88 lines
2.7 KiB
Agda
88 lines
2.7 KiB
Agda
open import Agda.Primitive using (Level; lsuc)
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open import Relation.Binary.PropositionalEquality using (_≡_)
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variable
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a : Level
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A : Set a
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module FirstAttempt where
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record Semigroup (A : Set a) : Set a where
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field
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_∙_ : A → A → A
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isAssociative : ∀ (a₁ a₂ a₃ : A) → a₁ ∙ (a₂ ∙ a₃) ≡ (a₁ ∙ a₂) ∙ a₃
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record Monoid (A : Set a) : Set a where
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field semigroup : Semigroup A
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open Semigroup semigroup public
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field
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zero : A
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isIdentityLeft : ∀ (a : A) → zero ∙ a ≡ a
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isIdentityRight : ∀ (a : A) → a ∙ zero ≡ a
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record ContrivedExample (A : Set a) : Set a where
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field
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-- first property
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monoid : Monoid A
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-- second property; Semigroup is a stand-in.
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semigroup : Semigroup A
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operationsEqual : Monoid._∙_ monoid ≡ Semigroup._∙_ semigroup
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module SecondAttempt where
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record IsSemigroup {A : Set a} (_∙_ : A → A → A) : Set a where
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field isAssociative : ∀ (a₁ a₂ a₃ : A) → a₁ ∙ (a₂ ∙ a₃) ≡ (a₁ ∙ a₂) ∙ a₃
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record IsMonoid {A : Set a} (zero : A) (_∙_ : A → A → A) : Set a where
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field
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isSemigroup : IsSemigroup _∙_
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isIdentityLeft : ∀ (a : A) → zero ∙ a ≡ a
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isIdentityRight : ∀ (a : A) → a ∙ zero ≡ a
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open IsSemigroup isSemigroup public
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record IsContrivedExample {A : Set a} (zero : A) (_∙_ : A → A → A) : Set a where
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field
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-- first property
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monoid : IsMonoid zero _∙_
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-- second property; Semigroup is a stand-in.
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semigroup : IsSemigroup _∙_
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record Semigroup (A : Set a) : Set a where
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field
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_∙_ : A → A → A
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isSemigroup : IsSemigroup _∙_
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record Monoid (A : Set a) : Set a where
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field
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zero : A
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_∙_ : A → A → A
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isMonoid : IsMonoid zero _∙_
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module ThirdAttempt {A : Set a} (_∙_ : A → A → A) where
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record IsSemigroup : Set a where
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field isAssociative : ∀ (a₁ a₂ a₃ : A) → a₁ ∙ (a₂ ∙ a₃) ≡ (a₁ ∙ a₂) ∙ a₃
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record IsMonoid (zero : A) : Set a where
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field
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isSemigroup : IsSemigroup
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isIdentityLeft : ∀ (a : A) → zero ∙ a ≡ a
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isIdentityRight : ∀ (a : A) → a ∙ zero ≡ a
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open IsSemigroup isSemigroup public
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record IsContrivedExample (zero : A) : Set a where
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field
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-- first property
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monoid : IsMonoid zero
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-- second property; Semigroup is a stand-in.
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semigroup : IsSemigroup
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