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[Refractor] contradiction over ⊥-elim in Data.*.Relation.Binary.Lex.* #2671

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merged 4 commits into from
Mar 19, 2025
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[Refractor] contradiction over ⊥-elim in Data.*.Relation.Binary.Lex.* #2671

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ec12f61
[Refractor] contradiction over ⊥-elim in
jmougeot Mar 13, 2025
34d3fe0
Modification
jmougeot Mar 17, 2025
93e3880
remove of useless import and replacing tt with _
jmougeot Mar 17, 2025
008e08c
add open import Data.Empty using (⊥; ⊥-elim)
jmougeot Mar 17, 2025
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6 changes: 3 additions & 3 deletions src/Data/List/Relation/Binary/Lex/Core.agda
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Expand Up @@ -8,13 +8,13 @@

module Data.List.Relation.Binary.Lex.Core where

open import Data.Empty using (⊥; ⊥-elim)
open import Data.Unit.Base using (⊤; tt)
open import Data.Empty using (⊥)
open import Data.Unit.Base using (⊤)
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.Nullary.Negation.Core using (¬_)
open import Relation.Binary.Core using (Rel)
open import Data.List.Relation.Binary.Pointwise.Base
using (Pointwise; []; _∷_; head; tail)
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8 changes: 3 additions & 5 deletions src/Data/List/Relation/Binary/Lex/Strict.agda
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Expand Up @@ -11,8 +11,6 @@

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)
Expand Down Expand Up @@ -146,7 +144,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where

≤-reflexive : (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) →
Pointwise _≈_ ⇒ Lex-≤ _≈_ _≺_
≤-reflexive _≈_ _≺_ [] = base tt
≤-reflexive _≈_ _≺_ [] = base _
≤-reflexive _≈_ _≺_ (x≈y ∷ xs≈ys) =
next x≈y (≤-reflexive _≈_ _≺_ xs≈ys)

Expand All @@ -168,7 +166,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
-- the following lemma.

≤-total : Symmetric _≈_ → Trichotomous _≈_ _≺_ → Total _≤_
≤-total _ _ [] [] = inj₁ (base tt)
≤-total _ _ [] [] = inj₁ (base _)
≤-total _ _ [] (x ∷ xs) = inj₁ halt
≤-total _ _ (x ∷ xs) [] = inj₂ halt
≤-total sym tri (x ∷ xs) (y ∷ ys) with tri x y
Expand All @@ -179,7 +177,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
... | inj₂ ys≲xs = inj₂ (next (sym x≈y) ys≲xs)

≤-decidable : Decidable _≈_ → Decidable _≺_ → Decidable _≤_
≤-decidable = Core.decidable (yes tt)
≤-decidable = Core.decidable (yes _)

≤-respects₂ : IsEquivalence _≈_ → _≺_ Respects₂ _≈_ →
_≤_ Respects₂ _≋_
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9 changes: 4 additions & 5 deletions src/Data/Vec/Relation/Binary/Lex/Core.agda
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Expand Up @@ -8,7 +8,6 @@

module Data.Vec.Relation.Binary.Lex.Core {a} {A : Set a} where

open import Data.Empty using (⊥; ⊥-elim)
open import Data.Nat.Base using (ℕ; suc)
import Data.Nat.Properties as ℕ using (_≟_; ≡-irrelevant)
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
Expand All @@ -27,7 +26,7 @@ open import Relation.Binary.PropositionalEquality.Core as ≡
using (_≡_; refl; cong)
import Relation.Nullary as Nullary
open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_; _⊎-dec_)
open import Relation.Nullary.Negation
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

private
variable
Expand Down Expand Up @@ -114,9 +113,9 @@ module _ {P : Set} {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where

antisym : ∀ {n} → Antisymmetric (_≋_ {n}) (_<ₗₑₓ_)
antisym (base _) (base _) = []
antisym (this x≺y m≡n) (this y≺x n≡m) = ⊥-elim (≺-asym x≺y y≺x)
antisym (this x≺y m≡n) (next y≈x ys<xs) = ⊥-elim (≺-irrefl (≈-sym y≈x) x≺y)
antisym (next x≈y xs<ys) (this y≺x m≡n) = ⊥-elim (≺-irrefl (≈-sym x≈y) y≺x)
antisym (this x≺y m≡n) (this y≺x n≡m) = contradiction y≺x (≺-asym x≺y)
antisym (this x≺y m≡n) (next y≈x ys<xs) = contradiction x≺y (≺-irrefl (≈-sym y≈x))
antisym (next x≈y xs<ys) (this y≺x m≡n) = contradiction y≺x (≺-irrefl (≈-sym x≈y))
antisym (next x≈y xs<ys) (next y≈x ys<xs) = x≈y ∷ (antisym xs<ys ys<xs)

module _ (≈-equiv : IsPartialEquivalence _≈_) (open IsPartialEquivalence ≈-equiv) where
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7 changes: 4 additions & 3 deletions src/Data/Vec/Relation/Binary/Lex/Strict.agda
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Expand Up @@ -23,7 +23,7 @@ open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
using (Pointwise; []; _∷_; head; tail)
open import Function.Base using (id; _on_; _∘_)
open import Induction.WellFounded
open import Relation.Nullary using (yes; no; ¬_)
open import Level using (Level; _⊔_)
open import Relation.Binary.Core using (REL; Rel; _⇒_)
open import Relation.Binary.Bundles
using (Poset; StrictPartialOrder; DecPoset; DecStrictPartialOrder
Expand All @@ -39,7 +39,8 @@ open import Relation.Binary.Definitions
open import Relation.Binary.Consequences using (asym⇒irr)
open import Relation.Binary.Construct.On as On using (wellFounded)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
open import Level using (Level; _⊔_)
open import Relation.Nullary.Decidable.Core using (yes; no)
open import Relation.Nullary.Negation.Core using (¬_; contradiction)

private
variable
Expand Down Expand Up @@ -139,7 +140,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
where

<-wellFounded : ∀ {n} → WellFounded (_<_ {n})
<-wellFounded {0} [] = acc λ ys<[] → ⊥-elim (xs≮[] ys<[])
<-wellFounded {0} [] = acc λ ys<[] → contradiction ys<[] xs≮[]

<-wellFounded {suc n} xs = Subrelation.wellFounded <⇒uncons-Lex uncons-Lex-wellFounded xs
where
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