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[ add ] Relation.Binary.Properties.PartialSetoid #2678

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Jun 17, 2025
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[ add ] Relation.Binary.Properties.PartialSetoid #2678

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a5a5bd2
add: properties of PERs
jamesmckinna Mar 17, 2025
f5d35af
use: `Relation.Binary.Definitions`
jamesmckinna Mar 27, 2025
4095af1
make prefix `partial-`
jamesmckinna May 27, 2025
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2 changes: 2 additions & 0 deletions CHANGELOG.md
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Expand Up @@ -120,6 +120,8 @@ New modules

* `Data.Sign.Show` to show a sign

* `Relation.Binary.Properties.PartialSetoid` to systematise properties of a PER

Additions to existing modules
-----------------------------

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47 changes: 47 additions & 0 deletions src/Relation/Binary/Properties/PartialSetoid.agda
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Additional properties for setoids
------------------------------------------------------------------------

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

open import Relation.Binary.Bundles using (PartialSetoid)

module Relation.Binary.Properties.PartialSetoid
{a ℓ} (S : PartialSetoid a ℓ) where

open import Data.Product.Base using (_,_; _×_)
open import Relation.Binary.Definitions using (LeftTrans; RightTrans)
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)

open PartialSetoid S

private
variable
x y z : Carrier

------------------------------------------------------------------------
-- Proofs for partial equivalence relations

trans-reflˡ : LeftTrans _≡_ _≈_
trans-reflˡ ≡.refl p = p
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Do we actually need to match on the proof here? Is this not just a variant of subst?

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@jamesmckinna jamesmckinna May 27, 2025 •
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Oh... yes, probably! er, no, actually, as the arguments are in the wrong order, and introducing a flip/≡.sym seemed like an indirection too far!?


trans-reflʳ : RightTrans _≈_ _≡_
trans-reflʳ p ≡.refl = p

p-reflˡ : x ≈ y → x ≈ x
p-reflˡ p = trans p (sym p)

p-reflʳ : x ≈ y → y ≈ y
p-reflʳ p = trans (sym p) p

p-refl : x ≈ y → x ≈ x × y ≈ y
p-refl p = p-reflˡ p , p-reflʳ p

p-reflexiveˡ : x ≈ y → x ≡ z → x ≈ z
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This is Trans _≈_ _≡_ _≈_ (and similarly below), suggesting that reflexive may not be quite the right name? Similarly is this not just a variant of subst?

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Hmmm... I see what you mean... Will ponder this for a bit!

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@jamesmckinna jamesmckinna May 27, 2025 •
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So... my reasoning (with a bit of post hoc rationalisation...) was as follows:

  • reflexive proofs are of the form x ≡ z → x ≈ z
  • these ones are 'partial', because preconditioned on having a proof of x ≈ y to licence them
  • the left/right designators then pick out which of x or y is being matched on

Possibly a bit convoluted, so I'm open to 'better' names!

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This is not Trans _≈_ _≡_ _≈_ because the second x would be a y in that case.

This is an extremely weird property which basically says: if x is related to something (anything) and we have evidence of that, and x and z are identical, then we can get evidence that x and z are related.

I'm at a loss for what to name it.

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@jamesmckinna jamesmckinna May 28, 2025 •
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I don't know about 'weird'... these properties are characteristic of being a PER rather than a full IsEquivalence!? Namely that the relation is reflexive only on its domain of definition, rather than on the whole type...

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@jamesmckinna jamesmckinna May 29, 2025 •
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As to names: I could call it (something like) conditional-reflexive, but then we would lose that the ≡.refl instantiation of such a lemma should be partial-refl... so those lemmas would also need to be renamed. And then... names are getting too long?

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Weird likely because I generically find PERs weird.

However, your comment hits the real issue on the head: only on its domain of definition. We really ought to define an 'is defined' construction (a la Trans and Symmetric, etc) and use it here. Then I would have understood that you were asking for some indirect evidence of being defined. Then we'll be in a decent position to think about names.

At least now I understand this combinator's meaning (when x is defined, then equality that involve it is reflexive), and that is no longer weird at all.

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Not sure I see clearly what the IsDefined construction would look like, and whether it would be 'free-standing', or tied to being in a PER?

Cf. Scott's "Identity and Existence in Intuitionistic Logic" (LNM 753) which explores variations on the theme of E t vs. t = t as expressions of 't is defined'/'t exists', but does seem tied to _=_ being an 'equality' (Leibniz-like) relation...?

p-reflexiveˡ p ≡.refl = p-reflˡ p

p-reflexiveʳ : x ≈ y → y ≡ z → y ≈ z
p-reflexiveʳ p ≡.refl = p-reflʳ p