24 lines
735 B
Agda
24 lines
735 B
Agda
module Properties.Equality where
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open import Agda.Builtin.Equality using (_≡_; refl)
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open import Properties.Contradiction using (¬)
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sym : ∀ {A : Set} {a b : A} → (a ≡ b) → (b ≡ a)
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sym refl = refl
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trans : ∀ {A : Set} {a b c : A} → (a ≡ b) → (b ≡ c) → (a ≡ c)
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trans refl refl = refl
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cong : ∀ {A B : Set} {a b : A} (f : A → B) → (a ≡ b) → (f a ≡ f b)
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cong f refl = refl
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subst₁ : ∀ {A : Set} {a b : A} (F : A → Set) → (a ≡ b) → (F a) → (F b)
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subst₁ F refl x = x
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subst₂ : ∀ {A B : Set} {a b : A} {c d : B} (F : A → B → Set) → (a ≡ b) → (c ≡ d) → (F a c) → (F b d)
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subst₂ F refl refl x = x
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_≢_ : ∀ {A : Set} → A → A → Set
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(a ≢ b) = ¬(a ≡ b)
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