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Hide more stuff (#124)
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Patrick Stevens
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@@ -12,17 +12,13 @@ open import Semirings.Definition
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open import Functions.Definition
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open import Groups.Isomorphisms.Definition
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open import Boolean.Definition
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open import Boolean.Lemmas
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module Groups.FreeGroup.Parity {a : _} {A : Set a} (decA : DecidableSet A) where
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open import Groups.FreeGroup.Word decA
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open import Groups.FreeGroup.Group decA
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xor : Bool → Bool → Bool
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xor BoolTrue BoolTrue = BoolFalse
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xor BoolTrue BoolFalse = BoolTrue
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xor BoolFalse b = b
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C2 : Group (reflSetoid Bool) xor
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Group.+WellDefined C2 refl refl = refl
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Group.0G C2 = BoolFalse
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@@ -39,14 +35,6 @@ Group.invLeft C2 {BoolFalse} = refl
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Group.invRight C2 {BoolTrue} = refl
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Group.invRight C2 {BoolFalse} = refl
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notNot : (x : Bool) → not (not x) ≡ x
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notNot BoolTrue = refl
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notNot BoolFalse = refl
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notWellDefined : {x y : Bool} → (x ≡ y) → not x ≡ not y
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notWellDefined {BoolTrue} {BoolTrue} x=y = refl
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notWellDefined {BoolFalse} {BoolFalse} x=y = refl
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parity : (x : A) → ReducedWord → Bool
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parity x empty = BoolFalse
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parity x (prependLetter (ofLetter y) w _) with decA x y
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@@ -56,108 +44,103 @@ parity x (prependLetter (ofInv y) w _) with decA x y
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parity x (prependLetter (ofInv y) w _) | inl _ = not (parity x w)
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parity x (prependLetter (ofInv y) w _) | inr _ = parity x w
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parityPrepend : (a : A) (w : ReducedWord) (l : A) → ((a ≡ l) → False) → parity a (prepend w (ofLetter l)) ≡ parity a w
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parityPrepend a empty l notEq with decA a l
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parityPrepend a empty l notEq | inl x = exFalso (notEq x)
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parityPrepend a empty l notEq | inr x = refl
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parityPrepend a (prependLetter (ofLetter r) w x) l notEq with decA a l
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parityPrepend a (prependLetter (ofLetter r) w x) l notEq | inl m = exFalso (notEq m)
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parityPrepend a (prependLetter (ofLetter r) w x) l notEq | inr _ = refl
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parityPrepend a (prependLetter (ofInv r) w x) l notEq with decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r with decA l r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inl l=r = exFalso (notEq (transitivity a=r (equalityCommutative l=r)))
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with decA a l
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inl x₁ = refl
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inr bad = exFalso (bad a=r)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r with decA l r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl x₁ = refl
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with decA a l
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inl a=r = exFalso (a!=r a=r)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inr x₁ = refl
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private
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parityPrepend : (a : A) (w : ReducedWord) (l : A) → ((a ≡ l) → False) → parity a (prepend w (ofLetter l)) ≡ parity a w
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parityPrepend a empty l notEq with decA a l
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parityPrepend a empty l notEq | inl x = exFalso (notEq x)
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parityPrepend a empty l notEq | inr x = refl
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parityPrepend a (prependLetter (ofLetter r) w x) l notEq with decA a l
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parityPrepend a (prependLetter (ofLetter r) w x) l notEq | inl m = exFalso (notEq m)
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parityPrepend a (prependLetter (ofLetter r) w x) l notEq | inr _ = refl
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parityPrepend a (prependLetter (ofInv r) w x) l notEq with decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r with decA l r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inl l=r = exFalso (notEq (transitivity a=r (equalityCommutative l=r)))
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with decA a l
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inl x₁ = refl
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inr bad = exFalso (bad a=r)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r with decA l r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl x₁ = refl
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with decA a l
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inl a=r = exFalso (a!=r a=r)
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inr x₁ = refl
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parityPrepend' : (a : A) (w : ReducedWord) (l : A) → ((a ≡ l) → False) → parity a (prepend w (ofInv l)) ≡ parity a w
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parityPrepend' a empty l notEq with decA a l
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parityPrepend' a empty l notEq | inl x = exFalso (notEq x)
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parityPrepend' a empty l notEq | inr x = refl
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq with decA l r
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inl m with decA a r
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... | inl a=r = exFalso (notEq (transitivity a=r (equalityCommutative m)))
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... | inr a!=r = refl
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inr l!=r with decA a l
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... | inl a=l = exFalso (notEq a=l)
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... | inr a!=l = refl
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq with decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r with decA l r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inl l=r = exFalso (notEq (transitivity a=r (equalityCommutative l=r)))
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with decA a l
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inl x₁ = refl
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inr bad = exFalso (bad a=r)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r with decA l r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r with decA a l
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inl a=l = exFalso (notEq a=l)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inr a!=l with decA a r
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... | inl a=r = exFalso (a!=r a=r)
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... | inr _ = refl
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with decA a l
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inl a=r = exFalso (a!=r a=r)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inr x₁ = refl
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parityPrepend' : (a : A) (w : ReducedWord) (l : A) → ((a ≡ l) → False) → parity a (prepend w (ofInv l)) ≡ parity a w
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parityPrepend' a empty l notEq with decA a l
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parityPrepend' a empty l notEq | inl x = exFalso (notEq x)
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parityPrepend' a empty l notEq | inr x = refl
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq with decA l r
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inl m with decA a r
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... | inl a=r = exFalso (notEq (transitivity a=r (equalityCommutative m)))
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... | inr a!=r = refl
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inr l!=r with decA a l
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... | inl a=l = exFalso (notEq a=l)
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... | inr a!=l = refl
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq with decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r with decA l r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inl l=r = exFalso (notEq (transitivity a=r (equalityCommutative l=r)))
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with decA a l
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inl x₁ = refl
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inr bad = exFalso (bad a=r)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r with decA l r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r with decA a l
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inl a=l = exFalso (notEq a=l)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inr a!=l with decA a r
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... | inl a=r = exFalso (a!=r a=r)
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... | inr _ = refl
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with decA a l
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inl a=l = exFalso (notEq a=l)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inl a=r = exFalso (a!=r a=r)
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inr x₁ = refl
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parityPrepend'' : (a : A) (w : ReducedWord) → parity a (prepend w (ofLetter a)) ≡ not (parity a w)
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parityPrepend'' a empty with decA a a
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... | inl _ = refl
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... | inr bad = exFalso (bad refl)
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parityPrepend'' a (prependLetter (ofLetter l) w x) with decA a a
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ with decA a l
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ | inl a=l = refl
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ | inr a!=l = refl
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inr bad = exFalso (bad refl)
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parityPrepend'' a (prependLetter (ofInv l) w x) with decA a l
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... | inl a=l = equalityCommutative (notNot (parity a w))
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l with decA a a
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inl _ with decA a l
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... | inl a=l = exFalso (a!=l a=l)
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... | inr _ = refl
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inr bad = exFalso (bad refl)
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parityPrepend'' : (a : A) (w : ReducedWord) → parity a (prepend w (ofLetter a)) ≡ not (parity a w)
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parityPrepend'' a empty with decA a a
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... | inl _ = refl
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... | inr bad = exFalso (bad refl)
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parityPrepend'' a (prependLetter (ofLetter l) w x) with decA a a
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ with decA a l
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ | inl a=l = refl
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ | inr a!=l = refl
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inr bad = exFalso (bad refl)
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parityPrepend'' a (prependLetter (ofInv l) w x) with decA a l
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... | inl a=l = equalityCommutative (notNot (parity a w))
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l with decA a a
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inl _ with decA a l
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... | inl a=l = exFalso (a!=l a=l)
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... | inr _ = refl
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inr bad = exFalso (bad refl)
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parityPrepend''' : (a : A) (w : ReducedWord) → parity a (prepend w (ofInv a)) ≡ not (parity a w)
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parityPrepend''' a empty with decA a a
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... | inl _ = refl
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... | inr bad = exFalso (bad refl)
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parityPrepend''' a (prependLetter (ofLetter l) w x) with decA a l
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... | inl a=l = equalityCommutative (notNot _)
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l with decA a a
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... | inl _ with decA a l
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inl _ | inl a=l = exFalso (a!=l a=l)
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inl _ | inr _ = refl
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inr bad = exFalso (bad refl)
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parityPrepend''' a (prependLetter (ofInv l) w x) with decA a a
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parityPrepend''' a (prependLetter (ofInv l) w x) | inl _ with decA a l
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... | inl a=l = refl
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... | inr a!=l = refl
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parityPrepend''' a (prependLetter (ofInv l) w x) | inr bad = exFalso (bad refl)
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parityPrepend''' : (a : A) (w : ReducedWord) → parity a (prepend w (ofInv a)) ≡ not (parity a w)
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parityPrepend''' a empty with decA a a
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... | inl _ = refl
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... | inr bad = exFalso (bad refl)
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parityPrepend''' a (prependLetter (ofLetter l) w x) with decA a l
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... | inl a=l = equalityCommutative (notNot _)
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l with decA a a
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... | inl _ with decA a l
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inl _ | inl a=l = exFalso (a!=l a=l)
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inl _ | inr _ = refl
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parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inr bad = exFalso (bad refl)
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parityPrepend''' a (prependLetter (ofInv l) w x) with decA a a
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parityPrepend''' a (prependLetter (ofInv l) w x) | inl _ with decA a l
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... | inl a=l = refl
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... | inr a!=l = refl
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parityPrepend''' a (prependLetter (ofInv l) w x) | inr bad = exFalso (bad refl)
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notXor : (x y : Bool) → not (xor x y) ≡ xor (not x) y
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notXor BoolTrue BoolTrue = refl
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notXor BoolTrue BoolFalse = refl
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notXor BoolFalse BoolTrue = refl
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notXor BoolFalse BoolFalse = refl
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parityHomIsHom : (a : A) → (x y : ReducedWord) → parity a (_+W_ x y) ≡ xor (parity a x) (parity a y)
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parityHomIsHom a empty y = refl
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parityHomIsHom a (prependLetter (ofLetter l) x _) y with decA a l
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parityHomIsHom a (prependLetter (ofLetter .a) x _) y | inl refl = transitivity (parityPrepend'' a (x +W y)) (transitivity (notWellDefined (parityHomIsHom a x y)) (notXor (parity a x) (parity a y)))
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parityHomIsHom a (prependLetter (ofLetter l) x _) y | inr a!=l = transitivity (parityPrepend a (_+W_ x y) l a!=l) (parityHomIsHom a x y)
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parityHomIsHom a (prependLetter (ofInv l) x _) y with decA a l
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parityHomIsHom a (prependLetter (ofInv .a) x _) y | inl refl = transitivity (parityPrepend''' a (x +W y)) (transitivity (notWellDefined (parityHomIsHom a x y)) (notXor (parity a x) (parity a y)))
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parityHomIsHom a (prependLetter (ofInv l) x _) y | inr a!=l = transitivity (parityPrepend' a (x +W y) l a!=l) (parityHomIsHom a x y)
|
||||
parityHomIsHom : (a : A) → (x y : ReducedWord) → parity a (_+W_ x y) ≡ xor (parity a x) (parity a y)
|
||||
parityHomIsHom a empty y = refl
|
||||
parityHomIsHom a (prependLetter (ofLetter l) x _) y with decA a l
|
||||
parityHomIsHom a (prependLetter (ofLetter .a) x _) y | inl refl = transitivity (parityPrepend'' a (x +W y)) (transitivity (applyEquality not (parityHomIsHom a x y)) (notXor (parity a x) (parity a y)))
|
||||
parityHomIsHom a (prependLetter (ofLetter l) x _) y | inr a!=l = transitivity (parityPrepend a (_+W_ x y) l a!=l) (parityHomIsHom a x y)
|
||||
parityHomIsHom a (prependLetter (ofInv l) x _) y with decA a l
|
||||
parityHomIsHom a (prependLetter (ofInv .a) x _) y | inl refl = transitivity (parityPrepend''' a (x +W y)) (transitivity (applyEquality not (parityHomIsHom a x y)) (notXor (parity a x) (parity a y)))
|
||||
parityHomIsHom a (prependLetter (ofInv l) x _) y | inr a!=l = transitivity (parityPrepend' a (x +W y) l a!=l) (parityHomIsHom a x y)
|
||||
|
||||
parityHom : (x : A) → GroupHom (freeGroup) C2 (parity x)
|
||||
GroupHom.groupHom (parityHom x) {y} {z} = parityHomIsHom x y z
|
||||
|
||||
@@ -19,60 +19,64 @@ open import Groups.FreeProduct.Setoid decidableIndex decidableGroups G
|
||||
open import Groups.FreeProduct.Addition decidableIndex decidableGroups G
|
||||
open import Groups.FreeProduct.Group decidableIndex decidableGroups G
|
||||
|
||||
universalPropertyFunction' : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {i : I} → ReducedSequenceBeginningWith i → C
|
||||
universalPropertyFunction' {_+_ = _+_} H fs homs {i} (ofEmpty .i g nonZero) = fs i g
|
||||
universalPropertyFunction' {_+_ = _+_} H fs homs {i} (prependLetter .i g nonZero x x₁) = (fs i g) + universalPropertyFunction' H fs homs x
|
||||
private
|
||||
universalPropertyFunction' : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {i : I} → ReducedSequenceBeginningWith i → C
|
||||
universalPropertyFunction' {_+_ = _+_} H fs homs {i} (ofEmpty .i g nonZero) = fs i g
|
||||
universalPropertyFunction' {_+_ = _+_} H fs homs {i} (prependLetter .i g nonZero x x₁) = (fs i g) + universalPropertyFunction' H fs homs x
|
||||
|
||||
universalPropertyFunction : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → ReducedSequence → C
|
||||
universalPropertyFunction H fs homs empty = Group.0G H
|
||||
universalPropertyFunction H fs homs (nonempty i x) = universalPropertyFunction' H fs homs x
|
||||
|
||||
upWellDefined' : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {m n : I} → (x : ReducedSequenceBeginningWith m) (y : ReducedSequenceBeginningWith n) → (eq : =RP' x y) → Setoid._∼_ T (universalPropertyFunction H fs homs (nonempty m x)) (universalPropertyFunction H fs homs (nonempty n y))
|
||||
upWellDefined' H fs homs (ofEmpty m g nonZero) (ofEmpty n g₁ nonZero₁) eq with decidableIndex m n
|
||||
... | inl refl = GroupHom.wellDefined (homs m) eq
|
||||
upWellDefined' H fs homs (prependLetter m g nonZero x x₁) (prependLetter n g₁ nonZero₁ y x₂) eq with decidableIndex m n
|
||||
... | inl refl = Group.+WellDefined H (GroupHom.wellDefined (homs m) (_&&_.fst eq)) (upWellDefined' H fs homs x y (_&&_.snd eq))
|
||||
private
|
||||
upWellDefined' : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {m n : I} → (x : ReducedSequenceBeginningWith m) (y : ReducedSequenceBeginningWith n) → (eq : =RP' x y) → Setoid._∼_ T (universalPropertyFunction H fs homs (nonempty m x)) (universalPropertyFunction H fs homs (nonempty n y))
|
||||
upWellDefined' H fs homs (ofEmpty m g nonZero) (ofEmpty n g₁ nonZero₁) eq with decidableIndex m n
|
||||
... | inl refl = GroupHom.wellDefined (homs m) eq
|
||||
upWellDefined' H fs homs (prependLetter m g nonZero x x₁) (prependLetter n g₁ nonZero₁ y x₂) eq with decidableIndex m n
|
||||
... | inl refl = Group.+WellDefined H (GroupHom.wellDefined (homs m) (_&&_.fst eq)) (upWellDefined' H fs homs x y (_&&_.snd eq))
|
||||
|
||||
upWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → (x : ReducedSequence) (y : ReducedSequence) → (eq : _=RP_ x y) → Setoid._∼_ T (universalPropertyFunction H fs homs x) (universalPropertyFunction H fs homs y)
|
||||
upWellDefined {T = T} H fs homs empty empty eq = Equivalence.reflexive (Setoid.eq T)
|
||||
upWellDefined H fs homs (nonempty i w1) (nonempty j w2) eq = upWellDefined' H fs homs w1 w2 eq
|
||||
upWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → (x : ReducedSequence) (y : ReducedSequence) → (eq : _=RP_ x y) → Setoid._∼_ T (universalPropertyFunction H fs homs x) (universalPropertyFunction H fs homs y)
|
||||
upWellDefined {T = T} H fs homs empty empty eq = Equivalence.reflexive (Setoid.eq T)
|
||||
upWellDefined H fs homs (nonempty i w1) (nonempty j w2) eq = upWellDefined' H fs homs w1 w2 eq
|
||||
|
||||
upPrepend : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {j : I} (y : ReducedSequence) → (g : A j) .(pr : _) → Setoid._∼_ T (universalPropertyFunction H fs homs (prepend j g pr y)) ((fs j g) + universalPropertyFunction H fs homs y)
|
||||
upPrepend {T = T} H fs homs empty g pr = Equivalence.symmetric (Setoid.eq T) (Group.identRight H)
|
||||
upPrepend {T = T} H fs homs {j} (nonempty i (ofEmpty .i h nonZero)) g pr with decidableIndex j i
|
||||
... | inr j!=i = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inl x = Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (imageOfIdentityIsIdentity (homs j))) (Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (GroupHom.wellDefined (homs j) x)) (GroupHom.groupHom (homs j)))
|
||||
... | inr x = GroupHom.groupHom (homs j)
|
||||
upPrepend {T = T} H fs homs {j} (nonempty k (prependLetter .k h nonZero y _)) g pr with decidableIndex j k
|
||||
... | inr j!=k = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inl x = transitive (symmetric (Group.identLeft H)) (transitive (Group.+WellDefined H (transitive (symmetric (imageOfIdentityIsIdentity (homs k))) (transitive (GroupHom.wellDefined (homs k) (Equivalence.symmetric (Setoid.eq (S k)) x)) (GroupHom.groupHom (homs k)))) reflexive) (symmetric (Group.+Associative H)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
... | inr x = transitive (Group.+WellDefined H (GroupHom.groupHom (homs k)) reflexive) (symmetric (Group.+Associative H))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
private
|
||||
upPrepend : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {j : I} (y : ReducedSequence) → (g : A j) .(pr : _) → Setoid._∼_ T (universalPropertyFunction H fs homs (prepend j g pr y)) ((fs j g) + universalPropertyFunction H fs homs y)
|
||||
upPrepend {T = T} H fs homs empty g pr = Equivalence.symmetric (Setoid.eq T) (Group.identRight H)
|
||||
upPrepend {T = T} H fs homs {j} (nonempty i (ofEmpty .i h nonZero)) g pr with decidableIndex j i
|
||||
... | inr j!=i = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inl x = Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (imageOfIdentityIsIdentity (homs j))) (Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (GroupHom.wellDefined (homs j) x)) (GroupHom.groupHom (homs j)))
|
||||
... | inr x = GroupHom.groupHom (homs j)
|
||||
upPrepend {T = T} H fs homs {j} (nonempty k (prependLetter .k h nonZero y _)) g pr with decidableIndex j k
|
||||
... | inr j!=k = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inl x = transitive (symmetric (Group.identLeft H)) (transitive (Group.+WellDefined H (transitive (symmetric (imageOfIdentityIsIdentity (homs k))) (transitive (GroupHom.wellDefined (homs k) (Equivalence.symmetric (Setoid.eq (S k)) x)) (GroupHom.groupHom (homs k)))) reflexive) (symmetric (Group.+Associative H)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
... | inr x = transitive (Group.+WellDefined H (GroupHom.groupHom (homs k)) reflexive) (symmetric (Group.+Associative H))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
upHom : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {i : I} (x : ReducedSequenceBeginningWith i) (y : ReducedSequence) → Setoid._∼_ T (universalPropertyFunction H fs homs (plus' x y)) (universalPropertyFunction' H fs homs x + universalPropertyFunction H fs homs y)
|
||||
upHom {T = T} H fs homs (ofEmpty _ g nonZero) empty = Equivalence.symmetric (Setoid.eq T) (Group.identRight H)
|
||||
upHom {T = T} H fs homs (ofEmpty j g nonZero) (nonempty i (ofEmpty .i h nonZero1)) with decidableIndex j i
|
||||
... | inr j!=i = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inl x = Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (imageOfIdentityIsIdentity (homs j))) (Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (GroupHom.wellDefined (homs j) x)) (GroupHom.groupHom (homs j)))
|
||||
... | inr x = GroupHom.groupHom (homs j)
|
||||
upHom {T = T} H fs homs (ofEmpty j g nonZero) (nonempty i (prependLetter .i h nonZero1 x x₁)) with decidableIndex j i
|
||||
... | inr j!=i = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inr _ = Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H (GroupHom.groupHom (homs j)) (Equivalence.reflexive (Setoid.eq T))) (Equivalence.symmetric (Setoid.eq T) (Group.+Associative H))
|
||||
... | inl eq1 = Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (Group.identLeft H)) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H (Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (imageOfIdentityIsIdentity (homs j))) (Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (homs j) (Equivalence.symmetric (Setoid.eq (S j)) eq1)) (GroupHom.groupHom (homs j)))) (Equivalence.reflexive (Setoid.eq T))) (Equivalence.symmetric (Setoid.eq T) (Group.+Associative H)))
|
||||
upHom {T = T} H fs homs (prependLetter j g nonZero {k} w k!=j) empty = Equivalence.transitive (Setoid.eq T) (Equivalence.transitive (Setoid.eq T) (upWellDefined H fs homs (plus' (prependLetter j g _ w k!=j) empty) (prepend j g _ (nonempty k w)) (prependWD' g nonZero (plus' w empty) (nonempty k w) (plusEmptyRight w))) (upPrepend H fs homs (nonempty k w) g nonZero)) (Equivalence.symmetric (Setoid.eq T) (Group.identRight H))
|
||||
upHom {T = T} H fs homs (prependLetter j g nonZero {k} m k!=j) (nonempty i x2) = transitive (upPrepend H fs homs (plus' m (nonempty i x2)) g nonZero) (transitive (Group.+WellDefined H reflexive (upHom H fs homs m (nonempty i x2))) (Group.+Associative H))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
private
|
||||
upHom : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {i : I} (x : ReducedSequenceBeginningWith i) (y : ReducedSequence) → Setoid._∼_ T (universalPropertyFunction H fs homs (plus' x y)) (universalPropertyFunction' H fs homs x + universalPropertyFunction H fs homs y)
|
||||
upHom {T = T} H fs homs (ofEmpty _ g nonZero) empty = Equivalence.symmetric (Setoid.eq T) (Group.identRight H)
|
||||
upHom {T = T} H fs homs (ofEmpty j g nonZero) (nonempty i (ofEmpty .i h nonZero1)) with decidableIndex j i
|
||||
... | inr j!=i = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inl x = Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (imageOfIdentityIsIdentity (homs j))) (Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (GroupHom.wellDefined (homs j) x)) (GroupHom.groupHom (homs j)))
|
||||
... | inr x = GroupHom.groupHom (homs j)
|
||||
upHom {T = T} H fs homs (ofEmpty j g nonZero) (nonempty i (prependLetter .i h nonZero1 x x₁)) with decidableIndex j i
|
||||
... | inr j!=i = Equivalence.reflexive (Setoid.eq T)
|
||||
... | inl refl with decidableGroups j ((j + g) h) (Group.0G (G j))
|
||||
... | inr _ = Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H (GroupHom.groupHom (homs j)) (Equivalence.reflexive (Setoid.eq T))) (Equivalence.symmetric (Setoid.eq T) (Group.+Associative H))
|
||||
... | inl eq1 = Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (Group.identLeft H)) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H (Equivalence.transitive (Setoid.eq T) (Equivalence.symmetric (Setoid.eq T) (imageOfIdentityIsIdentity (homs j))) (Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (homs j) (Equivalence.symmetric (Setoid.eq (S j)) eq1)) (GroupHom.groupHom (homs j)))) (Equivalence.reflexive (Setoid.eq T))) (Equivalence.symmetric (Setoid.eq T) (Group.+Associative H)))
|
||||
upHom {T = T} H fs homs (prependLetter j g nonZero {k} w k!=j) empty = Equivalence.transitive (Setoid.eq T) (Equivalence.transitive (Setoid.eq T) (upWellDefined H fs homs (plus' (prependLetter j g _ w k!=j) empty) (prepend j g _ (nonempty k w)) (prependWD' g nonZero (plus' w empty) (nonempty k w) (plusEmptyRight w))) (upPrepend H fs homs (nonempty k w) g nonZero)) (Equivalence.symmetric (Setoid.eq T) (Group.identRight H))
|
||||
upHom {T = T} H fs homs (prependLetter j g nonZero {k} m k!=j) (nonempty i x2) = transitive (upPrepend H fs homs (plus' m (nonempty i x2)) g nonZero) (transitive (Group.+WellDefined H reflexive (upHom H fs homs m (nonempty i x2))) (Group.+Associative H))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
universalPropertyHom : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → GroupHom FreeProductGroup H (universalPropertyFunction H fs homs)
|
||||
GroupHom.wellDefined (universalPropertyHom {T = T} H fs homs) {x} {y} eq = upWellDefined H fs homs x y eq
|
||||
@@ -86,29 +90,30 @@ GroupHom.groupHom (universalPropertyHom H fs homs) {nonempty i x} {nonempty j y}
|
||||
universalPropertyFunctionHasProperty : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → {i : I} → (g : A i) → (nz : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → Setoid._∼_ T (fs i g) (universalPropertyFunction H fs homs (injection g nz))
|
||||
universalPropertyFunctionHasProperty {T = T} H fs homs g nz = Equivalence.reflexive (Setoid.eq T)
|
||||
|
||||
universalPropertyFunctionUniquelyHasPropertyLemma : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → (otherFunction : ReducedSequence → C) → (isHom : GroupHom FreeProductGroup H otherFunction) → ({i : I} → (g : A i) → .(nz : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → Setoid._∼_ T (fs i g) (otherFunction (injection g nz))) → {k l : I} (neq : (k ≡ l) → False) (r : ReducedSequenceBeginningWith l) (g : A k) .(nz : (Setoid._∼_ (S k) g (Group.0G (G k)) → False)) → Setoid._∼_ T (otherFunction (nonempty k (prependLetter k g nz r neq))) (fs k g + universalPropertyFunction' H fs homs r)
|
||||
universalPropertyFunctionUniquelyHasPropertyLemma {T = T} H fs homs otherFunction hom x {k} {l} neq (ofEmpty .l g2 nonZero) g nz = transitive (GroupHom.wellDefined hom {nonempty k (prependLetter k g nz (ofEmpty l g2 nonZero) neq)} {nonempty _ (ofEmpty k g nz) +RP nonempty _ (ofEmpty l g2 nonZero)} t) (transitive (GroupHom.groupHom hom {nonempty k (ofEmpty k g nz)} {nonempty _ (ofEmpty l g2 nonZero)}) (Group.+WellDefined H (symmetric (x g nz)) (symmetric (x g2 nonZero))))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
t : Setoid._∼_ freeProductSetoid (nonempty k (prependLetter k g nz (ofEmpty l g2 nonZero) neq)) (prepend k g nz (nonempty l (ofEmpty l g2 nonZero)))
|
||||
t with decidableIndex k l
|
||||
... | inl p = exFalso (neq p)
|
||||
... | inr _ with decidableIndex k k
|
||||
... | inr bad = exFalso (bad refl)
|
||||
... | inl refl = Equivalence.reflexive (Setoid.eq (S k)) ,, =RP'reflex (ofEmpty l g2 _)
|
||||
universalPropertyFunctionUniquelyHasPropertyLemma {T = T} H fs homs otherFunction hom x {k} {l} neq (prependLetter .l h nonZero r pr) g nz = transitive (GroupHom.wellDefined hom {nonempty _ (prependLetter k g nz (prependLetter l h nonZero r pr) neq)} {(nonempty k (ofEmpty k g nz)) +RP (nonempty l (prependLetter l h nonZero r pr))} t) (transitive (GroupHom.groupHom hom {nonempty k (ofEmpty k g nz)} {nonempty l (prependLetter l h nonZero r pr)}) (Group.+WellDefined H (symmetric (x g nz)) (universalPropertyFunctionUniquelyHasPropertyLemma H fs homs otherFunction hom x pr r h nonZero)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
t : Setoid._∼_ freeProductSetoid (nonempty k (prependLetter k g nz (prependLetter l h nonZero r pr) neq)) (prepend k g nz (nonempty l (prependLetter l h nonZero r pr)))
|
||||
t with decidableIndex k l
|
||||
... | inl bad = exFalso (neq bad)
|
||||
... | inr k!=l with decidableIndex k k
|
||||
... | inr bad = exFalso (bad refl)
|
||||
... | inl refl with decidableIndex l l
|
||||
... | inr bad = exFalso (bad refl)
|
||||
... | inl refl = Equivalence.reflexive (Setoid.eq (S k)) ,, ((Equivalence.reflexive (Setoid.eq (S l))) ,, =RP'reflex r)
|
||||
private
|
||||
universalPropertyFunctionUniquelyHasPropertyLemma : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → (otherFunction : ReducedSequence → C) → (isHom : GroupHom FreeProductGroup H otherFunction) → ({i : I} → (g : A i) → .(nz : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → Setoid._∼_ T (fs i g) (otherFunction (injection g nz))) → {k l : I} (neq : (k ≡ l) → False) (r : ReducedSequenceBeginningWith l) (g : A k) .(nz : (Setoid._∼_ (S k) g (Group.0G (G k)) → False)) → Setoid._∼_ T (otherFunction (nonempty k (prependLetter k g nz r neq))) (fs k g + universalPropertyFunction' H fs homs r)
|
||||
universalPropertyFunctionUniquelyHasPropertyLemma {T = T} H fs homs otherFunction hom x {k} {l} neq (ofEmpty .l g2 nonZero) g nz = transitive (GroupHom.wellDefined hom {nonempty k (prependLetter k g nz (ofEmpty l g2 nonZero) neq)} {nonempty _ (ofEmpty k g nz) +RP nonempty _ (ofEmpty l g2 nonZero)} t) (transitive (GroupHom.groupHom hom {nonempty k (ofEmpty k g nz)} {nonempty _ (ofEmpty l g2 nonZero)}) (Group.+WellDefined H (symmetric (x g nz)) (symmetric (x g2 nonZero))))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
t : Setoid._∼_ freeProductSetoid (nonempty k (prependLetter k g nz (ofEmpty l g2 nonZero) neq)) (prepend k g nz (nonempty l (ofEmpty l g2 nonZero)))
|
||||
t with decidableIndex k l
|
||||
... | inl p = exFalso (neq p)
|
||||
... | inr _ with decidableIndex k k
|
||||
... | inr bad = exFalso (bad refl)
|
||||
... | inl refl = Equivalence.reflexive (Setoid.eq (S k)) ,, =RP'reflex (ofEmpty l g2 _)
|
||||
universalPropertyFunctionUniquelyHasPropertyLemma {T = T} H fs homs otherFunction hom x {k} {l} neq (prependLetter .l h nonZero r pr) g nz = transitive (GroupHom.wellDefined hom {nonempty _ (prependLetter k g nz (prependLetter l h nonZero r pr) neq)} {(nonempty k (ofEmpty k g nz)) +RP (nonempty l (prependLetter l h nonZero r pr))} t) (transitive (GroupHom.groupHom hom {nonempty k (ofEmpty k g nz)} {nonempty l (prependLetter l h nonZero r pr)}) (Group.+WellDefined H (symmetric (x g nz)) (universalPropertyFunctionUniquelyHasPropertyLemma H fs homs otherFunction hom x pr r h nonZero)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
t : Setoid._∼_ freeProductSetoid (nonempty k (prependLetter k g nz (prependLetter l h nonZero r pr) neq)) (prepend k g nz (nonempty l (prependLetter l h nonZero r pr)))
|
||||
t with decidableIndex k l
|
||||
... | inl bad = exFalso (neq bad)
|
||||
... | inr k!=l with decidableIndex k k
|
||||
... | inr bad = exFalso (bad refl)
|
||||
... | inl refl with decidableIndex l l
|
||||
... | inr bad = exFalso (bad refl)
|
||||
... | inl refl = Equivalence.reflexive (Setoid.eq (S k)) ,, ((Equivalence.reflexive (Setoid.eq (S l))) ,, =RP'reflex r)
|
||||
|
||||
universalPropertyFunctionUniquelyHasProperty : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+_ : C → C → C} (H : Group T _+_) → (fs : (i : I) → (A i → C)) → (homs : (i : I) → GroupHom (G i) H (fs i)) → (otherFunction : ReducedSequence → C) → (isHom : GroupHom FreeProductGroup H otherFunction) → ({i : I} → (g : A i) → .(nz : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → Setoid._∼_ T (fs i g) (otherFunction (injection g nz))) → (r : ReducedSequence) → Setoid._∼_ T (otherFunction r) (universalPropertyFunction H fs homs r)
|
||||
universalPropertyFunctionUniquelyHasProperty H fs homs otherFunction hom prop empty = imageOfIdentityIsIdentity hom
|
||||
|
||||
Reference in New Issue
Block a user