html/Data.List.Effectful.html
new file mode 100644
index 000000000..281ffb3a2
--- /dev/null
+++ b/site/static/agda_html/Data.List.Effectful.html
@@ -0,0 +1,392 @@
+
+Data.List.Effectful
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An effectful view of List

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Effectful where

+

+
open import Data.Bool.Base using (false; true)

+
open import Data.List.Base

+
  using (List; map; [_]; ap; []; _∷_; _++_; concat; concatMap)

+
open import Data.List.Properties

+
  using (++-identityʳ; ++-assoc; map-cong; concatMap-cong; map-concatMap;

+
    concatMap-pure)

+
open import Effect.Choice using (RawChoice)

+
open import Effect.Empty using (RawEmpty)

+
open import Effect.Functor using (RawFunctor)

+
open import Effect.Applicative

+
  using (RawApplicative; RawApplicativeZero; RawAlternative)

+
open import Effect.Monad

+
  using (RawMonad; module Join; RawMonadZero; RawMonadPlus)

+
open import Function.Base using (flip; _∘_; const; _$_; id; _∘′_; _$′_)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; _≗_; refl)

+
open import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
open ≡.≡-Reasoning

+

+
private

+
  variable

+
    ℓ : Level

+
    A : Set ℓ

+

+
------------------------------------------------------------------------

+
-- List applicative functor

+

+
functor : RawFunctor {ℓ} List

+
functor = record { _<$>_ = map }

+

+
applicative : RawApplicative {ℓ} List

+
applicative = record

+
  { rawFunctor = functor

+
  ; pure = [_]

+
  ; _<*>_  = ap

+
  }

+

+
empty : RawEmpty {ℓ} List

+
empty = record { empty = [] }

+

+
choice : RawChoice {ℓ} List

+
choice = record { _<|>_ = _++_ }

+

+
applicativeZero : RawApplicativeZero {ℓ} List

+
applicativeZero = record

+
  { rawApplicative = applicative

+
  ; rawEmpty = empty

+
  }

+

+
alternative : RawAlternative {ℓ} List

+
alternative = record

+
  { rawApplicativeZero = applicativeZero

+
  ; rawChoice = choice

+
  }

+

+
------------------------------------------------------------------------

+
-- List monad

+

+
monad : ∀ {ℓ} → RawMonad {ℓ} List

+
monad = record

+
  { rawApplicative = applicative

+
  ; _>>=_  = flip concatMap

+
  }

+

+
join : List (List A) → List A

+
join = Join.join monad

+

+
monadZero : ∀ {ℓ} → RawMonadZero {ℓ} List

+
monadZero = record

+
  { rawMonad = monad

+
  ; rawEmpty = empty

+
  }

+

+
monadPlus : ∀ {ℓ} → RawMonadPlus {ℓ} List

+
monadPlus = record

+
  { rawMonadZero = monadZero

+
  ; rawChoice  = choice

+
  }

+

+
------------------------------------------------------------------------

+
-- Get access to other monadic functions

+

+
module TraversableA {f g F} (App : RawApplicative {f} {g} F) where

+

+
  open RawApplicative App

+

+
  sequenceA : ∀ {A} → List (F A) → F (List A)

+
  sequenceA []       = pure []

+
  sequenceA (x ∷ xs) = _∷_ <$> x <*> sequenceA xs

+

+
  mapA : ∀ {a} {A : Set a} {B} → (A → F B) → List A → F (List B)

+
  mapA f = sequenceA ∘ map f

+

+
  forA : ∀ {a} {A : Set a} {B} → List A → (A → F B) → F (List B)

+
  forA = flip mapA

+

+
module TraversableM {m n M} (Mon : RawMonad {m} {n} M) where

+

+
  open RawMonad Mon

+

+
  open TraversableA rawApplicative public

+
    renaming

+
    ( sequenceA to sequenceM

+
    ; mapA      to mapM

+
    ; forA      to forM

+
    )

+

+
------------------------------------------------------------------------

+
-- The list monad.

+

+
private

+
  open module LMP {ℓ} = RawMonadPlus (monadPlus {ℓ = ℓ})

+

+
module MonadProperties where

+

+
  left-identity : ∀ {ℓ} {A B : Set ℓ} (x : A) (f : A → List B) →

+
                  (pure x >>= f) ≡ f x

+
  left-identity x f = ++-identityʳ (f x)

+

+
  right-identity : ∀ {ℓ} {A : Set ℓ} (xs : List A) →

+
                   (xs >>= pure) ≡ xs

+
  right-identity []       = refl

+
  right-identity (x ∷ xs) = ≡.cong (x ∷_) (right-identity xs)

+

+
  left-zero : ∀ {ℓ} {A B : Set ℓ} (f : A → List B) → (∅ >>= f) ≡ ∅

+
  left-zero f = refl

+

+
  right-zero : ∀ {ℓ} {A B : Set ℓ} (xs : List A) →

+
               (xs >>= const ∅) ≡ ∅ {A = B}

+
  right-zero []       = refl

+
  right-zero (x ∷ xs) = right-zero xs

+

+
  private

+

+
    not-left-distributive :

+
      let xs = true ∷ false ∷ []; f = pure; g = pure in

+
      (xs >>= λ x → f x ∣ g x) ≢ ((xs >>= f) ∣ (xs >>= g))

+
    not-left-distributive ()

+

+
  right-distributive : ∀ {ℓ} {A B : Set ℓ}

+
                       (xs ys : List A) (f : A → List B) →

+
                       (xs ∣ ys >>= f) ≡ ((xs >>= f) ∣ (ys >>= f))

+
  right-distributive []       ys f = refl

+
  right-distributive (x ∷ xs) ys f = begin

+
    f x ∣ (xs ∣ ys >>= f)              ≡⟨ ≡.cong (f x ∣_) $ right-distributive xs ys f ⟩

+
    f x ∣ ((xs >>= f) ∣ (ys >>= f))    ≡⟨ ≡.sym $ ++-assoc (f x) _ _ ⟩

+
    ((f x ∣ (xs >>= f)) ∣ (ys >>= f))  ∎

+

+
  associative : ∀ {ℓ} {A B C : Set ℓ}

+
                (xs : List A) (f : A → List B) (g : B → List C) →

+
                (xs >>= λ x → f x >>= g) ≡ (xs >>= f >>= g)

+
  associative []       f g = refl

+
  associative (x ∷ xs) f g = begin

+
    (f x >>= g) ∣ (xs >>= λ x → f x >>= g)  ≡⟨ ≡.cong ((f x >>= g) ∣_) $ associative xs f g ⟩

+
    (f x >>= g) ∣ (xs >>= f >>= g)          ≡⟨ ≡.sym $ right-distributive (f x) (xs >>= f) g ⟩

+
    (f x ∣ (xs >>= f) >>= g)                ∎

+

+
  cong : ∀ {ℓ} {A B : Set ℓ} {xs₁ xs₂} {f₁ f₂ : A → List B} →

+
         xs₁ ≡ xs₂ → f₁ ≗ f₂ → (xs₁ >>= f₁) ≡ (xs₂ >>= f₂)

+
  cong {xs₁ = xs} refl f₁≗f₂ = ≡.cong concat (map-cong f₁≗f₂ xs)

+

+
------------------------------------------------------------------------

+
-- The applicative functor derived from the list monad.

+

+
-- Note that these proofs (almost) show that RawIMonad.rawIApplicative

+
-- is correctly defined. The proofs can be reused if proof components

+
-- are ever added to RawIMonad and RawIApplicative.

+

+
module Applicative where

+

+
  private

+

+
    module MP = MonadProperties

+

+
    -- A variant of flip map.

+

+
    pam : ∀ {ℓ} {A B : Set ℓ} → List A → (A → B) → List B

+
    pam xs f = xs >>= pure ∘ f

+

+
  -- ∅ is a left zero for _⊛_.

+

+
  left-zero : ∀ {ℓ} {A B : Set ℓ} → (xs : List A) → (∅ ⊛ xs) ≡ ∅ {A = B}

+
  left-zero xs = begin

+
    ∅ ⊛ xs          ≡⟨⟩

+
    (∅ >>= pam xs)  ≡⟨ MonadProperties.left-zero (pam xs) ⟩

+
    ∅               ∎

+

+
  -- ∅ is a right zero for _⊛_.

+

+
  right-zero : ∀ {ℓ} {A B : Set ℓ} → (fs : List (A → B)) → (fs ⊛ ∅) ≡ ∅

+
  right-zero {ℓ} fs = begin

+
    fs ⊛ ∅            ≡⟨⟩

+
    (fs >>= pam ∅)    ≡⟨ (MP.cong (refl {x = fs}) λ f →

+
                          MP.left-zero (pure ∘ f)) ⟩

+
    (fs >>= λ _ → ∅)  ≡⟨ MP.right-zero fs ⟩

+
    ∅                 ∎

+

+
  unfold-<$> : ∀ {ℓ} {A B : Set ℓ} → (f : A → B) (as : List A) →

+
               (f <$> as) ≡ (pure f ⊛ as)

+
  unfold-<$> f as = ≡.sym (++-identityʳ (f <$> as))

+

+
  -- _⊛_ unfolds to binds.

+
  unfold-⊛ : ∀ {ℓ} {A B : Set ℓ} → (fs : List (A → B)) (as : List A) →

+
             (fs ⊛ as) ≡ (fs >>= pam as)

+
  unfold-⊛ fs as = begin

+
    fs ⊛ as

+
      ≡⟨ concatMap-cong (λ f → ≡.cong (map f) (concatMap-pure as)) fs ⟨

+
    concatMap (λ f → map f (concatMap pure as)) fs

+
      ≡⟨ concatMap-cong (λ f → map-concatMap f pure as) fs ⟩

+
    concatMap (λ f → concatMap (λ x → pure (f x)) as) fs

+
      ≡⟨⟩

+
    (fs >>= pam as)

+
      ∎

+

+
  -- _⊛_ distributes over _∣_ from the right.

+

+
  right-distributive : ∀ {ℓ} {A B : Set ℓ} (fs₁ fs₂ : List (A → B)) xs →

+
                       ((fs₁ ∣ fs₂) ⊛ xs) ≡ (fs₁ ⊛ xs ∣ fs₂ ⊛ xs)

+
  right-distributive fs₁ fs₂ xs = begin

+
    (fs₁ ∣ fs₂) ⊛ xs                     ≡⟨ unfold-⊛ (fs₁ ∣ fs₂) xs ⟩

+
    (fs₁ ∣ fs₂ >>= pam xs)               ≡⟨ MonadProperties.right-distributive fs₁ fs₂ (pam xs) ⟩

+
    (fs₁ >>= pam xs) ∣ (fs₂ >>= pam xs)  ≡⟨ ≡.cong₂ _∣_ (unfold-⊛ fs₁ xs) (unfold-⊛ fs₂ xs) ⟨

+
    (fs₁ ⊛ xs ∣ fs₂ ⊛ xs)                ∎

+

+
  -- _⊛_ does not distribute over _∣_ from the left.

+

+
  private

+

+
    not-left-distributive :

+
      let fs = id ∷ id ∷ []; xs₁ = true ∷ []; xs₂ = true ∷ false ∷ [] in

+
      (fs ⊛ (xs₁ ∣ xs₂)) ≢ (fs ⊛ xs₁ ∣ fs ⊛ xs₂)

+
    not-left-distributive ()

+

+
  -- Applicative functor laws.

+

+
  identity : ∀ {a} {A : Set a} (xs : List A) → (pure id ⊛ xs) ≡ xs

+
  identity xs = begin

+
    pure id ⊛ xs          ≡⟨ unfold-⊛ (pure id) xs ⟩

+
    (pure id >>= pam xs)  ≡⟨ MonadProperties.left-identity id (pam xs) ⟩

+
    (xs >>= pure)         ≡⟨ MonadProperties.right-identity xs ⟩

+
    xs                    ∎

+

+
  private

+

+
    pam-lemma : ∀ {ℓ} {A B C : Set ℓ}

+
                (xs : List A) (f : A → B) (fs : B → List C) →

+
                (pam xs f >>= fs) ≡ (xs >>= λ x → fs (f x))

+
    pam-lemma xs f fs = begin

+
      (pam xs f >>= fs)                 ≡⟨ MP.associative xs (pure ∘ f) fs ⟨

+
      (xs >>= λ x → pure (f x) >>= fs)  ≡⟨ MP.cong (refl {x = xs}) (λ x → MP.left-identity (f x) fs) ⟩

+
      (xs >>= λ x → fs (f x))           ∎

+

+
  composition : ∀ {ℓ} {A B C : Set ℓ}

+
                (fs : List (B → C)) (gs : List (A → B)) xs →

+
                (pure _∘′_ ⊛ fs ⊛ gs ⊛ xs) ≡ (fs ⊛ (gs ⊛ xs))

+
  composition {ℓ} fs gs xs = begin

+
    pure _∘′_ ⊛ fs ⊛ gs ⊛ xs

+
      ≡⟨ unfold-⊛ (pure _∘′_ ⊛ fs ⊛ gs) xs ⟩

+
    (pure _∘′_ ⊛ fs ⊛ gs >>= pam xs)

+
      ≡⟨ ≡.cong (_>>= pam xs) (unfold-⊛ (pure _∘′_ ⊛ fs) gs) ⟩

+
    (pure _∘′_ ⊛ fs >>= pam gs >>= pam xs)

+
      ≡⟨ ≡.cong (λ h → h >>= pam gs >>= pam xs) (unfold-⊛ (pure _∘′_) fs) ⟩

+
    (pure _∘′_ >>= pam fs >>= pam gs >>= pam xs)

+
      ≡⟨ MP.cong (MP.cong (MP.left-identity _∘′_ (pam fs))

+
                 (λ f → refl {x = pam gs f}))

+
                 (λ fg → refl {x = pam xs fg}) ⟩

+
    (pam fs _∘′_ >>= pam gs >>= pam xs)

+
      ≡⟨ MP.cong (pam-lemma fs _∘′_ (pam gs)) (λ _ → refl) ⟩

+
    ((fs >>= λ f → pam gs (f ∘′_)) >>= pam xs)

+
      ≡⟨ MP.associative fs (λ f → pam gs (_∘′_ f)) (pam xs) ⟨

+
    (fs >>= λ f → pam gs (f ∘′_) >>= pam xs)

+
      ≡⟨ MP.cong (refl {x = fs}) (λ f → pam-lemma gs (f ∘′_) (pam xs)) ⟩

+
    (fs >>= λ f → gs >>= λ g → pam xs (f ∘′ g))

+
      ≡⟨ (MP.cong (refl {x = fs}) λ f →

+
         MP.cong (refl {x = gs}) λ g →

+
         ≡.sym $ pam-lemma xs g (pure ∘ f)) ⟩

+
    (fs >>= λ f → gs >>= λ g → pam (pam xs g) f)

+
      ≡⟨ MP.cong (refl {x = fs}) (λ f → MP.associative gs (pam xs) (pure ∘ f)) ⟩

+
    (fs >>= pam (gs >>= pam xs))

+
      ≡⟨ unfold-⊛ fs (gs >>= pam xs) ⟨

+
    fs ⊛ (gs >>= pam xs)

+
      ≡⟨ ≡.cong (fs ⊛_) (unfold-⊛ gs xs) ⟨

+
    fs ⊛ (gs ⊛ xs)

+
      ∎

+

+
  homomorphism : ∀ {ℓ} {A B : Set ℓ} (f : A → B) x →

+
                 (pure f ⊛ pure x) ≡ pure (f x)

+
  homomorphism f x = begin

+
    pure f ⊛ pure x            ≡⟨⟩

+
    (pure f >>= pam (pure x))  ≡⟨ MP.left-identity f (pam (pure x)) ⟩

+
    pam (pure x) f             ≡⟨ MP.left-identity x (pure ∘ f) ⟩

+
    pure (f x)                 ∎

+

+
  interchange : ∀ {ℓ} {A B : Set ℓ} (fs : List (A → B)) {x} →

+
                (fs ⊛ pure x) ≡ (pure (_$′ x) ⊛ fs)

+
  interchange fs {x} = begin

+
    fs ⊛ pure x                ≡⟨⟩

+
    (fs >>= pam (pure x))      ≡⟨ (MP.cong (refl {x = fs}) λ f →

+
                                      MP.left-identity x (pure ∘ f)) ⟩

+
    (fs >>= λ f → pure (f x))  ≡⟨⟩

+
    (pam fs (_$′ x))           ≡⟨ ≡.sym $ MP.left-identity (_$′ x) (pam fs) ⟩

+
    (pure (_$′ x) >>= pam fs)  ≡⟨ unfold-⊛ (pure (_$′ x)) fs ⟨

+
    pure (_$′ x) ⊛ fs          ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Ext.Properties.html b/site/static/agda_html/Data.List.Ext.Properties.html
new file mode 100644
index 000000000..2020a5841
--- /dev/null
+++ b/site/static/agda_html/Data.List.Ext.Properties.html
@@ -0,0 +1,140 @@
+
+Data.List.Ext.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Data.List.Ext.Properties where

+

+
open import abstract-set-theory.Prelude hiding (lookup; map)

+

+
import Data.Product

+
import Data.Sum

+
import Function.Related.Propositional as R

+
open import Data.List.Membership.Propositional

+
open import Data.List.Membership.Propositional.Properties

+
open import Data.List.Relation.Binary.BagAndSetEquality

+
open import Data.List.Relation.Binary.Disjoint.Propositional

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
open import Data.List.Relation.Unary.AllPairs

+
open import Data.List.Relation.Unary.All

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.List.Relation.Unary.Unique.Propositional.Properties.WithK

+

+
-- TODO: stdlib?

+
_×-cong_ : ∀ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {k} → A R.∼[ k ] B → C R.∼[ k ] D → (A × C) R.∼[ k ] (B × D)

+
h ×-cong h' = (h M.×-cong h')

+
  where open import Data.Product.Function.NonDependent.Propositional as M

+

+
_⊎-cong_ : ∀ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {k} → A R.∼[ k ] B → C R.∼[ k ] D → (A ⊎ C) R.∼[ k ] (B ⊎ D)

+
h ⊎-cong h' = (h M.⊎-cong h')

+
  where open import Data.Sum.Function.Propositional as M

+

+
-- TODO: stdlib?

+
AllPairs⇒≡∨R∨Rᵒᵖ : ∀ {ℓ ℓ'} {A : Set ℓ} {R : A → A → Set ℓ'} {a b l}

+
                 → AllPairs R l → a ∈ l → b ∈ l → a ≡ b ⊎ R a b ⊎ R b a

+
AllPairs⇒≡∨R∨Rᵒᵖ (_ ∷ _) (here refl) (here refl) = inj₁ refl

+
AllPairs⇒≡∨R∨Rᵒᵖ (x ∷ _) (here refl) (there b∈l) = inj₂ (inj₁ (lookup x b∈l))

+
AllPairs⇒≡∨R∨Rᵒᵖ (x ∷ _) (there a∈l) (here refl) = inj₂ (inj₂ (lookup x a∈l))

+
AllPairs⇒≡∨R∨Rᵒᵖ (x ∷ h) (there a∈l) (there b∈l) = AllPairs⇒≡∨R∨Rᵒᵖ h a∈l b∈l

+

+
--------------------------------------------------------------

+
------- duplicate entries in lists and deduplication ---------

+
--------------------------------------------------------------

+
module _ {a} {A : Set a} ⦃ _ : DecEq A ⦄ where

+
  open import Data.List.Relation.Unary.Unique.DecPropositional.Properties {A = A} _≟_

+

+
  deduplicate≡ : List A → List A

+
  deduplicate≡ = deduplicate _≟_

+

+
  disj-on-dedup : ∀ {l l'} → Disjoint l l' → Disjoint (deduplicate≡ l) (deduplicate≡ l')

+
  disj-on-dedup = _∘ Data.Product.map (∈-deduplicate⁻ _≟_ _) (∈-deduplicate⁻ _≟_ _)

+

+
  ∈-dedup : ∀ {l a} → a ∈ l ⇔ a ∈ deduplicate≡ l

+
  ∈-dedup = mk⇔ (∈-deduplicate⁺ _≟_) (∈-deduplicate⁻ _≟_ _)

+

+
  -- TODO: stdlib?

+
  dedup-++-↭ : {l l' : List A} → Disjoint l l' → deduplicate≡ (l ++ l') ↭ deduplicate≡ l ++ deduplicate≡ l'

+
  dedup-++-↭ {l = l} {l'} disj = let dedup-unique = λ {l} → deduplicate-! l in ∼bag⇒↭ $

+
    unique∧set⇒bag dedup-unique (++⁺ dedup-unique dedup-unique (disj-on-dedup disj)) λ {a} →

+
      a ∈ deduplicate≡ (l ++ l')                 ∼⟨ R.SK-sym ∈-dedup ⟩

+
      a ∈ l ++ l'                                ∼⟨ helper ⟩

+
      (a ∈ l ⊎ a ∈ l')                           ∼⟨ ∈-dedup ⊎-cong ∈-dedup ⟩

+
      (a ∈ deduplicate≡ l ⊎ a ∈ deduplicate≡ l') ∼⟨ R.SK-sym helper ⟩

+
      a ∈ deduplicate≡ l ++ deduplicate≡ l'       ∎

+
    where open R.EquationalReasoning

+
          helper : ∀ {l l' a} → a ∈ l ++ l' ⇔ (a ∈ l ⊎ a ∈ l')

+
          helper = mk⇔ (∈-++⁻ _) Data.Sum.[ ∈-++⁺ˡ , ∈-++⁺ʳ _ ]

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Extrema.Core.html b/site/static/agda_html/Data.List.Extrema.Core.html
new file mode 100644
index 000000000..06bf19aff
--- /dev/null
+++ b/site/static/agda_html/Data.List.Extrema.Core.html
@@ -0,0 +1,198 @@
+
+Data.List.Extrema.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core lemmas needed to make list argmin/max functions work

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Definitions using (Trans)

+
open import Relation.Binary.Bundles using (TotalOrder; Setoid)

+

+
module Data.List.Extrema.Core

+
  {b ℓ₁ ℓ₂} (totalOrder : TotalOrder b ℓ₁ ℓ₂) where

+

+
open import Algebra.Core

+
open import Algebra.Definitions

+
import Algebra.Construct.NaturalChoice.Min as Min

+
import Algebra.Construct.NaturalChoice.Max as Max

+
open import Data.Product.Base using (_×_; _,_)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
open import Algebra.Construct.LiftedChoice

+

+
open TotalOrder totalOrder renaming (Carrier to B)

+
open import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_

+
  using (_<_; <-≤-trans; ≤-<-trans)

+

+
------------------------------------------------------------------------

+
-- Setup

+

+
-- open NonStrictToStrict totalOrder using (_<_; ≤-<-trans; <-≤-trans)

+

+
open Max totalOrder

+
open Min totalOrder

+

+
private

+

+
  variable

+
    a : Level

+
    A : Set a

+

+
  <-transʳ : Trans _≤_ _<_ _<_

+
  <-transʳ = ≤-<-trans trans antisym ≤-respˡ-≈

+

+
  <-transˡ : Trans _<_ _≤_ _<_

+
  <-transˡ = <-≤-trans Eq.sym trans antisym ≤-respʳ-≈

+

+
  module _ (f : A → B) where

+

+
    lemma₁ : ∀ {x y v} → f x ≤ v → f x ⊓ f y ≈ f y → f y ≤ v

+
    lemma₁ fx≤v fx⊓fy≈fy = trans (x⊓y≈y⇒y≤x fx⊓fy≈fy) fx≤v

+

+
    lemma₂ : ∀ {x y v} → f y ≤ v → f x ⊓ f y ≈ f x → f x ≤ v

+
    lemma₂ fy≤v fx⊓fy≈fx = trans (x⊓y≈x⇒x≤y fx⊓fy≈fx) fy≤v

+

+
    lemma₃ : ∀ {x y v} → f x < v → f x ⊓ f y ≈ f y → f y < v

+
    lemma₃ fx<v fx⊓fy≈fy = <-transʳ (x⊓y≈y⇒y≤x fx⊓fy≈fy) fx<v

+

+
    lemma₄ : ∀ {x y v} → f y < v → f x ⊓ f y ≈ f x → f x < v

+
    lemma₄ fx<v fx⊓fy≈fy = <-transʳ (x⊓y≈x⇒x≤y fx⊓fy≈fy) fx<v

+

+
------------------------------------------------------------------------

+
-- Definition of lifted max and min

+

+
⊓ᴸ : (A → B) → Op₂ A

+
⊓ᴸ = Lift _≈_ _⊓_ ⊓-sel

+

+
⊔ᴸ : (A → B) → Op₂ A

+
⊔ᴸ = Lift _≈_ _⊔_ ⊔-sel

+

+
------------------------------------------------------------------------

+
-- Properties of ⊓ᴸ

+

+
⊓ᴸ-sel : ∀ f → Selective {A = A} _≡_ (⊓ᴸ f)

+
⊓ᴸ-sel f = sel-≡ ⊓-isSelectiveMagma f

+

+
⊓ᴸ-presᵒ-≤v : ∀ f {v} (x y : A) → f x ≤ v ⊎ f y ≤ v → f (⊓ᴸ f x y) ≤ v

+
⊓ᴸ-presᵒ-≤v f = preservesᵒ ⊓-isSelectiveMagma f (lemma₁ f) (lemma₂ f)

+

+
⊓ᴸ-presᵒ-<v : ∀ f {v} (x y : A) → f x < v ⊎ f y < v → f (⊓ᴸ f x y) < v

+
⊓ᴸ-presᵒ-<v f = preservesᵒ ⊓-isSelectiveMagma f (lemma₃ f) (lemma₄ f)

+

+
⊓ᴸ-presᵇ-v≤ : ∀ f {v} {x y : A} → v ≤ f x → v ≤ f y → v ≤ f (⊓ᴸ f x y)

+
⊓ᴸ-presᵇ-v≤ f {v} = preservesᵇ ⊓-isSelectiveMagma {P = λ x → v ≤ f x} f

+

+
⊓ᴸ-presᵇ-v< : ∀ f {v} {x y : A} → v < f x → v < f y → v < f (⊓ᴸ f x y)

+
⊓ᴸ-presᵇ-v< f {v} = preservesᵇ ⊓-isSelectiveMagma {P = λ x → v < f x} f

+

+
⊓ᴸ-forcesᵇ-v≤ : ∀ f {v} (x y : A) → v ≤ f (⊓ᴸ f x y) → v ≤ f x × v ≤ f y

+
⊓ᴸ-forcesᵇ-v≤ f {v} = forcesᵇ ⊓-isSelectiveMagma f

+
  (λ v≤fx fx⊓fy≈fx → trans v≤fx (x⊓y≈x⇒x≤y fx⊓fy≈fx))

+
  (λ v≤fy fx⊓fy≈fy → trans v≤fy (x⊓y≈y⇒y≤x fx⊓fy≈fy))

+

+
------------------------------------------------------------------------

+
-- Properties of ⊔ᴸ

+

+
⊔ᴸ-sel : ∀ f → Selective {A = A} _≡_ (⊔ᴸ f)

+
⊔ᴸ-sel f = sel-≡ ⊔-isSelectiveMagma f

+

+
⊔ᴸ-presᵒ-v≤ : ∀ f {v} (x y : A) → v ≤ f x ⊎ v ≤ f y → v ≤ f (⊔ᴸ f x y)

+
⊔ᴸ-presᵒ-v≤ f {v} = preservesᵒ ⊔-isSelectiveMagma f

+
  (λ v≤fx fx⊔fy≈fy → trans v≤fx (x⊔y≈y⇒x≤y fx⊔fy≈fy))

+
  (λ v≤fy fx⊔fy≈fx → trans v≤fy (x⊔y≈x⇒y≤x fx⊔fy≈fx))

+

+
⊔ᴸ-presᵒ-v< : ∀ f {v} (x y : A) → v < f x ⊎ v < f y → v < f (⊔ᴸ f x y)

+
⊔ᴸ-presᵒ-v< f {v} = preservesᵒ ⊔-isSelectiveMagma f

+
  (λ v<fx fx⊔fy≈fy → <-transˡ v<fx (x⊔y≈y⇒x≤y fx⊔fy≈fy))

+
  (λ v<fy fx⊔fy≈fx → <-transˡ v<fy (x⊔y≈x⇒y≤x fx⊔fy≈fx))

+

+
⊔ᴸ-presᵇ-≤v : ∀ f {v} {x y : A} → f x ≤ v → f y ≤ v → f (⊔ᴸ f x y) ≤ v

+
⊔ᴸ-presᵇ-≤v f {v} = preservesᵇ ⊔-isSelectiveMagma {P = λ x → f x ≤ v} f

+

+
⊔ᴸ-presᵇ-<v : ∀ f {v} {x y : A} → f x < v → f y < v → f (⊔ᴸ f x y) < v

+
⊔ᴸ-presᵇ-<v f {v} = preservesᵇ ⊔-isSelectiveMagma {P = λ x → f x < v} f

+

+
⊔ᴸ-forcesᵇ-≤v : ∀ f {v} (x y : A) → f (⊔ᴸ f x y) ≤ v → f x ≤ v × f y ≤ v

+
⊔ᴸ-forcesᵇ-≤v f {v} = forcesᵇ ⊔-isSelectiveMagma f

+
  (λ fx≤v fx⊔fy≈fx → trans (x⊔y≈x⇒y≤x fx⊔fy≈fx) fx≤v)

+
  (λ fy≤v fx⊔fy≈fy → trans (x⊔y≈y⇒x≤y fx⊔fy≈fy) fy≤v)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Extrema.html b/site/static/agda_html/Data.List.Extrema.html
new file mode 100644
index 000000000..5278a4efa
--- /dev/null
+++ b/site/static/agda_html/Data.List.Extrema.html
@@ -0,0 +1,324 @@
+
+Data.List.Extrema
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Finding the maximum/minimum values in a list

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (TotalOrder; Setoid)

+

+
module Data.List.Extrema

+
  {b ℓ₁ ℓ₂} (totalOrder : TotalOrder b ℓ₁ ℓ₂) where

+

+
import Algebra.Construct.NaturalChoice.Min as Min

+
import Algebra.Construct.NaturalChoice.Max as Max

+
open import Data.List.Base using (List; foldr)

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.List.Relation.Unary.All using (All; []; _∷_; lookup; map; tabulate)

+
open import Data.List.Membership.Propositional using (_∈_; lose)

+
open import Data.List.Membership.Propositional.Properties

+
  using (foldr-selective)

+
open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_; _⊇_)

+
open import Data.List.Properties

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Function.Base using (id; flip; _on_; _∘_)

+
open import Level using (Level)

+
open import Relation.Unary using (Pred)

+
import Relation.Binary.Construct.NonStrictToStrict as NonStrictToStrict

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; sym; subst) renaming (refl to ≡-refl)

+
import Relation.Binary.Construct.On as On

+

+
------------------------------------------------------------------------

+
-- Setup

+

+
open TotalOrder totalOrder renaming (Carrier to B)

+
open NonStrictToStrict _≈_ _≤_ using (_<_)

+
open import Data.List.Extrema.Core totalOrder

+
  renaming (⊓ᴸ to ⊓-lift; ⊔ᴸ to ⊔-lift)

+

+
private

+
  variable

+
    a p : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- Functions

+

+
argmin : (A → B) → A → List A → A

+
argmin f = foldr (⊓-lift f)

+

+
argmax : (A → B) → A → List A → A

+
argmax f = foldr (⊔-lift f)

+

+
min : B → List B → B

+
min = argmin id

+

+
max : B → List B → B

+
max = argmax id

+

+
------------------------------------------------------------------------

+
-- Properties of argmin

+

+
module _ {f : A → B} where

+

+
  f[argmin]≤v⁺ : ∀ {v} ⊤ xs → (f ⊤ ≤ v) ⊎ (Any (λ x → f x ≤ v) xs) →

+
                f (argmin f ⊤ xs) ≤ v

+
  f[argmin]≤v⁺ = foldr-preservesᵒ (⊓ᴸ-presᵒ-≤v f)

+

+
  f[argmin]<v⁺ : ∀ {v} ⊤ xs → (f ⊤ < v) ⊎ (Any (λ x → f x < v) xs) →

+
                f (argmin f ⊤ xs) < v

+
  f[argmin]<v⁺ = foldr-preservesᵒ (⊓ᴸ-presᵒ-<v f)

+

+
  v≤f[argmin]⁺ : ∀ {v ⊤ xs} → v ≤ f ⊤ → All (λ x → v ≤ f x) xs →

+
                v ≤ f (argmin f ⊤ xs)

+
  v≤f[argmin]⁺ = foldr-preservesᵇ (⊓ᴸ-presᵇ-v≤ f)

+

+
  v<f[argmin]⁺ : ∀ {v ⊤ xs} → v < f ⊤ → All (λ x → v < f x) xs →

+
                v < f (argmin f ⊤ xs)

+
  v<f[argmin]⁺ = foldr-preservesᵇ (⊓ᴸ-presᵇ-v< f)

+

+
  f[argmin]≤f[⊤] : ∀ ⊤ xs → f (argmin f ⊤ xs) ≤ f ⊤

+
  f[argmin]≤f[⊤] ⊤ xs = f[argmin]≤v⁺ ⊤ xs (inj₁ refl)

+

+
  f[argmin]≤f[xs] : ∀ ⊤ xs → All (λ x → f (argmin f ⊤ xs) ≤ f x) xs

+
  f[argmin]≤f[xs] ⊤ xs = foldr-forcesᵇ (⊓ᴸ-forcesᵇ-v≤ f) ⊤ xs refl

+

+
  f[argmin]≈f[v]⁺ : ∀ {v ⊤ xs} → v ∈ xs → All (λ x → f v ≤ f x) xs → f v ≤ f ⊤ →

+
                    f (argmin f ⊤ xs) ≈ f v

+
  f[argmin]≈f[v]⁺ v∈xs fv≤fxs fv≤f⊤ = antisym

+
    (f[argmin]≤v⁺ _ _ (inj₂ (lose v∈xs refl)))

+
    (v≤f[argmin]⁺ fv≤f⊤ fv≤fxs)

+

+
argmin[xs]≤argmin[ys]⁺ : ∀ {f g : A → B} ⊤₁ {⊤₂} xs {ys : List A} →

+
                        (f ⊤₁ ≤ g ⊤₂) ⊎ Any (λ x → f x ≤ g ⊤₂) xs →

+
                        All (λ y → (f ⊤₁ ≤ g y) ⊎ Any (λ x → f x ≤ g y) xs) ys →

+
                        f (argmin f ⊤₁ xs) ≤ g (argmin g ⊤₂ ys)

+
argmin[xs]≤argmin[ys]⁺ ⊤₁ xs xs≤⊤₂ xs≤ys =

+
  v≤f[argmin]⁺ (f[argmin]≤v⁺ ⊤₁ _ xs≤⊤₂) (map (f[argmin]≤v⁺ ⊤₁ xs) xs≤ys)

+

+
argmin[xs]<argmin[ys]⁺ : ∀ {f g : A → B} ⊤₁ {⊤₂} xs {ys : List A} →

+
                        (f ⊤₁ < g ⊤₂) ⊎ Any (λ x → f x < g ⊤₂) xs →

+
                        All (λ y → (f ⊤₁ < g y) ⊎ Any (λ x → f x < g y) xs) ys →

+
                        f (argmin f ⊤₁ xs) < g (argmin g ⊤₂ ys)

+
argmin[xs]<argmin[ys]⁺ ⊤₁ xs xs<⊤₂ xs<ys =

+
  v<f[argmin]⁺ (f[argmin]<v⁺ ⊤₁ _ xs<⊤₂) (map (f[argmin]<v⁺ ⊤₁ xs) xs<ys)

+

+
argmin-sel : ∀ (f : A → B) ⊤ xs → (argmin f ⊤ xs ≡ ⊤) ⊎ (argmin f ⊤ xs ∈ xs)

+
argmin-sel f = foldr-selective (⊓ᴸ-sel f)

+

+
argmin-all : ∀ (f : A → B) {⊤ xs} {P : Pred A p} →

+
             P ⊤ → All P xs → P (argmin f ⊤ xs)

+
argmin-all f {⊤} {xs} {P = P}  p⊤ pxs with argmin-sel f ⊤ xs

+
... | inj₁ argmin≡⊤  = subst P (sym argmin≡⊤) p⊤

+
... | inj₂ argmin∈xs = lookup pxs argmin∈xs

+

+
------------------------------------------------------------------------

+
-- Properties of argmax

+

+
module _ {f : A → B} where

+

+
  v≤f[argmax]⁺ : ∀ {v} ⊥ xs → (v ≤ f ⊥) ⊎ (Any (λ x → v ≤ f x) xs) →

+
                v ≤ f (argmax f ⊥ xs)

+
  v≤f[argmax]⁺ = foldr-preservesᵒ (⊔ᴸ-presᵒ-v≤ f)

+

+
  v<f[argmax]⁺ : ∀ {v} ⊥ xs → (v < f ⊥) ⊎ (Any (λ x → v < f x) xs) →

+
                v < f (argmax f ⊥ xs)

+
  v<f[argmax]⁺ = foldr-preservesᵒ (⊔ᴸ-presᵒ-v< f)

+

+
  f[argmax]≤v⁺ : ∀ {v ⊥ xs} → f ⊥ ≤ v → All (λ x → f x ≤ v) xs →

+
                f (argmax f ⊥ xs) ≤ v

+
  f[argmax]≤v⁺ = foldr-preservesᵇ (⊔ᴸ-presᵇ-≤v f)

+

+
  f[argmax]<v⁺ : ∀ {v ⊥ xs} → f ⊥ < v → All (λ x → f x < v) xs →

+
                f (argmax f ⊥ xs) < v

+
  f[argmax]<v⁺ = foldr-preservesᵇ (⊔ᴸ-presᵇ-<v f)

+

+
  f[⊥]≤f[argmax] : ∀ ⊥ xs → f ⊥ ≤ f (argmax f ⊥ xs)

+
  f[⊥]≤f[argmax] ⊥ xs = v≤f[argmax]⁺ ⊥ xs (inj₁ refl)

+

+
  f[xs]≤f[argmax] : ∀ ⊥ xs → All (λ x → f x ≤ f (argmax f ⊥ xs)) xs

+
  f[xs]≤f[argmax] ⊥ xs = foldr-forcesᵇ (⊔ᴸ-forcesᵇ-≤v f) ⊥ xs refl

+

+
  f[argmax]≈f[v]⁺ : ∀ {v ⊥ xs} → v ∈ xs → All (λ x → f x ≤ f v) xs → f ⊥ ≤ f v →

+
                    f (argmax f ⊥ xs) ≈ f v

+
  f[argmax]≈f[v]⁺ v∈xs fxs≤fv f⊥≤fv = antisym

+
    (f[argmax]≤v⁺ f⊥≤fv fxs≤fv)

+
    (v≤f[argmax]⁺ _ _ (inj₂ (lose v∈xs refl)))

+

+
argmax[xs]≤argmax[ys]⁺ : ∀ {f g : A → B} {⊥₁} ⊥₂ {xs : List A} ys →

+
                         (f ⊥₁ ≤ g ⊥₂) ⊎ Any (λ y → f ⊥₁ ≤ g y) ys →

+
                         All (λ x → (f x ≤ g ⊥₂) ⊎ Any (λ y → f x ≤ g y) ys) xs →

+
                         f (argmax f ⊥₁ xs) ≤ g (argmax g ⊥₂ ys)

+
argmax[xs]≤argmax[ys]⁺ ⊥₂ ys ⊥₁≤ys xs≤ys =

+
  f[argmax]≤v⁺ (v≤f[argmax]⁺ ⊥₂ _ ⊥₁≤ys) (map (v≤f[argmax]⁺ ⊥₂ ys) xs≤ys)

+

+
argmax[xs]<argmax[ys]⁺ : ∀ {f g : A → B} {⊥₁} ⊥₂ {xs : List A} ys →

+
                         (f ⊥₁ < g ⊥₂) ⊎ Any (λ y → f ⊥₁ < g y) ys →

+
                         All (λ x → (f x < g ⊥₂) ⊎ Any (λ y → f x < g y) ys) xs →

+
                         f (argmax f ⊥₁ xs) < g (argmax g ⊥₂ ys)

+
argmax[xs]<argmax[ys]⁺ ⊥₂ ys ⊥₁<ys xs<ys =

+
  f[argmax]<v⁺ (v<f[argmax]⁺ ⊥₂ _ ⊥₁<ys) (map (v<f[argmax]⁺ ⊥₂ ys) xs<ys)

+

+
argmax-sel : ∀ (f : A → B) ⊥ xs → (argmax f ⊥ xs ≡ ⊥) ⊎ (argmax f ⊥ xs ∈ xs)

+
argmax-sel f = foldr-selective (⊔ᴸ-sel f)

+

+
argmax-all : ∀ (f : A → B) {P : Pred A p} {⊥ xs} →

+
             P ⊥ → All P xs → P (argmax f ⊥ xs)

+
argmax-all f {P = P} {⊥} {xs} p⊥ pxs with argmax-sel f ⊥ xs

+
... | inj₁ argmax≡⊥  = subst P (sym argmax≡⊥) p⊥

+
... | inj₂ argmax∈xs = lookup pxs argmax∈xs

+

+
------------------------------------------------------------------------

+
-- Properties of min

+

+
min≤v⁺ : ∀ {v} ⊤ xs → ⊤ ≤ v ⊎ Any (_≤ v) xs → min ⊤ xs ≤ v

+
min≤v⁺ = f[argmin]≤v⁺

+

+
min<v⁺ : ∀ {v} ⊤ xs → ⊤ < v ⊎ Any (_< v) xs → min ⊤ xs < v

+
min<v⁺ = f[argmin]<v⁺

+

+
v≤min⁺ : ∀ {v ⊤ xs} → v ≤ ⊤ → All (v ≤_) xs → v ≤ min ⊤ xs

+
v≤min⁺ = v≤f[argmin]⁺

+

+
v<min⁺ : ∀ {v ⊤ xs} → v < ⊤ → All (v <_) xs → v < min ⊤ xs

+
v<min⁺ = v<f[argmin]⁺

+

+
min≤⊤ : ∀ ⊤ xs → min ⊤ xs ≤ ⊤

+
min≤⊤ = f[argmin]≤f[⊤]

+

+
min≤xs : ∀ ⊥ xs → All (min ⊥ xs ≤_) xs

+
min≤xs = f[argmin]≤f[xs]

+

+
min≈v⁺ : ∀ {v ⊤ xs} → v ∈ xs → All (v ≤_) xs → v ≤ ⊤ → min ⊤ xs ≈ v

+
min≈v⁺ = f[argmin]≈f[v]⁺

+

+
min[xs]≤min[ys]⁺ : ∀ ⊤₁ {⊤₂} xs {ys} → (⊤₁ ≤ ⊤₂) ⊎ Any (_≤ ⊤₂) xs →

+
                   All (λ y → (⊤₁ ≤ y) ⊎ Any (λ x → x ≤ y) xs) ys →

+
                   min ⊤₁ xs ≤ min ⊤₂ ys

+
min[xs]≤min[ys]⁺ = argmin[xs]≤argmin[ys]⁺

+

+
min[xs]<min[ys]⁺ : ∀ ⊤₁ {⊤₂} xs {ys} → (⊤₁ < ⊤₂) ⊎ Any (_< ⊤₂) xs →

+
                   All (λ y → (⊤₁ < y) ⊎ Any (λ x → x < y) xs) ys →

+
                   min ⊤₁ xs < min ⊤₂ ys

+
min[xs]<min[ys]⁺ = argmin[xs]<argmin[ys]⁺

+

+
min-mono-⊆ : ∀ {⊥₁ ⊥₂ xs ys} → ⊥₁ ≤ ⊥₂ → xs ⊇ ys → min ⊥₁ xs ≤ min ⊥₂ ys

+
min-mono-⊆ ⊥₁≤⊥₂ ys⊆xs = min[xs]≤min[ys]⁺ _ _ (inj₁ ⊥₁≤⊥₂)

+
  (tabulate (inj₂ ∘ Any.map (λ {≡-refl → refl}) ∘ ys⊆xs))

+

+
------------------------------------------------------------------------

+
-- Properties of max

+

+
max≤v⁺ : ∀ {v ⊥ xs} → ⊥ ≤ v → All (_≤ v) xs → max ⊥ xs ≤ v

+
max≤v⁺ = f[argmax]≤v⁺

+

+
max<v⁺ : ∀ {v ⊥ xs} → ⊥ < v → All (_< v) xs → max ⊥ xs < v

+
max<v⁺ = f[argmax]<v⁺

+

+
v≤max⁺ : ∀ {v} ⊥ xs → v ≤ ⊥ ⊎ Any (v ≤_) xs → v ≤ max ⊥ xs

+
v≤max⁺ = v≤f[argmax]⁺

+

+
v<max⁺ : ∀ {v} ⊥ xs → v < ⊥ ⊎ Any (v <_) xs → v < max ⊥ xs

+
v<max⁺ = v<f[argmax]⁺

+

+
⊥≤max : ∀ ⊥ xs → ⊥ ≤ max ⊥ xs

+
⊥≤max = f[⊥]≤f[argmax]

+

+
xs≤max : ∀ ⊥ xs → All (_≤ max ⊥ xs) xs

+
xs≤max = f[xs]≤f[argmax]

+

+
max≈v⁺ : ∀ {v ⊤ xs} → v ∈ xs → All (_≤ v) xs → ⊤ ≤ v → max ⊤ xs ≈ v

+
max≈v⁺ = f[argmax]≈f[v]⁺

+

+
max[xs]≤max[ys]⁺ : ∀ {⊥₁} ⊥₂ {xs} ys → ⊥₁ ≤ ⊥₂ ⊎ Any (⊥₁ ≤_) ys →

+
                   All (λ x → x ≤ ⊥₂ ⊎ Any (x ≤_) ys) xs →

+
                   max ⊥₁ xs ≤ max ⊥₂ ys

+
max[xs]≤max[ys]⁺ = argmax[xs]≤argmax[ys]⁺

+

+
max[xs]<max[ys]⁺ : ∀ {⊥₁} ⊥₂ {xs} ys → ⊥₁ < ⊥₂ ⊎ Any (⊥₁ <_) ys →

+
                   All (λ x → x < ⊥₂ ⊎ Any (x <_) ys) xs →

+
                   max ⊥₁ xs < max ⊥₂ ys

+
max[xs]<max[ys]⁺ = argmax[xs]<argmax[ys]⁺

+

+
max-mono-⊆ : ∀ {⊥₁ ⊥₂ xs ys} → ⊥₁ ≤ ⊥₂ → xs ⊆ ys → max ⊥₁ xs ≤ max ⊥₂ ys

+
max-mono-⊆ ⊥₁≤⊥₂ xs⊆ys = max[xs]≤max[ys]⁺ _ _ (inj₁ ⊥₁≤⊥₂)

+
  (tabulate (inj₂ ∘ Any.map (λ {≡-refl → refl}) ∘ xs⊆ys))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.DecPropositional.html b/site/static/agda_html/Data.List.Membership.DecPropositional.html
new file mode 100644
index 000000000..ff81a4962
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.DecPropositional.html
@@ -0,0 +1,100 @@
+
+Data.List.Membership.DecPropositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Decidable propositional membership over lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Definitions using (DecidableEquality)

+

+
module Data.List.Membership.DecPropositional

+
  {a} {A : Set a} (_≟_ : DecidableEquality A) where

+

+
open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)

+

+
open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)

+

+
------------------------------------------------------------------------

+
-- Re-export contents of propositional membership

+

+
open import Data.List.Membership.Propositional {A = A} public

+
open import Data.List.Membership.DecSetoid (decSetoid _≟_) public

+
  using (_∈?_; _∉?_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.DecSetoid.html b/site/static/agda_html/Data.List.Membership.DecSetoid.html
new file mode 100644
index 000000000..1c2bcdbdb
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.DecSetoid.html
@@ -0,0 +1,110 @@
+
+Data.List.Membership.DecSetoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Decidable setoid membership over lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (DecSetoid)

+

+
module Data.List.Membership.DecSetoid {a ℓ} (DS : DecSetoid a ℓ) where

+

+
open import Data.List.Relation.Unary.Any using (any?)

+
open import Relation.Binary.Definitions using (Decidable)

+
open import Relation.Nullary.Decidable using (¬?)

+

+
open DecSetoid DS

+

+
------------------------------------------------------------------------

+
-- Re-export contents of propositional membership

+

+
open import Data.List.Membership.Setoid (DecSetoid.setoid DS) public

+

+
------------------------------------------------------------------------

+
-- Other operations

+

+
infix 4 _∈?_ _∉?_

+

+
_∈?_ : Decidable _∈_

+
x ∈? xs = any? (x ≟_) xs

+

+
_∉?_ : Decidable _∉_

+
x ∉? xs = ¬? (x ∈? xs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.Propositional.Properties.Core.html b/site/static/agda_html/Data.List.Membership.Propositional.Properties.Core.html
new file mode 100644
index 000000000..63b24713d
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.Propositional.Properties.Core.html
@@ -0,0 +1,162 @@
+
+Data.List.Membership.Propositional.Properties.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core properties related to propositional list membership.

+
------------------------------------------------------------------------

+

+
-- This file is needed to break the cyclic dependency with the proof

+
-- `Any-cong` in `Data.List.Relation.Unary.Any.Properties` which relies

+
-- on `Any↔` defined in this file.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Membership.Propositional.Properties.Core where

+

+
open import Data.List.Base using (List)

+
open import Data.List.Membership.Propositional

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.Product.Base as Product using (_,_; ∃; _×_)

+
open import Function.Base using (flip; id; _∘_)

+
open import Function.Bundles using (_↔_; mk↔ₛ′)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; resp)

+
open import Relation.Unary using (Pred; _⊆_)

+

+
private

+
  variable

+
    a p q : Level

+
    A : Set a

+
    x : A

+
    xs : List A

+

+
------------------------------------------------------------------------

+
-- find satisfies a simple equality when the predicate is a

+
-- propositional equality.

+

+
find-∈ : (x∈xs : x ∈ xs) → find x∈xs ≡ (x , x∈xs , refl)

+
find-∈ (here refl)  = refl

+
find-∈ (there x∈xs) rewrite find-∈ x∈xs = refl

+

+
------------------------------------------------------------------------

+
-- Lemmas relating map and find.

+

+
module _ {P : Pred A p} where

+

+
  map∘find : (p : Any P xs) → let x , x∈xs , px = find p in

+
             {f : (x ≡_) ⊆ P} → f refl ≡ px →

+
             Any.map f x∈xs ≡ p

+
  map∘find (here  p) hyp = cong here  hyp

+
  map∘find (there p) hyp = cong there (map∘find p hyp)

+

+
  find∘map : ∀ {Q : Pred A q} {xs} (p : Any P xs) (f : P ⊆ Q) →

+
             let x , x∈xs , px = find p in

+
             find (Any.map f p) ≡ (x , x∈xs , f px)

+
  find∘map (here  p) f = refl

+
  find∘map (there p) f rewrite find∘map p f = refl

+

+
------------------------------------------------------------------------

+
-- Any can be expressed using _∈_

+

+
module _ {P : Pred A p} where

+

+
  ∃∈-Any : (∃ λ x → x ∈ xs × P x) → Any P xs

+
  ∃∈-Any (x , x∈xs , px) = lose {P = P} x∈xs px

+

+
  ∃∈-Any∘find : (p : Any P xs) → ∃∈-Any (find p) ≡ p

+
  ∃∈-Any∘find p = map∘find p refl

+

+
  find∘∃∈-Any : (p : ∃ λ x → x ∈ xs × P x) → find (∃∈-Any p) ≡ p

+
  find∘∃∈-Any p@(x , x∈xs , px)

+
    rewrite find∘map x∈xs (flip (resp P) px) | find-∈ x∈xs = refl

+

+
  Any↔ : (∃ λ x → x ∈ xs × P x) ↔ Any P xs

+
  Any↔ = mk↔ₛ′ ∃∈-Any find ∃∈-Any∘find find∘∃∈-Any

+

+
------------------------------------------------------------------------

+
-- Hence, find and lose are inverses (more or less).

+

+
lose∘find : ∀ {P : Pred A p} {xs} (p : Any P xs) → ∃∈-Any (find p) ≡ p

+
lose∘find = ∃∈-Any∘find

+

+
find∘lose : ∀ (P : Pred A p) {x xs}

+
            (x∈xs : x ∈ xs) (px : P x) →

+
            find (lose {P = P} x∈xs px) ≡ (x , x∈xs , px)

+
find∘lose P {x} x∈xs px = find∘∃∈-Any (x , x∈xs , px)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.Propositional.Properties.html b/site/static/agda_html/Data.List.Membership.Propositional.Properties.html
new file mode 100644
index 000000000..b7b3e3ea9
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.Propositional.Properties.html
@@ -0,0 +1,480 @@
+
+Data.List.Membership.Propositional.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to propositional list membership

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Membership.Propositional.Properties where

+

+
open import Algebra.Core using (Op₂)

+
open import Algebra.Definitions using (Selective)

+
open import Data.Fin.Base using (Fin)

+
open import Data.List.Base as List

+
open import Data.List.Effectful using (monad)

+
open import Data.List.Membership.Propositional

+
  using (_∈_; _∉_; mapWith∈; _≢∈_)

+
import Data.List.Membership.Setoid.Properties as Membership

+
open import Data.List.Relation.Binary.Equality.Propositional

+
  using (_≋_; ≡⇒≋; ≋⇒≡)

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.List.Relation.Unary.Any.Properties

+
  using (map↔; concat↔; >>=↔; ⊛↔; Any-cong; ⊗↔′; ¬Any[])

+
open import Data.Nat.Base using (ℕ; suc; s≤s; _≤_; _<_; _≰_)

+
open import Data.Nat.Properties

+
  using (suc-injective; m≤n⇒m≤1+n; _≤?_; <⇒≢; ≰⇒>)

+
open import Data.Product.Base using (∃; ∃₂; _×_; _,_)

+
open import Data.Product.Properties using (×-≡,≡↔≡)

+
open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)

+
import Data.Product.Function.Dependent.Propositional as Σ

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Effect.Monad using (RawMonad)

+
open import Function.Base using (_∘_; _∘′_; _$_; id; flip; _⟨_⟩_)

+
open import Function.Definitions using (Injective)

+
import Function.Related.Propositional as Related

+
open import Function.Bundles using (_↔_; _↣_; Injection)

+
open import Function.Related.TypeIsomorphisms using (×-comm; ∃∃↔∃∃)

+
open import Function.Construct.Identity using (↔-id)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions as Binary hiding (Decidable)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; resp; _≗_)

+
open import Relation.Binary.PropositionalEquality.Properties as ≡ using (setoid)

+
import Relation.Binary.Properties.DecTotalOrder as DTOProperties

+
open import Relation.Nullary.Decidable.Core

+
  using (Dec; yes; no; ¬¬-excluded-middle)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Unary using (_⟨×⟩_; Decidable)

+

+
private

+
  open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})

+

+
  variable

+
    ℓ : Level

+
    A B C : Set ℓ

+

+
------------------------------------------------------------------------

+
-- Publicly re-export properties from Core

+

+
open import Data.List.Membership.Propositional.Properties.Core public

+

+
------------------------------------------------------------------------

+
-- Equality

+

+
∈-resp-≋ : ∀ {x : A} → (x ∈_) Respects _≋_

+
∈-resp-≋ = Membership.∈-resp-≋ (≡.setoid _)

+

+
∉-resp-≋ : ∀ {x : A} → (x ∉_) Respects _≋_

+
∉-resp-≋ = Membership.∉-resp-≋ (≡.setoid _)

+

+
------------------------------------------------------------------------

+
-- mapWith∈

+

+
mapWith∈-cong : ∀ (xs : List A) → (f g : ∀ {x} → x ∈ xs → B) →

+
                (∀ {x} → (x∈xs : x ∈ xs) → f x∈xs ≡ g x∈xs) →

+
                mapWith∈ xs f ≡ mapWith∈ xs g

+
mapWith∈-cong []       f g cong = refl

+
mapWith∈-cong (x ∷ xs) f g cong = cong₂ _∷_ (cong (here refl))

+
  (mapWith∈-cong xs (f ∘ there) (g ∘ there) (cong ∘ there))

+

+
mapWith∈≗map : ∀ (f : A → B) xs → mapWith∈ xs (λ {x} _ → f x) ≡ map f xs

+
mapWith∈≗map f xs =

+
  ≋⇒≡ (Membership.mapWith∈≗map (≡.setoid _) (≡.setoid _) f xs)

+

+
mapWith∈-id : (xs : List A) → mapWith∈ xs (λ {x} _ → x) ≡ xs

+
mapWith∈-id = Membership.mapWith∈-id (≡.setoid _)

+

+
map-mapWith∈ : (xs : List A) (f : ∀ {x} → x ∈ xs → B) (g : B → C) →

+
               map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)

+
map-mapWith∈ = Membership.map-mapWith∈ (≡.setoid _)

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ (f : A → B) where

+

+
  ∈-map⁺ : ∀ {x xs} → x ∈ xs → f x ∈ map f xs

+
  ∈-map⁺ = Membership.∈-map⁺ (≡.setoid A) (≡.setoid B) (cong f)

+

+
  ∈-map⁻ : ∀ {y xs} → y ∈ map f xs → ∃ λ x → x ∈ xs × y ≡ f x

+
  ∈-map⁻ = Membership.∈-map⁻ (≡.setoid A) (≡.setoid B)

+

+
  map-∈↔ : ∀ {y xs} → (∃ λ x → x ∈ xs × y ≡ f x) ↔ y ∈ map f xs

+
  map-∈↔ {y} {xs} =

+
    (∃ λ x → x ∈ xs × y ≡ f x)   ↔⟨ Any↔ ⟩

+
    Any (λ x → y ≡ f x) xs       ↔⟨ map↔ ⟩

+
    y ∈ List.map f xs            ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ {v : A} where

+

+
  ∈-++⁺ˡ : ∀ {xs ys} → v ∈ xs → v ∈ xs ++ ys

+
  ∈-++⁺ˡ = Membership.∈-++⁺ˡ (≡.setoid A)

+

+
  ∈-++⁺ʳ : ∀ xs {ys} → v ∈ ys → v ∈ xs ++ ys

+
  ∈-++⁺ʳ = Membership.∈-++⁺ʳ (≡.setoid A)

+

+
  ∈-++⁻ : ∀ xs {ys} → v ∈ xs ++ ys → (v ∈ xs) ⊎ (v ∈ ys)

+
  ∈-++⁻ = Membership.∈-++⁻ (≡.setoid A)

+

+
  ∈-insert : ∀ xs {ys} → v ∈ xs ++ [ v ] ++ ys

+
  ∈-insert xs = Membership.∈-insert (≡.setoid A) xs refl

+

+
  ∈-∃++ : ∀ {xs} → v ∈ xs → ∃₂ λ ys zs → xs ≡ ys ++ [ v ] ++ zs

+
  ∈-∃++ v∈xs

+
    with ys , zs , _ , refl , eq ← Membership.∈-∃++ (≡.setoid A) v∈xs

+
    = ys , zs , ≋⇒≡ eq

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {v : A} where

+

+
  ∈-concat⁺ : ∀ {xss} → Any (v ∈_) xss → v ∈ concat xss

+
  ∈-concat⁺ = Membership.∈-concat⁺ (≡.setoid A)

+

+
  ∈-concat⁻ : ∀ xss → v ∈ concat xss → Any (v ∈_) xss

+
  ∈-concat⁻ = Membership.∈-concat⁻ (≡.setoid A)

+

+
  ∈-concat⁺′ : ∀ {vs xss} → v ∈ vs → vs ∈ xss → v ∈ concat xss

+
  ∈-concat⁺′ v∈vs vs∈xss =

+
    Membership.∈-concat⁺′ (≡.setoid A) v∈vs (Any.map ≡⇒≋ vs∈xss)

+

+
  ∈-concat⁻′ : ∀ xss → v ∈ concat xss → ∃ λ xs → v ∈ xs × xs ∈ xss

+
  ∈-concat⁻′ xss v∈c =

+
    let xs , v∈xs , xs∈xss = Membership.∈-concat⁻′ (≡.setoid A) xss v∈c

+
    in xs , v∈xs , Any.map ≋⇒≡ xs∈xss

+

+
  concat-∈↔ : ∀ {xss : List (List A)} →

+
              (∃ λ xs → v ∈ xs × xs ∈ xss) ↔ v ∈ concat xss

+
  concat-∈↔ {xss} =

+
    (∃ λ xs → v ∈ xs × xs ∈ xss)  ↔⟨ Σ.cong (↔-id _) $ ×-comm _ _ ⟩

+
    (∃ λ xs → xs ∈ xss × v ∈ xs)  ↔⟨ Any↔ ⟩

+
    Any (Any (v ≡_)) xss          ↔⟨ concat↔ ⟩

+
    v ∈ concat xss                ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith

+

+
module _ (f : A → B → C) where

+

+
  ∈-cartesianProductWith⁺ : ∀ {xs ys a b} → a ∈ xs → b ∈ ys →

+
                            f a b ∈ cartesianProductWith f xs ys

+
  ∈-cartesianProductWith⁺ = Membership.∈-cartesianProductWith⁺

+
    (≡.setoid A) (≡.setoid B) (≡.setoid C) (cong₂ f)

+

+
  ∈-cartesianProductWith⁻ : ∀ xs ys {v} → v ∈ cartesianProductWith f xs ys →

+
                            ∃₂ λ a b → a ∈ xs × b ∈ ys × v ≡ f a b

+
  ∈-cartesianProductWith⁻ = Membership.∈-cartesianProductWith⁻

+
    (≡.setoid A) (≡.setoid B) (≡.setoid C) f

+

+
------------------------------------------------------------------------

+
-- cartesianProduct

+

+
∈-cartesianProduct⁺ : ∀ {x : A} {y : B} {xs ys} → x ∈ xs → y ∈ ys →

+
                      (x , y) ∈ cartesianProduct xs ys

+
∈-cartesianProduct⁺ = ∈-cartesianProductWith⁺ _,_

+

+
∈-cartesianProduct⁻ : ∀ xs ys {xy@(x , y) : A × B} →

+
                      xy ∈ cartesianProduct xs ys → x ∈ xs × y ∈ ys

+
∈-cartesianProduct⁻ xs ys xy∈p[xs,ys]

+
  with _ , _ , x∈xs , y∈ys , refl ← ∈-cartesianProductWith⁻ _,_ xs ys xy∈p[xs,ys]

+
  = x∈xs , y∈ys

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ (f : ℕ → A) where

+

+
  ∈-applyUpTo⁺ : ∀ {i n} → i < n → f i ∈ applyUpTo f n

+
  ∈-applyUpTo⁺ = Membership.∈-applyUpTo⁺ (≡.setoid _) f

+

+
  ∈-applyUpTo⁻ : ∀ {v n} → v ∈ applyUpTo f n →

+
                 ∃ λ i → i < n × v ≡ f i

+
  ∈-applyUpTo⁻ = Membership.∈-applyUpTo⁻ (≡.setoid _) f

+

+
------------------------------------------------------------------------

+
-- upTo

+

+
∈-upTo⁺ : ∀ {n i} → i < n → i ∈ upTo n

+
∈-upTo⁺ = ∈-applyUpTo⁺ id

+

+
∈-upTo⁻ : ∀ {n i} → i ∈ upTo n → i < n

+
∈-upTo⁻ p with _ , i<n , refl ← ∈-applyUpTo⁻ id p = i<n

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ (f : ℕ → A) where

+

+
  ∈-applyDownFrom⁺ : ∀ {i n} → i < n → f i ∈ applyDownFrom f n

+
  ∈-applyDownFrom⁺ = Membership.∈-applyDownFrom⁺ (≡.setoid _) f

+

+
  ∈-applyDownFrom⁻ : ∀ {v n} → v ∈ applyDownFrom f n →

+
                     ∃ λ i → i < n × v ≡ f i

+
  ∈-applyDownFrom⁻ = Membership.∈-applyDownFrom⁻ (≡.setoid _) f

+

+
------------------------------------------------------------------------

+
-- downFrom

+

+
∈-downFrom⁺ : ∀ {n i} → i < n → i ∈ downFrom n

+
∈-downFrom⁺ i<n = ∈-applyDownFrom⁺ id i<n

+

+
∈-downFrom⁻ : ∀ {n i} → i ∈ downFrom n → i < n

+
∈-downFrom⁻ p with _ , i<n , refl ← ∈-applyDownFrom⁻ id p = i<n

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {n} {f : Fin n → A} where

+

+
  ∈-tabulate⁺ : ∀ i → f i ∈ tabulate f

+
  ∈-tabulate⁺ = Membership.∈-tabulate⁺ (≡.setoid _)

+

+
  ∈-tabulate⁻ : ∀ {v} → v ∈ tabulate f → ∃ λ i → v ≡ f i

+
  ∈-tabulate⁻ = Membership.∈-tabulate⁻ (≡.setoid _)

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {p} {P : A → Set p} (P? : Decidable P) where

+

+
  ∈-filter⁺ : ∀ {x xs} → x ∈ xs → P x → x ∈ filter P? xs

+
  ∈-filter⁺ = Membership.∈-filter⁺ (≡.setoid A) P? (≡.resp P)

+

+
  ∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v

+
  ∈-filter⁻ = Membership.∈-filter⁻ (≡.setoid A) P? (≡.resp P)

+

+
------------------------------------------------------------------------

+
-- derun and deduplicate

+

+
module _ {r} {R : Rel A r} (R? : Binary.Decidable R) where

+

+
  ∈-derun⁻ : ∀ xs {z} → z ∈ derun R? xs → z ∈ xs

+
  ∈-derun⁻ xs z∈derun[R,xs] = Membership.∈-derun⁻ (≡.setoid A) R? xs z∈derun[R,xs]

+

+
  ∈-deduplicate⁻ : ∀ xs {z} → z ∈ deduplicate R? xs → z ∈ xs

+
  ∈-deduplicate⁻ xs z∈dedup[R,xs] = Membership.∈-deduplicate⁻ (≡.setoid A) R? xs z∈dedup[R,xs]

+

+
module _ (_≈?_ : DecidableEquality A) where

+

+
  ∈-derun⁺ : ∀ {xs z} → z ∈ xs → z ∈ derun _≈?_ xs

+
  ∈-derun⁺ z∈xs = Membership.∈-derun⁺ (≡.setoid A) _≈?_ (flip trans) z∈xs

+

+
  ∈-deduplicate⁺ : ∀ {xs z} → z ∈ xs → z ∈ deduplicate _≈?_ xs

+
  ∈-deduplicate⁺ z∈xs = Membership.∈-deduplicate⁺ (≡.setoid A) _≈?_ (λ c≡b a≡b → trans a≡b (sym c≡b)) z∈xs

+

+
------------------------------------------------------------------------

+
-- _>>=_

+

+
>>=-∈↔ : ∀ {xs} {f : A → List B} {y} →

+
         (∃ λ x → x ∈ xs × y ∈ f x) ↔ y ∈ (xs >>= f)

+
>>=-∈↔ {xs = xs} {f} {y} =

+
  (∃ λ x → x ∈ xs × y ∈ f x)  ↔⟨ Any↔ ⟩

+
  Any (Any (y ≡_) ∘ f) xs     ↔⟨ >>=↔ ⟩

+
  y ∈ (xs >>= f)              ∎

+
  where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _⊛_

+

+
⊛-∈↔ : ∀ (fs : List (A → B)) {xs y} →

+
       (∃₂ λ f x → f ∈ fs × x ∈ xs × y ≡ f x) ↔ y ∈ (fs ⊛ xs)

+
⊛-∈↔ fs {xs} {y} =

+
  (∃₂ λ f x → f ∈ fs × x ∈ xs × y ≡ f x)       ↔⟨ Σ.cong (↔-id _) (∃∃↔∃∃ _) ⟩

+
  (∃ λ f → f ∈ fs × ∃ λ x → x ∈ xs × y ≡ f x)  ↔⟨ Σ.cong (↔-id _) (↔-id _ ⟨ _×-cong_ ⟩ Any↔) ⟩

+
  (∃ λ f → f ∈ fs × Any (_≡_ y ∘ f) xs)        ↔⟨ Any↔ ⟩

+
  Any (λ f → Any (_≡_ y ∘ f) xs) fs            ↔⟨ ⊛↔ ⟩

+
  y ∈ (fs ⊛ xs)                                ∎

+
  where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _⊗_

+

+
⊗-∈↔ : ∀ {xs ys} {x : A} {y : B} →

+
       (x ∈ xs × y ∈ ys) ↔ (x , y) ∈ (xs ⊗ ys)

+
⊗-∈↔ {xs = xs} {ys} {x} {y} =

+
  (x ∈ xs × y ∈ ys)             ↔⟨ ⊗↔′ ⟩

+
  Any (x ≡_ ⟨×⟩ y ≡_) (xs ⊗ ys) ↔⟨ Any-cong (λ _ → ×-≡,≡↔≡) (↔-id _) ⟩

+
  (x , y) ∈ (xs ⊗ ys)           ∎

+
  where

+
  open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- length

+

+
∈-length : ∀ {x : A} {xs} → x ∈ xs → 0 < length xs

+
∈-length = Membership.∈-length (≡.setoid _)

+

+
------------------------------------------------------------------------

+
-- lookup

+

+
∈-lookup : ∀ {xs : List A} i → lookup xs i ∈ xs

+
∈-lookup {xs = xs} i = Membership.∈-lookup (≡.setoid _) xs i

+

+
------------------------------------------------------------------------

+
-- foldr

+

+
module _ {_•_ : Op₂ A} where

+

+
  foldr-selective : Selective _≡_ _•_ → ∀ e xs →

+
                    (foldr _•_ e xs ≡ e) ⊎ (foldr _•_ e xs ∈ xs)

+
  foldr-selective = Membership.foldr-selective (≡.setoid A)

+

+
------------------------------------------------------------------------

+
-- allFin

+

+
∈-allFin : ∀ {n} (k : Fin n) → k ∈ allFin n

+
∈-allFin = ∈-tabulate⁺

+

+
------------------------------------------------------------------------

+
-- inits

+

+
[]∈inits : ∀ {a} {A : Set a} (as : List A) → [] ∈ inits as

+
[]∈inits _ = here refl

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
-- Only a finite number of distinct elements can be members of a

+
-- given list.

+

+
finite : (inj : ℕ ↣ A) → ∀ xs → ¬ (∀ i → Injection.to inj i ∈ xs)

+
finite inj []       fᵢ∈[]   = ¬Any[] (fᵢ∈[] 0)

+
finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper

+
  where

+
  open Injection inj renaming (injective to f-inj)

+

+
  f : ℕ → _

+
  f = to

+

+
  not-x : ∀ {i} → f i ≢ x → f i ∈ xs

+
  not-x {i} fᵢ≢x with fᵢ∈x∷xs i

+
  ... | here  fᵢ≡x  = contradiction fᵢ≡x fᵢ≢x

+
  ... | there fᵢ∈xs = fᵢ∈xs

+

+
  helper : ¬ Dec (∃ λ i → f i ≡ x)

+
  helper (no fᵢ≢x)        = finite inj  xs (λ i → not-x (fᵢ≢x ∘ _,_ i))

+
  helper (yes (i , fᵢ≡x)) = finite f′-inj xs f′ⱼ∈xs

+
    where

+
    f′ : ℕ → _

+
    f′ j with i ≤? j

+
    ... | yes _ = f (suc j)

+
    ... | no  _ = f j

+

+
    ∈-if-not-i : ∀ {j} → i ≢ j → f j ∈ xs

+
    ∈-if-not-i i≢j = not-x (i≢j ∘ f-inj ∘ trans fᵢ≡x ∘ sym)

+

+
    lemma : ∀ {k j} → i ≤ j → i ≰ k → suc j ≢ k

+
    lemma i≤j i≰1+j refl = i≰1+j (m≤n⇒m≤1+n i≤j)

+

+
    f′ⱼ∈xs : ∀ j → f′ j ∈ xs

+
    f′ⱼ∈xs j with i ≤? j

+
    ... | yes i≤j = ∈-if-not-i (<⇒≢ (s≤s i≤j))

+
    ... | no  i≰j = ∈-if-not-i (<⇒≢ (≰⇒> i≰j) ∘ sym)

+

+
    f′-injective′ : Injective _≡_ _≡_ f′

+
    f′-injective′ {j} {k} eq with i ≤? j | i ≤? k

+
    ... | yes i≤j | yes i≤k = suc-injective (f-inj eq)

+
    ... | yes i≤j | no  i≰k = contradiction (f-inj eq) (lemma i≤j i≰k)

+
    ... | no  i≰j | yes i≤k = contradiction (f-inj eq) (lemma i≤k i≰j ∘ sym)

+
    ... | no  i≰j | no  i≰k = f-inj eq

+

+
    f′-inj : ℕ ↣ _

+
    f′-inj = record

+
      { to        = f′

+
      ; cong      = ≡.cong f′

+
      ; injective = f′-injective′

+
      }

+

+
------------------------------------------------------------------------

+
-- Different members

+

+
there-injective-≢∈ : ∀ {xs} {x y z : A} {x∈xs : x ∈ xs} {y∈xs : y ∈ xs} →

+
                     there {x = z} x∈xs ≢∈ there y∈xs →

+
                     x∈xs ≢∈ y∈xs

+
there-injective-≢∈ neq refl eq = neq refl (≡.cong there eq)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.Propositional.html b/site/static/agda_html/Data.List.Membership.Propositional.html
new file mode 100644
index 000000000..cc6b525c6
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.Propositional.html
@@ -0,0 +1,112 @@
+
+Data.List.Membership.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Data.List.Any.Membership instantiated with propositional equality,

+
-- along with some additional definitions.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Membership.Propositional {a} {A : Set a} where

+

+
open import Data.List.Relation.Unary.Any using (Any)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; resp; subst)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
import Data.List.Membership.Setoid as SetoidMembership

+

+
------------------------------------------------------------------------

+
-- Re-export contents of setoid membership

+

+
open SetoidMembership (setoid A) public hiding (lose)

+

+
------------------------------------------------------------------------

+
-- Different members

+

+
infix 4 _≢∈_

+

+
_≢∈_ : ∀ {x y : A} {xs} → x ∈ xs → y ∈ xs → Set _

+
_≢∈_ x∈xs y∈xs = ∀ x≡y → subst (_∈ _) x≡y x∈xs ≢ y∈xs

+

+
------------------------------------------------------------------------

+
-- Other operations

+

+
lose : ∀ {p} {P : A → Set p} {x xs} → x ∈ xs → P x → Any P xs

+
lose = SetoidMembership.lose (setoid A) (resp _)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.Setoid.Properties.html b/site/static/agda_html/Data.List.Membership.Setoid.Properties.html
new file mode 100644
index 000000000..7db029823
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.Setoid.Properties.html
@@ -0,0 +1,525 @@
+
+Data.List.Membership.Setoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to setoid list membership

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Membership.Setoid.Properties where

+

+
open import Algebra using (Op₂; Selective)

+
open import Data.Bool.Base using (true; false)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Fin.Properties using (suc-injective)

+
open import Data.List.Base hiding (find)

+
import Data.List.Membership.Setoid as Membership

+
import Data.List.Relation.Binary.Equality.Setoid as Equality

+
open import Data.List.Relation.Unary.All as All using (All)

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
import Data.List.Relation.Unary.Any.Properties as Any

+
import Data.List.Relation.Unary.Unique.Setoid as Unique

+
open import Data.Nat.Base using (suc; z<s; _<_)

+
open import Data.Product.Base as Product using (∃; _×_; _,_ ; ∃₂)

+
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Function.Base using (_$_; flip; _∘_; _∘′_; id)

+
open import Function.Bundles using (_↔_)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_; _Preserves_⟶_)

+
open import Relation.Binary.Definitions as Binary hiding (Decidable)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Nullary.Decidable using (does; _because_; yes; no)

+
open import Relation.Nullary.Negation using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Unary as Unary using (Decidable; Pred)

+

+
open Setoid using (Carrier)

+

+
private

+
  variable

+
    c c₁ c₂ c₃ p ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+

+
------------------------------------------------------------------------

+
-- Equality properties

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S

+
  open Equality S

+
  open Membership S

+

+
  -- _∈_ respects the underlying equality

+

+
  ∈-resp-≈ : ∀ {xs} → (_∈ xs) Respects _≈_

+
  ∈-resp-≈ x≈y x∈xs = Any.map (trans (sym x≈y)) x∈xs

+

+
  ∉-resp-≈ : ∀ {xs} → (_∉ xs) Respects _≈_

+
  ∉-resp-≈ v≈w v∉xs w∈xs = v∉xs (∈-resp-≈ (sym v≈w) w∈xs)

+

+
  ∈-resp-≋ : ∀ {x} → (x ∈_) Respects _≋_

+
  ∈-resp-≋ = Any.lift-resp (flip trans)

+

+
  ∉-resp-≋ : ∀ {x} → (x ∉_) Respects _≋_

+
  ∉-resp-≋ xs≋ys v∉xs v∈ys = v∉xs (∈-resp-≋ (≋-sym xs≋ys) v∈ys)

+

+
  -- index is injective in its first argument.

+

+
  index-injective : ∀ {x₁ x₂ xs} (x₁∈xs : x₁ ∈ xs) (x₂∈xs : x₂ ∈ xs) →

+
                    Any.index x₁∈xs ≡ Any.index x₂∈xs → x₁ ≈ x₂

+
  index-injective (here x₁≈x)   (here x₂≈x)   _  = trans x₁≈x (sym x₂≈x)

+
  index-injective (there x₁∈xs) (there x₂∈xs) eq = index-injective x₁∈xs x₂∈xs (suc-injective eq)

+

+
------------------------------------------------------------------------

+
-- Irrelevance

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S

+
  open Unique S

+
  open Membership S

+

+
  private

+
    ∉×∈⇒≉ : ∀ {x y xs} → All (y ≉_) xs → x ∈ xs → x ≉ y

+
    ∉×∈⇒≉ = All.lookupWith λ y≉z x≈z x≈y → y≉z (trans (sym x≈y) x≈z)

+

+
  unique⇒irrelevant : Binary.Irrelevant _≈_ → ∀ {xs} → Unique xs → Unary.Irrelevant (_∈ xs)

+
  unique⇒irrelevant ≈-irr _        (here p)  (here q)  =

+
    ≡.cong here (≈-irr p q)

+
  unique⇒irrelevant ≈-irr (_  ∷ u) (there p) (there q) =

+
    ≡.cong there (unique⇒irrelevant ≈-irr u p q)

+
  unique⇒irrelevant ≈-irr (≉s ∷ _) (here p)  (there q) =

+
    contradiction p (∉×∈⇒≉ ≉s q)

+
  unique⇒irrelevant ≈-irr (≉s ∷ _) (there p) (here q)  =

+
    contradiction q (∉×∈⇒≉ ≉s p)

+

+
------------------------------------------------------------------------

+
-- mapWith∈

+

+
module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where

+

+
  open Setoid S₁ renaming (Carrier to A₁; _≈_ to _≈₁_; refl to refl₁)

+
  open Setoid S₂ renaming (Carrier to A₂; _≈_ to _≈₂_; refl to refl₂)

+
  open Equality S₁ using ([]; _∷_) renaming (_≋_ to _≋₁_)

+
  open Equality S₂ using () renaming (_≋_ to _≋₂_)

+
  open Membership S₁

+

+
  mapWith∈-cong : ∀ {xs ys} → xs ≋₁ ys →

+
                  (f : ∀ {x} → x ∈ xs → A₂) →

+
                  (g : ∀ {y} → y ∈ ys → A₂) →

+
                  (∀ {x y} → x ≈₁ y → (x∈xs : x ∈ xs) (y∈ys : y ∈ ys) →

+
                     f x∈xs ≈₂ g y∈ys) →

+
                  mapWith∈ xs f ≋₂ mapWith∈ ys g

+
  mapWith∈-cong []            f g cong = []

+
  mapWith∈-cong (x≈y ∷ xs≋ys) f g cong =

+
    cong x≈y (here refl₁) (here refl₁) ∷

+
    mapWith∈-cong xs≋ys (f ∘ there) (g ∘ there)

+
      (λ x≈y x∈xs y∈ys → cong x≈y (there x∈xs) (there y∈ys))

+

+
  mapWith∈≗map : ∀ f xs → mapWith∈ xs (λ {x} _ → f x) ≋₂ map f xs

+
  mapWith∈≗map f []       = []

+
  mapWith∈≗map f (x ∷ xs) = refl₂ ∷ mapWith∈≗map f xs

+

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S

+
  open Membership S

+

+
  length-mapWith∈ : ∀ {a} {A : Set a} xs {f : ∀ {x} → x ∈ xs → A} →

+
                    length (mapWith∈ xs f) ≡ length xs

+
  length-mapWith∈ []       = ≡.refl

+
  length-mapWith∈ (x ∷ xs) = ≡.cong suc (length-mapWith∈ xs)

+

+
  mapWith∈-id : ∀ xs → mapWith∈ xs (λ {x} _ → x) ≡ xs

+
  mapWith∈-id []       = ≡.refl

+
  mapWith∈-id (x ∷ xs) = ≡.cong (x ∷_) (mapWith∈-id xs)

+

+
  map-mapWith∈ : ∀ {a b} {A : Set a} {B : Set b} →

+
                 ∀ xs (f : ∀ {x} → x ∈ xs → A) (g : A → B) →

+
                 map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)

+
  map-mapWith∈ []       f g = ≡.refl

+
  map-mapWith∈ (x ∷ xs) f g = ≡.cong (_ ∷_) (map-mapWith∈ xs (f ∘ there) g)

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where

+

+
  open Setoid S₁ renaming (_≈_ to _≈₁_)

+
  open Setoid S₂ renaming (_≈_ to _≈₂_)

+
  private module M₁ = Membership S₁; open M₁ using (find) renaming (_∈_ to _∈₁_)

+
  private module M₂ = Membership S₂; open M₂ using () renaming (_∈_ to _∈₂_)

+

+
  ∈-map⁺ : ∀ {f} → f Preserves _≈₁_ ⟶ _≈₂_ →

+
           ∀ {v xs} → v ∈₁ xs → f v ∈₂ map f xs

+
  ∈-map⁺ pres x∈xs = Any.map⁺ (Any.map pres x∈xs)

+

+
  ∈-map⁻ : ∀ {v xs f} → v ∈₂ map f xs →

+
           ∃ λ x → x ∈₁ xs × v ≈₂ f x

+
  ∈-map⁻ x∈map = find (Any.map⁻ x∈map)

+

+
  map-∷= : ∀ {f} (pres : f Preserves _≈₁_ ⟶ _≈₂_) →

+
           ∀ {xs x v} → (x∈xs : x ∈₁ xs) →

+
           map f (x∈xs M₁.∷= v) ≡ ∈-map⁺ pres x∈xs M₂.∷= f v

+
  map-∷= pres (here x≈y)   = ≡.refl

+
  map-∷= pres (there x∈xs) = ≡.cong (_ ∷_) (map-∷= pres x∈xs)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Membership S using (_∈_)

+
  open Setoid S

+
  open Equality S using (_≋_; _∷_; ≋-refl)

+

+
  ∈-++⁺ˡ : ∀ {v xs ys} → v ∈ xs → v ∈ xs ++ ys

+
  ∈-++⁺ˡ = Any.++⁺ˡ

+

+
  ∈-++⁺ʳ : ∀ {v} xs {ys} → v ∈ ys → v ∈ xs ++ ys

+
  ∈-++⁺ʳ = Any.++⁺ʳ

+

+
  ∈-++⁻ : ∀ {v} xs {ys} → v ∈ xs ++ ys → (v ∈ xs) ⊎ (v ∈ ys)

+
  ∈-++⁻ = Any.++⁻

+

+
  ∈-++⁺∘++⁻ : ∀ {v} xs {ys} (p : v ∈ xs ++ ys) →

+
              [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p

+
  ∈-++⁺∘++⁻ = Any.++⁺∘++⁻

+

+
  ∈-++⁻∘++⁺ : ∀ {v} xs {ys} (p : v ∈ xs ⊎ v ∈ ys) →

+
              ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p

+
  ∈-++⁻∘++⁺ = Any.++⁻∘++⁺

+

+
  ∈-++↔ : ∀ {v xs ys} → (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys

+
  ∈-++↔ = Any.++↔

+

+
  ∈-++-comm : ∀ {v} xs ys → v ∈ xs ++ ys → v ∈ ys ++ xs

+
  ∈-++-comm = Any.++-comm

+

+
  ∈-++-comm∘++-comm : ∀ {v} xs {ys} (p : v ∈ xs ++ ys) →

+
                      ∈-++-comm ys xs (∈-++-comm xs ys p) ≡ p

+
  ∈-++-comm∘++-comm = Any.++-comm∘++-comm

+

+
  ∈-++↔++ : ∀ {v} xs ys → v ∈ xs ++ ys ↔ v ∈ ys ++ xs

+
  ∈-++↔++ = Any.++↔++

+

+
  ∈-insert : ∀ xs {ys v w} → v ≈ w → v ∈ xs ++ [ w ] ++ ys

+
  ∈-insert xs = Any.++-insert xs

+

+
  ∈-∃++ : ∀ {v xs} → v ∈ xs → ∃₂ λ ys zs → ∃ λ w →

+
          v ≈ w × xs ≋ ys ++ [ w ] ++ zs

+
  ∈-∃++ (here px)        = [] , _ , _ , px , ≋-refl

+
  ∈-∃++ (there {d} v∈xs) =

+
    let hs , _ , _ , v≈v′ , eq = ∈-∃++ v∈xs

+
    in d ∷ hs , _ , _ , v≈v′ , refl ∷ eq

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S using (_≈_)

+
  open Membership S using (_∈_)

+
  open Equality S using (≋-setoid)

+
  open Membership ≋-setoid using (find) renaming (_∈_ to _∈ₗ_)

+

+
  ∈-concat⁺ : ∀ {v xss} → Any (v ∈_) xss → v ∈ concat xss

+
  ∈-concat⁺ = Any.concat⁺

+

+
  ∈-concat⁻ : ∀ {v} xss → v ∈ concat xss → Any (v ∈_) xss

+
  ∈-concat⁻ = Any.concat⁻

+

+
  ∈-concat⁺′ : ∀ {v vs xss} → v ∈ vs → vs ∈ₗ xss → v ∈ concat xss

+
  ∈-concat⁺′ v∈vs = ∈-concat⁺ ∘ Any.map (flip (∈-resp-≋ S) v∈vs)

+

+
  ∈-concat⁻′ : ∀ {v} xss → v ∈ concat xss → ∃ λ xs → v ∈ xs × xs ∈ₗ xss

+
  ∈-concat⁻′ xss v∈c[xss] =

+
    let xs , xs∈xss , v∈xs = find (∈-concat⁻ xss v∈c[xss]) in xs , v∈xs , xs∈xss

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Membership S using (_∈_)

+

+
  reverse⁺ : ∀ {x xs} → x ∈ xs → x ∈ reverse xs

+
  reverse⁺ = Any.reverse⁺

+

+
  reverse⁻ : ∀ {x xs} → x ∈ reverse xs → x ∈ xs

+
  reverse⁻ = Any.reverse⁻

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith

+

+
module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) (S₃ : Setoid c₃ ℓ₃) where

+

+
  open Setoid S₁ renaming (_≈_ to _≈₁_; refl to refl₁)

+
  open Setoid S₂ renaming (_≈_ to _≈₂_)

+
  open Setoid S₃ renaming (_≈_ to _≈₃_)

+
  open Membership S₁ renaming (_∈_ to _∈₁_)

+
  open Membership S₂ renaming (_∈_ to _∈₂_)

+
  open Membership S₃ renaming (_∈_ to _∈₃_)

+

+
  ∈-cartesianProductWith⁺ : ∀ {f} → f Preserves₂ _≈₁_ ⟶ _≈₂_ ⟶ _≈₃_ →

+
                            ∀ {xs ys a b} → a ∈₁ xs → b ∈₂ ys →

+
                            f a b ∈₃ cartesianProductWith f xs ys

+
  ∈-cartesianProductWith⁺ pres = Any.cartesianProductWith⁺ _ pres

+

+
  ∈-cartesianProductWith⁻ : ∀ f xs ys {v} → v ∈₃ cartesianProductWith f xs ys →

+
                            ∃₂ λ a b → a ∈₁ xs × b ∈₂ ys × v ≈₃ f a b

+
  ∈-cartesianProductWith⁻ f (x ∷ xs) ys v∈c with ∈-++⁻ S₃ (map (f x) ys) v∈c

+
  ... | inj₁ v∈map =

+
    let b , b∈ys , v≈fxb = ∈-map⁻ S₂ S₃ v∈map

+
    in x , b , here refl₁ , b∈ys , v≈fxb

+
  ... | inj₂ v∈com =

+
    let a , b , a∈xs , b∈ys , v≈fab = ∈-cartesianProductWith⁻ f xs ys v∈com

+
    in  a , b , there a∈xs , b∈ys , v≈fab

+

+
------------------------------------------------------------------------

+
-- cartesianProduct

+

+
module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where

+

+
  open Setoid S₁ renaming (Carrier to A)

+
  open Setoid S₂ renaming (Carrier to B)

+
  open Membership S₁ renaming (_∈_ to _∈₁_)

+
  open Membership S₂ renaming (_∈_ to _∈₂_)

+
  open Membership (S₁ ×ₛ S₂) renaming (_∈_ to _∈₁₂_)

+

+
  ∈-cartesianProduct⁺ : ∀ {x y xs ys} → x ∈₁ xs → y ∈₂ ys →

+
                        (x , y) ∈₁₂ cartesianProduct xs ys

+
  ∈-cartesianProduct⁺ = Any.cartesianProduct⁺

+

+
  ∈-cartesianProduct⁻ : ∀ xs ys {xy@(x , y) : A × B} →

+
                        xy ∈₁₂ cartesianProduct xs ys →

+
                        x ∈₁ xs × y ∈₂ ys

+
  ∈-cartesianProduct⁻ xs ys = Any.cartesianProduct⁻ xs ys

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S using (_≈_; refl)

+
  open Membership S using (_∈_)

+

+
  ∈-applyUpTo⁺ : ∀ f {i n} → i < n → f i ∈ applyUpTo f n

+
  ∈-applyUpTo⁺ f = Any.applyUpTo⁺ f refl

+

+
  ∈-applyUpTo⁻ : ∀ {v} f {n} → v ∈ applyUpTo f n →

+
                 ∃ λ i → i < n × v ≈ f i

+
  ∈-applyUpTo⁻ = Any.applyUpTo⁻

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
  ∈-applyDownFrom⁺ : ∀ f {i n} → i < n → f i ∈ applyDownFrom f n

+
  ∈-applyDownFrom⁺ f = Any.applyDownFrom⁺ f refl

+

+
  ∈-applyDownFrom⁻ : ∀ {v} f {n} → v ∈ applyDownFrom f n →

+
                     ∃ λ i → i < n × v ≈ f i

+
  ∈-applyDownFrom⁻ = Any.applyDownFrom⁻

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S using (_≈_; refl) renaming (Carrier to A)

+
  open Membership S using (_∈_)

+

+
  ∈-tabulate⁺ : ∀ {n} {f : Fin n → A} i → f i ∈ tabulate f

+
  ∈-tabulate⁺ i = Any.tabulate⁺ i refl

+

+
  ∈-tabulate⁻ : ∀ {n} {f : Fin n → A} {v} →

+
                v ∈ tabulate f → ∃ λ i → v ≈ f i

+
  ∈-tabulate⁻ = Any.tabulate⁻

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (S : Setoid c ℓ) {P : Pred (Carrier S) p}

+
         (P? : Decidable P) (resp : P Respects (Setoid._≈_ S)) where

+

+
  open Setoid S using (_≈_; sym)

+
  open Membership S using (_∈_)

+

+
  ∈-filter⁺ : ∀ {v xs} → v ∈ xs → P v → v ∈ filter P? xs

+
  ∈-filter⁺ {xs = x ∷ _} (here v≈x) Pv with P? x

+
  ... |  true because   _   = here v≈x

+
  ... | false because [¬Px] = contradiction (resp v≈x Pv) (invert [¬Px])

+
  ∈-filter⁺ {xs = x ∷ _} (there v∈xs) Pv with does (P? x)

+
  ... | true  = there (∈-filter⁺ v∈xs Pv)

+
  ... | false = ∈-filter⁺ v∈xs Pv

+

+
  ∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v

+
  ∈-filter⁻ {xs = x ∷ xs} v∈f[x∷xs] with P? x

+
  ... | false because  _   = Product.map there id (∈-filter⁻ v∈f[x∷xs])

+
  ... |  true because [Px] with v∈f[x∷xs]

+
  ...   | here  v≈x   = here v≈x , resp (sym v≈x) (invert [Px])

+
  ...   | there v∈fxs = Product.map there id (∈-filter⁻ v∈fxs)

+

+
------------------------------------------------------------------------

+
-- derun and deduplicate

+

+
module _ (S : Setoid c ℓ) {R : Rel (Carrier S) ℓ₂} (R? : Binary.Decidable R) where

+

+
  open Setoid S using (_≈_)

+
  open Membership S using (_∈_)

+

+
  ∈-derun⁺ : _≈_ Respectsʳ R → ∀ {xs z} → z ∈ xs → z ∈ derun R? xs

+
  ∈-derun⁺ ≈-resp-R z∈xs = Any.derun⁺ R? ≈-resp-R z∈xs

+

+
  ∈-deduplicate⁺ : _≈_ Respectsʳ (flip R) → ∀ {xs z} →

+
                   z ∈ xs → z ∈ deduplicate R? xs

+
  ∈-deduplicate⁺ ≈-resp-R z∈xs = Any.deduplicate⁺ R? ≈-resp-R z∈xs

+

+
  ∈-derun⁻ : ∀ xs {z} → z ∈ derun R? xs → z ∈ xs

+
  ∈-derun⁻ xs z∈derun[R,xs] = Any.derun⁻ R? z∈derun[R,xs]

+

+
  ∈-deduplicate⁻ : ∀ xs {z} → z ∈ deduplicate R? xs → z ∈ xs

+
  ∈-deduplicate⁻ xs z∈dedup[R,xs] = Any.deduplicate⁻ R? z∈dedup[R,xs]

+

+
------------------------------------------------------------------------

+
-- length

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Membership S using (_∈_)

+

+
  ∈-length : ∀ {x xs} → x ∈ xs → 0 < length xs

+
  ∈-length (here px)    = z<s

+
  ∈-length (there x∈xs) = z<s

+

+
------------------------------------------------------------------------

+
-- lookup

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S using (refl)

+
  open Membership S using (_∈_)

+

+
  ∈-lookup : ∀ xs i → lookup xs i ∈ xs

+
  ∈-lookup (x ∷ xs) zero    = here refl

+
  ∈-lookup (x ∷ xs) (suc i) = there (∈-lookup xs i)

+

+
------------------------------------------------------------------------

+
-- foldr

+

+
module _ (S : Setoid c ℓ) {_•_ : Op₂ (Carrier S)} where

+

+
  open Setoid S using (_≈_; refl; sym; trans)

+
  open Membership S using (_∈_)

+

+
  foldr-selective : Selective _≈_ _•_ → ∀ e xs →

+
                    (foldr _•_ e xs ≈ e) ⊎ (foldr _•_ e xs ∈ xs)

+
  foldr-selective •-sel i [] = inj₁ refl

+
  foldr-selective •-sel i (x ∷ xs) with •-sel x (foldr _•_ i xs)

+
  ... | inj₁ x•f≈x = inj₂ (here x•f≈x)

+
  ... | inj₂ x•f≈f with foldr-selective •-sel i xs

+
  ...   | inj₁ f≈i  = inj₁ (trans x•f≈f f≈i)

+
  ...   | inj₂ f∈xs = inj₂ (∈-resp-≈ S (sym x•f≈f) (there f∈xs))

+

+
------------------------------------------------------------------------

+
-- _∷=_

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S

+
  open Membership S

+

+
  ∈-∷=⁺-updated : ∀ {xs x v} (x∈xs : x ∈ xs) → v ∈ (x∈xs ∷= v)

+
  ∈-∷=⁺-updated (here  px)  = here refl

+
  ∈-∷=⁺-updated (there pxs) = there (∈-∷=⁺-updated pxs)

+

+
  ∈-∷=⁺-untouched : ∀ {xs x y v} (x∈xs : x ∈ xs) → (¬ x ≈ y) → y ∈ xs → y ∈ (x∈xs ∷= v)

+
  ∈-∷=⁺-untouched (here  x≈z)  x≉y (here  y≈z)  = contradiction (trans x≈z (sym y≈z)) x≉y

+
  ∈-∷=⁺-untouched (here  x≈z)  x≉y (there y∈xs) = there y∈xs

+
  ∈-∷=⁺-untouched (there x∈xs) x≉y (here  y≈z)  = here y≈z

+
  ∈-∷=⁺-untouched (there x∈xs) x≉y (there y∈xs) = there (∈-∷=⁺-untouched x∈xs x≉y y∈xs)

+

+
  ∈-∷=⁻ : ∀ {xs x y v} (x∈xs : x ∈ xs) → (¬ y ≈ v) → y ∈ (x∈xs ∷= v) → y ∈ xs

+
  ∈-∷=⁻ (here x≈z)   y≉v (here y≈v) = contradiction y≈v y≉v

+
  ∈-∷=⁻ (here x≈z)   y≉v (there y∈) = there y∈

+
  ∈-∷=⁻ (there x∈xs) y≉v (here y≈z) = here y≈z

+
  ∈-∷=⁻ (there x∈xs) y≉v (there y∈) = there (∈-∷=⁻ x∈xs y≉v y∈)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Membership.Setoid.html b/site/static/agda_html/Data.List.Membership.Setoid.html
new file mode 100644
index 000000000..abfd315dd
--- /dev/null
+++ b/site/static/agda_html/Data.List.Membership.Setoid.html
@@ -0,0 +1,133 @@
+
+Data.List.Membership.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- List membership and some related definitions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.List.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where

+

+
open import Data.List.Base using (List; []; _∷_)

+
open import Data.List.Relation.Unary.Any as Any

+
  using (Any; map; here; there)

+
open import Data.Product.Base as Product using (∃; _×_; _,_)

+
open import Function.Base using (_∘_; flip; const)

+
open import Relation.Binary.Definitions using (_Respects_)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Unary using (Pred)

+

+
open Setoid S renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
infix 4 _∈_ _∉_

+

+
_∈_ : A → List A → Set _

+
x ∈ xs = Any (x ≈_) xs

+

+
_∉_ : A → List A → Set _

+
x ∉ xs = ¬ x ∈ xs

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
_∷=_ = Any._∷=_ {A = A}

+
_─_ = Any._─_ {A = A}

+

+
mapWith∈ : ∀ {b} {B : Set b}

+
           (xs : List A) → (∀ {x} → x ∈ xs → B) → List B

+
mapWith∈ []       f = []

+
mapWith∈ (x ∷ xs) f = f (here refl) ∷ mapWith∈ xs (f ∘ there)

+

+
------------------------------------------------------------------------

+
-- Finding and losing witnesses

+

+
module _ {p} {P : Pred A p} where

+

+
  find : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x

+
  find (here px)   = _ , here refl , px

+
  find (there pxs) = let x , x∈xs , px = find pxs in x , there x∈xs , px

+

+
  lose : P Respects _≈_ →  ∀ {x xs} → x ∈ xs → P x → Any P xs

+
  lose resp x∈xs px = map (flip resp px) x∈xs

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.NonEmpty.Base.html b/site/static/agda_html/Data.List.NonEmpty.Base.html
new file mode 100644
index 000000000..2ba44959f
--- /dev/null
+++ b/site/static/agda_html/Data.List.NonEmpty.Base.html
@@ -0,0 +1,415 @@
+
+Data.List.NonEmpty.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Non-empty lists: base type and operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.NonEmpty.Base where

+

+
open import Level using (Level)

+
open import Data.Bool.Base using (Bool; false; true)

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.Maybe.Base using (Maybe ; nothing; just)

+
open import Data.Nat.Base as ℕ

+
open import Data.Product.Base as Prod using (∃; _×_; proj₁; proj₂; _,_; -,_)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Data.These.Base as These using (These; this; that; these)

+
open import Data.Vec.Base as Vec using (Vec; []; _∷_)

+
open import Function.Base

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl)

+
open import Relation.Unary using (Pred; Decidable; U; ∅)

+
open import Relation.Unary.Properties using (U?; ∅?)

+
open import Relation.Nullary.Decidable using (does)

+

+
private

+
  variable

+
    a p : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infixr 5 _∷_

+

+
record List⁺ (A : Set a) : Set a where

+
  constructor _∷_

+
  field

+
    head : A

+
    tail : List A

+

+
open List⁺ public

+

+
------------------------------------------------------------------------

+
-- Basic combinators

+

+
uncons : List⁺ A → A × List A

+
uncons (hd ∷ tl) = hd , tl

+

+
[_] : A → List⁺ A

+
[ x ] = x ∷ []

+

+
infixr 5 _∷⁺_

+

+
_∷⁺_ : A → List⁺ A → List⁺ A

+
x ∷⁺ y ∷ xs = x ∷ y ∷ xs

+

+
length : List⁺ A → ℕ

+
length (x ∷ xs) = suc (List.length xs)

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
toList : List⁺ A → List A

+
toList (x ∷ xs) = x ∷ xs

+

+
fromList : List A → Maybe (List⁺ A)

+
fromList []       = nothing

+
fromList (x ∷ xs) = just (x ∷ xs)

+

+
fromVec : ∀ {n} → Vec A (suc n) → List⁺ A

+
fromVec (x ∷ xs) = x ∷ Vec.toList xs

+

+
toVec : (xs : List⁺ A) → Vec A (length xs)

+
toVec (x ∷ xs) = x ∷ Vec.fromList xs

+

+
lift : (∀ {m} → Vec A (suc m) → ∃ λ n → Vec B (suc n)) →

+
       List⁺ A → List⁺ B

+
lift f xs = fromVec (proj₂ (f (toVec xs)))

+

+
------------------------------------------------------------------------

+
-- Other operations

+

+
map : (A → B) → List⁺ A → List⁺ B

+
map f (x ∷ xs) = (f x ∷ List.map f xs)

+

+
replicate : ∀ n → n ≢ 0 → A → List⁺ A

+
replicate n n≢0 a = a ∷ List.replicate (pred n) a

+

+
-- when dropping more than the size of the length of the list, the

+
-- last element remains

+
drop+ : ℕ → List⁺ A → List⁺ A

+
drop+ zero    xs           = xs

+
drop+ (suc n) (x ∷ [])     = x ∷ []

+
drop+ (suc n) (x ∷ y ∷ xs) = drop+ n (y ∷ xs)

+

+
-- Right fold. Note that s is only applied to the last element (see

+
-- the examples below).

+

+
foldr : (A → B → B) → (A → B) → List⁺ A → B

+
foldr {A = A} {B = B} c s (x ∷ xs) = foldr′ x xs

+
  where

+
  foldr′ : A → List A → B

+
  foldr′ x []       = s x

+
  foldr′ x (y ∷ xs) = c x (foldr′ y xs)

+

+
-- Right fold.

+

+
foldr₁ : (A → A → A) → List⁺ A → A

+
foldr₁ f = foldr f id

+

+
-- Left fold. Note that s is only applied to the first element (see

+
-- the examples below).

+

+
foldl : (B → A → B) → (A → B) → List⁺ A → B

+
foldl c s (x ∷ xs) = List.foldl c (s x) xs

+

+
-- Left fold.

+

+
foldl₁ : (A → A → A) → List⁺ A → A

+
foldl₁ f = foldl f id

+

+
-- Append (several variants).

+

+
infixr 5 _⁺++⁺_ _++⁺_ _⁺++_

+

+
_⁺++⁺_ : List⁺ A → List⁺ A → List⁺ A

+
(x ∷ xs) ⁺++⁺ (y ∷ ys) = x ∷ (xs List.++ y ∷ ys)

+

+
_⁺++_ : List⁺ A → List A → List⁺ A

+
(x ∷ xs) ⁺++ ys = x ∷ (xs List.++ ys)

+

+
_++⁺_ : List A → List⁺ A → List⁺ A

+
xs ++⁺ ys = List.foldr _∷⁺_ ys xs

+

+
concat : List⁺ (List⁺ A) → List⁺ A

+
concat (xs ∷ xss) = xs ⁺++ List.concat (List.map toList xss)

+

+
concatMap : (A → List⁺ B) → List⁺ A → List⁺ B

+
concatMap f = concat ∘′ map f

+

+
ap : List⁺ (A → B) → List⁺ A → List⁺ B

+
ap fs as = concatMap (λ f → map f as) fs

+

+
-- Inits

+

+
inits : List A → List⁺ (List A)

+
inits xs = [] ∷ List.Inits.tail xs

+

+
-- Tails

+

+
tails : List A → List⁺ (List A)

+
tails xs = xs ∷ List.Tails.tail xs

+

+
-- Reverse

+

+
reverse : List⁺ A → List⁺ A

+
reverse = lift (-,_ ∘′ Vec.reverse)

+

+
-- Align and Zip

+

+
alignWith : (These A B → C) → List⁺ A → List⁺ B → List⁺ C

+
alignWith f (a ∷ as) (b ∷ bs) = f (these a b) ∷ List.alignWith f as bs

+

+
zipWith : (A → B → C) → List⁺ A → List⁺ B → List⁺ C

+
zipWith f (a ∷ as) (b ∷ bs) = f a b ∷ List.zipWith f as bs

+

+
unalignWith : (A → These B C) → List⁺ A → These (List⁺ B) (List⁺ C)

+
unalignWith f = foldr (These.alignWith mcons mcons ∘′ f)

+
                    (These.map [_] [_] ∘′ f)

+

+
  where mcons : ∀ {e} {E : Set e} → These E (List⁺ E) → List⁺ E

+
        mcons = These.fold [_] id _∷⁺_

+

+
unzipWith : (A → B × C) → List⁺ A → List⁺ B × List⁺ C

+
unzipWith f (a ∷ as) = Prod.zip _∷_ _∷_ (f a) (List.unzipWith f as)

+

+
align : List⁺ A → List⁺ B → List⁺ (These A B)

+
align = alignWith id

+

+
zip : List⁺ A → List⁺ B → List⁺ (A × B)

+
zip = zipWith _,_

+

+
unalign : List⁺ (These A B) → These (List⁺ A) (List⁺ B)

+
unalign = unalignWith id

+

+
unzip : List⁺ (A × B) → List⁺ A × List⁺ B

+
unzip = unzipWith id

+

+
-- Snoc.

+

+
infixl 5 _∷ʳ_ _⁺∷ʳ_

+

+
_∷ʳ_ : List A → A → List⁺ A

+
[]       ∷ʳ y = [ y ]

+
(x ∷ xs) ∷ʳ y = x ∷ (xs List.∷ʳ y)

+

+
_⁺∷ʳ_ : List⁺ A → A → List⁺ A

+
xs ⁺∷ʳ x = toList xs ∷ʳ x

+

+
-- A snoc-view of non-empty lists.

+

+
infixl 5 _∷ʳ′_

+

+
data SnocView {A : Set a} : List⁺ A → Set a where

+
  _∷ʳ′_ : (xs : List A) (x : A) → SnocView (xs ∷ʳ x)

+

+
snocView : (xs : List⁺ A) → SnocView xs

+
snocView (x ∷ xs)              with List.initLast xs

+
snocView (x ∷ .[])             | []            = []       ∷ʳ′ x

+
snocView (x ∷ .(xs List.∷ʳ y)) | xs List.∷ʳ′ y = (x ∷ xs) ∷ʳ′ y

+

+
-- The last element in the list.

+

+
private

+
  last′ : ∀ {l} → SnocView {A = A} l → A

+
  last′ (_ ∷ʳ′ y) = y

+

+
last : List⁺ A → A

+
last = last′ ∘ snocView

+

+
-- Groups all contiguous elements for which the predicate returns the

+
-- same result into lists. The left sums are the ones for which the

+
-- predicate holds, the right ones are the ones for which it doesn't.

+
groupSeqsᵇ : (A → Bool) → List A → List (List⁺ A ⊎ List⁺ A)

+
groupSeqsᵇ p []       = []

+
groupSeqsᵇ p (x ∷ xs) with p x | groupSeqsᵇ p xs

+
... | true  | inj₁ xs′ ∷ xss = inj₁ (x ∷⁺ xs′) ∷ xss

+
... | true  | xss            = inj₁ [ x ]      ∷ xss

+
... | false | inj₂ xs′ ∷ xss = inj₂ (x ∷⁺ xs′) ∷ xss

+
... | false | xss            = inj₂ [ x ]      ∷ xss

+

+
-- Groups all contiguous elements /not/ satisfying the predicate into

+
-- lists. Elements satisfying the predicate are dropped.

+
wordsByᵇ : (A → Bool) → List A → List (List⁺ A)

+
wordsByᵇ p = List.mapMaybe Sum.[ const nothing , just ] ∘ groupSeqsᵇ p

+

+
groupSeqs : {P : Pred A p} → Decidable P → List A → List (List⁺ A ⊎ List⁺ A)

+
groupSeqs P? = groupSeqsᵇ (does ∘ P?)

+

+
wordsBy : {P : Pred A p} → Decidable P → List A → List (List⁺ A)

+
wordsBy P? = wordsByᵇ (does ∘ P?)

+

+
-- Inverse operation for groupSequences.

+
ungroupSeqs : List (List⁺ A ⊎ List⁺ A) → List A

+
ungroupSeqs = List.concat ∘ List.map Sum.[ toList , toList ]

+

+
------------------------------------------------------------------------

+
-- Examples

+

+
-- Note that these examples are simple unit tests, because the type

+
-- checker verifies them.

+

+
private

+
 module Examples {A B : Set}

+
                 (_⊕_ : A → B → B)

+
                 (_⊗_ : B → A → B)

+
                 (_⊙_ : A → A → A)

+
                 (f : A → B)

+
                 (a b c : A)

+
                 where

+

+
  hd : head (a ∷⁺ b ∷⁺ [ c ]) ≡ a

+
  hd = refl

+

+
  tl : tail (a ∷⁺ b ∷⁺ [ c ]) ≡ b ∷ c ∷ []

+
  tl = refl

+

+
  mp : map f (a ∷⁺ b ∷⁺ [ c ]) ≡ f a ∷⁺ f b ∷⁺ [ f c ]

+
  mp = refl

+

+
  right : foldr _⊕_ f (a ∷⁺ b ∷⁺ [ c ]) ≡ (a ⊕ (b ⊕ f c))

+
  right = refl

+

+
  right₁ : foldr₁ _⊙_ (a ∷⁺ b ∷⁺ [ c ]) ≡ (a ⊙ (b ⊙ c))

+
  right₁ = refl

+

+
  left : foldl _⊗_ f (a ∷⁺ b ∷⁺ [ c ]) ≡ ((f a ⊗ b) ⊗ c)

+
  left = refl

+

+
  left₁ : foldl₁ _⊙_ (a ∷⁺ b ∷⁺ [ c ]) ≡ ((a ⊙ b) ⊙ c)

+
  left₁ = refl

+

+
  ⁺app⁺ : (a ∷⁺ b ∷⁺ [ c ]) ⁺++⁺ (b ∷⁺ [ c ]) ≡

+
          a ∷⁺ b ∷⁺ c ∷⁺ b ∷⁺ [ c ]

+
  ⁺app⁺ = refl

+

+
  ⁺app : (a ∷⁺ b ∷⁺ [ c ]) ⁺++ (b ∷ c ∷ []) ≡

+
          a ∷⁺ b ∷⁺ c ∷⁺ b ∷⁺ [ c ]

+
  ⁺app = refl

+

+
  app⁺ : (a ∷ b ∷ c ∷ []) ++⁺ (b ∷⁺ [ c ]) ≡

+
          a ∷⁺ b ∷⁺ c ∷⁺ b ∷⁺ [ c ]

+
  app⁺ = refl

+

+
  conc : concat ((a ∷⁺ b ∷⁺ [ c ]) ∷⁺ [ b ∷⁺ [ c ] ]) ≡

+
         a ∷⁺ b ∷⁺ c ∷⁺ b ∷⁺ [ c ]

+
  conc = refl

+

+
  rev : reverse (a ∷⁺ b ∷⁺ [ c ]) ≡ c ∷⁺ b ∷⁺ [ a ]

+
  rev = refl

+

+
  snoc : (a ∷ b ∷ c ∷ []) ∷ʳ a ≡ a ∷⁺ b ∷⁺ c ∷⁺ [ a ]

+
  snoc = refl

+

+
  snoc⁺ : (a ∷⁺ b ∷⁺ [ c ]) ⁺∷ʳ a ≡ a ∷⁺ b ∷⁺ c ∷⁺ [ a ]

+
  snoc⁺ = refl

+

+
  groupSeqs-true : groupSeqs U? (a ∷ b ∷ c ∷ []) ≡

+
               inj₁ (a ∷⁺ b ∷⁺ [ c ]) ∷ []

+
  groupSeqs-true = refl

+

+
  groupSeqs-false : groupSeqs ∅? (a ∷ b ∷ c ∷ []) ≡

+
                inj₂ (a ∷⁺ b ∷⁺ [ c ]) ∷ []

+
  groupSeqs-false = refl

+

+
  groupSeqs-≡1 : groupSeqsᵇ (ℕ._≡ᵇ 1) (1 ∷ 2 ∷ 3 ∷ 1 ∷ 1 ∷ 2 ∷ 1 ∷ []) ≡

+
                 inj₁ [ 1 ] ∷

+
                 inj₂ (2 ∷⁺ [ 3 ]) ∷

+
                 inj₁ (1 ∷⁺ [ 1 ]) ∷

+
                 inj₂ [ 2 ] ∷

+
                 inj₁ [ 1 ] ∷

+
                 []

+
  groupSeqs-≡1 = refl

+

+
  wordsBy-true : wordsByᵇ (const true) (a ∷ b ∷ c ∷ []) ≡ []

+
  wordsBy-true = refl

+

+
  wordsBy-false : wordsByᵇ (const false) (a ∷ b ∷ c ∷ []) ≡

+
                  (a ∷⁺ b ∷⁺ [ c ]) ∷ []

+
  wordsBy-false = refl

+

+
  wordsBy-≡1 : wordsByᵇ (ℕ._≡ᵇ 1) (1 ∷ 2 ∷ 3 ∷ 1 ∷ 1 ∷ 2 ∷ 1 ∷ []) ≡

+
               (2 ∷⁺ [ 3 ]) ∷

+
               [ 2 ] ∷

+
               []

+
  wordsBy-≡1 = refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.NonEmpty.Effectful.html b/site/static/agda_html/Data.List.NonEmpty.Effectful.html
new file mode 100644
index 000000000..357b29e00
--- /dev/null
+++ b/site/static/agda_html/Data.List.NonEmpty.Effectful.html
@@ -0,0 +1,161 @@
+
+Data.List.NonEmpty.Effectful
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An effectful view of List⁺

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.NonEmpty.Effectful where

+

+
open import Agda.Builtin.List

+
import Data.List.Effectful as List

+
open import Data.List.NonEmpty.Base

+
open import Data.Product.Base using (uncurry)

+
open import Effect.Functor

+
open import Effect.Applicative

+
open import Effect.Monad

+
open import Effect.Comonad

+
open import Function.Base using (flip; _∘′_; _∘_)

+

+
------------------------------------------------------------------------

+
-- List⁺ applicative functor

+

+
functor : ∀ {f} → RawFunctor {f} List⁺

+
functor = record

+
  { _<$>_ = map

+
  }

+

+
applicative : ∀ {f} → RawApplicative {f} List⁺

+
applicative = record

+
  { rawFunctor = functor

+
  ; pure = [_]

+
  ; _<*>_  = ap

+
  }

+

+
------------------------------------------------------------------------

+
-- List⁺ monad

+

+
monad : ∀ {f} → RawMonad {f} List⁺

+
monad = record

+
  { rawApplicative = applicative

+
  ; _>>=_  = flip concatMap

+
  }

+

+
------------------------------------------------------------------------

+
-- List⁺ comonad

+

+
comonad : ∀ {f} → RawComonad {f} List⁺

+
comonad = record

+
  { extract = head

+
  ; extend  = λ f → uncurry (extend f) ∘′ uncons

+
  } where

+

+
  extend : ∀ {A B} → (List⁺ A → B) → A → List A → List⁺ B

+
  extend f x xs@[]       = f (x ∷ xs) ∷ []

+
  extend f x xs@(y ∷ ys) = f (x ∷ xs) ∷⁺ extend f y ys

+

+

+
------------------------------------------------------------------------

+
-- Get access to other monadic functions

+

+
module TraversableA {f g F} (App : RawApplicative {f} {g} F) where

+

+
  open RawApplicative App

+

+
  sequenceA : ∀ {A} → List⁺ (F A) → F (List⁺ A)

+
  sequenceA (x ∷ xs) = _∷_ <$> x ⊛ List.TraversableA.sequenceA App xs

+

+
  mapA : ∀ {a} {A : Set a} {B} → (A → F B) → List⁺ A → F (List⁺ B)

+
  mapA f = sequenceA ∘ map f

+

+
  forA : ∀ {a} {A : Set a} {B} → List⁺ A → (A → F B) → F (List⁺ B)

+
  forA = flip mapA

+

+
module TraversableM {m n M} (Mon : RawMonad {m} {n} M) where

+

+
  open RawMonad Mon

+

+
  open TraversableA rawApplicative public

+
    renaming

+
    ( sequenceA to sequenceM

+
    ; mapA      to mapM

+
    ; forA      to forM

+
    )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.NonEmpty.Properties.html b/site/static/agda_html/Data.List.NonEmpty.Properties.html
new file mode 100644
index 000000000..262be0728
--- /dev/null
+++ b/site/static/agda_html/Data.List.NonEmpty.Properties.html
@@ -0,0 +1,257 @@
+
+Data.List.NonEmpty.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of non-empty lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.NonEmpty.Properties where

+

+
open import Effect.Monad using (RawMonad)

+
open import Data.Nat.Base using (suc; _+_)

+
open import Data.Nat.Properties using (suc-injective)

+
open import Data.Maybe.Properties using (just-injective)

+
open import Data.Bool.Base using (Bool; true; false)

+
open import Data.List.Base as List using (List; []; _∷_; _++_)

+
open import Data.List.Effectful using () renaming (monad to listMonad)

+
open import Data.List.NonEmpty.Effectful using () renaming (monad to list⁺Monad)

+
open import Data.List.NonEmpty

+
  using (List⁺; _∷_; tail; head; toList; _⁺++_; _⁺++⁺_; _++⁺_; length; fromList;

+
    drop+; map; inits; tails; groupSeqs; ungroupSeqs)

+
open import Data.List.NonEmpty.Relation.Unary.All using (All; toList⁺; _∷_)

+
open import Data.List.Relation.Unary.All using ([]; _∷_) renaming (All to ListAll)

+
import Data.List.Properties as List

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Data.Sum.Relation.Unary.All using (inj₁; inj₂)

+
import Data.Sum.Relation.Unary.All as Sum using (All; inj₁; inj₂)

+
open import Level using (Level)

+
open import Function.Base using (_∘_; _$_)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂; _≗_)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+
open import Relation.Unary using (Pred; Decidable; ∁)

+
open import Relation.Nullary using (¬_; does; yes; no)

+

+
open ≡-Reasoning

+

+
private

+
  variable

+
    a p : Level

+
    A B C : Set a

+

+
  open module LMo {a} = RawMonad {f = a} listMonad

+
    using () renaming (_>>=_ to _⋆>>=_)

+
  open module L⁺Mo {a} = RawMonad {f = a} list⁺Monad

+

+
------------------------------------------------------------------------

+
-- toList

+

+
η : ∀ (xs : List⁺ A) → head xs ∷ tail xs ≡ toList xs

+
η _ = refl

+

+
toList-fromList : ∀ x (xs : List A) → x ∷ xs ≡ toList (x ∷ xs)

+
toList-fromList _ _ = refl

+

+
toList-⁺++ : ∀ (xs : List⁺ A) ys → toList xs ++ ys ≡ toList (xs ⁺++ ys)

+
toList-⁺++ _ _ = refl

+

+
toList-⁺++⁺ : ∀ (xs ys : List⁺ A) →

+
              toList xs ++ toList ys ≡ toList (xs ⁺++⁺ ys)

+
toList-⁺++⁺ _ _ = refl

+

+
toList->>= : ∀ (f : A → List⁺ B) (xs : List⁺ A) →

+
             (toList xs ⋆>>= toList ∘ f) ≡ toList (xs >>= f)

+
toList->>= f (x ∷ xs) = begin

+
  List.concat (List.map (toList ∘ f) (x ∷ xs))

+
    ≡⟨ cong List.concat $ List.map-∘ {g = toList} (x ∷ xs) ⟩

+
  List.concat (List.map toList (List.map f (x ∷ xs)))

+
    ∎

+

+
------------------------------------------------------------------------

+
-- _++⁺_

+

+
length-++⁺ : (xs : List A) (ys : List⁺ A) →

+
             length (xs ++⁺ ys) ≡ List.length xs + length ys

+
length-++⁺ [] ys                                = refl

+
length-++⁺ (x ∷ xs) ys rewrite length-++⁺ xs ys = refl

+

+
length-++⁺-tail : (xs : List A) (ys : List⁺ A) →

+
                  length (xs ++⁺ ys) ≡ suc (List.length xs + List.length (List⁺.tail ys))

+
length-++⁺-tail [] ys                                     = refl

+
length-++⁺-tail (x ∷ xs) ys rewrite length-++⁺-tail xs ys = refl

+

+
++-++⁺ : (xs : List A) → ∀ {ys zs} → (xs ++ ys) ++⁺ zs ≡ xs ++⁺ ys ++⁺ zs

+
++-++⁺ []      = refl

+
++-++⁺ (x ∷ l) = cong (x ∷_) (cong toList (++-++⁺ l))

+

+
++⁺-cancelˡ′ : ∀ xs ys {zs zs′ : List⁺ A} →

+
               xs ++⁺ zs ≡ ys ++⁺ zs′ →

+
               List.length xs ≡ List.length ys → zs ≡ zs′

+
++⁺-cancelˡ′ [] [] eq eqxs            = eq

+
++⁺-cancelˡ′ (x ∷ xs) (y ∷ ys) eq eql = ++⁺-cancelˡ′ xs ys

+
  (just-injective (cong fromList (cong List⁺.tail eq)))

+
  (suc-injective eql)

+

+
++⁺-cancelˡ : ∀ xs {ys zs : List⁺ A} → xs ++⁺ ys ≡ xs ++⁺ zs → ys ≡ zs

+
++⁺-cancelˡ xs eq = ++⁺-cancelˡ′ xs xs eq refl

+

+
drop-+-++⁺ : ∀ (xs : List A) ys → drop+ (List.length xs) (xs ++⁺ ys) ≡ ys

+
drop-+-++⁺ []       ys = refl

+
drop-+-++⁺ (x ∷ xs) ys = drop-+-++⁺ xs ys

+

+
map-++⁺ : ∀ (f : A → B) xs ys →

+
                  map f (xs ++⁺ ys) ≡ List.map f xs ++⁺ map f ys

+
map-++⁺ f [] ys       = refl

+
map-++⁺ f (x ∷ xs) ys = cong (λ zs → f x ∷ toList zs) (map-++⁺ f xs ys)

+

+
------------------------------------------------------------------------

+
-- map

+

+
length-map : ∀ (f : A → B) xs → length (map f xs) ≡ length xs

+
length-map f (_ ∷ xs) = cong suc (List.length-map f xs)

+

+
map-cong : ∀ {f g : A → B} → f ≗ g → map f ≗ map g

+
map-cong f≗g (x ∷ xs) = cong₂ _∷_ (f≗g x) (List.map-cong f≗g xs)

+

+
map-∘ : {g : B → C} {f : A → B} → map (g ∘ f) ≗ map g ∘ map f

+
map-∘ (x ∷ xs) = cong (_ ∷_) (List.map-∘ xs)

+

+
------------------------------------------------------------------------

+
-- inits

+

+
toList-inits : toList ∘ inits ≗ List.inits {A = A}

+
toList-inits _ = refl

+

+
------------------------------------------------------------------------

+
-- tails

+

+
toList-tails : toList ∘ tails ≗ List.tails {A = A}

+
toList-tails _ = refl

+

+
------------------------------------------------------------------------

+
-- groupSeqs

+

+
-- Groups all contiguous elements for which the predicate returns the

+
-- same result into lists.

+

+
module _ {P : Pred A p} (P? : Decidable P) where

+

+
  groupSeqs-groups : ∀ xs → ListAll (Sum.All (All P) (All (∁ P))) (groupSeqs P? xs)

+
  groupSeqs-groups []       = []

+
  groupSeqs-groups (x ∷ xs) with P? x | groupSeqs P? xs | groupSeqs-groups xs

+
  ... | yes px | []             | hyp             = inj₁ (px  ∷ []) ∷ hyp

+
  ... | yes px | inj₁ xs′ ∷ xss | inj₁ pxs ∷ pxss = inj₁ (px  ∷ toList⁺ pxs) ∷ pxss

+
  ... | yes px | inj₂ xs′ ∷ xss | inj₂ pxs ∷ pxss = inj₁ (px  ∷ []) ∷ inj₂ pxs ∷ pxss

+
  ... | no ¬px | []             | hyp             = inj₂ (¬px ∷ []) ∷ hyp

+
  ... | no ¬px | inj₂ xs′ ∷ xss | inj₂ pxs ∷ pxss = inj₂ (¬px ∷ toList⁺ pxs) ∷ pxss

+
  ... | no ¬px | inj₁ xs′ ∷ xss | inj₁ pxs ∷ pxss = inj₂ (¬px ∷ []) ∷ inj₁ pxs ∷ pxss

+

+
  ungroupSeqs-groupSeqs : ∀ xs → ungroupSeqs (groupSeqs P? xs) ≡ xs

+
  ungroupSeqs-groupSeqs []       = refl

+
  ungroupSeqs-groupSeqs (x ∷ xs)

+
    with does (P? x) | groupSeqs P? xs | ungroupSeqs-groupSeqs xs

+
  ... | true  | []         | hyp = cong (x ∷_) hyp

+
  ... | true  | inj₁ _ ∷ _ | hyp = cong (x ∷_) hyp

+
  ... | true  | inj₂ _ ∷ _ | hyp = cong (x ∷_) hyp

+
  ... | false | []         | hyp = cong (x ∷_) hyp

+
  ... | false | inj₁ _ ∷ _ | hyp = cong (x ∷_) hyp

+
  ... | false | inj₂ _ ∷ _ | hyp = cong (x ∷_) hyp

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
map-compose = map-∘

+
{-# WARNING_ON_USAGE map-compose

+
"Warning: map-compose was deprecated in v2.0.

+
Please use map-∘ instead."

+
#-}

+

+
map-++⁺-commute = map-++⁺

+
{-# WARNING_ON_USAGE map-++⁺-commute

+
"Warning: map-++⁺-commute was deprecated in v2.0.

+
Please use map-++⁺ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.NonEmpty.Relation.Unary.All.html b/site/static/agda_html/Data.List.NonEmpty.Relation.Unary.All.html
new file mode 100644
index 000000000..a0842727d
--- /dev/null
+++ b/site/static/agda_html/Data.List.NonEmpty.Relation.Unary.All.html
@@ -0,0 +1,116 @@
+
+Data.List.NonEmpty.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Non-empty lists where all elements satisfy a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.NonEmpty.Relation.Unary.All where

+

+
import Data.List.Relation.Unary.All as List

+
open import Data.List.Relation.Unary.All using ([]; _∷_)

+
open import Data.List.Base using ([]; _∷_)

+
open import Data.List.NonEmpty.Base using (List⁺; _∷_; toList)

+
open import Level

+
open import Relation.Unary using (Pred)

+

+
private

+
  variable

+
    a p : Level

+
    A : Set a

+
    P : Pred A p

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Given a predicate P, then All P xs means that every element in xs

+
-- satisfies P. See `Relation.Unary` for an explanation of predicates.

+

+
infixr 5 _∷_

+

+
data All {A : Set a} (P : Pred A p) : Pred (List⁺ A) (a ⊔ p) where

+
  _∷_ : ∀ {x xs} (px : P x) (pxs : List.All P xs) → All P (x ∷ xs)

+

+
------------------------------------------------------------------------

+
-- Functions

+

+
toList⁺ : ∀ {xs : List⁺ A} → All P xs → List.All P (toList xs)

+
toList⁺ (px ∷ pxs) = px ∷ pxs

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.NonEmpty.html b/site/static/agda_html/Data.List.NonEmpty.html
new file mode 100644
index 000000000..6a6b124b2
--- /dev/null
+++ b/site/static/agda_html/Data.List.NonEmpty.html
@@ -0,0 +1,117 @@
+
+Data.List.NonEmpty
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Non-empty lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.NonEmpty where

+

+
open import Level using (Level)

+
open import Data.List.Base as List using (List)

+

+
------------------------------------------------------------------------

+
-- Re-export basic type and operations

+

+
open import Data.List.NonEmpty.Base public

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
-- Version 1.4

+

+
infixl 5 _∷ʳ'_

+

+
_∷ʳ'_ : (xs : List A) (x : A) → SnocView (xs ∷ʳ x)

+
_∷ʳ'_ = SnocView._∷ʳ′_

+
{-# WARNING_ON_USAGE _∷ʳ'_

+
"Warning: _∷ʳ'_ (ending in an apostrophe) was deprecated in v1.4.

+
Please use _∷ʳ′_ (ending in a prime) instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Properties.html b/site/static/agda_html/Data.List.Properties.html
new file mode 100644
index 000000000..22cbbb0b1
--- /dev/null
+++ b/site/static/agda_html/Data.List.Properties.html
@@ -0,0 +1,1698 @@
+
+Data.List.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- List-related properties

+
------------------------------------------------------------------------

+

+
-- Note that the lemmas below could be generalised to work with other

+
-- equalities than _≡_.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+
{-# OPTIONS --warn=noUserWarning #-} -- for deprecated scans

+

+
module Data.List.Properties where

+

+
open import Algebra.Bundles

+
open import Algebra.Consequences.Propositional

+
 using (selfInverse⇒involutive; selfInverse⇒injective)

+
open import Algebra.Definitions as AlgebraicDefinitions using (SelfInverse; Involutive)

+
open import Algebra.Morphism.Structures using (IsMagmaHomomorphism; IsMonoidHomomorphism)

+
import Algebra.Structures as AlgebraicStructures

+
open import Data.Bool.Base using (Bool; false; true; not; if_then_else_)

+
open import Data.Fin.Base using (Fin; zero; suc; cast; toℕ)

+
open import Data.List.Base as List

+
open import Data.List.Membership.Propositional using (_∈_)

+
open import Data.List.Relation.Unary.All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)

+
open import Data.Maybe.Relation.Unary.Any using (just) renaming (Any to MAny)

+
open import Data.Nat.Base

+
open import Data.Nat.Divisibility using (_∣_; divides; ∣n⇒∣m*n)

+
open import Data.Nat.Properties

+
open import Data.Product.Base as Product

+
  using (_×_; _,_; uncurry; uncurry′; proj₁; proj₂; <_,_>)

+
import Data.Product.Relation.Unary.All as Product using (All)

+
open import Data.Sum using (_⊎_; inj₁; inj₂; isInj₁; isInj₂)

+
open import Data.These.Base as These using (These; this; that; these)

+
open import Data.Fin.Properties using (toℕ-cast)

+
open import Function.Base using (id; _∘_; _∘′_; _∋_; _-⟨_∣; ∣_⟩-_; _$_; const; flip)

+
open import Function.Definitions using (Injective)

+
open import Level using (Level)

+
open import Relation.Binary.Definitions as B using (DecidableEquality)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
open import Relation.Binary.PropositionalEquality.Properties as ≡

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no; contradiction)

+
open import Relation.Nullary.Decidable as Decidable using (isYes; map′; ⌊_⌋; ¬?; _×-dec_)

+
open import Relation.Unary using (Pred; Decidable; ∁)

+
open import Relation.Unary.Properties using (∁?)

+
import Data.Nat.GeneralisedArithmetic as ℕ

+

+
open ≡-Reasoning

+

+
private

+
  variable

+
    a b c d e p ℓ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+
    E : Set e

+
    x y z w : A

+
    xs ys zs ws : List A

+

+
------------------------------------------------------------------------

+
-- _∷_

+

+
∷-injective : x ∷ xs ≡ y List.∷ ys → x ≡ y × xs ≡ ys

+
∷-injective refl = refl , refl

+

+
∷-injectiveˡ : x ∷ xs ≡ y List.∷ ys → x ≡ y

+
∷-injectiveˡ refl = refl

+

+
∷-injectiveʳ : x ∷ xs ≡ y List.∷ ys → xs ≡ ys

+
∷-injectiveʳ refl = refl

+

+
∷-dec : Dec (x ≡ y) → Dec (xs ≡ ys) → Dec (x ∷ xs ≡ y List.∷ ys)

+
∷-dec x≟y xs≟ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≟y ×-dec xs≟ys)

+

+
≡-dec : DecidableEquality A → DecidableEquality (List A)

+
≡-dec _≟_ []       []       = yes refl

+
≡-dec _≟_ (x ∷ xs) []       = no λ()

+
≡-dec _≟_ []       (y ∷ ys) = no λ()

+
≡-dec _≟_ (x ∷ xs) (y ∷ ys) = ∷-dec (x ≟ y) (≡-dec _≟_ xs ys)

+

+
------------------------------------------------------------------------

+
-- map

+

+
map-id : map id ≗ id {A = List A}

+
map-id []       = refl

+
map-id (x ∷ xs) = cong (x ∷_) (map-id xs)

+

+
map-id-local : ∀ {f : A → A} {xs} → All (λ x → f x ≡ x) xs → map f xs ≡ xs

+
map-id-local []           = refl

+
map-id-local (fx≡x ∷ pxs) = cong₂ _∷_ fx≡x (map-id-local pxs)

+

+
map-++ : ∀ (f : A → B) xs ys →

+
                 map f (xs ++ ys) ≡ map f xs ++ map f ys

+
map-++ f []       ys = refl

+
map-++ f (x ∷ xs) ys = cong (f x ∷_) (map-++ f xs ys)

+

+
map-cong : ∀ {f g : A → B} → f ≗ g → map f ≗ map g

+
map-cong f≗g []       = refl

+
map-cong f≗g (x ∷ xs) = cong₂ _∷_ (f≗g x) (map-cong f≗g xs)

+

+
map-cong-local : ∀ {f g : A → B} {xs} →

+
            All (λ x → f x ≡ g x) xs → map f xs ≡ map g xs

+
map-cong-local []                = refl

+
map-cong-local (fx≡gx ∷ fxs≡gxs) = cong₂ _∷_ fx≡gx (map-cong-local fxs≡gxs)

+

+
length-map : ∀ (f : A → B) xs → length (map f xs) ≡ length xs

+
length-map f []       = refl

+
length-map f (x ∷ xs) = cong suc (length-map f xs)

+

+
map-∘ : {g : B → C} {f : A → B} → map (g ∘ f) ≗ map g ∘ map f

+
map-∘ []       = refl

+
map-∘ (x ∷ xs) = cong (_ ∷_) (map-∘ xs)

+

+
map-injective : ∀ {f : A → B} → Injective _≡_ _≡_ f → Injective _≡_ _≡_ (map f)

+
map-injective finj {[]} {[]} eq = refl

+
map-injective finj {x ∷ xs} {y ∷ ys} eq =

+
  let fx≡fy , fxs≡fys = ∷-injective eq in

+
  cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
length-++ : ∀ (xs : List A) {ys} →

+
            length (xs ++ ys) ≡ length xs + length ys

+
length-++ []       = refl

+
length-++ (x ∷ xs) = cong suc (length-++ xs)

+

+
module _ {A : Set a} where

+

+
  open AlgebraicDefinitions {A = List A} _≡_

+
  open AlgebraicStructures  {A = List A} _≡_

+

+
  ++-assoc : Associative _++_

+
  ++-assoc []       ys zs = refl

+
  ++-assoc (x ∷ xs) ys zs = cong (x ∷_) (++-assoc xs ys zs)

+

+
  ++-identityˡ : LeftIdentity [] _++_

+
  ++-identityˡ xs = refl

+

+
  ++-identityʳ : RightIdentity [] _++_

+
  ++-identityʳ []       = refl

+
  ++-identityʳ (x ∷ xs) = cong (x ∷_) (++-identityʳ xs)

+

+
  ++-identity : Identity [] _++_

+
  ++-identity = ++-identityˡ , ++-identityʳ

+

+
  ++-identityʳ-unique : ∀ (xs : List A) {ys} → xs ≡ xs ++ ys → ys ≡ []

+
  ++-identityʳ-unique []       refl = refl

+
  ++-identityʳ-unique (x ∷ xs) eq   =

+
    ++-identityʳ-unique xs (∷-injectiveʳ eq)

+

+
  ++-identityˡ-unique : ∀ {xs} (ys : List A) → xs ≡ ys ++ xs → ys ≡ []

+
  ++-identityˡ-unique               []       _  = refl

+
  ++-identityˡ-unique {xs = x ∷ xs} (y ∷ ys) eq

+
    with ++-identityˡ-unique (ys ++ [ x ]) (begin

+
         xs                  ≡⟨ ∷-injectiveʳ eq ⟩

+
         ys ++ x ∷ xs        ≡⟨ ++-assoc ys [ x ] xs ⟨

+
         (ys ++ [ x ]) ++ xs ∎)

+
  ++-identityˡ-unique {xs = x ∷ xs} (y ∷ []   ) eq | ()

+
  ++-identityˡ-unique {xs = x ∷ xs} (y ∷ _ ∷ _) eq | ()

+

+
  ++-cancelˡ : LeftCancellative _++_

+
  ++-cancelˡ []       _ _ ys≡zs             = ys≡zs

+
  ++-cancelˡ (x ∷ xs) _ _ x∷xs++ys≡x∷xs++zs = ++-cancelˡ xs _ _ (∷-injectiveʳ x∷xs++ys≡x∷xs++zs)

+

+
  ++-cancelʳ : RightCancellative _++_

+
  ++-cancelʳ _  []       []       _             = refl

+
  ++-cancelʳ xs []       (z ∷ zs) eq =

+
    contradiction (trans (cong length eq) (length-++ (z ∷ zs))) (m≢1+n+m (length xs))

+
  ++-cancelʳ xs (y ∷ ys) []       eq =

+
    contradiction (trans (sym (length-++ (y ∷ ys))) (cong length eq)) (m≢1+n+m (length xs) ∘ sym)

+
  ++-cancelʳ _  (y ∷ ys) (z ∷ zs) eq =

+
    cong₂ _∷_ (∷-injectiveˡ eq) (++-cancelʳ _ ys zs (∷-injectiveʳ eq))

+

+
  ++-cancel : Cancellative _++_

+
  ++-cancel = ++-cancelˡ , ++-cancelʳ

+

+
  ++-conicalˡ : ∀ (xs ys : List A) → xs ++ ys ≡ [] → xs ≡ []

+
  ++-conicalˡ []       _ refl = refl

+

+
  ++-conicalʳ : ∀ (xs ys : List A) → xs ++ ys ≡ [] → ys ≡ []

+
  ++-conicalʳ []       _ refl = refl

+

+
  ++-conical : Conical [] _++_

+
  ++-conical = ++-conicalˡ , ++-conicalʳ

+

+
  ++-isMagma : IsMagma _++_

+
  ++-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = cong₂ _++_

+
    }

+

+
  ++-isSemigroup : IsSemigroup _++_

+
  ++-isSemigroup = record

+
    { isMagma = ++-isMagma

+
    ; assoc   = ++-assoc

+
    }

+

+
  ++-isMonoid : IsMonoid _++_ []

+
  ++-isMonoid = record

+
    { isSemigroup = ++-isSemigroup

+
    ; identity    = ++-identity

+
    }

+

+
module _ (A : Set a) where

+

+
  ++-semigroup : Semigroup a a

+
  ++-semigroup = record

+
    { Carrier     = List A

+
    ; isSemigroup = ++-isSemigroup

+
    }

+

+
  ++-monoid : Monoid a a

+
  ++-monoid = record

+
    { Carrier  = List A

+
    ; isMonoid = ++-isMonoid

+
    }

+

+
module _ (A : Set a) where

+

+
  length-isMagmaHomomorphism : IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length

+
  length-isMagmaHomomorphism = record

+
    { isRelHomomorphism = record

+
      { cong = cong length

+
      }

+
    ; homo = λ xs ys → length-++ xs {ys}

+
    }

+

+
  length-isMonoidHomomorphism : IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length

+
  length-isMonoidHomomorphism = record

+
    { isMagmaHomomorphism = length-isMagmaHomomorphism

+
    ; ε-homo = refl

+
    }

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith

+

+
module _ (f : A → B → C) where

+

+
  private

+
    prod = cartesianProductWith f

+

+
  cartesianProductWith-zeroˡ : ∀ ys → prod [] ys ≡ []

+
  cartesianProductWith-zeroˡ _ = refl

+

+
  cartesianProductWith-zeroʳ : ∀ xs → prod xs [] ≡ []

+
  cartesianProductWith-zeroʳ []       = refl

+
  cartesianProductWith-zeroʳ (x ∷ xs) = cartesianProductWith-zeroʳ xs

+

+
  cartesianProductWith-distribʳ-++ : ∀ xs ys zs → prod (xs ++ ys) zs ≡ prod xs zs ++ prod ys zs

+
  cartesianProductWith-distribʳ-++ []       ys zs = refl

+
  cartesianProductWith-distribʳ-++ (x ∷ xs) ys zs = begin

+
    prod (x ∷ xs ++ ys) zs ≡⟨⟩

+
    map (f x) zs ++ prod (xs ++ ys) zs ≡⟨ cong (map (f x) zs ++_) (cartesianProductWith-distribʳ-++ xs ys zs) ⟩

+
    map (f x) zs ++ prod xs zs ++ prod ys zs ≡⟨ ++-assoc (map (f x) zs) (prod xs zs) (prod ys zs) ⟨

+
    (map (f x) zs ++ prod xs zs) ++ prod ys zs ≡⟨⟩

+
    prod (x ∷ xs) zs ++ prod ys zs ∎

+

+
------------------------------------------------------------------------

+
-- alignWith

+

+
module _ {f g : These A B → C} where

+

+
  alignWith-cong : f ≗ g → ∀ as → alignWith f as ≗ alignWith g as

+
  alignWith-cong f≗g []         bs       = map-cong (f≗g ∘ that) bs

+
  alignWith-cong f≗g as@(_ ∷ _) []       = map-cong (f≗g ∘ this) as

+
  alignWith-cong f≗g (a ∷ as)   (b ∷ bs) =

+
    cong₂ _∷_ (f≗g (these a b)) (alignWith-cong f≗g as bs)

+

+
module _ {f : These A B → C} where

+

+
  length-alignWith : ∀ xs ys →

+
                   length (alignWith f xs ys) ≡ length xs ⊔ length ys

+
  length-alignWith []         ys       = length-map (f ∘′ that) ys

+
  length-alignWith xs@(_ ∷ _) []       = length-map (f ∘′ this) xs

+
  length-alignWith (x ∷ xs)   (y ∷ ys) = cong suc (length-alignWith xs ys)

+

+
  alignWith-map : (g : D → A) (h : E → B) →

+
                  ∀ xs ys → alignWith f (map g xs) (map h ys) ≡

+
                            alignWith (f ∘′ These.map g h) xs ys

+
  alignWith-map g h []         ys     = sym (map-∘ ys)

+
  alignWith-map g h xs@(_ ∷ _) []     = sym (map-∘ xs)

+
  alignWith-map g h (x ∷ xs) (y ∷ ys) =

+
    cong₂ _∷_ refl (alignWith-map g h xs ys)

+

+
  map-alignWith : ∀ (g : C → D) → ∀ xs ys →

+
                  map g (alignWith f xs ys) ≡

+
                  alignWith (g ∘′ f) xs ys

+
  map-alignWith g []         ys     = sym (map-∘ ys)

+
  map-alignWith g xs@(_ ∷ _) []     = sym (map-∘ xs)

+
  map-alignWith g (x ∷ xs) (y ∷ ys) =

+
    cong₂ _∷_ refl (map-alignWith g xs ys)

+

+
  alignWith-flip : ∀ xs ys →

+
                 alignWith f xs ys ≡ alignWith (f ∘ These.swap) ys xs

+
  alignWith-flip []       []       = refl

+
  alignWith-flip []       (y ∷ ys) = refl

+
  alignWith-flip (x ∷ xs) []       = refl

+
  alignWith-flip (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (alignWith-flip xs ys)

+

+
module _ {f : These A A → B} where

+

+
  alignWith-comm : f ∘ These.swap ≗ f →

+
                 ∀ xs ys → alignWith f xs ys ≡ alignWith f ys xs

+
  alignWith-comm f-comm xs ys = begin

+
    alignWith f xs ys                ≡⟨ alignWith-flip xs ys ⟩

+
    alignWith (f ∘ These.swap) ys xs ≡⟨ alignWith-cong f-comm ys xs ⟩

+
    alignWith f ys xs                ∎

+

+
------------------------------------------------------------------------

+
-- align

+

+
align-map : ∀ (f : A → B) (g : C → D) →

+
            ∀ xs ys → align (map f xs) (map g ys) ≡

+
                      map (These.map f g) (align xs ys)

+
align-map f g xs ys = begin

+
  align (map f xs) (map g ys)       ≡⟨ alignWith-map f g xs ys ⟩

+
  alignWith (These.map f g) xs ys   ≡⟨ sym (map-alignWith (These.map f g) xs ys) ⟩

+
  map (These.map f g) (align xs ys) ∎

+

+
align-flip : ∀ (xs : List A) (ys : List B) →

+
             align xs ys ≡ map These.swap (align ys xs)

+
align-flip xs ys = begin

+
  align xs ys                  ≡⟨ alignWith-flip xs ys ⟩

+
  alignWith These.swap ys xs   ≡⟨ sym (map-alignWith These.swap ys xs) ⟩

+
  map These.swap (align ys xs) ∎

+

+
------------------------------------------------------------------------

+
-- zipWith

+

+
module _ {f g : A → B → C} where

+

+
  zipWith-cong : (∀ a b → f a b ≡ g a b) → ∀ as → zipWith f as ≗ zipWith g as

+
  zipWith-cong f≗g []         bs       = refl

+
  zipWith-cong f≗g as@(_ ∷ _) []       = refl

+
  zipWith-cong f≗g (a ∷ as)   (b ∷ bs) =

+
    cong₂ _∷_ (f≗g a b) (zipWith-cong f≗g as bs)

+

+
module _ (f : A → B → C) where

+

+
  zipWith-zeroˡ : ∀ xs → zipWith f [] xs ≡ []

+
  zipWith-zeroˡ []       = refl

+
  zipWith-zeroˡ (x ∷ xs) = refl

+

+
  zipWith-zeroʳ : ∀ xs → zipWith f xs [] ≡ []

+
  zipWith-zeroʳ []       = refl

+
  zipWith-zeroʳ (x ∷ xs) = refl

+

+
  length-zipWith : ∀ xs ys →

+
                   length (zipWith f xs ys) ≡ length xs ⊓ length ys

+
  length-zipWith []       []       = refl

+
  length-zipWith []       (y ∷ ys) = refl

+
  length-zipWith (x ∷ xs) []       = refl

+
  length-zipWith (x ∷ xs) (y ∷ ys) = cong suc (length-zipWith xs ys)

+

+
  zipWith-map : ∀ (g : D → A) (h : E → B) →

+
                ∀ xs ys → zipWith f (map g xs) (map h ys) ≡

+
                          zipWith (λ x y → f (g x) (h y)) xs ys

+
  zipWith-map g h []       []       = refl

+
  zipWith-map g h []       (y ∷ ys) = refl

+
  zipWith-map g h (x ∷ xs) []       = refl

+
  zipWith-map g h (x ∷ xs) (y ∷ ys) =

+
    cong₂ _∷_ refl (zipWith-map g h xs ys)

+

+
  map-zipWith : ∀ (g : C → D) → ∀ xs ys →

+
                map g (zipWith f xs ys) ≡

+
                zipWith (λ x y → g (f x y)) xs ys

+
  map-zipWith g []       []       = refl

+
  map-zipWith g []       (y ∷ ys) = refl

+
  map-zipWith g (x ∷ xs) []       = refl

+
  map-zipWith g (x ∷ xs) (y ∷ ys) =

+
    cong₂ _∷_ refl (map-zipWith g xs ys)

+

+
  zipWith-flip : ∀ xs ys → zipWith f xs ys ≡ zipWith (flip f) ys xs

+
  zipWith-flip []       []       = refl

+
  zipWith-flip []       (x ∷ ys) = refl

+
  zipWith-flip (x ∷ xs) []       = refl

+
  zipWith-flip (x ∷ xs) (y ∷ ys) = cong (f x y ∷_) (zipWith-flip xs ys)

+

+
module _ (f : A → A → B) where

+

+
  zipWith-comm : (∀ x y → f x y ≡ f y x) →

+
                 ∀ xs ys → zipWith f xs ys ≡ zipWith f ys xs

+
  zipWith-comm f-comm xs ys = begin

+
    zipWith f xs ys        ≡⟨ zipWith-flip f xs ys ⟩

+
    zipWith (flip f) ys xs ≡⟨ zipWith-cong (flip f-comm) ys xs ⟩

+
    zipWith f ys xs        ∎

+

+
------------------------------------------------------------------------

+
-- zip

+

+
zip-map : ∀ (f : A → B) (g : C → D) →

+
          ∀ xs ys → zip (map f xs) (map g ys) ≡

+
                    map (Product.map f g) (zip xs ys)

+
zip-map f g xs ys = begin

+
  zip (map f xs) (map g ys)         ≡⟨ zipWith-map _,_ f g xs ys ⟩

+
  zipWith (λ x y → f x , g y) xs ys ≡⟨ sym (map-zipWith _,_ (Product.map f g) xs ys) ⟩

+
  map (Product.map f g) (zip xs ys) ∎

+

+
zip-flip : ∀ (xs : List A) (ys : List B) →

+
           zip xs ys ≡ map Product.swap (zip ys xs)

+
zip-flip xs ys = begin

+
  zip xs ys                    ≡⟨ zipWith-flip _,_ xs ys ⟩

+
  zipWith (flip _,_) ys xs     ≡⟨ sym (map-zipWith _,_ Product.swap ys xs) ⟩

+
  map Product.swap (zip ys xs) ∎

+

+
------------------------------------------------------------------------

+
-- unalignWith

+

+
unalignWith-this : unalignWith ((A → These A B) ∋ this) ≗ (_, [])

+
unalignWith-this []       = refl

+
unalignWith-this (a ∷ as) = cong (Product.map₁ (a ∷_)) (unalignWith-this as)

+

+
unalignWith-that : unalignWith ((B → These A B) ∋ that) ≗ ([] ,_)

+
unalignWith-that []       = refl

+
unalignWith-that (b ∷ bs) = cong (Product.map₂ (b ∷_)) (unalignWith-that bs)

+

+
module _ {f g : C → These A B} where

+

+
  unalignWith-cong : f ≗ g → unalignWith f ≗ unalignWith g

+
  unalignWith-cong f≗g []       = refl

+
  unalignWith-cong f≗g (c ∷ cs) with f c | g c | f≗g c

+
  ... | this a    | ._ | refl = cong (Product.map₁ (a ∷_)) (unalignWith-cong f≗g cs)

+
  ... | that b    | ._ | refl = cong (Product.map₂ (b ∷_)) (unalignWith-cong f≗g cs)

+
  ... | these a b | ._ | refl = cong (Product.map (a ∷_) (b ∷_)) (unalignWith-cong f≗g cs)

+

+
module _ (f : C → These A B) where

+

+
  unalignWith-map : (g : D → C) → ∀ ds →

+
                    unalignWith f (map g ds) ≡ unalignWith (f ∘′ g) ds

+
  unalignWith-map g []       = refl

+
  unalignWith-map g (d ∷ ds) with f (g d)

+
  ... | this a    = cong (Product.map₁ (a ∷_)) (unalignWith-map g ds)

+
  ... | that b    = cong (Product.map₂ (b ∷_)) (unalignWith-map g ds)

+
  ... | these a b = cong (Product.map (a ∷_) (b ∷_)) (unalignWith-map g ds)

+

+
  map-unalignWith : (g : A → D) (h : B → E) →

+
    Product.map (map g) (map h) ∘′ unalignWith f ≗ unalignWith (These.map g h ∘′ f)

+
  map-unalignWith g h []       = refl

+
  map-unalignWith g h (c ∷ cs) with f c

+
  ... | this a    = cong (Product.map₁ (g a ∷_)) (map-unalignWith g h cs)

+
  ... | that b    = cong (Product.map₂ (h b ∷_)) (map-unalignWith g h cs)

+
  ... | these a b = cong (Product.map (g a ∷_) (h b ∷_)) (map-unalignWith g h cs)

+

+
  unalignWith-alignWith : (g : These A B → C) → f ∘′ g ≗ id → ∀ as bs →

+
                          unalignWith f (alignWith g as bs) ≡ (as , bs)

+
  unalignWith-alignWith g g∘f≗id []         bs = begin

+
    unalignWith f (map (g ∘′ that) bs) ≡⟨ unalignWith-map (g ∘′ that) bs ⟩

+
    unalignWith (f ∘′ g ∘′ that) bs    ≡⟨ unalignWith-cong (g∘f≗id ∘ that) bs ⟩

+
    unalignWith that bs                ≡⟨ unalignWith-that bs ⟩

+
    [] , bs                            ∎

+
  unalignWith-alignWith g g∘f≗id as@(_ ∷ _) [] = begin

+
    unalignWith f (map (g ∘′ this) as) ≡⟨ unalignWith-map (g ∘′ this) as ⟩

+
    unalignWith (f ∘′ g ∘′ this) as    ≡⟨ unalignWith-cong (g∘f≗id ∘ this) as ⟩

+
    unalignWith this as                ≡⟨ unalignWith-this as ⟩

+
    as , []                            ∎

+
  unalignWith-alignWith g g∘f≗id (a ∷ as)   (b ∷ bs)

+
    rewrite g∘f≗id (these a b) =

+
    cong (Product.map (a ∷_) (b ∷_)) (unalignWith-alignWith g g∘f≗id as bs)

+

+
------------------------------------------------------------------------

+
-- unzipWith

+

+
module _ {f g : A → B × C} where

+

+
  unzipWith-cong : f ≗ g → unzipWith f ≗ unzipWith g

+
  unzipWith-cong f≗g [] = refl

+
  unzipWith-cong f≗g (x ∷ xs) =

+
    cong₂ (Product.zip _∷_ _∷_) (f≗g x) (unzipWith-cong f≗g xs)

+

+
module _ (f : A → B × C) where

+

+
  length-unzipWith₁ : ∀ xys →

+
                     length (proj₁ (unzipWith f xys)) ≡ length xys

+
  length-unzipWith₁ []        = refl

+
  length-unzipWith₁ (x ∷ xys) = cong suc (length-unzipWith₁ xys)

+

+
  length-unzipWith₂ : ∀ xys →

+
                     length (proj₂ (unzipWith f xys)) ≡ length xys

+
  length-unzipWith₂ []        = refl

+
  length-unzipWith₂ (x ∷ xys) = cong suc (length-unzipWith₂ xys)

+

+
  zipWith-unzipWith : (g : B → C → A) → uncurry′ g ∘ f ≗ id →

+
                      uncurry′ (zipWith g) ∘ (unzipWith f)  ≗ id

+
  zipWith-unzipWith g g∘f≗id []       = refl

+
  zipWith-unzipWith g g∘f≗id (x ∷ xs) =

+
    cong₂ _∷_ (g∘f≗id x) (zipWith-unzipWith g g∘f≗id xs)

+

+
  unzipWith-zipWith : (g : B → C → A) → f ∘ uncurry′ g ≗ id →

+
                      ∀ xs ys → length xs ≡ length ys →

+
                      unzipWith f (zipWith g xs ys) ≡ (xs , ys)

+
  unzipWith-zipWith g f∘g≗id []       []       l≡l = refl

+
  unzipWith-zipWith g f∘g≗id (x ∷ xs) (y ∷ ys) l≡l  =

+
    cong₂ (Product.zip _∷_ _∷_) (f∘g≗id (x , y))

+
          (unzipWith-zipWith g f∘g≗id xs ys (suc-injective l≡l))

+

+
  unzipWith-map : (g : D → A) → unzipWith f ∘ map g ≗ unzipWith (f ∘ g)

+
  unzipWith-map g []       = refl

+
  unzipWith-map g (x ∷ xs) =

+
    cong (Product.zip _∷_ _∷_ (f (g x))) (unzipWith-map g xs)

+

+
  map-unzipWith : (g : B → D) (h : C → E) →

+
                  Product.map (map g) (map h) ∘ unzipWith f ≗

+
                  unzipWith (Product.map g h ∘ f)

+
  map-unzipWith g h []       = refl

+
  map-unzipWith g h (x ∷ xs) =

+
    cong (Product.zip _∷_ _∷_ _) (map-unzipWith g h xs)

+

+
  unzipWith-swap : unzipWith (Product.swap ∘ f) ≗

+
                   Product.swap ∘ unzipWith f

+
  unzipWith-swap []       = refl

+
  unzipWith-swap (x ∷ xs) =

+
    cong (Product.zip _∷_ _∷_ _) (unzipWith-swap xs)

+

+
  unzipWith-++ : ∀ xs ys →

+
                 unzipWith f (xs ++ ys) ≡

+
                 Product.zip _++_ _++_ (unzipWith f xs) (unzipWith f ys)

+
  unzipWith-++ []       ys = refl

+
  unzipWith-++ (x ∷ xs) ys =

+
    cong (Product.zip _∷_ _∷_ (f x)) (unzipWith-++ xs ys)

+

+
------------------------------------------------------------------------

+
-- unzip

+

+
unzip-map : ∀ (f : A → B) (g : C → D) →

+
            unzip ∘ map (Product.map f g) ≗

+
            Product.map (map f) (map g) ∘ unzip

+
unzip-map f g xs = begin

+
  unzip (map (Product.map f g) xs)       ≡⟨ unzipWith-map id (Product.map f g) xs ⟩

+
  unzipWith (Product.map f g) xs         ≡⟨ sym (map-unzipWith id f g xs) ⟩

+
  Product.map (map f) (map g) (unzip xs) ∎

+

+
unzip-swap : unzip ∘ map Product.swap ≗ Product.swap ∘ unzip {A = A} {B = B}

+
unzip-swap xs = begin

+
  unzip (map Product.swap xs) ≡⟨ unzipWith-map id Product.swap xs ⟩

+
  unzipWith Product.swap xs   ≡⟨ unzipWith-swap id xs ⟩

+
  Product.swap (unzip xs)     ∎

+

+
zip-unzip : uncurry′ zip ∘ unzip ≗ id {A = List (A × B)}

+
zip-unzip = zipWith-unzipWith id _,_ λ _ → refl

+

+
unzip-zip : ∀ (xs : List A) (ys : List B) →

+
            length xs ≡ length ys → unzip (zip xs ys) ≡ (xs , ys)

+
unzip-zip = unzipWith-zipWith id _,_ λ _ → refl

+

+
------------------------------------------------------------------------

+
-- foldr

+

+
foldr-universal : ∀ (h : List A → B) f e → (h [] ≡ e) →

+
                  (∀ x xs → h (x ∷ xs) ≡ f x (h xs)) →

+
                  h ≗ foldr f e

+
foldr-universal h f e base step []       = base

+
foldr-universal h f e base step (x ∷ xs) = begin

+
  h (x ∷ xs)          ≡⟨ step x xs ⟩

+
  f x (h xs)          ≡⟨ cong (f x) (foldr-universal h f e base step xs) ⟩

+
  f x (foldr f e xs)  ∎

+

+
foldr-cong : ∀ {f g : A → B → B} {d e : B} →

+
             (∀ x y → f x y ≡ g x y) → d ≡ e →

+
             foldr f d ≗ foldr g e

+
foldr-cong f≗g refl []      = refl

+
foldr-cong f≗g d≡e (x ∷ xs) rewrite foldr-cong f≗g d≡e xs = f≗g x _

+

+
foldr-fusion : ∀ (h : B → C) {f : A → B → B} {g : A → C → C} (e : B) →

+
               (∀ x y → h (f x y) ≡ g x (h y)) →

+
               h ∘ foldr f e ≗ foldr g (h e)

+
foldr-fusion h {f} {g} e fuse =

+
  foldr-universal (h ∘ foldr f e) g (h e) refl

+
                  (λ x xs → fuse x (foldr f e xs))

+

+
id-is-foldr : id {A = List A} ≗ foldr _∷_ []

+
id-is-foldr = foldr-universal id _∷_ [] refl (λ _ _ → refl)

+

+
++-is-foldr : (xs ys : List A) → xs ++ ys ≡ foldr _∷_ ys xs

+
++-is-foldr xs ys = begin

+
  xs ++ ys                ≡⟨ cong (_++ ys) (id-is-foldr xs) ⟩

+
  foldr _∷_ [] xs ++ ys   ≡⟨ foldr-fusion (_++ ys) [] (λ _ _ → refl) xs ⟩

+
  foldr _∷_ ([] ++ ys) xs ≡⟨⟩

+
  foldr _∷_ ys xs         ∎

+

+
foldr-++ : ∀ (f : A → B → B) x ys zs →

+
           foldr f x (ys ++ zs) ≡ foldr f (foldr f x zs) ys

+
foldr-++ f x []       zs = refl

+
foldr-++ f x (y ∷ ys) zs = cong (f y) (foldr-++ f x ys zs)

+

+
map-is-foldr : {f : A → B} → map f ≗ foldr (λ x ys → f x ∷ ys) []

+
map-is-foldr {f = f} xs = begin

+
  map f xs                        ≡⟨ cong (map f) (id-is-foldr xs) ⟩

+
  map f (foldr _∷_ [] xs)         ≡⟨ foldr-fusion (map f) [] (λ _ _ → refl) xs ⟩

+
  foldr (λ x ys → f x ∷ ys) [] xs ∎

+

+
foldr-∷ʳ : ∀ (f : A → B → B) x y ys →

+
           foldr f x (ys ∷ʳ y) ≡ foldr f (f y x) ys

+
foldr-∷ʳ f x y []       = refl

+
foldr-∷ʳ f x y (z ∷ ys) = cong (f z) (foldr-∷ʳ f x y ys)

+

+
foldr-map : ∀ (f : A → B → B) (g : C → A) x xs → foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs

+
foldr-map f g x []       = refl

+
foldr-map f g x (y ∷ xs) = cong (f (g y)) (foldr-map f g x xs)

+

+
-- Interaction with predicates

+

+
module _ {P : Pred A p} {f : A → A → A} where

+

+
  foldr-forcesᵇ : (∀ x y → P (f x y) → P x × P y) →

+
                  ∀ e xs → P (foldr f e xs) → All P xs

+
  foldr-forcesᵇ _      _ []       _     = []

+
  foldr-forcesᵇ forces _ (x ∷ xs) Pfold =

+
    let px , pfxs = forces _ _ Pfold in px ∷ foldr-forcesᵇ forces _ xs pfxs

+

+
  foldr-preservesᵇ : (∀ {x y} → P x → P y → P (f x y)) →

+
                     ∀ {e xs} → P e → All P xs → P (foldr f e xs)

+
  foldr-preservesᵇ _    Pe []         = Pe

+
  foldr-preservesᵇ pres Pe (px ∷ pxs) = pres px (foldr-preservesᵇ pres Pe pxs)

+

+
  foldr-preservesʳ : (∀ x {y} → P y → P (f x y)) →

+
                     ∀ {e} → P e → ∀ xs → P (foldr f e xs)

+
  foldr-preservesʳ pres Pe []       = Pe

+
  foldr-preservesʳ pres Pe (_ ∷ xs) = pres _ (foldr-preservesʳ pres Pe xs)

+

+
  foldr-preservesᵒ : (∀ x y → P x ⊎ P y → P (f x y)) →

+
                     ∀ e xs → P e ⊎ Any P xs → P (foldr f e xs)

+
  foldr-preservesᵒ pres e []       (inj₁ Pe)          = Pe

+
  foldr-preservesᵒ pres e (x ∷ xs) (inj₁ Pe)          =

+
    pres _ _ (inj₂ (foldr-preservesᵒ pres e xs (inj₁ Pe)))

+
  foldr-preservesᵒ pres e (x ∷ xs) (inj₂ (here px))   = pres _ _ (inj₁ px)

+
  foldr-preservesᵒ pres e (x ∷ xs) (inj₂ (there pxs)) =

+
    pres _ _ (inj₂ (foldr-preservesᵒ pres e xs (inj₂ pxs)))

+

+
------------------------------------------------------------------------

+
-- foldl

+

+
foldl-cong : ∀ {f g : B → A → B} → (∀ x y → f x y ≡ g x y) →

+
             ∀ x → foldl f x ≗ foldl g x

+
foldl-cong f≗g x []       = refl

+
foldl-cong f≗g x (y ∷ xs) rewrite f≗g x y = foldl-cong f≗g _ xs

+

+
foldl-++ : ∀ (f : A → B → A) x ys zs →

+
           foldl f x (ys ++ zs) ≡ foldl f (foldl f x ys) zs

+
foldl-++ f x []       zs = refl

+
foldl-++ f x (y ∷ ys) zs = foldl-++ f (f x y) ys zs

+

+
foldl-∷ʳ : ∀ (f : A → B → A) x y ys →

+
           foldl f x (ys ∷ʳ y) ≡ f (foldl f x ys) y

+
foldl-∷ʳ f x y []       = refl

+
foldl-∷ʳ f x y (z ∷ ys) = foldl-∷ʳ f (f x z) y ys

+

+
foldl-map : ∀ (f : A → B → A) (g : C → B) x xs → foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs

+
foldl-map f g x []       = refl

+
foldl-map f g x (y ∷ xs) = foldl-map f g (f x (g y)) xs

+

+
------------------------------------------------------------------------

+
-- concat

+

+
concat-map : ∀ {f : A → B} → concat ∘ map (map f) ≗ map f ∘ concat

+
concat-map {f = f} xss = begin

+
  concat (map (map f) xss)                   ≡⟨ cong concat (map-is-foldr xss) ⟩

+
  concat (foldr (λ xs → map f xs ∷_) [] xss) ≡⟨ foldr-fusion concat [] (λ _ _ → refl) xss ⟩

+
  foldr (λ ys → map f ys ++_) [] xss         ≡⟨ sym (foldr-fusion (map f) [] (map-++ f) xss) ⟩

+
  map f (concat xss)                         ∎

+

+
concat-++ : (xss yss : List (List A)) → concat xss ++ concat yss ≡ concat (xss ++ yss)

+
concat-++ [] yss = refl

+
concat-++ ([] ∷ xss) yss = concat-++ xss yss

+
concat-++ ((x ∷ xs) ∷ xss) yss = cong (x ∷_) (concat-++ (xs ∷ xss) yss)

+

+
concat-concat : concat {A = A} ∘ map concat ≗ concat ∘ concat

+
concat-concat [] = refl

+
concat-concat (xss ∷ xsss) = begin

+
  concat (map concat (xss ∷ xsss))   ≡⟨ cong (concat xss ++_) (concat-concat xsss) ⟩

+
  concat xss ++ concat (concat xsss) ≡⟨ concat-++ xss (concat xsss) ⟩

+
  concat (concat (xss ∷ xsss))       ∎

+

+
concat-[-] : concat {A = A} ∘ map [_] ≗ id

+
concat-[-] [] = refl

+
concat-[-] (x ∷ xs) = cong (x ∷_) (concat-[-] xs)

+

+
------------------------------------------------------------------------

+
-- concatMap

+

+
concatMap-cong : ∀ {f g : A → List B} → f ≗ g → concatMap f ≗ concatMap g

+
concatMap-cong eq xs = cong concat (map-cong eq xs)

+

+
concatMap-pure : concatMap {A = A} [_] ≗ id

+
concatMap-pure = concat-[-]

+

+
concatMap-map : (g : B → List C) → (f : A → B) → (xs : List A) →

+
                concatMap g (map f xs) ≡ concatMap (g ∘′ f) xs

+
concatMap-map g f xs

+
  = cong concat

+
      {x = map g (map f xs)}

+
      {y = map (g ∘′ f) xs}

+
      (sym $ map-∘ xs)

+

+
map-concatMap : (f : B → C) (g : A → List B) →

+
                map f ∘′ concatMap g ≗ concatMap (map f ∘′ g)

+
map-concatMap f g xs = begin

+
  map f (concatMap g xs)

+
    ≡⟨⟩

+
  map f (concat (map g xs))

+
    ≡⟨ concat-map (map g xs) ⟨

+
  concat (map (map f) (map g xs))

+
    ≡⟨ cong concat

+
         {x = map (map f) (map g xs)}

+
         {y = map (map f ∘′ g) xs}

+
         (sym $ map-∘ xs) ⟩

+
  concat (map (map f ∘′ g) xs)

+
    ≡⟨⟩

+
  concatMap (map f ∘′ g) xs

+
    ∎

+

+
------------------------------------------------------------------------

+
-- catMaybes

+

+
catMaybes-concatMap : catMaybes {A = A} ≗ concatMap fromMaybe

+
catMaybes-concatMap []             = refl

+
catMaybes-concatMap (just x  ∷ xs) = cong (x ∷_) $ catMaybes-concatMap xs

+
catMaybes-concatMap (nothing ∷ xs) = catMaybes-concatMap xs

+

+
length-catMaybes : ∀ xs → length (catMaybes {A = A} xs) ≤ length xs

+
length-catMaybes []             = ≤-refl

+
length-catMaybes (just _  ∷ xs) = s≤s $ length-catMaybes xs

+
length-catMaybes (nothing ∷ xs) = m≤n⇒m≤1+n $ length-catMaybes xs

+

+
catMaybes-++ : (xs ys : List (Maybe A)) →

+
               catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys

+
catMaybes-++ []             _  = refl

+
catMaybes-++ (just x  ∷ xs) ys = cong (x ∷_) $ catMaybes-++ xs ys

+
catMaybes-++ (nothing ∷ xs) ys = catMaybes-++ xs ys

+

+
map-catMaybes : (f : A → B) → map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)

+
map-catMaybes _ []             = refl

+
map-catMaybes f (just x  ∷ xs) = cong (f x ∷_) $ map-catMaybes f xs

+
map-catMaybes f (nothing ∷ xs) = map-catMaybes f xs

+

+
Any-catMaybes⁺ : ∀ {P : Pred A ℓ} {xs : List (Maybe A)} →

+
                 Any (MAny P) xs → Any P (catMaybes xs)

+
Any-catMaybes⁺ {xs = nothing ∷ xs} (there x∈)       = Any-catMaybes⁺ x∈

+
Any-catMaybes⁺ {xs = just x  ∷ xs} (here (just px)) = here px

+
Any-catMaybes⁺ {xs = just x  ∷ xs} (there x∈)       = there $ Any-catMaybes⁺ x∈

+

+
------------------------------------------------------------------------

+
-- mapMaybe

+

+
mapMaybe-cong : {f g : A → Maybe B} → f ≗ g → mapMaybe f ≗ mapMaybe g

+
mapMaybe-cong f≗g = cong catMaybes ∘ map-cong f≗g

+

+
mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs

+
mapMaybe-just []       = refl

+
mapMaybe-just (x ∷ xs) = cong (x ∷_) (mapMaybe-just xs)

+

+
mapMaybe-nothing : (xs : List A) →

+
                   mapMaybe {B = B} (λ _ → nothing) xs ≡ []

+
mapMaybe-nothing []       = refl

+
mapMaybe-nothing (x ∷ xs) = mapMaybe-nothing xs

+

+
module _ (f : A → Maybe B) where

+

+
  mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f)

+
  mapMaybe-concatMap xs = begin

+
    catMaybes (map f xs)            ≡⟨ catMaybes-concatMap (map f xs) ⟩

+
    concatMap fromMaybe (map f xs)  ≡⟨ concatMap-map fromMaybe f xs ⟩

+
    concatMap (fromMaybe ∘ f) xs    ∎

+

+
  length-mapMaybe : ∀ xs → length (mapMaybe f xs) ≤ length xs

+
  length-mapMaybe xs = ≤-begin

+
    length (mapMaybe f xs)      ≤⟨ length-catMaybes (map f xs) ⟩

+
    length (map f xs)           ≤⟨ ≤-reflexive (length-map f xs) ⟩

+
    length xs                   ≤-∎

+
    where open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎)

+

+
  mapMaybe-++ : ∀ xs ys →

+
                mapMaybe f (xs ++ ys) ≡ mapMaybe f xs ++ mapMaybe f ys

+
  mapMaybe-++ xs ys = begin

+
    catMaybes (map f (xs ++ ys))     ≡⟨ cong catMaybes (map-++ f xs ys) ⟩

+
    catMaybes (map f xs ++ map f ys) ≡⟨ catMaybes-++ (map f xs) (map f ys) ⟩

+
    mapMaybe f xs ++ mapMaybe f ys   ∎

+

+
module _ (f : B → Maybe C) (g : A → B) where

+

+
  mapMaybe-map : mapMaybe f ∘ map g ≗ mapMaybe (f ∘ g)

+
  mapMaybe-map = cong catMaybes ∘ sym ∘ map-∘

+

+
module _ (g : B → C) (f : A → Maybe B) where

+

+
  map-mapMaybe : map g ∘ mapMaybe f ≗ mapMaybe (Maybe.map g ∘ f)

+
  map-mapMaybe xs = begin

+
    map g (catMaybes (map f xs))       ≡⟨ map-catMaybes g (map f xs) ⟩

+
    mapMaybe (Maybe.map g) (map f xs)  ≡⟨ mapMaybe-map _ f xs ⟩

+
    mapMaybe (Maybe.map g ∘ f) xs      ∎

+

+
-- embedding-projection pairs

+
module _ {proj : B → Maybe A} {emb : A → B} where

+
  mapMaybe-map-retract : proj ∘ emb ≗ just → mapMaybe proj ∘ map emb ≗ id

+
  mapMaybe-map-retract retract xs = begin

+
    mapMaybe proj (map emb xs) ≡⟨ mapMaybe-map _ _ xs ⟩

+
    mapMaybe (proj ∘ emb) xs   ≡⟨ mapMaybe-cong retract xs ⟩

+
    mapMaybe just xs           ≡⟨ mapMaybe-just _ ⟩

+
    xs                         ∎

+

+
module _ {proj : C → Maybe B} {emb : A → C} where

+
  mapMaybe-map-none : proj ∘ emb ≗ const nothing → mapMaybe proj ∘ map emb ≗ const []

+
  mapMaybe-map-none retract xs = begin

+
    mapMaybe proj (map emb xs)  ≡⟨ mapMaybe-map _ _ xs ⟩

+
    mapMaybe (proj ∘ emb) xs    ≡⟨ mapMaybe-cong retract xs ⟩

+
    mapMaybe (const nothing) xs ≡⟨ mapMaybe-nothing xs ⟩

+
    []                          ∎

+

+
-- embedding-projection pairs on sums

+
mapMaybeIsInj₁∘mapInj₁ : (xs : List A) → mapMaybe (isInj₁ {B = B}) (map inj₁ xs) ≡ xs

+
mapMaybeIsInj₁∘mapInj₁ = mapMaybe-map-retract λ _ → refl

+

+
mapMaybeIsInj₁∘mapInj₂ : (xs : List B) → mapMaybe (isInj₁ {A = A}) (map inj₂ xs) ≡ []

+
mapMaybeIsInj₁∘mapInj₂ = mapMaybe-map-none λ _ → refl

+

+
mapMaybeIsInj₂∘mapInj₂ : (xs : List B) → mapMaybe (isInj₂ {A = A}) (map inj₂ xs) ≡ xs

+
mapMaybeIsInj₂∘mapInj₂ = mapMaybe-map-retract λ _ → refl

+

+
mapMaybeIsInj₂∘mapInj₁ : (xs : List A) → mapMaybe (isInj₂ {B = B}) (map inj₁ xs) ≡ []

+
mapMaybeIsInj₂∘mapInj₁ = mapMaybe-map-none λ _ → refl

+

+
------------------------------------------------------------------------

+
-- sum

+

+
sum-++ : ∀ xs ys → sum (xs ++ ys) ≡ sum xs + sum ys

+
sum-++ []       ys = refl

+
sum-++ (x ∷ xs) ys = begin

+
  x + sum (xs ++ ys)     ≡⟨ cong (x +_) (sum-++ xs ys) ⟩

+
  x + (sum xs + sum ys)  ≡⟨ sym (+-assoc x _ _) ⟩

+
  (x + sum xs) + sum ys  ∎

+

+
------------------------------------------------------------------------

+
-- product

+

+
∈⇒∣product : ∀ {n ns} → n ∈ ns → n ∣ product ns

+
∈⇒∣product {n} {n ∷ ns} (here  refl) = divides (product ns) (*-comm n (product ns))

+
∈⇒∣product {n} {m ∷ ns} (there n∈ns) = ∣n⇒∣m*n m (∈⇒∣product n∈ns)

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
length-applyUpTo : ∀ (f : ℕ → A) n → length (applyUpTo f n) ≡ n

+
length-applyUpTo f zero    = refl

+
length-applyUpTo f (suc n) = cong suc (length-applyUpTo (f ∘ suc) n)

+

+
lookup-applyUpTo : ∀ (f : ℕ → A) n i → lookup (applyUpTo f n) i ≡ f (toℕ i)

+
lookup-applyUpTo f (suc n) zero    = refl

+
lookup-applyUpTo f (suc n) (suc i) = lookup-applyUpTo (f ∘ suc) n i

+

+
applyUpTo-∷ʳ : ∀ (f : ℕ → A) n → applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)

+
applyUpTo-∷ʳ f zero = refl

+
applyUpTo-∷ʳ f (suc n) = cong (f 0 ∷_) (applyUpTo-∷ʳ (f ∘ suc) n)

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ (f : ℕ → A) where

+

+
  length-applyDownFrom : ∀ n → length (applyDownFrom f n) ≡ n

+
  length-applyDownFrom zero    = refl

+
  length-applyDownFrom (suc n) = cong suc (length-applyDownFrom n)

+

+
  lookup-applyDownFrom : ∀ n i → lookup (applyDownFrom f n) i ≡ f (n ∸ (suc (toℕ i)))

+
  lookup-applyDownFrom (suc n) zero    = refl

+
  lookup-applyDownFrom (suc n) (suc i) = lookup-applyDownFrom n i

+

+
  applyDownFrom-∷ʳ : ∀ n → applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)

+
  applyDownFrom-∷ʳ zero = refl

+
  applyDownFrom-∷ʳ (suc n) = cong (f (suc n) ∷_) (applyDownFrom-∷ʳ n)

+

+
------------------------------------------------------------------------

+
-- upTo

+

+
length-upTo : ∀ n → length (upTo n) ≡ n

+
length-upTo = length-applyUpTo id

+

+
lookup-upTo : ∀ n i → lookup (upTo n) i ≡ toℕ i

+
lookup-upTo = lookup-applyUpTo id

+

+
upTo-∷ʳ : ∀ n → upTo n ∷ʳ n ≡ upTo (suc n)

+
upTo-∷ʳ = applyUpTo-∷ʳ id

+

+
------------------------------------------------------------------------

+
-- downFrom

+

+
length-downFrom : ∀ n → length (downFrom n) ≡ n

+
length-downFrom = length-applyDownFrom id

+

+
lookup-downFrom : ∀ n i → lookup (downFrom n) i ≡ n ∸ (suc (toℕ i))

+
lookup-downFrom = lookup-applyDownFrom id

+

+
downFrom-∷ʳ : ∀ n → applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)

+
downFrom-∷ʳ = applyDownFrom-∷ʳ id

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
tabulate-cong : ∀ {n} {f g : Fin n → A} →

+
                f ≗ g → tabulate f ≡ tabulate g

+
tabulate-cong {n = zero}  p = refl

+
tabulate-cong {n = suc n} p = cong₂ _∷_ (p zero) (tabulate-cong (p ∘ suc))

+

+
tabulate-lookup : ∀ (xs : List A) → tabulate (lookup xs) ≡ xs

+
tabulate-lookup []       = refl

+
tabulate-lookup (x ∷ xs) = cong (_ ∷_) (tabulate-lookup xs)

+

+
length-tabulate : ∀ {n} → (f : Fin n → A) →

+
                  length (tabulate f) ≡ n

+
length-tabulate {n = zero} f = refl

+
length-tabulate {n = suc n} f = cong suc (length-tabulate (λ z → f (suc z)))

+

+
lookup-tabulate : ∀ {n} → (f : Fin n → A) →

+
                  ∀ i → let i′ = cast (sym (length-tabulate f)) i

+
                        in lookup (tabulate f) i′ ≡ f i

+
lookup-tabulate f zero    = refl

+
lookup-tabulate f (suc i) = lookup-tabulate (f ∘ suc) i

+

+
map-tabulate : ∀ {n} (g : Fin n → A) (f : A → B) →

+
               map f (tabulate g) ≡ tabulate (f ∘ g)

+
map-tabulate {n = zero}  g f = refl

+
map-tabulate {n = suc n} g f = cong (_ ∷_) (map-tabulate (g ∘ suc) f)

+

+
------------------------------------------------------------------------

+
-- _[_]%=_

+

+
length-%= : ∀ xs k (f : A → A) → length (xs [ k ]%= f) ≡ length xs

+
length-%= (x ∷ xs) zero    f = refl

+
length-%= (x ∷ xs) (suc k) f = cong suc (length-%= xs k f)

+

+
------------------------------------------------------------------------

+
-- _[_]∷=_

+

+
length-∷= : ∀ xs k (v : A) → length (xs [ k ]∷= v) ≡ length xs

+
length-∷= xs k v = length-%= xs k (const v)

+

+
map-∷= : ∀ xs k (v : A) (f : A → B) →

+
         let eq = sym (length-map f xs) in

+
         map f (xs [ k ]∷= v) ≡ map f xs [ cast eq k ]∷= f v

+
map-∷= (x ∷ xs) zero    v f = refl

+
map-∷= (x ∷ xs) (suc k) v f = cong (f x ∷_) (map-∷= xs k v f)

+

+
------------------------------------------------------------------------

+
-- insertAt

+

+
length-insertAt : ∀ (xs : List A) (i : Fin (suc (length xs))) v →

+
                  length (insertAt xs i v) ≡ suc (length xs)

+
length-insertAt xs       zero    v = refl

+
length-insertAt (x ∷ xs) (suc i) v = cong suc (length-insertAt xs i v)

+

+
------------------------------------------------------------------------

+
-- removeAt

+

+
length-removeAt : ∀ (xs : List A) k → length (removeAt xs k) ≡ pred (length xs)

+
length-removeAt (x ∷ xs) zero            = refl

+
length-removeAt (x ∷ xs@(_ ∷ _)) (suc k) = cong suc (length-removeAt xs k)

+

+
length-removeAt′ : ∀ (xs : List A) k → length xs ≡ suc (length (removeAt xs k))

+
length-removeAt′ xs@(_ ∷ _) k rewrite length-removeAt xs k = refl

+

+
map-removeAt : ∀ xs k (f : A → B) →

+
            let eq = sym (length-map f xs) in

+
            map f (removeAt xs k) ≡ removeAt (map f xs) (cast eq k)

+
map-removeAt (x ∷ xs) zero    f = refl

+
map-removeAt (x ∷ xs) (suc k) f = cong (f x ∷_) (map-removeAt xs k f)

+

+
------------------------------------------------------------------------

+
 -- insertAt and removeAt

+

+
removeAt-insertAt : ∀ (xs : List A) (i : Fin (suc (length xs))) v →

+
  removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs

+
removeAt-insertAt xs       zero    v = refl

+
removeAt-insertAt (x ∷ xs) (suc i) v = cong (_ ∷_) (removeAt-insertAt xs i v)

+

+
insertAt-removeAt : (xs : List A) (i : Fin (length xs)) →

+
  insertAt (removeAt xs i) (cast (length-removeAt′ xs i) i) (lookup xs i) ≡ xs

+
insertAt-removeAt (x ∷ xs) zero    = refl

+
insertAt-removeAt (x ∷ xs) (suc i) = cong (x ∷_) (insertAt-removeAt xs i)

+

+
------------------------------------------------------------------------

+
-- take

+

+
length-take : ∀ n (xs : List A) → length (take n xs) ≡ n ⊓ (length xs)

+
length-take zero    xs       = refl

+
length-take (suc n) []       = refl

+
length-take (suc n) (x ∷ xs) = cong suc (length-take n xs)

+

+
-- Take commutes with map.

+
take-map : ∀ {f : A → B} (n : ℕ) xs → take n (map f xs) ≡ map f (take n xs)

+
take-map zero xs = refl

+
take-map (suc s) [] = refl

+
take-map (suc s) (a ∷ xs) = cong (_ ∷_) (take-map s xs)

+

+
take-suc : (xs : List A) (i : Fin (length xs)) → let m = toℕ i in

+
           take (suc m) xs ≡ take m xs ∷ʳ lookup xs i

+
take-suc (x ∷ xs) zero    = refl

+
take-suc (x ∷ xs) (suc i) = cong (x ∷_) (take-suc xs i)

+

+
take-suc-tabulate : ∀ {n} (f : Fin n → A) (i : Fin n) → let m = toℕ i in

+
                    take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f i

+
take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym (lookup-tabulate f i)

+
  = take-suc (tabulate f) (cast _ i)

+

+
-- If you take at least as many elements from a list as it has, you get

+
-- the whole list.

+
take-all : (n : ℕ) (xs : List A) → n ≥ length xs → take n xs ≡ xs

+
take-all zero [] _ = refl

+
take-all (suc _) [] _ = refl

+
take-all (suc n) (x ∷ xs) (s≤s pf) = cong (x ∷_) (take-all n xs pf)

+

+
-- Taking from an empty list does nothing.

+
take-[] : ∀ m → take {A = A} m [] ≡ []

+
take-[] zero = refl

+
take-[] (suc m) = refl

+

+
-- Taking twice takes the minimum of both counts.

+
take-take : ∀ n m (xs : List A) → take n (take m xs) ≡ take (n ⊓ m) xs

+
take-take zero    m       xs       = refl

+
take-take (suc n) zero    xs       = refl

+
take-take (suc n) (suc m) []       = refl

+
take-take (suc n) (suc m) (x ∷ xs) = cong (x ∷_) (take-take n m xs)

+

+
-- Dropping m elements and then taking n is the same as

+
-- taking n + m elements and then dropping m.

+
take-drop : ∀ n m (xs : List A) →

+
            take n (drop m xs) ≡ drop m (take (m + n) xs)

+
take-drop n zero    xs       = refl

+
take-drop n (suc m) []       = take-[] n

+
take-drop n (suc m) (x ∷ xs) = take-drop n m xs

+

+
------------------------------------------------------------------------

+
-- drop

+

+
length-drop : ∀ n (xs : List A) → length (drop n xs) ≡ length xs ∸ n

+
length-drop zero    xs       = refl

+
length-drop (suc n) []       = refl

+
length-drop (suc n) (x ∷ xs) = length-drop n xs

+

+
-- Drop commutes with map.

+
drop-map : ∀ {f : A → B} (n : ℕ) xs → drop n (map f xs) ≡ map f (drop n xs)

+
drop-map zero xs = refl

+
drop-map (suc n) [] = refl

+
drop-map (suc n) (a ∷ xs) = drop-map n xs

+

+
-- Dropping from an empty list does nothing.

+
drop-[] : ∀ m → drop {A = A} m [] ≡ []

+
drop-[] zero = refl

+
drop-[] (suc m) = refl

+

+
take++drop≡id : ∀ n (xs : List A) → take n xs ++ drop n xs ≡ xs

+
take++drop≡id zero    xs       = refl

+
take++drop≡id (suc n) []       = refl

+
take++drop≡id (suc n) (x ∷ xs) = cong (x ∷_) (take++drop≡id n xs)

+

+
drop-take-suc : (xs : List A) (i : Fin (length xs)) → let m = toℕ i in

+
           drop m (take (suc m) xs) ≡ [ lookup xs i ]

+
drop-take-suc (x ∷ xs) zero    = refl

+
drop-take-suc (x ∷ xs) (suc i) = drop-take-suc xs i

+

+
drop-take-suc-tabulate : ∀ {n} (f : Fin n → A) (i : Fin n) → let m = toℕ i in

+
                  drop m (take (suc m) (tabulate f)) ≡ [ f i ]

+
drop-take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym (lookup-tabulate f i)

+
  = drop-take-suc (tabulate f) (cast _ i)

+

+
-- Dropping m elements and then n elements is same as dropping m+n elements

+
drop-drop : (m n : ℕ) → (xs : List A) → drop n (drop m xs) ≡ drop (m + n) xs

+
drop-drop zero n xs = refl

+
drop-drop (suc m) n [] = drop-[] n

+
drop-drop (suc m) n (x ∷ xs) = drop-drop m n xs

+

+
drop-all : (n : ℕ) (xs : List A) → n ≥ length xs → drop n xs ≡ []

+
drop-all n       []       _ = drop-[] n

+
drop-all (suc n) (x ∷ xs) p = drop-all n xs (s≤s⁻¹ p)

+

+
------------------------------------------------------------------------

+
-- replicate

+

+
length-replicate : ∀ n {x : A} → length (replicate n x) ≡ n

+
length-replicate zero    = refl

+
length-replicate (suc n) = cong suc (length-replicate n)

+

+
lookup-replicate : ∀ n (x : A) (i : Fin n) →

+
                   lookup (replicate n x) (cast (sym (length-replicate n)) i) ≡ x

+
lookup-replicate (suc n) x zero    = refl

+
lookup-replicate (suc n) x (suc i) = lookup-replicate n x i

+

+
map-replicate :  ∀ (f : A → B) n (x : A) →

+
                 map f (replicate n x) ≡ replicate n (f x)

+
map-replicate f zero    x = refl

+
map-replicate f (suc n) x = cong (_ ∷_) (map-replicate f n x)

+

+
zipWith-replicate : ∀ n (_⊕_ : A → B → C) (x : A) (y : B) →

+
                    zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)

+
zipWith-replicate zero    _⊕_ x y = refl

+
zipWith-replicate (suc n) _⊕_ x y = cong (x ⊕ y ∷_) (zipWith-replicate n _⊕_ x y)

+

+
------------------------------------------------------------------------

+
-- iterate

+

+
length-iterate : ∀ f (x : A) n → length (iterate f x n) ≡ n

+
length-iterate f x zero    = refl

+
length-iterate f x (suc n) = cong suc (length-iterate f (f x) n)

+

+
iterate-id : ∀ (x : A) n → iterate id x n ≡ replicate n x

+
iterate-id x zero    = refl

+
iterate-id x (suc n) = cong (_ ∷_) (iterate-id x n)

+

+
lookup-iterate : ∀ f (x : A) n (i : Fin n) →

+
  lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)

+
lookup-iterate f x (suc n) zero    = refl

+
lookup-iterate f x (suc n) (suc i) = lookup-iterate f (f x) n i

+

+
------------------------------------------------------------------------

+
-- splitAt

+

+
splitAt-defn : ∀ n → splitAt {A = A} n ≗ < take n , drop n >

+
splitAt-defn zero    xs       = refl

+
splitAt-defn (suc n) []       = refl

+
splitAt-defn (suc n) (x ∷ xs) = cong (Product.map (x ∷_) id) (splitAt-defn n xs)

+

+
module _ (f : A → B) where

+

+
  splitAt-map : ∀ n → splitAt n ∘ map f ≗

+
                      Product.map (map f) (map f) ∘ splitAt n

+
  splitAt-map zero    xs       = refl

+
  splitAt-map (suc n) []       = refl

+
  splitAt-map (suc n) (x ∷ xs) =

+
    cong (Product.map₁ (f x ∷_)) (splitAt-map n xs)

+

+
------------------------------------------------------------------------

+
-- takeWhile, dropWhile, and span

+

+
module _ {P : Pred A p} (P? : Decidable P) where

+

+
  takeWhile++dropWhile : ∀ xs → takeWhile P? xs ++ dropWhile P? xs ≡ xs

+
  takeWhile++dropWhile []       = refl

+
  takeWhile++dropWhile (x ∷ xs) with does (P? x)

+
  ... | true  = cong (x ∷_) (takeWhile++dropWhile xs)

+
  ... | false = refl

+

+
  span-defn : span P? ≗ < takeWhile P? , dropWhile P? >

+
  span-defn []       = refl

+
  span-defn (x ∷ xs) with does (P? x)

+
  ... | true  = cong (Product.map (x ∷_) id) (span-defn xs)

+
  ... | false = refl

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {P : Pred A p} (P? : Decidable P) where

+

+
  length-filter : ∀ xs → length (filter P? xs) ≤ length xs

+
  length-filter []       = z≤n

+
  length-filter (x ∷ xs) with ih ← length-filter xs | does (P? x)

+
  ... | false = m≤n⇒m≤1+n ih

+
  ... | true  = s≤s ih

+

+
  filter-all : ∀ {xs} → All P xs → filter P? xs ≡ xs

+
  filter-all {[]}     []         = refl

+
  filter-all {x ∷ xs} (px ∷ pxs) with P? x

+
  ... | no          ¬px = contradiction px ¬px

+
  ... | true  because _ = cong (x ∷_) (filter-all pxs)

+

+
  filter-notAll : ∀ xs → Any (∁ P) xs → length (filter P? xs) < length xs

+
  filter-notAll (x ∷ xs) (here ¬px) with P? x

+
  ... | false because _ = s≤s (length-filter xs)

+
  ... | yes          px = contradiction px ¬px

+
  filter-notAll (x ∷ xs) (there any) with ih ← filter-notAll xs any | does (P? x)

+
  ... | false = m≤n⇒m≤1+n ih

+
  ... | true  = s≤s ih

+

+
  filter-some : ∀ {xs} → Any P xs → 0 < length (filter P? xs)

+
  filter-some {x ∷ xs} (here px)   with P? x

+
  ... | true because _ = z<s

+
  ... | no         ¬px = contradiction px ¬px

+
  filter-some {x ∷ xs} (there pxs) with does (P? x)

+
  ... | true  = m≤n⇒m≤1+n (filter-some pxs)

+
  ... | false = filter-some pxs

+

+
  filter-none : ∀ {xs} → All (∁ P) xs → filter P? xs ≡ []

+
  filter-none {[]}     []           = refl

+
  filter-none {x ∷ xs} (¬px ∷ ¬pxs) with P? x

+
  ... | false because _ = filter-none ¬pxs

+
  ... | yes          px = contradiction px ¬px

+

+
  filter-complete : ∀ {xs} → length (filter P? xs) ≡ length xs →

+
                    filter P? xs ≡ xs

+
  filter-complete {[]}     eq = refl

+
  filter-complete {x ∷ xs} eq with does (P? x)

+
  ... | false = contradiction eq (<⇒≢ (s≤s (length-filter xs)))

+
  ... | true  = cong (x ∷_) (filter-complete (suc-injective eq))

+

+
  filter-accept : ∀ {x xs} → P x → filter P? (x ∷ xs) ≡ x ∷ (filter P? xs)

+
  filter-accept {x} Px with P? x

+
  ... | true because _ = refl

+
  ... | no         ¬Px = contradiction Px ¬Px

+

+
  filter-reject : ∀ {x xs} → ¬ P x → filter P? (x ∷ xs) ≡ filter P? xs

+
  filter-reject {x} ¬Px with P? x

+
  ... | yes          Px = contradiction Px ¬Px

+
  ... | false because _ = refl

+

+
  filter-idem : filter P? ∘ filter P? ≗ filter P?

+
  filter-idem []       = refl

+
  filter-idem (x ∷ xs) with does (P? x) in eq

+
  ... | false            = filter-idem xs

+
  ... | true  rewrite eq = cong (x ∷_) (filter-idem xs)

+

+
  filter-++ : ∀ xs ys → filter P? (xs ++ ys) ≡ filter P? xs ++ filter P? ys

+
  filter-++ []       ys = refl

+
  filter-++ (x ∷ xs) ys with ih ← filter-++ xs ys | does (P? x)

+
  ... | true  = cong (x ∷_) ih

+
  ... | false = ih

+

+
------------------------------------------------------------------------

+
-- derun and deduplicate

+

+
module _ {R : Rel A p} (R? : B.Decidable R) where

+

+
  length-derun : ∀ xs → length (derun R? xs) ≤ length xs

+
  length-derun [] = ≤-refl

+
  length-derun (x ∷ []) = ≤-refl

+
  length-derun (x ∷ y ∷ xs) with ih ← length-derun (y ∷ xs) | does (R? x y)

+
  ... | true  = m≤n⇒m≤1+n ih

+
  ... | false = s≤s ih

+

+
  length-deduplicate : ∀ xs → length (deduplicate R? xs) ≤ length xs

+
  length-deduplicate [] = z≤n

+
  length-deduplicate (x ∷ xs) = ≤-begin

+
    1 + length (filter (¬? ∘ R? x) r) ≤⟨ s≤s (length-filter (¬? ∘ R? x) r) ⟩

+
    1 + length r                      ≤⟨ s≤s (length-deduplicate xs) ⟩

+
    1 + length xs                     ≤-∎

+
    where

+
    open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎)

+
    r = deduplicate R? xs

+

+
  derun-reject : ∀ {x y} xs → R x y → derun R? (x ∷ y ∷ xs) ≡ derun R? (y ∷ xs)

+
  derun-reject {x} {y} xs Rxy with R? x y

+
  ... | yes _    = refl

+
  ... | no  ¬Rxy = contradiction Rxy ¬Rxy

+

+
  derun-accept : ∀ {x y} xs → ¬ R x y → derun R? (x ∷ y ∷ xs) ≡ x ∷ derun R? (y ∷ xs)

+
  derun-accept {x} {y} xs ¬Rxy with R? x y

+
  ... | yes Rxy = contradiction Rxy ¬Rxy

+
  ... | no  _   = refl

+

+
------------------------------------------------------------------------

+
-- partition

+

+
module _ {P : Pred A p} (P? : Decidable P) where

+

+
  partition-defn : partition P? ≗ < filter P? , filter (∁? P?) >

+
  partition-defn []       = refl

+
  partition-defn (x ∷ xs) with ih ← partition-defn xs | does (P? x)

+
  ...  | true  = cong (Product.map (x ∷_) id) ih

+
  ...  | false = cong (Product.map id (x ∷_)) ih

+

+
  length-partition : ∀ xs → (let (ys , zs) = partition P? xs) →

+
                     length ys ≤ length xs × length zs ≤ length xs

+
  length-partition []       = z≤n , z≤n

+
  length-partition (x ∷ xs) with ih ← length-partition xs | does (P? x)

+
  ...  | true  = Product.map s≤s m≤n⇒m≤1+n ih

+
  ...  | false = Product.map m≤n⇒m≤1+n s≤s ih

+

+
------------------------------------------------------------------------

+
-- _ʳ++_

+

+
ʳ++-defn : ∀ (xs : List A) {ys} → xs ʳ++ ys ≡ reverse xs ++ ys

+
ʳ++-defn [] = refl

+
ʳ++-defn (x ∷ xs) {ys} = begin

+
  (x ∷ xs)             ʳ++ ys   ≡⟨⟩

+
  xs         ʳ++   x     ∷ ys   ≡⟨⟩

+
  xs         ʳ++ [ x ]  ++ ys   ≡⟨ ʳ++-defn xs  ⟩

+
  reverse xs  ++ [ x ]  ++ ys   ≡⟨ sym (++-assoc (reverse xs) _ _) ⟩

+
  (reverse xs ++ [ x ]) ++ ys   ≡⟨ cong (_++ ys) (sym (ʳ++-defn xs)) ⟩

+
  (xs ʳ++ [ x ])        ++ ys   ≡⟨⟩

+
  reverse (x ∷ xs)      ++ ys   ∎

+

+
-- Reverse-append of append is reverse-append after reverse-append.

+

+
++-ʳ++ : ∀ (xs {ys zs} : List A) → (xs ++ ys) ʳ++ zs ≡ ys ʳ++ xs ʳ++ zs

+
++-ʳ++ [] = refl

+
++-ʳ++ (x ∷ xs) {ys} {zs} = begin

+
  (x ∷ xs ++ ys) ʳ++ zs   ≡⟨⟩

+
  (xs ++ ys) ʳ++ x ∷ zs   ≡⟨ ++-ʳ++ xs ⟩

+
  ys ʳ++ xs ʳ++ x ∷ zs    ≡⟨⟩

+
  ys ʳ++ (x ∷ xs) ʳ++ zs  ∎

+

+
-- Reverse-append of reverse-append is commuted reverse-append after append.

+

+
ʳ++-ʳ++ : ∀ (xs {ys zs} : List A) → (xs ʳ++ ys) ʳ++ zs ≡ ys ʳ++ xs ++ zs

+
ʳ++-ʳ++ [] = refl

+
ʳ++-ʳ++ (x ∷ xs) {ys} {zs} = begin

+
  ((x ∷ xs) ʳ++ ys) ʳ++ zs   ≡⟨⟩

+
  (xs ʳ++ x ∷ ys) ʳ++ zs     ≡⟨ ʳ++-ʳ++ xs ⟩

+
  (x ∷ ys) ʳ++ xs ++ zs      ≡⟨⟩

+
  ys ʳ++ (x ∷ xs) ++ zs      ∎

+

+
-- Length of reverse-append

+

+
length-ʳ++ : ∀ (xs {ys} : List A) →

+
             length (xs ʳ++ ys) ≡ length xs + length ys

+
length-ʳ++ [] = refl

+
length-ʳ++ (x ∷ xs) {ys} = begin

+
  length ((x ∷ xs) ʳ++ ys)      ≡⟨⟩

+
  length (xs ʳ++ x ∷ ys)        ≡⟨ length-ʳ++ xs ⟩

+
  length xs + length (x ∷ ys)   ≡⟨ +-suc _ _ ⟩

+
  length (x ∷ xs) + length ys   ∎

+

+
-- map distributes over reverse-append.

+

+
map-ʳ++ : (f : A → B) (xs {ys} : List A) →

+
          map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys

+
map-ʳ++ f []            = refl

+
map-ʳ++ f (x ∷ xs) {ys} = begin

+
  map f ((x ∷ xs) ʳ++ ys)         ≡⟨⟩

+
  map f (xs ʳ++ x ∷ ys)           ≡⟨ map-ʳ++ f xs ⟩

+
  map f xs ʳ++ map f (x ∷ ys)     ≡⟨⟩

+
  map f xs ʳ++ f x ∷ map f ys     ≡⟨⟩

+
  (f x ∷ map f xs) ʳ++ map f ys   ≡⟨⟩

+
  map f (x ∷ xs)   ʳ++ map f ys   ∎

+

+
-- A foldr after a reverse is a foldl.

+

+
foldr-ʳ++ : ∀ (f : A → B → B) b xs {ys} →

+
            foldr f b (xs ʳ++ ys) ≡ foldl (flip f) (foldr f b ys) xs

+
foldr-ʳ++ f b []       {_}  = refl

+
foldr-ʳ++ f b (x ∷ xs) {ys} = begin

+
  foldr f b ((x ∷ xs) ʳ++ ys)              ≡⟨⟩

+
  foldr f b (xs ʳ++ x ∷ ys)                ≡⟨ foldr-ʳ++ f b xs ⟩

+
  foldl (flip f) (foldr f b (x ∷ ys)) xs   ≡⟨⟩

+
  foldl (flip f) (f x (foldr f b ys)) xs   ≡⟨⟩

+
  foldl (flip f) (foldr f b ys) (x ∷ xs)   ∎

+

+
-- A foldl after a reverse is a foldr.

+

+
foldl-ʳ++ : ∀ (f : B → A → B) b xs {ys} →

+
            foldl f b (xs ʳ++ ys) ≡ foldl f (foldr (flip f) b xs) ys

+
foldl-ʳ++ f b []       {_}  = refl

+
foldl-ʳ++ f b (x ∷ xs) {ys} = begin

+
  foldl f b ((x ∷ xs) ʳ++ ys)              ≡⟨⟩

+
  foldl f b (xs ʳ++ x ∷ ys)                ≡⟨ foldl-ʳ++ f b xs ⟩

+
  foldl f (foldr (flip f) b xs) (x ∷ ys)   ≡⟨⟩

+
  foldl f (f (foldr (flip f) b xs) x) ys   ≡⟨⟩

+
  foldl f (foldr (flip f) b (x ∷ xs)) ys   ∎

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
-- reverse of cons is snoc of reverse.

+

+
unfold-reverse : ∀ (x : A) xs → reverse (x ∷ xs) ≡ reverse xs ∷ʳ x

+
unfold-reverse x xs = ʳ++-defn xs

+

+
-- reverse is an anti-homomorphism with respect to append.

+

+
reverse-++ : (xs ys : List A) →

+
                     reverse (xs ++ ys) ≡ reverse ys ++ reverse xs

+
reverse-++ xs ys = begin

+
  reverse (xs ++ ys)         ≡⟨⟩

+
  (xs ++ ys) ʳ++ []          ≡⟨ ++-ʳ++ xs ⟩

+
  ys ʳ++ xs ʳ++ []           ≡⟨⟩

+
  ys ʳ++ reverse xs          ≡⟨ ʳ++-defn ys ⟩

+
  reverse ys ++ reverse xs   ∎

+

+
-- reverse is self-inverse.

+

+
reverse-selfInverse : SelfInverse {A = List A} _≡_ reverse

+
reverse-selfInverse {x = xs} {y = ys} xs⁻¹≈ys = begin

+
  reverse ys         ≡⟨⟩

+
  ys ʳ++ []          ≡⟨ cong (_ʳ++ []) xs⁻¹≈ys ⟨

+
  reverse xs ʳ++ []  ≡⟨⟩

+
  (xs ʳ++ []) ʳ++ [] ≡⟨ ʳ++-ʳ++ xs ⟩

+
  [] ʳ++ xs ++ []    ≡⟨⟩

+
  xs ++ []           ≡⟨ ++-identityʳ xs ⟩

+
  xs                 ∎

+

+
-- reverse is involutive.

+

+
reverse-involutive : Involutive {A = List A} _≡_ reverse

+
reverse-involutive = selfInverse⇒involutive reverse-selfInverse

+

+
-- reverse is injective.

+

+
reverse-injective : Injective {A = List A} _≡_ _≡_ reverse

+
reverse-injective = selfInverse⇒injective reverse-selfInverse

+

+
-- reverse preserves length.

+

+
length-reverse : ∀ (xs : List A) → length (reverse xs) ≡ length xs

+
length-reverse xs = begin

+
  length (reverse xs)   ≡⟨⟩

+
  length (xs ʳ++ [])    ≡⟨ length-ʳ++ xs ⟩

+
  length xs + 0         ≡⟨ +-identityʳ _ ⟩

+
  length xs             ∎

+

+
reverse-map : (f : A → B) → map f ∘ reverse ≗ reverse ∘ map f

+
reverse-map f xs = begin

+
  map f (reverse xs) ≡⟨⟩

+
  map f (xs ʳ++ [])  ≡⟨ map-ʳ++ f xs ⟩

+
  map f xs ʳ++ []    ≡⟨⟩

+
  reverse (map f xs) ∎

+

+
reverse-foldr : ∀ (f : A → B → B) b →

+
                foldr f b ∘ reverse ≗ foldl (flip f) b

+
reverse-foldr f b xs = foldr-ʳ++ f b xs

+

+
reverse-foldl : ∀ (f : B → A → B) b xs →

+
                foldl f b (reverse xs) ≡ foldr (flip f) b xs

+
reverse-foldl f b xs = foldl-ʳ++ f b xs

+

+
------------------------------------------------------------------------

+
-- reverse, applyUpTo, and applyDownFrom

+

+
reverse-applyUpTo : ∀ (f : ℕ → A) n → reverse (applyUpTo f n) ≡ applyDownFrom f n

+
reverse-applyUpTo f zero = refl

+
reverse-applyUpTo f (suc n) = begin

+
  reverse (f 0 ∷ applyUpTo (f ∘ suc) n)  ≡⟨ reverse-++ [ f 0 ] (applyUpTo (f ∘ suc) n) ⟩

+
  reverse (applyUpTo (f ∘ suc) n) ∷ʳ f 0 ≡⟨ cong (_∷ʳ f 0) (reverse-applyUpTo (f ∘ suc) n) ⟩

+
  applyDownFrom (f ∘ suc) n ∷ʳ f 0       ≡⟨ applyDownFrom-∷ʳ f n ⟩

+
  applyDownFrom f (suc n)                ∎

+

+
reverse-upTo : ∀ n → reverse (upTo n) ≡ downFrom n

+
reverse-upTo = reverse-applyUpTo id

+

+
reverse-applyDownFrom : ∀ (f : ℕ → A) n → reverse (applyDownFrom f n) ≡ applyUpTo f n

+
reverse-applyDownFrom f zero = refl

+
reverse-applyDownFrom f (suc n) = begin

+
  reverse (f n ∷ applyDownFrom f n)  ≡⟨ reverse-++ [ f n ] (applyDownFrom f n) ⟩

+
  reverse (applyDownFrom f n) ∷ʳ f n ≡⟨ cong (_∷ʳ f n) (reverse-applyDownFrom f n) ⟩

+
  applyUpTo f n ∷ʳ f n               ≡⟨ applyUpTo-∷ʳ f n ⟩

+
  applyUpTo f (suc n)                ∎

+

+
reverse-downFrom : ∀ n → reverse (downFrom n) ≡ upTo n

+
reverse-downFrom = reverse-applyDownFrom id

+

+
------------------------------------------------------------------------

+
-- _∷ʳ_

+

+
∷ʳ-injective : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y

+
∷ʳ-injective []          []          refl = refl , refl

+
∷ʳ-injective (x ∷ xs)    (y  ∷ ys)   eq   with refl , eq′  ← ∷-injective eq

+
  = Product.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)

+
∷ʳ-injective []          (_ ∷ _ ∷ _) ()

+
∷ʳ-injective (_ ∷ _ ∷ _) []          ()

+

+
∷ʳ-injectiveˡ : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys

+
∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq)

+

+
∷ʳ-injectiveʳ : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y

+
∷ʳ-injectiveʳ xs ys eq = proj₂ (∷ʳ-injective xs ys eq)

+

+
∷ʳ-++ : ∀ xs (a : A) ys → xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys

+
∷ʳ-++ xs a ys = ++-assoc xs [ a ] ys

+

+
------------------------------------------------------------------------

+
-- uncons

+

+
module _ (f : A → B) where

+

+
  -- 'commute' List.uncons and List.map to obtain a Maybe.map and List.uncons.

+
  uncons-map : uncons ∘ map f ≗ Maybe.map (Product.map f (map f)) ∘ uncons

+
  uncons-map []       = refl

+
  uncons-map (x ∷ xs) = refl

+

+
------------------------------------------------------------------------

+
-- head

+

+
module _ {f : A → B} where

+

+
  -- 'commute' List.head and List.map to obtain a Maybe.map and List.head.

+
  head-map : head ∘ map f ≗ Maybe.map f ∘ head

+
  head-map []      = refl

+
  head-map (_ ∷ _) = refl

+

+
------------------------------------------------------------------------

+
-- last

+

+
module _ (f : A → B) where

+

+
  -- 'commute' List.last and List.map to obtain a Maybe.map and List.last.

+
  last-map : last ∘ map f ≗ Maybe.map f ∘ last

+
  last-map []               = refl

+
  last-map (x ∷ [])         = refl

+
  last-map (x ∷ xs@(_ ∷ _)) = last-map xs

+

+
------------------------------------------------------------------------

+
-- tail

+

+
module _ (f : A → B) where

+

+
  -- 'commute' List.tail and List.map to obtain a Maybe.map and List.tail.

+
  tail-map : tail ∘ map f ≗ Maybe.map (map f) ∘ tail

+
  tail-map []       = refl

+
  tail-map (x ∷ xs) = refl

+

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
map-id₂ = map-id-local

+
{-# WARNING_ON_USAGE map-id₂

+
"Warning: map-id₂ was deprecated in v2.0.

+
Please use map-id-local instead."

+
#-}

+

+
map-cong₂ = map-cong-local

+
{-# WARNING_ON_USAGE map-id₂

+
"Warning: map-cong₂ was deprecated in v2.0.

+
Please use map-cong-local instead."

+
#-}

+

+
map-compose = map-∘

+
{-# WARNING_ON_USAGE map-compose

+
"Warning: map-compose was deprecated in v2.0.

+
Please use map-∘ instead."

+
#-}

+

+
map-++-commute = map-++

+
{-# WARNING_ON_USAGE map-++-commute

+
"Warning: map-++-commute was deprecated in v2.0.

+
Please use map-++ instead."

+
#-}

+

+
sum-++-commute = sum-++

+
{-# WARNING_ON_USAGE sum-++-commute

+
"Warning: map-++-commute was deprecated in v2.0.

+
Please use map-++ instead."

+
#-}

+

+
reverse-map-commute = reverse-map

+
{-# WARNING_ON_USAGE reverse-map-commute

+
"Warning: reverse-map-commute was deprecated in v2.0.

+
Please use reverse-map instead."

+
#-}

+

+
reverse-++-commute = reverse-++

+
{-# WARNING_ON_USAGE reverse-++-commute

+
"Warning: reverse-++-commute was deprecated in v2.0.

+
Please use reverse-++ instead."

+
#-}

+

+
zipWith-identityˡ = zipWith-zeroˡ

+
{-# WARNING_ON_USAGE zipWith-identityˡ

+
"Warning: zipWith-identityˡ was deprecated in v2.0.

+
Please use zipWith-zeroˡ instead."

+
#-}

+

+
zipWith-identityʳ = zipWith-zeroʳ

+
{-# WARNING_ON_USAGE zipWith-identityʳ

+
"Warning: zipWith-identityʳ was deprecated in v2.0.

+
Please use zipWith-zeroʳ instead."

+
#-}

+

+
ʳ++-++ = ++-ʳ++

+
{-# WARNING_ON_USAGE ʳ++-++

+
"Warning: ʳ++-++ was deprecated in v2.0.

+
Please use ++-ʳ++ instead."

+
#-}

+

+
take++drop = take++drop≡id

+
{-# WARNING_ON_USAGE take++drop

+
"Warning: take++drop was deprecated in v2.0.

+
Please use take++drop≡id instead."

+
#-}

+

+
length-─ = length-removeAt

+
{-# WARNING_ON_USAGE length-─

+
"Warning: length-─ was deprecated in v2.0.

+
Please use length-removeAt instead."

+
#-}

+

+
map-─ = map-removeAt

+
{-# WARNING_ON_USAGE map-─

+
"Warning: map-─ was deprecated in v2.0.

+
Please use map-removeAt instead."

+
#-}

+

+
-- Version 2.1

+

+
scanr-defn : ∀ (f : A → B → B) (e : B) →

+
             scanr f e ≗ map (foldr f e) ∘ tails

+
scanr-defn f e []             = refl

+
scanr-defn f e (x ∷ [])       = refl

+
scanr-defn f e (x ∷ xs@(_ ∷ _))

+
  with eq ← scanr-defn f e xs

+
  with ys@(_ ∷ _) ← scanr f e xs

+
  = cong₂ (λ z → f x z ∷_) (∷-injectiveˡ eq) eq

+
{-# WARNING_ON_USAGE scanr-defn

+
"Warning: scanr-defn was deprecated in v2.1.

+
Please use Data.List.Scans.Properties.scanr-defn instead."

+
#-}

+

+
scanl-defn : ∀ (f : A → B → A) (e : A) →

+
             scanl f e ≗ map (foldl f e) ∘ inits

+
scanl-defn f e []       = refl

+
scanl-defn f e (x ∷ xs) = cong (e ∷_) (begin

+
   scanl f (f e x) xs

+
 ≡⟨ scanl-defn f (f e x) xs ⟩

+
   map (foldl f (f e x)) (inits xs)

+
 ≡⟨ refl ⟩

+
   map (foldl f e ∘ (x ∷_)) (inits xs)

+
 ≡⟨ map-∘ (inits xs) ⟩

+
   map (foldl f e) (map (x ∷_) (inits xs))

+
 ∎)

+
{-# WARNING_ON_USAGE scanl-defn

+
"Warning: scanl-defn was deprecated in v2.1.

+
Please use Data.List.Scans.Properties.scanl-defn instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.BagAndSetEquality.html b/site/static/agda_html/Data.List.Relation.Binary.BagAndSetEquality.html
new file mode 100644
index 000000000..60591c75d
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.BagAndSetEquality.html
@@ -0,0 +1,681 @@
+
+Data.List.Relation.Binary.BagAndSetEquality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bag and set equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.BagAndSetEquality where

+

+
open import Algebra.Bundles using (CommutativeMonoid)

+
open import Algebra.Definitions using (Idempotent)

+
open import Algebra.Structures.Biased using (isCommutativeMonoidˡ)

+
open import Effect.Monad using (RawMonad)

+
open import Data.Empty using (⊥; ⊥-elim)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.List.Base

+
  using (List; []; _∷_; map; _++_; concat; [_]; lookup; length)

+
open import Data.List.Effectful using (monad; module Applicative; module MonadProperties)

+
import Data.List.Properties as List

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.List.Relation.Unary.Any.Properties

+
  using (∷↔; map↔; Any-cong; ++↔; concat↔; >>=↔; ++↔++; ⊎↔; ⊥↔Any[])

+
open import Data.List.Membership.Propositional using (_∈_)

+
open import Data.List.Membership.Propositional.Properties

+
  using (∈-∃++)

+
open import Data.List.Relation.Binary.Subset.Propositional.Properties

+
  using (⊆-preorder)

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
  using (_↭_; ↭-sym; refl; module PermutationReasoning)

+
open import Data.List.Relation.Binary.Permutation.Propositional.Properties

+
  using (∈-resp-↭; ∈-resp-[σ⁻¹∘σ]; ↭-sym-involutive; shift; ++-comm)

+
open import Data.Product.Base as Product using (∃; _,_; proj₁; proj₂; _×_)

+
import Data.Product.Function.Dependent.Propositional as Σ

+
open import Data.Sum.Base as Sum using (_⊎_; [_,_]′; inj₁; inj₂)

+
open import Data.Sum.Properties using (inj₂-injective; inj₁-injective)

+
open import Data.Sum.Function.Propositional using (_⊎-cong_)

+
open import Data.Unit.Polymorphic.Base using (⊤)

+
open import Function.Base using (_∘_; _$_; id; _⟨_⟩_; case_of_)

+
open import Function.Bundles using (_↔_; Inverse; Equivalence; mk↔ₛ′; mk⇔)

+
open import Function.Related.Propositional as Related

+
  using (↔⇒; ⌊_⌋; ⌊_⌋→; ⇒→; K-refl; SK-sym)

+
open import Function.Related.TypeIsomorphisms using (×-identityʳ; ∃∃↔∃∃)

+
open import Function.Properties.Inverse using (↔-sym; ↔-trans; to-from)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Definitions using (Trans)

+
open import Relation.Binary.Bundles using (Preorder; Setoid)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
import Relation.Binary.Reasoning.Preorder as ≲-Reasoning

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; _≗_; refl)

+
open import Relation.Binary.PropositionalEquality.Properties as ≡

+
  using (module ≡-Reasoning)

+
open import Relation.Binary.Reasoning.Syntax

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+
    x y : A

+
    ws xs ys zs : List A

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
open Related public using (Kind; SymmetricKind) renaming

+
  ( implication         to subset

+
  ; reverseImplication  to superset

+
  ; equivalence         to set

+
  ; injection           to subbag

+
  ; reverseInjection    to superbag

+
  ; bijection           to bag

+
  )

+

+
[_]-Order : Kind → Set a → Preorder _ _ _

+
[ k ]-Order A = Related.InducedPreorder₂ {A = A} k _∈_

+

+
[_]-Equality : SymmetricKind → Set a → Setoid _ _

+
[ k ]-Equality A = Related.InducedEquivalence₂ {A = A} k _∈_

+

+
infix 4 _∼[_]_

+

+
_∼[_]_ : ∀ {a} {A : Set a} → List A → Kind → List A → Set _

+
_∼[_]_ {A = A} xs k ys = Preorder._≲_ ([ k ]-Order A) xs ys

+

+
private

+
  module Eq  {k a} {A : Set a} = Setoid ([ k ]-Equality A)

+
  module Ord {k a} {A : Set a} = Preorder ([ k ]-Order A)

+
  open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})

+
  module MP = MonadProperties

+

+
------------------------------------------------------------------------

+
-- Bag equality implies the other relations.

+

+
bag-=⇒ : ∀ {k} → xs ∼[ bag ] ys → xs ∼[ k ] ys

+
bag-=⇒ xs≈ys = ↔⇒ xs≈ys

+

+
------------------------------------------------------------------------

+
-- "Equational" reasoning for _⊆_ along with an additional relatedness

+

+
module ⊆-Reasoning {A : Set a} where

+
  private module Base = ≲-Reasoning (⊆-preorder A)

+

+
  open Base public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)

+
    renaming (≲-go to ⊆-go)

+

+
  open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → begin x) public

+
  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ ⊆-go public

+

+
  module _ {k : Related.ForwardKind} where

+
    ∼-go : Trans _∼[ ⌊ k ⌋→ ]_ _IsRelatedTo_ _IsRelatedTo_

+
    ∼-go eq = ⊆-go (⇒→ eq)

+

+
    open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public

+

+

+
------------------------------------------------------------------------

+
-- Congruence lemmas

+
------------------------------------------------------------------------

+
-- _∷_

+

+
module _ {k} {x y : A} {xs ys} where

+

+
  ∷-cong : x ≡ y → xs ∼[ k ] ys → x ∷ xs ∼[ k ] y ∷ ys

+
  ∷-cong refl xs≈ys {y} = begin

+
    y ∈ x ∷ xs        ↔⟨ SK-sym $ ∷↔ (y ≡_) ⟩

+
    (y ≡ x ⊎ y ∈ xs)  ∼⟨ K-refl ⊎-cong xs≈ys ⟩

+
    (y ≡ x ⊎ y ∈ ys)  ↔⟨ ∷↔ (y ≡_) ⟩

+
    y ∈ x ∷ ys        ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {k} {f g : A → B} {xs ys} where

+

+
  map-cong : f ≗ g → xs ∼[ k ] ys → map f xs ∼[ k ] map g ys

+
  map-cong f≗g xs≈ys {x} = begin

+
    x ∈ map f xs            ↔⟨ SK-sym $ map↔ ⟩

+
    Any (λ y → x ≡ f y) xs  ∼⟨ Any-cong (↔⇒ ∘ helper) xs≈ys ⟩

+
    Any (λ y → x ≡ g y) ys  ↔⟨ map↔ ⟩

+
    x ∈ map g ys            ∎

+
    where

+
    open Related.EquationalReasoning

+

+
    helper : ∀ y → x ≡ f y ↔ x ≡ g y

+
    helper y = mk↔ₛ′

+
      (λ x≡fy → ≡.trans x≡fy (        f≗g y))

+
      (λ x≡gy → ≡.trans x≡gy (≡.sym $ f≗g y))

+
      (λ { ≡.refl → ≡.trans-symˡ (f≗g y) })

+
      λ { ≡.refl → ≡.trans-symʳ (f≗g y) }

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ {k} {xs₁ xs₂ ys₁ ys₂ : List A} where

+

+
  ++-cong : xs₁ ∼[ k ] xs₂ → ys₁ ∼[ k ] ys₂ →

+
            xs₁ ++ ys₁ ∼[ k ] xs₂ ++ ys₂

+
  ++-cong xs₁≈xs₂ ys₁≈ys₂ {x} = begin

+
    x ∈ xs₁ ++ ys₁       ↔⟨ SK-sym $ ++↔ ⟩

+
    (x ∈ xs₁ ⊎ x ∈ ys₁)  ∼⟨ xs₁≈xs₂ ⊎-cong ys₁≈ys₂ ⟩

+
    (x ∈ xs₂ ⊎ x ∈ ys₂)  ↔⟨ ++↔ ⟩

+
    x ∈ xs₂ ++ ys₂       ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {k} {xss yss : List (List A)} where

+

+
  concat-cong : xss ∼[ k ] yss → concat xss ∼[ k ] concat yss

+
  concat-cong xss≈yss {x} = begin

+
    x ∈ concat xss        ↔⟨ SK-sym concat↔ ⟩

+
    Any (Any (x ≡_)) xss  ∼⟨ Any-cong (λ _ → _ ∎) xss≈yss ⟩

+
    Any (Any (x ≡_)) yss  ↔⟨ concat↔ ⟩

+
    x ∈ concat yss         ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _>>=_

+

+
module _ {k} {A B : Set a} {xs ys} {f g : A → List B} where

+

+
  >>=-cong : xs ∼[ k ] ys → (∀ x → f x ∼[ k ] g x) →

+
             (xs >>= f) ∼[ k ] (ys >>= g)

+
  >>=-cong xs≈ys f≈g {x} = begin

+
    x ∈ (xs >>= f)          ↔⟨ SK-sym >>=↔ ⟩

+
    Any (λ y → x ∈ f y) xs  ∼⟨ Any-cong (λ x → f≈g x) xs≈ys ⟩

+
    Any (λ y → x ∈ g y) ys  ↔⟨ >>=↔ ⟩

+
    x ∈ (ys >>= g)          ∎

+
    where open Related.EquationalReasoning

+

+

+
------------------------------------------------------------------------

+
-- _⊛_

+

+
module _ {k} {A B : Set a} {fs gs : List (A → B)} {xs ys} where

+

+
  ⊛-cong : fs ∼[ k ] gs → xs ∼[ k ] ys → (fs ⊛ xs) ∼[ k ] (gs ⊛ ys)

+
  ⊛-cong fs≈gs xs≈ys {x} = begin

+
    x ∈ (fs ⊛ xs)

+
      ≡⟨ ≡.cong (x ∈_) (Applicative.unfold-⊛ fs xs) ⟩

+
    x ∈ (fs >>= λ f → xs >>= λ x → pure (f x))

+
      ∼⟨ >>=-cong fs≈gs (λ f → >>=-cong xs≈ys λ x → K-refl) ⟩

+
    x ∈ (gs >>= λ g → ys >>= λ y → pure (g y))

+
      ≡⟨ ≡.cong (x ∈_) (Applicative.unfold-⊛ gs ys) ⟨

+
    x ∈ (gs ⊛ ys) ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _⊗_

+

+
module _ {ℓ k} {A B : Set ℓ} {xs₁ xs₂ : List A} {ys₁ ys₂ : List B} where

+

+
  ⊗-cong : xs₁ ∼[ k ] xs₂ → ys₁ ∼[ k ] ys₂ →

+
           (xs₁ ⊗ ys₁) ∼[ k ] (xs₂ ⊗ ys₂)

+
  ⊗-cong xs₁≈xs₂ ys₁≈ys₂ =

+
    ⊛-cong (map-cong (λ _ → refl) xs₁≈xs₂) ys₁≈ys₂

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
-- _++_ and [] form a commutative monoid, with either bag or set

+
-- equality as the underlying equality.

+

+
commutativeMonoid : SymmetricKind → Set a → CommutativeMonoid _ _

+
commutativeMonoid {a} k A = record

+
  { Carrier             = List A

+
  ; _≈_                 = _∼[ ⌊ k ⌋ ]_

+
  ; _∙_                 = _++_

+
  ; ε                   = []

+
  ; isCommutativeMonoid = isCommutativeMonoidˡ record

+
    { isSemigroup = record

+
      { isMagma = record

+
        { isEquivalence = Eq.isEquivalence

+
        ; ∙-cong        = ++-cong

+
        }

+
      ; assoc         = λ xs ys zs →

+
                          Eq.reflexive (List.++-assoc xs ys zs)

+
      }

+
    ; identityˡ = λ xs → K-refl

+
    ; comm      = λ xs ys → ↔⇒ (++↔++ xs ys)

+
    }

+
  }

+
  where open Related.EquationalReasoning

+

+
-- The only list which is bag or set equal to the empty list (or a

+
-- subset or subbag of the list) is the empty list itself.

+

+
empty-unique : ∀ {k} → xs ∼[ ⌊ k ⌋→ ] [] → xs ≡ []

+
empty-unique {xs = []}    _    = refl

+
empty-unique {xs = _ ∷ _} ∷∼[] with () ← ⇒→ ∷∼[] (here refl)

+

+
-- _++_ is idempotent (under set equality).

+

+
++-idempotent : Idempotent {A = List A} _∼[ set ]_ _++_

+
++-idempotent xs {x} =

+
  x ∈ xs ++ xs

+
    ∼⟨ mk⇔ ([ id , id ]′ ∘ (Inverse.from $ ++↔)) (Inverse.to ++↔ ∘ inj₁) ⟩

+
  x ∈ xs ∎

+
  where open Related.EquationalReasoning

+

+
-- The list monad's bind distributes from the left over _++_.

+

+
>>=-left-distributive :

+
  ∀ (xs : List A) {f g : A → List B} →

+
  (xs >>= λ x → f x ++ g x) ∼[ bag ] (xs >>= f) ++ (xs >>= g)

+
>>=-left-distributive {ℓ} xs {f} {g} {y} =

+
  y ∈ (xs >>= λ x → f x ++ g x)                      ↔⟨ SK-sym $ >>=↔ ⟩

+
  Any (λ x → y ∈ f x ++ g x) xs                      ↔⟨ SK-sym (Any-cong (λ _ → ++↔) (_ ∎)) ⟩

+
  Any (λ x → y ∈ f x ⊎ y ∈ g x) xs                   ↔⟨ SK-sym $ ⊎↔ ⟩

+
  (Any (λ x → y ∈ f x) xs ⊎ Any (λ x → y ∈ g x) xs)  ↔⟨ >>=↔ ⟨ _⊎-cong_ ⟩ >>=↔ ⟩

+
  (y ∈ (xs >>= f) ⊎ y ∈ (xs >>= g))                  ↔⟨ ++↔ ⟩

+
  y ∈ (xs >>= f) ++ (xs >>= g)                       ∎

+
  where open Related.EquationalReasoning

+

+
-- The same applies to _⊛_.

+

+
⊛-left-distributive :

+
  ∀ (fs : List (A → B)) xs₁ xs₂ →

+
  (fs ⊛ (xs₁ ++ xs₂)) ∼[ bag ] (fs ⊛ xs₁) ++ (fs ⊛ xs₂)

+
⊛-left-distributive {B = B} fs xs₁ xs₂ = begin

+
  fs ⊛ (xs₁ ++ xs₂)                       ≡⟨ Applicative.unfold-⊛ fs (xs₁ ++ xs₂) ⟩

+
  (fs >>= λ f → xs₁ ++ xs₂ >>= pure ∘ f)  ≡⟨ (MP.cong (refl {x = fs}) λ f →

+
                                                MP.right-distributive xs₁ xs₂ (pure ∘ f)) ⟩

+
  (fs >>= λ f → (xs₁ >>= pure ∘ f) ++

+
                (xs₂ >>= pure ∘ f))       ≈⟨ >>=-left-distributive fs ⟩

+

+
  (fs >>= λ f → xs₁ >>= pure ∘ f) ++

+
  (fs >>= λ f → xs₂ >>= pure ∘ f)         ≡⟨ ≡.cong₂ _++_ (Applicative.unfold-⊛ fs xs₁) (Applicative.unfold-⊛ fs xs₂) ⟨

+

+
  (fs ⊛ xs₁) ++ (fs ⊛ xs₂)                ∎

+
  where open ≈-Reasoning ([ bag ]-Equality B)

+

+
private

+

+
  -- If x ∷ xs is set equal to x ∷ ys, then xs and ys are not

+
  -- necessarily set equal.

+

+
  ¬-drop-cons : ∀ {x : A} →

+
    ¬ (∀ {xs ys} → x ∷ xs ∼[ set ] x ∷ ys → xs ∼[ set ] ys)

+
  ¬-drop-cons {x = x} drop-cons with Equivalence.to x∼[] (here refl)

+
    where

+
    x,x≈x :  (x ∷ x ∷ []) ∼[ set ] [ x ]

+
    x,x≈x = ++-idempotent [ x ]

+

+
    x∼[] : [ x ] ∼[ set ] []

+
    x∼[] = drop-cons x,x≈x

+
  ... | ()

+

+
-- However, the corresponding property does hold for bag equality.

+

+
drop-cons : ∀ {x : A} {xs ys} → x ∷ xs ∼[ bag ] x ∷ ys → xs ∼[ bag ] ys

+
drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =

+
  ⊎-left-cancellative

+
    (∼→⊎↔⊎ x∷xs≈x∷ys)

+
    (lemma x∷xs≈x∷ys)

+
    (lemma (SK-sym x∷xs≈x∷ys))

+

+
  where

+

+
  -- TODO: Some of the code below could perhaps be exposed to users.

+

+
  -- List membership can be expressed as "there is an index which

+
  -- points to the element".

+

+
  ∈-index : ∀ {z} (xs : List A) → z ∈ xs ↔ ∃ λ i → z ≡ lookup xs i

+
  ∈-index {z = z} [] =

+
    z ∈ []                              ↔⟨ SK-sym ⊥↔Any[] ⟩

+
    ⊥                                   ↔⟨ mk↔ₛ′ (λ ()) (λ { (() , _) }) (λ { (() , _) }) (λ ()) ⟩

+
    (∃ λ (i : Fin 0) → z ≡ lookup [] i)  ∎

+
    where

+
    open Related.EquationalReasoning

+
  ∈-index {z = z} (x ∷ xs) =

+
    z ∈ x ∷ xs                         ↔⟨ SK-sym (∷↔ _) ⟩

+
    (z ≡ x ⊎ z ∈ xs)                   ↔⟨ K-refl ⊎-cong ∈-index xs ⟩

+
    (z ≡ x ⊎ ∃ λ i → z ≡ lookup xs i)  ↔⟨ mk↔ₛ′ (λ { (inj₁ p) → zero , p; (inj₂ (i , p)) → suc i , p })

+
                                                  (λ { (zero , p) → inj₁ p; (suc i , p) → inj₂ (i , p) })

+
                                                  (λ { (zero , _) → refl; (suc _ , _) → refl })

+
                                                  (λ { (inj₁ _) → refl; (inj₂ _) → refl }) ⟩

+
    (∃ λ i → z ≡ lookup (x ∷ xs) i)    ∎

+
    where

+
    open Related.EquationalReasoning

+

+
  -- The index which points to the element.

+

+
  index-of : ∀ {a} {A : Set a} {z} {xs : List A} →

+
             z ∈ xs → Fin (length xs)

+
  index-of = proj₁ ∘ (Inverse.to (∈-index _))

+

+
  -- The type ∃ λ z → z ∈ xs is isomorphic to Fin n, where n is the

+
  -- length of xs.

+
  --

+
  -- Thierry Coquand pointed out that (a variant of) this statement is

+
  -- a generalisation of the fact that singletons are contractible.

+

+
  Fin-length : ∀ {a} {A : Set a}

+
               (xs : List A) → (∃ λ z → z ∈ xs) ↔ Fin (length xs)

+
  Fin-length xs =

+
    (∃ λ z → z ∈ xs)                   ↔⟨ Σ.cong K-refl (∈-index xs) ⟩

+
    (∃ λ z → ∃ λ i → z ≡ lookup xs i)  ↔⟨ ∃∃↔∃∃ _ ⟩

+
    (∃ λ i → ∃ λ z → z ≡ lookup xs i)  ↔⟨ Σ.cong K-refl (mk↔ₛ′ _ (λ _ → _ , refl) (λ _ → refl) (λ { (_ , refl) → refl })) ⟩

+
    (Fin (length xs) × ⊤)              ↔⟨ ×-identityʳ _ _ ⟩

+
    Fin (length xs)                    ∎

+
    where

+
    open Related.EquationalReasoning

+

+
  -- From this lemma we get that lists which are bag equivalent have

+
  -- related lengths.

+

+
  Fin-length-cong : ∀ {a} {A : Set a} {xs ys : List A} →

+
                    xs ∼[ bag ] ys → Fin (length xs) ↔ Fin (length ys)

+
  Fin-length-cong {xs = xs} {ys} xs≈ys =

+
    Fin (length xs)   ↔⟨ SK-sym $ Fin-length xs ⟩

+
    ∃ (λ z → z ∈ xs)  ↔⟨ Σ.cong K-refl xs≈ys ⟩

+
    ∃ (λ z → z ∈ ys)  ↔⟨ Fin-length ys ⟩

+
    Fin (length ys)   ∎

+
    where

+
    open Related.EquationalReasoning

+

+
  -- The index-of function commutes with applications of certain

+
  -- inverses.

+

+
  index-of-commutes :

+
    ∀ {a} {A : Set a} {z : A} {xs ys} →

+
    (xs≈ys : xs ∼[ bag ] ys) (p : z ∈ xs) →

+
    index-of (Inverse.to xs≈ys p) ≡

+
    Inverse.to (Fin-length-cong xs≈ys) (index-of p)

+
  index-of-commutes {z = z} {xs} {ys} xs≈ys p =

+
    index-of (to xs≈ys p)                                        ≡⟨ lemma z p ⟩

+

+
    index-of (to xs≈ys (proj₂

+
      (from (Fin-length xs) (to (Fin-length xs) (z , p)))))   ≡⟨⟩

+

+
    index-of (proj₂ (Product.map₂ (to xs≈ys)

+
      (from (Fin-length xs) (to (Fin-length xs) (z , p)))))   ≡⟨⟩

+

+
    to (Fin-length ys) (Product.map₂ (to xs≈ys)

+
      (from (Fin-length xs) (index-of p)))                    ≡⟨⟩

+

+
    to (Fin-length-cong xs≈ys) (index-of p)                   ∎

+
    where

+
    open ≡-Reasoning

+
    open Inverse

+

+
    lemma :

+
      ∀ z p →

+
      index-of (to xs≈ys p) ≡

+
      index-of (to xs≈ys (proj₂

+
        (from (Fin-length xs) (to (Fin-length xs) (z , p)))))

+
    lemma z p with to (Fin-length xs) (z , p)

+
                 | strictlyInverseʳ (Fin-length xs) (z , p)

+
    lemma .(lookup xs i) .(from (∈-index xs) (i , refl)) | i | refl =

+
      refl

+

+
  -- Bag equivalence isomorphisms preserve index equality. Note that

+
  -- this means that, even if the underlying equality is proof

+
  -- relevant, a bag equivalence isomorphism cannot map two distinct

+
  -- proofs, that point to the same position, to different positions.

+

+
  index-equality-preserved :

+
    ∀ {a} {A : Set a} {z : A} {xs ys} {p q : z ∈ xs}

+
    (xs≈ys : xs ∼[ bag ] ys) →

+
    index-of p ≡ index-of q →

+
    index-of (Inverse.to xs≈ys p) ≡

+
    index-of (Inverse.to xs≈ys q)

+
  index-equality-preserved {p = p} {q} xs≈ys eq =

+
    index-of (Inverse.to xs≈ys p)                   ≡⟨ index-of-commutes xs≈ys p ⟩

+
    Inverse.to (Fin-length-cong xs≈ys) (index-of p) ≡⟨ ≡.cong (Inverse.to (Fin-length-cong xs≈ys)) eq ⟩

+
    Inverse.to (Fin-length-cong xs≈ys) (index-of q) ≡⟨ ≡.sym $ index-of-commutes xs≈ys q ⟩

+
    index-of (Inverse.to xs≈ys q)                   ∎

+
    where

+
    open ≡-Reasoning

+

+
  -- The old inspect idiom.

+

+
  inspect : ∀ {a} {A : Set a} (x : A) → ∃ (x ≡_)

+
  inspect x = x , refl

+

+
  -- A function is "well-behaved" if any "left" element which is the

+
  -- image of a "right" element is in turn not mapped to another

+
  -- "left" element.

+

+
  Well-behaved : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →

+
                 (A ⊎ B → A ⊎ C) → Set _

+
  Well-behaved f =

+
    ∀ {b a a′} → f (inj₂ b) ≡ inj₁ a → f (inj₁ a) ≢ inj₁ a′

+

+
  -- The type constructor _⊎_ is left cancellative for certain

+
  -- well-behaved inverses.

+

+
  ⊎-left-cancellative :

+
    ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}

+
    (f : (A ⊎ B) ↔ (A ⊎ C)) →

+
    Well-behaved (Inverse.to   f) →

+
    Well-behaved (Inverse.from f) →

+
    B ↔ C

+
  ⊎-left-cancellative {A = A} = λ inv to-hyp from-hyp → mk↔ₛ′

+
    (g (to   inv) to-hyp)

+
    (g (from inv) from-hyp)

+
    (g∘g (SK-sym inv) from-hyp to-hyp)

+
    (g∘g         inv  to-hyp from-hyp)

+
    where

+
    open Inverse

+

+
    module _

+
      {a b c} {A : Set a} {B : Set b} {C : Set c}

+
      (f : A ⊎ B → A ⊎ C)

+
      (hyp : Well-behaved f)

+
      where

+

+
      mutual

+

+
        g : B → C

+
        g b = g′ (inspect (f (inj₂ b)))

+

+
        g′ : ∀ {b} → ∃ (f (inj₂ b) ≡_) → C

+
        g′ (inj₂ c , _)  = c

+
        g′ (inj₁ a , eq) = g″ eq (inspect (f (inj₁ a)))

+

+
        g″ : ∀ {a b} →

+
             f (inj₂ b) ≡ inj₁ a → ∃ (f (inj₁ a) ≡_) → C

+
        g″ _   (inj₂ c , _)   = c

+
        g″ eq₁ (inj₁ _ , eq₂) = ⊥-elim $ hyp eq₁ eq₂

+

+
    g∘g : ∀ {b c} {B : Set b} {C : Set c}

+
          (f : (A ⊎ B) ↔ (A ⊎ C)) →

+
          (to-hyp   : Well-behaved (to   f)) →

+
          (from-hyp : Well-behaved (from f)) →

+
          ∀ b → g (from f) from-hyp (g (to f) to-hyp b) ≡ b

+
    g∘g f to-hyp from-hyp b = g∘g′

+
      where

+
      open ≡-Reasoning

+

+
      g∘g′ : g (from f) from-hyp (g (to f) to-hyp b) ≡ b

+
      g∘g′ with inspect (to f (inj₂ b))

+
      g∘g′ | inj₂ c , eq₁ with inspect (from f (inj₂ c))

+
      ...   | inj₂ b′ , eq₂ = inj₂-injective (

+
        inj₂ b′            ≡⟨ ≡.sym eq₂ ⟩

+
        from f (inj₂ c)   ≡⟨ to-from f eq₁ ⟩

+
        inj₂ b            ∎)

+
      ...   | inj₁ a  , eq₂ with

+
        inj₁ a             ≡⟨ ≡.sym eq₂ ⟩

+
        from f (inj₂ c)    ≡⟨ to-from f eq₁ ⟩

+
        inj₂ b             ∎

+
      ... | ()

+
      g∘g′ | inj₁ a , eq₁ with inspect (to f (inj₁ a))

+
      g∘g′ | inj₁ a , eq₁ | inj₁ a′ , eq₂ = ⊥-elim $ to-hyp eq₁ eq₂

+
      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ with inspect (from f (inj₂ c))

+
      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₂ b′ , eq₃ with

+
        inj₁ a             ≡⟨ ≡.sym (to-from f eq₂) ⟩

+
        from f (inj₂ c)    ≡⟨ eq₃ ⟩

+
        inj₂ b′            ∎

+
      ... | ()

+
      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ with inspect (from f $ inj₁ a′)

+
      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₁ a″ , eq₄ = ⊥-elim $ from-hyp eq₃ eq₄

+
      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₂ b′ , eq₄ = inj₂-injective (

+
        let lemma =

+
              inj₁ a′            ≡⟨ ≡.sym eq₃ ⟩

+
              from f (inj₂ c)    ≡⟨ to-from f eq₂ ⟩

+
              inj₁ a             ∎

+
        in

+
        inj₂ b′             ≡⟨ ≡.sym eq₄ ⟩

+
        from f (inj₁ a′)    ≡⟨ ≡.cong (from f ∘ inj₁) $ inj₁-injective lemma ⟩

+
        from f (inj₁ a)     ≡⟨ to-from f eq₁ ⟩

+
        inj₂ b              ∎)

+

+
  -- Some final lemmas.

+

+
  ∼→⊎↔⊎ :

+
    ∀ {x : A} {xs ys} →

+
    x ∷ xs ∼[ bag ] x ∷ ys →

+
    ∀ {z} → (z ≡ x ⊎ z ∈ xs) ↔ (z ≡ x ⊎ z ∈ ys)

+
  ∼→⊎↔⊎ {x = x} {xs} {ys} x∷xs≈x∷ys {z} =

+
    (z ≡ x ⊎ z ∈ xs)  ↔⟨ ∷↔ _ ⟩

+
    z ∈ x ∷ xs        ↔⟨ x∷xs≈x∷ys ⟩

+
    z ∈ x ∷ ys        ↔⟨ SK-sym (∷↔ _) ⟩

+
    (z ≡ x ⊎ z ∈ ys)  ∎

+
    where

+
    open Related.EquationalReasoning

+

+
  lemma : ∀ {xs ys} (inv : x ∷ xs ∼[ bag ] x ∷ ys) {z} →

+
          Well-behaved (Inverse.to (∼→⊎↔⊎ inv {z}))

+
  lemma {xs} inv {b = z∈xs} {a = p} {a′ = q} hyp₁ hyp₂ = case contra of λ ()

+
    where

+
    open Inverse

+
    open ≡-Reasoning

+
    contra : zero ≡ suc (index-of {xs = xs} z∈xs)

+
    contra = begin

+
      zero

+
        ≡⟨⟩

+
      index-of {xs = x ∷ xs} (here p)

+
        ≡⟨⟩

+
      index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₁ p)

+
        ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ to-from (∼→⊎↔⊎ inv) {x = inj₁ p} hyp₂ ⟨

+
      index-of {xs = x ∷ xs} (to (∷↔ _) $ (from (∼→⊎↔⊎ inv) $ inj₁ q))

+
        ≡⟨ ≡.cong index-of $ strictlyInverseˡ (∷↔ _) (from inv (here q)) ⟩

+
      index-of {xs = x ∷ xs} (to (SK-sym inv) $ here q)

+
        ≡⟨ index-equality-preserved (SK-sym inv) refl ⟩

+
      index-of {xs = x ∷ xs} (to (SK-sym inv) $ here p)

+
        ≡⟨ ≡.cong index-of $ strictlyInverseˡ (∷↔ _) (from inv (here p)) ⟨

+
      index-of {xs = x ∷ xs} (to (∷↔ _) (from (∼→⊎↔⊎ inv) $ inj₁ p))

+
        ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ to-from (∼→⊎↔⊎ inv) {x = inj₂ z∈xs} hyp₁ ⟩

+
      index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₂ z∈xs)

+
        ≡⟨⟩

+
      index-of {xs = x ∷ xs} (there z∈xs)

+
        ≡⟨⟩

+
      suc (index-of {xs = xs} z∈xs)

+
        ∎

+

+

+
------------------------------------------------------------------------

+
-- Relationships to other relations

+

+
↭⇒∼bag : _↭_ {A = A} ⇒ _∼[ bag ]_

+
↭⇒∼bag xs↭ys {v} = mk↔ₛ′ (to xs↭ys) (from xs↭ys) (to∘from xs↭ys) (from∘to xs↭ys)

+
  where

+
  to : ∀ {xs ys} → xs ↭ ys → v ∈ xs → v ∈ ys

+
  to xs↭ys = ∈-resp-↭ xs↭ys

+

+
  from : ∀ {xs ys} → xs ↭ ys → v ∈ ys → v ∈ xs

+
  from xs↭ys = ∈-resp-↭ (↭-sym xs↭ys)

+

+
  from∘to : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ xs) → from p (to p q) ≡ q

+
  from∘to = ∈-resp-[σ⁻¹∘σ]

+

+
  to∘from : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ ys) → to p (from p q) ≡ q

+
  to∘from p with res ← from∘to (↭-sym p) rewrite ↭-sym-involutive p = res

+

+
∼bag⇒↭ : _∼[ bag ]_ ⇒ _↭_ {A = A}

+
∼bag⇒↭ {A = A} {[]} eq with refl ← empty-unique (↔-sym eq) = refl

+
∼bag⇒↭ {A = A} {x ∷ xs} eq

+
  with zs₁ , zs₂ , p ← ∈-∃++ (Inverse.to (eq {x}) (here ≡.refl)) rewrite p = begin

+
    x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (↔-trans eq (comm zs₁ (x ∷ zs₂)))) ⟩

+
    x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩

+
    x ∷ (zs₁ ++ zs₂) ↭⟨ shift x zs₁ zs₂ ⟨

+
    zs₁ ++ x ∷ zs₂   ∎

+
    where

+
    open CommutativeMonoid (commutativeMonoid bag A)

+
    open PermutationReasoning

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Propositional.html b/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Propositional.html
new file mode 100644
index 000000000..54796dd3b
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Propositional.html
@@ -0,0 +1,95 @@
+
+Data.List.Relation.Binary.Disjoint.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pairs of lists that share no common elements (propositional equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Disjoint.Propositional

+
  {a} {A : Set a} where

+

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+
open import Data.List.Relation.Binary.Disjoint.Setoid as DisjointUnique

+

+
------------------------------------------------------------------------

+
-- Re-export the contents of setoid uniqueness

+

+
open DisjointUnique (setoid A) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Setoid.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Setoid.Properties.html
new file mode 100644
index 000000000..a986e42ee
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Setoid.Properties.html
@@ -0,0 +1,133 @@
+
+Data.List.Relation.Binary.Disjoint.Setoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of disjoint lists (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Disjoint.Setoid.Properties where

+

+
open import Data.List.Base

+
open import Data.List.Relation.Binary.Disjoint.Setoid

+
import Data.List.Relation.Unary.Any as Any

+
open import Data.List.Relation.Unary.All as All

+
open import Data.List.Relation.Unary.All.Properties using (¬Any⇒All¬)

+
open import Data.List.Relation.Unary.Any.Properties using (++⁻)

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (Symmetric)

+
open import Relation.Nullary.Negation using (¬_)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+
------------------------------------------------------------------------

+

+
module _ {c ℓ} (S : Setoid c ℓ) where

+

+
  sym : Symmetric (Disjoint S)

+
  sym xs#ys (v∈ys , v∈xs) = xs#ys (v∈xs , v∈ys)

+

+
------------------------------------------------------------------------

+
-- Relationship with other predicates

+
------------------------------------------------------------------------

+

+
module _ {c ℓ} (S : Setoid c ℓ) where

+

+
  open Setoid S

+

+
  Disjoint⇒AllAll : ∀ {xs ys} → Disjoint S xs ys →

+
                    All (λ x → All (λ y → ¬ x ≈ y) ys) xs

+
  Disjoint⇒AllAll xs#ys = All.map (¬Any⇒All¬ _)

+
    (All.tabulate (λ v∈xs v∈ys → xs#ys (Any.map reflexive v∈xs , v∈ys)))

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {c ℓ} (S : Setoid c ℓ) where

+

+
  concat⁺ʳ : ∀ {vs xss} → All (Disjoint S vs) xss → Disjoint S vs (concat xss)

+
  concat⁺ʳ {xss = xs ∷ xss} (vs#xs ∷ vs#xss) (v∈vs , v∈xs++concatxss)

+
    with ++⁻ xs v∈xs++concatxss

+
  ... | inj₁ v∈xs  = vs#xs (v∈vs , v∈xs)

+
  ... | inj₂ v∈xss = concat⁺ʳ vs#xss (v∈vs , v∈xss)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Setoid.html b/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Setoid.html
new file mode 100644
index 000000000..f335e9f72
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Disjoint.Setoid.html
@@ -0,0 +1,114 @@
+
+Data.List.Relation.Binary.Disjoint.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pairs of lists that share no common elements (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.List.Relation.Binary.Disjoint.Setoid {c ℓ} (S : Setoid c ℓ) where

+

+
open import Level using (_⊔_)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Function.Base using (_∘_)

+
open import Data.List.Base using (List; []; [_]; _∷_)

+
open import Data.List.Relation.Unary.Any using (here; there)

+
open import Data.Product.Base using (_×_; _,_)

+

+
open Setoid S renaming (Carrier to A)

+
open import Data.List.Membership.Setoid S using (_∈_; _∉_)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
Disjoint : Rel (List A) (ℓ ⊔ c)

+
Disjoint xs ys = ∀ {v} → ¬ (v ∈ xs × v ∈ ys)

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
contractₗ : ∀ {x xs ys} → Disjoint (x ∷ xs) ys → Disjoint xs ys

+
contractₗ x∷xs∩ys=∅ (v∈xs , v∈ys) = x∷xs∩ys=∅ (there v∈xs , v∈ys)

+

+
contractᵣ : ∀ {xs y ys} → Disjoint xs (y ∷ ys) → Disjoint xs ys

+
contractᵣ xs#y∷ys (v∈xs , v∈ys) = xs#y∷ys (v∈xs , there v∈ys)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Equality.Propositional.html b/site/static/agda_html/Data.List.Relation.Binary.Equality.Propositional.html
new file mode 100644
index 000000000..a4facf66e
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Equality.Propositional.html
@@ -0,0 +1,112 @@
+
+Data.List.Relation.Binary.Equality.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise equality over lists using propositional equality

+
------------------------------------------------------------------------

+

+
-- Note think carefully about using this module as pointwise

+
-- propositional equality can usually be replaced with propositional

+
-- equality.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (_⇒_)

+

+
module Data.List.Relation.Binary.Equality.Propositional {a} {A : Set a} where

+

+
open import Data.List.Base

+
import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
------------------------------------------------------------------------

+
-- Re-export everything from setoid equality

+

+
open SetoidEquality (≡.setoid A) public

+

+
------------------------------------------------------------------------

+
-- ≋ is propositional

+

+
≋⇒≡ : _≋_ ⇒ _≡_

+
≋⇒≡ []             = refl

+
≋⇒≡ (refl ∷ xs≈ys) = cong (_ ∷_) (≋⇒≡ xs≈ys)

+

+
≡⇒≋ : _≡_ ⇒ _≋_

+
≡⇒≋ refl = ≋-refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Equality.Setoid.html b/site/static/agda_html/Data.List.Relation.Binary.Equality.Setoid.html
new file mode 100644
index 000000000..2aaa41325
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Equality.Setoid.html
@@ -0,0 +1,236 @@
+
+Data.List.Relation.Binary.Equality.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise equality over lists parameterised by a setoid

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.List.Relation.Binary.Equality.Setoid {a ℓ} (S : Setoid a ℓ) where

+
open import Algebra.Core using (Op₂)

+
open import Data.Fin.Base using (Fin)

+
open import Data.List.Base using (List; length; map; foldr; _++_; concat;

+
  tabulate; filter; _ʳ++_; reverse)

+
open import Data.List.Relation.Binary.Pointwise as PW using (Pointwise)

+
open import Data.List.Relation.Unary.Unique.Setoid S using (Unique)

+
open import Function.Base using (_∘_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_) renaming (Rel to Rel₂)

+
open import Relation.Binary.Definitions using (Transitive; Symmetric; Reflexive; _Respects_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Properties.Setoid S using (≉-resp₂)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Unary as U using (Pred)

+

+
open Setoid S renaming (Carrier to A)

+

+
private

+
  variable

+
    p q : Level

+

+
------------------------------------------------------------------------

+
-- Definition of equality

+
------------------------------------------------------------------------

+

+
infix 4 _≋_

+

+
_≋_ : Rel₂ (List A) (a ⊔ ℓ)

+
_≋_ = Pointwise _≈_

+

+
open PW public

+
  using ([]; _∷_)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+
------------------------------------------------------------------------

+

+
≋-refl : Reflexive _≋_

+
≋-refl = PW.refl refl

+

+
≋-reflexive : _≡_ ⇒ _≋_

+
≋-reflexive ≡.refl = ≋-refl

+

+
≋-sym : Symmetric _≋_

+
≋-sym = PW.symmetric sym

+

+
≋-trans : Transitive _≋_

+
≋-trans = PW.transitive trans

+

+
≋-isEquivalence : IsEquivalence _≋_

+
≋-isEquivalence = PW.isEquivalence isEquivalence

+

+
≋-setoid : Setoid _ _

+
≋-setoid = PW.setoid S

+

+
------------------------------------------------------------------------

+
-- Relationships to predicates

+
------------------------------------------------------------------------

+

+
open PW public

+
  using () renaming

+
  ( Any-resp-Pointwise      to Any-resp-≋

+
  ; All-resp-Pointwise      to All-resp-≋

+
  ; AllPairs-resp-Pointwise to AllPairs-resp-≋

+
  )

+

+
Unique-resp-≋ : Unique Respects _≋_

+
Unique-resp-≋ = AllPairs-resp-≋ ≉-resp₂

+

+
------------------------------------------------------------------------

+
-- List operations

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- length

+

+
≋-length : ∀ {xs ys} → xs ≋ ys → length xs ≡ length ys

+
≋-length = PW.Pointwise-length

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {b ℓ₂} (T : Setoid b ℓ₂) where

+

+
  open Setoid T using () renaming (_≈_ to _≈′_)

+
  private

+
    _≋′_ = Pointwise _≈′_

+

+
  map⁺ : ∀ {f} → f Preserves _≈_ ⟶ _≈′_ →

+
         ∀ {xs ys} → xs ≋ ys → map f xs ≋′ map f ys

+
  map⁺ {f} pres xs≋ys = PW.map⁺ f f (PW.map pres xs≋ys)

+

+
------------------------------------------------------------------------

+
-- foldr

+

+
foldr⁺ : ∀ {_•_ : Op₂ A} {_◦_ : Op₂ A} →

+
         (∀ {w x y z} → w ≈ x → y ≈ z → (w • y) ≈ (x ◦ z)) →

+
         ∀ {xs ys e f} → e ≈ f → xs ≋ ys →

+
         foldr _•_ e xs ≈ foldr _◦_ f ys

+
foldr⁺ ∙⇔◦ e≈f xs≋ys = PW.foldr⁺ ∙⇔◦ e≈f xs≋ys

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
++⁺ : ∀ {ws xs ys zs} → ws ≋ xs → ys ≋ zs → ws ++ ys ≋ xs ++ zs

+
++⁺ = PW.++⁺

+

+
++-cancelˡ : ∀ xs {ys zs} → xs ++ ys ≋ xs ++ zs → ys ≋ zs

+
++-cancelˡ xs = PW.++-cancelˡ xs

+

+
++-cancelʳ : ∀ {xs} ys zs → ys ++ xs ≋ zs ++ xs → ys ≋ zs

+
++-cancelʳ = PW.++-cancelʳ

+

+
------------------------------------------------------------------------

+
-- concat

+

+
concat⁺ : ∀ {xss yss} → Pointwise _≋_ xss yss → concat xss ≋ concat yss

+
concat⁺ = PW.concat⁺

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {n} {f g : Fin n → A}

+
  where

+

+
  tabulate⁺ : (∀ i → f i ≈ g i) → tabulate f ≋ tabulate g

+
  tabulate⁺ = PW.tabulate⁺

+

+
  tabulate⁻ : tabulate f ≋ tabulate g → (∀ i → f i ≈ g i)

+
  tabulate⁻ = PW.tabulate⁻

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {P : Pred A p} (P? : U.Decidable P) (resp : P Respects _≈_)

+
  where

+

+
  filter⁺ : ∀ {xs ys} → xs ≋ ys → filter P? xs ≋ filter P? ys

+
  filter⁺ xs≋ys = PW.filter⁺ P? P? resp (resp ∘ sym) xs≋ys

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
ʳ++⁺ : ∀{xs xs′ ys ys′} → xs ≋ xs′ → ys ≋ ys′ → xs ʳ++ ys ≋ xs′ ʳ++ ys′

+
ʳ++⁺ = PW.ʳ++⁺

+

+
reverse⁺ : ∀ {xs ys} → xs ≋ ys → reverse xs ≋ reverse ys

+
reverse⁺ = PW.reverse⁺

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Lex.Core.html b/site/static/agda_html/Data.List.Relation.Binary.Lex.Core.html
new file mode 100644
index 000000000..d099b5f04
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Lex.Core.html
@@ -0,0 +1,124 @@
+
+Data.List.Relation.Binary.Lex.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lexicographic ordering of lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Lex.Core where

+

+
open import Data.Empty using (⊥; ⊥-elim)

+
open import Data.Unit.Base using (⊤; tt)

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)

+
open import Data.List.Base using (List; []; _∷_)

+
open import Function.Base using (_∘_; flip; id)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Binary.Core using (Rel)

+
open import Data.List.Relation.Binary.Pointwise.Base

+
   using (Pointwise; []; _∷_; head; tail)

+

+
private

+
  variable

+
    a ℓ₁ ℓ₂ : Level

+

+
-- The lexicographic ordering itself can be either strict or non-strict,

+
-- depending on whether type P is inhabited.

+

+
data Lex {A : Set a} (P : Set)

+
         (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) :

+
         Rel (List A) (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  base : P                             → Lex P _≈_ _≺_ []       []

+
  halt : ∀ {y ys}                      → Lex P _≈_ _≺_ []       (y ∷ ys)

+
  this : ∀ {x xs y ys} (x≺y : x ≺ y)   → Lex P _≈_ _≺_ (x ∷ xs) (y ∷ ys)

+
  next : ∀ {x xs y ys} (x≈y : x ≈ y)

+
         (xs<ys : Lex P _≈_ _≺_ xs ys) → Lex P _≈_ _≺_ (x ∷ xs) (y ∷ ys)

+

+
------------------------------------------------------------------------

+
-- Lexicographic orderings, using a strict ordering as the base

+

+
Lex-< : {A : Set a} (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) →

+
        Rel (List A) (a ⊔ ℓ₁ ⊔ ℓ₂)

+
Lex-< = Lex ⊥

+

+
Lex-≤ : {A : Set a} (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) →

+
        Rel (List A) (a ⊔ ℓ₁ ⊔ ℓ₂)

+
Lex-≤ = Lex ⊤

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Lex.Strict.html b/site/static/agda_html/Data.List.Relation.Binary.Lex.Strict.html
new file mode 100644
index 000000000..d29500b3c
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Lex.Strict.html
@@ -0,0 +1,326 @@
+
+Data.List.Relation.Binary.Lex.Strict
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lexicographic ordering of lists

+
------------------------------------------------------------------------

+

+
-- The definitions of lexicographic ordering used here are suitable if

+
-- the argument order is a strict partial order.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Lex.Strict where

+

+
open import Data.Empty using (⊥)

+
open import Data.Unit.Base using (⊤; tt)

+
open import Function.Base using (_∘_; id)

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Data.List.Base using (List; []; _∷_)

+
open import Level using (_⊔_)

+
open import Relation.Nullary using (yes; no; ¬_)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Bundles

+
  using (StrictPartialOrder; StrictTotalOrder; Preorder; Poset; DecPoset; DecTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Irreflexive; Symmetric; _Respects₂_; Total; Asymmetric; Antisymmetric; Transitive; Trichotomous; Decidable; tri≈; tri<; tri>)

+
open import Relation.Binary.Consequences

+
open import Data.List.Relation.Binary.Pointwise as Pointwise

+
   using (Pointwise; []; _∷_; head; tail)

+

+
import Data.List.Relation.Binary.Lex as Core

+

+
------------------------------------------------------------------------

+
-- Re-exporting core definitions

+

+
open Core public

+
  using (Lex-<; Lex-≤; base; halt; this; next; ¬≤-this; ¬≤-next)

+

+
------------------------------------------------------------------------

+
-- Strict lexicographic ordering.

+

+
module _ {a ℓ₁ ℓ₂} {A : Set a} where

+

+
  -- Properties

+

+
  module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where

+

+
    private

+
      _≋_ = Pointwise _≈_

+
      _<_ = Lex-< _≈_ _≺_

+

+
    xs≮[] : ∀ {xs} → ¬ xs < []

+
    xs≮[] (base ())

+

+
    ¬[]<[] : ¬ [] < []

+
    ¬[]<[] = xs≮[]

+

+
    <-irreflexive : Irreflexive _≈_ _≺_ → Irreflexive _≋_ _<_

+
    <-irreflexive irr (x≈y ∷ xs≋ys) (this x<y)     = irr x≈y x<y

+
    <-irreflexive irr (x≈y ∷ xs≋ys) (next _ xs⊴ys) =

+
      <-irreflexive irr xs≋ys xs⊴ys

+

+
    <-asymmetric : Symmetric _≈_ → _≺_ Respects₂ _≈_ → Asymmetric _≺_ →

+
                   Asymmetric _<_

+
    <-asymmetric sym resp as = asym

+
      where

+
      irrefl : Irreflexive _≈_ _≺_

+
      irrefl = asym⇒irr resp sym as

+

+
      asym : Asymmetric _<_

+
      asym (base bot)       _                = bot

+
      asym (this x<y)       (this y<x)       = as x<y y<x

+
      asym (this x<y)       (next y≈x ys⊴xs) = irrefl (sym y≈x) x<y

+
      asym (next x≈y xs⊴ys) (this y<x)       = irrefl (sym x≈y) y<x

+
      asym (next x≈y xs⊴ys) (next y≈x ys⊴xs) = asym xs⊴ys ys⊴xs

+

+
    <-antisymmetric : Symmetric _≈_ → Irreflexive _≈_ _≺_ →

+
                      Asymmetric _≺_ → Antisymmetric _≋_ _<_

+
    <-antisymmetric = Core.antisymmetric

+

+
    <-transitive : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ →

+
                   Transitive _≺_ → Transitive _<_

+
    <-transitive = Core.transitive

+

+
    <-compare : Symmetric _≈_ → Trichotomous _≈_ _≺_ →

+
                Trichotomous _≋_ _<_

+
    <-compare sym tri []       []       = tri≈ ¬[]<[] []    ¬[]<[]

+
    <-compare sym tri []       (y ∷ ys) = tri< halt   (λ()) (λ())

+
    <-compare sym tri (x ∷ xs) []       = tri> (λ())  (λ()) halt

+
    <-compare sym tri (x ∷ xs) (y ∷ ys) with tri x y

+
    ... | tri< x<y x≉y y≮x =

+
          tri< (this x<y) (x≉y ∘ head) (¬≤-this (x≉y ∘ sym) y≮x)

+
    ... | tri> x≮y x≉y y<x =

+
          tri> (¬≤-this x≉y x≮y) (x≉y ∘ head) (this y<x)

+
    ... | tri≈ x≮y x≈y y≮x with <-compare sym tri xs ys

+
    ...   | tri< xs<ys xs≉ys ys≮xs =

+
            tri< (next x≈y xs<ys) (xs≉ys ∘ tail) (¬≤-next y≮x ys≮xs)

+
    ...   | tri≈ xs≮ys xs≈ys ys≮xs =

+
            tri≈ (¬≤-next x≮y xs≮ys) (x≈y ∷ xs≈ys) (¬≤-next y≮x ys≮xs)

+
    ...   | tri> xs≮ys xs≉ys ys<xs =

+
            tri> (¬≤-next x≮y xs≮ys) (xs≉ys ∘ tail) (next (sym x≈y) ys<xs)

+

+
    <-decidable : Decidable _≈_ → Decidable _≺_ → Decidable _<_

+
    <-decidable = Core.decidable (no id)

+

+
    <-respects₂ : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ →

+
                  _<_ Respects₂ _≋_

+
    <-respects₂ = Core.respects₂

+

+
    <-isStrictPartialOrder : IsStrictPartialOrder _≈_ _≺_ →

+
                             IsStrictPartialOrder _≋_ _<_

+
    <-isStrictPartialOrder spo = record

+
      { isEquivalence = Pointwise.isEquivalence isEquivalence

+
      ; irrefl        = <-irreflexive irrefl

+
      ; trans         = Core.transitive isEquivalence <-resp-≈ trans

+
      ; <-resp-≈      = Core.respects₂ isEquivalence <-resp-≈

+
      } where open IsStrictPartialOrder spo

+

+
    <-isStrictTotalOrder : IsStrictTotalOrder _≈_ _≺_ →

+
                           IsStrictTotalOrder _≋_ _<_

+
    <-isStrictTotalOrder sto = record

+
      { isStrictPartialOrder = <-isStrictPartialOrder isStrictPartialOrder

+
      ; compare              = <-compare Eq.sym compare

+
      } where open IsStrictTotalOrder sto

+

+
<-strictPartialOrder : ∀ {a ℓ₁ ℓ₂} → StrictPartialOrder a ℓ₁ ℓ₂ →

+
                       StrictPartialOrder _ _ _

+
<-strictPartialOrder spo = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder isStrictPartialOrder

+
  } where open StrictPartialOrder spo

+

+
<-strictTotalOrder : ∀ {a ℓ₁ ℓ₂} → StrictTotalOrder a ℓ₁ ℓ₂ →

+
                       StrictTotalOrder _ _ _

+
<-strictTotalOrder sto = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder isStrictTotalOrder

+
  } where open StrictTotalOrder sto

+

+
------------------------------------------------------------------------

+
-- Non-strict lexicographic ordering.

+

+
module _ {a ℓ₁ ℓ₂} {A : Set a} where

+

+
  -- Properties

+

+
  ≤-reflexive : (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) →

+
                Pointwise _≈_ ⇒ Lex-≤ _≈_ _≺_

+
  ≤-reflexive _≈_ _≺_ []            = base tt

+
  ≤-reflexive _≈_ _≺_ (x≈y ∷ xs≈ys) =

+
    next x≈y (≤-reflexive _≈_ _≺_ xs≈ys)

+

+
  module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where

+

+
    private

+
      _≋_ = Pointwise _≈_

+
      _≤_ = Lex-≤ _≈_ _≺_

+

+
    ≤-antisymmetric : Symmetric _≈_ → Irreflexive _≈_ _≺_ →

+
                      Asymmetric _≺_ → Antisymmetric _≋_ _≤_

+
    ≤-antisymmetric = Core.antisymmetric

+

+
    ≤-transitive : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ →

+
                   Transitive _≺_ → Transitive _≤_

+
    ≤-transitive = Core.transitive

+

+
    -- Note that trichotomy is an unnecessarily strong precondition for

+
    -- the following lemma.

+

+
    ≤-total : Symmetric _≈_ → Trichotomous _≈_ _≺_ → Total _≤_

+
    ≤-total _   _   []       []       = inj₁ (base tt)

+
    ≤-total _   _   []       (x ∷ xs) = inj₁ halt

+
    ≤-total _   _   (x ∷ xs) []       = inj₂ halt

+
    ≤-total sym tri (x ∷ xs) (y ∷ ys) with tri x y

+
    ... | tri< x<y _ _ = inj₁ (this x<y)

+
    ... | tri> _ _ y<x = inj₂ (this y<x)

+
    ... | tri≈ _ x≈y _ with ≤-total sym tri xs ys

+
    ...   | inj₁ xs≲ys = inj₁ (next      x≈y  xs≲ys)

+
    ...   | inj₂ ys≲xs = inj₂ (next (sym x≈y) ys≲xs)

+

+
    ≤-decidable : Decidable _≈_ → Decidable _≺_ → Decidable _≤_

+
    ≤-decidable = Core.decidable (yes tt)

+

+
    ≤-respects₂ : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ →

+
                  _≤_ Respects₂ _≋_

+
    ≤-respects₂ = Core.respects₂

+

+
    ≤-isPreorder : IsEquivalence _≈_ → Transitive _≺_ →

+
                   _≺_ Respects₂ _≈_ → IsPreorder _≋_ _≤_

+
    ≤-isPreorder eq tr resp = record

+
      { isEquivalence = Pointwise.isEquivalence eq

+
      ; reflexive     = ≤-reflexive _≈_ _≺_

+
      ; trans         = Core.transitive eq resp tr

+
      }

+

+
    ≤-isPartialOrder : IsStrictPartialOrder _≈_ _≺_ →

+
                       IsPartialOrder _≋_ _≤_

+
    ≤-isPartialOrder  spo = record

+
      { isPreorder = ≤-isPreorder isEquivalence trans <-resp-≈

+
      ; antisym    = Core.antisymmetric Eq.sym irrefl asym

+
      }

+
      where open IsStrictPartialOrder spo

+

+
    ≤-isDecPartialOrder : IsStrictTotalOrder _≈_ _≺_ →

+
                          IsDecPartialOrder _≋_ _≤_

+
    ≤-isDecPartialOrder sto = record

+
      { isPartialOrder = ≤-isPartialOrder isStrictPartialOrder

+
      ; _≟_            = Pointwise.decidable _≟_

+
      ; _≤?_           = ≤-decidable _≟_ _<?_

+
      } where open IsStrictTotalOrder sto

+

+
    ≤-isTotalOrder : IsStrictTotalOrder _≈_ _≺_ → IsTotalOrder _≋_ _≤_

+
    ≤-isTotalOrder sto = record

+
      { isPartialOrder = ≤-isPartialOrder isStrictPartialOrder

+
      ; total          = ≤-total Eq.sym compare

+
      }

+
      where open IsStrictTotalOrder sto

+

+
    ≤-isDecTotalOrder : IsStrictTotalOrder _≈_ _≺_ →

+
                        IsDecTotalOrder _≋_ _≤_

+
    ≤-isDecTotalOrder sto = record

+
      { isTotalOrder = ≤-isTotalOrder sto

+
      ; _≟_          = Pointwise.decidable _≟_

+
      ; _≤?_         = ≤-decidable _≟_ _<?_

+
      }

+
      where open IsStrictTotalOrder sto

+

+
≤-preorder : ∀ {a ℓ₁ ℓ₂} → Preorder a ℓ₁ ℓ₂ → Preorder _ _ _

+
≤-preorder pre = record

+
  { isPreorder = ≤-isPreorder isEquivalence trans ∼-resp-≈

+
  } where open Preorder pre

+

+
≤-partialOrder : ∀ {a ℓ₁ ℓ₂} → StrictPartialOrder a ℓ₁ ℓ₂ → Poset _ _ _

+
≤-partialOrder spo = record

+
  { isPartialOrder = ≤-isPartialOrder isStrictPartialOrder

+
  } where open StrictPartialOrder spo

+

+
≤-decPoset : ∀ {a ℓ₁ ℓ₂} → StrictTotalOrder a ℓ₁ ℓ₂ →

+
             DecPoset _ _ _

+
≤-decPoset sto = record

+
  { isDecPartialOrder = ≤-isDecPartialOrder isStrictTotalOrder

+
  } where open StrictTotalOrder sto

+

+

+
≤-decTotalOrder : ∀ {a ℓ₁ ℓ₂} → StrictTotalOrder a ℓ₁ ℓ₂ →

+
                  DecTotalOrder _ _ _

+
≤-decTotalOrder sto = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder isStrictTotalOrder

+
  } where open StrictTotalOrder sto

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Lex.html b/site/static/agda_html/Data.List.Relation.Binary.Lex.html
new file mode 100644
index 000000000..f67a0bb6d
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Lex.html
@@ -0,0 +1,195 @@
+
+Data.List.Relation.Binary.Lex
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lexicographic ordering of lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Lex where

+

+
open import Data.Empty using (⊥; ⊥-elim)

+
open import Data.Unit.Base using (⊤; tt)

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)

+
open import Data.List.Base using (List; []; _∷_)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])

+
open import Function.Base using (_∘_; flip; id)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Level using (_⊔_)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Nullary.Decidable as Dec

+
  using (Dec; yes; no; _×-dec_; _⊎-dec_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Symmetric; Transitive; Irreflexive; Asymmetric; Antisymmetric; Decidable; _Respects₂_; _Respects_)

+
open import Data.List.Relation.Binary.Pointwise.Base

+
   using (Pointwise; []; _∷_; head; tail)

+

+
------------------------------------------------------------------------

+
-- Re-exporting the core definitions and properties

+

+
open import Data.List.Relation.Binary.Lex.Core public

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ {a ℓ₁ ℓ₂} {A : Set a} {P : Set}

+
         {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where

+

+
  private

+
    _≋_ = Pointwise _≈_

+
    _<_ = Lex P _≈_ _≺_

+

+
  ¬≤-this : ∀ {x y xs ys} → ¬ (x ≈ y) → ¬ (x ≺ y) →

+
            ¬ (x ∷ xs) < (y ∷ ys)

+
  ¬≤-this x≉y x≮y (this x≺y)       = x≮y x≺y

+
  ¬≤-this x≉y x≮y (next x≈y xs<ys) = x≉y x≈y

+

+
  ¬≤-next : ∀ {x y xs ys} → ¬ x ≺ y → ¬ xs < ys →

+
            ¬ (x ∷ xs) < (y ∷ ys)

+
  ¬≤-next x≮y xs≮ys (this x≺y)     = x≮y x≺y

+
  ¬≤-next x≮y xs≮ys (next _ xs<ys) = xs≮ys xs<ys

+

+
  antisymmetric : Symmetric _≈_ → Irreflexive _≈_ _≺_ →

+
                  Asymmetric _≺_ → Antisymmetric _≋_ _<_

+
  antisymmetric sym ir asym = as

+
    where

+
    as : Antisymmetric _≋_ _<_

+
    as (base _)         (base _)         = []

+
    as (this x≺y)       (this y≺x)       = ⊥-elim (asym x≺y y≺x)

+
    as (this x≺y)       (next y≈x ys<xs) = ⊥-elim (ir (sym y≈x) x≺y)

+
    as (next x≈y xs<ys) (this y≺x)       = ⊥-elim (ir (sym x≈y) y≺x)

+
    as (next x≈y xs<ys) (next y≈x ys<xs) = x≈y ∷ as xs<ys ys<xs

+

+
  toSum : ∀ {x y xs ys} → (x ∷ xs) < (y ∷ ys) → (x ≺ y ⊎ (x ≈ y × xs < ys))

+
  toSum (this x≺y) = inj₁ x≺y

+
  toSum (next x≈y xs<ys) = inj₂ (x≈y , xs<ys)

+

+
  transitive : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ → Transitive _≺_ →

+
               Transitive _<_

+
  transitive eq resp tr = trans

+
    where

+
    trans : Transitive (Lex P _≈_ _≺_)

+
    trans (base p)         (base _)         = base p

+
    trans (base y)         halt             = halt

+
    trans halt             (this y≺z)       = halt

+
    trans halt             (next y≈z ys<zs) = halt

+
    trans (this x≺y)       (this y≺z)       = this (tr x≺y y≺z)

+
    trans (this x≺y)       (next y≈z ys<zs) = this (proj₁ resp y≈z x≺y)

+
    trans (next x≈y xs<ys) (this y≺z)       =

+
      this (proj₂ resp (IsEquivalence.sym eq x≈y) y≺z)

+
    trans (next x≈y xs<ys) (next y≈z ys<zs) =

+
      next (IsEquivalence.trans eq x≈y y≈z) (trans xs<ys ys<zs)

+

+
  respects₂ : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ → _<_ Respects₂ _≋_

+
  respects₂ eq (resp₁ , resp₂) = resp¹ , resp²

+
    where

+
    open IsEquivalence eq using (sym; trans)

+
    resp¹ : ∀ {xs} → Lex P _≈_ _≺_ xs Respects _≋_

+
    resp¹ []            xs<[]            = xs<[]

+
    resp¹ (_   ∷ _)     halt             = halt

+
    resp¹ (x≈y ∷ _)     (this z≺x)       = this (resp₁ x≈y z≺x)

+
    resp¹ (x≈y ∷ xs≋ys) (next z≈x zs<xs) =

+
      next (trans z≈x x≈y) (resp¹ xs≋ys zs<xs)

+

+
    resp² : ∀ {ys} → flip (Lex P _≈_ _≺_) ys Respects _≋_

+
    resp² []            []<ys            = []<ys

+
    resp² (x≈z ∷ _)     (this x≺y)       = this (resp₂ x≈z x≺y)

+
    resp² (x≈z ∷ xs≋zs) (next x≈y xs<ys) =

+
      next (trans (sym x≈z) x≈y) (resp² xs≋zs xs<ys)

+

+

+
  []<[]-⇔ : P ⇔ [] < []

+
  []<[]-⇔ = mk⇔ base (λ { (base p) → p })

+

+

+
  ∷<∷-⇔ : ∀ {x y xs ys} → (x ≺ y ⊎ (x ≈ y × xs < ys)) ⇔ (x ∷ xs) < (y ∷ ys)

+
  ∷<∷-⇔ = mk⇔ [ this , uncurry next ] toSum

+

+
  module _ (dec-P : Dec P) (dec-≈ : Decidable _≈_) (dec-≺ : Decidable _≺_)

+
    where

+

+
    decidable : Decidable _<_

+
    decidable []       []       = Dec.map []<[]-⇔ dec-P

+
    decidable []       (y ∷ ys) = yes halt

+
    decidable (x ∷ xs) []       = no λ()

+
    decidable (x ∷ xs) (y ∷ ys) =

+
      Dec.map ∷<∷-⇔ (dec-≺ x y ⊎-dec (dec-≈ x y ×-dec decidable xs ys))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Permutation.Homogeneous.html b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Homogeneous.html
new file mode 100644
index 000000000..9c65b8dd1
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Homogeneous.html
@@ -0,0 +1,138 @@
+
+Data.List.Relation.Binary.Permutation.Homogeneous
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- A definition for the permutation relation using setoid equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Permutation.Homogeneous where

+

+
open import Data.List.Base using (List; _∷_)

+
open import Data.List.Relation.Binary.Pointwise.Base as Pointwise

+
  using (Pointwise)

+
open import Data.List.Relation.Binary.Pointwise.Properties as Pointwise

+
  using (symmetric)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions using (Reflexive; Symmetric)

+

+
private

+
  variable

+
    a r s : Level

+
    A : Set a

+

+
data Permutation {A : Set a} (R : Rel A r) : Rel (List A) (a ⊔ r) where

+
  refl  : ∀ {xs ys} → Pointwise R xs ys → Permutation R xs ys

+
  prep  : ∀ {xs ys x y} (eq : R x y) → Permutation R xs ys → Permutation R (x ∷ xs) (y ∷ ys)

+
  swap  : ∀ {xs ys x y x′ y′} (eq₁ : R x x′) (eq₂ : R y y′) → Permutation R xs ys → Permutation R (x ∷ y ∷ xs) (y′ ∷ x′ ∷ ys)

+
  trans : ∀ {xs ys zs} → Permutation R xs ys → Permutation R ys zs → Permutation R xs zs

+

+
------------------------------------------------------------------------

+
-- The Permutation relation is an equivalence

+

+
module _ {R : Rel A r}  where

+

+
  sym : Symmetric R → Symmetric (Permutation R)

+
  sym R-sym (refl xs∼ys)           = refl (Pointwise.symmetric R-sym xs∼ys)

+
  sym R-sym (prep x∼x′ xs↭ys)      = prep (R-sym x∼x′) (sym R-sym xs↭ys)

+
  sym R-sym (swap x∼x′ y∼y′ xs↭ys) = swap (R-sym y∼y′) (R-sym x∼x′) (sym R-sym xs↭ys)

+
  sym R-sym (trans xs↭ys ys↭zs)    = trans (sym R-sym ys↭zs) (sym R-sym xs↭ys)

+

+
  isEquivalence : Reflexive R → Symmetric R → IsEquivalence (Permutation R)

+
  isEquivalence R-refl R-sym = record

+
    { refl  = refl (Pointwise.refl R-refl)

+
    ; sym   = sym R-sym

+
    ; trans = trans

+
    }

+

+
  setoid : Reflexive R → Symmetric R → Setoid _ _

+
  setoid R-refl R-sym = record

+
    { isEquivalence = isEquivalence R-refl R-sym

+
    }

+

+
map : ∀ {R : Rel A r} {S : Rel A s} →

+
      (R ⇒ S) → (Permutation R ⇒ Permutation S)

+
map R⇒S (refl xs∼ys)         = refl (Pointwise.map R⇒S xs∼ys)

+
map R⇒S (prep e xs∼ys)       = prep (R⇒S e) (map R⇒S xs∼ys)

+
map R⇒S (swap e₁ e₂ xs∼ys)   = swap (R⇒S e₁) (R⇒S e₂) (map R⇒S xs∼ys)

+
map R⇒S (trans xs∼ys ys∼zs)  = trans (map R⇒S xs∼ys) (map R⇒S ys∼zs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Permutation.Propositional.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Propositional.Properties.html
new file mode 100644
index 000000000..24e0aeeaa
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Propositional.Properties.html
@@ -0,0 +1,473 @@
+
+Data.List.Relation.Binary.Permutation.Propositional.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of permutation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Permutation.Propositional.Properties where

+

+
open import Algebra.Bundles

+
open import Algebra.Definitions

+
open import Algebra.Structures

+
open import Data.Bool.Base using (Bool; true; false)

+
open import Data.Nat.Base using (suc; _*_)

+
open import Data.Nat.Properties using (*-assoc; *-comm)

+
open import Data.Product.Base using (-,_; proj₂)

+
open import Data.List.Base as List

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.List.Relation.Unary.All using (All; []; _∷_)

+
open import Data.List.Membership.Propositional

+
open import Data.List.Membership.Propositional.Properties

+
import Data.List.Properties as Lₚ

+
open import Data.Product.Base using (_,_; _×_; ∃; ∃₂)

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Function.Base using (_∘_; _⟨_⟩_; _$_)

+
open import Level using (Level)

+
open import Relation.Unary using (Pred)

+
open import Relation.Binary.Core using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.Definitions using (_Respects_; Decidable)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_ ; refl ; cong; cong₂; _≢_)

+
open import Relation.Binary.PropositionalEquality.Properties using (module ≡-Reasoning)

+
open import Relation.Nullary

+

+
private

+
  variable

+
    a b p : Level

+
    A : Set a

+
    B : Set b

+
    xs ys : List A

+

+
------------------------------------------------------------------------

+
-- Permutations of empty and singleton lists

+

+
↭-empty-inv : ∀ {xs : List A} → xs ↭ [] → xs ≡ []

+
↭-empty-inv refl = refl

+
↭-empty-inv (trans p q) with refl ← ↭-empty-inv q = ↭-empty-inv p

+

+
¬x∷xs↭[] : ∀ {x} {xs : List A} → ¬ ((x ∷ xs) ↭ [])

+
¬x∷xs↭[] (trans s₁ s₂) with ↭-empty-inv s₂

+
... | refl = ¬x∷xs↭[] s₁

+

+
↭-singleton-inv : ∀ {x} {xs : List A} → xs ↭ [ x ] → xs ≡ [ x ]

+
↭-singleton-inv refl                                             = refl

+
↭-singleton-inv (prep _ ρ) with refl ← ↭-empty-inv ρ             = refl

+
↭-singleton-inv (trans ρ₁ ρ₂) with refl ← ↭-singleton-inv ρ₂ = ↭-singleton-inv ρ₁

+

+
------------------------------------------------------------------------

+
-- sym

+

+
↭-sym-involutive : ∀ {xs ys : List A} (p : xs ↭ ys) → ↭-sym (↭-sym p) ≡ p

+
↭-sym-involutive refl          = refl

+
↭-sym-involutive (prep x ↭)    = cong (prep x) (↭-sym-involutive ↭)

+
↭-sym-involutive (swap x y ↭)  = cong (swap x y) (↭-sym-involutive ↭)

+
↭-sym-involutive (trans ↭₁ ↭₂) =

+
  cong₂ trans (↭-sym-involutive ↭₁) (↭-sym-involutive ↭₂)

+

+
------------------------------------------------------------------------

+
-- Relationships to other predicates

+

+
All-resp-↭ : ∀ {P : Pred A p} → (All P) Respects _↭_

+
All-resp-↭ refl wit                     = wit

+
All-resp-↭ (prep x p) (px ∷ wit)        = px ∷ All-resp-↭ p wit

+
All-resp-↭ (swap x y p) (px ∷ py ∷ wit) = py ∷ px ∷ All-resp-↭ p wit

+
All-resp-↭ (trans p₁ p₂) wit            = All-resp-↭ p₂ (All-resp-↭ p₁ wit)

+

+
Any-resp-↭ : ∀ {P : Pred A p} → (Any P) Respects _↭_

+
Any-resp-↭ refl         wit                 = wit

+
Any-resp-↭ (prep x p)   (here px)           = here px

+
Any-resp-↭ (prep x p)   (there wit)         = there (Any-resp-↭ p wit)

+
Any-resp-↭ (swap x y p) (here px)           = there (here px)

+
Any-resp-↭ (swap x y p) (there (here px))   = here px

+
Any-resp-↭ (swap x y p) (there (there wit)) = there (there (Any-resp-↭ p wit))

+
Any-resp-↭ (trans p p₁) wit                 = Any-resp-↭ p₁ (Any-resp-↭ p wit)

+

+
∈-resp-↭ : ∀ {x : A} → (x ∈_) Respects _↭_

+
∈-resp-↭ = Any-resp-↭

+

+
Any-resp-[σ⁻¹∘σ] : {xs ys : List A} {P : Pred A p} →

+
                   (σ : xs ↭ ys) →

+
                   (ix : Any P xs) →

+
                   Any-resp-↭ (trans σ (↭-sym σ)) ix ≡ ix

+
Any-resp-[σ⁻¹∘σ] refl          ix               = refl

+
Any-resp-[σ⁻¹∘σ] (prep _ _)    (here _)         = refl

+
Any-resp-[σ⁻¹∘σ] (swap _ _ _)  (here _)         = refl

+
Any-resp-[σ⁻¹∘σ] (swap _ _ _)  (there (here _)) = refl

+
Any-resp-[σ⁻¹∘σ] (trans σ₁ σ₂) ix

+
  rewrite Any-resp-[σ⁻¹∘σ] σ₂ (Any-resp-↭ σ₁ ix)

+
  rewrite Any-resp-[σ⁻¹∘σ] σ₁ ix

+
  = refl

+
Any-resp-[σ⁻¹∘σ] (prep _ σ)    (there ix)

+
  rewrite Any-resp-[σ⁻¹∘σ] σ ix

+
  = refl

+
Any-resp-[σ⁻¹∘σ] (swap _ _ σ)  (there (there ix))

+
  rewrite Any-resp-[σ⁻¹∘σ] σ ix

+
  = refl

+

+
∈-resp-[σ⁻¹∘σ] : {xs ys : List A} {x : A} →

+
                 (σ : xs ↭ ys) →

+
                 (ix : x ∈ xs) →

+
                 ∈-resp-↭ (trans σ (↭-sym σ)) ix ≡ ix

+
∈-resp-[σ⁻¹∘σ] = Any-resp-[σ⁻¹∘σ]

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ (f : A → B) where

+

+
  map⁺ : ∀ {xs ys} → xs ↭ ys → map f xs ↭ map f ys

+
  map⁺ refl          = refl

+
  map⁺ (prep x p)    = prep _ (map⁺ p)

+
  map⁺ (swap x y p)  = swap _ _ (map⁺ p)

+
  map⁺ (trans p₁ p₂) = trans (map⁺ p₁) (map⁺ p₂)

+

+
  -- permutations preserve 'being a mapped list'

+
  ↭-map-inv : ∀ {xs ys} → map f xs ↭ ys → ∃ λ ys′ → ys ≡ map f ys′ × xs ↭ ys′

+
  ↭-map-inv {[]}     ρ                                                  = -, ↭-empty-inv (↭-sym ρ) , ↭-refl

+
  ↭-map-inv {x ∷ []} ρ                                                  = -, ↭-singleton-inv (↭-sym ρ) , ↭-refl

+
  ↭-map-inv {_ ∷ _ ∷ _} refl                                            = -, refl , ↭-refl

+
  ↭-map-inv {_ ∷ _ ∷ _} (prep _ ρ)    with _ , refl , ρ′ ← ↭-map-inv ρ  = -, refl , prep _ ρ′

+
  ↭-map-inv {_ ∷ _ ∷ _} (swap _ _ ρ)  with _ , refl , ρ′ ← ↭-map-inv ρ  = -, refl , swap _ _ ρ′

+
  ↭-map-inv {_ ∷ _ ∷ _} (trans ρ₁ ρ₂) with _ , refl , ρ₃ ← ↭-map-inv ρ₁

+
                                      with _ , refl , ρ₄ ← ↭-map-inv ρ₂ = -, refl , trans ρ₃ ρ₄

+

+
------------------------------------------------------------------------

+
-- length

+

+
↭-length : ∀ {xs ys : List A} → xs ↭ ys → length xs ≡ length ys

+
↭-length refl            = refl

+
↭-length (prep x lr)     = cong suc (↭-length lr)

+
↭-length (swap x y lr)   = cong (suc ∘ suc) (↭-length lr)

+
↭-length (trans lr₁ lr₂) = ≡.trans (↭-length lr₁) (↭-length lr₂)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
++⁺ˡ : ∀ xs {ys zs : List A} → ys ↭ zs → xs ++ ys ↭ xs ++ zs

+
++⁺ˡ []       ys↭zs = ys↭zs

+
++⁺ˡ (x ∷ xs) ys↭zs = prep x (++⁺ˡ xs ys↭zs)

+

+
++⁺ʳ : ∀ {xs ys : List A} zs → xs ↭ ys → xs ++ zs ↭ ys ++ zs

+
++⁺ʳ zs refl          = refl

+
++⁺ʳ zs (prep x ↭)    = prep x (++⁺ʳ zs ↭)

+
++⁺ʳ zs (swap x y ↭)  = swap x y (++⁺ʳ zs ↭)

+
++⁺ʳ zs (trans ↭₁ ↭₂) = trans (++⁺ʳ zs ↭₁) (++⁺ʳ zs ↭₂)

+

+
++⁺ : _++_ {A = A} Preserves₂ _↭_ ⟶ _↭_ ⟶ _↭_

+
++⁺ ws↭xs ys↭zs = trans (++⁺ʳ _ ws↭xs) (++⁺ˡ _ ys↭zs)

+

+
-- Some useful lemmas

+

+
zoom : ∀ h {t xs ys : List A} → xs ↭ ys → h ++ xs ++ t ↭ h ++ ys ++ t

+
zoom h {t} = ++⁺ˡ h ∘ ++⁺ʳ t

+

+
inject : ∀  (v : A) {ws xs ys zs} → ws ↭ ys → xs ↭ zs →

+
        ws ++ [ v ] ++ xs ↭ ys ++ [ v ] ++ zs

+
inject v ws↭ys xs↭zs = trans (++⁺ˡ _ (prep v xs↭zs)) (++⁺ʳ _ ws↭ys)

+

+
shift : ∀ v (xs ys : List A) → xs ++ [ v ] ++ ys ↭ v ∷ xs ++ ys

+
shift v []       ys = refl

+
shift v (x ∷ xs) ys = begin

+
  x ∷ (xs ++ [ v ] ++ ys) <⟨ shift v xs ys ⟩

+
  x ∷ v ∷ xs ++ ys        <<⟨ refl ⟩

+
  v ∷ x ∷ xs ++ ys        ∎

+
  where open PermutationReasoning

+

+
drop-mid-≡ : ∀ {x : A} ws xs {ys} {zs} →

+
             ws ++ [ x ] ++ ys ≡ xs ++ [ x ] ++ zs →

+
             ws ++ ys ↭ xs ++ zs

+
drop-mid-≡ []       []       eq   with cong tail eq

+
drop-mid-≡ []       []       eq   | refl = refl

+
drop-mid-≡ []       (x ∷ xs) refl = shift _ xs _

+
drop-mid-≡ (w ∷ ws) []       refl = ↭-sym (shift _ ws _)

+
drop-mid-≡ (w ∷ ws) (x ∷ xs) eq with Lₚ.∷-injective eq

+
... | refl , eq′ = prep w (drop-mid-≡ ws xs eq′)

+

+
drop-mid : ∀ {x : A} ws xs {ys zs} →

+
           ws ++ [ x ] ++ ys ↭ xs ++ [ x ] ++ zs →

+
           ws ++ ys ↭ xs ++ zs

+
drop-mid {A = A} {x} ws xs p = drop-mid′ p ws xs refl refl

+
  where

+
  drop-mid′ : ∀ {l′ l″ : List A} → l′ ↭ l″ →

+
              ∀ ws xs {ys zs} →

+
              ws ++ [ x ] ++ ys ≡ l′ →

+
              xs ++ [ x ] ++ zs ≡ l″ →

+
              ws ++ ys ↭ xs ++ zs

+
  drop-mid′ refl         ws           xs           refl eq   = drop-mid-≡ ws xs (≡.sym eq)

+
  drop-mid′ (prep x p)   []           []           refl eq   with cong tail eq

+
  drop-mid′ (prep x p)   []           []           refl eq   | refl = p

+
  drop-mid′ (prep x p)   []           (x ∷ xs)     refl refl = trans p (shift _ _ _)

+
  drop-mid′ (prep x p)   (w ∷ ws)     []           refl refl = trans (↭-sym (shift _ _ _)) p

+
  drop-mid′ (prep x p)   (w ∷ ws)     (x ∷ xs)     refl refl = prep _ (drop-mid′ p ws xs refl refl)

+
  drop-mid′ (swap y z p) []           []           refl refl = prep _ p

+
  drop-mid′ (swap y z p) []           (x ∷ [])     refl eq   with cong {B = List _}

+
                                                                       (λ { (x ∷ _ ∷ xs) → x ∷ xs

+
                                                                          ; _            → []

+
                                                                          })

+
                                                                       eq

+
  drop-mid′ (swap y z p) []           (x ∷ [])     refl eq   | refl = prep _ p

+
  drop-mid′ (swap y z p) []           (x ∷ _ ∷ xs) refl refl = prep _ (trans p (shift _ _ _))

+
  drop-mid′ (swap y z p) (w ∷ [])     []           refl eq   with cong tail eq

+
  drop-mid′ (swap y z p) (w ∷ [])     []           refl eq   | refl = prep _ p

+
  drop-mid′ (swap y z p) (w ∷ x ∷ ws) []           refl refl = prep _ (trans (↭-sym (shift _ _ _)) p)

+
  drop-mid′ (swap y y p) (y ∷ [])     (y ∷ [])     refl refl = prep _ p

+
  drop-mid′ (swap y z p) (y ∷ [])     (z ∷ y ∷ xs) refl refl = begin

+
      _ ∷ _             <⟨ p ⟩

+
      _ ∷ (xs ++ _ ∷ _) <⟨ shift _ _ _ ⟩

+
      _ ∷ _ ∷ xs ++ _   <<⟨ refl ⟩

+
      _ ∷ _ ∷ xs ++ _   ∎

+
      where open PermutationReasoning

+
  drop-mid′ (swap y z p) (y ∷ z ∷ ws) (z ∷ [])     refl refl = begin

+
      _ ∷ _ ∷ ws ++ _   <<⟨ refl ⟩

+
      _ ∷ (_ ∷ ws ++ _) <⟨ ↭-sym (shift _ _ _) ⟩

+
      _ ∷ (ws ++ _ ∷ _) <⟨ p ⟩

+
      _ ∷ _             ∎

+
      where open PermutationReasoning

+
  drop-mid′ (swap y z p) (y ∷ z ∷ ws) (z ∷ y ∷ xs) refl refl = swap y z (drop-mid′ p _ _ refl refl)

+
  drop-mid′ (trans p₁ p₂) ws  xs refl refl with ∈-∃++ (∈-resp-↭ p₁ (∈-insert ws))

+
  ... | (h , t , refl) = trans (drop-mid′ p₁ ws h refl refl) (drop-mid′ p₂ h xs refl refl)

+

+
-- Algebraic properties

+

+
++-identityˡ : LeftIdentity {A = List A} _↭_ [] _++_

+
++-identityˡ xs = refl

+

+
++-identityʳ : RightIdentity {A = List A} _↭_ [] _++_

+
++-identityʳ xs = ↭-reflexive (Lₚ.++-identityʳ xs)

+

+
++-identity : Identity {A = List A} _↭_ [] _++_

+
++-identity = ++-identityˡ , ++-identityʳ

+

+
++-assoc : Associative {A = List A} _↭_ _++_

+
++-assoc xs ys zs = ↭-reflexive (Lₚ.++-assoc xs ys zs)

+

+
++-comm : Commutative {A = List A} _↭_ _++_

+
++-comm []       ys = ↭-sym (++-identityʳ ys)

+
++-comm (x ∷ xs) ys = begin

+
  x ∷ xs ++ ys   <⟨ ++-comm xs ys ⟩

+
  x ∷ ys ++ xs   ↭⟨ shift x ys xs ⟨

+
  ys ++ (x ∷ xs) ∎

+
  where open PermutationReasoning

+

+
++-isMagma : IsMagma {A = List A} _↭_ _++_

+
++-isMagma = record

+
  { isEquivalence = ↭-isEquivalence

+
  ; ∙-cong        = ++⁺

+
  }

+

+
++-isSemigroup : IsSemigroup {A = List A} _↭_ _++_

+
++-isSemigroup = record

+
  { isMagma = ++-isMagma

+
  ; assoc   = ++-assoc

+
  }

+

+
++-isMonoid : IsMonoid {A = List A} _↭_ _++_ []

+
++-isMonoid = record

+
  { isSemigroup = ++-isSemigroup

+
  ; identity    = ++-identity

+
  }

+

+
++-isCommutativeMonoid : IsCommutativeMonoid {A = List A} _↭_ _++_ []

+
++-isCommutativeMonoid = record

+
  { isMonoid = ++-isMonoid

+
  ; comm     = ++-comm

+
  }

+

+
module _ {a} {A : Set a} where

+

+
  ++-magma : Magma _ _

+
  ++-magma = record

+
    { isMagma = ++-isMagma {A = A}

+
    }

+

+
  ++-semigroup : Semigroup a _

+
  ++-semigroup = record

+
    { isSemigroup = ++-isSemigroup {A = A}

+
    }

+

+
  ++-monoid : Monoid a _

+
  ++-monoid = record

+
    { isMonoid = ++-isMonoid {A = A}

+
    }

+

+
  ++-commutativeMonoid : CommutativeMonoid _ _

+
  ++-commutativeMonoid = record

+
    { isCommutativeMonoid = ++-isCommutativeMonoid {A = A}

+
    }

+

+
-- Another useful lemma

+

+
shifts : ∀ xs ys {zs : List A} → xs ++ ys ++ zs ↭ ys ++ xs ++ zs

+
shifts xs ys {zs} = begin

+
   xs ++ ys  ++ zs ↭⟨ ++-assoc xs ys zs ⟨

+
  (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩

+
  (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩

+
   ys ++ xs  ++ zs ∎

+
   where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- _∷_

+

+
drop-∷ : ∀ {x : A} {xs ys} → x ∷ xs ↭ x ∷ ys → xs ↭ ys

+
drop-∷ = drop-mid [] []

+

+
------------------------------------------------------------------------

+
-- _∷ʳ_

+

+
∷↭∷ʳ : ∀ (x : A) xs → x ∷ xs ↭ xs ∷ʳ x

+
∷↭∷ʳ x xs = ↭-sym (begin

+
  xs ++ [ x ]   ↭⟨ shift x xs [] ⟩

+
  x ∷ xs ++ []  ≡⟨ Lₚ.++-identityʳ _ ⟩

+
  x ∷ xs        ∎)

+
  where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- ʳ++

+

+
++↭ʳ++ : ∀ (xs ys : List A) → xs ++ ys ↭ xs ʳ++ ys

+
++↭ʳ++ []       ys = ↭-refl

+
++↭ʳ++ (x ∷ xs) ys = ↭-trans (↭-sym (shift x xs ys)) (++↭ʳ++ xs (x ∷ ys))

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
↭-reverse : (xs : List A) → reverse xs ↭ xs

+
↭-reverse [] = ↭-refl

+
↭-reverse (x ∷ xs) = begin

+
  reverse (x ∷ xs) ≡⟨ Lₚ.unfold-reverse x xs ⟩

+
  reverse xs ∷ʳ x ↭⟨ ∷↭∷ʳ x (reverse xs) ⟨

+
  x ∷ reverse xs   ↭⟨ prep x (↭-reverse xs) ⟩

+
  x ∷ xs ∎

+
  where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- merge

+

+
module _ {ℓ} {R : Rel A ℓ} (R? : Decidable R) where

+

+
  merge-↭ : ∀ xs ys → merge R? xs ys ↭ xs ++ ys

+
  merge-↭ []       []       = ↭-refl

+
  merge-↭ []       (y ∷ ys) = ↭-refl

+
  merge-↭ (x ∷ xs) []       = ↭-sym (++-identityʳ (x ∷ xs))

+
  merge-↭ (x ∷ xs) (y ∷ ys)

+
    with does (R? x y) | merge-↭ xs (y ∷ ys) | merge-↭ (x ∷ xs) ys

+
  ... | true  | rec | _   = prep x rec

+
  ... | false | _   | rec = begin

+
    y ∷ merge R? (x ∷ xs) ys <⟨ rec ⟩

+
    y ∷ x ∷ xs ++ ys         ↭⟨ shift y (x ∷ xs) ys ⟨

+
    (x ∷ xs) ++ y ∷ ys       ≡⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟨

+
    x ∷ xs ++ y ∷ ys         ∎

+
    where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- product

+

+
product-↭ : product Preserves _↭_ ⟶ _≡_

+
product-↭ refl = refl

+
product-↭ (prep x r) = cong (x *_) (product-↭ r)

+
product-↭ (trans r s) = ≡.trans (product-↭ r) (product-↭ s)

+
product-↭ (swap {xs} {ys} x y r) = begin

+
  x * (y * product xs) ≡˘⟨ *-assoc x y (product xs) ⟩

+
  (x * y) * product xs ≡⟨ cong₂ _*_ (*-comm x y) (product-↭ r) ⟩

+
  (y * x) * product ys ≡⟨ *-assoc y x (product ys) ⟩

+
  y * (x * product ys) ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- catMaybes

+

+
catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys

+
catMaybes-↭ refl                         = refl

+
catMaybes-↭ (trans xs↭ ↭ys)              = trans (catMaybes-↭ xs↭) (catMaybes-↭ ↭ys)

+
catMaybes-↭ (prep nothing  xs↭)          = catMaybes-↭ xs↭

+
catMaybes-↭ (prep (just x) xs↭)          = prep x $ catMaybes-↭ xs↭

+
catMaybes-↭ (swap nothing  nothing  xs↭) = catMaybes-↭ xs↭

+
catMaybes-↭ (swap nothing  (just y) xs↭) = prep y $ catMaybes-↭ xs↭

+
catMaybes-↭ (swap (just x) nothing  xs↭) = prep x $ catMaybes-↭ xs↭

+
catMaybes-↭ (swap (just x) (just y) xs↭) = swap x y $ catMaybes-↭ xs↭

+

+
------------------------------------------------------------------------

+
-- mapMaybe

+

+
mapMaybe-↭ : (f : A → Maybe B) → xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys

+
mapMaybe-↭ f = catMaybes-↭ ∘ map⁺ f

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Permutation.Propositional.html b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Propositional.html
new file mode 100644
index 000000000..8d342fba1
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Propositional.html
@@ -0,0 +1,177 @@
+
+Data.List.Relation.Binary.Permutation.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An inductive definition for the permutation relation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Permutation.Propositional

+
  {a} {A : Set a} where

+

+
open import Data.List.Base using (List; []; _∷_)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions using (Reflexive; Transitive)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)

+
import Relation.Binary.Reasoning.Setoid as EqReasoning

+
open import Relation.Binary.Reasoning.Syntax

+

+
------------------------------------------------------------------------

+
-- An inductive definition of permutation

+

+
-- Note that one would expect that this would be defined in terms of

+
-- `Permutation.Setoid`. This is not currently the case as it involves

+
-- adding in a bunch of trivial `_≡_` proofs to the constructors which

+
-- a) adds noise and b) prevents easy access to the variables `x`, `y`.

+
-- This may be changed in future when a better solution is found.

+

+
infix 3 _↭_

+

+
data _↭_ : Rel (List A) a where

+
  refl  : ∀ {xs}        → xs ↭ xs

+
  prep  : ∀ {xs ys} x   → xs ↭ ys → x ∷ xs ↭ x ∷ ys

+
  swap  : ∀ {xs ys} x y → xs ↭ ys → x ∷ y ∷ xs ↭ y ∷ x ∷ ys

+
  trans : ∀ {xs ys zs}  → xs ↭ ys → ys ↭ zs → xs ↭ zs

+

+
------------------------------------------------------------------------

+
-- _↭_ is an equivalence

+

+
↭-reflexive : _≡_ ⇒ _↭_

+
↭-reflexive refl = refl

+

+
↭-refl : Reflexive _↭_

+
↭-refl = refl

+

+
↭-sym : ∀ {xs ys} → xs ↭ ys → ys ↭ xs

+
↭-sym refl                = refl

+
↭-sym (prep x xs↭ys)      = prep x (↭-sym xs↭ys)

+
↭-sym (swap x y xs↭ys)    = swap y x (↭-sym xs↭ys)

+
↭-sym (trans xs↭ys ys↭zs) = trans (↭-sym ys↭zs) (↭-sym xs↭ys)

+

+
-- A smart version of trans that avoids unnecessary `refl`s (see #1113).

+
↭-trans : Transitive _↭_

+
↭-trans refl ρ₂ = ρ₂

+
↭-trans ρ₁ refl = ρ₁

+
↭-trans ρ₁ ρ₂   = trans ρ₁ ρ₂

+

+
↭-isEquivalence : IsEquivalence _↭_

+
↭-isEquivalence = record

+
  { refl  = refl

+
  ; sym   = ↭-sym

+
  ; trans = ↭-trans

+
  }

+

+
↭-setoid : Setoid _ _

+
↭-setoid = record

+
  { isEquivalence = ↭-isEquivalence

+
  }

+

+
------------------------------------------------------------------------

+
-- A reasoning API to chain permutation proofs and allow "zooming in"

+
-- to localised reasoning.

+

+
module PermutationReasoning where

+

+
  private module Base = EqReasoning ↭-setoid

+

+
  open Base public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+
    renaming (≈-go to ↭-go)

+

+
  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ ↭-go ↭-sym public

+

+
  -- Some extra combinators that allow us to skip certain elements

+

+
  infixr 2 step-swap step-prep

+

+
  -- Skip reasoning on the first element

+
  step-prep : ∀ x xs {ys zs : List A} → (x ∷ ys) IsRelatedTo zs →

+
              xs ↭ ys → (x ∷ xs) IsRelatedTo zs

+
  step-prep x xs rel xs↭ys = relTo (trans (prep x xs↭ys) (begin rel))

+

+
  -- Skip reasoning about the first two elements

+
  step-swap : ∀ x y xs {ys zs : List A} → (y ∷ x ∷ ys) IsRelatedTo zs →

+
              xs ↭ ys → (x ∷ y ∷ xs) IsRelatedTo zs

+
  step-swap x y xs rel xs↭ys = relTo (trans (swap x y xs↭ys) (begin rel))

+

+
  syntax step-prep x xs y↭z x↭y = x ∷ xs <⟨ x↭y ⟩ y↭z

+
  syntax step-swap x y xs y↭z x↭y = x ∷ y ∷ xs <<⟨ x↭y ⟩ y↭z

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Permutation.Setoid.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Setoid.Properties.html
new file mode 100644
index 000000000..68fe778f6
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Setoid.Properties.html
@@ -0,0 +1,582 @@
+
+Data.List.Relation.Binary.Permutation.Setoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of permutations using setoid equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core

+
  using (Rel; _⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions as B hiding (Decidable)

+

+
module Data.List.Relation.Binary.Permutation.Setoid.Properties

+
  {a ℓ} (S : Setoid a ℓ)

+
  where

+

+
open import Algebra

+
import Algebra.Properties.CommutativeMonoid as ACM

+
open import Data.Bool.Base using (true; false)

+
open import Data.List.Base as List hiding (head; tail)

+
open import Data.List.Relation.Binary.Pointwise as Pointwise

+
  using (Pointwise; head; tail)

+
import Data.List.Relation.Binary.Equality.Setoid as Equality

+
import Data.List.Relation.Binary.Permutation.Setoid as Permutation

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.AllPairs using (AllPairs; []; _∷_)

+
import Data.List.Relation.Unary.Unique.Setoid as Unique

+
import Data.List.Membership.Setoid as Membership

+
open import Data.List.Membership.Setoid.Properties using (∈-∃++; ∈-insert)

+
import Data.List.Properties as Lₚ

+
open import Data.Nat.Base using (ℕ; suc; _<_; z<s; _+_)

+
open import Data.Nat.Induction

+
open import Data.Nat.Properties

+
open import Data.Product.Base using (_,_; _×_; ∃; ∃₂; proj₁; proj₂)

+
open import Function.Base using (_∘_; _⟨_⟩_; flip)

+
open import Level using (Level; _⊔_)

+
open import Relation.Unary using (Pred; Decidable)

+
import Relation.Binary.Reasoning.Setoid as RelSetoid

+
open import Relation.Binary.Properties.Setoid S using (≉-resp₂)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_ ; refl; sym; cong; cong₂; subst; _≢_)

+
open import Relation.Nullary.Decidable using (yes; no; does)

+
open import Relation.Nullary.Negation using (contradiction)

+

+
private

+
  variable

+
    b p r : Level

+

+
open Setoid S using (_≈_) renaming (Carrier to A; refl to ≈-refl; sym to ≈-sym; trans to ≈-trans)

+
open Permutation S

+
open Membership S

+
open Unique S using (Unique)

+
open module ≋ = Equality S

+
  using (_≋_; []; _∷_; ≋-refl; ≋-sym; ≋-trans; All-resp-≋; Any-resp-≋; AllPairs-resp-≋)

+
open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- Relationships to other predicates

+
------------------------------------------------------------------------

+

+
All-resp-↭ : ∀ {P : Pred A p} → P Respects _≈_ → (All P) Respects _↭_

+
All-resp-↭ resp (refl xs≋ys)   pxs             = All-resp-≋ resp xs≋ys pxs

+
All-resp-↭ resp (prep x≈y p)   (px ∷ pxs)      = resp x≈y px ∷ All-resp-↭ resp p pxs

+
All-resp-↭ resp (swap ≈₁ ≈₂ p) (px ∷ py ∷ pxs) = resp ≈₂ py ∷ resp ≈₁ px ∷ All-resp-↭ resp p pxs

+
All-resp-↭ resp (trans p₁ p₂)  pxs             = All-resp-↭ resp p₂ (All-resp-↭ resp p₁ pxs)

+

+
Any-resp-↭ : ∀ {P : Pred A p} → P Respects _≈_ → (Any P) Respects _↭_

+
Any-resp-↭ resp (refl xs≋ys) pxs                 = Any-resp-≋ resp xs≋ys pxs

+
Any-resp-↭ resp (prep x≈y p) (here px)           = here (resp x≈y px)

+
Any-resp-↭ resp (prep x≈y p) (there pxs)         = there (Any-resp-↭ resp p pxs)

+
Any-resp-↭ resp (swap x y p) (here px)           = there (here (resp x px))

+
Any-resp-↭ resp (swap x y p) (there (here px))   = here (resp y px)

+
Any-resp-↭ resp (swap x y p) (there (there pxs)) = there (there (Any-resp-↭ resp p pxs))

+
Any-resp-↭ resp (trans p₁ p₂) pxs                = Any-resp-↭ resp p₂ (Any-resp-↭ resp p₁ pxs)

+

+
AllPairs-resp-↭ : ∀ {R : Rel A r} → Symmetric R → R Respects₂ _≈_ → (AllPairs R) Respects _↭_

+
AllPairs-resp-↭ sym resp (refl xs≋ys)     pxs             = AllPairs-resp-≋ resp xs≋ys pxs

+
AllPairs-resp-↭ sym resp (prep x≈y p)     (∼ ∷ pxs)       =

+
  All-resp-↭ (proj₁ resp) p (All.map (proj₂ resp x≈y) ∼) ∷

+
  AllPairs-resp-↭ sym resp p pxs

+
AllPairs-resp-↭ sym resp@(rʳ , rˡ) (swap eq₁ eq₂ p) ((∼₁ ∷ ∼₂) ∷ ∼₃ ∷ pxs) =

+
  (sym (rʳ eq₂ (rˡ eq₁ ∼₁)) ∷ All-resp-↭ rʳ p (All.map (rˡ eq₂) ∼₃)) ∷

+
  All-resp-↭ rʳ p (All.map (rˡ eq₁) ∼₂) ∷

+
  AllPairs-resp-↭ sym resp p pxs

+
AllPairs-resp-↭ sym resp (trans p₁ p₂)    pxs             =

+
  AllPairs-resp-↭ sym resp p₂ (AllPairs-resp-↭ sym resp p₁ pxs)

+

+
∈-resp-↭ : ∀ {x} → (x ∈_) Respects _↭_

+
∈-resp-↭ = Any-resp-↭ (flip ≈-trans)

+

+
Unique-resp-↭ : Unique Respects _↭_

+
Unique-resp-↭ = AllPairs-resp-↭ (_∘ ≈-sym) ≉-resp₂

+

+
------------------------------------------------------------------------

+
-- Relationships to other relations

+
------------------------------------------------------------------------

+

+
≋⇒↭ : _≋_ ⇒ _↭_

+
≋⇒↭ = refl

+

+
↭-respʳ-≋ : _↭_ Respectsʳ _≋_

+
↭-respʳ-≋ xs≋ys               (refl zs≋xs)         = refl (≋-trans zs≋xs xs≋ys)

+
↭-respʳ-≋ (x≈y ∷ xs≋ys)       (prep eq zs↭xs)      = prep (≈-trans eq x≈y) (↭-respʳ-≋ xs≋ys zs↭xs)

+
↭-respʳ-≋ (x≈y ∷ w≈z ∷ xs≋ys) (swap eq₁ eq₂ zs↭xs) = swap (≈-trans eq₁ w≈z) (≈-trans eq₂ x≈y) (↭-respʳ-≋ xs≋ys zs↭xs)

+
↭-respʳ-≋ xs≋ys               (trans ws↭zs zs↭xs)  = trans ws↭zs (↭-respʳ-≋ xs≋ys zs↭xs)

+

+
↭-respˡ-≋ : _↭_ Respectsˡ _≋_

+
↭-respˡ-≋ xs≋ys               (refl ys≋zs)         = refl (≋-trans (≋-sym xs≋ys) ys≋zs)

+
↭-respˡ-≋ (x≈y ∷ xs≋ys)       (prep eq zs↭xs)      = prep (≈-trans (≈-sym x≈y) eq) (↭-respˡ-≋ xs≋ys zs↭xs)

+
↭-respˡ-≋ (x≈y ∷ w≈z ∷ xs≋ys) (swap eq₁ eq₂ zs↭xs) = swap (≈-trans (≈-sym x≈y) eq₁) (≈-trans (≈-sym w≈z) eq₂) (↭-respˡ-≋ xs≋ys zs↭xs)

+
↭-respˡ-≋ xs≋ys               (trans ws↭zs zs↭xs)  = trans (↭-respˡ-≋ xs≋ys ws↭zs) zs↭xs

+

+
------------------------------------------------------------------------

+
-- Properties of steps

+
------------------------------------------------------------------------

+

+
0<steps : ∀ {xs ys} (xs↭ys : xs ↭ ys) → 0 < steps xs↭ys

+
0<steps (refl _)             = z<s

+
0<steps (prep eq xs↭ys)      = m<n⇒m<1+n (0<steps xs↭ys)

+
0<steps (swap eq₁ eq₂ xs↭ys) = m<n⇒m<1+n (0<steps xs↭ys)

+
0<steps (trans xs↭ys xs↭ys₁) =

+
  <-≤-trans (0<steps xs↭ys) (m≤m+n (steps xs↭ys) (steps xs↭ys₁))

+

+
steps-respˡ : ∀ {xs ys zs} (ys≋xs : ys ≋ xs) (ys↭zs : ys ↭ zs) →

+
              steps (↭-respˡ-≋ ys≋xs ys↭zs) ≡ steps ys↭zs

+
steps-respˡ _               (refl _)            = refl

+
steps-respˡ (_ ∷ ys≋xs)     (prep _ ys↭zs)      = cong suc (steps-respˡ ys≋xs ys↭zs)

+
steps-respˡ (_ ∷ _ ∷ ys≋xs) (swap _ _ ys↭zs)    = cong suc (steps-respˡ ys≋xs ys↭zs)

+
steps-respˡ ys≋xs           (trans ys↭ws ws↭zs) = cong (_+ steps ws↭zs) (steps-respˡ ys≋xs ys↭ws)

+

+
steps-respʳ : ∀ {xs ys zs} (xs≋ys : xs ≋ ys) (zs↭xs : zs ↭ xs) →

+
              steps (↭-respʳ-≋ xs≋ys zs↭xs) ≡ steps zs↭xs

+
steps-respʳ _               (refl _)            = refl

+
steps-respʳ (_ ∷ ys≋xs)     (prep _ ys↭zs)      = cong suc (steps-respʳ ys≋xs ys↭zs)

+
steps-respʳ (_ ∷ _ ∷ ys≋xs) (swap _ _ ys↭zs)    = cong suc (steps-respʳ ys≋xs ys↭zs)

+
steps-respʳ ys≋xs           (trans ys↭ws ws↭zs) = cong (steps ys↭ws +_) (steps-respʳ ys≋xs ws↭zs)

+

+
------------------------------------------------------------------------

+
-- Properties of list functions

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {ℓ} (T : Setoid b ℓ) where

+

+
  open Setoid T using () renaming (_≈_ to _≈′_)

+
  open Permutation T using () renaming (_↭_ to _↭′_)

+

+
  map⁺ : ∀ {f} → f Preserves _≈_ ⟶ _≈′_ →

+
         ∀ {xs ys} → xs ↭ ys → map f xs ↭′ map f ys

+
  map⁺ pres (refl xs≋ys)  = refl (Pointwise.map⁺ _ _ (Pointwise.map pres xs≋ys))

+
  map⁺ pres (prep x p)    = prep (pres x) (map⁺ pres p)

+
  map⁺ pres (swap x y p)  = swap (pres x) (pres y) (map⁺ pres p)

+
  map⁺ pres (trans p₁ p₂) = trans (map⁺ pres p₁) (map⁺ pres p₂)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
shift : ∀ {v w} → v ≈ w → (xs ys : List A) → xs ++ [ v ] ++ ys ↭ w ∷ xs ++ ys

+
shift {v} {w} v≈w []       ys = prep v≈w ↭-refl

+
shift {v} {w} v≈w (x ∷ xs) ys = begin

+
  x ∷ (xs ++ [ v ] ++ ys) <⟨ shift v≈w xs ys ⟩

+
  x ∷ w ∷ xs ++ ys        <<⟨ ↭-refl ⟩

+
  w ∷ x ∷ xs ++ ys        ∎

+

+
↭-shift : ∀ {v} (xs ys : List A) → xs ++ [ v ] ++ ys ↭ v ∷ xs ++ ys

+
↭-shift = shift ≈-refl

+

+
++⁺ˡ : ∀ xs {ys zs : List A} → ys ↭ zs → xs ++ ys ↭ xs ++ zs

+
++⁺ˡ []       ys↭zs = ys↭zs

+
++⁺ˡ (x ∷ xs) ys↭zs = ↭-prep _ (++⁺ˡ xs ys↭zs)

+

+
++⁺ʳ : ∀ {xs ys : List A} zs → xs ↭ ys → xs ++ zs ↭ ys ++ zs

+
++⁺ʳ zs (refl xs≋ys)  = refl (Pointwise.++⁺ xs≋ys ≋-refl)

+
++⁺ʳ zs (prep x ↭)    = prep x (++⁺ʳ zs ↭)

+
++⁺ʳ zs (swap x y ↭)  = swap x y (++⁺ʳ zs ↭)

+
++⁺ʳ zs (trans ↭₁ ↭₂) = trans (++⁺ʳ zs ↭₁) (++⁺ʳ zs ↭₂)

+

+
++⁺ : _++_ Preserves₂ _↭_ ⟶ _↭_ ⟶ _↭_

+
++⁺ ws↭xs ys↭zs = trans (++⁺ʳ _ ws↭xs) (++⁺ˡ _ ys↭zs)

+

+
-- Algebraic properties

+

+
++-identityˡ : LeftIdentity _↭_ [] _++_

+
++-identityˡ xs = ↭-refl

+

+
++-identityʳ : RightIdentity _↭_ [] _++_

+
++-identityʳ xs = ↭-reflexive (Lₚ.++-identityʳ xs)

+

+
++-identity : Identity _↭_ [] _++_

+
++-identity = ++-identityˡ , ++-identityʳ

+

+
++-assoc : Associative _↭_ _++_

+
++-assoc xs ys zs = ↭-reflexive (Lₚ.++-assoc xs ys zs)

+

+
++-comm : Commutative _↭_ _++_

+
++-comm []       ys = ↭-sym (++-identityʳ ys)

+
++-comm (x ∷ xs) ys = begin

+
  x ∷ xs ++ ys   <⟨ ++-comm xs ys ⟩

+
  x ∷ ys ++ xs   ↭⟨ ↭-shift ys xs ⟨

+
  ys ++ (x ∷ xs) ∎

+

+
-- Structures

+

+
++-isMagma : IsMagma _↭_ _++_

+
++-isMagma = record

+
  { isEquivalence = ↭-isEquivalence

+
  ; ∙-cong        = ++⁺

+
  }

+

+
++-isSemigroup : IsSemigroup _↭_ _++_

+
++-isSemigroup = record

+
  { isMagma = ++-isMagma

+
  ; assoc   = ++-assoc

+
  }

+

+
++-isMonoid : IsMonoid _↭_ _++_ []

+
++-isMonoid = record

+
  { isSemigroup = ++-isSemigroup

+
  ; identity    = ++-identity

+
  }

+

+
++-isCommutativeMonoid : IsCommutativeMonoid _↭_ _++_ []

+
++-isCommutativeMonoid = record

+
  { isMonoid = ++-isMonoid

+
  ; comm     = ++-comm

+
  }

+

+
-- Bundles

+

+
++-magma : Magma a (a ⊔ ℓ)

+
++-magma = record

+
  { isMagma = ++-isMagma

+
  }

+

+
++-semigroup : Semigroup a (a ⊔ ℓ)

+
++-semigroup = record

+
  { isSemigroup = ++-isSemigroup

+
  }

+

+
++-monoid : Monoid a (a ⊔ ℓ)

+
++-monoid = record

+
  { isMonoid = ++-isMonoid

+
  }

+

+
++-commutativeMonoid : CommutativeMonoid a (a ⊔ ℓ)

+
++-commutativeMonoid = record

+
  { isCommutativeMonoid = ++-isCommutativeMonoid

+
  }

+

+
-- Some other useful lemmas

+

+
zoom : ∀ h {t xs ys : List A} → xs ↭ ys → h ++ xs ++ t ↭ h ++ ys ++ t

+
zoom h {t} = ++⁺ˡ h ∘ ++⁺ʳ t

+

+
inject : ∀ (v : A) {ws xs ys zs} → ws ↭ ys → xs ↭ zs →

+
         ws ++ [ v ] ++ xs ↭ ys ++ [ v ] ++ zs

+
inject v ws↭ys xs↭zs = trans (++⁺ˡ _ (↭-prep _ xs↭zs)) (++⁺ʳ _ ws↭ys)

+

+
shifts : ∀ xs ys {zs : List A} → xs ++ ys ++ zs ↭ ys ++ xs ++ zs

+
shifts xs ys {zs} = begin

+
   xs ++ ys  ++ zs ↭⟨ ++-assoc xs ys zs ⟨

+
  (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩

+
  (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩

+
   ys ++ xs  ++ zs ∎

+

+
dropMiddleElement-≋ : ∀ {x} ws xs {ys} {zs} →

+
           ws ++ [ x ] ++ ys ≋ xs ++ [ x ] ++ zs →

+
           ws ++ ys ↭ xs ++ zs

+
dropMiddleElement-≋ []       []       (_   ∷ eq) = ≋⇒↭ eq

+
dropMiddleElement-≋ []       (x ∷ xs) (w≈v ∷ eq) = ↭-respˡ-≋ (≋-sym eq) (shift w≈v xs _)

+
dropMiddleElement-≋ (w ∷ ws) []       (w≈x ∷ eq) = ↭-respʳ-≋ eq (↭-sym (shift (≈-sym w≈x) ws _))

+
dropMiddleElement-≋ (w ∷ ws) (x ∷ xs) (w≈x ∷ eq) = prep w≈x (dropMiddleElement-≋ ws xs eq)

+

+
dropMiddleElement : ∀ {v} ws xs {ys zs} →

+
                    ws ++ [ v ] ++ ys ↭ xs ++ [ v ] ++ zs →

+
                    ws ++ ys ↭ xs ++ zs

+
dropMiddleElement {v} ws xs {ys} {zs} p = helper p ws xs ≋-refl ≋-refl

+
  where

+
  lemma : ∀ {w x y z} → w ≈ x → x ≈ y → z ≈ y → w ≈ z

+
  lemma w≈x x≈y z≈y = ≈-trans (≈-trans w≈x x≈y) (≈-sym z≈y)

+

+
  open PermutationReasoning

+

+
  -- The l′ & l″ could be eliminated at the cost of making the `trans` case

+
  -- much more difficult to prove. At the very least would require using `Acc`.

+
  helper : ∀ {l′ l″ : List A} → l′ ↭ l″ →

+
           ∀ ws xs {ys zs : List A} →

+
           ws ++ [ v ] ++ ys ≋ l′ →

+
           xs ++ [ v ] ++ zs ≋ l″ →

+
           ws ++ ys ↭ xs ++ zs

+
  helper {as}     {bs}     (refl eq3) ws xs {ys} {zs} eq1 eq2 =

+
    dropMiddleElement-≋ ws xs (≋-trans (≋-trans eq1 eq3) (≋-sym eq2))

+
  helper {_ ∷ as} {_ ∷ bs} (prep _ as↭bs) [] [] {ys} {zs} (_ ∷ ys≋as) (_ ∷ zs≋bs) = begin

+
    ys               ≋⟨  ys≋as ⟩

+
    as               ↭⟨  as↭bs ⟩

+
    bs               ≋⟨ zs≋bs ⟨

+
    zs               ∎

+
  helper {_ ∷ as} {_ ∷ bs} (prep a≈b as↭bs) [] (x ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    ys               ≋⟨  ≋₁ ⟩

+
    as               ↭⟨  as↭bs ⟩

+
    bs               ≋⟨ ≋₂ ⟨

+
    xs ++ v ∷ zs     ↭⟨  shift (lemma ≈₁ a≈b ≈₂) xs zs ⟩

+
    x ∷ xs ++ zs     ∎

+
  helper {_ ∷ as} {_ ∷ bs} (prep v≈w p) (w ∷ ws) [] {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    w ∷ ws ++ ys     ↭⟨  ↭-sym (shift (lemma ≈₂ (≈-sym v≈w) ≈₁) ws ys) ⟩

+
    ws ++ v ∷ ys     ≋⟨  ≋₁ ⟩

+
    as               ↭⟨  p ⟩

+
    bs               ≋⟨ ≋₂ ⟨

+
    zs               ∎

+
  helper {_ ∷ as} {_ ∷ bs} (prep w≈x p) (w ∷ ws) (x ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    w ∷ ws ++ ys     ↭⟨ prep (lemma ≈₁ w≈x ≈₂) (helper p ws xs ≋₁ ≋₂) ⟩

+
    x ∷ xs ++ zs     ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap v≈x y≈v p) [] [] {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    ys               ≋⟨  ≋₁ ⟩

+
    a ∷ as           ↭⟨  prep (≈-trans (≈-trans (≈-trans y≈v (≈-sym ≈₂)) ≈₁) v≈x) p ⟩

+
    b ∷ bs           ≋⟨ ≋₂ ⟨

+
    zs               ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap v≈w y≈w p) [] (x ∷ []) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    ys               ≋⟨  ≋₁ ⟩

+
    a ∷ as           ↭⟨  prep y≈w p ⟩

+
    _ ∷ bs           ≋⟨ ≈₂ ∷ tail ≋₂ ⟨

+
    x ∷ zs           ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap v≈w y≈x p) [] (x ∷ w ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    ys               ≋⟨ ≋₁ ⟩

+
    a ∷ as           ↭⟨ prep y≈x p ⟩

+
    _ ∷ bs           ≋⟨ ≋-sym (≈₂ ∷ tail ≋₂) ⟩

+
    x ∷ xs ++ v ∷ zs ↭⟨ prep ≈-refl (shift (lemma ≈₁ v≈w (head ≋₂)) xs zs) ⟩

+
    x ∷ w ∷ xs ++ zs ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap w≈x _ p) (w ∷ []) [] {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    w ∷ ys           ≋⟨ ≈₁ ∷ tail (≋₁) ⟩

+
    _ ∷ as           ↭⟨ prep w≈x p ⟩

+
    b ∷ bs           ≋⟨ ≋-sym ≋₂ ⟩

+
    zs               ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap w≈y x≈v p) (w ∷ x ∷ ws) [] {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    w ∷ x ∷ ws ++ ys ↭⟨ prep ≈-refl (↭-sym (shift (lemma ≈₂ (≈-sym x≈v) (head ≋₁)) ws ys)) ⟩

+
    w ∷ ws ++ v ∷ ys ≋⟨ ≈₁ ∷ tail ≋₁ ⟩

+
    _ ∷ as           ↭⟨ prep w≈y p ⟩

+
    b ∷ bs           ≋⟨ ≋-sym ≋₂ ⟩

+
    zs               ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap x≈v v≈y p) (x ∷ []) (y ∷ []) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    x ∷ ys           ≋⟨ ≈₁ ∷ tail ≋₁ ⟩

+
    _ ∷ as           ↭⟨ prep (≈-trans x≈v (≈-trans (≈-sym (head ≋₂)) (≈-trans (head ≋₁) v≈y))) p ⟩

+
    _ ∷ bs           ≋⟨ ≋-sym (≈₂ ∷ tail ≋₂) ⟩

+
    y ∷ zs           ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap y≈w v≈z p) (y ∷ []) (z ∷ w ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    y ∷ ys           ≋⟨ ≈₁ ∷ tail ≋₁ ⟩

+
    _ ∷ as           ↭⟨ prep y≈w p ⟩

+
    _ ∷ bs           ≋⟨ ≋-sym ≋₂ ⟩

+
    w ∷ xs ++ v ∷ zs ↭⟨ ↭-prep w (↭-shift xs zs) ⟩

+
    w ∷ v ∷ xs ++ zs ↭⟨ swap ≈-refl (lemma (head ≋₁) v≈z ≈₂) ↭-refl ⟩

+
    z ∷ w ∷ xs ++ zs ∎

+
  helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap y≈v w≈z p) (y ∷ w ∷ ws) (z ∷ []) {ys} {zs}    (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin

+
    y ∷ w ∷ ws ++ ys ↭⟨ swap (lemma ≈₁ y≈v (head ≋₂)) ≈-refl ↭-refl ⟩

+
    w ∷ v ∷ ws ++ ys ↭⟨ ↭-prep w (↭-sym (↭-shift ws ys)) ⟩

+
    w ∷ ws ++ v ∷ ys ≋⟨ ≋₁ ⟩

+
    _ ∷ as           ↭⟨ prep w≈z p ⟩

+
    _ ∷ bs           ≋⟨ ≋-sym (≈₂ ∷ tail ≋₂) ⟩

+
    z ∷ zs           ∎

+
  helper (swap x≈z y≈w p) (x ∷ y ∷ ws) (w ∷ z ∷ xs) {ys} {zs} (≈₁ ∷ ≈₃ ∷ ≋₁) (≈₂ ∷ ≈₄ ∷ ≋₂) = begin

+
    x ∷ y ∷ ws ++ ys ↭⟨ swap (lemma ≈₁ x≈z ≈₄) (lemma ≈₃ y≈w ≈₂) (helper p ws xs ≋₁ ≋₂) ⟩

+
    w ∷ z ∷ xs ++ zs ∎

+
  helper {as} {bs} (trans p₁ p₂) ws xs eq1 eq2

+
    with ∈-∃++ S (∈-resp-↭ (↭-respˡ-≋ (≋-sym eq1) p₁) (∈-insert S ws ≈-refl))

+
  ... | (h , t , w , v≈w , eq) = trans

+
    (helper p₁ ws h eq1 (≋-trans (≋.++⁺ ≋-refl (v≈w ∷ ≋-refl)) (≋-sym eq)))

+
    (helper p₂ h xs (≋-trans (≋.++⁺ ≋-refl (v≈w ∷ ≋-refl)) (≋-sym eq)) eq2)

+

+
dropMiddle : ∀ {vs} ws xs {ys zs} →

+
             ws ++ vs ++ ys ↭ xs ++ vs ++ zs →

+
             ws ++ ys ↭ xs ++ zs

+
dropMiddle {[]}     ws xs p = p

+
dropMiddle {v ∷ vs} ws xs p = dropMiddle ws xs (dropMiddleElement ws xs p)

+

+
split : ∀ (v : A) as bs {xs} → xs ↭ as ++ [ v ] ++ bs → ∃₂ λ ps qs → xs ≋ ps ++ [ v ] ++ qs

+
split v as bs p = helper as bs p (<-wellFounded (steps p))

+
  where

+
  helper : ∀ as bs {xs} (p : xs ↭ as ++ [ v ] ++ bs) → Acc _<_ (steps p) →

+
           ∃₂ λ ps qs → xs ≋ ps ++ [ v ] ++ qs

+
  helper []           bs (refl eq)    _ = []         , bs , eq

+
  helper (a ∷ [])     bs (refl eq)    _ = [ a ]      , bs , eq

+
  helper (a ∷ b ∷ as) bs (refl eq)    _ = a ∷ b ∷ as , bs , eq

+
  helper []           bs (prep v≈x _) _ = [] , _ , v≈x ∷ ≋-refl

+
  helper (a ∷ as)     bs (prep eq as↭xs) (acc rec) with helper as bs as↭xs (rec ≤-refl)

+
  ... | (ps , qs , eq₂) = a ∷ ps , qs , eq ∷ eq₂

+
  helper [] (b ∷ bs)     (swap x≈b y≈v _) _ = [ b ] , _     , x≈b ∷ y≈v ∷ ≋-refl

+
  helper (a ∷ [])     bs (swap x≈v y≈a ↭) _ = []    , a ∷ _ , x≈v ∷ y≈a ∷ ≋-refl

+
  helper (a ∷ b ∷ as) bs (swap x≈b y≈a as↭xs) (acc rec) with helper as bs as↭xs (rec ≤-refl)

+
  ... | (ps , qs , eq) = b ∷ a ∷ ps , qs , x≈b ∷ y≈a ∷ eq

+
  helper as           bs (trans ↭₁ ↭₂) (acc rec) with helper as bs ↭₂ (rec (m<n+m (steps ↭₂) (0<steps ↭₁)))

+
  ... | (ps , qs , eq) = helper ps qs (↭-respʳ-≋ eq ↭₁)

+
      (rec (subst (_< _) (sym (steps-respʳ eq ↭₁)) (m<m+n (steps ↭₁) (0<steps ↭₂))))

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {p} {P : Pred A p} (P? : Decidable P) (P≈ : P Respects _≈_) where

+

+
  filter⁺ : ∀ {xs ys : List A} → xs ↭ ys → filter P? xs ↭ filter P? ys

+
  filter⁺ (refl xs≋ys)        = refl (≋.filter⁺ P? P≈ xs≋ys)

+
  filter⁺ (trans xs↭zs zs↭ys) = trans (filter⁺ xs↭zs) (filter⁺ zs↭ys)

+
  filter⁺ {x ∷ xs} {y ∷ ys} (prep x≈y xs↭ys) with P? x | P? y

+
  ... | yes _  | yes _  = prep x≈y (filter⁺ xs↭ys)

+
  ... | yes Px | no ¬Py = contradiction (P≈ x≈y Px) ¬Py

+
  ... | no ¬Px | yes Py = contradiction (P≈ (≈-sym x≈y) Py) ¬Px

+
  ... | no  _  | no  _  = filter⁺ xs↭ys

+
  filter⁺ {x ∷ w ∷ xs} {y ∷ z ∷ ys} (swap x≈z w≈y xs↭ys) with P? x | P? y

+
  filter⁺ {x ∷ w ∷ xs} {y ∷ z ∷ ys} (swap x≈z w≈y xs↭ys) | no ¬Px | no ¬Py

+
    with P? z | P? w

+
  ... | _      | yes Pw = contradiction (P≈ w≈y Pw) ¬Py

+
  ... | yes Pz | _      = contradiction (P≈ (≈-sym x≈z) Pz) ¬Px

+
  ... | no _   | no  _  = filter⁺ xs↭ys

+
  filter⁺ {x ∷ w ∷ xs} {y ∷ z ∷ ys} (swap x≈z w≈y xs↭ys) | no ¬Px | yes Py

+
    with P? z | P? w

+
  ... | _      | no ¬Pw = contradiction (P≈ (≈-sym w≈y) Py) ¬Pw

+
  ... | yes Pz | _      = contradiction (P≈ (≈-sym x≈z) Pz) ¬Px

+
  ... | no _   | yes _  = prep w≈y (filter⁺ xs↭ys)

+
  filter⁺ {x ∷ w ∷ xs} {y ∷ z ∷ ys} (swap x≈z w≈y xs↭ys)  | yes Px | no ¬Py

+
    with P? z | P? w

+
  ... | no ¬Pz | _      = contradiction (P≈ x≈z Px) ¬Pz

+
  ... | _      | yes Pw = contradiction (P≈ w≈y Pw) ¬Py

+
  ... | yes _  | no _   = prep x≈z (filter⁺ xs↭ys)

+
  filter⁺ {x ∷ w ∷ xs} {y ∷ z ∷ ys} (swap x≈z w≈y xs↭ys) | yes Px | yes Py

+
    with P? z | P? w

+
  ... | no ¬Pz | _      = contradiction (P≈ x≈z Px) ¬Pz

+
  ... | _      | no ¬Pw = contradiction (P≈ (≈-sym w≈y) Py) ¬Pw

+
  ... | yes _  | yes _  = swap x≈z w≈y (filter⁺ xs↭ys)

+

+
------------------------------------------------------------------------

+
-- partition

+

+
module _ {p} {P : Pred A p} (P? : Decidable P) where

+

+
  partition-↭ : ∀ xs → (let ys , zs = partition P? xs) → xs ↭ ys ++ zs

+
  partition-↭ []       = ↭-refl

+
  partition-↭ (x ∷ xs) with does (P? x)

+
  ... | true  = ↭-prep x (partition-↭ xs)

+
  ... | false = ↭-trans (↭-prep x (partition-↭ xs)) (↭-sym (↭-shift _ _))

+
    where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- merge

+

+
module _ {ℓ} {R : Rel A ℓ} (R? : B.Decidable R) where

+

+
  merge-↭ : ∀ xs ys → merge R? xs ys ↭ xs ++ ys

+
  merge-↭ []       []       = ↭-refl

+
  merge-↭ []       (y ∷ ys) = ↭-refl

+
  merge-↭ (x ∷ xs) []       = ↭-sym (++-identityʳ (x ∷ xs))

+
  merge-↭ (x ∷ xs) (y ∷ ys)

+
    with does (R? x y) | merge-↭ xs (y ∷ ys) | merge-↭ (x ∷ xs) ys

+
  ... | true  | rec | _   = ↭-prep x rec

+
  ... | false | _   | rec = begin

+
    y ∷ merge R? (x ∷ xs) ys <⟨ rec ⟩

+
    y ∷ x ∷ xs ++ ys         ↭⟨ ↭-shift (x ∷ xs) ys ⟨

+
    (x ∷ xs) ++ y ∷ ys       ≡⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟨

+
    x ∷ xs ++ y ∷ ys         ∎

+
    where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- _∷ʳ_

+

+
∷↭∷ʳ : ∀ (x : A) xs → x ∷ xs ↭ xs ∷ʳ x

+
∷↭∷ʳ x xs = ↭-sym (begin

+
  xs ++ [ x ]   ↭⟨ ↭-shift xs [] ⟩

+
  x ∷ xs ++ []  ≡⟨ Lₚ.++-identityʳ _ ⟩

+
  x ∷ xs        ∎)

+
  where open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- ʳ++

+

+
++↭ʳ++ : ∀ (xs ys : List A) → xs ++ ys ↭ xs ʳ++ ys

+
++↭ʳ++ []       ys = ↭-refl

+
++↭ʳ++ (x ∷ xs) ys = ↭-trans (↭-sym (↭-shift xs ys)) (++↭ʳ++ xs (x ∷ ys))

+

+
------------------------------------------------------------------------

+
-- foldr of Commutative Monoid

+

+
module _ {_∙_ : Op₂ A} {ε : A} (isCmonoid : IsCommutativeMonoid _≈_ _∙_ ε) where

+
  open module CM = IsCommutativeMonoid isCmonoid

+

+
  private

+
    module S = RelSetoid setoid

+

+
    cmonoid : CommutativeMonoid _ _

+
    cmonoid = record { isCommutativeMonoid = isCmonoid }

+

+
  open ACM cmonoid

+

+
  foldr-commMonoid : ∀ {xs ys} → xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys

+
  foldr-commMonoid (refl []) = CM.refl

+
  foldr-commMonoid (refl (x≈y ∷ xs≈ys)) = ∙-cong x≈y (foldr-commMonoid (Permutation.refl xs≈ys))

+
  foldr-commMonoid (prep x≈y xs↭ys) = ∙-cong x≈y (foldr-commMonoid xs↭ys)

+
  foldr-commMonoid (swap {xs} {ys} {x} {y} {x′} {y′} x≈x′ y≈y′ xs↭ys) = S.begin

+
    x ∙ (y ∙ foldr _∙_ ε xs)   S.≈⟨ ∙-congˡ (∙-congˡ (foldr-commMonoid xs↭ys)) ⟩

+
    x ∙ (y ∙ foldr _∙_ ε ys)   S.≈⟨ assoc x y (foldr _∙_ ε ys) ⟨

+
    (x ∙ y) ∙ foldr _∙_ ε ys   S.≈⟨ ∙-congʳ (comm x y) ⟩

+
    (y ∙ x) ∙ foldr _∙_ ε ys   S.≈⟨ ∙-congʳ (∙-cong y≈y′ x≈x′) ⟩

+
    (y′ ∙ x′) ∙ foldr _∙_ ε ys S.≈⟨ assoc y′ x′ (foldr _∙_ ε ys) ⟩

+
    y′ ∙ (x′ ∙ foldr _∙_ ε ys) S.∎

+
  foldr-commMonoid (trans xs↭ys ys↭zs) = CM.trans (foldr-commMonoid xs↭ys) (foldr-commMonoid ys↭zs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Permutation.Setoid.html b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Setoid.html
new file mode 100644
index 000000000..2eba77925
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Permutation.Setoid.html
@@ -0,0 +1,192 @@
+
+Data.List.Relation.Binary.Permutation.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- A definition for the permutation relation using setoid equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Function.Base using (_∘′_)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive)

+
open import Relation.Binary.Reasoning.Syntax

+

+
module Data.List.Relation.Binary.Permutation.Setoid

+
  {a ℓ} (S : Setoid a ℓ) where

+

+
open import Data.List.Base using (List; _∷_)

+
import Data.List.Relation.Binary.Permutation.Homogeneous as Homogeneous

+
import Data.List.Relation.Binary.Pointwise.Properties as Pointwise using (refl)

+
open import Data.List.Relation.Binary.Equality.Setoid S

+
open import Data.Nat.Base using (ℕ; zero; suc; _+_)

+
open import Level using (_⊔_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
private

+
  module Eq = Setoid S

+
open Eq using (_≈_) renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open Homogeneous public

+
  using (refl; prep; swap; trans)

+

+
infix 3 _↭_

+

+
_↭_ : Rel (List A) (a ⊔ ℓ)

+
_↭_ = Homogeneous.Permutation _≈_

+

+
------------------------------------------------------------------------

+
-- Constructor aliases

+

+
-- These provide aliases for `swap` and `prep` when the elements being

+
-- swapped or prepended are propositionally equal

+

+
↭-prep : ∀ x {xs ys} → xs ↭ ys → x ∷ xs ↭ x ∷ ys

+
↭-prep x xs↭ys = prep Eq.refl xs↭ys

+

+
↭-swap : ∀ x y {xs ys} → xs ↭ ys → x ∷ y ∷ xs ↭ y ∷ x ∷ ys

+
↭-swap x y xs↭ys = swap Eq.refl Eq.refl xs↭ys

+

+
------------------------------------------------------------------------

+
-- Functions over permutations

+

+
steps : ∀ {xs ys} → xs ↭ ys → ℕ

+
steps (refl _)            = 1

+
steps (prep _ xs↭ys)      = suc (steps xs↭ys)

+
steps (swap _ _ xs↭ys)    = suc (steps xs↭ys)

+
steps (trans xs↭ys ys↭zs) = steps xs↭ys + steps ys↭zs

+

+
------------------------------------------------------------------------

+
-- _↭_ is an equivalence

+

+
↭-reflexive : _≡_ ⇒ _↭_

+
↭-reflexive refl = refl (Pointwise.refl Eq.refl)

+

+
↭-refl : Reflexive _↭_

+
↭-refl = ↭-reflexive refl

+

+
↭-sym : Symmetric _↭_

+
↭-sym = Homogeneous.sym Eq.sym

+

+
↭-trans : Transitive _↭_

+
↭-trans = trans

+

+
↭-isEquivalence : IsEquivalence _↭_

+
↭-isEquivalence = Homogeneous.isEquivalence Eq.refl Eq.sym

+

+
↭-setoid : Setoid _ _

+
↭-setoid = Homogeneous.setoid {R = _≈_} Eq.refl Eq.sym

+

+
------------------------------------------------------------------------

+
-- A reasoning API to chain permutation proofs

+

+
module PermutationReasoning where

+

+
  private module Base = ≈-Reasoning ↭-setoid

+

+
  open Base public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+
    renaming (≈-go to ↭-go)

+

+
  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ ↭-go ↭-sym public

+
  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ (↭-go ∘′ refl) ≋-sym public

+

+
  -- Some extra combinators that allow us to skip certain elements

+

+
  infixr 2 step-swap step-prep

+

+
  -- Skip reasoning on the first element

+
  step-prep : ∀ x xs {ys zs : List A} → (x ∷ ys) IsRelatedTo zs →

+
              xs ↭ ys → (x ∷ xs) IsRelatedTo zs

+
  step-prep x xs rel xs↭ys = relTo (trans (prep Eq.refl xs↭ys) (begin rel))

+

+
  -- Skip reasoning about the first two elements

+
  step-swap : ∀ x y xs {ys zs : List A} → (y ∷ x ∷ ys) IsRelatedTo zs →

+
              xs ↭ ys → (x ∷ y ∷ xs) IsRelatedTo zs

+
  step-swap x y xs rel xs↭ys = relTo (trans (swap Eq.refl Eq.refl xs↭ys) (begin rel))

+

+
  syntax step-prep x xs y↭z x↭y = x ∷ xs <⟨ x↭y ⟩ y↭z

+
  syntax step-swap x y xs y↭z x↭y = x ∷ y ∷ xs <<⟨ x↭y ⟩ y↭z

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Pointwise.Base.html b/site/static/agda_html/Data.List.Relation.Binary.Pointwise.Base.html
new file mode 100644
index 000000000..b490690ff
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Pointwise.Base.html
@@ -0,0 +1,142 @@
+
+Data.List.Relation.Binary.Pointwise.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise lifting of relations to lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Pointwise.Base where

+

+
open import Data.Product.Base as Product using (_×_; _,_; <_,_>; ∃-syntax)

+
open import Data.List.Base using (List; []; _∷_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (REL; _⇒_)

+
open import Relation.Binary.Construct.Composition using (_;_)

+

+
private

+
  variable

+
    a b c ℓ : Level

+
    A B : Set a

+
    x y : A

+
    xs ys : List A

+
    R S : REL A B ℓ

+

+
------------------------------------------------------------------------

+
-- Definition

+
------------------------------------------------------------------------

+

+
infixr 5 _∷_

+

+
data Pointwise {A : Set a} {B : Set b} (R : REL A B ℓ)

+
               : List A → List B → Set (a ⊔ b ⊔ ℓ) where

+
  []  : Pointwise R [] []

+
  _∷_ : (x∼y : R x y) (xs∼ys : Pointwise R xs ys) →

+
        Pointwise R (x ∷ xs) (y ∷ ys)

+

+
------------------------------------------------------------------------

+
-- Operations

+
------------------------------------------------------------------------

+

+
head : Pointwise R (x ∷ xs) (y ∷ ys) → R x y

+
head (x∼y ∷ xs∼ys) = x∼y

+

+
tail : Pointwise R (x ∷ xs) (y ∷ ys) → Pointwise R xs ys

+
tail (x∼y ∷ xs∼ys) = xs∼ys

+

+
uncons : Pointwise R (x ∷ xs) (y ∷ ys) → R x y × Pointwise R xs ys

+
uncons = < head , tail >

+

+
rec : ∀ (P : ∀ {xs ys} → Pointwise R xs ys → Set c) →

+
      (∀ {x y xs ys} {Rxsys : Pointwise R xs ys} →

+
        (Rxy : R x y) → P Rxsys → P (Rxy ∷ Rxsys)) →

+
      P [] →

+
      ∀ {xs ys} (Rxsys : Pointwise R xs ys) → P Rxsys

+
rec P c n []            = n

+
rec P c n (Rxy ∷ Rxsys) = c Rxy (rec P c n Rxsys)

+

+
map : R ⇒ S → Pointwise R ⇒ Pointwise S

+
map R⇒S []            = []

+
map R⇒S (Rxy ∷ Rxsys) = R⇒S Rxy ∷ map R⇒S Rxsys

+

+
unzip : Pointwise (R ; S) ⇒ (Pointwise R ; Pointwise S)

+
unzip [] = [] , [] , []

+
unzip ((y , r , s) ∷ xs∼ys) =

+
  Product.map (y ∷_) (Product.map (r ∷_) (s ∷_)) (unzip xs∼ys)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Pointwise.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Pointwise.Properties.html
new file mode 100644
index 000000000..7c96e349c
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Pointwise.Properties.html
@@ -0,0 +1,154 @@
+
+Data.List.Relation.Binary.Pointwise.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of pointwise lifting of relations to lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Pointwise.Properties where

+

+
open import Data.Product.Base using (_,_; uncurry)

+
open import Data.List.Base using (List; []; _∷_)

+
open import Level

+
open import Relation.Binary.Core using (REL; _⇒_)

+
open import Relation.Binary.Definitions

+
import Relation.Binary.PropositionalEquality.Core as ≡

+
open import Relation.Nullary using (yes; no; _×-dec_)

+
import Relation.Nullary.Decidable as Dec

+

+
open import Data.List.Relation.Binary.Pointwise.Base

+

+
private

+
  variable

+
    a b ℓ : Level

+
    A : Set a

+
    B : Set b

+
    R S T : REL A B ℓ

+

+
------------------------------------------------------------------------

+
-- Relational properties

+
------------------------------------------------------------------------

+

+
reflexive : R ⇒ S → Pointwise R ⇒ Pointwise S

+
reflexive = map

+

+
refl : Reflexive R → Reflexive (Pointwise R)

+
refl rfl {[]}     = []

+
refl rfl {x ∷ xs} = rfl ∷ refl rfl

+

+
symmetric : Sym R S → Sym (Pointwise R) (Pointwise S)

+
symmetric sym []            = []

+
symmetric sym (x∼y ∷ xs∼ys) = sym x∼y ∷ symmetric sym xs∼ys

+

+
transitive : Trans R S T →

+
             Trans (Pointwise R) (Pointwise S) (Pointwise T)

+
transitive trans []            []            = []

+
transitive trans (x∼y ∷ xs∼ys) (y∼z ∷ ys∼zs) =

+
  trans x∼y y∼z ∷ transitive trans xs∼ys ys∼zs

+

+
antisymmetric : Antisym R S T →

+
                Antisym (Pointwise R) (Pointwise S) (Pointwise T)

+
antisymmetric antisym []            []            = []

+
antisymmetric antisym (x∼y ∷ xs∼ys) (y∼x ∷ ys∼xs) =

+
  antisym x∼y y∼x ∷ antisymmetric antisym xs∼ys ys∼xs

+

+
respʳ : R Respectsʳ S → (Pointwise R) Respectsʳ (Pointwise S)

+
respʳ resp []            []            = []

+
respʳ resp (x≈y ∷ xs≈ys) (z∼x ∷ zs∼xs) = resp x≈y z∼x ∷ respʳ resp xs≈ys zs∼xs

+

+
respˡ : R Respectsˡ S → (Pointwise R) Respectsˡ (Pointwise S)

+
respˡ resp []            []            = []

+
respˡ resp (x≈y ∷ xs≈ys) (x∼z ∷ xs∼zs) = resp x≈y x∼z ∷ respˡ resp xs≈ys xs∼zs

+

+
respects₂ : R Respects₂ S → (Pointwise R) Respects₂ (Pointwise S)

+
respects₂ (rʳ , rˡ) = respʳ rʳ , respˡ rˡ

+

+
decidable : Decidable R → Decidable (Pointwise R)

+
decidable _  []       []       = yes []

+
decidable _  []       (y ∷ ys) = no λ()

+
decidable _  (x ∷ xs) []       = no λ()

+
decidable R? (x ∷ xs) (y ∷ ys) = Dec.map′ (uncurry _∷_) uncons

+
  (R? x y ×-dec decidable R? xs ys)

+

+
irrelevant : Irrelevant R → Irrelevant (Pointwise R)

+
irrelevant irr []       []         = ≡.refl

+
irrelevant irr (r ∷ rs) (r₁ ∷ rs₁) =

+
  ≡.cong₂ _∷_ (irr r r₁) (irrelevant irr rs rs₁)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Pointwise.html b/site/static/agda_html/Data.List.Relation.Binary.Pointwise.html
new file mode 100644
index 000000000..c4b859da8
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Pointwise.html
@@ -0,0 +1,354 @@
+
+Data.List.Relation.Binary.Pointwise
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise lifting of relations to lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Pointwise where

+

+
open import Algebra.Core using (Op₂)

+
open import Function.Base

+
open import Function.Bundles using (Inverse)

+
open import Data.Bool.Base using (true; false)

+
open import Data.Product.Base hiding (map)

+
open import Data.List.Base as List hiding (map; head; tail; uncons)

+
open import Data.List.Properties using (≡-dec; length-++)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.AllPairs using (AllPairs; []; _∷_)

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.Fin.Base using (Fin; toℕ; cast) renaming (zero to fzero; suc to fsuc)

+
open import Data.Nat.Base using (ℕ; zero; suc)

+
open import Data.Nat.Properties

+
open import Level

+
open import Relation.Nullary hiding (Irrelevant)

+
import Relation.Nullary.Decidable as Dec using (map′)

+
open import Relation.Unary as U using (Pred)

+
open import Relation.Binary.Core renaming (Rel to Rel₂)

+
open import Relation.Binary.Definitions using (_Respects_; _Respects₂_)

+
open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder; Poset)

+
open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence; IsPartialOrder; IsPreorder)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
private

+
  variable

+
    a b c d p q ℓ ℓ₁ ℓ₂ : Level

+
    A B C D : Set d

+
    x y z : A

+
    ws xs ys zs : List A

+
    R S T : REL A B ℓ

+

+
------------------------------------------------------------------------

+
-- Re-exporting the definition and basic operations

+
------------------------------------------------------------------------

+

+
open import Data.List.Relation.Binary.Pointwise.Base public

+
open import Data.List.Relation.Binary.Pointwise.Properties public

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isEquivalence : IsEquivalence R → IsEquivalence (Pointwise R)

+
isEquivalence eq = record

+
  { refl  = refl       Eq.refl

+
  ; sym   = symmetric  Eq.sym

+
  ; trans = transitive Eq.trans

+
  } where module Eq = IsEquivalence eq

+

+
isDecEquivalence : IsDecEquivalence R → IsDecEquivalence (Pointwise R)

+
isDecEquivalence eq = record

+
  { isEquivalence = isEquivalence DE.isEquivalence

+
  ; _≟_           = decidable     DE._≟_

+
  } where module DE = IsDecEquivalence eq

+

+
isPreorder : IsPreorder R S → IsPreorder (Pointwise R) (Pointwise S)

+
isPreorder pre = record

+
  { isEquivalence = isEquivalence Pre.isEquivalence

+
  ; reflexive     = reflexive     Pre.reflexive

+
  ; trans         = transitive    Pre.trans

+
  } where module Pre = IsPreorder pre

+

+
isPartialOrder : IsPartialOrder R S →

+
                 IsPartialOrder (Pointwise R) (Pointwise S)

+
isPartialOrder po = record

+
  { isPreorder = isPreorder    PO.isPreorder

+
  ; antisym    = antisymmetric PO.antisym

+
  } where module PO = IsPartialOrder po

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
setoid : Setoid a ℓ → Setoid a (a ⊔ ℓ)

+
setoid s = record

+
  { isEquivalence = isEquivalence (Setoid.isEquivalence s)

+
  }

+

+
decSetoid : DecSetoid a ℓ → DecSetoid a (a ⊔ ℓ)

+
decSetoid d = record

+
  { isDecEquivalence = isDecEquivalence (DecSetoid.isDecEquivalence d)

+
  }

+

+
preorder : Preorder a ℓ₁ ℓ₂ → Preorder _ _ _

+
preorder p = record

+
  { isPreorder = isPreorder (Preorder.isPreorder p)

+
  }

+

+
poset : Poset a ℓ₁ ℓ₂ → Poset _ _ _

+
poset p = record

+
  { isPartialOrder = isPartialOrder (Poset.isPartialOrder p)

+
  }

+

+
------------------------------------------------------------------------

+
-- Relationships to other list predicates

+
------------------------------------------------------------------------

+

+
All-resp-Pointwise : ∀ {P : Pred A p} → P Respects R →

+
                     (All P) Respects (Pointwise R)

+
All-resp-Pointwise resp []         []         = []

+
All-resp-Pointwise resp (x∼y ∷ xs) (px ∷ pxs) =

+
  resp x∼y px ∷ All-resp-Pointwise resp xs pxs

+

+
Any-resp-Pointwise : ∀ {P : Pred A p} → P Respects R →

+
                     (Any P) Respects (Pointwise R)

+
Any-resp-Pointwise resp (x∼y ∷ xs) (here px)   = here (resp x∼y px)

+
Any-resp-Pointwise resp (x∼y ∷ xs) (there pxs) =

+
  there (Any-resp-Pointwise resp xs pxs)

+

+
AllPairs-resp-Pointwise : R Respects₂ S →

+
                          (AllPairs R) Respects (Pointwise S)

+
AllPairs-resp-Pointwise _                    []         []         = []

+
AllPairs-resp-Pointwise resp@(respₗ , respᵣ) (x∼y ∷ xs) (px ∷ pxs) =

+
  All-resp-Pointwise respₗ xs (All.map (respᵣ x∼y) px) ∷

+
  (AllPairs-resp-Pointwise resp xs pxs)

+

+
------------------------------------------------------------------------

+
-- Relationship to functions over lists

+
------------------------------------------------------------------------

+
-- length

+

+
Pointwise-length : Pointwise R xs ys → length xs ≡ length ys

+
Pointwise-length []            = ≡.refl

+
Pointwise-length (x∼y ∷ xs∼ys) = ≡.cong ℕ.suc (Pointwise-length xs∼ys)

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
tabulate⁺ : ∀ {n} {f : Fin n → A} {g : Fin n → B} →

+
            (∀ i → R (f i) (g i)) → Pointwise R (tabulate f) (tabulate g)

+
tabulate⁺ {n = zero}  f∼g = []

+
tabulate⁺ {n = suc n} f∼g = f∼g fzero ∷ tabulate⁺ (f∼g ∘ fsuc)

+

+
tabulate⁻ : ∀ {n} {f : Fin n → A} {g : Fin n → B} →

+
            Pointwise R (tabulate f) (tabulate g) → (∀ i → R (f i) (g i))

+
tabulate⁻ {n = suc n} (x∼y ∷ xs∼ys) fzero    = x∼y

+
tabulate⁻ {n = suc n} (x∼y ∷ xs∼ys) (fsuc i) = tabulate⁻ xs∼ys i

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
++⁺ : Pointwise R ws xs → Pointwise R ys zs →

+
      Pointwise R (ws ++ ys) (xs ++ zs)

+
++⁺ []            ys∼zs = ys∼zs

+
++⁺ (w∼x ∷ ws∼xs) ys∼zs = w∼x ∷ ++⁺ ws∼xs ys∼zs

+

+
++-cancelˡ : ∀ xs → Pointwise R (xs ++ ys) (xs ++ zs) → Pointwise R ys zs

+
++-cancelˡ []       ys∼zs               = ys∼zs

+
++-cancelˡ (x ∷ xs) (_ ∷ xs++ys∼xs++zs) = ++-cancelˡ xs xs++ys∼xs++zs

+

+
++-cancelʳ : ∀ ys zs → Pointwise R (ys ++ xs) (zs ++ xs) → Pointwise R ys zs

+
++-cancelʳ []       []       _             = []

+
++-cancelʳ (y ∷ ys) (z ∷ zs) (y∼z ∷ ys∼zs) = y∼z ∷ (++-cancelʳ ys zs ys∼zs)

+
-- Impossible cases

+
++-cancelʳ {xs = xs}     []       (z ∷ zs) eq   =

+
  contradiction (≡.trans (Pointwise-length eq) (length-++ (z ∷ zs))) (m≢1+n+m (length xs))

+
++-cancelʳ {xs = xs}     (y ∷ ys) []       eq   =

+
  contradiction (≡.trans (≡.sym (length-++ (y ∷ ys))) (Pointwise-length eq)) (m≢1+n+m (length xs) ∘ ≡.sym)

+

+
------------------------------------------------------------------------

+
-- concat

+

+
concat⁺ : ∀ {xss yss} → Pointwise (Pointwise R) xss yss →

+
          Pointwise R (concat xss) (concat yss)

+
concat⁺ []                = []

+
concat⁺ (xs∼ys ∷ xss∼yss) = ++⁺ xs∼ys (concat⁺ xss∼yss)

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
reverseAcc⁺ : Pointwise R ws xs → Pointwise R ys zs →

+
              Pointwise R (reverseAcc ws ys) (reverseAcc xs zs)

+
reverseAcc⁺ rs′ []       = rs′

+
reverseAcc⁺ rs′ (r ∷ rs) = reverseAcc⁺ (r ∷ rs′) rs

+

+
ʳ++⁺ : Pointwise R ws xs → Pointwise R ys zs →

+
       Pointwise R (ws ʳ++ ys) (xs ʳ++ zs)

+
ʳ++⁺ rs rs′ = reverseAcc⁺ rs′ rs

+

+
reverse⁺ : Pointwise R xs ys → Pointwise R (reverse xs) (reverse ys)

+
reverse⁺ = reverseAcc⁺ []

+

+
------------------------------------------------------------------------

+
-- map

+

+
map⁺ : ∀ (f : A → C) (g : B → D) →

+
       Pointwise (λ a b → R (f a) (g b)) xs ys →

+
       Pointwise R (List.map f xs) (List.map g ys)

+
map⁺ f g []       = []

+
map⁺ f g (r ∷ rs) = r ∷ map⁺ f g rs

+

+
map⁻ : ∀ (f : A → C) (g : B → D) →

+
       Pointwise R (List.map f xs) (List.map g ys) →

+
       Pointwise (λ a b → R (f a) (g b)) xs ys

+
map⁻ {xs = []}    {[]}    f g [] = []

+
map⁻ {xs = _ ∷ _} {_ ∷ _} f g (r ∷ rs) = r ∷ map⁻ f g rs

+

+
------------------------------------------------------------------------

+
-- foldr

+

+
foldr⁺ : ∀ {_•_ : Op₂ A} {_◦_ : Op₂ B} →

+
         (∀ {w x y z} → R w x → R y z → R (w • y) (x ◦ z)) →

+
         ∀ {e f} → R e f → Pointwise R xs ys →

+
         R (foldr _•_ e xs) (foldr _◦_ f ys)

+
foldr⁺ _    e~f []            = e~f

+
foldr⁺ pres e~f (x~y ∷ xs~ys) = pres x~y (foldr⁺ pres e~f xs~ys)

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {P : Pred A p} {Q : Pred B q}

+
         (P? : U.Decidable P) (Q? : U.Decidable Q)

+
         (P⇒Q : ∀ {a b} → R a b → P a → Q b)

+
         (Q⇒P : ∀ {a b} → R a b → Q b → P a)

+
         where

+

+
  filter⁺ : Pointwise R xs ys →

+
            Pointwise R (filter P? xs) (filter Q? ys)

+
  filter⁺ []       = []

+
  filter⁺ {x ∷ _} {y ∷ _} (r ∷ rs) with P? x | Q? y

+
  ... | true  because _ | true  because _ = r ∷ filter⁺ rs

+
  ... | false because _ | false because _ = filter⁺ rs

+
  ... | yes p           | no ¬q = contradiction (P⇒Q r p) ¬q

+
  ... | no ¬p           | yes q = contradiction (Q⇒P r q) ¬p

+

+
------------------------------------------------------------------------

+
-- replicate

+

+
replicate⁺ : R x y → ∀ n → Pointwise R (replicate n x) (replicate n y)

+
replicate⁺ r 0       = []

+
replicate⁺ r (suc n) = r ∷ replicate⁺ r n

+

+
------------------------------------------------------------------------

+
-- lookup

+

+
lookup⁻ : length xs ≡ length ys →

+
          (∀ {i j} → toℕ i ≡ toℕ j → R (lookup xs i) (lookup ys j)) →

+
          Pointwise R xs ys

+
lookup⁻ {xs = []}    {ys = []}    _             _  = []

+
lookup⁻ {xs = _ ∷ _} {ys = _ ∷ _} |xs|≡|ys| eq = eq {fzero} ≡.refl ∷

+
  lookup⁻ (suc-injective |xs|≡|ys|) (eq ∘ ≡.cong ℕ.suc)

+

+
lookup⁺ : ∀ (Rxys : Pointwise R xs ys) →

+
          ∀ i → (let j = cast (Pointwise-length Rxys) i) →

+
          R (lookup xs i) (lookup ys j)

+
lookup⁺ (Rxy ∷ _)    fzero    = Rxy

+
lookup⁺ (_   ∷ Rxys) (fsuc i) = lookup⁺ Rxys i

+

+
------------------------------------------------------------------------

+
-- Properties of propositional pointwise

+
------------------------------------------------------------------------

+

+
Pointwise-≡⇒≡ : Pointwise {A = A} _≡_ ⇒ _≡_

+
Pointwise-≡⇒≡ []               = ≡.refl

+
Pointwise-≡⇒≡ (≡.refl ∷ xs∼ys) with Pointwise-≡⇒≡ xs∼ys

+
... | ≡.refl = ≡.refl

+

+
≡⇒Pointwise-≡ :  _≡_ ⇒ Pointwise {A = A} _≡_

+
≡⇒Pointwise-≡ ≡.refl = refl ≡.refl

+

+
Pointwise-≡↔≡ : Inverse (setoid (≡.setoid A)) (≡.setoid (List A))

+
Pointwise-≡↔≡ = record

+
  { to = id

+
  ; from = id

+
  ; to-cong = Pointwise-≡⇒≡

+
  ; from-cong = ≡⇒Pointwise-≡

+
  ; inverse = Pointwise-≡⇒≡ , ≡⇒Pointwise-≡

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Sublist.Heterogeneous.Core.html b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Heterogeneous.Core.html
new file mode 100644
index 000000000..22a112bb5
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Heterogeneous.Core.html
@@ -0,0 +1,107 @@
+
+Data.List.Relation.Binary.Sublist.Heterogeneous.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- This file contains some core definitions which are re-exported by

+
-- Data.List.Relation.Binary.Sublist.Heterogeneous.

+
------------------------------------------------------------------------

+

+
-- This module has R as explicit parameter, in contrast to the implicit

+
-- parameter R of the main module Sublist.Heterogeneous.

+

+
-- Parameterized data modules (https://github.com/agda/agda/issues/3210)

+
-- may simplify this setup, making this module obsolete.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (REL)

+

+
module Data.List.Relation.Binary.Sublist.Heterogeneous.Core

+
       {a b r} {A : Set a} {B : Set b} (R : REL A B r)

+
       where

+

+
open import Level using (_⊔_)

+
open import Data.List.Base using (List; []; _∷_)

+

+
infixr 5 _∷_ _∷ʳ_

+

+
data Sublist : REL (List A) (List B) (a ⊔ b ⊔ r) where

+
  []   : Sublist [] []

+
  _∷ʳ_ : ∀ {xs ys} → ∀ y → Sublist xs ys → Sublist xs (y ∷ ys)

+
  _∷_  : ∀ {x xs y ys} → R x y → Sublist xs ys → Sublist (x ∷ xs) (y ∷ ys)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Sublist.Heterogeneous.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Heterogeneous.Properties.html
new file mode 100644
index 000000000..0b8b51e0b
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Heterogeneous.Properties.html
@@ -0,0 +1,797 @@
+
+Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the heterogeneous sublist relation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Sublist.Heterogeneous.Properties where

+

+
open import Level

+

+
open import Data.Bool.Base using (true; false)

+
open import Data.List.Relation.Unary.All using (Null; []; _∷_)

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.List.Base as List hiding (map; _∷ʳ_)

+
import Data.List.Properties as List

+
open import Data.List.Relation.Unary.Any.Properties

+
  using (here-injective; there-injective)

+
open import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise; []; _∷_)

+
open import Data.List.Relation.Binary.Sublist.Heterogeneous

+

+
open import Data.Maybe.Relation.Unary.All as Maybe using (nothing; just)

+
open import Data.Nat.Base using (ℕ; _≤_; _≥_); open ℕ; open _≤_

+
import Data.Nat.Properties as ℕ

+
open import Data.Product.Base using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)

+

+
open import Function.Base

+
open import Function.Bundles using (_⤖_; _⇔_ ; mk⤖; mk⇔)

+
open import Function.Consequences.Propositional using (strictlySurjective⇒surjective)

+
open import Relation.Nullary.Decidable as Dec using (Dec; does; _because_; yes; no; ¬?)

+
open import Relation.Nullary.Negation using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Unary as U using (Pred)

+
open import Relation.Binary.Core using (Rel; REL; _⇒_)

+
open import Relation.Binary.Bundles using (Preorder; Poset; DecPoset)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Trans; Antisym; Decidable; Irrelevant; Irreflexive)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsPartialOrder; IsDecPartialOrder)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+

+
private

+
  variable

+
    a b c d e p q r s t : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+
    x : A

+
    y : B

+
    as cs xs : List A

+
    bs ds ys : List B

+
    ass : List (List A)

+
    bss : List (List B)

+
    m n : ℕ

+

+
------------------------------------------------------------------------

+
-- Injectivity of constructors

+

+
module _ {R : REL A B r} where

+

+
  ∷-injectiveˡ : {px qx : R x y} {pxs qxs : Sublist R xs ys} →

+
                 (Sublist R (x ∷ xs) (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → px ≡ qx

+
  ∷-injectiveˡ ≡.refl = ≡.refl

+

+
  ∷-injectiveʳ : {px qx : R x y} {pxs qxs : Sublist R xs ys} →

+
                 (Sublist R (x ∷ xs) (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → pxs ≡ qxs

+
  ∷-injectiveʳ ≡.refl = ≡.refl

+

+
  ∷ʳ-injective : {pxs qxs : Sublist R xs ys} →

+
                 (Sublist R xs (y ∷ ys) ∋ y ∷ʳ pxs) ≡ (y ∷ʳ qxs) → pxs ≡ qxs

+
  ∷ʳ-injective ≡.refl = ≡.refl

+

+
module _ {R : REL A B r} where

+

+
  length-mono-≤ : Sublist R as bs → length as ≤ length bs

+
  length-mono-≤ []        = z≤n

+
  length-mono-≤ (y ∷ʳ rs) = ℕ.m≤n⇒m≤1+n (length-mono-≤ rs)

+
  length-mono-≤ (r ∷ rs)  = s≤s (length-mono-≤ rs)

+

+
------------------------------------------------------------------------

+
-- Conversion to and from Pointwise (proto-reflexivity)

+

+
  fromPointwise : Pointwise R ⇒ Sublist R

+
  fromPointwise []       = []

+
  fromPointwise (p ∷ ps) = p ∷ fromPointwise ps

+

+
  toPointwise : length as ≡ length bs →

+
                Sublist R as bs → Pointwise R as bs

+
  toPointwise {bs = []}     eq []         = []

+
  toPointwise {bs = b ∷ bs} eq (r ∷ rs)   = r ∷ toPointwise (ℕ.suc-injective eq) rs

+
  toPointwise {bs = b ∷ bs} eq (b ∷ʳ rs) =

+
    contradiction (s≤s (length-mono-≤ rs)) (ℕ.<-irrefl eq)

+

+
------------------------------------------------------------------------

+
-- Various functions' outputs are sublists

+

+
-- These lemmas are generalisations of results of the form `f xs ⊆ xs`.

+
-- (where _⊆_ stands for Sublist R). If R is reflexive then we can

+
-- indeed obtain `f xs ⊆ xs` from `xs ⊆ ys → f xs ⊆ ys`. The other

+
-- direction is only true if R is both reflexive and transitive.

+

+
module _ {R : REL A B r} where

+

+
  tail-Sublist : Sublist R as bs →

+
                 Maybe.All (λ as → Sublist R as bs) (tail as)

+
  tail-Sublist []        = nothing

+
  tail-Sublist (b ∷ʳ ps) = Maybe.map (b ∷ʳ_) (tail-Sublist ps)

+
  tail-Sublist (p ∷ ps)  = just (_ ∷ʳ ps)

+

+
  take-Sublist : ∀ n → Sublist R as bs → Sublist R (take n as) bs

+
  take-Sublist n       (y ∷ʳ rs) = y ∷ʳ take-Sublist n rs

+
  take-Sublist zero    rs        = minimum _

+
  take-Sublist (suc n) []        = []

+
  take-Sublist (suc n) (r ∷ rs)  = r ∷ take-Sublist n rs

+

+
  drop-Sublist : ∀ n → Sublist R ⇒ (Sublist R ∘′ drop n)

+
  drop-Sublist n       (y ∷ʳ rs) = y ∷ʳ drop-Sublist n rs

+
  drop-Sublist zero    rs        = rs

+
  drop-Sublist (suc n) []        = []

+
  drop-Sublist (suc n) (r ∷ rs)  = _ ∷ʳ drop-Sublist n rs

+

+
module _ {R : REL A B r} {P : Pred A p} (P? : U.Decidable P) where

+

+
  takeWhile-Sublist : Sublist R as bs → Sublist R (takeWhile P? as) bs

+
  takeWhile-Sublist []        = []

+
  takeWhile-Sublist (y ∷ʳ rs) = y ∷ʳ takeWhile-Sublist rs

+
  takeWhile-Sublist {a ∷ as} (r ∷ rs) with does (P? a)

+
  ... | true  = r ∷ takeWhile-Sublist rs

+
  ... | false = minimum _

+

+
  dropWhile-Sublist : Sublist R as bs → Sublist R (dropWhile P? as) bs

+
  dropWhile-Sublist []        = []

+
  dropWhile-Sublist (y ∷ʳ rs) = y ∷ʳ dropWhile-Sublist rs

+
  dropWhile-Sublist {a ∷ as} (r ∷ rs) with does (P? a)

+
  ... | true  = _ ∷ʳ dropWhile-Sublist rs

+
  ... | false = r ∷ rs

+

+
  filter-Sublist : Sublist R as bs → Sublist R (filter P? as) bs

+
  filter-Sublist []        = []

+
  filter-Sublist (y ∷ʳ rs) = y ∷ʳ filter-Sublist rs

+
  filter-Sublist {a ∷ as} (r ∷ rs) with does (P? a)

+
  ... | true  = r ∷ filter-Sublist rs

+
  ... | false = _ ∷ʳ filter-Sublist rs

+

+
------------------------------------------------------------------------

+
-- Various functions are increasing wrt _⊆_

+

+
-- We write f⁺ for the proof that `xs ⊆ ys → f xs ⊆ f ys`

+
-- and f⁻ for the one that `f xs ⊆ f ys → xs ⊆ ys`.

+

+
module _ {R : REL A B r} where

+

+
------------------------------------------------------------------------

+
-- _∷_

+

+
  ∷ˡ⁻ : Sublist R (x ∷ xs) ys → Sublist R xs ys

+
  ∷ˡ⁻ (y ∷ʳ rs) = y ∷ʳ ∷ˡ⁻ rs

+
  ∷ˡ⁻ (r ∷  rs) = _ ∷ʳ rs

+

+
  ∷ʳ⁻ : ¬ R x y → Sublist R (x ∷ xs) (y ∷ ys) →

+
        Sublist R (x ∷ xs) ys

+
  ∷ʳ⁻ ¬r (y ∷ʳ rs) = rs

+
  ∷ʳ⁻ ¬r (r ∷  rs) = contradiction r ¬r

+

+
  ∷⁻ : Sublist R (x ∷ xs) (y ∷ ys) → Sublist R xs ys

+
  ∷⁻ (y ∷ʳ rs) = ∷ˡ⁻ rs

+
  ∷⁻ (x ∷  rs) = rs

+

+
module _ {R : REL C D r} where

+

+
------------------------------------------------------------------------

+
-- map

+

+
  map⁺ : (f : A → C) (g : B → D) →

+
         Sublist (λ a b → R (f a) (g b)) as bs →

+
         Sublist R (List.map f as) (List.map g bs)

+
  map⁺ f g []        = []

+
  map⁺ f g (y ∷ʳ rs) = g y ∷ʳ map⁺ f g rs

+
  map⁺ f g (r ∷ rs)  = r ∷ map⁺ f g rs

+

+
  map⁻ : (f : A → C) (g : B → D) →

+
         Sublist R (List.map f as) (List.map g bs) →

+
         Sublist (λ a b → R (f a) (g b)) as bs

+
  map⁻ {as = []}     {bs}     f g rs        = minimum _

+
  map⁻ {as = a ∷ as} {b ∷ bs} f g (_ ∷ʳ rs) = b ∷ʳ map⁻ f g rs

+
  map⁻ {as = a ∷ as} {b ∷ bs} f g (r ∷ rs)  = r ∷ map⁻ f g rs

+

+

+
module _ {R : REL A B r} where

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
  ++⁺ : Sublist R as bs → Sublist R cs ds →

+
        Sublist R (as ++ cs) (bs ++ ds)

+
  ++⁺ []         cds = cds

+
  ++⁺ (y ∷ʳ abs) cds = y ∷ʳ ++⁺ abs cds

+
  ++⁺ (ab ∷ abs) cds = ab ∷ ++⁺ abs cds

+

+
  ++⁻ : length as ≡ length bs →

+
        Sublist R (as ++ cs) (bs ++ ds) → Sublist R cs ds

+
  ++⁻ {as = []}     {[]}     eq rs = rs

+
  ++⁻ {as = a ∷ as} {b ∷ bs} eq rs = ++⁻ (ℕ.suc-injective eq) (∷⁻ rs)

+

+
  ++ˡ : (cs : List B) → Sublist R as bs → Sublist R as (cs ++ bs)

+
  ++ˡ zs = ++⁺ (minimum zs)

+

+
  ++ʳ : (cs : List B) → Sublist R as bs → Sublist R as (bs ++ cs)

+
  ++ʳ cs []        = minimum cs

+
  ++ʳ cs (y ∷ʳ rs) = y ∷ʳ ++ʳ cs rs

+
  ++ʳ cs (r ∷ rs)  = r ∷ ++ʳ cs rs

+

+

+
------------------------------------------------------------------------

+
-- concat

+

+
  concat⁺ : Sublist (Sublist R) ass bss →

+
            Sublist R (concat ass) (concat bss)

+
  concat⁺ []          = []

+
  concat⁺ (y  ∷ʳ rss) = ++ˡ y (concat⁺ rss)

+
  concat⁺ (rs ∷  rss) = ++⁺ rs (concat⁺ rss)

+

+
------------------------------------------------------------------------

+
-- take / drop

+

+
  take⁺ : m ≤ n → Pointwise R as bs →

+
          Sublist R (take m as) (take n bs)

+
  take⁺ z≤n       ps        = minimum _

+
  take⁺ (s≤s m≤n) []        = []

+
  take⁺ (s≤s m≤n) (p ∷  ps) = p ∷ take⁺ m≤n ps

+

+
  drop⁺ : m ≥ n → Sublist R as bs →

+
          Sublist R (drop m as) (drop n bs)

+
  drop⁺ (z≤n {m}) rs        = drop-Sublist m rs

+
  drop⁺ (s≤s m≥n) []        = []

+
  drop⁺ (s≤s m≥n) (y ∷ʳ rs) = drop⁺ (ℕ.m≤n⇒m≤1+n m≥n) rs

+
  drop⁺ (s≤s m≥n) (r ∷ rs)  = drop⁺ m≥n rs

+

+
  drop⁺-≥ : m ≥ n → Pointwise R as bs →

+
            Sublist R (drop m as) (drop n bs)

+
  drop⁺-≥ m≥n pw = drop⁺ m≥n (fromPointwise pw)

+

+
  drop⁺-⊆ : ∀ m → Sublist R as bs →

+
            Sublist R (drop m as) (drop m bs)

+
  drop⁺-⊆ m = drop⁺ (ℕ.≤-refl {m})

+

+
module _ {R : REL A B r} {P : Pred A p} {Q : Pred B q}

+
         (P? : U.Decidable P) (Q? : U.Decidable Q) where

+

+
  ⊆-takeWhile-Sublist : (∀ {a b} → R a b → P a → Q b) →

+
                        Pointwise R as bs →

+
                        Sublist R (takeWhile P? as) (takeWhile Q? bs)

+
  ⊆-takeWhile-Sublist rp⇒q [] = []

+
  ⊆-takeWhile-Sublist {a ∷ as} {b ∷ bs} rp⇒q (p ∷ ps) with P? a | Q? b

+
  ... | false because _ | _               = minimum _

+
  ... | true  because _ | true  because _ = p ∷ ⊆-takeWhile-Sublist rp⇒q ps

+
  ... | yes pa          | no ¬qb          = contradiction (rp⇒q p pa) ¬qb

+

+
  ⊇-dropWhile-Sublist : (∀ {a b} → R a b → Q b → P a) →

+
                        Pointwise R as bs →

+
                        Sublist R (dropWhile P? as) (dropWhile Q? bs)

+
  ⊇-dropWhile-Sublist rq⇒p [] = []

+
  ⊇-dropWhile-Sublist {a ∷ as} {b ∷ bs} rq⇒p (p ∷ ps) with P? a | Q? b

+
  ... | true  because _ | true  because _ = ⊇-dropWhile-Sublist rq⇒p ps

+
  ... | true  because _ | false because _ =

+
    b ∷ʳ dropWhile-Sublist P? (fromPointwise ps)

+
  ... | false because _ | false because _ = p ∷ fromPointwise ps

+
  ... | no ¬pa          | yes qb          = contradiction (rq⇒p p qb) ¬pa

+

+
  ⊆-filter-Sublist : (∀ {a b} → R a b → P a → Q b) →

+
                     Sublist R as bs → Sublist R (filter P? as) (filter Q? bs)

+
  ⊆-filter-Sublist rp⇒q [] = []

+
  ⊆-filter-Sublist rp⇒q (y ∷ʳ rs) with does (Q? y)

+
  ... | true  = y ∷ʳ ⊆-filter-Sublist rp⇒q rs

+
  ... | false = ⊆-filter-Sublist rp⇒q rs

+
  ⊆-filter-Sublist {a ∷ as} {b ∷ bs} rp⇒q (r ∷ rs) with P? a | Q? b

+
  ... | true  because _ | true  because _ = r ∷ ⊆-filter-Sublist rp⇒q rs

+
  ... | false because _ | true  because _ = _ ∷ʳ ⊆-filter-Sublist rp⇒q rs

+
  ... | false because _ | false because _ = ⊆-filter-Sublist rp⇒q rs

+
  ... | yes pa          | no ¬qb          = contradiction (rp⇒q r pa) ¬qb

+

+
module _ {R : Rel A r} {P : Pred A p} (P? : U.Decidable P) where

+

+
  takeWhile-filter : Pointwise R as as →

+
                     Sublist R (takeWhile P? as) (filter P? as)

+
  takeWhile-filter [] = []

+
  takeWhile-filter {a ∷ as} (p ∷ ps) with does (P? a)

+
  ... | true  = p ∷ takeWhile-filter ps

+
  ... | false = minimum _

+

+
  filter-dropWhile : Pointwise R as as →

+
                     Sublist R (filter P? as) (dropWhile (¬? ∘ P?) as)

+
  filter-dropWhile [] = []

+
  filter-dropWhile {a ∷ as} (p ∷ ps) with does (P? a)

+
  ... | true  = p ∷ filter-Sublist P? (fromPointwise ps)

+
  ... | false = filter-dropWhile ps

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
module _ {R : REL A B r} where

+

+
  reverseAcc⁺ : Sublist R as bs → Sublist R cs ds →

+
                Sublist R (reverseAcc cs as) (reverseAcc ds bs)

+
  reverseAcc⁺ []         cds = cds

+
  reverseAcc⁺ (y ∷ʳ abs) cds = reverseAcc⁺ abs (y ∷ʳ cds)

+
  reverseAcc⁺ (r ∷ abs)  cds = reverseAcc⁺ abs (r ∷ cds)

+

+
  ʳ++⁺ : Sublist R as bs → Sublist R cs ds →

+
         Sublist R (as ʳ++ cs) (bs ʳ++ ds)

+
  ʳ++⁺ = reverseAcc⁺

+

+
  reverse⁺ : Sublist R as bs → Sublist R (reverse as) (reverse bs)

+
  reverse⁺ rs = reverseAcc⁺ rs []

+

+
  reverse⁻ : Sublist R (reverse as) (reverse bs) → Sublist R as bs

+
  reverse⁻ {as = as} {bs = bs} p = cast (reverse⁺ p) where

+
    cast = ≡.subst₂ (Sublist R) (List.reverse-involutive as) (List.reverse-involutive bs)

+

+
------------------------------------------------------------------------

+
-- Inversion lemmas

+

+
module _ {R : REL A B r} where

+

+
  ∷⁻¹ : R x y → Sublist R xs ys ⇔ Sublist R (x ∷ xs) (y ∷ ys)

+
  ∷⁻¹ r = mk⇔ (r ∷_) ∷⁻

+

+
  ∷ʳ⁻¹ : ¬ R x y → Sublist R (x ∷ xs) ys ⇔ Sublist R (x ∷ xs) (y ∷ ys)

+
  ∷ʳ⁻¹ ¬r = mk⇔ (_ ∷ʳ_) (∷ʳ⁻ ¬r)

+

+
------------------------------------------------------------------------

+
-- Irrelevant special case

+

+
module _ {R : REL A B r} where

+

+
  Sublist-[]-irrelevant : U.Irrelevant (Sublist R [])

+
  Sublist-[]-irrelevant []       []        = ≡.refl

+
  Sublist-[]-irrelevant (y ∷ʳ p) (.y ∷ʳ q) = ≡.cong (y ∷ʳ_) (Sublist-[]-irrelevant p q)

+

+
------------------------------------------------------------------------

+
-- (to/from)Any is a bijection

+

+
  toAny-injective :{p q : Sublist R [ x ] xs} → toAny p ≡ toAny q → p ≡ q

+
  toAny-injective {p = y ∷ʳ p} {y ∷ʳ q} =

+
    ≡.cong (y ∷ʳ_) ∘′ toAny-injective ∘′ there-injective

+
  toAny-injective {p = _ ∷ p}  {_ ∷ q}  =

+
    ≡.cong₂ (flip _∷_) (Sublist-[]-irrelevant p q) ∘′ here-injective

+

+
  fromAny-injective : {p q : Any (R x) xs} →

+
                      fromAny {R = R} p ≡ fromAny q → p ≡ q

+
  fromAny-injective {p = here px} {here qx} = ≡.cong here ∘′ ∷-injectiveˡ

+
  fromAny-injective {p = there p} {there q} =

+
    ≡.cong there ∘′ fromAny-injective ∘′ ∷ʳ-injective

+

+
  toAny∘fromAny≗id : (p : Any (R x) xs) → toAny (fromAny {R = R} p) ≡ p

+
  toAny∘fromAny≗id (here px) = ≡.refl

+
  toAny∘fromAny≗id (there p) = ≡.cong there (toAny∘fromAny≗id p)

+

+
  Sublist-[x]-bijection : (Sublist R [ x ] xs) ⤖ (Any (R x) xs)

+
  Sublist-[x]-bijection = mk⤖ (toAny-injective , strictlySurjective⇒surjective < fromAny , toAny∘fromAny≗id >)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
module Reflexivity

+
    {R : Rel A r}

+
    (R-refl : Reflexive R) where

+

+
  reflexive : _≡_ ⇒ Sublist R

+
  reflexive ≡.refl = fromPointwise (Pw.refl R-refl)

+

+
  refl : Reflexive (Sublist R)

+
  refl = reflexive ≡.refl

+

+
open Reflexivity public

+

+
module Transitivity

+
    {R : REL A B r} {S : REL B C s} {T : REL A C t}

+
    (rs⇒t : Trans R S T) where

+

+
  trans : Trans (Sublist R) (Sublist S) (Sublist T)

+
  trans rs        (y ∷ʳ ss) = y ∷ʳ trans rs ss

+
  trans (y ∷ʳ rs) (s ∷ ss)  = _ ∷ʳ trans rs ss

+
  trans (r ∷ rs)  (s ∷ ss)  = rs⇒t r s ∷ trans rs ss

+
  trans []        []        = []

+

+
open Transitivity public

+

+
module Antisymmetry

+
    {R : REL A B r} {S : REL B A s} {E : REL A B e}

+
    (rs⇒e : Antisym R S E) where

+

+
  open ℕ.≤-Reasoning

+

+
  antisym : Antisym (Sublist R) (Sublist S) (Pointwise E)

+
  antisym []        []        = []

+
  antisym (r ∷ rs)  (s ∷ ss)  = rs⇒e r s ∷ antisym rs ss

+
  -- impossible cases

+
  antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷ʳ_ {ys₂} {zs} z ss) =

+
    contradiction (begin

+
    length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩

+
    length zs        ≤⟨ ℕ.n≤1+n (length zs) ⟩

+
    length (z ∷ zs)  ≤⟨ length-mono-≤ rs ⟩

+
    length ys₁       ∎) $ ℕ.<-irrefl ≡.refl

+
  antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷_ {y} {ys₂} {z} {zs} s ss)  =

+
    contradiction (begin

+
    length (z ∷ zs) ≤⟨ length-mono-≤ rs ⟩

+
    length ys₁      ≤⟨ length-mono-≤ ss ⟩

+
    length zs       ∎) $ ℕ.<-irrefl ≡.refl

+
  antisym (_∷_ {x} {xs} {y} {ys₁} r rs)  (_∷ʳ_ {ys₂} {zs} z ss) =

+
    contradiction (begin

+
    length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩

+
    length xs        ≤⟨ length-mono-≤ rs ⟩

+
    length ys₁       ∎) $ ℕ.<-irrefl ≡.refl

+

+
open Antisymmetry public

+

+
module _ {R : REL A B r} (R? : Decidable R) where

+

+
  sublist? : Decidable (Sublist R)

+
  sublist? []       ys       = yes (minimum ys)

+
  sublist? (x ∷ xs) []       = no λ ()

+
  sublist? (x ∷ xs) (y ∷ ys) with R? x y

+
  ... | true  because  [r] =

+
    Dec.map (∷⁻¹  (invert  [r])) (sublist? xs ys)

+
  ... | false because [¬r] =

+
    Dec.map (∷ʳ⁻¹ (invert [¬r])) (sublist? (x ∷ xs) ys)

+

+
module _ {E : Rel A e} {R : Rel A r} where

+

+
  isPreorder : IsPreorder E R → IsPreorder (Pointwise E) (Sublist R)

+
  isPreorder ER-isPreorder = record

+
    { isEquivalence = Pw.isEquivalence       ER.isEquivalence

+
    ; reflexive     = fromPointwise ∘ Pw.map ER.reflexive

+
    ; trans         = trans                  ER.trans

+
    } where module ER = IsPreorder ER-isPreorder

+

+
  isPartialOrder : IsPartialOrder E R → IsPartialOrder (Pointwise E) (Sublist R)

+
  isPartialOrder ER-isPartialOrder = record

+
    { isPreorder = isPreorder ER.isPreorder

+
    ; antisym    = antisym    ER.antisym

+
    } where module ER = IsPartialOrder ER-isPartialOrder

+

+
  isDecPartialOrder : IsDecPartialOrder E R →

+
                      IsDecPartialOrder (Pointwise E) (Sublist R)

+
  isDecPartialOrder ER-isDecPartialOrder = record

+
    { isPartialOrder = isPartialOrder ER.isPartialOrder

+
    ; _≟_            = Pw.decidable   ER._≟_

+
    ; _≤?_           = sublist?       ER._≤?_

+
    } where module ER = IsDecPartialOrder ER-isDecPartialOrder

+

+
preorder : Preorder a e r → Preorder _ _ _

+
preorder ER-preorder = record

+
  { isPreorder = isPreorder ER.isPreorder

+
  } where module ER = Preorder ER-preorder

+

+
poset : Poset a e r → Poset _ _ _

+
poset ER-poset = record

+
  { isPartialOrder = isPartialOrder ER.isPartialOrder

+
  } where module ER = Poset ER-poset

+

+
decPoset : DecPoset a e r → DecPoset _ _ _

+
decPoset ER-poset = record

+
  { isDecPartialOrder = isDecPartialOrder ER.isDecPartialOrder

+
  } where module ER = DecPoset ER-poset

+

+
------------------------------------------------------------------------

+
-- Properties of disjoint sublists

+

+
module Disjointness {a b r} {A : Set a} {B : Set b} {R : REL A B r} where

+

+
  private

+
    infix 4 _⊆_

+
    _⊆_ = Sublist R

+

+
  -- Forgetting the union

+

+
  DisjointUnion→Disjoint : ∀ {xs ys zs us} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : us ⊆ zs} →

+
    DisjointUnion τ₁ τ₂ τ → Disjoint τ₁ τ₂

+
  DisjointUnion→Disjoint []         = []

+
  DisjointUnion→Disjoint (y   ∷ₙ u) = y   ∷ₙ DisjointUnion→Disjoint u

+
  DisjointUnion→Disjoint (x≈y ∷ₗ u) = x≈y ∷ₗ DisjointUnion→Disjoint u

+
  DisjointUnion→Disjoint (x≈y ∷ᵣ u) = x≈y ∷ᵣ DisjointUnion→Disjoint u

+

+
  -- Reconstructing the union

+

+
  Disjoint→DisjointUnion : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
    Disjoint τ₁ τ₂ → ∃₂ λ us (τ : us ⊆ zs) → DisjointUnion τ₁ τ₂ τ

+
  Disjoint→DisjointUnion []         = _ , _ , []

+
  Disjoint→DisjointUnion (y   ∷ₙ u) = _ , _ , y   ∷ₙ proj₂ (proj₂ (Disjoint→DisjointUnion u))

+
  Disjoint→DisjointUnion (x≈y ∷ₗ u) = _ , _ , x≈y ∷ₗ proj₂ (proj₂ (Disjoint→DisjointUnion u))

+
  Disjoint→DisjointUnion (x≈y ∷ᵣ u) = _ , _ , x≈y ∷ᵣ proj₂ (proj₂ (Disjoint→DisjointUnion u))

+

+
  -- Disjoint is decidable

+

+
  ⊆-disjoint? : ∀ {xs ys zs} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) → Dec (Disjoint τ₁ τ₂)

+
  ⊆-disjoint? [] [] = yes []

+
  -- Present in both sublists: not disjoint.

+
  ⊆-disjoint? (x≈z ∷ τ₁) (y≈z ∷ τ₂) = no λ()

+
  -- Present in either sublist: ok.

+
  ⊆-disjoint? (y ∷ʳ τ₁) (x≈y ∷ τ₂) =

+
    Dec.map′ (x≈y ∷ᵣ_) (λ{ (_ ∷ᵣ d) → d }) (⊆-disjoint? τ₁ τ₂)

+
  ⊆-disjoint? (x≈y ∷ τ₁) (y ∷ʳ τ₂) =

+
    Dec.map′ (x≈y ∷ₗ_) (λ{ (_ ∷ₗ d) → d }) (⊆-disjoint? τ₁ τ₂)

+
  -- Present in neither sublist: ok.

+
  ⊆-disjoint? (y ∷ʳ τ₁) (.y ∷ʳ τ₂) =

+
    Dec.map′ (y ∷ₙ_) (λ{ (_ ∷ₙ d) → d }) (⊆-disjoint? τ₁ τ₂)

+

+
  -- Disjoint is proof-irrelevant

+

+
  Disjoint-irrelevant : ∀{xs ys zs} → Irrelevant (Disjoint {R = R} {xs} {ys} {zs})

+
  Disjoint-irrelevant [] [] = ≡.refl

+
  Disjoint-irrelevant (y   ∷ₙ d₁) (.y   ∷ₙ d₂) = ≡.cong (y ∷ₙ_) (Disjoint-irrelevant d₁ d₂)

+
  Disjoint-irrelevant (x≈y ∷ₗ d₁) (.x≈y ∷ₗ d₂) = ≡.cong (x≈y ∷ₗ_) (Disjoint-irrelevant d₁ d₂)

+
  Disjoint-irrelevant (x≈y ∷ᵣ d₁) (.x≈y ∷ᵣ d₂) = ≡.cong (x≈y ∷ᵣ_) (Disjoint-irrelevant d₁ d₂)

+

+
  -- Note: DisjointUnion is not proof-irrelevant unless the underlying relation R is.

+
  -- The proof is not entirely trivial, thus, we leave it for future work:

+
  --

+
  -- DisjointUnion-irrelevant : Irrelevant R →

+
  --                            ∀{xs ys us zs} {τ : us ⊆ zs} →

+
  --                            Irrelevant (λ (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) → DisjointUnion τ₁ τ₂ τ)

+

+
  -- Irreflexivity

+

+
  Disjoint-irrefl′ : ∀{xs ys} {τ : xs ⊆ ys} → Disjoint τ τ → Null xs

+
  Disjoint-irrefl′ []       = []

+
  Disjoint-irrefl′ (y ∷ₙ d) = Disjoint-irrefl′ d

+

+
  Disjoint-irrefl : ∀{x xs ys} → Irreflexive {A = x ∷ xs ⊆ ys } _≡_ Disjoint

+
  Disjoint-irrefl ≡.refl x with Disjoint-irrefl′ x

+
  ... | () ∷ _

+

+
  -- Symmetry

+

+
  DisjointUnion-sym : ∀ {xs ys xys} {zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : xys ⊆ zs} →

+
                            DisjointUnion τ₁ τ₂ τ → DisjointUnion τ₂ τ₁ τ

+
  DisjointUnion-sym []         = []

+
  DisjointUnion-sym (y   ∷ₙ d) = y ∷ₙ DisjointUnion-sym d

+
  DisjointUnion-sym (x≈y ∷ₗ d) = x≈y ∷ᵣ DisjointUnion-sym d

+
  DisjointUnion-sym (x≈y ∷ᵣ d) = x≈y ∷ₗ DisjointUnion-sym d

+

+
  Disjoint-sym : ∀ {xs ys} {zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
                 Disjoint τ₁ τ₂ → Disjoint τ₂ τ₁

+
  Disjoint-sym []         = []

+
  Disjoint-sym (y   ∷ₙ d) = y ∷ₙ Disjoint-sym d

+
  Disjoint-sym (x≈y ∷ₗ d) = x≈y ∷ᵣ Disjoint-sym d

+
  Disjoint-sym (x≈y ∷ᵣ d) = x≈y ∷ₗ Disjoint-sym d

+

+
  -- Empty sublist

+

+
  DisjointUnion-[]ˡ : ∀{xs ys} {ε : [] ⊆ ys} {τ : xs ⊆ ys} → DisjointUnion ε τ τ

+
  DisjointUnion-[]ˡ {ε = []}     {τ = []} = []

+
  DisjointUnion-[]ˡ {ε = y ∷ʳ ε} {τ = y  ∷ʳ τ} = y   ∷ₙ DisjointUnion-[]ˡ

+
  DisjointUnion-[]ˡ {ε = y ∷ʳ ε} {τ = x≈y ∷ τ} = x≈y ∷ᵣ DisjointUnion-[]ˡ

+

+
  DisjointUnion-[]ʳ : ∀{xs ys} {ε : [] ⊆ ys} {τ : xs ⊆ ys} → DisjointUnion τ ε τ

+
  DisjointUnion-[]ʳ {ε = []}     {τ = []} = []

+
  DisjointUnion-[]ʳ {ε = y ∷ʳ ε} {τ = y  ∷ʳ τ} = y   ∷ₙ DisjointUnion-[]ʳ

+
  DisjointUnion-[]ʳ {ε = y ∷ʳ ε} {τ = x≈y ∷ τ} = x≈y ∷ₗ DisjointUnion-[]ʳ

+

+
  -- A sublist τ : x∷xs ⊆ ys can be split into two disjoint sublists

+
  -- [x] ⊆ ys (canonical choice) and (∷ˡ⁻ τ) : xs ⊆ ys.

+

+
  DisjointUnion-fromAny∘toAny-∷ˡ⁻ : ∀ {x xs ys} (τ : (x ∷ xs) ⊆ ys) → DisjointUnion (fromAny (toAny τ)) (∷ˡ⁻ τ) τ

+
  DisjointUnion-fromAny∘toAny-∷ˡ⁻ (y  ∷ʳ τ) = y   ∷ₙ DisjointUnion-fromAny∘toAny-∷ˡ⁻ τ

+
  DisjointUnion-fromAny∘toAny-∷ˡ⁻ (xRy ∷ τ) = xRy ∷ₗ DisjointUnion-[]ˡ

+

+
  -- Disjoint union of three mutually disjoint lists.

+
  --

+
  -- τᵢⱼ denotes the disjoint union of τᵢ and τⱼ: DisjointUnion τᵢ τⱼ τᵢⱼ

+

+
  record DisjointUnion³

+
    {xs ys zs ts} (τ₁  : xs  ⊆ ts) (τ₂  : ys  ⊆ ts) (τ₃  : zs  ⊆ ts)

+
    {xys xzs yzs} (τ₁₂ : xys ⊆ ts) (τ₁₃ : xzs ⊆ ts) (τ₂₃ : yzs ⊆ ts) : Set (a ⊔ b ⊔ r) where

+
    field

+
      {union³} : List A

+
      sub³  : union³ ⊆ ts

+
      join₁ : DisjointUnion τ₁ τ₂₃ sub³

+
      join₂ : DisjointUnion τ₂ τ₁₃ sub³

+
      join₃ : DisjointUnion τ₃ τ₁₂ sub³

+

+
  infixr 5 _∷ʳ-DisjointUnion³_ _∷₁-DisjointUnion³_ _∷₂-DisjointUnion³_ _∷₃-DisjointUnion³_

+

+
  -- Weakening the target list ts of a disjoint union.

+

+
  _∷ʳ-DisjointUnion³_ :

+
    ∀ {xs ys zs ts} {τ₁ : xs ⊆ ts} {τ₂ : ys ⊆ ts} {τ₃ : zs ⊆ ts} →

+
    ∀ {xys xzs yzs} {τ₁₂ : xys ⊆ ts} {τ₁₃ : xzs ⊆ ts} {τ₂₃ : yzs ⊆ ts} →

+
    ∀ y →

+
    DisjointUnion³ τ₁ τ₂ τ₃ τ₁₂ τ₁₃ τ₂₃ →

+
    DisjointUnion³ (y ∷ʳ τ₁) (y ∷ʳ τ₂) (y ∷ʳ τ₃) (y ∷ʳ τ₁₂) (y ∷ʳ τ₁₃) (y ∷ʳ τ₂₃)

+
  y ∷ʳ-DisjointUnion³ record{ sub³ = σ ; join₁ = d₁ ; join₂ = d₂ ; join₃ = d₃ } = record

+
    { sub³  = y ∷ʳ σ

+
    ; join₁ = y ∷ₙ d₁

+
    ; join₂ = y ∷ₙ d₂

+
    ; join₃ = y ∷ₙ d₃

+
    }

+

+
  -- Adding an element to the first list.

+

+
  _∷₁-DisjointUnion³_ :

+
    ∀ {xs ys zs ts} {τ₁ : xs ⊆ ts} {τ₂ : ys ⊆ ts} {τ₃ : zs ⊆ ts} →

+
    ∀ {xys xzs yzs} {τ₁₂ : xys ⊆ ts} {τ₁₃ : xzs ⊆ ts} {τ₂₃ : yzs ⊆ ts} →

+
    ∀ {x y} (xRy : R x y) →

+
    DisjointUnion³ τ₁ τ₂ τ₃ τ₁₂ τ₁₃ τ₂₃ →

+
    DisjointUnion³ (xRy ∷ τ₁) (y ∷ʳ τ₂) (y ∷ʳ τ₃) (xRy ∷ τ₁₂) (xRy ∷ τ₁₃) (y ∷ʳ τ₂₃)

+
  xRy ∷₁-DisjointUnion³ record{ sub³ = σ ; join₁ = d₁ ; join₂ = d₂ ; join₃ = d₃ } = record

+
    { sub³  = xRy ∷ σ

+
    ; join₁ = xRy ∷ₗ d₁

+
    ; join₂ = xRy ∷ᵣ d₂

+
    ; join₃ = xRy ∷ᵣ d₃

+
    }

+

+
  -- Adding an element to the second list.

+

+
  _∷₂-DisjointUnion³_ :

+
    ∀ {xs ys zs ts} {τ₁ : xs ⊆ ts} {τ₂ : ys ⊆ ts} {τ₃ : zs ⊆ ts} →

+
    ∀ {xys xzs yzs} {τ₁₂ : xys ⊆ ts} {τ₁₃ : xzs ⊆ ts} {τ₂₃ : yzs ⊆ ts} →

+
    ∀ {x y} (xRy : R x y) →

+
    DisjointUnion³ τ₁ τ₂ τ₃ τ₁₂ τ₁₃ τ₂₃ →

+
    DisjointUnion³ (y ∷ʳ τ₁) (xRy ∷ τ₂) (y ∷ʳ τ₃) (xRy ∷ τ₁₂) (y ∷ʳ τ₁₃) (xRy ∷ τ₂₃)

+
  xRy ∷₂-DisjointUnion³ record{ sub³ = σ ; join₁ = d₁ ; join₂ = d₂ ; join₃ = d₃ } = record

+
    { sub³  = xRy ∷ σ

+
    ; join₁ = xRy ∷ᵣ d₁

+
    ; join₂ = xRy ∷ₗ d₂

+
    ; join₃ = xRy ∷ᵣ d₃

+
    }

+

+
  -- Adding an element to the third list.

+

+
  _∷₃-DisjointUnion³_ :

+
    ∀ {xs ys zs ts} {τ₁ : xs ⊆ ts} {τ₂ : ys ⊆ ts} {τ₃ : zs ⊆ ts} →

+
    ∀ {xys xzs yzs} {τ₁₂ : xys ⊆ ts} {τ₁₃ : xzs ⊆ ts} {τ₂₃ : yzs ⊆ ts} →

+
    ∀ {x y} (xRy : R x y) →

+
    DisjointUnion³ τ₁ τ₂ τ₃ τ₁₂ τ₁₃ τ₂₃ →

+
    DisjointUnion³ (y ∷ʳ τ₁) (y ∷ʳ τ₂) (xRy ∷ τ₃) (y ∷ʳ τ₁₂) (xRy ∷ τ₁₃) (xRy ∷ τ₂₃)

+
  xRy ∷₃-DisjointUnion³ record{ sub³ = σ ; join₁ = d₁ ; join₂ = d₂ ; join₃ = d₃ } = record

+
    { sub³  = xRy ∷ σ

+
    ; join₁ = xRy ∷ᵣ d₁

+
    ; join₂ = xRy ∷ᵣ d₂

+
    ; join₃ = xRy ∷ₗ d₃

+
    }

+

+
  -- Computing the disjoint union of three disjoint lists.

+

+
  disjointUnion³ : ∀{xs ys zs ts} {τ₁ : xs ⊆ ts} {τ₂ : ys ⊆ ts} {τ₃ : zs ⊆ ts}

+
    {xys xzs yzs} {τ₁₂ : xys ⊆ ts} {τ₁₃ : xzs ⊆ ts} {τ₂₃ : yzs ⊆ ts} →

+
    DisjointUnion τ₁ τ₂ τ₁₂ →

+
    DisjointUnion τ₁ τ₃ τ₁₃ →

+
    DisjointUnion τ₂ τ₃ τ₂₃ →

+
    DisjointUnion³ τ₁ τ₂ τ₃ τ₁₂ τ₁₃ τ₂₃

+
  disjointUnion³ [] [] [] = record { sub³ = [] ; join₁ = [] ; join₂ = [] ; join₃ = [] }

+
  disjointUnion³ (y   ∷ₙ d₁₂) (.y  ∷ₙ d₁₃) (.y   ∷ₙ d₂₃) = y ∷ʳ-DisjointUnion³ disjointUnion³ d₁₂ d₁₃ d₂₃

+
  disjointUnion³ (y   ∷ₙ d₁₂) (xRy ∷ᵣ d₁₃) (.xRy ∷ᵣ d₂₃) = xRy ∷₃-DisjointUnion³ disjointUnion³ d₁₂ d₁₃ d₂₃

+
  disjointUnion³ (xRy ∷ᵣ d₁₂) (y   ∷ₙ d₁₃) (.xRy ∷ₗ d₂₃) = xRy ∷₂-DisjointUnion³ disjointUnion³ d₁₂ d₁₃ d₂₃

+
  disjointUnion³ (xRy ∷ₗ d₁₂) (.xRy ∷ₗ d₁₃) (y    ∷ₙ d₂₃) = xRy ∷₁-DisjointUnion³ disjointUnion³ d₁₂ d₁₃ d₂₃

+
  disjointUnion³ (xRy ∷ᵣ d₁₂) (xRy′ ∷ᵣ d₁₃) ()

+

+
  -- If a sublist τ is disjoint to two lists σ₁ and σ₂,

+
  -- then also to their disjoint union σ.

+

+
  disjoint⇒disjoint-to-union : ∀{xs ys zs yzs ts}

+
    {τ : xs ⊆ ts} {σ₁ : ys ⊆ ts} {σ₂ : zs ⊆ ts} {σ : yzs ⊆ ts} →

+
    Disjoint τ σ₁ → Disjoint τ σ₂ → DisjointUnion σ₁ σ₂ σ → Disjoint τ σ

+
  disjoint⇒disjoint-to-union d₁ d₂ u = let

+
       _ , _ , u₁ = Disjoint→DisjointUnion d₁

+
       _ , _ , u₂ = Disjoint→DisjointUnion d₂

+
    in DisjointUnion→Disjoint (DisjointUnion³.join₁ (disjointUnion³ u₁ u₂ u))

+

+
open Disjointness public

+

+
-- Monotonicity of disjointness.

+

+
module DisjointnessMonotonicity

+
    {R : REL A B r} {S : REL B C s} {T : REL A C t}

+
    (rs⇒t : Trans R S T) where

+

+
  -- We can enlarge and convert the target list of a disjoint union.

+

+
  weakenDisjointUnion : ∀ {xs ys xys zs ws}

+
    {τ₁ : Sublist R xs zs}

+
    {τ₂ : Sublist R ys zs}

+
    {τ : Sublist R xys zs} (σ : Sublist S zs ws) →

+
    DisjointUnion τ₁ τ₂ τ →

+
    DisjointUnion (trans rs⇒t τ₁ σ) (trans rs⇒t τ₂ σ) (trans rs⇒t τ σ)

+
  weakenDisjointUnion [] [] = []

+
  weakenDisjointUnion (w  ∷ʳ σ) d         = w ∷ₙ weakenDisjointUnion σ d

+
  weakenDisjointUnion (_   ∷ σ) (y   ∷ₙ d) = _ ∷ₙ weakenDisjointUnion σ d

+
  weakenDisjointUnion (zSw ∷ σ) (xRz ∷ₗ d) = rs⇒t xRz zSw ∷ₗ weakenDisjointUnion σ d

+
  weakenDisjointUnion (zSw ∷ σ) (yRz ∷ᵣ d) = rs⇒t yRz zSw ∷ᵣ weakenDisjointUnion σ d

+

+
  weakenDisjoint : ∀ {xs ys zs ws}

+
    {τ₁ : Sublist R xs zs}

+
    {τ₂ : Sublist R ys zs} (σ : Sublist S zs ws) →

+
    Disjoint τ₁ τ₂ →

+
    Disjoint (trans rs⇒t τ₁ σ) (trans rs⇒t τ₂ σ)

+
  weakenDisjoint [] [] = []

+
  weakenDisjoint (w  ∷ʳ σ) d         = w ∷ₙ weakenDisjoint σ d

+
  weakenDisjoint (_   ∷ σ) (y   ∷ₙ d) = _ ∷ₙ weakenDisjoint σ d

+
  weakenDisjoint (zSw ∷ σ) (xRz ∷ₗ d) = rs⇒t xRz zSw ∷ₗ weakenDisjoint σ d

+
  weakenDisjoint (zSw ∷ σ) (yRz ∷ᵣ d) = rs⇒t yRz zSw ∷ᵣ weakenDisjoint σ d

+

+
  -- Lists remain disjoint when elements are removed.

+

+
  shrinkDisjoint : ∀ {us vs xs ys zs}

+
                      (σ₁ : Sublist R us xs) {τ₁ : Sublist S xs zs}

+
                      (σ₂ : Sublist R vs ys) {τ₂ : Sublist S ys zs} →

+
                      Disjoint τ₁ τ₂ →

+
                      Disjoint (trans rs⇒t σ₁ τ₁) (trans rs⇒t σ₂ τ₂)

+
  shrinkDisjoint σ₁         σ₂ (y   ∷ₙ d) = y ∷ₙ shrinkDisjoint σ₁ σ₂ d

+
  shrinkDisjoint (x  ∷ʳ σ₁) σ₂ (xSz ∷ₗ d) = _ ∷ₙ shrinkDisjoint σ₁ σ₂ d

+
  shrinkDisjoint (uRx ∷ σ₁) σ₂ (xSz ∷ₗ d) = rs⇒t uRx xSz ∷ₗ shrinkDisjoint σ₁ σ₂ d

+
  shrinkDisjoint σ₁ (y  ∷ʳ σ₂) (ySz ∷ᵣ d) = _ ∷ₙ shrinkDisjoint σ₁ σ₂ d

+
  shrinkDisjoint σ₁ (vRy ∷ σ₂) (ySz ∷ᵣ d) = rs⇒t vRy ySz ∷ᵣ shrinkDisjoint σ₁ σ₂ d

+
  shrinkDisjoint [] []         []         = []

+

+
open DisjointnessMonotonicity public

+

\ No newline at end of file
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@@ -0,0 +1,187 @@
+
+Data.List.Relation.Binary.Sublist.Heterogeneous
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An inductive definition of the heterogeneous sublist relation

+
-- This is a generalisation of what is commonly known as Order

+
-- Preserving Embeddings (OPE).

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.List.Base using (List; []; _∷_; [_])

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Core using (REL; _⇒_)

+
open import Relation.Binary.Definitions using (_⟶_Respects_; Min)

+
open import Relation.Unary using (Pred)

+

+
module Data.List.Relation.Binary.Sublist.Heterogeneous

+
  {a b r} {A : Set a} {B : Set b} {R : REL A B r}

+
  where

+

+
------------------------------------------------------------------------

+
-- Re-export core definitions

+

+
open import Data.List.Relation.Binary.Sublist.Heterogeneous.Core public

+

+
------------------------------------------------------------------------

+
-- Type and basic combinators

+

+
module _ {s} {S : REL A B s} where

+

+
  map : R ⇒ S → Sublist R ⇒ Sublist S

+
  map f []        = []

+
  map f (y ∷ʳ rs) = y ∷ʳ map f rs

+
  map f (r ∷ rs)  = f r ∷ map f rs

+

+
minimum : Min (Sublist R) []

+
minimum []       = []

+
minimum (x ∷ xs) = x ∷ʳ minimum xs

+

+
------------------------------------------------------------------------

+
-- Conversion to and from Any

+

+
-- Special case: Sublist R [ a ] bs → Any (R a) bs

+
toAny : ∀ {a as bs} → Sublist R (a ∷ as) bs → Any (R a) bs

+
toAny (y ∷ʳ rs) = there (toAny rs)

+
toAny (r ∷ rs)  = here r

+

+
fromAny : ∀ {a bs} → Any (R a) bs → Sublist R [ a ] bs

+
fromAny (here r)  = r ∷ minimum _

+
fromAny (there p) = _ ∷ʳ fromAny p

+

+
------------------------------------------------------------------------

+
-- Generalised lookup based on a proof of Any

+

+
module _ {p q} {P : Pred A p} {Q : Pred B q} (resp : P ⟶ Q Respects R) where

+

+
  lookup : ∀ {xs ys} → Sublist R xs ys → Any P xs → Any Q ys

+
  lookup (y ∷ʳ p)  k         = there (lookup p k)

+
  lookup (rxy ∷ p) (here px) = here (resp rxy px)

+
  lookup (rxy ∷ p) (there k) = there (lookup p k)

+

+
------------------------------------------------------------------------

+
-- Disjoint sublists xs,ys ⊆ zs

+
--

+
-- NB: This does not imply that xs and ys partition zs;

+
-- zs may have additional elements.

+

+
private

+
  infix 4 _⊆_

+
  _⊆_ = Sublist R

+

+
infixr 5 _∷ₙ_ _∷ₗ_ _∷ᵣ_

+

+
data Disjoint : ∀ {xs ys zs} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) → Set (a ⊔ b ⊔ r) where

+
  []   : Disjoint [] []

+

+
  -- Element y of zs is neither in xs nor in ys:

+
  _∷ₙ_ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
         (y : B)       → Disjoint τ₁ τ₂ → Disjoint (y  ∷ʳ τ₁) (y  ∷ʳ τ₂)

+

+
  -- Element y of zs is in xs as x with x≈y:

+
  _∷ₗ_ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {x y} →

+
         (x≈y : R x y) → Disjoint τ₁ τ₂ → Disjoint (x≈y ∷ τ₁) (y  ∷ʳ τ₂)

+

+
  -- Element y of zs is in ys as x with x≈y:

+
  _∷ᵣ_ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {x y} →

+
         (x≈y : R x y) → Disjoint τ₁ τ₂ → Disjoint (y  ∷ʳ τ₁) (x≈y ∷ τ₂)

+

+
------------------------------------------------------------------------

+
-- Disjoint union of two sublists xs,ys ⊆ zs

+
--

+
-- This is the Cover relation without overlap of Section 6 of

+
-- Conor McBride, Everybody's Got To Be Somewhere,

+
-- MSFP@FSCD 2018: 53-69.

+

+
data DisjointUnion : ∀ {xs ys zs us} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) (τ : us ⊆ zs) →  Set (a ⊔ b ⊔ r) where

+
  []   : DisjointUnion [] [] []

+

+
  -- Element y of zs is neither in xs nor in ys: skip.

+
  _∷ₙ_ : ∀ {xs ys zs us} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : us ⊆ zs} →

+
         (y : B)       → DisjointUnion τ₁ τ₂ τ → DisjointUnion (y  ∷ʳ τ₁) (y  ∷ʳ τ₂) (y ∷ʳ τ)

+

+
  -- Element y of zs is in xs as x with x≈y: add to us.

+
  _∷ₗ_ : ∀ {xs ys zs us} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : us ⊆ zs} {x y} →

+
         (x≈y : R x y) → DisjointUnion τ₁ τ₂ τ → DisjointUnion (x≈y ∷ τ₁) (y  ∷ʳ τ₂) (x≈y ∷ τ)

+

+
  -- Element y of zs is in ys as x with x≈y: add to us.

+
  _∷ᵣ_ : ∀ {xs ys zs us} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : us ⊆ zs} {x y} →

+
         (x≈y : R x y) → DisjointUnion τ₁ τ₂ τ → DisjointUnion (y  ∷ʳ τ₁) (x≈y ∷ τ₂) (x≈y ∷ τ)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Sublist.Propositional.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Propositional.Properties.html
new file mode 100644
index 000000000..6803fb546
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Propositional.Properties.html
@@ -0,0 +1,311 @@
+
+Data.List.Relation.Binary.Sublist.Propositional.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sublist-related properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Sublist.Propositional.Properties

+
  {a} {A : Set a} where

+

+
open import Data.List.Base using (List; []; _∷_;  map)

+
open import Data.List.Membership.Propositional using (_∈_)

+
open import Data.List.Relation.Unary.All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.List.Relation.Unary.Any.Properties

+
  using (here-injective; there-injective)

+
open import Data.List.Relation.Binary.Sublist.Propositional

+
  hiding (map)

+
import Data.List.Relation.Binary.Sublist.Setoid.Properties

+
  as SetoidProperties

+
open import Data.Product.Base using (∃; _,_; proj₂)

+
open import Function.Base using (id; _∘_; _∘′_)

+
open import Level using (Level)

+
open import Relation.Binary.Definitions using (_Respects_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; refl; cong; _≗_; trans)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; subst-injective; trans-reflʳ; trans-assoc)

+
open import Relation.Unary using (Pred)

+

+
private

+
  variable

+
    b ℓ : Level

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Re-exporting setoid properties

+

+
open SetoidProperties (setoid A) public

+
  hiding (map⁺; ⊆-trans-idˡ; ⊆-trans-idʳ; ⊆-trans-assoc)

+

+
map⁺ : ∀ {as bs} (f : A → B) → as ⊆ bs → map f as ⊆ map f bs

+
map⁺ {B = B} f = SetoidProperties.map⁺ (setoid A) (setoid B) (cong f)

+

+
------------------------------------------------------------------------

+
-- Category laws for _⊆_

+

+
⊆-trans-idˡ : ∀ {xs ys : List A} {τ : xs ⊆ ys} →

+
              ⊆-trans ⊆-refl τ ≡ τ

+
⊆-trans-idˡ {τ = τ} = SetoidProperties.⊆-trans-idˡ (setoid A) (λ _ → refl) τ

+

+
⊆-trans-idʳ : ∀ {xs ys : List A} {τ : xs ⊆ ys} →

+
              ⊆-trans τ ⊆-refl ≡ τ

+
⊆-trans-idʳ {τ = τ} = SetoidProperties.⊆-trans-idʳ (setoid A) trans-reflʳ τ

+

+
-- Note: The associativity law is oriented such that rewriting with it

+
-- may trigger reductions of ⊆-trans, which matches first on its

+
-- second argument and then on its first argument.

+

+
⊆-trans-assoc : ∀ {ws xs ys zs : List A}

+
                {τ₁ : ws ⊆ xs} {τ₂ : xs ⊆ ys} {τ₃ : ys ⊆ zs} →

+
                ⊆-trans τ₁ (⊆-trans τ₂ τ₃) ≡ ⊆-trans (⊆-trans τ₁ τ₂) τ₃

+
⊆-trans-assoc {τ₁ = τ₁} {τ₂ = τ₂} {τ₃ = τ₃} =

+
  SetoidProperties.⊆-trans-assoc (setoid A) (λ p _ _ → ≡.sym (trans-assoc p)) τ₁ τ₂ τ₃

+

+
------------------------------------------------------------------------

+
-- Laws concerning ⊆-trans and ∷ˡ⁻

+

+
⊆-trans-∷ˡ⁻ᵣ : ∀ {y} {xs ys zs : List A} {τ : xs ⊆ ys} {σ : (y ∷ ys) ⊆ zs} →

+
               ⊆-trans τ (∷ˡ⁻ σ) ≡ ⊆-trans (y ∷ʳ τ) σ

+
⊆-trans-∷ˡ⁻ᵣ {σ = x ∷ σ} = refl

+
⊆-trans-∷ˡ⁻ᵣ {σ = y ∷ʳ σ} = cong (y ∷ʳ_) ⊆-trans-∷ˡ⁻ᵣ

+

+
⊆-trans-∷ˡ⁻ₗ : ∀ {x} {xs ys zs : List A} {τ : (x ∷ xs) ⊆ ys} {σ : ys ⊆ zs} →

+
              ⊆-trans (∷ˡ⁻ τ) σ ≡ ∷ˡ⁻ (⊆-trans τ σ)

+
⊆-trans-∷ˡ⁻ₗ                {σ = y   ∷ʳ σ} = cong (y ∷ʳ_) ⊆-trans-∷ˡ⁻ₗ

+
⊆-trans-∷ˡ⁻ₗ {τ = y   ∷ʳ τ} {σ = refl ∷ σ} = cong (y ∷ʳ_) ⊆-trans-∷ˡ⁻ₗ

+
⊆-trans-∷ˡ⁻ₗ {τ = refl ∷ τ} {σ = refl ∷ σ} = refl

+

+
⊆-∷ˡ⁻trans-∷ : ∀ {y} {xs ys zs : List A} {τ : xs ⊆ ys} {σ : (y ∷ ys) ⊆ zs} →

+
               ∷ˡ⁻ (⊆-trans (refl ∷ τ) σ) ≡ ⊆-trans (y ∷ʳ τ) σ

+
⊆-∷ˡ⁻trans-∷ {σ = y   ∷ʳ σ} = cong (y ∷ʳ_) ⊆-∷ˡ⁻trans-∷

+
⊆-∷ˡ⁻trans-∷ {σ = refl ∷ σ} = refl

+

+
------------------------------------------------------------------------

+
-- Relationships to other predicates

+

+
-- All P is a contravariant functor from _⊆_ to Set.

+

+
All-resp-⊆ : {P : Pred A ℓ} → (All P) Respects _⊇_

+
All-resp-⊆ []          []       = []

+
All-resp-⊆ (_    ∷ʳ p) (_ ∷ xs) = All-resp-⊆ p xs

+
All-resp-⊆ (refl ∷  p) (x ∷ xs) = x ∷ All-resp-⊆ p xs

+

+
-- Any P is a covariant functor from _⊆_ to Set.

+

+
Any-resp-⊆ : {P : Pred A ℓ} → (Any P) Respects _⊆_

+
Any-resp-⊆ = lookup

+

+
------------------------------------------------------------------------

+
-- Functor laws for All-resp-⊆

+

+
-- First functor law: identity.

+

+
All-resp-⊆-refl : ∀ {P : Pred A ℓ} {xs : List A} →

+
                  All-resp-⊆ ⊆-refl ≗ id {A = All P xs}

+
All-resp-⊆-refl []       = refl

+
All-resp-⊆-refl (p ∷ ps) = cong (p ∷_) (All-resp-⊆-refl ps)

+

+
-- Second functor law: composition.

+

+
All-resp-⊆-trans : ∀ {P : Pred A ℓ} {xs ys zs} {τ : xs ⊆ ys} (τ′ : ys ⊆ zs) →

+
                   All-resp-⊆ {P = P} (⊆-trans τ τ′) ≗ All-resp-⊆ τ ∘ All-resp-⊆ τ′

+
All-resp-⊆-trans                (_    ∷ʳ τ′) (p ∷ ps) = All-resp-⊆-trans τ′ ps

+
All-resp-⊆-trans {τ = _ ∷ʳ _  } (refl ∷  τ′) (p ∷ ps) = All-resp-⊆-trans τ′ ps

+
All-resp-⊆-trans {τ = refl ∷ _} (refl ∷  τ′) (p ∷ ps) = cong (p ∷_) (All-resp-⊆-trans τ′ ps)

+
All-resp-⊆-trans {τ = []      } ([]        ) []       = refl

+

+
------------------------------------------------------------------------

+
-- Functor laws for Any-resp-⊆ / lookup

+

+
-- First functor law: identity.

+

+
Any-resp-⊆-refl : ∀ {P : Pred A ℓ} {xs} →

+
                  Any-resp-⊆ ⊆-refl ≗ id {A = Any P xs}

+
Any-resp-⊆-refl (here p)  = refl

+
Any-resp-⊆-refl (there i) = cong there (Any-resp-⊆-refl i)

+

+
lookup-⊆-refl = Any-resp-⊆-refl

+

+
-- Second functor law: composition.

+

+
Any-resp-⊆-trans : ∀ {P : Pred A ℓ} {xs ys zs} {τ : xs ⊆ ys} (τ′ : ys ⊆ zs) →

+
                   Any-resp-⊆ {P = P} (⊆-trans τ τ′) ≗ Any-resp-⊆ τ′ ∘ Any-resp-⊆ τ

+
Any-resp-⊆-trans                (_ ∷ʳ τ′) i         = cong there (Any-resp-⊆-trans τ′ i)

+
Any-resp-⊆-trans {τ = _   ∷ʳ _} (_ ∷  τ′) i         = cong there (Any-resp-⊆-trans τ′ i)

+
Any-resp-⊆-trans {τ = _    ∷ _} (_ ∷  τ′) (there i) = cong there (Any-resp-⊆-trans τ′ i)

+
Any-resp-⊆-trans {τ = refl ∷ _} (_ ∷  τ′) (here _)  = refl

+
Any-resp-⊆-trans {τ = []      } []        ()

+

+
lookup-⊆-trans = Any-resp-⊆-trans

+

+
------------------------------------------------------------------------

+
-- The `lookup` function for `xs ⊆ ys` is injective.

+
--

+
-- Note: `lookup` can be seen as a strictly increasing reindexing

+
-- function for indices into `xs`, producing indices into `ys`.

+

+
lookup-injective : ∀ {P : Pred A ℓ} {xs ys} {τ : xs ⊆ ys} {i j : Any P xs} →

+
                   lookup τ i ≡ lookup τ j → i ≡ j

+
lookup-injective {τ = _  ∷ʳ _}                     = lookup-injective ∘′ there-injective

+
lookup-injective {τ = x≡y ∷ _} {here  _} {here  _} = cong here ∘′ subst-injective x≡y ∘′ here-injective

+
  -- Note: instead of using subst-injective, we could match x≡y against refl on the lhs.

+
  -- However, this turns the following clause into a non-strict match.

+
lookup-injective {τ = _   ∷ _} {there _} {there _} = cong there ∘′ lookup-injective ∘′ there-injective

+

+
------------------------------------------------------------------------

+
-- from∈ ∘ to∈ turns a sublist morphism τ : x∷xs ⊆ ys into a morphism

+
-- [x] ⊆ ys.  The same morphism is obtained by pre-composing τ with

+
-- the canonial morphism [x] ⊆ x∷xs.

+
--

+
-- Note: This lemma does not hold for Sublist.Setoid, but could hold for

+
-- a hypothetical Sublist.Groupoid where trans refl = id.

+

+
from∈∘to∈ : ∀ {x : A} {xs ys} (τ : x ∷ xs ⊆ ys) →

+
            from∈ (to∈ τ) ≡ ⊆-trans (refl ∷ minimum xs) τ

+
from∈∘to∈ (x≡y ∷ τ) = cong (x≡y ∷_) ([]⊆-irrelevant _ _)

+
from∈∘to∈ (y  ∷ʳ τ) = cong (y  ∷ʳ_) (from∈∘to∈ τ)

+

+
from∈∘lookup : ∀{x : A} {xs ys} (τ : xs ⊆ ys) (i : x ∈ xs) →

+
               from∈ (lookup τ i) ≡ ⊆-trans (from∈ i) τ

+
from∈∘lookup (y   ∷ʳ τ)  i          = cong (y ∷ʳ_) (from∈∘lookup τ i)

+
from∈∘lookup (_    ∷ τ) (there i)   = cong (_ ∷ʳ_) (from∈∘lookup τ i)

+
from∈∘lookup (refl ∷ τ) (here refl) = cong (refl ∷_) ([]⊆-irrelevant _ _)

+

+
------------------------------------------------------------------------

+
-- Weak pushout (wpo)

+

+
-- A raw pushout is a weak pushout if the pushout square commutes.

+

+
IsWeakPushout : ∀{xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →

+
                RawPushout τ σ → Set a

+
IsWeakPushout {τ = τ} {σ = σ} rpo =

+
  ⊆-trans τ (RawPushout.leg₁ rpo) ≡

+
  ⊆-trans σ (RawPushout.leg₂ rpo)

+

+
-- Joining two list extensions with ⊆-pushout produces a weak pushout.

+

+
⊆-pushoutˡ-is-wpo : ∀{xs ys zs : List A} (τ : xs ⊆ ys) (σ : xs ⊆ zs) →

+
                    IsWeakPushout (⊆-pushoutˡ τ σ)

+
⊆-pushoutˡ-is-wpo [] σ

+
  rewrite ⊆-trans-idʳ {τ = σ}

+
        = ⊆-trans-idˡ {xs = []}

+
⊆-pushoutˡ-is-wpo (y   ∷ʳ τ) σ          = cong (y   ∷ʳ_) (⊆-pushoutˡ-is-wpo τ σ)

+
⊆-pushoutˡ-is-wpo (x≡y  ∷ τ) (z   ∷ʳ σ) = cong (z   ∷ʳ_) (⊆-pushoutˡ-is-wpo (x≡y ∷ τ) σ)

+
⊆-pushoutˡ-is-wpo (refl ∷ τ) (refl ∷ σ) = cong (refl ∷_) (⊆-pushoutˡ-is-wpo τ σ)

+

+
------------------------------------------------------------------------

+
-- Properties of disjointness

+

+
-- From τ₁ ⊎ τ₂ = τ, compute the injection ι₁ such that τ₁ = ⊆-trans ι₁ τ.

+

+
DisjointUnion-inj₁ : ∀ {xs ys zs xys : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : xys ⊆ zs} →

+
                      DisjointUnion τ₁ τ₂ τ → ∃ λ (ι₁ : xs ⊆ xys) → ⊆-trans ι₁ τ ≡ τ₁

+
DisjointUnion-inj₁ []         = []       , refl

+
DisjointUnion-inj₁ (y   ∷ₙ d) = _        , cong (y  ∷ʳ_) (proj₂ (DisjointUnion-inj₁ d))

+
DisjointUnion-inj₁ (x≈y ∷ₗ d) = refl ∷ _ , cong (x≈y ∷_) (proj₂ (DisjointUnion-inj₁ d))

+
DisjointUnion-inj₁ (x≈y ∷ᵣ d) = _ ∷ʳ _   , cong (_  ∷ʳ_) (proj₂ (DisjointUnion-inj₁ d))

+

+
-- From τ₁ ⊎ τ₂ = τ, compute the injection ι₂ such that τ₂ = ⊆-trans ι₂ τ.

+

+
DisjointUnion-inj₂ : ∀ {xs ys zs xys : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} {τ : xys ⊆ zs} →

+
                      DisjointUnion τ₁ τ₂ τ → ∃ λ (ι₂ : ys ⊆ xys) → ⊆-trans ι₂ τ ≡ τ₂

+
DisjointUnion-inj₂ []         = []       , refl

+
DisjointUnion-inj₂ (y   ∷ₙ d) = _        , cong (y  ∷ʳ_) (proj₂ (DisjointUnion-inj₂ d))

+
DisjointUnion-inj₂ (x≈y ∷ᵣ d) = refl ∷ _ , cong (x≈y ∷_) (proj₂ (DisjointUnion-inj₂ d))

+
DisjointUnion-inj₂ (x≈y ∷ₗ d) = _ ∷ʳ _   , cong (_  ∷ʳ_) (proj₂ (DisjointUnion-inj₂ d))

+

+
-- A sublist σ disjoint to both τ₁ and τ₂ is an equalizer

+
-- for the separators of τ₁ and τ₂.

+

+
equalize-separators : ∀ {us xs ys zs : List A}

+
  {σ : us ⊆ zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} (let s = separateˡ τ₁ τ₂) →

+
  Disjoint σ τ₁ → Disjoint σ τ₂ →

+
  ⊆-trans σ (Separation.separator₁ s) ≡

+
  ⊆-trans σ (Separation.separator₂ s)

+
equalize-separators [] [] = refl

+
equalize-separators (y    ∷ₙ d₁) (.y   ∷ₙ d₂) = cong (y ∷ʳ_) (equalize-separators d₁ d₂)

+
equalize-separators (y    ∷ₙ d₁) (refl ∷ᵣ d₂) = cong (y ∷ʳ_) (equalize-separators d₁ d₂)

+
equalize-separators (refl ∷ᵣ d₁) (y    ∷ₙ d₂) = cong (y ∷ʳ_) (equalize-separators d₁ d₂)

+
equalize-separators {τ₁ = refl ∷ _} {τ₂ = refl ∷ _}  -- match here to work around deficiency of Agda's forcing translation

+
                    (_    ∷ᵣ d₁) (_   ∷ᵣ d₂) = cong (_ ∷ʳ_) (cong (_ ∷ʳ_) (equalize-separators d₁ d₂))

+
equalize-separators (x≈y  ∷ₗ d₁) (.x≈y ∷ₗ d₂) = cong (trans x≈y refl ∷_) (equalize-separators d₁ d₂)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Sublist.Propositional.html b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Propositional.html
new file mode 100644
index 000000000..58438111c
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Propositional.html
@@ -0,0 +1,234 @@
+
+Data.List.Relation.Binary.Sublist.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An inductive definition of the sublist relation. This is commonly

+
-- known as Order Preserving Embeddings (OPE).

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Sublist.Propositional

+
  {a} {A : Set a} where

+

+
open import Data.List.Base using (List)

+
open import Data.List.Relation.Binary.Equality.Propositional using (≋⇒≡)

+
import Data.List.Relation.Binary.Sublist.Setoid as SetoidSublist

+
open import Data.List.Relation.Unary.Any using (Any)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles using (Preorder; Poset)

+
open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)

+
open import Relation.Binary.Definitions using (Antisymmetric)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (subst; _≡_; refl)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; isEquivalence)

+
open import Relation.Unary using (Pred)

+

+
------------------------------------------------------------------------

+
-- Re-export definition and operations from setoid sublists

+

+
open SetoidSublist (setoid A) public

+
  hiding

+
  (lookup; ⊆-reflexive; ⊆-antisym

+
  ; ⊆-isPreorder; ⊆-isPartialOrder

+
  ; ⊆-preorder; ⊆-poset

+
  )

+

+
------------------------------------------------------------------------

+
-- Additional operations

+

+
module _ {p} {P : Pred A p} where

+

+
  lookup : ∀ {xs ys} → xs ⊆ ys → Any P xs → Any P ys

+
  lookup = SetoidSublist.lookup (setoid A) (subst _)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
⊆-reflexive : _≡_ ⇒ _⊆_

+
⊆-reflexive refl = ⊆-refl

+

+
⊆-antisym : Antisymmetric _≡_ _⊆_

+
⊆-antisym xs⊆ys ys⊆xs = ≋⇒≡ (SetoidSublist.⊆-antisym (setoid A) xs⊆ys ys⊆xs)

+

+
⊆-isPreorder : IsPreorder _≡_ _⊆_

+
⊆-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ⊆-reflexive

+
  ; trans         = ⊆-trans

+
  }

+

+
⊆-isPartialOrder : IsPartialOrder _≡_ _⊆_

+
⊆-isPartialOrder = record

+
  { isPreorder = ⊆-isPreorder

+
  ; antisym    = ⊆-antisym

+
  }

+

+
⊆-preorder : Preorder a a a

+
⊆-preorder = record

+
  { isPreorder = ⊆-isPreorder

+
  }

+

+
⊆-poset : Poset a a a

+
⊆-poset = record

+
  { isPartialOrder = ⊆-isPartialOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Separating two sublists

+
--

+
-- Two possibly overlapping sublists τ : xs ⊆ zs and σ : ys ⊆ zs

+
-- can be turned into disjoint lists τρ : xs ⊆ zs and τρ : ys ⊆ zs′

+
-- by duplicating all entries of zs that occur both in xs and ys,

+
-- resulting in an extension ρ : zs ⊆ zs′ of zs.

+

+
record Separation {xs ys zs} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) : Set a where

+
  field

+
    {inflation} : List A

+
    separator₁  : zs ⊆ inflation

+
    separator₂  : zs ⊆ inflation

+
  separated₁ = ⊆-trans τ₁ separator₁

+
  separated₂ = ⊆-trans τ₂ separator₂

+
  field

+
    disjoint    : Disjoint separated₁ separated₂

+

+
infixr 5 _∷ₙ-Sep_ _∷ₗ-Sep_ _∷ᵣ-Sep_

+

+
-- Empty separation

+

+
[]-Sep : Separation [] []

+
[]-Sep = record { separator₁ = [] ; separator₂ = [] ; disjoint = [] }

+

+
-- Weaken a separation

+

+
_∷ₙ-Sep_ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
           ∀ z → Separation τ₁ τ₂ → Separation (z ∷ʳ τ₁) (z ∷ʳ τ₂)

+
z ∷ₙ-Sep record{ separator₁ = ρ₁; separator₂ = ρ₂; disjoint = d } = record

+
  { separator₁ = refl ∷ ρ₁

+
  ; separator₂ = refl ∷ ρ₂

+
  ; disjoint   = z   ∷ₙ d

+
  }

+

+
-- Extend a separation by an element of the first sublist.

+
--

+
-- Note: this requires a category law from the underlying equality,

+
-- trans x=z refl = x=z, thus, separation is not available for Sublist.Setoid.

+

+
_∷ₗ-Sep_ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
          ∀ {x z} (x≡z : x ≡ z) → Separation τ₁ τ₂ → Separation (x≡z ∷ τ₁) (z ∷ʳ τ₂)

+
refl ∷ₗ-Sep record{ separator₁ = ρ₁; separator₂ = ρ₂; disjoint = d } = record

+
  { separator₁ = refl ∷ ρ₁

+
  ; separator₂ = refl ∷ ρ₂

+
  ; disjoint   = refl ∷ₗ d

+
  }

+

+
-- Extend a separation by an element of the second sublist.

+

+
_∷ᵣ-Sep_ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
          ∀ {y z} (y≡z : y ≡ z) → Separation τ₁ τ₂ → Separation (z ∷ʳ τ₁) (y≡z ∷ τ₂)

+
refl ∷ᵣ-Sep record{ separator₁ = ρ₁; separator₂ = ρ₂; disjoint = d } = record

+
  { separator₁ = refl ∷ ρ₁

+
  ; separator₂ = refl ∷ ρ₂

+
  ; disjoint   = refl ∷ᵣ d

+
  }

+

+
-- Extend a separation by a common element of both sublists.

+
--

+
-- Left-biased: the left separator gets the first copy

+
-- of the common element.

+

+
∷-Sepˡ : ∀ {xs ys zs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →

+
        ∀ {x y z} (x≡z : x ≡ z) (y≡z : y ≡ z) →

+
        Separation τ₁ τ₂ → Separation (x≡z ∷ τ₁) (y≡z ∷ τ₂)

+
∷-Sepˡ refl refl record{ separator₁ = ρ₁; separator₂ = ρ₂; disjoint = d } = record

+
  { separator₁ = _ ∷ʳ refl ∷ ρ₁

+
  ; separator₂ = refl ∷ _ ∷ʳ ρ₂

+
  ; disjoint   = refl ∷ᵣ (refl ∷ₗ d)

+
  }

+

+
-- Left-biased separation of two sublists.  Of common elements,

+
-- the first sublist receives the first copy.

+

+
separateˡ : ∀ {xs ys zs} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) → Separation τ₁ τ₂

+
separateˡ [] [] = []-Sep

+
separateˡ (z  ∷ʳ τ₁) (z  ∷ʳ τ₂) = z   ∷ₙ-Sep separateˡ τ₁ τ₂

+
separateˡ (z  ∷ʳ τ₁) (y≡z ∷ τ₂) = y≡z ∷ᵣ-Sep separateˡ τ₁ τ₂

+
separateˡ (x≡z ∷ τ₁) (z  ∷ʳ τ₂) = x≡z ∷ₗ-Sep separateˡ τ₁ τ₂

+
separateˡ (x≡z ∷ τ₁) (y≡z ∷ τ₂) = ∷-Sepˡ x≡z y≡z (separateˡ τ₁ τ₂)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Sublist.Setoid.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Setoid.Properties.html
new file mode 100644
index 000000000..9974fea45
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Setoid.Properties.html
@@ -0,0 +1,428 @@
+
+Data.List.Relation.Binary.Sublist.Setoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the setoid sublist relation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel; _⇒_; _Preserves_⟶_)

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.List.Relation.Binary.Sublist.Setoid.Properties

+
  {c ℓ} (S : Setoid c ℓ) where

+

+
open import Data.List.Base hiding (_∷ʳ_)

+
open import Data.List.Relation.Unary.Any using (Any)

+
import Data.Maybe.Relation.Unary.All as Maybe

+
open import Data.Nat.Base using (ℕ; _≤_; _≥_)

+
import Data.Nat.Properties as ℕ

+
open import Data.Product.Base using (∃; _,_; proj₂)

+
open import Function.Base

+
open import Function.Bundles using (_⇔_; _⤖_)

+
open import Level

+
open import Relation.Binary.Definitions using () renaming (Decidable to Decidable₂)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl; cong; cong₂)

+
open import Relation.Binary.Structures using (IsDecTotalOrder)

+
open import Relation.Unary using (Pred; Decidable; Irrelevant)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Nullary.Decidable using (¬?; yes; no)

+

+
import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality

+
import Data.List.Relation.Binary.Sublist.Setoid as SetoidSublist

+
import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties

+
  as HeteroProperties

+
import Data.List.Membership.Setoid as SetoidMembership

+

+
open Setoid S using (_≈_; trans) renaming (Carrier to A; refl to ≈-refl)

+
open SetoidEquality S using (_≋_; ≋-refl)

+
open SetoidSublist S hiding (map)

+
open SetoidMembership S using (_∈_)

+

+
private

+
  variable

+
    p q r s t : Level

+
    a b x y : A

+
    as bs cs ds xs ys : List A

+
    P : Pred A p

+
    Q : Pred A q

+
    m n : ℕ

+

+

+
------------------------------------------------------------------------

+
-- Injectivity of constructors

+
------------------------------------------------------------------------

+

+
module _ where

+

+
  ∷-injectiveˡ : ∀ {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →

+
                 ((x ∷ xs) ⊆ (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → px ≡ qx

+
  ∷-injectiveˡ refl = refl

+

+
  ∷-injectiveʳ : ∀ {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →

+
                 ((x ∷ xs) ⊆ (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → pxs ≡ qxs

+
  ∷-injectiveʳ refl = refl

+

+
  ∷ʳ-injective : ∀ {pxs qxs : xs ⊆ ys} → y ∷ʳ pxs ≡ y ∷ʳ qxs → pxs ≡ qxs

+
  ∷ʳ-injective refl = refl

+

+
------------------------------------------------------------------------

+
-- Categorical properties

+
------------------------------------------------------------------------

+

+
module _ (trans-reflˡ : ∀ {x y} (p : x ≈ y) → trans ≈-refl p ≡ p) where

+

+
  ⊆-trans-idˡ : (pxs : xs ⊆ ys) → ⊆-trans ⊆-refl pxs ≡ pxs

+
  ⊆-trans-idˡ [] = refl

+
  ⊆-trans-idˡ (y ∷ʳ pxs) = cong (y ∷ʳ_) (⊆-trans-idˡ pxs)

+
  ⊆-trans-idˡ (x ∷ pxs) = cong₂ _∷_ (trans-reflˡ x) (⊆-trans-idˡ pxs)

+

+
module _ (trans-reflʳ : ∀ {x y} (p : x ≈ y) → trans p ≈-refl ≡ p) where

+

+
  ⊆-trans-idʳ : (pxs : xs ⊆ ys) → ⊆-trans pxs ⊆-refl ≡ pxs

+
  ⊆-trans-idʳ [] = refl

+
  ⊆-trans-idʳ (y ∷ʳ pxs) = cong (y ∷ʳ_) (⊆-trans-idʳ pxs)

+
  ⊆-trans-idʳ (x ∷ pxs) = cong₂ _∷_ (trans-reflʳ x) (⊆-trans-idʳ pxs)

+

+
module _ (≈-assoc : ∀ {w x y z} (p : w ≈ x) (q : x ≈ y) (r : y ≈ z) →

+
                    trans p (trans q r) ≡ trans (trans p q) r) where

+

+
  ⊆-trans-assoc : (ps : as ⊆ bs) (qs : bs ⊆ cs) (rs : cs ⊆ ds) →

+
            ⊆-trans ps (⊆-trans qs rs) ≡ ⊆-trans (⊆-trans ps qs) rs

+
  ⊆-trans-assoc ps qs (_ ∷ʳ rs) = cong (_ ∷ʳ_) (⊆-trans-assoc ps qs rs)

+
  ⊆-trans-assoc ps (_ ∷ʳ qs) (_ ∷ rs) = cong (_ ∷ʳ_) (⊆-trans-assoc ps qs rs)

+
  ⊆-trans-assoc (_ ∷ʳ ps) (_ ∷ qs) (_ ∷ rs) = cong (_ ∷ʳ_) (⊆-trans-assoc ps qs rs)

+
  ⊆-trans-assoc (p ∷ ps) (q ∷ qs) (r ∷ rs) = cong₂ _∷_ (≈-assoc p q r) (⊆-trans-assoc ps qs rs)

+
  ⊆-trans-assoc [] [] [] = refl

+

+

+
------------------------------------------------------------------------

+
-- Various functions' outputs are sublists

+
------------------------------------------------------------------------

+

+
tail-⊆ : ∀ xs → Maybe.All (_⊆ xs) (tail xs)

+
tail-⊆ xs = HeteroProperties.tail-Sublist ⊆-refl

+

+
take-⊆ : ∀ n xs → take n xs ⊆ xs

+
take-⊆ n xs = HeteroProperties.take-Sublist n ⊆-refl

+

+
drop-⊆ : ∀ n xs → drop n xs ⊆ xs

+
drop-⊆ n xs = HeteroProperties.drop-Sublist n ⊆-refl

+

+
module _ (P? : Decidable P) where

+

+
  takeWhile-⊆ : ∀ xs → takeWhile P? xs ⊆ xs

+
  takeWhile-⊆ xs = HeteroProperties.takeWhile-Sublist P? ⊆-refl

+

+
  dropWhile-⊆ : ∀ xs → dropWhile P? xs ⊆ xs

+
  dropWhile-⊆ xs = HeteroProperties.dropWhile-Sublist P? ⊆-refl

+

+
  filter-⊆ : ∀ xs → filter P? xs ⊆ xs

+
  filter-⊆ xs = HeteroProperties.filter-Sublist P? ⊆-refl

+

+
module _ (P? : Decidable P) where

+

+
  takeWhile⊆filter : ∀ xs → takeWhile P? xs ⊆ filter P? xs

+
  takeWhile⊆filter xs = HeteroProperties.takeWhile-filter P? {xs} ≋-refl

+

+
  filter⊆dropWhile : ∀ xs → filter P? xs ⊆ dropWhile (¬? ∘ P?) xs

+
  filter⊆dropWhile xs = HeteroProperties.filter-dropWhile P? {xs} ≋-refl

+

+
------------------------------------------------------------------------

+
-- Various list functions are increasing wrt _⊆_

+
------------------------------------------------------------------------

+
-- We write f⁺ for the proof that `xs ⊆ ys → f xs ⊆ f ys`

+
-- and f⁻ for the one that `f xs ⊆ f ys → xs ⊆ ys`.

+

+
module _ where

+

+
  ∷ˡ⁻ : a ∷ as ⊆ bs → as ⊆ bs

+
  ∷ˡ⁻ = HeteroProperties.∷ˡ⁻

+

+
  ∷ʳ⁻ : ¬ (a ≈ b) → a ∷ as ⊆ b ∷ bs → a ∷ as ⊆ bs

+
  ∷ʳ⁻ = HeteroProperties.∷ʳ⁻

+

+
  ∷⁻ : a ∷ as ⊆ b ∷ bs → as ⊆ bs

+
  ∷⁻ = HeteroProperties.∷⁻

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {b ℓ} (R : Setoid b ℓ) where

+

+
  open Setoid R using () renaming (Carrier to B; _≈_ to _≈′_)

+
  open SetoidSublist R using () renaming (_⊆_ to _⊆′_)

+

+
  map⁺ : ∀ {as bs} {f : A → B} → f Preserves _≈_ ⟶ _≈′_ →

+
         as ⊆ bs → map f as ⊆′ map f bs

+
  map⁺ {f = f} f-resp as⊆bs =

+
    HeteroProperties.map⁺ f f (SetoidSublist.map S f-resp as⊆bs)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ where

+

+
  ++⁺ˡ : ∀ cs → as ⊆ bs → as ⊆ cs ++ bs

+
  ++⁺ˡ = HeteroProperties.++ˡ

+

+
  ++⁺ʳ : ∀ cs → as ⊆ bs → as ⊆ bs ++ cs

+
  ++⁺ʳ = HeteroProperties.++ʳ

+

+
  ++⁺ : as ⊆ bs → cs ⊆ ds → as ++ cs ⊆ bs ++ ds

+
  ++⁺ = HeteroProperties.++⁺

+

+
  ++⁻ : length as ≡ length bs → as ++ cs ⊆ bs ++ ds → cs ⊆ ds

+
  ++⁻ = HeteroProperties.++⁻

+

+
------------------------------------------------------------------------

+
-- take

+

+
module _ where

+

+
  take⁺ : m ≤ n → take m xs ⊆ take n xs

+
  take⁺ m≤n = HeteroProperties.take⁺ m≤n ≋-refl

+

+
------------------------------------------------------------------------

+
-- drop

+

+
module _ where

+

+
  drop⁺ : m ≥ n → xs ⊆ ys → drop m xs ⊆ drop n ys

+
  drop⁺ = HeteroProperties.drop⁺

+

+
module _ where

+

+
  drop⁺-≥ : m ≥ n → drop m xs ⊆ drop n xs

+
  drop⁺-≥ m≥n = drop⁺ m≥n ⊆-refl

+

+
module _ where

+

+
  drop⁺-⊆ : ∀ n → xs ⊆ ys → drop n xs ⊆ drop n ys

+
  drop⁺-⊆ n xs⊆ys = drop⁺ {n} ℕ.≤-refl xs⊆ys

+

+
------------------------------------------------------------------------

+
-- takeWhile / dropWhile

+

+
module _ (P? : Decidable P) (Q? : Decidable Q) where

+

+
  takeWhile⁺ : ∀ {xs} → (∀ {a b} → a ≈ b → P a → Q b) →

+
               takeWhile P? xs ⊆ takeWhile Q? xs

+
  takeWhile⁺ {xs} P⇒Q = HeteroProperties.⊆-takeWhile-Sublist P? Q? {xs} P⇒Q ≋-refl

+

+
  dropWhile⁺ : ∀ {xs} → (∀ {a b} → a ≈ b → Q b → P a) →

+
               dropWhile P? xs ⊆ dropWhile Q? xs

+
  dropWhile⁺ {xs} P⇒Q = HeteroProperties.⊇-dropWhile-Sublist P? Q? {xs} P⇒Q ≋-refl

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (P? : Decidable P) (Q? : Decidable Q) where

+

+
  filter⁺ : (∀ {a b} → a ≈ b → P a → Q b) →

+
            as ⊆ bs → filter P? as ⊆ filter Q? bs

+
  filter⁺ = HeteroProperties.⊆-filter-Sublist P? Q?

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
module _ where

+

+
  reverseAcc⁺ : as ⊆ bs → cs ⊆ ds →

+
                reverseAcc cs as ⊆ reverseAcc ds bs

+
  reverseAcc⁺ = HeteroProperties.reverseAcc⁺

+

+
  ʳ++⁺ : as ⊆ bs → cs ⊆ ds →

+
         as ʳ++ cs ⊆ bs ʳ++ ds

+
  ʳ++⁺ = reverseAcc⁺

+

+
  reverse⁺ : as ⊆ bs → reverse as ⊆ reverse bs

+
  reverse⁺ = HeteroProperties.reverse⁺

+

+
  reverse⁻ : reverse as ⊆ reverse bs → as ⊆ bs

+
  reverse⁻ = HeteroProperties.reverse⁻

+

+
------------------------------------------------------------------------

+
-- merge

+

+
module _ {ℓ′} {_≤_ : Rel A ℓ′} (_≤?_ : Decidable₂ _≤_) where

+

+
  ⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys

+
  ⊆-mergeˡ []       ys = minimum ys

+
  ⊆-mergeˡ (x ∷ xs) [] = ⊆-refl

+
  ⊆-mergeˡ (x ∷ xs) (y ∷ ys)

+
   with x ≤? y  | ⊆-mergeˡ xs (y ∷ ys)

+
                      | ⊆-mergeˡ (x ∷ xs) ys

+
  ... | yes x≤y | rec | _   = ≈-refl ∷ rec

+
  ... | no  x≰y | _   | rec = y ∷ʳ rec

+

+
  ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys

+
  ⊆-mergeʳ [] ys =  ⊆-refl

+
  ⊆-mergeʳ (x ∷ xs) [] = minimum (merge _≤?_ (x ∷ xs) [])

+
  ⊆-mergeʳ (x ∷ xs) (y ∷ ys)

+
   with x ≤? y  | ⊆-mergeʳ xs (y ∷ ys)

+
                      | ⊆-mergeʳ (x ∷ xs) ys

+
  ... | yes x≤y | rec | _   = x ∷ʳ rec

+
  ... | no  x≰y | _   | rec = ≈-refl ∷ rec

+

+
------------------------------------------------------------------------

+
-- Inversion lemmas

+
------------------------------------------------------------------------

+

+
module _ where

+

+
  ∷⁻¹ : a ≈ b → as ⊆ bs ⇔ a ∷ as ⊆ b ∷ bs

+
  ∷⁻¹ = HeteroProperties.∷⁻¹

+

+
  ∷ʳ⁻¹ : ¬ (a ≈ b) → a ∷ as ⊆ bs ⇔ a ∷ as ⊆ b ∷ bs

+
  ∷ʳ⁻¹ = HeteroProperties.∷ʳ⁻¹

+

+
------------------------------------------------------------------------

+
-- Other

+
------------------------------------------------------------------------

+

+
module _ where

+

+
  length-mono-≤ : as ⊆ bs → length as ≤ length bs

+
  length-mono-≤ = HeteroProperties.length-mono-≤

+

+
------------------------------------------------------------------------

+
-- Conversion to and from list equality

+

+
  to-≋ : length as ≡ length bs → as ⊆ bs → as ≋ bs

+
  to-≋ = HeteroProperties.toPointwise

+

+
------------------------------------------------------------------------

+
-- Irrelevant special case

+

+
  []⊆-irrelevant : Irrelevant ([] ⊆_)

+
  []⊆-irrelevant = HeteroProperties.Sublist-[]-irrelevant

+

+
------------------------------------------------------------------------

+
-- (to/from)∈ is a bijection

+

+
module _ where

+

+
  to∈-injective : ∀ {p q : [ x ] ⊆ xs} → to∈ p ≡ to∈ q → p ≡ q

+
  to∈-injective = HeteroProperties.toAny-injective

+

+
  from∈-injective : ∀ {p q : x ∈ xs} → from∈ p ≡ from∈ q → p ≡ q

+
  from∈-injective = HeteroProperties.fromAny-injective

+

+
  to∈∘from∈≗id : ∀ (p : x ∈ xs) → to∈ (from∈ p) ≡ p

+
  to∈∘from∈≗id = HeteroProperties.toAny∘fromAny≗id

+

+
  [x]⊆xs⤖x∈xs : ([ x ] ⊆ xs) ⤖ (x ∈ xs)

+
  [x]⊆xs⤖x∈xs = HeteroProperties.Sublist-[x]-bijection

+

+
------------------------------------------------------------------------

+
-- Properties of Disjoint(ness) and DisjointUnion

+

+
open HeteroProperties.Disjointness {R = _≈_} public

+
open HeteroProperties.DisjointnessMonotonicity {R = _≈_} {S = _≈_} {T = _≈_} trans public

+

+
-- Shrinking one of two disjoint lists preserves disjointness.

+

+
-- We would have liked to define  shrinkDisjointˡ σ = shrinkDisjoint σ ⊆-refl

+
-- but alas, this is only possible for groupoids, not setoids in general.

+

+
shrinkDisjointˡ : ∀ {xs ys zs us} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} (σ : us ⊆ xs) →

+
    Disjoint τ₁ τ₂ →

+
    Disjoint (⊆-trans σ τ₁) τ₂

+
-- Not affected by σ:

+
shrinkDisjointˡ σ          (y   ∷ₙ d) = y             ∷ₙ shrinkDisjointˡ σ d

+
shrinkDisjointˡ σ          (y≈z ∷ᵣ d) = y≈z           ∷ᵣ shrinkDisjointˡ σ d

+
-- In σ: keep x.

+
shrinkDisjointˡ (u≈x ∷ σ)  (x≈z ∷ₗ d) = trans u≈x x≈z ∷ₗ shrinkDisjointˡ σ d

+
-- Not in σ: drop x.

+
shrinkDisjointˡ (x  ∷ʳ σ)  (x≈z ∷ₗ d) = _             ∷ₙ shrinkDisjointˡ σ d

+
shrinkDisjointˡ []         []         = []

+

+
shrinkDisjointʳ : ∀ {xs ys zs vs} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} (σ : vs ⊆ ys) →

+
  Disjoint τ₁ τ₂ →

+
  Disjoint τ₁ (⊆-trans σ τ₂)

+
-- Not affected by σ:

+
shrinkDisjointʳ σ          (y   ∷ₙ d) = y             ∷ₙ shrinkDisjointʳ σ d

+
shrinkDisjointʳ σ          (x≈z ∷ₗ d) = x≈z           ∷ₗ shrinkDisjointʳ σ d

+
-- In σ: keep y.

+
shrinkDisjointʳ (v≈y ∷ σ)  (y≈z ∷ᵣ d) = trans v≈y y≈z ∷ᵣ shrinkDisjointʳ σ d

+
-- Not in σ: drop y.

+
shrinkDisjointʳ (y  ∷ʳ σ)  (y≈z ∷ᵣ d) = _             ∷ₙ shrinkDisjointʳ σ d

+
shrinkDisjointʳ []         []         = []

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Sublist.Setoid.html b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Setoid.html
new file mode 100644
index 000000000..371fd9606
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Sublist.Setoid.html
@@ -0,0 +1,368 @@
+
+Data.List.Relation.Binary.Sublist.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An inductive definition of the sublist relation with respect to a

+
-- setoid. This is a generalisation of what is commonly known as Order

+
-- Preserving Embeddings (OPE).

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+
{-# OPTIONS --postfix-projections #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.List.Relation.Binary.Sublist.Setoid

+
  {c ℓ} (S : Setoid c ℓ) where

+

+
open import Level using (_⊔_)

+

+
open import Data.List.Base using (List; []; _∷_)

+
import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality

+
import Data.List.Relation.Binary.Sublist.Heterogeneous as Heterogeneous

+
import Data.List.Relation.Binary.Sublist.Heterogeneous.Core

+
  as HeterogeneousCore

+
import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties

+
  as HeterogeneousProperties

+
open import Data.Product.Base using (∃; ∃₂; _×_; _,_; proj₂)

+

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles using (Preorder; Poset)

+
open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Nullary using (¬_; Dec; yes; no)

+

+
open Setoid S renaming (Carrier to A)

+
open SetoidEquality S

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infix 4 _⊆_ _⊇_ _⊂_ _⊃_ _⊈_ _⊉_ _⊄_ _⊅_

+

+
_⊆_ : Rel (List A) (c ⊔ ℓ)

+
_⊆_ = Heterogeneous.Sublist _≈_

+

+
_⊇_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊇ ys = ys ⊆ xs

+

+
_⊂_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊂ ys = xs ⊆ ys × ¬ (xs ≋ ys)

+

+
_⊃_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊃ ys = ys ⊂ xs

+

+
_⊈_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊈ ys = ¬ (xs ⊆ ys)

+

+
_⊉_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊉ ys = ¬ (xs ⊇ ys)

+

+
_⊄_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊄ ys = ¬ (xs ⊂ ys)

+

+
_⊅_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊅ ys = ¬ (xs ⊃ ys)

+

+
------------------------------------------------------------------------

+
-- Re-export definitions and operations from heterogeneous sublists

+

+
open HeterogeneousCore _≈_ public

+
  using ([]; _∷_; _∷ʳ_)

+

+
open Heterogeneous {R = _≈_} public

+
  hiding (Sublist; []; _∷_; _∷ʳ_)

+
  renaming

+
  ( toAny   to to∈

+
  ; fromAny to from∈

+
  )

+

+
open Disjoint public

+
  using ([])

+

+
open DisjointUnion public

+
  using ([])

+

+
------------------------------------------------------------------------

+
-- Relational properties holding for Setoid case

+

+
⊆-reflexive : _≋_ ⇒ _⊆_

+
⊆-reflexive = HeterogeneousProperties.fromPointwise

+

+
open HeterogeneousProperties.Reflexivity {R = _≈_} refl public using ()

+
  renaming (refl to ⊆-refl)  -- ⊆-refl : Reflexive _⊆_

+

+
open HeterogeneousProperties.Transitivity {R = _≈_} {S = _≈_} {T = _≈_} trans public using ()

+
  renaming (trans to ⊆-trans)  -- ⊆-trans : Transitive _⊆_

+

+
open HeterogeneousProperties.Antisymmetry {R = _≈_} {S = _≈_} (λ x≈y _ → x≈y) public using ()

+
  renaming (antisym to ⊆-antisym)  -- ⊆-antisym : Antisymmetric _≋_ _⊆_

+

+
⊆-isPreorder : IsPreorder _≋_ _⊆_

+
⊆-isPreorder = record

+
  { isEquivalence = ≋-isEquivalence

+
  ; reflexive     = ⊆-reflexive

+
  ; trans         = ⊆-trans

+
  }

+

+
⊆-isPartialOrder : IsPartialOrder _≋_ _⊆_

+
⊆-isPartialOrder = record

+
  { isPreorder = ⊆-isPreorder

+
  ; antisym    = ⊆-antisym

+
  }

+

+
⊆-preorder : Preorder c (c ⊔ ℓ) (c ⊔ ℓ)

+
⊆-preorder = record

+
  { isPreorder = ⊆-isPreorder

+
  }

+

+
⊆-poset : Poset c (c ⊔ ℓ) (c ⊔ ℓ)

+
⊆-poset = record

+
  { isPartialOrder = ⊆-isPartialOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Raw pushout

+
--

+
-- The category _⊆_ does not have proper pushouts.  For instance consider:

+
--

+
--   τᵤ : [] ⊆ (u ∷ [])

+
--   τᵥ : [] ⊆ (v ∷ [])

+
--

+
-- Then, there are two unrelated upper bounds (u ∷ v ∷ []) and (v ∷ u ∷ []),

+
-- since _⊆_ does not include permutations.

+
--

+
-- Even though there are no unique least upper bounds, we can merge two

+
-- extensions of a list, producing a minimial superlist of both.

+
--

+
-- For the example, the left-biased merge would produce the pair:

+
--

+
--   τᵤ′ : (u ∷ []) ⊆ (u ∷ v ∷ [])

+
--   τᵥ′ : (v ∷ []) ⊆ (u ∷ v ∷ [])

+
--

+
-- We call such a pair a raw pushout.  It is then a weak pushout if the

+
-- resulting square commutes, i.e.:

+
--

+
--   ⊆-trans τᵤ τᵤ′ ~ ⊆-trans τᵥ τᵥ′

+
--

+
-- This requires a notion of equality _~_ on sublist morphisms.

+
--

+
-- Further, commutation requires a similar commutation property

+
-- for the underlying equality _≈_, namely

+
--

+
--   trans x≈y (sym x≈y) == trans x≈z (sym x≈z)

+
--

+
-- for some notion of equality _==_ for equality proofs _≈_.

+
-- Such a property is given e.g. if _≈_ is proof irrelevant

+
-- or forms a groupoid.

+

+
record RawPushout {xs ys zs : List A} (τ : xs ⊆ ys) (σ : xs ⊆ zs) : Set (c ⊔ ℓ) where

+
  field

+
    {upperBound} : List A

+
    leg₁         : ys ⊆ upperBound

+
    leg₂         : zs ⊆ upperBound

+

+
open RawPushout using (leg₁; leg₂)

+

+
------------------------------------------------------------------------

+
-- Extending corners of a raw pushout square

+

+
-- Extending the right upper corner.

+

+
infixr 5 _∷ʳ₁_ _∷ʳ₂_

+

+
_∷ʳ₁_ : ∀ {xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →

+
        (y : A) → RawPushout τ σ → RawPushout (y ∷ʳ τ) σ

+
y ∷ʳ₁ rpo = record

+
  { leg₁ = refl ∷ leg₁ rpo

+
  ; leg₂ = y   ∷ʳ leg₂ rpo

+
  }

+

+
-- Extending the left lower corner.

+

+
_∷ʳ₂_ : ∀ {xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →

+
        (z : A) → RawPushout τ σ → RawPushout τ (z ∷ʳ σ)

+
z ∷ʳ₂ rpo = record

+
  { leg₁ = z   ∷ʳ leg₁ rpo

+
  ; leg₂ = refl ∷ leg₂ rpo

+
  }

+

+
-- Extending both of these corners with equal elements.

+

+
∷-rpo : ∀ {x y z : A} {xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →

+
        (x≈y : x ≈ y) (x≈z : x ≈ z) → RawPushout τ σ → RawPushout (x≈y ∷ τ) (x≈z ∷ σ)

+
∷-rpo x≈y x≈z rpo = record

+
  { leg₁ = sym x≈y ∷ leg₁ rpo

+
  ; leg₂ = sym x≈z ∷ leg₂ rpo

+
  }

+

+
------------------------------------------------------------------------

+
-- Left-biased pushout: add elements of left extension first.

+

+
⊆-pushoutˡ : ∀ {xs ys zs : List A} →

+
             (τ : xs ⊆ ys) (σ : xs ⊆ zs) → RawPushout τ σ

+
⊆-pushoutˡ []        σ         = record { leg₁ = σ ; leg₂ = ⊆-refl }

+
⊆-pushoutˡ (y  ∷ʳ τ) σ         = y ∷ʳ₁ ⊆-pushoutˡ τ σ

+
⊆-pushoutˡ τ@(_ ∷ _) (z  ∷ʳ σ) = z ∷ʳ₂ ⊆-pushoutˡ τ σ

+
⊆-pushoutˡ (x≈y ∷ τ) (x≈z ∷ σ) = ∷-rpo x≈y x≈z (⊆-pushoutˡ τ σ)

+

+
-- Join two extensions, returning the upper bound and the diagonal

+
-- of the pushout square.

+

+
⊆-joinˡ : ∀ {xs ys zs : List A} →

+
          (τ : xs ⊆ ys) (σ : xs ⊆ zs) → ∃ λ us → xs ⊆ us

+
⊆-joinˡ τ σ = RawPushout.upperBound rpo , ⊆-trans τ (leg₁ rpo)

+
  where

+
  rpo = ⊆-pushoutˡ τ σ

+

+

+
------------------------------------------------------------------------

+
-- Upper bound of two sublists xs,ys ⊆ zs

+
--

+
-- Two sublists τ : xs ⊆ zs and σ : ys ⊆ zs

+
-- can be joined in a unique way if τ and σ are respected.

+
--

+
-- For instance, if τ : [x] ⊆ [x,y,x] and σ : [y] ⊆ [x,y,x]

+
-- then the union will be [x,y] or [y,x], depending on whether

+
-- τ picks the first x or the second one.

+
--

+
-- NB: If the content of τ and σ were ignored then the union would not

+
-- be unique.  Expressing uniqueness would require a notion of equality

+
-- of sublist proofs, which we do not (yet) have for the setoid case

+
-- (however, for the propositional case).

+

+
record UpperBound {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) : Set (c ⊔ ℓ) where

+
  field

+
    {theUpperBound} : List A

+
    sub  : theUpperBound ⊆ zs

+
    inj₁ : xs ⊆ theUpperBound

+
    inj₂ : ys ⊆ theUpperBound

+

+
open UpperBound

+

+
infixr 5 _∷ₗ-ub_ _∷ᵣ-ub_

+

+
∷ₙ-ub : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x} →

+
        UpperBound τ σ → UpperBound (x ∷ʳ τ) (x ∷ʳ σ)

+
∷ₙ-ub u = record

+
  { sub  = _ ∷ʳ u .sub

+
  ; inj₁ = u .inj₁

+
  ; inj₂ = u .inj₂

+
  }

+

+
_∷ₗ-ub_ : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x y} →

+
         (x≈y : x ≈ y) → UpperBound τ σ → UpperBound (x≈y ∷ τ) (y ∷ʳ σ)

+
x≈y ∷ₗ-ub u = record

+
  { sub = refl ∷ u .sub

+
  ; inj₁ = x≈y ∷ u .inj₁

+
  ; inj₂ = _  ∷ʳ u .inj₂

+
  }

+

+
_∷ᵣ-ub_ : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x y} →

+
         (x≈y : x ≈ y) → UpperBound τ σ → UpperBound (y ∷ʳ τ) (x≈y ∷ σ)

+
x≈y ∷ᵣ-ub u = record

+
  { sub  = refl ∷ u .sub

+
  ; inj₁ = _   ∷ʳ u .inj₁

+
  ; inj₂ = x≈y  ∷ u .inj₂

+
  }

+

+
_,_∷-ub_ : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x y z} →

+
         (x≈z : x ≈ z) (y≈z : y ≈ z) → UpperBound τ σ → UpperBound (x≈z ∷ τ) (y≈z ∷ σ)

+
x≈z , y≈z ∷-ub u = record

+
  { sub  =  refl ∷ u .sub

+
  ; inj₁ =  x≈z  ∷ u .inj₁

+
  ; inj₂ =  y≈z  ∷ u .inj₂

+
  }

+

+
⊆-upper-bound : ∀ {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) → UpperBound τ σ

+
⊆-upper-bound []        []        = record { sub = [] ; inj₁ = [] ; inj₂ = [] }

+
⊆-upper-bound (y ∷ʳ τ)  (.y ∷ʳ σ) = ∷ₙ-ub (⊆-upper-bound τ σ)

+
⊆-upper-bound (y ∷ʳ τ)  (x≈y ∷ σ) = x≈y ∷ᵣ-ub ⊆-upper-bound τ σ

+
⊆-upper-bound (x≈y ∷ τ) (y  ∷ʳ σ) = x≈y ∷ₗ-ub ⊆-upper-bound τ σ

+
⊆-upper-bound (x≈z ∷ τ) (y≈z ∷ σ) = x≈z , y≈z ∷-ub ⊆-upper-bound τ σ

+

+
------------------------------------------------------------------------

+
-- Disjoint union

+
--

+
-- Upper bound of two non-overlapping sublists.

+

+
⊆-disjoint-union : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} →

+
                   Disjoint τ σ → UpperBound τ σ

+
⊆-disjoint-union {τ = τ} {σ = σ} _ = ⊆-upper-bound τ σ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Subset.Propositional.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Subset.Propositional.Properties.html
new file mode 100644
index 000000000..1adabe2a9
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Subset.Propositional.Properties.html
@@ -0,0 +1,404 @@
+
+Data.List.Relation.Binary.Subset.Propositional.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the sublist relation over setoid equality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Subset.Propositional.Properties

+
  where

+

+
open import Data.Bool.Base using (Bool; true; false; T)

+
open import Data.List.Base using (List; map; _∷_; _++_; concat; applyUpTo;

+
  any; filter)

+
open import Data.List.Relation.Unary.Any using (Any; here; there)

+
open import Data.List.Relation.Unary.All using (All)

+
import Data.List.Relation.Unary.Any.Properties as Any hiding (filter⁺)

+
open import Data.List.Effectful using (monad)

+
open import Data.List.Relation.Unary.Any using (Any)

+
open import Data.List.Membership.Propositional using (_∈_; mapWith∈)

+
open import Data.List.Membership.Propositional.Properties

+
  using (map-∈↔; concat-∈↔; >>=-∈↔; ⊛-∈↔; ⊗-∈↔)

+
import Data.List.Relation.Binary.Subset.Setoid.Properties as Subset

+
open import Data.List.Relation.Binary.Subset.Propositional

+
  using (_⊆_; _⊇_)

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
  using (_↭_; ↭-sym; ↭-isEquivalence)

+
import Data.List.Relation.Binary.Permutation.Propositional.Properties as Permutation

+
open import Data.Nat.Base using (ℕ; _≤_)

+
import Data.Product.Base as Product

+
import Data.Sum.Base as Sum

+
open import Effect.Monad

+
open import Function.Base using (_∘_; _∘′_; id; _$_)

+
open import Function.Bundles using (_↔_; Inverse; Equivalence)

+
open import Level using (Level)

+
open import Relation.Nullary using (¬_; yes; no)

+
open import Relation.Unary using (Decidable; Pred) renaming (_⊆_ to _⋐_)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles using (Preorder)

+
open import Relation.Binary.Definitions hiding (Decidable)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≗_; subst; resp; refl)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (isEquivalence; setoid; module ≡-Reasoning)

+
open import Relation.Binary.Structures using (IsPreorder)

+
import Relation.Binary.Reasoning.Preorder as ≲-Reasoning

+

+
private

+
  open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})

+

+
  variable

+
    a b p q : Level

+
    A : Set a

+
    B : Set b

+
    ws xs ys zs : List A

+

+
------------------------------------------------------------------------

+
-- Relational properties with _≋_ (pointwise equality)

+
------------------------------------------------------------------------

+

+
⊆-reflexive : _≡_ {A = List A} ⇒ _⊆_

+
⊆-reflexive refl = id

+

+
⊆-refl : Reflexive {A = List A} _⊆_

+
⊆-refl x∈xs = x∈xs

+

+
⊆-trans : Transitive {A = List A} _⊆_

+
⊆-trans xs⊆ys ys⊆zs = ys⊆zs ∘ xs⊆ys

+

+
module _ (A : Set a) where

+

+
  ⊆-isPreorder : IsPreorder {A = List A} _≡_ _⊆_

+
  ⊆-isPreorder = record

+
    { isEquivalence = isEquivalence

+
    ; reflexive     = ⊆-reflexive

+
    ; trans         = ⊆-trans

+
    }

+

+
  ⊆-preorder : Preorder _ _ _

+
  ⊆-preorder = record

+
    { isPreorder = ⊆-isPreorder

+
    }

+

+
------------------------------------------------------------------------

+
-- Relational properties with _↭_ (permutation)

+
------------------------------------------------------------------------

+
-- See issue #1354 for why these proofs can't be taken from `Subset`

+

+
⊆-reflexive-↭ : _↭_ {A = A} ⇒ _⊆_

+
⊆-reflexive-↭ xs↭ys = Permutation.∈-resp-↭ xs↭ys

+

+
⊆-respʳ-↭ : _⊆_ {A = A} Respectsʳ _↭_

+
⊆-respʳ-↭ xs↭ys = Permutation.∈-resp-↭ xs↭ys ∘_

+

+
⊆-respˡ-↭ : _⊆_ {A = A} Respectsˡ _↭_

+
⊆-respˡ-↭ xs↭ys = _∘ Permutation.∈-resp-↭ (↭-sym xs↭ys)

+

+
module _ (A : Set a) where

+

+
  ⊆-↭-isPreorder : IsPreorder {A = List A} _↭_ _⊆_

+
  ⊆-↭-isPreorder = record

+
    { isEquivalence = ↭-isEquivalence

+
    ; reflexive     = ⊆-reflexive-↭

+
    ; trans         = ⊆-trans

+
    }

+

+
  ⊆-↭-preorder : Preorder _ _ _

+
  ⊆-↭-preorder = record

+
    { isPreorder = ⊆-↭-isPreorder

+
    }

+

+
------------------------------------------------------------------------

+
-- Reasoning over subsets

+
------------------------------------------------------------------------

+

+
module ⊆-Reasoning (A : Set a) where

+
  open Subset.⊆-Reasoning (setoid A) public

+
    hiding (step-≋; step-≋˘)

+

+
------------------------------------------------------------------------

+
-- Properties of _⊆_ and various list predicates

+
------------------------------------------------------------------------

+

+
Any-resp-⊆ : ∀ {P : Pred A p} → (Any P) Respects _⊆_

+
Any-resp-⊆ = Subset.Any-resp-⊆ (setoid _) (subst _)

+

+
All-resp-⊇ : ∀ {P : Pred A p} → (All P) Respects _⊇_

+
All-resp-⊇ = Subset.All-resp-⊇ (setoid _) (subst _)

+

+
------------------------------------------------------------------------

+
-- Properties relating _⊆_ to various list functions

+
------------------------------------------------------------------------

+
-- map

+

+
map⁺ : ∀ (f : A → B) → xs ⊆ ys → map f xs ⊆ map f ys

+
map⁺ f xs⊆ys =

+
  Inverse.to (map-∈↔ f) ∘

+
  Product.map₂ (Product.map₁ xs⊆ys) ∘

+
  Inverse.from (map-∈↔ f)

+

+
------------------------------------------------------------------------

+
-- ∷

+

+
xs⊆x∷xs : ∀ (xs : List A) x → xs ⊆ x ∷ xs

+
xs⊆x∷xs = Subset.xs⊆x∷xs (setoid _)

+

+
∷⁺ʳ : ∀ x → xs ⊆ ys → x ∷ xs ⊆ x ∷ ys

+
∷⁺ʳ = Subset.∷⁺ʳ (setoid _)

+

+
∈-∷⁺ʳ : ∀ {x} → x ∈ ys → xs ⊆ ys → x ∷ xs ⊆ ys

+
∈-∷⁺ʳ = Subset.∈-∷⁺ʳ (setoid _)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
xs⊆xs++ys : ∀ (xs ys : List A) → xs ⊆ xs ++ ys

+
xs⊆xs++ys = Subset.xs⊆xs++ys (setoid _)

+

+
xs⊆ys++xs : ∀ (xs ys : List A) → xs ⊆ ys ++ xs

+
xs⊆ys++xs = Subset.xs⊆ys++xs (setoid _)

+

+
++⁺ʳ : ∀ zs → xs ⊆ ys → zs ++ xs ⊆ zs ++ ys

+
++⁺ʳ = Subset.++⁺ʳ (setoid _)

+

+
++⁺ˡ : ∀ zs → xs ⊆ ys → xs ++ zs ⊆ ys ++ zs

+
++⁺ˡ = Subset.++⁺ˡ (setoid _)

+

+
++⁺ : ws ⊆ xs → ys ⊆ zs → ws ++ ys ⊆ xs ++ zs

+
++⁺ = Subset.++⁺ (setoid _)

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {xss yss : List (List A)} where

+

+
  concat⁺ : xss ⊆ yss → concat xss ⊆ concat yss

+
  concat⁺ xss⊆yss =

+
    Inverse.to concat-∈↔ ∘

+
    Product.map₂ (Product.map₂ xss⊆yss) ∘

+
    Inverse.from concat-∈↔

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
applyUpTo⁺ : ∀ (f : ℕ → A) {m n} → m ≤ n → applyUpTo f m ⊆ applyUpTo f n

+
applyUpTo⁺ = Subset.applyUpTo⁺ (setoid _)

+

+
------------------------------------------------------------------------

+
-- _>>=_

+

+
module _ {A B : Set a} (f g : A → List B) where

+

+
  >>=⁺ : xs ⊆ ys → (∀ {x} → f x ⊆ g x) → (xs >>= f) ⊆ (ys >>= g)

+
  >>=⁺ xs⊆ys f⊆g =

+
    Inverse.to >>=-∈↔ ∘

+
    Product.map₂ (Product.map xs⊆ys f⊆g) ∘

+
    Inverse.from >>=-∈↔

+

+
------------------------------------------------------------------------

+
-- _⊛_

+

+
module _ {A B : Set a} {fs gs : List (A → B)} where

+

+
  ⊛⁺ : fs ⊆ gs → xs ⊆ ys → (fs ⊛ xs) ⊆ (gs ⊛ ys)

+
  ⊛⁺ fs⊆gs xs⊆ys =

+
    (Inverse.to $ ⊛-∈↔ gs) ∘

+
    Product.map₂ (Product.map₂ (Product.map fs⊆gs (Product.map₁ xs⊆ys))) ∘

+
    (Inverse.from $ ⊛-∈↔ fs)

+

+
------------------------------------------------------------------------

+
-- _⊗_

+

+
module _ {A B : Set a} {ws xs : List A} {ys zs : List B} where

+

+
  ⊗⁺ : ws ⊆ xs → ys ⊆ zs → (ws ⊗ ys) ⊆ (xs ⊗ zs)

+
  ⊗⁺ ws⊆xs ys⊆zs =

+
    Inverse.to ⊗-∈↔ ∘

+
    Product.map ws⊆xs ys⊆zs ∘

+
    Inverse.from ⊗-∈↔

+

+
------------------------------------------------------------------------

+
-- any

+

+
module _ (p : A → Bool) {xs ys} where

+

+
  any⁺ : xs ⊆ ys → T (any p xs) → T (any p ys)

+
  any⁺ xs⊆ys =

+
    Equivalence.to Any.any⇔ ∘

+
    Any-resp-⊆ xs⊆ys ∘

+
    Equivalence.from Any.any⇔

+

+
------------------------------------------------------------------------

+
-- mapWith∈

+

+
module _ {xs : List A} {f : ∀ {x} → x ∈ xs → B}

+
         {ys : List A} {g : ∀ {x} → x ∈ ys → B}

+
         where

+

+
  mapWith∈⁺ : (xs⊆ys : xs ⊆ ys) → (∀ {x} → f {x} ≗ g ∘ xs⊆ys) →

+
                mapWith∈ xs f ⊆ mapWith∈ ys g

+
  mapWith∈⁺ xs⊆ys f≈g {x} =

+
    Inverse.to Any.mapWith∈↔ ∘

+
    Product.map₂ (Product.map xs⊆ys (λ {x∈xs} x≡fx∈xs → begin

+
      x               ≡⟨ x≡fx∈xs ⟩

+
      f x∈xs          ≡⟨ f≈g x∈xs ⟩

+
      g (xs⊆ys x∈xs)  ∎)) ∘

+
    Inverse.from Any.mapWith∈↔

+
    where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {P : Pred A p} (P? : Decidable P) where

+

+
  filter-⊆ : ∀ xs → filter P? xs ⊆ xs

+
  filter-⊆ = Subset.filter-⊆ (setoid A) P?

+

+
  module _ {Q : Pred A q} (Q? : Decidable Q) where

+

+
    filter⁺′ : P ⋐ Q → ∀ {xs ys} → xs ⊆ ys → filter P? xs ⊆ filter Q? ys

+
    filter⁺′ = Subset.filter⁺′ (setoid A) P? (resp P) Q? (resp Q)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+

+
-- Version 1.5

+

+
mono = Any-resp-⊆

+
{-# WARNING_ON_USAGE mono

+
"Warning: mono was deprecated in v1.5.

+
Please use Any-resp-⊆ instead."

+
#-}

+
map-mono = map⁺

+
{-# WARNING_ON_USAGE map-mono

+
"Warning: map-mono was deprecated in v1.5.

+
Please use map⁺ instead."

+
#-}

+
infix 4 _++-mono_

+
_++-mono_ = ++⁺

+
{-# WARNING_ON_USAGE _++-mono_

+
"Warning: _++-mono_ was deprecated in v1.5.

+
Please use ++⁺ instead."

+
#-}

+
concat-mono = concat⁺

+
{-# WARNING_ON_USAGE concat-mono

+
"Warning: concat-mono was deprecated in v1.5.

+
Please use concat⁺ instead."

+
#-}

+
>>=-mono = >>=⁺

+
{-# WARNING_ON_USAGE >>=-mono

+
"Warning: >>=-mono was deprecated in v1.5.

+
Please use >>=⁺ instead."

+
#-}

+
infix 4  _⊛-mono_

+
_⊛-mono_ = ⊛⁺

+
{-# WARNING_ON_USAGE _⊛-mono_

+
"Warning: _⊛-mono_ was deprecated in v1.5.

+
Please use ⊛⁺ instead."

+
#-}

+
infix 4  _⊗-mono_

+
_⊗-mono_ = ⊗⁺

+
{-# WARNING_ON_USAGE _⊗-mono_

+
"Warning: _⊗-mono_ was deprecated in v1.5.

+
Please use ⊗⁺ instead."

+
#-}

+
any-mono = any⁺

+
{-# WARNING_ON_USAGE any-mono

+
"Warning: any-mono was deprecated in v1.5.

+
Please use any⁺ instead."

+
#-}

+
map-with-∈-mono = mapWith∈⁺

+
{-# WARNING_ON_USAGE map-with-∈-mono

+
"Warning: map-with-∈-mono was deprecated in v1.5.

+
Please use mapWith∈⁺ instead."

+
#-}

+
map-with-∈⁺ = mapWith∈⁺

+
{-# WARNING_ON_USAGE map-with-∈⁺

+
"Warning: map-with-∈⁺ was deprecated in v2.0.

+
Please use mapWith∈⁺ instead."

+
#-}

+
filter⁺ = filter-⊆

+
{-# WARNING_ON_USAGE filter⁺

+
"Warning: filter⁺ was deprecated in v1.5.

+
Please use filter-⊆ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Subset.Propositional.html b/site/static/agda_html/Data.List.Relation.Binary.Subset.Propositional.html
new file mode 100644
index 000000000..cabde414d
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Subset.Propositional.html
@@ -0,0 +1,95 @@
+
+Data.List.Relation.Binary.Subset.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The sublist relation over propositional equality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Subset.Propositional

+
  {a} {A : Set a} where

+

+
import Data.List.Relation.Binary.Subset.Setoid as SetoidSubset

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
------------------------------------------------------------------------

+
-- Re-export parameterised definitions from setoid sublists

+

+
open SetoidSubset (setoid A) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Subset.Setoid.Properties.html b/site/static/agda_html/Data.List.Relation.Binary.Subset.Setoid.Properties.html
new file mode 100644
index 000000000..b1685619d
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Subset.Setoid.Properties.html
@@ -0,0 +1,393 @@
+
+Data.List.Relation.Binary.Subset.Setoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the extensional sublist relation over setoid equality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Binary.Subset.Setoid.Properties where

+

+
open import Data.Bool.Base using (Bool; true; false)

+
open import Data.List.Base hiding (_∷ʳ_; find)

+
import Data.List.Properties as List

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.List.Relation.Unary.All as All using (All)

+
import Data.List.Membership.Setoid as Membership

+
import Data.List.Membership.Setoid.Properties as Membershipₚ

+
open import Data.Nat.Base using (ℕ; s≤s; _≤_)

+
import Data.List.Relation.Binary.Subset.Setoid as Subset

+
import Data.List.Relation.Binary.Sublist.Setoid as Sublist

+
import Data.List.Relation.Binary.Equality.Setoid as Equality

+
import Data.List.Relation.Binary.Permutation.Setoid as Permutation

+
import Data.List.Relation.Binary.Permutation.Setoid.Properties as Permutationₚ

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_∘_; _∘₂_; _$_)

+
open import Level using (Level)

+
open import Relation.Nullary using (¬_; does; yes; no)

+
open import Relation.Nullary.Negation using (contradiction)

+
open import Relation.Unary using (Pred; Decidable) renaming (_⊆_ to _⋐_)

+
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Transitive; _Respectsʳ_; _Respectsˡ_; _Respects_)

+
open import Relation.Binary.Bundles using (Setoid; Preorder)

+
open import Relation.Binary.Structures using (IsPreorder)

+
import Relation.Binary.Reasoning.Preorder as ≲-Reasoning

+
open import Relation.Binary.Reasoning.Syntax

+

+
open Setoid using (Carrier)

+

+
private

+
  variable

+
    a b p q r ℓ : Level

+

+
------------------------------------------------------------------------

+
-- Relational properties with _≋_ (pointwise equality)

+
------------------------------------------------------------------------

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Subset S

+
  open Equality S

+
  open Membership S

+
  open Membershipₚ

+

+
  ⊆-reflexive : _≋_ ⇒ _⊆_

+
  ⊆-reflexive xs≋ys = ∈-resp-≋ S xs≋ys

+

+
  ⊆-refl : Reflexive _⊆_

+
  ⊆-refl x∈xs = x∈xs

+

+
  ⊆-trans : Transitive _⊆_

+
  ⊆-trans xs⊆ys ys⊆zs x∈xs = ys⊆zs (xs⊆ys x∈xs)

+

+
  ⊆-respʳ-≋ : _⊆_ Respectsʳ _≋_

+
  ⊆-respʳ-≋ xs≋ys = ∈-resp-≋ S xs≋ys ∘_

+

+
  ⊆-respˡ-≋ : _⊆_ Respectsˡ _≋_

+
  ⊆-respˡ-≋ xs≋ys = _∘ ∈-resp-≋ S (≋-sym xs≋ys)

+

+
  ⊆-isPreorder : IsPreorder _≋_ _⊆_

+
  ⊆-isPreorder = record

+
    { isEquivalence = ≋-isEquivalence

+
    ; reflexive     = ⊆-reflexive

+
    ; trans         = ⊆-trans

+
    }

+

+
  ⊆-preorder : Preorder _ _ _

+
  ⊆-preorder = record

+
    { isPreorder = ⊆-isPreorder

+
    }

+

+
------------------------------------------------------------------------

+
-- Relational properties with _↭_ (permutations)

+
------------------------------------------------------------------------

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Subset S

+
  open Permutation S

+
  open Membership S

+

+
  ⊆-reflexive-↭  : _↭_ ⇒ _⊆_

+
  ⊆-reflexive-↭ xs↭ys = Permutationₚ.∈-resp-↭ S xs↭ys

+

+
  ⊆-respʳ-↭ : _⊆_ Respectsʳ _↭_

+
  ⊆-respʳ-↭ xs↭ys = Permutationₚ.∈-resp-↭ S xs↭ys ∘_

+

+
  ⊆-respˡ-↭ : _⊆_ Respectsˡ _↭_

+
  ⊆-respˡ-↭ xs↭ys = _∘ Permutationₚ.∈-resp-↭ S (↭-sym xs↭ys)

+

+
  ⊆-↭-isPreorder : IsPreorder _↭_ _⊆_

+
  ⊆-↭-isPreorder = record

+
    { isEquivalence = ↭-isEquivalence

+
    ; reflexive     = ⊆-reflexive-↭

+
    ; trans         = ⊆-trans S

+
    }

+

+
  ⊆-↭-preorder : Preorder _ _ _

+
  ⊆-↭-preorder = record

+
    { isPreorder = ⊆-↭-isPreorder

+
    }

+

+
------------------------------------------------------------------------

+
-- Reasoning over subsets

+
------------------------------------------------------------------------

+

+
module ⊆-Reasoning (S : Setoid a ℓ) where

+
  open Membership S using (_∈_)

+

+
  private module Base = ≲-Reasoning (⊆-preorder S)

+

+
  open Base public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-≲; step-∼)

+
    renaming (≲-go to ⊆-go; ≈-go to ≋-go)

+

+
  open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → Base.begin x) public

+
  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ ⊆-go public

+
  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ ≋-go public

+

+
------------------------------------------------------------------------

+
-- Relationship with other binary relations

+
------------------------------------------------------------------------

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S

+
  open Subset S

+
  open Sublist S renaming (_⊆_ to _⊑_)

+

+
  Sublist⇒Subset : ∀ {xs ys} → xs ⊑ ys → xs ⊆ ys

+
  Sublist⇒Subset (x≈y ∷  xs⊑ys) (here v≈x)   = here (trans v≈x x≈y)

+
  Sublist⇒Subset (x≈y ∷  xs⊑ys) (there v∈xs) = there (Sublist⇒Subset xs⊑ys v∈xs)

+
  Sublist⇒Subset (y   ∷ʳ xs⊑ys) v∈xs         = there (Sublist⇒Subset xs⊑ys v∈xs)

+

+
------------------------------------------------------------------------

+
-- Relationship with predicates

+
------------------------------------------------------------------------

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S renaming (Carrier to A)

+
  open Subset S

+
  open Membership S

+

+
  Any-resp-⊆ : ∀ {P : Pred A p} → P Respects _≈_ → (Any P) Respects _⊆_

+
  Any-resp-⊆ resp ⊆ pxs with find pxs

+
  ... | (x , x∈xs , px) = lose resp (⊆ x∈xs) px

+

+
  All-resp-⊇ : ∀ {P : Pred A p} → P Respects _≈_ → (All P) Respects _⊇_

+
  All-resp-⊇ resp ⊇ pxs = All.tabulateₛ S (All.lookupₛ S resp pxs ∘ ⊇)

+

+
------------------------------------------------------------------------

+
-- Properties of list functions

+
------------------------------------------------------------------------

+
-- ∷

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S

+
  open Subset S

+
  open Membership S

+
  open Membershipₚ

+

+
  xs⊆x∷xs : ∀ xs x → xs ⊆ x ∷ xs

+
  xs⊆x∷xs xs x = there

+

+
  ∷⁺ʳ : ∀ {xs ys} x → xs ⊆ ys → x ∷ xs ⊆ x ∷ ys

+
  ∷⁺ʳ x xs⊆ys (here  p) = here p

+
  ∷⁺ʳ x xs⊆ys (there p) = there (xs⊆ys p)

+

+
  ∈-∷⁺ʳ : ∀ {xs ys x} → x ∈ ys → xs ⊆ ys → x ∷ xs ⊆ ys

+
  ∈-∷⁺ʳ x∈ys _  (here  v≈x)  = ∈-resp-≈ S (sym v≈x) x∈ys

+
  ∈-∷⁺ʳ _ xs⊆ys (there x∈xs) = xs⊆ys x∈xs

+

+
------------------------------------------------------------------------

+
-- ++

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Subset S

+
  open Membership S

+
  open Membershipₚ

+

+
  xs⊆xs++ys : ∀ xs ys → xs ⊆ xs ++ ys

+
  xs⊆xs++ys xs ys = ∈-++⁺ˡ S

+

+
  xs⊆ys++xs : ∀ xs ys → xs ⊆ ys ++ xs

+
  xs⊆ys++xs xs ys = ∈-++⁺ʳ S _

+

+
  ++⁺ʳ : ∀ {xs ys} zs → xs ⊆ ys → zs ++ xs ⊆ zs ++ ys

+
  ++⁺ʳ []       xs⊆ys           = xs⊆ys

+
  ++⁺ʳ (x ∷ zs) xs⊆ys (here p)  = here p

+
  ++⁺ʳ (x ∷ zs) xs⊆ys (there p) = there (++⁺ʳ zs xs⊆ys p)

+

+
  ++⁺ˡ : ∀ {xs ys} zs → xs ⊆ ys → xs ++ zs ⊆ ys ++ zs

+
  ++⁺ˡ {[]}     {ys} zs xs⊆ys           = xs⊆ys++xs zs ys

+
  ++⁺ˡ {x ∷ xs} {ys} zs xs⊆ys (here  p) = xs⊆xs++ys ys zs (xs⊆ys (here p))

+
  ++⁺ˡ {x ∷ xs} {ys} zs xs⊆ys (there p) = ++⁺ˡ zs (xs⊆ys ∘ there) p

+

+
  ++⁺ : ∀ {ws xs ys zs} → ws ⊆ xs → ys ⊆ zs → ws ++ ys ⊆ xs ++ zs

+
  ++⁺ ws⊆xs ys⊆zs = ⊆-trans S (++⁺ˡ _ ws⊆xs) (++⁺ʳ _ ys⊆zs)

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ (S : Setoid a ℓ) (R : Setoid b r) where

+

+
  private

+
    module S = Setoid S

+
    module R = Setoid R

+

+
    module S⊆ = Subset S

+
    module R⊆ = Subset R

+

+
  open Membershipₚ

+

+
  map⁺ : ∀ {as bs} {f : S.Carrier → R.Carrier} →

+
         f Preserves S._≈_ ⟶ R._≈_ →

+
         as S⊆.⊆ bs → map f as R⊆.⊆ map f bs

+
  map⁺ {f = f} f-pres as⊆bs v∈f[as] =

+
    let x , x∈as , v≈f[x] = ∈-map⁻ S R v∈f[as] in

+
    ∈-resp-≈ R (R.sym v≈f[x]) (∈-map⁺ S R f-pres (as⊆bs x∈as))

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S renaming (Carrier to A)

+
  open Subset S

+

+
  reverse-selfAdjoint : ∀ {as bs} → as ⊆ reverse bs → reverse as ⊆ bs

+
  reverse-selfAdjoint rs = reverse⁻ ∘ rs ∘ reverse⁻

+
    where reverse⁻ = Membershipₚ.reverse⁻ S

+

+
-- NB. the unit and counit of this adjunction are given by:

+
-- reverse-η : ∀ {xs} → xs ⊆ reverse xs

+
-- reverse-η = Membershipₚ.reverse⁺ S

+
-- reverse-ε : ∀ {xs} → reverse xs ⊆ xs

+
-- reverse-ε = Membershipₚ.reverse⁻ S

+

+
  reverse⁺ : ∀ {as bs} → as ⊆ bs → reverse as ⊆ reverse bs

+
  reverse⁺ {as} {bs} rs = reverse-selfAdjoint $ begin

+
    as                   ⊆⟨ rs ⟩

+
    bs                   ≡⟨ List.reverse-involutive bs ⟨

+
    reverse (reverse bs) ∎

+
    where open ⊆-Reasoning S

+

+
  reverse⁻ : ∀ {as bs} → reverse as ⊆ reverse bs → as ⊆ bs

+
  reverse⁻ {as} {bs} rs = begin

+
    as                   ≡⟨ List.reverse-involutive as ⟨

+
    reverse (reverse as) ⊆⟨ reverse-selfAdjoint rs ⟩

+
    bs                   ∎

+
    where open ⊆-Reasoning S

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S renaming (Carrier to A)

+
  open Subset S

+
  open Membershipₚ

+

+
  filter-⊆ : ∀ {P : Pred A p} (P? : Decidable P) →

+
             ∀ xs → filter P? xs ⊆ xs

+
  filter-⊆ P? (x ∷ xs) y∈f[x∷xs] with does (P? x)

+
  ... | false = there (filter-⊆ P? xs y∈f[x∷xs])

+
  ... | true  with y∈f[x∷xs]

+
  ...   | here  y≈x     = here y≈x

+
  ...   | there y∈f[xs] = there (filter-⊆ P? xs y∈f[xs])

+

+
  -- Should be known as `filter⁺` (no prime) but `filter-⊆` used

+
  -- to be called this so for backwards compatability reasons, the

+
  -- correct name will have to wait until the deprecated name is

+
  -- removed.

+
  filter⁺′ : ∀ {P : Pred A p} (P? : Decidable P) → P Respects _≈_ →

+
             ∀ {Q : Pred A q} (Q? : Decidable Q) → Q Respects _≈_ →

+
             P ⋐ Q → ∀ {xs ys} → xs ⊆ ys → filter P? xs ⊆ filter Q? ys

+
  filter⁺′ P? P-resp Q? Q-resp P⋐Q xs⊆ys v∈fxs with ∈-filter⁻ S P? P-resp v∈fxs

+
  ... | v∈xs , Pv = ∈-filter⁺ S Q? Q-resp (xs⊆ys v∈xs) (P⋐Q Pv)

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S renaming (Carrier to A)

+
  open Subset S

+

+
  applyUpTo⁺ : ∀ (f : ℕ → A) {m n} → m ≤ n → applyUpTo f m ⊆ applyUpTo f n

+
  applyUpTo⁺ _ (s≤s m≤n) (here  f≡f[0]) = here f≡f[0]

+
  applyUpTo⁺ _ (s≤s m≤n) (there v∈xs)   = there (applyUpTo⁺ _ m≤n v∈xs)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+

+
-- Version 1.5

+

+
filter⁺ = filter-⊆

+
{-# WARNING_ON_USAGE filter⁺

+
"Warning: filter⁺ was deprecated in v1.5.

+
Please use filter-⊆ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Binary.Subset.Setoid.html b/site/static/agda_html/Data.List.Relation.Binary.Subset.Setoid.html
new file mode 100644
index 000000000..c7a7493c6
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Binary.Subset.Setoid.html
@@ -0,0 +1,115 @@
+
+Data.List.Relation.Binary.Subset.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The extensional sublist relation over setoid equality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Data.List.Relation.Binary.Subset.Setoid

+
  {c ℓ} (S : Setoid c ℓ) where

+

+
open import Data.List.Base using (List)

+
open import Data.List.Membership.Setoid S using (_∈_)

+
open import Function.Base using (flip)

+
open import Level using (_⊔_)

+
open import Relation.Nullary.Negation using (¬_)

+

+
open Setoid S renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
infix 4 _⊆_ _⊇_ _⊈_ _⊉_

+

+
_⊆_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊆ ys = ∀ {x} → x ∈ xs → x ∈ ys

+

+
_⊇_ : Rel (List A) (c ⊔ ℓ)

+
_⊇_ = flip _⊆_

+

+
_⊈_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊈ ys = ¬ xs ⊆ ys

+

+
_⊉_ : Rel (List A) (c ⊔ ℓ)

+
xs ⊉ ys = ¬ xs ⊇ ys

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.All.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.All.Properties.html
new file mode 100644
index 000000000..d76d9512e
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.All.Properties.html
@@ -0,0 +1,881 @@
+
+Data.List.Relation.Unary.All.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to All

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.All.Properties where

+

+
open import Axiom.Extensionality.Propositional using (Extensionality)

+
open import Data.Bool.Base using (Bool; T; true; false)

+
open import Data.Bool.Properties using (T-∧)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.List.Base as List hiding (lookup; updateAt)

+
open import Data.List.Membership.Propositional using (_∈_; _≢∈_)

+
open import Data.List.Membership.Propositional.Properties

+
  using (there-injective-≢∈; ∈-filter⁻)

+
import Data.List.Membership.Setoid as SetoidMembership

+
import Data.List.Properties as List

+
import Data.List.Relation.Binary.Equality.Setoid as ≋

+
open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)

+
open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)

+
open import Data.List.Relation.Unary.All as All using

+
  ( All; []; _∷_; lookup; updateAt

+
  ; _[_]=_; here; there

+
  ; Null

+
  )

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)

+
open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing; fromAny)

+
open import Data.Maybe.Relation.Unary.Any as Maybe using (just)

+
open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)

+
open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)

+
open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)

+
open import Function.Base using (_∘_; _$_; id; case_of_; flip)

+
open import Function.Bundles using (_↠_; mk↠ₛ; _⇔_; mk⇔; _↔_; mk↔ₛ′; Equivalence)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (REL)

+
open import Relation.Binary.Bundles using (Setoid)

+
import Relation.Binary.Definitions as B

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂; _≗_)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Nullary.Decidable

+
  using (Dec; does; yes; no; _because_; ¬?; decidable-stable)

+
open import Relation.Unary

+
  using (Decidable; Pred; Universal; ∁; _∩_; _⟨×⟩_) renaming (_⊆_ to _⋐_)

+
open import Relation.Unary.Properties using (∁?)

+

+
private

+
  variable

+
    a b c p q r ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    P : Pred A p

+
    Q : Pred B q

+
    R : Pred C r

+
    x y : A

+
    xs ys : List A

+

+
------------------------------------------------------------------------

+
-- Properties regarding Null

+

+
Null⇒null : Null xs → T (null xs)

+
Null⇒null [] = _

+

+
null⇒Null : T (null xs) → Null xs

+
null⇒Null {xs = []   } _ = []

+
null⇒Null {xs = _ ∷ _} ()

+

+
------------------------------------------------------------------------

+
-- Properties of the "points-to" relation _[_]=_

+

+
-- Relation _[_]=_ is deterministic: each index points to a single value.

+

+
[]=-injective : ∀ {px qx : P x} {pxs : All P xs} {i : x ∈ xs} →

+
                pxs [ i ]= px →

+
                pxs [ i ]= qx →

+
                px ≡ qx

+
[]=-injective here          here          = refl

+
[]=-injective (there x↦px) (there x↦qx) = []=-injective x↦px x↦qx

+

+
-- See also Data.List.Relation.Unary.All.Properties.WithK.[]=-irrelevant.

+

+
------------------------------------------------------------------------

+
-- Lemmas relating Any, All and negation.

+

+
¬Any⇒All¬ : ∀ xs → ¬ Any P xs → All (¬_ ∘ P) xs

+
¬Any⇒All¬ []       ¬p = []

+
¬Any⇒All¬ (x ∷ xs) ¬p = ¬p ∘ here ∷ ¬Any⇒All¬ xs (¬p ∘ there)

+

+
All¬⇒¬Any : ∀ {xs} → All (¬_ ∘ P) xs → ¬ Any P xs

+
All¬⇒¬Any (¬p ∷ _)  (here  p) = ¬p p

+
All¬⇒¬Any (_  ∷ ¬p) (there p) = All¬⇒¬Any ¬p p

+

+
¬All⇒Any¬ : Decidable P → ∀ xs → ¬ All P xs → Any (¬_ ∘ P) xs

+
¬All⇒Any¬ dec []       ¬∀ = contradiction [] ¬∀

+
¬All⇒Any¬ dec (x ∷ xs) ¬∀ with dec x

+
... |  true because  [p] = there (¬All⇒Any¬ dec xs (¬∀ ∘ _∷_ (invert [p])))

+
... | false because [¬p] = here (invert [¬p])

+

+
Any¬⇒¬All : ∀ {xs} → Any (¬_ ∘ P) xs → ¬ All P xs

+
Any¬⇒¬All (here  ¬p) = ¬p           ∘ All.head

+
Any¬⇒¬All (there ¬p) = Any¬⇒¬All ¬p ∘ All.tail

+

+
¬Any↠All¬ : ∀ {xs} → (¬ Any P xs) ↠ All (¬_ ∘ P) xs

+
¬Any↠All¬ = mk↠ₛ {to = ¬Any⇒All¬ _} (λ y → All¬⇒¬Any y , to∘from y)

+
  where

+
  to∘from : ∀ {xs} (¬p : All (¬_ ∘ P) xs) → ¬Any⇒All¬ xs (All¬⇒¬Any ¬p) ≡ ¬p

+
  to∘from []         = refl

+
  to∘from (¬p ∷ ¬ps) = cong₂ _∷_ refl (to∘from ¬ps)

+

+
  -- If equality of functions were extensional, then the surjection

+
  -- could be strengthened to a bijection.

+

+
  from∘to : Extensionality _ _ →

+
            ∀ xs → (¬p : ¬ Any P xs) → All¬⇒¬Any (¬Any⇒All¬ xs ¬p) ≡ ¬p

+
  from∘to ext []       ¬p = ext λ ()

+
  from∘to ext (x ∷ xs) ¬p = ext λ

+
    { (here p)  → refl

+
    ; (there p) → cong (λ f → f p) $ from∘to ext xs (¬p ∘ there)

+
    }

+

+
Any¬⇔¬All : ∀ {xs} → Decidable P → Any (¬_ ∘ P) xs ⇔ (¬ All P xs)

+
Any¬⇔¬All dec = mk⇔ Any¬⇒¬All (¬All⇒Any¬ dec _)

+

+
private

+
  -- If equality of functions were extensional, then the logical

+
  -- equivalence could be strengthened to a surjection.

+
  to∘from : Extensionality _ _ → (dec : Decidable P) →

+
            (¬∀ : ¬ All P xs) → Any¬⇒¬All (¬All⇒Any¬ dec xs ¬∀) ≡ ¬∀

+
  to∘from ext P ¬∀ = ext λ ∀P → contradiction ∀P ¬∀

+

+
module _ {_~_ : REL A B ℓ} where

+

+
  All-swap : ∀ {xs ys} →

+
             All (λ x → All (x ~_) ys) xs →

+
             All (λ y → All (_~ y) xs) ys

+
  All-swap {ys = []}     _   = []

+
  All-swap {ys = y ∷ ys} []  = All.universal (λ _ → []) (y ∷ ys)

+
  All-swap {ys = y ∷ ys} ((x~y ∷ x~ys) ∷ pxs) =

+
    (x~y ∷ (All.map All.head pxs)) ∷

+
    All-swap (x~ys ∷ (All.map All.tail pxs))

+

+
------------------------------------------------------------------------

+
-- Defining properties of lookup and _[_]=_

+
--

+
-- pxs [ i ]= px  if and only if  lookup pxs i = px.

+

+
-- `i` points to `lookup pxs i` in `pxs`.

+

+
[]=lookup : (pxs : All P xs) (i : x ∈ xs) →

+
            pxs [ i ]= lookup pxs i

+
[]=lookup (px ∷ pxs) (here refl) = here

+
[]=lookup (px ∷ pxs) (there i)   = there ([]=lookup pxs i)

+

+
-- If `i` points to `px` in `pxs`, then `lookup pxs i ≡ px`.

+

+
[]=⇒lookup : ∀ {px : P x} {pxs : All P xs} {i : x ∈ xs} →

+
             pxs [ i ]= px →

+
             lookup pxs i ≡ px

+
[]=⇒lookup x↦px = []=-injective ([]=lookup _ _) x↦px

+

+
-- If `lookup pxs i ≡ px`, then `i` points to `px` in `pxs`.

+

+
lookup⇒[]= : ∀ {px : P x} (pxs : All P xs) (i : x ∈ xs) →

+
             lookup pxs i ≡ px →

+
             pxs [ i ]= px

+
lookup⇒[]= pxs i refl = []=lookup pxs i

+

+
------------------------------------------------------------------------

+
-- Properties of operations over `All`

+
------------------------------------------------------------------------

+
-- map

+

+
map-cong : ∀ {f : P ⋐ Q} {g : P ⋐ Q} (pxs : All P xs) →

+
           (∀ {x} → f {x} ≗ g) → All.map f pxs ≡ All.map g pxs

+
map-cong []         _   = refl

+
map-cong (px ∷ pxs) feq = cong₂ _∷_ (feq px) (map-cong pxs feq)

+

+
map-id : ∀ (pxs : All P xs) → All.map id pxs ≡ pxs

+
map-id []         = refl

+
map-id (px ∷ pxs) = cong (px ∷_)  (map-id pxs)

+

+
map-∘ : ∀ {f : P ⋐ Q} {g : Q ⋐ R} (pxs : All P xs) →

+
        All.map g (All.map f pxs) ≡ All.map (g ∘ f) pxs

+
map-∘ []         = refl

+
map-∘ (px ∷ pxs) = cong (_ ∷_) (map-∘ pxs)

+

+
lookup-map : ∀ {f : P ⋐ Q} (pxs : All P xs) (i : x ∈ xs) →

+
             lookup (All.map f pxs) i ≡ f (lookup pxs i)

+
lookup-map (px ∷ pxs) (here refl) = refl

+
lookup-map (px ∷ pxs) (there i)   = lookup-map pxs i

+

+
------------------------------------------------------------------------

+
-- _[_]%=_ / updateAt

+

+
  -- Defining properties of updateAt:

+

+
-- (+) updateAt actually updates the element at the given index.

+

+
updateAt-updates : ∀ (i : x ∈ xs) {f : P x → P x} {px : P x} (pxs : All P xs) →

+
                   pxs              [ i ]= px →

+
                   updateAt i f pxs [ i ]= f px

+
updateAt-updates (here  refl) (px ∷ pxs) here         = here

+
updateAt-updates (there i)    (px ∷ pxs) (there x↦px) =

+
  there (updateAt-updates i pxs x↦px)

+

+
-- (-) updateAt i does not touch the elements at other indices.

+

+
updateAt-minimal : ∀ (i : x ∈ xs) (j : y ∈ xs) →

+
                   ∀ {f : P y → P y} {px : P x} (pxs : All P xs) →

+
                   i ≢∈ j →

+
                   pxs              [ i ]= px →

+
                   updateAt j f pxs [ i ]= px

+
updateAt-minimal (here .refl) (here refl) (px ∷ pxs) i≢j here        =

+
  contradiction refl (i≢j refl)

+
updateAt-minimal (here .refl) (there j)   (px ∷ pxs) i≢j here        = here

+
updateAt-minimal (there i)    (here refl) (px ∷ pxs) i≢j (there val) = there val

+
updateAt-minimal (there i)    (there j)   (px ∷ pxs) i≢j (there val) =

+
  there (updateAt-minimal i j pxs (there-injective-≢∈ i≢j) val)

+

+
-- lookup after updateAt reduces.

+

+
-- For same index this is an easy consequence of updateAt-updates

+
-- using []=↔lookup.

+

+
lookup∘updateAt : ∀ (pxs : All P xs) (i : x ∈ xs) {f : P x → P x} →

+
                  lookup (updateAt i f pxs) i ≡ f (lookup pxs i)

+
lookup∘updateAt pxs i =

+
  []=⇒lookup (updateAt-updates i pxs (lookup⇒[]= pxs i refl))

+

+
-- For different indices it easily follows from updateAt-minimal.

+

+
lookup∘updateAt′ : ∀ (i : x ∈ xs) (j : y ∈ xs) →

+
                   ∀ {f : P y → P y} {px : P x} (pxs : All P xs) →

+
                   i ≢∈ j →

+
                   lookup (updateAt j f pxs) i ≡ lookup pxs i

+
lookup∘updateAt′ i j pxs i≢j =

+
  []=⇒lookup (updateAt-minimal i j pxs i≢j (lookup⇒[]= pxs i refl))

+

+
-- The other properties are consequences of (+) and (-).

+
-- We spell the most natural properties out.

+
-- Direct inductive proofs are in most cases easier than just using

+
-- the defining properties.

+

+
-- In the explanations, we make use of shorthand  f = g ↾ x

+
-- meaning that f and g agree locally at point x, i.e.  f x ≡ g x.

+

+
-- updateAt (i : x ∈ xs)  is a morphism

+
-- from the monoid of endofunctions  P x → P x

+
-- to the monoid of endofunctions  All P xs → All P xs.

+

+
-- 1a. local identity:  f = id ↾ (lookup pxs i)

+
--             implies  updateAt i f = id ↾ pxs

+

+
updateAt-id-local : ∀ (i : x ∈ xs) {f : P x → P x} (pxs : All P xs) →

+
                    f (lookup pxs i) ≡ lookup pxs i →

+
                    updateAt i f pxs ≡ pxs

+
updateAt-id-local (here refl)(px ∷ pxs) eq = cong (_∷ pxs) eq

+
updateAt-id-local (there i)  (px ∷ pxs) eq = cong (px ∷_) (updateAt-id-local i pxs eq)

+

+
-- 1b. identity:  updateAt i id ≗ id

+

+
updateAt-id : ∀ (i : x ∈ xs) (pxs : All P xs) → updateAt i id pxs ≡ pxs

+
updateAt-id i pxs = updateAt-id-local i pxs refl

+

+
-- 2a. relative composition:  f ∘ g = h ↾ (lookup i pxs)

+
--                   implies  updateAt i f ∘ updateAt i g = updateAt i h ↾ pxs

+

+
updateAt-∘-local : ∀ (i : x ∈ xs) {f g h : P x → P x} (pxs : All P xs) →

+
                   f (g (lookup pxs i)) ≡ h (lookup pxs i) →

+
                   updateAt i f (updateAt i g pxs) ≡ updateAt i h pxs

+
updateAt-∘-local (here refl) (px ∷ pxs) fg=h = cong (_∷ pxs) fg=h

+
updateAt-∘-local (there i)   (px ∷ pxs) fg=h = cong (px ∷_) (updateAt-∘-local i pxs fg=h)

+

+
-- 2b. composition:  updateAt i f ∘ updateAt i g ≗ updateAt i (f ∘ g)

+

+
updateAt-∘ : ∀ (i : x ∈ xs) {f g : P x → P x} →

+
             updateAt {P = P} i f ∘ updateAt i g ≗ updateAt i (f ∘ g)

+
updateAt-∘ i pxs = updateAt-∘-local i pxs refl

+

+
-- 3. congruence:  updateAt i  is a congruence wrt. extensional equality.

+

+
-- 3a.  If    f = g ↾ (lookup pxs i)

+
--      then  updateAt i f = updateAt i g ↾ pxs

+

+
updateAt-cong-local : ∀ (i : x ∈ xs) {f g : P x → P x} (pxs : All P xs) →

+
                      f (lookup pxs i) ≡ g (lookup pxs i) →

+
                      updateAt i f pxs ≡ updateAt i g pxs

+
updateAt-cong-local (here refl) (px ∷ pxs) f=g = cong (_∷ pxs) f=g

+
updateAt-cong-local (there i)   (px ∷ pxs) f=g = cong (px ∷_) (updateAt-cong-local i pxs f=g)

+

+
-- 3b. congruence:  f ≗ g → updateAt i f ≗ updateAt i g

+

+
updateAt-cong : ∀ (i : x ∈ xs) {f g : P x → P x} →

+
                f ≗ g → updateAt {P = P} i f ≗ updateAt i g

+
updateAt-cong i f≗g pxs = updateAt-cong-local i pxs (f≗g (lookup pxs i))

+

+
-- The order of updates at different indices i ≢ j does not matter.

+

+
-- This a consequence of updateAt-updates and updateAt-minimal

+
-- but easier to prove inductively.

+

+
updateAt-commutes : ∀ (i : x ∈ xs) (j : y ∈ xs) →

+
                    ∀ {f : P x → P x} {g : P y → P y} →

+
                    i ≢∈ j →

+
                    updateAt {P = P} i f ∘ updateAt j g ≗ updateAt j g ∘ updateAt i f

+
updateAt-commutes (here refl) (here refl) i≢j (px ∷ pxs) =

+
  contradiction refl (i≢j refl)

+
updateAt-commutes (here refl) (there j)   i≢j (px ∷ pxs) = refl

+
updateAt-commutes (there i)   (here refl) i≢j (px ∷ pxs) = refl

+
updateAt-commutes (there i)   (there j)   i≢j (px ∷ pxs) =

+
  cong (px ∷_) (updateAt-commutes i j (there-injective-≢∈ i≢j) pxs)

+

+
map-updateAt : ∀ {f : P ⋐ Q} {g : P x → P x} {h : Q x → Q x}

+
               (pxs : All P xs) (i : x ∈ xs) →

+
               f (g (lookup pxs i)) ≡ h (f (lookup pxs i)) →

+
               All.map f (pxs All.[ i ]%= g) ≡ (All.map f pxs) All.[ i ]%= h

+
map-updateAt (px ∷ pxs) (here refl) = cong (_∷ _)

+
map-updateAt (px ∷ pxs) (there i) feq = cong (_ ∷_) (map-updateAt pxs i feq)

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- singleton

+

+
singleton⁻ : All P [ x ] → P x

+
singleton⁻ (px ∷ []) = px

+

+
-- head

+

+
head⁺ : All P xs → Maybe.All P (head xs)

+
head⁺ []       = nothing

+
head⁺ (px ∷ _) = just px

+

+
-- tail

+

+
tail⁺ : All P xs → Maybe.All (All P) (tail xs)

+
tail⁺ []        = nothing

+
tail⁺ (_ ∷ pxs) = just pxs

+

+
-- last

+

+
last⁺ : All P xs → Maybe.All P (last xs)

+
last⁺ []                 = nothing

+
last⁺ (px ∷ [])          = just px

+
last⁺ (px ∷ pxs@(_ ∷ _)) = last⁺ pxs

+

+
-- uncons

+

+
uncons⁺ : All P xs → Maybe.All (P ⟨×⟩ All P) (uncons xs)

+
uncons⁺ []         = nothing

+
uncons⁺ (px ∷ pxs) = just (px , pxs)

+

+
uncons⁻ : Maybe.All (P ⟨×⟩ All P) (uncons xs) → All P xs

+
uncons⁻ {xs = []}     nothing           = []

+
uncons⁻ {xs = x ∷ xs} (just (px , pxs)) = px ∷ pxs

+

+
-- map

+

+
map⁺ : ∀ {f : A → B} → All (P ∘ f) xs → All P (map f xs)

+
map⁺ []       = []

+
map⁺ (p ∷ ps) = p ∷ map⁺ ps

+

+
map⁻ : ∀ {f : A → B} → All P (map f xs) → All (P ∘ f) xs

+
map⁻ {xs = []}    []       = []

+
map⁻ {xs = _ ∷ _} (p ∷ ps) = p ∷ map⁻ ps

+

+
-- A variant of All.map.

+

+
gmap⁺ : ∀ {f : A → B} → P ⋐ Q ∘ f → All P ⋐ All Q ∘ map f

+
gmap⁺ g = map⁺ ∘ All.map g

+

+
gmap⁻ : ∀ {f : A → B} → Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P

+
gmap⁻ g = All.map g ∘ map⁻

+

+
------------------------------------------------------------------------

+
-- mapMaybe

+

+
mapMaybe⁺ : ∀ {f : A → Maybe B} →

+
            All (Maybe.All P) (map f xs) → All P (mapMaybe f xs)

+
mapMaybe⁺ {xs = []}     {f = f} []         = []

+
mapMaybe⁺ {xs = x ∷ xs} {f = f} (px ∷ pxs) with f x

+
... | nothing = mapMaybe⁺ pxs

+
... | just v with just pv ← px = pv ∷ mapMaybe⁺ pxs

+

+
------------------------------------------------------------------------

+
-- catMaybes

+

+
All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)

+
All-catMaybes⁺ [] = []

+
All-catMaybes⁺ (just px ∷ pxs) = px ∷ All-catMaybes⁺ pxs

+
All-catMaybes⁺ (nothing ∷ pxs) = All-catMaybes⁺ pxs

+

+
Any-catMaybes⁺ : All (Maybe.Any P) xs → All P (catMaybes xs)

+
Any-catMaybes⁺ = All-catMaybes⁺ ∘ All.map fromAny

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
++⁺ : All P xs → All P ys → All P (xs ++ ys)

+
++⁺ []         pys = pys

+
++⁺ (px ∷ pxs) pys = px ∷ ++⁺ pxs pys

+

+
++⁻ˡ : ∀ xs {ys} → All P (xs ++ ys) → All P xs

+
++⁻ˡ []       p          = []

+
++⁻ˡ (x ∷ xs) (px ∷ pxs) = px ∷ (++⁻ˡ _ pxs)

+

+
++⁻ʳ : ∀ xs {ys} → All P (xs ++ ys) → All P ys

+
++⁻ʳ []       p          = p

+
++⁻ʳ (x ∷ xs) (px ∷ pxs) = ++⁻ʳ xs pxs

+

+
++⁻ : ∀ xs {ys} → All P (xs ++ ys) → All P xs × All P ys

+
++⁻ []       p          = [] , p

+
++⁻ (x ∷ xs) (px ∷ pxs) = Product.map (px ∷_) id (++⁻ _ pxs)

+

+
++↔ : (All P xs × All P ys) ↔ All P (xs ++ ys)

+
++↔ {xs = zs} = mk↔ₛ′ (uncurry ++⁺) (++⁻ zs) (++⁺∘++⁻ zs) ++⁻∘++⁺

+
  where

+
  ++⁺∘++⁻ : ∀ xs (p : All P (xs ++ ys)) → uncurry′ ++⁺ (++⁻ xs p) ≡ p

+
  ++⁺∘++⁻ []       p          = refl

+
  ++⁺∘++⁻ (x ∷ xs) (px ∷ pxs) = cong (_∷_ px) $ ++⁺∘++⁻ xs pxs

+

+
  ++⁻∘++⁺ : ∀ (p : All P xs × All P ys) → ++⁻ xs (uncurry ++⁺ p) ≡ p

+
  ++⁻∘++⁺ ([]       , pys) = refl

+
  ++⁻∘++⁺ (px ∷ pxs , pys) rewrite ++⁻∘++⁺ (pxs , pys) = refl

+

+
------------------------------------------------------------------------

+
-- concat

+

+
concat⁺ : ∀ {xss} → All (All P) xss → All P (concat xss)

+
concat⁺ []           = []

+
concat⁺ (pxs ∷ pxss) = ++⁺ pxs (concat⁺ pxss)

+

+
concat⁻ : ∀ {xss} → All P (concat xss) → All (All P) xss

+
concat⁻ {xss = []}       []  = []

+
concat⁻ {xss = xs ∷ xss} pxs = ++⁻ˡ xs pxs ∷ concat⁻ (++⁻ʳ xs pxs)

+

+
------------------------------------------------------------------------

+
-- snoc

+

+
∷ʳ⁺ : All P xs → P x → All P (xs ∷ʳ x)

+
∷ʳ⁺ pxs px = ++⁺ pxs (px ∷ [])

+

+
∷ʳ⁻ : All P (xs ∷ʳ x) → All P xs × P x

+
∷ʳ⁻ pxs = Product.map₂ singleton⁻ $ ++⁻ _ pxs

+

+
-- unsnoc

+

+
unsnoc⁺ : All P xs → Maybe.All (All P ⟨×⟩ P) (unsnoc xs)

+
unsnoc⁺ {xs = xs} pxs with initLast xs

+
unsnoc⁺ {xs = .[]}        pxs | []       = nothing

+
unsnoc⁺ {xs = .(xs ∷ʳ x)} pxs | xs ∷ʳ′ x = just (∷ʳ⁻ pxs)

+

+
unsnoc⁻ : Maybe.All (All P ⟨×⟩ P) (unsnoc xs) → All P xs

+
unsnoc⁻ {xs = xs} pxs with initLast xs

+
unsnoc⁻ {xs = .[]}        nothing           | []       = []

+
unsnoc⁻ {xs = .(xs ∷ʳ x)} (just (pxs , px)) | xs ∷ʳ′ x = ∷ʳ⁺ pxs px

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith and cartesianProduct

+

+
module _ (S₁ : Setoid a ℓ₁) (S₂ : Setoid b ℓ₂) where

+
  open SetoidMembership S₁ using () renaming (_∈_ to _∈₁_)

+
  open SetoidMembership S₂ using () renaming (_∈_ to _∈₂_)

+

+
  cartesianProductWith⁺ : ∀ f xs ys →

+
                          (∀ {x y} → x ∈₁ xs → y ∈₂ ys → P (f x y)) →

+
                          All P (cartesianProductWith f xs ys)

+
  cartesianProductWith⁺ f []       ys pres = []

+
  cartesianProductWith⁺ f (x ∷ xs) ys pres = ++⁺

+
    (map⁺ (All.tabulateₛ S₂ (pres (here (Setoid.refl S₁)))))

+
    (cartesianProductWith⁺ f xs ys (pres ∘ there))

+

+
  cartesianProduct⁺ : ∀ xs ys →

+
                      (∀ {x y} → x ∈₁ xs → y ∈₂ ys → P (x , y)) →

+
                      All P (cartesianProduct xs ys)

+
  cartesianProduct⁺ = cartesianProductWith⁺ _,_

+

+
------------------------------------------------------------------------

+
-- take and drop

+

+
drop⁺ : ∀ n → All P xs → All P (drop n xs)

+
drop⁺ zero    pxs        = pxs

+
drop⁺ (suc n) []         = []

+
drop⁺ (suc n) (px ∷ pxs) = drop⁺ n pxs

+

+
dropWhile⁺ : (Q? : Decidable Q) → All P xs → All P (dropWhile Q? xs)

+
dropWhile⁺               Q? []         = []

+
dropWhile⁺ {xs = x ∷ xs} Q? (px ∷ pxs) with does (Q? x)

+
... | true  = dropWhile⁺ Q? pxs

+
... | false = px ∷ pxs

+

+
dropWhile⁻ : (P? : Decidable P) → dropWhile P? xs ≡ [] → All P xs

+
dropWhile⁻ {xs = []}     P? eq = []

+
dropWhile⁻ {xs = x ∷ xs} P? eq with P? x

+
... | yes px = px ∷ (dropWhile⁻ P? eq)

+
... | no ¬px = case eq of λ ()

+

+
all-head-dropWhile : (P? : Decidable P) →

+
                     ∀ xs → Maybe.All (∁ P) (head (dropWhile P? xs))

+
all-head-dropWhile P? []       = nothing

+
all-head-dropWhile P? (x ∷ xs) with P? x

+
... | yes px = all-head-dropWhile P? xs

+
... | no ¬px = just ¬px

+

+
take⁺ : ∀ n → All P xs → All P (take n xs)

+
take⁺ zero    pxs        = []

+
take⁺ (suc n) []         = []

+
take⁺ (suc n) (px ∷ pxs) = px ∷ take⁺ n pxs

+

+
takeWhile⁺ : (Q? : Decidable Q) → All P xs → All P (takeWhile Q? xs)

+
takeWhile⁺               Q? []         = []

+
takeWhile⁺ {xs = x ∷ xs} Q? (px ∷ pxs) with does (Q? x)

+
... | true  = px ∷ takeWhile⁺ Q? pxs

+
... | false = []

+

+
takeWhile⁻ : (P? : Decidable P) → takeWhile P? xs ≡ xs → All P xs

+
takeWhile⁻ {xs = []}     P? eq = []

+
takeWhile⁻ {xs = x ∷ xs} P? eq with P? x

+
... | yes px = px ∷ takeWhile⁻ P? (List.∷-injectiveʳ eq)

+
... | no ¬px = case eq of λ ()

+

+
all-takeWhile : (P? : Decidable P) → ∀ xs → All P (takeWhile P? xs)

+
all-takeWhile P? []       = []

+
all-takeWhile P? (x ∷ xs) with P? x

+
... | yes px = px ∷ all-takeWhile P? xs

+
... | no ¬px = []

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
applyUpTo⁺₁ : ∀ f n → (∀ {i} → i < n → P (f i)) → All P (applyUpTo f n)

+
applyUpTo⁺₁ f zero    Pf = []

+
applyUpTo⁺₁ f (suc n) Pf = Pf z<s ∷ applyUpTo⁺₁ (f ∘ suc) n (Pf ∘ s<s)

+

+
applyUpTo⁺₂ : ∀ f n → (∀ i → P (f i)) → All P (applyUpTo f n)

+
applyUpTo⁺₂ f n Pf = applyUpTo⁺₁ f n (λ _ → Pf _)

+

+
applyUpTo⁻ : ∀ f n → All P (applyUpTo f n) → ∀ {i} → i < n → P (f i)

+
applyUpTo⁻ f (suc n) (px ∷ _)   z<s       = px

+
applyUpTo⁻ f (suc n) (_  ∷ pxs) (s<s i<n@(s≤s _)) =

+
  applyUpTo⁻ (f ∘ suc) n pxs i<n

+

+
------------------------------------------------------------------------

+
-- upTo

+

+
all-upTo : ∀ n → All (_< n) (upTo n)

+
all-upTo n = applyUpTo⁺₁ id n id

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
applyDownFrom⁺₁ : ∀ f n → (∀ {i} → i < n → P (f i)) → All P (applyDownFrom f n)

+
applyDownFrom⁺₁ f zero    Pf = []

+
applyDownFrom⁺₁ f (suc n) Pf = Pf ≤-refl ∷ applyDownFrom⁺₁ f n (Pf ∘ m≤n⇒m≤1+n)

+

+
applyDownFrom⁺₂ : ∀ f n → (∀ i → P (f i)) → All P (applyDownFrom f n)

+
applyDownFrom⁺₂ f n Pf = applyDownFrom⁺₁ f n (λ _ → Pf _)

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
tabulate⁺ : ∀ {n} {f : Fin n → A} →

+
            (∀ i → P (f i)) → All P (tabulate f)

+
tabulate⁺ {n = zero}  Pf = []

+
tabulate⁺ {n = suc _} Pf = Pf zero ∷ tabulate⁺ (Pf ∘ suc)

+

+
tabulate⁻ : ∀ {n} {f : Fin n → A} →

+
            All P (tabulate f) → (∀ i → P (f i))

+
tabulate⁻ (px ∷ _) zero    = px

+
tabulate⁻ (_ ∷ pf) (suc i) = tabulate⁻ pf i

+

+
------------------------------------------------------------------------

+
-- remove

+

+
─⁺ : ∀ (p : Any P xs) → All Q xs → All Q (xs Any.─ p)

+
─⁺ (here px) (_ ∷ qs) = qs

+
─⁺ (there p) (q ∷ qs) = q ∷ ─⁺ p qs

+

+
─⁻ : ∀ (p : Any P xs) → Q (Any.lookup p) → All Q (xs Any.─ p) → All Q xs

+
─⁻ (here px) q qs        = q ∷ qs

+
─⁻ (there p) q (q′ ∷ qs) = q′ ∷ ─⁻ p q qs

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (P? : Decidable P) where

+

+
  all-filter : ∀ xs → All P (filter P? xs)

+
  all-filter []       = []

+
  all-filter (x ∷ xs) with P? x

+
  ... | true  because [Px] = invert [Px] ∷ all-filter xs

+
  ... | false because  _   = all-filter xs

+

+
  filter⁺ : All Q xs → All Q (filter P? xs)

+
  filter⁺ {xs = _}     [] = []

+
  filter⁺ {xs = x ∷ _} (Qx ∷ Qxs) with does (P? x)

+
  ... | false = filter⁺ Qxs

+
  ... | true  = Qx ∷ filter⁺ Qxs

+

+
  filter⁻ : All Q (filter P? xs) → All Q (filter (¬? ∘ P?) xs) → All Q xs

+
  filter⁻ {xs = []}          []          []                         = []

+
  filter⁻ {xs = x ∷ _}       all⁺        all⁻ with P? x  | ¬? (P? x)

+
  filter⁻ {xs = x ∷ _}       all⁺        all⁻  | yes  Px | yes  ¬Px = contradiction Px ¬Px

+
  filter⁻ {xs = x ∷ _} (qx ∷ all⁺)       all⁻  | yes  Px | no  ¬¬Px = qx ∷ filter⁻ all⁺ all⁻

+
  filter⁻ {xs = x ∷ _}       all⁺  (qx ∷ all⁻) | no    _ | yes  ¬Px = qx ∷ filter⁻ all⁺ all⁻

+
  filter⁻ {xs = x ∷ _}       all⁺        all⁻  | no  ¬Px | no  ¬¬Px = contradiction ¬Px ¬¬Px

+

+
------------------------------------------------------------------------

+
-- partition

+

+
module _ {P : A → Set p} (P? : Decidable P) where

+

+
  partition-All : ∀ xs → (let ys , zs = partition P? xs) →

+
                  All P ys × All (∁ P) zs

+
  partition-All xs rewrite List.partition-defn P? xs =

+
    all-filter P? xs , all-filter (∁? P?) xs

+

+
------------------------------------------------------------------------

+
-- derun and deduplicate

+

+
module _ {R : A → A → Set q} (R? : B.Decidable R) where

+

+
  derun⁺ : All P xs → All P (derun R? xs)

+
  derun⁺ {xs = []}         []                 = []

+
  derun⁺ {xs = x ∷ []}     (px ∷ [])          = px ∷ []

+
  derun⁺ {xs = x ∷ y ∷ xs} (px ∷ all[P,y∷xs]) with does (R? x y)

+
  ... | false = px ∷ derun⁺ all[P,y∷xs]

+
  ... | true  = derun⁺ all[P,y∷xs]

+

+
  deduplicate⁺ : All P xs → All P (deduplicate R? xs)

+
  deduplicate⁺ []         = []

+
  deduplicate⁺ (px ∷ pxs) = px ∷ filter⁺ (¬? ∘ R? _) (deduplicate⁺ pxs)

+

+
  derun⁻ : P B.Respects (flip R) → ∀ xs → All P (derun R? xs) → All P xs

+
  derun⁻ {P = P} P-resp-R []       []          = []

+
  derun⁻ {P = P} P-resp-R (x ∷ xs) all[P,x∷xs] = aux x xs all[P,x∷xs]

+
    where

+
    aux : ∀ x xs → All P (derun R? (x ∷ xs)) → All P (x ∷ xs)

+
    aux x []       (px ∷ []) = px ∷ []

+
    aux x (y ∷ xs) all[P,x∷y∷xs] with R? x y

+
    aux x (y ∷ xs) all[P,y∷xs]        | yes Rxy

+
      with r@(py ∷ _) ← aux y xs all[P,y∷xs] = P-resp-R Rxy py ∷ r

+
    aux x (y ∷ xs) (px ∷ all[P,y∷xs]) | no _ = px ∷ aux y xs all[P,y∷xs]

+

+
  deduplicate⁻ : P B.Respects R → ∀ xs → All P (deduplicate R? xs) → All P xs

+
  deduplicate⁻ {P = P} resp []       [] = []

+
  deduplicate⁻ {P = P} resp (x ∷ xs) (px ∷ pxs!) =

+
    px ∷ deduplicate⁻ resp xs (filter⁻ (¬? ∘ R? x) pxs! (All.tabulate aux))

+
    where

+
    aux : ∀ {z} → z ∈ filter (¬? ∘ ¬? ∘ R? x) (deduplicate R? xs) → P z

+
    aux {z = z} z∈filter = resp (decidable-stable (R? x z)

+
      (Product.proj₂ (∈-filter⁻ (¬? ∘ ¬? ∘ R? x) {z} {deduplicate R? xs} z∈filter))) px

+

+
------------------------------------------------------------------------

+
-- zipWith

+

+
zipWith⁺ : ∀ (f : A → B → C) → Pointwise (λ x y → P (f x y)) xs ys →

+
           All P (zipWith f xs ys)

+
zipWith⁺ f []              = []

+
zipWith⁺ f (Pfxy ∷ Pfxsys) = Pfxy ∷ zipWith⁺ f Pfxsys

+

+
------------------------------------------------------------------------

+
-- Operations for constructing lists

+
------------------------------------------------------------------------

+
-- fromMaybe

+

+
fromMaybe⁺ : ∀ {mx} → Maybe.All P mx → All P (fromMaybe mx)

+
fromMaybe⁺ (just px) = px ∷ []

+
fromMaybe⁺ nothing   = []

+

+
fromMaybe⁻ : ∀ mx → All P (fromMaybe mx) → Maybe.All P mx

+
fromMaybe⁻ (just x) (px ∷ []) = just px

+
fromMaybe⁻ nothing  p         = nothing

+

+
------------------------------------------------------------------------

+
-- replicate

+

+
replicate⁺ : ∀ n → P x → All P (replicate n x)

+
replicate⁺ zero    px = []

+
replicate⁺ (suc n) px = px ∷ replicate⁺ n px

+

+
replicate⁻ : ∀ {n} → All P (replicate (suc n) x) → P x

+
replicate⁻ (px ∷ _) = px

+

+
------------------------------------------------------------------------

+
-- inits

+

+
inits⁺ : All P xs → All (All P) (inits xs)

+
inits⁺ []         = [] ∷ []

+
inits⁺ (px ∷ pxs) = [] ∷ gmap⁺ (px ∷_) (inits⁺ pxs)

+

+
inits⁻ : ∀ xs → All (All P) (inits xs) → All P xs

+
inits⁻ []               pxs                   = []

+
inits⁻ (x ∷ [])         ([] ∷ p[x] ∷ [])      = p[x]

+
inits⁻ (x ∷ xs@(_ ∷ _)) ([] ∷ pxs@(p[x] ∷ _)) =

+
  singleton⁻ p[x] ∷ inits⁻ xs (All.map (drop⁺ 1) (map⁻ pxs))

+

+
------------------------------------------------------------------------

+
-- tails

+

+
tails⁺ : All P xs → All (All P) (tails xs)

+
tails⁺ []             = [] ∷ []

+
tails⁺ pxxs@(_ ∷ pxs) = pxxs ∷ tails⁺ pxs

+

+
tails⁻ : ∀ xs → All (All P) (tails xs) → All P xs

+
tails⁻ []       pxs        = []

+
tails⁻ (x ∷ xs) (pxxs ∷ _) = pxxs

+

+
------------------------------------------------------------------------

+
-- all

+

+
module _ (p : A → Bool) where

+

+
  all⁺ : ∀ xs → T (all p xs) → All (T ∘ p) xs

+
  all⁺ []       _      = []

+
  all⁺ (x ∷ xs) px∷pxs =

+
    let px , pxs = Equivalence.to (T-∧ {p x}) px∷pxs

+
    in px ∷ all⁺ xs pxs

+

+
  all⁻ : All (T ∘ p) xs → T (all p xs)

+
  all⁻ []         = _

+
  all⁻ (px ∷ pxs) = Equivalence.from T-∧ (px , all⁻ pxs)

+

+
------------------------------------------------------------------------

+
-- All is anti-monotone.

+

+
anti-mono : xs ⊆ ys → All P ys → All P xs

+
anti-mono xs⊆ys pys = All.tabulate (lookup pys ∘ xs⊆ys)

+

+
all-anti-mono : ∀ (p : A → Bool) → xs ⊆ ys → T (all p ys) → T (all p xs)

+
all-anti-mono p xs⊆ys = all⁻ p ∘ anti-mono xs⊆ys ∘ all⁺ p _

+

+
------------------------------------------------------------------------

+
-- Interactions with pointwise equality

+
------------------------------------------------------------------------

+

+
module _ (S : Setoid c ℓ) where

+

+
  open Setoid S

+
  open ≋ S

+

+
  respects : P B.Respects _≈_ → (All P) B.Respects _≋_

+
  respects p≈ []            []         = []

+
  respects p≈ (x≈y ∷ xs≈ys) (px ∷ pxs) = p≈ x≈y px ∷ respects p≈ xs≈ys pxs

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.3

+

+
Any¬→¬All = Any¬⇒¬All

+
{-# WARNING_ON_USAGE Any¬→¬All

+
"Warning: Any¬→¬All was deprecated in v1.3.

+
Please use Any¬⇒¬All instead."

+
#-}

+

+
-- Version 2.0

+

+
updateAt-id-relative = updateAt-id-local

+
{-# WARNING_ON_USAGE updateAt-id-relative

+
"Warning: updateAt-id-relative was deprecated in v2.0.

+
Please use updateAt-id-local instead."

+
#-}

+

+
updateAt-compose-relative = updateAt-∘-local

+
{-# WARNING_ON_USAGE updateAt-compose-relative

+
"Warning: updateAt-compose-relative was deprecated in v2.0.

+
Please use updateAt-∘-local instead."

+
#-}

+

+
updateAt-compose = updateAt-∘

+
{-# WARNING_ON_USAGE updateAt-compose

+
"Warning: updateAt-compose was deprecated in v2.0.

+
Please use updateAt-∘ instead."

+
#-}

+

+
updateAt-cong-relative = updateAt-cong-local

+
{-# WARNING_ON_USAGE updateAt-cong-relative

+
"Warning: updateAt-cong-relative was deprecated in v2.0.

+
Please use updateAt-cong-local instead."

+
#-}

+

+
gmap = gmap⁺

+
{-# WARNING_ON_USAGE gmap

+
"Warning: gmap was deprecated in v2.0.

+
Please use gmap⁺ instead."

+
#-}

+

+
-- Version 2.1

+

+
map-compose = map-∘

+
{-# WARNING_ON_USAGE map-compose

+
"Warning: map-compose was deprecated in v2.1.

+
Please use map-∘ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.All.html b/site/static/agda_html/Data.List.Relation.Unary.All.html
new file mode 100644
index 000000000..465c4d933
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.All.html
@@ -0,0 +1,324 @@
+
+Data.List.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists where all elements satisfy a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.All where

+

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.List.Membership.Propositional renaming (_∈_ to _∈ₚ_)

+
import Data.List.Membership.Setoid as SetoidMembership

+
open import Data.Product.Base as Product

+
  using (∃; -,_; _×_; _,_; proj₁; proj₂; uncurry)

+
open import Data.Sum.Base as Sum using (inj₁; inj₂)

+
open import Effect.Applicative

+
open import Effect.Monad

+
open import Function.Base using (_∘_; _∘′_; id; const)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary hiding (Irrelevant)

+
import Relation.Nullary.Decidable as Dec

+
open import Relation.Unary hiding (_∈_)

+
import Relation.Unary.Properties as Unary

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (_Respects_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
private

+
  variable

+
    a b p q r ℓ : Level

+
    A : Set a

+
    B : Set b

+
    P Q R : Pred A p

+
    x : A

+
    xs : List A

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
-- Given a predicate P, then All P xs means that every element in xs

+
-- satisfies P. See `Relation.Unary` for an explanation of predicates.

+

+
infixr 5 _∷_

+

+
data All {A : Set a} (P : Pred A p) : Pred (List A) (a ⊔ p) where

+
  []  : All P []

+
  _∷_ : ∀ {x xs} (px : P x) (pxs : All P xs) → All P (x ∷ xs)

+

+
-- All P xs is a finite map from indices x ∈ xs to content P x.

+
-- Relation pxs [ i ]= px states that, in map pxs, key i : x ∈ xs points

+
-- to value px.

+

+
infix 4 _[_]=_

+

+
data _[_]=_ {A : Set a} {P : Pred A p} :

+
            ∀ {x xs} → All P xs → x ∈ₚ xs → P x → Set (a ⊔ p) where

+

+
  here  : ∀ {x xs} {px : P x} {pxs : All P xs} →

+
          px ∷ pxs [ here refl ]= px

+

+
  there : ∀ {x xs y} {px : P x} {pxs : All P xs} {py : P y} {i : x ∈ₚ xs} →

+
          pxs [ i ]= px →

+
          py ∷ pxs [ there i ]= px

+

+
-- A list is empty if having an element is impossible.

+

+
Null : Pred (List A) _

+
Null = All ∅

+

+
------------------------------------------------------------------------

+
-- Operations on All

+

+
uncons : All P (x ∷ xs) → P x × All P xs

+
uncons (px ∷ pxs) = px , pxs

+

+
head : All P (x ∷ xs) → P x

+
head = proj₁ ∘ uncons

+

+
tail : All P (x ∷ xs) → All P xs

+
tail = proj₂ ∘ uncons

+

+
reduce : (f : ∀ {x} → P x → B) → All P xs → List B

+
reduce f []         = []

+
reduce f (px ∷ pxs) = f px ∷ reduce f pxs

+

+
construct : (f : B → ∃ P) (xs : List B) → ∃ (All P)

+
construct f []       = [] , []

+
construct f (x ∷ xs) = Product.zip _∷_ _∷_ (f x) (construct f xs)

+

+
fromList : (xs : List (∃ P)) → All P (List.map proj₁ xs)

+
fromList []              = []

+
fromList ((x , p) ∷ xps) = p ∷ fromList xps

+

+
toList : All P xs → List (∃ P)

+
toList pxs = reduce (λ {x} px → x , px) pxs

+

+
map : P ⊆ Q → All P ⊆ All Q

+
map g []         = []

+
map g (px ∷ pxs) = g px ∷ map g pxs

+

+
zipWith : P ∩ Q ⊆ R → All P ∩ All Q ⊆ All R

+
zipWith f ([] , [])             = []

+
zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)

+

+
unzipWith : R ⊆ P ∩ Q → All R ⊆ All P ∩ All Q

+
unzipWith f []         = [] , []

+
unzipWith f (rx ∷ rxs) = Product.zip _∷_ _∷_ (f rx) (unzipWith f rxs)

+

+
zip : All P ∩ All Q ⊆ All (P ∩ Q)

+
zip = zipWith id

+

+
unzip : All (P ∩ Q) ⊆ All P ∩ All Q

+
unzip = unzipWith id

+

+
module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where

+
  open Setoid S renaming (refl to ≈-refl)

+
  open SetoidMembership S

+

+
  tabulateₛ : (∀ {x} → x ∈ xs → P x) → All P xs

+
  tabulateₛ {[]}    hyp = []

+
  tabulateₛ {_ ∷ _} hyp = hyp (here ≈-refl) ∷ tabulateₛ (hyp ∘ there)

+

+
tabulate : (∀ {x} → x ∈ₚ xs → P x) → All P xs

+
tabulate = tabulateₛ (≡.setoid _)

+

+
self : ∀ {xs} → All (const A) xs

+
self = tabulate (λ {x} _ → x)

+

+
------------------------------------------------------------------------

+
-- (weak) updateAt

+

+
infixl 6 _[_]%=_ _[_]≔_

+

+
updateAt : x ∈ₚ xs → (P x → P x) → All P xs → All P xs

+
updateAt () f []

+
updateAt (here refl) f (px ∷ pxs) = f px ∷ pxs

+
updateAt (there i)   f (px ∷ pxs) =   px ∷ updateAt i f pxs

+

+
_[_]%=_ : All P xs → x ∈ₚ xs → (P x → P x) → All P xs

+
pxs [ i ]%= f = updateAt i f pxs

+

+
_[_]≔_ : All P xs → x ∈ₚ xs → P x → All P xs

+
pxs [ i ]≔ px = pxs [ i ]%= const px

+

+
------------------------------------------------------------------------

+
-- Traversable-like functions

+

+
module _ (p : Level) {A : Set a} {P : Pred A (a ⊔ p)}

+
         {F : Set (a ⊔ p) → Set (a ⊔ p)}

+
         (App : RawApplicative F)

+
         where

+

+
  open RawApplicative App

+

+
  sequenceA : All (F ∘′ P) ⊆ F ∘′ All P

+
  sequenceA []       = pure []

+
  sequenceA (x ∷ xs) = _∷_ <$> x <*> sequenceA xs

+

+
  mapA : ∀ {Q : Pred A q} → (Q ⊆ F ∘′ P) → All Q ⊆ (F ∘′ All P)

+
  mapA f = sequenceA ∘′ map f

+

+
  forA : ∀ {Q : Pred A q} → All Q xs → (Q ⊆ F ∘′ P) → F (All P xs)

+
  forA qxs f = mapA f qxs

+

+
module _ (p : Level) {A : Set a} {P : Pred A (a ⊔ p)}

+
         {M : Set (a ⊔ p) → Set (a ⊔ p)}

+
         (Mon : RawMonad M)

+
         where

+

+
  private App = RawMonad.rawApplicative Mon

+

+
  sequenceM : All (M ∘′ P) ⊆ M ∘′ All P

+
  sequenceM = sequenceA p App

+

+
  mapM : ∀ {Q : Pred A q} → (Q ⊆ M ∘′ P) → All Q ⊆ (M ∘′ All P)

+
  mapM = mapA p App

+

+
  forM : ∀ {Q : Pred A q} → All Q xs → (Q ⊆ M ∘′ P) → M (All P xs)

+
  forM = forA p App

+

+
------------------------------------------------------------------------

+
-- Generalised lookup based on a proof of Any

+

+
lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)

+
lookupAny (px ∷ pxs) (here qx) = px , qx

+
lookupAny (px ∷ pxs) (there i) = lookupAny pxs i

+

+
lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)

+
lookupWith f pxs i = Product.uncurry f (lookupAny pxs i)

+

+
lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)

+
lookup pxs = lookupWith (λ { px refl → px }) pxs

+

+
module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where

+
  open Setoid S renaming (sym to ≈-sym)

+
  open SetoidMembership S

+

+
  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)

+
  lookupₛ resp pxs = lookupWith (λ py x≈y → resp (≈-sym x≈y) py) pxs

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by All

+

+
all? : Decidable P → Decidable (All P)

+
all? p []       = yes []

+
all? p (x ∷ xs) = Dec.map′ (uncurry _∷_) uncons (p x ×-dec all? p xs)

+

+
universal : Universal P → Universal (All P)

+
universal u []       = []

+
universal u (x ∷ xs) = u x ∷ universal u xs

+

+
universal-U : Universal (All {A = A} U)

+
universal-U = universal Unary.U-Universal

+

+
irrelevant : Irrelevant P → Irrelevant (All P)

+
irrelevant irr []           []           = ≡.refl

+
irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =

+
  ≡.cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)

+

+
satisfiable : Satisfiable (All P)

+
satisfiable = [] , []

+

+
------------------------------------------------------------------------

+
-- Generalised decidability procedure

+

+
decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]

+
decide p∪q [] = inj₁ []

+
decide p∪q (x ∷ xs) with p∪q x

+
... | inj₂ qx = inj₂ (here qx)

+
... | inj₁ px = Sum.map (px ∷_) there (decide p∪q xs)

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.4

+

+
all = all?

+
{-# WARNING_ON_USAGE all

+
"Warning: all was deprecated in v1.4.

+
Please use all? instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.AllPairs.Core.html b/site/static/agda_html/Data.List.Relation.Unary.AllPairs.Core.html
new file mode 100644
index 000000000..10c507259
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.AllPairs.Core.html
@@ -0,0 +1,111 @@
+
+Data.List.Relation.Unary.AllPairs.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists where every pair of elements are related (symmetrically)

+
------------------------------------------------------------------------

+

+
-- Core modules are not meant to be used directly outside of the

+
-- standard library.

+

+
-- This module should be removable if and when Agda issue

+
-- https://github.com/agda/agda/issues/3210 is fixed

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+

+
module Data.List.Relation.Unary.AllPairs.Core

+
  {a ℓ} {A : Set a} (R : Rel A ℓ) where

+

+
open import Level

+
open import Data.List.Base

+
open import Data.List.Relation.Unary.All

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- AllPairs R xs means that every pair of elements (x , y) in xs is a

+
-- member of relation R (as long as x comes before y in the list).

+

+
infixr 5 _∷_

+

+
data AllPairs : List A → Set (a ⊔ ℓ) where

+
  []  : AllPairs []

+
  _∷_ : ∀ {x xs} → All (R x) xs → AllPairs xs → AllPairs (x ∷ xs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.AllPairs.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.AllPairs.Properties.html
new file mode 100644
index 000000000..9d54ed783
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.AllPairs.Properties.html
@@ -0,0 +1,228 @@
+
+Data.List.Relation.Unary.AllPairs.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to AllPairs

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.AllPairs.Properties where

+

+
open import Data.List.Base using (List; []; _∷_; map; _++_; concat; take; drop;

+
  applyUpTo; applyDownFrom; tabulate; filter; catMaybes)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.All.Properties as All using (Any-catMaybes⁺)

+
open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)

+
open import Data.Bool.Base using (true; false)

+
open import Data.Maybe.Base using (Maybe; nothing; just)

+
open import Data.Maybe.Relation.Binary.Pointwise using (pointwise⊆any; Pointwise)

+
open import Data.Fin.Base as F using (Fin)

+
open import Data.Fin.Properties using (suc-injective; <⇒≢)

+
open import Data.Nat.Base using (zero; suc; _<_; z≤n; s≤s; z<s; s<s)

+
open import Data.Nat.Properties using (≤-refl; m<n⇒m<1+n)

+
open import Function.Base using (_∘_; flip)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (DecSetoid)

+
open import Relation.Binary.PropositionalEquality.Core using (_≢_)

+
open import Relation.Unary using (Pred; Decidable; _⊆_)

+
open import Relation.Nullary.Decidable.Core using (does)

+

+
private

+
  variable

+
    a b c p ℓ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {R : Rel A ℓ} {f : B → A} where

+

+
  map⁺ : ∀ {xs} → AllPairs (λ x y → R (f x) (f y)) xs →

+
         AllPairs R (map f xs)

+
  map⁺ []           = []

+
  map⁺ (x∉xs ∷ xs!) = All.map⁺ x∉xs ∷ map⁺ xs!

+

+
------------------------------------------------------------------------

+
-- ++

+

+
module _ {R : Rel A ℓ} where

+

+
  ++⁺ : ∀ {xs ys} → AllPairs R xs → AllPairs R ys →

+
        All (λ x → All (R x) ys) xs → AllPairs R (xs ++ ys)

+
  ++⁺ []         Rys _              = Rys

+
  ++⁺ (px ∷ Rxs) Rys (Rxys ∷ Rxsys) = All.++⁺ px Rxys ∷ ++⁺ Rxs Rys Rxsys

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {R : Rel A ℓ} where

+

+
  concat⁺ : ∀ {xss} → All (AllPairs R) xss →

+
            AllPairs (λ xs ys → All (λ x → All (R x) ys) xs) xss →

+
            AllPairs R (concat xss)

+
  concat⁺ []           []              = []

+
  concat⁺ (pxs ∷ pxss) (Rxsxss ∷ Rxss) = ++⁺ pxs (concat⁺ pxss Rxss)

+
    (All.map All.concat⁺ (All.All-swap Rxsxss))

+

+
------------------------------------------------------------------------

+
-- take and drop

+

+
module _ {R : Rel A ℓ} where

+

+
  take⁺ : ∀ {xs} n → AllPairs R xs → AllPairs R (take n xs)

+
  take⁺ zero    pxs        = []

+
  take⁺ (suc n) []         = []

+
  take⁺ (suc n) (px ∷ pxs) = All.take⁺ n px ∷ take⁺ n pxs

+

+
  drop⁺ : ∀ {xs} n → AllPairs R xs → AllPairs R (drop n xs)

+
  drop⁺ zero    pxs       = pxs

+
  drop⁺ (suc n) []        = []

+
  drop⁺ (suc n) (_ ∷ pxs) = drop⁺ n pxs

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ {R : Rel A ℓ} where

+

+
  applyUpTo⁺₁ : ∀ f n → (∀ {i j} → i < j → j < n → R (f i) (f j)) → AllPairs R (applyUpTo f n)

+
  applyUpTo⁺₁ f zero    Rf = []

+
  applyUpTo⁺₁ f (suc n) Rf =

+
    All.applyUpTo⁺₁ _ n (Rf (s≤s z≤n) ∘ s≤s) ∷

+
    applyUpTo⁺₁ _ n (λ i≤j j<n → Rf (s≤s i≤j) (s≤s j<n))

+

+
  applyUpTo⁺₂ : ∀ f n → (∀ i j → R (f i) (f j)) → AllPairs R (applyUpTo f n)

+
  applyUpTo⁺₂ f n Rf = applyUpTo⁺₁ f n (λ _ _ → Rf _ _)

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ {R : Rel A ℓ} where

+

+
  applyDownFrom⁺₁ : ∀ f n → (∀ {i j} → j < i → i < n → R (f i) (f j)) →

+
                    AllPairs R (applyDownFrom f n)

+
  applyDownFrom⁺₁ f zero    Rf = []

+
  applyDownFrom⁺₁ f (suc n) Rf =

+
    All.applyDownFrom⁺₁ _ n (flip Rf ≤-refl)  ∷

+
    applyDownFrom⁺₁ f n (λ j<i i<n → Rf j<i (m<n⇒m<1+n i<n))

+

+
  applyDownFrom⁺₂ : ∀ f n → (∀ i j → R (f i) (f j)) → AllPairs R (applyDownFrom f n)

+
  applyDownFrom⁺₂ f n Rf = applyDownFrom⁺₁ f n (λ _ _ → Rf _ _)

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {R : Rel A ℓ} where

+

+
  tabulate⁺-< : ∀ {n} {f : Fin n → A} → (∀ {i j} → i F.< j → R (f i) (f j)) →

+
              AllPairs R (tabulate f)

+
  tabulate⁺-< {zero}  fᵢ~fⱼ = []

+
  tabulate⁺-< {suc n} fᵢ~fⱼ =

+
    All.tabulate⁺ (λ _ → fᵢ~fⱼ z<s) ∷

+
    tabulate⁺-< (fᵢ~fⱼ ∘ s<s)

+

+
  tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → i ≢ j → R (f i) (f j)) →

+
              AllPairs R (tabulate f)

+
  tabulate⁺ fᵢ~fⱼ = tabulate⁺-< (fᵢ~fⱼ ∘ <⇒≢)

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {R : Rel A ℓ} {P : Pred A p} (P? : Decidable P) where

+

+
  filter⁺ : ∀ {xs} → AllPairs R xs → AllPairs R (filter P? xs)

+
  filter⁺ {_}      []           = []

+
  filter⁺ {x ∷ xs} (x∉xs ∷ xs!) with does (P? x)

+
  ... | false = filter⁺ xs!

+
  ... | true  = All.filter⁺ P? x∉xs ∷ filter⁺ xs!

+

+
------------------------------------------------------------------------

+
-- catMaybes

+

+
module _ {R : Rel A ℓ} where

+

+
  catMaybes⁺ : {xs : List (Maybe A)} → AllPairs (Pointwise R) xs → AllPairs R (catMaybes xs)

+
  catMaybes⁺ {xs = []} [] = []

+
  catMaybes⁺ {xs = nothing ∷  _} (x∼xs ∷ pxs) = catMaybes⁺ pxs

+
  catMaybes⁺ {xs = just x  ∷ xs} (x∼xs ∷ pxs) = Any-catMaybes⁺ (All.map pointwise⊆any x∼xs) ∷ catMaybes⁺ pxs

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.AllPairs.html b/site/static/agda_html/Data.List.Relation.Unary.AllPairs.html
new file mode 100644
index 000000000..4dc011208
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.AllPairs.html
@@ -0,0 +1,157 @@
+
+Data.List.Relation.Unary.AllPairs
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists where every pair of elements are related (symmetrically)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel; _⇒_)

+

+
module Data.List.Relation.Unary.AllPairs

+
       {a ℓ} {A : Set a} {R : Rel A ℓ} where

+

+
open import Data.List.Base using (List; []; _∷_)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)

+
open import Function.Base using (id; _∘_)

+
open import Level using (_⊔_)

+
open import Relation.Binary.Definitions as B

+
open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)

+
open import Relation.Binary.PropositionalEquality.Core using (refl; cong₂)

+
open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)

+
open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open import Data.List.Relation.Unary.AllPairs.Core public

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
head : ∀ {x xs} → AllPairs R (x ∷ xs) → All (R x) xs

+
head (px ∷ pxs) = px

+

+
tail : ∀ {x xs} → AllPairs R (x ∷ xs) → AllPairs R xs

+
tail (px ∷ pxs) = pxs

+

+
uncons : ∀ {x xs} → AllPairs R (x ∷ xs) → All (R x) xs × AllPairs R xs

+
uncons = < head , tail >

+

+
module _ {q} {S : Rel A q} where

+

+
  map : R ⇒ S → AllPairs R ⊆ AllPairs S

+
  map ~₁⇒~₂ [] = []

+
  map ~₁⇒~₂ (x~xs ∷ pxs) = All.map ~₁⇒~₂ x~xs ∷ (map ~₁⇒~₂ pxs)

+

+
module _ {s t} {S : Rel A s} {T : Rel A t} where

+

+
  zipWith : R ∩ᵇ S ⇒ T → AllPairs R ∩ᵘ AllPairs S ⊆ AllPairs T

+
  zipWith f ([] , [])             = []

+
  zipWith f (px ∷ pxs , qx ∷ qxs) = All.zipWith f (px , qx) ∷ zipWith f (pxs , qxs)

+

+
  unzipWith : T ⇒ R ∩ᵇ S → AllPairs T ⊆ AllPairs R ∩ᵘ AllPairs S

+
  unzipWith f []         = [] , []

+
  unzipWith f (rx ∷ rxs) = Prod.zip _∷_ _∷_ (All.unzipWith f rx) (unzipWith f rxs)

+

+
module _ {s} {S : Rel A s} where

+

+
  zip : AllPairs R ∩ᵘ AllPairs S ⊆ AllPairs (R ∩ᵇ S)

+
  zip = zipWith id

+

+
  unzip : AllPairs (R ∩ᵇ S) ⊆ AllPairs R ∩ᵘ AllPairs S

+
  unzip = unzipWith id

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by AllPairs

+

+
allPairs? : B.Decidable R → U.Decidable (AllPairs R)

+
allPairs? R? []       = yes []

+
allPairs? R? (x ∷ xs) =

+
  Dec.map′ (uncurry _∷_) uncons (All.all? (R? x) xs ×-dec allPairs? R? xs)

+

+
irrelevant : B.Irrelevant R → U.Irrelevant (AllPairs R)

+
irrelevant irr []           []           = refl

+
irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =

+
  cong₂ _∷_ (All.irrelevant irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)

+

+
satisfiable : U.Satisfiable (AllPairs R)

+
satisfiable = [] , []

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Any.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Any.Properties.html
new file mode 100644
index 000000000..39223ac54
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Any.Properties.html
@@ -0,0 +1,826 @@
+
+Data.List.Relation.Unary.Any.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to Any

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Any.Properties where

+

+
open import Data.Bool.Base using (Bool; false; true; T)

+
open import Data.Bool.Properties using (T-∨; T-≡)

+
open import Data.Empty using (⊥)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.List.Base as List hiding (find)

+
open import Data.List.Effectful as List using (monad)

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.List.Membership.Propositional

+
open import Data.List.Membership.Propositional.Properties.Core

+
  using (Any↔; find∘map; map∘find; lose∘find)

+
open import Data.List.Relation.Binary.Pointwise

+
  using (Pointwise; []; _∷_)

+
open import Data.Nat.Base using (zero; suc; _<_; z<s; s<s; s≤s)

+
open import Data.Nat.Properties using (_≟_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1+n)

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Data.Maybe.Relation.Unary.Any as MAny using (just)

+
open import Data.Product.Base as Product

+
  using (_×_; _,_; ∃; ∃₂; proj₁; proj₂)

+
open import Data.Product.Function.NonDependent.Propositional

+
  using (_×-cong_)

+
import Data.Product.Function.Dependent.Propositional as Σ

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Data.Sum.Function.Propositional using (_⊎-cong_)

+
open import Effect.Monad using (RawMonad)

+
open import Function.Base using (_$_; _∘_; flip; const; id; _∘′_)

+
open import Function.Bundles

+
import Function.Properties.Inverse as Inverse

+
open import Function.Related.Propositional as Related using (Kind; Related)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel; REL)

+
open import Relation.Binary.Definitions as B

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; sym; trans; cong; cong₂; resp)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+
open import Relation.Unary as U

+
  using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)

+
open import Relation.Nullary.Decidable.Core

+
  using (¬?; _because_; does; yes; no; decidable-stable)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (invert)

+

+
private

+
  open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})

+

+
private

+
  variable

+
    a b c p q r ℓ : Level

+
    A B C : Set a

+
    P Q R : Pred A p

+
    x y : A

+
    xs ys : List A

+

+
------------------------------------------------------------------------

+
-- Equality properties

+

+
lift-resp : ∀ {_≈_ : Rel A ℓ} → P Respects _≈_ →

+
            (Any P) Respects (Pointwise _≈_)

+
lift-resp resp (x≈y ∷ xs≈ys) (here px)   = here (resp x≈y px)

+
lift-resp resp (x≈y ∷ xs≈ys) (there pxs) = there (lift-resp resp xs≈ys pxs)

+

+
here-injective : ∀ {p q : P x} → here {P = P} {xs = xs} p ≡ here q → p ≡ q

+
here-injective refl = refl

+

+
there-injective : ∀ {p q : Any P xs} → there {x = x} p ≡ there q → p ≡ q

+
there-injective refl = refl

+

+
------------------------------------------------------------------------

+
-- Misc

+

+
¬Any[] : ¬ Any P []

+
¬Any[] ()

+

+
------------------------------------------------------------------------

+
-- Any is a congruence

+

+
Any-cong : ∀ {k : Kind} → (∀ x → Related k (P x) (Q x)) →

+
           (∀ {z} → Related k (z ∈ xs) (z ∈ ys)) →

+
           Related k (Any P xs) (Any Q ys)

+
Any-cong {P = P} {Q = Q} {xs = xs} {ys} P↔Q xs≈ys =

+
  Any P xs                ↔⟨ Related.SK-sym Any↔ ⟩

+
  (∃ λ x → x ∈ xs × P x)  ∼⟨ Σ.cong Inverse.↔-refl (xs≈ys ×-cong P↔Q _) ⟩

+
  (∃ λ x → x ∈ ys × Q x)  ↔⟨ Any↔ ⟩

+
  Any Q ys                ∎

+
  where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- Any.map

+

+
map-cong : (f g : P ⋐ Q) → (∀ {x} (p : P x) → f p ≡ g p) →

+
          (p : Any P xs) → Any.map f p ≡ Any.map g p

+
map-cong f g hyp (here  p) = cong here (hyp p)

+
map-cong f g hyp (there p) = cong there $ map-cong f g hyp p

+

+
map-id : ∀ (f : P ⋐ P) → (∀ {x} (p : P x) → f p ≡ p) →

+
         (p : Any P xs) → Any.map f p ≡ p

+
map-id f hyp (here  p) = cong here (hyp p)

+
map-id f hyp (there p) = cong there $ map-id f hyp p

+

+
map-∘ : ∀ (f : Q ⋐ R) (g : P ⋐ Q) (p : Any P xs) →

+
        Any.map (f ∘ g) p ≡ Any.map f (Any.map g p)

+
map-∘ f g (here  p) = refl

+
map-∘ f g (there p) = cong there $ map-∘ f g p

+

+
------------------------------------------------------------------------

+
-- Any.lookup

+

+
lookup-result : ∀ (p : Any P xs) → P (Any.lookup p)

+
lookup-result (here px) = px

+
lookup-result (there p) = lookup-result p

+

+
lookup-index : ∀ (p : Any P xs) → P (lookup xs (Any.index p))

+
lookup-index (here px)   = px

+
lookup-index (there pxs) = lookup-index pxs

+

+
------------------------------------------------------------------------

+
-- Swapping

+

+
-- Nested occurrences of Any can sometimes be swapped. See also ×↔.

+

+
module _ {R : REL A B ℓ} where

+

+
  swap : Any (λ x → Any (R x) ys) xs → Any (λ y → Any (flip R y) xs) ys

+
  swap (here  pys)  = Any.map here pys

+
  swap (there pxys) = Any.map there (swap pxys)

+

+
  swap-there : (any : Any (λ x → Any (R x) ys) xs) →

+
               swap (Any.map (there {x = x}) any) ≡ there (swap any)

+
  swap-there (here  pys)  = refl

+
  swap-there (there pxys) = cong (Any.map there) (swap-there pxys)

+

+
module _ {R : REL A B ℓ} where

+

+
  swap-invol : (any : Any (λ x → Any (R x) ys) xs) →

+
               swap (swap any) ≡ any

+
  swap-invol (here (here px))   = refl

+
  swap-invol (here (there pys)) =

+
    cong (Any.map there) (swap-invol (here pys))

+
  swap-invol (there pxys)       =

+
    trans (swap-there (swap pxys)) (cong there (swap-invol pxys))

+

+
module _ {R : REL A B ℓ} where

+

+
  swap↔ : Any (λ x → Any (R x) ys) xs ↔ Any (λ y → Any (flip R y) xs) ys

+
  swap↔ = mk↔ₛ′ swap swap swap-invol swap-invol

+

+
------------------------------------------------------------------------

+
-- Lemmas relating Any to ⊥

+

+
⊥↔Any⊥ : ⊥ ↔ Any (const ⊥) xs

+
⊥↔Any⊥ = mk↔ₛ′ (λ()) (λ p → from p) (λ p → from p) (λ())

+
  where

+
  from : Any (const ⊥) xs → B

+
  from (there p) = from p

+

+
⊥↔Any[] : ⊥ ↔ Any P []

+
⊥↔Any[] = mk↔ₛ′ (λ()) (λ()) (λ()) (λ())

+

+
------------------------------------------------------------------------

+
-- Lemmas relating Any to ⊤

+

+
-- These introduction and elimination rules are not inverses, though.

+

+
any⁺ : ∀ (p : A → Bool) → Any (T ∘ p) xs → T (any p xs)

+
any⁺ p (here  px)          = Equivalence.from T-∨ (inj₁ px)

+
any⁺ p (there {x = x} pxs) with p x

+
... | true  = _

+
... | false = any⁺ p pxs

+

+
any⁻ : ∀ (p : A → Bool) xs → T (any p xs) → Any (T ∘ p) xs

+
any⁻ p (x ∷ xs) px∷xs with p x in eq

+
... | true  = here (Equivalence.from T-≡ eq)

+
... | false = there (any⁻ p xs px∷xs)

+

+
any⇔ : ∀ {p : A → Bool} → Any (T ∘ p) xs ⇔ T (any p xs)

+
any⇔ = mk⇔ (any⁺ _) (any⁻ _ _)

+

+
------------------------------------------------------------------------

+
-- Sums commute with Any

+

+
Any-⊎⁺ : Any P xs ⊎ Any Q xs → Any (λ x → P x ⊎ Q x) xs

+
Any-⊎⁺ = [ Any.map inj₁ , Any.map inj₂ ]′

+

+
Any-⊎⁻ : Any (λ x → P x ⊎ Q x) xs → Any P xs ⊎ Any Q xs

+
Any-⊎⁻ (here (inj₁ p)) = inj₁ (here p)

+
Any-⊎⁻ (here (inj₂ q)) = inj₂ (here q)

+
Any-⊎⁻ (there p)       = Sum.map there there (Any-⊎⁻ p)

+

+
⊎↔ : (Any P xs ⊎ Any Q xs) ↔ Any (λ x → P x ⊎ Q x) xs

+
⊎↔ {P = P} {Q = Q} = mk↔ₛ′ Any-⊎⁺ Any-⊎⁻ to∘from from∘to

+
  where

+
  from∘to : (p : Any P xs ⊎ Any Q xs) → Any-⊎⁻ (Any-⊎⁺ p) ≡ p

+
  from∘to (inj₁ (here  p)) = refl

+
  from∘to (inj₁ (there p)) rewrite from∘to (inj₁ p) = refl

+
  from∘to (inj₂ (here  q)) = refl

+
  from∘to (inj₂ (there q)) rewrite from∘to (inj₂ q) = refl

+

+
  to∘from : (p : Any (λ x → P x ⊎ Q x) xs) → Any-⊎⁺ (Any-⊎⁻ p) ≡ p

+
  to∘from (here (inj₁ p)) = refl

+
  to∘from (here (inj₂ q)) = refl

+
  to∘from (there p) with Any-⊎⁻ p | to∘from p

+
  ... | inj₁ p | refl = refl

+
  ... | inj₂ q | refl = refl

+

+
------------------------------------------------------------------------

+
-- Products "commute" with Any.

+

+
Any-×⁺ : Any P xs × Any Q ys → Any (λ x → Any (λ y → P x × Q y) ys) xs

+
Any-×⁺ (p , q) = Any.map (λ p → Any.map (λ q → (p , q)) q) p

+

+
Any-×⁻ : Any (λ x → Any (λ y → P x × Q y) ys) xs →

+
         Any P xs × Any Q ys

+
Any-×⁻ pq = let x , x∈xs , pq′ = find pq in

+
            let y , y∈ys , p , q = find pq′ in

+
            lose x∈xs p , lose y∈ys q

+

+
×↔ : ∀ {xs ys} →

+
     (Any P xs × Any Q ys) ↔ Any (λ x → Any (λ y → P x × Q y) ys) xs

+
×↔ {P = P} {Q = Q} {xs} {ys} = mk↔ₛ′ Any-×⁺ Any-×⁻ to∘from from∘to

+
  where

+
  open ≡-Reasoning

+

+
  from∘to : ∀ pq → Any-×⁻ (Any-×⁺ pq) ≡ pq

+
  from∘to (p , q) = let x , x∈xs , px = find p in

+

+
    Any-×⁻ (Any-×⁺ (p , q))

+

+
     ≡⟨⟩

+

+
    (let (x , x∈xs , pq)    = find (Any-×⁺ (p , q))

+
         (y , y∈ys , p , q) = find pq

+
     in  lose x∈xs p , lose y∈ys q)

+

+
     ≡⟨ cong (λ • → let (x , x∈xs , pq)    = •

+
                        (y , y∈ys , p , q) = find pq

+
                    in  lose x∈xs p , lose y∈ys q)

+
             (find∘map p (λ p → Any.map (p ,_) q)) ⟩

+

+
    (let (x , x∈xs , p)     = find p

+
         (y , y∈ys , p , q) = find (Any.map (p ,_) q)

+
     in  lose x∈xs p , lose y∈ys q)

+

+
     ≡⟨ cong (λ • → let (x , x∈xs , _)     = find p

+
                        (y , y∈ys , p , q) = •

+
                    in  lose x∈xs p , lose y∈ys q)

+
             (find∘map q (px ,_)) ⟩

+

+
    (let (x , x∈xs , p) = find p

+
         (y , y∈ys , q) = find q

+
     in  lose x∈xs p , lose y∈ys q)

+

+
     ≡⟨ cong₂ _,_ (lose∘find p) (lose∘find q) ⟩

+

+
    (p , q) ∎

+

+
  to∘from : ∀ pq → Any-×⁺ (Any-×⁻ pq) ≡ pq

+
  to∘from pq =

+
    let x , x∈xs , pq′ = find pq

+
        y , y∈ys , px , qy = find pq′

+

+
        h : P ⋐ λ x → Any (λ y → (P x) × (Q y)) ys

+
        h p = Any.map (p ,_) (lose y∈ys qy)

+

+
        helper : h px ≡ pq′

+
        helper = begin

+
          Any.map (px ,_) (lose y∈ys qy)

+
            ≡⟨ map-∘ (px ,_) (λ z → resp Q z qy) y∈ys ⟨

+
          Any.map (λ z → px , resp Q z qy) y∈ys

+
            ≡⟨ map∘find pq′ refl ⟩

+
          pq′

+
            ∎

+

+
    in  begin

+
      Any-×⁺ (Any-×⁻ pq)

+
        ≡⟨⟩

+
      Any.map h (lose x∈xs px)

+
        ≡⟨ map-∘ h (λ z → resp P z px) x∈xs ⟨

+
      Any.map (λ z → Any.map (resp P z px ,_) (lose y∈ys qy)) x∈xs

+
        ≡⟨ map∘find pq helper ⟩

+
      pq

+
        ∎

+

+
------------------------------------------------------------------------

+
-- Half-applied product commutes with Any.

+

+
module _ {_~_ : REL A B r} where

+

+
  Any-Σ⁺ʳ : (∃ λ x → Any (_~ x) xs) → Any (∃ ∘ _~_) xs

+
  Any-Σ⁺ʳ (b , here px) = here (b , px)

+
  Any-Σ⁺ʳ (b , there pxs) = there (Any-Σ⁺ʳ (b , pxs))

+

+
  Any-Σ⁻ʳ : Any (∃ ∘ _~_) xs → ∃ λ x → Any (_~ x) xs

+
  Any-Σ⁻ʳ (here (b , x)) = b , here x

+
  Any-Σ⁻ʳ (there xs) = Product.map₂ there $ Any-Σ⁻ʳ xs

+

+
------------------------------------------------------------------------

+
-- Invertible introduction (⁺) and elimination (⁻) rules for various

+
-- list functions

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- singleton

+

+
singleton⁺ : P x → Any P [ x ]

+
singleton⁺ Px = here Px

+

+
singleton⁻ : Any P [ x ] → P x

+
singleton⁻ (here Px) = Px

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {f : A → B} where

+

+
  map⁺ : Any (P ∘ f) xs → Any P (List.map f xs)

+
  map⁺ (here p)  = here p

+
  map⁺ (there p) = there $ map⁺ p

+

+
  map⁻ : Any P (List.map f xs) → Any (P ∘ f) xs

+
  map⁻ {xs = x ∷ xs} (here p)  = here p

+
  map⁻ {xs = x ∷ xs} (there p) = there $ map⁻ p

+

+
  map⁺∘map⁻ : (p : Any P (List.map f xs)) → map⁺ (map⁻ p) ≡ p

+
  map⁺∘map⁻ {xs = x ∷ xs} (here  p) = refl

+
  map⁺∘map⁻ {xs = x ∷ xs} (there p) = cong there (map⁺∘map⁻ p)

+

+
  map⁻∘map⁺ : ∀ (P : Pred B p) →

+
              (p : Any (P ∘ f) xs) → map⁻ {P = P} (map⁺ p) ≡ p

+
  map⁻∘map⁺ P (here  p) = refl

+
  map⁻∘map⁺ P (there p) = cong there (map⁻∘map⁺ P p)

+

+
  map↔ : Any (P ∘ f) xs ↔ Any P (List.map f xs)

+
  map↔ = mk↔ₛ′ map⁺ map⁻ map⁺∘map⁻ (map⁻∘map⁺ _)

+

+
  gmap : P ⋐ Q ∘ f → Any P ⋐ Any Q ∘ map f

+
  gmap g = map⁺ ∘ Any.map g

+

+
------------------------------------------------------------------------

+
-- mapMaybe

+

+
module _ (f : A → Maybe B) where

+

+
  mapMaybe⁺ : ∀ xs → Any (MAny.Any P) (map f xs) → Any P (mapMaybe f xs)

+
  mapMaybe⁺ (x ∷ xs) ps with f x | ps

+
  ... | nothing | there pxs      = mapMaybe⁺ xs pxs

+
  ... | just _  | here (just py) = here py

+
  ... | just _  | there pxs      = there (mapMaybe⁺ xs pxs)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ {P : A → Set p} where

+

+
  ++⁺ˡ : Any P xs → Any P (xs ++ ys)

+
  ++⁺ˡ (here p)  = here p

+
  ++⁺ˡ (there p) = there (++⁺ˡ p)

+

+
  ++⁺ʳ : ∀ xs {ys} → Any P ys → Any P (xs ++ ys)

+
  ++⁺ʳ []       p = p

+
  ++⁺ʳ (x ∷ xs) p = there (++⁺ʳ xs p)

+

+
  ++⁻ : ∀ xs {ys} → Any P (xs ++ ys) → Any P xs ⊎ Any P ys

+
  ++⁻ []       p         = inj₂ p

+
  ++⁻ (x ∷ xs) (here p)  = inj₁ (here p)

+
  ++⁻ (x ∷ xs) (there p) = Sum.map there id (++⁻ xs p)

+

+
  ++⁺∘++⁻ : ∀ xs {ys} (p : Any P (xs ++ ys)) → [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs p) ≡ p

+
  ++⁺∘++⁻ []       p         = refl

+
  ++⁺∘++⁻ (x ∷ xs) (here  p) = refl

+
  ++⁺∘++⁻ (x ∷ xs) (there p) with ih ← ++⁺∘++⁻ xs p | ++⁻ xs p

+
  ... | inj₁ _ = cong there ih

+
  ... | inj₂ _ = cong there ih

+

+
  ++⁻∘++⁺ : ∀ xs {ys} (p : Any P xs ⊎ Any P ys) →

+
            ++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p) ≡ p

+
  ++⁻∘++⁺ []            (inj₂ p)         = refl

+
  ++⁻∘++⁺ (x ∷ xs)      (inj₁ (here  p)) = refl

+
  ++⁻∘++⁺ (x ∷ xs) {ys} (inj₁ (there p)) rewrite ++⁻∘++⁺ xs {ys} (inj₁ p) = refl

+
  ++⁻∘++⁺ (x ∷ xs)      (inj₂ p)         rewrite ++⁻∘++⁺ xs      (inj₂ p) = refl

+

+
  ++↔ : ∀ {xs ys} → (Any P xs ⊎ Any P ys) ↔ Any P (xs ++ ys)

+
  ++↔ {xs = xs} = mk↔ₛ′ [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁺∘++⁻ xs) (++⁻∘++⁺ xs)

+

+
  ++-comm : ∀ xs ys → Any P (xs ++ ys) → Any P (ys ++ xs)

+
  ++-comm xs ys = [ ++⁺ʳ ys , ++⁺ˡ ]′ ∘ ++⁻ xs

+

+
  ++-comm∘++-comm : ∀ xs {ys} (p : Any P (xs ++ ys)) →

+
                    ++-comm ys xs (++-comm xs ys p) ≡ p

+
  ++-comm∘++-comm [] {ys} p

+
    rewrite ++⁻∘++⁺ ys {ys = []} (inj₁ p) = refl

+
  ++-comm∘++-comm (x ∷ xs) {ys} (here p)

+
    rewrite ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₂ (here p)) = refl

+
  ++-comm∘++-comm (x ∷ xs)      (there p) with ++⁻ xs p | ++-comm∘++-comm xs p

+
  ++-comm∘++-comm (x ∷ xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ʳ ys p))))

+
    | inj₁ p | refl

+
    rewrite ++⁻∘++⁺ ys (inj₂                 p)

+
          | ++⁻∘++⁺ ys (inj₂ $ there {x = x} p) = refl

+
  ++-comm∘++-comm (x ∷ xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ˡ p))))

+
    | inj₂ p | refl

+
    rewrite ++⁻∘++⁺ ys {ys =     xs} (inj₁ p)

+
          | ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₁ p) = refl

+

+
  ++↔++ : ∀ xs ys → Any P (xs ++ ys) ↔ Any P (ys ++ xs)

+
  ++↔++ xs ys = mk↔ₛ′ (++-comm xs ys) (++-comm ys xs)

+
                        (++-comm∘++-comm ys) (++-comm∘++-comm xs)

+

+
  ++-insert : ∀ xs {ys} → P x → Any P (xs ++ [ x ] ++ ys)

+
  ++-insert xs Px = ++⁺ʳ xs (++⁺ˡ (singleton⁺ Px))

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {P : A → Set p} where

+

+
  concat⁺ : ∀ {xss} → Any (Any P) xss → Any P (concat xss)

+
  concat⁺ (here p)           = ++⁺ˡ p

+
  concat⁺ (there {x = xs} p) = ++⁺ʳ xs (concat⁺ p)

+

+
  concat⁻ : ∀ xss → Any P (concat xss) → Any (Any P) xss

+
  concat⁻ ([]       ∷ xss) p         = there $ concat⁻ xss p

+
  concat⁻ ((x ∷ xs) ∷ xss) (here  p) = here (here p)

+
  concat⁻ ((x ∷ xs) ∷ xss) (there p) with concat⁻ (xs ∷ xss) p

+
  ... | here  p′ = here (there p′)

+
  ... | there p′ = there p′

+

+
  concat⁻∘++⁺ˡ : ∀ {xs} xss (p : Any P xs) →

+
                 concat⁻ (xs ∷ xss) (++⁺ˡ p) ≡ here p

+
  concat⁻∘++⁺ˡ xss (here  p) = refl

+
  concat⁻∘++⁺ˡ xss (there p) rewrite concat⁻∘++⁺ˡ xss p = refl

+

+
  concat⁻∘++⁺ʳ : ∀ xs xss (p : Any P (concat xss)) →

+
                   concat⁻ (xs ∷ xss) (++⁺ʳ xs p) ≡ there (concat⁻ xss p)

+
  concat⁻∘++⁺ʳ []       xss p = refl

+
  concat⁻∘++⁺ʳ (x ∷ xs) xss p rewrite concat⁻∘++⁺ʳ xs xss p = refl

+

+
  concat⁺∘concat⁻ : ∀ xss (p : Any P (concat xss)) →

+
                      concat⁺ (concat⁻ xss p) ≡ p

+
  concat⁺∘concat⁻ ([]       ∷ xss) p         = concat⁺∘concat⁻ xss p

+
  concat⁺∘concat⁻ ((x ∷ xs) ∷ xss) (here p)  = refl

+
  concat⁺∘concat⁻ ((x ∷ xs) ∷ xss) (there p)

+
    with p | concat⁻ (xs ∷ xss) p | concat⁺∘concat⁻ (xs ∷ xss) p

+
  ... | .(++⁺ˡ p′)              | here  p′ | refl = refl

+
  ... | .(++⁺ʳ xs (concat⁺ p′)) | there p′ | refl = refl

+

+
  concat⁻∘concat⁺ : ∀ {xss} (p : Any (Any P) xss) → concat⁻ xss (concat⁺ p) ≡ p

+
  concat⁻∘concat⁺ (here                      p) = concat⁻∘++⁺ˡ _ p

+
  concat⁻∘concat⁺ (there {x = xs} {xs = xss} p)

+
    rewrite concat⁻∘++⁺ʳ xs xss (concat⁺ p) =

+
      cong there $ concat⁻∘concat⁺ p

+

+
  concat↔ : ∀ {xss} → Any (Any P) xss ↔ Any P (concat xss)

+
  concat↔ {xss} = mk↔ₛ′ concat⁺ (concat⁻ xss) (concat⁺∘concat⁻ xss) concat⁻∘concat⁺

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith

+

+
module _ (f : A → B → C) where

+

+
  cartesianProductWith⁺ : (∀ {x y} → P x → Q y → R (f x y)) →

+
                          Any P xs → Any Q ys →

+
                          Any R (cartesianProductWith f xs ys)

+
  cartesianProductWith⁺ pres (here  px)  qys = ++⁺ˡ (map⁺ (Any.map (pres px) qys))

+
  cartesianProductWith⁺ pres (there qxs) qys = ++⁺ʳ _ (cartesianProductWith⁺ pres qxs qys)

+

+
  cartesianProductWith⁻ : (∀ {x y} → R (f x y) → P x × Q y) → ∀ xs ys →

+
                          Any R (cartesianProductWith f xs ys) →

+
                          Any P xs × Any Q ys

+
  cartesianProductWith⁻ resp (x ∷ xs) ys Rxsys with ++⁻ (map (f x) ys) Rxsys

+
  ... | inj₁ Rfxys = let Rxys = map⁻ Rfxys

+
    in here (proj₁ (resp (proj₂ (Any.satisfied Rxys)))) , Any.map (proj₂ ∘ resp) Rxys

+
  ... | inj₂ Rc    = let pxs , qys = cartesianProductWith⁻ resp xs ys Rc

+
    in there pxs , qys

+

+
------------------------------------------------------------------------

+
-- cartesianProduct

+

+
cartesianProduct⁺ : Any P xs → Any Q ys →

+
                    Any (P ⟨×⟩ Q) (cartesianProduct xs ys)

+
cartesianProduct⁺ = cartesianProductWith⁺ _,_ _,_

+

+
cartesianProduct⁻ : ∀ xs ys → Any (P ⟨×⟩ Q) (cartesianProduct xs ys) →

+
                    Any P xs × Any Q ys

+
cartesianProduct⁻ = cartesianProductWith⁻ _,_ id

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
applyUpTo⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyUpTo f n)

+
applyUpTo⁺ _ p z<s       = here p

+
applyUpTo⁺ f p (s<s i<n@(s≤s _)) =

+
  there (applyUpTo⁺ (f ∘ suc) p i<n)

+

+
applyUpTo⁻ : ∀ f {n} → Any P (applyUpTo f n) →

+
             ∃ λ i → i < n × P (f i)

+
applyUpTo⁻ f {suc n} (here p)  = zero , z<s , p

+
applyUpTo⁻ f {suc n} (there p) =

+
  let i , i<n , q = applyUpTo⁻ (f ∘ suc) p in suc i , s<s i<n , q

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
applyDownFrom⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyDownFrom f n)

+
applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≟ n

+
... | yes refl = here p

+
... | no  i≢n  = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))

+

+
applyDownFrom⁻ : ∀ f {n} → Any P (applyDownFrom f n) →

+
                 ∃ λ i → i < n × P (f i)

+
applyDownFrom⁻ f {suc n} (here p)  = n , ≤-refl , p

+
applyDownFrom⁻ f {suc n} (there p) =

+
  let i , i<n , q = applyDownFrom⁻ f p in i , m<n⇒m<1+n i<n , q

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
tabulate⁺ : ∀ {n} {f : Fin n → A} i → P (f i) → Any P (tabulate f)

+
tabulate⁺ zero    p = here p

+
tabulate⁺ (suc i) p = there (tabulate⁺ i p)

+

+
tabulate⁻ : ∀ {n} {f : Fin n → A} → Any P (tabulate f) → ∃ λ i → P (f i)

+
tabulate⁻ {n = suc _} (here p)  = zero , p

+
tabulate⁻ {n = suc _} (there p) = Product.map suc id (tabulate⁻ p)

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (Q? : U.Decidable Q) where

+

+
  filter⁺ : (p : Any P xs) → Any P (filter Q? xs) ⊎ ¬ Q (Any.lookup p)

+
  filter⁺ {xs = x ∷ _} (here px) with Q? x

+
  ... | true  because _     = inj₁ (here px)

+
  ... | false because [¬Qx] = inj₂ (invert [¬Qx])

+
  filter⁺ {xs = x ∷ _} (there p) with does (Q? x)

+
  ... | true  = Sum.map₁ there (filter⁺ p)

+
  ... | false = filter⁺ p

+

+
  filter⁻ : Any P (filter Q? xs) → Any P xs

+
  filter⁻ {xs = x ∷ xs} p with does (Q? x) | p

+
  ... | true  | here px   = here px

+
  ... | true  | there pxs = there (filter⁻ pxs)

+
  ... | false | pxs       = there (filter⁻ pxs)

+

+
------------------------------------------------------------------------

+
-- derun and deduplicate

+

+
module _ {R : Rel A r} (R? : B.Decidable R) where

+

+
  private

+
    derun⁺-aux : ∀ x xs → P Respects R → P x → Any P (derun R? (x ∷ xs))

+
    derun⁺-aux x [] resp Px = here Px

+
    derun⁺-aux x (y ∷ xs) resp Px with R? x y

+
    ... | true  because [Rxy] = derun⁺-aux y xs resp (resp (invert [Rxy]) Px)

+
    ... | false because _     = here Px

+

+
  derun⁺ : P Respects R → Any P xs → Any P (derun R? xs)

+
  derun⁺ {xs = x ∷ xs}     resp (here px)   = derun⁺-aux x xs resp px

+
  derun⁺ {xs = x ∷ y ∷ xs} resp (there pxs) with does (R? x y)

+
  ... | true  = derun⁺ resp pxs

+
  ... | false = there (derun⁺ resp pxs)

+

+
  deduplicate⁺ : ∀ {xs} → P Respects (flip R) → Any P xs → Any P (deduplicate R? xs)

+
  deduplicate⁺ {xs = x ∷ xs} resp (here px)   = here px

+
  deduplicate⁺ {xs = x ∷ xs} resp (there pxs)

+
    with filter⁺ (¬? ∘ R? x) (deduplicate⁺ resp pxs)

+
  ... | inj₁ p   = there p

+
  ... | inj₂ ¬¬q =

+
    let q = decidable-stable (R? x (Any.lookup (deduplicate⁺ resp pxs))) ¬¬q

+
    in  here (resp q (lookup-result (deduplicate⁺ resp pxs)))

+

+
  private

+
    derun⁻-aux : Any P (derun R? (x ∷ xs)) → Any P (x ∷ xs)

+
    derun⁻-aux {x = x} {[]}    (here px) = here px

+
    derun⁻-aux {x = x} {y ∷ _} p[x∷y∷xs] with does (R? x y) | p[x∷y∷xs]

+
    ... | true  | p[y∷xs]        = there (derun⁻-aux p[y∷xs])

+
    ... | false | here px        = here px

+
    ... | false | there p[y∷xs]! = there (derun⁻-aux p[y∷xs]!)

+

+
  derun⁻ : Any P (derun R? xs) → Any P xs

+
  derun⁻ {xs = x ∷ xs} p[x∷xs]! = derun⁻-aux p[x∷xs]!

+

+
  deduplicate⁻ : Any P (deduplicate R? xs) → Any P xs

+
  deduplicate⁻ {xs = x ∷ _} (here px)    = here px

+
  deduplicate⁻ {xs = x ∷ _} (there pxs!) = there (deduplicate⁻ (filter⁻ (¬? ∘ R? x) pxs!))

+

+
------------------------------------------------------------------------

+
-- mapWith∈.

+

+
module _ {P : B → Set p} where

+

+
  mapWith∈⁺ : ∀ {xs : List A} (f : ∀ {x} → x ∈ xs → B) →

+
                (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) →

+
                Any P (mapWith∈ xs f)

+
  mapWith∈⁺ f (_ , here refl  , p) = here p

+
  mapWith∈⁺ f (_ , there x∈xs , p) =

+
    there $ mapWith∈⁺ (f ∘ there) (_ , x∈xs , p)

+

+
  mapWith∈⁻ : ∀ (xs : List A) (f : ∀ {x} → x ∈ xs → B) →

+
                Any P (mapWith∈ xs f) →

+
                ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)

+
  mapWith∈⁻ (y ∷ xs) f (here  p) = (y , here refl , p)

+
  mapWith∈⁻ (y ∷ xs) f (there p) =

+
    Product.map₂ (Product.map there id) $ mapWith∈⁻ xs (f ∘ there) p

+

+
  mapWith∈↔ : ∀ {xs : List A} {f : ∀ {x} → x ∈ xs → B} →

+
                (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) ↔ Any P (mapWith∈ xs f)

+
  mapWith∈↔ = mk↔ₛ′ (mapWith∈⁺ _) (mapWith∈⁻ _ _) (to∘from _ _) (from∘to _)

+
    where

+
    from∘to : ∀ {xs : List A} (f : ∀ {x} → x ∈ xs → B)

+
              (p : ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) →

+
              mapWith∈⁻ xs f (mapWith∈⁺ f p) ≡ p

+
    from∘to f (_ , here refl  , p) = refl

+
    from∘to f (_ , there x∈xs , p)

+
      rewrite from∘to (f ∘ there) (_ , x∈xs , p) = refl

+

+
    to∘from : ∀ (xs : List A) (f : ∀ {x} → x ∈ xs → B)

+
              (p : Any P (mapWith∈ xs f)) →

+
              mapWith∈⁺ f (mapWith∈⁻ xs f p) ≡ p

+
    to∘from (y ∷ xs) f (here  p) = refl

+
    to∘from (y ∷ xs) f (there p) =

+
      cong there $ to∘from xs (f ∘ there) p

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
reverseAcc⁺ : ∀ acc xs → Any P acc ⊎ Any P xs → Any P (reverseAcc acc xs)

+
reverseAcc⁺ acc []       (inj₁ ps)        = ps

+
reverseAcc⁺ acc (x ∷ xs) (inj₁ ps)        = reverseAcc⁺ (x ∷ acc) xs (inj₁ (there ps))

+
reverseAcc⁺ acc (x ∷ xs) (inj₂ (here px)) = reverseAcc⁺ (x ∷ acc) xs (inj₁ (here px))

+
reverseAcc⁺ acc (x ∷ xs) (inj₂ (there y)) = reverseAcc⁺ (x ∷ acc) xs (inj₂ y)

+

+
reverseAcc⁻ : ∀ acc xs → Any P (reverseAcc acc xs) → Any P acc ⊎ Any P xs

+
reverseAcc⁻ acc []       ps = inj₁ ps

+
reverseAcc⁻ acc (x ∷ xs) ps with reverseAcc⁻ (x ∷ acc) xs ps

+
... | inj₁ (here px) = inj₂ (here px)

+
... | inj₁ (there pxs) = inj₁ pxs

+
... | inj₂ pxs = inj₂ (there pxs)

+

+
reverse⁺ : Any P xs → Any P (reverse xs)

+
reverse⁺ ps = reverseAcc⁺ [] _ (inj₂ ps)

+

+
reverse⁻ : Any P (reverse xs) → Any P xs

+
reverse⁻ ps with inj₂ pxs ← reverseAcc⁻ [] _ ps = pxs

+

+
------------------------------------------------------------------------

+
-- pure

+

+
pure⁺ : P x → Any P (pure x)

+
pure⁺ = here

+

+
pure⁻ : Any P (pure x) → P x

+
pure⁻ (here p) = p

+

+
pure⁺∘pure⁻ : (p : Any P (pure x)) → pure⁺ (pure⁻ p) ≡ p

+
pure⁺∘pure⁻ (here p) = refl

+

+
pure⁻∘pure⁺ : (p : P x) → pure⁻ {P = P} (pure⁺ p) ≡ p

+
pure⁻∘pure⁺ p = refl

+

+
pure↔ : P x ↔ Any P (pure x)

+
pure↔ {P = P} = mk↔ₛ′ pure⁺ pure⁻ pure⁺∘pure⁻ (pure⁻∘pure⁺ {P = P})

+

+
------------------------------------------------------------------------

+
-- _∷_

+

+
∷↔ : (P : Pred A p) → (P x ⊎ Any P xs) ↔ Any P (x ∷ xs)

+
∷↔ {x = x} {xs} P =

+
  (P x         ⊎ Any P xs)  ↔⟨ pure↔ ⊎-cong (Any P xs ∎) ⟩

+
  (Any P [ x ] ⊎ Any P xs)  ↔⟨ ++↔ ⟩

+
  Any P (x ∷ xs)            ∎

+
  where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _>>=_

+

+
module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where

+

+
  >>=↔ : Any (Any P ∘ f) xs ↔ Any P (xs >>= f)

+
  >>=↔ {xs = xs} =

+
    Any (Any P ∘ f) xs           ↔⟨ map↔ ⟩

+
    Any (Any P) (List.map f xs)  ↔⟨ concat↔ ⟩

+
    Any P (xs >>= f)             ∎

+
    where open Related.EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- _⊛_

+

+
⊛↔ : ∀ {P : B → Set ℓ} {fs : List (A → B)} {xs : List A} →

+
     Any (λ f → Any (P ∘ f) xs) fs ↔ Any P (fs ⊛ xs)

+
⊛↔ {P = P} {fs} {xs} =

+
  Any (λ f → Any (P ∘ f) xs) fs                ↔⟨ Any-cong (λ _ → Any-cong (λ _ → pure↔) (_ ∎)) (_ ∎) ⟩

+
  Any (λ f → Any (Any P ∘ pure ∘ f) xs) fs     ↔⟨ Any-cong (λ _ → >>=↔ ) (_ ∎) ⟩

+
  Any (λ f → Any P (xs >>= pure ∘ f)) fs       ↔⟨ >>=↔ ⟩

+
  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡⟨ cong (Any P) (List.Applicative.unfold-⊛ fs xs) ⟨

+
  Any P (fs ⊛ xs)                               ∎

+
  where open Related.EquationalReasoning

+

+

+
-- An alternative introduction rule for _⊛_

+

+
⊛⁺′ : ∀ {P : Pred A ℓ} {Q : Pred B ℓ} {fs : List (A → B)} {xs} →

+
      Any (P ⟨→⟩ Q) fs → Any P xs → Any Q (fs ⊛ xs)

+
⊛⁺′ pq p = Inverse.to ⊛↔ (Any.map (λ pq → Any.map (λ {x} → pq {x}) p) pq)

+

+
------------------------------------------------------------------------

+
-- _⊗_

+

+
⊗↔ : {P : A × B → Set ℓ} {xs : List A} {ys : List B} →

+
     Any (λ x → Any (λ y → P (x , y)) ys) xs ↔ Any P (xs ⊗ ys)

+
⊗↔ {P = P} {xs} {ys} =

+
  Any (λ x → Any (λ y → P (x , y)) ys) xs                           ↔⟨ pure↔ ⟩

+
  Any (λ _,_ → Any (λ x → Any (λ y → P (x , y)) ys) xs) (pure _,_)  ↔⟨ ⊛↔ ⟩

+
  Any (λ x, → Any (P ∘ x,) ys) (pure _,_ ⊛ xs)                      ↔⟨ ⊛↔ ⟩

+
  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡⟨ cong (Any P ∘′ (_⊛ ys)) (List.Applicative.unfold-<$> _,_ xs) ⟨

+
  Any P (xs ⊗ ys)                                                   ∎

+
  where open Related.EquationalReasoning

+

+
⊗↔′ : {P : A → Set ℓ} {Q : B → Set ℓ} {xs : List A} {ys : List B} →

+
      (Any P xs × Any Q ys) ↔ Any (P ⟨×⟩ Q) (xs ⊗ ys)

+
⊗↔′ {P = P} {Q} {xs} {ys} =

+
  (Any P xs × Any Q ys)                    ↔⟨ ×↔ ⟩

+
  Any (λ x → Any (λ y → P x × Q y) ys) xs  ↔⟨ ⊗↔ ⟩

+
  Any (P ⟨×⟩ Q) (xs ⊗ ys)                  ∎

+
  where open Related.EquationalReasoning

+

+
map-with-∈⁺ = mapWith∈⁺

+
{-# WARNING_ON_USAGE map-with-∈⁺

+
"Warning: map-with-∈⁺ was deprecated in v2.0.

+
Please use mapWith∈⁺ instead."

+
#-}

+
map-with-∈⁻ = mapWith∈⁻

+
{-# WARNING_ON_USAGE map-with-∈⁻

+
"Warning: map-with-∈⁻ was deprecated in v2.0.

+
Please use mapWith∈⁻ instead."

+
#-}

+
map-with-∈↔ = mapWith∈↔

+
{-# WARNING_ON_USAGE map-with-∈↔

+
"Warning: map-with-∈↔ was deprecated in v2.0.

+
Please use mapWith∈↔ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Any.html b/site/static/agda_html/Data.List.Relation.Unary.Any.html
new file mode 100644
index 000000000..69f299ec8
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Any.html
@@ -0,0 +1,186 @@
+
+Data.List.Relation.Unary.Any
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists where at least one element satisfies a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Any where

+

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.List.Base as List using (List; []; [_]; _∷_; removeAt)

+
open import Data.Product.Base as Product using (∃; _,_)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary.Decidable.Core as Dec using (no; _⊎-dec_)

+
open import Relation.Nullary.Negation using (¬_; contradiction)

+
open import Relation.Unary using (Pred; _⊆_; Decidable; Satisfiable)

+

+
private

+
  variable

+
    a p q : Level

+
    A : Set a

+
    P Q : Pred A p

+
    x : A

+
    xs : List A

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Given a predicate P, then Any P xs means that at least one element

+
-- in xs satisfies P. See `Relation.Unary` for an explanation of

+
-- predicates.

+

+
data Any {A : Set a} (P : Pred A p) : Pred (List A) (a ⊔ p) where

+
  here  : ∀ {x xs} (px  : P x)      → Any P (x ∷ xs)

+
  there : ∀ {x xs} (pxs : Any P xs) → Any P (x ∷ xs)

+

+
------------------------------------------------------------------------

+
-- Operations on Any

+

+
head : ¬ Any P xs → Any P (x ∷ xs) → P x

+
head ¬pxs (here px)   = px

+
head ¬pxs (there pxs) = contradiction pxs ¬pxs

+

+
tail : ¬ P x → Any P (x ∷ xs) → Any P xs

+
tail ¬px (here  px)  = contradiction px ¬px

+
tail ¬px (there pxs) = pxs

+

+
map : P ⊆ Q → Any P ⊆ Any Q

+
map g (here px)   = here (g px)

+
map g (there pxs) = there (map g pxs)

+

+
-- `index x∈xs` is the list position (zero-based) which `x∈xs` points to.

+

+
index : Any P xs → Fin (List.length xs)

+
index (here  px)  = zero

+
index (there pxs) = suc (index pxs)

+

+
lookup : {P : Pred A p} → Any P xs → A

+
lookup {xs = xs} p = List.lookup xs (index p)

+

+
infixr 5 _∷=_

+

+
_∷=_ : {P : Pred A p} → Any P xs → A → List A

+
_∷=_ {xs = xs} x∈xs v = xs List.[ index x∈xs ]∷= v

+

+
infixl 4 _─_

+
_─_ : {P : Pred A p} → ∀ xs → Any P xs → List A

+
xs ─ x∈xs = removeAt xs (index x∈xs)

+

+
-- If any element satisfies P, then P is satisfied.

+

+
satisfied : Any P xs → ∃ P

+
satisfied (here px)   = _ , px

+
satisfied (there pxs) = satisfied pxs

+

+
toSum : Any P (x ∷ xs) → P x ⊎ Any P xs

+
toSum (here px)   = inj₁ px

+
toSum (there pxs) = inj₂ pxs

+

+
fromSum : P x ⊎ Any P xs → Any P (x ∷ xs)

+
fromSum (inj₁ px)  = here px

+
fromSum (inj₂ pxs) = there pxs

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by Any

+

+
any? : Decidable P → Decidable (Any P)

+
any? P? []       = no λ()

+
any? P? (x ∷ xs) = Dec.map′ fromSum toSum (P? x ⊎-dec any? P? xs)

+

+
satisfiable : Satisfiable P → Satisfiable (Any P)

+
satisfiable (x , Px) = [ x ] , here Px

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.4

+

+
any = any?

+
{-# WARNING_ON_USAGE any

+
"Warning: any was deprecated in v1.4.

+
Please use any? instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.First.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.First.Properties.html
new file mode 100644
index 000000000..56a0db22d
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.First.Properties.html
@@ -0,0 +1,194 @@
+
+Data.List.Relation.Unary.First.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of First

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.First.Properties where

+

+
open import Data.Fin.Base using (suc)

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.Any as Any using (here; there)

+
open import Data.List.Relation.Unary.First

+
import Data.Sum.Base as Sum

+
open import Function.Base using (_∘′_; _∘_; id)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl; _≗_)

+
import Relation.Nullary.Decidable.Core as Dec

+
open import Relation.Nullary.Negation.Core using (contradiction)

+
open import Relation.Unary using (Pred; _⊆_; ∁; Irrelevant; Decidable)

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {a b p q} {A : Set a} {B : Set b} {P : Pred B p} {Q : Pred B q} where

+

+
  map⁺ : {f : A → B} → First (P ∘′ f) (Q ∘′ f) ⊆ First P Q ∘′ List.map f

+
  map⁺ [ qfx ]        = [ qfx ]

+
  map⁺ (pfxs ∷ pqfxs) = pfxs ∷ map⁺ pqfxs

+

+
  map⁻ : {f : A → B} → First P Q ∘′ List.map f ⊆ First (P ∘′ f) (Q ∘′ f)

+
  map⁻ {f} {x ∷ xs} [ qfx ]       = [ qfx ]

+
  map⁻ {f} {x ∷ xs} (pfx ∷ pqfxs) = pfx ∷ map⁻ pqfxs

+

+
------------------------------------------------------------------------

+
-- (++)

+

+
module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where

+

+
  ++⁺ : ∀ {xs ys} → All P xs → First P Q ys → First P Q (xs List.++ ys)

+
  ++⁺ []         pqys = pqys

+
  ++⁺ (px ∷ pxs) pqys = px ∷ ++⁺ pxs pqys

+

+
  ⁺++ : ∀ {xs} → First P Q xs → ∀ ys → First P Q (xs List.++ ys)

+
  ⁺++ [ qx ]      ys = [ qx ]

+
  ⁺++ (px ∷ pqxs) ys = px ∷ ⁺++ pqxs ys

+

+
------------------------------------------------------------------------

+
-- Relationship to All

+

+
module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where

+

+
  All⇒¬First : P ⊆ ∁ Q → All P ⊆ ∁ (First P Q)

+
  All⇒¬First p⇒¬q (px ∷ pxs) [ qx ]   = contradiction qx (p⇒¬q px)

+
  All⇒¬First p⇒¬q (_ ∷ pxs)  (_ ∷ hf) = All⇒¬First p⇒¬q pxs hf

+

+
  First⇒¬All : Q ⊆ ∁ P → First P Q ⊆ ∁ (All P)

+
  First⇒¬All q⇒¬p [ qx ]     (px ∷ pxs) = q⇒¬p qx px

+
  First⇒¬All q⇒¬p (_ ∷ pqxs) (_ ∷ pxs)  = First⇒¬All q⇒¬p pqxs pxs

+

+
------------------------------------------------------------------------

+
-- Irrelevance

+

+
  unique-index : ∀ {xs} → P ⊆ ∁ Q → (f₁ f₂ : First P Q xs) → index f₁ ≡ index f₂

+
  unique-index p⇒¬q [ _ ]    [ _ ]    = refl

+
  unique-index p⇒¬q [ qx ]   (px ∷ _) = contradiction qx (p⇒¬q px)

+
  unique-index p⇒¬q (px ∷ _) [ qx ]   = contradiction qx (p⇒¬q px)

+
  unique-index p⇒¬q (_ ∷ f₁) (_ ∷ f₂) = ≡.cong suc (unique-index p⇒¬q f₁ f₂)

+

+
  irrelevant : P ⊆ ∁ Q → Irrelevant P → Irrelevant Q → Irrelevant (First P Q)

+
  irrelevant p⇒¬q p-irr q-irr [ px ]    [ qx ]    = ≡.cong [_] (q-irr px qx)

+
  irrelevant p⇒¬q p-irr q-irr [ qx ]    (px ∷ _)  = contradiction qx (p⇒¬q px)

+
  irrelevant p⇒¬q p-irr q-irr (px ∷ _)  [ qx ]    = contradiction qx (p⇒¬q px)

+
  irrelevant p⇒¬q p-irr q-irr (px ∷ f)  (qx ∷ g) =

+
    ≡.cong₂ _∷_ (p-irr px qx) (irrelevant p⇒¬q p-irr q-irr f g)

+

+
------------------------------------------------------------------------

+
-- Decidability

+

+
module _ {a p} {A : Set a} {P : Pred A p} where

+

+
  first? : Decidable P → Decidable (First P (∁ P))

+
  first? P? = Dec.fromSum

+
            ∘ Sum.map₂ (All⇒¬First contradiction)

+
            ∘ first (Dec.toSum ∘ P?)

+

+
  cofirst? : Decidable P → Decidable (First (∁ P) P)

+
  cofirst? P? = Dec.fromSum

+
              ∘ Sum.map₂ (All⇒¬First id)

+
              ∘ first (Sum.swap ∘ Dec.toSum ∘ P?)

+

+
------------------------------------------------------------------------

+
-- Conversion to Any

+

+
module _ {a p} {A : Set a} {P : Pred A p} where

+

+
  fromAny∘toAny≗id : ∀ {xs} → fromAny {Q = P} {x = xs} ∘′ toAny ≗ id

+
  fromAny∘toAny≗id [ qx ]      = refl

+
  fromAny∘toAny≗id (px ∷ pqxs) = ≡.cong (_ ∷_) (fromAny∘toAny≗id pqxs)

+

+
  toAny∘fromAny≗id : ∀ {xs} → toAny {Q = P} ∘′ fromAny {x = xs} ≗ id

+
  toAny∘fromAny≗id (here px) = refl

+
  toAny∘fromAny≗id (there v) = ≡.cong there (toAny∘fromAny≗id v)

+

+
------------------------------------------------------------------------

+
-- Equivalence between the inductive definition and the view

+

+
module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where

+

+
  toView : ∀ {as} → First P Q as → FirstView P Q as

+
  toView [ qx ] = [] ++ qx ∷ _

+
  toView (px ∷ pqxs) with toView pqxs

+
  ... | pxs ++  qy ∷ ys = (px ∷ pxs) ++ qy ∷ ys

+

+
  fromView : ∀ {as} → FirstView P Q as → First P Q as

+
  fromView (pxs ++ qy ∷ ys) = ++⁺ pxs [ qy ]

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.First.html b/site/static/agda_html/Data.List.Relation.Unary.First.html
new file mode 100644
index 000000000..7940bbba5
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.First.html
@@ -0,0 +1,185 @@
+
+Data.List.Relation.Unary.First
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- First generalizes the idea that an element is the first in a list to

+
-- satisfy a predicate.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.First {a} {A : Set a} where

+

+
open import Level using (_⊔_)

+
open import Data.Empty

+
open import Data.Fin.Base as Fin using (Fin; zero; suc)

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.Product.Base using (∃; -,_; _,_)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Function.Base using (id; _∘′_)

+
open import Relation.Unary

+
open import Relation.Nullary

+

+
------------------------------------------------------------------------

+
-- Basic type.

+

+
module _ {p q} (P : Pred A p) (Q : Pred A q) where

+

+
  infix 1 _++_∷_

+
  infixr 5 _∷_

+

+
  data First : Pred (List A) (a ⊔ p ⊔ q) where

+
    [_] : ∀ {x xs} → Q x            → First (x ∷ xs)

+
    _∷_ : ∀ {x xs} → P x → First xs → First (x ∷ xs)

+

+
  data FirstView : Pred (List A) (a ⊔ p ⊔ q) where

+
    _++_∷_ : ∀ {xs y} → All P xs → Q y → ∀ ys → FirstView (xs List.++ y ∷ ys)

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {p q r s} {P : Pred A p} {Q : Pred A q} {R : Pred A r} {S : Pred A s} where

+

+
  map : P ⊆ R → Q ⊆ S → First P Q ⊆ First R S

+
  map p⇒r q⇒r [ qx ]      = [ q⇒r qx ]

+
  map p⇒r q⇒r (px ∷ pqxs) = p⇒r px ∷ map p⇒r q⇒r pqxs

+

+
module _ {p q r} {P : Pred A p} {Q : Pred A q} {R : Pred A r} where

+

+
  map₁ : P ⊆ R → First P Q ⊆ First R Q

+
  map₁ p⇒r = map p⇒r id

+

+
  map₂ : Q ⊆ R → First P Q ⊆ First P R

+
  map₂ = map id

+

+
  refine : P ⊆ Q ∪ R → First P Q ⊆ First R Q

+
  refine f [ qx ]      = [ qx ]

+
  refine f (px ∷ pqxs) with f px

+
  ... | inj₁ qx = [ qx ]

+
  ... | inj₂ rx = rx ∷ refine f pqxs

+

+
module _ {p q} {P : Pred A p} {Q : Pred A q} where

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
  empty : ¬ First P Q []

+
  empty ()

+

+
  tail : ∀ {x xs} → ¬ Q x → First P Q (x ∷ xs) → First P Q xs

+
  tail ¬qx [ qx ]      = ⊥-elim (¬qx qx)

+
  tail ¬qx (px ∷ pqxs) = pqxs

+

+
  index : First P Q ⊆ (Fin ∘′ List.length)

+
  index [ qx ]     = zero

+
  index (_ ∷ pqxs) = suc (index pqxs)

+

+
  index-satisfied : ∀ {xs} (pqxs : First P Q xs) → Q (List.lookup xs (index pqxs))

+
  index-satisfied [ qx ]     = qx

+
  index-satisfied (_ ∷ pqxs) = index-satisfied pqxs

+

+
  satisfied : ∀ {xs} → First P Q xs → ∃ Q

+
  satisfied pqxs = -, index-satisfied pqxs

+

+
  satisfiable : Satisfiable Q → Satisfiable (First P Q)

+
  satisfiable (x , qx) = List.[ x ] , [ qx ]

+

+
------------------------------------------------------------------------

+
-- Decidability results

+

+
  first : Π[ P ∪ Q ] → Π[ First P Q ∪ All P ]

+
  first p⊎q []       = inj₂ []

+
  first p⊎q (x ∷ xs) with p⊎q x

+
  ... | inj₁ px = Sum.map (px ∷_) (px ∷_) (first p⊎q xs)

+
  ... | inj₂ qx = inj₁ [ qx ]

+

+
------------------------------------------------------------------------

+
-- Relationship with Any

+

+
module _ {q} {Q : Pred A q} where

+

+
  fromAny : Any Q ⊆ First U Q

+
  fromAny (here qx)   = [ qx ]

+
  fromAny (there any) = _ ∷ fromAny any

+

+
  toAny : ∀ {p} {P : Pred A p} → First P Q ⊆ Any Q

+
  toAny [ qx ]     = here qx

+
  toAny (_ ∷ pqxs) = there (toAny pqxs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Linked.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Linked.Properties.html
new file mode 100644
index 000000000..4b4527909
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Linked.Properties.html
@@ -0,0 +1,228 @@
+
+Data.List.Relation.Unary.Linked.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to Linked

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Linked.Properties where

+

+
open import Data.Bool.Base using (true; false)

+
open import Data.List.Base hiding (any)

+
open import Data.List.Relation.Unary.AllPairs as AllPairs

+
  using (AllPairs; []; _∷_)

+
import Data.List.Relation.Unary.AllPairs.Properties as AllPairs

+
open import Data.List.Relation.Unary.All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.Linked as Linked

+
  using (Linked; []; [-]; _∷_)

+
open import Data.Nat.Base using (zero; suc; _<_; z<s; s<s)

+
open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)

+
open import Data.Maybe.Relation.Binary.Connected

+
  using (Connected; just; nothing; just-nothing; nothing-just)

+
open import Level using (Level)

+
open import Function.Base using (_∘_; flip; _on_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (DecSetoid)

+
open import Relation.Binary.Definitions using (Transitive)

+
open import Relation.Binary.PropositionalEquality.Core using (_≢_)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Nullary.Decidable using (yes; no; does)

+

+
private

+
  variable

+
    a b p ℓ : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Relationship to other predicates

+
------------------------------------------------------------------------

+

+
module _ {R : Rel A ℓ} where

+

+
  AllPairs⇒Linked : ∀ {xs} → AllPairs R xs → Linked R xs

+
  AllPairs⇒Linked []                    = []

+
  AllPairs⇒Linked (px ∷ [])             = [-]

+
  AllPairs⇒Linked ((px ∷ _) ∷ py ∷ pxs) =

+
    px ∷ (AllPairs⇒Linked (py ∷ pxs))

+

+
module _ {R : Rel A ℓ} (trans : Transitive R) where

+

+
  Linked⇒All : ∀ {v x xs} → R v x →

+
               Linked R (x ∷ xs) → All (R v) (x ∷ xs)

+
  Linked⇒All Rvx [-]         = Rvx ∷ []

+
  Linked⇒All Rvx (Rxy ∷ Rxs) = Rvx ∷ Linked⇒All (trans Rvx Rxy) Rxs

+

+
  Linked⇒AllPairs : ∀ {xs} → Linked R xs → AllPairs R xs

+
  Linked⇒AllPairs []          = []

+
  Linked⇒AllPairs [-]         = [] ∷ []

+
  Linked⇒AllPairs (Rxy ∷ Rxs) = Linked⇒All Rxy Rxs ∷ Linked⇒AllPairs Rxs

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {R : Rel A ℓ} {f : B → A} where

+

+
  map⁺ : ∀ {xs} → Linked (R on f) xs → Linked R (map f xs)

+
  map⁺ []           = []

+
  map⁺ [-]          = [-]

+
  map⁺ (Rxy ∷ Rxs)  = Rxy ∷ map⁺ Rxs

+

+
  map⁻ : ∀ {xs} → Linked R (map f xs) → Linked (R on f) xs

+
  map⁻ {[]}         []           = []

+
  map⁻ {x ∷ []}     [-]          = [-]

+
  map⁻ {x ∷ y ∷ xs} (Rxy ∷ Rxs)  = Rxy ∷ map⁻ Rxs

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ {R : Rel A ℓ} where

+

+
  ++⁺ : ∀ {xs ys} →

+
        Linked R xs →

+
        Connected R (last xs) (head ys) →

+
        Linked R ys →

+
        Linked R (xs ++ ys)

+
  ++⁺ []          _          Rys         = Rys

+
  ++⁺ [-]         _          []          = [-]

+
  ++⁺ [-]         (just Rxy) [-]         = Rxy ∷ [-]

+
  ++⁺ [-]         (just Rxy) (Ryz ∷ Rys) = Rxy ∷ Ryz ∷ Rys

+
  ++⁺ (Rxy ∷ Rxs) Rxsys      Rys         = Rxy ∷ ++⁺ Rxs Rxsys Rys

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ {R : Rel A ℓ} where

+

+
  applyUpTo⁺₁ : ∀ f n → (∀ {i} → suc i < n → R (f i) (f (suc i))) →

+
                Linked R (applyUpTo f n)

+
  applyUpTo⁺₁ f 0               Rf = []

+
  applyUpTo⁺₁ f 1               Rf = [-]

+
  applyUpTo⁺₁ f (suc n@(suc _)) Rf =

+
    Rf (s<s z<s) ∷ (applyUpTo⁺₁ (f ∘ suc) n (Rf ∘ s<s))

+

+
  applyUpTo⁺₂ : ∀ f n → (∀ i → R (f i) (f (suc i))) →

+
                Linked R (applyUpTo f n)

+
  applyUpTo⁺₂ f n Rf = applyUpTo⁺₁ f n (λ _ → Rf _)

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ {R : Rel A ℓ} where

+

+
  applyDownFrom⁺₁ : ∀ f n → (∀ {i} → suc i < n → R (f (suc i)) (f i)) →

+
                    Linked R (applyDownFrom f n)

+
  applyDownFrom⁺₁ f 0               Rf = []

+
  applyDownFrom⁺₁ f 1               Rf = [-]

+
  applyDownFrom⁺₁ f (suc n@(suc _)) Rf =

+
    Rf ≤-refl ∷ applyDownFrom⁺₁ f n (Rf ∘ m≤n⇒m≤1+n)

+

+
  applyDownFrom⁺₂ : ∀ f n → (∀ i → R (f (suc i)) (f i)) →

+
                    Linked R (applyDownFrom f n)

+
  applyDownFrom⁺₂ f n Rf = applyDownFrom⁺₁ f n (λ _ → Rf _)

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {P : Pred A p} (P? : Decidable P)

+
         {R : Rel A ℓ} (trans : Transitive R)

+
         where

+

+
  ∷-filter⁺ : ∀ {x xs} → Linked R (x ∷ xs) → Linked R (x ∷ filter P? xs)

+
  ∷-filter⁺ [-] = [-]

+
  ∷-filter⁺ {xs = y ∷ _} (r ∷ [-]) with does (P? y)

+
  ... | false = [-]

+
  ... | true  = r ∷ [-]

+
  ∷-filter⁺ {x = x} {xs = y ∷ ys} (r ∷ r′ ∷ rs)

+
    with does (P? y) | ∷-filter⁺ {xs = ys} | ∷-filter⁺ (r′ ∷ rs)

+
  ... | false | ihf | _   = ihf (trans r r′ ∷ rs)

+
  ... | true  | _   | iht = r ∷ iht

+

+
  filter⁺   : ∀ {xs} → Linked R xs → Linked R (filter P? xs)

+
  filter⁺ [] = []

+
  filter⁺ {xs = x ∷ []} [-] with does (P? x)

+
  ... | false = []

+
  ... | true  = [-]

+
  filter⁺ {xs = x ∷ _} (r ∷ rs) with does (P? x)

+
  ... | false = filter⁺ rs

+
  ... | true  = ∷-filter⁺ (r ∷ rs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Linked.html b/site/static/agda_html/Data.List.Relation.Unary.Linked.html
new file mode 100644
index 000000000..26cc699ca
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Linked.html
@@ -0,0 +1,190 @@
+
+Data.List.Relation.Unary.Linked
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists where every consecutative pair of elements is related.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Linked {a} {A : Set a} where

+

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)

+
open import Data.Maybe.Base using (just)

+
open import Data.Maybe.Relation.Binary.Connected

+
  using (Connected; just; just-nothing)

+
open import Function.Base using (id; _∘_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Definitions as B

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)

+
open import Relation.Binary.PropositionalEquality.Core using (refl; cong₂)

+
open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)

+
open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)

+

+
private

+
  variable

+
    p q r ℓ : Level

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Linked R xs means that every consecutative pair of elements in xs is

+
-- a member of relation R.

+

+
infixr 5 _∷_

+

+
data Linked (R : Rel A ℓ) : List A → Set (a ⊔ ℓ) where

+
  []  : Linked R []

+
  [-] : ∀ {x} → Linked R (x ∷ [])

+
  _∷_ : ∀ {x y xs} → R x y → Linked R (y ∷ xs) → Linked R (x ∷ y ∷ xs)

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
module _ {R : Rel A p} where

+

+
  head : ∀ {x y xs} → Linked R (x ∷ y ∷ xs) → R x y

+
  head (Rxy ∷ Rxs) = Rxy

+

+
  tail : ∀ {x xs} → Linked R (x ∷ xs) → Linked R xs

+
  tail [-]       = []

+
  tail (_ ∷ Rxs) = Rxs

+

+
  head′ : ∀ {x xs} → Linked R (x ∷ xs) → Connected R (just x) (List.head xs)

+
  head′ [-]       = just-nothing

+
  head′ (Rxy ∷ _) = just Rxy

+

+
  infixr 5 _∷′_

+

+
  _∷′_ : ∀ {x xs} →

+
         Connected R (just x) (List.head xs) →

+
         Linked R xs →

+
         Linked R (x ∷ xs)

+
  _∷′_ {xs = []}     _  _            = [-]

+
  _∷′_ {xs = y ∷ xs} (just Rxy) Ryxs = Rxy ∷ Ryxs

+

+
module _ {R : Rel A p} {S : Rel A q} where

+

+
  map : R ⇒ S → Linked R ⊆ Linked S

+
  map R⇒S []           = []

+
  map R⇒S [-]          = [-]

+
  map R⇒S (x~xs ∷ pxs) = R⇒S x~xs ∷ map R⇒S pxs

+

+
module _ {P : Rel A p} {Q : Rel A q} {R : Rel A r} where

+

+
  zipWith : P ∩ᵇ Q ⇒ R → Linked P ∩ᵘ Linked Q ⊆ Linked R

+
  zipWith f ([] , [])             = []

+
  zipWith f ([-] , [-])           = [-]

+
  zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)

+

+
  unzipWith : R ⇒ P ∩ᵇ Q → Linked R ⊆ Linked P ∩ᵘ Linked Q

+
  unzipWith f []         = [] , []

+
  unzipWith f [-]        = [-] , [-]

+
  unzipWith f (rx ∷ rxs) = Prod.zip _∷_ _∷_ (f rx) (unzipWith f rxs)

+

+
module _ {P : Rel A p} {Q : Rel A q} where

+

+
  zip : Linked P ∩ᵘ Linked Q ⊆ Linked (P ∩ᵇ Q)

+
  zip = zipWith id

+

+
  unzip : Linked (P ∩ᵇ Q) ⊆ Linked P ∩ᵘ Linked Q

+
  unzip = unzipWith id

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by Linked

+

+
module _ {R : Rel A ℓ} where

+

+
  linked? : B.Decidable R → U.Decidable (Linked R)

+
  linked? R? []           = yes []

+
  linked? R? (x ∷ [])     = yes [-]

+
  linked? R? (x ∷ y ∷ xs) =

+
    map′ (uncurry _∷_) < head , tail > (R? x y ×-dec linked? R? (y ∷ xs))

+

+
  irrelevant : B.Irrelevant R → U.Irrelevant (Linked R)

+
  irrelevant irr []           []           = refl

+
  irrelevant irr [-]          [-]          = refl

+
  irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =

+
    cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)

+

+
  satisfiable : U.Satisfiable (Linked R)

+
  satisfiable = [] , []

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Sorted.TotalOrder.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Sorted.TotalOrder.Properties.html
new file mode 100644
index 000000000..c2ae9c96f
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Sorted.TotalOrder.Properties.html
@@ -0,0 +1,229 @@
+
+Data.List.Relation.Unary.Sorted.TotalOrder.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sorted lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Sorted.TotalOrder.Properties where

+

+
open import Data.List.Base

+
open import Data.List.Relation.Unary.All using (All)

+
open import Data.List.Relation.Unary.AllPairs using (AllPairs)

+
open import Data.List.Relation.Unary.Linked as Linked

+
  using (Linked; []; [-]; _∷_; _∷′_; head′; tail)

+
import Data.List.Relation.Unary.Linked.Properties as Linked

+
import Data.List.Relation.Binary.Sublist.Setoid as Sublist

+
import Data.List.Relation.Binary.Sublist.Setoid.Properties as SublistProperties

+
open import Data.List.Relation.Unary.Sorted.TotalOrder hiding (head)

+

+
open import Data.Maybe.Base using (just; nothing)

+
open import Data.Maybe.Relation.Binary.Connected using (Connected; just)

+
open import Data.Nat.Base using (ℕ; zero; suc; _<_)

+

+
open import Level using (Level)

+
open import Relation.Binary.Core using (_Preserves_⟶_)

+
open import Relation.Binary.Bundles using (TotalOrder; DecTotalOrder; Preorder)

+
import Relation.Binary.Properties.TotalOrder as TotalOrderProperties

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Nullary.Decidable using (yes; no)

+

+
private

+
  variable

+
    a b p ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level

+

+
------------------------------------------------------------------------

+
-- Relationship to other predicates

+
------------------------------------------------------------------------

+

+
module _ (O : TotalOrder a ℓ₁ ℓ₂) where

+
  open TotalOrder O

+

+
  AllPairs⇒Sorted : ∀ {xs} → AllPairs _≤_ xs → Sorted O xs

+
  AllPairs⇒Sorted = Linked.AllPairs⇒Linked

+

+
  Sorted⇒AllPairs : ∀ {xs} → Sorted O xs → AllPairs _≤_ xs

+
  Sorted⇒AllPairs = Linked.Linked⇒AllPairs trans

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- map

+

+
module _ (O₁ : TotalOrder a ℓ₁ ℓ₂) (O₂ : TotalOrder a ℓ₁ ℓ₂) where

+
  private

+
    module O₁ = TotalOrder O₁

+
    module O₂ = TotalOrder O₂

+

+
  map⁺ : ∀ {f xs} → f Preserves O₁._≤_ ⟶ O₂._≤_ →

+
         Sorted O₁ xs → Sorted O₂ (map f xs)

+
  map⁺ pres xs↗ = Linked.map⁺ (Linked.map pres xs↗)

+

+
  map⁻ : ∀ {f xs} → (∀ {x y} → f x O₂.≤ f y → x O₁.≤ y) →

+
         Sorted O₂ (map f xs) → Sorted O₁ xs

+
  map⁻ pres fxs↗ = Linked.map pres (Linked.map⁻ fxs↗)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ (O : TotalOrder a ℓ₁ ℓ₂) where

+
  open TotalOrder O

+

+
  ++⁺ : ∀ {xs ys} → Sorted O xs → Connected _≤_ (last xs) (head ys) →

+
        Sorted O ys → Sorted O (xs ++ ys)

+
  ++⁺ = Linked.++⁺

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ (O : TotalOrder a ℓ₁ ℓ₂) where

+
  open TotalOrder O

+

+
  applyUpTo⁺₁ : ∀ f n → (∀ {i} → suc i < n → f i ≤ f (suc i)) →

+
                Sorted O (applyUpTo f n)

+
  applyUpTo⁺₁ = Linked.applyUpTo⁺₁

+

+
  applyUpTo⁺₂ : ∀ f n → (∀ i → f i ≤ f (suc i)) →

+
                Sorted O (applyUpTo f n)

+
  applyUpTo⁺₂ = Linked.applyUpTo⁺₂

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ (O : TotalOrder a ℓ₁ ℓ₂) where

+
  open TotalOrder O

+

+
  applyDownFrom⁺₁ : ∀ f n → (∀ {i} → suc i < n → f (suc i) ≤ f i) →

+
                    Sorted O (applyDownFrom f n)

+
  applyDownFrom⁺₁ = Linked.applyDownFrom⁺₁

+

+
  applyDownFrom⁺₂ : ∀ f n → (∀ i → f (suc i) ≤ f i) →

+
                    Sorted O (applyDownFrom f n)

+
  applyDownFrom⁺₂ = Linked.applyDownFrom⁺₂

+

+

+
------------------------------------------------------------------------

+
-- merge

+

+
module _ (DO : DecTotalOrder a ℓ₁ ℓ₂) where

+
  open DecTotalOrder DO using (_≤_; _≤?_) renaming (totalOrder to O)

+
  open TotalOrderProperties O using (≰⇒≥)

+

+
  private

+
    merge-con : ∀ {v xs ys} →

+
                Connected _≤_ (just v) (head xs) →

+
                Connected _≤_ (just v) (head ys) →

+
                Connected _≤_ (just v) (head (merge _≤?_ xs ys))

+
    merge-con {xs = []}              cxs cys = cys

+
    merge-con {xs = x ∷ xs} {[]}     cxs cys = cxs

+
    merge-con {xs = x ∷ xs} {y ∷ ys} cxs cys with x ≤? y

+
    ... | yes x≤y = cxs

+
    ... | no  x≰y = cys

+

+
  merge⁺ : ∀ {xs ys} → Sorted O xs → Sorted O ys → Sorted O (merge _≤?_ xs ys)

+
  merge⁺ {[]}              rxs rys = rys

+
  merge⁺ {x ∷ xs} {[]}     rxs rys = rxs

+
  merge⁺ {x ∷ xs} {y ∷ ys} rxs rys

+
   with x ≤? y  | merge⁺ (Linked.tail rxs) rys

+
                      | merge⁺ rxs (Linked.tail rys)

+
  ... | yes x≤y | rec | _   = merge-con (head′ rxs)      (just x≤y)  ∷′ rec

+
  ... | no  x≰y | _   | rec = merge-con (just (≰⇒≥ x≰y)) (head′ rys) ∷′ rec

+

+
  -- Reexport ⊆-mergeˡʳ

+

+
  S = Preorder.Eq.setoid (DecTotalOrder.preorder DO)

+
  open Sublist S using (_⊆_)

+
  module SP = SublistProperties S

+

+
  ⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys

+
  ⊆-mergeˡ = SP.⊆-mergeˡ _≤?_

+

+
  ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys

+
  ⊆-mergeʳ = SP.⊆-mergeʳ _≤?_

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (O : TotalOrder a ℓ₁ ℓ₂) {P : Pred _ p} (P? : Decidable P) where

+
  open TotalOrder O

+

+
  filter⁺ : ∀ {xs} → Sorted O xs → Sorted O (filter P? xs)

+
  filter⁺ = Linked.filter⁺ P? trans

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Sorted.TotalOrder.html b/site/static/agda_html/Data.List.Relation.Unary.Sorted.TotalOrder.html
new file mode 100644
index 000000000..59ce40c07
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Sorted.TotalOrder.html
@@ -0,0 +1,126 @@
+
+Data.List.Relation.Unary.Sorted.TotalOrder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sorted lists

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (TotalOrder)

+

+
module Data.List.Relation.Unary.Sorted.TotalOrder

+
  {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂) where

+

+
open TotalOrder totalOrder renaming (Carrier to A)

+

+
open import Data.List.Base using (List; []; _∷_)

+
open import Data.List.Relation.Unary.Linked as Linked using (Linked)

+
open import Level using (_⊔_)

+
open import Relation.Unary as U using (Pred; _⊆_)

+
open import Relation.Binary.Definitions as B

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
Sorted : Pred (List A) (a ⊔ ℓ₂)

+
Sorted xs = Linked _≤_ xs

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
module _ {x y xs} where

+

+
  head : Sorted (x ∷ y ∷ xs) → x ≤ y

+
  head = Linked.head

+

+
  tail : Sorted (x ∷ xs) → Sorted xs

+
  tail = Linked.tail

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by Sorted

+

+
sorted? : B.Decidable _≤_ → U.Decidable Sorted

+
sorted? = Linked.linked?

+

+
irrelevant : B.Irrelevant _≤_ → U.Irrelevant Sorted

+
irrelevant = Linked.irrelevant

+

+
satisfiable : U.Satisfiable Sorted

+
satisfiable = Linked.satisfiable

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.DecPropositional.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecPropositional.Properties.html
new file mode 100644
index 000000000..880a0b196
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecPropositional.Properties.html
@@ -0,0 +1,107 @@
+
+Data.List.Relation.Unary.Unique.DecPropositional.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of lists made up entirely of decidably unique elements

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Definitions using (DecidableEquality)

+

+
module Data.List.Relation.Unary.Unique.DecPropositional.Properties

+
  {a} {A : Set a} (_≟_ : DecidableEquality A) where

+

+
open import Data.List.Base using (deduplicate)

+
open import Data.List.Relation.Unary.All.Properties using (all-filter)

+
open import Data.List.Relation.Unary.Unique.DecPropositional _≟_

+
import Data.List.Relation.Unary.Unique.DecSetoid.Properties as Setoid

+
open import Level

+
open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)

+

+
------------------------------------------------------------------------

+
-- Re-export propositional properties

+

+
open import Data.List.Relation.Unary.Unique.Propositional.Properties public

+

+
------------------------------------------------------------------------

+
-- deduplicate

+

+
deduplicate-! : ∀ xs → Unique (deduplicate _≟_ xs)

+
deduplicate-! = Setoid.deduplicate-! (decSetoid _≟_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.DecPropositional.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecPropositional.html
new file mode 100644
index 000000000..bfe1aee73
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecPropositional.html
@@ -0,0 +1,96 @@
+
+Data.List.Relation.Unary.Unique.DecPropositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists made up entirely of unique elements (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (DecSetoid)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)

+

+
module Data.List.Relation.Unary.Unique.DecPropositional

+
  {a} {A : Set a} (_≟_ : DecidableEquality A) where

+

+
------------------------------------------------------------------------

+
-- Re-export setoid definition

+

+
open import Data.List.Relation.Unary.Unique.DecSetoid (decSetoid _≟_) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.DecSetoid.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecSetoid.Properties.html
new file mode 100644
index 000000000..2c048a167
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecSetoid.Properties.html
@@ -0,0 +1,109 @@
+
+Data.List.Relation.Unary.Unique.DecSetoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of lists made up entirely of decidably unique elements

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.List.Base using ([]; _∷_; deduplicate)

+
import Data.List.Relation.Unary.Unique.DecSetoid as Unique

+
open import Data.List.Relation.Unary.All.Properties using (all-filter)

+
open import Data.List.Relation.Unary.Unique.Setoid.Properties

+
open import Level

+
open import Relation.Binary.Bundles using (DecSetoid)

+

+
module Data.List.Relation.Unary.Unique.DecSetoid.Properties where

+

+
private

+
  variable

+
    a ℓ : Level

+

+
------------------------------------------------------------------------

+
-- deduplicate

+

+
module _ (DS : DecSetoid a ℓ) where

+

+
  open DecSetoid DS renaming (setoid to S)

+
  open Unique DS

+

+
  deduplicate-! : ∀ xs → Unique (deduplicate _≟_ xs)

+
  deduplicate-! []       = []

+
  deduplicate-! (x ∷ xs) = all-filter _ (deduplicate _≟_ xs) ∷ filter⁺ S _ (deduplicate-! xs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.DecSetoid.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecSetoid.html
new file mode 100644
index 000000000..3178e12c3
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.DecSetoid.html
@@ -0,0 +1,105 @@
+
+Data.List.Relation.Unary.Unique.DecSetoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists made up entirely of unique elements (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (DecSetoid)

+
import Data.List.Relation.Unary.AllPairs as AllPairs

+
open import Relation.Unary using (Decidable)

+
open import Relation.Nullary.Decidable using (¬?)

+

+
module Data.List.Relation.Unary.Unique.DecSetoid

+
  {a ℓ} (DS : DecSetoid a ℓ) where

+

+
open DecSetoid DS renaming (setoid to S)

+

+
------------------------------------------------------------------------

+
-- Re-export setoid definition

+

+
open import Data.List.Relation.Unary.Unique.Setoid S public

+

+
------------------------------------------------------------------------

+
-- Additional properties

+

+
unique? : Decidable Unique

+
unique? = AllPairs.allPairs? (λ x y → ¬? (x ≟ y))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.Properties.WithK.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.Properties.WithK.html
new file mode 100644
index 000000000..798d5a3cd
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.Properties.WithK.html
@@ -0,0 +1,125 @@
+
+Data.List.Relation.Unary.Unique.Propositional.Properties.WithK
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of unique lists (setoid equality) using K

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --safe --with-K #-}

+

+
module Data.List.Relation.Unary.Unique.Propositional.Properties.WithK where

+

+
open import Data.Empty

+
open import Data.List using (List)

+
open import Data.List.Membership.Propositional

+
open import Data.List.Relation.Binary.BagAndSetEquality

+
open import Data.List.Relation.Unary.All.Properties

+
open import Data.List.Relation.Unary.Any

+
open import Data.List.Relation.Unary.Unique.Propositional

+
open import Data.Product

+
open import Function.Bundles

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality

+
open import Relation.Nullary

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+
    l l' : List A

+

+
------------------------------------------------------------------------

+
-- membership

+

+
unique⇒∈-prop : Unique l → (x : A) → Irrelevant (x ∈ l)

+
unique⇒∈-prop (hx ∷ h) x (here refl) (here refl) = refl

+
unique⇒∈-prop (hx ∷ h) x (here refl) (there b)   = ⊥-elim (All¬⇒¬Any hx b)

+
unique⇒∈-prop (hx ∷ h) x (there a)   (here refl) = ⊥-elim (All¬⇒¬Any hx a)

+
unique⇒∈-prop (hx ∷ h) x (there a)   (there b) rewrite unique⇒∈-prop h x a b = refl

+

+
unique∧set⇒bag : Unique l → Unique l' → l ∼[ set ] l' → l ∼[ bag ] l'

+
unique∧set⇒bag h h' eq {a} = record

+
  { to = to

+
  ; from = from

+
  ; to-cong = to-cong

+
  ; from-cong = from-cong

+
  ; inverse = (λ {x} {y} _ → unique⇒∈-prop h' a (to y) x)

+
            , (λ {x} {y} _ → unique⇒∈-prop h a (from y) x)

+
  }

+
  where open Equivalence (eq {a})

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.Properties.html
new file mode 100644
index 000000000..011ddaed7
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.Properties.html
@@ -0,0 +1,233 @@
+
+Data.List.Relation.Unary.Unique.Propositional.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of unique lists (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Unique.Propositional.Properties where

+

+
open import Data.List.Base using (map; _++_; concat; cartesianProductWith;

+
  cartesianProduct; drop; take; applyUpTo; upTo; applyDownFrom; downFrom;

+
  tabulate; allFin; filter)

+
open import Data.List.Relation.Binary.Disjoint.Propositional

+
  using (Disjoint)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs)

+
open import Data.List.Relation.Unary.Unique.Propositional using (Unique)

+
import Data.List.Relation.Unary.Unique.Setoid.Properties as Setoid

+
open import Data.Fin.Base using (Fin)

+
open import Data.Nat.Base using (_<_)

+
open import Data.Nat.Properties using (<⇒≢)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)

+
open import Function.Base using (id; _∘_)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (refl; _≡_; _≢_; sym)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Nullary.Negation using (¬_)

+

+
private

+
  variable

+
    a b c p : Level

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {A : Set a} {B : Set b} where

+

+
  map⁺ : ∀ {f} → (∀ {x y} → f x ≡ f y → x ≡ y) →

+
         ∀ {xs} → Unique xs → Unique (map f xs)

+
  map⁺ = Setoid.map⁺ (setoid A) (setoid B)

+

+
------------------------------------------------------------------------

+
-- ++

+

+
module _ {A : Set a} {xs ys} where

+

+
  ++⁺ : Unique xs → Unique ys → Disjoint xs ys → Unique (xs ++ ys)

+
  ++⁺ = Setoid.++⁺ (setoid A)

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {A : Set a} {xss} where

+

+
  concat⁺ : All Unique xss → AllPairs Disjoint xss → Unique (concat xss)

+
  concat⁺ = Setoid.concat⁺ (setoid A)

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith

+

+
module _ {A : Set a} {B : Set b} {C : Set c} {xs ys} where

+

+
  cartesianProductWith⁺ : ∀ f → (∀ {w x y z} → f w y ≡ f x z → w ≡ x × y ≡ z) →

+
                          Unique xs → Unique ys →

+
                          Unique (cartesianProductWith f xs ys)

+
  cartesianProductWith⁺ = Setoid.cartesianProductWith⁺ (setoid A) (setoid B) (setoid C)

+

+
------------------------------------------------------------------------

+
-- cartesianProduct

+

+
module _ {A : Set a} {B : Set b} where

+

+
  cartesianProduct⁺ : ∀ {xs ys} → Unique xs → Unique ys →

+
                      Unique (cartesianProduct xs ys)

+
  cartesianProduct⁺ xs ys = AllPairs.map (_∘ ≡⇒≡×≡)

+
    (Setoid.cartesianProduct⁺ (setoid A) (setoid B) xs ys)

+

+
------------------------------------------------------------------------

+
-- take & drop

+

+
module _ {A : Set a} where

+

+
  drop⁺ : ∀ {xs} n → Unique xs → Unique (drop n xs)

+
  drop⁺ = Setoid.drop⁺ (setoid A)

+

+
  take⁺ : ∀ {xs} n → Unique xs → Unique (take n xs)

+
  take⁺ = Setoid.take⁺ (setoid A)

+

+
------------------------------------------------------------------------

+
-- applyUpTo & upTo

+

+
module _ {A : Set a} where

+

+
  applyUpTo⁺₁ : ∀ f n → (∀ {i j} → i < j → j < n → f i ≢ f j) →

+
                Unique (applyUpTo f n)

+
  applyUpTo⁺₁ = Setoid.applyUpTo⁺₁ (setoid A)

+

+
  applyUpTo⁺₂ : ∀ f n → (∀ i j → f i ≢ f j) →

+
                Unique (applyUpTo f n)

+
  applyUpTo⁺₂ = Setoid.applyUpTo⁺₂  (setoid A)

+

+
------------------------------------------------------------------------

+
-- upTo

+

+
upTo⁺ : ∀ n → Unique (upTo n)

+
upTo⁺ n = applyUpTo⁺₁ id n (λ i<j _ → <⇒≢ i<j)

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ {A : Set a} where

+

+
  applyDownFrom⁺₁ : ∀ f n → (∀ {i j} → j < i → i < n → f i ≢ f j) →

+
                    Unique (applyDownFrom f n)

+
  applyDownFrom⁺₁ = Setoid.applyDownFrom⁺₁ (setoid A)

+

+
  applyDownFrom⁺₂ : ∀ f n → (∀ i j → f i ≢ f j) →

+
                    Unique (applyDownFrom f n)

+
  applyDownFrom⁺₂ = Setoid.applyDownFrom⁺₂ (setoid A)

+

+
------------------------------------------------------------------------

+
-- downFrom

+

+
downFrom⁺ : ∀ n → Unique (downFrom n)

+
downFrom⁺ n = applyDownFrom⁺₁ id n (λ j<i _ → <⇒≢ j<i ∘ sym)

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {A : Set a} where

+

+
  tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → f i ≡ f j → i ≡ j) →

+
              Unique (tabulate f)

+
  tabulate⁺ = Setoid.tabulate⁺ (setoid A)

+

+
------------------------------------------------------------------------

+
-- allFin

+

+
allFin⁺ : ∀ n → Unique (allFin n)

+
allFin⁺ n = tabulate⁺ id

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ {A : Set a} {P : Pred _ p} (P? : Decidable P) where

+

+
  filter⁺ : ∀ {xs} → Unique xs → Unique (filter P? xs)

+
  filter⁺ = Setoid.filter⁺ (setoid A) P?

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.html
new file mode 100644
index 000000000..764cc092a
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.Propositional.html
@@ -0,0 +1,94 @@
+
+Data.List.Relation.Unary.Unique.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists made up entirely of unique elements (propositional equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Unique.Propositional {a} {A : Set a} where

+

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+
open import Data.List.Relation.Unary.Unique.Setoid as SetoidUnique

+

+
------------------------------------------------------------------------

+
-- Re-export the contents of setoid uniqueness

+

+
open SetoidUnique (setoid A) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.Setoid.Properties.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.Setoid.Properties.html
new file mode 100644
index 000000000..9e3fd41a2
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.Setoid.Properties.html
@@ -0,0 +1,235 @@
+
+Data.List.Relation.Unary.Unique.Setoid.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of unique lists (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Relation.Unary.Unique.Setoid.Properties where

+

+
open import Data.List.Base

+
open import Data.List.Membership.Setoid.Properties

+
open import Data.List.Relation.Binary.Disjoint.Setoid

+
open import Data.List.Relation.Binary.Disjoint.Setoid.Properties

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.List.Relation.Unary.All.Properties using (All¬⇒¬Any)

+
open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs)

+
open import Data.List.Relation.Unary.Unique.Setoid

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)

+
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)

+
import Data.List.Relation.Unary.AllPairs.Properties as AllPairs

+
open import Data.Fin.Base using (Fin)

+
open import Data.Nat.Base using (_<_)

+
open import Function.Base using (_∘_; id)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Nullary.Negation using (contraposition)

+

+
private

+
  variable

+
    a b c p ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for list operations

+
------------------------------------------------------------------------

+
-- map

+

+
module _ (S : Setoid a ℓ₁) (R : Setoid b ℓ₂) where

+

+
  open Setoid S renaming (_≈_ to _≈₁_)

+
  open Setoid R renaming (_≈_ to _≈₂_)

+

+
  map⁺ : ∀ {f} → (∀ {x y} → f x ≈₂ f y → x ≈₁ y) →

+
         ∀ {xs} → Unique S xs → Unique R (map f xs)

+
  map⁺ inj xs! = AllPairs.map⁺ (AllPairs.map (contraposition inj) xs!)

+

+
------------------------------------------------------------------------

+
-- ++

+

+
module _ (S : Setoid a ℓ) where

+

+
  ++⁺ : ∀ {xs ys} → Unique S xs → Unique S ys → Disjoint S xs ys → Unique S (xs ++ ys)

+
  ++⁺ xs! ys! xs#ys = AllPairs.++⁺ xs! ys! (Disjoint⇒AllAll S xs#ys)

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ (S : Setoid a ℓ) where

+

+
  concat⁺ : ∀ {xss} → All (Unique S) xss → AllPairs (Disjoint S) xss → Unique S (concat xss)

+
  concat⁺ xss! xss# = AllPairs.concat⁺ xss! (AllPairs.map (Disjoint⇒AllAll S) xss#)

+

+
------------------------------------------------------------------------

+
-- cartesianProductWith

+

+
module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) (U : Setoid c ℓ₃) where

+

+
  open Setoid S using () renaming (_≈_ to _≈₁_)

+
  open Setoid T using () renaming (_≈_ to _≈₂_)

+
  open Setoid U using () renaming (_≈_ to _≈₃_; sym to sym₃; trans to trans₃)

+

+
  cartesianProductWith⁺ : ∀ {xs ys} f → (∀ {w x y z} → f w y ≈₃ f x z → w ≈₁ x × y ≈₂ z) →

+
                          Unique S xs → Unique T ys →

+
                          Unique U (cartesianProductWith f xs ys)

+
  cartesianProductWith⁺ {_}      {_}  f f-inj  []          ys! = [] {S = U}

+
  cartesianProductWith⁺ {x ∷ xs} {ys} f f-inj (x∉xs ∷ xs!) ys! = ++⁺ U

+
    (map⁺ T U (proj₂ ∘ f-inj) ys!)

+
    (cartesianProductWith⁺ f f-inj xs! ys!)

+
    map#cartesianProductWith

+
    where

+
    map#cartesianProductWith : Disjoint U (map (f x) ys) (cartesianProductWith f xs ys)

+
    map#cartesianProductWith (v∈map , v∈com) with

+
      ∈-map⁻ T U v∈map | ∈-cartesianProductWith⁻ S T U f xs ys v∈com

+
    ... | (c , _ , v≈fxc) | (a , b , a∈xs , _ , v≈fab) =

+
      All¬⇒¬Any x∉xs (∈-resp-≈ S (proj₁ (f-inj (trans₃ (sym₃ v≈fab) v≈fxc))) a∈xs)

+

+
------------------------------------------------------------------------

+
-- cartesianProduct

+

+
module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) {xs ys} where

+

+
  cartesianProduct⁺ : Unique S xs → Unique T ys →

+
                      Unique (S ×ₛ T) (cartesianProduct xs ys)

+
  cartesianProduct⁺ = cartesianProductWith⁺ S T (S ×ₛ T) _,_ id

+

+
------------------------------------------------------------------------

+
-- take & drop

+

+
module _ (S : Setoid a ℓ) where

+

+
  drop⁺ : ∀ {xs} n → Unique S xs → Unique S (drop n xs)

+
  drop⁺ = AllPairs.drop⁺

+

+
  take⁺ : ∀ {xs} n → Unique S xs → Unique S (take n xs)

+
  take⁺ = AllPairs.take⁺

+

+
------------------------------------------------------------------------

+
-- applyUpTo

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S

+

+
  applyUpTo⁺₁ : ∀ f n → (∀ {i j} → i < j → j < n → f i ≉ f j) →

+
                Unique S (applyUpTo f n)

+
  applyUpTo⁺₁ = AllPairs.applyUpTo⁺₁

+

+
  applyUpTo⁺₂ : ∀ f n → (∀ i j → f i ≉ f j) →

+
                Unique S (applyUpTo f n)

+
  applyUpTo⁺₂ = AllPairs.applyUpTo⁺₂

+

+
------------------------------------------------------------------------

+
-- applyDownFrom

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S

+

+
  applyDownFrom⁺₁ : ∀ f n → (∀ {i j} → j < i → i < n → f i ≉ f j) →

+
                    Unique S (applyDownFrom f n)

+
  applyDownFrom⁺₁ = AllPairs.applyDownFrom⁺₁

+

+
  applyDownFrom⁺₂ : ∀ f n → (∀ i j → f i ≉ f j) →

+
                    Unique S (applyDownFrom f n)

+
  applyDownFrom⁺₂ = AllPairs.applyDownFrom⁺₂

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S renaming (Carrier to A)

+

+
  tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → f i ≈ f j → i ≡ j) →

+
              Unique S (tabulate f)

+
  tabulate⁺ f-inj = AllPairs.tabulate⁺ (_∘ f-inj)

+

+
------------------------------------------------------------------------

+
-- filter

+

+
module _ (S : Setoid a ℓ) {P : Pred _ p} (P? : Decidable P) where

+

+
  filter⁺ : ∀ {xs} → Unique S xs → Unique S (filter P? xs)

+
  filter⁺ = AllPairs.filter⁺ P?

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Relation.Unary.Unique.Setoid.html b/site/static/agda_html/Data.List.Relation.Unary.Unique.Setoid.html
new file mode 100644
index 000000000..334234ab8
--- /dev/null
+++ b/site/static/agda_html/Data.List.Relation.Unary.Unique.Setoid.html
@@ -0,0 +1,105 @@
+
+Data.List.Relation.Unary.Unique.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists made up entirely of unique elements (setoid equality)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Nullary.Negation using (¬_)

+

+
module Data.List.Relation.Unary.Unique.Setoid {a ℓ} (S : Setoid a ℓ) where

+

+
open Setoid S renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
private

+
  Distinct : Rel A ℓ

+
  Distinct x y = ¬ (x ≈ y)

+

+
open import Data.List.Relation.Unary.AllPairs.Core Distinct public

+
  renaming (AllPairs to Unique)

+

+
open import Data.List.Relation.Unary.AllPairs {R = Distinct} public

+
  using (head; tail)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Scans.Base.html b/site/static/agda_html/Data.List.Scans.Base.html
new file mode 100644
index 000000000..2d47e7ca3
--- /dev/null
+++ b/site/static/agda_html/Data.List.Scans.Base.html
@@ -0,0 +1,126 @@
+
+Data.List.Scans.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- List scans: definitions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.List.Scans.Base where

+

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.List.NonEmpty.Base as List⁺ using (List⁺; _∷_; toList)

+
open import Function.Base using (_∘_)

+
open import Level using (Level)

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
-- Scanr

+

+
module _ (f : A → B → B) where

+

+
  scanr⁺ : (e : B) → List A → List⁺ B

+
  scanr⁺ e []       = e ∷ []

+
  scanr⁺ e (x ∷ xs) = let y ∷ ys = scanr⁺ e xs in f x y ∷ y ∷ ys

+

+
  scanr : (e : B) → List A → List B

+
  scanr e = toList ∘ scanr⁺ e

+

+
-- Scanl

+

+
module _ (f : A → B → A) where

+

+
  scanl⁺ : A → List B → List⁺ A

+
  scanl⁺ e xs = e ∷ go e xs

+
    where

+
    go : A → List B → List A

+
    go _ []       = []

+
    go e (x ∷ xs) = let fex = f e x in fex ∷ go fex xs

+

+
  scanl : A → List B → List A

+
  scanl e = toList ∘ scanl⁺ e

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Sort.Base.html b/site/static/agda_html/Data.List.Sort.Base.html
new file mode 100644
index 000000000..df94d9afa
--- /dev/null
+++ b/site/static/agda_html/Data.List.Sort.Base.html
@@ -0,0 +1,109 @@
+
+Data.List.Sort.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The core definition of a sorting algorithm

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.List.Base using (List)

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
open import Level using (_⊔_)

+
open import Relation.Binary.Bundles using (TotalOrder)

+

+
module Data.List.Sort.Base

+
  {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂)

+
  where

+

+
open TotalOrder totalOrder renaming (Carrier to A)

+
open import Data.List.Relation.Unary.Sorted.TotalOrder totalOrder

+

+
------------------------------------------------------------------------

+
-- Definition of a sorting algorithm

+

+
record SortingAlgorithm : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    sort   : List A → List A

+

+
    -- The output of `sort` is a permutation of the input

+
    sort-↭ : ∀ xs → sort xs ↭ xs

+

+
    -- The output of `sort` is sorted.

+
    sort-↗ : ∀ xs → Sorted (sort xs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Sort.MergeSort.html b/site/static/agda_html/Data.List.Sort.MergeSort.html
new file mode 100644
index 000000000..d7cbbe3c9
--- /dev/null
+++ b/site/static/agda_html/Data.List.Sort.MergeSort.html
@@ -0,0 +1,194 @@
+
+Data.List.Sort.MergeSort
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An implementation of merge sort along with proofs of correctness.

+
------------------------------------------------------------------------

+

+
-- Unless you are need a particular property of MergeSort, you should

+
-- import and use the sorting algorithm from `Data.List.Sort` instead

+
-- of this file.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (DecTotalOrder)

+

+
module Data.List.Sort.MergeSort

+
  {a ℓ₁ ℓ₂} (O : DecTotalOrder a ℓ₁ ℓ₂) where

+
open import Data.Bool.Base using (true; false)

+
open import Data.List.Base

+
  using (List; []; _∷_; merge; length; map; [_]; concat; _++_)

+
open import Data.List.Properties using (length-partition; ++-assoc; concat-[-])

+
open import Data.List.Relation.Unary.Linked using ([]; [-])

+
import Data.List.Relation.Unary.Sorted.TotalOrder.Properties as Sorted

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
import Data.List.Relation.Unary.All.Properties as All

+
open import Data.List.Relation.Binary.Permutation.Propositional

+
import Data.List.Relation.Binary.Permutation.Propositional.Properties as Perm

+
open import Data.Maybe.Base using (just)

+
open import Data.Nat.Base using (_<_; _>_; z<s; s<s)

+
open import Data.Nat.Induction

+
open import Data.Nat.Properties using (m<n⇒m<1+n)

+
open import Data.Product.Base as Product using (_,_)

+
open import Function.Base using (_∘_)

+
open import Relation.Nullary.Negation.Core using (¬_)

+
open import Relation.Nullary.Decidable.Core using (does)

+

+
open DecTotalOrder O renaming (Carrier to A)

+

+
open import Data.List.Sort.Base totalOrder

+
open import Data.List.Relation.Unary.Sorted.TotalOrder totalOrder hiding (head)

+
open import Relation.Binary.Properties.DecTotalOrder O using (≰⇒≥; ≰-respˡ-≈)

+

+
open PermutationReasoning

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
mergePairs : List (List A) → List (List A)

+
mergePairs (xs ∷ ys ∷ yss) = merge _≤?_ xs ys ∷ mergePairs yss

+
mergePairs xss             = xss

+

+
private

+
  length-mergePairs : ∀ xs ys yss → let zss = xs ∷ ys ∷ yss in

+
                      length (mergePairs zss) < length zss

+
  length-mergePairs _ _ []              = s<s z<s

+
  length-mergePairs _ _ (xs ∷ [])       = s<s (s<s z<s)

+
  length-mergePairs _ _ (xs ∷ ys ∷ yss) = s<s (m<n⇒m<1+n (length-mergePairs xs ys yss))

+

+
mergeAll : (xss : List (List A)) → Acc _<_ (length xss) → List A

+
mergeAll []        _               = []

+
mergeAll (xs ∷ []) _               = xs

+
mergeAll xss@(xs ∷ ys ∷ yss) (acc rec) = mergeAll

+
  (mergePairs xss) (rec (length-mergePairs xs ys yss))

+

+
sort : List A → List A

+
sort xs = mergeAll (map [_] xs) (<-wellFounded-fast _)

+

+
------------------------------------------------------------------------

+
-- Permutation property

+

+
mergePairs-↭ : ∀ xss → concat (mergePairs xss) ↭ concat xss

+
mergePairs-↭ []              = ↭-refl

+
mergePairs-↭ (xs ∷ [])       = ↭-refl

+
mergePairs-↭ (xs ∷ ys ∷ xss) = begin

+
  merge _ xs ys ++ concat (mergePairs xss) ↭⟨ Perm.++⁺ (Perm.merge-↭ _ xs ys) (mergePairs-↭ xss) ⟩

+
  (xs ++ ys)    ++ concat xss              ≡⟨ ++-assoc xs ys (concat xss) ⟩

+
  xs ++ ys      ++ concat xss              ∎

+

+
mergeAll-↭ : ∀ xss (rec : Acc _<_ (length xss)) → mergeAll xss rec ↭ concat xss

+
mergeAll-↭ []              _ = ↭-refl

+
mergeAll-↭ (xs ∷ [])       _ = ↭-sym (Perm.++-identityʳ xs)

+
mergeAll-↭ (xs ∷ ys ∷ xss) (acc rec) = begin

+
  mergeAll (mergePairs (xs ∷ ys ∷ xss)) _ ↭⟨ mergeAll-↭ (mergePairs (xs ∷ ys ∷ xss)) _ ⟩

+
  concat   (mergePairs (xs ∷ ys ∷ xss))   ↭⟨ mergePairs-↭ (xs ∷ ys ∷ xss) ⟩

+
  concat   (xs ∷ ys ∷ xss)                ∎

+

+
sort-↭ : ∀ xs → sort xs ↭ xs

+
sort-↭ xs = begin

+
  mergeAll (map [_] xs) _ ↭⟨ mergeAll-↭ (map [_] xs) _ ⟩

+
  concat (map [_] xs)     ≡⟨ concat-[-] xs ⟩

+
  xs                      ∎

+

+
------------------------------------------------------------------------

+
-- Sorted property

+

+
mergePairs-↗ : ∀ {xss} → All Sorted xss → All Sorted (mergePairs xss)

+
mergePairs-↗ []                 = []

+
mergePairs-↗ (xs↗ ∷ [])         = xs↗ ∷ []

+
mergePairs-↗ (xs↗ ∷ ys↗ ∷ xss↗) = Sorted.merge⁺ O xs↗ ys↗ ∷ mergePairs-↗ xss↗

+

+
mergeAll-↗ : ∀ {xss} (rec : Acc _<_ (length xss)) →

+
             All Sorted xss → Sorted (mergeAll xss rec)

+
mergeAll-↗ rec       []                 = []

+
mergeAll-↗ rec       (xs↗ ∷ [])         = xs↗

+
mergeAll-↗ (acc rec) (xs↗ ∷ ys↗ ∷ xss↗) = mergeAll-↗ _ (mergePairs-↗ (xs↗ ∷ ys↗ ∷ xss↗))

+

+
sort-↗ : ∀ xs → Sorted (sort xs)

+
sort-↗ xs = mergeAll-↗ _ (All.map⁺ (All.universal (λ _ → [-]) xs))

+

+
------------------------------------------------------------------------

+
-- Algorithm

+

+
mergeSort : SortingAlgorithm

+
mergeSort = record

+
  { sort   = sort

+
  ; sort-↭ = sort-↭

+
  ; sort-↗ = sort-↗

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.Sort.html b/site/static/agda_html/Data.List.Sort.html
new file mode 100644
index 000000000..a91d225a7
--- /dev/null
+++ b/site/static/agda_html/Data.List.Sort.html
@@ -0,0 +1,118 @@
+
+Data.List.Sort
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Functions and definitions for sorting lists

+
------------------------------------------------------------------------

+

+
-- See `Data.List.Relation.Unary.Sorted` for the property of a list

+
-- being sorted.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.List.Base using (List)

+
open import Relation.Binary.Bundles using (DecTotalOrder)

+

+
module Data.List.Sort

+
  {a ℓ₁ ℓ₂} (O : DecTotalOrder a ℓ₁ ℓ₂)

+
  where

+

+
open DecTotalOrder O renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Re-export core definitions

+

+
open import Data.List.Sort.Base totalOrder public

+
  using (SortingAlgorithm)

+

+
------------------------------------------------------------------------

+
-- An instance of a sorting algorithm

+

+
open import Data.List.Sort.MergeSort O using (mergeSort)

+

+
abstract

+
  sortingAlgorithm : SortingAlgorithm

+
  sortingAlgorithm = mergeSort

+

+
open SortingAlgorithm sortingAlgorithm public

+
  using

+
  ( sort   -- : List A → List A

+
  ; sort-↭ -- : ∀ xs → sort xs ↭ xs

+
  ; sort-↗ -- : ∀ xs → Sorted (sort xs)

+
  )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.List.html b/site/static/agda_html/Data.List.html
new file mode 100644
index 000000000..db0443b8a
--- /dev/null
+++ b/site/static/agda_html/Data.List.html
@@ -0,0 +1,98 @@
+
+Data.List
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lists

+
------------------------------------------------------------------------

+

+
-- See README.Data.List for examples of how to use and reason about

+
-- lists.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+
{-# OPTIONS --warn=noUserWarning #-} -- for deprecated scans

+

+
module Data.List where

+

+
------------------------------------------------------------------------

+
-- Types and basic operations

+

+
open import Data.List.Base public

+
  hiding (scanr; scanl)

+
open import Data.List.Scans.Base public

+
  using (scanr; scanl)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Base.html b/site/static/agda_html/Data.Maybe.Base.html
new file mode 100644
index 000000000..ffb9e65e4
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Base.html
@@ -0,0 +1,222 @@
+
+Data.Maybe.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The Maybe type and some operations

+
------------------------------------------------------------------------

+

+
-- The definitions in this file are reexported by Data.Maybe.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Base where

+

+
open import Level using (Level; Lift)

+
open import Data.Bool.Base using (Bool; true; false; not)

+
open import Data.Unit.Base using (⊤)

+
open import Data.These.Base using (These; this; that; these)

+
open import Data.Product.Base as Prod using (_×_; _,_)

+
open import Function.Base using (_∘_; id; const)

+
import Relation.Nullary.Decidable.Core as Dec

+

+
private

+
  variable

+
    a b c : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open import Agda.Builtin.Maybe public

+
  using (Maybe; just; nothing)

+

+
------------------------------------------------------------------------

+
-- Some operations

+

+
boolToMaybe : Bool → Maybe ⊤

+
boolToMaybe true  = just _

+
boolToMaybe false = nothing

+

+
is-just : Maybe A → Bool

+
is-just (just _) = true

+
is-just nothing  = false

+

+
is-nothing : Maybe A → Bool

+
is-nothing = not ∘ is-just

+

+
-- A dependent eliminator.

+

+
maybe : ∀ {A : Set a} {B : Maybe A → Set b} →

+
        ((x : A) → B (just x)) → B nothing → (x : Maybe A) → B x

+
maybe j n (just x) = j x

+
maybe j n nothing  = n

+

+
-- A non-dependent eliminator.

+

+
maybe′ : (A → B) → B → Maybe A → B

+
maybe′ = maybe

+

+
-- A defaulting mechanism

+

+
fromMaybe : A → Maybe A → A

+
fromMaybe = maybe′ id

+

+
-- A safe variant of "fromJust". If the value is nothing, then the

+
-- return type is the unit type.

+

+
module _ {a} {A : Set a} where

+

+
  From-just : Maybe A → Set a

+
  From-just (just _) = A

+
  From-just nothing  = Lift a ⊤

+

+
  from-just : (x : Maybe A) → From-just x

+
  from-just (just x) = x

+
  from-just nothing  = _

+

+
-- Functoriality: map

+

+
map : (A → B) → Maybe A → Maybe B

+
map f = maybe (just ∘ f) nothing

+

+
-- Applicative: ap

+

+
ap : Maybe (A → B) → Maybe A → Maybe B

+
ap nothing  = const nothing

+
ap (just f) = map f

+

+
-- Monad: bind

+

+
infixl 1 _>>=_

+
_>>=_ : Maybe A → (A → Maybe B) → Maybe B

+
nothing >>= f = nothing

+
just a  >>= f = f a

+

+
-- Alternative: <∣>

+

+
infixr 6 _<∣>_

+
_<∣>_ : Maybe A → Maybe A → Maybe A

+
just x  <∣> my = just x

+
nothing <∣> my = my

+

+
-- Just when the boolean is true

+

+
when : Bool → A → Maybe A

+
when b c = map (const c) (boolToMaybe b)

+

+
------------------------------------------------------------------------

+
-- Aligning and zipping

+

+
alignWith : (These A B → C) → Maybe A → Maybe B → Maybe C

+
alignWith f (just a) (just b) = just (f (these a b))

+
alignWith f (just a) nothing  = just (f (this a))

+
alignWith f nothing  (just b) = just (f (that b))

+
alignWith f nothing  nothing  = nothing

+

+
zipWith : (A → B → C) → Maybe A → Maybe B → Maybe C

+
zipWith f (just a) (just b) = just (f a b)

+
zipWith _ _        _        = nothing

+

+
align : Maybe A → Maybe B → Maybe (These A B)

+
align = alignWith id

+

+
zip : Maybe A → Maybe B → Maybe (A × B)

+
zip = zipWith _,_

+

+
------------------------------------------------------------------------

+
-- Injections.

+

+
thisM : A → Maybe B → These A B

+
thisM a = maybe′ (these a) (this a)

+

+
thatM : Maybe A → B → These A B

+
thatM = maybe′ these that

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.1

+
-- decToMaybe

+

+
open Dec using (decToMaybe) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Effectful.html b/site/static/agda_html/Data.Maybe.Effectful.html
new file mode 100644
index 000000000..ddeea6336
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Effectful.html
@@ -0,0 +1,185 @@
+
+Data.Maybe.Effectful
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An effectful view of Maybe

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Effectful where

+

+
open import Level

+
open import Data.Maybe.Base

+

+
open import Effect.Choice

+
open import Effect.Empty

+
open import Effect.Functor

+
open import Effect.Applicative

+
open import Effect.Monad

+

+
open import Function.Base

+

+
private

+
  variable

+
    a b f g m n : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Maybe applicative functor

+

+
functor : RawFunctor {f} Maybe

+
functor = record

+
  { _<$>_ = map

+
  }

+

+
applicative : RawApplicative {f} Maybe

+
applicative = record

+
  { rawFunctor = functor

+
  ; pure       = just

+
  ; _<*>_      = maybe map (const nothing)

+
  }

+

+
empty : RawEmpty {f} Maybe

+
empty = record { empty = nothing }

+

+
choice : RawChoice {f} Maybe

+
choice = record { _<|>_ = _<∣>_ }

+

+
applicativeZero : RawApplicativeZero {f} Maybe

+
applicativeZero = record

+
  { rawApplicative = applicative

+
  ; rawEmpty       = empty

+
  }

+

+
alternative : RawAlternative {f} Maybe

+
alternative = record

+
  { rawApplicativeZero = applicativeZero

+
  ; rawChoice          = choice

+
  }

+

+
------------------------------------------------------------------------

+
-- Maybe monad

+

+
monad : RawMonad {f} Maybe

+
monad = record

+
  { rawApplicative = applicative

+
  ; _>>=_ = _>>=_

+
  }

+

+
join : Maybe (Maybe A) → Maybe A

+
join = Join.join monad

+

+
monadZero : RawMonadZero {f} Maybe

+
monadZero = record

+
  { rawMonad = monad

+
  ; rawEmpty = empty

+
  }

+

+
monadPlus : RawMonadPlus {f} Maybe

+
monadPlus {f} = record

+
  { rawMonadZero = monadZero

+
  ; rawChoice    = choice

+
  }

+

+
module TraversableA {F} (App : RawApplicative {f} {g} F) where

+

+
  open RawApplicative App

+

+
  sequenceA : Maybe (F A) → F (Maybe A)

+
  sequenceA nothing  = pure nothing

+
  sequenceA (just x) = just <$> x

+

+
  mapA : (A → F B) → Maybe A → F (Maybe B)

+
  mapA f = sequenceA ∘ map f

+

+
  forA : Maybe A → (A → F B) → F (Maybe B)

+
  forA = flip mapA

+

+
module TraversableM {M} (Mon : RawMonad {m} {n} M) where

+

+
  open RawMonad Mon

+

+
  open TraversableA rawApplicative public

+
    renaming

+
    ( sequenceA to sequenceM

+
    ; mapA      to mapM

+
    ; forA      to forM

+
    )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Properties.html b/site/static/agda_html/Data.Maybe.Properties.html
new file mode 100644
index 000000000..a332ad3fd
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Properties.html
@@ -0,0 +1,265 @@
+
+Data.Maybe.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Maybe-related properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Properties where

+

+
open import Algebra.Bundles using (Semigroup; Monoid)

+
import Algebra.Structures as Structures

+
import Algebra.Definitions as Definitions

+
open import Data.Maybe.Base using (Maybe; just; nothing; map; _<∣>_;

+
  maybe; maybe′)

+
open import Data.Maybe.Relation.Unary.All using (All; just; nothing)

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_∋_; id; _∘_; _∘′_)

+
open import Function.Definitions using (Injective)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂; _≗_)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (isEquivalence)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Nullary.Decidable using (yes; no)

+
open import Relation.Nullary.Decidable using (map′)

+

+
private

+
  variable

+
    a b c : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Equality

+

+
just-injective : ∀ {x y} → (Maybe A ∋ just x) ≡ just y → x ≡ y

+
just-injective refl = refl

+

+
≡-dec : DecidableEquality A → DecidableEquality (Maybe A)

+
≡-dec _≟_ nothing  nothing  = yes refl

+
≡-dec _≟_ (just x) nothing  = no λ()

+
≡-dec _≟_ nothing  (just y) = no λ()

+
≡-dec _≟_ (just x) (just y) = map′ (cong just) just-injective (x ≟ y)

+

+
------------------------------------------------------------------------

+
-- map

+

+
map-id : map id ≗ id {A = Maybe A}

+
map-id (just x) = refl

+
map-id nothing  = refl

+

+
map-id-local : ∀ {f : A → A} {mx} → All (λ x → f x ≡ x) mx → map f mx ≡ mx

+
map-id-local (just eq) = cong just eq

+
map-id-local nothing   = refl

+

+
map-<∣> : ∀ (f : A → B) mx my →

+
  map f (mx <∣> my) ≡ map f mx <∣> map f my

+
map-<∣> f (just x) my = refl

+
map-<∣> f nothing  my = refl

+

+
map-cong : {f g : A → B} → f ≗ g → map f ≗ map g

+
map-cong f≗g (just x) = cong just (f≗g x)

+
map-cong f≗g nothing  = refl

+

+
map-cong-local : ∀ {f g : A → B} {mx} →

+
  All (λ x → f x ≡ g x) mx → map f mx ≡ map g mx

+
map-cong-local (just eq) = cong just eq

+
map-cong-local nothing   = refl

+

+
map-injective : ∀ {f : A → B} → Injective _≡_ _≡_ f → Injective _≡_ _≡_ (map f)

+
map-injective f-inj {nothing} {nothing} p = refl

+
map-injective f-inj {just x} {just y} p = cong just (f-inj (just-injective p))

+

+
map-∘ : {g : B → C} {f : A → B} → map (g ∘ f) ≗ map g ∘ map f

+
map-∘ (just x) = refl

+
map-∘ nothing  = refl

+

+
map-nothing : ∀ {f : A → B} {ma} → ma ≡ nothing → map f ma ≡ nothing

+
map-nothing refl = refl

+

+
map-just : ∀ {f : A → B} {ma a} → ma ≡ just a → map f ma ≡ just (f a)

+
map-just refl = refl

+

+
------------------------------------------------------------------------

+
-- maybe

+

+
maybe-map : ∀ {C : Maybe B → Set c}

+
            (j : (x : B) → C (just x)) (n : C nothing) (f : A → B) ma →

+
            maybe {B = C} j n (map f ma) ≡ maybe {B = C ∘ map f} (j ∘ f) n ma

+
maybe-map j n f (just x) = refl

+
maybe-map j n f nothing  = refl

+

+
maybe′-map : ∀ j (n : C) (f : A → B) ma →

+
             maybe′ j n (map f ma) ≡ maybe′ (j ∘′ f) n ma

+
maybe′-map = maybe-map

+

+
------------------------------------------------------------------------

+
-- _<∣>_

+

+
module _ {A : Set a} where

+

+
  open Definitions {A = Maybe A} _≡_

+

+
  <∣>-assoc : Associative _<∣>_

+
  <∣>-assoc (just x) y z = refl

+
  <∣>-assoc nothing  y z = refl

+

+
  <∣>-identityˡ : LeftIdentity nothing _<∣>_

+
  <∣>-identityˡ (just x) = refl

+
  <∣>-identityˡ nothing  = refl

+

+
  <∣>-identityʳ : RightIdentity nothing _<∣>_

+
  <∣>-identityʳ (just x) = refl

+
  <∣>-identityʳ nothing  = refl

+

+
  <∣>-identity : Identity nothing _<∣>_

+
  <∣>-identity = <∣>-identityˡ , <∣>-identityʳ

+

+
  <∣>-idem : Idempotent _<∣>_

+
  <∣>-idem (just x) = refl

+
  <∣>-idem nothing = refl

+

+
module _ (A : Set a) where

+

+
  open Structures {A = Maybe A} _≡_

+

+
  <∣>-isMagma : IsMagma _<∣>_

+
  <∣>-isMagma = record

+
    { isEquivalence = isEquivalence

+
    ; ∙-cong        = cong₂ _<∣>_

+
    }

+

+
  <∣>-isSemigroup : IsSemigroup _<∣>_

+
  <∣>-isSemigroup = record

+
    { isMagma = <∣>-isMagma

+
    ; assoc   = <∣>-assoc

+
    }

+

+
  <∣>-isMonoid : IsMonoid _<∣>_ nothing

+
  <∣>-isMonoid = record

+
    { isSemigroup = <∣>-isSemigroup

+
    ; identity    = <∣>-identity

+
    }

+

+
  <∣>-semigroup : Semigroup a a

+
  <∣>-semigroup = record

+
    { isSemigroup = <∣>-isSemigroup

+
    }

+

+
  <∣>-monoid : Monoid a a

+
  <∣>-monoid = record

+
    { isMonoid = <∣>-isMonoid

+
    }

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
map-id₂ = map-id-local

+
{-# WARNING_ON_USAGE map-id₂

+
"Warning: map-id₂ was deprecated in v2.0.

+
Please use map-id-local instead."

+
#-}

+

+
map-cong₂ = map-cong-local

+
{-# WARNING_ON_USAGE map-id₂

+
"Warning: map-cong₂ was deprecated in v2.0.

+
Please use map-cong-local instead."

+
#-}

+

+
map-compose = map-∘

+
{-# WARNING_ON_USAGE map-compose

+
"Warning: map-compose was deprecated in v2.0.

+
Please use map-∘ instead."

+
#-}

+

+
map-<∣>-commute = map-<∣>

+
{-# WARNING_ON_USAGE map-<∣>-commute

+
"Warning: map-<∣>-commute was deprecated in v2.0.

+
Please use map-<∣> instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Relation.Binary.Connected.html b/site/static/agda_html/Data.Maybe.Relation.Binary.Connected.html
new file mode 100644
index 000000000..6edc0bda8
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Relation.Binary.Connected.html
@@ -0,0 +1,144 @@
+
+Data.Maybe.Relation.Binary.Connected
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lifting a relation such that `nothing` is also related to `just`

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Relation.Binary.Connected where

+

+
open import Level

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Relation.Binary.Core using (REL; _⇒_)

+
open import Relation.Binary.Definitions using (Reflexive; Sym; Decidable)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Nullary

+
import Relation.Nullary.Decidable as Dec

+

+
private

+
  variable

+
    a b ℓ : Level

+
    A : Set a

+
    B : Set b

+
    R S T : REL A B ℓ

+
    x y : A

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Connected {A : Set a} {B : Set b} (R : REL A B ℓ)

+
               : REL (Maybe A) (Maybe B) (a ⊔ b ⊔ ℓ) where

+
  just         : R x y → Connected R (just x) (just y)

+
  just-nothing : Connected R (just x) nothing

+
  nothing-just : Connected R nothing (just y)

+
  nothing      : Connected R nothing nothing

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
drop-just : Connected R (just x) (just y) → R x y

+
drop-just (just p) = p

+

+
just-equivalence : R x y ⇔ Connected R (just x) (just y)

+
just-equivalence = mk⇔ just drop-just

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
refl : Reflexive R → Reflexive (Connected R)

+
refl R-refl {just _}  = just R-refl

+
refl R-refl {nothing} = nothing

+

+
reflexive : _≡_ ⇒ R → _≡_ ⇒ Connected R

+
reflexive reflexive ≡.refl = refl (reflexive ≡.refl)

+

+
sym : Sym R S → Sym (Connected R) (Connected S)

+
sym R-sym (just p)     = just (R-sym p)

+
sym R-sym nothing-just = just-nothing

+
sym R-sym just-nothing = nothing-just

+
sym R-sym nothing      = nothing

+

+
connected? : Decidable R → Decidable (Connected R)

+
connected? R? (just x) (just y) = Dec.map just-equivalence (R? x y)

+
connected? R? (just x) nothing  = yes just-nothing

+
connected? R? nothing  (just y) = yes nothing-just

+
connected? R? nothing  nothing  = yes nothing

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Relation.Binary.Pointwise.html b/site/static/agda_html/Data.Maybe.Relation.Binary.Pointwise.html
new file mode 100644
index 000000000..3efa813cf
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Relation.Binary.Pointwise.html
@@ -0,0 +1,188 @@
+
+Data.Maybe.Relation.Binary.Pointwise
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise lifting of relations to maybes

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Relation.Binary.Pointwise where

+

+
open import Level

+
open import Data.Product.Base using (∃; _×_; -,_; _,_)

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Data.Maybe.Relation.Unary.Any using (Any; just)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Relation.Binary.Core using (REL; Rel; _⇒_)

+
open import Relation.Binary.Bundles using (Setoid; DecSetoid)

+
open import Relation.Binary.Definitions using (Reflexive; Sym; Trans; Decidable)

+
open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Nullary

+
open import Relation.Unary using (_⊆_)

+
import Relation.Nullary.Decidable as Dec

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Pointwise

+
       {a b ℓ} {A : Set a} {B : Set b}

+
       (R : REL A B ℓ) : REL (Maybe A) (Maybe B) (a ⊔ b ⊔ ℓ) where

+
  just    : ∀ {x y} → R x y → Pointwise R (just x) (just y)

+
  nothing : Pointwise R nothing nothing

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ {a b ℓ} {A : Set a} {B : Set b} {R : REL A B ℓ} where

+

+
  drop-just : ∀ {x y} → Pointwise R (just x) (just y) → R x y

+
  drop-just (just p) = p

+

+
  just-equivalence : ∀ {x y} → R x y ⇔ Pointwise R (just x) (just y)

+
  just-equivalence = mk⇔ just drop-just

+

+
  nothing-inv : ∀ {x} → Pointwise R nothing x → x ≡ nothing

+
  nothing-inv nothing = ≡.refl

+

+
  just-inv : ∀ {x y} → Pointwise R (just x) y → ∃ λ z → y ≡ just z × R x z

+
  just-inv (just r) = -, ≡.refl , r

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
module _ {a r} {A : Set a} {R : Rel A r} where

+

+
  refl : Reflexive R → Reflexive (Pointwise R)

+
  refl R-refl {just _}  = just R-refl

+
  refl R-refl {nothing} = nothing

+

+
  reflexive : _≡_ ⇒ R → _≡_ ⇒ Pointwise R

+
  reflexive reflexive ≡.refl = refl (reflexive ≡.refl)

+

+
module _ {a b r₁ r₂} {A : Set a} {B : Set b}

+
         {R : REL A B r₁} {S : REL B A r₂} where

+

+
  sym : Sym R S → Sym (Pointwise R) (Pointwise S)

+
  sym R-sym (just p) = just (R-sym p)

+
  sym R-sym nothing  = nothing

+

+
module _ {a b c r₁ r₂ r₃} {A : Set a} {B : Set b} {C : Set c}

+
         {R : REL A B r₁} {S : REL B C r₂} {T : REL A C r₃} where

+

+
  trans : Trans R S T → Trans (Pointwise R) (Pointwise S) (Pointwise T)

+
  trans R-trans (just p) (just q) = just (R-trans p q)

+
  trans R-trans nothing  nothing  = nothing

+

+
module _ {a r} {A : Set a} {R : Rel A r} where

+

+
  dec : Decidable R → Decidable (Pointwise R)

+
  dec R-dec (just x) (just y) = Dec.map just-equivalence (R-dec x y)

+
  dec R-dec (just x) nothing  = no (λ ())

+
  dec R-dec nothing  (just y) = no (λ ())

+
  dec R-dec nothing  nothing  = yes nothing

+

+
  isEquivalence : IsEquivalence R → IsEquivalence (Pointwise R)

+
  isEquivalence R-isEquivalence = record

+
    { refl  = refl R.refl

+
    ; sym   = sym R.sym

+
    ; trans = trans R.trans

+
    } where module R = IsEquivalence R-isEquivalence

+

+
  isDecEquivalence : IsDecEquivalence R → IsDecEquivalence (Pointwise R)

+
  isDecEquivalence R-isDecEquivalence = record

+
    { isEquivalence = isEquivalence R.isEquivalence

+
    ; _≟_           = dec R._≟_

+
    } where module R = IsDecEquivalence R-isDecEquivalence

+

+
  pointwise⊆any : ∀ {x} → Pointwise R (just x) ⊆ Any (R x)

+
  pointwise⊆any (just Rxy) = just Rxy

+

+
module _ {c ℓ} where

+

+
  setoid : Setoid c ℓ → Setoid c (c ⊔ ℓ)

+
  setoid S = record

+
    { isEquivalence = isEquivalence S.isEquivalence

+
    } where module S = Setoid S

+

+
  decSetoid : DecSetoid c ℓ → DecSetoid c (c ⊔ ℓ)

+
  decSetoid S = record

+
    { isDecEquivalence = isDecEquivalence S.isDecEquivalence

+
    } where module S = DecSetoid S

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Relation.Unary.All.Properties.html b/site/static/agda_html/Data.Maybe.Relation.Unary.All.Properties.html
new file mode 100644
index 000000000..5ef27cfcf
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Relation.Unary.All.Properties.html
@@ -0,0 +1,147 @@
+
+Data.Maybe.Relation.Unary.All.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to All

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Relation.Unary.All.Properties where

+

+
open import Data.Maybe.Base

+
open import Data.Maybe.Relation.Unary.All as All

+
  using (All; nothing; just)

+
open import Data.Maybe.Relation.Binary.Connected

+
open import Data.Product.Base using (_×_; _,_)

+
open import Function.Base using (_∘_)

+
open import Level

+
open import Relation.Unary

+
open import Relation.Binary.Core

+

+
private

+
  variable

+
    a b p q ℓ : Level

+
    A : Set a

+
    B : Set b

+
    P : Pred A p

+
    Q : Pred B q

+

+
------------------------------------------------------------------------

+
-- Relationship with other combinators

+
------------------------------------------------------------------------

+

+
All⇒Connectedˡ : ∀ {R : Rel A ℓ} {x y} →

+
                 All (R x) y → Connected R (just x) y

+
All⇒Connectedˡ (just x) = just x

+
All⇒Connectedˡ nothing  = just-nothing

+

+
All⇒Connectedʳ : ∀ {R : Rel A ℓ} {x y} →

+
                 All (λ v → R v y) x → Connected R x (just y)

+
All⇒Connectedʳ (just x) = just x

+
All⇒Connectedʳ nothing  = nothing-just

+

+
------------------------------------------------------------------------

+
-- Introduction (⁺) and elimination (⁻) rules for maybe operations

+
------------------------------------------------------------------------

+
-- map

+

+
map⁺ : ∀ {f : A → B} {mx} → All (P ∘ f) mx → All P (map f mx)

+
map⁺ (just p) = just p

+
map⁺ nothing  = nothing

+

+
map⁻ : ∀ {f : A → B} {mx} → All P (map f mx) → All (P ∘ f) mx

+
map⁻ {mx = just x}  (just px) = just px

+
map⁻ {mx = nothing} nothing   = nothing

+

+
-- A variant of All.map.

+

+
gmap : ∀ {f : A → B} → P ⊆ Q ∘ f → All P ⊆ All Q ∘ map f

+
gmap g = map⁺ ∘ All.map g

+

+
------------------------------------------------------------------------

+
-- _<∣>_

+

+
<∣>⁺ : ∀ {mx my} → All P mx → All P my → All P (mx <∣> my)

+
<∣>⁺ (just px) pmy = just px

+
<∣>⁺ nothing   pmy = pmy

+

+
<∣>⁻ : ∀ mx {my} → All P (mx <∣> my) → All P mx

+
<∣>⁻ (just x) pmxy = pmxy

+
<∣>⁻ nothing  pmxy = nothing

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Relation.Unary.All.html b/site/static/agda_html/Data.Maybe.Relation.Unary.All.html
new file mode 100644
index 000000000..a0c7b2272
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Relation.Unary.All.html
@@ -0,0 +1,197 @@
+
+Data.Maybe.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Maybes where all the elements satisfy a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Relation.Unary.All where

+

+
open import Effect.Applicative

+
open import Effect.Monad

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Data.Maybe.Relation.Unary.Any using (Any; just)

+
open import Data.Product.Base as Product using (_,_)

+
open import Function.Base using (id; _∘′_)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Level

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)

+
open import Relation.Unary

+
open import Relation.Nullary hiding (Irrelevant)

+
import Relation.Nullary.Decidable as Dec

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data All {a p} {A : Set a} (P : Pred A p) : Pred (Maybe A) (a ⊔ p) where

+
  just    : ∀ {x} → P x → All P (just x)

+
  nothing : All P nothing

+

+
------------------------------------------------------------------------

+
-- Basic operations

+

+
module _ {a p} {A : Set a} {P : Pred A p} where

+

+
  drop-just : ∀ {x} → All P (just x) → P x

+
  drop-just (just px) = px

+

+
  just-equivalence : ∀ {x} → P x ⇔ All P (just x)

+
  just-equivalence = mk⇔ just drop-just

+

+
  map : ∀ {q} {Q : Pred A q} → P ⊆ Q → All P ⊆ All Q

+
  map f (just px) = just (f px)

+
  map f nothing   = nothing

+

+
  fromAny : Any P ⊆ All P

+
  fromAny (just px) = just px

+

+
------------------------------------------------------------------------

+
-- (un/)zip(/With)

+

+
module _ {a p q r} {A : Set a} {P : Pred A p} {Q : Pred A q} {R : Pred A r} where

+

+
  zipWith : P ∩ Q ⊆ R → All P ∩ All Q ⊆ All R

+
  zipWith f (just px , just qx) = just (f (px , qx))

+
  zipWith f (nothing , nothing) = nothing

+

+
  unzipWith : P ⊆ Q ∩ R → All P ⊆ All Q ∩ All R

+
  unzipWith f (just px) = Product.map just just (f px)

+
  unzipWith f nothing   = nothing , nothing

+

+
module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where

+

+
  zip : All P ∩ All Q ⊆ All (P ∩ Q)

+
  zip = zipWith id

+

+
  unzip : All (P ∩ Q) ⊆ All P ∩ All Q

+
  unzip = unzipWith id

+

+
------------------------------------------------------------------------

+
-- Traversable-like functions

+

+
module _ {a f} p {A : Set a} {P : Pred A (a ⊔ p)} {F}

+
         (App : RawApplicative {a ⊔ p} {f} F) where

+

+
  open RawApplicative App

+

+
  sequenceA : All (F ∘′ P) ⊆ F ∘′ All P

+
  sequenceA nothing   = pure nothing

+
  sequenceA (just px) = just <$> px

+

+
  mapA : ∀ {q} {Q : Pred A q} → (Q ⊆ F ∘′ P) → All Q ⊆ (F ∘′ All P)

+
  mapA f = sequenceA ∘′ map f

+

+
  forA : ∀ {q} {Q : Pred A q} {xs} → All Q xs → (Q ⊆ F ∘′ P) → F (All P xs)

+
  forA qxs f = mapA f qxs

+

+
module _ {a f} p {A : Set a} {P : Pred A (a ⊔ p)} {M}

+
         (Mon : RawMonad {a ⊔ p} {f} M) where

+

+
  private App = RawMonad.rawApplicative Mon

+

+
  sequenceM : All (M ∘′ P) ⊆ M ∘′ All P

+
  sequenceM = sequenceA p App

+

+
  mapM : ∀ {q} {Q : Pred A q} → (Q ⊆ M ∘′ P) → All Q ⊆ (M ∘′ All P)

+
  mapM = mapA p App

+

+
  forM : ∀ {q} {Q : Pred A q} {xs} → All Q xs → (Q ⊆ M ∘′ P) → M (All P xs)

+
  forM = forA p App

+

+
------------------------------------------------------------------------

+
-- Seeing All as a predicate transformer

+

+
module _ {a p} {A : Set a} {P : Pred A p} where

+

+
  dec : Decidable P → Decidable (All P)

+
  dec P-dec nothing  = yes nothing

+
  dec P-dec (just x) = Dec.map just-equivalence (P-dec x)

+

+
  universal : Universal P → Universal (All P)

+
  universal P-universal (just x) = just (P-universal x)

+
  universal P-universal nothing  = nothing

+

+
  irrelevant : Irrelevant P → Irrelevant (All P)

+
  irrelevant P-irrelevant (just p) (just q) = cong just (P-irrelevant p q)

+
  irrelevant P-irrelevant nothing  nothing  = refl

+

+
  satisfiable : Satisfiable (All P)

+
  satisfiable = nothing , nothing

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.Relation.Unary.Any.html b/site/static/agda_html/Data.Maybe.Relation.Unary.Any.html
new file mode 100644
index 000000000..6a4621fc0
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.Relation.Unary.Any.html
@@ -0,0 +1,153 @@
+
+Data.Maybe.Relation.Unary.Any
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Maybes where one of the elements satisfies a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe.Relation.Unary.Any where

+

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Data.Product.Base as Product using (∃; _,_; -,_)

+
open import Function.Base using (id)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Level

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)

+
open import Relation.Unary

+
open import Relation.Nullary hiding (Irrelevant)

+
import Relation.Nullary.Decidable as Dec

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Any {a p} {A : Set a} (P : Pred A p) : Pred (Maybe A) (a ⊔ p) where

+
  just : ∀ {x} → P x → Any P (just x)

+

+
------------------------------------------------------------------------

+
-- Basic operations

+

+
module _ {a p} {A : Set a} {P : Pred A p} where

+

+
  drop-just : ∀ {x} → Any P (just x) → P x

+
  drop-just (just px) = px

+

+
  just-equivalence : ∀ {x} → P x ⇔ Any P (just x)

+
  just-equivalence = mk⇔ just drop-just

+

+
  map : ∀ {q} {Q : Pred A q} → P ⊆ Q → Any P ⊆ Any Q

+
  map f (just px) = just (f px)

+

+
  satisfied : ∀ {x} → Any P x → ∃ P

+
  satisfied (just p) = -, p

+

+
------------------------------------------------------------------------

+
-- (un/)zip(/With)

+

+
module _ {a p q r} {A : Set a} {P : Pred A p} {Q : Pred A q} {R : Pred A r} where

+

+
  zipWith : P ∩ Q ⊆ R → Any P ∩ Any Q ⊆ Any R

+
  zipWith f (just px , just qx) = just (f (px , qx))

+

+
  unzipWith : P ⊆ Q ∩ R → Any P ⊆ Any Q ∩ Any R

+
  unzipWith f (just px) = Product.map just just (f px)

+

+
module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where

+

+
  zip : Any P ∩ Any Q ⊆ Any (P ∩ Q)

+
  zip = zipWith id

+

+
  unzip : Any (P ∩ Q) ⊆ Any P ∩ Any Q

+
  unzip = unzipWith id

+

+
------------------------------------------------------------------------

+
-- Seeing Any as a predicate transformer

+

+
module _ {a p} {A : Set a} {P : Pred A p} where

+

+
  dec : Decidable P → Decidable (Any P)

+
  dec P-dec nothing  = no λ ()

+
  dec P-dec (just x) = Dec.map just-equivalence (P-dec x)

+

+
  irrelevant : Irrelevant P → Irrelevant (Any P)

+
  irrelevant P-irrelevant (just p) (just q) = cong just (P-irrelevant p q)

+

+
  satisfiable : Satisfiable P → Satisfiable (Any P)

+
  satisfiable P-satisfiable = Product.map just just P-satisfiable

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Maybe.html b/site/static/agda_html/Data.Maybe.html
new file mode 100644
index 000000000..f51ab6b4c
--- /dev/null
+++ b/site/static/agda_html/Data.Maybe.html
@@ -0,0 +1,118 @@
+
+Data.Maybe
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The Maybe type

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Maybe where

+

+
open import Data.Empty using (⊥)

+
open import Data.Unit.Base using (⊤)

+
open import Data.Bool.Base using (T)

+
open import Data.Maybe.Relation.Unary.All using (All)

+
open import Data.Maybe.Relation.Unary.Any using (Any; just)

+
open import Level using (Level)

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- The base type and some operations

+

+
open import Data.Maybe.Base public

+

+
------------------------------------------------------------------------

+
-- Using Any and All to define Is-just and Is-nothing

+

+
Is-just : Maybe A → Set _

+
Is-just = Any (λ _ → ⊤)

+

+
Is-nothing : Maybe A → Set _

+
Is-nothing = All (λ _ → ⊥)

+

+
to-witness : ∀ {m : Maybe A} → Is-just m → A

+
to-witness (just {x = p} _) = p

+

+
to-witness-T : ∀ (m : Maybe A) → T (is-just m) → A

+
to-witness-T (just p) _  = p

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Base.html b/site/static/agda_html/Data.Nat.Base.html
new file mode 100644
index 000000000..82b91cf53
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Base.html
@@ -0,0 +1,521 @@
+
+Data.Nat.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Natural numbers, basic types and operations

+
------------------------------------------------------------------------

+

+
-- See README.Data.Nat for examples of how to use and reason about

+
-- naturals.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Base where

+

+
open import Algebra.Bundles.Raw using (RawMagma; RawMonoid; RawNearSemiring; RawSemiring)

+
open import Algebra.Definitions.RawMagma using (_∣ˡ_; _,_)

+
open import Data.Bool.Base using (Bool; true; false; T; not)

+
open import Data.Parity.Base using (Parity; 0ℙ; 1ℙ)

+
open import Level using (0ℓ)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Unary using (Pred)

+

+
------------------------------------------------------------------------

+
-- Types

+

+
open import Agda.Builtin.Nat public

+
  using (zero; suc) renaming (Nat to ℕ)

+

+
--smart constructor

+
pattern 2+ n = suc (suc n)

+

+
------------------------------------------------------------------------

+
-- Boolean equality relation

+

+
open import Agda.Builtin.Nat public

+
  using () renaming (_==_ to _≡ᵇ_)

+

+
------------------------------------------------------------------------

+
-- Boolean ordering relation

+

+
open import Agda.Builtin.Nat public

+
  using () renaming (_<_ to _<ᵇ_)

+

+
infix 4 _≤ᵇ_

+
_≤ᵇ_ : (m n : ℕ) → Bool

+
zero  ≤ᵇ n = true

+
suc m ≤ᵇ n = m <ᵇ n

+

+
------------------------------------------------------------------------

+
-- Standard ordering relations

+

+
infix 4 _≤_ _<_ _≥_ _>_ _≰_ _≮_ _≱_ _≯_

+

+
data _≤_ : Rel ℕ 0ℓ where

+
  z≤n : ∀ {n}                 → zero  ≤ n

+
  s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n

+

+
_<_ : Rel ℕ 0ℓ

+
m < n = suc m ≤ n

+

+
-- Smart constructors of _<_

+

+
pattern z<s     {n}     = s≤s (z≤n {n})

+
pattern s<s {m} {n} m<n = s≤s {m} {n} m<n

+
pattern sz<ss   {n}     = s<s (z<s {n})

+

+
-- Smart destructors of _≤_, _<_

+

+
s≤s⁻¹ : ∀ {m n} → suc m ≤ suc n → m ≤ n

+
s≤s⁻¹ (s≤s m≤n) = m≤n

+

+
s<s⁻¹ : ∀ {m n} → suc m < suc n → m < n

+
s<s⁻¹ (s<s m<n) = m<n

+

+

+
------------------------------------------------------------------------

+
-- Other derived ordering relations

+

+
_≥_ : Rel ℕ 0ℓ

+
m ≥ n = n ≤ m

+

+
_>_ : Rel ℕ 0ℓ

+
m > n = n < m

+

+
_≰_ : Rel ℕ 0ℓ

+
a ≰ b = ¬ a ≤ b

+

+
_≮_ : Rel ℕ 0ℓ

+
a ≮ b = ¬ a < b

+

+
_≱_ : Rel ℕ 0ℓ

+
a ≱ b = ¬ a ≥ b

+

+
_≯_ : Rel ℕ 0ℓ

+
a ≯ b = ¬ a > b

+

+
------------------------------------------------------------------------

+
-- Simple predicates

+

+
-- Defining these predicates in terms of `T` and therefore ultimately

+
-- `⊤` and `⊥` allows Agda to automatically infer them for any natural

+
-- of the correct form. Consequently in many circumstances this

+
-- eliminates the need to explicitly pass a proof when the predicate

+
-- argument is either an implicit or an instance argument. See `_/_`

+
-- and `_%_` further down this file for examples.

+
--

+
-- Furthermore, defining these predicates as single-field records

+
-- (rather defining them directly as the type of their field) is

+
-- necessary as the current version of Agda is far better at

+
-- reconstructing meta-variable values for the record parameters.

+

+
-- A predicate saying that a number is not equal to 0.

+

+
record NonZero (n : ℕ) : Set where

+
  field

+
    nonZero : T (not (n ≡ᵇ 0))

+

+
-- Instances

+

+
instance

+
  nonZero : ∀ {n} → NonZero (suc n)

+
  nonZero = _

+

+
-- Constructors

+

+
≢-nonZero : ∀ {n} → n ≢ 0 → NonZero n

+
≢-nonZero {zero}  0≢0 = contradiction refl 0≢0

+
≢-nonZero {suc n} n≢0 = _

+

+
>-nonZero : ∀ {n} → n > 0 → NonZero n

+
>-nonZero z<s = _

+

+
-- Destructors

+

+
≢-nonZero⁻¹ : ∀ n → .{{NonZero n}} → n ≢ 0

+
≢-nonZero⁻¹ (suc n) ()

+

+
>-nonZero⁻¹ : ∀ n → .{{NonZero n}} → n > 0

+
>-nonZero⁻¹ (suc n) = z<s

+

+
-- The property of being a non-zero, non-unit

+

+
record NonTrivial (n : ℕ) : Set where

+
  field

+
    nonTrivial : T (1 <ᵇ n)

+

+
-- Instances

+

+
instance

+
  nonTrivial : ∀ {n} → NonTrivial (2+ n)

+
  nonTrivial = _

+

+
-- Constructors

+

+
n>1⇒nonTrivial : ∀ {n} → n > 1 → NonTrivial n

+
n>1⇒nonTrivial sz<ss = _

+

+
-- Destructors

+

+
nonTrivial⇒nonZero : ∀ n → .{{NonTrivial n}} → NonZero n

+
nonTrivial⇒nonZero (2+ _) = _

+

+
nonTrivial⇒n>1 : ∀ n → .{{NonTrivial n}} → n > 1

+
nonTrivial⇒n>1 (2+ _) = sz<ss

+

+
nonTrivial⇒≢1 : ∀ {n} → .{{NonTrivial n}} → n ≢ 1

+
nonTrivial⇒≢1 {{()}} refl

+

+
------------------------------------------------------------------------

+
-- Raw bundles

+

+
open import Agda.Builtin.Nat public

+
  using (_+_; _*_) renaming (_-_ to _∸_)

+

+
+-rawMagma : RawMagma 0ℓ 0ℓ

+
+-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  }

+

+
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ

+
+-0-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  ; ε   = 0

+
  }

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  ; ε = 1

+
  }

+

+
+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ

+
+-*-rawNearSemiring = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0

+
  }

+

+
+-*-rawSemiring : RawSemiring 0ℓ 0ℓ

+
+-*-rawSemiring = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0

+
  ; 1# = 1

+
  }

+

+
------------------------------------------------------------------------

+
-- Arithmetic

+

+
open import Agda.Builtin.Nat

+
  using (div-helper; mod-helper)

+

+
pred : ℕ → ℕ

+
pred n = n ∸ 1

+

+
infix  8 _!

+
infixl 7 _⊓_ _⊓′_ _/_ _%_

+
infixl 6 _+⋎_ _⊔_ _⊔′_

+

+
-- Argument-swapping addition. Used by Data.Vec._⋎_.

+

+
_+⋎_ : ℕ → ℕ → ℕ

+
zero  +⋎ n = n

+
suc m +⋎ n = suc (n +⋎ m)

+

+
-- Max.

+

+
_⊔_ : ℕ → ℕ → ℕ

+
zero  ⊔ n     = n

+
suc m ⊔ zero  = suc m

+
suc m ⊔ suc n = suc (m ⊔ n)

+

+
-- Max defined in terms of primitive operations.

+
-- This is much faster than `_⊔_` but harder to reason about. For proofs

+
-- involving this function, convert it to `_⊔_` with `Data.Nat.Properties.⊔≡⊔‵`.

+
_⊔′_ : ℕ → ℕ → ℕ

+
m ⊔′ n with m <ᵇ n

+
... | false = m

+
... | true  = n

+

+
-- Min.

+

+
_⊓_ : ℕ → ℕ → ℕ

+
zero  ⊓ n     = zero

+
suc m ⊓ zero  = zero

+
suc m ⊓ suc n = suc (m ⊓ n)

+

+
-- Min defined in terms of primitive operations.

+
-- This is much faster than `_⊓_` but harder to reason about. For proofs

+
-- involving this function, convert it to `_⊓_` wtih `Data.Nat.properties.⊓≡⊓′`.

+
_⊓′_ : ℕ → ℕ → ℕ

+
m ⊓′ n with m <ᵇ n

+
... | false = n

+
... | true  = m

+

+
-- Parity

+

+
parity : ℕ → Parity

+
parity 0             = 0ℙ

+
parity 1             = 1ℙ

+
parity (suc (suc n)) = parity n

+

+
-- Division by 2, rounded downwards.

+

+
⌊_/2⌋ : ℕ → ℕ

+
⌊ 0 /2⌋           = 0

+
⌊ 1 /2⌋           = 0

+
⌊ suc (suc n) /2⌋ = suc ⌊ n /2⌋

+

+
-- Division by 2, rounded upwards.

+

+
⌈_/2⌉ : ℕ → ℕ

+
⌈ n /2⌉ = ⌊ suc n /2⌋

+

+
-- Naïve exponentiation

+

+
infixr 8 _^_

+

+
_^_ : ℕ → ℕ → ℕ

+
x ^ zero  = 1

+
x ^ suc n = x * x ^ n

+

+
-- Distance

+

+
∣_-_∣ : ℕ → ℕ → ℕ

+
∣ zero  - y     ∣ = y

+
∣ x     - zero  ∣ = x

+
∣ suc x - suc y ∣ = ∣ x - y ∣

+

+
-- Distance in terms of primitive operations.

+
-- This is much faster than `∣_-_∣` but harder to reason about.

+
-- For proofs involving this function, convert it to `∣_-_∣` with

+
-- `Data.Nat.Properties.∣-∣≡∣-∣′`.

+
∣_-_∣′ : ℕ → ℕ → ℕ

+
∣ x - y ∣′ with x <ᵇ y

+
... | false = x ∸ y

+
... | true  = y ∸ x

+

+
-- Division

+
-- Note properties of these are in `Nat.DivMod` not `Nat.Properties`

+

+
_/_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ

+
m / (suc n) = div-helper 0 n m n

+

+
-- Remainder/modulus

+
-- Note properties of these are in `Nat.DivMod` not `Nat.Properties`

+

+
_%_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ

+
m % (suc n) = mod-helper 0 n m n

+

+
-- Factorial

+

+
_! : ℕ → ℕ

+
zero  ! = 1

+
suc n ! = suc n * n !

+

+
------------------------------------------------------------------------

+
-- Extensionally equivalent alternative definitions of _≤_/_<_ etc.

+

+
-- _≤′_: this definition is more suitable for well-founded induction

+
-- (see Data.Nat.Induction)

+

+
infix 4 _≤′_ _<′_ _≥′_ _>′_

+

+
data _≤′_ (m : ℕ) : ℕ → Set where

+
  ≤′-refl :                         m ≤′ m

+
  ≤′-step : ∀ {n} (m≤′n : m ≤′ n) → m ≤′ suc n

+

+
_<′_ : Rel ℕ 0ℓ

+
m <′ n = suc m ≤′ n

+

+
-- Smart constructors of _<′_

+

+
pattern <′-base          = ≤′-refl

+
pattern <′-step {n} m<′n = ≤′-step {n} m<′n

+

+
_≥′_ : Rel ℕ 0ℓ

+
m ≥′ n = n ≤′ m

+

+
_>′_ : Rel ℕ 0ℓ

+
m >′ n = n <′ m

+

+
-- _≤″_: this definition of _≤_ is used for proof-irrelevant ‵DivMod`

+
-- and is a specialised instance of a general algebraic construction

+

+
infix 4 _≤″_ _<″_ _≥″_ _>″_

+

+
_≤″_ : (m n : ℕ)  → Set

+
_≤″_ = _∣ˡ_ +-rawMagma

+

+
_<″_ : Rel ℕ 0ℓ

+
m <″ n = suc m ≤″ n

+

+
_≥″_ : Rel ℕ 0ℓ

+
m ≥″ n = n ≤″ m

+

+
_>″_ : Rel ℕ 0ℓ

+
m >″ n = n <″ m

+

+
-- Smart destructor of _<″_

+

+
s<″s⁻¹ : ∀ {m n} → suc m <″ suc n → m <″ n

+
s<″s⁻¹ (k , refl) = k , refl

+

+
-- _≤‴_: this definition is useful for induction with an upper bound.

+

+
data _≤‴_ : ℕ → ℕ → Set where

+
  ≤‴-refl : ∀{m} → m ≤‴ m

+
  ≤‴-step : ∀{m n} → suc m ≤‴ n → m ≤‴ n

+

+
infix 4 _≤‴_ _<‴_ _≥‴_ _>‴_

+

+
_<‴_ : Rel ℕ 0ℓ

+
m <‴ n = suc m ≤‴ n

+

+
_≥‴_ : Rel ℕ 0ℓ

+
m ≥‴ n = n ≤‴ m

+

+
_>‴_ : Rel ℕ 0ℓ

+
m >‴ n = n <‴ m

+

+
------------------------------------------------------------------------

+
-- A comparison view. Taken from "View from the left"

+
-- (McBride/McKinna); details may differ.

+

+
data Ordering : Rel ℕ 0ℓ where

+
  less    : ∀ m k → Ordering m (suc (m + k))

+
  equal   : ∀ m   → Ordering m m

+
  greater : ∀ m k → Ordering (suc (m + k)) m

+

+
compare : ∀ m n → Ordering m n

+
compare zero    zero    = equal   zero

+
compare (suc m) zero    = greater zero m

+
compare zero    (suc n) = less    zero n

+
compare (suc m) (suc n) with compare m n

+
... | less    m k = less (suc m) k

+
... | equal   m   = equal (suc m)

+
... | greater n k = greater (suc n) k

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.1

+

+
-- Smart constructors of _≤″_ and _<″_

+
pattern less-than-or-equal {k} eq = k , eq

+
{-# WARNING_ON_USAGE less-than-or-equal

+
"Warning: less-than-or-equal was deprecated in v2.1. Please match directly on proofs of ≤″ using constructor Algebra.Definitions.RawMagma._∣ˡ_._,_ instead. "

+
#-}

+
pattern ≤″-offset k = k , refl

+
{-# WARNING_ON_USAGE ≤″-offset

+
"Warning: ≤″-offset was deprecated in v2.1. Please match directly on proofs of ≤″ using pattern (_, refl) from Algebra.Definitions.RawMagma._∣ˡ_ instead. "

+
#-}

+
pattern <″-offset k = k , refl

+
{-# WARNING_ON_USAGE <″-offset

+
"Warning: <″-offset was deprecated in v2.1. Please match directly on proofs of ≤″ using pattern (_, refl) from Algebra.Definitions.RawMagma._∣ˡ_ instead. "

+
#-}

+

+
-- Smart destructors of _<″_

+

+
s≤″s⁻¹ : ∀ {m n} → suc m ≤″ suc n → m ≤″ n

+
s≤″s⁻¹ (k , refl) = k , refl

+
{-# WARNING_ON_USAGE s≤″s⁻¹

+
"Warning: s≤″s⁻¹ was deprecated in v2.1. Please match directly on proofs of ≤″ using pattern (_, refl) from Algebra.Definitions.RawMagma._∣ˡ_ instead. "

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Binary.Base.html b/site/static/agda_html/Data.Nat.Binary.Base.html
new file mode 100644
index 000000000..47cc61ae6
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Binary.Base.html
@@ -0,0 +1,271 @@
+
+Data.Nat.Binary.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Natural numbers represented in binary.

+
------------------------------------------------------------------------

+

+
-- This module contains an alternative formulation of ℕ that is

+
-- still reasonably computationally efficient without having to use

+
-- built-in functions.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Binary.Base where

+

+
open import Algebra.Bundles.Raw using (RawMagma; RawMonoid)

+
open import Algebra.Core using (Op₂)

+
open import Data.Bool.Base using (if_then_else_)

+
open import Data.Nat.Base as ℕ using (ℕ)

+
open import Data.Sum.Base using (_⊎_)

+
open import Function.Base using (_on_)

+
open import Level using (0ℓ)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+
open import Relation.Nullary.Negation using (¬_)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data ℕᵇ : Set where

+
  zero   : ℕᵇ

+
  2[1+_] : ℕᵇ → ℕᵇ    -- n → 2*(1+n) = nonzero even numbers

+
  1+[2_] : ℕᵇ → ℕᵇ    -- n → 1 + 2*n = odd numbers

+

+
------------------------------------------------------------------------

+
-- Ordering relations

+

+
infix 4 _<_ _>_ _≤_ _≥_ _≮_ _≯_ _≰_ _≱_

+

+
data _<_ : Rel ℕᵇ 0ℓ  where

+
  0<even    : ∀ {x} → zero < 2[1+ x ]

+
  0<odd     : ∀ {x} → zero < 1+[2 x ]

+
  even<even : ∀ {x y} → x < y → 2[1+ x ] < 2[1+ y ]

+
  even<odd  : ∀ {x y} → x < y → 2[1+ x ] < 1+[2 y ]

+
  odd<even  : ∀ {x y} → x < y ⊎ x ≡ y → 1+[2 x ] < 2[1+ y ]

+
  odd<odd   : ∀ {x y} → x < y → 1+[2 x ] < 1+[2 y ]

+
  -- In these constructors "even" stands for nonzero even.

+

+
_>_ : Rel ℕᵇ 0ℓ

+
x > y = y < x

+

+
_≤_ : Rel ℕᵇ 0ℓ

+
x ≤ y = x < y ⊎ x ≡ y

+

+
_≥_ : Rel ℕᵇ 0ℓ

+
x ≥ y = y ≤ x

+

+
_≮_ : Rel ℕᵇ 0ℓ

+
x ≮ y = ¬ (x < y)

+

+
_≯_ : Rel ℕᵇ 0ℓ

+
x ≯ y = ¬ (x > y)

+

+
_≰_ : Rel ℕᵇ 0ℓ

+
x ≰ y = ¬ (x ≤ y)

+

+
_≱_ : Rel ℕᵇ 0ℓ

+
x ≱ y = ¬ (x ≥ y)

+

+
------------------------------------------------------------------------

+
-- Basic operations

+

+
double : ℕᵇ → ℕᵇ

+
double zero     = zero

+
double 2[1+ x ] = 2[1+ 1+[2 x ] ]

+
double 1+[2 x ] = 2[1+ (double x) ]

+

+
suc : ℕᵇ → ℕᵇ

+
suc zero     =  1+[2 zero ]

+
suc 2[1+ x ] =  1+[2 (suc x) ]

+
suc 1+[2 x ] =  2[1+ x ]

+

+
pred : ℕᵇ → ℕᵇ

+
pred zero     = zero

+
pred 2[1+ x ] = 1+[2 x ]

+
pred 1+[2 x ] = double x

+

+
------------------------------------------------------------------------

+
-- Addition, multiplication and certain related functions

+

+
infixl 6 _+_

+
infixl 7 _*_

+

+
_+_ : Op₂ ℕᵇ

+
zero     + y        =  y

+
x        + zero     =  x

+
2[1+ x ] + 2[1+ y ] =  2[1+ suc (x + y) ]

+
2[1+ x ] + 1+[2 y ] =  suc 2[1+ (x + y) ]

+
1+[2 x ] + 2[1+ y ] =  suc 2[1+ (x + y) ]

+
1+[2 x ] + 1+[2 y ] =  suc 1+[2 (x + y) ]

+

+
_*_ : Op₂ ℕᵇ

+
zero     * _        =  zero

+
_        * zero     =  zero

+
2[1+ x ] * 2[1+ y ] =  double 2[1+ x + (y + x * y) ]

+
2[1+ x ] * 1+[2 y ] =  2[1+ x + y * 2[1+ x ] ]

+
1+[2 x ] * 2[1+ y ] =  2[1+ y + x * 2[1+ y ] ]

+
1+[2 x ] * 1+[2 y ] =  1+[2 x + y * 1+[2 x ] ]

+

+
------------------------------------------------------------------------

+
-- Conversion between ℕᵇ and ℕ

+

+
toℕ : ℕᵇ → ℕ

+
toℕ zero     =  0

+
toℕ 2[1+ x ] =  2 ℕ.* (ℕ.suc (toℕ x))

+
toℕ 1+[2 x ] =  ℕ.suc (2 ℕ.* (toℕ x))

+

+
fromℕ : ℕ → ℕᵇ

+
fromℕ n = helper n n

+
  module fromℕ where

+
  helper : ℕ → ℕ → ℕᵇ

+
  helper 0 _ = zero

+
  helper (ℕ.suc n) (ℕ.suc w) =

+
    if (n ℕ.% 2 ℕ.≡ᵇ 0)

+
      then 1+[2 helper (n ℕ./ 2) w ]

+
      else 2[1+ helper (n ℕ./ 2) w ]

+
  -- Impossible case

+
  helper _ 0 = zero

+

+
-- An alternative slower definition

+
fromℕ' : ℕ → ℕᵇ

+
fromℕ' 0 = zero

+
fromℕ' (ℕ.suc n) = suc (fromℕ' n)

+

+
-- An alternative ordering lifted from ℕ

+

+
infix 4 _<ℕ_

+

+
_<ℕ_ :  Rel ℕᵇ 0ℓ

+
_<ℕ_ =  ℕ._<_ on toℕ

+

+
------------------------------------------------------------------------

+
-- Other functions

+

+
-- Useful in some termination proofs.

+
size : ℕᵇ → ℕ

+
size zero     = 0

+
size 2[1+ x ] = ℕ.suc (size x)

+
size 1+[2 x ] = ℕ.suc (size x)

+

+
------------------------------------------------------------------------

+
-- Constants

+

+
0ᵇ = zero

+
1ᵇ = suc 0ᵇ

+
2ᵇ = suc 1ᵇ

+
3ᵇ = suc 2ᵇ

+
4ᵇ = suc 3ᵇ

+
5ᵇ = suc 4ᵇ

+
6ᵇ = suc 5ᵇ

+
7ᵇ = suc 6ᵇ

+
8ᵇ = suc 7ᵇ

+
9ᵇ = suc 8ᵇ

+

+
------------------------------------------------------------------------

+
-- Raw bundles for _+_

+

+
+-rawMagma : RawMagma 0ℓ 0ℓ

+
+-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  }

+

+
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ

+
+-0-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  ; ε   = 0ᵇ

+
  }

+

+
------------------------------------------------------------------------

+
-- Raw bundles for _*_

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  ; ε   = 1ᵇ

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Binary.Properties.html b/site/static/agda_html/Data.Nat.Binary.Properties.html
new file mode 100644
index 000000000..acab9d3b7
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Binary.Properties.html
@@ -0,0 +1,1617 @@
+
+Data.Nat.Binary.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic properties of ℕᵇ

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Binary.Properties where

+

+
open import Algebra.Bundles

+
open import Algebra.Morphism.Structures

+
import Algebra.Morphism.MonoidMonomorphism as MonoidMonomorphism

+
open import Algebra.Consequences.Propositional using (comm∧distrˡ⇒distrʳ)

+
open import Data.Bool.Base using (if_then_else_; Bool; true; false)

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Data.Nat.Binary.Base

+
open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s; s<s⁻¹)

+
open import Data.Nat.DivMod using (_%_; _/_; m/n≤m; +-distrib-/-∣ˡ)

+
open import Data.Nat.Divisibility using (∣-refl)

+
import Data.Nat.Properties as ℕ

+
open import Data.Nat.Solver using (module +-*-Solver)

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Function.Base using (_∘_; _$_; id)

+
open import Function.Definitions using (Injective; Surjective;

+
  Inverseˡ; Inverseʳ; Inverseᵇ)

+
open import Function.Consequences.Propositional

+
 using (strictlySurjective⇒surjective; strictlyInverseˡ⇒inverseˡ;

+
        strictlyInverseʳ⇒inverseʳ)

+
open import Level using (0ℓ)

+
open import Relation.Binary

+
open import Relation.Binary.Consequences using (trans∧irr⇒asym; tri⇒dec<)

+
open import Relation.Binary.Morphism

+
 using (IsRelHomomorphism; IsOrderHomomorphism; IsOrderMonomorphism)

+
import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphism

+
open import Relation.Binary.PropositionalEquality.Algebra

+
  using (magma; isMagma)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl; cong; cong₂; sym; _≗_; trans; ≢-sym; subst₂;

+
        subst; resp₂)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (isDecEquivalence; setoid; decSetoid; module ≡-Reasoning;

+
         isEquivalence)

+
open import Relation.Nullary using (¬_; yes; no)

+
import Relation.Nullary.Decidable as Dec

+
open import Relation.Nullary.Negation.Core using (contradiction)

+

+
open import Algebra.Definitions {A = ℕᵇ} _≡_

+
open import Algebra.Structures {A = ℕᵇ} _≡_

+
import Algebra.Properties.CommutativeSemigroup as CommSemigProp

+
import Algebra.Properties.CommutativeSemigroup ℕ.+-commutativeSemigroup

+
  as ℕ-+-semigroupProperties

+
import Relation.Binary.Construct.StrictToNonStrict _≡_ _<_

+
  as StrictToNonStrict

+
open +-*-Solver

+

+
private

+
  variable

+
    x : ℕᵇ

+

+
infix 4  _<?_ _≟_ _≤?_

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+
------------------------------------------------------------------------

+

+
2[1+x]≢0 : 2[1+ x ] ≢ 0ᵇ

+
2[1+x]≢0 ()

+

+
1+[2x]≢0 : 1+[2 x ] ≢ 0ᵇ

+
1+[2x]≢0 ()

+

+
2[1+_]-injective : Injective _≡_ _≡_ 2[1+_]

+
2[1+_]-injective refl = refl

+

+
1+[2_]-injective : Injective _≡_ _≡_ 1+[2_]

+
1+[2_]-injective refl = refl

+

+
_≟_ : DecidableEquality ℕᵇ

+
zero     ≟ zero     =  yes refl

+
zero     ≟ 2[1+ _ ] =  no λ()

+
zero     ≟ 1+[2 _ ] =  no λ()

+
2[1+ _ ] ≟ zero     =  no λ()

+
2[1+ x ] ≟ 2[1+ y ] =  Dec.map′ (cong 2[1+_]) 2[1+_]-injective (x ≟ y)

+
2[1+ _ ] ≟ 1+[2 _ ] =  no λ()

+
1+[2 _ ] ≟ zero     =  no λ()

+
1+[2 _ ] ≟ 2[1+ _ ] =  no λ()

+
1+[2 x ] ≟ 1+[2 y ] =  Dec.map′ (cong 1+[2_]) 1+[2_]-injective (x ≟ y)

+

+
≡-isDecEquivalence : IsDecEquivalence {A = ℕᵇ} _≡_

+
≡-isDecEquivalence = isDecEquivalence _≟_

+

+
≡-setoid : Setoid 0ℓ 0ℓ

+
≡-setoid = setoid ℕᵇ

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Properties of toℕ & fromℕ

+
------------------------------------------------------------------------

+

+
toℕ-double : ∀ x → toℕ (double x) ≡ 2 ℕ.* (toℕ x)

+
toℕ-double zero     =  refl

+
toℕ-double 1+[2 x ] =  cong ((2 ℕ.*_) ∘ ℕ.suc) (toℕ-double x)

+
toℕ-double 2[1+ x ] =  cong (2 ℕ.*_) (sym (ℕ.*-distribˡ-+ 2 1 (toℕ x)))

+

+
toℕ-suc : ∀ x → toℕ (suc x) ≡ ℕ.suc (toℕ x)

+
toℕ-suc zero     =  refl

+
toℕ-suc 2[1+ x ] =  cong (ℕ.suc ∘ (2 ℕ.*_)) (toℕ-suc x)

+
toℕ-suc 1+[2 x ] =  ℕ.*-distribˡ-+ 2 1 (toℕ x)

+

+
toℕ-pred : ∀ x → toℕ (pred x) ≡ ℕ.pred (toℕ x)

+
toℕ-pred zero     =  refl

+
toℕ-pred 2[1+ x ] =  cong ℕ.pred $ sym $ ℕ.*-distribˡ-+ 2 1 (toℕ x)

+
toℕ-pred 1+[2 x ] =  toℕ-double x

+

+
toℕ-fromℕ' : toℕ ∘ fromℕ' ≗ id

+
toℕ-fromℕ' 0         = refl

+
toℕ-fromℕ' (ℕ.suc n) = begin

+
  toℕ (fromℕ' (ℕ.suc n))   ≡⟨⟩

+
  toℕ (suc (fromℕ' n))     ≡⟨ toℕ-suc (fromℕ' n) ⟩

+
  ℕ.suc (toℕ (fromℕ' n))   ≡⟨ cong ℕ.suc (toℕ-fromℕ' n) ⟩

+
  ℕ.suc n                 ∎

+
  where open ≡-Reasoning

+

+
fromℕ≡fromℕ' : fromℕ ≗ fromℕ'

+
fromℕ≡fromℕ' n = fromℕ-helper≡fromℕ' n n ℕ.≤-refl

+
  where

+
  split : ℕᵇ → Maybe Bool × ℕᵇ

+
  split zero     = nothing , zero

+
  split 2[1+ n ] = just false , n

+
  split 1+[2 n ] = just true , n

+

+
  head = proj₁ ∘ split

+
  tail = proj₂ ∘ split

+

+
  split-injective : Injective _≡_ _≡_ split

+
  split-injective {zero} {zero} refl = refl

+
  split-injective {2[1+ _ ]} {2[1+ _ ]} refl = refl

+
  split-injective {1+[2 _ ]} {1+[2 _ ]} refl = refl

+

+
  split-if : ∀ x xs → split (if x then 1+[2 xs ] else 2[1+ xs ]) ≡ (just x , xs)

+
  split-if false xs = refl

+
  split-if true xs = refl

+

+
  head-suc : ∀ n → head (suc (suc (suc n))) ≡ head (suc n)

+
  head-suc zero = refl

+
  head-suc 2[1+ n ] = refl

+
  head-suc 1+[2 n ] = refl

+

+
  tail-suc : ∀ n → suc (tail (suc n)) ≡ tail (suc (suc (suc n)))

+
  tail-suc zero = refl

+
  tail-suc 2[1+ n ] = refl

+
  tail-suc 1+[2 n ] = refl

+

+
  head-homo : ∀ n → head (suc (fromℕ' n)) ≡ just (n % 2 ℕ.≡ᵇ 0)

+
  head-homo ℕ.zero = refl

+
  head-homo (ℕ.suc ℕ.zero) = refl

+
  head-homo (ℕ.suc (ℕ.suc n)) = trans (head-suc (fromℕ' n)) (head-homo n)

+

+
  open ≡-Reasoning

+

+
  tail-homo : ∀ n → tail (suc (fromℕ' n)) ≡ fromℕ' (n / 2)

+
  tail-homo ℕ.zero = refl

+
  tail-homo (ℕ.suc ℕ.zero) = refl

+
  tail-homo (ℕ.suc (ℕ.suc n)) = begin

+
    tail (suc (fromℕ' (ℕ.suc (ℕ.suc n))))  ≡⟨ tail-suc (fromℕ' n) ⟨

+
    suc (tail (suc (fromℕ' n)))            ≡⟨ cong suc (tail-homo n) ⟩

+
    fromℕ' (ℕ.suc (n / 2))                 ≡⟨ cong fromℕ' (+-distrib-/-∣ˡ {2} n ∣-refl) ⟨

+
    fromℕ' (ℕ.suc (ℕ.suc n) / 2)           ∎

+

+
  fromℕ-helper≡fromℕ' : ∀ n w → n ℕ.≤ w → fromℕ.helper n n w ≡ fromℕ' n

+
  fromℕ-helper≡fromℕ' ℕ.zero w p = refl

+
  fromℕ-helper≡fromℕ' (ℕ.suc n) (ℕ.suc w) (s≤s n≤w) =

+
    split-injective (begin

+
      split (fromℕ.helper n (ℕ.suc n) (ℕ.suc w))         ≡⟨ split-if _ _ ⟩

+
      just (n % 2 ℕ.≡ᵇ 0) , fromℕ.helper n (n / 2) w     ≡⟨ cong (_ ,_) rec-n/2 ⟩

+
      just (n % 2 ℕ.≡ᵇ 0) , fromℕ' (n / 2)               ≡⟨ cong₂ _,_ (head-homo n) (tail-homo n) ⟨

+
      head (fromℕ' (ℕ.suc n)) , tail (fromℕ' (ℕ.suc n))  ≡⟨⟩

+
      split (fromℕ' (ℕ.suc n))                           ∎)

+
    where rec-n/2 = fromℕ-helper≡fromℕ' (n / 2) w (ℕ.≤-trans (m/n≤m n 2) n≤w)

+

+
toℕ-fromℕ : toℕ ∘ fromℕ ≗ id

+
toℕ-fromℕ n rewrite fromℕ≡fromℕ' n = toℕ-fromℕ' n

+

+
toℕ-injective : Injective _≡_ _≡_ toℕ

+
toℕ-injective {zero}     {zero}     _               =  refl

+
toℕ-injective {2[1+ x ]} {2[1+ y ]} 2[1+xN]≡2[1+yN] =  cong 2[1+_] x≡y

+
  where

+
  1+xN≡1+yN = ℕ.*-cancelˡ-≡ (ℕ.suc _) (ℕ.suc _) 2 2[1+xN]≡2[1+yN]

+
  xN≡yN     = cong ℕ.pred 1+xN≡1+yN

+
  x≡y       = toℕ-injective xN≡yN

+

+
toℕ-injective {2[1+ x ]} {1+[2 y ]} 2[1+xN]≡1+2yN =

+
  contradiction 2[1+xN]≡1+2yN (ℕ.even≢odd (ℕ.suc (toℕ x)) (toℕ y))

+

+
toℕ-injective {1+[2 x ]} {2[1+ y ]} 1+2xN≡2[1+yN] =

+
  contradiction (sym 1+2xN≡2[1+yN]) (ℕ.even≢odd (ℕ.suc (toℕ y)) (toℕ x))

+

+
toℕ-injective {1+[2 x ]} {1+[2 y ]} 1+2xN≡1+2yN =  cong 1+[2_] x≡y

+
  where

+
  2xN≡2yN = cong ℕ.pred 1+2xN≡1+2yN

+
  xN≡yN   = ℕ.*-cancelˡ-≡ _ _ 2 2xN≡2yN

+
  x≡y     = toℕ-injective xN≡yN

+

+
toℕ-surjective : Surjective _≡_ _≡_ toℕ

+
toℕ-surjective = strictlySurjective⇒surjective (λ n → fromℕ n , toℕ-fromℕ n)

+

+
toℕ-isRelHomomorphism : IsRelHomomorphism _≡_ _≡_ toℕ

+
toℕ-isRelHomomorphism = record

+
  { cong = cong toℕ

+
  }

+

+
fromℕ-injective : Injective _≡_ _≡_ fromℕ

+
fromℕ-injective {x} {y} f[x]≡f[y] = begin

+
  x             ≡⟨ sym (toℕ-fromℕ x) ⟩

+
  toℕ (fromℕ x) ≡⟨ cong toℕ f[x]≡f[y] ⟩

+
  toℕ (fromℕ y) ≡⟨ toℕ-fromℕ y ⟩

+
  y             ∎

+
  where open ≡-Reasoning

+

+
fromℕ-toℕ : fromℕ ∘ toℕ ≗ id

+
fromℕ-toℕ = toℕ-injective ∘ toℕ-fromℕ ∘ toℕ

+

+
toℕ-inverseˡ : Inverseˡ _≡_ _≡_ toℕ fromℕ

+
toℕ-inverseˡ = strictlyInverseˡ⇒inverseˡ {f⁻¹ = fromℕ} toℕ toℕ-fromℕ

+

+
toℕ-inverseʳ : Inverseʳ _≡_ _≡_ toℕ fromℕ

+
toℕ-inverseʳ = strictlyInverseʳ⇒inverseʳ toℕ fromℕ-toℕ

+

+
toℕ-inverseᵇ : Inverseᵇ _≡_ _≡_ toℕ fromℕ

+
toℕ-inverseᵇ = toℕ-inverseˡ , toℕ-inverseʳ

+

+
fromℕ-pred : ∀ n → fromℕ (ℕ.pred n) ≡ pred (fromℕ n)

+
fromℕ-pred n = begin

+
  fromℕ (ℕ.pred n)        ≡⟨ cong (fromℕ ∘ ℕ.pred) (sym (toℕ-fromℕ n)) ⟩

+
  fromℕ (ℕ.pred (toℕ y))  ≡⟨ cong fromℕ (sym (toℕ-pred y)) ⟩

+
  fromℕ (toℕ (pred y))    ≡⟨ fromℕ-toℕ (pred y) ⟩

+
  pred y                  ≡⟨ refl ⟩

+
  pred (fromℕ n)          ∎

+
  where open ≡-Reasoning;  y = fromℕ n

+

+
x≡0⇒toℕ[x]≡0 : x ≡ zero → toℕ x ≡ 0

+
x≡0⇒toℕ[x]≡0 {zero} _ = refl

+

+
toℕ[x]≡0⇒x≡0 : toℕ x ≡ 0 → x ≡ zero

+
toℕ[x]≡0⇒x≡0 {zero} _ = refl

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+
------------------------------------------------------------------------

+
-- Basic properties

+

+
x≮0 : x ≮ zero

+
x≮0 ()

+

+
x≢0⇒x>0 : x ≢ zero → x > zero

+
x≢0⇒x>0 {zero}     0≢0 =  contradiction refl 0≢0

+
x≢0⇒x>0 {2[1+ _ ]} _   =  0<even

+
x≢0⇒x>0 {1+[2 _ ]} _   =  0<odd

+

+
1+[2x]<2[1+x] : ∀ x → 1+[2 x ] < 2[1+ x ]

+
1+[2x]<2[1+x] x =  odd<even (inj₂ refl)

+

+
<⇒≢ : _<_ ⇒ _≢_

+
<⇒≢ (even<even x<x) refl = <⇒≢ x<x refl

+
<⇒≢ (odd<odd   x<x) refl = <⇒≢ x<x refl

+

+
>⇒≢ : _>_ ⇒ _≢_

+
>⇒≢ y<x =  ≢-sym (<⇒≢ y<x)

+

+
≡⇒≮ : _≡_ ⇒ _≮_

+
≡⇒≮ x≡y x<y =  <⇒≢ x<y x≡y

+

+
≡⇒≯ : _≡_ ⇒ _≯_

+
≡⇒≯ x≡y x>y =  >⇒≢ x>y x≡y

+

+
<⇒≯ : _<_ ⇒ _≯_

+
<⇒≯ (even<even x<y)        (even<even y<x)        = <⇒≯ x<y y<x

+
<⇒≯ (even<odd x<y)         (odd<even (inj₁ y<x))  = <⇒≯ x<y y<x

+
<⇒≯ (even<odd x<y)         (odd<even (inj₂ refl)) = <⇒≢ x<y refl

+
<⇒≯ (odd<even (inj₁ x<y))  (even<odd y<x)         = <⇒≯ x<y y<x

+
<⇒≯ (odd<even (inj₂ refl)) (even<odd y<x)         = <⇒≢ y<x refl

+
<⇒≯ (odd<odd x<y)          (odd<odd y<x)          = <⇒≯ x<y y<x

+

+
>⇒≮ : _>_ ⇒ _≮_

+
>⇒≮ y<x =  <⇒≯ y<x

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ = inj₁

+

+
------------------------------------------------------------------------

+
-- Properties of _<_ and toℕ & fromℕ.

+

+
toℕ-mono-< : toℕ Preserves _<_ ⟶ ℕ._<_

+
toℕ-mono-< {zero}     {2[1+ _ ]} _               =  ℕ.0<1+n

+
toℕ-mono-< {zero}     {1+[2 _ ]} _               =  ℕ.0<1+n

+
toℕ-mono-< {2[1+ x ]} {2[1+ y ]} (even<even x<y) =  begin

+
  ℕ.suc (2 ℕ.* (ℕ.suc xN))    ≤⟨ ℕ.+-monoʳ-≤ 1 (ℕ.*-monoʳ-≤ 2 xN<yN) ⟩

+
  ℕ.suc (2 ℕ.* yN)            ≤⟨ ℕ.n≤1+n _ ⟩

+
  2 ℕ.+ (2 ℕ.* yN)            ≡⟨ sym (ℕ.*-distribˡ-+ 2 1 yN) ⟩

+
  2 ℕ.* (ℕ.suc yN)            ∎

+
  where open ℕ.≤-Reasoning;  xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y

+
toℕ-mono-< {2[1+ x ]} {1+[2 y ]} (even<odd x<y) =

+
  ℕ.+-monoʳ-≤ 1 (ℕ.*-monoʳ-≤ 2 (toℕ-mono-< x<y))

+
toℕ-mono-< {1+[2 x ]} {2[1+ y ]} (odd<even (inj₁ x<y)) =   begin

+
  ℕ.suc (ℕ.suc (2 ℕ.* xN))    ≡⟨⟩

+
  2 ℕ.+ (2 ℕ.* xN)            ≡⟨ sym (ℕ.*-distribˡ-+ 2 1 xN) ⟩

+
  2 ℕ.* (ℕ.suc xN)            ≤⟨ ℕ.*-monoʳ-≤ 2 xN<yN ⟩

+
  2 ℕ.* yN                    ≤⟨ ℕ.*-monoʳ-≤ 2 (ℕ.n≤1+n _) ⟩

+
  2 ℕ.* (ℕ.suc yN)            ∎

+
  where open ℕ.≤-Reasoning;  xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y

+
toℕ-mono-< {1+[2 x ]} {2[1+ .x ]} (odd<even (inj₂ refl)) =

+
  ℕ.≤-reflexive (sym (ℕ.*-distribˡ-+ 2 1 (toℕ x)))

+
toℕ-mono-< {1+[2 x ]} {1+[2 y ]} (odd<odd x<y) =  ℕ.+-monoʳ-< 1 (ℕ.*-monoʳ-< 2 xN<yN)

+
  where xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y

+

+
toℕ-cancel-< : ∀ {x y} → toℕ x ℕ.< toℕ y → x < y

+
toℕ-cancel-< {zero}     {2[1+ y ]} x<y = 0<even

+
toℕ-cancel-< {zero}     {1+[2 y ]} x<y = 0<odd

+
toℕ-cancel-< {2[1+ x ]} {2[1+ y ]} x<y =

+
  even<even (toℕ-cancel-< (s<s⁻¹ (ℕ.*-cancelˡ-< 2 _ _ x<y)))

+
toℕ-cancel-< {2[1+ x ]} {1+[2 y ]} x<y

+
  rewrite ℕ.*-distribˡ-+ 2 1 (toℕ x) =

+
  even<odd (toℕ-cancel-< (ℕ.*-cancelˡ-< 2 _ _ (ℕ.≤-trans (s≤s (ℕ.n≤1+n _)) (s<s⁻¹ x<y))))

+
toℕ-cancel-< {1+[2 x ]} {2[1+ y ]} x<y with toℕ x ℕ.≟ toℕ y

+
... | yes x≡y = odd<even (inj₂ (toℕ-injective x≡y))

+
... | no  x≢y

+
  rewrite ℕ.+-suc (toℕ y) (toℕ y ℕ.+ 0) =

+
  odd<even (inj₁ (toℕ-cancel-< (ℕ.≤∧≢⇒< (ℕ.*-cancelˡ-≤ 2 (ℕ.+-cancelˡ-≤ 2 _ _ x<y)) x≢y)))

+
toℕ-cancel-< {1+[2 x ]} {1+[2 y ]} x<y =

+
  odd<odd (toℕ-cancel-< (ℕ.*-cancelˡ-< 2 _ _ (s<s⁻¹ x<y)))

+

+
fromℕ-cancel-< : ∀ {x y} → fromℕ x < fromℕ y → x ℕ.< y

+
fromℕ-cancel-< = subst₂ ℕ._<_ (toℕ-fromℕ _) (toℕ-fromℕ _) ∘ toℕ-mono-<

+

+
fromℕ-mono-< : fromℕ Preserves ℕ._<_ ⟶ _<_

+
fromℕ-mono-< = toℕ-cancel-< ∘ subst₂ ℕ._<_ (sym (toℕ-fromℕ _)) (sym (toℕ-fromℕ _))

+

+
toℕ-isHomomorphism-< : IsOrderHomomorphism _≡_ _≡_ _<_ ℕ._<_ toℕ

+
toℕ-isHomomorphism-< = record

+
  { cong = cong toℕ

+
  ; mono = toℕ-mono-<

+
  }

+

+
toℕ-isMonomorphism-< : IsOrderMonomorphism _≡_ _≡_ _<_ ℕ._<_ toℕ

+
toℕ-isMonomorphism-< = record

+
  { isOrderHomomorphism = toℕ-isHomomorphism-<

+
  ; injective           = toℕ-injective

+
  ; cancel              = toℕ-cancel-<

+
  }

+

+
------------------------------------------------------------------------

+
-- Relational properties of _<_

+

+
<-irrefl : Irreflexive _≡_ _<_

+
<-irrefl refl (even<even x<x) =  <-irrefl refl x<x

+
<-irrefl refl (odd<odd x<x)   =  <-irrefl refl x<x

+

+
<-trans : Transitive _<_

+
<-trans {zero} {_}      {2[1+ _ ]} _  _        =  0<even

+
<-trans {zero} {_}      {1+[2 _ ]} _  _        =  0<odd

+
<-trans (even<even x<y) (even<even y<z)        =  even<even (<-trans x<y y<z)

+
<-trans (even<even x<y) (even<odd y<z)         =  even<odd (<-trans x<y y<z)

+
<-trans (even<odd x<y)  (odd<even (inj₁ y<z))  =  even<even (<-trans x<y y<z)

+
<-trans (even<odd x<y)  (odd<even (inj₂ refl)) =  even<even x<y

+
<-trans (even<odd x<y)  (odd<odd y<z)          =  even<odd (<-trans x<y y<z)

+
<-trans (odd<even (inj₁ x<y))  (even<even y<z) =  odd<even (inj₁ (<-trans x<y y<z))

+
<-trans (odd<even (inj₂ refl)) (even<even x<z) =  odd<even (inj₁ x<z)

+
<-trans (odd<even (inj₁ x<y))  (even<odd y<z)  =  odd<odd (<-trans x<y y<z)

+
<-trans (odd<even (inj₂ refl)) (even<odd x<z)  =  odd<odd x<z

+
<-trans (odd<odd x<y) (odd<even (inj₁ y<z))    =  odd<even (inj₁ (<-trans x<y y<z))

+
<-trans (odd<odd x<y) (odd<even (inj₂ refl))   =  odd<even (inj₁ x<y)

+
<-trans (odd<odd x<y) (odd<odd y<z)            =  odd<odd (<-trans x<y y<z)

+

+
<-asym : Asymmetric _<_

+
<-asym {x} {y} = trans∧irr⇒asym refl <-trans <-irrefl {x} {y}

+

+
-- Should not be implemented via the morphism `toℕ` in order to

+
-- preserve O(log n) time requirement.

+
<-cmp : Trichotomous _≡_ _<_

+
<-cmp zero     zero      = tri≈ x≮0    refl  x≮0

+
<-cmp zero     2[1+ _ ]  = tri< 0<even (λ()) x≮0

+
<-cmp zero     1+[2 _ ]  = tri< 0<odd  (λ()) x≮0

+
<-cmp 2[1+ _ ] zero      = tri> (λ())  (λ()) 0<even

+
<-cmp 2[1+ x ] 2[1+ y ]  with <-cmp x y

+
... | tri< x<y _    _   =  tri< lt (<⇒≢ lt) (<⇒≯ lt)  where lt = even<even x<y

+
... | tri≈ _   refl _   =  tri≈ (<-irrefl refl) refl (<-irrefl refl)

+
... | tri> _   _    x>y =  tri> (>⇒≮ gt) (>⇒≢ gt) gt  where gt = even<even x>y

+
<-cmp 2[1+ x ] 1+[2 y ]  with <-cmp x y

+
... | tri< x<y _    _ =  tri< lt (<⇒≢ lt) (<⇒≯ lt)  where lt = even<odd x<y

+
... | tri≈ _   refl _ =  tri> (>⇒≮ gt) (>⇒≢ gt) gt

+
  where

+
  gt = subst (_< 2[1+ x ]) refl (1+[2x]<2[1+x] x)

+
... | tri> _ _ y<x =  tri> (>⇒≮ gt) (>⇒≢ gt) gt  where gt = odd<even (inj₁ y<x)

+
<-cmp 1+[2 _ ] zero =  tri> (>⇒≮ gt) (>⇒≢ gt) gt  where gt = 0<odd

+
<-cmp 1+[2 x ] 2[1+ y ]  with <-cmp x y

+
... | tri< x<y _ _ =  tri< lt (<⇒≢ lt) (<⇒≯ lt)  where lt = odd<even (inj₁ x<y)

+
... | tri≈ _ x≡y _ =  tri< lt (<⇒≢ lt) (<⇒≯ lt)  where lt = odd<even (inj₂ x≡y)

+
... | tri> _ _ x>y =  tri> (>⇒≮ gt) (>⇒≢ gt) gt  where gt = even<odd x>y

+
<-cmp 1+[2 x ] 1+[2 y ]  with <-cmp x y

+
... | tri< x<y _  _   =  tri< lt (<⇒≢ lt) (<⇒≯ lt)  where lt = odd<odd x<y

+
... | tri≈ _ refl _   =  tri≈ (≡⇒≮ refl) refl (≡⇒≯ refl)

+
... | tri> _ _    x>y =  tri> (>⇒≮ gt) (>⇒≢ gt) gt  where gt = odd<odd x>y

+

+
_<?_ : Decidable _<_

+
_<?_ = tri⇒dec< <-cmp

+

+
------------------------------------------------------------------------

+
-- Structures for _<_

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = resp₂ _<_

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles for _<_

+

+
<-strictPartialOrder : StrictPartialOrder _ _ _

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder _ _ _

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder =  <-isStrictTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _<_

+

+
x<2[1+x] : ∀ x → x < 2[1+ x ]

+
x<1+[2x] : ∀ x → x < 1+[2 x ]

+

+
x<2[1+x] zero     = 0<even

+
x<2[1+x] 2[1+ x ] = even<even (x<2[1+x] x)

+
x<2[1+x] 1+[2 x ] = odd<even (inj₁ (x<1+[2x] x))

+

+
x<1+[2x] zero     = 0<odd

+
x<1+[2x] 2[1+ x ] = even<odd (x<2[1+x] x)

+
x<1+[2x] 1+[2 x ] = odd<odd (x<1+[2x] x)

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+
------------------------------------------------------------------------

+
-- Basic properties

+

+
<⇒≱ : _<_ ⇒ _≱_

+
<⇒≱ x<y (inj₁ y<x) =  contradiction y<x (<⇒≯ x<y)

+
<⇒≱ x<y (inj₂ y≡x) =  contradiction (sym y≡x) (<⇒≢ x<y)

+

+
≤⇒≯ : _≤_ ⇒ _≯_

+
≤⇒≯ x≤y x>y =  <⇒≱ x>y x≤y

+

+
≮⇒≥ : _≮_ ⇒ _≥_

+
≮⇒≥ {x} {y} x≮y  with <-cmp x y

+
... | tri< lt _  _   =  contradiction lt x≮y

+
... | tri≈ _  eq _   =  inj₂ (sym eq)

+
... | tri> _  _  y<x =  <⇒≤ y<x

+

+
≰⇒> : _≰_ ⇒ _>_

+
≰⇒> {x} {y} x≰y  with <-cmp x y

+
... | tri< lt _  _   =  contradiction (<⇒≤ lt) x≰y

+
... | tri≈ _  eq _   =  contradiction (inj₂ eq) x≰y

+
... | tri> _  _  x>y =  x>y

+

+
≤∧≢⇒< : ∀ {x y} → x ≤ y → x ≢ y → x < y

+
≤∧≢⇒< (inj₁ x<y) _   =  x<y

+
≤∧≢⇒< (inj₂ x≡y) x≢y =  contradiction x≡y x≢y

+

+
0≤x : ∀ x → zero ≤ x

+
0≤x zero     =  inj₂ refl

+
0≤x 2[1+ _ ] =  inj₁ 0<even

+
0≤x 1+[2 x ] =  inj₁ 0<odd

+

+
x≤0⇒x≡0 : x ≤ zero → x ≡ zero

+
x≤0⇒x≡0 (inj₂ x≡0) = x≡0

+

+
------------------------------------------------------------------------

+
-- Properties of _<_ and toℕ & fromℕ.

+

+
fromℕ-mono-≤ : fromℕ Preserves ℕ._≤_ ⟶ _≤_

+
fromℕ-mono-≤ m≤n  with ℕ.m≤n⇒m<n∨m≡n m≤n

+
... | inj₁ m<n =  inj₁ (fromℕ-mono-< m<n)

+
... | inj₂ m≡n =  inj₂ (cong fromℕ m≡n)

+

+
toℕ-mono-≤ : toℕ Preserves _≤_ ⟶ ℕ._≤_

+
toℕ-mono-≤ (inj₁ x<y)  =  ℕ.<⇒≤ (toℕ-mono-< x<y)

+
toℕ-mono-≤ (inj₂ refl) =  ℕ.≤-reflexive refl

+

+
toℕ-cancel-≤ : ∀ {x y} → toℕ x ℕ.≤ toℕ y → x ≤ y

+
toℕ-cancel-≤ = subst₂ _≤_ (fromℕ-toℕ _) (fromℕ-toℕ _) ∘ fromℕ-mono-≤

+

+
fromℕ-cancel-≤ : ∀ {x y} → fromℕ x ≤ fromℕ y → x ℕ.≤ y

+
fromℕ-cancel-≤ = subst₂ ℕ._≤_ (toℕ-fromℕ _) (toℕ-fromℕ _) ∘ toℕ-mono-≤

+

+
toℕ-isHomomorphism-≤ : IsOrderHomomorphism _≡_ _≡_ _≤_ ℕ._≤_ toℕ

+
toℕ-isHomomorphism-≤ = record

+
  { cong = cong toℕ

+
  ; mono = toℕ-mono-≤

+
  }

+

+
toℕ-isMonomorphism-≤ : IsOrderMonomorphism _≡_ _≡_ _≤_ ℕ._≤_ toℕ

+
toℕ-isMonomorphism-≤ = record

+
  { isOrderHomomorphism = toℕ-isHomomorphism-≤

+
  ; injective           = toℕ-injective

+
  ; cancel              = toℕ-cancel-≤

+
  }

+

+
------------------------------------------------------------------------

+
-- Relational properties of _≤_

+

+
≤-refl : Reflexive _≤_

+
≤-refl =  inj₂ refl

+

+
≤-reflexive : _≡_ ⇒ _≤_

+
≤-reflexive {x} {_} refl =  ≤-refl {x}

+

+
≤-trans : Transitive _≤_

+
≤-trans = StrictToNonStrict.trans isEquivalence (resp₂ _<_) <-trans

+

+
<-≤-trans : ∀ {x y z} → x < y → y ≤ z → x < z

+
<-≤-trans x<y (inj₁ y<z)  =  <-trans x<y y<z

+
<-≤-trans x<y (inj₂ refl) =  x<y

+

+
≤-<-trans : ∀ {x y z} → x ≤ y → y < z → x < z

+
≤-<-trans (inj₁ x<y)  y<z =  <-trans x<y y<z

+
≤-<-trans (inj₂ refl) y<z =  y<z

+

+
≤-antisym : Antisymmetric _≡_ _≤_

+
≤-antisym = StrictToNonStrict.antisym isEquivalence <-trans <-irrefl

+

+
≤-total : Total _≤_

+
≤-total x y with <-cmp x y

+
... | tri< x<y _  _   = inj₁ (<⇒≤ x<y)

+
... | tri≈ _  x≡y _   = inj₁ (≤-reflexive x≡y)

+
... | tri> _  _   y<x = inj₂ (<⇒≤ y<x)

+

+
-- Should not be implemented via the morphism `toℕ` in order to

+
-- preserve O(log n) time requirement.

+
_≤?_ : Decidable _≤_

+
x ≤? y with <-cmp x y

+
... | tri< x<y _   _   = yes (<⇒≤ x<y)

+
... | tri≈ _   x≡y _   = yes (≤-reflexive x≡y)

+
... | tri> _   _   y<x = no (<⇒≱ y<x)

+

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≤-isPreorder : IsPreorder _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder _≡_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≤-preorder : Preorder 0ℓ 0ℓ 0ℓ

+
≤-preorder = record

+
  { isPreorder = ≤-isPreorder

+
  }

+

+
≤-partialOrder : Poset 0ℓ 0ℓ 0ℓ

+
≤-partialOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  }

+

+
≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ

+
≤-totalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  }

+

+
≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ

+
≤-decTotalOrder = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Equational reasoning for _≤_ and _<_

+

+
module ≤-Reasoning where

+

+
  open import Relation.Binary.Reasoning.Base.Triple

+
    ≤-isPreorder

+
    <-asym

+
    <-trans

+
    (resp₂ _<_)

+
    <⇒≤

+
    <-≤-trans

+
    ≤-<-trans

+
    public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+

+
------------------------------------------------------------------------

+
-- Properties of _<ℕ_

+
------------------------------------------------------------------------

+

+
<⇒<ℕ : ∀ {x y} → x < y → x <ℕ y

+
<⇒<ℕ x<y = toℕ-mono-< x<y

+

+
<ℕ⇒< : ∀ {x y} → x <ℕ y → x < y

+
<ℕ⇒< {x} {y} t[x]<t[y] = begin-strict

+
  x             ≡⟨ sym (fromℕ-toℕ x) ⟩

+
  fromℕ (toℕ x) <⟨ fromℕ-mono-< t[x]<t[y] ⟩

+
  fromℕ (toℕ y) ≡⟨ fromℕ-toℕ y ⟩

+
  y             ∎

+
  where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _+_

+

+
-- toℕ/fromℕ are homomorphisms for _+_

+

+
toℕ-homo-+ : ∀ x y → toℕ (x + y) ≡ toℕ x ℕ.+ toℕ y

+
toℕ-homo-+ zero     _        = refl

+
toℕ-homo-+ 2[1+ x ] zero     = cong ℕ.suc (sym (ℕ.+-identityʳ _))

+
toℕ-homo-+ 1+[2 x ] zero     = cong ℕ.suc (sym (ℕ.+-identityʳ _))

+
toℕ-homo-+ 2[1+ x ] 2[1+ y ] = begin

+
  toℕ (2[1+ x ] + 2[1+ y ])          ≡⟨⟩

+
  toℕ 2[1+ (suc (x + y)) ]           ≡⟨⟩

+
  2 ℕ.* (1 ℕ.+ (toℕ (suc (x + y))))  ≡⟨ cong ((2 ℕ.*_) ∘ ℕ.suc) (toℕ-suc (x + y)) ⟩

+
  2 ℕ.* (2 ℕ.+ toℕ (x + y))          ≡⟨ cong ((2 ℕ.*_) ∘ (2 ℕ.+_)) (toℕ-homo-+ x y) ⟩

+
  2 ℕ.* (2 ℕ.+ (toℕ x ℕ.+ toℕ y))    ≡⟨ solve 2 (λ m n → con 2 :* (con 2 :+ (m :+ n))  :=

+
                                          con 2 :* (con 1 :+ m) :+ con 2 :* (con 1 :+ n))

+
                                          refl (toℕ x) (toℕ y) ⟩

+
  toℕ 2[1+ x ] ℕ.+ toℕ 2[1+ y ]      ∎

+
  where open ≡-Reasoning

+

+
toℕ-homo-+ 2[1+ x ] 1+[2 y ] = begin

+
  toℕ (2[1+ x ] + 1+[2 y ])             ≡⟨⟩

+
  toℕ (suc 2[1+ (x + y) ])              ≡⟨ toℕ-suc 2[1+ (x + y) ] ⟩

+
  ℕ.suc (toℕ 2[1+ (x + y) ])            ≡⟨⟩

+
  ℕ.suc (2 ℕ.* (ℕ.suc (toℕ (x + y))))   ≡⟨ cong (λ v → ℕ.suc (2 ℕ.* ℕ.suc v)) (toℕ-homo-+ x y) ⟩

+
  ℕ.suc (2 ℕ.* (ℕ.suc (m ℕ.+ n)))       ≡⟨ solve 2 (λ m n → con 1 :+ (con 2 :* (con 1 :+ (m :+ n))) :=

+
                                             con 2 :* (con 1 :+ m) :+ (con 1 :+ (con 2 :* n)))

+
                                             refl m n ⟩

+
  (2 ℕ.* ℕ.suc m) ℕ.+ (ℕ.suc (2 ℕ.* n)) ≡⟨⟩

+
  toℕ 2[1+ x ] ℕ.+ toℕ 1+[2 y ]         ∎

+
  where open ≡-Reasoning;  m = toℕ x; n = toℕ y

+

+
toℕ-homo-+ 1+[2 x ] 2[1+ y ] = begin

+
  toℕ (1+[2 x ] + 2[1+ y ])                 ≡⟨⟩

+
  toℕ (suc 2[1+ (x + y) ])                  ≡⟨ toℕ-suc 2[1+ (x + y) ] ⟩

+
  ℕ.suc (toℕ 2[1+ (x + y) ])                ≡⟨⟩

+
  ℕ.suc (2 ℕ.* (ℕ.suc (toℕ (x + y))))       ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_) ∘ ℕ.suc) (toℕ-homo-+ x y) ⟩

+
  ℕ.suc (2 ℕ.* (ℕ.suc (m ℕ.+ n)))           ≡⟨ solve 2 (λ m n → con 1 :+ (con 2 :* (con 1 :+ (m :+ n))) :=

+
                                                 (con 1 :+ (con 2 :* m)) :+ (con 2 :* (con 1 :+ n)))

+
                                                 refl m n ⟩

+
  (ℕ.suc (2 ℕ.* m)) ℕ.+ (2 ℕ.* (ℕ.suc n))   ≡⟨⟩

+
  toℕ 1+[2 x ] ℕ.+ toℕ 2[1+ y ]             ∎

+
  where open ≡-Reasoning;  m = toℕ x;  n = toℕ y

+

+
toℕ-homo-+ 1+[2 x ] 1+[2 y ] = begin

+
  toℕ (1+[2 x ] + 1+[2 y ])               ≡⟨⟩

+
  toℕ (suc 1+[2 (x + y) ])                ≡⟨ toℕ-suc 1+[2 (x + y) ] ⟩

+
  ℕ.suc (toℕ 1+[2 (x + y) ])              ≡⟨⟩

+
  ℕ.suc (ℕ.suc (2 ℕ.* (toℕ (x + y))))     ≡⟨ cong (ℕ.suc ∘ ℕ.suc ∘ (2 ℕ.*_)) (toℕ-homo-+ x y) ⟩

+
  ℕ.suc (ℕ.suc (2 ℕ.* (m ℕ.+ n)))         ≡⟨ solve 2 (λ m n → con 1 :+ (con 1 :+ (con 2 :* (m :+ n))) :=

+
                                               (con 1 :+ (con 2 :* m)) :+ (con 1 :+ (con 2 :* n)))

+
                                               refl m n ⟩

+
  (ℕ.suc (2 ℕ.* m)) ℕ.+ (ℕ.suc (2 ℕ.* n)) ≡⟨⟩

+
  toℕ 1+[2 x ] ℕ.+ toℕ 1+[2 y ]           ∎

+
  where open ≡-Reasoning;  m = toℕ x;  n = toℕ y

+

+
toℕ-isMagmaHomomorphism-+ : IsMagmaHomomorphism +-rawMagma ℕ.+-rawMagma toℕ

+
toℕ-isMagmaHomomorphism-+ = record

+
  { isRelHomomorphism = toℕ-isRelHomomorphism

+
  ; homo              = toℕ-homo-+

+
  }

+

+
toℕ-isMonoidHomomorphism-+ : IsMonoidHomomorphism +-0-rawMonoid ℕ.+-0-rawMonoid toℕ

+
toℕ-isMonoidHomomorphism-+ = record

+
  { isMagmaHomomorphism = toℕ-isMagmaHomomorphism-+

+
  ; ε-homo              = refl

+
  }

+

+
toℕ-isMonoidMonomorphism-+ : IsMonoidMonomorphism +-0-rawMonoid ℕ.+-0-rawMonoid toℕ

+
toℕ-isMonoidMonomorphism-+ = record

+
  { isMonoidHomomorphism = toℕ-isMonoidHomomorphism-+

+
  ; injective            = toℕ-injective

+
  }

+

+
suc≗1+ : suc ≗ 1ᵇ +_

+
suc≗1+ zero     = refl

+
suc≗1+ 2[1+ _ ] = refl

+
suc≗1+ 1+[2 _ ] = refl

+

+
suc-+ : ∀ x y → suc x + y ≡ suc (x + y)

+
suc-+ zero        y     =  sym (suc≗1+ y)

+
suc-+ 2[1+ x ] zero     =  refl

+
suc-+ 1+[2 x ] zero     =  refl

+
suc-+ 2[1+ x ] 2[1+ y ] =  cong (suc ∘ 2[1+_]) (suc-+ x y)

+
suc-+ 2[1+ x ] 1+[2 y ] =  cong (suc ∘ 1+[2_]) (suc-+ x y)

+
suc-+ 1+[2 x ] 2[1+ y ] =  refl

+
suc-+ 1+[2 x ] 1+[2 y ] =  refl

+

+
1+≗suc : (1ᵇ +_) ≗ suc

+
1+≗suc = suc-+ zero

+

+
fromℕ'-homo-+ : ∀ m n → fromℕ' (m ℕ.+ n) ≡ fromℕ' m + fromℕ' n

+
fromℕ'-homo-+ 0         _ = refl

+
fromℕ'-homo-+ (ℕ.suc m) n = begin

+
  fromℕ' ((ℕ.suc m) ℕ.+ n)         ≡⟨⟩

+
  suc (fromℕ' (m ℕ.+ n))           ≡⟨ cong suc (fromℕ'-homo-+ m n) ⟩

+
  suc (a + b)                      ≡⟨ sym (suc-+ a b) ⟩

+
  (suc a) + b                      ≡⟨⟩

+
  (fromℕ' (ℕ.suc m)) + (fromℕ' n)  ∎

+
  where open ≡-Reasoning;  a = fromℕ' m;  b = fromℕ' n

+

+
fromℕ-homo-+ : ∀ m n → fromℕ (m ℕ.+ n) ≡ fromℕ m + fromℕ n

+
fromℕ-homo-+ m n rewrite fromℕ≡fromℕ' (m ℕ.+ n) | fromℕ≡fromℕ' m | fromℕ≡fromℕ' n =

+
  fromℕ'-homo-+ m n

+

+
------------------------------------------------------------------------

+
-- Algebraic properties of _+_

+

+
-- Mostly proved by using the isomorphism between `ℕ` and `ℕᵇ` provided

+
-- by `toℕ`/`fromℕ`.

+

+
private

+
  module +-Monomorphism = MonoidMonomorphism toℕ-isMonoidMonomorphism-+

+

+
+-assoc : Associative _+_

+
+-assoc = +-Monomorphism.assoc ℕ.+-isMagma ℕ.+-assoc

+

+
+-comm : Commutative _+_

+
+-comm = +-Monomorphism.comm ℕ.+-isMagma ℕ.+-comm

+

+
+-identityˡ : LeftIdentity zero _+_

+
+-identityˡ _ = refl

+

+
+-identityʳ : RightIdentity zero _+_

+
+-identityʳ = +-Monomorphism.identityʳ ℕ.+-isMagma ℕ.+-identityʳ

+

+
+-identity : Identity zero _+_

+
+-identity = +-identityˡ , +-identityʳ

+

+
+-cancelˡ-≡ : LeftCancellative _+_

+
+-cancelˡ-≡ = +-Monomorphism.cancelˡ ℕ.+-isMagma ℕ.+-cancelˡ-≡

+

+
+-cancelʳ-≡ : RightCancellative _+_

+
+-cancelʳ-≡ = +-Monomorphism.cancelʳ ℕ.+-isMagma ℕ.+-cancelʳ-≡

+

+
------------------------------------------------------------------------

+
-- Structures for _+_

+

+
+-isMagma : IsMagma _+_

+
+-isMagma = isMagma _+_

+

+
+-isSemigroup : IsSemigroup _+_

+
+-isSemigroup = +-Monomorphism.isSemigroup ℕ.+-isSemigroup

+

+
+-isCommutativeSemigroup : IsCommutativeSemigroup _+_

+
+-isCommutativeSemigroup = record

+
  { isSemigroup = +-isSemigroup

+
  ; comm        = +-comm

+
  }

+

+
+-0-isMonoid : IsMonoid _+_ 0ᵇ

+
+-0-isMonoid = +-Monomorphism.isMonoid ℕ.+-0-isMonoid

+

+
+-0-isCommutativeMonoid : IsCommutativeMonoid _+_ 0ᵇ

+
+-0-isCommutativeMonoid = +-Monomorphism.isCommutativeMonoid ℕ.+-0-isCommutativeMonoid

+

+
------------------------------------------------------------------------

+
-- Bundles for _+_

+

+
+-magma : Magma 0ℓ 0ℓ

+
+-magma = magma _+_

+

+
+-semigroup : Semigroup 0ℓ 0ℓ

+
+-semigroup = record

+
  { isSemigroup = +-isSemigroup

+
  }

+

+
+-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ

+
+-commutativeSemigroup = record

+
  { isCommutativeSemigroup = +-isCommutativeSemigroup

+
  }

+

+
+-0-monoid : Monoid 0ℓ 0ℓ

+
+-0-monoid = record

+
  { ε        = zero

+
  ; isMonoid = +-0-isMonoid

+
  }

+

+
+-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
+-0-commutativeMonoid = record

+
  { isCommutativeMonoid = +-0-isCommutativeMonoid

+
  }

+

+
module Bin+CSemigroup = CommSemigProp +-commutativeSemigroup

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _≤_

+

+
+-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
+-mono-≤ {x} {x′} {y} {y′} x≤x′ y≤y′ =  begin

+
  x + y                 ≡⟨ sym $ cong₂ _+_ (fromℕ-toℕ x) (fromℕ-toℕ y) ⟩

+
  fromℕ m + fromℕ n     ≡⟨ sym (fromℕ-homo-+ m n) ⟩

+
  fromℕ (m ℕ.+ n)       ≤⟨ fromℕ-mono-≤ (ℕ.+-mono-≤ m≤m′ n≤n′) ⟩

+
  fromℕ (m′ ℕ.+ n′)     ≡⟨ fromℕ-homo-+ m′ n′ ⟩

+
  fromℕ m′ + fromℕ n′   ≡⟨ cong₂ _+_ (fromℕ-toℕ x′) (fromℕ-toℕ y′) ⟩

+
  x′ + y′               ∎

+
  where

+
  open ≤-Reasoning

+
  m    = toℕ x;             m′   = toℕ x′

+
  n    = toℕ y;             n′   = toℕ y′

+
  m≤m′ = toℕ-mono-≤ x≤x′;   n≤n′ = toℕ-mono-≤ y≤y′

+

+
+-monoˡ-≤ : ∀ x → (_+ x) Preserves _≤_ ⟶ _≤_

+
+-monoˡ-≤ x y≤z =  +-mono-≤ y≤z (≤-refl {x})

+

+
+-monoʳ-≤ : ∀ x → (x +_) Preserves _≤_ ⟶ _≤_

+
+-monoʳ-≤ x y≤z =  +-mono-≤ (≤-refl {x}) y≤z

+

+
+-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_

+
+-mono-<-≤ {x} {x′} {y} {y′} x<x′ y≤y′ =  begin-strict

+
  x + y                  ≡⟨ sym $ cong₂ _+_ (fromℕ-toℕ x) (fromℕ-toℕ y) ⟩

+
  fromℕ m + fromℕ n      ≡⟨ sym (fromℕ-homo-+ m n) ⟩

+
  fromℕ (m ℕ.+ n)        <⟨ fromℕ-mono-< (ℕ.+-mono-<-≤ m<m′ n≤n′) ⟩

+
  fromℕ (m′ ℕ.+ n′)      ≡⟨ fromℕ-homo-+ m′ n′ ⟩

+
  fromℕ m′ + fromℕ n′    ≡⟨ cong₂ _+_ (fromℕ-toℕ x′) (fromℕ-toℕ y′) ⟩

+
  x′ + y′                ∎

+
  where

+
  open ≤-Reasoning

+
  m    = toℕ x;             n    = toℕ y

+
  m′   = toℕ x′;            n′   = toℕ y′

+
  m<m′ = toℕ-mono-< x<x′;   n≤n′ = toℕ-mono-≤ y≤y′

+

+
+-mono-≤-< : _+_ Preserves₂ _≤_ ⟶ _<_ ⟶ _<_

+
+-mono-≤-< {x} {x′} {y} {y′} x≤x′ y<y′ =  subst₂ _<_ (+-comm y x) (+-comm y′ x′) y+x<y′+x′

+
  where

+
  y+x<y′+x′ =  +-mono-<-≤ y<y′ x≤x′

+

+
+-monoˡ-< : ∀ x → (_+ x) Preserves _<_ ⟶ _<_

+
+-monoˡ-< x y<z =  +-mono-<-≤ y<z (≤-refl {x})

+

+
+-monoʳ-< : ∀ x → (x +_) Preserves _<_ ⟶ _<_

+
+-monoʳ-< x y<z =  +-mono-≤-< (≤-refl {x}) y<z

+

+
x≤y+x : ∀ x y → x ≤ y + x

+
x≤y+x x y =  begin

+
  x        ≡⟨ sym (+-identityˡ x) ⟩

+
  0ᵇ + x   ≤⟨ +-monoˡ-≤ x (0≤x y) ⟩

+
  y + x    ∎

+
  where open ≤-Reasoning

+

+
x≤x+y : ∀ x y → x ≤ x + y

+
x≤x+y x y =  begin

+
  x        ≤⟨ x≤y+x x y ⟩

+
  y + x    ≡⟨ +-comm y x ⟩

+
  x + y    ∎

+
  where open ≤-Reasoning

+

+
x<x+y : ∀ x {y} → y > 0ᵇ → x < x + y

+
x<x+y x {y} y>0 = begin-strict

+
  x                             ≡⟨ sym (fromℕ-toℕ x) ⟩

+
  fromℕ (toℕ x)                 <⟨ fromℕ-mono-< (ℕ.m<m+n (toℕ x) (toℕ-mono-< y>0)) ⟩

+
  fromℕ (toℕ x ℕ.+ toℕ y)       ≡⟨ fromℕ-homo-+ (toℕ x) (toℕ y) ⟩

+
  fromℕ (toℕ x) + fromℕ (toℕ y) ≡⟨ cong₂ _+_ (fromℕ-toℕ x) (fromℕ-toℕ y) ⟩

+
  x + y                         ∎

+
  where open ≤-Reasoning

+

+
x<x+1 : ∀ x → x < x + 1ᵇ

+
x<x+1 x = x<x+y x 0<odd

+

+
x<1+x : ∀ x → x < 1ᵇ + x

+
x<1+x x rewrite +-comm 1ᵇ x = x<x+1 x

+

+
x<1⇒x≡0 : x < 1ᵇ → x ≡ zero

+
x<1⇒x≡0 0<odd = refl

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
x≢0⇒x+y≢0 : ∀ {x} (y : ℕᵇ) → x ≢ zero → x + y ≢ zero

+
x≢0⇒x+y≢0 {2[1+ _ ]} zero _   =  λ()

+
x≢0⇒x+y≢0 {zero}     _    0≢0 =  contradiction refl 0≢0

+

+
------------------------------------------------------------------------

+
-- Properties of _*_

+

+
-- toℕ/fromℕ are homomorphisms for _*_

+

+
private  2*ₙ2*ₙ =  (2 ℕ.*_) ∘ (2 ℕ.*_)

+

+
toℕ-homo-* : ∀ x y → toℕ (x * y) ≡ toℕ x ℕ.* toℕ y

+
toℕ-homo-* x y =  aux x y (size x ℕ.+ size y) ℕ.≤-refl

+
  where

+
  aux : (x y : ℕᵇ) → (cnt : ℕ) → (size x ℕ.+ size y ℕ.≤ cnt) →  toℕ (x * y) ≡ toℕ x ℕ.* toℕ y

+
  aux zero     _        _ _ = refl

+
  aux 2[1+ x ] zero     _ _ = sym (ℕ.*-zeroʳ (toℕ x ℕ.+ (ℕ.suc (toℕ x ℕ.+ 0))))

+
  aux 1+[2 x ] zero     _ _ = sym (ℕ.*-zeroʳ (toℕ x ℕ.+ (toℕ x ℕ.+ 0)))

+
  aux 2[1+ x ] 2[1+ y ] (ℕ.suc cnt) (s≤s |x|+1+|y|≤cnt) = begin

+
    toℕ (2[1+ x ] * 2[1+ y ])                ≡⟨⟩

+
    toℕ (double 2[1+ (x + (y + xy)) ])       ≡⟨ toℕ-double 2[1+ (x + (y + xy)) ] ⟩

+
    2 ℕ.* (toℕ 2[1+ (x + (y + xy)) ])        ≡⟨⟩

+
    2*ₙ2*ₙ (ℕ.suc (toℕ (x + (y + xy))))      ≡⟨ cong (2*ₙ2*ₙ ∘ ℕ.suc) (toℕ-homo-+ x (y + xy)) ⟩

+
    2*ₙ2*ₙ (ℕ.suc (m ℕ.+ (toℕ (y + xy))))    ≡⟨ cong (2*ₙ2*ₙ ∘ ℕ.suc ∘ (m ℕ.+_)) (toℕ-homo-+ y xy) ⟩

+
    2*ₙ2*ₙ (ℕ.suc (m ℕ.+ (n ℕ.+ toℕ xy)))    ≡⟨ cong (2*ₙ2*ₙ ∘ ℕ.suc ∘ (m ℕ.+_) ∘ (n ℕ.+_))

+
                                                  (aux x y cnt |x|+|y|≤cnt) ⟩

+
    2*ₙ2*ₙ (ℕ.suc (m ℕ.+ (n ℕ.+ (m ℕ.* n)))) ≡⟨ solve 2 (λ m n → con 2 :* (con 2 :* (con 1 :+ (m :+ (n :+ m :* n)))) :=

+
                                                  (con 2 :* (con 1 :+ m)) :* (con 2 :* (con 1 :+ n)))

+
                                                  refl m n ⟩

+
    (2 ℕ.* (1 ℕ.+ m)) ℕ.* (2 ℕ.* (1 ℕ.+ n))  ≡⟨⟩

+
    toℕ 2[1+ x ] ℕ.* toℕ 2[1+ y ]            ∎

+
    where

+
    open ≡-Reasoning;  m = toℕ x;  n = toℕ y;  xy = x * y

+

+
    |x|+|y|≤cnt = ℕ.≤-trans (ℕ.+-monoʳ-≤ (size x) (ℕ.n≤1+n (size y))) |x|+1+|y|≤cnt

+

+
  aux 2[1+ x ] 1+[2 y ] (ℕ.suc cnt) (s≤s |x|+1+|y|≤cnt) = begin

+
    toℕ (2[1+ x ] * 1+[2 y ])                  ≡⟨⟩

+
    toℕ (2[1+ (x + y * 2[1+ x ]) ])            ≡⟨⟩

+
    2 ℕ.* (ℕ.suc (toℕ (x + y * 2[1+ x ])))     ≡⟨ cong ((2 ℕ.*_) ∘ ℕ.suc) (toℕ-homo-+ x _) ⟩

+
    2 ℕ.* (ℕ.suc (m ℕ.+ (toℕ (y * 2[1+ x ])))) ≡⟨ cong ((2 ℕ.*_) ∘ ℕ.suc ∘ (m ℕ.+_))

+
                                                    (aux y 2[1+ x ] cnt |y|+1+|x|≤cnt) ⟩

+
    2 ℕ.* (1+m ℕ.+ (n ℕ.* (toℕ 2[1+ x ])))     ≡⟨⟩

+
    2 ℕ.* (1+m ℕ.+ (n ℕ.* 2[1+m]))             ≡⟨ solve 2 (λ m n →

+
                                                    con 2 :* ((con 1 :+ m) :+ (n :* (con 2 :* (con 1 :+ m)))) :=

+
                                                    (con 2 :* (con 1 :+ m)) :* (con 1 :+ con 2 :* n))

+
                                                    refl m n ⟩

+
    2[1+m] ℕ.* (ℕ.suc (2 ℕ.* n))               ≡⟨⟩

+
    toℕ 2[1+ x ] ℕ.* toℕ 1+[2 y ]              ∎

+
    where

+
    open ≡-Reasoning;   m = toℕ x;  n = toℕ y;  1+m = ℕ.suc m;  2[1+m] = 2 ℕ.* (ℕ.suc m)

+

+
    eq : size x ℕ.+ (ℕ.suc (size y)) ≡ size y ℕ.+ (ℕ.suc (size x))

+
    eq = ℕ-+-semigroupProperties.x∙yz≈z∙yx (size x) 1 _

+

+
    |y|+1+|x|≤cnt =  subst (ℕ._≤ cnt) eq |x|+1+|y|≤cnt

+

+
  aux 1+[2 x ] 2[1+ y ] (ℕ.suc cnt) (s≤s |x|+1+|y|≤cnt) = begin

+
    toℕ (1+[2 x ] * 2[1+ y ])                  ≡⟨⟩

+
    toℕ 2[1+ (y + x * 2[1+ y ]) ]              ≡⟨⟩

+
    2 ℕ.* (ℕ.suc (toℕ (y + x * 2[1+ y ])))     ≡⟨ cong ((2 ℕ.*_) ∘ ℕ.suc)

+
                                                    (toℕ-homo-+ y (x * 2[1+ y ])) ⟩

+
    2 ℕ.* (ℕ.suc (n ℕ.+ (toℕ (x * 2[1+ y ])))) ≡⟨ cong ((2 ℕ.*_) ∘ ℕ.suc ∘ (n ℕ.+_))

+
                                                    (aux x 2[1+ y ] cnt |x|+1+|y|≤cnt) ⟩

+
    2 ℕ.* (1+n ℕ.+ (m ℕ.* toℕ 2[1+ y ]))       ≡⟨⟩

+
    2 ℕ.* (1+n ℕ.+ (m ℕ.* 2[1+n]))             ≡⟨ solve 2 (λ m n →

+
                                                    con 2 :* ((con 1 :+ n) :+ (m :* (con 2 :* (con 1 :+ n)))) :=

+
                                                    (con 1 :+ (con 2 :* m)) :* (con 2 :* (con 1 :+ n)))

+
                                                    refl m n ⟩

+
    (ℕ.suc 2m) ℕ.* 2[1+n]                      ≡⟨⟩

+
    toℕ 1+[2 x ] ℕ.* toℕ 2[1+ y ]              ∎

+
    where

+
    open ≡-Reasoning

+
    m  = toℕ x;     n      = toℕ y;            1+n = ℕ.suc n

+
    2m = 2 ℕ.* m;   2[1+n] = 2 ℕ.* (ℕ.suc n)

+

+
  aux 1+[2 x ] 1+[2 y ] (ℕ.suc cnt) (s≤s |x|+1+|y|≤cnt) = begin

+
    toℕ (1+[2 x ] * 1+[2 y ])               ≡⟨⟩

+
    toℕ 1+[2 (x + y * 1+2x) ]               ≡⟨⟩

+
    ℕ.suc (2 ℕ.* (toℕ (x + y * 1+2x)))      ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_)) (toℕ-homo-+ x (y * 1+2x)) ⟩

+
    ℕ.suc (2 ℕ.* (m ℕ.+ (toℕ (y * 1+2x))))  ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_) ∘ (m ℕ.+_))

+
                                                 (aux y 1+2x cnt |y|+1+|x|≤cnt) ⟩

+
    ℕ.suc (2 ℕ.* (m ℕ.+ (n ℕ.* [1+2x]′)))   ≡⟨ cong ℕ.suc $ ℕ.*-distribˡ-+ 2 m (n ℕ.* [1+2x]′) ⟩

+
    ℕ.suc (2m ℕ.+ (2 ℕ.* (n ℕ.* [1+2x]′)))  ≡⟨ cong (ℕ.suc ∘ (2m ℕ.+_)) (sym (ℕ.*-assoc 2 n _)) ⟩

+
    (ℕ.suc 2m) ℕ.+ 2n ℕ.* [1+2x]′           ≡⟨⟩

+
    [1+2x]′ ℕ.+ 2n ℕ.* [1+2x]′              ≡⟨ cong (ℕ._+ (2n ℕ.* [1+2x]′)) $

+
                                                    sym (ℕ.*-identityˡ [1+2x]′) ⟩

+
    1 ℕ.* [1+2x]′ ℕ.+ 2n ℕ.* [1+2x]′        ≡⟨ sym (ℕ.*-distribʳ-+ [1+2x]′ 1 2n) ⟩

+
    (ℕ.suc 2n) ℕ.* [1+2x]′                  ≡⟨ ℕ.*-comm (ℕ.suc 2n) [1+2x]′ ⟩

+
    toℕ 1+[2 x ] ℕ.* toℕ 1+[2 y ]           ∎

+
    where

+
    open ≡-Reasoning

+
    m    = toℕ x;      n       = toℕ y;     2m = 2 ℕ.* m;    2n = 2 ℕ.* n

+
    1+2x = 1+[2 x ];   [1+2x]′ = toℕ 1+2x

+

+
    eq : size x ℕ.+ (ℕ.suc (size y)) ≡ size y ℕ.+ (ℕ.suc (size x))

+
    eq = ℕ-+-semigroupProperties.x∙yz≈z∙yx (size x) 1 _

+

+
    |y|+1+|x|≤cnt = subst (ℕ._≤ cnt) eq |x|+1+|y|≤cnt

+

+

+
toℕ-isMagmaHomomorphism-* : IsMagmaHomomorphism *-rawMagma ℕ.*-rawMagma toℕ

+
toℕ-isMagmaHomomorphism-* = record

+
  { isRelHomomorphism = toℕ-isRelHomomorphism

+
  ; homo              = toℕ-homo-*

+
  }

+

+
toℕ-isMonoidHomomorphism-* : IsMonoidHomomorphism *-1-rawMonoid ℕ.*-1-rawMonoid toℕ

+
toℕ-isMonoidHomomorphism-* = record

+
  { isMagmaHomomorphism = toℕ-isMagmaHomomorphism-*

+
  ; ε-homo              = refl

+
  }

+

+
toℕ-isMonoidMonomorphism-* : IsMonoidMonomorphism *-1-rawMonoid ℕ.*-1-rawMonoid toℕ

+
toℕ-isMonoidMonomorphism-* = record

+
  { isMonoidHomomorphism = toℕ-isMonoidHomomorphism-*

+
  ; injective            = toℕ-injective

+
  }

+

+
fromℕ-homo-* : ∀ m n → fromℕ (m ℕ.* n) ≡ fromℕ m * fromℕ n

+
fromℕ-homo-* m n = begin

+
  fromℕ (m ℕ.* n)           ≡⟨ cong fromℕ (cong₂ ℕ._*_ m≡aN n≡bN) ⟩

+
  fromℕ (toℕ a ℕ.* toℕ b)   ≡⟨ cong fromℕ (sym (toℕ-homo-* a b)) ⟩

+
  fromℕ (toℕ (a * b))       ≡⟨ fromℕ-toℕ (a * b) ⟩

+
  a * b                     ∎

+
  where

+
  open ≡-Reasoning

+
  a    = fromℕ m;             b    = fromℕ n

+
  m≡aN = sym (toℕ-fromℕ m);   n≡bN = sym (toℕ-fromℕ n)

+

+
private

+
  module *-Monomorphism = MonoidMonomorphism toℕ-isMonoidMonomorphism-*

+

+
------------------------------------------------------------------------

+
-- Algebraic properties of _*_

+

+
-- Mostly proved by using the isomorphism between `ℕ` and `ℕᵇ` provided

+
-- by `toℕ`/`fromℕ`.

+

+
*-assoc : Associative _*_

+
*-assoc = *-Monomorphism.assoc ℕ.*-isMagma ℕ.*-assoc

+

+
*-comm : Commutative _*_

+
*-comm = *-Monomorphism.comm ℕ.*-isMagma ℕ.*-comm

+

+
*-identityˡ : LeftIdentity 1ᵇ _*_

+
*-identityˡ = *-Monomorphism.identityˡ ℕ.*-isMagma ℕ.*-identityˡ

+

+
*-identityʳ : RightIdentity 1ᵇ _*_

+
*-identityʳ x =  trans (*-comm x 1ᵇ) (*-identityˡ x)

+

+
*-identity : Identity 1ᵇ _*_

+
*-identity = (*-identityˡ , *-identityʳ)

+

+
*-zeroˡ : LeftZero zero _*_

+
*-zeroˡ _ = refl

+

+
*-zeroʳ : RightZero zero _*_

+
*-zeroʳ zero     = refl

+
*-zeroʳ 2[1+ _ ] = refl

+
*-zeroʳ 1+[2 _ ] = refl

+

+
*-zero : Zero zero _*_

+
*-zero = *-zeroˡ , *-zeroʳ

+

+
*-distribˡ-+ : _*_ DistributesOverˡ _+_

+
*-distribˡ-+ a b c = begin

+
  a * (b + c)                          ≡⟨ sym (fromℕ-toℕ (a * (b + c))) ⟩

+
  fromℕ (toℕ (a * (b + c)))            ≡⟨ cong fromℕ (toℕ-homo-* a (b + c)) ⟩

+
  fromℕ (k ℕ.* (toℕ (b + c)))          ≡⟨ cong (fromℕ ∘ (k ℕ.*_)) (toℕ-homo-+ b c) ⟩

+
  fromℕ (k ℕ.* (m ℕ.+ n))              ≡⟨ cong fromℕ (ℕ.*-distribˡ-+ k m n) ⟩

+
  fromℕ (k ℕ.* m ℕ.+ k ℕ.* n)          ≡⟨ cong fromℕ $ sym $

+
                                            cong₂ ℕ._+_ (toℕ-homo-* a b) (toℕ-homo-* a c) ⟩

+
  fromℕ (toℕ (a * b) ℕ.+ toℕ (a * c))  ≡⟨ cong fromℕ (sym (toℕ-homo-+ (a * b) (a * c))) ⟩

+
  fromℕ (toℕ (a * b + a * c))          ≡⟨ fromℕ-toℕ (a * b + a * c) ⟩

+
  a * b + a * c                        ∎

+
  where open ≡-Reasoning;  k = toℕ a;  m = toℕ b;  n = toℕ c

+

+
*-distribʳ-+ : _*_ DistributesOverʳ _+_

+
*-distribʳ-+ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-+

+

+
*-distrib-+ : _*_ DistributesOver _+_

+
*-distrib-+ = *-distribˡ-+ , *-distribʳ-+

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
*-isMagma : IsMagma _*_

+
*-isMagma = isMagma _*_

+

+
*-isSemigroup : IsSemigroup _*_

+
*-isSemigroup = *-Monomorphism.isSemigroup ℕ.*-isSemigroup

+

+
*-1-isMonoid : IsMonoid _*_ 1ᵇ

+
*-1-isMonoid = *-Monomorphism.isMonoid ℕ.*-1-isMonoid

+

+
*-1-isCommutativeMonoid : IsCommutativeMonoid _*_ 1ᵇ

+
*-1-isCommutativeMonoid = *-Monomorphism.isCommutativeMonoid ℕ.*-1-isCommutativeMonoid

+

+
+-*-isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _+_ _*_ zero 1ᵇ

+
+-*-isSemiringWithoutAnnihilatingZero = record

+
  { +-isCommutativeMonoid = +-0-isCommutativeMonoid

+
  ; *-cong                = cong₂ _*_

+
  ; *-assoc               = *-assoc

+
  ; *-identity            = *-identity

+
  ; distrib               = *-distrib-+

+
  }

+

+
+-*-isSemiring : IsSemiring _+_ _*_ zero 1ᵇ

+
+-*-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = +-*-isSemiringWithoutAnnihilatingZero

+
  ; zero                              = *-zero

+
  }

+

+
+-*-isCommutativeSemiring : IsCommutativeSemiring _+_ _*_ zero 1ᵇ

+
+-*-isCommutativeSemiring = record

+
  { isSemiring = +-*-isSemiring

+
  ; *-comm     = *-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
*-magma : Magma 0ℓ 0ℓ

+
*-magma = record

+
  { isMagma = *-isMagma

+
  }

+

+
*-semigroup : Semigroup 0ℓ 0ℓ

+
*-semigroup = record

+
  { isSemigroup = *-isSemigroup

+
  }

+

+
*-1-monoid : Monoid 0ℓ 0ℓ

+
*-1-monoid = record

+
  { isMonoid = *-1-isMonoid

+
  }

+

+
*-1-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
*-1-commutativeMonoid = record

+
  { isCommutativeMonoid = *-1-isCommutativeMonoid

+
  }

+

+
+-*-semiring : Semiring 0ℓ 0ℓ

+
+-*-semiring = record

+
  { isSemiring = +-*-isSemiring

+
  }

+

+
+-*-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ

+
+-*-commutativeSemiring = record

+
  { isCommutativeSemiring = +-*-isCommutativeSemiring

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _≤_ & _<_

+

+
*-mono-≤ : _*_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
*-mono-≤ {x} {u} {y} {v} x≤u y≤v = toℕ-cancel-≤ (begin

+
  toℕ (x * y)      ≡⟨ toℕ-homo-* x y ⟩

+
  toℕ x ℕ.* toℕ y  ≤⟨ ℕ.*-mono-≤ (toℕ-mono-≤ x≤u) (toℕ-mono-≤ y≤v) ⟩

+
  toℕ u ℕ.* toℕ v  ≡⟨ sym (toℕ-homo-* u v) ⟩

+
  toℕ (u * v)      ∎)

+
  where open ℕ.≤-Reasoning

+

+
*-monoʳ-≤ : ∀ x → (x *_) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤ x y≤y′ = *-mono-≤ (≤-refl {x}) y≤y′

+

+
*-monoˡ-≤ : ∀ x → (_* x) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤ x y≤y′ = *-mono-≤ y≤y′ (≤-refl {x})

+

+
*-mono-< : _*_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
*-mono-< {x} {u} {y} {v} x<u y<v = toℕ-cancel-< (begin-strict

+
  toℕ (x * y)      ≡⟨ toℕ-homo-* x y ⟩

+
  toℕ x ℕ.* toℕ y  <⟨ ℕ.*-mono-< (toℕ-mono-< x<u) (toℕ-mono-< y<v) ⟩

+
  toℕ u ℕ.* toℕ v  ≡⟨ sym (toℕ-homo-* u v) ⟩

+
  toℕ (u * v)      ∎)

+
  where open ℕ.≤-Reasoning

+

+
*-monoʳ-< : ∀ x → ((1ᵇ + x) *_) Preserves _<_ ⟶ _<_

+
*-monoʳ-< x {y} {z} y<z = begin-strict

+
  (1ᵇ + x) * y    ≡⟨ *-distribʳ-+ y 1ᵇ x ⟩

+
  1ᵇ * y + x * y  ≡⟨ cong (_+ x * y) (*-identityˡ y) ⟩

+
  y      + x * y  <⟨ +-mono-<-≤ y<z (*-monoʳ-≤ x (<⇒≤ y<z)) ⟩

+
  z      + x * z  ≡⟨ cong (_+ x * z) (sym (*-identityˡ z)) ⟩

+
  1ᵇ * z + x * z  ≡⟨ sym (*-distribʳ-+ z 1ᵇ x) ⟩

+
  (1ᵇ + x) * z    ∎

+
  where open ≤-Reasoning

+

+
*-monoˡ-< : ∀ x → (_* (1ᵇ + x)) Preserves _<_ ⟶ _<_

+
*-monoˡ-< x {y} {z} y<z = begin-strict

+
  y * (1ᵇ + x)   ≡⟨ *-comm y (1ᵇ + x) ⟩

+
  (1ᵇ + x) * y   <⟨ *-monoʳ-< x y<z ⟩

+
  (1ᵇ + x) * z   ≡⟨ *-comm (1ᵇ + x) z ⟩

+
  z * (1ᵇ + x)   ∎

+
  where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Other properties of _*_

+

+
x*y≡0⇒x≡0∨y≡0 : ∀ x {y} → x * y ≡ zero → x ≡ zero ⊎ y ≡ zero

+
x*y≡0⇒x≡0∨y≡0 zero {_}    _ =  inj₁ refl

+
x*y≡0⇒x≡0∨y≡0 _    {zero} _ =  inj₂ refl

+

+
x≢0∧y≢0⇒x*y≢0 : ∀ {x y} → x ≢ zero → y ≢ zero → x * y ≢ zero

+
x≢0∧y≢0⇒x*y≢0 {x} {_} x≢0 y≢0 xy≡0  with x*y≡0⇒x≡0∨y≡0 x xy≡0

+
... | inj₁ x≡0 =  x≢0 x≡0

+
... | inj₂ y≡0 =  y≢0 y≡0

+

+
2*x≡x+x : ∀ x → 2ᵇ * x ≡ x + x

+
2*x≡x+x x = begin

+
  2ᵇ * x            ≡⟨⟩

+
  (1ᵇ + 1ᵇ) * x     ≡⟨ *-distribʳ-+ x 1ᵇ 1ᵇ ⟩

+
  1ᵇ * x + 1ᵇ * x   ≡⟨ cong₂ _+_ (*-identityˡ x) (*-identityˡ x) ⟩

+
  x + x             ∎

+
  where open ≡-Reasoning

+

+
1+-* : ∀ x y → (1ᵇ + x) * y ≡ y + x * y

+
1+-* x y = begin

+
  (1ᵇ + x) * y     ≡⟨ *-distribʳ-+ y 1ᵇ x ⟩

+
  1ᵇ * y + x * y   ≡⟨ cong (_+ x * y) (*-identityˡ y) ⟩

+
  y + x * y        ∎

+
  where open ≡-Reasoning

+

+
*-1+ : ∀ x y → y * (1ᵇ + x) ≡ y + y * x

+
*-1+ x y = begin

+
  y * (1ᵇ + x)     ≡⟨ *-distribˡ-+ y 1ᵇ x ⟩

+
  y * 1ᵇ + y * x   ≡⟨ cong (_+ y * x) (*-identityʳ y) ⟩

+
  y + y * x        ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of double

+
------------------------------------------------------------------------

+

+
double[x]≡0⇒x≡0 : double x ≡ zero → x ≡ zero

+
double[x]≡0⇒x≡0 {zero} _ = refl

+

+
x≡0⇒double[x]≡0 : x ≡ 0ᵇ → double x ≡ 0ᵇ

+
x≡0⇒double[x]≡0 = cong double

+

+
x≢0⇒double[x]≢0 : x ≢ zero → double x ≢ zero

+
x≢0⇒double[x]≢0 x≢0 = x≢0 ∘ double[x]≡0⇒x≡0

+

+
double≢1 : double x ≢ 1ᵇ

+
double≢1 {zero} ()

+

+
double≗2* : double ≗ 2ᵇ *_

+
double≗2* x =  toℕ-injective $ begin

+
  toℕ (double x)  ≡⟨ toℕ-double x ⟩

+
  2 ℕ.* (toℕ x)   ≡⟨ sym (toℕ-homo-* 2ᵇ x) ⟩

+
  toℕ (2ᵇ * x)    ∎

+
  where open ≡-Reasoning

+

+
double-*-assoc : ∀ x y → (double x) * y ≡ double (x * y)

+
double-*-assoc x y = begin

+
  (double x) * y     ≡⟨ cong (_* y) (double≗2* x) ⟩

+
  (2ᵇ * x) * y       ≡⟨ *-assoc 2ᵇ x y ⟩

+
  2ᵇ * (x * y)       ≡⟨ sym (double≗2* (x * y)) ⟩

+
  double (x * y)     ∎

+
  where open ≡-Reasoning

+

+
double[x]≡x+x : ∀ x → double x ≡ x + x

+
double[x]≡x+x x =  trans (double≗2* x) (2*x≡x+x x)

+

+
double-distrib-+ : ∀ x y → double (x + y) ≡ double x + double y

+
double-distrib-+ x y = begin

+
  double (x + y)         ≡⟨ double≗2* (x + y) ⟩

+
  2ᵇ * (x + y)           ≡⟨ *-distribˡ-+ 2ᵇ x y ⟩

+
  (2ᵇ * x) + (2ᵇ * y)    ≡⟨ sym (cong₂ _+_ (double≗2* x) (double≗2* y)) ⟩

+
  double x + double y    ∎

+
  where open ≡-Reasoning

+

+
double-mono-≤ : double Preserves _≤_ ⟶ _≤_

+
double-mono-≤ {x} {y} x≤y =  begin

+
  double x   ≡⟨ double≗2* x ⟩

+
  2ᵇ * x     ≤⟨ *-monoʳ-≤ 2ᵇ x≤y ⟩

+
  2ᵇ * y     ≡⟨ sym (double≗2* y) ⟩

+
  double y   ∎

+
  where open ≤-Reasoning

+

+
double-mono-< : double Preserves _<_ ⟶ _<_

+
double-mono-< {x} {y} x<y =  begin-strict

+
  double x   ≡⟨ double≗2* x ⟩

+
  2ᵇ * x     <⟨ *-monoʳ-< 1ᵇ x<y ⟩

+
  2ᵇ * y     ≡⟨ sym (double≗2* y) ⟩

+
  double y   ∎

+
  where open ≤-Reasoning

+

+
double-cancel-≤ : ∀ {x y} → double x ≤ double y → x ≤ y

+
double-cancel-≤ {x} {y} 2x≤2y  with <-cmp x y

+
... | tri< x<y _   _   =  <⇒≤ x<y

+
... | tri≈ _   x≡y _   =  ≤-reflexive x≡y

+
... | tri> _   _   x>y =  contradiction 2x≤2y (<⇒≱ (double-mono-< x>y))

+

+
double-cancel-< : ∀ {x y} → double x < double y → x < y

+
double-cancel-< {x} {y} 2x<2y  with <-cmp x y

+
... | tri< x<y _    _   =  x<y

+
... | tri≈ _   refl _   =  contradiction 2x<2y (<-irrefl refl)

+
... | tri> _   _    x>y =  contradiction (double-mono-< x>y) (<⇒≯ 2x<2y)

+

+
x<double[x] : ∀ x → x ≢ zero → x < double x

+
x<double[x] x x≢0 = begin-strict

+
  x                 <⟨ x<x+y x (x≢0⇒x>0 x≢0) ⟩

+
  x + x             ≡⟨ sym (double[x]≡x+x x) ⟩

+
  double x          ∎

+
  where open ≤-Reasoning

+

+
x≤double[x] : ∀ x → x ≤ double x

+
x≤double[x] x =  begin

+
  x         ≤⟨ x≤x+y x x ⟩

+
  x + x     ≡⟨ sym (double[x]≡x+x x) ⟩

+
  double x  ∎

+
  where open ≤-Reasoning

+

+
double-suc : ∀ x → double (suc x) ≡ 2ᵇ + double x

+
double-suc x = begin

+
  double (suc x)   ≡⟨ cong double (suc≗1+ x) ⟩

+
  double (1ᵇ + x)  ≡⟨ double-distrib-+ 1ᵇ x ⟩

+
  2ᵇ + double x    ∎

+
  where open ≡-Reasoning

+

+

+

+
------------------------------------------------------------------------

+
-- Properties of suc

+
------------------------------------------------------------------------

+

+
suc≢0 : suc x ≢ zero

+
suc≢0 {zero}     ()

+
suc≢0 {2[1+ _ ]} ()

+
suc≢0 {1+[2 _ ]} ()

+

+
suc-injective : Injective _≡_ _≡_ suc

+
suc-injective {zero} {zero} p = refl

+
suc-injective {zero} {2[1+ y ]} p = contradiction 1+[2 p ]-injective (suc≢0 ∘ sym)

+
suc-injective {2[1+ x ]} {zero} p = contradiction 1+[2 p ]-injective suc≢0

+
suc-injective {2[1+ x ]} {2[1+ y ]} p = cong 2[1+_] (suc-injective 1+[2 p ]-injective)

+
suc-injective {1+[2 x ]} {1+[2 y ]} refl = refl

+

+
2[1+_]-double-suc : 2[1+_] ≗ double ∘ suc

+
2[1+_]-double-suc zero     = refl

+
2[1+_]-double-suc 2[1+ x ] = cong 2[1+_] (2[1+_]-double-suc x)

+
2[1+_]-double-suc 1+[2 x ] = refl

+

+
1+[2_]-suc-double : 1+[2_] ≗ suc ∘ double

+
1+[2_]-suc-double zero     = refl

+
1+[2_]-suc-double 2[1+ x ] = refl

+
1+[2_]-suc-double 1+[2 x ] = begin

+
  1+[2 1+[2 x ] ]         ≡⟨ cong 1+[2_] (1+[2_]-suc-double x) ⟩

+
  1+[2 (suc 2x) ]         ≡⟨⟩

+
  suc 2[1+ 2x ]           ≡⟨ cong suc (2[1+_]-double-suc 2x) ⟩

+
  suc (double (suc 2x))   ≡⟨ cong (suc ∘ double) (sym (1+[2_]-suc-double x)) ⟩

+
  suc (double 1+[2 x ])   ∎

+
  where open ≡-Reasoning;  2x = double x

+

+
x+suc[y]≡suc[x]+y : ∀ x y → x + suc y ≡ suc x + y

+
x+suc[y]≡suc[x]+y x y = begin

+
  x + suc y     ≡⟨ +-comm x _ ⟩

+
  suc y + x     ≡⟨ suc-+ y x ⟩

+
  suc (y + x)   ≡⟨ cong suc (+-comm y x) ⟩

+
  suc (x + y)   ≡⟨ sym (suc-+ x y) ⟩

+
  suc x + y     ∎

+
  where open ≡-Reasoning

+

+
0<suc : ∀ x → zero < suc x

+
0<suc x =  x≢0⇒x>0 (suc≢0 {x})

+

+
x<suc[x] : ∀ x → x < suc x

+
x<suc[x] x = begin-strict

+
  x        <⟨ x<1+x x ⟩

+
  1ᵇ + x   ≡⟨ sym (suc≗1+ x) ⟩

+
  suc x    ∎

+
  where open ≤-Reasoning

+

+
x≤suc[x] : ∀ x → x ≤ suc x

+
x≤suc[x] x = <⇒≤ (x<suc[x] x)

+

+
x≢suc[x] : ∀ x → x ≢ suc x

+
x≢suc[x] x = <⇒≢ (x<suc[x] x)

+

+
suc-mono-≤ : suc Preserves _≤_ ⟶ _≤_

+
suc-mono-≤ {x} {y} x≤y =  begin

+
  suc x     ≡⟨ suc≗1+ x ⟩

+
  1ᵇ + x    ≤⟨ +-monoʳ-≤ 1ᵇ x≤y ⟩

+
  1ᵇ + y    ≡⟨ sym (suc≗1+ y) ⟩

+
  suc y     ∎

+
  where open ≤-Reasoning

+

+
suc[x]≤y⇒x<y : ∀ {x y} → suc x ≤ y → x < y

+
suc[x]≤y⇒x<y {x} (inj₁ sx<y) = <-trans (x<suc[x] x) sx<y

+
suc[x]≤y⇒x<y {x} (inj₂ refl) = x<suc[x] x

+

+
x<y⇒suc[x]≤y : ∀ {x y} → x < y → suc x ≤ y

+
x<y⇒suc[x]≤y {x} {y} x<y = begin

+
  suc x                  ≡⟨ sym (fromℕ-toℕ (suc x)) ⟩

+
  fromℕ (toℕ (suc x))    ≡⟨ cong fromℕ (toℕ-suc x) ⟩

+
  fromℕ (ℕ.suc (toℕ x))  ≤⟨ fromℕ-mono-≤ (toℕ-mono-< x<y) ⟩

+
  fromℕ (toℕ y)          ≡⟨ fromℕ-toℕ y ⟩

+
  y                      ∎

+
  where open ≤-Reasoning

+

+
suc-* : ∀ x y → suc x * y ≡ y + x * y

+
suc-* x y = begin

+
  suc x * y     ≡⟨ cong (_* y) (suc≗1+ x) ⟩

+
  (1ᵇ + x) * y  ≡⟨ 1+-* x y ⟩

+
  y + x * y     ∎

+
  where open ≡-Reasoning

+

+
*-suc : ∀ x y → x * suc y ≡ x + x * y

+
*-suc x y = begin

+
  x * suc y     ≡⟨ cong (x *_) (suc≗1+ y) ⟩

+
  x * (1ᵇ + y)  ≡⟨ *-1+ y x ⟩

+
  x + x * y     ∎

+
  where open ≡-Reasoning

+

+
x≤suc[y]*x : ∀ x y → x ≤ (suc y) * x

+
x≤suc[y]*x x y = begin

+
  x             ≤⟨ x≤x+y x (y * x) ⟩

+
  x + y * x     ≡⟨ sym (suc-* y x) ⟩

+
  (suc y) * x   ∎

+
  where open ≤-Reasoning

+

+
suc[x]≤double[x] : ∀ x → x ≢ zero → suc x ≤ double x

+
suc[x]≤double[x] x =  x<y⇒suc[x]≤y {x} {double x} ∘ x<double[x] x

+

+
suc[x]<2[1+x] : ∀ x → suc x < 2[1+ x ]

+
suc[x]<2[1+x] x =  begin-strict

+
  suc x           <⟨ x<double[x] (suc x) suc≢0  ⟩

+
  double (suc x)  ≡⟨ sym (2[1+_]-double-suc x) ⟩

+
  2[1+ x ]        ∎

+
  where open ≤-Reasoning

+

+
double[x]<1+[2x] : ∀ x → double x < 1+[2 x ]

+
double[x]<1+[2x] x = begin-strict

+
  double x         <⟨ x<suc[x] (double x) ⟩

+
  suc (double x)   ≡⟨ sym (1+[2_]-suc-double x) ⟩

+
  1+[2 x ]         ∎

+
  where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of pred

+
------------------------------------------------------------------------

+

+
pred-suc : pred ∘ suc ≗ id

+
pred-suc zero     =  refl

+
pred-suc 2[1+ x ] =  sym (2[1+_]-double-suc x)

+
pred-suc 1+[2 x ] =  refl

+

+
suc-pred : x ≢ zero → suc (pred x) ≡ x

+
suc-pred {zero}     0≢0 =  contradiction refl 0≢0

+
suc-pred {2[1+ _ ]} _   =  refl

+
suc-pred {1+[2 x ]} _   =  sym (1+[2_]-suc-double x)

+

+
pred-mono-≤ : pred Preserves _≤_ ⟶ _≤_

+
pred-mono-≤ {x} {y} x≤y = begin

+
  pred x             ≡⟨ cong pred (sym (fromℕ-toℕ x)) ⟩

+
  pred (fromℕ m)     ≡⟨ sym (fromℕ-pred m) ⟩

+
  fromℕ (ℕ.pred m)   ≤⟨ fromℕ-mono-≤ (ℕ.pred-mono-≤ (toℕ-mono-≤ x≤y)) ⟩

+
  fromℕ (ℕ.pred n)   ≡⟨ fromℕ-pred n ⟩

+
  pred (fromℕ n)     ≡⟨ cong pred (fromℕ-toℕ y) ⟩

+
  pred y             ∎

+
  where

+
  open ≤-Reasoning;  m = toℕ x;  n = toℕ y

+

+
pred[x]<x : x ≢ zero → pred x < x

+
pred[x]<x {x} x≢0 = begin-strict

+
  pred x       <⟨ x<suc[x] (pred x) ⟩

+
  suc (pred x) ≡⟨ suc-pred x≢0 ⟩

+
  x            ∎

+
  where open ≤-Reasoning

+

+
pred[x]+y≡x+pred[y] : ∀ {x y} → x ≢ 0ᵇ → y ≢ 0ᵇ → (pred x) + y ≡ x + pred y

+
pred[x]+y≡x+pred[y] {x} {y} x≢0 y≢0 = begin

+
  px + y           ≡⟨ cong (px +_) (sym (suc-pred y≢0)) ⟩

+
  px + suc py      ≡⟨ cong (px +_) (suc≗1+ py) ⟩

+
  px + (1ᵇ + py)   ≡⟨ Bin+CSemigroup.x∙yz≈yx∙z px 1ᵇ py ⟩

+
  (1ᵇ + px) + py   ≡⟨ cong (_+ py) (sym (suc≗1+ px)) ⟩

+
  (suc px) + py    ≡⟨ cong (_+ py) (suc-pred x≢0) ⟩

+
  x + py           ∎

+
  where open ≡-Reasoning;  px = pred x;  py = pred y

+

+

+
------------------------------------------------------------------------

+
-- Properties of size

+
------------------------------------------------------------------------

+

+
|x|≡0⇒x≡0 : size x ≡ 0 → x ≡ 0ᵇ

+
|x|≡0⇒x≡0 {zero} refl =  refl

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.4

+

+
*-+-isSemiringWithoutAnnihilatingZero = +-*-isSemiringWithoutAnnihilatingZero

+
{-# WARNING_ON_USAGE *-+-isSemiringWithoutAnnihilatingZero

+
"Warning: *-+-isSemiringWithoutAnnihilatingZero was deprecated in v1.4.

+
Please use +-*-isSemiringWithoutAnnihilatingZero instead."

+
#-}

+
*-+-isSemiring = +-*-isSemiring

+
{-# WARNING_ON_USAGE *-+-isSemiring

+
"Warning: *-+-isSemiring was deprecated in v1.4.

+
Please use +-*-isSemiring instead."

+
#-}

+
*-+-isCommutativeSemiring = +-*-isCommutativeSemiring

+
{-# WARNING_ON_USAGE *-+-isCommutativeSemiring

+
"Warning: *-+-isCommutativeSemiring was deprecated in v1.4.

+
Please use +-*-isCommutativeSemiring instead."

+
#-}

+
*-+-semiring = +-*-semiring

+
{-# WARNING_ON_USAGE *-+-semiring

+
"Warning: *-+-semiring was deprecated in v1.4.

+
Please use +-*-semiring instead."

+
#-}

+
*-+-commutativeSemiring = +-*-commutativeSemiring

+
{-# WARNING_ON_USAGE *-+-commutativeSemiring

+
"Warning: *-+-commutativeSemiring was deprecated in v1.4.

+
Please use +-*-commutativeSemiring instead."

+
#-}

+

+
-- Version 2.0

+

+
{- issue1858/issue1755: raw bundles have moved to `Data.X.Base` -}

+
open Data.Nat.Binary.Base public

+
  using (+-rawMagma; +-0-rawMonoid; *-rawMagma; *-1-rawMonoid)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Binary.html b/site/static/agda_html/Data.Nat.Binary.html
new file mode 100644
index 000000000..ce91c0973
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Binary.html
@@ -0,0 +1,89 @@
+
+Data.Nat.Binary
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Natural numbers represented in binary natively in Agda.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Binary where

+

+
open import Data.Nat.Binary.Base public

+
open import Data.Nat.Binary.Properties public using (_≟_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Coprimality.html b/site/static/agda_html/Data.Nat.Coprimality.html
new file mode 100644
index 000000000..338587585
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Coprimality.html
@@ -0,0 +1,213 @@
+
+Data.Nat.Coprimality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Coprimality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Coprimality where

+

+
open import Data.Nat.Base

+
open import Data.Nat.Divisibility

+
open import Data.Nat.GCD

+
open import Data.Nat.GCD.Lemmas

+
open import Data.Nat.Primality

+
open import Data.Nat.Properties

+
open import Data.Nat.DivMod

+
open import Data.Product.Base as Prod

+
open import Data.Sum.Base as Sum using (inj₁; inj₂)

+
open import Function.Base using (_∘_)

+
open import Level using (0ℓ)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; refl; trans; cong; subst)

+
open import Relation.Nullary as Nullary using (¬_; contradiction; map′)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions using (Symmetric; Decidable)

+

+
private

+
  variable d m n o p : ℕ

+

+
open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Definition

+
--

+
-- Coprime m n is inhabited iff m and n are coprime (relatively

+
-- prime), i.e. if their only common divisor is 1.

+

+
Coprime : Rel ℕ 0ℓ

+
Coprime m n = ∀ {d} → d ∣ m × d ∣ n → d ≡ 1

+

+
------------------------------------------------------------------------

+
-- Relationship between GCD and coprimality

+

+
coprime⇒GCD≡1 : Coprime m n → GCD m n 1

+
coprime⇒GCD≡1 {m} {n} coprime = GCD.is (1∣ m , 1∣ n) (∣-reflexive ∘ coprime)

+

+
GCD≡1⇒coprime : GCD m n 1 → Coprime m n

+
GCD≡1⇒coprime g cd with divides q eq ← GCD.greatest g cd

+
  = m*n≡1⇒n≡1 q _ (≡.sym eq)

+

+
coprime⇒gcd≡1 : Coprime m n → gcd m n ≡ 1

+
coprime⇒gcd≡1 coprime = GCD.unique (gcd-GCD _ _) (coprime⇒GCD≡1 coprime)

+

+
gcd≡1⇒coprime : gcd m n ≡ 1 → Coprime m n

+
gcd≡1⇒coprime gcd≡1 = GCD≡1⇒coprime (subst (GCD _ _) gcd≡1 (gcd-GCD _ _))

+

+
coprime-/gcd : ∀ m n .{{_ : NonZero (gcd m n)}} →

+
               Coprime (m / gcd m n) (n / gcd m n)

+
coprime-/gcd m n = GCD≡1⇒coprime (GCD-/gcd m n)

+

+
------------------------------------------------------------------------

+
-- Relational properties of Coprime

+

+
sym : Symmetric Coprime

+
sym c = c ∘ swap

+

+
coprime? : Decidable Coprime

+
coprime? m n = map′ gcd≡1⇒coprime coprime⇒gcd≡1 (gcd m n ≟ 1)

+

+
------------------------------------------------------------------------

+
-- Other basic properties

+

+
-- Everything is coprime to 1.

+

+
1-coprimeTo : ∀ m → Coprime 1 m

+
1-coprimeTo m = ∣1⇒≡1 ∘ proj₁

+

+
-- Nothing except for 1 is coprime to 0.

+

+
0-coprimeTo-m⇒m≡1 : Coprime 0 m → m ≡ 1

+
0-coprimeTo-m⇒m≡1 {m} coprime = coprime (m ∣0 , ∣-refl)

+

+
¬0-coprimeTo-2+ : .{{NonTrivial n}} → ¬ Coprime 0 n

+
¬0-coprimeTo-2+ {n} coprime = contradiction (0-coprimeTo-m⇒m≡1 coprime) (nonTrivial⇒≢1 {n})

+

+
-- If m and n are coprime, then n + m and n are also coprime.

+

+
coprime-+ : Coprime m n → Coprime (n + m) n

+
coprime-+ coprime (d₁ , d₂) = coprime (∣m+n∣m⇒∣n d₁ d₂ , d₂)

+

+
-- Recomputable

+

+
recompute : .(Coprime n d) → Coprime n d

+
recompute {n} {d} coprime = Nullary.recompute (coprime? n d) coprime

+

+
------------------------------------------------------------------------

+
-- Relationship with Bezout's lemma

+

+
-- If the "gcd" in Bézout's identity is non-zero, then the "other"

+
-- divisors are coprime.

+

+
Bézout-coprime : .{{NonZero d}} →

+
                 Bézout.Identity d (m * d) (n * d) → Coprime m n

+
Bézout-coprime {d = suc _} (Bézout.+- x y eq) (divides-refl q₁ , divides-refl q₂) =

+
  lem₁₀ y q₂ x q₁ eq

+
Bézout-coprime {d = suc _} (Bézout.-+ x y eq) (divides-refl q₁ , divides-refl q₂) =

+
  lem₁₀ x q₁ y q₂ eq

+

+
-- Coprime numbers satisfy Bézout's identity.

+

+
coprime-Bézout : Coprime m n → Bézout.Identity 1 m n

+
coprime-Bézout = Bézout.identity ∘ coprime⇒GCD≡1

+

+
-- If m divides n*o and is coprime to n, then it divides o.

+

+
coprime-divisor : Coprime m n → m ∣ n * o → m ∣ o

+
coprime-divisor {o = o} c (divides q eq′) with coprime-Bézout c

+
... | Bézout.+- x y eq = divides (x * o ∸ y * q) (lem₈ x y eq eq′)

+
... | Bézout.-+ x y eq = divides (y * q ∸ x * o) (lem₉ x y eq eq′)

+

+
-- If d is a common divisor of m*o and n*o, and m and n are coprime,

+
-- then d divides o.

+

+
coprime-factors : Coprime m n → d ∣ m * o × d ∣ n * o → d ∣ o

+
coprime-factors c (divides q₁ eq₁ , divides q₂ eq₂) with coprime-Bézout c

+
... | Bézout.+- x y eq = divides (x * q₁ ∸ y * q₂) (lem₁₁ x y eq eq₁ eq₂)

+
... | Bézout.-+ x y eq = divides (y * q₂ ∸ x * q₁) (lem₁₁ y x eq eq₂ eq₁)

+

+
------------------------------------------------------------------------

+
-- Primality implies coprimality.

+

+
prime⇒coprime : Prime p → .{{NonZero n}} → n < p → Coprime p n

+
prime⇒coprime p n<p (d∣p , d∣n) with prime⇒irreducible p d∣p

+
... | inj₁ d≡1      = d≡1

+
... | inj₂ d≡p@refl = contradiction n<p (≤⇒≯ (∣⇒≤ d∣n))

+

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+
+Data.Nat.DivMod.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core lemmas for division and modulus operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.DivMod.Core where

+

+
open import Agda.Builtin.Nat using ()

+
  renaming (div-helper to divₕ; mod-helper to modₕ)

+

+
open import Data.Nat.Base using (zero; suc; _+_; _*_; _∸_; _≤_; _<_;

+
 _≥_; z≤n; s≤s)

+
open import Data.Nat.Properties

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; sym; subst; trans)

+
open import Relation.Nullary.Decidable using (yes; no)

+
open import Relation.Nullary.Negation using (contradiction)

+

+
open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Helper lemmas that have no interpretation for _%_, only for modₕ

+

+
private

+

+
  mod-cong₃ : ∀ {c n a₁ a₂ b} → a₁ ≡ a₂ → modₕ c n a₁ b ≡ modₕ c n a₂ b

+
  mod-cong₃ refl = refl

+

+
  modₕ-skipTo0 : ∀ acc n a b → modₕ acc n (b + a) a ≡ modₕ (a + acc) n b 0

+
  modₕ-skipTo0 acc n zero    b = cong (λ v → modₕ acc n v 0) (+-identityʳ b)

+
  modₕ-skipTo0 acc n (suc a) b = begin-equality

+
    modₕ acc n (b + suc a) (suc a) ≡⟨ mod-cong₃ (+-suc b a) ⟩

+
    modₕ acc n (suc b + a) (suc a) ≡⟨⟩

+
    modₕ (suc acc) n (b + a) a     ≡⟨ modₕ-skipTo0 (suc acc) n a b ⟩

+
    modₕ (a + suc acc) n b 0       ≡⟨ cong (λ v → modₕ v n b 0) (+-suc a acc) ⟩

+
    modₕ (suc a + acc) n b 0       ∎

+

+
------------------------------------------------------------------------

+
-- Lemmas for modₕ that also have an interpretation for _%_

+

+
a[modₕ]1≡0 : ∀ a → modₕ 0 0 a 0 ≡ 0

+
a[modₕ]1≡0 zero    = refl

+
a[modₕ]1≡0 (suc a) = a[modₕ]1≡0 a

+

+
n[modₕ]n≡0 : ∀ acc v → modₕ acc (acc + v) (suc v) v ≡ 0

+
n[modₕ]n≡0 acc v = modₕ-skipTo0 acc (acc + v) v 1

+

+
a[modₕ]n<n : ∀ acc d n → modₕ acc (acc + n) d n ≤ acc + n

+
a[modₕ]n<n acc zero    n       = m≤m+n acc n

+
a[modₕ]n<n acc (suc d) zero    = a[modₕ]n<n zero d (acc + 0)

+
a[modₕ]n<n acc (suc d) (suc n) rewrite +-suc acc n = a[modₕ]n<n (suc acc) d n

+

+
a[modₕ]n≤a : ∀ acc a n → modₕ acc (acc + n) a n ≤ acc + a

+
a[modₕ]n≤a acc zero    n       = ≤-reflexive (sym (+-identityʳ acc))

+
a[modₕ]n≤a acc (suc a) (suc n) = begin

+
  modₕ acc (acc + suc n) (suc a) (suc n) ≡⟨ cong (λ v → modₕ acc v (suc a) (suc n)) (+-suc acc n) ⟩

+
  modₕ acc (suc acc + n) (suc a) (suc n) ≤⟨ a[modₕ]n≤a (suc acc) a n ⟩

+
  suc acc + a                            ≡⟨ sym (+-suc acc a) ⟩

+
  acc + suc a                            ∎

+
a[modₕ]n≤a acc (suc a) zero    = begin

+
  modₕ acc (acc + 0) (suc a) 0 ≡⟨ cong (λ v → modₕ acc v (suc a) 0) (+-identityʳ acc) ⟩

+
  modₕ acc acc (suc a) 0       ≤⟨ a[modₕ]n≤a 0 a acc ⟩

+
  a                            ≤⟨ n≤1+n a ⟩

+
  suc a                        ≤⟨ m≤n+m (suc a) acc ⟩

+
  acc + suc a                  ∎

+

+
a≤n⇒a[modₕ]n≡a : ∀ acc n a b → modₕ acc n a (a + b) ≡ acc + a

+
a≤n⇒a[modₕ]n≡a acc n zero    b = sym (+-identityʳ acc)

+
a≤n⇒a[modₕ]n≡a acc n (suc a) b = begin-equality

+
  modₕ (suc acc) n a (a + b) ≡⟨ a≤n⇒a[modₕ]n≡a (suc acc) n a b ⟩

+
  suc acc + a                ≡⟨ sym (+-suc acc a) ⟩

+
  acc + suc a                ∎

+

+
modₕ-idem : ∀ acc a n → modₕ 0 (acc + n) (modₕ acc (acc + n) a n) (acc + n) ≡ modₕ acc (acc + n) a n

+
modₕ-idem acc zero    n       = a≤n⇒a[modₕ]n≡a 0 (acc + n) acc n

+
modₕ-idem acc (suc a) zero    rewrite +-identityʳ acc = modₕ-idem 0 a acc

+
modₕ-idem acc (suc a) (suc n) rewrite +-suc acc n = modₕ-idem (suc acc) a n

+

+
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 : ∀ acc l n → modₕ acc (acc + l) (suc n) l ≡ 0 → modₕ acc (acc + l) n l ≡ acc + l

+
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 acc zero    zero    eq rewrite +-identityʳ acc = refl

+
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 acc zero    (suc n) eq rewrite +-identityʳ acc = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 acc n eq

+
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 acc (suc l) (suc n) eq rewrite +-suc acc l     = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 (suc acc) l n eq

+

+
k<1+a[modₕ]n⇒k≤a[modₕ]n  : ∀ acc k n l → suc k ≤ modₕ acc (acc + l) (suc n) l → k ≤ modₕ acc (acc + l) n l

+
k<1+a[modₕ]n⇒k≤a[modₕ]n acc k zero    (suc l) (s≤s leq) = leq

+
k<1+a[modₕ]n⇒k≤a[modₕ]n acc k (suc n) zero    leq rewrite +-identityʳ acc = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 k n acc leq

+
k<1+a[modₕ]n⇒k≤a[modₕ]n acc k (suc n) (suc l) leq rewrite +-suc acc l     = k<1+a[modₕ]n⇒k≤a[modₕ]n (suc acc) k n l leq

+

+
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k : ∀ acc k n l → 0 < modₕ acc (acc + l) (suc n) l →

+
                            modₕ acc (acc + l) (suc n) l ≤ suc k → modₕ acc (acc + l) n l ≤ k

+
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k acc k zero    (suc l) 0<mod (s≤s leq) = leq

+
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k acc k (suc n) zero    0<mod leq rewrite +-identityʳ acc = 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 k n acc 0<mod leq

+
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k acc k (suc n) (suc l) 0<mod leq rewrite +-suc acc l = 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k (suc acc) k n l 0<mod leq

+

+
a+n[modₕ]n≡a[modₕ]n : ∀ acc a n → modₕ acc (acc + n) (acc + a + suc n) n ≡ modₕ acc (acc + n) a n

+
a+n[modₕ]n≡a[modₕ]n acc zero n rewrite +-identityʳ acc = begin-equality

+
  modₕ acc (acc + n) (acc + suc n) n   ≡⟨ mod-cong₃ (+-suc acc n) ⟩

+
  modₕ acc (acc + n) (suc acc + n) n   ≡⟨ modₕ-skipTo0 acc (acc + n) n (suc acc) ⟩

+
  modₕ (acc + n) (acc + n) (suc acc) 0 ≡⟨⟩

+
  modₕ 0 (acc + n) acc (acc + n)       ≡⟨ a≤n⇒a[modₕ]n≡a 0 (acc + n) acc n ⟩

+
  acc                                  ∎

+
a+n[modₕ]n≡a[modₕ]n acc (suc a) zero rewrite +-identityʳ acc = begin-equality

+
  modₕ acc acc (acc + suc a + 1)   0   ≡⟨ mod-cong₃ (+-comm (acc + suc a) 1) ⟩

+
  modₕ acc acc (1 + (acc + suc a)) 0   ≡⟨⟩

+
  modₕ 0   acc (acc + suc a)       acc ≡⟨ mod-cong₃ (+-comm acc (suc a)) ⟩

+
  modₕ 0   acc (suc a + acc)       acc ≡⟨ mod-cong₃ (sym (+-suc a acc)) ⟩

+
  modₕ 0   acc (a + suc acc)       acc ≡⟨ a+n[modₕ]n≡a[modₕ]n 0 a acc ⟩

+
  modₕ 0   acc a                   acc ∎

+
a+n[modₕ]n≡a[modₕ]n acc (suc a) (suc n) rewrite +-suc acc n = begin-equality

+
  mod₁ (acc + suc a + (2 + n)) (suc n) ≡⟨ cong (λ v → mod₁ (v + suc (suc n)) (suc n)) (+-suc acc a) ⟩

+
  mod₁ (suc acc + a + (2 + n)) (suc n) ≡⟨⟩

+
  mod₂ (acc + a + (2 + n))     n       ≡⟨ mod-cong₃ (sym (+-assoc (acc + a) 1 (suc n))) ⟩

+
  mod₂ (acc + a + 1 + suc n)   n       ≡⟨ mod-cong₃ (cong (_+ suc n) (+-comm (acc + a) 1)) ⟩

+
  mod₂ (suc acc + a + suc n)   n       ≡⟨ a+n[modₕ]n≡a[modₕ]n (suc acc) a n ⟩

+
  mod₂ a                       n       ∎

+
  where

+
  mod₁ = modₕ acc       (suc acc + n)

+
  mod₂ = modₕ (suc acc) (suc acc + n)

+

+
------------------------------------------------------------------------

+
-- Helper lemmas that have no interpretation for `_/_`, only for `divₕ`

+

+
private

+

+
  div-cong₃ : ∀ {c n a₁ a₂ b} → a₁ ≡ a₂ → divₕ c n a₁ b ≡ divₕ c n a₂ b

+
  div-cong₃ refl = refl

+

+
  acc≤divₕ[acc] : ∀ {acc} d n j → acc ≤ divₕ acc d n j

+
  acc≤divₕ[acc] {acc} d zero    j       = ≤-refl

+
  acc≤divₕ[acc] {acc} d (suc n) zero    = ≤-trans (n≤1+n acc) (acc≤divₕ[acc] d n d)

+
  acc≤divₕ[acc] {acc} d (suc n) (suc j) = acc≤divₕ[acc] d n j

+

+
  pattern inj₂′ x = inj₂ (inj₁ x)

+
  pattern inj₃  x = inj₂ (inj₂ x)

+

+
  -- This hideous lemma details the conditions needed for two divisions to

+
  -- be equal when the two offsets (i.e. the 4ᵗʰ parameters) are different.

+
  -- It may be that this triple sum has an elegant simplification to a

+
  -- set of inequalities involving the modulus but I can't find it.

+
  divₕ-offsetEq : ∀ {acc₁ acc₂} d n j k → j ≤ d → k ≤ d →

+
                  (acc₁ ≡ acc₂     × j ≤ k × k < modₕ 0 d n d) ⊎

+
                  (acc₁ ≡ acc₂     × modₕ 0 d n d ≤ j × j ≤ k) ⊎

+
                  (acc₁ ≡ suc acc₂ × k < modₕ 0 d n d × modₕ 0 d n d ≤ j) →

+
                  divₕ acc₁ d n j ≡ divₕ acc₂ d n k

+
  divₕ-offsetEq d zero    j       k       j≤d k≤d (inj₁  (refl , _)) = refl

+
  divₕ-offsetEq d zero    j       k       j≤d k≤d (inj₂′ (refl , _)) = refl

+
  divₕ-offsetEq d zero    j       k       j≤d k≤d (inj₃ (eq , () , _))

+
  -- (0 , 0) cases

+
  divₕ-offsetEq d (suc n) zero    zero    j≤d k≤d (inj₁ (refl , _)) =

+
    divₕ-offsetEq d n d d ≤-refl ≤-refl (inj₂′ (refl , a[modₕ]n<n 0 n d , ≤-refl))

+
  divₕ-offsetEq d (suc n) zero    zero    j≤d k≤d (inj₂′ (refl , _)) =

+
    divₕ-offsetEq d n d d ≤-refl ≤-refl (inj₂′ (refl , a[modₕ]n<n 0 n d , ≤-refl))

+
  divₕ-offsetEq d (suc n) zero    zero    j≤d k≤d (inj₃ (_ , 0<mod , mod≤0)) =

+
    contradiction (<-≤-trans 0<mod mod≤0) λ()

+
  -- (0 , suc) cases

+
  divₕ-offsetEq d (suc n) zero (suc k)    j≤d k≤d (inj₁  (refl , _ , 1+k<mod)) =

+
    divₕ-offsetEq d n d k ≤-refl (<⇒≤ k≤d) (inj₃ (refl , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d 1+k<mod , a[modₕ]n<n 0 n d))

+
  divₕ-offsetEq d (suc n) zero (suc k)    j≤d k≤d (inj₂′ (refl , mod≤0 , _)) =

+
    divₕ-offsetEq d n d k ≤-refl (<⇒≤ k≤d) (inj₃ (refl , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n (n≤0⇒n≡0 mod≤0))) k≤d , a[modₕ]n<n 0 n d))

+
  divₕ-offsetEq d (suc n) zero (suc k)    j≤d k≤d (inj₃  (_ , 1+k<mod , mod≤0)) =

+
    contradiction (<-≤-trans 1+k<mod mod≤0) λ()

+
  -- (suc , 0) cases

+
  divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₁  (_ , () , _))

+
  divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₂′ (_ , _ , ()))

+
  divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₃  (eq , 0<mod , mod≤1+j)) =

+
    divₕ-offsetEq d n j d (<⇒≤ j≤d) ≤-refl (inj₂′ (eq , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d 0<mod mod≤1+j , <⇒≤ j≤d))

+
  -- (suc , suc) cases

+
  divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₁ (eq , s≤s j≤k , 1+k<mod)) =

+
    divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d 1+k<mod))

+
  divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₂′ (eq , mod≤1+j , (s≤s j≤k))) with modₕ 0 d (suc n) d ≟ 0

+
  ... | yes mod≡0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n mod≡0)) k≤d))

+
  ... | no  mod≢0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₂′ (eq , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (n≢0⇒n>0 mod≢0) mod≤1+j , j≤k))

+
  divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₃  (eq , k<mod , mod≤1+j)) =

+
    divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₃ (eq , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d k<mod , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (≤-<-trans z≤n k<mod) mod≤1+j))

+

+
------------------------------------------------------------------------

+
-- Lemmas for divₕ that also have an interpretation for _/_

+

+
-- The quotient and remainder are related to the dividend and

+
-- divisor in the right way.

+

+
div-mod-lemma : ∀ accᵐ accᵈ d n →

+
    accᵐ + accᵈ * suc (accᵐ + n) + d ≡

+
    modₕ accᵐ (accᵐ + n) d n + divₕ accᵈ (accᵐ + n) d n * suc (accᵐ + n)

+
div-mod-lemma accᵐ accᵈ zero    n    = +-identityʳ _

+
div-mod-lemma accᵐ accᵈ (suc d) zero rewrite +-identityʳ accᵐ = begin-equality

+
  accᵐ + accᵈ * suc accᵐ + suc d          ≡⟨ +-suc _ d ⟩

+
  suc accᵈ * suc accᵐ + d                 ≡⟨ div-mod-lemma zero (suc accᵈ) d accᵐ ⟩

+
  modₕ 0          accᵐ d accᵐ +

+
  divₕ (suc accᵈ) accᵐ d accᵐ * suc accᵐ  ≡⟨⟩

+
  modₕ accᵐ accᵐ (suc d) 0 +

+
  divₕ accᵈ accᵐ (suc d) 0 * suc accᵐ     ∎

+
div-mod-lemma accᵐ accᵈ (suc d) (suc n) rewrite +-suc accᵐ n = begin-equality

+
  accᵐ + accᵈ * m + suc d             ≡⟨ +-suc _ d ⟩

+
  suc (accᵐ + accᵈ * m + d)           ≡⟨ div-mod-lemma (suc accᵐ) accᵈ d n ⟩

+
  modₕ _ _ d n + divₕ accᵈ _ d n * m  ∎

+
  where

+
  m = 2 + accᵐ + n

+

+
divₕ-restart : ∀ {acc} d n j → j < n → divₕ acc d n j ≡ divₕ (suc acc) d (n ∸ suc j) d

+
divₕ-restart d (suc n) zero    j<n       = refl

+
divₕ-restart d (suc n) (suc j) (s≤s j<n) = divₕ-restart d n j j<n

+

+
divₕ-extractAcc : ∀ acc d n j → divₕ acc d n j ≡ acc + divₕ 0 d n j

+
divₕ-extractAcc acc d zero j          = sym (+-identityʳ acc)

+
divₕ-extractAcc acc d (suc n) (suc j) = divₕ-extractAcc acc d n j

+
divₕ-extractAcc acc d (suc n) zero = begin-equality

+
  divₕ (suc acc)    d n d  ≡⟨ divₕ-extractAcc (suc acc) d n d ⟩

+
  suc acc +  divₕ 0 d n d  ≡⟨ sym (+-suc acc _) ⟩

+
  acc + suc (divₕ 0 d n d) ≡⟨ cong (acc +_) (sym (divₕ-extractAcc 1 d n d)) ⟩

+
  acc +      divₕ 1 d n d  ∎

+

+
divₕ-finish : ∀ {acc} d n j → j ≥ n → divₕ acc d n j ≡ acc

+
divₕ-finish d zero    j       j≥n       = refl

+
divₕ-finish d (suc n) (suc j) (s≤s j≥n) = divₕ-finish d n j j≥n

+

+
n[divₕ]n≡1 : ∀ n m → divₕ 0 n (suc m) m ≡ 1

+
n[divₕ]n≡1 n zero    = refl

+
n[divₕ]n≡1 n (suc m) = n[divₕ]n≡1 n m

+

+
a[divₕ]1≡a : ∀ acc a → divₕ acc 0 a 0 ≡ acc + a

+
a[divₕ]1≡a acc zero    = sym (+-identityʳ acc)

+
a[divₕ]1≡a acc (suc a) = trans (a[divₕ]1≡a (suc acc) a) (sym (+-suc acc a))

+

+
a*n[divₕ]n≡a : ∀ acc a n → divₕ acc n (a * suc n) n ≡ acc + a

+
a*n[divₕ]n≡a acc zero    n = sym (+-identityʳ acc)

+
a*n[divₕ]n≡a acc (suc a) n = begin-equality

+
  divₕ acc       n (suc a * suc n)             n ≡⟨ divₕ-restart n (suc a * suc n) n (m≤m+n (suc n) _) ⟩

+
  divₕ (suc acc) n (suc a * suc n ∸ suc n)     n ≡⟨⟩

+
  divₕ (suc acc) n (suc n + a * suc n ∸ suc n) n ≡⟨ div-cong₃ (m+n∸m≡n (suc n) (a * suc n)) ⟩

+
  divₕ (suc acc) n (a * suc n)                 n ≡⟨ a*n[divₕ]n≡a (suc acc) a n ⟩

+
  suc acc + a                                    ≡⟨ sym (+-suc acc a) ⟩

+
  acc + suc a                                    ∎

+

+
+-distrib-divₕ : ∀ acc k m n j → modₕ k (k + j) m j + modₕ 0 (k + j) n (k + j) < suc (k + j) →

+
                 divₕ acc (k + j) (m + n) j ≡ divₕ acc (k + j) m j + divₕ 0 (k + j) n (k + j)

+
+-distrib-divₕ acc k (suc m) n zero leq rewrite +-identityʳ k = +-distrib-divₕ (suc acc) 0 m n k leq

+
+-distrib-divₕ acc k (suc m) n (suc j) leq rewrite +-suc k j = +-distrib-divₕ acc (suc k) m n j leq

+
+-distrib-divₕ acc k zero n j leq = begin-equality

+
  divₕ acc (k + j) n j           ≡⟨ divₕ-extractAcc acc (k + j) n j ⟩

+
  acc + divₕ 0 (k + j) n j       ≡⟨ cong (acc +_) (divₕ-offsetEq _ n j _ (m≤n+m j k) ≤-refl case) ⟩

+
  acc + divₕ 0 (k + j) n (k + j) ∎

+
  where

+
  case = inj₂′ (refl , +-cancelˡ-≤ (suc k) _ _ leq , m≤n+m j k)

+

+
divₕ-mono-≤ : ∀ {acc} k {m n o p} → m ≤ n → p ≤ o → divₕ acc (k + o) m o ≤ divₕ acc (k + p) n p

+
divₕ-mono-≤ {acc} k {0} {n} {_} {p} z≤n p≤o = acc≤divₕ[acc] (k + p) n p

+
divₕ-mono-≤ {acc} k {_} {_} {suc o} {suc p} (s≤s m≤n) (s≤s p≤o)

+
  rewrite +-suc k o | +-suc k p = divₕ-mono-≤ (suc k) m≤n p≤o

+
divₕ-mono-≤ {acc} k {suc m} {suc n} {o} {0}      (s≤s m≤n) z≤n with o <? suc m

+
... | no  o≮1+m rewrite +-identityʳ k = begin

+
  divₕ acc (k + o) (suc m) o ≡⟨ divₕ-finish (k + o) (suc m) o (≮⇒≥ o≮1+m) ⟩

+
  acc                        ≤⟨ n≤1+n acc ⟩

+
  suc acc                    ≤⟨ acc≤divₕ[acc] k n k ⟩

+
  divₕ (suc acc) k n k       ∎

+
... | yes o<1+m rewrite +-identityʳ k = begin

+
  divₕ acc       (k + o) (suc m) o       ≡⟨ divₕ-restart (k + o) (suc m) o o<1+m ⟩

+
  divₕ (suc acc) (k + o) (m ∸ o) (k + o) ≤⟨ divₕ-mono-≤ 0 (≤-trans (m∸n≤m m o) m≤n) (m≤m+n k o) ⟩

+
  divₕ (suc acc) k       n       k       ∎

+

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+
+Data.Nat.DivMod
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Natural number division

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.DivMod where

+

+
open import Agda.Builtin.Nat using (div-helper; mod-helper)

+

+
open import Data.Fin.Base using (Fin; toℕ; fromℕ<)

+
open import Data.Fin.Properties using (nonZeroIndex; toℕ-fromℕ<)

+
open import Data.Nat.Base

+
open import Data.Nat.DivMod.Core

+
open import Data.Nat.Divisibility.Core

+
open import Data.Nat.Induction

+
open import Data.Nat.Properties

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Function.Base using (_$_; _∘_)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; cong; cong₂; refl; trans; _≢_; sym)

+
open import Relation.Nullary.Negation using (contradiction)

+

+
open ≤-Reasoning

+

+
private

+
  variable

+
    m n o p : ℕ

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
open import Data.Nat.Base public

+
  using (_%_; _/_)

+

+
------------------------------------------------------------------------

+
-- Relationship between _%_ and _/_

+

+
m≡m%n+[m/n]*n : ∀ m n .{{_ : NonZero n}} → m ≡ m % n + (m / n) * n

+
m≡m%n+[m/n]*n m (suc n) = div-mod-lemma 0 0 m n

+

+
m%n≡m∸m/n*n : ∀ m n .{{_ : NonZero n}} → m % n ≡ m ∸ (m / n) * n

+
m%n≡m∸m/n*n m n = begin-equality

+
  m % n                  ≡⟨ m+n∸n≡m (m % n) m/n*n ⟨

+
  m % n + m/n*n ∸ m/n*n  ≡⟨ cong (_∸ m/n*n) (m≡m%n+[m/n]*n m n) ⟨

+
  m ∸ m/n*n              ∎

+
  where m/n*n = (m / n) * n

+

+
------------------------------------------------------------------------

+
-- Properties of _%_

+

+
%-congˡ : .{{_ : NonZero o}} → m ≡ n → m % o ≡ n % o

+
%-congˡ refl = refl

+

+
%-congʳ : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ≡ n →

+
          o % m ≡ o % n

+
%-congʳ refl = refl

+

+
n%1≡0 : ∀ n → n % 1 ≡ 0

+
n%1≡0 = a[modₕ]1≡0

+

+
n%n≡0 : ∀ n .{{_ : NonZero n}} → n % n ≡ 0

+
n%n≡0 (suc n-1) = n[modₕ]n≡0 0 n-1

+

+
m%n%n≡m%n : ∀ m n .{{_ : NonZero n}} → m % n % n ≡ m % n

+
m%n%n≡m%n m (suc n-1) = modₕ-idem 0 m n-1

+

+
[m+n]%n≡m%n : ∀ m n .{{_ : NonZero n}} → (m + n) % n ≡ m % n

+
[m+n]%n≡m%n m (suc n-1) = a+n[modₕ]n≡a[modₕ]n 0 m n-1

+

+
[m+kn]%n≡m%n : ∀ m k n .{{_ : NonZero n}} → (m + k * n) % n ≡ m % n

+
[m+kn]%n≡m%n m zero    n = cong (_% n) (+-identityʳ m)

+
[m+kn]%n≡m%n m (suc k) n = begin-equality

+
  (m + (n + k * n)) % n ≡⟨ cong (_% n) (+-assoc m n (k * n)) ⟨

+
  (m + n + k * n)   % n ≡⟨ [m+kn]%n≡m%n (m + n) k n ⟩

+
  (m + n)           % n ≡⟨ [m+n]%n≡m%n m n ⟩

+
  m                 % n ∎

+

+
m≤n⇒[n∸m]%m≡n%m : .{{_ : NonZero m}} → m ≤ n →

+
                  (n ∸ m) % m ≡ n % m

+
m≤n⇒[n∸m]%m≡n%m {m} {n} m≤n = begin-equality

+
  (n ∸ m) % m     ≡⟨ [m+n]%n≡m%n (n ∸ m) m ⟨

+
  (n ∸ m + m) % m ≡⟨ cong (_% m) (m∸n+n≡m m≤n) ⟩

+
  n % m           ∎

+

+
m*n≤o⇒[o∸m*n]%n≡o%n : ∀ m {n o} .{{_ : NonZero n}} → m * n ≤ o →

+
                      (o ∸ m * n) % n ≡ o % n

+
m*n≤o⇒[o∸m*n]%n≡o%n m {n} {o} m*n≤o = begin-equality

+
  (o ∸ m * n) % n         ≡⟨ [m+kn]%n≡m%n (o ∸ m * n) m n ⟨

+
  (o ∸ m * n + m * n) % n ≡⟨ cong (_% n) (m∸n+n≡m m*n≤o) ⟩

+
  o % n                   ∎

+

+
m*n%n≡0 : ∀ m n .{{_ : NonZero n}} → (m * n) % n ≡ 0

+
m*n%n≡0 m n@(suc _) = [m+kn]%n≡m%n 0 m n

+

+
m%n<n : ∀ m n .{{_ : NonZero n}} → m % n < n

+
m%n<n m (suc n-1) = s≤s (a[modₕ]n<n 0 m n-1)

+

+
m%n≤n : ∀ m n .{{_ : NonZero n}} → m % n ≤ n

+
m%n≤n m n = <⇒≤ (m%n<n m n)

+

+
m%n≤m : ∀ m n .{{_ : NonZero n}} → m % n ≤ m

+
m%n≤m m (suc n-1) = a[modₕ]n≤a 0 m n-1

+

+
m≤n⇒m%n≡m : m ≤ n → m % suc n ≡ m

+
m≤n⇒m%n≡m {m = m} m≤n with k , refl ← m≤n⇒∃[o]m+o≡n m≤n

+
  = a≤n⇒a[modₕ]n≡a 0 (m + k) m k

+

+
m<n⇒m%n≡m : .{{_ : NonZero n}} → m < n → m % n ≡ m

+
m<n⇒m%n≡m {n = suc _} m<n = m≤n⇒m%n≡m (<⇒≤pred m<n)

+

+
%-pred-≡0 : ∀ {m n} .{{_ : NonZero n}} → (suc m % n) ≡ 0 → (m % n) ≡ n ∸ 1

+
%-pred-≡0 {m} {suc n-1} eq = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 n-1 m eq

+

+
m<[1+n%d]⇒m≤[n%d] : ∀ {m} n d .{{_ : NonZero d}} → m < suc n % d → m ≤ n % d

+
m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-1

+

+
[1+m%d]≤1+n⇒[m%d]≤n : ∀ m n d .{{_ : NonZero d}} → 0 < suc m % d → suc m % d ≤ suc n → m % d ≤ n

+
[1+m%d]≤1+n⇒[m%d]≤n m n (suc d-1) leq = 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 n m d-1 leq

+

+
%-distribˡ-+ : ∀ m n d .{{_ : NonZero d}} → (m + n) % d ≡ ((m % d) + (n % d)) % d

+
%-distribˡ-+ m n d@(suc d-1) = begin-equality

+
  (m + n)                         % d ≡⟨ cong (λ v → (v + n) % d) (m≡m%n+[m/n]*n m d) ⟩

+
  (m % d +  m / d * d + n)        % d ≡⟨ cong (_% d) (+-assoc (m % d) _ n) ⟩

+
  (m % d + (m / d * d + n))       % d ≡⟨ cong (λ v → (m % d + v) % d) (+-comm _ n) ⟩

+
  (m % d + (n + m / d * d))       % d ≡⟨ cong (_% d) (+-assoc (m % d) n _) ⟨

+
  (m % d +  n + m / d * d)        % d ≡⟨ [m+kn]%n≡m%n (m % d + n) (m / d) d ⟩

+
  (m % d +  n)                    % d ≡⟨ cong (λ v → (m % d + v) % d) (m≡m%n+[m/n]*n n d) ⟩

+
  (m % d + (n % d + (n / d) * d)) % d ≡⟨ cong (_% d) (+-assoc (m % d) (n % d) _) ⟨

+
  (m % d +  n % d + (n / d) * d)  % d ≡⟨ [m+kn]%n≡m%n (m % d + n % d) (n / d) d ⟩

+
  (m % d +  n % d)                % d ∎

+

+
%-distribˡ-* : ∀ m n d .{{_ : NonZero d}} → (m * n) % d ≡ ((m % d) * (n % d)) % d

+
%-distribˡ-* m n d = begin-equality

+
  (m * n)                                             % d ≡⟨ cong (λ h → (h * n) % d) (m≡m%n+[m/n]*n m d) ⟩

+
  ((m′ + k * d) * n)                                  % d ≡⟨ cong (λ h → ((m′ + k * d) * h) % d) (m≡m%n+[m/n]*n n d) ⟩

+
  ((m′ + k * d) * (n′ + j * d))                       % d ≡⟨ cong (_% d) lemma ⟩

+
  (m′ * n′ + (m′ * j + (n′ + j * d) * k) * d)         % d ≡⟨ [m+kn]%n≡m%n (m′ * n′) (m′ * j + (n′ + j * d) * k) d ⟩

+
  (m′ * n′)                                           % d ≡⟨⟩

+
  ((m % d) * (n % d)) % d ∎

+
  where

+
  m′ = m % d

+
  n′ = n % d

+
  k  = m / d

+
  j  = n / d

+
  lemma : (m′ + k * d) * (n′ + j * d) ≡ m′ * n′ + (m′ * j + (n′ + j * d) * k) * d

+
  lemma = begin-equality

+
    (m′ + k * d) * (n′ + j * d)                       ≡⟨ *-distribʳ-+ (n′ + j * d) m′ (k * d) ⟩

+
    m′ * (n′ + j * d) + (k * d) * (n′ + j * d)        ≡⟨ cong₂ _+_ (*-distribˡ-+ m′ n′ (j * d)) (*-comm (k * d) (n′ + j * d)) ⟩

+
    (m′ * n′ + m′ * (j * d)) + (n′ + j * d) * (k * d) ≡⟨ +-assoc (m′ * n′) (m′ * (j * d)) ((n′ + j * d) * (k * d)) ⟩

+
    m′ * n′ + (m′ * (j * d) + (n′ + j * d) * (k * d)) ≡⟨ cong (m′ * n′ +_) (cong₂ _+_ (*-assoc m′ j d) (*-assoc (n′ + j * d) k d)) ⟨

+
    m′ * n′ + ((m′ * j) * d + ((n′ + j * d) * k) * d) ≡⟨ cong (m′ * n′ +_) (*-distribʳ-+ d (m′ * j) ((n′ + j * d) * k)) ⟨

+
    m′ * n′ + (m′ * j + (n′ + j * d) * k) * d         ∎

+

+
%-remove-+ˡ : ∀ {m} n {d} .{{_ : NonZero d}} → d ∣ m → (m + n) % d ≡ n % d

+
%-remove-+ˡ {m@.(p * d)} n {d} (divides-refl p) = begin-equality

+
  (m + n)     % d ≡⟨⟩

+
  (p * d + n) % d ≡⟨ cong (_% d) (+-comm (p * d) n) ⟩

+
  (n + p * d) % d ≡⟨ [m+kn]%n≡m%n n p d ⟩

+
  n           % d ∎

+

+
%-remove-+ʳ : ∀ m {n d} .{{_ : NonZero d}} → d ∣ n → (m + n) % d ≡ m % d

+
%-remove-+ʳ m {n} rewrite +-comm m n = %-remove-+ˡ {n} m

+

+
------------------------------------------------------------------------

+
-- Properties of _/_

+

+
/-congˡ : .{{_ : NonZero o}} → m ≡ n → m / o ≡ n / o

+
/-congˡ refl = refl

+

+
/-congʳ : .{{_ : NonZero n}} .{{_ : NonZero o}} → n ≡ o → m / n ≡ m / o

+
/-congʳ refl = refl

+

+
0/n≡0 : ∀ n .{{_ : NonZero n}} → 0 / n ≡ 0

+
0/n≡0 (suc _) = refl

+

+
n/1≡n : ∀ n → n / 1 ≡ n

+
n/1≡n n = a[divₕ]1≡a 0 n

+

+
n/n≡1 : ∀ n .{{_ : NonZero n}} → n / n ≡ 1

+
n/n≡1 (suc n-1) = n[divₕ]n≡1 n-1 n-1

+

+
m*n/n≡m : ∀ m n .{{_ : NonZero n}} → m * n / n ≡ m

+
m*n/n≡m m (suc n-1) = a*n[divₕ]n≡a 0 m n-1

+

+
m/n*n≡m : ∀ {m n} .{{_ : NonZero n}} → n ∣ m → m / n * n ≡ m

+
m/n*n≡m {n = n} (divides-refl q) = cong (_* n) (m*n/n≡m q n)

+

+
m*[n/m]≡n : .{{_ : NonZero m}} → m ∣ n → m * (n / m) ≡ n

+
m*[n/m]≡n {m} m∣n = trans (*-comm m (_ / m)) (m/n*n≡m m∣n)

+

+
m/n*n≤m : ∀ m n .{{_ : NonZero n}} → (m / n) * n ≤ m

+
m/n*n≤m m n = begin

+
  (m / n) * n          ≤⟨ m≤m+n ((m / n) * n) (m % n) ⟩

+
  (m / n) * n + m % n  ≡⟨ +-comm _ (m % n) ⟩

+
  m % n + (m / n) * n  ≡⟨ m≡m%n+[m/n]*n m n ⟨

+
  m                    ∎

+

+
m/n≤m : ∀ m n .{{_ : NonZero n}} → (m / n) ≤ m

+
m/n≤m m n = *-cancelʳ-≤ (m / n) m n (begin

+
  (m / n) * n ≤⟨ m/n*n≤m m n ⟩

+
  m           ≤⟨ m≤m*n m n ⟩

+
  m * n       ∎)

+

+
m/n<m : ∀ m n .{{_ : NonZero m}} .{{_ : NonZero n}} →

+
        1 < n → m / n < m

+
m/n<m m n 1<n = *-cancelʳ-< _ (m / n) m $ begin-strict

+
  m / n * n ≤⟨ m/n*n≤m m n ⟩

+
  m         <⟨ m<m*n m n 1<n ⟩

+
  m * n     ∎

+

+
/-mono-≤ : .{{_ : NonZero o}} .{{_ : NonZero p}} →

+
           m ≤ n → o ≥ p → m / o ≤ n / p

+
/-mono-≤ m≤n (s≤s o≥p) = divₕ-mono-≤ 0 m≤n o≥p

+

+
/-monoˡ-≤ : ∀ o .{{_ : NonZero o}} → m ≤ n → m / o ≤ n / o

+
/-monoˡ-≤ o m≤n = /-mono-≤ m≤n (≤-refl {o})

+

+
/-monoʳ-≤ : ∀ m {n o} .{{_ : NonZero n}} .{{_ : NonZero o}} →

+
            n ≥ o → m / n ≤ m / o

+
/-monoʳ-≤ m n≥o = /-mono-≤ ≤-refl n≥o

+

+
/-cancelʳ-≡ : ∀ {m n o} .{{_ : NonZero o}} →

+
              o ∣ m → o ∣ n → m / o ≡ n / o → m ≡ n

+
/-cancelʳ-≡ {m} {n} {o} o∣m o∣n m/o≡n/o = begin-equality

+
  m           ≡⟨ m*[n/m]≡n {o} {m} o∣m ⟨

+
  o * (m / o) ≡⟨ cong (o *_) m/o≡n/o ⟩

+
  o * (n / o) ≡⟨ m*[n/m]≡n {o} {n} o∣n ⟩

+
  n           ∎

+

+
m<n⇒m/n≡0 : ∀ {m n} .{{_ : NonZero n}} → m < n → m / n ≡ 0

+
m<n⇒m/n≡0 {m} {suc n-1} (s≤s m≤n) = divₕ-finish n-1 m n-1 m≤n

+

+
m≥n⇒m/n>0 : ∀ {m n} .{{_ : NonZero n}} → m ≥ n → m / n > 0

+
m≥n⇒m/n>0 {m@(suc _)} {n@(suc _)} m≥n = begin

+
  1     ≡⟨ n/n≡1 m ⟨

+
  m / m ≤⟨ /-monoʳ-≤ m m≥n ⟩

+
  m / n ∎

+

+
m/n≡0⇒m<n : ∀ {m n} .{{_ : NonZero n}} → m / n ≡ 0 → m < n

+
m/n≡0⇒m<n {m} {n@(suc _)} m/n≡0  with <-≤-connex m n

+
... | inj₁ m<n = m<n

+
... | inj₂ n≤m = contradiction m/n≡0 (≢-nonZero⁻¹ _)

+
  where instance _ =  >-nonZero (m≥n⇒m/n>0 n≤m)

+

+
m/n≢0⇒n≤m : ∀ {m n} .{{_ : NonZero n}} → m / n ≢ 0 → n ≤ m

+
m/n≢0⇒n≤m {m} {n@(suc _)} m/n≢0 with <-≤-connex m n

+
... | inj₁ m<n = contradiction (m<n⇒m/n≡0 m<n) m/n≢0

+
... | inj₂ n≤m = n≤m

+

+
+-distrib-/ : ∀ m n {d} .{{_ : NonZero d}} → m % d + n % d < d →

+
              (m + n) / d ≡ m / d + n / d

+
+-distrib-/ m n {suc d-1} leq = +-distrib-divₕ 0 0 m n d-1 leq

+

+
+-distrib-/-∣ˡ : ∀ {m} n {d} .{{_ : NonZero d}} →

+
                 d ∣ m → (m + n) / d ≡ m / d + n / d

+
+-distrib-/-∣ˡ {m@.(p * d)} n {d} (divides-refl p) = +-distrib-/ m n $ begin-strict

+
  m % d + n % d     ≡⟨⟩

+
  p * d % d + n % d ≡⟨ cong (_+ n % d) (m*n%n≡0 p d) ⟩

+
  n % d             <⟨ m%n<n n d ⟩

+
  d                 ∎

+

+
+-distrib-/-∣ʳ : ∀ m {n} {d} .{{_ : NonZero d}} →

+
                 d ∣ n → (m + n) / d ≡ m / d + n / d

+
+-distrib-/-∣ʳ m {n@.(p * d)} {d} (divides-refl p) = +-distrib-/ m n $ begin-strict

+
  m % d + n % d     ≡⟨⟩

+
  m % d + p * d % d ≡⟨ cong (m % d +_) (m*n%n≡0 p d) ⟩

+
  m % d + 0         ≡⟨ +-identityʳ _ ⟩

+
  m % d             <⟨ m%n<n m d ⟩

+
  d                 ∎

+

+
m/n≡1+[m∸n]/n : ∀ {m n} .{{_ : NonZero n}} → m ≥ n → m / n ≡ 1 + ((m ∸ n) / n)

+
m/n≡1+[m∸n]/n {m@(suc m-1)} {n@(suc n-1)} m≥n = begin-equality

+
  m / n                              ≡⟨⟩

+
  div-helper 0 n-1 m n-1             ≡⟨ divₕ-restart n-1 m n-1 m≥n ⟩

+
  div-helper 1 n-1 (m ∸ n) n-1       ≡⟨ divₕ-extractAcc 1 n-1 (m ∸ n) n-1 ⟩

+
  1 + (div-helper 0 n-1 (m ∸ n) n-1) ≡⟨⟩

+
  1 + (m ∸ n) / n                    ∎

+

+
[m∸n]/n≡m/n∸1 : ∀ m n .{{_ : NonZero n}} → (m ∸ n) / n ≡ pred (m / n)

+
[m∸n]/n≡m/n∸1 m n with <-≤-connex m n

+
... | inj₁ m<n = begin-equality

+
  (m ∸ n) / n  ≡⟨ m<n⇒m/n≡0 (≤-<-trans (m∸n≤m m n) m<n) ⟩

+
  0            ≡⟨⟩

+
  pred 0       ≡⟨ cong pred (m<n⇒m/n≡0 m<n) ⟨

+
  pred (m / n) ∎

+
... | inj₂ n≥m = begin-equality

+
  (m ∸ n) / n            ≡⟨⟩

+
  pred (1 + (m ∸ n) / n) ≡⟨ cong pred (m/n≡1+[m∸n]/n n≥m) ⟨

+
  pred (m / n)           ∎

+

+
m∣n⇒o%n%m≡o%m : ∀ m n o .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣ n →

+
                o % n % m ≡ o % m

+
m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality

+
  o % n % m                ≡⟨⟩

+
  o % pm % m               ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩

+
  (o ∸ o / pm * pm) % m    ≡⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟨

+
  (o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩

+
  o % m                    ∎

+
  where

+
  pm = p * m

+

+
  lem : o / pm * p * m ≤ o

+
  lem = begin

+
    o / pm * p * m       ≡⟨ *-assoc (o / pm) p m ⟩

+
    o / pm * pm          ≤⟨ m/n*n≤m o pm ⟩

+
    o                    ∎

+

+
m*n/m*o≡n/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (m * o)}} →

+
              (m * n) / (m * o) ≡ n / o

+
m*n/m*o≡n/o m n o = helper (<-wellFounded n)

+
  where

+
  instance _ = m*n≢0 m o

+
  helper : ∀ {n} → Acc _<_ n → (m * n) / (m * o) ≡ n / o

+
  helper {n} (acc rec) with <-≤-connex n o

+
  ... | inj₁ n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o))

+
    where instance _ = m*n≢0⇒m≢0 m

+
  ... | inj₂ n≥o = begin-equality

+
    (m * n) / (m * o)             ≡⟨ m/n≡1+[m∸n]/n (*-monoʳ-≤ m n≥o) ⟩

+
    1 + (m * n ∸ m * o) / (m * o) ≡⟨ cong (suc ∘ (_/ (m * o))) (*-distribˡ-∸ m n o) ⟨

+
    1 + (m * (n ∸ o)) / (m * o)   ≡⟨ cong suc (helper (rec n∸o<n)) ⟩

+
    1 + (n ∸ o) / o               ≡⟨ m/n≡1+[m∸n]/n n≥o ⟨

+
    n / o                         ∎

+
    where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o)) n≥o

+

+
m*n/o*n≡m/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (o * n)}} →

+
              m * n / (o * n) ≡ m / o

+
m*n/o*n≡m/o m n o = begin-equality

+
  m * n / (o * n) ≡⟨ /-congˡ (*-comm m n) ⟩

+
  n * m / (o * n) ≡⟨ /-congʳ (*-comm o n) ⟩

+
  n * m / (n * o) ≡⟨ m*n/m*o≡n/o n m o ⟩

+
  m / o           ∎

+
  where instance

+
    _ : NonZero n

+
    _ = m*n≢0⇒n≢0 o

+
    _ : NonZero (n * o)

+
    _ = m*n≢0 n o

+

+
m<n*o⇒m/o<n : ∀ {m n o} .{{_ : NonZero o}} → m < n * o → m / o < n

+
m<n*o⇒m/o<n {m} {1} {o} m<o rewrite *-identityˡ o = begin-strict

+
  m / o ≡⟨ m<n⇒m/n≡0 m<o ⟩

+
  0     <⟨ z<s ⟩

+
  1 ∎

+
m<n*o⇒m/o<n {m} {suc n@(suc _)} {o} m<n*o = pred-cancel-< $ begin-strict

+
  pred (m / o) ≡⟨ [m∸n]/n≡m/n∸1 m o ⟨

+
  (m ∸ o) / o  <⟨ m<n*o⇒m/o<n (m<n+o⇒m∸n<o m o m<n*o) ⟩

+
  n ∎

+
  where instance _ = m*n≢0 n o

+

+
[m∸n*o]/o≡m/o∸n : ∀ m n o .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n

+
[m∸n*o]/o≡m/o∸n m zero    o = refl

+
[m∸n*o]/o≡m/o∸n m (suc n) o = begin-equality

+
  (m ∸ (o + n * o)) / o ≡⟨ /-congˡ (∸-+-assoc m o (n * o)) ⟨

+
  (m ∸ o ∸ n * o) / o   ≡⟨ [m∸n*o]/o≡m/o∸n (m ∸ o) n o ⟩

+
  (m ∸ o) / o ∸ n       ≡⟨ cong (_∸ n) ([m∸n]/n≡m/n∸1 m o) ⟩

+
  m / o ∸ 1 ∸ n         ≡⟨ ∸-+-assoc (m / o) 1 n ⟩

+
  m / o ∸ suc n         ∎

+

+
m/n/o≡m/[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}}

+
                .{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o)

+
m/n/o≡m/[n*o] m n o = begin-equality

+
  m / n / o                             ≡⟨ /-congˡ {o = o} (/-congˡ (m≡m%n+[m/n]*n m n*o)) ⟩

+
  (m % n*o + m / n*o * n*o) / n / o     ≡⟨ /-congˡ (+-distrib-/-∣ʳ (m % n*o) lem₁) ⟩

+
  (m % n*o / n + m / n*o * n*o / n) / o ≡⟨ cong (λ # → (m % n*o / n + #) / o) lem₂ ⟩

+
  (m % n*o / n + m / n*o * o) / o       ≡⟨ +-distrib-/-∣ʳ (m % n*o / n) (divides-refl (m / n*o)) ⟩

+
  m % n*o / n / o + m / n*o * o / o     ≡⟨ cong (m % n*o / n / o +_) (m*n/n≡m (m / n*o) o) ⟩

+
  m % n*o / n / o + m / n*o             ≡⟨ cong (_+ m / n*o) (m<n⇒m/n≡0 (m<n*o⇒m/o<n {n = o} lem₃)) ⟩

+
  m / n*o                               ∎

+
  where

+
  n*o = n * o

+
  o*n = o * n

+

+
  lem₁ : n ∣ m / n*o * n*o

+
  lem₁ = divides (m / n*o * o) $ begin-equality

+
    m / n*o * n*o   ≡⟨ cong (m / n*o *_) (*-comm n o) ⟩

+
    m / n*o * o*n   ≡⟨ *-assoc (m / n*o) o n ⟨

+
    m / n*o * o * n ∎

+

+
  lem₂ : m / n*o * n*o / n ≡ m / n*o * o

+
  lem₂ = begin-equality

+
    m / n*o * n*o / n   ≡⟨ cong (λ # → m / n*o * # / n) (*-comm n o) ⟩

+
    m / n*o * o*n / n   ≡⟨ /-congˡ (*-assoc (m / n*o) o n) ⟨

+
    m / n*o * o * n / n ≡⟨ m*n/n≡m (m / n*o * o) n ⟩

+
    m / n*o * o         ∎

+

+
  lem₃ : m % n*o < o*n

+
  lem₃ = begin-strict

+
    m % n*o <⟨ m%n<n m n*o ⟩

+
    n*o     ≡⟨ *-comm n o ⟩

+
    o*n     ∎

+

+
*-/-assoc : ∀ m {n d} .{{_ : NonZero d}} → d ∣ n → m * n / d ≡ m * (n / d)

+
*-/-assoc zero    {_} {d} d∣n = 0/n≡0 d

+
*-/-assoc (suc m) {n} {d} d∣n = begin-equality

+
  (n + m * n) / d     ≡⟨ +-distrib-/-∣ˡ _ d∣n ⟩

+
  n / d + (m * n) / d ≡⟨ cong (n / d +_) (*-/-assoc m d∣n) ⟩

+
  n / d + m * (n / d) ∎

+

+
/-*-interchange : .{{_ : NonZero o}} .{{_ : NonZero p}} .{{_ : NonZero (o * p)}} →

+
                  o ∣ m → p ∣ n → (m * n) / (o * p) ≡ (m / o) * (n / p)

+
/-*-interchange {o} {p} {m@.(q * o)} {n@.(r * p)} (divides-refl q) (divides-refl r)

+
  = begin-equality

+
  (m * n) / (o * p) ≡⟨⟩

+
  q * o * (r * p) / (o * p) ≡⟨ /-congˡ ([m*n]*[o*p]≡[m*o]*[n*p] q o r p) ⟩

+
  q * r * (o * p) / (o * p) ≡⟨ m*n/n≡m (q * r) (o * p) ⟩

+
  q * r                     ≡⟨ cong₂ _*_ (m*n/n≡m q o) (m*n/n≡m r p) ⟨

+
  (q * o / o) * (r * p / p) ≡⟨⟩

+
  (m / o) * (n / p)         ∎

+

+
m*n/m!≡n/[m∸1]! : ∀ m n .{{_ : NonZero m}} →

+
                  let instance _ = m !≢0 ; instance _ = (pred m) !≢0 in

+
                  (m * n / m !) ≡ (n / (pred m) !)

+
m*n/m!≡n/[m∸1]! m′@(suc m) n = m*n/m*o≡n/o m′ n (m !)

+
  where instance

+
    _ = m !≢0

+
    _ = m′ !≢0

+

+
m%[n*o]/o≡m/o%n : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} →

+
                  {{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n

+
m%[n*o]/o≡m/o%n m n o = begin-equality

+
  m % (n * o) / o                   ≡⟨ /-congˡ (m%n≡m∸m/n*n m (n * o)) ⟩

+
  (m ∸ (m / (n * o) * (n * o))) / o ≡⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟨

+
  (m ∸ (m / (n * o) * n * o)) / o   ≡⟨ [m∸n*o]/o≡m/o∸n m (m / (n * o) * n) o ⟩

+
  m / o ∸ m / (n * o) * n           ≡⟨ cong (λ # → m / o ∸ # * n) (/-congʳ (*-comm n o)) ⟩

+
  m / o ∸ m / (o * n) * n           ≡⟨ cong (λ # → m / o ∸ # * n) (m/n/o≡m/[n*o] m o n ) ⟨

+
  m / o ∸ m / o / n * n             ≡⟨ m%n≡m∸m/n*n (m / o) n ⟨

+
  m / o % n                         ∎

+
  where instance _ = m*n≢0 o n

+

+
m%n*o≡m*o%[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero (n * o)}} →

+
                  m % n * o ≡ m * o % (n * o)

+
m%n*o≡m*o%[n*o] m n o = begin-equality

+
  m % n * o                         ≡⟨ cong (_* o) (m%n≡m∸m/n*n m n) ⟩

+
  (m ∸ m / n * n) * o               ≡⟨ *-distribʳ-∸ o m (m / n * n) ⟩

+
  m * o ∸ m / n * n * o             ≡⟨ cong (λ # → m * o ∸ # * n * o) (m*n/o*n≡m/o m o n) ⟨

+
  m * o ∸ m * o / (n * o) * n * o   ≡⟨ cong (m * o ∸_) (*-assoc (m * o / (n * o)) n o) ⟩

+
  m * o ∸ m * o / (n * o) * (n * o) ≡⟨ m%n≡m∸m/n*n (m * o) (n * o) ⟨

+
  m * o % (n * o)                   ∎

+

+
[m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p .{{_ : NonZero (p * n)}} → o < n →

+
                              (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o

+
[m*n+o]%[p*n]≡[m*n]%[p*n]+o m {n} {o} p@(suc p-1) o<n = begin-equality

+
  (mn + o) % pn           ≡⟨ %-distribˡ-+ mn o pn ⟩

+
  (mn % pn + o % pn) % pn ≡⟨ cong (λ # → (mn % pn + #) % pn) (m<n⇒m%n≡m (m<n⇒m<o*n p o<n)) ⟩

+
  (mn % pn + o) % pn      ≡⟨ m<n⇒m%n≡m lem₂ ⟩

+
  mn % pn + o             ∎

+
  where

+
  mn = m * n

+
  pn = p * n

+

+
  lem₁ : mn % pn ≤ p-1 * n

+
  lem₁ = begin

+
    mn % pn     ≡⟨ m%n*o≡m*o%[n*o] m p n ⟨

+
    (m % p) * n ≤⟨ *-monoˡ-≤ n (m<1+n⇒m≤n (m%n<n m p)) ⟩

+
    p-1 * n     ∎

+

+
  lem₂ : mn % pn + o < pn

+
  lem₂ = begin-strict

+
    mn % pn + o <⟨ +-mono-≤-< lem₁ o<n ⟩

+
    p-1 * n + n ≡⟨ +-comm (p-1 * n) n ⟩

+
    pn          ∎

+

+
------------------------------------------------------------------------

+
--  A specification of integer division.

+

+
record DivMod (dividend divisor : ℕ) : Set where

+
  constructor result

+
  field

+
    quotient  : ℕ

+
    remainder : Fin divisor

+
    property  : dividend ≡ toℕ remainder + quotient * divisor

+

+
  nonZeroDivisor : NonZero divisor

+
  nonZeroDivisor = nonZeroIndex remainder

+

+

+
infixl 7 _div_ _mod_ _divMod_

+

+
_div_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ

+
_div_ = _/_

+

+
_mod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → Fin divisor

+
m mod n = fromℕ< (m%n<n m n)

+

+
_divMod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} →

+
           DivMod dividend divisor

+
m divMod n = result (m / n) (m mod n) $ begin-equality

+
  m                               ≡⟨  m≡m%n+[m/n]*n m n ⟩

+
  m % n                + [m/n]*n  ≡⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< [m%n]<n) ⟨

+
  toℕ (fromℕ< [m%n]<n) + [m/n]*n  ∎

+
  where [m/n]*n = m / n * n ; [m%n]<n = m%n<n m n

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Divisibility.Core.html b/site/static/agda_html/Data.Nat.Divisibility.Core.html
new file mode 100644
index 000000000..002c7014b
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Divisibility.Core.html
@@ -0,0 +1,158 @@
+
+Data.Nat.Divisibility.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core definition of divisibility

+
------------------------------------------------------------------------

+

+
-- The definition of divisibility is split out from

+
-- `Data.Nat.Divisibility` to avoid a dependency cycle with

+
-- `Data.Nat.DivMod`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Divisibility.Core where

+

+
open import Data.Nat.Base using (ℕ; _*_; _<_; NonTrivial)

+
open import Data.Nat.Properties

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl)

+

+

+
private

+
  variable m n o p : ℕ

+

+
------------------------------------------------------------------------

+
-- Main definition

+
--

+
-- m ∣ n is inhabited iff m divides n. Some sources, like Hardy and

+
-- Wright's "An Introduction to the Theory of Numbers", require m to

+
-- be non-zero. However, some things become a bit nicer if m is

+
-- allowed to be zero. For instance, _∣_ becomes a partial order, and

+
-- the gcd of 0 and 0 becomes defined.

+

+
infix 4 _∣_ _∤_

+

+
record _∣_ (m n : ℕ) : Set where

+
  constructor divides

+
  field quotient : ℕ

+
        equality : n ≡ quotient * m

+

+
_∤_ : Rel ℕ _

+
m ∤ n = ¬ (m ∣ n)

+

+
-- Smart constructor

+

+
pattern divides-refl q = divides q refl

+

+
open _∣_ using (quotient) public

+

+
------------------------------------------------------------------------

+
-- Restricted divisor relation

+

+
-- Relation for having a non-trivial divisor below a given bound.

+
-- Useful when reasoning about primality.

+
infix 10 _HasNonTrivialDivisorLessThan_

+

+
record _HasNonTrivialDivisorLessThan_ (m n : ℕ) : Set where

+
  constructor hasNonTrivialDivisor

+
  field

+
    {divisor}       : ℕ

+
    .{{nontrivial}} : NonTrivial divisor

+
    divisor-<       : divisor < n

+
    divisor-∣       : divisor ∣ m

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.1

+

+
*-pres-∣ : o ∣ m → p ∣ n → o * p ∣ m * n

+
*-pres-∣ {o} {m@.(q * o)} {p} {n@.(r * p)} (divides-refl q) (divides-refl r) =

+
  divides (q * r) ([m*n]*[o*p]≡[m*o]*[n*p] q o r p)

+

+
{-# WARNING_ON_USAGE *-pres-∣

+
"Warning: *-pres-∣ was deprecated in v2.1.

+
 Please use Data.Nat.Divisibility.*-pres-∣ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Divisibility.html b/site/static/agda_html/Data.Nat.Divisibility.html
new file mode 100644
index 000000000..1415ddacd
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Divisibility.html
@@ -0,0 +1,437 @@
+
+Data.Nat.Divisibility
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Divisibility

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Divisibility where

+

+
open import Data.Nat.Base

+
open import Data.Nat.DivMod

+
  using (m≡m%n+[m/n]*n; m%n≡m∸m/n*n; m*n/n≡m; m*n%n≡0; *-/-assoc)

+
open import Data.Nat.Properties

+
open import Function.Base using (_∘′_; _$_; flip)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Level using (0ℓ)

+
open import Relation.Nullary.Decidable as Dec using (yes; no)

+
open import Relation.Nullary.Negation.Core using (contradiction)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles using (Preorder; Poset)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsPartialOrder)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Transitive; Antisymmetric; Decidable)

+
import Relation.Binary.Reasoning.Preorder as ≲-Reasoning

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl; sym; cong; subst)

+
open import Relation.Binary.Reasoning.Syntax

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (isEquivalence; module ≡-Reasoning)

+

+
private

+
  variable d m n o p : ℕ

+

+

+
------------------------------------------------------------------------

+
-- Definition and derived properties

+

+
open import Data.Nat.Divisibility.Core public hiding (*-pres-∣)

+

+
quotient≢0 : (m∣n : m ∣ n) → .{{NonZero n}} → NonZero (quotient m∣n)

+
quotient≢0 m∣n rewrite _∣_.equality m∣n = m*n≢0⇒m≢0 (quotient m∣n)

+

+
m∣n⇒n≡quotient*m : (m∣n : m ∣ n) → n ≡ (quotient m∣n) * m

+
m∣n⇒n≡quotient*m m∣n = _∣_.equality m∣n

+

+
m∣n⇒n≡m*quotient : (m∣n : m ∣ n) → n ≡ m * (quotient m∣n)

+
m∣n⇒n≡m*quotient {m = m} m∣n rewrite _∣_.equality m∣n = *-comm (quotient m∣n) m

+

+
quotient-∣ : (m∣n : m ∣ n) → (quotient m∣n) ∣ n

+
quotient-∣ {m = m} m∣n = divides m (m∣n⇒n≡m*quotient m∣n)

+

+
quotient>1 : (m∣n : m ∣ n) → m < n → 1 < quotient m∣n

+
quotient>1 {m} {n} m∣n m<n = *-cancelˡ-< m 1 (quotient m∣n) $ begin-strict

+
    m * 1        ≡⟨ *-identityʳ m ⟩

+
    m            <⟨ m<n ⟩

+
    n            ≡⟨ m∣n⇒n≡m*quotient m∣n ⟩

+
    m * quotient m∣n ∎

+
  where open ≤-Reasoning

+

+
quotient-< : (m∣n : m ∣ n) → .{{NonTrivial m}} → .{{NonZero n}} → quotient m∣n < n

+
quotient-< {m} {n} m∣n = begin-strict

+
  quotient m∣n     <⟨ m<m*n (quotient m∣n) m (nonTrivial⇒n>1 m) ⟩

+
  quotient m∣n * m ≡⟨ _∣_.equality m∣n ⟨

+
  n                ∎

+
  where open ≤-Reasoning; instance _ = quotient≢0 m∣n

+

+
------------------------------------------------------------------------

+
-- Relating _/_ and quotient

+

+
n/m≡quotient : (m∣n : m ∣ n) .{{_ : NonZero m}} → n / m ≡ quotient m∣n

+
n/m≡quotient {m = m} (divides-refl q) = m*n/n≡m q m

+

+
------------------------------------------------------------------------

+
-- Relationship with _%_

+

+
m%n≡0⇒n∣m : ∀ m n .{{_ : NonZero n}} → m % n ≡ 0 → n ∣ m

+
m%n≡0⇒n∣m m n eq = divides (m / n) $ begin

+
  m                ≡⟨ m≡m%n+[m/n]*n m n ⟩

+
  m % n + [m/n]*n  ≡⟨ cong (_+ [m/n]*n) eq ⟩

+
  [m/n]*n          ∎

+
  where open ≡-Reasoning; [m/n]*n = m / n * n

+

+
n∣m⇒m%n≡0 : ∀ m n .{{_ : NonZero n}} → n ∣ m → m % n ≡ 0

+
n∣m⇒m%n≡0 .(q * n) n (divides-refl q) = m*n%n≡0 q n

+

+
m%n≡0⇔n∣m : ∀ m n .{{_ : NonZero n}} → m % n ≡ 0 ⇔ n ∣ m

+
m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and _≤_

+

+
∣⇒≤ : .{{_ : NonZero n}} → m ∣ n → m ≤ n

+
∣⇒≤ {n@.(q * m)} {m} (divides-refl q@(suc p)) = m≤m+n m (p * m)

+

+
>⇒∤ : .{{_ : NonZero n}} → m > n → m ∤ n

+
>⇒∤ {n@(suc _)} n<m@(s<s _) m∣n = contradiction (∣⇒≤ m∣n) (<⇒≱ n<m)

+

+
------------------------------------------------------------------------

+
-- _∣_ is a partial order

+

+
-- these could/should inherit from Algebra.Properties.Monoid.Divisibility

+

+
∣-reflexive : _≡_ ⇒ _∣_

+
∣-reflexive {n} refl = divides 1 (sym (*-identityˡ n))

+

+
∣-refl : Reflexive _∣_

+
∣-refl = ∣-reflexive refl

+

+
∣-trans : Transitive _∣_

+
∣-trans (divides-refl p) (divides-refl q) =

+
  divides (q * p) (sym (*-assoc q p _))

+

+
∣-antisym : Antisymmetric _≡_ _∣_

+
∣-antisym {m}     {zero}   _  q∣p = m∣n⇒n≡m*quotient q∣p

+
∣-antisym {zero}  {n}     p∣q  _  = sym (m∣n⇒n≡m*quotient p∣q)

+
∣-antisym {suc m} {suc n} p∣q q∣p = ≤-antisym (∣⇒≤ p∣q) (∣⇒≤ q∣p)

+

+
infix 4 _∣?_

+

+
_∣?_ : Decidable _∣_

+
zero  ∣? zero   = yes (divides-refl 0)

+
zero  ∣? suc m  = no ((λ()) ∘′ ∣-antisym (divides-refl 0))

+
n@(suc _) ∣? m  = Dec.map (m%n≡0⇔n∣m m n) (m % n ≟ 0)

+

+
∣-isPreorder : IsPreorder _≡_ _∣_

+
∣-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ∣-reflexive

+
  ; trans         = ∣-trans

+
  }

+

+
∣-isPartialOrder : IsPartialOrder _≡_ _∣_

+
∣-isPartialOrder = record

+
  { isPreorder = ∣-isPreorder

+
  ; antisym    = ∣-antisym

+
  }

+

+
∣-preorder : Preorder 0ℓ 0ℓ 0ℓ

+
∣-preorder = record

+
  { isPreorder = ∣-isPreorder

+
  }

+

+
∣-poset : Poset 0ℓ 0ℓ 0ℓ

+
∣-poset = record

+
  { isPartialOrder = ∣-isPartialOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- A reasoning module for the _∣_ relation

+

+
module ∣-Reasoning where

+
  private module Base = ≲-Reasoning ∣-preorder

+

+
  open Base public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)

+
    renaming (≲-go to ∣-go)

+

+
  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public

+

+

+
------------------------------------------------------------------------

+
-- Simple properties of _∣_

+

+
infix 10 _∣0 1∣_

+

+
_∣0 : ∀ n → n ∣ 0

+
n ∣0 = divides-refl 0

+

+
0∣⇒≡0 : 0 ∣ n → n ≡ 0

+
0∣⇒≡0 {n} 0∣n = ∣-antisym (n ∣0) 0∣n

+

+
1∣_ : ∀ n → 1 ∣ n

+
1∣ n = divides n (sym (*-identityʳ n))

+

+
∣1⇒≡1 : n ∣ 1 → n ≡ 1

+
∣1⇒≡1 {n} n∣1 = ∣-antisym n∣1 (1∣ n)

+

+
n∣n : n ∣ n

+
n∣n = ∣-refl

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and _+_

+

+
∣m∣n⇒∣m+n : d ∣ m → d ∣ n → d ∣ m + n

+
∣m∣n⇒∣m+n (divides-refl p) (divides-refl q) =

+
  divides (p + q) (sym (*-distribʳ-+ _ p q))

+

+
∣m+n∣m⇒∣n : d ∣ m + n → d ∣ m → d ∣ n

+
∣m+n∣m⇒∣n {d} {m} {n} (divides p m+n≡p*d) (divides-refl q) =

+
  divides (p ∸ q) $ begin-equality

+
    n             ≡⟨ m+n∸n≡m n m ⟨

+
    n + m ∸ m     ≡⟨ cong (_∸ m) (+-comm n m) ⟩

+
    m + n ∸ m     ≡⟨ cong (_∸ m) m+n≡p*d ⟩

+
    p * d ∸ q * d ≡⟨ *-distribʳ-∸ d p q ⟨

+
    (p ∸ q) * d   ∎

+
    where open ∣-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and _*_

+

+
n∣m*n : ∀ m {n} → n ∣ m * n

+
n∣m*n m = divides m refl

+

+
m∣m*n : ∀ {m} n → m ∣ m * n

+
m∣m*n n = divides n (*-comm _ n)

+

+
n∣m*n*o : ∀ m {n} o → n ∣ m * n * o

+
n∣m*n*o m o = ∣-trans (n∣m*n m) (m∣m*n o)

+

+
∣m⇒∣m*n : ∀ n → d ∣ m → d ∣ m * n

+
∣m⇒∣m*n n (divides-refl q) = ∣-trans (n∣m*n q) (m∣m*n n)

+

+
∣n⇒∣m*n : ∀ m {n} → d ∣ n → d ∣ m * n

+
∣n⇒∣m*n m {n} rewrite *-comm m n = ∣m⇒∣m*n m

+

+
m*n∣⇒m∣ : ∀ m n → m * n ∣ d → m ∣ d

+
m*n∣⇒m∣ m n (divides-refl q) = ∣n⇒∣m*n q (m∣m*n n)

+

+
m*n∣⇒n∣ : ∀ m n → m * n ∣ d → n ∣ d

+
m*n∣⇒n∣ m n rewrite *-comm m n = m*n∣⇒m∣ n m

+

+
*-pres-∣ : o ∣ m → p ∣ n → o * p ∣ m * n

+
*-pres-∣ {o} {m@.(q * o)} {p} {n@.(r * p)} (divides-refl q) (divides-refl r) =

+
  divides (q * r) ([m*n]*[o*p]≡[m*o]*[n*p] q o r p)

+

+
*-monoʳ-∣ : ∀ o → m ∣ n → o * m ∣ o * n

+
*-monoʳ-∣ o = *-pres-∣ (∣-refl {o})

+

+
*-monoˡ-∣ : ∀ o → m ∣ n → m * o ∣ n * o

+
*-monoˡ-∣ o = flip *-pres-∣ (∣-refl {o})

+

+
*-cancelˡ-∣ : ∀ o .{{_ : NonZero o}} → o * m ∣ o * n → m ∣ n

+
*-cancelˡ-∣ {m} {n} o o*m∣o*n = divides q $ *-cancelˡ-≡ n (q * m) o $ begin-equality

+
  o * n       ≡⟨ m∣n⇒n≡m*quotient o*m∣o*n ⟩

+
  o * m * q   ≡⟨ *-assoc o m q ⟩

+
  o * (m * q) ≡⟨ cong (o *_) (*-comm q m) ⟨

+
  o * (q * m) ∎

+
  where

+
  open ∣-Reasoning

+
  q = quotient o*m∣o*n

+

+
*-cancelʳ-∣ : ∀ o .{{_ : NonZero o}} → m * o ∣ n * o → m ∣ n

+
*-cancelʳ-∣ {m} {n} o rewrite *-comm m o | *-comm n o = *-cancelˡ-∣ o

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and _∸_

+

+
∣m∸n∣n⇒∣m : ∀ d → n ≤ m → d ∣ m ∸ n → d ∣ n → d ∣ m

+
∣m∸n∣n⇒∣m {n} {m} d n≤m (divides p m∸n≡p*d) (divides-refl q) =

+
  divides (p + q) $ begin-equality

+
    m             ≡⟨ m+[n∸m]≡n n≤m ⟨

+
    n + (m ∸ n)   ≡⟨ +-comm n (m ∸ n) ⟩

+
    m ∸ n + n     ≡⟨ cong (_+ n) m∸n≡p*d ⟩

+
    p * d + q * d ≡⟨ *-distribʳ-+ d p q ⟨

+
    (p + q) * d   ∎

+
  where open ∣-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and _/_

+

+
m/n∣m : .{{_ : NonZero n}} → n ∣ m → m / n ∣ m

+
m/n∣m {n} {m} n∣m = begin

+
  m / n        ≡⟨ n/m≡quotient n∣m ⟩

+
  quotient n∣m ∣⟨ quotient-∣ n∣m ⟩

+
  m            ∎

+
  where open ∣-Reasoning

+

+
m*n∣o⇒m∣o/n : ∀ m n .{{_ : NonZero n}} → m * n ∣ o → m ∣ o / n

+
m*n∣o⇒m∣o/n m n (divides-refl p) = divides p $ begin-equality

+
  p * (m * n) / n   ≡⟨ *-/-assoc p (n∣m*n m) ⟩

+
  p * ((m * n) / n) ≡⟨ cong (p *_) (m*n/n≡m m n) ⟩

+
  p * m ∎

+
  where open ∣-Reasoning

+

+
m*n∣o⇒n∣o/m : ∀ m n .{{_ : NonZero m}} → m * n ∣ o → n ∣ (o / m)

+
m*n∣o⇒n∣o/m m n rewrite *-comm m n = m*n∣o⇒m∣o/n n m

+

+
m∣n/o⇒m*o∣n : .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → m * o ∣ n

+
m∣n/o⇒m*o∣n {o} {n@.(p * o)} {m} (divides-refl p) m∣p*o/o = begin

+
  m * o ∣⟨ *-monoˡ-∣ o (subst (m ∣_) (m*n/n≡m p o) m∣p*o/o) ⟩

+
  p * o ∎

+
  where open ∣-Reasoning

+

+
m∣n/o⇒o*m∣n : .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → o * m ∣ n

+
m∣n/o⇒o*m∣n {o} {_} {m} rewrite *-comm o m = m∣n/o⇒m*o∣n

+

+
m/n∣o⇒m∣o*n : .{{_ : NonZero n}} → n ∣ m → m / n ∣ o → m ∣ o * n

+
m/n∣o⇒m∣o*n {n} {m@.(p * n)} {o} (divides-refl p) p*n/n∣o = begin

+
  p * n ∣⟨ *-monoˡ-∣ n (subst (_∣ o) (m*n/n≡m p n) p*n/n∣o) ⟩

+
  o * n ∎

+
  where open ∣-Reasoning

+

+
m∣n*o⇒m/n∣o : .{{_ : NonZero n}} → n ∣ m → m ∣ o * n → m / n ∣ o

+
m∣n*o⇒m/n∣o {n} {m@.(p * n)} {o} (divides-refl p) pn∣on = begin

+
  m / n     ≡⟨⟩

+
  p * n / n ≡⟨ m*n/n≡m p n ⟩

+
  p         ∣⟨ *-cancelʳ-∣ n pn∣on ⟩

+
  o         ∎

+
  where open ∣-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and _%_

+

+
∣n∣m%n⇒∣m : .{{_ : NonZero n}} → d ∣ n → d ∣ m % n → d ∣ m

+
∣n∣m%n⇒∣m {n@.(p * d)} {d} {m} (divides-refl p) (divides q m%n≡qd) =

+
  divides (q + (m / n) * p) $ begin-equality

+
    m                         ≡⟨ m≡m%n+[m/n]*n m n ⟩

+
    m % n + (m / n) * n       ≡⟨ cong (_+ (m / n) * n) m%n≡qd ⟩

+
    q * d + (m / n) * n       ≡⟨⟩

+
    q * d + (m / n) * (p * d) ≡⟨ cong (q * d +_) (*-assoc (m / n) p d) ⟨

+
    q * d + ((m / n) * p) * d ≡⟨ *-distribʳ-+ d q _ ⟨

+
    (q + (m / n) * p) * d     ∎

+
  where open ∣-Reasoning

+

+
%-presˡ-∣ : .{{_ : NonZero n}} → d ∣ m → d ∣ n → d ∣ m % n

+
%-presˡ-∣ {n} {d} {m@.(p * d)} (divides-refl p) (divides q 1+n≡qd) =

+
  divides (p ∸ m / n * q) $ begin-equality

+
    m % n                   ≡⟨ m%n≡m∸m/n*n m n ⟩

+
    m ∸ m / n * n           ≡⟨ cong (λ v → m ∸ m / n * v) 1+n≡qd ⟩

+
    m ∸ m / n * (q * d)     ≡⟨ cong (m ∸_) (*-assoc (m / n) q d) ⟨

+
    m  ∸ (m / n * q) * d    ≡⟨⟩

+
    p * d ∸ (m / n * q) * d ≡⟨ *-distribʳ-∸ d p (m / n * q) ⟨

+
    (p ∸ m / n * q) * d     ∎

+
  where open ∣-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _∣_ and !_

+

+
m≤n⇒m!∣n! : m ≤ n → m ! ∣ n !

+
m≤n⇒m!∣n! m≤n = help (≤⇒≤′ m≤n)

+
  where

+
  help : m ≤′ n → m ! ∣ n !

+
  help         ≤′-refl       = ∣-refl

+
  help {n = n} (≤′-step m≤n) = ∣n⇒∣m*n n (help m≤n)

+

+
------------------------------------------------------------------------

+
-- Properties of _HasNonTrivialDivisorLessThan_

+

+
-- Smart constructor

+

+
hasNonTrivialDivisor-≢ : .{{NonTrivial d}} → .{{NonZero n}} →

+
                         d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n

+
hasNonTrivialDivisor-≢ d≢n d∣n

+
  = hasNonTrivialDivisor (≤∧≢⇒< (∣⇒≤ d∣n) d≢n) d∣n

+

+
-- Monotonicity wrt ∣

+

+
hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o →

+
                         o HasNonTrivialDivisorLessThan n

+
hasNonTrivialDivisor-∣ (hasNonTrivialDivisor d<n d∣m) m∣o

+
  = hasNonTrivialDivisor d<n (∣-trans d∣m m∣o)

+

+
-- Monotonicity wrt ≤

+

+
hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o →

+
                         m HasNonTrivialDivisorLessThan o

+
hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) n≤o

+
  = hasNonTrivialDivisor (<-≤-trans d<n n≤o) d∣m

+

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+Data.Nat.GCD.Lemmas
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
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+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
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+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Boring lemmas used in Data.Nat.GCD and Data.Nat.Coprimality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.GCD.Lemmas where

+

+
open import Data.Nat.Base

+
open import Data.Nat.Properties

+
open import Function.Base using (_$_)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂; sym)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+

+
open ≡-Reasoning

+

+
private

+
  comm-factor : ∀ x k n → x * k + x * n ≡ x * (n + k)

+
  comm-factor x k n = begin

+
    x * k + x * n ≡⟨ +-comm (x * k) _ ⟩

+
    x * n + x * k ≡⟨ *-distribˡ-+ x n k ⟨

+
    x * (n + k)   ∎

+

+
  distrib-comm₂ : ∀ d x k n → d + x * (n + k) ≡ d + x * k + x * n

+
  distrib-comm₂ d x k n = begin

+
    d + x * (n + k)     ≡⟨ cong (d +_) (comm-factor x k n) ⟨

+
    d + (x * k + x * n) ≡⟨ +-assoc d _ _ ⟨

+
    d + x * k + x * n   ∎

+

+
-- Other properties

+
-- TODO: Can this proof be simplified? An automatic solver which can

+
-- handle ∸ would be nice...

+
lem₀ : ∀ i j m n → i * m ≡ j * m + n → (i ∸ j) * m ≡ n

+
lem₀ i j m n eq = begin

+
  (i ∸ j) * m            ≡⟨ *-distribʳ-∸ m i j ⟩

+
  (i * m) ∸ (j * m)      ≡⟨ cong (_∸ j * m) eq ⟩

+
  (j * m + n) ∸ (j * m)  ≡⟨ cong (_∸ j * m) (+-comm (j * m) n) ⟩

+
  (n + j * m) ∸ (j * m)  ≡⟨ m+n∸n≡m n (j * m) ⟩

+
  n                      ∎

+

+
lem₁ : ∀ i j → 2 + i ≤′ 2 + j + i

+
lem₁ i j = ≤⇒≤′ $ s≤s $ s≤s $ m≤n+m i j

+

+
private

+
  times2 : ∀ x → x + x ≡ 2 * x

+
  times2 x = cong (x +_) (sym (+-identityʳ x))

+

+
  times2′ : ∀ x y → x * y + x * y ≡ 2 * x * y

+
  times2′ x y = begin

+
    x * y + x * y ≡⟨ times2 (x * y) ⟩

+
    2 * (x * y)   ≡⟨ *-assoc 2 x y ⟨

+
    2 * x * y     ∎

+

+
lem₂ : ∀ d x {k n} →

+
       d + x * k ≡ x * n → d + x * (n + k) ≡ 2 * x * n

+
lem₂ d x {k} {n} eq = begin

+
  d + x * (n + k)    ≡⟨ distrib-comm₂ d x k n ⟩

+
  d + x * k + x * n  ≡⟨ cong (_+ x * n) eq ⟩

+
  x * n + x * n      ≡⟨ times2′ x n ⟩

+
  2 * x * n          ∎

+

+
private

+
  distrib₃ : ∀ a b c x → (a + b + c) * x ≡ a * x + b * x + c * x

+
  distrib₃ a b c x = begin

+
    (a + b + c) * x       ≡⟨ *-distribʳ-+ x (a + b) c ⟩

+
    (a + b) * x + c * x   ≡⟨ cong (_+ c * x) (*-distribʳ-+ x a b) ⟩

+
    a * x + b * x + c * x ∎

+

+
  lem₃₁ : ∀ a b c → a + (b + c) ≡ b + a + c

+
  lem₃₁ a b c = begin

+
    a + (b + c) ≡⟨ +-assoc a b c ⟨

+
    (a + b) + c ≡⟨ cong (_+ c) (+-comm a b) ⟩

+
    b + a + c   ∎

+

+
  +-assoc-comm : ∀ a b c d → a + (b + c + d) ≡ (a + c) + (b + d)

+
  +-assoc-comm a b c d = begin

+
    a + (b + c + d)   ≡⟨ cong (a +_) (cong (_+ d) (+-comm b c)) ⟩

+
    a + (c + b + d)   ≡⟨ cong (a +_) (+-assoc c b d) ⟩

+
    a + (c + (b + d)) ≡⟨ +-assoc a c _ ⟨

+
    (a + c) + (b + d) ∎

+

+
  *-on-right : ∀ a b c {d} → b * c ≡ d → a * b * c ≡ a * d

+
  *-on-right a b c {d} eq = begin

+
    a * b * c   ≡⟨ *-assoc a b c ⟩

+
    a * (b * c) ≡⟨ cong (a *_) eq ⟩

+
    a * d       ∎

+

+
  *-on-left : ∀ a b c {d} → a * b ≡ d → a * (b * c) ≡ d * c

+
  *-on-left a b c {d} eq = begin

+
    a * (b * c) ≡⟨ *-assoc a b c ⟨

+
    a * b * c   ≡⟨ cong (_* c) eq ⟩

+
    d * c       ∎

+

+
  +-on-right : ∀ a b c {d} → b + c ≡ d → a + b + c ≡ a + d

+
  +-on-right a b c {d} eq = begin

+
    a + b + c   ≡⟨ +-assoc a b c ⟩

+
    a + (b + c) ≡⟨ cong (a +_) eq ⟩

+
    a + d       ∎

+

+
  +-on-left : ∀ a b c d → a + b ≡ d → a + (b + c) ≡ d + c

+
  +-on-left a b c d eq = begin

+
    a + (b + c) ≡⟨ +-assoc a b c ⟨

+
    a + b + c   ≡⟨ cong (_+ c) eq ⟩

+
    d + c       ∎

+

+
  +-focus-mid : ∀ a b c d → a + b + c + d ≡ a + (b + c) + d

+
  +-focus-mid a b c d = begin

+
    a + b + c + d     ≡⟨ cong (_+ d) (+-assoc a b c) ⟩

+
    a + (b + c) + d   ∎

+

+
  +-assoc-comm′ : ∀ a b c d → a + b + c + d ≡ a + (b + d) + c

+
  +-assoc-comm′ a b c d = begin

+
    a + b + c + d     ≡⟨ +-on-left a (b + c) d (a + b + c) (sym $ +-assoc a b c) ⟨

+
    a + (b + c + d)   ≡⟨ cong (a +_) (+-on-right b c d (+-comm c d)) ⟩

+
    a + (b + (d + c)) ≡⟨ cong (a +_) (+-assoc b d c) ⟨

+
    a + (b + d + c)   ≡⟨ +-assoc a _ c ⟨

+
    a + (b + d) + c   ∎

+

+
  lem₃₂ : ∀ a b c n → a * n + (b * n + a * n + c * n) ≡ (a + a + (b + c)) * n

+
  lem₃₂ a b c n = begin

+
    a * n + (b * n + a * n + c * n) ≡⟨ cong (a * n +_) (distrib₃ b a c n) ⟨

+
    a * n + (b + a + c) * n         ≡⟨ *-distribʳ-+ n a _ ⟨

+
    (a + (b + a + c)) * n           ≡⟨ cong (_* n) (+-assoc-comm a b a c) ⟩

+
    (a + a + (b + c)) * n           ∎

+

+
  mid-to-right : ∀ a b c → a + 2 * b + c ≡ (a + b + c) + b

+
  mid-to-right a b c = begin

+
    a + 2 * b + c   ≡⟨ cong (λ x → a + x + c) (times2 b) ⟨

+
    a + (b + b) + c ≡⟨ +-assoc-comm′ a b c b ⟨

+
    a + b + c + b   ∎

+

+
  mid-to-left : ∀ a b c → a + 2 * b + c ≡ b + (a + b + c)

+
  mid-to-left a b c = begin

+
    a + 2 * b + c   ≡⟨ cong (λ x → a + x + c) (times2 b) ⟨

+
    a + (b + b) + c ≡⟨ cong (_+ c) (+-on-left a b b _ (+-comm a b)) ⟩

+
    b + a + b + c   ≡⟨ +-on-left b (a + b) c (b + a + b) (sym $ +-assoc b a b) ⟨

+
    b + (a + b + c) ∎

+

+
lem₃ : ∀ d x {i k n} →

+
       d + (1 + x + i) * k ≡ x * n →

+
       d + (1 + x + i) * (n + k) ≡ (1 + 2 * x + i) * n

+
lem₃ d x {i} {k} {n} eq = begin

+
  d + y * (n + k)                     ≡⟨ distrib-comm₂ d y k n ⟩

+
  d + y * k + y * n                   ≡⟨ cong (_+ y * n) eq ⟩

+
  x * n + y * n                       ≡⟨ cong (x * n +_) (distrib₃ 1 x i n) ⟩

+
  x * n + (1 * n + x * n + i * n)     ≡⟨ lem₃₂ x 1 i n  ⟩

+
  (x + x + (1 + i)) * n               ≡⟨ cong (_* n) (cong (_+ (1 + i)) (times2 x)) ⟩

+
  (2 * x + (1 + i)) * n               ≡⟨ cong (_* n) (lem₃₁ (2 * x) 1 i) ⟩

+
  (1 + 2 * x + i) * n                 ∎

+
  where y = 1 + x + i

+

+
lem₄ : ∀ d y {k i} n →

+
       d + y * k ≡ (1 + y + i) * n →

+
       d + y * (n + k) ≡ (1 + 2 * y + i) * n

+
lem₄ d y {k} {i} n eq = begin

+
  d + y * (n + k)          ≡⟨ distrib-comm₂ d y k n ⟩

+
  d + y * k + y * n        ≡⟨ cong (_+ y * n) eq ⟩

+
  (1 + y + i) * n + y * n  ≡⟨ *-distribʳ-+ n (1 + y + i) y ⟨

+
  (1 + y + i + y) * n      ≡⟨ cong (_* n) (mid-to-right 1 y i) ⟨

+
  (1 + 2 * y + i) * n      ∎

+

+
lem₅ : ∀ d x {n k} →

+
       d + x * n ≡ x * k →

+
       d + 2 * x * n ≡ x * (n + k)

+
lem₅ d x {n} {k} eq = begin

+
  d + 2 * x * n       ≡⟨ cong (d +_) (times2′ x n) ⟨

+
  d + (x * n + x * n) ≡⟨ +-assoc d (x * n) _ ⟨

+
  d + x * n + x * n   ≡⟨ cong (_+ x * n) eq ⟩

+
  x * k + x * n       ≡⟨ comm-factor x k n ⟩

+
  x * (n + k)         ∎

+

+
lem₆ : ∀ d x {n i k} →

+
       d + x * n ≡ (1 + x + i) * k →

+
       d + (1 + 2 * x + i) * n ≡ (1 + x + i) * (n + k)

+
lem₆ d x {n} {i} {k} eq = begin

+
  d + (1 + 2 * x + i) * n  ≡⟨ cong (λ z → d + z * n) (mid-to-left 1 x i) ⟩

+
  d + (x + y) * n          ≡⟨ cong (d +_) (*-distribʳ-+ n x y) ⟩

+
  d + (x * n + y * n)      ≡⟨ +-on-left d _ _ _ eq ⟩

+
  y * k + y * n            ≡⟨ comm-factor y k n ⟩

+
  y * (n + k)              ∎

+
  where y = 1 + x + i

+

+
lem₇ : ∀ d y {i} n {k} →

+
       d + (1 + y + i) * n ≡ y * k →

+
       d + (1 + 2 * y + i) * n ≡ y * (n + k)

+
lem₇ d y {i} n {k} eq = begin

+
  d + (1 + 2 * y + i) * n       ≡⟨ cong (λ z → d + z * n) (mid-to-right 1 y i) ⟩

+
  d + (1 + y + i + y) * n       ≡⟨ cong (d +_) (*-distribʳ-+ n (1 + y + i) y) ⟩

+
  d + ((1 + y + i) * n + y * n) ≡⟨ +-on-left d _ _ _ eq ⟩

+
  y * k + y * n                 ≡⟨ comm-factor y k n ⟩

+
  y * (n + k)                   ∎

+

+
lem₈ : ∀ {i j k q} x y →

+
       1 + y * j ≡ x * i → j * k ≡ q * i →

+
       k ≡ (x * k ∸ y * q) * i

+
lem₈ {i} {j} {k} {q} x y eq eq′ =

+
  sym (lem₀ (x * k) (y * q) i k lemma)

+
  where

+
  lemma = begin

+
    x * k * i        ≡⟨ *-on-right x k i (*-comm k i) ⟩

+
    x * (i * k)      ≡⟨ *-on-left x i k (sym eq) ⟩

+
    (1 + y * j) * k  ≡⟨ +-comm k _ ⟩

+
    (y * j) * k + k  ≡⟨ cong (_+ k) (*-assoc y j k) ⟩

+
    y * (j * k) + k  ≡⟨ cong (λ n → y * n + k) eq′ ⟩

+
    y * (q * i) + k  ≡⟨ cong (_+ k) (*-assoc y q i)  ⟨

+
    y *  q * i  + k  ∎

+

+
lem₉ : ∀ {i j k q} x y →

+
       1 + x * i ≡ y * j → j * k ≡ q * i →

+
       k ≡ (y * q ∸ x * k) * i

+
lem₉ {i} {j} {k} {q} x y eq eq′ =

+
  sym (lem₀ (y * q) (x * k) i k lemma)

+
  where

+
  lem : ∀ a b c → a * b * c ≡ b * c * a

+
  lem a b c = begin

+
    a * b * c   ≡⟨ *-assoc a b c ⟩

+
    a * (b * c) ≡⟨ *-comm a _ ⟩

+
    b * c * a   ∎

+
  lemma = begin

+
    y * q * i        ≡⟨ lem y q i ⟩

+
    q * i * y        ≡⟨ cong (λ n → n * y) eq′ ⟨

+
    j * k * y        ≡⟨ lem y j k ⟨

+
    y * j * k        ≡⟨ cong (λ n → n * k) eq ⟨

+
    (1 + x * i) * k  ≡⟨ +-comm k _ ⟩

+
    x * i * k + k    ≡⟨ cong (_+ k) (*-on-right x i k (*-comm i k)) ⟩

+
    x * (k * i) + k  ≡⟨ cong (_+ k) (*-assoc x k i) ⟨

+
    x * k * i + k    ∎

+

+
lem₁₀ : ∀ {a′} b c {d} e f → let a = suc a′ in

+
        a + b * (c * d * a) ≡ e * (f * d * a) →

+
        d ≡ 1

+
lem₁₀ {a′} b c {d} e f eq =

+
  m*n≡1⇒n≡1 (e * f ∸ b * c) d

+
    (lem₀ (e * f) (b * c) d 1

+
       (*-cancelʳ-≡ (e * f * d) (b * c * d + 1) _ (begin

+
          e * f * d * a           ≡⟨ *-assoc₄₃ e f d a ⟩

+
          e * (f * d * a)         ≡⟨ eq ⟨

+
          a + b * (c * d * a)     ≡⟨ cong (a +_) (*-assoc₄₃ b c d a) ⟨

+
          a + b * c * d * a       ≡⟨ +-comm a _ ⟩

+
          b * c * d * a + a       ≡⟨ cong (b * c * d * a +_) (+-identityʳ a) ⟨

+
          b * c * d * a + (a + 0) ≡⟨ *-distribʳ-+ a (b * c * d) 1 ⟨

+
          (b * c * d + 1) * a     ∎)))

+
  where a = suc a′

+
        *-assoc₄₃ : ∀ w x y z → w * x * y * z ≡ w * (x * y * z)

+
        *-assoc₄₃ w x y z = begin

+
          w * x * y * z   ≡⟨ cong (_* z) (*-assoc w x y) ⟩

+
          w * (x * y) * z ≡⟨ *-assoc w _ z ⟩

+
          w * (x * y * z) ∎

+

+
lem₁₁ : ∀ {i j m n k d} x y →

+
       1 + y * j ≡ x * i → i * k ≡ m * d → j * k ≡ n * d →

+
       k ≡ (x * m ∸ y * n) * d

+
lem₁₁ {i} {j} {m} {n} {k} {d} x y eq eq₁ eq₂ =

+
  sym (lem₀ (x * m) (y * n) d k (begin

+
    x * m * d        ≡⟨ *-on-right x m d (sym eq₁) ⟩

+
    x * (i * k)      ≡⟨ *-on-left x i k (sym eq) ⟩

+
    (1 + y * j) * k  ≡⟨ +-comm k _ ⟩

+
    y * j * k + k    ≡⟨ cong (_+ k) (*-assoc y j k) ⟩

+
    y * (j * k) + k  ≡⟨ cong (λ p → y * p + k) eq₂ ⟩

+
    y * (n * d) + k  ≡⟨ cong (_+ k) (*-assoc y n d) ⟨

+
    y * n * d + k    ∎))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.GCD.html b/site/static/agda_html/Data.Nat.GCD.html
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@@ -0,0 +1,481 @@
+
+Data.Nat.GCD
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Greatest common divisor

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.GCD where

+

+
open import Data.Nat.Base

+
open import Data.Nat.Divisibility

+
open import Data.Nat.DivMod

+
open import Data.Nat.GCD.Lemmas

+
open import Data.Nat.Properties

+
open import Data.Nat.Induction

+
  using (Acc; acc; <′-Rec; <′-recBuilder; <-wellFounded-fast)

+
open import Data.Product.Base

+
  using (_×_; _,_; proj₂; proj₁; swap; uncurry′; ∃; map)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Function.Base using (_$_; _∘_)

+
open import Induction using (build)

+
open import Induction.Lexicographic using (_⊗_; [_⊗_])

+
open import Relation.Binary.Definitions using (tri<; tri>; tri≈; Symmetric)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_; _≢_; subst; cong)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+
open import Relation.Nullary.Decidable using (Dec)

+
open import Relation.Nullary.Negation using (contradiction)

+
import Relation.Nullary.Decidable as Dec

+

+
open import Algebra.Definitions {A = ℕ} _≡_ as Algebra

+
  using (Associative; Commutative; LeftIdentity; RightIdentity; LeftZero; RightZero; Zero)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Calculated via Euclid's algorithm. In order to show progress,

+
-- avoiding the initial step where the first argument may increase, it

+
-- is necessary to first define a version `gcd′` which assumes that the

+
-- second argument is strictly smaller than the first. The full `gcd`

+
-- function then compares the two arguments and applies `gcd′`

+
-- accordingly.

+

+
gcd′ : ∀ m n → Acc _<_ m → n < m → ℕ

+
gcd′ m zero      _         _   = m

+
gcd′ m n@(suc _) (acc rec) n<m = gcd′ n (m % n) (rec n<m) (m%n<n m n)

+

+
gcd : ℕ → ℕ → ℕ

+
gcd m n with <-cmp m n

+
... | tri< m<n _ _ = gcd′ n m (<-wellFounded-fast n) m<n

+
... | tri≈ _ _ _   = m

+
... | tri> _ _ n<m = gcd′ m n (<-wellFounded-fast m) n<m

+

+
------------------------------------------------------------------------

+
-- Core properties of gcd′

+

+
gcd′[m,n]∣m,n : ∀ {m n} rec n<m → gcd′ m n rec n<m ∣ m × gcd′ m n rec n<m ∣ n

+
gcd′[m,n]∣m,n {m} {zero}      rec       n<m = ∣-refl , m ∣0

+
gcd′[m,n]∣m,n {m} {n@(suc _)} (acc rec) n<m

+
  with gcd∣n , gcd∣m%n ← gcd′[m,n]∣m,n (rec n<m) (m%n<n m n)

+
  = ∣n∣m%n⇒∣m gcd∣n gcd∣m%n , gcd∣n

+

+
gcd′-greatest : ∀ {m n c} rec n<m → c ∣ m → c ∣ n → c ∣ gcd′ m n rec n<m

+
gcd′-greatest {m} {zero}      rec       n<m c∣m c∣n = c∣m

+
gcd′-greatest {m} {n@(suc _)} (acc rec) n<m c∣m c∣n =

+
  gcd′-greatest (rec n<m) (m%n<n m n) c∣n (%-presˡ-∣ c∣m c∣n)

+

+
------------------------------------------------------------------------

+
-- Core properties of gcd

+

+
gcd[m,n]∣m : ∀ m n → gcd m n ∣ m

+
gcd[m,n]∣m m n with <-cmp m n

+
... | tri< n<m _ _ = proj₂ (gcd′[m,n]∣m,n {n} {m} _ _)

+
... | tri≈ _ _ _   = ∣-refl

+
... | tri> _ _ m<n = proj₁ (gcd′[m,n]∣m,n {m} {n} _ _)

+

+
gcd[m,n]∣n : ∀ m n → gcd m n ∣ n

+
gcd[m,n]∣n m n with <-cmp m n

+
... | tri< n<m _    _ = proj₁ (gcd′[m,n]∣m,n {n} {m} _ _)

+
... | tri≈ _ ≡.refl _ = ∣-refl

+
... | tri> _ _    m<n = proj₂ (gcd′[m,n]∣m,n {m} {n} _ _)

+

+
gcd-greatest : ∀ {m n c} → c ∣ m → c ∣ n → c ∣ gcd m n

+
gcd-greatest {m} {n} c∣m c∣n with <-cmp m n

+
... | tri< n<m _ _ = gcd′-greatest _ _ c∣n c∣m

+
... | tri≈ _ _ _   = c∣m

+
... | tri> _ _ m<n = gcd′-greatest _ _ c∣m c∣n

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
-- Note that all other properties of `gcd` should be inferable from the

+
-- 3 core properties above.

+

+
gcd[0,0]≡0 : gcd 0 0 ≡ 0

+
gcd[0,0]≡0 = ∣-antisym (gcd 0 0 ∣0) (gcd-greatest (0 ∣0) (0 ∣0))

+

+
gcd[m,n]≢0 : ∀ m n → m ≢ 0 ⊎ n ≢ 0 → gcd m n ≢ 0

+
gcd[m,n]≢0 m n (inj₁ m≢0) eq = m≢0 (0∣⇒≡0 (subst (_∣ m) eq (gcd[m,n]∣m m n)))

+
gcd[m,n]≢0 m n (inj₂ n≢0) eq = n≢0 (0∣⇒≡0 (subst (_∣ n) eq (gcd[m,n]∣n m n)))

+

+
gcd[m,n]≡0⇒m≡0 : ∀ {m n} → gcd m n ≡ 0 → m ≡ 0

+
gcd[m,n]≡0⇒m≡0 {zero}  {n} eq = ≡.refl

+
gcd[m,n]≡0⇒m≡0 {suc m} {n} eq = contradiction eq (gcd[m,n]≢0 (suc m) n (inj₁ λ()))

+

+
gcd[m,n]≡0⇒n≡0 : ∀ m {n} → gcd m n ≡ 0 → n ≡ 0

+
gcd[m,n]≡0⇒n≡0 m {zero}  eq = ≡.refl

+
gcd[m,n]≡0⇒n≡0 m {suc n} eq = contradiction eq (gcd[m,n]≢0 m (suc n) (inj₂ λ()))

+

+
gcd-comm : Commutative gcd

+
gcd-comm m n = ∣-antisym

+
  (gcd-greatest (gcd[m,n]∣n m n) (gcd[m,n]∣m m n))

+
  (gcd-greatest (gcd[m,n]∣n n m) (gcd[m,n]∣m n m))

+

+
gcd-assoc : Associative gcd

+
gcd-assoc m n p = ∣-antisym

+
  (gcd-greatest gcd[gcd[m,n],p]|m (gcd-greatest gcd[gcd[m,n],p]∣n gcd[gcd[m,n],p]∣p))

+
  (gcd-greatest (gcd-greatest gcd[m,gcd[n,p]]∣m gcd[m,gcd[n,p]]∣n) gcd[m,gcd[n,p]]∣p)

+
  where

+
    open ∣-Reasoning

+
    gcd[gcd[m,n],p]|m = begin

+
      gcd (gcd m n) p ∣⟨ gcd[m,n]∣m (gcd m n) p ⟩

+
      gcd m n         ∣⟨ gcd[m,n]∣m m n ⟩

+
      m               ∎

+
    gcd[gcd[m,n],p]∣n = begin

+
      gcd (gcd m n) p ∣⟨ gcd[m,n]∣m (gcd m n) p ⟩

+
      gcd m n         ∣⟨ gcd[m,n]∣n m n ⟩

+
      n               ∎

+
    gcd[gcd[m,n],p]∣p = gcd[m,n]∣n (gcd m n) p

+
    gcd[m,gcd[n,p]]∣m = gcd[m,n]∣m m (gcd n p)

+
    gcd[m,gcd[n,p]]∣n = begin

+
      gcd m (gcd n p) ∣⟨ gcd[m,n]∣n m (gcd n p) ⟩

+
      gcd n p         ∣⟨ gcd[m,n]∣m n p ⟩

+
      n               ∎

+
    gcd[m,gcd[n,p]]∣p = begin

+
      gcd m (gcd n p) ∣⟨ gcd[m,n]∣n m (gcd n p) ⟩

+
      gcd n p         ∣⟨ gcd[m,n]∣n n p ⟩

+
      p               ∎

+

+
gcd-identityˡ : LeftIdentity 0 gcd

+
gcd-identityˡ zero = ≡.refl

+
gcd-identityˡ (suc _) = ≡.refl

+

+
gcd-identityʳ : RightIdentity 0 gcd

+
gcd-identityʳ zero = ≡.refl

+
gcd-identityʳ (suc _) = ≡.refl

+

+
gcd-identity : Algebra.Identity 0 gcd

+
gcd-identity = gcd-identityˡ , gcd-identityʳ

+

+
gcd-zeroˡ : LeftZero 1 gcd

+
gcd-zeroˡ n = ∣-antisym gcd[1,n]∣1 1∣gcd[1,n]

+
  where

+
    gcd[1,n]∣1 = gcd[m,n]∣m 1 n

+
    1∣gcd[1,n] = 1∣ gcd 1 n

+

+
gcd-zeroʳ : RightZero 1 gcd

+
gcd-zeroʳ n = ∣-antisym gcd[n,1]∣1 1∣gcd[n,1]

+
  where

+
    gcd[n,1]∣1 = gcd[m,n]∣n n 1

+
    1∣gcd[n,1] = 1∣ gcd n 1

+

+
gcd-zero : Zero 1 gcd

+
gcd-zero = gcd-zeroˡ , gcd-zeroʳ

+

+
gcd-universality : ∀ {m n g} →

+
                   (∀ {d} → d ∣ m × d ∣ n → d ∣ g) →

+
                   (∀ {d} → d ∣ g → d ∣ m × d ∣ n) →

+
                   g ≡ gcd m n

+
gcd-universality {m} {n} forwards backwards with back₁ , back₂ ← backwards ∣-refl

+
  = ∣-antisym (gcd-greatest back₁ back₂) (forwards (gcd[m,n]∣m m n , gcd[m,n]∣n m n))

+

+
-- This could be simplified with some nice backwards/forwards reasoning

+
-- after the new function hierarchy is up and running.

+
gcd[cm,cn]/c≡gcd[m,n] : ∀ c m n .{{_ : NonZero c}} → gcd (c * m) (c * n) / c ≡ gcd m n

+
gcd[cm,cn]/c≡gcd[m,n] c m n = gcd-universality forwards backwards

+
  where

+
  forwards : ∀ {d : ℕ} → d ∣ m × d ∣ n → d ∣ gcd (c * m) (c * n) / c

+
  forwards {d} (d∣m , d∣n) = m*n∣o⇒n∣o/m c d (gcd-greatest (*-monoʳ-∣ c d∣m) (*-monoʳ-∣ c d∣n))

+

+
  backwards : ∀ {d : ℕ} → d ∣ gcd (c * m) (c * n) / c → d ∣ m × d ∣ n

+
  backwards {d} d∣gcd[cm,cn]/c

+
    with cd∣gcd[cm,n] ← m∣n/o⇒o*m∣n (gcd-greatest (m∣m*n m) (m∣m*n n)) d∣gcd[cm,cn]/c

+
    = *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣m (c * m) _)) ,

+
      *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣n (c * m) _))

+

+
c*gcd[m,n]≡gcd[cm,cn] : ∀ c m n → c * gcd m n ≡ gcd (c * m) (c * n)

+
c*gcd[m,n]≡gcd[cm,cn] zero      m n = ≡.sym gcd[0,0]≡0

+
c*gcd[m,n]≡gcd[cm,cn] c@(suc _) m n = begin

+
  c * gcd m n                   ≡⟨ cong (c *_) (≡.sym (gcd[cm,cn]/c≡gcd[m,n] c m n)) ⟩

+
  c * (gcd (c * m) (c * n) / c) ≡⟨ m*[n/m]≡n (gcd-greatest (m∣m*n m) (m∣m*n n)) ⟩

+
  gcd (c * m) (c * n)           ∎

+
  where open ≡-Reasoning

+

+
gcd[m,n]≤n : ∀ m n .{{_ : NonZero n}} → gcd m n ≤ n

+
gcd[m,n]≤n m n = ∣⇒≤ (gcd[m,n]∣n m n)

+

+
n/gcd[m,n]≢0 : ∀ m n .{{_ : NonZero n}} .{{gcd≢0 : NonZero (gcd m n)}} →

+
               n / gcd m n ≢ 0

+
n/gcd[m,n]≢0 m n = m<n⇒n≢0 (m≥n⇒m/n>0 {n} {gcd m n} (gcd[m,n]≤n m n))

+

+
m/gcd[m,n]≢0 : ∀ m n .{{_ : NonZero m}} .{{gcd≢0 : NonZero (gcd m n)}} →

+
               m / gcd m n ≢ 0

+
m/gcd[m,n]≢0 m n rewrite gcd-comm m n = n/gcd[m,n]≢0 n m

+

+
------------------------------------------------------------------------

+
-- A formal specification of GCD

+

+
module GCD where

+

+
  -- Specification of the greatest common divisor (gcd) of two natural

+
  -- numbers.

+

+
  record GCD (m n gcd : ℕ) : Set where

+
    constructor is

+
    field

+
      -- The gcd is a common divisor.

+
      commonDivisor : gcd ∣ m × gcd ∣ n

+

+
      -- All common divisors divide the gcd, i.e. the gcd is the

+
      -- greatest common divisor according to the partial order _∣_.

+
      greatest : ∀ {d} → d ∣ m × d ∣ n → d ∣ gcd

+

+
    gcd∣m : gcd ∣ m

+
    gcd∣m = proj₁ commonDivisor

+

+
    gcd∣n : gcd ∣ n

+
    gcd∣n = proj₂ commonDivisor

+

+
  open GCD public

+

+

+
  -- The gcd is unique.

+

+
  unique : ∀ {d₁ d₂ m n} → GCD m n d₁ → GCD m n d₂ → d₁ ≡ d₂

+
  unique d₁ d₂ = ∣-antisym (GCD.greatest d₂ (GCD.commonDivisor d₁))

+
                           (GCD.greatest d₁ (GCD.commonDivisor d₂))

+

+
  -- The gcd relation is "symmetric".

+

+
  sym : ∀ {d m n} → GCD m n d → GCD n m d

+
  sym g = is (swap $ GCD.commonDivisor g) (GCD.greatest g ∘ swap)

+

+
  -- The gcd relation is "reflexive".

+

+
  refl : ∀ {n} → GCD n n n

+
  refl = is (∣-refl , ∣-refl) proj₁

+

+
  -- The GCD of 0 and n is n.

+

+
  base : ∀ {n} → GCD 0 n n

+
  base {n} = is (n ∣0 , ∣-refl) proj₂

+

+
  -- If d is the gcd of n and k, then it is also the gcd of n and

+
  -- n + k.

+

+
  step : ∀ {n k d} → GCD n k d → GCD n (n + k) d

+
  step {n} {k} {d} g with d₁ , d₂ ← GCD.commonDivisor g

+
    = is (d₁ , ∣m∣n⇒∣m+n d₁ d₂) greatest′

+
    where

+
    greatest′ : ∀ {d′} → d′ ∣ n × d′ ∣ n + k → d′ ∣ d

+
    greatest′ (d₁ , d₂) = GCD.greatest g (d₁ , ∣m+n∣m⇒∣n d₂ d₁)

+

+
open GCD public using (GCD) hiding (module GCD)

+

+
-- The function gcd fulfils the conditions required of GCD

+

+
gcd-GCD : ∀ m n → GCD m n (gcd m n)

+
gcd-GCD m n = record

+
  { commonDivisor = gcd[m,n]∣m m n , gcd[m,n]∣n m n

+
  ; greatest      = uncurry′ gcd-greatest

+
  }

+

+
-- Calculates the gcd of the arguments.

+

+
mkGCD : ∀ m n → ∃ λ d → GCD m n d

+
mkGCD m n = gcd m n , gcd-GCD m n

+

+
-- gcd as a proposition is decidable

+

+
gcd? : (m n d : ℕ) → Dec (GCD m n d)

+
gcd? m n d =

+
  Dec.map′ (λ { ≡.refl → gcd-GCD m n }) (GCD.unique (gcd-GCD m n))

+
           (gcd m n ≟ d)

+

+
GCD-* : ∀ {m n d c} .{{_ : NonZero c}} → GCD (m * c) (n * c) (d * c) → GCD m n d

+
GCD-* {c = suc _} (GCD.is (dc∣nc , dc∣mc) dc-greatest) =

+
  GCD.is (*-cancelʳ-∣ _ dc∣nc , *-cancelʳ-∣ _ dc∣mc)

+
  λ {_} → *-cancelʳ-∣ _ ∘ dc-greatest ∘ map (*-monoˡ-∣ _) (*-monoˡ-∣ _)

+

+
GCD-/ : ∀ {m n d c} .{{_ : NonZero c}} → c ∣ m → c ∣ n → c ∣ d →

+
        GCD m n d → GCD (m / c) (n / c) (d / c)

+
GCD-/ {m} {n} {d} {c} {{x}}

+
  (divides-refl p) (divides-refl q) (divides-refl r) gcd

+
  rewrite m*n/n≡m p c {{x}} | m*n/n≡m q c {{x}} | m*n/n≡m r c {{x}} = GCD-* gcd

+

+
GCD-/gcd : ∀ m n .{{_ : NonZero (gcd m n)}} → GCD (m / gcd m n) (n / gcd m n) 1

+
GCD-/gcd m n rewrite ≡.sym (n/n≡1 (gcd m n)) =

+
  GCD-/ (gcd[m,n]∣m m n) (gcd[m,n]∣n m n) ∣-refl (gcd-GCD m n)

+

+
------------------------------------------------------------------------

+
-- Calculating the gcd

+

+
-- The calculation also proves Bézout's lemma.

+

+
module Bézout where

+

+
  module Identity where

+

+
    -- If m and n have greatest common divisor d, then one of the

+
    -- following two equations is satisfied, for some numbers x and y.

+
    -- The proof is "lemma" below (Bézout's lemma).

+
    --

+
    -- (If this identity was stated using integers instead of natural

+
    -- numbers, then it would not be necessary to have two equations.)

+

+
    data Identity (d m n : ℕ) : Set where

+
      +- : (x y : ℕ) (eq : d + y * n ≡ x * m) → Identity d m n

+
      -+ : (x y : ℕ) (eq : d + x * m ≡ y * n) → Identity d m n

+

+
    -- Various properties about Identity.

+

+
    sym : ∀ {d} → Symmetric (Identity d)

+
    sym (+- x y eq) = -+ y x eq

+
    sym (-+ x y eq) = +- y x eq

+

+
    refl : ∀ {d} → Identity d d d

+
    refl = -+ 0 1 ≡.refl

+

+
    base : ∀ {d} → Identity d 0 d

+
    base = -+ 0 1 ≡.refl

+

+
    private

+
      infixl 7 _⊕_

+

+
      _⊕_ : ℕ → ℕ → ℕ

+
      m ⊕ n = 1 + m + n

+

+
    step : ∀ {d n k} → Identity d n k → Identity d n (n + k)

+
    step {d} {n} (+-  x  y       eq) with compare x y

+
    ... | equal x     = +- (2 * x)     x       (lem₂ d x   eq)

+
    ... | less x i    = +- (2 * x ⊕ i) (x ⊕ i) (lem₃ d x   eq)

+
    ... | greater y i = +- (2 * y ⊕ i) y       (lem₄ d y n eq)

+
    step {d} {n} (-+  x  y       eq) with compare x y

+
    ... | equal x     = -+ (2 * x)     x       (lem₅ d x   eq)

+
    ... | less x i    = -+ (2 * x ⊕ i) (x ⊕ i) (lem₆ d x   eq)

+
    ... | greater y i = -+ (2 * y ⊕ i) y       (lem₇ d y n eq)

+

+
  open Identity public using (Identity; +-; -+) hiding (module Identity)

+

+
  module Lemma where

+

+
    -- This type packs up the gcd, the proof that it is a gcd, and the

+
    -- proof that it satisfies Bézout's identity.

+

+
    data Lemma (m n : ℕ) : Set where

+
      result : (d : ℕ) (g : GCD m n d) (b : Identity d m n) → Lemma m n

+

+
    -- Various properties about Lemma.

+

+
    sym : Symmetric Lemma

+
    sym (result d g b) = result d (GCD.sym g) (Identity.sym b)

+

+
    base : ∀ d → Lemma 0 d

+
    base d = result d GCD.base Identity.base

+

+
    refl : ∀ d → Lemma d d

+
    refl d = result d GCD.refl Identity.refl

+

+
    stepˡ : ∀ {n k} → Lemma n (suc k) → Lemma n (suc (n + k))

+
    stepˡ {n} {k} (result d g b) =

+
      subst (Lemma n) (+-suc n k) $

+
        result d (GCD.step g) (Identity.step b)

+

+
    stepʳ : ∀ {n k} → Lemma (suc k) n → Lemma (suc (n + k)) n

+
    stepʳ = sym ∘ stepˡ ∘ sym

+

+
  open Lemma public using (Lemma; result) hiding (module Lemma)

+

+
  -- Bézout's lemma proved using some variant of the extended

+
  -- Euclidean algorithm.

+

+
  lemma : (m n : ℕ) → Lemma m n

+
  lemma m n = build [ <′-recBuilder ⊗ <′-recBuilder ] P gcd″ (m , n)

+
    where

+
    P : ℕ × ℕ → Set

+
    P (m , n) = Lemma m n

+

+
    gcd″ : ∀ p → (<′-Rec ⊗ <′-Rec) P p → P p

+
    gcd″ (zero      , n)     rec = Lemma.base n

+
    gcd″ (m@(suc _) , zero)  rec = Lemma.sym (Lemma.base m)

+
    gcd″ (m′@(suc m) , n′@(suc n)) rec with compare m n

+
    ... | equal m     = Lemma.refl m′

+
    ... | less m k    = Lemma.stepˡ $ proj₁ rec (lem₁ k m)

+
                      -- "gcd (suc m) (suc k)"

+
    ... | greater n k = Lemma.stepʳ $ proj₂ rec (lem₁ k n) n′

+
                      -- "gcd (suc k) (suc n)"

+

+
  -- Bézout's identity can be recovered from the GCD.

+

+
  identity : ∀ {m n d} → GCD m n d → Identity d m n

+
  identity {m} {n} g with result d g′ b ← lemma m n rewrite GCD.unique g g′ = b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.GeneralisedArithmetic.html b/site/static/agda_html/Data.Nat.GeneralisedArithmetic.html
new file mode 100644
index 000000000..75c63032f
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.GeneralisedArithmetic.html
@@ -0,0 +1,185 @@
+
+Data.Nat.GeneralisedArithmetic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- A generalisation of the arithmetic operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.GeneralisedArithmetic where

+

+
open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_; _^_)

+
open import Data.Nat.Properties using (+-comm; +-assoc; *-identityˡ;

+
  *-assoc)

+
open import Function.Base using (_∘′_; _∘_; id)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; sym)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+
open ≡-Reasoning

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
fold : A → (A → A) → ℕ → A

+
fold z s zero    = z

+
fold z s (suc n) = s (fold z s n)

+

+
iterate : (A → A) → A → ℕ → A

+
iterate f x zero    = x

+
iterate f x (suc n) = iterate f (f x) n

+

+
add : (0# : A) (1+ : A → A) → ℕ → A → A

+
add 0# 1+ n z = fold z 1+ n

+

+
mul : (0# : A) (1+ : A → A) → (+ : A → A → A) → (ℕ → A → A)

+
mul 0# 1+ _+_ n x = fold 0# (λ s → x + s) n

+

+
-- Properties

+

+
fold-+ : ∀ (z : A) (s : A → A) m {n} →

+
         fold z s (m + n) ≡ fold (fold z s n) s m

+
fold-+ z s zero    = refl

+
fold-+ z s (suc m) = cong s (fold-+ z s m)

+

+
fold-k : ∀ (z : A) (s : A → A) {k} m →

+
         fold k (s ∘′_) m z ≡ fold (k z) s m

+
fold-k z s zero    = refl

+
fold-k z s (suc m) = cong s (fold-k z s m)

+

+
fold-* : ∀ (z : A) (s : A → A) m {n} →

+
         fold z s (m * n) ≡ fold z (fold id (s ∘_) n) m

+
fold-* z s zero        = refl

+
fold-* z s (suc m) {n} = let +n = fold id (s ∘′_) n in begin

+
  fold z s (n + m * n)        ≡⟨ fold-+ z s n ⟩

+
  fold (fold z s (m * n)) s n ≡⟨ cong (λ z → fold z s n) (fold-* z s m) ⟩

+
  fold (fold z +n m) s n      ≡⟨ sym (fold-k _ s n) ⟩

+
  fold z +n (suc m)           ∎

+

+
fold-pull : ∀ (z : A) (s : A → A) (g : A → A → A) (p : A)

+
            (eqz : g z p ≡ p)

+
            (eqs : ∀ l → s (g l p) ≡ g (s l) p) →

+
            ∀ m → fold p s m ≡ g (fold z s m) p

+
fold-pull z s _ _ eqz _ zero    = sym eqz

+
fold-pull z s g p eqz eqs (suc m) = begin

+
  s (fold p s m)       ≡⟨ cong s (fold-pull z s g p eqz eqs m) ⟩

+
  s (g (fold z s m) p) ≡⟨ eqs (fold z s m) ⟩

+
  g (s (fold z s m)) p ∎

+

+
iterate-is-fold : ∀ (z : A) s m → fold z s m ≡ iterate s z m

+
iterate-is-fold z s zero    = refl

+
iterate-is-fold z s (suc m) = begin

+
  fold z s (suc m)  ≡⟨ cong (fold z s) (+-comm 1 m) ⟩

+
  fold z s (m + 1)  ≡⟨ fold-+ z s m ⟩

+
  fold (s z) s m    ≡⟨ iterate-is-fold (s z) s m ⟩

+
  iterate s (s z) m ∎

+

+
id-is-fold : ∀ m → fold zero suc m ≡ m

+
id-is-fold zero    = refl

+
id-is-fold (suc m) = cong suc (id-is-fold m)

+

+
+-is-fold : ∀ m {n} → fold n suc m ≡ m + n

+
+-is-fold zero    = refl

+
+-is-fold (suc m) = cong suc (+-is-fold m)

+

+
*-is-fold : ∀ m {n} → fold zero (n +_) m ≡ m * n

+
*-is-fold zero        = refl

+
*-is-fold (suc m) {n} = cong (n +_) (*-is-fold m)

+

+
^-is-fold : ∀ {m} n → fold 1 (m *_) n ≡ m ^ n

+
^-is-fold     zero    = refl

+
^-is-fold {m} (suc n) = cong (m *_) (^-is-fold n)

+

+
*+-is-fold : ∀ m n {p} → fold p (n +_) m ≡ m * n + p

+
*+-is-fold m n {p} = begin

+
  fold p (n +_) m     ≡⟨ fold-pull _ _ _+_ p refl

+
                         (λ l → sym (+-assoc n l p)) m ⟩

+
  fold 0 (n +_) m + p ≡⟨ cong (_+ p) (*-is-fold m) ⟩

+
  m * n + p           ∎

+

+
^*-is-fold : ∀ m n {p} → fold p (m *_) n ≡ m ^ n * p

+
^*-is-fold m n {p} = begin

+
  fold p (m *_) n     ≡⟨ fold-pull _ _ _*_ p (*-identityˡ p)

+
                         (λ l → sym (*-assoc m l p)) n ⟩

+
  fold 1 (m *_) n * p ≡⟨ cong (_* p) (^-is-fold n) ⟩

+
  m ^ n * p           ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Induction.html b/site/static/agda_html/Data.Nat.Induction.html
new file mode 100644
index 000000000..fb0105769
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Induction.html
@@ -0,0 +1,187 @@
+
+Data.Nat.Induction
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Various forms of induction for natural numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Induction where

+

+
open import Data.Nat.Base

+
open import Data.Nat.Properties using (<⇒<′)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Data.Unit.Polymorphic.Base

+
open import Induction

+
open import Induction.WellFounded as WF

+
open import Level using (Level)

+

+
private

+
  variable

+
    ℓ : Level

+

+
------------------------------------------------------------------------

+
-- Re-export accessability

+

+
open WF public using (Acc; acc)

+

+
------------------------------------------------------------------------

+
-- Ordinary induction

+

+
Rec : ∀ ℓ → RecStruct ℕ ℓ ℓ

+
Rec ℓ P zero    = ⊤

+
Rec ℓ P (suc n) = P n

+

+
recBuilder : RecursorBuilder (Rec ℓ)

+
recBuilder P f zero    = _

+
recBuilder P f (suc n) = f n (recBuilder P f n)

+

+
rec : Recursor (Rec ℓ)

+
rec = build recBuilder

+

+
------------------------------------------------------------------------

+
-- Complete induction

+

+
CRec : ∀ ℓ → RecStruct ℕ ℓ ℓ

+
CRec ℓ P zero    = ⊤

+
CRec ℓ P (suc n) = P n × CRec ℓ P n

+

+
cRecBuilder : RecursorBuilder (CRec ℓ)

+
cRecBuilder P f zero    = _

+
cRecBuilder P f (suc n) = f n ih , ih

+
  where ih = cRecBuilder P f n

+

+
cRec : Recursor (CRec ℓ)

+
cRec = build cRecBuilder

+

+
------------------------------------------------------------------------

+
-- Complete induction based on _<′_

+

+
<′-Rec : RecStruct ℕ ℓ ℓ

+
<′-Rec = WfRec _<′_

+

+
-- mutual definition

+

+
<′-wellFounded : WellFounded _<′_

+
<′-wellFounded′ : ∀ n → <′-Rec (Acc _<′_) n

+

+
<′-wellFounded n = acc (<′-wellFounded′ n)

+

+
<′-wellFounded′ (suc n) <′-base       = <′-wellFounded n

+
<′-wellFounded′ (suc n) (<′-step m<n) = <′-wellFounded′ n m<n

+

+
module _ {ℓ : Level} where

+
  open WF.All <′-wellFounded ℓ public

+
    renaming ( wfRecBuilder to <′-recBuilder

+
             ; wfRec        to <′-rec

+
             )

+
    hiding (wfRec-builder)

+

+
------------------------------------------------------------------------

+
-- Complete induction based on _<_

+

+
<-Rec : RecStruct ℕ ℓ ℓ

+
<-Rec = WfRec _<_

+

+
<-wellFounded : WellFounded _<_

+
<-wellFounded = Subrelation.wellFounded <⇒<′ <′-wellFounded

+

+
-- A version of `<-wellFounded` that cheats by skipping building

+
-- the first billion proofs. Use this when you require the function

+
-- using the proof of well-foundedness to evaluate fast.

+
--

+
-- IMPORTANT: You have to be a little bit careful when using this to

+
-- always make the function be strict in some other argument than the

+
-- accessibility proof, otherwise you will have neutral terms unfolding

+
-- a billion times before getting stuck.

+
<-wellFounded-fast : WellFounded _<_

+
<-wellFounded-fast = <-wellFounded-skip 1000000000

+
  where

+
  <-wellFounded-skip : ∀ (k : ℕ) → WellFounded _<_

+
  <-wellFounded-skip zero    n       = <-wellFounded n

+
  <-wellFounded-skip (suc k) zero    = <-wellFounded 0

+
  <-wellFounded-skip (suc k) (suc n) = acc λ {m} _ → <-wellFounded-skip k m

+

+
module _ {ℓ : Level} where

+
  open WF.All <-wellFounded ℓ public

+
    renaming ( wfRecBuilder to <-recBuilder

+
             ; wfRec        to <-rec

+
             )

+
    hiding (wfRec-builder)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Primality.html b/site/static/agda_html/Data.Nat.Primality.html
new file mode 100644
index 000000000..3b117f824
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Primality.html
@@ -0,0 +1,502 @@
+
+Data.Nat.Primality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Primality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Primality where

+

+
open import Data.List.Base using ([]; _∷_; product)

+
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)

+
open import Data.Nat.Base

+
open import Data.Nat.Divisibility

+
open import Data.Nat.GCD using (module GCD; module Bézout)

+
open import Data.Nat.Properties

+
open import Data.Product.Base using (∃-syntax; _×_; map₂; _,_)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Function.Base using (flip; _∘_; _∘′_)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Relation.Nullary.Decidable as Dec

+
  using (yes; no; from-yes; from-no; ¬?; _×-dec_; _⊎-dec_; _→-dec_; decidable-stable)

+
open import Relation.Nullary.Negation using (¬_; contradiction; contradiction₂)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl; cong)

+

+
private

+
  variable

+
    d m n o p : ℕ

+

+
  recompute-nonTrivial : .{{NonTrivial n}} → NonTrivial n

+
  recompute-nonTrivial {n} {{nontrivial}} =

+
    Dec.recompute (nonTrivial? n) nontrivial

+

+
------------------------------------------------------------------------

+
-- Definitions

+
------------------------------------------------------------------------

+

+
-- The positive/existential relation `BoundedNonTrivialDivisor` is

+
-- the basis for the whole development, as it captures the possible

+
-- non-trivial divisors of a given number; its complement, `Rough`,

+
-- therefore sets *lower* bounds on any possible such divisors.

+

+
-- The predicate `Composite` is then defined as the 'diagonal' instance

+
-- of `BoundedNonTrivialDivisor`, while `Prime` is essentially defined as

+
-- the complement of `Composite`. Finally, `Irreducible` is the positive

+
-- analogue of `Prime`.

+

+
------------------------------------------------------------------------

+
-- Roughness

+

+
-- A number is m-rough if all its non-trivial divisors are bounded below

+
-- by m.

+
infix 10 _Rough_

+

+
_Rough_ : ℕ → Pred ℕ _

+
m Rough n = ¬ (n HasNonTrivialDivisorLessThan m)

+

+
------------------------------------------------------------------------

+
-- Compositeness

+

+
-- A number is composite if it has a proper non-trivial divisor.

+
Composite : Pred ℕ _

+
Composite n = n HasNonTrivialDivisorLessThan n

+

+
-- A shorter pattern synonym for the record constructor producing a

+
-- witness for `Composite`.

+
pattern

+
  composite {d} d<n d∣n = hasNonTrivialDivisor {divisor = d} d<n d∣n

+

+
------------------------------------------------------------------------

+
-- Primality

+

+
-- Prime as the complement of Composite (and hence the diagonal of Rough

+
-- as defined above). The constructor `prime` takes a proof `notComposite`

+
-- that NonTrivial p is not composite and thereby enforces that:

+
-- * p is a fortiori NonZero and NonUnit

+
-- * p is p-Rough, i.e. any proper divisor must be at least p, i.e. p itself

+
record Prime (p : ℕ) : Set where

+
  constructor prime

+
  field

+
    .{{nontrivial}} : NonTrivial p

+
    notComposite    : ¬ Composite p

+

+
------------------------------------------------------------------------

+
-- Irreducibility

+

+
Irreducible : Pred ℕ _

+
Irreducible n = ∀ {d} → d ∣ n → d ≡ 1 ⊎ d ≡ n

+

+
------------------------------------------------------------------------

+
-- Properties

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Roughness

+

+
-- 1 is always n-rough

+
rough-1 : ∀ n → n Rough 1

+
rough-1 _ (hasNonTrivialDivisor _ d∣1) =

+
  contradiction (∣1⇒≡1 d∣1) nonTrivial⇒≢1

+

+
-- Any number is 0-, 1- and 2-rough,

+
-- because no proper divisor d can be strictly less than 0, 1, or 2

+
0-rough : 0 Rough n

+
0-rough (hasNonTrivialDivisor () _)

+

+
1-rough : 1 Rough n

+
1-rough (hasNonTrivialDivisor {{()}} z<s _)

+

+
2-rough : 2 Rough n

+
2-rough (hasNonTrivialDivisor {{()}} (s<s z<s) _)

+

+
-- If a number n > 1 is m-rough, then m ≤ n

+
rough⇒≤ : .{{NonTrivial n}} → m Rough n → m ≤ n

+
rough⇒≤ rough = ≮⇒≥ n≮m

+
  where n≮m = λ m>n → rough (hasNonTrivialDivisor m>n ∣-refl)

+

+
-- If a number n is m-rough, and m ∤ n, then n is (suc m)-rough

+
∤⇒rough-suc : m ∤ n → m Rough n → (suc m) Rough n

+
∤⇒rough-suc m∤n r (hasNonTrivialDivisor d<1+m d∣n)

+
  with m<1+n⇒m<n∨m≡n d<1+m

+
... | inj₁ d<m      = r (hasNonTrivialDivisor d<m d∣n)

+
... | inj₂ d≡m@refl = contradiction d∣n m∤n

+

+
-- If a number is m-rough, then so are all of its divisors

+
rough∧∣⇒rough : m Rough o → n ∣ o → m Rough n

+
rough∧∣⇒rough r n∣o bntd = r (hasNonTrivialDivisor-∣ bntd n∣o)

+

+
------------------------------------------------------------------------

+
-- Compositeness

+

+
-- Smart constructors

+

+
composite-≢ : ∀ d → .{{NonTrivial d}} → .{{NonZero n}} →

+
              d ≢ n → d ∣ n → Composite n

+
composite-≢ d = hasNonTrivialDivisor-≢ {d}

+

+
composite-∣ : .{{NonZero n}} → Composite m → m ∣ n → Composite n

+
composite-∣ (composite {d} d<m d∣n) m∣n@(divides-refl q)

+
  = composite (*-monoʳ-< q d<m) (*-monoʳ-∣ q d∣n)

+
  where instance

+
    _ = m≢0∧n>1⇒m*n>1 q d

+
    _ = m*n≢0⇒m≢0 q

+

+
-- Basic (counter-)examples of Composite

+

+
¬composite[0] : ¬ Composite 0

+
¬composite[0] = 0-rough

+

+
¬composite[1] : ¬ Composite 1

+
¬composite[1] = 1-rough

+

+
composite[4] : Composite 4

+
composite[4] = composite-≢ 2 (λ()) (divides-refl 2)

+

+
composite[6] : Composite 6

+
composite[6] = composite-≢ 3 (λ()) (divides-refl 2)

+

+
composite⇒nonZero : Composite n → NonZero n

+
composite⇒nonZero {suc _} _ = _

+

+
composite⇒nonTrivial : Composite n → NonTrivial n

+
composite⇒nonTrivial {1}    composite[1] =

+
  contradiction composite[1] ¬composite[1]

+
composite⇒nonTrivial {2+ _} _            = _

+

+
composite? : Decidable Composite

+
composite? n = Dec.map CompositeUpTo⇔Composite (compositeUpTo? n)

+
  where

+
  -- For technical reasons, in order to be able to prove decidability

+
  -- via the `all?` and `any?` combinators for *bounded* predicates on

+
  -- `ℕ`, we further define the bounded counterparts to predicates

+
  -- `P...` as `P...UpTo` and show the equivalence of the two.

+

+
  -- Equivalent bounded predicate definition

+
  CompositeUpTo : Pred ℕ _

+
  CompositeUpTo n = ∃[ d ] d < n × NonTrivial d × d ∣ n

+

+
  -- Proof of equivalence

+
  comp-upto⇒comp : CompositeUpTo n → Composite n

+
  comp-upto⇒comp (_ , d<n , ntd , d∣n) = composite d<n d∣n

+
    where instance _ = ntd

+

+
  comp⇒comp-upto : Composite n → CompositeUpTo n

+
  comp⇒comp-upto (composite d<n d∣n) = _ , d<n , recompute-nonTrivial , d∣n

+

+
  CompositeUpTo⇔Composite : CompositeUpTo n ⇔ Composite n

+
  CompositeUpTo⇔Composite = mk⇔ comp-upto⇒comp comp⇒comp-upto

+

+
  -- Proof of decidability

+
  compositeUpTo? : Decidable CompositeUpTo

+
  compositeUpTo? n = anyUpTo? (λ d → nonTrivial? d ×-dec d ∣? n) n

+

+
------------------------------------------------------------------------

+
-- Primality

+

+
-- Basic (counter-)examples

+

+
¬prime[0] : ¬ Prime 0

+
¬prime[0] ()

+

+
¬prime[1] : ¬ Prime 1

+
¬prime[1] ()

+

+
prime[2] : Prime 2

+
prime[2] = prime 2-rough

+

+
prime⇒nonZero : Prime p → NonZero p

+
prime⇒nonZero _ = nonTrivial⇒nonZero _

+

+
prime⇒nonTrivial : Prime p → NonTrivial p

+
prime⇒nonTrivial _ = recompute-nonTrivial

+

+
prime? : Decidable Prime

+
prime? 0        = no ¬prime[0]

+
prime? 1        = no ¬prime[1]

+
prime? n@(2+ _) = Dec.map PrimeUpTo⇔Prime (primeUpTo? n)

+
  where

+
  -- Equivalent bounded predicate definition

+
  PrimeUpTo : Pred ℕ _

+
  PrimeUpTo n = ∀ {d} → d < n → NonTrivial d → d ∤ n

+

+
  -- Proof of equivalence

+
  prime⇒prime-upto : Prime n → PrimeUpTo n

+
  prime⇒prime-upto (prime p) {d} d<n ntd d∣n

+
    = p (composite d<n d∣n) where instance _ = ntd

+

+
  prime-upto⇒prime : .{{NonTrivial n}} → PrimeUpTo n → Prime n

+
  prime-upto⇒prime upto = prime

+
    λ (composite d<n d∣n) → upto d<n recompute-nonTrivial d∣n

+

+
  PrimeUpTo⇔Prime : .{{NonTrivial n}} → PrimeUpTo n ⇔ Prime n

+
  PrimeUpTo⇔Prime = mk⇔ prime-upto⇒prime prime⇒prime-upto

+

+
  -- Proof of decidability

+
  primeUpTo? : Decidable PrimeUpTo

+
  primeUpTo? n = allUpTo? (λ d → nonTrivial? d →-dec ¬? (d ∣? n)) n

+

+
-- Euclid's lemma - for p prime, if p ∣ m * n, then either p ∣ m or p ∣ n.

+
--

+
-- This demonstrates that the usual definition of prime numbers matches

+
-- the ring theoretic definition of a prime element of the semiring ℕ.

+
-- This is useful for proving many other theorems involving prime numbers.

+
euclidsLemma : ∀ m n {p} → Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n

+
euclidsLemma m n {p} pp@(prime pr) p∣m*n = result

+
  where

+
  open ∣-Reasoning

+
  instance _ = prime⇒nonZero pp

+

+
  p∣rmn : ∀ r → p ∣ r * m * n

+
  p∣rmn r = begin

+
    p           ∣⟨ p∣m*n ⟩

+
    m * n       ∣⟨ n∣m*n r ⟩

+
    r * (m * n) ≡⟨ *-assoc r m n ⟨

+
    r * m * n   ∎

+

+
  result : p ∣ m ⊎ p ∣ n

+
  result with Bézout.lemma m p

+
  -- if the GCD of m and p is zero then p must be zero, which is

+
  -- impossible as p is a prime.

+
  -- note: this should be a typechecker-rejectable case!?

+
  ... | Bézout.result 0 g _ =

+
    contradiction (0∣⇒≡0 (GCD.gcd∣n g)) (≢-nonZero⁻¹ _)

+

+
  -- if the GCD of m and p is one then m and p are coprime, and we know

+
  -- that for some integers s and r, sm + rp = 1. We can use this fact

+
  -- to determine that p divides n

+
  ... | Bézout.result 1 _ (Bézout.+- r s 1+sp≡rm) =

+
    inj₂ (flip ∣m+n∣m⇒∣n (n∣m*n*o s n) (begin

+
      p             ∣⟨ p∣rmn r ⟩

+
      r * m * n     ≡⟨ cong (_* n) 1+sp≡rm ⟨

+
      n + s * p * n ≡⟨ +-comm n (s * p * n) ⟩

+
      s * p * n + n ∎))

+

+
  ... | Bézout.result 1 _ (Bézout.-+ r s 1+rm≡sp) =

+
    inj₂ (flip ∣m+n∣m⇒∣n (p∣rmn r) (begin

+
      p             ∣⟨ n∣m*n*o s n ⟩

+
      s * p * n     ≡⟨ cong (_* n) 1+rm≡sp ⟨

+
      n + r * m * n ≡⟨ +-comm n (r * m * n) ⟩

+
      r * m * n + n ∎))

+

+
  -- if the GCD of m and p is greater than one, then it must be p and

+
  -- hence p ∣ m.

+
  ... | Bézout.result d@(2+ _) g _ with d ≟ p

+
  ...   | yes d≡p@refl = inj₁ (GCD.gcd∣m g)

+
  ...   | no  d≢p = contradiction (composite-≢ d d≢p (GCD.gcd∣n g)) pr

+

+
-- Relationship between roughness and primality.

+
prime⇒rough : Prime p → p Rough p

+
prime⇒rough (prime pr) = pr

+

+
-- If a number n is p-rough, and p > 1 divides n, then p must be prime

+
rough∧∣⇒prime : .{{NonTrivial p}} → p Rough n → p ∣ n → Prime p

+
rough∧∣⇒prime r p∣n = prime (rough∧∣⇒rough r p∣n)

+

+
-- If a number n is m-rough, and m * m > n, then n must be prime.

+
rough∧square>⇒prime : .{{NonTrivial n}} → m Rough n → m * m > n → Prime n

+
rough∧square>⇒prime rough m*m>n = prime ¬composite

+
  where

+
    ¬composite : ¬ Composite _

+
    ¬composite (composite d<n d∣n) = contradiction (m∣n⇒n≡quotient*m d∣n)

+
      (<⇒≢ (<-≤-trans m*m>n (*-mono-≤

+
        (rough⇒≤ (rough∧∣⇒rough rough (quotient-∣ d∣n)))

+
        (rough⇒≤ (rough∧∣⇒rough rough d∣n)))))

+
      where instance _ = n>1⇒nonTrivial (quotient>1 d∣n d<n)

+

+
-- Relationship between compositeness and primality.

+
composite⇒¬prime : Composite n → ¬ Prime n

+
composite⇒¬prime composite[d] (prime p) = p composite[d]

+

+
¬composite⇒prime : .{{NonTrivial n}} → ¬ Composite n → Prime n

+
¬composite⇒prime = prime

+

+
prime⇒¬composite : Prime n → ¬ Composite n

+
prime⇒¬composite (prime p) = p

+

+
-- Note that this has to recompute the factor!

+
¬prime⇒composite : .{{NonTrivial n}} → ¬ Prime n → Composite n

+
¬prime⇒composite {n} ¬prime[n] =

+
  decidable-stable (composite? n) (¬prime[n] ∘′ ¬composite⇒prime)

+

+
productOfPrimes≢0 : ∀ {as} → All Prime as → NonZero (product as)

+
productOfPrimes≢0 pas = product≢0 (All.map prime⇒nonZero pas)

+
  where

+
  product≢0 : ∀ {ns} → All NonZero ns → NonZero (product ns)

+
  product≢0 [] = _

+
  product≢0 {n ∷ ns} (nzn ∷ nzns) = m*n≢0 n _ {{nzn}} {{product≢0 nzns}}

+

+
productOfPrimes≥1 : ∀ {as} → All Prime as → product as ≥ 1

+
productOfPrimes≥1 {as} pas = >-nonZero⁻¹ _ {{productOfPrimes≢0 pas}}

+

+
------------------------------------------------------------------------

+
-- Basic (counter-)examples of Irreducible

+

+
¬irreducible[0] : ¬ Irreducible 0

+
¬irreducible[0] irr[0] = contradiction₂ 2≡1⊎2≡0 (λ ()) (λ ())

+
  where 2≡1⊎2≡0 = irr[0] {2} (divides-refl 0)

+

+
irreducible[1] : Irreducible 1

+
irreducible[1] m∣1 = inj₁ (∣1⇒≡1 m∣1)

+

+
irreducible[2] : Irreducible 2

+
irreducible[2] {zero}  0∣2 with () ← 0∣⇒≡0 0∣2

+
irreducible[2] {suc _} d∣2 with ∣⇒≤ d∣2

+
... | z<s     = inj₁ refl

+
... | s<s z<s = inj₂ refl

+

+
irreducible⇒nonZero : Irreducible n → NonZero n

+
irreducible⇒nonZero {zero}    = flip contradiction ¬irreducible[0]

+
irreducible⇒nonZero {suc _} _ = _

+

+
irreducible? : Decidable Irreducible

+
irreducible? zero      = no ¬irreducible[0]

+
irreducible? n@(suc _) =

+
  Dec.map IrreducibleUpTo⇔Irreducible (irreducibleUpTo? n)

+
  where

+
  -- Equivalent bounded predicate definition

+
  IrreducibleUpTo : Pred ℕ _

+
  IrreducibleUpTo n = ∀ {d} → d < n → d ∣ n → d ≡ 1 ⊎ d ≡ n

+

+
  -- Proof of equivalence

+
  irr-upto⇒irr : .{{NonZero n}} → IrreducibleUpTo n → Irreducible n

+
  irr-upto⇒irr irr-upto m∣n

+
    = [ flip irr-upto m∣n , inj₂ ]′ (m≤n⇒m<n∨m≡n (∣⇒≤ m∣n))

+

+
  irr⇒irr-upto : Irreducible n → IrreducibleUpTo n

+
  irr⇒irr-upto irr m<n m∣n = irr m∣n

+

+
  IrreducibleUpTo⇔Irreducible : .{{NonZero n}} →

+
                                 IrreducibleUpTo n ⇔ Irreducible n

+
  IrreducibleUpTo⇔Irreducible = mk⇔ irr-upto⇒irr irr⇒irr-upto

+

+
  -- Decidability

+
  irreducibleUpTo? : Decidable IrreducibleUpTo

+
  irreducibleUpTo? n = allUpTo?

+
    (λ m → (m ∣? n) →-dec (m ≟ 1 ⊎-dec m ≟ n)) n

+

+
-- Relationship between primality and irreducibility.

+
prime⇒irreducible : Prime p → Irreducible p

+
prime⇒irreducible {p} pp@(prime pr) = irr

+
  where

+
  instance _ = prime⇒nonZero pp

+
  irr : .{{NonZero p}} → Irreducible p

+
  irr {0}    0∣p = contradiction (0∣⇒≡0 0∣p) (≢-nonZero⁻¹ p)

+
  irr {1}    1∣p = inj₁ refl

+
  irr {2+ _} d∣p = inj₂ (≤∧≮⇒≡ (∣⇒≤ d∣p) d≮p)

+
    where d≮p = λ d<p → pr (composite d<p d∣p)

+

+

+
irreducible⇒prime : .{{NonTrivial p}} → Irreducible p → Prime p

+
irreducible⇒prime irr = prime

+
  λ (composite d<p d∣p) → [ nonTrivial⇒≢1 , (<⇒≢ d<p) ]′ (irr d∣p)

+

+
------------------------------------------------------------------------

+
-- Using decidability

+

+
-- Once we have the above decision procedures, then instead of

+
-- constructing proofs of e.g. Prime-ness by hand, we call the

+
-- appropriate function, and use the witness extraction functions

+
-- `from-yes`, `from-no` to return the checked proofs.

+

+
private

+

+
  -- Example: 2 is prime, but not-composite.

+
  2-is-prime : Prime 2

+
  2-is-prime = from-yes (prime? 2)

+

+
  2-is-not-composite : ¬ Composite 2

+
  2-is-not-composite = from-no (composite? 2)

+

+
  -- Example: 4 and 6 are composite, hence not-prime

+
  4-is-composite : Composite 4

+
  4-is-composite = from-yes (composite? 4)

+

+
  4-is-not-prime : ¬ Prime 4

+
  4-is-not-prime = from-no (prime? 4)

+

+
  6-is-composite : Composite 6

+
  6-is-composite = from-yes (composite? 6)

+

+
  6-is-not-prime : ¬ Prime 6

+
  6-is-not-prime = from-no (prime? 6)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Properties.html b/site/static/agda_html/Data.Nat.Properties.html
new file mode 100644
index 000000000..b4eba1419
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Properties.html
@@ -0,0 +1,2491 @@
+
+Data.Nat.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- A bunch of properties about natural number operations

+
------------------------------------------------------------------------

+

+
-- See README.Data.Nat for some examples showing how this module can be

+
-- used.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Properties where

+

+
open import Axiom.UniquenessOfIdentityProofs using (module Decidable⇒UIP)

+
open import Algebra.Bundles using (Magma; Semigroup; CommutativeSemigroup;

+
  CommutativeMonoid; Monoid; Semiring; CommutativeSemiring; CommutativeSemiringWithoutOne)

+
open import Algebra.Definitions.RawMagma using (_,_)

+
open import Algebra.Morphism

+
open import Algebra.Consequences.Propositional

+
  using (comm+cancelˡ⇒cancelʳ; comm∧distrʳ⇒distrˡ; comm∧distrˡ⇒distrʳ)

+
open import Algebra.Construct.NaturalChoice.Base

+
  using (MinOperator; MaxOperator)

+
import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp

+
import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp

+
import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties

+
open import Data.Bool.Base using (Bool; false; true; T)

+
open import Data.Bool.Properties using (T?)

+
open import Data.Nat.Base

+
open import Data.Product.Base using (∃; _×_; _,_)

+
open import Data.Sum.Base as Sum using (inj₁; inj₂; _⊎_; [_,_]′)

+
open import Data.Unit.Base using (tt)

+
open import Function.Base using (_∘_; flip; _$_; id; _∘′_; _$′_)

+
open import Function.Bundles using (_↣_)

+
open import Function.Metric.Nat using (TriangleInequality; IsProtoMetric; IsPreMetric;

+
  IsQuasiSemiMetric; IsSemiMetric; IsMetric; PreMetric; QuasiSemiMetric;

+
  SemiMetric; Metric)

+
open import Level using (0ℓ)

+
open import Relation.Unary as U using (Pred)

+
open import Relation.Binary.Core

+
  using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary

+
open import Relation.Binary.Consequences using (flip-Connex)

+
open import Relation.Binary.PropositionalEquality

+
open import Relation.Nullary hiding (Irrelevant)

+
open import Relation.Nullary.Decidable using (True; via-injection; map′; recompute)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (fromEquivalence)

+

+
open import Algebra.Definitions {A = ℕ} _≡_

+
  hiding (LeftCancellative; RightCancellative; Cancellative)

+
open import Algebra.Definitions

+
  using (LeftCancellative; RightCancellative; Cancellative)

+
open import Algebra.Structures {A = ℕ} _≡_

+

+

+
private

+
  variable

+
    m n o k : ℕ

+

+

+
------------------------------------------------------------------------

+
-- Properties of NonZero

+
------------------------------------------------------------------------

+

+
nonZero? : U.Decidable NonZero

+
nonZero? zero    = no NonZero.nonZero

+
nonZero? (suc n) = yes _

+

+
------------------------------------------------------------------------

+
-- Properties of NonTrivial

+
------------------------------------------------------------------------

+

+
nonTrivial? : U.Decidable NonTrivial

+
nonTrivial? 0      = no λ()

+
nonTrivial? 1      = no λ()

+
nonTrivial? (2+ n) = yes _

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+
------------------------------------------------------------------------

+

+
suc-injective : suc m ≡ suc n → m ≡ n

+
suc-injective = cong pred

+

+
≡ᵇ⇒≡ : ∀ m n → T (m ≡ᵇ n) → m ≡ n

+
≡ᵇ⇒≡ zero    zero    _  = refl

+
≡ᵇ⇒≡ (suc m) (suc n) eq = cong suc (≡ᵇ⇒≡ m n eq)

+

+
≡⇒≡ᵇ : ∀ m n → m ≡ n → T (m ≡ᵇ n)

+
≡⇒≡ᵇ zero    zero    eq = _

+
≡⇒≡ᵇ (suc m) (suc n) eq = ≡⇒≡ᵇ m n (suc-injective eq)

+

+
-- NB: we use the builtin function `_≡ᵇ_` here so that the function

+
-- quickly decides whether to return `yes` or `no`. It still takes

+
-- a linear amount of time to generate the proof if it is inspected.

+
-- We expect the main benefit to be visible in compiled code as the

+
-- backend erases proofs.

+

+
infix 4 _≟_

+
_≟_ : DecidableEquality ℕ

+
m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))

+

+
≡-irrelevant : Irrelevant {A = ℕ} _≡_

+
≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≟_

+

+
≟-diag : (eq : m ≡ n) → (m ≟ n) ≡ yes eq

+
≟-diag = ≡-≟-identity _≟_

+

+
≡-isDecEquivalence : IsDecEquivalence (_≡_ {A = ℕ})

+
≡-isDecEquivalence = record

+
  { isEquivalence = isEquivalence

+
  ; _≟_           = _≟_

+
  }

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = record

+
  { Carrier          = ℕ

+
  ; _≈_              = _≡_

+
  ; isDecEquivalence = ≡-isDecEquivalence

+
  }

+

+
0≢1+n : 0 ≢ suc n

+
0≢1+n ()

+

+
1+n≢0 : suc n ≢ 0

+
1+n≢0 ()

+

+
1+n≢n : suc n ≢ n

+
1+n≢n {suc n} = 1+n≢n ∘ suc-injective

+

+
------------------------------------------------------------------------

+
-- Properties of _<ᵇ_

+
------------------------------------------------------------------------

+

+
<ᵇ⇒< : ∀ m n → T (m <ᵇ n) → m < n

+
<ᵇ⇒< zero    (suc n) m<n = z<s

+
<ᵇ⇒< (suc m) (suc n) m<n = s<s (<ᵇ⇒< m n m<n)

+

+
<⇒<ᵇ : m < n → T (m <ᵇ n)

+
<⇒<ᵇ z<s               = tt

+
<⇒<ᵇ (s<s m<n@(s≤s _)) = <⇒<ᵇ m<n

+

+
<ᵇ-reflects-< : ∀ m n → Reflects (m < n) (m <ᵇ n)

+
<ᵇ-reflects-< m n = fromEquivalence (<ᵇ⇒< m n) <⇒<ᵇ

+

+
------------------------------------------------------------------------

+
-- Properties of _≤ᵇ_

+
------------------------------------------------------------------------

+

+
≤ᵇ⇒≤ : ∀ m n → T (m ≤ᵇ n) → m ≤ n

+
≤ᵇ⇒≤ zero    n m≤n = z≤n

+
≤ᵇ⇒≤ (suc m) n m≤n = <ᵇ⇒< m n m≤n

+

+
≤⇒≤ᵇ : m ≤ n → T (m ≤ᵇ n)

+
≤⇒≤ᵇ z≤n         = tt

+
≤⇒≤ᵇ m≤n@(s≤s _) = <⇒<ᵇ m≤n

+

+
≤ᵇ-reflects-≤ : ∀ m n → Reflects (m ≤ n) (m ≤ᵇ n)

+
≤ᵇ-reflects-≤ m n = fromEquivalence (≤ᵇ⇒≤ m n) ≤⇒≤ᵇ

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Relational properties of _≤_

+

+
≤-reflexive : _≡_ ⇒ _≤_

+
≤-reflexive {zero}  refl = z≤n

+
≤-reflexive {suc m} refl = s≤s (≤-reflexive refl)

+

+
≤-refl : Reflexive _≤_

+
≤-refl = ≤-reflexive refl

+

+
≤-antisym : Antisymmetric _≡_ _≤_

+
≤-antisym z≤n       z≤n       = refl

+
≤-antisym (s≤s m≤n) (s≤s n≤m) = cong suc (≤-antisym m≤n n≤m)

+

+
≤-trans : Transitive _≤_

+
≤-trans z≤n       _         = z≤n

+
≤-trans (s≤s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)

+

+
≤-total : Total _≤_

+
≤-total zero    _       = inj₁ z≤n

+
≤-total _       zero    = inj₂ z≤n

+
≤-total (suc m) (suc n) = Sum.map s≤s s≤s (≤-total m n)

+

+
≤-irrelevant : Irrelevant _≤_

+
≤-irrelevant z≤n        z≤n        = refl

+
≤-irrelevant (s≤s m≤n₁) (s≤s m≤n₂) = cong s≤s (≤-irrelevant m≤n₁ m≤n₂)

+

+
-- NB: we use the builtin function `_<ᵇ_` here so that the function

+
-- quickly decides whether to return `yes` or `no`. It still takes

+
-- a linear amount of time to generate the proof if it is inspected.

+
-- We expect the main benefit to be visible in compiled code as the

+
-- backend erases proofs.

+

+
infix 4 _≤?_ _≥?_

+

+
_≤?_ : Decidable _≤_

+
m ≤? n = map′ (≤ᵇ⇒≤ m n) ≤⇒≤ᵇ (T? (m ≤ᵇ n))

+

+
_≥?_ : Decidable _≥_

+
_≥?_ = flip _≤?_

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≤-isPreorder : IsPreorder _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isTotalPreorder : IsTotalPreorder _≡_ _≤_

+
≤-isTotalPreorder = record

+
  { isPreorder = ≤-isPreorder

+
  ; total      = ≤-total

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder _≡_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≤-preorder : Preorder 0ℓ 0ℓ 0ℓ

+
≤-preorder = record

+
  { isPreorder = ≤-isPreorder

+
  }

+

+
≤-totalPreorder : TotalPreorder 0ℓ 0ℓ 0ℓ

+
≤-totalPreorder = record

+
  { isTotalPreorder = ≤-isTotalPreorder

+
  }

+

+
≤-poset : Poset 0ℓ 0ℓ 0ℓ

+
≤-poset = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  }

+

+
≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ

+
≤-totalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  }

+

+
≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ

+
≤-decTotalOrder = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _≤_

+

+
s≤s-injective : {p q : m ≤ n} → s≤s p ≡ s≤s q → p ≡ q

+
s≤s-injective refl = refl

+

+
≤-pred : suc m ≤ suc n → m ≤ n

+
≤-pred = s≤s⁻¹

+

+
m≤n⇒m≤1+n : m ≤ n → m ≤ 1 + n

+
m≤n⇒m≤1+n z≤n       = z≤n

+
m≤n⇒m≤1+n (s≤s m≤n) = s≤s (m≤n⇒m≤1+n m≤n)

+

+
n≤1+n : ∀ n → n ≤ 1 + n

+
n≤1+n _ = m≤n⇒m≤1+n ≤-refl

+

+
1+n≰n : 1 + n ≰ n

+
1+n≰n (s≤s 1+n≤n) = 1+n≰n 1+n≤n

+

+
n≤0⇒n≡0 : n ≤ 0 → n ≡ 0

+
n≤0⇒n≡0 z≤n = refl

+

+
n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1

+
n≤1⇒n≡0∨n≡1 z≤n       = inj₁ refl

+
n≤1⇒n≡0∨n≡1 (s≤s z≤n) = inj₂ refl

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+
------------------------------------------------------------------------

+

+
-- Relationships between the various relations

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ z<s               = z≤n

+
<⇒≤ (s<s m<n@(s≤s _)) = s≤s (<⇒≤ m<n)

+

+
<⇒≢ : _<_ ⇒ _≢_

+
<⇒≢ m<n refl = 1+n≰n m<n

+

+
>⇒≢ : _>_ ⇒ _≢_

+
>⇒≢ = ≢-sym ∘ <⇒≢

+

+
≤⇒≯ : _≤_ ⇒ _≯_

+
≤⇒≯ (s≤s m≤n) (s≤s n≤m) = ≤⇒≯ m≤n n≤m

+

+
<⇒≱ : _<_ ⇒ _≱_

+
<⇒≱ (s≤s m+1≤n) (s≤s n≤m) = <⇒≱ m+1≤n n≤m

+

+
<⇒≯ : _<_ ⇒ _≯_

+
<⇒≯ (s≤s m<n) (s≤s n<m) = <⇒≯ m<n n<m

+

+
≰⇒≮ : _≰_ ⇒ _≮_

+
≰⇒≮ m≰n 1+m≤n = m≰n (<⇒≤ 1+m≤n)

+

+
≰⇒> : _≰_ ⇒ _>_

+
≰⇒> {zero}          z≰n = contradiction z≤n z≰n

+
≰⇒> {suc m} {zero}  _   = z<s

+
≰⇒> {suc m} {suc n} m≰n = s<s (≰⇒> (m≰n ∘ s≤s))

+

+
≰⇒≥ : _≰_ ⇒ _≥_

+
≰⇒≥ = <⇒≤ ∘ ≰⇒>

+

+
≮⇒≥ : _≮_ ⇒ _≥_

+
≮⇒≥ {_}     {zero}  _       = z≤n

+
≮⇒≥ {zero}  {suc j} 1≮j+1   = contradiction z<s 1≮j+1

+
≮⇒≥ {suc i} {suc j} i+1≮j+1 = s≤s (≮⇒≥ (i+1≮j+1 ∘ s<s))

+

+
≤∧≢⇒< : ∀ {m n} → m ≤ n → m ≢ n → m < n

+
≤∧≢⇒< {_} {zero}  z≤n       m≢n     = contradiction refl m≢n

+
≤∧≢⇒< {_} {suc n} z≤n       m≢n     = z<s

+
≤∧≢⇒< {_} {suc n} (s≤s m≤n) 1+m≢1+n =

+
  s<s (≤∧≢⇒< m≤n (1+m≢1+n ∘ cong suc))

+

+
≤∧≮⇒≡ : ∀ {m n} → m ≤ n → m ≮ n → m ≡ n

+
≤∧≮⇒≡ m≤n m≮n = ≤-antisym m≤n (≮⇒≥ m≮n)

+

+
≤-<-connex : Connex _≤_ _<_

+
≤-<-connex m n with m ≤? n

+
... | yes m≤n = inj₁ m≤n

+
... | no ¬m≤n = inj₂ (≰⇒> ¬m≤n)

+

+
≥->-connex : Connex _≥_ _>_

+
≥->-connex = flip ≤-<-connex

+

+
<-≤-connex : Connex _<_ _≤_

+
<-≤-connex = flip-Connex ≤-<-connex

+

+
>-≥-connex : Connex _>_ _≥_

+
>-≥-connex = flip-Connex ≥->-connex

+

+
------------------------------------------------------------------------

+
-- Relational properties of _<_

+

+
<-irrefl : Irreflexive _≡_ _<_

+
<-irrefl refl (s<s n<n) = <-irrefl refl n<n

+

+
<-asym : Asymmetric _<_

+
<-asym (s<s n<m) (s<s m<n) = <-asym n<m m<n

+

+
<-trans : Transitive _<_

+
<-trans (s≤s i≤j) (s≤s j<k) = s≤s (≤-trans i≤j (≤-trans (n≤1+n _) j<k))

+

+
≤-<-trans : LeftTrans _≤_ _<_

+
≤-<-trans m≤n (s<s n≤o) = s≤s (≤-trans m≤n n≤o)

+

+
<-≤-trans : RightTrans _<_ _≤_

+
<-≤-trans (s<s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)

+

+
-- NB: we use the builtin function `_<ᵇ_` here so that the function

+
-- quickly decides which constructor to return. It still takes a

+
-- linear amount of time to generate the proof if it is inspected.

+
-- We expect the main benefit to be visible in compiled code as the

+
-- backend erases proofs.

+

+
<-cmp : Trichotomous _≡_ _<_

+
<-cmp m n with m ≟ n | T? (m <ᵇ n)

+
... | yes m≡n | _       = tri≈ (<-irrefl m≡n) m≡n (<-irrefl (sym m≡n))

+
... | no  m≢n | yes m<n = tri< (<ᵇ⇒< m n m<n) m≢n (<⇒≯ (<ᵇ⇒< m n m<n))

+
... | no  m≢n | no  m≮n = tri> (m≮n ∘ <⇒<ᵇ)   m≢n (≤∧≢⇒< (≮⇒≥ (m≮n ∘ <⇒<ᵇ)) (m≢n ∘ sym))

+

+
infix 4 _<?_ _>?_

+

+
_<?_ : Decidable _<_

+
m <? n = suc m ≤? n

+

+
_>?_ : Decidable _>_

+
_>?_ = flip _<?_

+

+
<-irrelevant : Irrelevant _<_

+
<-irrelevant = ≤-irrelevant

+

+
<-resp₂-≡ : _<_ Respects₂ _≡_

+
<-resp₂-≡ = subst (_ <_) , subst (_< _)

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = <-resp₂-≡

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_

+
<-isStrictTotalOrder = isStrictTotalOrderᶜ record

+
  { isEquivalence = isEquivalence

+
  ; trans         = <-trans

+
  ; compare       = <-cmp

+
  }

+

+
<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _<_

+

+
s<s-injective : {p q : m < n} → s<s p ≡ s<s q → p ≡ q

+
s<s-injective refl = refl

+

+
<-pred : suc m < suc n → m < n

+
<-pred = s<s⁻¹

+

+
m<n⇒m<1+n : m < n → m < 1 + n

+
m<n⇒m<1+n z<s               = z<s

+
m<n⇒m<1+n (s<s m<n@(s≤s _)) = s<s (m<n⇒m<1+n m<n)

+

+
n≮0 : n ≮ 0

+
n≮0 ()

+

+
n≮n : ∀ n → n ≮ n -- implicit?

+
n≮n n = <-irrefl (refl {x = n})

+

+
0<1+n : 0 < suc n

+
0<1+n = z<s

+

+
n<1+n : ∀ n → n < suc n

+
n<1+n n = ≤-refl

+

+
n<1⇒n≡0 : n < 1 → n ≡ 0

+
n<1⇒n≡0 (s≤s n≤0) = n≤0⇒n≡0 n≤0

+

+
n>0⇒n≢0 : n > 0 → n ≢ 0

+
n>0⇒n≢0 {suc n} _ ()

+

+
n≢0⇒n>0 : n ≢ 0 → n > 0

+
n≢0⇒n>0 {zero}  0≢0 =  contradiction refl 0≢0

+
n≢0⇒n>0 {suc n} _   =  0<1+n

+

+
m<n⇒0<n : m < n → 0 < n

+
m<n⇒0<n = ≤-trans 0<1+n

+

+
m<n⇒n≢0 : m < n → n ≢ 0

+
m<n⇒n≢0 (s≤s m≤n) ()

+

+
m<n⇒m≤1+n : m < n → m ≤ suc n

+
m<n⇒m≤1+n = m≤n⇒m≤1+n ∘ <⇒≤

+

+
m<1+n⇒m<n∨m≡n :  ∀ {m n} → m < suc n → m < n ⊎ m ≡ n

+
m<1+n⇒m<n∨m≡n {0}     {0}     _           = inj₂ refl

+
m<1+n⇒m<n∨m≡n {0}     {suc n} _           = inj₁ 0<1+n

+
m<1+n⇒m<n∨m≡n {suc m} {suc n} (s<s m<1+n) = Sum.map s<s (cong suc) (m<1+n⇒m<n∨m≡n m<1+n)

+

+
m≤n⇒m<n∨m≡n : m ≤ n → m < n ⊎ m ≡ n

+
m≤n⇒m<n∨m≡n m≤n = m<1+n⇒m<n∨m≡n (s≤s m≤n)

+

+
m<1+n⇒m≤n : m < suc n → m ≤ n

+
m<1+n⇒m≤n (s≤s m≤n) = m≤n

+

+
∀[m≤n⇒m≢o]⇒n<o : ∀ n o → (∀ {m} → m ≤ n → m ≢ o) → n < o

+
∀[m≤n⇒m≢o]⇒n<o _       zero    m≤n⇒n≢0 = contradiction refl (m≤n⇒n≢0 z≤n)

+
∀[m≤n⇒m≢o]⇒n<o zero    (suc o) _       = 0<1+n

+
∀[m≤n⇒m≢o]⇒n<o (suc n) (suc o) m≤n⇒n≢o = s≤s (∀[m≤n⇒m≢o]⇒n<o n o rec)

+
  where

+
  rec : ∀ {m} → m ≤ n → m ≢ o

+
  rec m≤n refl = m≤n⇒n≢o (s≤s m≤n) refl

+

+
∀[m<n⇒m≢o]⇒n≤o : ∀ n o → (∀ {m} → m < n → m ≢ o) → n ≤ o

+
∀[m<n⇒m≢o]⇒n≤o zero    n       _       = z≤n

+
∀[m<n⇒m≢o]⇒n≤o (suc n) zero    m<n⇒m≢0 = contradiction refl (m<n⇒m≢0 0<1+n)

+
∀[m<n⇒m≢o]⇒n≤o (suc n) (suc o) m<n⇒m≢o = s≤s (∀[m<n⇒m≢o]⇒n≤o n o rec)

+
  where

+
  rec : ∀ {m} → m < n → m ≢ o

+
  rec o<n refl = m<n⇒m≢o (s<s o<n) refl

+

+
------------------------------------------------------------------------

+
-- A module for reasoning about the _≤_ and _<_ relations

+
------------------------------------------------------------------------

+

+
module ≤-Reasoning where

+
  open import Relation.Binary.Reasoning.Base.Triple

+
    ≤-isPreorder

+
    <-asym

+
    <-trans

+
    (resp₂ _<_)

+
    <⇒≤

+
    <-≤-trans

+
    ≤-<-trans

+
    public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+

+
open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _+_

+
------------------------------------------------------------------------

+

+
+-suc : ∀ m n → m + suc n ≡ suc (m + n)

+
+-suc zero    n = refl

+
+-suc (suc m) n = cong suc (+-suc m n)

+

+
------------------------------------------------------------------------

+
-- Algebraic properties of _+_

+

+
+-assoc : Associative _+_

+
+-assoc zero    _ _ = refl

+
+-assoc (suc m) n o = cong suc (+-assoc m n o)

+

+
+-identityˡ : LeftIdentity 0 _+_

+
+-identityˡ _ = refl

+

+
+-identityʳ : RightIdentity 0 _+_

+
+-identityʳ zero    = refl

+
+-identityʳ (suc n) = cong suc (+-identityʳ n)

+

+
+-identity : Identity 0 _+_

+
+-identity = +-identityˡ , +-identityʳ

+

+
+-comm : Commutative _+_

+
+-comm zero    n = sym (+-identityʳ n)

+
+-comm (suc m) n = begin-equality

+
  suc m + n   ≡⟨⟩

+
  suc (m + n) ≡⟨ cong suc (+-comm m n) ⟩

+
  suc (n + m) ≡⟨ sym (+-suc n m) ⟩

+
  n + suc m   ∎

+

+
+-cancelˡ-≡ : LeftCancellative _≡_ _+_

+
+-cancelˡ-≡ zero    _ _ eq = eq

+
+-cancelˡ-≡ (suc m) _ _ eq = +-cancelˡ-≡ m _ _ (cong pred eq)

+

+
+-cancelʳ-≡ : RightCancellative _≡_ _+_

+
+-cancelʳ-≡ = comm+cancelˡ⇒cancelʳ +-comm +-cancelˡ-≡

+

+
+-cancel-≡ : Cancellative _≡_ _+_

+
+-cancel-≡ = +-cancelˡ-≡ , +-cancelʳ-≡

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
+-isMagma : IsMagma _+_

+
+-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _+_

+
  }

+

+
+-isSemigroup : IsSemigroup _+_

+
+-isSemigroup = record

+
  { isMagma = +-isMagma

+
  ; assoc   = +-assoc

+
  }

+

+
+-isCommutativeSemigroup : IsCommutativeSemigroup _+_

+
+-isCommutativeSemigroup = record

+
  { isSemigroup = +-isSemigroup

+
  ; comm        = +-comm

+
  }

+

+
+-0-isMonoid : IsMonoid _+_ 0

+
+-0-isMonoid = record

+
  { isSemigroup = +-isSemigroup

+
  ; identity    = +-identity

+
  }

+

+
+-0-isCommutativeMonoid : IsCommutativeMonoid _+_ 0

+
+-0-isCommutativeMonoid = record

+
  { isMonoid = +-0-isMonoid

+
  ; comm     = +-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
+-magma : Magma 0ℓ 0ℓ

+
+-magma = record

+
  { isMagma = +-isMagma

+
  }

+

+
+-semigroup : Semigroup 0ℓ 0ℓ

+
+-semigroup = record

+
  { isSemigroup = +-isSemigroup

+
  }

+

+
+-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ

+
+-commutativeSemigroup = record

+
  { isCommutativeSemigroup = +-isCommutativeSemigroup

+
  }

+

+
+-0-monoid : Monoid 0ℓ 0ℓ

+
+-0-monoid = record

+
  { isMonoid = +-0-isMonoid

+
  }

+

+
+-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
+-0-commutativeMonoid = record

+
  { isCommutativeMonoid = +-0-isCommutativeMonoid

+
  }

+

+
∸-magma : Magma 0ℓ 0ℓ

+
∸-magma = magma _∸_

+

+

+
------------------------------------------------------------------------

+
-- Other properties of _+_ and _≡_

+

+
m≢1+m+n : ∀ m {n} → m ≢ suc (m + n)

+
m≢1+m+n (suc m) eq = m≢1+m+n m (cong pred eq)

+

+
m≢1+n+m : ∀ m {n} → m ≢ suc (n + m)

+
m≢1+n+m m m≡1+n+m = m≢1+m+n m (trans m≡1+n+m (cong suc (+-comm _ m)))

+

+
m+1+n≢m : ∀ m {n} → m + suc n ≢ m

+
m+1+n≢m (suc m) = (m+1+n≢m m) ∘ suc-injective

+

+
m+1+n≢n : ∀ m {n} → m + suc n ≢ n

+
m+1+n≢n m {n} rewrite +-suc m n = ≢-sym (m≢1+n+m n)

+

+
m+1+n≢0 : ∀ m {n} → m + suc n ≢ 0

+
m+1+n≢0 m {n} rewrite +-suc m n = λ()

+

+
m+n≡0⇒m≡0 : ∀ m {n} → m + n ≡ 0 → m ≡ 0

+
m+n≡0⇒m≡0 zero eq = refl

+

+
m+n≡0⇒n≡0 : ∀ m {n} → m + n ≡ 0 → n ≡ 0

+
m+n≡0⇒n≡0 m {n} m+n≡0 = m+n≡0⇒m≡0 n (trans (+-comm n m) (m+n≡0))

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _≤_/_<_

+

+
+-cancelˡ-≤ : LeftCancellative _≤_ _+_

+
+-cancelˡ-≤ zero    _ _ le       = le

+
+-cancelˡ-≤ (suc m) _ _ (s≤s le) = +-cancelˡ-≤ m _ _ le

+

+
+-cancelʳ-≤ : RightCancellative _≤_ _+_

+
+-cancelʳ-≤ m n o le =

+
  +-cancelˡ-≤ m _ _ (subst₂ _≤_ (+-comm n m) (+-comm o m) le)

+

+
+-cancel-≤ : Cancellative _≤_ _+_

+
+-cancel-≤ = +-cancelˡ-≤ , +-cancelʳ-≤

+

+
+-cancelˡ-< : LeftCancellative _<_ _+_

+
+-cancelˡ-< m n o = +-cancelˡ-≤ m (suc n) o ∘ subst (_≤ m + o) (sym (+-suc m n))

+

+
+-cancelʳ-< : RightCancellative _<_ _+_

+
+-cancelʳ-< m n o n+m<o+m = +-cancelʳ-≤ m (suc n) o n+m<o+m

+

+
+-cancel-< : Cancellative _<_ _+_

+
+-cancel-< = +-cancelˡ-< , +-cancelʳ-<

+

+
m≤n⇒m≤o+n : ∀ o → m ≤ n → m ≤ o + n

+
m≤n⇒m≤o+n zero    m≤n = m≤n

+
m≤n⇒m≤o+n (suc o) m≤n = m≤n⇒m≤1+n (m≤n⇒m≤o+n o m≤n)

+

+
m≤n⇒m≤n+o : ∀ o → m ≤ n → m ≤ n + o

+
m≤n⇒m≤n+o {m} o m≤n = subst (m ≤_) (+-comm o _) (m≤n⇒m≤o+n o m≤n)

+

+
m≤m+n : ∀ m n → m ≤ m + n

+
m≤m+n zero    n = z≤n

+
m≤m+n (suc m) n = s≤s (m≤m+n m n)

+

+
m≤n+m : ∀ m n → m ≤ n + m

+
m≤n+m m n = subst (m ≤_) (+-comm m n) (m≤m+n m n)

+

+
m+n≤o⇒m≤o : ∀ m {n o} → m + n ≤ o → m ≤ o

+
m+n≤o⇒m≤o zero    m+n≤o       = z≤n

+
m+n≤o⇒m≤o (suc m) (s≤s m+n≤o) = s≤s (m+n≤o⇒m≤o m m+n≤o)

+

+
m+n≤o⇒n≤o : ∀ m {n o} → m + n ≤ o → n ≤ o

+
m+n≤o⇒n≤o zero    n≤o   = n≤o

+
m+n≤o⇒n≤o (suc m) m+n<o = m+n≤o⇒n≤o m (<⇒≤ m+n<o)

+

+
+-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
+-mono-≤ {_} {m} z≤n       o≤p = ≤-trans o≤p (m≤n+m _ m)

+
+-mono-≤ {_} {_} (s≤s m≤n) o≤p = s≤s (+-mono-≤ m≤n o≤p)

+

+
+-monoˡ-≤ : ∀ n → (_+ n) Preserves _≤_ ⟶ _≤_

+
+-monoˡ-≤ n m≤o = +-mono-≤ m≤o (≤-refl {n})

+

+
+-monoʳ-≤ : ∀ n → (n +_) Preserves _≤_ ⟶ _≤_

+
+-monoʳ-≤ n m≤o = +-mono-≤ (≤-refl {n}) m≤o

+

+
+-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_

+
+-mono-<-≤ {_} {suc n} z<s               o≤p = s≤s (m≤n⇒m≤o+n n o≤p)

+
+-mono-<-≤ {_} {_}     (s<s m<n@(s≤s _)) o≤p = s≤s (+-mono-<-≤ m<n o≤p)

+

+
+-mono-≤-< : _+_ Preserves₂ _≤_ ⟶ _<_ ⟶ _<_

+
+-mono-≤-< {_} {n} z≤n       o<p = ≤-trans o<p (m≤n+m _ n)

+
+-mono-≤-< {_} {_} (s≤s m≤n) o<p = s≤s (+-mono-≤-< m≤n o<p)

+

+
+-mono-< : _+_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
+-mono-< m≤n = +-mono-≤-< (<⇒≤ m≤n)

+

+
+-monoˡ-< : ∀ n → (_+ n) Preserves _<_ ⟶ _<_

+
+-monoˡ-< n = +-monoˡ-≤ n

+

+
+-monoʳ-< : ∀ n → (n +_) Preserves _<_ ⟶ _<_

+
+-monoʳ-< zero    m≤o = m≤o

+
+-monoʳ-< (suc n) m≤o = s≤s (+-monoʳ-< n m≤o)

+

+
m+1+n≰m : ∀ m {n} → m + suc n ≰ m

+
m+1+n≰m (suc m) m+1+n≤m = m+1+n≰m m (s≤s⁻¹ m+1+n≤m)

+

+
m<m+n : ∀ m {n} → n > 0 → m < m + n

+
m<m+n zero    n>0 = n>0

+
m<m+n (suc m) n>0 = s<s (m<m+n m n>0)

+

+
m<n+m : ∀ m {n} → n > 0 → m < n + m

+
m<n+m m {n} n>0 rewrite +-comm n m = m<m+n m n>0

+

+
m+n≮n : ∀ m n → m + n ≮ n

+
m+n≮n zero    n                = n≮n n

+
m+n≮n (suc m) n@(suc _) sm+n<n = m+n≮n m n (m<n⇒m<1+n (s<s⁻¹ sm+n<n))

+

+
m+n≮m : ∀ m n → m + n ≮ m

+
m+n≮m m n = subst (_≮ m) (+-comm n m) (m+n≮n n m)

+

+
------------------------------------------------------------------------

+
-- Properties of _*_

+
------------------------------------------------------------------------

+

+
*-suc : ∀ m n → m * suc n ≡ m + m * n

+
*-suc zero    n = refl

+
*-suc (suc m) n = begin-equality

+
  suc m * suc n         ≡⟨⟩

+
  suc n + m * suc n     ≡⟨ cong (suc n +_) (*-suc m n) ⟩

+
  suc n + (m + m * n)   ≡⟨⟩

+
  suc (n + (m + m * n)) ≡⟨ cong suc (sym (+-assoc n m (m * n))) ⟩

+
  suc (n + m + m * n)   ≡⟨ cong (λ x → suc (x + m * n)) (+-comm n m) ⟩

+
  suc (m + n + m * n)   ≡⟨ cong suc (+-assoc m n (m * n)) ⟩

+
  suc (m + (n + m * n)) ≡⟨⟩

+
  suc m + suc m * n     ∎

+

+
------------------------------------------------------------------------

+
-- Algebraic properties of _*_

+

+
*-identityˡ : LeftIdentity 1 _*_

+
*-identityˡ n = +-identityʳ n

+

+
*-identityʳ : RightIdentity 1 _*_

+
*-identityʳ zero    = refl

+
*-identityʳ (suc n) = cong suc (*-identityʳ n)

+

+
*-identity : Identity 1 _*_

+
*-identity = *-identityˡ , *-identityʳ

+

+
*-zeroˡ : LeftZero 0 _*_

+
*-zeroˡ _ = refl

+

+
*-zeroʳ : RightZero 0 _*_

+
*-zeroʳ zero    = refl

+
*-zeroʳ (suc n) = *-zeroʳ n

+

+
*-zero : Zero 0 _*_

+
*-zero = *-zeroˡ , *-zeroʳ

+

+
*-comm : Commutative _*_

+
*-comm zero    n = sym (*-zeroʳ n)

+
*-comm (suc m) n = begin-equality

+
  suc m * n  ≡⟨⟩

+
  n + m * n  ≡⟨ cong (n +_) (*-comm m n) ⟩

+
  n + n * m  ≡⟨ sym (*-suc n m) ⟩

+
  n * suc m  ∎

+

+
*-distribʳ-+ : _*_ DistributesOverʳ _+_

+
*-distribʳ-+ m zero    o = refl

+
*-distribʳ-+ m (suc n) o = begin-equality

+
  (suc n + o) * m     ≡⟨⟩

+
  m + (n + o) * m     ≡⟨ cong (m +_) (*-distribʳ-+ m n o) ⟩

+
  m + (n * m + o * m) ≡⟨ sym (+-assoc m (n * m) (o * m)) ⟩

+
  m + n * m + o * m   ≡⟨⟩

+
  suc n * m + o * m   ∎

+

+
*-distribˡ-+ : _*_ DistributesOverˡ _+_

+
*-distribˡ-+ = comm∧distrʳ⇒distrˡ *-comm *-distribʳ-+

+

+
*-distrib-+ : _*_ DistributesOver _+_

+
*-distrib-+ = *-distribˡ-+ , *-distribʳ-+

+

+
*-assoc : Associative _*_

+
*-assoc zero    n o = refl

+
*-assoc (suc m) n o = begin-equality

+
  (suc m * n) * o     ≡⟨⟩

+
  (n + m * n) * o     ≡⟨ *-distribʳ-+ o n (m * n) ⟩

+
  n * o + (m * n) * o ≡⟨ cong (n * o +_) (*-assoc m n o) ⟩

+
  n * o + m * (n * o) ≡⟨⟩

+
  suc m * (n * o)     ∎

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
*-isMagma : IsMagma _*_

+
*-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _*_

+
  }

+

+
*-isSemigroup : IsSemigroup _*_

+
*-isSemigroup = record

+
  { isMagma = *-isMagma

+
  ; assoc   = *-assoc

+
  }

+

+
*-isCommutativeSemigroup : IsCommutativeSemigroup _*_

+
*-isCommutativeSemigroup = record

+
  { isSemigroup = *-isSemigroup

+
  ; comm        = *-comm

+
  }

+

+
*-1-isMonoid : IsMonoid _*_ 1

+
*-1-isMonoid = record

+
  { isSemigroup = *-isSemigroup

+
  ; identity    = *-identity

+
  }

+

+
*-1-isCommutativeMonoid : IsCommutativeMonoid _*_ 1

+
*-1-isCommutativeMonoid = record

+
  { isMonoid = *-1-isMonoid

+
  ; comm     = *-comm

+
  }

+

+
+-*-isSemiring : IsSemiring _+_ _*_ 0 1

+
+-*-isSemiring = record

+
  { isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = +-0-isCommutativeMonoid

+
    ; *-cong                = cong₂ _*_

+
    ; *-assoc               = *-assoc

+
    ; *-identity            = *-identity

+
    ; distrib               = *-distrib-+

+
    }

+
  ; zero = *-zero

+
  }

+

+
+-*-isCommutativeSemiring : IsCommutativeSemiring _+_ _*_ 0 1

+
+-*-isCommutativeSemiring = record

+
  { isSemiring = +-*-isSemiring

+
  ; *-comm     = *-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
*-magma : Magma 0ℓ 0ℓ

+
*-magma = record

+
  { isMagma = *-isMagma

+
  }

+

+
*-semigroup : Semigroup 0ℓ 0ℓ

+
*-semigroup = record

+
  { isSemigroup = *-isSemigroup

+
  }

+

+
*-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ

+
*-commutativeSemigroup = record

+
  { isCommutativeSemigroup = *-isCommutativeSemigroup

+
  }

+

+
*-1-monoid : Monoid 0ℓ 0ℓ

+
*-1-monoid = record

+
  { isMonoid = *-1-isMonoid

+
  }

+

+
*-1-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
*-1-commutativeMonoid = record

+
  { isCommutativeMonoid = *-1-isCommutativeMonoid

+
  }

+

+
+-*-semiring : Semiring 0ℓ 0ℓ

+
+-*-semiring = record

+
  { isSemiring = +-*-isSemiring

+
  }

+

+
+-*-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ

+
+-*-commutativeSemiring = record

+
  { isCommutativeSemiring = +-*-isCommutativeSemiring

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _*_ and _≡_

+

+
*-cancelʳ-≡ : ∀ m n o .{{_ : NonZero o}} → m * o ≡ n * o → m ≡ n

+
*-cancelʳ-≡ zero    zero    (suc o) eq = refl

+
*-cancelʳ-≡ (suc m) (suc n) (suc o) eq =

+
  cong suc (*-cancelʳ-≡ m n (suc o) (+-cancelˡ-≡ (suc o) (m * suc o) (n * suc o) eq))

+

+
*-cancelˡ-≡ : ∀ m n o .{{_ : NonZero o}} → o * m ≡ o * n → m ≡ n

+
*-cancelˡ-≡ m n o rewrite *-comm o m | *-comm o n = *-cancelʳ-≡ m n o

+

+
m*n≡0⇒m≡0∨n≡0 : ∀ m {n} → m * n ≡ 0 → m ≡ 0 ⊎ n ≡ 0

+
m*n≡0⇒m≡0∨n≡0 zero    {n}     eq = inj₁ refl

+
m*n≡0⇒m≡0∨n≡0 (suc m) {zero}  eq = inj₂ refl

+

+
m*n≢0 : ∀ m n .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)

+
m*n≢0 (suc m) (suc n) = _

+

+
m*n≢0⇒m≢0 : ∀ m {n} → .{{NonZero (m * n)}} → NonZero m

+
m*n≢0⇒m≢0 (suc _) = _

+

+
m*n≢0⇒n≢0 : ∀ m {n} → .{{NonZero (m * n)}} → NonZero n

+
m*n≢0⇒n≢0 m {n} rewrite *-comm m n = m*n≢0⇒m≢0 n {m}

+

+
m*n≡0⇒m≡0 : ∀ m n .{{_ : NonZero n}} → m * n ≡ 0 → m ≡ 0

+
m*n≡0⇒m≡0 zero (suc _) eq = refl

+

+
m*n≡1⇒m≡1 : ∀ m n → m * n ≡ 1 → m ≡ 1

+
m*n≡1⇒m≡1 (suc zero)    n          _  = refl

+
m*n≡1⇒m≡1 (suc (suc m)) (suc zero) ()

+
m*n≡1⇒m≡1 (suc (suc m)) zero       eq =

+
  contradiction (trans (sym $ *-zeroʳ m) eq) λ()

+

+
m*n≡1⇒n≡1 : ∀ m n → m * n ≡ 1 → n ≡ 1

+
m*n≡1⇒n≡1 m n eq = m*n≡1⇒m≡1 n m (trans (*-comm n m) eq)

+

+
[m*n]*[o*p]≡[m*o]*[n*p] : ∀ m n o p → (m * n) * (o * p) ≡ (m * o) * (n * p)

+
[m*n]*[o*p]≡[m*o]*[n*p] m n o p = begin-equality

+
  (m * n) * (o * p) ≡⟨  *-assoc m n (o * p) ⟩

+
  m * (n * (o * p)) ≡⟨  cong (m *_) (x∙yz≈y∙xz n o p) ⟩

+
  m * (o * (n * p)) ≡⟨ *-assoc m o (n * p) ⟨

+
  (m * o) * (n * p) ∎

+
  where open CommSemigroupProperties *-commutativeSemigroup

+

+
m≢0∧n>1⇒m*n>1 : ∀ m n .{{_ : NonZero m}} .{{_ : NonTrivial n}} → NonTrivial (m * n)

+
m≢0∧n>1⇒m*n>1 (suc m) (2+ n) = _

+

+
n≢0∧m>1⇒m*n>1 : ∀ m n .{{_ : NonZero n}} .{{_ : NonTrivial m}} → NonTrivial (m * n)

+
n≢0∧m>1⇒m*n>1 m n rewrite *-comm m n = m≢0∧n>1⇒m*n>1 n m

+

+
------------------------------------------------------------------------

+
-- Other properties of _*_ and _≤_/_<_

+

+
*-cancelʳ-≤ : ∀ m n o .{{_ : NonZero o}} → m * o ≤ n * o → m ≤ n

+
*-cancelʳ-≤ zero    _       _         _  = z≤n

+
*-cancelʳ-≤ (suc m) (suc n) o@(suc _) le =

+
  s≤s (*-cancelʳ-≤ m n o (+-cancelˡ-≤ _ _ _ le))

+

+
*-cancelˡ-≤ : ∀ o .{{_ : NonZero o}} → o * m ≤ o * n → m ≤ n

+
*-cancelˡ-≤ {m} {n} o rewrite *-comm o m | *-comm o n = *-cancelʳ-≤ m n o

+

+
*-mono-≤ : _*_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
*-mono-≤ z≤n       _   = z≤n

+
*-mono-≤ (s≤s m≤n) u≤v = +-mono-≤ u≤v (*-mono-≤ m≤n u≤v)

+

+
*-monoˡ-≤ : ∀ n → (_* n) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤ n m≤o = *-mono-≤ m≤o (≤-refl {n})

+

+
*-monoʳ-≤ : ∀ n → (n *_) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤ n m≤o = *-mono-≤ (≤-refl {n}) m≤o

+

+
*-mono-< : _*_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
*-mono-< z<s               u<v@(s≤s _) = 0<1+n

+
*-mono-< (s<s m<n@(s≤s _)) u<v@(s≤s _) = +-mono-< u<v (*-mono-< m<n u<v)

+

+
*-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_* n) Preserves _<_ ⟶ _<_

+
*-monoˡ-< n@(suc _) z<s               = 0<1+n

+
*-monoˡ-< n@(suc _) (s<s m<o@(s≤s _)) = +-mono-≤-< ≤-refl (*-monoˡ-< n m<o)

+

+
*-monoʳ-< : ∀ n .{{_ : NonZero n}} → (n *_) Preserves _<_ ⟶ _<_

+
*-monoʳ-< (suc zero)      m<o@(s≤s _) = +-mono-≤ m<o z≤n

+
*-monoʳ-< (suc n@(suc _)) m<o@(s≤s _) = +-mono-≤ m<o (<⇒≤ (*-monoʳ-< n m<o))

+

+
m≤m*n : ∀ m n .{{_ : NonZero n}} → m ≤ m * n

+
m≤m*n m n@(suc _) = begin

+
  m     ≡⟨ sym (*-identityʳ m) ⟩

+
  m * 1 ≤⟨ *-monoʳ-≤ m 0<1+n ⟩

+
  m * n ∎

+

+
m≤n*m : ∀ m n .{{_ : NonZero n}} → m ≤ n * m

+
m≤n*m m n@(suc _) = begin

+
  m     ≤⟨ m≤m*n m n ⟩

+
  m * n ≡⟨ *-comm m n ⟩

+
  n * m ∎

+

+
m<m*n : ∀ m n .{{_ : NonZero m}} → 1 < n → m < m * n

+
m<m*n m@(suc m-1) n@(suc (suc n-2)) (s≤s (s≤s _)) = begin-strict

+
  m           <⟨ s≤s (s≤s (m≤n+m m-1 n-2)) ⟩

+
  n + m-1     ≤⟨ +-monoʳ-≤ n (m≤m*n m-1 n) ⟩

+
  n + m-1 * n ≡⟨⟩

+
  m * n       ∎

+

+
m<n⇒m<n*o : ∀ o .{{_ : NonZero o}} → m < n → m < n * o

+
m<n⇒m<n*o {n = n} o m<n = <-≤-trans m<n (m≤m*n n o)

+

+
m<n⇒m<o*n : ∀ {m n} o .{{_ : NonZero o}} → m < n → m < o * n

+
m<n⇒m<o*n {m} {n} o m<n = begin-strict

+
  m     <⟨ m<n⇒m<n*o o m<n ⟩

+
  n * o ≡⟨ *-comm n o ⟩

+
  o * n ∎

+

+
*-cancelʳ-< : RightCancellative _<_ _*_

+
*-cancelʳ-< zero    zero    (suc o) _     = 0<1+n

+
*-cancelʳ-< (suc m) zero    (suc o) _     = 0<1+n

+
*-cancelʳ-< m       (suc n) (suc o) nm<om =

+
  s≤s (*-cancelʳ-< m n o (+-cancelˡ-< m _ _ nm<om))

+

+
*-cancelˡ-< : LeftCancellative _<_ _*_

+
*-cancelˡ-< x y z rewrite *-comm x y | *-comm x z = *-cancelʳ-< x y z

+

+
*-cancel-< : Cancellative _<_ _*_

+
*-cancel-< = *-cancelˡ-< , *-cancelʳ-<

+

+
------------------------------------------------------------------------

+
-- Properties of _^_

+
------------------------------------------------------------------------

+

+
^-identityʳ : RightIdentity 1 _^_

+
^-identityʳ zero    = refl

+
^-identityʳ (suc n) = cong suc (^-identityʳ n)

+

+
^-zeroˡ : LeftZero 1 _^_

+
^-zeroˡ zero    = refl

+
^-zeroˡ (suc n) = begin-equality

+
  1 ^ suc n   ≡⟨⟩

+
  1 * (1 ^ n) ≡⟨ *-identityˡ (1 ^ n) ⟩

+
  1 ^ n       ≡⟨ ^-zeroˡ n ⟩

+
  1           ∎

+

+
^-distribˡ-+-* : ∀ m n o → m ^ (n + o) ≡ m ^ n * m ^ o

+
^-distribˡ-+-* m zero    o = sym (+-identityʳ (m ^ o))

+
^-distribˡ-+-* m (suc n) o = begin-equality

+
  m * (m ^ (n + o))       ≡⟨ cong (m *_) (^-distribˡ-+-* m n o) ⟩

+
  m * ((m ^ n) * (m ^ o)) ≡⟨ sym (*-assoc m _ _) ⟩

+
  (m * (m ^ n)) * (m ^ o) ∎

+

+
^-semigroup-morphism : ∀ {n} → (n ^_) Is +-semigroup -Semigroup⟶ *-semigroup

+
^-semigroup-morphism = record

+
  { ⟦⟧-cong = cong (_ ^_)

+
  ; ∙-homo  = ^-distribˡ-+-* _

+
  }

+

+
^-monoid-morphism : ∀ {n} → (n ^_) Is +-0-monoid -Monoid⟶ *-1-monoid

+
^-monoid-morphism = record

+
  { sm-homo = ^-semigroup-morphism

+
  ; ε-homo  = refl

+
  }

+

+
^-*-assoc : ∀ m n o → (m ^ n) ^ o ≡ m ^ (n * o)

+
^-*-assoc m n zero    = cong (m ^_) (sym $ *-zeroʳ n)

+
^-*-assoc m n (suc o) = begin-equality

+
  (m ^ n) * ((m ^ n) ^ o) ≡⟨ cong ((m ^ n) *_) (^-*-assoc m n o) ⟩

+
  (m ^ n) * (m ^ (n * o)) ≡⟨ sym (^-distribˡ-+-* m n (n * o)) ⟩

+
  m ^ (n + n * o)         ≡⟨ cong (m ^_) (sym (*-suc n o)) ⟩

+
  m ^ (n * (suc o)) ∎

+

+
m^n≡0⇒m≡0 : ∀ m n → m ^ n ≡ 0 → m ≡ 0

+
m^n≡0⇒m≡0 m (suc n) eq = [ id , m^n≡0⇒m≡0 m n ]′ (m*n≡0⇒m≡0∨n≡0 m eq)

+

+
m^n≡1⇒n≡0∨m≡1 : ∀ m n → m ^ n ≡ 1 → n ≡ 0 ⊎ m ≡ 1

+
m^n≡1⇒n≡0∨m≡1 m zero    _  = inj₁ refl

+
m^n≡1⇒n≡0∨m≡1 m (suc n) eq = inj₂ (m*n≡1⇒m≡1 m (m ^ n) eq)

+

+
m^n≢0 : ∀ m n .{{_ : NonZero m}} → NonZero (m ^ n)

+
m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)

+

+
m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0

+
m^n>0 m n = >-nonZero⁻¹ (m ^ n) {{m^n≢0 m n}}

+

+
^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_

+
^-monoˡ-≤ zero m≤o = s≤s z≤n

+
^-monoˡ-≤ (suc n) m≤o = *-mono-≤ m≤o (^-monoˡ-≤ n m≤o)

+

+
^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_

+
^-monoʳ-≤ m {_} {o} z≤n = n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{m^n≢0 m o}})

+
^-monoʳ-≤ m (s≤s n≤o) = *-monoʳ-≤ m (^-monoʳ-≤ m n≤o)

+

+
^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_

+
^-monoˡ-< (suc zero)      m<o = *-monoˡ-< 1 m<o

+
^-monoˡ-< (suc n@(suc _)) m<o = *-mono-< m<o (^-monoˡ-< n m<o)

+

+
^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_

+
^-monoʳ-< m@(suc _) 1<m {zero}  {suc o} z<s       = *-mono-≤ 1<m (m^n>0 m o)

+
^-monoʳ-< m@(suc _) 1<m {suc n} {suc o} (s<s n<o) = *-monoʳ-< m (^-monoʳ-< m 1<m n<o)

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_ and _⊔_

+
------------------------------------------------------------------------

+
-- Basic specification in terms of _≤_

+

+
m≤n⇒m⊔n≡n : m ≤ n → m ⊔ n ≡ n

+
m≤n⇒m⊔n≡n {zero}  _         = refl

+
m≤n⇒m⊔n≡n {suc m} (s≤s m≤n) = cong suc (m≤n⇒m⊔n≡n m≤n)

+

+
m≥n⇒m⊔n≡m : m ≥ n → m ⊔ n ≡ m

+
m≥n⇒m⊔n≡m {zero}  {zero}  z≤n       = refl

+
m≥n⇒m⊔n≡m {suc m} {zero}  z≤n       = refl

+
m≥n⇒m⊔n≡m {suc m} {suc n} (s≤s m≥n) = cong suc (m≥n⇒m⊔n≡m m≥n)

+

+
m≤n⇒m⊓n≡m : m ≤ n → m ⊓ n ≡ m

+
m≤n⇒m⊓n≡m {zero}  z≤n       = refl

+
m≤n⇒m⊓n≡m {suc m} (s≤s m≤n) = cong suc (m≤n⇒m⊓n≡m m≤n)

+

+
m≥n⇒m⊓n≡n : m ≥ n → m ⊓ n ≡ n

+
m≥n⇒m⊓n≡n {zero}  {zero}  z≤n       = refl

+
m≥n⇒m⊓n≡n {suc m} {zero}  z≤n       = refl

+
m≥n⇒m⊓n≡n {suc m} {suc n} (s≤s m≤n) = cong suc (m≥n⇒m⊓n≡n m≤n)

+

+
⊓-operator : MinOperator ≤-totalPreorder

+
⊓-operator = record

+
  { x≤y⇒x⊓y≈x = m≤n⇒m⊓n≡m

+
  ; x≥y⇒x⊓y≈y = m≥n⇒m⊓n≡n

+
  }

+

+
⊔-operator : MaxOperator ≤-totalPreorder

+
⊔-operator = record

+
  { x≤y⇒x⊔y≈y = m≤n⇒m⊔n≡n

+
  ; x≥y⇒x⊔y≈x = m≥n⇒m⊔n≡m

+
  }

+

+
------------------------------------------------------------------------

+
-- Equality to their counterparts defined in terms of primitive operations

+

+
⊔≡⊔′ : ∀ m n → m ⊔ n ≡ m ⊔′ n

+
⊔≡⊔′ m n with m <ᵇ n in eq

+
... | false = m≥n⇒m⊔n≡m (≮⇒≥ (λ m<n → subst T eq (<⇒<ᵇ m<n)))

+
... | true  = m≤n⇒m⊔n≡n (<⇒≤ (<ᵇ⇒< m n (subst T (sym eq) _)))

+

+
⊓≡⊓′ : ∀ m n → m ⊓ n ≡ m ⊓′ n

+
⊓≡⊓′ m n with m <ᵇ n in eq

+
... | false = m≥n⇒m⊓n≡n (≮⇒≥ (λ m<n → subst T eq (<⇒<ᵇ m<n)))

+
... | true  = m≤n⇒m⊓n≡m (<⇒≤ (<ᵇ⇒< m n (subst T (sym eq) _)))

+

+
------------------------------------------------------------------------

+
-- Derived properties of _⊓_ and _⊔_

+

+
private

+
  module ⊓-⊔-properties        = MinMaxOp        ⊓-operator ⊔-operator

+
  module ⊓-⊔-latticeProperties = LatticeMinMaxOp ⊓-operator ⊔-operator

+

+
open ⊓-⊔-properties public

+
  using

+
  ( ⊓-idem                    -- : Idempotent _⊓_

+
  ; ⊓-sel                     -- : Selective _⊓_

+
  ; ⊓-assoc                   -- : Associative _⊓_

+
  ; ⊓-comm                    -- : Commutative _⊓_

+

+
  ; ⊔-idem                    -- : Idempotent _⊔_

+
  ; ⊔-sel                     -- : Selective _⊔_

+
  ; ⊔-assoc                   -- : Associative _⊔_

+
  ; ⊔-comm                    -- : Commutative _⊔_

+

+
  ; ⊓-distribˡ-⊔              -- : _⊓_ DistributesOverˡ _⊔_

+
  ; ⊓-distribʳ-⊔              -- : _⊓_ DistributesOverʳ _⊔_

+
  ; ⊓-distrib-⊔               -- : _⊓_ DistributesOver  _⊔_

+
  ; ⊔-distribˡ-⊓              -- : _⊔_ DistributesOverˡ _⊓_

+
  ; ⊔-distribʳ-⊓              -- : _⊔_ DistributesOverʳ _⊓_

+
  ; ⊔-distrib-⊓               -- : _⊔_ DistributesOver  _⊓_

+
  ; ⊓-absorbs-⊔               -- : _⊓_ Absorbs _⊔_

+
  ; ⊔-absorbs-⊓               -- : _⊔_ Absorbs _⊓_

+
  ; ⊔-⊓-absorptive            -- : Absorptive _⊔_ _⊓_

+
  ; ⊓-⊔-absorptive            -- : Absorptive _⊓_ _⊔_

+

+
  ; ⊓-isMagma                 -- : IsMagma _⊓_

+
  ; ⊓-isSemigroup             -- : IsSemigroup _⊓_

+
  ; ⊓-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊓_

+
  ; ⊓-isBand                  -- : IsBand _⊓_

+
  ; ⊓-isSelectiveMagma        -- : IsSelectiveMagma _⊓_

+

+
  ; ⊔-isMagma                 -- : IsMagma _⊔_

+
  ; ⊔-isSemigroup             -- : IsSemigroup _⊔_

+
  ; ⊔-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊔_

+
  ; ⊔-isBand                  -- : IsBand _⊔_

+
  ; ⊔-isSelectiveMagma        -- : IsSelectiveMagma _⊔_

+

+
  ; ⊓-magma                   -- : Magma _ _

+
  ; ⊓-semigroup               -- : Semigroup _ _

+
  ; ⊓-band                    -- : Band _ _

+
  ; ⊓-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊓-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊔-magma                   -- : Magma _ _

+
  ; ⊔-semigroup               -- : Semigroup _ _

+
  ; ⊔-band                    -- : Band _ _

+
  ; ⊔-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊔-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊓-glb                     -- : ∀ {m n o} → m ≥ o → n ≥ o → m ⊓ n ≥ o

+
  ; ⊓-triangulate             -- : ∀ m n o → m ⊓ n ⊓ o ≡ (m ⊓ n) ⊓ (n ⊓ o)

+
  ; ⊓-mono-≤                  -- : _⊓_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊓-monoˡ-≤                 -- : ∀ n → (_⊓ n) Preserves _≤_ ⟶ _≤_

+
  ; ⊓-monoʳ-≤                 -- : ∀ n → (n ⊓_) Preserves _≤_ ⟶ _≤_

+

+
  ; ⊔-lub                     -- : ∀ {m n o} → m ≤ o → n ≤ o → m ⊔ n ≤ o

+
  ; ⊔-triangulate             -- : ∀ m n o → m ⊔ n ⊔ o ≡ (m ⊔ n) ⊔ (n ⊔ o)

+
  ; ⊔-mono-≤                  -- : _⊔_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊔-monoˡ-≤                 -- : ∀ n → (_⊔ n) Preserves _≤_ ⟶ _≤_

+
  ; ⊔-monoʳ-≤                 -- : ∀ n → (n ⊔_) Preserves _≤_ ⟶ _≤_

+
  )

+
  renaming

+
  ( x⊓y≈y⇒y≤x to m⊓n≡n⇒n≤m    -- : ∀ {m n} → m ⊓ n ≡ n → n ≤ m

+
  ; x⊓y≈x⇒x≤y to m⊓n≡m⇒m≤n    -- : ∀ {m n} → m ⊓ n ≡ m → m ≤ n

+
  ; x⊓y≤x     to m⊓n≤m        -- : ∀ m n → m ⊓ n ≤ m

+
  ; x⊓y≤y     to m⊓n≤n        -- : ∀ m n → m ⊓ n ≤ n

+
  ; x≤y⇒x⊓z≤y to m≤n⇒m⊓o≤n    -- : ∀ {m n} o → m ≤ n → m ⊓ o ≤ n

+
  ; x≤y⇒z⊓x≤y to m≤n⇒o⊓m≤n    -- : ∀ {m n} o → m ≤ n → o ⊓ m ≤ n

+
  ; x≤y⊓z⇒x≤y to m≤n⊓o⇒m≤n    -- : ∀ {m} n o → m ≤ n ⊓ o → m ≤ n

+
  ; x≤y⊓z⇒x≤z to m≤n⊓o⇒m≤o    -- : ∀ {m} n o → m ≤ n ⊓ o → m ≤ o

+

+
  ; x⊔y≈y⇒x≤y to m⊔n≡n⇒m≤n    -- : ∀ {m n} → m ⊔ n ≡ n → m ≤ n

+
  ; x⊔y≈x⇒y≤x to m⊔n≡m⇒n≤m    -- : ∀ {m n} → m ⊔ n ≡ m → n ≤ m

+
  ; x≤x⊔y     to m≤m⊔n        -- : ∀ m n → m ≤ m ⊔ n

+
  ; x≤y⊔x     to m≤n⊔m        -- : ∀ m n → m ≤ n ⊔ m

+
  ; x≤y⇒x≤y⊔z to m≤n⇒m≤n⊔o    -- : ∀ {m n} o → m ≤ n → m ≤ n ⊔ o

+
  ; x≤y⇒x≤z⊔y to m≤n⇒m≤o⊔n    -- : ∀ {m n} o → m ≤ n → m ≤ o ⊔ n

+
  ; x⊔y≤z⇒x≤z to m⊔n≤o⇒m≤o    -- : ∀ m n {o} → m ⊔ n ≤ o → m ≤ o

+
  ; x⊔y≤z⇒y≤z to m⊔n≤o⇒n≤o    -- : ∀ m n {o} → m ⊔ n ≤ o → n ≤ o

+

+
  ; x⊓y≤x⊔y   to m⊓n≤m⊔n      -- : ∀ m n → m ⊓ n ≤ m ⊔ n

+
  )

+

+
open ⊓-⊔-latticeProperties public

+
  using

+
  ( ⊓-isSemilattice           -- : IsSemilattice _⊓_

+
  ; ⊔-isSemilattice           -- : IsSemilattice _⊔_

+
  ; ⊔-⊓-isLattice             -- : IsLattice _⊔_ _⊓_

+
  ; ⊓-⊔-isLattice             -- : IsLattice _⊓_ _⊔_

+
  ; ⊔-⊓-isDistributiveLattice -- : IsDistributiveLattice _⊔_ _⊓_

+
  ; ⊓-⊔-isDistributiveLattice -- : IsDistributiveLattice _⊓_ _⊔_

+

+
  ; ⊓-semilattice             -- : Semilattice _ _

+
  ; ⊔-semilattice             -- : Semilattice _ _

+
  ; ⊔-⊓-lattice               -- : Lattice _ _

+
  ; ⊓-⊔-lattice               -- : Lattice _ _

+
  ; ⊔-⊓-distributiveLattice   -- : DistributiveLattice _ _

+
  ; ⊓-⊔-distributiveLattice   -- : DistributiveLattice _ _

+
  )

+

+
------------------------------------------------------------------------

+
-- Automatically derived properties of _⊓_ and _⊔_

+

+
⊔-identityˡ : LeftIdentity 0 _⊔_

+
⊔-identityˡ _ = refl

+

+
⊔-identityʳ : RightIdentity 0 _⊔_

+
⊔-identityʳ zero    = refl

+
⊔-identityʳ (suc n) = refl

+

+
⊔-identity : Identity 0 _⊔_

+
⊔-identity = ⊔-identityˡ , ⊔-identityʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
⊔-0-isMonoid : IsMonoid _⊔_ 0

+
⊔-0-isMonoid = record

+
  { isSemigroup = ⊔-isSemigroup

+
  ; identity    = ⊔-identity

+
  }

+

+
⊔-0-isCommutativeMonoid : IsCommutativeMonoid _⊔_ 0

+
⊔-0-isCommutativeMonoid = record

+
  { isMonoid = ⊔-0-isMonoid

+
  ; comm     = ⊔-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
⊔-0-monoid : Monoid 0ℓ 0ℓ

+
⊔-0-monoid = record

+
  { isMonoid = ⊔-0-isMonoid

+
  }

+

+
⊔-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
⊔-0-commutativeMonoid = record

+
  { isCommutativeMonoid = ⊔-0-isCommutativeMonoid

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊔_ and _≤_/_<_

+

+
mono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ m n → f (m ⊔ n) ≡ f m ⊔ f n

+
mono-≤-distrib-⊔ {f} = ⊓-⊔-properties.mono-≤-distrib-⊔ (cong f)

+

+
mono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ m n → f (m ⊓ n) ≡ f m ⊓ f n

+
mono-≤-distrib-⊓ {f} = ⊓-⊔-properties.mono-≤-distrib-⊓ (cong f)

+

+
antimono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ m n → f (m ⊓ n) ≡ f m ⊔ f n

+
antimono-≤-distrib-⊓ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊓ (cong f)

+

+
antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ m n → f (m ⊔ n) ≡ f m ⊓ f n

+
antimono-≤-distrib-⊔ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊔ (cong f)

+

+
m<n⇒m<n⊔o : ∀ o → m < n → m < n ⊔ o

+
m<n⇒m<n⊔o = m≤n⇒m≤n⊔o

+

+
m<n⇒m<o⊔n : ∀ o → m < n → m < o ⊔ n

+
m<n⇒m<o⊔n = m≤n⇒m≤o⊔n

+

+
m⊔n<o⇒m<o : ∀ m n {o} → m ⊔ n < o → m < o

+
m⊔n<o⇒m<o m n m⊔n<o = ≤-<-trans (m≤m⊔n m n) m⊔n<o

+

+
m⊔n<o⇒n<o : ∀ m n {o} → m ⊔ n < o → n < o

+
m⊔n<o⇒n<o m n m⊔n<o = ≤-<-trans (m≤n⊔m m n) m⊔n<o

+

+
⊔-mono-< : _⊔_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
⊔-mono-< = ⊔-mono-≤

+

+
⊔-pres-<m : n < m → o < m → n ⊔ o < m

+
⊔-pres-<m {m = m} n<m o<m = subst (_ <_) (⊔-idem m) (⊔-mono-< n<m o<m)

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊔_ and _+_

+

+
+-distribˡ-⊔ : _+_ DistributesOverˡ _⊔_

+
+-distribˡ-⊔ zero    n o = refl

+
+-distribˡ-⊔ (suc m) n o = cong suc (+-distribˡ-⊔ m n o)

+

+
+-distribʳ-⊔ : _+_ DistributesOverʳ _⊔_

+
+-distribʳ-⊔ = comm∧distrˡ⇒distrʳ +-comm +-distribˡ-⊔

+

+
+-distrib-⊔ : _+_ DistributesOver _⊔_

+
+-distrib-⊔ = +-distribˡ-⊔ , +-distribʳ-⊔

+

+
m⊔n≤m+n : ∀ m n → m ⊔ n ≤ m + n

+
m⊔n≤m+n m n with ⊔-sel m n

+
... | inj₁ m⊔n≡m rewrite m⊔n≡m = m≤m+n m n

+
... | inj₂ m⊔n≡n rewrite m⊔n≡n = m≤n+m n m

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊔_ and _*_

+

+
*-distribˡ-⊔ : _*_ DistributesOverˡ _⊔_

+
*-distribˡ-⊔ m zero o = sym (cong (_⊔ m * o) (*-zeroʳ m))

+
*-distribˡ-⊔ m (suc n) zero = begin-equality

+
  m * (suc n ⊔ zero)         ≡⟨⟩

+
  m * suc n                  ≡⟨ ⊔-identityʳ (m * suc n) ⟨

+
  m * suc n ⊔ zero           ≡⟨ cong (m * suc n ⊔_) (*-zeroʳ m) ⟨

+
  m * suc n ⊔ m * zero       ∎

+
*-distribˡ-⊔ m (suc n) (suc o) = begin-equality

+
  m * (suc n ⊔ suc o)        ≡⟨⟩

+
  m * suc (n ⊔ o)            ≡⟨ *-suc m (n ⊔ o) ⟩

+
  m + m * (n ⊔ o)            ≡⟨ cong (m +_) (*-distribˡ-⊔ m n o) ⟩

+
  m + (m * n ⊔ m * o)        ≡⟨ +-distribˡ-⊔ m (m * n) (m * o) ⟩

+
  (m + m * n) ⊔ (m + m * o)  ≡⟨ cong₂ _⊔_ (*-suc m n) (*-suc m o) ⟨

+
  (m * suc n) ⊔ (m * suc o)  ∎

+

+
*-distribʳ-⊔ : _*_ DistributesOverʳ _⊔_

+
*-distribʳ-⊔ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-⊔

+

+
*-distrib-⊔ : _*_ DistributesOver _⊔_

+
*-distrib-⊔ = *-distribˡ-⊔ , *-distribʳ-⊔

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
⊓-zeroˡ : LeftZero 0 _⊓_

+
⊓-zeroˡ _ = refl

+

+
⊓-zeroʳ : RightZero 0 _⊓_

+
⊓-zeroʳ zero    = refl

+
⊓-zeroʳ (suc n) = refl

+

+
⊓-zero : Zero 0 _⊓_

+
⊓-zero = ⊓-zeroˡ , ⊓-zeroʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
⊔-⊓-isSemiringWithoutOne : IsSemiringWithoutOne _⊔_ _⊓_ 0

+
⊔-⊓-isSemiringWithoutOne = record

+
  { +-isCommutativeMonoid = ⊔-0-isCommutativeMonoid

+
  ; *-cong                = cong₂ _⊓_

+
  ; *-assoc               = ⊓-assoc

+
  ; distrib               = ⊓-distrib-⊔

+
  ; zero                  = ⊓-zero

+
  }

+

+
⊔-⊓-isCommutativeSemiringWithoutOne

+
  : IsCommutativeSemiringWithoutOne _⊔_ _⊓_ 0

+
⊔-⊓-isCommutativeSemiringWithoutOne = record

+
  { isSemiringWithoutOne = ⊔-⊓-isSemiringWithoutOne

+
  ; *-comm               = ⊓-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
⊔-⊓-commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne 0ℓ 0ℓ

+
⊔-⊓-commutativeSemiringWithoutOne = record

+
  { isCommutativeSemiringWithoutOne =

+
      ⊔-⊓-isCommutativeSemiringWithoutOne

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_ and _≤_/_<_

+

+
m<n⇒m⊓o<n : ∀ o → m < n → m ⊓ o < n

+
m<n⇒m⊓o<n o m<n = ≤-<-trans (m⊓n≤m _ o) m<n

+

+
m<n⇒o⊓m<n : ∀ o → m < n → o ⊓ m < n

+
m<n⇒o⊓m<n o m<n = ≤-<-trans (m⊓n≤n o _) m<n

+

+
m<n⊓o⇒m<n : ∀ n o → m < n ⊓ o → m < n

+
m<n⊓o⇒m<n = m≤n⊓o⇒m≤n

+

+
m<n⊓o⇒m<o : ∀ n o → m < n ⊓ o → m < o

+
m<n⊓o⇒m<o = m≤n⊓o⇒m≤o

+

+
⊓-mono-< : _⊓_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
⊓-mono-< = ⊓-mono-≤

+

+
⊓-pres-m< : m < n → m < o → m < n ⊓ o

+
⊓-pres-m< {m} m<n m<o = subst (_< _) (⊓-idem m) (⊓-mono-< m<n m<o)

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_ and _+_

+

+
+-distribˡ-⊓ : _+_ DistributesOverˡ _⊓_

+
+-distribˡ-⊓ zero    n o = refl

+
+-distribˡ-⊓ (suc m) n o = cong suc (+-distribˡ-⊓ m n o)

+

+
+-distribʳ-⊓ : _+_ DistributesOverʳ _⊓_

+
+-distribʳ-⊓ = comm∧distrˡ⇒distrʳ +-comm +-distribˡ-⊓

+

+
+-distrib-⊓ : _+_ DistributesOver _⊓_

+
+-distrib-⊓ = +-distribˡ-⊓ , +-distribʳ-⊓

+

+
m⊓n≤m+n : ∀ m n → m ⊓ n ≤ m + n

+
m⊓n≤m+n m n with ⊓-sel m n

+
... | inj₁ m⊓n≡m rewrite m⊓n≡m = m≤m+n m n

+
... | inj₂ m⊓n≡n rewrite m⊓n≡n = m≤n+m n m

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_ and _*_

+

+
*-distribˡ-⊓ : _*_ DistributesOverˡ _⊓_

+
*-distribˡ-⊓ m 0 o = begin-equality

+
  m * (0 ⊓ o)               ≡⟨⟩

+
  m * 0                     ≡⟨ *-zeroʳ m ⟩

+
  0                         ≡⟨⟩

+
  0 ⊓ (m * o)               ≡⟨ cong (_⊓ (m * o)) (*-zeroʳ m) ⟨

+
  (m * 0) ⊓ (m * o)         ∎

+
*-distribˡ-⊓ m (suc n) 0 = begin-equality

+
  m * (suc n ⊓ 0)           ≡⟨⟩

+
  m * 0                     ≡⟨ *-zeroʳ m ⟩

+
  0                         ≡⟨ ⊓-zeroʳ (m * suc n) ⟨

+
  (m * suc n) ⊓ 0           ≡⟨ cong (m * suc n ⊓_) (*-zeroʳ m) ⟨

+
  (m * suc n) ⊓ (m * 0)     ∎

+
*-distribˡ-⊓ m (suc n) (suc o) = begin-equality

+
  m * (suc n ⊓ suc o)       ≡⟨⟩

+
  m * suc (n ⊓ o)           ≡⟨ *-suc m (n ⊓ o) ⟩

+
  m + m * (n ⊓ o)           ≡⟨ cong (m +_) (*-distribˡ-⊓ m n o) ⟩

+
  m + (m * n) ⊓ (m * o)     ≡⟨ +-distribˡ-⊓ m (m * n) (m * o) ⟩

+
  (m + m * n) ⊓ (m + m * o) ≡⟨ cong₂ _⊓_ (*-suc m n) (*-suc m o) ⟨

+
  (m * suc n) ⊓ (m * suc o) ∎

+

+
*-distribʳ-⊓ : _*_ DistributesOverʳ _⊓_

+
*-distribʳ-⊓ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-⊓

+

+
*-distrib-⊓ : _*_ DistributesOver _⊓_

+
*-distrib-⊓ = *-distribˡ-⊓ , *-distribʳ-⊓

+

+
------------------------------------------------------------------------

+
-- Properties of _∸_

+
------------------------------------------------------------------------

+

+
0∸n≡0 : LeftZero zero _∸_

+
0∸n≡0 zero    = refl

+
0∸n≡0 (suc _) = refl

+

+
n∸n≡0 : ∀ n → n ∸ n ≡ 0

+
n∸n≡0 zero    = refl

+
n∸n≡0 (suc n) = n∸n≡0 n

+

+
------------------------------------------------------------------------

+
-- Properties of _∸_ and pred

+

+
pred[m∸n]≡m∸[1+n] : ∀ m n → pred (m ∸ n) ≡ m ∸ suc n

+
pred[m∸n]≡m∸[1+n] zero    zero    = refl

+
pred[m∸n]≡m∸[1+n] (suc m) zero    = refl

+
pred[m∸n]≡m∸[1+n] zero (suc n)    = refl

+
pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n

+

+
------------------------------------------------------------------------

+
-- Properties of _∸_ and _≤_/_<_

+

+
m∸n≤m : ∀ m n → m ∸ n ≤ m

+
m∸n≤m n       zero    = ≤-refl

+
m∸n≤m zero    (suc n) = ≤-refl

+
m∸n≤m (suc m) (suc n) = ≤-trans (m∸n≤m m n) (n≤1+n m)

+

+
m≮m∸n : ∀ m n → m ≮ m ∸ n

+
m≮m∸n m       zero    = n≮n m

+
m≮m∸n (suc m) (suc n) = m≮m∸n m n ∘ ≤-trans (n≤1+n (suc m))

+

+
1+m≢m∸n : ∀ {m} n → suc m ≢ m ∸ n

+
1+m≢m∸n {m} n eq = m≮m∸n m n (≤-reflexive eq)

+

+
∸-mono : _∸_ Preserves₂ _≤_ ⟶ _≥_ ⟶ _≤_

+
∸-mono z≤n         (s≤s n₁≥n₂)    = z≤n

+
∸-mono (s≤s m₁≤m₂) (s≤s n₁≥n₂)    = ∸-mono m₁≤m₂ n₁≥n₂

+
∸-mono m₁≤m₂       (z≤n {n = n₁}) = ≤-trans (m∸n≤m _ n₁) m₁≤m₂

+

+
∸-monoˡ-≤ : ∀ o → m ≤ n → m ∸ o ≤ n ∸ o

+
∸-monoˡ-≤ o m≤n = ∸-mono {u = o} m≤n ≤-refl

+

+
∸-monoʳ-≤ : ∀ o → m ≤ n → o ∸ m ≥ o ∸ n

+
∸-monoʳ-≤ _ m≤n = ∸-mono ≤-refl m≤n

+

+
∸-monoˡ-< : ∀ {m n o} → m < o → n ≤ m → m ∸ n < o ∸ n

+
∸-monoˡ-< {m}     {zero}  {o}     m<o       n≤m       = m<o

+
∸-monoˡ-< {suc m} {suc n} {suc o} (s≤s m<o) (s≤s n≤m) = ∸-monoˡ-< m<o n≤m

+

+
∸-monoʳ-< : ∀ {m n o} → o < n → n ≤ m → m ∸ n < m ∸ o

+
∸-monoʳ-< {n = suc n} {zero}  (s≤s o<n) (s≤s n<m) = s≤s (m∸n≤m _ n)

+
∸-monoʳ-< {n = suc n} {suc o} (s≤s o<n) (s≤s n<m) = ∸-monoʳ-< o<n n<m

+

+
∸-cancelʳ-≤ : ∀ {m n o} → m ≤ o → o ∸ n ≤ o ∸ m → m ≤ n

+
∸-cancelʳ-≤ {_}     {_}     z≤n       _       = z≤n

+
∸-cancelʳ-≤ {suc m} {zero}  (s≤s _)   o<o∸m   = contradiction o<o∸m (m≮m∸n _ m)

+
∸-cancelʳ-≤ {suc m} {suc n} (s≤s m≤o) o∸n<o∸m = s≤s (∸-cancelʳ-≤ m≤o o∸n<o∸m)

+

+
∸-cancelʳ-< : ∀ {m n o} → o ∸ m < o ∸ n → n < m

+
∸-cancelʳ-< {zero}  {n}     {o}     o<o∸n   = contradiction o<o∸n (m≮m∸n o n)

+
∸-cancelʳ-< {suc m} {zero}  {_}     o∸n<o∸m = 0<1+n

+
∸-cancelʳ-< {suc m} {suc n} {suc o} o∸n<o∸m = s≤s (∸-cancelʳ-< o∸n<o∸m)

+

+
∸-cancelˡ-≡ :  n ≤ m → o ≤ m → m ∸ n ≡ m ∸ o → n ≡ o

+
∸-cancelˡ-≡ {_}         z≤n       z≤n       _  = refl

+
∸-cancelˡ-≡ {o = suc o} z≤n       (s≤s _)   eq = contradiction eq (1+m≢m∸n o)

+
∸-cancelˡ-≡ {n = suc n} (s≤s _)   z≤n       eq = contradiction (sym eq) (1+m≢m∸n n)

+
∸-cancelˡ-≡ {_}         (s≤s n≤m) (s≤s o≤m) eq = cong suc (∸-cancelˡ-≡ n≤m o≤m eq)

+

+
∸-cancelʳ-≡ :  o ≤ m → o ≤ n → m ∸ o ≡ n ∸ o → m ≡ n

+
∸-cancelʳ-≡  z≤n       z≤n      eq = eq

+
∸-cancelʳ-≡ (s≤s o≤m) (s≤s o≤n) eq = cong suc (∸-cancelʳ-≡ o≤m o≤n eq)

+

+
m∸n≡0⇒m≤n : m ∸ n ≡ 0 → m ≤ n

+
m∸n≡0⇒m≤n {zero}  {_}    _   = z≤n

+
m∸n≡0⇒m≤n {suc m} {suc n} eq = s≤s (m∸n≡0⇒m≤n eq)

+

+
m≤n⇒m∸n≡0 : m ≤ n → m ∸ n ≡ 0

+
m≤n⇒m∸n≡0 {n = n} z≤n      = 0∸n≡0 n

+
m≤n⇒m∸n≡0 {_}    (s≤s m≤n) = m≤n⇒m∸n≡0 m≤n

+

+
m<n⇒0<n∸m : m < n → 0 < n ∸ m

+
m<n⇒0<n∸m {zero}  {suc n} _         = 0<1+n

+
m<n⇒0<n∸m {suc m} {suc n} (s≤s m<n) = m<n⇒0<n∸m m<n

+

+
m∸n≢0⇒n<m : m ∸ n ≢ 0 → n < m

+
m∸n≢0⇒n<m {m} {n} m∸n≢0 with n <? m

+
... | yes n<m = n<m

+
... | no  n≮m = contradiction (m≤n⇒m∸n≡0 (≮⇒≥ n≮m)) m∸n≢0

+

+
m>n⇒m∸n≢0 : m > n → m ∸ n ≢ 0

+
m>n⇒m∸n≢0 {n = suc n} (s≤s m>n) = m>n⇒m∸n≢0 m>n

+

+
m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n

+
m≤n⇒n∸m≤n z≤n       = ≤-refl

+
m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)

+

+
------------------------------------------------------------------------

+
-- Properties of _∸_ and _+_

+

+
+-∸-comm : ∀ {m} n {o} → o ≤ m → (m + n) ∸ o ≡ (m ∸ o) + n

+
+-∸-comm {zero}  _ {zero}  _         = refl

+
+-∸-comm {suc m} _ {zero}  _         = refl

+
+-∸-comm {suc m} n {suc o} (s≤s o≤m) = +-∸-comm n o≤m

+

+
∸-+-assoc : ∀ m n o → (m ∸ n) ∸ o ≡ m ∸ (n + o)

+
∸-+-assoc zero zero o = refl

+
∸-+-assoc zero (suc n) o = 0∸n≡0 o

+
∸-+-assoc (suc m) zero o = refl

+
∸-+-assoc (suc m) (suc n) o = ∸-+-assoc m n o

+

+
+-∸-assoc : ∀ m {n o} → o ≤ n → (m + n) ∸ o ≡ m + (n ∸ o)

+
+-∸-assoc m (z≤n {n = n})             = begin-equality m + n ∎

+
+-∸-assoc m (s≤s {m = o} {n = n} o≤n) = begin-equality

+
  (m + suc n) ∸ suc o  ≡⟨ cong (_∸ suc o) (+-suc m n) ⟩

+
  suc (m + n) ∸ suc o  ≡⟨⟩

+
  (m + n) ∸ o          ≡⟨ +-∸-assoc m o≤n ⟩

+
  m + (n ∸ o)          ∎

+

+
m≤n+o⇒m∸n≤o : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o

+
m≤n+o⇒m∸n≤o      m  zero    le = le

+
m≤n+o⇒m∸n≤o zero    (suc n)  _ = z≤n

+
m≤n+o⇒m∸n≤o (suc m) (suc n) le = m≤n+o⇒m∸n≤o m n (s≤s⁻¹ le)

+

+
m<n+o⇒m∸n<o : ∀ m n {o} → .{{NonZero o}} → m < n + o → m ∸ n < o

+
m<n+o⇒m∸n<o      m  zero                lt = lt

+
m<n+o⇒m∸n<o zero    (suc n) {o@(suc _)} lt = z<s

+
m<n+o⇒m∸n<o (suc m) (suc n)             lt = m<n+o⇒m∸n<o m n  (s<s⁻¹ lt)

+

+
m+n≤o⇒m≤o∸n : ∀ m {n o} → m + n ≤ o → m ≤ o ∸ n

+
m+n≤o⇒m≤o∸n zero    le       = z≤n

+
m+n≤o⇒m≤o∸n (suc m) (s≤s le)

+
  rewrite +-∸-assoc 1 (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m le)

+

+
m≤o∸n⇒m+n≤o : ∀ m {n o} (n≤o : n ≤ o) → m ≤ o ∸ n → m + n ≤ o

+
m≤o∸n⇒m+n≤o m         z≤n       le rewrite +-identityʳ m = le

+
m≤o∸n⇒m+n≤o m {suc n} (s≤s n≤o) le rewrite +-suc m n = s≤s (m≤o∸n⇒m+n≤o m n≤o le)

+

+
m≤n+m∸n : ∀ m n → m ≤ n + (m ∸ n)

+
m≤n+m∸n zero    n       = z≤n

+
m≤n+m∸n (suc m) zero    = ≤-refl

+
m≤n+m∸n (suc m) (suc n) = s≤s (m≤n+m∸n m n)

+

+
m+n∸n≡m : ∀ m n → m + n ∸ n ≡ m

+
m+n∸n≡m m n = begin-equality

+
  (m + n) ∸ n  ≡⟨ +-∸-assoc m (≤-refl {x = n}) ⟩

+
  m + (n ∸ n)  ≡⟨ cong (m +_) (n∸n≡0 n) ⟩

+
  m + 0        ≡⟨ +-identityʳ m ⟩

+
  m            ∎

+

+
m+n∸m≡n : ∀ m n → m + n ∸ m ≡ n

+
m+n∸m≡n m n = trans (cong (_∸ m) (+-comm m n)) (m+n∸n≡m n m)

+

+
m+[n∸m]≡n : m ≤ n → m + (n ∸ m) ≡ n

+
m+[n∸m]≡n {m} {n} m≤n = begin-equality

+
  m + (n ∸ m)  ≡⟨ sym $ +-∸-assoc m m≤n ⟩

+
  (m + n) ∸ m  ≡⟨ cong (_∸ m) (+-comm m n) ⟩

+
  (n + m) ∸ m  ≡⟨ m+n∸n≡m n m ⟩

+
  n            ∎

+

+
m∸n+n≡m : ∀ {m n} → n ≤ m → (m ∸ n) + n ≡ m

+
m∸n+n≡m {m} {n} n≤m = begin-equality

+
  (m ∸ n) + n ≡⟨ sym (+-∸-comm n n≤m) ⟩

+
  (m + n) ∸ n ≡⟨ m+n∸n≡m m n ⟩

+
  m           ∎

+

+
m∸[m∸n]≡n : ∀ {m n} → n ≤ m → m ∸ (m ∸ n) ≡ n

+
m∸[m∸n]≡n {m}     {_}     z≤n       = n∸n≡0 m

+
m∸[m∸n]≡n {suc m} {suc n} (s≤s n≤m) = begin-equality

+
  suc m ∸ (m ∸ n)   ≡⟨ +-∸-assoc 1 (m∸n≤m m n) ⟩

+
  suc (m ∸ (m ∸ n)) ≡⟨ cong suc (m∸[m∸n]≡n n≤m) ⟩

+
  suc n             ∎

+

+
[m+n]∸[m+o]≡n∸o : ∀ m n o → (m + n) ∸ (m + o) ≡ n ∸ o

+
[m+n]∸[m+o]≡n∸o zero    n o = refl

+
[m+n]∸[m+o]≡n∸o (suc m) n o = [m+n]∸[m+o]≡n∸o m n o

+

+
------------------------------------------------------------------------

+
-- Properties of _∸_ and _*_

+

+
*-distribʳ-∸ : _*_ DistributesOverʳ _∸_

+
*-distribʳ-∸ m       zero    zero    = refl

+
*-distribʳ-∸ zero    zero    (suc o) = sym (0∸n≡0 (o * zero))

+
*-distribʳ-∸ (suc m) zero    (suc o) = refl

+
*-distribʳ-∸ m       (suc n) zero    = refl

+
*-distribʳ-∸ m       (suc n) (suc o) = begin-equality

+
  (n ∸ o) * m             ≡⟨ *-distribʳ-∸ m n o ⟩

+
  n * m ∸ o * m           ≡⟨ sym $ [m+n]∸[m+o]≡n∸o m _ _ ⟩

+
  m + n * m ∸ (m + o * m) ∎

+

+
*-distribˡ-∸ : _*_ DistributesOverˡ _∸_

+
*-distribˡ-∸ = comm∧distrʳ⇒distrˡ *-comm *-distribʳ-∸

+

+
*-distrib-∸ : _*_ DistributesOver _∸_

+
*-distrib-∸ = *-distribˡ-∸ , *-distribʳ-∸

+

+
even≢odd :  ∀ m n → 2 * m ≢ suc (2 * n)

+
even≢odd (suc m) zero    eq = contradiction (suc-injective eq) (m+1+n≢0 m)

+
even≢odd (suc m) (suc n) eq = even≢odd m n (suc-injective (begin-equality

+
  suc (2 * m)         ≡⟨ sym (+-suc m _) ⟩

+
  m + suc (m + 0)     ≡⟨ suc-injective eq ⟩

+
  suc n + suc (n + 0) ≡⟨ cong suc (+-suc n _) ⟩

+
  suc (suc (2 * n))   ∎))

+

+
------------------------------------------------------------------------

+
-- Properties of _∸_ and _⊓_ and _⊔_

+

+
m⊓n+n∸m≡n : ∀ m n → (m ⊓ n) + (n ∸ m) ≡ n

+
m⊓n+n∸m≡n zero    n       = refl

+
m⊓n+n∸m≡n (suc m) zero    = refl

+
m⊓n+n∸m≡n (suc m) (suc n) = cong suc $ m⊓n+n∸m≡n m n

+

+
[m∸n]⊓[n∸m]≡0 : ∀ m n → (m ∸ n) ⊓ (n ∸ m) ≡ 0

+
[m∸n]⊓[n∸m]≡0 zero zero       = refl

+
[m∸n]⊓[n∸m]≡0 zero (suc n)    = refl

+
[m∸n]⊓[n∸m]≡0 (suc m) zero    = refl

+
[m∸n]⊓[n∸m]≡0 (suc m) (suc n) = [m∸n]⊓[n∸m]≡0 m n

+

+
∸-distribˡ-⊓-⊔ : ∀ m n o → m ∸ (n ⊓ o) ≡ (m ∸ n) ⊔ (m ∸ o)

+
∸-distribˡ-⊓-⊔ m n o = antimono-≤-distrib-⊓ (∸-monoʳ-≤ m) n o

+

+
∸-distribʳ-⊓ : _∸_ DistributesOverʳ _⊓_

+
∸-distribʳ-⊓ m n o = mono-≤-distrib-⊓ (∸-monoˡ-≤ m) n o

+

+
∸-distribˡ-⊔-⊓ : ∀ m n o → m ∸ (n ⊔ o) ≡ (m ∸ n) ⊓ (m ∸ o)

+
∸-distribˡ-⊔-⊓ m n o = antimono-≤-distrib-⊔ (∸-monoʳ-≤ m) n o

+

+
∸-distribʳ-⊔ : _∸_ DistributesOverʳ _⊔_

+
∸-distribʳ-⊔ m n o = mono-≤-distrib-⊔ (∸-monoˡ-≤ m) n o

+

+
------------------------------------------------------------------------

+
-- Properties of pred

+
------------------------------------------------------------------------

+

+
pred[n]≤n : pred n ≤ n

+
pred[n]≤n {zero}  = z≤n

+
pred[n]≤n {suc n} = n≤1+n n

+

+
≤pred⇒≤ : m ≤ pred n → m ≤ n

+
≤pred⇒≤ {n = zero}  le = le

+
≤pred⇒≤ {n = suc n} le = m≤n⇒m≤1+n le

+

+
≤⇒pred≤ : m ≤ n → pred m ≤ n

+
≤⇒pred≤ {zero}  le = le

+
≤⇒pred≤ {suc m} le = ≤-trans (n≤1+n m) le

+

+
<⇒≤pred : m < n → m ≤ pred n

+
<⇒≤pred (s≤s le) = le

+

+
suc-pred : ∀ n .{{_ : NonZero n}} → suc (pred n) ≡ n

+
suc-pred (suc n) = refl

+

+
pred-mono-≤ : pred Preserves _≤_ ⟶ _≤_

+
pred-mono-≤ {zero}          _   = z≤n

+
pred-mono-≤ {suc _} {suc _} m≤n = s≤s⁻¹ m≤n

+

+
pred-mono-< : .{{NonZero m}} → m < n → pred m < pred n

+
pred-mono-< {m = suc _} {n = suc _} = s<s⁻¹

+

+
pred-cancel-≤ : pred m ≤ pred n → (m ≡ 1 × n ≡ 0) ⊎ m ≤ n

+
pred-cancel-≤ {m = zero}  {n = zero}  _  = inj₂ z≤n

+
pred-cancel-≤ {m = zero}  {n = suc _} _  = inj₂ z≤n

+
pred-cancel-≤ {m = suc _} {n = zero} z≤n = inj₁ (refl , refl)

+
pred-cancel-≤ {m = suc _} {n = suc _} le = inj₂ (s≤s le)

+

+
pred-cancel-< : pred m < pred n → m < n

+
pred-cancel-< {m = zero}  {n = suc _} _ = z<s

+
pred-cancel-< {m = suc _} {n = suc _}   = s<s

+

+
pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n

+
pred-injective {suc m} {suc n} = cong suc

+

+
pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n

+
pred-cancel-≡ {m = zero}  {n = zero}  _    = inj₂ refl

+
pred-cancel-≡ {m = zero}  {n = suc _} refl = inj₁ (inj₁ (refl , refl))

+
pred-cancel-≡ {m = suc _} {n = zero}  refl = inj₁ (inj₂ (refl , refl))

+
pred-cancel-≡ {m = suc _} {n = suc _}      = inj₂ ∘ pred-injective

+

+
------------------------------------------------------------------------

+
-- Properties of ∣_-_∣

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Basic

+

+
m≡n⇒∣m-n∣≡0 : m ≡ n → ∣ m - n ∣ ≡ 0

+
m≡n⇒∣m-n∣≡0 {zero}  refl = refl

+
m≡n⇒∣m-n∣≡0 {suc m} refl = m≡n⇒∣m-n∣≡0 {m} refl

+

+
∣m-n∣≡0⇒m≡n :  ∣ m - n ∣ ≡ 0 → m ≡ n

+
∣m-n∣≡0⇒m≡n {zero}  {zero}  eq = refl

+
∣m-n∣≡0⇒m≡n {suc m} {suc n} eq = cong suc (∣m-n∣≡0⇒m≡n eq)

+

+
m≤n⇒∣n-m∣≡n∸m : m ≤ n → ∣ n - m ∣ ≡ n ∸ m

+
m≤n⇒∣n-m∣≡n∸m {n = zero}  z≤n       = refl

+
m≤n⇒∣n-m∣≡n∸m {n = suc n} z≤n       = refl

+
m≤n⇒∣n-m∣≡n∸m {n = _}     (s≤s m≤n) = m≤n⇒∣n-m∣≡n∸m m≤n

+

+
m≤n⇒∣m-n∣≡n∸m : m ≤ n → ∣ m - n ∣ ≡ n ∸ m

+
m≤n⇒∣m-n∣≡n∸m {n = zero}  z≤n       = refl

+
m≤n⇒∣m-n∣≡n∸m {n = suc n} z≤n       = refl

+
m≤n⇒∣m-n∣≡n∸m {n = _}     (s≤s m≤n) = m≤n⇒∣m-n∣≡n∸m m≤n

+

+
∣m-n∣≡m∸n⇒n≤m : ∣ m - n ∣ ≡ m ∸ n → n ≤ m

+
∣m-n∣≡m∸n⇒n≤m {zero}  {zero}  eq = z≤n

+
∣m-n∣≡m∸n⇒n≤m {suc m} {zero}  eq = z≤n

+
∣m-n∣≡m∸n⇒n≤m {suc m} {suc n} eq = s≤s (∣m-n∣≡m∸n⇒n≤m eq)

+

+
∣n-n∣≡0 : ∀ n → ∣ n - n ∣ ≡ 0

+
∣n-n∣≡0 n = m≡n⇒∣m-n∣≡0 {n} refl

+

+
∣m-m+n∣≡n : ∀ m n → ∣ m - m + n ∣ ≡ n

+
∣m-m+n∣≡n zero    n = refl

+
∣m-m+n∣≡n (suc m) n = ∣m-m+n∣≡n m n

+

+
∣m+n-m+o∣≡∣n-o∣ : ∀ m n o → ∣ m + n - m + o ∣ ≡ ∣ n - o ∣

+
∣m+n-m+o∣≡∣n-o∣ zero    n o = refl

+
∣m+n-m+o∣≡∣n-o∣ (suc m) n o = ∣m+n-m+o∣≡∣n-o∣ m n o

+

+
m∸n≤∣m-n∣ : ∀ m n → m ∸ n ≤ ∣ m - n ∣

+
m∸n≤∣m-n∣ m n with ≤-total m n

+
... | inj₁ m≤n = subst (_≤ ∣ m - n ∣) (sym (m≤n⇒m∸n≡0 m≤n)) z≤n

+
... | inj₂ n≤m = subst (m ∸ n ≤_) (sym (m≤n⇒∣n-m∣≡n∸m n≤m)) ≤-refl

+

+
∣m-n∣≤m⊔n : ∀ m n → ∣ m - n ∣ ≤ m ⊔ n

+
∣m-n∣≤m⊔n zero    m       = ≤-refl

+
∣m-n∣≤m⊔n (suc m) zero    = ≤-refl

+
∣m-n∣≤m⊔n (suc m) (suc n) = m≤n⇒m≤1+n (∣m-n∣≤m⊔n m n)

+

+
∣-∣-identityˡ : LeftIdentity 0 ∣_-_∣

+
∣-∣-identityˡ x = refl

+

+
∣-∣-identityʳ : RightIdentity 0 ∣_-_∣

+
∣-∣-identityʳ zero    = refl

+
∣-∣-identityʳ (suc x) = refl

+

+
∣-∣-identity : Identity 0 ∣_-_∣

+
∣-∣-identity = ∣-∣-identityˡ , ∣-∣-identityʳ

+

+
∣-∣-comm : Commutative ∣_-_∣

+
∣-∣-comm zero    zero    = refl

+
∣-∣-comm zero    (suc n) = refl

+
∣-∣-comm (suc m) zero    = refl

+
∣-∣-comm (suc m) (suc n) = ∣-∣-comm m n

+

+
∣m-n∣≡[m∸n]∨[n∸m] : ∀ m n → (∣ m - n ∣ ≡ m ∸ n) ⊎ (∣ m - n ∣ ≡ n ∸ m)

+
∣m-n∣≡[m∸n]∨[n∸m] m n with ≤-total m n

+
... | inj₂ n≤m = inj₁ $ m≤n⇒∣n-m∣≡n∸m n≤m

+
... | inj₁ m≤n = inj₂ $ begin-equality

+
  ∣ m - n ∣ ≡⟨ ∣-∣-comm m n ⟩

+
  ∣ n - m ∣ ≡⟨ m≤n⇒∣n-m∣≡n∸m m≤n ⟩

+
  n ∸ m     ∎

+

+
private

+

+
  *-distribˡ-∣-∣-aux : ∀ a m n → m ≤ n → a * ∣ n - m ∣ ≡ ∣ a * n - a * m ∣

+
  *-distribˡ-∣-∣-aux a m n m≤n = begin-equality

+
    a * ∣ n - m ∣     ≡⟨ cong (a *_) (m≤n⇒∣n-m∣≡n∸m m≤n) ⟩

+
    a * (n ∸ m)       ≡⟨ *-distribˡ-∸ a n m ⟩

+
    a * n ∸ a * m     ≡⟨ sym $′ m≤n⇒∣n-m∣≡n∸m (*-monoʳ-≤ a m≤n) ⟩

+
    ∣ a * n - a * m ∣ ∎

+

+
*-distribˡ-∣-∣ : _*_ DistributesOverˡ ∣_-_∣

+
*-distribˡ-∣-∣ a m n with ≤-total m n

+
... | inj₂ n≤m = *-distribˡ-∣-∣-aux a n m n≤m

+
... | inj₁ m≤n = begin-equality

+
  a * ∣ m - n ∣     ≡⟨ cong (a *_) (∣-∣-comm m n) ⟩

+
  a * ∣ n - m ∣     ≡⟨ *-distribˡ-∣-∣-aux a m n m≤n ⟩

+
  ∣ a * n - a * m ∣ ≡⟨ ∣-∣-comm (a * n) (a * m) ⟩

+
  ∣ a * m - a * n ∣ ∎

+

+
*-distribʳ-∣-∣ : _*_ DistributesOverʳ ∣_-_∣

+
*-distribʳ-∣-∣ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-∣-∣

+

+
*-distrib-∣-∣ : _*_ DistributesOver ∣_-_∣

+
*-distrib-∣-∣ = *-distribˡ-∣-∣ , *-distribʳ-∣-∣

+

+
m≤n+∣n-m∣ : ∀ m n → m ≤ n + ∣ n - m ∣

+
m≤n+∣n-m∣ zero    n       = z≤n

+
m≤n+∣n-m∣ (suc m) zero    = ≤-refl

+
m≤n+∣n-m∣ (suc m) (suc n) = s≤s (m≤n+∣n-m∣ m n)

+

+
m≤n+∣m-n∣ : ∀ m n → m ≤ n + ∣ m - n ∣

+
m≤n+∣m-n∣ m n = subst (m ≤_) (cong (n +_) (∣-∣-comm n m)) (m≤n+∣n-m∣ m n)

+

+
m≤∣m-n∣+n : ∀ m n → m ≤ ∣ m - n ∣ + n

+
m≤∣m-n∣+n m n = subst (m ≤_) (+-comm n _) (m≤n+∣m-n∣ m n)

+

+
∣-∣-triangle : TriangleInequality ∣_-_∣

+
∣-∣-triangle zero    y       z       = m≤n+∣n-m∣ z y

+
∣-∣-triangle x       zero    z       = begin

+
  ∣ x - z ∣     ≤⟨ ∣m-n∣≤m⊔n x z ⟩

+
  x ⊔ z         ≤⟨ m⊔n≤m+n x z ⟩

+
  x + z         ≡⟨ cong₂ _+_ (sym (∣-∣-identityʳ x)) refl ⟩

+
  ∣ x - 0 ∣ + z ∎

+
  where open ≤-Reasoning

+
∣-∣-triangle x       y       zero    = begin

+
  ∣ x - 0 ∣             ≡⟨ ∣-∣-identityʳ x ⟩

+
  x                     ≤⟨ m≤∣m-n∣+n x y ⟩

+
  ∣ x - y ∣ + y         ≡⟨ cong₂ _+_ refl (sym (∣-∣-identityʳ y)) ⟩

+
  ∣ x - y ∣ + ∣ y - 0 ∣ ∎

+
  where open ≤-Reasoning

+
∣-∣-triangle (suc x) (suc y) (suc z) = ∣-∣-triangle x y z

+

+
∣-∣≡∣-∣′ : ∀ m n → ∣ m - n ∣ ≡ ∣ m - n ∣′

+
∣-∣≡∣-∣′ m n with m <ᵇ n in eq

+
... | false = m≤n⇒∣n-m∣≡n∸m {n} {m} (≮⇒≥ (λ m<n → subst T eq (<⇒<ᵇ m<n)))

+
... | true  = m≤n⇒∣m-n∣≡n∸m {m} {n} (<⇒≤ (<ᵇ⇒< m n (subst T (sym eq) _)))

+

+
------------------------------------------------------------------------

+
-- Metric structures

+

+
∣-∣-isProtoMetric : IsProtoMetric _≡_ ∣_-_∣

+
∣-∣-isProtoMetric = record

+
  { isPartialOrder  = ≤-isPartialOrder

+
  ; ≈-isEquivalence = isEquivalence

+
  ; cong            = cong₂ ∣_-_∣

+
  ; nonNegative     = z≤n

+
  }

+

+
∣-∣-isPreMetric : IsPreMetric _≡_ ∣_-_∣

+
∣-∣-isPreMetric = record

+
  { isProtoMetric = ∣-∣-isProtoMetric

+
  ; ≈⇒0           = m≡n⇒∣m-n∣≡0

+
  }

+

+
∣-∣-isQuasiSemiMetric : IsQuasiSemiMetric _≡_ ∣_-_∣

+
∣-∣-isQuasiSemiMetric = record

+
  { isPreMetric = ∣-∣-isPreMetric

+
  ; 0⇒≈         = ∣m-n∣≡0⇒m≡n

+
  }

+

+
∣-∣-isSemiMetric : IsSemiMetric _≡_ ∣_-_∣

+
∣-∣-isSemiMetric = record

+
  { isQuasiSemiMetric = ∣-∣-isQuasiSemiMetric

+
  ; sym               = ∣-∣-comm

+
  }

+

+
∣-∣-isMetric : IsMetric _≡_ ∣_-_∣

+
∣-∣-isMetric = record

+
  { isSemiMetric = ∣-∣-isSemiMetric

+
  ; triangle     = ∣-∣-triangle

+
  }

+

+
------------------------------------------------------------------------

+
-- Metric bundles

+

+
∣-∣-quasiSemiMetric : QuasiSemiMetric 0ℓ 0ℓ

+
∣-∣-quasiSemiMetric = record

+
  { isQuasiSemiMetric = ∣-∣-isQuasiSemiMetric

+
  }

+

+
∣-∣-semiMetric : SemiMetric 0ℓ 0ℓ

+
∣-∣-semiMetric = record

+
  { isSemiMetric = ∣-∣-isSemiMetric

+
  }

+

+
∣-∣-preMetric : PreMetric 0ℓ 0ℓ

+
∣-∣-preMetric = record

+
  { isPreMetric = ∣-∣-isPreMetric

+
  }

+

+
∣-∣-metric : Metric 0ℓ 0ℓ

+
∣-∣-metric = record

+
  { isMetric = ∣-∣-isMetric

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of ⌊_/2⌋ and ⌈_/2⌉

+
------------------------------------------------------------------------

+

+
⌊n/2⌋-mono : ⌊_/2⌋ Preserves _≤_ ⟶ _≤_

+
⌊n/2⌋-mono z≤n             = z≤n

+
⌊n/2⌋-mono (s≤s z≤n)       = z≤n

+
⌊n/2⌋-mono (s≤s (s≤s m≤n)) = s≤s (⌊n/2⌋-mono m≤n)

+

+
⌈n/2⌉-mono : ⌈_/2⌉ Preserves _≤_ ⟶ _≤_

+
⌈n/2⌉-mono m≤n = ⌊n/2⌋-mono (s≤s m≤n)

+

+
⌊n/2⌋≤⌈n/2⌉ : ∀ n → ⌊ n /2⌋ ≤ ⌈ n /2⌉

+
⌊n/2⌋≤⌈n/2⌉ zero          = z≤n

+
⌊n/2⌋≤⌈n/2⌉ (suc zero)    = z≤n

+
⌊n/2⌋≤⌈n/2⌉ (suc (suc n)) = s≤s (⌊n/2⌋≤⌈n/2⌉ n)

+

+
⌊n/2⌋+⌈n/2⌉≡n : ∀ n → ⌊ n /2⌋ + ⌈ n /2⌉ ≡ n

+
⌊n/2⌋+⌈n/2⌉≡n zero    = refl

+
⌊n/2⌋+⌈n/2⌉≡n (suc n) = begin-equality

+
  ⌊ suc n /2⌋ + suc ⌊ n /2⌋   ≡⟨ +-comm ⌊ suc n /2⌋ (suc ⌊ n /2⌋) ⟩

+
  suc ⌊ n /2⌋ + ⌊ suc n /2⌋   ≡⟨⟩

+
  suc (⌊ n /2⌋ + ⌊ suc n /2⌋) ≡⟨ cong suc (⌊n/2⌋+⌈n/2⌉≡n n) ⟩

+
  suc n                       ∎

+

+
⌊n/2⌋≤n : ∀ n → ⌊ n /2⌋ ≤ n

+
⌊n/2⌋≤n zero          = z≤n

+
⌊n/2⌋≤n (suc zero)    = z≤n

+
⌊n/2⌋≤n (suc (suc n)) = s≤s (m≤n⇒m≤1+n (⌊n/2⌋≤n n))

+

+
⌊n/2⌋<n : ∀ n → ⌊ suc n /2⌋ < suc n

+
⌊n/2⌋<n zero    = z<s

+
⌊n/2⌋<n (suc n) = s<s (s≤s (⌊n/2⌋≤n n))

+

+
n≡⌊n+n/2⌋ : ∀ n → n ≡ ⌊ n + n /2⌋

+
n≡⌊n+n/2⌋ zero          = refl

+
n≡⌊n+n/2⌋ (suc zero)    = refl

+
n≡⌊n+n/2⌋ (suc n′@(suc n)) =

+
  cong suc (trans (n≡⌊n+n/2⌋ _) (cong ⌊_/2⌋ (sym (+-suc n n′))))

+

+
⌈n/2⌉≤n : ∀ n → ⌈ n /2⌉ ≤ n

+
⌈n/2⌉≤n zero    = z≤n

+
⌈n/2⌉≤n (suc n) = s≤s (⌊n/2⌋≤n n)

+

+
⌈n/2⌉<n : ∀ n → ⌈ suc (suc n) /2⌉ < suc (suc n)

+
⌈n/2⌉<n n = s<s (⌊n/2⌋<n n)

+

+
n≡⌈n+n/2⌉ : ∀ n → n ≡ ⌈ n + n /2⌉

+
n≡⌈n+n/2⌉ zero            = refl

+
n≡⌈n+n/2⌉ (suc zero)      = refl

+
n≡⌈n+n/2⌉ (suc n′@(suc n)) =

+
  cong suc (trans (n≡⌈n+n/2⌉ _) (cong ⌈_/2⌉ (sym (+-suc n n′))))

+

+
------------------------------------------------------------------------

+
-- Properties of !_

+

+
1≤n! : ∀ n → 1 ≤ n !

+
1≤n! zero    = ≤-refl

+
1≤n! (suc n) = *-mono-≤ (m≤m+n 1 n) (1≤n! n)

+

+
infix 4 _!≢0 _!*_!≢0

+

+
_!≢0 : ∀ n → NonZero (n !)

+
n !≢0 = >-nonZero (1≤n! n)

+

+
_!*_!≢0 : ∀ m n → NonZero (m ! * n !)

+
m !* n !≢0 = m*n≢0 _ _ {{m !≢0}} {{n !≢0}}

+

+
------------------------------------------------------------------------

+
-- Properties of _≤′_ and _<′_

+

+
≤′-trans : Transitive _≤′_

+
≤′-trans m≤n ≤′-refl       = m≤n

+
≤′-trans m≤n (≤′-step n≤o) = ≤′-step (≤′-trans m≤n n≤o)

+

+
z≤′n : zero ≤′ n

+
z≤′n {zero}  = ≤′-refl

+
z≤′n {suc n} = ≤′-step z≤′n

+

+
s≤′s : m ≤′ n → suc m ≤′ suc n

+
s≤′s ≤′-refl        = ≤′-refl

+
s≤′s (≤′-step m≤′n) = ≤′-step (s≤′s m≤′n)

+

+
≤′⇒≤ : _≤′_ ⇒ _≤_

+
≤′⇒≤ ≤′-refl        = ≤-refl

+
≤′⇒≤ (≤′-step m≤′n) = m≤n⇒m≤1+n (≤′⇒≤ m≤′n)

+

+
≤⇒≤′ : _≤_ ⇒ _≤′_

+
≤⇒≤′ z≤n       = z≤′n

+
≤⇒≤′ (s≤s m≤n) = s≤′s (≤⇒≤′ m≤n)

+

+
≤′-step-injective : {p q : m ≤′ n} → ≤′-step p ≡ ≤′-step q → p ≡ q

+
≤′-step-injective refl = refl

+

+
------------------------------------------------------------------------

+
-- Properties of _<′_ and _<_

+
------------------------------------------------------------------------

+

+
z<′s : zero <′ suc n

+
z<′s {zero}  = <′-base

+
z<′s {suc n} = <′-step (z<′s {n})

+

+
s<′s : m <′ n → suc m <′ suc n

+
s<′s <′-base        = <′-base

+
s<′s (<′-step m<′n) = <′-step (s<′s m<′n)

+

+
<⇒<′ : m < n → m <′ n

+
<⇒<′ z<s               = z<′s

+
<⇒<′ (s<s m<n@(s≤s _)) = s<′s (<⇒<′ m<n)

+

+
<′⇒< : m <′ n → m < n

+
<′⇒< <′-base        = n<1+n _

+
<′⇒< (<′-step m<′n) = m<n⇒m<1+n (<′⇒< m<′n)

+

+
m<1+n⇒m<n∨m≡n′ : m < suc n → m < n ⊎ m ≡ n

+
m<1+n⇒m<n∨m≡n′ m<n with <⇒<′ m<n

+
... | <′-base      = inj₂ refl

+
... | <′-step m<′n = inj₁ (<′⇒< m<′n)

+

+
------------------------------------------------------------------------

+
-- Other properties of _≤′_ and _<′_

+
------------------------------------------------------------------------

+

+
infix 4 _≤′?_ _<′?_ _≥′?_ _>′?_

+

+
_≤′?_ : Decidable _≤′_

+
m ≤′? n = map′ ≤⇒≤′ ≤′⇒≤ (m ≤? n)

+

+
_<′?_ : Decidable _<′_

+
m <′? n = suc m ≤′? n

+

+
_≥′?_ : Decidable _≥′_

+
_≥′?_ = flip _≤′?_

+

+
_>′?_ : Decidable _>′_

+
_>′?_ = flip _<′?_

+

+
m≤′m+n : ∀ m n → m ≤′ m + n

+
m≤′m+n m n = ≤⇒≤′ (m≤m+n m n)

+

+
n≤′m+n : ∀ m n → n ≤′ m + n

+
n≤′m+n zero    n = ≤′-refl

+
n≤′m+n (suc m) n = ≤′-step (n≤′m+n m n)

+

+
⌈n/2⌉≤′n : ∀ n → ⌈ n /2⌉ ≤′ n

+
⌈n/2⌉≤′n zero          = ≤′-refl

+
⌈n/2⌉≤′n (suc zero)    = ≤′-refl

+
⌈n/2⌉≤′n (suc (suc n)) = s≤′s (≤′-step (⌈n/2⌉≤′n n))

+

+
⌊n/2⌋≤′n : ∀ n → ⌊ n /2⌋ ≤′ n

+
⌊n/2⌋≤′n zero    = ≤′-refl

+
⌊n/2⌋≤′n (suc n) = ≤′-step (⌈n/2⌉≤′n n)

+

+
------------------------------------------------------------------------

+
-- Properties of _≤″_ and _<″_

+
------------------------------------------------------------------------

+

+
-- equivalence of  _≤″_ to _≤_

+

+
≤⇒≤″ : _≤_ ⇒ _≤″_

+
≤⇒≤″ = (_ ,_) ∘ m+[n∸m]≡n

+

+
<⇒<″ : _<_ ⇒ _<″_

+
<⇒<″ = ≤⇒≤″

+

+
≤″⇒≤ : _≤″_ ⇒ _≤_

+
≤″⇒≤ (k , refl) = m≤m+n _ k

+

+
-- equivalence to the old definition of _≤″_

+

+
≤″-proof : (le : m ≤″ n) → let k , _ = le in m + k ≡ n

+
≤″-proof (_ , prf) = prf

+

+
-- yielding analogous proof for _≤_

+

+
m≤n⇒∃[o]m+o≡n : .(m ≤ n) → ∃ λ k → m + k ≡ n

+
m≤n⇒∃[o]m+o≡n m≤n = _ , m+[n∸m]≡n (recompute (_ ≤? _) m≤n)

+

+
-- whose witness is equal to monus

+

+
guarded-∸≗∸ : ∀ {m n} → .(m≤n : m ≤ n) →

+
              let k , _ = m≤n⇒∃[o]m+o≡n m≤n in k ≡ n ∸ m

+
guarded-∸≗∸ m≤n = refl

+

+
-- equivalence of _<″_ to _<ᵇ_

+

+
m<ᵇn⇒1+m+[n-1+m]≡n : ∀ m n → T (m <ᵇ n) → suc m + (n ∸ suc m) ≡ n

+
m<ᵇn⇒1+m+[n-1+m]≡n m n lt = m+[n∸m]≡n (<ᵇ⇒< m n lt)

+

+
m<ᵇ1+m+n : ∀ m {n} → T (m <ᵇ suc (m + n))

+
m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)

+

+
<ᵇ⇒<″ : T (m <ᵇ n) → m <″ n

+
<ᵇ⇒<″ {m} {n} = <⇒<″ ∘ (<ᵇ⇒< m n)

+

+
<″⇒<ᵇ : ∀ {m n} → m <″ n → T (m <ᵇ n)

+
<″⇒<ᵇ {m} (k , refl) = <⇒<ᵇ (m≤m+n (suc m) k)

+

+
-- NB: we use the builtin function `_<ᵇ_ : (m n : ℕ) → Bool` here so

+
-- that the function quickly decides whether to return `yes` or `no`.

+
-- It still takes a linear amount of time to generate the proof if it

+
-- is inspected. We expect the main benefit to be visible for compiled

+
-- code: the backend erases proofs.

+

+
infix 4 _<″?_ _≤″?_ _≥″?_ _>″?_

+

+
_<″?_ : Decidable _<″_

+
m <″? n = map′ <ᵇ⇒<″ <″⇒<ᵇ (T? (m <ᵇ n))

+

+
_≤″?_ : Decidable _≤″_

+
zero  ≤″? n = yes (n , refl)

+
suc m ≤″? n = m <″? n

+

+
_≥″?_ : Decidable _≥″_

+
_≥″?_ = flip _≤″?_

+

+
_>″?_ : Decidable _>″_

+
_>″?_ = flip _<″?_

+

+
≤″-irrelevant : Irrelevant _≤″_

+
≤″-irrelevant {m} (_ , eq₁) (_ , eq₂)

+
  with refl ← +-cancelˡ-≡ m _ _ (trans eq₁ (sym eq₂))

+
  = cong (_ ,_) (≡-irrelevant eq₁ eq₂)

+

+
<″-irrelevant : Irrelevant _<″_

+
<″-irrelevant = ≤″-irrelevant

+

+
>″-irrelevant : Irrelevant _>″_

+
>″-irrelevant = ≤″-irrelevant

+

+
≥″-irrelevant : Irrelevant _≥″_

+
≥″-irrelevant = ≤″-irrelevant

+

+
------------------------------------------------------------------------

+
-- Properties of _≤‴_

+
------------------------------------------------------------------------

+

+
≤‴⇒≤″ : ∀{m n} → m ≤‴ n → m ≤″ n

+
≤‴⇒≤″ {m = m} ≤‴-refl       = 0 , +-identityʳ m

+
≤‴⇒≤″ {m = m} (≤‴-step m≤n) = _ , trans (+-suc m _) (≤″-proof (≤‴⇒≤″ m≤n))

+

+
m≤‴m+k : ∀{m n k} → m + k ≡ n → m ≤‴ n

+
m≤‴m+k {m} {k = zero}  refl = subst (λ z → m ≤‴ z) (sym (+-identityʳ m)) (≤‴-refl {m})

+
m≤‴m+k {m} {k = suc k} prf  = ≤‴-step (m≤‴m+k {k = k} (trans (sym (+-suc m _)) prf))

+

+
≤″⇒≤‴ : ∀{m n} → m ≤″ n → m ≤‴ n

+
≤″⇒≤‴ m≤n = m≤‴m+k (≤″-proof m≤n)

+

+
0≤‴n : 0 ≤‴ n

+
0≤‴n = m≤‴m+k refl

+

+
<ᵇ⇒<‴ : T (m <ᵇ n) → m <‴ n

+
<ᵇ⇒<‴ leq = ≤″⇒≤‴ (<ᵇ⇒<″ leq)

+

+
<‴⇒<ᵇ : ∀ {m n} → m <‴ n → T (m <ᵇ n)

+
<‴⇒<ᵇ leq = <″⇒<ᵇ (≤‴⇒≤″ leq)

+

+
infix 4 _<‴?_ _≤‴?_ _≥‴?_ _>‴?_

+

+
_<‴?_ : Decidable _<‴_

+
m <‴? n = map′ <ᵇ⇒<‴ <‴⇒<ᵇ (T? (m <ᵇ n))

+

+
_≤‴?_ : Decidable _≤‴_

+
zero ≤‴? n = yes 0≤‴n

+
suc m ≤‴? n = m <‴? n

+

+
_≥‴?_ : Decidable _≥‴_

+
_≥‴?_ = flip _≤‴?_

+

+
_>‴?_ : Decidable _>‴_

+
_>‴?_ = flip _<‴?_

+

+
≤⇒≤‴ : _≤_ ⇒ _≤‴_

+
≤⇒≤‴ = ≤″⇒≤‴ ∘ ≤⇒≤″

+

+
≤‴⇒≤ : _≤‴_ ⇒ _≤_

+
≤‴⇒≤ = ≤″⇒≤ ∘ ≤‴⇒≤″

+

+
------------------------------------------------------------------------

+
-- Other properties

+
------------------------------------------------------------------------

+

+
-- If there is an injection from a type to ℕ, then the type has

+
-- decidable equality.

+

+
eq? : ∀ {a} {A : Set a} → A ↣ ℕ → DecidableEquality A

+
eq? inj = via-injection inj _≟_

+

+
-- It's possible to decide existential and universal predicates up to

+
-- a limit.

+

+
module _ {p} {P : Pred ℕ p} (P? : U.Decidable P) where

+

+
  anyUpTo? : ∀ v → Dec (∃ λ n → n < v × P n)

+
  anyUpTo? zero    = no λ {(_ , () , _)}

+
  anyUpTo? (suc v) with P? v | anyUpTo? v

+
  ... | yes Pv | _                  = yes (v , ≤-refl , Pv)

+
  ... | _      | yes (n , n<v , Pn) = yes (n , m≤n⇒m≤1+n n<v , Pn)

+
  ... | no ¬Pv | no ¬Pn<v           = no ¬Pn<1+v

+
    where

+
    ¬Pn<1+v : ¬ (∃ λ n → n < suc v × P n)

+
    ¬Pn<1+v (n , s≤s n≤v , Pn) with n ≟ v

+
    ... | yes refl = ¬Pv Pn

+
    ... | no  n≢v  = ¬Pn<v (n , ≤∧≢⇒< n≤v n≢v , Pn)

+

+
  allUpTo? : ∀ v → Dec (∀ {n} → n < v → P n)

+
  allUpTo? zero    = yes λ()

+
  allUpTo? (suc v) with P? v | allUpTo? v

+
  ... | no ¬Pv | _        = no λ prf → ¬Pv   (prf ≤-refl)

+
  ... | _      | no ¬Pn<v = no λ prf → ¬Pn<v (prf ∘ m≤n⇒m≤1+n)

+
  ... | yes Pn | yes Pn<v = yes Pn<1+v

+
    where

+
      Pn<1+v : ∀ {n} → n < suc v → P n

+
      Pn<1+v {n} (s≤s n≤v) with n ≟ v

+
      ... | yes refl = Pn

+
      ... | no  n≢v  = Pn<v (≤∧≢⇒< n≤v n≢v)

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.3

+

+
∀[m≤n⇒m≢o]⇒o<n : ∀ n o → (∀ {m} → m ≤ n → m ≢ o) → n < o

+
∀[m≤n⇒m≢o]⇒o<n = ∀[m≤n⇒m≢o]⇒n<o

+
{-# WARNING_ON_USAGE ∀[m≤n⇒m≢o]⇒o<n

+
"Warning: ∀[m≤n⇒m≢o]⇒o<n was deprecated in v1.3.

+
Please use ∀[m≤n⇒m≢o]⇒n<o instead."

+
#-}

+
∀[m<n⇒m≢o]⇒o≤n : ∀ n o → (∀ {m} → m < n → m ≢ o) → n ≤ o

+
∀[m<n⇒m≢o]⇒o≤n = ∀[m<n⇒m≢o]⇒n≤o

+
{-# WARNING_ON_USAGE ∀[m<n⇒m≢o]⇒o≤n

+
"Warning: ∀[m<n⇒m≢o]⇒o≤n was deprecated in v1.3.

+
Please use ∀[m<n⇒m≢o]⇒n≤o instead."

+
#-}

+

+
-- Version 1.4

+

+
*-+-isSemiring = +-*-isSemiring

+
{-# WARNING_ON_USAGE *-+-isSemiring

+
"Warning: *-+-isSemiring was deprecated in v1.4.

+
Please use +-*-isSemiring instead."

+
#-}

+
*-+-isCommutativeSemiring = +-*-isCommutativeSemiring

+
{-# WARNING_ON_USAGE *-+-isCommutativeSemiring

+
"Warning: *-+-isCommutativeSemiring was deprecated in v1.4.

+
Please use +-*-isCommutativeSemiring instead."

+
#-}

+
*-+-semiring = +-*-semiring

+
{-# WARNING_ON_USAGE *-+-semiring

+
"Warning: *-+-semiring was deprecated in v1.4.

+
Please use +-*-semiring instead."

+
#-}

+
*-+-commutativeSemiring = +-*-commutativeSemiring

+
{-# WARNING_ON_USAGE *-+-commutativeSemiring

+
"Warning: *-+-commutativeSemiring was deprecated in v1.4.

+
Please use +-*-commutativeSemiring instead."

+
#-}

+

+
-- Version 1.6

+

+
∣m+n-m+o∣≡∣n-o| = ∣m+n-m+o∣≡∣n-o∣

+
{-# WARNING_ON_USAGE ∣m+n-m+o∣≡∣n-o|

+
"Warning: ∣m+n-m+o∣≡∣n-o| was deprecated in v1.6.

+
Please use ∣m+n-m+o∣≡∣n-o∣ instead. Note the final is a \\| rather than a |"

+
#-}

+
m≤n⇒n⊔m≡n = m≥n⇒m⊔n≡m

+
{-# WARNING_ON_USAGE m≤n⇒n⊔m≡n

+
"Warning: m≤n⇒n⊔m≡n was deprecated in v1.6. Please use m≥n⇒m⊔n≡m instead."

+
#-}

+
m≤n⇒n⊓m≡m = m≥n⇒m⊓n≡n

+
{-# WARNING_ON_USAGE m≤n⇒n⊓m≡m

+
"Warning: m≤n⇒n⊓m≡m was deprecated in v1.6. Please use m≥n⇒m⊓n≡n instead."

+
#-}

+
n⊔m≡m⇒n≤m = m⊔n≡n⇒m≤n

+
{-# WARNING_ON_USAGE n⊔m≡m⇒n≤m

+
"Warning: n⊔m≡m⇒n≤m was deprecated in v1.6. Please use m⊔n≡n⇒m≤n instead."

+
#-}

+
n⊔m≡n⇒m≤n = m⊔n≡m⇒n≤m

+
{-# WARNING_ON_USAGE n⊔m≡n⇒m≤n

+
"Warning: n⊔m≡n⇒m≤n was deprecated in v1.6. Please use m⊔n≡m⇒n≤m instead."

+
#-}

+
n≤m⊔n = m≤n⊔m

+
{-# WARNING_ON_USAGE n≤m⊔n

+
"Warning: n≤m⊔n was deprecated in v1.6. Please use m≤n⊔m instead."

+
#-}

+
⊔-least = ⊔-lub

+
{-# WARNING_ON_USAGE ⊔-least

+
"Warning: ⊔-least was deprecated in v1.6. Please use ⊔-lub instead."

+
#-}

+
⊓-greatest = ⊓-glb

+
{-# WARNING_ON_USAGE ⊓-greatest

+
"Warning: ⊓-greatest was deprecated in v1.6. Please use ⊓-glb instead."

+
#-}

+
⊔-pres-≤m = ⊔-lub

+
{-# WARNING_ON_USAGE ⊔-pres-≤m

+
"Warning: ⊔-pres-≤m was deprecated in v1.6. Please use ⊔-lub instead."

+
#-}

+
⊓-pres-m≤ = ⊓-glb

+
{-# WARNING_ON_USAGE ⊓-pres-m≤

+
"Warning: ⊓-pres-m≤ was deprecated in v1.6. Please use ⊓-glb instead."

+
#-}

+
⊔-abs-⊓ = ⊔-absorbs-⊓

+
{-# WARNING_ON_USAGE ⊔-abs-⊓

+
"Warning: ⊔-abs-⊓ was deprecated in v1.6. Please use ⊔-absorbs-⊓ instead."

+
#-}

+
⊓-abs-⊔ = ⊓-absorbs-⊔

+
{-# WARNING_ON_USAGE ⊓-abs-⊔

+
"Warning: ⊓-abs-⊔ was deprecated in v1.6. Please use ⊓-absorbs-⊔ instead."

+
#-}

+

+
-- Version 2.0

+

+
suc[pred[n]]≡n : n ≢ 0 → suc (pred n) ≡ n

+
suc[pred[n]]≡n {zero}  0≢0 = contradiction refl 0≢0

+
suc[pred[n]]≡n {suc n} _   = refl

+
{-# WARNING_ON_USAGE suc[pred[n]]≡n

+
"Warning: suc[pred[n]]≡n was deprecated in v2.0. Please use suc-pred instead. Note that the proof now uses instance arguments"

+
#-}

+

+
≤-step = m≤n⇒m≤1+n

+
{-# WARNING_ON_USAGE ≤-step

+
"Warning: ≤-step was deprecated in v2.0. Please use m≤n⇒m≤1+n instead. "

+
#-}

+

+
≤-stepsˡ = m≤n⇒m≤o+n

+
{-# WARNING_ON_USAGE ≤-stepsˡ

+
"Warning: ≤-stepsˡ was deprecated in v2.0. Please use m≤n⇒m≤o+n instead. "

+
#-}

+

+
≤-stepsʳ = m≤n⇒m≤n+o

+
{-# WARNING_ON_USAGE ≤-stepsʳ

+
"Warning: ≤-stepsʳ was deprecated in v2.0. Please use m≤n⇒m≤n+o instead. "

+
#-}

+

+
<-step = m<n⇒m<1+n

+
{-# WARNING_ON_USAGE <-step

+
"Warning: <-step was deprecated in v2.0. Please use m<n⇒m<1+n instead. "

+
#-}

+

+
pred-mono = pred-mono-≤

+
{-# WARNING_ON_USAGE pred-mono

+
"Warning: pred-mono was deprecated in v2.0. Please use pred-mono-≤ instead. "

+
#-}

+

+
{- issue1844/issue1755: raw bundles have moved to `Data.X.Base` -}

+
open Data.Nat.Base public

+
  using (*-rawMagma; *-1-rawMonoid)

+

+
<-transʳ = ≤-<-trans

+
{-# WARNING_ON_USAGE <-transʳ

+
"Warning: <-transʳ was deprecated in v2.0. Please use ≤-<-trans instead. "

+
#-}

+

+
<-transˡ = <-≤-trans

+
{-# WARNING_ON_USAGE <-transˡ

+
"Warning: <-transˡ was deprecated in v2.0. Please use <-≤-trans instead. "

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Show.html b/site/static/agda_html/Data.Nat.Show.html
new file mode 100644
index 000000000..a33b4517c
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Show.html
@@ -0,0 +1,162 @@
+
+Data.Nat.Show
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Showing natural numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Show where

+

+
open import Data.Bool.Base using (_∧_)

+
open import Data.Char.Base as Char using (Char)

+
open import Data.Digit using (showDigit; toDigits; toNatDigits)

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.List.Effectful using (module TraversableA)

+
open import Data.Maybe.Base as Maybe using (Maybe; nothing; _<∣>_; when)

+
import Data.Maybe.Effectful as Maybe

+
open import Data.Nat

+
open import Data.Product.Base using (proj₁)

+
open import Data.String.Base using (toList; fromList; String)

+
open import Function.Base using (_∘′_; _∘_)

+
open import Relation.Nullary.Decidable using (True)

+

+
------------------------------------------------------------------------

+
-- Read

+

+
readMaybe : ∀ base {base≤16 : True (base ≤? 16)} → String → Maybe ℕ

+
readMaybe _ "" = nothing

+
readMaybe base = Maybe.map convert

+
              ∘′ TraversableA.mapA Maybe.applicative readDigit

+
              ∘′ toList

+

+
  where

+

+
    convert : List ℕ → ℕ

+
    convert = List.foldl (λ acc d → base * acc + d) 0

+

+
    char0 = Char.toℕ '0'

+
    char9 = Char.toℕ '9'

+
    chara = Char.toℕ 'a'

+
    charf = Char.toℕ 'f'

+

+
    readDigit : Char → Maybe ℕ

+
    readDigit c = digit Maybe.>>= λ n → when (n <ᵇ base) n where

+

+
      charc = Char.toℕ c

+

+
      dec = when ((char0 ≤ᵇ charc) ∧ (charc ≤ᵇ char9)) (charc ∸ char0)

+
      hex = when ((chara ≤ᵇ charc) ∧ (charc ≤ᵇ charf)) (10 + charc ∸ chara)

+
      digit = dec <∣> hex

+

+
------------------------------------------------------------------------

+
-- Show

+

+
-- Decimal notation

+
-- Time complexity is O(log₁₀(n))

+

+
toDigitChar : ℕ → Char

+
toDigitChar n = Char.fromℕ (n + Char.toℕ '0')

+

+
toDecimalChars : ℕ → List Char

+
toDecimalChars = List.map toDigitChar ∘′ toNatDigits 10

+

+
show : ℕ → String

+
show = fromList ∘′ toDecimalChars

+

+
-- Arbitrary base betwen 2 & 16.

+
-- Warning: when compiled the time complexity of `showInBase b n` is

+
-- O(n) instead of the expected O(log(n)).

+

+
charsInBase : (base : ℕ)

+
              {base≥2 : True (2 ≤? base)}

+
              {base≤16 : True (base ≤? 16)} →

+
              ℕ → List Char

+
charsInBase base {base≥2} {base≤16} = List.map (showDigit {base≤16 = base≤16})

+
                                    ∘ List.reverse

+
                                    ∘ proj₁

+
                                    ∘ toDigits base {base≥2 = base≥2}

+

+
showInBase : (base : ℕ)

+
             {base≥2 : True (2 ≤? base)}

+
             {base≤16 : True (base ≤? 16)} →

+
             ℕ → String

+
showInBase base {base≥2} {base≤16} = fromList

+
                                   ∘ charsInBase base {base≥2} {base≤16}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.Solver.html b/site/static/agda_html/Data.Nat.Solver.html
new file mode 100644
index 000000000..f64cc741b
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.Solver.html
@@ -0,0 +1,99 @@
+
+Data.Nat.Solver
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Automatic solvers for equations over naturals

+
------------------------------------------------------------------------

+

+
-- See README.Data.Nat for examples of how to use this solver

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat.Solver where

+

+
import Algebra.Solver.Ring.Simple as Solver

+
import Algebra.Solver.Ring.AlmostCommutativeRing as ACR

+
open import Data.Nat.Properties

+

+
------------------------------------------------------------------------

+
-- A module for automatically solving propositional equivalences

+
-- containing _+_ and _*_

+

+
module +-*-Solver =

+
  Solver (ACR.fromCommutativeSemiring +-*-commutativeSemiring) _≟_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Nat.html b/site/static/agda_html/Data.Nat.html
new file mode 100644
index 000000000..b34ac2613
--- /dev/null
+++ b/site/static/agda_html/Data.Nat.html
@@ -0,0 +1,126 @@
+
+Data.Nat
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Natural numbers

+
------------------------------------------------------------------------

+

+
-- See README.Data.Nat for examples of how to use and reason about

+
-- naturals.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Nat where

+

+
------------------------------------------------------------------------

+
-- Publicly re-export the contents of the base module

+

+
open import Data.Nat.Base public

+

+
------------------------------------------------------------------------

+
-- Publicly re-export queries

+

+
open import Data.Nat.Properties public

+
  using

+
  -- key values

+
  ( nonZero?

+
  -- equalities

+
  ; _≟_ ; eq?

+
  -- standard orders & their relationship

+
  ; _≤?_ ; _≥?_ ; _<?_ ; _>?_

+
  ; ≤-<-connex ; ≥->-connex ; <-≤-connex ; >-≥-connex

+
  ; <-cmp

+
  -- alternative definitions of the orders

+
  ; _≤′?_; _≥′?_; _<′?_; _>′?_

+
  ; _≤″?_; _<″?_; _≥″?_; _>″?_

+
  ; _<‴?_; _≤‴?_; _≥‴?_; _>‴?_

+
  -- bounded predicates

+
  ; anyUpTo? ; allUpTo?

+
  )

+

+
------------------------------------------------------------------------

+
-- Deprecated

+

+
-- Version 0.17

+

+
-- Version 2.0

+

+
-- solely for the re-export of this name, formerly in `Data.Nat.Properties.Core`

+
open import Data.Nat.Properties public

+
  using (≤-pred)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Parity.Base.html b/site/static/agda_html/Data.Parity.Base.html
new file mode 100644
index 000000000..184338ef2
--- /dev/null
+++ b/site/static/agda_html/Data.Parity.Base.html
@@ -0,0 +1,194 @@
+
+Data.Parity.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Parity

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Parity.Base where

+

+
open import Algebra.Bundles.Raw

+
  using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring)

+
open import Data.Sign.Base using (Sign; +; -)

+
open import Level using (0ℓ)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Parity : Set where

+
  0ℙ : Parity

+
  1ℙ : Parity

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
-- The opposite parity.

+

+
infix 8 _⁻¹

+

+
_⁻¹ : Parity → Parity

+
1ℙ ⁻¹ = 0ℙ

+
0ℙ ⁻¹ = 1ℙ

+

+
-- Addition.

+

+
infixl 7 _+_

+

+
_+_ : Parity → Parity → Parity

+
0ℙ + p = p

+
1ℙ + p = p ⁻¹

+

+
-- Multiplication.

+

+
infixl 7 _*_

+

+
_*_ : Parity → Parity → Parity

+
0ℙ * p = 0ℙ

+
1ℙ * p = p

+

+
------------------------------------------------------------------------

+
-- Raw Bundles

+

+
+-rawMagma : RawMagma 0ℓ 0ℓ

+
+-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  }

+

+
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ

+
+-0-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  ; ε = 0ℙ

+
  }

+

+
+-0-rawGroup : RawGroup 0ℓ 0ℓ

+
+-0-rawGroup = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  ; _⁻¹ = _⁻¹

+
  ; ε = 0ℙ

+
  }

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  ; ε = 1ℙ

+
  }

+

+
+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ

+
+-*-rawNearSemiring = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0ℙ

+
  }

+

+
+-*-rawSemiring : RawSemiring 0ℓ 0ℓ

+
+-*-rawSemiring = record

+
  { Carrier = _

+
  ; _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0ℙ

+
  ; 1# = 1ℙ

+
  }

+

+

+
------------------------------------------------------------------------

+
-- Homomorphisms between Parity and Sign

+

+
toSign : Parity → Sign

+
toSign 0ℙ = +

+
toSign 1ℙ = -

+

+
fromSign : Sign → Parity

+
fromSign + = 0ℙ

+
fromSign - = 1ℙ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Algebra.html b/site/static/agda_html/Data.Product.Algebra.html
new file mode 100644
index 000000000..3ecd03116
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Algebra.html
@@ -0,0 +1,256 @@
+
+Data.Product.Algebra
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Algebraic properties of products

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Algebra where

+

+
open import Algebra

+
open import Data.Bool.Base using (true; false)

+
open import Data.Empty.Polymorphic using (⊥; ⊥-elim)

+
open import Data.Product.Base

+
open import Data.Product.Properties

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Data.Sum.Algebra

+
open import Data.Unit.Polymorphic using (⊤; tt)

+
open import Function.Base using (_∘′_)

+
open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)

+
open import Function.Properties.Inverse using (↔-isEquivalence)

+
open import Level using (Level; suc)

+
open import Relation.Binary.PropositionalEquality.Core

+

+
import Function.Definitions as FuncDef

+

+
------------------------------------------------------------------------

+

+
private

+
  variable

+
    a b c d p : Level

+
    A B C D : Set a

+

+
------------------------------------------------------------------------

+
-- Properties of Σ

+

+
-- Σ is associative

+
Σ-assoc : {B : A → Set b} {C : (a : A) → B a → Set c} →

+
          Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))

+
Σ-assoc = mk↔ₛ′ assocʳ assocˡ cong′ cong′

+

+
-- Σ is associative, alternate formulation

+
Σ-assoc-alt : {B : A → Set b} {C : Σ A B → Set c} →

+
          Σ (Σ A B) C ↔ Σ A (λ a → Σ (B a) (curry C a))

+
Σ-assoc-alt = mk↔ₛ′ assocʳ-curried assocˡ-curried cong′ cong′

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
-- × is a congruence

+
×-cong : A ↔ B → C ↔ D → (A × C) ↔ (B × D)

+
×-cong i j = mk↔ₛ′ (map I.to J.to) (map I.from J.from)

+
  (λ {(a , b) → cong₂ _,_ (I.strictlyInverseˡ a) (J.strictlyInverseˡ b)})

+
  (λ {(a , b) → cong₂ _,_ (I.strictlyInverseʳ a) (J.strictlyInverseʳ b)})

+
  where module I = Inverse i; module J = Inverse j

+

+
-- × is commutative.

+
-- (we don't use Commutative because it isn't polymorphic enough)

+
×-comm : (A : Set a) (B : Set b) → (A × B) ↔ (B × A)

+
×-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive

+

+
module _ (ℓ : Level) where

+

+
  -- × is associative

+
  ×-assoc : Associative {ℓ = ℓ} _↔_ _×_

+
  ×-assoc _ _ _ = mk↔ₛ′ assocʳ′ assocˡ′ cong′ cong′

+

+
  -- ⊤ is the identity for ×

+
  ×-identityˡ : LeftIdentity {ℓ = ℓ} _↔_ ⊤ _×_

+
  ×-identityˡ _ = mk↔ₛ′ proj₂ (tt ,_) cong′ cong′

+

+
  ×-identityʳ : RightIdentity {ℓ = ℓ} _↔_ ⊤ _×_

+
  ×-identityʳ _ = mk↔ₛ′ proj₁ (_, tt) cong′ cong′

+

+
  ×-identity : Identity _↔_ ⊤ _×_

+
  ×-identity = ×-identityˡ , ×-identityʳ

+

+
  -- ⊥ is the zero for ×

+
  ×-zeroˡ : LeftZero {ℓ = ℓ} _↔_ ⊥ _×_

+
  ×-zeroˡ A = mk↔ₛ′ proj₁ ⊥-elim ⊥-elim λ ()

+

+
  ×-zeroʳ : RightZero {ℓ = ℓ} _↔_ ⊥ _×_

+
  ×-zeroʳ A = mk↔ₛ′ proj₂ ⊥-elim ⊥-elim λ ()

+

+
  ×-zero : Zero _↔_ ⊥ _×_

+
  ×-zero = ×-zeroˡ , ×-zeroʳ

+

+
  -- × distributes over ⊎

+
  ×-distribˡ-⊎ : _DistributesOverˡ_ {ℓ = ℓ} _↔_ _×_ _⊎_

+
  ×-distribˡ-⊎ _ _ _ = mk↔ₛ′

+
    (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)

+
    [ map₂ inj₁ , map₂ inj₂ ]′

+
    Sum.[ cong′ , cong′ ]

+
    (uncurry λ _ → Sum.[ cong′ , cong′ ])

+

+
  ×-distribʳ-⊎ : _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_

+
  ×-distribʳ-⊎ _ _ _ = mk↔ₛ′

+
    (uncurry [ curry inj₁ , curry inj₂ ]′)

+
    [ map₁ inj₁ , map₁ inj₂ ]′

+
    Sum.[ cong′ , cong′ ]

+
    (uncurry Sum.[ (λ _ → cong′) , (λ _ → cong′) ])

+

+
  ×-distrib-⊎ : _DistributesOver_ {ℓ = ℓ} _↔_ _×_ _⊎_

+
  ×-distrib-⊎ = ×-distribˡ-⊎ , ×-distribʳ-⊎

+

+
------------------------------------------------------------------------

+
-- Algebraic structures

+

+
  ×-isMagma : IsMagma {ℓ = ℓ} _↔_ _×_

+
  ×-isMagma = record

+
    { isEquivalence = ↔-isEquivalence

+
    ; ∙-cong        = ×-cong

+
    }

+

+
  ×-isSemigroup : IsSemigroup _↔_ _×_

+
  ×-isSemigroup = record

+
    { isMagma = ×-isMagma

+
    ; assoc   = λ _ _ _ → Σ-assoc

+
    }

+

+
  ×-isMonoid : IsMonoid _↔_ _×_ ⊤

+
  ×-isMonoid = record

+
    { isSemigroup = ×-isSemigroup

+
    ; identity    = ×-identityˡ , ×-identityʳ

+
    }

+

+
  ×-isCommutativeMonoid : IsCommutativeMonoid _↔_ _×_ ⊤

+
  ×-isCommutativeMonoid = record

+
    { isMonoid = ×-isMonoid

+
    ; comm     = ×-comm

+
    }

+

+
  ⊎-×-isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _↔_ _⊎_ _×_ ⊥ ⊤

+
  ⊎-×-isSemiringWithoutAnnihilatingZero = record

+
    { +-isCommutativeMonoid = ⊎-isCommutativeMonoid ℓ

+
    ; *-cong                = ×-cong

+
    ; *-assoc               = ×-assoc

+
    ; *-identity            = ×-identity

+
    ; distrib               = ×-distrib-⊎

+
    }

+

+
  ⊎-×-isSemiring : IsSemiring _↔_ _⊎_ _×_ ⊥ ⊤

+
  ⊎-×-isSemiring = record

+
    { isSemiringWithoutAnnihilatingZero = ⊎-×-isSemiringWithoutAnnihilatingZero

+
    ; zero                              = ×-zero

+
    }

+

+
  ⊎-×-isCommutativeSemiring : IsCommutativeSemiring _↔_ _⊎_ _×_ ⊥ ⊤

+
  ⊎-×-isCommutativeSemiring = record

+
    { isSemiring = ⊎-×-isSemiring

+
    ; *-comm     = ×-comm

+
    }

+
------------------------------------------------------------------------

+
-- Algebraic bundles

+

+
  ×-magma : Magma (suc ℓ) ℓ

+
  ×-magma = record

+
    { isMagma = ×-isMagma

+
    }

+

+
  ×-semigroup : Semigroup (suc ℓ) ℓ

+
  ×-semigroup = record

+
    { isSemigroup = ×-isSemigroup

+
    }

+

+
  ×-monoid : Monoid (suc ℓ) ℓ

+
  ×-monoid = record

+
    { isMonoid = ×-isMonoid

+
    }

+

+
  ×-commutativeMonoid : CommutativeMonoid (suc ℓ) ℓ

+
  ×-commutativeMonoid = record

+
    { isCommutativeMonoid = ×-isCommutativeMonoid

+
    }

+

+
  ×-⊎-commutativeSemiring : CommutativeSemiring (suc ℓ) ℓ

+
  ×-⊎-commutativeSemiring = record

+
    { isCommutativeSemiring = ⊎-×-isCommutativeSemiring

+
    }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Base.html b/site/static/agda_html/Data.Product.Base.html
new file mode 100644
index 000000000..4bce9d0c9
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Base.html
@@ -0,0 +1,270 @@
+
+Data.Product.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Products

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Base where

+

+
open import Function.Base

+
open import Level using (Level; _⊔_)

+

+
private

+
  variable

+
    a b c d e f ℓ p q r : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+
    E : Set e

+
    F : Set f

+

+
------------------------------------------------------------------------

+
-- Definition of dependent products

+

+
open import Agda.Builtin.Sigma public

+
  renaming (fst to proj₁; snd to proj₂)

+
  hiding (module Σ)

+

+
module Σ = Agda.Builtin.Sigma.Σ

+
  renaming (fst to proj₁; snd to proj₂)

+

+
------------------------------------------------------------------------

+
-- Existential quantifiers

+

+
∃ : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)

+
∃ = Σ _

+

+
∃₂ : ∀ {A : Set a} {B : A → Set b}

+
     (C : (x : A) → B x → Set c) → Set (a ⊔ b ⊔ c)

+
∃₂ C = ∃ λ a → ∃ λ b → C a b

+

+
------------------------------------------------------------------------

+
-- Syntaxes

+

+
-- The syntax declaration below is attached to Σ-syntax, to make it

+
-- easy to import Σ without the special syntax.

+

+
infix 2 Σ-syntax

+

+
Σ-syntax : (A : Set a) → (A → Set b) → Set (a ⊔ b)

+
Σ-syntax = Σ

+

+
syntax Σ-syntax A (λ x → B) = Σ[ x ∈ A ] B

+

+
infix 2 ∃-syntax

+

+
∃-syntax : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)

+
∃-syntax = ∃

+

+
syntax ∃-syntax (λ x → B) = ∃[ x ] B

+

+
------------------------------------------------------------------------

+
-- Definition of non-dependent products

+

+
infixr 4 _,′_

+
infixr 2 _×_

+

+
_×_ : ∀ (A : Set a) (B : Set b) → Set (a ⊔ b)

+
A × B = Σ[ x ∈ A ] B

+

+
_,′_ : A → B → A × B

+
_,′_ = _,_

+

+
------------------------------------------------------------------------

+
-- Operations over dependent products

+

+
infix  4 -,_

+
infixr 2 _-×-_ _-,-_

+
infixl 2 _<*>_

+

+
-- Sometimes the first component can be inferred.

+

+
-,_ : ∀ {A : Set a} {B : A → Set b} {x} → B x → Σ _ B

+
-, y = _ , y

+

+
<_,_> : ∀ {A : Set a} {B : A → Set b} {C : ∀ {x} → B x → Set c}

+
        (f : (x : A) → B x) → ((x : A) → C (f x)) →

+
        ((x : A) → Σ (B x) C)

+
< f , g > x = (f x , g x)

+

+
map : ∀ {P : A → Set p} {Q : B → Set q} →

+
      (f : A → B) → (∀ {x} → P x → Q (f x)) →

+
      Σ A P → Σ B Q

+
map f g (x , y) = (f x , g y)

+

+
map₁ : (A → B) → A × C → B × C

+
map₁ f = map f id

+

+
map₂ : ∀ {A : Set a} {B : A → Set b} {C : A → Set c} →

+
       (∀ {x} → B x → C x) → Σ A B → Σ A C

+
map₂ f = map id f

+

+
-- A version of map where the output can depend on the input

+
dmap : ∀ {B : A → Set b} {P : A → Set p} {Q : ∀ {a} → P a → B a → Set q} →

+
       (f : (a : A) → B a) → (∀ {a} (b : P a) → Q b (f a)) →

+
       ((a , b) : Σ A P) → Σ (B a) (Q b)

+
dmap f g (x , y) = f x , g y

+

+
zip : ∀ {P : A → Set p} {Q : B → Set q} {R : C → Set r} →

+
      (_∙_ : A → B → C) →

+
      (∀ {x y} → P x → Q y → R (x ∙ y)) →

+
      Σ A P → Σ B Q → Σ C R

+
zip _∙_ _∘_ (a , p) (b , q) = ((a ∙ b) , (p ∘ q))

+

+
curry : ∀ {A : Set a} {B : A → Set b} {C : Σ A B → Set c} →

+
        ((p : Σ A B) → C p) →

+
        ((x : A) → (y : B x) → C (x , y))

+
curry f x y = f (x , y)

+

+
uncurry : ∀ {A : Set a} {B : A → Set b} {C : Σ A B → Set c} →

+
          ((x : A) → (y : B x) → C (x , y)) →

+
          ((p : Σ A B) → C p)

+
uncurry f (x , y) = f x y

+

+
-- Rewriting dependent products

+
assocʳ : {B : A → Set b} {C : (a : A) → B a → Set c} →

+
          Σ (Σ A B) (uncurry C) → Σ A (λ a → Σ (B a) (C a))

+
assocʳ ((a , b) , c) = (a , (b , c))

+

+
assocˡ : {B : A → Set b} {C : (a : A) → B a → Set c} →

+
          Σ A (λ a → Σ (B a) (C a)) → Σ (Σ A B) (uncurry C)

+
assocˡ (a , (b , c)) = ((a , b) , c)

+

+
-- Alternate form of associativity for dependent products

+
-- where the C parameter is uncurried.

+
assocʳ-curried : {B : A → Set b} {C : Σ A B → Set c} →

+
                 Σ (Σ A B) C → Σ A (λ a → Σ (B a) (curry C a))

+
assocʳ-curried ((a , b) , c) = (a , (b , c))

+

+
assocˡ-curried : {B : A → Set b} {C : Σ A B → Set c} →

+
          Σ A (λ a → Σ (B a) (curry C a)) → Σ (Σ A B) C

+
assocˡ-curried (a , (b , c)) = ((a , b) , c)

+

+
------------------------------------------------------------------------

+
-- Operations for non-dependent products

+

+
-- Any of the above operations for dependent products will also work for

+
-- non-dependent products but sometimes Agda has difficulty inferring

+
-- the non-dependency. Primed (′ = \prime) versions of the operations

+
-- are therefore provided below that sometimes have better inference

+
-- properties.

+

+
zip′ : (A → B → C) → (D → E → F) → A × D → B × E → C × F

+
zip′ f g = zip f g

+

+
curry′ : (A × B → C) → (A → B → C)

+
curry′ = curry

+

+
uncurry′ : (A → B → C) → (A × B → C)

+
uncurry′ = uncurry

+

+
map₂′ : (B → C) → A × B → A × C

+
map₂′ f = map₂ f

+

+
dmap′ : ∀ {x y} {X : A → Set x} {Y : B → Set y} →

+
        ((a : A) → X a) → ((b : B) → Y b) →

+
        ((a , b) : A × B) → X a × Y b

+
dmap′ f g = dmap f g

+

+
_<*>_ : ∀ {x y} {X : A → Set x} {Y : B → Set y} →

+
        ((a : A) → X a) × ((b : B) → Y b) →

+
        ((a , b) : A × B) → X a × Y b

+
_<*>_ = uncurry dmap′

+

+
-- Operations that can only be defined for non-dependent products

+

+
swap : A × B → B × A

+
swap (x , y) = (y , x)

+

+
_-×-_ : (A → B → Set p) → (A → B → Set q) → (A → B → Set _)

+
f -×- g = f -⟪ _×_ ⟫- g

+

+
_-,-_ : (A → B → C) → (A → B → D) → (A → B → C × D)

+
f -,- g = f -⟪ _,_ ⟫- g

+

+
-- Rewriting non-dependent products

+
assocʳ′ : (A × B) × C → A × (B × C)

+
assocʳ′ ((a , b) , c) = (a , (b , c))

+

+
assocˡ′ : A × (B × C) → (A × B) × C

+
assocˡ′ (a , (b , c)) = ((a , b) , c)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Function.Dependent.Propositional.html b/site/static/agda_html/Data.Product.Function.Dependent.Propositional.html
new file mode 100644
index 000000000..f627b5cc6
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Function.Dependent.Propositional.html
@@ -0,0 +1,395 @@
+
+Data.Product.Function.Dependent.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Dependent product combinators for propositional equality

+
-- preserving functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Function.Dependent.Propositional where

+

+
open import Data.Product.Base as Product using (Σ; map; proj₂; _,_)

+
open import Data.Product.Properties using (Σ-≡,≡→≡; Σ-≡,≡↔≡; Σ-≡,≡←≡)

+
open import Level using (Level; 0ℓ)

+
open import Function.Related.Propositional

+
  using (_∼[_]_; module EquationalReasoning; K-reflexive;

+
         implication; reverseImplication; equivalence; injection;

+
         reverseInjection; leftInverse; surjection; bijection)

+
open import Function.Base using (_$_; _∘_; _∘′_)

+
open import Function.Properties.Inverse using (↔⇒↠; ↔⇒⟶; ↔⇒⟵; ↔-sym; ↔⇒↩)

+
open import Function.Properties.RightInverse using (↩⇒↪; ↪⇒↩)

+
open import Function.Properties.Inverse.HalfAdjointEquivalence

+
  using (↔⇒≃; _≃_; ≃⇒↔)

+
open import Function.Consequences.Propositional

+
  using (inverseʳ⇒injective; strictlySurjective⇒surjective)

+
open import Function.Definitions using (Inverseˡ; Inverseʳ; Injective; StrictlySurjective)

+
open import Function.Bundles

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.PropositionalEquality.Properties as ≡

+
  using (module ≡-Reasoning)

+

+
private

+
  variable

+
    i a b c d : Level

+
    I J : Set i

+
    A B : I → Set a

+

+
------------------------------------------------------------------------

+
-- Functions

+

+
module _ where

+
  open Func

+

+
  Σ-⟶ : (I⟶J : I ⟶ J) →

+
         (∀ {i} → A i ⟶ B (to I⟶J i)) →

+
         Σ I A ⟶ Σ J B

+
  Σ-⟶ I⟶J A⟶B = mk⟶ $ Product.map (to I⟶J) (to A⟶B)

+

+
------------------------------------------------------------------------

+
-- Equivalences

+

+
module _ where

+
  open Surjection

+

+
  Σ-⇔ : (I↠J : I ↠ J) →

+
         (∀ {i} → A i ⇔ B (to I↠J i)) →

+
         Σ I A ⇔ Σ J B

+
  Σ-⇔ {B = B} I↠J A⇔B = mk⇔

+
    (map (to  I↠J) (Equivalence.to A⇔B))

+
    (map (to⁻ I↠J) (Equivalence.from A⇔B ∘ ≡.subst B (≡.sym (proj₂ (surjective I↠J _) ≡.refl))))

+

+
  -- See also Data.Product.Relation.Binary.Pointwise.Dependent.WithK.↣.

+

+
------------------------------------------------------------------------

+
-- Injections

+

+
module _ where

+

+
  Σ-↣ : (I↔J : I ↔ J) →

+
         (∀ {i} → A i ↣ B (Inverse.to I↔J i)) →

+
         Σ I A ↣ Σ J B

+
  Σ-↣ {I = I} {J = J} {A = A} {B = B} I↔J A↣B = mk↣ to-injective

+
    where

+
    open ≡.≡-Reasoning

+

+
    I≃J = ↔⇒≃ I↔J

+

+
    subst-application′ :

+
      let open _≃_ I≃J in

+
      {x₁ x₂ : I} {y : A (from (to x₁))}

+
      (g : ∀ x → A (from (to x)) → B (to x))

+
      (eq : to x₁ ≡ to x₂) →

+
      ≡.subst B eq (g x₁ y) ≡ g x₂ (≡.subst A (≡.cong from eq) y)

+
    subst-application′ {x₁} {x₂} {y} g eq =

+
      ≡.subst B eq (g x₁ y)                      ≡⟨ ≡.cong (≡.subst B eq) (≡.sym (g′-lemma _ _)) ⟩

+
      ≡.subst B eq (g′ (to x₁) y)                ≡⟨ ≡.subst-application A g′ eq ⟩

+
      g′ (to x₂) (≡.subst A (≡.cong from eq) y)  ≡⟨ g′-lemma _ _ ⟩

+
      g x₂ (≡.subst A (≡.cong from eq) y)        ∎

+
      where

+
      open _≃_ I≃J

+

+
      g′ : ∀ x → A (from x) → B x

+
      g′ x =

+
        ≡.subst B (right-inverse-of x) ∘

+
        g (from x) ∘

+
        ≡.subst A (≡.sym (≡.cong from (right-inverse-of x)))

+

+
      g′-lemma : ∀ x y → g′ (to x) y ≡ g x y

+
      g′-lemma x y =

+
        ≡.subst B (right-inverse-of (to x))

+
          (g (from (to x)) $

+
           ≡.subst A (≡.sym (≡.cong from (right-inverse-of (to x)))) y)  ≡⟨ ≡.cong (λ p → ≡.subst B p (g (from (to x))

+
                                                                                                           (≡.subst A (≡.sym (≡.cong from p)) y)))

+
                                                                               (≡.sym (left-right x)) ⟩

+
        ≡.subst B (≡.cong to (left-inverse-of x))

+
          (g (from (to x)) $

+
           ≡.subst A

+
             (≡.sym (≡.cong from (≡.cong to (left-inverse-of x))))

+
             y)                                                           ≡⟨ lemma _ ⟩

+

+
        g x y                                                             ∎

+
        where

+
        lemma : ∀ {x′} eq {y : A (from (to x′))} →

+
                  ≡.subst B (≡.cong to eq)

+
                    (g (from (to x))

+
                      (≡.subst A (≡.sym (≡.cong from (≡.cong to eq))) y)) ≡

+
                  g x′ y

+
        lemma ≡.refl = ≡.refl

+

+
    open Injection

+

+
    to′ : Σ I A → Σ J B

+
    to′ = Product.map (_≃_.to I≃J) (to A↣B)

+

+
    to-injective : Injective _≡_ _≡_ to′

+
    to-injective {(x₁ , x₂)} {(y₁ , y₂)} =

+

+
      Σ-≡,≡→≡ ∘′

+

+
      map (_≃_.injective I≃J) (λ {eq₁} eq₂ → injective A↣B (

+
              to A↣B (≡.subst A (_≃_.injective I≃J eq₁) x₂)             ≡⟨⟩

+

+
              (let eq =

+
                      ≡.trans (≡.sym (_≃_.left-inverse-of I≃J x₁))

+
                        (≡.trans (≡.cong (_≃_.from I≃J) eq₁)

+
                          (≡.trans (_≃_.left-inverse-of I≃J y₁)

+
                            ≡.refl)) in

+
              to A↣B (≡.subst A eq x₂))                                   ≡⟨ ≡.cong (λ p → to A↣B

+
                                                                                             (≡.subst A

+
                                                                                               (≡.trans (≡.sym (_≃_.left-inverse-of I≃J _))

+
                                                                                                  (≡.trans (≡.cong (_≃_.from I≃J) eq₁) p))

+
                                                                                               x₂))

+
                                                                               (≡.trans-reflʳ _) ⟩

+

+
              (let eq = ≡.trans (≡.sym (_≃_.left-inverse-of I≃J x₁))

+
                          (≡.trans (≡.cong (_≃_.from I≃J) eq₁)

+
                            (_≃_.left-inverse-of I≃J y₁)) in

+
              to A↣B (≡.subst A eq x₂))                                  ≡⟨ ≡.cong (to A↣B)

+
                                                                               (≡.sym (≡.subst-subst (≡.sym (_≃_.left-inverse-of I≃J _)))) ⟩

+

+
              to A↣B ((≡.subst A (≡.trans (≡.cong (_≃_.from I≃J) eq₁)

+
                             (_≃_.left-inverse-of I≃J y₁)) $

+
               ≡.subst A (≡.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ ≡.cong (to A↣B)

+
                                                                               (≡.sym (≡.subst-subst (≡.cong (_≃_.from I≃J) eq₁))) ⟩

+
              to A↣B (

+
              (≡.subst A (_≃_.left-inverse-of I≃J y₁) $

+
               ≡.subst A (≡.cong (_≃_.from I≃J) eq₁) $

+
               ≡.subst A (≡.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ ≡.sym (subst-application′

+
                                                                                      (λ x y → to A↣B

+
                                                                                                 (≡.subst A (_≃_.left-inverse-of I≃J x) y))

+
                                                                                      eq₁) ⟩

+
              ≡.subst B eq₁ (to A↣B $

+
                 (≡.subst A (_≃_.left-inverse-of I≃J x₁) $

+
                  ≡.subst A (≡.sym (_≃_.left-inverse-of I≃J x₁)) x₂))  ≡⟨ ≡.cong (≡.subst B eq₁ ∘ to A↣B)

+
                                                                               (≡.subst-subst (≡.sym (_≃_.left-inverse-of I≃J _))) ⟩

+

+
              (let eq = ≡.trans (≡.sym (_≃_.left-inverse-of I≃J x₁))

+
                          (_≃_.left-inverse-of I≃J x₁) in

+
              ≡.subst B eq₁ (to A↣B (≡.subst A eq x₂)))                  ≡⟨ ≡.cong (λ p → ≡.subst B eq₁ (to A↣B (≡.subst A p x₂)))

+
                                                                              (≡.trans-symˡ (_≃_.left-inverse-of I≃J _)) ⟩

+
              ≡.subst B eq₁ (to A↣B (≡.subst A ≡.refl x₂))               ≡⟨⟩

+

+
              ≡.subst B eq₁ (to A↣B x₂)                                  ≡⟨ eq₂ ⟩

+

+
              to A↣B y₂                                                  ∎

+

+
        )) ∘

+

+
      Σ-≡,≡←≡

+

+
------------------------------------------------------------------------

+
-- Surjections

+

+
module _ where

+
  open Surjection

+

+
  Σ-↠ : (I↠J : I ↠ J) →

+
       (∀ {x} → A x ↠ B (to I↠J x)) →

+
       Σ I A ↠ Σ J B

+
  Σ-↠ {I = I} {J = J} {A = A} {B = B} I↠J A↠B =

+
    mk↠ₛ strictlySurjective′

+
    where

+
    to′ : Σ I A → Σ J B

+
    to′ = map (to I↠J) (to A↠B)

+

+
    backcast : ∀ {i} → B i → B (to I↠J (to⁻ I↠J i))

+
    backcast = ≡.subst B (≡.sym (to∘to⁻ I↠J _))

+

+
    to⁻′ : Σ J B → Σ I A

+
    to⁻′ = map (to⁻ I↠J) (Surjection.to⁻ A↠B ∘ backcast)

+

+
    strictlySurjective′ : StrictlySurjective _≡_ to′

+
    strictlySurjective′ (x , y) = to⁻′ (x , y) , Σ-≡,≡→≡

+
      ( to∘to⁻ I↠J x

+
      , (≡.subst B (to∘to⁻ I↠J x) (to A↠B (to⁻ A↠B (backcast y))) ≡⟨ ≡.cong (≡.subst B _) (to∘to⁻ A↠B _) ⟩

+
         ≡.subst B (to∘to⁻ I↠J x) (backcast y)                      ≡⟨ ≡.subst-subst-sym (to∘to⁻ I↠J x) ⟩

+
         y                                                          ∎)

+
      ) where open ≡.≡-Reasoning

+

+

+
------------------------------------------------------------------------

+
-- Left inverses

+

+
module _ where

+
  open LeftInverse

+

+
  Σ-↩ : (I↩J : I ↩ J) →

+
         (∀ {i} → A i ↩ B (to I↩J i)) →

+
         Σ I A ↩ Σ J B

+
  Σ-↩ {I = I} {J = J} {A = A} {B = B} I↩J A↩B = mk↩ {to = to′ } {from = from′} inv

+
    where

+
    to′ : Σ I A → Σ J B

+
    to′ = map (to I↩J) (to A↩B)

+

+
    backcast : ∀ {j} → B j → B (to I↩J (from I↩J j))

+
    backcast = ≡.subst B (≡.sym (inverseˡ I↩J ≡.refl))

+

+
    from′ : Σ J B → Σ I A

+
    from′ = map (from I↩J) (from A↩B ∘ backcast)

+

+
    inv : Inverseˡ _≡_ _≡_ to′ from′

+
    inv {j , b} ≡.refl = Σ-≡,≡→≡ (strictlyInverseˡ I↩J j  , (

+
      begin

+
        ≡.subst B (inverseˡ I↩J ≡.refl) (to A↩B (from A↩B (backcast b))) ≡⟨ ≡.cong (≡.subst B _) (inverseˡ A↩B ≡.refl) ⟩

+
        ≡.subst B (inverseˡ I↩J ≡.refl) (backcast b)                       ≡⟨ ≡.subst-subst-sym (inverseˡ I↩J _) ⟩

+
        b                                                                  ∎)) where open ≡.≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Right inverses

+

+
------------------------------------------------------------------------

+
-- Inverses

+

+
module _ where

+
  open Inverse

+

+
  Σ-↔ : (I↔J : I ↔ J) →

+
      (∀ {x} → A x ↔ B (to I↔J x)) →

+
      Σ I A ↔ Σ J B

+
  Σ-↔ {I = I} {J = J} {A = A} {B = B} I↔J A↔B = mk↔ₛ′

+
    (Surjection.to surjection′)

+
    (Surjection.to⁻ surjection′)

+
    (Surjection.to∘to⁻ surjection′)

+
    left-inverse-of

+
    where

+
    open ≡.≡-Reasoning

+

+
    I≃J = ↔⇒≃ I↔J

+

+
    surjection′ : Σ I A ↠ Σ J B

+
    surjection′ = Σ-↠ (↔⇒↠ (≃⇒↔ I≃J)) (↔⇒↠ A↔B)

+

+
    left-inverse-of : ∀ p → Surjection.to⁻ surjection′ (Surjection.to surjection′ p) ≡ p

+
    left-inverse-of (x , y) = to Σ-≡,≡↔≡

+
      ( _≃_.left-inverse-of I≃J x

+
      , (≡.subst A (_≃_.left-inverse-of I≃J x)

+
           (from A↔B

+
              (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J

+
                                    (_≃_.to I≃J x)))

+
                 (to A↔B y)))                   ≡⟨ ≡.subst-application B (λ _ → from A↔B) _ ⟩

+

+
         from A↔B

+
           (≡.subst B (≡.cong (_≃_.to I≃J)

+
                          (_≃_.left-inverse-of I≃J x))

+
              (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J

+
                                    (_≃_.to I≃J x)))

+
                 (to A↔B y)))                   ≡⟨ ≡.cong (λ eq → from A↔B (≡.subst B eq

+
                                                                                  (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J _)) _)))

+
                                                                   (_≃_.left-right I≃J _) ⟩

+
         from A↔B

+
           (≡.subst B (_≃_.right-inverse-of I≃J

+
                          (_≃_.to I≃J x))

+
              (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J

+
                                    (_≃_.to I≃J x)))

+
                 (to A↔B y)))                   ≡⟨ ≡.cong (from A↔B)

+
                                                                   (≡.subst-subst-sym (_≃_.right-inverse-of I≃J _)) ⟩

+

+
         from A↔B (to A↔B y)      ≡⟨ Inverse.strictlyInverseʳ A↔B _ ⟩

+

+
         y                                                    ∎)

+
      )

+

+

+
private module _ where

+
  open Inverse

+

+
  swap-coercions : ∀ {k} (B : J → Set b)

+
    (I↔J : _↔_ I J) →

+
    (∀ {x} → A x ∼[ k ] B (to I↔J x)) →

+
    ∀ {x} → A (from I↔J x) ∼[ k ] B x

+
  swap-coercions {A = A} B I↔J eq {x} =

+
    A (from I↔J x)           ∼⟨ eq ⟩

+
    B (to I↔J (from I↔J x)) ↔⟨ K-reflexive (≡.cong B $ strictlyInverseˡ I↔J x) ⟩

+
    B x                       ∎

+
    where open EquationalReasoning

+

+

+
cong : ∀ {k} (I↔J : I ↔ J) →

+
       (∀ {x} → A x ∼[ k ] B (Inverse.to I↔J x)) →

+
       Σ I A ∼[ k ] Σ J B

+
cong {k = implication}                I↔J A⟶B = Σ-⟶ (↔⇒⟶ I↔J) A⟶B

+
cong {B = B} {k = reverseImplication} I↔J A⟵B = Σ-⟶ (↔⇒⟵ I↔J) (swap-coercions {k = reverseImplication} B I↔J A⟵B)

+
cong {k = equivalence}                I↔J A⇔B = Σ-⇔ (↔⇒↠ I↔J) A⇔B

+
cong {k = injection}                  I↔J A↣B = Σ-↣ I↔J A↣B

+
cong {B = B} {k = reverseInjection}   I↔J A↢B = Σ-↣ (↔-sym I↔J) (swap-coercions {k = reverseInjection} B I↔J A↢B)

+
cong {B = B} {k = leftInverse}        I↔J A↩B = ↩⇒↪ (Σ-↩ (↔⇒↩ (↔-sym I↔J)) (↪⇒↩ (swap-coercions {k = leftInverse} B I↔J A↩B)))

+
cong {k = surjection}                 I↔J A↠B = Σ-↠ (↔⇒↠ I↔J) A↠B

+
cong {k = bijection}                  I↔J A↔B = Σ-↔ I↔J A↔B

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Function.NonDependent.Propositional.html b/site/static/agda_html/Data.Product.Function.NonDependent.Propositional.html
new file mode 100644
index 000000000..82d3a7e74
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Function.NonDependent.Propositional.html
@@ -0,0 +1,164 @@
+
+Data.Product.Function.NonDependent.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Non-dependent product combinators for propositional equality

+
-- preserving functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Function.NonDependent.Propositional where

+

+
open import Data.Product.Base using (_×_; map)

+
open import Data.Product.Function.NonDependent.Setoid

+
open import Data.Product.Relation.Binary.Pointwise.NonDependent

+
  using (_×ₛ_; Pointwise-≡↔≡)

+
open import Function.Base using (id)

+
open import Function.Bundles

+
  using (Inverse; _⟶_; _⇔_; _↣_; _↠_; _⤖_; _↩_; _↪_; _↔_)

+
open import Function.Properties.Inverse as Inv

+
  using (Inverse⇒Equivalence; Inverse⇒Injection; Inverse⇒Surjection;

+
         Inverse⇒Bijection)

+
open import Function.Related.Propositional

+
  using (_∼[_]_; implication; reverseImplication; equivalence; injection;

+
         reverseInjection; leftInverse; surjection; bijection)

+
import Function.Construct.Composition as Compose

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary hiding (_⇔_)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
private

+
  variable

+
    a b c d : Level

+
    A B C D : Set a

+

+
------------------------------------------------------------------------

+
-- Helper lemma

+

+
private

+
  liftViaInverse : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →

+
                   (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →

+
                   (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →

+
                   (R (setoid A) (setoid C) → R (setoid B) (setoid D) → R (setoid A ×ₛ setoid B) (setoid C ×ₛ setoid D)) →

+
                   R (setoid A) (setoid C) → R (setoid B) (setoid D) →

+
                   R (setoid (A × B)) (setoid (C × D))

+
  liftViaInverse trans inv⇒R lift RAC RBD =

+
    Inv.transportVia trans inv⇒R (Inv.sym Pointwise-≡↔≡) (lift RAC RBD) Pointwise-≡↔≡

+

+
------------------------------------------------------------------------

+
-- Combinators for various function types

+

+
infixr 2 _×-⟶_ _×-⇔_ _×-↣_ _×-↠_ _×-⤖_ _×-↩_ _×-↪_ _×-↔_

+

+
_×-⟶_ : A ⟶ B → C ⟶ D → (A × C) ⟶ (B × D)

+
_×-⟶_ = liftViaInverse Compose.function Inv.toFunction _×-function_

+

+
_×-⇔_ : A ⇔ B → C ⇔ D → (A × C) ⇔ (B × D)

+
_×-⇔_ = liftViaInverse Compose.equivalence Inverse⇒Equivalence _×-equivalence_

+

+
_×-↣_ : A ↣ B → C ↣ D → (A × C) ↣ (B × D)

+
_×-↣_ = liftViaInverse Compose.injection Inverse⇒Injection _×-injection_

+

+
_×-↠_ : A ↠ B → C ↠ D → (A × C) ↠ (B × D)

+
_×-↠_ = liftViaInverse Compose.surjection Inverse⇒Surjection _×-surjection_

+

+
_×-⤖_ : A ⤖ B → C ⤖ D → (A × C) ⤖ (B × D)

+
_×-⤖_ = liftViaInverse Compose.bijection Inverse⇒Bijection _×-bijection_

+

+
_×-↩_ : A ↩ B → C ↩ D → (A × C) ↩ (B × D)

+
_×-↩_ = liftViaInverse Compose.leftInverse Inverse.leftInverse _×-leftInverse_

+

+
_×-↪_ : A ↪ B → C ↪ D → (A × C) ↪ (B × D)

+
_×-↪_ = liftViaInverse Compose.rightInverse Inverse.rightInverse _×-rightInverse_

+

+
_×-↔_ : A ↔ B → C ↔ D → (A × C) ↔ (B × D)

+
_×-↔_ = liftViaInverse Compose.inverse id _×-inverse_

+

+
infixr 2 _×-cong_

+

+
_×-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A × C) ∼[ k ] (B × D)

+
_×-cong_ {k = implication}         = _×-⟶_

+
_×-cong_ {k = reverseImplication}  = _×-⟶_

+
_×-cong_ {k = equivalence}         = _×-⇔_

+
_×-cong_ {k = injection}           = _×-↣_

+
_×-cong_ {k = reverseInjection}    = _×-↣_

+
_×-cong_ {k = leftInverse}         = _×-↪_

+
_×-cong_ {k = surjection}          = _×-↠_

+
_×-cong_ {k = bijection}           = _×-↔_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Function.NonDependent.Setoid.html b/site/static/agda_html/Data.Product.Function.NonDependent.Setoid.html
new file mode 100644
index 000000000..dccf51574
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Function.NonDependent.Setoid.html
@@ -0,0 +1,213 @@
+
+Data.Product.Function.NonDependent.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Non-dependent product combinators for setoid equality preserving

+
-- functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Function.NonDependent.Setoid where

+

+
open import Data.Product.Base as Product

+
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)

+
open import Level using (Level)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Function.Bundles

+
  using (Func; Equivalence; Injection; Surjection; Bijection; LeftInverse;

+
         RightInverse; Inverse)

+

+
private

+
  variable

+
    a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂ : Level

+
    a ℓ : Level

+
    A B C D : Setoid a ℓ

+

+
------------------------------------------------------------------------

+
-- Combinators for equality preserving functions

+

+
proj₁ₛ : Func (A ×ₛ B) A

+
proj₁ₛ = record { to = proj₁ ; cong = proj₁ }

+

+
proj₂ₛ : Func (A ×ₛ B) B

+
proj₂ₛ = record { to = proj₂ ; cong = proj₂ }

+

+
<_,_>ₛ : Func A B → Func A C → Func A (B ×ₛ C)

+
< f , g >ₛ = record

+
  { to   = < to   f , to   g >

+
  ; cong = < cong f , cong g >

+
  } where open Func

+

+
swapₛ : Func (A ×ₛ B) (B ×ₛ A)

+
swapₛ = < proj₂ₛ , proj₁ₛ >ₛ

+

+
------------------------------------------------------------------------

+
-- Function bundles

+

+
_×-function_ : Func A B → Func C D → Func (A ×ₛ C) (B ×ₛ D)

+
f ×-function g = record

+
  { to    = Product.map (to f) (to g)

+
  ; cong  = Product.map (cong f) (cong g)

+
  } where open Func

+

+
infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_

+

+
_×-equivalence_ : Equivalence A B → Equivalence C D →

+
                  Equivalence (A ×ₛ C) (B ×ₛ D)

+
_×-equivalence_ f g = record

+
  { to        = Product.map (to f) (to g)

+
  ; from      = Product.map (from f) (from g)

+
  ; to-cong   = Product.map (to-cong f) (to-cong g)

+
  ; from-cong = Product.map (from-cong f) (from-cong g)

+
  } where open Equivalence

+

+
_×-injection_ : Injection A B → Injection C D →

+
                Injection (A ×ₛ C) (B ×ₛ D)

+
f ×-injection g = record

+
  { to        = Product.map (to f) (to g)

+
  ; cong      = Product.map (cong f) (cong g)

+
  ; injective = Product.map (injective f) (injective g)

+
  } where open Injection

+

+
_×-surjection_ : Surjection A B → Surjection C D →

+
                 Surjection (A ×ₛ C) (B ×ₛ D)

+
f ×-surjection g = record

+
  { to         = Product.map (to f) (to g)

+
  ; cong       = Product.map (cong f) (cong g)

+
  ; surjective = λ y → Product.zip _,_ (λ ff gg x₂ → (ff (proj₁ x₂)) , (gg (proj₂ x₂))) (surjective f (proj₁ y)) (surjective g (proj₂ y))

+
  } where open Surjection

+

+
_×-bijection_ : Bijection A B → Bijection C D →

+
                Bijection (A ×ₛ C) (B ×ₛ D)

+
f ×-bijection g = record

+
  { to         = Product.map (to f) (to g)

+
  ; cong       = Product.map (cong f) (cong g)

+
  ; bijective  = Product.map (injective f) (injective g) ,

+
                 λ { (y₀ , y₁) → Product.zip _,_ (λ {ff gg (x₀ , x₁) → ff x₀ , gg x₁}) (surjective f y₀) (surjective g y₁)}

+
  } where open Bijection

+

+
_×-leftInverse_ : LeftInverse A B → LeftInverse C D →

+
                  LeftInverse (A ×ₛ C) (B ×ₛ D)

+
f ×-leftInverse g = record

+
  { to        = Product.map (to f) (to g)

+
  ; from      = Product.map (from f) (from g)

+
  ; to-cong   = Product.map (to-cong f) (to-cong g)

+
  ; from-cong = Product.map (from-cong f) (from-cong g)

+
  ; inverseˡ   = λ x → inverseˡ f (proj₁ x) , inverseˡ g (proj₂ x)

+
  } where open LeftInverse

+

+
_×-rightInverse_ : RightInverse A B → RightInverse C D →

+
                   RightInverse (A ×ₛ C) (B ×ₛ D)

+
f ×-rightInverse g = record

+
  { to        = Product.map (to f) (to g)

+
  ; from      = Product.map (from f) (from g)

+
  ; to-cong   = Product.map (to-cong f) (to-cong g)

+
  ; from-cong = Product.map (from-cong f) (from-cong g)

+
  ; inverseʳ   = λ x → inverseʳ f (proj₁ x) , inverseʳ g (proj₂ x)

+
  } where open RightInverse

+

+
infixr 2 _×-surjection_ _×-inverse_

+

+
_×-inverse_ : Inverse A B → Inverse C D →

+
              Inverse (A ×ₛ C) (B ×ₛ D)

+
f ×-inverse g = record

+
  { to        = Product.map (to f) (to g)

+
  ; from      = Product.map (from f) (from g)

+
  ; to-cong   = Product.map (to-cong f) (to-cong g)

+
  ; from-cong = Product.map (from-cong f) (from-cong g)

+
  ; inverse   = (λ x → inverseˡ f (proj₁ x) , inverseˡ g (proj₂ x)) ,

+
                (λ x → inverseʳ f (proj₁ x) , inverseʳ g (proj₂ x))

+
  } where open Inverse

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
_×-left-inverse_ = _×-leftInverse_

+
{-# WARNING_ON_USAGE _×-left-inverse_

+
"Warning: _×-left-inverse_ was deprecated in v2.0.

+
Please use _×-leftInverse_ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Nary.NonDependent.html b/site/static/agda_html/Data.Product.Nary.NonDependent.html
new file mode 100644
index 000000000..3d4dfbbc2
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Nary.NonDependent.html
@@ -0,0 +1,324 @@
+
+Data.Product.Nary.NonDependent
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Nondependent heterogeneous N-ary products

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Nary.NonDependent where

+

+
------------------------------------------------------------------------

+
-- Concrete examples can be found in README.Nary. This file's comments

+
-- are more focused on the implementation details and the motivations

+
-- behind the design decisions.

+
------------------------------------------------------------------------

+

+
open import Level using (Level)

+
open import Agda.Builtin.Unit

+
open import Data.Product.Base as Prod

+
import Data.Product.Properties as Prodₚ

+
open import Data.Sum.Base using (_⊎_)

+
open import Data.Nat.Base using (ℕ; zero; suc; pred)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Function.Base using (const; _∘′_; _∘_)

+
open import Relation.Nullary.Decidable.Core using (Dec; yes; no; _×-dec_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)

+

+
open import Function.Nary.NonDependent.Base

+

+
-- Provided n Levels and a corresponding "vector" of `n` Sets, we can

+
-- build a big right-nested product type packing a value for each one

+
-- of these Sets.

+
-- We have two distinct but equivalent definitions:

+
-- the first which is always ⊤-terminated

+
-- the other which has a special case for n = 1 because we want our

+
-- `(un)curryₙ` functions to work for user-written functions and

+
-- products and they rarely are ⊤-terminated.

+

+
Product⊤ : ∀ n {ls} → Sets n ls → Set (⨆ n ls)

+
Product⊤ zero    as       = ⊤

+
Product⊤ (suc n) (a , as) = a × Product⊤ n as

+

+
Product : ∀ n {ls} → Sets n ls → Set (⨆ n ls)

+
Product 0       _        = ⊤

+
Product 1       (a , _)  = a

+
Product (suc n) (a , as) = a × Product n as

+

+
-- Pointwise lifting of a relation on products

+

+
Allₙ : (∀ {a} {A : Set a} → Rel A a) →

+
        ∀ n {ls} {as : Sets n ls} (l r : Product n as) → Sets n ls

+
Allₙ R 0               l       r       = _

+
Allₙ R 1               a       b       = R a b , _

+
Allₙ R (suc n@(suc _)) (a , l) (b , r) = R a b , Allₙ R n l r

+

+
Equalₙ : ∀ n {ls} {as : Sets n ls} (l r : Product n as) → Sets n ls

+
Equalₙ = Allₙ _≡_

+

+
------------------------------------------------------------------------

+
-- Generic Programs

+

+
-- Once we have these type definitions, we can write generic programs

+
-- over them. They will typically be split into two or three definitions:

+

+
-- 1. action on the vector of n levels (if any)

+
-- 2. action on the corresponding vector of n Sets

+
-- 3. actual program, typed thank to the function defined in step 2.

+
------------------------------------------------------------------------

+

+
-- see Relation.Binary.PropositionalEquality for congₙ and substₙ, two

+
-- equality-related generic programs.

+

+
------------------------------------------------------------------------

+
-- equivalence of Product and Product⊤

+

+
toProduct : ∀ n {ls} {as : Sets n ls} → Product⊤ n as → Product n as

+
toProduct 0               _        = _

+
toProduct 1               (v , _)  = v

+
toProduct (suc n@(suc _)) (v , vs) = v , toProduct n vs

+

+
toProduct⊤ : ∀ n {ls} {as : Sets n ls} → Product n as → Product⊤ n as

+
toProduct⊤ 0               _        = _

+
toProduct⊤ 1               v        = v , _

+
toProduct⊤ (suc n@(suc _)) (v , vs) = v , toProduct⊤ n vs

+

+
------------------------------------------------------------------------

+
-- (un)curry

+

+
-- We start by defining `curryₙ` and `uncurryₙ` converting back and forth

+
-- between `A₁ → ⋯ → Aₙ → B` and `(A₁ × ⋯ × Aₙ) → B` by induction on `n`.

+

+
curryₙ : ∀ n {ls} {as : Sets n ls} {r} {b : Set r} →

+
         (Product n as → b) → as ⇉ b

+
curryₙ 0               f = f _

+
curryₙ 1               f = f

+
curryₙ (suc n@(suc _)) f = curryₙ n ∘′ curry f

+

+
uncurryₙ : ∀ n {ls} {as : Sets n ls} {r} {b : Set r} →

+
           as ⇉ b → (Product n as → b)

+
uncurryₙ 0               f = const f

+
uncurryₙ 1               f = f

+
uncurryₙ (suc n@(suc _)) f = uncurry (uncurryₙ n ∘′ f)

+

+
-- Variants for Product⊤

+

+
curry⊤ₙ : ∀ n {ls} {as : Sets n ls} {r} {b : Set r} →

+
          (Product⊤ n as → b) → as ⇉ b

+
curry⊤ₙ zero    f = f _

+
curry⊤ₙ (suc n) f = curry⊤ₙ n ∘′ curry f

+

+
uncurry⊤ₙ : ∀ n {ls} {as : Sets n ls} {r} {b : Set r} →

+
            as ⇉ b → (Product⊤ n as → b)

+
uncurry⊤ₙ zero    f = const f

+
uncurry⊤ₙ (suc n) f = uncurry (uncurry⊤ₙ n ∘′ f)

+

+
------------------------------------------------------------------------

+
-- decidability

+

+
Product⊤-dec : ∀ n {ls} {as : Sets n ls} → Product⊤ n (Dec <$> as) → Dec (Product⊤ n as)

+
Product⊤-dec zero    _          = yes _

+
Product⊤-dec (suc n) (p? , ps?) = p? ×-dec Product⊤-dec n ps?

+

+
Product-dec : ∀ n {ls} {as : Sets n ls} → Product n (Dec <$> as) → Dec (Product n as)

+
Product-dec 0               _          = yes _

+
Product-dec 1               p?         = p?

+
Product-dec (suc n@(suc _)) (p? , ps?) = p? ×-dec Product-dec n ps?

+

+
------------------------------------------------------------------------

+
-- pointwise liftings

+

+
toEqualₙ : ∀ n {ls} {as : Sets n ls} {l r : Product n as} →

+
           l ≡ r → Product n (Equalₙ n l r)

+
toEqualₙ 0               eq = _

+
toEqualₙ 1               eq = eq

+
toEqualₙ (suc n@(suc _)) eq = Prod.map₂ (toEqualₙ n) (Prodₚ.,-injective eq)

+

+
fromEqualₙ : ∀ n {ls} {as : Sets n ls} {l r : Product n as} →

+
             Product n (Equalₙ n l r) → l ≡ r

+
fromEqualₙ 0               eq = refl

+
fromEqualₙ 1               eq = eq

+
fromEqualₙ (suc n@(suc _)) eq = uncurry (cong₂ _,_) (Prod.map₂ (fromEqualₙ n) eq)

+

+
------------------------------------------------------------------------

+
-- projection of the k-th component

+

+
-- To know at which Set level the k-th projection out of an n-ary

+
-- product lives, we need to extract said level, by induction on k.

+

+
Levelₙ : ∀ {n} → Levels n → Fin n → Level

+
Levelₙ (l , _)  zero    = l

+
Levelₙ (_ , ls) (suc k) = Levelₙ ls k

+

+
-- Once we have the Sets used in the product, we can extract the one we

+
-- are interested in, once more by induction on k.

+

+
Projₙ : ∀ {n ls} → Sets n ls → ∀ k → Set (Levelₙ ls k)

+
Projₙ (a , _)  zero    = a

+
Projₙ (_ , as) (suc k) = Projₙ as k

+

+
-- Finally, provided a Product of these sets, we can extract the k-th

+
-- value. `projₙ` takes both `n` and `k` explicitly because we expect

+
-- the user will be using a concrete `k` (potentially manufactured

+
-- using `Data.Fin`'s `#_`) and it will not be possible to infer `n`

+
-- from it.

+

+
projₙ : ∀ n {ls} {as : Sets n ls} k → Product n as → Projₙ as k

+
projₙ 1               zero    v        = v

+
projₙ (suc n@(suc _)) zero    (v , _)  = v

+
projₙ (suc n@(suc _)) (suc k) (_ , vs) = projₙ n k vs

+
projₙ 1 (suc ()) v

+

+
------------------------------------------------------------------------

+
-- zip

+

+
zipWith : ∀ n {lsa lsb lsc}

+
          {as : Sets n lsa} {bs : Sets n lsb} {cs : Sets n lsc} →

+
          (∀ k → Projₙ as k → Projₙ bs k → Projₙ cs k) →

+
          Product n as → Product n bs → Product n cs

+
zipWith 0               f _        _        = _

+
zipWith 1               f v        w        = f zero v w

+
zipWith (suc n@(suc _)) f (v , vs) (w , ws) =

+
  f zero v w , zipWith n (λ k → f (suc k)) vs ws

+

+
------------------------------------------------------------------------

+
-- removal of the k-th component

+

+
Levelₙ⁻ : ∀ {n} → Levels n → Fin n → Levels (pred n)

+
Levelₙ⁻               (_ , ls) zero    = ls

+
Levelₙ⁻ {suc (suc _)} (l , ls) (suc k) = l , Levelₙ⁻ ls k

+
Levelₙ⁻ {1} _ (suc ())

+

+
Removeₙ : ∀ {n ls} → Sets n ls → ∀ k → Sets (pred n) (Levelₙ⁻ ls k)

+
Removeₙ               (_ , as) zero    = as

+
Removeₙ {suc (suc _)} (a , as) (suc k) = a , Removeₙ as k

+
Removeₙ {1} _ (suc ())

+

+
removeₙ : ∀ n {ls} {as : Sets n ls} k →

+
          Product n as → Product (pred n) (Removeₙ as k)

+
removeₙ (suc zero)          zero    _        = _

+
removeₙ (suc (suc _))       zero    (_ , vs) = vs

+
removeₙ (suc (suc zero))    (suc k) (v , _)  = v

+
removeₙ (suc (suc (suc _))) (suc k) (v , vs) = v , removeₙ _ k vs

+
removeₙ (suc zero) (suc ()) _

+

+
------------------------------------------------------------------------

+
-- insertion of a k-th component

+

+
Levelₙ⁺ : ∀ {n} → Levels n → Fin (suc n) → Level → Levels (suc n)

+
Levelₙ⁺         ls       zero    l⁺ = l⁺ , ls

+
Levelₙ⁺ {suc _} (l , ls) (suc k) l⁺ = l , Levelₙ⁺ ls k l⁺

+
Levelₙ⁺ {0} _ (suc ())

+

+
Insertₙ : ∀ {n ls l⁺} → Sets n ls → ∀ k (a⁺ : Set l⁺) → Sets (suc n) (Levelₙ⁺ ls k l⁺)

+
Insertₙ         as       zero    a⁺ = a⁺ , as

+
Insertₙ {suc _} (a , as) (suc k) a⁺ = a , Insertₙ as k a⁺

+
Insertₙ {zero} _ (suc ()) _

+

+
insertₙ : ∀ n {ls l⁺} {as : Sets n ls} {a⁺ : Set l⁺} k (v⁺ : a⁺) →

+
          Product n as → Product (suc n) (Insertₙ as k a⁺)

+
insertₙ 0               zero    v⁺ vs       = v⁺

+
insertₙ (suc n)         zero    v⁺ vs       = v⁺ , vs

+
insertₙ 1               (suc k) v⁺ vs       = vs , insertₙ 0 k v⁺ _

+
insertₙ (suc n@(suc _)) (suc k) v⁺ (v , vs) = v , insertₙ n k v⁺ vs

+
insertₙ 0 (suc ()) _ _

+

+
------------------------------------------------------------------------

+
-- update of a k-th component

+

+
Levelₙᵘ : ∀ {n} → Levels n → Fin n → Level → Levels n

+
Levelₙᵘ (_ , ls) zero    lᵘ = lᵘ , ls

+
Levelₙᵘ (l , ls) (suc k) lᵘ = l , Levelₙᵘ ls k lᵘ

+

+
Updateₙ : ∀ {n ls lᵘ} (as : Sets n ls) k (aᵘ : Set lᵘ) → Sets n (Levelₙᵘ ls k lᵘ)

+
Updateₙ (_ , as) zero    aᵘ = aᵘ , as

+
Updateₙ (a , as) (suc k) aᵘ = a , Updateₙ as k aᵘ

+

+
updateₙ : ∀ n {ls lᵘ} {as : Sets n ls} k {aᵘ : _ → Set lᵘ} (f : ∀ v → aᵘ v)

+
          (vs : Product n as) → Product n (Updateₙ as k (aᵘ (projₙ n k vs)))

+
updateₙ 1               zero    f v        = f v

+
updateₙ (suc (suc _))   zero    f (v , vs) = f v , vs

+
updateₙ (suc n@(suc _)) (suc k) f (v , vs) = v , updateₙ n k f vs

+
updateₙ 1 (suc ()) _ _

+

+
updateₙ′ : ∀ n {ls lᵘ} {as : Sets n ls} k {aᵘ : Set lᵘ} (f : Projₙ as k → aᵘ) →

+
           Product n as → Product n (Updateₙ as k aᵘ)

+
updateₙ′ n k = updateₙ n k

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Properties.Ext.html b/site/static/agda_html/Data.Product.Properties.Ext.html
new file mode 100644
index 000000000..8815e37dc
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Properties.Ext.html
@@ -0,0 +1,122 @@
+
+Data.Product.Properties.Ext
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Data.Product.Properties.Ext where

+

+
open import Data.Product.Base as Prod

+
open import Data.Product.Function.Dependent.Propositional renaming (cong to ∃-cong)

+
open import Data.Sum.Base

+
open import Function

+
open import Function.Related.Propositional

+
open import Level

+
open import Relation.Binary.PropositionalEquality hiding (cong; [_])

+
open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)

+

+
-- TODO: remove this once it's in the stdlib

+

+
private variable

+
  ℓ ℓ′ : Level

+
  A B C : Set ℓ

+
  a a₁ a₂ b₁ b₂ : Level

+
  k : Kind

+

+
∃-cong′ : {A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂}

+
        → (∀ {x} → B₁ x ∼[ k ] B₂ x)

+
        → Σ A B₁ ∼[ k ] Σ A B₂

+
∃-cong′ = ∃-cong (mk↔ (id , id))

+

+
∃-≡ : ∀ {A : Set a₁} (P : A → Set a₂) {x} → P x ⇔ (∃[ y ] y ≡ x × P y)

+
∃-≡ P {x} = mk⇔ (λ Px → x , refl , Px) (λ where (y , (refl , Py)) → Py)

+

+
∃-distrib-⊎ : {P : A → Set ℓ} {Q : A → Set ℓ′} → (∃[ a ] (P a ⊎ Q a)) ↔ (∃[ a ] P a ⊎ ∃[ a ] Q a)

+
∃-distrib-⊎ = mk↔ₛ′

+
  (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)

+
  [ Prod.map₂ inj₁ , Prod.map₂ inj₂ ]′

+
  [ (λ _ → P.refl) , (λ _ → P.refl) ]

+
  (uncurry λ _ → [ (λ _ → P.refl) , (λ _ → P.refl) ])

+

+
-- the version in Function.Related.TypeIsomorphisms isn't level-polymorphic enough

+
×-distribˡ-⊎' : (A × (B ⊎ C)) ↔ (A × B ⊎ A × C)

+
×-distribˡ-⊎' = ∃-distrib-⊎

+

+
×-⇔-swap : (A × B × C) ⇔ (B × A × C)

+
×-⇔-swap {A = A}{B = B}{C = C} = let open EquationalReasoning in

+
  (A × B × C) ∼⟨ mk⇔ (λ x → (proj₁ ∘ proj₂) x , proj₁ x , (proj₂ ∘ proj₂) x)

+
                     (λ x → (proj₁ ∘ proj₂) x , proj₁ x , (proj₂ ∘ proj₂) x) ⟩

+
  (B × A × C) ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Properties.html b/site/static/agda_html/Data.Product.Properties.html
new file mode 100644
index 000000000..9185edbeb
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Properties.html
@@ -0,0 +1,190 @@
+
+Data.Product.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of products

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Properties where

+

+
open import Axiom.UniquenessOfIdentityProofs using (UIP; module Decidable⇒UIP)

+
open import Data.Product.Base using (_,_; Σ; _×_; map; swap; ∃; ∃₂)

+
open import Function.Base using (_∋_; _∘_; id)

+
open import Function.Bundles using (_↔_; mk↔ₛ′)

+
open import Level using (Level)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; _≗_; subst; cong; cong₂; cong′)

+
open import Relation.Nullary.Decidable as Dec using (Dec; yes; no)

+

+
private

+
  variable

+
    a b c d ℓ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+

+
------------------------------------------------------------------------

+
-- Equality (dependent)

+

+
module _ {B : A → Set b} where

+

+
  ,-injectiveˡ : ∀ {a c} {b : B a} {d : B c} → (a , b) ≡ (c , d) → a ≡ c

+
  ,-injectiveˡ refl = refl

+

+
  ,-injectiveʳ-≡ : ∀ {a b} {c : B a} {d : B b} → UIP A → (a , c) ≡ (b , d) → (q : a ≡ b) → subst B q c ≡ d

+
  ,-injectiveʳ-≡ {c = c} u refl q = cong (λ x → subst B x c) (u q refl)

+

+
  ,-injectiveʳ-UIP : ∀ {a} {b c : B a} → UIP A → (Σ A B ∋ (a , b)) ≡ (a , c) → b ≡ c

+
  ,-injectiveʳ-UIP u p = ,-injectiveʳ-≡ u p refl

+

+
  ≡-dec : DecidableEquality A → (∀ {a} → DecidableEquality (B a)) →

+
          DecidableEquality (Σ A B)

+
  ≡-dec dec₁ dec₂ (a , x) (b , y) with dec₁ a b

+
  ... | no [a≢b] = no ([a≢b] ∘ ,-injectiveˡ)

+
  ... | yes refl = Dec.map′ (cong (a ,_)) (,-injectiveʳ-UIP (Decidable⇒UIP.≡-irrelevant dec₁)) (dec₂ x y)

+

+
------------------------------------------------------------------------

+
-- Equality (non-dependent)

+

+
,-injectiveʳ : ∀ {a c : A} {b d : B} → (a , b) ≡ (c , d) → b ≡ d

+
,-injectiveʳ refl = refl

+

+
,-injective : ∀ {a c : A} {b d : B} → (a , b) ≡ (c , d) → a ≡ c × b ≡ d

+
,-injective refl = refl , refl

+

+
map-cong : ∀ {f g : A → C} {h i : B → D} → f ≗ g → h ≗ i → map f h ≗ map g i

+
map-cong f≗g h≗i (x , y) = cong₂ _,_ (f≗g x) (h≗i y)

+

+
-- The following properties are definitionally true (because of η)

+
-- but for symmetry with ⊎ it is convenient to define and name them.

+

+
swap-involutive : swap {A = A} {B = B} ∘ swap ≗ id

+
swap-involutive _ = refl

+

+
------------------------------------------------------------------------

+
-- Equality between pairs can be expressed as a pair of equalities

+

+
module _ {A : Set a} {B : A → Set b} {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : Σ A B} where

+
  Σ-≡,≡→≡ : Σ (a₁ ≡ a₂) (λ p → subst B p b₁ ≡ b₂) → p₁ ≡ p₂

+
  Σ-≡,≡→≡ (refl , refl) = refl

+

+
  Σ-≡,≡←≡ : p₁ ≡ p₂ → Σ (a₁ ≡ a₂) (λ p → subst B p b₁ ≡ b₂)

+
  Σ-≡,≡←≡ refl = refl , refl

+

+
  private

+
    left-inverse-of : (p : Σ (a₁ ≡ a₂) (λ x → subst B x b₁ ≡ b₂)) →

+
                      Σ-≡,≡←≡ (Σ-≡,≡→≡ p) ≡ p

+
    left-inverse-of (refl , refl) = refl

+

+
    right-inverse-of : (p : p₁ ≡ p₂) → Σ-≡,≡→≡ (Σ-≡,≡←≡ p) ≡ p

+
    right-inverse-of refl = refl

+

+
  Σ-≡,≡↔≡ : (∃ λ (p : a₁ ≡ a₂) → subst B p b₁ ≡ b₂) ↔ p₁ ≡ p₂

+
  Σ-≡,≡↔≡ = mk↔ₛ′ Σ-≡,≡→≡ Σ-≡,≡←≡ right-inverse-of left-inverse-of

+

+
-- the non-dependent case. Proofs are exactly as above, and straightforward.

+
module _ {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : A × B} where

+
  ×-≡,≡→≡ : (a₁ ≡ a₂ × b₁ ≡ b₂) → p₁ ≡ p₂

+
  ×-≡,≡→≡ (refl , refl) = refl

+

+
  ×-≡,≡←≡ : p₁ ≡ p₂ → (a₁ ≡ a₂ × b₁ ≡ b₂)

+
  ×-≡,≡←≡ refl = refl , refl

+

+
  ×-≡,≡↔≡ : (a₁ ≡ a₂ × b₁ ≡ b₂) ↔ p₁ ≡ p₂

+
  ×-≡,≡↔≡ = mk↔ₛ′

+
    ×-≡,≡→≡

+
    ×-≡,≡←≡

+
    (λ { refl          → refl        })

+
    (λ { (refl , refl) → refl        })

+

+
------------------------------------------------------------------------

+
-- The order of ∃₂ can be swapped

+

+
∃∃↔∃∃ : (R : A → B → Set ℓ) → (∃₂ λ x y → R x y) ↔ (∃₂ λ y x → R x y)

+
∃∃↔∃∃ R = mk↔ₛ′ to from cong′ cong′

+
  where

+
  to : (∃₂ λ x y → R x y) → (∃₂ λ y x → R x y)

+
  to (x , y , Rxy) = (y , x , Rxy)

+

+
  from : (∃₂ λ y x → R x y) → (∃₂ λ x y → R x y)

+
  from (y , x , Rxy) = (x , y , Rxy)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Relation.Binary.Pointwise.NonDependent.html b/site/static/agda_html/Data.Product.Relation.Binary.Pointwise.NonDependent.html
new file mode 100644
index 000000000..30b1c2217
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Relation.Binary.Pointwise.NonDependent.html
@@ -0,0 +1,283 @@
+
+Data.Product.Relation.Binary.Pointwise.NonDependent
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise products of binary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Relation.Binary.Pointwise.NonDependent where

+

+
open import Data.Product.Base as Product

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Level using (Level; _⊔_; 0ℓ)

+
open import Function.Base using (id)

+
open import Function.Bundles using (Inverse)

+
open import Relation.Nullary.Decidable using (_×-dec_)

+
open import Relation.Binary.Core using (REL; Rel; _⇒_)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Preorder; Poset; StrictPartialOrder)

+
open import Relation.Binary.Definitions

+
open import Relation.Binary.Structures

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level

+
    A B C D : Set a

+
    R S ≈₁ ≈₂ : Rel A ℓ₁

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
Pointwise : REL A B ℓ₁ → REL C D ℓ₂ → REL (A × C) (B × D) (ℓ₁ ⊔ ℓ₂)

+
Pointwise R S (a , c) (b , d) = (R a b) × (S c d)

+

+
------------------------------------------------------------------------

+
-- Pointwise preserves many relational properties

+

+
×-reflexive : ≈₁ ⇒ R → ≈₂ ⇒ S → Pointwise ≈₁ ≈₂ ⇒ Pointwise R S

+
×-reflexive refl₁ refl₂ = Product.map refl₁ refl₂

+

+
×-refl : Reflexive R → Reflexive S → Reflexive (Pointwise R S)

+
×-refl refl₁ refl₂ = refl₁ , refl₂

+

+
×-irreflexive₁ : Irreflexive ≈₁ R →

+
                 Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
×-irreflexive₁ ir x≈y x<y = ir (proj₁ x≈y) (proj₁ x<y)

+

+
×-irreflexive₂ : Irreflexive ≈₂ S →

+
                 Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
×-irreflexive₂ ir x≈y x<y = ir (proj₂ x≈y) (proj₂ x<y)

+

+
×-symmetric : Symmetric R → Symmetric S → Symmetric (Pointwise R S)

+
×-symmetric sym₁ sym₂ = Product.map sym₁ sym₂

+

+
×-transitive : Transitive R → Transitive S → Transitive (Pointwise R S)

+
×-transitive trans₁ trans₂ = Product.zip trans₁ trans₂

+

+
×-antisymmetric : Antisymmetric ≈₁ R → Antisymmetric ≈₂ S →

+
                  Antisymmetric (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
×-antisymmetric antisym₁ antisym₂ = Product.zip antisym₁ antisym₂

+

+
×-asymmetric₁ : Asymmetric R → Asymmetric (Pointwise R S)

+
×-asymmetric₁ asym₁ x<y y<x = asym₁ (proj₁ x<y) (proj₁ y<x)

+

+
×-asymmetric₂ : Asymmetric S → Asymmetric (Pointwise R S)

+
×-asymmetric₂ asym₂ x<y y<x = asym₂ (proj₂ x<y) (proj₂ y<x)

+

+
×-respectsʳ : R Respectsʳ ≈₁ → S Respectsʳ ≈₂ →

+
             (Pointwise R S) Respectsʳ (Pointwise ≈₁ ≈₂)

+
×-respectsʳ resp₁ resp₂ = Product.zip resp₁ resp₂

+

+
×-respectsˡ : R Respectsˡ ≈₁ → S Respectsˡ ≈₂ →

+
             (Pointwise R S) Respectsˡ (Pointwise ≈₁ ≈₂)

+
×-respectsˡ resp₁ resp₂ = Product.zip resp₁ resp₂

+

+
×-respects₂ : R Respects₂ ≈₁ → S Respects₂ ≈₂ →

+
              (Pointwise R S) Respects₂ (Pointwise ≈₁ ≈₂)

+
×-respects₂ = Product.zip ×-respectsʳ ×-respectsˡ

+

+
×-total : Symmetric R → Total R → Total S → Total (Pointwise R S)

+
×-total sym₁ total₁ total₂ (x₁ , x₂) (y₁ , y₂)

+
  with total₁ x₁ y₁ | total₂ x₂ y₂

+
... | inj₁ x₁∼y₁ | inj₁ x₂∼y₂ = inj₁ (     x₁∼y₁ , x₂∼y₂)

+
... | inj₁ x₁∼y₁ | inj₂ y₂∼x₂ = inj₂ (sym₁ x₁∼y₁ , y₂∼x₂)

+
... | inj₂ y₁∼x₁ | inj₂ y₂∼x₂ = inj₂ (     y₁∼x₁ , y₂∼x₂)

+
... | inj₂ y₁∼x₁ | inj₁ x₂∼y₂ = inj₁ (sym₁ y₁∼x₁ , x₂∼y₂)

+

+
×-decidable : Decidable R → Decidable S → Decidable (Pointwise R S)

+
×-decidable _≟₁_ _≟₂_ (x₁ , x₂) (y₁ , y₂) = (x₁ ≟₁ y₁) ×-dec (x₂ ≟₂ y₂)

+

+
------------------------------------------------------------------------

+
-- Structures can also be combined.

+

+
-- Some collections of properties which are preserved by ×-Rel.

+

+
×-isEquivalence : IsEquivalence R → IsEquivalence S →

+
                  IsEquivalence (Pointwise R S)

+
×-isEquivalence {R = R} {S = S} eq₁ eq₂ = record

+
  { refl  = ×-refl {R = R} {S = S} (refl eq₁) (refl eq₂)

+
  ; sym   = ×-symmetric {R = R} {S = S} (sym eq₁) (sym eq₂)

+
  ; trans = ×-transitive {R = R} {S = S} (trans eq₁) (trans eq₂)

+
  } where open IsEquivalence

+

+
×-isDecEquivalence : IsDecEquivalence R → IsDecEquivalence S →

+
                     IsDecEquivalence (Pointwise R S)

+
×-isDecEquivalence eq₁ eq₂ = record

+
  { isEquivalence = ×-isEquivalence

+
                      (isEquivalence eq₁) (isEquivalence eq₂)

+
  ; _≟_           = ×-decidable (_≟_ eq₁) (_≟_ eq₂)

+
  } where open IsDecEquivalence

+

+
×-isPreorder : IsPreorder ≈₁ R → IsPreorder ≈₂ S →

+
               IsPreorder (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
×-isPreorder {R = R} {S = S} pre₁ pre₂ = record

+
  { isEquivalence = ×-isEquivalence

+
                      (isEquivalence pre₁) (isEquivalence pre₂)

+
  ; reflexive     = ×-reflexive {R = R} {S = S}

+
                      (reflexive pre₁) (reflexive pre₂)

+
  ; trans         = ×-transitive {R = R} {S = S}

+
                      (trans pre₁) (trans pre₂)

+
  } where open IsPreorder

+

+
×-isPartialOrder : IsPartialOrder ≈₁ R → IsPartialOrder ≈₂ S →

+
                   IsPartialOrder (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
×-isPartialOrder {R = R} {S = S} po₁ po₂ = record

+
  { isPreorder = ×-isPreorder (isPreorder po₁) (isPreorder po₂)

+
  ; antisym    = ×-antisymmetric {R = R} {S = S}

+
                   (antisym po₁) (antisym po₂)

+
  } where open IsPartialOrder

+

+
×-isStrictPartialOrder : IsStrictPartialOrder ≈₁ R →

+
                         IsStrictPartialOrder ≈₂ S →

+
                         IsStrictPartialOrder (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
×-isStrictPartialOrder {R = R} {≈₂ = ≈₂} {S = S} spo₁ spo₂ = record

+
  { isEquivalence = ×-isEquivalence

+
                      (isEquivalence spo₁) (isEquivalence spo₂)

+
  ; irrefl        = ×-irreflexive₁ {R = R} {≈₂ = ≈₂} {S = S}

+
                      (irrefl spo₁)

+
  ; trans         = ×-transitive {R = R} {S = S}

+
                      (trans spo₁) (trans spo₂)

+
  ; <-resp-≈      = ×-respects₂ (<-resp-≈ spo₁) (<-resp-≈ spo₂)

+
  } where open IsStrictPartialOrder

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
×-setoid : Setoid a ℓ₁ → Setoid b ℓ₂ → Setoid _ _

+
×-setoid s₁ s₂ = record

+
  { isEquivalence =

+
      ×-isEquivalence (isEquivalence s₁) (isEquivalence s₂)

+
  } where open Setoid

+

+
×-decSetoid : DecSetoid a ℓ₁ → DecSetoid b ℓ₂ → DecSetoid _ _

+
×-decSetoid s₁ s₂ = record

+
  { isDecEquivalence =

+
      ×-isDecEquivalence (isDecEquivalence s₁) (isDecEquivalence s₂)

+
  } where open DecSetoid

+

+
×-preorder : Preorder a ℓ₁ ℓ₂ → Preorder b ℓ₃ ℓ₄ → Preorder _ _ _

+
×-preorder p₁ p₂ = record

+
  { isPreorder = ×-isPreorder (isPreorder p₁) (isPreorder p₂)

+
  } where open Preorder

+

+
×-poset : Poset a ℓ₁ ℓ₂ → Poset b ℓ₃ ℓ₄ → Poset _ _ _

+
×-poset s₁ s₂ = record

+
  { isPartialOrder = ×-isPartialOrder (isPartialOrder s₁)

+
                     (isPartialOrder s₂)

+
  } where open Poset

+

+
×-strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →

+
                       StrictPartialOrder b ℓ₃ ℓ₄ →

+
                       StrictPartialOrder _ _ _

+
×-strictPartialOrder s₁ s₂ = record

+
  { isStrictPartialOrder = ×-isStrictPartialOrder

+
                             (isStrictPartialOrder s₁)

+
                             (isStrictPartialOrder s₂)

+
  } where open StrictPartialOrder

+

+
------------------------------------------------------------------------

+
-- Additional notation

+

+
-- Infix combining setoids

+
infix 4 _×ₛ_

+
_×ₛ_ : Setoid a ℓ₁ → Setoid b ℓ₂ → Setoid _ _

+
_×ₛ_ = ×-setoid

+

+
------------------------------------------------------------------------

+
-- The propositional equality setoid over products can be

+
-- decomposed using ×-Rel

+

+
≡×≡⇒≡ : Pointwise _≡_ _≡_ ⇒ _≡_ {A = A × B}

+
≡×≡⇒≡ (≡.refl , ≡.refl) = ≡.refl

+

+
≡⇒≡×≡ : _≡_ {A = A × B} ⇒ Pointwise _≡_ _≡_

+
≡⇒≡×≡ ≡.refl = (≡.refl , ≡.refl)

+

+
Pointwise-≡↔≡ : Inverse (≡.setoid A ×ₛ ≡.setoid B) (≡.setoid (A × B))

+
Pointwise-≡↔≡ = record

+
  { to         = id

+
  ; from       = id

+
  ; to-cong    = ≡×≡⇒≡

+
  ; from-cong  = ≡⇒≡×≡

+
  ; inverse    = ≡×≡⇒≡ , ≡⇒≡×≡

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.Relation.Unary.All.html b/site/static/agda_html/Data.Product.Relation.Unary.All.html
new file mode 100644
index 000000000..b3542650a
--- /dev/null
+++ b/site/static/agda_html/Data.Product.Relation.Unary.All.html
@@ -0,0 +1,98 @@
+
+Data.Product.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lifting of two predicates

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product.Relation.Unary.All where

+

+
open import Level using (Level; _⊔_)

+
open import Data.Product.Base using (_×_; _,_)

+

+
private

+
  variable

+
    a b p q : Level

+
    A : Set a

+
    B : Set b

+

+
All : (A → Set p) → (B → Set q) → (A × B → Set (p ⊔ q))

+
All P Q (a , b) = P a × Q b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Product.html b/site/static/agda_html/Data.Product.html
new file mode 100644
index 000000000..d7b10162d
--- /dev/null
+++ b/site/static/agda_html/Data.Product.html
@@ -0,0 +1,140 @@
+
+Data.Product
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Products

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Product where

+

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
private

+
  variable

+
    a b c ℓ p q r s : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Definition of dependent products

+

+
open import Data.Product.Base public

+

+
-- These are here as they are not 'basic' but instead "very dependent",

+
-- i.e. the result type depends on the full input 'point' (x , y).

+
map-Σ : {B : A → Set b} {P : A → Set p} {Q : {x : A} → P x → B x → Set q} →

+
   (f : (x : A) → B x) → (∀ {x} → (y : P x) → Q y (f x)) →

+
   ((x , y) : Σ A P) → Σ (B x) (Q y)

+
map-Σ f g (x , y) = (f x , g y)

+

+
-- This is a "non-dependent" version of map-Σ whereby the input is actually

+
-- a pair (i.e. _×_ ) but the output type still depends on the input 'point' (x , y).

+
map-Σ′ : {B : A → Set b} {P : Set p} {Q : P → Set q} →

+
  (f : (x : A) → B x) → ((x : P) → Q x) → ((x , y) : A × P) → B x × Q y

+
map-Σ′ f g (x , y) = (f x , g y)

+

+
-- This is a generic zipWith for Σ where different functions are applied to each

+
-- component pair, and recombined.

+
zipWith : {P : A → Set p} {Q : B → Set q} {R : C → Set r} {S : (x : C) → R x → Set s}

+
  (_∙_ : A → B → C) → (_∘_ : ∀ {x y} → P x → Q y → R (x ∙ y)) →

+
  (_*_ : (x : C) → (y : R x) → S x y) →

+
  ((a , p) : Σ A P) → ((b , q) : Σ B Q) → S (a ∙ b) (p ∘ q)

+
zipWith _∙_ _∘_ _*_ (a , p) (b , q) = (a ∙ b) * (p ∘ q)

+

+
------------------------------------------------------------------------

+
-- Negation of existential quantifier

+

+
∄ : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)

+
∄ P = ¬ ∃ P

+

+
-- Unique existence (parametrised by an underlying equality).

+

+
∃! : {A : Set a} → (A → A → Set ℓ) → (A → Set b) → Set (a ⊔ b ⊔ ℓ)

+
∃! _≈_ B = ∃ λ x → B x × (∀ {y} → B y → x ≈ y)

+

+
-- Syntax

+

+
infix 2 ∄-syntax

+

+
∄-syntax : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)

+
∄-syntax = ∄

+

+
syntax ∄-syntax (λ x → B) = ∄[ x ] B

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Rational.Base.html b/site/static/agda_html/Data.Rational.Base.html
new file mode 100644
index 000000000..75d743796
--- /dev/null
+++ b/site/static/agda_html/Data.Rational.Base.html
@@ -0,0 +1,446 @@
+
+Data.Rational.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Rational numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Rational.Base where

+

+
open import Algebra.Bundles.Raw

+
open import Data.Bool.Base using (Bool; true; false; if_then_else_)

+
open import Data.Integer.Base as ℤ

+
  using (ℤ; +_; +0; +[1+_]; -[1+_])

+
  hiding (module ℤ)

+
open import Data.Nat.GCD

+
open import Data.Nat.Coprimality as ℕ

+
  using (Coprime; Bézout-coprime; coprime-/gcd; coprime?; ¬0-coprimeTo-2+)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc; 2+) hiding (module ℕ)

+
open import Data.Rational.Unnormalised.Base as ℚᵘ using (ℚᵘ; mkℚᵘ)

+
open import Data.Sum.Base using (inj₂)

+
open import Function.Base using (id)

+
open import Level using (0ℓ)

+
open import Relation.Nullary.Decidable.Core using (recompute)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Unary using (Pred)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl)

+

+
------------------------------------------------------------------------

+
-- Rational numbers in reduced form. Note that there is exactly one

+
-- way to represent every rational number.

+

+
record ℚ : Set where

+
  -- We add "no-eta-equality; pattern" to the record to stop Agda

+
  -- automatically unfolding rationals when arithmetic operations are

+
  -- applied to them (see definition of operators below and Issue #1753

+
  -- for details).

+
  no-eta-equality; pattern

+

+
  constructor mkℚ

+
  field

+
    numerator     : ℤ

+
    denominator-1 : ℕ

+
    .isCoprime    : Coprime ℤ.∣ numerator ∣ (suc denominator-1)

+

+
  denominatorℕ : ℕ

+
  denominatorℕ = suc denominator-1

+

+
  denominator : ℤ

+
  denominator = + denominatorℕ

+

+
open ℚ public using ()

+
  renaming

+
  ( numerator    to ↥_

+
  ; denominator  to ↧_

+
  ; denominatorℕ to ↧ₙ_

+
  )

+

+
mkℚ+ : ∀ n d → .{{_ : ℕ.NonZero d}} → .(Coprime n d) → ℚ

+
mkℚ+ n (suc d) coprime = mkℚ (+ n) d coprime

+

+
------------------------------------------------------------------------

+
-- Equality of rational numbers (coincides with _≡_)

+

+
infix 4 _≃_

+

+
data _≃_ : Rel ℚ 0ℓ where

+
  *≡* : ∀ {p q} → (↥ p ℤ.* ↧ q) ≡ (↥ q ℤ.* ↧ p) → p ≃ q

+

+
_≄_ : Rel ℚ 0ℓ

+
p ≄ q = ¬ (p ≃ q)

+

+
------------------------------------------------------------------------

+
-- Ordering of rationals

+

+
infix 4 _≤_ _<_ _≥_ _>_ _≰_ _≱_ _≮_ _≯_

+

+
data _≤_ : Rel ℚ 0ℓ where

+
  *≤* : ∀ {p q} → (↥ p ℤ.* ↧ q) ℤ.≤ (↥ q ℤ.* ↧ p) → p ≤ q

+

+
data _<_ : Rel ℚ 0ℓ where

+
  *<* : ∀ {p q} → (↥ p ℤ.* ↧ q) ℤ.< (↥ q ℤ.* ↧ p) → p < q

+

+
_≥_ : Rel ℚ 0ℓ

+
x ≥ y = y ≤ x

+

+
_>_ : Rel ℚ 0ℓ

+
x > y = y < x

+

+
_≰_ : Rel ℚ 0ℓ

+
x ≰ y = ¬ (x ≤ y)

+

+
_≱_ : Rel ℚ 0ℓ

+
x ≱ y = ¬ (x ≥ y)

+

+
_≮_ : Rel ℚ 0ℓ

+
x ≮ y = ¬ (x < y)

+

+
_≯_ : Rel ℚ 0ℓ

+
x ≯ y = ¬ (x > y)

+

+
------------------------------------------------------------------------

+
-- Boolean ordering

+

+
infix 4 _≤ᵇ_

+

+
_≤ᵇ_ : ℚ → ℚ → Bool

+
p ≤ᵇ q = (↥ p ℤ.* ↧ q) ℤ.≤ᵇ (↥ q ℤ.* ↧ p)

+

+
------------------------------------------------------------------------

+
-- Negation

+

+
-_ : ℚ → ℚ

+
- mkℚ -[1+ n ] d prf = mkℚ +[1+ n ] d prf

+
- mkℚ +0       d prf = mkℚ +0       d prf

+
- mkℚ +[1+ n ] d prf = mkℚ -[1+ n ] d prf

+

+
------------------------------------------------------------------------

+
-- Constructing rationals

+

+
-- A constructor for ℚ that takes two natural numbers, say 6 and 21,

+
-- and returns them in a normalized form, e.g. say 2 and 7

+

+
normalize : ∀ (m n : ℕ) .{{_ : ℕ.NonZero n}} → ℚ

+
normalize m n = mkℚ+ (m ℕ./ gcd m n) (n ℕ./ gcd m n) (coprime-/gcd m n)

+
  where

+
    instance

+
      g≢0   = ℕ.≢-nonZero (gcd[m,n]≢0 m n (inj₂ (ℕ.≢-nonZero⁻¹ n)))

+
      n/g≢0 = ℕ.≢-nonZero (n/gcd[m,n]≢0 m n {{gcd≢0 = g≢0}})

+

+
-- A constructor for ℚ that (unlike mkℚ) automatically normalises it's

+
-- arguments. See the constants section below for how to use this operator.

+

+
infixl 7 _/_

+

+
_/_ : (n : ℤ) (d : ℕ) → .{{_ : ℕ.NonZero d}} → ℚ

+
(+ n      / d) =   normalize n       d

+
(-[1+ n ] / d) = - normalize (suc n) d

+

+
------------------------------------------------------------------------

+
-- Conversion to and from unnormalized rationals

+

+
toℚᵘ : ℚ → ℚᵘ

+
toℚᵘ (mkℚ n d-1 _) = mkℚᵘ n d-1

+

+
fromℚᵘ : ℚᵘ → ℚ

+
fromℚᵘ (mkℚᵘ n d-1) = n / suc d-1

+

+
------------------------------------------------------------------------

+
-- Some constants

+

+
0ℚ : ℚ

+
0ℚ = + 0 / 1

+

+
1ℚ : ℚ

+
1ℚ = + 1 / 1

+

+
½ : ℚ

+
½ = + 1 / 2

+

+
-½ : ℚ

+
-½ = - ½

+

+
------------------------------------------------------------------------

+
-- Simple predicates

+

+
NonZero : Pred ℚ 0ℓ

+
NonZero p = ℚᵘ.NonZero (toℚᵘ p)

+

+
Positive : Pred ℚ 0ℓ

+
Positive p = ℚᵘ.Positive (toℚᵘ p)

+

+
Negative : Pred ℚ 0ℓ

+
Negative p = ℚᵘ.Negative (toℚᵘ p)

+

+
NonPositive : Pred ℚ 0ℓ

+
NonPositive p = ℚᵘ.NonPositive (toℚᵘ p)

+

+
NonNegative : Pred ℚ 0ℓ

+
NonNegative p = ℚᵘ.NonNegative (toℚᵘ p)

+

+
-- Instances

+

+
open ℤ public

+
  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)

+

+
-- Constructors

+

+
≢-nonZero : ∀ {p} → p ≢ 0ℚ → NonZero p

+
≢-nonZero {mkℚ -[1+ _ ] _         _} _   = _

+
≢-nonZero {mkℚ +[1+ _ ] _         _} _   = _

+
≢-nonZero {mkℚ +0       zero      _} p≢0 = contradiction refl p≢0

+
≢-nonZero {mkℚ +0       d@(suc m) c} p≢0 =

+
  contradiction (λ {d} → ℕ.recompute c {d}) (¬0-coprimeTo-2+ {{ℕ.nonTrivial {m}}})

+

+
>-nonZero : ∀ {p} → p > 0ℚ → NonZero p

+
>-nonZero {p@(mkℚ _ _ _)} (*<* p<q) = ℚᵘ.>-nonZero {toℚᵘ p} (ℚᵘ.*<* p<q)

+

+
<-nonZero : ∀ {p} → p < 0ℚ → NonZero p

+
<-nonZero {p@(mkℚ _ _ _)} (*<* p<q) = ℚᵘ.<-nonZero {toℚᵘ p} (ℚᵘ.*<* p<q)

+

+
positive : ∀ {p} → p > 0ℚ → Positive p

+
positive {p@(mkℚ _ _ _)} (*<* p<q) = ℚᵘ.positive {toℚᵘ p} (ℚᵘ.*<* p<q)

+

+
negative : ∀ {p} → p < 0ℚ → Negative p

+
negative {p@(mkℚ _ _ _)} (*<* p<q) = ℚᵘ.negative {toℚᵘ p} (ℚᵘ.*<* p<q)

+

+
nonPositive : ∀ {p} → p ≤ 0ℚ → NonPositive p

+
nonPositive {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonPositive {toℚᵘ p} (ℚᵘ.*≤* p≤q)

+

+
nonNegative : ∀ {p} → p ≥ 0ℚ → NonNegative p

+
nonNegative {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonNegative {toℚᵘ p} (ℚᵘ.*≤* p≤q)

+

+
------------------------------------------------------------------------

+
-- Operations on rationals

+

+
-- For explanation of the `@record{}` annotations see notes in the

+
-- equivalent place in `Data.Rational.Unnormalised.Base`.

+

+
infix  8 -_ 1/_

+
infixl 7 _*_ _÷_ _⊓_

+
infixl 6 _-_ _+_ _⊔_

+

+
-- addition

+
_+_ : ℚ → ℚ → ℚ

+
p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)

+

+
-- multiplication

+
_*_ : ℚ → ℚ → ℚ

+
p@record{} * q@record{} = (↥ p ℤ.* ↥ q) / (↧ₙ p ℕ.* ↧ₙ q)

+

+
-- subtraction

+
_-_ : ℚ → ℚ → ℚ

+
p - q = p + (- q)

+

+
-- reciprocal: requires a proof that the numerator is not zero

+
1/_ : (p : ℚ) → .{{_ : NonZero p}} → ℚ

+
1/ mkℚ +[1+ n ] d prf = mkℚ +[1+ d ] n (ℕ.sym prf)

+
1/ mkℚ -[1+ n ] d prf = mkℚ -[1+ d ] n (ℕ.sym prf)

+

+
-- division: requires a proof that the denominator is not zero

+
_÷_ : (p q : ℚ) → .{{_ : NonZero q}} → ℚ

+
p ÷ q = p * (1/ q)

+

+
-- max

+
_⊔_ : (p q : ℚ) → ℚ

+
p@record{} ⊔ q@record{} = if p ≤ᵇ q then q else p

+

+
-- min

+
_⊓_ : (p q : ℚ) → ℚ

+
p@record{} ⊓ q@record{} = if p ≤ᵇ q then p else q

+

+
-- absolute value

+
∣_∣ : ℚ → ℚ

+
∣ mkℚ n d c ∣ = mkℚ (+ ℤ.∣ n ∣) d c

+

+
------------------------------------------------------------------------

+
-- Rounding functions

+

+
-- Floor (round towards -∞)

+
floor : ℚ → ℤ

+
floor p@record{} = ↥ p ℤ./ ↧ p

+

+
-- Ceiling (round towards +∞)

+
ceiling : ℚ → ℤ

+
ceiling p@record{} = ℤ.- floor (- p)

+

+
-- Truncate  (round towards 0)

+
truncate : ℚ → ℤ

+
truncate p = if p ≤ᵇ 0ℚ then ceiling p else floor p

+

+
-- Round (to nearest integer)

+
round : ℚ → ℤ

+
round p = if p ≤ᵇ 0ℚ then ceiling (p - ½) else floor (p + ½)

+

+
-- Fractional part (remainder after floor)

+
fracPart : ℚ → ℚ

+
fracPart p@record{} = ∣ p - truncate p / 1 ∣

+

+
-- Extra notations  ⌊ ⌋ floor,  ⌈ ⌉ ceiling,  [ ] truncate

+
syntax floor p = ⌊ p ⌋

+
syntax ceiling p = ⌈ p ⌉

+
syntax truncate p = [ p ]

+

+
------------------------------------------------------------------------

+
-- Raw bundles

+

+
+-rawMagma : RawMagma 0ℓ 0ℓ

+
+-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  }

+

+
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ

+
+-0-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  ; ε   = 0ℚ

+
  }

+

+
+-0-rawGroup : RawGroup 0ℓ 0ℓ

+
+-0-rawGroup = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _+_

+
  ; ε   = 0ℚ

+
  ; _⁻¹ = -_

+
  }

+

+
+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ

+
+-*-rawNearSemiring = record

+
  { _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0#  = 0ℚ

+
  }

+

+
+-*-rawSemiring : RawSemiring 0ℓ 0ℓ

+
+-*-rawSemiring = record

+
  { _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0#  = 0ℚ

+
  ; 1#  = 1ℚ

+
  }

+

+
+-*-rawRing : RawRing 0ℓ 0ℓ

+
+-*-rawRing = record

+
  { _≈_ = _≡_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; -_  = -_

+
  ; 0#  = 0ℚ

+
  ; 1#  = 1ℚ

+
  }

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  ; ε   = 1ℚ

+
  }

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
+-rawMonoid = +-0-rawMonoid

+
{-# WARNING_ON_USAGE +-rawMonoid

+
"Warning: +-rawMonoid was deprecated in v2.0

+
Please use +-0-rawMonoid instead."

+
#-}

+
*-rawMonoid = *-1-rawMonoid

+
{-# WARNING_ON_USAGE *-rawMonoid

+
"Warning: *-rawMonoid was deprecated in v2.0

+
Please use *-1-rawMonoid instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Rational.Properties.html b/site/static/agda_html/Data.Rational.Properties.html
new file mode 100644
index 000000000..1b2300145
--- /dev/null
+++ b/site/static/agda_html/Data.Rational.Properties.html
@@ -0,0 +1,1865 @@
+
+Data.Rational.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of Rational numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+
{-# OPTIONS --warn=noUserWarning #-} -- for +-rawMonoid, *-rawMonoid (issue #1865, #1844, #1755)

+

+
module Data.Rational.Properties where

+

+
open import Algebra.Apartness

+
open import Algebra.Construct.NaturalChoice.Base

+
import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp

+
import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp

+
open import Algebra.Consequences.Propositional

+
open import Algebra.Morphism

+
open import Algebra.Bundles

+
import Algebra.Morphism.MagmaMonomorphism  as MagmaMonomorphisms

+
import Algebra.Morphism.MonoidMonomorphism as MonoidMonomorphisms

+
import Algebra.Morphism.GroupMonomorphism  as GroupMonomorphisms

+
import Algebra.Morphism.RingMonomorphism   as RingMonomorphisms

+
import Algebra.Lattice.Morphism.LatticeMonomorphism as LatticeMonomorphisms

+
import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties

+
import Algebra.Properties.Group as GroupProperties

+
open import Data.Bool.Base using (T; true; false)

+
open import Data.Integer.Base as ℤ using (ℤ; +_; -[1+_]; +[1+_]; +0; 0ℤ; 1ℤ; _◃_)

+
open import Data.Integer.Coprimality using (coprime-divisor)

+
import Data.Integer.Properties as ℤ

+
open import Data.Integer.GCD using (gcd; gcd[i,j]≡0⇒i≡0; gcd[i,j]≡0⇒j≡0)

+
open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)

+
import Data.Nat.Properties as ℕ

+
open import Data.Nat.Coprimality as C using (Coprime; coprime?)

+
open import Data.Nat.Divisibility using (_∣_; divides; ∣-antisym; *-pres-∣)

+
import Data.Nat.GCD as ℕ

+
import Data.Nat.DivMod as ℕ

+
open import Data.Product.Base using (proj₁; proj₂; _×_; _,_; uncurry)

+
open import Data.Rational.Base

+
open import Data.Rational.Unnormalised.Base as ℚᵘ

+
  using (ℚᵘ; mkℚᵘ; *≡*; *≤*; *<*)

+
  renaming

+
  ( ↥_ to ↥ᵘ_; ↧_ to ↧ᵘ_; ↧ₙ_ to ↧ₙᵘ_

+
  ; _≃_ to _≃ᵘ_; _≤_ to _≤ᵘ_; _<_ to _<ᵘ_

+
  ; _+_ to _+ᵘ_

+
  )

+
import Data.Rational.Unnormalised.Properties as ℚᵘ

+
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′; _⊎_)

+
import Data.Sign.Base as Sign

+
open import Function.Base using (_∘_; _∘′_; _∘₂_; _$_; flip)

+
open import Function.Definitions using (Injective)

+
open import Level using (0ℓ)

+
open import Relation.Binary

+
open import Relation.Binary.Morphism.Structures

+
import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms

+
import Relation.Binary.Properties.DecSetoid as DecSetoidProperties

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂; sym; trans; _≢_; subst; subst₂; resp₂)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; decSetoid; module ≡-Reasoning; isEquivalence)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
open import Relation.Binary.Reasoning.Syntax using (module ≃-syntax)

+
open import Relation.Nullary.Decidable.Core as Dec

+
  using (yes; no; recompute; map′; _×-dec_)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+

+
open import Algebra.Definitions {A = ℚ} _≡_

+
open import Algebra.Structures  {A = ℚ} _≡_

+

+
private

+
  variable

+
    p q r : ℚ

+

+
------------------------------------------------------------------------

+
-- Propositional equality

+
------------------------------------------------------------------------

+

+
mkℚ-cong : ∀ {n₁ n₂ d₁ d₂}

+
           .{c₁ : Coprime ℤ.∣ n₁ ∣ (suc d₁)}

+
           .{c₂ : Coprime ℤ.∣ n₂ ∣ (suc d₂)} →

+
           n₁ ≡ n₂ → d₁ ≡ d₂ → mkℚ n₁ d₁ c₁ ≡ mkℚ n₂ d₂ c₂

+
mkℚ-cong refl refl = refl

+

+
mkℚ-injective : ∀ {n₁ n₂ d₁ d₂}

+
                .{c₁ : Coprime ℤ.∣ n₁ ∣ (suc d₁)}

+
                .{c₂ : Coprime ℤ.∣ n₂ ∣ (suc d₂)} →

+
                mkℚ n₁ d₁ c₁ ≡ mkℚ n₂ d₂ c₂ → n₁ ≡ n₂ × d₁ ≡ d₂

+
mkℚ-injective refl = refl , refl

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality ℚ

+
mkℚ n₁ d₁ _ ≟ mkℚ n₂ d₂ _ = map′

+
  (uncurry mkℚ-cong)

+
  mkℚ-injective

+
  (n₁ ℤ.≟ n₂ ×-dec d₁ ℕ.≟ d₂)

+

+
≡-setoid : Setoid 0ℓ 0ℓ

+
≡-setoid = setoid ℚ

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = decSetoid _≟_

+

+
1≢0 : 1ℚ ≢ 0ℚ

+
1≢0 = λ ()

+

+
------------------------------------------------------------------------

+
-- mkℚ+

+
------------------------------------------------------------------------

+

+
mkℚ+-cong : ∀ {n₁ n₂ d₁ d₂} .{{_ : ℕ.NonZero d₁}} .{{_ : ℕ.NonZero d₂}}

+
           .{c₁ : Coprime n₁ d₁}

+
           .{c₂ : Coprime n₂ d₂} →

+
           n₁ ≡ n₂ → d₁ ≡ d₂ →

+
           mkℚ+ n₁ d₁ c₁ ≡ mkℚ+ n₂ d₂ c₂

+
mkℚ+-cong refl refl = refl

+

+
mkℚ+-injective : ∀ {n₁ n₂ d₁ d₂} .{{_ : ℕ.NonZero d₁}} .{{_ : ℕ.NonZero d₂}}

+
           .{c₁ : Coprime n₁ d₁}

+
           .{c₂ : Coprime n₂ d₂} →

+
           mkℚ+ n₁ d₁ c₁ ≡ mkℚ+ n₂ d₂ c₂ →

+
           n₁ ≡ n₂ × d₁ ≡ d₂

+
mkℚ+-injective {d₁ = suc _} {suc _} refl = refl , refl

+

+
↥-mkℚ+ : ∀ n d .{{_ : ℕ.NonZero d}} .{c : Coprime n d} → ↥ (mkℚ+ n d c) ≡ + n

+
↥-mkℚ+ n (suc d) = refl

+

+
↧-mkℚ+ : ∀ n d .{{_ : ℕ.NonZero d}} .{c : Coprime n d} → ↧ (mkℚ+ n d c) ≡ + d

+
↧-mkℚ+ n (suc d) = refl

+

+
mkℚ+-nonNeg : ∀ n d .{{_ : ℕ.NonZero d}} .{c : Coprime n d} →

+
              NonNegative (mkℚ+ n d c)

+
mkℚ+-nonNeg n (suc d) = _

+

+
mkℚ+-pos : ∀ n d .{{_ : ℕ.NonZero n}} .{{_ : ℕ.NonZero d}}

+
           .{c : Coprime n d} → Positive (mkℚ+ n d c)

+
mkℚ+-pos (suc n) (suc d) = _

+

+
------------------------------------------------------------------------

+
-- Numerator and denominator equality

+
------------------------------------------------------------------------

+

+
drop-*≡* : p ≃ q → ↥ p ℤ.* ↧ q ≡ ↥ q ℤ.* ↧ p

+
drop-*≡* (*≡* eq) = eq

+

+
≡⇒≃ : _≡_ ⇒ _≃_

+
≡⇒≃ refl = *≡* refl

+

+
≃⇒≡ : _≃_ ⇒ _≡_

+
≃⇒≡ {x = mkℚ n₁ d₁ c₁} {y = mkℚ n₂ d₂ c₂} (*≡* eq) = helper

+
  where

+
  open ≡-Reasoning

+

+
  1+d₁∣1+d₂ : suc d₁ ∣ suc d₂

+
  1+d₁∣1+d₂ = coprime-divisor (+ suc d₁) n₁ (+ suc d₂)

+
    (C.sym (C.recompute c₁)) $

+
    divides ℤ.∣ n₂ ∣ $ begin

+
      ℤ.∣ n₁ ℤ.* + suc d₂ ∣  ≡⟨ cong ℤ.∣_∣ eq ⟩

+
      ℤ.∣ n₂ ℤ.* + suc d₁ ∣  ≡⟨ ℤ.abs-* n₂ (+ suc d₁) ⟩

+
      ℤ.∣ n₂ ∣ ℕ.* suc d₁    ∎

+

+
  1+d₂∣1+d₁ : suc d₂ ∣ suc d₁

+
  1+d₂∣1+d₁ = coprime-divisor (+ suc d₂) n₂ (+ suc d₁)

+
    (C.sym (C.recompute c₂)) $

+
    divides ℤ.∣ n₁ ∣ (begin

+
      ℤ.∣ n₂ ℤ.* + suc d₁ ∣  ≡⟨ cong ℤ.∣_∣ (sym eq) ⟩

+
      ℤ.∣ n₁ ℤ.* + suc d₂ ∣  ≡⟨ ℤ.abs-* n₁ (+ suc d₂) ⟩

+
      ℤ.∣ n₁ ∣ ℕ.* suc d₂    ∎)

+

+
  helper : mkℚ n₁ d₁ c₁ ≡ mkℚ n₂ d₂ c₂

+
  helper with ∣-antisym 1+d₁∣1+d₂ 1+d₂∣1+d₁

+
  ... | refl with ℤ.*-cancelʳ-≡ n₁ n₂ (+ suc d₁) eq

+
  ...   | refl = refl

+

+
≃-sym : Symmetric _≃_

+
≃-sym = ≡⇒≃ ∘′ sym ∘′ ≃⇒≡

+

+
------------------------------------------------------------------------

+
-- Properties of ↥

+
------------------------------------------------------------------------

+

+
↥p≡0⇒p≡0 : ∀ p → ↥ p ≡ 0ℤ → p ≡ 0ℚ

+
↥p≡0⇒p≡0 (mkℚ +0 d-1 0-coprime-d) ↥p≡0 = mkℚ-cong refl d-1≡0

+
  where d-1≡0 = ℕ.suc-injective (C.0-coprimeTo-m⇒m≡1 (C.recompute 0-coprime-d))

+

+
p≡0⇒↥p≡0 : ∀ p → p ≡ 0ℚ → ↥ p ≡ 0ℤ

+
p≡0⇒↥p≡0 p refl = refl

+

+
↥p≡↥q≡0⇒p≡q : ∀ p q → ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≡ q

+
↥p≡↥q≡0⇒p≡q p q ↥p≡0 ↥q≡0 = trans (↥p≡0⇒p≡0 p ↥p≡0) (sym (↥p≡0⇒p≡0 q ↥q≡0))

+

+
------------------------------------------------------------------------

+
-- Basic properties of sign predicates

+
------------------------------------------------------------------------

+

+
nonNeg≢neg : ∀ p q → .{{NonNegative p}} → .{{Negative q}} → p ≢ q

+
nonNeg≢neg (mkℚ (+ _) _ _) (mkℚ -[1+ _ ] _ _) ()

+

+
pos⇒nonNeg : ∀ p → .{{Positive p}} → NonNegative p

+
pos⇒nonNeg p = ℚᵘ.pos⇒nonNeg (toℚᵘ p)

+

+
neg⇒nonPos : ∀ p → .{{Negative p}} → NonPositive p

+
neg⇒nonPos p = ℚᵘ.neg⇒nonPos (toℚᵘ p)

+

+
nonNeg∧nonZero⇒pos : ∀ p → .{{NonNegative p}} → .{{NonZero p}} → Positive p

+
nonNeg∧nonZero⇒pos (mkℚ +[1+ _ ] _ _) = _

+

+
pos⇒nonZero : ∀ p → .{{Positive p}} → NonZero p

+
pos⇒nonZero (mkℚ +[1+ _ ] _ _) = _

+

+
neg⇒nonZero : ∀ p → .{{Negative p}} → NonZero p

+
neg⇒nonZero (mkℚ -[1+ _ ] _ _) = _

+

+
------------------------------------------------------------------------

+
-- Properties of -_

+
------------------------------------------------------------------------

+

+
↥-neg : ∀ p → ↥ (- p) ≡ ℤ.- (↥ p)

+
↥-neg (mkℚ -[1+ _ ] _ _) = refl

+
↥-neg (mkℚ +0       _ _) = refl

+
↥-neg (mkℚ +[1+ _ ] _ _) = refl

+

+
↧-neg : ∀ p → ↧ (- p) ≡ ↧ p

+
↧-neg (mkℚ -[1+ _ ] _ _) = refl

+
↧-neg (mkℚ +0       _ _) = refl

+
↧-neg (mkℚ +[1+ _ ] _ _) = refl

+

+
neg-injective : - p ≡ - q → p ≡ q

+
neg-injective {mkℚ +[1+ m ] _ _} {mkℚ +[1+ n ] _ _} refl = refl

+
neg-injective {mkℚ +0       _ _} {mkℚ +0       _ _} refl = refl

+
neg-injective {mkℚ -[1+ m ] _ _} {mkℚ -[1+ n ] _ _} refl = refl

+
neg-injective {mkℚ +[1+ m ] _ _} {mkℚ -[1+ n ] _ _} ()

+
neg-injective {mkℚ +0       _ _} {mkℚ -[1+ n ] _ _} ()

+
neg-injective {mkℚ -[1+ m ] _ _} {mkℚ +[1+ n ] _ _} ()

+
neg-injective {mkℚ -[1+ m ] _ _} {mkℚ +0       _ _} ()

+

+
neg-pos : Positive p → Negative (- p)

+
neg-pos {mkℚ +[1+ _ ] _ _} _ = _

+

+
------------------------------------------------------------------------

+
-- Properties of normalize

+
------------------------------------------------------------------------

+

+
normalize-coprime : ∀ {n d-1} .(c : Coprime n (suc d-1)) →

+
                    normalize n (suc d-1) ≡ mkℚ (+ n) d-1 c

+
normalize-coprime {n} {d-1} c = begin

+
  normalize n d                                  ≡⟨⟩

+
  mkℚ+ ((n ℕ./ g) {{g≢0}}) ((d ℕ./ g) {{g≢0}}) _ ≡⟨ mkℚ+-cong {c₂ = c₂} (ℕ./-congʳ {{g≢0}} g≡1) (ℕ./-congʳ {{g≢0}} g≡1) ⟩

+
  mkℚ+ (n ℕ./ 1) (d ℕ./ 1) _                     ≡⟨ mkℚ+-cong {c₂ = c} (ℕ.n/1≡n n) (ℕ.n/1≡n d) ⟩

+
  mkℚ+ n d _                                     ≡⟨⟩

+
  mkℚ (+ n) d-1 _                                ∎

+
  where

+
  open ≡-Reasoning; d = suc d-1; g = ℕ.gcd n d

+
  c′ = C.recompute c

+
  c₂ : Coprime (n ℕ./ 1) (d ℕ./ 1)

+
  c₂ = subst₂ Coprime (sym (ℕ.n/1≡n n)) (sym (ℕ.n/1≡n d)) c′

+
  g≡1 = C.coprime⇒gcd≡1 c′

+
  instance

+
    g≢0   = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0 n d (inj₂ λ()))

+
    n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 n d {{_}} {{g≢0}})

+
    d/1≢0 = ℕ.≢-nonZero (subst (_≢ 0) (sym (ℕ.n/1≡n d)) λ())

+

+
↥-normalize : ∀ i n .{{_ : ℕ.NonZero n}} → ↥ (normalize i n) ℤ.* gcd (+ i) (+ n) ≡ + i

+
↥-normalize i n = begin

+
  ↥ (normalize i n) ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↥-mkℚ+ _ (n ℕ./ g)) ⟩

+
  + i/g     ℤ.* + g          ≡⟨⟩

+
  Sign.+ ◃ i/g ℕ.* g         ≡⟨ cong (Sign.+ ◃_) (ℕ.m/n*n≡m (ℕ.gcd[m,n]∣m i n)) ⟩

+
  Sign.+ ◃ i                 ≡⟨ ℤ.+◃n≡+n i ⟩

+
  + i                        ∎

+
  where

+
  open ≡-Reasoning

+
  g     = ℕ.gcd i n

+
  instance g≢0 = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0 i n (inj₂ (ℕ.≢-nonZero⁻¹ n)))

+
  instance n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 i n {{gcd≢0 = g≢0}})

+
  i/g = (i ℕ./ g) {{g≢0}}

+

+
↧-normalize : ∀ i n .{{_ : ℕ.NonZero n}} → ↧ (normalize i n) ℤ.* gcd (+ i) (+ n) ≡ + n

+
↧-normalize i n = begin

+
  ↧ (normalize i n) ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↧-mkℚ+ _ (n ℕ./ g)) ⟩

+
  + (n ℕ./ g)       ℤ.* + g  ≡⟨⟩

+
  Sign.+ ◃ n ℕ./ g     ℕ.* g ≡⟨ cong (Sign.+ ◃_) (ℕ.m/n*n≡m (ℕ.gcd[m,n]∣n i n)) ⟩

+
  Sign.+ ◃ n                 ≡⟨ ℤ.+◃n≡+n n ⟩

+
  + n                        ∎

+
  where

+
  open ≡-Reasoning

+
  g = ℕ.gcd i n

+
  instance g≢0   = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0   i n (inj₂ (ℕ.≢-nonZero⁻¹ n)))

+
  instance n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 i n {{gcd≢0 = g≢0}})

+

+
normalize-cong : ∀ {m₁ n₁ m₂ n₂} .{{_ : ℕ.NonZero n₁}} .{{_ : ℕ.NonZero n₂}} →

+
                 m₁ ≡ m₂ → n₁ ≡ n₂ → normalize m₁ n₁ ≡ normalize m₂ n₂

+
normalize-cong {m} {n} refl refl =

+
  mkℚ+-cong (ℕ./-congʳ {n = g} refl) (ℕ./-congʳ {n = g} refl)

+
  where

+
  g = ℕ.gcd m n

+
  instance

+
    g≢0   = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0 m n (inj₂ (ℕ.≢-nonZero⁻¹ n)))

+
    n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 m n {{gcd≢0 = g≢0}})

+

+
normalize-nonNeg : ∀ m n .{{_ : ℕ.NonZero n}} → NonNegative (normalize m n)

+
normalize-nonNeg m n = mkℚ+-nonNeg (m ℕ./ g) (n ℕ./ g)

+
  where

+
  g = ℕ.gcd m n

+
  instance

+
    g≢0   = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0 m n (inj₂ (ℕ.≢-nonZero⁻¹ n)))

+
    n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 m n {{gcd≢0 = g≢0}})

+

+
normalize-pos : ∀ m n .{{_ : ℕ.NonZero n}} .{{_ : ℕ.NonZero m}} → Positive (normalize m n)

+
normalize-pos m n = mkℚ+-pos (m ℕ./ ℕ.gcd m n) (n ℕ./ ℕ.gcd m n)

+
  where

+
  g = ℕ.gcd m n

+
  instance

+
    g≢0   = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0 m n (inj₂ (ℕ.≢-nonZero⁻¹ n)))

+
    n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 m n {{gcd≢0 = g≢0}})

+
    m/g≢0 = ℕ.≢-nonZero (ℕ.m/gcd[m,n]≢0 m n {{gcd≢0 = g≢0}})

+

+
normalize-injective-≃ : ∀ m n c d {{_ : ℕ.NonZero c}} {{_ : ℕ.NonZero d}} →

+
                        normalize m c ≡ normalize n d →

+
                        m ℕ.* d ≡ n ℕ.* c

+
normalize-injective-≃ m n c d eq = ℕ./-cancelʳ-≡

+
  md∣gcd[m,c]gcd[n,d]

+
  nc∣gcd[m,c]gcd[n,d]

+
  $ begin

+
    (m ℕ.* d) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ≡⟨  ℕ./-*-interchange gcd[m,c]∣m gcd[n,d]∣d ⟩

+
    (m ℕ./ gcd[m,c]) ℕ.* (d ℕ./ gcd[n,d]) ≡⟨  cong₂ ℕ._*_ m/gcd[m,c]≡n/gcd[n,d] (sym c/gcd[m,c]≡d/gcd[n,d]) ⟩

+
    (n ℕ./ gcd[n,d]) ℕ.* (c ℕ./ gcd[m,c]) ≡⟨ ℕ./-*-interchange gcd[n,d]∣n gcd[m,c]∣c ⟨

+
    (n ℕ.* c) ℕ./ (gcd[n,d] ℕ.* gcd[m,c]) ≡⟨ ℕ./-congʳ (ℕ.*-comm gcd[n,d] gcd[m,c]) ⟩

+
    (n ℕ.* c) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ∎

+
  where

+
  open ≡-Reasoning

+
  gcd[m,c]   = ℕ.gcd m c

+
  gcd[n,d]   = ℕ.gcd n d

+
  gcd[m,c]∣m = ℕ.gcd[m,n]∣m m c

+
  gcd[m,c]∣c = ℕ.gcd[m,n]∣n m c

+
  gcd[n,d]∣n = ℕ.gcd[m,n]∣m n d

+
  gcd[n,d]∣d = ℕ.gcd[m,n]∣n n d

+
  md∣gcd[m,c]gcd[n,d] = *-pres-∣ gcd[m,c]∣m gcd[n,d]∣d

+
  nc∣gcd[n,d]gcd[m,c] = *-pres-∣ gcd[n,d]∣n gcd[m,c]∣c

+
  nc∣gcd[m,c]gcd[n,d] = subst (_∣ n ℕ.* c) (ℕ.*-comm gcd[n,d] gcd[m,c]) nc∣gcd[n,d]gcd[m,c]

+

+
  gcd[m,c]≢0′          = ℕ.gcd[m,n]≢0 m c (inj₂ (ℕ.≢-nonZero⁻¹ c))

+
  gcd[n,d]≢0′          = ℕ.gcd[m,n]≢0 n d (inj₂ (ℕ.≢-nonZero⁻¹ d))

+
  instance

+
    gcd[m,c]≢0   = ℕ.≢-nonZero gcd[m,c]≢0′

+
    gcd[n,d]≢0   = ℕ.≢-nonZero gcd[n,d]≢0′

+
    c/gcd[m,c]≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 m c {{gcd≢0 = gcd[m,c]≢0}})

+
    d/gcd[n,d]≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 n d {{gcd≢0 = gcd[n,d]≢0}})

+
    gcd[m,c]*gcd[n,d]≢0 = ℕ.m*n≢0 gcd[m,c] gcd[n,d]

+
    gcd[n,d]*gcd[m,c]≢0 = ℕ.m*n≢0 gcd[n,d] gcd[m,c]

+

+
  div = mkℚ+-injective eq

+
  m/gcd[m,c]≡n/gcd[n,d] = proj₁ div

+
  c/gcd[m,c]≡d/gcd[n,d] = proj₂ div

+

+
------------------------------------------------------------------------

+
-- Properties of _/_

+
------------------------------------------------------------------------

+

+
↥-/ : ∀ i n .{{_ : ℕ.NonZero n}} → ↥ (i / n) ℤ.* gcd i (+ n) ≡ i

+
↥-/ (+ m)    n = ↥-normalize m n

+
↥-/ -[1+ m ] n = begin-equality

+
  ↥ (- norm)   ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↥-neg norm) ⟩

+
  ℤ.- (↥ norm) ℤ.* + g  ≡⟨ sym (ℤ.neg-distribˡ-* (↥ norm) (+ g)) ⟩

+
  ℤ.- (↥ norm  ℤ.* + g) ≡⟨ cong (ℤ.-_) (↥-normalize (suc m) n) ⟩

+
  Sign.- ◃ suc m        ≡⟨⟩

+
  -[1+ m ]              ∎

+
  where

+
  open ℤ.≤-Reasoning

+
  g    = ℕ.gcd (suc m) n

+
  norm = normalize (suc m) n

+

+
↧-/ : ∀ i n .{{_ : ℕ.NonZero n}} → ↧ (i / n) ℤ.* gcd i (+ n) ≡ + n

+
↧-/ (+ m)    n = ↧-normalize m n

+
↧-/ -[1+ m ] n = begin-equality

+
  ↧ (- norm) ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↧-neg norm) ⟩

+
  ↧ norm     ℤ.* + g  ≡⟨ ↧-normalize (suc m) n ⟩

+
  + n                 ∎

+
  where

+
  open ℤ.≤-Reasoning

+
  g    = ℕ.gcd (suc m) n

+
  norm = normalize (suc m) n

+

+
↥p/↧p≡p : ∀ p → ↥ p / ↧ₙ p ≡ p

+
↥p/↧p≡p (mkℚ (+ n)    d-1 prf) = normalize-coprime prf

+
↥p/↧p≡p (mkℚ -[1+ n ] d-1 prf) = cong (-_) (normalize-coprime prf)

+

+
0/n≡0 : ∀ n .{{_ : ℕ.NonZero n}} → 0ℤ / n ≡ 0ℚ

+
0/n≡0 n@(suc n-1) {{n≢0}} = mkℚ+-cong {{n/n≢0}} {c₂ = 0-cop-1} (ℕ.0/n≡0 (ℕ.gcd 0 n)) (ℕ.n/n≡1 n)

+
  where

+
  0-cop-1 = C.sym (C.1-coprimeTo 0)

+
  n/n≢0   = ℕ.>-nonZero (subst (ℕ._> 0) (sym (ℕ.n/n≡1 n)) (ℕ.z<s))

+

+
/-cong : ∀ {p₁ q₁ p₂ q₂} .{{_ : ℕ.NonZero q₁}} .{{_ : ℕ.NonZero q₂}} →

+
         p₁ ≡ p₂ → q₁ ≡ q₂ → p₁ / q₁ ≡ p₂ / q₂

+
/-cong {+ n}       refl = normalize-cong {n} refl

+
/-cong { -[1+ n ]} refl = cong -_ ∘′ normalize-cong {suc n} refl

+

+
private

+
  /-injective-≃-helper : ∀ {m n c d} .{{_ : ℕ.NonZero c}} .{{_ : ℕ.NonZero d}} →

+
                         - normalize (suc m) c ≡ normalize n d →

+
                          mkℚᵘ -[1+ m ] (ℕ.pred c) ≃ᵘ mkℚᵘ (+ n) (ℕ.pred d)

+
  /-injective-≃-helper {m} {n} {c} {d} -norm≡norm = contradiction

+
    (sym -norm≡norm)

+
    (nonNeg≢neg (normalize n d) (- normalize (suc m) c))

+
    where instance

+
      _ : NonNegative (normalize n d)

+
      _ = normalize-nonNeg n d

+

+
      _ : Negative (- normalize (suc m) c)

+
      _ = neg-pos {normalize (suc m) c} (normalize-pos (suc m) c)

+

+
/-injective-≃ : ∀ p q → ↥ᵘ p / ↧ₙᵘ p ≡ ↥ᵘ q / ↧ₙᵘ q → p ≃ᵘ q

+
/-injective-≃ (mkℚᵘ (+ m)    c-1) (mkℚᵘ (+ n)    d-1) eq =

+
  *≡* (cong (Sign.+ ◃_) (normalize-injective-≃ m n _ _ eq))

+
/-injective-≃ (mkℚᵘ (+ m)    c-1) (mkℚᵘ -[1+ n ] d-1) eq =

+
  ℚᵘ.≃-sym (/-injective-≃-helper (sym eq))

+
/-injective-≃ (mkℚᵘ -[1+ m ] c-1) (mkℚᵘ (+ n)    d-1) eq =

+
  /-injective-≃-helper eq

+
/-injective-≃ (mkℚᵘ -[1+ m ] c-1) (mkℚᵘ -[1+ n ] d-1) eq =

+
  *≡* (cong (Sign.- ◃_) (normalize-injective-≃ (suc m) (suc n) _ _ (neg-injective eq)))

+

+
------------------------------------------------------------------------

+
-- Properties of toℚ/fromℚ

+
------------------------------------------------------------------------

+

+
↥ᵘ-toℚᵘ : ∀ p → ↥ᵘ (toℚᵘ p) ≡ ↥ p

+
↥ᵘ-toℚᵘ p@record{} = refl

+

+
↧ᵘ-toℚᵘ : ∀ p → ↧ᵘ (toℚᵘ p) ≡ ↧ p

+
↧ᵘ-toℚᵘ p@record{} = refl

+

+
toℚᵘ-injective : Injective _≡_ _≃ᵘ_ toℚᵘ

+
toℚᵘ-injective {x@record{}} {y@record{}} (*≡* eq) = ≃⇒≡ (*≡* eq)

+

+
fromℚᵘ-injective : Injective _≃ᵘ_ _≡_ fromℚᵘ

+
fromℚᵘ-injective {p@record{}} {q@record{}} = /-injective-≃ p q

+

+
fromℚᵘ-toℚᵘ : ∀ p → fromℚᵘ (toℚᵘ p) ≡ p

+
fromℚᵘ-toℚᵘ (mkℚ (+ n)      d-1 c) = normalize-coprime c

+
fromℚᵘ-toℚᵘ (mkℚ (-[1+ n ]) d-1 c) = cong (-_) (normalize-coprime c)

+

+
toℚᵘ-fromℚᵘ : ∀ p → toℚᵘ (fromℚᵘ p) ≃ᵘ p

+
toℚᵘ-fromℚᵘ p = fromℚᵘ-injective (fromℚᵘ-toℚᵘ (fromℚᵘ p))

+

+
toℚᵘ-cong : toℚᵘ Preserves _≡_ ⟶ _≃ᵘ_

+
toℚᵘ-cong refl = *≡* refl

+

+
fromℚᵘ-cong : fromℚᵘ Preserves _≃ᵘ_ ⟶ _≡_

+
fromℚᵘ-cong {p} {q} p≃q = toℚᵘ-injective (begin-equality

+
  toℚᵘ (fromℚᵘ p)  ≃⟨  toℚᵘ-fromℚᵘ p ⟩

+
  p                ≃⟨  p≃q ⟩

+
  q                ≃⟨ toℚᵘ-fromℚᵘ q ⟨

+
  toℚᵘ (fromℚᵘ q)  ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
toℚᵘ-isRelHomomorphism : IsRelHomomorphism _≡_ _≃ᵘ_ toℚᵘ

+
toℚᵘ-isRelHomomorphism = record

+
  { cong = toℚᵘ-cong

+
  }

+

+
toℚᵘ-isRelMonomorphism : IsRelMonomorphism _≡_ _≃ᵘ_ toℚᵘ

+
toℚᵘ-isRelMonomorphism = record

+
  { isHomomorphism = toℚᵘ-isRelHomomorphism

+
  ; injective      = toℚᵘ-injective

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+
------------------------------------------------------------------------

+

+
drop-*≤* : p ≤ q → (↥ p ℤ.* ↧ q) ℤ.≤ (↥ q ℤ.* ↧ p)

+
drop-*≤* (*≤* pq≤qp) = pq≤qp

+

+
------------------------------------------------------------------------

+
-- toℚᵘ is a isomorphism

+

+
toℚᵘ-mono-≤ : p ≤ q → toℚᵘ p ≤ᵘ toℚᵘ q

+
toℚᵘ-mono-≤ {p@record{}} {q@record{}} (*≤* p≤q) = *≤* p≤q

+

+
toℚᵘ-cancel-≤ : toℚᵘ p ≤ᵘ toℚᵘ q → p ≤ q

+
toℚᵘ-cancel-≤ {p@record{}} {q@record{}} (*≤* p≤q) = *≤* p≤q

+

+
toℚᵘ-isOrderHomomorphism-≤ : IsOrderHomomorphism _≡_ _≃ᵘ_ _≤_ _≤ᵘ_ toℚᵘ

+
toℚᵘ-isOrderHomomorphism-≤ = record

+
  { cong = toℚᵘ-cong

+
  ; mono = toℚᵘ-mono-≤

+
  }

+

+
toℚᵘ-isOrderMonomorphism-≤ : IsOrderMonomorphism _≡_ _≃ᵘ_ _≤_ _≤ᵘ_ toℚᵘ

+
toℚᵘ-isOrderMonomorphism-≤ = record

+
  { isOrderHomomorphism = toℚᵘ-isOrderHomomorphism-≤

+
  ; injective           = toℚᵘ-injective

+
  ; cancel              = toℚᵘ-cancel-≤

+
  }

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
private

+
  module ≤-Monomorphism = OrderMonomorphisms toℚᵘ-isOrderMonomorphism-≤

+

+
≤-reflexive : _≡_ ⇒ _≤_

+
≤-reflexive refl = *≤* ℤ.≤-refl

+

+
≤-refl : Reflexive _≤_

+
≤-refl = ≤-reflexive refl

+

+
≤-trans : Transitive _≤_

+
≤-trans = ≤-Monomorphism.trans ℚᵘ.≤-trans

+

+
≤-antisym : Antisymmetric _≡_ _≤_

+
≤-antisym (*≤* le₁) (*≤* le₂) = ≃⇒≡ (*≡* (ℤ.≤-antisym le₁ le₂))

+

+
≤-total : Total _≤_

+
≤-total p q = [ inj₁ ∘ *≤* , inj₂ ∘ *≤* ]′ (ℤ.≤-total (↥ p ℤ.* ↧ q) (↥ q ℤ.* ↧ p))

+

+
infix 4 _≤?_ _≥?_

+

+
_≤?_ : Decidable _≤_

+
p ≤? q = Dec.map′ *≤* drop-*≤* (↥ p ℤ.* ↧ q ℤ.≤? ↥ q ℤ.* ↧ p)

+

+
_≥?_ : Decidable _≥_

+
_≥?_ = flip _≤?_

+

+
≤-irrelevant : Irrelevant _≤_

+
≤-irrelevant (*≤* p≤q₁) (*≤* p≤q₂) = cong *≤* (ℤ.≤-irrelevant p≤q₁ p≤q₂)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≤-isPreorder : IsPreorder _≡_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isTotalPreorder : IsTotalPreorder _≡_ _≤_

+
≤-isTotalPreorder = record

+
  { isPreorder = ≤-isPreorder

+
  ; total      = ≤-total

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≡_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder _≡_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≤-totalPreorder : TotalPreorder 0ℓ 0ℓ 0ℓ

+
≤-totalPreorder = record

+
  { isTotalPreorder = ≤-isTotalPreorder

+
  }

+

+
≤-decTotalOrder : DecTotalOrder _ _ _

+
≤-decTotalOrder = record

+
  { Carrier         = ℚ

+
  ; _≈_             = _≡_

+
  ; _≤_             = _≤_

+
  ; isDecTotalOrder = ≤-isDecTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+
------------------------------------------------------------------------

+

+
drop-*<* : p < q → (↥ p ℤ.* ↧ q) ℤ.< (↥ q ℤ.* ↧ p)

+
drop-*<* (*<* pq<qp) = pq<qp

+

+
------------------------------------------------------------------------

+
-- toℚᵘ is a isomorphism

+

+
toℚᵘ-mono-< : p < q → toℚᵘ p <ᵘ toℚᵘ q

+
toℚᵘ-mono-< {p@record{}} {q@record{}} (*<* p<q) = *<* p<q

+

+
toℚᵘ-cancel-< : toℚᵘ p <ᵘ toℚᵘ q → p < q

+
toℚᵘ-cancel-< {p@record{}} {q@record{}} (*<* p<q) = *<* p<q

+

+
toℚᵘ-isOrderHomomorphism-< : IsOrderHomomorphism _≡_ _≃ᵘ_ _<_ _<ᵘ_ toℚᵘ

+
toℚᵘ-isOrderHomomorphism-< = record

+
  { cong = toℚᵘ-cong

+
  ; mono = toℚᵘ-mono-<

+
  }

+

+
toℚᵘ-isOrderMonomorphism-< : IsOrderMonomorphism _≡_ _≃ᵘ_ _<_ _<ᵘ_ toℚᵘ

+
toℚᵘ-isOrderMonomorphism-< = record

+
  { isOrderHomomorphism = toℚᵘ-isOrderHomomorphism-<

+
  ; injective           = toℚᵘ-injective

+
  ; cancel              = toℚᵘ-cancel-<

+
  }

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ (*<* p<q) = *≤* (ℤ.<⇒≤ p<q)

+

+
≮⇒≥ : _≮_ ⇒ _≥_

+
≮⇒≥ {p} {q} p≮q = *≤* (ℤ.≮⇒≥ (p≮q ∘ *<*))

+

+
≰⇒> : _≰_ ⇒ _>_

+
≰⇒> {p} {q} p≰q = *<* (ℤ.≰⇒> (p≰q ∘ *≤*))

+

+
<⇒≢ : _<_ ⇒ _≢_

+
<⇒≢ {p} {q} (*<* p<q) = ℤ.<⇒≢ p<q ∘ drop-*≡* ∘ ≡⇒≃

+

+
<-irrefl : Irreflexive _≡_ _<_

+
<-irrefl refl (*<* p<p) = ℤ.<-irrefl refl p<p

+

+
<-asym : Asymmetric _<_

+
<-asym (*<* p<q) (*<* q<p) = ℤ.<-asym p<q q<p

+

+
<-dense : Dense _<_

+
<-dense {p} {q} p<q = let

+
    m , p<ᵘm , m<ᵘq = ℚᵘ.<-dense (toℚᵘ-mono-< p<q)

+

+
    m≃m : m ≃ᵘ toℚᵘ (fromℚᵘ m)

+
    m≃m = ℚᵘ.≃-sym (toℚᵘ-fromℚᵘ m)

+

+
    p<m : p < fromℚᵘ m

+
    p<m = toℚᵘ-cancel-< (ℚᵘ.<-respʳ-≃ m≃m p<ᵘm)

+

+
    m<q : fromℚᵘ m < q

+
    m<q = toℚᵘ-cancel-< (ℚᵘ.<-respˡ-≃ m≃m m<ᵘq)

+
  in fromℚᵘ m , p<m , m<q

+

+
<-≤-trans : Trans _<_ _≤_ _<_

+
<-≤-trans {p} {q} {r} (*<* p<q) (*≤* q≤r) = *<*

+
  (ℤ.*-cancelʳ-<-nonNeg _ (begin-strict

+
  let n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; sd₁ = ↧ p; sd₂ = ↧ q; sd₃ = ↧ r in

+
  (n₁  ℤ.* sd₃) ℤ.* sd₂  ≡⟨ ℤ.*-assoc n₁ sd₃ sd₂ ⟩

+
  n₁   ℤ.* (sd₃ ℤ.* sd₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm sd₃ sd₂) ⟩

+
  n₁   ℤ.* (sd₂ ℤ.* sd₃) ≡⟨ sym (ℤ.*-assoc n₁ sd₂ sd₃) ⟩

+
  (n₁  ℤ.* sd₂) ℤ.* sd₃  <⟨ ℤ.*-monoʳ-<-pos (↧ r) p<q ⟩

+
  (n₂  ℤ.* sd₁) ℤ.* sd₃  ≡⟨ cong (ℤ._* sd₃) (ℤ.*-comm n₂ sd₁) ⟩

+
  (sd₁ ℤ.* n₂)  ℤ.* sd₃  ≡⟨ ℤ.*-assoc sd₁ n₂ sd₃ ⟩

+
  sd₁  ℤ.* (n₂  ℤ.* sd₃) ≤⟨ ℤ.*-monoˡ-≤-nonNeg (↧ p) q≤r ⟩

+
  sd₁  ℤ.* (n₃  ℤ.* sd₂) ≡⟨ sym (ℤ.*-assoc sd₁ n₃ sd₂) ⟩

+
  (sd₁ ℤ.* n₃)  ℤ.* sd₂  ≡⟨ cong (ℤ._* sd₂) (ℤ.*-comm sd₁ n₃) ⟩

+
  (n₃  ℤ.* sd₁) ℤ.* sd₂  ∎))

+
  where open ℤ.≤-Reasoning

+

+
≤-<-trans : Trans _≤_ _<_ _<_

+
≤-<-trans {p} {q} {r} (*≤* p≤q) (*<* q<r) = *<*

+
  (ℤ.*-cancelʳ-<-nonNeg _ (begin-strict

+
  let n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; sd₁ = ↧ p; sd₂ = ↧ q; sd₃ = ↧ r in

+
  (n₁  ℤ.* sd₃) ℤ.* sd₂  ≡⟨ ℤ.*-assoc n₁ sd₃ sd₂ ⟩

+
  n₁   ℤ.* (sd₃ ℤ.* sd₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm sd₃ sd₂) ⟩

+
  n₁   ℤ.* (sd₂ ℤ.* sd₃) ≡⟨ sym (ℤ.*-assoc n₁ sd₂ sd₃) ⟩

+
  (n₁  ℤ.* sd₂) ℤ.* sd₃  ≤⟨ ℤ.*-monoʳ-≤-nonNeg (↧ r) p≤q ⟩

+
  (n₂  ℤ.* sd₁) ℤ.* sd₃  ≡⟨ cong (ℤ._* sd₃) (ℤ.*-comm n₂ sd₁) ⟩

+
  (sd₁ ℤ.* n₂)  ℤ.* sd₃  ≡⟨ ℤ.*-assoc sd₁ n₂ sd₃ ⟩

+
  sd₁  ℤ.* (n₂  ℤ.* sd₃) <⟨ ℤ.*-monoˡ-<-pos (↧ p) q<r ⟩

+
  sd₁  ℤ.* (n₃  ℤ.* sd₂) ≡⟨ sym (ℤ.*-assoc sd₁ n₃ sd₂) ⟩

+
  (sd₁ ℤ.* n₃)  ℤ.* sd₂  ≡⟨ cong (ℤ._* sd₂) (ℤ.*-comm sd₁ n₃) ⟩

+
  (n₃  ℤ.* sd₁) ℤ.* sd₂  ∎))

+
  where open ℤ.≤-Reasoning

+

+
<-trans : Transitive _<_

+
<-trans p<q = ≤-<-trans (<⇒≤ p<q)

+

+
infix 4 _<?_ _>?_

+

+
_<?_ : Decidable _<_

+
p <? q = Dec.map′ *<* drop-*<* ((↥ p ℤ.* ↧ q) ℤ.<? (↥ q ℤ.* ↧ p))

+

+
_>?_ : Decidable _>_

+
_>?_ = flip _<?_

+

+
<-cmp : Trichotomous _≡_ _<_

+
<-cmp p q with ℤ.<-cmp (↥ p ℤ.* ↧ q) (↥ q ℤ.* ↧ p)

+
... | tri< < ≢ ≯ = tri< (*<* <)        (≢ ∘ drop-*≡* ∘ ≡⇒≃) (≯ ∘ drop-*<*)

+
... | tri≈ ≮ ≡ ≯ = tri≈ (≮ ∘ drop-*<*) (≃⇒≡ (*≡* ≡))   (≯ ∘ drop-*<*)

+
... | tri> ≮ ≢ > = tri> (≮ ∘ drop-*<*) (≢ ∘ drop-*≡* ∘ ≡⇒≃) (*<* >)

+

+
<-irrelevant : Irrelevant _<_

+
<-irrelevant (*<* p<q₁) (*<* p<q₂) = cong *<* (ℤ.<-irrelevant p<q₁ p<q₂)

+

+
<-respʳ-≡ : _<_ Respectsʳ _≡_

+
<-respʳ-≡ = subst (_ <_)

+

+
<-respˡ-≡ : _<_ Respectsˡ _≡_

+
<-respˡ-≡ = subst (_< _)

+

+
<-resp-≡ : _<_ Respects₂ _≡_

+
<-resp-≡ = <-respʳ-≡ , <-respˡ-≡

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = <-resp-≡

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
<-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_

+
<-isDenseLinearOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  ; dense              = <-dense

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
<-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ

+
<-denseLinearOrder = record

+
  { isDenseLinearOrder = <-isDenseLinearOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- A specialised module for reasoning about the _≤_ and _<_ relations

+
------------------------------------------------------------------------

+

+
module ≤-Reasoning where

+
  import Relation.Binary.Reasoning.Base.Triple

+
    ≤-isPreorder

+
    <-asym

+
    <-trans

+
    (resp₂ _<_)

+
    <⇒≤

+
    <-≤-trans

+
    ≤-<-trans

+
    as Triple

+

+
  open Triple public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+

+
  ≃-go : Trans _≃_ _IsRelatedTo_ _IsRelatedTo_

+
  ≃-go = Triple.≈-go ∘′ ≃⇒≡

+

+
  open ≃-syntax _IsRelatedTo_ _IsRelatedTo_ ≃-go (λ {x y} → ≃-sym {x} {y}) public

+

+
------------------------------------------------------------------------

+
-- Properties of Positive/NonPositive/Negative/NonNegative and _≤_/_<_

+

+
positive⁻¹ : ∀ p → .{{Positive p}} → p > 0ℚ

+
positive⁻¹ p = toℚᵘ-cancel-< (ℚᵘ.positive⁻¹ (toℚᵘ p))

+

+
nonNegative⁻¹ : ∀ p → .{{NonNegative p}} → p ≥ 0ℚ

+
nonNegative⁻¹ p = toℚᵘ-cancel-≤ (ℚᵘ.nonNegative⁻¹ (toℚᵘ p))

+

+
negative⁻¹ : ∀ p → .{{Negative p}} → p < 0ℚ

+
negative⁻¹ p = toℚᵘ-cancel-< (ℚᵘ.negative⁻¹ (toℚᵘ p))

+

+
nonPositive⁻¹ : ∀ p → .{{NonPositive p}} → p ≤ 0ℚ

+
nonPositive⁻¹ p = toℚᵘ-cancel-≤ (ℚᵘ.nonPositive⁻¹ (toℚᵘ p))

+

+
neg<pos : ∀ p q → .{{Negative p}} → .{{Positive q}} → p < q

+
neg<pos p q = toℚᵘ-cancel-< (ℚᵘ.neg<pos (toℚᵘ p) (toℚᵘ q))

+

+
------------------------------------------------------------------------

+
-- Properties of -_ and _≤_/_<_

+

+
neg-antimono-< : -_ Preserves _<_ ⟶ _>_

+
neg-antimono-< {mkℚ -[1+ _ ] _ _} {mkℚ -[1+ _ ] _ _} (*<* (ℤ.-<- n<m)) = *<* (ℤ.+<+ (ℕ.s<s n<m))

+
neg-antimono-< {mkℚ -[1+ _ ] _ _} {mkℚ +0       _ _} (*<* ℤ.-<+)       = *<* (ℤ.+<+ ℕ.z<s)

+
neg-antimono-< {mkℚ -[1+ _ ] _ _} {mkℚ +[1+ _ ] _ _} (*<* ℤ.-<+)       = *<* ℤ.-<+

+
neg-antimono-< {mkℚ +0       _ _} {mkℚ +0       _ _} (*<* (ℤ.+<+ m<n)) = *<* (ℤ.+<+ m<n)

+
neg-antimono-< {mkℚ +0       _ _} {mkℚ +[1+ _ ] _ _} (*<* (ℤ.+<+ m<n)) = *<* ℤ.-<+

+
neg-antimono-< {mkℚ +[1+ _ ] _ _} {mkℚ +0       _ _} (*<* (ℤ.+<+ ()))

+
neg-antimono-< {mkℚ +[1+ _ ] _ _} {mkℚ +[1+ _ ] _ _} (*<* (ℤ.+<+ (ℕ.s<s m<n))) = *<* (ℤ.-<- m<n)

+

+
neg-antimono-≤ : -_ Preserves _≤_ ⟶ _≥_

+
neg-antimono-≤ {mkℚ -[1+ _ ] _ _} {mkℚ -[1+ _ ] _ _} (*≤* (ℤ.-≤- n≤m)) = *≤* (ℤ.+≤+ (ℕ.s≤s n≤m))

+
neg-antimono-≤ {mkℚ -[1+ _ ] _ _} {mkℚ +0       _ _} (*≤* ℤ.-≤+)       = *≤* (ℤ.+≤+ ℕ.z≤n)

+
neg-antimono-≤ {mkℚ -[1+ _ ] _ _} {mkℚ +[1+ _ ] _ _} (*≤* ℤ.-≤+)       = *≤* ℤ.-≤+

+
neg-antimono-≤ {mkℚ +0       _ _} {mkℚ +0       _ _} (*≤* (ℤ.+≤+ m≤n)) = *≤* (ℤ.+≤+ m≤n)

+
neg-antimono-≤ {mkℚ +0       _ _} {mkℚ +[1+ _ ] _ _} (*≤* (ℤ.+≤+ m≤n)) = *≤* ℤ.-≤+

+
neg-antimono-≤ {mkℚ +[1+ _ ] _ _} {mkℚ +0       _ _} (*≤* (ℤ.+≤+ ()))

+
neg-antimono-≤ {mkℚ +[1+ _ ] _ _} {mkℚ +[1+ _ ] _ _} (*≤* (ℤ.+≤+ (ℕ.s≤s m≤n))) = *≤* (ℤ.-≤- m≤n)

+

+
------------------------------------------------------------------------

+
-- Properties of _≤ᵇ_

+
------------------------------------------------------------------------

+

+
≤ᵇ⇒≤ : T (p ≤ᵇ q) → p ≤ q

+
≤ᵇ⇒≤ = *≤* ∘′ ℤ.≤ᵇ⇒≤

+

+
≤⇒≤ᵇ : p ≤ q → T (p ≤ᵇ q)

+
≤⇒≤ᵇ = ℤ.≤⇒≤ᵇ ∘′ drop-*≤*

+

+
------------------------------------------------------------------------

+
-- Properties of _+_

+
------------------------------------------------------------------------

+

+
private

+
  ↥+ᵘ : ℚ → ℚ → ℤ

+
  ↥+ᵘ p q = ↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p

+

+
  ↧+ᵘ : ℚ → ℚ → ℤ

+
  ↧+ᵘ p q = ↧ p ℤ.* ↧ q

+

+
  +-nf : ℚ → ℚ → ℤ

+
  +-nf p q = gcd (↥+ᵘ p q) (↧+ᵘ p q)

+

+
↥-+ : ∀ p q → ↥ (p + q) ℤ.* +-nf p q ≡ ↥+ᵘ p q

+
↥-+ p@record{} q@record{} = ↥-/ (↥+ᵘ p q) (↧ₙ p ℕ.* ↧ₙ q)

+

+
↧-+ : ∀ p q → ↧ (p + q) ℤ.* +-nf p q ≡ ↧+ᵘ p q

+
↧-+ p@record{} q@record{} = ↧-/ (↥+ᵘ p q) (↧ₙ p ℕ.* ↧ₙ q)

+

+

+
------------------------------------------------------------------------

+
-- Monomorphic to unnormalised _+_

+

+
open Definitions ℚ ℚᵘ ℚᵘ._≃_

+

+
toℚᵘ-homo-+ : Homomorphic₂ toℚᵘ _+_ ℚᵘ._+_

+
toℚᵘ-homo-+ p@record{} q@record{} with +-nf p q ℤ.≟ 0ℤ

+
... | yes nf[p,q]≡0 = *≡* $ begin

+
  ↥ᵘ (toℚᵘ (p + q)) ℤ.* ↧+ᵘ p q   ≡⟨ cong (ℤ._* ↧+ᵘ p q) (↥ᵘ-toℚᵘ (p + q)) ⟩

+
  ↥ (p + q) ℤ.* ↧+ᵘ p q           ≡⟨ cong (ℤ._* ↧+ᵘ p q) eq ⟩

+
  0ℤ        ℤ.* ↧+ᵘ p q           ≡⟨⟩

+
  0ℤ        ℤ.* ↧ (p + q)         ≡⟨ cong (ℤ._* ↧ (p + q)) (sym eq2) ⟩

+
  ↥+ᵘ p q   ℤ.* ↧ (p + q)         ≡⟨ cong (↥+ᵘ p q ℤ.*_) (sym (↧ᵘ-toℚᵘ (p + q))) ⟩

+
  ↥+ᵘ p q   ℤ.* ↧ᵘ (toℚᵘ (p + q)) ∎

+
  where

+
  open ≡-Reasoning

+
  eq2 : ↥+ᵘ p q ≡ 0ℤ

+
  eq2 = gcd[i,j]≡0⇒i≡0 (↥+ᵘ p q) (↧+ᵘ p q) nf[p,q]≡0

+

+
  eq : ↥ (p + q) ≡ 0ℤ

+
  eq rewrite eq2 = cong ↥_ (0/n≡0 (↧ₙ p ℕ.* ↧ₙ q))

+

+
... | no  nf[p,q]≢0 = *≡* (ℤ.*-cancelʳ-≡ _ _ (+-nf p q) {{ℤ.≢-nonZero nf[p,q]≢0}} $ begin

+
    (↥ᵘ (toℚᵘ (p + q))) ℤ.* ↧+ᵘ p q  ℤ.* +-nf p q ≡⟨ cong (λ v → v ℤ.* ↧+ᵘ p q ℤ.* +-nf p q) (↥ᵘ-toℚᵘ (p + q)) ⟩

+
    ↥ (p + q) ℤ.* ↧+ᵘ p q ℤ.* +-nf p q            ≡⟨ xy∙z≈xz∙y (↥ (p + q)) _ _ ⟩

+
    ↥ (p + q) ℤ.* +-nf p q ℤ.* ↧+ᵘ p q            ≡⟨ cong (ℤ._* ↧+ᵘ p q) (↥-+ p q) ⟩

+
    ↥+ᵘ p q ℤ.* ↧+ᵘ p q                           ≡⟨ cong (↥+ᵘ p q ℤ.*_) (sym (↧-+ p q)) ⟩

+
    ↥+ᵘ p q ℤ.* (↧ (p + q) ℤ.* +-nf p q)          ≡⟨ x∙yz≈xy∙z (↥+ᵘ p q) _ _ ⟩

+
    ↥+ᵘ p q ℤ.* ↧ (p + q)  ℤ.* +-nf p q           ≡⟨ cong (λ v → ↥+ᵘ p q ℤ.* v ℤ.* +-nf p q) (↧ᵘ-toℚᵘ (p + q)) ⟨

+
    ↥+ᵘ p q ℤ.* ↧ᵘ (toℚᵘ (p + q)) ℤ.* +-nf p q    ∎)

+
  where open ≡-Reasoning; open CommSemigroupProperties ℤ.*-commutativeSemigroup

+

+
toℚᵘ-isMagmaHomomorphism-+ : IsMagmaHomomorphism +-rawMagma ℚᵘ.+-rawMagma toℚᵘ

+
toℚᵘ-isMagmaHomomorphism-+ = record

+
  { isRelHomomorphism = toℚᵘ-isRelHomomorphism

+
  ; homo              = toℚᵘ-homo-+

+
  }

+

+
toℚᵘ-isMonoidHomomorphism-+ : IsMonoidHomomorphism +-0-rawMonoid ℚᵘ.+-0-rawMonoid toℚᵘ

+
toℚᵘ-isMonoidHomomorphism-+ = record

+
  { isMagmaHomomorphism = toℚᵘ-isMagmaHomomorphism-+

+
  ; ε-homo              = ℚᵘ.≃-refl

+
  }

+

+
toℚᵘ-isMonoidMonomorphism-+ : IsMonoidMonomorphism +-0-rawMonoid ℚᵘ.+-0-rawMonoid toℚᵘ

+
toℚᵘ-isMonoidMonomorphism-+ = record

+
  { isMonoidHomomorphism = toℚᵘ-isMonoidHomomorphism-+

+
  ; injective            = toℚᵘ-injective

+
  }

+

+
------------------------------------------------------------------------

+
-- Monomorphic to unnormalised -_

+

+
toℚᵘ-homo‿- : Homomorphic₁ toℚᵘ (-_) (ℚᵘ.-_)

+
toℚᵘ-homo‿- (mkℚ +0       _ _) = *≡* refl

+
toℚᵘ-homo‿- (mkℚ +[1+ _ ] _ _) = *≡* refl

+
toℚᵘ-homo‿- (mkℚ -[1+ _ ] _ _) = *≡* refl

+

+
toℚᵘ-isGroupHomomorphism-+ : IsGroupHomomorphism +-0-rawGroup ℚᵘ.+-0-rawGroup toℚᵘ

+
toℚᵘ-isGroupHomomorphism-+ = record

+
  { isMonoidHomomorphism = toℚᵘ-isMonoidHomomorphism-+

+
  ; ⁻¹-homo              = toℚᵘ-homo‿-

+
  }

+

+
toℚᵘ-isGroupMonomorphism-+ : IsGroupMonomorphism +-0-rawGroup ℚᵘ.+-0-rawGroup toℚᵘ

+
toℚᵘ-isGroupMonomorphism-+ = record

+
  { isGroupHomomorphism = toℚᵘ-isGroupHomomorphism-+

+
  ; injective           = toℚᵘ-injective

+
  }

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
private

+
  module +-Monomorphism = GroupMonomorphisms toℚᵘ-isGroupMonomorphism-+

+

+
+-assoc : Associative _+_

+
+-assoc = +-Monomorphism.assoc ℚᵘ.+-isMagma ℚᵘ.+-assoc

+

+
+-comm : Commutative _+_

+
+-comm = +-Monomorphism.comm ℚᵘ.+-isMagma ℚᵘ.+-comm

+

+
+-identityˡ : LeftIdentity 0ℚ _+_

+
+-identityˡ = +-Monomorphism.identityˡ ℚᵘ.+-isMagma ℚᵘ.+-identityˡ

+

+
+-identityʳ : RightIdentity 0ℚ _+_

+
+-identityʳ = +-Monomorphism.identityʳ ℚᵘ.+-isMagma ℚᵘ.+-identityʳ

+

+
+-identity : Identity 0ℚ _+_

+
+-identity = +-identityˡ , +-identityʳ

+

+
+-inverseˡ : LeftInverse 0ℚ -_ _+_

+
+-inverseˡ = +-Monomorphism.inverseˡ ℚᵘ.+-isMagma ℚᵘ.+-inverseˡ

+

+
+-inverseʳ : RightInverse 0ℚ -_ _+_

+
+-inverseʳ = +-Monomorphism.inverseʳ ℚᵘ.+-isMagma ℚᵘ.+-inverseʳ

+

+
+-inverse : Inverse 0ℚ -_ _+_

+
+-inverse = +-Monomorphism.inverse ℚᵘ.+-isMagma ℚᵘ.+-inverse

+

+
-‿cong : Congruent₁ (-_)

+
-‿cong = +-Monomorphism.⁻¹-cong ℚᵘ.+-isMagma ℚᵘ.-‿cong

+

+
neg-distrib-+ : ∀ p q → - (p + q) ≡ (- p) + (- q)

+
neg-distrib-+ = +-Monomorphism.⁻¹-distrib-∙ ℚᵘ.+-0-isAbelianGroup (ℚᵘ.≃-reflexive ∘₂ ℚᵘ.neg-distrib-+)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
+-isMagma : IsMagma _+_

+
+-isMagma = +-Monomorphism.isMagma ℚᵘ.+-isMagma

+

+
+-isSemigroup : IsSemigroup _+_

+
+-isSemigroup = +-Monomorphism.isSemigroup ℚᵘ.+-isSemigroup

+

+
+-0-isMonoid : IsMonoid _+_ 0ℚ

+
+-0-isMonoid = +-Monomorphism.isMonoid ℚᵘ.+-0-isMonoid

+

+
+-0-isCommutativeMonoid : IsCommutativeMonoid _+_ 0ℚ

+
+-0-isCommutativeMonoid = +-Monomorphism.isCommutativeMonoid ℚᵘ.+-0-isCommutativeMonoid

+

+
+-0-isGroup : IsGroup _+_ 0ℚ (-_)

+
+-0-isGroup = +-Monomorphism.isGroup ℚᵘ.+-0-isGroup

+

+
+-0-isAbelianGroup : IsAbelianGroup _+_ 0ℚ (-_)

+
+-0-isAbelianGroup = +-Monomorphism.isAbelianGroup ℚᵘ.+-0-isAbelianGroup

+

+
------------------------------------------------------------------------

+
-- Packages

+

+
+-magma : Magma 0ℓ 0ℓ

+
+-magma = record

+
  { isMagma = +-isMagma

+
  }

+

+
+-semigroup : Semigroup 0ℓ 0ℓ

+
+-semigroup = record

+
  { isSemigroup = +-isSemigroup

+
  }

+

+
+-0-monoid : Monoid 0ℓ 0ℓ

+
+-0-monoid = record

+
  { isMonoid = +-0-isMonoid

+
  }

+

+
+-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
+-0-commutativeMonoid = record

+
  { isCommutativeMonoid = +-0-isCommutativeMonoid

+
  }

+

+
+-0-group : Group 0ℓ 0ℓ

+
+-0-group = record

+
  { isGroup = +-0-isGroup

+
  }

+

+
+-0-abelianGroup : AbelianGroup 0ℓ 0ℓ

+
+-0-abelianGroup = record

+
  { isAbelianGroup = +-0-isAbelianGroup

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _≤_

+

+
+-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
+-mono-≤ {p} {q} {r} {s} p≤q r≤s = toℚᵘ-cancel-≤ (begin

+
  toℚᵘ(p + r)          ≃⟨ toℚᵘ-homo-+ p r ⟩

+
  toℚᵘ(p) ℚᵘ.+ toℚᵘ(r) ≤⟨ ℚᵘ.+-mono-≤ (toℚᵘ-mono-≤ p≤q) (toℚᵘ-mono-≤ r≤s) ⟩

+
  toℚᵘ(q) ℚᵘ.+ toℚᵘ(s) ≃⟨ ℚᵘ.≃-sym (toℚᵘ-homo-+ q s) ⟩

+
  toℚᵘ(q + s)          ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
+-monoˡ-≤ : ∀ r → (_+ r) Preserves _≤_ ⟶ _≤_

+
+-monoˡ-≤ r p≤q = +-mono-≤ p≤q (≤-refl {r})

+

+
+-monoʳ-≤ : ∀ r → (_+_ r) Preserves _≤_ ⟶ _≤_

+
+-monoʳ-≤ r p≤q = +-mono-≤ (≤-refl {r}) p≤q

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _<_

+

+
+-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_

+
+-mono-<-≤ {p} {q} {r} {s} p<q r≤s = toℚᵘ-cancel-< (begin-strict

+
  toℚᵘ(p + r)          ≃⟨ toℚᵘ-homo-+ p r ⟩

+
  toℚᵘ(p) ℚᵘ.+ toℚᵘ(r) <⟨ ℚᵘ.+-mono-<-≤ (toℚᵘ-mono-< p<q) (toℚᵘ-mono-≤ r≤s) ⟩

+
  toℚᵘ(q) ℚᵘ.+ toℚᵘ(s) ≃⟨ ℚᵘ.≃-sym (toℚᵘ-homo-+ q s) ⟩

+
  toℚᵘ(q + s)          ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
+-mono-≤-< : _+_ Preserves₂ _≤_ ⟶ _<_ ⟶ _<_

+
+-mono-≤-< {p} {q} {r} {s} p≤q r<s rewrite +-comm p r | +-comm q s = +-mono-<-≤ r<s p≤q

+

+
+-mono-< : _+_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
+-mono-< {p} {q} {r} {s} p<q r<s = <-trans (+-mono-<-≤ p<q (≤-refl {r})) (+-mono-≤-< (≤-refl {q}) r<s)

+

+
+-monoˡ-< : ∀ r → (_+ r) Preserves _<_ ⟶ _<_

+
+-monoˡ-< r p<q = +-mono-<-≤ p<q (≤-refl {r})

+

+
+-monoʳ-< : ∀ r → (_+_ r) Preserves _<_ ⟶ _<_

+
+-monoʳ-< r p<q = +-mono-≤-< (≤-refl {r}) p<q

+

+
------------------------------------------------------------------------

+
-- Properties of _*_

+
------------------------------------------------------------------------

+

+
private

+
  *-nf : ℚ → ℚ → ℤ

+
  *-nf p q = gcd (↥ p ℤ.* ↥ q) (↧ p ℤ.* ↧ q)

+

+
↥-* : ∀ p q → ↥ (p * q) ℤ.* *-nf p q ≡ ↥ p ℤ.* ↥ q

+
↥-* p@record{} q@record{} = ↥-/ (↥ p ℤ.* ↥ q) (↧ₙ p ℕ.* ↧ₙ q)

+

+
↧-* : ∀ p q → ↧ (p * q) ℤ.* *-nf p q ≡ ↧ p ℤ.* ↧ q

+
↧-* p@record{} q@record{} = ↧-/ (↥ p ℤ.* ↥ q) (↧ₙ p ℕ.* ↧ₙ q)

+

+
------------------------------------------------------------------------

+
-- Monomorphic to unnormalised _*_

+

+
toℚᵘ-homo-* : Homomorphic₂ toℚᵘ _*_ ℚᵘ._*_

+
toℚᵘ-homo-* p@record{} q@record{} with *-nf p q ℤ.≟ 0ℤ

+
... | yes nf[p,q]≡0 = *≡* $ begin

+
  ↥ᵘ (toℚᵘ (p * q)) ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) (↥ᵘ-toℚᵘ (p * q)) ⟩

+
  ↥ (p * q)         ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) eq ⟩

+
  0ℤ                ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨⟩

+
  0ℤ                ℤ.* ↧ (p * q)         ≡⟨ cong (ℤ._* ↧ (p * q)) (sym eq2) ⟩

+
  (↥ p ℤ.* ↥ q)     ℤ.* ↧ (p * q)         ≡⟨ cong ((↥ p ℤ.* ↥ q) ℤ.*_) (sym (↧ᵘ-toℚᵘ (p * q))) ⟩

+
  (↥ p ℤ.* ↥ q)     ℤ.* ↧ᵘ (toℚᵘ (p * q)) ∎

+
  where

+
  open ≡-Reasoning

+
  eq2 : ↥ p ℤ.* ↥ q ≡ 0ℤ

+
  eq2 = gcd[i,j]≡0⇒i≡0 (↥ p ℤ.* ↥ q) (↧ p ℤ.* ↧ q) nf[p,q]≡0

+

+
  eq : ↥ (p * q) ≡ 0ℤ

+
  eq rewrite eq2 = cong ↥_ (0/n≡0 (↧ₙ p ℕ.* ↧ₙ q))

+
... | no nf[p,q]≢0 = *≡* (ℤ.*-cancelʳ-≡ _ _ (*-nf p q) {{ℤ.≢-nonZero nf[p,q]≢0}} $ begin

+
  ↥ᵘ (toℚᵘ (p * q)) ℤ.* (↧ p ℤ.* ↧ q) ℤ.* *-nf p q     ≡⟨ cong (λ v → v ℤ.* (↧ p ℤ.* ↧ q) ℤ.* *-nf p q) (↥ᵘ-toℚᵘ (p * q)) ⟩

+
  ↥ (p * q)         ℤ.* (↧ p ℤ.* ↧ q) ℤ.* *-nf p q     ≡⟨ xy∙z≈xz∙y (↥ (p * q)) _ _ ⟩

+
  ↥ (p * q)         ℤ.* *-nf p q ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) (↥-* p q) ⟩

+
  (↥ p ℤ.* ↥ q)     ℤ.* (↧ p ℤ.* ↧ q)                  ≡⟨ cong ((↥ p ℤ.* ↥ q) ℤ.*_) (sym (↧-* p q)) ⟩

+
  (↥ p ℤ.* ↥ q)     ℤ.* (↧ (p * q) ℤ.* *-nf p q)       ≡⟨ x∙yz≈xy∙z (↥ p ℤ.* ↥ q) _ _ ⟩

+
  (↥ p ℤ.* ↥ q)     ℤ.* ↧ (p * q)  ℤ.* *-nf p q        ≡⟨ cong (λ v → (↥ p ℤ.* ↥ q) ℤ.* v ℤ.* *-nf p q) (↧ᵘ-toℚᵘ (p * q)) ⟨

+
  (↥ p ℤ.* ↥ q)     ℤ.* ↧ᵘ (toℚᵘ (p * q)) ℤ.* *-nf p q ∎)

+
  where open ≡-Reasoning; open CommSemigroupProperties ℤ.*-commutativeSemigroup

+

+
toℚᵘ-homo-1/ : ∀ p .{{_ : NonZero p}} → toℚᵘ (1/ p) ℚᵘ.≃ (ℚᵘ.1/ toℚᵘ p)

+
toℚᵘ-homo-1/ (mkℚ +[1+ _ ] _ _) = ℚᵘ.≃-refl

+
toℚᵘ-homo-1/ (mkℚ -[1+ _ ] _ _) = ℚᵘ.≃-refl

+

+
toℚᵘ-isMagmaHomomorphism-* : IsMagmaHomomorphism *-rawMagma ℚᵘ.*-rawMagma toℚᵘ

+
toℚᵘ-isMagmaHomomorphism-* = record

+
  { isRelHomomorphism = toℚᵘ-isRelHomomorphism

+
  ; homo              = toℚᵘ-homo-*

+
  }

+

+
toℚᵘ-isMonoidHomomorphism-* : IsMonoidHomomorphism *-1-rawMonoid ℚᵘ.*-1-rawMonoid toℚᵘ

+
toℚᵘ-isMonoidHomomorphism-* = record

+
  { isMagmaHomomorphism = toℚᵘ-isMagmaHomomorphism-*

+
  ; ε-homo              = ℚᵘ.≃-refl

+
  }

+

+
toℚᵘ-isMonoidMonomorphism-* : IsMonoidMonomorphism *-1-rawMonoid ℚᵘ.*-1-rawMonoid toℚᵘ

+
toℚᵘ-isMonoidMonomorphism-* = record

+
  { isMonoidHomomorphism = toℚᵘ-isMonoidHomomorphism-*

+
  ; injective            = toℚᵘ-injective

+
  }

+

+
toℚᵘ-isNearSemiringHomomorphism-+-* : IsNearSemiringHomomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ

+
toℚᵘ-isNearSemiringHomomorphism-+-* = record

+
  { +-isMonoidHomomorphism = toℚᵘ-isMonoidHomomorphism-+

+
  ; *-homo                 = toℚᵘ-homo-*

+
  }

+

+
toℚᵘ-isNearSemiringMonomorphism-+-* : IsNearSemiringMonomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ

+
toℚᵘ-isNearSemiringMonomorphism-+-* = record

+
  { isNearSemiringHomomorphism = toℚᵘ-isNearSemiringHomomorphism-+-*

+
  ; injective                  = toℚᵘ-injective

+
  }

+

+
toℚᵘ-isSemiringHomomorphism-+-* : IsSemiringHomomorphism +-*-rawSemiring ℚᵘ.+-*-rawSemiring toℚᵘ

+
toℚᵘ-isSemiringHomomorphism-+-* = record

+
  { isNearSemiringHomomorphism = toℚᵘ-isNearSemiringHomomorphism-+-*

+
  ; 1#-homo                    = ℚᵘ.≃-refl

+
  }

+

+
toℚᵘ-isSemiringMonomorphism-+-* : IsSemiringMonomorphism +-*-rawSemiring ℚᵘ.+-*-rawSemiring toℚᵘ

+
toℚᵘ-isSemiringMonomorphism-+-* = record

+
  { isSemiringHomomorphism = toℚᵘ-isSemiringHomomorphism-+-*

+
  ; injective              = toℚᵘ-injective

+
  }

+

+
toℚᵘ-isRingHomomorphism-+-* : IsRingHomomorphism +-*-rawRing ℚᵘ.+-*-rawRing toℚᵘ

+
toℚᵘ-isRingHomomorphism-+-* = record

+
  { isSemiringHomomorphism = toℚᵘ-isSemiringHomomorphism-+-*

+
  ; -‿homo                 = toℚᵘ-homo‿-

+
  }

+

+
toℚᵘ-isRingMonomorphism-+-* : IsRingMonomorphism +-*-rawRing ℚᵘ.+-*-rawRing toℚᵘ

+
toℚᵘ-isRingMonomorphism-+-* = record

+
  { isRingHomomorphism = toℚᵘ-isRingHomomorphism-+-*

+
  ; injective          = toℚᵘ-injective

+
  }

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
private

+
  module +-*-Monomorphism = RingMonomorphisms toℚᵘ-isRingMonomorphism-+-*

+

+
*-assoc : Associative _*_

+
*-assoc = +-*-Monomorphism.*-assoc ℚᵘ.*-isMagma ℚᵘ.*-assoc

+

+
*-comm : Commutative _*_

+
*-comm = +-*-Monomorphism.*-comm ℚᵘ.*-isMagma ℚᵘ.*-comm

+

+
*-identityˡ : LeftIdentity 1ℚ _*_

+
*-identityˡ = +-*-Monomorphism.*-identityˡ ℚᵘ.*-isMagma ℚᵘ.*-identityˡ

+

+
*-identityʳ : RightIdentity 1ℚ _*_

+
*-identityʳ = +-*-Monomorphism.*-identityʳ ℚᵘ.*-isMagma ℚᵘ.*-identityʳ

+

+
*-identity : Identity 1ℚ _*_

+
*-identity = *-identityˡ , *-identityʳ

+

+
*-zeroˡ : LeftZero 0ℚ _*_

+
*-zeroˡ = +-*-Monomorphism.zeroˡ ℚᵘ.+-0-isGroup ℚᵘ.*-isMagma ℚᵘ.*-zeroˡ

+

+
*-zeroʳ : RightZero 0ℚ _*_

+
*-zeroʳ = +-*-Monomorphism.zeroʳ ℚᵘ.+-0-isGroup ℚᵘ.*-isMagma ℚᵘ.*-zeroʳ

+

+
*-zero : Zero 0ℚ _*_

+
*-zero = *-zeroˡ , *-zeroʳ

+

+
*-distribˡ-+ : _*_ DistributesOverˡ _+_

+
*-distribˡ-+ = +-*-Monomorphism.distribˡ ℚᵘ.+-0-isGroup ℚᵘ.*-isMagma ℚᵘ.*-distribˡ-+

+

+
*-distribʳ-+ : _*_ DistributesOverʳ _+_

+
*-distribʳ-+ = +-*-Monomorphism.distribʳ ℚᵘ.+-0-isGroup ℚᵘ.*-isMagma ℚᵘ.*-distribʳ-+

+

+
*-distrib-+ : _*_ DistributesOver _+_

+
*-distrib-+ = *-distribˡ-+ , *-distribʳ-+

+

+
*-inverseˡ : ∀ p .{{_ : NonZero p}} → (1/ p) * p ≡ 1ℚ

+
*-inverseˡ p = toℚᵘ-injective (begin-equality

+
  toℚᵘ (1/ p * p)             ≃⟨ toℚᵘ-homo-* (1/ p) p ⟩

+
  toℚᵘ (1/ p) ℚᵘ.* toℚᵘ p     ≃⟨ ℚᵘ.*-congʳ (toℚᵘ-homo-1/ p) ⟩

+
  ℚᵘ.1/ (toℚᵘ p) ℚᵘ.* toℚᵘ p  ≃⟨ ℚᵘ.*-inverseˡ (toℚᵘ p) ⟩

+
  ℚᵘ.1ℚᵘ                      ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
*-inverseʳ : ∀ p .{{_ : NonZero p}} → p * (1/ p) ≡ 1ℚ

+
*-inverseʳ p = trans (*-comm p (1/ p)) (*-inverseˡ p)

+

+
neg-distribˡ-* : ∀ p q → - (p * q) ≡ - p * q

+
neg-distribˡ-* = +-*-Monomorphism.neg-distribˡ-* ℚᵘ.+-0-isGroup ℚᵘ.*-isMagma ℚᵘ.neg-distribˡ-*

+

+
neg-distribʳ-* : ∀ p q → - (p * q) ≡ p * - q

+
neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-isMagma ℚᵘ.neg-distribʳ-*

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
*-isMagma : IsMagma _*_

+
*-isMagma = +-*-Monomorphism.*-isMagma ℚᵘ.*-isMagma

+

+
*-isSemigroup : IsSemigroup _*_

+
*-isSemigroup = +-*-Monomorphism.*-isSemigroup ℚᵘ.*-isSemigroup

+

+
*-1-isMonoid : IsMonoid _*_ 1ℚ

+
*-1-isMonoid = +-*-Monomorphism.*-isMonoid ℚᵘ.*-1-isMonoid

+

+
*-1-isCommutativeMonoid : IsCommutativeMonoid _*_ 1ℚ

+
*-1-isCommutativeMonoid = +-*-Monomorphism.*-isCommutativeMonoid ℚᵘ.*-1-isCommutativeMonoid

+

+
+-*-isRing : IsRing _+_ _*_ -_ 0ℚ 1ℚ

+
+-*-isRing = +-*-Monomorphism.isRing ℚᵘ.+-*-isRing

+

+
+-*-isCommutativeRing : IsCommutativeRing _+_ _*_ -_ 0ℚ 1ℚ

+
+-*-isCommutativeRing = +-*-Monomorphism.isCommutativeRing ℚᵘ.+-*-isCommutativeRing

+

+
------------------------------------------------------------------------

+
-- Packages

+

+
*-magma : Magma 0ℓ 0ℓ

+
*-magma = record

+
  { isMagma = *-isMagma

+
  }

+

+
*-semigroup : Semigroup 0ℓ 0ℓ

+
*-semigroup = record

+
  { isSemigroup = *-isSemigroup

+
  }

+

+
*-1-monoid : Monoid 0ℓ 0ℓ

+
*-1-monoid = record

+
  { isMonoid = *-1-isMonoid

+
  }

+

+
*-1-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
*-1-commutativeMonoid = record

+
  { isCommutativeMonoid = *-1-isCommutativeMonoid

+
  }

+

+
+-*-ring : Ring 0ℓ 0ℓ

+
+-*-ring = record

+
  { isRing = +-*-isRing

+
  }

+

+
+-*-commutativeRing : CommutativeRing 0ℓ 0ℓ

+
+-*-commutativeRing = record

+
  { isCommutativeRing = +-*-isCommutativeRing

+
  }

+

+

+
------------------------------------------------------------------------

+
-- HeytingField structures and bundles

+

+
module _ where

+
  open CommutativeRing +-*-commutativeRing

+
    using (+-group; zeroˡ; *-congʳ; isCommutativeRing)

+

+
  open GroupProperties +-group

+
  open DecSetoidProperties ≡-decSetoid

+

+
  #⇒invertible : p ≢ q → Invertible 1ℚ _*_ (p - q)

+
  #⇒invertible {p} {q} p≢q = let r = p - q in 1/ r , *-inverseˡ r , *-inverseʳ r

+
    where instance _ = ≢-nonZero (p≢q ∘ (x∙y⁻¹≈ε⇒x≈y p q))

+

+
  invertible⇒# : Invertible 1ℚ _*_ (p - q) → p ≢ q

+
  invertible⇒# {p} {q} (1/[p-q] , _ , [p-q]/[p-q]≡1) p≡q = contradiction 1≡0 1≢0

+
    where

+
    open ≈-Reasoning ≡-setoid

+
    1≡0 : 1ℚ ≡ 0ℚ

+
    1≡0 = begin

+
      1ℚ                 ≈⟨ [p-q]/[p-q]≡1 ⟨

+
      (p - q) * 1/[p-q]  ≈⟨ *-congʳ (x≈y⇒x∙y⁻¹≈ε p≡q) ⟩

+
      0ℚ * 1/[p-q]       ≈⟨ zeroˡ 1/[p-q] ⟩

+
      0ℚ                 ∎

+

+
  isHeytingCommutativeRing : IsHeytingCommutativeRing _≡_ _≢_ _+_ _*_ -_ 0ℚ 1ℚ

+
  isHeytingCommutativeRing = record

+
    { isCommutativeRing = isCommutativeRing

+
    ; isApartnessRelation = ≉-isApartnessRelation

+
    ; #⇒invertible = #⇒invertible

+
    ; invertible⇒# = invertible⇒#

+
    }

+

+
  isHeytingField : IsHeytingField _≡_ _≢_ _+_ _*_ -_ 0ℚ 1ℚ

+
  isHeytingField = record

+
    { isHeytingCommutativeRing = isHeytingCommutativeRing

+
    ; tight = ≉-tight

+
    }

+

+
  heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ

+
  heytingCommutativeRing = record { isHeytingCommutativeRing = isHeytingCommutativeRing }

+

+
  heytingField : HeytingField 0ℓ 0ℓ 0ℓ

+
  heytingField = record { isHeytingField = isHeytingField }

+

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _≤_

+

+
*-cancelʳ-≤-pos : ∀ r .{{_ : Positive r}} → p * r ≤ q * r → p ≤ q

+
*-cancelʳ-≤-pos {p} {q} r pr≤qr = toℚᵘ-cancel-≤ (ℚᵘ.*-cancelʳ-≤-pos (toℚᵘ r) (begin

+
  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨

+
  toℚᵘ (p * r)        ≤⟨  toℚᵘ-mono-≤ pr≤qr ⟩

+
  toℚᵘ (q * r)        ≃⟨  toℚᵘ-homo-* q r ⟩

+
  toℚᵘ q ℚᵘ.* toℚᵘ r  ∎))

+
  where open ℚᵘ.≤-Reasoning

+

+
*-cancelˡ-≤-pos : ∀ r .{{_ : Positive r}} → r * p ≤ r * q → p ≤ q

+
*-cancelˡ-≤-pos {p} {q} r rewrite *-comm r p | *-comm r q = *-cancelʳ-≤-pos r

+

+
*-monoʳ-≤-nonNeg : ∀ r .{{_ : NonNegative r}} → (_* r) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤-nonNeg r {p} {q} p≤q = toℚᵘ-cancel-≤ (begin

+
  toℚᵘ (p * r)        ≃⟨  toℚᵘ-homo-* p r ⟩

+
  toℚᵘ p ℚᵘ.* toℚᵘ r  ≤⟨  ℚᵘ.*-monoˡ-≤-nonNeg (toℚᵘ r) (toℚᵘ-mono-≤ p≤q) ⟩

+
  toℚᵘ q ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* q r ⟨

+
  toℚᵘ (q * r)        ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
*-monoˡ-≤-nonNeg : ∀ r .{{_ : NonNegative r}} → (r *_) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤-nonNeg r {p} {q} rewrite *-comm r p | *-comm r q = *-monoʳ-≤-nonNeg r

+

+
*-monoʳ-≤-nonPos : ∀ r .{{_ : NonPositive r}} → (_* r) Preserves _≤_ ⟶ _≥_

+
*-monoʳ-≤-nonPos r {p} {q} p≤q = toℚᵘ-cancel-≤ (begin

+
  toℚᵘ (q * r)        ≃⟨ toℚᵘ-homo-* q r ⟩

+
  toℚᵘ q ℚᵘ.* toℚᵘ r  ≤⟨ ℚᵘ.*-monoˡ-≤-nonPos (toℚᵘ r) (toℚᵘ-mono-≤ p≤q) ⟩

+
  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨

+
  toℚᵘ (p * r)        ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
*-monoˡ-≤-nonPos : ∀ r .{{_ : NonPositive r}} → (r *_) Preserves _≤_ ⟶ _≥_

+
*-monoˡ-≤-nonPos r {p} {q} rewrite *-comm r p | *-comm r q = *-monoʳ-≤-nonPos r

+

+
*-cancelʳ-≤-neg : ∀ r .{{_ : Negative r}} → p * r ≤ q * r → p ≥ q

+
*-cancelʳ-≤-neg {p} {q} r pr≤qr = toℚᵘ-cancel-≤ (ℚᵘ.*-cancelʳ-≤-neg _ (begin

+
  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨

+
  toℚᵘ (p * r)        ≤⟨ toℚᵘ-mono-≤ pr≤qr ⟩

+
  toℚᵘ (q * r)        ≃⟨ toℚᵘ-homo-* q r ⟩

+
  toℚᵘ q ℚᵘ.* toℚᵘ r  ∎))

+
  where open ℚᵘ.≤-Reasoning

+

+
*-cancelˡ-≤-neg : ∀ r .{{_ : Negative r}} → r * p ≤ r * q → p ≥ q

+
*-cancelˡ-≤-neg {p} {q} r rewrite *-comm r p | *-comm r q = *-cancelʳ-≤-neg r

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _<_

+

+
*-monoˡ-<-pos : ∀ r .{{_ : Positive r}} → (_* r) Preserves _<_ ⟶ _<_

+
*-monoˡ-<-pos r {p} {q} p<q = toℚᵘ-cancel-< (begin-strict

+
  toℚᵘ (p * r)        ≃⟨ toℚᵘ-homo-* p r ⟩

+
  toℚᵘ p ℚᵘ.* toℚᵘ r  <⟨ ℚᵘ.*-monoˡ-<-pos (toℚᵘ r) (toℚᵘ-mono-< p<q) ⟩

+
  toℚᵘ q ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* q r ⟨

+
  toℚᵘ (q * r)        ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
*-monoʳ-<-pos : ∀ r .{{_ : Positive r}} → (r *_) Preserves _<_ ⟶ _<_

+
*-monoʳ-<-pos r {p} {q} rewrite *-comm r p | *-comm r q = *-monoˡ-<-pos r

+

+
*-cancelˡ-<-nonNeg : ∀ r .{{_ : NonNegative r}} → ∀ {p q} → r * p < r * q → p < q

+
*-cancelˡ-<-nonNeg r {p} {q} rp<rq = toℚᵘ-cancel-< (ℚᵘ.*-cancelˡ-<-nonNeg (toℚᵘ r) (begin-strict

+
  toℚᵘ r ℚᵘ.* toℚᵘ p  ≃⟨ toℚᵘ-homo-* r p ⟨

+
  toℚᵘ (r * p)        <⟨ toℚᵘ-mono-< rp<rq ⟩

+
  toℚᵘ (r * q)        ≃⟨ toℚᵘ-homo-* r q ⟩

+
  toℚᵘ r ℚᵘ.* toℚᵘ q  ∎))

+
  where open ℚᵘ.≤-Reasoning

+

+
*-cancelʳ-<-nonNeg : ∀ r .{{_ : NonNegative r}} → ∀ {p q} → p * r < q * r → p < q

+
*-cancelʳ-<-nonNeg r {p} {q} rewrite *-comm p r | *-comm q r = *-cancelˡ-<-nonNeg r

+

+
*-monoˡ-<-neg : ∀ r .{{_ : Negative r}} → (_* r) Preserves _<_ ⟶ _>_

+
*-monoˡ-<-neg r {p} {q} p<q = toℚᵘ-cancel-< (begin-strict

+
  toℚᵘ (q * r)        ≃⟨ toℚᵘ-homo-* q r ⟩

+
  toℚᵘ q ℚᵘ.* toℚᵘ r  <⟨ ℚᵘ.*-monoˡ-<-neg (toℚᵘ r) (toℚᵘ-mono-< p<q) ⟩

+
  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨

+
  toℚᵘ (p * r)        ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
*-monoʳ-<-neg : ∀ r .{{_ : Negative r}} → (r *_) Preserves _<_ ⟶ _>_

+
*-monoʳ-<-neg r {p} {q} rewrite *-comm r p | *-comm r q = *-monoˡ-<-neg r

+

+
*-cancelˡ-<-nonPos : ∀ r .{{_ : NonPositive r}} → r * p < r * q → p > q

+
*-cancelˡ-<-nonPos {p} {q} r rp<rq = toℚᵘ-cancel-< (ℚᵘ.*-cancelˡ-<-nonPos (toℚᵘ r) (begin-strict

+
  toℚᵘ r ℚᵘ.* toℚᵘ p  ≃⟨ toℚᵘ-homo-* r p ⟨

+
  toℚᵘ (r * p)        <⟨  toℚᵘ-mono-< rp<rq ⟩

+
  toℚᵘ (r * q)        ≃⟨  toℚᵘ-homo-* r q ⟩

+
  toℚᵘ r ℚᵘ.* toℚᵘ q  ∎))

+
  where open ℚᵘ.≤-Reasoning

+

+
*-cancelʳ-<-nonPos : ∀ r .{{_ : NonPositive r}} → p * r < q * r → p > q

+
*-cancelʳ-<-nonPos {p} {q} r rewrite *-comm p r | *-comm q r = *-cancelˡ-<-nonPos r

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_

+
------------------------------------------------------------------------

+

+
p≤q⇒p⊔q≡q : p ≤ q → p ⊔ q ≡ q

+
p≤q⇒p⊔q≡q {p@record{}} {q@record{}} p≤q with p ≤ᵇ q in p≰q

+
... | true  = refl

+
... | false = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())

+

+
p≥q⇒p⊔q≡p : p ≥ q → p ⊔ q ≡ p

+
p≥q⇒p⊔q≡p {p@record{}} {q@record{}} p≥q with p ≤ᵇ q in p≤q

+
... | true  = ≤-antisym p≥q (≤ᵇ⇒≤ (subst T (sym p≤q) _))

+
... | false = refl

+

+
p≤q⇒p⊓q≡p : p ≤ q → p ⊓ q ≡ p

+
p≤q⇒p⊓q≡p {p@record{}} {q@record{}} p≤q with p ≤ᵇ q in p≰q

+
... | true  = refl

+
... | false = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())

+

+
p≥q⇒p⊓q≡q : p ≥ q → p ⊓ q ≡ q

+
p≥q⇒p⊓q≡q {p@record{}} {q@record{}} p≥q with p ≤ᵇ q in p≤q

+
... | true  = ≤-antisym (≤ᵇ⇒≤ (subst T (sym p≤q) _)) p≥q

+
... | false = refl

+

+
⊓-operator : MinOperator ≤-totalPreorder

+
⊓-operator = record

+
  { x≤y⇒x⊓y≈x = p≤q⇒p⊓q≡p

+
  ; x≥y⇒x⊓y≈y = p≥q⇒p⊓q≡q

+
  }

+

+
⊔-operator : MaxOperator ≤-totalPreorder

+
⊔-operator = record

+
  { x≤y⇒x⊔y≈y = p≤q⇒p⊔q≡q

+
  ; x≥y⇒x⊔y≈x = p≥q⇒p⊔q≡p

+
  }

+

+
------------------------------------------------------------------------

+
-- Automatically derived properties of _⊓_ and _⊔_

+

+
private

+
  module ⊓-⊔-properties        = MinMaxOp        ⊓-operator ⊔-operator

+
  module ⊓-⊔-latticeProperties = LatticeMinMaxOp ⊓-operator ⊔-operator

+

+
open ⊓-⊔-properties public

+
  using

+
  ( ⊓-idem                    -- : Idempotent _⊓_

+
  ; ⊓-sel                     -- : Selective _⊓_

+
  ; ⊓-assoc                   -- : Associative _⊓_

+
  ; ⊓-comm                    -- : Commutative _⊓_

+

+
  ; ⊔-idem                    -- : Idempotent _⊔_

+
  ; ⊔-sel                     -- : Selective _⊔_

+
  ; ⊔-assoc                   -- : Associative _⊔_

+
  ; ⊔-comm                    -- : Commutative _⊔_

+

+
  ; ⊓-distribˡ-⊔              -- : _⊓_ DistributesOverˡ _⊔_

+
  ; ⊓-distribʳ-⊔              -- : _⊓_ DistributesOverʳ _⊔_

+
  ; ⊓-distrib-⊔               -- : _⊓_ DistributesOver  _⊔_

+
  ; ⊔-distribˡ-⊓              -- : _⊔_ DistributesOverˡ _⊓_

+
  ; ⊔-distribʳ-⊓              -- : _⊔_ DistributesOverʳ _⊓_

+
  ; ⊔-distrib-⊓               -- : _⊔_ DistributesOver  _⊓_

+
  ; ⊓-absorbs-⊔               -- : _⊓_ Absorbs _⊔_

+
  ; ⊔-absorbs-⊓               -- : _⊔_ Absorbs _⊓_

+
  ; ⊔-⊓-absorptive            -- : Absorptive _⊔_ _⊓_

+
  ; ⊓-⊔-absorptive            -- : Absorptive _⊓_ _⊔_

+

+
  ; ⊓-isMagma                 -- : IsMagma _⊓_

+
  ; ⊓-isSemigroup             -- : IsSemigroup _⊓_

+
  ; ⊓-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊓_

+
  ; ⊓-isBand                  -- : IsBand _⊓_

+
  ; ⊓-isSelectiveMagma        -- : IsSelectiveMagma _⊓_

+

+
  ; ⊔-isMagma                 -- : IsMagma _⊔_

+
  ; ⊔-isSemigroup             -- : IsSemigroup _⊔_

+
  ; ⊔-isCommutativeSemigroup  -- : IsCommutativeSemigroup _⊔_

+
  ; ⊔-isBand                  -- : IsBand _⊔_

+
  ; ⊔-isSelectiveMagma        -- : IsSelectiveMagma _⊔_

+

+
  ; ⊓-magma                   -- : Magma _ _

+
  ; ⊓-semigroup               -- : Semigroup _ _

+
  ; ⊓-band                    -- : Band _ _

+
  ; ⊓-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊓-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊔-magma                   -- : Magma _ _

+
  ; ⊔-semigroup               -- : Semigroup _ _

+
  ; ⊔-band                    -- : Band _ _

+
  ; ⊔-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊔-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊓-glb                     -- : ∀ {p q r} → p ≥ r → q ≥ r → p ⊓ q ≥ r

+
  ; ⊓-triangulate             -- : ∀ p q r → p ⊓ q ⊓ r ≡ (p ⊓ q) ⊓ (q ⊓ r)

+
  ; ⊓-mono-≤                  -- : _⊓_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊓-monoˡ-≤                 -- : ∀ p → (_⊓ p) Preserves _≤_ ⟶ _≤_

+
  ; ⊓-monoʳ-≤                 -- : ∀ p → (p ⊓_) Preserves _≤_ ⟶ _≤_

+

+
  ; ⊔-lub                     -- : ∀ {p q r} → p ≤ r → q ≤ r → p ⊔ q ≤ r

+
  ; ⊔-triangulate             -- : ∀ p q r → p ⊔ q ⊔ r ≡ (p ⊔ q) ⊔ (q ⊔ r)

+
  ; ⊔-mono-≤                  -- : _⊔_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊔-monoˡ-≤                 -- : ∀ p → (_⊔ p) Preserves _≤_ ⟶ _≤_

+
  ; ⊔-monoʳ-≤                 -- : ∀ p → (p ⊔_) Preserves _≤_ ⟶ _≤_

+
  )

+
  renaming

+
  ( x⊓y≈y⇒y≤x to p⊓q≡q⇒q≤p    -- : ∀ {p q} → p ⊓ q ≡ q → q ≤ p

+
  ; x⊓y≈x⇒x≤y to p⊓q≡p⇒p≤q    -- : ∀ {p q} → p ⊓ q ≡ p → p ≤ q

+
  ; x⊓y≤x     to p⊓q≤p        -- : ∀ p q → p ⊓ q ≤ p

+
  ; x⊓y≤y     to p⊓q≤q        -- : ∀ p q → p ⊓ q ≤ q

+
  ; x≤y⇒x⊓z≤y to p≤q⇒p⊓r≤q    -- : ∀ {p q} r → p ≤ q → p ⊓ r ≤ q

+
  ; x≤y⇒z⊓x≤y to p≤q⇒r⊓p≤q    -- : ∀ {p q} r → p ≤ q → r ⊓ p ≤ q

+
  ; x≤y⊓z⇒x≤y to p≤q⊓r⇒p≤q    -- : ∀ {p} q r → p ≤ q ⊓ r → p ≤ q

+
  ; x≤y⊓z⇒x≤z to p≤q⊓r⇒p≤r    -- : ∀ {p} q r → p ≤ q ⊓ r → p ≤ r

+

+
  ; x⊔y≈y⇒x≤y to p⊔q≡q⇒p≤q    -- : ∀ {p q} → p ⊔ q ≡ q → p ≤ q

+
  ; x⊔y≈x⇒y≤x to p⊔q≡p⇒q≤p    -- : ∀ {p q} → p ⊔ q ≡ p → q ≤ p

+
  ; x≤x⊔y     to p≤p⊔q        -- : ∀ p q → p ≤ p ⊔ q

+
  ; x≤y⊔x     to p≤q⊔p        -- : ∀ p q → p ≤ q ⊔ p

+
  ; x≤y⇒x≤y⊔z to p≤q⇒p≤q⊔r    -- : ∀ {p q} r → p ≤ q → p ≤ q ⊔ r

+
  ; x≤y⇒x≤z⊔y to p≤q⇒p≤r⊔q    -- : ∀ {p q} r → p ≤ q → p ≤ r ⊔ q

+
  ; x⊔y≤z⇒x≤z to p⊔q≤r⇒p≤r    -- : ∀ p q {r} → p ⊔ q ≤ r → p ≤ r

+
  ; x⊔y≤z⇒y≤z to p⊔q≤r⇒q≤r    -- : ∀ p q {r} → p ⊔ q ≤ r → q ≤ r

+

+
  ; x⊓y≤x⊔y   to p⊓q≤p⊔q      -- : ∀ p q → p ⊓ q ≤ p ⊔ q

+
  )

+

+
open ⊓-⊔-latticeProperties public

+
  using

+
  ( ⊓-isSemilattice           -- : IsSemilattice _⊓_

+
  ; ⊔-isSemilattice           -- : IsSemilattice _⊔_

+
  ; ⊔-⊓-isLattice             -- : IsLattice _⊔_ _⊓_

+
  ; ⊓-⊔-isLattice             -- : IsLattice _⊓_ _⊔_

+
  ; ⊔-⊓-isDistributiveLattice -- : IsDistributiveLattice _⊔_ _⊓_

+
  ; ⊓-⊔-isDistributiveLattice -- : IsDistributiveLattice _⊓_ _⊔_

+

+
  ; ⊓-semilattice             -- : Semilattice _ _

+
  ; ⊔-semilattice             -- : Semilattice _ _

+
  ; ⊔-⊓-lattice               -- : Lattice _ _

+
  ; ⊓-⊔-lattice               -- : Lattice _ _

+
  ; ⊔-⊓-distributiveLattice   -- : DistributiveLattice _ _

+
  ; ⊓-⊔-distributiveLattice   -- : DistributiveLattice _ _

+
  )

+

+
------------------------------------------------------------------------

+
-- Other properties of _⊓_ and _⊔_

+

+
mono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ p q → f (p ⊔ q) ≡ f p ⊔ f q

+
mono-≤-distrib-⊔ {f} = ⊓-⊔-properties.mono-≤-distrib-⊔ (cong f)

+

+
mono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ p q → f (p ⊓ q) ≡ f p ⊓ f q

+
mono-≤-distrib-⊓ {f} = ⊓-⊔-properties.mono-≤-distrib-⊓ (cong f)

+

+
mono-<-distrib-⊓ : ∀ {f} → f Preserves _<_ ⟶ _<_ →

+
                   ∀ p q → f (p ⊓ q) ≡ f p ⊓ f q

+
mono-<-distrib-⊓ {f} f-mono-< p q with <-cmp p q

+
... | tri< p<q p≢r  p≯q = begin

+
  f (p ⊓ q)  ≡⟨ cong f (p≤q⇒p⊓q≡p (<⇒≤ p<q)) ⟩

+
  f p        ≡⟨ p≤q⇒p⊓q≡p (<⇒≤ (f-mono-< p<q)) ⟨

+
  f p ⊓ f q  ∎

+
  where open ≡-Reasoning

+
... | tri≈ p≮q refl p≯q = begin

+
  f (p ⊓ q)  ≡⟨ cong f (⊓-idem p) ⟩

+
  f p        ≡⟨ ⊓-idem (f p) ⟨

+
  f p ⊓ f q  ∎

+
  where open ≡-Reasoning

+
... | tri> p≮q p≡r  p>q = begin

+
  f (p ⊓ q)  ≡⟨ cong f (p≥q⇒p⊓q≡q (<⇒≤ p>q)) ⟩

+
  f q        ≡⟨ p≥q⇒p⊓q≡q (<⇒≤ (f-mono-< p>q)) ⟨

+
  f p ⊓ f q  ∎

+
  where open ≡-Reasoning

+

+
mono-<-distrib-⊔ : ∀ {f} → f Preserves _<_ ⟶ _<_ →

+
                   ∀ p q → f (p ⊔ q) ≡ f p ⊔ f q

+
mono-<-distrib-⊔ {f} f-mono-< p q with <-cmp p q

+
... | tri< p<q p≢r  p≯q = begin

+
  f (p ⊔ q)  ≡⟨ cong f (p≤q⇒p⊔q≡q (<⇒≤ p<q)) ⟩

+
  f q        ≡⟨ p≤q⇒p⊔q≡q (<⇒≤ (f-mono-< p<q)) ⟨

+
  f p ⊔ f q  ∎

+
  where open ≡-Reasoning

+
... | tri≈ p≮q refl p≯q = begin

+
  f (p ⊔ q)  ≡⟨ cong f (⊔-idem p) ⟩

+
  f q        ≡⟨ ⊔-idem (f p) ⟨

+
  f p ⊔ f q  ∎

+
  where open ≡-Reasoning

+
... | tri> p≮q p≡r  p>q = begin

+
  f (p ⊔ q)  ≡⟨ cong f (p≥q⇒p⊔q≡p (<⇒≤ p>q)) ⟩

+
  f p        ≡⟨ p≥q⇒p⊔q≡p (<⇒≤ (f-mono-< p>q)) ⟨

+
  f p ⊔ f q  ∎

+
  where open ≡-Reasoning

+

+
antimono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ p q → f (p ⊓ q) ≡ f p ⊔ f q

+
antimono-≤-distrib-⊓ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊓ (cong f)

+

+
antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ p q → f (p ⊔ q) ≡ f p ⊓ f q

+
antimono-≤-distrib-⊔ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊔ (cong f)

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_ and _*_

+

+
*-distribˡ-⊓-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊓ (p * r)

+
*-distribˡ-⊓-nonNeg p = mono-≤-distrib-⊓ (*-monoˡ-≤-nonNeg p)

+

+
*-distribʳ-⊓-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊓ (r * p)

+
*-distribʳ-⊓-nonNeg p = mono-≤-distrib-⊓ (*-monoʳ-≤-nonNeg p)

+

+
*-distribˡ-⊔-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊔ (p * r)

+
*-distribˡ-⊔-nonNeg p = mono-≤-distrib-⊔ (*-monoˡ-≤-nonNeg p)

+

+
*-distribʳ-⊔-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊔ (r * p)

+
*-distribʳ-⊔-nonNeg p = mono-≤-distrib-⊔ (*-monoʳ-≤-nonNeg p)

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_, _⊔_ and _*_

+

+
*-distribˡ-⊔-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊓ (p * r)

+
*-distribˡ-⊔-nonPos p = antimono-≤-distrib-⊔ (*-monoˡ-≤-nonPos p)

+

+
*-distribʳ-⊔-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊓ (r * p)

+
*-distribʳ-⊔-nonPos p = antimono-≤-distrib-⊔ (*-monoʳ-≤-nonPos p)

+

+
*-distribˡ-⊓-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊔ (p * r)

+
*-distribˡ-⊓-nonPos p = antimono-≤-distrib-⊓ (*-monoˡ-≤-nonPos p)

+

+
*-distribʳ-⊓-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊔ (r * p)

+
*-distribʳ-⊓-nonPos p = antimono-≤-distrib-⊓ (*-monoʳ-≤-nonPos p)

+

+
------------------------------------------------------------------------

+
-- Properties of 1/_

+
------------------------------------------------------------------------

+

+
nonZero⇒1/nonZero : ∀ p .{{_ : NonZero p}} → NonZero (1/ p)

+
nonZero⇒1/nonZero (mkℚ +[1+ _ ] _ _) = _

+
nonZero⇒1/nonZero (mkℚ -[1+ _ ] _ _) = _

+

+
1/-involutive : ∀ p .{{_ : NonZero p}} → (1/ (1/ p)) {{nonZero⇒1/nonZero p}} ≡ p

+
1/-involutive (mkℚ +[1+ n ] d-1 _) = refl

+
1/-involutive (mkℚ -[1+ n ] d-1 _) = refl

+

+
1/pos⇒pos : ∀ p .{{_ : Positive p}} → Positive ((1/ p) {{pos⇒nonZero p}})

+
1/pos⇒pos (mkℚ +[1+ _ ] _ _) = _

+

+
1/neg⇒neg : ∀ p .{{_ : Negative p}} → Negative ((1/ p) {{neg⇒nonZero p}})

+
1/neg⇒neg (mkℚ -[1+ _ ] _ _) = _

+

+
pos⇒1/pos : ∀ p .{{_ : NonZero p}} .{{_ : Positive (1/ p)}} → Positive p

+
pos⇒1/pos p = subst Positive (1/-involutive p) (1/pos⇒pos (1/ p))

+

+
neg⇒1/neg : ∀ p .{{_ : NonZero p}} .{{_ : Negative (1/ p)}} → Negative p

+
neg⇒1/neg p = subst Negative (1/-involutive p) (1/neg⇒neg (1/ p))

+

+
------------------------------------------------------------------------

+
-- Properties of ∣_∣

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Monomorphic to unnormalised -_

+

+
toℚᵘ-homo-∣-∣ : Homomorphic₁ toℚᵘ ∣_∣ ℚᵘ.∣_∣

+
toℚᵘ-homo-∣-∣ (mkℚ +[1+ _ ] _ _) = *≡* refl

+
toℚᵘ-homo-∣-∣ (mkℚ +0       _ _) = *≡* refl

+
toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
∣p∣≡0⇒p≡0 : ∀ p → ∣ p ∣ ≡ 0ℚ → p ≡ 0ℚ

+
∣p∣≡0⇒p≡0 (mkℚ +0 zero _) ∣p∣≡0 = refl

+

+
0≤∣p∣ : ∀ p → 0ℚ ≤ ∣ p ∣

+
0≤∣p∣ p@record{} = *≤* (begin

+
  (↥ 0ℚ) ℤ.* (↧ ∣ p ∣)  ≡⟨ ℤ.*-zeroˡ (↧ ∣ p ∣) ⟩

+
  0ℤ                    ≤⟨ ℤ.+≤+ ℕ.z≤n ⟩

+
  ↥ ∣ p ∣               ≡⟨ ℤ.*-identityʳ (↥ ∣ p ∣) ⟨

+
  ↥ ∣ p ∣ ℤ.* 1ℤ        ∎)

+
  where open ℤ.≤-Reasoning

+

+
0≤p⇒∣p∣≡p : 0ℚ ≤ p → ∣ p ∣ ≡ p

+
0≤p⇒∣p∣≡p {p@record{}} 0≤p = toℚᵘ-injective (ℚᵘ.0≤p⇒∣p∣≃p (toℚᵘ-mono-≤ 0≤p))

+

+
∣-p∣≡∣p∣ : ∀ p → ∣ - p ∣ ≡ ∣ p ∣

+
∣-p∣≡∣p∣ (mkℚ +[1+ n ] d-1 _) = refl

+
∣-p∣≡∣p∣ (mkℚ +0       d-1 _) = refl

+
∣-p∣≡∣p∣ (mkℚ -[1+ n ] d-1 _) = refl

+

+
∣p∣≡p⇒0≤p : ∀ {p} → ∣ p ∣ ≡ p → 0ℚ ≤ p

+
∣p∣≡p⇒0≤p {p} ∣p∣≡p = toℚᵘ-cancel-≤ (ℚᵘ.∣p∣≃p⇒0≤p (begin-equality

+
  ℚᵘ.∣ toℚᵘ p ∣  ≃⟨ ℚᵘ.≃-sym (toℚᵘ-homo-∣-∣ p) ⟩

+
  toℚᵘ ∣ p ∣     ≡⟨ cong toℚᵘ ∣p∣≡p ⟩

+
  toℚᵘ p         ∎))

+
  where open ℚᵘ.≤-Reasoning

+

+
∣p∣≡p∨∣p∣≡-p : ∀ p → ∣ p ∣ ≡ p ⊎ ∣ p ∣ ≡ - p

+
∣p∣≡p∨∣p∣≡-p (mkℚ (+ n) d-1 _) = inj₁ refl

+
∣p∣≡p∨∣p∣≡-p (mkℚ (-[1+ n ]) d-1 _) = inj₂ refl

+

+
∣p+q∣≤∣p∣+∣q∣ : ∀ p q → ∣ p + q ∣ ≤ ∣ p ∣ + ∣ q ∣

+
∣p+q∣≤∣p∣+∣q∣ p q = toℚᵘ-cancel-≤ (begin

+
  toℚᵘ ∣ p + q ∣                    ≃⟨ toℚᵘ-homo-∣-∣ (p + q) ⟩

+
  ℚᵘ.∣ toℚᵘ (p + q) ∣               ≃⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-+ p q) ⟩

+
  ℚᵘ.∣ toℚᵘ p ℚᵘ.+ toℚᵘ q ∣         ≤⟨ ℚᵘ.∣p+q∣≤∣p∣+∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩

+
  ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.+ ℚᵘ.∣ toℚᵘ q ∣  ≃⟨ ℚᵘ.+-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟨

+
  toℚᵘ ∣ p ∣ ℚᵘ.+ toℚᵘ ∣ q ∣        ≃⟨ toℚᵘ-homo-+ ∣ p ∣ ∣ q ∣ ⟨

+
  toℚᵘ (∣ p ∣ + ∣ q ∣)              ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
∣p-q∣≤∣p∣+∣q∣ : ∀ p q → ∣ p - q ∣ ≤ ∣ p ∣ + ∣ q ∣

+
∣p-q∣≤∣p∣+∣q∣ p@record{} q@record{} = begin

+
  ∣ p   -     q ∣  ≤⟨ ∣p+q∣≤∣p∣+∣q∣ p (- q) ⟩

+
  ∣ p ∣ + ∣ - q ∣  ≡⟨ cong (λ h → ∣ p ∣ + h) (∣-p∣≡∣p∣ q) ⟩

+
  ∣ p ∣ + ∣   q ∣  ∎

+
  where open ≤-Reasoning

+

+
∣p*q∣≡∣p∣*∣q∣ : ∀ p q → ∣ p * q ∣ ≡ ∣ p ∣ * ∣ q ∣

+
∣p*q∣≡∣p∣*∣q∣ p q = toℚᵘ-injective (begin-equality

+
  toℚᵘ ∣ p * q ∣                    ≃⟨ toℚᵘ-homo-∣-∣ (p * q) ⟩

+
  ℚᵘ.∣ toℚᵘ (p * q) ∣               ≃⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-* p q) ⟩

+
  ℚᵘ.∣ toℚᵘ p ℚᵘ.* toℚᵘ q ∣         ≃⟨ ℚᵘ.∣p*q∣≃∣p∣*∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩

+
  ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.* ℚᵘ.∣ toℚᵘ q ∣  ≃⟨ ℚᵘ.*-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟨

+
  toℚᵘ ∣ p ∣ ℚᵘ.* toℚᵘ ∣ q ∣        ≃⟨ toℚᵘ-homo-* ∣ p ∣ ∣ q ∣ ⟨

+
  toℚᵘ (∣ p ∣ * ∣ q ∣)              ∎)

+
  where open ℚᵘ.≤-Reasoning

+

+
∣-∣-nonNeg : ∀ p → NonNegative ∣ p ∣

+
∣-∣-nonNeg (mkℚ +[1+ _ ] _ _) = _

+
∣-∣-nonNeg (mkℚ +0       _ _) = _

+
∣-∣-nonNeg (mkℚ -[1+ _ ] _ _) = _

+

+
∣∣p∣∣≡∣p∣ : ∀ p → ∣ ∣ p ∣ ∣ ≡ ∣ p ∣

+
∣∣p∣∣≡∣p∣ p = 0≤p⇒∣p∣≡p (0≤∣p∣ p)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
*-monoʳ-≤-neg : ∀ r → Negative r → (_* r) Preserves _≤_ ⟶ _≥_

+
*-monoʳ-≤-neg r@(mkℚ -[1+ _ ] _ _) _ = *-monoʳ-≤-nonPos r

+
{-# WARNING_ON_USAGE *-monoʳ-≤-neg

+
"Warning: *-monoʳ-≤-neg was deprecated in v2.0.

+
Please use *-monoʳ-≤-nonPos instead."

+
#-}

+
*-monoˡ-≤-neg : ∀ r → Negative r → (r *_) Preserves _≤_ ⟶ _≥_

+
*-monoˡ-≤-neg r@(mkℚ -[1+ _ ] _ _) _ = *-monoˡ-≤-nonPos r

+
{-# WARNING_ON_USAGE *-monoˡ-≤-neg

+
"Warning: *-monoˡ-≤-neg was deprecated in v2.0.

+
Please use *-monoˡ-≤-nonPos instead."

+
#-}

+
*-monoʳ-≤-pos : ∀ r → Positive r → (_* r) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤-pos r@(mkℚ +[1+ _ ] _ _) _ = *-monoʳ-≤-nonNeg r

+
{-# WARNING_ON_USAGE *-monoʳ-≤-pos

+
"Warning: *-monoʳ-≤-pos was deprecated in v2.0.

+
Please use *-monoʳ-≤-nonNeg instead."

+
#-}

+
*-monoˡ-≤-pos : ∀ r → Positive r → (r *_) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤-pos r@(mkℚ +[1+ _ ] _ _) _ = *-monoˡ-≤-nonNeg r

+
{-# WARNING_ON_USAGE *-monoˡ-≤-pos

+
"Warning: *-monoˡ-≤-pos was deprecated in v2.0.

+
Please use *-monoˡ-≤-nonNeg instead."

+
#-}

+
*-cancelˡ-<-pos : ∀ r → Positive r → ∀ {p q} → r * p < r * q → p < q

+
*-cancelˡ-<-pos r@(mkℚ +[1+ _ ] _ _) _ = *-cancelˡ-<-nonNeg r

+
{-# WARNING_ON_USAGE *-cancelˡ-<-pos

+
"Warning: *-cancelˡ-<-pos was deprecated in v2.0.

+
Please use *-cancelˡ-<-nonNeg instead."

+
#-}

+
*-cancelʳ-<-pos : ∀ r → Positive r → ∀ {p q} → p * r < q * r → p < q

+
*-cancelʳ-<-pos r@(mkℚ +[1+ _ ] _ _) _ = *-cancelʳ-<-nonNeg r

+
{-# WARNING_ON_USAGE *-cancelʳ-<-pos

+
"Warning: *-cancelʳ-<-pos was deprecated in v2.0.

+
Please use *-cancelʳ-<-nonNeg instead."

+
#-}

+
*-cancelˡ-<-neg : ∀ r → Negative r → ∀ {p q} → r * p < r * q → p > q

+
*-cancelˡ-<-neg r@(mkℚ -[1+ _ ] _ _) _ = *-cancelˡ-<-nonPos r

+
{-# WARNING_ON_USAGE *-cancelˡ-<-neg

+
"Warning: *-cancelˡ-<-neg was deprecated in v2.0.

+
Please use *-cancelˡ-<-nonPos instead."

+
#-}

+
*-cancelʳ-<-neg : ∀ r → Negative r → ∀ {p q} → p * r < q * r → p > q

+
*-cancelʳ-<-neg r@(mkℚ -[1+ _ ] _ _) _ = *-cancelʳ-<-nonPos r

+
{-# WARNING_ON_USAGE *-cancelʳ-<-neg

+
"Warning: *-cancelʳ-<-neg was deprecated in v2.0.

+
Please use *-cancelʳ-<-nonPos instead."

+
#-}

+
negative<positive : Negative p → Positive q → p < q

+
negative<positive {p} {q} p<0 q>0 = neg<pos p q {{p<0}} {{q>0}}

+
{-# WARNING_ON_USAGE negative<positive

+
"Warning: negative<positive was deprecated in v2.0.

+
Please use neg<pos instead."

+
#-}

+
{- issue1865/issue1755: raw bundles have moved to `Data.X.Base` -}

+
open Data.Rational.Base public

+
  using (+-rawMagma; +-0-rawGroup; *-rawMagma; +-*-rawNearSemiring; +-*-rawSemiring; +-*-rawRing)

+
  renaming (+-0-rawMonoid to +-rawMonoid; *-1-rawMonoid to *-rawMonoid)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Rational.Unnormalised.Base.html b/site/static/agda_html/Data.Rational.Unnormalised.Base.html
new file mode 100644
index 000000000..c4f1b6d01
--- /dev/null
+++ b/site/static/agda_html/Data.Rational.Unnormalised.Base.html
@@ -0,0 +1,464 @@
+
+Data.Rational.Unnormalised.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Rational numbers in non-reduced form.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Rational.Unnormalised.Base where

+

+
open import Algebra.Bundles.Raw

+
  using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)

+
open import Data.Bool.Base using (Bool; true; false; if_then_else_)

+
open import Data.Integer.Base as ℤ

+
  using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)

+
  hiding (module ℤ)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)

+
open import Level using (0ℓ)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Unary using (Pred)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; refl)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Here we define rationals that are not necessarily in reduced form.

+
-- Consequently there are multiple ways of representing a given rational

+
-- number, and the performance of the arithmetic operations may suffer

+
-- due to blowup of the numerator and denominator.

+

+
-- Nonetheless they are much easier to reason about. In general proofs

+
-- are first proved for these unnormalised rationals and then translated

+
-- into the normalised rationals.

+

+
record ℚᵘ : Set where

+
  -- We add "no-eta-equality; pattern" to the record to stop Agda

+
  -- automatically unfolding rationals when arithmetic operations are

+
  -- applied to them (see definition of operators below and Issue #1753

+
  -- for details).

+
  no-eta-equality; pattern

+

+
  constructor mkℚᵘ

+
  field

+
    numerator     : ℤ

+
    denominator-1 : ℕ

+

+
  denominatorℕ : ℕ

+
  denominatorℕ = suc denominator-1

+

+
  denominator : ℤ

+
  denominator = + denominatorℕ

+

+
open ℚᵘ public using ()

+
  renaming

+
  ( numerator    to ↥_

+
  ; denominator  to ↧_

+
  ; denominatorℕ to ↧ₙ_

+
  )

+

+
------------------------------------------------------------------------

+
-- Equality of rational numbers (does not coincide with _≡_)

+

+
infix 4 _≃_ _≠_

+

+
data _≃_ : Rel ℚᵘ 0ℓ where

+
  *≡* : ∀ {p q} → (↥ p ℤ.* ↧ q) ≡ (↥ q ℤ.* ↧ p) → p ≃ q

+

+
_≄_ : Rel ℚᵘ 0ℓ

+
p ≄ q = ¬ (p ≃ q)

+

+
------------------------------------------------------------------------

+
-- Ordering of rationals

+

+
infix 4 _≤_ _<_ _≥_ _>_ _≰_ _≱_ _≮_ _≯_

+

+
data _≤_ : Rel ℚᵘ 0ℓ where

+
  *≤* : ∀ {p q} → (↥ p ℤ.* ↧ q) ℤ.≤ (↥ q ℤ.* ↧ p) → p ≤ q

+

+
data _<_ : Rel ℚᵘ 0ℓ where

+
  *<* : ∀ {p q} → (↥ p ℤ.* ↧ q) ℤ.< (↥ q ℤ.* ↧ p) → p < q

+

+
_≥_ : Rel ℚᵘ 0ℓ

+
x ≥ y = y ≤ x

+

+
_>_ : Rel ℚᵘ 0ℓ

+
x > y = y < x

+

+
_≰_ : Rel ℚᵘ 0ℓ

+
x ≰ y = ¬ (x ≤ y)

+

+
_≱_ : Rel ℚᵘ 0ℓ

+
x ≱ y = ¬ (x ≥ y)

+

+
_≮_ : Rel ℚᵘ 0ℓ

+
x ≮ y = ¬ (x < y)

+

+
_≯_ : Rel ℚᵘ 0ℓ

+
x ≯ y = ¬ (x > y)

+

+
------------------------------------------------------------------------

+
-- Boolean ordering

+

+
infix 4 _≤ᵇ_

+

+
_≤ᵇ_ : ℚᵘ → ℚᵘ → Bool

+
p ≤ᵇ q = (↥ p ℤ.* ↧ q) ℤ.≤ᵇ (↥ q ℤ.* ↧ p)

+

+
------------------------------------------------------------------------

+
-- Constructing rationals

+

+
-- An alternative constructor for ℚᵘ. See the constants section below

+
-- for examples of how to use this operator.

+

+
infixl 7 _/_

+

+
_/_ : (n : ℤ) (d : ℕ) .{{_ : ℕ.NonZero d}} → ℚᵘ

+
n / suc d = mkℚᵘ n d

+

+
------------------------------------------------------------------------

+
-- Some constants

+

+
0ℚᵘ : ℚᵘ

+
0ℚᵘ = + 0 / 1

+

+
1ℚᵘ : ℚᵘ

+
1ℚᵘ = + 1 / 1

+

+
½ : ℚᵘ

+
½ = + 1 / 2

+

+
-½ : ℚᵘ

+
-½ = ℤ.- (+ 1) / 2

+

+
------------------------------------------------------------------------

+
-- Simple predicates

+

+
NonZero : Pred ℚᵘ 0ℓ

+
NonZero p = ℤ.NonZero (↥ p)

+

+
Positive : Pred ℚᵘ 0ℓ

+
Positive p = ℤ.Positive (↥ p)

+

+
Negative : Pred ℚᵘ 0ℓ

+
Negative p = ℤ.Negative (↥ p)

+

+
NonPositive : Pred ℚᵘ 0ℓ

+
NonPositive p = ℤ.NonPositive (↥ p)

+

+
NonNegative : Pred ℚᵘ 0ℓ

+
NonNegative p = ℤ.NonNegative (↥ p)

+

+
-- Instances

+

+
open ℤ public

+
  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)

+

+
-- Constructors and destructors

+

+
-- Note: these could be proved more elegantly using the constructors

+
-- from ℤ but it requires importing `Data.Integer.Properties` which

+
-- we would like to avoid doing.

+

+
≢-nonZero : ∀ {p} → p ≄ 0ℚᵘ → NonZero p

+
≢-nonZero {mkℚᵘ -[1+ _ ] _      } _   = _

+
≢-nonZero {mkℚᵘ +[1+ _ ] _      } _   = _

+
≢-nonZero {mkℚᵘ +0       zero   } p≢0 = contradiction (*≡* refl) p≢0

+
≢-nonZero {mkℚᵘ +0       (suc d)} p≢0 = contradiction (*≡* refl) p≢0

+

+
>-nonZero : ∀ {p} → p > 0ℚᵘ → NonZero p

+
>-nonZero {mkℚᵘ +0       _} (*<* (+<+ ()))

+
>-nonZero {mkℚᵘ +[1+ n ] _} (*<* _) = _

+

+
<-nonZero : ∀ {p} → p < 0ℚᵘ → NonZero p

+
<-nonZero {mkℚᵘ +[1+ n ] _} (*<* _) = _

+
<-nonZero {mkℚᵘ +0 _}       (*<* (+<+ ()))

+
<-nonZero {mkℚᵘ -[1+ n ] _} (*<* _) = _

+

+
positive : ∀ {p} → p > 0ℚᵘ → Positive p

+
positive {mkℚᵘ +[1+ n ]   _} (*<* _) = _

+
positive {mkℚᵘ +0         _} (*<* (+<+ ()))

+
positive {mkℚᵘ (-[1+_] n) _} (*<* ())

+

+
negative : ∀ {p} → p < 0ℚᵘ → Negative p

+
negative {mkℚᵘ +[1+ n ]   _} (*<* (+<+ ()))

+
negative {mkℚᵘ +0         _} (*<* (+<+ ()))

+
negative {mkℚᵘ (-[1+_] n) _} (*<* _       ) = _

+

+
nonPositive : ∀ {p} → p ≤ 0ℚᵘ → NonPositive p

+
nonPositive {mkℚᵘ +[1+ n ] _} (*≤* (+≤+ ()))

+
nonPositive {mkℚᵘ +0       _} (*≤* _) = _

+
nonPositive {mkℚᵘ -[1+ n ] _} (*≤* _) = _

+

+
nonNegative : ∀ {p} → p ≥ 0ℚᵘ → NonNegative p

+
nonNegative {mkℚᵘ +0       _} (*≤* _) = _

+
nonNegative {mkℚᵘ +[1+ n ] _} (*≤* _) = _

+

+
------------------------------------------------------------------------

+
-- Operations on rationals

+

+
-- Explanation for `@record{}` everywhere: combined with no-eta-equality

+
-- on the record definition of ℚᵘ above, these annotations prevent the

+
-- operations from automatically expanding unless their arguments are

+
-- explicitly pattern matched on.

+
--

+
-- For example prior to their addition, `p + q` would often be

+
-- normalised by Agda to `(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`.

+
-- While in this small example this isn't a big problem, it leads to an

+
-- exponential blowup when you have large arithmetic expressions which

+
-- would often choke both type-checking and the display code. For

+
-- example, the normalised form of `p + q + r + s + t + u` would be

+
-- ~300 lines long.

+
--

+
-- This is fundementally a problem with Agda, so if over-eager

+
-- normalisation is ever fixed in Agda (e.g. with glued representation

+
-- of terms) these annotations can be removed.

+

+
infix  8 -_ 1/_

+
infixl 7 _*_ _÷_ _⊓_

+
infixl 6 _-_ _+_ _⊔_

+

+
-- negation

+

+
-_ : ℚᵘ → ℚᵘ

+
- mkℚᵘ n d = mkℚᵘ (ℤ.- n) d

+

+
-- addition

+

+
_+_ : ℚᵘ → ℚᵘ → ℚᵘ

+
p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)

+

+
-- multiplication

+

+
_*_ : ℚᵘ → ℚᵘ → ℚᵘ

+
p@record{} * q@record{} = (↥ p ℤ.* ↥ q) / (↧ₙ p ℕ.* ↧ₙ q)

+

+
-- subtraction

+

+
_-_ : ℚᵘ → ℚᵘ → ℚᵘ

+
p - q = p + (- q)

+

+
-- reciprocal: requires a proof that the numerator is not zero

+

+
1/_ : (p : ℚᵘ) → .{{_ : NonZero p}} → ℚᵘ

+
1/ mkℚᵘ +[1+ n ] d = mkℚᵘ +[1+ d ] n

+
1/ mkℚᵘ -[1+ n ] d = mkℚᵘ -[1+ d ] n

+

+
-- division: requires a proof that the denominator is not zero

+

+
_÷_ : (p q : ℚᵘ) → .{{_ : NonZero q}} → ℚᵘ

+
p@record{} ÷ q@record{} = p * (1/ q)

+

+
-- max

+
_⊔_ : (p q : ℚᵘ) → ℚᵘ

+
p@record{} ⊔ q@record{} = if p ≤ᵇ q then q else p

+

+
-- min

+
_⊓_ : (p q : ℚᵘ) → ℚᵘ

+
p@record{} ⊓ q@record{} = if p ≤ᵇ q then p else q

+

+
-- absolute value

+
∣_∣ : ℚᵘ → ℚᵘ

+
∣ mkℚᵘ p q ∣ = mkℚᵘ (+ ℤ.∣ p ∣) q

+

+
------------------------------------------------------------------------

+
-- Rounding functions

+

+
-- Floor (round towards -∞)

+
floor : ℚᵘ → ℤ

+
floor p@record{} = ↥ p ℤ./ ↧ p

+

+
-- Ceiling (round towards +∞)

+
ceiling : ℚᵘ → ℤ

+
ceiling p@record{} = ℤ.- floor (- p)

+

+
-- Truncate  (round towards 0)

+
truncate : ℚᵘ → ℤ

+
truncate p = if p ≤ᵇ 0ℚᵘ then ceiling p else floor p

+

+
-- Round (to nearest integer)

+
round : ℚᵘ → ℤ

+
round p = if p ≤ᵇ 0ℚᵘ then ceiling (p - ½) else floor (p + ½)

+

+
-- Fractional part (remainder after floor)

+
fracPart : ℚᵘ → ℚᵘ

+
fracPart p@record{} = ∣ p - truncate p / 1 ∣

+

+
-- Extra notations  ⌊ ⌋ floor,  ⌈ ⌉ ceiling,  [ ] truncate

+
syntax floor    p = ⌊ p ⌋

+
syntax ceiling  p = ⌈ p ⌉

+
syntax truncate p = [ p ]

+

+
------------------------------------------------------------------------

+
-- Raw bundles for _+_

+

+
+-rawMagma : RawMagma 0ℓ 0ℓ

+
+-rawMagma = record

+
  { _≈_ = _≃_

+
  ; _∙_ = _+_

+
  }

+

+
+-0-rawMonoid : RawMonoid 0ℓ 0ℓ

+
+-0-rawMonoid = record

+
  { _≈_ = _≃_

+
  ; _∙_ = _+_

+
  ; ε   = 0ℚᵘ

+
  }

+

+
+-0-rawGroup : RawGroup 0ℓ 0ℓ

+
+-0-rawGroup = record

+
  { Carrier = ℚᵘ

+
  ; _≈_ = _≃_

+
  ; _∙_ = _+_

+
  ; ε = 0ℚᵘ

+
  ; _⁻¹ = -_

+
  }

+

+
+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ

+
+-*-rawNearSemiring = record

+
  { Carrier = ℚᵘ

+
  ; _≈_ = _≃_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0ℚᵘ

+
  }

+

+
+-*-rawSemiring : RawSemiring 0ℓ 0ℓ

+
+-*-rawSemiring = record

+
  { Carrier = ℚᵘ

+
  ; _≈_ = _≃_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; 0# = 0ℚᵘ

+
  ; 1# = 1ℚᵘ

+
  }

+

+
+-*-rawRing : RawRing 0ℓ 0ℓ

+
+-*-rawRing = record

+
  { Carrier = ℚᵘ

+
  ; _≈_ = _≃_

+
  ; _+_ = _+_

+
  ; _*_ = _*_

+
  ; -_ = -_

+
  ; 0# = 0ℚᵘ

+
  ; 1# = 1ℚᵘ

+
  }

+

+
------------------------------------------------------------------------

+
-- Raw bundles for _*_

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record

+
  { _≈_ = _≃_

+
  ; _∙_ = _*_

+
  }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record

+
  { _≈_ = _≃_

+
  ; _∙_ = _*_

+
  ; ε   = 1ℚᵘ

+
  }

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
+-rawMonoid = +-0-rawMonoid

+
{-# WARNING_ON_USAGE +-rawMonoid

+
"Warning: +-rawMonoid was deprecated in v2.0

+
Please use +-0-rawMonoid instead."

+
#-}

+
*-rawMonoid = *-1-rawMonoid

+
{-# WARNING_ON_USAGE *-rawMonoid

+
"Warning: *-rawMonoid was deprecated in v2.0

+
Please use *-1-rawMonoid instead."

+
#-}

+
_≠_ = _≄_

+
{-# WARNING_ON_USAGE _≠_

+
"Warning: _≠_ was deprecated in v2.0

+
Please use _≄_ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Rational.Unnormalised.Properties.html b/site/static/agda_html/Data.Rational.Unnormalised.Properties.html
new file mode 100644
index 000000000..62b05252a
--- /dev/null
+++ b/site/static/agda_html/Data.Rational.Unnormalised.Properties.html
@@ -0,0 +1,2067 @@
+
+Data.Rational.Unnormalised.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
-----------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of unnormalized Rational numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+
{-# OPTIONS --warn=noUserWarning #-} -- for +-rawMonoid, *-rawMonoid (issue #1865, #1844, #1755)

+

+
module Data.Rational.Unnormalised.Properties where

+

+
open import Algebra

+
open import Algebra.Apartness

+
open import Algebra.Lattice

+
import Algebra.Consequences.Setoid as Consequences

+
open import Algebra.Consequences.Propositional

+
open import Algebra.Construct.NaturalChoice.Base

+
import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp

+
import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp

+
open import Data.Bool.Base using (T; true; false)

+
open import Data.Nat.Base as ℕ using (suc; pred)

+
import Data.Nat.Properties as ℕ

+
open import Data.Integer.Base as ℤ using (ℤ; +0; +[1+_]; -[1+_]; 0ℤ; 1ℤ; -1ℤ)

+
open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)

+
import Data.Integer.Properties as ℤ

+
open import Data.Rational.Unnormalised.Base

+
open import Data.Product.Base using (_,_; proj₁; proj₂)

+
open import Data.Sum.Base using (_⊎_; [_,_]′; inj₁; inj₂)

+
import Data.Sign as Sign

+
open import Function.Base using (_on_; _$_; _∘_; flip)

+
open import Level using (0ℓ)

+
open import Relation.Nullary.Decidable.Core as Dec using (yes; no)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; DenseLinearOrder)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence; IsApartnessRelation; IsTotalPreorder; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsDenseLinearOrder)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive; Cotransitive; Tight; Decidable; Antisymmetric; Asymmetric; Dense; Total; Trans; Trichotomous; Irreflexive; Irrelevant; _Respectsˡ_; _Respectsʳ_; _Respects₂_; tri≈; tri<; tri>)

+
import Relation.Binary.Consequences as BC

+
open import Relation.Binary.PropositionalEquality

+
import Relation.Binary.Properties.Poset as PosetProperties

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
open import Relation.Binary.Reasoning.Syntax

+

+
open import Algebra.Properties.CommutativeSemigroup ℤ.*-commutativeSemigroup

+

+
private

+
  variable

+
    p q r : ℚᵘ

+

+
------------------------------------------------------------------------

+
-- Properties of ↥_ and ↧_

+
------------------------------------------------------------------------

+

+
↥↧≡⇒≡ : ∀ {p q} → ↥ p ≡ ↥ q → ↧ₙ p ≡ ↧ₙ q → p ≡ q

+
↥↧≡⇒≡ {mkℚᵘ _ _} {mkℚᵘ _ _} refl refl = refl

+

+
------------------------------------------------------------------------

+
-- Properties of _/_

+
------------------------------------------------------------------------

+

+
/-cong : ∀ {n₁ d₁ n₂ d₂} .{{_ : ℕ.NonZero d₁}} .{{_ : ℕ.NonZero d₂}} →

+
         n₁ ≡ n₂ → d₁ ≡ d₂ → n₁ / d₁ ≡ n₂ / d₂

+
/-cong refl refl = refl

+

+
↥[n/d]≡n : ∀ n d .{{_ : ℕ.NonZero d}} → ↥ (n / d) ≡ n

+
↥[n/d]≡n n (suc d) = refl

+

+
↧[n/d]≡d : ∀ n d .{{_ : ℕ.NonZero d}} → ↧ (n / d) ≡ ℤ.+ d

+
↧[n/d]≡d n (suc d) = refl

+

+
------------------------------------------------------------------------

+
-- Properties of _≃_

+
------------------------------------------------------------------------

+

+
drop-*≡* : ∀ {p q} → p ≃ q → ↥ p ℤ.* ↧ q ≡ ↥ q ℤ.* ↧ p

+
drop-*≡* (*≡* eq) = eq

+

+
≃-refl : Reflexive _≃_

+
≃-refl = *≡* refl

+

+
≃-reflexive : _≡_ ⇒ _≃_

+
≃-reflexive refl = *≡* refl

+

+
≃-sym : Symmetric _≃_

+
≃-sym (*≡* eq) = *≡* (sym eq)

+

+
≃-trans : Transitive _≃_

+
≃-trans {x} {y} {z} (*≡* ad≡cb) (*≡* cf≡ed) =

+
  *≡* (ℤ.*-cancelʳ-≡ (↥ x ℤ.* ↧ z) (↥ z ℤ.* ↧ x) (↧ y) (begin

+
     ↥ x ℤ.* ↧ z ℤ.* ↧ y ≡⟨ xy∙z≈xz∙y (↥ x) _ _ ⟩

+
     ↥ x ℤ.* ↧ y ℤ.* ↧ z ≡⟨ cong (ℤ._* ↧ z) ad≡cb ⟩

+
     ↥ y ℤ.* ↧ x ℤ.* ↧ z ≡⟨ xy∙z≈xz∙y (↥ y) _ _ ⟩

+
     ↥ y ℤ.* ↧ z ℤ.* ↧ x ≡⟨ cong (ℤ._* ↧ x) cf≡ed ⟩

+
     ↥ z ℤ.* ↧ y ℤ.* ↧ x ≡⟨ xy∙z≈xz∙y (↥ z) _ _ ⟩

+
     ↥ z ℤ.* ↧ x ℤ.* ↧ y ∎))

+
  where open ≡-Reasoning

+

+
infix 4 _≃?_

+

+
_≃?_ : Decidable _≃_

+
p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* ↧ p)

+

+
0≄1 : 0ℚᵘ ≄ 1ℚᵘ

+
0≄1 = Dec.from-no (0ℚᵘ ≃? 1ℚᵘ)

+

+
≃-≄-irreflexive : Irreflexive _≃_ _≄_

+
≃-≄-irreflexive x≃y x≄y = x≄y x≃y

+

+
≄-symmetric : Symmetric _≄_

+
≄-symmetric x≄y y≃x = x≄y (≃-sym y≃x)

+

+
≄-cotransitive : Cotransitive _≄_

+
≄-cotransitive {x} {y} x≄y z with x ≃? z | z ≃? y

+
... | no  x≄z | _       = inj₁ x≄z

+
... | yes _   | no z≄y  = inj₂ z≄y

+
... | yes x≃z | yes z≃y = contradiction (≃-trans x≃z z≃y) x≄y

+

+
≃-isEquivalence : IsEquivalence _≃_

+
≃-isEquivalence = record

+
  { refl  = ≃-refl

+
  ; sym   = ≃-sym

+
  ; trans = ≃-trans

+
  }

+

+
≃-isDecEquivalence : IsDecEquivalence _≃_

+
≃-isDecEquivalence = record

+
  { isEquivalence = ≃-isEquivalence

+
  ; _≟_           = _≃?_

+
  }

+

+
≄-isApartnessRelation : IsApartnessRelation _≃_ _≄_

+
≄-isApartnessRelation = record

+
  { irrefl  = ≃-≄-irreflexive

+
  ; sym     = ≄-symmetric

+
  ; cotrans = ≄-cotransitive

+
  }

+

+
≄-tight : Tight _≃_ _≄_

+
proj₁ (≄-tight p q) ¬p≄q = Dec.decidable-stable (p ≃? q) ¬p≄q

+
proj₂ (≄-tight p q) p≃q p≄q = p≄q p≃q

+

+
≃-setoid : Setoid 0ℓ 0ℓ

+
≃-setoid = record

+
  { isEquivalence = ≃-isEquivalence

+
  }

+

+
≃-decSetoid : DecSetoid 0ℓ 0ℓ

+
≃-decSetoid = record

+
  { isDecEquivalence = ≃-isDecEquivalence

+
  }

+

+
module ≃-Reasoning = ≈-Reasoning ≃-setoid

+

+
↥p≡0⇒p≃0 : ∀ p → ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ

+
↥p≡0⇒p≃0 p ↥p≡0 = *≡* (cong (ℤ._* (↧ 0ℚᵘ)) ↥p≡0)

+

+
p≃0⇒↥p≡0 : ∀ p → p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ

+
p≃0⇒↥p≡0 p (*≡* eq) = begin

+
  ↥ p          ≡⟨ ℤ.*-identityʳ (↥ p) ⟨

+
  ↥ p ℤ.* 1ℤ  ≡⟨ eq ⟩

+
  0ℤ           ∎

+
  where open ≡-Reasoning

+

+
↥p≡↥q≡0⇒p≃q : ∀ p q → ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q

+
↥p≡↥q≡0⇒p≃q p q ↥p≡0 ↥q≡0 = ≃-trans (↥p≡0⇒p≃0 p ↥p≡0) (≃-sym (↥p≡0⇒p≃0 _ ↥q≡0))

+

+
------------------------------------------------------------------------

+
-- Properties of -_

+
------------------------------------------------------------------------

+

+
neg-involutive-≡ : Involutive _≡_ (-_)

+
neg-involutive-≡ (mkℚᵘ n d) = cong (λ n → mkℚᵘ n d) (ℤ.neg-involutive n)

+

+
neg-involutive : Involutive _≃_ (-_)

+
neg-involutive p rewrite neg-involutive-≡ p = ≃-refl

+

+
-‿cong : Congruent₁ _≃_ (-_)

+
-‿cong {p@record{}} {q@record{}} (*≡* p≡q) = *≡* (begin

+
  ↥(- p) ℤ.* ↧ q            ≡⟨ ℤ.*-identityˡ (ℤ.- (↥ p) ℤ.* ↧ q) ⟨

+
  1ℤ ℤ.* (↥(- p) ℤ.* ↧ q)   ≡⟨ ℤ.*-assoc 1ℤ (↥ (- p)) (↧ q) ⟨

+
  (1ℤ ℤ.* ℤ.-(↥ p)) ℤ.* ↧ q ≡⟨ cong (ℤ._* ↧ q) (ℤ.neg-distribʳ-* 1ℤ (↥ p)) ⟨

+
  ℤ.-(1ℤ ℤ.* ↥ p) ℤ.* ↧ q   ≡⟨  cong (ℤ._* ↧ q) (ℤ.neg-distribˡ-* 1ℤ (↥ p)) ⟩

+
  (-1ℤ ℤ.* ↥ p) ℤ.* ↧ q     ≡⟨  ℤ.*-assoc ℤ.-1ℤ (↥ p) (↧ q) ⟩

+
  -1ℤ ℤ.* (↥ p ℤ.* ↧ q)     ≡⟨  cong (ℤ.-1ℤ ℤ.*_) p≡q ⟩

+
  -1ℤ ℤ.* (↥ q ℤ.* ↧ p)     ≡⟨ ℤ.*-assoc ℤ.-1ℤ (↥ q) (↧ p) ⟨

+
  (-1ℤ ℤ.* ↥ q) ℤ.* ↧ p     ≡⟨ cong (ℤ._* ↧ p) (ℤ.neg-distribˡ-* 1ℤ (↥ q)) ⟨

+
  ℤ.-(1ℤ ℤ.* ↥ q) ℤ.* ↧ p   ≡⟨  cong (ℤ._* ↧ p) (ℤ.neg-distribʳ-* 1ℤ (↥ q)) ⟩

+
  (1ℤ ℤ.* ↥(- q)) ℤ.* ↧ p   ≡⟨  ℤ.*-assoc 1ℤ (ℤ.- (↥ q)) (↧ p) ⟩

+
  1ℤ ℤ.* (↥(- q) ℤ.* ↧ p)   ≡⟨  ℤ.*-identityˡ (↥ (- q) ℤ.* ↧ p) ⟩

+
  ↥(- q) ℤ.* ↧ p            ∎)

+
  where open ≡-Reasoning

+

+
neg-mono-< : -_ Preserves  _<_ ⟶ _>_

+
neg-mono-< {p@record{}} {q@record{}} (*<* p<q) = *<* $ begin-strict

+
  ℤ.-  ↥ q ℤ.* ↧ p     ≡⟨ ℤ.neg-distribˡ-* (↥ q) (↧ p) ⟨

+
  ℤ.- (↥ q ℤ.* ↧ p)    <⟨ ℤ.neg-mono-< p<q ⟩

+
  ℤ.- (↥ p ℤ.* ↧ q)    ≡⟨ ℤ.neg-distribˡ-* (↥ p) (↧ q) ⟩

+
  ↥ (- p) ℤ.* ↧ (- q)  ∎

+
  where open ℤ.≤-Reasoning

+

+
neg-cancel-< : ∀ {p q} → - p < - q → q < p

+
neg-cancel-< {p@record{}} {q@record{}} (*<* -↥p↧q<-↥q↧p) = *<* $ begin-strict

+
  ↥ q ℤ.* ↧ p              ≡⟨ ℤ.neg-involutive (↥ q ℤ.* ↧ p) ⟨

+
  ℤ.- ℤ.- (↥ q ℤ.* ↧ p)    ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ q) (↧ p)) ⟩

+
  ℤ.- ((ℤ.- ↥ q) ℤ.* ↧ p)  <⟨ ℤ.neg-mono-< -↥p↧q<-↥q↧p ⟩

+
  ℤ.- ((ℤ.- ↥ p) ℤ.* ↧ q)  ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ p) (↧ q)) ⟨

+
  ℤ.- ℤ.- (↥ p ℤ.* ↧ q)    ≡⟨ ℤ.neg-involutive (↥ p ℤ.* ↧ q) ⟩

+
  ↥ p ℤ.* ↧ q              ∎

+
  where open ℤ.≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _≤_

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
drop-*≤* : p ≤ q → (↥ p ℤ.* ↧ q) ℤ.≤ (↥ q ℤ.* ↧ p)

+
drop-*≤* (*≤* pq≤qp) = pq≤qp

+

+
≤-reflexive : _≃_ ⇒ _≤_

+
≤-reflexive (*≡* eq) = *≤* (ℤ.≤-reflexive eq)

+

+
≤-refl : Reflexive _≤_

+
≤-refl = ≤-reflexive ≃-refl

+

+
≤-reflexive-≡ : _≡_ ⇒ _≤_

+
≤-reflexive-≡ refl = ≤-refl

+

+
≤-trans : Transitive _≤_

+
≤-trans {p} {q} {r} (*≤* eq₁) (*≤* eq₂)

+
  = let n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; d₁ = ↧ p; d₂ = ↧ q; d₃ = ↧ r in *≤* $

+
  ℤ.*-cancelʳ-≤-pos (n₁ ℤ.* d₃) (n₃ ℤ.* d₁) d₂ $ begin

+
  (n₁  ℤ.* d₃) ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩

+
  n₁   ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩

+
  n₁   ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨

+
  (n₁  ℤ.* d₂) ℤ.* d₃  ≤⟨ ℤ.*-monoʳ-≤-nonNeg d₃ eq₁ ⟩

+
  (n₂  ℤ.* d₁) ℤ.* d₃  ≡⟨ cong (ℤ._* d₃) (ℤ.*-comm n₂ d₁) ⟩

+
  (d₁ ℤ.* n₂)  ℤ.* d₃  ≡⟨ ℤ.*-assoc d₁ n₂ d₃ ⟩

+
  d₁  ℤ.* (n₂  ℤ.* d₃) ≤⟨ ℤ.*-monoˡ-≤-nonNeg d₁ eq₂ ⟩

+
  d₁  ℤ.* (n₃  ℤ.* d₂) ≡⟨ ℤ.*-assoc d₁ n₃ d₂ ⟨

+
  (d₁ ℤ.* n₃)  ℤ.* d₂  ≡⟨ cong (ℤ._* d₂) (ℤ.*-comm d₁ n₃) ⟩

+
  (n₃  ℤ.* d₁) ℤ.* d₂  ∎ where open ℤ.≤-Reasoning

+

+
≤-antisym : Antisymmetric _≃_ _≤_

+
≤-antisym (*≤* le₁) (*≤* le₂) = *≡* (ℤ.≤-antisym le₁ le₂)

+

+
≤-total : Total _≤_

+
≤-total p q = [ inj₁ ∘ *≤* , inj₂ ∘ *≤* ]′ (ℤ.≤-total

+
  (↥ p ℤ.* ↧ q)

+
  (↥ q ℤ.* ↧ p))

+

+
≤-respˡ-≃ : _≤_ Respectsˡ _≃_

+
≤-respˡ-≃ x≈y = ≤-trans (≤-reflexive (≃-sym x≈y))

+

+
≤-respʳ-≃ : _≤_ Respectsʳ _≃_

+
≤-respʳ-≃ x≈y z≤x = ≤-trans z≤x (≤-reflexive x≈y)

+

+
≤-resp₂-≃ : _≤_ Respects₂ _≃_

+
≤-resp₂-≃ = ≤-respʳ-≃ , ≤-respˡ-≃

+

+
infix 4 _≤?_ _≥?_

+

+
_≤?_ : Decidable _≤_

+
p ≤? q = Dec.map′ *≤* drop-*≤* (↥ p ℤ.* ↧ q ℤ.≤? ↥ q ℤ.* ↧ p)

+

+
_≥?_ : Decidable _≥_

+
_≥?_ = flip _≤?_

+

+
≤-irrelevant : Irrelevant _≤_

+
≤-irrelevant (*≤* p≤q₁) (*≤* p≤q₂) = cong *≤* (ℤ.≤-irrelevant p≤q₁ p≤q₂)

+

+
------------------------------------------------------------------------

+
-- Structures over _≃_

+

+
≤-isPreorder : IsPreorder _≃_ _≤_

+
≤-isPreorder = record

+
  { isEquivalence = ≃-isEquivalence

+
  ; reflexive     = ≤-reflexive

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isTotalPreorder : IsTotalPreorder _≃_ _≤_

+
≤-isTotalPreorder = record

+
  { isPreorder = ≤-isPreorder

+
  ; total      = ≤-total

+
  }

+

+
≤-isPartialOrder : IsPartialOrder _≃_ _≤_

+
≤-isPartialOrder = record

+
  { isPreorder = ≤-isPreorder

+
  ; antisym    = ≤-antisym

+
  }

+

+
≤-isTotalOrder : IsTotalOrder _≃_ _≤_

+
≤-isTotalOrder = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  ; total          = ≤-total

+
  }

+

+
≤-isDecTotalOrder : IsDecTotalOrder _≃_ _≤_

+
≤-isDecTotalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  ; _≟_          = _≃?_

+
  ; _≤?_         = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles over _≃_

+

+
≤-preorder : Preorder 0ℓ 0ℓ 0ℓ

+
≤-preorder = record

+
  { isPreorder = ≤-isPreorder

+
  }

+

+
≤-totalPreorder : TotalPreorder 0ℓ 0ℓ 0ℓ

+
≤-totalPreorder = record

+
  { isTotalPreorder = ≤-isTotalPreorder

+
  }

+

+
≤-poset : Poset 0ℓ 0ℓ 0ℓ

+
≤-poset = record

+
  { isPartialOrder = ≤-isPartialOrder

+
  }

+

+
≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ

+
≤-totalOrder = record

+
  { isTotalOrder = ≤-isTotalOrder

+
  }

+

+
≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ

+
≤-decTotalOrder = record

+
  { isDecTotalOrder = ≤-isDecTotalOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Structures over _≡_

+

+
≤-isPreorder-≡ : IsPreorder _≡_ _≤_

+
≤-isPreorder-≡ = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = ≤-reflexive-≡

+
  ; trans         = ≤-trans

+
  }

+

+
≤-isTotalPreorder-≡ : IsTotalPreorder _≡_ _≤_

+
≤-isTotalPreorder-≡ = record

+
  { isPreorder = ≤-isPreorder-≡

+
  ; total      = ≤-total

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles over _≡_

+

+
≤-preorder-≡ : Preorder 0ℓ 0ℓ 0ℓ

+
≤-preorder-≡ = record

+
  { isPreorder = ≤-isPreorder-≡

+
  }

+

+
≤-totalPreorder-≡ : TotalPreorder 0ℓ 0ℓ 0ℓ

+
≤-totalPreorder-≡ = record

+
  { isTotalPreorder = ≤-isTotalPreorder-≡

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties of _≤_

+

+
mono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≤_ → f Preserves _≃_ ⟶ _≃_

+
mono⇒cong = BC.mono⇒cong _≃_ _≃_ ≃-sym ≤-reflexive ≤-antisym

+

+
antimono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≥_ → f Preserves _≃_ ⟶ _≃_

+
antimono⇒cong = BC.antimono⇒cong _≃_ _≃_ ≃-sym ≤-reflexive ≤-antisym

+

+
------------------------------------------------------------------------

+
-- Properties of _≤ᵇ_

+
------------------------------------------------------------------------

+

+
≤ᵇ⇒≤ : T (p ≤ᵇ q) → p ≤ q

+
≤ᵇ⇒≤ = *≤* ∘ ℤ.≤ᵇ⇒≤

+

+
≤⇒≤ᵇ : p ≤ q → T (p ≤ᵇ q)

+
≤⇒≤ᵇ = ℤ.≤⇒≤ᵇ ∘ drop-*≤*

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+
------------------------------------------------------------------------

+

+
drop-*<* : p < q → (↥ p ℤ.* ↧ q) ℤ.< (↥ q ℤ.* ↧ p)

+
drop-*<* (*<* pq<qp) = pq<qp

+

+
------------------------------------------------------------------------

+
-- Relationship between other operators

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ (*<* p<q) = *≤* (ℤ.<⇒≤ p<q)

+

+
<⇒≢ : _<_ ⇒ _≢_

+
<⇒≢ (*<* x<y) refl = ℤ.<⇒≢ x<y refl

+

+
<⇒≱ : _<_ ⇒ _≱_

+
<⇒≱ (*<* x<y) = ℤ.<⇒≱ x<y ∘ drop-*≤*

+

+
≰⇒> : _≰_ ⇒ _>_

+
≰⇒> p≰q = *<* (ℤ.≰⇒> (p≰q ∘ *≤*))

+

+
≮⇒≥ : _≮_ ⇒ _≥_

+
≮⇒≥ p≮q = *≤* (ℤ.≮⇒≥ (p≮q ∘ *<*))

+

+
≰⇒≥ : _≰_ ⇒ _≥_

+
≰⇒≥ = <⇒≤ ∘ ≰⇒>

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
<-irrefl-≡ : Irreflexive _≡_ _<_

+
<-irrefl-≡ refl (*<* x<x) = ℤ.<-irrefl refl x<x

+

+
<-irrefl : Irreflexive _≃_ _<_

+
<-irrefl (*≡* x≡y) (*<* x<y) = ℤ.<-irrefl x≡y x<y

+

+
<-asym : Asymmetric _<_

+
<-asym (*<* x<y) = ℤ.<-asym x<y ∘ drop-*<*

+

+
<-dense : Dense _<_

+
<-dense {p} {q} (*<* p<q) = m , p<m , m<q

+
  where

+
  open ℤ.≤-Reasoning

+
  m : ℚᵘ

+
  m = mkℚᵘ (↥ p ℤ.+ ↥ q) (pred (↧ₙ p ℕ.+ ↧ₙ q))

+

+
  p<m : p < m

+
  p<m = *<* (begin-strict

+
    ↥ p ℤ.* ↧ m

+
      ≡⟨⟩

+
    ↥ p ℤ.* (↧ p ℤ.+ ↧ q)

+
      ≡⟨ ℤ.*-distribˡ-+ (↥ p) (↧ p) (↧ q) ⟩

+
    ↥ p ℤ.* ↧ p ℤ.+ ↥ p ℤ.* ↧ q

+
      <⟨ ℤ.+-monoʳ-< (↥ p ℤ.* ↧ p) p<q ⟩

+
    ↥ p ℤ.* ↧ p ℤ.+ ↥ q ℤ.* ↧ p

+
      ≡⟨ ℤ.*-distribʳ-+ (↧ p) (↥ p) (↥ q) ⟨

+
    (↥ p ℤ.+ ↥ q) ℤ.* ↧ p

+
      ≡⟨⟩

+
    ↥ m ℤ.* ↧ p ∎)

+

+
  m<q : m < q

+
  m<q = *<* (begin-strict

+
    ↥ m ℤ.* ↧ q

+
      ≡⟨ ℤ.*-distribʳ-+ (↧ q) (↥ p) (↥ q) ⟩

+
    ↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ q

+
      <⟨ ℤ.+-monoˡ-< (↥ q ℤ.* ↧ q) p<q ⟩

+
    ↥ q ℤ.* ↧ p ℤ.+ ↥ q ℤ.* ↧ q

+
      ≡⟨ ℤ.*-distribˡ-+ (↥ q) (↧ p) (↧ q) ⟨

+
    ↥ q ℤ.* (↧ p ℤ.+ ↧ q)

+
      ≡⟨⟩

+
    ↥ q ℤ.* ↧ m ∎)

+

+
≤-<-trans : Trans _≤_ _<_ _<_

+
≤-<-trans {p} {q} {r} (*≤* p≤q) (*<* q<r) = *<* $

+
  ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict

+
  n₁ ℤ.*  d₃ ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩

+
  n₁ ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩

+
  n₁ ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨

+
  n₁ ℤ.*  d₂ ℤ.* d₃  ≤⟨ ℤ.*-monoʳ-≤-nonNeg (↧ r) p≤q ⟩

+
  n₂ ℤ.*  d₁ ℤ.* d₃  ≡⟨ cong (ℤ._* d₃) (ℤ.*-comm n₂ d₁) ⟩

+
  d₁ ℤ.*  n₂ ℤ.* d₃  ≡⟨ ℤ.*-assoc d₁ n₂ d₃ ⟩

+
  d₁ ℤ.* (n₂ ℤ.* d₃) <⟨ ℤ.*-monoˡ-<-pos (↧ p) q<r ⟩

+
  d₁ ℤ.* (n₃ ℤ.* d₂) ≡⟨ ℤ.*-assoc d₁ n₃ d₂ ⟨

+
  d₁ ℤ.*  n₃ ℤ.* d₂  ≡⟨ cong (ℤ._* d₂) (ℤ.*-comm d₁ n₃) ⟩

+
  n₃ ℤ.*  d₁ ℤ.* d₂  ∎

+
  where open ℤ.≤-Reasoning

+
        n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; d₁ = ↧ p; d₂ = ↧ q; d₃ = ↧ r

+

+
<-≤-trans : Trans _<_ _≤_ _<_

+
<-≤-trans {p} {q} {r} (*<* p<q) (*≤* q≤r) = *<* $

+
  ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict

+
  n₁ ℤ.*  d₃ ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩

+
  n₁ ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩

+
  n₁ ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨

+
  n₁ ℤ.*  d₂ ℤ.* d₃  <⟨ ℤ.*-monoʳ-<-pos (↧ r) p<q ⟩

+
  n₂ ℤ.*  d₁ ℤ.* d₃  ≡⟨ cong (ℤ._* d₃) (ℤ.*-comm n₂ d₁) ⟩

+
  d₁ ℤ.*  n₂ ℤ.* d₃  ≡⟨ ℤ.*-assoc d₁ n₂ d₃ ⟩

+
  d₁ ℤ.* (n₂ ℤ.* d₃) ≤⟨ ℤ.*-monoˡ-≤-nonNeg (↧ p) q≤r ⟩

+
  d₁ ℤ.* (n₃ ℤ.* d₂) ≡⟨ ℤ.*-assoc d₁ n₃ d₂ ⟨

+
  d₁ ℤ.*  n₃ ℤ.* d₂  ≡⟨ cong (ℤ._* d₂) (ℤ.*-comm d₁ n₃) ⟩

+
  n₃ ℤ.*  d₁ ℤ.* d₂  ∎

+
  where open ℤ.≤-Reasoning

+
        n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; d₁ = ↧ p; d₂ = ↧ q; d₃ = ↧ r

+

+
<-trans : Transitive _<_

+
<-trans = ≤-<-trans ∘ <⇒≤

+

+
<-cmp : Trichotomous _≃_ _<_

+
<-cmp p q with ℤ.<-cmp (↥ p ℤ.* ↧ q) (↥ q ℤ.* ↧ p)

+
... | tri< x<y x≉y x≯y = tri< (*<* x<y) (x≉y ∘ drop-*≡*) (x≯y ∘ drop-*<*)

+
... | tri≈ x≮y x≈y x≯y = tri≈ (x≮y ∘ drop-*<*) (*≡* x≈y) (x≯y ∘ drop-*<*)

+
... | tri> x≮y x≉y x>y = tri> (x≮y ∘ drop-*<*) (x≉y ∘ drop-*≡*) (*<* x>y)

+

+
infix 4 _<?_ _>?_

+

+
_<?_ : Decidable _<_

+
p <? q = Dec.map′ *<* drop-*<* (↥ p ℤ.* ↧ q ℤ.<? ↥ q ℤ.* ↧ p)

+

+
_>?_ : Decidable _>_

+
_>?_ = flip _<?_

+

+
<-irrelevant : Irrelevant _<_

+
<-irrelevant (*<* p<q₁) (*<* p<q₂) = cong *<* (ℤ.<-irrelevant p<q₁ p<q₂)

+

+
<-respʳ-≃ : _<_ Respectsʳ _≃_

+
<-respʳ-≃ {p} {q} {r} (*≡* q≡r) (*<* p<q) = *<* $

+
  ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict

+
  n₁ ℤ.*  d₃ ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩

+
  n₁ ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩

+
  n₁ ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨

+
  n₁ ℤ.*  d₂ ℤ.* d₃  <⟨ ℤ.*-monoʳ-<-pos (↧ r) p<q ⟩

+
  n₂ ℤ.*  d₁ ℤ.* d₃  ≡⟨ ℤ.*-assoc n₂ d₁ d₃ ⟩

+
  n₂ ℤ.* (d₁ ℤ.* d₃) ≡⟨ cong (n₂ ℤ.*_) (ℤ.*-comm d₁ d₃) ⟩

+
  n₂ ℤ.* (d₃ ℤ.* d₁) ≡⟨ ℤ.*-assoc n₂ d₃ d₁ ⟨

+
  n₂ ℤ.*  d₃ ℤ.* d₁  ≡⟨ cong (ℤ._* d₁) q≡r ⟩

+
  n₃ ℤ.*  d₂ ℤ.* d₁  ≡⟨ ℤ.*-assoc n₃ d₂ d₁ ⟩

+
  n₃ ℤ.* (d₂ ℤ.* d₁) ≡⟨ cong (n₃ ℤ.*_) (ℤ.*-comm d₂ d₁) ⟩

+
  n₃ ℤ.* (d₁ ℤ.* d₂) ≡⟨ ℤ.*-assoc n₃ d₁ d₂ ⟨

+
  n₃ ℤ.*  d₁ ℤ.* d₂  ∎

+
  where open ℤ.≤-Reasoning

+
        n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; d₁ = ↧ p; d₂ = ↧ q; d₃ = ↧ r

+

+
<-respˡ-≃ : _<_ Respectsˡ _≃_

+
<-respˡ-≃ q≃r q<p

+
  = subst (_< _) (neg-involutive-≡ _)

+
  $ subst (_ <_) (neg-involutive-≡ _)

+
  $ neg-mono-< (<-respʳ-≃ (-‿cong q≃r) (neg-mono-< q<p))

+

+
<-resp-≃ : _<_ Respects₂ _≃_

+
<-resp-≃ = <-respʳ-≃ , <-respˡ-≃

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
<-isStrictPartialOrder-≡ : IsStrictPartialOrder _≡_ _<_

+
<-isStrictPartialOrder-≡ = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl-≡

+
  ; trans         = <-trans

+
  ; <-resp-≈      = subst (_ <_) , subst (_< _)

+
  }

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≃_ _<_

+
<-isStrictPartialOrder = record

+
  { isEquivalence = ≃-isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans

+
  ; <-resp-≈      = <-resp-≃

+
  }

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≃_ _<_

+
<-isStrictTotalOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  ; compare              = <-cmp

+
  }

+

+
<-isDenseLinearOrder : IsDenseLinearOrder _≃_ _<_

+
<-isDenseLinearOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  ; dense              = <-dense

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
<-strictPartialOrder-≡ : StrictPartialOrder 0ℓ 0ℓ 0ℓ

+
<-strictPartialOrder-≡ = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder-≡

+
  }

+

+
<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
<-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ

+
<-denseLinearOrder = record

+
  { isDenseLinearOrder = <-isDenseLinearOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- A specialised module for reasoning about the _≤_ and _<_ relations

+
------------------------------------------------------------------------

+

+
module ≤-Reasoning where

+
  import Relation.Binary.Reasoning.Base.Triple

+
    ≤-isPreorder

+
    <-asym

+
    <-trans

+
    <-resp-≃

+
    <⇒≤

+
    <-≤-trans

+
    ≤-<-trans

+
    as Triple

+

+
  open Triple public

+
    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)

+
    renaming (≈-go to ≃-go)

+

+
  open ≃-syntax _IsRelatedTo_ _IsRelatedTo_ ≃-go ≃-sym public

+

+
------------------------------------------------------------------------

+
-- Properties of ↥_/↧_

+

+
≥0⇒↥≥0 : ∀ {n dm} → mkℚᵘ n dm ≥ 0ℚᵘ → n ℤ.≥ 0ℤ

+
≥0⇒↥≥0 {n} {dm} r≥0 = ℤ.≤-trans (drop-*≤* r≥0)

+
                                (ℤ.≤-reflexive $ ℤ.*-identityʳ n)

+

+
>0⇒↥>0 : ∀ {n dm} → mkℚᵘ n dm > 0ℚᵘ → n ℤ.> 0ℤ

+
>0⇒↥>0 {n} {dm} r>0 = ℤ.<-≤-trans (drop-*<* r>0)

+
                                  (ℤ.≤-reflexive $ ℤ.*-identityʳ n)

+

+
------------------------------------------------------------------------

+
-- Properties of sign predicates

+

+
positive⁻¹ : ∀ p → .{{Positive p}} → p > 0ℚᵘ

+
positive⁻¹ (mkℚᵘ +[1+ n ] _) = *<* (ℤ.+<+ ℕ.z<s)

+

+
nonNegative⁻¹ : ∀ p → .{{NonNegative p}} → p ≥ 0ℚᵘ

+
nonNegative⁻¹ (mkℚᵘ +0       _) = *≤* (ℤ.+≤+ ℕ.z≤n)

+
nonNegative⁻¹ (mkℚᵘ +[1+ n ] _) = *≤* (ℤ.+≤+ ℕ.z≤n)

+

+
negative⁻¹ : ∀ p → .{{Negative p}} → p < 0ℚᵘ

+
negative⁻¹ (mkℚᵘ -[1+ n ] _) = *<* ℤ.-<+

+

+
nonPositive⁻¹ : ∀ p → .{{NonPositive p}} → p ≤ 0ℚᵘ

+
nonPositive⁻¹ (mkℚᵘ +0       _) = *≤* (ℤ.+≤+ ℕ.z≤n)

+
nonPositive⁻¹ (mkℚᵘ -[1+ n ] _) = *≤* ℤ.-≤+

+

+
pos⇒nonNeg : ∀ p → .{{Positive p}} → NonNegative p

+
pos⇒nonNeg (mkℚᵘ +0       _) = _

+
pos⇒nonNeg (mkℚᵘ +[1+ n ] _) = _

+

+
neg⇒nonPos : ∀ p → .{{Negative p}} → NonPositive p

+
neg⇒nonPos (mkℚᵘ +0       _) = _

+
neg⇒nonPos (mkℚᵘ -[1+ n ] _) = _

+

+
neg<pos : ∀ p q → .{{Negative p}} → .{{Positive q}} → p < q

+
neg<pos p q = <-trans (negative⁻¹ p) (positive⁻¹ q)

+

+
pos⇒nonZero : ∀ p → .{{Positive p}} → NonZero p

+
pos⇒nonZero (mkℚᵘ (+[1+ _ ]) _) = _

+

+
nonNeg∧nonPos⇒0 : ∀ p → .{{NonNegative p}} → .{{NonPositive p}} → p ≃ 0ℚᵘ

+
nonNeg∧nonPos⇒0 (mkℚᵘ +0 _) = *≡* refl

+

+
nonNeg<⇒pos : ∀ {p q} .{{_ : NonNegative p}} → p < q → Positive q

+
nonNeg<⇒pos {p} p<q = positive (≤-<-trans (nonNegative⁻¹ p) p<q)

+

+
nonNeg≤⇒nonNeg : ∀ {p q} .{{_ : NonNegative p}} → p ≤ q → NonNegative q

+
nonNeg≤⇒nonNeg {p} p≤q = nonNegative (≤-trans (nonNegative⁻¹ p) p≤q)

+

+
neg⇒nonZero : ∀ p → .{{Negative p}} → NonZero p

+
neg⇒nonZero (mkℚᵘ (-[1+ _ ]) _) = _

+

+
------------------------------------------------------------------------

+
-- Properties of _+_

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
-- Congruence

+

+
+-cong : Congruent₂ _≃_ _+_

+
+-cong {x@record{}} {y@record{}} {u@record{}} {v@record{}} (*≡* ab′∼a′b) (*≡* cd′∼c′d) = *≡* (begin

+
  (↥x ℤ.* ↧u ℤ.+ ↥u ℤ.* ↧x) ℤ.* (↧y ℤ.* ↧v)               ≡⟨ solve 6 (λ ↥x ↧x ↧y ↥u ↧u ↧v →

+
                                                             (↥x :* ↧u :+ ↥u :* ↧x) :* (↧y :* ↧v) :=

+
                                                             (↥x :* ↧y :* (↧u :* ↧v)) :+ ↥u :* ↧v :* (↧x :* ↧y))

+
                                                             refl (↥ x) (↧ x) (↧ y) (↥ u) (↧ u) (↧ v) ⟩

+
  ↥x ℤ.* ↧y ℤ.* (↧u ℤ.* ↧v) ℤ.+ ↥u ℤ.* ↧v ℤ.* (↧x ℤ.* ↧y) ≡⟨ cong₂ ℤ._+_ (cong (ℤ._* (↧u ℤ.* ↧v)) ab′∼a′b)

+
                                                                         (cong (ℤ._* (↧x ℤ.* ↧y)) cd′∼c′d) ⟩

+
  ↥y ℤ.* ↧x ℤ.* (↧u ℤ.* ↧v) ℤ.+ ↥v ℤ.* ↧u ℤ.* (↧x ℤ.* ↧y) ≡⟨ solve 6 (λ ↧x ↥y ↧y ↧u ↥v ↧v →

+
                                                             (↥y :* ↧x :* (↧u :* ↧v)) :+ ↥v :* ↧u :* (↧x :* ↧y) :=

+
                                                             (↥y :* ↧v :+ ↥v :* ↧y) :* (↧x :* ↧u))

+
                                                             refl (↧ x) (↥ y) (↧ y) (↧ u) (↥ v) (↧ v) ⟩

+
  (↥y ℤ.* ↧v ℤ.+ ↥v ℤ.* ↧y) ℤ.* (↧x ℤ.* ↧u)               ∎)

+
  where

+
  ↥x = ↥ x; ↧x = ↧ x; ↥y = ↥ y; ↧y = ↧ y; ↥u = ↥ u; ↧u = ↧ u; ↥v = ↥ v; ↧v = ↧ v

+
  open ≡-Reasoning

+
  open ℤ-solver

+

+
+-congʳ : ∀ p → q ≃ r → p + q ≃ p + r

+
+-congʳ p q≃r = +-cong (≃-refl {p}) q≃r

+

+
+-congˡ : ∀ p → q ≃ r → q + p ≃ r + p

+
+-congˡ p q≃r = +-cong q≃r (≃-refl {p})

+

+
-- Associativity

+

+
+-assoc-↥ : Associative (_≡_ on ↥_) _+_

+
+-assoc-↥ p@record{} q@record{} r@record{} = solve 6 (λ ↥p ↧p ↥q ↧q ↥r ↧r →

+
    (↥p :* ↧q :+ ↥q :* ↧p) :* ↧r :+ ↥r :* (↧p :* ↧q) :=

+
    ↥p :* (↧q :* ↧r) :+ (↥q :* ↧r :+ ↥r :* ↧q) :* ↧p)

+
  refl (↥ p) (↧ p) (↥ q) (↧ q) (↥ r) (↧ r)

+
  where open ℤ-solver

+

+
+-assoc-↧ : Associative (_≡_ on ↧ₙ_) _+_

+
+-assoc-↧ p@record{} q@record{} r@record{} = ℕ.*-assoc (↧ₙ p) (↧ₙ q) (↧ₙ r)

+

+
+-assoc-≡ : Associative _≡_ _+_

+
+-assoc-≡ p q r = ↥↧≡⇒≡ (+-assoc-↥ p q r) (+-assoc-↧ p q r)

+

+
+-assoc : Associative _≃_ _+_

+
+-assoc p q r = ≃-reflexive (+-assoc-≡ p q r)

+

+
-- Commutativity

+

+
+-comm-↥ : Commutative (_≡_ on ↥_) _+_

+
+-comm-↥ p@record{} q@record{} = ℤ.+-comm (↥ p ℤ.* ↧ q) (↥ q ℤ.* ↧ p)

+

+
+-comm-↧ : Commutative (_≡_ on ↧ₙ_) _+_

+
+-comm-↧ p@record{} q@record{} = ℕ.*-comm (↧ₙ p) (↧ₙ q)

+

+
+-comm-≡ : Commutative _≡_ _+_

+
+-comm-≡ p q = ↥↧≡⇒≡ (+-comm-↥ p q) (+-comm-↧ p q)

+

+
+-comm : Commutative _≃_ _+_

+
+-comm p q = ≃-reflexive (+-comm-≡ p q)

+

+
-- Identities

+

+
+-identityˡ-↥ : LeftIdentity (_≡_ on ↥_) 0ℚᵘ _+_

+
+-identityˡ-↥ p@record{} = begin

+
  0ℤ ℤ.* ↧ p ℤ.+ ↥ p ℤ.* 1ℤ ≡⟨ cong₂ ℤ._+_ (ℤ.*-zeroˡ (↧ p)) (ℤ.*-identityʳ (↥ p)) ⟩

+
  0ℤ ℤ.+ ↥ p                ≡⟨ ℤ.+-identityˡ (↥ p) ⟩

+
  ↥ p                       ∎ where open ≡-Reasoning

+

+
+-identityˡ-↧ : LeftIdentity (_≡_ on ↧ₙ_) 0ℚᵘ _+_

+
+-identityˡ-↧ p@record{} = ℕ.+-identityʳ (↧ₙ p)

+

+
+-identityˡ-≡ : LeftIdentity _≡_ 0ℚᵘ _+_

+
+-identityˡ-≡ p = ↥↧≡⇒≡ (+-identityˡ-↥ p) (+-identityˡ-↧ p)

+

+
+-identityˡ : LeftIdentity _≃_ 0ℚᵘ _+_

+
+-identityˡ p = ≃-reflexive (+-identityˡ-≡ p)

+

+
+-identityʳ-≡ : RightIdentity _≡_ 0ℚᵘ _+_

+
+-identityʳ-≡ = comm+idˡ⇒idʳ +-comm-≡ {e = 0ℚᵘ} +-identityˡ-≡

+

+
+-identityʳ : RightIdentity _≃_ 0ℚᵘ _+_

+
+-identityʳ p = ≃-reflexive (+-identityʳ-≡ p)

+

+
+-identity-≡ : Identity _≡_ 0ℚᵘ _+_

+
+-identity-≡ = +-identityˡ-≡ , +-identityʳ-≡

+

+
+-identity : Identity _≃_ 0ℚᵘ _+_

+
+-identity = +-identityˡ , +-identityʳ

+

+
+-inverseˡ : LeftInverse _≃_ 0ℚᵘ -_ _+_

+
+-inverseˡ p@record{} = *≡* let n = ↥ p; d = ↧ p in

+
  ((ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d) ℤ.* 1ℤ ≡⟨ ℤ.*-identityʳ ((ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d) ⟩

+
  (ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d          ≡⟨ cong (ℤ._+ (n ℤ.* d)) (ℤ.neg-distribˡ-* n d) ⟨

+
  ℤ.- (n ℤ.* d) ℤ.+ n ℤ.* d          ≡⟨ ℤ.+-inverseˡ (n ℤ.* d) ⟩

+
  0ℤ                                 ∎ where open ≡-Reasoning

+

+
+-inverseʳ : RightInverse _≃_ 0ℚᵘ -_ _+_

+
+-inverseʳ p@record{} = *≡* let n = ↥ p; d = ↧ p in

+
  (n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d) ℤ.* 1ℤ ≡⟨ ℤ.*-identityʳ (n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d) ⟩

+
  n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d          ≡⟨ cong (λ n+d → n ℤ.* d ℤ.+ n+d) (ℤ.neg-distribˡ-* n d) ⟨

+
  n ℤ.* d ℤ.+ ℤ.- (n ℤ.* d)          ≡⟨ ℤ.+-inverseʳ (n ℤ.* d) ⟩

+
  0ℤ                                 ∎ where open ≡-Reasoning

+

+
+-inverse : Inverse _≃_ 0ℚᵘ -_ _+_

+
+-inverse = +-inverseˡ , +-inverseʳ

+

+
+-cancelˡ : ∀ {r p q} → r + p ≃ r + q → p ≃ q

+
+-cancelˡ {r} {p} {q} r+p≃r+q = begin-equality

+
  p            ≃⟨ +-identityʳ p ⟨

+
  p + 0ℚᵘ      ≃⟨ +-congʳ p (+-inverseʳ r) ⟨

+
  p + (r - r)  ≃⟨ +-assoc p r (- r) ⟨

+
  (p + r) - r  ≃⟨ +-congˡ (- r) (+-comm p r) ⟩

+
  (r + p) - r  ≃⟨ +-congˡ (- r) r+p≃r+q ⟩

+
  (r + q) - r  ≃⟨ +-congˡ (- r) (+-comm r q) ⟩

+
  (q + r) - r  ≃⟨ +-assoc q r (- r) ⟩

+
  q + (r - r)  ≃⟨ +-congʳ q (+-inverseʳ r) ⟩

+
  q + 0ℚᵘ      ≃⟨ +-identityʳ q ⟩

+
  q            ∎ where open ≤-Reasoning

+

+
+-cancelʳ : ∀ {r p q} → p + r ≃ q + r → p ≃ q

+
+-cancelʳ {r} {p} {q} p+r≃q+r = +-cancelˡ {r} $ begin-equality

+
  r + p ≃⟨ +-comm r p ⟩

+
  p + r ≃⟨ p+r≃q+r ⟩

+
  q + r ≃⟨ +-comm q r ⟩

+
  r + q ∎ where open ≤-Reasoning

+

+
p+p≃0⇒p≃0 : ∀ p → p + p ≃ 0ℚᵘ → p ≃ 0ℚᵘ

+
p+p≃0⇒p≃0 (mkℚᵘ ℤ.+0 _) (*≡* _) = *≡* refl

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and -_

+

+
neg-distrib-+ : ∀ p q → - (p + q) ≡ (- p) + (- q)

+
neg-distrib-+ p@record{} q@record{} = ↥↧≡⇒≡ (begin

+
    ℤ.- (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p)       ≡⟨ ℤ.neg-distrib-+ (↥ p ℤ.* ↧ q) _ ⟩

+
    ℤ.- (↥ p ℤ.* ↧ q) ℤ.+ ℤ.- (↥ q ℤ.* ↧ p) ≡⟨ cong₂ ℤ._+_ (ℤ.neg-distribˡ-* (↥ p) _)

+
                                                           (ℤ.neg-distribˡ-* (↥ q) _) ⟩

+
    (ℤ.- ↥ p) ℤ.* ↧ q ℤ.+ (ℤ.- ↥ q) ℤ.* ↧ p ∎

+
  ) refl

+
  where open ≡-Reasoning

+

+
p≃-p⇒p≃0 : ∀ p → p ≃ - p → p ≃ 0ℚᵘ

+
p≃-p⇒p≃0 p p≃-p = p+p≃0⇒p≃0 p (begin-equality

+
  p + p  ≃⟨ +-congʳ p p≃-p ⟩

+
  p - p  ≃⟨ +-inverseʳ p ⟩

+
  0ℚᵘ    ∎)

+
  where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _≤_

+

+
private

+
  lemma : ∀ r p q → (↥ r ℤ.* ↧ p ℤ.+ ↥ p ℤ.* ↧ r) ℤ.* (↧ r ℤ.* ↧ q)

+
                    ≡ (↥ r ℤ.* ↧ r) ℤ.* (↧ p ℤ.* ↧ q) ℤ.+ (↧ r ℤ.* ↧ r) ℤ.* (↥ p ℤ.* ↧ q)

+
  lemma r p q = solve 5 (λ ↥r ↧r ↧p ↥p ↧q →

+
                          (↥r :* ↧p :+ ↥p :* ↧r) :* (↧r :* ↧q) :=

+
                          (↥r :* ↧r) :* (↧p :* ↧q) :+ (↧r :* ↧r) :* (↥p :* ↧q))

+
                      refl (↥ r) (↧ r) (↧ p) (↥ p) (↧ q)

+
    where open ℤ-solver

+

+
+-monoʳ-≤ : ∀ r → (r +_) Preserves _≤_ ⟶ _≤_

+
+-monoʳ-≤ r@record{} {p@record{}} {q@record{}} (*≤* x≤y) = *≤* $ begin

+
  ↥ (r + p) ℤ.* ↧ (r + q)                                  ≡⟨ lemma r p q ⟩

+
  r₂ ℤ.* (↧ p ℤ.* ↧ q) ℤ.+ (↧ r ℤ.* ↧ r) ℤ.* (↥ p ℤ.* ↧ q) ≤⟨ leq ⟩

+
  r₂ ℤ.* (↧ q ℤ.* ↧ p) ℤ.+ (↧ r ℤ.* ↧ r) ℤ.* (↥ q ℤ.* ↧ p) ≡⟨ sym $ lemma r q p ⟩

+
  ↥ (r + q) ℤ.* (↧ (r + p))                                ∎

+
  where

+
  open ℤ.≤-Reasoning; r₂ = ↥ r ℤ.* ↧ r

+
  leq = ℤ.+-mono-≤

+
    (ℤ.≤-reflexive $ cong (r₂ ℤ.*_) (ℤ.*-comm (↧ p) (↧ q)))

+
    (ℤ.*-monoˡ-≤-nonNeg (↧ r ℤ.* ↧ r) x≤y)

+

+
+-monoˡ-≤ : ∀ r → (_+ r) Preserves _≤_ ⟶ _≤_

+
+-monoˡ-≤ r {p} {q} rewrite +-comm-≡ p r | +-comm-≡ q r = +-monoʳ-≤ r

+

+
+-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
+-mono-≤ {p} {q} {u} {v} p≤q u≤v = ≤-trans (+-monoˡ-≤ u p≤q) (+-monoʳ-≤ q u≤v)

+

+
p≤q⇒p≤r+q : ∀ r .{{_ : NonNegative r}} → p ≤ q → p ≤ r + q

+
p≤q⇒p≤r+q {p} {q} r p≤q = subst (_≤ r + q) (+-identityˡ-≡ p) (+-mono-≤ (nonNegative⁻¹ r) p≤q)

+

+
p≤q+p : ∀ p q .{{_ : NonNegative q}} → p ≤ q + p

+
p≤q+p p q = p≤q⇒p≤r+q q ≤-refl

+

+
p≤p+q : ∀ p q .{{_ : NonNegative q}} → p ≤ p + q

+
p≤p+q p q rewrite +-comm-≡ p q = p≤q+p p q

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and _<_

+

+
+-monoʳ-< : ∀ r → (r +_) Preserves _<_ ⟶ _<_

+
+-monoʳ-< r@record{} {p@record{}} {q@record{}} (*<* x<y) = *<* $ begin-strict

+
  ↥ (r + p) ℤ.* (↧ (r + q))                          ≡⟨ lemma r p q ⟩

+
  ↥r↧r ℤ.* (↧ p ℤ.* ↧ q) ℤ.+ ↧r↧r ℤ.* (↥ p ℤ.* ↧ q)  <⟨ leq ⟩

+
  ↥r↧r ℤ.* (↧ q ℤ.* ↧ p) ℤ.+ ↧r↧r ℤ.* (↥ q ℤ.* ↧ p)  ≡⟨ sym $ lemma r q p ⟩

+
  ↥ (r + q) ℤ.* (↧ (r + p))                          ∎

+
  where

+
  open ℤ.≤-Reasoning; ↥r↧r = ↥ r ℤ.* ↧ r; ↧r↧r = ↧ r ℤ.* ↧ r

+
  leq = ℤ.+-mono-≤-<

+
    (ℤ.≤-reflexive $ cong (↥r↧r ℤ.*_) (ℤ.*-comm (↧ p) (↧ q)))

+
    (ℤ.*-monoˡ-<-pos ↧r↧r x<y)

+

+
+-monoˡ-< : ∀ r → (_+ r) Preserves _<_ ⟶ _<_

+
+-monoˡ-< r {p} {q} rewrite +-comm-≡ p r | +-comm-≡ q r = +-monoʳ-< r

+

+
+-mono-< : _+_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
+-mono-< {p} {q} {u} {v} p<q u<v = <-trans (+-monoˡ-< u p<q) (+-monoʳ-< q u<v)

+

+
+-mono-≤-< : _+_ Preserves₂ _≤_ ⟶ _<_ ⟶ _<_

+
+-mono-≤-< {p} {q} {r} p≤q q<r = ≤-<-trans (+-monoˡ-≤ r p≤q) (+-monoʳ-< q q<r)

+

+
+-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_

+
+-mono-<-≤ {p} {q} {r} p<q q≤r = <-≤-trans (+-monoˡ-< r p<q) (+-monoʳ-≤ q q≤r)

+

+
------------------------------------------------------------------------

+
-- Properties of _+_ and predicates

+

+
pos+pos⇒pos : ∀ p .{{_ : Positive p}} →

+
              ∀ q .{{_ : Positive q}} →

+
              Positive (p + q)

+
pos+pos⇒pos p q = positive (+-mono-< (positive⁻¹ p) (positive⁻¹ q))

+

+
nonNeg+nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} →

+
                       ∀ q .{{_ : NonNegative q}} →

+
                       NonNegative (p + q)

+
nonNeg+nonNeg⇒nonNeg p q = nonNegative

+
  (+-mono-≤ (nonNegative⁻¹ p) (nonNegative⁻¹ q))

+

+
------------------------------------------------------------------------

+
-- Properties of _-_

+

+
+-minus-telescope : ∀ p q r → (p - q) + (q - r) ≃ p - r

+
+-minus-telescope p q r = begin-equality

+
  (p - q) + (q - r)   ≃⟨ ≃-sym (+-assoc (p - q) q (- r)) ⟩

+
  (p - q) + q - r     ≃⟨ +-congˡ (- r) (+-assoc p (- q) q) ⟩

+
  (p + (- q + q)) - r ≃⟨ +-congˡ (- r) (+-congʳ p (+-inverseˡ q)) ⟩

+
  (p + 0ℚᵘ) - r       ≃⟨ +-congˡ (- r) (+-identityʳ p) ⟩

+
  p - r               ∎ where open ≤-Reasoning

+

+
p≃q⇒p-q≃0 : ∀ p q → p ≃ q → p - q ≃ 0ℚᵘ

+
p≃q⇒p-q≃0 p q p≃q = begin-equality

+
  p - q ≃⟨ +-congˡ (- q) p≃q ⟩

+
  q - q ≃⟨ +-inverseʳ q ⟩

+
  0ℚᵘ   ∎ where open ≤-Reasoning

+

+
p-q≃0⇒p≃q : ∀ p q → p - q ≃ 0ℚᵘ → p ≃ q

+
p-q≃0⇒p≃q p q p-q≃0 = begin-equality

+
  p             ≡⟨ +-identityʳ-≡ p ⟨

+
  p + 0ℚᵘ       ≃⟨ +-congʳ p (≃-sym (+-inverseˡ q)) ⟩

+
  p + (- q + q) ≡⟨ +-assoc-≡ p (- q) q ⟨

+
  (p - q) + q   ≃⟨ +-congˡ q p-q≃0 ⟩

+
  0ℚᵘ + q       ≡⟨ +-identityˡ-≡ q ⟩

+
  q             ∎ where open ≤-Reasoning

+

+
neg-mono-≤ : -_ Preserves _≤_ ⟶ _≥_

+
neg-mono-≤ {p@record{}} {q@record{}} (*≤* p≤q) = *≤* $ begin

+
  ℤ.- ↥ q ℤ.* ↧ p   ≡⟨ ℤ.neg-distribˡ-* (↥ q) (↧ p) ⟨

+
  ℤ.- (↥ q ℤ.* ↧ p) ≤⟨ ℤ.neg-mono-≤ p≤q ⟩

+
  ℤ.- (↥ p ℤ.* ↧ q) ≡⟨ ℤ.neg-distribˡ-* (↥ p) (↧ q) ⟩

+
  ℤ.- ↥ p ℤ.* ↧ q   ∎ where open ℤ.≤-Reasoning

+

+
neg-cancel-≤ : ∀ {p q} → - p ≤ - q → q ≤ p

+
neg-cancel-≤ {p@record{}} {q@record{}} (*≤* -↥p↧q≤-↥q↧p) = *≤* $ begin

+
  ↥ q ℤ.* ↧ p             ≡⟨ ℤ.neg-involutive (↥ q ℤ.* ↧ p) ⟨

+
  ℤ.- ℤ.- (↥ q ℤ.* ↧ p)   ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ q) (↧ p)) ⟩

+
  ℤ.- ((ℤ.- ↥ q) ℤ.* ↧ p) ≤⟨ ℤ.neg-mono-≤ -↥p↧q≤-↥q↧p ⟩

+
  ℤ.- ((ℤ.- ↥ p) ℤ.* ↧ q) ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ p) (↧ q)) ⟨

+
  ℤ.- ℤ.- (↥ p ℤ.* ↧ q)   ≡⟨ ℤ.neg-involutive (↥ p ℤ.* ↧ q) ⟩

+
  ↥ p ℤ.* ↧ q             ∎

+
  where

+
    open ℤ.≤-Reasoning

+

+
p≤q⇒p-q≤0 : ∀ {p q} → p ≤ q → p - q ≤ 0ℚᵘ

+
p≤q⇒p-q≤0 {p} {q} p≤q = begin

+
  p - q ≤⟨ +-monoˡ-≤ (- q) p≤q ⟩

+
  q - q ≃⟨ +-inverseʳ q ⟩

+
  0ℚᵘ   ∎ where open ≤-Reasoning

+

+
p-q≤0⇒p≤q : ∀ {p q} → p - q ≤ 0ℚᵘ → p ≤ q

+
p-q≤0⇒p≤q {p} {q} p-q≤0 = begin

+
  p             ≡⟨ +-identityʳ-≡ p ⟨

+
  p + 0ℚᵘ       ≃⟨ +-congʳ p (≃-sym (+-inverseˡ q)) ⟩

+
  p + (- q + q) ≡⟨ +-assoc-≡ p (- q) q ⟨

+
  (p - q) + q   ≤⟨ +-monoˡ-≤ q p-q≤0 ⟩

+
  0ℚᵘ + q       ≡⟨ +-identityˡ-≡ q ⟩

+
  q             ∎ where open ≤-Reasoning

+

+
p≤q⇒0≤q-p : ∀ {p q} → p ≤ q → 0ℚᵘ ≤ q - p

+
p≤q⇒0≤q-p {p} {q} p≤q = begin

+
  0ℚᵘ   ≃⟨ ≃-sym (+-inverseʳ p) ⟩

+
  p - p ≤⟨ +-monoˡ-≤ (- p) p≤q ⟩

+
  q - p ∎ where open ≤-Reasoning

+

+
0≤q-p⇒p≤q : ∀ {p q} → 0ℚᵘ ≤ q - p → p ≤ q

+
0≤q-p⇒p≤q {p} {q} 0≤p-q = begin

+
  p             ≡⟨ +-identityˡ-≡ p ⟨

+
  0ℚᵘ + p       ≤⟨ +-monoˡ-≤ p 0≤p-q ⟩

+
  q - p + p     ≡⟨ +-assoc-≡ q (- p) p ⟩

+
  q + (- p + p) ≃⟨ +-congʳ q (+-inverseˡ p) ⟩

+
  q + 0ℚᵘ       ≡⟨ +-identityʳ-≡ q ⟩

+
  q             ∎ where open ≤-Reasoning

+

+
------------------------------------------------------------------------

+
-- Algebraic structures

+

+
+-isMagma : IsMagma _≃_ _+_

+
+-isMagma = record

+
  { isEquivalence = ≃-isEquivalence

+
  ; ∙-cong        = +-cong

+
  }

+

+
+-isSemigroup : IsSemigroup _≃_ _+_

+
+-isSemigroup = record

+
  { isMagma = +-isMagma

+
  ; assoc   = +-assoc

+
  }

+

+
+-0-isMonoid : IsMonoid _≃_ _+_ 0ℚᵘ

+
+-0-isMonoid = record

+
  { isSemigroup = +-isSemigroup

+
  ; identity    = +-identity

+
  }

+

+
+-0-isCommutativeMonoid : IsCommutativeMonoid _≃_ _+_ 0ℚᵘ

+
+-0-isCommutativeMonoid = record

+
  { isMonoid = +-0-isMonoid

+
  ; comm     = +-comm

+
  }

+

+
+-0-isGroup : IsGroup _≃_ _+_ 0ℚᵘ (-_)

+
+-0-isGroup = record

+
  { isMonoid = +-0-isMonoid

+
  ; inverse  = +-inverse

+
  ; ⁻¹-cong  = -‿cong

+
  }

+

+
+-0-isAbelianGroup : IsAbelianGroup _≃_ _+_ 0ℚᵘ (-_)

+
+-0-isAbelianGroup = record

+
  { isGroup = +-0-isGroup

+
  ; comm    = +-comm

+
  }

+

+
------------------------------------------------------------------------

+
-- Algebraic bundles

+

+
+-magma : Magma 0ℓ 0ℓ

+
+-magma = record

+
  { isMagma = +-isMagma

+
  }

+

+
+-semigroup : Semigroup 0ℓ 0ℓ

+
+-semigroup = record

+
  { isSemigroup = +-isSemigroup

+
  }

+

+
+-0-monoid : Monoid 0ℓ 0ℓ

+
+-0-monoid = record

+
  { isMonoid = +-0-isMonoid

+
  }

+

+
+-0-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
+-0-commutativeMonoid = record

+
  { isCommutativeMonoid = +-0-isCommutativeMonoid

+
  }

+

+
+-0-group : Group 0ℓ 0ℓ

+
+-0-group = record

+
  { isGroup = +-0-isGroup

+
  }

+

+
+-0-abelianGroup : AbelianGroup 0ℓ 0ℓ

+
+-0-abelianGroup = record

+
  { isAbelianGroup = +-0-isAbelianGroup

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _*_

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
*-cong : Congruent₂ _≃_ _*_

+
*-cong {x@record{}} {y@record{}} {u@record{}} {v@record{}} (*≡* ↥x↧y≡↥y↧x) (*≡* ↥u↧v≡↥v↧u) = *≡* (begin

+
  (↥ x ℤ.* ↥ u) ℤ.* (↧ y ℤ.* ↧ v) ≡⟨ solve 4 (λ ↥x ↥u ↧y ↧v →

+
                                       (↥x :* ↥u) :* (↧y :* ↧v) :=

+
                                       (↥u :* ↧v) :* (↥x :* ↧y))

+
                                       refl (↥ x) (↥ u) (↧ y) (↧ v) ⟩

+
  (↥ u ℤ.* ↧ v) ℤ.* (↥ x ℤ.* ↧ y) ≡⟨ cong₂ ℤ._*_ ↥u↧v≡↥v↧u ↥x↧y≡↥y↧x ⟩

+
  (↥ v ℤ.* ↧ u) ℤ.* (↥ y ℤ.* ↧ x) ≡⟨ solve 4 (λ ↥v ↧u ↥y ↧x →

+
                                       (↥v :* ↧u) :* (↥y :* ↧x) :=

+
                                       (↥y :* ↥v) :* (↧x :* ↧u))

+
                                       refl (↥ v) (↧ u) (↥ y) (↧ x) ⟩

+
  (↥ y ℤ.* ↥ v) ℤ.* (↧ x ℤ.* ↧ u) ∎)

+
  where open ≡-Reasoning; open ℤ-solver

+

+
*-congˡ : LeftCongruent _≃_ _*_

+
*-congˡ {p} q≃r = *-cong (≃-refl {p}) q≃r

+

+
*-congʳ : RightCongruent _≃_ _*_

+
*-congʳ {p} q≃r = *-cong q≃r (≃-refl {p})

+

+
-- Associativity

+

+
*-assoc-↥ : Associative (_≡_ on ↥_) _*_

+
*-assoc-↥ p@record{} q@record{} r@record{} = ℤ.*-assoc (↥ p) (↥ q) (↥ r)

+

+
*-assoc-↧ : Associative (_≡_ on ↧ₙ_) _*_

+
*-assoc-↧ p@record{} q@record{} r@record{} = ℕ.*-assoc (↧ₙ p) (↧ₙ q) (↧ₙ r)

+

+
*-assoc-≡ : Associative _≡_ _*_

+
*-assoc-≡ p q r = ↥↧≡⇒≡ (*-assoc-↥ p q r) (*-assoc-↧ p q r)

+

+
*-assoc : Associative _≃_ _*_

+
*-assoc p q r = ≃-reflexive (*-assoc-≡ p q r)

+

+
-- Commutativity

+

+
*-comm-↥ : Commutative (_≡_ on ↥_) _*_

+
*-comm-↥ p@record{} q@record{} = ℤ.*-comm (↥ p) (↥ q)

+

+
*-comm-↧ : Commutative (_≡_ on ↧ₙ_) _*_

+
*-comm-↧ p@record{} q@record{} = ℕ.*-comm (↧ₙ p) (↧ₙ q)

+

+
*-comm-≡ : Commutative _≡_ _*_

+
*-comm-≡ p q = ↥↧≡⇒≡ (*-comm-↥ p q) (*-comm-↧ p q)

+

+
*-comm : Commutative _≃_ _*_

+
*-comm p q = ≃-reflexive (*-comm-≡ p q)

+

+
-- Identities

+

+
*-identityˡ-≡ : LeftIdentity _≡_ 1ℚᵘ _*_

+
*-identityˡ-≡ p@record{} = ↥↧≡⇒≡ (ℤ.*-identityˡ (↥ p)) (ℕ.+-identityʳ (↧ₙ p))

+

+
*-identityʳ-≡ : RightIdentity _≡_ 1ℚᵘ _*_

+
*-identityʳ-≡ = comm+idˡ⇒idʳ *-comm-≡ {e = 1ℚᵘ} *-identityˡ-≡

+

+
*-identity-≡ : Identity _≡_ 1ℚᵘ _*_

+
*-identity-≡ = *-identityˡ-≡ , *-identityʳ-≡

+

+
*-identityˡ : LeftIdentity _≃_ 1ℚᵘ _*_

+
*-identityˡ p = ≃-reflexive (*-identityˡ-≡ p)

+

+
*-identityʳ : RightIdentity _≃_ 1ℚᵘ _*_

+
*-identityʳ p = ≃-reflexive (*-identityʳ-≡ p)

+

+
*-identity : Identity _≃_ 1ℚᵘ _*_

+
*-identity = *-identityˡ , *-identityʳ

+

+
*-inverseˡ : ∀ p .{{_ : NonZero p}} → (1/ p) * p ≃ 1ℚᵘ

+
*-inverseˡ p@(mkℚᵘ -[1+ n ] d) = *-inverseˡ (mkℚᵘ +[1+ n ] d)

+
*-inverseˡ p@(mkℚᵘ +[1+ n ] d) = *≡* $ cong +[1+_] $ begin

+
  (n ℕ.+ d ℕ.* suc n) ℕ.* 1 ≡⟨ ℕ.*-identityʳ _ ⟩

+
  n ℕ.+ d ℕ.* suc n         ≡⟨ cong (n ℕ.+_) (ℕ.*-suc d n) ⟩

+
  n ℕ.+ (d ℕ.+ d ℕ.* n)     ≡⟨ trans (sym $ ℕ.+-assoc n d _) (trans

+
                                      (cong₂ ℕ._+_ (ℕ.+-comm n d) (ℕ.*-comm d n))

+
                                      (ℕ.+-assoc d n _)) ⟩

+
  d ℕ.+ (n ℕ.+ n ℕ.* d)     ≡⟨ cong (d ℕ.+_) (sym (ℕ.*-suc n d)) ⟩

+
  d ℕ.+ n ℕ.* suc d         ≡⟨ ℕ.+-identityʳ _ ⟨

+
  d ℕ.+ n ℕ.* suc d ℕ.+ 0   ∎

+
  where open ≡-Reasoning

+

+
*-inverseʳ : ∀ p .{{_ : NonZero p}} → p * 1/ p ≃ 1ℚᵘ

+
*-inverseʳ p = ≃-trans (*-comm p (1/ p)) (*-inverseˡ p)

+

+
≄⇒invertible : p ≄ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)

+
≄⇒invertible {p} {q} p≄q = _ , *-inverseˡ (p - q) , *-inverseʳ (p - q)

+
  where instance

+
  _ : NonZero (p - q)

+
  _ = ≢-nonZero (p≄q ∘ p-q≃0⇒p≃q p q)

+

+
*-zeroˡ : LeftZero _≃_ 0ℚᵘ _*_

+
*-zeroˡ p@record{} = *≡* refl

+

+
*-zeroʳ : RightZero _≃_ 0ℚᵘ _*_

+
*-zeroʳ = Consequences.comm+zeˡ⇒zeʳ ≃-setoid *-comm *-zeroˡ

+

+
*-zero : Zero _≃_ 0ℚᵘ _*_

+
*-zero = *-zeroˡ , *-zeroʳ

+

+
invertible⇒≄ : Invertible _≃_ 1ℚᵘ _*_ (p - q) → p ≄ q

+
invertible⇒≄ {p} {q} (1/p-q , 1/x*x≃1 , x*1/x≃1) p≃q = 0≄1 (begin

+
  0ℚᵘ             ≈⟨ *-zeroˡ 1/p-q ⟨

+
  0ℚᵘ * 1/p-q     ≈⟨ *-congʳ (p≃q⇒p-q≃0 p q p≃q) ⟨

+
  (p - q) * 1/p-q ≈⟨ x*1/x≃1 ⟩

+
  1ℚᵘ             ∎)

+
  where open ≃-Reasoning

+

+
*-distribˡ-+ : _DistributesOverˡ_ _≃_ _*_ _+_

+
*-distribˡ-+ p@record{} q@record{} r@record{} =

+
  let ↥p = ↥ p; ↧p = ↧ p

+
      ↥q = ↥ q; ↧q = ↧ q

+
      ↥r = ↥ r; ↧r = ↧ r

+
      eq : (↥p ℤ.* (↥q ℤ.* ↧r ℤ.+ ↥r ℤ.* ↧q)) ℤ.* (↧p ℤ.* ↧q ℤ.* (↧p ℤ.* ↧r)) ≡

+
           (↥p ℤ.* ↥q ℤ.* (↧p ℤ.* ↧r) ℤ.+ ↥p ℤ.* ↥r ℤ.* (↧p ℤ.* ↧q)) ℤ.* (↧p ℤ.* (↧q ℤ.* ↧r))

+
      eq = solve 6 (λ ↥p ↧p ↥q d e f →

+
           (↥p :* (↥q :* f :+ e :* d)) :* (↧p :* d :* (↧p :* f)) :=

+
           (↥p :* ↥q :* (↧p :* f) :+ ↥p :* e :* (↧p :* d)) :* (↧p :* (d :* f)))

+
           refl ↥p ↧p ↥q ↧q ↥r ↧r

+
  in *≡* eq where open ℤ-solver

+

+
*-distribʳ-+ : _DistributesOverʳ_ _≃_ _*_ _+_

+
*-distribʳ-+ = Consequences.comm+distrˡ⇒distrʳ ≃-setoid +-cong *-comm *-distribˡ-+

+

+
*-distrib-+ : _DistributesOver_ _≃_ _*_ _+_

+
*-distrib-+ = *-distribˡ-+ , *-distribʳ-+

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and -_

+

+
neg-distribˡ-* : ∀ p q → - (p * q) ≃ - p * q

+
neg-distribˡ-* p@record{} q@record{} =

+
  *≡* $ cong (ℤ._* (↧ p ℤ.* ↧ q)) $ ℤ.neg-distribˡ-* (↥ p) (↥ q)

+

+
neg-distribʳ-* : ∀ p q → - (p * q) ≃ p * - q

+
neg-distribʳ-* p@record{} q@record{} =

+
  *≡* $ cong (ℤ._* (↧ p ℤ.* ↧ q)) $ ℤ.neg-distribʳ-* (↥ p) (↥ q)

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _/_

+

+
*-cancelˡ-/ : ∀ p {q r} .{{_ : ℕ.NonZero r}} .{{_ : ℕ.NonZero (p ℕ.* r)}} →

+
              ((ℤ.+ p ℤ.* q) / (p ℕ.* r)) ≃ (q / r)

+
*-cancelˡ-/ p {q} {r} = *≡* (begin-equality

+
  (↥ ((ℤ.+ p ℤ.* q) / (p ℕ.* r))) ℤ.* (↧ (q / r)) ≡⟨  cong (ℤ._* ↧ (q / r)) (↥[n/d]≡n (ℤ.+ p ℤ.* q) (p ℕ.* r)) ⟩

+
  (ℤ.+ p ℤ.* q) ℤ.* (↧ (q / r))                   ≡⟨  cong ((ℤ.+ p ℤ.* q) ℤ.*_) (↧[n/d]≡d q r) ⟩

+
  (ℤ.+ p ℤ.* q) ℤ.* ℤ.+ r                         ≡⟨  xy∙z≈y∙xz (ℤ.+ p) q (ℤ.+ r) ⟩

+
  (q ℤ.* (ℤ.+ p ℤ.* ℤ.+ r))                       ≡⟨ cong (ℤ._* (ℤ.+ p ℤ.* ℤ.+ r)) (↥[n/d]≡n q r) ⟨

+
  (↥ (q / r)) ℤ.* (ℤ.+ p ℤ.* ℤ.+ r)               ≡⟨  cong (↥ (q / r) ℤ.*_) (ℤ.pos-* p r) ⟨

+
  (↥ (q / r)) ℤ.* (ℤ.+ (p ℕ.* r))                 ≡⟨ cong (↥ (q / r) ℤ.*_) (↧[n/d]≡d (ℤ.+ p ℤ.* q) (p ℕ.* r)) ⟨

+
  (↥ (q / r)) ℤ.* (↧ ((ℤ.+ p ℤ.* q) / (p ℕ.* r))) ∎)

+
  where open ℤ.≤-Reasoning

+

+
*-cancelʳ-/ : ∀ p {q r} .{{_ : ℕ.NonZero r}} .{{_ : ℕ.NonZero (r ℕ.* p)}} →

+
              ((q ℤ.* ℤ.+ p) / (r ℕ.* p)) ≃ (q / r)

+
*-cancelʳ-/ p {q} {r} rewrite ℕ.*-comm r p | ℤ.*-comm q (ℤ.+ p) = *-cancelˡ-/ p

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _≤_

+

+
private

+
  reorder₁ : ∀ a b c d → a ℤ.* b ℤ.* (c ℤ.* d) ≡ a ℤ.* c ℤ.* b ℤ.* d

+
  reorder₁ = solve 4 (λ a b c d → (a :* b) :* (c :* d) := (a :* c) :* b :* d) refl

+
    where open ℤ-solver

+

+
  reorder₂ : ∀ a b c d → a ℤ.* b ℤ.* (c ℤ.* d) ≡ a ℤ.* c ℤ.* (b ℤ.* d)

+
  reorder₂ = solve 4 (λ a b c d → (a :* b) :* (c :* d) := (a :* c) :* (b :* d)) refl

+
    where open ℤ-solver

+

+
  +▹-nonNeg : ∀ n → ℤ.NonNegative (Sign.+ ℤ.◃ n)

+
  +▹-nonNeg 0       = _

+
  +▹-nonNeg (suc _) = _

+

+
*-cancelʳ-≤-pos : ∀ r .{{_ : Positive r}} → p * r ≤ q * r → p ≤ q

+
*-cancelʳ-≤-pos {p@record{}} {q@record{}} r@(mkℚᵘ +[1+ _ ] _) (*≤* x≤y) =

+
 *≤* $ ℤ.*-cancelʳ-≤-pos _ _ (↥ r ℤ.* ↧ r) $ begin

+
    (↥ p ℤ.* ↧ q) ℤ.* (↥ r ℤ.* ↧ r)  ≡⟨ reorder₂ (↥ p) _ _ (↧ r) ⟩

+
    (↥ p ℤ.* ↥ r) ℤ.* (↧ q ℤ.* ↧ r)  ≤⟨ x≤y ⟩

+
    (↥ q ℤ.* ↥ r) ℤ.* (↧ p ℤ.* ↧ r)  ≡⟨ reorder₂ (↥ q) _ _ (↧ r) ⟩

+
    (↥ q ℤ.* ↧ p) ℤ.* (↥ r ℤ.* ↧ r)  ∎ where open ℤ.≤-Reasoning

+

+
*-cancelˡ-≤-pos : ∀ r .{{_ : Positive r}} → r * p ≤ r * q → p ≤ q

+
*-cancelˡ-≤-pos {p} {q} r rewrite *-comm-≡ r p | *-comm-≡ r q = *-cancelʳ-≤-pos r

+

+
*-cancelʳ-≤-neg : ∀ r .{{_ : Negative r}} → p * r ≤ q * r → q ≤ p

+
*-cancelʳ-≤-neg {p} {q} r@(mkℚᵘ -[1+ _ ] _) pr≤qr = neg-cancel-≤ (*-cancelʳ-≤-pos (- r) (begin

+
  - p * - r    ≃⟨ neg-distribˡ-* p (- r) ⟨

+
  - (p * - r)  ≃⟨ -‿cong (neg-distribʳ-* p r) ⟨

+
  - - (p * r)  ≃⟨ neg-involutive (p * r) ⟩

+
  p * r        ≤⟨ pr≤qr ⟩

+
  q * r        ≃⟨ neg-involutive (q * r) ⟨

+
  - - (q * r)  ≃⟨ -‿cong (neg-distribʳ-* q r) ⟩

+
  - (q * - r)  ≃⟨ neg-distribˡ-* q (- r) ⟩

+
  - q * - r    ∎))

+
  where open ≤-Reasoning

+

+
*-cancelˡ-≤-neg : ∀ r .{{_ : Negative r}} → r * p ≤ r * q → q ≤ p

+
*-cancelˡ-≤-neg {p} {q} r rewrite *-comm-≡ r p | *-comm-≡ r q = *-cancelʳ-≤-neg r

+

+
*-monoˡ-≤-nonNeg : ∀ r .{{_ : NonNegative r}} → (_* r) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤-nonNeg r@(mkℚᵘ (ℤ.+ n) _) {p@record{}} {q@record{}} (*≤* x<y) = *≤* $ begin

+
  ↥ p ℤ.* ↥ r ℤ.* (↧ q   ℤ.* ↧ r)  ≡⟨  reorder₂ (↥ p) _ _ _ ⟩

+
  l₁          ℤ.* (ℤ.+ n ℤ.* ↧ r)  ≡⟨  cong (l₁ ℤ.*_) (ℤ.pos-* n _) ⟨

+
  l₁          ℤ.* ℤ.+ (n ℕ.* ↧ₙ r) ≤⟨  ℤ.*-monoʳ-≤-nonNeg (ℤ.+ (n ℕ.* ↧ₙ r)) x<y ⟩

+
  l₂          ℤ.* ℤ.+ (n ℕ.* ↧ₙ r) ≡⟨ cong (l₂ ℤ.*_) (ℤ.pos-* n _) ⟩

+
  l₂          ℤ.* (ℤ.+ n ℤ.* ↧ r)  ≡⟨  reorder₂ (↥ q) _ _ _ ⟩

+
  ↥ q ℤ.* ↥ r ℤ.* (↧ p   ℤ.* ↧ r)  ∎

+
  where open ℤ.≤-Reasoning; l₁ = ↥ p ℤ.* ↧ q ; l₂ = ↥ q ℤ.* ↧ p

+

+
*-monoʳ-≤-nonNeg : ∀ r .{{_ :  NonNegative r}} → (r *_) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤-nonNeg r {p} {q} rewrite *-comm-≡ r p | *-comm-≡ r q = *-monoˡ-≤-nonNeg r

+

+
*-mono-≤-nonNeg : ∀ {p q r s} .{{_ : NonNegative p}} .{{_ : NonNegative r}} →

+
                  p ≤ q → r ≤ s → p * r ≤ q * s

+
*-mono-≤-nonNeg {p} {q} {r} {s} p≤q r≤s = begin

+
  p * r ≤⟨ *-monoˡ-≤-nonNeg r p≤q ⟩

+
  q * r ≤⟨ *-monoʳ-≤-nonNeg q {{nonNeg≤⇒nonNeg p≤q}} r≤s ⟩

+
  q * s ∎

+
  where open ≤-Reasoning

+

+
*-monoˡ-≤-nonPos : ∀ r .{{_ : NonPositive r}} → (_* r) Preserves _≤_ ⟶ _≥_

+
*-monoˡ-≤-nonPos r {p} {q} p≤q = begin

+
  q * r        ≃⟨ neg-involutive (q * r) ⟨

+
  - - (q * r)  ≃⟨  -‿cong (neg-distribʳ-* q r) ⟩

+
  - (q * - r)  ≤⟨  neg-mono-≤ (*-monoˡ-≤-nonNeg (- r) {{ -r≥0}} p≤q) ⟩

+
  - (p * - r)  ≃⟨ -‿cong (neg-distribʳ-* p r) ⟨

+
  - - (p * r)  ≃⟨  neg-involutive (p * r) ⟩

+
  p * r        ∎

+
  where open ≤-Reasoning; -r≥0 = nonNegative (neg-mono-≤ (nonPositive⁻¹ r))

+

+
*-monoʳ-≤-nonPos : ∀ r .{{_ :  NonPositive r}} → (r *_) Preserves _≤_ ⟶ _≥_

+
*-monoʳ-≤-nonPos r {p} {q} rewrite *-comm-≡ r q | *-comm-≡ r p = *-monoˡ-≤-nonPos r

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and _<_

+

+
*-monoˡ-<-pos : ∀ r .{{_ : Positive r}} → (_* r) Preserves _<_ ⟶ _<_

+
*-monoˡ-<-pos r@record{} {p@record{}} {q@record{}} (*<* x<y) = *<* $ begin-strict

+
  ↥ p ℤ.*  ↥ r ℤ.* (↧ q  ℤ.* ↧ r) ≡⟨ reorder₁ (↥ p) _ _ _ ⟩

+
  ↥ p ℤ.*  ↧ q ℤ.*  ↥ r  ℤ.* ↧ r  <⟨ ℤ.*-monoʳ-<-pos (↧ r) (ℤ.*-monoʳ-<-pos (↥ r) x<y) ⟩

+
  ↥ q ℤ.*  ↧ p ℤ.*  ↥ r  ℤ.* ↧ r  ≡⟨ reorder₁ (↥ q) _ _ _ ⟨

+
  ↥ q ℤ.*  ↥ r ℤ.* (↧ p  ℤ.* ↧ r) ∎ where open ℤ.≤-Reasoning

+

+
*-monoʳ-<-pos : ∀ r .{{_ : Positive r}} → (r *_) Preserves _<_ ⟶ _<_

+
*-monoʳ-<-pos r {p} {q} rewrite *-comm-≡ r p | *-comm-≡ r q = *-monoˡ-<-pos r

+

+
*-mono-<-nonNeg : ∀ {p q r s} .{{_ : NonNegative p}} .{{_ : NonNegative r}} →

+
                  p < q → r < s → p * r < q * s

+
*-mono-<-nonNeg {p} {q} {r} {s} p<q r<s = begin-strict

+
  p * r ≤⟨ *-monoˡ-≤-nonNeg r (<⇒≤ p<q) ⟩

+
  q * r <⟨ *-monoʳ-<-pos q {{nonNeg<⇒pos p<q}} r<s ⟩

+
  q * s ∎

+
  where open ≤-Reasoning

+

+
*-cancelʳ-<-nonNeg : ∀ r .{{_ : NonNegative r}} → p * r < q * r → p < q

+
*-cancelʳ-<-nonNeg {p@record{}} {q@record{}} r@(mkℚᵘ (ℤ.+ _) _) (*<* x<y) =

+
  *<* $ ℤ.*-cancelʳ-<-nonNeg (↥ r ℤ.* ↧ r) {{+▹-nonNeg _}} $ begin-strict

+
    (↥ p ℤ.* ↧ q) ℤ.* (↥ r ℤ.* ↧ r)  ≡⟨ reorder₂ (↥ p) _ _ (↧ r) ⟩

+
    (↥ p ℤ.* ↥ r) ℤ.* (↧ q ℤ.* ↧ r)  <⟨ x<y ⟩

+
    (↥ q ℤ.* ↥ r) ℤ.* (↧ p ℤ.* ↧ r)  ≡⟨ reorder₂ (↥ q) _ _ (↧ r) ⟩

+
    (↥ q ℤ.* ↧ p) ℤ.* (↥ r ℤ.* ↧ r)  ∎ where open ℤ.≤-Reasoning

+

+
*-cancelˡ-<-nonNeg : ∀ r .{{_ : NonNegative r}} → r * p < r * q → p < q

+
*-cancelˡ-<-nonNeg {p} {q} r rewrite *-comm-≡ r p | *-comm-≡ r q = *-cancelʳ-<-nonNeg r

+

+
*-monoˡ-<-neg : ∀ r .{{_ :  Negative r}} → (_* r) Preserves _<_ ⟶ _>_

+
*-monoˡ-<-neg r {p} {q} p<q = begin-strict

+
  q * r        ≃⟨ neg-involutive (q * r) ⟨

+
  - - (q * r)  ≃⟨ -‿cong (neg-distribʳ-* q r) ⟩

+
  - (q * - r)  <⟨ neg-mono-< (*-monoˡ-<-pos (- r) {{ -r>0}} p<q) ⟩

+
  - (p * - r)  ≃⟨ -‿cong (neg-distribʳ-* p r) ⟨

+
  - - (p * r)  ≃⟨ neg-involutive (p * r) ⟩

+
  p * r        ∎

+
  where open ≤-Reasoning; -r>0 = positive (neg-mono-< (negative⁻¹ r))

+

+
*-monoʳ-<-neg : ∀ r .{{_ : Negative r}} → (r *_) Preserves _<_ ⟶ _>_

+
*-monoʳ-<-neg r {p} {q} rewrite *-comm-≡ r q | *-comm-≡ r p = *-monoˡ-<-neg r

+

+
*-cancelˡ-<-nonPos : ∀ r .{{_ : NonPositive r}} → r * p < r * q → q < p

+
*-cancelˡ-<-nonPos {p} {q} r rp<rq =

+
  *-cancelˡ-<-nonNeg (- r) {{ -r≥0}} $ begin-strict

+
    - r * q    ≃⟨ neg-distribˡ-* r q ⟨

+
    - (r * q)  <⟨ neg-mono-< rp<rq ⟩

+
    - (r * p)  ≃⟨ neg-distribˡ-* r p ⟩

+
    - r * p    ∎

+
  where open ≤-Reasoning; -r≥0 = nonNegative (neg-mono-≤ (nonPositive⁻¹ r))

+

+
*-cancelʳ-<-nonPos : ∀ r .{{_ : NonPositive r}} → p * r < q * r → q < p

+
*-cancelʳ-<-nonPos {p} {q} r rewrite *-comm-≡ p r | *-comm-≡ q r = *-cancelˡ-<-nonPos r

+

+
------------------------------------------------------------------------

+
-- Properties of _*_ and predicates

+

+
pos*pos⇒pos : ∀ p .{{_ : Positive p}} →

+
              ∀ q .{{_ : Positive q}} →

+
              Positive (p * q)

+
pos*pos⇒pos p q = positive

+
  (*-mono-<-nonNeg (positive⁻¹ p) (positive⁻¹ q))

+

+
nonNeg*nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} →

+
                       ∀ q .{{_ : NonNegative q}} →

+
                       NonNegative (p * q)

+
nonNeg*nonNeg⇒nonNeg p q = nonNegative

+
  (*-mono-≤-nonNeg (nonNegative⁻¹ p) (nonNegative⁻¹ q))

+

+
------------------------------------------------------------------------

+
-- Algebraic structures

+

+
*-isMagma : IsMagma _≃_ _*_

+
*-isMagma = record

+
  { isEquivalence = ≃-isEquivalence

+
  ; ∙-cong        = *-cong

+
  }

+

+
*-isSemigroup : IsSemigroup _≃_ _*_

+
*-isSemigroup = record

+
  { isMagma = *-isMagma

+
  ; assoc   = *-assoc

+
  }

+

+
*-1-isMonoid : IsMonoid _≃_ _*_ 1ℚᵘ

+
*-1-isMonoid = record

+
  { isSemigroup = *-isSemigroup

+
  ; identity    = *-identity

+
  }

+

+
*-1-isCommutativeMonoid : IsCommutativeMonoid _≃_ _*_ 1ℚᵘ

+
*-1-isCommutativeMonoid = record

+
  { isMonoid = *-1-isMonoid

+
  ; comm     = *-comm

+
  }

+

+
+-*-isRing : IsRing _≃_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ

+
+-*-isRing = record

+
  { +-isAbelianGroup = +-0-isAbelianGroup

+
  ; *-cong           = *-cong

+
  ; *-assoc          = *-assoc

+
  ; *-identity       = *-identity

+
  ; distrib          = *-distrib-+

+
  }

+

+
+-*-isCommutativeRing : IsCommutativeRing _≃_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ

+
+-*-isCommutativeRing = record

+
  { isRing = +-*-isRing

+
  ; *-comm = *-comm

+
  }

+

+
+-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ

+
+-*-isHeytingCommutativeRing = record

+
  { isCommutativeRing   = +-*-isCommutativeRing

+
  ; isApartnessRelation = ≄-isApartnessRelation

+
  ; #⇒invertible        = ≄⇒invertible

+
  ; invertible⇒#        = invertible⇒≄

+
  }

+

+
+-*-isHeytingField : IsHeytingField _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ

+
+-*-isHeytingField = record

+
  { isHeytingCommutativeRing = +-*-isHeytingCommutativeRing

+
  ; tight                    = ≄-tight

+
  }

+

+
------------------------------------------------------------------------

+
-- Algebraic bundles

+

+
*-magma : Magma 0ℓ 0ℓ

+
*-magma = record

+
  { isMagma = *-isMagma

+
  }

+

+
*-semigroup : Semigroup 0ℓ 0ℓ

+
*-semigroup = record

+
  { isSemigroup = *-isSemigroup

+
  }

+

+
*-1-monoid : Monoid 0ℓ 0ℓ

+
*-1-monoid = record

+
  { isMonoid = *-1-isMonoid

+
  }

+

+
*-1-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
*-1-commutativeMonoid = record

+
  { isCommutativeMonoid = *-1-isCommutativeMonoid

+
  }

+

+
+-*-ring : Ring 0ℓ 0ℓ

+
+-*-ring = record

+
  { isRing = +-*-isRing

+
  }

+

+
+-*-commutativeRing : CommutativeRing 0ℓ 0ℓ

+
+-*-commutativeRing = record

+
  { isCommutativeRing = +-*-isCommutativeRing

+
  }

+

+
+-*-heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ

+
+-*-heytingCommutativeRing = record

+
  { isHeytingCommutativeRing = +-*-isHeytingCommutativeRing

+
  }

+

+
+-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ

+
+-*-heytingField = record

+
  { isHeytingField = +-*-isHeytingField

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of 1/_

+
------------------------------------------------------------------------

+

+
private

+
  p>1⇒p≢0 : p > 1ℚᵘ → NonZero p

+
  p>1⇒p≢0 {p} p>1 = pos⇒nonZero p {{positive (<-trans (*<* (ℤ.+<+ ℕ.≤-refl)) p>1)}}

+

+
1/nonZero⇒nonZero : ∀ p .{{_ : NonZero p}} → NonZero (1/ p)

+
1/nonZero⇒nonZero (mkℚᵘ (+[1+ _ ]) _) = _

+
1/nonZero⇒nonZero (mkℚᵘ (-[1+ _ ]) _) = _

+

+
1/-involutive-≡ : ∀ p .{{_ : NonZero p}} →

+
                  (1/ (1/ p)) {{1/nonZero⇒nonZero p}} ≡ p

+
1/-involutive-≡ (mkℚᵘ +[1+ n ] d-1) = refl

+
1/-involutive-≡ (mkℚᵘ -[1+ n ] d-1) = refl

+

+
1/-involutive : ∀ p .{{_ : NonZero p}} →

+
                (1/ (1/ p)) {{1/nonZero⇒nonZero p}} ≃ p

+
1/-involutive p = ≃-reflexive (1/-involutive-≡ p)

+

+
1/pos⇒pos : ∀ p .{{p>0 : Positive p}} → Positive ((1/ p) {{pos⇒nonZero p}})

+
1/pos⇒pos (mkℚᵘ +[1+ n ] d-1) = _

+

+
1/neg⇒neg : ∀ p .{{p<0 : Negative p}} → Negative ((1/ p) {{neg⇒nonZero p}})

+
1/neg⇒neg (mkℚᵘ -[1+ n ] d-1) = _

+

+
p>1⇒1/p<1 : ∀ {p} → (p>1 : p > 1ℚᵘ) → (1/ p) {{p>1⇒p≢0 p>1}} < 1ℚᵘ

+
p>1⇒1/p<1 {p} p>1 = lemma′ p (p>1⇒p≢0 p>1) p>1

+
  where

+
  lemma′ : ∀ p p≢0 → p > 1ℚᵘ → (1/ p) {{p≢0}} < 1ℚᵘ

+
  lemma′ (mkℚᵘ n@(+[1+ _ ]) d-1) _ (*<* ↥p1>1↧p) = *<* (begin-strict

+
    ↥ (1/ mkℚᵘ n d-1) ℤ.* 1ℤ         ≡⟨⟩

+
    +[1+ d-1 ] ℤ.* 1ℤ                ≡⟨ ℤ.*-comm +[1+ d-1 ] 1ℤ ⟩

+
    1ℤ ℤ.* +[1+ d-1 ]                <⟨ ↥p1>1↧p ⟩

+
    n  ℤ.* 1ℤ                        ≡⟨ ℤ.*-comm n 1ℤ ⟩

+
    1ℤ ℤ.* n                         ≡⟨⟩

+
    (↥ 1ℚᵘ) ℤ.* (↧ (1/ mkℚᵘ n d-1))  ∎)

+
    where open ℤ.≤-Reasoning

+

+
1/-antimono-≤-pos : ∀ {p q} .{{_ : Positive p}} .{{_ : Positive q}} →

+
                    p ≤ q → (1/ q) {{pos⇒nonZero q}} ≤ (1/ p) {{pos⇒nonZero p}}

+
1/-antimono-≤-pos {p} {q} p≤q = begin

+
  1/q              ≃⟨ *-identityˡ 1/q ⟨

+
  1ℚᵘ * 1/q        ≃⟨ *-congʳ (*-inverseˡ p) ⟨

+
  (1/p * p) * 1/q  ≤⟨  *-monoˡ-≤-nonNeg 1/q (*-monoʳ-≤-nonNeg 1/p p≤q) ⟩

+
  (1/p * q) * 1/q  ≃⟨  *-assoc 1/p q 1/q ⟩

+
  1/p * (q * 1/q)  ≃⟨  *-congˡ {1/p} (*-inverseʳ q) ⟩

+
  1/p * 1ℚᵘ        ≃⟨  *-identityʳ (1/p) ⟩

+
  1/p              ∎

+
  where

+
  open ≤-Reasoning

+
  instance

+
    _ = pos⇒nonZero p

+
    _ = pos⇒nonZero q

+
  1/p = 1/ p

+
  1/q = 1/ q

+
  instance

+
    1/p≥0 : NonNegative 1/p

+
    1/p≥0 = pos⇒nonNeg 1/p {{1/pos⇒pos p}}

+

+
    1/q≥0 : NonNegative 1/q

+
    1/q≥0 = pos⇒nonNeg 1/q {{1/pos⇒pos q}}

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_ and _⊔_

+
------------------------------------------------------------------------

+
-- Basic specification in terms of _≤_

+

+
p≤q⇒p⊔q≃q : p ≤ q → p ⊔ q ≃ q

+
p≤q⇒p⊔q≃q {p@record{}} {q@record{}} p≤q with p ≤ᵇ q | inspect (p ≤ᵇ_) q

+
... | true  | _       = ≃-refl

+
... | false | [ p≰q ] = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())

+

+
p≥q⇒p⊔q≃p : p ≥ q → p ⊔ q ≃ p

+
p≥q⇒p⊔q≃p {p@record{}} {q@record{}} p≥q with p ≤ᵇ q | inspect (p ≤ᵇ_) q

+
... | true  | [ p≤q ] = ≤-antisym p≥q (≤ᵇ⇒≤ (subst T (sym p≤q) _))

+
... | false | [ p≤q ] = ≃-refl

+

+
p≤q⇒p⊓q≃p : p ≤ q → p ⊓ q ≃ p

+
p≤q⇒p⊓q≃p {p@record{}} {q@record{}} p≤q with p ≤ᵇ q | inspect (p ≤ᵇ_) q

+
... | true  | _       = ≃-refl

+
... | false | [ p≰q ] = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())

+

+
p≥q⇒p⊓q≃q : p ≥ q → p ⊓ q ≃ q

+
p≥q⇒p⊓q≃q {p@record{}} {q@record{}} p≥q with p ≤ᵇ q | inspect (p ≤ᵇ_) q

+
... | true  | [ p≤q ] = ≤-antisym (≤ᵇ⇒≤ (subst T (sym p≤q) _)) p≥q

+
... | false | [ p≤q ] = ≃-refl

+

+
⊓-operator : MinOperator ≤-totalPreorder

+
⊓-operator = record

+
  { x≤y⇒x⊓y≈x = p≤q⇒p⊓q≃p

+
  ; x≥y⇒x⊓y≈y = p≥q⇒p⊓q≃q

+
  }

+

+
⊔-operator : MaxOperator ≤-totalPreorder

+
⊔-operator = record

+
  { x≤y⇒x⊔y≈y = p≤q⇒p⊔q≃q

+
  ; x≥y⇒x⊔y≈x = p≥q⇒p⊔q≃p

+
  }

+

+
------------------------------------------------------------------------

+
-- Derived properties of _⊓_ and _⊔_

+

+
private

+
  module ⊓-⊔-properties        = MinMaxOp        ⊓-operator ⊔-operator

+
  module ⊓-⊔-latticeProperties = LatticeMinMaxOp ⊓-operator ⊔-operator

+

+
open ⊓-⊔-properties public

+
  using

+
  ( ⊓-congˡ                   -- : LeftCongruent _≃_ _⊓_

+
  ; ⊓-congʳ                   -- : RightCongruent _≃_ _⊓_

+
  ; ⊓-cong                    -- : Congruent₂ _≃_ _⊓_

+
  ; ⊓-idem                    -- : Idempotent _≃_ _⊓_

+
  ; ⊓-sel                     -- : Selective _≃_ _⊓_

+
  ; ⊓-assoc                   -- : Associative _≃_ _⊓_

+
  ; ⊓-comm                    -- : Commutative _≃_ _⊓_

+

+
  ; ⊔-congˡ                   -- : LeftCongruent _≃_ _⊔_

+
  ; ⊔-congʳ                   -- : RightCongruent _≃_ _⊔_

+
  ; ⊔-cong                    -- : Congruent₂ _≃_ _⊔_

+
  ; ⊔-idem                    -- : Idempotent _≃_ _⊔_

+
  ; ⊔-sel                     -- : Selective _≃_ _⊔_

+
  ; ⊔-assoc                   -- : Associative _≃_ _⊔_

+
  ; ⊔-comm                    -- : Commutative _≃_ _⊔_

+

+
  ; ⊓-distribˡ-⊔              -- : _DistributesOverˡ_ _≃_ _⊓_ _⊔_

+
  ; ⊓-distribʳ-⊔              -- : _DistributesOverʳ_ _≃_ _⊓_ _⊔_

+
  ; ⊓-distrib-⊔               -- : _DistributesOver_  _≃_ _⊓_ _⊔_

+
  ; ⊔-distribˡ-⊓              -- : _DistributesOverˡ_ _≃_ _⊔_ _⊓_

+
  ; ⊔-distribʳ-⊓              -- : _DistributesOverʳ_ _≃_ _⊔_ _⊓_

+
  ; ⊔-distrib-⊓               -- : _DistributesOver_  _≃_ _⊔_ _⊓_

+
  ; ⊓-absorbs-⊔               -- : _Absorbs_ _≃_ _⊓_ _⊔_

+
  ; ⊔-absorbs-⊓               -- : _Absorbs_ _≃_ _⊔_ _⊓_

+
  ; ⊔-⊓-absorptive            -- : Absorptive _≃_ _⊔_ _⊓_

+
  ; ⊓-⊔-absorptive            -- : Absorptive _≃_ _⊓_ _⊔_

+

+
  ; ⊓-isMagma                 -- : IsMagma _≃_ _⊓_

+
  ; ⊓-isSemigroup             -- : IsSemigroup _≃_ _⊓_

+
  ; ⊓-isCommutativeSemigroup  -- : IsCommutativeSemigroup _≃_ _⊓_

+
  ; ⊓-isBand                  -- : IsBand _≃_ _⊓_

+
  ; ⊓-isSelectiveMagma        -- : IsSelectiveMagma _≃_ _⊓_

+

+
  ; ⊔-isMagma                 -- : IsMagma _≃_ _⊔_

+
  ; ⊔-isSemigroup             -- : IsSemigroup _≃_ _⊔_

+
  ; ⊔-isCommutativeSemigroup  -- : IsCommutativeSemigroup _≃_ _⊔_

+
  ; ⊔-isBand                  -- : IsBand _≃_ _⊔_

+
  ; ⊔-isSelectiveMagma        -- : IsSelectiveMagma _≃_ _⊔_

+

+
  ; ⊓-magma                   -- : Magma _ _

+
  ; ⊓-semigroup               -- : Semigroup _ _

+
  ; ⊓-band                    -- : Band _ _

+
  ; ⊓-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊓-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊔-magma                   -- : Magma _ _

+
  ; ⊔-semigroup               -- : Semigroup _ _

+
  ; ⊔-band                    -- : Band _ _

+
  ; ⊔-commutativeSemigroup    -- : CommutativeSemigroup _ _

+
  ; ⊔-selectiveMagma          -- : SelectiveMagma _ _

+

+
  ; ⊓-triangulate             -- : ∀ p q r → p ⊓ q ⊓ r ≃ (p ⊓ q) ⊓ (q ⊓ r)

+
  ; ⊔-triangulate             -- : ∀ p q r → p ⊔ q ⊔ r ≃ (p ⊔ q) ⊔ (q ⊔ r)

+

+
  ; ⊓-glb                     -- : ∀ {p q r} → p ≥ r → q ≥ r → p ⊓ q ≥ r

+
  ; ⊓-mono-≤                  -- : _⊓_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊓-monoˡ-≤                 -- : ∀ p → (_⊓ p) Preserves _≤_ ⟶ _≤_

+
  ; ⊓-monoʳ-≤                 -- : ∀ p → (p ⊓_) Preserves _≤_ ⟶ _≤_

+

+
  ; ⊔-lub                     -- : ∀ {p q r} → p ≤ r → q ≤ r → p ⊔ q ≤ r

+
  ; ⊔-mono-≤                  -- : _⊔_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
  ; ⊔-monoˡ-≤                 -- : ∀ p → (_⊔ p) Preserves _≤_ ⟶ _≤_

+
  ; ⊔-monoʳ-≤                 -- : ∀ p → (p ⊔_) Preserves _≤_ ⟶ _≤_

+
  )

+
  renaming

+
  ( x⊓y≈y⇒y≤x  to p⊓q≃q⇒q≤p      -- : ∀ {p q} → p ⊓ q ≃ q → q ≤ p

+
  ; x⊓y≈x⇒x≤y  to p⊓q≃p⇒p≤q      -- : ∀ {p q} → p ⊓ q ≃ p → p ≤ q

+
  ; x⊔y≈y⇒x≤y  to p⊔q≃q⇒p≤q      -- : ∀ {p q} → p ⊔ q ≃ q → p ≤ q

+
  ; x⊔y≈x⇒y≤x  to p⊔q≃p⇒q≤p      -- : ∀ {p q} → p ⊔ q ≃ p → q ≤ p

+

+
  ; x⊓y≤x      to p⊓q≤p          -- : ∀ p q → p ⊓ q ≤ p

+
  ; x⊓y≤y      to p⊓q≤q          -- : ∀ p q → p ⊓ q ≤ q

+
  ; x≤y⇒x⊓z≤y  to p≤q⇒p⊓r≤q      -- : ∀ {p q} r → p ≤ q → p ⊓ r ≤ q

+
  ; x≤y⇒z⊓x≤y  to p≤q⇒r⊓p≤q      -- : ∀ {p q} r → p ≤ q → r ⊓ p ≤ q

+
  ; x≤y⊓z⇒x≤y  to p≤q⊓r⇒p≤q      -- : ∀ {p} q r → p ≤ q ⊓ r → p ≤ q

+
  ; x≤y⊓z⇒x≤z  to p≤q⊓r⇒p≤r      -- : ∀ {p} q r → p ≤ q ⊓ r → p ≤ r

+

+
  ; x≤x⊔y      to p≤p⊔q          -- : ∀ p q → p ≤ p ⊔ q

+
  ; x≤y⊔x      to p≤q⊔p          -- : ∀ p q → p ≤ q ⊔ p

+
  ; x≤y⇒x≤y⊔z  to p≤q⇒p≤q⊔r      -- : ∀ {p q} r → p ≤ q → p ≤ q ⊔ r

+
  ; x≤y⇒x≤z⊔y  to p≤q⇒p≤r⊔q      -- : ∀ {p q} r → p ≤ q → p ≤ r ⊔ q

+
  ; x⊔y≤z⇒x≤z  to p⊔q≤r⇒p≤r      -- : ∀ p q {r} → p ⊔ q ≤ r → p ≤ r

+
  ; x⊔y≤z⇒y≤z  to p⊔q≤r⇒q≤r      -- : ∀ p q {r} → p ⊔ q ≤ r → q ≤ r

+

+
  ; x⊓y≤x⊔y    to p⊓q≤p⊔q        -- : ∀ p q → p ⊓ q ≤ p ⊔ q

+
  )

+

+
open ⊓-⊔-latticeProperties public

+
  using

+
  ( ⊓-semilattice             -- : Semilattice _ _

+
  ; ⊔-semilattice             -- : Semilattice _ _

+
  ; ⊔-⊓-lattice               -- : Lattice _ _

+
  ; ⊓-⊔-lattice               -- : Lattice _ _

+
  ; ⊔-⊓-distributiveLattice   -- : DistributiveLattice _ _

+
  ; ⊓-⊔-distributiveLattice   -- : DistributiveLattice _ _

+

+
  ; ⊓-isSemilattice           -- : IsSemilattice _≃_ _⊓_

+
  ; ⊔-isSemilattice           -- : IsSemilattice _≃_ _⊔_

+
  ; ⊔-⊓-isLattice             -- : IsLattice _≃_ _⊔_ _⊓_

+
  ; ⊓-⊔-isLattice             -- : IsLattice _≃_ _⊓_ _⊔_

+
  ; ⊔-⊓-isDistributiveLattice -- : IsDistributiveLattice _≃_ _⊔_ _⊓_

+
  ; ⊓-⊔-isDistributiveLattice -- : IsDistributiveLattice _≃_ _⊓_ _⊔_

+
  )

+

+
------------------------------------------------------------------------

+
-- Raw bundles

+

+
⊓-rawMagma : RawMagma _ _

+
⊓-rawMagma = Magma.rawMagma ⊓-magma

+

+
⊔-rawMagma : RawMagma _ _

+
⊔-rawMagma = Magma.rawMagma ⊔-magma

+

+
⊔-⊓-rawLattice : RawLattice _ _

+
⊔-⊓-rawLattice = Lattice.rawLattice ⊔-⊓-lattice

+

+
------------------------------------------------------------------------

+
-- Monotonic or antimonotic functions distribute over _⊓_ and _⊔_

+

+
mono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ m n → f (m ⊔ n) ≃ f m ⊔ f n

+
mono-≤-distrib-⊔ pres = ⊓-⊔-properties.mono-≤-distrib-⊔ (mono⇒cong pres) pres

+

+
mono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≤_ →

+
                   ∀ m n → f (m ⊓ n) ≃ f m ⊓ f n

+
mono-≤-distrib-⊓ pres = ⊓-⊔-properties.mono-≤-distrib-⊓ (mono⇒cong pres) pres

+

+
antimono-≤-distrib-⊓ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ m n → f (m ⊓ n) ≃ f m ⊔ f n

+
antimono-≤-distrib-⊓ pres = ⊓-⊔-properties.antimono-≤-distrib-⊓ (antimono⇒cong pres) pres

+

+
antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →

+
                       ∀ m n → f (m ⊔ n) ≃ f m ⊓ f n

+
antimono-≤-distrib-⊔ pres = ⊓-⊔-properties.antimono-≤-distrib-⊔ (antimono⇒cong pres) pres

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_, _⊔_ and -_

+

+
neg-distrib-⊔-⊓ : ∀ p q → - (p ⊔ q) ≃ - p ⊓ - q

+
neg-distrib-⊔-⊓ = antimono-≤-distrib-⊔ neg-mono-≤

+

+
neg-distrib-⊓-⊔ : ∀ p q → - (p ⊓ q) ≃ - p ⊔ - q

+
neg-distrib-⊓-⊔ = antimono-≤-distrib-⊓ neg-mono-≤

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_, _⊔_ and _*_

+

+
*-distribˡ-⊓-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊓ (p * r)

+
*-distribˡ-⊓-nonNeg p = mono-≤-distrib-⊓ (*-monoʳ-≤-nonNeg p)

+

+
*-distribʳ-⊓-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊓ (r * p)

+
*-distribʳ-⊓-nonNeg p = mono-≤-distrib-⊓ (*-monoˡ-≤-nonNeg p)

+

+
*-distribˡ-⊔-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊔ (p * r)

+
*-distribˡ-⊔-nonNeg p = mono-≤-distrib-⊔ (*-monoʳ-≤-nonNeg p)

+

+
*-distribʳ-⊔-nonNeg : ∀ p .{{_ : NonNegative p}} → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊔ (r * p)

+
*-distribʳ-⊔-nonNeg p = mono-≤-distrib-⊔ (*-monoˡ-≤-nonNeg p)

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_, _⊔_ and _*_

+

+
*-distribˡ-⊔-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊓ (p * r)

+
*-distribˡ-⊔-nonPos p = antimono-≤-distrib-⊔ (*-monoʳ-≤-nonPos p)

+

+
*-distribʳ-⊔-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊓ (r * p)

+
*-distribʳ-⊔-nonPos p = antimono-≤-distrib-⊔ (*-monoˡ-≤-nonPos p)

+

+
*-distribˡ-⊓-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊔ (p * r)

+
*-distribˡ-⊓-nonPos p = antimono-≤-distrib-⊓ (*-monoʳ-≤-nonPos p)

+

+
*-distribʳ-⊓-nonPos : ∀ p .{{_ : NonPositive p}} → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊔ (r * p)

+
*-distribʳ-⊓-nonPos p = antimono-≤-distrib-⊓ (*-monoˡ-≤-nonPos p)

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_, _⊔_ and _<_

+

+
⊓-mono-< : _⊓_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
⊓-mono-< {p} {r} {q} {s} p<r q<s with ⊓-sel r s

+
... | inj₁ r⊓s≃r = <-respʳ-≃ (≃-sym r⊓s≃r) (≤-<-trans (p⊓q≤p p q) p<r)

+
... | inj₂ r⊓s≃s = <-respʳ-≃ (≃-sym r⊓s≃s) (≤-<-trans (p⊓q≤q p q) q<s)

+

+
⊔-mono-< : _⊔_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

+
⊔-mono-< {p} {r} {q} {s} p<r q<s with ⊔-sel p q

+
... | inj₁ p⊔q≃p = <-respˡ-≃ (≃-sym p⊔q≃p) (<-≤-trans p<r (p≤p⊔q r s))

+
... | inj₂ p⊔q≃q = <-respˡ-≃ (≃-sym p⊔q≃q) (<-≤-trans q<s (p≤q⊔p r s))

+

+
------------------------------------------------------------------------

+
-- Properties of _⊓_, _⊔_ and predicates

+

+
pos⊓pos⇒pos : ∀ p .{{_ : Positive p}} →

+
              ∀ q .{{_ : Positive q}} →

+
              Positive (p ⊓ q)

+
pos⊓pos⇒pos p q = positive (⊓-mono-< (positive⁻¹ p) (positive⁻¹ q))

+

+
pos⊔pos⇒pos : ∀ p .{{_ : Positive p}} →

+
              ∀ q .{{_ : Positive q}} →

+
              Positive (p ⊔ q)

+
pos⊔pos⇒pos p q = positive (⊔-mono-< (positive⁻¹ p) (positive⁻¹ q))

+

+
------------------------------------------------------------------------

+
-- Properties of ∣_∣

+
------------------------------------------------------------------------

+

+
∣-∣-cong : p ≃ q → ∣ p ∣ ≃ ∣ q ∣

+
∣-∣-cong p@{mkℚᵘ +[1+ _ ] _} q@{mkℚᵘ +[1+ _ ] _} (*≡* ↥p↧q≡↥q↧p) = *≡* ↥p↧q≡↥q↧p

+
∣-∣-cong p@{mkℚᵘ +0       _} q@{mkℚᵘ +0       _} (*≡* ↥p↧q≡↥q↧p) = *≡* ↥p↧q≡↥q↧p

+
∣-∣-cong p@{mkℚᵘ -[1+ _ ] _} q@{mkℚᵘ +0       _} (*≡* ())

+
∣-∣-cong p@{mkℚᵘ -[1+ _ ] _} q@{mkℚᵘ -[1+ _ ] _} (*≡* ↥p↧q≡↥q↧p) = *≡* (begin

+
  ↥ ∣ p ∣ ℤ.* ↧ q            ≡⟨ ℤ.neg-involutive _ ⟩

+
  ℤ.- ℤ.- (↥ ∣ p ∣ ℤ.* ↧ q)  ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ ∣ p ∣) (↧ q)) ⟩

+
  ℤ.- (↥ p ℤ.* ↧ q)          ≡⟨ cong ℤ.-_ ↥p↧q≡↥q↧p ⟩

+
  ℤ.- (↥ q ℤ.* ↧ p)          ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ ∣ q ∣) (↧ p)) ⟩

+
  ℤ.- ℤ.- (↥ ∣ q ∣ ℤ.* ↧ p)  ≡⟨ ℤ.neg-involutive _ ⟨

+
  ↥ ∣ q ∣ ℤ.* ↧ p            ∎)

+
  where open ≡-Reasoning

+

+
∣p∣≃0⇒p≃0 : ∣ p ∣ ≃ 0ℚᵘ → p ≃ 0ℚᵘ

+
∣p∣≃0⇒p≃0 {mkℚᵘ (ℤ.+ n)  d-1} p≃0ℚ = p≃0ℚ

+
∣p∣≃0⇒p≃0 {mkℚᵘ -[1+ n ] d-1} (*≡* ())

+

+
0≤∣p∣ : ∀ p → 0ℚᵘ ≤ ∣ p ∣

+
0≤∣p∣ (mkℚᵘ +0       _) = *≤* (ℤ.+≤+ ℕ.z≤n)

+
0≤∣p∣ (mkℚᵘ +[1+ _ ] _) = *≤* (ℤ.+≤+ ℕ.z≤n)

+
0≤∣p∣ (mkℚᵘ -[1+ _ ] _) = *≤* (ℤ.+≤+ ℕ.z≤n)

+

+
∣-p∣≡∣p∣ : ∀ p → ∣ - p ∣ ≡ ∣ p ∣

+
∣-p∣≡∣p∣ (mkℚᵘ +[1+ n ] d) = refl

+
∣-p∣≡∣p∣ (mkℚᵘ +0       d) = refl

+
∣-p∣≡∣p∣ (mkℚᵘ -[1+ n ] d) = refl

+

+
∣-p∣≃∣p∣ : ∀ p → ∣ - p ∣ ≃ ∣ p ∣

+
∣-p∣≃∣p∣ = ≃-reflexive ∘ ∣-p∣≡∣p∣

+

+
0≤p⇒∣p∣≡p : 0ℚᵘ ≤ p → ∣ p ∣ ≡ p

+
0≤p⇒∣p∣≡p {mkℚᵘ (ℤ.+ n)  d-1} 0≤p = refl

+
0≤p⇒∣p∣≡p {mkℚᵘ -[1+ n ] d-1} 0≤p = contradiction 0≤p (<⇒≱ (*<* ℤ.-<+))

+

+
0≤p⇒∣p∣≃p : 0ℚᵘ ≤ p → ∣ p ∣ ≃ p

+
0≤p⇒∣p∣≃p {p} = ≃-reflexive ∘ 0≤p⇒∣p∣≡p {p}

+

+
∣p∣≡p⇒0≤p : ∣ p ∣ ≡ p → 0ℚᵘ ≤ p

+
∣p∣≡p⇒0≤p {mkℚᵘ (ℤ.+ n) d-1} ∣p∣≡p = *≤* (begin

+
  0ℤ ℤ.* +[1+ d-1 ]  ≡⟨ ℤ.*-zeroˡ (ℤ.+ d-1) ⟩

+
  0ℤ                 ≤⟨ ℤ.+≤+ ℕ.z≤n ⟩

+
  ℤ.+ n              ≡⟨ ℤ.*-identityʳ (ℤ.+ n) ⟨

+
  ℤ.+ n ℤ.* 1ℤ       ∎)

+
  where open ℤ.≤-Reasoning

+

+
∣p∣≡p∨∣p∣≡-p : ∀ p → (∣ p ∣ ≡ p) ⊎ (∣ p ∣ ≡ - p)

+
∣p∣≡p∨∣p∣≡-p (mkℚᵘ (ℤ.+ n)    d-1) = inj₁ refl

+
∣p∣≡p∨∣p∣≡-p (mkℚᵘ (-[1+ n ]) d-1) = inj₂ refl

+

+
∣p∣≃p⇒0≤p : ∣ p ∣ ≃ p → 0ℚᵘ ≤ p

+
∣p∣≃p⇒0≤p {p} ∣p∣≃p with ∣p∣≡p∨∣p∣≡-p p

+
... | inj₁ ∣p∣≡p  = ∣p∣≡p⇒0≤p ∣p∣≡p

+
... | inj₂ ∣p∣≡-p rewrite ∣p∣≡-p = ≤-reflexive (≃-sym (p≃-p⇒p≃0 p (≃-sym ∣p∣≃p)))

+

+
∣p+q∣≤∣p∣+∣q∣ : ∀ p q → ∣ p + q ∣ ≤ ∣ p ∣ + ∣ q ∣

+
∣p+q∣≤∣p∣+∣q∣ p@record{} q@record{} = *≤* (begin

+
  ↥ ∣ p + q ∣ ℤ.* ↧ (∣ p ∣ + ∣ q ∣)                ≡⟨⟩

+
  ↥ ∣ (↥p↧q ℤ.+ ↥q↧p) / ↧p↧q ∣ ℤ.* ℤ.+ ↧p↧q        ≡⟨⟩

+
  ↥ (ℤ.+ ℤ.∣ ↥p↧q ℤ.+ ↥q↧p ∣ / ↧p↧q) ℤ.* ℤ.+ ↧p↧q  ≡⟨ cong (ℤ._* ℤ.+ ↧p↧q) (↥[n/d]≡n (ℤ.+ ℤ.∣ ↥p↧q ℤ.+ ↥q↧p ∣) ↧p↧q) ⟩

+
  ℤ.+ ℤ.∣ ↥p↧q ℤ.+ ↥q↧p ∣ ℤ.* ℤ.+ ↧p↧q             ≤⟨ ℤ.*-monoʳ-≤-nonNeg (ℤ.+ ↧p↧q) (ℤ.+≤+ (ℤ.∣i+j∣≤∣i∣+∣j∣ ↥p↧q ↥q↧p)) ⟩

+
  (ℤ.+ ℤ.∣ ↥p↧q ∣ ℤ.+ ℤ.+ ℤ.∣ ↥q↧p ∣) ℤ.* ℤ.+ ↧p↧q ≡⟨ cong₂ (λ h₁ h₂ → (h₁ ℤ.+ h₂) ℤ.* ℤ.+ ↧p↧q) ∣↥p∣↧q≡∣↥p↧q∣ ∣↥q∣↧p≡∣↥q↧p∣ ⟨

+
  (∣↥p∣↧q ℤ.+ ∣↥q∣↧p) ℤ.* ℤ.+ ↧p↧q                 ≡⟨⟩

+
  (↥∣p∣↧q ℤ.+ ↥∣q∣↧p) ℤ.* ℤ.+ ↧p↧q                 ≡⟨ cong (ℤ._* ℤ.+ ↧p↧q) (↥[n/d]≡n (↥∣p∣↧q ℤ.+ ↥∣q∣↧p) ↧p↧q) ⟩

+
  ↥ ((↥∣p∣↧q ℤ.+ ↥∣q∣↧p) / ↧p↧q) ℤ.* ℤ.+ ↧p↧q      ≡⟨⟩

+
  ↥ (∣ p ∣ + ∣ q ∣) ℤ.* ↧ ∣ p + q ∣ ∎)

+
  where

+
    open ℤ.≤-Reasoning

+
    ↥p↧q = ↥ p ℤ.* ↧ q

+
    ↥q↧p = ↥ q ℤ.* ↧ p

+
    ↥∣p∣↧q = ↥ ∣ p ∣ ℤ.* ↧ q

+
    ↥∣q∣↧p = ↥ ∣ q ∣ ℤ.* ↧ p

+
    ∣↥p∣↧q = ℤ.+ ℤ.∣ ↥ p ∣ ℤ.* ↧ q

+
    ∣↥q∣↧p = ℤ.+ ℤ.∣ ↥ q ∣ ℤ.* ↧ p

+
    ↧p↧q = ↧ₙ p ℕ.* ↧ₙ q

+
    ∣m∣n≡∣mn∣ : ∀ m n → ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ n ≡ ℤ.+ ℤ.∣ m ℤ.* ℤ.+ n ∣

+
    ∣m∣n≡∣mn∣ m n = begin-equality

+
      ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ n                        ≡⟨⟩

+
      ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ ℤ.∣ ℤ.+ n ∣              ≡⟨ ℤ.pos-* ℤ.∣ m ∣ ℤ.∣ ℤ.+ n ∣ ⟨

+
      ℤ.+ (ℤ.∣ m ∣ ℕ.* n)                          ≡⟨⟩

+
      ℤ.+ (ℤ.∣ m ∣ ℕ.* ℤ.∣ ℤ.+ n ∣)                ≡⟨ cong ℤ.+_ (ℤ.∣i*j∣≡∣i∣*∣j∣ m (ℤ.+ n)) ⟨

+
      ℤ.+ (ℤ.∣ m ℤ.* ℤ.+ n ∣)                      ∎

+
    ∣↥p∣↧q≡∣↥p↧q∣ : ∣↥p∣↧q ≡ ℤ.+ ℤ.∣ ↥p↧q ∣

+
    ∣↥p∣↧q≡∣↥p↧q∣ = ∣m∣n≡∣mn∣ (↥ p) (↧ₙ q)

+
    ∣↥q∣↧p≡∣↥q↧p∣ : ∣↥q∣↧p ≡ ℤ.+ ℤ.∣ ↥q↧p ∣

+
    ∣↥q∣↧p≡∣↥q↧p∣ = ∣m∣n≡∣mn∣ (↥ q) (↧ₙ p)

+

+
∣p-q∣≤∣p∣+∣q∣ : ∀ p q → ∣ p - q ∣ ≤ ∣ p ∣ + ∣ q ∣

+
∣p-q∣≤∣p∣+∣q∣ p q = begin

+
  ∣ p   -     q ∣  ≤⟨ ∣p+q∣≤∣p∣+∣q∣ p (- q) ⟩

+
  ∣ p ∣ + ∣ - q ∣  ≡⟨ cong (∣ p ∣ +_) (∣-p∣≡∣p∣ q) ⟩

+
  ∣ p ∣ + ∣   q ∣  ∎

+
  where open ≤-Reasoning

+

+
∣p*q∣≡∣p∣*∣q∣ : ∀ p q → ∣ p * q ∣ ≡ ∣ p ∣ * ∣ q ∣

+
∣p*q∣≡∣p∣*∣q∣ p@record{} q@record{} = begin

+
  ∣ p * q ∣                                           ≡⟨⟩

+
  ∣ (↥ p ℤ.* ↥ q) / (↧ₙ p ℕ.* ↧ₙ q) ∣                 ≡⟨⟩

+
  ℤ.+ ℤ.∣ ↥ p ℤ.* ↥ q ∣ / (↧ₙ p ℕ.* ↧ₙ q)             ≡⟨ cong (λ h → ℤ.+ h / ((↧ₙ p) ℕ.* (↧ₙ q))) (ℤ.∣i*j∣≡∣i∣*∣j∣ (↥ p) (↥ q)) ⟩

+
  ℤ.+ (ℤ.∣ ↥ p ∣ ℕ.* ℤ.∣ ↥ q ∣) / (↧ₙ p ℕ.* ↧ₙ q)     ≡⟨ cong (_/ (↧ₙ p ℕ.* ↧ₙ q)) (ℤ.pos-* ℤ.∣ ↥ p ∣ ℤ.∣ ↥ q ∣) ⟩

+
  (ℤ.+ ℤ.∣ ↥ p ∣ ℤ.* ℤ.+ ℤ.∣ ↥ q ∣) / (↧ₙ p ℕ.* ↧ₙ q) ≡⟨⟩

+
  (ℤ.+ ℤ.∣ ↥ p ∣ / ↧ₙ p) * (ℤ.+ ℤ.∣ ↥ q ∣ / ↧ₙ q)     ≡⟨⟩

+
  ∣ p ∣ * ∣ q ∣                                       ∎

+
  where open ≡-Reasoning

+

+
∣p*q∣≃∣p∣*∣q∣ : ∀ p q → ∣ p * q ∣ ≃ ∣ p ∣ * ∣ q ∣

+
∣p*q∣≃∣p∣*∣q∣ p q = ≃-reflexive (∣p*q∣≡∣p∣*∣q∣ p q)

+

+
∣∣p∣∣≡∣p∣ : ∀ p → ∣ ∣ p ∣ ∣ ≡ ∣ p ∣

+
∣∣p∣∣≡∣p∣ p = 0≤p⇒∣p∣≡p (0≤∣p∣ p)

+

+
∣∣p∣∣≃∣p∣ : ∀ p → ∣ ∣ p ∣ ∣ ≃ ∣ p ∣

+
∣∣p∣∣≃∣p∣ p = ≃-reflexive (∣∣p∣∣≡∣p∣ p)

+

+
∣-∣-nonNeg : ∀ p → NonNegative ∣ p ∣

+
∣-∣-nonNeg (mkℚᵘ +[1+ _ ] _) = _

+
∣-∣-nonNeg (mkℚᵘ +0       _) = _

+
∣-∣-nonNeg (mkℚᵘ -[1+ _ ] _) = _

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.5

+

+
neg-mono-<-> = neg-mono-<

+
{-# WARNING_ON_USAGE neg-mono-<->

+
"Warning: neg-mono-<-> was deprecated in v1.5.

+
Please use neg-mono-< instead."

+
#-}

+

+
-- Version 2.0

+

+
↥[p/q]≡p = ↥[n/d]≡n

+
{-# WARNING_ON_USAGE ↥[p/q]≡p

+
"Warning: ↥[p/q]≡p was deprecated in v2.0.

+
Please use ↥[n/d]≡n instead."

+
#-}

+
↧[p/q]≡q = ↧[n/d]≡d

+
{-# WARNING_ON_USAGE ↧[p/q]≡q

+
"Warning: ↧[p/q]≡q was deprecated in v2.0.

+
Please use ↧[n/d]≡d instead."

+
#-}

+
*-monoʳ-≤-pos : ∀ {r} → Positive r → (r *_) Preserves _≤_ ⟶ _≤_

+
*-monoʳ-≤-pos r@{mkℚᵘ +[1+ _ ] _} _ = *-monoʳ-≤-nonNeg r

+
{-# WARNING_ON_USAGE *-monoʳ-≤-pos

+
"Warning: *-monoʳ-≤-pos was deprecated in v2.0.

+
Please use *-monoʳ-≤-nonNeg instead."

+
#-}

+
*-monoˡ-≤-pos : ∀ {r} → Positive r → (_* r) Preserves _≤_ ⟶ _≤_

+
*-monoˡ-≤-pos r@{mkℚᵘ +[1+ _ ] _} _ = *-monoˡ-≤-nonNeg r

+
{-# WARNING_ON_USAGE *-monoˡ-≤-pos

+
"Warning: *-monoˡ-≤-nonNeg was deprecated in v2.0.

+
Please use *-monoˡ-≤-nonNeg instead."

+
#-}

+
≤-steps = p≤q⇒p≤r+q

+
{-# WARNING_ON_USAGE ≤-steps

+
"Warning: ≤-steps was deprecated in v2.0

+
Please use p≤q⇒p≤r+q instead."

+
#-}

+
*-monoˡ-≤-neg : ∀ r → Negative r → (_* r) Preserves _≤_ ⟶ _≥_

+
*-monoˡ-≤-neg r@(mkℚᵘ -[1+ _ ] _) _ = *-monoˡ-≤-nonPos r

+
{-# WARNING_ON_USAGE *-monoˡ-≤-neg

+
"Warning: *-monoˡ-≤-neg was deprecated in v2.0.

+
Please use *-monoˡ-≤-nonPos instead."

+
#-}

+
*-monoʳ-≤-neg : ∀ r → Negative r → (r *_) Preserves _≤_ ⟶ _≥_

+
*-monoʳ-≤-neg r@(mkℚᵘ -[1+ _ ] _) _ = *-monoʳ-≤-nonPos r

+
{-# WARNING_ON_USAGE *-monoʳ-≤-neg

+
"Warning: *-monoʳ-≤-neg was deprecated in v2.0.

+
Please use *-monoʳ-≤-nonPos instead."

+
#-}

+
*-cancelˡ-<-pos : ∀ r → Positive r → ∀ {p q} → r * p < r * q → p < q

+
*-cancelˡ-<-pos r@(mkℚᵘ +[1+ _ ] _) r>0 = *-cancelˡ-<-nonNeg r

+
{-# WARNING_ON_USAGE *-cancelˡ-<-pos

+
"Warning: *-cancelˡ-<-pos was deprecated in v2.0.

+
Please use *-cancelˡ-<-nonNeg instead."

+
#-}

+
*-cancelʳ-<-pos : ∀ r → Positive r → ∀ {p q} → p * r < q * r → p < q

+
*-cancelʳ-<-pos r@(mkℚᵘ +[1+ _ ] _) r>0 = *-cancelʳ-<-nonNeg r

+
{-# WARNING_ON_USAGE *-cancelʳ-<-pos

+
"Warning: *-cancelʳ-<-pos was deprecated in v2.0.

+
Please use *-cancelʳ-<-nonNeg instead."

+
#-}

+
*-cancelˡ-<-neg : ∀ r → Negative r → ∀ {p q} → r * p < r * q → q < p

+
*-cancelˡ-<-neg r@(mkℚᵘ -[1+ _ ] _) _ = *-cancelˡ-<-nonPos r

+
{-# WARNING_ON_USAGE *-cancelˡ-<-neg

+
"Warning: *-cancelˡ-<-neg was deprecated in v2.0.

+
Please use *-cancelˡ-<-nonPos instead."

+
#-}

+
*-cancelʳ-<-neg : ∀ r → Negative r → ∀ {p q} → p * r < q * r → q < p

+
*-cancelʳ-<-neg r@(mkℚᵘ -[1+ _ ] _) _ = *-cancelʳ-<-nonPos r

+
{-# WARNING_ON_USAGE *-cancelʳ-<-neg

+
"Warning: *-cancelʳ-<-neg was deprecated in v2.0.

+
Please use *-cancelʳ-<-nonPos instead."

+
#-}

+
positive⇒nonNegative : ∀ {p} → Positive p → NonNegative p

+
positive⇒nonNegative {p} p>0 = pos⇒nonNeg p {{p>0}}

+
{-# WARNING_ON_USAGE positive⇒nonNegative

+
"Warning: positive⇒nonNegative was deprecated in v2.0.

+
Please use pos⇒nonNeg instead."

+
#-}

+
negative⇒nonPositive : ∀ {p} → Negative p → NonPositive p

+
negative⇒nonPositive {p} p<0 = neg⇒nonPos p {{p<0}}

+
{-# WARNING_ON_USAGE negative⇒nonPositive

+
"Warning: negative⇒nonPositive was deprecated in v2.0.

+
Please use neg⇒nonPos instead."

+
#-}

+
negative<positive : ∀ {p q} → .(Negative p) → .(Positive q) → p < q

+
negative<positive {p} {q} p<0 q>0 = neg<pos p q {{p<0}} {{q>0}}

+
{-# WARNING_ON_USAGE negative<positive

+
"Warning: negative<positive was deprecated in v2.0.

+
Please use neg<pos instead."

+
#-}

+

+
{- issue1865/issue1755: raw bundles have moved to `Data.X.Base` -}

+
open Data.Rational.Unnormalised.Base public

+
  using (+-rawMagma; +-0-rawGroup; *-rawMagma; +-*-rawNearSemiring; +-*-rawSemiring; +-*-rawRing)

+
  renaming (+-0-rawMonoid to +-rawMonoid; *-1-rawMonoid to *-rawMonoid)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Rational.html b/site/static/agda_html/Data.Rational.html
new file mode 100644
index 000000000..10a7eb1be
--- /dev/null
+++ b/site/static/agda_html/Data.Rational.html
@@ -0,0 +1,114 @@
+
+Data.Rational
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Rational numbers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Rational where

+

+
------------------------------------------------------------------------

+
-- Publicly re-export contents of core module and queries

+

+
open import Data.Rational.Base public

+
open import Data.Rational.Properties public

+
  using (_≟_; _≤?_; _<?_; _≥?_; _>?_)

+

+
------------------------------------------------------------------------

+
-- Deprecated

+

+
-- Version 1.0

+

+
open import Data.Rational.Properties public

+
  using (drop-*≤*; ≃⇒≡; ≡⇒≃)

+

+
-- Version 1.5

+

+
import Data.Integer.Show as ℤ

+
open import Data.String.Base using (String; _++_)

+

+
show : ℚ → String

+
show p = ℤ.show (↥ p) ++ "/" ++ ℤ.show (↧ p)

+

+
{-# WARNING_ON_USAGE show

+
"Warning: show was deprecated in v1.5.

+
Please use Data.Rational.Show's show instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Record.html b/site/static/agda_html/Data.Record.html
new file mode 100644
index 000000000..e25ecff17
--- /dev/null
+++ b/site/static/agda_html/Data.Record.html
@@ -0,0 +1,255 @@
+
+Data.Record
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Record types with manifest fields and "with", based on Randy

+
-- Pollack's "Dependently Typed Records in Type Theory"

+
------------------------------------------------------------------------

+

+
-- For an example of how this module can be used, see README.Record.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.Bool.Base using (true; false; if_then_else_)

+
open import Data.Empty using (⊥)

+
open import Data.List.Base using (List; []; _∷_; foldr)

+
open import Data.Product.Base hiding (proj₁; proj₂)

+
open import Data.Unit.Polymorphic using (⊤)

+
open import Function.Base using (id; _∘_)

+
open import Level using (suc; _⊔_)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Nullary.Decidable using (does)

+

+
-- The module is parametrised by the type of labels, which should come

+
-- with decidable equality.

+

+
module Data.Record {ℓ} (Label : Set ℓ) (_≟_ : DecidableEquality Label) where

+

+
------------------------------------------------------------------------

+
-- A Σ-type with a manifest field

+

+
-- A variant of Σ where the value of the second field is "manifest"

+
-- (given by the first).

+

+
infix 4 _,

+

+
record Manifest-Σ {a b} (A : Set a) {B : A → Set b}

+
                  (f : (x : A) → B x) : Set a where

+
  constructor _,

+
  field proj₁ : A

+

+
  proj₂ : B proj₁

+
  proj₂ = f proj₁

+

+
------------------------------------------------------------------------

+
-- Signatures and records

+

+
mutual

+

+
  infixl 5 _,_∶_ _,_≔_

+

+
  data Signature s : Set (suc s ⊔ ℓ) where

+
    ∅     : Signature s

+
    _,_∶_ : (Sig : Signature s)

+
            (ℓ : Label)

+
            (A : Record Sig → Set s) →

+
            Signature s

+
    _,_≔_ : (Sig : Signature s)

+
            (ℓ : Label)

+
            {A : Record Sig → Set s}

+
            (a : (r : Record Sig) → A r) →

+
            Signature s

+

+
  -- Record is a record type to ensure that the signature can be

+
  -- inferred from a value of type Record Sig.

+

+
  record Record {s} (Sig : Signature s) : Set s where

+
    eta-equality

+
    inductive

+
    constructor rec

+
    field fun : Record-fun Sig

+

+
  Record-fun : ∀ {s} → Signature s → Set s

+
  Record-fun ∅             = ⊤

+
  Record-fun (Sig , ℓ ∶ A) =          Σ (Record Sig) A

+
  Record-fun (Sig , ℓ ≔ a) = Manifest-Σ (Record Sig) a

+

+
------------------------------------------------------------------------

+
-- Labels

+

+
-- A signature's labels, starting with the last one.

+

+
labels : ∀ {s} → Signature s → List Label

+
labels ∅             = []

+
labels (Sig , ℓ ∶ A) = ℓ ∷ labels Sig

+
labels (Sig , ℓ ≔ a) = ℓ ∷ labels Sig

+

+
-- Inhabited if the label is part of the signature.

+

+
infix 4 _∈_

+

+
_∈_ : ∀ {s} → Label → Signature s → Set

+
ℓ ∈ Sig =

+
  foldr (λ ℓ′ → if does (ℓ ≟ ℓ′) then (λ _ → ⊤) else id) ⊥ (labels Sig)

+

+
------------------------------------------------------------------------

+
-- Projections

+

+
-- Signature restriction and projection. (Restriction means removal of

+
-- a given field and all subsequent fields.)

+

+
Restrict : ∀ {s} (Sig : Signature s) (ℓ : Label) → ℓ ∈ Sig →

+
           Signature s

+
Restrict ∅              ℓ ()

+
Restrict (Sig , ℓ′ ∶ A) ℓ ℓ∈ with does (ℓ ≟ ℓ′)

+
... | true  = Sig

+
... | false = Restrict Sig ℓ ℓ∈

+
Restrict (Sig , ℓ′ ≔ a) ℓ ℓ∈ with does (ℓ ≟ ℓ′)

+
... | true  = Sig

+
... | false = Restrict Sig ℓ ℓ∈

+

+
Restricted : ∀ {s} (Sig : Signature s) (ℓ : Label) → ℓ ∈ Sig → Set s

+
Restricted Sig ℓ ℓ∈ = Record (Restrict Sig ℓ ℓ∈)

+

+
Proj : ∀ {s} (Sig : Signature s) (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} →

+
       Restricted Sig ℓ ℓ∈ → Set s

+
Proj ∅              ℓ {}

+
Proj (Sig , ℓ′ ∶ A) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)

+
... | true  = A

+
... | false = Proj Sig ℓ {ℓ∈}

+
Proj (_,_≔_ Sig ℓ′ {A = A} a) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)

+
... | true  = A

+
... | false = Proj Sig ℓ {ℓ∈}

+

+
-- Record restriction and projection.

+

+
infixl 5 _∣_

+

+
_∣_ : ∀ {s} {Sig : Signature s} → Record Sig →

+
      (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} → Restricted Sig ℓ ℓ∈

+
_∣_ {Sig = ∅}            r       ℓ {}

+
_∣_ {Sig = Sig , ℓ′ ∶ A} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)

+
... | true  = Σ.proj₁ r

+
... | false = _∣_ (Σ.proj₁ r) ℓ {ℓ∈}

+
_∣_ {Sig = Sig , ℓ′ ≔ a} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)

+
... | true  = Manifest-Σ.proj₁ r

+
... | false = _∣_ (Manifest-Σ.proj₁ r) ℓ {ℓ∈}

+

+
infixl 5 _·_

+

+
_·_ : ∀ {s} {Sig : Signature s} (r : Record Sig)

+
      (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} →

+
      Proj Sig ℓ {ℓ∈} (r ∣ ℓ)

+
_·_ {Sig = ∅}            r       ℓ {}

+
_·_ {Sig = Sig , ℓ′ ∶ A} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)

+
... | true  = Σ.proj₂ r

+
... | false = _·_ (Σ.proj₁ r) ℓ {ℓ∈}

+
_·_ {Sig = Sig , ℓ′ ≔ a} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)

+
... | true  = Manifest-Σ.proj₂ r

+
... | false = _·_ (Manifest-Σ.proj₁ r) ℓ {ℓ∈}

+

+
------------------------------------------------------------------------

+
-- With

+

+
-- Sig With ℓ ≔ a is the signature Sig, but with the ℓ field set to a.

+

+
mutual

+

+
  infixl 5 _With_≔_

+

+
  _With_≔_ : ∀ {s} (Sig : Signature s) (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} →

+
             ((r : Restricted Sig ℓ ℓ∈) → Proj Sig ℓ r) → Signature s

+
  _With_≔_ ∅ ℓ {} a

+
  _With_≔_ (Sig , ℓ′ ∶ A)   ℓ {ℓ∈} a with does (ℓ ≟ ℓ′)

+
  ... | true  = Sig                   , ℓ′ ≔ a

+
  ... | false = _With_≔_ Sig ℓ {ℓ∈} a , ℓ′ ∶ A ∘ drop-With

+
  _With_≔_  (Sig , ℓ′ ≔ a′) ℓ {ℓ∈} a with does (ℓ ≟ ℓ′)

+
  ... | true  = Sig                   , ℓ′ ≔ a

+
  ... | false = _With_≔_ Sig ℓ {ℓ∈} a , ℓ′ ≔ a′ ∘ drop-With

+

+
  drop-With : ∀ {s} {Sig : Signature s} {ℓ : Label} {ℓ∈ : ℓ ∈ Sig}

+
              {a : (r : Restricted Sig ℓ ℓ∈) → Proj Sig ℓ r} →

+
              Record (_With_≔_ Sig ℓ {ℓ∈} a) → Record Sig

+
  drop-With {Sig = ∅} {ℓ∈ = ()}      r

+
  drop-With {Sig = Sig , ℓ′ ∶ A} {ℓ} (rec r) with does (ℓ ≟ ℓ′)

+
  ... | true  = rec (Manifest-Σ.proj₁ r , Manifest-Σ.proj₂ r)

+
  ... | false = rec (drop-With (Σ.proj₁ r) , Σ.proj₂ r)

+
  drop-With {Sig = Sig , ℓ′ ≔ a} {ℓ} (rec r) with does (ℓ ≟ ℓ′)

+
  ... | true  = rec (Manifest-Σ.proj₁ r ,)

+
  ... | false = rec (drop-With (Manifest-Σ.proj₁ r) ,)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Refinement.html b/site/static/agda_html/Data.Refinement.html
new file mode 100644
index 000000000..0519d3502
--- /dev/null
+++ b/site/static/agda_html/Data.Refinement.html
@@ -0,0 +1,123 @@
+
+Data.Refinement
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Refinement type: a value together with a proof irrelevant witness.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Refinement where

+

+
open import Level

+
open import Data.Irrelevant as Irrelevant using (Irrelevant)

+
open import Function.Base

+
open import Relation.Unary using (IUniversal; _⇒_; _⊢_)

+

+
private

+
  variable

+
    a b p q : Level

+
    A : Set a

+
    B : Set b

+

+
record Refinement {a p} (A : Set a) (P : A → Set p) : Set (a ⊔ p) where

+
  constructor _,_

+
  field value : A

+
        proof : Irrelevant (P value)

+
infixr 4 _,_

+
open Refinement public

+

+
-- The syntax declaration below is meant to mimick set comprehension.

+
-- It is attached to Refinement-syntax, to make it easy to import

+
-- Data.Refinement without the special syntax.

+

+
infix 2 Refinement-syntax

+
Refinement-syntax = Refinement

+
syntax Refinement-syntax A (λ x → P) = [ x ∈ A ∣ P ]

+

+
module _ {P : A → Set p} {Q : B → Set q} where

+

+
  map : (f : A → B) → ∀[ P ⇒ f ⊢ Q ] →

+
        [ a ∈ A ∣ P a ] → [ b ∈ B ∣ Q b ]

+
  map f prf (a , p) = f a , Irrelevant.map prf p

+

+
module _ {P : A → Set p} {Q : A → Set q} where

+

+
  refine : ∀[ P ⇒ Q ] → [ a ∈ A ∣ P a ] → [ a ∈ A ∣ Q a ]

+
  refine = map id

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sign.Base.html b/site/static/agda_html/Data.Sign.Base.html
new file mode 100644
index 000000000..525fb8aeb
--- /dev/null
+++ b/site/static/agda_html/Data.Sign.Base.html
@@ -0,0 +1,139 @@
+
+Data.Sign.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Signs

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sign.Base where

+

+
open import Algebra.Bundles.Raw using (RawMagma; RawMonoid; RawGroup)

+
open import Level using (0ℓ)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Sign : Set where

+
  - : Sign

+
  + : Sign

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
-- The opposite sign.

+

+
opposite : Sign → Sign

+
opposite - = +

+
opposite + = -

+

+
-- "Multiplication".

+

+
infixl 7 _*_

+

+
_*_ : Sign → Sign → Sign

+
+ * s₂ = s₂

+
- * s₂ = opposite s₂

+

+
------------------------------------------------------------------------

+
-- Raw Bundles

+

+
*-rawMagma : RawMagma 0ℓ 0ℓ

+
*-rawMagma = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  }

+

+
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ

+
*-1-rawMonoid = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  ; ε = +

+
  }

+

+
*-1-rawGroup : RawGroup 0ℓ 0ℓ

+
*-1-rawGroup = record

+
  { _≈_ = _≡_

+
  ; _∙_ = _*_

+
  ; _⁻¹ = opposite

+
  ; ε = +

+
  }

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sign.Properties.html b/site/static/agda_html/Data.Sign.Properties.html
new file mode 100644
index 000000000..5200ec5e9
--- /dev/null
+++ b/site/static/agda_html/Data.Sign.Properties.html
@@ -0,0 +1,286 @@
+
+Data.Sign.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some properties about signs

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sign.Properties where

+

+
open import Algebra.Bundles using (Magma; Semigroup; CommutativeSemigroup;

+
  Monoid; CommutativeMonoid; Group; AbelianGroup)

+
open import Data.Sign.Base using (Sign; opposite; _*_; +; -)

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_$_; id)

+
open import Function.Definitions using (Injective)

+
open import Level using (0ℓ)

+
open import Relation.Binary

+
  using (Decidable; DecidableEquality; Setoid; DecSetoid; IsDecEquivalence)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; _≢_; sym; cong₂)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; decSetoid; isDecEquivalence; isEquivalence)

+
open import Relation.Nullary.Decidable using (yes; no)

+
open import Relation.Nullary.Negation.Core using (contradiction)

+

+
open import Algebra.Structures {A = Sign} _≡_

+
open import Algebra.Definitions {A = Sign} _≡_

+
open import Algebra.Consequences.Propositional

+
  using (selfInverse⇒involutive; selfInverse⇒injective)

+

+
------------------------------------------------------------------------

+
-- Equality

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Sign

+
- ≟ - = yes refl

+
- ≟ + = no λ()

+
+ ≟ - = no λ()

+
+ ≟ + = yes refl

+

+
≡-setoid : Setoid 0ℓ 0ℓ

+
≡-setoid = setoid Sign

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = decSetoid _≟_

+

+
≡-isDecEquivalence : IsDecEquivalence _≡_

+
≡-isDecEquivalence = isDecEquivalence _≟_

+

+
------------------------------------------------------------------------

+
-- opposite

+

+
-- Algebraic properties of opposite

+

+
opposite-selfInverse : SelfInverse opposite

+
opposite-selfInverse { - } { + } refl = refl

+
opposite-selfInverse { + } { - } refl = refl

+

+
opposite-involutive : Involutive opposite

+
opposite-involutive = selfInverse⇒involutive opposite-selfInverse

+

+
opposite-injective : Injective _≡_ _≡_ opposite

+
opposite-injective = selfInverse⇒injective opposite-selfInverse

+

+

+
------------------------------------------------------------------------

+
-- other properties of opposite

+

+
s≢opposite[s] : ∀ s → s ≢ opposite s

+
s≢opposite[s] - ()

+
s≢opposite[s] + ()

+

+
------------------------------------------------------------------------

+
-- _*_

+

+
-- Algebraic properties of _*_

+

+
s*s≡+ : ∀ s → s * s ≡ +

+
s*s≡+ + = refl

+
s*s≡+ - = refl

+

+
*-identityˡ : LeftIdentity + _*_

+
*-identityˡ _ = refl

+

+
*-identityʳ : RightIdentity + _*_

+
*-identityʳ - = refl

+
*-identityʳ + = refl

+

+
*-identity : Identity + _*_

+
*-identity = *-identityˡ  , *-identityʳ

+

+
*-comm : Commutative _*_

+
*-comm + + = refl

+
*-comm + - = refl

+
*-comm - + = refl

+
*-comm - - = refl

+

+
*-assoc : Associative _*_

+
*-assoc + + _ = refl

+
*-assoc + - _ = refl

+
*-assoc - + _ = refl

+
*-assoc - - + = refl

+
*-assoc - - - = refl

+

+
*-cancelʳ-≡ : RightCancellative _*_

+
*-cancelʳ-≡ _ - - _  = refl

+
*-cancelʳ-≡ _ - + eq = contradiction (sym eq) (s≢opposite[s] _)

+
*-cancelʳ-≡ _ + - eq = contradiction eq (s≢opposite[s] _)

+
*-cancelʳ-≡ _ + + _  = refl

+

+
*-cancelˡ-≡ : LeftCancellative _*_

+
*-cancelˡ-≡ - _ _ eq = opposite-injective eq

+
*-cancelˡ-≡ + _ _ eq = eq

+

+
*-cancel-≡ : Cancellative _*_

+
*-cancel-≡ = *-cancelˡ-≡ , *-cancelʳ-≡

+

+
*-inverse : Inverse + id _*_

+
*-inverse = s*s≡+ , s*s≡+

+

+
*-isMagma : IsMagma _*_

+
*-isMagma = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _*_

+
  }

+

+
*-magma : Magma 0ℓ 0ℓ

+
*-magma = record

+
  { isMagma = *-isMagma

+
  }

+

+
*-isSemigroup : IsSemigroup _*_

+
*-isSemigroup = record

+
  { isMagma = *-isMagma

+
  ; assoc   = *-assoc

+
  }

+

+
*-semigroup : Semigroup 0ℓ 0ℓ

+
*-semigroup = record

+
  { isSemigroup = *-isSemigroup

+
  }

+

+
*-isCommutativeSemigroup : IsCommutativeSemigroup _*_

+
*-isCommutativeSemigroup = record

+
  { isSemigroup = *-isSemigroup

+
  ; comm = *-comm

+
  }

+

+
*-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ

+
*-commutativeSemigroup = record

+
  { isCommutativeSemigroup = *-isCommutativeSemigroup

+
  }

+

+
*-isMonoid : IsMonoid _*_ +

+
*-isMonoid = record

+
  { isSemigroup = *-isSemigroup

+
  ; identity    = *-identity

+
  }

+

+
*-monoid : Monoid 0ℓ 0ℓ

+
*-monoid = record

+
  { isMonoid = *-isMonoid

+
  }

+

+
*-isCommutativeMonoid : IsCommutativeMonoid _*_ +

+
*-isCommutativeMonoid = record

+
   { isMonoid = *-isMonoid

+
   ; comm = *-comm

+
   }

+

+
*-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ

+
*-commutativeMonoid = record

+
  { isCommutativeMonoid = *-isCommutativeMonoid

+
  }

+

+
*-isGroup : IsGroup _*_ + id

+
*-isGroup = record

+
  { isMonoid = *-isMonoid

+
  ; inverse = *-inverse

+
  ; ⁻¹-cong = id

+
  }

+

+
*-group : Group 0ℓ 0ℓ

+
*-group = record

+
  { isGroup = *-isGroup

+
  }

+

+
*-isAbelianGroup : IsAbelianGroup _*_ + id

+
*-isAbelianGroup = record

+
  { isGroup = *-isGroup

+
  ; comm = *-comm

+
  }

+

+
*-abelianGroup : AbelianGroup 0ℓ 0ℓ

+
*-abelianGroup = record

+
  { isAbelianGroup = *-isAbelianGroup

+
  }

+

+
-- Other properties of _*_

+

+
s*opposite[s]≡- : ∀ s → s * opposite s ≡ -

+
s*opposite[s]≡- + = refl

+
s*opposite[s]≡- - = refl

+

+
opposite[s]*s≡- : ∀ s → opposite s * s ≡ -

+
opposite[s]*s≡- + = refl

+
opposite[s]*s≡- - = refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sign.html b/site/static/agda_html/Data.Sign.html
new file mode 100644
index 000000000..372f63430
--- /dev/null
+++ b/site/static/agda_html/Data.Sign.html
@@ -0,0 +1,93 @@
+
+Data.Sign
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Signs

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sign where

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open import Data.Sign.Base public

+
open import Data.Sign.Properties public

+
  using (_≟_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.String.Base.html b/site/static/agda_html/Data.String.Base.html
new file mode 100644
index 000000000..3d9e23dbf
--- /dev/null
+++ b/site/static/agda_html/Data.String.Base.html
@@ -0,0 +1,273 @@
+
+Data.String.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Strings: builtin type and basic operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.String.Base where

+

+
open import Data.Bool.Base using (Bool; true; false; if_then_else_)

+
open import Data.Char.Base as Char using (Char)

+
open import Data.List.Base as List using (List; [_]; _∷_; [])

+
open import Data.List.NonEmpty.Base as NE using (List⁺)

+
open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise)

+
open import Data.List.Relation.Binary.Lex.Core using (Lex-<; Lex-≤)

+
open import Data.Maybe.Base as Maybe using (Maybe)

+
open import Data.Nat.Base using (ℕ; _∸_; ⌊_/2⌋; ⌈_/2⌉; _≡ᵇ_)

+
open import Data.Product.Base using (proj₁; proj₂)

+
open import Function.Base using (_on_; _∘′_; _∘_)

+
open import Level using (Level; 0ℓ)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Nullary.Decidable.Core using (does; T?)

+

+
------------------------------------------------------------------------

+
-- From Agda.Builtin: type and renamed primitives

+

+
-- Note that we do not re-export primStringAppend because we want to

+
-- give it an infix definition and be able to assign it a level.

+

+
import Agda.Builtin.String as String

+

+
open String public using ( String )

+
  renaming

+
  ( primStringUncons   to uncons

+
  ; primStringToList   to toList

+
  ; primStringFromList to fromList

+
  ; primShowString     to show

+
  )

+

+
------------------------------------------------------------------------

+
-- Relations

+

+
-- Pointwise equality on Strings

+

+
infix 4 _≈_

+
_≈_ : Rel String 0ℓ

+
_≈_ = Pointwise _≡_ on toList

+

+
-- Lexicographic ordering on Strings

+

+
infix 4 _<_

+
_<_ : Rel String 0ℓ

+
_<_ = Lex-< _≡_ Char._<_ on toList

+

+
infix 4 _≤_

+
_≤_ : Rel String 0ℓ

+
_≤_ = Lex-≤ _≡_ Char._<_ on toList

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
-- List-like operations

+

+
head : String → Maybe Char

+
head = Maybe.map proj₁ ∘′ uncons

+

+
tail : String → Maybe String

+
tail = Maybe.map proj₂ ∘′ uncons

+

+
-- Additional conversion functions

+

+
fromChar : Char → String

+
fromChar = fromList ∘′ [_]

+

+
fromList⁺ : List⁺ Char → String

+
fromList⁺ = fromList ∘′ NE.toList

+

+
-- List-like functions

+

+
infixr 5 _++_

+
_++_ : String → String → String

+
_++_ = String.primStringAppend

+

+
length : String → ℕ

+
length = List.length ∘ toList

+

+
replicate : ℕ → Char → String

+
replicate n = fromList ∘ List.replicate n

+

+
concat : List String → String

+
concat = List.foldr _++_ ""

+

+
intersperse : String → List String → String

+
intersperse sep = concat ∘′ (List.intersperse sep)

+

+
unwords : List String → String

+
unwords = intersperse " "

+

+
unlines : List String → String

+
unlines = intersperse "\n"

+

+
between : String → String → String → String

+
between left right middle = left ++ middle ++ right

+

+
parens : String → String

+
parens = between "(" ")"

+

+
braces : String → String

+
braces = between "{" "}"

+

+
-- append that also introduces spaces, if necessary

+
infixr 5 _<+>_

+
_<+>_ : String → String → String

+
"" <+> b = b

+
a <+> "" = a

+
a <+> b = a ++ " " ++ b

+

+
------------------------------------------------------------------------

+
-- Padding

+

+
-- Each one of the padding functions should verify the following

+
-- invariant:

+
--   If length str ≤ n then length (padLeft c n str) ≡ n

+
--   and otherwise padLeft c n str ≡ str.

+

+
-- Appending an empty string is expensive (append for Haskell's

+
-- Text creates a fresh Text value in which both contents are

+
-- copied) so we precompute `n ∸ length str` and check whether

+
-- it is equal to 0.

+

+
padLeft : Char → ℕ → String → String

+
padLeft c n str =

+
  let l = n ∸ length str in

+
  if l ≡ᵇ 0 then str else replicate l c ++ str

+

+
padRight : Char → ℕ → String → String

+
padRight c n str =

+
  let l = n ∸ length str in

+
  if l ≡ᵇ 0 then str else str ++ replicate l c

+

+
padBoth : Char → Char → ℕ → String → String

+
padBoth cₗ cᵣ n str =

+
  let l = n ∸ length str in

+
  if l ≡ᵇ 0 then str else replicate ⌊ l /2⌋ cₗ ++ str ++ replicate ⌈ l /2⌉ cᵣ

+

+
------------------------------------------------------------------------

+
-- Alignment

+

+
-- We can align a String left, center or right in a column of a given

+
-- width by padding it with whitespace.

+

+
data Alignment : Set where

+
  Left Center Right : Alignment

+

+
fromAlignment : Alignment → ℕ → String → String

+
fromAlignment Left   = padRight ' '

+
fromAlignment Center = padBoth ' ' ' '

+
fromAlignment Right  = padLeft ' '

+

+
------------------------------------------------------------------------

+
-- Splitting strings

+

+
wordsBy : ∀ {p} {P : Pred Char p} → Decidable P → String → List String

+
wordsBy P? = List.map fromList ∘ List.wordsBy P? ∘ toList

+

+
wordsByᵇ : (Char → Bool) → String → List String

+
wordsByᵇ p = wordsBy (T? ∘ p)

+

+
words : String → List String

+
words = wordsByᵇ Char.isSpace

+

+
-- `words` ignores contiguous whitespace

+
_ : words " abc  b   " ≡ "abc" ∷ "b" ∷ []

+
_ = refl

+

+
linesBy : ∀ {p} {P : Pred Char p} → Decidable P → String → List String

+
linesBy P? = List.map fromList ∘ List.linesBy P? ∘ toList

+

+
linesByᵇ : (Char → Bool) → String → List String

+
linesByᵇ p = linesBy (T? ∘ p)

+

+
lines : String → List String

+
lines = linesByᵇ ('\n' Char.≈ᵇ_)

+

+
-- `lines` preserves empty lines

+
_ : lines "\nabc\n\nb\n\n\n" ≡ "" ∷ "abc" ∷ "" ∷ "b" ∷ "" ∷ "" ∷ []

+
_ = refl

+

+
map : (Char → Char) → String → String

+
map f = fromList ∘ List.map f ∘ toList

+

+
_ : map Char.toUpper "abc" ≡ "ABC"

+
_ = refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.String.Properties.html b/site/static/agda_html/Data.String.Properties.html
new file mode 100644
index 000000000..6379d38de
--- /dev/null
+++ b/site/static/agda_html/Data.String.Properties.html
@@ -0,0 +1,254 @@
+
+Data.String.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of operations on strings

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.String.Properties where

+

+
open import Data.Bool.Base using (Bool)

+
import Data.Char.Properties as Char

+
import Data.List.Properties as List

+
import Data.List.Relation.Binary.Pointwise as Pointwise

+
import Data.List.Relation.Binary.Lex.Strict as StrictLex

+
open import Data.String.Base

+
open import Function.Base

+
open import Relation.Nullary.Decidable using (yes; no)

+
open import Relation.Nullary.Decidable using (map′; isYes)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; StrictPartialOrder; StrictTotalOrder; DecTotalOrder; DecPoset)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsDecPartialOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive; Substitutive; Decidable; DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core

+
import Relation.Binary.Construct.On as On

+
import Relation.Binary.PropositionalEquality.Properties as PropEq

+

+
------------------------------------------------------------------------

+
-- Primitive properties

+

+
open import Agda.Builtin.String.Properties public

+
  renaming ( primStringToListInjective to toList-injective)

+

+
------------------------------------------------------------------------

+
-- Properties of _≈_

+

+
≈⇒≡ : _≈_ ⇒ _≡_

+
≈⇒≡ = toList-injective _ _

+
    ∘ Pointwise.Pointwise-≡⇒≡

+

+
≈-reflexive : _≡_ ⇒ _≈_

+
≈-reflexive = Pointwise.≡⇒Pointwise-≡

+
            ∘ cong toList

+

+
≈-refl : Reflexive _≈_

+
≈-refl {x} = ≈-reflexive {x} {x} refl

+

+
≈-sym : Symmetric _≈_

+
≈-sym = Pointwise.symmetric sym

+

+
≈-trans : Transitive _≈_

+
≈-trans = Pointwise.transitive trans

+

+
≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ

+
≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p

+

+
infix 4 _≈?_

+
_≈?_ : Decidable _≈_

+
x ≈? y = Pointwise.decidable Char._≟_ (toList x) (toList y)

+

+
≈-isEquivalence : IsEquivalence _≈_

+
≈-isEquivalence = record

+
  { refl  = λ {i} → ≈-refl {i}

+
  ; sym   = λ {i j} → ≈-sym {i} {j}

+
  ; trans = λ {i j k} → ≈-trans {i} {j} {k}

+
  }

+

+
≈-setoid : Setoid _ _

+
≈-setoid = record

+
  { isEquivalence = ≈-isEquivalence

+
  }

+

+
≈-isDecEquivalence : IsDecEquivalence _≈_

+
≈-isDecEquivalence = record

+
  { isEquivalence = ≈-isEquivalence

+
  ; _≟_           = _≈?_

+
  }

+

+
≈-decSetoid : DecSetoid _ _

+
≈-decSetoid = record

+
  { isDecEquivalence = ≈-isDecEquivalence

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality String

+
x ≟ y = map′ ≈⇒≡ ≈-reflexive $ x ≈? y

+

+
≡-setoid : Setoid _ _

+
≡-setoid = PropEq.setoid String

+

+
≡-decSetoid : DecSetoid _ _

+
≡-decSetoid = PropEq.decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+

+
infix 4 _<?_

+
_<?_ : Decidable _<_

+
x <? y = StrictLex.<-decidable Char._≟_ Char._<?_ (toList x) (toList y)

+

+
<-isStrictPartialOrder-≈ : IsStrictPartialOrder _≈_ _<_

+
<-isStrictPartialOrder-≈ =

+
  On.isStrictPartialOrder

+
    toList

+
    (StrictLex.<-isStrictPartialOrder Char.<-isStrictPartialOrder)

+

+
<-isStrictTotalOrder-≈ : IsStrictTotalOrder _≈_ _<_

+
<-isStrictTotalOrder-≈ =

+
  On.isStrictTotalOrder

+
    toList

+
    (StrictLex.<-isStrictTotalOrder Char.<-isStrictTotalOrder)

+

+
<-strictPartialOrder-≈ : StrictPartialOrder _ _ _

+
<-strictPartialOrder-≈ =

+
  On.strictPartialOrder

+
    (StrictLex.<-strictPartialOrder Char.<-strictPartialOrder)

+
    toList

+

+
<-strictTotalOrder-≈ : StrictTotalOrder _ _ _

+
<-strictTotalOrder-≈ =

+
  On.strictTotalOrder

+
    (StrictLex.<-strictTotalOrder Char.<-strictTotalOrder)

+
    toList

+

+
≤-isDecPartialOrder-≈ : IsDecPartialOrder _≈_ _≤_

+
≤-isDecPartialOrder-≈ =

+
  On.isDecPartialOrder

+
    toList

+
    (StrictLex.≤-isDecPartialOrder Char.<-isStrictTotalOrder)

+

+
≤-isDecTotalOrder-≈ : IsDecTotalOrder _≈_ _≤_

+
≤-isDecTotalOrder-≈ =

+
  On.isDecTotalOrder

+
    toList

+
    (StrictLex.≤-isDecTotalOrder Char.<-isStrictTotalOrder)

+

+
≤-decTotalOrder-≈ :  DecTotalOrder _ _ _

+
≤-decTotalOrder-≈ =

+
  On.decTotalOrder

+
    (StrictLex.≤-decTotalOrder Char.<-strictTotalOrder)

+
    toList

+

+
≤-decPoset-≈ : DecPoset _ _ _

+
≤-decPoset-≈ =

+
  On.decPoset

+
    (StrictLex.≤-decPoset Char.<-strictTotalOrder)

+
    toList

+

+
------------------------------------------------------------------------

+
-- Alternative Boolean equality test.

+
--

+
-- Why is the definition _==_ = primStringEquality not used? One

+
-- reason is that the present definition can sometimes improve type

+
-- inference, at least with the version of Agda that is current at the

+
-- time of writing: see unit-test below.

+

+
infix 4 _==_

+
_==_ : String → String → Bool

+
s₁ == s₂ = isYes (s₁ ≟ s₂)

+

+
private

+

+
  -- The following unit test does not type-check (at the time of

+
  -- writing) if _==_ is replaced by primStringEquality.

+

+
  data P : (String → Bool) → Set where

+
    p : (c : String) → P (_==_ c)

+

+
  unit-test : P (_==_ "")

+
  unit-test = p _

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.String.html b/site/static/agda_html/Data.String.html
new file mode 100644
index 000000000..28d0d772a
--- /dev/null
+++ b/site/static/agda_html/Data.String.html
@@ -0,0 +1,147 @@
+
+Data.String
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Strings

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.String where

+

+
open import Data.Bool.Base using (if_then_else_)

+
open import Data.Char.Base as Char using (Char)

+
open import Function.Base using (_∘_; _$_)

+
open import Data.Nat.Base as ℕ using (ℕ)

+
import Data.Nat.Properties as ℕ

+
open import Data.List.Base as List using (List)

+
open import Data.List.Extrema ℕ.≤-totalOrder

+
open import Data.Vec.Base as Vec using (Vec)

+
open import Data.Char.Base as Char using (Char)

+
import Data.Char.Properties as Char using (_≟_)

+
open import Relation.Nullary.Decidable.Core using (does)

+

+
open import Data.List.Membership.DecPropositional Char._≟_

+

+

+
------------------------------------------------------------------------

+
-- Re-export contents of base, and decidability of equality

+

+
open import Data.String.Base public

+
open import Data.String.Properties using (_≈?_; _≟_; _<?_; _==_) public

+

+
------------------------------------------------------------------------

+
-- Conversion functions

+

+
toVec : (s : String) → Vec Char (length s)

+
toVec s = Vec.fromList (toList s)

+

+
fromVec : ∀ {n} → Vec Char n → String

+
fromVec = fromList ∘ Vec.toList

+

+

+
-- enclose string with parens if it contains a space character

+
parensIfSpace : String → String

+
parensIfSpace s = if does (' ' ∈? toList s) then parens s else s

+

+

+
------------------------------------------------------------------------

+
-- Rectangle

+

+
-- Build a rectangular column by:

+
-- Given a vector of cells and a padding function for each one

+
-- Compute the max of the widths, and pad the strings accordingly.

+

+
rectangle : ∀ {n} → Vec (ℕ → String → String) n →

+
            Vec String n → Vec String n

+
rectangle pads cells = Vec.zipWith (λ p c → p width c) pads cells where

+

+
  sizes = List.map length (Vec.toList cells)

+
  width = max 0 sizes

+

+
-- Special cases for left, center, and right alignment

+

+
rectangleˡ : ∀ {n} → Char → Vec String n → Vec String n

+
rectangleˡ c = rectangle (Vec.replicate _ $ padLeft c)

+

+
rectangleʳ : ∀ {n} → Char → Vec String n → Vec String n

+
rectangleʳ c = rectangle (Vec.replicate _ $ padRight c)

+

+
rectangleᶜ : ∀ {n} → Char → Char → Vec String n → Vec String n

+
rectangleᶜ cₗ cᵣ = rectangle (Vec.replicate _ $ padBoth cₗ cᵣ)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Algebra.html b/site/static/agda_html/Data.Sum.Algebra.html
new file mode 100644
index 000000000..af87359cb
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Algebra.html
@@ -0,0 +1,197 @@
+
+Data.Sum.Algebra
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Algebraic properties of sums (disjoint unions)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Algebra where

+

+
open import Algebra.Bundles

+
  using (Magma; Semigroup; Monoid; CommutativeMonoid)

+
open import Algebra.Definitions

+
open import Algebra.Structures

+
  using (IsMagma; IsSemigroup; IsMonoid; IsCommutativeMonoid)

+
open import Data.Empty.Polymorphic using (⊥)

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; map; [_,_]; swap; assocʳ; assocˡ)

+
open import Data.Sum.Properties using (swap-involutive)

+
open import Data.Unit.Polymorphic.Base using (⊤; tt)

+
open import Function.Base using (id; _∘_)

+
open import Function.Properties.Inverse using (↔-isEquivalence)

+
open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)

+
open import Level using (Level; suc)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong′)

+

+
import Function.Definitions as FuncDef

+

+
------------------------------------------------------------------------

+
-- Setup

+

+
private

+
  variable

+
    a b c d : Level

+
    A B C D : Set a

+

+
  ♯ : {B : ⊥ {a} → Set b} → (w : ⊥) → B w

+
  ♯ ()

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
⊎-cong : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)

+
⊎-cong i j = mk↔ₛ′ (map I.to J.to) (map I.from J.from)

+
  [ cong inj₁ ∘ I.strictlyInverseˡ , cong inj₂ ∘ J.strictlyInverseˡ ]

+
  [ cong inj₁ ∘ I.strictlyInverseʳ , cong inj₂ ∘ J.strictlyInverseʳ ]

+
  where module I = Inverse i; module J = Inverse j

+

+
-- ⊎ is commutative.

+
-- We don't use Commutative because it isn't polymorphic enough.

+
⊎-comm : (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)

+
⊎-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive

+

+
module _ (ℓ : Level) where

+

+
  -- ⊎ is associative

+
  ⊎-assoc : Associative {ℓ = ℓ} _↔_ _⊎_

+
  ⊎-assoc _ _ _ = mk↔ₛ′ assocʳ assocˡ

+
    [ cong′ , [ cong′ , cong′ ] ] [ [ cong′ , cong′ ] , cong′ ]

+

+
  -- ⊥ is an identity for ⊎

+
  ⊎-identityˡ : LeftIdentity {ℓ = ℓ} _↔_ ⊥ _⊎_

+
  ⊎-identityˡ A = mk↔ₛ′ [ ♯ , id ] inj₂ cong′ [ ♯ , cong′ ]

+

+
  ⊎-identityʳ : RightIdentity {ℓ = ℓ} _↔_ ⊥ _⊎_

+
  ⊎-identityʳ _ = mk↔ₛ′ [ id , ♯ ] inj₁ cong′ [ cong′ , ♯ ]

+

+
  ⊎-identity : Identity _↔_ ⊥ _⊎_

+
  ⊎-identity = ⊎-identityˡ , ⊎-identityʳ

+

+
------------------------------------------------------------------------

+
-- Algebraic structures

+

+
  ⊎-isMagma : IsMagma {ℓ = ℓ} _↔_ _⊎_

+
  ⊎-isMagma = record

+
    { isEquivalence = ↔-isEquivalence

+
    ; ∙-cong        = ⊎-cong

+
    }

+

+
  ⊎-isSemigroup : IsSemigroup _↔_ _⊎_

+
  ⊎-isSemigroup = record

+
    { isMagma = ⊎-isMagma

+
    ; assoc   = ⊎-assoc

+
    }

+

+
  ⊎-isMonoid : IsMonoid _↔_ _⊎_ ⊥

+
  ⊎-isMonoid = record

+
    { isSemigroup = ⊎-isSemigroup

+
    ; identity    = ⊎-identityˡ , ⊎-identityʳ

+
    }

+

+
  ⊎-isCommutativeMonoid : IsCommutativeMonoid _↔_ _⊎_ ⊥

+
  ⊎-isCommutativeMonoid = record

+
    { isMonoid = ⊎-isMonoid

+
    ; comm     = ⊎-comm

+
    }

+

+
------------------------------------------------------------------------

+
-- Algebraic bundles

+

+
  ⊎-magma : Magma (suc ℓ) ℓ

+
  ⊎-magma = record

+
    { isMagma = ⊎-isMagma

+
    }

+

+
  ⊎-semigroup : Semigroup (suc ℓ) ℓ

+
  ⊎-semigroup = record

+
    { isSemigroup = ⊎-isSemigroup

+
    }

+

+
  ⊎-monoid : Monoid (suc ℓ) ℓ

+
  ⊎-monoid = record

+
    { isMonoid = ⊎-isMonoid

+
    }

+

+
  ⊎-commutativeMonoid : CommutativeMonoid (suc ℓ) ℓ

+
  ⊎-commutativeMonoid = record

+
    { isCommutativeMonoid = ⊎-isCommutativeMonoid

+
    }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Base.html b/site/static/agda_html/Data.Sum.Base.html
new file mode 100644
index 000000000..52e6d3e63
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Base.html
@@ -0,0 +1,151 @@
+
+Data.Sum.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sums (disjoint unions)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Base where

+

+
open import Data.Bool.Base using (true; false)

+
open import Function.Base using (_∘_; _∘′_; _-⟪_⟫-_ ; id)

+
open import Level using (Level; _⊔_)

+

+
private

+
  variable

+
    a b c d : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infixr 1 _⊎_

+

+
data _⊎_ (A : Set a) (B : Set b) : Set (a ⊔ b) where

+
  inj₁ : (x : A) → A ⊎ B

+
  inj₂ : (y : B) → A ⊎ B

+

+
------------------------------------------------------------------------

+
-- Functions

+

+
[_,_] : ∀ {C : A ⊎ B → Set c} →

+
        ((x : A) → C (inj₁ x)) → ((x : B) → C (inj₂ x)) →

+
        ((x : A ⊎ B) → C x)

+
[ f , g ] (inj₁ x) = f x

+
[ f , g ] (inj₂ y) = g y

+

+
[_,_]′ : (A → C) → (B → C) → (A ⊎ B → C)

+
[_,_]′ = [_,_]

+

+
fromInj₁ : (B → A) → A ⊎ B → A

+
fromInj₁ = [ id ,_]′

+

+
fromInj₂ : (A → B) → A ⊎ B → B

+
fromInj₂ = [_, id ]′

+

+
reduce : A ⊎ A → A

+
reduce = [ id , id ]′

+

+
swap : A ⊎ B → B ⊎ A

+
swap (inj₁ x) = inj₂ x

+
swap (inj₂ x) = inj₁ x

+

+
map : (A → C) → (B → D) → (A ⊎ B → C ⊎ D)

+
map f g = [ inj₁ ∘ f , inj₂ ∘ g ]′

+

+
map₁ : (A → C) → (A ⊎ B → C ⊎ B)

+
map₁ f = map f id

+

+
map₂ : (B → D) → (A ⊎ B → A ⊎ D)

+
map₂ = map id

+

+
assocʳ : (A ⊎ B) ⊎ C → A ⊎ B ⊎ C

+
assocʳ = [ map₂ inj₁ , inj₂ ∘′ inj₂ ]′

+

+
assocˡ : A ⊎ B ⊎ C → (A ⊎ B) ⊎ C

+
assocˡ = [ inj₁ ∘′ inj₁ , map₁ inj₂ ]′

+

+
infixr 1 _-⊎-_

+
_-⊎-_ : (A → B → Set c) → (A → B → Set d) → (A → B → Set (c ⊔ d))

+
f -⊎- g = f -⟪ _⊎_ ⟫- g

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Effectful.Left.html b/site/static/agda_html/Data.Sum.Effectful.Left.html
new file mode 100644
index 000000000..c5c854170
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Effectful.Left.html
@@ -0,0 +1,171 @@
+
+Data.Sum.Effectful.Left
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An effectful view of the Sum type (Left-biased)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Level

+

+
module Data.Sum.Effectful.Left {a} (A : Set a) (b : Level) where

+

+
open import Data.Sum.Base

+
open import Effect.Choice

+
open import Effect.Empty

+
open import Effect.Functor

+
open import Effect.Applicative

+
open import Effect.Monad

+
open import Function.Base

+

+
-- To minimize the universe level of the RawFunctor, we require that

+
-- elements of B are "lifted" to a copy of B at a higher universe level

+
-- (a ⊔ b). See the examples for how this is done.

+

+
------------------------------------------------------------------------

+
-- Left-biased monad instance for _⊎_

+

+
Sumₗ : Set (a ⊔ b) → Set (a ⊔ b)

+
Sumₗ B = A ⊎ B

+

+
functor : RawFunctor Sumₗ

+
functor = record { _<$>_ = map₂ }

+

+
applicative : RawApplicative Sumₗ

+
applicative = record

+
  { rawFunctor = functor

+
  ; pure = inj₂

+
  ; _<*>_ = [ const ∘ inj₁ , map₂ ]′

+
  }

+

+
empty : A → RawEmpty Sumₗ

+
empty a = record { empty = inj₁ a }

+

+
choice : RawChoice Sumₗ

+
choice = record { _<|>_ = [ flip const , const ∘ inj₂ ]′ }

+

+
applicativeZero : A → RawApplicativeZero Sumₗ

+
applicativeZero a = record

+
  { rawApplicative = applicative

+
  ; rawEmpty = empty a

+
  }

+

+
alternative : A → RawAlternative Sumₗ

+
alternative a = record

+
  { rawApplicativeZero = applicativeZero a

+
  ; rawChoice = choice

+
  }

+

+
monad : RawMonad Sumₗ

+
monad = record

+
  { rawApplicative = applicative

+
  ; _>>=_ = [ const ∘′ inj₁ , _|>′_ ]′

+
  }

+

+
join : {B : Set (a ⊔ b)} → Sumₗ (Sumₗ B) → Sumₗ B

+
join = Join.join monad

+

+
------------------------------------------------------------------------

+
-- Get access to other monadic functions

+

+
module TraversableA {F} (App : RawApplicative {a ⊔ b} {a ⊔ b} F) where

+

+
  open RawApplicative App

+

+
  sequenceA : ∀ {A} → Sumₗ (F A) → F (Sumₗ A)

+
  sequenceA (inj₁ a) = pure (inj₁ a)

+
  sequenceA (inj₂ x) = inj₂ <$> x

+

+
  mapA : ∀ {A B} → (A → F B) → Sumₗ A → F (Sumₗ B)

+
  mapA f = sequenceA ∘ map₂ f

+

+
  forA : ∀ {A B} → Sumₗ A → (A → F B) → F (Sumₗ B)

+
  forA = flip mapA

+

+
module TraversableM {M} (Mon : RawMonad {a ⊔ b} {a ⊔ b} M) where

+

+
  open RawMonad Mon

+

+
  open TraversableA rawApplicative public

+
    renaming

+
    ( sequenceA to sequenceM

+
    ; mapA      to mapM

+
    ; forA      to forM

+
    )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Function.Propositional.html b/site/static/agda_html/Data.Sum.Function.Propositional.html
new file mode 100644
index 000000000..6fc3b5a44
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Function.Propositional.html
@@ -0,0 +1,163 @@
+
+Data.Sum.Function.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sum combinators for propositional equality preserving functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Function.Propositional where

+

+
open import Data.Sum.Base using (_⊎_)

+
open import Data.Sum.Function.Setoid

+
open import Data.Sum.Relation.Binary.Pointwise using (Pointwise-≡↔≡; _⊎ₛ_)

+
open import Function.Construct.Composition as Compose

+
open import Function.Related.Propositional

+
  using (_∼[_]_; implication; reverseImplication; equivalence; injection;

+
    reverseInjection; leftInverse; surjection; bijection)

+
open import Function.Base using (id)

+
open import Function.Bundles

+
  using (Inverse; _⟶_; _⇔_; _↣_; _↠_; _↩_; _↪_; _⤖_; _↔_)

+
open import Function.Properties.Inverse as Inv

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (REL)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
private

+
  variable

+
    a b c d : Level

+
    A B C D : Set a

+

+

+
------------------------------------------------------------------------

+
-- Helper lemma

+

+
private

+
  liftViaInverse : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →

+
                   (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →

+
                   (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →

+
                   (R (setoid A) (setoid C) → R (setoid B) (setoid D) → R (setoid A ⊎ₛ setoid B) (setoid C ⊎ₛ setoid D)) →

+
                   R (setoid A) (setoid C) → R (setoid B) (setoid D) →

+
                   R (setoid (A ⊎ B)) (setoid (C ⊎ D))

+
  liftViaInverse trans inv⇒R lift RAC RBD =

+
    Inv.transportVia trans inv⇒R (Inv.sym (Pointwise-≡↔≡ _ _)) (lift RAC RBD) (Pointwise-≡↔≡ _ _)

+

+
------------------------------------------------------------------------

+
-- Combinators for various function types

+

+
infixr 1 _⊎-⟶_ _⊎-⇔_ _⊎-↣_ _⊎-↩_ _⊎-↪_ _⊎-↔_

+

+
_⊎-⟶_ : A ⟶ B → C ⟶ D → (A ⊎ C) ⟶ (B ⊎ D)

+
_⊎-⟶_ = liftViaInverse Compose.function Inv.toFunction _⊎-function_

+

+

+
_⊎-⇔_ : A ⇔ B → C ⇔ D → (A ⊎ C) ⇔ (B ⊎ D)

+
_⊎-⇔_ = liftViaInverse Compose.equivalence Inverse⇒Equivalence _⊎-equivalence_

+

+
_⊎-↣_ : A ↣ B → C ↣ D → (A ⊎ C) ↣ (B ⊎ D)

+
_⊎-↣_ = liftViaInverse Compose.injection Inverse⇒Injection _⊎-injection_

+

+
_⊎-↠_ : A ↠ B → C ↠ D → (A ⊎ C) ↠ (B ⊎ D)

+
_⊎-↠_ = liftViaInverse Compose.surjection Inverse⇒Surjection _⊎-surjection_

+

+
_⊎-↩_ : A ↩ B → C ↩ D → (A ⊎ C) ↩ (B ⊎ D)

+
_⊎-↩_ = liftViaInverse Compose.leftInverse Inverse.leftInverse _⊎-leftInverse_

+

+
_⊎-↪_ : A ↪ B → C ↪ D → (A ⊎ C) ↪ (B ⊎ D)

+
_⊎-↪_ = liftViaInverse Compose.rightInverse Inverse.rightInverse _⊎-rightInverse_

+

+
_⊎-⤖_ : A ⤖ B → C ⤖ D → (A ⊎ C) ⤖ (B ⊎ D)

+
_⊎-⤖_ = liftViaInverse Compose.bijection Inverse⇒Bijection _⊎-bijection_

+

+
_⊎-↔_ : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)

+
_⊎-↔_ = liftViaInverse Compose.inverse id _⊎-inverse_

+

+
infixr 1 _⊎-cong_

+

+
_⊎-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A ⊎ C) ∼[ k ] (B ⊎ D)

+
_⊎-cong_ {k = implication}         = _⊎-⟶_

+
_⊎-cong_ {k = reverseImplication}  = _⊎-⟶_

+
_⊎-cong_ {k = equivalence}         = _⊎-⇔_

+
_⊎-cong_ {k = injection}           = _⊎-↣_

+
_⊎-cong_ {k = reverseInjection}    = _⊎-↣_

+
_⊎-cong_ {k = leftInverse}         = _⊎-↪_

+
_⊎-cong_ {k = surjection}          = _⊎-↠_

+
_⊎-cong_ {k = bijection}           = _⊎-↔_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Function.Setoid.html b/site/static/agda_html/Data.Sum.Function.Setoid.html
new file mode 100644
index 000000000..8e8357662
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Function.Setoid.html
@@ -0,0 +1,251 @@
+
+Data.Sum.Function.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sum combinators for setoid equality preserving functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Function.Setoid where

+

+
open import Data.Product.Base as Product using (_,_)

+
open import Data.Sum.Base as Sum

+
open import Data.Sum.Relation.Binary.Pointwise as Pointwise

+
open import Relation.Binary

+
open import Function.Base

+
open import Function.Bundles

+
open import Function.Definitions

+
open import Level

+

+
private

+
  variable

+
    a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂ : Level

+
    a ℓ : Level

+
    A B C D : Set a

+
    ≈₁ ≈₂ ≈₃ ≈₄ : Rel A ℓ

+
    S T U V : Setoid a ℓ

+

+
------------------------------------------------------------------------

+
-- Combinators for equality preserving functions

+

+
inj₁ₛ : Func S (S ⊎ₛ T)

+
inj₁ₛ = record { to = inj₁ ; cong = inj₁ }

+

+
inj₂ₛ : Func T (S ⊎ₛ T)

+
inj₂ₛ = record { to = inj₂ ; cong = inj₂ }

+

+
[_,_]ₛ : Func S U → Func T U → Func (S ⊎ₛ T) U

+
[ f , g ]ₛ = record

+
  { to   = [ to f , to g ]

+
  ; cong = λ where

+
    (inj₁ x∼₁y) → cong f x∼₁y

+
    (inj₂ x∼₂y) → cong g x∼₂y

+
  } where open Func

+

+
swapₛ : Func (S ⊎ₛ T) (T ⊎ₛ S)

+
swapₛ = [ inj₂ₛ , inj₁ₛ ]ₛ

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
⊎-injective : ∀ {f g} →

+
              Injective ≈₁ ≈₂ f →

+
              Injective ≈₃ ≈₄ g →

+
              Injective (Pointwise ≈₁ ≈₃) (Pointwise ≈₂ ≈₄) (Sum.map f g)

+
⊎-injective f-inj g-inj {inj₁ x} {inj₁ y} (inj₁ x∼₁y) = inj₁ (f-inj x∼₁y)

+
⊎-injective f-inj g-inj {inj₂ x} {inj₂ y} (inj₂ x∼₂y) = inj₂ (g-inj x∼₂y)

+

+
⊎-strictlySurjective : ∀ {f : A → B} {g : C → D} →

+
              StrictlySurjective ≈₁ f →

+
              StrictlySurjective ≈₂ g →

+
              StrictlySurjective (Pointwise ≈₁ ≈₂) (Sum.map f g)

+
⊎-strictlySurjective f-sur g-sur =

+
  [ Product.map inj₁ inj₁ ∘ f-sur

+
  , Product.map inj₂ inj₂ ∘ g-sur

+
  ]

+

+
⊎-surjective : ∀ {f : A → B} {g : C → D} →

+
              Surjective ≈₁ ≈₂ f →

+
              Surjective ≈₃ ≈₄ g →

+
              Surjective (Pointwise ≈₁ ≈₃) (Pointwise ≈₂ ≈₄) (Sum.map f g)

+
⊎-surjective f-sur g-sur =

+
  [ Product.map inj₁ (λ { fwd (inj₁ x) → inj₁ (fwd x)}) ∘ f-sur

+
  , Product.map inj₂ (λ { fwd (inj₂ y) → inj₂ (fwd y)}) ∘ g-sur

+
  ]

+

+

+
infixr 1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_

+

+
------------------------------------------------------------------------

+
-- Function bundles

+

+
_⊎-function_ : Func S T → Func U V → Func (S ⊎ₛ U) (T ⊎ₛ V)

+
S→T ⊎-function U→V = record

+
  { to    = Sum.map (to S→T) (to U→V)

+
  ; cong  = Pointwise.map (cong S→T) (cong U→V)

+
  } where open Func

+

+
_⊎-equivalence_ : Equivalence S T → Equivalence U V →

+
                  Equivalence (S ⊎ₛ U) (T ⊎ₛ V)

+
S⇔T ⊎-equivalence U⇔V = record

+
  { to        = Sum.map (to S⇔T) (to U⇔V)

+
  ; from      = Sum.map (from S⇔T) (from U⇔V)

+
  ; to-cong   = Pointwise.map (to-cong S⇔T) (to-cong U⇔V)

+
  ; from-cong = Pointwise.map (from-cong S⇔T) (from-cong U⇔V)

+
  } where open Equivalence

+

+
_⊎-injection_ : Injection S T → Injection U V →

+
                Injection (S ⊎ₛ U) (T ⊎ₛ V)

+
S↣T ⊎-injection U↣V = record

+
  { to        = Sum.map (to S↣T) (to U↣V)

+
  ; cong      = Pointwise.map (cong S↣T) (cong U↣V)

+
  ; injective = ⊎-injective (injective S↣T) (injective U↣V)

+
  } where open Injection

+

+
infixr 1 _⊎-surjection_ _⊎-inverse_

+
_⊎-surjection_ : Surjection S T → Surjection U V →

+
                 Surjection (S ⊎ₛ U) (T ⊎ₛ V)

+
S↠T ⊎-surjection U↠V = record

+
  { to              = Sum.map (to S↠T) (to U↠V)

+
  ; cong            = Pointwise.map (cong S↠T) (cong U↠V)

+
  ; surjective      = ⊎-surjective (surjective S↠T) (surjective U↠V)

+
  } where open Surjection

+

+
_⊎-bijection_ : Bijection S T → Bijection U V →

+
                 Bijection (S ⊎ₛ U) (T ⊎ₛ V)

+
S⤖T ⊎-bijection U⤖V = record

+
  { to        = Sum.map (to S⤖T) (to U⤖V)

+
  ; cong      = Pointwise.map (cong S⤖T) (cong U⤖V)

+
  ; bijective = ⊎-injective (injective S⤖T) (injective U⤖V) ,

+
                ⊎-surjective (surjective S⤖T) (surjective U⤖V)

+
  } where open Bijection

+

+
_⊎-leftInverse_ : LeftInverse S T → LeftInverse U V →

+
                  LeftInverse (S ⊎ₛ U) (T ⊎ₛ V)

+
S↩T ⊎-leftInverse U↩V = record

+
  { to              = Sum.map (to S↩T) (to U↩V)

+
  ; from            = Sum.map (from S↩T) (from U↩V)

+
  ; to-cong         = Pointwise.map (to-cong S↩T) (to-cong U↩V)

+
  ; from-cong       = Pointwise.map (from-cong S↩T) (from-cong U↩V)

+
  ; inverseˡ        = λ { {inj₁ _} {.(inj₁ _)} (inj₁ x) → inj₁ (inverseˡ S↩T x)

+
                        ; {inj₂ _} {.(inj₂ _)} (inj₂ x) → inj₂ (inverseˡ U↩V x)}

+
  } where open LeftInverse

+

+
_⊎-rightInverse_ : RightInverse S T → RightInverse U V →

+
                   RightInverse (S ⊎ₛ U) (T ⊎ₛ V)

+
S↪T ⊎-rightInverse U↪V = record

+
  { to              = Sum.map (to S↪T) (to U↪V)

+
  ; from            = Sum.map (from S↪T) (from U↪V)

+
  ; to-cong         = Pointwise.map (to-cong S↪T) (to-cong U↪V)

+
  ; from-cong       = Pointwise.map (from-cong S↪T) (from-cong U↪V)

+
  ; inverseʳ        = λ { {inj₁ _} (inj₁ x) → inj₁ (inverseʳ S↪T x)

+
                        ; {inj₂ _} (inj₂ x) → inj₂ (inverseʳ U↪V x)

+
                        }

+
  } where open RightInverse

+

+
_⊎-inverse_ : Inverse S T → Inverse U V →

+
              Inverse (S ⊎ₛ U) (T ⊎ₛ V)

+
S↔T ⊎-inverse U↔V = record

+
  { to        = Sum.map (to S↔T) (to U↔V)

+
  ; from      = Sum.map (from S↔T) (from U↔V)

+
  ; to-cong   = Pointwise.map (to-cong S↔T) (to-cong U↔V)

+
  ; from-cong = Pointwise.map (from-cong S↔T) (from-cong U↔V)

+
  ; inverse   = (λ { {inj₁ _} (inj₁ x) → inj₁ (inverseˡ S↔T x)

+
                   ; {inj₂ _} (inj₂ x) → inj₂ (inverseˡ U↔V x)}) ,

+
                 λ { {inj₁ _} (inj₁ x) → inj₁ (inverseʳ S↔T x)

+
                   ; {inj₂ _} (inj₂ x) → inj₂ (inverseʳ U↔V x)

+
                   }

+
  } where open Inverse

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
_⊎-left-inverse_ = _⊎-leftInverse_

+
{-# WARNING_ON_USAGE _⊎-left-inverse_

+
"Warning: _⊎-left-inverse_ was deprecated in v2.0.

+
Please use _⊎-leftInverse_ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Properties.html b/site/static/agda_html/Data.Sum.Properties.html
new file mode 100644
index 000000000..159363d58
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Properties.html
@@ -0,0 +1,229 @@
+
+Data.Sum.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of sums (disjoint unions)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Properties where

+

+
open import Level using (Level)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; swap; [_,_]; map;

+
  map₁; map₂; assocˡ; assocʳ)

+
open import Function.Base using (_∋_; _∘_; id)

+
open import Function.Bundles using (mk↔ₛ′; _↔_)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≗_; refl; cong)

+
open import Relation.Nullary.Decidable.Core using (yes; no; map′)

+

+
private

+
  variable

+
    a b c d e f : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+
    E : Set e

+
    F : Set f

+

+
inj₁-injective : ∀ {x y} → (A ⊎ B ∋ inj₁ x) ≡ inj₁ y → x ≡ y

+
inj₁-injective refl = refl

+

+
inj₂-injective : ∀ {x y} → (A ⊎ B ∋ inj₂ x) ≡ inj₂ y → x ≡ y

+
inj₂-injective refl = refl

+

+
module _ (dec₁ : DecidableEquality A)

+
         (dec₂ : DecidableEquality B) where

+

+
  ≡-dec : DecidableEquality (A ⊎ B)

+
  ≡-dec (inj₁ x) (inj₁ y) = map′ (cong inj₁) inj₁-injective (dec₁ x y)

+
  ≡-dec (inj₁ x) (inj₂ y) = no λ()

+
  ≡-dec (inj₂ x) (inj₁ y) = no λ()

+
  ≡-dec (inj₂ x) (inj₂ y) = map′ (cong inj₂) inj₂-injective (dec₂ x y)

+

+
swap-involutive : swap {A = A} {B = B} ∘ swap ≗ id

+
swap-involutive = [ (λ _ → refl) , (λ _ → refl) ]

+

+
swap-↔ : (A ⊎ B) ↔ (B ⊎ A)

+
swap-↔ = mk↔ₛ′ swap swap swap-involutive swap-involutive

+

+
map-id : map {A = A} {B = B} id id ≗ id

+
map-id (inj₁ _) = refl

+
map-id (inj₂ _) = refl

+

+
[,]-∘ : (f : A → B)

+
              {g : C → A} {h : D → A} →

+
              f ∘ [ g , h ] ≗ [ f ∘ g , f ∘ h ]

+
[,]-∘ _ (inj₁ _) = refl

+
[,]-∘ _ (inj₂ _) = refl

+

+
[,]-map : {f : A → B}  {g : C → D}

+
          {f′ : B → E} {g′ : D → E} →

+
          [ f′ , g′ ] ∘ map f g ≗ [ f′ ∘ f , g′ ∘ g ]

+
[,]-map (inj₁ _) = refl

+
[,]-map (inj₂ _) = refl

+

+
map-map : {f : A → B}  {g : C → D}

+
              {f′ : B → E} {g′ : D → F} →

+
              map f′ g′ ∘ map f g ≗ map (f′ ∘ f) (g′ ∘ g)

+
map-map (inj₁ _) = refl

+
map-map (inj₂ _) = refl

+

+
map₁₂-map₂₁ : {f : A → B} {g : C → D} →

+
                map₁ f ∘ map₂ g ≗ map₂ g ∘ map₁ f

+
map₁₂-map₂₁ (inj₁ _) = refl

+
map₁₂-map₂₁ (inj₂ _) = refl

+

+
map-assocˡ : (f : A → C) (g : B → D) (h : C → F) →

+
  map (map f g) h ∘ assocˡ ≗ assocˡ ∘ map f (map g h)

+
map-assocˡ _ _ _ (inj₁       x ) = refl

+
map-assocˡ _ _ _ (inj₂ (inj₁ y)) = refl

+
map-assocˡ _ _ _ (inj₂ (inj₂ z)) = refl

+

+
map-assocʳ : (f : A → C) (g : B → D) (h : C → F) →

+
  map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h

+
map-assocʳ _ _ _ (inj₁ (inj₁ x)) = refl

+
map-assocʳ _ _ _ (inj₁ (inj₂ y)) = refl

+
map-assocʳ _ _ _ (inj₂       z ) = refl

+

+
[,]-cong : {f f′ : A → B} {g g′ : C → B} →

+
           f ≗ f′ → g ≗ g′ →

+
           [ f , g ] ≗ [ f′ , g′ ]

+
[,]-cong = [_,_]

+

+
[-,]-cong : {f f′ : A → B} {g : C → B} →

+
            f ≗ f′ →

+
            [ f , g ] ≗ [ f′ , g ]

+
[-,]-cong = [_, (λ _ → refl) ]

+

+
[,-]-cong : {f : A → B} {g g′ : C → B} →

+
            g ≗ g′ →

+
            [ f , g ] ≗ [ f , g′ ]

+
[,-]-cong = [ (λ _ → refl) ,_]

+

+
map-cong : {f f′ : A → B} {g g′ : C → D} →

+
           f ≗ f′ → g ≗ g′ →

+
           map f g ≗ map f′ g′

+
map-cong f≗f′ g≗g′ (inj₁ x) = cong inj₁ (f≗f′ x)

+
map-cong f≗f′ g≗g′ (inj₂ x) = cong inj₂ (g≗g′ x)

+

+
map₁-cong : {f f′ : A → B} →

+
            f ≗ f′ →

+
            map₁ {B = C} f ≗ map₁ f′

+
map₁-cong f≗f′ = [-,]-cong ((cong inj₁) ∘ f≗f′)

+

+
map₂-cong : {g g′ : C → D} →

+
            g ≗ g′ →

+
            map₂ {A = A} g ≗ map₂ g′

+
map₂-cong g≗g′ = [,-]-cong ((cong inj₂) ∘ g≗g′)

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
[,]-∘-distr = [,]-∘

+
{-# WARNING_ON_USAGE [,]-∘-distr

+
"Warning: [,]-∘-distr was deprecated in v2.0.

+
Please use [,]-∘ instead."

+
#-}

+

+
[,]-map-commute = [,]-map

+
{-# WARNING_ON_USAGE [,]-map-commute

+
"Warning: [,]-map-commute was deprecated in v2.0.

+
Please use [,]-map instead."

+
#-}

+

+
map-commute = map-map

+
{-# WARNING_ON_USAGE map-commute

+
"Warning: map-commute was deprecated in v2.0.

+
Please use map-map instead."

+
#-}

+

+
map₁₂-commute = map₁₂-map₂₁

+
{-# WARNING_ON_USAGE map₁₂-commute

+
"Warning: map₁₂-commute was deprecated in v2.0.

+
Please use map₁₂-map₂₁ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Relation.Binary.Pointwise.html b/site/static/agda_html/Data.Sum.Relation.Binary.Pointwise.html
new file mode 100644
index 000000000..0e1bfdf28
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Relation.Binary.Pointwise.html
@@ -0,0 +1,295 @@
+
+Data.Sum.Relation.Binary.Pointwise
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Pointwise sum

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Relation.Binary.Pointwise where

+

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)

+
open import Data.Sum.Properties

+
open import Level using (Level; _⊔_)

+
open import Function.Base using (const; _∘_; id)

+
open import Function.Bundles using (Inverse; mk↔)

+
open import Relation.Nullary

+
import Relation.Nullary.Decidable as Dec

+
open import Relation.Binary

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+

+
private

+
  variable

+
    a b c d ℓ₁ ℓ₂ ℓ₃ ℓ : Level

+
    A B C D : Set ℓ

+
    R S T U : REL A B ℓ

+
    ≈₁ ≈₂ : Rel A ℓ

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Pointwise {A : Set a} {B : Set b} {C : Set c} {D : Set d}

+
               (R : REL A C ℓ₁) (S : REL B D ℓ₂)

+
               : REL (A ⊎ B) (C ⊎ D) (a ⊔ b ⊔ c ⊔ d ⊔ ℓ₁ ⊔ ℓ₂) where

+
  inj₁ : ∀ {a c} → R a c → Pointwise R S (inj₁ a) (inj₁ c)

+
  inj₂ : ∀ {b d} → S b d → Pointwise R S (inj₂ b) (inj₂ d)

+

+
----------------------------------------------------------------------

+
-- Functions

+

+
map : ∀ {f : A → C} {g : B → D} →

+
      R =[ f ]⇒ T → S =[ g ]⇒ U →

+
      Pointwise R S =[ Sum.map f g ]⇒ Pointwise T U

+
map R⇒T _ (inj₁ x) = inj₁ (R⇒T x)

+
map _ S⇒U (inj₂ x) = inj₂ (S⇒U x)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
drop-inj₁ : ∀ {x y} → Pointwise R S (inj₁ x) (inj₁ y) → R x y

+
drop-inj₁ (inj₁ x) = x

+

+
drop-inj₂ : ∀ {x y} → Pointwise R S (inj₂ x) (inj₂ y) → S x y

+
drop-inj₂ (inj₂ x) = x

+

+
⊎-refl : Reflexive R → Reflexive S → Reflexive (Pointwise R S)

+
⊎-refl refl₁ refl₂ {inj₁ x} = inj₁ refl₁

+
⊎-refl refl₁ refl₂ {inj₂ y} = inj₂ refl₂

+

+
⊎-symmetric : Symmetric R → Symmetric S →

+
              Symmetric (Pointwise R S)

+
⊎-symmetric sym₁ sym₂ (inj₁ x) = inj₁ (sym₁ x)

+
⊎-symmetric sym₁ sym₂ (inj₂ x) = inj₂ (sym₂ x)

+

+
⊎-transitive : Transitive R → Transitive S →

+
               Transitive (Pointwise R S)

+
⊎-transitive trans₁ trans₂ (inj₁ x) (inj₁ y) = inj₁ (trans₁ x y)

+
⊎-transitive trans₁ trans₂ (inj₂ x) (inj₂ y) = inj₂ (trans₂ x y)

+

+
⊎-asymmetric : Asymmetric R → Asymmetric S →

+
               Asymmetric (Pointwise R S)

+
⊎-asymmetric asym₁ asym₂ (inj₁ x) = λ { (inj₁ y) → asym₁ x y }

+
⊎-asymmetric asym₁ asym₂ (inj₂ x) = λ { (inj₂ y) → asym₂ x y }

+

+
⊎-substitutive : Substitutive R ℓ₃ → Substitutive S ℓ₃ →

+
                 Substitutive (Pointwise R S) ℓ₃

+
⊎-substitutive subst₁ subst₂ P (inj₁ x) = subst₁ (P ∘ inj₁) x

+
⊎-substitutive subst₁ subst₂ P (inj₂ x) = subst₂ (P ∘ inj₂) x

+

+
⊎-decidable : Decidable R → Decidable S → Decidable (Pointwise R S)

+
⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₁ y) = Dec.map′ inj₁ drop-inj₁ (x ≟₁ y)

+
⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₂ y) = no λ()

+
⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₁ y) = no λ()

+
⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₂ y) = Dec.map′ inj₂ drop-inj₂ (x ≟₂ y)

+

+
⊎-reflexive : ≈₁ ⇒ R → ≈₂ ⇒ S →

+
              (Pointwise ≈₁ ≈₂) ⇒ (Pointwise R S)

+
⊎-reflexive refl₁ refl₂ (inj₁ x) = inj₁ (refl₁ x)

+
⊎-reflexive refl₁ refl₂ (inj₂ x) = inj₂ (refl₂ x)

+

+
⊎-irreflexive : Irreflexive ≈₁ R → Irreflexive ≈₂ S →

+
                Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
⊎-irreflexive irrefl₁ irrefl₂ (inj₁ x) (inj₁ y) = irrefl₁ x y

+
⊎-irreflexive irrefl₁ irrefl₂ (inj₂ x) (inj₂ y) = irrefl₂ x y

+

+
⊎-antisymmetric : Antisymmetric ≈₁ R → Antisymmetric ≈₂ S →

+
                  Antisymmetric (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
⊎-antisymmetric antisym₁ antisym₂ (inj₁ x) (inj₁ y) = inj₁ (antisym₁ x y)

+
⊎-antisymmetric antisym₁ antisym₂ (inj₂ x) (inj₂ y) = inj₂ (antisym₂ x y)

+

+
⊎-respectsˡ : R Respectsˡ ≈₁ → S Respectsˡ ≈₂ →

+
              (Pointwise R S) Respectsˡ (Pointwise ≈₁ ≈₂)

+
⊎-respectsˡ resp₁ resp₂ (inj₁ x) (inj₁ y) = inj₁ (resp₁ x y)

+
⊎-respectsˡ resp₁ resp₂ (inj₂ x) (inj₂ y) = inj₂ (resp₂ x y)

+

+
⊎-respectsʳ : R Respectsʳ ≈₁ → S Respectsʳ ≈₂ →

+
              (Pointwise R S) Respectsʳ (Pointwise ≈₁ ≈₂)

+
⊎-respectsʳ resp₁ resp₂ (inj₁ x) (inj₁ y) = inj₁ (resp₁ x y)

+
⊎-respectsʳ resp₁ resp₂ (inj₂ x) (inj₂ y) = inj₂ (resp₂ x y)

+

+
⊎-respects₂ : R Respects₂ ≈₁ → S Respects₂ ≈₂ →

+
              (Pointwise R S) Respects₂ (Pointwise ≈₁ ≈₂)

+
⊎-respects₂ (r₁ , l₁) (r₂ , l₂) = ⊎-respectsʳ r₁ r₂ , ⊎-respectsˡ l₁ l₂

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
⊎-isEquivalence : IsEquivalence ≈₁ → IsEquivalence ≈₂ →

+
                  IsEquivalence (Pointwise ≈₁ ≈₂)

+
⊎-isEquivalence eq₁ eq₂ = record

+
  { refl  = ⊎-refl       (refl  eq₁) (refl  eq₂)

+
  ; sym   = ⊎-symmetric  (sym   eq₁) (sym   eq₂)

+
  ; trans = ⊎-transitive (trans eq₁) (trans eq₂)

+
  } where open IsEquivalence

+

+
⊎-isDecEquivalence : IsDecEquivalence ≈₁ → IsDecEquivalence ≈₂ →

+
                     IsDecEquivalence (Pointwise ≈₁ ≈₂)

+
⊎-isDecEquivalence eq₁ eq₂ = record

+
  { isEquivalence =

+
      ⊎-isEquivalence (isEquivalence eq₁) (isEquivalence eq₂)

+
  ; _≟_           = ⊎-decidable (_≟_ eq₁) (_≟_ eq₂)

+
  } where open IsDecEquivalence

+

+
⊎-isPreorder : IsPreorder ≈₁ R → IsPreorder ≈₂ S →

+
               IsPreorder (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
⊎-isPreorder pre₁ pre₂ = record

+
  { isEquivalence =

+
      ⊎-isEquivalence (isEquivalence pre₁) (isEquivalence pre₂)

+
  ; reflexive     = ⊎-reflexive (reflexive pre₁) (reflexive pre₂)

+
  ; trans         = ⊎-transitive (trans pre₁) (trans pre₂)

+
  } where open IsPreorder

+

+
⊎-isPartialOrder : IsPartialOrder ≈₁ R → IsPartialOrder ≈₂ S →

+
                   IsPartialOrder

+
                     (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
⊎-isPartialOrder po₁ po₂ = record

+
  { isPreorder = ⊎-isPreorder (isPreorder po₁) (isPreorder po₂)

+
  ; antisym    = ⊎-antisymmetric (antisym po₁) (antisym    po₂)

+
  } where open IsPartialOrder

+

+
⊎-isStrictPartialOrder : IsStrictPartialOrder ≈₁ R →

+
                         IsStrictPartialOrder ≈₂ S →

+
                         IsStrictPartialOrder

+
                           (Pointwise ≈₁ ≈₂) (Pointwise R S)

+
⊎-isStrictPartialOrder spo₁ spo₂ = record

+
  { isEquivalence =

+
      ⊎-isEquivalence (isEquivalence spo₁) (isEquivalence spo₂)

+
  ; irrefl        = ⊎-irreflexive (irrefl   spo₁) (irrefl   spo₂)

+
  ; trans         = ⊎-transitive  (trans    spo₁) (trans    spo₂)

+
  ; <-resp-≈      = ⊎-respects₂   (<-resp-≈ spo₁) (<-resp-≈ spo₂)

+
  } where open IsStrictPartialOrder

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
⊎-setoid : Setoid a b → Setoid c d → Setoid _ _

+
⊎-setoid s₁ s₂ = record

+
  { isEquivalence =

+
      ⊎-isEquivalence (isEquivalence s₁) (isEquivalence s₂)

+
  } where open Setoid

+

+
⊎-decSetoid : DecSetoid a b → DecSetoid c d → DecSetoid _ _

+
⊎-decSetoid ds₁ ds₂ = record

+
  { isDecEquivalence =

+
      ⊎-isDecEquivalence (isDecEquivalence ds₁) (isDecEquivalence ds₂)

+
  } where open DecSetoid

+

+
⊎-preorder : Preorder a b ℓ₁ → Preorder c d ℓ₂ → Preorder _ _ _

+
⊎-preorder p₁ p₂ = record

+
  { isPreorder   =

+
      ⊎-isPreorder (isPreorder p₁) (isPreorder p₂)

+
  } where open Preorder

+

+
⊎-poset : Poset a b c → Poset a b c → Poset _ _ _

+
⊎-poset po₁ po₂ = record

+
  { isPartialOrder =

+
      ⊎-isPartialOrder (isPartialOrder po₁) (isPartialOrder po₂)

+
  } where open Poset

+

+
------------------------------------------------------------------------

+
-- Additional notation

+

+
-- Infix combining setoids

+
infix 4 _⊎ₛ_

+
_⊎ₛ_ : Setoid a b → Setoid c d → Setoid _ _

+
_⊎ₛ_ = ⊎-setoid

+

+
------------------------------------------------------------------------

+
-- The propositional equality setoid over products can be

+
-- decomposed using Pointwise

+

+
Pointwise-≡⇒≡ : (Pointwise _≡_ _≡_) ⇒ _≡_ {A = A ⊎ B}

+
Pointwise-≡⇒≡ (inj₁ x) = ≡.cong inj₁ x

+
Pointwise-≡⇒≡ (inj₂ x) = ≡.cong inj₂ x

+

+
≡⇒Pointwise-≡ : _≡_ {A = A ⊎ B} ⇒ (Pointwise _≡_ _≡_)

+
≡⇒Pointwise-≡ ≡.refl = ⊎-refl ≡.refl ≡.refl

+

+
Pointwise-≡↔≡ : (A : Set a) (B : Set b) →

+
                 Inverse (≡.setoid A ⊎ₛ ≡.setoid B) (≡.setoid (A ⊎ B))

+
Pointwise-≡↔≡ _ _ = record

+
  { to        = id

+
  ; from      = id

+
  ; to-cong   = Pointwise-≡⇒≡

+
  ; from-cong = ≡⇒Pointwise-≡

+
  ; inverse   = Pointwise-≡⇒≡ , ≡⇒Pointwise-≡

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.Relation.Unary.All.html b/site/static/agda_html/Data.Sum.Relation.Unary.All.html
new file mode 100644
index 000000000..ddd7e4d44
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.Relation.Unary.All.html
@@ -0,0 +1,116 @@
+
+Data.Sum.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Heterogeneous `All` predicate for disjoint sums

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum.Relation.Unary.All where

+

+
open import Data.Sum.Base  using (_⊎_; inj₁; inj₂)

+
open import Level          using (Level; _⊔_)

+
open import Relation.Unary using (Pred)

+

+
private

+
  variable

+
    a b c p q : Level

+
    A B : Set _

+
    P Q : Pred A p

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data All {A : Set a} {B : Set b} (P : Pred A p) (Q : Pred B q)

+
         : Pred (A ⊎ B) (a ⊔ b ⊔ p ⊔ q) where

+
  inj₁ : ∀ {a} → P a → All P Q (inj₁ a)

+
  inj₂ : ∀ {b} → Q b → All P Q (inj₂ b)

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
-- Elimination

+

+
[_,_] : ∀ {C : (x : A ⊎ B) → All P Q x → Set c} →

+
        ((x : A) (y : P x) → C (inj₁ x) (inj₁ y)) →

+
        ((x : B) (y : Q x) → C (inj₂ x) (inj₂ y)) →

+
        (x : A ⊎ B) (y : All P Q x) → C x y

+
[ f , g ] (inj₁ x) (inj₁ y) = f x y

+
[ f , g ] (inj₂ x) (inj₂ y) = g x y

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Sum.html b/site/static/agda_html/Data.Sum.html
new file mode 100644
index 000000000..7c16aaf2b
--- /dev/null
+++ b/site/static/agda_html/Data.Sum.html
@@ -0,0 +1,141 @@
+
+Data.Sum
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Sums (disjoint unions)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Sum where

+

+
open import Data.Unit.Polymorphic.Base using (⊤; tt)

+
open import Data.Maybe.Base using (Maybe; just; nothing)

+
open import Level

+
import Relation.Nullary.Decidable.Core as Dec

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- Re-export content from base module

+

+
open import Data.Sum.Base public

+

+
------------------------------------------------------------------------

+
-- Additional functions

+

+
module _ {A : Set a} {B : Set b} where

+

+
  isInj₁ : A ⊎ B → Maybe A

+
  isInj₁ (inj₁ x) = just x

+
  isInj₁ (inj₂ y) = nothing

+

+
  isInj₂ : A ⊎ B → Maybe B

+
  isInj₂ (inj₁ x) = nothing

+
  isInj₂ (inj₂ y) = just y

+

+
  From-inj₁ : A ⊎ B → Set a

+
  From-inj₁ (inj₁ _) = A

+
  From-inj₁ (inj₂ _) = ⊤

+

+
  from-inj₁ : (x : A ⊎ B) → From-inj₁ x

+
  from-inj₁ (inj₁ x) = x

+
  from-inj₁ (inj₂ _) = _

+

+
  From-inj₂ : A ⊎ B → Set b

+
  From-inj₂ (inj₁ _) = ⊤

+
  From-inj₂ (inj₂ _) = B

+

+
  from-inj₂ : (x : A ⊎ B) → From-inj₂ x

+
  from-inj₂ (inj₁ _) = _

+
  from-inj₂ (inj₂ x) = x

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.1

+

+
open Dec public using (fromDec; toDec)

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.These.Base.html b/site/static/agda_html/Data.These.Base.html
new file mode 100644
index 000000000..efc83b0e8
--- /dev/null
+++ b/site/static/agda_html/Data.These.Base.html
@@ -0,0 +1,157 @@
+
+Data.These.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An either-or-both data type, basic type and operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.These.Base where

+

+
open import Level

+
open import Data.Sum.Base using (_⊎_; [_,_]′)

+
open import Function.Base

+

+
private

+
  variable

+
    a b c d e f : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+
    E : Set e

+
    F : Set f

+

+
data These {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where

+
  this  : A     → These A B

+
  that  :     B → These A B

+
  these : A → B → These A B

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
-- injection

+

+
fromSum : A ⊎ B → These A B

+
fromSum = [ this , that ]′

+

+
-- map

+

+
map : (f : A → B) (g : C → D) → These A C → These B D

+
map f g (this a)    = this (f a)

+
map f g (that b)    = that (g b)

+
map f g (these a b) = these (f a) (g b)

+

+
map₁ : (f : A → B) → These A C → These B C

+
map₁ f = map f id

+

+
map₂ : (g : B → C) → These A B → These A C

+
map₂ = map id

+

+
-- fold

+

+
fold : (A → C) → (B → C) → (A → B → C) → These A B → C

+
fold l r lr (this a)    = l a

+
fold l r lr (that b)    = r b

+
fold l r lr (these a b) = lr a b

+

+
foldWithDefaults : A → B → (A → B → C) → These A B → C

+
foldWithDefaults a b lr = fold (flip lr b) (lr a) lr

+

+
-- swap

+

+
swap : These A B → These B A

+
swap = fold that this (flip these)

+

+
-- align

+

+
alignWith : (These A C → E) → (These B D → F) → These A B → These C D → These E F

+
alignWith f g (this a)    (this c)    = this (f (these a c))

+
alignWith f g (this a)    (that d)    = these (f (this a)) (g (that d))

+
alignWith f g (this a)    (these c d) = these (f (these a c)) (g (that d))

+
alignWith f g (that b)    (this c)    = these (f (that c)) (g (this b))

+
alignWith f g (that b)    (that d)    = that (g (these b d))

+
alignWith f g (that b)    (these c d) = these (f (that c)) (g (these b d))

+
alignWith f g (these a b) (this c)    = these (f (these a c)) (g (this b))

+
alignWith f g (these a b) (that d)    = these (f (this a)) (g (these b d))

+
alignWith f g (these a b) (these c d) = these (f (these a c)) (g (these b d))

+

+
align : These A B → These C D → These (These A C) (These B D)

+
align = alignWith id id

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.These.Properties.html b/site/static/agda_html/Data.These.Properties.html
new file mode 100644
index 000000000..6142dd84e
--- /dev/null
+++ b/site/static/agda_html/Data.These.Properties.html
@@ -0,0 +1,128 @@
+
+Data.These.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of These

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.These.Properties where

+

+
open import Data.Product.Base using (_×_; _,_; <_,_>; uncurry)

+
open import Data.These.Base using (These; this; that; these)

+
open import Function.Base using (_∘_)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂)

+
open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)

+

+
------------------------------------------------------------------------

+
-- Equality

+

+
module _ {a b} {A : Set a} {B : Set b} where

+

+
  this-injective : ∀ {x y : A} → this {B = B} x ≡ this y → x ≡ y

+
  this-injective refl = refl

+

+
  that-injective : ∀ {a b : B} → that {A = A} a ≡ that b → a ≡ b

+
  that-injective refl = refl

+

+
  these-injectiveˡ : ∀ {x y : A} {a b : B} → these x a ≡ these y b → x ≡ y

+
  these-injectiveˡ refl = refl

+

+
  these-injectiveʳ : ∀ {x y : A} {a b : B} → these x a ≡ these y b → a ≡ b

+
  these-injectiveʳ refl = refl

+

+
  these-injective : ∀ {x y : A} {a b : B} → these x a ≡ these y b → x ≡ y × a ≡ b

+
  these-injective = < these-injectiveˡ , these-injectiveʳ >

+

+
  ≡-dec : DecidableEquality A → DecidableEquality B → DecidableEquality (These A B)

+
  ≡-dec dec₁ dec₂ (this x)    (this y) =

+
    map′ (cong this) this-injective (dec₁ x y)

+
  ≡-dec dec₁ dec₂ (this x)    (that y)    = no λ()

+
  ≡-dec dec₁ dec₂ (this x)    (these y b) = no λ()

+
  ≡-dec dec₁ dec₂ (that x)    (this y)    = no λ()

+
  ≡-dec dec₁ dec₂ (that x)    (that y) =

+
    map′ (cong that) that-injective (dec₂ x y)

+
  ≡-dec dec₁ dec₂ (that x)    (these y b) = no λ()

+
  ≡-dec dec₁ dec₂ (these x a) (this y)    = no λ()

+
  ≡-dec dec₁ dec₂ (these x a) (that y)    = no λ()

+
  ≡-dec dec₁ dec₂ (these x a) (these y b) =

+
    map′ (uncurry (cong₂ these)) these-injective (dec₁ x y ×-dec dec₂ a b)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.These.html b/site/static/agda_html/Data.These.html
new file mode 100644
index 000000000..2cc79c86a
--- /dev/null
+++ b/site/static/agda_html/Data.These.html
@@ -0,0 +1,131 @@
+
+Data.These
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An either-or-both data type

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.These where

+

+
open import Level

+
open import Data.Maybe.Base using (Maybe; just; nothing; maybe′)

+
open import Data.Sum.Base using (_⊎_; [_,_]′)

+
open import Function.Base using (const; _∘′_; id; constᵣ)

+

+

+
------------------------------------------------------------------------

+
-- Re-exporting the datatype and its operations

+

+
open import Data.These.Base public

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Additional operations

+

+
-- projections

+

+
fromThis : These A B → Maybe A

+
fromThis = fold just (const nothing) (const ∘′ just)

+

+
fromThat : These A B → Maybe B

+
fromThat = fold (const nothing) just (const just)

+

+
leftMost : These A A → A

+
leftMost = fold id id const

+

+
rightMost : These A A → A

+
rightMost = fold id id constᵣ

+

+
mergeThese : (A → A → A) → These A A → A

+
mergeThese = fold id id

+

+
-- deletions

+

+
deleteThis : These A B → Maybe (These A B)

+
deleteThis = fold (const nothing) (just ∘′ that) (const (just ∘′ that))

+

+
deleteThat : These A B → Maybe (These A B)

+
deleteThat = fold (just ∘′ this) (const nothing) (const ∘′ just ∘′ this)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Unit.Base.html b/site/static/agda_html/Data.Unit.Base.html
new file mode 100644
index 000000000..4d6a635b3
--- /dev/null
+++ b/site/static/agda_html/Data.Unit.Base.html
@@ -0,0 +1,98 @@
+
+Data.Unit.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The unit type and the total relation on unit

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Unit.Base where

+

+
------------------------------------------------------------------------

+
-- A unit type defined as a record type

+

+
-- Note that by default the unit type is not universe polymorphic as it

+
-- often results in unsolved metas. See `Data.Unit.Polymorphic` for a

+
-- universe polymorphic variant.

+

+
-- Note also that the name of this type is "\top", not T.

+

+
open import Agda.Builtin.Unit public

+
  using (⊤; tt)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Unit.Polymorphic.Base.html b/site/static/agda_html/Data.Unit.Polymorphic.Base.html
new file mode 100644
index 000000000..7f2301470
--- /dev/null
+++ b/site/static/agda_html/Data.Unit.Polymorphic.Base.html
@@ -0,0 +1,98 @@
+
+Data.Unit.Polymorphic.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- A universe polymorphic unit type, as a Lift of the Level 0 one.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Unit.Polymorphic.Base where

+

+
open import Level

+
import Data.Unit.Base as ⊤

+

+
------------------------------------------------------------------------

+
-- A unit type defined as a synonym

+

+
⊤ : {ℓ : Level} → Set ℓ

+
⊤ {ℓ} = Lift ℓ ⊤.⊤

+

+
tt : {ℓ : Level} → ⊤ {ℓ}

+
tt = lift ⊤.tt

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Unit.Polymorphic.Properties.html b/site/static/agda_html/Data.Unit.Polymorphic.Properties.html
new file mode 100644
index 000000000..4f47bb691
--- /dev/null
+++ b/site/static/agda_html/Data.Unit.Polymorphic.Properties.html
@@ -0,0 +1,189 @@
+
+Data.Unit.Polymorphic.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the polymorphic unit type

+
-- Defines Decidable Equality and Decidable Ordering as well

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Unit.Polymorphic.Properties where

+

+
open import Level using (Level)

+
open import Function.Bundles using (_↔_; mk↔)

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base using (inj₁)

+
open import Data.Unit.Base renaming (⊤ to ⊤*)

+
open import Data.Unit.Polymorphic.Base using (⊤; tt)

+
open import Relation.Nullary.Decidable using (yes)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (DecidableEquality; Antisymmetric; Total)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; trans)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (decSetoid; setoid; isEquivalence)

+

+
private

+
  variable

+
    ℓ : Level

+

+
------------------------------------------------------------------------

+
-- Equality

+
------------------------------------------------------------------------

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality (⊤ {ℓ})

+
_ ≟ _ = yes refl

+

+
≡-setoid : ∀ ℓ → Setoid ℓ ℓ

+
≡-setoid _ = setoid ⊤

+

+
≡-decSetoid : ∀ ℓ → DecSetoid ℓ ℓ

+
≡-decSetoid _ = decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Ordering

+
------------------------------------------------------------------------

+

+
≡-total : Total {A = ⊤ {ℓ}} _≡_

+
≡-total _ _ = inj₁ refl

+

+
≡-antisym : Antisymmetric {A = ⊤ {ℓ}} _≡_ _≡_

+
≡-antisym p _ = p

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≡-isPreorder : ∀ ℓ → IsPreorder {ℓ} {_} {⊤} _≡_ _≡_

+
≡-isPreorder ℓ = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = λ x → x

+
  ; trans         = trans

+
  }

+

+
≡-isPartialOrder : ∀ ℓ → IsPartialOrder {ℓ} _≡_ _≡_

+
≡-isPartialOrder ℓ = record

+
  { isPreorder = ≡-isPreorder ℓ

+
  ; antisym    = ≡-antisym

+
  }

+

+
≡-isTotalOrder : ∀ ℓ → IsTotalOrder {ℓ} _≡_ _≡_

+
≡-isTotalOrder ℓ = record

+
  { isPartialOrder = ≡-isPartialOrder ℓ

+
  ; total          = ≡-total

+
  }

+

+
≡-isDecTotalOrder : ∀ ℓ → IsDecTotalOrder {ℓ} _≡_ _≡_

+
≡-isDecTotalOrder ℓ = record

+
  { isTotalOrder = ≡-isTotalOrder ℓ

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≟_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≡-preorder : ∀ ℓ → Preorder ℓ ℓ ℓ

+
≡-preorder ℓ = record

+
  { isPreorder = ≡-isPreorder ℓ

+
  }

+

+
≡-poset : ∀ ℓ → Poset ℓ ℓ ℓ

+
≡-poset ℓ = record

+
  { isPartialOrder = ≡-isPartialOrder ℓ

+
  }

+

+
≡-totalOrder : ∀ ℓ → TotalOrder ℓ ℓ ℓ

+
≡-totalOrder ℓ = record

+
  { isTotalOrder = ≡-isTotalOrder ℓ

+
  }

+

+
≡-decTotalOrder : ∀ ℓ → DecTotalOrder ℓ ℓ ℓ

+
≡-decTotalOrder ℓ = record

+
  { isDecTotalOrder = ≡-isDecTotalOrder ℓ

+
  }

+

+
⊤↔⊤* : ⊤ {ℓ} ↔ ⊤*

+
⊤↔⊤* = mk↔ ((λ _ → refl) , (λ _ → refl))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Unit.Polymorphic.html b/site/static/agda_html/Data.Unit.Polymorphic.html
new file mode 100644
index 000000000..e0b11e0b4
--- /dev/null
+++ b/site/static/agda_html/Data.Unit.Polymorphic.html
@@ -0,0 +1,96 @@
+
+Data.Unit.Polymorphic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The universe polymorphic unit type and the total relation on unit

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Unit.Polymorphic where

+

+
------------------------------------------------------------------------

+
-- Re-export contents of Base module

+

+
open import Data.Unit.Polymorphic.Base public

+

+
------------------------------------------------------------------------

+
-- Re-export query operations

+

+
open import Data.Unit.Polymorphic.Properties public using (_≟_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Unit.Properties.html b/site/static/agda_html/Data.Unit.Properties.html
new file mode 100644
index 000000000..0b49b5c95
--- /dev/null
+++ b/site/static/agda_html/Data.Unit.Properties.html
@@ -0,0 +1,171 @@
+
+Data.Unit.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the unit type

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Unit.Properties where

+

+
open import Data.Sum.Base using (inj₁)

+
open import Data.Unit.Base using (⊤)

+
open import Level using (0ℓ)

+
open import Relation.Nullary using (Irrelevant; yes)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Poset; DecTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions using (DecidableEquality; Total; Antisymmetric)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; trans)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; decSetoid; isEquivalence)

+

+
------------------------------------------------------------------------

+
-- Irrelevancy

+

+
⊤-irrelevant : Irrelevant ⊤

+
⊤-irrelevant _ _ = refl

+

+
------------------------------------------------------------------------

+
-- Equality

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality ⊤

+
_ ≟ _ = yes refl

+

+
≡-setoid : Setoid 0ℓ 0ℓ

+
≡-setoid = setoid ⊤

+

+
≡-decSetoid : DecSetoid 0ℓ 0ℓ

+
≡-decSetoid = decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
≡-total : Total {A = ⊤} _≡_

+
≡-total _ _ = inj₁ refl

+

+
≡-antisym : Antisymmetric {A = ⊤} _≡_ _≡_

+
≡-antisym eq _ = eq

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
≡-isPreorder : IsPreorder {A = ⊤} _≡_ _≡_

+
≡-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = λ x → x

+
  ; trans         = trans

+
  }

+

+
≡-isPartialOrder : IsPartialOrder _≡_ _≡_

+
≡-isPartialOrder = record

+
  { isPreorder = ≡-isPreorder

+
  ; antisym    = ≡-antisym

+
  }

+

+
≡-isTotalOrder : IsTotalOrder _≡_ _≡_

+
≡-isTotalOrder = record

+
  { isPartialOrder = ≡-isPartialOrder

+
  ; total          = ≡-total

+
  }

+

+
≡-isDecTotalOrder : IsDecTotalOrder _≡_ _≡_

+
≡-isDecTotalOrder = record

+
  { isTotalOrder = ≡-isTotalOrder

+
  ; _≟_          = _≟_

+
  ; _≤?_         = _≟_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
≡-poset : Poset 0ℓ 0ℓ 0ℓ

+
≡-poset = record

+
  { isPartialOrder = ≡-isPartialOrder

+
  }

+

+
≡-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ

+
≡-decTotalOrder = record

+
  { isDecTotalOrder = ≡-isDecTotalOrder

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Unit.html b/site/static/agda_html/Data.Unit.html
new file mode 100644
index 000000000..a6b936842
--- /dev/null
+++ b/site/static/agda_html/Data.Unit.html
@@ -0,0 +1,97 @@
+
+Data.Unit
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The unit type, Level-monomorphic version

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Unit where

+

+
------------------------------------------------------------------------

+
-- Re-export contents of base module

+

+
open import Data.Unit.Base public

+

+
------------------------------------------------------------------------

+
-- Re-export query operations

+

+
open import Data.Unit.Properties public

+
  using (_≟_)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Universe.html b/site/static/agda_html/Data.Universe.html
new file mode 100644
index 000000000..488f91fff
--- /dev/null
+++ b/site/static/agda_html/Data.Universe.html
@@ -0,0 +1,96 @@
+
+Data.Universe
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Universes

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Universe where

+

+
open import Level

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
record Universe u e : Set (suc (u ⊔ e)) where

+
  field

+
    U : Set u       -- Codes.

+
    El : U → Set e  -- Decoding function.

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Base.html b/site/static/agda_html/Data.Vec.Base.html
new file mode 100644
index 000000000..b3fa17f15
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Base.html
@@ -0,0 +1,450 @@
+
+Data.Vec.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Vectors, basic types and operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Base where

+

+
open import Data.Bool.Base using (Bool; true; false; if_then_else_)

+
open import Data.Nat.Base

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.List.Base as List using (List)

+
open import Data.Product.Base as Product using (∃; ∃₂; _×_; _,_; proj₁; proj₂)

+
open import Data.These.Base as These using (These; this; that; these)

+
open import Function.Base using (const; _∘′_; id; _∘_; _$_)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; trans; cong)

+
open import Relation.Nullary.Decidable.Core using (does; T?)

+
open import Relation.Unary using (Pred; Decidable)

+

+
private

+
  variable

+
    a b c p : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    m n : ℕ

+

+
------------------------------------------------------------------------

+
-- Types

+

+
infixr 5 _∷_

+

+
data Vec (A : Set a) : ℕ → Set a where

+
  []  : Vec A zero

+
  _∷_ : ∀ (x : A) (xs : Vec A n) → Vec A (suc n)

+

+
infix 4 _[_]=_

+

+
data _[_]=_ {A : Set a} : Vec A n → Fin n → A → Set a where

+
  here  : ∀     {x}   {xs : Vec A n} → x ∷ xs [ zero ]= x

+
  there : ∀ {i} {x y} {xs : Vec A n}

+
    (xs[i]=x : xs [ i ]= x) → y ∷ xs [ suc i ]= x

+

+
------------------------------------------------------------------------

+
-- Basic operations

+

+
length : Vec A n → ℕ

+
length {n = n} _ = n

+

+
head : Vec A (1 + n) → A

+
head (x ∷ xs) = x

+

+
tail : Vec A (1 + n) → Vec A n

+
tail (x ∷ xs) = xs

+

+
lookup : Vec A n → Fin n → A

+
lookup (x ∷ xs) zero    = x

+
lookup (x ∷ xs) (suc i) = lookup xs i

+

+
iterate : (A → A) → A → ∀ n → Vec A n

+
iterate s z zero    = []

+
iterate s z (suc n) = z ∷ iterate s (s z) n

+

+
insertAt : Vec A n → Fin (suc n) → A → Vec A (suc n)

+
insertAt xs       zero     v = v ∷ xs

+
insertAt (x ∷ xs) (suc i)  v = x ∷ insertAt xs i v

+

+
removeAt : Vec A (suc n) → Fin (suc n) → Vec A n

+
removeAt (x ∷ xs)         zero    = xs

+
removeAt (x ∷ xs@(_ ∷ _)) (suc i) = x ∷ removeAt xs i

+

+
updateAt : Vec A n → Fin n → (A → A) → Vec A n

+
updateAt (x ∷ xs) zero    f = f x ∷ xs

+
updateAt (x ∷ xs) (suc i) f = x   ∷ updateAt xs i f

+

+
-- xs [ i ]%= f  modifies the i-th element of xs according to f

+

+
infixl 6 _[_]%=_ _[_]≔_

+

+
_[_]%=_ : Vec A n → Fin n → (A → A) → Vec A n

+
xs [ i ]%= f = updateAt xs i f

+

+
-- xs [ i ]≔ y  overwrites the i-th element of xs with y

+

+
_[_]≔_ : Vec A n → Fin n → A → Vec A n

+
xs [ i ]≔ y = xs [ i ]%= const y

+

+
------------------------------------------------------------------------

+
-- Operations for transforming vectors

+

+
-- See README.Data.Vec.Relation.Binary.Equality.Cast for the reasoning

+
-- system of `cast`-ed equality.

+
cast : .(eq : m ≡ n) → Vec A m → Vec A n

+
cast {n = zero}  eq []       = []

+
cast {n = suc _} eq (x ∷ xs) = x ∷ cast (cong pred eq) xs

+

+
map : (A → B) → Vec A n → Vec B n

+
map f []       = []

+
map f (x ∷ xs) = f x ∷ map f xs

+

+
-- Concatenation.

+

+
infixr 5 _++_

+

+
_++_ : Vec A m → Vec A n → Vec A (m + n)

+
[]       ++ ys = ys

+
(x ∷ xs) ++ ys = x ∷ (xs ++ ys)

+

+
concat : Vec (Vec A m) n → Vec A (n * m)

+
concat []         = []

+
concat (xs ∷ xss) = xs ++ concat xss

+

+
-- Align, Restrict, and Zip.

+

+
alignWith : (These A B → C) → Vec A m → Vec B n → Vec C (m ⊔ n)

+
alignWith f []         bs       = map (f ∘′ that) bs

+
alignWith f as@(_ ∷ _) []       = map (f ∘′ this) as

+
alignWith f (a ∷ as)   (b ∷ bs) = f (these a b) ∷ alignWith f as bs

+

+
restrictWith : (A → B → C) → Vec A m → Vec B n → Vec C (m ⊓ n)

+
restrictWith f []       bs       = []

+
restrictWith f (_ ∷ _)  []       = []

+
restrictWith f (a ∷ as) (b ∷ bs) = f a b ∷ restrictWith f as bs

+

+
zipWith : (A → B → C) → Vec A n → Vec B n → Vec C n

+
zipWith f []       []       = []

+
zipWith f (x ∷ xs) (y ∷ ys) = f x y ∷ zipWith f xs ys

+

+
unzipWith : (A → B × C) → Vec A n → Vec B n × Vec C n

+
unzipWith f []       = [] , []

+
unzipWith f (a ∷ as) = Product.zip _∷_ _∷_ (f a) (unzipWith f as)

+

+
align : Vec A m → Vec B n → Vec (These A B) (m ⊔ n)

+
align = alignWith id

+

+
restrict : Vec A m → Vec B n → Vec (A × B) (m ⊓ n)

+
restrict = restrictWith _,_

+

+
zip : Vec A n → Vec B n → Vec (A × B) n

+
zip = zipWith _,_

+

+
unzip : Vec (A × B) n → Vec A n × Vec B n

+
unzip = unzipWith id

+

+
-- Interleaving.

+

+
infixr 5 _⋎_

+

+
_⋎_ : Vec A m → Vec A n → Vec A (m +⋎ n)

+
[]       ⋎ ys = ys

+
(x ∷ xs) ⋎ ys = x ∷ (ys ⋎ xs)

+

+
-- Pointwise application

+

+
infixl 4 _⊛_

+

+
_⊛_ : Vec (A → B) n → Vec A n → Vec B n

+
[]       ⊛ []       = []

+
(f ∷ fs) ⊛ (x ∷ xs) = f x ∷ (fs ⊛ xs)

+

+
-- Multiplication

+

+
module CartesianBind where

+
  infixl 1 _>>=_

+

+
  _>>=_ : Vec A m → (A → Vec B n) → Vec B (m * n)

+
  xs >>= f = concat (map f xs)

+

+
infixl 4 _⊛*_

+

+
_⊛*_ : Vec (A → B) m → Vec A n → Vec B (m * n)

+
fs ⊛* xs = fs CartesianBind.>>= λ f → map f xs

+

+
allPairs : Vec A m → Vec B n → Vec (A × B) (m * n)

+
allPairs xs ys = map _,_ xs ⊛* ys

+

+
-- Diagonal

+

+
diagonal : Vec (Vec A n) n → Vec A n

+
diagonal [] = []

+
diagonal (xs ∷ xss) = head xs ∷ diagonal (map tail xss)

+

+
module DiagonalBind where

+
  infixl 1 _>>=_

+

+
  _>>=_ : Vec A n → (A → Vec B n) → Vec B n

+
  xs >>= f = diagonal (map f xs)

+

+

+
------------------------------------------------------------------------

+
-- Operations for reducing vectors

+

+
-- Dependent folds

+

+
module _ (A : Set a) (B : ℕ → Set b) where

+

+
  FoldrOp = ∀ {n} → A → B n → B (suc n)

+
  FoldlOp = ∀ {n} → B n → A → B (suc n)

+

+
foldr : ∀ (B : ℕ → Set b) → FoldrOp A B → B zero → Vec A n → B n

+
foldr B _⊕_ e []       = e

+
foldr B _⊕_ e (x ∷ xs) = x ⊕ foldr B _⊕_ e xs

+

+
foldl : ∀ (B : ℕ → Set b) → FoldlOp A B → B zero → Vec A n → B n

+
foldl B _⊕_ e []       = e

+
foldl B _⊕_ e (x ∷ xs) = foldl (B ∘ suc) _⊕_ (e ⊕ x) xs

+

+
-- Non-dependent folds

+

+
foldr′ : (A → B → B) → B → Vec A n → B

+
foldr′ _⊕_ = foldr _ _⊕_

+

+
foldl′ : (B → A → B) → B → Vec A n → B

+
foldl′ _⊕_ = foldl _ _⊕_

+

+
-- Non-empty folds

+

+
foldr₁ : (A → A → A) → Vec A (suc n) → A

+
foldr₁ _⊕_ (x ∷ [])     = x

+
foldr₁ _⊕_ (x ∷ y ∷ ys) = x ⊕ foldr₁ _⊕_ (y ∷ ys)

+

+
foldl₁ : (A → A → A) → Vec A (suc n) → A

+
foldl₁ _⊕_ (x ∷ xs) = foldl _ _⊕_ x xs

+

+
-- Special folds

+

+
sum : Vec ℕ n → ℕ

+
sum = foldr _ _+_ 0

+

+
count : ∀ {P : Pred A p} → Decidable P → Vec A n → ℕ

+
count P? []       = zero

+
count P? (x ∷ xs) = if does (P? x) then suc else id $ count P? xs

+

+
countᵇ : (A → Bool) → Vec A n → ℕ

+
countᵇ p = count (T? ∘ p)

+

+
------------------------------------------------------------------------

+
-- Operations for building vectors

+

+
[_] : A → Vec A 1

+
[ x ] = x ∷ []

+

+
replicate : (n : ℕ) → A → Vec A n

+
replicate zero    x = []

+
replicate (suc n) x = x ∷ replicate n x

+

+
tabulate : (Fin n → A) → Vec A n

+
tabulate {n = zero}  f = []

+
tabulate {n = suc n} f = f zero ∷ tabulate (f ∘ suc)

+

+
allFin : ∀ n → Vec (Fin n) n

+
allFin _ = tabulate id

+

+
------------------------------------------------------------------------

+
-- Operations for dividing vectors

+

+
splitAt : ∀ m {n} (xs : Vec A (m + n)) →

+
          ∃₂ λ (ys : Vec A m) (zs : Vec A n) → xs ≡ ys ++ zs

+
splitAt zero    xs                = [] , xs , refl

+
splitAt (suc m) (x ∷ xs) =

+
  let ys , zs , eq = splitAt m xs in x ∷ ys , zs , cong (x ∷_) eq

+

+
take : ∀ m {n} → Vec A (m + n) → Vec A m

+
take m xs = proj₁ (splitAt m xs)

+

+
drop : ∀ m {n} → Vec A (m + n) → Vec A n

+
drop m xs = proj₁ (proj₂ (splitAt m xs))

+

+
group : ∀ n k (xs : Vec A (n * k)) →

+
        ∃ λ (xss : Vec (Vec A k) n) → xs ≡ concat xss

+
group zero    k []                  = ([] , refl)

+
group (suc n) k xs  =

+
  let ys , zs , eq-split = splitAt k xs in

+
  let zss , eq-group     = group n k zs in

+
   (ys ∷ zss) , trans eq-split (cong (ys ++_) eq-group)

+

+
split : Vec A n → Vec A ⌈ n /2⌉ × Vec A ⌊ n /2⌋

+
split []           = ([]     , [])

+
split (x ∷ [])     = (x ∷ [] , [])

+
split (x ∷ y ∷ xs) = Product.map (x ∷_) (y ∷_) (split xs)

+

+
uncons : Vec A (suc n) → A × Vec A n

+
uncons (x ∷ xs) = x , xs

+

+
------------------------------------------------------------------------

+
-- Operations involving ≤

+

+
-- Take the first 'm' elements of a vector.

+
truncate : ∀ {m n} → m ≤ n → Vec A n → Vec A m

+
truncate {m = zero} _ _    = []

+
truncate (s≤s le) (x ∷ xs) = x ∷ (truncate le xs)

+

+
-- Pad out a vector with extra elements.

+
padRight : ∀ {m n} → m ≤ n → A → Vec A m → Vec A n

+
padRight z≤n      a xs       = replicate _ a

+
padRight (s≤s le) a (x ∷ xs) = x ∷ padRight le a xs

+

+
------------------------------------------------------------------------

+
-- Operations for converting between lists

+

+
toList : Vec A n → List A

+
toList []       = List.[]

+
toList (x ∷ xs) = List._∷_ x (toList xs)

+

+
fromList : (xs : List A) → Vec A (List.length xs)

+
fromList List.[]         = []

+
fromList (List._∷_ x xs) = x ∷ fromList xs

+

+
------------------------------------------------------------------------

+
-- Operations for reversing vectors

+

+
-- snoc

+

+
infixl 5 _∷ʳ_

+

+
_∷ʳ_ : Vec A n → A → Vec A (suc n)

+
[]       ∷ʳ y = [ y ]

+
(x ∷ xs) ∷ʳ y = x ∷ (xs ∷ʳ y)

+

+
-- vanilla reverse

+

+
reverse : Vec A n → Vec A n

+
reverse = foldl (Vec _) (λ rev x → x ∷ rev) []

+

+
-- reverse-append

+

+
infix 5 _ʳ++_

+

+
_ʳ++_ : Vec A m → Vec A n → Vec A (m + n)

+
xs ʳ++ ys = foldl (Vec _ ∘ (_+ _)) (λ rev x → x ∷ rev) ys xs

+

+
-- init and last

+

+
initLast : ∀ (xs : Vec A (1 + n)) → ∃₂ λ ys y → xs ≡ ys ∷ʳ y

+
initLast {n = zero}  (x ∷ []) = [] , x , refl

+
initLast {n = suc n} (x ∷ xs) =

+
  let ys , y , eq = initLast xs in

+
  x ∷ ys , y , cong (x ∷_) eq

+

+
init : Vec A (1 + n) → Vec A n

+
init xs = proj₁ (initLast xs)

+

+
last : Vec A (1 + n) → A

+
last xs = proj₁ (proj₂ (initLast xs))

+

+
------------------------------------------------------------------------

+
-- Other operations

+

+
transpose : Vec (Vec A n) m → Vec (Vec A m) n

+
transpose {n = n} []          = replicate n []

+
transpose {n = n} (as ∷ ass) = ((replicate n _∷_) ⊛ as) ⊛ transpose ass

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
remove = removeAt

+
{-# WARNING_ON_USAGE remove

+
"Warning: remove was deprecated in v2.0.

+
Please use removeAt instead."

+
#-}

+
insert = insertAt

+
{-# WARNING_ON_USAGE insert

+
"Warning: insert was deprecated in v2.0.

+
Please use insertAt instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Bounded.Base.html b/site/static/agda_html/Data.Vec.Bounded.Base.html
new file mode 100644
index 000000000..0d6fa2f5c
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Bounded.Base.html
@@ -0,0 +1,240 @@
+
+Data.Vec.Bounded.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bounded vectors, basic types and operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Bounded.Base where

+

+
open import Data.Nat.Base

+
import Data.Nat.Properties as ℕ

+
open import Data.List.Base as List using (List)

+
open import Data.List.Extrema ℕ.≤-totalOrder

+
open import Data.List.Relation.Unary.All as All using (All)

+
import Data.List.Relation.Unary.All.Properties as All

+
open import Data.List.Membership.Propositional using (mapWith∈)

+
open import Data.Product.Base using (∃; _×_; _,_; proj₁; proj₂)

+
open import Data.Vec.Base as Vec using (Vec)

+
open import Data.These.Base as These using (These)

+
open import Function.Base using (_∘_; id; _$_)

+
open import Level using (Level)

+
open import Relation.Nullary.Decidable.Core using (recompute)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+

+
private

+
  variable

+
    a b c p : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    m n : ℕ

+

+
------------------------------------------------------------------------

+
-- Types

+

+
infix 4 _,_

+
record Vec≤ (A : Set a) (n : ℕ) : Set a where

+
  constructor _,_

+
  field {length} : ℕ

+
        vec      : Vec A length

+
        .bound   : length ≤ n

+

+
-- projection to recompute irrelevant field

+
isBounded : (as : Vec≤ A n) → Vec≤.length as ≤ n

+
isBounded as@(_ , m≤n) = recompute (_ ℕ.≤? _) m≤n

+

+
------------------------------------------------------------------------

+
-- Conversion functions

+

+
toVec : (as : Vec≤ A n) → Vec A (Vec≤.length as)

+
toVec as@(vs , _) = vs

+

+
fromVec : Vec A n → Vec≤ A n

+
fromVec v = v , ℕ.≤-refl

+

+
padRight : A → Vec≤ A n → Vec A n

+
padRight a as@(vs , m≤n)

+
  with k , refl ← ℕ.m≤n⇒∃[o]m+o≡n m≤n

+
  = vs Vec.++ Vec.replicate k a

+

+
padLeft : A → Vec≤ A n → Vec A n

+
padLeft a record { length = m ; vec = vs ; bound = m≤n }

+
  with k , refl ← ℕ.m≤n⇒∃[o]m+o≡n m≤n

+
  rewrite ℕ.+-comm m k

+
  = Vec.replicate k a Vec.++ vs

+

+
private

+
  split : ∀ m k → m + k ≡ ⌊ k /2⌋ + (m + ⌈ k /2⌉)

+
  split m k = begin

+
    m + k                   ≡⟨ ≡.cong (m +_) (ℕ.⌊n/2⌋+⌈n/2⌉≡n k) ⟨

+
    m + (⌊ k /2⌋ + ⌈ k /2⌉) ≡⟨ ≡.cong (m +_) (ℕ.+-comm ⌊ k /2⌋ ⌈ k /2⌉) ⟩

+
    m + (⌈ k /2⌉ + ⌊ k /2⌋) ≡⟨ ℕ.+-assoc m ⌈ k /2⌉ ⌊ k /2⌋ ⟨

+
    m + ⌈ k /2⌉ + ⌊ k /2⌋   ≡⟨ ℕ.+-comm _ ⌊ k /2⌋ ⟩

+
    ⌊ k /2⌋ + (m + ⌈ k /2⌉) ∎

+
    where open ≡-Reasoning

+

+
padBoth : A → A → Vec≤ A n → Vec A n

+
padBoth aₗ aᵣ record { length = m ; vec = vs ; bound = m≤n }

+
  with k , refl ← ℕ.m≤n⇒∃[o]m+o≡n m≤n

+
  rewrite split m k

+
  = Vec.replicate ⌊ k /2⌋ aₗ

+
      Vec.++ vs

+
      Vec.++ Vec.replicate ⌈ k /2⌉ aᵣ

+

+

+
fromList : (as : List A) → Vec≤ A (List.length as)

+
fromList = fromVec ∘ Vec.fromList

+

+
toList : Vec≤ A n → List A

+
toList = Vec.toList ∘ toVec

+

+
------------------------------------------------------------------------

+
-- Creating new Vec≤ vectors

+

+
replicate : .(m≤n : m ≤ n) → A → Vec≤ A n

+
replicate m≤n a = Vec.replicate _ a , m≤n

+

+
[] : Vec≤ A n

+
[] = Vec.[] , z≤n

+

+
infixr 5 _∷_

+
_∷_ : A → Vec≤ A n → Vec≤ A (suc n)

+
a ∷ (as , p) = a Vec.∷ as , s≤s p

+

+
------------------------------------------------------------------------

+
-- Modifying Vec≤ vectors

+

+
≤-cast : .(m≤n : m ≤ n) → Vec≤ A m → Vec≤ A n

+
≤-cast m≤n (v , p) = v , ℕ.≤-trans p m≤n

+

+
≡-cast : .(eq : m ≡ n) → Vec≤ A m → Vec≤ A n

+
≡-cast m≡n = ≤-cast (ℕ.≤-reflexive m≡n)

+

+
map : (A → B) → Vec≤ A n → Vec≤ B n

+
map f (v , p) = Vec.map f v , p

+

+
reverse : Vec≤ A n → Vec≤ A n

+
reverse (v , p) = Vec.reverse v , p

+

+
-- Align and Zip.

+

+
alignWith : (These A B → C) → Vec≤ A n → Vec≤ B n → Vec≤ C n

+
alignWith f (as , p) (bs , q) = Vec.alignWith f as bs , ℕ.⊔-lub p q

+

+
zipWith : (A → B → C) → Vec≤ A n → Vec≤ B n → Vec≤ C n

+
zipWith f (as , p) (bs , q) = Vec.restrictWith f as bs , ℕ.m≤n⇒m⊓o≤n _ p

+

+
zip : Vec≤ A n → Vec≤ B n → Vec≤ (A × B) n

+
zip = zipWith _,_

+

+
align : Vec≤ A n → Vec≤ B n → Vec≤ (These A B) n

+
align = alignWith id

+

+
-- take and drop

+

+
take : ∀ n → Vec≤ A m → Vec≤ A (n ⊓ m)

+
take             zero    _                = []

+
take             (suc n) (Vec.[] , p)     = []

+
take {m = suc m} (suc n) (a Vec.∷ as , p) = a ∷ take n (as , s≤s⁻¹ p)

+

+
drop : ∀ n → Vec≤ A m → Vec≤ A (m ∸ n)

+
drop             zero    v                = v

+
drop             (suc n) (Vec.[] , p)     = []

+
drop {m = suc m} (suc n) (a Vec.∷ as , p) = drop n (as , s≤s⁻¹ p)

+

+
------------------------------------------------------------------------

+
-- Lifting a collection of bounded vectors to the same size

+

+
rectangle : List (∃ (Vec≤ A)) → ∃ (List ∘ Vec≤ A)

+
rectangle {A = A} rows = width , padded where

+

+
  sizes = List.map proj₁ rows

+
  width = max 0 sizes

+

+
  all≤ : All (λ v → proj₁ v ≤ width) rows

+
  all≤ = All.map⁻ (xs≤max 0 sizes)

+

+
  padded : List (Vec≤ A width)

+
  padded = mapWith∈ rows $ λ {x} x∈rows →

+
    ≤-cast (All.lookup all≤ x∈rows) (proj₂ x)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Functional.html b/site/static/agda_html/Data.Vec.Functional.html
new file mode 100644
index 000000000..7263c39d1
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Functional.html
@@ -0,0 +1,253 @@
+
+Data.Vec.Functional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Vectors defined as functions from a finite set to a type.

+
------------------------------------------------------------------------

+

+
-- This implementation is designed for reasoning about fixed-size

+
-- vectors where ease of retrieval of elements is prioritised.

+

+
-- See `Data.Vec` for an alternative implementation using inductive

+
-- data-types, which is more suitable for reasoning about vectors that

+
-- will grow or shrink in size.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Functional where

+

+
open import Data.Fin.Base

+
open import Data.List.Base as L using (List)

+
open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero; pred)

+
open import Data.Product.Base using (Σ; ∃; _×_; _,_; proj₁; proj₂; uncurry)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])

+
open import Data.Vec.Base as V using (Vec)

+
open import Function.Base using (_∘_; const; flip; _ˢ_; id)

+
open import Level using (Level)

+

+
infixr 5 _∷_ _++_

+
infixl 4 _⊛_

+
infixl 1 _>>=_

+

+
private

+
  variable

+
    a b c : Level

+
    A B C : Set a

+
    m n : ℕ

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
Vector : Set a → ℕ → Set a

+
Vector A n = Fin n → A

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
toVec : Vector A n → Vec A n

+
toVec = V.tabulate

+

+
fromVec : Vec A n → Vector A n

+
fromVec = V.lookup

+

+
toList : Vector A n → List A

+
toList = L.tabulate

+

+
fromList : ∀ (xs : List A) → Vector A (L.length xs)

+
fromList = L.lookup

+

+
------------------------------------------------------------------------

+
-- Basic operations

+

+
[] : Vector A zero

+
[] ()

+

+
_∷_ : A → Vector A n → Vector A (suc n)

+
(x ∷ xs) zero    = x

+
(x ∷ xs) (suc i) = xs i

+

+
length : Vector A n → ℕ

+
length {n = n} _ = n

+

+
head : Vector A (suc n) → A

+
head xs = xs zero

+

+
tail : Vector A (suc n) → Vector A n

+
tail xs = xs ∘ suc

+

+
uncons : Vector A (suc n) → A × Vector A n

+
uncons xs = head xs , tail xs

+

+
replicate : (n : ℕ) → A → Vector A n

+
replicate n = const

+

+
insertAt : Vector A n → Fin (suc n) → A → Vector A (suc n)

+
insertAt {n = n}     xs zero    v zero    = v

+
insertAt {n = n}     xs zero    v (suc j) = xs j

+
insertAt {n = suc n} xs (suc i) v zero    = head xs

+
insertAt {n = suc n} xs (suc i) v (suc j) = insertAt (tail xs) i v j

+

+
removeAt : Vector A (suc n) → Fin (suc n) → Vector A n

+
removeAt t i = t ∘ punchIn i

+

+
updateAt : Vector A n → Fin n → (A → A) → Vector A n

+
updateAt {n = suc n} xs zero    f zero    = f (head xs)

+
updateAt {n = suc n} xs zero    f (suc j) = xs (suc j)

+
updateAt {n = suc n} xs (suc i) f zero    = head xs

+
updateAt {n = suc n} xs (suc i) f (suc j) = updateAt (tail xs) i f j

+

+
------------------------------------------------------------------------

+
-- Transformations

+

+
map : (A → B) → ∀ {n} → Vector A n → Vector B n

+
map f xs = f ∘ xs

+

+
_++_ : Vector A m → Vector A n → Vector A (m ℕ.+ n)

+
_++_ {m = m} xs ys i = [ xs , ys ] (splitAt m i)

+

+
concat : Vector (Vector A m) n → Vector A (n ℕ.* m)

+
concat {m = m} xss i = uncurry (flip xss) (quotRem m i)

+

+
foldr : (A → B → B) → B → ∀ {n} → Vector A n → B

+
foldr f z {n = zero}  xs = z

+
foldr f z {n = suc n} xs = f (head xs) (foldr f z (tail xs))

+

+
foldl : (B → A → B) → B → ∀ {n} → Vector A n → B

+
foldl f z {n = zero}  xs = z

+
foldl f z {n = suc n} xs = foldl f (f z (head xs)) (tail xs)

+

+
rearrange : (Fin m → Fin n) → Vector A n → Vector A m

+
rearrange r xs = xs ∘ r

+

+
_⊛_ : Vector (A → B) n → Vector A n → Vector B n

+
_⊛_ = _ˢ_

+

+
_>>=_ : Vector A m → (A → Vector B n) → Vector B (m ℕ.* n)

+
xs >>= f = concat (map f xs)

+

+
zipWith : (A → B → C) → ∀ {n} → Vector A n → Vector B n → Vector C n

+
zipWith f xs ys i = f (xs i) (ys i)

+

+
unzipWith : (A → B × C) → Vector A n → Vector B n × Vector C n

+
unzipWith f xs = proj₁ ∘ f ∘ xs , proj₂ ∘ f ∘ xs

+

+
zip : Vector A n → Vector B n → Vector (A × B) n

+
zip = zipWith _,_

+

+
unzip : Vector (A × B) n → Vector A n × Vector B n

+
unzip = unzipWith id

+

+
take : ∀ m {n} → Vector A (m ℕ.+ n) → Vector A m

+
take _ {n = n} xs = xs ∘ (_↑ˡ n)

+

+
drop : ∀ m {n} → Vector A (m ℕ.+ n) → Vector A n

+
drop m xs = xs ∘ (m ↑ʳ_)

+

+
reverse : Vector A n → Vector A n

+
reverse xs = xs ∘ opposite

+

+
init : Vector A (suc n) → Vector A n

+
init xs = xs ∘ inject₁

+

+
last : Vector A (suc n) → A

+
last {n = n} xs = xs (fromℕ n)

+

+
transpose : Vector (Vector A n) m → Vector (Vector A m) n

+
transpose = flip

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
remove : Fin (suc n) → Vector A (suc n) → Vector A n

+
remove = flip removeAt

+
{-# WARNING_ON_USAGE remove

+
"Warning: remove was deprecated in v2.0.

+
Please use removeAt instead.

+
NOTE: argument order has been flipped."

+
#-}

+
insert = insertAt

+
{-# WARNING_ON_USAGE insert

+
"Warning: insert was deprecated in v2.0.

+
Please use insertAt instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Membership.Propositional.html b/site/static/agda_html/Data.Vec.Membership.Propositional.html
new file mode 100644
index 000000000..e5106bd79
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Membership.Propositional.html
@@ -0,0 +1,105 @@
+
+Data.Vec.Membership.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Data.Vec.Any.Membership instantiated with propositional equality,

+
-- along with some additional definitions.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Membership.Propositional {a} {A : Set a} where

+

+
open import Data.Vec.Base using (Vec)

+
open import Data.Vec.Relation.Unary.Any using (Any)

+
open import Relation.Binary.PropositionalEquality.Core using (subst)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
import Data.Vec.Membership.Setoid as SetoidMembership

+

+
------------------------------------------------------------------------

+
-- Re-export contents of setoid membership

+

+
open SetoidMembership (setoid A) public hiding (lose)

+

+
------------------------------------------------------------------------

+
-- Other operations

+

+
lose : ∀ {p} {P : A → Set p} {x n} {xs : Vec A n} → x ∈ xs → P x → Any P xs

+
lose = SetoidMembership.lose (setoid A) (subst _)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Membership.Setoid.html b/site/static/agda_html/Data.Vec.Membership.Setoid.html
new file mode 100644
index 000000000..1c9a3fd68
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Membership.Setoid.html
@@ -0,0 +1,135 @@
+
+Data.Vec.Membership.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Membership of vectors, along with some additional definitions.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (_Respects_)

+

+
module Data.Vec.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where

+

+
open import Function.Base using (_∘_; flip)

+
open import Data.Vec.Base as Vec using (Vec; []; _∷_)

+
open import Data.Vec.Relation.Unary.Any as Any

+
  using (Any; here; there; index)

+
open import Data.Product.Base using (∃; _×_; _,_)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Unary using (Pred)

+

+
open Setoid S renaming (Carrier to A)

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
infix 4 _∈_ _∉_

+

+
_∈_ : A → ∀ {n} → Vec A n → Set _

+
x ∈ xs = Any (x ≈_) xs

+

+
_∉_ : A → ∀ {n} → Vec A n → Set _

+
x ∉ xs = ¬ x ∈ xs

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
mapWith∈ : ∀ {b} {B : Set b} {n}

+
           (xs : Vec A n) → (∀ {x} → x ∈ xs → B) → Vec B n

+
mapWith∈ []       f = []

+
mapWith∈ (x ∷ xs) f = f (here refl) ∷ mapWith∈ xs (f ∘ there)

+

+
infixr 5 _∷=_

+

+
_∷=_ : ∀ {n} {xs : Vec A n} {x} → x ∈ xs → A → Vec A n

+
_∷=_ {xs = xs} x∈xs v = xs Vec.[ index x∈xs ]≔ v

+

+
------------------------------------------------------------------------

+
-- Finding and losing witnesses

+

+
module _ {p} {P : Pred A p} where

+

+
  find : ∀ {n} {xs : Vec A n} → Any P xs → ∃ λ x → x ∈ xs × P x

+
  find (here px)   = _ , here refl , px

+
  find (there pxs) = let x , x∈xs , px = find pxs in x , there x∈xs , px

+

+
  lose : P Respects _≈_ → ∀ {x n} {xs : Vec A n} → x ∈ xs → P x → Any P xs

+
  lose resp x∈xs px = Any.map (flip resp px) x∈xs

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.N-ary.html b/site/static/agda_html/Data.Vec.N-ary.html
new file mode 100644
index 000000000..c70ded9fa
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.N-ary.html
@@ -0,0 +1,265 @@
+
+Data.Vec.N-ary
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Code for converting Vec A n → B to and from n-ary functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.N-ary where

+

+
open import Axiom.Extensionality.Propositional using (Extensionality)

+
open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)

+
open import Data.Nat.Base hiding (_⊔_)

+
open import Data.Product.Base as Product using (∃; _,_)

+
open import Data.Vec.Base using (Vec; []; _∷_; head; tail)

+
open import Function.Base using (_∘_; id; flip; constᵣ)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (REL)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- N-ary functions

+

+
N-ary-level : Level → Level → ℕ → Level

+
N-ary-level ℓ₁ ℓ₂ zero    = ℓ₂

+
N-ary-level ℓ₁ ℓ₂ (suc n) = ℓ₁ ⊔ N-ary-level ℓ₁ ℓ₂ n

+

+
N-ary : ∀ (n : ℕ) → Set ℓ₁ → Set ℓ₂ → Set (N-ary-level ℓ₁ ℓ₂ n)

+
N-ary zero    A B = B

+
N-ary (suc n) A B = A → N-ary n A B

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
curryⁿ : ∀ {n} → (Vec A n → B) → N-ary n A B

+
curryⁿ {n = zero}  f = f []

+
curryⁿ {n = suc n} f = λ x → curryⁿ (f ∘ _∷_ x)

+

+
infix -1 _$ⁿ_

+

+
_$ⁿ_ : ∀ {n} → N-ary n A B → (Vec A n → B)

+
f $ⁿ []       = f

+
f $ⁿ (x ∷ xs) = f x $ⁿ xs

+

+
------------------------------------------------------------------------

+
-- Quantifiers

+

+
module _ {A : Set a} where

+

+
  -- Universal quantifier.

+

+
  ∀ⁿ : ∀ n → N-ary n A (Set ℓ) → Set (N-ary-level a ℓ n)

+
  ∀ⁿ zero    P = P

+
  ∀ⁿ (suc n) P = ∀ x → ∀ⁿ n (P x)

+

+
  -- Universal quantifier with implicit (hidden) arguments.

+

+
  ∀ⁿʰ : ∀ n → N-ary n A (Set ℓ) → Set (N-ary-level a ℓ n)

+
  ∀ⁿʰ zero    P = P

+
  ∀ⁿʰ (suc n) P = ∀ {x} → ∀ⁿʰ n (P x)

+

+
  -- Existential quantifier.

+

+
  ∃ⁿ : ∀ n → N-ary n A (Set ℓ) → Set (N-ary-level a ℓ n)

+
  ∃ⁿ zero    P = P

+
  ∃ⁿ (suc n) P = ∃ λ x → ∃ⁿ n (P x)

+

+
------------------------------------------------------------------------

+
-- N-ary function equality

+

+
Eq : ∀ {A : Set a} {B : Set b} {C : Set c} n →

+
     REL B C ℓ → REL (N-ary n A B) (N-ary n A C) (N-ary-level a ℓ n)

+
Eq n _∼_ f g = ∀ⁿ n (curryⁿ {n = n} λ xs → (f $ⁿ xs) ∼ (g $ⁿ xs))

+

+
-- A variant where all the arguments are implicit (hidden).

+

+
Eqʰ : ∀ {A : Set a} {B : Set b} {C : Set c} n →

+
      REL B C ℓ → REL (N-ary n A B) (N-ary n A C) (N-ary-level a ℓ n)

+
Eqʰ n _∼_ f g = ∀ⁿʰ n (curryⁿ {n = n} λ xs → (f $ⁿ xs) ∼ (g $ⁿ xs))

+

+
------------------------------------------------------------------------

+
-- Some lemmas

+

+
-- The functions curryⁿ and _$ⁿ_ are inverses.

+

+
left-inverse : ∀ {n} (f : Vec A n → B) →

+
               ∀ xs → (curryⁿ f $ⁿ xs) ≡ f xs

+
left-inverse f []       = refl

+
left-inverse f (x ∷ xs) = left-inverse (f ∘ _∷_ x) xs

+

+
right-inverse : ∀ n (f : N-ary n A B) →

+
                Eq n _≡_ (curryⁿ (_$ⁿ_ {n = n} f)) f

+
right-inverse zero    f = refl

+
right-inverse (suc n) f = λ x → right-inverse n (f x)

+

+
-- ∀ⁿ can be expressed in an "uncurried" way.

+

+
uncurry-∀ⁿ : ∀ n {P : N-ary n A (Set ℓ)} →

+
             ∀ⁿ n P ⇔ (∀ (xs : Vec A n) → P $ⁿ xs)

+
uncurry-∀ⁿ {a} {A} {ℓ} n = mk⇔ (⇒ n) (⇐ n)

+
  where

+
  ⇒ : ∀ n {P : N-ary n A (Set ℓ)} →

+
      ∀ⁿ n P → (∀ (xs : Vec A n) → P $ⁿ xs)

+
  ⇒ zero    p []       = p

+
  ⇒ (suc n) p (x ∷ xs) = ⇒ n (p x) xs

+

+
  ⇐ : ∀ n {P : N-ary n A (Set ℓ)} →

+
      (∀ (xs : Vec A n) → P $ⁿ xs) → ∀ⁿ n P

+
  ⇐ zero    p = p []

+
  ⇐ (suc n) p = λ x → ⇐ n (p ∘ _∷_ x)

+

+
-- ∃ⁿ can be expressed in an "uncurried" way.

+

+
uncurry-∃ⁿ : ∀ n {P : N-ary n A (Set ℓ)} →

+
             ∃ⁿ n P ⇔ (∃ λ (xs : Vec A n) → P $ⁿ xs)

+
uncurry-∃ⁿ {a} {A} {ℓ} n = mk⇔ (⇒ n) (⇐ n)

+
  where

+
  ⇒ : ∀ n {P : N-ary n A (Set ℓ)} →

+
      ∃ⁿ n P → (∃ λ (xs : Vec A n) → P $ⁿ xs)

+
  ⇒ zero    p       = ([] , p)

+
  ⇒ (suc n) (x , p) = Product.map (_∷_ x) id (⇒ n p)

+

+
  ⇐ : ∀ n {P : N-ary n A (Set ℓ)} →

+
      (∃ λ (xs : Vec A n) → P $ⁿ xs) → ∃ⁿ n P

+
  ⇐ zero    ([] , p)     = p

+
  ⇐ (suc n) (x ∷ xs , p) = (x , ⇐ n (xs , p))

+

+
-- Conversion preserves equality.

+

+
module _ (_∼_ : REL B C ℓ) where

+

+
  curryⁿ-cong : ∀ {n} (f : Vec A n → B) (g : Vec A n → C) →

+
                (∀ xs → f xs ∼ g xs) →

+
                Eq n _∼_ (curryⁿ f) (curryⁿ g)

+
  curryⁿ-cong {n = zero}  f g hyp = hyp []

+
  curryⁿ-cong {n = suc n} f g hyp = λ x →

+
    curryⁿ-cong (f ∘ _∷_ x) (g ∘ _∷_ x) (λ xs → hyp (x ∷ xs))

+

+
  curryⁿ-cong⁻¹ : ∀ {n} (f : Vec A n → B) (g : Vec A n → C) →

+
                  Eq n _∼_ (curryⁿ f) (curryⁿ g) →

+
                  ∀ xs → f xs ∼ g xs

+
  curryⁿ-cong⁻¹ f g hyp []       = hyp

+
  curryⁿ-cong⁻¹ f g hyp (x ∷ xs) =

+
    curryⁿ-cong⁻¹ (f ∘ _∷_ x) (g ∘ _∷_ x) (hyp x) xs

+

+
  appⁿ-cong : ∀ {n} (f : N-ary n A B) (g : N-ary n A C) →

+
              Eq n _∼_ f g →

+
              (xs : Vec A n) → (f $ⁿ xs) ∼ (g $ⁿ xs)

+
  appⁿ-cong f g hyp []       = hyp

+
  appⁿ-cong f g hyp (x ∷ xs) = appⁿ-cong (f x) (g x) (hyp x) xs

+

+
  appⁿ-cong⁻¹ : ∀ {n} (f : N-ary n A B) (g : N-ary n A C) →

+
                ((xs : Vec A n) → (f $ⁿ xs) ∼ (g $ⁿ xs)) →

+
                Eq n _∼_ f g

+
  appⁿ-cong⁻¹ {n = zero}  f g hyp = hyp []

+
  appⁿ-cong⁻¹ {n = suc n} f g hyp = λ x →

+
    appⁿ-cong⁻¹ (f x) (g x) (λ xs → hyp (x ∷ xs))

+

+
-- Eq and Eqʰ are equivalent.

+

+
Eq-to-Eqʰ : ∀ n (_∼_ : REL B C ℓ) {f : N-ary n A B} {g : N-ary n A C} →

+
            Eq n _∼_ f g → Eqʰ n _∼_ f g

+
Eq-to-Eqʰ zero    _∼_ eq = eq

+
Eq-to-Eqʰ (suc n) _∼_ eq = Eq-to-Eqʰ n _∼_ (eq _)

+

+
Eqʰ-to-Eq : ∀ n (_∼_ : REL B C ℓ) {f : N-ary n A B} {g : N-ary n A C} →

+
            Eqʰ n _∼_ f g → Eq n _∼_ f g

+
Eqʰ-to-Eq zero    _∼_ eq = eq

+
Eqʰ-to-Eq (suc n) _∼_ eq = λ _ → Eqʰ-to-Eq n _∼_ eq

+

+
module _ (ext : ∀ {a b} → Extensionality a b) where

+

+
  Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B

+
  Vec↔N-ary zero = mk↔ₛ′ (λ vxs → vxs []) (flip constᵣ) (λ _ → refl)

+
    (λ vxs → ext λ where [] → refl)

+
  Vec↔N-ary (suc n) = let open Inverse (Vec↔N-ary n) in

+
    mk↔ₛ′ (λ vxs x → to λ xs → vxs (x ∷ xs))

+
    (λ any xs → from (any (head xs)) (tail xs))

+
    (λ any → ext λ x → strictlyInverseˡ _)

+
    (λ vxs → ext λ where (x ∷ xs) → cong (λ f → f xs) (strictlyInverseʳ (λ ys → vxs (x ∷ ys))))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Properties.html b/site/static/agda_html/Data.Vec.Properties.html
new file mode 100644
index 000000000..702438d79
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Properties.html
@@ -0,0 +1,1530 @@
+
+Data.Vec.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some Vec-related properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Properties where

+

+
open import Algebra.Definitions

+
open import Data.Bool.Base using (true; false)

+
open import Data.Fin.Base as Fin

+
  using (Fin; zero; suc; toℕ; fromℕ<; _↑ˡ_; _↑ʳ_)

+
open import Data.List.Base as List using (List)

+
import Data.List.Properties as List

+
open import Data.Nat.Base using (ℕ; zero; suc; _+_; _≤_; _<_; s≤s; pred; s<s⁻¹; _≥_;

+
  s≤s⁻¹; z≤n)

+
open import Data.Nat.Properties

+
  using (+-assoc; m≤n⇒m≤1+n; m≤m+n; ≤-refl; ≤-trans; ≤-irrelevant; ≤⇒≤″; suc-injective; +-comm; +-suc)

+
open import Data.Product.Base as Product

+
  using (_×_; _,_; proj₁; proj₂; <_,_>; uncurry)

+
open import Data.Sum.Base using ([_,_]′)

+
open import Data.Sum.Properties using ([,]-map)

+
open import Data.Vec.Base

+
open import Data.Vec.Relation.Binary.Equality.Cast as VecCast

+
  using (_≈[_]_; ≈-sym; module CastReasoning)

+
open import Function.Base using (_∘_; id; _$_; const; _ˢ_; flip)

+
open import Function.Bundles using (_↔_; mk↔ₛ′)

+
open import Level using (Level)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; _≢_; _≗_; refl; sym; trans; cong; cong₂; subst)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+
open import Relation.Unary using (Pred; Decidable)

+
open import Relation.Nullary.Decidable.Core using (Dec; does; yes; _×-dec_; map′)

+
open import Relation.Nullary.Negation.Core using (contradiction)

+
import Data.Nat.GeneralisedArithmetic as ℕ

+

+
private

+
  variable

+
    a b c d p : Level

+
    A B C D : Set a

+
    w x y z : A

+
    m n o : ℕ

+
    ws xs ys zs : Vec A n

+

+
------------------------------------------------------------------------

+
-- Properties of propositional equality over vectors

+

+
∷-injectiveˡ : x ∷ xs ≡ y ∷ ys → x ≡ y

+
∷-injectiveˡ refl = refl

+

+
∷-injectiveʳ : x ∷ xs ≡ y ∷ ys → xs ≡ ys

+
∷-injectiveʳ refl = refl

+

+
∷-injective : x ∷ xs ≡ y ∷ ys → x ≡ y × xs ≡ ys

+
∷-injective refl = refl , refl

+

+
≡-dec : DecidableEquality A → DecidableEquality (Vec A n)

+
≡-dec _≟_ []       []       = yes refl

+
≡-dec _≟_ (x ∷ xs) (y ∷ ys) = map′ (uncurry (cong₂ _∷_))

+
  ∷-injective (x ≟ y ×-dec ≡-dec _≟_ xs ys)

+

+
------------------------------------------------------------------------

+
-- _[_]=_

+

+
[]=-injective : ∀ {i} → xs [ i ]= x → xs [ i ]= y → x ≡ y

+
[]=-injective here          here          = refl

+
[]=-injective (there xsᵢ≡x) (there xsᵢ≡y) = []=-injective xsᵢ≡x xsᵢ≡y

+

+
-- See also Data.Vec.Properties.WithK.[]=-irrelevant.

+

+
------------------------------------------------------------------------

+
-- take

+

+
unfold-take : ∀ n x (xs : Vec A (n + m)) → take (suc n) (x ∷ xs) ≡ x ∷ take n xs

+
unfold-take n x xs = refl

+

+
take-zipWith : ∀ (f : A → B → C) →

+
               (xs : Vec A (m + n)) (ys : Vec B (m + n)) →

+
               take m (zipWith f xs ys) ≡ zipWith f (take m xs) (take m ys)

+
take-zipWith {m = zero}  f xs       ys       = refl

+
take-zipWith {m = suc m} f (x ∷ xs) (y ∷ ys) = cong (f x y ∷_) (take-zipWith f xs ys)

+

+
take-map : ∀ (f : A → B) (m : ℕ) (xs : Vec A (m + n)) →

+
           take m (map f xs) ≡ map f (take m xs)

+
take-map f zero    xs       = refl

+
take-map f (suc m) (x ∷ xs) = cong (f x ∷_) (take-map f m xs)

+

+
------------------------------------------------------------------------

+
-- drop

+

+
unfold-drop : ∀ n x (xs : Vec A (n + m)) →

+
              drop (suc n) (x ∷ xs) ≡ drop n xs

+
unfold-drop n x xs = refl

+

+
drop-zipWith : (f : A → B → C) →

+
               (xs : Vec A (m + n)) (ys : Vec B (m + n)) →

+
               drop m (zipWith f xs ys) ≡ zipWith f (drop m xs) (drop m ys)

+
drop-zipWith {m = zero}  f   xs       ys     = refl

+
drop-zipWith {m = suc m} f (x ∷ xs) (y ∷ ys) = drop-zipWith f xs ys

+

+
drop-map : ∀ (f : A → B) (m : ℕ) (xs : Vec A (m + n)) →

+
           drop m (map f xs) ≡ map f (drop m xs)

+
drop-map f zero    xs       = refl

+
drop-map f (suc m) (x ∷ xs) = drop-map f m xs

+

+
------------------------------------------------------------------------

+
-- take and drop together

+

+
take++drop≡id : ∀ (m : ℕ) (xs : Vec A (m + n)) → take m xs ++ drop m xs ≡ xs

+
take++drop≡id zero    xs       = refl

+
take++drop≡id (suc m) (x ∷ xs) = cong (x ∷_) (take++drop≡id m xs)

+

+
------------------------------------------------------------------------

+
-- truncate

+

+
truncate-refl : (xs : Vec A n) → truncate ≤-refl xs ≡ xs

+
truncate-refl []       = refl

+
truncate-refl (x ∷ xs) = cong (x ∷_) (truncate-refl xs)

+

+
truncate-trans : ∀ {p} (m≤n : m ≤ n) (n≤p : n ≤ p) (xs : Vec A p) →

+
                 truncate (≤-trans m≤n n≤p) xs ≡ truncate m≤n (truncate n≤p xs)

+
truncate-trans z≤n       n≤p       xs = refl

+
truncate-trans (s≤s m≤n) (s≤s n≤p) (x ∷ xs) = cong (x ∷_) (truncate-trans m≤n n≤p xs)

+

+
truncate-irrelevant : (m≤n₁ m≤n₂ : m ≤ n) → truncate {A = A} m≤n₁ ≗ truncate m≤n₂

+
truncate-irrelevant m≤n₁ m≤n₂ xs = cong (λ m≤n → truncate m≤n xs) (≤-irrelevant m≤n₁ m≤n₂)

+

+
truncate≡take : (m≤n : m ≤ n) (xs : Vec A n) .(eq : n ≡ m + o) →

+
                truncate m≤n xs ≡ take m (cast eq xs)

+
truncate≡take z≤n       _        eq = refl

+
truncate≡take (s≤s m≤n) (x ∷ xs) eq = cong (x ∷_) (truncate≡take m≤n xs (suc-injective eq))

+

+
take≡truncate : ∀ m (xs : Vec A (m + n)) →

+
                take m xs ≡ truncate (m≤m+n m n) xs

+
take≡truncate zero    _        = refl

+
take≡truncate (suc m) (x ∷ xs) = cong (x ∷_) (take≡truncate m xs)

+

+
------------------------------------------------------------------------

+
-- pad

+

+
padRight-refl : (a : A) (xs : Vec A n) → padRight ≤-refl a xs ≡ xs

+
padRight-refl a []       = refl

+
padRight-refl a (x ∷ xs) = cong (x ∷_) (padRight-refl a xs)

+

+
padRight-replicate : (m≤n : m ≤ n) (a : A) → replicate n a ≡ padRight m≤n a (replicate m a)

+
padRight-replicate z≤n       a = refl

+
padRight-replicate (s≤s m≤n) a = cong (a ∷_) (padRight-replicate m≤n a)

+

+
padRight-trans : ∀ {p} (m≤n : m ≤ n) (n≤p : n ≤ p) (a : A) (xs : Vec A m) →

+
            padRight (≤-trans m≤n n≤p) a xs ≡ padRight n≤p a (padRight m≤n a xs)

+
padRight-trans z≤n       n≤p       a []       = padRight-replicate n≤p a

+
padRight-trans (s≤s m≤n) (s≤s n≤p) a (x ∷ xs) = cong (x ∷_) (padRight-trans m≤n n≤p a xs)

+

+
------------------------------------------------------------------------

+
-- truncate and padRight together

+

+
truncate-padRight : (m≤n : m ≤ n) (a : A) (xs : Vec A m) →

+
                    truncate m≤n (padRight m≤n a xs) ≡ xs

+
truncate-padRight z≤n       a []       = refl

+
truncate-padRight (s≤s m≤n) a (x ∷ xs) = cong (x ∷_) (truncate-padRight m≤n a xs)

+

+
------------------------------------------------------------------------

+
-- lookup

+

+
[]=⇒lookup : ∀ {i} → xs [ i ]= x → lookup xs i ≡ x

+
[]=⇒lookup here            = refl

+
[]=⇒lookup (there xs[i]=x) = []=⇒lookup xs[i]=x

+

+
lookup⇒[]= : ∀ (i : Fin n) xs → lookup xs i ≡ x → xs [ i ]= x

+
lookup⇒[]= zero    (_ ∷ _)  refl = here

+
lookup⇒[]= (suc i) (_ ∷ xs) p    = there (lookup⇒[]= i xs p)

+

+
[]=↔lookup : ∀ {i} → xs [ i ]= x ↔ lookup xs i ≡ x

+
[]=↔lookup {xs = ys} {i = i} =

+
  mk↔ₛ′ []=⇒lookup (lookup⇒[]= i ys) ([]=⇒lookup∘lookup⇒[]= _ i) lookup⇒[]=∘[]=⇒lookup

+
  where

+
  lookup⇒[]=∘[]=⇒lookup : ∀ {i} (p : xs [ i ]= x) →

+
                          lookup⇒[]= i xs ([]=⇒lookup p) ≡ p

+
  lookup⇒[]=∘[]=⇒lookup here      = refl

+
  lookup⇒[]=∘[]=⇒lookup (there p) = cong there (lookup⇒[]=∘[]=⇒lookup p)

+

+
  []=⇒lookup∘lookup⇒[]= : ∀ xs (i : Fin n) (p : lookup xs i ≡ x) →

+
                          []=⇒lookup (lookup⇒[]= i xs p) ≡ p

+
  []=⇒lookup∘lookup⇒[]= (x ∷ xs) zero    refl = refl

+
  []=⇒lookup∘lookup⇒[]= (x ∷ xs) (suc i) p    = []=⇒lookup∘lookup⇒[]= xs i p

+

+
lookup-truncate : (m≤n : m ≤ n) (xs : Vec A n) (i : Fin m) →

+
                  lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)

+
lookup-truncate (s≤s m≤m+n) (_ ∷ _)  zero    = refl

+
lookup-truncate (s≤s m≤m+n) (_ ∷ xs) (suc i) = lookup-truncate m≤m+n xs i

+

+
lookup-take-inject≤ : (xs : Vec A (m + n)) (i : Fin m) →

+
                      lookup (take m xs) i ≡ lookup xs (Fin.inject≤ i (m≤m+n m n))

+
lookup-take-inject≤ {m = m} {n = n} xs i = begin

+
  lookup (take _ xs) i

+
    ≡⟨ cong (λ ys → lookup ys i) (take≡truncate m xs) ⟩

+
  lookup (truncate _ xs) i

+
    ≡⟨ lookup-truncate (m≤m+n m n) xs i ⟩

+
  lookup xs (Fin.inject≤ i (m≤m+n m n))

+
    ∎ where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- updateAt (_[_]%=_)

+

+
-- (+) updateAt i actually updates the element at index i.

+

+
updateAt-updates : ∀ (i : Fin n) {f : A → A} (xs : Vec A n) →

+
                   xs [ i ]= x → (updateAt xs i f) [ i ]= f x

+
updateAt-updates zero    (x ∷ xs) here        = here

+
updateAt-updates (suc i) (x ∷ xs) (there loc) = there (updateAt-updates i xs loc)

+

+
-- (-) updateAt i does not touch the elements at other indices.

+

+
updateAt-minimal : ∀ (i j : Fin n) {f : A → A} (xs : Vec A n) →

+
                   i ≢ j → xs [ i ]= x → (updateAt xs j f) [ i ]= x

+
updateAt-minimal zero    zero    (x ∷ xs) 0≢0 here        = contradiction refl 0≢0

+
updateAt-minimal zero    (suc j) (x ∷ xs) _   here        = here

+
updateAt-minimal (suc i) zero    (x ∷ xs) _   (there loc) = there loc

+
updateAt-minimal (suc i) (suc j) (x ∷ xs) i≢j (there loc) =

+
  there (updateAt-minimal i j xs (i≢j ∘ cong suc) loc)

+

+
-- The other properties are consequences of (+) and (-).

+
-- We spell the most natural properties out.

+
-- Direct inductive proofs are in most cases easier than just using

+
-- the defining properties.

+

+
-- In the explanations, we make use of shorthand  f = g ↾ x

+
-- meaning that f and g agree locally at point x, i.e.  f x ≡ g x.

+

+
-- updateAt i  is a morphism from the monoid of endofunctions  A → A

+
-- to the monoid of endofunctions  Vec A n → Vec A n

+

+
-- 1a. local identity:  f = id ↾ (lookup xs i)

+
--             implies  updateAt i f = id ↾ xs

+

+
updateAt-id-local : ∀ (i : Fin n) {f : A → A} (xs : Vec A n) →

+
                    f (lookup xs i) ≡ lookup xs i →

+
                    updateAt xs i f ≡ xs

+
updateAt-id-local zero    (x ∷ xs) eq = cong (_∷ xs) eq

+
updateAt-id-local (suc i) (x ∷ xs) eq = cong (x ∷_) (updateAt-id-local i xs eq)

+

+
-- 1b. identity:  updateAt i id ≗ id

+

+
updateAt-id : ∀ (i : Fin n) (xs : Vec A n) → updateAt xs i id ≡ xs

+
updateAt-id i xs = updateAt-id-local i xs refl

+

+
-- 2a. local composition:  f ∘ g = h ↾ (lookup xs i)

+
--                implies  updateAt i f ∘ updateAt i g = updateAt i h ↾ xs

+

+
updateAt-updateAt-local : ∀ (i : Fin n) {f g h : A → A} (xs : Vec A n) →

+
                          f (g (lookup xs i)) ≡ h (lookup xs i) →

+
                          updateAt (updateAt xs i g) i f ≡ updateAt xs i h

+
updateAt-updateAt-local zero    (x ∷ xs) fg=h = cong (_∷ xs) fg=h

+
updateAt-updateAt-local (suc i) (x ∷ xs) fg=h = cong (x ∷_) (updateAt-updateAt-local i xs fg=h)

+

+
-- 2b. composition:  updateAt i f ∘ updateAt i g ≗ updateAt i (f ∘ g)

+

+
updateAt-updateAt : ∀ (i : Fin n) {f g : A → A} (xs : Vec A n) →

+
                    updateAt (updateAt xs i g) i f ≡ updateAt xs i (f ∘ g)

+
updateAt-updateAt i xs = updateAt-updateAt-local i xs refl

+

+
-- 3. congruence:  updateAt i  is a congruence wrt. extensional equality.

+

+
-- 3a.  If    f = g ↾ (lookup xs i)

+
--      then  updateAt i f = updateAt i g ↾ xs

+

+
updateAt-cong-local : ∀ (i : Fin n) {f g : A → A} (xs : Vec A n) →

+
                      f (lookup xs i) ≡ g (lookup xs i) →

+
                      updateAt xs i f ≡ updateAt xs i g

+
updateAt-cong-local zero    (x ∷ xs) f=g = cong (_∷ xs) f=g

+
updateAt-cong-local (suc i) (x ∷ xs) f=g = cong (x ∷_) (updateAt-cong-local i xs f=g)

+

+
-- 3b. congruence:  f ≗ g → updateAt i f ≗ updateAt i g

+

+
updateAt-cong : ∀ (i : Fin n) {f g : A → A} → f ≗ g → (xs : Vec A n) →

+
                updateAt xs i f ≡ updateAt xs i g

+
updateAt-cong i f≗g xs = updateAt-cong-local i xs (f≗g (lookup xs i))

+

+
-- The order of updates at different indices i ≢ j does not matter.

+

+
-- This a consequence of updateAt-updates and updateAt-minimal

+
-- but easier to prove inductively.

+

+
updateAt-commutes : ∀ (i j : Fin n) {f g : A → A} → i ≢ j → (xs : Vec A n) →

+
                    updateAt (updateAt xs j g) i f ≡ updateAt (updateAt xs i f) j g

+
updateAt-commutes zero    zero    0≢0 (x ∷ xs) = contradiction refl 0≢0

+
updateAt-commutes zero    (suc j) i≢j (x ∷ xs) = refl

+
updateAt-commutes (suc i) zero    i≢j (x ∷ xs) = refl

+
updateAt-commutes (suc i) (suc j) i≢j (x ∷ xs) =

+
   cong (x ∷_) (updateAt-commutes i j (i≢j ∘ cong suc) xs)

+

+
-- lookup after updateAt reduces.

+

+
-- For same index this is an easy consequence of updateAt-updates

+
-- using []=↔lookup.

+

+
lookup∘updateAt : ∀ (i : Fin n) {f : A → A} xs →

+
                  lookup (updateAt xs i f) i ≡ f (lookup xs i)

+
lookup∘updateAt i xs =

+
  []=⇒lookup (updateAt-updates i xs (lookup⇒[]= i _ refl))

+

+
-- For different indices it easily follows from updateAt-minimal.

+

+
lookup∘updateAt′ : ∀ (i j : Fin n) {f : A → A} → i ≢ j → ∀ xs →

+
                   lookup (updateAt xs j f) i ≡ lookup xs i

+
lookup∘updateAt′ i j xs i≢j =

+
  []=⇒lookup (updateAt-minimal i j i≢j xs (lookup⇒[]= i _ refl))

+

+
-- Aliases for notation _[_]%=_

+

+
[]%=-id : ∀ (xs : Vec A n) (i : Fin n) → xs [ i ]%= id ≡ xs

+
[]%=-id xs i = updateAt-id i xs

+

+
[]%=-∘ : ∀ (xs : Vec A n) (i : Fin n) {f g : A → A} →

+
      xs [ i ]%= f

+
         [ i ]%= g

+
    ≡ xs [ i ]%= g ∘ f

+
[]%=-∘ xs i = updateAt-updateAt i xs

+

+

+
------------------------------------------------------------------------

+
-- _[_]≔_ (update)

+
--

+
-- _[_]≔_ is defined in terms of updateAt, and all of its properties

+
-- are special cases of the ones for updateAt.

+

+
[]≔-idempotent : ∀ (xs : Vec A n) (i : Fin n) →

+
                  (xs [ i ]≔ x) [ i ]≔ y ≡ xs [ i ]≔ y

+
[]≔-idempotent xs i = updateAt-updateAt i xs

+

+
[]≔-commutes : ∀ (xs : Vec A n) (i j : Fin n) → i ≢ j →

+
                (xs [ i ]≔ x) [ j ]≔ y ≡ (xs [ j ]≔ y) [ i ]≔ x

+
[]≔-commutes xs i j i≢j = updateAt-commutes j i (i≢j ∘ sym) xs

+

+
[]≔-updates : ∀ (xs : Vec A n) (i : Fin n) → (xs [ i ]≔ x) [ i ]= x

+
[]≔-updates xs i = updateAt-updates i xs (lookup⇒[]= i xs refl)

+

+
[]≔-minimal : ∀ (xs : Vec A n) (i j : Fin n) → i ≢ j →

+
              xs [ i ]= x → (xs [ j ]≔ y) [ i ]= x

+
[]≔-minimal xs i j i≢j loc = updateAt-minimal i j xs i≢j loc

+

+
[]≔-lookup : ∀ (xs : Vec A n) (i : Fin n) → xs [ i ]≔ lookup xs i ≡ xs

+
[]≔-lookup xs i = updateAt-id-local i xs refl

+

+
[]≔-++-↑ˡ : ∀ (xs : Vec A m) (ys : Vec A n) i →

+
            (xs ++ ys) [ i ↑ˡ n ]≔ x ≡ (xs [ i ]≔ x) ++ ys

+
[]≔-++-↑ˡ (x ∷ xs) ys zero    = refl

+
[]≔-++-↑ˡ (x ∷ xs) ys (suc i) =

+
  cong (x ∷_) $ []≔-++-↑ˡ xs ys i

+

+
[]≔-++-↑ʳ : ∀ (xs : Vec A m) (ys : Vec A n) i →

+
            (xs ++ ys) [ m ↑ʳ i ]≔ y ≡ xs ++ (ys [ i ]≔ y)

+
[]≔-++-↑ʳ {m = zero}     []    (y ∷ ys) i = refl

+
[]≔-++-↑ʳ {m = suc n} (x ∷ xs) (y ∷ ys) i = cong (x ∷_) $ []≔-++-↑ʳ xs (y ∷ ys) i

+

+
lookup∘update : ∀ (i : Fin n) (xs : Vec A n) x →

+
                lookup (xs [ i ]≔ x) i ≡ x

+
lookup∘update i xs x = lookup∘updateAt i xs

+

+
lookup∘update′ : ∀ {i j} → i ≢ j → ∀ (xs : Vec A n) y →

+
                 lookup (xs [ j ]≔ y) i ≡ lookup xs i

+
lookup∘update′ {i = i} {j} i≢j xs y = lookup∘updateAt′ i j i≢j xs

+

+
------------------------------------------------------------------------

+
-- cast

+

+
open VecCast public

+
  using (cast-is-id; cast-trans)

+

+
subst-is-cast : (eq : m ≡ n) (xs : Vec A m) → subst (Vec A) eq xs ≡ cast eq xs

+
subst-is-cast refl xs = sym (cast-is-id refl xs)

+

+
cast-sym : .(eq : m ≡ n) {xs : Vec A m} {ys : Vec A n} →

+
           cast eq xs ≡ ys → cast (sym eq) ys ≡ xs

+
cast-sym eq {xs = []}     {ys = []}     _ = refl

+
cast-sym eq {xs = x ∷ xs} {ys = y ∷ ys} xxs[eq]≡yys =

+
  let x≡y , xs[eq]≡ys = ∷-injective xxs[eq]≡yys

+
  in cong₂ _∷_ (sym x≡y) (cast-sym (suc-injective eq) xs[eq]≡ys)

+

+
------------------------------------------------------------------------

+
-- map

+

+
map-id : map id ≗ id {A = Vec A n}

+
map-id []       = refl

+
map-id (x ∷ xs) = cong (x ∷_) (map-id xs)

+

+
map-const : ∀ (xs : Vec A n) (y : B) → map (const y) xs ≡ replicate n y

+
map-const []       _ = refl

+
map-const (_ ∷ xs) y = cong (y ∷_) (map-const xs y)

+

+
map-cast : (f : A → B) .(eq : m ≡ n) (xs : Vec A m) →

+
           map f (cast eq xs) ≡ cast eq (map f xs)

+
map-cast {n = zero}  f eq []       = refl

+
map-cast {n = suc _} f eq (x ∷ xs)

+
  = cong (f x ∷_) (map-cast f (suc-injective eq) xs)

+

+
map-++ : ∀ (f : A → B) (xs : Vec A m) (ys : Vec A n) →

+
         map f (xs ++ ys) ≡ map f xs ++ map f ys

+
map-++ f []       ys = refl

+
map-++ f (x ∷ xs) ys = cong (f x ∷_) (map-++ f xs ys)

+

+
map-cong : ∀ {f g : A → B} → f ≗ g → map {n = n} f ≗ map g

+
map-cong f≗g []       = refl

+
map-cong f≗g (x ∷ xs) = cong₂ _∷_ (f≗g x) (map-cong f≗g xs)

+

+
map-∘ : ∀ (f : B → C) (g : A → B) →

+
        map {n = n} (f ∘ g) ≗ map f ∘ map g

+
map-∘ f g []       = refl

+
map-∘ f g (x ∷ xs) = cong (f (g x) ∷_) (map-∘ f g xs)

+

+
lookup-map : ∀ (i : Fin n) (f : A → B) (xs : Vec A n) →

+
             lookup (map f xs) i ≡ f (lookup xs i)

+
lookup-map zero    f (x ∷ xs) = refl

+
lookup-map (suc i) f (x ∷ xs) = lookup-map i f xs

+

+
map-updateAt : ∀ {f : A → B} {g : A → A} {h : B → B}

+
               (xs : Vec A n) (i : Fin n) →

+
               f (g (lookup xs i)) ≡ h (f (lookup xs i)) →

+
               map f (updateAt xs i g) ≡ updateAt (map f xs) i h

+
map-updateAt (x ∷ xs) zero    eq = cong (_∷ _) eq

+
map-updateAt (x ∷ xs) (suc i) eq = cong (_ ∷_) (map-updateAt xs i eq)

+

+
map-insertAt : ∀ (f : A → B) (x : A) (xs : Vec A n) (i : Fin (suc n)) →

+
             map f (insertAt xs i x) ≡ insertAt (map f xs) i (f x)

+
map-insertAt f _ []        Fin.zero = refl

+
map-insertAt f _ (x' ∷ xs) Fin.zero = refl

+
map-insertAt f x (x' ∷ xs) (Fin.suc i) = cong (_ ∷_) (map-insertAt f x xs i)

+

+
map-[]≔ : ∀ (f : A → B) (xs : Vec A n) (i : Fin n) →

+
          map f (xs [ i ]≔ x) ≡ map f xs [ i ]≔ f x

+
map-[]≔ f xs i = map-updateAt xs i refl

+

+
map-⊛ : ∀ (f : A → B → C) (g : A → B) (xs : Vec A n) →

+
        (map f xs ⊛ map g xs) ≡ map (f ˢ g) xs

+
map-⊛ f g []       = refl

+
map-⊛ f g (x ∷ xs) = cong (f x (g x) ∷_) (map-⊛ f g xs)

+

+
toList-map : ∀ (f : A → B) (xs : Vec A n) →

+
             toList (map f xs) ≡ List.map f (toList xs)

+
toList-map f [] = refl

+
toList-map f (x ∷ xs) = cong (f x List.∷_) (toList-map f xs)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
-- See also Data.Vec.Properties.WithK.++-assoc.

+

+
++-injectiveˡ : ∀ {n} (ws xs : Vec A n) → ws ++ ys ≡ xs ++ zs → ws ≡ xs

+
++-injectiveˡ []       []         _  = refl

+
++-injectiveˡ (_ ∷ ws) (_ ∷ xs) eq =

+
  cong₂ _∷_ (∷-injectiveˡ eq) (++-injectiveˡ _ _ (∷-injectiveʳ eq))

+

+
++-injectiveʳ : ∀ {n} (ws xs : Vec A n) → ws ++ ys ≡ xs ++ zs → ys ≡ zs

+
++-injectiveʳ []       []         eq = eq

+
++-injectiveʳ (x ∷ ws) (x′ ∷ xs) eq =

+
  ++-injectiveʳ ws xs (∷-injectiveʳ eq)

+

+
++-injective  : ∀ (ws xs : Vec A n) →

+
                ws ++ ys ≡ xs ++ zs → ws ≡ xs × ys ≡ zs

+
++-injective ws xs eq =

+
  (++-injectiveˡ ws xs eq , ++-injectiveʳ ws xs eq)

+

+
++-assoc : ∀ .(eq : (m + n) + o ≡ m + (n + o)) (xs : Vec A m) (ys : Vec A n) (zs : Vec A o) →

+
           cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)

+
++-assoc eq []       ys zs = cast-is-id eq (ys ++ zs)

+
++-assoc eq (x ∷ xs) ys zs = cong (x ∷_) (++-assoc (cong pred eq) xs ys zs)

+

+
++-identityʳ : ∀ .(eq : n + zero ≡ n) (xs : Vec A n) → cast eq (xs ++ []) ≡ xs

+
++-identityʳ eq []       = refl

+
++-identityʳ eq (x ∷ xs) = cong (x ∷_) (++-identityʳ (cong pred eq) xs)

+

+
cast-++ˡ : ∀ .(eq : m ≡ o) (xs : Vec A m) {ys : Vec A n} →

+
           cast (cong (_+ n) eq) (xs ++ ys) ≡ cast eq xs ++ ys

+
cast-++ˡ {o = zero}  eq []       {ys} = cast-is-id refl (cast eq [] ++ ys)

+
cast-++ˡ {o = suc o} eq (x ∷ xs) {ys} = cong (x ∷_) (cast-++ˡ (cong pred eq) xs)

+

+
cast-++ʳ : ∀ .(eq : n ≡ o) (xs : Vec A m) {ys : Vec A n} →

+
           cast (cong (m +_) eq) (xs ++ ys) ≡ xs ++ cast eq ys

+
cast-++ʳ {m = zero}  eq []       {ys} = refl

+
cast-++ʳ {m = suc m} eq (x ∷ xs) {ys} = cong (x ∷_) (cast-++ʳ eq xs)

+

+
lookup-++-< : ∀ (xs : Vec A m) (ys : Vec A n) →

+
              ∀ i (i<m : toℕ i < m) →

+
              lookup (xs ++ ys) i  ≡ lookup xs (Fin.fromℕ< i<m)

+
lookup-++-< (x ∷ xs) ys zero    _     = refl

+
lookup-++-< (x ∷ xs) ys (suc i) si<sm = lookup-++-< xs ys i (s<s⁻¹ si<sm)

+

+
lookup-++-≥ : ∀ (xs : Vec A m) (ys : Vec A n) →

+
              ∀ i (i≥m : toℕ i ≥ m) →

+
              lookup (xs ++ ys) i ≡ lookup ys (Fin.reduce≥ i i≥m)

+
lookup-++-≥ []       ys i       _     = refl

+
lookup-++-≥ (x ∷ xs) ys (suc i) si≥sm = lookup-++-≥ xs ys i (s≤s⁻¹ si≥sm)

+

+
lookup-++ˡ : ∀ (xs : Vec A m) (ys : Vec A n) i →

+
             lookup (xs ++ ys) (i ↑ˡ n) ≡ lookup xs i

+
lookup-++ˡ (x ∷ xs) ys zero    = refl

+
lookup-++ˡ (x ∷ xs) ys (suc i) = lookup-++ˡ xs ys i

+

+
lookup-++ʳ : ∀ (xs : Vec A m) (ys : Vec A n) i →

+
             lookup (xs ++ ys) (m ↑ʳ i) ≡ lookup ys i

+
lookup-++ʳ []       ys       zero    = refl

+
lookup-++ʳ []       (y ∷ xs) (suc i) = lookup-++ʳ [] xs i

+
lookup-++ʳ (x ∷ xs) ys       i       = lookup-++ʳ xs ys i

+

+
lookup-splitAt : ∀ m (xs : Vec A m) (ys : Vec A n) i →

+
                lookup (xs ++ ys) i ≡ [ lookup xs , lookup ys ]′

+
                (Fin.splitAt m i)

+
lookup-splitAt zero    []       ys i       = refl

+
lookup-splitAt (suc m) (x ∷ xs) ys zero    = refl

+
lookup-splitAt (suc m) (x ∷ xs) ys (suc i) = trans

+
  (lookup-splitAt m xs ys i)

+
  (sym ([,]-map (Fin.splitAt m i)))

+

+
toList-++ : (xs : Vec A n) (ys : Vec A m) →

+
            toList (xs ++ ys) ≡ toList xs List.++ toList ys

+
toList-++ []       ys = refl

+
toList-++ (x ∷ xs) ys = cong (x List.∷_) (toList-++ xs ys)

+

+
------------------------------------------------------------------------

+
-- concat

+

+
lookup-cast : .(eq : m ≡ n) (xs : Vec A m) (i : Fin m) →

+
              lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i

+
lookup-cast {n = suc _} eq (x ∷ _)  zero    = refl

+
lookup-cast {n = suc _} eq (_ ∷ xs) (suc i) =

+
  lookup-cast (suc-injective eq) xs i

+

+
lookup-cast₁ : .(eq : m ≡ n) (xs : Vec A m) (i : Fin n) →

+
               lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)

+
lookup-cast₁ eq (x ∷ _)  zero    = refl

+
lookup-cast₁ eq (_ ∷ xs) (suc i) =

+
  lookup-cast₁ (suc-injective eq) xs i

+

+
lookup-cast₂ : .(eq : m ≡ n) (xs : Vec A n) (i : Fin m) →

+
               lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i

+
lookup-cast₂ eq (x ∷ _)  zero    = refl

+
lookup-cast₂ eq (_ ∷ xs) (suc i) =

+
  lookup-cast₂ (suc-injective eq) xs i

+

+
lookup-concat : ∀ (xss : Vec (Vec A m) n) i j →

+
                lookup (concat xss) (Fin.combine i j) ≡ lookup (lookup xss i) j

+
lookup-concat (xs ∷ xss) zero j = lookup-++ˡ xs (concat xss) j

+
lookup-concat (xs ∷ xss) (suc i) j = begin

+
  lookup (concat (xs ∷ xss)) (Fin.combine (suc i) j)

+
    ≡⟨ lookup-++ʳ xs (concat xss) (Fin.combine i j) ⟩

+
  lookup (concat xss) (Fin.combine i j)

+
    ≡⟨ lookup-concat xss i j ⟩

+
  lookup (lookup (xs ∷ xss) (suc i)) j

+
    ∎ where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- zipWith

+

+
module _ {f : A → A → A} where

+

+
  zipWith-assoc : Associative _≡_ f →

+
                  Associative _≡_ (zipWith {n = n} f)

+
  zipWith-assoc assoc []       []       []       = refl

+
  zipWith-assoc assoc (x ∷ xs) (y ∷ ys) (z ∷ zs) =

+
    cong₂ _∷_ (assoc x y z) (zipWith-assoc assoc xs ys zs)

+

+
  zipWith-idem : Idempotent _≡_ f →

+
                 Idempotent _≡_ (zipWith {n = n} f)

+
  zipWith-idem idem []       = refl

+
  zipWith-idem idem (x ∷ xs) =

+
    cong₂ _∷_ (idem x) (zipWith-idem idem xs)

+

+
module _ {f : A → A → A} {e : A} where

+

+
  zipWith-identityˡ : LeftIdentity _≡_ e f →

+
                      LeftIdentity _≡_ (replicate n e) (zipWith f)

+
  zipWith-identityˡ idˡ []       = refl

+
  zipWith-identityˡ idˡ (x ∷ xs) =

+
    cong₂ _∷_ (idˡ x) (zipWith-identityˡ idˡ xs)

+

+
  zipWith-identityʳ : RightIdentity _≡_ e f →

+
                      RightIdentity _≡_ (replicate n e) (zipWith f)

+
  zipWith-identityʳ idʳ []       = refl

+
  zipWith-identityʳ idʳ (x ∷ xs) =

+
    cong₂ _∷_ (idʳ x) (zipWith-identityʳ idʳ xs)

+

+
  zipWith-zeroˡ : LeftZero _≡_ e f →

+
                  LeftZero _≡_ (replicate n e) (zipWith f)

+
  zipWith-zeroˡ zeˡ []       = refl

+
  zipWith-zeroˡ zeˡ (x ∷ xs) =

+
    cong₂ _∷_ (zeˡ x) (zipWith-zeroˡ zeˡ xs)

+

+
  zipWith-zeroʳ : RightZero _≡_ e f →

+
                  RightZero _≡_ (replicate n e) (zipWith f)

+
  zipWith-zeroʳ zeʳ []       = refl

+
  zipWith-zeroʳ zeʳ (x ∷ xs) =

+
    cong₂ _∷_ (zeʳ x) (zipWith-zeroʳ zeʳ xs)

+

+
module _ {f : A → A → A} {e : A} {⁻¹ : A → A} where

+

+
  zipWith-inverseˡ : LeftInverse _≡_ e ⁻¹ f →

+
                     LeftInverse _≡_ (replicate n e) (map ⁻¹) (zipWith f)

+
  zipWith-inverseˡ invˡ []       = refl

+
  zipWith-inverseˡ invˡ (x ∷ xs) =

+
    cong₂ _∷_ (invˡ x) (zipWith-inverseˡ invˡ xs)

+

+
  zipWith-inverseʳ : RightInverse _≡_ e ⁻¹ f →

+
                     RightInverse _≡_ (replicate n e) (map ⁻¹) (zipWith f)

+
  zipWith-inverseʳ invʳ []       = refl

+
  zipWith-inverseʳ invʳ (x ∷ xs) =

+
    cong₂ _∷_ (invʳ x) (zipWith-inverseʳ invʳ xs)

+

+
module _ {f g : A → A → A} where

+

+
  zipWith-distribˡ : _DistributesOverˡ_ _≡_ f g →

+
                     _DistributesOverˡ_ _≡_ (zipWith {n = n} f) (zipWith g)

+
  zipWith-distribˡ distribˡ []        []      []       = refl

+
  zipWith-distribˡ distribˡ (x ∷ xs) (y ∷ ys) (z ∷ zs) =

+
    cong₂ _∷_ (distribˡ x y z) (zipWith-distribˡ distribˡ xs ys zs)

+

+
  zipWith-distribʳ : _DistributesOverʳ_ _≡_ f g →

+
                     _DistributesOverʳ_ _≡_ (zipWith {n = n} f) (zipWith g)

+
  zipWith-distribʳ distribʳ []        []      []       = refl

+
  zipWith-distribʳ distribʳ (x ∷ xs) (y ∷ ys) (z ∷ zs) =

+
    cong₂ _∷_ (distribʳ x y z) (zipWith-distribʳ distribʳ xs ys zs)

+

+
  zipWith-absorbs : _Absorbs_ _≡_ f g →

+
                   _Absorbs_ _≡_ (zipWith {n = n} f) (zipWith g)

+
  zipWith-absorbs abs []       []       = refl

+
  zipWith-absorbs abs (x ∷ xs) (y ∷ ys) =

+
    cong₂ _∷_ (abs x y) (zipWith-absorbs abs xs ys)

+

+
module _ {f : A → B → C} {g : B → A → C} where

+

+
  zipWith-comm : ∀ (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys →

+
                 zipWith f xs ys ≡ zipWith g ys xs

+
  zipWith-comm comm []       []       = refl

+
  zipWith-comm comm (x ∷ xs) (y ∷ ys) =

+
    cong₂ _∷_ (comm x y) (zipWith-comm comm xs ys)

+

+
zipWith-map₁ : ∀ (_⊕_ : B → C → D) (f : A → B)

+
               (xs : Vec A n) (ys : Vec C n) →

+
               zipWith _⊕_ (map f xs) ys ≡ zipWith (λ x y → f x ⊕ y) xs ys

+
zipWith-map₁ _⊕_ f []       []       = refl

+
zipWith-map₁ _⊕_ f (x ∷ xs) (y ∷ ys) =

+
  cong (f x ⊕ y ∷_) (zipWith-map₁ _⊕_ f xs ys)

+

+
zipWith-map₂ : ∀ (_⊕_ : A → C → D) (f : B → C)

+
               (xs : Vec A n) (ys : Vec B n) →

+
               zipWith _⊕_ xs (map f ys) ≡ zipWith (λ x y → x ⊕ f y) xs ys

+
zipWith-map₂ _⊕_ f []       []       = refl

+
zipWith-map₂ _⊕_ f (x ∷ xs) (y ∷ ys) =

+
  cong (x ⊕ f y ∷_) (zipWith-map₂ _⊕_ f xs ys)

+

+
lookup-zipWith : ∀ (f : A → B → C) (i : Fin n) xs ys →

+
                 lookup (zipWith f xs ys) i ≡ f (lookup xs i) (lookup ys i)

+
lookup-zipWith _ zero    (x ∷ _)  (y ∷ _)   = refl

+
lookup-zipWith _ (suc i) (_ ∷ xs) (_ ∷ ys)  = lookup-zipWith _ i xs ys

+

+
zipWith-++ : ∀ (f : A → B → C)

+
             (xs : Vec A n) (ys : Vec A m)

+
             (xs' : Vec B n) (ys' : Vec B m) →

+
             zipWith f (xs ++ ys) (xs' ++ ys') ≡

+
             zipWith f xs xs' ++ zipWith f ys ys'

+
zipWith-++ f []       ys []         ys' = refl

+
zipWith-++ f (x ∷ xs) ys (x' ∷ xs') ys' =

+
  cong (_ ∷_) (zipWith-++ f xs ys xs' ys')

+

+
------------------------------------------------------------------------

+
-- zip

+

+
lookup-zip : ∀ (i : Fin n) (xs : Vec A n) (ys : Vec B n) →

+
             lookup (zip xs ys) i ≡ (lookup xs i , lookup ys i)

+
lookup-zip = lookup-zipWith _,_

+

+
-- map lifts projections to vectors of products.

+

+
map-proj₁-zip : ∀ (xs : Vec A n) (ys : Vec B n) →

+
                map proj₁ (zip xs ys) ≡ xs

+
map-proj₁-zip []       []       = refl

+
map-proj₁-zip (x ∷ xs) (y ∷ ys) = cong (x ∷_) (map-proj₁-zip xs ys)

+

+
map-proj₂-zip : ∀ (xs : Vec A n) (ys : Vec B n) →

+
                map proj₂ (zip xs ys) ≡ ys

+
map-proj₂-zip []       []       = refl

+
map-proj₂-zip (x ∷ xs) (y ∷ ys) = cong (y ∷_) (map-proj₂-zip xs ys)

+

+
-- map lifts pairing to vectors of products.

+

+
map-<,>-zip : ∀ (f : A → B) (g : A → C) (xs : Vec A n) →

+
              map < f , g > xs ≡ zip (map f xs) (map g xs)

+
map-<,>-zip f g []       = refl

+
map-<,>-zip f g (x ∷ xs) = cong (_ ∷_) (map-<,>-zip f g xs)

+

+
map-zip : ∀ (f : A → B) (g : C → D) (xs : Vec A n) (ys : Vec C n) →

+
          map (Product.map f g) (zip xs ys) ≡ zip (map f xs) (map g ys)

+
map-zip f g []       []       = refl

+
map-zip f g (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (map-zip f g xs ys)

+

+
------------------------------------------------------------------------

+
-- unzip

+

+
lookup-unzip : ∀ (i : Fin n) (xys : Vec (A × B) n) →

+
               let xs , ys = unzip xys

+
               in (lookup xs i , lookup ys i) ≡ lookup xys i

+
lookup-unzip ()      []

+
lookup-unzip zero    ((x , y) ∷ xys) = refl

+
lookup-unzip (suc i) ((x , y) ∷ xys) = lookup-unzip i xys

+

+
map-unzip : ∀ (f : A → B) (g : C → D) (xys : Vec (A × C) n) →

+
            let xs , ys = unzip xys

+
            in (map f xs , map g ys) ≡ unzip (map (Product.map f g) xys)

+
map-unzip f g []              = refl

+
map-unzip f g ((x , y) ∷ xys) =

+
  cong (Product.map (f x ∷_) (g y ∷_)) (map-unzip f g xys)

+

+
-- Products of vectors are isomorphic to vectors of products.

+

+
unzip∘zip : ∀ (xs : Vec A n) (ys : Vec B n) →

+
            unzip (zip xs ys) ≡ (xs , ys)

+
unzip∘zip [] []             = refl

+
unzip∘zip (x ∷ xs) (y ∷ ys) =

+
  cong (Product.map (x ∷_) (y ∷_)) (unzip∘zip xs ys)

+

+
zip∘unzip : ∀ (xys : Vec (A × B) n) → uncurry zip (unzip xys) ≡ xys

+
zip∘unzip []         = refl

+
zip∘unzip (xy ∷ xys) = cong (xy ∷_) (zip∘unzip xys)

+

+
×v↔v× : (Vec A n × Vec B n) ↔ Vec (A × B) n

+
×v↔v× = mk↔ₛ′ (uncurry zip) unzip zip∘unzip (uncurry unzip∘zip)

+

+
------------------------------------------------------------------------

+
-- _⊛_

+

+
lookup-⊛ : ∀ i (fs : Vec (A → B) n) (xs : Vec A n) →

+
           lookup (fs ⊛ xs) i ≡ (lookup fs i $ lookup xs i)

+
lookup-⊛ zero    (f ∷ fs) (x ∷ xs) = refl

+
lookup-⊛ (suc i) (f ∷ fs) (x ∷ xs) = lookup-⊛ i fs xs

+

+
map-is-⊛ : ∀ (f : A → B) (xs : Vec A n) →

+
           map f xs ≡ (replicate n f ⊛ xs)

+
map-is-⊛ f []       = refl

+
map-is-⊛ f (x ∷ xs) = cong (_ ∷_) (map-is-⊛ f xs)

+

+
⊛-is-zipWith : ∀ (fs : Vec (A → B) n) (xs : Vec A n) →

+
               (fs ⊛ xs) ≡ zipWith _$_ fs xs

+
⊛-is-zipWith []       []       = refl

+
⊛-is-zipWith (f ∷ fs) (x ∷ xs) = cong (f x ∷_) (⊛-is-zipWith fs xs)

+

+
zipWith-is-⊛ : ∀ (f : A → B → C) (xs : Vec A n) (ys : Vec B n) →

+
               zipWith f xs ys ≡ (replicate n f ⊛ xs ⊛ ys)

+
zipWith-is-⊛ f []       []       = refl

+
zipWith-is-⊛ f (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (zipWith-is-⊛ f xs ys)

+

+
⊛-is->>= : ∀ (fs : Vec (A → B) n) (xs : Vec A n) →

+
           (fs ⊛ xs) ≡ (fs DiagonalBind.>>= flip map xs)

+
⊛-is->>= []       []       = refl

+
⊛-is->>= (f ∷ fs) (x ∷ xs) = cong (f x ∷_) $ begin

+
  fs ⊛ xs                                          ≡⟨ ⊛-is->>= fs xs ⟩

+
  diagonal (map (flip map xs) fs)                  ≡⟨⟩

+
  diagonal (map (tail ∘ flip map (x ∷ xs)) fs)     ≡⟨ cong diagonal (map-∘ _ _ _) ⟩

+
  diagonal (map tail (map (flip map (x ∷ xs)) fs)) ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- _⊛*_

+

+
lookup-⊛* : ∀ (fs : Vec (A → B) m) (xs : Vec A n) i j →

+
            lookup (fs ⊛* xs) (Fin.combine i j) ≡ (lookup fs i $ lookup xs j)

+
lookup-⊛* (f ∷ fs) xs zero j = trans (lookup-++ˡ (map f xs) _ j) (lookup-map j f xs)

+
lookup-⊛* (f ∷ fs) xs (suc i) j = trans (lookup-++ʳ (map f xs) _ (Fin.combine i j)) (lookup-⊛* fs xs i j)

+

+
------------------------------------------------------------------------

+
-- foldl

+

+
-- The (uniqueness part of the) universality property for foldl.

+

+
foldl-universal : ∀ (B : ℕ → Set b) (f : FoldlOp A B) e

+
                  (h : ∀ {c} (C : ℕ → Set c) (g : FoldlOp A C) (e : C zero) →

+
                       ∀ {n} → Vec A n → C n) →

+
                  (∀ {c} {C} {g : FoldlOp A C} e → h {c} C g e [] ≡ e) →

+
                  (∀ {c} {C} {g : FoldlOp A C} e {n} x →

+
                   (h {c} C g e {suc n}) ∘ (x ∷_) ≗ h (C ∘ suc) g (g e x)) →

+
                  h B f e ≗ foldl {n = n} B f e

+
foldl-universal B f e h base step []       = base e

+
foldl-universal B f e h base step (x ∷ xs) = begin

+
  h B f e (x ∷ xs)             ≡⟨ step e x xs ⟩

+
  h (B ∘ suc) f (f e x) xs     ≡⟨ foldl-universal _ f (f e x) h base step xs ⟩

+
  foldl (B ∘ suc) f (f e x) xs ≡⟨⟩

+
  foldl B f e (x ∷ xs)         ∎

+
  where open ≡-Reasoning

+

+
foldl-fusion : ∀ {B : ℕ → Set b} {C : ℕ → Set c}

+
               (h : ∀ {n} → B n → C n) →

+
               {f : FoldlOp A B} {d : B zero} →

+
               {g : FoldlOp A C} {e : C zero} →

+
               (h d ≡ e) →

+
               (∀ {n} b x → h (f {n} b x) ≡ g (h b) x) →

+
               h ∘ foldl {n = n} B f d ≗ foldl C g e

+
foldl-fusion h {f} {d} {g} {e} base fuse []       = base

+
foldl-fusion h {f} {d} {g} {e} base fuse (x ∷ xs) =

+
  foldl-fusion h eq fuse xs

+
  where

+
    open ≡-Reasoning

+
    eq : h (f d x) ≡ g e x

+
    eq = begin

+
      h (f d x) ≡⟨ fuse d x ⟩

+
      g (h d) x ≡⟨ cong (λ e → g e x) base ⟩

+
      g e x     ∎

+

+
foldl-[] : ∀ (B : ℕ → Set b) (f : FoldlOp A B) {e} → foldl B f e [] ≡ e

+
foldl-[] _ _ = refl

+

+
------------------------------------------------------------------------

+
-- foldr

+

+
-- See also Data.Vec.Properties.WithK.foldr-cong.

+

+
-- The (uniqueness part of the) universality property for foldr.

+

+
module _ (B : ℕ → Set b) (f : FoldrOp A B) {e : B zero} where

+

+
  foldr-universal : (h : ∀ {n} → Vec A n → B n) →

+
                    h [] ≡ e →

+
                    (∀ {n} x → h ∘ (x ∷_) ≗ f {n} x ∘ h) →

+
                    h ≗ foldr {n = n} B f e

+
  foldr-universal h base step []       = base

+
  foldr-universal h base step (x ∷ xs) = begin

+
    h (x ∷ xs)           ≡⟨ step x xs ⟩

+
    f x (h xs)           ≡⟨ cong (f x) (foldr-universal h base step xs) ⟩

+
    f x (foldr B f e xs) ∎

+
    where open ≡-Reasoning

+

+
  foldr-[] : foldr B f e [] ≡ e

+
  foldr-[] = refl

+

+
  foldr-++ : ∀ (xs : Vec A m) →

+
             foldr B f e (xs ++ ys) ≡ foldr (B ∘ (_+ n)) f (foldr B f e ys) xs

+
  foldr-++ []       = refl

+
  foldr-++ (x ∷ xs) = cong (f x) (foldr-++ xs)

+

+
-- fusion and identity as consequences of universality

+

+
foldr-fusion : ∀ {B : ℕ → Set b} {f : FoldrOp A B} e

+
                 {C : ℕ → Set c} {g : FoldrOp A C}

+
               (h : ∀ {n} → B n → C n) →

+
               (∀ {n} x → h ∘ f {n} x ≗ g x ∘ h) →

+
               h ∘ foldr {n = n} B f e ≗ foldr C g (h e)

+
foldr-fusion {B = B} {f} e {C} h fuse =

+
  foldr-universal C _ _ refl (λ x xs → fuse x (foldr B f e xs))

+

+
id-is-foldr : id ≗ foldr {n = n} (Vec A) _∷_ []

+
id-is-foldr = foldr-universal _ _ id refl (λ _ _ → refl)

+

+
map-is-foldr : ∀ (f : A → B) →

+
               map {n = n} f ≗ foldr (Vec B) (λ x ys → f x ∷ ys) []

+
map-is-foldr f = foldr-universal (Vec _) (λ x ys → f x ∷ ys) (map f) refl (λ _ _ → refl)

+

+
++-is-foldr : ∀ (xs : Vec A m) →

+
              xs ++ ys ≡ foldr (Vec A ∘ (_+ n)) _∷_ ys xs

+
++-is-foldr {A = A} {n = n} {ys} xs =

+
  foldr-universal (Vec A ∘ (_+ n)) _∷_ (_++ ys) refl (λ _ _ → refl) xs

+

+
------------------------------------------------------------------------

+
-- _∷ʳ_

+

+
-- snoc is snoc

+

+
unfold-∷ʳ : ∀ .(eq : suc n ≡ n + 1) x (xs : Vec A n) → cast eq (xs ∷ʳ x) ≡ xs ++ [ x ]

+
unfold-∷ʳ eq x []       = refl

+
unfold-∷ʳ eq x (y ∷ xs) = cong (y ∷_) (unfold-∷ʳ (cong pred eq) x xs)

+

+
∷ʳ-injective : ∀ (xs ys : Vec A n) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y

+
∷ʳ-injective []       []        refl = (refl , refl)

+
∷ʳ-injective (x ∷ xs) (y  ∷ ys) eq   with ∷-injective eq

+
... | refl , eq′ = Product.map₁ (cong (x ∷_)) (∷ʳ-injective xs ys eq′)

+

+
∷ʳ-injectiveˡ : ∀ (xs ys : Vec A n) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys

+
∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq)

+

+
∷ʳ-injectiveʳ : ∀ (xs ys : Vec A n) → xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y

+
∷ʳ-injectiveʳ xs ys eq = proj₂ (∷ʳ-injective xs ys eq)

+

+
foldl-∷ʳ : ∀ (B : ℕ → Set b) (f : FoldlOp A B) e y (ys : Vec A n) →

+
           foldl B f e (ys ∷ʳ y) ≡ f (foldl B f e ys) y

+
foldl-∷ʳ B f e y []       = refl

+
foldl-∷ʳ B f e y (x ∷ xs) = foldl-∷ʳ (B ∘ suc) f (f e x) y xs

+

+
foldr-∷ʳ : ∀ (B : ℕ → Set b) (f : FoldrOp A B) {e} y (ys : Vec A n) →

+
           foldr B f e (ys ∷ʳ y) ≡ foldr (B ∘ suc) f (f y e) ys

+
foldr-∷ʳ B f y []       = refl

+
foldr-∷ʳ B f y (x ∷ xs) = cong (f x) (foldr-∷ʳ B f y xs)

+

+
-- init, last and _∷ʳ_

+

+
init-∷ʳ : ∀ x (xs : Vec A n) → init (xs ∷ʳ x) ≡ xs

+
init-∷ʳ x []       = refl

+
init-∷ʳ x (y ∷ xs) = cong (y ∷_) (init-∷ʳ x xs)

+

+
last-∷ʳ : ∀ x (xs : Vec A n) → last (xs ∷ʳ x) ≡ x

+
last-∷ʳ x []       = refl

+
last-∷ʳ x (y ∷ xs) = last-∷ʳ x xs

+

+
-- map and _∷ʳ_

+

+
map-∷ʳ : ∀ (f : A → B) x (xs : Vec A n) → map f (xs ∷ʳ x) ≡ map f xs ∷ʳ f x

+
map-∷ʳ f x []       = refl

+
map-∷ʳ f x (y ∷ xs) = cong (f y ∷_) (map-∷ʳ f x xs)

+

+
-- cast and _∷ʳ_

+

+
cast-∷ʳ : ∀ .(eq : suc n ≡ suc m) x (xs : Vec A n) →

+
          cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x

+
cast-∷ʳ {m = zero}  eq x []       = refl

+
cast-∷ʳ {m = suc m} eq x (y ∷ xs) = cong (y ∷_) (cast-∷ʳ (cong pred eq) x xs)

+

+
-- _++_ and _∷ʳ_

+

+
++-∷ʳ : ∀ .(eq : suc (m + n) ≡ m + suc n) z (xs : Vec A m) (ys : Vec A n) →

+
        cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)

+
++-∷ʳ {m = zero}  eq z []       []       = refl

+
++-∷ʳ {m = zero}  eq z []       (y ∷ ys) = cong (y ∷_) (++-∷ʳ refl z [] ys)

+
++-∷ʳ {m = suc m} eq z (x ∷ xs) ys       = cong (x ∷_) (++-∷ʳ (cong pred eq) z xs ys)

+

+
∷ʳ-++ : ∀ .(eq : (suc n) + m ≡ n + suc m) a (xs : Vec A n) {ys} →

+
        cast eq ((xs ∷ʳ a) ++ ys) ≡ xs ++ (a ∷ ys)

+
∷ʳ-++ eq a []       {ys} = cong (a ∷_) (cast-is-id (cong pred eq) ys)

+
∷ʳ-++ eq a (x ∷ xs) {ys} = cong (x ∷_) (∷ʳ-++ (cong pred eq) a xs)

+

+
------------------------------------------------------------------------

+
-- reverse

+

+
-- reverse of cons is snoc of reverse.

+

+
reverse-∷ : ∀ x (xs : Vec A n) → reverse (x ∷ xs) ≡ reverse xs ∷ʳ x

+
reverse-∷ x xs = sym (foldl-fusion (_∷ʳ x) refl (λ b x → refl) xs)

+

+
-- foldl after a reverse is a foldr

+

+
foldl-reverse : ∀ (B : ℕ → Set b) (f : FoldlOp A B) {e} →

+
                foldl {n = n} B f e ∘ reverse ≗ foldr B (flip f) e

+
foldl-reverse _ _ {e} []       = refl

+
foldl-reverse B f {e} (x ∷ xs) = begin

+
  foldl B f e (reverse (x ∷ xs)) ≡⟨ cong (foldl B f e) (reverse-∷ x xs) ⟩

+
  foldl B f e (reverse xs ∷ʳ x)  ≡⟨ foldl-∷ʳ B f e x (reverse xs) ⟩

+
  f (foldl B f e (reverse xs)) x ≡⟨ cong (flip f x) (foldl-reverse B f xs) ⟩

+
  f (foldr B (flip f) e xs) x    ≡⟨⟩

+
  foldr B (flip f) e (x ∷ xs)    ∎

+
  where open ≡-Reasoning

+

+
-- foldr after a reverse is a foldl

+

+
foldr-reverse : ∀ (B : ℕ → Set b) (f : FoldrOp A B) {e} →

+
                foldr {n = n} B f e ∘ reverse ≗ foldl B (flip f) e

+
foldr-reverse B f {e} xs = foldl-fusion (foldr B f e) refl (λ _ _ → refl) xs

+

+
-- reverse is involutive.

+

+
reverse-involutive : Involutive {A = Vec A n} _≡_ reverse

+
reverse-involutive xs = begin

+
  reverse (reverse xs)    ≡⟨  foldl-reverse (Vec _) (flip _∷_) xs ⟩

+
  foldr (Vec _) _∷_ [] xs ≡⟨ id-is-foldr xs ⟨

+
  xs                      ∎

+
  where open ≡-Reasoning

+

+
reverse-reverse : reverse xs ≡ ys → reverse ys ≡ xs

+
reverse-reverse {xs = xs} {ys} eq =  begin

+
  reverse ys           ≡⟨ cong reverse eq ⟨

+
  reverse (reverse xs) ≡⟨  reverse-involutive xs ⟩

+
  xs                   ∎

+
  where open ≡-Reasoning

+

+
-- reverse is injective.

+

+
reverse-injective : reverse xs ≡ reverse ys → xs ≡ ys

+
reverse-injective {xs = xs} {ys} eq = begin

+
  xs                   ≡⟨ reverse-reverse eq ⟨

+
  reverse (reverse ys) ≡⟨  reverse-involutive ys ⟩

+
  ys                   ∎

+
  where open ≡-Reasoning

+

+
-- init and last of reverse

+

+
init-reverse : init ∘ reverse ≗ reverse ∘ tail {A = A} {n = n}

+
init-reverse (x ∷ xs) = begin

+
  init (reverse (x ∷ xs))   ≡⟨ cong init (reverse-∷ x xs) ⟩

+
  init (reverse xs ∷ʳ x)    ≡⟨ init-∷ʳ x (reverse xs) ⟩

+
  reverse xs                ∎

+
  where open ≡-Reasoning

+

+
last-reverse : last ∘ reverse ≗ head {A = A} {n = n}

+
last-reverse (x ∷ xs) = begin

+
  last (reverse (x ∷ xs))   ≡⟨ cong last (reverse-∷ x xs) ⟩

+
  last (reverse xs ∷ʳ x)    ≡⟨ last-∷ʳ x (reverse xs) ⟩

+
  x                         ∎

+
  where open ≡-Reasoning

+

+
-- map and reverse

+

+
map-reverse : ∀ (f : A → B) (xs : Vec A n) →

+
              map f (reverse xs) ≡ reverse (map f xs)

+
map-reverse f []       = refl

+
map-reverse f (x ∷ xs) = begin

+
  map f (reverse (x ∷ xs))  ≡⟨  cong (map f) (reverse-∷ x xs) ⟩

+
  map f (reverse xs ∷ʳ x)   ≡⟨  map-∷ʳ f x (reverse xs) ⟩

+
  map f (reverse xs) ∷ʳ f x ≡⟨  cong (_∷ʳ f x) (map-reverse f xs ) ⟩

+
  reverse (map f xs) ∷ʳ f x ≡⟨ reverse-∷ (f x) (map f xs) ⟨

+
  reverse (f x ∷ map f xs)  ≡⟨⟩

+
  reverse (map f (x ∷ xs))  ∎

+
  where open ≡-Reasoning

+

+
-- append and reverse

+

+
reverse-++ : ∀ .(eq : m + n ≡ n + m) (xs : Vec A m) (ys : Vec A n) →

+
             cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs

+
reverse-++ {m = zero}  {n = n} eq []       ys = ≈-sym (++-identityʳ (sym eq) (reverse ys))

+
reverse-++ {m = suc m} {n = n} eq (x ∷ xs) ys = begin

+
  reverse (x ∷ xs ++ ys)              ≂⟨ reverse-∷ x (xs ++ ys) ⟩

+
  reverse (xs ++ ys) ∷ʳ x             ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (+-comm m n)) x (reverse (xs ++ ys)))

+
                                                (reverse-++ _ xs ys) ⟩

+
  (reverse ys ++ reverse xs) ∷ʳ x     ≈⟨ ++-∷ʳ (sym (+-suc n m)) x (reverse ys) (reverse xs) ⟩

+
  reverse ys ++ (reverse xs ∷ʳ x)     ≂⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟨

+
  reverse ys ++ (reverse (x ∷ xs))    ∎

+
  where open CastReasoning

+

+
cast-reverse : ∀ .(eq : m ≡ n) → cast eq ∘ reverse {A = A} {n = m} ≗ reverse ∘ cast eq

+
cast-reverse {n = zero}  eq []       = refl

+
cast-reverse {n = suc n} eq (x ∷ xs) = begin

+
  reverse (x ∷ xs)           ≂⟨ reverse-∷ x xs ⟩

+
  reverse xs ∷ʳ x            ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ eq x (reverse xs))

+
                                       (cast-reverse (cong pred eq) xs) ⟩

+
  reverse (cast _ xs) ∷ʳ x   ≂⟨ reverse-∷ x (cast (cong pred eq) xs) ⟨

+
  reverse (x ∷ cast _ xs)    ≈⟨⟩

+
  reverse (cast eq (x ∷ xs)) ∎

+
  where open CastReasoning

+

+
------------------------------------------------------------------------

+
-- _ʳ++_

+

+
-- reverse-append is append of reverse.

+

+
unfold-ʳ++ : ∀ (xs : Vec A m) (ys : Vec A n) → xs ʳ++ ys ≡ reverse xs ++ ys

+
unfold-ʳ++ xs ys = sym (foldl-fusion (_++ ys) refl (λ b x → refl) xs)

+

+
-- foldr after a reverse-append is a foldl

+

+
foldr-ʳ++ : ∀ (B : ℕ → Set b) (f : FoldrOp A B) {e}

+
            (xs : Vec A m) {ys : Vec A n} →

+
            foldr B f e (xs ʳ++ ys) ≡

+
            foldl (B ∘ (_+ n)) (flip f) (foldr B f e ys) xs

+
foldr-ʳ++ B f {e} xs = foldl-fusion (foldr B f e) refl (λ _ _ → refl) xs

+

+
-- map and _ʳ++_

+

+
map-ʳ++ : ∀ (f : A → B) (xs : Vec A m) →

+
          map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys

+
map-ʳ++ {ys = ys} f xs = begin

+
  map f (xs ʳ++ ys)              ≡⟨  cong (map f) (unfold-ʳ++ xs ys) ⟩

+
  map f (reverse xs ++ ys)       ≡⟨  map-++ f (reverse xs) ys ⟩

+
  map f (reverse xs) ++ map f ys ≡⟨  cong (_++ map f ys) (map-reverse f xs) ⟩

+
  reverse (map f xs) ++ map f ys ≡⟨ unfold-ʳ++ (map f xs) (map f ys) ⟨

+
  map f xs ʳ++ map f ys          ∎

+
  where open ≡-Reasoning

+

+
∷-ʳ++ : ∀ .(eq : (suc m) + n ≡ m + suc n) a (xs : Vec A m) {ys} →

+
        cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)

+
∷-ʳ++ eq a xs {ys} = begin

+
  (a ∷ xs) ʳ++ ys         ≂⟨ unfold-ʳ++ (a ∷ xs) ys ⟩

+
  reverse (a ∷ xs) ++ ys  ≂⟨ cong (_++ ys) (reverse-∷ a xs) ⟩

+
  (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩

+
  reverse xs ++ (a ∷ ys)  ≂⟨ unfold-ʳ++ xs (a ∷ ys) ⟨

+
  xs ʳ++ (a ∷ ys)         ∎

+
  where open CastReasoning

+

+
++-ʳ++ : ∀ .(eq : m + n + o ≡ n + (m + o)) (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →

+
         cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)

+
++-ʳ++ {m = m} {n} {o} eq xs {ys} {zs} = begin

+
  ((xs ++ ys) ʳ++ zs)              ≂⟨ unfold-ʳ++ (xs ++ ys) zs ⟩

+
  reverse (xs ++ ys) ++ zs         ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (xs ++ ys)))

+
                                             (reverse-++ (+-comm m n) xs ys) ⟩

+
  (reverse ys ++ reverse xs) ++ zs ≈⟨ ++-assoc (trans (cong (_+ o) (+-comm n m)) eq) (reverse ys) (reverse xs) zs ⟩

+
  reverse ys ++ (reverse xs ++ zs) ≂⟨ cong (reverse ys ++_) (unfold-ʳ++ xs zs) ⟨

+
  reverse ys ++ (xs ʳ++ zs)        ≂⟨ unfold-ʳ++ ys (xs ʳ++ zs) ⟨

+
  ys ʳ++ (xs ʳ++ zs)               ∎

+
  where open CastReasoning

+

+
ʳ++-ʳ++ : ∀ .(eq : (m + n) + o ≡ n + (m + o)) (xs : Vec A m) {ys : Vec A n} {zs} →

+
          cast eq ((xs ʳ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ++ zs)

+
ʳ++-ʳ++ {m = m} {n} {o} eq xs {ys} {zs} = begin

+
  (xs ʳ++ ys) ʳ++ zs                         ≂⟨ cong (_ʳ++ zs) (unfold-ʳ++ xs ys) ⟩

+
  (reverse xs ++ ys) ʳ++ zs                  ≂⟨ unfold-ʳ++ (reverse xs ++ ys) zs ⟩

+
  reverse (reverse xs ++ ys) ++ zs           ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (reverse xs ++ ys)))

+
                                                       (reverse-++ (+-comm m n) (reverse xs) ys) ⟩

+
  (reverse ys ++ reverse (reverse xs)) ++ zs ≂⟨ cong ((_++ zs) ∘ (reverse ys ++_)) (reverse-involutive xs) ⟩

+
  (reverse ys ++ xs) ++ zs                   ≈⟨ ++-assoc (+-assoc n m o) (reverse ys) xs zs ⟩

+
  reverse ys ++ (xs ++ zs)                   ≂⟨ unfold-ʳ++ ys (xs ++ zs) ⟨

+
  ys ʳ++ (xs ++ zs)                          ∎

+
  where open CastReasoning

+

+
------------------------------------------------------------------------

+
-- sum

+

+
sum-++ : ∀ (xs : Vec ℕ m) → sum (xs ++ ys) ≡ sum xs + sum ys

+
sum-++ {_}       []       = refl

+
sum-++ {ys = ys} (x ∷ xs) = begin

+
  x + sum (xs ++ ys)     ≡⟨  cong (x +_) (sum-++ xs) ⟩

+
  x + (sum xs + sum ys)  ≡⟨ +-assoc x (sum xs) (sum ys) ⟨

+
  sum (x ∷ xs) + sum ys  ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- replicate

+

+
lookup-replicate : ∀ (i : Fin n) (x : A) → lookup (replicate n x) i ≡ x

+
lookup-replicate zero    x = refl

+
lookup-replicate (suc i) x = lookup-replicate i x

+

+
map-replicate :  ∀ (f : A → B) (x : A) n →

+
                 map f (replicate n x) ≡ replicate n (f x)

+
map-replicate f x zero = refl

+
map-replicate f x (suc n) = cong (f x ∷_) (map-replicate f x n)

+

+
transpose-replicate : ∀ (xs : Vec A m) →

+
                      transpose (replicate n xs) ≡ map (replicate n) xs

+
transpose-replicate {n = zero}  _  = sym (map-const _ [])

+
transpose-replicate {n = suc n} xs = begin

+
  transpose (replicate (suc n) xs)                    ≡⟨⟩

+
  (replicate _ _∷_ ⊛ xs ⊛ transpose (replicate _ xs)) ≡⟨ cong₂ _⊛_ (sym (map-is-⊛ _∷_ xs)) (transpose-replicate xs) ⟩

+
  (map _∷_ xs ⊛ map (replicate n) xs)                 ≡⟨ map-⊛ _∷_ (replicate n) xs ⟩

+
  map (replicate (suc n)) xs                          ∎

+
  where open ≡-Reasoning

+

+
zipWith-replicate : ∀ (_⊕_ : A → B → C) (x : A) (y : B) →

+
                    zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)

+
zipWith-replicate {n = zero}  _⊕_ x y = refl

+
zipWith-replicate {n = suc n} _⊕_ x y = cong (x ⊕ y ∷_) (zipWith-replicate _⊕_ x y)

+

+
zipWith-replicate₁ : ∀ (_⊕_ : A → B → C) (x : A) (ys : Vec B n) →

+
                     zipWith _⊕_ (replicate n x) ys ≡ map (x ⊕_) ys

+
zipWith-replicate₁ _⊕_ x []       = refl

+
zipWith-replicate₁ _⊕_ x (y ∷ ys) =

+
  cong (x ⊕ y ∷_) (zipWith-replicate₁ _⊕_ x ys)

+

+
zipWith-replicate₂ : ∀ (_⊕_ : A → B → C) (xs : Vec A n) (y : B) →

+
                     zipWith _⊕_ xs (replicate n y) ≡ map (_⊕ y) xs

+
zipWith-replicate₂ _⊕_ []       y = refl

+
zipWith-replicate₂ _⊕_ (x ∷ xs) y =

+
  cong (x ⊕ y ∷_) (zipWith-replicate₂ _⊕_ xs y)

+

+
toList-replicate : ∀ (n : ℕ) (x : A) →

+
                   toList (replicate n x) ≡ List.replicate n x

+
toList-replicate zero    x = refl

+
toList-replicate (suc n) x = cong (_ List.∷_) (toList-replicate n x)

+

+
------------------------------------------------------------------------

+
-- iterate

+

+
iterate-id : ∀ (x : A) n → iterate id x n ≡ replicate n x

+
iterate-id x zero    = refl

+
iterate-id x (suc n) = cong (_ ∷_) (iterate-id (id x) n)

+

+
take-iterate : ∀ n f (x : A) → take n (iterate f x (n + m)) ≡ iterate f x n

+
take-iterate zero    f x = refl

+
take-iterate (suc n) f x = cong (_ ∷_) (take-iterate n f (f x))

+

+
drop-iterate : ∀ n f (x : A) → drop n (iterate f x (n + zero)) ≡ []

+
drop-iterate zero    f x = refl

+
drop-iterate (suc n) f x = drop-iterate n f (f x)

+

+
lookup-iterate :  ∀ f (x : A) (i : Fin n) → lookup (iterate f x n) i ≡ ℕ.iterate f x (toℕ i)

+
lookup-iterate f x zero    = refl

+
lookup-iterate f x (suc i) = lookup-iterate f (f x) i

+

+
toList-iterate : ∀ f (x : A) n → toList (iterate f x n) ≡ List.iterate f x n

+
toList-iterate f x zero    = refl

+
toList-iterate f x (suc n) = cong (_ List.∷_) (toList-iterate f (f x) n)

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
lookup∘tabulate : ∀ (f : Fin n → A) (i : Fin n) →

+
                  lookup (tabulate f) i ≡ f i

+
lookup∘tabulate f zero    = refl

+
lookup∘tabulate f (suc i) = lookup∘tabulate (f ∘ suc) i

+

+
tabulate∘lookup : ∀ (xs : Vec A n) → tabulate (lookup xs) ≡ xs

+
tabulate∘lookup []       = refl

+
tabulate∘lookup (x ∷ xs) = cong (x ∷_) (tabulate∘lookup xs)

+

+
tabulate-∘ : ∀ (f : A → B) (g : Fin n → A) →

+
             tabulate (f ∘ g) ≡ map f (tabulate g)

+
tabulate-∘ {n = zero}  f g = refl

+
tabulate-∘ {n = suc n} f g = cong (f (g zero) ∷_) (tabulate-∘ f (g ∘ suc))

+

+
tabulate-cong : ∀ {f g : Fin n → A} → f ≗ g → tabulate f ≡ tabulate g

+
tabulate-cong {n = zero}  p = refl

+
tabulate-cong {n = suc n} p = cong₂ _∷_ (p zero) (tabulate-cong (p ∘ suc))

+

+
------------------------------------------------------------------------

+
-- allFin

+

+
lookup-allFin : ∀ (i : Fin n) → lookup (allFin n) i ≡ i

+
lookup-allFin = lookup∘tabulate id

+

+
allFin-map : ∀ n → allFin (suc n) ≡ zero ∷ map suc (allFin n)

+
allFin-map n = cong (zero ∷_) $ tabulate-∘ suc id

+

+
tabulate-allFin : ∀ (f : Fin n → A) → tabulate f ≡ map f (allFin n)

+
tabulate-allFin f = tabulate-∘ f id

+

+
-- If you look up every possible index, in increasing order, then you

+
-- get back the vector you started with.

+

+
map-lookup-allFin : ∀ (xs : Vec A n) → map (lookup xs) (allFin n) ≡ xs

+
map-lookup-allFin {n = n} xs = begin

+
  map (lookup xs) (allFin n) ≡⟨ tabulate-∘ (lookup xs) id ⟨

+
  tabulate (lookup xs)       ≡⟨ tabulate∘lookup xs ⟩

+
  xs                         ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- count

+

+
module _ {P : Pred A p} (P? : Decidable P) where

+

+
  count≤n : ∀ (xs : Vec A n) → count P? xs ≤ n

+
  count≤n []       = z≤n

+
  count≤n (x ∷ xs) with does (P? x)

+
  ... | true  = s≤s (count≤n xs)

+
  ... | false = m≤n⇒m≤1+n (count≤n xs)

+

+
------------------------------------------------------------------------

+
-- length

+

+
length-toList : (xs : Vec A n) → List.length (toList xs) ≡ length xs

+
length-toList []       = refl

+
length-toList (x ∷ xs) = cong suc (length-toList xs)

+

+
------------------------------------------------------------------------

+
-- insertAt

+

+
insertAt-lookup : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →

+
                  lookup (insertAt xs i v) i ≡ v

+
insertAt-lookup xs       zero     v = refl

+
insertAt-lookup (x ∷ xs) (suc i)  v = insertAt-lookup xs i v

+

+
insertAt-punchIn : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) (j : Fin n) →

+
                   lookup (insertAt xs i v) (Fin.punchIn i j) ≡ lookup xs j

+
insertAt-punchIn xs       zero     v j       = refl

+
insertAt-punchIn (x ∷ xs) (suc i)  v zero    = refl

+
insertAt-punchIn (x ∷ xs) (suc i)  v (suc j) = insertAt-punchIn xs i v j

+

+
toList-insertAt : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →

+
                  toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v

+
toList-insertAt xs       zero    v = refl

+
toList-insertAt (x ∷ xs) (suc i) v = cong (_ List.∷_) (toList-insertAt xs i v)

+

+
------------------------------------------------------------------------

+
-- removeAt

+

+
removeAt-punchOut : ∀ (xs : Vec A (suc n)) {i} {j} (i≢j : i ≢ j) →

+
                  lookup (removeAt xs i) (Fin.punchOut i≢j) ≡ lookup xs j

+
removeAt-punchOut (x ∷ xs)     {zero}  {zero}  i≢j = contradiction refl i≢j

+
removeAt-punchOut (x ∷ xs)     {zero}  {suc j} i≢j = refl

+
removeAt-punchOut (x ∷ y ∷ xs) {suc i} {zero}  i≢j = refl

+
removeAt-punchOut (x ∷ y ∷ xs) {suc i} {suc j} i≢j =

+
  removeAt-punchOut (y ∷ xs) (i≢j ∘ cong suc)

+

+
------------------------------------------------------------------------

+
-- insertAt and removeAt

+

+
removeAt-insertAt : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →

+
                    removeAt (insertAt xs i v) i ≡ xs

+
removeAt-insertAt xs               zero           v = refl

+
removeAt-insertAt (x ∷ xs)         (suc zero)     v = refl

+
removeAt-insertAt (x ∷ xs@(_ ∷ _)) (suc (suc i))  v =

+
  cong (x ∷_) (removeAt-insertAt xs (suc i) v)

+

+
insertAt-removeAt : ∀ (xs : Vec A (suc n)) (i : Fin (suc n)) →

+
                    insertAt (removeAt xs i) i (lookup xs i) ≡ xs

+
insertAt-removeAt (x ∷ xs)         zero     = refl

+
insertAt-removeAt (x ∷ xs@(_ ∷ _)) (suc i)  =

+
  cong (x ∷_) (insertAt-removeAt xs i)

+

+
------------------------------------------------------------------------

+
-- Conversion function

+

+
toList∘fromList : (xs : List A) → toList (fromList xs) ≡ xs

+
toList∘fromList List.[]       = refl

+
toList∘fromList (x List.∷ xs) = cong (x List.∷_) (toList∘fromList xs)

+

+
toList-cast : ∀ .(eq : m ≡ n) (xs : Vec A m) → toList (cast eq xs) ≡ toList xs

+
toList-cast {n = zero}  eq []       = refl

+
toList-cast {n = suc _} eq (x ∷ xs) =

+
  cong (x List.∷_) (toList-cast (cong pred eq) xs)

+

+
cast-fromList : ∀ {xs ys : List A} (eq : xs ≡ ys) →

+
                cast (cong List.length eq) (fromList xs) ≡ fromList ys

+
cast-fromList {xs = List.[]}     {ys = List.[]}     eq = refl

+
cast-fromList {xs = x List.∷ xs} {ys = y List.∷ ys} eq =

+
  let x≡y , xs≡ys = List.∷-injective eq in begin

+
  x ∷ cast (cong (pred ∘ List.length) eq) (fromList xs) ≡⟨ cong (_ ∷_) (cast-fromList xs≡ys) ⟩

+
  x ∷ fromList ys                                       ≡⟨ cong (_∷ _) x≡y ⟩

+
  y ∷ fromList ys                                       ∎

+
  where open ≡-Reasoning

+

+
fromList-map : ∀ (f : A → B) (xs : List A) →

+
               cast (List.length-map f xs) (fromList (List.map f xs)) ≡ map f (fromList xs)

+
fromList-map f List.[]       = refl

+
fromList-map f (x List.∷ xs) = cong (f x ∷_) (fromList-map f xs)

+

+
fromList-++ : ∀ (xs : List A) {ys : List A} →

+
              cast (List.length-++ xs) (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys

+
fromList-++ List.[]       {ys} = cast-is-id refl (fromList ys)

+
fromList-++ (x List.∷ xs) {ys} = cong (x ∷_) (fromList-++ xs)

+

+
fromList-reverse : (xs : List A) → cast (List.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)

+
fromList-reverse List.[] = refl

+
fromList-reverse (x List.∷ xs) = begin

+
  fromList (List.reverse (x List.∷ xs))         ≈⟨ cast-fromList (List.ʳ++-defn xs) ⟩

+
  fromList (List.reverse xs List.++ List.[ x ]) ≈⟨ fromList-++ (List.reverse xs) ⟩

+
  fromList (List.reverse xs) ++ [ x ]           ≈⟨ unfold-∷ʳ (+-comm 1 _) x (fromList (List.reverse xs)) ⟨

+
  fromList (List.reverse xs) ∷ʳ x               ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (List.length-reverse xs)) _ _)

+
                                                          (fromList-reverse xs) ⟩

+
  reverse (fromList xs) ∷ʳ x                    ≂⟨ reverse-∷ x (fromList xs) ⟨

+
  reverse (x ∷ fromList xs)                     ≈⟨⟩

+
  reverse (fromList (x List.∷ xs))              ∎

+
  where open CastReasoning

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
updateAt-id-relative = updateAt-id-local

+
{-# WARNING_ON_USAGE updateAt-id-relative

+
"Warning: updateAt-id-relative was deprecated in v2.0.

+
Please use updateAt-id-local instead."

+
#-}

+

+
updateAt-compose-relative = updateAt-updateAt-local

+
{-# WARNING_ON_USAGE updateAt-compose-relative

+
"Warning: updateAt-compose-relative was deprecated in v2.0.

+
Please use updateAt-updateAt-local instead."

+
#-}

+

+
updateAt-compose = updateAt-updateAt

+
{-# WARNING_ON_USAGE updateAt-compose

+
"Warning: updateAt-compose was deprecated in v2.0.

+
Please use updateAt-updateAt instead."

+
#-}

+

+
updateAt-cong-relative = updateAt-cong-local

+
{-# WARNING_ON_USAGE updateAt-cong-relative

+
"Warning: updateAt-cong-relative was deprecated in v2.0.

+
Please use updateAt-cong-local instead."

+
#-}

+

+
[]%=-compose = []%=-∘

+
{-# WARNING_ON_USAGE []%=-compose

+
"Warning: []%=-compose was deprecated in v2.0.

+
Please use []%=-∘ instead."

+
#-}

+

+
[]≔-++-inject+ : ∀ {m n x} (xs : Vec A m) (ys : Vec A n) i →

+
                 (xs ++ ys) [ i ↑ˡ n ]≔ x ≡ (xs [ i ]≔ x) ++ ys

+
[]≔-++-inject+ = []≔-++-↑ˡ

+
{-# WARNING_ON_USAGE []≔-++-inject+

+
"Warning: []≔-++-inject+ was deprecated in v2.0.

+
Please use []≔-++-↑ˡ instead."

+
#-}

+
idIsFold = id-is-foldr

+
{-# WARNING_ON_USAGE idIsFold

+
"Warning: idIsFold was deprecated in v2.0.

+
Please use id-is-foldr instead."

+
#-}

+
sum-++-commute = sum-++

+
{-# WARNING_ON_USAGE sum-++-commute

+
"Warning: sum-++-commute was deprecated in v2.0.

+
Please use sum-++ instead."

+
#-}

+
take-drop-id = take++drop≡id

+
{-# WARNING_ON_USAGE take-drop-id

+
"Warning: take-drop-id was deprecated in v2.0.

+
Please use take++drop≡id instead."

+
#-}

+
take-distr-zipWith = take-zipWith

+
{-# WARNING_ON_USAGE take-distr-zipWith

+
"Warning: take-distr-zipWith was deprecated in v2.0.

+
Please use take-zipWith instead."

+
#-}

+
take-distr-map = take-map

+
{-# WARNING_ON_USAGE take-distr-map

+
"Warning: take-distr-map was deprecated in v2.0.

+
Please use take-map instead."

+
#-}

+
drop-distr-zipWith = drop-zipWith

+
{-# WARNING_ON_USAGE drop-distr-zipWith

+
"Warning: drop-distr-zipWith was deprecated in v2.0.

+
Please use drop-zipWith instead."

+
#-}

+
drop-distr-map = drop-map

+
{-# WARNING_ON_USAGE drop-distr-map

+
"Warning: drop-distr-map was deprecated in v2.0.

+
Please use drop-map instead."

+
#-}

+

+
map-insert = map-insertAt

+
{-# WARNING_ON_USAGE map-insert

+
"Warning: map-insert was deprecated in v2.0.

+
Please use map-insertAt instead."

+
#-}

+
insert-lookup = insertAt-lookup

+
{-# WARNING_ON_USAGE insert-lookup

+
"Warning: insert-lookup was deprecated in v2.0.

+
Please use insertAt-lookup instead."

+
#-}

+
insert-punchIn = insertAt-punchIn

+
{-# WARNING_ON_USAGE insert-punchIn

+
"Warning: insert-punchIn was deprecated in v2.0.

+
Please use insertAt-punchIn instead."

+
#-}

+
remove-PunchOut = removeAt-punchOut

+
{-# WARNING_ON_USAGE remove-PunchOut

+
"Warning: remove-PunchOut was deprecated in v2.0.

+
Please use removeAt-punchOut instead."

+
#-}

+
remove-insert = removeAt-insertAt

+
{-# WARNING_ON_USAGE remove-insert

+
"Warning: remove-insert was deprecated in v2.0.

+
Please use removeAt-insertAt instead."

+
#-}

+
insert-remove = insertAt-removeAt

+
{-# WARNING_ON_USAGE insert-remove

+
"Warning: insert-remove was deprecated in v2.0.

+
Please use insertAt-removeAt instead."

+
#-}

+

+
lookup-inject≤-take : ∀ m (m≤m+n : m ≤ m + n) (i : Fin m) (xs : Vec A (m + n)) →

+
                      lookup xs (Fin.inject≤ i m≤m+n) ≡ lookup (take m xs) i

+
lookup-inject≤-take m m≤m+n i xs = sym (begin

+
  lookup (take m xs) i

+
    ≡⟨ lookup-take-inject≤ xs i ⟩

+
  lookup xs (Fin.inject≤ i _)

+
    ≡⟨⟩

+
  lookup xs (Fin.inject≤ i m≤m+n)

+
    ∎) where open ≡-Reasoning

+
{-# WARNING_ON_USAGE lookup-inject≤-take

+
"Warning: lookup-inject≤-take was deprecated in v2.0.

+
Please use lookup-take-inject≤ or lookup-truncate, take≡truncate instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Relation.Binary.Equality.Cast.html b/site/static/agda_html/Data.Vec.Relation.Binary.Equality.Cast.html
new file mode 100644
index 000000000..3f1b48a9c
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Relation.Binary.Equality.Cast.html
@@ -0,0 +1,205 @@
+
+Data.Vec.Relation.Binary.Equality.Cast
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An equational reasoning library for propositional equality over

+
-- vectors of different indices using cast.

+
--

+
-- See README.Data.Vec.Relation.Binary.Equality.Cast for

+
-- documentation and examples.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Relation.Binary.Equality.Cast {a} {A : Set a} where

+

+
open import Data.Nat.Base using (ℕ; zero; suc)

+
open import Data.Nat.Properties using (suc-injective)

+
open import Data.Vec.Base

+
open import Relation.Binary.Core using (REL; _⇒_)

+
open import Relation.Binary.Definitions using (Sym; Trans)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; trans; sym; cong)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+

+
private

+
  variable

+
    l m n o : ℕ

+
    xs ys zs : Vec A n

+

+

+
cast-is-id : .(eq : m ≡ m) (xs : Vec A m) → cast eq xs ≡ xs

+
cast-is-id eq []       = refl

+
cast-is-id eq (x ∷ xs) = cong (x ∷_) (cast-is-id (suc-injective eq) xs)

+

+
cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (xs : Vec A m) →

+
             cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs

+
cast-trans {m = zero}  {n = zero}  {o = zero}  eq₁ eq₂ [] = refl

+
cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (x ∷ xs) =

+
  cong (x ∷_) (cast-trans (suc-injective eq₁) (suc-injective eq₂) xs)

+

+

+
infix 3 _≈[_]_

+

+
_≈[_]_ : ∀ {n m} → Vec A n → .(eq : n ≡ m) → Vec A m → Set a

+
xs ≈[ eq ] ys = cast eq xs ≡ ys

+

+
------------------------------------------------------------------------

+
-- _≈[_]_ is ‘reflexive’, ‘symmetric’ and ‘transitive’

+

+
≈-reflexive : ∀ {n} → _≡_ ⇒ (λ xs ys → _≈[_]_ {n} xs refl ys)

+
≈-reflexive {x = x} eq = trans (cast-is-id refl x) eq

+

+
≈-sym : .{m≡n : m ≡ n} → Sym _≈[ m≡n ]_ _≈[ sym m≡n ]_

+
≈-sym {m≡n = m≡n} {xs} {ys} xs≈ys = begin

+
  cast (sym m≡n) ys             ≡⟨ cong (cast (sym m≡n)) xs≈ys ⟨

+
  cast (sym m≡n) (cast m≡n xs)  ≡⟨ cast-trans m≡n (sym m≡n) xs ⟩

+
  cast (trans m≡n (sym m≡n)) xs ≡⟨ cast-is-id (trans m≡n (sym m≡n)) xs ⟩

+
  xs                            ∎

+
  where open ≡-Reasoning

+

+
≈-trans : ∀ .{m≡n : m ≡ n} .{n≡o : n ≡ o} → Trans _≈[ m≡n ]_ _≈[ n≡o ]_ _≈[ trans m≡n n≡o ]_

+
≈-trans {m≡n = m≡n} {n≡o} {xs} {ys} {zs} xs≈ys ys≈zs = begin

+
  cast (trans m≡n n≡o) xs ≡⟨ cast-trans m≡n n≡o xs ⟨

+
  cast n≡o (cast m≡n xs)  ≡⟨ cong (cast n≡o) xs≈ys ⟩

+
  cast n≡o ys             ≡⟨ ys≈zs ⟩

+
  zs                      ∎

+
  where open ≡-Reasoning

+

+
------------------------------------------------------------------------

+
-- Reasoning combinators

+

+
module CastReasoning where

+

+
  open ≡-Reasoning public

+
    renaming (begin_ to begin-≡_; _∎ to _≡-∎)

+

+
  begin_ : ∀ .{m≡n : m ≡ n} {xs : Vec A m} {ys} → xs ≈[ m≡n ] ys → cast m≡n xs ≡ ys

+
  begin xs≈ys = xs≈ys

+

+
  _∎ : (xs : Vec A n) → cast refl xs ≡ xs

+
  _∎ xs = ≈-reflexive refl

+

+
  _≈⟨⟩_ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys} → xs ≈[ m≡n ] ys → xs ≈[ m≡n ] ys

+
  xs ≈⟨⟩ xs≈ys = xs≈ys

+

+
  -- composition of _≈[_]_

+
  step-≈-⟩ : ∀ .{m≡n : m ≡ n}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →

+
           ys ≈[ trans (sym m≡n) m≡o ] zs → xs ≈[ m≡n ] ys → xs ≈[ m≡o ] zs

+
  step-≈-⟩ xs ys≈zs xs≈ys = ≈-trans xs≈ys ys≈zs

+

+
  step-≈-⟨ : ∀ .{n≡m : n ≡ m}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →

+
           ys ≈[ trans n≡m m≡o ] zs → ys ≈[ n≡m ] xs → xs ≈[ m≡o ] zs

+
  step-≈-⟨ xs ys≈zs ys≈xs = step-≈-⟩ xs ys≈zs (≈-sym ys≈xs)

+

+
  -- composition of the equality type on the right-hand side of _≈[_]_,

+
  -- or escaping to ordinary _≡_

+
  step-≃-⟩ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → xs ≈[ m≡n ] ys → xs ≈[ m≡n ] zs

+
  step-≃-⟩ xs ys≡zs xs≈ys = ≈-trans xs≈ys (≈-reflexive ys≡zs)

+

+
  step-≃-⟨ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → ys ≈[ sym m≡n ] xs → xs ≈[ m≡n ] zs

+
  step-≃-⟨ xs ys≡zs ys≈xs = step-≃-⟩ xs ys≡zs (≈-sym ys≈xs)

+

+
  -- composition of the equality type on the left-hand side of _≈[_]_

+
  step-≂-⟩ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → xs ≡ ys → xs ≈[ m≡n ] zs

+
  step-≂-⟩ xs ys≈zs xs≡ys = ≈-trans (≈-reflexive xs≡ys) ys≈zs

+

+
  step-≂-⟨ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → ys ≡ xs → xs ≈[ m≡n ] zs

+
  step-≂-⟨ xs ys≈zs ys≡xs = step-≂-⟩ xs ys≈zs (sym ys≡xs)

+

+
  -- `cong` after a `_≈[_]_` step that exposes the `cast` to the `cong`

+
  -- operation

+
  ≈-cong : ∀ .{l≡o : l ≡ o} .{m≡n : m ≡ n} {xs : Vec A m} {ys zs} (f : Vec A o → Vec A n) →

+
           xs ≈[ m≡n ] f (cast l≡o ys) → ys ≈[ l≡o ] zs → xs ≈[ m≡n ] f zs

+
  ≈-cong f xs≈fys ys≈zs = trans xs≈fys (cong f ys≈zs)

+

+
  ------------------------------------------------------------------------

+
  -- convenient syntax for ‘equational’ reasoning

+

+
  infix 1 begin_

+
  infixr 2 step-≃-⟩ step-≃-⟨ step-≂-⟩ step-≂-⟨ step-≈-⟩ step-≈-⟨ _≈⟨⟩_ ≈-cong

+
  infix 3 _∎

+

+
  syntax step-≃-⟩ xs ys≡zs xs≈ys  = xs ≃⟨ xs≈ys ⟩ ys≡zs

+
  syntax step-≃-⟨ xs ys≡zs xs≈ys  = xs ≃⟨ xs≈ys ⟨ ys≡zs

+
  syntax step-≂-⟩ xs ys≈zs xs≡ys  = xs ≂⟨ xs≡ys ⟩ ys≈zs

+
  syntax step-≂-⟨ xs ys≈zs ys≡xs  = xs ≂⟨ ys≡xs ⟨ ys≈zs

+
  syntax step-≈-⟩ xs ys≈zs xs≈ys  = xs ≈⟨ xs≈ys ⟩ ys≈zs

+
  syntax step-≈-⟨ xs ys≈zs ys≈xs  = xs ≈⟨ ys≈xs ⟨ ys≈zs

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Relation.Binary.Pointwise.Inductive.html b/site/static/agda_html/Data.Vec.Relation.Binary.Pointwise.Inductive.html
new file mode 100644
index 000000000..002f3ad47
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Relation.Binary.Pointwise.Inductive.html
@@ -0,0 +1,348 @@
+
+Data.Vec.Relation.Binary.Pointwise.Inductive
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Inductive pointwise lifting of relations to vectors

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Relation.Binary.Pointwise.Inductive where

+

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Nat.Base using (ℕ; zero; suc)

+
open import Data.Product.Base using (_×_; _,_; uncurry; <_,_>)

+
open import Data.Vec.Base as Vec hiding ([_]; head; tail; map; lookup; uncons)

+
open import Data.Vec.Relation.Unary.All using (All; []; _∷_)

+
open import Level using (Level; _⊔_)

+
open import Function.Base using (_∘_)

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Relation.Binary.Core using (REL; Rel; _⇒_)

+
open import Relation.Binary.Bundles using (Setoid; DecSetoid)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Trans; Decidable; Reflexive; Sym)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Nullary.Decidable using (yes; no; _×-dec_; map′)

+
open import Relation.Unary using (Pred)

+

+
private

+
  variable

+
    a b c d ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infixr 5 _∷_

+

+
data Pointwise {a b ℓ} {A : Set a} {B : Set b} (_∼_ : REL A B ℓ) :

+
               ∀ {m n} (xs : Vec A m) (ys : Vec B n) → Set (a ⊔ b ⊔ ℓ)

+
               where

+
  []  : Pointwise _∼_ [] []

+
  _∷_ : ∀ {m n x y} {xs : Vec A m} {ys : Vec B n}

+
        (x∼y : x ∼ y) (xs∼ys : Pointwise _∼_ xs ys) →

+
        Pointwise _∼_ (x ∷ xs) (y ∷ ys)

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
length-equal : ∀ {m n} {_∼_ : REL A B ℓ} {xs : Vec A m} {ys : Vec B n} →

+
               Pointwise _∼_ xs ys → m ≡ n

+
length-equal []          = ≡.refl

+
length-equal (_ ∷ xs∼ys) = ≡.cong suc (length-equal xs∼ys)

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
module _ {_∼_ : REL A B ℓ} where

+

+
  head : ∀ {m n x y} {xs : Vec A m} {ys : Vec B n} →

+
         Pointwise _∼_ (x ∷ xs) (y ∷ ys) → x ∼ y

+
  head (x∼y ∷ xs∼ys) = x∼y

+

+
  tail : ∀ {m n x y} {xs : Vec A m} {ys : Vec B n} →

+
         Pointwise _∼_ (x ∷ xs) (y ∷ ys) → Pointwise _∼_ xs ys

+
  tail (x∼y ∷ xs∼ys) = xs∼ys

+

+
  uncons : ∀ {m n x y} {xs : Vec A m} {ys : Vec B n} →

+
           Pointwise _∼_ (x ∷ xs) (y ∷ ys) → x ∼ y × Pointwise _∼_ xs ys

+
  uncons = < head , tail >

+

+
  lookup : ∀ {n} {xs : Vec A n} {ys : Vec B n} → Pointwise _∼_ xs ys →

+
           ∀ i → (Vec.lookup xs i) ∼ (Vec.lookup ys i)

+
  lookup (x∼y ∷ _)     zero    = x∼y

+
  lookup (_   ∷ xs∼ys) (suc i) = lookup xs∼ys i

+

+
  map : ∀ {ℓ₂} {_≈_ : REL A B ℓ₂} →

+
        _≈_ ⇒ _∼_ → ∀ {m n} → Pointwise _≈_ ⇒ Pointwise _∼_ {m} {n}

+
  map ∼₁⇒∼₂ []             = []

+
  map ∼₁⇒∼₂ (x∼y ∷ xs∼ys) = ∼₁⇒∼₂ x∼y ∷ map ∼₁⇒∼₂ xs∼ys

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
refl : ∀ {_∼_ : Rel A ℓ} {n} →

+
       Reflexive _∼_ → Reflexive (Pointwise _∼_ {n})

+
refl ∼-refl {[]}      = []

+
refl ∼-refl {x ∷ xs} = ∼-refl ∷ refl ∼-refl

+

+
sym : ∀ {P : REL A B ℓ} {Q : REL B A ℓ} {m n} →

+
      Sym P Q → Sym (Pointwise P) (Pointwise Q {m} {n})

+
sym sm []             = []

+
sym sm (x∼y ∷ xs∼ys) = sm x∼y ∷ sym sm xs∼ys

+

+
trans : ∀ {P : REL A B ℓ} {Q : REL B C ℓ} {R : REL A C ℓ} {m n o} →

+
        Trans P Q R →

+
        Trans (Pointwise P {m}) (Pointwise Q {n} {o}) (Pointwise R)

+
trans trns []             []             = []

+
trans trns (x∼y ∷ xs∼ys) (y∼z ∷ ys∼zs) =

+
  trns x∼y y∼z ∷ trans trns xs∼ys ys∼zs

+

+
decidable : ∀ {_∼_ : REL A B ℓ} →

+
            Decidable _∼_ → ∀ {m n} → Decidable (Pointwise _∼_ {m} {n})

+
decidable dec []       []       = yes []

+
decidable dec []       (y ∷ ys) = no λ()

+
decidable dec (x ∷ xs) []       = no λ()

+
decidable dec (x ∷ xs) (y ∷ ys) =

+
  map′ (uncurry _∷_) uncons (dec x y ×-dec decidable dec xs ys)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ {_∼_ : Rel A ℓ} where

+

+
  isEquivalence : IsEquivalence _∼_ → ∀ n →

+
                  IsEquivalence (Pointwise _∼_ {n})

+
  isEquivalence equiv n = record

+
    { refl  = refl  Eq.refl

+
    ; sym   = sym   Eq.sym

+
    ; trans = trans Eq.trans

+
    } where module Eq = IsEquivalence equiv

+

+
  isDecEquivalence : IsDecEquivalence _∼_ → ∀ n →

+
                     IsDecEquivalence (Pointwise _∼_ {n})

+
  isDecEquivalence decEquiv n = record

+
    { isEquivalence = isEquivalence Eq.isEquivalence n

+
    ; _≟_           = decidable Eq._≟_

+
    } where module Eq = IsDecEquivalence decEquiv

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
setoid : Setoid a ℓ → ℕ → Setoid a (a ⊔ ℓ)

+
setoid S n = record

+
   { isEquivalence = isEquivalence Eq.isEquivalence n

+
   } where module Eq = Setoid S

+

+
decSetoid : DecSetoid a ℓ → ℕ → DecSetoid a (a ⊔ ℓ)

+
decSetoid S n = record

+
   { isDecEquivalence = isDecEquivalence Eq.isDecEquivalence n

+
   } where module Eq = DecSetoid S

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {_∼₁_ : REL A B ℓ₁} {_∼₂_ : REL C D ℓ₂}

+
         {f : A → C} {g : B → D}

+
         where

+

+
  map⁺ : (∀ {x y} → x ∼₁ y → f x ∼₂ g y) →

+
         ∀ {m n xs ys} → Pointwise _∼₁_ {m} {n} xs ys →

+
         Pointwise _∼₂_ (Vec.map f xs) (Vec.map g ys)

+
  map⁺ ∼₁⇒∼₂ []             = []

+
  map⁺ ∼₁⇒∼₂ (x∼y ∷ xs∼ys) = ∼₁⇒∼₂ x∼y ∷ map⁺ ∼₁⇒∼₂ xs∼ys

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ {_∼_ : REL A B ℓ} where

+

+
  ++⁺ : ∀ {m n p q}

+
        {ws : Vec A m} {xs : Vec B p} {ys : Vec A n} {zs : Vec B q} →

+
        Pointwise _∼_ ws xs → Pointwise _∼_ ys zs →

+
        Pointwise _∼_ (ws ++ ys) (xs ++ zs)

+
  ++⁺ []            ys∼zs = ys∼zs

+
  ++⁺ (w∼x ∷ ws∼xs) ys∼zs = w∼x ∷ (++⁺ ws∼xs ys∼zs)

+

+
  ++ˡ⁻ : ∀ {m n}

+
         (ws : Vec A m) (xs : Vec B m) {ys : Vec A n} {zs : Vec B n} →

+
         Pointwise _∼_ (ws ++ ys) (xs ++ zs) → Pointwise _∼_ ws xs

+
  ++ˡ⁻ []       []        _                    = []

+
  ++ˡ⁻ (w ∷ ws) (x ∷ xs) (w∼x ∷ ps) = w∼x ∷ ++ˡ⁻ ws xs ps

+

+
  ++ʳ⁻ : ∀ {m n}

+
         (ws : Vec A m) (xs : Vec B m) {ys : Vec A n} {zs : Vec B n} →

+
         Pointwise _∼_ (ws ++ ys) (xs ++ zs) → Pointwise _∼_ ys zs

+
  ++ʳ⁻ [] [] ys∼zs = ys∼zs

+
  ++ʳ⁻ (w ∷ ws) (x ∷ xs) (_ ∷ ps) = ++ʳ⁻ ws xs ps

+

+
  ++⁻ : ∀ {m n}

+
        (ws : Vec A m) (xs : Vec B m) {ys : Vec A n} {zs : Vec B n} →

+
        Pointwise _∼_ (ws ++ ys) (xs ++ zs) →

+
        Pointwise _∼_ ws xs × Pointwise _∼_ ys zs

+
  ++⁻ ws xs ps = ++ˡ⁻ ws xs ps , ++ʳ⁻ ws xs ps

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {_∼_ : REL A B ℓ} where

+

+
  concat⁺ : ∀ {m n p q}

+
            {xss : Vec (Vec A m) n} {yss : Vec (Vec B p) q} →

+
            Pointwise (Pointwise _∼_) xss yss →

+
            Pointwise _∼_ (concat xss) (concat yss)

+
  concat⁺ []           = []

+
  concat⁺ (xs∼ys ∷ ps) = ++⁺ xs∼ys (concat⁺ ps)

+

+
  concat⁻ : ∀ {m n} (xss : Vec (Vec A m) n) (yss : Vec (Vec B m) n) →

+
            Pointwise _∼_ (concat xss) (concat yss) →

+
            Pointwise (Pointwise _∼_) xss yss

+
  concat⁻ []         []         [] = []

+
  concat⁻ (xs ∷ xss) (ys ∷ yss) ps =

+
    ++ˡ⁻ xs ys ps ∷ concat⁻ xss yss (++ʳ⁻ xs ys ps)

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {_∼_ : REL A B ℓ} where

+

+
  tabulate⁺ : ∀ {n} {f : Fin n → A} {g : Fin n → B} →

+
              (∀ i → f i ∼ g i) →

+
              Pointwise _∼_ (tabulate f) (tabulate g)

+
  tabulate⁺ {zero}  f∼g = []

+
  tabulate⁺ {suc n} f∼g = f∼g zero ∷ tabulate⁺ (f∼g ∘ suc)

+

+
  tabulate⁻ : ∀ {n} {f : Fin n → A} {g : Fin n → B} →

+
              Pointwise _∼_ (tabulate f) (tabulate g) →

+
              (∀ i → f i ∼ g i)

+
  tabulate⁻ (f₀∼g₀ ∷ _)   zero    = f₀∼g₀

+
  tabulate⁻ (_     ∷ f∼g) (suc i) = tabulate⁻ f∼g i

+

+
------------------------------------------------------------------------

+
-- cong

+

+
module _ {_∼_ : Rel A ℓ} (refl : Reflexive _∼_) where

+
  cong-[_]≔ : ∀ {n} i p {xs} {ys} →

+
              Pointwise _∼_ {n} xs ys →

+
              Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)

+
  cong-[ zero ]≔  p (_   ∷ eqn) = refl ∷ eqn

+
  cong-[ suc i ]≔ p (x∼y ∷ eqn) = x∼y  ∷ cong-[ i ]≔ p eqn

+

+
------------------------------------------------------------------------

+
-- Degenerate pointwise relations

+

+
module _ {P : Pred A ℓ} where

+

+
  Pointwiseˡ⇒All : ∀ {m n} {xs : Vec A m} {ys : Vec B n} →

+
                   Pointwise (λ x y → P x) xs ys → All P xs

+
  Pointwiseˡ⇒All []       = []

+
  Pointwiseˡ⇒All (p ∷ ps) = p ∷ Pointwiseˡ⇒All ps

+

+
  Pointwiseʳ⇒All : ∀ {n} {xs : Vec B n} {ys : Vec A n} →

+
                   Pointwise (λ x y → P y) xs ys → All P ys

+
  Pointwiseʳ⇒All []       = []

+
  Pointwiseʳ⇒All (p ∷ ps) = p ∷ Pointwiseʳ⇒All ps

+

+
  All⇒Pointwiseˡ : ∀ {n} {xs : Vec A n} {ys : Vec B n} →

+
                   All P xs → Pointwise (λ x y → P x) xs ys

+
  All⇒Pointwiseˡ {ys = []}    []       = []

+
  All⇒Pointwiseˡ {ys = _ ∷ _} (p ∷ ps) = p ∷ All⇒Pointwiseˡ ps

+

+
  All⇒Pointwiseʳ : ∀ {n} {xs : Vec B n} {ys : Vec A n} →

+
                   All P ys → Pointwise (λ x y → P y) xs ys

+
  All⇒Pointwiseʳ {xs = []}    []       = []

+
  All⇒Pointwiseʳ {xs = _ ∷ _} (p ∷ ps) = p ∷ All⇒Pointwiseʳ ps

+

+
------------------------------------------------------------------------

+
-- Pointwise _≡_ is equivalent to _≡_

+

+
Pointwise-≡⇒≡ : ∀ {n} {xs ys : Vec A n} → Pointwise _≡_ xs ys → xs ≡ ys

+
Pointwise-≡⇒≡ []               = ≡.refl

+
Pointwise-≡⇒≡ (≡.refl ∷ xs∼ys) = ≡.cong (_ ∷_) (Pointwise-≡⇒≡ xs∼ys)

+

+
≡⇒Pointwise-≡ : ∀ {n} {xs ys : Vec A n} → xs ≡ ys → Pointwise _≡_ xs ys

+
≡⇒Pointwise-≡ ≡.refl = refl ≡.refl

+

+
Pointwise-≡↔≡ : ∀ {n} {xs ys : Vec A n} → Pointwise _≡_ xs ys ⇔ xs ≡ ys

+
Pointwise-≡↔≡ = mk⇔ Pointwise-≡⇒≡ ≡⇒Pointwise-≡

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Relation.Unary.All.Properties.html b/site/static/agda_html/Data.Vec.Relation.Unary.All.Properties.html
new file mode 100644
index 000000000..af7db9f1a
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Relation.Unary.All.Properties.html
@@ -0,0 +1,258 @@
+
+Data.Vec.Relation.Unary.All.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to All

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Relation.Unary.All.Properties where

+

+
open import Data.Nat.Base using (ℕ; zero; suc; _+_)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.List.Base using ([]; _∷_)

+
open import Data.List.Relation.Unary.All as List using ([]; _∷_)

+
open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)

+
open import Data.Vec.Base as Vec using (Vec; []; _∷_; map; _++_; concat;

+
  tabulate; drop; take; toList; fromList)

+
open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)

+
open import Level using (Level)

+
open import Function.Base using (_∘_; id)

+
open import Function.Bundles using (_↔_; mk↔ₛ′)

+
open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_)

+
open import Relation.Binary.Core using (REL)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; cong₂)

+

+
private

+
  variable

+
    a b p q : Level

+
    A : Set a

+
    B : Set b

+
    P : Pred A p

+
    Q : Pred B q

+
    m n : ℕ

+
    xs : Vec A n

+

+
------------------------------------------------------------------------

+
-- lookup

+

+
lookup⁺ : All P xs → ∀ i → P (Vec.lookup xs i)

+
lookup⁺ (px ∷ _)  zero    = px

+
lookup⁺ (_ ∷ pxs) (suc i) = lookup⁺ pxs i

+

+
lookup⁻ : (∀ i → P (Vec.lookup xs i)) → All P xs

+
lookup⁻ {xs = []}    pxs = []

+
lookup⁻ {xs = _ ∷ _} pxs = pxs zero ∷ lookup⁻ (pxs ∘ suc)

+

+
------------------------------------------------------------------------

+
-- map

+

+
map⁺ : {f : A → B} → All (P ∘ f) xs → All P (map f xs)

+
map⁺ []         = []

+
map⁺ (px ∷ pxs) = px ∷ map⁺ pxs

+

+
map⁻ : {f : A → B} → All P (map f xs) → All (P ∘ f) xs

+
map⁻ {xs = []}    []       = []

+
map⁻ {xs = _ ∷ _} (px ∷ pxs) = px ∷ map⁻ pxs

+

+
-- A variant of All.map

+

+
gmap : {f : A → B} → P ⋐ Q ∘ f → All P {n} ⋐ All Q {n} ∘ map f

+
gmap g = map⁺ ∘ All.map g

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
++⁺ : {xs : Vec A m} {ys : Vec A n} →

+
      All P xs → All P ys → All P (xs ++ ys)

+
++⁺ []         pys = pys

+
++⁺ (px ∷ pxs) pys = px ∷ ++⁺ pxs pys

+

+
++ˡ⁻ : (xs : Vec A m) {ys : Vec A n} →

+
       All P (xs ++ ys) → All P xs

+
++ˡ⁻ []       _          = []

+
++ˡ⁻ (x ∷ xs) (px ∷ pxs) = px ∷ ++ˡ⁻ xs pxs

+

+
++ʳ⁻ : (xs : Vec A m) {ys : Vec A n} →

+
       All P (xs ++ ys) → All P ys

+
++ʳ⁻ []       pys        = pys

+
++ʳ⁻ (x ∷ xs) (px ∷ pxs) = ++ʳ⁻ xs pxs

+

+
++⁻ : (xs : Vec A m) {ys : Vec A n} →

+
      All P (xs ++ ys) → All P xs × All P ys

+
++⁻ []       p          = [] , p

+
++⁻ (x ∷ xs) (px ∷ pxs) = Product.map₁ (px ∷_) (++⁻ _ pxs)

+

+
++⁺∘++⁻ : (xs : Vec A m) {ys : Vec A n} →

+
          (p : All P (xs ++ ys)) →

+
          uncurry′ ++⁺ (++⁻ xs p) ≡ p

+
++⁺∘++⁻ []       p          = refl

+
++⁺∘++⁻ (x ∷ xs) (px ∷ pxs) = cong (px ∷_) (++⁺∘++⁻ xs pxs)

+

+
++⁻∘++⁺ : {xs : Vec A m} {ys : Vec A n} →

+
          (p : All P xs × All P ys) →

+
          ++⁻ xs (uncurry ++⁺ p) ≡ p

+
++⁻∘++⁺ ([]       , pys) = refl

+
++⁻∘++⁺ (px ∷ pxs , pys) rewrite ++⁻∘++⁺ (pxs , pys) = refl

+

+
++↔ : {xs : Vec A m} {ys : Vec A n} →

+
      (All P xs × All P ys) ↔ All P (xs ++ ys)

+
++↔ {xs = xs} = mk↔ₛ′ (uncurry ++⁺) (++⁻ xs) (++⁺∘++⁻ xs) ++⁻∘++⁺

+

+
------------------------------------------------------------------------

+
-- concat

+

+
concat⁺ : {xss : Vec (Vec A m) n} →

+
          All (All P) xss → All P (concat xss)

+
concat⁺ []           = []

+
concat⁺ (pxs ∷ pxss) = ++⁺ pxs (concat⁺ pxss)

+

+
concat⁻ : (xss : Vec (Vec A m) n) →

+
          All P (concat xss) → All (All P) xss

+
concat⁻ []         []   = []

+
concat⁻ (xs ∷ xss) pxss = ++ˡ⁻ xs pxss ∷ concat⁻ xss (++ʳ⁻ xs pxss)

+

+
------------------------------------------------------------------------

+
-- swap

+

+
module _ {_~_ : REL A B p} where

+

+
  All-swap : ∀ {n m xs ys} →

+
             All (λ x → All (x ~_) ys) {n} xs →

+
             All (λ y → All (_~ y) xs) {m} ys

+
  All-swap {ys = []}     _   = []

+
  All-swap {ys = y ∷ ys} []  = All.universal (λ _ → []) (y ∷ ys)

+
  All-swap {ys = y ∷ ys} ((x~y ∷ x~ys) ∷ pxs) =

+
    (x~y ∷ (All.map All.head pxs)) ∷

+
    All-swap (x~ys ∷ (All.map All.tail pxs))

+

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {P : A → Set p} where

+

+
  tabulate⁺ : ∀ {n} {f : Fin n → A} →

+
              (∀ i → P (f i)) → All P (tabulate f)

+
  tabulate⁺ {zero}  Pf = []

+
  tabulate⁺ {suc n} Pf = Pf zero ∷ tabulate⁺ (Pf ∘ suc)

+

+
  tabulate⁻ : ∀ {n} {f : Fin n → A} →

+
              All P (tabulate f) → (∀ i → P (f i))

+
  tabulate⁻ (px ∷ _) zero    = px

+
  tabulate⁻ (_ ∷ pf) (suc i) = tabulate⁻ pf i

+

+
------------------------------------------------------------------------

+
-- take and drop

+

+
drop⁺ : ∀ m {xs} → All P {m + n} xs → All P {n} (drop m xs)

+
drop⁺ zero pxs = pxs

+
drop⁺ (suc m) {x ∷ xs} (px ∷ pxs) = drop⁺ m pxs

+

+
take⁺ : ∀ m {xs} → All P {m + n} xs → All P {m} (take m xs)

+
take⁺ zero pxs = []

+
take⁺ (suc m) {x ∷ xs} (px ∷ pxs) = px ∷ take⁺ m pxs

+

+
------------------------------------------------------------------------

+
-- toList

+

+
module _ {P : Pred A p} where

+

+
  toList⁺ : ∀ {n} {xs : Vec A n} → All P xs → List.All P (toList xs)

+
  toList⁺ []         = []

+
  toList⁺ (px ∷ pxs) = px ∷ toList⁺ pxs

+

+
  toList⁻ : ∀ {n} {xs : Vec A n} → List.All P (toList xs) → All P xs

+
  toList⁻ {xs = []}     []         = []

+
  toList⁻ {xs = x ∷ xs} (px ∷ pxs) = px ∷ toList⁻ pxs

+

+
------------------------------------------------------------------------

+
-- fromList

+

+
module _ {P : Pred A p} where

+

+
  fromList⁺ : ∀ {xs} → List.All P xs → All P (fromList xs)

+
  fromList⁺ []         = []

+
  fromList⁺ (px ∷ pxs) = px ∷ fromList⁺ pxs

+

+
  fromList⁻ : ∀ {xs} → All P (fromList xs) → List.All P xs

+
  fromList⁻ {[]}     []         = []

+
  fromList⁻ {x ∷ xs} (px ∷ pxs) = px ∷ (fromList⁻ pxs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Relation.Unary.All.html b/site/static/agda_html/Data.Vec.Relation.Unary.All.html
new file mode 100644
index 000000000..94067b121
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Relation.Unary.All.html
@@ -0,0 +1,217 @@
+
+Data.Vec.Relation.Unary.All
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Vectors where all elements satisfy a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Relation.Unary.All where

+

+
open import Data.Nat.Base using (ℕ; zero; suc; NonZero)

+
open import Data.Product.Base as Product using (_×_; _,_; uncurry; <_,_>)

+
open import Data.Sum.Base as Sum using (inj₁; inj₂)

+
open import Data.Vec.Base as Vec using (Vec; []; _∷_)

+
open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.Vec.Membership.Propositional renaming (_∈_ to _∈ₚ_)

+
import Data.Vec.Membership.Setoid as SetoidMembership

+
open import Function.Base using (_∘_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no)

+
open import Relation.Unary hiding (_∈_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (_Respects_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)

+

+
private

+
  variable

+
    a b c p q r ℓ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    P : Pred A p

+
    Q : Pred A q

+
    R : Pred A r

+
    n : ℕ

+
    x : A

+
    xs : Vec A n

+

+
------------------------------------------------------------------------

+
-- All P xs means that all elements in xs satisfy P.

+

+
infixr 5 _∷_

+

+
data All {A : Set a} (P : Pred A p) : Vec A n → Set (p ⊔ a) where

+
  []  : All P []

+
  _∷_ : (px : P x) (pxs : All P xs) → All P (x ∷ xs)

+

+
------------------------------------------------------------------------

+
-- Operations on All

+

+
head : All P (x ∷ xs) → P x

+
head (px ∷ pxs) = px

+

+
tail : All P (x ∷ xs) → All P xs

+
tail (px ∷ pxs) = pxs

+

+
reduce : (f : ∀ {x} → P x → B) → ∀ {n} {xs : Vec A n} → All P xs → Vec B n

+
reduce f []         = []

+
reduce f (px ∷ pxs) = f px ∷ reduce f pxs

+

+
uncons : All P (x ∷ xs) → P x × All P xs

+
uncons = < head , tail >

+

+
map : P ⊆ Q → All P ⊆ All Q {n}

+
map g []         = []

+
map g (px ∷ pxs) = g px ∷ map g pxs

+

+
zip : All P ∩ All Q ⊆ All (P ∩ Q) {n}

+
zip ([] , [])             = []

+
zip (px ∷ pxs , qx ∷ qxs) = (px , qx) ∷ zip (pxs , qxs)

+

+
unzip : All (P ∩ Q) {n} ⊆ All P ∩ All Q

+
unzip []           = [] , []

+
unzip (pqx ∷ pqxs) = Product.zip _∷_ _∷_ pqx (unzip pqxs)

+

+
module _ {P : Pred A p} {Q : Pred B q} {R : Pred C r} where

+

+
  zipWith : ∀ {_⊕_ : A → B → C} →

+
            (∀ {x y} → P x → Q y → R (x ⊕ y)) →

+
            ∀ {n xs ys} → All P {n} xs → All Q {n} ys →

+
            All R {n} (Vec.zipWith _⊕_ xs ys)

+
  zipWith _⊕_ {xs = []}     {[]}     []         []         = []

+
  zipWith _⊕_ {xs = x ∷ xs} {y ∷ ys} (px ∷ pxs) (qy ∷ qys) =

+
    px ⊕ qy ∷ zipWith _⊕_ pxs qys

+

+
------------------------------------------------------------------------

+
-- Generalised lookup based on a proof of Any

+

+
lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)

+
lookupAny (px ∷ pxs) (here qx) = px , qx

+
lookupAny (px ∷ pxs) (there i) = lookupAny pxs i

+

+
lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)

+
lookupWith f pxs i = Product.uncurry f (lookupAny pxs i)

+

+
lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)

+
lookup pxs = lookupWith (λ { px refl → px }) pxs

+

+
module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where

+
  open Setoid S renaming (sym to sym₁)

+
  open SetoidMembership S

+

+
  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)

+
  lookupₛ resp pxs = lookupWith (λ py x=y → resp (sym₁ x=y) py) pxs

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by All

+

+
all? : ∀ {n} → Decidable P → Decidable (All P {n})

+
all? P? []       = yes []

+
all? P? (x ∷ xs) = Dec.map′ (uncurry _∷_) uncons (P? x ×-dec all? P? xs)

+

+
universal : Universal P → ∀ {n} → Universal (All P {n})

+
universal u []       = []

+
universal u (x ∷ xs) = u x ∷ universal u xs

+

+
irrelevant : Irrelevant P → ∀ {n} → Irrelevant (All P {n})

+
irrelevant irr []           []           = refl

+
irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =

+
  cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)

+

+
satisfiable : Satisfiable P → ∀ {n} → Satisfiable (All P {n})

+
satisfiable (x , p) {zero}  = [] , []

+
satisfiable (x , p) {suc n} = Product.map (x ∷_) (p ∷_) (satisfiable (x , p))

+

+
------------------------------------------------------------------------

+
-- Generalised decidability procedure

+

+
decide :  Π[ P ∪ Q ] → Π[ All P {n} ∪ Any Q ]

+
decide p∪q [] = inj₁ []

+
decide p∪q (x ∷ xs) with p∪q x

+
... | inj₂ qx = inj₂ (here qx)

+
... | inj₁ px = Sum.map (px ∷_) there (decide p∪q xs)

+

+

+
all = all?

+
{-# WARNING_ON_USAGE all

+
"Warning: all was deprecated in v1.4.

+
Please use all? instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Relation.Unary.Any.Properties.html b/site/static/agda_html/Data.Vec.Relation.Unary.Any.Properties.html
new file mode 100644
index 000000000..fe5d867fd
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Relation.Unary.Any.Properties.html
@@ -0,0 +1,500 @@
+
+Data.Vec.Relation.Unary.Any.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of vector's Any

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Relation.Unary.Any.Properties where

+

+
open import Data.Nat.Base using (suc; zero)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Empty using (⊥)

+
open import Data.List.Base using ([]; _∷_)

+
import Data.List.Relation.Unary.Any as List

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Data.Sum.Function.Propositional using (_⊎-cong_)

+
open import Data.Product.Base as Product using (∃; ∃₂; _×_; _,_; proj₁; proj₂)

+
open import Data.Vec.Base hiding (here; there)

+
open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)

+
open import Data.Vec.Membership.Propositional

+
  using (_∈_; mapWith∈; find; lose)

+
open import Data.Vec.Relation.Binary.Pointwise.Inductive

+
  using (Pointwise; []; _∷_)

+
open import Function.Base

+
open import Function.Bundles using (_↔_; mk↔ₛ′)

+
open import Function.Properties.Inverse using (↔-refl; ↔-trans)

+
open import Level using (Level)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Relation.Unary hiding (_∈_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions using (_Respects_)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; sym; trans; cong)

+

+
private

+
  variable

+
    a b p q r ℓ : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Equality properties

+

+
module _ {P : Pred A p} {_≈_ : Rel A ℓ} where

+

+
  lift-resp : ∀ {n} → P Respects _≈_ → (Any P {n}) Respects (Pointwise _≈_)

+
  lift-resp resp (x∼y ∷ xs∼ys) (here px)   = here (resp x∼y px)

+
  lift-resp resp (x∼y ∷ xs∼ys) (there pxs) = there (lift-resp resp xs∼ys pxs)

+

+
module _ {P : Pred A p} where

+

+
  here-injective : ∀ {n x xs} {p q : P x} →

+
                   here {P = P} {n = n} {xs = xs} p ≡ here q → p ≡ q

+
  here-injective refl = refl

+

+
  there-injective : ∀ {n x xs} {p q : Any P xs} →

+
                    there {n = n} {x = x} p ≡ there q → p ≡ q

+
  there-injective refl = refl

+

+
------------------------------------------------------------------------

+
-- Misc

+

+
  ¬Any[] : ¬ Any P []

+
  ¬Any[] ()

+

+
  lookup-index : ∀ {m} {xs : Vec A m} (p : Any P xs) →

+
                 P (lookup xs (Any.index p))

+
  lookup-index (here px) = px

+
  lookup-index (there p) = lookup-index p

+

+
------------------------------------------------------------------------

+
-- Convert from/to List.Any

+

+
  fromList⁺ : ∀ {xs} → List.Any P xs → Any P (fromList xs)

+
  fromList⁺ (List.here px) = here px

+
  fromList⁺ (List.there v) = there (fromList⁺ v)

+

+
  fromList⁻ : ∀ {xs} → Any P (fromList xs) → List.Any P xs

+
  fromList⁻ {x ∷ xs} (here px)   = List.here px

+
  fromList⁻ {x ∷ xs} (there pxs) = List.there (fromList⁻ pxs)

+

+
  toList⁺ : ∀ {n} {xs : Vec A n} → Any P xs → List.Any P (toList xs)

+
  toList⁺ (here px) = List.here px

+
  toList⁺ (there v) = List.there (toList⁺ v)

+

+
  toList⁻ : ∀ {n} {xs : Vec A n} → List.Any P (toList xs) → Any P xs

+
  toList⁻ {xs = x ∷ xs} (List.here px)   = here px

+
  toList⁻ {xs = x ∷ xs} (List.there pxs) = there (toList⁻ pxs)

+

+
------------------------------------------------------------------------

+
-- map

+

+
map-id : ∀ {P : Pred A p} (f : P ⊆ P) {n xs} →

+
         (∀ {x} (p : P x) → f p ≡ p) →

+
         (p : Any P {n} xs) → Any.map f p ≡ p

+
map-id f hyp (here  p) = cong here (hyp p)

+
map-id f hyp (there p) = cong there $ map-id f hyp p

+

+
map-∘ : ∀ {P : Pred A p} {Q : A → Set q} {R : A → Set r}

+
        (f : Q ⊆ R) (g : P ⊆ Q)

+
        {n xs} (p : Any P {n} xs) →

+
        Any.map (f ∘ g) p ≡ Any.map f (Any.map g p)

+
map-∘ f g (here  p) = refl

+
map-∘ f g (there p) = cong there $ map-∘ f g p

+

+
------------------------------------------------------------------------

+
-- Swapping

+

+
module _ {P : A → B → Set ℓ} where

+

+
  swap : ∀ {n m} {xs : Vec A n} {ys : Vec B m} →

+
         Any (λ x → Any (P x) ys) xs →

+
         Any (λ y → Any (flip P y) xs) ys

+
  swap (here  pys)  = Any.map here pys

+
  swap (there pxys) = Any.map there (swap pxys)

+

+
  swap-there : ∀ {n m x xs ys} → (any : Any (λ x → Any (P x) {n} ys) {m} xs) →

+
               swap (Any.map (there {x = x}) any) ≡ there (swap any)

+
  swap-there (here  pys)  = refl

+
  swap-there (there pxys) = cong (Any.map there) (swap-there pxys)

+

+
module _ {P : A → B → Set ℓ} where

+

+
  swap-invol : ∀ {n m} {xs : Vec A n} {ys : Vec B m} →

+
               (any : Any (λ x → Any (P x) ys) xs) →

+
               swap (swap any) ≡ any

+
  swap-invol (here (here _)) = refl

+
  swap-invol (here (there pys)) = cong (Any.map there) (swap-invol (here pys))

+
  swap-invol (there pxys) = trans (swap-there (swap pxys))

+
                          $ cong there (swap-invol pxys)

+

+
module _ {P : A → B → Set ℓ} where

+

+
  swap↔ : ∀ {n m} {xs : Vec A n} {ys : Vec B m} →

+
          Any (λ x → Any (P x) ys) xs ↔ Any (λ y → Any (flip P y) xs) ys

+
  swap↔ = mk↔ₛ′ swap swap swap-invol swap-invol

+

+
------------------------------------------------------------------------

+
-- Lemmas relating Any to ⊥

+

+
⊥↔Any⊥ : ∀ {n} {xs : Vec A n} → ⊥ ↔ Any (const ⊥) xs

+
⊥↔Any⊥ = mk↔ₛ′ (λ ()) (λ p → from p) (λ p → from p) (λ ())

+
  where

+
  from : ∀ {n xs} → Any (const ⊥) {n} xs → ∀ {b} {B : Set b} → B

+
  from (there p) = from p

+

+
⊥↔Any[] : ∀ {P : Pred A p} → ⊥ ↔ Any P []

+
⊥↔Any[] = mk↔ₛ′ (λ()) (λ()) (λ()) (λ())

+

+
------------------------------------------------------------------------

+
-- Sums commute with Any

+

+
module _ {P : Pred A p} {Q : A → Set q} where

+

+
  Any-⊎⁺ : ∀ {n} {xs : Vec A n} → Any P xs ⊎ Any Q xs → Any (λ x → P x ⊎ Q x) xs

+
  Any-⊎⁺ = [ Any.map inj₁ , Any.map inj₂ ]′

+

+
  Any-⊎⁻ : ∀ {n} {xs : Vec A n} → Any (λ x → P x ⊎ Q x) xs → Any P xs ⊎ Any Q xs

+
  Any-⊎⁻ (here (inj₁ p)) = inj₁ (here p)

+
  Any-⊎⁻ (here (inj₂ q)) = inj₂ (here q)

+
  Any-⊎⁻ (there p)       = Sum.map there there (Any-⊎⁻ p)

+

+
  ⊎↔ : ∀ {n} {xs : Vec A n} → (Any P xs ⊎ Any Q xs) ↔ Any (λ x → P x ⊎ Q x) xs

+
  ⊎↔ = mk↔ₛ′ Any-⊎⁺ Any-⊎⁻ to∘from from∘to

+
    where

+
    from∘to : ∀ {n} {xs : Vec A n} (p : Any P xs ⊎ Any Q xs) → Any-⊎⁻ (Any-⊎⁺ p) ≡ p

+
    from∘to (inj₁ (here  p)) = refl

+
    from∘to (inj₁ (there p)) rewrite from∘to (inj₁ p) = refl

+
    from∘to (inj₂ (here  q)) = refl

+
    from∘to (inj₂ (there q)) rewrite from∘to (inj₂ q) = refl

+

+
    to∘from : ∀ {n} {xs : Vec A n} (p : Any (λ x → P x ⊎ Q x) xs) →

+
              Any-⊎⁺ (Any-⊎⁻ p) ≡ p

+
    to∘from (here (inj₁ p)) = refl

+
    to∘from (here (inj₂ q)) = refl

+
    to∘from (there p) with Any-⊎⁻ p | to∘from p

+
    to∘from (there .(Any.map inj₁ p)) | inj₁ p | refl = refl

+
    to∘from (there .(Any.map inj₂ q)) | inj₂ q | refl = refl

+

+
------------------------------------------------------------------------

+
-- Products "commute" with Any.

+

+
module _ {P : Pred A p} {Q : Pred B q} where

+

+
  Any-×⁺ : ∀ {n m} {xs : Vec A n} {ys : Vec B m} → Any P xs × Any Q ys →

+
           Any (λ x → Any (λ y → P x × Q y) ys) xs

+
  Any-×⁺ (p , q) = Any.map (λ p → Any.map (p ,_) q) p

+

+
  Any-×⁻ : ∀ {n m} {xs : Vec A n} {ys : Vec B m} →

+
           Any (λ x → Any (λ y → P x × Q y) ys) xs →

+
           Any P xs × Any Q ys

+
  Any-×⁻ pq with find pq

+
  ... | x , x∈xs , pxys with find pxys

+
  ...   | y , y∈ys , px , py = lose x∈xs px , lose y∈ys py

+

+
------------------------------------------------------------------------

+
-- Invertible introduction (⁺) and elimination (⁻) rules for various

+
-- vector functions

+

+
------------------------------------------------------------------------

+
-- Singleton ([_])

+

+
module _ {P : Pred A p} where

+

+
  singleton⁺ : ∀ {x} → P x → Any P [ x ]

+
  singleton⁺ Px = here Px

+

+
  singleton⁻ : ∀ {x} → Any P [ x ] → P x

+
  singleton⁻ (here Px) = Px

+

+
  singleton⁺∘singleton⁻ : ∀ {x} (p : Any P [ x ]) →

+
                          singleton⁺ (singleton⁻ p) ≡ p

+
  singleton⁺∘singleton⁻ (here px) = refl

+

+
  singleton⁻∘singleton⁺ : ∀ {x} (p : P x) →

+
                          singleton⁻ (singleton⁺ p) ≡ p

+
  singleton⁻∘singleton⁺ p = refl

+

+
  singleton↔ : ∀ {x} → P x ↔ Any P [ x ]

+
  singleton↔ = mk↔ₛ′ singleton⁺ singleton⁻ singleton⁺∘singleton⁻ singleton⁻∘singleton⁺

+

+
------------------------------------------------------------------------

+
-- map

+

+
module _ {f : A → B} where

+

+
  map⁺ : ∀ {P : Pred B p} {n} {xs : Vec A n} →

+
         Any (P ∘ f) xs → Any P (map f xs)

+
  map⁺ (here p)  = here p

+
  map⁺ (there p) = there $ map⁺ p

+

+
  map⁻ : ∀ {P : Pred B p} {n} {xs : Vec A n} →

+
         Any P (map f xs) → Any (P ∘ f) xs

+
  map⁻ {xs = x ∷ xs} (here p)  = here p

+
  map⁻ {xs = x ∷ xs} (there p) = there $ map⁻ p

+

+
  map⁺∘map⁻ : ∀ {P : Pred B p} {n} {xs : Vec A n} →

+
              (p : Any P (map f xs)) → map⁺ (map⁻ p) ≡ p

+
  map⁺∘map⁻ {xs = x ∷ xs} (here  p) = refl

+
  map⁺∘map⁻ {xs = x ∷ xs} (there p) = cong there (map⁺∘map⁻ p)

+

+
  map⁻∘map⁺ : ∀ (P : Pred B p) {n} {xs : Vec A n} →

+
              (p : Any (P ∘ f) xs) → map⁻ {P = P} (map⁺ p) ≡ p

+
  map⁻∘map⁺ P (here  p) = refl

+
  map⁻∘map⁺ P (there p) = cong there (map⁻∘map⁺ P p)

+

+
  map↔ : ∀ {P : Pred B p} {n} {xs : Vec A n} →

+
         Any (P ∘ f) xs ↔ Any P (map f xs)

+
  map↔ = mk↔ₛ′ map⁺ map⁻ map⁺∘map⁻ (map⁻∘map⁺ _)

+

+
------------------------------------------------------------------------

+
-- _++_

+

+
module _ {P : Pred A p} where

+

+
  ++⁺ˡ : ∀ {n m} {xs : Vec A n} {ys : Vec A m} → Any P xs → Any P (xs ++ ys)

+
  ++⁺ˡ (here p)  = here p

+
  ++⁺ˡ (there p) = there (++⁺ˡ p)

+

+
  ++⁺ʳ : ∀ {n m} (xs : Vec A n) {ys : Vec A m} → Any P ys → Any P (xs ++ ys)

+
  ++⁺ʳ []       p = p

+
  ++⁺ʳ (x ∷ xs) p = there (++⁺ʳ xs p)

+

+
  ++⁻ : ∀ {n m} (xs : Vec A n) {ys : Vec A m} → Any P (xs ++ ys) → Any P xs ⊎ Any P ys

+
  ++⁻ []       p         = inj₂ p

+
  ++⁻ (x ∷ xs) (here p)  = inj₁ (here p)

+
  ++⁻ (x ∷ xs) (there p) = Sum.map there id (++⁻ xs p)

+

+
  ++⁺∘++⁻ : ∀ {n m} (xs : Vec A n) {ys : Vec A m} (p : Any P (xs ++ ys)) →

+
           [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs p) ≡ p

+
  ++⁺∘++⁻ []       p         = refl

+
  ++⁺∘++⁻ (x ∷ xs) (here  p) = refl

+
  ++⁺∘++⁻ (x ∷ xs) (there p) with ++⁻ xs p | ++⁺∘++⁻ xs p

+
  ++⁺∘++⁻ (x ∷ xs) (there p) | inj₁ p′ | ih = cong there ih

+
  ++⁺∘++⁻ (x ∷ xs) (there p) | inj₂ p′ | ih = cong there ih

+

+
  ++⁻∘++⁺ : ∀ {n m} (xs : Vec A n) {ys : Vec A m} (p : Any P xs ⊎ Any P ys) →

+
            ++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p) ≡ p

+
  ++⁻∘++⁺ []            (inj₂ p)         = refl

+
  ++⁻∘++⁺ (x ∷ xs)      (inj₁ (here  p)) = refl

+
  ++⁻∘++⁺ (x ∷ xs) {ys} (inj₁ (there p)) rewrite ++⁻∘++⁺ xs {ys} (inj₁ p) = refl

+
  ++⁻∘++⁺ (x ∷ xs)      (inj₂ p)         rewrite ++⁻∘++⁺ xs      (inj₂ p) = refl

+

+
  ++↔ : ∀ {n m} {xs : Vec A n} {ys : Vec A m} →

+
        (Any P xs ⊎ Any P ys) ↔ Any P (xs ++ ys)

+
  ++↔ {xs = xs} = mk↔ₛ′ [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁺∘++⁻ xs) (++⁻∘++⁺ xs)

+

+
  ++-comm : ∀ {n m} (xs : Vec A n) (ys : Vec A m) →

+
            Any P (xs ++ ys) → Any P (ys ++ xs)

+
  ++-comm xs ys = [ ++⁺ʳ ys , ++⁺ˡ ]′ ∘ ++⁻ xs

+

+
  ++-comm∘++-comm : ∀ {n m} (xs : Vec A n) {ys : Vec A m} (p : Any P (xs ++ ys)) →

+
                    ++-comm ys xs (++-comm xs ys p) ≡ p

+
  ++-comm∘++-comm [] {ys} p

+
    rewrite ++⁻∘++⁺ ys {ys = []} (inj₁ p) = refl

+
  ++-comm∘++-comm (x ∷ xs) {ys} (here p)

+
    rewrite ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₂ (here p)) = refl

+
  ++-comm∘++-comm (x ∷ xs)      (there p) with ++⁻ xs p | ++-comm∘++-comm xs p

+
  ++-comm∘++-comm (x ∷ xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ʳ ys p))))

+
    | inj₁ p | refl

+
    rewrite ++⁻∘++⁺ ys (inj₂                 p)

+
            | ++⁻∘++⁺ ys (inj₂ $ there {x = x} p) = refl

+
  ++-comm∘++-comm (x ∷ xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ˡ p))))

+
    | inj₂ p | refl

+
    rewrite ++⁻∘++⁺ ys {ys =     xs} (inj₁ p)

+
            | ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₁ p) = refl

+

+
  ++↔++ : ∀ {n m} (xs : Vec A n) (ys : Vec A m) → Any P (xs ++ ys) ↔ Any P (ys ++ xs)

+
  ++↔++ xs ys = mk↔ₛ′ (++-comm xs ys) (++-comm ys xs)

+
                        (++-comm∘++-comm ys) (++-comm∘++-comm xs)

+

+
  ++-insert : ∀ {n m x} (xs : Vec A n) {ys : Vec A m} → P x → Any P (xs ++ [ x ] ++ ys)

+
  ++-insert xs Px = ++⁺ʳ xs (++⁺ˡ (singleton⁺ Px))

+

+
------------------------------------------------------------------------

+
-- concat

+

+
module _ {P : Pred A p} where

+

+
  concat⁺ : ∀ {n m} {xss : Vec (Vec A n) m} → Any (Any P) xss → Any P (concat xss)

+
  concat⁺ (here p)           = ++⁺ˡ p

+
  concat⁺ (there {x = xs} p) = ++⁺ʳ xs (concat⁺ p)

+

+
  concat⁻ : ∀ {n m} (xss : Vec (Vec A n) m) → Any P (concat xss) → Any (Any P) xss

+
  concat⁻ (xs ∷ xss) p = [ here , there ∘ concat⁻ xss ]′ (++⁻ xs p)

+

+
  concat⁻∘++⁺ˡ : ∀ {n m xs} (xss : Vec (Vec A n) m) (p : Any P xs) →

+
                concat⁻ (xs ∷ xss) (++⁺ˡ p) ≡ here p

+
  concat⁻∘++⁺ˡ xss p rewrite ++⁻∘++⁺ _ {concat xss} (inj₁ p) = refl

+

+
  concat⁻∘++⁺ʳ : ∀ {n m} xs (xss : Vec (Vec A n) m) (p : Any P (concat xss)) →

+
                concat⁻ (xs ∷ xss) (++⁺ʳ xs p) ≡ there (concat⁻ xss p)

+
  concat⁻∘++⁺ʳ xs xss p rewrite ++⁻∘++⁺ xs (inj₂ p) = refl

+

+
  concat⁺∘concat⁻ : ∀ {n m} (xss : Vec (Vec A n) m) (p : Any P (concat xss)) →

+
                   concat⁺ (concat⁻ xss p) ≡ p

+
  concat⁺∘concat⁻ (xs ∷ xss) p  with ++⁻ xs p in eq

+
  ... | inj₁ pxs

+
    = trans (cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (sym eq))

+
    $ ++⁺∘++⁻ xs p

+
  ... | inj₂ pxss rewrite concat⁺∘concat⁻ xss pxss

+
    = trans (cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (sym eq))

+
    $ ++⁺∘++⁻ xs p

+

+
  concat⁻∘concat⁺ : ∀ {n m} {xss : Vec (Vec A n) m} (p : Any (Any P) xss) →

+
                    concat⁻ xss (concat⁺ p) ≡ p

+
  concat⁻∘concat⁺ {xss = xs ∷ xss} (here p)

+
    rewrite ++⁻∘++⁺ xs {concat xss} (inj₁ p) = refl

+
  concat⁻∘concat⁺ {xss = xs ∷ xss} (there p)

+
    rewrite ++⁻∘++⁺ xs {concat xss} (inj₂ (concat⁺ p))

+
            | concat⁻∘concat⁺ p = refl

+

+
  concat↔ : ∀ {n m} {xss : Vec (Vec A n) m} → Any (Any P) xss ↔ Any P (concat xss)

+
  concat↔ {xss = xss} = mk↔ₛ′ concat⁺ (concat⁻ xss) (concat⁺∘concat⁻ xss) concat⁻∘concat⁺

+

+
------------------------------------------------------------------------

+
-- tabulate

+

+
module _ {P : Pred A p} where

+

+
  tabulate⁺ : ∀ {n} {f : Fin n → A} i → P (f i) → Any P (tabulate f)

+
  tabulate⁺ zero    p = here p

+
  tabulate⁺ (suc i) p = there (tabulate⁺ i p)

+

+
  tabulate⁻ : ∀ {n} {f : Fin n → A} →

+
              Any P (tabulate f) → ∃ λ i → P (f i)

+
  tabulate⁻ (here p)  = zero , p

+
  tabulate⁻ (there p) = Product.map suc id (tabulate⁻ p)

+

+
------------------------------------------------------------------------

+
-- mapWith∈

+

+
module _ {P : Pred B p} where

+

+
  mapWith∈⁺ : ∀ {n} {xs : Vec A n} (f : ∀ {x} → x ∈ xs → B) →

+
              (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) →

+
              Any P (mapWith∈ xs f)

+
  mapWith∈⁺ f (_ , here refl  , p) = here p

+
  mapWith∈⁺ f (_ , there x∈xs , p) =

+
    there $ mapWith∈⁺ (f ∘ there) (_ , x∈xs , p)

+

+
  mapWith∈⁻ : ∀ {n} (xs : Vec A n) (f : ∀ {x} → x ∈ xs → B) →

+
              Any P (mapWith∈ xs f) →

+
              ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)

+
  mapWith∈⁻ (y ∷ xs) f (here  p) = (y , here refl , p)

+
  mapWith∈⁻ (y ∷ xs) f (there p) =

+
    Product.map id (Product.map there id) $ mapWith∈⁻ xs (f ∘ there) p

+

+
  mapWith∈↔ : ∀ {n} {xs : Vec A n} {f : ∀ {x} → x ∈ xs → B} →

+
    (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) ↔ Any P (mapWith∈ xs f)

+
  mapWith∈↔ = mk↔ₛ′ (mapWith∈⁺ _) (mapWith∈⁻ _ _) (to∘from _ _) (from∘to _)

+
    where

+
    from∘to : ∀ {n} {xs : Vec A n} (f : ∀ {x} → x ∈ xs → B)

+
              (p : ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) →

+
              mapWith∈⁻ xs f (mapWith∈⁺ f p) ≡ p

+
    from∘to f (_ , here refl  , p) = refl

+
    from∘to f (_ , there x∈xs , p)

+
      rewrite from∘to (f ∘ there) (_ , x∈xs , p) = refl

+

+
    to∘from : ∀ {n} (xs : Vec A n) (f : ∀ {x} → x ∈ xs → B)

+
              (p : Any P (mapWith∈ xs f)) →

+
              mapWith∈⁺ f (mapWith∈⁻ xs f p) ≡ p

+
    to∘from (y ∷ xs) f (here  p) = refl

+
    to∘from (y ∷ xs) f (there p) =

+
      cong there $ to∘from xs (f ∘ there) p

+

+
------------------------------------------------------------------------

+
-- _∷_

+

+
∷↔ : ∀ {n} (P : Pred A p) {x} {xs : Vec A n} →

+
     (P x ⊎ Any P xs) ↔ Any P (x ∷ xs)

+
∷↔ P {x} {xs} = ↔-trans (singleton↔ ⊎-cong ↔-refl) ++↔

+

+
------------------------------------------------------------------------

+
-- _>>=_

+

+
module _ {A B : Set a} {P : Pred B p} {m} {f : A → Vec B m} where

+

+
  open CartesianBind

+

+
  >>=↔ : ∀ {n} {xs : Vec A n} → Any (Any P ∘ f) xs ↔ Any P (xs >>= f)

+
  >>=↔ = ↔-trans map↔ concat↔

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.Relation.Unary.Any.html b/site/static/agda_html/Data.Vec.Relation.Unary.Any.html
new file mode 100644
index 000000000..5d4d7753b
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.Relation.Unary.Any.html
@@ -0,0 +1,165 @@
+
+Data.Vec.Relation.Unary.Any
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Vectors where at least one element satisfies a given property

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec.Relation.Unary.Any {a} {A : Set a} where

+

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Nat.Base using (ℕ; zero; suc; NonZero)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)

+
open import Data.Vec.Base as Vec using (Vec; []; [_]; _∷_)

+
open import Data.Product.Base as Product using (∃; _,_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary.Negation using (¬_; contradiction)

+
open import Relation.Nullary.Decidable as Dec using (no; _⊎-dec_)

+
open import Relation.Unary

+

+
private

+
  variable

+
    p q : Level

+
    P : Pred A p

+
    Q : Pred A q

+
    n : ℕ

+
    xs : Vec A n

+

+
------------------------------------------------------------------------

+
-- Any P xs means that at least one element in xs satisfies P.

+

+
data Any (P : Pred A p) : ∀ {n} → Vec A n → Set (a ⊔ p) where

+
  here  : ∀ {n x} {xs : Vec A n} (px  : P x)      → Any P (x ∷ xs)

+
  there : ∀ {n x} {xs : Vec A n} (pxs : Any P xs) → Any P (x ∷ xs)

+

+
------------------------------------------------------------------------

+
-- Operations on Any

+

+
-- If the tail does not satisfy the predicate, then the head will.

+
head : ∀ {x} → ¬ Any P xs → Any P (x ∷ xs) → P x

+
head ¬pxs (here px)   = px

+
head ¬pxs (there pxs) = contradiction pxs ¬pxs

+

+
-- If the head does not satisfy the predicate, then the tail will.

+
tail : ∀ {x} → ¬ P x → Any P (x ∷ xs) → Any P xs

+
tail ¬px (here  px)  = contradiction px ¬px

+
tail ¬px (there pxs) = pxs

+

+
-- Convert back and forth with sum

+
toSum : ∀ {x} → Any P (x ∷ xs) → P x ⊎ Any P xs

+
toSum (here px)   = inj₁ px

+
toSum (there pxs) = inj₂ pxs

+

+
fromSum : ∀ {x} → P x ⊎ Any P xs → Any P (x ∷ xs)

+
fromSum = [ here , there ]′

+

+
map : P ⊆ Q → ∀ {n} → Any P {n} ⊆ Any Q {n}

+
map g (here px)   = here (g px)

+
map g (there pxs) = there (map g pxs)

+

+
index : Any P {n} xs → Fin n

+
index (here  px)  = zero

+
index (there pxs) = suc (index pxs)

+

+
lookup : Any P xs → A

+
lookup {xs = xs} p = Vec.lookup xs (index p)

+

+
-- If any element satisfies P, then P is satisfied.

+
satisfied : Any P xs → ∃ P

+
satisfied (here px)   = _ , px

+
satisfied (there pxs) = satisfied pxs

+

+
------------------------------------------------------------------------

+
-- Properties of predicates preserved by Any

+

+
any? : Decidable P → ∀ {n} → Decidable (Any P {n})

+
any? P? []       = no λ()

+
any? P? (x ∷ xs) = Dec.map′ fromSum toSum (P? x ⊎-dec any? P? xs)

+

+
satisfiable : Satisfiable P → ∀ {n} → Satisfiable (Any P {suc n})

+
satisfiable (x , p) {zero}  = x ∷ [] , here p

+
satisfiable (x , p) {suc n} = Product.map (x ∷_) there (satisfiable (x , p))

+

+
any = any?

+
{-# WARNING_ON_USAGE any

+
"Warning: any was deprecated in v1.4.

+
Please use any? instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Vec.html b/site/static/agda_html/Data.Vec.html
new file mode 100644
index 000000000..6be70c03b
--- /dev/null
+++ b/site/static/agda_html/Data.Vec.html
@@ -0,0 +1,132 @@
+
+Data.Vec
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Vectors

+
------------------------------------------------------------------------

+

+
-- This implementation is designed for reasoning about dynamic

+
-- vectors which may increase or decrease in size.

+

+
-- See `Data.Vec.Functional` for an alternative implementation as

+
-- functions from finite sets, which is more suitable for reasoning

+
-- about fixed sized vectors and for when ease of retrieval is

+
-- important.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Vec where

+

+
open import Level

+
open import Data.Bool.Base

+
import Data.Nat.Properties as ℕ

+
open import Data.Vec.Bounded.Base as Vec≤

+
  using (Vec≤; ≤-cast; fromVec)

+
open import Function.Base using (_$_)

+
open import Relation.Nullary

+
open import Relation.Unary

+

+
private

+
  variable

+
    a p : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- Publicly re-export the contents of the base module

+

+
open import Data.Vec.Base public

+

+
------------------------------------------------------------------------

+
-- Additional operations

+

+
module _ {P : A → Set p} (P? : Decidable P) where

+

+
  filter : ∀ {n} → Vec A n → Vec≤ A n

+
  filter []       = Vec≤.[]

+
  filter (a ∷ as) = if does (P? a) then a Vec≤.∷_ else ≤-cast (ℕ.n≤1+n _) $ filter as

+

+
  takeWhile : ∀ {n} → Vec A n → Vec≤ A n

+
  takeWhile []       = Vec≤.[]

+
  takeWhile (a ∷ as) = if does (P? a) then a Vec≤.∷ takeWhile as else Vec≤.[]

+

+
  dropWhile : ∀ {n} → Vec A n → Vec≤ A n

+
  dropWhile Vec.[]       = Vec≤.[]

+
  dropWhile (a Vec.∷ as) =

+
    if does (P? a) then ≤-cast (ℕ.n≤1+n _) (dropWhile as)

+
    else fromVec (a Vec.∷ as)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.W.html b/site/static/agda_html/Data.W.html
new file mode 100644
index 000000000..710295989
--- /dev/null
+++ b/site/static/agda_html/Data.W.html
@@ -0,0 +1,143 @@
+
+Data.W
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- W-types

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.W where

+

+
open import Level using (Level; _⊔_)

+
open import Function.Base using (_$_; _∘_; const)

+
open import Data.Product.Base using (_,_; -,_; proj₂)

+
open import Data.Container.Core using (Container; ⟦_⟧; Shape; Position; _⇒_; ⟪_⟫)

+
open import Data.Container.Relation.Unary.All using (□; all)

+
open import Relation.Nullary.Negation using (¬_)

+
open import Agda.Builtin.Equality using (_≡_; refl)

+

+
private

+
  variable

+
    s p s₁ s₂ p₁ p₂ ℓ : Level

+
    C : Container s p

+
    C₁ : Container s₁ p₁

+
    C₂ : Container s₂ p₂

+

+
-- The family of W-types.

+

+
data W (C : Container s p) : Set (s ⊔ p) where

+
  sup : ⟦ C ⟧ (W C) → W C

+

+
sup-injective₁ : ∀ {s t : Shape C} {f : Position C s → W C} {g} →

+
                 sup (s , f) ≡ sup (t , g) → s ≡ t

+
sup-injective₁ refl = refl

+

+
-- See also Data.W.WithK.sup-injective₂.

+

+
-- Projections.

+

+
head : W C → Shape C

+
head (sup (x , f)) = x

+

+
tail : (x : W C) → Position C (head x) → W C

+
tail (sup (x , f)) = f

+

+
-- map

+

+
map : (m : C₁ ⇒ C₂) → W C₁ → W C₂

+
map m (sup (x , f)) = sup (⟪ m ⟫ (x , λ p → map m (f p)))

+

+
-- induction

+

+
module _ (P : W C → Set ℓ)

+
         (alg : ∀ {t} → □ C P t → P (sup t)) where

+

+
 induction : (w : W C) → P w

+
 induction (sup (s , f)) = alg $ all (induction ∘ f)

+

+
module _ {P : Set ℓ} (alg : ⟦ C ⟧ P → P) where

+

+
 foldr : W C → P

+
 foldr = induction (const P) (alg ∘ -,_ ∘ □.proof)

+

+
-- If Position is always inhabited, then W_C is empty.

+

+
inhabited⇒empty : (∀ s → Position C s) → ¬ W C

+
inhabited⇒empty b = foldr ((_$ b _) ∘ proj₂)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Word64.Base.html b/site/static/agda_html/Data.Word64.Base.html
new file mode 100644
index 000000000..fef7b92ef
--- /dev/null
+++ b/site/static/agda_html/Data.Word64.Base.html
@@ -0,0 +1,118 @@
+
+Data.Word64.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Machine words: basic type and conversion functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Word64.Base where

+

+
open import Algebra.Core using (Op₂)

+
open import Data.Nat.Base as ℕ using (ℕ)

+
open import Function.Base using (_on_; _∘₂′_)

+
open import Level using (zero)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
------------------------------------------------------------------------

+
-- Re-export built-ins publicly

+

+
open import Agda.Builtin.Word public

+
  using (Word64)

+
  renaming

+
  ( primWord64ToNat   to toℕ

+
  ; primWord64FromNat to fromℕ

+
  )

+

+
liftOp₂ : Op₂ ℕ → Op₂ Word64

+
liftOp₂ op = fromℕ ∘₂′ op on toℕ

+

+
infix 4 _≈_

+
_≈_ : Rel Word64 zero

+
_≈_ = _≡_ on toℕ

+

+
infix 4 _<_

+
_<_ : Rel Word64 zero

+
_<_ = ℕ._<_ on toℕ

+

+
infix 4 _≤_

+
_≤_ : Rel Word64 zero

+
_≤_ = ℕ._≤_ on toℕ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Word64.Properties.html b/site/static/agda_html/Data.Word64.Properties.html
new file mode 100644
index 000000000..365eb2f2b
--- /dev/null
+++ b/site/static/agda_html/Data.Word64.Properties.html
@@ -0,0 +1,185 @@
+
+Data.Word64.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of operations on machine words

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Word64.Properties where

+

+
import Data.Nat.Base as ℕ

+
open import Data.Bool.Base using (Bool)

+
open import Data.Word64.Base using (_≈_; toℕ; Word64; _<_)

+
import Data.Nat.Properties as ℕ

+
open import Relation.Nullary.Decidable.Core using (map′; ⌊_⌋)

+
open import Relation.Binary

+
  using ( _⇒_; Reflexive; Symmetric; Transitive; Substitutive

+
        ; Decidable; DecidableEquality; IsEquivalence; IsDecEquivalence

+
        ; Setoid; DecSetoid; StrictTotalOrder)

+
import Relation.Binary.Construct.On as On

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; cong; sym; trans; subst)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid; decSetoid)

+

+
------------------------------------------------------------------------

+
-- Primitive properties

+

+
open import Agda.Builtin.Word.Properties

+
  renaming (primWord64ToNatInjective to toℕ-injective)

+
  public

+

+
------------------------------------------------------------------------

+
-- Properties of _≈_

+

+
≈⇒≡ : _≈_ ⇒ _≡_

+
≈⇒≡ = toℕ-injective _ _

+

+
≈-reflexive : _≡_ ⇒ _≈_

+
≈-reflexive = cong toℕ

+

+
≈-refl : Reflexive _≈_

+
≈-refl = refl

+

+
≈-sym : Symmetric _≈_

+
≈-sym = sym

+

+
≈-trans : Transitive _≈_

+
≈-trans = trans

+

+
≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ

+
≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p

+

+
infix 4 _≈?_

+
_≈?_ : Decidable _≈_

+
x ≈? y = toℕ x ℕ.≟ toℕ y

+

+
≈-isEquivalence : IsEquivalence _≈_

+
≈-isEquivalence = record

+
  { refl  = λ {i} → ≈-refl {i}

+
  ; sym   = λ {i j} → ≈-sym {i} {j}

+
  ; trans = λ {i j k} → ≈-trans {i} {j} {k}

+
  }

+

+
≈-setoid : Setoid _ _

+
≈-setoid = record

+
  { isEquivalence = ≈-isEquivalence

+
  }

+

+
≈-isDecEquivalence : IsDecEquivalence _≈_

+
≈-isDecEquivalence = record

+
  { isEquivalence = ≈-isEquivalence

+
  ; _≟_           = _≈?_

+
  }

+

+
≈-decSetoid : DecSetoid _ _

+
≈-decSetoid = record

+
  { isDecEquivalence = ≈-isDecEquivalence

+
  }

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+

+
infix 4 _≟_

+
_≟_ : DecidableEquality Word64

+
x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)

+

+
≡-setoid : Setoid _ _

+
≡-setoid = setoid Word64

+

+
≡-decSetoid : DecSetoid _ _

+
≡-decSetoid = decSetoid _≟_

+

+
------------------------------------------------------------------------

+
-- Boolean equality test.

+

+
infix 4 _==_

+
_==_ : Word64 → Word64 → Bool

+
w₁ == w₂ = ⌊ w₁ ≟ w₂ ⌋

+

+
------------------------------------------------------------------------

+
-- Properties of _<_

+

+
infix 4 _<?_

+
_<?_ : Decidable _<_

+
_<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_

+

+
<-strictTotalOrder-≈ : StrictTotalOrder _ _ _

+
<-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Data.Word64.html b/site/static/agda_html/Data.Word64.html
new file mode 100644
index 000000000..afbf92a40
--- /dev/null
+++ b/site/static/agda_html/Data.Word64.html
@@ -0,0 +1,92 @@
+
+Data.Word64
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Machine words

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Data.Word64 where

+

+
------------------------------------------------------------------------

+
-- Re-export base definitions and decidability of equality

+

+
open import Data.Word64.Base public

+
open import Data.Word64.Properties using (_≈?_; _<?_; _≟_; _==_) public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Effect.Applicative.html b/site/static/agda_html/Effect.Applicative.html
new file mode 100644
index 000000000..6fafdd48e
--- /dev/null
+++ b/site/static/agda_html/Effect.Applicative.html
@@ -0,0 +1,210 @@
+
+Effect.Applicative
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Applicative functors

+
------------------------------------------------------------------------

+

+
-- Note that currently the applicative functor laws are not included

+
-- here.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Effect.Applicative where

+

+
open import Data.Bool.Base using (Bool; true; false)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Data.Unit.Polymorphic.Base using (⊤)

+

+
open import Effect.Choice using (RawChoice)

+
open import Effect.Empty using (RawEmpty)

+
open import Effect.Functor as Fun using (RawFunctor)

+

+
open import Function.Base using (const; flip; _∘′_)

+
open import Level using (Level; suc; _⊔_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
private

+
  variable

+
    f g : Level

+
    A B C : Set f

+
------------------------------------------------------------------------

+
-- The type of raw applicatives

+

+
record RawApplicative (F : Set f → Set g) : Set (suc f ⊔ g) where

+
  infixl 4 _<*>_ _<*_ _*>_

+
  infixl 4 _⊛_ _<⊛_ _⊛>_

+
  infix  4 _⊗_

+
  field

+
    rawFunctor : RawFunctor F

+
    pure : A → F A

+
    _<*>_ : F (A → B) → F A → F B

+

+
  open RawFunctor rawFunctor public

+

+
  _<*_ : F A → F B → F A

+
  a <* b = const <$> a <*> b

+

+
  _*>_ : F A → F B → F B

+
  a *> b = flip const <$> a <*> b

+

+
  zipWith : (A → B → C) → F A → F B → F C

+
  zipWith f x y = f <$> x <*> y

+

+
  zip : F A → F B → F (A × B)

+
  zip = zipWith _,_

+

+
  -- Haskell-style alternative name for pure

+
  return : A → F A

+
  return = pure

+

+
  -- backwards compatibility: unicode variants

+
  _⊛_ : F (A → B) → F A → F B

+
  _⊛_ = _<*>_

+

+
  _<⊛_ : F A → F B → F A

+
  _<⊛_ = _<*_

+

+
  _⊛>_ : F A → F B → F B

+
  _⊛>_ = _*>_

+

+
  _⊗_ : F A → F B → F (A × B)

+
  _⊗_ = zip

+

+
module _ where

+

+
  open RawApplicative

+
  open RawFunctor

+

+
  -- Smart constructor

+
  mkRawApplicative :

+
    (F : Set f → Set g) →

+
    (pure : ∀ {A} → A → F A) →

+
    (app : ∀ {A B} → F (A → B) → F A → F B) →

+
    RawApplicative F

+
  mkRawApplicative F pure app .rawFunctor ._<$>_ = app ∘′ pure

+
  mkRawApplicative F pure app .pure = pure

+
  mkRawApplicative F pure app ._<*>_ = app

+

+
------------------------------------------------------------------------

+
-- The type of raw applicatives with a zero

+

+
record RawApplicativeZero (F : Set f → Set g) : Set (suc f ⊔ g) where

+
  field

+
    rawApplicative : RawApplicative F

+
    rawEmpty : RawEmpty F

+

+
  open RawApplicative rawApplicative public

+
  open RawEmpty rawEmpty public

+

+
  guard : Bool → F ⊤

+
  guard true = pure _

+
  guard false = empty

+

+
------------------------------------------------------------------------

+
-- The type of raw alternative applicatives

+

+
record RawAlternative (F : Set f → Set g) : Set (suc f ⊔ g) where

+
  field

+
    rawApplicativeZero : RawApplicativeZero F

+
    rawChoice : RawChoice F

+

+
  open RawApplicativeZero rawApplicativeZero public

+
  open RawChoice rawChoice public

+

+
------------------------------------------------------------------------

+
-- The type of applicative morphisms

+

+
record Morphism {F₁ F₂ : Set f → Set g}

+
                (A₁ : RawApplicative F₁)

+
                (A₂ : RawApplicative F₂) : Set (suc f ⊔ g) where

+
  module A₁ = RawApplicative A₁

+
  module A₂ = RawApplicative A₂

+
  field

+
    functorMorphism : Fun.Morphism A₁.rawFunctor A₂.rawFunctor

+

+
  open Fun.Morphism functorMorphism public

+

+
  field

+
    op-pure : (x : A) → op (A₁.pure x) ≡ A₂.pure x

+
    op-<*>  : (f : F₁ (A → B)) (x : F₁ A) →

+
              op (f A₁.⊛ x) ≡ (op f A₂.⊛ op x)

+

+
  -- backwards compatibility: unicode variants

+
  op-⊛ = op-<*>

+

\ No newline at end of file
diff --git a/site/static/agda_html/Effect.Choice.html b/site/static/agda_html/Effect.Choice.html
new file mode 100644
index 000000000..61ff48cea
--- /dev/null
+++ b/site/static/agda_html/Effect.Choice.html
@@ -0,0 +1,103 @@
+
+Effect.Choice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Type constructors giving rise to a semigroup at every type

+
-- e.g. (List, _++_)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Effect.Choice where

+

+
open import Level

+

+
private

+
  variable

+
    ℓ ℓ′ : Level

+
    A  : Set ℓ

+

+
record RawChoice (F : Set ℓ → Set ℓ′) : Set (suc ℓ ⊔ ℓ′) where

+
  infixr 3 _<|>_ _∣_

+
  field

+
    _<|>_ : F A → F A → F A

+

+
  -- backwards compatibility: unicode variants

+
  _∣_ : F A → F A → F A

+
  _∣_ = _<|>_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Effect.Comonad.html b/site/static/agda_html/Effect.Comonad.html
new file mode 100644
index 000000000..b88875b01
--- /dev/null
+++ b/site/static/agda_html/Effect.Comonad.html
@@ -0,0 +1,125 @@
+
+Effect.Comonad
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Comonads

+
------------------------------------------------------------------------

+

+
-- Note that currently the monad laws are not included here.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Effect.Comonad where

+

+
open import Level

+
open import Function.Base using (id; _∘′_; flip)

+

+
private

+
  variable

+
    a b c f : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
record RawComonad (W : Set f → Set f) : Set (suc f) where

+

+
  infixl 1 _=>>_ _=>=_

+
  infixr 1 _<<=_ _=<=_

+

+
  field

+
    extract : W A → A

+
    extend  : (W A → B) → (W A → W B)

+

+
  duplicate : W A → W (W A)

+
  duplicate = extend id

+

+
  liftW : (A → B) → W A → W B

+
  liftW f = extend (f ∘′ extract)

+

+
  _=>>_ : W A → (W A → B) → W B

+
  _=>>_ = flip extend

+

+
  _=>=_ : (W A → B) → (W B → C) → W A → C

+
  f =>= g = g ∘′ extend f

+

+
  _<<=_ : (W A → B) → W A → W B

+
  _<<=_ = extend

+

+
  _=<=_ : (W B → C) → (W A → B) → W A → C

+
  _=<=_ = flip _=>=_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Effect.Empty.html b/site/static/agda_html/Effect.Empty.html
new file mode 100644
index 000000000..16fede096
--- /dev/null
+++ b/site/static/agda_html/Effect.Empty.html
@@ -0,0 +1,101 @@
+
+Effect.Empty
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Empty values (e.g. [] for List, nothing for Maybe)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Effect.Empty where

+

+
open import Level

+

+
private

+
  variable

+
    ℓ ℓ′ : Level

+
    A  : Set ℓ

+

+
record RawEmpty (F : Set ℓ → Set ℓ′) : Set (suc ℓ ⊔ ℓ′) where

+
  field

+
    empty : F A

+

+
  -- backwards compatibility: unicode variants

+
  ∅ : F A

+
  ∅ = empty

+

\ No newline at end of file
diff --git a/site/static/agda_html/Effect.Functor.html b/site/static/agda_html/Effect.Functor.html
new file mode 100644
index 000000000..c8a9598c1
--- /dev/null
+++ b/site/static/agda_html/Effect.Functor.html
@@ -0,0 +1,126 @@
+
+Effect.Functor
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Functors

+
------------------------------------------------------------------------

+

+
-- Note that currently the functor laws are not included here.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Effect.Functor where

+

+
open import Data.Unit.Polymorphic.Base using (⊤)

+
open import Function.Base using (const; flip)

+
open import Level

+

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
private

+
  variable

+
    ℓ ℓ′ ℓ″ : Level

+
    A B X Y : Set ℓ

+

+
record RawFunctor (F : Set ℓ → Set ℓ′) : Set (suc ℓ ⊔ ℓ′) where

+
  infixl 4 _<$>_ _<$_

+
  infixl 1 _<&>_

+
  field

+
    _<$>_ : (A → B) → F A → F B

+

+
  _<$_ : A → F B → F A

+
  x <$ y = const x <$> y

+

+
  _<&>_ : F A → (A → B) → F B

+
  _<&>_ = flip _<$>_

+

+
  ignore : F A → F ⊤

+
  ignore = _ <$_

+

+
-- A functor morphism from F₁ to F₂ is an operation op such that

+
-- op (F₁ f x) ≡ F₂ f (op x)

+

+
record Morphism {F₁ : Set ℓ → Set ℓ′} {F₂ : Set ℓ → Set ℓ″}

+
                (fun₁ : RawFunctor F₁)

+
                (fun₂ : RawFunctor F₂) : Set (suc ℓ ⊔ ℓ′ ⊔ ℓ″) where

+
  open RawFunctor

+
  field

+
    op     : F₁ X → F₂ X

+
    op-<$> : (f : X → Y) (x : F₁ X) →

+
             op (fun₁ ._<$>_ f x) ≡ fun₂ ._<$>_ f (op x)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Effect.Monad.html b/site/static/agda_html/Effect.Monad.html
new file mode 100644
index 000000000..75d6f2d66
--- /dev/null
+++ b/site/static/agda_html/Effect.Monad.html
@@ -0,0 +1,211 @@
+
+Effect.Monad
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Monads

+
------------------------------------------------------------------------

+

+
-- Note that currently the monad laws are not included here.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Effect.Monad where

+

+
open import Data.Bool.Base using (Bool; true; false; not)

+
open import Data.Unit.Polymorphic.Base using (⊤)

+

+
open import Effect.Choice

+
open import Effect.Empty

+
open import Effect.Applicative

+
open import Function.Base using (id; flip; _$′_; _∘′_)

+
open import Level using (Level; suc; _⊔_)

+

+
private

+
  variable

+
    f g g₁ g₂ : Level

+
    A B C : Set f

+

+
------------------------------------------------------------------------

+
-- The type of raw monads

+

+
record RawMonad (F : Set f → Set g) : Set (suc f ⊔ g) where

+
  infixl 1 _>>=_ _>>_ _>=>_

+
  infixr 1 _=<<_ _<=<_

+
  field

+
    rawApplicative : RawApplicative F

+
    _>>=_ : F A → (A → F B) → F B

+

+
  open RawApplicative rawApplicative public

+

+
  _>>_ : F A → F B → F B

+
  _>>_ = _*>_

+

+
  _=<<_ : (A → F B) → F A → F B

+
  _=<<_ = flip _>>=_

+

+
  Kleisli : Set f → Set f → Set (f ⊔ g)

+
  Kleisli A B = A → F B

+

+
  _>=>_ : Kleisli A B → Kleisli B C → Kleisli A C

+
  (f >=> g) a = f a >>= g

+

+
  _<=<_ : Kleisli B C → Kleisli A B → Kleisli A C

+
  _<=<_ = flip _>=>_

+

+
  when : Bool → F ⊤ → F ⊤

+
  when true m = m

+
  when false m = pure _

+

+
  unless : Bool → F ⊤ → F ⊤

+
  unless = when ∘′ not

+

+
-- When level g=f, a join/μ operator is definable

+

+
module Join {F : Set f → Set f} (M : RawMonad F) where

+
  open RawMonad M

+

+
  join : F (F A) → F A

+
  join = _>>= id

+

+
-- Smart constructor

+

+
module _ where

+

+
  open RawMonad

+
  open RawApplicative

+

+
  mkRawMonad :

+
    (F : Set f → Set g) →

+
    (pure : ∀ {A} → A → F A) →

+
    (bind : ∀ {A B} → F A → (A → F B) → F B) →

+
    RawMonad F

+
  mkRawMonad F pure _>>=_ .rawApplicative =

+
    mkRawApplicative _ pure $′ λ mf mx → do

+
      f ← mf

+
      x ← mx

+
      pure (f x)

+
  mkRawMonad F pure _>>=_ ._>>=_ = _>>=_

+

+
------------------------------------------------------------------------

+
-- The type of raw monads with a zero

+

+
record RawMonadZero (F : Set f → Set g) : Set (suc f ⊔ g) where

+
  field

+
    rawMonad : RawMonad F

+
    rawEmpty : RawEmpty F

+

+
  open RawMonad rawMonad public

+
  open RawEmpty rawEmpty public

+

+
  rawApplicativeZero : RawApplicativeZero F

+
  rawApplicativeZero = record

+
    { rawApplicative = rawApplicative

+
    ; rawEmpty = rawEmpty

+
    }

+

+
------------------------------------------------------------------------

+
-- The type of raw monadplus

+

+
record RawMonadPlus (F : Set f → Set g) : Set (suc f ⊔ g) where

+
  field

+
    rawMonadZero : RawMonadZero F

+
    rawChoice    : RawChoice F

+

+
  open RawMonadZero rawMonadZero public

+
  open RawChoice rawChoice public

+

+
  rawAlternative : RawAlternative F

+
  rawAlternative = record

+
    { rawApplicativeZero = rawApplicativeZero

+
    ; rawChoice = rawChoice

+
    }

+

+
------------------------------------------------------------------------

+
-- The type of raw monad transformer

+

+
-- F has been RawMonadT'd as TF

+
record RawMonadTd (F : Set f → Set g₁) (TF : Set f → Set g₂) : Set (suc f ⊔ g₁ ⊔ g₂) where

+
  field

+
    lift : F A → TF A

+
    rawMonad : RawMonad TF

+

+
  open RawMonad rawMonad public

+

+
RawMonadT : (T : (Set f → Set g₁) → (Set f → Set g₂)) → Set (suc f ⊔ suc g₁ ⊔ g₂)

+
RawMonadT T = ∀ {M} → RawMonad M → RawMonadTd M (T M)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Everything.html b/site/static/agda_html/Everything.html
new file mode 100644
index 000000000..1ac7054b0
--- /dev/null
+++ b/site/static/agda_html/Everything.html
@@ -0,0 +1,108 @@
+
+Everything
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Everything where

+

+
open import CategoricalCrypto

+
open import Class.Computational22

+
open import Leios.Abstract

+
open import Leios.Base

+
open import Leios.Blocks

+
open import Leios.Config

+
open import Leios.Defaults

+
open import Leios.FFD

+
open import Leios.Foreign.BaseTypes

+
open import Leios.Foreign.HsTypes

+
open import Leios.Foreign.Types

+
open import Leios.Foreign.Util

+
open import Leios.KeyRegistration

+
open import Leios.Network

+
open import Leios.Prelude

+
open import Leios.Protocol

+
open import Leios.Short

+
open import Leios.Short.Decidable

+
-- open import Leios.Short.Deterministic

+
open import Leios.Short.Trace.Verifier

+
open import Leios.Short.Trace.Verifier.Test

+
open import Leios.Simplified

+
open import Leios.Simplified.Deterministic

+
-- open import Leios.Simplified.Deterministic.Test

+
open import Leios.SpecStructure

+
open import Leios.Traces

+
open import Leios.Voting

+
open import Leios.VRF

+
open import StateMachine

+

\ No newline at end of file
diff --git a/site/static/agda_html/Foreign.Haskell.Coerce.html b/site/static/agda_html/Foreign.Haskell.Coerce.html
new file mode 100644
index 000000000..f7c00b5b3
--- /dev/null
+++ b/site/static/agda_html/Foreign.Haskell.Coerce.html
@@ -0,0 +1,248 @@
+
+Foreign.Haskell.Coerce
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Zero-cost coercion to cross the FFI boundary

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible #-}

+

+
module Foreign.Haskell.Coerce where

+

+
------------------------------------------------------------------------

+
-- Motivation

+

+
-- Problem: No COMPILE directives for common inductive types

+

+
-- In order to guarantee that the vast majority of the libary is

+
-- considered safe by Agda, we had to remove the COMPILE pragmas

+
-- associated to common inductive types.

+
-- These directives cannot possibly be checked by the compiler and

+
-- the user could define unsound mappings (e.g. swapping Bool's

+
-- true and false).

+

+
-- Solution: Essentially identical definitions + zero-cost coercions

+

+
-- To solve this problem, we have introduced a number of essentially

+
-- identical definitions in the Foreign.Haskell.* modules. However

+
-- converting back and forth between the FFI-friendly type and its

+
-- safe counterpart would take linear time.

+
-- This module defines zero cost coercions between these types.

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
open import Level using (Level; _⊔_)

+
open import Agda.Builtin.Nat

+
open import Agda.Builtin.Int

+

+
import IO.Primitive.Core       as STD

+
import Data.List.Base          as STD

+
import Data.List.NonEmpty.Base as STD

+
import Data.Maybe.Base         as STD

+
import Data.Product.Base       as STD

+
import Data.Sum.Base           as STD

+

+
import Foreign.Haskell.Pair          as FFI

+
import Foreign.Haskell.Either        as FFI

+
import Foreign.Haskell.List.NonEmpty as FFI

+

+
private

+
  variable

+
    a b c d e f : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+

+
-- We define a simple indexed datatype `Coercible`. A value of

+
-- `Coercible A B` is a proof that `A` and `B` have the same underlying

+
-- runtime representation. The only possible proof is an incantation

+
-- from the implementer begging to be trusted.

+

+
-- We need this type to be concrete so that overlapping instances can be

+
-- checked for equality: we do not care what proof we get as long as we

+
-- get one.

+

+
-- We need for it to be a data type rather than a record type so that

+
-- Agda does not mistakenly build arbitrary instances by η-expansion.

+

+
data Coercible (A : Set a) (B : Set b) : Set where

+
  TrustMe : Coercible A B

+

+
{-# FOREIGN GHC data AgdaCoercible l1 l2 a b = TrustMe #-}

+
{-# COMPILE GHC Coercible = data AgdaCoercible (TrustMe) #-}

+

+
-- Once we get our hands on a proof that `Coercible A B` we postulate

+
-- that it is safe to convert an `A` into a `B`. This is done under the

+
-- hood by using `unsafeCoerce`.

+

+
postulate coerce : {{_ : Coercible A B}} → A → B

+

+
{-# FOREIGN GHC import Unsafe.Coerce #-}

+
{-# COMPILE GHC coerce = \ _ _ _ _ _ -> unsafeCoerce #-}

+

+
------------------------------------------------------------------------

+
-- Unary and binary variants for covariant type constructors

+

+
Coercible₁ : ∀ a c → (T : Set a → Set b) (U : Set c → Set d) → Set _

+
Coercible₁ _ _ T U = ∀ {A B} → {{_ : Coercible A B}} → Coercible (T A) (U B)

+

+
Coercible₂ : ∀ a b d e → (T : Set a → Set b → Set c) (U : Set d → Set e → Set f) → Set _

+
Coercible₂ _ _ _ _ T U = ∀ {A B} → {{_ : Coercible A B}} → Coercible₁ _ _ (T A) (U B)

+

+
------------------------------------------------------------------------

+
-- Instances

+

+
-- Nat

+

+
-- Our first instance reveals one of Agda's secrets: natural numbers are

+
-- represented by (arbitrary precision) integers at runtime! Note that

+
-- we may only coerce in one direction: arbitrary integers may actually

+
-- be negative and will not do as mere natural numbers.

+

+
instance

+

+
  nat-toInt : Coercible Nat Int

+
  nat-toInt = TrustMe

+

+
-- We then proceed to state that data types from the standard library

+
-- can be converted to their FFI equivalents which are bound to actual

+
-- Haskell types.

+

+
-- Product

+

+
  pair-toFFI : Coercible₂ a b c d STD._×_ FFI.Pair

+
  pair-toFFI = TrustMe

+

+
  pair-fromFFI : Coercible₂ a b c d FFI.Pair STD._×_

+
  pair-fromFFI = TrustMe

+

+
-- Sum

+

+
  either-toFFI : Coercible₂ a b c d STD._⊎_ FFI.Either

+
  either-toFFI = TrustMe

+

+
  either-fromFFI : Coercible₂ a b c d FFI.Either STD._⊎_

+
  either-fromFFI = TrustMe

+

+
-- NonEmpty

+

+
  nonEmpty-toFFI : Coercible₁ a b STD.List⁺ FFI.NonEmpty

+
  nonEmpty-toFFI = TrustMe

+

+
  nonEmpty-fromFFI : Coercible₁ a b FFI.NonEmpty STD.List⁺

+
  nonEmpty-fromFFI = TrustMe

+

+
-- We follow up with purely structural rules for builtin data types

+
-- which already have known low-level representations.

+

+
-- Maybe

+

+
  coerce-maybe : Coercible₁ a b STD.Maybe STD.Maybe

+
  coerce-maybe = TrustMe

+

+
-- List

+

+
  coerce-list : Coercible₁ a b STD.List STD.List

+
  coerce-list = TrustMe

+

+
-- IO

+

+
  coerce-IO : Coercible₁ a b STD.IO STD.IO

+
  coerce-IO = TrustMe

+

+
-- Function

+
-- Note that functions are contravariant in their domain.

+

+
  coerce-fun : {{_ : Coercible A B}} → Coercible₁ c d (λ C → B → C) (λ D → A → D)

+
  coerce-fun = TrustMe

+

+
-- Finally we add a reflexivity proof to discharge all the dangling

+
-- constraints involving type variables and concrete builtin types such

+
-- as `Bool`.

+

+
-- This rule overlaps with the purely structural ones: when attempting

+
-- to prove `Coercible (List A) (List A)`, should Agda use the proof

+
-- obtained by `coerce-refl` or the one obtained by `coerce-list coerce-refl`?

+
-- Because we are using a datatype with a single constructor these

+
-- distinctions do not matter: both proofs are definitionally equal.

+

+
  coerce-refl : Coercible A A

+
  coerce-refl = TrustMe

+

\ No newline at end of file
diff --git a/site/static/agda_html/Foreign.Haskell.Either.html b/site/static/agda_html/Foreign.Haskell.Either.html
new file mode 100644
index 000000000..fd805e462
--- /dev/null
+++ b/site/static/agda_html/Foreign.Haskell.Either.html
@@ -0,0 +1,124 @@
+
+Foreign.Haskell.Either
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The Either type which calls out to Haskell via the FFI

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible #-}

+

+
module Foreign.Haskell.Either where

+

+
open import Level

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data Either (A : Set a) (B : Set b) : Set (a ⊔ b) where

+
  left  : A → Either A B

+
  right : B → Either A B

+

+
{-# FOREIGN GHC type AgdaEither l1 l2 a b = Either a b #-}

+
{-# COMPILE GHC Either = data MAlonzo.Code.Foreign.Haskell.Either.AgdaEither (Left | Right) #-}

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
toForeign : A ⊎ B → Either A B

+
toForeign (inj₁ a) = left a

+
toForeign (inj₂ b) = right b

+
{-# WARNING_ON_USAGE toForeign

+
"Warning: toForeign was deprecated in 2.0.

+
Please use Foreign.Haskell.Coerce.coerce instead."

+
#-}

+

+
fromForeign : Either A B → A ⊎ B

+
fromForeign (left a)  = inj₁ a

+
fromForeign (right b) = inj₂ b

+
{-# WARNING_ON_USAGE fromForeign

+
"Warning: fromForeign was deprecated in 2.0.

+
Please use Foreign.Haskell.Coerce.coerce instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Foreign.Haskell.List.NonEmpty.html b/site/static/agda_html/Foreign.Haskell.List.NonEmpty.html
new file mode 100644
index 000000000..06f9dc6ad
--- /dev/null
+++ b/site/static/agda_html/Foreign.Haskell.List.NonEmpty.html
@@ -0,0 +1,108 @@
+
+Foreign.Haskell.List.NonEmpty
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The NonEmpty type which calls out to Haskell via the FFI

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible #-}

+

+
module Foreign.Haskell.List.NonEmpty where

+

+
open import Level

+
open import Data.List.NonEmpty.Base as Data using (_∷_)

+
open import Data.List using (List)

+

+
{-# FOREIGN GHC import qualified Data.List.NonEmpty as NE #-}

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data NonEmpty (A : Set a) : Set a where

+
  _∷_ : A → List A → NonEmpty A

+

+
infixr 5 _∷_

+

+
{-# FOREIGN GHC type AgdaNonEmpty l a = NE.NonEmpty a #-}

+
{-# COMPILE GHC NonEmpty = data AgdaNonEmpty ((NE.:|)) #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Foreign.Haskell.Pair.html b/site/static/agda_html/Foreign.Haskell.Pair.html
new file mode 100644
index 000000000..4099809d8
--- /dev/null
+++ b/site/static/agda_html/Foreign.Haskell.Pair.html
@@ -0,0 +1,125 @@
+
+Foreign.Haskell.Pair
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The Pair type which calls out to Haskell via the FFI

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible #-}

+

+
module Foreign.Haskell.Pair where

+

+
open import Level

+
open import Data.Product.Base using (_×_; _,_)

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
record Pair (A : Set a) (B : Set b) : Set (a ⊔ b) where

+
  constructor _,_

+
  field  fst : A

+
         snd : B

+
infixr 4 _,_

+
open Pair public

+

+
{-# FOREIGN GHC type AgdaPair l1 l2 a b = (a , b) #-}

+
{-# COMPILE GHC Pair = data MAlonzo.Code.Foreign.Haskell.Pair.AgdaPair ((,)) #-}

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
toForeign : A × B → Pair A B

+
toForeign (a , b) = (a , b)

+
{-# WARNING_ON_USAGE toForeign

+
"Warning: toForeign was deprecated in 2.0.

+
Please use Foreign.Haskell.Coerce.coerce instead."

+
#-}

+

+
fromForeign : Pair A B → A × B

+
fromForeign (a , b) = (a , b)

+
{-# WARNING_ON_USAGE fromForeign

+
"Warning: fromForeign was deprecated in 2.0.

+
Please use Foreign.Haskell.Coerce.coerce instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Foreign.Haskell.html b/site/static/agda_html/Foreign.Haskell.html
new file mode 100644
index 000000000..d4a0d7a63
--- /dev/null
+++ b/site/static/agda_html/Foreign.Haskell.html
@@ -0,0 +1,106 @@
+
+Foreign.Haskell
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Type(s) used (only) when calling out to Haskell via the FFI

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible #-}

+

+
module Foreign.Haskell where

+

+
open import Level

+

+
------------------------------------------------------------------------

+
-- Pairs

+

+
open import Foreign.Haskell.Pair public

+
  renaming

+
  ( toForeign   to toForeignPair

+
  ; fromForeign to fromForeignPair

+
  )

+

+
------------------------------------------------------------------------

+
-- Sums

+

+
open import Foreign.Haskell.Either public

+
  renaming

+
  ( toForeign   to toForeignEither

+
  ; fromForeign to fromForeignEither

+
  )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Base.html b/site/static/agda_html/Function.Base.html
new file mode 100644
index 000000000..f0fc323c3
--- /dev/null
+++ b/site/static/agda_html/Function.Base.html
@@ -0,0 +1,347 @@
+
+Function.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Simple combinators working solely on and with functions

+
------------------------------------------------------------------------

+

+
-- The contents of this module is also accessible via the `Function`

+
-- module. See `Function.Strict` for strict versions of these

+
-- combinators.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Base where

+

+
open import Level using (Level)

+

+
private

+
  variable

+
    a b c d e : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+
    D : Set d

+
    E : Set e

+

+
------------------------------------------------------------------------

+
-- Some simple functions

+

+
id : A → A

+
id x = x

+

+
const : A → B → A

+
const x = λ _ → x

+

+
constᵣ : A → B → B

+
constᵣ _ = id

+

+
------------------------------------------------------------------------

+
-- Operations on dependent functions

+

+
-- These are functions whose output has a type that depends on the

+
-- value of the input to the function.

+

+
infixr 9 _∘_ _∘₂_

+
infixl 8 _ˢ_

+
infixl 0 _|>_

+
infix  0 case_return_of_

+
infixr -1 _$_

+

+
-- Composition

+

+
_∘_ : ∀ {A : Set a} {B : A → Set b} {C : {x : A} → B x → Set c} →

+
      (∀ {x} (y : B x) → C y) → (g : (x : A) → B x) →

+
      ((x : A) → C (g x))

+
f ∘ g = λ x → f (g x)

+
{-# INLINE _∘_ #-}

+

+
_∘₂_ : ∀ {A₁ : Set a} {A₂ : A₁ → Set d}

+
         {B : (x : A₁) → A₂ x → Set b}

+
         {C : {x : A₁} → {y : A₂ x} → B x y → Set c} →

+
       ({x : A₁} → {y : A₂ x} → (z : B x y) → C z) →

+
       (g : (x : A₁) → (y : A₂ x) → B x y) →

+
       ((x : A₁) → (y : A₂ x) → C (g x y))

+
f ∘₂ g = λ x y → f (g x y)

+

+
-- Flipping order of arguments

+

+
flip : ∀ {A : Set a} {B : Set b} {C : A → B → Set c} →

+
       ((x : A) (y : B) → C x y) → ((y : B) (x : A) → C x y)

+
flip f = λ y x → f x y

+
{-# INLINE flip #-}

+

+
-- Application - note that _$_ is right associative, as in Haskell.

+
-- If you want a left associative infix application operator, use

+
-- RawFunctor._<$>_ from Effect.Functor.

+

+
_$_ : ∀ {A : Set a} {B : A → Set b} →

+
      ((x : A) → B x) → ((x : A) → B x)

+
f $ x = f x

+
{-# INLINE _$_ #-}

+

+
-- Flipped application (aka pipe-forward)

+

+
_|>_ : ∀ {A : Set a} {B : A → Set b} →

+
       (a : A) → (∀ a → B a) → B a

+
_|>_ = flip _$_

+
{-# INLINE _|>_ #-}

+

+
-- The S combinator - written infix as in Conor McBride's paper

+
-- "Outrageous but Meaningful Coincidences: Dependent type-safe syntax

+
-- and evaluation".

+

+
_ˢ_ : ∀ {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →

+
      ((x : A) (y : B x) → C x y) →

+
      (g : (x : A) → B x) →

+
      ((x : A) → C x (g x))

+
f ˢ g = λ x → f x (g x)

+
{-# INLINE _ˢ_ #-}

+

+
-- Converting between implicit and explicit function spaces.

+

+
_$- : ∀ {A : Set a} {B : A → Set b} → ((x : A) → B x) → ({x : A} → B x)

+
f $- = f _

+
{-# INLINE _$- #-}

+

+
λ- : ∀ {A : Set a} {B : A → Set b} → ({x : A} → B x) → ((x : A) → B x)

+
λ- f = λ x → f

+
{-# INLINE λ- #-}

+

+
-- Case expressions (to be used with pattern-matching lambdas, see

+
-- README.Case).

+

+
case_returning_of_ : ∀ {A : Set a} (x : A) (B : A → Set b) →

+
                  ((x : A) → B x) → B x

+
case x returning B of f = f x

+
{-# INLINE case_returning_of_ #-}

+

+
------------------------------------------------------------------------

+
-- Non-dependent versions of dependent operations

+

+
-- Any of the above operations for dependent functions will also work

+
-- for non-dependent functions but sometimes Agda has difficulty

+
-- inferring the non-dependency. Primed (′ = \prime) versions of the

+
-- operations are therefore provided below that sometimes have better

+
-- inference properties.

+

+
infixr 9 _∘′_ _∘₂′_

+
infixl 0 _|>′_

+
infix  0 case_of_

+
infixr -1 _$′_

+

+
-- Composition

+

+
_∘′_ : (B → C) → (A → B) → (A → C)

+
f ∘′ g = _∘_ f g

+

+
_∘₂′_ : (C → D) → (A → B → C) → (A → B → D)

+
f ∘₂′ g = _∘₂_ f g

+

+
-- Flipping order of arguments

+

+
flip′ : (A → B → C) → (B → A → C)

+
flip′ = flip

+

+
-- Application

+

+
_$′_ : (A → B) → (A → B)

+
_$′_ = _$_

+

+
-- Flipped application (aka pipe-forward)

+

+
_|>′_ : A → (A → B) → B

+
_|>′_ = _|>_

+

+
-- Case expressions (to be used with pattern-matching lambdas, see

+
-- README.Case).

+

+
case_of_ : A → (A → B) → B

+
case x of f = case x returning _ of f

+
{-# INLINE case_of_ #-}

+

+
------------------------------------------------------------------------

+
-- Operations that are only defined for non-dependent functions

+

+
infixl 1 _⟨_⟩_

+
infixl 0 _∋_

+

+
-- Binary application

+

+
_⟨_⟩_ : A → (A → B → C) → B → C

+
x ⟨ f ⟩ y = f x y

+

+
-- In Agda you cannot annotate every subexpression with a type

+
-- signature. This function can be used instead.

+

+
_∋_ : (A : Set a) → A → A

+
A ∋ x = x

+

+
-- Conversely it is sometimes useful to be able to extract the

+
-- type of a given expression.

+

+
typeOf : {A : Set a} → A → Set a

+
typeOf {A = A} _ = A

+

+
-- Construct an element of the given type by instance search.

+

+
it : {A : Set a} → {{A}} → A

+
it {{x}} = x

+

+
------------------------------------------------------------------------

+
-- Composition of a binary function with other functions

+

+
infixr 0 _-⟪_⟫-_ _-⟨_⟫-_

+
infixl 0 _-⟪_⟩-_

+
infixr 1 _-⟨_⟩-_ ∣_⟫-_ ∣_⟩-_

+
infixl 1 _on_ _on₂_ _-⟪_∣ _-⟨_∣

+

+
-- Two binary functions

+

+
_-⟪_⟫-_ : (A → B → C) → (C → D → E) → (A → B → D) → (A → B → E)

+
f -⟪ _*_ ⟫- g = λ x y → f x y * g x y

+

+
-- A single binary function on the left

+

+
_-⟪_∣ : (A → B → C) → (C → B → D) → (A → B → D)

+
f -⟪ _*_ ∣ = f -⟪ _*_ ⟫- constᵣ

+

+
-- A single binary function on the right

+

+
∣_⟫-_ : (A → C → D) → (A → B → C) → (A → B → D)

+
∣ _*_ ⟫- g = const -⟪ _*_ ⟫- g

+

+
-- A single unary function on the left

+

+
_-⟨_∣ : (A → C) → (C → B → D) → (A → B → D)

+
f -⟨ _*_ ∣ = f ∘₂ const -⟪ _*_ ∣

+

+
-- A single unary function on the right

+

+
∣_⟩-_ : (A → C → D) → (B → C) → (A → B → D)

+
∣ _*_ ⟩- g = ∣ _*_ ⟫- g ∘₂ constᵣ

+

+
-- A binary function and a unary function

+

+
_-⟪_⟩-_ : (A → B → C) → (C → D → E) → (B → D) → (A → B → E)

+
f -⟪ _*_ ⟩- g = f -⟪ _*_ ⟫- ∣ constᵣ ⟩- g

+

+
-- A unary function and a binary function

+

+
_-⟨_⟫-_ : (A → C) → (C → D → E) → (A → B → D) → (A → B → E)

+
f -⟨ _*_ ⟫- g = f -⟨ const ∣ -⟪ _*_ ⟫- g

+

+
-- Two unary functions

+

+
_-⟨_⟩-_ : (A → C) → (C → D → E) → (B → D) → (A → B → E)

+
f -⟨ _*_ ⟩- g = f -⟨ const ∣ -⟪ _*_ ⟫- ∣ constᵣ ⟩- g

+

+
-- A single binary function on both sides

+

+
_on₂_ : (C → C → D) → (A → B → C) → (A → B → D)

+
_*_ on₂ f = f -⟪ _*_ ⟫- f

+

+
-- A single unary function on both sides

+

+
_on_ : (B → B → C) → (A → B) → (A → A → C)

+
_*_ on f = f -⟨ _*_ ⟩- f

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.4

+

+
_-[_]-_ = _-⟪_⟫-_

+
{-# WARNING_ON_USAGE _-[_]-_

+
"Warning: Function._-[_]-_ was deprecated in v1.4.

+
Please use _-⟪_⟫-_ instead."

+
#-}

+

+
-- Version 2.0

+

+
case_return_of_ = case_returning_of_

+
{-# WARNING_ON_USAGE case_return_of_

+
"case_return_of_ was deprecated in v2.0.

+
Please use case_returning_of_ instead."

+
#-}

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Bundles.html b/site/static/agda_html/Function.Bundles.html
new file mode 100644
index 000000000..5c917146b
--- /dev/null
+++ b/site/static/agda_html/Function.Bundles.html
@@ -0,0 +1,628 @@
+
+Function.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for types of functions

+
------------------------------------------------------------------------

+

+
-- The contents of this file should usually be accessed from `Function`.

+

+
-- Note that these bundles differ from those found elsewhere in other

+
-- library hierarchies as they take Setoids as parameters. This is

+
-- because a function is of no use without knowing what its domain and

+
-- codomain is, as well which equalities are being considered over them.

+
-- One consequence of this is that they are not built from the

+
-- definitions found in `Function.Structures` as is usually the case in

+
-- other library hierarchies, as this would duplicate the equality

+
-- axioms.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Bundles where

+

+
open import Function.Base using (_∘_)

+
open import Function.Definitions

+
import Function.Structures as FunctionStructures

+
open import Level using (Level; _⊔_; suc)

+
open import Data.Product.Base using (_,_; proj₁; proj₂)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Core using (_Preserves_⟶_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+
open import Function.Consequences.Propositional

+
open Setoid using (isEquivalence)

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+

+
------------------------------------------------------------------------

+
-- Setoid bundles

+
------------------------------------------------------------------------

+

+
module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where

+

+
  open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)

+
  open Setoid To   using () renaming (Carrier to B; _≈_ to _≈₂_)

+
  open FunctionStructures _≈₁_ _≈₂_

+

+
------------------------------------------------------------------------

+
-- Bundles with one element

+

+
  -- Called `Func` rather than `Function` in order to avoid clashing

+
  -- with the top-level module.

+
  record Func : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to   : A → B

+
      cong : Congruent _≈₁_ _≈₂_ to

+

+
    isCongruent : IsCongruent to

+
    isCongruent = record

+
      { cong           = cong

+
      ; isEquivalence₁ = isEquivalence From

+
      ; isEquivalence₂ = isEquivalence To

+
      }

+

+
    open IsCongruent isCongruent public

+
      using (module Eq₁; module Eq₂)

+

+

+
  record Injection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to          : A → B

+
      cong        : Congruent _≈₁_ _≈₂_ to

+
      injective   : Injective _≈₁_ _≈₂_ to

+

+
    function : Func

+
    function = record

+
      { to   = to

+
      ; cong = cong

+
      }

+

+
    open Func function public

+
      hiding (to; cong)

+

+
    isInjection : IsInjection to

+
    isInjection = record

+
      { isCongruent = isCongruent

+
      ; injective   = injective

+
      }

+

+

+
  record Surjection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to         : A → B

+
      cong       : Congruent _≈₁_ _≈₂_ to

+
      surjective : Surjective _≈₁_ _≈₂_ to

+

+
    function : Func

+
    function = record

+
      { to   = to

+
      ; cong = cong

+
      }

+

+
    open Func function public

+
      hiding (to; cong)

+

+
    isSurjection : IsSurjection to

+
    isSurjection = record

+
      { isCongruent = isCongruent

+
      ; surjective  = surjective

+
      }

+

+
    open IsSurjection isSurjection public

+
      using

+
      ( strictlySurjective

+
      )

+

+
    to⁻ : B → A

+
    to⁻ = proj₁ ∘ surjective

+

+
    to∘to⁻ : ∀ x → to (to⁻ x) ≈₂ x

+
    to∘to⁻ = proj₂ ∘ strictlySurjective

+

+

+
  record Bijection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to        : A → B

+
      cong      : Congruent _≈₁_ _≈₂_ to

+
      bijective : Bijective _≈₁_ _≈₂_ to

+

+
    injective : Injective _≈₁_ _≈₂_ to

+
    injective = proj₁ bijective

+

+
    surjective : Surjective _≈₁_ _≈₂_ to

+
    surjective = proj₂ bijective

+

+
    injection : Injection

+
    injection = record

+
      { cong      = cong

+
      ; injective = injective

+
      }

+

+
    surjection : Surjection

+
    surjection = record

+
      { cong       = cong

+
      ; surjective = surjective

+
      }

+

+
    open Injection  injection  public using (isInjection)

+
    open Surjection surjection public using (isSurjection; to⁻;  strictlySurjective)

+

+
    isBijection : IsBijection to

+
    isBijection = record

+
      { isInjection = isInjection

+
      ; surjective  = surjective

+
      }

+

+
    open IsBijection isBijection public using (module Eq₁; module Eq₂)

+

+

+
------------------------------------------------------------------------

+
-- Bundles with two elements

+

+
module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where

+

+
  open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)

+
  open Setoid To   using () renaming (Carrier to B; _≈_ to _≈₂_)

+
  open FunctionStructures _≈₁_ _≈₂_

+

+
  record Equivalence : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to        : A → B

+
      from      : B → A

+
      to-cong   : Congruent _≈₁_ _≈₂_ to

+
      from-cong : Congruent _≈₂_ _≈₁_ from

+

+
    toFunction : Func From To

+
    toFunction = record

+
      { to = to

+
      ; cong = to-cong

+
      }

+

+
    open Func toFunction public

+
      using (module Eq₁; module Eq₂)

+
      renaming (isCongruent to to-isCongruent)

+

+
    fromFunction : Func To From

+
    fromFunction = record

+
      { to = from

+
      ; cong = from-cong

+
      }

+

+
    open Func fromFunction public

+
      using ()

+
      renaming (isCongruent to from-isCongruent)

+

+

+
  record LeftInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to        : A → B

+
      from      : B → A

+
      to-cong   : Congruent _≈₁_ _≈₂_ to

+
      from-cong : Congruent _≈₂_ _≈₁_ from

+
      inverseˡ  : Inverseˡ _≈₁_ _≈₂_ to from

+

+
    isCongruent : IsCongruent to

+
    isCongruent = record

+
      { cong           = to-cong

+
      ; isEquivalence₁ = isEquivalence From

+
      ; isEquivalence₂ = isEquivalence To

+
      }

+

+
    isLeftInverse : IsLeftInverse to from

+
    isLeftInverse = record

+
      { isCongruent = isCongruent

+
      ; from-cong   = from-cong

+
      ; inverseˡ    = inverseˡ

+
      }

+

+
    open IsLeftInverse isLeftInverse public

+
      using (module Eq₁; module Eq₂; strictlyInverseˡ; isSurjection)

+

+
    equivalence : Equivalence

+
    equivalence = record

+
      { to-cong   = to-cong

+
      ; from-cong = from-cong

+
      }

+

+
    isSplitSurjection : IsSplitSurjection to

+
    isSplitSurjection = record

+
      { from = from

+
      ; isLeftInverse = isLeftInverse

+
      }

+

+
    surjection : Surjection From To

+
    surjection = record

+
      { to = to

+
      ; cong = to-cong

+
      ; surjective = λ y → from y , inverseˡ

+
      }

+

+

+

+
  record RightInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to        : A → B

+
      from      : B → A

+
      to-cong   : Congruent _≈₁_ _≈₂_ to

+
      from-cong : from Preserves _≈₂_ ⟶ _≈₁_

+
      inverseʳ  : Inverseʳ _≈₁_ _≈₂_ to from

+

+
    isCongruent : IsCongruent to

+
    isCongruent = record

+
      { cong           = to-cong

+
      ; isEquivalence₁ = isEquivalence From

+
      ; isEquivalence₂ = isEquivalence To

+
      }

+

+
    isRightInverse : IsRightInverse to from

+
    isRightInverse = record

+
      { isCongruent = isCongruent

+
      ; from-cong   = from-cong

+
      ; inverseʳ    = inverseʳ

+
      }

+

+
    open IsRightInverse isRightInverse public

+
      using (module Eq₁; module Eq₂; strictlyInverseʳ)

+

+
    equivalence : Equivalence

+
    equivalence = record

+
      { to-cong   = to-cong

+
      ; from-cong = from-cong

+
      }

+

+

+
  record Inverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to        : A → B

+
      from      : B → A

+
      to-cong   : Congruent _≈₁_ _≈₂_ to

+
      from-cong : Congruent _≈₂_ _≈₁_ from

+
      inverse   : Inverseᵇ _≈₁_ _≈₂_ to from

+

+
    inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from

+
    inverseˡ = proj₁ inverse

+

+
    inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from

+
    inverseʳ = proj₂ inverse

+

+
    leftInverse : LeftInverse

+
    leftInverse = record

+
      { to-cong   = to-cong

+
      ; from-cong = from-cong

+
      ; inverseˡ  = inverseˡ

+
      }

+

+
    rightInverse : RightInverse

+
    rightInverse = record

+
      { to-cong   = to-cong

+
      ; from-cong = from-cong

+
      ; inverseʳ  = inverseʳ

+
      }

+

+
    open LeftInverse leftInverse   public using (isLeftInverse; strictlyInverseˡ)

+
    open RightInverse rightInverse public using (isRightInverse; strictlyInverseʳ)

+

+
    isInverse : IsInverse to from

+
    isInverse = record

+
      { isLeftInverse = isLeftInverse

+
      ; inverseʳ      = inverseʳ

+
      }

+

+
    open IsInverse isInverse public using (module Eq₁; module Eq₂)

+

+

+
------------------------------------------------------------------------

+
-- Bundles with three elements

+

+
  record BiEquivalence : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to         : A → B

+
      from₁      : B → A

+
      from₂      : B → A

+
      to-cong    : Congruent _≈₁_ _≈₂_ to

+
      from₁-cong : Congruent _≈₂_ _≈₁_ from₁

+
      from₂-cong : Congruent _≈₂_ _≈₁_ from₂

+

+

+
  record BiInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to          : A → B

+
      from₁       : B → A

+
      from₂       : B → A

+
      to-cong     : Congruent _≈₁_ _≈₂_ to

+
      from₁-cong  : Congruent _≈₂_ _≈₁_ from₁

+
      from₂-cong  : Congruent _≈₂_ _≈₁_ from₂

+
      inverseˡ  : Inverseˡ _≈₁_ _≈₂_ to from₁

+
      inverseʳ  : Inverseʳ _≈₁_ _≈₂_ to from₂

+

+
    to-isCongruent : IsCongruent to

+
    to-isCongruent = record

+
      { cong           = to-cong

+
      ; isEquivalence₁ = isEquivalence From

+
      ; isEquivalence₂ = isEquivalence To

+
      }

+

+
    isBiInverse : IsBiInverse to from₁ from₂

+
    isBiInverse = record

+
      { to-isCongruent = to-isCongruent

+
      ; from₁-cong     = from₁-cong

+
      ; from₂-cong     = from₂-cong

+
      ; inverseˡ       = inverseˡ

+
      ; inverseʳ       = inverseʳ

+
      }

+

+
    biEquivalence : BiEquivalence

+
    biEquivalence = record

+
      { to-cong    = to-cong

+
      ; from₁-cong = from₁-cong

+
      ; from₂-cong = from₂-cong

+
      }

+

+
------------------------------------------------------------------------

+
-- Other

+

+
  -- A left inverse is also known as a “split surjection”.

+
  --

+
  -- As the name implies, a split surjection is a special kind of

+
  -- surjection where the witness generated in the domain in the

+
  -- function for elements `x₁` and `x₂` are equal if `x₁ ≈ x₂` .

+
  --

+
  -- The difference is the `from-cong` law --- generally, the section

+
  -- (called `Surjection.to⁻` or `SplitSurjection.from`) of a surjection

+
  -- need not respect equality, whereas it must in a split surjection.

+
  --

+
  -- The two notions coincide when the equivalence relation on `B` is

+
  -- propositional equality (because all functions respect propositional

+
  -- equality).

+
  --

+
  -- For further background on (split) surjections, one may consult any

+
  -- general mathematical references which work without the principle

+
  -- of choice. For example:

+
  --

+
  --   https://ncatlab.org/nlab/show/split+epimorphism.

+
  --

+
  -- The connection to set-theoretic notions with the same names is

+
  -- justified by the setoid type theory/homotopy type theory

+
  -- observation/definition that (∃x : A. P) = ∥ Σx : A. P ∥ --- i.e.,

+
  -- we can read set-theoretic ∃ as squashed/propositionally truncated Σ.

+
  --

+
  -- We see working with setoids as working in the MLTT model of a setoid

+
  -- type theory, in which ∥ X ∥ is interpreted as the setoid with carrier

+
  -- set X and the equivalence relation that relates all elements.

+
  -- All maps into ∥ X ∥ respect equality, so in the idiomatic definitions

+
  -- here, we drop the corresponding trivial `cong` field completely.

+

+
  SplitSurjection : Set _

+
  SplitSurjection = LeftInverse

+

+
  module SplitSurjection (splitSurjection : SplitSurjection) =

+
    LeftInverse splitSurjection

+

+
------------------------------------------------------------------------

+
-- Infix abbreviations for oft-used items

+
------------------------------------------------------------------------

+

+
-- Same naming convention as used for propositional equality below, with

+
-- appended ₛ (for 'S'etoid).

+

+
infixr 0 _⟶ₛ_

+
_⟶ₛ_ : Setoid a ℓ₁ → Setoid b ℓ₂ → Set _

+
_⟶ₛ_ = Func

+

+
------------------------------------------------------------------------

+
-- Bundles specialised for propositional equality

+
------------------------------------------------------------------------

+

+
infix 3 _⟶_ _↣_ _↠_ _⤖_ _⇔_ _↩_ _↪_ _↩↪_ _↔_

+
_⟶_ : Set a → Set b → Set _

+
A ⟶ B = Func (≡.setoid A) (≡.setoid B)

+

+
_↣_ : Set a → Set b → Set _

+
A ↣ B = Injection (≡.setoid A) (≡.setoid B)

+

+
_↠_ : Set a → Set b → Set _

+
A ↠ B = Surjection (≡.setoid A) (≡.setoid B)

+

+
_⤖_ : Set a → Set b → Set _

+
A ⤖ B = Bijection (≡.setoid A) (≡.setoid B)

+

+
_⇔_ : Set a → Set b → Set _

+
A ⇔ B = Equivalence (≡.setoid A) (≡.setoid B)

+

+
_↩_ : Set a → Set b → Set _

+
A ↩ B = LeftInverse (≡.setoid A) (≡.setoid B)

+

+
_↪_ : Set a → Set b → Set _

+
A ↪ B = RightInverse (≡.setoid A) (≡.setoid B)

+

+
_↩↪_ : Set a → Set b → Set _

+
A ↩↪ B = BiInverse (≡.setoid A) (≡.setoid B)

+

+
_↔_ : Set a → Set b → Set _

+
A ↔ B = Inverse (≡.setoid A) (≡.setoid B)

+

+
-- We now define some constructors for the above that

+
-- automatically provide the required congruency proofs.

+

+
module _ {A : Set a} {B : Set b} where

+

+
  mk⟶ : (A → B) → A ⟶ B

+
  mk⟶ to = record

+
    { to        = to

+
    ; cong      = ≡.cong to

+
    }

+

+
  mk↣ : ∀ {to : A → B} → Injective _≡_ _≡_ to → A ↣ B

+
  mk↣ {to} inj = record

+
    { to         = to

+
    ; cong      = ≡.cong to

+
    ; injective = inj

+
    }

+

+
  mk↠ : ∀ {to : A → B} → Surjective _≡_ _≡_ to → A ↠ B

+
  mk↠ {to} surj = record

+
    { to         = to

+
    ; cong       = ≡.cong to

+
    ; surjective = surj

+
    }

+

+
  mk⤖ : ∀ {to : A → B} → Bijective _≡_ _≡_ to → A ⤖ B

+
  mk⤖ {to} bij = record

+
    { to        = to

+
    ; cong      = ≡.cong to

+
    ; bijective = bij

+
    }

+

+
  mk⇔ : ∀ (to : A → B) (from : B → A) → A ⇔ B

+
  mk⇔ to from = record

+
    { to        = to

+
    ; from      = from

+
    ; to-cong   = ≡.cong to

+
    ; from-cong = ≡.cong from

+
    }

+

+
  mk↩ : ∀ {to : A → B} {from : B → A} → Inverseˡ _≡_ _≡_ to from → A ↩ B

+
  mk↩ {to} {from} invˡ = record

+
    { to        = to

+
    ; from      = from

+
    ; to-cong   = ≡.cong to

+
    ; from-cong = ≡.cong from

+
    ; inverseˡ  = invˡ

+
    }

+

+
  mk↪ : ∀ {to : A → B} {from : B → A} → Inverseʳ _≡_ _≡_ to from → A ↪ B

+
  mk↪ {to} {from} invʳ = record

+
    { to        = to

+
    ; from      = from

+
    ; to-cong   = ≡.cong to

+
    ; from-cong = ≡.cong from

+
    ; inverseʳ  = invʳ

+
    }

+

+
  mk↩↪ : ∀ {to : A → B} {from₁ : B → A} {from₂ : B → A} →

+
         Inverseˡ _≡_ _≡_ to from₁ → Inverseʳ _≡_ _≡_ to from₂ → A ↩↪ B

+
  mk↩↪ {to} {from₁} {from₂} invˡ invʳ = record

+
    { to         = to

+
    ; from₁      = from₁

+
    ; from₂      = from₂

+
    ; to-cong    = ≡.cong to

+
    ; from₁-cong = ≡.cong from₁

+
    ; from₂-cong = ≡.cong from₂

+
    ; inverseˡ   = invˡ

+
    ; inverseʳ   = invʳ

+
    }

+

+
  mk↔ : ∀ {to : A → B} {from : B → A} → Inverseᵇ _≡_ _≡_ to from → A ↔ B

+
  mk↔ {to} {from} inv = record

+
    { to        = to

+
    ; from      = from

+
    ; to-cong   = ≡.cong to

+
    ; from-cong = ≡.cong from

+
    ; inverse   = inv

+
    }

+

+

+
  -- Strict variant of the above.

+
  mk↠ₛ : ∀ {to : A → B} → StrictlySurjective _≡_ to → A ↠ B

+
  mk↠ₛ = mk↠ ∘ strictlySurjective⇒surjective

+

+
  mk↔ₛ′ : ∀ (to : A → B) (from : B → A) →

+
          StrictlyInverseˡ _≡_ to from →

+
          StrictlyInverseʳ _≡_ to from →

+
          A ↔ B

+
  mk↔ₛ′ to from invˡ invʳ = mk↔ {to} {from}

+
    ( strictlyInverseˡ⇒inverseˡ to invˡ

+
    , strictlyInverseʳ⇒inverseʳ to invʳ

+
    )

+

+
------------------------------------------------------------------------

+
-- Other

+
------------------------------------------------------------------------

+

+
-- Alternative syntax for the application of functions

+

+
module _ {From : Setoid a ℓ₁} {To : Setoid b ℓ₂} where

+
  open Setoid

+

+
  infixl 5 _⟨$⟩_

+
  _⟨$⟩_ : Func From To → Carrier From → Carrier To

+
  _⟨$⟩_ = Func.to

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Consequences.Propositional.html b/site/static/agda_html/Function.Consequences.Propositional.html
new file mode 100644
index 000000000..19f2ebd02
--- /dev/null
+++ b/site/static/agda_html/Function.Consequences.Propositional.html
@@ -0,0 +1,130 @@
+
+Function.Consequences.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Relationships between properties of functions where the equality

+
-- over both the domain and codomain is assumed to be _≡_

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Consequences.Propositional

+
  {a b} {A : Set a} {B : Set b}

+
  where

+

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; cong)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (setoid)

+
open import Function.Definitions

+
open import Relation.Nullary.Negation.Core using (contraposition)

+

+
import Function.Consequences.Setoid (setoid A) (setoid B) as Setoid

+

+
------------------------------------------------------------------------

+
-- Re-export setoid properties

+

+
open Setoid public

+
  hiding

+
  ( strictlySurjective⇒surjective

+
  ; strictlyInverseˡ⇒inverseˡ

+
  ; strictlyInverseʳ⇒inverseʳ

+
  )

+

+
------------------------------------------------------------------------

+
-- Properties that rely on congruence

+

+
private

+
  variable

+
    f : A → B

+
    f⁻¹ : B → A

+

+
strictlySurjective⇒surjective : StrictlySurjective _≡_ f →

+
                                 Surjective _≡_ _≡_ f

+
strictlySurjective⇒surjective =

+
 Setoid.strictlySurjective⇒surjective (cong _)

+

+
strictlyInverseˡ⇒inverseˡ : ∀ f → StrictlyInverseˡ _≡_ f f⁻¹ →

+
                            Inverseˡ _≡_ _≡_ f f⁻¹

+
strictlyInverseˡ⇒inverseˡ f =

+
  Setoid.strictlyInverseˡ⇒inverseˡ (cong _)

+

+
strictlyInverseʳ⇒inverseʳ : ∀ f → StrictlyInverseʳ _≡_ f f⁻¹ →

+
                            Inverseʳ _≡_ _≡_ f f⁻¹

+
strictlyInverseʳ⇒inverseʳ f =

+
  Setoid.strictlyInverseʳ⇒inverseʳ (cong _)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Consequences.Setoid.html b/site/static/agda_html/Function.Consequences.Setoid.html
new file mode 100644
index 000000000..38df8ef83
--- /dev/null
+++ b/site/static/agda_html/Function.Consequences.Setoid.html
@@ -0,0 +1,169 @@
+
+Function.Consequences.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Relationships between properties of functions where the equality

+
-- over both the domain and codomain are assumed to be setoids.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Setoid)

+

+
module Function.Consequences.Setoid

+
  {a b ℓ₁ ℓ₂}

+
  (S : Setoid a ℓ₁)

+
  (T : Setoid b ℓ₂)

+
  where

+

+
open import Function.Definitions

+
open import Relation.Nullary.Negation.Core

+

+
import Function.Consequences as C

+

+
private

+
  open module S = Setoid S using () renaming (Carrier to A; _≈_ to ≈₁)

+
  open module T = Setoid T using () renaming (Carrier to B; _≈_ to ≈₂)

+

+
  variable

+
    f : A → B

+
    f⁻¹ : B → A

+

+
------------------------------------------------------------------------

+
-- Injective

+

+
contraInjective : Injective ≈₁ ≈₂ f →

+
                  ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))

+
contraInjective = C.contraInjective ≈₂

+

+
------------------------------------------------------------------------

+
-- Inverseˡ

+

+
inverseˡ⇒surjective : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Surjective ≈₁ ≈₂ f

+
inverseˡ⇒surjective = C.inverseˡ⇒surjective ≈₂

+

+
------------------------------------------------------------------------

+
-- Inverseʳ

+

+
inverseʳ⇒injective : ∀ f → Inverseʳ ≈₁ ≈₂ f f⁻¹ → Injective ≈₁ ≈₂ f

+
inverseʳ⇒injective f = C.inverseʳ⇒injective ≈₂ f T.refl S.sym S.trans

+

+
------------------------------------------------------------------------

+
-- Inverseᵇ

+

+
inverseᵇ⇒bijective : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Bijective ≈₁ ≈₂ f

+
inverseᵇ⇒bijective = C.inverseᵇ⇒bijective ≈₂ T.refl S.sym S.trans

+

+
------------------------------------------------------------------------

+
-- StrictlySurjective

+

+
surjective⇒strictlySurjective : Surjective ≈₁ ≈₂ f →

+
                                 StrictlySurjective ≈₂ f

+
surjective⇒strictlySurjective =

+
  C.surjective⇒strictlySurjective ≈₂ S.refl

+

+
strictlySurjective⇒surjective : Congruent ≈₁ ≈₂ f →

+
                                 StrictlySurjective ≈₂ f →

+
                                 Surjective ≈₁ ≈₂ f

+
strictlySurjective⇒surjective =

+
  C.strictlySurjective⇒surjective T.trans

+

+
------------------------------------------------------------------------

+
-- StrictlyInverseˡ

+

+
inverseˡ⇒strictlyInverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ →

+
                            StrictlyInverseˡ ≈₂ f f⁻¹

+
inverseˡ⇒strictlyInverseˡ = C.inverseˡ⇒strictlyInverseˡ ≈₁ ≈₂ S.refl

+

+
strictlyInverseˡ⇒inverseˡ : Congruent ≈₁ ≈₂ f →

+
                            StrictlyInverseˡ ≈₂ f f⁻¹ →

+
                            Inverseˡ ≈₁ ≈₂ f f⁻¹

+
strictlyInverseˡ⇒inverseˡ = C.strictlyInverseˡ⇒inverseˡ T.trans

+

+
------------------------------------------------------------------------

+
-- StrictlyInverseʳ

+

+
inverseʳ⇒strictlyInverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ →

+
                            StrictlyInverseʳ ≈₁ f f⁻¹

+
inverseʳ⇒strictlyInverseʳ = C.inverseʳ⇒strictlyInverseʳ ≈₁ ≈₂ T.refl

+

+
strictlyInverseʳ⇒inverseʳ : Congruent ≈₂ ≈₁ f⁻¹ →

+
                            StrictlyInverseʳ ≈₁ f f⁻¹ →

+
                            Inverseʳ ≈₁ ≈₂ f f⁻¹

+
strictlyInverseʳ⇒inverseʳ = C.strictlyInverseʳ⇒inverseʳ S.trans

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Consequences.html b/site/static/agda_html/Function.Consequences.html
new file mode 100644
index 000000000..2b41ae236
--- /dev/null
+++ b/site/static/agda_html/Function.Consequences.html
@@ -0,0 +1,191 @@
+
+Function.Consequences
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Relationships between properties of functions. See

+
-- `Function.Consequences.Propositional` for specialisations to

+
-- propositional equality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Consequences where

+

+
open import Data.Product.Base as Product

+
open import Function.Definitions

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)

+
open import Relation.Nullary.Negation.Core using (¬_; contraposition)

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+
    A B : Set a

+
    ≈₁ ≈₂ : Rel A ℓ₁

+
    f f⁻¹ : A → B

+

+
------------------------------------------------------------------------

+
-- Injective

+

+
contraInjective : ∀ (≈₂ : Rel B ℓ₂) → Injective ≈₁ ≈₂ f →

+
                  ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))

+
contraInjective _ inj p = contraposition inj p

+

+
------------------------------------------------------------------------

+
-- Inverseˡ

+

+
inverseˡ⇒surjective : ∀ (≈₂ : Rel B ℓ₂) →

+
                      Inverseˡ ≈₁ ≈₂ f f⁻¹ →

+
                      Surjective ≈₁ ≈₂ f

+
inverseˡ⇒surjective ≈₂ invˡ y = (_ , invˡ)

+

+
------------------------------------------------------------------------

+
-- Inverseʳ

+

+
inverseʳ⇒injective : ∀ (≈₂ : Rel B ℓ₂) f →

+
                     Reflexive ≈₂ →

+
                     Symmetric ≈₁ →

+
                     Transitive ≈₁ →

+
                     Inverseʳ ≈₁ ≈₂ f f⁻¹ →

+
                     Injective ≈₁ ≈₂ f

+
inverseʳ⇒injective ≈₂ f refl sym trans invʳ {x} {y} fx≈fy =

+
  trans (sym (invʳ refl)) (invʳ fx≈fy)

+

+
------------------------------------------------------------------------

+
-- Inverseᵇ

+

+
inverseᵇ⇒bijective : ∀ (≈₂ : Rel B ℓ₂) →

+
                     Reflexive ≈₂ →

+
                     Symmetric ≈₁ →

+
                     Transitive ≈₁ →

+
                     Inverseᵇ ≈₁ ≈₂ f f⁻¹ →

+
                     Bijective ≈₁ ≈₂ f

+
inverseᵇ⇒bijective {f = f} ≈₂ refl sym trans (invˡ , invʳ) =

+
  (inverseʳ⇒injective ≈₂ f refl sym trans invʳ , inverseˡ⇒surjective ≈₂ invˡ)

+

+
------------------------------------------------------------------------

+
-- StrictlySurjective

+

+
surjective⇒strictlySurjective : ∀ (≈₂ : Rel B ℓ₂) →

+
                                 Reflexive ≈₁ →

+
                                 Surjective ≈₁ ≈₂ f →

+
                                 StrictlySurjective ≈₂ f

+
surjective⇒strictlySurjective _ refl surj x =

+
  Product.map₂ (λ v → v refl) (surj x)

+

+
strictlySurjective⇒surjective : Transitive ≈₂ →

+
                                 Congruent ≈₁ ≈₂ f →

+
                                 StrictlySurjective ≈₂ f →

+
                                 Surjective ≈₁ ≈₂ f

+
strictlySurjective⇒surjective trans cong surj x =

+
  Product.map₂ (λ fy≈x z≈y → trans (cong z≈y) fy≈x) (surj x)

+

+
------------------------------------------------------------------------

+
-- StrictlyInverseˡ

+

+
inverseˡ⇒strictlyInverseˡ : ∀ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) →

+
                            Reflexive ≈₁ →

+
                            Inverseˡ ≈₁ ≈₂ f f⁻¹ →

+
                            StrictlyInverseˡ ≈₂ f f⁻¹

+
inverseˡ⇒strictlyInverseˡ _ _ refl sinv x = sinv refl

+

+
strictlyInverseˡ⇒inverseˡ : Transitive ≈₂ →

+
                            Congruent ≈₁ ≈₂ f →

+
                            StrictlyInverseˡ ≈₂ f f⁻¹ →

+
                            Inverseˡ ≈₁ ≈₂ f f⁻¹

+
strictlyInverseˡ⇒inverseˡ trans cong sinv {x} y≈f⁻¹x =

+
  trans (cong y≈f⁻¹x) (sinv x)

+

+
------------------------------------------------------------------------

+
-- StrictlyInverseʳ

+

+
inverseʳ⇒strictlyInverseʳ : ∀ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) →

+
                            Reflexive ≈₂ →

+
                            Inverseʳ ≈₁ ≈₂ f f⁻¹ →

+
                            StrictlyInverseʳ ≈₁ f f⁻¹

+
inverseʳ⇒strictlyInverseʳ _ _ refl sinv x = sinv refl

+

+
strictlyInverseʳ⇒inverseʳ : Transitive ≈₁ →

+
                            Congruent ≈₂ ≈₁ f⁻¹ →

+
                            StrictlyInverseʳ ≈₁ f f⁻¹ →

+
                            Inverseʳ ≈₁ ≈₂ f f⁻¹

+
strictlyInverseʳ⇒inverseʳ trans cong sinv {x} y≈f⁻¹x =

+
  trans (cong y≈f⁻¹x) (sinv x)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Construct.Composition.html b/site/static/agda_html/Function.Construct.Composition.html
new file mode 100644
index 000000000..9277317d0
--- /dev/null
+++ b/site/static/agda_html/Function.Construct.Composition.html
@@ -0,0 +1,364 @@
+
+Function.Construct.Composition
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Composition of functional properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Construct.Composition where

+

+
open import Data.Product.Base as Product using (_,_)

+
open import Function.Base using (_∘_; flip)

+
open import Function.Bundles

+
open import Function.Definitions

+
open import Function.Structures

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (Transitive)

+

+
private

+
  variable

+
    a b c ℓ₁ ℓ₂ ℓ₃ : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)

+
         {f : A → B} {g : B → C}

+
         where

+

+
  congruent : Congruent ≈₁ ≈₂ f → Congruent ≈₂ ≈₃ g →

+
              Congruent ≈₁ ≈₃ (g ∘ f)

+
  congruent f-cong g-cong = g-cong ∘ f-cong

+

+
  injective : Injective ≈₁ ≈₂ f → Injective ≈₂ ≈₃ g →

+
              Injective ≈₁ ≈₃ (g ∘ f)

+
  injective f-inj g-inj = f-inj ∘ g-inj

+

+
  surjective : Surjective ≈₁ ≈₂ f → Surjective ≈₂ ≈₃ g →

+
               Surjective ≈₁ ≈₃ (g ∘ f)

+
  surjective f-sur g-sur x with g-sur x

+
  ... | y , gproof with f-sur y

+
  ...   | z , fproof = z , gproof ∘ fproof

+

+
  bijective : Bijective ≈₁ ≈₂ f → Bijective ≈₂ ≈₃ g →

+
              Bijective ≈₁ ≈₃ (g ∘ f)

+
  bijective = Product.zip′ injective surjective

+

+
module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)

+
         {f : A → B} {f⁻¹ : B → A} {g : B → C} {g⁻¹ : C → B}

+
         where

+

+
  inverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₃ g g⁻¹ →

+
             Inverseˡ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)

+
  inverseˡ f-inv g-inv = g-inv ∘ f-inv

+

+
  inverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₃ g g⁻¹ →

+
             Inverseʳ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)

+
  inverseʳ f-inv g-inv = f-inv ∘ g-inv

+

+
  inverseᵇ : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₃ g g⁻¹ →

+
             Inverseᵇ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)

+
  inverseᵇ = Product.zip′ inverseˡ inverseʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}

+
         {f : A → B} {g : B → C}

+
         where

+

+
  isCongruent : IsCongruent ≈₁ ≈₂ f → IsCongruent ≈₂ ≈₃ g →

+
                IsCongruent ≈₁ ≈₃ (g ∘ f)

+
  isCongruent f-cong g-cong = record

+
    { cong           = G.cong ∘ F.cong

+
    ; isEquivalence₁ = F.isEquivalence₁

+
    ; isEquivalence₂ = G.isEquivalence₂

+
    } where module F = IsCongruent f-cong; module G = IsCongruent g-cong

+

+
  isInjection : IsInjection ≈₁ ≈₂ f → IsInjection ≈₂ ≈₃ g →

+
                IsInjection ≈₁ ≈₃ (g ∘ f)

+
  isInjection f-inj g-inj = record

+
    { isCongruent = isCongruent F.isCongruent G.isCongruent

+
    ; injective   = injective ≈₁ ≈₂ ≈₃ F.injective G.injective

+
    } where module F = IsInjection f-inj; module G = IsInjection g-inj

+

+
  isSurjection : IsSurjection ≈₁ ≈₂ f → IsSurjection ≈₂ ≈₃ g →

+
                 IsSurjection ≈₁ ≈₃ (g ∘ f)

+
  isSurjection f-surj g-surj = record

+
    { isCongruent = isCongruent F.isCongruent G.isCongruent

+
    ; surjective  = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective

+
    } where module F = IsSurjection f-surj; module G = IsSurjection g-surj

+

+
  isBijection : IsBijection ≈₁ ≈₂ f → IsBijection ≈₂ ≈₃ g →

+
                IsBijection ≈₁ ≈₃ (g ∘ f)

+
  isBijection f-bij g-bij = record

+
    { isInjection = isInjection F.isInjection G.isInjection

+
    ; surjective  = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective

+
    } where module F = IsBijection f-bij; module G = IsBijection g-bij

+

+
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}

+
         {f : A → B} {g : B → C} {f⁻¹ : B → A} {g⁻¹ : C → B}

+
         where

+

+
  isLeftInverse : IsLeftInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₃ g g⁻¹ →

+
                  IsLeftInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)

+
  isLeftInverse f-invˡ g-invˡ = record

+
    { isCongruent = isCongruent F.isCongruent G.isCongruent

+
    ; from-cong   = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong

+
    ; inverseˡ    = inverseˡ ≈₁ ≈₂ ≈₃ F.inverseˡ G.inverseˡ

+
    } where module F = IsLeftInverse f-invˡ; module G = IsLeftInverse g-invˡ

+

+
  isRightInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₃ g g⁻¹ →

+
                   IsRightInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)

+
  isRightInverse f-invʳ g-invʳ = record

+
    { isCongruent = isCongruent F.isCongruent G.isCongruent

+
    ; from-cong   = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong

+
    ; inverseʳ    = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ

+
    } where module F = IsRightInverse f-invʳ; module G = IsRightInverse g-invʳ

+

+
  isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₃ g g⁻¹ →

+
              IsInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)

+
  isInverse f-inv g-inv = record

+
    { isLeftInverse = isLeftInverse F.isLeftInverse G.isLeftInverse

+
    ; inverseʳ      = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ

+
    } where module F = IsInverse f-inv; module G = IsInverse g-inv

+

+
------------------------------------------------------------------------

+
-- Setoid bundles

+

+
module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where

+

+
  open Setoid renaming (_≈_ to ≈)

+

+
  function : Func R S → Func S T → Func R T

+
  function f g = record

+
    { to   = G.to ∘ F.to

+
    ; cong = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong

+
    } where module F = Func f; module G = Func g

+

+
  injection : Injection R S → Injection S T → Injection R T

+
  injection inj₁ inj₂ = record

+
    { to        = G.to ∘ F.to

+
    ; cong      = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong

+
    ; injective = injective (≈ R) (≈ S) (≈ T) F.injective G.injective

+
    } where module F = Injection inj₁; module G = Injection inj₂

+

+
  surjection : Surjection R S → Surjection S T → Surjection R T

+
  surjection surj₁ surj₂ = record

+
    { to         = G.to ∘ F.to

+
    ; cong       = congruent  (≈ R) (≈ S) (≈ T) F.cong G.cong

+
    ; surjective = surjective (≈ R) (≈ S) (≈ T) F.surjective G.surjective

+
    } where module F = Surjection surj₁; module G = Surjection surj₂

+

+
  bijection : Bijection R S → Bijection S T → Bijection R T

+
  bijection bij₁ bij₂ = record

+
    { to        = G.to ∘ F.to

+
    ; cong      = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong

+
    ; bijective = bijective (≈ R) (≈ S) (≈ T) F.bijective G.bijective

+
    } where module F = Bijection bij₁; module G = Bijection bij₂

+

+
  equivalence : Equivalence R S → Equivalence S T → Equivalence R T

+
  equivalence equiv₁ equiv₂ = record

+
    { to        = G.to ∘ F.to

+
    ; from      = F.from ∘ G.from

+
    ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong

+
    ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong

+
    } where module F = Equivalence equiv₁; module G = Equivalence equiv₂

+

+
  leftInverse : LeftInverse R S → LeftInverse S T → LeftInverse R T

+
  leftInverse invˡ₁ invˡ₂ = record

+
    { to        = G.to ∘ F.to

+
    ; from      = F.from ∘ G.from

+
    ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong

+
    ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong

+
    ; inverseˡ  = inverseˡ  (≈ R) (≈ S) (≈ T) F.inverseˡ G.inverseˡ

+
    } where module F = LeftInverse invˡ₁; module G = LeftInverse invˡ₂

+

+
  rightInverse : RightInverse R S → RightInverse S T → RightInverse R T

+
  rightInverse invʳ₁ invʳ₂ = record

+
    { to        = G.to ∘ F.to

+
    ; from      = F.from ∘ G.from

+
    ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong

+
    ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong

+
    ; inverseʳ  = inverseʳ  (≈ R) (≈ S) (≈ T) F.inverseʳ G.inverseʳ

+
    } where module F = RightInverse invʳ₁; module G = RightInverse invʳ₂

+

+
  inverse : Inverse R S → Inverse S T → Inverse R T

+
  inverse inv₁ inv₂ = record

+
    { to        = G.to ∘ F.to

+
    ; from      = F.from ∘ G.from

+
    ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong

+
    ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong

+
    ; inverse   = inverseᵇ  (≈ R) (≈ S) (≈ T) F.inverse G.inverse

+
    } where module F = Inverse inv₁; module G = Inverse inv₂

+

+
------------------------------------------------------------------------

+
-- Propositional bundles

+

+
-- Notice the flipped order of the arguments to mirror composition.

+

+
infix 8 _⟶-∘_ _↣-∘_ _↠-∘_ _⤖-∘_ _⇔-∘_ _↩-∘_ _↪-∘_ _↔-∘_

+

+
_⟶-∘_ : (B ⟶ C) → (A ⟶ B) → (A ⟶ C)

+
_⟶-∘_ = flip function

+

+
_↣-∘_ : B ↣ C → A ↣ B → A ↣ C

+
_↣-∘_ = flip injection

+

+
_↠-∘_ : B ↠ C → A ↠ B → A ↠ C

+
_↠-∘_ = flip surjection

+

+
_⤖-∘_ : B ⤖ C → A ⤖ B → A ⤖ C

+
_⤖-∘_ = flip bijection

+

+
_⇔-∘_ : B ⇔ C → A ⇔ B → A ⇔ C

+
_⇔-∘_ = flip equivalence

+

+
_↩-∘_ : B ↩ C → A ↩ B → A ↩ C

+
_↩-∘_ = flip leftInverse

+

+
_↪-∘_  : B ↪ C → A ↪ B → A ↪ C

+
_↪-∘_ = flip rightInverse

+

+
_↔-∘_ : B ↔ C → A ↔ B → A ↔ C

+
_↔-∘_ = flip inverse

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version v2.0

+

+
infix 8 _∘-⟶_ _∘-↣_ _∘-↠_ _∘-⤖_ _∘-⇔_ _∘-↩_ _∘-↪_ _∘-↔_

+

+
_∘-⟶_ = _⟶-∘_

+
{-# WARNING_ON_USAGE _∘-⟶_

+
"Warning: _∘-⟶_ was deprecated in v2.0.

+
Please use _⟶-∘_ instead."

+
#-}

+

+
_∘-↣_ = _↣-∘_

+
{-# WARNING_ON_USAGE _∘-↣_

+
"Warning: _∘-↣_ was deprecated in v2.0.

+
Please use _↣-∘_ instead."

+
#-}

+

+
_∘-↠_ = _↠-∘_

+
{-# WARNING_ON_USAGE _∘-↠_

+
"Warning: _∘-↠_ was deprecated in v2.0.

+
Please use _↠-∘_ instead."

+
#-}

+

+
_∘-⤖_ = _⤖-∘_

+
{-# WARNING_ON_USAGE _∘-⤖_

+
"Warning: _∘-⤖_ was deprecated in v2.0.

+
Please use _⤖-∘_ instead."

+
#-}

+

+
_∘-⇔_ = _⇔-∘_

+
{-# WARNING_ON_USAGE _∘-⇔_

+
"Warning: _∘-⇔_ was deprecated in v2.0.

+
Please use _⇔-∘_ instead."

+
#-}

+

+
_∘-↩_ = _↩-∘_

+
{-# WARNING_ON_USAGE _∘-↩_

+
"Warning: _∘-↩_ was deprecated in v2.0.

+
Please use _↩-∘_ instead."

+
#-}

+

+
_∘-↪_ = _↪-∘_

+
{-# WARNING_ON_USAGE _∘-↪_

+
"Warning: _∘-↪_ was deprecated in v2.0.

+
Please use _↪-∘_ instead."

+
#-}

+

+
_∘-↔_ = _↔-∘_

+
{-# WARNING_ON_USAGE _∘-↔_

+
"Warning: _∘-↔_ was deprecated in v2.0.

+
Please use _↔-∘_ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Construct.Identity.html b/site/static/agda_html/Function.Construct.Identity.html
new file mode 100644
index 000000000..35bd24606
--- /dev/null
+++ b/site/static/agda_html/Function.Construct.Identity.html
@@ -0,0 +1,340 @@
+
+Function.Construct.Identity
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The identity function

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Construct.Identity where

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (id)

+
open import Function.Bundles

+
import Function.Definitions as Definitions

+
import Function.Structures as Structures

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures as B hiding (IsEquivalence)

+
open import Relation.Binary.Definitions using (Reflexive)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
private

+
  variable

+
    a ℓ : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (_≈_ : Rel A ℓ) where

+

+
  open Definitions

+

+
  congruent : Congruent _≈_ _≈_ id

+
  congruent = id

+

+
  injective : Injective _≈_ _≈_ id

+
  injective = id

+

+
  surjective : Surjective _≈_ _≈_ id

+
  surjective x = x , id

+

+
  bijective : Bijective _≈_ _≈_ id

+
  bijective = injective , surjective

+

+
  inverseˡ : Inverseˡ _≈_ _≈_ id id

+
  inverseˡ = id

+

+
  inverseʳ : Inverseʳ _≈_ _≈_ id id

+
  inverseʳ = id

+

+
  inverseᵇ : Inverseᵇ _≈_ _≈_ id id

+
  inverseᵇ = inverseˡ , inverseʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ {_≈_ : Rel A ℓ} (isEq : B.IsEquivalence _≈_) where

+

+
  open Structures _≈_ _≈_

+
  open B.IsEquivalence isEq

+

+
  isCongruent : IsCongruent id

+
  isCongruent = record

+
    { cong           = id

+
    ; isEquivalence₁ = isEq

+
    ; isEquivalence₂ = isEq

+
    }

+

+
  isInjection : IsInjection id

+
  isInjection = record

+
    { isCongruent = isCongruent

+
    ; injective   = injective _≈_

+
    }

+

+
  isSurjection : IsSurjection id

+
  isSurjection = record

+
    { isCongruent = isCongruent

+
    ; surjective  = surjective _≈_

+
    }

+

+
  isBijection : IsBijection id

+
  isBijection = record

+
    { isInjection = isInjection

+
    ; surjective  = surjective _≈_

+
    }

+

+
  isLeftInverse : IsLeftInverse id id

+
  isLeftInverse = record

+
    { isCongruent = isCongruent

+
    ; from-cong   = id

+
    ; inverseˡ    = inverseˡ _≈_

+
    }

+

+
  isRightInverse : IsRightInverse id id

+
  isRightInverse = record

+
    { isCongruent = isCongruent

+
    ; from-cong   = id

+
    ; inverseʳ    = inverseʳ _≈_

+
    }

+

+
  isInverse : IsInverse id id

+
  isInverse = record

+
    { isLeftInverse = isLeftInverse

+
    ; inverseʳ      = inverseʳ _≈_

+
    }

+

+
------------------------------------------------------------------------

+
-- Setoid bundles

+

+
module _ (S : Setoid a ℓ) where

+

+
  open Setoid S

+

+
  function : Func S S

+
  function = record

+
    { to   = id

+
    ; cong = id

+
    }

+

+
  injection : Injection S S

+
  injection = record

+
    { to        = id

+
    ; cong      = id

+
    ; injective = injective _≈_

+
    }

+

+
  surjection : Surjection S S

+
  surjection = record

+
    { to         = id

+
    ; cong       = id

+
    ; surjective = surjective _≈_

+
    }

+

+
  bijection : Bijection S S

+
  bijection = record

+
    { to        = id

+
    ; cong      = id

+
    ; bijective = bijective _≈_

+
    }

+

+
  equivalence : Equivalence S S

+
  equivalence = record

+
    { to        = id

+
    ; from      = id

+
    ; to-cong   = id

+
    ; from-cong = id

+
    }

+

+
  leftInverse : LeftInverse S S

+
  leftInverse = record

+
    { to        = id

+
    ; from      = id

+
    ; to-cong   = id

+
    ; from-cong = id

+
    ; inverseˡ  = inverseˡ _≈_

+
    }

+

+
  rightInverse : RightInverse S S

+
  rightInverse = record

+
    { to        = id

+
    ; from      = id

+
    ; to-cong   = id

+
    ; from-cong = id

+
    ; inverseʳ  = inverseʳ _≈_

+
    }

+

+
  inverse : Inverse S S

+
  inverse = record

+
    { to        = id

+
    ; from      = id

+
    ; to-cong   = id

+
    ; from-cong = id

+
    ; inverse   = inverseᵇ _≈_

+
    }

+

+
------------------------------------------------------------------------

+
-- Propositional bundles

+

+
module _ (A : Set a) where

+

+
  ⟶-id : A ⟶ A

+
  ⟶-id = function (setoid A)

+

+
  ↣-id : A ↣ A

+
  ↣-id = injection (setoid A)

+

+
  ↠-id : A ↠ A

+
  ↠-id = surjection (setoid A)

+

+
  ⤖-id : A ⤖ A

+
  ⤖-id = bijection (setoid A)

+

+
  ⇔-id : A ⇔ A

+
  ⇔-id = equivalence (setoid A)

+

+
  ↩-id : A ↩ A

+
  ↩-id = leftInverse (setoid A)

+

+
  ↪-id : A ↪ A

+
  ↪-id = rightInverse (setoid A)

+

+
  ↔-id : A ↔ A

+
  ↔-id = inverse (setoid A)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version v2.0

+

+
id-⟶ = ⟶-id

+
{-# WARNING_ON_USAGE id-⟶

+
"Warning: id-⟶ was deprecated in v2.0.

+
Please use ⟶-id instead."

+
#-}

+

+
id-↣ = ↣-id

+
{-# WARNING_ON_USAGE id-↣

+
"Warning: id-↣ was deprecated in v2.0.

+
Please use ↣-id instead."

+
#-}

+

+
id-↠ = ↠-id

+
{-# WARNING_ON_USAGE id-↠

+
"Warning: id-↠ was deprecated in v2.0.

+
Please use ↠-id instead."

+
#-}

+

+
id-⤖ = ⤖-id

+
{-# WARNING_ON_USAGE id-⤖

+
"Warning: id-⤖ was deprecated in v2.0.

+
Please use ⤖-id instead."

+
#-}

+

+
id-⇔ = ⇔-id

+
{-# WARNING_ON_USAGE id-⇔

+
"Warning: id-⇔ was deprecated in v2.0.

+
Please use ⇔-id instead."

+
#-}

+

+
id-↩ = ↩-id

+
{-# WARNING_ON_USAGE id-↩

+
"Warning: id-↩ was deprecated in v2.0.

+
Please use ↩-id instead."

+
#-}

+

+
id-↪ = ↪-id

+
{-# WARNING_ON_USAGE id-↪

+
"Warning: id-↪ was deprecated in v2.0.

+
Please use ↪-id instead."

+
#-}

+

+
id-↔ = ↔-id

+
{-# WARNING_ON_USAGE id-↔

+
"Warning: id-↔ was deprecated in v2.0.

+
Please use ↔-id instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Construct.Symmetry.html b/site/static/agda_html/Function.Construct.Symmetry.html
new file mode 100644
index 000000000..c19fe6898
--- /dev/null
+++ b/site/static/agda_html/Function.Construct.Symmetry.html
@@ -0,0 +1,322 @@
+
+Function.Construct.Symmetry
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some functional properties are symmetric

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Construct.Symmetry where

+

+
open import Data.Product.Base using (_,_; swap; proj₁; proj₂)

+
open import Function.Base using (_∘_)

+
open import Function.Definitions

+
  using (Bijective; Injective; Surjective; Inverseˡ; Inverseʳ; Inverseᵇ; Congruent)

+
open import Function.Structures

+
  using (IsBijection; IsCongruent; IsRightInverse; IsLeftInverse; IsInverse)

+
open import Function.Bundles

+
  using (Bijection; Equivalence; LeftInverse; RightInverse; Inverse; _⤖_; _⇔_; _↩_; _↪_; _↔_)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+

+
private

+
  variable

+
    a b c ℓ₁ ℓ₂ ℓ₃ : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B}

+
         ((inj , surj) : Bijective ≈₁ ≈₂ f)

+
         where

+

+
  private

+
    f⁻¹      = proj₁ ∘ surj

+
    f∘f⁻¹≡id = proj₂ ∘ surj

+

+
  injective : Reflexive ≈₁ → Symmetric ≈₂ → Transitive ≈₂ →

+
              Congruent ≈₁ ≈₂ f → Injective ≈₂ ≈₁ f⁻¹

+
  injective refl sym trans cong gx≈gy =

+
    trans (trans (sym (f∘f⁻¹≡id _ refl)) (cong gx≈gy)) (f∘f⁻¹≡id _ refl)

+

+
  surjective : Reflexive ≈₁ → Transitive ≈₂ → Surjective ≈₂ ≈₁ f⁻¹

+
  surjective refl trans x = f x , inj ∘ trans (f∘f⁻¹≡id _ refl)

+

+
  bijective : Reflexive ≈₁ → Symmetric ≈₂ → Transitive ≈₂ →

+
              Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹

+
  bijective refl sym trans cong = injective refl sym trans cong , surjective refl trans

+

+
module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {f : A → B} {f⁻¹ : B → A} where

+

+
  inverseʳ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₁ f⁻¹ f

+
  inverseʳ inv = inv

+

+
  inverseˡ : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₁ f⁻¹ f

+
  inverseˡ inv = inv

+

+
  inverseᵇ : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₁ f⁻¹ f

+
  inverseᵇ (invˡ , invʳ) = (invʳ , invˡ)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂}

+
         {f : A → B} (isBij : IsBijection ≈₁ ≈₂ f)

+
         where

+

+
  private

+
    module IB = IsBijection isBij

+
    f⁻¹       = proj₁ ∘ IB.surjective

+

+
  -- We can only flip a bijection if the witness produced by the

+
  -- surjection proof respects the equality on the codomain.

+
  isBijection : Congruent ≈₂ ≈₁ f⁻¹ → IsBijection ≈₂ ≈₁ f⁻¹

+
  isBijection f⁻¹-cong = record

+
    { isInjection = record

+
      { isCongruent = record

+
        { cong           = f⁻¹-cong

+
        ; isEquivalence₁ = IB.Eq₂.isEquivalence

+
        ; isEquivalence₂ = IB.Eq₁.isEquivalence

+
        }

+
      ; injective = injective IB.bijective IB.Eq₁.refl IB.Eq₂.sym IB.Eq₂.trans IB.cong

+
      }

+
    ; surjective = surjective IB.bijective IB.Eq₁.refl IB.Eq₂.trans

+
    }

+

+
module _ {≈₁ : Rel A ℓ₁} {f : A → B} (isBij : IsBijection ≈₁ _≡_ f) where

+

+
  -- We can always flip a bijection if using the equality over the

+
  -- codomain is propositional equality.

+
  isBijection-≡ : IsBijection _≡_ ≈₁ _

+
  isBijection-≡ = isBijection isBij (IB.Eq₁.reflexive ∘ cong _)

+
    where module IB = IsBijection isBij

+

+
module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B} {f⁻¹ : B → A} where

+

+
  isCongruent : IsCongruent ≈₁ ≈₂ f → Congruent ≈₂ ≈₁ f⁻¹ → IsCongruent ≈₂ ≈₁ f⁻¹

+
  isCongruent ic cg = record

+
    { cong           = cg

+
    ; isEquivalence₁ = F.isEquivalence₂

+
    ; isEquivalence₂ = F.isEquivalence₁

+
    } where module F = IsCongruent ic

+

+
  isLeftInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₁ f⁻¹ f

+
  isLeftInverse inv = record

+
    { isCongruent = isCongruent F.isCongruent F.from-cong

+
    ; from-cong   = F.to-cong

+
    ; inverseˡ    = inverseˡ ≈₁ ≈₂ F.inverseʳ

+
    } where module F = IsRightInverse inv

+

+
  isRightInverse : IsLeftInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₁ f⁻¹ f

+
  isRightInverse inv = record

+
    { isCongruent = isCongruent F.isCongruent F.from-cong

+
    ; from-cong   = F.to-cong

+
    ; inverseʳ    = inverseʳ ≈₁ ≈₂ F.inverseˡ

+
    } where module F = IsLeftInverse inv

+

+
  isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₁ f⁻¹ f

+
  isInverse f-inv = record

+
    { isLeftInverse = isLeftInverse F.isRightInverse

+
    ; inverseʳ      = inverseʳ ≈₁ ≈₂ F.inverseˡ

+
    } where module F = IsInverse f-inv

+

+
------------------------------------------------------------------------

+
-- Setoid bundles

+

+
module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} (bij : Bijection R S) where

+

+
  private

+
    module IB = Bijection bij

+
    from      = proj₁ ∘ IB.surjective

+

+
  -- We can only flip a bijection if the witness produced by the

+
  -- surjection proof respects the equality on the codomain.

+
  bijection : Congruent IB.Eq₂._≈_ IB.Eq₁._≈_ from → Bijection S R

+
  bijection cong = record

+
    { to        = from

+
    ; cong      = cong

+
    ; bijective = bijective IB.bijective IB.Eq₁.refl IB.Eq₂.sym IB.Eq₂.trans IB.cong

+
    }

+

+
-- We can always flip a bijection if using the equality over the

+
-- codomain is propositional equality.

+
bijection-≡ : {R : Setoid a ℓ₁} {B : Set b} →

+
              Bijection R (setoid B) → Bijection (setoid B) R

+
bijection-≡ bij = bijection bij (B.Eq₁.reflexive ∘ cong _)

+
 where module B = Bijection bij

+

+
module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} where

+

+
  equivalence : Equivalence R S → Equivalence S R

+
  equivalence equiv = record

+
    { to        = E.from

+
    ; from      = E.to

+
    ; to-cong   = E.from-cong

+
    ; from-cong = E.to-cong

+
    } where module E = Equivalence equiv

+

+
  rightInverse : LeftInverse R S → RightInverse S R

+
  rightInverse left = record

+
    { to         = L.from

+
    ; from       = L.to

+
    ; to-cong    = L.from-cong

+
    ; from-cong  = L.to-cong

+
    ; inverseʳ   = L.inverseˡ

+
    } where module L = LeftInverse left

+

+
  leftInverse : RightInverse R S → LeftInverse S R

+
  leftInverse right = record

+
    { to        = R.from

+
    ; from      = R.to

+
    ; to-cong   = R.from-cong

+
    ; from-cong = R.to-cong

+
    ; inverseˡ  = R.inverseʳ

+
    } where module R = RightInverse right

+

+
  inverse : Inverse R S → Inverse S R

+
  inverse inv = record

+
    { to        = I.from

+
    ; from      = I.to

+
    ; to-cong   = I.from-cong

+
    ; from-cong = I.to-cong

+
    ; inverse   = swap I.inverse

+
    } where module I = Inverse inv

+

+
------------------------------------------------------------------------

+
-- Propositional bundles

+

+
⤖-sym : A ⤖ B → B ⤖ A

+
⤖-sym b = bijection b (cong _)

+

+
⇔-sym : A ⇔ B → B ⇔ A

+
⇔-sym = equivalence

+

+
↩-sym : A ↩ B → B ↪ A

+
↩-sym = rightInverse

+

+
↪-sym : A ↪ B → B ↩ A

+
↪-sym = leftInverse

+

+
↔-sym : A ↔ B → B ↔ A

+
↔-sym = inverse

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version v2.0

+

+
sym-⤖ = ⤖-sym

+
{-# WARNING_ON_USAGE sym-⤖

+
"Warning: sym-⤖ was deprecated in v2.0.

+
Please use ⤖-sym instead."

+
#-}

+

+
sym-⇔ = ⇔-sym

+
{-# WARNING_ON_USAGE sym-⇔

+
"Warning: sym-⇔ was deprecated in v2.0.

+
Please use ⇔-sym instead."

+
#-}

+

+
sym-↩ = ↩-sym

+
{-# WARNING_ON_USAGE sym-↩

+
"Warning: sym-↩ was deprecated in v2.0.

+
Please use ↩-sym instead."

+
#-}

+

+
sym-↪ = ↪-sym

+
{-# WARNING_ON_USAGE sym-↪

+
"Warning: sym-↪ was deprecated in v2.0.

+
Please use ↪-sym instead."

+
#-}

+

+
sym-↔ = ↔-sym

+
{-# WARNING_ON_USAGE sym-↔

+
"Warning: sym-↔ was deprecated in v2.0.

+
Please use ↔-sym instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Core.html b/site/static/agda_html/Function.Core.html
new file mode 100644
index 000000000..8385e1a6b
--- /dev/null
+++ b/site/static/agda_html/Function.Core.html
@@ -0,0 +1,105 @@
+
+Function.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core definitions for Functions

+
------------------------------------------------------------------------

+

+
-- The contents of this file should always be accessed from `Function`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Core where

+

+
open import Level using (_⊔_)

+

+
------------------------------------------------------------------------

+
-- Types

+

+
Fun₁ : ∀ {a} → Set a → Set a

+
Fun₁ A = A → A

+

+
Fun₂ : ∀ {a} → Set a → Set a

+
Fun₂ A = A → A → A

+

+
------------------------------------------------------------------------

+
-- Morphism

+

+
Morphism : ∀ {a} → ∀ {b} → Set a → Set b → Set (a ⊔ b)

+
Morphism A B = A → B

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Definitions.html b/site/static/agda_html/Function.Definitions.html
new file mode 100644
index 000000000..b1eeb8b04
--- /dev/null
+++ b/site/static/agda_html/Function.Definitions.html
@@ -0,0 +1,141 @@
+
+Function.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions for types of functions.

+
------------------------------------------------------------------------

+

+
-- The contents of this file should usually be accessed from `Function`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Definitions where

+

+
open import Data.Product.Base using (∃; _×_)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel)

+

+
private

+
  variable

+
    a ℓ₁ ℓ₂ : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- Basic definitions

+

+
module _

+
  (_≈₁_ : Rel A ℓ₁) -- Equality over the domain

+
  (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain

+
  where

+

+
  Congruent : (A → B) → Set _

+
  Congruent f = ∀ {x y} → x ≈₁ y → f x ≈₂ f y

+

+
  Injective : (A → B) → Set _

+
  Injective f = ∀ {x y} → f x ≈₂ f y → x ≈₁ y

+

+
  Surjective : (A → B) → Set _

+
  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y

+

+
  Bijective : (A → B) → Set _

+
  Bijective f = Injective f × Surjective f

+

+
  Inverseˡ : (A → B) → (B → A) → Set _

+
  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x

+

+
  Inverseʳ : (A → B) → (B → A) → Set _

+
  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x

+

+
  Inverseᵇ : (A → B) → (B → A) → Set _

+
  Inverseᵇ f g = Inverseˡ f g × Inverseʳ f g

+

+
------------------------------------------------------------------------

+
-- Strict definitions

+

+
-- These are often easier to use once but much harder to compose and

+
-- reason about.

+

+
StrictlySurjective : Rel B ℓ₂ → (A → B) → Set _

+
StrictlySurjective _≈₂_ f = ∀ y → ∃ λ x → f x ≈₂ y

+

+
StrictlyInverseˡ : Rel B ℓ₂ → (A → B) → (B → A) → Set _

+
StrictlyInverseˡ _≈₂_ f g = ∀ y → f (g y) ≈₂ y

+

+
StrictlyInverseʳ : Rel A ℓ₁ → (A → B) → (B → A) → Set _

+
StrictlyInverseʳ _≈₁_ f g = ∀ x → g (f x) ≈₁ x

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Dependent.Bundles.html b/site/static/agda_html/Function.Dependent.Bundles.html
new file mode 100644
index 000000000..aca4bd77a
--- /dev/null
+++ b/site/static/agda_html/Function.Dependent.Bundles.html
@@ -0,0 +1,127 @@
+
+Function.Dependent.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for types of functions

+
------------------------------------------------------------------------

+

+
-- The contents of this file should usually be accessed from `Function`.

+

+
-- Note that these bundles differ from those found elsewhere in other

+
-- library hierarchies as they take Setoids as parameters. This is

+
-- because a function is of no use without knowing what its domain and

+
-- codomain is, as well which equalities are being considered over them.

+
-- One consequence of this is that they are not built from the

+
-- definitions found in `Function.Structures` as is usually the case in

+
-- other library hierarchies, as this would duplicate the equality

+
-- axioms.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Dependent.Bundles where

+

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+

+
------------------------------------------------------------------------

+
-- Setoid bundles

+
------------------------------------------------------------------------

+

+
module _

+
  (From : Setoid a ℓ₁)

+
  (To : IndexedSetoid (Setoid.Carrier From) b ℓ₂)

+
  where

+

+
  open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)

+
  open IndexedSetoid To using () renaming (Carrier to B; _≈_ to _≈₂_)

+

+
------------------------------------------------------------------------

+
-- Bundles with one element

+

+
  -- Called `Func` rather than `Function` in order to avoid clashing

+
  -- with the top-level module.

+
  record Func : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    field

+
      to   : (x : A) → B x

+
      cong : ∀ {x y} → x ≈₁ y → to x ≈₂ to y

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Identity.Effectful.html b/site/static/agda_html/Function.Identity.Effectful.html
new file mode 100644
index 000000000..81156a020
--- /dev/null
+++ b/site/static/agda_html/Function.Identity.Effectful.html
@@ -0,0 +1,124 @@
+
+Function.Identity.Effectful
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An effectful view of the identity function

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Identity.Effectful where

+

+
open import Effect.Functor

+
open import Effect.Applicative

+
open import Effect.Monad

+
open import Effect.Comonad

+
open import Function.Base using (id; _∘′_; _|>′_; _$′_; flip)

+
open import Level

+

+
private

+
  variable

+
    ℓ : Level

+

+
Identity : (A : Set ℓ) → Set ℓ

+
Identity A = A

+

+
functor : RawFunctor {ℓ} Identity

+
functor = record

+
  { _<$>_ = id

+
  }

+

+
applicative : RawApplicative {ℓ} Identity

+
applicative = record

+
  { rawFunctor = functor

+
  ; pure = id

+
  ; _<*>_  = _$′_

+
  }

+

+
monad : RawMonad {ℓ} Identity

+
monad = record

+
  { rawApplicative = applicative

+
  ; _>>=_  = _|>′_

+
  }

+

+
comonad : RawComonad {ℓ} Identity

+
comonad = record

+
  { extract = id

+
  ; extend  = _$′_

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Indexed.Relation.Binary.Equality.html b/site/static/agda_html/Function.Indexed.Relation.Binary.Equality.html
new file mode 100644
index 000000000..0aaf69e52
--- /dev/null
+++ b/site/static/agda_html/Function.Indexed.Relation.Binary.Equality.html
@@ -0,0 +1,104 @@
+
+Function.Indexed.Relation.Binary.Equality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Function setoids and related constructions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Indexed.Relation.Binary.Equality where

+

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)

+

+
-- A variant of setoid which uses the propositional equality setoid

+
-- for the domain, and a more convenient definition of _≈_.

+

+
≡-setoid : ∀ {f t₁ t₂} (From : Set f) → IndexedSetoid From t₁ t₂ → Setoid _ _

+
≡-setoid From To = record

+
  { Carrier       = (x : From) → Carrier x

+
  ; _≈_           = λ f g → ∀ x → f x ≈ g x

+
  ; isEquivalence = record

+
    { refl  = λ {f} x → refl

+
    ; sym   = λ f∼g x → sym (f∼g x)

+
    ; trans = λ f∼g g∼h x → trans (f∼g x) (g∼h x)

+
    }

+
  } where open IndexedSetoid To

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Bundles.html b/site/static/agda_html/Function.Metric.Bundles.html
new file mode 100644
index 000000000..4c9dd299e
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Bundles.html
@@ -0,0 +1,226 @@
+
+Function.Metric.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for metrics

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Function.Metric`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Bundles  where

+

+
open import Algebra.Core using (Op₂)

+
open import Level using (Level; suc; _⊔_)

+
open import Relation.Binary.Core using (Rel)

+

+
open import Function.Metric.Structures

+
open import Function.Metric.Core

+

+
------------------------------------------------------------------------

+
-- ProtoMetric

+

+
record ProtoMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)

+
                 : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where

+
  infix 4 _≈_ _≈ᵢ_ _≤_

+
  field

+
    Carrier       : Set a

+
    Image         : Set i

+
    _≈_           : Rel Carrier ℓ₁

+
    _≈ᵢ_          : Rel Image ℓ₂

+
    _≤_           : Rel Image ℓ₃

+
    0#            : Image

+
    d             : DistanceFunction Carrier Image

+
    isProtoMetric : IsProtoMetric _≈_ _≈ᵢ_ _≤_ 0# d

+

+
  open IsProtoMetric isProtoMetric public

+

+
------------------------------------------------------------------------

+
-- PreMetric

+

+
record PreMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)

+
               : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where

+
  infix 4 _≈_ _≈ᵢ_ _≤_

+
  field

+
    Carrier     : Set a

+
    Image       : Set i

+
    _≈_         : Rel Carrier ℓ₁

+
    _≈ᵢ_        : Rel Image ℓ₂

+
    _≤_         : Rel Image ℓ₃

+
    0#          : Image

+
    d           : DistanceFunction Carrier Image

+
    isPreMetric : IsPreMetric _≈_ _≈ᵢ_ _≤_ 0# d

+

+
  open IsPreMetric isPreMetric public

+

+
  protoMetric : ProtoMetric a i ℓ₁ ℓ₂ ℓ₃

+
  protoMetric = record

+
    { isProtoMetric = isProtoMetric

+
    }

+

+
------------------------------------------------------------------------

+
-- QuasiSemiMetric

+

+
record QuasiSemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)

+
                     : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where

+

+
  infix 4 _≈_ _≈ᵢ_ _≤_

+
  field

+
    Carrier           : Set a

+
    Image             : Set i

+
    _≈_               : Rel Carrier ℓ₁

+
    _≈ᵢ_              : Rel Image ℓ₂

+
    _≤_               : Rel Image ℓ₃

+
    0#                : Image

+
    d                 : DistanceFunction Carrier Image

+
    isQuasiSemiMetric : IsQuasiSemiMetric _≈_ _≈ᵢ_ _≤_ 0# d

+

+
  open IsQuasiSemiMetric isQuasiSemiMetric public

+

+
  preMetric : PreMetric a i ℓ₁ ℓ₂ ℓ₃

+
  preMetric = record

+
    { isPreMetric = isPreMetric

+
    }

+

+
  open PreMetric preMetric public

+
    using (protoMetric)

+

+
------------------------------------------------------------------------

+
-- SemiMetric

+

+
record SemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)

+
                : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where

+
  infix 4 _≈_ _≈ᵢ_ _≤_

+
  field

+
    Carrier      : Set a

+
    Image        : Set i

+
    _≈_          : Rel Carrier ℓ₁

+
    _≈ᵢ_         : Rel Image ℓ₂

+
    _≤_          : Rel Image ℓ₃

+
    0#           : Image

+
    d            : DistanceFunction Carrier Image

+
    isSemiMetric : IsSemiMetric _≈_ _≈ᵢ_ _≤_ 0# d

+

+
  open IsSemiMetric isSemiMetric public

+

+
  quasiSemiMetric : QuasiSemiMetric a i ℓ₁ ℓ₂ ℓ₃

+
  quasiSemiMetric = record

+
    { isQuasiSemiMetric = isQuasiSemiMetric

+
    }

+

+
  open QuasiSemiMetric quasiSemiMetric public

+
    using (protoMetric; preMetric)

+

+
------------------------------------------------------------------------

+
-- GeneralMetric

+

+
-- Note that this package is not necessarily a metric in the classical

+
-- sense as there is no way to ensure that the _∙_ operator really

+
-- represents addition. See `Function.Metric.Nat` and

+
-- `Function.Metric.Rational` for more specialised `Metric` and

+
-- `UltraMetric` packages.

+

+
-- See the discussion accompanying the `IsGeneralMetric` structure for

+
-- more details.

+

+
record GeneralMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)

+
                   : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where

+
  infix 4 _≈_ _≈ᵢ_ _≤_

+
  infixl 6 _∙_

+
  field

+
    Carrier         : Set a

+
    Image           : Set i

+
    _≈_             : Rel Carrier ℓ₁

+
    _≈ᵢ_            : Rel Image ℓ₂

+
    _≤_             : Rel Image ℓ₃

+
    0#              : Image

+
    _∙_             : Op₂ Image

+
    d               : DistanceFunction Carrier Image

+
    isGeneralMetric : IsGeneralMetric _≈_ _≈ᵢ_ _≤_ 0# _∙_ d

+

+
  open IsGeneralMetric isGeneralMetric public

+

+
  semiMetric : SemiMetric a i ℓ₁ ℓ₂ ℓ₃

+
  semiMetric = record

+
    { isSemiMetric = isSemiMetric

+
    }

+

+
  open SemiMetric semiMetric public

+
    using (protoMetric; preMetric; quasiSemiMetric)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Core.html b/site/static/agda_html/Function.Metric.Core.html
new file mode 100644
index 000000000..3d792d00b
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Core.html
@@ -0,0 +1,97 @@
+
+Function.Metric.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core metric definitions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Core where

+

+
open import Level using (Level)

+
private

+
  variable

+
    a i : Level

+

+
------------------------------------------------------------------------

+
-- Distance functions

+

+
DistanceFunction : Set a → Set i → Set _

+
DistanceFunction A I = A → A → I

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Definitions.html b/site/static/agda_html/Function.Metric.Definitions.html
new file mode 100644
index 000000000..ca0fdf729
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Definitions.html
@@ -0,0 +1,140 @@
+
+Function.Metric.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions of properties over distance functions

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Function.Metric`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Definitions where

+

+
open import Algebra.Core using (Op₂)

+
open import Data.Product.Base using (∃)

+
open import Function.Metric.Core using (DistanceFunction)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)

+
open import Relation.Nullary.Negation using (¬_)

+

+
private

+
  variable

+
    a i ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    I : Set i

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
Congruent : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → Set _

+
Congruent _≈ₐ_ _≈ᵢ_ d = d Preserves₂ _≈ₐ_ ⟶ _≈ₐ_ ⟶ _≈ᵢ_

+

+
Indiscernable : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → I → Set _

+
Indiscernable _≈ₐ_ _≈ᵢ_ d 0# = ∀ {x y} → d x y ≈ᵢ 0# → x ≈ₐ y

+

+
Definite : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → I → Set _

+
Definite _≈ₐ_ _≈ᵢ_ d 0# = ∀ {x y} → x ≈ₐ y → d x y ≈ᵢ 0#

+

+
NonNegative : Rel I ℓ₂ → DistanceFunction A I → I → Set _

+
NonNegative _≤_ d 0# = ∀ {x y} → 0# ≤ d x y

+

+
Symmetric : Rel I ℓ → DistanceFunction A I → Set _

+
Symmetric _≈_ d = ∀ x y → d x y ≈ d y x

+

+
TriangleInequality : Rel I ℓ → Op₂ I → DistanceFunction A I → _

+
TriangleInequality _≤_ _∙_ d = ∀ x y z → d x z ≤ (d x y ∙ d y z)

+

+
Bounded : Rel I ℓ → DistanceFunction A I → Set _

+
Bounded _≤_ d = ∃ λ n → ∀ x y → d x y ≤ n

+

+
TranslationInvariant : Rel I ℓ₂ → Op₂ A → DistanceFunction A I → Set _

+
TranslationInvariant _≈_ _∙_ d = ∀ {x y a} → d (x ∙ a) (y ∙ a) ≈ d x y

+

+
Contracting : Rel I ℓ → (A → A) → DistanceFunction A I → Set _

+
Contracting _≤_ f d = ∀ x y → d (f x) (f y) ≤ d x y

+

+
ContractingOnOrbits : Rel I ℓ → (A → A) → DistanceFunction A I → Set _

+
ContractingOnOrbits _≤_ f d = ∀ x → d (f x) (f (f x)) ≤ d x (f x)

+

+
StrictlyContracting : Rel A ℓ₁ → Rel I ℓ₂ → (A → A) → DistanceFunction A I → Set _

+
StrictlyContracting _≈_ _<_ f d = ∀ {x y} → ¬ (y ≈ x) → d (f x) (f y) < d x y

+

+
StrictlyContractingOnOrbits : Rel A ℓ₁ → Rel I ℓ₂ → (A → A) → DistanceFunction A I → Set _

+
StrictlyContractingOnOrbits _≈_ _<_ f d = ∀ {x} → ¬ (f x ≈ x) → d (f x) (f (f x)) < d x (f x)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Nat.Bundles.html b/site/static/agda_html/Function.Metric.Nat.Bundles.html
new file mode 100644
index 000000000..58be53a78
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Nat.Bundles.html
@@ -0,0 +1,217 @@
+
+Function.Metric.Nat.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for metrics over ℕ

+
------------------------------------------------------------------------

+

+
-- Unfortunately, unlike definitions and structures, the bundles over

+
-- general metric spaces cannot be reused as it is impossible to

+
-- constrain the image set to ℕ.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Nat.Bundles where

+

+
open import Data.Nat.Base hiding (suc; _⊔_)

+
open import Function.Base using (const)

+
open import Level using (Level; suc; _⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+
open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)

+

+
open import Function.Metric.Nat.Core using (DistanceFunction)

+
open import Function.Metric.Nat.Structures

+
open import Function.Metric.Bundles as Base

+
  using (GeneralMetric)

+

+
------------------------------------------------------------------------

+
-- Proto-metric

+

+
record ProtoMetric a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier       : Set a

+
    _≈_           : Rel Carrier ℓ

+
    d             : DistanceFunction Carrier

+
    isProtoMetric : IsProtoMetric _≈_ d

+

+
  open IsProtoMetric isProtoMetric public

+

+
------------------------------------------------------------------------

+
-- PreMetric

+

+
record PreMetric a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier     : Set a

+
    _≈_         : Rel Carrier ℓ

+
    d           : DistanceFunction Carrier

+
    isPreMetric : IsPreMetric _≈_ d

+

+
  open IsPreMetric isPreMetric public

+

+
  protoMetric : ProtoMetric a ℓ

+
  protoMetric = record

+
    { isProtoMetric = isProtoMetric

+
    }

+

+
------------------------------------------------------------------------

+
-- QuasiSemiMetric

+

+
record QuasiSemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier           : Set a

+
    _≈_               : Rel Carrier ℓ

+
    d                 : DistanceFunction Carrier

+
    isQuasiSemiMetric : IsQuasiSemiMetric _≈_ d

+

+
  open IsQuasiSemiMetric isQuasiSemiMetric public

+

+
  preMetric : PreMetric a ℓ

+
  preMetric = record

+
    { isPreMetric = isPreMetric

+
    }

+

+
  open PreMetric preMetric public

+
    using (protoMetric)

+

+
------------------------------------------------------------------------

+
-- SemiMetric

+

+
record SemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier      : Set a

+
    _≈_          : Rel Carrier ℓ

+
    d            : DistanceFunction Carrier

+
    isSemiMetric : IsSemiMetric _≈_ d

+

+
  open IsSemiMetric isSemiMetric public

+

+
  quasiSemiMetric : QuasiSemiMetric a ℓ

+
  quasiSemiMetric = record

+
    { isQuasiSemiMetric = isQuasiSemiMetric

+
    }

+

+
  open QuasiSemiMetric quasiSemiMetric public

+
    using (protoMetric; preMetric)

+

+
------------------------------------------------------------------------

+
-- Metrics

+

+
record Metric a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier  : Set a

+
    _≈_      : Rel Carrier ℓ

+
    d        : DistanceFunction Carrier

+
    isMetric : IsMetric _≈_ d

+

+
  open IsMetric isMetric public

+

+
  semiMetric : SemiMetric a ℓ

+
  semiMetric = record

+
    { isSemiMetric = isSemiMetric

+
    }

+

+
  open SemiMetric semiMetric public

+
    using (protoMetric; preMetric; quasiSemiMetric)

+

+
------------------------------------------------------------------------

+
-- UltraMetrics

+

+
record UltraMetric a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier       : Set a

+
    _≈_           : Rel Carrier ℓ

+
    d             : DistanceFunction Carrier

+
    isUltraMetric : IsUltraMetric _≈_ d

+

+
  open IsUltraMetric isUltraMetric public

+

+
  semiMetric : SemiMetric a ℓ

+
  semiMetric = record

+
    { isSemiMetric = isSemiMetric

+
    }

+

+
  open SemiMetric semiMetric public

+
    using (protoMetric; preMetric; quasiSemiMetric)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Nat.Core.html b/site/static/agda_html/Function.Metric.Nat.Core.html
new file mode 100644
index 000000000..6d9790bc0
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Nat.Core.html
@@ -0,0 +1,95 @@
+
+Function.Metric.Nat.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core definitions for metrics over ℕ

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Nat.Core where

+

+
open import Data.Nat.Base using (ℕ)

+
import Function.Metric.Core as Base

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
DistanceFunction : ∀ {a} → Set a → Set a

+
DistanceFunction A = Base.DistanceFunction A ℕ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Nat.Definitions.html b/site/static/agda_html/Function.Metric.Nat.Definitions.html
new file mode 100644
index 000000000..22146ed34
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Nat.Definitions.html
@@ -0,0 +1,145 @@
+
+Function.Metric.Nat.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core definitions for metrics over ℕ

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Nat.Definitions where

+

+
open import Algebra.Core using (Op₂)

+
open import Data.Nat.Base

+
open import Level using (Level)

+
open import Relation.Binary.Core

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
open import Function.Metric.Nat.Core

+
import Function.Metric.Definitions as Base

+

+
private

+
  variable

+
    a ℓ : Level

+
    A   : Set a

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
-- Basic

+

+
Congruent : Rel A ℓ → DistanceFunction A → Set _

+
Congruent _≈ₐ_ d = Base.Congruent _≈ₐ_ _≡_ d

+

+
Indiscernable : Rel A ℓ → DistanceFunction A → Set _

+
Indiscernable _≈ₐ_ d = Base.Indiscernable _≈ₐ_ _≡_ d 0

+

+
Definite : Rel A ℓ → DistanceFunction A → Set _

+
Definite _≈ₐ_ d = Base.Definite _≈ₐ_ _≡_ d 0

+

+
Symmetric : DistanceFunction A → Set _

+
Symmetric = Base.Symmetric _≡_

+

+
Bounded : DistanceFunction A → Set _

+
Bounded = Base.Bounded _≤_

+

+
TranslationInvariant : Op₂ A → DistanceFunction A → Set _

+
TranslationInvariant = Base.TranslationInvariant _≡_

+

+
-- Inequalities

+

+
TriangleInequality : DistanceFunction A → Set _

+
TriangleInequality = Base.TriangleInequality _≤_ _+_

+

+
MaxTriangleInequality : DistanceFunction A → Set _

+
MaxTriangleInequality = Base.TriangleInequality _≤_ _⊔_

+

+
-- Contractions

+

+
Contracting : (A → A) → DistanceFunction A → Set _

+
Contracting = Base.Contracting _≤_

+

+
ContractingOnOrbits : (A → A) → DistanceFunction A → Set _

+
ContractingOnOrbits = Base.ContractingOnOrbits _≤_

+

+
StrictlyContracting : Rel A ℓ → (A → A) → DistanceFunction A → Set _

+
StrictlyContracting _≈_ = Base.StrictlyContracting _≈_ _<_

+

+
StrictlyContractingOnOrbits : Rel A ℓ → (A → A) → DistanceFunction A → Set _

+
StrictlyContractingOnOrbits _≈_ = Base.StrictlyContractingOnOrbits _≈_ _<_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Nat.Structures.html b/site/static/agda_html/Function.Metric.Nat.Structures.html
new file mode 100644
index 000000000..7c7e28ab3
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Nat.Structures.html
@@ -0,0 +1,153 @@
+
+Function.Metric.Nat.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core definitions for metrics over ℕ

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Nat.Structures where

+

+
open import Data.Nat.Base hiding (suc)

+
open import Function.Base using (const)

+
open import Level using (Level; suc)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+

+
open import Function.Metric.Nat.Core

+
open import Function.Metric.Nat.Definitions

+
import Function.Metric.Structures as Base

+

+
private

+
  variable

+
    a ℓ : Level

+
    A   : Set a

+

+
------------------------------------------------------------------------

+
-- Proto-metrics

+

+
IsProtoMetric : Rel A ℓ → DistanceFunction A → Set _

+
IsProtoMetric _≈_ = Base.IsProtoMetric _≈_ _≡_ _≤_ 0

+

+
open Base using (module IsProtoMetric) public

+

+
------------------------------------------------------------------------

+
-- Pre-metrics

+

+
IsPreMetric : Rel A ℓ → DistanceFunction A → Set _

+
IsPreMetric _≈_ = Base.IsPreMetric _≈_ _≡_ _≤_ 0

+

+
open Base using (module IsPreMetric) public

+

+
------------------------------------------------------------------------

+
-- Quasi-semi-metrics

+

+
IsQuasiSemiMetric : Rel A ℓ → DistanceFunction A → Set _

+
IsQuasiSemiMetric _≈_ = Base.IsQuasiSemiMetric _≈_ _≡_ _≤_ 0

+

+
open Base using (module IsQuasiSemiMetric) public

+

+
------------------------------------------------------------------------

+
-- Semi-metrics

+

+
IsSemiMetric : Rel A ℓ → DistanceFunction A → Set _

+
IsSemiMetric _≈_ = Base.IsSemiMetric _≈_ _≡_ _≤_ 0

+

+
open Base using (module IsSemiMetric) public

+

+
------------------------------------------------------------------------

+
-- Metrics

+

+
IsMetric : Rel A ℓ → DistanceFunction A → Set _

+
IsMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_ 0 _+_

+

+
module IsMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}

+
                (M : IsMetric _≈_ d) where

+
  open Base.IsGeneralMetric M public

+

+
------------------------------------------------------------------------

+
-- Ultra-metrics

+

+
IsUltraMetric : Rel A ℓ → DistanceFunction A → Set _

+
IsUltraMetric _≈_ = Base.IsGeneralMetric _≈_ _≡_ _≤_ 0 _⊔_

+

+
module IsUltraMetric {_≈_ : Rel A ℓ} {d : DistanceFunction A}

+
                     (UM : IsUltraMetric _≈_ d) where

+
  open Base.IsGeneralMetric UM public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Nat.html b/site/static/agda_html/Function.Metric.Nat.html
new file mode 100644
index 000000000..a31aa2142
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Nat.html
@@ -0,0 +1,91 @@
+
+Function.Metric.Nat
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Metrics with ℕ as the codomain of the metric function

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Metric.Nat where

+

+
open import Function.Metric.Nat.Core public

+
open import Function.Metric.Nat.Definitions public

+
open import Function.Metric.Nat.Structures public

+
open import Function.Metric.Nat.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Metric.Structures.html b/site/static/agda_html/Function.Metric.Structures.html
new file mode 100644
index 000000000..9fe3fd363
--- /dev/null
+++ b/site/static/agda_html/Function.Metric.Structures.html
@@ -0,0 +1,174 @@
+
+Function.Metric.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some metric structures (not packed up with sets, operations, etc.)

+
------------------------------------------------------------------------

+

+
-- The contents of this module should usually be accessed via

+
-- `Function.Metric`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Structures using (IsPartialOrder; IsEquivalence)

+

+
module Function.Metric.Structures

+
  {a i ℓ₁ ℓ₂ ℓ₃} {A : Set a} {I : Set i}

+
  (_≈ₐ_ : Rel A ℓ₁) (_≈ᵢ_ : Rel I ℓ₂) (_≤_ : Rel I ℓ₃) (0# : I) where

+

+
open import Algebra.Core using (Op₂)

+
open import Function.Metric.Core

+
open import Function.Metric.Definitions

+
open import Level using (_⊔_)

+

+
------------------------------------------------------------------------

+
-- Proto-metrics

+

+
-- We do not insist that the ordering relation is total as otherwise

+
-- we would exclude the real numbers.

+

+
record IsProtoMetric (d : DistanceFunction A I)

+
                   : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  field

+
    isPartialOrder   : IsPartialOrder _≈ᵢ_ _≤_

+
    ≈-isEquivalence  : IsEquivalence _≈ₐ_

+
    cong             : Congruent _≈ₐ_ _≈ᵢ_ d

+
    nonNegative      : NonNegative _≤_ d 0#

+

+
  open IsPartialOrder isPartialOrder public

+
    renaming (module Eq to EqI)

+

+
  module EqC = IsEquivalence ≈-isEquivalence

+

+
------------------------------------------------------------------------

+
-- Pre-metrics

+

+
record IsPreMetric (d : DistanceFunction A I)

+
                 : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  field

+
    isProtoMetric : IsProtoMetric d

+
    ≈⇒0           : Definite _≈ₐ_ _≈ᵢ_ d 0#

+

+
  open IsProtoMetric isProtoMetric public

+

+
------------------------------------------------------------------------

+
-- Quasi-semi-metrics

+

+
record IsQuasiSemiMetric (d : DistanceFunction A I)

+
                       : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  field

+
    isPreMetric : IsPreMetric d

+
    0⇒≈         : Indiscernable _≈ₐ_ _≈ᵢ_ d 0#

+

+
  open IsPreMetric isPreMetric public

+

+
------------------------------------------------------------------------

+
-- Semi-metrics

+

+
record IsSemiMetric (d : DistanceFunction A I)

+
                  : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  field

+
    isQuasiSemiMetric : IsQuasiSemiMetric d

+
    sym               : Symmetric _≈ᵢ_ d

+

+
  open IsQuasiSemiMetric isQuasiSemiMetric public

+

+
------------------------------------------------------------------------

+
-- General metrics

+

+
-- A general metric obeys a generalised form of the triangle inequality.

+
-- It can be specialised to a standard metric/ultrametric/inframetric

+
-- etc. by providing the correct operator.

+
--

+
-- Furthermore we do not assume that _∙_ & 0# form a monoid as

+
-- associativity does not hold for p-relaxed metrics/p-inframetrics and

+
-- the identity laws do not hold for ultrametrics over negative

+
-- codomains.

+
--

+
-- See "Properties of distance spaces with power triangle inequalities"

+
-- by Daniel J. Greenhoe, 2016 (arXiv)

+

+
record IsGeneralMetric (_∙_ : Op₂ I) (d : DistanceFunction A I)

+
                     : Set (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  field

+
    isSemiMetric : IsSemiMetric d

+
    triangle     : TriangleInequality _≤_ _∙_ d

+

+
  open IsSemiMetric isSemiMetric public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Nary.NonDependent.Base.html b/site/static/agda_html/Function.Nary.NonDependent.Base.html
new file mode 100644
index 000000000..a473be944
--- /dev/null
+++ b/site/static/agda_html/Function.Nary.NonDependent.Base.html
@@ -0,0 +1,220 @@
+
+Function.Nary.NonDependent.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Heterogeneous N-ary Functions: basic types and operations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Nary.NonDependent.Base where

+

+
------------------------------------------------------------------------

+
-- Concrete examples can be found in README.Nary. This file's comments

+
-- are more focused on the implementation details and the motivations

+
-- behind the design decisions.

+
------------------------------------------------------------------------

+

+
open import Level using (Level; 0ℓ; _⊔_)

+
open import Data.Nat.Base using (ℕ; zero; suc)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Data.Unit.Polymorphic.Base

+
open import Function.Base using (_∘′_; _$′_; const; flip)

+

+
private

+
  variable

+
    a b c : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Type Definitions

+

+
-- We want to define n-ary function spaces and generic n-ary operations

+
-- on them such as (un)currying, zipWith, alignWith, etc.

+

+
-- We want users to be able to use these seamlessly whenever n is concrete.

+

+
-- In other words, we want Agda to infer the sets `A₁, ⋯, Aₙ` when we

+
-- write `uncurryₙ n f` where `f` has type `A₁ → ⋯ → Aₙ → B`. For this

+
-- to happen, we need the structure in which these Sets are stored to

+
-- effectively η-expand to `A₁, ⋯, Aₙ` when the parameter `n` is known.

+

+
-- Hence the following definitions:

+
------------------------------------------------------------------------

+

+
-- First, a "vector" of `n` Levels (defined by induction on n so that it

+
-- may be built by η-expansion and unification). Each Level will be that

+
-- of one of the Sets we want to take the n-ary product of.

+

+
Levels : ℕ → Set

+
Levels zero    = ⊤

+
Levels (suc n) = Level × Levels n

+

+
-- The overall Level of a `n` Sets of respective levels `l₁, ⋯, lₙ` will

+
-- be the least upper bound `l₁ ⊔ ⋯ ⊔ lₙ` of all of the Levels involved.

+
-- Hence the following definition of n-ary least upper bound:

+

+
⨆ : ∀ n → Levels n → Level

+
⨆ zero    _        = Level.zero

+
⨆ (suc n) (l , ls) = l ⊔ (⨆ n ls)

+

+
-- Second, a "vector" of `n` Sets whose respective Levels are determined

+
-- by the `Levels n` input.

+

+
Sets : ∀ n (ls : Levels n) → Set (Level.suc (⨆ n ls))

+
Sets zero    _        = ⊤

+
Sets (suc n) (l , ls) = Set l × Sets n ls

+

+
-- Third, a function type whose domains are given by a "vector" of `n`

+
-- Sets `A₁, ⋯, Aₙ` and whose codomain is `B`. `Arrows` forms such a

+
-- type of shape `A₁ → ⋯ → Aₙ → B` by induction on `n`.

+

+
Arrows : ∀ n {r ls} → Sets n ls → Set r → Set (r ⊔ (⨆ n ls))

+
Arrows zero    _        b = b

+
Arrows (suc n) (a , as) b = a → Arrows n as b

+

+
-- We introduce a notation for this definition

+

+
infixr 0 _⇉_

+
_⇉_ : ∀ {n ls r} → Sets n ls → Set r → Set (r ⊔ (⨆ n ls))

+
_⇉_ = Arrows _

+

+
------------------------------------------------------------------------

+
-- Operations on Sets

+

+
-- Level-respecting map

+

+
infixr -1 _<$>_

+

+
_<$>_ : (∀ {l} → Set l → Set l) →

+
        ∀ {n ls} → Sets n ls → Sets n ls

+
_<$>_ f {zero}   as        = _

+
_<$>_ f {suc n}  (a , as)  = f a , (f <$> as)

+

+
-- Level-modifying generalised map

+

+
lmap : (Level → Level) → ∀ n → Levels n → Levels n

+
lmap f zero    ls       = _

+
lmap f (suc n) (l , ls) = f l , lmap f n ls

+

+
smap : ∀ f → (∀ {l} → Set l → Set (f l)) →

+
       ∀ n {ls} → Sets n ls → Sets n (lmap f n ls)

+
smap f F zero    as       = _

+
smap f F (suc n) (a , as) = F a , smap f F n as

+

+
------------------------------------------------------------------------

+
-- Operations on Functions

+

+
-- mapping under n arguments

+

+
mapₙ : ∀ n {ls} {as : Sets n ls} → (B → C) → as ⇉ B → as ⇉ C

+
mapₙ zero    f v = f v

+
mapₙ (suc n) f g = mapₙ n f ∘′ g

+

+
-- compose function at the n-th position

+

+
infix 1 _%=_⊢_

+

+
_%=_⊢_ : ∀ n {ls} {as : Sets n ls} → (A → B) → as ⇉ (B → C) → as ⇉ (A → C)

+
n %= f ⊢ g = mapₙ n (_∘′ f) g

+

+
-- partially apply function at the n-th position

+

+
infix 1 _∷=_⊢_

+

+
_∷=_⊢_ : ∀ n {ls} {as : Sets n ls} → A → as ⇉ (A → B) → as ⇉ B

+
n ∷= x ⊢ g = mapₙ n (_$′ x) g

+

+
-- hole at the n-th position

+

+
holeₙ : ∀ n {ls} {as : Sets n ls} → (A → as ⇉ B) → as ⇉ (A → B)

+
holeₙ zero    f = f

+
holeₙ (suc n) f = holeₙ n ∘′ flip f

+

+
-- function constant in its n first arguments

+

+
-- Note that its type will only be inferred if it is used in a context

+
-- specifying what the type of the function ought to be. Just like the

+
-- usual const: there is no way to infer its domain from its argument.

+

+
constₙ : ∀ n {ls r} {as : Sets n ls} {b : Set r} → b → as ⇉ b

+
constₙ zero    v = v

+
constₙ (suc n) v = const (constₙ n v)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Nary.NonDependent.html b/site/static/agda_html/Function.Nary.NonDependent.html
new file mode 100644
index 000000000..e67500ccc
--- /dev/null
+++ b/site/static/agda_html/Function.Nary.NonDependent.html
@@ -0,0 +1,177 @@
+
+Function.Nary.NonDependent
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Heterogeneous N-ary Functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Nary.NonDependent where

+

+
------------------------------------------------------------------------

+
-- Concrete examples can be found in README.Nary. This file's comments

+
-- are more focused on the implementation details and the motivations

+
-- behind the design decisions.

+
------------------------------------------------------------------------

+

+
open import Level using (Level; 0ℓ; _⊔_; Lift)

+
open import Data.Nat.Base using (ℕ; zero; suc)

+
open import Data.Fin.Base using (Fin; zero; suc)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Data.Product.Nary.NonDependent

+
  using (Product; uncurryₙ; Equalₙ; curryₙ; fromEqualₙ; toEqualₙ)

+
open import Function.Base using (_∘′_; _$′_; const; flip)

+
open import Relation.Unary using (IUniversal)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; cong)

+

+
private

+
  variable

+
    a b r : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Re-exporting the basic operations

+

+
open import Function.Nary.NonDependent.Base public

+

+
------------------------------------------------------------------------

+
-- Additional operations on Levels

+

+
ltabulate : ∀ n → (Fin n → Level) → Levels n

+
ltabulate zero    f = _

+
ltabulate (suc n) f = f zero , ltabulate n (f ∘′ suc)

+

+
lreplicate : ∀ n → Level → Levels n

+
lreplicate n ℓ = ltabulate n (const ℓ)

+

+
0ℓs : ∀[ Levels ]

+
0ℓs = lreplicate _ 0ℓ

+

+
------------------------------------------------------------------------

+
-- Congruence

+

+
module _ n {ls} {as : Sets n ls} {R : Set r} (f : as ⇉ R) where

+

+
  private

+
    g : Product n as → R

+
    g = uncurryₙ n f

+

+
-- Congruentₙ : ∀ n. ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →

+
--                   a₁₁ ≡ a₁₂ → ⋯ → aₙ₁ ≡ aₙ₂ →

+
--                   f a₁₁ ⋯ aₙ₁ ≡ f a₁₂ ⋯ aₙ₂

+

+
  Congruentₙ : Set (r Level.⊔ ⨆ n ls)

+
  Congruentₙ = ∀ {l r} → Equalₙ n l r ⇉ (g l ≡ g r)

+

+
  congₙ : Congruentₙ

+
  congₙ = curryₙ n (cong g ∘′ fromEqualₙ n)

+

+
-- Congruence at a specific location

+

+
module _ m n {ls ls′} {as : Sets m ls} {bs : Sets n ls′}

+
         (f : as ⇉ (A → bs ⇉ B)) where

+

+
  private

+
    g : Product m as → A → Product n bs → B

+
    g vs a ws = uncurryₙ n (uncurryₙ m f vs a) ws

+

+
  congAt : ∀ {vs ws a₁ a₂} → a₁ ≡ a₂ → g vs a₁ ws ≡ g vs a₂ ws

+
  congAt {vs} {ws} = cong (λ a → g vs a ws)

+

+
------------------------------------------------------------------------

+
-- Injectivity

+

+
module _ n {ls} {as : Sets n ls} {R : Set r} (con : as ⇉ R) where

+

+
-- Injectiveₙ : ∀ n. ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →

+
--                   con a₁₁ ⋯ aₙ₁ ≡ con a₁₂ ⋯ aₙ₂ →

+
--                   a₁₁ ≡ a₁₂ × ⋯ × aₙ₁ ≡ aₙ₂

+

+
  private

+
    c : Product n as → R

+
    c = uncurryₙ n con

+

+
  Injectiveₙ : Set (r Level.⊔ ⨆ n ls)

+
  Injectiveₙ = ∀ {l r} → c l ≡ c r → Product n (Equalₙ n l r)

+

+
  injectiveₙ : (∀ {l r} → c l ≡ c r → l ≡ r) → Injectiveₙ

+
  injectiveₙ con-inj eq = toEqualₙ n (con-inj eq)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Properties.Bijection.html b/site/static/agda_html/Function.Properties.Bijection.html
new file mode 100644
index 000000000..e830f77a7
--- /dev/null
+++ b/site/static/agda_html/Function.Properties.Bijection.html
@@ -0,0 +1,155 @@
+
+Function.Properties.Bijection
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some basic properties of bijections.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Properties.Bijection where

+

+
open import Function.Bundles using (Bijection; Inverse; Equivalence;

+
  _⤖_; _↔_; _⇔_)

+
open import Level using (Level)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions using (Reflexive; Trans)

+
open import Relation.Binary.PropositionalEquality.Properties using (setoid)

+
open import Data.Product.Base using (_,_; proj₁; proj₂)

+
open import Function.Base using (_∘_)

+
open import Function.Properties.Surjection using (injective⇒to⁻-cong)

+
open import Function.Properties.Inverse using (Inverse⇒Equivalence)

+

+
import Function.Construct.Identity as Identity

+
import Function.Construct.Symmetry as Symmetry

+
import Function.Construct.Composition as Composition

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+
    A B : Set a

+
    T S : Setoid a ℓ

+

+
------------------------------------------------------------------------

+
-- Setoid properties

+

+
refl : Reflexive (Bijection {a} {ℓ})

+
refl = Identity.bijection _

+

+
-- Can't prove full symmetry as we have no proof that the witness

+
-- produced by the surjection proof preserves equality

+
sym-≡ : Bijection S (setoid B) → Bijection (setoid B) S

+
sym-≡ = Symmetry.bijection-≡

+

+
trans : Trans (Bijection {a} {ℓ₁} {b} {ℓ₂}) (Bijection {b} {ℓ₂} {c} {ℓ₃}) Bijection

+
trans = Composition.bijection

+

+
------------------------------------------------------------------------

+
-- Propositional properties

+

+
⤖-isEquivalence : IsEquivalence {ℓ = ℓ} _⤖_

+
⤖-isEquivalence = record

+
  { refl  = refl

+
  ; sym   = sym-≡

+
  ; trans = trans

+
  }

+

+
------------------------------------------------------------------------

+
-- Conversion functions

+

+
Bijection⇒Inverse : Bijection S T → Inverse S T

+
Bijection⇒Inverse bij = record

+
  { to        = to

+
  ; from      = to⁻

+
  ; to-cong   = cong

+
  ; from-cong = injective⇒to⁻-cong surjection injective

+
  ; inverse   = (λ y≈to⁻[x] → Eq₂.trans (cong y≈to⁻[x]) (to∘to⁻ _)) ,

+
                (λ y≈to[x] → injective (Eq₂.trans (to∘to⁻ _) y≈to[x]))

+
  }

+
  where open Bijection bij; to∘to⁻ = proj₂ ∘ strictlySurjective

+

+
Bijection⇒Equivalence : Bijection T S → Equivalence T S

+
Bijection⇒Equivalence = Inverse⇒Equivalence ∘ Bijection⇒Inverse

+

+
⤖⇒↔ : A ⤖ B → A ↔ B

+
⤖⇒↔ = Bijection⇒Inverse

+

+
⤖⇒⇔ : A ⤖ B → A ⇔ B

+
⤖⇒⇔ = Bijection⇒Equivalence

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Properties.Inverse.HalfAdjointEquivalence.html b/site/static/agda_html/Function.Properties.Inverse.HalfAdjointEquivalence.html
new file mode 100644
index 000000000..b2e264964
--- /dev/null
+++ b/site/static/agda_html/Function.Properties.Inverse.HalfAdjointEquivalence.html
@@ -0,0 +1,200 @@
+
+Function.Properties.Inverse.HalfAdjointEquivalence
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Half adjoint equivalences

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Properties.Inverse.HalfAdjointEquivalence where

+

+
open import Function.Base using (id; _∘_)

+
open import Function.Bundles using (Inverse; _↔_; mk↔ₛ′)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.PropositionalEquality

+
  using (_≡_; refl; cong; sym; trans; trans-reflʳ; cong-≡id; cong-∘; naturality;

+
         cong-id; trans-assoc; trans-symˡ; module ≡-Reasoning)

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+

+
-- Half adjoint equivalences (see the HoTT book).

+
--

+
-- They are inverses with an extra coherence condition that the left

+
-- and right inversion proofs interact the right way with `cong`.

+

+
infix 4 _≃_

+

+
record _≃_ (A : Set a) (B : Set b) : Set (a ⊔ b) where

+
  field

+
    to               : A → B

+
    from             : B → A

+
    left-inverse-of  : ∀ x → from (to x) ≡ x

+
    right-inverse-of : ∀ x → to (from x) ≡ x

+
    left-right       : ∀ x → cong to (left-inverse-of x) ≡ right-inverse-of (to x)

+

+
  -- The forward direction of a half adjoint equivalence is injective.

+

+
  injective : ∀ {x y} → to x ≡ to y → x ≡ y

+
  injective {x} {y} to-x≡to-y =

+
    x            ≡⟨ sym (left-inverse-of _) ⟩

+
    from (to x)  ≡⟨ cong from to-x≡to-y ⟩

+
    from (to y)  ≡⟨ left-inverse-of _ ⟩

+
    y            ∎

+
    where open ≡-Reasoning

+

+
-- Half adjoint equivalences can be turned into inverses.

+

+
≃⇒↔ : A ≃ B → A ↔ B

+
≃⇒↔ A≃B = mk↔ₛ′ to from right-inverse-of left-inverse-of

+
  where open _≃_ A≃B

+

+
-- Inverses can be turned into half adjoint equivalences.

+
--

+
-- (This proof is based on one in the HoTT book.)

+

+
↔⇒≃ : A ↔ B → A ≃ B

+
↔⇒≃ A↔B = record

+
  { to               = to

+
  ; from             = from

+
  ; left-inverse-of  = strictlyInverseʳ

+
  ; right-inverse-of = right-inverse-of

+
  ; left-right       = left-right

+
  }

+
  where

+
  open ≡-Reasoning

+
  open module A↔B = Inverse A↔B

+

+
  right-inverse-of : ∀ x → to (from x) ≡ x

+
  right-inverse-of x =

+
    to (from x)               ≡⟨ sym (A↔B.strictlyInverseˡ _) ⟩

+
    to (from (to (from x)))   ≡⟨ cong to (strictlyInverseʳ  _) ⟩

+
    to (from x)               ≡⟨ A↔B.strictlyInverseˡ _ ⟩

+
    x                         ∎

+

+
  left-right :

+
    ∀ x →

+
    cong to (strictlyInverseʳ x) ≡ right-inverse-of (to x)

+
  left-right x =

+
    cong to (strictlyInverseʳ x)               ≡⟨⟩

+

+
    trans refl (cong to (strictlyInverseʳ _))  ≡⟨ cong (λ p → trans p (cong to (strictlyInverseʳ _)))

+
                                                          (sym (trans-symˡ (A↔B.strictlyInverseˡ _))) ⟩

+
    trans (trans (sym (A↔B.strictlyInverseˡ _))

+
               (A↔B.strictlyInverseˡ _))

+
      (cong to (strictlyInverseʳ _))           ≡⟨ trans-assoc (sym (A↔B.strictlyInverseˡ _)) ⟩

+

+
    trans (sym (A↔B.strictlyInverseˡ _))

+
      (trans (A↔B.strictlyInverseˡ _)

+
         (cong to (strictlyInverseʳ _)))       ≡⟨ cong (trans (sym (A↔B.strictlyInverseˡ _))) lemma ⟩

+

+
    trans (sym (A↔B.strictlyInverseˡ _))

+
      (trans (cong to (strictlyInverseʳ _))

+
         (trans (A↔B.strictlyInverseˡ _) refl))      ≡⟨⟩

+

+
    right-inverse-of (to x)                      ∎

+
    where

+
    lemma =

+
      trans (A↔B.strictlyInverseˡ _)

+
        (cong to (strictlyInverseʳ _))             ≡⟨ cong (trans (A↔B.strictlyInverseˡ _)) (sym (cong-id _)) ⟩

+

+
      trans (A↔B.strictlyInverseˡ _)

+
        (cong id (cong to (strictlyInverseʳ _)))   ≡⟨ sym (naturality A↔B.strictlyInverseˡ) ⟩

+

+
      trans (cong (to ∘ from)

+
                 (cong to (strictlyInverseʳ _)))

+
        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (λ p → trans p (A↔B.strictlyInverseˡ _))

+
                                                              (sym (cong-∘ _)) ⟩

+
      trans (cong (to ∘ from ∘ to)

+
                      (strictlyInverseʳ _))

+
        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (λ p → trans p (A↔B.strictlyInverseˡ _))

+
                                                              (cong-∘ _) ⟩

+
      trans (cong to

+
                 (cong (from ∘ to)

+
                    (strictlyInverseʳ _)))

+
        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (λ p → trans (cong to p) (strictlyInverseˡ (to x)))

+
                                                              (cong-≡id strictlyInverseʳ) ⟩

+
      trans (cong to (strictlyInverseʳ _))

+
        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (trans (cong to (strictlyInverseʳ _)))

+
                                                              (sym (trans-reflʳ _)) ⟩

+
      trans (cong to (strictlyInverseʳ _))

+
        (trans (A↔B.strictlyInverseˡ _) refl)            ∎

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Properties.Inverse.html b/site/static/agda_html/Function.Properties.Inverse.html
new file mode 100644
index 000000000..f8b07a377
--- /dev/null
+++ b/site/static/agda_html/Function.Properties.Inverse.html
@@ -0,0 +1,236 @@
+
+Function.Properties.Inverse
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of inverses.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Properties.Inverse where

+

+
open import Axiom.Extensionality.Propositional using (Extensionality)

+
open import Data.Product.Base using (_,_; proj₁; proj₂)

+
open import Function.Bundles

+
import Function.Properties.RightInverse as RightInverse

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (REL)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
import Function.Consequences.Setoid as Consequences

+

+
import Function.Construct.Identity as Identity

+
import Function.Construct.Symmetry as Symmetry

+
import Function.Construct.Composition as Composition

+

+
private

+
  variable

+
    a b ℓ ℓ₁ ℓ₂ : Level

+
    A B C D : Set a

+
    S T U V : Setoid a ℓ

+

+
------------------------------------------------------------------------

+
-- Setoid bundles

+

+
open Identity    public using () renaming (inverse to refl)

+
open Symmetry    public using () renaming (inverse to sym)

+
open Composition public using () renaming (inverse to trans)

+

+
isEquivalence : IsEquivalence (Inverse {a} {b})

+
isEquivalence = record

+
  { refl  = λ {x} → Identity.inverse x

+
  ; sym   = sym

+
  ; trans = trans

+
  }

+

+
------------------------------------------------------------------------

+
-- Propositional bundles

+

+
↔-refl : A ↔ A

+
↔-refl = Identity.↔-id _

+

+
↔-sym : A ↔ B → B ↔ A

+
↔-sym = Symmetry.↔-sym

+

+
↔-trans : A ↔ B → B ↔ C → A ↔ C

+
↔-trans = Composition.inverse

+

+
-- need to η-expand for everything to line up properly

+
↔-isEquivalence : IsEquivalence {ℓ = ℓ} _↔_

+
↔-isEquivalence = record

+
  { refl  = ↔-refl

+
  ; sym   = ↔-sym

+
  ; trans = ↔-trans

+
  }

+

+
------------------------------------------------------------------------

+
-- Conversion functions

+

+
toFunction : Inverse S T → Func S T

+
toFunction I = record { to = to ; cong = to-cong }

+
  where open Inverse I

+

+
fromFunction : Inverse S T → Func T S

+
fromFunction I = record { to = from ; cong = from-cong }

+
  where open Inverse I

+

+
Inverse⇒Injection : Inverse S T → Injection S T

+
Inverse⇒Injection {S = S} {T = T} I = record

+
  { to = to

+
  ; cong = to-cong

+
  ; injective = inverseʳ⇒injective to inverseʳ

+
  } where open Inverse I; open Consequences S T

+

+
Inverse⇒Surjection : Inverse S T → Surjection S T

+
Inverse⇒Surjection {S = S} {T = T} I = record

+
  { to = to

+
  ; cong = to-cong

+
  ; surjective = inverseˡ⇒surjective inverseˡ

+
  } where open Inverse I; open Consequences S T

+

+
Inverse⇒Bijection : Inverse S T → Bijection S T

+
Inverse⇒Bijection {S = S} {T = T} I = record

+
  { to        = to

+
  ; cong      = to-cong

+
  ; bijective = inverseᵇ⇒bijective inverse

+
  } where open Inverse I; open Consequences S T

+

+
Inverse⇒Equivalence : Inverse S T → Equivalence S T

+
Inverse⇒Equivalence I = record

+
  { to        = to

+
  ; from      = from

+
  ; to-cong   = to-cong

+
  ; from-cong = from-cong

+
  } where open Inverse I

+

+
↔⇒⟶ : A ↔ B → A ⟶ B

+
↔⇒⟶ = toFunction

+

+
↔⇒⟵ : A ↔ B → B ⟶ A

+
↔⇒⟵ = fromFunction

+

+
↔⇒↣ : A ↔ B → A ↣ B

+
↔⇒↣ = Inverse⇒Injection

+

+
↔⇒↠ : A ↔ B → A ↠ B

+
↔⇒↠ = Inverse⇒Surjection

+

+
↔⇒⤖ : A ↔ B → A ⤖ B

+
↔⇒⤖ = Inverse⇒Bijection

+

+
↔⇒⇔ : A ↔ B → A ⇔ B

+
↔⇒⇔ = Inverse⇒Equivalence

+

+
↔⇒↩ : A ↔ B → A ↩ B

+
↔⇒↩ = Inverse.leftInverse

+

+
↔⇒↪ : A ↔ B → A ↪ B

+
↔⇒↪ = Inverse.rightInverse

+

+
-- The functions above can be combined with the following lemma to

+
-- transport an arbitrary relation R (e.g. Injection) across

+
-- inverses.

+
transportVia : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →

+
               (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →

+
               (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →

+
               Inverse S T → R T U → Inverse U V → R S V

+
transportVia R-trans inv⇒R IBA RBC ICD =

+
  R-trans (inv⇒R IBA) (R-trans RBC (inv⇒R ICD))

+

+
------------------------------------------------------------------------

+
-- Other

+

+
module _ (ext : ∀ {a b} → Extensionality a b) where

+

+
  ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)

+
  ↔-fun A↔B C↔D = mk↔ₛ′

+
    (λ a→c b → to C↔D (a→c (from A↔B b)))

+
    (λ b→d a → from C↔D (b→d (to A↔B a)))

+
    (λ b→d → ext λ _ → ≡.trans (strictlyInverseˡ C↔D _ ) (≡.cong b→d (strictlyInverseˡ A↔B _)))

+
    (λ a→c → ext λ _ → ≡.trans (strictlyInverseʳ C↔D _ ) (≡.cong a→c (strictlyInverseʳ A↔B _)))

+
    where open Inverse

+

+
module _ (I : Inverse S T) where

+
  open Inverse I

+

+
  to-from : ∀ {x y} → to x Eq₂.≈ y → from y Eq₁.≈ x

+
  to-from = RightInverse.to-from rightInverse

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Properties.RightInverse.html b/site/static/agda_html/Function.Properties.RightInverse.html
new file mode 100644
index 000000000..43f7f4ceb
--- /dev/null
+++ b/site/static/agda_html/Function.Properties.RightInverse.html
@@ -0,0 +1,158 @@
+
+Function.Properties.RightInverse
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of right inverses

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Properties.RightInverse where

+

+
open import Function.Base

+
open import Function.Definitions

+
open import Function.Bundles

+
open import Function.Consequences using (inverseˡ⇒surjective)

+
open import Level using (Level)

+
open import Data.Product.Base using (_,_)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
private

+
  variable

+
    ℓ₁ ℓ₂ a b : Level

+
    A B : Set a

+
    S T : Setoid a ℓ₁

+

+
------------------------------------------------------------------------

+
-- Constructors

+

+
mkRightInverse : (e : Equivalence S T) (open Equivalence e) →

+
                 Inverseʳ Eq₁._≈_ Eq₂._≈_ to from →

+
                 RightInverse S T

+
mkRightInverse eq invʳ = record

+
  { Equivalence eq

+
  ; inverseʳ = invʳ

+
  }

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
RightInverse⇒LeftInverse : RightInverse S T → LeftInverse T S

+
RightInverse⇒LeftInverse I = record

+
  { to         = from

+
  ; from       = to

+
  ; to-cong    = from-cong

+
  ; from-cong  = to-cong

+
  ; inverseˡ   = inverseʳ

+
  } where open RightInverse I

+

+
LeftInverse⇒RightInverse : LeftInverse S T → RightInverse T S

+
LeftInverse⇒RightInverse I = record

+
  { to         = from

+
  ; from       = to

+
  ; to-cong    = from-cong

+
  ; from-cong  = to-cong

+
  ; inverseʳ    = inverseˡ

+
  } where open LeftInverse I

+

+
RightInverse⇒Surjection : RightInverse S T → Surjection T S

+
RightInverse⇒Surjection I = record

+
  { to         = from

+
  ; cong       = from-cong

+
  ; surjective = inverseˡ⇒surjective Eq₁._≈_ inverseʳ

+
  } where open RightInverse I

+

+
↪⇒↠ : B ↪ A → A ↠ B

+
↪⇒↠ = RightInverse⇒Surjection

+

+
↪⇒↩ : B ↪ A → A ↩ B

+
↪⇒↩ = RightInverse⇒LeftInverse

+

+
↩⇒↪ : B ↩ A → A ↪ B

+
↩⇒↪ = LeftInverse⇒RightInverse

+

+
------------------------------------------------------------------------

+
-- Other

+

+
module _ (R : RightInverse S T) where

+
  open RightInverse R

+

+
  to-from : ∀ {x y} → to x Eq₂.≈ y → from y Eq₁.≈ x

+
  to-from eq = Eq₁.trans (from-cong (Eq₂.sym eq)) (strictlyInverseʳ _)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Properties.Surjection.html b/site/static/agda_html/Function.Properties.Surjection.html
new file mode 100644
index 000000000..ed8c2cce3
--- /dev/null
+++ b/site/static/agda_html/Function.Properties.Surjection.html
@@ -0,0 +1,158 @@
+
+Function.Properties.Surjection
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of surjections

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Properties.Surjection where

+

+
open import Function.Base using (_∘_; _$_)

+
open import Function.Definitions using (Surjective; Injective; Congruent)

+
open import Function.Bundles using (Func; Surjection; _↠_; _⟶_; _↪_; mk↪;

+
  _⇔_; mk⇔)

+
import Function.Construct.Identity as Identity

+
import Function.Construct.Composition as Compose

+
open import Level using (Level)

+
open import Data.Product.Base using (proj₁; proj₂)

+
import Relation.Binary.PropositionalEquality.Core as ≡

+
open import Relation.Binary.Definitions using (Reflexive; Trans)

+
open import Relation.Binary.Bundles using (Setoid)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+
    A B : Set a

+
    T S : Setoid a ℓ

+

+
------------------------------------------------------------------------

+
-- Constructors

+

+
mkSurjection : (f : Func S T) (open Func f) →

+
              Surjective Eq₁._≈_ Eq₂._≈_ to  →

+
              Surjection S T

+
mkSurjection f surjective = record

+
  { Func f

+
  ; surjective = surjective

+
  }

+

+
------------------------------------------------------------------------

+
-- Conversion functions

+

+
↠⇒⟶ : A ↠ B → A ⟶ B

+
↠⇒⟶ = Surjection.function

+

+
↠⇒↪ : A ↠ B → B ↪ A

+
↠⇒↪ s = mk↪ {from = to} λ { ≡.refl → proj₂ (strictlySurjective _)}

+
  where open Surjection s

+

+
↠⇒⇔ : A ↠ B → A ⇔ B

+
↠⇒⇔ s = mk⇔ to (proj₁ ∘ surjective)

+
  where open Surjection s

+

+
------------------------------------------------------------------------

+
-- Setoid properties

+

+
refl : Reflexive (Surjection {a} {ℓ})

+
refl {x = x} = Identity.surjection x

+

+
trans : Trans (Surjection {a} {ℓ₁} {b} {ℓ₂})

+
              (Surjection {b} {ℓ₂} {c} {ℓ₃})

+
              (Surjection {a} {ℓ₁} {c} {ℓ₃})

+
trans = Compose.surjection

+

+
------------------------------------------------------------------------

+
-- Other

+

+
injective⇒to⁻-cong : (surj : Surjection S T) →

+
                      (open Surjection surj) →

+
                      Injective Eq₁._≈_ Eq₂._≈_ to →

+
                      Congruent Eq₂._≈_ Eq₁._≈_ to⁻

+
injective⇒to⁻-cong {T = T} surj injective {x} {y} x≈y = injective $ begin

+
  to (to⁻ x) ≈⟨ to∘to⁻ x ⟩

+
  x          ≈⟨ x≈y ⟩

+
  y          ≈⟨ to∘to⁻ y ⟨

+
  to (to⁻ y) ∎

+
  where

+
  open ≈-Reasoning T

+
  open Surjection surj

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Related.Propositional.html b/site/static/agda_html/Function.Related.Propositional.html
new file mode 100644
index 000000000..74691533d
--- /dev/null
+++ b/site/static/agda_html/Function.Related.Propositional.html
@@ -0,0 +1,468 @@
+
+Function.Related.Propositional
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Relatedness for the function hierarchy

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Related.Propositional where

+

+
open import Level

+
open import Relation.Binary

+
  using (Rel; REL; Sym; Reflexive; Trans; IsEquivalence; Setoid; IsPreorder; Preorder)

+
open import Function.Bundles

+
open import Function.Base

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+
open import Relation.Binary.Reasoning.Syntax

+

+
open import Function.Properties.Surjection   using (↠⇒↪; ↠⇒⇔)

+
open import Function.Properties.RightInverse using (↪⇒↠)

+
open import Function.Properties.Bijection    using (⤖⇒↔; ⤖⇒⇔)

+
open import Function.Properties.Inverse      using (↔⇒⤖; ↔⇒⇔; ↔⇒↣; ↔⇒↠)

+

+
import Function.Construct.Symmetry    as Symmetry

+
import Function.Construct.Identity    as Identity

+
import Function.Construct.Composition as Composition

+

+
------------------------------------------------------------------------

+
-- Relatedness

+

+
-- There are several kinds of "relatedness".

+

+
-- The idea to include kinds other than equivalence and bijection came

+
-- from Simon Thompson and Bengt Nordström. /NAD

+

+
data Kind : Set where

+
  implication        : Kind

+
  reverseImplication : Kind

+
  equivalence        : Kind

+
  injection          : Kind

+
  reverseInjection   : Kind

+
  leftInverse        : Kind

+
  surjection         : Kind

+
  bijection          : Kind

+

+
private

+
  variable

+
    a b c p : Level

+
    A B C : Set a

+
    k : Kind

+

+
-- Interpretation of the codes above. The code "bijection" is

+
-- interpreted as Inverse rather than Bijection; the two types are

+
-- equivalent.

+

+
infix 4 _∼[_]_

+

+
_∼[_]_ : Set a → Kind → Set b → Set _

+
A ∼[ implication        ] B = A ⟶ B

+
A ∼[ reverseImplication ] B = B ⟶ A

+
A ∼[ equivalence        ] B = A ⇔ B

+
A ∼[ injection          ] B = A ↣ B

+
A ∼[ reverseInjection   ] B = B ↣ A

+
A ∼[ leftInverse        ] B = A ↪ B

+
A ∼[ surjection         ] B = A ↠ B

+
A ∼[ bijection          ] B = A ↔ B

+

+
-- A non-infix synonym.

+

+
Related : Kind → Set a → Set b → Set _

+
Related k A B = A ∼[ k ] B

+

+
-- The bijective equality implies any kind of relatedness.

+

+
↔⇒ : A ∼[ bijection ] B → A ∼[ k ] B

+
↔⇒ {k = implication}        = mk⟶ ∘ Inverse.to

+
↔⇒ {k = reverseImplication} = mk⟶ ∘ Inverse.from

+
↔⇒ {k = equivalence}        = ↔⇒⇔

+
↔⇒ {k = injection}          = ↔⇒↣

+
↔⇒ {k = reverseInjection}   = ↔⇒↣ ∘ Symmetry.inverse

+
↔⇒ {k = leftInverse}        = Inverse.rightInverse

+
↔⇒ {k = surjection}         = ↔⇒↠

+
↔⇒ {k = bijection}          = id

+

+
-- Propositional equality also implies any kind of relatedness.

+

+
≡⇒ : A ≡ B → A ∼[ k ] B

+
≡⇒ ≡.refl = ↔⇒ (Identity.↔-id _)

+

+
------------------------------------------------------------------------

+
-- Special kinds of kinds

+

+
-- Kinds whose interpretation is symmetric.

+

+
data SymmetricKind : Set where

+
  equivalence : SymmetricKind

+
  bijection   : SymmetricKind

+

+
-- Forgetful map.

+

+
⌊_⌋ : SymmetricKind → Kind

+
⌊ equivalence ⌋ = equivalence

+
⌊ bijection   ⌋ = bijection

+

+
-- The proof of symmetry can be found below.

+

+
-- Kinds whose interpretation include a function which "goes in the

+
-- forward direction".

+

+
data ForwardKind : Set where

+
  implication : ForwardKind

+
  equivalence : ForwardKind

+
  injection   : ForwardKind

+
  leftInverse : ForwardKind

+
  surjection  : ForwardKind

+
  bijection   : ForwardKind

+

+
-- Forgetful map.

+

+
⌊_⌋→ : ForwardKind → Kind

+
⌊ implication  ⌋→ = implication

+
⌊ equivalence  ⌋→ = equivalence

+
⌊ injection    ⌋→ = injection

+
⌊ leftInverse  ⌋→ = leftInverse

+
⌊ surjection   ⌋→ = surjection

+
⌊ bijection    ⌋→ = bijection

+

+
-- The function.

+

+
⇒→ : ∀ {k} → A ∼[ ⌊ k ⌋→ ] B → A → B

+
⇒→ {k = implication} = Func.to

+
⇒→ {k = equivalence} = Equivalence.to

+
⇒→ {k = injection}   = Injection.to

+
⇒→ {k = leftInverse} = RightInverse.to

+
⇒→ {k = surjection}  = Surjection.to

+
⇒→ {k = bijection}   = Inverse.to

+

+
-- Kinds whose interpretation include a function which "goes backwards".

+

+
data BackwardKind : Set where

+
  reverseImplication : BackwardKind

+
  equivalence        : BackwardKind

+
  reverseInjection   : BackwardKind

+
  leftInverse        : BackwardKind

+
  surjection         : BackwardKind

+
  bijection          : BackwardKind

+

+
-- Forgetful map.

+

+
⌊_⌋← : BackwardKind → Kind

+
⌊ reverseImplication ⌋← = reverseImplication

+
⌊ equivalence        ⌋← = equivalence

+
⌊ reverseInjection   ⌋← = reverseInjection

+
⌊ leftInverse        ⌋← = leftInverse

+
⌊ surjection         ⌋← = surjection

+
⌊ bijection          ⌋← = bijection

+

+
-- The function.

+

+
⇒← : ∀ {k} → A ∼[ ⌊ k ⌋← ] B → B → A

+
⇒← {k = reverseImplication} = Func.to

+
⇒← {k = equivalence}        = Equivalence.from

+
⇒← {k = reverseInjection}   = Injection.to

+
⇒← {k = leftInverse}        = RightInverse.from

+
⇒← {k = surjection}         = RightInverse.to ∘ ↠⇒↪

+
⇒← {k = bijection}          = Inverse.from

+

+
-- Kinds whose interpretation include functions going in both

+
-- directions.

+

+
data EquivalenceKind : Set where

+
  equivalence : EquivalenceKind

+
  leftInverse : EquivalenceKind

+
  surjection  : EquivalenceKind

+
  bijection   : EquivalenceKind

+

+
-- Forgetful map.

+

+
⌊_⌋⇔ : EquivalenceKind → Kind

+
⌊ equivalence ⌋⇔ = equivalence

+
⌊ leftInverse ⌋⇔ = leftInverse

+
⌊ surjection  ⌋⇔ = surjection

+
⌊ bijection   ⌋⇔ = bijection

+

+
-- The functions.

+

+
⇒⇔ : ∀ {k} → A ∼[ ⌊ k ⌋⇔ ] B → A ∼[ equivalence ] B

+
⇒⇔ {k = equivalence} = id

+
⇒⇔ {k = leftInverse} = RightInverse.equivalence

+
⇒⇔ {k = surjection}  = ↠⇒⇔

+
⇒⇔ {k = bijection}   = ↔⇒⇔

+

+
-- Conversions between special kinds.

+

+
⇔⌊_⌋ : SymmetricKind → EquivalenceKind

+
⇔⌊ equivalence ⌋ = equivalence

+
⇔⌊ bijection   ⌋ = bijection

+

+
→⌊_⌋ : EquivalenceKind → ForwardKind

+
→⌊ equivalence ⌋ = equivalence

+
→⌊ leftInverse ⌋ = leftInverse

+
→⌊ surjection  ⌋ = surjection

+
→⌊ bijection   ⌋ = bijection

+

+
←⌊_⌋ : EquivalenceKind → BackwardKind

+
←⌊ equivalence ⌋ = equivalence

+
←⌊ leftInverse ⌋ = leftInverse

+
←⌊ surjection  ⌋ = surjection

+
←⌊ bijection   ⌋ = bijection

+

+
------------------------------------------------------------------------

+
-- Opposites

+

+
-- For every kind there is an opposite kind.

+

+
_op : Kind → Kind

+
implication        op = reverseImplication

+
reverseImplication op = implication

+
equivalence        op = equivalence

+
injection          op = reverseInjection

+
reverseInjection   op = injection

+
leftInverse        op = surjection

+
surjection         op = leftInverse

+
bijection          op = bijection

+

+
-- For every morphism there is a corresponding reverse morphism of the

+
-- opposite kind.

+

+
reverse : A ∼[ k ] B → B ∼[ k op ] A

+
reverse {k = implication}        = id

+
reverse {k = reverseImplication} = id

+
reverse {k = equivalence}        = Symmetry.⇔-sym

+
reverse {k = injection}          = id

+
reverse {k = reverseInjection}   = id

+
reverse {k = leftInverse}        = ↪⇒↠

+
reverse {k = surjection}         = ↠⇒↪

+
reverse {k = bijection}          = Symmetry.↔-sym

+

+
------------------------------------------------------------------------

+
-- For a fixed universe level every kind is a preorder and each

+
-- symmetric kind is an equivalence

+

+
K-refl : Reflexive (Related {a} k)

+
K-refl {k = implication}        = Identity.⟶-id _

+
K-refl {k = reverseImplication} = Identity.⟶-id _

+
K-refl {k = equivalence}        = Identity.⇔-id _

+
K-refl {k = injection}          = Identity.↣-id _

+
K-refl {k = reverseInjection}   = Identity.↣-id _

+
K-refl {k = leftInverse}        = Identity.↪-id _

+
K-refl {k = surjection}         = Identity.↠-id _

+
K-refl {k = bijection}          = Identity.↔-id _

+

+
K-reflexive : _≡_ Relation.Binary.⇒ Related {a} k

+
K-reflexive ≡.refl = K-refl

+

+
K-trans : Trans (Related {a} {b} k)

+
                (Related {b} {c} k)

+
                (Related {a} {c} k)

+
K-trans {k = implication}        = flip Composition._⟶-∘_

+
K-trans {k = reverseImplication} = Composition._⟶-∘_

+
K-trans {k = equivalence}        = flip Composition._⇔-∘_

+
K-trans {k = injection}          = flip Composition._↣-∘_

+
K-trans {k = reverseInjection}   = Composition._↣-∘_

+
K-trans {k = leftInverse}        = flip Composition._↪-∘_

+
K-trans {k = surjection}         = flip Composition._↠-∘_

+
K-trans {k = bijection}          = flip Composition._↔-∘_

+

+
SK-sym : ∀ {k} → Sym (Related {a} {b} ⌊ k ⌋)

+
                     (Related {b} {a} ⌊ k ⌋)

+
SK-sym {k = equivalence} = reverse

+
SK-sym {k = bijection}   = reverse

+

+
SK-isEquivalence : ∀ k → IsEquivalence {ℓ = a} (Related ⌊ k ⌋)

+
SK-isEquivalence k = record

+
  { refl  = K-refl

+
  ; sym   = SK-sym

+
  ; trans = K-trans

+
  }

+

+
SK-setoid : SymmetricKind → (ℓ : Level) → Setoid _ _

+
SK-setoid k ℓ = record { isEquivalence = SK-isEquivalence {ℓ} k }

+

+
K-isPreorder : ∀ k → IsPreorder {ℓ = a} _↔_ (Related k)

+
K-isPreorder k = record

+
  { isEquivalence = SK-isEquivalence bijection

+
  ; reflexive     = ↔⇒

+
  ; trans         = K-trans

+
  }

+

+
K-preorder : Kind → (ℓ : Level) → Preorder _ ℓ _

+
K-preorder k ℓ = record { isPreorder = K-isPreorder k }

+

+
------------------------------------------------------------------------

+
-- Equational reasoning

+

+
-- Equational reasoning for related things. Note that we don't use

+
-- the `Relation.Binary.Reasoning.Syntax` for this as this relation

+
-- is heterogeneous.

+

+
module EquationalReasoning {k : Kind} where

+

+
  -- Combinators with one heterogeneous relation

+
  module _ {a b : Level} where

+
    open begin-syntax (Related {a} {b} k) id public

+
    open ≡-noncomputing-syntax (Related {a} {b} k) public

+

+
  -- Combinators with two heterogeneous relations

+
  module _ {a b c : Level} where

+
    private

+
      rel1 = Related {b} {c} k

+
      rel2 = Related {a} {c} k

+

+
    open ∼-syntax rel1 rel2 K-trans public

+
    open ⤖-syntax rel1 rel2 (K-trans ∘′ ↔⇒ ∘′ ⤖⇒↔) Symmetry.⤖-sym public

+
    open ↔-syntax rel1 rel2 (K-trans ∘′ ↔⇒) Symmetry.↔-sym public

+

+
  -- Combinators with homogeneous relations

+
  module _ {a : Level} where

+
    open end-syntax (Related {a} k) K-refl public

+

+

+
  infixr 2 _↔⟨⟩_

+
  _↔⟨⟩_ : (A : Set a) → A ∼[ k ] B → A ∼[ k ] B

+
  A ↔⟨⟩ A⇔B = A⇔B

+
  {-# WARNING_ON_USAGE _↔⟨⟩_

+
  "Warning: _↔⟨⟩_ was deprecated in v2.0.

+
  Please use _≡⟨⟩_ instead. "

+
  #-}

+

+
------------------------------------------------------------------------

+
-- Every unary relation induces a preorder and, for symmetric kinds,

+
-- an equivalence. (No claim is made that these relations are unique.)

+

+
InducedRelation₁ : Kind → (P : A → Set p) → A → A → Set _

+
InducedRelation₁ k P = λ x y → P x ∼[ k ] P y

+

+
InducedPreorder₁ : Kind → (P : A → Set p) → Preorder _ _ _

+
InducedPreorder₁ k P = record

+
  { _≈_        = _≡_

+
  ; _≲_        = InducedRelation₁ k P

+
  ; isPreorder = record

+
    { isEquivalence = ≡.isEquivalence

+
    ; reflexive     = reflexive ∘

+
                      K-reflexive ∘

+
                      ≡.cong P

+
    ; trans         = K-trans

+
    }

+
  } where open Preorder (K-preorder _ _)

+

+
InducedEquivalence₁ : SymmetricKind → (P : A → Set p) → Setoid _ _

+
InducedEquivalence₁ k P = record

+
  { _≈_           = InducedRelation₁ ⌊ k ⌋ P

+
  ; isEquivalence = record

+
    { refl  = K-refl

+
    ; sym   = SK-sym

+
    ; trans = K-trans

+
    }

+
  }

+

+
------------------------------------------------------------------------

+
-- Every binary relation induces a preorder and, for symmetric kinds,

+
-- an equivalence. (No claim is made that these relations are unique.)

+

+
InducedRelation₂ : Kind → ∀ {s} → (A → B → Set s) → B → B → Set _

+
InducedRelation₂ k _S_ = λ x y → ∀ {z} → (z S x) ∼[ k ] (z S y)

+

+
InducedPreorder₂ : Kind → ∀ {s} → (A → B → Set s) → Preorder _ _ _

+
InducedPreorder₂ k _S_ = record

+
  { _≈_        = _≡_

+
  ; _≲_        = InducedRelation₂ k _S_

+
  ; isPreorder = record

+
    { isEquivalence = ≡.isEquivalence

+
    ; reflexive     = λ x≡y {z} →

+
                        reflexive $

+
                        K-reflexive $

+
                        ≡.cong (_S_ z) x≡y

+

+
    ; trans         = λ i↝j j↝k → K-trans i↝j j↝k

+
    }

+
  } where open Preorder (K-preorder _ _)

+

+
InducedEquivalence₂ : SymmetricKind → ∀ {s} → (A → B → Set s) → Setoid _ _

+
InducedEquivalence₂ k _S_ = record

+
  { _≈_           = InducedRelation₂ ⌊ k ⌋ _S_

+
  ; isEquivalence = record

+
    { refl  = refl

+
    ; sym   = λ i↝j → sym i↝j

+
    ; trans = λ i↝j j↝k → trans i↝j j↝k

+
    }

+
  } where open Setoid (SK-setoid _ _)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Related.TypeIsomorphisms.html b/site/static/agda_html/Function.Related.TypeIsomorphisms.html
new file mode 100644
index 000000000..8379f0502
--- /dev/null
+++ b/site/static/agda_html/Function.Related.TypeIsomorphisms.html
@@ -0,0 +1,411 @@
+
+Function.Related.TypeIsomorphisms
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic lemmas showing that various types are related (isomorphic or

+
-- equivalent or…)

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Related.TypeIsomorphisms where

+

+
open import Algebra

+
open import Algebra.Structures.Biased using (isCommutativeSemiringˡ)

+
open import Axiom.Extensionality.Propositional using (Extensionality)

+
open import Data.Bool.Base using (true; false)

+
open import Data.Empty.Polymorphic using (⊥; ⊥-elim)

+
open import Data.Product.Base as Product

+
  using (_×_; Σ; curry; uncurry; _,_; -,_; <_,_>; proj₁; proj₂; ∃₂; ∃)

+
open import Data.Product.Function.NonDependent.Propositional

+
open import Data.Sum.Base as Sum

+
open import Data.Sum.Properties using (swap-involutive)

+
open import Data.Sum.Function.Propositional using (_⊎-cong_)

+
open import Data.Unit.Polymorphic.Base using (⊤)

+
open import Level using (Level; Lift; 0ℓ; suc)

+
open import Function.Base

+
open import Function.Bundles

+
open import Function.Related.Propositional

+
import Function.Construct.Identity as Identity

+
open import Relation.Binary hiding (_⇔_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)

+
open import Relation.Binary.PropositionalEquality.Properties

+
  using (module ≡-Reasoning)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ; contradiction)

+
import Relation.Nullary.Indexed as I

+
open import Relation.Nullary.Decidable using (True)

+

+
private

+
  variable

+
    a b c d : Level

+
    A B C D : Set a

+

+
------------------------------------------------------------------------

+
-- Properties of Σ and _×_

+

+
-- Σ is associative

+
Σ-assoc : ∀ {A : Set a} {B : A → Set b} {C : (a : A) → B a → Set c} →

+
          Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))

+
Σ-assoc = mk↔ₛ′ Product.assocʳ Product.assocˡ (λ _ → refl) (λ _ → refl)

+

+
-- × is commutative

+

+
×-comm : ∀ (A : Set a) (B : Set b) → (A × B) ↔ (B × A)

+
×-comm _ _ = mk↔ₛ′ Product.swap Product.swap (λ _ → refl) λ _ → refl

+

+
-- × has ⊤ as its identity

+

+
×-identityˡ : ∀ ℓ → LeftIdentity  {ℓ = ℓ} _↔_ ⊤ _×_

+
×-identityˡ _ _ = mk↔ₛ′ proj₂ -,_ (λ _ → refl) (λ _ → refl)

+

+
×-identityʳ : ∀ ℓ → RightIdentity  {ℓ = ℓ} _↔_ ⊤ _×_

+
×-identityʳ _ _ = mk↔ₛ′ proj₁ (_, _) (λ _ → refl) (λ _ → refl)

+

+
×-identity : ∀ ℓ → Identity _↔_ ⊤ _×_

+
×-identity ℓ = ×-identityˡ ℓ , ×-identityʳ ℓ

+

+
-- × has ⊥ has its zero

+

+
×-zeroˡ : ∀ ℓ → LeftZero {ℓ = ℓ} _↔_ ⊥ _×_

+
×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥-elim > (λ _ → refl) (λ { () })

+

+
×-zeroʳ : ∀ ℓ → RightZero {ℓ = ℓ} _↔_ ⊥ _×_

+
×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥-elim , id > (λ _ → refl) (λ { () })

+

+
×-zero : ∀ ℓ → Zero _↔_ ⊥ _×_

+
×-zero ℓ  = ×-zeroˡ ℓ , ×-zeroʳ ℓ

+

+
------------------------------------------------------------------------

+
-- Properties of ⊎

+

+
-- ⊎ is associative

+

+
⊎-assoc : ∀ ℓ → Associative {ℓ = ℓ} _↔_ _⊎_

+
⊎-assoc ℓ _ _ _ = mk↔ₛ′

+
  [ [ inj₁ , inj₂ ∘′ inj₁ ]′ , inj₂ ∘′ inj₂ ]′

+
  [ inj₁ ∘′ inj₁ , [ inj₁ ∘′ inj₂ , inj₂ ]′ ]′

+
  [ (λ _ → refl) , [ (λ _ → refl) , (λ _ → refl) ] ]

+
  [ [ (λ _ → refl) , (λ _ → refl) ] , (λ _ → refl) ]

+

+
-- ⊎ is commutative

+

+
⊎-comm : ∀ (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)

+
⊎-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive

+

+
-- ⊎ has ⊥ as its identity

+

+
⊎-identityˡ : ∀ ℓ → LeftIdentity _↔_ (⊥ {ℓ}) _⊎_

+
⊎-identityˡ _ _ = mk↔ₛ′ [ (λ ()) , id ]′ inj₂ (λ _ → refl)

+
                          [ (λ ()) , (λ _ → refl) ]

+

+
⊎-identityʳ : ∀ ℓ → RightIdentity _↔_ (⊥ {ℓ}) _⊎_

+
⊎-identityʳ _ _ = mk↔ₛ′ [ id , (λ ()) ]′ inj₁ (λ _ → refl)

+
                          [ (λ _ → refl) , (λ ()) ]

+

+
⊎-identity : ∀ ℓ → Identity _↔_ ⊥ _⊎_

+
⊎-identity ℓ = ⊎-identityˡ ℓ , ⊎-identityʳ ℓ

+

+
------------------------------------------------------------------------

+
-- Properties of × and ⊎

+

+
-- × distributes over ⊎

+

+
×-distribˡ-⊎ : ∀ ℓ → _DistributesOverˡ_ {ℓ = ℓ} _↔_ _×_ _⊎_

+
×-distribˡ-⊎ ℓ _ _ _ = mk↔ₛ′

+
  (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)

+
  [ Product.map₂ inj₁ , Product.map₂ inj₂ ]′

+
  [ (λ _ → refl) , (λ _ → refl) ]

+
  (uncurry λ _ → [ (λ _ → refl) , (λ _ → refl) ])

+

+
×-distribʳ-⊎ : ∀ ℓ → _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_

+
×-distribʳ-⊎ ℓ _ _ _ = mk↔ₛ′

+
  (uncurry [ curry inj₁ , curry inj₂ ]′)

+
  [ Product.map₁ inj₁ , Product.map₁ inj₂ ]′

+
  [ (λ _ → refl) , (λ _ → refl) ]

+
  (uncurry [ (λ _ _ → refl) , (λ _ _ → refl) ])

+

+
×-distrib-⊎ : ∀ ℓ → _DistributesOver_ {ℓ = ℓ} _↔_ _×_ _⊎_

+
×-distrib-⊎ ℓ = ×-distribˡ-⊎ ℓ , ×-distribʳ-⊎ ℓ

+

+
------------------------------------------------------------------------

+
-- ⊥, ⊤, _×_ and _⊎_ form a commutative semiring

+

+
-- ⊤, _×_ form a commutative monoid

+

+
×-isMagma : ∀ k ℓ → IsMagma {Level.suc ℓ} (Related ⌊ k ⌋) _×_

+
×-isMagma k ℓ = record

+
  { isEquivalence = SK-isEquivalence k

+
  ; ∙-cong        = _×-cong_

+
  }

+

+
×-magma : SymmetricKind → (ℓ : Level) → Magma _ _

+
×-magma k ℓ = record

+
  { isMagma = ×-isMagma k ℓ

+
  }

+

+
×-isSemigroup : ∀ k ℓ → IsSemigroup {Level.suc ℓ} (Related ⌊ k ⌋) _×_

+
×-isSemigroup k ℓ = record

+
  { isMagma = ×-isMagma k ℓ

+
  ; assoc   = λ _ _ _ → ↔⇒ Σ-assoc

+
  }

+

+
×-semigroup : SymmetricKind → (ℓ : Level) → Semigroup _ _

+
×-semigroup k ℓ = record

+
  { isSemigroup = ×-isSemigroup k ℓ

+
  }

+

+
×-isMonoid : ∀ k ℓ → IsMonoid (Related ⌊ k ⌋) _×_ ⊤

+
×-isMonoid k ℓ = record

+
  { isSemigroup = ×-isSemigroup k ℓ

+
  ; identity    = (↔⇒ ∘ ×-identityˡ ℓ) , (↔⇒ ∘ ×-identityʳ ℓ)

+
  }

+

+
×-monoid : SymmetricKind → (ℓ : Level) → Monoid _ _

+
×-monoid k ℓ = record

+
  { isMonoid = ×-isMonoid k ℓ

+
  }

+

+
×-isCommutativeMonoid : ∀ k ℓ → IsCommutativeMonoid (Related ⌊ k ⌋) _×_ ⊤

+
×-isCommutativeMonoid k ℓ = record

+
  { isMonoid = ×-isMonoid k ℓ

+
  ; comm     = λ _ _ → ↔⇒ (×-comm _ _)

+
  }

+

+
×-commutativeMonoid : SymmetricKind → (ℓ : Level) → CommutativeMonoid _ _

+
×-commutativeMonoid k ℓ = record

+
  { isCommutativeMonoid = ×-isCommutativeMonoid k ℓ

+
  }

+

+
-- ⊥, _⊎_ form a commutative monoid

+

+
⊎-isMagma : ∀ k ℓ → IsMagma {Level.suc ℓ} (Related ⌊ k ⌋) _⊎_

+
⊎-isMagma k ℓ = record

+
  { isEquivalence = SK-isEquivalence k

+
  ; ∙-cong        = _⊎-cong_

+
  }

+

+
⊎-magma : SymmetricKind → (ℓ : Level) → Magma _ _

+
⊎-magma k ℓ = record

+
  { isMagma = ⊎-isMagma k ℓ

+
  }

+

+
⊎-isSemigroup : ∀ k ℓ → IsSemigroup {Level.suc ℓ} (Related ⌊ k ⌋) _⊎_

+
⊎-isSemigroup k ℓ = record

+
  { isMagma = ⊎-isMagma k ℓ

+
  ; assoc   = λ A B C → ↔⇒ (⊎-assoc ℓ A B C)

+
  }

+

+
⊎-semigroup : SymmetricKind → (ℓ : Level) → Semigroup _ _

+
⊎-semigroup k ℓ = record

+
  { isSemigroup = ⊎-isSemigroup k ℓ

+
  }

+

+
⊎-isMonoid : ∀ k ℓ → IsMonoid (Related ⌊ k ⌋) _⊎_ ⊥

+
⊎-isMonoid k ℓ = record

+
  { isSemigroup = ⊎-isSemigroup k ℓ

+
  ; identity    = (↔⇒ ∘ ⊎-identityˡ ℓ) , (↔⇒ ∘ ⊎-identityʳ ℓ)

+
  }

+

+
⊎-monoid : SymmetricKind → (ℓ : Level) → Monoid _ _

+
⊎-monoid k ℓ = record

+
  { isMonoid = ⊎-isMonoid k ℓ

+
  }

+

+
⊎-isCommutativeMonoid : ∀ k ℓ → IsCommutativeMonoid (Related ⌊ k ⌋) _⊎_ ⊥

+
⊎-isCommutativeMonoid k ℓ = record

+
  { isMonoid = ⊎-isMonoid k ℓ

+
  ; comm     = λ _ _ → ↔⇒ (⊎-comm _ _)

+
  }

+

+
⊎-commutativeMonoid : SymmetricKind → (ℓ : Level) →

+
                      CommutativeMonoid _ _

+
⊎-commutativeMonoid k ℓ = record

+
  { isCommutativeMonoid = ⊎-isCommutativeMonoid k ℓ

+
  }

+

+
×-⊎-isCommutativeSemiring : ∀ k ℓ →

+
  IsCommutativeSemiring (Related ⌊ k ⌋) _⊎_ _×_ ⊥ ⊤

+
×-⊎-isCommutativeSemiring k ℓ = isCommutativeSemiringˡ record

+
  { +-isCommutativeMonoid = ⊎-isCommutativeMonoid k ℓ

+
  ; *-isCommutativeMonoid = ×-isCommutativeMonoid k ℓ

+
  ; distribʳ              = λ A B C → ↔⇒ (×-distribʳ-⊎ ℓ A B C)

+
  ; zeroˡ                 = ↔⇒ ∘ ×-zeroˡ ℓ

+
  }

+

+
×-⊎-commutativeSemiring : SymmetricKind → (ℓ : Level) →

+
                          CommutativeSemiring (Level.suc ℓ) ℓ

+
×-⊎-commutativeSemiring k ℓ = record

+
  { isCommutativeSemiring = ×-⊎-isCommutativeSemiring k ℓ

+
  }

+

+
------------------------------------------------------------------------

+
-- Some reordering lemmas

+

+
ΠΠ↔ΠΠ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →

+
        ((x : A) (y : B) → P x y) ↔ ((y : B) (x : A) → P x y)

+
ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip (λ _ → refl) (λ _ → refl)

+

+
∃∃↔∃∃ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →

+
        (∃₂ λ x y → P x y) ↔ (∃₂ λ y x → P x y)

+
∃∃↔∃∃ P = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)

+
  where

+
  to : (∃₂ λ x y → P x y) → (∃₂ λ y x → P x y)

+
  to (x , y , Pxy) = (y , x , Pxy)

+

+
  from : (∃₂ λ y x → P x y) → (∃₂ λ x y → P x y)

+
  from (y , x , Pxy) = (x , y , Pxy)

+

+
------------------------------------------------------------------------

+
-- Implicit and explicit function spaces are isomorphic

+

+
Π↔Π : ∀ {A : Set a} {B : A → Set b} →

+
      ((x : A) → B x) ↔ ({x : A} → B x)

+
Π↔Π = mk↔ₛ′ _$- λ- (λ _ → refl) (λ _ → refl)

+

+
------------------------------------------------------------------------

+
-- _→_ preserves the symmetric relations

+

+
→-cong-⇔ : A ⇔ B → C ⇔ D → (A → C) ⇔ (B → D)

+
→-cong-⇔ A⇔B C⇔D = mk⇔

+
  (λ f → to C⇔D ∘ f ∘ from A⇔B)

+
  (λ f → from C⇔D ∘ f ∘ to A⇔B)

+
  where open Equivalence

+

+
→-cong-↔ : Extensionality a c → Extensionality b d →

+
             {A : Set a} {B : Set b} {C : Set c} {D : Set d} →

+
             A ↔ B → C ↔ D → (A → C) ↔ (B → D)

+
→-cong-↔ ext₁ ext₂ A↔B C↔D = mk↔ₛ′

+
  (λ f → to C↔D ∘ f ∘ from A↔B)

+
  (λ f → from C↔D ∘ f ∘ to A↔B)

+
  (λ f → ext₂ λ x → begin

+
    to C↔D (from C↔D (f (to A↔B (from A↔B x)))) ≡⟨ strictlyInverseˡ C↔D _ ⟩

+
    f (to A↔B (from A↔B x))                     ≡⟨ cong f $ strictlyInverseˡ A↔B x ⟩

+
    f x                                         ∎)

+
  (λ f → ext₁ λ x → begin

+
    from C↔D (to C↔D (f (from A↔B (to A↔B x)))) ≡⟨ strictlyInverseʳ C↔D _ ⟩

+
    f (from A↔B (to A↔B x))                     ≡⟨ cong f $ strictlyInverseʳ A↔B x ⟩

+
    f x                                         ∎)

+
  where open Inverse; open ≡-Reasoning

+

+
→-cong :  Extensionality a c → Extensionality b d →

+
          ∀ {k} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →

+
          A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A → C) ∼[ ⌊ k ⌋ ] (B → D)

+
→-cong extAC extBD {equivalence} = →-cong-⇔

+
→-cong extAC extBD {bijection}   = →-cong-↔ extAC extBD

+

+
------------------------------------------------------------------------

+
-- ¬_ (at Level 0) preserves the symmetric relations

+

+
¬-cong-⇔ : A ⇔ B → (¬ A) ⇔ (¬ B)

+
¬-cong-⇔ A⇔B = →-cong-⇔ A⇔B (Identity.⇔-id _)

+

+
¬-cong : Extensionality a 0ℓ → Extensionality b 0ℓ →

+
         ∀ {k} {A : Set a} {B : Set b} →

+
         A ∼[ ⌊ k ⌋ ] B → (¬ A) ∼[ ⌊ k ⌋ ] (¬ B)

+
¬-cong extA extB A≈B = →-cong extA extB A≈B (K-reflexive refl)

+

+
------------------------------------------------------------------------

+
-- _⇔_ preserves _⇔_

+

+
-- The type of the following proof is a bit more general.

+

+
Related-cong :

+
  ∀ {k} →

+
  A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A ∼[ ⌊ k ⌋ ] C) ⇔ (B ∼[ ⌊ k ⌋ ] D)

+
Related-cong {A = A} {B = B} {C = C} {D = D} A≈B C≈D = mk⇔

+
  (λ A≈C → B  ∼⟨ SK-sym A≈B ⟩

+
            A  ∼⟨ A≈C ⟩

+
            C  ∼⟨ C≈D ⟩

+
            D  ∎)

+
  (λ B≈D → A  ∼⟨ A≈B ⟩

+
            B  ∼⟨ B≈D ⟩

+
            D  ∼⟨ SK-sym C≈D ⟩

+
            C  ∎)

+
  where open EquationalReasoning

+

+
------------------------------------------------------------------------

+
-- A lemma relating True dec and P, where dec : Dec P

+

+
True↔ : ∀ {p} {P : Set p}

+
        (dec : Dec P) → ((p₁ p₂ : P) → p₁ ≡ p₂) → True dec ↔ P

+
True↔ ( true because  [p]) irr =

+
  mk↔ₛ′ (λ _ → invert [p]) (λ _ → _) (irr _) (λ _ → refl)

+
True↔ (false because ofⁿ ¬p) _ =

+
  mk↔ₛ′ (λ()) (invert (ofⁿ ¬p)) (λ x → flip contradiction ¬p x) (λ ())

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Strict.html b/site/static/agda_html/Function.Strict.html
new file mode 100644
index 000000000..fbd1a4192
--- /dev/null
+++ b/site/static/agda_html/Function.Strict.html
@@ -0,0 +1,152 @@
+
+Function.Strict
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Strict combinators (i.e. that use call-by-value)

+
------------------------------------------------------------------------

+

+
-- The contents of this module is also accessible via the `Function`

+
-- module.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function.Strict where

+

+
open import Agda.Builtin.Equality using (_≡_)

+
open import Function.Base using (flip)

+
open import Level using (Level)

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+

+
infixl 0 _!|>_ _!|>′_

+
infixr -1 _$!_ _$!′_

+

+
------------------------------------------------------------------------

+
-- Dependent combinators

+

+
-- These are functions whose output has a type that depends on the

+
-- value of the input to the function.

+

+
open import Agda.Builtin.Strict public

+
  renaming

+
  ( primForce      to force

+
  ; primForceLemma to force-≡

+
  )

+

+
-- Application

+
_$!_ : ∀ {A : Set a} {B : A → Set b} →

+
       ((x : A) → B x) → ((x : A) → B x)

+
f $! x = force x f

+

+
-- Flipped application

+
_!|>_ : ∀ {A : Set a} {B : A → Set b} →

+
       (a : A) → (∀ a → B a) → B a

+
_!|>_ = flip _$!_

+

+
------------------------------------------------------------------------

+
-- Non-dependent combinators

+

+
-- Any of the above operations for dependent functions will also work

+
-- for non-dependent functions but sometimes Agda has difficulty

+
-- inferring the non-dependency. Primed (′ = \prime) versions of the

+
-- operations are therefore provided below that sometimes have better

+
-- inference properties.

+

+
seq : A → B → B

+
seq a b = force a (λ _ → b)

+

+
seq-≡ : (a : A) (b : B) → seq a b ≡ b

+
seq-≡ a b = force-≡ a (λ _ → b)

+

+
force′ : A → (A → B) → B

+
force′ = force

+

+
force′-≡ : (a : A) (f : A → B) → force′ a f ≡ f a

+
force′-≡ = force-≡

+

+
-- Application

+
_$!′_ : (A → B) → (A → B)

+
_$!′_ = _$!_

+

+
-- Flipped application

+
_!|>′_ : A → (A → B) → B

+
_!|>′_ = _!|>_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Structures.Biased.html b/site/static/agda_html/Function.Structures.Biased.html
new file mode 100644
index 000000000..9f13d677e
--- /dev/null
+++ b/site/static/agda_html/Function.Structures.Biased.html
@@ -0,0 +1,204 @@
+
+Function.Structures.Biased
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Ways to give instances of certain structures where some fields can

+
-- be given in terms of others.

+
-- The contents of this file should usually be accessed from `Function`.

+
------------------------------------------------------------------------

+

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
module Function.Structures.Biased {a b ℓ₁ ℓ₂}

+
  {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain

+
  {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain

+
  where

+

+
open import Data.Product.Base as Product using (∃; _×_; _,_)

+
open import Function.Base

+
open import Function.Definitions

+
open import Function.Structures _≈₁_ _≈₂_

+
open import Function.Consequences.Setoid

+
open import Level using (_⊔_)

+

+
------------------------------------------------------------------------

+
-- Surjection

+
------------------------------------------------------------------------

+

+
record IsStrictSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent : IsCongruent f

+
    strictlySurjective : StrictlySurjective _≈₂_ f

+

+
  open IsCongruent isCongruent public

+

+
  isSurjection : IsSurjection f

+
  isSurjection = record

+
    { isCongruent = isCongruent

+
    ; surjective = strictlySurjective⇒surjective

+
        Eq₁.setoid Eq₂.setoid cong strictlySurjective

+
    }

+

+
open IsStrictSurjection public

+
  using () renaming (isSurjection to isStrictSurjection)

+

+
------------------------------------------------------------------------

+
-- Bijection

+
------------------------------------------------------------------------

+

+
record IsStrictBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isInjection : IsInjection f

+
    strictlySurjective  : StrictlySurjective _≈₂_ f

+

+
  isBijection : IsBijection f

+
  isBijection = record

+
    { isInjection = isInjection

+
    ; surjective = strictlySurjective⇒surjective

+
        Eq₁.setoid Eq₂.setoid cong strictlySurjective

+
    } where open IsInjection isInjection

+

+
open IsStrictBijection public

+
  using () renaming (isBijection to isStrictBijection)

+

+
------------------------------------------------------------------------

+
-- Left inverse

+
------------------------------------------------------------------------

+

+
record IsStrictLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent  : IsCongruent to

+
    from-cong    : Congruent _≈₂_ _≈₁_ from

+
    strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from

+

+
  isLeftInverse : IsLeftInverse to from

+
  isLeftInverse = record

+
    { isCongruent = isCongruent

+
    ; from-cong = from-cong

+
    ; inverseˡ = strictlyInverseˡ⇒inverseˡ

+
        Eq₁.setoid Eq₂.setoid cong strictlyInverseˡ

+
    } where open IsCongruent isCongruent

+

+
open IsStrictLeftInverse public

+
  using () renaming (isLeftInverse to isStrictLeftInverse)

+

+
------------------------------------------------------------------------

+
-- Right inverse

+
------------------------------------------------------------------------

+

+
record IsStrictRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent : IsCongruent to

+
    from-cong   : Congruent _≈₂_ _≈₁_ from

+
    strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from

+

+
  isRightInverse : IsRightInverse to from

+
  isRightInverse = record

+
    { isCongruent = isCongruent

+
    ; from-cong = from-cong

+
    ; inverseʳ = strictlyInverseʳ⇒inverseʳ

+
        Eq₁.setoid Eq₂.setoid from-cong strictlyInverseʳ

+
    } where open IsCongruent isCongruent

+

+
open IsStrictRightInverse public

+
  using () renaming (isRightInverse to isStrictRightInverse)

+

+
------------------------------------------------------------------------

+
-- Inverse

+
------------------------------------------------------------------------

+

+
record IsStrictInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isLeftInverse : IsLeftInverse to from

+
    strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from

+

+
  isInverse : IsInverse to from

+
  isInverse = record

+
    { isLeftInverse = isLeftInverse

+
    ; inverseʳ      = strictlyInverseʳ⇒inverseʳ

+
        Eq₁.setoid Eq₂.setoid from-cong strictlyInverseʳ

+
    } where open IsLeftInverse isLeftInverse

+

+
open IsStrictInverse public

+
  using () renaming (isInverse to isStrictInverse)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.Structures.html b/site/static/agda_html/Function.Structures.html
new file mode 100644
index 000000000..1732e3b1f
--- /dev/null
+++ b/site/static/agda_html/Function.Structures.html
@@ -0,0 +1,266 @@
+
+Function.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Structures for types of functions

+
------------------------------------------------------------------------

+

+
-- The contents of this file should usually be accessed from `Function`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Structures using (IsEquivalence)

+

+
module Function.Structures {a b ℓ₁ ℓ₂}

+
  {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain

+
  {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain

+
  where

+

+
open import Data.Product.Base as Product using (∃; _×_; _,_)

+
open import Function.Base

+
open import Function.Definitions

+
open import Level using (_⊔_)

+

+
------------------------------------------------------------------------

+
-- One element structures

+
------------------------------------------------------------------------

+

+
record IsCongruent (to : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    cong           : Congruent _≈₁_ _≈₂_ to

+
    isEquivalence₁ : IsEquivalence _≈₁_

+
    isEquivalence₂ : IsEquivalence _≈₂_

+

+
  module Eq₁ where

+

+
    setoid : Setoid a ℓ₁

+
    setoid = record

+
      { isEquivalence = isEquivalence₁

+
      }

+

+
    open Setoid setoid public

+

+
  module Eq₂ where

+

+
    setoid : Setoid b ℓ₂

+
    setoid = record

+
      { isEquivalence = isEquivalence₂

+
      }

+

+
    open Setoid setoid public

+

+

+
record IsInjection (to : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent : IsCongruent to

+
    injective   : Injective _≈₁_ _≈₂_ to

+

+
  open IsCongruent isCongruent public

+

+

+
record IsSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent : IsCongruent f

+
    surjective  : Surjective _≈₁_ _≈₂_ f

+

+
  open IsCongruent isCongruent public

+

+
  strictlySurjective : StrictlySurjective _≈₂_ f

+
  strictlySurjective x = Product.map₂ (λ v → v Eq₁.refl) (surjective x)

+

+

+
record IsBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isInjection : IsInjection f

+
    surjective  : Surjective _≈₁_ _≈₂_ f

+

+
  open IsInjection isInjection public

+

+
  bijective : Bijective _≈₁_ _≈₂_ f

+
  bijective = injective , surjective

+

+
  isSurjection : IsSurjection f

+
  isSurjection = record

+
    { isCongruent = isCongruent

+
    ; surjective  = surjective

+
    }

+

+
  open IsSurjection isSurjection public

+
    using (strictlySurjective)

+

+

+
------------------------------------------------------------------------

+
-- Two element structures

+
------------------------------------------------------------------------

+

+
record IsLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent  : IsCongruent to

+
    from-cong    : Congruent _≈₂_ _≈₁_ from

+
    inverseˡ     : Inverseˡ _≈₁_ _≈₂_ to from

+

+
  open IsCongruent isCongruent public

+
    renaming (cong to to-cong)

+

+
  strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from

+
  strictlyInverseˡ x = inverseˡ Eq₁.refl

+

+
  isSurjection : IsSurjection to

+
  isSurjection = record

+
    { isCongruent = isCongruent

+
    ; surjective = λ y → from y , inverseˡ

+
    }

+

+

+
record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isCongruent : IsCongruent to

+
    from-cong   : Congruent _≈₂_ _≈₁_ from

+
    inverseʳ    : Inverseʳ _≈₁_ _≈₂_ to from

+

+
  open IsCongruent isCongruent public

+
    renaming (cong to to-cong)

+

+
  strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from

+
  strictlyInverseʳ x = inverseʳ Eq₂.refl

+

+

+
record IsInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isLeftInverse : IsLeftInverse to from

+
    inverseʳ      : Inverseʳ _≈₁_ _≈₂_ to from

+

+
  open IsLeftInverse isLeftInverse public

+

+
  isRightInverse : IsRightInverse to from

+
  isRightInverse = record

+
    { isCongruent = isCongruent

+
    ; from-cong   = from-cong

+
    ; inverseʳ    = inverseʳ

+
    }

+

+
  open IsRightInverse isRightInverse public

+
    using (strictlyInverseʳ)

+

+
  inverse : Inverseᵇ _≈₁_ _≈₂_ to from

+
  inverse = inverseˡ , inverseʳ

+

+

+
------------------------------------------------------------------------

+
-- Three element structures

+
------------------------------------------------------------------------

+

+
record IsBiEquivalence

+
  (to : A → B) (from₁ : B → A) (from₂ : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    to-isCongruent : IsCongruent to

+
    from₁-cong    : Congruent _≈₂_ _≈₁_ from₁

+
    from₂-cong    : Congruent _≈₂_ _≈₁_ from₂

+

+
  open IsCongruent to-isCongruent public

+
    renaming (cong to to-cong₁)

+

+

+
record IsBiInverse

+
  (to : A → B) (from₁ : B → A) (from₂ : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    to-isCongruent : IsCongruent to

+
    from₁-cong     : Congruent _≈₂_ _≈₁_ from₁

+
    from₂-cong     : Congruent _≈₂_ _≈₁_ from₂

+
    inverseˡ       : Inverseˡ _≈₁_ _≈₂_ to from₁

+
    inverseʳ       : Inverseʳ _≈₁_ _≈₂_ to from₂

+

+
  open IsCongruent to-isCongruent public

+
    renaming (cong to to-cong)

+

+

+
------------------------------------------------------------------------

+
-- Other

+
------------------------------------------------------------------------

+

+
-- See the comment on `SplitSurjection` in `Function.Bundles` for an

+
-- explanation of (split) surjections.

+
record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    from : B → A

+
    isLeftInverse : IsLeftInverse f from

+

+
  open IsLeftInverse isLeftInverse public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Function.html b/site/static/agda_html/Function.html
new file mode 100644
index 000000000..29c6db904
--- /dev/null
+++ b/site/static/agda_html/Function.html
@@ -0,0 +1,94 @@
+
+Function
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Functions

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Function where

+

+
open import Function.Core public

+
open import Function.Base public

+
open import Function.Strict public

+
open import Function.Definitions public

+
open import Function.Structures public

+
open import Function.Structures.Biased public

+
open import Function.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/IO.Primitive.Core.html b/site/static/agda_html/IO.Primitive.Core.html
new file mode 100644
index 000000000..fe4c24b01
--- /dev/null
+++ b/site/static/agda_html/IO.Primitive.Core.html
@@ -0,0 +1,118 @@
+
+IO.Primitive.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Primitive IO: simple bindings to Haskell types and functions

+
------------------------------------------------------------------------

+

+
-- NOTE: the contents of this module should be accessed via `IO`.

+

+
{-# OPTIONS --cubical-compatible #-}

+

+
module IO.Primitive.Core where

+

+
open import Level using (Level)

+
private

+
  variable

+
    a : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- The IO monad

+

+
open import Agda.Builtin.IO public

+
  using (IO)

+

+
infixl 1 _>>=_

+

+
postulate

+
  pure : A → IO A

+
  _>>=_  : IO A → (A → IO B) → IO B

+

+
{-# COMPILE GHC pure = \_ _ -> return    #-}

+
{-# COMPILE GHC _>>=_  = \_ _ _ _ -> (>>=) #-}

+
{-# COMPILE UHC pure = \_ _ x -> UHC.Agda.Builtins.primReturn x #-}

+
{-# COMPILE UHC _>>=_  = \_ _ _ _ x y -> UHC.Agda.Builtins.primBind x y #-}

+

+
-- Haskell-style alternative syntax

+
return : A → IO A

+
return = pure

+

+
_>>_ : IO A → IO B → IO B

+
a >> b = a >>= λ _ → b

+

\ No newline at end of file
diff --git a/site/static/agda_html/Induction.Lexicographic.html b/site/static/agda_html/Induction.Lexicographic.html
new file mode 100644
index 000000000..c9b80c40b
--- /dev/null
+++ b/site/static/agda_html/Induction.Lexicographic.html
@@ -0,0 +1,160 @@
+
+Induction.Lexicographic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Lexicographic induction

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Induction.Lexicographic where

+

+
open import Data.Product.Base using (Σ; _,_; _×_)

+
open import Induction

+
open import Level

+

+
-- The structure of lexicographic induction.

+

+
Σ-Rec : ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : A → Set b} →

+
        RecStruct A (ℓ₁ ⊔ b) ℓ₂ → (∀ x → RecStruct (B x) ℓ₁ ℓ₃) →

+
        RecStruct (Σ A B) _ _

+
Σ-Rec RecA RecB P (x , y) =

+
  -- Either x is constant and y is "smaller", ...

+
  RecB x (λ y′ → P (x , y′)) y

+
    ×

+
  -- ...or x is "smaller" and y is arbitrary.

+
  RecA (λ x′ → ∀ y′ → P (x′ , y′)) x

+

+
infixr 2 _⊗_

+

+
_⊗_ : ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : Set b} →

+
      RecStruct A (ℓ₁ ⊔ b) ℓ₂ → RecStruct B ℓ₁ ℓ₃ →

+
      RecStruct (A × B) _ _

+
RecA ⊗ RecB = Σ-Rec RecA (λ _ → RecB)

+

+
-- Constructs a recursor builder for lexicographic induction.

+

+
Σ-rec-builder :

+
  ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : A → Set b}

+
    {RecA : RecStruct A (ℓ₁ ⊔ b) ℓ₂}

+
    {RecB : ∀ x → RecStruct (B x) ℓ₁ ℓ₃} →

+
  RecursorBuilder RecA → (∀ x → RecursorBuilder (RecB x)) →

+
  RecursorBuilder (Σ-Rec RecA RecB)

+
Σ-rec-builder {RecA = RecA} {RecB = RecB} recA recB P f (x , y) =

+
  (p₁ x y p₂x , p₂x)

+
  where

+
  p₁ : ∀ x y →

+
       RecA (λ x′ → ∀ y′ → P (x′ , y′)) x →

+
       RecB x (λ y′ → P (x , y′)) y

+
  p₁ x y x-rec = recB x

+
                      (λ y′ → P (x , y′))

+
                      (λ y y-rec → f (x , y) (y-rec , x-rec))

+
                      y

+

+
  p₂ : ∀ x → RecA (λ x′ → ∀ y′ → P (x′ , y′)) x

+
  p₂ = recA (λ x → ∀ y → P (x , y))

+
            (λ x x-rec y → f (x , y) (p₁ x y x-rec , x-rec))

+

+
  p₂x = p₂ x

+

+
[_⊗_] : ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : Set b}

+
          {RecA : RecStruct A (ℓ₁ ⊔ b) ℓ₂} {RecB : RecStruct B ℓ₁ ℓ₃} →

+
        RecursorBuilder RecA → RecursorBuilder RecB →

+
        RecursorBuilder (RecA ⊗ RecB)

+
[ recA ⊗ recB ] = Σ-rec-builder recA (λ _ → recB)

+

+
------------------------------------------------------------------------

+
-- Example

+

+
private

+

+
  open import Data.Nat.Base

+
  open import Data.Nat.Induction as ℕ

+

+
  -- The Ackermann function à la Rózsa Péter.

+

+
  ackermann : ℕ → ℕ → ℕ

+
  ackermann m n =

+
    build [ ℕ.recBuilder ⊗ ℕ.recBuilder ]

+
          (λ _ → ℕ)

+
          (λ { (zero  , n)     _                   → 1 + n

+
             ; (suc m , zero)  (_         , ackm•) → ackm• 1

+
             ; (suc m , suc n) (ack[1+m]n , ackm•) → ackm• ack[1+m]n

+
             })

+
          (m , n)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Induction.WellFounded.html b/site/static/agda_html/Induction.WellFounded.html
new file mode 100644
index 000000000..d36f57498
--- /dev/null
+++ b/site/static/agda_html/Induction.WellFounded.html
@@ -0,0 +1,342 @@
+
+Induction.WellFounded
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Well-founded induction

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Induction.WellFounded where

+

+
open import Data.Product.Base using (Σ; _,_; proj₁; proj₂)

+
open import Function.Base using (_∘_; flip; _on_)

+
open import Induction

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Definitions

+
  using (Symmetric; Asymmetric; Irreflexive; _Respects₂_;

+
    _Respectsʳ_; _Respects_)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)

+
open import Relation.Binary.Consequences using (asym⇒irr)

+
open import Relation.Unary

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
private

+
  variable

+
    a b ℓ ℓ₁ ℓ₂ r : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
-- When using well-founded recursion you can recurse arbitrarily, as

+
-- long as the arguments become smaller, and "smaller" is

+
-- well-founded.

+

+
WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _

+
WfRec _<_ P x = ∀ {y} → y < x → P y

+

+
-- The accessibility predicate: x is accessible if everything which is

+
-- smaller than x is also accessible (inductively).

+

+
data Acc {A : Set a} (_<_ : Rel A ℓ) (x : A) : Set (a ⊔ ℓ) where

+
  acc : (rs : WfRec _<_ (Acc _<_) x) → Acc _<_ x

+

+
-- The accessibility predicate encodes what it means to be

+
-- well-founded; if all elements are accessible, then _<_ is

+
-- well-founded.

+

+
WellFounded : Rel A ℓ → Set _

+
WellFounded _<_ = ∀ x → Acc _<_ x

+

+
------------------------------------------------------------------------

+
-- Basic properties

+

+
acc-inverse : ∀ {_<_ : Rel A ℓ} {x : A} (q : Acc _<_ x) →

+
              WfRec _<_ (Acc _<_) x

+
acc-inverse (acc rs) y<x = rs y<x

+

+
module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where

+

+
  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_

+
  Acc-resp-flip-≈ respʳ x≈y (acc rec) = acc λ z<y → rec (respʳ x≈y z<y)

+

+
  Acc-resp-≈ : Symmetric _≈_ → _<_ Respectsʳ _≈_ → (Acc _<_) Respects _≈_

+
  Acc-resp-≈ sym respʳ x≈y wf = Acc-resp-flip-≈ (respʳ ∘ sym) x≈y wf

+

+
------------------------------------------------------------------------

+
-- Well-founded induction for the subset of accessible elements:

+

+
module Some {_<_ : Rel A r} {ℓ} where

+

+
  wfRecBuilder : SubsetRecursorBuilder (Acc _<_) (WfRec _<_ {ℓ = ℓ})

+
  wfRecBuilder P f x (acc rs) = λ y<x → f _ (wfRecBuilder P f _ (rs y<x))

+

+
  wfRec : SubsetRecursor (Acc _<_) (WfRec _<_)

+
  wfRec = subsetBuild wfRecBuilder

+

+
  unfold-wfRec : (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P) {x : A} (q : Acc _<_ x) →

+
                 wfRec P f x q ≡ f x λ y<x → wfRec P f _ (acc-inverse q y<x)

+
  unfold-wfRec P f (acc rs) = refl

+

+

+
------------------------------------------------------------------------

+
-- Well-founded induction for all elements, assuming they are all

+
-- accessible:

+

+
module All {_<_ : Rel A r} (wf : WellFounded _<_) ℓ where

+

+
  wfRecBuilder : RecursorBuilder (WfRec _<_ {ℓ = ℓ})

+
  wfRecBuilder P f x = Some.wfRecBuilder P f x (wf x)

+

+
  wfRec : Recursor (WfRec _<_)

+
  wfRec = build wfRecBuilder

+

+
  wfRec-builder = wfRecBuilder

+

+
module FixPoint

+
  {_<_ : Rel A r} (wf : WellFounded _<_)

+
  (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P)

+
  (f-ext : (x : A) {IH IH′ : WfRec _<_ P x} →

+
           (∀ {y} y<x → IH {y} y<x ≡ IH′ y<x) →

+
           f x IH ≡ f x IH′)

+
  where

+

+
  some-wfrec-Irrelevant : Pred A _

+
  some-wfrec-Irrelevant x = ∀ q q′ → Some.wfRec P f x q ≡ Some.wfRec P f x q′

+

+
  some-wfRec-irrelevant : ∀ x → some-wfrec-Irrelevant x

+
  some-wfRec-irrelevant = All.wfRec wf _ some-wfrec-Irrelevant

+
    λ { x IH (acc rs) (acc rs′) → f-ext x λ y<x → IH y<x (rs y<x) (rs′ y<x) }

+

+
  open All wf ℓ

+

+
  wfRecBuilder-wfRec : ∀ {x y} y<x → wfRecBuilder P f x y<x ≡ wfRec P f y

+
  wfRecBuilder-wfRec {x} {y} y<x with acc rs ← wf x

+
   = some-wfRec-irrelevant y (rs y<x) (wf y)

+

+
  unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x λ _ → wfRec P f _

+
  unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec

+

+
------------------------------------------------------------------------

+
-- Well-founded relations are asymmetric and irreflexive.

+

+
module _ {_<_ : Rel A r} where

+
  acc⇒asym : ∀ {x y} → Acc _<_ x → x < y → ¬ (y < x)

+
  acc⇒asym {x} hx =

+
    Some.wfRec (λ x → ∀ {y} → x < y → ¬ (y < x)) (λ _ hx x<y y<x → hx y<x y<x x<y) _ hx

+

+
  wf⇒asym : WellFounded _<_ → Asymmetric _<_

+
  wf⇒asym wf = acc⇒asym (wf _)

+

+
  wf⇒irrefl : {_≈_ : Rel A ℓ} → _<_ Respects₂ _≈_ →

+
              Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_

+
  wf⇒irrefl r s wf = asym⇒irr r s (wf⇒asym wf)

+

+
------------------------------------------------------------------------

+
-- It might be useful to establish proofs of Acc or Well-founded using

+
-- combinators such as the ones below (see, for instance,

+
-- "Constructing Recursion Operators in Intuitionistic Type Theory" by

+
-- Lawrence C Paulson).

+

+
module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂}

+
                   (<₁⇒<₂ : ∀ {x y} → x <₁ y → x <₂ y) where

+

+
  accessible : Acc _<₂_ ⊆ Acc _<₁_

+
  accessible (acc rs) = acc λ y<x → accessible (rs (<₁⇒<₂ y<x))

+

+
  wellFounded : WellFounded _<₂_ → WellFounded _<₁_

+
  wellFounded wf = λ x → accessible (wf x)

+

+

+
-- DEPRECATED in v1.4.

+
-- Please use proofs in `Relation.Binary.Construct.On` instead.

+
module InverseImage {_<_ : Rel B ℓ} (f : A → B) where

+

+
  accessible : ∀ {x} → Acc _<_ (f x) → Acc (_<_ on f) x

+
  accessible (acc rs) = acc λ fy<fx → accessible (rs fy<fx)

+

+
  wellFounded : WellFounded _<_ → WellFounded (_<_ on f)

+
  wellFounded wf = λ x → accessible (wf (f x))

+

+
  well-founded = wellFounded

+
  {-# WARNING_ON_USAGE accessible

+
  "Warning: accessible was deprecated in v1.4.

+
\ \Please use accessible from `Relation.Binary.Construct.On` instead."

+
  #-}

+
  {-# WARNING_ON_USAGE wellFounded

+
  "Warning: wellFounded was deprecated in v1.4.

+
\ \Please use wellFounded from `Relation.Binary.Construct.On` instead."

+
  #-}

+

+

+
-- DEPRECATED in v1.5.

+
-- Please use `TransClosure` from `Relation.Binary.Construct.Closure.Transitive` instead.

+
module TransitiveClosure {A : Set a} (_<_ : Rel A ℓ) where

+

+
  infix 4 _<⁺_

+

+
  data _<⁺_ : Rel A (a ⊔ ℓ) where

+
    [_]   : ∀ {x y} (x<y : x < y) → x <⁺ y

+
    trans : ∀ {x y z} (x<y : x <⁺ y) (y<z : y <⁺ z) → x <⁺ z

+

+
  downwardsClosed : ∀ {x y} → Acc _<⁺_ y → x <⁺ y → Acc _<⁺_ x

+
  downwardsClosed (acc rs) x<y = acc λ z<x → rs (trans z<x x<y)

+

+
  mutual

+

+
    accessible : ∀ {x} → Acc _<_ x → Acc _<⁺_ x

+
    accessible acc-x = acc (accessible′ acc-x)

+

+
    accessible′ : ∀ {x} → Acc _<_ x → WfRec _<⁺_ (Acc _<⁺_) x

+
    accessible′ (acc rs) [ y<x ]         = accessible (rs y<x)

+
    accessible′ acc-x    (trans y<z z<x) =

+
      downwardsClosed (accessible′ acc-x z<x) y<z

+

+
  wellFounded : WellFounded _<_ → WellFounded _<⁺_

+
  wellFounded wf = λ x → accessible (wf x)

+

+
  {-# WARNING_ON_USAGE _<⁺_

+
  "Warning: _<⁺_ was deprecated in v1.5.

+
\ \Please use TransClosure from Relation.Binary.Construct.Closure.Transitive instead."

+
  #-}

+

+

+
-- DEPRECATED in v1.3.

+
-- Please use `×-Lex` from `Data.Product.Relation.Binary.Lex.Strict` instead.

+
module Lexicographic {A : Set a} {B : A → Set b}

+
                     (RelA : Rel A ℓ₁)

+
                     (RelB : ∀ x → Rel (B x) ℓ₂) where

+

+
  infix 4 _<_

+
  data _<_ : Rel (Σ A B) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
    left  : ∀ {x₁ y₁ x₂ y₂} (x₁<x₂ : RelA   x₁ x₂) → (x₁ , y₁) < (x₂ , y₂)

+
    right : ∀ {x y₁ y₂}     (y₁<y₂ : RelB x y₁ y₂) → (x  , y₁) < (x  , y₂)

+

+
  mutual

+

+
    accessible : ∀ {x y} →

+
                 Acc RelA x → (∀ {x} → WellFounded (RelB x)) →

+
                 Acc _<_ (x , y)

+
    accessible accA wfB = acc (accessible′ accA (wfB _) wfB)

+

+

+
    accessible′ :

+
      ∀ {x y} →

+
      Acc RelA x → Acc (RelB x) y → (∀ {x} → WellFounded (RelB x)) →

+
      WfRec _<_ (Acc _<_) (x , y)

+
    accessible′ (acc rsA) _    wfB (left  x′<x) = accessible (rsA x′<x) wfB

+
    accessible′ accA (acc rsB) wfB (right y′<y) =

+
      acc (accessible′ accA (rsB y′<y) wfB)

+

+
  wellFounded : WellFounded RelA → (∀ {x} → WellFounded (RelB x)) →

+
                WellFounded _<_

+
  wellFounded wfA wfB p = accessible (wfA (proj₁ p)) wfB

+

+
  well-founded = wellFounded

+

+
  {-# WARNING_ON_USAGE _<_

+
  "Warning: _<_ was deprecated in v1.3.

+
\ \Please use `×-Lex` from `Data.Product.Relation.Binary.Lex.Strict` instead."

+
  #-}

+
  {-# WARNING_ON_USAGE accessible

+
  "Warning: accessible was deprecated in v1.3."

+
  #-}

+
  {-# WARNING_ON_USAGE accessible′

+
  "Warning: accessible′ was deprecated in v1.3."

+
  #-}

+
  {-# WARNING_ON_USAGE wellFounded

+
  "Warning: wellFounded was deprecated in v1.3.

+
\ \Please use `×-wellFounded` from `Data.Product.Relation.Binary.Lex.Strict` instead."

+
  #-}

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.0

+

+
module Inverse-image = InverseImage

+
module Transitive-closure = TransitiveClosure

+

\ No newline at end of file
diff --git a/site/static/agda_html/Induction.html b/site/static/agda_html/Induction.html
new file mode 100644
index 000000000..3b8fabddc
--- /dev/null
+++ b/site/static/agda_html/Induction.html
@@ -0,0 +1,141 @@
+
+Induction
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- An abstraction of various forms of recursion/induction

+
------------------------------------------------------------------------

+

+
-- The idea underlying Induction.* comes from Epigram 1, see Section 4

+
-- of "The view from the left" by McBride and McKinna.

+

+
-- Note: The types in this module can perhaps be easier to understand

+
-- if they are normalised. Note also that Agda can do the

+
-- normalisation for you.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Induction where

+

+
open import Level

+
open import Relation.Unary

+
open import Relation.Unary.PredicateTransformer using (PT)

+

+
private

+
  variable

+
    a ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    Q : Pred A ℓ

+
    Rec : PT A A ℓ₁ ℓ₂

+

+

+
------------------------------------------------------------------------

+
-- A RecStruct describes the allowed structure of recursion. The

+
-- examples in Data.Nat.Induction should explain what this is all

+
-- about.

+

+
RecStruct : Set a → (ℓ₁ ℓ₂ : Level) → Set _

+
RecStruct A = PT A A

+

+
-- A recursor builder constructs an instance of a recursion structure

+
-- for a given input.

+

+
RecursorBuilder : RecStruct A ℓ₁ ℓ₂ → Set _

+
RecursorBuilder Rec = ∀ P → Rec P ⊆′ P → Universal (Rec P)

+

+
-- A recursor can be used to actually compute/prove something useful.

+

+
Recursor : RecStruct A ℓ₁ ℓ₂ → Set _

+
Recursor Rec = ∀ P → Rec P ⊆′ P → Universal P

+

+
-- And recursors can be constructed from recursor builders.

+

+
build : RecursorBuilder Rec → Recursor Rec

+
build builder P f x = f x (builder P f x)

+

+
-- We can repeat the exercise above for subsets of the type we are

+
-- recursing over.

+

+
SubsetRecursorBuilder : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _

+
SubsetRecursorBuilder Q Rec = ∀ P → Rec P ⊆′ P → Q ⊆′ Rec P

+

+
SubsetRecursor : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _

+
SubsetRecursor Q Rec = ∀ P → Rec P ⊆′ P → Q ⊆′ P

+

+
subsetBuild : SubsetRecursorBuilder Q Rec → SubsetRecursor Q Rec

+
subsetBuild builder P f x q = f x (builder P f x q)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Abstract.html b/site/static/agda_html/Leios.Abstract.html
new file mode 100644
index 000000000..2d3b3bb54
--- /dev/null
+++ b/site/static/agda_html/Leios.Abstract.html
@@ -0,0 +1,99 @@
+
+Leios.Abstract
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Leios.Abstract where

+

+
open import Leios.Prelude

+

+
record LeiosAbstract : Type₁ where

+
  field Tx : Type

+
        ⦃ DecEq-Tx ⦄ : DecEq Tx

+
        PoolID : Type

+
        ⦃ DecEq-PoolID ⦄ : DecEq PoolID

+
        BodyHash VrfPf PrivKey Sig Hash : Type -- these could have been byte strings, but this is safer

+
        ⦃ DecEq-Hash ⦄ : DecEq Hash

+
        ⦃ DecEq-VrfPf ⦄ : DecEq VrfPf

+
        ⦃ DecEq-Sig ⦄ : DecEq Sig

+
        Vote : Type

+
        ⦃ DecEq-Vote ⦄ : DecEq Vote

+
        vote : PrivKey → Hash → Vote

+
        sign : PrivKey → Hash → Sig

+
        ⦃ Hashable-Txs ⦄ : Hashable (List Tx) Hash

+
        L : ℕ

+
        ⦃ NonZero-L ⦄ : NonZero L

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Base.html b/site/static/agda_html/Leios.Base.html
new file mode 100644
index 000000000..fa840fc24
--- /dev/null
+++ b/site/static/agda_html/Leios.Base.html
@@ -0,0 +1,118 @@
+
+Leios.Base
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude

+
open import Leios.Abstract

+
open import Leios.VRF

+

+
module Leios.Base (a : LeiosAbstract) (open LeiosAbstract a) (vrf' : LeiosVRF a)

+
  (let open LeiosVRF vrf') where

+

+
open import Leios.Blocks a using (EndorserBlock)

+

+
StakeDistr : Type

+
StakeDistr = TotalMap PoolID ℕ

+

+
RankingBlock : Type

+
RankingBlock = These EndorserBlock (List Tx)

+

+
record BaseAbstract : Type₁ where

+
  field Cert : Type

+
        VTy : Type

+
        initSlot : VTy → ℕ

+
        V-chkCerts : List PubKey → EndorserBlock × Cert → Bool

+

+
  data Input : Type where

+
    INIT   : (EndorserBlock × Cert → Bool) → Input

+
    SUBMIT : RankingBlock → Input

+
    FTCH-LDG : Input

+

+
  data Output : Type where

+
    STAKE : StakeDistr → Output

+
    EMPTY : Output

+
    BASE-LDG : List RankingBlock → Output

+

+
  record Functionality : Type₁ where

+
    field State : Type

+
          _-⟦_/_⟧⇀_ : State → Input → Output → State → Type

+
          ⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : {s : State} → {i : Input} → {o : Output} → {s' : State} → (s -⟦ i / o ⟧⇀ s') ⁇

+
          SUBMIT-total : ∀ {s b} → ∃[ s' ] s -⟦ SUBMIT b / EMPTY ⟧⇀ s'

+

+
    open Input public

+
    open Output public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Blocks.html b/site/static/agda_html/Leios.Blocks.html
new file mode 100644
index 000000000..edb620073
--- /dev/null
+++ b/site/static/agda_html/Leios.Blocks.html
@@ -0,0 +1,247 @@
+
+Leios.Blocks
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude

+
open import Leios.Abstract

+
open import Leios.FFD

+
open import Tactic.Defaults

+
open import Tactic.Derive.DecEq

+

+
module Leios.Blocks (a : LeiosAbstract) (let open LeiosAbstract a) where

+

+
-- IsBlock typeclass (could do a closed-world approach instead)

+
-- Q: should votes have an instance of this class?

+
record IsBlock (B : Type) : Type where

+
  field slotNumber : B → ℕ

+
        producerID : B → PoolID

+
        lotteryPf  : B → VrfPf

+
        signature  : B → Sig

+

+
  infix 4 _∈ᴮ_

+

+
  _∈ᴮ_ : B → ℙ ℕ → Type

+
  b ∈ᴮ X = slotNumber b ∈ X

+

+
open IsBlock ⦃...⦄ public

+

+
IBRef = Hash

+
EBRef = Hash

+

+
--------------------------------------------------------------------------------

+
-- Input Blocks

+
--------------------------------------------------------------------------------

+

+
record IBHeaderOSig (sig : Type) : Type where

+
  field slotNumber : ℕ

+
        producerID : PoolID

+
        lotteryPf  : VrfPf

+
        bodyHash   : Hash

+
        signature  : sig

+

+
IBHeader    = IBHeaderOSig Sig

+
PreIBHeader = IBHeaderOSig ⊤

+

+
record IBBody : Type where

+
  field txs : List Tx

+

+
record InputBlock : Type where

+
  field header : IBHeader

+
        body   : IBBody

+

+
  open IBHeaderOSig header public

+

+
unquoteDecl DecEq-IBBody DecEq-IBHeaderOSig DecEq-InputBlock =

+
  derive-DecEq (

+
      (quote IBBody , DecEq-IBBody)

+
    ∷ (quote IBHeaderOSig , DecEq-IBHeaderOSig)

+
    ∷ (quote InputBlock , DecEq-InputBlock) ∷ [])

+

+
instance

+
  IsBlock-IBHeader : IsBlock IBHeader

+
  IsBlock-IBHeader = record { IBHeaderOSig }

+

+
  Hashable-IBBody : Hashable IBBody Hash

+
  Hashable-IBBody .hash b = hash (b .IBBody.txs)

+

+
  Hashable-IBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → Hashable IBHeader Hash

+
  Hashable-IBHeader .hash b = hash {T = PreIBHeader}

+
    record { IBHeaderOSig b hiding (signature) ; signature = _ }

+

+
  IsBlock-InputBlock : IsBlock InputBlock

+
  IsBlock-InputBlock = record { InputBlock }

+

+
mkIBHeader : ⦃ Hashable PreIBHeader Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List Tx → IBHeader

+
mkIBHeader slot id π pKey txs = record { signature = sign pKey (hash h) ; IBHeaderOSig h }

+
  where

+
    h : IBHeaderOSig ⊤

+
    h = record { slotNumber = slot

+
               ; producerID = id

+
               ; lotteryPf  = π

+
               ; bodyHash   = hash txs

+
               ; signature  = _

+
               }

+

+
getIBRef : ⦃ Hashable PreIBHeader Hash ⦄ → InputBlock → IBRef

+
getIBRef = hash ∘ InputBlock.header

+

+
--------------------------------------------------------------------------------

+
-- Endorser Blocks

+
--------------------------------------------------------------------------------

+

+
record EndorserBlockOSig (sig : Type) : Type where

+
  field slotNumber : ℕ

+
        producerID : PoolID

+
        lotteryPf  : VrfPf

+
        ibRefs     : List IBRef

+
        ebRefs     : List EBRef

+
        signature  : sig

+

+
EndorserBlock    = EndorserBlockOSig Sig

+
PreEndorserBlock = EndorserBlockOSig ⊤

+

+
instance

+
  Hashable-EndorserBlock : ⦃ Hashable PreEndorserBlock Hash ⦄ → Hashable EndorserBlock Hash

+
  Hashable-EndorserBlock .hash b = hash {T = PreEndorserBlock}

+
    record { EndorserBlockOSig b hiding (signature) ; signature = _ }

+

+
  IsBlock-EndorserBlock : IsBlock EndorserBlock

+
  IsBlock-EndorserBlock = record { EndorserBlockOSig }

+

+
unquoteDecl DecEq-EndorserBlockOSig = derive-DecEq ((quote EndorserBlockOSig , DecEq-EndorserBlockOSig) ∷ [])

+

+
mkEB : ⦃ Hashable PreEndorserBlock Hash ⦄ → ℕ → PoolID → VrfPf → PrivKey → List IBRef → List EBRef → EndorserBlock

+
mkEB slot id π pKey LI LE = record { signature = sign pKey (hash b) ; EndorserBlockOSig b }

+
  where

+
    b : PreEndorserBlock

+
    b = record { slotNumber = slot

+
               ; producerID = id

+
               ; lotteryPf  = π

+
               ; ibRefs     = LI

+
               ; ebRefs     = LE

+
               ; signature  = _

+
               }

+

+
getEBRef : ⦃ Hashable PreEndorserBlock Hash ⦄ → EndorserBlock → EBRef

+
getEBRef = hash

+

+
--------------------------------------------------------------------------------

+
-- Votes

+
--------------------------------------------------------------------------------

+

+

+

+
--------------------------------------------------------------------------------

+
-- FFD for Leios Blocks

+
--------------------------------------------------------------------------------

+

+
module GenFFD ⦃ _ : IsBlock (List Vote) ⦄ where

+
  data Header : Type where

+
    ibHeader : IBHeader → Header

+
    ebHeader : EndorserBlock → Header

+
    vtHeader : List Vote → Header

+

+
  data Body : Type where

+
    ibBody : IBBody → Body

+

+
  unquoteDecl DecEq-Header DecEq-Body =

+
    derive-DecEq ((quote Header , DecEq-Header) ∷ (quote Body , DecEq-Body) ∷ [])

+

+
  ID : Type

+
  ID = ℕ × PoolID

+

+
  matchIB : IBHeader → IBBody → Type

+
  matchIB h b = bodyHash ≡ hash b

+
    where open IBHeaderOSig h; open IBBody b

+

+
  matchIB? :  ∀ (h : IBHeader) → (b : IBBody) → Dec (matchIB h b)

+
  matchIB? h b = bodyHash ≟ hash b

+
    where open IBHeaderOSig h; open IBBody b

+

+
  match : Header → Body → Type

+
  match (ibHeader h) (ibBody b) = matchIB h b

+
  match _ _ = ⊥

+

+
  -- can we express uniqueness wrt pipelines as a property?

+
  msgID : Header → ID

+
  msgID (ibHeader h) = (slotNumber h , producerID h)

+
  msgID (ebHeader h) = (slotNumber h , producerID h) -- NOTE: this isn't in the paper

+
  msgID (vtHeader  h) = (slotNumber h , producerID h) -- NOTE: this isn't in the paper

+

+
ffdAbstract : ⦃ _ : IsBlock (List Vote) ⦄ → FFDAbstract

+
ffdAbstract = record { GenFFD }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Config.html b/site/static/agda_html/Leios.Config.html
new file mode 100644
index 000000000..cebed5e7f
--- /dev/null
+++ b/site/static/agda_html/Leios.Config.html
@@ -0,0 +1,95 @@
+
+Leios.Config
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Leios.Prelude

+
open import Tactic.Defaults

+
open import Tactic.Derive.DecEq

+

+
module Leios.Config where

+

+
data BlockType : Type where

+
  IB EB VT : BlockType

+

+
unquoteDecl DecEq-BlockType = derive-DecEq ((quote BlockType , DecEq-BlockType) ∷ [])

+

+
record Params : Type where

+
  field numberOfParties : ℕ

+
        sutId : Fin numberOfParties

+
        stakeDistribution : TotalMap (Fin numberOfParties) ℕ

+
        stageLength : ℕ

+
        ⦃ NonZero-stageLength ⦄ : NonZero stageLength

+
        winning-slots : ℙ (BlockType × ℕ)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Defaults.html b/site/static/agda_html/Leios.Defaults.html
new file mode 100644
index 000000000..42f957dd6
--- /dev/null
+++ b/site/static/agda_html/Leios.Defaults.html
@@ -0,0 +1,351 @@
+
+Leios.Defaults
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Leios.Prelude

+
open import Leios.Abstract

+
open import Leios.Config

+
open import Leios.SpecStructure

+

+
open import Axiom.Set.Properties th

+
open import Data.Nat.Show as N

+
open import Data.Integer hiding (_≟_)

+
open import Data.String as S using (intersperse)

+
open import Function.Related.TypeIsomorphisms

+
open import Relation.Binary.Structures

+

+
open import Tactic.Defaults

+
open import Tactic.Derive.DecEq

+

+
open Equivalence

+

+
-- The module contains very simple implementations for the functionalities

+
-- that allow to build examples for traces for the different Leios variants

+
module Leios.Defaults (params : Params) (let open Params params) where

+

+
instance

+
  htx : Hashable (List ℕ) (List ℕ)

+
  htx = record { hash = id }

+

+
d-Abstract : LeiosAbstract

+
d-Abstract =

+
  record

+
    { Tx       = ℕ

+
    ; PoolID   = Fin numberOfParties

+
    ; BodyHash = List ℕ

+
    ; VrfPf    = ⊤

+
    ; PrivKey  = BlockType × ⊤

+
    ; Sig      = ⊤

+
    ; Hash     = List ℕ

+
    ; Vote     = ⊤

+
    ; vote     = λ _ _ → tt

+
    ; sign     = λ _ _ → tt

+
    ; L        = stageLength

+
    }

+

+
open LeiosAbstract d-Abstract public

+

+
open import Leios.VRF d-Abstract public

+

+
sutStake : ℕ

+
sutStake = TotalMap.lookup stakeDistribution sutId

+

+
sortition : BlockType → ℕ → ℕ

+
sortition b n with (b , n) ∈? winning-slots

+
... | yes _ = 0

+
... | no _ = sutStake

+

+
d-VRF : LeiosVRF

+
d-VRF =

+
  record

+
    { PubKey     = Fin numberOfParties × ⊤

+
    ; vrf        =

+
        record

+
          { isKeyPair = λ _ _ → ⊤

+
          ; eval      = λ (b , _) y → sortition b y , tt

+
          ; verify    = λ _ _ _ _ → ⊤

+
          ; verify?   = λ _ _ _ _ → yes tt

+
          }

+
    ; genIBInput = id

+
    ; genEBInput = id

+
    ; genVInput  = id

+
    ; genV1Input = id

+
    ; genV2Input = id

+
    ; poolID     = proj₁

+
    ; verifySig  = λ _ _ → ⊤

+
    ; verifySig? = λ _ _ → yes tt

+
    }

+

+
open LeiosVRF d-VRF public

+

+
open import Leios.Blocks d-Abstract public

+
open import Leios.KeyRegistration d-Abstract d-VRF public

+

+
d-KeyRegistration : KeyRegistrationAbstract

+
d-KeyRegistration = _

+

+
d-KeyRegistrationFunctionality : KeyRegistrationAbstract.Functionality d-KeyRegistration

+
d-KeyRegistrationFunctionality =

+
  record

+
    { State     = ⊤

+
    ; _-⟦_/_⟧⇀_ = λ _ _ _ _ → ⊤

+
    }

+

+
open import Leios.Base d-Abstract d-VRF public

+

+
d-Base : BaseAbstract

+
d-Base =

+
  record

+
    { Cert       = ⊤

+
    ; VTy        = ⊤

+
    ; initSlot   = λ _ → 0

+
    ; V-chkCerts = λ _ _ → true

+
    }

+

+
d-BaseFunctionality : BaseAbstract.Functionality d-Base

+
d-BaseFunctionality =

+
  record

+
    { State         = ⊤

+
    ; _-⟦_/_⟧⇀_     = λ _ _ _ _ → ⊤

+
    ; Dec-_-⟦_/_⟧⇀_ = ⁇ (yes tt)

+
    ; SUBMIT-total  = tt , tt

+
    }

+

+
open import Leios.FFD public

+

+
instance

+
  isb : IsBlock (List ⊤)

+
  isb =

+
    record

+
      { slotNumber = λ _ → 0

+
      ; producerID = λ _ → sutId

+
      ; lotteryPf  = λ _ → tt

+
      }

+

+
  hhs : Hashable PreIBHeader (List ℕ)

+
  hhs .hash = IBHeaderOSig.bodyHash

+

+
  hpe : Hashable PreEndorserBlock (List ℕ)

+
  hpe .hash = L.concat ∘ EndorserBlockOSig.ibRefs

+

+
record FFDBuffers : Type where

+
  field inIBs : List InputBlock

+
        inEBs : List EndorserBlock

+
        inVTs : List (List Vote)

+

+
        outIBs : List InputBlock

+
        outEBs : List EndorserBlock

+
        outVTs : List (List Vote)

+

+
unquoteDecl DecEq-FFDBuffers = derive-DecEq ((quote FFDBuffers , DecEq-FFDBuffers) ∷ [])

+

+
open GenFFD.Header

+
open GenFFD.Body

+
open FFDBuffers

+

+
flushIns : FFDBuffers → List (GenFFD.Header ⊎ GenFFD.Body)

+
flushIns record { inIBs = ibs ; inEBs = ebs ; inVTs = vts } =

+
  flushIBs ibs ++ L.map (inj₁ ∘ ebHeader) ebs ++ L.map (inj₁ ∘ vtHeader) vts

+
  where

+
    flushIBs : List InputBlock → List (GenFFD.Header ⊎ GenFFD.Body)

+
    flushIBs [] = []

+
    flushIBs (record {header = h; body = b} ∷ ibs) = inj₁ (ibHeader h) ∷ inj₂ (ibBody b) ∷ flushIBs ibs

+

+
data SimpleFFD : FFDBuffers → FFDAbstract.Input ffdAbstract → FFDAbstract.Output ffdAbstract → FFDBuffers → Type where

+
  SendIB : ∀ {s h b}    → SimpleFFD s (FFDAbstract.Send (ibHeader h) (just (ibBody b))) FFDAbstract.SendRes (record s { outIBs = record {header = h; body = b} ∷ outIBs s})

+
  SendEB : ∀ {s eb}     → SimpleFFD s (FFDAbstract.Send (ebHeader eb) nothing) FFDAbstract.SendRes (record s { outEBs = eb ∷ outEBs s})

+
  SendVS : ∀ {s vs}     → SimpleFFD s (FFDAbstract.Send (vtHeader vs) nothing) FFDAbstract.SendRes (record s { outVTs = vs ∷ outVTs s})

+

+
  BadSendIB : ∀ {s h}   → SimpleFFD s (FFDAbstract.Send (ibHeader h) nothing) FFDAbstract.SendRes s

+
  BadSendEB : ∀ {s h b} → SimpleFFD s (FFDAbstract.Send (ebHeader h) (just b)) FFDAbstract.SendRes s

+
  BadSendVS : ∀ {s h b} → SimpleFFD s (FFDAbstract.Send (vtHeader h) (just b)) FFDAbstract.SendRes s

+

+
  Fetch : ∀ {s}         → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes (flushIns s)) (record s { inIBs = [] ; inEBs = [] ; inVTs = [] })

+

+
send-total : ∀ {s h b} → ∃[ s' ] (SimpleFFD s (FFDAbstract.Send h b) FFDAbstract.SendRes s')

+
send-total {s} {ibHeader h} {just (ibBody b)} = record s { outIBs = record {header = h; body = b} ∷ outIBs s} , SendIB

+
send-total {s} {ebHeader eb} {nothing}        = record s { outEBs = eb ∷ outEBs s} , SendEB

+
send-total {s} {vtHeader vs} {nothing}        = record s { outVTs = vs ∷ outVTs s} , SendVS

+

+
send-total {s} {ibHeader h} {nothing} = s , BadSendIB

+
send-total {s} {ebHeader eb} {just _} = s , BadSendEB

+
send-total {s} {vtHeader vs} {just _} = s , BadSendVS

+

+
fetch-total : ∀ {s} → ∃[ x ] (∃[ s' ] (SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes x) s'))

+
fetch-total {s} = flushIns s , (record s { inIBs = [] ; inEBs = [] ; inVTs = [] } , Fetch)

+

+
send-complete : ∀ {s h b s'} → SimpleFFD s (FFDAbstract.Send h b) FFDAbstract.SendRes s' → s' ≡ proj₁ (send-total {s} {h} {b})

+
send-complete SendIB    = refl

+
send-complete SendEB    = refl

+
send-complete SendVS    = refl

+
send-complete BadSendIB = refl

+
send-complete BadSendEB = refl

+
send-complete BadSendVS = refl

+

+
fetch-complete₁ : ∀ {s r s'} → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes r) s' → s' ≡ proj₁ (proj₂ (fetch-total {s}))

+
fetch-complete₁ Fetch = refl

+

+
fetch-complete₂ : ∀ {s r s'} → SimpleFFD s FFDAbstract.Fetch (FFDAbstract.FetchRes r) s' → r ≡ proj₁ (fetch-total {s})

+
fetch-complete₂ Fetch = refl

+

+
instance

+
  Dec-SimpleFFD : ∀ {s i o s'} → SimpleFFD s i o s' ⁇

+
  Dec-SimpleFFD {s} {FFDAbstract.Send h b} {FFDAbstract.SendRes} {s'} with s' ≟ proj₁ (send-total {s} {h} {b})

+
  ... | yes p rewrite p = ⁇ yes (proj₂ send-total)

+
  ... | no ¬p = ⁇ no λ x → ⊥-elim (¬p (send-complete x))

+
  Dec-SimpleFFD {_} {FFDAbstract.Send _ _} {FFDAbstract.FetchRes _} {_} = ⁇ no λ ()

+
  Dec-SimpleFFD {s} {FFDAbstract.Fetch} {FFDAbstract.FetchRes r} {s'}

+
    with s' ≟ proj₁ (proj₂ (fetch-total {s}))

+
      | r ≟ proj₁ (fetch-total {s})

+
  ... | yes p | yes q rewrite p rewrite q = ⁇ yes (proj₂ (proj₂ (fetch-total {s})))

+
  ... | yes p | no ¬q = ⁇ no λ x → ⊥-elim (¬q (fetch-complete₂ x))

+
  ... | no ¬p | _ = ⁇ no λ x → ⊥-elim (¬p (fetch-complete₁ x))

+
  Dec-SimpleFFD {_} {FFDAbstract.Fetch} {FFDAbstract.SendRes} {_} = ⁇ no λ ()

+

+
d-FFDFunctionality : FFDAbstract.Functionality ffdAbstract

+
d-FFDFunctionality =

+
  record

+
    { State         = FFDBuffers

+
    ; initFFDState  = record { inIBs = []; inEBs = []; inVTs = []; outIBs = []; outEBs = []; outVTs = [] }

+
    ; _-⟦_/_⟧⇀_     = SimpleFFD

+
    ; Dec-_-⟦_/_⟧⇀_ = Dec-SimpleFFD

+
    ; Send-total    = send-total

+
    ; Fetch-total   = fetch-total

+
    }

+

+
open import Leios.Voting public

+

+
d-VotingAbstract : VotingAbstract (Fin 1 × EndorserBlock)

+
d-VotingAbstract =

+
  record

+
    { VotingState     = ⊤

+
    ; initVotingState = tt

+
    ; isVoteCertified = λ _ _ → ⊤

+
    }

+

+
d-VotingAbstract-2 : VotingAbstract (Fin 2 × EndorserBlock)

+
d-VotingAbstract-2 =

+
  record

+
    { VotingState     = ⊤

+
    ; initVotingState = tt

+
    ; isVoteCertified = λ _ _ → ⊤

+
    }

+

+
d-SpecStructure : SpecStructure 1

+
d-SpecStructure = record

+
      { a                         = d-Abstract

+
      ; Hashable-PreIBHeader      = hhs

+
      ; Hashable-PreEndorserBlock = hpe

+
      ; id                        = sutId

+
      ; FFD'                      = d-FFDFunctionality

+
      ; vrf'                      = d-VRF

+
      ; sk-IB                     = IB , tt

+
      ; sk-EB                     = EB , tt

+
      ; sk-VT                     = VT , tt

+
      ; pk-IB                     = sutId , tt

+
      ; pk-EB                     = sutId , tt

+
      ; pk-VT                     = sutId , tt

+
      ; B'                        = d-Base

+
      ; BF                        = d-BaseFunctionality

+
      ; initBaseState             = tt

+
      ; K'                        = d-KeyRegistration

+
      ; KF                        = d-KeyRegistrationFunctionality

+
      ; va                        = d-VotingAbstract

+
      }

+

+
d-SpecStructure-2 : SpecStructure 2

+
d-SpecStructure-2 = record

+
      { a                         = d-Abstract

+
      ; Hashable-PreIBHeader      = hhs

+
      ; Hashable-PreEndorserBlock = hpe

+
      ; id                        = sutId

+
      ; FFD'                      = d-FFDFunctionality

+
      ; vrf'                      = d-VRF

+
      ; sk-IB                     = IB , tt

+
      ; sk-EB                     = EB , tt

+
      ; sk-VT                     = VT , tt

+
      ; pk-IB                     = sutId , tt

+
      ; pk-EB                     = sutId , tt

+
      ; pk-VT                     = sutId , tt

+
      ; B'                        = d-Base

+
      ; BF                        = d-BaseFunctionality

+
      ; initBaseState             = tt

+
      ; K'                        = d-KeyRegistration

+
      ; KF                        = d-KeyRegistrationFunctionality

+
      ; va                        = d-VotingAbstract-2

+
      }

+

+
open import Leios.Short d-SpecStructure public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.FFD.html b/site/static/agda_html/Leios.FFD.html
new file mode 100644
index 000000000..f9f0d896a
--- /dev/null
+++ b/site/static/agda_html/Leios.FFD.html
@@ -0,0 +1,108 @@
+
+Leios.FFD
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Leios.FFD where

+

+
open import Leios.Prelude

+

+
record FFDAbstract : Type₁ where

+
  field Header Body ID : Type

+
        ⦃ DecEq-Header ⦄ : DecEq Header

+
        ⦃ DecEq-Body ⦄ : DecEq Body

+
        match : Header → Body → Type

+
        msgID : Header → ID

+

+
  data Input : Type where

+
    Send  : Header → Maybe Body → Input

+
    Fetch : Input

+

+
  data Output : Type where

+
    SendRes  : Output

+
    FetchRes : List (Header ⊎ Body) → Output

+

+
  record Functionality : Type₁ where

+
    field State : Type

+
          initFFDState : State

+
          _-⟦_/_⟧⇀_ : State → Input → Output → State → Type

+
          ⦃ Dec-_-⟦_/_⟧⇀_ ⦄ : {s : State} → {i : Input} → {o : Output} → {s' : State} → (s -⟦ i / o ⟧⇀ s') ⁇

+
          Send-total : ∀ {ffds h b} → ∃[ ffds' ] ffds -⟦ Send h b / SendRes ⟧⇀ ffds'

+
          Fetch-total : ∀ {ffds} → ∃[ r ] (∃[ ffds' ] (ffds -⟦ Fetch / FetchRes r ⟧⇀ ffds'))

+

+
    open Input public

+
    open Output public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Foreign.BaseTypes.html b/site/static/agda_html/Leios.Foreign.BaseTypes.html
new file mode 100644
index 000000000..843e2973b
--- /dev/null
+++ b/site/static/agda_html/Leios.Foreign.BaseTypes.html
@@ -0,0 +1,196 @@
+
+Leios.Foreign.BaseTypes
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Leios.Foreign.BaseTypes where

+

+
-- TODO: copied from the formal-ledger project for now

+
-- Added: * TotalMap

+

+
open import Data.Rational

+

+
open import Leios.Prelude

+

+
open import Data.Fin

+
open import Function.Related.TypeIsomorphisms

+
open import Relation.Binary.Structures

+

+
open import Tactic.Derive.Convertible

+
open import Tactic.Derive.HsType

+

+
open import Class.Convertible

+
open import Class.Decidable.Instances

+
open import Class.HasHsType

+

+
open import Leios.Foreign.HsTypes as F

+
open import Leios.Foreign.Util

+
open import Foreign.Haskell

+

+
instance

+
  iConvTop    = Convertible-Refl {⊤}

+
  iConvNat    = Convertible-Refl {ℕ}

+
  iConvString = Convertible-Refl {String}

+
  iConvBool   = Convertible-Refl {Bool}

+

+
instance

+

+
  -- * Unit and empty

+

+
  HsTy-⊥ = MkHsType ⊥ F.Empty

+
  Conv-⊥ = autoConvert ⊥

+

+
  HsTy-⊤ = MkHsType ⊤ ⊤

+

+
  -- * Rational numbers

+

+
  HsTy-Rational = MkHsType ℚ F.Rational

+
  Conv-Rational : HsConvertible ℚ

+
  Conv-Rational = λ where

+
    .to (mkℚ n d _)       → n F., suc d

+
    .from (n F., zero)    → 0ℚ -- TODO is there a safer way to do this?

+
    .from (n F., (suc d)) → n Data.Rational./ suc d

+

+
  -- * Maps and Sets

+

+
  HsTy-HSSet : ∀ {A} → ⦃ HasHsType A ⦄ → HasHsType (ℙ A)

+
  HsTy-HSSet {A} = MkHsType _ (F.HSSet (HsType A))

+

+
  Conv-HSSet : ∀ {A} ⦃ _ : HasHsType A ⦄

+
             → ⦃ HsConvertible A ⦄

+
             → HsConvertible (ℙ A)

+
  Conv-HSSet = λ where

+
    .to   → F.MkHSSet ∘ to ∘ setToList

+
    .from → fromList ∘ from ∘ F.HSSet.elems

+

+
  Convertible-FinSet : Convertible₁ ℙ_ List

+
  Convertible-FinSet = λ where

+
    .to   → map to   ∘ setToList

+
    .from → fromList ∘ map from

+

+
  Convertible-Map : ∀ {K K' V V'} → ⦃ DecEq K ⦄

+
    → ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄

+
    → Convertible (K ⇀ V) (List $ Pair K' V')

+
  Convertible-Map = λ where

+
    .to   → to        ∘ proj₁

+
    .from → fromListᵐ ∘ map from

+

+
  HsTy-Map : ∀ {A B} → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (A ⇀ B)

+
  HsTy-Map {A} {B} = MkHsType _ (F.HSMap (HsType A) (HsType B))

+

+
  Conv-HSMap : ∀ {A B} ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄

+
    → ⦃ DecEq A ⦄

+
    → ⦃ HsConvertible A ⦄

+
    → ⦃ HsConvertible B ⦄

+
    → HsConvertible (A ⇀ B)

+
  Conv-HSMap = λ where

+
    .to   → F.MkHSMap ∘ to

+
    .from → from ∘ F.HSMap.assocList

+

+
  Convertible-TotalMap : ∀ {K K' V V'} → ⦃ DecEq K ⦄ → ⦃ Listable K ⦄

+
    → ⦃ Convertible K K' ⦄ → ⦃ Convertible V V' ⦄

+
    → Convertible (TotalMap K V) (List $ Pair K' V')

+
  Convertible-TotalMap {K} = λ where

+
    .to   → to ∘ TotalMap.rel

+
    .from → λ x →

+
      let (r , l) = fromListᵐ (map from x)

+
      in case (¿ total r ¿) of λ where

+
           (yes p) → record { rel = r ; left-unique-rel = l ; total-rel = p }

+
           (no p) → error "Expected total map"

+

+
  HsTy-TotalMap : ∀ {A B} → ⦃ HasHsType A ⦄ → ⦃ HasHsType B ⦄ → HasHsType (TotalMap A B)

+
  HsTy-TotalMap {A} {B} = MkHsType _ (F.HSMap (HsType A) (HsType B))

+

+
  Conv-HSTotalMap : ∀ {A B} ⦃ _ : HasHsType A ⦄ ⦃ _ : HasHsType B ⦄

+
    → ⦃ DecEq A ⦄

+
    → ⦃ Listable A ⦄

+
    → ⦃ HsConvertible A ⦄

+
    → ⦃ HsConvertible B ⦄

+
    → HsConvertible (TotalMap A B)

+
  Conv-HSTotalMap = λ where

+
    .to   → MkHSMap ∘ to

+
    .from → from ∘ F.HSMap.assocList

+

+
  -- * ComputationResult

+

+
  open import Class.Computational as C

+

+
  HsTy-ComputationResult : ∀ {l} {Err} {A : Type l}

+
                           → ⦃ HasHsType Err ⦄ → ⦃ HasHsType A ⦄

+
                           → HasHsType (C.ComputationResult Err A)

+
  HsTy-ComputationResult {Err = Err} {A} = MkHsType _ (F.ComputationResult (HsType Err) (HsType A))

+

+
  Conv-ComputationResult : ConvertibleType C.ComputationResult F.ComputationResult

+
  Conv-ComputationResult = autoConvertible

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Foreign.HsTypes.html b/site/static/agda_html/Leios.Foreign.HsTypes.html
new file mode 100644
index 000000000..a469dad3b
--- /dev/null
+++ b/site/static/agda_html/Leios.Foreign.HsTypes.html
@@ -0,0 +1,153 @@
+
+Leios.Foreign.HsTypes
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Leios.Foreign.HsTypes where

+

+
{-# FOREIGN GHC

+
  {-# LANGUAGE DeriveGeneric #-}

+
  {-# LANGUAGE DeriveFunctor #-}

+
#-}

+

+
open import Leios.Prelude

+

+
open import Foreign.Haskell

+
open import Foreign.Haskell.Coerce

+
open import Foreign.Haskell.Either

+
open import Data.Rational.Base

+

+
{-# FOREIGN GHC

+
  import GHC.Generics (Generic)

+
  import Data.Void (Void)

+
  import Prelude hiding (Rational)

+
  import GHC.Real (Ratio(..))

+
#-}

+

+
-- * The empty type

+

+
data Empty : Type where

+
{-# COMPILE GHC Empty = data Void () #-}

+

+
-- * Rational

+

+
data Rational : Type where

+
  _,_ : ℤ → ℕ → Rational

+
{-# COMPILE GHC Rational = data Rational ((:%)) #-}

+

+
-- We'll generate code with qualified references to Rational in this

+
-- module, so make sure to define it.

+
{-# FOREIGN GHC type Rational = Ratio Integer #-}

+

+
-- * Maps and Sets

+

+
record HSMap K V : Type where

+
  constructor MkHSMap

+
  field assocList : List (Pair K V)

+

+
record HSSet A : Type where

+
  constructor MkHSSet

+
  field elems : List A

+

+
{-# FOREIGN GHC

+
  newtype HSMap k v = MkHSMap [(k, v)]

+
    deriving (Generic, Show, Eq, Ord)

+
  newtype HSSet a = MkHSSet [a]

+
    deriving (Generic, Show, Eq, Ord)

+
#-}

+
{-# COMPILE GHC HSMap = data HSMap (MkHSMap) #-}

+
{-# COMPILE GHC HSSet = data HSSet (MkHSSet) #-}

+

+
-- * ComputationResult

+

+
data ComputationResult E A : Type where

+
  Success : A → ComputationResult E A

+
  Failure : E → ComputationResult E A

+

+
{-# FOREIGN GHC

+
  data ComputationResult e a = Success a | Failure e

+
    deriving (Functor, Eq, Show, Generic)

+

+
  instance Applicative (ComputationResult e) where

+
    pure = Success

+
    (Success f) <*> x = f <$> x

+
    (Failure e) <*> _ = Failure e

+

+
  instance Monad (ComputationResult e) where

+
    return = pure

+
    (Success a) >>= m = m a

+
    (Failure e) >>= _ = Failure e

+
#-}

+
{-# COMPILE GHC ComputationResult = data ComputationResult (Success | Failure) #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Foreign.Types.html b/site/static/agda_html/Leios.Foreign.Types.html
new file mode 100644
index 000000000..b5dd35f90
--- /dev/null
+++ b/site/static/agda_html/Leios.Foreign.Types.html
@@ -0,0 +1,241 @@
+
+Leios.Foreign.Types
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Data.Char.Base as C using (Char)

+
import Data.String as S

+
open import Data.Integer using (+_; ∣_∣)

+

+
open import Class.Convertible

+
open import Class.HasHsType

+
open import Tactic.Derive.Convertible

+
open import Tactic.Derive.HsType

+

+
open import Leios.Prelude

+

+
open import Leios.Foreign.BaseTypes

+
open import Leios.Foreign.HsTypes

+
open import Leios.Foreign.Util

+

+
module Leios.Foreign.Types where

+

+
{-# FOREIGN GHC

+
  {-# LANGUAGE DuplicateRecordFields #-}

+
#-}

+

+
-- TODO: Get rid of hardcoded parameters in this module

+

+
{-

+
numberOfParties : ℕ

+
numberOfParties = 2

+

+
open import Leios.Defaults numberOfParties fzero

+
  renaming (EndorserBlock to EndorserBlockAgda; IBHeader to IBHeaderAgda)

+

+
dropDash : S.String → S.String

+
dropDash = S.concat ∘ S.wordsByᵇ ('-' C.≈ᵇ_)

+

+
prefix : S.String → S.String → S.String

+
prefix = S._++_

+

+
instance

+
  HsTy-SlotUpkeep = autoHsType SlotUpkeep ⊣ onConstructors dropDash

+
  Conv-SlotUpkeep = autoConvert SlotUpkeep

+

+
record IBHeader : Type where

+
  field slotNumber : ℕ

+
        producerID : ℕ

+
        bodyHash : List ℕ

+

+
{-# FOREIGN GHC

+
data IBHeader = IBHeader {slotNumber :: Integer, producerID :: Integer, bodyHash :: Data.Text.Text }

+
  deriving (Show, Eq, Generic)

+
#-}

+

+
{-# COMPILE GHC IBHeader = data IBHeader (IBHeader) #-}

+

+
instance

+
  HsTy-IBHeader = MkHsType IBHeaderAgda IBHeader

+

+
  Conv-IBHeader : Convertible IBHeaderAgda IBHeader

+
  Conv-IBHeader = record

+
    { to = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →

+
        record { slotNumber = s ; producerID = toℕ p ; bodyHash = h}

+
    ; from = λ (record { slotNumber = s ; producerID = p ; bodyHash = h }) →

+
        case p <? numberOfParties of λ where

+
          (yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; bodyHash = h ; signature = tt }

+
          (no _) → error "Conversion to Fin not possible!"

+
    }

+

+
  HsTy-IBBody = autoHsType IBBody

+
  Conv-IBBody = autoConvert IBBody

+

+
  HsTy-InputBlock = autoHsType InputBlock

+
  Conv-InputBlock = autoConvert InputBlock

+

+
  Conv-ℕ : HsConvertible ℕ

+
  Conv-ℕ = Convertible-Refl

+

+
record EndorserBlock : Type where

+
  field slotNumber : ℕ

+
        producerID : ℕ

+
        ibRefs     : List (List IBRef)

+

+
{-# FOREIGN GHC

+
data EndorserBlock = EndorserBlock { slotNumber :: Integer, producerID :: Integer, ibRefs :: [Data.Text.Text] }

+
  deriving (Show, Eq, Generic)

+
#-}

+

+
{-# COMPILE GHC EndorserBlock = data EndorserBlock (EndorserBlock) #-}

+

+
instance

+
  HsTy-EndorserBlock = MkHsType EndorserBlockAgda EndorserBlock

+

+
  Conv-EndorserBlock : Convertible EndorserBlockAgda EndorserBlock

+
  Conv-EndorserBlock =

+
    record

+
      { to = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →

+
          record { slotNumber = s ; producerID = toℕ p ; ibRefs = refs }

+
      ; from = λ (record { slotNumber = s ; producerID = p ; ibRefs = refs }) →

+
        case p <? numberOfParties of λ where

+
          (yes q) → record { slotNumber = s ; producerID = #_ p {numberOfParties} {fromWitness q} ; lotteryPf = tt ; signature = tt ; ibRefs = refs ; ebRefs = [] }

+
          (no _) → error "Conversion to Fin not possible!"

+
      }

+

+
  HsTy-FFDState = autoHsType FFDState

+
  Conv-FFDState = autoConvert FFDState

+

+
  HsTy-Fin : ∀ {n} → HasHsType (Fin n)

+
  HsTy-Fin .HasHsType.HsType = ℕ

+

+
  Conv-Fin : ∀ {n} → HsConvertible (Fin n)

+
  Conv-Fin {n} =

+
    record

+
      { to = toℕ

+
      ; from = λ m →

+
          case m <? n of λ where

+
            (yes p) → #_ m {n} {fromWitness p}

+
            (no _) → error "Conversion to Fin not possible!"

+
      }

+

+
  HsTy-LeiosState = autoHsType LeiosState

+
  Conv-LeiosState = autoConvert LeiosState

+

+
  HsTy-LeiosInput = autoHsType LeiosInput ⊣ onConstructors (prefix "I_" ∘ dropDash)

+
  Conv-LeiosInput = autoConvert LeiosInput

+

+
  HsTy-LeiosOutput = autoHsType LeiosOutput ⊣ onConstructors (prefix "O_" ∘ dropDash)

+
  Conv-LeiosOutput = autoConvert LeiosOutput

+

+
open import Class.Computational as C

+
open import Class.Computational22

+

+
open Computational22

+
open BaseAbstract

+
open FFDAbstract

+

+
open GenFFD.Header using (ibHeader; ebHeader; vtHeader)

+
open GenFFD.Body using (ibBody)

+
open FFDState

+

+
open import Leios.Short.Deterministic d-SpecStructure public

+

+
open FunTot (completeFin numberOfParties) (maximalFin numberOfParties)

+

+
d-StakeDistribution : TotalMap (Fin numberOfParties) ℕ

+
d-StakeDistribution = Fun⇒TotalMap (const 100000000)

+

+
instance

+
  Computational-B : Computational22 (BaseAbstract.Functionality._-⟦_/_⟧⇀_ d-BaseFunctionality) String

+
  Computational-B .computeProof s (INIT x) = success ((STAKE d-StakeDistribution , tt) , tt)

+
  Computational-B .computeProof s (SUBMIT x) = success ((EMPTY , tt) , tt)

+
  Computational-B .computeProof s FTCH-LDG = success (((BASE-LDG []) , tt) , tt)

+
  Computational-B .completeness _ _ _ _ _ = {!!} -- TODO: Completeness proof

+

+
  Computational-FFD : Computational22 (FFDAbstract.Functionality._-⟦_/_⟧⇀_ d-FFDFunctionality) String

+
  Computational-FFD .computeProof s (Send (ibHeader h) (just (ibBody b))) = success ((SendRes , record s {outIBs = record {header = h; body = b} ∷ outIBs s}) , SendIB)

+
  Computational-FFD .computeProof s (Send (ebHeader h) nothing) = success ((SendRes , record s {outEBs = h ∷ outEBs s}) , SendEB)

+
  Computational-FFD .computeProof s (Send (vtHeader h) nothing) = success ((SendRes , record s {outVTs = h ∷ outVTs s}) , SendVS)

+
  Computational-FFD .computeProof s Fetch = success ((FetchRes (flushIns s) , record s {inIBs = []; inEBs = []; inVTs = []}) , Fetch)

+

+
  Computational-FFD .computeProof _ _ = failure "FFD error"

+
  Computational-FFD .completeness _ _ _ _ _ = {!!} -- TODO:Completeness proof

+

+
stepHs : HsType (LeiosState → LeiosInput → C.ComputationResult String (LeiosOutput × LeiosState))

+
stepHs = to (compute Computational--⟦/⟧⇀)

+

+
{-# COMPILE GHC stepHs as step #-}

+
-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Foreign.Util.html b/site/static/agda_html/Leios.Foreign.Util.html
new file mode 100644
index 000000000..c5eb6885a
--- /dev/null
+++ b/site/static/agda_html/Leios.Foreign.Util.html
@@ -0,0 +1,85 @@
+
+Leios.Foreign.Util
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Leios.Foreign.Util where

+

+
open import Leios.Prelude

+

+
postulate

+
  error : {A : Set} → String → A

+
{-# FOREIGN GHC import Data.Text #-}

+
{-# COMPILE GHC error = \ _ s -> error (unpack s) #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.KeyRegistration.html b/site/static/agda_html/Leios.KeyRegistration.html
new file mode 100644
index 000000000..326e6c139
--- /dev/null
+++ b/site/static/agda_html/Leios.KeyRegistration.html
@@ -0,0 +1,100 @@
+
+Leios.KeyRegistration
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude

+
open import Leios.Abstract

+
open import Leios.VRF

+

+
module Leios.KeyRegistration (a : LeiosAbstract) (open LeiosAbstract a)

+
  (vrf : LeiosVRF a) (let open LeiosVRF vrf) where

+

+
record KeyRegistrationAbstract : Type₁ where

+

+
  data Input : Type₁ where

+
    INIT : PubKey → PubKey → PubKey → Input

+

+
  data Output : Type where

+
    PUBKEYS : List PubKey → Output

+

+
  record Functionality : Type₁ where

+
    field State : Type

+
          _-⟦_/_⟧⇀_ : State → Input → Output → State → Type

+

+
    open Input public

+
    open Output public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Network.html b/site/static/agda_html/Leios.Network.html
new file mode 100644
index 000000000..1a525a666
--- /dev/null
+++ b/site/static/agda_html/Leios.Network.html
@@ -0,0 +1,149 @@
+
+Leios.Network
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Leios.Network where

+

+
open import abstract-set-theory.Prelude hiding (_∘_; _⊗_)

+
open import abstract-set-theory.FiniteSetTheory using (ℙ_; _∈_; _∪_; ❴_❵; _∉_)

+

+
open import CategoricalCrypto

+

+
record Abstract : Set₁ where

+
  field Header Body ID : Set

+
        match : Header → Body → Set

+
        msgID : Header → ID

+

+
module Broadcast (M Peer : Set) where

+
  open Channel

+

+
  C : Channel

+
  C .P = Peer

+
  C .rcvType _ = Peer × M

+
  C .sndType _ = M

+

+
  postulate Functionality : Machine I C

+

+
  Single : Channel

+
  Single .P = ⊤

+
  Single .rcvType _ = Peer × M

+
  Single .sndType _ = M

+

+
  postulate SingleFunctionality : Machine I Single

+

+
  -- connectWithBroadcast : ∀ {A} → (Peer → Machine Single A) → Machine I A

+
  -- connectWithBroadcast = {!!}

+

+
module HeaderDiffusion (a : Abstract) (Peer : Set) (self : Peer) where

+
  open Channel

+
  open Abstract a

+
  module B = Broadcast Header Peer

+

+
  data Port : Set where

+
    Send    : Port -- we want to send a header

+
    Forward : Port -- we want to forward a header

+

+
  C : Channel

+
  C .P = Port

+
  C .sndType _ = Header

+
  C .rcvType Forward = Header

+
  C .rcvType Send = ⊥

+

+
  data Input : Set where

+
    S : Header → Input

+
    F : Header → Input

+
    R : Peer → Header → Input

+

+
  data Output : Set where

+
    Verify : Header → Output

+

+
  private variable

+
    h : Header

+
    s : ℙ ID

+

+
  data Step : ∃ (rcvType (B.Single ⊗ C ᵀ)) → ℙ ID → ℙ ID × Maybe (∃ (sndType (B.Single ⊗ C ᵀ))) → Set where

+
    Init : Step (inj₂ Send , h) s (s ∪ ❴ msgID h ❵ , just (inj₁ _ , h))

+
    Receive1  : ∀ {p} → Step (inj₁ _ , p , h) s (s , just (inj₂ Forward , h))

+
    Receive2  : msgID h ∉ s → Step (inj₂ Forward , h) s (s ∪ ❴ msgID h ❵ , just (inj₁ _ , h))

+
    Receive2' : msgID h ∈ s → Step (inj₂ Forward , h) s (s , nothing)

+

+
  step : ∃ (rcvType (B.Single ⊗ C ᵀ)) → ∃ (sndType (B.Single ⊗ C ᵀ))

+
  step (inj₁ _   , _ , h) = (inj₂ Forward , h)

+
  step (inj₂ Forward , h) = (inj₁ _ , h)

+
  step (inj₂ Send    , h) = (inj₁ _ , h)

+

+
  Functionality : Machine I C

+
  Functionality = MkMachine' Step ∘ B.SingleFunctionality

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Prelude.html b/site/static/agda_html/Leios.Prelude.html
new file mode 100644
index 000000000..5a27b3d2b
--- /dev/null
+++ b/site/static/agda_html/Leios.Prelude.html
@@ -0,0 +1,202 @@
+
+Leios.Prelude
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module Leios.Prelude where

+

+
open import abstract-set-theory.FiniteSetTheory public

+
open import abstract-set-theory.Prelude public

+
open import Data.List using (upTo)

+

+
open import Class.HasAdd public

+
open import Class.HasOrder public

+
open import Class.Hashable public

+
open import Prelude.InferenceRules public

+

+
module T where

+
  open import Data.These public

+
open T public using (These; this; that)

+

+
module L where

+
  open import Data.List public

+
open L public using (List; []; _∷_; _++_; catMaybes; head; length; sum; and; or; any)

+

+
module A where

+
  open import Data.List.Relation.Unary.Any public

+
open A public using (here; there)

+

+
module N where

+
  open import Data.Nat public

+
  open import Data.Nat.Properties public

+
open N public using (ℕ; zero; suc)

+

+
module F where

+
  open import Data.Fin public

+
  open import Data.Fin.Properties public

+
open F public using (Fin; toℕ; #_) renaming (zero to fzero; suc to fsuc)

+

+
fromTo : ℕ → ℕ → List ℕ

+
fromTo m n = map (_+ m) (upTo (n ∸ m))

+

+
slice : (L : ℕ) → ⦃ NonZero L ⦄ → ℕ → ℕ → ℙ ℕ

+
slice L s x = fromList (fromTo s' (s' + (L ∸ 1)))

+
  where s' = ((s / L) ∸ x) * L -- equivalent to the formula in the paper

+

+
filter : {A : Set} → (P : A → Type) ⦃ _ : P ⁇¹ ⦄ → List A → List A

+
filter P = L.filter ¿ P ¿¹

+

+
instance

+
  IsSet-List : {A : Set} → IsSet (List A) A

+
  IsSet-List .toSet A = fromList A

+

+
open import Data.List.Relation.Unary.Any

+
open Properties

+

+
finite⇒A≡∅⊎∃a∈A : {X : Type} → {A : ℙ X} → finite A → (A ≡ᵉ ∅) ⊎ Σ[ a ∈ X ] a ∈ A

+
finite⇒A≡∅⊎∃a∈A ([]    , h) = inj₁ (∅-least (λ a∈A → ⊥-elim (case Equivalence.to h a∈A of λ ())))

+
finite⇒A≡∅⊎∃a∈A (x ∷ _ , h) = inj₂ (x , Equivalence.from h (here refl))

+

+
completeFin : ∀ (n : ℕ) → ℙ (Fin n)

+
completeFin zero = ∅

+
completeFin (ℕ.suc n) = singleton (F.fromℕ n) ∪ mapˢ F.inject₁ (completeFin n)

+

+
m≤n∧n≤m⇒m≡n : ∀ {n m : ℕ} → n N.≤ m → m N.≤ n → m ≡ n

+
m≤n∧n≤m⇒m≡n z≤n z≤n = refl

+
m≤n∧n≤m⇒m≡n (s≤s n≤m) (s≤s m≤n) = cong N.suc (m≤n∧n≤m⇒m≡n n≤m m≤n)

+

+
toℕ-fromℕ : ∀ {n} {a : Fin (N.suc n)} → toℕ a ≡ n → a ≡ F.fromℕ n

+
toℕ-fromℕ {zero} {fzero} x = refl

+
toℕ-fromℕ {N.suc n} {fsuc a} x = cong fsuc (toℕ-fromℕ {n} {a} (N.suc-injective x))

+

+
open Equivalence

+

+
maximalFin : ∀ (n : ℕ) → isMaximal (completeFin n)

+
maximalFin (ℕ.suc n) {a} with toℕ a N.<? n

+
... | yes p =

+
  let n≢toℕ = ≢-sym (N.<⇒≢ p)

+
      fn = F.lower₁ a n≢toℕ

+
      fn≡a = F.inject₁-lower₁ a n≢toℕ

+
  in (to ∈-∪) (inj₂ ((to ∈-map) (fn , (sym fn≡a , maximalFin n))))

+
... | no ¬p with a F.≟ F.fromℕ n

+
... | yes q = (to ∈-∪) (inj₁ ((to ∈-singleton) q))

+
... | no ¬q =

+
  let n≢toℕ = N.≰⇒> ¬p

+
      a<sucn = F.toℕ<n a

+
  in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (N.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))

+

+
record Listable (A : Type) : Type where

+
  field

+
    listing  : ℙ A

+
    complete : ∀ {a : A} → a ∈ listing

+

+
totalDec : ∀ {A B : Type} → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → {R : Rel A B} → Dec (total R)

+
totalDec {A} {B} {R} with all? (_∈? dom R)

+
... | yes p = yes λ {a} → p {a} ((Listable.complete it) {a})

+
... | no ¬p = no λ x → ¬p λ {a} _ → x {a}

+

+
instance

+
  total? : ∀ {A B : Type} → ⦃ DecEq A ⦄ → ⦃ Listable A ⦄ → {R : Rel A B} → ({a : A} → a ∈ dom R) ⁇

+
  total? = ⁇ totalDec

+

+
  Listable-Fin : ∀ {n} → Listable (Fin n)

+
  Listable-Fin {zero} = record { listing = ∅ ; complete = λ {a} → ⊥-elim $ (Inverse.to F.0↔⊥) a }

+
  Listable-Fin {suc n} =

+
    let record { listing = l ; complete = c } = Listable-Fin {n}

+
    in record

+
         { listing = singleton (F.fromℕ n) ∪ mapˢ F.inject₁ l

+
         ; complete = complete

+
         }

+
       where

+
         complete : ∀ {a} → a ∈ singleton (F.fromℕ n) ∪ mapˢ F.inject₁ (let record { listing = l } = Listable-Fin {n} in l)

+
         complete {a} with F.toℕ a N.<? n

+
         ... | yes p =

+
           let record { listing = l ; complete = c } = Listable-Fin {n}

+
               n≢toℕ = ≢-sym (N.<⇒≢ p)

+
               fn = F.lower₁ a n≢toℕ

+
               fn≡a = F.inject₁-lower₁ a n≢toℕ

+
           in (Equivalence.to ∈-∪) (inj₂ ((Equivalence.to ∈-map) (fn , (sym fn≡a , c))))

+
         ... | no ¬p with a F.≟ F.fromℕ n

+
         ... | yes q = (Equivalence.to ∈-∪) (inj₁ ((Equivalence.to ∈-singleton) q))

+
         ... | no ¬q =

+
           let n≢toℕ = N.≰⇒> ¬p

+
               a<sucn = F.toℕ<n a

+
           in ⊥-elim $ (¬q ∘ toℕ-fromℕ) (N.suc-injective (m≤n∧n≤m⇒m≡n n≢toℕ a<sucn))

+

+
completeFinL : ∀ (n : ℕ) → List (Fin n)

+
completeFinL zero = []

+
completeFinL (ℕ.suc n) = F.fromℕ n ∷ L.map F.inject₁ (completeFinL n)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Protocol.html b/site/static/agda_html/Leios.Protocol.html
new file mode 100644
index 000000000..815e1a94e
--- /dev/null
+++ b/site/static/agda_html/Leios.Protocol.html
@@ -0,0 +1,336 @@
+
+Leios.Protocol
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude hiding (id)

+
open import Leios.FFD

+
open import Leios.SpecStructure

+

+
module Leios.Protocol {n} (⋯ : SpecStructure n) (let open SpecStructure ⋯) (SlotUpkeep : Type) where

+

+
open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)

+
open GenFFD

+

+
-- High level structure:

+

+

+
--                                      (simple) Leios

+
--                                        /         |

+
-- +-------------------------------------+          |

+
-- | Header Diffusion     Body Diffusion |          |

+
-- +-------------------------------------+       Base Protocol

+
--                                        \      /

+
--                                        Network

+

+
data LeiosInput : Type where

+
  INIT     : VTy → LeiosInput

+
  SUBMIT   : EndorserBlock ⊎ List Tx → LeiosInput

+
  SLOT     : LeiosInput

+
  FTCH-LDG : LeiosInput

+

+
data LeiosOutput : Type where

+
  FTCH-LDG : List Tx → LeiosOutput

+
  EMPTY    : LeiosOutput

+

+
record LeiosState : Type where

+
  field V           : VTy

+
        SD          : StakeDistr

+
        FFDState    : FFD.State

+
        Ledger      : List Tx

+
        ToPropose   : List Tx

+
        IBs         : List InputBlock

+
        EBs         : List EndorserBlock

+
        Vs          : List (List Vote)

+
        slot        : ℕ

+
        IBHeaders   : List IBHeader

+
        IBBodies    : List IBBody

+
        Upkeep      : ℙ SlotUpkeep

+
        BaseState   : B.State

+
        votingState : VotingState

+
        PubKeys     : List PubKey

+

+
  lookupEB : EBRef → Maybe EndorserBlock

+
  lookupEB r = find (λ b → getEBRef b ≟ r) EBs

+

+
  lookupIB : IBRef → Maybe InputBlock

+
  lookupIB r = find (λ b → getIBRef b ≟ r) IBs

+

+
  lookupTxs : EndorserBlock → List Tx

+
  lookupTxs eb = do

+
    eb′ ← mapMaybe lookupEB $ ebRefs eb

+
    ib  ← mapMaybe lookupIB $ ibRefs eb′

+
    txs $ body ib

+
    where open EndorserBlockOSig

+
          open IBBody

+
          open InputBlock

+

+
  constructLedger : List RankingBlock → List Tx

+
  constructLedger = L.concat ∘ L.map (T.mergeThese L._++_ ∘ T.map₁ lookupTxs)

+

+
  needsUpkeep : SlotUpkeep → Set

+
  needsUpkeep = _∉ Upkeep

+

+
  Dec-needsUpkeep : ∀ {u : SlotUpkeep} → ⦃ DecEq SlotUpkeep ⦄ → needsUpkeep u ⁇

+
  Dec-needsUpkeep {u} .dec = ¬? (u ∈? Upkeep)

+

+
addUpkeep : LeiosState → SlotUpkeep → LeiosState

+
addUpkeep s u = let open LeiosState s in record s { Upkeep = Upkeep ∪ ❴ u ❵ }

+
{-# INJECTIVE_FOR_INFERENCE addUpkeep #-}

+

+
initLeiosState : VTy → StakeDistr → B.State → List PubKey → LeiosState

+
initLeiosState V SD bs pks = record

+
  { V           = V

+
  ; SD          = SD

+
  ; FFDState    = FFD.initFFDState

+
  ; Ledger      = []

+
  ; ToPropose   = []

+
  ; IBs         = []

+
  ; EBs         = []

+
  ; Vs          = []

+
  ; slot        = initSlot V

+
  ; IBHeaders   = []

+
  ; IBBodies    = []

+
  ; Upkeep      = ∅

+
  ; BaseState   = bs

+
  ; votingState = initVotingState

+
  ; PubKeys     = pks

+
  }

+

+
stake' : PoolID → LeiosState → ℕ

+
stake' pid record { SD = SD } = TotalMap.lookup SD pid

+

+
stake'' : PubKey → LeiosState → ℕ

+
stake'' pk = stake' (poolID pk)

+

+
stake : LeiosState → ℕ

+
stake = stake' id

+

+
lookupPubKeyAndStake : ∀ {B} → ⦃ _ : IsBlock B ⦄ → LeiosState → B → Maybe (PubKey × ℕ)

+
lookupPubKeyAndStake s b =

+
  L.head $

+
    L.map (λ pk → (pk , stake'' pk s)) $

+
      L.filter (λ pk → producerID b ≟ poolID pk) (LeiosState.PubKeys s)

+

+
module _ (s : LeiosState)  where

+

+
  record ibHeaderValid (h : IBHeader) (pk : PubKey) (st : ℕ) : Type where

+
    field lotteryPfValid : verify pk (slotNumber h) st (lotteryPf h)

+
          signatureValid : verifySig pk (signature h)

+

+
  record ibBodyValid (b : IBBody) : Type where

+

+
  ibHeaderValid? : (h : IBHeader) (pk : PubKey) (st : ℕ) → Dec (ibHeaderValid h pk st)

+
  ibHeaderValid? h pk st

+
    with verify? pk (slotNumber h) st (lotteryPf h)

+
  ... | no ¬p = no (¬p ∘ ibHeaderValid.lotteryPfValid)

+
  ... | yes p

+
    with verifySig? pk (signature h)

+
  ... | yes q = yes (record { lotteryPfValid = p ; signatureValid = q })

+
  ... | no ¬q = no (¬q ∘ ibHeaderValid.signatureValid)

+

+
  ibBodyValid? : (b : IBBody) → Dec (ibBodyValid b)

+
  ibBodyValid? _ = yes record {}

+

+
  ibValid : InputBlock → Type

+
  ibValid record { header = h ; body = b }

+
    with lookupPubKeyAndStake s h

+
  ... | just (pk , pid) = ibHeaderValid h pk (stake'' pk s) × ibBodyValid b

+
  ... | nothing = ⊥

+

+
  ibValid? : (ib : InputBlock) → Dec (ibValid ib)

+
  ibValid? record { header = h ; body = b }

+
    with lookupPubKeyAndStake s h

+
  ... | just (pk , pid) = ibHeaderValid? h pk (stake'' pk s) ×-dec ibBodyValid? b

+
  ... | nothing = no λ x → x

+

+
  record ebValid (eb : EndorserBlock) (pk : PubKey) (st : ℕ) : Type where

+
    field lotteryPfValid : verify pk (slotNumber eb) st (lotteryPf eb)

+
          signatureValid : verifySig pk (signature eb)

+
    -- TODO

+
    -- ibRefsValid : ?

+
    -- ebRefsValid : ?

+

+
  ebValid? : (eb : EndorserBlock) (pk : PubKey) (st : ℕ) → Dec (ebValid eb pk st)

+
  ebValid? h pk st

+
    with verify? pk (slotNumber h) st (lotteryPf h)

+
  ... | no ¬p = no (¬p ∘ ebValid.lotteryPfValid)

+
  ... | yes p

+
    with verifySig? pk (signature h)

+
  ... | yes q = yes (record { lotteryPfValid = p ; signatureValid = q })

+
  ... | no ¬q = no (¬q ∘ ebValid.signatureValid)

+

+
  -- TODO

+
  record vsValid (vs : List Vote) : Type where

+

+
  vsValid? : (vs : List Vote) → Dec (vsValid vs)

+
  vsValid? _ = yes record {}

+

+
  headerValid : Header → Type

+
  headerValid (ibHeader h)

+
    with lookupPubKeyAndStake s h

+
  ... | just (pk , pid) = ibHeaderValid h pk (stake'' pk s)

+
  ... | nothing = ⊥

+
  headerValid (ebHeader h)

+
    with lookupPubKeyAndStake s h

+
  ... | just (pk , pid) = ebValid h pk (stake'' pk s)

+
  ... | nothing = ⊥

+
  headerValid (vtHeader h) = vsValid h

+

+
  headerValid? : (h : Header) → Dec (headerValid h)

+
  headerValid? (ibHeader h)

+
    with lookupPubKeyAndStake s h

+
  ... | just (pk , pid) = ibHeaderValid? h pk (stake'' pk s)

+
  ... | nothing = no λ x → x

+
  headerValid? (ebHeader h)

+
    with lookupPubKeyAndStake s h

+
  ... | just (pk , pid) = ebValid? h pk (stake'' pk s)

+
  ... | nothing = no λ x → x

+
  headerValid? (vtHeader h) = vsValid? h

+

+
  bodyValid : Body → Type

+
  bodyValid (ibBody b) = ibBodyValid b

+

+
  bodyValid? : (b : Body) → Dec (bodyValid b)

+
  bodyValid? (ibBody b) = ibBodyValid? b

+

+
  isValid : Header ⊎ Body → Type

+
  isValid (inj₁ h) = headerValid h

+
  isValid (inj₂ b) = bodyValid b

+

+
  isValid? : ∀ (x : Header ⊎ Body) → Dec (isValid x)

+
  isValid? (inj₁ h) = headerValid? h

+
  isValid? (inj₂ b) = bodyValid? b

+

+
-- some predicates about EBs

+
module _ (s : LeiosState) (eb : EndorserBlock) where

+
  open EndorserBlockOSig eb

+
  open LeiosState s

+

+
  allIBRefsKnown : Type

+
  allIBRefsKnown = ∀[ ref ∈ fromList ibRefs ] ref ∈ˡ map getIBRef IBs

+

+
module _ (s : LeiosState) where

+

+
  open LeiosState s

+

+
  upd : Header ⊎ Body → LeiosState

+
  upd (inj₁ (ebHeader eb)) = record s { EBs = eb ∷ EBs }

+
  upd (inj₁ (vtHeader vs)) = record s { Vs = vs ∷ Vs }

+
  upd (inj₁ (ibHeader h)) with A.any? (matchIB? h) IBBodies

+
  ... | yes p =

+
    record s

+
      { IBs = record { header = h ; body = A.lookup p } ∷ IBs

+
      ; IBBodies = IBBodies A.─ p

+
      }

+
  ... | no _ =

+
    record s

+
      { IBHeaders = h ∷ IBHeaders

+
      }

+
  upd (inj₂ (ibBody b)) with A.any? (flip matchIB? b) IBHeaders

+
  ... | yes p =

+
    record s

+
      { IBs = record { header = A.lookup p ; body = b } ∷ IBs

+
      ; IBHeaders = IBHeaders A.─ p

+
      }

+
  ... | no _ =

+
    record s

+
      { IBBodies = b ∷ IBBodies

+
      }

+

+
module _ {s s'} where

+
  open LeiosState s'

+

+
  upd-preserves-Upkeep : ∀ {x} → LeiosState.Upkeep s ≡ LeiosState.Upkeep s'

+
                               → LeiosState.Upkeep s ≡ LeiosState.Upkeep (upd s' x)

+
  upd-preserves-Upkeep {inj₁ (ibHeader x)} refl with A.any? (matchIB? x) IBBodies

+
  ... | yes p = refl

+
  ... | no ¬p = refl

+
  upd-preserves-Upkeep {inj₁ (ebHeader x)} refl = refl

+
  upd-preserves-Upkeep {inj₁ (vtHeader x)} refl = refl

+
  upd-preserves-Upkeep {inj₂ (ibBody x)} refl with A.any? (flip matchIB? x) IBHeaders

+
  ... | yes p = refl

+
  ... | no ¬p = refl

+

+
infix 25 _↑_

+
_↑_ : LeiosState → List (Header ⊎ Body) → LeiosState

+
_↑_ = foldr (flip upd)

+

+
↑-preserves-Upkeep : ∀ {s x} → LeiosState.Upkeep s ≡ LeiosState.Upkeep (s ↑ x)

+
↑-preserves-Upkeep {x = []} = refl

+
↑-preserves-Upkeep {s = s} {x = x ∷ x₁} =

+
  upd-preserves-Upkeep {s = s} {x = x} (↑-preserves-Upkeep {x = x₁})

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Short.Decidable.html b/site/static/agda_html/Leios.Short.Decidable.html
new file mode 100644
index 000000000..477cf1708
--- /dev/null
+++ b/site/static/agda_html/Leios.Short.Decidable.html
@@ -0,0 +1,184 @@
+
+Leios.Short.Decidable
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Leios.Prelude hiding (id)

+
open import Leios.SpecStructure using (SpecStructure)

+

+
module Leios.Short.Decidable (⋯ : SpecStructure 1) (let open SpecStructure ⋯) where

+

+
open import Leios.Short ⋯ renaming (isVoteCertified to isVoteCertified')

+
open B hiding (_-⟦_/_⟧⇀_)

+
open FFD hiding (_-⟦_/_⟧⇀_)

+

+
module _ {s : LeiosState} (let open LeiosState s renaming (FFDState to ffds; BaseState to bs)) where

+

+
  IB-Role? : ∀ {π ffds'} →

+
           let b = GenFFD.ibBody (record { txs = ToPropose })

+
               h = GenFFD.ibHeader (mkIBHeader slot id π sk-IB ToPropose)

+
           in

+
           { _ : auto∶ needsUpkeep IB-Role }

+
           { _ : auto∶ canProduceIB slot sk-IB (stake s) π }

+
           { _ : auto∶ ffds FFD.-⟦ FFD.Send h (just b) / FFD.SendRes ⟧⇀ ffds' } →

+
           ─────────────────────────────────────────────────────────────────────────

+
           s ↝ addUpkeep record s { FFDState = ffds' } IB-Role

+
  IB-Role? {_} {_} {p} {q} {r} = IB-Role (toWitness p) (toWitness q) (toWitness r)

+

+
  {-

+
  No-IB-Role? :

+
              { _ : auto∶ needsUpkeep IB-Role }

+
              { _ : auto∶ ∀ π → ¬ canProduceIB slot sk-IB (stake s) π } →

+
              ─────────────────────────────────────────────

+
              s ↝ addUpkeep s IB-Role

+
  No-IB-Role? {p} {q} = No-IB-Role (toWitness p) (toWitness q)

+
  -}

+

+
  EB-Role? : ∀ {π ffds'} →

+
           let LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs

+
               h = mkEB slot id π sk-EB LI []

+
           in

+
           { _ : auto∶ needsUpkeep EB-Role }

+
           { _ : auto∶ canProduceEB slot sk-EB (stake s) π }

+
           { _ : auto∶ ffds FFD.-⟦ FFD.Send (GenFFD.ebHeader h) nothing / FFD.SendRes ⟧⇀ ffds' } →

+
           ─────────────────────────────────────────────────────────────────────────

+
           s ↝ addUpkeep record s { FFDState = ffds' } EB-Role

+
  EB-Role? {_} {_} {p} {q} {r} = EB-Role (toWitness p) (toWitness q) (toWitness r)

+

+
  {-

+
  No-EB-Role? :

+
              { _ : auto∶ needsUpkeep EB-Role }

+
              { _ : auto∶ ∀ π → ¬ canProduceEB slot sk-EB (stake s) π } →

+
              ─────────────────────────────────────────────

+
              s ↝ addUpkeep s EB-Role

+
  No-EB-Role? {_} {p} {q} = No-EB-Role (toWitness p) (toWitness q)

+
  -}

+

+
  V-Role? : ∀ {ffds'} →

+
          let EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs

+
              votes = map (vote sk-VT ∘ hash) EBs'

+
          in

+
          { _ : auto∶ needsUpkeep VT-Role }

+
          { _ : auto∶ canProduceV slot sk-VT (stake s) }

+
          { _ : auto∶ ffds FFD.-⟦ FFD.Send (GenFFD.vtHeader votes) nothing / FFD.SendRes ⟧⇀ ffds' } →

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } VT-Role

+
  V-Role? {_} {p} {q} {r} = VT-Role (toWitness p) (toWitness q) (toWitness r)

+

+
  No-V-Role? :

+
             { _ : auto∶ needsUpkeep VT-Role }

+
             { _ : auto∶ ¬ canProduceV slot sk-VT (stake s) } →

+
             ─────────────────────────────────────────────

+
             s ↝ addUpkeep s VT-Role

+
  No-V-Role? {p} {q} = No-VT-Role (toWitness p) (toWitness q)

+

+
  {-

+
  Init? : ∀ {ks pks ks' SD bs' V} →

+
        { _ : auto∶ ks K.-⟦ K.INIT pk-IB pk-EB pk-V / K.PUBKEYS pks ⟧⇀ ks' }

+
        { _ : initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs' } →

+
        ────────────────────────────────────────────────────────────────

+
        nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs'

+
  Init? = ?

+
  -}

+

+
  Base₂a? : ∀ {eb bs'} →

+
          { _ : auto∶ needsUpkeep Base }

+
          { _ : auto∶ eb ∈ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2) EBs }

+
          { _ : auto∶ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs' } →

+
          ───────────────────────────────────────────────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base

+
  Base₂a? {_} {_} {p} {q} {r} = Base₂a (toWitness p) (toWitness q) (toWitness r)

+

+
  Base₂b? : ∀ {bs'} →

+
          { _ : auto∶ needsUpkeep Base }

+
          { _ : auto∶ [] ≡ filter (λ eb → isVoteCertified' s eb × eb ∈ᴮ slice L slot 2) EBs }

+
          { _ : auto∶ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs' } →

+
          ───────────────────────────────────────────────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base

+
  Base₂b? {_} {p} {q} {r} = Base₂b (toWitness p) (toWitness q) (toWitness r)

+

+
  Slot? : ∀ {rbs bs' msgs ffds'} →

+
         { _ : auto∶ allDone s }

+
         { _ : auto∶ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs' }

+
         { _ : auto∶ ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds' } →

+
         ───────────────────────────────────────────────────────────────────────

+
         just s -⟦ SLOT / EMPTY ⟧⇀ record s

+
             { FFDState  = ffds'

+
             ; BaseState = bs'

+
             ; Ledger    = constructLedger rbs

+
             ; slot      = suc slot

+
             ; Upkeep    = ∅

+
             } ↑ L.filter (isValid? s) msgs

+
  Slot? {_} {_} {_} {_} {p} {q} {r} = Slot (toWitness p) (toWitness q) (toWitness r)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Short.Trace.Verifier.Test.html b/site/static/agda_html/Leios.Short.Trace.Verifier.Test.html
new file mode 100644
index 000000000..dac975276
--- /dev/null
+++ b/site/static/agda_html/Leios.Short.Trace.Verifier.Test.html
@@ -0,0 +1,210 @@
+
+Leios.Short.Trace.Verifier.Test
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Leios.Prelude hiding (id)

+
open import Leios.Config

+

+
module Leios.Short.Trace.Verifier.Test where

+

+
params : Params

+
params =

+
  record

+
    { numberOfParties = 2

+
    ; sutId = fzero

+
    ; stakeDistribution =

+
        let open FunTot (completeFin 2) (maximalFin 2)

+
        in Fun⇒TotalMap (const 100000000)

+
    ; stageLength = 2

+
    ; winning-slots = fromList $

+
        (IB , 0) ∷ (EB , 0) ∷ (VT , 0) ∷

+
        (IB , 1) ∷ (EB , 1) ∷ (VT , 1) ∷

+
        (IB , 2) ∷ (EB , 2) ∷ (VT , 2) ∷

+
        (IB , 3) ∷ (EB , 3) ∷ (VT , 3) ∷

+
        (IB , 4) ∷ (EB , 4) ∷ (VT , 4) ∷

+
        (IB , 5) ∷ (EB , 5) ∷ (VT , 5) ∷

+
                              (VT , 6) ∷

+
        (IB , 7) ∷ (EB , 7) ∷ (VT , 7) ∷

+
        (IB , 8) ∷ (EB , 8) ∷ (VT , 8) ∷

+
        []

+
    }

+

+
open Params params

+
open import Leios.Short.Trace.Verifier params

+

+
private

+
  opaque

+
    unfolding List-Model

+

+
    test₁ : Bool

+
    test₁ = ¿ ValidTrace (inj₁ (IB-Role-Action 0 , SLOT) ∷ []) ¿ᵇ

+

+
    _ : test₁ ≡ true

+
    _ = refl

+

+
    test-valid-ib : Bool

+
    test-valid-ib =

+
      let h = record { slotNumber = 1

+
                     ; producerID = fsuc fzero

+
                     ; lotteryPf = tt

+
                     ; bodyHash = 0 ∷ 1 ∷ 2 ∷ []

+
                     ; signature = tt

+
                     }

+
          b = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }

+
          ib = record { header = h ; body = b }

+
          pks = L.zip (completeFinL numberOfParties) (L.replicate numberOfParties tt)

+
          s = initLeiosState tt stakeDistribution tt pks

+
      in isYes (ibValid? s ib)

+

+
    _ : test-valid-ib ≡ true

+
    _ = refl

+

+
    test₂ : Bool

+
    test₂ =

+
      let t = L.reverse $

+
            -- slot 0

+
              inj₁ (IB-Role-Action 0    , SLOT)

+
            ∷ inj₁ (EB-Role-Action 0 [] , SLOT)

+
            ∷ inj₁ (VT-Role-Action 0    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    0    , SLOT)

+
            -- slot 1

+
            ∷ inj₁ (IB-Role-Action 1    , SLOT)

+
            ∷ inj₁ (VT-Role-Action 1    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    1    , SLOT)

+
            -- slot 2

+
            ∷ inj₂ (IB-Recv-Update

+
                (record { header =

+
                  record { slotNumber = 1

+
                         ; producerID = fsuc fzero

+
                         ; lotteryPf = tt

+
                         ; bodyHash = 0 ∷ 1 ∷ 2 ∷ []

+
                         ; signature = tt

+
                         }

+
                        ; body = record { txs = 0 ∷ 1 ∷ 2 ∷ [] }}))

+
            ∷ inj₁ (IB-Role-Action 2    , SLOT)

+
            ∷ inj₁ (EB-Role-Action 2 [] , SLOT)

+
            ∷ inj₁ (VT-Role-Action 2    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    2    , SLOT)

+
            -- slot 3

+
            ∷ inj₂ (IB-Recv-Update

+
                (record { header =

+
                  record { slotNumber = 2

+
                         ; producerID = fsuc fzero

+
                         ; lotteryPf = tt

+
                         ; bodyHash = 3 ∷ 4 ∷ 5 ∷ []

+
                         ; signature = tt

+
                         }

+
                        ; body = record { txs = 3 ∷ 4 ∷ 5 ∷ [] }}))

+
            ∷ inj₁ (IB-Role-Action 3    , SLOT)

+
            ∷ inj₁ (VT-Role-Action 3    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    3    , SLOT)

+
            -- slot 4

+
            ∷ inj₁ (IB-Role-Action 4    , SLOT)

+
            ∷ inj₁ (EB-Role-Action 4 [] , SLOT)

+
            ∷ inj₁ (VT-Role-Action 4    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    4    , SLOT)

+
            -- slot 5

+
            ∷ inj₁ (IB-Role-Action 5    , SLOT)

+
            ∷ inj₁ (VT-Role-Action 5    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    5    , SLOT)

+
            -- slot 6

+
            ∷ inj₁ (No-IB-Role-Action   , SLOT)

+
            ∷ inj₁ (No-EB-Role-Action   , SLOT)

+
            ∷ inj₁ (VT-Role-Action 6    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    6    , SLOT)

+
            -- slot 7

+
            ∷ inj₁ (IB-Role-Action 7    , SLOT)

+
            ∷ inj₁ (VT-Role-Action 7    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    7    , SLOT)

+
            -- slot 8

+
            ∷ inj₁ (IB-Role-Action 8    , SLOT)

+
            ∷ inj₁ (EB-Role-Action 8 ((3 ∷ 4 ∷ 5 ∷ []) ∷ []) , SLOT)

+
            ∷ inj₁ (VT-Role-Action 8    , SLOT)

+
            ∷ inj₁ (Base₂b-Action       , SLOT)

+
            ∷ inj₁ (Slot-Action    8    , SLOT)

+
            ∷ []

+
      in ¿ ValidTrace t ¿ᵇ

+

+
    _ : test₂ ≡ true

+
    _ = refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Short.Trace.Verifier.html b/site/static/agda_html/Leios.Short.Trace.Verifier.html
new file mode 100644
index 000000000..69c353f14
--- /dev/null
+++ b/site/static/agda_html/Leios.Short.Trace.Verifier.html
@@ -0,0 +1,518 @@
+
+Leios.Short.Trace.Verifier
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Leios.Prelude hiding (id)

+
open import Leios.Config

+

+
-- TODO: SpecStructure as parameter

+
module Leios.Short.Trace.Verifier (params : Params) (let open Params params) where

+

+
open import Leios.Defaults params

+
  using (LeiosState; initLeiosState; isb; hpe; hhs; htx; SendIB; FFDBuffers; Dec-SimpleFFD)

+
  renaming (d-SpecStructure to traceSpecStructure) public

+

+
open import Leios.SpecStructure using (SpecStructure)

+
open SpecStructure traceSpecStructure hiding (Hashable-IBHeader; Hashable-EndorserBlock; isVoteCertified) public

+

+
open import Leios.Short traceSpecStructure hiding (LeiosState; initLeiosState) public

+
open import Prelude.Closures _↝_

+
open GenFFD

+

+
data FFDUpdate : Type where

+
  IB-Recv-Update : InputBlock → FFDUpdate

+
  EB-Recv-Update : EndorserBlock → FFDUpdate

+
  VT-Recv-Update : List Vote → FFDUpdate

+

+
data Action : Type where

+
  IB-Role-Action : ℕ → Action

+
  EB-Role-Action : ℕ → List IBRef → Action

+
  VT-Role-Action : ℕ → Action

+
  No-IB-Role-Action No-EB-Role-Action No-VT-Role-Action : Action

+
  Ftch-Action : Action

+
  Slot-Action : ℕ → Action

+
  Base₁-Action : Action

+
  Base₂a-Action : EndorserBlock → Action

+
  Base₂b-Action : Action

+

+
Actions = List (Action × LeiosInput)

+

+
private variable

+
  s s′ : LeiosState

+
  α : Action

+

+
data ValidUpdate : FFDUpdate → LeiosState → Type where

+

+
  IB-Recv : ∀ {ib} →

+
    ValidUpdate (IB-Recv-Update ib) s

+

+
  EB-Recv : ∀ {eb} →

+
    ValidUpdate (EB-Recv-Update eb) s

+

+
  VT-Recv : ∀ {vt} →

+
    ValidUpdate (VT-Recv-Update vt) s

+

+
data ValidAction : Action → LeiosState → LeiosInput → Type where

+

+
  IB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                b = record { txs = ToPropose }

+
                h = mkIBHeader slot id tt sk-IB ToPropose

+
                ffds' = proj₁ (FFD.Send-total {ffds} {ibHeader h} {just (ibBody b)})

+
            in .(needsUpkeep IB-Role) →

+
               .(canProduceIB slot sk-IB (stake s) tt) →

+
               .(ffds FFD.-⟦ FFD.Send (ibHeader h) (just (ibBody b)) / FFD.SendRes ⟧⇀ ffds') →

+
               ValidAction (IB-Role-Action slot) s SLOT

+

+
  EB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs

+
                h = mkEB slot id tt sk-EB LI []

+
                ffds' = proj₁ (FFD.Send-total {ffds} {ebHeader h} {nothing})

+
            in .(needsUpkeep EB-Role) →

+
               .(canProduceEB slot sk-EB (stake s) tt) →

+
               .(ffds FFD.-⟦ FFD.Send (ebHeader h) nothing / FFD.SendRes ⟧⇀ ffds') →

+
               ValidAction (EB-Role-Action slot LI) s SLOT

+

+
  VT-Role : let open LeiosState s renaming (FFDState to ffds)

+
                EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs

+
                votes = map (vote sk-VT ∘ hash) EBs'

+
                ffds' = proj₁ (FFD.Send-total {ffds} {vtHeader votes} {nothing})

+
            in .(needsUpkeep VT-Role) →

+
               .(canProduceV slot sk-VT (stake s)) →

+
               .(ffds FFD.-⟦ FFD.Send (vtHeader votes) nothing / FFD.SendRes ⟧⇀ ffds') →

+
               ValidAction (VT-Role-Action slot) s SLOT

+

+
  No-IB-Role : let open LeiosState s

+
               in needsUpkeep IB-Role →

+
                  (∀ π → ¬ canProduceIB slot sk-IB (stake s) π) →

+
                  ValidAction No-IB-Role-Action s SLOT

+

+
  No-EB-Role : let open LeiosState s

+
               in needsUpkeep EB-Role →

+
                  (∀ π → ¬ canProduceEB slot sk-EB (stake s) π) →

+
                  ValidAction No-EB-Role-Action s SLOT

+

+
  No-VT-Role : let open LeiosState s

+
               in needsUpkeep VT-Role →

+
                  (¬ canProduceV slot sk-VT (stake s)) →

+
                  ValidAction No-VT-Role-Action s SLOT

+

+
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs)

+
             (msgs , (ffds' , _)) = FFD.Fetch-total {ffds}

+
         in .(allDone s) →

+
            .(bs B.-⟦ B.FTCH-LDG / B.BASE-LDG [] ⟧⇀ tt) →

+
            .(ffds FFD.-⟦ FFD.Fetch / FFD.FetchRes msgs ⟧⇀ ffds') →

+
            ValidAction (Slot-Action slot) s SLOT

+

+
  Ftch : ValidAction Ftch-Action s FTCH-LDG

+

+
  Base₁ : ∀ {txs} → ValidAction Base₁-Action s (SUBMIT (inj₂ txs))

+

+
  Base₂a : ∀ {eb} → let open LeiosState s renaming (BaseState to bs)

+
           in .(needsUpkeep Base) →

+
              .(eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs) →

+
              .(bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ tt) →

+
              ValidAction (Base₂a-Action eb) s SLOT

+

+
  Base₂b : let open LeiosState s renaming (BaseState to bs)

+
           in .(needsUpkeep Base) →

+
              .([] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs) →

+
              .(bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ tt) →

+
              ValidAction Base₂b-Action s SLOT

+

+
private variable

+
  i : LeiosInput

+
  o : LeiosOutput

+

+
⟦_⟧ : ValidAction α s i → LeiosState × LeiosOutput

+
⟦ IB-Role {s} _ _ _ ⟧ =

+
  let open LeiosState s renaming (FFDState to ffds)

+
      b = record { txs = ToPropose }

+
      h = mkIBHeader slot id tt sk-IB ToPropose

+
      ffds' = proj₁ (FFD.Send-total {ffds} {ibHeader h} {just (ibBody b)})

+
  in addUpkeep record s { FFDState = ffds' } IB-Role , EMPTY

+
⟦ EB-Role {s} _ _ _ ⟧ =

+
  let open LeiosState s renaming (FFDState to ffds)

+
      LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs

+
      h = mkEB slot id tt sk-EB LI []

+
      ffds' = proj₁ (FFD.Send-total {ffds} {ebHeader h} {nothing})

+
  in addUpkeep record s { FFDState = ffds' } EB-Role , EMPTY

+
⟦ VT-Role {s} _ _ _ ⟧ =

+
  let open LeiosState s renaming (FFDState to ffds)

+
      EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs

+
      votes = map (vote sk-VT ∘ hash) EBs'

+
      ffds' = proj₁ (FFD.Send-total {ffds} {vtHeader votes} {nothing})

+
  in addUpkeep record s { FFDState = ffds' } VT-Role , EMPTY

+
⟦ No-IB-Role {s} _ _ ⟧ = addUpkeep s IB-Role , EMPTY

+
⟦ No-EB-Role {s} _ _ ⟧ = addUpkeep s EB-Role , EMPTY

+
⟦ No-VT-Role {s} _ _ ⟧ = addUpkeep s VT-Role , EMPTY

+
⟦ Slot {s} _ _ _ ⟧ =

+
  let open LeiosState s renaming (FFDState to ffds)

+
      (msgs , (ffds' , _)) = FFD.Fetch-total {ffds}

+
  in

+
  (record s

+
     { FFDState  = ffds'

+
     ; BaseState = tt

+
     ; Ledger    = constructLedger []

+
     ; slot      = suc slot

+
     ; Upkeep    = ∅

+
     } ↑ L.filter (isValid? s) msgs

+
  , EMPTY)

+
⟦ Ftch {s} ⟧ = s , FTCH-LDG (LeiosState.Ledger s)

+
⟦ Base₁ {s} {txs} ⟧ = record s { ToPropose = txs } , EMPTY

+
⟦ Base₂a {s} _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY

+
⟦ Base₂b {s} _ _ _ ⟧ = addUpkeep record s { BaseState = tt } Base , EMPTY

+

+
open LeiosState

+
open FFDBuffers

+

+
ValidAction→Eq-Slot : ∀ {s sl} → ValidAction (Slot-Action sl) s SLOT → sl ≡ slot s

+
ValidAction→Eq-Slot (Slot _ _ _) = refl

+

+
ValidAction→Eq-IB : ∀ {s sl} → ValidAction (IB-Role-Action sl) s SLOT → sl ≡ slot s

+
ValidAction→Eq-IB (IB-Role _ _ _) = refl

+

+
ValidAction→Eq-EB : ∀ {s sl ibs} → ValidAction (EB-Role-Action sl ibs) s SLOT → sl ≡ slot s × ibs ≡ (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))

+
ValidAction→Eq-EB (EB-Role _ _ _) = refl , refl

+

+
ValidAction→Eq-VT : ∀ {s sl} → ValidAction (VT-Role-Action sl) s SLOT → sl ≡ slot s

+
ValidAction→Eq-VT (VT-Role _ _ _) = refl

+

+
instance

+
  Dec-ValidAction : ValidAction ⁇³

+
  Dec-ValidAction {IB-Role-Action sl} {s} {SLOT} .dec

+
    with sl ≟ slot s

+
  ... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-IB x))

+
  ... | yes p rewrite p

+
    with dec | dec | dec

+
  ... | yes x | yes y | yes z = yes (IB-Role x y z)

+
  ... | no ¬p | _ | _ = no λ where (IB-Role p _ _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | no ¬p | _ = no λ where (IB-Role _ p _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | _ | no ¬p = no λ where (IB-Role _ _ p) → ⊥-elim (¬p (recompute dec p))

+
  Dec-ValidAction {IB-Role-Action _} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {IB-Role-Action _} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {IB-Role-Action _} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {EB-Role-Action sl ibs} {s} {SLOT} .dec

+
    with sl ≟ slot s | ibs ≟ (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))

+
  ... | no ¬p | _ = no λ x → ⊥-elim (¬p (proj₁ $ ValidAction→Eq-EB x))

+
  ... | _ | no ¬q = no λ x → ⊥-elim (¬q (proj₂ $ ValidAction→Eq-EB x))

+
  ... | yes p | yes q rewrite p rewrite q

+
    with dec | dec | dec

+
  ... | yes x | yes y | yes z = yes (EB-Role x y z)

+
  ... | no ¬p | _ | _ = no λ where (EB-Role p _ _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | no ¬p | _ = no λ where (EB-Role _ p _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | _ | no ¬p = no λ where (EB-Role _ _ p) → ⊥-elim (¬p (recompute dec p))

+
  Dec-ValidAction {EB-Role-Action _ _} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {EB-Role-Action _ _} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {EB-Role-Action _ _} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {VT-Role-Action sl} {s} {SLOT} .dec

+
    with sl ≟ slot s

+
  ... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-VT x))

+
  ... | yes p rewrite p

+
    with dec | dec | dec

+
  ... | yes x | yes y | yes z = yes (VT-Role x y z)

+
  ... | no ¬p | _ | _ = no λ where (VT-Role p _ _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | no ¬p | _ = no λ where (VT-Role _ p _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | _ | no ¬p = no λ where (VT-Role _ _ p) → ⊥-elim (¬p (recompute dec p))

+
  Dec-ValidAction {VT-Role-Action _} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {VT-Role-Action _} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {VT-Role-Action _} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {No-IB-Role-Action} {s} {SLOT} .dec

+
    with dec | dec

+
  ... | yes p | yes q = yes (No-IB-Role p q)

+
  ... | no ¬p | _ = no λ where (No-IB-Role p _) → ⊥-elim (¬p p)

+
  ... | _ | no ¬q = no λ where (No-IB-Role _ q) → ⊥-elim (¬q q)

+
  Dec-ValidAction {No-IB-Role-Action} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {No-IB-Role-Action} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {No-IB-Role-Action} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {No-EB-Role-Action} {s} {SLOT} .dec

+
    with dec | dec

+
  ... | yes p | yes q = yes (No-EB-Role p q)

+
  ... | no ¬p | _ = no λ where (No-EB-Role p _) → ⊥-elim (¬p p)

+
  ... | _ | no ¬q = no λ where (No-EB-Role _ q) → ⊥-elim (¬q q)

+
  Dec-ValidAction {No-EB-Role-Action} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {No-EB-Role-Action} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {No-EB-Role-Action} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {No-VT-Role-Action} {s} {SLOT} .dec

+
    with dec | dec

+
  ... | yes p | yes q = yes (No-VT-Role p q)

+
  ... | no ¬p | _ = no λ where (No-VT-Role p _) → ⊥-elim (¬p p)

+
  ... | _ | no ¬q = no λ where (No-VT-Role _ q) → ⊥-elim (¬q q)

+
  Dec-ValidAction {No-VT-Role-Action} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {No-VT-Role-Action} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {No-VT-Role-Action} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {Slot-Action sl} {s} {SLOT} .dec

+
    with sl ≟ slot s

+
  ... | no ¬p = no λ x → ⊥-elim (¬p (ValidAction→Eq-Slot x))

+
  ... | yes p rewrite p

+
    with dec | dec | dec

+
  ... | yes x | yes y | yes z = yes (Slot x y z)

+
  ... | no ¬p | _ | _ = no λ where (Slot p _ _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | no ¬p | _ = no λ where (Slot _ p _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | _ | no ¬p = no λ where (Slot _ _ p) → ⊥-elim (¬p (recompute dec p))

+
  Dec-ValidAction {Slot-Action _} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {Slot-Action _} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {Slot-Action _} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {Ftch-Action} {s} {FTCH-LDG} .dec = yes Ftch

+
  Dec-ValidAction {Ftch-Action} {s} {SLOT} .dec = no λ ()

+
  Dec-ValidAction {Ftch-Action} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {Ftch-Action} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {Base₁-Action} {s} {SUBMIT (inj₁ ebs)} .dec = no λ ()

+
  Dec-ValidAction {Base₁-Action} {s} {SUBMIT (inj₂ txs)} .dec = yes (Base₁ {s} {txs})

+
  Dec-ValidAction {Base₁-Action} {s} {SLOT} .dec = no λ ()

+
  Dec-ValidAction {Base₁-Action} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {Base₁-Action} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {Base₂a-Action eb} {s} {SLOT} .dec

+
    with dec | dec | dec

+
  ... | yes x | yes y | yes z = yes (Base₂a x y z)

+
  ... | no ¬p | _ | _ = no λ where (Base₂a p _ _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | no ¬p | _ = no λ where (Base₂a {s} {eb} _ p _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | _ | no ¬p = no λ where (Base₂a _ _ p) → ⊥-elim (¬p (recompute dec p))

+
  Dec-ValidAction {Base₂a-Action _} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {Base₂a-Action _} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {Base₂a-Action _} {s} {INIT _} .dec = no λ ()

+
  Dec-ValidAction {Base₂b-Action} {s} {SLOT} .dec

+
    with dec | dec | dec

+
  ... | yes x | yes y | yes z = yes (Base₂b x y z)

+
  ... | no ¬p | _ | _ = no λ where (Base₂b p _ _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | no ¬p | _ = no λ where (Base₂b _ p _) → ⊥-elim (¬p (recompute dec p))

+
  ... | _ | _ | no ¬p = no λ where (Base₂b _ _ p) → ⊥-elim (¬p (recompute dec p))

+
  Dec-ValidAction {Base₂b-Action} {s} {SUBMIT _} .dec = no λ ()

+
  Dec-ValidAction {Base₂b-Action} {s} {FTCH-LDG} .dec = no λ ()

+
  Dec-ValidAction {Base₂b-Action} {s} {INIT _} .dec = no λ ()

+

+
instance

+
  Dec-ValidUpdate : ValidUpdate ⁇²

+
  Dec-ValidUpdate {IB-Recv-Update _} .dec = yes IB-Recv

+
  Dec-ValidUpdate {EB-Recv-Update _} .dec = yes EB-Recv

+
  Dec-ValidUpdate {VT-Recv-Update _} .dec = yes VT-Recv

+

+
mutual

+
  data ValidTrace : List ((Action × LeiosInput) ⊎ FFDUpdate) → Type where

+
    [] :

+
      ─────────────

+
      ValidTrace []

+

+
    _/_∷_⊣_ : ∀ α i {αs} →

+
      ∀ (tr : ValidTrace αs) →

+
      ∙ ValidAction α (proj₁ ⟦ tr ⟧∗) i

+
        ───────────────────

+
        ValidTrace (inj₁ (α , i) ∷ αs)

+

+
    _↥_ : ∀ {f αs} →

+
      ∀ (tr : ValidTrace αs) →

+
        (vu : ValidUpdate f (proj₁ ⟦ tr ⟧∗)) →

+
        ───────────────────

+
        ValidTrace (inj₂ f ∷ αs)

+

+

+
  ⟦_⟧∗ : ∀ {αs : List ((Action × LeiosInput) ⊎ FFDUpdate)} → ValidTrace αs → LeiosState × LeiosOutput

+
  ⟦ [] ⟧∗ = initLeiosState tt stakeDistribution tt pks , EMPTY

+
    where pks = L.zip (completeFinL numberOfParties) (L.replicate numberOfParties tt)

+
  ⟦ _ / _ ∷ _ ⊣ vα ⟧∗ = ⟦ vα ⟧

+
  ⟦ _↥_ {IB-Recv-Update ib} tr vu ⟧∗ =

+
    let (s , o) = ⟦ tr ⟧∗

+
    in record s { FFDState = record (FFDState s) { inIBs = ib ∷ inIBs (FFDState s)}} , o

+
  ⟦ _↥_ {EB-Recv-Update eb} tr vu ⟧∗ =

+
    let (s , o) = ⟦ tr ⟧∗

+
    in record s { FFDState = record (FFDState s) { inEBs = eb ∷ inEBs (FFDState s)}} , o

+
  ⟦ _↥_ {VT-Recv-Update vt} tr vu ⟧∗ =

+
    let (s , o) = ⟦ tr ⟧∗

+
    in record s { FFDState = record (FFDState s) { inVTs = vt ∷ inVTs (FFDState s)}} , o

+

+
Irr-ValidAction : Irrelevant (ValidAction α s i)

+
Irr-ValidAction (IB-Role _ _ _) (IB-Role _ _ _)   = refl

+
Irr-ValidAction (EB-Role _ _ _) (EB-Role _ _ _)   = refl

+
Irr-ValidAction (VT-Role _ _ _) (VT-Role _ _ _)   = refl

+
Irr-ValidAction (No-IB-Role _ _) (No-IB-Role _ _) = refl

+
Irr-ValidAction (No-EB-Role _ _) (No-EB-Role _ _) = refl

+
Irr-ValidAction (No-VT-Role _ _) (No-VT-Role _ _) = refl

+
Irr-ValidAction (Slot _ _ _) (Slot _ _ _)         = refl

+
Irr-ValidAction Ftch Ftch                         = refl

+
Irr-ValidAction Base₁ Base₁                       = refl

+
Irr-ValidAction (Base₂a _ _ _) (Base₂a _ _ _)     = refl

+
Irr-ValidAction (Base₂b _ _ _) (Base₂b _ _ _)     = refl

+

+
Irr-ValidUpdate : ∀ {f} → Irrelevant (ValidUpdate f s)

+
Irr-ValidUpdate IB-Recv IB-Recv = refl

+
Irr-ValidUpdate EB-Recv EB-Recv = refl

+
Irr-ValidUpdate VT-Recv VT-Recv = refl

+

+
Irr-ValidTrace : ∀ {αs} → Irrelevant (ValidTrace αs)

+
Irr-ValidTrace [] [] = refl

+
Irr-ValidTrace (α / i ∷ vαs ⊣ vα) (.α / .i ∷ vαs′ ⊣ vα′)

+
  rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidAction vα vα′

+
  = refl

+
Irr-ValidTrace (vαs ↥ u) (vαs′ ↥ u′)

+
  rewrite Irr-ValidTrace vαs vαs′ | Irr-ValidUpdate u u′

+
  = refl

+

+
instance

+
  Dec-ValidTrace : ValidTrace ⁇¹

+
  Dec-ValidTrace {tr} .dec with tr

+
  ... | [] = yes []

+
  ... | inj₁ (α , i) ∷ αs

+
    with ¿ ValidTrace αs ¿

+
  ... | no ¬vαs = no λ where (_ / _ ∷ vαs ⊣ _) → ¬vαs vαs

+
  ... | yes vαs

+
    with ¿ ValidAction α (proj₁ ⟦ vαs ⟧∗) i ¿

+
  ... | no ¬vα = no λ where

+
    (_ / _ ∷ tr ⊣ vα) → ¬vα

+
                  $ subst (λ x → ValidAction α x i) (cong (proj₁ ∘ ⟦_⟧∗) $ Irr-ValidTrace tr vαs) vα

+
  ... | yes vα = yes $ _ / _ ∷ vαs ⊣ vα

+
  Dec-ValidTrace {tr} .dec | inj₂ u ∷ αs

+
    with ¿ ValidTrace αs ¿

+
  ... | no ¬vαs = no λ where (vαs ↥ _) → ¬vαs vαs

+
  ... | yes vαs

+
    with ¿ ValidUpdate u (proj₁ ⟦ vαs ⟧∗) ¿

+
  ... | yes vu = yes (vαs ↥ vu)

+
  ... | no ¬vu = no λ where

+
    (tr ↥ vu) → ¬vu $ subst (λ x → ValidUpdate u x) (cong (proj₁ ∘ ⟦_⟧∗) $ Irr-ValidTrace tr vαs) vu

+

+
data _⇑_ : LeiosState → LeiosState → Type where

+

+
  UpdateIB : ∀ {s ib} → let open LeiosState s renaming (FFDState to ffds) in

+
    s ⇑ record s { FFDState = record ffds { inIBs = ib ∷ inIBs ffds } }

+

+
  UpdateEB : ∀ {s eb} → let open LeiosState s renaming (FFDState to ffds) in

+
    s ⇑ record s { FFDState = record ffds { inEBs = eb ∷ inEBs ffds } }

+

+
  UpdateVT : ∀ {s vt} → let open LeiosState s renaming (FFDState to ffds) in

+
    s ⇑ record s { FFDState = record ffds { inVTs = vt ∷ inVTs ffds } }

+

+
data LocalStep : LeiosState → LeiosState → Type where

+

+
  StateStep : ∀ {s i o s′} →

+
    ∙ just s -⟦ i / o ⟧⇀ s′

+
      ───────────────────

+
      LocalStep s s′

+

+
  UpdateState : ∀ {s s′} →

+
    ∙ s ⇑ s′

+
      ───────────────────

+
      LocalStep s s′

+

+
  -- TODO: add base layer update

+

+
getLabel : just s -⟦ i / o ⟧⇀ s′ → Action

+
getLabel (Slot {s} _ _ _)            = Slot-Action (slot s)

+
getLabel Ftch                        = Ftch-Action

+
getLabel Base₁                       = Base₁-Action

+
getLabel (Base₂a {s} {eb} _ _ _)     = Base₂a-Action eb

+
getLabel (Base₂b _ _ _)              = Base₂b-Action

+
getLabel (Roles (IB-Role {s} _ _ _)) = IB-Role-Action (slot s)

+
getLabel (Roles (EB-Role {s} _ _ _)) = EB-Role-Action (slot s) (map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s))

+
getLabel (Roles (VT-Role {s} _ _ _)) = VT-Role-Action (slot s)

+
getLabel (Roles (No-IB-Role _ _))    = No-IB-Role-Action

+
getLabel (Roles (No-EB-Role _ _))    = No-EB-Role-Action

+
getLabel (Roles (No-VT-Role _ _))    = No-VT-Role-Action

+

+
ValidAction-sound : (vα : ValidAction α s i) → let (s′ , o) = ⟦ vα ⟧ in just s -⟦ i / o ⟧⇀ s′

+
ValidAction-sound (Slot x x₁ x₂)    = Slot {rbs = []} (recompute dec x) (recompute dec x₁) (recompute dec x₂)

+
ValidAction-sound Ftch              = Ftch

+
ValidAction-sound Base₁             = Base₁

+
ValidAction-sound (Base₂a x x₁ x₂)  = Base₂a (recompute dec x) (recompute dec x₁) (recompute dec x₂)

+
ValidAction-sound (Base₂b x x₁ x₂)  = Base₂b (recompute dec x) (recompute dec x₁) (recompute dec x₂)

+
ValidAction-sound (IB-Role x x₁ x₂) = Roles (IB-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))

+
ValidAction-sound (EB-Role x x₁ x₂) = Roles (EB-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))

+
ValidAction-sound (VT-Role x x₁ x₂) = Roles (VT-Role (recompute dec x) (recompute dec x₁) (recompute dec x₂))

+
ValidAction-sound (No-IB-Role x x₁) = Roles (No-IB-Role x x₁)

+
ValidAction-sound (No-EB-Role x x₁) = Roles (No-EB-Role x x₁)

+
ValidAction-sound (No-VT-Role x x₁) = Roles (No-VT-Role x x₁)

+

+
ValidAction-complete : (st : just s -⟦ i / o ⟧⇀ s′) → ValidAction (getLabel st) s i

+
ValidAction-complete {s} {s′} (Roles (IB-Role {s} {π} {ffds'} x x₁ _)) =

+
  let b  = record { txs = ToPropose s }

+
      h  = mkIBHeader (slot s) id tt sk-IB (ToPropose s)

+
      pr = proj₂ (FFD.Send-total {FFDState s} {ibHeader h} {just (ibBody b)})

+
  in IB-Role {s} x x₁ pr

+
ValidAction-complete {s} (Roles (EB-Role x x₁ _)) =

+
  let LI = map getIBRef $ filter (_∈ᴮ slice L (slot s) 3) (IBs s)

+
      h  = mkEB (slot s) id tt sk-EB LI []

+
      pr = proj₂ (FFD.Send-total {FFDState s} {ebHeader h} {nothing})

+
  in EB-Role {s} x x₁ pr

+
ValidAction-complete {s} (Roles (VT-Role x x₁ _))  =

+
  let EBs'  = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L (slot s) 1) (EBs s)

+
      votes = map (vote sk-VT ∘ hash) EBs'

+
      pr    = proj₂ (FFD.Send-total {FFDState s} {vtHeader votes} {nothing})

+
  in VT-Role {s} x x₁ pr

+
ValidAction-complete (Roles (No-IB-Role x x₁)) = No-IB-Role x x₁

+
ValidAction-complete (Roles (No-EB-Role x x₁)) = No-EB-Role x x₁

+
ValidAction-complete (Roles (No-VT-Role x x₁)) = No-VT-Role x x₁

+
ValidAction-complete Ftch                      = Ftch

+
ValidAction-complete Base₁                     = Base₁

+
ValidAction-complete (Base₂a x x₁ x₂)          = Base₂a x x₁ x₂

+
ValidAction-complete (Base₂b x x₁ x₂)          = Base₂b x x₁ x₂

+
ValidAction-complete {s} (Slot x x₁ _)         = Slot x x₁ (proj₂ (proj₂ (FFD.Fetch-total {FFDState s})))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Short.html b/site/static/agda_html/Leios.Short.html
new file mode 100644
index 000000000..f85c65ca6
--- /dev/null
+++ b/site/static/agda_html/Leios.Short.html
@@ -0,0 +1,291 @@
+
+
+
+  
+  
+  
+  Leios.Short
+  
+  
+
+
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
+
Short-Pipeline Leios

+
+
This document is a specification of Short-Pipeline Leios, usually
+abbreviated as Short Leios. On a high level, the pipeline looks like
+this:

+
    
+
  1. If elected, propose IB
    2. 
+
  2. Wait
    3. 
+
  3. Wait
    4. 
+
  4. If elected, propose EB
    5. 
+
  5. If elected, vote If elected, propose RB
    6. 
+

+
Upkeep

+
A node that never produces a block even though it could is not
+supposed to be an honest node, and we prevent that by tracking whether a
+node has checked if it can make a block in a particular slot.
+LeiosState contains a set of SlotUpkeep and we
+ensure that this set contains all elements before we can advance to the
+next slot, resetting this field to the empty set.

+
data SlotUpkeep : Type where

+
  Base IB-Role EB-Role VT-Role : SlotUpkeep

+

+
+
Block/Vote production rules

+
We now define the rules for block production given by the relation
+_↝_. These are split in two:

+
    
+
  1. Positive rules, when we do need to create a block.
    2. 
+
  2. Negative rules, when we cannot create a block.
    3. 
+

+
The purpose of the negative rules is to properly adjust the upkeep if
+we cannot make a block.

+
Note that _↝_, starting with an empty upkeep can always
+make exactly three steps corresponding to the three types of Leios
+specific blocks.

+
data _↝_ : LeiosState → LeiosState → Type where

+

+
Positive rules

+
  IB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                b = ibBody (record { txs = ToPropose })

+
                h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)

+
          in

+
          ∙ needsUpkeep IB-Role

+
          ∙ canProduceIB slot sk-IB (stake s) π

+
          ∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } IB-Role

+

+
  EB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs

+
                h = mkEB slot id π sk-EB LI []

+
          in

+
          ∙ needsUpkeep EB-Role

+
          ∙ canProduceEB slot sk-EB (stake s) π

+
          ∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } EB-Role

+

+
  VT-Role : let open LeiosState s renaming (FFDState to ffds)

+
                EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs

+
                votes = map (vote sk-VT ∘ hash) EBs'

+
          in

+
          ∙ needsUpkeep VT-Role

+
          ∙ canProduceV slot sk-VT (stake s)

+
          ∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } VT-Role

+

+
Negative rules

+
  No-IB-Role : let open LeiosState s in

+
             ∙ needsUpkeep IB-Role

+
             ∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)

+
             ─────────────────────────────────────────────

+
             s ↝ addUpkeep s IB-Role

+

+
  No-EB-Role : let open LeiosState s in

+
             ∙ needsUpkeep EB-Role

+
             ∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)

+
             ─────────────────────────────────────────────

+
             s ↝ addUpkeep s EB-Role

+

+
  No-VT-Role : let open LeiosState s in

+
             ∙ needsUpkeep VT-Role

+
             ∙ ¬ canProduceV slot sk-VT (stake s)

+
             ─────────────────────────────────────────────

+
             s ↝ addUpkeep s VT-Role

+

+
Uniform short-pipeline

+
stage : ℕ → ⦃ _ : NonZero L ⦄ → ℕ

+
stage s = s / L

+

+
beginningOfStage : ℕ → Type

+
beginningOfStage s = stage s * L ≡ s

+

+
allDone : LeiosState → Type

+
allDone s =

+
  let open LeiosState s

+
  in   (beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ EB-Role ∷ VT-Role ∷ Base ∷ []))

+
   ⊎ (¬ beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ VT-Role ∷ Base ∷ []))

+

+
data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where

+

+
Initialization

+
  Init :

+
       ∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'

+
       ∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'

+
       ────────────────────────────────────────────────────────────────

+
       nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks

+

+
Network and Ledger

+
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in

+
       ∙ allDone s

+
       ∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'

+
       ∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'

+
       ───────────────────────────────────────────────────────────────────────

+
       just s -⟦ SLOT / EMPTY ⟧⇀ record s

+
           { FFDState  = ffds'

+
           ; BaseState = bs'

+
           ; Ledger    = constructLedger rbs

+
           ; slot      = suc slot

+
           ; Upkeep    = ∅

+
           } ↑ L.filter (isValid? s) msgs

+

+
  Ftch :

+
       ────────────────────────────────────────────────────────

+
       just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s

+

+
Base chain

+Note: Submitted data to the base chain is only taken into account if the
+party submitting is the block producer on the base chain for the given
+slot
+
  Base₁   :

+
          ───────────────────────────────────────────────────────────────────

+
          just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }

+

+
  Base₂a  : let open LeiosState s renaming (BaseState to bs) in

+
          ∙ needsUpkeep Base

+
          ∙ eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs

+
          ∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'

+
          ───────────────────────────────────────────────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base

+

+
  Base₂b  : let open LeiosState s renaming (BaseState to bs) in

+
          ∙ needsUpkeep Base

+
          ∙ [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs

+
          ∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'

+
          ───────────────────────────────────────────────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base

+

+
Protocol rules

+
  Roles :

+
        ∙ s ↝ s'

+
        ─────────────────────────────

+
        just s -⟦ SLOT / EMPTY ⟧⇀ s'

+

+

+
diff --git a/site/static/agda_html/Leios.Short.md b/site/static/agda_html/Leios.Short.md
new file mode 100644
index 000000000..ec841be73
--- /dev/null
+++ b/site/static/agda_html/Leios.Short.md
@@ -0,0 +1,182 @@
+## Short-Pipeline Leios
+
+
+
+This document is a specification of Short-Pipeline Leios, usually
+abbreviated as Short Leios. On a high level, the pipeline looks like this:
+
+1. If elected, propose IB
+2. Wait
+3. Wait
+4. If elected, propose EB
+5. If elected, vote
+   If elected, propose RB
+
+### Upkeep
+
+A node that never produces a block even though it could is not
+supposed to be an honest node, and we prevent that by tracking whether
+a node has checked if it can make a block in a particular slot.
+`LeiosState` contains a set of `SlotUpkeep` and we ensure that this
+set contains all elements before we can advance to the next slot,
+resetting this field to the empty set.
+
+
data SlotUpkeep : Type where
+  Base IB-Role EB-Role VT-Role : SlotUpkeep
+

+
+### Block/Vote production rules
+
+We now define the rules for block production given by the relation `_↝_`. These are split in two:
+
+1. Positive rules, when we do need to create a block.
+2. Negative rules, when we cannot create a block.
+
+The purpose of the negative rules is to properly adjust the upkeep if
+we cannot make a block.
+
+Note that `_↝_`, starting with an empty upkeep can always make exactly
+three steps corresponding to the three types of Leios specific blocks.
+
+
data _↝_ : LeiosState → LeiosState → Type where
+
#### Positive rules
+
  IB-Role : let open LeiosState s renaming (FFDState to ffds)
+                b = ibBody (record { txs = ToPropose })
+                h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)
+          in
+          ∙ needsUpkeep IB-Role
+          ∙ canProduceIB slot sk-IB (stake s) π
+          ∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'
+          ─────────────────────────────────────────────────────────────────────────
+          s ↝ addUpkeep record s { FFDState = ffds' } IB-Role
+
  EB-Role : let open LeiosState s renaming (FFDState to ffds)
+                LI = map getIBRef $ filter (_∈ᴮ slice L slot 3) IBs
+                h = mkEB slot id π sk-EB LI []
+          in
+          ∙ needsUpkeep EB-Role
+          ∙ canProduceEB slot sk-EB (stake s) π
+          ∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'
+          ─────────────────────────────────────────────────────────────────────────
+          s ↝ addUpkeep record s { FFDState = ffds' } EB-Role
+
  VT-Role : let open LeiosState s renaming (FFDState to ffds)
+                EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot 1) EBs
+                votes = map (vote sk-VT ∘ hash) EBs'
+          in
+          ∙ needsUpkeep VT-Role
+          ∙ canProduceV slot sk-VT (stake s)
+          ∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'
+          ─────────────────────────────────────────────────────────────────────────
+          s ↝ addUpkeep record s { FFDState = ffds' } VT-Role
+
#### Negative rules
+
  No-IB-Role : let open LeiosState s in
+             ∙ needsUpkeep IB-Role
+             ∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)
+             ─────────────────────────────────────────────
+             s ↝ addUpkeep s IB-Role
+
  No-EB-Role : let open LeiosState s in
+             ∙ needsUpkeep EB-Role
+             ∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)
+             ─────────────────────────────────────────────
+             s ↝ addUpkeep s EB-Role
+
  No-VT-Role : let open LeiosState s in
+             ∙ needsUpkeep VT-Role
+             ∙ ¬ canProduceV slot sk-VT (stake s)
+             ─────────────────────────────────────────────
+             s ↝ addUpkeep s VT-Role
+
### Uniform short-pipeline
+
stage : ℕ → ⦃ _ : NonZero L ⦄ → ℕ
+stage s = s / L
+
+beginningOfStage : ℕ → Type
+beginningOfStage s = stage s * L ≡ s
+
+allDone : LeiosState → Type
+allDone s =
+  let open LeiosState s
+  in   (beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ EB-Role ∷ VT-Role ∷ Base ∷ []))
+   ⊎ (¬ beginningOfStage slot × Upkeep ≡ᵉ fromList (IB-Role ∷ VT-Role ∷ Base ∷ []))
+
+data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where
+
#### Initialization
+
  Init :
+       ∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'
+       ∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'
+       ────────────────────────────────────────────────────────────────
+       nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks
+
#### Network and Ledger
+
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in
+       ∙ allDone s
+       ∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'
+       ∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'
+       ───────────────────────────────────────────────────────────────────────
+       just s -⟦ SLOT / EMPTY ⟧⇀ record s
+           { FFDState  = ffds'
+           ; BaseState = bs'
+           ; Ledger    = constructLedger rbs
+           ; slot      = suc slot
+           ; Upkeep    = ∅
+           } ↑ L.filter (isValid? s) msgs
+
  Ftch :
+       ────────────────────────────────────────────────────────
+       just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s
+
#### Base chain
+
+Note: Submitted data to the base chain is only taken into account
+      if the party submitting is the block producer on the base chain
+      for the given slot
+
  Base₁   :
+          ───────────────────────────────────────────────────────────────────
+          just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }
+
  Base₂a  : let open LeiosState s renaming (BaseState to bs) in
+          ∙ needsUpkeep Base
+          ∙ eb ∈ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs
+          ∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'
+          ───────────────────────────────────────────────────────────────────────
+          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base
+
+  Base₂b  : let open LeiosState s renaming (BaseState to bs) in
+          ∙ needsUpkeep Base
+          ∙ [] ≡ filter (λ eb → isVoteCertified s eb × eb ∈ᴮ slice L slot 2) EBs
+          ∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'
+          ───────────────────────────────────────────────────────────────────────
+          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base
+
#### Protocol rules
+
  Roles :
+        ∙ s ↝ s'
+        ─────────────────────────────
+        just s -⟦ SLOT / EMPTY ⟧⇀ s'
+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Simplified.Deterministic.html b/site/static/agda_html/Leios.Simplified.Deterministic.html
new file mode 100644
index 000000000..f526c012d
--- /dev/null
+++ b/site/static/agda_html/Leios.Simplified.Deterministic.html
@@ -0,0 +1,434 @@
+
+Leios.Simplified.Deterministic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
--{-# OPTIONS --safe #-}

+
{-# OPTIONS --allow-unsolved-metas #-}

+

+
--------------------------------------------------------------------------------

+
-- Deterministic variant of simple Leios

+
--------------------------------------------------------------------------------

+

+
open import Leios.Prelude hiding (id)

+
open import Prelude.Init using (∃₂-syntax)

+
open import Leios.FFD

+

+
import Data.List as L

+
open import Data.List.Relation.Unary.Any using (here)

+

+
open import Leios.SpecStructure

+

+
module Leios.Simplified.Deterministic (⋯ : SpecStructure 2) (let open SpecStructure ⋯) (Λ μ : ℕ) where

+

+
import Leios.Simplified

+
open import Leios.Simplified ⋯ Λ μ hiding (_-⟦_/_⟧⇀_)

+
module ND = Leios.Simplified ⋯ Λ μ

+

+
open import Class.Computational

+
open import Class.Computational22

+
open import StateMachine

+

+
open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)

+
open GenFFD

+

+
open FFD hiding (_-⟦_/_⟧⇀_)

+

+
private variable s s' s0 s1 s2 s3 s4 s5 : LeiosState

+
                 i      : LeiosInput

+
                 o      : LeiosOutput

+
                 ffds'  : FFD.State

+
                 π      : VrfPf

+
                 bs'    : B.State

+
                 ks ks' : K.State

+
                 msgs   : List (FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract)

+
                 eb     : EndorserBlock

+
                 rbs    : List RankingBlock

+
                 txs    : List Tx

+
                 V      : VTy

+
                 SD     : StakeDistr

+
                 pks    : List PubKey

+

+
lemma : ∀ {u} → u ∈ LeiosState.Upkeep (addUpkeep s u)

+
lemma = to ∈-∪ (inj₂ (to ∈-singleton refl))

+
  where open Equivalence

+

+
addUpkeep⇒¬needsUpkeep : ∀ {u} → ¬ LeiosState.needsUpkeep (addUpkeep s u) u

+
addUpkeep⇒¬needsUpkeep {s = s} = λ x → x (lemma {s = s})

+

+
data _⊢_ : LeiosInput → LeiosState → Type where

+
  Init :

+
       ∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'

+
       ∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'

+
       ────────────────────────────────────────────────────────────────

+
       INIT V ⊢ initLeiosState V SD bs' pks

+

+
data _-⟦Base⟧⇀_ : LeiosState → LeiosState → Type where

+

+
  Base₂a  : ∀ {ebs} → let open LeiosState s renaming (BaseState to bs) in

+
          ∙ eb ∷ ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs

+
          ∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'

+
          ───────────────────────────────────────────────────────────────────────

+
          s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base

+

+
  Base₂b  : let open LeiosState s renaming (BaseState to bs) in

+
          ∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs

+
          ∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'

+
          ───────────────────────────────────────────────────────────────────────

+
          s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs' } Base

+

+
Base⇒ND : LeiosState.needsUpkeep s Base → s -⟦Base⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'

+
Base⇒ND u (Base₂a x₁ x₂) = Base₂a u (subst (_ ∈_) x₁ (Equivalence.to ∈-fromList (here refl))) x₂

+
Base⇒ND u (Base₂b x₁ x₂) = Base₂b u x₁ x₂

+

+
Base-Upkeep : ∀ {u} → u ≢ Base → LeiosState.needsUpkeep s u → s -⟦Base⟧⇀ s'

+
                  → LeiosState.needsUpkeep s' u

+
Base-Upkeep u≢Base h (Base₂a _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢Base (Equivalence.from ∈-singleton y)

+
Base-Upkeep u≢Base h (Base₂b _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢Base (Equivalence.from ∈-singleton y)

+

+
opaque

+
  Base-total : ∃[ s' ] s -⟦Base⟧⇀ s'

+
  Base-total {s = s} with

+
    (let open LeiosState s in filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs)

+
    in eq

+
  ... | []    = -, Base₂b (sym eq) (proj₂ B.SUBMIT-total)

+
  ... | x ∷ l = -, Base₂a (sym eq) (proj₂ B.SUBMIT-total)

+

+
  Base-total' : ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄

+
              → ∃[ bs ] s -⟦Base⟧⇀ addUpkeep record s { BaseState = bs } Base

+
  Base-total' {s = s} = let open LeiosState s in

+
    case ∃[ ebs ] ebs ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs ∋ -, refl

+
      of λ where

+
        (eb ∷ _ , eq) → -, Base₂a eq (proj₂ B.SUBMIT-total)

+
        ([]     , eq) → -, Base₂b eq (proj₂ B.SUBMIT-total)

+

+
data _-⟦IB-Role⟧⇀_ : LeiosState → LeiosState → Type where

+

+
  IB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                b = ibBody (record { txs = ToPropose })

+
                h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)

+
          in

+
          ∙ canProduceIB slot sk-IB (stake s) π

+
          ∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'

+
          ────────────────────────────────────────────────────────────────────

+
          s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } IB-Role

+

+
  No-IB-Role : let open LeiosState s in

+
          ∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)

+
          ────────────────────────────────────────

+
          s -⟦IB-Role⟧⇀ addUpkeep s IB-Role

+

+
IB-Role⇒ND : LeiosState.needsUpkeep s IB-Role → s -⟦IB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'

+
IB-Role⇒ND u (IB-Role x₁ x₂) = Roles (IB-Role u x₁ x₂)

+
IB-Role⇒ND u (No-IB-Role x₁) = Roles (No-IB-Role u x₁)

+

+
IB-Role-Upkeep : ∀ {u} → u ≢ IB-Role → LeiosState.needsUpkeep s u → s -⟦IB-Role⟧⇀ s'

+
                  → LeiosState.needsUpkeep s' u

+
IB-Role-Upkeep u≢IB-Role h (IB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢IB-Role (Equivalence.from ∈-singleton y)

+
IB-Role-Upkeep u≢IB-Role h (No-IB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢IB-Role (Equivalence.from ∈-singleton y)

+

+
opaque

+
  IB-Role-total : ∃[ s' ] s -⟦IB-Role⟧⇀ s'

+
  IB-Role-total {s = s} = let open LeiosState s in case Dec-canProduceIB of λ where

+
    (inj₁ (π , pf)) → -, IB-Role    pf (proj₂ FFD.Send-total)

+
    (inj₂ pf)       → -, No-IB-Role pf

+

+
  IB-Role-total' : ∃[ ffds ] s -⟦IB-Role⟧⇀ addUpkeep record s { FFDState = ffds } IB-Role

+
  IB-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceIB of λ where

+
    (inj₁ (π , pf)) → -, IB-Role    pf (proj₂ FFD.Send-total)

+
    (inj₂ pf)       → -, No-IB-Role pf

+

+
data _-⟦EB-Role⟧⇀_ : LeiosState → LeiosState → Type where

+

+
  EB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                LI = map getIBRef $ filter (_∈ᴮ slice L slot (Λ + 1)) IBs

+
                LE = map getEBRef $ filter (isVote1Certified s) $

+
                           filter (_∈ᴮ slice L slot (μ + 2)) EBs

+
                h = mkEB slot id π sk-EB LI LE

+
          in

+
          ∙ canProduceEB slot sk-EB (stake s) π

+
          ∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'

+
          ────────────────────────────────────────────────────────────────────

+
          s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds' } EB-Role

+

+
  No-EB-Role : let open LeiosState s in

+
          ∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)

+
          ────────────────────────────────────────

+
          s -⟦EB-Role⟧⇀ addUpkeep s EB-Role

+

+
EB-Role⇒ND : LeiosState.needsUpkeep s EB-Role → s -⟦EB-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'

+
EB-Role⇒ND u (EB-Role x₁ x₂) = Roles (EB-Role u x₁ x₂)

+
EB-Role⇒ND u (No-EB-Role x₁) = Roles (No-EB-Role u x₁)

+

+
EB-Role-Upkeep : ∀ {u} → u ≢ EB-Role → LeiosState.needsUpkeep s u → s -⟦EB-Role⟧⇀ s'

+
                  → LeiosState.needsUpkeep s' u

+
EB-Role-Upkeep u≢EB-Role h (EB-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢EB-Role (Equivalence.from ∈-singleton y)

+
EB-Role-Upkeep u≢EB-Role h (No-EB-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢EB-Role (Equivalence.from ∈-singleton y)

+

+
opaque

+
  EB-Role-total : ∃[ s' ] s -⟦EB-Role⟧⇀ s'

+
  EB-Role-total {s = s} = let open LeiosState s in case Dec-canProduceEB of λ where

+
    (inj₁ (π , pf)) → -, EB-Role    pf (proj₂ FFD.Send-total)

+
    (inj₂ pf)       → -, No-EB-Role pf

+

+
  EB-Role-total' : ∃[ ffds ] s -⟦EB-Role⟧⇀ addUpkeep record s { FFDState = ffds } EB-Role

+
  EB-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceEB of λ where

+
    (inj₁ (π , pf)) → -, EB-Role    pf (proj₂ FFD.Send-total)

+
    (inj₂ pf)       → -, No-EB-Role pf

+

+
data _-⟦V1-Role⟧⇀_ : LeiosState → LeiosState → Type where

+

+
  V1-Role : let open LeiosState s renaming (FFDState to ffds)

+
                EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot (μ + 1)) EBs

+
                votes = map (vote sk-VT ∘ hash) EBs'

+
          in

+
          ∙ canProduceV1 slot sk-VT (stake s)

+
          ∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'

+
          ────────────────────────────────────────────────────────────────────

+
          s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V1-Role

+

+
  No-V1-Role : let open LeiosState s in

+
          ∙ ¬ canProduceV1 slot sk-VT (stake s)

+
          ────────────────────────────────────────

+
          s -⟦V1-Role⟧⇀ addUpkeep s V1-Role

+

+
V1-Role⇒ND : LeiosState.needsUpkeep s V1-Role → s -⟦V1-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'

+
V1-Role⇒ND u (V1-Role x₁ x₂) = Roles (V1-Role u x₁ x₂)

+
V1-Role⇒ND u (No-V1-Role x₁) = Roles (No-V1-Role u x₁)

+

+
V1-Role-Upkeep : ∀ {u} → u ≢ V1-Role → LeiosState.needsUpkeep s u → s -⟦V1-Role⟧⇀ s'

+
                  → LeiosState.needsUpkeep s' u

+
V1-Role-Upkeep u≢V1-Role h (V1-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢V1-Role (Equivalence.from ∈-singleton y)

+
V1-Role-Upkeep u≢V1-Role h (No-V1-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢V1-Role (Equivalence.from ∈-singleton y)

+

+
opaque

+
  V1-Role-total : ∃[ s' ] s -⟦V1-Role⟧⇀ s'

+
  V1-Role-total {s = s} = let open LeiosState s in case Dec-canProduceV1 of λ where

+
    (yes p) → -, V1-Role p (proj₂ FFD.Send-total)

+
    (no ¬p) → -, No-V1-Role ¬p

+

+
  V1-Role-total' : ∃[ ffds ] s -⟦V1-Role⟧⇀ addUpkeep record s { FFDState = ffds } V1-Role

+
  V1-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceV1 of λ where

+
    (yes p) → -, V1-Role    p (proj₂ FFD.Send-total)

+
    (no ¬p) → -, No-V1-Role ¬p

+

+
data _-⟦V2-Role⟧⇀_ : LeiosState → LeiosState → Type where

+

+
  V2-Role : let open LeiosState s renaming (FFDState to ffds)

+
                EBs' = filter (vote2Eligible s) $ filter (_∈ᴮ slice L slot 1) EBs

+
                votes = map (vote sk-VT ∘ hash) EBs'

+
          in

+
          ∙ canProduceV2 slot sk-VT (stake s)

+
          ∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'

+
          ────────────────────────────────────────────────────────────────────

+
          s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds' } V2-Role

+

+
  No-V2-Role : let open LeiosState s in

+
          ∙ ¬ canProduceV2 slot sk-VT (stake s)

+
          ────────────────────────────────────────

+
          s -⟦V2-Role⟧⇀ addUpkeep s V2-Role

+

+
V2-Role⇒ND : LeiosState.needsUpkeep s V2-Role → s -⟦V2-Role⟧⇀ s' → just s ND.-⟦ SLOT / EMPTY ⟧⇀ s'

+
V2-Role⇒ND u (V2-Role x₁ x₂) = Roles (V2-Role u x₁ x₂)

+
V2-Role⇒ND u (No-V2-Role x₁) = Roles (No-V2-Role u x₁)

+

+
V2-Role-Upkeep : ∀ {u} → u ≢ V2-Role → LeiosState.needsUpkeep s u → s -⟦V2-Role⟧⇀ s'

+
                  → LeiosState.needsUpkeep s' u

+
V2-Role-Upkeep u≢V2-Role h (V2-Role _ _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢V2-Role (Equivalence.from ∈-singleton y)

+
V2-Role-Upkeep u≢V2-Role h (No-V2-Role _) u∈su = case Equivalence.from ∈-∪ u∈su of λ where

+
  (inj₁ x) → h x

+
  (inj₂ y) → u≢V2-Role (Equivalence.from ∈-singleton y)

+

+
opaque

+
  V2-Role-total : ∃[ s' ] s -⟦V2-Role⟧⇀ s'

+
  V2-Role-total {s = s} = let open LeiosState s in case Dec-canProduceV2 of λ where

+
    (yes p) → -, V2-Role p (proj₂ FFD.Send-total)

+
    (no ¬p) → -, No-V2-Role ¬p

+

+
  V2-Role-total' : ∃[ ffds ] s -⟦V2-Role⟧⇀ addUpkeep record s { FFDState = ffds } V2-Role

+
  V2-Role-total' {s = s} = let open LeiosState s in case Dec-canProduceV2 of λ where

+
    (yes p) → -, V2-Role    p (proj₂ FFD.Send-total)

+
    (no ¬p) → -, No-V2-Role ¬p

+

+
data _-⟦_/_⟧⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where

+

+
  -- Network and Ledger

+

+
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs)

+
             s0 = record s

+
                    { FFDState = ffds'

+
                    ; Ledger   = constructLedger rbs

+
                    ; slot     = suc slot

+
                    ; Upkeep   = ∅

+
                    ; BaseState = bs'

+
                    } ↑ L.filter (isValid? s) msgs

+
       in

+
       ∙ Upkeep ≡ᵉ allUpkeep

+
       ∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'

+
       ∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'

+
       ∙ s0 -⟦Base⟧⇀    s1

+
       ∙ s1 -⟦IB-Role⟧⇀ s2

+
       ∙ s2 -⟦EB-Role⟧⇀ s3

+
       ∙ s3 -⟦V1-Role⟧⇀ s4

+
       ∙ s4 -⟦V2-Role⟧⇀ s5

+
       ───────────────────────────────────────────────────────────────────────

+
       s -⟦ SLOT / EMPTY ⟧⇀ s5

+

+
  Ftch :

+
       ───────────────────────────────────────────────────

+
       s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s

+

+
  -- Base chain

+
  --

+
  -- Note: Submitted data to the base chain is only taken into account

+
  --       if the party submitting is the block producer on the base chain

+
  --       for the given slot

+

+
  Base₁   :

+
          ──────────────────────────────────────────────────────────────

+
          s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }

+

+
  -- Protocol rules

+

+
_-⟦_/_⟧ⁿᵈ⇀_ : LeiosState → LeiosInput → LeiosOutput → LeiosState → Type

+
s -⟦ i / o ⟧ⁿᵈ⇀ s' = just s ND.-⟦ i / o ⟧⇀ s'

+

+
_-⟦_/_⟧ⁿᵈ*⇀_ : LeiosState → List LeiosInput → List LeiosOutput → LeiosState → Type

+
_-⟦_/_⟧ⁿᵈ*⇀_ = ReflexiveTransitiveClosure _-⟦_/_⟧ⁿᵈ⇀_

+

+
-- Key fact: stepping with the deterministic relation means we can

+
-- also step with the non-deterministic one

+
-- TODO: this is a lot like a weak simulation, can we do something prettier?

+
-⟦/⟧⇀⇒ND : s -⟦ i / o ⟧⇀ s' → ∃₂[ i , o ] (s -⟦ i / o ⟧ⁿᵈ*⇀ s')

+
-⟦/⟧⇀⇒ND (Slot {s = s} {msgs = msgs} {s1 = s1} {s2 = s2} {s3 = s3} {s4 = s4} x x₁ x₂ hB hIB hEB hV1 hV2) = replicate 6 SLOT , replicate 6 EMPTY ,

+
  let

+
    s0 = _

+
    upkeep≡∅ : LeiosState.Upkeep s0 ≡ ∅

+
    upkeep≡∅ = sym (↑-preserves-Upkeep {x = L.filter (isValid? s) msgs})

+
    needsAllUpkeep : ∀ {u} → LeiosState.needsUpkeep s0 u

+
    needsAllUpkeep {u} = subst (u ∉_) (sym upkeep≡∅) Properties.∉-∅

+
    needsUpkeep1 : ∀ {u} → u ≢ Base → LeiosState.needsUpkeep s1 u

+
    needsUpkeep1 h1 = Base-Upkeep h1 needsAllUpkeep hB

+
    needsUpkeep2 : ∀ {u} → u ≢ Base → u ≢ IB-Role → LeiosState.needsUpkeep s2 u

+
    needsUpkeep2 h1 h2 = IB-Role-Upkeep h2 (needsUpkeep1 h1) hIB

+
    needsUpkeep3 : ∀ {u} → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → LeiosState.needsUpkeep s3 u

+
    needsUpkeep3 h1 h2 h3 = EB-Role-Upkeep h3 (needsUpkeep2 h1 h2) hEB

+
    needsUpkeep4 : ∀ {u} → u ≢ Base → u ≢ IB-Role → u ≢ EB-Role → u ≢ V1-Role → LeiosState.needsUpkeep s4 u

+
    needsUpkeep4 h1 h2 h3 h4 = V1-Role-Upkeep h4 (needsUpkeep3 h1 h2 h3) hV1

+
  in (BS-ind (ND.Slot x x₁ x₂) $

+
      BS-ind (Base⇒ND {s = s0} needsAllUpkeep hB) $

+
      BS-ind (IB-Role⇒ND (needsUpkeep1 (λ ())) hIB) $

+
      BS-ind (EB-Role⇒ND (needsUpkeep2 (λ ()) (λ ())) hEB) $

+
      BS-ind (V1-Role⇒ND (needsUpkeep3 (λ ()) (λ ()) (λ ())) hV1) $

+
      STS⇒RTC (V2-Role⇒ND (needsUpkeep4 (λ ()) (λ ()) (λ ()) (λ ())) hV2))

+
-⟦/⟧⇀⇒ND Ftch = _ , _ , STS⇒RTC Ftch

+
-⟦/⟧⇀⇒ND Base₁ = _ , _ , STS⇒RTC Base₁

+

+
open Computational22 ⦃...⦄

+

+
module _ ⦃ Computational-B : Computational22 B._-⟦_/_⟧⇀_ String ⦄

+
         ⦃ Computational-FFD : Computational22 FFD._-⟦_/_⟧⇀_ String ⦄ where

+

+
  instance

+
    Computational--⟦/⟧⇀ : Computational22 _-⟦_/_⟧⇀_ String

+
    Computational--⟦/⟧⇀ .computeProof s (INIT x) = failure "No handling of INIT here"

+
    Computational--⟦/⟧⇀ .computeProof s (SUBMIT (inj₁ eb)) = failure "Cannot submit EB here"

+
    Computational--⟦/⟧⇀ .computeProof s (SUBMIT (inj₂ txs)) = success (-, Base₁)

+
    Computational--⟦/⟧⇀ .computeProof s* SLOT = let open LeiosState s* in

+
      case (¿ Upkeep ≡ᵉ allUpkeep ¿ ,′ computeProof BaseState B.FTCH-LDG ,′ computeProof FFDState FFD.Fetch) of λ where

+
        (yes p , success ((B.BASE-LDG l , bs) , p₁) , success ((FFD.FetchRes msgs , ffds) , p₂)) →

+
          success ((_ , (Slot p p₁ p₂ (proj₂ Base-total) (proj₂ IB-Role-total) (proj₂ EB-Role-total) (proj₂ V1-Role-total) (proj₂ V2-Role-total))))

+
        (yes p , _ , _) → failure "Subsystem failed"

+
        (no ¬p , _) → failure "Upkeep incorrect"

+
    Computational--⟦/⟧⇀ .computeProof s FTCH-LDG = success (-, Ftch)

+
    Computational--⟦/⟧⇀ .completeness = {!!}

+

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+
+Leios.Simplified
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude hiding (id)

+
open import Leios.FFD

+
open import Leios.SpecStructure

+
open import Data.Fin.Patterns

+

+
open import Tactic.Defaults

+
open import Tactic.Derive.DecEq

+

+
module Leios.Simplified (⋯ : SpecStructure 2) (let open SpecStructure ⋯) (Λ μ : ℕ) where

+

+
data SlotUpkeep : Type where

+
  Base IB-Role EB-Role V1-Role V2-Role : SlotUpkeep

+

+
unquoteDecl DecEq-SlotUpkeep = derive-DecEq ((quote SlotUpkeep , DecEq-SlotUpkeep) ∷ [])

+

+
allUpkeep : ℙ SlotUpkeep

+
allUpkeep = fromList (Base ∷ IB-Role ∷ EB-Role ∷ V1-Role ∷ V2-Role ∷ [])

+

+
open import Leios.Protocol (⋯) SlotUpkeep public

+

+
open BaseAbstract B' using (Cert; V-chkCerts; VTy; initSlot)

+
open FFD hiding (_-⟦_/_⟧⇀_)

+
open GenFFD

+

+
isVote1Certified : LeiosState → EndorserBlock → Type

+
isVote1Certified s eb = isVoteCertified (LeiosState.votingState s) (0F , eb)

+

+
isVote2Certified : LeiosState → EndorserBlock → Type

+
isVote2Certified s eb = isVoteCertified (LeiosState.votingState s) (1F , eb)

+

+
-- Predicates about EBs

+
module _ (s : LeiosState) (eb : EndorserBlock) where

+
  open EndorserBlockOSig eb

+
  open LeiosState s

+

+
  vote2Eligible : Type

+
  vote2Eligible = length ebRefs ≥ lengthˢ candidateEBs / 2 -- should this be `>`?

+
                × fromList ebRefs ⊆ candidateEBs

+
    where candidateEBs : ℙ Hash

+
          candidateEBs = mapˢ getEBRef $ filterˢ (_∈ᴮ slice L slot (μ + 3)) (fromList EBs)

+

+
private variable s s'   : LeiosState

+
                 ffds'  : FFD.State

+
                 π      : VrfPf

+
                 bs'    : B.State

+
                 ks ks' : K.State

+
                 msgs   : List (FFDAbstract.Header ffdAbstract ⊎ FFDAbstract.Body ffdAbstract)

+
                 eb     : EndorserBlock

+
                 rbs    : List RankingBlock

+
                 txs    : List Tx

+
                 V      : VTy

+
                 SD     : StakeDistr

+
                 pks    : List PubKey

+

+
data _↝_ : LeiosState → LeiosState → Type where

+

+
  IB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                b = ibBody (record { txs = ToPropose })

+
                h = ibHeader (mkIBHeader slot id π sk-IB ToPropose)

+
          in

+
          ∙ needsUpkeep IB-Role

+
          ∙ canProduceIB slot sk-IB (stake s) π

+
          ∙ ffds FFD.-⟦ Send h (just b) / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } IB-Role

+

+
  EB-Role : let open LeiosState s renaming (FFDState to ffds)

+
                LI = map getIBRef $ filter (_∈ᴮ slice L slot (Λ + 1)) IBs

+
                LE = map getEBRef $ filter (isVote1Certified s) $

+
                           filter (_∈ᴮ slice L slot (μ + 2)) EBs

+
                h = mkEB slot id π sk-EB LI LE

+
          in

+
          ∙ needsUpkeep EB-Role

+
          ∙ canProduceEB slot sk-EB (stake s) π

+
          ∙ ffds FFD.-⟦ Send (ebHeader h) nothing / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } EB-Role

+

+
  V1-Role : let open LeiosState s renaming (FFDState to ffds)

+
                EBs' = filter (allIBRefsKnown s) $ filter (_∈ᴮ slice L slot (μ + 1)) EBs

+
                votes = map (vote sk-VT ∘ hash) EBs'

+
          in

+
          ∙ needsUpkeep V1-Role

+
          ∙ canProduceV1 slot sk-VT (stake s)

+
          ∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } V1-Role

+

+
  V2-Role : let open LeiosState s renaming (FFDState to ffds)

+
                EBs' = filter (vote2Eligible s) $ filter (_∈ᴮ slice L slot 1) EBs

+
                votes = map (vote sk-VT ∘ hash) EBs'

+
          in

+
          ∙ needsUpkeep V2-Role

+
          ∙ canProduceV2 slot sk-VT (stake s)

+
          ∙ ffds FFD.-⟦ Send (vtHeader votes) nothing / SendRes ⟧⇀ ffds'

+
          ─────────────────────────────────────────────────────────────────────────

+
          s ↝ addUpkeep record s { FFDState = ffds' } V2-Role

+

+
  -- Note: Base doesn't need a negative rule, since it can always be invoked

+

+
  No-IB-Role : let open LeiosState s in

+
          ∙ needsUpkeep IB-Role

+
          ∙ (∀ π → ¬ canProduceIB slot sk-IB (stake s) π)

+
          ─────────────────────────────────────────────

+
          s ↝ addUpkeep s IB-Role

+

+
  No-EB-Role : let open LeiosState s in

+
          ∙ needsUpkeep EB-Role

+
          ∙ (∀ π → ¬ canProduceEB slot sk-EB (stake s) π)

+
          ─────────────────────────────────────────────

+
          s ↝ addUpkeep s EB-Role

+

+
  No-V1-Role : let open LeiosState s in

+
          ∙ needsUpkeep V1-Role

+
          ∙ ¬ canProduceV1 slot sk-VT (stake s)

+
          ─────────────────────────────────────────────

+
          s ↝ addUpkeep s V1-Role

+

+
  No-V2-Role : let open LeiosState s in

+
          ∙ needsUpkeep V2-Role

+
          ∙ ¬ canProduceV2 slot sk-VT (stake s)

+
          ─────────────────────────────────────────────

+
          s ↝ addUpkeep s V2-Role

+

+
data _-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type where

+

+
  -- Initialization

+

+
  Init :

+
       ∙ ks K.-⟦ K.INIT pk-IB pk-EB pk-VT / K.PUBKEYS pks ⟧⇀ ks'

+
       ∙ initBaseState B.-⟦ B.INIT (V-chkCerts pks) / B.STAKE SD ⟧⇀ bs'

+
       ────────────────────────────────────────────────────────────────

+
       nothing -⟦ INIT V / EMPTY ⟧⇀ initLeiosState V SD bs' pks

+

+
  -- Network and Ledger

+

+
  Slot : let open LeiosState s renaming (FFDState to ffds; BaseState to bs) in

+
       ∙ Upkeep ≡ᵉ allUpkeep

+
       ∙ bs B.-⟦ B.FTCH-LDG / B.BASE-LDG rbs ⟧⇀ bs'

+
       ∙ ffds FFD.-⟦ Fetch / FetchRes msgs ⟧⇀ ffds'

+
       ───────────────────────────────────────────────────────────────────────

+
       just s -⟦ SLOT / EMPTY ⟧⇀ record s

+
           { FFDState  = ffds'

+
           ; Ledger    = constructLedger rbs

+
           ; slot      = suc slot

+
           ; Upkeep    = ∅

+
           ; BaseState = bs'

+
           } ↑ L.filter (isValid? s) msgs

+

+
  Ftch :

+
       ────────────────────────────────────────────────────────

+
       just s -⟦ FTCH-LDG / FTCH-LDG (LeiosState.Ledger s) ⟧⇀ s

+

+
  -- Base chain

+
  --

+
  -- Note: Submitted data to the base chain is only taken into account

+
  --       if the party submitting is the block producer on the base chain

+
  --       for the given slot

+

+
  Base₁   :

+
          ───────────────────────────────────────────────────────────────────

+
          just s -⟦ SUBMIT (inj₂ txs) / EMPTY ⟧⇀ record s { ToPropose = txs }

+

+
  Base₂a  : let open LeiosState s renaming (BaseState to bs) in

+
          ∙ needsUpkeep Base

+
          ∙ eb ∈ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs

+
          ∙ bs B.-⟦ B.SUBMIT (this eb) / B.EMPTY ⟧⇀ bs'

+
          ───────────────────────────────────────────────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base

+

+
  Base₂b  : let open LeiosState s renaming (BaseState to bs) in

+
          ∙ needsUpkeep Base

+
          ∙ [] ≡ filter (λ eb → isVote2Certified s eb × eb ∈ᴮ slice L slot 2) EBs

+
          ∙ bs B.-⟦ B.SUBMIT (that ToPropose) / B.EMPTY ⟧⇀ bs'

+
          ───────────────────────────────────────────────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ addUpkeep record s { BaseState = bs' } Base

+

+
  -- Protocol rules

+

+
  Roles : ∙ s ↝ s'

+
          ─────────────────────────────

+
          just s -⟦ SLOT / EMPTY ⟧⇀ s'

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.SpecStructure.html b/site/static/agda_html/Leios.SpecStructure.html
new file mode 100644
index 000000000..0dfbbba76
--- /dev/null
+++ b/site/static/agda_html/Leios.SpecStructure.html
@@ -0,0 +1,130 @@
+
+Leios.SpecStructure
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude hiding (id)

+
open import Leios.Abstract

+
open import Leios.FFD

+
open import Leios.VRF

+

+
import Leios.Base

+
import Leios.Blocks

+
import Leios.KeyRegistration

+
import Leios.Voting

+

+
open import Data.Fin

+

+
module Leios.SpecStructure where

+

+
record SpecStructure (rounds : ℕ) : Type₁ where

+
  field a : LeiosAbstract

+

+
  open LeiosAbstract a public

+
  open Leios.Blocks a public

+

+
  field ⦃ IsBlock-Vote ⦄ : IsBlock (List Vote)

+
        ⦃ Hashable-PreIBHeader ⦄ : Hashable PreIBHeader Hash

+
        ⦃ Hashable-PreEndorserBlock ⦄ : Hashable PreEndorserBlock Hash

+
        id : PoolID

+
        FFD' : FFDAbstract.Functionality ffdAbstract

+
        vrf' : LeiosVRF a

+

+
  open LeiosVRF vrf' public

+

+
  field sk-IB sk-EB sk-VT : PrivKey

+
        pk-IB pk-EB pk-VT : PubKey

+

+
  open Leios.Base a vrf' public

+

+
  field B' : BaseAbstract

+
        BF : BaseAbstract.Functionality B'

+
        initBaseState : BaseAbstract.Functionality.State BF

+

+
  open Leios.KeyRegistration a vrf' public

+

+
  field K' : KeyRegistrationAbstract

+
        KF : KeyRegistrationAbstract.Functionality K'

+

+
  module B   = BaseAbstract.Functionality BF

+
  module K   = KeyRegistrationAbstract.Functionality KF

+
  module FFD = FFDAbstract.Functionality FFD'

+

+
  open Leios.Voting public

+

+
  field va : VotingAbstract (Fin rounds × EndorserBlock)

+
  open VotingAbstract va public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Traces.html b/site/static/agda_html/Leios.Traces.html
new file mode 100644
index 000000000..dac5f6c5a
--- /dev/null
+++ b/site/static/agda_html/Leios.Traces.html
@@ -0,0 +1,97 @@
+
+Leios.Traces
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude

+
open import Leios.SpecStructure

+

+
import Leios.Protocol

+

+
module Leios.Traces {n} (⋯ : SpecStructure n) {u : Type} (let open Leios.Protocol ⋯ u)

+
  (_-⟦_/_⟧⇀_ : Maybe LeiosState → LeiosInput → LeiosOutput → LeiosState → Type)

+
  where

+

+
_⇉_ : LeiosState → LeiosState → Type

+
s₁ ⇉ s₂ = Σ[ (i , o) ∈ LeiosInput × LeiosOutput ] (just s₁ -⟦ i / o ⟧⇀ s₂)

+

+
_⇉[_]_ : LeiosState → ℕ → LeiosState → Type

+
s₁ ⇉[ zero ] s₂ = s₁ ≡ s₂

+
s₁ ⇉[ suc m ] sₙ = Σ[ s₂ ∈ LeiosState ] (s₁ ⇉ s₂ × s₂ ⇉[ m ] sₙ)

+

+
_⇉⋆_ : LeiosState → LeiosState → Type

+
s₁ ⇉⋆ sₙ = Σ[ n ∈  ℕ ] (s₁ ⇉[ n ] sₙ)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.VRF.html b/site/static/agda_html/Leios.VRF.html
new file mode 100644
index 000000000..397f1b9c5
--- /dev/null
+++ b/site/static/agda_html/Leios.VRF.html
@@ -0,0 +1,147 @@
+
+Leios.VRF
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude

+
open import Leios.Abstract

+

+
module Leios.VRF (a : LeiosAbstract) (let open LeiosAbstract a) where

+

+
record VRF (Dom Range PubKey : Type) : Type₁ where

+
  field isKeyPair : PubKey → PrivKey → Type

+
        eval : PrivKey → Dom → Range × VrfPf

+
        verify : PubKey → Dom → Range → VrfPf → Type

+
        verify? : (pk : PubKey) → (d : Dom) → (r : Range) → (pf : VrfPf) → Dec (verify pk d r pf)

+

+
record LeiosVRF : Type₁ where

+
  field PubKey : Type

+
        poolID : PubKey → PoolID

+
        verifySig : PubKey → Sig → Type

+
        verifySig? : (pk : PubKey) → (sig : Sig) → Dec (verifySig pk sig)

+

+
        vrf : VRF ℕ ℕ PubKey

+

+
  open VRF vrf public

+

+
  -- transforming slot numbers into VRF seeds

+
  field genIBInput genEBInput genVInput genV1Input genV2Input : ℕ → ℕ

+

+
  canProduceIB : ℕ → PrivKey → ℕ → VrfPf → Type

+
  canProduceIB slot k stake π = let (val , pf) = eval k (genIBInput slot) in val < stake × pf ≡ π

+

+
  Dec-canProduceIB : ∀ {slot k stake} → (∃[ π ] canProduceIB slot k stake π) ⊎ (∀ π → ¬ canProduceIB slot k stake π)

+
  Dec-canProduceIB {slot} {k} {stake} with eval k (genIBInput slot)

+
  ... | (val , pf) = case ¿ val < stake ¿ of λ where

+
    (yes p) → inj₁ (pf , p , refl)

+
    (no ¬p) → inj₂ (λ π (h , _) → ¬p h)

+

+
  canProduceIBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type

+
  canProduceIBPub slot val k pf stake = verify k (genIBInput slot) val pf × val < stake

+

+
  canProduceEB : ℕ → PrivKey → ℕ → VrfPf → Type

+
  canProduceEB slot k stake π = let (val , pf) = eval k (genEBInput slot) in val < stake × pf ≡ π

+

+
  Dec-canProduceEB : ∀ {slot k stake} → (∃[ π ] canProduceEB slot k stake π) ⊎ (∀ π → ¬ canProduceEB slot k stake π)

+
  Dec-canProduceEB {slot} {k} {stake} with eval k (genEBInput slot)

+
  ... | (val , pf) = case ¿ val < stake ¿ of λ where

+
    (yes p) → inj₁ (pf , p , refl)

+
    (no ¬p) → inj₂ (λ π (h , _) → ¬p h)

+

+
  canProduceEBPub : ℕ → ℕ → PubKey → VrfPf → ℕ → Type

+
  canProduceEBPub slot val k pf stake = verify k (genEBInput slot) val pf × val < stake

+

+
  canProduceV : ℕ → PrivKey → ℕ → Type

+
  canProduceV slot k stake = proj₁ (eval k (genVInput slot)) < stake

+

+
  Dec-canProduceV : ∀ {slot k stake} → Dec (canProduceV slot k stake)

+
  Dec-canProduceV {slot} {k} {stake} with eval k (genVInput slot)

+
  ... | (val , pf) = ¿ val < stake ¿

+

+
  canProduceV1 : ℕ → PrivKey → ℕ → Type

+
  canProduceV1 slot k stake = proj₁ (eval k (genV1Input slot)) < stake

+

+
  Dec-canProduceV1 : ∀ {slot k stake} → Dec (canProduceV1 slot k stake)

+
  Dec-canProduceV1 {slot} {k} {stake} with eval k (genV1Input slot)

+
  ... | (val , pf) = ¿ val < stake ¿

+

+
  canProduceV2 : ℕ → PrivKey → ℕ → Type

+
  canProduceV2 slot k stake = proj₁ (eval k (genV2Input slot)) < stake

+

+
  Dec-canProduceV2 : ∀ {slot k stake} → Dec (canProduceV2 slot k stake)

+
  Dec-canProduceV2 {slot} {k} {stake} with eval k (genV2Input slot)

+
  ... | (val , pf) = ¿ val < stake ¿

+

\ No newline at end of file
diff --git a/site/static/agda_html/Leios.Voting.html b/site/static/agda_html/Leios.Voting.html
new file mode 100644
index 000000000..64fa33701
--- /dev/null
+++ b/site/static/agda_html/Leios.Voting.html
@@ -0,0 +1,89 @@
+
+Leios.Voting
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
open import Leios.Prelude

+

+
module Leios.Voting where

+

+
record VotingAbstract (X : Type) : Type₁ where

+
  field VotingState     : Type

+
        initVotingState : VotingState

+
        isVoteCertified : VotingState → X → Type

+

+
        ⦃ isVoteCertified⁇ ⦄ : ∀ {vs x} → isVoteCertified vs x ⁇

+

\ No newline at end of file
diff --git a/site/static/agda_html/Level.html b/site/static/agda_html/Level.html
new file mode 100644
index 000000000..3afcedc40
--- /dev/null
+++ b/site/static/agda_html/Level.html
@@ -0,0 +1,111 @@
+
+Level
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

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+                 

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+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Universe levels

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Level where

+

+
-- Levels.

+

+
open import Agda.Primitive as Prim public

+
  using    (Level; _⊔_; Setω)

+
  renaming (lzero to zero; lsuc to suc)

+

+
-- Lifting.

+

+
record Lift {a} ℓ (A : Set a) : Set (a ⊔ ℓ) where

+
  constructor lift

+
  field lower : A

+

+
open Lift public

+

+
-- Synonyms

+

+
0ℓ : Level

+
0ℓ = zero

+

+
levelOfType : ∀ {a} → Set a → Level

+
levelOfType {a} _ = a

+

+
levelOfTerm : ∀ {a} {A : Set a} → A → Level

+
levelOfTerm {a} _ = a

+

\ No newline at end of file
diff --git a/site/static/agda_html/Meta.Init.html b/site/static/agda_html/Meta.Init.html
new file mode 100644
index 000000000..f9a158291
--- /dev/null
+++ b/site/static/agda_html/Meta.Init.html
@@ -0,0 +1,92 @@
+
+Meta.Init
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
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+               

+             

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+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module Meta.Init where

+

+
open import Reflection.Debug public

+
open import Reflection.TCI public

+
open import Reflection.Syntax public

+
open import Reflection.AST.Term public

+
  using (vΠ[_∶_]_)

+

+
instance

+
  iMonad-TC = Monad-TC

+
  iMonadTC-TCI = MonadTC-TCI

+
  iMonadReader-TC = MonadReader-TC

+
  iMonadError-TC = MonadError-TC

+

\ No newline at end of file
diff --git a/site/static/agda_html/Meta.Prelude.html b/site/static/agda_html/Meta.Prelude.html
new file mode 100644
index 000000000..3ed5527e9
--- /dev/null
+++ b/site/static/agda_html/Meta.Prelude.html
@@ -0,0 +1,108 @@
+
+Meta.Prelude
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

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+                   
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+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module Meta.Prelude where

+

+
open import Level renaming (_⊔_ to _⊔ˡ_; suc to sucˡ; zero to zeroˡ) public

+
open import Function public

+

+
open import Data.Bool hiding (_≟_; _≤_; _≤?_; _<_; _<?_) public

+
open import Data.Empty public

+
open import Data.List hiding (align; alignWith; fromMaybe; map; zip; zipWith) public

+
open import Data.Maybe hiding (_>>=_; ap; align; alignWith; fromMaybe; map; zip; zipWith) public

+
open import Data.Unit hiding (_≟_) public

+
open import Data.Sum hiding (assocʳ; assocˡ; map; map₁; map₂; reduce; swap) public

+
open import Data.Product hiding (assocʳ; assocˡ; map; map₁; map₂; swap; zip; zipWith) public

+
open import Data.Nat hiding (_≟_; _≤_; _≤?_; _<_; _<?_; _≤ᵇ_; _≡ᵇ_) public

+
open import Data.String using (String; _<+>_) public

+

+
open import Relation.Binary.PropositionalEquality hiding (preorder; setoid; [_]; module ≡-Reasoning; decSetoid) public

+

+
lookupᵇ : {A B : Set} → (A → A → Bool) → List (A × B) → A → Maybe B

+
lookupᵇ f [] n = nothing

+
lookupᵇ f ((k , v) ∷ l) n = if f k n then just v else lookupᵇ f l n

+

+
open import Data.Fin

+
open import Data.List

+

+
zipWithIndex : {A B : Set} → (ℕ → A → B) → List A → List B

+
zipWithIndex f l = zipWith f (upTo $ length l) l

+

+
enumerate : {A : Set} → List A → List (ℕ × A)

+
enumerate = zipWithIndex _,_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Prelude.Closures.html b/site/static/agda_html/Prelude.Closures.html
new file mode 100644
index 000000000..14abd0cbe
--- /dev/null
+++ b/site/static/agda_html/Prelude.Closures.html
@@ -0,0 +1,99 @@
+
+Prelude.Closures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
open import Prelude.Init

+

+
module Prelude.Closures {A : Type ℓ} (_—→_ : Rel A ℓ) where

+

+
private variable x y z : A

+

+
-- left-biased

+
infix  3 _∎

+
infixr 2 _—→⟨_⟩_ _—↠⟨_⟩_

+
infix  1 begin_; pattern begin_ x = x

+
infix -1 _—↠_

+

+
data _—↠_ : Rel A ℓ where

+
  _∎ : ∀ x → x —↠ x

+
  _—→⟨_⟩_ : ∀ x → x —→ y → y —↠ z → x —↠ z

+

+
—↠-trans : Transitive _—↠_

+
—↠-trans (x ∎) xz = xz

+
—↠-trans (_ —→⟨ r ⟩ xy) yz = _ —→⟨ r ⟩ —↠-trans xy yz

+

+
_—↠⟨_⟩_ : ∀ x → x —↠ y → y —↠ z → x —↠ z

+
_—↠⟨_⟩_ _ = —↠-trans

+

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+Prelude.InferenceRules
+             
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+               

+                 Back
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+                 
+               

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+               

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+                 

+               

+             

+             

+               
+               

+             
module Prelude.InferenceRules where

+

+
open import Prelude.Init

+

+
-- syntax for inference rules (separating hypotheses from conclusion)

+
infix -50

+
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+
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+
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+
  _────────_

+
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+
  _──────_

+
  _─────_

+
  _────_

+
  _───_

+
  : Type ℓ → Type ℓ′ → Type (ℓ ⊔ₗ ℓ′)

+
A ────────────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A  ───────────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A   ──────────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A    ─────────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A     ────────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A      ───────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A       ──────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A        ─────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A         ────────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A          ───────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A           ──────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A            ─────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A             ────────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A              ───────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A               ──────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                ─────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                 ────────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                  ───────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                   ──────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                    ─────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                     ────────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                      ───────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                       ──────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                        ─────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                         ────────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                          ───────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                           ──────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                            ─────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                             ────────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                              ───────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                               ──────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                ─────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                 ────────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                  ───────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                   ──────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                    ─────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                     ────────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                      ───────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                       ──────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                        ─────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                         ────────────────────────────────────────────────────────────────────────── B = A → B

+
A                                          ───────────────────────────────────────────────────────────────────────── B = A → B

+
A                                           ──────────────────────────────────────────────────────────────────────── B = A → B

+
A                                            ─────────────────────────────────────────────────────────────────────── B = A → B

+
A                                             ────────────────────────────────────────────────────────────────────── B = A → B

+
A                                              ───────────────────────────────────────────────────────────────────── B = A → B

+
A                                               ──────────────────────────────────────────────────────────────────── B = A → B

+
A                                                ─────────────────────────────────────────────────────────────────── B = A → B

+
A                                                 ────────────────────────────────────────────────────────────────── B = A → B

+
A                                                  ───────────────────────────────────────────────────────────────── B = A → B

+
A                                                   ──────────────────────────────────────────────────────────────── B = A → B

+
A                                                    ─────────────────────────────────────────────────────────────── B = A → B

+
A                                                     ────────────────────────────────────────────────────────────── B = A → B

+
A                                                      ───────────────────────────────────────────────────────────── B = A → B

+
A                                                       ──────────────────────────────────────────────────────────── B = A → B

+
A                                                        ─────────────────────────────────────────────────────────── B = A → B

+
A                                                         ────────────────────────────────────────────────────────── B = A → B

+
A                                                          ───────────────────────────────────────────────────────── B = A → B

+
A                                                           ──────────────────────────────────────────────────────── B = A → B

+
A                                                            ─────────────────────────────────────────────────────── B = A → B

+
A                                                             ────────────────────────────────────────────────────── B = A → B

+
A                                                              ───────────────────────────────────────────────────── B = A → B

+
A                                                               ──────────────────────────────────────────────────── B = A → B

+
A                                                                ─────────────────────────────────────────────────── B = A → B

+
A                                                                 ────────────────────────────────────────────────── B = A → B

+
A                                                                  ───────────────────────────────────────────────── B = A → B

+
A                                                                   ──────────────────────────────────────────────── B = A → B

+
A                                                                    ─────────────────────────────────────────────── B = A → B

+
A                                                                     ────────────────────────────────────────────── B = A → B

+
A                                                                      ───────────────────────────────────────────── B = A → B

+
A                                                                       ──────────────────────────────────────────── B = A → B

+
A                                                                        ─────────────────────────────────────────── B = A → B

+
A                                                                         ────────────────────────────────────────── B = A → B

+
A                                                                          ───────────────────────────────────────── B = A → B

+
A                                                                           ──────────────────────────────────────── B = A → B

+
A                                                                            ─────────────────────────────────────── B = A → B

+
A                                                                             ────────────────────────────────────── B = A → B

+
A                                                                              ───────────────────────────────────── B = A → B

+
A                                                                               ──────────────────────────────────── B = A → B

+
A                                                                                ─────────────────────────────────── B = A → B

+
A                                                                                 ────────────────────────────────── B = A → B

+
A                                                                                  ───────────────────────────────── B = A → B

+
A                                                                                   ──────────────────────────────── B = A → B

+
A                                                                                    ─────────────────────────────── B = A → B

+
A                                                                                     ────────────────────────────── B = A → B

+
A                                                                                      ───────────────────────────── B = A → B

+
A                                                                                       ──────────────────────────── B = A → B

+
A                                                                                        ─────────────────────────── B = A → B

+
A                                                                                         ────────────────────────── B = A → B

+
A                                                                                          ───────────────────────── B = A → B

+
A                                                                                           ──────────────────────── B = A → B

+
A                                                                                            ─────────────────────── B = A → B

+
A                                                                                             ────────────────────── B = A → B

+
A                                                                                              ───────────────────── B = A → B

+
A                                                                                               ──────────────────── B = A → B

+
A                                                                                                ─────────────────── B = A → B

+
A                                                                                                 ────────────────── B = A → B

+
A                                                                                                  ───────────────── B = A → B

+
A                                                                                                   ──────────────── B = A → B

+
A                                                                                                    ─────────────── B = A → B

+
A                                                                                                     ────────────── B = A → B

+
A                                                                                                      ───────────── B = A → B

+
A                                                                                                       ──────────── B = A → B

+
A                                                                                                        ─────────── B = A → B

+
A                                                                                                         ────────── B = A → B

+
A                                                                                                          ───────── B = A → B

+
A                                                                                                           ──────── B = A → B

+
A                                                                                                            ─────── B = A → B

+
A                                                                                                             ────── B = A → B

+
A                                                                                                              ───── B = A → B

+
A                                                                                                               ──── B = A → B

+
A                                                                                                                ─── B = A → B

+

+
infix -100

+
  ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
  ─────────────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
  ────────────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
  ───────────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
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+
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+
  ────────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
  ───────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
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+
  ─────────────────────────────────────────────────────────────────────────────────────────────────────────_

+
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+
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+
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+
  ─────────────────────────────────────────────────────────────────────────────────────────────────────_

+
  ────────────────────────────────────────────────────────────────────────────────────────────────────_

+
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+
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+
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+
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+
  ───────────────────────────────────────────────────────────────────────────────────────────────_

+
  ──────────────────────────────────────────────────────────────────────────────────────────────_

+
  ─────────────────────────────────────────────────────────────────────────────────────────────_

+
  ────────────────────────────────────────────────────────────────────────────────────────────_

+
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+
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+
  ─────────────────────────────────────────────────────────────────────────────────────────_

+
  ────────────────────────────────────────────────────────────────────────────────────────_

+
  ───────────────────────────────────────────────────────────────────────────────────────_

+
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+
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+
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+
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+
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+
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+
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+
  ───────────────────────────────────────────────────────────────────────────────_

+
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+
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+
  ────────────────────────────────────────────────────────────────────────────_

+
  ───────────────────────────────────────────────────────────────────────────_

+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
                                                                                                              ────_

+
                                                                                                               ───_

+
  : Type ℓ → Type ℓ

+
────────────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────────── A = A

+
────────────────────────────────────────────────── A = A

+
───────────────────────────────────────────────── A = A

+
──────────────────────────────────────────────── A = A

+
─────────────────────────────────────────────── A = A

+
────────────────────────────────────────────── A = A

+
───────────────────────────────────────────── A = A

+
──────────────────────────────────────────── A = A

+
─────────────────────────────────────────── A = A

+
────────────────────────────────────────── A = A

+
───────────────────────────────────────── A = A

+
──────────────────────────────────────── A = A

+
─────────────────────────────────────── A = A

+
────────────────────────────────────── A = A

+
───────────────────────────────────── A = A

+
──────────────────────────────────── A = A

+
─────────────────────────────────── A = A

+
────────────────────────────────── A = A

+
───────────────────────────────── A = A

+
──────────────────────────────── A = A

+
─────────────────────────────── A = A

+
────────────────────────────── A = A

+
───────────────────────────── A = A

+
──────────────────────────── A = A

+
─────────────────────────── A = A

+
────────────────────────── A = A

+
───────────────────────── A = A

+
──────────────────────── A = A

+
─────────────────────── A = A

+
────────────────────── A = A

+
───────────────────── A = A

+
──────────────────── A = A

+
─────────────────── A = A

+
────────────────── A = A

+
───────────────── A = A

+
──────────────── A = A

+
─────────────── A = A

+
────────────── A = A

+
───────────── A = A

+
──────────── A = A

+
─────────── A = A

+
────────── A = A

+
───────── A = A

+
──────── A = A

+
─────── A = A

+
────── A = A

+
───── A = A

+
──── A = A

+
─── A = A

+

+
infixr -100 _∙_

+
_∙_ : Type ℓ → Type ℓ′ → Type (ℓ ⊔ₗ ℓ′)

+
A ∙ B = A → B

+

+
infix -150 ∙_

+
∙_ : Type ℓ → Type ℓ

+
∙ A = A

+

+
-- ** equivalences

+

+
-- syntax for inference rules (separating hypotheses from conclusion)

+
infix -50

+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
  _═══════════════════════════════════════════════_

+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
  _═══════════════════════════════════_

+
  _══════════════════════════════════_

+
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+
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+
  _═══════════════════════════════_

+
  _══════════════════════════════_

+
  _═════════════════════════════_

+
  _════════════════════════════_

+
  _═══════════════════════════_

+
  _══════════════════════════_

+
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+
  _════════════════════════_

+
  _═══════════════════════_

+
  _══════════════════════_

+
  _═════════════════════_

+
  _════════════════════_

+
  _═══════════════════_

+
  _══════════════════_

+
  _═════════════════_

+
  _════════════════_

+
  _═══════════════_

+
  _══════════════_

+
  _═════════════_

+
  _════════════_

+
  _═══════════_

+
  _══════════_

+
  _═════════_

+
  _════════_

+
  _═══════_

+
  _══════_

+
  _═════_

+
  _════_

+
  _═══_

+
  : Type ℓ → Type ℓ′ → Type (ℓ ⊔ₗ ℓ′)

+
A ══════════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A  ═════════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A   ════════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A    ═══════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A     ══════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A      ═════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A       ════════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A        ═══════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A         ══════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A          ═════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A           ════════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A            ═══════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A             ══════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A              ═════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A               ════════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                ═══════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                 ══════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                  ═════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                   ════════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                    ═══════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                     ══════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                      ═════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                       ════════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                        ═══════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                         ══════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                          ═════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                           ════════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                            ═══════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                             ══════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                              ═════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                               ════════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                ═══════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                 ══════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                  ═════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                   ════════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                    ═══════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                     ══════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                      ═════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                       ════════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                        ═══════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                         ══════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                          ═════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                           ════════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                            ═══════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                             ══════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                              ═════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                               ════════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                ═══════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                 ══════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                  ═════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                   ════════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                    ═══════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                     ══════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                      ═════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                       ════════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                        ═══════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                         ══════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                          ═════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                           ════════════════════════════════════════════════════════ B = A ↔ B

+
A                                                            ═══════════════════════════════════════════════════════ B = A ↔ B

+
A                                                             ══════════════════════════════════════════════════════ B = A ↔ B

+
A                                                              ═════════════════════════════════════════════════════ B = A ↔ B

+
A                                                               ════════════════════════════════════════════════════ B = A ↔ B

+
A                                                                ═══════════════════════════════════════════════════ B = A ↔ B

+
A                                                                 ══════════════════════════════════════════════════ B = A ↔ B

+
A                                                                  ═════════════════════════════════════════════════ B = A ↔ B

+
A                                                                   ════════════════════════════════════════════════ B = A ↔ B

+
A                                                                    ═══════════════════════════════════════════════ B = A ↔ B

+
A                                                                     ══════════════════════════════════════════════ B = A ↔ B

+
A                                                                      ═════════════════════════════════════════════ B = A ↔ B

+
A                                                                       ════════════════════════════════════════════ B = A ↔ B

+
A                                                                        ═══════════════════════════════════════════ B = A ↔ B

+
A                                                                         ══════════════════════════════════════════ B = A ↔ B

+
A                                                                          ═════════════════════════════════════════ B = A ↔ B

+
A                                                                           ════════════════════════════════════════ B = A ↔ B

+
A                                                                            ═══════════════════════════════════════ B = A ↔ B

+
A                                                                             ══════════════════════════════════════ B = A ↔ B

+
A                                                                              ═════════════════════════════════════ B = A ↔ B

+
A                                                                               ════════════════════════════════════ B = A ↔ B

+
A                                                                                ═══════════════════════════════════ B = A ↔ B

+
A                                                                                 ══════════════════════════════════ B = A ↔ B

+
A                                                                                  ═════════════════════════════════ B = A ↔ B

+
A                                                                                   ════════════════════════════════ B = A ↔ B

+
A                                                                                    ═══════════════════════════════ B = A ↔ B

+
A                                                                                     ══════════════════════════════ B = A ↔ B

+
A                                                                                      ═════════════════════════════ B = A ↔ B

+
A                                                                                       ════════════════════════════ B = A ↔ B

+
A                                                                                        ═══════════════════════════ B = A ↔ B

+
A                                                                                         ══════════════════════════ B = A ↔ B

+
A                                                                                          ═════════════════════════ B = A ↔ B

+
A                                                                                           ════════════════════════ B = A ↔ B

+
A                                                                                            ═══════════════════════ B = A ↔ B

+
A                                                                                             ══════════════════════ B = A ↔ B

+
A                                                                                              ═════════════════════ B = A ↔ B

+
A                                                                                               ════════════════════ B = A ↔ B

+
A                                                                                                ═══════════════════ B = A ↔ B

+
A                                                                                                 ══════════════════ B = A ↔ B

+
A                                                                                                  ═════════════════ B = A ↔ B

+
A                                                                                                   ════════════════ B = A ↔ B

+
A                                                                                                    ═══════════════ B = A ↔ B

+
A                                                                                                     ══════════════ B = A ↔ B

+
A                                                                                                      ═════════════ B = A ↔ B

+
A                                                                                                       ════════════ B = A ↔ B

+
A                                                                                                        ═══════════ B = A ↔ B

+
A                                                                                                         ══════════ B = A ↔ B

+
A                                                                                                          ═════════ B = A ↔ B

+
A                                                                                                           ════════ B = A ↔ B

+
A                                                                                                            ═══════ B = A ↔ B

+
A                                                                                                             ══════ B = A ↔ B

+
A                                                                                                              ═════ B = A ↔ B

+
A                                                                                                               ════ B = A ↔ B

+
A                                                                                                                ═══ B = A ↔ B

+

+

+
private

+
  _ : Type

+
  _ = ∀ {l : Level} →

+
    ─────────────────

+
    l ≡ 0ℓ

+

+
  _ : Type

+
  _ = ∀ {l : Level} →

+
    ∙ l ≢ 1ℓ

+
    ∙ l ≢ 2ℓ

+
    ∙ l ≢ 3ℓ

+
      ──────

+
      l ≡ 0ℓ

+

+
  _ : Type

+
  _ = ∀ {l : Level} →

+
    -- [LIMITATION] cannot have dependent function space, aka Π-types

+
    -- ∙ (∃x : ∃ (_> l))

+
    -- ∙ (∃y : ∃ (_≤ l))

+
    ∀ (∃x : ∃ (_≡ l)) →

+
    ∙ l ≢ 1ℓ

+
    → (∃y : ∃ (_≢ l))

+
    → l ≢ 2ℓ

+
    ∙ l ≢ 3ℓ

+
    → let x , _ = ∃x; y , _ = ∃y in

+
    ∙ x ≢ y

+
      ─────────────────────────────

+
      l ≡ 0ℓ

+

+
  -- making sure precedence works as expected

+
  data X : Level → Type where

+
    ruleX :

+
      ∙ ℓ ≢ 1ℓ

+
      ∙ ℓ ≢ 2ℓ

+
        ───────

+
        X ℓ

+
  data Y : Level → Type where

+
    ruleY :

+
        ℓ ≢ 1ℓ

+
      → ℓ ≢ 2ℓ

+
        --—————

+
      → Y ℓ

+

+
  X⇔Y :

+
    X ℓ

+
    ═══

+
    Y ℓ

+
  X⇔Y = ↝ , ↜

+
    where ↝ = λ where (ruleX p q) → ruleY p q

+
          ↜ = λ where (ruleY p q) → ruleX p q

+

\ No newline at end of file
diff --git a/site/static/agda_html/Prelude.Init.html b/site/static/agda_html/Prelude.Init.html
new file mode 100644
index 000000000..34d003a5f
--- /dev/null
+++ b/site/static/agda_html/Prelude.Init.html
@@ -0,0 +1,253 @@
+
+Prelude.Init
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Prelude.Init where

+

+
open import Agda.Primitive public

+
  using ()

+
  renaming (Set to Type; Setω to Typeω)

+
open import Level public

+
  using (Level; 0ℓ; Setω)

+
  renaming (suc to lsuc; _⊔_ to _⊔ₗ_)

+
1ℓ = lsuc 0ℓ; 2ℓ = lsuc 1ℓ; 3ℓ = lsuc 2ℓ; 4ℓ = lsuc 3ℓ

+
variable ℓ ℓ′ ℓ″ ℓ‴ ℓ₀ ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level

+

+
open import Function public

+
  using (_∋_; _∘_; _∘′_; _∘₂_; _$_; flip; it; id; _on_; const; case_of_)

+
open import Data.Empty public

+
  using (⊥; ⊥-elim)

+
open import Data.Unit public

+
  using (⊤; tt)

+
open import Data.Bool public

+
  using (Bool; true; false; T; not; _∧_; _∨_; if_then_else_)

+
open import Data.Product public

+
  using ( _×_; _,_; proj₁; proj₂; Σ; Σ-syntax; ∃; ∃-syntax; -,_

+
        ; map₁; map₂; curry; uncurry )

+
open import Data.Sum public

+
  using (_⊎_; inj₁; inj₂)

+
module Bin where

+
  open import Data.Nat.Binary public

+
  open import Data.Nat.Binary.Properties public

+
open Bin public

+
  using () renaming (ℕᵇ to Bitstring)

+

+
module Nat where

+
  open import Data.Nat public

+
  open import Data.Nat.Properties public

+
  open import Data.Nat.DivMod public

+
  open import Data.Nat.Induction public

+
open Nat public

+
  using (ℕ; suc; _≤_; _<_; _<ᵇ_; _≥_; _>_; _+_; _∸_; _*_; _/_)

+

+
module Fi where

+
  open import Data.Fin public

+
  open import Data.Fin.Properties public

+
open Fi public

+
  using (Fin; fromℕ<; pred)

+
  renaming (zero to fzero; suc to fsuc)

+

+
module Fl where

+
  open import Data.Float public

+
  open import Data.Float.Properties public

+
open Fl public

+
  using (Float)

+

+
module Int where

+
  open import Data.Integer public

+
  open import Data.Integer.Properties public

+
open Int public

+
  using (ℤ)

+

+
module M where

+
  open import Data.Maybe public

+
  open import Data.Maybe.Properties public

+
  module Any where

+
    open import Data.Maybe.Relation.Unary.Any public

+
  module All where

+
    open import Data.Maybe.Relation.Unary.All public

+
    open import Data.Maybe.Relation.Unary.All.Properties public

+
open import Data.Maybe public

+
  using ( Maybe; just; nothing; maybe; fromMaybe; is-just; is-nothing

+
        ; decToMaybe; boolToMaybe )

+
  renaming (map to mapMaybe)

+

+
module V where

+
  open import Data.Vec public

+
  open import Data.Vec.Properties public

+
  module All where

+
    open import Data.Vec.Relation.Unary.All public

+
    open import Data.Vec.Relation.Unary.All.Properties public

+
  module Any where

+
    open import Data.Vec.Relation.Unary.Any public

+
    open import Data.Vec.Relation.Unary.Any.Properties public

+
open V public

+
  using (Vec; []; _∷_)

+

+
module L where

+
  open import Data.List public

+
  open import Data.List.Properties public

+
  module All where

+
    open import Data.List.Relation.Unary.All public

+
    open import Data.List.Relation.Unary.All.Properties public

+
  module Any where

+
    open import Data.List.Relation.Unary.Any public

+
    open import Data.List.Relation.Unary.Any.Properties public

+
  module Mem where

+
    open import Data.List.Membership.Propositional public

+
    open import Data.List.Membership.Propositional.Properties public

+
  module First where

+
    open import Data.List.Relation.Unary.First public

+
    open import Data.List.Relation.Unary.First.Properties public

+
  module NE where

+
    open import Data.List.NonEmpty public

+
    open import Data.List.NonEmpty.Properties public

+
  module SubL where

+
    open import Data.List.Relation.Binary.Sublist.Propositional public

+
    open import Data.List.Relation.Binary.Sublist.Propositional.Properties public

+
  module SubS where

+
    open import Data.List.Relation.Binary.Subset.Propositional public

+
    open import Data.List.Relation.Binary.Subset.Propositional.Properties public

+
  module Unique where

+
    open import Data.List.Relation.Unary.Unique.Propositional public

+
    open import Data.List.Relation.Unary.Unique.Propositional.Properties public

+
open L public

+
  using (List; []; _∷_; _++_; map; catMaybes; head; filter; length; sum; and; or

+
        ; any; _[_]∷=_)

+
open L.NE public

+
  using (List⁺; _∷_; _∷⁺_)

+
open L.All public

+
  using ([]; _∷_)

+
open L.Any public

+
  using (here; there; _─_)

+
open L.First public

+
  using (First; _∷_)

+
open L.Mem public

+
  using (_∈_; _∉_)

+
open L.SubL public

+
  using (_⊆_; []; _∷ʳ_; _∷_)

+
open L.SubS public

+
  using ()

+
  renaming (_⊆_ to _⊆ˢ_)

+
open L.Unique public

+
  using (Unique)

+

+
open import Relation.Nullary public

+
  using (¬_; Dec; yes; no; Irrelevant)

+
open import Relation.Nullary.Decidable public

+
  using ( ⌊_⌋; True; False; toWitness; toWitnessFalse; fromWitness

+
        ; dec-yes; dec-no; dec-true; dec-false ; ¬?; _⊎-dec_; _×-dec_)

+
  renaming (map′ to mapDec)

+
open import Relation.Nullary.Negation public

+
  using (contradiction; contraposition)

+
open import Relation.Unary public

+
  using (Pred)

+
  renaming (Decidable to Decidable¹; Irrelevant to Irrelevant¹)

+
open import Relation.Binary public

+
  using (Rel; REL; DecidableEquality; Transitive)

+
  renaming (Decidable to Decidable²; Irrelevant to Irrelevant²)

+
open import Relation.Binary.PropositionalEquality public

+
  using (_≡_; _≢_; refl; cong; subst; sym; module ≡-Reasoning)

+

+
open import Algebra.Core public

+
  using (Op₁; Op₂)

+

+
-- ** extras

+

+
-- uniquely exists

+
∃! : {A : Type ℓ} → Pred A ℓ′ → Type _

+
∃! = P.∃! _≡_

+
  where import Data.Product as P

+

+
∃₂-syntax : ∀ {a b c} {A : Set a} {B : Set b}

+
     (C : A → B → Set c) → Set _

+
∃₂-syntax C = ∃ λ a → ∃ λ b → C a b

+

+
infix 2 ∃₂-syntax

+
syntax ∃₂-syntax (λ x y → C) = ∃₂[ x , y ] C

+

+
-- iff

+
_↔_ : Type ℓ → Type ℓ′ → Type (ℓ ⊔ₗ ℓ′)

+
A ↔ B = (A → B) × (B → A)

+

+
-- singleton types

+
data Is {A : Type ℓ} : A → Type ℓ where

+
  ⟫_ : (x : A) → Is x

+
infix -100 ⟫_

+

+
-- get the type of a given term

+
typeOf : ∀ {A : Type ℓ} → (a : A) → Type ℓ

+
typeOf {A = A} _ = A

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Abstraction.html b/site/static/agda_html/Reflection.AST.Abstraction.html
new file mode 100644
index 000000000..f1de40f13
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Abstraction.html
@@ -0,0 +1,142 @@
+
+Reflection.AST.Abstraction
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Abstractions used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Abstraction where

+

+
open import Data.String.Base as String                 using (String)

+
open import Data.String.Properties as String           using (_≟_)

+
open import Data.Product.Base                          using (_×_; <_,_>; uncurry)

+
open import Level using (Level)

+
open import Relation.Nullary.Decidable.Core            using (Dec; map′; _×-dec_)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+
    s t : String

+
    x y : A

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public

+
  using (Abs)

+

+
open Abs public

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
map : (A → B) → Abs A → Abs B

+
map f (abs s x) = abs s (f x)

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
abs-injective₁ : abs s x ≡ abs t y → s ≡ t

+
abs-injective₁ refl = refl

+

+
abs-injective₂ : abs s x ≡ abs t y → x ≡ y

+
abs-injective₂ refl = refl

+

+
abs-injective : abs s x ≡ abs t y → s ≡ t × x ≡ y

+
abs-injective = < abs-injective₁ , abs-injective₂ >

+

+
-- We often need decidability of equality for Abs A when implementing it

+
-- for A. Unfortunately ≡-dec makes the termination checker unhappy.

+
-- Instead, we can match on both Abs A and use unAbs-dec for an

+
-- obviously decreasing recursive call.

+

+
unAbs : Abs A → A

+
unAbs (abs s a) = a

+

+
unAbs-dec : {abs1 abs2 : Abs A} → Dec (unAbs abs1 ≡ unAbs abs2) → Dec (abs1 ≡ abs2)

+
unAbs-dec {abs1 = abs s a} {abs t a′} a≟a′ =

+
  map′ (uncurry (cong₂ abs)) abs-injective (s String.≟ t ×-dec a≟a′)

+

+
≡-dec : DecidableEquality A → DecidableEquality (Abs A)

+
≡-dec _≟_ x y = unAbs-dec (unAbs x ≟ unAbs y)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Argument.Information.html b/site/static/agda_html/Reflection.AST.Argument.Information.html
new file mode 100644
index 000000000..86b4a8a13
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Argument.Information.html
@@ -0,0 +1,135 @@
+
+Reflection.AST.Argument.Information
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Argument information used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Argument.Information where

+

+
open import Data.Product.Base                          using (_×_; <_,_>; uncurry)

+
open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)

+

+
open import Reflection.AST.Argument.Modality as Modality using (Modality)

+
open import Reflection.AST.Argument.Visibility as Visibility using (Visibility)

+

+
private

+
  variable

+
    v v′ : Visibility

+
    m m′ : Modality

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public using (ArgInfo)

+
open ArgInfo public

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
visibility : ArgInfo → Visibility

+
visibility (arg-info v _) = v

+

+
modality : ArgInfo → Modality

+
modality (arg-info _ m) = m

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
arg-info-injective₁ : arg-info v m ≡ arg-info v′ m′ → v ≡ v′

+
arg-info-injective₁ refl = refl

+

+
arg-info-injective₂ : arg-info v m ≡ arg-info v′ m′ → m ≡ m′

+
arg-info-injective₂ refl = refl

+

+
arg-info-injective : arg-info v m ≡ arg-info v′ m′ → v ≡ v′ × m ≡ m′

+
arg-info-injective = < arg-info-injective₁ , arg-info-injective₂ >

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality ArgInfo

+
arg-info v m ≟ arg-info v′ m′ =

+
  map′

+
    (uncurry (cong₂ arg-info))

+
    arg-info-injective

+
    (v Visibility.≟ v′ ×-dec m Modality.≟ m′)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Argument.Modality.html b/site/static/agda_html/Reflection.AST.Argument.Modality.html
new file mode 100644
index 000000000..aa3c7fef6
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Argument.Modality.html
@@ -0,0 +1,135 @@
+
+Reflection.AST.Argument.Modality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Modalities used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Argument.Modality where

+

+
open import Data.Product.Base                          using (_×_; <_,_>; uncurry)

+
open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)

+

+
open import Reflection.AST.Argument.Relevance as Relevance using (Relevance)

+
open import Reflection.AST.Argument.Quantity as Quantity using (Quantity)

+

+
private

+
  variable

+
    r r′ : Relevance

+
    q q′ : Quantity

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public using (Modality)

+
open Modality public

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
relevance : Modality → Relevance

+
relevance (modality r _) = r

+

+
quantity : Modality → Quantity

+
quantity (modality _ q) = q

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
modality-injective₁ : modality r q ≡ modality r′ q′ → r ≡ r′

+
modality-injective₁ refl = refl

+

+
modality-injective₂ : modality r q ≡ modality r′ q′ → q ≡ q′

+
modality-injective₂ refl = refl

+

+
modality-injective : modality r q ≡ modality r′ q′ → r ≡ r′ × q ≡ q′

+
modality-injective = < modality-injective₁ , modality-injective₂ >

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Modality

+
modality r q ≟ modality r′ q′ =

+
  map′

+
    (uncurry (cong₂ modality))

+
    modality-injective

+
    (r Relevance.≟ r′ ×-dec q Quantity.≟ q′)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Argument.Quantity.html b/site/static/agda_html/Reflection.AST.Argument.Quantity.html
new file mode 100644
index 000000000..e66221f54
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Argument.Quantity.html
@@ -0,0 +1,107 @@
+
+Reflection.AST.Argument.Quantity
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Argument quantities used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Argument.Quantity where

+

+
open import Relation.Nullary.Decidable.Core           using (yes; no)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (refl)

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public using (Quantity)

+
open Quantity public

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Quantity

+
quantity-ω ≟ quantity-ω = yes refl

+
quantity-0 ≟ quantity-0 = yes refl

+
quantity-ω ≟ quantity-0 = no λ()

+
quantity-0 ≟ quantity-ω = no λ()

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Argument.Relevance.html b/site/static/agda_html/Reflection.AST.Argument.Relevance.html
new file mode 100644
index 000000000..fc8081e40
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Argument.Relevance.html
@@ -0,0 +1,107 @@
+
+Reflection.AST.Argument.Relevance
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Argument relevance used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Argument.Relevance where

+

+
open import Relation.Nullary.Decidable.Core            using (yes; no)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (refl)

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public using (Relevance)

+
open Relevance public

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Relevance

+
relevant   ≟ relevant   = yes refl

+
irrelevant ≟ irrelevant = yes refl

+
relevant   ≟ irrelevant = no λ()

+
irrelevant ≟ relevant   = no λ()

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Argument.Visibility.html b/site/static/agda_html/Reflection.AST.Argument.Visibility.html
new file mode 100644
index 000000000..656ade829
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Argument.Visibility.html
@@ -0,0 +1,112 @@
+
+Reflection.AST.Argument.Visibility
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Argument visibility used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Argument.Visibility where

+

+
open import Relation.Nullary.Decidable.Core            using (yes; no)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (refl)

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public using (Visibility)

+
open Visibility public

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Visibility

+
visible   ≟ visible   = yes refl

+
hidden    ≟ hidden    = yes refl

+
instance′ ≟ instance′ = yes refl

+
visible   ≟ hidden    = no λ()

+
visible   ≟ instance′ = no λ()

+
hidden    ≟ visible   = no λ()

+
hidden    ≟ instance′ = no λ()

+
instance′ ≟ visible   = no λ()

+
instance′ ≟ hidden    = no λ()

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Argument.html b/site/static/agda_html/Reflection.AST.Argument.html
new file mode 100644
index 000000000..943ea4fbd
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Argument.html
@@ -0,0 +1,170 @@
+
+Reflection.AST.Argument
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Arguments used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Argument where

+

+
open import Data.List.Base as List                     using (List; []; _∷_)

+
open import Data.Product.Base                          using (_×_; <_,_>; uncurry)

+
open import Relation.Nullary.Decidable.Core            using (Dec; map′; _×-dec_)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)

+
open import Level

+

+
open import Reflection.AST.Argument.Visibility

+
open import Reflection.AST.Argument.Relevance

+
open import Reflection.AST.Argument.Quantity

+
open import Reflection.AST.Argument.Modality

+
open import Reflection.AST.Argument.Information as Information

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+
    i j : ArgInfo

+
    x y : A

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtins publicly

+

+
open import Agda.Builtin.Reflection public using (Arg)

+
open Arg public

+

+
-- Pattern synonyms

+

+
pattern defaultModality = modality relevant quantity-ω

+

+
pattern vArg ty = arg (arg-info visible   defaultModality) ty

+
pattern hArg ty = arg (arg-info hidden    defaultModality) ty

+
pattern iArg ty = arg (arg-info instance′ defaultModality) ty

+

+
------------------------------------------------------------------------

+
-- Lists of arguments

+

+
Args : (A : Set a) → Set a

+
Args A = List (Arg A)

+

+
-- Pattern for appending a visible argument

+
infixr 5 _⟨∷⟩_

+
pattern _⟨∷⟩_ x args = vArg x ∷ args

+

+
-- Pattern for appending a hidden argument

+
infixr 5 _⟅∷⟆_

+
pattern _⟅∷⟆_ x args = hArg x ∷ args

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
map : (A → B) → Arg A → Arg B

+
map f (arg i x) = arg i (f x)

+

+
map-Args : (A → B) → Args A → Args B

+
map-Args f xs = List.map (map f) xs

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
arg-injective₁ : arg i x ≡ arg j y → i ≡ j

+
arg-injective₁ refl = refl

+

+
arg-injective₂ : arg i x ≡ arg j y → x ≡ y

+
arg-injective₂ refl = refl

+

+
arg-injective : arg i x ≡ arg j y → i ≡ j × x ≡ y

+
arg-injective = < arg-injective₁ , arg-injective₂ >

+

+
-- We often need decidability of equality for Arg A when implementing it

+
-- for A. Unfortunately ≡-dec makes the termination checker unhappy.

+
-- Instead, we can match on both Arg A and use unArg-dec for an

+
-- obviously decreasing recursive call.

+

+
unArg : Arg A → A

+
unArg (arg i a) = a

+

+
unArg-dec : {arg1 arg2 : Arg A} → Dec (unArg arg1 ≡ unArg arg2) → Dec (arg1 ≡ arg2)

+
unArg-dec {arg1 = arg i x} {arg j y} arg1≟arg2 =

+
  map′ (uncurry (cong₂ arg)) arg-injective (i Information.≟ j ×-dec arg1≟arg2)

+

+
≡-dec : DecidableEquality A → DecidableEquality (Arg A)

+
≡-dec _≟_ x y = unArg-dec (unArg x ≟ unArg y)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.DeBruijn.html b/site/static/agda_html/Reflection.AST.DeBruijn.html
new file mode 100644
index 000000000..d8f70f3a5
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.DeBruijn.html
@@ -0,0 +1,211 @@
+
+Reflection.AST.DeBruijn
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Weakening, strengthening and free variable check for reflected terms.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.DeBruijn where

+

+
open import Data.Bool.Base  using (Bool; true; false; _∨_; if_then_else_)

+
open import Data.Nat.Base   using (ℕ; zero; suc; _+_; _∸_; _<ᵇ_; _≡ᵇ_)

+
open import Data.List.Base  using (List; []; _∷_; _++_)

+
open import Data.Maybe.Base using (Maybe; nothing; just)

+
import Data.Maybe.Effectful as Maybe

+
import Function.Identity.Effectful as Identity

+
open import Effect.Applicative using (RawApplicative; mkRawApplicative)

+

+
open import Reflection

+
open import Reflection.AST.Argument.Visibility using (Visibility)

+
import Reflection.AST.Traversal as Traverse

+

+
private

+
  variable A : Set

+

+
------------------------------------------------------------------------

+
-- Weakening

+

+
module _ where

+

+
  open Traverse Identity.applicative

+

+
  private

+

+
    wkVar : ℕ → Cxt → ℕ → ℕ

+
    wkVar by (from , _) i = if i <ᵇ from then i else i + by

+

+
    actions : ℕ → Actions

+
    actions k = record defaultActions { onVar = wkVar k }

+

+
  weakenFrom′ : (Actions → Cxt → A → A) → (from by : ℕ) → A → A

+
  weakenFrom′ trav from by = trav (actions by) (from , []) -- not using the context part

+

+
  weakenFrom : (from by : ℕ) → Term → Term

+
  weakenFrom = weakenFrom′ traverseTerm

+

+
  weaken : (by : ℕ) → Term → Term

+
  weaken = weakenFrom 0

+

+
  weakenArgs : (by : ℕ) → Args Term → Args Term

+
  weakenArgs = weakenFrom′ traverseArgs 0

+

+
  weakenClauses : (by : ℕ) → Clauses → Clauses

+
  weakenClauses = weakenFrom′ traverseClauses 0

+

+

+
------------------------------------------------------------------------

+
-- η-expansion

+

+
private

+
  η : Visibility → (Args Term → Term) → Args Term → Term

+
  η h f args =

+
    lam h (abs "x" (f (weakenArgs 1 args ++

+
                       arg (arg-info h defaultModality) (var 0 []) ∷

+
                       [])))

+

+
η-expand : Visibility → Term → Term

+
η-expand h (var x      args) = η h (var (suc x)) args

+
η-expand h (con c      args) = η h (con c) args

+
η-expand h (def f      args) = η h (def f) args

+
η-expand h (pat-lam cs args) = η h (pat-lam (weakenClauses 1 cs)) args

+
η-expand h (meta x     args) = η h (meta x) args

+
η-expand h t@(lam _ _)       = t

+
η-expand h t@(pi _ _)        = t

+
η-expand h t@(agda-sort _)   = t

+
η-expand h t@(lit _)         = t

+
η-expand h t@unknown         = t

+

+

+
------------------------------------------------------------------------

+
-- Strengthening

+

+
module _ where

+

+
  open Traverse Maybe.applicative

+

+
  private

+
    strVar : ℕ → Cxt → ℕ → Maybe ℕ

+
    strVar by (from , Γ) i with i <ᵇ from | i <ᵇ from + by

+
    ... | true | _    = just i

+
    ... | _    | true = nothing

+
    ... | _    | _    = just (i ∸ by)

+

+
    actions : ℕ → Actions

+
    actions by = record defaultActions { onVar = strVar by }

+

+
  strengthenFromBy′ : (Actions → Cxt → A → Maybe A) → (from by : ℕ) → A → Maybe A

+
  strengthenFromBy′ trav from by = trav (actions by) (from , []) -- not using the context part

+

+
  strengthenFromBy : (from by : ℕ) → Term → Maybe Term

+
  strengthenFromBy = strengthenFromBy′ traverseTerm

+

+
  strengthenBy : (by : ℕ) → Term → Maybe Term

+
  strengthenBy = strengthenFromBy 0

+

+
  strengthenFrom : (from : ℕ) → Term → Maybe Term

+
  strengthenFrom from = strengthenFromBy from 1

+

+
  strengthen : Term → Maybe Term

+
  strengthen = strengthenFromBy 0 1

+

+

+
------------------------------------------------------------------------

+
-- Free variable check

+

+
module _ where

+

+
  private

+
    anyApplicative : ∀ {ℓ} → RawApplicative {ℓ} (λ _ → Bool)

+
    anyApplicative = mkRawApplicative _ (λ _ → false)  _∨_

+

+
  open Traverse anyApplicative

+

+
  private

+
    fvVar : ℕ → Cxt → ℕ → Bool

+
    fvVar i (n , _) x = i + n ≡ᵇ x

+

+
    actions : ℕ → Actions

+
    actions i = record defaultActions { onVar = fvVar i }

+

+
  infix 4 _∈FV_

+

+
  _∈FV_ : ℕ → Term → Bool

+
  i ∈FV t = traverseTerm (actions i) (0 , []) t

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Definition.html b/site/static/agda_html/Reflection.AST.Definition.html
new file mode 100644
index 000000000..30f07248e
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Definition.html
@@ -0,0 +1,193 @@
+
+Reflection.AST.Definition
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Definition where

+

+
import Data.List.Properties as List                    using (≡-dec)

+
import Data.Nat.Properties as ℕ                        using (_≟_)

+
open import Data.Product.Base                          using (_×_; <_,_>; uncurry)

+
open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_; yes; no)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)

+

+
import Reflection.AST.Argument          as Arg

+
import Reflection.AST.Argument.Quantity as Quantity

+
import Reflection.AST.Name              as Name

+
import Reflection.AST.Term              as Term

+

+
------------------------------------------------------------------------

+
-- Re-exporting type publically

+

+
open import Agda.Builtin.Reflection public

+
  using    ( Definition

+
           ; function

+
           ; data-type

+
           ; axiom

+
           )

+
  renaming ( record-type to record′

+
           ; data-cons   to constructor′

+
           ; prim-fun    to primitive′ )

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
function-injective : ∀ {cs cs′} → function cs ≡ function cs′ → cs ≡ cs′

+
function-injective refl = refl

+

+
data-type-injective₁ : ∀ {pars pars′ cs cs′} → data-type pars cs ≡ data-type pars′ cs′ → pars ≡ pars′

+
data-type-injective₁ refl = refl

+

+
data-type-injective₂ : ∀ {pars pars′ cs cs′} → data-type pars cs ≡ data-type pars′ cs′ → cs ≡ cs′

+
data-type-injective₂ refl = refl

+

+
data-type-injective : ∀ {pars pars′ cs cs′} → data-type pars cs ≡ data-type pars′ cs′ → pars ≡ pars′ × cs ≡ cs′

+
data-type-injective = < data-type-injective₁ , data-type-injective₂ >

+

+
record′-injective₁ : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → c ≡ c′

+
record′-injective₁ refl = refl

+

+
record′-injective₂ : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → fs ≡ fs′

+
record′-injective₂ refl = refl

+

+
record′-injective : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → c ≡ c′ × fs ≡ fs′

+
record′-injective = < record′-injective₁ , record′-injective₂ >

+

+
constructor′-injective₁ : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → c ≡ c′

+
constructor′-injective₁ refl = refl

+

+
constructor′-injective₂ : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → q ≡ q′

+
constructor′-injective₂ refl = refl

+

+
constructor′-injective : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → c ≡ c′ × q ≡ q′

+
constructor′-injective = < constructor′-injective₁ , constructor′-injective₂ >

+

+
infix 4 _≟_

+

+
_≟_ : DecidableEquality Definition

+
function cs       ≟ function cs′        =

+
  map′ (cong function) function-injective (cs Term.≟-Clauses cs′)

+
data-type pars cs ≟ data-type pars′ cs′ =

+
  map′ (uncurry (cong₂ data-type)) data-type-injective

+
           (pars ℕ.≟ pars′ ×-dec List.≡-dec Name._≟_ cs cs′)

+
record′ c fs      ≟ record′ c′ fs′      =

+
  map′ (uncurry (cong₂ record′)) record′-injective

+
           (c Name.≟ c′ ×-dec List.≡-dec (Arg.≡-dec Name._≟_) fs fs′)

+
constructor′ d q  ≟ constructor′ d′ q′  =

+
  map′ (uncurry (cong₂ constructor′)) constructor′-injective

+
           (d Name.≟ d′ ×-dec q Quantity.≟ q′)

+
axiom             ≟ axiom               = yes refl

+
primitive′        ≟ primitive′          = yes refl

+

+
-- antidiagonal

+
function cs ≟ data-type pars cs₁ = no (λ ())

+
function cs ≟ record′ c fs = no (λ ())

+
function cs ≟ constructor′ d q = no (λ ())

+
function cs ≟ axiom = no (λ ())

+
function cs ≟ primitive′ = no (λ ())

+
data-type pars cs ≟ function cs₁ = no (λ ())

+
data-type pars cs ≟ record′ c fs = no (λ ())

+
data-type pars cs ≟ constructor′ d q = no (λ ())

+
data-type pars cs ≟ axiom = no (λ ())

+
data-type pars cs ≟ primitive′ = no (λ ())

+
record′ c fs ≟ function cs = no (λ ())

+
record′ c fs ≟ data-type pars cs = no (λ ())

+
record′ c fs ≟ constructor′ d q = no (λ ())

+
record′ c fs ≟ axiom = no (λ ())

+
record′ c fs ≟ primitive′ = no (λ ())

+
constructor′ d q ≟ function cs = no (λ ())

+
constructor′ d q ≟ data-type pars cs = no (λ ())

+
constructor′ d q ≟ record′ c fs = no (λ ())

+
constructor′ d q ≟ axiom = no (λ ())

+
constructor′ d q ≟ primitive′ = no (λ ())

+
axiom ≟ function cs = no (λ ())

+
axiom ≟ data-type pars cs = no (λ ())

+
axiom ≟ record′ c fs = no (λ ())

+
axiom ≟ constructor′ d q = no (λ ())

+
axiom ≟ primitive′ = no (λ ())

+
primitive′ ≟ function cs = no (λ ())

+
primitive′ ≟ data-type pars cs = no (λ ())

+
primitive′ ≟ record′ c fs = no (λ ())

+
primitive′ ≟ constructor′ d q = no (λ ())

+
primitive′ ≟ axiom = no (λ ())

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Literal.html b/site/static/agda_html/Reflection.AST.Literal.html
new file mode 100644
index 000000000..a08338e1f
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Literal.html
@@ -0,0 +1,186 @@
+
+Reflection.AST.Literal
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Literals used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Literal where

+

+
open import Data.Bool.Base using (Bool; true; false)

+
import Data.Char.Properties as Char

+
import Data.Float.Properties as Float

+
import Data.Nat.Properties as ℕ

+
import Data.String.Properties as String

+
import Data.Word64.Properties as Word64

+
import Reflection.AST.Meta as Meta

+
import Reflection.AST.Name as Name

+
open import Relation.Nullary.Decidable.Core            using (yes; no; map′; isYes)

+
open import Relation.Binary.Definitions                using (DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtin type and constructors

+

+
open import Agda.Builtin.Reflection public

+
  using ( Literal )

+
open Literal public

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
meta-injective : ∀ {x y} → meta x ≡ meta y → x ≡ y

+
meta-injective refl = refl

+

+
nat-injective : ∀ {x y} → nat x ≡ nat y → x ≡ y

+
nat-injective refl = refl

+

+
word64-injective : ∀ {x y} → word64 x ≡ word64 y → x ≡ y

+
word64-injective refl = refl

+

+
float-injective : ∀ {x y} → float x ≡ float y → x ≡ y

+
float-injective refl = refl

+

+
char-injective : ∀ {x y} → char x ≡ char y → x ≡ y

+
char-injective refl = refl

+

+
string-injective : ∀ {x y} → string x ≡ string y → x ≡ y

+
string-injective refl = refl

+

+
name-injective : ∀ {x y} → name x ≡ name y → x ≡ y

+
name-injective refl = refl

+

+
infix 4 _≟_

+
_≟_ : DecidableEquality Literal

+
nat x ≟ nat x₁ = map′ (cong nat) nat-injective (x ℕ.≟ x₁)

+
nat x ≟ word64 x₁ = no (λ ())

+
nat x ≟ float x₁ = no (λ ())

+
nat x ≟ char x₁ = no (λ ())

+
nat x ≟ string x₁ = no (λ ())

+
nat x ≟ name x₁ = no (λ ())

+
nat x ≟ meta x₁ = no (λ ())

+
word64 x ≟ word64 x₁ = map′ (cong word64) word64-injective (x Word64.≟ x₁)

+
word64 x ≟ nat x₁ = no (λ ())

+
word64 x ≟ float x₁ = no (λ ())

+
word64 x ≟ char x₁ = no (λ ())

+
word64 x ≟ string x₁ = no (λ ())

+
word64 x ≟ name x₁ = no (λ ())

+
word64 x ≟ meta x₁ = no (λ ())

+
float x ≟ nat x₁ = no (λ ())

+
float x ≟ word64 x₁ = no (λ ())

+
float x ≟ float x₁ = map′ (cong float) float-injective (x Float.≟ x₁)

+
float x ≟ char x₁ = no (λ ())

+
float x ≟ string x₁ = no (λ ())

+
float x ≟ name x₁ = no (λ ())

+
float x ≟ meta x₁ = no (λ ())

+
char x ≟ nat x₁ = no (λ ())

+
char x ≟ word64 x₁ = no (λ ())

+
char x ≟ float x₁ = no (λ ())

+
char x ≟ char x₁ = map′ (cong char) char-injective (x Char.≟ x₁)

+
char x ≟ string x₁ = no (λ ())

+
char x ≟ name x₁ = no (λ ())

+
char x ≟ meta x₁ = no (λ ())

+
string x ≟ nat x₁ = no (λ ())

+
string x ≟ word64 x₁ = no (λ ())

+
string x ≟ float x₁ = no (λ ())

+
string x ≟ char x₁ = no (λ ())

+
string x ≟ string x₁ = map′ (cong string) string-injective (x String.≟ x₁)

+
string x ≟ name x₁ = no (λ ())

+
string x ≟ meta x₁ = no (λ ())

+
name x ≟ nat x₁ = no (λ ())

+
name x ≟ word64 x₁ = no (λ ())

+
name x ≟ float x₁ = no (λ ())

+
name x ≟ char x₁ = no (λ ())

+
name x ≟ string x₁ = no (λ ())

+
name x ≟ name x₁ = map′ (cong name) name-injective (x Name.≟ x₁)

+
name x ≟ meta x₁ = no (λ ())

+
meta x ≟ nat x₁ = no (λ ())

+
meta x ≟ word64 x₁ = no (λ ())

+
meta x ≟ float x₁ = no (λ ())

+
meta x ≟ char x₁ = no (λ ())

+
meta x ≟ string x₁ = no (λ ())

+
meta x ≟ name x₁ = no (λ ())

+
meta x ≟ meta x₁ = map′ (cong meta) meta-injective (x Meta.≟ x₁)

+

+
infix 4 _≡ᵇ_

+

+
_≡ᵇ_ : Literal → Literal → Bool

+
l ≡ᵇ l′ = isYes (l ≟ l′)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Meta.html b/site/static/agda_html/Reflection.AST.Meta.html
new file mode 100644
index 000000000..e589093fa
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Meta.html
@@ -0,0 +1,113 @@
+
+Reflection.AST.Meta
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Metavariables used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Meta where

+

+
import Data.Nat.Properties as ℕ

+
open import Function.Base                              using (_on_)

+
open import Relation.Nullary.Decidable.Core            using (map′)

+
open import Relation.Binary.Core                       using (Rel)

+
open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)

+
import Relation.Binary.Construct.On as On

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)

+

+
open import Agda.Builtin.Reflection public

+
  using (Meta) renaming (primMetaToNat to toℕ; primMetaEquality to _≡ᵇ_)

+

+
open import Agda.Builtin.Reflection.Properties public

+
  renaming (primMetaToNatInjective to toℕ-injective)

+

+
-- Equality of metas is decidable.

+

+
infix 4 _≈?_ _≟_ _≈_

+

+
_≈_ : Rel Meta _

+
_≈_ = _≡_ on toℕ

+

+
_≈?_ : Decidable _≈_

+
_≈?_ = On.decidable toℕ _≡_ ℕ._≟_

+

+
_≟_ : DecidableEquality Meta

+
m ≟ n = map′ (toℕ-injective _ _) (cong toℕ) (m ≈? n)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Name.html b/site/static/agda_html/Reflection.AST.Name.html
new file mode 100644
index 000000000..5349b44f3
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Name.html
@@ -0,0 +1,127 @@
+
+Reflection.AST.Name
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Names used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Name where

+

+
open import Data.List.Base              using (List)

+
import Data.Product.Properties as Prodₚ using (≡-dec)

+
import Data.Word64.Properties as Wₚ       using (_≟_)

+
open import Function.Base               using (_on_)

+
open import Relation.Nullary.Decidable.Core            using (map′)

+
open import Relation.Binary.Core                       using (Rel)

+
open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)

+
open import Relation.Binary.Construct.On               using (decidable)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)

+

+
------------------------------------------------------------------------

+
-- Re-export built-ins

+

+
open import Agda.Builtin.Reflection public

+
  using (Name) renaming (primQNameToWord64s to toWords; primQNameEquality to _≡ᵇ_)

+

+
open import Agda.Builtin.Reflection.Properties public

+
  renaming (primQNameToWord64sInjective to toWords-injective)

+

+
------------------------------------------------------------------------

+
-- More definitions

+
------------------------------------------------------------------------

+

+
Names : Set

+
Names = List Name

+

+
------------------------------------------------------------------------

+
-- Decidable equality for names

+
------------------------------------------------------------------------

+

+
infix 4 _≈?_ _≟_ _≈_

+

+
_≈_ : Rel Name _

+
_≈_ = _≡_ on toWords

+

+
_≈?_ : Decidable _≈_

+
_≈?_ = decidable toWords _≡_ (Prodₚ.≡-dec Wₚ._≟_ Wₚ._≟_)

+

+
_≟_ : DecidableEquality Name

+
m ≟ n = map′ (toWords-injective _ _) (cong toWords) (m ≈? n)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Pattern.html b/site/static/agda_html/Reflection.AST.Pattern.html
new file mode 100644
index 000000000..ef74fb5c1
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Pattern.html
@@ -0,0 +1,106 @@
+
+Reflection.AST.Pattern
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Patterns used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Pattern where

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtin type and constructors

+

+
open import Agda.Builtin.Reflection public using (Pattern)

+
open Pattern public

+

+
------------------------------------------------------------------------

+
-- Re-exporting definitions that used to be here

+

+
open import Reflection.AST.Term public

+
  using    ( proj-injective )

+
  renaming ( pat-con-injective₁ to con-injective₁

+
           ; pat-con-injective₂ to con-injective₂

+
           ; pat-con-injective  to con-injective

+
           ; pat-var-injective  to var-injective

+
           ; pat-lit-injective  to lit-injective

+
           ; _≟-Patterns_       to _≟s_

+
           ; _≟-Pattern_        to _≟_

+
           )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Show.html b/site/static/agda_html/Reflection.AST.Show.html
new file mode 100644
index 000000000..a51e9f248
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Show.html
@@ -0,0 +1,231 @@
+
+Reflection.AST.Show
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Converting reflection machinery to strings

+
------------------------------------------------------------------------

+

+
-- Note that Reflection.termErr can also be used directly in tactic

+
-- error messages.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Show where

+

+
import Data.Char.Base as Char

+
import Data.Float.Base as Float

+
open import Data.List.Base hiding (_++_; intersperse)

+
import Data.Nat.Show as ℕ

+
open import Data.String.Base as String using (String; _++_; intersperse; braces; parens; _<+>_)

+
open import Data.String as String using (parensIfSpace)

+
open import Data.Product.Base using (_×_; _,_)

+
import Data.Word64.Base as Word64

+
open import Function.Base using (id; _∘′_; case_of_)

+
open import Relation.Nullary.Decidable.Core using (yes; no)

+

+
open import Reflection.AST.Abstraction hiding (map)

+
open import Reflection.AST.Argument hiding (map)

+
open import Reflection.AST.Argument.Quantity

+
open import Reflection.AST.Argument.Relevance

+
open import Reflection.AST.Argument.Visibility

+
open import Reflection.AST.Argument.Modality

+
open import Reflection.AST.Argument.Information

+
open import Reflection.AST.Definition

+
open import Reflection.AST.Literal

+
open import Reflection.AST.Pattern

+
open import Reflection.AST.Term

+

+
------------------------------------------------------------------------

+
-- Re-export primitive show functions

+

+
open import Agda.Builtin.Reflection public

+
  using () renaming

+
  ( primShowMeta  to showMeta

+
  ; primShowQName to showName

+
  )

+

+
------------------------------------------------------------------------

+
-- Non-primitive show functions

+

+
showRelevance : Relevance → String

+
showRelevance relevant   = "relevant"

+
showRelevance irrelevant = "irrelevant"

+

+
showRel : Relevance → String

+
showRel relevant   = ""

+
showRel irrelevant = "."

+

+
showVisibility : Visibility → String

+
showVisibility visible   = "visible"

+
showVisibility hidden    = "hidden"

+
showVisibility instance′ = "instance"

+

+
showQuantity : Quantity → String

+
showQuantity quantity-0 = "quantity-0"

+
showQuantity quantity-ω = "quantity-ω"

+

+
showLiteral : Literal → String

+
showLiteral (nat x)    = ℕ.show x

+
showLiteral (word64 x) = ℕ.show (Word64.toℕ x)

+
showLiteral (float x)  = Float.show x

+
showLiteral (char x)   = Char.show x

+
showLiteral (string x) = String.show x

+
showLiteral (name x)   = showName x

+
showLiteral (meta x)   = showMeta x

+

+
private

+
  -- add appropriate parens depending on the given visibility

+
  visibilityParen : Visibility → String → String

+
  visibilityParen visible   s = parensIfSpace s

+
  visibilityParen hidden    s = braces s

+
  visibilityParen instance′ s = braces (braces s)

+

+
mutual

+

+
  showTerms : List (Arg Term) → String

+
  showTerms []             = ""

+
  showTerms (arg i t ∷ ts) = visibilityParen (visibility i) (showTerm t) <+> showTerms ts

+

+
  showTerm : Term → String

+
  showTerm (var x args)         = "var" <+> ℕ.show x <+> showTerms args

+
  showTerm (con c args)         = showName c <+> showTerms args

+
  showTerm (def f args)         = showName f <+> showTerms args

+
  showTerm (lam v (abs s x))    = "λ" <+> visibilityParen v s <+> "→" <+> showTerm x

+
  showTerm (pat-lam cs args)    =

+
    "λ {" <+> showClauses cs <+> "}" <+> showTerms args

+
  showTerm (Π[ x ∶ arg i a ] b) =

+
    "Π (" ++ visibilityParen (visibility i) x <+> ":" <+>

+
    parensIfSpace (showTerm a) ++ ")" <+> parensIfSpace (showTerm b)

+
  showTerm (sort s)             = showSort s

+
  showTerm (lit l)              = showLiteral l

+
  showTerm (meta x args)        = showMeta x <+> showTerms args

+
  showTerm unknown              = "unknown"

+

+
  showSort : Sort → String

+
  showSort (set t) = "Set" <+> parensIfSpace (showTerm t)

+
  showSort (lit n) = "Set" ++ ℕ.show n -- no space to disambiguate from set t

+
  showSort (prop t) = "Prop" <+> parensIfSpace (showTerm t)

+
  showSort (propLit n) = "Prop" ++ ℕ.show n -- no space to disambiguate from prop t

+
  showSort (inf n) = "Setω" ++ ℕ.show n

+
  showSort unknown = "unknown"

+

+
  showPatterns : List (Arg Pattern) → String

+
  showPatterns []       = ""

+
  showPatterns (a ∷ ps) = showArg a <+> showPatterns ps

+
    where

+
    -- Quantities are ignored.

+
    showArg : Arg Pattern → String

+
    showArg (arg (arg-info h (modality r _)) p) =

+
      braces? (showRel r ++ showPattern p)

+
      where

+
      braces? = case h of λ where

+
        visible   → id

+
        hidden    → braces

+
        instance′ → braces ∘′ braces

+

+
  showPattern : Pattern → String

+
  showPattern (con c []) = showName c

+
  showPattern (con c ps) = parens (showName c <+> showPatterns ps)

+
  showPattern (dot t)    = "." ++ parens (showTerm t)

+
  showPattern (var x)    = "pat-var" <+> ℕ.show x

+
  showPattern (lit l)    = showLiteral l

+
  showPattern (proj f)   = showName f

+
  showPattern (absurd _) = "()"

+

+
  showClause : Clause → String

+
  showClause (clause tel ps t)      = "[" <+> showTel tel <+> "]" <+> showPatterns ps <+> "→" <+> showTerm t

+
  showClause (absurd-clause tel ps) = "[" <+> showTel tel <+> "]" <+> showPatterns ps

+

+
  showClauses : List Clause → String

+
  showClauses []       = ""

+
  showClauses (c ∷ cs) = showClause c <+> ";" <+> showClauses cs

+

+
  showTel : List (String × Arg Type) → String

+
  showTel [] = ""

+
  showTel ((x , arg i t) ∷ tel) = visibilityParen (visibility i) (x <+> ":" <+> showTerm t) ++ showTel tel

+

+
showDefinition : Definition → String

+
showDefinition (function cs)       = "function" <+> braces (showClauses cs)

+
showDefinition (data-type pars cs) =

+
 "datatype" <+> ℕ.show pars <+> braces (intersperse ", " (map showName cs))

+
showDefinition (record′ c fs)      =

+
 "record" <+> showName c <+> braces (intersperse ", " (map (showName ∘′ unArg) fs))

+
showDefinition (constructor′ d q)  = "constructor" <+> showName d <+> showQuantity q

+
showDefinition axiom               = "axiom"

+
showDefinition primitive′          = "primitive"

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Term.html b/site/static/agda_html/Reflection.AST.Term.html
new file mode 100644
index 000000000..5e5608f2c
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Term.html
@@ -0,0 +1,560 @@
+
+Reflection.AST.Term
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Terms used in the reflection machinery

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST.Term where

+

+
open import Data.List.Base as List                     hiding (_++_)

+
open import Data.List.Properties                       using (∷-dec)

+
open import Data.Nat.Base                              using (ℕ; zero; suc)

+
import Data.Nat.Properties as ℕ

+
open import Data.Product.Base                          using (_×_; _,_; <_,_>; uncurry; map₁)

+
open import Data.Product.Properties                    using (,-injective)

+
open import Data.Maybe.Base                            using (Maybe; just; nothing)

+
open import Data.String.Base                           using (String)

+
open import Data.String.Properties as String           hiding (_≟_)

+
open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)

+
open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)

+

+
open import Reflection.AST.Abstraction

+
open import Reflection.AST.Argument

+
open import Reflection.AST.Argument.Information using (visibility)

+
open import Reflection.AST.Argument.Visibility as Visibility hiding (_≟_)

+
import Reflection.AST.Literal as Literal

+
import Reflection.AST.Meta as Meta

+
open import Reflection.AST.Name as Name using (Name)

+

+
------------------------------------------------------------------------

+
-- Re-exporting the builtin type and constructors

+

+
open import Agda.Builtin.Reflection as Builtin public

+
  using (Sort; Type; Term; Clause; Pattern)

+

+
open Term public

+
  renaming (agda-sort to sort)

+

+
open Sort    public

+
open Clause  public

+
open Pattern public

+

+
------------------------------------------------------------------------

+
-- Handy synonyms

+

+
Clauses : Set

+
Clauses = List Clause

+

+
Telescope : Set

+
Telescope = List (String × Arg Type)

+

+
-- Pattern synonyms for more compact presentation

+

+
pattern vLam s t           = lam visible   (abs s t)

+
pattern hLam s t           = lam hidden    (abs s t)

+
pattern iLam s t           = lam instance′ (abs s t)

+
pattern Π[_∶_]_ s a ty     = pi a (abs s ty)

+
pattern vΠ[_∶_]_ s a ty    = Π[ s ∶ (vArg a) ] ty

+
pattern hΠ[_∶_]_ s a ty    = Π[ s ∶ (hArg a) ] ty

+
pattern iΠ[_∶_]_ s a ty    = Π[ s ∶ (iArg a) ] ty

+

+
------------------------------------------------------------------------

+
-- Utility functions

+

+
getName : Term → Maybe Name

+
getName (con c args) = just c

+
getName (def f args) = just f

+
getName _            = nothing

+

+
-- "n ⋯⟨∷⟩ xs" prepends "n" visible unknown arguments to the list of

+
-- arguments. Useful when constructing the list of arguments for a

+
-- function with initial inferrable arguments.

+
infixr 5 _⋯⟨∷⟩_

+
_⋯⟨∷⟩_ : ℕ → Args Term → Args Term

+
zero  ⋯⟨∷⟩ xs = xs

+
suc i ⋯⟨∷⟩ xs = unknown ⟨∷⟩ (i ⋯⟨∷⟩ xs)

+
{-# INLINE _⋯⟨∷⟩_ #-}

+

+
-- "n ⋯⟅∷⟆ xs" prepends "n" hidden unknown arguments to the list of

+
-- arguments. Useful when constructing the list of arguments for a

+
-- function with initial implicit arguments.

+
infixr 5 _⋯⟅∷⟆_

+
_⋯⟅∷⟆_ : ℕ → Args Term → Args Term

+
zero  ⋯⟅∷⟆ xs = xs

+
suc i ⋯⟅∷⟆ xs = unknown ⟅∷⟆ (i ⋯⟅∷⟆ xs)

+
{-# INLINE _⋯⟅∷⟆_ #-}

+

+
-- Strips off any pi bindings returning the list of variables removed

+
-- and the eventual body of the expression, e.g.

+
--

+
--   stripPis `∀ x {y} → f x y` = (["x", "y"], f x y)

+
stripPis : Term → List (String × Arg Type) × Term

+
stripPis (Π[ s ∶ t ] x) = map₁ ((s , t) ∷_) (stripPis x)

+
stripPis x              = [] , x

+

+
prependLams : List (String × Visibility) → Term → Term

+
prependLams xs t = foldr (λ {(s , v) t → lam v (abs s t)}) t (reverse xs)

+

+
prependHLams : List String → Term → Term

+
prependHLams vs = prependLams (List.map (_, hidden) vs)

+

+
prependVLams : List String → Term → Term

+
prependVLams vs = prependLams (List.map (_, visible) vs)

+

+
------------------------------------------------------------------------

+
-- Decidable equality

+

+
clause-injective₁ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → tel ≡ tel′

+
clause-injective₁ refl = refl

+

+
clause-injective₂ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → ps ≡ ps′

+
clause-injective₂ refl = refl

+

+
clause-injective₃ : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → b ≡ b′

+
clause-injective₃ refl = refl

+

+
clause-injective : ∀ {tel tel′ ps ps′ b b′} → clause tel ps b ≡ clause tel′ ps′ b′ → tel ≡ tel′ × ps ≡ ps′ × b ≡ b′

+
clause-injective = < clause-injective₁ , < clause-injective₂ , clause-injective₃ > >

+

+
absurd-clause-injective₁ : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′

+
absurd-clause-injective₁ refl = refl

+

+
absurd-clause-injective₂ : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → ps ≡ ps′

+
absurd-clause-injective₂ refl = refl

+

+
absurd-clause-injective : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′ × ps ≡ ps′

+
absurd-clause-injective = < absurd-clause-injective₁ , absurd-clause-injective₂ >

+

+
infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_

+
        _≟-Clause_ _≟-Clauses_ _≟_

+
        _≟-Sort_ _≟-Pattern_ _≟-Patterns_ _≟-Telescope_

+

+
_≟-AbsTerm_  : DecidableEquality (Abs Term)

+
_≟-AbsType_  : DecidableEquality (Abs Type)

+
_≟-ArgTerm_  : DecidableEquality (Arg Term)

+
_≟-ArgType_  : DecidableEquality (Arg Type)

+
_≟-Args_     : DecidableEquality (Args Term)

+
_≟-Clause_   : DecidableEquality Clause

+
_≟-Clauses_  : DecidableEquality Clauses

+
_≟_          : DecidableEquality Term

+
_≟-Sort_     : DecidableEquality Sort

+
_≟-Patterns_ : DecidableEquality (Args Pattern)

+
_≟-Pattern_  : DecidableEquality Pattern

+

+
-- Decidable equality 'transformers'

+
-- We need to inline these because the terms are not sized so

+
-- termination would not obvious if we were to use higher-order

+
-- functions such as Data.List.Properties' ≡-dec

+

+
abs s a ≟-AbsTerm abs s′ a′ = unAbs-dec (a ≟ a′)

+
abs s a ≟-AbsType abs s′ a′ = unAbs-dec (a ≟ a′)

+
arg i a ≟-ArgTerm arg i′ a′ = unArg-dec (a ≟ a′)

+
arg i a ≟-ArgType arg i′ a′ = unArg-dec (a ≟ a′)

+

+
[]       ≟-Args []       = yes refl

+
(x ∷ xs) ≟-Args (y ∷ ys) = ∷-dec (x ≟-ArgTerm y) (xs ≟-Args ys)

+
[]       ≟-Args (_ ∷ _)  = no λ()

+
(_ ∷ _)  ≟-Args []       = no λ()

+

+
[]       ≟-Clauses []       = yes refl

+
(x ∷ xs) ≟-Clauses (y ∷ ys) = ∷-dec (x ≟-Clause y) (xs ≟-Clauses ys)

+
[]       ≟-Clauses (_ ∷ _)  = no λ()

+
(_ ∷ _)  ≟-Clauses []       = no λ()

+

+
_≟-Telescope_ : DecidableEquality Telescope

+
[] ≟-Telescope [] = yes refl

+
((x , t) ∷ tel) ≟-Telescope ((x′ , t′) ∷ tel′) = ∷-dec

+
  (map′ (uncurry (cong₂ _,_)) ,-injective ((x String.≟ x′) ×-dec (t ≟-ArgTerm t′)))

+
  (tel ≟-Telescope tel′)

+
[] ≟-Telescope (_ ∷ _) = no λ ()

+
(_ ∷ _) ≟-Telescope [] = no λ ()

+

+
clause tel ps b ≟-Clause clause tel′ ps′ b′ =

+
  map′ (λ (tel≡tel′ , ps≡ps′ , b≡b′) → cong₂ (uncurry clause) (cong₂ _,_ tel≡tel′ ps≡ps′) b≡b′)

+
           clause-injective

+
           (tel ≟-Telescope tel′ ×-dec ps ≟-Patterns ps′ ×-dec b ≟ b′)

+
absurd-clause tel ps ≟-Clause absurd-clause tel′ ps′ =

+
  map′ (uncurry (cong₂ absurd-clause))

+
           absurd-clause-injective

+
           (tel ≟-Telescope tel′ ×-dec ps ≟-Patterns ps′)

+
clause _ _ _      ≟-Clause absurd-clause _ _ = no λ()

+
absurd-clause _ _ ≟-Clause clause _ _ _      = no λ()

+

+
var-injective₁ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → x ≡ x′

+
var-injective₁ refl = refl

+

+
var-injective₂ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → args ≡ args′

+
var-injective₂ refl = refl

+

+
var-injective :  ∀ {x x′ args args′} → var x args ≡ var x′ args′ → x ≡ x′ × args ≡ args′

+
var-injective = < var-injective₁ , var-injective₂ >

+

+
con-injective₁ : ∀ {c c′ args args′} → Term.con c args ≡ con c′ args′ → c ≡ c′

+
con-injective₁ refl = refl

+

+
con-injective₂ : ∀ {c c′ args args′} → Term.con c args ≡ con c′ args′ → args ≡ args′

+
con-injective₂ refl = refl

+

+
con-injective : ∀ {c c′ args args′} → con c args ≡ con c′ args′ → c ≡ c′ × args ≡ args′

+
con-injective = < con-injective₁ , con-injective₂ >

+

+
def-injective₁ : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → f ≡ f′

+
def-injective₁ refl = refl

+

+
def-injective₂ : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → args ≡ args′

+
def-injective₂ refl = refl

+

+
def-injective : ∀ {f f′ args args′} → def f args ≡ def f′ args′ → f ≡ f′ × args ≡ args′

+
def-injective = < def-injective₁ , def-injective₂ >

+

+
meta-injective₁ : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → x ≡ x′

+
meta-injective₁ refl = refl

+

+
meta-injective₂ : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → args ≡ args′

+
meta-injective₂ refl = refl

+

+
meta-injective : ∀ {x x′ args args′} → meta x args ≡ meta x′ args′ → x ≡ x′ × args ≡ args′

+
meta-injective = < meta-injective₁ , meta-injective₂ >

+

+
lam-injective₁ : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → v ≡ v′

+
lam-injective₁ refl = refl

+

+
lam-injective₂ : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → t ≡ t′

+
lam-injective₂ refl = refl

+

+
lam-injective : ∀ {v v′ t t′} → lam v t ≡ lam v′ t′ → v ≡ v′ × t ≡ t′

+
lam-injective = < lam-injective₁ , lam-injective₂ >

+

+
pat-lam-injective₁ : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → cs ≡ cs′

+
pat-lam-injective₁ refl = refl

+

+
pat-lam-injective₂ : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → args ≡ args′

+
pat-lam-injective₂ refl = refl

+

+
pat-lam-injective : ∀ {cs cs′ args args′} → pat-lam cs args ≡ pat-lam cs′ args′ → cs ≡ cs′ × args ≡ args′

+
pat-lam-injective = < pat-lam-injective₁ , pat-lam-injective₂ >

+

+
pi-injective₁ : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₁ ≡ t₁′

+
pi-injective₁ refl = refl

+

+
pi-injective₂ : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₂ ≡ t₂′

+
pi-injective₂ refl = refl

+

+
pi-injective : ∀ {t₁ t₁′ t₂ t₂′} → pi t₁ t₂ ≡ pi t₁′ t₂′ → t₁ ≡ t₁′ × t₂ ≡ t₂′

+
pi-injective = < pi-injective₁ , pi-injective₂ >

+

+
sort-injective : ∀ {x y} → sort x ≡ sort y → x ≡ y

+
sort-injective refl = refl

+

+
lit-injective : ∀ {x y} → lit x ≡ lit y → x ≡ y

+
lit-injective refl = refl

+

+
set-injective : ∀ {x y} → set x ≡ set y → x ≡ y

+
set-injective refl = refl

+

+
slit-injective : ∀ {x y} → Sort.lit x ≡ lit y → x ≡ y

+
slit-injective refl = refl

+

+
prop-injective : ∀ {x y} → prop x ≡ prop y → x ≡ y

+
prop-injective refl = refl

+

+
propLit-injective : ∀ {x y} → propLit x ≡ propLit y → x ≡ y

+
propLit-injective refl = refl

+

+
inf-injective : ∀ {x y} → inf x ≡ inf y → x ≡ y

+
inf-injective refl = refl

+

+
var x args ≟ var x′ args′ =

+
  map′ (uncurry (cong₂ var)) var-injective (x ℕ.≟ x′ ×-dec args ≟-Args args′)

+
con c args ≟ con c′ args′ =

+
  map′ (uncurry (cong₂ con)) con-injective (c Name.≟ c′ ×-dec args ≟-Args args′)

+
def f args ≟ def f′ args′ =

+
  map′ (uncurry (cong₂ def)) def-injective (f Name.≟ f′ ×-dec args ≟-Args args′)

+
meta x args ≟ meta x′ args′ =

+
  map′ (uncurry (cong₂ meta)) meta-injective (x Meta.≟ x′   ×-dec args ≟-Args args′)

+
lam v t    ≟ lam v′ t′    =

+
  map′ (uncurry (cong₂ lam)) lam-injective (v Visibility.≟ v′ ×-dec t ≟-AbsTerm t′)

+
pat-lam cs args ≟ pat-lam cs′ args′ =

+
  map′ (uncurry (cong₂ pat-lam)) pat-lam-injective (cs ≟-Clauses cs′ ×-dec args ≟-Args args′)

+
pi t₁ t₂   ≟ pi t₁′ t₂′   =

+
  map′ (uncurry (cong₂ pi))  pi-injective (t₁ ≟-ArgType t₁′  ×-dec t₂ ≟-AbsType t₂′)

+
sort s     ≟ sort s′      = map′ (cong sort)  sort-injective (s ≟-Sort s′)

+
lit l      ≟ lit l′       = map′ (cong lit)   lit-injective (l Literal.≟ l′)

+
unknown    ≟ unknown      = yes refl

+

+
var x args ≟ con c args′ = no λ()

+
var x args ≟ def f args′ = no λ()

+
var x args ≟ lam v t     = no λ()

+
var x args ≟ pi t₁ t₂    = no λ()

+
var x args ≟ sort _      = no λ()

+
var x args ≟ lit _      = no λ()

+
var x args ≟ meta _ _    = no λ()

+
var x args ≟ unknown     = no λ()

+
con c args ≟ var x args′ = no λ()

+
con c args ≟ def f args′ = no λ()

+
con c args ≟ lam v t     = no λ()

+
con c args ≟ pi t₁ t₂    = no λ()

+
con c args ≟ sort _      = no λ()

+
con c args ≟ lit _      = no λ()

+
con c args ≟ meta _ _    = no λ()

+
con c args ≟ unknown     = no λ()

+
def f args ≟ var x args′ = no λ()

+
def f args ≟ con c args′ = no λ()

+
def f args ≟ lam v t     = no λ()

+
def f args ≟ pi t₁ t₂    = no λ()

+
def f args ≟ sort _      = no λ()

+
def f args ≟ lit _      = no λ()

+
def f args ≟ meta _ _    = no λ()

+
def f args ≟ unknown     = no λ()

+
lam v t    ≟ var x args  = no λ()

+
lam v t    ≟ con c args  = no λ()

+
lam v t    ≟ def f args  = no λ()

+
lam v t    ≟ pi t₁ t₂    = no λ()

+
lam v t    ≟ sort _      = no λ()

+
lam v t    ≟ lit _      = no λ()

+
lam v t    ≟ meta _ _    = no λ()

+
lam v t    ≟ unknown     = no λ()

+
pi t₁ t₂   ≟ var x args  = no λ()

+
pi t₁ t₂   ≟ con c args  = no λ()

+
pi t₁ t₂   ≟ def f args  = no λ()

+
pi t₁ t₂   ≟ lam v t     = no λ()

+
pi t₁ t₂   ≟ sort _      = no λ()

+
pi t₁ t₂   ≟ lit _      = no λ()

+
pi t₁ t₂   ≟ meta _ _    = no λ()

+
pi t₁ t₂   ≟ unknown     = no λ()

+
sort _     ≟ var x args  = no λ()

+
sort _     ≟ con c args  = no λ()

+
sort _     ≟ def f args  = no λ()

+
sort _     ≟ lam v t     = no λ()

+
sort _     ≟ pi t₁ t₂    = no λ()

+
sort _     ≟ lit _       = no λ()

+
sort _     ≟ meta _ _    = no λ()

+
sort _     ≟ unknown     = no λ()

+
lit _     ≟ var x args  = no λ()

+
lit _     ≟ con c args  = no λ()

+
lit _     ≟ def f args  = no λ()

+
lit _     ≟ lam v t     = no λ()

+
lit _     ≟ pi t₁ t₂    = no λ()

+
lit _     ≟ sort _      = no λ()

+
lit _     ≟ meta _ _    = no λ()

+
lit _     ≟ unknown     = no λ()

+
meta _ _   ≟ var x args  = no λ()

+
meta _ _   ≟ con c args  = no λ()

+
meta _ _   ≟ def f args  = no λ()

+
meta _ _   ≟ lam v t     = no λ()

+
meta _ _   ≟ pi t₁ t₂    = no λ()

+
meta _ _   ≟ sort _      = no λ()

+
meta _ _   ≟ lit _       = no λ()

+
meta _ _   ≟ unknown     = no λ()

+
unknown    ≟ var x args  = no λ()

+
unknown    ≟ con c args  = no λ()

+
unknown    ≟ def f args  = no λ()

+
unknown    ≟ lam v t     = no λ()

+
unknown    ≟ pi t₁ t₂    = no λ()

+
unknown    ≟ sort _      = no λ()

+
unknown    ≟ lit _       = no λ()

+
unknown    ≟ meta _ _    = no λ()

+
pat-lam _ _ ≟ var x args  = no λ()

+
pat-lam _ _ ≟ con c args  = no λ()

+
pat-lam _ _ ≟ def f args  = no λ()

+
pat-lam _ _ ≟ lam v t     = no λ()

+
pat-lam _ _ ≟ pi t₁ t₂    = no λ()

+
pat-lam _ _ ≟ sort _      = no λ()

+
pat-lam _ _ ≟ lit _       = no λ()

+
pat-lam _ _ ≟ meta _ _    = no λ()

+
pat-lam _ _ ≟ unknown     = no λ()

+
var x args  ≟ pat-lam _ _ = no λ()

+
con c args  ≟ pat-lam _ _ = no λ()

+
def f args  ≟ pat-lam _ _ = no λ()

+
lam v t     ≟ pat-lam _ _ = no λ()

+
pi t₁ t₂    ≟ pat-lam _ _ = no λ()

+
sort _      ≟ pat-lam _ _ = no λ()

+
lit _       ≟ pat-lam _ _ = no λ()

+
meta _ _    ≟ pat-lam _ _ = no λ()

+
unknown     ≟ pat-lam _ _ = no λ()

+

+
set t   ≟-Sort set t′  = map′ (cong set) set-injective (t ≟ t′)

+
lit n   ≟-Sort lit n′  = map′ (cong lit) slit-injective (n ℕ.≟ n′)

+
prop t   ≟-Sort prop t′  = map′ (cong prop) prop-injective (t ≟ t′)

+
propLit n   ≟-Sort propLit n′  = map′ (cong propLit) propLit-injective (n ℕ.≟ n′)

+
inf n   ≟-Sort inf n′  = map′ (cong inf) inf-injective (n ℕ.≟ n′)

+
unknown ≟-Sort unknown = yes refl

+
set _   ≟-Sort lit _   = no λ()

+
set _   ≟-Sort prop _   = no λ()

+
set _   ≟-Sort propLit _   = no λ()

+
set _   ≟-Sort inf _   = no λ()

+
set _   ≟-Sort unknown = no λ()

+
lit _   ≟-Sort set _   = no λ()

+
lit _   ≟-Sort prop _   = no λ()

+
lit _   ≟-Sort propLit _   = no λ()

+
lit _   ≟-Sort inf _   = no λ()

+
lit _   ≟-Sort unknown = no λ()

+
prop _  ≟-Sort set _   = no λ()

+
prop _  ≟-Sort lit _   = no λ()

+
prop _  ≟-Sort propLit _   = no λ()

+
prop _  ≟-Sort inf _   = no λ()

+
prop _  ≟-Sort unknown   = no λ()

+
propLit _  ≟-Sort set _   = no λ()

+
propLit _  ≟-Sort lit _   = no λ()

+
propLit _  ≟-Sort prop _   = no λ()

+
propLit _  ≟-Sort inf _   = no λ()

+
propLit _  ≟-Sort unknown   = no λ()

+
inf _  ≟-Sort set _   = no λ()

+
inf _  ≟-Sort lit _   = no λ()

+
inf _  ≟-Sort prop _   = no λ()

+
inf _  ≟-Sort propLit _   = no λ()

+
inf _  ≟-Sort unknown   = no λ()

+
unknown ≟-Sort set _   = no λ()

+
unknown ≟-Sort lit _   = no λ()

+
unknown ≟-Sort prop _   = no λ()

+
unknown ≟-Sort propLit _   = no λ()

+
unknown ≟-Sort inf _   = no λ()

+

+

+
pat-con-injective₁ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′

+
pat-con-injective₁ refl = refl

+

+
pat-con-injective₂ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → args ≡ args′

+
pat-con-injective₂ refl = refl

+

+
pat-con-injective : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′ × args ≡ args′

+
pat-con-injective = < pat-con-injective₁ , pat-con-injective₂ >

+

+
pat-var-injective : ∀ {x y} → var x ≡ var y → x ≡ y

+
pat-var-injective refl = refl

+

+
pat-lit-injective : ∀ {x y} → Pattern.lit x ≡ lit y → x ≡ y

+
pat-lit-injective refl = refl

+

+
proj-injective : ∀ {x y} → proj x ≡ proj y → x ≡ y

+
proj-injective refl = refl

+

+
dot-injective : ∀ {x y} → dot x ≡ dot y → x ≡ y

+
dot-injective refl = refl

+

+
absurd-injective : ∀ {x y} → absurd x ≡ absurd y → x ≡ y

+
absurd-injective refl = refl

+

+
con c ps ≟-Pattern con c′ ps′ = map′ (uncurry (cong₂ con)) pat-con-injective (c Name.≟ c′ ×-dec ps ≟-Patterns ps′)

+
var x    ≟-Pattern var x′     = map′ (cong var) pat-var-injective (x ℕ.≟ x′)

+
lit l    ≟-Pattern lit l′     = map′ (cong lit) pat-lit-injective (l Literal.≟ l′)

+
proj a   ≟-Pattern proj a′    = map′ (cong proj) proj-injective (a Name.≟ a′)

+
dot t    ≟-Pattern dot t′     = map′ (cong dot) dot-injective (t ≟ t′)

+
absurd x ≟-Pattern absurd x′  = map′ (cong absurd) absurd-injective (x ℕ.≟ x′)

+

+
con x x₁ ≟-Pattern dot x₂ = no (λ ())

+
con x x₁ ≟-Pattern var x₂ = no (λ ())

+
con x x₁ ≟-Pattern lit x₂ = no (λ ())

+
con x x₁ ≟-Pattern proj x₂ = no (λ ())

+
con x x₁ ≟-Pattern absurd x₂ = no (λ ())

+
dot x ≟-Pattern con x₁ x₂ = no (λ ())

+
dot x ≟-Pattern var x₁ = no (λ ())

+
dot x ≟-Pattern lit x₁ = no (λ ())

+
dot x ≟-Pattern proj x₁ = no (λ ())

+
dot x ≟-Pattern absurd x₁ = no (λ ())

+
var s ≟-Pattern con x x₁ = no (λ ())

+
var s ≟-Pattern dot x = no (λ ())

+
var s ≟-Pattern lit x = no (λ ())

+
var s ≟-Pattern proj x = no (λ ())

+
var s ≟-Pattern absurd x = no (λ ())

+
lit x ≟-Pattern con x₁ x₂ = no (λ ())

+
lit x ≟-Pattern dot x₁ = no (λ ())

+
lit x ≟-Pattern var _ = no (λ ())

+
lit x ≟-Pattern proj x₁ = no (λ ())

+
lit x ≟-Pattern absurd x₁ = no (λ ())

+
proj x ≟-Pattern con x₁ x₂ = no (λ ())

+
proj x ≟-Pattern dot x₁ = no (λ ())

+
proj x ≟-Pattern var _ = no (λ ())

+
proj x ≟-Pattern lit x₁ = no (λ ())

+
proj x ≟-Pattern absurd x₁ = no (λ ())

+
absurd x ≟-Pattern con x₁ x₂ = no (λ ())

+
absurd x ≟-Pattern dot x₁ = no (λ ())

+
absurd x ≟-Pattern var _ = no (λ ())

+
absurd x ≟-Pattern lit x₁ = no (λ ())

+
absurd x ≟-Pattern proj x₁ = no (λ ())

+

+
[]             ≟-Patterns []             = yes refl

+
(arg i p ∷ xs) ≟-Patterns (arg j q ∷ ys) = ∷-dec (unArg-dec (p ≟-Pattern q)) (xs ≟-Patterns ys)

+

+
[]      ≟-Patterns (_ ∷ _) = no λ()

+
(_ ∷ _) ≟-Patterns []      = no λ()

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.Traversal.html b/site/static/agda_html/Reflection.AST.Traversal.html
new file mode 100644
index 000000000..17eaa9638
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.Traversal.html
@@ -0,0 +1,210 @@
+
+Reflection.AST.Traversal
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- de Bruijn-aware generic traversal of reflected terms.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Effect.Applicative using (RawApplicative)

+

+
module Reflection.AST.Traversal

+
  {F : Set → Set} (AppF : RawApplicative F) where

+

+
open import Data.Nat.Base     using (ℕ; zero; suc; _+_)

+
open import Data.List.Base    using (List; []; _∷_; _++_; reverse; length)

+
open import Data.String.Base  using (String)

+
open import Data.Product.Base using (_×_; _,_)

+
open import Function.Base     using (_∘_)

+
open import Reflection        hiding (pure)

+

+
open RawApplicative AppF

+

+
------------------------------------------------------------------------

+
-- Context representation

+

+
-- Track both number of variables and actual context, to avoid having to

+
-- compute the length of the context everytime it's needed.

+
record Cxt : Set where

+
  constructor _,_

+
  pattern

+
  field

+
    len     : ℕ

+
    context : List (String × Arg Term)

+
infixr 4 _,_

+

+
private

+
  _∷cxt_ : String × Arg Term → Cxt → Cxt

+
  e ∷cxt (n , Γ) = (suc n , e ∷ Γ)

+

+
  _++cxt_ : List (String × Arg Term) → Cxt → Cxt

+
  es ++cxt (n , Γ) = (length es + n , es ++ Γ)

+

+
------------------------------------------------------------------------

+
-- Actions

+

+
Action : Set → Set

+
Action A = Cxt → A → F A

+

+
-- A traversal gets to operate on variables, metas, and names.

+
record Actions : Set where

+
  field

+
    onVar  : Action ℕ

+
    onMeta : Action Meta

+
    onCon  : Action Name

+
    onDef  : Action Name

+

+
-- Default action: do nothing.

+
defaultActions : Actions

+
defaultActions .Actions.onVar  _ = pure

+
defaultActions .Actions.onMeta _ = pure

+
defaultActions .Actions.onCon  _ = pure

+
defaultActions .Actions.onDef  _ = pure

+

+
------------------------------------------------------------------------

+
-- Traversal functions

+

+
module _ (actions : Actions) where

+

+
  open Actions actions

+

+
  traverseTerm    : Action Term

+
  traverseSort    : Action Sort

+
  traversePattern : Action Pattern

+
  traverseArgs    : Action (List (Arg Term))

+
  traverseArg     : Action (Arg Term)

+
  traversePats    : Action (List (Arg Pattern))

+
  traverseAbs     : Arg Term → Action (Abs Term)

+
  traverseClauses : Action Clauses

+
  traverseClause  : Action Clause

+
  traverseTel     : Action (List (String × Arg Term))

+

+
  traverseTerm Γ (var x args)      = var       <$> onVar Γ x ⊛ traverseArgs Γ args

+
  traverseTerm Γ (con c args)      = con       <$> onCon Γ c ⊛ traverseArgs Γ args

+
  traverseTerm Γ (def f args)      = def       <$> onDef Γ f ⊛ traverseArgs Γ args

+
  traverseTerm Γ (pat-lam cs args) = pat-lam   <$> traverseClauses Γ cs ⊛ traverseArgs Γ args

+
  traverseTerm Γ (pi a b)          = pi        <$> traverseArg Γ a ⊛ traverseAbs a Γ b

+
  traverseTerm Γ (agda-sort s)     = agda-sort <$> traverseSort Γ s

+
  traverseTerm Γ (meta x args)     = meta      <$> onMeta Γ x ⊛ traverseArgs Γ args

+
  traverseTerm Γ t@(lit _)         = pure t

+
  traverseTerm Γ t@unknown         = pure t

+
  traverseTerm Γ (lam v t)         = lam v     <$> traverseAbs (arg (arg-info v m) unknown) Γ t

+
    where

+
    m = defaultModality

+

+
  traverseArg Γ (arg i t) = arg i <$> traverseTerm Γ t

+
  traverseArgs Γ []       = pure []

+
  traverseArgs Γ (a ∷ as) = _∷_ <$> traverseArg Γ a ⊛ traverseArgs Γ as

+

+
  traverseAbs ty Γ (abs x t) = abs x <$> traverseTerm ((x , ty) ∷cxt Γ) t

+

+
  traverseClauses Γ []       = pure []

+
  traverseClauses Γ (c ∷ cs) = _∷_ <$> traverseClause Γ c ⊛ traverseClauses Γ cs

+

+
  traverseClause Γ (Clause.clause tel ps t) =

+
      Clause.clause <$> traverseTel Γ tel

+
                     ⊛  traversePats Γ′ ps

+
                     ⊛ traverseTerm Γ′ t

+
    where Γ′ = reverse tel ++cxt Γ

+
  traverseClause Γ (Clause.absurd-clause tel ps) =

+
      Clause.absurd-clause <$> traverseTel Γ tel

+
                            ⊛  traversePats Γ′ ps

+
    where Γ′ = reverse tel ++cxt Γ

+

+
  traverseTel Γ [] = pure []

+
  traverseTel Γ ((x , ty) ∷ tel) =

+
    _∷_ ∘ (x ,_) <$> traverseArg Γ ty ⊛ traverseTel ((x , ty) ∷cxt Γ) tel

+

+
  traverseSort Γ (Sort.set t)       = Sort.set <$> traverseTerm Γ t

+
  traverseSort Γ t@(Sort.lit _)     = pure t

+
  traverseSort Γ (Sort.prop t)      = Sort.prop <$> traverseTerm Γ t

+
  traverseSort Γ t@(Sort.propLit _) = pure t

+
  traverseSort Γ t@(Sort.inf _)     = pure t

+
  traverseSort Γ t@Sort.unknown     = pure t

+

+
  traversePattern Γ (Pattern.con c ps) = Pattern.con <$> onCon Γ c ⊛ traversePats Γ ps

+
  traversePattern Γ (Pattern.dot t)    = Pattern.dot <$> traverseTerm Γ t

+
  traversePattern Γ (Pattern.var x)    = Pattern.var <$> onVar Γ x

+
  traversePattern Γ p@(Pattern.lit _)  = pure p

+
  traversePattern Γ p@(Pattern.proj _) = pure p

+
  traversePattern Γ (Pattern.absurd x) = Pattern.absurd <$> onVar Γ x

+

+
  traversePats Γ [] = pure []

+
  traversePats Γ (arg i p ∷ ps) = _∷_ ∘ arg i <$> traversePattern Γ p ⊛ traversePats Γ ps

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.AST.html b/site/static/agda_html/Reflection.AST.html
new file mode 100644
index 000000000..f17a2d8a5
--- /dev/null
+++ b/site/static/agda_html/Reflection.AST.html
@@ -0,0 +1,136 @@
+
+Reflection.AST
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The reflected abstract syntax tree

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.AST where

+

+
import Agda.Builtin.Reflection as Builtin

+

+
------------------------------------------------------------------------

+
-- Names, Metas, and Literals re-exported publicly

+

+
open import Reflection.AST.Abstraction as Abstraction public

+
  using (Abs; abs)

+
open import Reflection.AST.Argument as Argument public

+
  using (Arg; arg; Args; vArg; hArg; iArg; defaultModality)

+
open import Reflection.AST.Definition as Definition  public

+
  using (Definition)

+
open import Reflection.AST.Meta as Meta public

+
  using (Meta)

+
open import Reflection.AST.Name as Name public

+
  using (Name; Names)

+
open import Reflection.AST.Literal as Literal public

+
  using (Literal)

+
open import Reflection.AST.Pattern as Pattern public

+
  using (Pattern)

+
open import Reflection.AST.Term as Term public

+
  using (Term; Type; Clause; Clauses; Sort)

+

+
import Reflection.AST.Argument.Modality    as Modality

+
import Reflection.AST.Argument.Quantity    as Quantity

+
import Reflection.AST.Argument.Relevance   as Relevance

+
import Reflection.AST.Argument.Visibility  as Visibility

+
import Reflection.AST.Argument.Information as Information

+

+
open Definition.Definition public

+
open Information.ArgInfo public

+
open Literal.Literal public

+
open Modality.Modality public

+
open Quantity.Quantity public

+
open Relevance.Relevance public

+
open Term.Term public

+
open Visibility.Visibility public

+

+
open import Reflection.AST.Show public

+

+
------------------------------------------------------------------------

+
-- Fixity

+

+
open Builtin public

+
  using (non-assoc; related; unrelated; fixity)

+
  renaming

+
  ( left-assoc      to assocˡ

+
  ; right-assoc     to assocʳ

+
  ; primQNameFixity to getFixity

+
  )

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.Debug.html b/site/static/agda_html/Reflection.Debug.html
new file mode 100644
index 000000000..1b72f122b
--- /dev/null
+++ b/site/static/agda_html/Reflection.Debug.html
@@ -0,0 +1,197 @@
+
+Reflection.Debug
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module Reflection.Debug where

+

+
open import Meta.Prelude hiding (⊤; _∧_; _∨_; filter)

+

+
import Data.Bool as B

+
import Data.String as S

+
open import Data.Char

+
open import Data.List using (map)

+

+
open import Relation.Nullary.Decidable using (⌊_⌋)

+

+
open import Reflection

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
record IsErrorPart (A : Set) : Set where

+
  field toErrorPart : A → ErrorPart

+

+
open IsErrorPart ⦃...⦄ public

+

+
instance

+
  IsErrorPart-String : IsErrorPart String

+
  IsErrorPart-String .toErrorPart = ErrorPart.strErr

+

+
  IsErrorPart-Term : IsErrorPart Term

+
  IsErrorPart-Term .toErrorPart = ErrorPart.termErr

+

+
  IsErrorPart-Name : IsErrorPart Name

+
  IsErrorPart-Name .toErrorPart = ErrorPart.nameErr

+

+
  IsErrorPart-Clause : IsErrorPart Clause

+
  IsErrorPart-Clause .toErrorPart c = ErrorPart.termErr (pat-lam [ c ] [])

+

+
  IsErrorPart-ListClause : IsErrorPart (List Clause)

+
  IsErrorPart-ListClause .toErrorPart cs = ErrorPart.termErr (pat-lam cs [])

+

+
infixr 5 _∷ᵈ_ _++ᵈ_

+
_∷ᵈ_ : A → ⦃ _ : IsErrorPart A ⦄ → List ErrorPart → List ErrorPart

+
e ∷ᵈ es = toErrorPart e ∷ es

+

+
_++ᵈ_ : List A → ⦃ _ : IsErrorPart A ⦄ → List ErrorPart → List ErrorPart

+
es ++ᵈ es' = map toErrorPart es ++ es'

+

+
_ᵈ : ⦃ _ : IsErrorPart A ⦄ → List A → List ErrorPart

+
_ᵈ = map toErrorPart

+

+
data DebugSelection : Set where

+
  FullPath : DebugSelection

+
  Last     : DebugSelection

+
  All      : DebugSelection

+
  Custom   : (List String → String) → DebugSelection

+

+
-- If All is selected, this pragma shows all debug info

+
-- {-# OPTIONS -v allTactics:100 #-}

+

+
Filter : Set

+
Filter = List String → Bool

+

+
module Filter where

+
  open import Algebra.Lattice

+
  open import Data.Bool.Properties

+
  import Algebra.Function

+

+
  Filter-Alg : BooleanAlgebra _ _

+
  Filter-Alg = Algebra.Function.hom (List String) ∨-∧-booleanAlgebra

+

+
  open BooleanAlgebra Filter-Alg public

+

+
  private

+
    _≣_ : String → String → Bool

+
    s ≣ s' = ⌊ s S.≟ s' ⌋

+

+
  contains : String → Filter

+
  contains s l = foldl (λ acc s' → acc B.∨ s ≣ s') false l

+

+
  noneOf : List String → Filter

+
  noneOf [] = ⊤

+
  noneOf (x ∷ l) = ¬ contains x ∧ noneOf l

+

+
  endsWith : String → Filter

+
  endsWith s l with last l

+
  ... | just x  = s ≣ x

+
  ... | nothing = false

+

+
  beginsWith : String → Filter

+
  beginsWith s l with head l

+
  ... | just x  = s ≣ x

+
  ... | nothing = false

+

+
record DebugOptions : Set where

+
  field

+
    path      : List String

+
    selection : DebugSelection

+
    filter    : Filter

+
    level     : ℕ

+
    prefix    : Char

+

+
defaultDebugOptions : DebugOptions

+
defaultDebugOptions = record

+
  { path = []; selection = All; filter = Filter.⊤; level = 100; prefix = '|' }

+

+
specializeDebugOptions : DebugOptions → DebugOptions → DebugOptions

+
specializeDebugOptions record { path = path₁ } opts@record { path = path₂ } =

+
  record opts { path = path₁ ++ path₂ }

+

+
debugOptionsPath : DebugOptions → String

+
debugOptionsPath record { path = path ; selection = FullPath } = S.intersperse "/" path

+
debugOptionsPath record { path = path ; selection = Last } with last path

+
... | just x  = x

+
... | nothing = ""

+
debugOptionsPath record { path = path ; selection = All } = "allTactics"

+
debugOptionsPath record { path = path ; selection = Custom f } = f path

+

+
debugPrintPrefix : DebugOptions → ErrorPart

+
debugPrintPrefix o = let open DebugOptions o in strErr (S.replicate (length path) prefix)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.Syntax.html b/site/static/agda_html/Reflection.Syntax.html
new file mode 100644
index 000000000..800083d2d
--- /dev/null
+++ b/site/static/agda_html/Reflection.Syntax.html
@@ -0,0 +1,188 @@
+
+Reflection.Syntax
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module Reflection.Syntax where

+

+
open import Meta.Prelude

+

+
open import Reflection.AST.Argument public

+
  hiding (map)

+
open import Reflection.AST.Term public

+
  hiding (_≟_; getName)

+
open import Reflection.AST.Name public

+
  hiding (_≟_; _≡ᵇ_; _≈_; _≈?_)

+
open import Reflection.AST.Definition public

+
  hiding (_≟_)

+
open import Reflection.AST.Meta public

+
  hiding (_≟_; _≡ᵇ_; _≈_; _≈?_)

+
open import Reflection.AST.Abstraction public

+
  using (unAbs)

+

+
open import Agda.Builtin.Reflection public

+
  using (ArgInfo; Modality; Visibility; Literal; Meta)

+

+
open import Reflection.AST.Argument public

+
  using (Arg; arg)

+
open import Reflection.AST.Argument.Visibility public

+
  using (Visibility; visible; hidden; instance′)

+
open import Reflection.AST.Argument.Information public

+
  using (ArgInfo; arg-info)

+
open import Reflection.AST.Abstraction public

+
  using (Abs; abs; unAbs)

+
open import Reflection.AST.Argument public

+
  using (vArg; hArg; iArg; unArg; _⟨∷⟩_; map-Args)

+
open import Reflection public

+
  using ( Name; Meta; Literal; Pattern; Clause

+
        ; ErrorPart; strErr

+
        ; Term; Type; pi; var; con; def; lam; pat-lam; agda-sort; lit; meta; unknown

+
        ; Definition; data-cons; data-type; record-type

+
        )

+

+
open import Reflection using (hidden; instance′; TC)

+

+
-- ** Smart constructors

+

+
-- arguments

+
pattern hArg? = hArg unknown

+
pattern vArg? = vArg unknown

+
pattern iArg? = iArg unknown

+

+
-- variables

+
pattern ♯ n = var n []

+
pattern ♯_⟦_⟧ n x = var n (vArg x ∷ [])

+
pattern ♯_⟦_∣_⟧ n x y = var n (vArg x ∷ vArg y ∷ [])

+

+
-- patterns

+
pattern `_ x = Pattern.var x

+
pattern `∅   = Pattern.absurd 0

+

+
-- clauses

+
pattern ⟦⦅_⦆∅⟧ tel = absurd-clause tel (vArg `∅ ∷ [])

+
pattern ⟦∅⟧ = absurd-clause [] (vArg `∅ ∷ [])

+

+
pattern ⟦⇒_⟧ k = clause [] [] k

+
pattern ⟦⦅_⦆⇒_⟧ tel k = clause tel [] k

+

+
pattern ⟦_⇒_⟧ x k = clause [] (vArg x ∷ []) k

+
pattern ⟦_⦅_⦆⇒_⟧ x tel k = clause tel (vArg x ∷ []) k

+

+
pattern ⟦_∣_⇒_⟧ x y k = clause [] (vArg x ∷ vArg y ∷ []) k

+
pattern ⟦_∣_⦅_⦆⇒_⟧ x y tel k = clause tel (vArg x ∷ vArg y ∷ []) k

+

+
pattern ⟦_∣_∣_⇒_⟧ x y z k = Clause.clause [] (vArg x ∷ vArg y ∷ vArg z ∷ []) k

+
pattern ⟦_∣_∣_⦅_⦆⇒_⟧ x y z tel k = Clause.clause tel (vArg x ∷ vArg y ∷ vArg z ∷ []) k

+

+
-- lambdas

+
pattern `λ_⇒_       x     k = lam visible (abs x k)

+
pattern `λ⟦_∣_⇒_⟧   x y   k = `λ x ⇒ `λ y ⇒ k

+
pattern `λ⟦_∣_∣_⇒_⟧ x y z k = `λ x ⇒ `λ y ⇒ `λ z ⇒ k

+

+
pattern `λ⟅_⟆⇒_     x k = lam hidden (abs x k)

+
pattern `λ⦃_⦄⇒_     x k = lam instance′ (abs x k)

+

+
pattern `λ⦅_⦆∅ tel = pat-lam (⟦⦅ tel ⦆∅⟧ ∷ []) []

+
pattern `λ∅ = pat-lam (⟦∅⟧ ∷ []) []

+

+
pattern `λ⟦_⇒_⟧ p k = pat-lam (⟦ p ⇒ k ⟧ ∷ []) []

+
pattern `λ⟦_⦅_⦆⇒_⟧ p tel k = pat-lam (⟦ p ⦅ tel ⦆⇒ k ⟧ ∷ []) []

+

+
pattern `λ⟦_⇒_∣_⇒_⟧ p₁ k₁ p₂ k₂ = pat-lam (⟦ p₁ ⇒ k₁ ⟧ ∷ ⟦ p₂ ⇒ k₂ ⟧ ∷ []) []

+
pattern `λ⟦_⦅_⦆⇒_∣_⦅_⦆⇒_⟧ p₁ tel₁ k₁ p₂ tel₂ k₂ = pat-lam (⟦ p₁ ⦅ tel₁ ⦆⇒ k₁ ⟧ ∷ ⟦ p₂ ⦅ tel₂ ⦆⇒ k₂ ⟧ ∷ []) []

+

+
-- function application

+
pattern _∙ n = def n []

+
pattern _∙⟦_⟧ n x = def n (vArg x ∷ [])

+
pattern _∙⟦_∣_⟧ n x y = def n (vArg x ∷ vArg y ∷ [])

+
pattern _∙⟦_∣_∣_⟧ n x y z = def n (vArg x ∷ vArg y ∷ vArg z ∷ [])

+
pattern _∙⟦_∣_∣_∣_⟧ n x y z w = def n (vArg x ∷ vArg y ∷ vArg z ∷ vArg w ∷ [])

+

+
pattern _◆ n = con n []

+
pattern _◆⟦_⟧ n x = con n (vArg x ∷ [])

+
pattern _◆⟦_∣_⟧ n x y = con n (vArg x ∷ vArg y ∷ [])

+

+
pattern _◇ n = Pattern.con n []

+
pattern _◇⟦_⟧ n x = Pattern.con n (vArg x ∷ [])

+
pattern _◇⟦_∣_⟧ n x y = Pattern.con n (vArg x ∷ vArg y ∷ [])

+

+
-- ** useful type aliases

+
PatTelescope = Telescope

+
Context      = Args Type

+
TTerm        = Term × Type

+
Hole         = Term

+
THole        = Hole × Type

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.TCI.html b/site/static/agda_html/Reflection.TCI.html
new file mode 100644
index 000000000..a348fb214
--- /dev/null
+++ b/site/static/agda_html/Reflection.TCI.html
@@ -0,0 +1,212 @@
+
+Reflection.TCI
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
--------------------------------------------------------------------------------

+
-- Improved TC

+
--------------------------------------------------------------------------------

+

+
module Reflection.TCI where

+

+
open import Meta.Prelude

+

+
open import Data.List using (map)

+

+
import Agda.Builtin.Reflection as R'

+
import Reflection as R

+
open import Reflection.Syntax

+

+
open import Class.Monad hiding (Monad-TC)

+
open import Class.MonadError using (MonadError)

+
open import Class.MonadReader

+
open import Class.MonadTC

+

+
open Monad

+

+
private variable a f : Level

+
                 A B C D : Set a

+

+
TC : Set a → Set a

+
TC = ReaderT TCEnv R.TC

+

+
Monad-TC : Monad TC

+
Monad-TC = Monad-ReaderT ⦃ Class.Monad.Monad-TC ⦄

+

+
MonadReader-TC : MonadReader TCEnv TC ⦃ Monad-TC ⦄

+
MonadReader-TC = MonadReader-ReaderT ⦃ Class.Monad.Monad-TC ⦄

+

+
MonadError-TC : MonadError (List ErrorPart) TC

+
MonadError-TC = MonadError-ReaderT ⦃ Class.Monad.Monad-TC ⦄ ⦃ Class.MonadError.MonadError-TC ⦄

+

+
applyReductionOptions : TC A → TC A

+
applyReductionOptions x r@record { reduction = onlyReduce red } = R'.withReduceDefs (true ,  red) (x r)

+
applyReductionOptions x r@record { reduction = dontReduce red } = R'.withReduceDefs (false , red) (x r)

+

+
applyNormalisation : TC A → TC A

+
applyNormalisation x r@record { normalisation = n } = R.withNormalisation n (applyReductionOptions x r)

+

+
applyReconstruction : TC A → TC A

+
applyReconstruction x r@record { reconstruction = false } = x r

+
applyReconstruction x r@record { reconstruction = true  } = R'.withReconstructed true (x r)

+

+
applyNoConstraints : TC A → TC A

+
applyNoConstraints x r@record { noConstraints = false } = x r

+
applyNoConstraints x r@record { noConstraints = true  } = R'.noConstraints (x r)

+

+
applyExtContext : Telescope → R.TC A → R.TC A

+
applyExtContext [] x       = x

+
applyExtContext (t ∷ ts) x = applyExtContext ts $ (uncurry R.extendContext) t x

+

+
private

+
  liftTC : R.TC A → TC A

+
  liftTC x = λ r → applyExtContext (r .TCEnv.localContext) x

+

+
  liftTC1 : (A → R.TC B) → A → TC B

+
  liftTC1 f a = liftTC (f a)

+

+
  liftTC2 : (A → B → R.TC C) → A → B → TC C

+
  liftTC2 f a b = liftTC (f a b)

+

+
  liftTC3 : (A → B → C → R.TC D) → A → B → C → TC D

+
  liftTC3 f a b c = liftTC (f a b c)

+

+
module MonadTCI where

+
  unify             : Term → Term → TC ⊤

+
  unify             = applyNoConstraints ∘₂ liftTC2 R.unify

+

+
  typeError         : List ErrorPart → TC A

+
  typeError         = liftTC1 R.typeError

+

+
  inferType         : Term → TC Type

+
  inferType         = applyReconstruction ∘ applyNormalisation ∘ liftTC1 R.inferType

+

+
  checkType         : Term → Type → TC Term

+
  checkType         = (applyReconstruction ∘ applyNormalisation) ∘₂ liftTC2 R.checkType

+

+
  normalise         : Term → TC Term

+
  normalise         = applyReductionOptions ∘ applyReconstruction ∘ liftTC1 R.normalise

+

+
  reduce            : Term → TC Term

+
  reduce            = applyReductionOptions ∘ applyReconstruction ∘ liftTC1 R.reduce

+

+
  quoteTC           : A → TC Term

+
  quoteTC           = applyNormalisation ∘ liftTC1 R.quoteTC

+

+
  unquoteTC         : Term → TC A

+
  unquoteTC         = liftTC1 R.unquoteTC

+

+
  quoteωTC          : ∀ {A : Setω} → A → TC Term

+
  quoteωTC          = λ a → liftTC (R'.quoteωTC a)

+

+
  freshName         : String → TC Name

+
  freshName         = liftTC1 R.freshName

+

+
  declareDef        : Arg Name → Type → TC ⊤

+
  declareDef        = liftTC2 R.declareDef

+

+
  declarePostulate  : Arg Name → Type → TC ⊤

+
  declarePostulate  = liftTC2 R.declarePostulate

+

+
  defineFun         : Name → List Clause → TC ⊤

+
  defineFun         = liftTC2 R.defineFun

+

+
  getType           : Name → TC Type

+
  getType           = applyReconstruction ∘ liftTC1 R.getType

+

+
  getDefinition     : Name → TC Definition

+
  getDefinition     = applyReconstruction ∘ liftTC1 R.getDefinition

+

+
  blockOnMeta       : Meta → TC A

+
  blockOnMeta       = liftTC1 R.blockOnMeta

+

+
  commitTC          : TC ⊤

+
  commitTC          = liftTC R.commitTC

+

+
  isMacro           : Name → TC Bool

+
  isMacro           = liftTC1 R.isMacro

+

+
  debugPrint        : String → ℕ → List ErrorPart → TC ⊤

+
  debugPrint        = liftTC3 R.debugPrint

+

+
  runSpeculative    : TC (A × Bool) → TC A

+
  runSpeculative    = R.runSpeculative ∘_

+

+
  getInstances      : Meta → TC (List Term)

+
  getInstances      = liftTC1 R'.getInstances

+

+
MonadTC-TCI : MonadTC TC ⦃ Monad-TC ⦄ ⦃ MonadError-TC ⦄

+
MonadTC-TCI = record { MonadTCI }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.TCM.Format.html b/site/static/agda_html/Reflection.TCM.Format.html
new file mode 100644
index 000000000..ba262a829
--- /dev/null
+++ b/site/static/agda_html/Reflection.TCM.Format.html
@@ -0,0 +1,162 @@
+
+Reflection.TCM.Format
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Printf-style versions of typeError and debugPrint

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.TCM.Format where

+

+
open import Agda.Builtin.Reflection using (Name; Term; ErrorPart; termErr; nameErr; strErr;

+
                                           TC; typeError; debugPrint)

+
open import Level                   using (Level)

+
open import Function.Base           using (_∘_)

+
open import Data.List.Base          using (List; [_]; concat)

+
open import Data.Maybe.Base         using (Maybe; nothing; just)

+
open import Data.Nat.Base           using (ℕ)

+
open import Data.String.Base        using (String)

+
open import Data.Sum.Base           using (_⊎_; inj₁; inj₂)

+
open import Data.Unit.Base          using (⊤)

+

+
import Text.Format as StdFormat using (formatSpec)

+
import Text.Printf as StdPrintf using (printfSpec)

+

+
open import Text.Format.Generic

+
open import Text.Printf.Generic

+

+
private

+
  variable

+
    ℓ : Level

+
    A : Set ℓ

+

+
------------------------------------------------------------------------

+
-- Format specification.

+
-- This extends the formats from Text.Printf with

+
--   %t  - Term

+
--   %q  - Name

+
--   %e  - List ErrorPart

+
-- Rendering goes to List ErrorPart.

+

+
module Specification where

+

+
  data ErrorChunk : Set where

+
    `Term `Name `Parts : ErrorChunk

+

+
  errorSpec : FormatSpec

+
  errorSpec .FormatSpec.ArgChunk = ErrorChunk

+
  errorSpec .FormatSpec.ArgType `Term  = Term

+
  errorSpec .FormatSpec.ArgType `Name  = Name

+
  errorSpec .FormatSpec.ArgType `Parts = List ErrorPart

+
  errorSpec .FormatSpec.lexArg 't'     = just `Term

+
  errorSpec .FormatSpec.lexArg 'q'     = just `Name

+
  errorSpec .FormatSpec.lexArg 'e'     = just `Parts

+
  errorSpec .FormatSpec.lexArg  _      = nothing

+

+
  formatSpec : FormatSpec

+
  formatSpec = unionSpec errorSpec StdFormat.formatSpec

+

+
  open PrintfSpec

+

+
  printfSpec : PrintfSpec formatSpec (List ErrorPart)

+
  printfSpec .renderArg (inj₁ `Term)  t  = [ termErr t ]

+
  printfSpec .renderArg (inj₁ `Name)  n  = [ nameErr n ]

+
  printfSpec .renderArg (inj₁ `Parts) es = es

+
  printfSpec .renderArg (inj₂ arg)    x  = [ strErr (StdPrintf.printfSpec .renderArg arg x) ]

+
  printfSpec .renderStr               s  = [ strErr s ]

+

+
open Specification

+

+
open Format formatSpec

+
open Type   formatSpec renaming (map to mapPrintf)

+
open Render printfSpec

+

+
------------------------------------------------------------------------

+
-- Printf versions of typeError and debugPrint

+

+
typeErrorFmt : (fmt : String) → Printf (lexer fmt) (TC A)

+
typeErrorFmt fmt = mapPrintf (lexer fmt) (typeError ∘ concat) (printf fmt)

+

+
debugPrintFmt : String → ℕ → (fmt : String) → Printf (lexer fmt) (TC ⊤)

+
debugPrintFmt tag lvl fmt = mapPrintf (lexer fmt) (debugPrint tag lvl ∘ concat) (printf fmt)

+

+
-- Combine with "%e" format for nested format calls.

+
errorPartFmt : (fmt : String) → Printf (lexer fmt) (List ErrorPart)

+
errorPartFmt fmt = mapPrintf (lexer fmt) concat (printf fmt)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.TCM.Syntax.html b/site/static/agda_html/Reflection.TCM.Syntax.html
new file mode 100644
index 000000000..0fa79e232
--- /dev/null
+++ b/site/static/agda_html/Reflection.TCM.Syntax.html
@@ -0,0 +1,129 @@
+
+Reflection.TCM.Syntax
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Monad syntax for the TC monad

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.TCM.Syntax where

+

+
open import Agda.Builtin.Reflection

+
open import Level using (Level)

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Monad syntax

+

+
infixl 3 _<|>_

+
_<|>_ : TC A → TC A → TC A

+
_<|>_ = catchTC

+
{-# INLINE _<|>_ #-}

+

+
infixl 1 _>>=_ _>>_ _<&>_

+
_>>=_ : TC A → (A → TC B) → TC B

+
_>>=_ = bindTC

+
{-# INLINE _>>=_ #-}

+

+
_>>_ : TC A → TC B → TC B

+
xs >> ys = bindTC xs (λ _ → ys)

+
{-# INLINE _>>_ #-}

+

+
infixl 4 _<$>_ _<*>_ _<$_

+
_<*>_ : TC (A → B) → TC A → TC B

+
fs <*> xs = bindTC fs (λ f → bindTC xs (λ x → returnTC (f x)))

+
{-# INLINE _<*>_ #-}

+

+
_<$>_ : (A → B) → TC A → TC B

+
f <$> xs = bindTC xs (λ x → returnTC (f x))

+
{-# INLINE _<$>_ #-}

+

+
_<$_ : A → TC B → TC A

+
x <$ xs = bindTC xs (λ _ → returnTC x)

+
{-# INLINE _<$_ #-}

+

+
_<&>_ : TC A → (A → B) → TC B

+
xs <&> f = bindTC xs (λ x → returnTC (f x))

+
{-# INLINE _<&>_ #-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.TCM.html b/site/static/agda_html/Reflection.TCM.html
new file mode 100644
index 000000000..671577523
--- /dev/null
+++ b/site/static/agda_html/Reflection.TCM.html
@@ -0,0 +1,122 @@
+
+Reflection.TCM
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The TC (Type Checking) monad

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection.TCM where

+

+
import Agda.Builtin.Reflection as Builtin

+

+
open import Reflection.AST.Term

+
import Reflection.TCM.Format as Format

+

+
------------------------------------------------------------------------

+
-- Type errors

+

+
open Builtin public

+
  using (ErrorPart; strErr; termErr; nameErr)

+

+
------------------------------------------------------------------------

+
-- The monad

+

+
open Builtin public

+
  using

+
  ( TC; bindTC; unify; typeError; inferType; checkType

+
  ; normalise; reduce

+
  ; catchTC; quoteTC; unquoteTC

+
  ; getContext; extendContext; inContext; freshName

+
  ; declareDef; declarePostulate; defineFun; getType; getDefinition

+
  ; blockOnMeta; commitTC; isMacro; withNormalisation

+
  ; debugPrint; noConstraints; runSpeculative

+
  ; Blocker; blockerMeta; blockerAny; blockerAll; blockTC

+
  )

+
  renaming (returnTC to pure)

+

+
open Format public

+
  using (typeErrorFmt; debugPrintFmt; errorPartFmt)

+

+
------------------------------------------------------------------------

+
-- Utility functions

+

+
newMeta : Type → TC Term

+
newMeta = checkType unknown

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.Tactic.html b/site/static/agda_html/Reflection.Tactic.html
new file mode 100644
index 000000000..07f2d3f9e
--- /dev/null
+++ b/site/static/agda_html/Reflection.Tactic.html
@@ -0,0 +1,126 @@
+
+Reflection.Tactic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
--------------------------------------------------------------------------------

+
-- Types & functions for interacting with tactics/metaprograms

+
--------------------------------------------------------------------------------

+

+
module Reflection.Tactic where

+

+
import Reflection as R

+
open import Class.Monad hiding (Monad-TC)

+
open import Class.MonadTC

+
open import Meta.Prelude

+
open import Meta.Init

+

+
open MonadTC ⦃...⦄

+

+
private variable

+
  a : Level

+
  A : Set a

+

+
Tactic = Term → R.TC ⊤

+
UnquoteDecl = R.TC ⊤

+

+
ITactic : Set

+
ITactic = TC ⊤

+

+
initTacEnv : (TCEnv → TCEnv) → ITactic → Tactic

+
initTacEnv f tac goal = (initTCEnvWithGoal goal) R.>>= tac ∘ f

+

+
initTacOpts : ITactic → TCOptions → Tactic

+
initTacOpts tac opts = initTacEnv (λ env → record env { options = opts }) tac

+

+
module _ ⦃ opts : TCOptions ⦄ where

+

+
  initTac : ITactic → Tactic

+
  initTac tac = initTacOpts tac opts

+

+
  initUnquoteWithGoal : Term → TC ⊤ → UnquoteDecl

+
  initUnquoteWithGoal g unq = (initTCEnvWithGoal g) R.>>= unq

+
                            ∘ (λ env → record env { options = opts })

+

+
  initUnquote : TC ⊤ → UnquoteDecl

+
  initUnquote unq = initUnquoteWithGoal unknown unq

+

+
  macro

+
    byTC : TC A → Tactic

+
    byTC comp = initTac ((comp >>= quoteTC) >>= unifyWithGoal)

+

+
    by : ITactic → Tactic

+
    by = initTac

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.Utils.Substitute.html b/site/static/agda_html/Reflection.Utils.Substitute.html
new file mode 100644
index 000000000..61a0dd193
--- /dev/null
+++ b/site/static/agda_html/Reflection.Utils.Substitute.html
@@ -0,0 +1,123 @@
+
+Reflection.Utils.Substitute
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Reflection.Utils.Substitute where

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
-- There aren't any nice substitution functions (that I can find) in the standard library or

+
-- stdlib-meta. This one is cheating and only works for closed terms at either never gets

+
-- applied, or where we can safely throw away the arguments (e.g. `unknown`).

+

+
Subst : Set → Set

+
Subst A = ℕ → Term → A → A

+

+
substTerm : Subst Term

+
substArgs : Subst (Args Term)

+
substArg  : Subst (Arg Term)

+
substAbs  : Subst (Abs Term)

+
substSort : Subst Sort

+

+
substTerm x s (var y args) =

+
  case compare x y of λ where

+
    (less _ _)    → var (y ∸ 1) (substArgs x s args)

+
    (equal _)     → s  -- cheating and dropping the args!

+
    (greater _ _) → var y (substArgs x s args)

+
substTerm x s (con c args) = con c (substArgs x s args)

+
substTerm x s (def f args) = def f (substArgs x s args)

+
substTerm x s (lam v t) = lam v (substAbs x s t)

+
substTerm x s (pat-lam cs args₁) = unknown  -- ignoring for now

+
substTerm x s (pi a b) = pi (substArg x s a) (substAbs x s b)

+
substTerm x s (agda-sort s₁) = agda-sort (substSort x s s₁)

+
substTerm x s (lit l) = lit l

+
substTerm x s (meta m args) = meta m (substArgs x s args)

+
substTerm x s unknown = unknown

+

+
substArgs x s [] = []

+
substArgs x s (a ∷ as) = substArg x s a ∷ substArgs x s as

+

+
substArg x s (arg i t) = arg i (substTerm x s t)

+

+
substAbs x s (abs z t) = abs z (substTerm (ℕ.suc x) s t)

+

+
substSort x s (set t) = set (substTerm x s t)

+
substSort x s (lit n) = lit n

+
substSort x s (prop t) = prop (substTerm x s t)

+
substSort x s (propLit n) = propLit n

+
substSort x s (inf n) = inf n

+
substSort x s unknown = unknown

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.Utils.TCI.html b/site/static/agda_html/Reflection.Utils.TCI.html
new file mode 100644
index 000000000..5f14b1927
--- /dev/null
+++ b/site/static/agda_html/Reflection.Utils.TCI.html
@@ -0,0 +1,377 @@
+
+Reflection.Utils.TCI
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
open import Class.Monad

+
open import Class.MonadError.Instances

+
open import Class.MonadReader.Instances

+
open import Class.MonadTC.Instances

+

+
module Reflection.Utils.TCI

+
  {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ ⦃ me : MonadError (List ErrorPart) M ⦄

+
  ⦃ mre : MonadReader TCEnv M ⦄ ⦃ _ : MonadTC M ⦄ where

+

+
import Data.Sum

+
open import Data.List using (map; zipWith)

+
open import Reflection.Utils

+
open import Relation.Nullary.Decidable using (⌊_⌋)

+

+
open import Class.DecEq

+
open import Class.Functor

+
open import Class.Traversable

+

+
private variable

+
  a b : Level

+
  A : Set a

+
  B : Set b

+

+
instance _ = Functor-M

+

+
fresh-level : M Level

+
fresh-level = newMeta (quote Level ∙) >>= unquoteTC

+

+
withHole : Type → (Hole → M ⊤) → M Hole

+
withHole ty k = do

+
  hole′ ← newMeta ty

+
  k hole′

+
  return hole′

+

+
apply : Type → Term → List (Arg Term) → M (Type × Term)

+
apply A t []       = return (A , t)

+
apply A t (a ∷ as) = do

+
  A ← reduce A

+
  A , t ← apply₁ A t a

+
  apply A t as

+
  where

+
    apply₁ : Type → Term → Arg Term → M (Type × Term)

+
    apply₁ (pi (arg i₁@(arg-info k _) A) B) t₁ (arg i₂ t₂) = do

+
      a ← fresh-level

+
      b ← fresh-level

+
      A ← unquoteTC {A = Set a} A

+
      B ← unquoteTC {A = A → Set b} (lam visible B)

+
      t₂ ← unquoteTC {A = A} t₂

+
      Bt₂ ← quoteTC (B t₂)

+
      case k of λ where

+
        visible → do

+
          t₁ ← unquoteTC {A = ∀ (x : A) → B x} t₁

+
          Bt₂ ,_ <$> quoteTC (t₁ t₂)

+
        hidden → do

+
          t₁ ← unquoteTC {A = ∀ {x : A} → B x} t₁

+
          Bt₂ ,_ <$> quoteTC (t₁ {x = t₂})

+
        instance′ → do

+
          t₁ ← unquoteTC {A = ∀ ⦃ x : A ⦄ → B x} t₁

+
          Bt₂ ,_ <$> quoteTC (t₁ ⦃ x = t₂ ⦄)

+
    apply₁ (meta x _) _ _ = blockOnMeta x

+
    apply₁ A          _ _ = error1 "apply: not a Π-type"

+

+
_-∙-_ : Term → Term → M Term

+
f -∙- x = do

+
  ty ← inferType f

+
  proj₂ <$> apply ty f [ vArg x ]

+

+
_-∗-_ : Term → List Term → M Term

+
f -∗- []       = return f

+
f -∗- (x ∷ xs) = f -∙- x >>= _-∗- xs

+

+
instantiate : Hole → M Term

+
instantiate = reduce >=> normalise

+

+
ensureNoMetas : Term → M ⊤

+
ensureNoMetas = λ where

+
  (var x args) → noMetaArgs args

+
  (con c args) → noMetaArgs args

+
  (def f args) → noMetaArgs args

+
  (lam v (abs _ t)) → ensureNoMetas t

+
  (pat-lam cs args) → noMetaClauses cs >> noMetaArgs args

+
  (pi a b) → noMetaArg a >> noMetaAbs b

+
  (agda-sort s) → noMetaSort s

+
  (lit l) → return _

+
  -- (meta x _) → errorP "meta found!"

+
  (meta x _) → blockOnMeta x

+
  unknown → return _

+
   where

+
    noMetaArg : Arg Term → M ⊤

+
    noMetaArg (arg _ v) = ensureNoMetas v

+

+
    noMetaArgs : List (Arg Term) → M ⊤

+
    noMetaArgs [] = return _

+
    noMetaArgs (v ∷ vs) = noMetaArg v >> noMetaArgs vs

+

+
    noMetaClause : Clause → M ⊤

+
    noMetaClause (clause _ ps t) = ensureNoMetas t

+
    noMetaClause (absurd-clause _ ps) = return _

+

+
    noMetaClauses : List Clause → M ⊤

+
    noMetaClauses [] = return _

+
    noMetaClauses (c ∷ cs) = noMetaClause c >> noMetaClauses cs

+

+
    noMetaAbs : Abs Term → M ⊤

+
    noMetaAbs (abs _ v) = ensureNoMetas v

+

+
    noMetaSort : Sort → M ⊤

+
    noMetaSort (set t) = ensureNoMetas t

+
    noMetaSort _       = return _

+

+
module NewMeta where

+
  unifyStrict : THole → Term → M ⊤

+
  unifyStrict (hole , ty) x = withAppendDebugPath "unifyStrict" do

+
    debugLog (hole ∷ᵈ " :=? " ∷ᵈ x ∷ᵈ [])

+
    m ← newMeta ty

+
    noConstraints $ unify m x >> unify hole m

+
    debugLog (hole ∷ᵈ " :=  " ∷ᵈ x ∷ᵈ [])

+

+
printTerm : String → Term → M ⊤

+
printTerm s t = do

+
  ty ← inferType t

+
  debugLog

+
    ( s ∷ᵈ ": {"

+
    ∷ᵈ ty

+
    ∷ᵈ " ∋ "

+
    ∷ᵈ t

+
    ∷ᵈ "}\n"

+
    ∷ᵈ [])

+

+
module NoMeta where

+
  unifyStrict : THole → Term → M ⊤

+
  unifyStrict (hole , ty) x = withAppendDebugPath "unifyStrictNoMeta" do

+
    -- unify hole x

+
    -- instantiate hole >>= ensureNoMetas

+

+
    debugLog ("———————————————————————————————————————\nx" ∷ᵈ x ∷ᵈ [])

+
    unify hole x

+
    hole ← normalise hole

+
    printTerm "hole′" hole

+
    -- (x ∷ hole ∷ []) ← mapM instantiate (x ∷ hole ∷ [])

+
    --   where _ → _IMPOSSIBLE_

+
    -- printTerm "x′" x

+
    ensureNoMetas hole

+
    debugLog1 "No metas found :)\n"

+

+
open NewMeta public

+
-- open NoMeta public

+

+
import Data.List.NonEmpty as NE

+

+
unifyStricts : THole → List Term → M ⊤

+
unifyStricts ht = NE.foldl₁ (λ x y → catch x (λ _ → y))

+
                ∘ (NE._∷ʳ error1 "∅")

+
                ∘ map ({-noConstraints ∘ -}unifyStrict ht)

+

+
compatible? : Type → Type → M Bool

+
compatible? ty ty′ = do

+
  -- print $ show ty ◇ " ≈? " ◇ show ty′

+
  b ← runSpeculative $ (_, false) <$>

+
    catch (unify (varsToUnknown ty) (varsToUnknown ty′) >> return true)

+
          (λ _ → return false)

+
  -- print $ "  ——→ " ◇ show b

+
  return b

+

+
logTelescope : List (Maybe String × Arg Type) → M ⊤

+
logTelescope l = withAppendDebugPath "context" do

+
  catch (helper (length l) l) (λ _ → debugLog1 "Error while printing the context!")

+
  where

+
    helper : ℕ → List (Maybe String × Arg Type) → M ⊤

+
    helper n [] = return _

+
    helper n ((x , ty@(arg _ t)) ∷ l) = do

+
      let name = case x of λ where

+
        nothing   → "?"

+
        (just "") → "_"

+
        (just x)  → x

+
      debugLog ("  " ∷ᵈ name ∷ᵈ " : "  ∷ᵈ mapVars (_∸ (n ∸ (1 + length l))) t ∷ᵈ [])

+
      extendContext (name , ty) $ helper n l

+

+
logContext : Telescope → M ⊤

+
logContext l = withAppendDebugPath "context" do

+
  debugLog1 "Context:"

+
  catch (helper (length l) l) (λ _ → debugLog1 "Error while printing the context!")

+
  catch (do

+
    goalTy ← goalTy

+
    debugLog ("  ⊢ " ∷ᵈ goalTy ∷ᵈ []))

+
    λ _ → do

+
      inj₁ t ← reader TCEnv.goal

+
        where _ → error1 "Bug in logContext!"

+
      debugLog ("Error while infering the goal type! Goal: " ∷ᵈ t ∷ᵈ [])

+
  where

+
    helper : ℕ → Telescope → M ⊤

+
    helper n [] = return _

+
    helper n ((_ , arg _ t) ∷ l) = do

+
      helper n l

+
      debugLog ("  " ∷ᵈ ♯ (n ∸ (length l + 1)) ∷ᵈ " : " ∷ᵈ mapVars (_+ (n ∸ length l)) t ∷ᵈ [])

+

+
logCurrentContext : M ⊤

+
logCurrentContext = markDontFail "logCurrentContext" (logContext =<< getContext)

+

+
inDebugPath : String → M A → M A

+
inDebugPath path x = withAppendDebugPath path do

+
  debugLog (("*****" <+> path <+> "*****") ∷ᵈ [])

+
  -- This throws an internal error if the macro is run in a type signature

+
  -- (Agda issue #7187)

+
  -- logCurrentContext

+
  x

+

+
viewAndReduceTy : Type → M TypeView

+
viewAndReduceTy ty = helper ty =<< (length ∘ proj₁ ∘ viewTy) <$> normalise ty

+
  where

+
    helper : Type → ℕ → M TypeView

+
    helper ty    zero = return $ viewTy ty

+
    helper ty (suc k) with viewTy ty

+
    ... | (tel , ty') = extendContext' (map (λ where (abs s t) → s , t) tel) $ do

+
      rTy ← reduce ty'

+
      (tel' , rTy') ← helper rTy k

+
      return (tel ++ tel' , rTy')

+

+
getType' : Name → M TypeView

+
getType' n = viewAndReduceTy =<< getType n

+

+
getDataDef : Name → M DataDef

+
getDataDef n = inDebugPath "getDataDef" do

+
  debugLog ("Find details for datatype: " ∷ᵈ n ∷ᵈ [])

+
  (data-type pars cs) ← getDefinition n

+
    where _ → error1 "Not a data definition!"

+
  debugLogᵐ ("Constructor names: " ∷ᵈᵐ cs ᵛ ∷ᵈᵐ []ᵐ)

+
  cs' ← traverse ⦃ Functor-List ⦄ (λ n → (n ,_) <$> getType' n) cs

+
  debugLogᵐ ("Result: " ∷ᵈᵐ cs' ᵛⁿ ∷ᵈᵐ []ᵐ)

+
  args ← proj₁ <$> getType' n

+
  return record { name = n ; constructors = cs' ; params = take pars args ; indices = drop pars args }

+

+
getRecordDef : Name → M RecordDef

+
getRecordDef n = do

+
  (record-type c fs) ← getDefinition n

+
    where _ → error1 "Not a record definition!"

+
  args ← proj₁ <$> getType' n

+
  return (record { name = c ; fields = fs ; params = args })

+

+
getDataOrRecordDef : Name → M (DataDef ⊎ RecordDef)

+
getDataOrRecordDef n =

+
  catch (inj₁ <$> getDataDef n)

+
    λ _ → catch (inj₂ <$> getRecordDef n)

+
    λ _ → error1 "Neither a data not a record definition!"

+

+
getParams : Name → M (List (Abs (Arg Type)))

+
getParams n = Data.Sum.[ DataDef.params , RecordDef.params ] <$> getDataOrRecordDef n

+

+
getParamsAndIndices : Name → M (List (Abs (Arg Type)))

+
getParamsAndIndices n =

+
  Data.Sum.[ (λ d → DataDef.params d ++ DataDef.indices d) , RecordDef.params ] <$> getDataOrRecordDef n

+

+
getFuel : String → M ℕ

+
getFuel s = do

+
  fuels ← reader (λ where record { options = record { fuel = f } } → f )

+
  case lookupᵇ (λ s s' → ⌊ s ≟ s' ⌋) fuels s of λ where

+
    nothing  → error1 ("Key" <+> s <+> "doesn't exist in the fuels map")

+
    (just f) → return f

+

+
isSort : Term → M Bool

+
isSort t = do

+
  (agda-sort _) ← normalise t

+
    where _ → return false

+
  return true

+

+
isNArySort : ℕ → Term → M Bool

+
isNArySort n t = do

+
  (tel , ty) ← viewAndReduceTy t

+
  b ← isSort ty

+
  return (⌊ (length tel) ≟ n ⌋ ∧ b)

+

+
isDefT : Name → Term → M Bool

+
isDefT n t = do

+
  (def n' _) ← normalise t

+
    where _ → return false

+
  return ⌊ n ≟ n' ⌋

+

+
-- run a computation that returns a term, resetting the TC state but

+
-- ensuring that the term doesn't contain any metavariables

+
withSafeReset : M Term → M Term

+
withSafeReset x = runAndReset $ do

+
  t ← x

+
  let m = findMetas t

+
  if null m

+
    then return t

+
    else do

+
      debugLogᵐ ("Remaining metavariables:" ∷ᵈᵐ m ᵛⁿ ∷ᵈᵐ []ᵐ)

+
      debugLog ("In term: " ∷ᵈ t ∷ᵈ [])

+
      error1 "Unsolved metavariables remaining in withSafeReset!"

+

+
-- apply a list of terms to a name, picking the correct constructor & visibility

+
applyWithVisibility : Name → List Term → M Term

+
applyWithVisibility n l = do

+
  (argTys , _) ← getType' n

+
  nameConstr n (zipWith (λ where (abs _ (arg i _)) → arg i) argTys l)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.Utils.html b/site/static/agda_html/Reflection.Utils.html
new file mode 100644
index 000000000..612f95d63
--- /dev/null
+++ b/site/static/agda_html/Reflection.Utils.html
@@ -0,0 +1,330 @@
+
+Reflection.Utils
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
module Reflection.Utils where

+

+
open import Meta.Prelude

+
open import Meta.Init hiding (toℕ)

+

+
import Reflection.AST.Abstraction as Abs

+
import Reflection.AST.Name as Name

+
open import Reflection.AST.Argument as Arg hiding (map)

+
open import Reflection.AST.Argument.Information

+
open import Reflection.AST.Argument.Visibility as Vis

+

+
open import Data.Product using (map₁)

+
open import Data.List using (map; zip; zipWith)

+
open import Data.Nat using (_≤ᵇ_)

+
open import Data.Fin using (toℕ)

+

+
open import Relation.Nullary using (Dec)

+
open import Relation.Nullary.Decidable using (⌊_⌋)

+

+
private variable

+
  ℓ : Level

+
  A B : Set ℓ

+

+
-- alternative view of function types as a pair of a list of arguments and a return type

+
TypeView = List (Abs (Arg Type)) × Type

+

+
record DataDef : Set where

+
  field

+
    name : Name

+
    constructors : List (Name × TypeView)

+
    params : List (Abs (Arg Type))

+
    indices : List (Abs (Arg Type))

+

+
record RecordDef : Set where

+
  field

+
    name : Name

+
    fields : List (Arg Name)

+
    params : List (Abs (Arg Type))

+

+
absName : Abs A → String

+
absName (abs s x) = s

+

+
getVisibility : ∀ {a} {A : Set a} → Arg A → Visibility

+
getVisibility (arg (arg-info v _) _) = v

+

+
findMetas : Term → List Term

+
findMetas' : List (Arg Term) → List Term

+
findMetasCl : List Clause → List Term

+

+
findMetas (var x args) = findMetas' args

+
findMetas (con c args) = findMetas' args

+
findMetas (def f args) = findMetas' args

+
findMetas (lam v (abs _ x)) = findMetas x

+
findMetas (pat-lam cs args) = findMetasCl cs Data.List.++ findMetas' args

+
findMetas (pi (arg _ a) (abs _ b)) = findMetas a Data.List.++ findMetas b

+
findMetas (agda-sort s) = []

+
findMetas (lit l) = []

+
findMetas m@(meta x args) = m ∷ findMetas' args

+
findMetas unknown = []

+

+
findMetas' [] = []

+
findMetas' ((arg _ t) ∷ ts) = findMetas t Data.List.++ findMetas' ts

+

+
findMetasCl [] = []

+
findMetasCl (Clause.clause tel ps t ∷ c) = findMetas t Data.List.++ findMetasCl c

+
findMetasCl (Clause.absurd-clause tel ps ∷ c) = findMetasCl c

+

+
isMeta : Term → Bool

+
isMeta (meta _ _) = true

+
isMeta _ = false

+

+
unArgs : Args A → List A

+
unArgs = map unArg

+

+
mapVariables : (ℕ → ℕ) → (Pattern → Pattern)

+
mapVariables f (Pattern.var n)      = Pattern.var (f n)

+
mapVariables f (Pattern.con c args) = Pattern.con c $ go args

+
  where

+
    go : List (Arg Pattern) → List (Arg Pattern)

+
    go [] = []

+
    go (arg i p ∷ l) = arg i (mapVariables f p) ∷ go l

+
mapVariables _ p                    = p

+

+
viewTy : Type → TypeView

+
viewTy (Π[ s ∶ a ] ty) = map₁ ((abs s a) ∷_) (viewTy ty)

+
viewTy ty              = [] , ty

+

+
tyView : TypeView → Type

+
tyView ([] , ty) = ty

+
tyView (abs s a ∷ as , ty) = Π[ s ∶ a ] tyView (as , ty)

+

+
argumentWise : (Type → Type) → Type → Type

+
argumentWise f ty =

+
  let

+
    as , r = viewTy ty

+
    as′ = map (Abs.map $ Arg.map f) as

+
  in tyView (as′ , r)

+

+
viewTy′ : Type → Args Type × Type

+
viewTy′ (Π[ _ ∶ a ] ty) = map₁ (a ∷_) (viewTy′ ty)

+
viewTy′ ty              = [] , ty

+

+
argTys : Type → Args Type

+
argTys = proj₁ ∘ viewTy′

+

+
resultTy : Type → Type

+
resultTy = proj₂ ∘ viewTy′

+

+
tyName : Type → Maybe Name

+
tyName (con n _) = just n

+
tyName (def n _) = just n

+
tyName _         = nothing

+

+
args : Term → Args Term

+
args (var _ xs)  = xs

+
args (def _ xs)  = xs

+
args (con _ xs)  = xs

+
args _           = []

+

+
args′ : Term → List Term

+
args′ = unArgs ∘ args

+

+
module _ (f : ℕ → ℕ) where

+

+
  private

+
    _⁇_ : ℕ → ℕ → ℕ

+
    bound ⁇ x = if bound ≤ᵇ x then f x else x

+

+
  mutual

+
    mapFreeVars : ℕ → (Term → Term)

+
    mapFreeVars bound = λ where

+
      (var x as)

+
        → var (bound ⁇ x) (mapFreeVars∗ bound as)

+
      (def c as)

+
        → def c (mapFreeVars∗ bound as)

+
      (con c as)

+
        → con c (mapFreeVars∗ bound as)

+
      (lam v (abs x t))

+
        → lam v (abs x (mapFreeVars (suc bound) t))

+
      (pat-lam cs as)

+
        → pat-lam (mapFreeVarsᶜ∗ bound cs) (mapFreeVars∗ bound as)

+
      (pi (arg i t) (abs x t′))

+
        → pi (arg i (mapFreeVars bound t)) (abs x (mapFreeVars (suc bound) t′))

+
      (agda-sort s)

+
        → agda-sort (mapFreeVarsˢ bound s)

+
      (meta x as)

+
        → meta x (mapFreeVars∗ bound as)

+
      t → t

+
    mapFreeVars∗ : ℕ → (Args Term → Args Term)

+
    mapFreeVars∗ b = λ where

+
      [] → []

+
      (arg i t ∷ ts) → arg i (mapFreeVars b t) ∷ mapFreeVars∗ b ts

+

+
    mapFreeVarsᵖ : ℕ → (Pattern → Pattern)

+
    mapFreeVarsᵖ b = λ where

+
      (con c ps) → con c (mapFreeVarsᵖ∗ b ps)

+
      (dot t)    → dot (mapFreeVars b t)

+
      (absurd x) → absurd (b ⁇ x)

+
      p → p

+
    mapFreeVarsᵖ∗ : ℕ → (Args Pattern → Args Pattern)

+
    mapFreeVarsᵖ∗ b = λ where

+
      [] → []

+
      (arg i p ∷ ps) → arg i (mapFreeVarsᵖ b p) ∷ mapFreeVarsᵖ∗ (suc b) ps

+

+
    mapFreeVarsᵗ : ℕ → (Telescope → Telescope)

+
    mapFreeVarsᵗ b = λ where

+
      [] → []

+
      ((s , arg i t) ∷ ts) → (s , arg i (mapFreeVars b t)) ∷ mapFreeVarsᵗ (suc b) ts

+

+
    mapFreeVarsᶜ : ℕ → (Clause → Clause)

+
    mapFreeVarsᶜ b = λ where

+
      (clause tel ps t) → clause (mapFreeVarsᵗ b tel) (mapFreeVarsᵖ∗ b ps)

+
                                 (mapFreeVars (length tel + b) t)

+
      (absurd-clause tel ps) → absurd-clause (mapFreeVarsᵗ b tel) (mapFreeVarsᵖ∗ b ps)

+
    mapFreeVarsᶜ∗ : ℕ → (List Clause → List Clause)

+
    mapFreeVarsᶜ∗ b = λ where

+
      [] → []

+
      (c ∷ cs) → mapFreeVarsᶜ b c ∷ mapFreeVarsᶜ∗ b cs

+

+
    mapFreeVarsˢ : ℕ → (Sort → Sort)

+
    mapFreeVarsˢ b = λ where

+
      (set t) → set (mapFreeVars b t)

+
      (prop t) → prop (mapFreeVars b t)

+
      s → s

+

+
  mapVars : Term → Term

+
  mapVars = mapFreeVars 0

+

+
varsToUnknown : Type → Type

+
varsToUnknown′ : Args Type → Args Type

+

+
varsToUnknown (var _ _)  = unknown

+
varsToUnknown (def c xs) = def c (varsToUnknown′ xs)

+
varsToUnknown (con c xs) = con c (varsToUnknown′ xs)

+
varsToUnknown ty         = ty

+

+
varsToUnknown′ []              = []

+
varsToUnknown′ (arg i ty ∷ xs) = arg i (varsToUnknown ty) ∷ varsToUnknown′ xs

+

+
parameters : Definition → ℕ

+
parameters (data-type pars _) = pars

+
parameters _                  = 0

+

+
vArgs : Args A → List A

+
vArgs [] = []

+
vArgs (vArg x ∷ xs) = x ∷ vArgs xs

+
vArgs (_      ∷ xs) = vArgs xs

+

+
argInfo : Arg A → ArgInfo

+
argInfo (arg i _) = i

+

+
isVisible? : (a : Arg A) → Dec (visibility (argInfo a) ≡ visible)

+
isVisible? a = visibility (argInfo a) Vis.≟ visible

+

+
isInstance? : (a : Arg A) → Dec (visibility (argInfo a) ≡ instance′)

+
isInstance? a = visibility (argInfo a) Vis.≟ instance′

+

+
isHidden? : (a : Arg A) → Dec (visibility (argInfo a) ≡ hidden)

+
isHidden? a = visibility (argInfo a) Vis.≟ hidden

+

+
remove-iArgs : Args A → Args A

+
remove-iArgs [] = []

+
remove-iArgs (iArg x ∷ xs) = remove-iArgs xs

+
remove-iArgs (x      ∷ xs) = x ∷ remove-iArgs xs

+

+
hide : Arg A → Arg A

+
hide (vArg x) = hArg x

+
hide (hArg x) = hArg x

+
hide (iArg x) = iArg x

+
hide a        = a

+

+
∀indices⋯ : Args Type → Type → Type

+
∀indices⋯ []       ty = ty

+
∀indices⋯ (i ∷ is) ty = Π[ "_" ∶ hide i ] (∀indices⋯ is ty)

+

+
apply⋯ : Args Type → Name → Type

+
apply⋯ is n = def n $ remove-iArgs $

+
  map (λ{ (n , arg i _) → arg i (♯ (length is ∸ suc (toℕ n)))}) (zip (allFin $ length is) is)

+

+
-- ** records

+

+
mkRecord : List (Name × Term) → Term

+
mkRecord fs = pat-lam (map (λ where (fn , e) → clause [] [ vArg (proj fn) ] e) fs) []

+

+
updateField : List Name → Term → Name → Term → Term

+
updateField fs rexp fn fexp =

+
  pat-lam (flip map fs $ λ f →

+
    if ⌊ f Name.≟ fn ⌋ then

+
      clause [] [ vArg (proj fn) ] fexp

+
    else

+
      clause [] [ vArg (proj f) ] (f ∙⟦ rexp ⟧)

+
    ) []

+

\ No newline at end of file
diff --git a/site/static/agda_html/Reflection.html b/site/static/agda_html/Reflection.html
new file mode 100644
index 000000000..01b2ed57f
--- /dev/null
+++ b/site/static/agda_html/Reflection.html
@@ -0,0 +1,262 @@
+
+Reflection
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Support for reflection

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Reflection where

+

+
------------------------------------------------------------------------

+
-- Re-export contents publicly

+

+
open import Reflection.AST public

+
open import Reflection.TCM public

+
open import Reflection.TCM.Syntax public

+
  using (_>>=_; _>>_)

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
import Reflection.AST.Abstraction as Abstraction

+
import Reflection.AST.Argument as Argument

+
import Reflection.AST.Definition as Definition

+
import Reflection.AST.Meta as Meta

+
import Reflection.AST.Name as Name

+
import Reflection.AST.Literal as Literal

+
import Reflection.AST.Pattern as Pattern

+
import Reflection.AST.Term as Term

+
import Reflection.AST.Argument.Modality as Modality

+
import Reflection.AST.Argument.Quantity as Quantity

+
import Reflection.AST.Argument.Relevance as Relevance

+
import Reflection.AST.Argument.Visibility as Visibility

+
import Reflection.AST.Argument.Information as Information

+

+
-- Version 1.3

+

+
Arg-info = Information.ArgInfo

+
{-# WARNING_ON_USAGE Arg-info

+
"Warning: Arg-info was deprecated in v1.3.

+
Please use Reflection.AST.Argument.Information's ArgInfo instead."

+
#-}

+

+
infix 4 _≟-Lit_ _≟-Name_ _≟-Meta_ _≟-Visibility_ _≟-Relevance_ _≟-Arg-info_

+
        _≟-Pattern_ _≟-ArgPatterns_

+

+
_≟-Lit_ = Literal._≟_

+
{-# WARNING_ON_USAGE _≟-Lit_

+
"Warning: _≟-Lit_ was deprecated in v1.3.

+
Please use Reflection.AST.Literal's _≟_ instead."

+
#-}

+

+
_≟-Name_ = Name._≟_

+
{-# WARNING_ON_USAGE _≟-Name_

+
"Warning: _≟-Name_ was deprecated in v1.3.

+
Please use Reflection.AST.Name's _≟_ instead."

+
#-}

+

+
_≟-Meta_ = Meta._≟_

+
{-# WARNING_ON_USAGE _≟-Meta_

+
"Warning: _≟-Meta_ was deprecated in v1.3.

+
Please use Reflection.AST.Meta's _≟_ instead."

+
#-}

+

+
_≟-Visibility_ = Visibility._≟_

+
{-# WARNING_ON_USAGE _≟-Visibility_

+
"Warning: _≟-Visibility_ was deprecated in v1.3.

+
Please use Reflection.AST.Argument.Visibility's _≟_ instead."

+
#-}

+

+
_≟-Relevance_ = Relevance._≟_

+
{-# WARNING_ON_USAGE _≟-Relevance_

+
"Warning: _≟-Relevance_ was deprecated in v1.3.

+
Please use Reflection.AST.Argument.Relevance's _≟_ instead."

+
#-}

+

+
_≟-Arg-info_ = Information._≟_

+
{-# WARNING_ON_USAGE _≟-Arg-info_

+
"Warning: _≟-Arg-info_ was deprecated in v1.3.

+
Please use Reflection.AST.Argument.Information's _≟_ instead."

+
#-}

+

+
_≟-Pattern_ = Pattern._≟_

+
{-# WARNING_ON_USAGE _≟-Pattern_

+
"Warning: _≟-Pattern_ was deprecated in v1.3.

+
Please use Reflection.AST.Pattern's _≟_ instead."

+
#-}

+

+
_≟-ArgPatterns_ = Pattern._≟s_

+
{-# WARNING_ON_USAGE _≟-ArgPatterns_

+
"Warning: _≟-ArgPatterns_ was deprecated in v1.3.

+
Please use Reflection.AST.Pattern's _≟s_ instead."

+
#-}

+

+
map-Abs = Abstraction.map

+
{-# WARNING_ON_USAGE map-Abs

+
"Warning: map-Abs was deprecated in v1.3.

+
Please use Reflection.AST.Abstraction's map instead."

+
#-}

+

+
map-Arg = Argument.map

+
{-# WARNING_ON_USAGE map-Arg

+
"Warning: map-Arg was deprecated in v1.3.

+
Please use Reflection.AST.Argument's map instead."

+
#-}

+

+
map-Args = Argument.map-Args

+
{-# WARNING_ON_USAGE map-Args

+
"Warning: map-Args was deprecated in v1.3.

+
Please use Reflection.AST.Argument's map-Args instead."

+
#-}

+

+
visibility = Information.visibility

+
{-# WARNING_ON_USAGE visibility

+
"Warning: visibility was deprecated in v1.3.

+
Please use Reflection.AST.Argument.Information's visibility instead."

+
#-}

+

+
relevance = Modality.relevance

+
{-# WARNING_ON_USAGE relevance

+
"Warning: relevance was deprecated in v1.3.

+
Please use Reflection.AST.Argument.Modality's relevance instead."

+
#-}

+

+
infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_

+
        _≟-Clause_ _≟-Clauses_ _≟_

+
        _≟-Sort_

+

+
_≟-AbsTerm_ = Term._≟-AbsTerm_

+
{-# WARNING_ON_USAGE _≟-AbsTerm_

+
"Warning: _≟-AbsTerm_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-AbsTerm_ instead."

+
#-}

+

+
_≟-AbsType_ = Term._≟-AbsType_

+
{-# WARNING_ON_USAGE _≟-AbsType_

+
"Warning: _≟-AbsType_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-AbsType_ instead."

+
#-}

+

+
_≟-ArgTerm_ = Term._≟-ArgTerm_

+
{-# WARNING_ON_USAGE _≟-ArgTerm_

+
"Warning: _≟-ArgTerm_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-ArgTerm_ instead."

+
#-}

+

+
_≟-ArgType_ = Term._≟-ArgType_

+
{-# WARNING_ON_USAGE _≟-ArgType_

+
"Warning: _≟-ArgType_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-ArgType_ instead."

+
#-}

+

+
_≟-Args_    = Term._≟-Args_

+
{-# WARNING_ON_USAGE _≟-Args_

+
"Warning: _≟-Args_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-Args_ instead."

+
#-}

+

+
_≟-Clause_  = Term._≟-Clause_

+
{-# WARNING_ON_USAGE _≟-Clause_

+
"Warning: _≟-Clause_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-Clause_ instead."

+
#-}

+

+
_≟-Clauses_ = Term._≟-Clauses_

+
{-# WARNING_ON_USAGE _≟-Clauses_

+
"Warning: _≟-Clauses_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-Clauses_ instead."

+
#-}

+

+
_≟_         = Term._≟_

+
{-# WARNING_ON_USAGE _≟_

+
"Warning: _≟_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟_ instead."

+
#-}

+

+
_≟-Sort_    = Term._≟-Sort_

+
{-# WARNING_ON_USAGE _≟-Sort_

+
"Warning: _≟-Sort_ was deprecated in v1.3.

+
Please use Reflection.AST.Term's _≟-Sort_ instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Bundles.html b/site/static/agda_html/Relation.Binary.Bundles.html
new file mode 100644
index 000000000..b401025dd
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Bundles.html
@@ -0,0 +1,469 @@
+
+Relation.Binary.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for homogeneous binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Relation.Binary`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Bundles where

+

+
open import Function.Base using (flip)

+
open import Level using (Level; suc; _⊔_)

+
open import Relation.Nullary.Negation.Core using (¬_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Structures -- most of it

+

+
------------------------------------------------------------------------

+
-- Setoids

+
------------------------------------------------------------------------

+

+
record PartialSetoid a ℓ : Set (suc (a ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier              : Set a

+
    _≈_                  : Rel Carrier ℓ

+
    isPartialEquivalence : IsPartialEquivalence _≈_

+

+
  open IsPartialEquivalence isPartialEquivalence public

+

+
  infix 4 _≉_

+
  _≉_ : Rel Carrier _

+
  x ≉ y = ¬ (x ≈ y)

+

+

+
record Setoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier       : Set c

+
    _≈_           : Rel Carrier ℓ

+
    isEquivalence : IsEquivalence _≈_

+

+
  open IsEquivalence isEquivalence public

+
    using (refl; reflexive; isPartialEquivalence)

+

+
  partialSetoid : PartialSetoid c ℓ

+
  partialSetoid = record

+
    { isPartialEquivalence = isPartialEquivalence

+
    }

+

+
  open PartialSetoid partialSetoid public

+
    hiding (Carrier; _≈_; isPartialEquivalence)

+

+

+
record DecSetoid c ℓ : Set (suc (c ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ

+
    isDecEquivalence : IsDecEquivalence _≈_

+

+
  open IsDecEquivalence isDecEquivalence public

+
    using (_≟_; isEquivalence)

+

+
  setoid : Setoid c ℓ

+
  setoid = record

+
    { isEquivalence = isEquivalence

+
    }

+

+
  open Setoid setoid public

+
    hiding (Carrier; _≈_; isEquivalence)

+

+
------------------------------------------------------------------------

+
-- Preorders

+
------------------------------------------------------------------------

+

+
record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≲_

+
  field

+
    Carrier    : Set c

+
    _≈_        : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≲_        : Rel Carrier ℓ₂  -- The relation.

+
    isPreorder : IsPreorder _≈_ _≲_

+

+
  open IsPreorder isPreorder public

+
    hiding (module Eq)

+

+
  module Eq where

+
    setoid : Setoid c ℓ₁

+
    setoid = record

+
      { isEquivalence = isEquivalence

+
      }

+

+
    open Setoid setoid public

+

+
  infix 4 _⋦_

+
  _⋦_ : Rel Carrier _

+
  x ⋦ y = ¬ (x ≲ y)

+

+
  infix 4 _≳_

+
  _≳_ = flip _≲_

+

+
  infix 4 _⋧_

+
  _⋧_ = flip _⋦_

+

+
  -- Deprecated.

+
  infix 4 _∼_

+
  _∼_ = _≲_

+
  {-# WARNING_ON_USAGE _∼_

+
  "Warning: _∼_ was deprecated in v2.0.

+
  Please use _≲_ instead. "

+
  #-}

+

+

+
record TotalPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≲_

+
  field

+
    Carrier         : Set c

+
    _≈_             : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≲_             : Rel Carrier ℓ₂  -- The relation.

+
    isTotalPreorder : IsTotalPreorder _≈_ _≲_

+

+
  open IsTotalPreorder isTotalPreorder public

+
    using (total; isPreorder)

+

+
  preorder : Preorder c ℓ₁ ℓ₂

+
  preorder = record

+
    { isPreorder = isPreorder

+
    }

+

+
  open Preorder preorder public

+
    hiding (Carrier; _≈_; _≲_; isPreorder)

+

+
------------------------------------------------------------------------

+
-- Partial orders

+
------------------------------------------------------------------------

+

+
record Poset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≤_

+
  field

+
    Carrier        : Set c

+
    _≈_            : Rel Carrier ℓ₁

+
    _≤_            : Rel Carrier ℓ₂

+
    isPartialOrder : IsPartialOrder _≈_ _≤_

+

+
  open IsPartialOrder isPartialOrder public

+
    using (antisym; isPreorder)

+

+
  preorder : Preorder c ℓ₁ ℓ₂

+
  preorder = record

+
    { isPreorder = isPreorder

+
    }

+

+
  open Preorder preorder public

+
    hiding (Carrier; _≈_; _≲_; isPreorder)

+
    renaming

+
    ( _⋦_ to _≰_; _≳_ to _≥_; _⋧_ to _≱_

+
    ; ≲-respˡ-≈ to ≤-respˡ-≈

+
    ; ≲-respʳ-≈ to ≤-respʳ-≈

+
    ; ≲-resp-≈  to ≤-resp-≈

+
    )

+

+

+
record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≤_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ₁

+
    _≤_               : Rel Carrier ℓ₂

+
    isDecPartialOrder : IsDecPartialOrder _≈_ _≤_

+

+
  private module DPO = IsDecPartialOrder isDecPartialOrder

+

+
  open DPO public

+
    using (_≟_; _≤?_; isPartialOrder)

+

+
  poset : Poset c ℓ₁ ℓ₂

+
  poset = record

+
    { isPartialOrder = isPartialOrder

+
    }

+

+
  open Poset poset public

+
    hiding (Carrier; _≈_; _≤_; isPartialOrder; module Eq)

+

+
  module Eq where

+
    decSetoid : DecSetoid c ℓ₁

+
    decSetoid = record

+
      { isDecEquivalence = DPO.Eq.isDecEquivalence

+
      }

+

+
    open DecSetoid decSetoid public

+

+

+
record StrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _<_

+
  field

+
    Carrier              : Set c

+
    _≈_                  : Rel Carrier ℓ₁

+
    _<_                  : Rel Carrier ℓ₂

+
    isStrictPartialOrder : IsStrictPartialOrder _≈_ _<_

+

+
  open IsStrictPartialOrder isStrictPartialOrder public

+
    hiding (module Eq)

+

+
  module Eq where

+
    setoid : Setoid c ℓ₁

+
    setoid = record

+
      { isEquivalence = isEquivalence

+
      }

+

+
    open Setoid setoid public

+

+
  infix 4 _≮_

+
  _≮_ : Rel Carrier _

+
  x ≮ y = ¬ (x < y)

+

+
  infix 4 _>_

+
  _>_ = flip _<_

+

+
  infix 4 _≯_

+
  _≯_ = flip _≮_

+

+

+
record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _<_

+
  field

+
    Carrier                 : Set c

+
    _≈_                     : Rel Carrier ℓ₁

+
    _<_                     : Rel Carrier ℓ₂

+
    isDecStrictPartialOrder : IsDecStrictPartialOrder _≈_ _<_

+

+
  private module DSPO = IsDecStrictPartialOrder isDecStrictPartialOrder

+

+
  open DSPO public

+
    using (_<?_; _≟_; isStrictPartialOrder)

+

+
  strictPartialOrder : StrictPartialOrder c ℓ₁ ℓ₂

+
  strictPartialOrder = record

+
    { isStrictPartialOrder = isStrictPartialOrder

+
    }

+

+
  open StrictPartialOrder strictPartialOrder public

+
    hiding (Carrier; _≈_; _<_; isStrictPartialOrder; module Eq)

+

+
  module Eq where

+

+
    decSetoid : DecSetoid c ℓ₁

+
    decSetoid = record

+
      { isDecEquivalence = DSPO.Eq.isDecEquivalence

+
      }

+

+
    open DecSetoid decSetoid public

+

+

+
------------------------------------------------------------------------

+
-- Total orders

+
------------------------------------------------------------------------

+

+
record TotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≤_

+
  field

+
    Carrier      : Set c

+
    _≈_          : Rel Carrier ℓ₁

+
    _≤_          : Rel Carrier ℓ₂

+
    isTotalOrder : IsTotalOrder _≈_ _≤_

+

+
  open IsTotalOrder isTotalOrder public

+
    using (total; isPartialOrder; isTotalPreorder)

+

+
  poset : Poset c ℓ₁ ℓ₂

+
  poset = record

+
    { isPartialOrder = isPartialOrder

+
    }

+

+
  open Poset poset public

+
    hiding (Carrier; _≈_; _≤_; isPartialOrder)

+

+
  totalPreorder : TotalPreorder c ℓ₁ ℓ₂

+
  totalPreorder = record

+
    { isTotalPreorder = isTotalPreorder

+
    }

+

+

+
record DecTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≤_

+
  field

+
    Carrier         : Set c

+
    _≈_             : Rel Carrier ℓ₁

+
    _≤_             : Rel Carrier ℓ₂

+
    isDecTotalOrder : IsDecTotalOrder _≈_ _≤_

+

+
  private module DTO = IsDecTotalOrder isDecTotalOrder

+

+
  open DTO public

+
    using (_≟_; _≤?_; isTotalOrder; isDecPartialOrder)

+

+
  totalOrder : TotalOrder c ℓ₁ ℓ₂

+
  totalOrder = record

+
    { isTotalOrder = isTotalOrder

+
    }

+

+
  open TotalOrder totalOrder public

+
    hiding (Carrier; _≈_; _≤_; isTotalOrder; module Eq)

+

+
  decPoset : DecPoset c ℓ₁ ℓ₂

+
  decPoset = record

+
    { isDecPartialOrder = isDecPartialOrder

+
    }

+

+
  open DecPoset decPoset public

+
    using (module Eq)

+

+

+
-- Note that these orders are decidable. The current implementation

+
-- of `Trichotomous` subsumes irreflexivity and asymmetry. Any reasonable

+
-- definition capturing these three properties implies decidability

+
-- as `Trichotomous` necessarily separates out the equality case.

+

+
record StrictTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _<_

+
  field

+
    Carrier            : Set c

+
    _≈_                : Rel Carrier ℓ₁

+
    _<_                : Rel Carrier ℓ₂

+
    isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_

+

+
  open IsStrictTotalOrder isStrictTotalOrder public

+
    using

+
    ( _≟_; _<?_; compare; isStrictPartialOrder

+
    ; isDecStrictPartialOrder; isDecEquivalence

+
    )

+

+
  strictPartialOrder : StrictPartialOrder c ℓ₁ ℓ₂

+
  strictPartialOrder = record

+
    { isStrictPartialOrder = isStrictPartialOrder

+
    }

+

+
  open StrictPartialOrder strictPartialOrder public

+
    hiding (Carrier; _≈_; _<_; isStrictPartialOrder; module Eq)

+

+
  decStrictPartialOrder : DecStrictPartialOrder c ℓ₁ ℓ₂

+
  decStrictPartialOrder = record

+
    { isDecStrictPartialOrder = isDecStrictPartialOrder

+
    }

+

+
  open DecStrictPartialOrder decStrictPartialOrder public

+
    using (module Eq)

+

+
  decSetoid : DecSetoid c ℓ₁

+
  decSetoid = record

+
    { isDecEquivalence = Eq.isDecEquivalence

+
    }

+
  {-# WARNING_ON_USAGE decSetoid

+
  "Warning: decSetoid was deprecated in v1.3.

+
  Please use Eq.decSetoid instead."

+
  #-}

+

+

+
record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _<_

+
  field

+
    Carrier            : Set c

+
    _≈_                : Rel Carrier ℓ₁

+
    _<_                : Rel Carrier ℓ₂

+
    isDenseLinearOrder : IsDenseLinearOrder _≈_ _<_

+

+
  open IsDenseLinearOrder isDenseLinearOrder public

+
    using (isStrictTotalOrder; dense)

+

+
  strictTotalOrder : StrictTotalOrder c ℓ₁ ℓ₂

+
  strictTotalOrder = record

+
    { isStrictTotalOrder = isStrictTotalOrder

+
    }

+

+
  open StrictTotalOrder strictTotalOrder public

+
    hiding (Carrier; _≈_; _<_; isStrictTotalOrder)

+

+

+
------------------------------------------------------------------------

+
-- Apartness relations

+
------------------------------------------------------------------------

+

+
record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _#_

+
  field

+
    Carrier             : Set c

+
    _≈_                 : Rel Carrier ℓ₁

+
    _#_                 : Rel Carrier ℓ₂

+
    isApartnessRelation : IsApartnessRelation _≈_ _#_

+

+
  open IsApartnessRelation isApartnessRelation public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Consequences.html b/site/static/agda_html/Relation.Binary.Consequences.html
new file mode 100644
index 000000000..99d49f769
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Consequences.html
@@ -0,0 +1,392 @@
+
+Relation.Binary.Consequences
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some properties imply others

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Consequences where

+

+
open import Data.Empty using (⊥-elim)

+
open import Data.Product.Base using (_,_)

+
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)

+
open import Function.Base using (_∘_; _∘₂_; _$_; flip)

+
open import Level using (Level)

+
open import Relation.Binary.Core

+
open import Relation.Binary.Definitions

+
open import Relation.Nullary.Negation.Core using (¬_)

+
open import Relation.Nullary.Decidable.Core

+
  using (yes; no; recompute; map′; dec⇒maybe)

+
open import Relation.Unary using (∁; Pred)

+

+
private

+
  variable

+
    a ℓ ℓ₁ ℓ₂ ℓ₃ ℓ₄ p : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- Substitutive properties

+

+
module _ {_∼_ : Rel A ℓ} (R : Rel A p) where

+

+
  subst⇒respˡ : Substitutive _∼_ p → R Respectsˡ _∼_

+
  subst⇒respˡ subst {y} x′∼x Px′y = subst (flip R y) x′∼x Px′y

+

+
  subst⇒respʳ : Substitutive _∼_ p → R Respectsʳ _∼_

+
  subst⇒respʳ subst {x} y′∼y Pxy′ = subst (R x) y′∼y Pxy′

+

+
  subst⇒resp₂ : Substitutive _∼_ p → R Respects₂ _∼_

+
  subst⇒resp₂ subst = subst⇒respʳ subst , subst⇒respˡ subst

+

+
module _ {_∼_ : Rel A ℓ} {P : Pred A p} where

+

+
  resp⇒¬-resp : Symmetric _∼_ → P Respects _∼_ → (∁ P) Respects _∼_

+
  resp⇒¬-resp sym resp x∼y ¬Px Py = ¬Px (resp (sym x∼y) Py)

+

+
------------------------------------------------------------------------

+
-- Proofs for negation

+

+
module _ {_∼_ : Rel A ℓ} where

+

+
  sym⇒¬-sym : Symmetric _∼_ → Symmetric (¬_ ∘₂ _∼_)

+
  sym⇒¬-sym sym≁ x≁y y∼x = x≁y (sym≁ y∼x)

+

+
  -- N.B. the implicit arguments to Cotransitive are permuted w.r.t.

+
  -- those of Transitive

+
  cotrans⇒¬-trans : Cotransitive _∼_ → Transitive (¬_ ∘₂ _∼_)

+
  cotrans⇒¬-trans cotrans {j = z} x≁z z≁y x∼y =

+
    [ x≁z , z≁y ]′ (cotrans x∼y z)

+

+
------------------------------------------------------------------------

+
-- Proofs for Irreflexive relations

+

+
module _ {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} where

+

+
  irrefl⇒¬-refl : Reflexive _≈_ → Irreflexive _≈_ _∼_ →

+
                  Reflexive (¬_ ∘₂ _∼_)

+
  irrefl⇒¬-refl re irr = irr re

+

+
------------------------------------------------------------------------

+
-- Proofs for non-strict orders

+

+
module _ {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} where

+

+
  total⇒refl : _≤_ Respects₂ _≈_ → Symmetric _≈_ →

+
               Total _≤_ → _≈_ ⇒ _≤_

+
  total⇒refl (respʳ , respˡ) sym total {x} {y} x≈y with total x y

+
  ... | inj₁ x∼y = x∼y

+
  ... | inj₂ y∼x = respʳ x≈y (respˡ (sym x≈y) y∼x)

+

+
  total∧dec⇒dec : _≈_ ⇒ _≤_ → Antisymmetric _≈_ _≤_ →

+
                  Total _≤_ → Decidable _≈_ → Decidable _≤_

+
  total∧dec⇒dec refl antisym total _≟_ x y with total x y

+
  ... | inj₁ x≤y = yes x≤y

+
  ... | inj₂ y≤x = map′ refl (flip antisym y≤x) (x ≟ y)

+

+
module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {≤₁ : Rel A ℓ₃} {≤₂ : Rel B ℓ₄} where

+

+
  mono⇒cong : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ →

+
              ∀ {f} → f Preserves ≤₁ ⟶ ≤₂ → f Preserves ≈₁ ⟶ ≈₂

+
  mono⇒cong sym reflexive antisym mono x≈y = antisym

+
    (mono (reflexive x≈y))

+
    (mono (reflexive (sym x≈y)))

+

+
  antimono⇒cong : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ →

+
                  ∀ {f} → f Preserves ≤₁ ⟶ (flip ≤₂) → f Preserves ≈₁ ⟶ ≈₂

+
  antimono⇒cong sym reflexive antisym antimono p≈q = antisym

+
    (antimono (reflexive (sym p≈q)))

+
    (antimono (reflexive p≈q))

+

+
  mono₂⇒cong₂ : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ → ∀ {f} →

+
                f Preserves₂ ≤₁ ⟶ ≤₁ ⟶ ≤₂ →

+
                f Preserves₂ ≈₁ ⟶ ≈₁ ⟶ ≈₂

+
  mono₂⇒cong₂ sym reflexive antisym mono x≈y u≈v = antisym

+
    (mono (reflexive x≈y) (reflexive u≈v))

+
    (mono (reflexive (sym x≈y)) (reflexive (sym u≈v)))

+

+
------------------------------------------------------------------------

+
-- Proofs for strict orders

+

+
module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where

+

+
  trans∧irr⇒asym : Reflexive _≈_ → Transitive _<_ →

+
                   Irreflexive _≈_ _<_ → Asymmetric _<_

+
  trans∧irr⇒asym refl trans irrefl x<y y<x =

+
    irrefl refl (trans x<y y<x)

+

+
  irr∧antisym⇒asym : Irreflexive _≈_ _<_ → Antisymmetric _≈_ _<_ →

+
                     Asymmetric _<_

+
  irr∧antisym⇒asym irrefl antisym x<y y<x =

+
    irrefl (antisym x<y y<x) x<y

+

+
  asym⇒antisym : Asymmetric _<_ → Antisymmetric _≈_ _<_

+
  asym⇒antisym asym x<y y<x = ⊥-elim (asym x<y y<x)

+

+
  asym⇒irr : _<_ Respects₂ _≈_ → Symmetric _≈_ →

+
             Asymmetric _<_ → Irreflexive _≈_ _<_

+
  asym⇒irr (respʳ , respˡ) sym asym {x} {y} x≈y x<y =

+
    asym x<y (respʳ (sym x≈y) (respˡ x≈y x<y))

+

+
  tri⇒asym : Trichotomous _≈_ _<_ → Asymmetric _<_

+
  tri⇒asym tri {x} {y} x<y x>y with tri x y

+
  ... | tri< _   _ x≯y = x≯y x>y

+
  ... | tri≈ _   _ x≯y = x≯y x>y

+
  ... | tri> x≮y _ _   = x≮y x<y

+

+
  tri⇒irr : Trichotomous _≈_ _<_ → Irreflexive _≈_ _<_

+
  tri⇒irr compare {x} {y} x≈y x<y with compare x y

+
  ... | tri< _   x≉y y≮x = x≉y x≈y

+
  ... | tri> x≮y x≉y y<x = x≉y x≈y

+
  ... | tri≈ x≮y _   y≮x = x≮y x<y

+

+
  tri⇒dec≈ : Trichotomous _≈_ _<_ → Decidable _≈_

+
  tri⇒dec≈ compare x y with compare x y

+
  ... | tri< _ x≉y _ = no  x≉y

+
  ... | tri≈ _ x≈y _ = yes x≈y

+
  ... | tri> _ x≉y _ = no  x≉y

+

+
  tri⇒dec< : Trichotomous _≈_ _<_ → Decidable _<_

+
  tri⇒dec< compare x y with compare x y

+
  ... | tri< x<y _ _ = yes x<y

+
  ... | tri≈ x≮y _ _ = no  x≮y

+
  ... | tri> x≮y _ _ = no  x≮y

+

+
  trans∧tri⇒respʳ : Symmetric _≈_ → Transitive _≈_ →

+
                    Transitive _<_ → Trichotomous _≈_ _<_ →

+
                    _<_ Respectsʳ _≈_

+
  trans∧tri⇒respʳ sym ≈-tr <-tr tri {x} {y} {z} y≈z x<y with tri x z

+
  ... | tri< x<z _ _ = x<z

+
  ... | tri≈ _ x≈z _ = ⊥-elim (tri⇒irr tri (≈-tr x≈z (sym y≈z)) x<y)

+
  ... | tri> _ _ z<x = ⊥-elim (tri⇒irr tri (sym y≈z) (<-tr z<x x<y))

+

+
  trans∧tri⇒respˡ : Transitive _≈_ →

+
                    Transitive _<_ → Trichotomous _≈_ _<_ →

+
                    _<_ Respectsˡ _≈_

+
  trans∧tri⇒respˡ ≈-tr <-tr tri {z} {_} {y} x≈y x<z with tri y z

+
  ... | tri< y<z _ _ = y<z

+
  ... | tri≈ _ y≈z _ = ⊥-elim (tri⇒irr tri (≈-tr x≈y y≈z) x<z)

+
  ... | tri> _ _ z<y = ⊥-elim (tri⇒irr tri x≈y (<-tr x<z z<y))

+

+
  trans∧tri⇒resp : Symmetric _≈_ → Transitive _≈_ →

+
                   Transitive _<_ → Trichotomous _≈_ _<_ →

+
                   _<_ Respects₂ _≈_

+
  trans∧tri⇒resp sym ≈-tr <-tr tri =

+
    trans∧tri⇒respʳ sym ≈-tr <-tr tri ,

+
    trans∧tri⇒respˡ ≈-tr <-tr tri

+

+
------------------------------------------------------------------------

+
-- Without Loss of Generality

+

+
module _  {_R_ : Rel A ℓ₁} {Q : Rel A ℓ₂} where

+

+
  wlog : Total _R_ → Symmetric Q →

+
         (∀ a b → a R b → Q a b) →

+
         ∀ a b → Q a b

+
  wlog r-total q-sym prf a b with r-total a b

+
  ... | inj₁ aRb = prf a b aRb

+
  ... | inj₂ bRa = q-sym (prf b a bRa)

+

+
------------------------------------------------------------------------

+
-- Other proofs

+

+
module _ {R : REL A B p} where

+

+
  dec⇒weaklyDec : Decidable R → WeaklyDecidable R

+
  dec⇒weaklyDec dec x y = dec⇒maybe (dec x y)

+

+
  dec⇒recomputable : Decidable R → Recomputable R

+
  dec⇒recomputable dec {a} {b} = recompute $ dec a b

+

+
module _ {R : REL A B ℓ₁} {S : REL A B ℓ₂} where

+

+
  map-NonEmpty : R ⇒ S → NonEmpty R → NonEmpty S

+
  map-NonEmpty f x = nonEmpty (f (NonEmpty.proof x))

+

+
module _ {R : REL A B ℓ₁} {S : REL B A ℓ₂} where

+

+
  flip-Connex : Connex R S → Connex S R

+
  flip-Connex f x y = Sum.swap (f y x)

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.6

+

+
subst⟶respˡ = subst⇒respˡ

+
{-# WARNING_ON_USAGE subst⟶respˡ

+
"Warning: subst⟶respˡ was deprecated in v1.6.

+
Please use subst⇒respˡ instead."

+
#-}

+
subst⟶respʳ = subst⇒respʳ

+
{-# WARNING_ON_USAGE subst⟶respʳ

+
"Warning: subst⟶respʳ was deprecated in v1.6.

+
Please use subst⇒respʳ instead."

+
#-}

+
subst⟶resp₂ = subst⇒resp₂

+
{-# WARNING_ON_USAGE subst⟶resp₂

+
"Warning: subst⟶resp₂ was deprecated in v1.6.

+
Please use subst⇒resp₂ instead."

+
#-}

+
P-resp⟶¬P-resp = resp⇒¬-resp

+
{-# WARNING_ON_USAGE P-resp⟶¬P-resp

+
"Warning: P-resp⟶¬P-resp was deprecated in v1.6.

+
Please use resp⇒¬-resp instead."

+
#-}

+
total⟶refl = total⇒refl

+
{-# WARNING_ON_USAGE total⟶refl

+
"Warning: total⟶refl was deprecated in v1.6.

+
Please use total⇒refl instead."

+
#-}

+
total+dec⟶dec = total∧dec⇒dec

+
{-# WARNING_ON_USAGE total+dec⟶dec

+
"Warning: total+dec⟶dec was deprecated in v1.6.

+
Please use total∧dec⇒dec instead."

+
#-}

+
trans∧irr⟶asym = trans∧irr⇒asym

+
{-# WARNING_ON_USAGE trans∧irr⟶asym

+
"Warning: trans∧irr⟶asym was deprecated in v1.6.

+
Please use trans∧irr⇒asym instead."

+
#-}

+
irr∧antisym⟶asym = irr∧antisym⇒asym

+
{-# WARNING_ON_USAGE irr∧antisym⟶asym

+
"Warning: irr∧antisym⟶asym was deprecated in v1.6.

+
Please use irr∧antisym⇒asym instead."

+
#-}

+
asym⟶antisym = asym⇒antisym

+
{-# WARNING_ON_USAGE asym⟶antisym

+
"Warning: asym⟶antisym was deprecated in v1.6.

+
Please use asym⇒antisym instead."

+
#-}

+
asym⟶irr = asym⇒irr

+
{-# WARNING_ON_USAGE asym⟶irr

+
"Warning: asym⟶irr was deprecated in v1.6.

+
Please use asym⇒irr instead."

+
#-}

+
tri⟶asym = tri⇒asym

+
{-# WARNING_ON_USAGE tri⟶asym

+
"Warning: tri⟶asym was deprecated in v1.6.

+
Please use tri⇒asym instead."

+
#-}

+
tri⟶irr = tri⇒irr

+
{-# WARNING_ON_USAGE tri⟶irr

+
"Warning: tri⟶irr was deprecated in v1.6.

+
Please use tri⇒irr instead."

+
#-}

+
tri⟶dec≈ = tri⇒dec≈

+
{-# WARNING_ON_USAGE tri⟶dec≈

+
"Warning: tri⟶dec≈ was deprecated in v1.6.

+
Please use tri⇒dec≈ instead."

+
#-}

+
tri⟶dec< = tri⇒dec<

+
{-# WARNING_ON_USAGE tri⟶dec<

+
"Warning: tri⟶dec< was deprecated in v1.6.

+
Please use tri⇒dec< instead."

+
#-}

+
trans∧tri⟶respʳ≈ = trans∧tri⇒respʳ

+
{-# WARNING_ON_USAGE trans∧tri⟶respʳ≈

+
"Warning: trans∧tri⟶respʳ≈ was deprecated in v1.6.

+
Please use trans∧tri⇒respʳ instead."

+
#-}

+
trans∧tri⟶respˡ≈ = trans∧tri⇒respˡ

+
{-# WARNING_ON_USAGE trans∧tri⟶respˡ≈

+
"Warning: trans∧tri⟶respˡ≈ was deprecated in v1.6.

+
Please use trans∧tri⇒respˡ instead."

+
#-}

+
trans∧tri⟶resp≈ = trans∧tri⇒resp

+
{-# WARNING_ON_USAGE trans∧tri⟶resp≈

+
"Warning: trans∧tri⟶resp≈ was deprecated in v1.6.

+
Please use trans∧tri⇒resp instead."

+
#-}

+
dec⟶weaklyDec = dec⇒weaklyDec

+
{-# WARNING_ON_USAGE dec⟶weaklyDec

+
"Warning: dec⟶weaklyDec was deprecated in v1.6.

+
Please use dec⇒weaklyDec instead."

+
#-}

+
dec⟶recomputable = dec⇒recomputable

+
{-# WARNING_ON_USAGE dec⟶recomputable

+
"Warning: dec⟶recomputable was deprecated in v1.6.

+
Please use dec⇒recomputable instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Closure.Reflexive.Properties.html b/site/static/agda_html/Relation.Binary.Construct.Closure.Reflexive.Properties.html
new file mode 100644
index 000000000..2e023e195
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Closure.Reflexive.Properties.html
@@ -0,0 +1,224 @@
+
+Relation.Binary.Construct.Closure.Reflexive.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Some properties of reflexive closures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --safe --cubical-compatible #-}

+

+
module Relation.Binary.Construct.Closure.Reflexive.Properties where

+

+
open import Data.Product.Base as Product

+
open import Data.Sum.Base as Sum

+
open import Function.Bundles using (_⇔_; mk⇔)

+
open import Function.Base using (id)

+
open import Level

+
open import Relation.Binary.Core using (Rel; REL; _=[_]⇒_)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsDecStrictPartialOrder; IsDecPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Symmetric; Transitive; Reflexive; Asymmetric; Antisymmetric; Trichotomous; Total; Decidable; DecidableEquality; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects_; _Respects₂_)

+
open import Relation.Binary.Construct.Closure.Reflexive

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)

+
import Relation.Binary.PropositionalEquality.Properties as ≡

+
open import Relation.Nullary.Negation.Core using (contradiction)

+
open import Relation.Nullary.Decidable as Dec using (yes; no)

+
open import Relation.Unary using (Pred)

+

+
private

+
  variable

+
    a b ℓ p q : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
module _ {P : Rel A p} {Q : Rel B q} where

+

+
  =[]⇒ : ∀ {f : A → B} → P =[ f ]⇒ Q → ReflClosure P =[ f ]⇒ ReflClosure Q

+
  =[]⇒ x [ x∼y ] = [ x x∼y ]

+
  =[]⇒ x refl    = refl

+

+
module _ {_~_ : Rel A ℓ} where

+

+
  private

+
    _~ᵒ_ = ReflClosure _~_

+

+
  fromSum : ∀ {x y} → x ≡ y ⊎ x ~ y → x ~ᵒ y

+
  fromSum (inj₁ refl) = refl

+
  fromSum (inj₂ y) = [ y ]

+

+
  toSum : ∀ {x y} → x ~ᵒ y → x ≡ y ⊎ x ~ y

+
  toSum [ x∼y ] = inj₂ x∼y

+
  toSum refl = inj₁ refl

+

+
  ⊎⇔Refl : ∀ {a b} → (a ≡ b ⊎ a ~ b) ⇔ a ~ᵒ b

+
  ⊎⇔Refl = mk⇔ fromSum toSum

+

+
  sym : Symmetric _~_ → Symmetric _~ᵒ_

+
  sym ~-sym [ x∼y ] = [ ~-sym x∼y ]

+
  sym ~-sym refl    = refl

+

+
  trans : Transitive _~_ → Transitive _~ᵒ_

+
  trans ~-trans [ x∼y ] [ x∼y₁ ] = [ ~-trans x∼y x∼y₁ ]

+
  trans ~-trans [ x∼y ] refl     = [ x∼y ]

+
  trans ~-trans refl    [ x∼y ]  = [ x∼y ]

+
  trans ~-trans refl    refl     = refl

+

+
  antisym : (_≈_ : Rel A p) → Reflexive _≈_ →

+
            Asymmetric _~_ → Antisymmetric _≈_ _~ᵒ_

+
  antisym _≈_ ref asym [ x∼y ] [ y∼x ] = contradiction x∼y (asym y∼x)

+
  antisym _≈_ ref asym [ x∼y ] refl    = ref

+
  antisym _≈_ ref asym refl    _       = ref

+

+
  total : Trichotomous _≡_ _~_ → Total _~ᵒ_

+
  total compare x y with compare x y

+
  ... | tri< a _    _ = inj₁ [ a ]

+
  ... | tri≈ _ refl _ = inj₁ refl

+
  ... | tri> _ _    c = inj₂ [ c ]

+

+
  dec : DecidableEquality A → Decidable _~_ → Decidable _~ᵒ_

+
  dec ≡-dec ~-dec a b = Dec.map ⊎⇔Refl (≡-dec a b Dec.⊎-dec ~-dec a b)

+

+
  decidable : Trichotomous _≡_ _~_ → Decidable _~ᵒ_

+
  decidable ~-tri a b with ~-tri a b

+
  ... | tri< a~b  _  _ = yes [ a~b ]

+
  ... | tri≈ _  refl _ = yes refl

+
  ... | tri> ¬a ¬b   _ = no λ { refl → ¬b refl ; [ p ] → ¬a p }

+

+
  respˡ : ∀ {P : REL A B p} → P Respectsˡ _~_ → P Respectsˡ _~ᵒ_

+
  respˡ p-respˡ-~ [ x∼y ] = p-respˡ-~ x∼y

+
  respˡ _         refl    = id

+

+
  respʳ : ∀ {P : REL B A p} → P Respectsʳ _~_ → P Respectsʳ _~ᵒ_

+
  respʳ = respˡ

+

+
module _ {_~_ : Rel A ℓ} {P : Pred A p} where

+

+
  resp : P Respects _~_ → P Respects (ReflClosure _~_)

+
  resp p-resp-~ [ x∼y ] = p-resp-~ x∼y

+
  resp _        refl    = id

+

+
module _ {_~_ : Rel A ℓ} {P : Rel A p} where

+

+
  resp₂ : P Respects₂ _~_ → P Respects₂ (ReflClosure _~_)

+
  resp₂ = Product.map respˡ respʳ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ {_~_ : Rel A ℓ} where

+

+
  private

+
    _~ᵒ_ = ReflClosure _~_

+

+
  isPreorder : Transitive _~_ → IsPreorder _≡_ _~ᵒ_

+
  isPreorder ~-trans = record

+
    { isEquivalence = ≡.isEquivalence

+
    ; reflexive     = λ { refl → refl }

+
    ; trans         = trans ~-trans

+
    }

+

+
  isPartialOrder : IsStrictPartialOrder _≡_ _~_ → IsPartialOrder _≡_ _~ᵒ_

+
  isPartialOrder O = record

+
    { isPreorder = isPreorder O.trans

+
    ; antisym    = antisym _≡_ refl O.asym

+
    } where module O = IsStrictPartialOrder O

+

+
  isDecPartialOrder : IsDecStrictPartialOrder _≡_ _~_ → IsDecPartialOrder _≡_ _~ᵒ_

+
  isDecPartialOrder O = record

+
    { isPartialOrder = isPartialOrder O.isStrictPartialOrder

+
    ; _≟_            = O._≟_

+
    ; _≤?_           = dec O._≟_ O._<?_

+
    } where module O = IsDecStrictPartialOrder O

+

+
  isTotalOrder : IsStrictTotalOrder _≡_ _~_ → IsTotalOrder _≡_ _~ᵒ_

+
  isTotalOrder O = record

+
    { isPartialOrder = isPartialOrder isStrictPartialOrder

+
    ; total          = total compare

+
    } where open IsStrictTotalOrder O

+

+
  isDecTotalOrder : IsStrictTotalOrder _≡_ _~_ → IsDecTotalOrder _≡_ _~ᵒ_

+
  isDecTotalOrder O = record

+
    { isTotalOrder = isTotalOrder O

+
    ; _≟_          = _≟_

+
    ; _≤?_         = dec _≟_ _<?_

+
    } where open IsStrictTotalOrder O

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Closure.Reflexive.html b/site/static/agda_html/Relation.Binary.Construct.Closure.Reflexive.html
new file mode 100644
index 000000000..72a501bba
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Closure.Reflexive.html
@@ -0,0 +1,159 @@
+
+Relation.Binary.Construct.Closure.Reflexive
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Reflexive closures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Construct.Closure.Reflexive where

+

+
open import Data.Unit.Base

+
open import Level

+
open import Function.Base using (_∋_)

+
open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)

+
open import Relation.Binary.Definitions using (Reflexive)

+
open import Relation.Binary.Construct.Constant.Core using (Const)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_ ; refl)

+

+
private

+
  variable

+
    a ℓ ℓ₁ ℓ₂ : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
data ReflClosure {A : Set a} (_∼_ : Rel A ℓ) : Rel A (a ⊔ ℓ) where

+
  refl : Reflexive (ReflClosure _∼_)

+
  [_]  : ∀ {x y} (x∼y : x ∼ y) → ReflClosure _∼_ x y

+

+
------------------------------------------------------------------------

+
-- Operations

+

+
map : ∀ {R : Rel A ℓ₁} {S : Rel B ℓ₂} {f : A → B} →

+
      R =[ f ]⇒ S → ReflClosure R =[ f ]⇒ ReflClosure S

+
map R⇒S [ xRy ] = [ R⇒S xRy ]

+
map R⇒S refl    = refl

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
-- The reflexive closure has no effect on reflexive relations.

+
drop-refl : {R : Rel A ℓ} → Reflexive R → ReflClosure R ⇒ R

+
drop-refl rfl [ xRy ] = xRy

+
drop-refl rfl refl    = rfl

+

+
reflexive : {R : Rel A ℓ} → _≡_ ⇒ ReflClosure R

+
reflexive refl = refl

+

+
[]-injective : {R : Rel A ℓ} → ∀ {x y p q} →

+
               (ReflClosure R x y ∋ [ p ]) ≡ [ q ] → p ≡ q

+
[]-injective refl = refl

+

+
------------------------------------------------------------------------

+
-- Example usage

+

+
private

+
  module Maybe where

+
    Maybe : Set a → Set a

+
    Maybe A = ReflClosure (Const A) tt tt

+

+
    nothing : Maybe A

+
    nothing = refl

+

+
    just : A → Maybe A

+
    just = [_]

+

+

+

+
------------------------------------------------------------------------

+
-- Deprecations

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- v1.5

+

+
Refl = ReflClosure

+
{-# WARNING_ON_USAGE Refl

+
"Warning: Refl was deprecated in v1.5.

+
Please use ReflClosure instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Composition.html b/site/static/agda_html/Relation.Binary.Construct.Composition.html
new file mode 100644
index 000000000..d976f7408
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Composition.html
@@ -0,0 +1,164 @@
+
+Relation.Binary.Construct.Composition
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Composition of two binary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Construct.Composition where

+

+
open import Data.Product.Base using (∃; _×_; _,_)

+
open import Function.Base

+
open import Level

+
open import Relation.Binary.Core using (Rel; REL; _⇒_)

+
open import Relation.Binary.Structures using (IsPreorder)

+
open import Relation.Binary.Definitions

+
  using (_Respects_; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Reflexive; Transitive)

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infixr 9 _;_

+

+
_;_ : {A : Set a} {B : Set b} {C : Set c} →

+
      REL A B ℓ₁ → REL B C ℓ₂ → REL A C (b ⊔ ℓ₁ ⊔ ℓ₂)

+
L ; R = λ i j → ∃ λ k → L i k × R k j

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (L : Rel A ℓ₁) (R : Rel A ℓ₂) where

+

+
  reflexive : Reflexive L → Reflexive R → Reflexive (L ; R)

+
  reflexive L-refl R-refl = _ , L-refl , R-refl

+

+
  respects : ∀ {p} {P : A → Set p} →

+
             P Respects L → P Respects R → P Respects (L ; R)

+
  respects resp-L resp-R (k , Lik , Rkj) = resp-R Rkj ∘ resp-L Lik

+

+
module _ {≈ : Rel A ℓ} (L : REL A B ℓ₁) (R : REL B C ℓ₂) where

+

+
  respectsˡ : L Respectsˡ ≈ → (L ; R) Respectsˡ ≈

+
  respectsˡ Lˡ i≈i′ (k , Lik , Rkj) = k , Lˡ i≈i′ Lik , Rkj

+

+
module _ {≈ : Rel C ℓ} (L : REL A B ℓ₁) (R : REL B C ℓ₂) where

+

+
  respectsʳ : R Respectsʳ ≈ → (L ; R) Respectsʳ ≈

+
  respectsʳ Rʳ j≈j′ (k , Lik , Rkj) = k , Lik , Rʳ j≈j′ Rkj

+

+
module _ {≈ : Rel A ℓ} (L : REL A B ℓ₁) (R : REL B A ℓ₂) where

+

+
  respects₂ :  L Respectsˡ ≈ → R Respectsʳ ≈ → (L ; R) Respects₂ ≈

+
  respects₂ Lˡ Rʳ = respectsʳ L R Rʳ , respectsˡ L R Lˡ

+

+
module _ {≈ : REL A B ℓ} (L : REL A B ℓ₁) (R : Rel B ℓ₂) where

+

+
  impliesˡ : Reflexive R → (≈ ⇒ L) → (≈ ⇒ L ; R)

+
  impliesˡ R-refl ≈⇒L {i} {j} i≈j = j , ≈⇒L i≈j , R-refl

+

+
module _ {≈ : REL A B ℓ} (L : Rel A ℓ₁) (R : REL A B ℓ₂) where

+

+
  impliesʳ : Reflexive L → (≈ ⇒ R) → (≈ ⇒ L ; R)

+
  impliesʳ L-refl ≈⇒R {i} {j} i≈j = i , L-refl , ≈⇒R i≈j

+

+
module _ (L : Rel A ℓ₁) (R : Rel A ℓ₂) (comm : R ; L ⇒ L ; R) where

+

+
  transitive : Transitive L → Transitive R → Transitive (L ; R)

+
  transitive L-trans R-trans {i} {j} {k} (x , Lix , Rxj) (y , Ljy , Ryk) =

+
    let z , Lxz , Rzy = comm (j , Rxj , Ljy) in z , L-trans Lix  Lxz , R-trans Rzy Ryk

+

+
  isPreorder : {≈ : Rel A ℓ} → IsPreorder ≈ L → IsPreorder ≈ R → IsPreorder ≈ (L ; R)

+
  isPreorder Oˡ Oʳ = record

+
    { isEquivalence = Oˡ.isEquivalence

+
    ; reflexive     = impliesˡ L R Oʳ.refl Oˡ.reflexive

+
    ; trans         = transitive Oˡ.trans Oʳ.trans

+
    }

+
    where module Oˡ = IsPreorder Oˡ; module Oʳ = IsPreorder Oʳ

+

+
transitive⇒≈;≈⊆≈ : (≈ : Rel A ℓ) → Transitive ≈ → (≈ ; ≈) ⇒ ≈

+
transitive⇒≈;≈⊆≈ _ trans (_ , l , r) = trans l r

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Constant.Core.html b/site/static/agda_html/Relation.Binary.Construct.Constant.Core.html
new file mode 100644
index 000000000..c52d8c648
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Constant.Core.html
@@ -0,0 +1,101 @@
+
+Relation.Binary.Construct.Constant.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The binary relation defined by a constant

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Construct.Constant.Core where

+

+
open import Level

+
open import Relation.Binary.Core using (REL)

+

+
private

+
  variable

+
    a b c : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
Const : Set c → REL A B c

+
Const I = λ _ _ → I

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Flip.EqAndOrd.html b/site/static/agda_html/Relation.Binary.Construct.Flip.EqAndOrd.html
new file mode 100644
index 000000000..3e3bda984
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Flip.EqAndOrd.html
@@ -0,0 +1,277 @@
+
+Relation.Binary.Construct.Flip.EqAndOrd
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Many properties which hold for `∼` also hold for `flip ∼`. Unlike

+
-- the module `Relation.Binary.Construct.Flip.Ord` this module does not

+
-- flip the underlying equality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel; REL; _⇒_)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; TotalPreorder)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsTotalPreorder)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive; Asymmetric; Total; _Respects_; _Respects₂_; Minimum; Maximum; Irreflexive; Antisymmetric; Trichotomous; Decidable; tri<; tri>; tri≈)

+

+
module Relation.Binary.Construct.Flip.EqAndOrd where

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (flip; _∘_)

+
open import Level using (Level)

+

+
private

+
  variable

+
    a b p ℓ ℓ₁ ℓ₂ : Level

+
    A B : Set a

+
    ≈ ∼ ≤ < : Rel A ℓ

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (∼ : Rel A ℓ) where

+

+
  refl : Reflexive ∼ → Reflexive (flip ∼)

+
  refl refl = refl

+

+
  sym : Symmetric ∼ → Symmetric (flip ∼)

+
  sym sym = sym

+

+
  trans : Transitive ∼ → Transitive (flip ∼)

+
  trans trans = flip trans

+

+
  asym : Asymmetric ∼ → Asymmetric (flip ∼)

+
  asym asym = asym

+

+
  total : Total ∼ → Total (flip ∼)

+
  total total x y = total y x

+

+
  resp : ∀ {p} (P : A → Set p) → Symmetric ∼ →

+
             P Respects ∼ → P Respects (flip ∼)

+
  resp _ sym resp ∼ = resp (sym ∼)

+

+
  max : ∀ {⊥} → Minimum ∼ ⊥ → Maximum (flip ∼) ⊥

+
  max min = min

+

+
  min : ∀ {⊤} → Maximum ∼ ⊤ → Minimum (flip ∼) ⊤

+
  min max = max

+

+
module _ {≈ : Rel A ℓ₁} (∼ : Rel A ℓ₂) where

+

+
  reflexive : Symmetric ≈ → (≈ ⇒ ∼) → (≈ ⇒ flip ∼)

+
  reflexive sym impl = impl ∘ sym

+

+
  irrefl : Symmetric ≈ → Irreflexive ≈ ∼ → Irreflexive ≈ (flip ∼)

+
  irrefl sym irrefl x≈y y∼x = irrefl (sym x≈y) y∼x

+

+
  antisym : Antisymmetric ≈ ∼ → Antisymmetric ≈ (flip ∼)

+
  antisym antisym = flip antisym

+

+
  compare : Trichotomous ≈ ∼ → Trichotomous ≈ (flip ∼)

+
  compare cmp x y with cmp x y

+
  ... | tri< x<y x≉y y≮x = tri> y≮x x≉y x<y

+
  ... | tri≈ x≮y x≈y y≮x = tri≈ y≮x x≈y x≮y

+
  ... | tri> x≮y x≉y y<x = tri< y<x x≉y x≮y

+

+
module _ (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) where

+

+
  resp₂ : ∼₁ Respects₂ ∼₂ → (flip ∼₁) Respects₂ ∼₂

+
  resp₂ (resp₁ , resp₂) = resp₂ , resp₁

+

+
module _ (∼ : REL A B ℓ) where

+

+
  dec : Decidable ∼ → Decidable (flip ∼)

+
  dec dec = flip dec

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isEquivalence : IsEquivalence ≈ → IsEquivalence (flip ≈)

+
isEquivalence {≈ = ≈} eq = record

+
  { refl  = refl  ≈ Eq.refl

+
  ; sym   = sym   ≈ Eq.sym

+
  ; trans = trans ≈ Eq.trans

+
  } where module Eq = IsEquivalence eq

+

+
isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)

+
isDecEquivalence {≈ = ≈} eq = record

+
  { isEquivalence = isEquivalence Dec.isEquivalence

+
  ; _≟_           = dec ≈ Dec._≟_

+
  } where module Dec = IsDecEquivalence eq

+

+
isPreorder : IsPreorder ≈ ∼ → IsPreorder ≈ (flip ∼)

+
isPreorder {≈ = ≈} {∼ = ∼} O = record

+
  { isEquivalence = O.isEquivalence

+
  ; reflexive     = reflexive ∼ O.Eq.sym O.reflexive

+
  ; trans         = trans ∼ O.trans

+
  } where module O = IsPreorder O

+

+
isTotalPreorder : IsTotalPreorder ≈ ∼ → IsTotalPreorder ≈ (flip ∼)

+
isTotalPreorder O = record

+
  { isPreorder = isPreorder O.isPreorder

+
  ; total      = total _ O.total

+
  } where module O = IsTotalPreorder O

+

+
isPartialOrder : IsPartialOrder ≈ ≤ → IsPartialOrder ≈ (flip ≤)

+
isPartialOrder {≤ = ≤} O = record

+
  { isPreorder = isPreorder O.isPreorder

+
  ; antisym    = antisym ≤ O.antisym

+
  } where module O = IsPartialOrder O

+

+
isTotalOrder : IsTotalOrder ≈ ≤ → IsTotalOrder ≈ (flip ≤)

+
isTotalOrder O = record

+
  { isPartialOrder = isPartialOrder O.isPartialOrder

+
  ; total          = total _ O.total

+
  } where module O = IsTotalOrder O

+

+
isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder ≈ (flip ≤)

+
isDecTotalOrder O = record

+
  { isTotalOrder = isTotalOrder O.isTotalOrder

+
  ; _≟_          = O._≟_

+
  ; _≤?_         = dec _ O._≤?_

+
  } where module O = IsDecTotalOrder O

+

+
isStrictPartialOrder : IsStrictPartialOrder ≈ < →

+
                       IsStrictPartialOrder ≈ (flip <)

+
isStrictPartialOrder {< = <} O = record

+
  { isEquivalence = O.isEquivalence

+
  ; irrefl        = irrefl < O.Eq.sym O.irrefl

+
  ; trans         = trans < O.trans

+
  ; <-resp-≈      = resp₂ _ _ O.<-resp-≈

+
  } where module O = IsStrictPartialOrder O

+

+
isStrictTotalOrder : IsStrictTotalOrder ≈ < →

+
                     IsStrictTotalOrder ≈ (flip <)

+
isStrictTotalOrder {< = <} O = record

+
  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder

+
  ; compare              = compare _ O.compare

+
  } where module O = IsStrictTotalOrder O

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
setoid : Setoid a ℓ → Setoid a ℓ

+
setoid S = record

+
  { isEquivalence = isEquivalence S.isEquivalence

+
  } where module S = Setoid S

+

+
decSetoid : DecSetoid a ℓ → DecSetoid a ℓ

+
decSetoid S = record

+
  { isDecEquivalence = isDecEquivalence S.isDecEquivalence

+
  } where module S = DecSetoid S

+

+
preorder : Preorder a ℓ₁ ℓ₂ → Preorder a ℓ₁ ℓ₂

+
preorder O = record

+
  { isPreorder = isPreorder O.isPreorder

+
  } where module O = Preorder O

+

+
totalPreorder : TotalPreorder a ℓ₁ ℓ₂ → TotalPreorder a ℓ₁ ℓ₂

+
totalPreorder O = record

+
  { isTotalPreorder = isTotalPreorder O.isTotalPreorder

+
  } where module O = TotalPreorder O

+

+
poset : Poset a ℓ₁ ℓ₂ → Poset a ℓ₁ ℓ₂

+
poset O = record

+
  { isPartialOrder = isPartialOrder O.isPartialOrder

+
  } where module O = Poset O

+

+
totalOrder : TotalOrder a ℓ₁ ℓ₂ → TotalOrder a ℓ₁ ℓ₂

+
totalOrder O = record

+
  { isTotalOrder = isTotalOrder O.isTotalOrder

+
  } where module O = TotalOrder O

+

+
decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ → DecTotalOrder a ℓ₁ ℓ₂

+
decTotalOrder O = record

+
  { isDecTotalOrder = isDecTotalOrder O.isDecTotalOrder

+
  } where module O = DecTotalOrder O

+

+
strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →

+
                     StrictPartialOrder a ℓ₁ ℓ₂

+
strictPartialOrder O = record

+
  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder

+
  } where module O = StrictPartialOrder O

+

+
strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂ →

+
                   StrictTotalOrder a ℓ₁ ℓ₂

+
strictTotalOrder O = record

+
  { isStrictTotalOrder = isStrictTotalOrder O.isStrictTotalOrder

+
  } where module O = StrictTotalOrder O

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Intersection.html b/site/static/agda_html/Relation.Binary.Construct.Intersection.html
new file mode 100644
index 000000000..6181a124d
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Intersection.html
@@ -0,0 +1,221 @@
+
+Relation.Binary.Construct.Intersection
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Intersection of two binary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Construct.Intersection where

+

+
open import Data.Product.Base

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])

+
open import Function.Base using (_∘_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Core using (Rel; REL; _⇒_)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsStrictPartialOrder)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive; Antisymmetric; Decidable; _Respects_; _Respectsˡ_; _Respectsʳ_; _Respects₂_; Minimum; Maximum; Irreflexive)

+
open import Relation.Nullary.Decidable using (yes; no; _×-dec_)

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ ℓ₃ : Level

+
    A B : Set a

+
    ≈ L R : Rel A ℓ₁

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infixl 6 _∩_

+

+
_∩_ : REL A B ℓ₁ → REL A B ℓ₂ → REL A B (ℓ₁ ⊔ ℓ₂)

+
L ∩ R = λ i j → L i j × R i j

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
module _ (L : Rel A ℓ₁) (R : Rel A ℓ₂) where

+

+
  reflexive : Reflexive L → Reflexive R → Reflexive (L ∩ R)

+
  reflexive L-refl R-refl = L-refl , R-refl

+

+
  symmetric : Symmetric L → Symmetric R → Symmetric (L ∩ R)

+
  symmetric L-sym R-sym = map L-sym R-sym

+

+
  transitive : Transitive L → Transitive R → Transitive (L ∩ R)

+
  transitive L-trans R-trans = zip L-trans R-trans

+

+
  respects : ∀ {p} (P : A → Set p) →

+
             P Respects L ⊎ P Respects R → P Respects (L ∩ R)

+
  respects P resp (Lxy , Rxy) = [ (λ x → x Lxy) , (λ x → x Rxy) ] resp

+

+
  min : ∀ {⊤} → Minimum L ⊤ → Minimum R ⊤ → Minimum (L ∩ R) ⊤

+
  min L-min R-min = < L-min , R-min >

+

+
  max : ∀ {⊥} → Maximum L ⊥ → Maximum R ⊥ → Maximum (L ∩ R) ⊥

+
  max L-max R-max = < L-max , R-max >

+

+
module _ (≈ : REL A B ℓ₁) {L : REL A B ℓ₂} {R : REL A B ℓ₃} where

+

+
  implies : (≈ ⇒ L) → (≈ ⇒ R) → ≈ ⇒ (L ∩ R)

+
  implies ≈⇒L ≈⇒R = < ≈⇒L , ≈⇒R >

+

+
module _ (≈ : REL A B ℓ₁) (L : REL A B ℓ₂) (R : REL A B ℓ₃) where

+

+
  irreflexive : Irreflexive ≈ L ⊎ Irreflexive ≈ R → Irreflexive ≈ (L ∩ R)

+
  irreflexive irrefl x≈y (Lxy , Rxy) = [ (λ x → x x≈y Lxy) , (λ x → x x≈y Rxy) ] irrefl

+

+
module _ (≈ : Rel A ℓ₁) (L : Rel A ℓ₂) (R : Rel A ℓ₃) where

+

+
  respectsˡ : L Respectsˡ ≈ → R Respectsˡ ≈ → (L ∩ R) Respectsˡ ≈

+
  respectsˡ L-resp R-resp x≈y = map (L-resp x≈y) (R-resp x≈y)

+

+
  respectsʳ : L Respectsʳ ≈ → R Respectsʳ ≈ → (L ∩ R) Respectsʳ ≈

+
  respectsʳ L-resp R-resp x≈y = map (L-resp x≈y) (R-resp x≈y)

+

+
  respects₂ : L Respects₂ ≈ → R Respects₂ ≈ → (L ∩ R) Respects₂ ≈

+
  respects₂ (Lʳ , Lˡ) (Rʳ , Rˡ) = respectsʳ Lʳ Rʳ , respectsˡ Lˡ Rˡ

+

+
  antisymmetric : Antisymmetric ≈ L ⊎ Antisymmetric ≈ R → Antisymmetric ≈ (L ∩ R)

+
  antisymmetric (inj₁ L-antisym) (Lxy , _) (Lyx , _) = L-antisym Lxy Lyx

+
  antisymmetric (inj₂ R-antisym) (_ , Rxy) (_ , Ryx) = R-antisym Rxy Ryx

+

+
module _ {L : REL A B ℓ₁} {R : REL A B ℓ₂} where

+

+
  decidable : Decidable L → Decidable R → Decidable (L ∩ R)

+
  decidable L? R? x y = L? x y ×-dec R? x y

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isEquivalence : IsEquivalence L → IsEquivalence R → IsEquivalence (L ∩ R)

+
isEquivalence {L = L} {R = R} eqₗ eqᵣ = record

+
  { refl  = reflexive  L R L.refl  R.refl

+
  ; sym   = symmetric  L R L.sym   R.sym

+
  ; trans = transitive L R L.trans R.trans

+
  } where module L = IsEquivalence eqₗ; module R = IsEquivalence eqᵣ

+

+
isDecEquivalence : IsDecEquivalence L → IsDecEquivalence R → IsDecEquivalence (L ∩ R)

+
isDecEquivalence eqₗ eqᵣ = record

+
  { isEquivalence = isEquivalence L.isEquivalence R.isEquivalence

+
  ; _≟_           = decidable L._≟_ R._≟_

+
  } where module L = IsDecEquivalence eqₗ; module R = IsDecEquivalence eqᵣ

+

+
isPreorder : IsPreorder ≈ L → IsPreorder ≈ R → IsPreorder ≈ (L ∩ R)

+
isPreorder {≈ = ≈} {L = L} {R = R} Oₗ Oᵣ = record

+
  { isEquivalence = Oₗ.isEquivalence

+
  ; reflexive     = implies ≈ Oₗ.reflexive Oᵣ.reflexive

+
  ; trans         = transitive L R Oₗ.trans Oᵣ.trans

+
  }

+
  where module Oₗ = IsPreorder Oₗ; module Oᵣ = IsPreorder Oᵣ

+

+
isPartialOrderˡ : IsPartialOrder ≈ L → IsPreorder ≈ R → IsPartialOrder ≈ (L ∩ R)

+
isPartialOrderˡ {≈ = ≈} {L = L} {R = R} Oₗ Oᵣ = record

+
  { isPreorder = isPreorder Oₗ.isPreorder Oᵣ

+
  ; antisym    = antisymmetric ≈ L R (inj₁ Oₗ.antisym)

+
  } where module Oₗ = IsPartialOrder Oₗ; module Oᵣ = IsPreorder Oᵣ

+

+
isPartialOrderʳ : IsPreorder ≈ L → IsPartialOrder ≈ R → IsPartialOrder ≈ (L ∩ R)

+
isPartialOrderʳ {≈ = ≈} {L = L} {R = R} Oₗ Oᵣ = record

+
  { isPreorder = isPreorder Oₗ Oᵣ.isPreorder

+
  ; antisym    = antisymmetric ≈ L R (inj₂ Oᵣ.antisym)

+
  } where module Oₗ = IsPreorder Oₗ; module Oᵣ = IsPartialOrder Oᵣ

+

+
isStrictPartialOrderˡ : IsStrictPartialOrder ≈ L →

+
                        Transitive R → R Respects₂ ≈ →

+
                        IsStrictPartialOrder ≈ (L ∩ R)

+
isStrictPartialOrderˡ {≈ = ≈} {L = L} {R = R}  Oₗ transᵣ respᵣ = record

+
  { isEquivalence = Oₗ.isEquivalence

+
  ; irrefl        = irreflexive ≈ L R (inj₁ Oₗ.irrefl)

+
  ; trans         = transitive L R Oₗ.trans transᵣ

+
  ; <-resp-≈      = respects₂ ≈ L R Oₗ.<-resp-≈ respᵣ

+
  } where module Oₗ = IsStrictPartialOrder Oₗ

+

+
isStrictPartialOrderʳ : Transitive L → L Respects₂ ≈ →

+
                        IsStrictPartialOrder ≈ R →

+
                        IsStrictPartialOrder ≈ (L ∩ R)

+
isStrictPartialOrderʳ {L = L} {≈ = ≈} {R = R} transₗ respₗ Oᵣ = record

+
  { isEquivalence = Oᵣ.isEquivalence

+
  ; irrefl        = irreflexive ≈ L R (inj₂ Oᵣ.irrefl)

+
  ; trans         = transitive L R transₗ Oᵣ.trans

+
  ; <-resp-≈      = respects₂ ≈ L R respₗ Oᵣ.<-resp-≈

+
  } where module Oᵣ = IsStrictPartialOrder Oᵣ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.NaturalOrder.Left.html b/site/static/agda_html/Relation.Binary.Construct.NaturalOrder.Left.html
new file mode 100644
index 000000000..d8c17f7a6
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.NaturalOrder.Left.html
@@ -0,0 +1,266 @@
+
+Relation.Binary.Construct.NaturalOrder.Left
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Conversion of binary operators to binary relations via the left

+
-- natural order.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Core

+
open import Data.Product.Base using (_,_; _×_)

+
open import Data.Sum.Base using (inj₁; inj₂; map)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Bundles

+
  using (Preorder; Poset; DecPoset; TotalOrder; DecTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Symmetric; Transitive; Reflexive; Antisymmetric; Total; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Decidable)

+
open import Relation.Nullary.Negation using (¬_)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+
open import Relation.Binary.Lattice using (Infimum)

+

+
module Relation.Binary.Construct.NaturalOrder.Left

+
  {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) (_∙_ : Op₂ A) where

+

+
open import Algebra.Definitions _≈_

+
open import Algebra.Structures _≈_

+
open import Algebra.Lattice.Structures _≈_

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
infix 4 _≤_

+

+
_≤_ : Rel A ℓ

+
x ≤ y = x ≈ (x ∙ y)

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
reflexive : IsMagma _∙_ → Idempotent _∙_ → _≈_ ⇒ _≤_

+
reflexive magma idem {x} {y} x≈y = begin

+
  x     ≈⟨ sym (idem x) ⟩

+
  x ∙ x ≈⟨ ∙-cong refl x≈y ⟩

+
  x ∙ y ∎

+
  where open IsMagma magma; open ≈-Reasoning setoid

+

+
refl : Symmetric _≈_ → Idempotent _∙_ → Reflexive _≤_

+
refl sym idem {x} = sym (idem x)

+

+
antisym : IsEquivalence _≈_ → Commutative _∙_ → Antisymmetric _≈_ _≤_

+
antisym isEq comm {x} {y} x≤y y≤x = begin

+
  x     ≈⟨ x≤y ⟩

+
  x ∙ y ≈⟨ comm x y ⟩

+
  y ∙ x ≈⟨ sym y≤x ⟩

+
  y     ∎

+
  where open IsEquivalence isEq; open ≈-Reasoning (record { isEquivalence = isEq })

+

+
total : Symmetric _≈_ → Transitive _≈_ → Selective _∙_ → Commutative _∙_ → Total _≤_

+
total sym trans sel comm x y = map sym (λ x∙y≈y → trans (sym x∙y≈y) (comm x y)) (sel x y)

+

+
trans : IsSemigroup _∙_ → Transitive _≤_

+
trans semi {x} {y} {z} x≤y y≤z = begin

+
  x           ≈⟨ x≤y ⟩

+
  x ∙ y       ≈⟨ ∙-cong S.refl y≤z ⟩

+
  x ∙ (y ∙ z) ≈⟨ sym (assoc x y z) ⟩

+
  (x ∙ y) ∙ z ≈⟨ ∙-cong (sym x≤y) S.refl ⟩

+
  x ∙ z       ∎

+
  where open module S = IsSemigroup semi; open ≈-Reasoning S.setoid

+

+
respʳ : IsMagma _∙_ → _≤_ Respectsʳ _≈_

+
respʳ magma {x} {y} {z} y≈z x≤y = begin

+
  x     ≈⟨ x≤y ⟩

+
  x ∙ y ≈⟨ ∙-cong M.refl y≈z ⟩

+
  x ∙ z ∎

+
  where open module M = IsMagma magma; open ≈-Reasoning M.setoid

+

+
respˡ : IsMagma _∙_ → _≤_ Respectsˡ _≈_

+
respˡ magma {x} {y} {z} y≈z y≤x = begin

+
  z     ≈⟨ sym y≈z ⟩

+
  y     ≈⟨ y≤x ⟩

+
  y ∙ x ≈⟨ ∙-cong y≈z M.refl ⟩

+
  z ∙ x ∎

+
  where open module M = IsMagma magma; open ≈-Reasoning M.setoid

+

+
resp₂ : IsMagma _∙_ →  _≤_ Respects₂ _≈_

+
resp₂ magma = respʳ magma , respˡ magma

+

+
dec : Decidable _≈_ → Decidable _≤_

+
dec _≟_ x y = x ≟ (x ∙ y)

+

+
module _ (semi : IsSemilattice _∙_) where

+

+
  private open module S = IsSemilattice semi

+
  open ≈-Reasoning setoid

+

+
  x∙y≤x : ∀ x y → (x ∙ y) ≤ x

+
  x∙y≤x x y = begin

+
    x ∙ y       ≈⟨ ∙-cong (sym (idem x)) S.refl ⟩

+
    (x ∙ x) ∙ y ≈⟨ assoc x x y ⟩

+
    x ∙ (x ∙ y) ≈⟨ comm x (x ∙ y) ⟩

+
    (x ∙ y) ∙ x ∎

+

+
  x∙y≤y : ∀ x y → (x ∙ y) ≤ y

+
  x∙y≤y x y = begin

+
    x ∙ y        ≈⟨ ∙-cong S.refl (sym (idem y)) ⟩

+
    x ∙ (y ∙ y)  ≈⟨ sym (assoc x y y) ⟩

+
    (x ∙ y) ∙ y  ∎

+

+
  ∙-presʳ-≤ : ∀ {x y} z → z ≤ x → z ≤ y → z ≤ (x ∙ y)

+
  ∙-presʳ-≤ {x} {y} z z≤x z≤y = begin

+
    z            ≈⟨ z≤y ⟩

+
    z ∙ y        ≈⟨ ∙-cong z≤x S.refl ⟩

+
    (z ∙ x) ∙ y  ≈⟨ assoc z x y ⟩

+
    z ∙ (x ∙ y)  ∎

+

+
  infimum : Infimum _≤_ _∙_

+
  infimum x y = x∙y≤x x y , x∙y≤y x y , ∙-presʳ-≤

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isPreorder : IsBand _∙_ → IsPreorder _≈_ _≤_

+
isPreorder band = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = reflexive isMagma idem

+
  ; trans         = trans isSemigroup

+
  }

+
  where open IsBand band hiding (reflexive; trans)

+

+
isPartialOrder : IsSemilattice _∙_ → IsPartialOrder _≈_ _≤_

+
isPartialOrder semilattice = record

+
  { isPreorder = isPreorder isBand

+
  ; antisym    = antisym isEquivalence comm

+
  }

+
  where open IsSemilattice semilattice

+

+
isDecPartialOrder : IsSemilattice _∙_ → Decidable _≈_ →

+
                    IsDecPartialOrder _≈_ _≤_

+
isDecPartialOrder semilattice _≟_ = record

+
  { isPartialOrder = isPartialOrder semilattice

+
  ; _≟_            = _≟_

+
  ; _≤?_           = dec _≟_

+
  }

+

+
isTotalOrder : IsSemilattice _∙_ → Selective _∙_ → IsTotalOrder _≈_ _≤_

+
isTotalOrder latt sel  = record

+
  { isPartialOrder = isPartialOrder latt

+
  ; total          = total sym S.trans sel comm

+
  }

+
  where open module S = IsSemilattice latt

+

+
isDecTotalOrder : IsSemilattice _∙_ → Selective _∙_ →

+
                  Decidable _≈_ → IsDecTotalOrder _≈_ _≤_

+
isDecTotalOrder latt sel _≟_ = record

+
  { isTotalOrder = isTotalOrder latt sel

+
  ; _≟_          = _≟_

+
  ; _≤?_         = dec _≟_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
preorder : IsBand _∙_ → Preorder a ℓ ℓ

+
preorder band = record

+
  { isPreorder = isPreorder band

+
  }

+

+
poset : IsSemilattice _∙_ → Poset a ℓ ℓ

+
poset latt = record

+
  { isPartialOrder = isPartialOrder latt

+
  }

+

+
decPoset : IsSemilattice _∙_ → Decidable _≈_ → DecPoset a ℓ ℓ

+
decPoset latt dec = record

+
  { isDecPartialOrder = isDecPartialOrder latt dec

+
  }

+

+
totalOrder : IsSemilattice _∙_ → Selective _∙_ → TotalOrder a ℓ ℓ

+
totalOrder latt sel = record

+
  { isTotalOrder = isTotalOrder latt sel

+
  }

+

+
decTotalOrder : IsSemilattice _∙_ → Selective _∙_ →

+
                Decidable _≈_ → DecTotalOrder a ℓ ℓ

+
decTotalOrder latt sel dec = record

+
  { isDecTotalOrder = isDecTotalOrder latt sel dec

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.NonStrictToStrict.html b/site/static/agda_html/Relation.Binary.Construct.NonStrictToStrict.html
new file mode 100644
index 000000000..d1b707d4d
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.NonStrictToStrict.html
@@ -0,0 +1,225 @@
+
+Relation.Binary.Construct.NonStrictToStrict
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Conversion of _≤_ to _<_

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Structures

+
  using (IsPartialOrder; IsEquivalence; IsStrictPartialOrder; IsDecPartialOrder; IsDecStrictPartialOrder; IsTotalOrder; IsStrictTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Trichotomous; Antisymmetric; Symmetric; Total; Decidable; Irreflexive; Transitive; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Trans; Asymmetric; tri≈; tri<; tri>)

+

+
module Relation.Binary.Construct.NonStrictToStrict

+
  {a ℓ₁ ℓ₂} {A : Set a} (_≈_ : Rel A ℓ₁) (_≤_ : Rel A ℓ₂) where

+

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Function.Base using (_∘_; flip)

+
open import Relation.Nullary using (¬_; yes; no)

+
open import Relation.Nullary.Negation using (contradiction)

+
open import Relation.Nullary.Decidable using (¬?; _×-dec_)

+

+
private

+
  _≉_ : Rel A ℓ₁

+
  x ≉ y = ¬ (x ≈ y)

+

+
------------------------------------------------------------------------

+
-- _≤_ can be turned into _<_ as follows:

+

+
infix 4  _<_

+

+
_<_ : Rel A _

+
x < y = x ≤ y × x ≉ y

+

+
------------------------------------------------------------------------

+
-- Relationship between relations

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ = proj₁

+

+
<⇒≉ : ∀ {x y} → x < y → x ≉ y

+
<⇒≉ = proj₂

+

+
≤∧≉⇒< : ∀ {x y} → x ≤ y → x ≉ y → x < y

+
≤∧≉⇒< = _,_

+

+
<⇒≱ : Antisymmetric _≈_ _≤_ → ∀ {x y} → x < y → ¬ (y ≤ x)

+
<⇒≱ antisym (x≤y , x≉y) y≤x = x≉y (antisym x≤y y≤x)

+

+
≤⇒≯ : Antisymmetric _≈_ _≤_ → ∀ {x y} → x ≤ y → ¬ (y < x)

+
≤⇒≯ antisym x≤y y<x = <⇒≱ antisym y<x x≤y

+

+
≰⇒> : Symmetric _≈_ → (_≈_ ⇒ _≤_) → Total _≤_ →

+
      ∀ {x y} → ¬ (x ≤ y) → y < x

+
≰⇒> sym refl total {x} {y} x≰y with total x y

+
... | inj₁ x≤y = contradiction x≤y x≰y

+
... | inj₂ y≤x = y≤x , x≰y ∘ refl ∘ sym

+

+
≮⇒≥ : Symmetric _≈_ → Decidable _≈_ → _≈_ ⇒ _≤_ → Total _≤_ →

+
      ∀ {x y} → ¬ (x < y) → y ≤ x

+
≮⇒≥ sym _≟_ ≤-refl _≤?_ {x} {y} x≮y with x ≟ y | y ≤? x

+
... | yes x≈y  | _        = ≤-refl (sym x≈y)

+
... | _        | inj₁ y≤x = y≤x

+
... | no  x≉y  | inj₂ x≤y = contradiction (x≤y , x≉y) x≮y

+

+
------------------------------------------------------------------------

+
-- Relational properties

+

+
<-irrefl : Irreflexive _≈_ _<_

+
<-irrefl x≈y (_ , x≉y) = x≉y x≈y

+

+
<-trans : IsPartialOrder _≈_ _≤_ → Transitive _<_

+
<-trans po (x≤y , x≉y) (y≤z , y≉z) =

+
  (trans x≤y y≤z , x≉y ∘ antisym x≤y ∘ trans y≤z ∘ reflexive ∘ Eq.sym)

+
  where open IsPartialOrder po

+

+
<-≤-trans : Symmetric _≈_ → Transitive _≤_ → Antisymmetric _≈_ _≤_ →

+
           _≤_ Respectsʳ _≈_ → Trans _<_ _≤_ _<_

+
<-≤-trans sym trans antisym respʳ (x≤y , x≉y) y≤z =

+
  trans x≤y y≤z , (λ x≈z → x≉y (antisym x≤y (respʳ (sym x≈z) y≤z)))

+

+
≤-<-trans : Transitive _≤_ → Antisymmetric _≈_ _≤_ →

+
           _≤_ Respectsˡ _≈_ → Trans _≤_ _<_ _<_

+
≤-<-trans trans antisym respʳ x≤y (y≤z , y≉z) =

+
  trans x≤y y≤z , (λ x≈z → y≉z (antisym y≤z (respʳ x≈z x≤y)))

+

+
<-asym : Antisymmetric _≈_ _≤_ → Asymmetric _<_

+
<-asym antisym (x≤y , x≉y) (y≤x , _) = x≉y (antisym x≤y y≤x)

+

+
<-respˡ-≈ : Transitive _≈_ → _≤_ Respectsˡ _≈_ → _<_ Respectsˡ _≈_

+
<-respˡ-≈ trans respˡ y≈z (y≤x , y≉x) =

+
  respˡ y≈z y≤x , y≉x ∘ trans y≈z

+

+
<-respʳ-≈ : Symmetric _≈_ → Transitive _≈_ →

+
            _≤_ Respectsʳ _≈_ → _<_ Respectsʳ _≈_

+
<-respʳ-≈ sym trans respʳ y≈z (x≤y , x≉y) =

+
  (respʳ y≈z x≤y) , λ x≈z → x≉y (trans x≈z (sym y≈z))

+

+
<-resp-≈ : IsEquivalence _≈_ → _≤_ Respects₂ _≈_ → _<_ Respects₂ _≈_

+
<-resp-≈ eq (respʳ , respˡ) =

+
  <-respʳ-≈ sym trans respʳ , <-respˡ-≈ trans respˡ

+
  where open IsEquivalence eq

+

+
<-trichotomous : Symmetric _≈_ → Decidable _≈_ →

+
                 Antisymmetric _≈_ _≤_ → Total _≤_ →

+
                 Trichotomous _≈_ _<_

+
<-trichotomous ≈-sym _≟_ antisym total x y with x ≟ y

+
... | yes x≈y = tri≈ (<-irrefl x≈y) x≈y (<-irrefl (≈-sym x≈y))

+
... | no  x≉y with total x y

+
...   | inj₁ x≤y = tri< (x≤y , x≉y) x≉y (x≉y ∘ antisym x≤y ∘ proj₁)

+
...   | inj₂ y≤x = tri> (x≉y ∘ flip antisym y≤x ∘ proj₁) x≉y (y≤x , x≉y ∘ ≈-sym)

+

+
<-decidable : Decidable _≈_ → Decidable _≤_ → Decidable _<_

+
<-decidable _≟_ _≤?_ x y = x ≤? y ×-dec ¬? (x ≟ y)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
<-isStrictPartialOrder : IsPartialOrder _≈_ _≤_ →

+
                         IsStrictPartialOrder _≈_ _<_

+
<-isStrictPartialOrder po = record

+
  { isEquivalence = isEquivalence

+
  ; irrefl        = <-irrefl

+
  ; trans         = <-trans po

+
  ; <-resp-≈      = <-resp-≈ isEquivalence ≤-resp-≈

+
  } where open IsPartialOrder po

+

+
<-isDecStrictPartialOrder : IsDecPartialOrder _≈_ _≤_ →

+
                            IsDecStrictPartialOrder _≈_ _<_

+
<-isDecStrictPartialOrder dpo = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder

+
  ; _≟_ = _≟_

+
  ; _<?_ = <-decidable _≟_ _≤?_

+
  } where open IsDecPartialOrder dpo

+

+
<-isStrictTotalOrder₁ : Decidable _≈_ → IsTotalOrder _≈_ _≤_ →

+
                        IsStrictTotalOrder _≈_ _<_

+
<-isStrictTotalOrder₁ ≟ tot = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder

+
  ; compare              = <-trichotomous Eq.sym ≟ antisym total

+
  } where open IsTotalOrder tot

+

+
<-isStrictTotalOrder₂ : IsDecTotalOrder _≈_ _≤_ →

+
                        IsStrictTotalOrder _≈_ _<_

+
<-isStrictTotalOrder₂ dtot = <-isStrictTotalOrder₁ _≟_ isTotalOrder

+
  where open IsDecTotalOrder dtot

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.On.html b/site/static/agda_html/Relation.Binary.Construct.On.html
new file mode 100644
index 000000000..3e90bc9e5
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.On.html
@@ -0,0 +1,301 @@
+
+Relation.Binary.Construct.On
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Many properties which hold for `_∼_` also hold for `_∼_ on f`

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Construct.On where

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_on_; _∘_)

+
open import Induction.WellFounded using (WellFounded; Acc; acc)

+
open import Level using (Level)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Bundles

+
  using (Preorder; Setoid; DecSetoid; Poset; DecPoset; StrictPartialOrder; TotalOrder; DecTotalOrder; StrictTotalOrder)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsStrictPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Irreflexive; Symmetric; Transitive; Antisymmetric; Asymmetric; Decidable; Total; Trichotomous; _Respects_; _Respects₂_)

+

+
private

+
  variable

+
    a b p ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
module _ (f : B → A) where

+

+
  implies : (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) →

+
            ≈ ⇒ ∼ → (≈ on f) ⇒ (∼ on f)

+
  implies _ _ impl = impl

+

+
  reflexive : (∼ : Rel A ℓ) → Reflexive ∼ → Reflexive (∼ on f)

+
  reflexive _ refl = refl

+

+
  irreflexive : (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) →

+
                Irreflexive ≈ ∼ → Irreflexive (≈ on f) (∼ on f)

+
  irreflexive _ _ irrefl = irrefl

+

+
  symmetric : (∼ : Rel A ℓ) → Symmetric ∼ → Symmetric (∼ on f)

+
  symmetric _ sym = sym

+

+
  transitive : (∼ : Rel A ℓ) → Transitive ∼ → Transitive (∼ on f)

+
  transitive _ trans = trans

+

+
  antisymmetric : (≈ : Rel A ℓ₁) (≤ : Rel A ℓ₂) →

+
                  Antisymmetric ≈ ≤ → Antisymmetric (≈ on f) (≤ on f)

+
  antisymmetric _ _ antisym = antisym

+

+
  asymmetric : (< : Rel A ℓ) → Asymmetric < → Asymmetric (< on f)

+
  asymmetric _ asym = asym

+

+
  respects : (∼ : Rel A ℓ) (P : A → Set p) →

+
             P Respects ∼ → (P ∘ f) Respects (∼ on f)

+
  respects _ _ resp = resp

+

+
  respects₂ : (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) →

+
              ∼₁ Respects₂ ∼₂ → (∼₁ on f) Respects₂ (∼₂ on f)

+
  respects₂ _ _ (resp₁ , resp₂) = (resp₁ , resp₂)

+

+
  decidable : (∼ : Rel A ℓ) → Decidable ∼ → Decidable (∼ on f)

+
  decidable _ dec x y = dec (f x) (f y)

+

+
  total : (∼ : Rel A ℓ) → Total ∼ → Total (∼ on f)

+
  total _ tot x y = tot (f x) (f y)

+

+
  trichotomous : (≈ : Rel A ℓ₁) (< : Rel A ℓ₂) →

+
                 Trichotomous ≈ < → Trichotomous (≈ on f) (< on f)

+
  trichotomous _ _ compare x y = compare (f x) (f y)

+

+
  accessible : ∀ {∼ : Rel A ℓ} {x} → Acc ∼ (f x) → Acc (∼ on f) x

+
  accessible (acc rs) = acc (λ fy<fx → accessible (rs fy<fx))

+

+
  wellFounded : {∼ : Rel A ℓ} → WellFounded ∼ → WellFounded (∼ on f)

+
  wellFounded wf x = accessible (wf (f x))

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ (f : B → A) {≈ : Rel A ℓ₁} where

+

+
  isEquivalence : IsEquivalence ≈ →

+
                  IsEquivalence (≈ on f)

+
  isEquivalence eq = record

+
    { refl  = reflexive  f ≈ Eq.refl

+
    ; sym   = symmetric  f ≈ Eq.sym

+
    ; trans = transitive f ≈ Eq.trans

+
    } where module Eq = IsEquivalence eq

+

+
  isDecEquivalence : IsDecEquivalence ≈ →

+
                     IsDecEquivalence (≈ on f)

+
  isDecEquivalence dec = record

+
    { isEquivalence = isEquivalence Dec.isEquivalence

+
    ; _≟_           = decidable f ≈ Dec._≟_

+
    } where module Dec = IsDecEquivalence dec

+

+
module _ (f : B → A) {≈ : Rel A ℓ₁} {∼ : Rel A ℓ₂} where

+

+
  isPreorder : IsPreorder ≈ ∼ → IsPreorder (≈ on f) (∼ on f)

+
  isPreorder pre = record

+
    { isEquivalence = isEquivalence f Pre.isEquivalence

+
    ; reflexive     = implies f ≈ ∼ Pre.reflexive

+
    ; trans         = transitive f ∼ Pre.trans

+
    } where module Pre = IsPreorder pre

+

+
  isPartialOrder : IsPartialOrder ≈ ∼ →

+
                   IsPartialOrder (≈ on f) (∼ on f)

+
  isPartialOrder po = record

+
    { isPreorder = isPreorder Po.isPreorder

+
    ; antisym    = antisymmetric f ≈ ∼ Po.antisym

+
    } where module Po = IsPartialOrder po

+

+
  isDecPartialOrder : IsDecPartialOrder ≈ ∼ →

+
                      IsDecPartialOrder (≈ on f) (∼ on f)

+
  isDecPartialOrder dpo = record

+
    { isPartialOrder = isPartialOrder DPO.isPartialOrder

+
    ; _≟_            = decidable f _ DPO._≟_

+
    ; _≤?_           = decidable f _ DPO._≤?_

+
    } where module DPO = IsDecPartialOrder dpo

+

+
  isStrictPartialOrder : IsStrictPartialOrder ≈ ∼ →

+
                         IsStrictPartialOrder (≈ on f) (∼ on f)

+
  isStrictPartialOrder spo = record

+
    { isEquivalence = isEquivalence f Spo.isEquivalence

+
    ; irrefl        = irreflexive f ≈ ∼ Spo.irrefl

+
    ; trans         = transitive f ∼ Spo.trans

+
    ; <-resp-≈      = respects₂ f ∼ ≈ Spo.<-resp-≈

+
    } where module Spo = IsStrictPartialOrder spo

+

+
  isTotalOrder : IsTotalOrder ≈ ∼ →

+
                 IsTotalOrder (≈ on f) (∼ on f)

+
  isTotalOrder to = record

+
    { isPartialOrder = isPartialOrder To.isPartialOrder

+
    ; total          = total f ∼ To.total

+
    } where module To = IsTotalOrder to

+

+
  isDecTotalOrder : IsDecTotalOrder ≈ ∼ →

+
                    IsDecTotalOrder (≈ on f) (∼ on f)

+
  isDecTotalOrder dec = record

+
    { isTotalOrder = isTotalOrder Dec.isTotalOrder

+
    ; _≟_          = decidable f ≈ Dec._≟_

+
    ; _≤?_         = decidable f ∼ Dec._≤?_

+
    } where module Dec = IsDecTotalOrder dec

+

+
  isStrictTotalOrder : IsStrictTotalOrder ≈ ∼ →

+
                       IsStrictTotalOrder (≈ on f) (∼ on f)

+
  isStrictTotalOrder sto = record

+
    { isStrictPartialOrder = isStrictPartialOrder Sto.isStrictPartialOrder

+
    ; compare              = trichotomous _ _ _ Sto.compare

+
    } where module Sto = IsStrictTotalOrder sto

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
preorder : (P : Preorder a ℓ₁ ℓ₂) →

+
           (B → Preorder.Carrier P) →

+
           Preorder _ _ _

+
preorder P f = record

+
  { isPreorder = isPreorder f (Preorder.isPreorder P)

+
  }

+

+
setoid : (S : Setoid a ℓ) →

+
         (B → Setoid.Carrier S) →

+
         Setoid _ _

+
setoid S f = record

+
  { isEquivalence = isEquivalence f (Setoid.isEquivalence S)

+
  }

+

+
decSetoid : (D : DecSetoid a ℓ) →

+
            (B → DecSetoid.Carrier D) →

+
            DecSetoid _ _

+
decSetoid D f = record

+
  { isDecEquivalence = isDecEquivalence f (DecSetoid.isDecEquivalence D)

+
  }

+

+
poset : ∀ (P : Poset a ℓ₁ ℓ₂) →

+
        (B → Poset.Carrier P) →

+
        Poset _ _ _

+
poset P f = record

+
  { isPartialOrder = isPartialOrder f (Poset.isPartialOrder P)

+
  }

+

+
decPoset : (D : DecPoset a ℓ₁ ℓ₂) →

+
           (B → DecPoset.Carrier D) →

+
           DecPoset _ _ _

+
decPoset D f = record

+
  { isDecPartialOrder =

+
      isDecPartialOrder f (DecPoset.isDecPartialOrder D)

+
  }

+

+
strictPartialOrder : (S : StrictPartialOrder a ℓ₁ ℓ₂) →

+
                     (B → StrictPartialOrder.Carrier S) →

+
                     StrictPartialOrder _ _ _

+
strictPartialOrder S f = record

+
  { isStrictPartialOrder =

+
      isStrictPartialOrder f (StrictPartialOrder.isStrictPartialOrder S)

+
  }

+

+
totalOrder : (T : TotalOrder a ℓ₁ ℓ₂) →

+
             (B → TotalOrder.Carrier T) →

+
             TotalOrder _ _ _

+
totalOrder T f = record

+
  { isTotalOrder = isTotalOrder f (TotalOrder.isTotalOrder T)

+
  }

+

+
decTotalOrder : (D : DecTotalOrder a ℓ₁ ℓ₂) →

+
                (B → DecTotalOrder.Carrier D) →

+
                DecTotalOrder _ _ _

+
decTotalOrder D f = record

+
  { isDecTotalOrder = isDecTotalOrder f (DecTotalOrder.isDecTotalOrder D)

+
  }

+

+
strictTotalOrder : (S : StrictTotalOrder a ℓ₁ ℓ₂) →

+
                   (B → StrictTotalOrder.Carrier S) →

+
                   StrictTotalOrder _ _ _

+
strictTotalOrder S f = record

+
  { isStrictTotalOrder =

+
      isStrictTotalOrder f (StrictTotalOrder.isStrictTotalOrder S)

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.StrictToNonStrict.html b/site/static/agda_html/Relation.Binary.Construct.StrictToNonStrict.html
new file mode 100644
index 000000000..c8f5eb682
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.StrictToNonStrict.html
@@ -0,0 +1,241 @@
+
+Relation.Binary.Construct.StrictToNonStrict
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Conversion of < to ≤, along with a number of properties

+
------------------------------------------------------------------------

+

+
-- Possible TODO: Prove that a conversion ≤ → < → ≤ returns a

+
-- relation equivalent to the original one (and similarly for

+
-- < → ≤ → <).

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Transitive; Symmetric; Irreflexive; Antisymmetric; Trichotomous; Decidable; Trans; Total; _Respects₂_; _Respectsʳ_; _Respectsˡ_; tri<; tri≈; tri>)

+

+
module Relation.Binary.Construct.StrictToNonStrict

+
  {a ℓ₁ ℓ₂} {A : Set a}

+
  (_≈_ : Rel A ℓ₁) (_<_ : Rel A ℓ₂)

+
  where

+

+
open import Data.Product.Base

+
open import Data.Sum.Base

+
open import Data.Empty

+
open import Function.Base

+
open import Relation.Binary.Consequences

+
open import Relation.Nullary.Decidable using (_⊎-dec_; yes; no)

+

+
------------------------------------------------------------------------

+
-- Conversion

+

+
-- _<_ can be turned into _≤_ as follows:

+
infix 4  _≤_

+

+
_≤_ : Rel A _

+
x ≤ y = (x < y) ⊎ (x ≈ y)

+

+
------------------------------------------------------------------------

+
-- The converted relations have certain properties

+
-- (if the original relations have certain other properties)

+

+
<⇒≤ : _<_ ⇒ _≤_

+
<⇒≤ = inj₁

+

+
reflexive : _≈_ ⇒ _≤_

+
reflexive = inj₂

+

+
antisym : IsEquivalence _≈_ → Transitive _<_ → Irreflexive _≈_ _<_ →

+
          Antisymmetric _≈_ _≤_

+
antisym eq trans irrefl = as

+
  where

+
  module Eq = IsEquivalence eq

+

+
  as : Antisymmetric _≈_ _≤_

+
  as (inj₂ x≈y) _          = x≈y

+
  as (inj₁ _)   (inj₂ y≈x) = Eq.sym y≈x

+
  as (inj₁ x<y) (inj₁ y<x) =

+
    ⊥-elim (trans∧irr⇒asym {_≈_ = _≈_} Eq.refl trans irrefl x<y y<x)

+

+
trans : IsEquivalence _≈_ → _<_ Respects₂ _≈_ → Transitive _<_ →

+
        Transitive _≤_

+
trans eq (respʳ , respˡ) <-trans = tr

+
  where

+
  module Eq = IsEquivalence eq

+

+
  tr : Transitive _≤_

+
  tr (inj₁ x<y) (inj₁ y<z) = inj₁ $ <-trans x<y y<z

+
  tr (inj₁ x<y) (inj₂ y≈z) = inj₁ $ respʳ y≈z x<y

+
  tr (inj₂ x≈y) (inj₁ y<z) = inj₁ $ respˡ (Eq.sym x≈y) y<z

+
  tr (inj₂ x≈y) (inj₂ y≈z) = inj₂ $ Eq.trans x≈y y≈z

+

+
<-≤-trans : Transitive _<_ → _<_ Respectsʳ _≈_ → Trans _<_ _≤_ _<_

+
<-≤-trans trans respʳ x<y (inj₁ y<z) = trans x<y y<z

+
<-≤-trans trans respʳ x<y (inj₂ y≈z) = respʳ y≈z x<y

+

+
≤-<-trans : Symmetric _≈_ → Transitive _<_ → _<_ Respectsˡ _≈_ → Trans _≤_ _<_ _<_

+
≤-<-trans sym trans respˡ (inj₁ x<y) y<z = trans x<y y<z

+
≤-<-trans sym trans respˡ (inj₂ x≈y) y<z = respˡ (sym x≈y) y<z

+

+
≤-respʳ-≈ : Transitive _≈_ → _<_ Respectsʳ _≈_ → _≤_ Respectsʳ _≈_

+
≤-respʳ-≈ trans respʳ y′≈y (inj₁ x<y′) = inj₁ (respʳ y′≈y x<y′)

+
≤-respʳ-≈ trans respʳ y′≈y (inj₂ x≈y′) = inj₂ (trans x≈y′ y′≈y)

+

+
≤-respˡ-≈ : Symmetric _≈_ → Transitive _≈_ → _<_ Respectsˡ _≈_ → _≤_ Respectsˡ _≈_

+
≤-respˡ-≈ sym trans respˡ x′≈x (inj₁ x′<y) = inj₁ (respˡ x′≈x x′<y)

+
≤-respˡ-≈ sym trans respˡ x′≈x (inj₂ x′≈y) = inj₂ (trans (sym x′≈x) x′≈y)

+

+
≤-resp-≈ : IsEquivalence _≈_ → _<_ Respects₂ _≈_ → _≤_ Respects₂ _≈_

+
≤-resp-≈ eq (respʳ , respˡ) = ≤-respʳ-≈ Eq.trans respʳ , ≤-respˡ-≈ Eq.sym Eq.trans respˡ

+
  where module Eq = IsEquivalence eq

+

+
total : Trichotomous _≈_ _<_ → Total _≤_

+
total <-tri x y with <-tri x y

+
... | tri< x<y x≉y x≯y = inj₁ (inj₁ x<y)

+
... | tri≈ x≮y x≈y x≯y = inj₁ (inj₂ x≈y)

+
... | tri> x≮y x≉y x>y = inj₂ (inj₁ x>y)

+

+
decidable : Decidable _≈_ → Decidable _<_ → Decidable _≤_

+
decidable ≈-dec <-dec x y = <-dec x y ⊎-dec ≈-dec x y

+

+
decidable′ : Trichotomous _≈_ _<_ → Decidable _≤_

+
decidable′ compare x y with compare x y

+
... | tri< x<y _   _ = yes (inj₁ x<y)

+
... | tri≈ _   x≈y _ = yes (inj₂ x≈y)

+
... | tri> x≮y x≉y _ = no [ x≮y , x≉y ]′

+

+
------------------------------------------------------------------------

+
-- Converting structures

+

+
isPreorder₁ : IsPreorder _≈_ _<_ → IsPreorder _≈_ _≤_

+
isPreorder₁ PO = record

+
  { isEquivalence = S.isEquivalence

+
  ; reflexive     = reflexive

+
  ; trans         = trans S.isEquivalence S.∼-resp-≈ S.trans

+
  }

+
  where module S = IsPreorder PO

+

+
isPreorder₂ : IsStrictPartialOrder _≈_ _<_ → IsPreorder _≈_ _≤_

+
isPreorder₂ SPO = record

+
  { isEquivalence = S.isEquivalence

+
  ; reflexive     = reflexive

+
  ; trans         = trans S.isEquivalence S.<-resp-≈ S.trans

+
  }

+
  where module S = IsStrictPartialOrder SPO

+

+
isPartialOrder : IsStrictPartialOrder _≈_ _<_ → IsPartialOrder _≈_ _≤_

+
isPartialOrder SPO = record

+
  { isPreorder = isPreorder₂ SPO

+
  ; antisym    = antisym S.isEquivalence S.trans S.irrefl

+
  }

+
  where module S = IsStrictPartialOrder SPO

+

+
isTotalOrder : IsStrictTotalOrder _≈_ _<_ → IsTotalOrder _≈_ _≤_

+
isTotalOrder STO = record

+
  { isPartialOrder = isPartialOrder S.isStrictPartialOrder

+
  ; total          = total S.compare

+
  }

+
  where module S = IsStrictTotalOrder STO

+

+
isDecTotalOrder : IsStrictTotalOrder _≈_ _<_ → IsDecTotalOrder _≈_ _≤_

+
isDecTotalOrder STO = record

+
  { isTotalOrder = isTotalOrder STO

+
  ; _≟_          = S._≟_

+
  ; _≤?_         = decidable′ S.compare

+
  }

+
  where module S = IsStrictTotalOrder STO

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 1.4

+

+
decidable' : Trichotomous _≈_ _<_ → Decidable _≤_

+
decidable' = decidable′

+
{-# WARNING_ON_USAGE decidable'

+
"Warning: decidable' (ending in an apostrophe) was deprecated in v1.4.

+
Please use decidable′ (ending in a prime) instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Construct.Subst.Equality.html b/site/static/agda_html/Relation.Binary.Construct.Subst.Equality.html
new file mode 100644
index 000000000..588ed4c4d
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Construct.Subst.Equality.html
@@ -0,0 +1,121 @@
+
+Relation.Binary.Construct.Subst.Equality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Substituting equalities for binary relations

+
------------------------------------------------------------------------

+

+
-- For more general transformations between binary relations

+
-- see `Relation.Binary.Morphisms`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.Product.Base using (_,_)

+
open import Relation.Binary.Core using (Rel; _⇔_)

+
open import Relation.Binary.Structures using (IsEquivalence)

+
open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)

+

+
module Relation.Binary.Construct.Subst.Equality

+
  {a ℓ₁ ℓ₂} {A : Set a} {≈₁ : Rel A ℓ₁} {≈₂ : Rel A ℓ₂}

+
  (equiv@(to , from) : ≈₁ ⇔ ≈₂)

+
  where

+

+
open import Function.Base

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
refl : Reflexive ≈₁ → Reflexive ≈₂

+
refl refl = to refl

+

+
sym : Symmetric ≈₁ → Symmetric ≈₂

+
sym sym = to ∘′ sym ∘′ from

+

+
trans : Transitive ≈₁ → Transitive ≈₂

+
trans trans x≈y y≈z = to (trans (from x≈y) (from y≈z))

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isEquivalence : IsEquivalence ≈₁ → IsEquivalence ≈₂

+
isEquivalence E = record

+
  { refl  = refl  E.refl

+
  ; sym   = sym   E.sym

+
  ; trans = trans E.trans

+
  } where module E = IsEquivalence E

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Core.html b/site/static/agda_html/Relation.Binary.Core.html
new file mode 100644
index 000000000..c2a750bc2
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Core.html
@@ -0,0 +1,143 @@
+
+Relation.Binary.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Relation.Binary`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Core where

+

+
open import Data.Product.Base using (_×_)

+
open import Function.Base using (_on_)

+
open import Level using (Level; _⊔_; suc)

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Definitions

+
------------------------------------------------------------------------

+

+
-- Heterogeneous binary relations

+

+
REL : Set a → Set b → (ℓ : Level) → Set (a ⊔ b ⊔ suc ℓ)

+
REL A B ℓ = A → B → Set ℓ

+

+
-- Homogeneous binary relations

+

+
Rel : Set a → (ℓ : Level) → Set (a ⊔ suc ℓ)

+
Rel A ℓ = REL A A ℓ

+

+
------------------------------------------------------------------------

+
-- Relationships between relations

+
------------------------------------------------------------------------

+

+
infix 4 _⇒_ _⇔_ _=[_]⇒_

+

+
-- Implication/containment - could also be written _⊆_.

+
-- and corresponding notion of equivalence

+

+
_⇒_ : REL A B ℓ₁ → REL A B ℓ₂ → Set _

+
P ⇒ Q = ∀ {x y} → P x y → Q x y

+

+
_⇔_ : REL A B ℓ₁ → REL A B ℓ₂ → Set _

+
P ⇔ Q = P ⇒ Q × Q ⇒ P

+

+
-- Generalised implication - if P ≡ Q it can be read as "f preserves P".

+

+
_=[_]⇒_ : Rel A ℓ₁ → (A → B) → Rel B ℓ₂ → Set _

+
P =[ f ]⇒ Q = P ⇒ (Q on f)

+

+
-- A synonym for _=[_]⇒_.

+

+
_Preserves_⟶_ : (A → B) → Rel A ℓ₁ → Rel B ℓ₂ → Set _

+
f Preserves P ⟶ Q = P =[ f ]⇒ Q

+

+
-- A binary variant of _Preserves_⟶_.

+

+
_Preserves₂_⟶_⟶_ : (A → B → C) → Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → Set _

+
_∙_ Preserves₂ P ⟶ Q ⟶ R = ∀ {x y u v} → P x y → Q u v → R (x ∙ u) (y ∙ v)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Definitions.html b/site/static/agda_html/Relation.Binary.Definitions.html
new file mode 100644
index 000000000..2309fadc8
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Definitions.html
@@ -0,0 +1,336 @@
+
+Relation.Binary.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Relation.Binary`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Definitions where

+

+
open import Agda.Builtin.Equality using (_≡_)

+

+
open import Data.Product.Base using (_×_; ∃-syntax)

+
open import Data.Sum.Base using (_⊎_)

+
open import Function.Base using (_on_; flip)

+
open import Level

+
open import Relation.Binary.Core

+
open import Relation.Nullary as Nullary using (¬_; Dec)

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Definitions

+
------------------------------------------------------------------------

+

+
-- Reflexivity - defined without an underlying equality. It could

+
-- alternatively be defined as `_≈_ ⇒ _∼_` for some equality `_≈_`.

+

+
-- Confusingly the convention in the library is to use the name "refl"

+
-- for proofs of Reflexive and `reflexive` for proofs of type `_≈_ ⇒ _∼_`,

+
-- e.g. in the definition of `IsEquivalence` later in this file. This

+
-- convention is a legacy from the early days of the library.

+

+
Reflexive : Rel A ℓ → Set _

+
Reflexive _∼_ = ∀ {x} → x ∼ x

+

+
-- Generalised symmetry.

+

+
Sym : REL A B ℓ₁ → REL B A ℓ₂ → Set _

+
Sym P Q = P ⇒ flip Q

+

+
-- Symmetry.

+

+
Symmetric : Rel A ℓ → Set _

+
Symmetric _∼_ = Sym _∼_ _∼_

+

+
-- Generalised transitivity.

+

+
Trans : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _

+
Trans P Q R = ∀ {i j k} → P i j → Q j k → R i k

+

+
RightTrans : REL A B ℓ₁ → REL B B ℓ₂ → Set _

+
RightTrans R S = Trans R S R

+

+
LeftTrans : REL A A ℓ₁ → REL A B ℓ₂ → Set _

+
LeftTrans S R = Trans S R R

+

+
-- A flipped variant of generalised transitivity.

+

+
TransFlip : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _

+
TransFlip P Q R = ∀ {i j k} → Q j k → P i j → R i k

+

+
-- Transitivity.

+

+
Transitive : Rel A ℓ → Set _

+
Transitive _∼_ = Trans _∼_ _∼_ _∼_

+

+
-- Generalised antisymmetry

+

+
Antisym : REL A B ℓ₁ → REL B A ℓ₂ → REL A B ℓ₃ → Set _

+
Antisym R S E = ∀ {i j} → R i j → S j i → E i j

+

+
-- Antisymmetry.

+

+
Antisymmetric : Rel A ℓ₁ → Rel A ℓ₂ → Set _

+
Antisymmetric _≈_ _≤_ = Antisym _≤_ _≤_ _≈_

+

+
-- Irreflexivity - this is defined terms of the underlying equality.

+

+
Irreflexive : REL A B ℓ₁ → REL A B ℓ₂ → Set _

+
Irreflexive _≈_ _<_ = ∀ {x y} → x ≈ y → ¬ (x < y)

+

+
-- Asymmetry.

+

+
Asymmetric : Rel A ℓ → Set _

+
Asymmetric _<_ = ∀ {x y} → x < y → ¬ (y < x)

+

+
-- Density

+

+
Dense : Rel A ℓ → Set _

+
Dense _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y

+

+
-- Generalised connex - at least one of the two relations holds.

+

+
Connex : REL A B ℓ₁ → REL B A ℓ₂ → Set _

+
Connex P Q = ∀ x y → P x y ⊎ Q y x

+

+
-- Totality.

+

+
Total : Rel A ℓ → Set _

+
Total _∼_ = Connex _∼_ _∼_

+

+
-- Generalised trichotomy - exactly one of three types has a witness.

+

+
data Tri (A : Set a) (B : Set b) (C : Set c) : Set (a ⊔ b ⊔ c) where

+
  tri< : ( a :   A) (¬b : ¬ B) (¬c : ¬ C) → Tri A B C

+
  tri≈ : (¬a : ¬ A) ( b :   B) (¬c : ¬ C) → Tri A B C

+
  tri> : (¬a : ¬ A) (¬b : ¬ B) ( c :   C) → Tri A B C

+

+
-- Trichotomy.

+

+
Trichotomous : Rel A ℓ₁ → Rel A ℓ₂ → Set _

+
Trichotomous _≈_ _<_ = ∀ x y → Tri (x < y) (x ≈ y) (x > y)

+
  where _>_ = flip _<_

+

+
-- Generalised maximum element.

+

+
Max : REL A B ℓ → B → Set _

+
Max _≤_ T = ∀ x → x ≤ T

+

+
-- Maximum element.

+

+
Maximum : Rel A ℓ → A → Set _

+
Maximum = Max

+

+
-- Generalised minimum element.

+

+
Min : REL A B ℓ → A → Set _

+
Min R = Max (flip R)

+

+
-- Minimum element.

+

+
Minimum : Rel A ℓ → A → Set _

+
Minimum = Min

+

+
-- Definitions for apartness relations

+

+
-- Note that Cotransitive's arguments are permuted with respect to Transitive's.

+
Cotransitive : Rel A ℓ → Set _

+
Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)

+

+
Tight : Rel A ℓ₁ → Rel A ℓ₂ → Set _

+
Tight _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)

+

+
-- Properties of order morphisms, aka order-preserving maps

+

+
Monotonic₁ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _

+
Monotonic₁ _≤_ _⊑_ f = f Preserves _≤_ ⟶ _⊑_

+

+
Antitonic₁ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _

+
Antitonic₁ _≤_ _⊑_ f = f Preserves (flip _≤_) ⟶ _⊑_

+

+
Monotonic₂ : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _

+
Monotonic₂ _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ _⊑_ ⟶ _≼_

+

+
MonotonicAntitonic : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _

+
MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_

+

+
AntitonicMonotonic : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _

+
AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_

+

+
Antitonic₂ : Rel A ℓ₁ → Rel B ℓ₂ → Rel C ℓ₃ → (A → B → C) → Set _

+
Antitonic₂ _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_

+

+
Adjoint : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → (B → A) → Set _

+
Adjoint _≤_ _⊑_ f g = ∀ {x y} → (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)

+

+
-- Unary relations respecting a binary relation.

+

+
_⟶_Respects_ : (A → Set ℓ₁) → (B → Set ℓ₂) → REL A B ℓ₃ → Set _

+
P ⟶ Q Respects _∼_ = ∀ {x y} → x ∼ y → P x → Q y

+

+
-- Unary relation respects a binary relation.

+

+
_Respects_ : (A → Set ℓ₁) → Rel A ℓ₂ → Set _

+
P Respects _∼_ = P ⟶ P Respects _∼_

+

+
-- Right respecting - relatedness is preserved on the right by equality.

+

+
_Respectsʳ_ : REL A B ℓ₁ → Rel B ℓ₂ → Set _

+
_∼_ Respectsʳ _≈_ = ∀ {x} → (x ∼_) Respects _≈_

+

+
-- Left respecting - relatedness is preserved on the left by equality.

+

+
_Respectsˡ_ : REL A B ℓ₁ → Rel A ℓ₂ → Set _

+
P Respectsˡ _∼_ = ∀ {y} → (flip P y) Respects _∼_

+

+
-- Respecting - relatedness is preserved on both sides by equality

+

+
_Respects₂_ : Rel A ℓ₁ → Rel A ℓ₂ → Set _

+
P Respects₂ _∼_ = (P Respectsʳ _∼_) × (P Respectsˡ _∼_)

+

+
-- Substitutivity - any two related elements satisfy exactly the same

+
-- set of unary relations. Note that only the various derivatives

+
-- of propositional equality can satisfy this property.

+

+
Substitutive : Rel A ℓ₁ → (ℓ₂ : Level) → Set _

+
Substitutive {A = A} _∼_ p = (P : A → Set p) → P Respects _∼_

+

+
-- Irrelevancy - all proofs that a given pair of elements are related

+
-- are indistinguishable.

+

+
Irrelevant : REL A B ℓ → Set _

+
Irrelevant _∼_ = ∀ {x y} → Nullary.Irrelevant (x ∼ y)

+

+
-- Recomputability - we can rebuild a relevant proof given an

+
-- irrelevant one.

+

+
Recomputable : REL A B ℓ → Set _

+
Recomputable _∼_ = ∀ {x y} → Nullary.Recomputable (x ∼ y)

+

+
-- Stability

+

+
Stable : REL A B ℓ → Set _

+
Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)

+

+
-- Weak decidability - it is sometimes possible to determine if a given

+
-- pair of elements are related.

+

+
WeaklyDecidable : REL A B ℓ → Set _

+
WeaklyDecidable _∼_ = ∀ x y → Nullary.WeaklyDecidable (x ∼ y)

+

+
-- Decidability - it is possible to determine whether a given pair of

+
-- elements are related.

+

+
Decidable : REL A B ℓ → Set _

+
Decidable _∼_ = ∀ x y → Dec (x ∼ y)

+

+
-- Propositional equality is decidable for the type.

+

+
DecidableEquality : (A : Set a) → Set _

+
DecidableEquality A = Decidable {A = A} _≡_

+

+
-- Universal - all pairs of elements are related

+

+
Universal : REL A B ℓ → Set _

+
Universal _∼_ = ∀ x y → x ∼ y

+

+
-- Empty - no elements are related

+

+
Empty : REL A B ℓ → Set _

+
Empty _∼_ = ∀ {x y} → ¬ (x ∼ y)

+

+
-- Non-emptiness - at least one pair of elements are related.

+

+
record NonEmpty {A : Set a} {B : Set b}

+
                (T : REL A B ℓ) : Set (a ⊔ b ⊔ ℓ) where

+
  constructor nonEmpty

+
  field

+
    {x}   : A

+
    {y}   : B

+
    proof : T x y

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Bundles.html b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Bundles.html
new file mode 100644
index 000000000..fa24e3743
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Bundles.html
@@ -0,0 +1,134 @@
+
+Relation.Binary.Indexed.Heterogeneous.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Indexed binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Indexed.Heterogeneous`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Indexed.Heterogeneous.Bundles where

+

+
open import Level using (suc; _⊔_)

+
open import Relation.Binary.Indexed.Heterogeneous.Core

+
open import Relation.Binary.Indexed.Heterogeneous.Structures

+

+
------------------------------------------------------------------------

+
-- Definitions

+

+
record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where

+
  infix 4 _≈_

+
  field

+
    Carrier       : I → Set c

+
    _≈_           : IRel Carrier ℓ

+
    isEquivalence : IsIndexedEquivalence Carrier _≈_

+

+
  open IsIndexedEquivalence isEquivalence public

+

+

+
record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :

+
                       Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix 4 _≈_ _≲_

+
  field

+
    Carrier    : I → Set c

+
    _≈_        : IRel Carrier ℓ₁  -- The underlying equality.

+
    _≲_        : IRel Carrier ℓ₂  -- The relation.

+
    isPreorder : IsIndexedPreorder Carrier _≈_ _≲_

+

+
  open IsIndexedPreorder isPreorder public

+

+
  infix 4 _∼_

+
  _∼_ = _≲_

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
{-# WARNING_ON_USAGE IndexedPreorder._∼_

+
"Warning: IndexedPreorder._∼_ was deprecated in v2.0. Please use IndexedPreorder._≲_ instead. "

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html
new file mode 100644
index 000000000..35143e530
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html
@@ -0,0 +1,136 @@
+
+Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Creates trivially indexed records from their non-indexed counterpart.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Indexed.Heterogeneous.Construct.Trivial

+
  {i} {I : Set i} where

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Setoid; Preorder)

+
open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)

+
open import Relation.Binary.Indexed.Heterogeneous

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
module _ {a} {A : Set a} where

+

+
  private

+
    Aᵢ : I → Set a

+
    Aᵢ i = A

+

+
  isIndexedEquivalence : ∀ {ℓ} {_≈_ : Rel A ℓ} → IsEquivalence _≈_ →

+
                         IsIndexedEquivalence Aᵢ _≈_

+
  isIndexedEquivalence isEq = record

+
    { refl  = refl

+
    ; sym   = sym

+
    ; trans = trans

+
    }

+
    where open IsEquivalence isEq

+

+
  isIndexedPreorder : ∀ {ℓ₁ ℓ₂} {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} →

+
                      IsPreorder _≈_ _∼_ →

+
                      IsIndexedPreorder Aᵢ _≈_ _∼_

+
  isIndexedPreorder isPreorder = record

+
    { isEquivalence = isIndexedEquivalence isEquivalence

+
    ; reflexive     = reflexive

+
    ; trans         = trans

+
    }

+
    where open IsPreorder isPreorder

+

+
------------------------------------------------------------------------

+
-- Bundles

+

+
indexedSetoid : ∀ {a ℓ} → Setoid a ℓ → IndexedSetoid I a ℓ

+
indexedSetoid S = record

+
  { isEquivalence = isIndexedEquivalence isEquivalence

+
  }

+
  where open Setoid S

+

+
indexedPreorder : ∀ {a ℓ₁ ℓ₂} → Preorder a ℓ₁ ℓ₂ →

+
                  IndexedPreorder I a ℓ₁ ℓ₂

+
indexedPreorder O = record

+
  { isPreorder = isIndexedPreorder isPreorder

+
  }

+
  where open Preorder O

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Core.html b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Core.html
new file mode 100644
index 000000000..32e0bb8ae
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Core.html
@@ -0,0 +1,115 @@
+
+Relation.Binary.Indexed.Heterogeneous.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Indexed binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Indexed.Heterogeneous`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Indexed.Heterogeneous.Core where

+

+
open import Level

+
import Relation.Binary.Core as B

+

+
------------------------------------------------------------------------

+
-- Indexed binary relations

+

+
-- Heterogeneous types

+

+
IREL : ∀ {i₁ i₂ a₁ a₂} {I₁ : Set i₁} {I₂ : Set i₂} →

+
      (I₁ → Set a₁) → (I₂ → Set a₂) → (ℓ : Level) → Set _

+
IREL A₁ A₂ ℓ = ∀ {i₁ i₂} → A₁ i₁ → A₂ i₂ → Set ℓ

+

+
-- Homogeneous types

+

+
IRel : ∀ {i a} {I : Set i} → (I → Set a) → (ℓ : Level) → Set _

+
IRel A ℓ = IREL A A ℓ

+

+
------------------------------------------------------------------------

+
-- Generalised implication.

+

+
infixr 4 _=[_]⇒_

+

+
_=[_]⇒_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : A → Set b} →

+
          B.Rel A ℓ₁ → ((x : A) → B x) → IRel B ℓ₂ → Set _

+
P =[ f ]⇒ Q = ∀ {i j} → P i j → Q (f i) (f j)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Definitions.html b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Definitions.html
new file mode 100644
index 000000000..444f53e30
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Definitions.html
@@ -0,0 +1,110 @@
+
+Relation.Binary.Indexed.Heterogeneous.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Indexed binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Indexed.Heterogeneous`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Indexed.Heterogeneous.Definitions where

+

+
open import Level

+
import Relation.Binary.Definitions as B

+
open import Relation.Binary.Indexed.Heterogeneous.Core

+

+
private

+
  variable

+
    i a ℓ : Level

+
    I : Set i

+

+
------------------------------------------------------------------------

+
-- Simple properties of indexed binary relations

+

+
Reflexive : (A : I → Set a) → IRel A ℓ → Set _

+
Reflexive _ _∼_ = ∀ {i} → B.Reflexive (_∼_ {i})

+

+
Symmetric : (A : I → Set a) → IRel A ℓ → Set _

+
Symmetric _ _∼_ = ∀ {i j} → B.Sym (_∼_ {i} {j}) _∼_

+

+
Transitive : (A : I → Set a) → IRel A ℓ → Set _

+
Transitive _ _∼_ = ∀ {i j k} → B.Trans _∼_ (_∼_ {j}) (_∼_ {i} {k})

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Structures.html b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Structures.html
new file mode 100644
index 000000000..b6209b05e
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.Structures.html
@@ -0,0 +1,123 @@
+
+Relation.Binary.Indexed.Heterogeneous.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Indexed binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Indexed.Heterogeneous`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Indexed.Heterogeneous.Core

+

+
module Relation.Binary.Indexed.Heterogeneous.Structures

+
  {i a ℓ} {I : Set i} (A : I → Set a) (_≈_ : IRel A ℓ)

+
  where

+

+
open import Function.Base

+
open import Level using (suc; _⊔_)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Indexed.Heterogeneous.Definitions

+

+
------------------------------------------------------------------------

+
-- Equivalences

+

+
record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where

+
  field

+
    refl  : Reflexive  A _≈_

+
    sym   : Symmetric  A _≈_

+
    trans : Transitive A _≈_

+

+
  reflexive : ∀ {i} → _≡_ ⟨ _⇒_ ⟩ _≈_ {i}

+
  reflexive ≡.refl = refl

+

+

+
record IsIndexedPreorder {ℓ₂} (_≲_ : IRel A ℓ₂) : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isEquivalence : IsIndexedEquivalence

+
    reflexive     : ∀ {i j} → (_≈_ {i} {j}) ⟨ _⇒_ ⟩ _≲_

+
    trans         : Transitive A _≲_

+

+
  module Eq = IsIndexedEquivalence isEquivalence

+

+
  refl : Reflexive A _≲_

+
  refl = reflexive Eq.refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.html b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.html
new file mode 100644
index 000000000..4b8451815
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Indexed.Heterogeneous.html
@@ -0,0 +1,94 @@
+
+Relation.Binary.Indexed.Heterogeneous
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Heterogeneously-indexed binary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Indexed.Heterogeneous where

+

+
------------------------------------------------------------------------

+
-- Publicly export core definitions

+

+
open import Relation.Binary.Indexed.Heterogeneous.Core public

+
open import Relation.Binary.Indexed.Heterogeneous.Definitions public

+
open import Relation.Binary.Indexed.Heterogeneous.Structures public

+
open import Relation.Binary.Indexed.Heterogeneous.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Lattice.Bundles.html b/site/static/agda_html/Relation.Binary.Lattice.Bundles.html
new file mode 100644
index 000000000..1ffc65fcd
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Lattice.Bundles.html
@@ -0,0 +1,305 @@
+
+Relation.Binary.Lattice.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for order-theoretic lattices

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Lattice`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Lattice.Bundles where

+

+
open import Algebra.Core

+
open import Level using (suc; _⊔_)

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Bundles using (Poset; Setoid)

+
open import Relation.Binary.Lattice.Structures

+

+
------------------------------------------------------------------------

+
-- Join semilattices

+

+
record JoinSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 6 _∨_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_               : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_               : Op₂ Carrier     -- The join operation.

+
    isJoinSemilattice : IsJoinSemilattice _≈_ _≤_ _∨_

+

+
  open IsJoinSemilattice isJoinSemilattice public

+

+
  poset : Poset c ℓ₁ ℓ₂

+
  poset = record { isPartialOrder = isPartialOrder }

+

+
  open Poset poset public using (preorder)

+

+

+
record BoundedJoinSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 6 _∨_

+
  field

+
    Carrier                  : Set c

+
    _≈_                      : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_                      : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_                      : Op₂ Carrier     -- The join operation.

+
    ⊥                        : Carrier         -- The minimum.

+
    isBoundedJoinSemilattice : IsBoundedJoinSemilattice _≈_ _≤_ _∨_ ⊥

+

+
  open IsBoundedJoinSemilattice isBoundedJoinSemilattice public

+

+
  joinSemilattice : JoinSemilattice c ℓ₁ ℓ₂

+
  joinSemilattice = record { isJoinSemilattice = isJoinSemilattice }

+

+
  open JoinSemilattice joinSemilattice public using (preorder; poset)

+

+
------------------------------------------------------------------------

+
-- Meet semilattices

+

+
record MeetSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 7 _∧_

+
  field

+
    Carrier           : Set c

+
    _≈_               : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_               : Rel Carrier ℓ₂  -- The partial order.

+
    _∧_               : Op₂ Carrier     -- The meet operation.

+
    isMeetSemilattice : IsMeetSemilattice _≈_ _≤_ _∧_

+

+
  open IsMeetSemilattice isMeetSemilattice public

+

+
  poset : Poset c ℓ₁ ℓ₂

+
  poset = record { isPartialOrder = isPartialOrder }

+

+
  open Poset poset public using (preorder)

+

+
record BoundedMeetSemilattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 7 _∧_

+
  field

+
    Carrier                  : Set c

+
    _≈_                      : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_                      : Rel Carrier ℓ₂  -- The partial order.

+
    _∧_                      : Op₂ Carrier     -- The join operation.

+
    ⊤                        : Carrier         -- The maximum.

+
    isBoundedMeetSemilattice : IsBoundedMeetSemilattice _≈_ _≤_ _∧_ ⊤

+

+
  open IsBoundedMeetSemilattice isBoundedMeetSemilattice public

+

+
  meetSemilattice : MeetSemilattice c ℓ₁ ℓ₂

+
  meetSemilattice = record { isMeetSemilattice = isMeetSemilattice }

+

+
  open MeetSemilattice meetSemilattice public using (preorder; poset)

+

+
------------------------------------------------------------------------

+
-- Lattices

+

+
record Lattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 6 _∨_

+
  infixr 7 _∧_

+
  field

+
    Carrier   : Set c

+
    _≈_       : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_       : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_       : Op₂ Carrier     -- The join operation.

+
    _∧_       : Op₂ Carrier     -- The meet operation.

+
    isLattice : IsLattice _≈_ _≤_ _∨_ _∧_

+

+
  open IsLattice isLattice public

+

+
  setoid : Setoid c ℓ₁

+
  setoid = record { isEquivalence = isEquivalence }

+

+
  joinSemilattice : JoinSemilattice c ℓ₁ ℓ₂

+
  joinSemilattice = record { isJoinSemilattice = isJoinSemilattice }

+

+
  meetSemilattice : MeetSemilattice c ℓ₁ ℓ₂

+
  meetSemilattice = record { isMeetSemilattice = isMeetSemilattice }

+

+
  open JoinSemilattice joinSemilattice public using (poset; preorder)

+

+
record DistributiveLattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 6 _∨_

+
  infixr 7 _∧_

+
  field

+
    Carrier : Set c

+
    _≈_     : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_     : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_     : Op₂ Carrier     -- The join operation.

+
    _∧_     : Op₂ Carrier     -- The meet operation.

+
    isDistributiveLattice : IsDistributiveLattice _≈_ _≤_ _∨_ _∧_

+

+
  open IsDistributiveLattice isDistributiveLattice using (∧-distribˡ-∨) public

+
  open IsDistributiveLattice isDistributiveLattice using (isLattice)

+

+
  lattice : Lattice c ℓ₁ ℓ₂

+
  lattice = record { isLattice = isLattice }

+

+
  open Lattice lattice hiding (Carrier; _≈_; _≤_; _∨_; _∧_) public

+

+
record BoundedLattice c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 6 _∨_

+
  infixr 7 _∧_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_              : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_              : Op₂ Carrier     -- The join operation.

+
    _∧_              : Op₂ Carrier     -- The meet operation.

+
    ⊤                : Carrier         -- The maximum.

+
    ⊥                : Carrier         -- The minimum.

+
    isBoundedLattice : IsBoundedLattice _≈_ _≤_ _∨_ _∧_ ⊤ ⊥

+

+
  open IsBoundedLattice isBoundedLattice public

+

+
  boundedJoinSemilattice : BoundedJoinSemilattice c ℓ₁ ℓ₂

+
  boundedJoinSemilattice = record

+
    { isBoundedJoinSemilattice = isBoundedJoinSemilattice }

+

+
  boundedMeetSemilattice : BoundedMeetSemilattice c ℓ₁ ℓ₂

+
  boundedMeetSemilattice = record

+
    { isBoundedMeetSemilattice = isBoundedMeetSemilattice }

+

+
  lattice : Lattice c ℓ₁ ℓ₂

+
  lattice = record { isLattice = isLattice }

+

+
  open Lattice lattice public

+
    using (joinSemilattice; meetSemilattice; poset; preorder; setoid)

+

+
------------------------------------------------------------------------

+
-- Heyting algebras (a bounded lattice with exponential operator)

+

+
record HeytingAlgebra c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 5 _⇨_

+
  infixr 6 _∨_

+
  infixr 7 _∧_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_              : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_              : Op₂ Carrier     -- The join operation.

+
    _∧_              : Op₂ Carrier     -- The meet operation.

+
    _⇨_              : Op₂ Carrier     -- The exponential operation.

+
    ⊤                : Carrier         -- The maximum.

+
    ⊥                : Carrier         -- The minimum.

+
    isHeytingAlgebra : IsHeytingAlgebra _≈_ _≤_ _∨_ _∧_ _⇨_ ⊤ ⊥

+

+
  boundedLattice : BoundedLattice c ℓ₁ ℓ₂

+
  boundedLattice = record

+
    { isBoundedLattice = IsHeytingAlgebra.isBoundedLattice isHeytingAlgebra }

+

+
  open IsHeytingAlgebra isHeytingAlgebra

+
    using (exponential; transpose-⇨; transpose-∧) public

+
  open BoundedLattice boundedLattice

+
    hiding (Carrier; _≈_; _≤_; _∨_; _∧_; ⊤; ⊥) public

+

+
------------------------------------------------------------------------

+
-- Boolean algebras (a specialized Heyting algebra)

+

+
record BooleanAlgebra c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

+
  infix  4 _≈_ _≤_

+
  infixr 6 _∨_

+
  infixr 7 _∧_

+
  infix 8 ¬_

+
  field

+
    Carrier          : Set c

+
    _≈_              : Rel Carrier ℓ₁  -- The underlying equality.

+
    _≤_              : Rel Carrier ℓ₂  -- The partial order.

+
    _∨_              : Op₂ Carrier     -- The join operation.

+
    _∧_              : Op₂ Carrier     -- The meet operation.

+
    ¬_               : Op₁ Carrier     -- The negation operation.

+
    ⊤                : Carrier         -- The maximum.

+
    ⊥                : Carrier         -- The minimum.

+
    isBooleanAlgebra : IsBooleanAlgebra _≈_ _≤_ _∨_ _∧_ ¬_ ⊤ ⊥

+

+
  open IsBooleanAlgebra isBooleanAlgebra using (isHeytingAlgebra)

+

+
  heytingAlgebra : HeytingAlgebra c ℓ₁ ℓ₂

+
  heytingAlgebra = record { isHeytingAlgebra = isHeytingAlgebra }

+

+
  open HeytingAlgebra heytingAlgebra public

+
    hiding (Carrier; _≈_; _≤_; _∨_; _∧_; ⊤; ⊥)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Lattice.Definitions.html b/site/static/agda_html/Relation.Binary.Lattice.Definitions.html
new file mode 100644
index 000000000..d9907ce6c
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Lattice.Definitions.html
@@ -0,0 +1,114 @@
+
+Relation.Binary.Lattice.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Definitions for order-theoretic lattices

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Lattice`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Lattice.Definitions where

+

+
open import Algebra.Core

+
open import Data.Product.Base using (_×_; _,_)

+
open import Function.Base using (flip)

+
open import Relation.Binary.Core using (Rel)

+
open import Level using (Level)

+

+
private

+
  variable

+
    a ℓ : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- Relationships between orders and operators

+

+
Supremum : Rel A ℓ → Op₂ A → Set _

+
Supremum _≤_ _∨_ =

+
  ∀ x y → x ≤ (x ∨ y) × y ≤ (x ∨ y) × ∀ z → x ≤ z → y ≤ z → (x ∨ y) ≤ z

+

+
Infimum : Rel A ℓ → Op₂ A → Set _

+
Infimum _≤_ = Supremum (flip _≤_)

+

+
Exponential : Rel A ℓ → Op₂ A → Op₂ A → Set _

+
Exponential _≤_ _∧_ _⇨_ =

+
  ∀ w x y → ((w ∧ x) ≤ y → w ≤ (x ⇨ y)) × (w ≤ (x ⇨ y) → (w ∧ x) ≤ y)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Lattice.Properties.BoundedJoinSemilattice.html b/site/static/agda_html/Relation.Binary.Lattice.Properties.BoundedJoinSemilattice.html
new file mode 100644
index 000000000..fe85429df
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Lattice.Properties.BoundedJoinSemilattice.html
@@ -0,0 +1,131 @@
+
+Relation.Binary.Lattice.Properties.BoundedJoinSemilattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties satisfied by bounded join semilattices

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Lattice

+

+
module Relation.Binary.Lattice.Properties.BoundedJoinSemilattice

+
  {c ℓ₁ ℓ₂} (J : BoundedJoinSemilattice c ℓ₁ ℓ₂) where

+

+
open BoundedJoinSemilattice J

+

+
open import Algebra.Definitions _≈_

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_∘_; flip)

+
open import Relation.Binary.Properties.Poset poset

+
open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice

+
  using (∨-comm)

+

+
-- Bottom is an identity of the meet operation.

+

+
identityˡ : LeftIdentity ⊥ _∨_

+
identityˡ x =

+
  let _ , x≤⊥∨x , least = supremum ⊥ x

+
  in antisym (least x (minimum x) refl) x≤⊥∨x

+

+
identityʳ : RightIdentity ⊥ _∨_

+
identityʳ x =

+
  let x≤x∨⊥ , _ , least = supremum x ⊥

+
  in antisym (least x refl (minimum x)) x≤x∨⊥

+

+
identity : Identity ⊥ _∨_

+
identity = identityˡ , identityʳ

+

+

+
-- The dual construction is a bounded meet semilattice.

+

+
dualIsBoundedMeetSemilattice : IsBoundedMeetSemilattice _≈_ (flip _≤_) _∨_ ⊥

+
dualIsBoundedMeetSemilattice = record

+
  { isMeetSemilattice = record

+
    { isPartialOrder  = ≥-isPartialOrder

+
    ; infimum         = supremum

+
    }

+
  ; maximum           = minimum

+
  }

+

+
dualBoundedMeetSemilattice : BoundedMeetSemilattice c ℓ₁ ℓ₂

+
dualBoundedMeetSemilattice = record

+
  { ⊤                        = ⊥

+
  ; isBoundedMeetSemilattice = dualIsBoundedMeetSemilattice

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Lattice.Properties.JoinSemilattice.html b/site/static/agda_html/Relation.Binary.Lattice.Properties.JoinSemilattice.html
new file mode 100644
index 000000000..01e2f3653
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Lattice.Properties.JoinSemilattice.html
@@ -0,0 +1,203 @@
+
+Relation.Binary.Lattice.Properties.JoinSemilattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties satisfied by join semilattices

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Lattice

+

+
module Relation.Binary.Lattice.Properties.JoinSemilattice

+
  {c ℓ₁ ℓ₂} (J : JoinSemilattice c ℓ₁ ℓ₂) where

+

+
open JoinSemilattice J

+

+
import Algebra.Lattice as Alg

+
import Algebra.Structures as Alg

+
open import Algebra.Definitions _≈_

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_∘_; flip)

+
open import Relation.Binary.Core using (_Preserves₂_⟶_⟶_)

+
open import Relation.Binary.Structures using (IsDecPartialOrder)

+
open import Relation.Binary.Definitions using (Decidable)

+
open import Relation.Binary.Properties.Poset poset

+
open import Relation.Nullary using (¬_; yes; no)

+
open import Relation.Nullary.Negation using (contraposition)

+

+
import Relation.Binary.Reasoning.PartialOrder as PoR

+

+
------------------------------------------------------------------------

+
-- Algebraic properties

+

+
-- The join operation is monotonic.

+

+
∨-monotonic : _∨_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_

+
∨-monotonic {x} {y} {u} {v} x≤y u≤v =

+
  let _     , _     , least  = supremum x u

+
      y≤y∨v , v≤y∨v , _      = supremum y v

+
  in least (y ∨ v) (trans x≤y y≤y∨v) (trans u≤v v≤y∨v)

+

+
∨-cong : _∨_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_

+
∨-cong x≈y u≈v = antisym (∨-monotonic (reflexive x≈y) (reflexive u≈v))

+
                         (∨-monotonic (reflexive (Eq.sym x≈y))

+
                                      (reflexive (Eq.sym u≈v)))

+

+
-- The join operation is commutative, associative and idempotent.

+

+
∨-comm : Commutative _∨_

+
∨-comm x y =

+
  let x≤x∨y , y≤x∨y , least  = supremum x y

+
      y≤y∨x , x≤y∨x , least′ = supremum y x

+
  in antisym (least (y ∨ x) x≤y∨x y≤y∨x) (least′ (x ∨ y) y≤x∨y x≤x∨y)

+

+
∨-assoc : Associative _∨_

+
∨-assoc x y z =

+
  let x∨y≤[x∨y]∨z , z≤[x∨y]∨z   , least  = supremum (x ∨ y) z

+
      x≤x∨[y∨z]   , y∨z≤x∨[y∨z] , least′ = supremum x (y ∨ z)

+
      y≤y∨z       , z≤y∨z       , _      = supremum y z

+
      x≤x∨y       , y≤x∨y       , _      = supremum x y

+
  in antisym (least  (x ∨ (y ∨ z)) (∨-monotonic refl y≤y∨z)

+
                     (trans z≤y∨z y∨z≤x∨[y∨z]))

+
             (least′ ((x ∨ y) ∨ z) (trans x≤x∨y x∨y≤[x∨y]∨z)

+
                     (∨-monotonic y≤x∨y refl))

+

+
∨-idempotent : Idempotent _∨_

+
∨-idempotent x =

+
  let x≤x∨x , _ , least = supremum x x

+
  in antisym (least x refl refl) x≤x∨x

+

+
x≤y⇒x∨y≈y : ∀ {x y} → x ≤ y → x ∨ y ≈ y

+
x≤y⇒x∨y≈y {x} {y} x≤y = antisym

+
  (begin x ∨ y ≤⟨ ∨-monotonic x≤y refl ⟩

+
         y ∨ y ≈⟨ ∨-idempotent _ ⟩

+
         y ∎)

+
  (y≤x∨y _ _)

+
  where open PoR poset

+

+
-- Every order-theoretic semilattice can be turned into an algebraic one.

+

+
isAlgSemilattice : Alg.IsSemilattice _≈_ _∨_

+
isAlgSemilattice = record

+
  { isBand = record

+
    { isSemigroup = record

+
      { isMagma   = record

+
        { isEquivalence = isEquivalence

+
        ; ∙-cong        = ∨-cong

+
        }

+
      ; assoc  = ∨-assoc

+
      }

+
    ; idem     = ∨-idempotent

+
    }

+
  ; comm       = ∨-comm

+
  }

+

+
algSemilattice : Alg.Semilattice c ℓ₁

+
algSemilattice = record { isSemilattice = isAlgSemilattice }

+

+
------------------------------------------------------------------------

+
-- The dual construction is a meet semilattice.

+

+
dualIsMeetSemilattice : IsMeetSemilattice _≈_ (flip _≤_) _∨_

+
dualIsMeetSemilattice = record

+
  { isPartialOrder = ≥-isPartialOrder

+
  ; infimum        = supremum

+
  }

+

+
dualMeetSemilattice : MeetSemilattice c ℓ₁ ℓ₂

+
dualMeetSemilattice = record

+
  { _∧_               = _∨_

+
  ; isMeetSemilattice = dualIsMeetSemilattice

+
  }

+

+
------------------------------------------------------------------------

+
-- If ≈ is decidable then so is ≤

+

+
≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_

+
≈-dec⇒≤-dec _≟_ x y with (x ∨ y) ≟ y

+
... | yes x∨y≈y = yes (trans (x≤x∨y x y) (reflexive x∨y≈y))

+
... | no  x∨y≉y = no (contraposition x≤y⇒x∨y≈y x∨y≉y)

+

+
≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_

+
≈-dec⇒isDecPartialOrder _≟_ = record

+
  { isPartialOrder = isPartialOrder

+
  ; _≟_            = _≟_

+
  ; _≤?_           = ≈-dec⇒≤-dec _≟_

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Lattice.Structures.html b/site/static/agda_html/Relation.Binary.Lattice.Structures.html
new file mode 100644
index 000000000..0b0ccc9ed
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Lattice.Structures.html
@@ -0,0 +1,263 @@
+
+Relation.Binary.Lattice.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Structures for order-theoretic lattices

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via

+
-- `Relation.Binary.Lattice`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Structures using (IsPartialOrder)

+
open import Relation.Binary.Definitions using (Minimum; Maximum)

+

+
module Relation.Binary.Lattice.Structures

+
 {a ℓ₁ ℓ₂} {A : Set a}

+
 (_≈_ : Rel A ℓ₁) -- The underlying equality.

+
 (_≤_ : Rel A ℓ₂) -- The partial order.

+
 where

+

+
open import Algebra.Core

+
open import Algebra.Definitions

+
open import Data.Product.Base using (_×_; _,_)

+
open import Level using (suc; _⊔_)

+

+
open import Relation.Binary.Lattice.Definitions

+

+
------------------------------------------------------------------------

+
-- Join semilattices

+

+
record IsJoinSemilattice (_∨_ : Op₂ A)    -- The join operation.

+
                         : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isPartialOrder : IsPartialOrder _≈_ _≤_

+
    supremum       : Supremum _≤_ _∨_

+

+
  x≤x∨y : ∀ x y → x ≤ (x ∨ y)

+
  x≤x∨y x y = let pf , _ , _ = supremum x y in pf

+

+
  y≤x∨y : ∀ x y → y ≤ (x ∨ y)

+
  y≤x∨y x y = let _ , pf , _ = supremum x y in pf

+

+
  ∨-least : ∀ {x y z} → x ≤ z → y ≤ z → (x ∨ y) ≤ z

+
  ∨-least {x} {y} {z} = let _ , _ , pf = supremum x y in pf z

+

+
  open IsPartialOrder isPartialOrder public

+

+
record IsBoundedJoinSemilattice (_∨_ : Op₂ A)    -- The join operation.

+
                                (⊥   : A)        -- The minimum.

+
                                : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isJoinSemilattice : IsJoinSemilattice _∨_

+
    minimum           : Minimum _≤_ ⊥

+

+
  open IsJoinSemilattice isJoinSemilattice public

+

+
------------------------------------------------------------------------

+
-- Meet semilattices

+

+
record IsMeetSemilattice (_∧_ : Op₂ A)    -- The meet operation.

+
                         : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isPartialOrder : IsPartialOrder _≈_ _≤_

+
    infimum        : Infimum _≤_ _∧_

+

+
  x∧y≤x : ∀ x y → (x ∧ y) ≤ x

+
  x∧y≤x x y = let pf , _ , _ = infimum x y in pf

+

+
  x∧y≤y : ∀ x y → (x ∧ y) ≤ y

+
  x∧y≤y x y = let _ , pf , _ = infimum x y in pf

+

+
  ∧-greatest : ∀ {x y z} → x ≤ y → x ≤ z → x ≤ (y ∧ z)

+
  ∧-greatest {x} {y} {z} = let _ , _ , pf = infimum y z in pf x

+

+
  open IsPartialOrder isPartialOrder public

+

+
record IsBoundedMeetSemilattice (_∧_ : Op₂ A)    -- The join operation.

+
                                (⊤   : A)        -- The maximum.

+
                                : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isMeetSemilattice : IsMeetSemilattice _∧_

+
    maximum           : Maximum _≤_ ⊤

+

+
  open IsMeetSemilattice isMeetSemilattice public

+

+
------------------------------------------------------------------------

+
-- Lattices

+

+
record IsLattice (_∨_ : Op₂ A)    -- The join operation.

+
                 (_∧_ : Op₂ A)    -- The meet operation.

+
                 : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isPartialOrder : IsPartialOrder _≈_ _≤_

+
    supremum       : Supremum _≤_ _∨_

+
    infimum        : Infimum _≤_ _∧_

+

+
  isJoinSemilattice : IsJoinSemilattice  _∨_

+
  isJoinSemilattice = record

+
    { isPartialOrder = isPartialOrder

+
    ; supremum       = supremum

+
    }

+

+
  isMeetSemilattice : IsMeetSemilattice  _∧_

+
  isMeetSemilattice = record

+
    { isPartialOrder = isPartialOrder

+
    ; infimum        = infimum

+
    }

+

+
  open IsJoinSemilattice isJoinSemilattice public

+
    using (x≤x∨y; y≤x∨y; ∨-least)

+
  open IsMeetSemilattice isMeetSemilattice public

+
    using (x∧y≤x; x∧y≤y; ∧-greatest)

+
  open IsPartialOrder isPartialOrder public

+

+
record IsDistributiveLattice (_∨_ : Op₂ A)    -- The join operation.

+
                             (_∧_ : Op₂ A)    -- The meet operation.

+
                             : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isLattice    : IsLattice _∨_ _∧_

+
    ∧-distribˡ-∨ : _DistributesOverˡ_ _≈_ _∧_ _∨_

+

+
  open IsLattice isLattice public

+

+
record IsBoundedLattice (_∨_ : Op₂ A)    -- The join operation.

+
                        (_∧_ : Op₂ A)    -- The meet operation.

+
                        (⊤   : A)        -- The maximum.

+
                        (⊥   : A)        -- The minimum.

+
                        : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isLattice : IsLattice _∨_ _∧_

+
    maximum   : Maximum _≤_ ⊤

+
    minimum   : Minimum _≤_ ⊥

+

+
  open IsLattice isLattice public

+

+
  isBoundedJoinSemilattice : IsBoundedJoinSemilattice  _∨_ ⊥

+
  isBoundedJoinSemilattice = record

+
    { isJoinSemilattice = isJoinSemilattice

+
    ; minimum           = minimum

+
    }

+

+
  isBoundedMeetSemilattice : IsBoundedMeetSemilattice _∧_ ⊤

+
  isBoundedMeetSemilattice = record

+
    { isMeetSemilattice = isMeetSemilattice

+
    ; maximum           = maximum

+
    }

+

+
------------------------------------------------------------------------

+
-- Heyting algebras (a bounded lattice with exponential operator)

+

+
record IsHeytingAlgebra (_∨_ : Op₂ A)    -- The join operation.

+
                        (_∧_ : Op₂ A)    -- The meet operation.

+
                        (_⇨_ : Op₂ A)    -- The exponential operation.

+
                        (⊤   : A)        -- The maximum.

+
                        (⊥   : A)        -- The minimum.

+
                        : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isBoundedLattice : IsBoundedLattice _∨_ _∧_ ⊤ ⊥

+
    exponential      : Exponential _≤_ _∧_ _⇨_

+

+
  transpose-⇨ : ∀ {w x y} → (w ∧ x) ≤ y → w ≤ (x ⇨ y)

+
  transpose-⇨ {w} {x} {y} = let pf , _ = exponential w x y in pf

+

+
  transpose-∧ : ∀ {w x y} → w ≤ (x ⇨ y) → (w ∧ x) ≤ y

+
  transpose-∧ {w} {x} {y} = let _ , pf = exponential w x y in pf

+

+
  open IsBoundedLattice isBoundedLattice public

+

+
------------------------------------------------------------------------

+
-- Boolean algebras (a specialized Heyting algebra)

+

+
record IsBooleanAlgebra (_∨_ : Op₂ A)    -- The join operation.

+
                        (_∧_ : Op₂ A)    -- The meet operation.

+
                        (¬_ : Op₁ A)     -- The negation operation.

+
                        (⊤   : A)        -- The maximum.

+
                        (⊥   : A)        -- The minimum.

+
                        : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  infixr 5 _⇨_

+
  _⇨_ : Op₂ A

+
  x ⇨ y = (¬ x) ∨ y

+

+
  field

+
    isHeytingAlgebra : IsHeytingAlgebra _∨_ _∧_ _⇨_ ⊤ ⊥

+

+
  open IsHeytingAlgebra isHeytingAlgebra public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Lattice.html b/site/static/agda_html/Relation.Binary.Lattice.html
new file mode 100644
index 000000000..668a5c41f
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Lattice.html
@@ -0,0 +1,93 @@
+
+Relation.Binary.Lattice
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Order-theoretic lattices

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.Lattice where

+

+
------------------------------------------------------------------------

+
-- Re-export various components of the lattice hierarchy

+

+
open import Relation.Binary.Lattice.Definitions public

+
open import Relation.Binary.Lattice.Structures public

+
open import Relation.Binary.Lattice.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Morphism.Bundles.html b/site/static/agda_html/Relation.Binary.Morphism.Bundles.html
new file mode 100644
index 000000000..2acbc67f7
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Morphism.Bundles.html
@@ -0,0 +1,181 @@
+
+Relation.Binary.Morphism.Bundles
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Bundles for morphisms between binary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Level

+
open import Relation.Binary.Core using (_Preserves_⟶_)

+
open import Relation.Binary.Bundles

+
open import Relation.Binary.Morphism.Structures

+
open import Relation.Binary.Consequences using (mono⇒cong)

+

+
module Relation.Binary.Morphism.Bundles where

+

+
private

+
  variable

+
    ℓ₁ ℓ₂ ℓ₃ ℓ₄ ℓ₅ ℓ₆ : Level

+

+
------------------------------------------------------------------------

+
-- Setoids

+
------------------------------------------------------------------------

+

+
module _ (S₁ : Setoid ℓ₁ ℓ₂) (S₂ : Setoid ℓ₃ ℓ₄) where

+

+
  record SetoidHomomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where

+
    open Setoid

+
    field

+
      ⟦_⟧               : Carrier S₁ → Carrier S₂

+
      isRelHomomorphism : IsRelHomomorphism (_≈_ S₁) (_≈_ S₂) ⟦_⟧

+

+
    open IsRelHomomorphism isRelHomomorphism public

+

+

+
  record SetoidMonomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where

+
    open Setoid

+
    field

+
      ⟦_⟧               : Carrier S₁ → Carrier S₂

+
      isRelMonomorphism : IsRelMonomorphism (_≈_ S₁) (_≈_ S₂) ⟦_⟧

+

+
    open IsRelMonomorphism isRelMonomorphism public

+

+
    homomorphism : SetoidHomomorphism

+
    homomorphism = record { isRelHomomorphism = isHomomorphism }

+

+

+
  record SetoidIsomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄) where

+
    open Setoid

+
    field

+
      ⟦_⟧              : Carrier S₁ → Carrier S₂

+
      isRelIsomorphism : IsRelIsomorphism (_≈_ S₁) (_≈_ S₂) ⟦_⟧

+

+
    open IsRelIsomorphism isRelIsomorphism public

+

+
    monomorphism : SetoidMonomorphism

+
    monomorphism = record { isRelMonomorphism = isMonomorphism }

+

+
    open SetoidMonomorphism monomorphism public

+
      using (homomorphism)

+

+

+
------------------------------------------------------------------------

+
-- Preorders

+
------------------------------------------------------------------------

+

+
record PreorderHomomorphism (S₁ : Preorder ℓ₁ ℓ₂ ℓ₃)

+
                            (S₂ : Preorder ℓ₄ ℓ₅ ℓ₆)

+
                            : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄ ⊔ ℓ₅ ⊔ ℓ₆) where

+
  open Preorder

+
  field

+
    ⟦_⟧                 : Carrier S₁ → Carrier S₂

+
    isOrderHomomorphism : IsOrderHomomorphism (_≈_ S₁) (_≈_ S₂) (_≲_ S₁) (_≲_ S₂) ⟦_⟧

+

+
  open IsOrderHomomorphism isOrderHomomorphism public

+

+
------------------------------------------------------------------------

+
-- Posets

+
------------------------------------------------------------------------

+

+
module _ (P : Poset ℓ₁ ℓ₂ ℓ₃) (Q : Poset ℓ₄ ℓ₅ ℓ₆) where

+

+
  private

+
    module P = Poset P

+
    module Q = Poset Q

+

+
  record PosetHomomorphism : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄ ⊔ ℓ₅ ⊔ ℓ₆) where

+
    field

+
      ⟦_⟧                 : P.Carrier → Q.Carrier

+
      isOrderHomomorphism : IsOrderHomomorphism P._≈_ Q._≈_ P._≤_ Q._≤_ ⟦_⟧

+

+
    open IsOrderHomomorphism isOrderHomomorphism public

+

+

+
  -- Smart constructor that automatically constructs the congruence

+
  -- proof from the monotonicity proof

+
  mkPosetHomo : ∀ f → f Preserves P._≤_ ⟶ Q._≤_ → PosetHomomorphism

+
  mkPosetHomo f mono = record

+
    { ⟦_⟧ = f

+
    ; isOrderHomomorphism = record

+
      { cong = mono⇒cong P._≈_ Q._≈_ P.Eq.sym P.reflexive Q.antisym mono

+
      ; mono = mono

+
      }

+
    }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Morphism.Definitions.html b/site/static/agda_html/Relation.Binary.Morphism.Definitions.html
new file mode 100644
index 000000000..8fcc219aa
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Morphism.Definitions.html
@@ -0,0 +1,109 @@
+
+Relation.Binary.Morphism.Definitions
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Basic definitions for morphisms between algebraic structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core

+

+
module Relation.Binary.Morphism.Definitions

+
  {a} (A : Set a)     -- The domain of the morphism

+
  {b} (B : Set b)     -- The codomain of the morphism

+
  where

+

+
open import Level using (Level)

+

+
private

+
  variable

+
    ℓ₁ ℓ₂ : Level

+

+
------------------------------------------------------------------------

+
-- Morphism definition in Function.Core

+

+
open import Function.Core public

+
  using (Morphism)

+

+
------------------------------------------------------------------------

+
-- Basic definitions

+

+
Homomorphic₂ : Rel A ℓ₁ → Rel B ℓ₂ → (A → B) → Set _

+
Homomorphic₂ _∼₁_ _∼₂_ ⟦_⟧ = ∀ {x y} → x ∼₁ y → ⟦ x ⟧ ∼₂ ⟦ y ⟧

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Morphism.OrderMonomorphism.html b/site/static/agda_html/Relation.Binary.Morphism.OrderMonomorphism.html
new file mode 100644
index 000000000..619969a1d
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Morphism.OrderMonomorphism.html
@@ -0,0 +1,187 @@
+
+Relation.Binary.Morphism.OrderMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between orders

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Algebra.Morphism.Definitions

+
open import Function.Base

+
open import Data.Product.Base using (_,_; map)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Structures

+
  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)

+
open import Relation.Binary.Definitions

+
  using (Irreflexive; Antisymmetric; Trichotomous; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects₂_)

+
open import Relation.Binary.Morphism

+
import Relation.Binary.Morphism.RelMonomorphism as RawRelation

+

+
module Relation.Binary.Morphism.OrderMonomorphism

+
  {a b ℓ₁ ℓ₂ ℓ₃ ℓ₄} {A : Set a} {B : Set b}

+
  {_≈₁_ : Rel A ℓ₁} {_≈₂_ : Rel B ℓ₃}

+
  {_≲₁_ : Rel A ℓ₂} {_≲₂_ : Rel B ℓ₄}

+
  {⟦_⟧ : A → B}

+
  (isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧)

+
  where

+

+
open IsOrderMonomorphism isOrderMonomorphism

+

+
------------------------------------------------------------------------

+
-- Re-export equivalence proofs

+

+
module EqM = RawRelation Eq.isRelMonomorphism

+

+
open RawRelation isRelMonomorphism public

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
reflexive : _≈₂_ ⇒ _≲₂_ → _≈₁_ ⇒ _≲₁_

+
reflexive refl x≈y = cancel (refl (cong x≈y))

+

+
irrefl : Irreflexive _≈₂_ _≲₂_ → Irreflexive _≈₁_ _≲₁_

+
irrefl irrefl x≈y x∼y = irrefl (cong x≈y) (mono x∼y)

+

+
antisym : Antisymmetric _≈₂_ _≲₂_ → Antisymmetric _≈₁_ _≲₁_

+
antisym antisym x∼y y∼x = injective (antisym (mono x∼y) (mono y∼x))

+

+
compare : Trichotomous _≈₂_ _≲₂_ → Trichotomous _≈₁_ _≲₁_

+
compare compare x y with compare ⟦ x ⟧ ⟦ y ⟧

+
... | tri< a ¬b ¬c = tri< (cancel a) (¬b ∘ cong) (¬c ∘ mono)

+
... | tri≈ ¬a b ¬c = tri≈ (¬a ∘ mono) (injective b) (¬c ∘ mono)

+
... | tri> ¬a ¬b c = tri> (¬a ∘ mono) (¬b ∘ cong) (cancel c)

+

+
respˡ : _≲₂_ Respectsˡ _≈₂_ → _≲₁_ Respectsˡ _≈₁_

+
respˡ resp x≈y x∼z = cancel (resp (cong x≈y) (mono x∼z))

+

+
respʳ : _≲₂_ Respectsʳ _≈₂_ → _≲₁_ Respectsʳ _≈₁_

+
respʳ resp x≈y y∼z = cancel (resp (cong x≈y) (mono y∼z))

+

+
resp : _≲₂_ Respects₂ _≈₂_ → _≲₁_ Respects₂ _≈₁_

+
resp = map respʳ respˡ

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isPreorder : IsPreorder _≈₂_ _≲₂_ → IsPreorder _≈₁_ _≲₁_

+
isPreorder O = record

+
  { isEquivalence = EqM.isEquivalence O.isEquivalence

+
  ; reflexive     = reflexive O.reflexive

+
  ; trans         = trans O.trans

+
  } where module O = IsPreorder O

+

+
isPartialOrder : IsPartialOrder _≈₂_ _≲₂_ → IsPartialOrder _≈₁_ _≲₁_

+
isPartialOrder O = record

+
  { isPreorder = isPreorder O.isPreorder

+
  ; antisym    = antisym O.antisym

+
  } where module O = IsPartialOrder O

+

+
isTotalOrder : IsTotalOrder _≈₂_ _≲₂_ → IsTotalOrder _≈₁_ _≲₁_

+
isTotalOrder O = record

+
  { isPartialOrder = isPartialOrder O.isPartialOrder

+
  ; total          = total O.total

+
  } where module O = IsTotalOrder O

+

+
isDecTotalOrder : IsDecTotalOrder _≈₂_ _≲₂_ → IsDecTotalOrder _≈₁_ _≲₁_

+
isDecTotalOrder O = record

+
  { isTotalOrder = isTotalOrder O.isTotalOrder

+
  ; _≟_          = EqM.dec O._≟_

+
  ; _≤?_         = dec O._≤?_

+
  } where module O = IsDecTotalOrder O

+

+
isStrictPartialOrder : IsStrictPartialOrder _≈₂_ _≲₂_ →

+
                       IsStrictPartialOrder _≈₁_ _≲₁_

+
isStrictPartialOrder O = record

+
  { isEquivalence = EqM.isEquivalence O.isEquivalence

+
  ; irrefl        = irrefl O.irrefl

+
  ; trans         = trans O.trans

+
  ; <-resp-≈      = resp O.<-resp-≈

+
  } where module O = IsStrictPartialOrder O

+

+
isStrictTotalOrder : IsStrictTotalOrder _≈₂_ _≲₂_ →

+
                     IsStrictTotalOrder _≈₁_ _≲₁_

+
isStrictTotalOrder O = record

+
  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder

+
  ; compare              = compare O.compare

+
  } where module O = IsStrictTotalOrder O

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Morphism.RelMonomorphism.html b/site/static/agda_html/Relation.Binary.Morphism.RelMonomorphism.html
new file mode 100644
index 000000000..2f71f4458
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Morphism.RelMonomorphism.html
@@ -0,0 +1,143 @@
+
+Relation.Binary.Morphism.RelMonomorphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Consequences of a monomorphism between binary relations

+
------------------------------------------------------------------------

+

+
-- See Data.Nat.Binary.Properties for examples of how this and similar

+
-- modules can be used to easily translate properties between types.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Function.Base

+
open import Relation.Binary.Core using (Rel)

+
open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)

+
open import Relation.Binary.Definitions

+
  using (Reflexive; Symmetric; Transitive; Total; Asymmetric; Decidable)

+
open import Relation.Binary.Morphism

+

+
module Relation.Binary.Morphism.RelMonomorphism

+
  {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}

+
  {_∼₁_ : Rel A ℓ₁} {_∼₂_ : Rel B ℓ₂}

+
  {⟦_⟧ : A → B} (isMonomorphism : IsRelMonomorphism _∼₁_ _∼₂_ ⟦_⟧)

+
  where

+

+
open import Data.Sum.Base as Sum

+
open import Relation.Nullary.Decidable using (yes; no)

+
open import Relation.Nullary.Decidable

+

+
open IsRelMonomorphism isMonomorphism

+

+
------------------------------------------------------------------------

+
-- Properties

+

+
refl : Reflexive _∼₂_ → Reflexive _∼₁_

+
refl refl = injective refl

+

+
sym : Symmetric _∼₂_ → Symmetric _∼₁_

+
sym sym x∼y = injective (sym (cong x∼y))

+

+
trans : Transitive _∼₂_ → Transitive _∼₁_

+
trans trans x∼y y∼z = injective (trans (cong x∼y) (cong y∼z))

+

+
total : Total _∼₂_ → Total _∼₁_

+
total total x y = Sum.map injective injective (total ⟦ x ⟧ ⟦ y ⟧)

+

+
asym : Asymmetric _∼₂_ → Asymmetric _∼₁_

+
asym asym x∼y y∼x = asym (cong x∼y) (cong y∼x)

+

+
dec : Decidable _∼₂_ → Decidable _∼₁_

+
dec _∼?_ x y = map′ injective cong (⟦ x ⟧ ∼? ⟦ y ⟧)

+

+
------------------------------------------------------------------------

+
-- Structures

+

+
isEquivalence : IsEquivalence _∼₂_ → IsEquivalence _∼₁_

+
isEquivalence isEq = record

+
  { refl  = refl  E.refl

+
  ; sym   = sym   E.sym

+
  ; trans = trans E.trans

+
  } where module E = IsEquivalence isEq

+

+
isDecEquivalence : IsDecEquivalence _∼₂_ → IsDecEquivalence _∼₁_

+
isDecEquivalence isDecEq = record

+
  { isEquivalence  = isEquivalence E.isEquivalence

+
  ; _≟_            = dec E._≟_

+
  } where module E = IsDecEquivalence isDecEq

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Morphism.Structures.html b/site/static/agda_html/Relation.Binary.Morphism.Structures.html
new file mode 100644
index 000000000..ca0404100
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Morphism.Structures.html
@@ -0,0 +1,194 @@
+
+Relation.Binary.Morphism.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Order morphisms

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+

+
module Relation.Binary.Morphism.Structures

+
  {a b} {A : Set a} {B : Set b}

+
  where

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Definitions

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Morphism.Definitions A B

+

+
private

+
  variable

+
    ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level

+

+
------------------------------------------------------------------------

+
-- Relations

+
------------------------------------------------------------------------

+

+
record IsRelHomomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)

+
                         (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    cong : Homomorphic₂ _∼₁_ _∼₂_ ⟦_⟧

+

+

+
record IsRelMonomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)

+
                         (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isHomomorphism : IsRelHomomorphism _∼₁_ _∼₂_ ⟦_⟧

+
    injective      : Injective _∼₁_ _∼₂_ ⟦_⟧

+

+
  open IsRelHomomorphism isHomomorphism public

+

+

+
record IsRelIsomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)

+
                        (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where

+
  field

+
    isMonomorphism : IsRelMonomorphism _∼₁_ _∼₂_ ⟦_⟧

+
    surjective     : Surjective _∼₁_ _∼₂_ ⟦_⟧

+

+
  open IsRelMonomorphism isMonomorphism public

+

+
  bijective : Bijective _∼₁_ _∼₂_ ⟦_⟧

+
  bijective = injective , surjective

+

+

+
------------------------------------------------------------------------

+
-- Orders

+
------------------------------------------------------------------------

+

+
record IsOrderHomomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)

+
                           (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)

+
                           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)

+
                           where

+
  field

+
    cong  : Homomorphic₂ _≈₁_ _≈₂_ ⟦_⟧

+
    mono  : Homomorphic₂ _≲₁_ _≲₂_ ⟦_⟧

+

+
  module Eq where

+
    isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelHomomorphism = record { cong = cong }

+

+
  isRelHomomorphism : IsRelHomomorphism _≲₁_ _≲₂_ ⟦_⟧

+
  isRelHomomorphism = record { cong = mono }

+

+

+
record IsOrderMonomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)

+
                           (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)

+
                           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)

+
                           where

+
  field

+
    isOrderHomomorphism : IsOrderHomomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧

+
    injective           : Injective _≈₁_ _≈₂_ ⟦_⟧

+
    cancel              : Injective _≲₁_ _≲₂_ ⟦_⟧

+

+
  open IsOrderHomomorphism isOrderHomomorphism public

+
    hiding (module Eq)

+

+
  module Eq where

+
    isRelMonomorphism : IsRelMonomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelMonomorphism = record

+
      { isHomomorphism = IsOrderHomomorphism.Eq.isRelHomomorphism isOrderHomomorphism

+
      ; injective      = injective

+
      }

+

+
  isRelMonomorphism : IsRelMonomorphism _≲₁_ _≲₂_ ⟦_⟧

+
  isRelMonomorphism = record

+
    { isHomomorphism = isRelHomomorphism

+
    ; injective      = cancel

+
    }

+

+

+
record IsOrderIsomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)

+
                          (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)

+
                          (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)

+
                          where

+
  field

+
    isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧

+
    surjective          : Surjective _≈₁_ _≈₂_ ⟦_⟧

+

+
  open IsOrderMonomorphism isOrderMonomorphism public

+
    hiding (module Eq)

+

+
  module Eq where

+
    isRelIsomorphism : IsRelIsomorphism _≈₁_ _≈₂_ ⟦_⟧

+
    isRelIsomorphism = record

+
      { isMonomorphism = IsOrderMonomorphism.Eq.isRelMonomorphism isOrderMonomorphism

+
      ; surjective     = surjective

+
      }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Morphism.html b/site/static/agda_html/Relation.Binary.Morphism.html
new file mode 100644
index 000000000..e8891b699
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Morphism.html
@@ -0,0 +1,95 @@
+
+Relation.Binary.Morphism
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Order morphisms

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core

+

+
module Relation.Binary.Morphism where

+

+
------------------------------------------------------------------------

+
-- Re-export contents of morphisms

+

+
open import Relation.Binary.Morphism.Definitions public

+
open import Relation.Binary.Morphism.Structures public

+
open import Relation.Binary.Morphism.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.ApartnessRelation.html b/site/static/agda_html/Relation.Binary.Properties.ApartnessRelation.html
new file mode 100644
index 000000000..2046b121c
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.ApartnessRelation.html
@@ -0,0 +1,106 @@
+
+Relation.Binary.Properties.ApartnessRelation
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Apartness properties

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core using (Rel)

+

+
module Relation.Binary.Properties.ApartnessRelation

+
  {a ℓ₁ ℓ₂} {A : Set a}

+
  {_≈_ : Rel A ℓ₁}

+
  {_#_ : Rel A ℓ₂}

+
  where

+

+
open import Function.Base using (_∘₂_)

+
open import Relation.Binary.Definitions using (Reflexive)

+
open import Relation.Binary.Consequences using (sym⇒¬-sym; cotrans⇒¬-trans)

+
open import Relation.Binary.Structures using (IsEquivalence; IsApartnessRelation)

+
open import Relation.Nullary.Negation using (¬_)

+

+
¬#-isEquivalence : Reflexive _≈_ → IsApartnessRelation _≈_ _#_ →

+
                   IsEquivalence (¬_ ∘₂ _#_)

+
¬#-isEquivalence re apart = record

+
  { refl = irrefl re

+
  ; sym = λ {a} {b} → sym⇒¬-sym sym {a} {b}

+
  ; trans = cotrans⇒¬-trans cotrans

+
  } where open IsApartnessRelation apart

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.DecSetoid.html b/site/static/agda_html/Relation.Binary.Properties.DecSetoid.html
new file mode 100644
index 000000000..e2c8630de
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.DecSetoid.html
@@ -0,0 +1,120 @@
+
+Relation.Binary.Properties.DecSetoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Every decidable setoid induces tight apartness relation.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (DecSetoid; ApartnessRelation)

+

+
module Relation.Binary.Properties.DecSetoid {c ℓ} (S : DecSetoid c ℓ) where

+

+
open import Data.Product using (_,_)

+
open import Data.Sum using (inj₁; inj₂)

+
open import Relation.Binary.Definitions

+
  using (Cotransitive; Tight)

+
import Relation.Binary.Properties.Setoid as SetoidProperties

+
open import Relation.Binary.Structures

+
  using (IsApartnessRelation; IsDecEquivalence)

+
open import Relation.Nullary.Decidable.Core

+
  using (yes; no; decidable-stable)

+

+
open DecSetoid S using (_≈_; _≉_; _≟_; setoid; trans)

+
open SetoidProperties setoid

+

+
≉-cotrans : Cotransitive _≉_

+
≉-cotrans {x} {y} x≉y z with x ≟ z | z ≟ y

+
... | _ | no z≉y = inj₂ z≉y

+
... | no x≉z | _ = inj₁ x≉z

+
... | yes x≈z | yes z≈y = inj₁ λ _ → x≉y (trans x≈z z≈y)

+

+
≉-isApartnessRelation : IsApartnessRelation _≈_ _≉_

+
≉-isApartnessRelation = record

+
  { irrefl = ≉-irrefl

+
  ; sym = ≉-sym

+
  ; cotrans = ≉-cotrans

+
  }

+

+
≉-apartnessRelation : ApartnessRelation c ℓ ℓ

+
≉-apartnessRelation = record { isApartnessRelation = ≉-isApartnessRelation }

+

+
≉-tight : Tight _≈_ _≉_

+
≉-tight x y = decidable-stable (x ≟ y) , ≉-irrefl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.DecTotalOrder.html b/site/static/agda_html/Relation.Binary.Properties.DecTotalOrder.html
new file mode 100644
index 000000000..5c6b3ebed
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.DecTotalOrder.html
@@ -0,0 +1,173 @@
+
+Relation.Binary.Properties.DecTotalOrder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties satisfied by decidable total orders

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Structures

+
  using (IsDecTotalOrder; IsStrictTotalOrder)

+
open import Relation.Binary.Bundles

+
  using (DecTotalOrder; StrictTotalOrder)

+

+
module Relation.Binary.Properties.DecTotalOrder

+
  {d₁ d₂ d₃} (DTO : DecTotalOrder d₁ d₂ d₃) where

+

+
open DecTotalOrder DTO hiding (trans)

+

+
import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd

+
import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict

+
import Relation.Binary.Properties.TotalOrder totalOrder as TotalOrderProperties

+
open import Relation.Nullary.Negation using (¬_)

+

+
------------------------------------------------------------------------

+
-- _≥_ - the flipped relation is also a total order

+

+
open TotalOrderProperties public

+
  using

+
  ( ≥-refl

+
  ; ≥-reflexive

+
  ; ≥-trans

+
  ; ≥-antisym

+
  ; ≥-total

+
  ; ≥-isPreorder

+
  ; ≥-isPartialOrder

+
  ; ≥-isTotalOrder

+
  ; ≥-preorder

+
  ; ≥-poset

+
  ; ≥-totalOrder

+
  )

+

+
≥-isDecTotalOrder : IsDecTotalOrder _≈_ _≥_

+
≥-isDecTotalOrder = EqAndOrd.isDecTotalOrder isDecTotalOrder

+

+
≥-decTotalOrder : DecTotalOrder _ _ _

+
≥-decTotalOrder = record

+
  { isDecTotalOrder = ≥-isDecTotalOrder

+
  }

+

+
open DecTotalOrder ≥-decTotalOrder public

+
  using () renaming (_≤?_ to _≥?_)

+

+
------------------------------------------------------------------------

+
-- _<_ - the strict version is a strict total order

+

+
open TotalOrderProperties public

+
  using

+
  ( _<_

+
  ; <-resp-≈

+
  ; <-respʳ-≈

+
  ; <-respˡ-≈

+
  ; <-irrefl

+
  ; <-asym

+
  ; <-trans

+
  ; <-isStrictPartialOrder

+
  ; <-strictPartialOrder

+
  ; <⇒≉

+
  ; ≤∧≉⇒<

+
  ; <⇒≱

+
  ; ≤⇒≯

+
  )

+

+
<-isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_

+
<-isStrictTotalOrder = ToStrict.<-isStrictTotalOrder₂ isDecTotalOrder

+

+
<-strictTotalOrder : StrictTotalOrder _ _ _

+
<-strictTotalOrder = record

+
  { isStrictTotalOrder = <-isStrictTotalOrder

+
  }

+

+
open StrictTotalOrder <-strictTotalOrder public

+
  using (_≮_) renaming (compare to <-compare)

+

+
------------------------------------------------------------------------

+
-- _≰_ - the negated order

+

+
open TotalOrderProperties public

+
  using

+
  ( ≰-respʳ-≈

+
  ; ≰-respˡ-≈

+
  ; ≰⇒>

+
  ; ≰⇒≥

+
  )

+

+
≮⇒≥ : ∀ {x y} → x ≮ y → y ≤ x

+
≮⇒≥ = ToStrict.≮⇒≥ Eq.sym _≟_ reflexive total

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.Poset.html b/site/static/agda_html/Relation.Binary.Properties.Poset.html
new file mode 100644
index 000000000..7d435bdf9
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.Poset.html
@@ -0,0 +1,203 @@
+
+Relation.Binary.Properties.Poset
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties satisfied by posets

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (flip; _∘_)

+
open import Relation.Binary.Core using (Rel; _Preserves_⟶_)

+
open import Relation.Binary.Bundles using (Poset; StrictPartialOrder)

+
open import Relation.Binary.Structures

+
  using (IsPartialOrder; IsStrictPartialOrder; IsDecPartialOrder)

+
open import Relation.Binary.Definitions

+
  using (_Respectsˡ_; _Respectsʳ_; Decidable)

+
import Relation.Binary.Consequences as Consequences

+
open import Relation.Nullary using (¬_; yes; no)

+
open import Relation.Nullary.Negation using (contradiction)

+

+
module Relation.Binary.Properties.Poset

+
   {p₁ p₂ p₃} (P : Poset p₁ p₂ p₃) where

+

+
open Poset P renaming (Carrier to A)

+

+
import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict

+
import Relation.Binary.Properties.Preorder preorder as PreorderProperties

+
open Eq using (_≉_)

+

+
------------------------------------------------------------------------

+
-- The _≥_ relation is also a poset.

+

+
open PreorderProperties public

+
  using () renaming

+
  ( converse-isPreorder to ≥-isPreorder

+
  ; converse-preorder   to ≥-preorder

+
  )

+

+
≥-isPartialOrder : IsPartialOrder _≈_ _≥_

+
≥-isPartialOrder = record

+
  { isPreorder   = ≥-isPreorder

+
  ; antisym      = flip antisym

+
  }

+

+
≥-poset : Poset p₁ p₂ p₃

+
≥-poset = record

+
  { isPartialOrder = ≥-isPartialOrder

+
  }

+

+
open Poset ≥-poset public

+
  using () renaming

+
  ( refl      to ≥-refl

+
  ; reflexive to ≥-reflexive

+
  ; trans     to ≥-trans

+
  ; antisym   to ≥-antisym

+
  )

+

+
------------------------------------------------------------------------

+
-- Negated order

+

+
≰-respˡ-≈ : _≰_ Respectsˡ _≈_

+
≰-respˡ-≈ x≈y = _∘ ≤-respˡ-≈ (Eq.sym x≈y)

+

+
≰-respʳ-≈ : _≰_ Respectsʳ _≈_

+
≰-respʳ-≈ x≈y = _∘ ≤-respʳ-≈ (Eq.sym x≈y)

+

+
------------------------------------------------------------------------

+
-- Partial orders can be turned into strict partial orders

+

+
infix 4 _<_

+

+
_<_ : Rel A _

+
_<_ = ToStrict._<_

+

+
<-isStrictPartialOrder : IsStrictPartialOrder _≈_ _<_

+
<-isStrictPartialOrder = ToStrict.<-isStrictPartialOrder isPartialOrder

+

+
<-strictPartialOrder : StrictPartialOrder _ _ _

+
<-strictPartialOrder = record

+
  { isStrictPartialOrder = <-isStrictPartialOrder

+
  }

+

+
open StrictPartialOrder <-strictPartialOrder public

+
  using (_≮_; <-resp-≈; <-respʳ-≈; <-respˡ-≈)

+
  renaming

+
  ( irrefl to <-irrefl

+
  ; asym   to <-asym

+
  ; trans  to <-trans

+
  )

+

+
<⇒≉ : ∀ {x y} → x < y → x ≉ y

+
<⇒≉ = ToStrict.<⇒≉

+

+
≤∧≉⇒< : ∀ {x y} → x ≤ y → x ≉ y → x < y

+
≤∧≉⇒< = ToStrict.≤∧≉⇒<

+

+
<⇒≱ : ∀ {x y} → x < y → y ≰ x

+
<⇒≱ = ToStrict.<⇒≱ antisym

+

+
≤⇒≯ : ∀ {x y} → x ≤ y → y ≮ x

+
≤⇒≯ = ToStrict.≤⇒≯ antisym

+

+
------------------------------------------------------------------------

+
-- If ≤ is decidable then so is ≈

+

+
≤-dec⇒≈-dec : Decidable _≤_ → Decidable _≈_

+
≤-dec⇒≈-dec _≤?_ x y with x ≤? y | y ≤? x

+
... | yes x≤y | yes y≤x = yes (antisym x≤y y≤x)

+
... | yes x≤y | no  y≰x = no λ x≈y → contradiction (reflexive (Eq.sym x≈y)) y≰x

+
... | no  x≰y | _       = no λ x≈y → contradiction (reflexive x≈y) x≰y

+

+
≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_

+
≤-dec⇒isDecPartialOrder _≤?_ = record

+
  { isPartialOrder = isPartialOrder

+
  ; _≟_            = ≤-dec⇒≈-dec _≤?_

+
  ; _≤?_           = _≤?_

+
  }

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
mono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≤_ → f Preserves _≈_ ⟶ _≈_

+
mono⇒cong = Consequences.mono⇒cong _≈_ _≈_ Eq.sym reflexive antisym

+

+
antimono⇒cong : ∀ {f} → f Preserves _≤_ ⟶ _≥_ → f Preserves _≈_ ⟶ _≈_

+
antimono⇒cong = Consequences.antimono⇒cong _≈_ _≈_ Eq.sym reflexive antisym

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.Preorder.html b/site/static/agda_html/Relation.Binary.Properties.Preorder.html
new file mode 100644
index 000000000..3b4d83259
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.Preorder.html
@@ -0,0 +1,140 @@
+
+Relation.Binary.Properties.Preorder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties satisfied by preorders

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Preorder; Setoid)

+
open import Relation.Binary.Structures using (IsPreorder)

+

+
module Relation.Binary.Properties.Preorder

+
  {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where

+

+
open import Function.Base using (flip)

+
open import Data.Product.Base as Product using (_×_; _,_; swap)

+
import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd

+

+
open Preorder P

+

+

+
------------------------------------------------------------------------

+
-- The converse relation is also a preorder.

+

+
converse-isPreorder : IsPreorder _≈_ _≳_

+
converse-isPreorder = EqAndOrd.isPreorder isPreorder

+

+
converse-preorder : Preorder p₁ p₂ p₃

+
converse-preorder = EqAndOrd.preorder P

+

+
------------------------------------------------------------------------

+
-- For every preorder there is an induced equivalence

+

+
InducedEquivalence : Setoid _ _

+
InducedEquivalence = record

+
  { _≈_           = λ x y → x ≲ y × x ≳ y

+
  ; isEquivalence = record

+
    { refl  = (refl , refl)

+
    ; sym   = swap

+
    ; trans = Product.zip trans (flip trans)

+
    }

+
  }

+

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
invIsPreorder = converse-isPreorder

+
{-# WARNING_ON_USAGE invIsPreorder

+
"Warning: invIsPreorder was deprecated in v2.0.

+
Please use converse-isPreorder instead."

+
#-}

+
invPreorder = converse-preorder

+
{-# WARNING_ON_USAGE invPreorder

+
"Warning: invPreorder was deprecated in v2.0.

+
Please use converse-preorder instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.Setoid.html b/site/static/agda_html/Relation.Binary.Properties.Setoid.html
new file mode 100644
index 000000000..9eb2f8c73
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.Setoid.html
@@ -0,0 +1,179 @@
+
+Relation.Binary.Properties.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Additional properties for setoids

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_∘_; id; _$_; flip)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Binary.Core using (_⇒_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)

+
open import Relation.Binary.Definitions

+
  using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_; Irreflexive)

+
open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)

+
open import Relation.Binary.Construct.Composition

+
  using (_;_; impliesˡ; transitive⇒≈;≈⊆≈)

+

+
module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where

+

+
open Setoid S

+

+
------------------------------------------------------------------------

+
-- Every setoid is a preorder and partial order with respect to

+
-- propositional equality

+

+
isPreorder : IsPreorder _≡_ _≈_

+
isPreorder = record

+
  { isEquivalence = record

+
    { refl  = ≡.refl

+
    ; sym   = ≡.sym

+
    ; trans = ≡.trans

+
    }

+
  ; reflexive     = reflexive

+
  ; trans         = trans

+
  }

+

+
≈-isPreorder : IsPreorder _≈_ _≈_

+
≈-isPreorder = record

+
  { isEquivalence = isEquivalence

+
  ; reflexive     = id

+
  ; trans         = trans

+
  }

+

+
≈-isPartialOrder : IsPartialOrder _≈_ _≈_

+
≈-isPartialOrder = record

+
  { isPreorder = ≈-isPreorder

+
  ; antisym    = λ i≈j _ → i≈j

+
  }

+

+
preorder : Preorder a a ℓ

+
preorder = record

+
  { isPreorder = isPreorder

+
  }

+

+
≈-preorder : Preorder a ℓ ℓ

+
≈-preorder = record

+
  { isPreorder = ≈-isPreorder

+
  }

+

+
≈-poset : Poset a ℓ ℓ

+
≈-poset = record

+
  { isPartialOrder = ≈-isPartialOrder

+
  }

+

+
------------------------------------------------------------------------

+
-- Properties of _≉_

+

+
≉-sym :  Symmetric _≉_

+
≉-sym x≉y =  x≉y ∘ sym

+

+
≉-respˡ : _≉_ Respectsˡ _≈_

+
≉-respˡ x≈x′ x≉y = x≉y ∘ trans x≈x′

+

+
≉-respʳ : _≉_ Respectsʳ _≈_

+
≉-respʳ y≈y′ x≉y x≈y′ = x≉y $ trans x≈y′ (sym y≈y′)

+

+
≉-resp₂ : _≉_ Respects₂ _≈_

+
≉-resp₂ = ≉-respʳ , ≉-respˡ

+

+
≉-irrefl : Irreflexive _≈_ _≉_

+
≉-irrefl x≈y x≉y = contradiction x≈y x≉y

+

+
------------------------------------------------------------------------

+
-- Equality is closed under composition

+

+
≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_

+
≈;≈⇒≈ = transitive⇒≈;≈⊆≈ _ trans

+

+
≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_

+
≈⇒≈;≈ = impliesˡ _≈_ _≈_ refl id

+

+
------------------------------------------------------------------------

+
-- Other properties

+

+
respʳ-flip : _≈_ Respectsʳ (flip _≈_)

+
respʳ-flip y≈z x≈z = trans x≈z (sym y≈z)

+

+
respˡ-flip : _≈_ Respectsˡ (flip _≈_)

+
respˡ-flip = trans

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Properties.TotalOrder.html b/site/static/agda_html/Relation.Binary.Properties.TotalOrder.html
new file mode 100644
index 000000000..eef11d184
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Properties.TotalOrder.html
@@ -0,0 +1,177 @@
+
+Relation.Binary.Properties.TotalOrder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties satisfied by total orders

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (TotalOrder; DecTotalOrder)

+
open import Relation.Binary.Definitions using (Decidable)

+
open import Relation.Binary.Structures using (IsTotalOrder)

+

+
module Relation.Binary.Properties.TotalOrder

+
  {t₁ t₂ t₃} (T : TotalOrder t₁ t₂ t₃) where

+

+
open TotalOrder T

+

+
open import Data.Product.Base using (proj₁)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd

+
import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict

+
import Relation.Binary.Properties.Poset poset as PosetProperties

+
open import Relation.Binary.Consequences

+

+
------------------------------------------------------------------------

+
-- Total orders are almost decidable total orders

+

+
decTotalOrder : Decidable _≈_ → DecTotalOrder _ _ _

+
decTotalOrder ≟ = record

+
  { isDecTotalOrder = record

+
    { isTotalOrder = isTotalOrder

+
    ; _≟_          = ≟

+
    ; _≤?_         = total∧dec⇒dec reflexive antisym total ≟

+
    }

+
  }

+

+
------------------------------------------------------------------------

+
-- _≥_ - the flipped relation is also a total order

+

+
open PosetProperties public

+
  using

+
  ( ≥-refl

+
  ; ≥-reflexive

+
  ; ≥-trans

+
  ; ≥-antisym

+
  ; ≥-isPreorder

+
  ; ≥-isPartialOrder

+
  ; ≥-preorder

+
  ; ≥-poset

+
  )

+

+
≥-isTotalOrder : IsTotalOrder _≈_ _≥_

+
≥-isTotalOrder = EqAndOrd.isTotalOrder isTotalOrder

+

+
≥-totalOrder : TotalOrder _ _ _

+
≥-totalOrder = record

+
  { isTotalOrder = ≥-isTotalOrder

+
  }

+

+
open TotalOrder ≥-totalOrder public

+
  using () renaming (total to ≥-total)

+

+
------------------------------------------------------------------------

+
-- _<_ - the strict version is a strict partial order

+

+
-- Note that total orders can NOT be turned into strict total orders as

+
-- in order to distinguish between the _≤_ and _<_ cases we must have

+
-- decidable equality _≈_.

+

+
open PosetProperties public

+
  using

+
  ( _<_

+
  ; <-resp-≈

+
  ; <-respʳ-≈

+
  ; <-respˡ-≈

+
  ; <-irrefl

+
  ; <-asym

+
  ; <-trans

+
  ; <-isStrictPartialOrder

+
  ; <-strictPartialOrder

+
  ; <⇒≉

+
  ; ≤∧≉⇒<

+
  ; <⇒≱

+
  ; ≤⇒≯

+
  )

+

+
------------------------------------------------------------------------

+
-- _≰_ - the negated order

+

+
open PosetProperties public

+
  using

+
  ( ≰-respʳ-≈

+
  ; ≰-respˡ-≈

+
  )

+

+
≰⇒> : ∀ {x y} → x ≰ y → y < x

+
≰⇒> = ToStrict.≰⇒> Eq.sym reflexive total

+

+
≰⇒≥ : ∀ {x y} → x ≰ y → y ≤ x

+
≰⇒≥ x≰y = proj₁ (≰⇒> x≰y)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.PropositionalEquality.Algebra.html b/site/static/agda_html/Relation.Binary.PropositionalEquality.Algebra.html
new file mode 100644
index 000000000..b103046da
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.PropositionalEquality.Algebra.html
@@ -0,0 +1,112 @@
+
+Relation.Binary.PropositionalEquality.Algebra
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Propositional (intensional) equality - Algebraic structures

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.PropositionalEquality.Algebra where

+

+
open import Algebra.Bundles using (Magma)

+
open import Algebra.Core using (Op₂)

+
open import Algebra.Structures using (IsMagma)

+
open import Level using (Level)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong₂)

+
open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)

+

+
private

+
  variable

+
    a : Level

+
    A : Set a

+

+
------------------------------------------------------------------------

+
-- Any operation forms a magma over _≡_

+

+
isMagma : (_∙_ : Op₂ A) → IsMagma _≡_ _∙_

+
isMagma _∙_ = record

+
  { isEquivalence = isEquivalence

+
  ; ∙-cong        = cong₂ _∙_

+
  }

+

+
magma : (_∙_ : Op₂ A) → Magma _ _

+
magma _∙_ = record

+
  { isMagma = isMagma _∙_

+
  }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.PropositionalEquality.Core.html b/site/static/agda_html/Relation.Binary.PropositionalEquality.Core.html
new file mode 100644
index 000000000..60e0f1c82
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.PropositionalEquality.Core.html
@@ -0,0 +1,179 @@
+
+Relation.Binary.PropositionalEquality.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Propositional equality

+
--

+
-- This file contains some core definitions which are re-exported by

+
-- Relation.Binary.PropositionalEquality.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.PropositionalEquality.Core where

+

+
open import Data.Product.Base using (_,_)

+
open import Function.Base using (_∘_)

+
open import Level

+
open import Relation.Binary.Core

+
open import Relation.Binary.Definitions

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
private

+
  variable

+
    a b ℓ : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Propositional equality

+

+
open import Agda.Builtin.Equality public

+

+
infix 4 _≢_

+
_≢_ : {A : Set a} → Rel A a

+
x ≢ y = ¬ x ≡ y

+

+
------------------------------------------------------------------------

+
-- Pointwise equality

+

+
infix 4 _≗_

+

+
_≗_ : (f g : A → B) → Set _

+
_≗_ {A = A} {B = B} f g = ∀ x → f x ≡ g x

+

+

+
------------------------------------------------------------------------

+
-- A variant of `refl` where the argument is explicit

+

+
pattern erefl x = refl {x = x}

+

+
------------------------------------------------------------------------

+
-- Congruence lemmas

+

+
cong : ∀ (f : A → B) {x y} → x ≡ y → f x ≡ f y

+
cong f refl = refl

+

+
cong′ : ∀ {f : A → B} x → f x ≡ f x

+
cong′ _ = refl

+

+
icong : ∀ {f : A → B} {x y} → x ≡ y → f x ≡ f y

+
icong = cong _

+

+
icong′ : ∀ {f : A → B} x → f x ≡ f x

+
icong′ _ = refl

+

+
cong₂ : ∀ (f : A → B → C) {x y u v} → x ≡ y → u ≡ v → f x u ≡ f y v

+
cong₂ f refl refl = refl

+

+
cong-app : ∀ {A : Set a} {B : A → Set b} {f g : (x : A) → B x} →

+
           f ≡ g → (x : A) → f x ≡ g x

+
cong-app refl x = refl

+

+
------------------------------------------------------------------------

+
-- Properties of _≡_

+

+
sym : Symmetric {A = A} _≡_

+
sym refl = refl

+

+
trans : Transitive {A = A} _≡_

+
trans refl eq = eq

+

+
subst : Substitutive {A = A} _≡_ ℓ

+
subst P refl p = p

+

+
subst₂ : ∀ (_∼_ : REL A B ℓ) {x y u v} → x ≡ y → u ≡ v → x ∼ u → y ∼ v

+
subst₂ _ refl refl p = p

+

+
resp : ∀ (P : A → Set ℓ) → P Respects _≡_

+
resp P refl p = p

+

+
respˡ : ∀ (∼ : Rel A ℓ) → ∼ Respectsˡ _≡_

+
respˡ _∼_ refl x∼y = x∼y

+

+
respʳ : ∀ (∼ : Rel A ℓ) → ∼ Respectsʳ _≡_

+
respʳ _∼_ refl x∼y = x∼y

+

+
resp₂ : ∀ (∼ : Rel A ℓ) → ∼ Respects₂ _≡_

+
resp₂ _∼_ = respʳ _∼_ , respˡ _∼_

+

+
------------------------------------------------------------------------

+
-- Properties of _≢_

+

+
≢-sym : Symmetric {A = A} _≢_

+
≢-sym x≢y =  x≢y ∘ sym

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.PropositionalEquality.Properties.html b/site/static/agda_html/Relation.Binary.PropositionalEquality.Properties.html
new file mode 100644
index 000000000..10c997cc6
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.PropositionalEquality.Properties.html
@@ -0,0 +1,282 @@
+
+Relation.Binary.PropositionalEquality.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Propositional equality

+
--

+
-- This file contains some core properies of propositional equality

+
-- which are re-exported by Relation.Binary.PropositionalEquality. They

+
-- are ``equality rearrangement'' lemmas.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.PropositionalEquality.Properties where

+

+
open import Function.Base using (id; _∘_)

+
open import Level using (Level)

+
open import Relation.Binary.Bundles

+
  using (Setoid; DecSetoid; Preorder; Poset)

+
open import Relation.Binary.Structures

+
  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder)

+
open import Relation.Binary.Definitions

+
  using (DecidableEquality)

+
import Relation.Binary.Properties.Setoid as Setoid

+
open import Relation.Binary.PropositionalEquality.Core

+
open import Relation.Unary using (Pred)

+
open import Relation.Binary.Reasoning.Syntax

+

+

+
private

+
  variable

+
    a b c p : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Standard eliminator for the propositional equality type

+

+
J : {A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b)

+
    {y : A} (p : x ≡ y) → B x refl → B y p

+
J B refl b = b

+

+
------------------------------------------------------------------------

+
-- Binary and/or dependent versions of standard operations on equality

+

+
dcong : ∀ {A : Set a} {B : A → Set b} (f : (x : A) → B x) {x y}

+
      → (p : x ≡ y) → subst B p (f x) ≡ f y

+
dcong f refl = refl

+

+
dcong₂ : ∀ {A : Set a} {B : A → Set b} {C : Set c}

+
         (f : (x : A) → B x → C) {x₁ x₂ y₁ y₂}

+
       → (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂

+
       → f x₁ y₁ ≡ f x₂ y₂

+
dcong₂ f refl refl = refl

+

+
dsubst₂ : ∀ {A : Set a} {B : A → Set b} (C : (x : A) → B x → Set c)

+
          {x₁ x₂ y₁ y₂} (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂

+
        → C x₁ y₁ → C x₂ y₂

+
dsubst₂ C refl refl c = c

+

+
ddcong₂ : ∀ {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c}

+
         (f : (x : A) (y : B x) → C x y) {x₁ x₂ y₁ y₂}

+
         (p : x₁ ≡ x₂) (q : subst B p y₁ ≡ y₂)

+
       → dsubst₂ C p q (f x₁ y₁) ≡ f x₂ y₂

+
ddcong₂ f refl refl = refl

+

+
------------------------------------------------------------------------

+
-- Various equality rearrangement lemmas

+

+
trans-reflʳ : ∀ {x y : A} (p : x ≡ y) → trans p refl ≡ p

+
trans-reflʳ refl = refl

+

+
trans-assoc : ∀ {x y z u : A} (p : x ≡ y) {q : y ≡ z} {r : z ≡ u} →

+
  trans (trans p q) r ≡ trans p (trans q r)

+
trans-assoc refl = refl

+

+
trans-symˡ : ∀ {x y : A} (p : x ≡ y) → trans (sym p) p ≡ refl

+
trans-symˡ refl = refl

+

+
trans-symʳ : ∀ {x y : A} (p : x ≡ y) → trans p (sym p) ≡ refl

+
trans-symʳ refl = refl

+

+
trans-injectiveˡ : ∀ {x y z : A} {p₁ p₂ : x ≡ y} (q : y ≡ z) →

+
                   trans p₁ q ≡ trans p₂ q → p₁ ≡ p₂

+
trans-injectiveˡ refl = subst₂ _≡_ (trans-reflʳ _) (trans-reflʳ _)

+

+
trans-injectiveʳ : ∀ {x y z : A} (p : x ≡ y) {q₁ q₂ : y ≡ z} →

+
                   trans p q₁ ≡ trans p q₂ → q₁ ≡ q₂

+
trans-injectiveʳ refl eq = eq

+

+
cong-id : ∀ {x y : A} (p : x ≡ y) → cong id p ≡ p

+
cong-id refl = refl

+

+
cong-∘ : ∀ {x y : A} {f : B → C} {g : A → B} (p : x ≡ y) →

+
         cong (f ∘ g) p ≡ cong f (cong g p)

+
cong-∘ refl = refl

+

+
sym-cong : ∀ {x y : A} {f : A → B} (p : x ≡ y) → sym (cong f p) ≡ cong f (sym p)

+
sym-cong refl = refl

+

+
trans-cong : ∀ {x y z : A} {f : A → B} (p : x ≡ y) {q : y ≡ z} →

+
             trans (cong f p) (cong f q) ≡ cong f (trans p q)

+
trans-cong refl = refl

+

+
cong₂-reflˡ : ∀ {_∙_ : A → B → C} {x u v} → (p : u ≡ v) →

+
              cong₂ _∙_ refl p ≡ cong (x ∙_) p

+
cong₂-reflˡ refl = refl

+

+
cong₂-reflʳ : ∀ {_∙_ : A → B → C} {x y u} → (p : x ≡ y) →

+
              cong₂ _∙_ p refl ≡ cong (_∙ u) p

+
cong₂-reflʳ refl = refl

+

+
module _ {P : Pred A p} {x y : A} where

+

+
  subst-injective : ∀ (x≡y : x ≡ y) {p q : P x} →

+
                    subst P x≡y p ≡ subst P x≡y q → p ≡ q

+
  subst-injective refl p≡q = p≡q

+

+
  subst-subst : ∀ {z} (x≡y : x ≡ y) {y≡z : y ≡ z} {p : P x} →

+
                subst P y≡z (subst P x≡y p) ≡ subst P (trans x≡y y≡z) p

+
  subst-subst refl = refl

+

+
  subst-subst-sym : (x≡y : x ≡ y) {p : P y} →

+
                    subst P x≡y (subst P (sym x≡y) p) ≡ p

+
  subst-subst-sym refl = refl

+

+
  subst-sym-subst : (x≡y : x ≡ y) {p : P x} →

+
                    subst P (sym x≡y) (subst P x≡y p) ≡ p

+
  subst-sym-subst refl = refl

+

+
subst-∘ : ∀ {x y : A} {P : Pred B p} {f : A → B}

+
          (x≡y : x ≡ y) {p : P (f x)} →

+
          subst (P ∘ f) x≡y p ≡ subst P (cong f x≡y) p

+
subst-∘ refl = refl

+

+
-- Lemma 2.3.11 in the HoTT book, and `transport_map` in the UniMath

+
-- library

+
subst-application′ : ∀ {a b₁ b₂} {A : Set a}

+
                     (B₁ : A → Set b₁) {B₂ : A → Set b₂}

+
                     {x₁ x₂ : A} {y : B₁ x₁}

+
                     (g : ∀ x → B₁ x → B₂ x) (eq : x₁ ≡ x₂) →

+
                     subst B₂ eq (g x₁ y) ≡ g x₂ (subst B₁ eq y)

+
subst-application′ _ _ refl = refl

+

+
subst-application : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}

+
                    (B₁ : A₁ → Set b₁) {B₂ : A₂ → Set b₂}

+
                    {f : A₂ → A₁} {x₁ x₂ : A₂} {y : B₁ (f x₁)}

+
                    (g : ∀ x → B₁ (f x) → B₂ x) (eq : x₁ ≡ x₂) →

+
                    subst B₂ eq (g x₁ y) ≡ g x₂ (subst B₁ (cong f eq) y)

+
subst-application _ _ refl = refl

+

+
------------------------------------------------------------------------

+
-- Structure of equality as a binary relation

+

+
isEquivalence : IsEquivalence {A = A} _≡_

+
isEquivalence = record

+
  { refl  = refl

+
  ; sym   = sym

+
  ; trans = trans

+
  }

+

+
isDecEquivalence : DecidableEquality A → IsDecEquivalence _≡_

+
isDecEquivalence _≟_ = record

+
  { isEquivalence = isEquivalence

+
  ; _≟_           = _≟_

+
  }

+

+
setoid : Set a → Setoid _ _

+
setoid A = record

+
  { Carrier       = A

+
  ; _≈_           = _≡_

+
  ; isEquivalence = isEquivalence

+
  }

+

+
decSetoid : DecidableEquality A → DecSetoid _ _

+
decSetoid _≟_ = record

+
  { _≈_              = _≡_

+
  ; isDecEquivalence = isDecEquivalence _≟_

+
  }

+

+
------------------------------------------------------------------------

+
-- Bundles for equality as a binary relation

+

+
isPreorder : IsPreorder {A = A} _≡_ _≡_

+
isPreorder = Setoid.≈-isPreorder (setoid _)

+

+
isPartialOrder : IsPartialOrder {A = A} _≡_ _≡_

+
isPartialOrder = Setoid.≈-isPartialOrder (setoid _)

+

+
preorder : Set a → Preorder _ _ _

+
preorder A = Setoid.≈-preorder (setoid A)

+

+
poset : Set a → Poset _ _ _

+
poset A = Setoid.≈-poset (setoid A)

+

+
------------------------------------------------------------------------

+
-- Reasoning

+

+
-- This is a special instance of `Relation.Binary.Reasoning.Setoid`.

+
-- Rather than instantiating the latter with (setoid A), we reimplement

+
-- equation chains from scratch since then goals are printed much more

+
-- readably.

+
module ≡-Reasoning {a} {A : Set a} where

+

+
  open begin-syntax {A = A} _≡_ id public

+
  open ≡-syntax {A = A} _≡_ trans public

+
  open end-syntax {A = A} _≡_ refl public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.PropositionalEquality.html b/site/static/agda_html/Relation.Binary.PropositionalEquality.html
new file mode 100644
index 000000000..603896ada
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.PropositionalEquality.html
@@ -0,0 +1,209 @@
+
+Relation.Binary.PropositionalEquality
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Propositional (intensional) equality

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary.PropositionalEquality where

+

+
open import Axiom.UniquenessOfIdentityProofs

+
open import Function.Base using (id; _∘_)

+
import Function.Dependent.Bundles as Dependent

+
open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary using (Irrelevant)

+
open import Relation.Nullary.Decidable using (yes; no; dec-yes-irr; dec-no)

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Definitions using (DecidableEquality)

+
open import Relation.Binary.Indexed.Heterogeneous

+
  using (IndexedSetoid)

+
import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial

+
  as Trivial

+

+
private

+
  variable

+
    a b c ℓ p : Level

+
    A B C : Set a

+

+
------------------------------------------------------------------------

+
-- Re-export contents modules that make up the parts

+

+
open import Relation.Binary.PropositionalEquality.Core public

+
open import Relation.Binary.PropositionalEquality.Properties public

+
open import Relation.Binary.PropositionalEquality.Algebra public

+

+
------------------------------------------------------------------------

+
-- Pointwise equality

+

+
_→-setoid_ : ∀ (A : Set a) (B : Set b) → Setoid _ _

+
A →-setoid B = ≡-setoid A (Trivial.indexedSetoid (setoid B))

+

+
:→-to-Π : ∀ {A : Set a} {B : IndexedSetoid A b ℓ} →

+
          ((x : A) → IndexedSetoid.Carrier B x) →

+
          Dependent.Func (setoid A) B

+
:→-to-Π {B = B} f = record

+
  { to = f

+
  ; cong = λ { refl → IndexedSetoid.refl B }

+
  }

+

+
→-to-⟶ : ∀ {A : Set a} {B : Setoid b ℓ} →

+
         (A → Setoid.Carrier B) →

+
         Dependent.Func (setoid A) (Trivial.indexedSetoid B)

+
→-to-⟶ = :→-to-Π

+

+
------------------------------------------------------------------------

+
-- More complex rearrangement lemmas

+

+
-- A lemma that is very similar to Lemma 2.4.3 from the HoTT book.

+

+
naturality : ∀ {x y} {x≡y : x ≡ y} {f g : A → B}

+
             (f≡g : ∀ x → f x ≡ g x) →

+
             trans (cong f x≡y) (f≡g y) ≡ trans (f≡g x) (cong g x≡y)

+
naturality {x = x} {x≡y = refl} f≡g =

+
  f≡g x               ≡⟨ sym (trans-reflʳ _) ⟩

+
  trans (f≡g x) refl  ∎

+
  where open ≡-Reasoning

+

+
-- A lemma that is very similar to Corollary 2.4.4 from the HoTT book.

+

+
cong-≡id : ∀ {f : A → A} {x : A} (f≡id : ∀ x → f x ≡ x) →

+
           cong f (f≡id x) ≡ f≡id (f x)

+
cong-≡id {f = f} {x} f≡id = begin

+
  cong f fx≡x                                    ≡⟨ sym (trans-reflʳ _) ⟩

+
  trans (cong f fx≡x) refl                       ≡⟨ cong (trans _) (sym (trans-symʳ fx≡x)) ⟩

+
  trans (cong f fx≡x) (trans fx≡x (sym fx≡x))    ≡⟨ sym (trans-assoc (cong f fx≡x)) ⟩

+
  trans (trans (cong f fx≡x) fx≡x) (sym fx≡x)    ≡⟨ cong (λ p → trans p (sym _)) (naturality f≡id) ⟩

+
  trans (trans f²x≡x (cong id fx≡x)) (sym fx≡x)  ≡⟨ cong (λ p → trans (trans f²x≡x p) (sym fx≡x)) (cong-id _) ⟩

+
  trans (trans f²x≡x fx≡x) (sym fx≡x)            ≡⟨ trans-assoc f²x≡x ⟩

+
  trans f²x≡x (trans fx≡x (sym fx≡x))            ≡⟨ cong (trans _) (trans-symʳ fx≡x) ⟩

+
  trans f²x≡x refl                               ≡⟨ trans-reflʳ _ ⟩

+
  f≡id (f x)                                     ∎

+
  where open ≡-Reasoning; fx≡x = f≡id x; f²x≡x = f≡id (f x)

+

+
module _ (_≟_ : DecidableEquality A) {x y : A} where

+

+
  ≡-≟-identity : (eq : x ≡ y) → x ≟ y ≡ yes eq

+
  ≡-≟-identity eq = dec-yes-irr (x ≟ y) (Decidable⇒UIP.≡-irrelevant _≟_) eq

+

+
  ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y

+
  ≢-≟-identity = dec-no (x ≟ y)

+

+

+
------------------------------------------------------------------------

+
-- Inspect

+

+
-- Inspect can be used when you want to pattern match on the result r

+
-- of some expression e, and you also need to "remember" that r ≡ e.

+

+
-- See README.Inspect for an explanation of how/why to use this.

+

+
-- Normally (but not always) the new `with ... in` syntax described at

+
-- https://agda.readthedocs.io/en/v2.6.4/language/with-abstraction.html#with-abstraction-equality

+
-- can be used instead."

+

+
record Reveal_·_is_ {A : Set a} {B : A → Set b}

+
                    (f : (x : A) → B x) (x : A) (y : B x) :

+
                    Set (a ⊔ b) where

+
  constructor [_]

+
  field eq : f x ≡ y

+

+
inspect : ∀ {A : Set a} {B : A → Set b}

+
          (f : (x : A) → B x) (x : A) → Reveal f · x is f x

+
inspect f x = [ refl ]

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
isPropositional : Set a → Set a

+
isPropositional = Irrelevant

+

+
{-# WARNING_ON_USAGE isPropositional

+
"Warning: isPropositional was deprecated in v2.0.

+
Please use Relation.Nullary.Irrelevant instead. "

+
#-}

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.Base.Double.html b/site/static/agda_html/Relation.Binary.Reasoning.Base.Double.html
new file mode 100644
index 000000000..2ef25343f
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.Base.Double.html
@@ -0,0 +1,178 @@
+
+Relation.Binary.Reasoning.Base.Double
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The basic code for equational reasoning with two relations:

+
-- equality and some other ordering.

+
------------------------------------------------------------------------

+
--

+
-- See `Data.Nat.Properties` or `Relation.Binary.Reasoning.PartialOrder`

+
-- for examples of how to instantiate this module.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Level using (_⊔_)

+
open import Function.Base using (case_of_)

+
open import Relation.Nullary.Decidable.Core using (Dec; yes; no)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Definitions using (Reflexive; Trans)

+
open import Relation.Binary.Structures using (IsPreorder)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Reasoning.Syntax

+

+

+
module Relation.Binary.Reasoning.Base.Double {a ℓ₁ ℓ₂} {A : Set a}

+
  {_≈_ : Rel A ℓ₁} {_≲_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _≲_)

+
  where

+

+
open IsPreorder isPreorder

+

+
------------------------------------------------------------------------

+
-- A datatype to hide the current relation type

+

+
infix 4 _IsRelatedTo_

+

+
data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  nonstrict : (x≲y : x ≲ y) → x IsRelatedTo y

+
  equals    : (x≈y : x ≈ y) → x IsRelatedTo y

+

+
start : _IsRelatedTo_ ⇒ _≲_

+
start (equals x≈y) = reflexive x≈y

+
start (nonstrict x≲y) = x≲y

+

+
≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_

+
≡-go x≡y (equals y≈z) = equals (case x≡y of λ where ≡.refl → y≈z)

+
≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where ≡.refl → y≤z)

+

+
≲-go : Trans _≲_ _IsRelatedTo_ _IsRelatedTo_

+
≲-go x≲y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x≲y)

+
≲-go x≲y (nonstrict y≲z) = nonstrict (trans x≲y y≲z)

+

+
≈-go : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_

+
≈-go x≈y (equals y≈z) = equals (Eq.trans x≈y y≈z)

+
≈-go x≈y (nonstrict y≲z) = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y≲z)

+

+
stop : Reflexive _IsRelatedTo_

+
stop = equals Eq.refl

+

+
------------------------------------------------------------------------

+
-- A record that is used to ensure that the final relation proved by the

+
-- chain of reasoning can be converted into the required relation.

+

+
data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂) where

+
  isEquality : ∀ x≈y → IsEquality (equals x≈y)

+

+
IsEquality? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsEquality x≲y)

+
IsEquality? (nonstrict _) = no λ()

+
IsEquality? (equals x≈y)  = yes (isEquality x≈y)

+

+
extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y

+
extractEquality (isEquality x≈y) = x≈y

+

+
equalitySubRelation : SubRelation  _IsRelatedTo_ _ _

+
equalitySubRelation = record

+
  { IsS = IsEquality

+
  ; IsS? = IsEquality?

+
  ; extract = extractEquality

+
  }

+

+
------------------------------------------------------------------------

+
-- Reasoning combinators

+

+
open begin-syntax  _IsRelatedTo_ start public

+
open begin-equality-syntax  _IsRelatedTo_ equalitySubRelation public

+
open ≡-syntax _IsRelatedTo_ ≡-go public

+
open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go Eq.sym public

+
open ≲-syntax _IsRelatedTo_ _IsRelatedTo_ ≲-go public

+
open end-syntax _IsRelatedTo_ stop public

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ≲-go public

+
{-# WARNING_ON_USAGE step-∼

+
"Warning: step-∼ and _∼⟨_⟩_ syntax was deprecated in v2.0.

+
Please use step-≲ and _≲⟨_⟩_ instead. "

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.Base.Single.html b/site/static/agda_html/Relation.Binary.Reasoning.Base.Single.html
new file mode 100644
index 000000000..79794a6ab
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.Base.Single.html
@@ -0,0 +1,127 @@
+
+Relation.Binary.Reasoning.Base.Single
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The basic code for equational reasoning with a single relation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Level using (_⊔_)

+
open import Function.Base using (case_of_)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Definitions using (Reflexive; Transitive; Trans)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Reasoning.Syntax

+

+
module Relation.Binary.Reasoning.Base.Single

+
  {a ℓ} {A : Set a} (_∼_ : Rel A ℓ)

+
  (refl : Reflexive _∼_) (trans : Transitive _∼_)

+
  where

+

+
------------------------------------------------------------------------

+
-- Definition of "related to"

+

+
-- This seemingly unnecessary type is used to make it possible to

+
-- infer arguments even if the underlying equality evaluates.

+

+
infix 4 _IsRelatedTo_

+

+
data _IsRelatedTo_ (x y : A) : Set ℓ where

+
  relTo : (x∼y : x ∼ y) → x IsRelatedTo y

+

+
start : _IsRelatedTo_ ⇒ _∼_

+
start (relTo x∼y) = x∼y

+

+
∼-go : Trans _∼_ _IsRelatedTo_ _IsRelatedTo_

+
∼-go x∼y (relTo y∼z) = relTo (trans x∼y y∼z)

+

+
≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_

+
≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where ≡.refl → y∼z)

+

+
stop : Reflexive _IsRelatedTo_

+
stop = relTo refl

+

+
------------------------------------------------------------------------

+
-- Reasoning combinators

+

+
open begin-syntax _IsRelatedTo_ start public

+
open ≡-syntax _IsRelatedTo_ ≡-go public

+
open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public

+
open end-syntax _IsRelatedTo_ stop public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.Base.Triple.html b/site/static/agda_html/Relation.Binary.Reasoning.Base.Triple.html
new file mode 100644
index 000000000..abc74ad76
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.Base.Triple.html
@@ -0,0 +1,205 @@
+
+Relation.Binary.Reasoning.Base.Triple
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- The basic code for equational reasoning with three relations:

+
-- equality, strict ordering and non-strict ordering.

+
------------------------------------------------------------------------

+
--

+
-- See `Data.Nat.Properties` or `Relation.Binary.Reasoning.PartialOrder`

+
-- for examples of how to instantiate this module.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.Product.Base using (proj₁; proj₂)

+
open import Level using (_⊔_)

+
open import Function.Base using (case_of_)

+
open import Relation.Nullary.Decidable.Core

+
  using (Dec; yes; no)

+
open import Relation.Binary.Core using (Rel; _⇒_)

+
open import Relation.Binary.Structures using (IsPreorder)

+
open import Relation.Binary.Definitions

+
  using (Transitive; _Respects₂_; Reflexive; Trans; Irreflexive; Asymmetric)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Reasoning.Syntax

+

+
module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a}

+
  {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} {_<_ : Rel A ℓ₃}

+
  (isPreorder : IsPreorder _≈_ _≤_)

+
  (<-asym : Asymmetric _<_) (<-trans : Transitive _<_) (<-resp-≈ : _<_ Respects₂ _≈_)

+
  (<⇒≤ : _<_ ⇒ _≤_)

+
  (<-≤-trans : Trans _<_ _≤_ _<_) (≤-<-trans : Trans _≤_ _<_ _<_)

+
  where

+

+
open IsPreorder isPreorder

+

+
------------------------------------------------------------------------

+
-- A datatype to abstract over the current relation

+

+
infix 4 _IsRelatedTo_

+

+
data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  strict    : (x<y : x < y) → x IsRelatedTo y

+
  nonstrict : (x≤y : x ≤ y) → x IsRelatedTo y

+
  equals    : (x≈y : x ≈ y) → x IsRelatedTo y

+

+
start : _IsRelatedTo_ ⇒ _≤_

+
start (equals x≈y) = reflexive x≈y

+
start (nonstrict x≤y) = x≤y

+
start (strict x<y) = <⇒≤ x<y

+

+
≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_

+
≡-go x≡y (equals y≈z) = equals (case x≡y of λ where ≡.refl → y≈z)

+
≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where ≡.refl → y≤z)

+
≡-go x≡y (strict y<z) = strict (case x≡y of λ where ≡.refl → y<z)

+

+
≈-go : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_

+
≈-go x≈y (equals y≈z) = equals (Eq.trans x≈y y≈z)

+
≈-go x≈y (nonstrict y≤z) = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y≤z)

+
≈-go x≈y (strict y<z) = strict (proj₂ <-resp-≈ (Eq.sym x≈y) y<z)

+

+
≤-go : Trans _≤_ _IsRelatedTo_ _IsRelatedTo_

+
≤-go x≤y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x≤y)

+
≤-go x≤y (nonstrict y≤z) = nonstrict (trans x≤y y≤z)

+
≤-go x≤y (strict y<z) = strict (≤-<-trans x≤y y<z)

+

+
<-go : Trans _<_ _IsRelatedTo_ _IsRelatedTo_

+
<-go x<y (equals y≈z) = strict (proj₁ <-resp-≈ y≈z x<y)

+
<-go x<y (nonstrict y≤z) = strict (<-≤-trans x<y y≤z)

+
<-go x<y (strict y<z) = strict (<-trans x<y y<z)

+

+
stop : Reflexive _IsRelatedTo_

+
stop = equals Eq.refl

+

+

+
------------------------------------------------------------------------

+
-- Types that are used to ensure that the final relation proved by the

+
-- chain of reasoning can be converted into the required relation.

+

+
data IsStrict {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  isStrict : ∀ x<y → IsStrict (strict x<y)

+

+
IsStrict? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsStrict x≲y)

+
IsStrict? (strict    x<y) = yes (isStrict x<y)

+
IsStrict? (nonstrict _)   = no λ()

+
IsStrict? (equals    _)   = no λ()

+

+
extractStrict : ∀ {x y} {x≲y : x IsRelatedTo y} → IsStrict x≲y → x < y

+
extractStrict (isStrict x<y) = x<y

+

+
strictRelation : SubRelation _IsRelatedTo_ _ _

+
strictRelation = record

+
  { IsS = IsStrict

+
  ; IsS? = IsStrict?

+
  ; extract = extractStrict

+
  }

+

+
------------------------------------------------------------------------

+
-- Equality sub-relation

+

+
data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where

+
  isEquality : ∀ x≈y → IsEquality (equals x≈y)

+

+
IsEquality? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsEquality x≲y)

+
IsEquality? (strict    _) = no λ()

+
IsEquality? (nonstrict _) = no λ()

+
IsEquality? (equals x≈y)  = yes (isEquality x≈y)

+

+
extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y

+
extractEquality (isEquality x≈y) = x≈y

+

+
eqRelation : SubRelation _IsRelatedTo_ _ _

+
eqRelation = record

+
  { IsS = IsEquality

+
  ; IsS? = IsEquality?

+
  ; extract = extractEquality

+
  }

+

+
------------------------------------------------------------------------

+
-- Reasoning combinators

+

+
open begin-syntax _IsRelatedTo_ start public

+
open begin-equality-syntax _IsRelatedTo_ eqRelation public

+
open begin-strict-syntax _IsRelatedTo_ strictRelation public

+
open begin-contradiction-syntax _IsRelatedTo_ strictRelation <-asym public

+
open ≡-syntax _IsRelatedTo_ ≡-go public

+
open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go Eq.sym public

+
open ≤-syntax _IsRelatedTo_ _IsRelatedTo_ ≤-go public

+
open <-syntax _IsRelatedTo_ _IsRelatedTo_ <-go public

+
open end-syntax _IsRelatedTo_ stop public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.PartialOrder.html b/site/static/agda_html/Relation.Binary.Reasoning.PartialOrder.html
new file mode 100644
index 000000000..940529e05
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.PartialOrder.html
@@ -0,0 +1,140 @@
+
+Relation.Binary.Reasoning.PartialOrder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Convenient syntax for "equational reasoning" using a partial order

+
------------------------------------------------------------------------

+

+
-- Example uses:

+
--

+
--    u≤x : u ≤ x

+
--    u≤x = begin

+
--      u  ≈⟨ u≈v ⟩

+
--      v  ≡⟨ v≡w ⟩

+
--      w  <⟨ w≤x ⟩

+
--      x  ∎

+
--

+
--    u<x : u < x

+
--    u<x = begin-strict

+
--      u  ≈⟨ u≈v ⟩

+
--      v  ≡⟨ v≡w ⟩

+
--      w  <⟨ w≤x ⟩

+
--      x  ∎

+
--

+
--    u<e : u < e

+
--    u<e = begin-strict

+
--      u  ≈⟨ u≈v ⟩

+
--      v  ≡⟨ v≡w ⟩

+
--      w  <⟨ w<x ⟩

+
--      x  ≤⟨ x≤y ⟩

+
--      y  <⟨ y<z ⟩

+
--      z  ≡⟨ d≡z ⟨

+
--      d  ≈⟨ e≈d ⟨

+
--      e  ∎

+
--

+
--    u≈w : u ≈ w

+
--    u≈w = begin-equality

+
--      u  ≈⟨ u≈v ⟩

+
--      v  ≡⟨ v≡w ⟩

+
--      w  ∎

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Poset)

+

+
module Relation.Binary.Reasoning.PartialOrder

+
  {p₁ p₂ p₃} (P : Poset p₁ p₂ p₃) where

+

+
open Poset P

+
open import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_

+
  as Strict

+
  using (_<_)

+

+
------------------------------------------------------------------------

+
-- Re-export contents of base module

+

+
open import Relation.Binary.Reasoning.Base.Triple

+
  isPreorder

+
  (Strict.<-asym antisym)

+
  (Strict.<-trans isPartialOrder)

+
  (Strict.<-resp-≈ isEquivalence ≤-resp-≈)

+
  Strict.<⇒≤

+
  (Strict.<-≤-trans Eq.sym trans antisym ≤-respʳ-≈)

+
  (Strict.≤-<-trans trans antisym ≤-respˡ-≈)

+
  public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.Preorder.html b/site/static/agda_html/Relation.Binary.Reasoning.Preorder.html
new file mode 100644
index 000000000..25661a00b
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.Preorder.html
@@ -0,0 +1,113 @@
+
+Relation.Binary.Reasoning.Preorder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
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+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Convenient syntax for "equational reasoning" using a preorder

+
------------------------------------------------------------------------

+

+
-- Example uses:

+
--

+
--    u∼y : u ∼ y

+
--    u∼y = begin

+
--      u  ≈⟨ u≈v ⟩

+
--      v  ≡⟨ v≡w ⟩

+
--      w  ∼⟨ w∼y ⟩

+
--      y  ≈⟨ z≈y ⟩

+
--      z  ∎

+
--

+
--    u≈w : u ≈ w

+
--    u≈w = begin-equality

+
--      u  ≈⟨ u≈v ⟩

+
--      v  ≡⟨ v≡w ⟩

+
--      w  ≡⟨ x≡w ⟨

+
--      x  ∎

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Preorder)

+

+
module Relation.Binary.Reasoning.Preorder

+
  {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where

+

+
open Preorder P

+

+
------------------------------------------------------------------------

+
-- Publicly re-export the contents of the base module

+

+
open import Relation.Binary.Reasoning.Base.Double isPreorder public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.Setoid.html b/site/static/agda_html/Relation.Binary.Reasoning.Setoid.html
new file mode 100644
index 000000000..64a3fbbc9
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.Setoid.html
@@ -0,0 +1,120 @@
+
+Relation.Binary.Reasoning.Setoid
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Convenient syntax for reasoning with a setoid

+
------------------------------------------------------------------------

+

+
-- Example use:

+

+
-- n*0≡0 : ∀ n → n * 0 ≡ 0

+
-- n*0≡0 zero    = refl

+
-- n*0≡0 (suc n) = begin

+
--   suc n * 0 ≈⟨ refl ⟩

+
--   n * 0 + 0 ≈⟨ ... ⟩

+
--   n * 0     ≈⟨ n*0≡0 n ⟩

+
--   0         ∎

+

+
-- Module `≡-Reasoning` in `Relation.Binary.PropositionalEquality`

+
-- is recommended for equational reasoning when the underlying equality

+
-- is `_≡_`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Bundles using (Setoid)

+
open import Relation.Binary.Reasoning.Syntax using (module ≈-syntax)

+

+
module Relation.Binary.Reasoning.Setoid {s₁ s₂} (S : Setoid s₁ s₂) where

+

+
open Setoid S

+

+
import Relation.Binary.Reasoning.Base.Single _≈_ refl trans

+
  as SingleRelReasoning

+

+
------------------------------------------------------------------------

+
-- Reasoning combinators

+

+
-- Export the combinators for single relation reasoning, hiding the

+
-- single misnamed combinator.

+
open SingleRelReasoning public

+
  hiding (step-∼)

+
  renaming (∼-go to ≈-go)

+

+
-- Re-export the equality-based combinators instead

+
open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go sym public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reasoning.Syntax.html b/site/static/agda_html/Relation.Binary.Reasoning.Syntax.html
new file mode 100644
index 000000000..e84529b60
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reasoning.Syntax.html
@@ -0,0 +1,522 @@
+
+Relation.Binary.Reasoning.Syntax
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Syntax for the building blocks of equational reasoning modules

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Level using (Level; _⊔_; suc)

+
open import Relation.Nullary.Decidable.Core

+
  using (Dec; True; toWitness)

+
open import Relation.Nullary.Negation.Core using (contradiction)

+
open import Relation.Binary.Core using (Rel; REL; _⇒_)

+
open import Relation.Binary.Definitions

+
  using (_Respectsʳ_; Asymmetric; Trans; Sym; Reflexive)

+
open import Relation.Binary.PropositionalEquality.Core as ≡

+
  using (_≡_)

+

+
-- List of `Reasoning` modules that do not use this framework and so

+
-- need to be updated manually if the syntax changes.

+
--

+
--   Data/Vec/Relation/Binary/Equality/Cast

+
--   Relation/Binary/HeterogeneousEquality

+
--   Effect/Monad/Partiality

+
--   Effect/Monad/Partiality/All

+
--   Codata/Guarded/Stream/Relation/Binary/Pointwise

+
--   Function/Reasoning

+

+
module Relation.Binary.Reasoning.Syntax where

+

+
private

+
  variable

+
    a ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level

+
    A B C : Set a

+
    x y z : A

+

+
------------------------------------------------------------------------

+
-- Syntax for beginning a reasoning chain

+
------------------------------------------------------------------------

+

+
------------------------------------------------------------------------

+
-- Basic begin syntax

+

+
module begin-syntax

+
  (R : REL A B ℓ₁)

+
  {S : REL A B ℓ₂}

+
  (reflexive : R ⇒ S)

+
  where

+

+
  infix 1 begin_

+

+
  begin_ : R x y → S x y

+
  begin_ = reflexive

+

+
------------------------------------------------------------------------

+
-- Begin subrelation syntax

+

+
-- Sometimes we want to support sub-relations with the

+
-- same reasoning operators as the main relations (e.g. perform equality

+
-- proofs with non-strict reasoning operators). This record bundles all

+
-- the parts needed to extract the sub-relation proofs.

+
record SubRelation {A : Set a} (R : Rel A ℓ₁) ℓ₂ ℓ₃ : Set (a ⊔ ℓ₁ ⊔ suc ℓ₂ ⊔ suc ℓ₃) where

+
  field

+
    S : Rel A ℓ₂

+
    IsS : R x y → Set ℓ₃

+
    IsS? : ∀ (xRy : R x y) → Dec (IsS xRy)

+
    extract : ∀ {xRy : R x y} → IsS xRy → S x y

+

+
module begin-subrelation-syntax

+
  (R : Rel A ℓ₁)

+
  (sub : SubRelation R ℓ₂ ℓ₃)

+
  where

+
  open SubRelation sub

+

+
  infix 1 begin_

+

+
  begin_ : ∀ {x y} (xRy : R x y) → {s : True (IsS? xRy)} → S x y

+
  begin_ r {s} = extract (toWitness s)

+

+
-- Begin equality syntax

+
module begin-equality-syntax

+
  (R : Rel A ℓ₁)

+
  (sub : SubRelation R ℓ₂ ℓ₃) where

+

+
  open begin-subrelation-syntax R sub public

+
    renaming (begin_ to begin-equality_)

+

+
-- Begin apartness syntax

+
module begin-apartness-syntax

+
  (R : Rel A ℓ₁)

+
  (sub : SubRelation R ℓ₂ ℓ₃) where

+

+
  open begin-subrelation-syntax R sub public

+
    renaming (begin_ to begin-apartness_)

+

+
-- Begin strict syntax

+
module begin-strict-syntax

+
  (R : Rel A ℓ₁)

+
  (sub : SubRelation R ℓ₂ ℓ₃) where

+

+
  open begin-subrelation-syntax R sub public

+
    renaming (begin_ to begin-strict_)

+

+
------------------------------------------------------------------------

+
-- Begin membership syntax

+

+
module begin-membership-syntax

+
  (R : Rel A ℓ₁)

+
  (_∈_ : REL B A ℓ₂)

+
  (resp : _∈_ Respectsʳ R) where

+

+
  infix  1 step-∈

+

+
  step-∈ : ∀ (x : B) {xs ys} → R xs ys → x ∈ xs → x ∈ ys

+
  step-∈ x = resp

+

+
  syntax step-∈ x  xs⊆ys x∈xs  = x ∈⟨ x∈xs ⟩ xs⊆ys

+

+
------------------------------------------------------------------------

+
-- Begin contradiction syntax

+

+
-- Used with asymmetric subrelations to derive a contradiction from a

+
-- proof that an element is related to itself.

+
module begin-contradiction-syntax

+
  (R : Rel A ℓ₁)

+
  (sub : SubRelation R ℓ₂ ℓ₃)

+
  (asym : Asymmetric (SubRelation.S sub))

+
  where

+

+
  open SubRelation sub

+

+
  infix 1 begin-contradiction_

+

+
  begin-contradiction_ : ∀ (xRx : R x x) {s : True (IsS? xRx)} →

+
                         ∀ {b} {B : Set b} → B

+
  begin-contradiction_ {x} r {s} = contradiction x<x (asym x<x)

+
    where

+
    x<x : S x x

+
    x<x = extract (toWitness s)

+

+
------------------------------------------------------------------------

+
-- Syntax for continuing a chain of reasoning steps

+
------------------------------------------------------------------------

+

+
-- Note that the arguments to the `step`s are not provided in their

+
-- "natural" order and syntax declarations are later used to re-order

+
-- them. This is because the `step` ordering allows the type-checker to

+
-- better infer the middle argument `y` from the `_IsRelatedTo_`

+
-- argument (see issue 622).

+
--

+
-- This has two practical benefits. First it speeds up type-checking by

+
-- approximately a factor of 5. Secondly it allows the combinators to be

+
-- used with macros that use reflection, e.g. `Tactic.RingSolver`, where

+
-- they need to be able to extract `y` using reflection.

+

+
------------------------------------------------------------------------

+
-- Syntax for unidirectional relations

+

+
-- See https://github.com/agda/agda-stdlib/issues/2150 for a possible

+
-- simplification.

+

+
module _

+
  {R : REL A B ℓ₂}

+
  (S : REL B C ℓ₁)

+
  (T : REL A C ℓ₃)

+
  (step : Trans R S T)

+
  where

+

+
  forward : ∀ (x : A) {y z} → S y z → R x y → T x z

+
  forward x yRz x∼y = step {x} x∼y yRz

+

+
  -- Arbitrary relation syntax

+
  module ∼-syntax where

+
    infixr 2 step-∼

+
    step-∼ = forward

+
    syntax step-∼ x yRz x∼y = x ∼⟨ x∼y ⟩ yRz

+

+

+
  -- Preorder syntax

+
  module ≲-syntax where

+
    infixr 2 step-≲

+
    step-≲ = forward

+
    syntax step-≲ x yRz x≲y = x ≲⟨ x≲y ⟩ yRz

+

+

+
  -- Partial order syntax

+
  module ≤-syntax where

+
    infixr 2 step-≤

+
    step-≤ = forward

+
    syntax step-≤ x yRz x≤y = x ≤⟨ x≤y ⟩ yRz

+

+

+
  -- Strict partial order syntax

+
  module <-syntax where

+
    infixr 2 step-<

+
    step-< = forward

+
    syntax step-< x yRz x<y = x <⟨ x<y ⟩ yRz

+

+

+
  -- Subset order syntax

+
  module ⊆-syntax where

+
    infixr 2 step-⊆

+
    step-⊆ = forward

+
    syntax step-⊆ x yRz x⊆y = x ⊆⟨ x⊆y ⟩ yRz

+

+

+
  -- Strict subset order syntax

+
  module ⊂-syntax where

+
    infixr 2 step-⊂

+
    step-⊂ = forward

+
    syntax step-⊂ x yRz x⊂y = x ⊂⟨ x⊂y ⟩ yRz

+

+

+
  -- Square subset order syntax

+
  module ⊑-syntax where

+
    infixr 2 step-⊑

+
    step-⊑ = forward

+
    syntax step-⊑ x yRz x⊑y = x ⊑⟨ x⊑y ⟩ yRz

+

+

+
  -- Strict square subset order syntax

+
  module ⊏-syntax where

+
    infixr 2 step-⊏

+
    step-⊏ = forward

+
    syntax step-⊏ x yRz x⊏y = x ⊏⟨ x⊏y ⟩ yRz

+

+

+
  -- Divisibility syntax

+
  module ∣-syntax where

+
    infixr 2 step-∣

+
    step-∣ = forward

+
    syntax step-∣ x yRz x∣y = x ∣⟨ x∣y ⟩ yRz

+

+

+
  -- Single-step syntax

+
  module ⟶-syntax where

+
    infixr 2 step-⟶

+
    step-⟶ = forward

+
    syntax step-⟶ x yRz x∣y = x ⟶⟨ x∣y ⟩ yRz

+

+

+
  -- Multi-step syntax

+
  module ⟶*-syntax where

+
    infixr 2 step-⟶*

+
    step-⟶* = forward

+
    syntax step-⟶* x yRz x∣y = x ⟶*⟨ x∣y ⟩ yRz

+

+

+
------------------------------------------------------------------------

+
-- Syntax for bidirectional relations

+

+
  module _

+
    {U : REL B A ℓ₄}

+
    (sym : Sym U R)

+
    where

+

+
    backward : ∀ x {y z} → S y z → U y x → T x z

+
    backward x yRz x≈y = forward x yRz (sym x≈y)

+

+
    -- Setoid equality syntax

+
    module ≈-syntax where

+
      infixr 2 step-≈-⟩ step-≈-⟨

+
      step-≈-⟩ = forward

+
      step-≈-⟨ = backward

+
      syntax step-≈-⟩ x yRz x≈y = x ≈⟨ x≈y ⟩ yRz

+
      syntax step-≈-⟨ x yRz y≈x = x ≈⟨ y≈x ⟨ yRz

+

+
      -- Deprecated

+
      infixr 2 step-≈ step-≈˘

+
      step-≈ = step-≈-⟩

+
      {-# WARNING_ON_USAGE step-≈

+
      "Warning: step-≈ was deprecated in v2.0.

+
      Please use step-≈-⟩ instead."

+
      #-}

+
      step-≈˘ = step-≈-⟨

+
      {-# WARNING_ON_USAGE step-≈˘

+
      "Warning: step-≈˘ and _≈˘⟨_⟩_ was deprecated in v2.0.

+
      Please use step-≈-⟨ and _≈⟨_⟨_ instead."

+
      #-}

+
      syntax step-≈˘ x yRz y≈x = x ≈˘⟨ y≈x ⟩ yRz

+

+

+
    -- Container equality syntax

+
    module ≋-syntax where

+
      infixr 2 step-≋-⟩ step-≋-⟨

+
      step-≋-⟩ = forward

+
      step-≋-⟨ = backward

+
      syntax step-≋-⟩ x yRz x≋y = x ≋⟨ x≋y ⟩ yRz

+
      syntax step-≋-⟨ x yRz y≋x = x ≋⟨ y≋x ⟨ yRz

+

+

+
      -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.

+
      infixr 2 step-≋ step-≋˘

+
      step-≋ = step-≋-⟩

+
      {-# WARNING_ON_USAGE step-≋

+
      "Warning: step-≋ was deprecated in v2.0.

+
      Please use step-≋-⟩ instead."

+
      #-}

+
      step-≋˘ = step-≋-⟨

+
      {-# WARNING_ON_USAGE step-≋˘

+
      "Warning: step-≋˘ and _≋˘⟨_⟩_ was deprecated in v2.0.

+
      Please use step-≋-⟨ and _≋⟨_⟨_ instead."

+
      #-}

+
      syntax step-≋˘ x yRz y≋x = x ≋˘⟨ y≋x ⟩ yRz

+

+

+
    -- Other equality syntax

+
    module ≃-syntax where

+
      infixr 2 step-≃-⟩ step-≃-⟨

+
      step-≃-⟩ = forward

+
      step-≃-⟨ = backward

+
      syntax step-≃-⟩ x yRz x≃y = x ≃⟨ x≃y ⟩ yRz

+
      syntax step-≃-⟨ x yRz y≃x = x ≃⟨ y≃x ⟨ yRz

+

+

+
    -- Apartness relation syntax

+
    module #-syntax where

+
      infixr 2 step-#-⟩ step-#-⟨

+
      step-#-⟩ = forward

+
      step-#-⟨ = backward

+
      syntax step-#-⟩ x yRz x#y = x #⟨ x#y ⟩ yRz

+
      syntax step-#-⟨ x yRz y#x = x #⟨ y#x ⟨ yRz

+

+
      -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.

+
      infixr 2 step-# step-#˘

+
      step-# = step-#-⟩

+
      {-# WARNING_ON_USAGE step-#

+
      "Warning: step-# was deprecated in v2.0.

+
      Please use step-#-⟩ instead."

+
      #-}

+
      step-#˘ = step-#-⟨

+
      {-# WARNING_ON_USAGE step-#˘

+
      "Warning: step-#˘ and _#˘⟨_⟩_ was deprecated in v2.0.

+
      Please use step-#-⟨ and _#⟨_⟨_ instead."

+
      #-}

+
      syntax step-#˘ x yRz y#x = x #˘⟨ y#x ⟩ yRz

+

+

+
    -- Bijection syntax

+
    module ⤖-syntax where

+
      infixr 2 step-⤖ step-⬻

+
      step-⤖ = forward

+
      step-⬻ = backward

+
      syntax step-⤖ x yRz x⤖y = x ⤖⟨ x⤖y ⟩ yRz

+
      syntax step-⬻ x yRz y⤖x = x ⬻⟨ y⤖x ⟩ yRz

+

+

+
    -- Inverse syntax

+
    module ↔-syntax where

+
      infixr 2 step-↔-⟩ step-↔-⟨

+
      step-↔-⟩ = forward

+
      step-↔-⟨ = backward

+
      syntax step-↔-⟩ x yRz x↔y = x ↔⟨ x↔y ⟩ yRz

+
      syntax step-↔-⟨ x yRz y↔x = x ↔⟨ y↔x ⟨ yRz

+

+

+
    -- Inverse syntax

+
    module ↭-syntax where

+
      infixr 2 step-↭-⟩ step-↭-⟨

+
      step-↭-⟩ = forward

+
      step-↭-⟨ = backward

+
      syntax step-↭-⟩ x yRz x↭y = x ↭⟨ x↭y ⟩ yRz

+
      syntax step-↭-⟨ x yRz y↭x = x ↭⟨ y↭x ⟨ yRz

+

+

+
      -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.

+
      infixr 2 step-↭ step-↭˘

+
      step-↭ = forward

+
      {-# WARNING_ON_USAGE step-↭

+
      "Warning: step-↭ was deprecated in v2.0.

+
      Please use step-↭-⟩ instead."

+
      #-}

+
      step-↭˘ = backward

+
      {-# WARNING_ON_USAGE step-↭˘

+
      "Warning: step-↭˘ and _↭˘⟨_⟩_ was deprecated in v2.0.

+
      Please use step-↭-⟨ and _↭⟨_⟨_ instead."

+
      #-}

+
      syntax step-↭˘ x yRz y↭x = x ↭˘⟨ y↭x ⟩ yRz

+

+
------------------------------------------------------------------------

+
-- Propositional equality

+

+
-- Crucially often the step function cannot just be `subst` or pattern

+
-- match on `refl` as we often want to compute which constructor the

+
-- relation begins with, in order for the implicit subrelation

+
-- arguments to resolve. See `≡-noncomputable-syntax` below if this

+
-- is not required.

+
module ≡-syntax

+
  (R : REL A B ℓ₁)

+
  (step : Trans _≡_ R R)

+
  where

+

+
  infixr 2 step-≡-⟩  step-≡-∣ step-≡-⟨

+
  step-≡-⟩ = forward R R step

+

+
  step-≡-∣ : ∀ x {y} → R x y → R x y

+
  step-≡-∣ x xRy = xRy

+

+
  step-≡-⟨ = backward R R step ≡.sym

+

+
  syntax step-≡-⟩ x yRz x≡y = x ≡⟨ x≡y ⟩ yRz

+
  syntax step-≡-∣ x xRy     = x ≡⟨⟩ xRy

+
  syntax step-≡-⟨ x yRz y≡x = x ≡⟨ y≡x ⟨ yRz

+

+

+
  -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.

+
  infixr 2 step-≡ step-≡˘

+
  step-≡ = step-≡-⟩

+
  {-# WARNING_ON_USAGE step-≡

+
  "Warning: step-≡ was deprecated in v2.0.

+
  Please use step-≡-⟩ instead."

+
  #-}

+
  step-≡˘ = step-≡-⟨

+
  {-# WARNING_ON_USAGE step-≡˘

+
  "Warning: step-≡˘ and _≡˘⟨_⟩_ was deprecated in v2.0.

+
  Please use step-≡-⟨ and _≡⟨_⟨_ instead."

+
  #-}

+
  syntax step-≡˘ x yRz y≡x = x ≡˘⟨ y≡x ⟩ yRz

+

+

+
-- Unlike ≡-syntax above, chains of reasoning using this syntax will not

+
-- reduce when proofs of propositional equality which are not definitionally

+
-- equal to `refl` are passed.

+
module ≡-noncomputing-syntax (R : REL A B ℓ₁) where

+

+
  private

+
    step : Trans _≡_ R R

+
    step ≡.refl xRy = xRy

+

+
  open ≡-syntax R step public

+

+
------------------------------------------------------------------------

+
-- Syntax for ending a chain of reasoning

+
------------------------------------------------------------------------

+

+
module end-syntax

+
  (R : Rel A ℓ₁)

+
  (reflexive : Reflexive R)

+
  where

+

+
  infix 3 _∎

+

+
  _∎ : ∀ x → R x x

+
  x ∎ = reflexive

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Reflection.html b/site/static/agda_html/Relation.Binary.Reflection.html
new file mode 100644
index 000000000..0d47309fc
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Reflection.html
@@ -0,0 +1,180 @@
+
+Relation.Binary.Reflection
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Helpers intended to ease the development of "tactics" which use

+
-- proof by reflection

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Data.Fin.Base using (Fin)

+
open import Data.Nat.Base using (ℕ)

+
open import Data.Vec.Base as Vec using (Vec; allFin)

+
open import Function.Base using (id; _⟨_⟩_)

+
open import Function.Bundles using (module Equivalence)

+
open import Level using (Level)

+
open import Relation.Binary.Bundles using (Setoid)

+
import Relation.Binary.PropositionalEquality.Core as ≡

+

+
-- Think of the parameters as follows:

+
--

+
-- * Expr:    A representation of code.

+
-- * var:     The Expr type should support a notion of variables.

+
-- * ⟦_⟧:     Computes the semantics of an expression. Takes an

+
--            environment mapping variables to something.

+
-- * ⟦_⇓⟧:    Computes the semantics of the normal form of the

+
--            expression.

+
-- * correct: Normalisation preserves the semantics.

+
--

+
-- Given these parameters two "tactics" are returned, prove and solve.

+
--

+
-- For an example of the use of this module, see Algebra.RingSolver.

+

+
module Relation.Binary.Reflection

+
         {e a s}

+
         {Expr : ℕ → Set e} {A : Set a}

+
         (Sem : Setoid a s)

+
         (var : ∀ {n} → Fin n → Expr n)

+
         (⟦_⟧ ⟦_⇓⟧ : ∀ {n} → Expr n → Vec A n → Setoid.Carrier Sem)

+
         (correct : ∀ {n} (e : Expr n) ρ →

+
                    ⟦ e ⇓⟧ ρ ⟨ Setoid._≈_ Sem ⟩ ⟦ e ⟧ ρ)

+
         where

+

+
open import Data.Vec.N-ary

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)

+
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

+

+
open Setoid Sem

+
open ≈-Reasoning Sem

+

+
-- If two normalised expressions are semantically equal, then their

+
-- non-normalised forms are also equal.

+

+
prove : ∀ {n} (ρ : Vec A n) e₁ e₂ →

+
        ⟦ e₁ ⇓⟧ ρ ≈ ⟦ e₂ ⇓⟧ ρ →

+
        ⟦ e₁  ⟧ ρ ≈ ⟦ e₂  ⟧ ρ

+
prove ρ e₁ e₂ hyp = begin

+
  ⟦ e₁  ⟧ ρ ≈⟨ sym (correct e₁ ρ) ⟩

+
  ⟦ e₁ ⇓⟧ ρ ≈⟨ hyp ⟩

+
  ⟦ e₂ ⇓⟧ ρ ≈⟨ correct e₂ ρ ⟩

+
  ⟦ e₂  ⟧ ρ ∎

+

+
-- Applies the function to all possible "variables".

+

+
close : ∀ {A : Set e} n → N-ary n (Expr n) A → A

+
close n f = f $ⁿ Vec.map var (allFin n)

+

+
-- A variant of prove which should in many cases be easier to use,

+
-- because variables and environments are handled in a less explicit

+
-- way.

+
--

+
-- If the type signature of solve is a bit daunting, then it may be

+
-- helpful to instantiate n with a small natural number and normalise

+
-- the remainder of the type.

+

+
solve : ∀ n (f : N-ary n (Expr n) (Expr n × Expr n)) →

+
  Eqʰ n _≈_ (curryⁿ ⟦ proj₁ (close n f) ⇓⟧) (curryⁿ ⟦ proj₂ (close n f) ⇓⟧) →

+
  Eq  n _≈_ (curryⁿ ⟦ proj₁ (close n f)  ⟧) (curryⁿ ⟦ proj₂ (close n f)  ⟧)

+
solve n f hyp =

+
  curryⁿ-cong _≈_ ⟦ proj₁ (close n f) ⟧ ⟦ proj₂ (close n f) ⟧

+
    (λ ρ → prove ρ (proj₁ (close n f)) (proj₂ (close n f))

+
             (curryⁿ-cong⁻¹ _≈_

+
                ⟦ proj₁ (close n f) ⇓⟧ ⟦ proj₂ (close n f) ⇓⟧

+
                (Eqʰ-to-Eq n _≈_ hyp) ρ))

+

+
-- A variant of solve which does not require that the normal form

+
-- equality is proved for an arbitrary environment.

+

+
solve₁ : ∀ n (f : N-ary n (Expr n) (Expr n × Expr n)) →

+
         ∀ⁿ n (curryⁿ λ ρ →

+
                 ⟦ proj₁ (close n f) ⇓⟧ ρ ≈ ⟦ proj₂ (close n f) ⇓⟧ ρ →

+
                 ⟦ proj₁ (close n f)  ⟧ ρ ≈ ⟦ proj₂ (close n f)  ⟧ ρ)

+
solve₁ n f =

+
  Equivalence.from (uncurry-∀ⁿ n) λ ρ →

+
    ≡.subst id (≡.sym (left-inverse (λ _ → _ ≈ _ → _ ≈ _) ρ))

+
      (prove ρ (proj₁ (close n f)) (proj₂ (close n f)))

+

+
-- A variant of _,_ which is intended to make uses of solve and solve₁

+
-- look a bit nicer.

+

+
infix 4 _⊜_

+

+
_⊜_ : ∀ {n} → Expr n → Expr n → Expr n × Expr n

+
_⊜_ = _,_

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Structures.Biased.html b/site/static/agda_html/Relation.Binary.Structures.Biased.html
new file mode 100644
index 000000000..585277a52
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Structures.Biased.html
@@ -0,0 +1,126 @@
+
+Relation.Binary.Structures.Biased
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Ways to give instances of certain structures where some fields can

+
-- be given in terms of others

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Relation.Binary`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core

+

+
module Relation.Binary.Structures.Biased

+
  {a ℓ} {A : Set a} -- The underlying set

+
  (_≈_ : Rel A ℓ)   -- The underlying equality relation

+
  where

+

+
open import Level using (Level; _⊔_)

+
open import Relation.Binary.Consequences

+
open import Relation.Binary.Definitions

+
open import Relation.Binary.Structures _≈_

+

+
private

+
  variable

+
    ℓ₂ : Level

+

+
-- To construct a StrictTotalOrder you only need to prove transitivity and

+
-- trichotomy as the current implementation of `Trichotomous` subsumes

+
-- irreflexivity and asymmetry.

+
record IsStrictTotalOrderᶜ (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isEquivalence : IsEquivalence

+
    trans         : Transitive _<_

+
    compare       : Trichotomous _≈_ _<_

+

+
  isStrictTotalOrderᶜ : IsStrictTotalOrder _<_

+
  isStrictTotalOrderᶜ = record

+
    { isStrictPartialOrder = record

+
      { isEquivalence = isEquivalence

+
      ; irrefl = tri⇒irr compare

+
      ; trans = trans

+
      ; <-resp-≈ = trans∧tri⇒resp Eq.sym Eq.trans trans compare

+
      }

+
    ; compare = compare

+
    } where module Eq = IsEquivalence isEquivalence

+

+
open IsStrictTotalOrderᶜ public

+
  using (isStrictTotalOrderᶜ)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.Structures.html b/site/static/agda_html/Relation.Binary.Structures.html
new file mode 100644
index 000000000..32ce514dd
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.Structures.html
@@ -0,0 +1,395 @@
+
+Relation.Binary.Structures
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Structures for homogeneous binary relations

+
------------------------------------------------------------------------

+

+
-- The contents of this module should be accessed via `Relation.Binary`.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
open import Relation.Binary.Core

+

+
module Relation.Binary.Structures

+
  {a ℓ} {A : Set a} -- The underlying set

+
  (_≈_ : Rel A ℓ)   -- The underlying equality relation

+
  where

+

+
open import Data.Product.Base using (proj₁; proj₂; _,_)

+
open import Level using (Level; _⊔_)

+
open import Relation.Nullary.Negation.Core using (¬_)

+
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

+
open import Relation.Binary.Consequences

+
open import Relation.Binary.Definitions

+

+
private

+
  variable

+
    ℓ₂ : Level

+

+
------------------------------------------------------------------------

+
-- Equivalences

+
------------------------------------------------------------------------

+
-- Note all the following equivalences refer to the equality provided

+
-- as a module parameter at the top of this file.

+

+
record IsPartialEquivalence : Set (a ⊔ ℓ) where

+
  field

+
    sym   : Symmetric _≈_

+
    trans : Transitive _≈_

+

+
-- The preorders of this library are defined in terms of an underlying

+
-- equivalence relation, and hence equivalence relations are not

+
-- defined in terms of preorders.

+

+
-- To preserve backwards compatability, equivalence relations are

+
-- not defined in terms of their partial counterparts.

+

+
record IsEquivalence : Set (a ⊔ ℓ) where

+
  field

+
    refl  : Reflexive _≈_

+
    sym   : Symmetric _≈_

+
    trans : Transitive _≈_

+

+
  reflexive : _≡_ ⇒ _≈_

+
  reflexive ≡.refl = refl

+

+
  isPartialEquivalence : IsPartialEquivalence

+
  isPartialEquivalence = record

+
    { sym = sym

+
    ; trans = trans

+
    }

+

+

+
record IsDecEquivalence : Set (a ⊔ ℓ) where

+
  infix 4 _≟_

+
  field

+
    isEquivalence : IsEquivalence

+
    _≟_           : Decidable _≈_

+

+
  open IsEquivalence isEquivalence public

+

+

+
------------------------------------------------------------------------

+
-- Preorders

+
------------------------------------------------------------------------

+

+
record IsPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isEquivalence : IsEquivalence

+
    -- Reflexivity is expressed in terms of the underlying equality:

+
    reflexive     : _≈_ ⇒ _≲_

+
    trans         : Transitive _≲_

+

+
  module Eq = IsEquivalence isEquivalence

+

+
  refl : Reflexive _≲_

+
  refl = reflexive Eq.refl

+

+
  ≲-respˡ-≈ : _≲_ Respectsˡ _≈_

+
  ≲-respˡ-≈ x≈y x∼z = trans (reflexive (Eq.sym x≈y)) x∼z

+

+
  ≲-respʳ-≈ : _≲_ Respectsʳ _≈_

+
  ≲-respʳ-≈ x≈y z∼x = trans z∼x (reflexive x≈y)

+

+
  ≲-resp-≈ : _≲_ Respects₂ _≈_

+
  ≲-resp-≈ = ≲-respʳ-≈ , ≲-respˡ-≈

+

+
  ∼-respˡ-≈ = ≲-respˡ-≈

+
  {-# WARNING_ON_USAGE ∼-respˡ-≈

+
  "Warning: ∼-respˡ-≈ was deprecated in v2.0.

+
  Please use ≲-respˡ-≈ instead. "

+
  #-}

+
  ∼-respʳ-≈ = ≲-respʳ-≈

+
  {-# WARNING_ON_USAGE ∼-respʳ-≈

+
  "Warning: ∼-respʳ-≈ was deprecated in v2.0.

+
  Please use ≲-respʳ-≈ instead. "

+
  #-}

+
  ∼-resp-≈ = ≲-resp-≈

+
  {-# WARNING_ON_USAGE ∼-resp-≈

+
  "Warning: ∼-resp-≈ was deprecated in v2.0.

+
  Please use ≲-resp-≈ instead. "

+
  #-}

+

+

+
record IsTotalPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isPreorder : IsPreorder _≲_

+
    total      : Total _≲_

+

+
  open IsPreorder isPreorder public

+

+

+
------------------------------------------------------------------------

+
-- Partial orders

+
------------------------------------------------------------------------

+

+
record IsPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isPreorder : IsPreorder _≤_

+
    antisym    : Antisymmetric _≈_ _≤_

+

+
  open IsPreorder isPreorder public

+
    renaming

+
    ( ∼-respˡ-≈ to ≤-respˡ-≈

+
    ; ∼-respʳ-≈ to ≤-respʳ-≈

+
    ; ∼-resp-≈  to ≤-resp-≈

+
    )

+

+

+
record IsDecPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  infix 4 _≟_ _≤?_

+
  field

+
    isPartialOrder : IsPartialOrder _≤_

+
    _≟_            : Decidable _≈_

+
    _≤?_           : Decidable _≤_

+

+
  open IsPartialOrder isPartialOrder public

+
    hiding (module Eq)

+

+
  module Eq where

+

+
    isDecEquivalence : IsDecEquivalence

+
    isDecEquivalence = record

+
      { isEquivalence = isEquivalence

+
      ; _≟_           = _≟_

+
      }

+

+
    open IsDecEquivalence isDecEquivalence public

+

+

+
record IsStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isEquivalence : IsEquivalence

+
    irrefl        : Irreflexive _≈_ _<_

+
    trans         : Transitive _<_

+
    <-resp-≈      : _<_ Respects₂ _≈_

+

+
  module Eq = IsEquivalence isEquivalence

+

+
  asym : Asymmetric _<_

+
  asym {x} {y} = trans∧irr⇒asym Eq.refl trans irrefl {x = x} {y}

+

+
  <-respʳ-≈ : _<_ Respectsʳ _≈_

+
  <-respʳ-≈ = proj₁ <-resp-≈

+

+
  <-respˡ-≈ : _<_ Respectsˡ _≈_

+
  <-respˡ-≈ = proj₂ <-resp-≈

+

+

+
record IsDecStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  infix 4 _≟_ _<?_

+
  field

+
    isStrictPartialOrder : IsStrictPartialOrder _<_

+
    _≟_                  : Decidable _≈_

+
    _<?_                 : Decidable _<_

+

+
  private

+
    module SPO = IsStrictPartialOrder isStrictPartialOrder

+

+
  open SPO public hiding (module Eq)

+

+
  module Eq where

+

+
    isDecEquivalence : IsDecEquivalence

+
    isDecEquivalence = record

+
      { isEquivalence = SPO.isEquivalence

+
      ; _≟_           = _≟_

+
      }

+

+
    open IsDecEquivalence isDecEquivalence public

+

+

+
------------------------------------------------------------------------

+
-- Total orders

+
------------------------------------------------------------------------

+

+
record IsTotalOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isPartialOrder : IsPartialOrder _≤_

+
    total          : Total _≤_

+

+
  open IsPartialOrder isPartialOrder public

+

+
  isTotalPreorder : IsTotalPreorder _≤_

+
  isTotalPreorder = record

+
    { isPreorder = isPreorder

+
    ; total      = total

+
    }

+

+

+
record IsDecTotalOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  infix 4 _≟_ _≤?_

+
  field

+
    isTotalOrder : IsTotalOrder _≤_

+
    _≟_          : Decidable _≈_

+
    _≤?_         : Decidable _≤_

+

+
  open IsTotalOrder isTotalOrder public

+
    hiding (module Eq)

+

+
  isDecPartialOrder : IsDecPartialOrder _≤_

+
  isDecPartialOrder = record

+
    { isPartialOrder = isPartialOrder

+
    ; _≟_            = _≟_

+
    ; _≤?_           = _≤?_

+
    }

+

+
  module Eq where

+

+
    isDecEquivalence : IsDecEquivalence

+
    isDecEquivalence = record

+
      { isEquivalence = isEquivalence

+
      ; _≟_           = _≟_

+
      }

+

+
    open IsDecEquivalence isDecEquivalence public

+

+

+
-- Note that these orders are decidable. The current implementation

+
-- of `Trichotomous` subsumes irreflexivity and asymmetry. See

+
-- `Relation.Binary.Structures.Biased` for ways of constructing this

+
-- record without having to prove `isStrictPartialOrder`.

+

+
record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isStrictPartialOrder : IsStrictPartialOrder _<_

+
    compare              : Trichotomous _≈_ _<_

+

+
  open IsStrictPartialOrder isStrictPartialOrder public

+
    hiding (module Eq)

+

+
  -- `Trichotomous` necessarily separates out the equality case so

+
  --  it implies decidability.

+
  infix 4 _≟_ _<?_

+

+
  _≟_ : Decidable _≈_

+
  _≟_ = tri⇒dec≈ compare

+

+
  _<?_ : Decidable _<_

+
  _<?_ = tri⇒dec< compare

+

+
  isDecStrictPartialOrder : IsDecStrictPartialOrder _<_

+
  isDecStrictPartialOrder = record

+
    { isStrictPartialOrder = isStrictPartialOrder

+
    ; _≟_                  = _≟_

+
    ; _<?_                 = _<?_

+
    }

+

+
  -- Redefine the `Eq` module to include decidability proofs

+
  module Eq where

+

+
    isDecEquivalence : IsDecEquivalence

+
    isDecEquivalence = record

+
      { isEquivalence = isEquivalence

+
      ; _≟_           = _≟_

+
      }

+

+
    open IsDecEquivalence isDecEquivalence public

+

+
  isDecEquivalence : IsDecEquivalence

+
  isDecEquivalence = record

+
    { isEquivalence = isEquivalence

+
    ; _≟_           = _≟_

+
    }

+
  {-# WARNING_ON_USAGE isDecEquivalence

+
  "Warning: isDecEquivalence was deprecated in v2.0.

+
  Please use Eq.isDecEquivalence instead. "

+
  #-}

+

+
record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    isStrictTotalOrder : IsStrictTotalOrder _<_

+
    dense              : Dense _<_

+

+
  open IsStrictTotalOrder isStrictTotalOrder public

+

+

+
------------------------------------------------------------------------

+
-- Apartness relations

+
------------------------------------------------------------------------

+

+
record IsApartnessRelation (_#_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

+
  field

+
    irrefl  : Irreflexive _≈_ _#_

+
    sym     : Symmetric _#_

+
    cotrans : Cotransitive _#_

+

+
  _¬#_ : A → A → Set _

+
  x ¬# y = ¬ (x # y)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Binary.html b/site/static/agda_html/Relation.Binary.html
new file mode 100644
index 000000000..f355315c1
--- /dev/null
+++ b/site/static/agda_html/Relation.Binary.html
@@ -0,0 +1,95 @@
+
+Relation.Binary
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of homogeneous binary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Binary where

+

+
------------------------------------------------------------------------

+
-- Re-export various components of the binary relation hierarchy

+

+
open import Relation.Binary.Core public

+
open import Relation.Binary.Definitions public

+
open import Relation.Binary.Structures public

+
open import Relation.Binary.Structures.Biased public

+
open import Relation.Binary.Bundles public

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Decidable.Core.html b/site/static/agda_html/Relation.Nullary.Decidable.Core.html
new file mode 100644
index 000000000..8927e74bd
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Decidable.Core.html
@@ -0,0 +1,322 @@
+
+Relation.Nullary.Decidable.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Operations on and properties of decidable relations

+
--

+
-- This file contains some core definitions which are re-exported by

+
-- Relation.Nullary.Decidable

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Decidable.Core where

+

+
-- decToMaybe was deprecated in v2.1 #2330/#2336

+
-- this can go through `Data.Maybe.Base` once that deprecation is fully done.

+
open import Agda.Builtin.Maybe using (Maybe; just; nothing)

+

+
open import Agda.Builtin.Equality using (_≡_)

+
open import Level using (Level)

+
open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)

+
open import Data.Unit.Polymorphic.Base using (⊤)

+
open import Data.Product.Base using (_×_)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Function.Base using (_∘_; const; _$_; flip)

+
open import Relation.Nullary.Recomputable as Recomputable hiding (recompute-constant)

+
open import Relation.Nullary.Reflects as Reflects hiding (recompute; recompute-constant)

+
open import Relation.Nullary.Negation.Core

+
  using (¬_; Stable; negated-stable; contradiction; DoubleNegation)

+

+

+
private

+
  variable

+
    a b : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- Definition.

+

+
-- Decidability proofs have two parts: the `does` term which contains

+
-- the boolean result and the `proof` term which contains a proof that

+
-- reflects the boolean result. This definition allows the boolean

+
-- part of the decision procedure to compute independently from the

+
-- proof. This leads to better computational behaviour when we only care

+
-- about the result and not the proof. See README.Design.Decidability

+
-- for further details.

+

+
infix 2 _because_

+

+
record Dec (A : Set a) : Set a where

+
  constructor _because_

+
  field

+
    does  : Bool

+
    proof : Reflects A does

+

+
open Dec public

+

+
pattern yes a =  true because ofʸ  a

+
pattern no ¬a = false because ofⁿ ¬a

+

+
------------------------------------------------------------------------

+
-- Flattening

+

+
module _ {A : Set a} where

+

+
  From-yes : Dec A → Set a

+
  From-yes (true  because _) = A

+
  From-yes (false because _) = ⊤

+

+
  From-no : Dec A → Set a

+
  From-no (false because _) = ¬ A

+
  From-no (true  because _) = ⊤

+

+
------------------------------------------------------------------------

+
-- Recompute

+

+
-- Given an irrelevant proof of a decidable type, a proof can

+
-- be recomputed and subsequently used in relevant contexts.

+

+
recompute : Dec A → Recomputable A

+
recompute = Reflects.recompute ∘ proof

+

+
recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q

+
recompute-constant = Recomputable.recompute-constant ∘ recompute

+

+
------------------------------------------------------------------------

+
-- Interaction with negation, sum, product etc.

+

+
infixr 1 _⊎-dec_

+
infixr 2 _×-dec_ _→-dec_

+

+
T? : ∀ x → Dec (T x)

+
T? x = x because T-reflects x

+

+
¬? : Dec A → Dec (¬ A)

+
does  (¬? a?) = not (does a?)

+
proof (¬? a?) = ¬-reflects (proof a?)

+

+
_×-dec_ : Dec A → Dec B → Dec (A × B)

+
does  (a? ×-dec b?) = does a? ∧ does b?

+
proof (a? ×-dec b?) = proof a? ×-reflects proof b?

+

+
_⊎-dec_ : Dec A → Dec B → Dec (A ⊎ B)

+
does  (a? ⊎-dec b?) = does a? ∨ does b?

+
proof (a? ⊎-dec b?) = proof a? ⊎-reflects proof b?

+

+
_→-dec_ : Dec A → Dec B → Dec (A → B)

+
does  (a? →-dec b?) = not (does a?) ∨ does b?

+
proof (a? →-dec b?) = proof a? →-reflects proof b?

+

+
------------------------------------------------------------------------

+
-- Relationship with Maybe

+

+
dec⇒maybe : Dec A → Maybe A

+
dec⇒maybe ( true because [a]) = just (invert [a])

+
dec⇒maybe (false because  _ ) = nothing

+

+
------------------------------------------------------------------------

+
-- Relationship with Sum

+

+
toSum : Dec A → A ⊎ ¬ A

+
toSum ( true because  [p]) = inj₁ (invert  [p])

+
toSum (false because [¬p]) = inj₂ (invert [¬p])

+

+
fromSum : A ⊎ ¬ A → Dec A

+
fromSum (inj₁ p)  = yes p

+
fromSum (inj₂ ¬p) = no ¬p

+

+
------------------------------------------------------------------------

+
-- Relationship with booleans

+

+
-- `isYes` is a stricter version of `does`. The lack of computation

+
-- means that we can recover the proposition `P` from `isYes a?` by

+
-- unification. This is useful when we are using the decision procedure

+
-- for proof automation.

+

+
isYes : Dec A → Bool

+
isYes (true  because _) = true

+
isYes (false because _) = false

+

+
isNo : Dec A → Bool

+
isNo = not ∘ isYes

+

+
True : Dec A → Set

+
True = T ∘ isYes

+

+
False : Dec A → Set

+
False = T ∘ isNo

+

+
-- The traditional name for isYes is ⌊_⌋, indicating the stripping of evidence.

+
⌊_⌋ = isYes

+

+
------------------------------------------------------------------------

+
-- Witnesses

+

+
-- Gives a witness to the "truth".

+
toWitness : {a? : Dec A} → True a? → A

+
toWitness {a? = true  because [a]} _  = invert [a]

+
toWitness {a? = false because  _ } ()

+

+
-- Establishes a "truth", given a witness.

+
fromWitness : {a? : Dec A} → A → True a?

+
fromWitness {a? = true  because   _ } = const _

+
fromWitness {a? = false because [¬a]} = invert [¬a]

+

+
-- Variants for False.

+
toWitnessFalse : {a? : Dec A} → False a? → ¬ A

+
toWitnessFalse {a? = true  because   _ } ()

+
toWitnessFalse {a? = false because [¬a]} _  = invert [¬a]

+

+
fromWitnessFalse : {a? : Dec A} → ¬ A → False a?

+
fromWitnessFalse {a? = true  because [a]} = flip _$_ (invert [a])

+
fromWitnessFalse {a? = false because  _ } = const _

+

+
-- If a decision procedure returns "yes", then we can extract the

+
-- proof using from-yes.

+
from-yes : (a? : Dec A) → From-yes a?

+
from-yes (true  because [a]) = invert [a]

+
from-yes (false because _ ) = _

+

+
-- If a decision procedure returns "no", then we can extract the proof

+
-- using from-no.

+
from-no : (a? : Dec A) → From-no a?

+
from-no (false because [¬a]) = invert [¬a]

+
from-no (true  because   _ ) = _

+

+
------------------------------------------------------------------------

+
-- Maps

+

+
map′ : (A → B) → (B → A) → Dec A → Dec B

+
does  (map′ A→B B→A a?)                   = does a?

+
proof (map′ A→B B→A (true  because  [a])) = of (A→B (invert [a]))

+
proof (map′ A→B B→A (false because [¬a])) = of (invert [¬a] ∘ B→A)

+

+
------------------------------------------------------------------------

+
-- Relationship with double-negation

+

+
-- Decidable predicates are stable.

+

+
decidable-stable : Dec A → Stable A

+
decidable-stable (true  because  [a]) ¬¬a = invert [a]

+
decidable-stable (false because [¬a]) ¬¬a = contradiction (invert [¬a]) ¬¬a

+

+
¬-drop-Dec : Dec (¬ ¬ A) → Dec (¬ A)

+
¬-drop-Dec ¬¬a? = map′ negated-stable contradiction (¬? ¬¬a?)

+

+
-- A double-negation-translated variant of excluded middle (or: every

+
-- nullary relation is decidable in the double-negation monad).

+

+
¬¬-excluded-middle : DoubleNegation (Dec A)

+
¬¬-excluded-middle ¬?a = ¬?a (no (λ a → ¬?a (yes a)))

+

+

+
------------------------------------------------------------------------

+
-- DEPRECATED NAMES

+
------------------------------------------------------------------------

+
-- Please use the new names as continuing support for the old names is

+
-- not guaranteed.

+

+
-- Version 2.0

+

+
excluded-middle = ¬¬-excluded-middle

+
{-# WARNING_ON_USAGE excluded-middle

+
"Warning: excluded-middle was deprecated in v2.0.

+
Please use ¬¬-excluded-middle instead."

+
#-}

+

+
-- Version 2.1

+

+
decToMaybe = dec⇒maybe

+
{-# WARNING_ON_USAGE decToMaybe

+
"Warning: decToMaybe was deprecated in v2.1.

+
Please use Relation.Nullary.Decidable.Core.dec⇒maybe instead."

+
#-}

+

+
fromDec = toSum

+
{-# WARNING_ON_USAGE fromDec

+
"Warning: fromDec was deprecated in v2.1.

+
Please use Relation.Nullary.Decidable.Core.toSum instead."

+
#-}

+

+
toDec = fromSum

+
{-# WARNING_ON_USAGE toDec

+
"Warning: toDec was deprecated in v2.1.

+
Please use Relation.Nullary.Decidable.Core.fromSum instead."

+
#-}

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Decidable.html b/site/static/agda_html/Relation.Nullary.Decidable.html
new file mode 100644
index 000000000..f15bffd0e
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Decidable.html
@@ -0,0 +1,159 @@
+
+Relation.Nullary.Decidable
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Operations on and properties of decidable relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Decidable where

+

+
open import Level using (Level)

+
open import Data.Bool.Base using (true; false)

+
open import Data.Product.Base using (∃; _,_)

+
open import Function.Bundles using

+
  (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔ₛ′)

+
open import Relation.Binary.Bundles using (Setoid; module Setoid)

+
open import Relation.Binary.Definitions using (Decidable)

+
open import Relation.Nullary using (Irrelevant)

+
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

+
open import Relation.Nullary.Reflects using (invert)

+
open import Relation.Binary.PropositionalEquality.Core

+
  using (_≡_; refl; sym; trans; cong′)

+

+
private

+
  variable

+
    a b ℓ₁ ℓ₂ : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- Re-exporting the core definitions

+

+
open import Relation.Nullary.Decidable.Core public

+

+
------------------------------------------------------------------------

+
-- Maps

+

+
map : A ⇔ B → Dec A → Dec B

+
map A⇔B = map′ to from

+
  where open Equivalence A⇔B

+

+
-- If there is an injection from one setoid to another, and the

+
-- latter's equivalence relation is decidable, then the former's

+
-- equivalence relation is also decidable.

+
via-injection : {S : Setoid a ℓ₁} {T : Setoid b ℓ₂}

+
                (inj : Injection S T) (open Injection inj) →

+
                Decidable Eq₂._≈_ → Decidable Eq₁._≈_

+
via-injection inj _≟_ x y = map′ injective cong (to x ≟ to y)

+
  where open Injection inj

+

+
------------------------------------------------------------------------

+
-- A lemma relating True and Dec

+

+
True-↔ : (a? : Dec A) → Irrelevant A → True a? ↔ A

+
True-↔ (true  because [a]) irr = let a = invert [a] in mk↔ₛ′ (λ _ → a) _ (irr a) cong′

+
True-↔ (false because [¬a]) _  = let ¬a = invert [¬a] in mk↔ₛ′ (λ ()) ¬a (λ a → contradiction a ¬a) λ ()

+

+
------------------------------------------------------------------------

+
-- Result of decidability

+

+
isYes≗does : (a? : Dec A) → isYes a? ≡ does a?

+
isYes≗does (true  because _) = refl

+
isYes≗does (false because _) = refl

+

+
dec-true : (a? : Dec A) → A → does a? ≡ true

+
dec-true (true  because   _ ) a = refl

+
dec-true (false because [¬a]) a = contradiction a (invert [¬a])

+

+
dec-false : (a? : Dec A) → ¬ A → does a? ≡ false

+
dec-false (false because  _ ) ¬a = refl

+
dec-false (true  because [a]) ¬a = contradiction (invert [a]) ¬a

+

+
dec-yes : (a? : Dec A) → A → ∃ λ a → a? ≡ yes a

+
dec-yes a? a with yes a′ ← a? | refl ← dec-true a? a = a′ , refl

+

+
dec-no : (a? : Dec A) (¬a : ¬ A) → a? ≡ no ¬a

+
dec-no a? ¬a with no _ ← a? | refl ← dec-false a? ¬a = refl

+

+
dec-yes-irr : (a? : Dec A) → Irrelevant A → (a : A) → a? ≡ yes a

+
dec-yes-irr a? irr a with a′ , eq ← dec-yes a? a rewrite irr a a′ = eq

+

+
⌊⌋-map′ : ∀ t f (a? : Dec A) → ⌊ map′ {B = B} t f a? ⌋ ≡ ⌊ a? ⌋

+
⌊⌋-map′ t f a? = trans (isYes≗does (map′ t f a?)) (sym (isYes≗does a?))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Indexed.html b/site/static/agda_html/Relation.Nullary.Indexed.html
new file mode 100644
index 000000000..be4305877
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Indexed.html
@@ -0,0 +1,96 @@
+
+Relation.Nullary.Indexed
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Negation indexed by a Level

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Indexed where

+

+
open import Data.Empty hiding (⊥-elim)

+
open import Level

+

+
------------------------------------------------------------------------

+
-- Negation.

+

+
-- level polymorphic version of ¬

+
¬ : ∀ {ℓ} (b : Level) → Set ℓ → Set (ℓ ⊔ b)

+
¬ b P = P → Lift b ⊥

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Negation.Core.html b/site/static/agda_html/Relation.Nullary.Negation.Core.html
new file mode 100644
index 000000000..e15b47c34
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Negation.Core.html
@@ -0,0 +1,155 @@
+
+Relation.Nullary.Negation.Core
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Core properties related to negation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Negation.Core where

+

+
open import Data.Empty using (⊥; ⊥-elim-irr)

+
open import Data.Sum.Base using (_⊎_; [_,_]; inj₁; inj₂)

+
open import Function.Base using (flip; _∘_; const)

+
open import Level

+

+
private

+
  variable

+
    a p q w : Level

+
    A B C : Set a

+
    Whatever : Set w

+

+
------------------------------------------------------------------------

+
-- Negation.

+

+
infix 3 ¬_

+
¬_ : Set a → Set a

+
¬ A = A → ⊥

+

+
------------------------------------------------------------------------

+
-- Stability.

+

+
-- Double-negation

+
DoubleNegation : Set a → Set a

+
DoubleNegation A = ¬ ¬ A

+

+
-- Stability under double-negation.

+
Stable : Set a → Set a

+
Stable A = ¬ ¬ A → A

+

+
------------------------------------------------------------------------

+
-- Relationship to sum

+

+
infixr 1 _¬-⊎_

+
_¬-⊎_ : ¬ A → ¬ B → ¬ (A ⊎ B)

+
_¬-⊎_ = [_,_]

+

+
------------------------------------------------------------------------

+
-- Uses of negation

+

+
contradiction-irr : .A → ¬ A → Whatever

+
contradiction-irr a ¬a = ⊥-elim-irr (¬a a)

+

+
contradiction : A → ¬ A → Whatever

+
contradiction a = contradiction-irr a

+

+
contradiction₂ : A ⊎ B → ¬ A → ¬ B → Whatever

+
contradiction₂ (inj₁ a) ¬a ¬b = contradiction a ¬a

+
contradiction₂ (inj₂ b) ¬a ¬b = contradiction b ¬b

+

+
contraposition : (A → B) → ¬ B → ¬ A

+
contraposition f ¬b a = contradiction (f a) ¬b

+

+
-- Everything is stable in the double-negation monad.

+
stable : ¬ ¬ Stable A

+
stable ¬[¬¬a→a] = ¬[¬¬a→a] (contradiction (¬[¬¬a→a] ∘ const))

+

+
-- Negated predicates are stable.

+
negated-stable : Stable (¬ A)

+
negated-stable ¬¬¬a a = ¬¬¬a (contradiction a)

+

+
¬¬-map : (A → B) → ¬ ¬ A → ¬ ¬ B

+
¬¬-map f = contraposition (contraposition f)

+

+
-- Note also the following use of flip:

+
private

+
  note : (A → ¬ B) → B → ¬ A

+
  note = flip

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Negation.html b/site/static/agda_html/Relation.Nullary.Negation.html
new file mode 100644
index 000000000..c5c7097a2
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Negation.html
@@ -0,0 +1,185 @@
+
+Relation.Nullary.Negation
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties related to negation

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Negation where

+

+
open import Data.Bool.Base using (Bool; false; true; if_then_else_)

+
open import Data.Product.Base as Product using (_,_; Σ; Σ-syntax; ∃; curry; uncurry)

+
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])

+
open import Effect.Monad using (RawMonad; mkRawMonad)

+
open import Function.Base using (flip; _∘_; const; _∘′_)

+
open import Level using (Level)

+
open import Relation.Nullary.Decidable.Core using (Dec; yes; no; ¬¬-excluded-middle)

+
open import Relation.Unary using (Universal; Pred)

+

+
private

+
  variable

+
    a b c d p w : Level

+
    A B C D : Set a

+
    P : Pred A p

+
    Whatever : Set w

+

+
------------------------------------------------------------------------

+
-- Re-export public definitions

+

+
open import Relation.Nullary.Negation.Core public

+

+
------------------------------------------------------------------------

+
-- Quantifier juggling

+

+
∃⟶¬∀¬ : ∃ P → ¬ (∀ x → ¬ P x)

+
∃⟶¬∀¬ = flip uncurry

+

+
∀⟶¬∃¬ : (∀ x → P x) → ¬ ∃ λ x → ¬ P x

+
∀⟶¬∃¬ ∀xPx (x , ¬Px) = ¬Px (∀xPx x)

+

+
¬∃⟶∀¬ : ¬ ∃ (λ x → P x) → ∀ x → ¬ P x

+
¬∃⟶∀¬ = curry

+

+
∀¬⟶¬∃ : (∀ x → ¬ P x) → ¬ ∃ (λ x → P x)

+
∀¬⟶¬∃ = uncurry

+

+
∃¬⟶¬∀ : ∃ (λ x → ¬ P x) → ¬ (∀ x → P x)

+
∃¬⟶¬∀ = flip ∀⟶¬∃¬

+

+
------------------------------------------------------------------------

+
-- Double Negation

+

+
-- Double-negation is a monad (if we assume that all elements of ¬ ¬ P

+
-- are equal).

+

+
¬¬-Monad : RawMonad {a} DoubleNegation

+
¬¬-Monad = mkRawMonad

+
  DoubleNegation

+
  contradiction

+
  (λ x f → negated-stable (¬¬-map f x))

+

+
¬¬-push : DoubleNegation Π[ P ] → Π[ DoubleNegation ∘ P ]

+
¬¬-push ¬¬∀P a ¬Pa = ¬¬∀P (λ ∀P → ¬Pa (∀P a))

+

+
-- If Whatever is instantiated with ¬ ¬ something, then this function

+
-- is call with current continuation in the double-negation monad, or,

+
-- if you will, a double-negation translation of Peirce's law.

+
--

+
-- In order to prove ¬ ¬ P one can assume ¬ P and prove ⊥. However,

+
-- sometimes it is nice to avoid leaving the double-negation monad; in

+
-- that case this function can be used (with Whatever instantiated to

+
-- ⊥).

+

+
call/cc : ((A → Whatever) → DoubleNegation A) → DoubleNegation A

+
call/cc hyp ¬a = hyp (flip contradiction ¬a) ¬a

+

+
-- The "independence of premise" rule, in the double-negation monad.

+
-- It is assumed that the index set (A) is inhabited.

+

+
independence-of-premise : A → (B → Σ A P) → DoubleNegation (Σ[ x ∈ A ] (B → P x))

+
independence-of-premise {A = A} {B = B} {P = P} q f = ¬¬-map helper ¬¬-excluded-middle

+
  where

+
  helper : Dec B → Σ[ x ∈ A ] (B → P x)

+
  helper (yes p) = Product.map₂ const (f p)

+
  helper (no ¬p) = (q , flip contradiction ¬p)

+

+
-- The independence of premise rule for binary sums.

+

+
independence-of-premise-⊎ : (A → B ⊎ C) → DoubleNegation ((A → B) ⊎ (A → C))

+
independence-of-premise-⊎ {A = A} {B = B} {C = C} f = ¬¬-map helper ¬¬-excluded-middle

+
  where

+
  helper : Dec A → (A → B) ⊎ (A → C)

+
  helper (yes p) = Sum.map const const (f p)

+
  helper (no ¬p) = inj₁ (flip contradiction ¬p)

+

+
private

+

+
  -- Note that independence-of-premise-⊎ is a consequence of

+
  -- independence-of-premise (for simplicity it is assumed that Q and

+
  -- R have the same type here):

+

+
  corollary : {B C : Set b} → (A → B ⊎ C) → DoubleNegation ((A → B) ⊎ (A → C))

+
  corollary {A = A} {B = B} {C = C} f =

+
    ¬¬-map helper (independence-of-premise true ([ _,_ true , _,_ false ] ∘′ f))

+
    where

+
    helper : ∃ (λ b → A → if b then B else C) → (A → B) ⊎ (A → C)

+
    helper (true  , f) = inj₁ f

+
    helper (false , f) = inj₂ f

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Recomputable.html b/site/static/agda_html/Relation.Nullary.Recomputable.html
new file mode 100644
index 000000000..d4c0af38e
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Recomputable.html
@@ -0,0 +1,140 @@
+
+Relation.Nullary.Recomputable
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Recomputable types and their algebra as Harrop formulas

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Recomputable where

+

+
open import Agda.Builtin.Equality using (_≡_; refl)

+
open import Data.Empty using (⊥)

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)

+
open import Level using (Level)

+
open import Relation.Nullary.Negation.Core using (¬_)

+

+
private

+
  variable

+
    a b : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- Definition

+
--

+
-- The idea of being 'recomputable' is that, given an *irrelevant* proof

+
-- of a proposition `A` (signalled by being a value of type `.A`, all of

+
-- whose inhabitants are identified up to definitional equality, and hence

+
-- do *not* admit pattern-matching), one may 'promote' such a value to a

+
-- 'genuine' value of `A`, available for subsequent eg. pattern-matching.

+

+
Recomputable : (A : Set a) → Set a

+
Recomputable A = .A → A

+

+
------------------------------------------------------------------------

+
-- Fundamental property: 'promotion' is a constant function

+

+
recompute-constant : (r : Recomputable A) (p q : A) → r p ≡ r q

+
recompute-constant r p q = refl

+

+
------------------------------------------------------------------------

+
-- Constructions

+

+
⊥-recompute : Recomputable ⊥

+
⊥-recompute ()

+

+
_×-recompute_ : Recomputable A → Recomputable B → Recomputable (A × B)

+
(rA ×-recompute rB) p = rA (p .proj₁) , rB (p .proj₂)

+

+
_→-recompute_ : (A : Set a) → Recomputable B → Recomputable (A → B)

+
(A →-recompute rB) f a = rB (f a)

+

+
Π-recompute : (B : A → Set b) → (∀ x → Recomputable (B x)) → Recomputable (∀ x → B x)

+
Π-recompute B rB f a = rB a (f a)

+

+
∀-recompute : (B : A → Set b) → (∀ {x} → Recomputable (B x)) → Recomputable (∀ {x} → B x)

+
∀-recompute B rB f = rB f

+

+
-- corollary: negated propositions are Recomputable

+

+
¬-recompute : Recomputable (¬ A)

+
¬-recompute {A = A} = A →-recompute ⊥-recompute

+

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.Reflects.html b/site/static/agda_html/Relation.Nullary.Reflects.html
new file mode 100644
index 000000000..5c4692e27
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.Reflects.html
@@ -0,0 +1,194 @@
+
+Relation.Nullary.Reflects
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of the `Reflects` construct

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary.Reflects where

+

+
open import Agda.Builtin.Equality

+

+
open import Data.Bool.Base

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)

+
open import Level using (Level)

+
open import Function.Base using (_$_; _∘_; const; id)

+
open import Relation.Nullary.Negation.Core

+
  using (¬_; contradiction-irr; contradiction; _¬-⊎_)

+
open import Relation.Nullary.Recomputable as Recomputable using (Recomputable)

+

+
private

+
  variable

+
    a : Level

+
    A B : Set a

+

+
------------------------------------------------------------------------

+
-- `Reflects` idiom.

+

+
-- The truth value of A is reflected by a boolean value.

+
-- `Reflects A b` is equivalent to `if b then A else ¬ A`.

+

+
data Reflects (A : Set a) : Bool → Set a where

+
  ofʸ : ( a :   A) → Reflects A true

+
  ofⁿ : (¬a : ¬ A) → Reflects A false

+

+
------------------------------------------------------------------------

+
-- Constructors and destructors

+

+
-- These lemmas are intended to be used mostly when `b` is a value, so

+
-- that the `if` expressions have already been evaluated away.

+
-- In this case, `of` works like the relevant constructor (`ofⁿ` or

+
-- `ofʸ`), and `invert` strips off the constructor to just give either

+
-- the proof of `A` or the proof of `¬ A`.

+

+
of : ∀ {b} → if b then A else ¬ A → Reflects A b

+
of {b = false} ¬a = ofⁿ ¬a

+
of {b = true }  a = ofʸ a

+

+
invert : ∀ {b} → Reflects A b → if b then A else ¬ A

+
invert (ofʸ  a) = a

+
invert (ofⁿ ¬a) = ¬a

+

+
------------------------------------------------------------------------

+
-- recompute

+

+
-- Given an irrelevant proof of a reflected type, a proof can

+
-- be recomputed and subsequently used in relevant contexts.

+

+
recompute : ∀ {b} → Reflects A b → Recomputable A

+
recompute (ofʸ  a) _ = a

+
recompute (ofⁿ ¬a) a = contradiction-irr a ¬a

+

+
recompute-constant : ∀ {b} (r : Reflects A b) (p q : A) →

+
                     recompute r p ≡ recompute r q

+
recompute-constant = Recomputable.recompute-constant ∘ recompute

+

+
------------------------------------------------------------------------

+
-- Interaction with negation, product, sums etc.

+

+
infixr 1 _⊎-reflects_

+
infixr 2 _×-reflects_ _→-reflects_

+

+
T-reflects : ∀ b → Reflects (T b) b

+
T-reflects true  = of _

+
T-reflects false = of id

+

+
-- If we can decide A, then we can decide its negation.

+
¬-reflects : ∀ {b} → Reflects A b → Reflects (¬ A) (not b)

+
¬-reflects (ofʸ  a) = of (_$ a)

+
¬-reflects (ofⁿ ¬a) = of ¬a

+

+
-- If we can decide A and Q then we can decide their product

+
_×-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →

+
               Reflects (A × B) (a ∧ b)

+
ofʸ  a ×-reflects ofʸ  b = of (a , b)

+
ofʸ  a ×-reflects ofⁿ ¬b = of (¬b ∘ proj₂)

+
ofⁿ ¬a ×-reflects _      = of (¬a ∘ proj₁)

+

+
_⊎-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →

+
               Reflects (A ⊎ B) (a ∨ b)

+
ofʸ  a ⊎-reflects      _ = of (inj₁ a)

+
ofⁿ ¬a ⊎-reflects ofʸ  b = of (inj₂ b)

+
ofⁿ ¬a ⊎-reflects ofⁿ ¬b = of (¬a ¬-⊎ ¬b)

+

+
_→-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →

+
                Reflects (A → B) (not a ∨ b)

+
ofʸ  a →-reflects ofʸ  b = of (const b)

+
ofʸ  a →-reflects ofⁿ ¬b = of (¬b ∘ (_$ a))

+
ofⁿ ¬a →-reflects _      = of (λ a → contradiction a ¬a)

+

+
------------------------------------------------------------------------

+
-- Other lemmas

+

+
fromEquivalence : ∀ {b} → (T b → A) → (A → T b) → Reflects A b

+
fromEquivalence {b = true}  sound complete = of (sound _)

+
fromEquivalence {b = false} sound complete = of complete

+

+
-- `Reflects` is deterministic.

+
det : ∀ {b b′} → Reflects A b → Reflects A b′ → b ≡ b′

+
det (ofʸ  a) (ofʸ  _) = refl

+
det (ofʸ  a) (ofⁿ ¬a) = contradiction a ¬a

+
det (ofⁿ ¬a) (ofʸ  a) = contradiction a ¬a

+
det (ofⁿ ¬a) (ofⁿ  _) = refl

+

+
T-reflects-elim : ∀ {a b} → Reflects (T a) b → b ≡ a

+
T-reflects-elim {a} r = det r (T-reflects a)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Nullary.html b/site/static/agda_html/Relation.Nullary.html
new file mode 100644
index 000000000..907757dde
--- /dev/null
+++ b/site/static/agda_html/Relation.Nullary.html
@@ -0,0 +1,119 @@
+
+Relation.Nullary
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Operations on nullary relations (like negation and decidability)

+
------------------------------------------------------------------------

+

+
-- Some operations on/properties of nullary relations, i.e. sets.

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Nullary where

+

+
open import Agda.Builtin.Equality using (_≡_)

+
open import Agda.Builtin.Maybe using (Maybe)

+
open import Level using (Level)

+

+
private

+
  variable

+
    p : Level

+
    P : Set p

+

+

+
------------------------------------------------------------------------

+
-- Re-exports

+

+
open import Relation.Nullary.Recomputable public using (Recomputable)

+
open import Relation.Nullary.Negation.Core public

+
open import Relation.Nullary.Reflects public hiding (recompute; recompute-constant)

+
open import Relation.Nullary.Decidable.Core public

+

+
------------------------------------------------------------------------

+
-- Irrelevant types

+

+
Irrelevant : Set p → Set p

+
Irrelevant P = ∀ (p₁ p₂ : P) → p₁ ≡ p₂

+

+
------------------------------------------------------------------------

+
-- Weak decidability

+
-- `nothing` is 'don't know'/'give up'; `just` is `yes`/`definitely`

+

+
WeaklyDecidable : Set p → Set p

+
WeaklyDecidable = Maybe

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Unary.PredicateTransformer.html b/site/static/agda_html/Relation.Unary.PredicateTransformer.html
new file mode 100644
index 000000000..73de96b0e
--- /dev/null
+++ b/site/static/agda_html/Relation.Unary.PredicateTransformer.html
@@ -0,0 +1,204 @@
+
+Relation.Unary.PredicateTransformer
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Predicate transformers

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Unary.PredicateTransformer where

+

+
open import Data.Product.Base using (∃)

+
open import Function.Base using (_∘_)

+
open import Level hiding (_⊔_)

+
open import Relation.Nullary

+
open import Relation.Unary

+
open import Relation.Binary.Core using (REL)

+

+
private

+
  variable

+
    a b c i ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Heterogeneous and homogeneous predicate transformers

+

+
PT : Set a → Set b → (ℓ₁ ℓ₂ : Level) → Set _

+
PT A B ℓ₁ ℓ₂ = Pred A ℓ₁ → Pred B ℓ₂

+

+
Pt : Set a → (ℓ : Level) → Set _

+
Pt A ℓ = PT A A ℓ ℓ

+

+
------------------------------------------------------------------------

+
-- Composition and identity

+

+
infixr 9 _⍮_

+

+
_⍮_ : PT B C ℓ₂ ℓ₃ → PT A B ℓ₁ ℓ₂ → PT A C ℓ₁ _

+
S ⍮ T = S ∘ T

+

+
skip : PT A A ℓ ℓ

+
skip P = P

+

+
------------------------------------------------------------------------

+
-- Operations on predicates extend pointwise to predicate transformers

+

+
-- The bottom and the top of the predicate transformer lattice.

+

+
abort : PT A B 0ℓ 0ℓ

+
abort = λ _ → ∅

+

+
magic : PT A B 0ℓ 0ℓ

+
magic = λ _ → U

+

+
-- Negation.

+

+
infix 8 ∼_

+

+
∼_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂

+
∼ T = ∁ ∘ T

+

+
-- Refinement.

+

+
infix 4 _⊑_ _⊒_ _⊑′_ _⊒′_

+

+
_⊑_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _

+
S ⊑ T = ∀ {X} → S X ⊆ T X

+

+
_⊑′_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _

+
S ⊑′ T = ∀ X → S X ⊆ T X

+

+
_⊒_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _

+
T ⊒ S = T ⊑ S

+

+
_⊒′_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _

+
T ⊒′ S = S ⊑′ T

+

+
-- The dual of refinement.

+

+
infix 4 _⋢_

+

+
_⋢_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → Set _

+
S ⋢ T = ∃ λ X → S X ≬ T X

+

+
-- Union.

+

+
infixl 6 _⊓_

+

+
_⊓_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂

+
S ⊓ T = λ X → S X ∪ T X

+

+
-- Intersection.

+

+
infixl 7 _⊔_

+

+
_⊔_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂

+
S ⊔ T = λ X → S X ∩ T X

+

+
-- Implication.

+

+
infixl 8 _⇛_

+

+
_⇛_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂

+
S ⇛ T = λ X → S X ⇒ T X

+

+
-- Infinitary union and intersection.

+

+
infix 9 ⨆ ⨅

+

+
⨆ : ∀ (I : Set i) → (I → PT A B ℓ₁ ℓ₂) → PT A B ℓ₁ _

+
⨆ I T = λ X → ⋃[ i ∶ I ] T i X

+

+
syntax ⨆ I (λ i → T) = ⨆[ i ∶ I ] T

+

+
⨅ : ∀ (I : Set i) → (I → PT A B ℓ₁ ℓ₂) → PT A B ℓ₁ _

+
⨅ I T = λ X → ⋂[ i ∶ I ] T i X

+

+
syntax ⨅ I (λ i → T) = ⨅[ i ∶ I ] T

+

+
-- Angelic and demonic update.

+

+
⟨_⟩ : REL A B ℓ → PT B A ℓ _

+
⟨ R ⟩ P = λ x → R x ≬ P

+

+
[_] : REL A B ℓ → PT B A ℓ _

+
[ R ] P = λ x → R x ⊆ P

+

\ No newline at end of file
diff --git a/site/static/agda_html/Relation.Unary.Properties.html b/site/static/agda_html/Relation.Unary.Properties.html
new file mode 100644
index 000000000..a539afeff
--- /dev/null
+++ b/site/static/agda_html/Relation.Unary.Properties.html
@@ -0,0 +1,320 @@
+
+Relation.Unary.Properties
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Properties of constructions over unary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Unary.Properties where

+

+
open import Data.Product.Base as Product using (_×_; _,_; swap; proj₁; zip′)

+
open import Data.Sum.Base using (inj₁; inj₂)

+
open import Data.Unit.Base using (tt)

+
open import Level using (Level)

+
open import Relation.Binary.Core as Binary

+
open import Relation.Binary.Definitions

+
  hiding (Decidable; Universal; Irrelevant; Empty)

+
open import Relation.Binary.PropositionalEquality.Core using (refl)

+
open import Relation.Unary

+
open import Relation.Nullary.Decidable using (yes; no; _⊎-dec_; _×-dec_; ¬?)

+
open import Function.Base using (id; _$_; _∘_)

+

+
private

+
  variable

+
    a b ℓ ℓ₁ ℓ₂ ℓ₃ : Level

+
    A : Set a

+
    B : Set b

+

+
------------------------------------------------------------------------

+
-- The empty set

+

+
∅? : Decidable {A = A} ∅

+
∅? _ = no λ()

+

+
∅-Empty : Empty {A = A} ∅

+
∅-Empty x ()

+

+
∁∅-Universal : Universal {A = A} (∁ ∅)

+
∁∅-Universal = λ x x∈∅ → x∈∅

+

+
------------------------------------------------------------------------

+
-- The universe

+

+
U? : Decidable {A = A} U

+
U? _ = yes tt

+

+
U-Universal : Universal {A = A} U

+
U-Universal = λ _ → _

+

+
∁U-Empty : Empty {A = A} (∁ U)

+
∁U-Empty = λ x x∈∁U → x∈∁U _

+

+
------------------------------------------------------------------------

+
-- Subset properties

+

+
∅-⊆ : (P : Pred A ℓ) → ∅ ⊆ P

+
∅-⊆ P ()

+

+
⊆-U : (P : Pred A ℓ) → P ⊆ U

+
⊆-U P _ = _

+

+
⊆-refl : Reflexive {A = Pred A ℓ} _⊆_

+
⊆-refl x∈P = x∈P

+

+
⊆-reflexive : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _⊆_

+
⊆-reflexive (P⊆Q , Q⊆P) = P⊆Q

+

+
⊆-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆_ _⊆_ _⊆_

+
⊆-trans P⊆Q Q⊆R x∈P = Q⊆R (P⊆Q x∈P)

+

+
⊆-antisym : Antisymmetric {A = Pred A ℓ} _≐_ _⊆_

+
⊆-antisym = _,_

+

+
⊆-min : Min {B = Pred A ℓ} _⊆_ ∅

+
⊆-min = ∅-⊆

+

+
⊆-max : Max {A = Pred A ℓ} _⊆_ U

+
⊆-max = ⊆-U

+

+
⊂⇒⊆ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _⊆_

+
⊂⇒⊆ = proj₁

+

+
⊂-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂_ _⊂_ _⊂_

+
⊂-trans (P⊆Q , Q⊈P) (Q⊆R , R⊈Q) = (λ x∈P → Q⊆R (P⊆Q x∈P)) , (λ R⊆P → R⊈Q (λ x∈R → P⊆Q (R⊆P x∈R)))

+

+
⊂-⊆-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂_ _⊆_ _⊂_

+
⊂-⊆-trans (P⊆Q , Q⊈P) Q⊆R = (λ x∈P → Q⊆R (P⊆Q x∈P)) , (λ R⊆P → Q⊈P (λ x∈Q → R⊆P (Q⊆R x∈Q)))

+

+
⊆-⊂-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆_ _⊂_ _⊂_

+
⊆-⊂-trans P⊆Q (Q⊆R , R⊈Q) = (λ x∈P → Q⊆R (P⊆Q x∈P)) , (λ R⊆P → R⊈Q (λ R⊆Q → P⊆Q (R⊆P R⊆Q)))

+

+
⊂-respʳ-≐ : _Respectsʳ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _≐_

+
⊂-respʳ-≐ (Q⊆R , _) P⊂Q = ⊂-⊆-trans P⊂Q Q⊆R

+

+
⊂-respˡ-≐ : _Respectsˡ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _≐_

+
⊂-respˡ-≐ (_ , R⊆Q) P⊂Q = ⊆-⊂-trans R⊆Q P⊂Q

+

+
⊂-resp-≐ : _Respects₂_ {A = Pred A ℓ} _⊂_ _≐_

+
⊂-resp-≐ = ⊂-respʳ-≐ , ⊂-respˡ-≐

+

+
⊂-irrefl : Irreflexive {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _⊂_

+
⊂-irrefl (_ , Q⊆P) (_ , Q⊈P) = Q⊈P Q⊆P

+

+
⊂-antisym : Antisymmetric {A = Pred A ℓ} _≐_ _⊂_

+
⊂-antisym (P⊆Q , _) (Q⊆P , _) = ⊆-antisym P⊆Q Q⊆P

+

+
⊂-asym : Asymmetric {A = Pred A ℓ} _⊂_

+
⊂-asym (_ , Q⊈P) = Q⊈P ∘ proj₁

+

+
∅-⊆′ : (P : Pred A ℓ) → ∅ ⊆′ P

+
∅-⊆′ _ _ = λ ()

+

+
⊆′-U : (P : Pred A ℓ) → P ⊆′ U

+
⊆′-U _ _ _ = _

+

+
⊆′-refl : Reflexive {A = Pred A ℓ} _⊆′_

+
⊆′-refl x x∈P = x∈P

+

+
⊆′-reflexive : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _⊆′_

+
⊆′-reflexive (P⊆Q , Q⊆P) = P⊆Q

+

+
⊆′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆′_ _⊆′_ _⊆′_

+
⊆′-trans P⊆Q Q⊆R x x∈P = Q⊆R x (P⊆Q x x∈P)

+

+
⊆′-antisym : Antisymmetric {A = Pred A ℓ} _≐′_ _⊆′_

+
⊆′-antisym = _,_

+

+
⊆′-min : Min {B = Pred A ℓ} _⊆′_ ∅

+
⊆′-min = ∅-⊆′

+

+
⊆′-max : Max {A = Pred A ℓ} _⊆′_ U

+
⊆′-max = ⊆′-U

+

+
⊂′⇒⊆′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _⊆′_

+
⊂′⇒⊆′ = proj₁

+

+
⊂′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂′_ _⊂′_ _⊂′_

+
⊂′-trans (P⊆Q , Q⊈P) (Q⊆R , R⊈Q) = ⊆′-trans P⊆Q Q⊆R , λ R⊆P → R⊈Q (⊆′-trans R⊆P P⊆Q)

+

+
⊂′-⊆′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊂′_ _⊆′_ _⊂′_

+
⊂′-⊆′-trans (P⊆Q , Q⊈P) Q⊆R = ⊆′-trans P⊆Q Q⊆R , λ R⊆P → Q⊈P (⊆′-trans Q⊆R R⊆P)

+

+
⊆′-⊂′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _⊆′_ _⊂′_ _⊂′_

+
⊆′-⊂′-trans P⊆Q (Q⊆R , R⊈Q) = ⊆′-trans P⊆Q Q⊆R , λ R⊆P → R⊈Q (⊆′-trans R⊆P P⊆Q)

+

+
⊂′-respʳ-≐′ : _Respectsʳ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _≐′_

+
⊂′-respʳ-≐′ (Q⊆R , _) P⊂Q = ⊂′-⊆′-trans P⊂Q Q⊆R

+

+
⊂′-respˡ-≐′ : _Respectsˡ_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _≐′_

+
⊂′-respˡ-≐′ (_ , R⊆Q) P⊂Q = ⊆′-⊂′-trans R⊆Q P⊂Q

+

+
⊂′-resp-≐′ : _Respects₂_ {A = Pred A ℓ₁} _⊂′_ _≐′_

+
⊂′-resp-≐′ = ⊂′-respʳ-≐′ , ⊂′-respˡ-≐′

+

+
⊂′-irrefl : Irreflexive {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _⊂′_

+
⊂′-irrefl (_ , Q⊆P) (_ , Q⊈P) = Q⊈P Q⊆P

+

+
⊂′-antisym : Antisymmetric {A = Pred A ℓ} _≐′_ _⊂′_

+
⊂′-antisym (P⊆Q , _) (Q⊆P , _) = ⊆′-antisym P⊆Q Q⊆P

+

+
⊆⇒⊆′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆_ _⊆′_

+
⊆⇒⊆′ P⊆Q _ x∈P = P⊆Q x∈P

+

+
⊆′⇒⊆ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊆′_ _⊆_

+
⊆′⇒⊆ P⊆Q x∈P = P⊆Q _ x∈P

+

+
⊂⇒⊂′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂_ _⊂′_

+
⊂⇒⊂′ = Product.map ⊆⇒⊆′ (_∘ ⊆′⇒⊆)

+

+
⊂′⇒⊂ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _⊂_

+
⊂′⇒⊂ = Product.map ⊆′⇒⊆ (_∘ ⊆⇒⊆′)

+

+
------------------------------------------------------------------------

+
-- Equality properties

+

+
≐-refl : Reflexive {A = Pred A ℓ} _≐_

+
≐-refl = id , id

+

+
≐-sym : Sym {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _≐_

+
≐-sym = swap

+

+
≐-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _≐_ _≐_ _≐_

+
≐-trans = zip′ (λ P⊆Q Q⊆R x∈P → Q⊆R (P⊆Q x∈P)) (λ Q⊆P R⊆Q x∈R → Q⊆P (R⊆Q x∈R))

+

+
≐′-refl : Reflexive {A = Pred A ℓ} _≐′_

+
≐′-refl = (λ _ → id) , (λ _ → id)

+

+
≐′-sym : Sym {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _≐′_

+
≐′-sym = swap

+

+
≐′-trans : Trans {A = Pred A ℓ₁} {B = Pred A ℓ₂} {C = Pred A ℓ₃} _≐′_ _≐′_ _≐′_

+
≐′-trans = zip′ (λ P⊆Q Q⊆R x x∈P → Q⊆R x (P⊆Q x x∈P)) λ Q⊆P R⊆Q x x∈R → Q⊆P x (R⊆Q x x∈R)

+

+
≐⇒≐′ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐_ _≐′_

+
≐⇒≐′ = Product.map ⊆⇒⊆′ ⊆⇒⊆′

+

+
≐′⇒≐ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _≐_

+
≐′⇒≐ = Product.map ⊆′⇒⊆ ⊆′⇒⊆

+

+
------------------------------------------------------------------------

+
-- Decidability properties

+

+
∁? : {P : Pred A ℓ} → Decidable P → Decidable (∁ P)

+
∁? P? x = ¬? (P? x)

+

+
infix 2 _×?_ _⊙?_

+
infix 10 _~?

+
infixr 1 _⊎?_

+
infixr 7 _∩?_

+
infixr 6 _∪?_

+

+
_∪?_ : {P : Pred A ℓ₁} {Q : Pred A ℓ₂} →

+
       Decidable P → Decidable Q → Decidable (P ∪ Q)

+
_∪?_ P? Q? x = (P? x) ⊎-dec (Q? x)

+

+
_∩?_ : {P : Pred A ℓ₁} {Q : Pred A ℓ₂} →

+
       Decidable P → Decidable Q → Decidable (P ∩ Q)

+
_∩?_ P? Q? x = (P? x) ×-dec (Q? x)

+

+
_×?_ : {P : Pred A ℓ₁} {Q : Pred B ℓ₂} →

+
       Decidable P → Decidable Q → Decidable (P ⟨×⟩ Q)

+
_×?_ P? Q? (a , b) = (P? a) ×-dec (Q? b)

+

+
_⊙?_ : {P : Pred A ℓ₁} {Q : Pred B ℓ₂} →

+
       Decidable P → Decidable Q → Decidable (P ⟨⊙⟩ Q)

+
_⊙?_ P? Q? (a , b) = (P? a) ⊎-dec (Q? b)

+

+
_⊎?_ : {P : Pred A ℓ} {Q : Pred B ℓ} →

+
       Decidable P → Decidable Q → Decidable (P ⟨⊎⟩ Q)

+
_⊎?_ P? Q? (inj₁ a) = P? a

+
_⊎?_ P? Q? (inj₂ b) = Q? b

+

+
_~? : {P : Pred (A × B) ℓ} → Decidable P → Decidable (P ~)

+
_~? P? = P? ∘ swap

+

+
------------------------------------------------------------------------

+
-- Irrelevant properties

+

+
U-irrelevant : Irrelevant {A = A} U

+
U-irrelevant a b = refl

+

+
∁-irrelevant : (P : Pred A ℓ) → Irrelevant (∁ P)

+
∁-irrelevant P a b = refl

+

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+
+Relation.Unary
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Unary relations

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Relation.Unary where

+

+
open import Data.Empty using (⊥)

+
open import Data.Unit.Base using (⊤)

+
open import Data.Product.Base using (_×_; _,_; Σ-syntax; ∃; uncurry; swap)

+
open import Data.Sum.Base using (_⊎_; [_,_])

+
open import Function.Base using (_∘_; _|>_)

+
open import Level using (Level; _⊔_; 0ℓ; suc; Lift)

+
open import Relation.Binary.PropositionalEquality.Core using (_≡_)

+
open import Relation.Nullary as Nullary using (¬_; Dec; True)

+

+
private

+
  variable

+
    a b c ℓ ℓ₁ ℓ₂ : Level

+
    A : Set a

+
    B : Set b

+
    C : Set c

+

+
------------------------------------------------------------------------

+
-- Definition

+

+
-- Unary relations are known as predicates and `Pred A ℓ` can be viewed

+
-- as some property that elements of type A might satisfy.

+

+
-- Consequently `P : Pred A ℓ` can also be seen as a subset of A

+
-- containing all the elements of A that satisfy property P. This view

+
-- informs much of the notation used below.

+

+
Pred : ∀ {a} → Set a → (ℓ : Level) → Set (a ⊔ suc ℓ)

+
Pred A ℓ = A → Set ℓ

+

+
------------------------------------------------------------------------

+
-- Special sets

+

+
-- The empty set.

+
-- Explicitly not level polymorphic as this often causes unsolved metas;

+
-- see `Relation.Unary.Polymorphic` for a level-polymorphic version.

+

+
∅ : Pred A 0ℓ

+
∅ = λ _ → ⊥

+

+
-- The singleton set.

+

+
｛_｝ : A → Pred A _

+
｛ x ｝ = x ≡_

+

+
-- The universal set.

+
-- Explicitly not level polymorphic (see comments for `∅` for more details)

+

+
U : Pred A 0ℓ

+
U = λ _ → ⊤

+

+
------------------------------------------------------------------------

+
-- Membership

+

+
infix 4 _∈_ _∉_

+

+
_∈_ : A → Pred A ℓ → Set _

+
x ∈ P = P x

+

+
_∉_ : A → Pred A ℓ → Set _

+
x ∉ P = ¬ x ∈ P

+

+
------------------------------------------------------------------------

+
-- Subset relations

+

+
infix 4 _⊆_ _⊇_ _⊈_ _⊉_ _⊂_ _⊃_ _⊄_ _⊅_ _≐_ _≐′_

+

+
_⊆_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊆ Q = ∀ {x} → x ∈ P → x ∈ Q

+

+
_⊇_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊇ Q = Q ⊆ P

+

+
_⊈_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊈ Q = ¬ (P ⊆ Q)

+

+
_⊉_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊉ Q = ¬ (P ⊇ Q)

+

+
_⊂_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊂ Q = P ⊆ Q × Q ⊈ P

+

+
_⊃_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊃ Q = Q ⊂ P

+

+
_⊄_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊄ Q = ¬ (P ⊂ Q)

+

+
_⊅_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊅ Q = ¬ (P ⊃ Q)

+

+
_≐_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ≐ Q = (P ⊆ Q) × (Q ⊆ P)

+

+
-- The following primed variants of _⊆_ can be used when 'x' can't

+
-- be inferred from 'x ∈ P'.

+

+
infix 4 _⊆′_ _⊇′_ _⊈′_ _⊉′_ _⊂′_ _⊃′_ _⊄′_ _⊅′_

+

+
_⊆′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊆′ Q = ∀ x → x ∈ P → x ∈ Q

+

+
_⊇′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
Q ⊇′ P = P ⊆′ Q

+

+
_⊈′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊈′ Q = ¬ (P ⊆′ Q)

+

+
_⊉′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊉′ Q = ¬ (P ⊇′ Q)

+

+
_⊂′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊂′ Q = P ⊆′ Q × Q ⊈′ P

+

+
_⊃′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊃′ Q = Q ⊂′ P

+

+
_⊄′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊄′ Q = ¬ (P ⊂′ Q)

+

+
_⊅′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ⊅′ Q = ¬ (P ⊃′ Q)

+

+
_≐′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ≐′ Q = (P ⊆′ Q) × (Q ⊆′ P)

+

+
------------------------------------------------------------------------

+
-- Properties of sets

+

+
infix 10 Satisfiable Universal IUniversal

+

+
-- Emptiness - no element satisfies P.

+

+
Empty : Pred A ℓ → Set _

+
Empty P = ∀ x → x ∉ P

+

+
-- Satisfiable - at least one element satisfies P.

+

+
Satisfiable : Pred A ℓ → Set _

+
Satisfiable P = ∃ λ x → x ∈ P

+

+
syntax Satisfiable P = ∃⟨ P ⟩

+

+
-- Universality - all elements satisfy P.

+

+
Universal : Pred A ℓ → Set _

+
Universal P = ∀ x → x ∈ P

+

+
syntax Universal  P = Π[ P ]

+

+
-- Implicit universality - all elements satisfy P.

+

+
IUniversal : Pred A ℓ → Set _

+
IUniversal P = ∀ {x} → x ∈ P

+

+
syntax IUniversal P = ∀[ P ]

+

+
-- Irrelevance - any two proofs that an element satifies P are

+
-- indistinguishable.

+

+
Irrelevant : Pred A ℓ → Set _

+
Irrelevant P = ∀ {x} → Nullary.Irrelevant (P x)

+

+
-- Recomputability - we can rebuild a relevant proof given an

+
-- irrelevant one.

+

+
Recomputable : Pred A ℓ → Set _

+
Recomputable P = ∀ {x} → Nullary.Recomputable (P x)

+

+
-- Stability - instances of P are stable wrt double negation

+

+
Stable : Pred A ℓ → Set _

+
Stable P = ∀ x → Nullary.Stable (P x)

+

+
-- Weak Decidability

+

+
WeaklyDecidable : Pred A ℓ → Set _

+
WeaklyDecidable P = ∀ x → Nullary.WeaklyDecidable (P x)

+

+
-- Decidability - it is possible to determine if an arbitrary element

+
-- satisfies P.

+

+
Decidable : Pred A ℓ → Set _

+
Decidable P = ∀ x → Dec (P x)

+

+
-- Erasure: A decidable predicate gives rise to another one, more

+
-- amenable to η-expansion

+

+
⌊_⌋ : {P : Pred A ℓ} → Decidable P → Pred A ℓ

+
⌊ P? ⌋ a = Lift _ (True (P? a))

+

+
------------------------------------------------------------------------

+
-- Operations on sets

+

+
infix 10 ⋃ ⋂

+
infixr 9 _⊢_

+
infixr 8 _⇒_

+
infixr 7 _∩_

+
infixr 6 _∪_

+
infixr 6 _∖_

+
infix 4 _≬_

+

+
-- Complement.

+

+
∁ : Pred A ℓ → Pred A ℓ

+
∁ P = λ x → x ∉ P

+

+
-- Implication.

+

+
_⇒_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _

+
P ⇒ Q = λ x → x ∈ P → x ∈ Q

+

+
-- Union.

+

+
_∪_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _

+
P ∪ Q = λ x → x ∈ P ⊎ x ∈ Q

+

+
-- Intersection.

+

+
_∩_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _

+
P ∩ Q = λ x → x ∈ P × x ∈ Q

+

+
-- Difference.

+

+
_∖_ : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _

+
P ∖ Q = λ x → x ∈ P × x ∉ Q

+

+
-- Infinitary union.

+

+
⋃ : ∀ {i} (I : Set i) → (I → Pred A ℓ) → Pred A _

+
⋃ I P = λ x → Σ[ i ∈ I ] P i x

+

+
syntax ⋃ I (λ i → P) = ⋃[ i ∶ I ] P

+

+
-- Infinitary intersection.

+

+
⋂ : ∀ {i} (I : Set i) → (I → Pred A ℓ) → Pred A _

+
⋂ I P = λ x → (i : I) → P i x

+

+
syntax ⋂ I (λ i → P) = ⋂[ i ∶ I ] P

+

+
-- Positive version of non-disjointness, dual to inclusion.

+

+
_≬_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _

+
P ≬ Q = ∃ λ x → x ∈ P × x ∈ Q

+

+
-- Update.

+

+
_⊢_ : (A → B) → Pred B ℓ → Pred A ℓ

+
f ⊢ P = λ x → P (f x)

+

+
------------------------------------------------------------------------

+
-- Predicate combinators

+

+
-- These differ from the set operations above, as the carrier set of the

+
-- resulting predicates are not the same as the carrier set of the

+
-- component predicates.

+

+
infixr  2 _⟨×⟩_

+
infixr  2 _⟨⊙⟩_

+
infixr  1 _⟨⊎⟩_

+
infixr  0 _⟨→⟩_

+
infixl  9 _⟨·⟩_

+
infix  10 _~

+
infixr  9 _⟨∘⟩_

+
infixr  2 _//_ _\\_

+

+
-- Product.

+

+
_⟨×⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A × B) _

+
(P ⟨×⟩ Q) (x , y) = x ∈ P × y ∈ Q

+

+
-- Sum over one element.

+

+
_⟨⊎⟩_ : Pred A ℓ → Pred B ℓ → Pred (A ⊎ B) _

+
P ⟨⊎⟩ Q = [ P , Q ]

+

+
-- Sum over two elements.

+

+
_⟨⊙⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A × B) _

+
(P ⟨⊙⟩ Q) (x , y) = x ∈ P ⊎ y ∈ Q

+

+
-- Implication.

+

+
_⟨→⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A → B) _

+
(P ⟨→⟩ Q) f = P ⊆ Q ∘ f

+

+
-- Product.

+

+
_⟨·⟩_ : (P : Pred A ℓ₁) (Q : Pred B ℓ₂) →

+
        (P ⟨×⟩ (P ⟨→⟩ Q)) ⊆ Q ∘ uncurry _|>_

+
(P ⟨·⟩ Q) (p , f) = f p

+

+
-- Converse.

+

+
_~ : Pred (A × B) ℓ → Pred (B × A) ℓ

+
P ~ = P ∘ swap

+

+
-- Composition.

+

+
_⟨∘⟩_ : Pred (A × B) ℓ₁ → Pred (B × C) ℓ₂ → Pred (A × C) _

+
(P ⟨∘⟩ Q) (x , z) = ∃ λ y → (x , y) ∈ P × (y , z) ∈ Q

+

+
-- Post-division.

+

+
_//_ : Pred (A × C) ℓ₁ → Pred (B × C) ℓ₂ → Pred (A × B) _

+
(P // Q) (x , y) = Q ∘ (y ,_) ⊆ P ∘ (x ,_)

+

+
-- Pre-division.

+

+
_\\_ : Pred (A × C) ℓ₁ → Pred (A × B) ℓ₂ → Pred (B × C) _

+
P \\ Q = (P ~ // Q ~) ~

+

\ No newline at end of file
diff --git a/site/static/agda_html/StateMachine.html b/site/static/agda_html/StateMachine.html
new file mode 100644
index 000000000..50c515474
--- /dev/null
+++ b/site/static/agda_html/StateMachine.html
@@ -0,0 +1,133 @@
+
+StateMachine
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module StateMachine where

+

+
open import Agda.Primitive using () renaming (Set to Type)

+

+
open import Prelude.Init hiding (map)

+
open import Prelude.InferenceRules

+

+
open import Class.Bifunctor

+

+
private

+
  variable S I O : Type

+
           s s' s'' : S

+
           i : I

+
           is : List I

+
           o : O

+
           os : List O

+
           STS : S → I → O → S → Type

+

+
module _ (_-⟦_/_⟧⇀_ : S → I → O → S → Type) where

+
  data _-⟦_/_⟧*⇀_ : S → List I → List O → S → Type where

+

+
    BS-base :

+
        ─────────────────

+
        s -⟦ [] / [] ⟧*⇀ s

+

+
    BS-ind :

+
      ∙ s  -⟦ i      / o      ⟧⇀  s'

+
      ∙ s' -⟦ is     / os     ⟧*⇀ s''

+
        ───────────────────────────────────────

+
        s  -⟦ i ∷ is / o ∷ os ⟧*⇀ s''

+

+
ReflexiveTransitiveClosure = _-⟦_/_⟧*⇀_

+

+
data IdSTS {S} : S → ⊤ → ⊤ → S → Type where

+
  Id-nop : IdSTS s _ _ s

+

+
Invariant : (S → I → O → S → Type) → (S → Type) → Type

+
Invariant _-⟦_/_⟧⇀_ P = ∀ {s i o s'} → s -⟦ i / o ⟧⇀ s' → P s → P s'

+

+
Total : (S → I → O → S → Type) → Type

+
Total _-⟦_/_⟧⇀_ = ∀ {s i} → ∃₂[ o , s' ] s -⟦ i / o ⟧⇀ s'

+

+
STS⇒RTC : STS s i o s' → ReflexiveTransitiveClosure STS s (i ∷ []) (o ∷ []) s'

+
STS⇒RTC sts = BS-ind sts BS-base

+

+
RTC-preserves-inv : ∀ {P} → Invariant STS P → Invariant (ReflexiveTransitiveClosure STS) P

+
RTC-preserves-inv inv (BS-base)      = id

+
RTC-preserves-inv inv (BS-ind p₁ p₂) = RTC-preserves-inv inv p₂ ∘ inv p₁

+

+
RTC-total : Total STS → Total (ReflexiveTransitiveClosure STS)

+
RTC-total SS-total {s} {[]}     = ([] , s , BS-base)

+
RTC-total SS-total {s} {i ∷ is} = case SS-total of λ where

+
  (o , s' , H) → case RTC-total SS-total of λ where

+
    (os , s'' , H') → (o ∷ os , s'' , BS-ind H H')

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.AnyOf.html b/site/static/agda_html/Tactic.AnyOf.html
new file mode 100644
index 000000000..6780fe37e
--- /dev/null
+++ b/site/static/agda_html/Tactic.AnyOf.html
@@ -0,0 +1,113 @@
+
+Tactic.AnyOf
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
--------------------------------------------------------------------------------

+
-- anyOf: take the first term that type checks

+
--------------------------------------------------------------------------------

+

+
module Tactic.AnyOf where

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
open import Reflection.Utils.TCI

+
open import Reflection.Tactic

+

+
open import Class.Monad

+
open import Class.MonadError.Instances

+
open import Class.MonadTC.Instances

+

+
anyOf' : List Term → ITactic

+
anyOf' = inDebugPath "anyOf" ∘

+
  foldl (λ x t → catch (debugLog ("Attempting: " ∷ᵈ t ∷ᵈ []) >> unifyWithGoal t >> debugLog ("Success with: " ∷ᵈ t ∷ᵈ [])) (λ _ → x))

+
    (logAndError1 "None of the provded terms solve the goal!")

+

+
anyOfⁿ : List Name → ITactic

+
anyOfⁿ = inDebugPath "anyOf" ∘

+
  foldl (λ x n → catch (debugLog ("Attempting: " ∷ᵈ n ∷ᵈ []) >> unifyWithGoal (n ∙) >> debugLog ("Success with: " ∷ᵈ n ∷ᵈ [])) (λ _ → x))

+
    (logAndError1 "None of the provded terms solve the goal!")

+

+
module _ ⦃ _ : TCOptions ⦄ where

+
  anyOfᵗ : List Term → Tactic

+
  anyOfᵗ l = initTac $ anyOf' l

+

+
  anyOfⁿᵗ : List Name → Tactic

+
  anyOfⁿᵗ l = initTac $ anyOfⁿ l

+

+
  macro

+
    anyOf = anyOfᵗ

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Assumption.html b/site/static/agda_html/Tactic.Assumption.html
new file mode 100644
index 000000000..40e355670
--- /dev/null
+++ b/site/static/agda_html/Tactic.Assumption.html
@@ -0,0 +1,129 @@
+
+Tactic.Assumption
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
--------------------------------------------------------------------------------

+
-- assumption: try to solve the goal with one of the available assumptions

+
--------------------------------------------------------------------------------

+

+
module Tactic.Assumption where

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
open import Class.Functor

+
open import Class.Monad

+
open import Class.MonadError.Instances

+
open import Class.MonadReader.Instances

+
open import Class.MonadTC.Instances

+

+
open import Reflection.Tactic

+
open import Reflection.Utils

+
open import Reflection.Utils.TCI

+

+
instance

+
  _ = Functor-M

+

+
solve : ⦃ TCOptions ⦄ → Term → Tactic

+
solve t = initTac $ runSpeculative $ do

+
  inj₁ goal ← reader TCEnv.goal

+
    where _ → error1 "solve: Goal is not a term!"

+
  metas ← findMetas <$> checkType t goal

+
  if null metas

+
    then (_, true)  <$> unify goal t

+
    else (_, false) <$> error1 "Unsolved metavariables remaining!"

+

+
assumption' : ITactic

+
assumption' = inDebugPath "assumption" do

+
  c ← getContext

+
  foldl (λ x k → do

+
    catch (unifyWithGoal (♯ k) >> debugLog ("Success with: " ∷ᵈ ♯ k ∷ᵈ [])) (λ _ → x))

+
    (logAndError1 "No valid assumption!") (downFrom $ length c)

+

+
module _ ⦃ _ : TCOptions ⦄ where

+
  macro

+
    assumption = initTac assumption'

+
    assumptionOpts = initTacOpts assumption'

+

+
private

+
  open import Tactic.Defaults

+
  module Test where

+
    test₁ : {A B : Set} → A → B → A

+
    test₁ a b = assumption

+

+
    test₂ : {A B : Set} → A → B → B

+
    test₂ a b = assumption

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.ByEq.html b/site/static/agda_html/Tactic.ByEq.html
new file mode 100644
index 000000000..4b43e895f
--- /dev/null
+++ b/site/static/agda_html/Tactic.ByEq.html
@@ -0,0 +1,135 @@
+
+Tactic.ByEq
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
module Tactic.ByEq where

+

+
open import Meta.Init hiding (TC)

+
open import Meta.Prelude

+
open import Class.Functor.Core; open import Class.Functor.Instances

+
open import Class.Monad.Core; open import Class.Monad.Instances

+
open import Reflection using (TC; withNormalisation; inferType; unify)

+
open import Reflection.Utils

+

+
-- Introduce as many arguments as possible and then:

+
--   1. for those of type `_ ≡ _`, unify with  `refl`

+
--   2. ignore the rest of the arguments (unify with `_`)

+
--   3. unify the hole with `refl`

+
by-eq : Hole → TC ⊤

+
by-eq hole = do

+
  ty ← withNormalisation true $ inferType hole

+
  let ps : Args Pattern

+
      ps = argTys ty <&> fmap λ {(def (quote _≡_) _) → quote refl ◇; _ → dot unknown}

+
  unify hole $ pat-lam [ clause [] ps (quote refl ◆) ] []

+

+
macro $by-eq = by-eq

+

+
private

+
  -- test that macro works

+
  _ : ∀ {n m k : ℕ} → n ≡ m → m ≡ k → n ≡ k

+
  _ = $by-eq

+

+
  _ : ∀ {n m k x y : ℕ} → n ≡ m → x ≡ y → m ≡ n

+
  _ = $by-eq

+

+
  -- test that tactic arguments work

+
  f : {@(tactic by-eq) _ : ∀ {n m : ℕ} → n ≡ m → m ≡ n} → Bool → Bool

+
  f = id

+

+
  _ : f {λ where refl → refl} true ≡ true

+
  _ = refl

+

+
  _ : f {$by-eq} true ≡ true

+
  _ = refl

+

+
  _ : f true ≡ true

+
  _ = refl

+

+
  -- test that normalisation works

+
  Sym = ∀ {n m : ℕ} → n ≡ m → m ≡ n

+

+
  g : {@(tactic by-eq) _ : Sym} → Bool → Bool

+
  g = id

+

+
  _ : g {λ where refl → refl} true ≡ true

+
  _ = refl

+

+
  _ : g {$by-eq} true ≡ true

+
  _ = refl

+

+
  _ : g true ≡ true

+
  _ = refl

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.ClauseBuilder.html b/site/static/agda_html/Tactic.ClauseBuilder.html
new file mode 100644
index 000000000..43b219fcc
--- /dev/null
+++ b/site/static/agda_html/Tactic.ClauseBuilder.html
@@ -0,0 +1,419 @@
+
+Tactic.ClauseBuilder
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
module Tactic.ClauseBuilder where

+

+
open import Meta.Prelude hiding ([_,_])

+
open import Meta.Init hiding (sort)

+

+
import Data.List.NonEmpty as NE

+
open import Data.List using (zipWith)

+
open import Data.Nat.Properties using (≤-totalOrder; ≤-decTotalOrder)

+
open import Data.Product using (map₁)

+

+
-- Note: using sort from Data.List.Sort directly doesn't work for

+
-- metaprogramming, since it's `abstract` for whatever reason

+
open import Data.List.Sort using (SortingAlgorithm)

+
open import Data.List.Sort.MergeSort using (mergeSort)

+
open SortingAlgorithm ≤-decTotalOrder (mergeSort ≤-decTotalOrder) public

+

+
open import Reflection.Utils

+
open import Reflection.Utils.TCI

+

+
open import Class.DecEq

+
open import Class.Functor

+
open import Class.MonadError.Instances

+
open import Class.MonadReader.Instances

+
open import Class.MonadTC.Instances

+
open import Class.Traversable

+

+
module _ where

+
  open import Class.Monad hiding (Monad-List) public

+

+
import Class.Monad

+

+
private variable

+
  a b : Level

+
  A : Set a

+

+
record ClauseBuilder (M : Set → Set) : Set₁ where

+
  field

+
    Base : Set → Set

+
    liftBase : Base A → M A

+
    addPattern : List (String × Arg Type) → Arg Pattern → M A → M A

+
    toClause : M (Maybe Term) → Base Clause

+

+
open ClauseBuilder {{...}} public

+

+
SinglePattern : Set

+
SinglePattern = List (String × Arg Type) × Arg Pattern

+

+
typedVarSinglePattern : String → Arg Type → SinglePattern

+
typedVarSinglePattern n t@(arg i _) = ([ n , t ] , arg i (` 0))

+

+
varSinglePattern : Arg String → SinglePattern

+
varSinglePattern (arg i n) = ([ n , arg i unknown ] , arg i (` 0))

+

+
multiSinglePattern : List String → Arg Pattern → SinglePattern

+
multiSinglePattern s p = ((_, vArg unknown) <$> s) , p

+

+
findIndexDefault : List ℕ → ℕ → ℕ → ℕ

+
findIndexDefault l d a with filter (λ where (i , x) → x ≟ a) (zipWithIndex _,_ l)

+
... | []          = d

+
... | (i , _) ∷ _ = i

+

+
singlePatternFromPattern : Arg Pattern → SinglePattern

+
singlePatternFromPattern (arg i p) =

+
  replicate (length (appearingIndices p)) ("" , vArg unknown) , arg i (replacePatternIndex p)

+
  where

+
    appearingIndices : Pattern → List ℕ

+
    appearingIndicesHelper : List (Arg Pattern) → List ℕ

+

+
    appearingIndices (Pattern.con c ps) = appearingIndicesHelper ps

+
    appearingIndices (Pattern.dot t) = [] -- TODO: this is probably wrong

+
    appearingIndices (` x) = [ x ]

+
    appearingIndices (Pattern.lit l) = []

+
    appearingIndices (Pattern.proj f) = []

+
    appearingIndices (Pattern.absurd x) = []

+

+
    appearingIndicesHelper [] = []

+
    appearingIndicesHelper (arg _ p ∷ ps) = appearingIndices p ++ appearingIndicesHelper ps

+

+
    normalisedIndexList : List ℕ

+
    normalisedIndexList = sort $ deduplicate _≟_ $ appearingIndices p

+

+
    replacePatternIndex : Pattern → Pattern

+
    replacePatternIndexHelper : List (Arg Pattern) → List (Arg Pattern)

+

+
    replacePatternIndex (Pattern.con c ps) = Pattern.con c (replacePatternIndexHelper ps)

+
    replacePatternIndex (Pattern.dot t) = Pattern.dot t

+
    replacePatternIndex (` x) = ` findIndexDefault normalisedIndexList 0 x

+
    replacePatternIndex (Pattern.lit l) = Pattern.lit l

+
    replacePatternIndex (Pattern.proj f) = Pattern.proj f

+
    replacePatternIndex (Pattern.absurd x) = Pattern.absurd x

+

+
    replacePatternIndexHelper [] = []

+
    replacePatternIndexHelper (arg i p ∷ ps) = (arg i (replacePatternIndex p)) ∷ (replacePatternIndexHelper ps)

+

+
module _ {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ ⦃ me : MonadError (List ErrorPart) M ⦄ ⦃ mre : MonadReader TCEnv M ⦄ ⦃ _ : MonadTC M ⦄ where

+

+
  instance _ = Functor-M {M = M}

+

+
  ctxSinglePatterns : M (List SinglePattern)

+
  ctxSinglePatterns = do

+
    ctx ← getContext

+
    return (singlePatternFromPattern <$> zipWithIndex (λ where k (_ , (arg i _)) → arg i (` k)) ctx)

+

+
  -- TODO: add the ability to match initial hidden arguments

+
  -- TODO: add dot patterns

+
  constrToPattern : Name → Type → M SinglePattern

+
  constrToPattern n ty = do

+
    (introTel , _) ← viewTy <$> (runAndReset $ local (λ env → record env { normalisation = true }) $ inferType (n ◆))

+
    let patternTel = zipWith (λ where (abs _ (arg i _)) k → arg i (` k)) introTel $ downFrom $ length introTel

+
    return (((λ where (abs s (arg i t)) → (s , arg i unknown)) <$> introTel) , (vArg $ Pattern.con n patternTel))

+

+
  -- like constrToPattern, but applied to parameters

+
  constrToPatternTyped : Name → List Term → M SinglePattern

+
  constrToPatternTyped n ps = do

+
    t ← applyWithVisibility n ps

+
    debugLogᵐ (t ᵛⁿ ∷ᵈᵐ []ᵐ)

+
    (introTel , _) ← viewTy <$> (local (λ env → record env { normalisation = true }) $ inferType t)

+
    let patternTel = zipWith (λ where (abs _ (arg i _)) k → arg i (` k)) introTel $ downFrom $ length introTel

+
    return (((λ where (abs s (arg i t)) → (s , arg i t)) <$> introTel) , (vArg $ Pattern.con n patternTel))

+

+
  -- all possible patterns for an inductive type

+
  constructorPatterns : Arg Type → M (List SinglePattern)

+
  constructorPatterns (arg i ty) = do

+
    constrs ← getConstrsForType ty

+
    return $ constrs <&> λ where

+
      (n , t) → let argInfo = proj₁ $ viewTy t

+
        in (((λ where (abs n' t') → n' , t') <$> argInfo)

+
           , arg i (Pattern.con n (zipWithIndex (λ where k (abs _ (arg i _)) → arg i (` (length argInfo ∸ ℕ.suc k))) argInfo)))

+

+
  -- all possible patterns for an inductive type

+
  constructorPatterns' : Type → M (List SinglePattern)

+
  constructorPatterns' ty = do

+
    constrs ← getConstrsForType ty

+
    traverse (λ (n , _) → constrToPattern n ty) constrs

+

+
  -- all possible patterns for an inductive type

+
  constructorPatternsTyped : Type → List Type → M (List SinglePattern)

+
  constructorPatternsTyped ty ps = do

+
    constrs ← getConstrsForType ty

+
    traverse (λ (n , _) → constrToPatternTyped n ps) constrs

+

+
ClauseInfo : Set

+
ClauseInfo = List SinglePattern

+

+
data ClauseExpr : Set where

+
  MatchExpr : List (SinglePattern × (ClauseExpr ⊎ Maybe Term)) → ClauseExpr

+

+
multiClauseExpr : List (NE.List⁺ SinglePattern × (ClauseExpr ⊎ Maybe Term)) → ClauseExpr

+
multiClauseExpr = MatchExpr ∘ Data.List.map helper

+
  where

+
    helper' : (List SinglePattern × (ClauseExpr ⊎ Maybe Term)) → ClauseExpr ⊎ Maybe Term

+
    helper' ([]     , e) = e

+
    helper' (p ∷ ps , e) = inj₁ (MatchExpr [ p , helper' (ps , e) ])

+

+
    helper : (NE.List⁺ SinglePattern × (ClauseExpr ⊎ Maybe Term)) → SinglePattern × (ClauseExpr ⊎ Maybe Term)

+
    helper (p NE.∷ ps , e) = p , helper' (ps , e)

+

+
clauseExprToClauseInfo : ClauseExpr → List (ClauseInfo × Maybe Term)

+
clauseExprToClauseInfo (MatchExpr []) = []

+
clauseExprToClauseInfo (MatchExpr ((p , inj₁ e) ∷ xs)) =

+
  (map₁ (p ∷_) <$> clauseExprToClauseInfo e) ++ clauseExprToClauseInfo (MatchExpr xs)

+
clauseExprToClauseInfo (MatchExpr ((p , inj₂ t) ∷ xs)) = ([ p ] , t) ∷ clauseExprToClauseInfo (MatchExpr xs)

+

+
clauseInfoToClauseArgs : ClauseInfo → List (String × Arg Type) × List (Arg Pattern)

+
clauseInfoToClauseArgs i =

+
  case helper 0 i of λ where (tel , ps , _) → (tel , ps)

+
  where

+
    helper : ℕ → ClauseInfo → List (String × Arg Type) × List (Arg Pattern) × ℕ

+
    helper k [] = ([] , [] , k)

+
    helper k ((tel' , arg h p) ∷ i) with helper k i

+
    ... | (tel , ps , k') = (tel' ++ tel , arg h (mapVariables (_+ k') p) ∷ ps , length tel' + k')

+

+
clauseInfoToClause : ClauseInfo → Maybe Term → Clause

+
clauseInfoToClause i t with clauseInfoToClauseArgs i | t

+
... | tel , ps | just x  = Clause.clause tel ps x

+
... | tel , ps | nothing = Clause.absurd-clause tel ps

+

+
clauseExprToClauses : ClauseExpr → List Clause

+
clauseExprToClauses e = (λ { (i , t) → clauseInfoToClause i t }) <$> clauseExprToClauseInfo e

+

+
nonBindingClause : Term → Clause

+
nonBindingClause = Clause.clause [] []

+

+
clauseExprToPatLam : ClauseExpr → Term

+
clauseExprToPatLam e = pat-lam (clauseExprToClauses e) []

+

+
record ContextMonad (M : ∀ {a} → Set a → Set a) ⦃ _ : Monad M ⦄ : Setω where

+
  field

+
    introPatternM : SinglePattern → M A → M A

+

+
  extendContextM : Arg Type → M A → M A

+
  extendContextM l x = introPatternM (typedVarSinglePattern "" l) x

+

+
open ContextMonad ⦃...⦄

+

+
Monad-Id : Monad id

+
Monad-Id .return = id

+
Monad-Id ._>>=_  = flip _$_

+

+
-- No context

+
ContextMonad-Id : ContextMonad id ⦃ Monad-Id ⦄

+
ContextMonad-Id .introPatternM _ a = a

+

+
module _ {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ ⦃ me : MonadError (List ErrorPart) M ⦄ ⦃ mre : MonadReader TCEnv M ⦄ ⦃ _ : MonadTC M ⦄ where

+

+
  instance _ = Functor-M {M = M}

+

+
  refineWithSingle : (Term → Term) → M Term → M Term

+
  refineWithSingle ref x = do

+
    goalTy ← goalTy

+
    m ← newMeta unknown

+
    res ← checkType (ref m) goalTy

+
    x' ← runWithHole m x

+
    unify m x'

+
    return $ ref x'

+

+
  caseMatch : Term → M ClauseExpr → M Term

+
  caseMatch t expr = debugLog ("Match" ∷ᵈ t ∷ᵈ []) >> (refineWithSingle (quote case_of_ ∙⟦ t ∣_⟧) $

+
    (λ expr' → pat-lam (clauseExprToClauses expr') []) <$> expr)

+

+
  currentTyConstrPatterns : M (List SinglePattern)

+
  currentTyConstrPatterns = do

+
    (ty ∷ _ , _) ← viewTy′ <$> goalTy

+
      where _ → error1 "currentTyConstrPatterns: Goal type is not a forall!"

+
    constructorPatterns' (unArg ty)

+

+
stripMetaLambdas : Term → Term

+
stripMetaLambdas = helper 0

+
  where

+
    helper : ℕ → Term → Term

+
    helper k (lam _ (abs _ t)) = helper (k + 1) t

+
    helper k (meta x as) = meta x $ map-Args (mapVars (_∸ k)) $ take (length as ∸ k) as

+
    helper _ _ = unknown

+

+
module _ {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ ⦃ me : MonadError (List ErrorPart) M ⦄ ⦃ mre : MonadReader TCEnv M ⦄ ⦃ _ : MonadTC M ⦄ where

+

+
  isProj : Pattern → Bool

+
  isProj (Pattern.proj _) = true

+
  isProj _                = false

+

+
  -- if the goal is of type (a : A) → B, return the type of the branch of pattern p and new context

+
  specializeType : SinglePattern → Type → M (Type × Telescope)

+
  specializeType p@(t , arg i p') goalTy = markDontFail "specializeType" $ inDebugPath "specializeType" $ runAndReset do

+
    debugLog ("Goal type to specialize: " ∷ᵈ goalTy ∷ᵈ [])

+
    cls@((Clause.clause tel _ _) ∷ _) ← return $ clauseExprToClauses $ MatchExpr $

+
        (p , inj₂ (just unknown)) ∷

+
        (if isProj p' then [] else [ varSinglePattern (arg i "_") , inj₂ (just unknown) ])

+
      where _ → error1 "BUG"

+
    debugLog ("With pattern: " ∷ᵈ cls ∷ᵈ [])

+
    (pat-lam (cl@(Clause.clause tel' _ (meta x ap)) ∷ _) []) ← checkType (pat-lam cls []) goalTy

+
      where t → debugLog ("BUG in specializeType:" ∷ᵈ t ∷ᵈ "\nWith pattern:" ∷ᵈ cls

+
                  ∷ᵈ "\nWith type:" ∷ᵈ goalTy ∷ᵈ "\nSinglePattern:" -- ∷ᵈ {!p!}

+
                  ∷ᵈ []) >> error1 "BUG"

+
    let varsToUnbind = 0

+
    let newCtx = tel'

+
    let m = meta x (map-Args (mapVars (_∸ varsToUnbind)) $ take (length ap ∸ varsToUnbind) ap)

+
    debugLog1 "New context:"

+
    logContext newCtx

+
    goalTy' ← extendContext' newCtx $ inferType m

+
    return (goalTy' , newCtx)

+

+
  ContextMonad-MonadTC : ContextMonad M

+
  ContextMonad-MonadTC .introPatternM pat x = do

+
    goalTy ← goalTy

+
    (newGoalTy , newContext) ← specializeType pat goalTy

+
    extendContext' newContext (runWithGoalTy newGoalTy x)

+

+
module ClauseExprM {M : ∀ {a} → Set a → Set a} ⦃ _ : Monad M ⦄ ⦃ _ : ContextMonad M ⦄ where

+

+
  instance _ = Functor-M {M = M}

+

+
  -- Construct a ClauseExpr in M and extend the context appropriately

+
  matchExprM : List (SinglePattern × M (ClauseExpr ⊎ Maybe Term)) → M ClauseExpr

+
  matchExprM = _<$>_ MatchExpr ∘ traverse (λ (a , b) → (a ,_) <$> introPatternM a b)

+

+
  multiMatchExprM : List (NE.List⁺ SinglePattern × M (ClauseExpr ⊎ Maybe Term)) → M ClauseExpr

+
  multiMatchExprM = matchExprM ∘ Data.List.map helper

+
    where

+
      helper' : (List SinglePattern × M (ClauseExpr ⊎ Maybe Term)) → M (ClauseExpr ⊎ Maybe Term)

+
      helper' ([]     , e) = e

+
      helper' (p ∷ ps , e) = inj₁ <$> (matchExprM [ p , helper' (ps , e) ])

+

+
      helper : (NE.List⁺ SinglePattern × M (ClauseExpr ⊎ Maybe Term)) → SinglePattern × M (ClauseExpr ⊎ Maybe Term)

+
      helper (p NE.∷ ps , e) = (p , helper' (ps , e))

+

+
  singleMatchExpr : SinglePattern → M (ClauseExpr ⊎ Maybe Term) → M ClauseExpr

+
  singleMatchExpr p x = matchExprM [ p , x ]

+

+
  singleTelescopeMatchExpr : NE.List⁺ SinglePattern → M (ClauseExpr ⊎ Maybe Term) → M ClauseExpr

+
  singleTelescopeMatchExpr (p NE.∷ ps) x = helper p ps x

+
    where

+
      helper : SinglePattern → List SinglePattern → M (ClauseExpr ⊎ Maybe Term) → M ClauseExpr

+
      helper p [] x = singleMatchExpr p x

+
      helper p (p' ∷ ps) x = singleMatchExpr p $ inj₁ <$> helper p' ps x

+

+
  introExpr : Arg String → M (ClauseExpr ⊎ Maybe Term) → M ClauseExpr

+
  introExpr n x = singleMatchExpr (varSinglePattern n) x

+

+
  introsExpr : NE.List⁺ (Arg String) → M (ClauseExpr ⊎ Maybe Term) → M ClauseExpr

+
  introsExpr (p NE.∷ ps) x = helper p ps x

+
    where

+
      helper : Arg String → List (Arg String) → M (ClauseExpr ⊎ Maybe Term) → M ClauseExpr

+
      helper n [] x = introExpr n x

+
      helper n (n' ∷ ns) x = introExpr n $ inj₁ <$> helper n ns x

+

+
  contMatch : M ClauseExpr → M (ClauseExpr ⊎ Maybe Term)

+
  contMatch expr = inj₁ <$> expr

+

+
  finishMatch : M Term → M (ClauseExpr ⊎ Maybe Term)

+
  finishMatch t = (inj₂ ∘ just) <$> t

+

+
  bindCtxMatchExpr : ⦃ me : MonadError (List ErrorPart) M ⦄ ⦃ mre : MonadReader TCEnv M ⦄ ⦃ _ : MonadTC M ⦄

+
                   → M ClauseExpr → M ClauseExpr

+
  bindCtxMatchExpr x = do

+
    e ← ctxSinglePatterns

+
    case NE.fromList e of λ where

+
      (just e') → singleTelescopeMatchExpr e' $ contMatch x

+
      nothing → x

+

+
clauseTelescope : Clause → List (String × Arg Type)

+
clauseTelescope (Clause.clause tel _ _) = tel

+
clauseTelescope (Clause.absurd-clause tel _) = tel

+

+
module _ where

+
  open ClauseExprM ⦃ Monad-Id ⦄ ⦃ ContextMonad-Id ⦄

+

+
  instanciatePattern : SinglePattern → List (Arg Type)

+
  instanciatePattern p = proj₂ <$>

+
    (clauseTelescope $ from-just $ head $ clauseExprToClauses $ singleMatchExpr p $ finishMatch unknown)

+

+
  instanciatePatterns : List SinglePattern → Term → List Clause

+
  instanciatePatterns [] t = [ Clause.clause [] [] t ]

+
  instanciatePatterns (x ∷ ps) t =

+
    clauseExprToClauses $ singleTelescopeMatchExpr (x NE.∷ ps) (finishMatch t)

+

+
ctxBindingClause : Term → TC Clause

+
ctxBindingClause t = do

+
  pats ← ctxSinglePatterns

+
  (c ∷ _) ← return $ instanciatePatterns (reverse pats) t

+
    where _ → error1 "Bug in ctxBindingClause"

+
  return c

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Defaults.html b/site/static/agda_html/Tactic.Defaults.html
new file mode 100644
index 000000000..63b4a0e87
--- /dev/null
+++ b/site/static/agda_html/Tactic.Defaults.html
@@ -0,0 +1,95 @@
+
+Tactic.Defaults
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
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+             

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+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
module Tactic.Defaults where

+

+
open import Meta.Prelude

+

+
open import Class.MonadTC

+
open import Reflection.Debug

+

+
-- There should only ever be one instance, with this being convenient

+
-- to tweak all at once

+
instance

+
  defaultTCOptionsI = record

+
    { debug = record defaultDebugOptions

+
      { prefix = '┃' -- ┃⎸

+
      ; filter = Filter.⊥

+
      }

+
    ; fuel  = ("reduceDec/constrs" , 5) ∷ []

+
    }

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Derive.Convertible.html b/site/static/agda_html/Tactic.Derive.Convertible.html
new file mode 100644
index 000000000..a81b9b715
--- /dev/null
+++ b/site/static/agda_html/Tactic.Derive.Convertible.html
@@ -0,0 +1,262 @@
+
+Tactic.Derive.Convertible
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
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+               
+               

+             
module Tactic.Derive.Convertible where

+

+
open import Level

+
open import Meta.Prelude

+
open import Meta.Init

+

+
import Data.List as L

+
import Data.List.NonEmpty as NE

+
import Data.String as S

+
import Data.Product as P

+

+
open import Relation.Nullary

+
open import Relation.Nullary.Decidable

+

+
open import Reflection.Tactic

+
open import Reflection.AST.Term

+
open import Reflection.AST.DeBruijn

+
import Reflection.TCM as R

+
open import Reflection.Utils

+
open import Reflection.Utils.TCI

+
import Function.Identity.Effectful as Identity

+

+
open import Class.DecEq

+
open import Class.DecEq

+
open import Class.Functor

+
open import Class.Monad

+
open import Class.MonadTC.Instances

+
open import Class.Traversable

+
open import Class.Show

+
open import Class.MonadReader

+

+
open import Reflection.Utils.Substitute

+
open import Class.Convertible

+
open import Class.HasHsType

+
open import Tactic.Derive.HsType

+

+
private instance

+
  _ = Functor-M {TC}

+

+
-- TODO: move to agda-stdlib-meta

+
liftTC : ∀ {a} {A : Set a} → R.TC A → TC A

+
liftTC m _ = m

+

+
private

+

+
  open MonadReader ⦃...⦄

+

+
  variable

+
    A B C : Set

+

+
  TyViewTel = List (Abs (Arg Type))

+

+
  substTel : Subst TyViewTel

+
  substTel x s [] = []

+
  substTel x s (abs z t ∷ tel) = abs z (substArg x s t) ∷ (substTel (ℕ.suc x) s tel)

+
    -- Note: `abs` is abused in TyViewTel and doesn't actually scope over the `t`

+

+
  -- Substitute leading level parameters with lzero

+
  smashLevels : TyViewTel → ℕ × TyViewTel

+
  smashLevels (abs s (arg i (def (quote Level) [])) ∷ tel) =

+
    P.map ℕ.suc (substTel 0 (quote 0ℓ ∙)) $ smashLevels tel

+
  smashLevels tel = (0 , tel)

+

+
  tyViewToTel : TyViewTel → Telescope

+
  tyViewToTel = L.map λ where (abs s a) → s , a

+

+
  hideTyView : Abs (Arg A) → Abs (Arg A)

+
  hideTyView (abs s (arg (arg-info _ m) x)) = abs s (arg (arg-info hidden m) x)

+

+
  -- The type of a Convertible instance. For parameterised types adds the appropriate instance

+
  -- arguments and instantiates level arguments to lzero. For instance,

+
  -- instanceType _⊎_ Hs.Either = {A B : Set} {a b : Set} → ⦃ Convertible A a ⦄ → ⦃ Convertible B b ⦄

+
  --                              Convertible (A ⊎ B) (Hs.Either a b)

+
  instanceType : (agdaName hsName : Name) → TC TypeView

+
  instanceType agdaName hsName = do

+
    aLvls , agdaParams ← smashLevels <$> getParamsAndIndices agdaName

+
    hLvls , hsParams   ← smashLevels <$> getParamsAndIndices hsName

+
    true ← return (length agdaParams == length hsParams)

+
      where false → liftTC $ R.typeErrorFmt "%q and %q have different number of parameters" agdaName hsName

+
    let n = length agdaParams

+
        l₀ = quote 0ℓ ∙

+
    agdaTy ← applyWithVisibility agdaName $ L.replicate aLvls l₀ ++ L.map ♯ (take n (downFrom (n + n)))

+
    hsTy   ← applyWithVisibility hsName   $ L.replicate hLvls l₀ ++ L.map ♯ (downFrom n)

+
    let instHead = weaken n $ quote Convertible ∙⟦ agdaTy ∣ hsTy ⟧

+
        tel      = L.map hideTyView (agdaParams ++ hsParams) ++

+
                   L.replicate n (abs "_" (iArg (quote Convertible ∙⟦ ♯ (n + n ∸ 1) ∣ ♯ (n ∸ 1) ⟧)))

+
    return $ tel , instHead

+

+
  -- Compute one clause of the Convertible instance. For instance,

+
  -- conversionClause Convertible.to to ((c₁ , _) , (c₂ , _)) generates

+
  -- .to (c₁ x₁ .. xn) = c₂ (to x₁) .. (to xn)

+
  -- where the xi are the visible constructor arguments.

+
  conversionClause : Name → Name → (Name × Type) × (Name × Type) → TC Clause

+
  conversionClause prjP prjE ((fromC , fromTy) , (toC , toTy)) = do

+
    let isVis   = λ { (abs _ (arg (arg-info visible _) _)) → true; _ → false }

+
        fromTel = L.filterᵇ isVis (proj₁ (viewTy fromTy))

+
        toTel   = L.filterᵇ isVis (proj₁ (viewTy toTy))

+
        n       = length fromTel

+
        mkCon  c mkArg = Term.con c    $ L.map (vArg ∘ mkArg ∘ ♯)  (downFrom n)

+
        mkConP c mkArg = Pattern.con c $ L.map (vArg ∘ mkArg ∘ `_) (downFrom n)

+
    true ← return (n == length toTel)

+
      where false → liftTC $ R.typeErrorFmt "%q and %q have different number of arguments" fromC toC

+
    return $ clause (tyViewToTel $ L.map (λ where (abs x (arg i _)) → abs x (arg i unknown)) fromTel)

+
                    (vArg (proj prjP) ∷ vArg (mkConP fromC id) ∷ [])

+
                    (mkCon toC (prjE ∙⟦_⟧))

+

+
  -- Compute the clauses of a convertible instance.

+
  instanceClauses : (agdaName hsName : Name) → TC (List Clause)

+
  instanceClauses agdaName hsName = do

+
    agdaCons ← getConstrs agdaName

+
    hsCons   ← getConstrs hsName

+
    agdaPars ← length <$> getParamsAndIndices agdaName

+
    hsPars   ← length <$> getParamsAndIndices hsName

+
    true ← return (length agdaCons == length hsCons)

+
      where false → liftTC $ R.typeErrorFmt "%q and %q have different number of constructors" agdaName hsName

+
    toClauses   ← mapM (conversionClause (quote Convertible.to)   (quote to)  ) (L.zip agdaCons hsCons)

+
    fromClauses ← mapM (conversionClause (quote Convertible.from) (quote from)) (L.zip hsCons agdaCons)

+
    return $ toClauses ++ fromClauses

+

+
  absurdClause : Name → Clause

+
  absurdClause prj = absurd-clause (("x" , vArg unknown) ∷ []) (vArg (proj prj) ∷ vArg (absurd 0) ∷ [])

+

+
  -- Compute conversion clauses for the current goal and wrap them in a pattern lambda.

+
  patternLambda : TC Term

+
  patternLambda = do

+
    quote Convertible ∙⟦ `A ∣ `B ⟧ ← reduce =<< goalTy

+
      where t → liftTC $ R.typeErrorFmt "Expected Convertible A B, got %t" t

+
    agdaCons ← getConstrsForType `A

+
    hsCons   ← getConstrsForType `B

+
    toClauses   ← mapM (conversionClause (quote Convertible.to)   (quote to)  ) (L.zip agdaCons hsCons)

+
    fromClauses ← mapM (conversionClause (quote Convertible.from) (quote from)) (L.zip hsCons agdaCons)

+
    case toClauses ++ fromClauses of λ where

+
      []  → return $ pat-lam (absurdClause (quote Convertible.to) ∷ absurdClause (quote Convertible.from) ∷ []) []

+
      cls → return $ pat-lam cls []

+

+
doPatternLambda : Term → R.TC Term

+
doPatternLambda hole = patternLambda =<< initTCEnvWithGoal hole

+

+
-- Deriving a Convertible instance. Usage

+
--   unquoteDecl iName = deriveConvertible iName (quote AgdaTy) (quote HsTy)

+
deriveConvertible : Name → Name → Name → R.TC ⊤

+
deriveConvertible instName agdaName hsName = initUnquoteWithGoal ⦃ defaultTCOptions ⦄ (agda-sort (lit 0)) do

+
  agdaDef ← getDefinition agdaName

+
  hsDef   ← getDefinition hsName

+
  -- instName ← freshName $ "Convertible" S.++ show hsName

+
  instTel , instTy ← instanceType agdaName hsName

+
  inst    ← declareDef (iArg instName) (tyView (instTel , instTy))

+
  clauses ← instanceClauses agdaName hsName

+
  defineFun instName clauses

+
  return _

+

+
-- Macros providing an alternative interface. Usage

+
--   iName : ConvertibleType AgdaTy HsTy

+
--   iName = autoConvertible

+
-- or

+
--   iName = autoConvert AgdaTy

+
macro

+
  ConvertibleType : Name → Name → Tactic

+
  ConvertibleType agdaName hsName = initTac ⦃ defaultTCOptions ⦄ $

+
    unifyWithGoal ∘ tyView =<< instanceType agdaName hsName

+

+
  autoConvertible : Tactic

+
  autoConvertible = initTac ⦃ defaultTCOptions ⦄ $

+
    unifyWithGoal =<< patternLambda

+

+
  autoConvert : Name → Tactic

+
  autoConvert d hole = do

+
    hsTyMeta ← R.newMeta `Set

+
    R.checkType hole $ quote Convertible ∙⟦ d ∙ ∣ hsTyMeta ⟧

+
    hsTy ← solveHsType (d ∙)

+
    R.unify hsTyMeta hsTy

+
    R.unify hole =<< doPatternLambda hole

+

+
-- Tests

+

+
private

+

+
  record Test : Set where

+
    field f : ℕ

+
          g : Maybe ℕ

+
          h : List ℕ

+

+
  instance

+
    HsTy-Test = autoHsType Test ⊣ withConstructor "MkTest" • fieldPrefix "t"

+
    Conv-Test = autoConvert Test

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Derive.DecEq.html b/site/static/agda_html/Tactic.Derive.DecEq.html
new file mode 100644
index 000000000..9eeb27e0e
--- /dev/null
+++ b/site/static/agda_html/Tactic.Derive.DecEq.html
@@ -0,0 +1,203 @@
+
+Tactic.Derive.DecEq
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
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+             

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+                   
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+                 

+               

+             

+             

+               
+               

+             
-- Deriving decidable equality. This works in several cases that use

+
-- mutual recursion, examples are at the bottom.

+
--

+
-- TODO: This breaks with:

+
-- - dependent records, e.g. Product

+
-- - anything listed in Tactic.Derive

+
-- - maybe more

+

+
{-# OPTIONS -v allTactics:100 #-}

+
{-# OPTIONS --safe #-}

+
module Tactic.Derive.DecEq where

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
import Data.List as L

+
import Data.List.NonEmpty as NE

+

+
open import Relation.Nullary

+
open import Relation.Nullary.Decidable

+

+
open import Reflection.Tactic

+
open import Reflection.AST.Term using (_≟-Pattern_)

+
open import Reflection.Utils

+
open import Reflection.Utils.TCI

+

+
open import Class.DecEq.Core

+
open import Class.Functor

+
open import Class.Monad

+
open import Class.MonadTC.Instances

+
open import Class.Traversable

+

+
open import Tactic.ClauseBuilder

+
open import Tactic.Derive (quote DecEq) (quote _≟_)

+

+
instance

+
  _ = ContextMonad-MonadTC

+

+
open ClauseExprM

+

+
`yes `no : Term → Term

+
`yes x = quote _because_ ◆⟦ quote true  ◆ ∣ quote ofʸ ◆⟦ x ⟧ ⟧

+
`no  x = quote _because_ ◆⟦ quote false ◆ ∣ quote ofⁿ ◆⟦ x ⟧ ⟧

+

+
-- We take the Dec P argument first to improve type checking performance.

+
-- It's easy to infer the type of P from this argument and we need to know

+
-- P to be able to check the pattern lambda generated for the P → Q direction

+
-- of the isomorphism. Having the isomorphism first would cause the type checker

+
-- to go back and forth between the pattern lambda and the Dec P argument,

+
-- inferring just enough of the type of make progress on the lambda.

+
map' : ∀ {p q} {P : Set p} {Q : Set q} → Dec P → P ⇔ Q → Dec Q

+
map' d record { to = to ; from = from } = map′ to from d

+

+
module _ (transName : Name → Maybe Name) where

+

+
  eqFromTerm : Term → Term → Term → Term

+
  eqFromTerm (def n _) t t' with transName n

+
  ... | just n'     = def (quote _≟_) (iArg (n' ∙) ∷ vArg t ∷ vArg t' ∷ [])

+
  ... | nothing     = quote _≟_ ∙⟦ t ∣ t' ⟧

+
  eqFromTerm _ t t' = quote _≟_ ∙⟦ t ∣ t' ⟧

+

+
  toDecEqName : SinglePattern → List (Term → Term → Term)

+
  toDecEqName (l , _) = L.map (λ where (_ , arg _ t) → eqFromTerm t) l

+

+
  -- on the diagonal we have one pattern, outside we don't care

+
  -- assume that the types in the pattern are properly normalized

+
  mapDiag : Maybe SinglePattern → TC Term

+
  mapDiag nothing          = return $ `no `λ⦅ [ ("" , vArg?) ] ⦆∅

+
  mapDiag (just p@(l , _)) = let k = length l in do

+
    typeList ← traverse ⦃ Functor-List ⦄ inferType (applyDownFrom ♯ (length l))

+
    return $ quote map' ∙⟦ genPf k (L.map eqFromTerm typeList) ∣ genEquiv k ⟧

+
    where

+
      genPf : ℕ → List (Term → Term → Term) → Term

+
      genPf k []      = `yes (quote tt ◆)

+
      genPf k (n ∷ l) = quote _×-dec_ ∙⟦ genPf k l ∣ n (♯ (length l)) (♯ (length l + k)) ⟧

+

+
      -- c x1 .. xn ≡ c y1 .. yn ⇔ x1 ≡ y1 .. xn ≡ yn

+
      genEquiv : ℕ → Term

+
      genEquiv n = quote mk⇔ ∙⟦ `λ⟦ reflPattern n ⇒ quote refl ◆ ⟧ ∣ `λ⟦ quote refl ◇ ⇒ reflTerm n ⟧ ⟧

+
        where

+
          reflPattern : ℕ → Pattern

+
          reflPattern 0       = quote tt ◇

+
          reflPattern (suc n) = quote _,_ ◇⟦ reflPattern n ∣ quote refl ◇ ⟧

+

+
          reflTerm : ℕ → Term

+
          reflTerm 0       = quote tt ◆

+
          reflTerm (suc n) = quote _,_ ◆⟦ reflTerm n ∣ quote refl ◆ ⟧

+

+
  toMapDiag : SinglePattern → SinglePattern → NE.List⁺ SinglePattern × TC (ClauseExpr ⊎ Maybe Term)

+
  toMapDiag p@(_ , arg _ p₁) p'@(_ , arg _ p₂) =

+
    (p NE.∷ [ p' ] , finishMatch (if ⌊ p₁ ≟-Pattern p₂ ⌋ then mapDiag (just p) else mapDiag nothing))

+

+
module _ ⦃ _ : TCOptions ⦄ where

+
  derive-DecEq : List (Name × Name) → UnquoteDecl

+
  derive-DecEq = derive-Class 0 (λ transName ps → cartesianProductWith (toMapDiag transName) ps ps)

+

+
private

+
  open import Tactic.Derive.TestTypes

+
  open import Tactic.Defaults

+

+
  unquoteDecl DecEq-These = derive-DecEq [ (quote These , DecEq-These) ]

+

+
  unquoteDecl DecEq-⊎ = derive-DecEq [ (quote _⊎_ , DecEq-⊎) ]

+

+
  unquoteDecl DecEq-Bool DecEq-ℤ DecEq-List DecEq-Maybe DecEq-ℕ DecEq-Sign DecEq-⊤ = derive-DecEq ((quote Bool , DecEq-Bool) ∷ (quote ℤ , DecEq-ℤ) ∷ (quote List , DecEq-List) ∷ (quote Maybe , DecEq-Maybe) ∷ (quote ℕ , DecEq-ℕ) ∷ (quote Sign , DecEq-Sign) ∷ (quote ⊤ , DecEq-⊤) ∷ [])

+

+
  unquoteDecl DecEq-Fin = derive-DecEq [ (quote Fin , DecEq-Fin) ]

+

+
  -- this doesn't work since the clause builder machinery doesn't deal

+
  -- with absurd patterns yet

+

+
  --unquoteDecl DecEq-Vec = derive-DecEq [ (quote Vec , DecEq-Vec) ]

+

+
  unquoteDecl DecEq-E1 = derive-DecEq [ (quote E1 , DecEq-E1) ]

+
  unquoteDecl DecEq-E2 = derive-DecEq [ (quote E2 , DecEq-E2) ]

+

+
  -- this uses mutual recursion with List E3

+
  unquoteDecl DecEq-E3 = derive-DecEq [ (quote E3 , DecEq-E3) ]

+
  -- unquoteDecl DecEq-E4 = derive-DecEq [ (quote E4 , DecEq-E4) ]

+

+
  unquoteDecl DecEq-R1 = derive-DecEq [ (quote R1 , DecEq-R1) ]

+
  unquoteDecl DecEq-R2 = derive-DecEq [ (quote R2 , DecEq-R2) ]

+

+
  unquoteDecl DecEq-M₁ DecEq-M₂ = derive-DecEq $ (quote M₁ , DecEq-M₁) ∷ (quote M₂ , DecEq-M₂) ∷ []

+

+
  -- Expected: DecEq-Term DecEq-Product

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Derive.HsType.html b/site/static/agda_html/Tactic.Derive.HsType.html
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+
+Tactic.Derive.HsType
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
module Tactic.Derive.HsType where

+

+
open import Meta.Init hiding (TC; Monad-TC; MonadError-TC)

+

+
open import Level using (Level; 0ℓ)

+
open import Agda.Builtin.Reflection using (declareData; defineData; pragmaForeign; pragmaCompile;

+
                                           solveInstanceConstraints)

+
open import Reflection as R hiding (showName; _>>=_; _>>_)

+
open import Reflection.AST hiding (showName)

+
open import Reflection.AST.DeBruijn

+
open import Data.Maybe using (Maybe; nothing; just; fromMaybe; maybe′; _<∣>_)

+
open import Data.Unit using (⊤; tt)

+
open import Data.Integer.Base using (ℤ)

+
open import Data.Sum using (_⊎_; inj₁; inj₂)

+
open import Data.String using (String) renaming (_++_ to _&_)

+
open import Data.Product hiding (map; zip; zipWith)

+
import Data.String as String

+
open import Data.Bool

+
open import Data.Nat

+
open import Data.List hiding (lookup; fromMaybe)

+
open import Data.Char using (toUpper; toLower)

+
open import Foreign.Haskell

+
open import Function

+
open import Text.Printf

+

+
open import Class.DecEq

+
open import Class.Functor

+
open import Class.Monad

+
open import Class.MonadError

+
open import Class.MonadReader

+
open import Class.Show

+
open import Class.Show.Instances

+
open import Class.MonadTC using (TCEnv; dontReduce; defaultTCOptions)

+
open import Tactic.Derive.Show using (showName)

+

+
open import Reflection.Utils

+
open import Reflection.Utils.TCI

+
open import Foreign.Haskell.Pair using (Pair)

+
open import Class.HasHsType

+

+
{-

+
Need to generate:

+
  - a data type that can be compiled to a Haskell data type

+
  - the corresponding Haskell type (in a FOREIGN pragma)

+
  - the COMPILE pragma binding them together

+
-}

+

+
private variable

+
  l : Level

+
  A B : Set l

+

+
-- TODO: somewhere else

+
`Set = agda-sort (Sort.set (quote 0ℓ ∙))

+

+
-- * Controlling the names of the generated types

+

+
record NameEnv : Set where

+
  field

+
    customNames : List (Name × String)

+
    tName       : Name → Maybe String

+
    cName       : Name → Maybe String

+
    fName       : Name → Maybe String

+

+
private

+
  mapHead : (A → A) → List A → List A

+
  mapHead f []       = []

+
  mapHead f (x ∷ xs) = f x ∷ xs

+

+
capitalize : String → String

+
capitalize = String.fromList ∘ mapHead toUpper ∘ String.toList

+

+
uncapitalize : String → String

+
uncapitalize = String.fromList ∘ mapHead toLower ∘ String.toList

+

+
private

+
  lookup : ⦃ DecEq A ⦄ → A → List (A × B) → Maybe B

+
  lookup x xs = proj₂ <$> findᵇ ((x ==_) ∘ proj₁) xs

+

+
  lookupEnv : (NameEnv → Name → Maybe String) → NameEnv → Name → Maybe String

+
  lookupEnv fn env x = lookup x (NameEnv.customNames env) <∣> fn env x

+

+
  lookupTypeName  = lookupEnv NameEnv.tName

+
  lookupConName   = lookupEnv NameEnv.cName

+
  lookupFieldName = lookupEnv NameEnv.fName

+

+
  emptyEnv : NameEnv

+
  emptyEnv = record{ customNames = []

+
                   ; tName = const nothing

+
                   ; cName = const nothing

+
                   ; fName = const nothing }

+

+
  customName : Name → String → NameEnv

+
  customName x s = record emptyEnv { customNames = (x , s) ∷ [] }

+

+
onTypes : (String → String) → NameEnv

+
onTypes f = record emptyEnv { tName = just ∘ f ∘ showName }

+

+
onConstructors : (String → String) → NameEnv

+
onConstructors f = record emptyEnv { cName = just ∘ f ∘ showName }

+

+
withName : String → NameEnv

+
withName t = onTypes (const t)

+

+
-- Only use for single constructor types obviously

+
withConstructor : String → NameEnv

+
withConstructor c = onConstructors (const c)

+

+
onFields : (String → String) → NameEnv

+
onFields f = record emptyEnv { fName = just ∘ f ∘ showName }

+

+
fieldPrefix : String → NameEnv

+
fieldPrefix pre = onFields $ (pre String.++_) ∘ capitalize

+

+
infixr 5 _•_

+
_•_ : NameEnv → NameEnv → NameEnv

+
env • env₁ = record

+
  { customNames = env.customNames ++ env₁.customNames

+
  ; tName       = λ x → env.tName x <∣> env₁.tName x

+
  ; cName       = λ x → env.cName x <∣> env₁.cName x

+
  ; fName       = λ x → env.fName x <∣> env₁.fName x

+
  }

+
  where

+
    module env  = NameEnv env

+
    module env₁ = NameEnv env₁

+

+

+
-- * The inner workings

+

+
solveHsType : Term → TC Term

+
solveHsType tm = do

+
  hsTy ← checkType (quote HsType ∙⟦ tm ⟧) `Set

+
  solveInstanceConstraints

+
  normalise hsTy >>= λ where

+
    (def (quote HsType) _) → typeErrorFmt "Failed to solve HsType %t" tm

+
    hsTy                   → return hsTy

+

+
private

+
  debug = debugPrintFmt "tactic.hs-types"

+

+
  _‼_ : List A → ℕ → Maybe A

+
  []       ‼ i     = nothing

+
  (x ∷ xs) ‼ zero  = just x

+
  (x ∷ xs) ‼ suc i = xs ‼ i

+

+
  specialHsTypes : List (Name × String)

+
  specialHsTypes = (quote ⊤      , "()")

+
                 ∷ (quote ℕ      , "Integer")

+
                 ∷ (quote ℤ      , "Integer")

+
                 ∷ (quote Bool   , "Bool")

+
                 ∷ (quote List   , "[]")

+
                 ∷ (quote Pair   , "(,)")

+
                 ∷ (quote Maybe  , "Maybe")

+
                 ∷ (quote Either , "Either")

+
                 ∷ (quote String , "Data.Text.Text")

+
                 ∷ []

+

+
  hsTypeName : NameEnv → Name → String

+
  hsTypeName env d = fromMaybe (capitalize $ showName d) (lookupTypeName env d)

+

+
  freshHsTypeName : NameEnv → Name → TC Name

+
  freshHsTypeName env d = freshName (hsTypeName env d)

+

+
  hsConName : NameEnv → Name → String

+
  hsConName env c = fromMaybe (capitalize $ showName c) (lookupConName env c)

+

+
  hsFieldName : NameEnv → Name → String

+
  hsFieldName env f = fromMaybe (uncapitalize $ showName f) (lookupFieldName env f)

+

+
  freshHsConName : NameEnv → Name → Name → TC Name

+
  freshHsConName env tyName c =

+
    if showName c == "constructor"

+
    then freshName (hsConName env tyName) -- Unnamed record constructor: use type name

+
    else freshName (hsConName env c)

+

+
  isThis : Name → Term → Bool

+
  isThis f (def g _) = f == g

+
  isThis _ _ = false

+

+
  computeHsType : Name → Name → Term → TC Term

+
  computeHsType aThis hThis tm with isThis aThis tm

+
  ... | true = pure (hThis ∙)

+
  computeHsType aThis hThis (Π[ x ∶ arg (arg-info visible _) a ] b) | false = do

+
    dom ← computeHsType aThis hThis a

+
    rng ← extendContext x (vArg a) $ computeHsType aThis hThis b

+
    pure (vΠ[ x ∶ dom ] rng)

+
  computeHsType aThis hThis (Π[ x ∶ a ] b) | false = do

+
    ty ← extendContext x a $ computeHsType aThis hThis b

+
    just ty′ ← pure (strengthen ty)

+
      where nothing → extendContext x a $ typeErrorFmt "%s free in computed HsType %t" x ty

+
    pure ty′

+
  -- Hack to get recursive occurrences under lists to work

+
  computeHsType aThis hThis (def (quote List) (_ ∷ vArg a ∷ [])) | false = do

+
    ty ← computeHsType aThis hThis a

+
    pure (quote List ∙⟦ ty ⟧)

+
  computeHsType _ _ tm | false = do

+
    debug 10 "solving HsType %t" tm

+
    ty ← solveHsType tm

+
    debug 10 "HsType %t = %t" tm ty

+
    pure ty

+

+
  makeHsCon : NameEnv → Name → Name → Name → TC (Name × Agda.Builtin.Reflection.Quantity × Type)

+
  makeHsCon env agdaName hsName c = do

+
    debug 10 "Making constructor %q : %q" c agdaName

+
    def agdaName' _ ← normalise (def agdaName [])

+
      where _ → typeErrorFmt "Failed to compute source type for %q" agdaName

+
    hsC  ← freshHsConName env hsName c

+
    cTy  ← getType c

+
    debug 10 "cTy = %t" cTy

+
    hsTy ← computeHsType agdaName' hsName cTy

+
    debug 10 "hsTy = %t" hsTy

+
    pure (hsC , quantity-ω , hsTy)

+

+
  makeHsData : NameEnv → Name → ℕ → List Name → TC Name

+
  makeHsData env agdaName nPars constrs = do

+
    hsName ← freshHsTypeName env agdaName

+
    declareData hsName 0 `Set

+
    hsCons ← mapM (makeHsCon env agdaName hsName) constrs

+
    defineData hsName hsCons

+
    pure hsName

+

+
  makeHsType : NameEnv → Name → TC Name

+
  makeHsType env d = getDefinition d >>= λ where

+
      (data-type pars cs) → makeHsData env d pars cs

+
      (record-type c fs)  → makeHsData env d 0 (c ∷ [])

+
      _                   → typeErrorFmt "%q is not a data or record type" d

+

+
  joinStrings : String → List String → String

+
  joinStrings sep ss = foldr _&_ "" $ intersperse sep ss

+

+
  compilePragma : Name → List Name → String

+
  compilePragma d cs = printf "= data %s (%s)" (showName d) (joinStrings " | " (map showName cs))

+

+
  renderHsTypeName : Name → String

+
  renderHsTypeName d = fromMaybe ("MAlonzo.Code." String.++ R.showName d) (lookup d specialHsTypes)

+

+
  renderHsType : Term → String

+
  renderHsArgs : List (Arg Term) → List String

+

+
  renderHsType (def (quote List) (_ ∷ vArg a ∷ [])) = printf "[%s]" (renderHsType a)

+
  renderHsType (def (quote Pair) (_ ∷ _ ∷ vArg a ∷ vArg b ∷ [])) = printf "(%s, %s)" (renderHsType a) (renderHsType b)

+
  renderHsType (def d []) = renderHsTypeName d

+
  renderHsType (def d vs) = printf "(%s %s)" (renderHsTypeName d) (joinStrings " " (renderHsArgs vs))

+
  renderHsType t = printf "(TODO: renderHsType %s)" (show t)

+

+
  renderHsArgs [] = []

+
  renderHsArgs (vArg a ∷ as) = renderHsType a ∷ renderHsArgs as

+
  renderHsArgs (_ ∷ as) = renderHsArgs as

+

+
  foreignPragma : Name → List Name → TC String

+
  foreignPragma d cs = do

+
    cons ← forM cs λ c → do

+
            tel , _ ← viewTy <$> getType c

+
            let args = map unAbs tel

+
            pure $ printf "%s %s" (showName c) (joinStrings " " $ renderHsArgs args)

+
    pure $ printf "data %s = %s\n  deriving (Show, Eq, Generic)"

+
                  (showName d) (joinStrings " | " cons)

+

+
  -- Record types

+
  foreignPragmaRec : NameEnv → Name → List Name → List Name → TC String

+
  foreignPragmaRec _ d [] _ = typeErrorFmt "impossible: %q is a record type with no constructors" d

+
  foreignPragmaRec env d (c ∷ _) fs = do

+
    tel , _ ← viewTy <$> withNormalisation true (getType c)

+
    let fNames = map (hsFieldName env) fs

+
    let args = map unAbs tel

+
        renderField f ty = printf "%s :: %s" f (renderHsType $ unArg ty)

+
        con = printf "%s {%s}" (showName c) (joinStrings ", " $ zipWith renderField fNames args)

+
    pure $ printf "data %s = %s\n  deriving (Show, Eq, Generic)"

+
                  (showName d) con

+

+
  hsImports : String

+
  hsImports = "import GHC.Generics (Generic)"

+

+
  -- Take the name of a simple data type and generate the COMPILE and

+
  -- FOREIGN pragmas to bind to Haskell.

+
  bindHsType : NameEnv → Name → Name → TC ⊤

+
  bindHsType env agdaName hsName = getDefinition hsName >>= λ where

+
    (data-type pars cs) → do

+
      pragmaForeign "GHC" hsImports

+
      pragmaCompile "GHC" hsName $ compilePragma hsName cs

+
      getDefinition agdaName >>= λ where

+
        (data-type _ _)    → pragmaForeign "GHC" =<< foreignPragma hsName cs

+
        (record-type _ fs) → pragmaForeign "GHC" =<< foreignPragmaRec env hsName cs (map unArg fs)

+
        _ → typeErrorFmt "%q is not a data or record type" agdaName

+
    _ → typeErrorFmt "%q is not a data type (impossible)" hsName

+

+
  computeProjections : ℕ → Name → TC (List Term)

+
  computeProjections npars c = do

+
    argTys , _ ← viewTy <$> getType c

+
    let is  = downFrom (length argTys)

+
        tel = map (λ where (abs x ty) → x , ty) argTys

+
        lhs = vArg (con c (map (λ where (i , abs x (arg info _)) → arg info (var i))

+
                               (drop npars (zip is argTys)))) ∷ []

+
    return $ map (λ i → pat-lam (clause tel lhs (var i []) ∷ []) []) (drop npars is)

+

+
  makeTypeAlias : Name → NameEnv → Term → TC ⊤

+
  makeTypeAlias agdaName env hole = do

+
    hsTy ← solveHsType (def agdaName [])

+
    pragmaForeign "GHC" $ printf "type %s = %s" (hsTypeName env agdaName) (renderHsType hsTy)

+
    unify hole (quote return ∙⟦ con (quote tt) [] ⟧)

+

+
-- * Exported macros

+

+
doAutoHsType : NameEnv → Name → Term → TC Term

+
doAutoHsType env d hole = do

+
  checkType hole (quote HasHsType ∙⟦ d ∙ ⟧)

+
  hs ← makeHsType env d

+
  debug 50 "  HsType %q = %q" d hs

+
  bindHsType env d hs

+
  unify hole (`λ⟦ proj (quote HasHsType.HsType) ⇒ hs ∙ ⟧)

+
  pure (hs ∙)

+

+
macro

+
  autoHsType : Name → Term → TC ⊤

+
  autoHsType d hole = _ <$ doAutoHsType emptyEnv d hole

+

+
  infix 0 autoHsType_⊣_

+
  autoHsType_⊣_ : Name → NameEnv → Term → TC ⊤

+
  autoHsType_⊣_ d env hole = _ <$ doAutoHsType env d hole

+

+
  infix 9 _↦_

+
  _↦_ : Name → String → Term → TC ⊤

+
  x ↦ s = unify (quote customName ∙⟦ lit (name x) ∣ lit (string s) ⟧)

+

+
  -- Generate a Haskell type synonym for the HsType of the given type

+
  -- Usage `unquoteDecl = do hsTypeSynonym Foo; hsTypeSynonym Bar`

+
  hsTypeAlias : Name → Term → TC ⊤

+
  hsTypeAlias agdaName = makeTypeAlias agdaName emptyEnv

+

+
  -- The only NameEnv that's useful here is `withName`.

+
  hsTypeAlias_⊣_ : Name → NameEnv → Term → TC ⊤

+
  hsTypeAlias agdaName ⊣ env = makeTypeAlias agdaName env

+

+
  -- * Macros for constructing and deconstructing generated types

+

+
  hsCon : Term → ℕ → Term → TC ⊤

+
  hsCon agdaTy i hole = do

+
    hsTy@(def hsName _) ← solveHsType agdaTy

+
      where _ → typeErrorFmt "Failed to compute HsType of %t" agdaTy

+
    cs ← getDefinition hsName >>= λ where

+
      (data-type _ cs)  → return cs

+
      _ → typeErrorFmt "Expected HsType %t to be a data type, but got %t" agdaTy hsTy

+
    c ← maybe′ return  (typeErrorFmt "%q has only %u constructors" hsName (length cs)) (cs ‼ i)

+
    unify hole (con c [])

+

+
  hsProj : Term → ℕ → Term → TC ⊤

+
  hsProj agdaTy i hole = do

+
    hsTy@(def hsName _) ← solveHsType agdaTy

+
      where _ → typeErrorFmt "Failed to compute HsType of %t" agdaTy

+
    prjs ← getDefinition hsName >>= λ where

+
      (data-type npars (c ∷ [])) → computeProjections npars c

+
      _ → typeErrorFmt "Expected HsType %t to be a single constructor data type, but got %t" agdaTy hsTy

+
    prj ← maybe′ return (typeErrorFmt "%q has only %u fields" hsName (length prjs)) (prjs ‼ i)

+
    target ← newMeta `Set

+
    checkType hole (pi (vArg hsTy) (abs "_" target))

+
    unify hole prj

+

+
  hsTyName : Term → Term → TC ⊤

+
  hsTyName agdaTy hole = do

+
    hsTy@(def hsName _) ← solveHsType agdaTy

+
      where _ → typeErrorFmt "Failed to compute HsType of %t" agdaTy

+
    unify hole (lit (name hsName))

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Derive.Show.html b/site/static/agda_html/Tactic.Derive.Show.html
new file mode 100644
index 000000000..aa0669514
--- /dev/null
+++ b/site/static/agda_html/Tactic.Derive.Show.html
@@ -0,0 +1,169 @@
+
+Tactic.Derive.Show
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
-- Deriving show.

+

+
{-# OPTIONS -v allTactics:100 #-}

+
{-# OPTIONS --safe #-}

+
module Tactic.Derive.Show where

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
open import Agda.Builtin.Reflection using (primShowQName)

+

+
import Data.List as L

+
import Data.List.NonEmpty as NE

+
import Reflection as R

+
open import Data.String using (fromList; toList) renaming (_++_ to _++S_)

+
open import Reflection.Tactic

+
open import Relation.Nullary.Negation

+
open import Relation.Nullary.Decidable

+

+
open import Class.DecEq

+
open import Class.Functor

+
open import Class.MonadTC.Instances

+
open import Class.Show.Core

+
open import Class.Traversable

+

+
open import Tactic.ClauseBuilder

+
open import Tactic.Derive (quote Show) (quote show)

+

+
instance

+
  _ = ContextMonad-MonadTC

+

+
open ClauseExprM

+

+
sLit = Term.lit ∘ Agda.Builtin.Reflection.Literal.string

+

+
showName : Name → String

+
showName n = let n' = R.showName n in dropModulePrefix n'

+
  where

+
    dropModulePrefix : String → String

+
    dropModulePrefix s = fromList $ reverse $ takeWhile (¬? ∘ ('.' ≟_)) $ reverse $ toList s

+

+
wrapWithPars : String → String

+
wrapWithPars s = "(" ++S s ++S ")"

+

+
genPars : Term → Term

+
genPars t = quote wrapWithPars ∙⟦ t ⟧

+

+
module _ (transName : Name → Maybe Name) where

+
  showFromTerm : Term → Term → Term

+
  showFromTerm (def n _) t with transName n

+
  ... | just n'    = def (quote show) (iArg (n' ∙) ∷ vArg t ∷ [])

+
  ... | nothing    = quote show ∙⟦ t ⟧

+
  showFromTerm _ t = quote show ∙⟦ t ⟧

+

+
  genShow : Name → List Term → Term

+
  genShow n [] = sLit (showName n)

+
  genShow n (x ∷ l) = quote _<+>_ ∙⟦ genShow n l ∣ genPars x ⟧

+

+
  patternToClause : SinglePattern → NE.List⁺ SinglePattern × TC (ClauseExpr ⊎ Maybe Term)

+
  patternToClause p@(l , arg _ (Pattern.con n _)) = NE.[ p ] , finishMatch do

+
    typeList ← traverse ⦃ Functor-List ⦄ (λ t → do T ← inferType t; return (T , t)) (applyDownFrom ♯ (length l))

+
    return $ genShow n (L.map (uncurry showFromTerm) $ reverse typeList)

+
  patternToClause p = NE.[ p ] , error1 "Error: not a con!"

+

+
module _ ⦃ _ : TCOptions ⦄ where

+
  derive-Show : List (Name × Name) → UnquoteDecl

+
  derive-Show = derive-Class 0 (λ transName → L.map (patternToClause transName))

+

+
private

+
  open import Tactic.Derive.TestTypes

+
  import Data.Nat.Show

+
  open import Tactic.Defaults

+

+
  unquoteDecl Show-Bool Show-ℤ Show-List Show-Maybe Show-ℕ Show-Sign Show-⊤ = derive-Show ((quote Bool , Show-Bool) ∷ (quote ℤ , Show-ℤ) ∷ (quote List , Show-List) ∷ (quote Maybe , Show-Maybe) ∷ (quote ℕ , Show-ℕ) ∷ (quote Sign , Show-Sign) ∷ (quote ⊤ , Show-⊤) ∷ [])

+

+
  unquoteDecl Show-Fin = derive-Show [ (quote Fin , Show-Fin) ]

+
  unquoteDecl Show-Vec = derive-Show [ (quote Vec , Show-Vec) ]

+

+
  unquoteDecl Show-These = derive-Show [ (quote These , Show-These) ]

+
  unquoteDecl Show-⊎ = derive-Show [ (quote _⊎_ , Show-⊎) ]

+

+
  unquoteDecl Show-E1 Show-E2 Show-E3 = derive-Show

+
    ((quote E1 , Show-E1) ∷ (quote E2 , Show-E2) ∷ (quote E3 , Show-E3) ∷ [])

+

+
  -- unquoteDecl Show-E4 = derive-Show [ (quote E4 , Show-E4) ]

+

+
  unquoteDecl Show-R1 Show-R2 = derive-Show

+
    ((quote R1 , Show-R1) ∷ (quote R2 , Show-R2) ∷ [])

+

+
  unquoteDecl Show-M₁ Show-M₂ = derive-Show $ (quote M₁ , Show-M₁) ∷ (quote M₂ , Show-M₂) ∷ []

+

+
  -- Expected: Show-Product Show-Term

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Derive.TestTypes.html b/site/static/agda_html/Tactic.Derive.TestTypes.html
new file mode 100644
index 000000000..e3193f146
--- /dev/null
+++ b/site/static/agda_html/Tactic.Derive.TestTypes.html
@@ -0,0 +1,161 @@
+
+Tactic.Derive.TestTypes
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+
{-# OPTIONS -v allTactics:100 #-}

+
module Tactic.Derive.TestTypes where

+

+
open import Data.Fin

+

+
open import Meta.Prelude

+
open import Meta.Init

+

+
data E0 : Set where

+

+
data E1 : Set where

+
  c1E1 : E1

+
  c2E1 : E1

+
  c3E1 : E1

+
  c4E1 : E1

+
  c5E1 : E1

+
  c6E1 : E1

+
  c7E1 : E1

+

+
data E2 {a} (A : Set a) : Set a where

+
  c1E2 : A → E2 A

+
  c2E2 : E2 A → E2 A

+

+
data E3 {a} (A : Set a) : Set a where

+
  c1E3 : List (E3 A) → Maybe (E3 A) → E3 A

+
  c2E3 : E3 A

+

+
data E4 : {n : ℕ} → Fin n → Set where

+
  c1E4 : ∀ {k} → E4 {suc k} zero

+
  c2E4 : ∀ {k} {l} → E4 {suc k} (suc l)

+

+
record R1 : Set where

+
  field f1R1 : E1

+
        f2R1 : E2 ℕ

+

+
record R2 {a} (A : Set a) : Set a where

+
  field f1R2 : E1

+
        f2R2 : E2 ℕ

+
        f3R2 : R1

+
        f4R2 : R1

+
        f5R2 : A

+
        f6R2 : A

+

+
data M₁ : Set

+
data M₂ : Set

+
data M₁ where

+
  m₁ : M₁

+
  m₂→₁ : M₂ → M₁

+
data M₂ where

+
  m₂ : M₂

+
  m₁→₂ : M₁ → M₂

+

+
AllTestTypes : List Name

+
AllTestTypes = quote E0 ∷ quote E1 ∷ quote E2 ∷ quote E3 ∷ quote R1 ∷ quote R2 ∷ quote M₁ ∷ quote M₂ ∷ []

+

+
open import Data.Bool using (Bool) public

+
open import Data.Char using (Char) public

+
open import Data.Container using (Container) public

+
open import Data.Container using (Container) public

+
open import Data.Digit using (Digit) public

+
open import Data.Empty using (⊥) public

+
open import Data.Fin using (Fin) public

+
open import Data.Float using (Float) public

+
open import Data.Integer using (ℤ) public

+
open import Data.List using (List) public

+
open import Data.Maybe using (Maybe) public

+
open import Data.Nat using (ℕ) public

+
open import Data.Product using (Σ) public

+
open import Data.Rational using (ℚ) public

+
open import Data.Record using (Record) public

+
open import Data.Refinement using (Refinement) public

+
open import Data.Sign using (Sign) public

+
open import Data.String using (String) public

+
open import Data.Sum using (_⊎_) public

+
open import Data.These using (These) public

+
open import Data.Unit using (⊤) public

+
open import Data.Universe using (Universe) public

+
open import Data.Vec using (Vec) public

+
open import Data.W using (W) public

+
open import Data.Word64 using (Word64) public

+

+
stdlibTypes : List Name

+
stdlibTypes = quote Bool ∷ quote Container ∷ quote Fin ∷ quote ℤ ∷ quote List ∷ quote Maybe ∷ quote ℕ ∷ quote Σ ∷ quote ℚ ∷ quote Record ∷ quote Refinement ∷ quote Sign ∷ quote _⊎_ ∷ quote These ∷ quote ⊤ ∷ quote Universe ∷ quote Vec ∷ quote W ∷ []

+

\ No newline at end of file
diff --git a/site/static/agda_html/Tactic.Derive.html b/site/static/agda_html/Tactic.Derive.html
new file mode 100644
index 000000000..7f555fcab
--- /dev/null
+++ b/site/static/agda_html/Tactic.Derive.html
@@ -0,0 +1,198 @@
+
+Tactic.Derive
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
-- Generic type class dervations. Works with mutually recursive types

+
-- and can use mutual recursion to satisfy the termination

+
-- checker. Writing an actual derivation strategy then does not

+
-- require dealing with any mutual recursion, it is all handled here.

+
--

+
-- TODO: This breaks with:

+
-- - mutual recursion that nests too deep (i.e. deeper than 1, e.g. Term)

+
-- - indexed datatypes that require absurd clauses (e.g. Vec)

+
--

+
-- TODO: support type classes with more than one field

+
--

+
-- TODO: enable using existing code for instance when mutually recursing

+
--

+
-- TODO: Requires K to pass the termination checker. Maybe we can do without somehow.

+

+
{-# OPTIONS -v allTactics:100 #-}

+
{-# OPTIONS --safe --without-K #-}

+
open import Meta.Init

+

+
module Tactic.Derive (className : Name) (projName : Name) where

+

+
open import Meta.Prelude

+

+
open import Agda.Builtin.Reflection using () renaming (primShowQName to showName)

+

+
import Data.List as L

+
import Data.List.NonEmpty as NE

+
import Data.String as S

+
open import Data.Maybe using (fromMaybe)

+
open import Reflection.Tactic

+
open import Reflection.Utils

+
open import Reflection.Utils.TCI

+
open import Relation.Nullary.Decidable

+
open import Tactic.ClauseBuilder

+

+
open import Class.DecEq

+
open import Class.Monad

+
open import Class.Traversable

+
open import Class.Functor

+
open import Class.MonadReader.Instances

+
open import Class.MonadTC.Instances

+

+
instance

+
  _ = ContextMonad-MonadTC

+
  _ = Functor-M

+

+
open ClauseExprM

+

+
genClassType : ℕ → Name → Maybe Name → TC Type

+
genClassType arity dName wName = do

+
  params ← getParamsAndIndices dName

+
  adjParams ← adjustParams $ take (length params ∸ arity) params

+
  debugLog1 "AdjustedParams: "

+
  logTelescope (L.map ((λ where (abs s x) → just s , x) ∘ proj₁) adjParams)

+
  ty ← applyWithVisibility dName (L.map ♯ (trueIndices adjParams))

+
  return $ modifyClassType wName (L.map proj₁ adjParams , ty)

+
  where

+
    adjustParams : List (Abs (Arg Type)) → TC (List ((Abs (Arg Type)) × Bool))

+
    adjustParams [] = return []

+
    adjustParams (abs x (arg _ t) ∷ l) = do

+
      a ← (if_then [ (abs "_" (iArg (className ∙⟦ ♯ 0 ⟧)) , false) ] else []) <$> isNArySort arity t

+
      ps ← extendContext (x , hArg t) (adjustParams l)

+
      let ps' = flip L.map ps λ where

+
        (abs s (arg i t) , b) → (abs s (arg i (mapVars (_+ (if b then length a else 0)) t)) , b)

+
      return (((abs x (hArg t) , true) ∷ a) ++ ps')

+

+
    trueIndices : {A : Set} → List (A × Bool) → List ℕ

+
    trueIndices [] = []

+
    trueIndices (x ∷ l) = if proj₂ x then length l ∷ trueIndices l else trueIndices l

+

+
    modifyClassType : Maybe Name → TypeView → Type

+
    modifyClassType nothing  (tel , ty) = tyView (tel , className ∙⟦ ty ⟧)

+
    modifyClassType (just n) (tel , ty) = tyView (tel , className ∙⟦ n ∙⟦ ty ⟧ ⟧)

+

+
lookupName : List (Name × Name) → Name → Maybe Name

+
lookupName = lookupᵇ (λ n n' → ⌊ n ≟ n' ⌋)

+

+
-- Look at the constructors of the argument and return all types that

+
-- recursively contain it. This isn't very clever right now.

+
genMutualHelpers : Name → TC (List Name)

+
genMutualHelpers n = do

+
  tys ← L.map (unArg ∘ unAbs) <$> (L.concatMap (proj₁ ∘ viewTy ∘ proj₂) <$> getConstrs n)

+
  return $ deduplicate _≟_ $ L.mapMaybe helper tys

+
  where

+
    helper : Type → Maybe Name

+
    helper (def n' args) =

+
      if L.any (λ where (arg _ (def n'' _)) → ⌊ n ≟ n'' ⌋ ; _ → false) args

+
      then just n' else nothing

+
    helper _ = nothing

+

+
module _ (arity : ℕ) (genCe : (Name → Maybe Name) → List SinglePattern → List (NE.List⁺ SinglePattern × TC (ClauseExpr ⊎ Maybe Term))) where

+
  -- Generate the declaration & definition of a particular derivation.

+
  --

+
  -- Takes a dictionary (for mutual recursion), a wrapper (also for

+
  -- mutual recursion), the name of the original type we want to derive

+
  -- Show for and the name we want to define Show originally at.

+
  deriveSingle : List (Name × Name) → Name → Name → Maybe Name → TC (Arg Name × Type × List Clause)

+
  deriveSingle transName dName iName wName = inDebugPath "DeriveSingle" do

+
    debugLog ("For: " ∷ᵈ dName ∷ᵈ [])

+
    goalTy ← genClassType arity dName wName

+
    ps ← constructorPatterns' (fromMaybe dName wName ∙)

+
    -- TODO: find a good way of printing this

+
    --debugLogᵐ ("Constrs: " ∷ᵈᵐ ps ᵛⁿ ∷ᵈᵐ []ᵐ)

+
    cs ← local (λ c → record c { goal = inj₂ goalTy }) $

+
      singleMatchExpr ([] , iArg (Pattern.proj projName)) $ contMatch $ multiMatchExprM $

+
        genCe (lookupName transName) ps

+
    let defName = maybe (maybe vArg (iArg iName) ∘ lookupName transName) (iArg iName) wName

+
    return (defName , goalTy , clauseExprToClauses cs)

+

+
  deriveMulti : Name × Name × List Name → TC (List (Arg Name × Type × List Clause))

+
  deriveMulti (dName , iName , hClasses) = do

+
    hClassNames ← traverse ⦃ Functor-List ⦄

+
      (λ cn → freshName (showName className S.++ "-" S.++ showName cn S.++ showName dName)) hClasses

+
    traverse ⦃ Functor-List ⦄ (deriveSingle (L.zip hClasses hClassNames) dName iName) (nothing ∷ L.map just hClasses)

+

+
  derive-Class : ⦃ TCOptions ⦄ → List (Name × Name) → UnquoteDecl

+
  derive-Class l = initUnquoteWithGoal (className ∙) $

+
    declareAndDefineFuns =<< concat <$> traverse ⦃ Functor-List ⦄ helper l

+
    where

+
      helper : Name × Name → TC (List (Arg Name × Type × List Clause))

+
      helper (a , b) = do hs ← genMutualHelpers a ; deriveMulti (a , b , hs)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Text.Format.Generic.html b/site/static/agda_html/Text.Format.Generic.html
new file mode 100644
index 000000000..991c415aa
--- /dev/null
+++ b/site/static/agda_html/Text.Format.Generic.html
@@ -0,0 +1,204 @@
+
+Text.Format.Generic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Format strings for Printf and Scanf

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Text.Format.Generic where

+

+
open import Level using (0ℓ)

+
open import Effect.Applicative

+
open import Data.Char.Base using (Char)

+
open import Data.List.Base as List

+
open import Data.Maybe.Base as Maybe

+
open import Data.Nat.Base

+
open import Data.Product.Base using (_,_)

+
open import Data.Product.Nary.NonDependent

+
open import Data.Sum.Base

+
open import Data.String.Base

+
import Data.Sum.Effectful.Left as Sumₗ

+
open import Function.Base

+
open import Function.Nary.NonDependent using (0ℓs; Sets)

+
open import Function.Strict

+

+
------------------------------------------------------------------------

+
-- Format specifications.

+
-- Defines the supported %-codes.

+

+
record FormatSpec : Set₁ where

+
  field

+
    ArgChunk : Set

+
    ArgType  : ArgChunk → Set

+
    lexArg   : Char → Maybe ArgChunk

+

+
module _ where

+
  open FormatSpec

+

+
  -- Left-biased union of format specs.

+
  unionSpec : FormatSpec → FormatSpec → FormatSpec

+
  unionSpec spec₁ spec₂ .ArgChunk  = spec₁ .ArgChunk ⊎ spec₂ .ArgChunk

+
  unionSpec spec₁ spec₂ .ArgType (inj₁ a) = spec₁ .ArgType a

+
  unionSpec spec₁ spec₂ .ArgType (inj₂ a) = spec₂ .ArgType a

+
  unionSpec spec₁ spec₂ .lexArg  c =

+
    Maybe.map inj₁ (spec₁ .lexArg c) <∣>

+
    Maybe.map inj₂ (spec₂ .lexArg c)

+

+
module Format (spec : FormatSpec) where

+

+
  open FormatSpec spec

+

+
  ----------------------------------------------------------------------

+
  -- Basic types

+

+
  data Chunk : Set where

+
    Arg : ArgChunk → Chunk

+
    Raw : String → Chunk

+

+
  Format : Set

+
  Format = List Chunk

+

+
  ----------------------------------------------------------------------

+
  -- Semantics

+

+
  size : Format → ℕ

+
  size = List.sum ∘′ List.map λ { (Raw _) → 0; _ → 1 }

+

+
  -- Meaning of a format as a list of value types

+

+
  ⟦_⟧ : (fmt : Format) → Sets (size fmt) 0ℓs

+
  ⟦ []          ⟧ = _

+
  ⟦ Arg a  ∷ cs ⟧ = ArgType a , ⟦ cs ⟧

+
  ⟦ Raw _  ∷ cs ⟧ =             ⟦ cs ⟧

+

+
  ----------------------------------------------------------------------

+
  -- Lexer: from Strings to Formats

+

+
  -- Lexing may fail. To have a useful error message, we defined the

+
  -- following enumerated type

+

+
  data Error : Set where

+
    UnexpectedEndOfString : String → Error

+
    -- ^ expected a type declaration; found an empty string

+
    InvalidType : String → Char → String → Error

+
    -- ^ invalid type declaration

+
    --   return a focus: prefix processed, character causing failure, rest

+

+
  lexer : String → Error ⊎ List Chunk

+
  lexer input = loop [] [] (toList input) where

+

+
    open RawApplicative (Sumₗ.applicative Error 0ℓ)

+

+
    -- Type synonyms used locally to document the code

+
    RevWord = List Char -- Mere characters accumulated so far

+
    Prefix  = RevWord   -- Prefix of the input String already read

+

+
    toRevString : RevWord → String

+
    toRevString = fromList ∘′ reverse

+

+
    -- Push a Raw token if we have accumulated some mere characters

+
    push : RevWord → List Chunk → List Chunk

+
    push [] ks = ks

+
    push cs ks = Raw (toRevString cs) ∷ ks

+

+
    -- Main loop

+
    loop : RevWord → Prefix → List Char → Error ⊎ List Chunk

+
    type :           Prefix → List Char → Error ⊎ List Chunk

+

+
    loop acc bef []               = pure (push acc [])

+
    -- escaped '%' character: treat like a mere character

+
    loop acc bef ('%' ∷ '%' ∷ cs) = loop ('%' ∷ acc) ('%' ∷ '%' ∷ bef) cs

+
    -- non escaped '%': type declaration following

+
    loop acc bef ('%' ∷ cs)       = push acc <$> type ('%' ∷ bef) cs

+
    -- mere character: push onto the accumulator

+
    loop acc bef (c ∷ cs)         = loop (c ∷ acc) (c ∷ bef) cs

+

+
    type bef []       = inj₁ (UnexpectedEndOfString input)

+
    type bef (c ∷ cs) = _∷_ <$> chunk c ⊛ loop [] (c ∷ bef) cs where

+

+
      chunk : Char → Error ⊎ Chunk

+
      chunk c =

+
        case lexArg c of λ where

+
          (just ch) → pure (Arg ch)

+
          nothing   →

+
            force′ (toRevString bef) λ prefix →

+
            force′ (fromList cs)     λ suffix →

+
            inj₁ (InvalidType prefix c suffix)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Text.Format.html b/site/static/agda_html/Text.Format.html
new file mode 100644
index 000000000..a1e28ea94
--- /dev/null
+++ b/site/static/agda_html/Text.Format.html
@@ -0,0 +1,134 @@
+
+Text.Format
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Format strings for Printf and Scanf

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Text.Format where

+

+
open import Data.Maybe.Base

+
open import Text.Format.Generic

+

+
-- Formatted types

+
open import Data.Char.Base using (Char)

+
open import Data.Integer.Base using (ℤ)

+
open import Data.Float.Base using (Float)

+
open import Data.Nat.Base using (ℕ)

+
open import Data.String.Base using (String)

+

+
------------------------------------------------------------------------

+
-- Basic types

+

+
data ArgChunk : Set where

+
  ℕArg ℤArg FloatArg CharArg StringArg : ArgChunk

+

+
------------------------------------------------------------------------

+
-- Semantics

+

+
ArgType : (fmt : ArgChunk) → Set

+
ArgType ℕArg      = ℕ

+
ArgType ℤArg      = ℤ

+
ArgType FloatArg  = Float

+
ArgType CharArg   = Char

+
ArgType StringArg = String

+

+
lexArg : Char → Maybe ArgChunk

+
lexArg 'd' = just ℤArg

+
lexArg 'i' = just ℤArg

+
lexArg 'u' = just ℕArg

+
lexArg 'f' = just FloatArg

+
lexArg 'c' = just CharArg

+
lexArg 's' = just StringArg

+
lexArg _   = nothing

+

+
formatSpec : FormatSpec

+
formatSpec .FormatSpec.ArgChunk = ArgChunk

+
formatSpec .FormatSpec.ArgType  = ArgType

+
formatSpec .FormatSpec.lexArg   = lexArg

+

+
open Format formatSpec public

+

+
pattern `ℕ      = Arg ℕArg

+
pattern `ℤ      = Arg ℤArg

+
pattern `Float  = Arg FloatArg

+
pattern `Char   = Arg CharArg

+
pattern `String = Arg StringArg

+

\ No newline at end of file
diff --git a/site/static/agda_html/Text.Printf.Generic.html b/site/static/agda_html/Text.Printf.Generic.html
new file mode 100644
index 000000000..dd1e8710c
--- /dev/null
+++ b/site/static/agda_html/Text.Printf.Generic.html
@@ -0,0 +1,159 @@
+
+Text.Printf.Generic
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Generic printf function.

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Text.Printf.Generic where

+

+
open import Level using (Level; 0ℓ; _⊔_; Lift)

+
open import Data.List.Base as List using (List; []; _∷_)

+
open import Data.Char.Base

+
open import Data.Maybe.Base hiding (map)

+
open import Data.Nat.Base using (ℕ)

+
open import Data.Product.Base hiding (map)

+
open import Data.Product.Nary.NonDependent

+
open import Data.String.Base using (String)

+
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)

+
open import Data.Unit.Base using (⊤)

+
open import Function.Nary.NonDependent

+
open import Function.Base

+

+
open import Text.Format.Generic

+

+
private

+
  variable

+
    ℓ : Level

+
    A B : Set ℓ

+

+
------------------------------------------------------------------------

+
-- Printf argument specifications.

+
-- Defines the rendering of chunks.

+

+
record PrintfSpec {ℓ} (spec : FormatSpec) (A : Set ℓ) : Set (Level.suc 0ℓ ⊔ ℓ) where

+

+
  open FormatSpec spec public

+

+
  field

+
    renderArg : ∀ arg → ArgType arg → A

+
    renderStr : String → A

+

+
module Type (spec : FormatSpec) where

+

+
  open Format spec renaming (Error to FormatError)

+

+
  record Error (e : FormatError) : Set ℓ where

+

+
  private

+

+
    Size : FormatError ⊎ Format → ℕ

+
    Size (inj₁ err) = 0

+
    Size (inj₂ fmt) = size fmt

+

+
  Printf : ∀ pr → Set ℓ → Set (ℓ ⊔ ⨆ (Size pr) 0ℓs)

+
  Printf (inj₁ err) _ = Error err

+
  Printf (inj₂ fmt) B = Arrows _ ⟦ fmt ⟧ B

+

+
  map : ∀ pr → (A → B) → Printf pr A → Printf pr B

+
  map (inj₁ err) f p = _

+
  map (inj₂ fmt) f p = mapₙ _ f p

+

+
module Render {spec : FormatSpec} (render : PrintfSpec spec A) where

+

+
  open PrintfSpec render

+
  open Type spec

+

+
  open Format spec renaming (Error to FormatError)

+

+
  assemble : ∀ fmt → Product⊤ _ ⟦ fmt ⟧ → List A

+
  assemble [] vs                    = []

+
  assemble (Arg a   ∷ fmt) (x , vs) = renderArg a x ∷ assemble fmt vs

+
  assemble (Raw str ∷ fmt) vs       = renderStr str ∷ assemble fmt vs

+

+
  private

+

+
    printf′ : ∀ pr → Printf pr (List A)

+
    printf′ (inj₁ err) = _

+
    printf′ (inj₂ fmt) = curry⊤ₙ _ (assemble fmt)

+

+
  printf : (input : String) → Printf (lexer input) (List A)

+
  printf input = printf′ (lexer input)

+

\ No newline at end of file
diff --git a/site/static/agda_html/Text.Printf.html b/site/static/agda_html/Text.Printf.html
new file mode 100644
index 000000000..b5beb0829
--- /dev/null
+++ b/site/static/agda_html/Text.Printf.html
@@ -0,0 +1,111 @@
+
+Text.Printf
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
------------------------------------------------------------------------

+
-- The Agda standard library

+
--

+
-- Printf

+
------------------------------------------------------------------------

+

+
{-# OPTIONS --cubical-compatible --safe #-}

+

+
module Text.Printf where

+

+
open import Data.String.Base using (String; fromChar; concat)

+
open import Function.Base using (id)

+

+
import Data.Integer.Show as ℤ

+
import Data.Float.Base   as Float

+
import Data.Nat.Show     as ℕ

+

+
open import Text.Format as Format hiding (Error)

+
open import Text.Printf.Generic

+

+
printfSpec : PrintfSpec formatSpec String

+
printfSpec .PrintfSpec.renderArg ℕArg      = ℕ.show

+
printfSpec .PrintfSpec.renderArg ℤArg      = ℤ.show

+
printfSpec .PrintfSpec.renderArg FloatArg  = Float.show

+
printfSpec .PrintfSpec.renderArg CharArg   = fromChar

+
printfSpec .PrintfSpec.renderArg StringArg = id

+
printfSpec .PrintfSpec.renderStr           = id

+

+
module Printf = Type formatSpec

+
open Printf public hiding (map)

+
open Render printfSpec public renaming (printf to gprintf)

+

+
printf : (fmt : String) → Printf (lexer fmt) String

+
printf fmt = Printf.map (lexer fmt) concat (gprintf fmt)

+

\ No newline at end of file
diff --git a/site/static/agda_html/abstract-set-theory.FiniteSetTheory.html b/site/static/agda_html/abstract-set-theory.FiniteSetTheory.html
new file mode 100644
index 000000000..85f34eab4
--- /dev/null
+++ b/site/static/agda_html/abstract-set-theory.FiniteSetTheory.html
@@ -0,0 +1,205 @@
+
+abstract-set-theory.FiniteSetTheory
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe #-}

+

+
module abstract-set-theory.FiniteSetTheory where

+

+
open import abstract-set-theory.Prelude

+

+
import Algebra

+
import Axiom.Set.List as L

+
open import Axiom.Set

+
open import Relation.Binary using (_Preserves_⟶_)

+

+
opaque

+
  List-Model : Theory {0ℓ}

+
  List-Model = L.List-Model

+
  List-Modelᶠ : Theoryᶠ

+
  List-Modelᶠ = L.List-Modelᶠ

+
  List-Modelᵈ : Theoryᵈ

+
  List-Modelᵈ = L.List-Modelᵈ

+

+
private variable

+
  A A' B C : Set

+

+
open Theoryᵈ List-Modelᵈ public

+
  renaming (Set to ℙ_; filter to filterˢ?; map to mapˢ; ∅ to ∅ˢ; ❴_❵ to ❴_❵ˢ)

+
  hiding (_∈_; _∉_)

+

+
open import Axiom.Set.Map th public

+
  renaming ( Map to infixr 1 _⇀_

+
           ; filterᵐ to filterᵐ?; filterKeys to filterKeys?; _∣^'_ to _∣^'?_ )

+

+
open import Axiom.Set.Factor List-Model public

+
open import Axiom.Set.Map.Dec List-Modelᵈ public

+
open import Axiom.Set.Rel th public hiding (_∣'_; _∣^'_; dom; range)

+
open import Axiom.Set.Sum th public

+
open import Axiom.Set.TotalMap th public

+
open import Axiom.Set.TotalMapOn th

+
open import Class.IsSet th public

+
open import Class.HasEmptySet th public

+
open import Class.HasSingleton th public

+

+
open import Axiom.Set.Properties th using (card-≡ᵉ)

+

+
module _ ⦃ _ : DecEq A ⦄ where

+
  open Restriction {A} ∈-sp public

+
    renaming (_∣_ to _∣ʳ_; _∣_ᶜ to _∣ʳ_ᶜ)

+

+
  open Corestriction {A} ∈-sp public

+
    renaming (_∣^_ to _∣^ʳ_; _∣^_ᶜ to _∣^ʳ_ᶜ) public

+

+
  open Restrictionᵐ {A} ∈-sp public

+
  open Corestrictionᵐ {A} ∈-sp public

+
  open Unionᵐ {A} ∈-sp public

+
  open Intersection {A} ∈-sp public

+
  open Lookupᵐ {A} ∈-sp public

+
  open Lookupᵐᵈ {A} ∈-sp public

+

+
module _ ⦃ _ : DecEq A ⦄ ⦃ _ : DecEq B ⦄ where

+
  open Intersectionᵐ {A} {B} ∈-sp public

+
  open IndexedSumUnionᵐ {A} {B} ∈-sp (_∈? _) public

+

+
module Properties where

+
  open import Axiom.Set.Properties th public

+
  module _ ⦃ _ : DecEq A ⦄ where

+
    open Intersectionᵖ {A} ∈-sp public

+

+
opaque

+
  unfolding List-Model

+

+
  to-sp : {A : Type} (P : A → Type) → ⦃ P ⁇¹ ⦄ → specProperty P

+
  to-sp _ = dec¹

+

+
  finiteness : ∀ (X : Theory.Set th A) → finite X

+
  finiteness = Theoryᶠ.finiteness List-Modelᶠ

+

+
  lengthˢ : ∀ {𝕊} ⦃ _ : DecEq A ⦄ ⦃ _ : IsSet 𝕊 A ⦄ → 𝕊 → ℕ

+
  lengthˢ X = Theoryᶠ.lengthˢ List-Modelᶠ (toSet X)

+

+
  lengthˢ-≡ᵉ :  ∀ {𝕊} ⦃ _ : DecEq A ⦄ ⦃ _ : IsSet 𝕊 A ⦄ → (X Y : 𝕊)

+
    → toSet X ≡ᵉ toSet Y

+
    → lengthˢ X ≡ lengthˢ Y

+
  lengthˢ-≡ᵉ X Y X≡Y =

+
    card-≡ᵉ (-, Theoryᶠ.DecEq⇒strongly-finite List-Modelᶠ (toSet X))

+
            (-, Theoryᶠ.DecEq⇒strongly-finite List-Modelᶠ (toSet Y)) X≡Y

+

+
  setToList : ℙ A → List A

+
  setToList = id

+

+
  instance

+
    DecEq-ℙ : ⦃ _ : DecEq A ⦄ → DecEq (ℙ A)

+
    DecEq-ℙ = L.Decˡ.DecEq-Set

+

+
    Show-ℙ : ⦃ _ : Show A ⦄ → Show (ℙ A)

+
    Show-ℙ .show = λ x → Show-finite .show (x , (finiteness x))

+

+
_ᶠᵐ : A ⇀ B → FinMap A B

+
(R , uniq) ᶠᵐ = (R , uniq , finiteness _)

+

+
_ᶠˢ : ℙ A → FinSet A

+
X ᶠˢ = X , finiteness _

+

+
filterˢ : (P : A → Type) ⦃ _ : P ⁇¹ ⦄ → ℙ A → ℙ A

+
filterˢ P = filterˢ? (to-sp P)

+

+
filterᵐ : (P : A × B → Type) ⦃ _ : P ⁇¹ ⦄ → (A ⇀ B) → (A ⇀ B)

+
filterᵐ P = filterᵐ? (to-sp P)

+

+
filterKeys : (P : A → Type) ⦃ _ : P ⁇¹ ⦄ → (A ⇀ B) → (A ⇀ B)

+
filterKeys P = filterKeys? (to-sp P)

+

+
_∣^'_ : A ⇀ B → (P : B → Type) ⦃ _ : P ⁇¹ ⦄ → A ⇀ B

+
s ∣^' P = s ∣^'? to-sp P

+

+
indexedSumᵛ' : ⦃ DecEq A ⦄ → ⦃ DecEq B ⦄ → ⦃ CommutativeMonoid 0ℓ 0ℓ C ⦄ → (B → C) → A ⇀ B → C

+
indexedSumᵛ' f m = indexedSumᵛ f (m ᶠᵐ)

+

+
indexedSum' : ⦃ DecEq A ⦄ → ⦃ CommutativeMonoid 0ℓ 0ℓ B ⦄ → (A → B) → ℙ A → B

+
indexedSum' f s = indexedSum f (s ᶠˢ)

+

+
syntax indexedSumᵛ' (λ a → x) m = ∑[ a ← m ] x

+
syntax indexedSum'  (λ a → x) m = ∑ˢ[ a ← m ] x

+

+
module _ ⦃ _ : DecEq A ⦄ ⦃ _ : DecEq B ⦄ ⦃ _ : CommutativeMonoid 0ℓ 0ℓ C ⦄ where

+

+
  aggregateBy : ⦃ DecEq C ⦄ → Rel A B → A ⇀ C → B ⇀ C

+
  aggregateBy R m = mapFromFun (λ b → ∑[ x ← m ∣ dom (R ∣^ʳ ❴ b ❵) ] x) (range R)

+

+
  indexedSumᵛ'-cong : ∀ {f : B → C} → indexedSumᵛ' f Preserves _≡ᵉ_ on proj₁ ⟶ CommutativeMonoid._≈_ it

+
  indexedSumᵛ'-cong {x = x} {y} = indexedSum-cong {A = A × B} {x = (x ˢ) ᶠˢ} {(y ˢ) ᶠˢ}

+

\ No newline at end of file
diff --git a/site/static/agda_html/abstract-set-theory.Prelude.html b/site/static/agda_html/abstract-set-theory.Prelude.html
new file mode 100644
index 000000000..c2ea2a4ed
--- /dev/null
+++ b/site/static/agda_html/abstract-set-theory.Prelude.html
@@ -0,0 +1,139 @@
+
+abstract-set-theory.Prelude
+             
+             
+             

+               

+                 Back
+                 
+               

+               

+                 
+                   
+                     
+                   
+                 
+                 
+               

+             

+             

+               

+                 

+                   
+                   

+                 

+               

+             

+             

+               
+               

+             
{-# OPTIONS --safe --without-K #-}

+

+
module abstract-set-theory.Prelude where

+

+
open import Agda.Primitive using (lzero) renaming (Set to Type) public

+

+
open import Level public

+
  hiding (lower)

+
  renaming (_⊔_ to _⊔ˡ_; suc to sucˡ; zero to zeroˡ)

+
open import Function public

+

+
open import Data.Bool public

+
  hiding (_≟_; _≤_; _≤?_; _<_; _<?_; if_then_else_)

+
open import Data.Empty public

+
open import Data.List public

+
  hiding (align; alignWith; filter; fromMaybe; map; zip; zipWith)

+
open import Data.List.Membership.Propositional public

+
  using () renaming (_∈_ to _∈ˡ_; _∉_ to _∉ˡ_)

+
open import Data.Maybe public

+
  hiding (_>>=_; align; alignWith; ap; fromMaybe; map; zip; zipWith)

+
open import Data.Unit public

+
  using (⊤; tt)

+
open import Data.Unit.Polymorphic public

+
  using ()

+
  renaming (⊤ to ⊤↑; tt to tt↑)

+
instance Poly-tt = tt↑

+
open import Data.Sum public

+
  hiding (assocʳ; assocˡ; map; map₁; map₂; reduce; swap)

+
open import Data.Product public

+
  hiding (assocʳ; assocˡ; map; map₁; map₂; map₂′; swap; _<*>_)

+
open import Data.Nat public

+
  hiding (_≟_; _≤_; _≤?_; _<_; _<?_; _≤ᵇ_; _≡ᵇ_; _≥_; _>_; less-than-or-equal)

+
  renaming (_+_ to _+ℕ_)

+
open import Data.Integer as ℤ public

+
  using (ℤ)

+
  renaming (_+_ to _+ℤ_)

+
open import Data.String public

+
  using (String; _<+>_)

+

+
open import Relation.Nullary public

+
open import Relation.Nullary.Negation public

+
open import Relation.Nullary.Decidable public

+
  using (Dec; yes; no; dec-yes; dec-no; ⌊_⌋; ¬?; toWitness; fromWitness)

+
  renaming (map′ to mapDec)

+
open import Relation.Unary public

+
  using (Pred) renaming (Decidable to Decidable¹)

+
open import Relation.Binary public

+
  using () renaming (Decidable to Decidable²)

+
open import Relation.Binary.PropositionalEquality public

+
  hiding (preorder; isPreorder; setoid; [_])

+

+
open import Class.Functor public

+
  renaming (fmap to map)

+
open import Class.Bifunctor public

+
open import Class.Semigroup public

+
open import Class.Monoid public

+
open import Class.CommutativeMonoid public

+
open import Class.Applicative public

+
open import Class.Monad public

+
open import Class.DecEq public; instance DecEq-×′ = DecEq-×

+
open import Class.Decidable public

+
open import Class.Show public

+

\ No newline at end of file
diff --git a/site/static/agda_html/agda.js b/site/static/agda_html/agda.js
index 81e777d34..e45a8282c 100644
--- a/site/static/agda_html/agda.js
+++ b/site/static/agda_html/agda.js
@@ -81,50 +81,54 @@ if (searchInput && searchResults) {
 
   // Function to scroll to a specific line and highlight terms
   function scrollToLine(lineNumber, searchTerm) {
-    // Wait for the next frame to ensure content is rendered
-    requestAnimationFrame(() => {
-      const targetElement = document.querySelector('#L' + lineNumber);
-      if (targetElement) {
-        // Add a small delay to ensure content is fully rendered
-        setTimeout(() => {
-          targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' });
-          targetElement.classList.add('highlight-line');
-          setTimeout(() => targetElement.classList.remove('highlight-line'), 2000);
-          
-          // Highlight all matching terms on the page
-          const content = document.querySelector('.agda-content');
-          if (content && searchTerm) {
-            const regex = new RegExp(searchTerm, 'gi');
-            const walker = document.createTreeWalker(content, NodeFilter.SHOW_TEXT, null, false);
-            const nodesToHighlight = [];
-            
-            while (walker.nextNode()) {
-              const node = walker.currentNode;
-              if (node.textContent.match(regex)) {
-                nodesToHighlight.push(node);
-              }
+    const targetElement = document.querySelector('.Agda .line:nth-child(' + (lineNumber - 1) + ')');
+    if (targetElement) {
+        // Remove any existing highlights
+        document.querySelectorAll('.line.highlight-line').forEach(line => {
+            line.classList.remove('highlight-line');
+        });
+
+        // Add highlight to target line
+        targetElement.classList.add('highlight-line');
+        
+        // Scroll to the line
+        targetElement.scrollIntoView({ behavior: 'smooth', block: 'center' });
+
+        // Highlight matching terms
+        if (searchTerm) {
+            const content = document.querySelector('.agda-content');
+            if (content) {
+                const regex = new RegExp(searchTerm, 'gi');
+                const walker = document.createTreeWalker(content, NodeFilter.SHOW_TEXT, null, false);
+                const nodesToHighlight = [];
+                
+                while (walker.nextNode()) {
+                    const node = walker.currentNode;
+                    if (node.textContent.match(regex)) {
+                        nodesToHighlight.push(node);
+                    }
+                }
+                
+                nodesToHighlight.forEach(node => {
+                    const span = document.createElement('span');
+                    span.className = 'search-term-highlight';
+                    span.innerHTML = node.textContent.replace(regex, match => '' + match + '');
+                    node.parentNode.replaceChild(span, node);
+                });
+                
+                // Remove highlights after 5 seconds
+                setTimeout(() => {
+                    const highlights = document.querySelectorAll('.search-term-highlight');
+                    highlights.forEach(highlight => {
+                        const text = highlight.textContent;
+                        const textNode = document.createTextNode(text);
+                        highlight.parentNode.replaceChild(textNode, highlight);
+                    });
+                    targetElement.classList.remove('highlight-line');
+                }, 5000);
             }
-            
-            nodesToHighlight.forEach(node => {
-              const span = document.createElement('span');
-              span.className = 'search-term-highlight';
-              span.innerHTML = node.textContent.replace(regex, match => '' + match + '');
-              node.parentNode.replaceChild(span, node);
-            });
-            
-            // Remove highlights after 5 seconds
-            setTimeout(() => {
-              const highlights = document.querySelectorAll('.search-term-highlight');
-              highlights.forEach(highlight => {
-                const text = highlight.textContent;
-                const textNode = document.createTextNode(text);
-                highlight.parentNode.replaceChild(textNode, highlight);
-              });
-            }, 5000);
-          }
-        }, 100); // Small delay to ensure content is rendered
-      }
-    });
+        }
+    }
   }
 
   // Check for stored line number and search term on page load
@@ -142,6 +146,17 @@ if (searchInput && searchResults) {
     }, 100);
   });
 
+  // Check for line number in URL hash on page load
+  window.addEventListener('load', () => {
+    const hash = window.location.hash;
+    if (hash.startsWith('#L')) {
+        const lineNumber = parseInt(hash.substring(2));
+        if (!isNaN(lineNumber)) {
+            scrollToLine(lineNumber);
+        }
+    }
+  });
+
   // Close search when clicking outside or pressing Escape
   document.addEventListener('click', (e) => {
     if (e.target === searchOverlay) {

From 995763467aa2cd78c56fe26e378d00cb4f72a3f7 Mon Sep 17 00:00:00 2001
From: William Wolff 
Date: Thu, 15 May 2025 12:29:26 +0200
Subject: [PATCH 10/10] git: add workflow for uupdating formal spec

---
 .../workflows/formal-spec-integration.yaml    | 59 +++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100644 .github/workflows/formal-spec-integration.yaml

diff --git a/.github/workflows/formal-spec-integration.yaml b/.github/workflows/formal-spec-integration.yaml
new file mode 100644
index 000000000..c60d0dd9a
--- /dev/null
+++ b/.github/workflows/formal-spec-integration.yaml
@@ -0,0 +1,59 @@
+name: "formal-spec-integration"
+
+on:
+  workflow_run:
+    workflows: ["formal-spec"]
+    types:
+      - completed
+    branches:
+      - main
+
+jobs:
+  integrate-formal-spec:
+    name: "Integrate Formal Spec"
+    runs-on: ubuntu-22.04
+    if: ${{ github.event.workflow_run.conclusion == 'success' }}
+    steps:
+      - name: 📥 Checkout Leios site
+        uses: actions/checkout@v4
+        with:
+          repository: input-output-hk/ouroboros-leios
+          token: ${{ secrets.GH_PAT }}
+
+      - name: 📥 Download formal spec HTML
+        uses: dawidd6/action-download-artifact@v2
+        with:
+          workflow: formal-spec.yaml
+          name: formal-spec-html
+          repo: input-output-hk/ouroboros-leios-formal-spec
+          token: ${{ secrets.GH_PAT }}
+
+      - name: 🏗️ Setup Node.js
+        uses: actions/setup-node@v4
+        with:
+          node-version: 22
+          cache: "yarn"
+          cache-dependency-path: ./site/yarn.lock
+
+      - name: 📦 Install dependencies
+        working-directory: site
+        run: yarn install
+
+      - name: 📝 Update formal spec
+        run: |
+          # Create formal spec directory if it doesn't exist
+          mkdir -p site/static/formal-spec
+          # Copy the HTML files
+          cp -r formal-spec-html/* site/static/formal-spec/
+
+      - name: 🏗️ Build Docusaurus site
+        working-directory: site
+        run: |
+          yarn build
+
+      - name: 🚀 Publish GitHub Pages
+        uses: peaceiris/actions-gh-pages@v4
+        with:
+          github_token: ${{ secrets.GITHUB_TOKEN }}
+          publish_dir: ./site/build
+          cname: leios.cardano-scaling.org 
\ No newline at end of file