mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-17 09:28:40 +00:00
58 lines
2.6 KiB
Agda
58 lines
2.6 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Sets.FinSet.Definition
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open import Groups.Definition
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open import Groups.SymmetricGroups.Definition
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open import Decidable.Sets
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module Groups.FreeGroup.Definition where
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data FreeCompletion {a : _} (A : Set a) : Set a where
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ofLetter : A → FreeCompletion A
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ofInv : A → FreeCompletion A
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freeInverse : {a : _} {A : Set a} (l : FreeCompletion A) → FreeCompletion A
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freeInverse (ofLetter x) = ofInv x
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freeInverse (ofInv x) = ofLetter x
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ofLetterInjective : {a : _} {A : Set a} {x y : A} → (ofLetter x ≡ ofLetter y) → x ≡ y
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ofLetterInjective refl = refl
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ofInvInjective : {a : _} {A : Set a} {x y : A} → (ofInv x ≡ ofInv y) → x ≡ y
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ofInvInjective refl = refl
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decidableFreeCompletion : {a : _} {A : Set a} → DecidableSet A → DecidableSet (FreeCompletion A)
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decidableFreeCompletion {A = A} record { eq = dec } = record { eq = pr }
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where
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pr : (a b : FreeCompletion A) → (a ≡ b) || (a ≡ b → False)
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pr (ofLetter x) (ofLetter y) with dec x y
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... | inl refl = inl refl
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... | inr x!=y = inr λ p → x!=y (ofLetterInjective p)
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pr (ofLetter x) (ofInv y) = inr λ ()
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pr (ofInv x) (ofLetter y) = inr λ ()
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pr (ofInv x) (ofInv y) with dec x y
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... | inl refl = inl refl
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... | inr x!=y = inr λ p → x!=y (ofInvInjective p)
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freeCompletionEqual : {a : _} {A : Set a} (dec : DecidableSet A) (x y : FreeCompletion A) → Bool
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freeCompletionEqual dec x y with DecidableSet.eq (decidableFreeCompletion dec) x y
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freeCompletionEqual dec x y | inl x₁ = BoolTrue
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freeCompletionEqual dec x y | inr x₁ = BoolFalse
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freeCompletionEqualFalse : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → ((x ≡ y) → False) → (freeCompletionEqual dec x y) ≡ BoolFalse
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freeCompletionEqualFalse dec {x = x} {y} x!=y with DecidableSet.eq (decidableFreeCompletion dec) x y
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freeCompletionEqualFalse dec {x} {y} x!=y | inl x=y = exFalso (x!=y x=y)
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freeCompletionEqualFalse dec {x} {y} x!=y | inr _ = refl
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freeCompletionEqualFalse' : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → .((freeCompletionEqual dec x y) ≡ BoolFalse) → (x ≡ y) → False
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freeCompletionEqualFalse' dec {x} {y} pr with DecidableSet.eq (decidableFreeCompletion dec) x y
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freeCompletionEqualFalse' dec {x} {y} () | inl x₁
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freeCompletionEqualFalse' dec {x} {y} pr | inr ans = ans
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