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[Refractor] contradiction over ⊥-elim in antisymmetric def #2665

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jamesmckinna merged 2 commits into from
Mar 15, 2025
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[Refractor] contradiction over ⊥-elim in antisymmetric def #2665

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[Refractor] contradiction over ⊥-elim in antisymmetric def
jmougeot Mar 12, 2025
38846f3
Update src/Data/List/Relation/Binary/Lex.agda
jamesmckinna Mar 15, 2025
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17 changes: 9 additions & 8 deletions src/Data/List/Relation/Binary/Lex.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -8,23 +8,24 @@

module Data.List.Relation.Binary.Lex where

open import Data.Empty using (⊥; ⊥-elim)
open import Data.Unit.Base using (⊤; tt)
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
open import Data.List.Base using (List; []; _∷_)
open import Data.List.Relation.Binary.Pointwise.Base
using (Pointwise; []; _∷_; head; tail)
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
open import Function.Base using (_∘_; flip; id)
open import Function.Bundles using (_⇔_; mk⇔)
open import Level using (_⊔_)
open import Relation.Nullary.Negation using (¬_)
open import Relation.Nullary.Negation.Core using (¬_; contradiction)
open import Relation.Nullary.Decidable as Dec
using (Dec; yes; no; _×-dec_; _⊎-dec_)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Structures using (IsEquivalence)
open import Relation.Binary.Definitions
using (Symmetric; Transitive; Irreflexive; Asymmetric; Antisymmetric; Decidable; _Respects₂_; _Respects_)
open import Data.List.Relation.Binary.Pointwise.Base
using (Pointwise; []; _∷_; head; tail)
using (Symmetric; Transitive; Irreflexive; Asymmetric; Antisymmetric
; Decidable; _Respects₂_; _Respects_)


------------------------------------------------------------------------
-- Re-exporting the core definitions and properties
Expand Down Expand Up @@ -57,9 +58,9 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} {P : Set}
where
as : Antisymmetric _≋_ _<_
as (base _) (base _) = []
as (this x≺y) (this y≺x) = ⊥-elim (asym x≺y y≺x)
as (this x≺y) (next y≈x ys<xs) = ⊥-elim (ir (sym y≈x) x≺y)
as (next x≈y xs<ys) (this y≺x) = ⊥-elim (ir (sym x≈y) y≺x)
as (this x≺y) (this y≺x) = contradiction y≺x (asym x≺y)
as (this x≺y) (next y≈x ys<xs) = contradiction x≺y (ir (sym y≈x))
as (next x≈y xs<ys) (this y≺x) = contradiction y≺x (ir (sym x≈y))
as (next x≈y xs<ys) (next y≈x ys<xs) = x≈y ∷ as xs<ys ys<xs

toSum : ∀ {x y xs ys} → (x ∷ xs) < (y ∷ ys) → (x ≺ y ⊎ (x ≈ y × xs < ys))
Expand Down