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Added Irreflexivity and Asymmetry of WellFounded Relations #2119

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Merged
MatthewDaggitt 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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8 changes: 8 additions & 0 deletions CHANGELOG.md
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Expand Up @@ -3514,6 +3514,14 @@ 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-asymm : (x : A) → (Acc _<_ x) → (y : A) → x < y → ¬ (y < x)
Wf-asymm : WellFounded _<_ → Asymmetric _<_

Acc-irrefl : {_≈_ : Rel A ℓ} → Symmetric _≈_ → _<_ Respects₂ _≈_ →
(x : A) → Acc _<_ x → (y : A) → x ≈ y → ¬ (x < y)
Wf-irrefl : WellFounded _<_ → {_≈_ : Rel A ℓ} → Symmetric _≈_ →
_<_ Respects₂ _≈_ → Irreflexive _≈_ _<_
```

* Added new file `Relation.Binary.Reasoning.Base.Apartness`
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21 changes: 20 additions & 1 deletion src/Induction/WellFounded.agda
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Expand Up @@ -8,7 +8,7 @@

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; _⊔_)
Expand All @@ -17,6 +17,7 @@ open import Relation.Binary.Definitions
using (Symmetric; _Respectsʳ_; _Respects_)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
open import Relation.Unary
open import Relation.Nullary.Negation.Core using (¬_)

private
variable
Expand Down Expand Up @@ -112,6 +113,24 @@ module FixPoint
unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x (λ y _ → wfRec P f y)
unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec

------------------------------------------------------------------------
-- Well-founded relations are asymmetric and irreflexive.

module _ {_<_ : Rel A r} where
Acc-asymm : (x : A) → (Acc _<_ x) → (y : A) → x < y → ¬ (y < x)
Acc-asymm = Some.wfRec _ λ x hx y x<y y<x → hx y y<x x y<x x<y

Wf-asymm : WellFounded _<_ → Asymmetric _<_
Wf-asymm wf x<y y<x = Acc-asymm _ (wf _) _ x<y y<x

Acc-irrefl : {_≈_ : Rel A ℓ} → Symmetric _≈_ → _<_ Respects₂ _≈_ →
(x : A) → Acc _<_ x → (y : A) → x ≈ y → ¬ (x < y)
Acc-irrefl s r x p y x≈y x<y =
Acc-asymm x p y x<y (proj₂ r x≈y (proj₁ r (s x≈y) x<y))

Wf-irrefl : WellFounded _<_ → {_≈_ : Rel A ℓ} → Symmetric _≈_ →
_<_ Respects₂ _≈_ → Irreflexive _≈_ _<_
Wf-irrefl wf s r = Acc-irrefl s r _ (wf _) _

------------------------------------------------------------------------
-- It might be useful to establish proofs of Acc or Well-founded using
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