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Free-group lemmas (#106)

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Patrick Stevens
2020-04-05 11:09:12 +01:00
committed by GitHub
parent 61b5f8acc5
commit 2f07a8e972
18 changed files with 327 additions and 188 deletions
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15
Groups/FreeGroup/Definition.agda
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@@ -17,6 +17,10 @@ data FreeCompletion {a : _} (A : Set a) : Set a where
ofLetter : A → FreeCompletion A
ofInv : A → FreeCompletion A
freeCompletionMap : {a b : _} {A : Set a} {B : Set b} (f : A → B) (w : FreeCompletion A) → FreeCompletion B
freeCompletionMap f (ofLetter x) = ofLetter (f x)
freeCompletionMap f (ofInv x) = ofInv (f x)
freeInverse : {a : _} {A : Set a} (l : FreeCompletion A) → FreeCompletion A
freeInverse (ofLetter x) = ofInv x
freeInverse (ofInv x) = ofLetter x
@@ -27,8 +31,11 @@ ofLetterInjective refl = refl
ofInvInjective : {a : _} {A : Set a} {x y : A} → (ofInv x ≡ ofInv y) → x ≡ y
ofInvInjective refl = refl
ofLetterOfInv : {a : _} {A : Set a} {x y : A} → ofLetter x ≡ ofInv y → False
ofLetterOfInv ()
decidableFreeCompletion : {a : _} {A : Set a} → DecidableSet A → DecidableSet (FreeCompletion A)
decidableFreeCompletion {A = A} record { eq = dec } = record { eq = pr }
decidableFreeCompletion {A = A} dec = pr
where
pr : (a b : FreeCompletion A) → (a ≡ b) || (a ≡ b → False)
pr (ofLetter x) (ofLetter y) with dec x y
@@ -41,17 +48,17 @@ decidableFreeCompletion {A = A} record { eq = dec } = record { eq = pr }
... | inr x!=y = inr λ p → x!=y (ofInvInjective p)
freeCompletionEqual : {a : _} {A : Set a} (dec : DecidableSet A) (x y : FreeCompletion A) → Bool
freeCompletionEqual dec x y with DecidableSet.eq (decidableFreeCompletion dec) x y
freeCompletionEqual dec x y with decidableFreeCompletion dec x y
freeCompletionEqual dec x y | inl x₁ = BoolTrue
freeCompletionEqual dec x y | inr x₁ = BoolFalse
freeCompletionEqualFalse : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → ((x ≡ y) → False) → (freeCompletionEqual dec x y) ≡ BoolFalse
freeCompletionEqualFalse dec {x = x} {y} x!=y with DecidableSet.eq (decidableFreeCompletion dec) x y
freeCompletionEqualFalse dec {x = x} {y} x!=y with decidableFreeCompletion dec x y
freeCompletionEqualFalse dec {x} {y} x!=y | inl x=y = exFalso (x!=y x=y)
freeCompletionEqualFalse dec {x} {y} x!=y | inr _ = refl
freeCompletionEqualFalse' : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → .((freeCompletionEqual dec x y) ≡ BoolFalse) → (x ≡ y) → False
freeCompletionEqualFalse' dec {x} {y} pr with DecidableSet.eq (decidableFreeCompletion dec) x y
freeCompletionEqualFalse' dec {x} {y} pr with decidableFreeCompletion dec x y
freeCompletionEqualFalse' dec {x} {y} () | inl x₁
freeCompletionEqualFalse' dec {x} {y} pr | inr ans = ans