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Define the free product (#104)
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Patrick Stevens
GitHub
163
Groups/FreeGroup/Parity.agda
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163
Groups/FreeGroup/Parity.agda
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{-# OPTIONS --safe --warning=error #-}
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open import Sets.EquivalenceRelations
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open import Setoids.Setoids
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open import Groups.FreeGroup.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Decidable.Sets
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open import Numbers.Naturals.Order
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open import LogicalFormulae
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open import Semirings.Definition
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open import Functions
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open import Groups.Isomorphisms.Definition
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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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Group.inverse C2 = id
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Group.+Associative C2 {BoolTrue} {BoolTrue} {BoolTrue} = refl
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Group.+Associative C2 {BoolTrue} {BoolTrue} {BoolFalse} = refl
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Group.+Associative C2 {BoolTrue} {BoolFalse} {z} = refl
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Group.+Associative C2 {BoolFalse} {y} {z} = refl
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Group.identRight C2 {BoolTrue} = refl
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Group.identRight C2 {BoolFalse} = refl
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Group.identLeft C2 {x} = refl
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Group.invLeft C2 {BoolTrue} = refl
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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 DecidableSet.eq decA x y
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parity x (prependLetter (ofLetter y) w _) | inl _ = not (parity x w)
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parity x (prependLetter (ofLetter y) w _) | inr _ = parity x w
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parity x (prependLetter (ofInv y) w _) with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA a r
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parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA l r
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parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inl m with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA a r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA l r
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parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA a a
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parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA a a
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parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inl _ with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq decA a a
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... | inl _ with DecidableSet.eq 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 DecidableSet.eq decA a a
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parityPrepend''' a (prependLetter (ofInv l) w x) | inl _ with DecidableSet.eq 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 DecidableSet.eq 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 DecidableSet.eq 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)
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parityHom : (x : A) → GroupHom (freeGroup) C2 (parity x)
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GroupHom.groupHom (parityHom x) {y} {z} = parityHomIsHom x y z
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GroupHom.wellDefined (parityHom x) refl = refl
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