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* refactored from #2350 * enriched the `module` contents in response to review comments
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src/Data/List/Relation/Binary/Equality/Setoid/Properties.agda
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------------------------------------------------------------------------ | ||
-- The Agda standard library | ||
-- | ||
-- Properties of List modulo ≋ | ||
------------------------------------------------------------------------ | ||
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{-# OPTIONS --cubical-compatible --safe #-} | ||
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open import Relation.Binary.Bundles using (Setoid) | ||
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module Data.List.Relation.Binary.Equality.Setoid.Properties | ||
{c ℓ} (S : Setoid c ℓ) | ||
where | ||
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open import Algebra.Bundles using (Magma; Semigroup; Monoid) | ||
import Algebra.Structures as Structures | ||
open import Data.List.Base using (List; []; _++_) | ||
import Data.List.Properties as List | ||
import Data.List.Relation.Binary.Equality.Setoid as ≋ | ||
open import Data.Product.Base using (_,_) | ||
open import Function.Base using (_∘_) | ||
open import Level using (_⊔_) | ||
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open ≋ S using (_≋_; ≋-refl; ≋-reflexive; ≋-isEquivalence; ++⁺) | ||
open Structures _≋_ using (IsMagma; IsSemigroup; IsMonoid) | ||
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------------------------------------------------------------------------ | ||
-- The []-++-Monoid | ||
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-- Structures | ||
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isMagma : IsMagma _++_ | ||
isMagma = record | ||
{ isEquivalence = ≋-isEquivalence | ||
; ∙-cong = ++⁺ | ||
} | ||
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isSemigroup : IsSemigroup _++_ | ||
isSemigroup = record | ||
{ isMagma = isMagma | ||
; assoc = λ xs ys zs → ≋-reflexive (List.++-assoc xs ys zs) | ||
} | ||
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isMonoid : IsMonoid _++_ [] | ||
isMonoid = record | ||
{ isSemigroup = isSemigroup | ||
; identity = (λ _ → ≋-refl) , ≋-reflexive ∘ List.++-identityʳ | ||
} | ||
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-- Bundles | ||
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magma : Magma c (c ⊔ ℓ) | ||
magma = record { isMagma = isMagma } | ||
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semigroup : Semigroup c (c ⊔ ℓ) | ||
semigroup = record { isSemigroup = isSemigroup } | ||
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monoid : Monoid c (c ⊔ ℓ) | ||
monoid = record { isMonoid = isMonoid } |