[ refactor ] Data.List.Relation.Binary.Sublist.Propositional.Properties

CHANGELOG.md

@@ -72,6 +72,12 @@ Additions to existing modules
   ≟-≢          : (i≢j : i ≢ j) → (i ≟ j) ≡ no i≢j
   ```
 
+* In `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
+  ```agda
+  All-resp-⊆ : (P Respects _≈_) → (All P) Respects _⊇_
+  Any-resp-⊆ : (P Respects _≈_) → (Any P) Respects _⊆_
+  ```
+
 * In `Data.Nat.Properties`:
   ```agda
   ≟-≢   : (m≢n : m ≢ n) → (m ≟ n) ≡ no m≢n

src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda

@@ -43,7 +43,7 @@ private
 
 module _ {A : Set a} where
   open SetoidProperties (setoid A) public
-    hiding (map⁺; ⊆-trans-idˡ; ⊆-trans-idʳ; ⊆-trans-assoc)
+    hiding (map⁺; All-resp-⊆; Any-resp-⊆; ⊆-trans-idˡ; ⊆-trans-idʳ; ⊆-trans-assoc)
 
 ------------------------------------------------------------------------
 -- Relationship between _⊆_ and Setoid._⊆_
@@ -103,17 +103,17 @@ map⁺ f = SetoidProperties.map⁺ (setoid _) (setoid _) (cong f)
 ------------------------------------------------------------------------
 -- Relationships to other predicates
 
+module _ {P : Pred A ℓ} where
+
 -- 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
+  All-resp-⊆ : (All P) Respects _⊇_
+  All-resp-⊆ = SetoidProperties.All-resp-⊆ (setoid _) (≡.resp P)
 
 -- Any P is a covariant functor from _⊆_ to Set.
 
-Any-resp-⊆ : {P : Pred A ℓ} → (Any P) Respects _⊆_
-Any-resp-⊆ = lookup
+  Any-resp-⊆ : (Any P) Respects _⊆_
+  Any-resp-⊆ = SetoidProperties.Any-resp-⊆ (setoid _) (≡.resp P)
 
 ------------------------------------------------------------------------
 -- Functor laws for All-resp-⊆

src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda

@@ -12,18 +12,20 @@ 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.Base hiding (_∷ʳ_; lookup)
 open import Data.List.Properties using (++-identityʳ)
 open import Data.List.Relation.Unary.Any using (Any)
-open import Data.List.Relation.Unary.All using (All; tabulateₛ)
+open import Data.List.Relation.Unary.All
+  using (All; []; _∷_; tabulateₛ)
 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.Definitions
+  using (_Respects_) renaming (Decidable to Decidable₂)
 import Relation.Binary.Properties.Setoid as SetoidProperties
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; refl; sym; cong; cong₂)
@@ -42,7 +44,8 @@ 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 Setoid S using (_≈_; trans)
+  renaming (Carrier to A; refl to ≈-refl; sym to ≈-sym)
 open SetoidEquality S using (_≋_; ≋-refl; ≋-reflexive; ≋-setoid)
 open SetoidSublist S hiding (map)
 open SetoidProperties S using (≈-preorder)
@@ -106,6 +109,23 @@ module _ (≈-assoc : ∀ {w x y z} (p : w ≈ x) (q : x ≈ y) (r : y ≈ z) 
   ⊆-trans-assoc [] [] [] = refl
 
 
+------------------------------------------------------------------------
+-- Relationships to other predicates
+
+module _ {P : Pred A p} (resp : P Respects _≈_) where
+
+-- All P is a contravariant functor from _⊆_ to Set.
+
+  All-resp-⊆ : (All P) Respects _⊇_
+  All-resp-⊆ []        []       = []
+  All-resp-⊆ (_  ∷ʳ p) (x ∷ xs) = All-resp-⊆ p xs
+  All-resp-⊆ (x≈y ∷ p) (x ∷ xs) = resp (≈-sym x≈y) x ∷ All-resp-⊆ p xs
+
+-- Any P is a covariant functor from _⊆_ to Set.
+
+  Any-resp-⊆ : (Any P) Respects _⊆_
+  Any-resp-⊆ = lookup resp
+
 ------------------------------------------------------------------------
 -- Reasoning over sublists
 ------------------------------------------------------------------------