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Added Irreflexivity and Asymmetry of WellFounded Relations #2119

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Merged
merged 6 commits into from
Oct 6, 2023
Merged

Added Irreflexivity and Asymmetry of WellFounded Relations #2119

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0b67efc
Added Irreflexicity and Asymmetry of WellFounded
jmarkakis Oct 2, 2023
935ed98
Renamed wf-asym, acc-asym and made arguments implicit
jmarkakis Oct 3, 2023
0413180
Simplified wf-irrefl, and changed CHANGElOG
jmarkakis Oct 3, 2023
43f9178
Update CHANGELOG.md
jmarkakis Oct 3, 2023
8ec63d0
Merge branch 'master' into WellFoundedProperties
MatthewDaggitt Oct 5, 2023
285902a
Changed names + solved a merge conflict
jmarkakis Oct 5, 2023
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5 changes: 5 additions & 0 deletions CHANGELOG.md
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Expand Up @@ -3745,6 +3745,11 @@ This is a full list of proofs that have changed form to use irrelevant instance
* Added new proof to `Induction.WellFounded`
```agda
Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_

acc-asym : Acc _<_ x → x < y → ¬ (y < x)
wf-asym : WellFounded _<_ → Asymmetric _<_
wf-irrefl : _<_ Respects₂ _≈_ → Symmetric _≈_ →
WellFounded _<_ → Irreflexive _≈_ _<_
```

* Added new file `Relation.Binary.Reasoning.Base.Apartness`
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22 changes: 20 additions & 2 deletions src/Induction/WellFounded.agda
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Expand Up @@ -8,15 +8,18 @@

module Induction.WellFounded where

open import Data.Product.Base using (Σ; _,_; proj₁)
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; _Respectsʳ_; _Respects_)
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
Expand Down Expand Up @@ -116,6 +119,21 @@ module FixPoint
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} {y} hx =
(Some.wfRec _ λ x hx y x<y y<x → hx y y<x x y<x x<y) x hx y

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,
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