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https://github.com/Smaug123/agdaproofs
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A bit of cleanup (#69)
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@@ -9,28 +9,14 @@ open import Sets.FinSet
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open import Groups.Groups
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open import Groups.Definition
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open import Sets.EquivalenceRelations
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open import Setoids.Functions.Extension
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module Groups.SymmetricGroups.Definition where
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data SymmetryGroupElements {a b : _} {A : Set a} (S : Setoid {a} {b} A) : Set (a ⊔ b) where
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sym : {f : A → A} → SetoidBijection S S f → SymmetryGroupElements S
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WellDefined : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) → Set (a ⊔ c ⊔ d)
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WellDefined {A = A} S T f = {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)
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data ExtensionallyEqual {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f g : A → B) (fWd : WellDefined S T f) (gWd : WellDefined S T g) : Set (a ⊔ b ⊔ c ⊔ d) where
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eq : ({x : A} → Setoid._∼_ T (f x) (g x)) → ExtensionallyEqual S T f g fWd gWd
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extensionallyEqualReflexive : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) (fWD1 fWD2 : WellDefined S T f) → ExtensionallyEqual S T f f fWD1 fWD2
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extensionallyEqualReflexive S T f fWD1 _ = eq (Equivalence.reflexive (Setoid.eq T))
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extensionallyEqualSymmetric : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f g : A → B) (fWD : WellDefined S T f) (gWD : WellDefined S T g) → ExtensionallyEqual S T f g fWD gWD → ExtensionallyEqual S T g f gWD fWD
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extensionallyEqualSymmetric S T f g fWD gWD (eq pr) = eq (Equivalence.symmetric (Setoid.eq T) pr)
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extensionallyEqualTransitive : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f g h : A → B) (fWD : WellDefined S T f) (gWD : WellDefined S T g) (hWD : WellDefined S T h) → ExtensionallyEqual S T f g fWD gWD → ExtensionallyEqual S T g h gWD hWD → ExtensionallyEqual S T f h fWD hWD
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extensionallyEqualTransitive S T f g h fWD gWD hWD (eq pr1) (eq pr2) = eq (Equivalence.transitive (Setoid.eq T) pr1 pr2)
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symmetricSetoid : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → Setoid (SymmetryGroupElements S)
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Setoid._∼_ (symmetricSetoid S) (sym {f} bijF) (sym {g} bijG) = ExtensionallyEqual S S f g (SetoidBijection.wellDefined bijF) (SetoidBijection.wellDefined bijG)
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Setoid._∼_ (symmetricSetoid S) (sym {f} bijF) (sym {g} bijG) = ExtensionallyEqual {S = S} {S} (SetoidBijection.wellDefined bijF) (SetoidBijection.wellDefined bijG)
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Equivalence.reflexive (Setoid.eq (symmetricSetoid S)) {sym {f} bijF} = extensionallyEqualReflexive S S f (SetoidBijection.wellDefined bijF) (SetoidBijection.wellDefined bijF)
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Equivalence.symmetric (Setoid.eq (symmetricSetoid S)) {sym {f} bijF} {sym {g} bijG} f~g = extensionallyEqualSymmetric S S f g (SetoidBijection.wellDefined bijF) (SetoidBijection.wellDefined bijG) f~g
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Equivalence.transitive (Setoid.eq (symmetricSetoid S)) {sym {f} bijF} {sym {g} bijG} {sym {h} bijH} f~g g~h = extensionallyEqualTransitive S S f g h (SetoidBijection.wellDefined bijF) (SetoidBijection.wellDefined bijG) (SetoidBijection.wellDefined bijH) f~g g~h
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@@ -43,27 +29,27 @@ module Groups.SymmetricGroups.Definition where
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... | record { inverse = inverse ; inverseWellDefined = iwd ; isLeft = isLeft ; isRight = isRight } = sym (setoidInvertibleImpliesBijective (record { fWellDefined = iwd ; inverse = f ; inverseWellDefined = SetoidInjection.wellDefined (SetoidBijection.inj bijF) ; isLeft = λ b → isRight b ; isRight = λ b → isLeft b }))
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symmetricGroupInvIsLeft : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → {x : SymmetryGroupElements S} → Setoid._∼_ (symmetricSetoid S) (symmetricGroupOp (symmetricGroupInv S x) x) (sym setoidIdIsBijective)
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symmetricGroupInvIsLeft {A = A} S {sym {f = f} fBij} = ExtensionallyEqual.eq ans
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symmetricGroupInvIsLeft {A = A} S {sym {f = f} fBij} = ans
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where
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ans : {x : A} → Setoid._∼_ S (SetoidInvertible.inverse (setoidBijectiveImpliesInvertible fBij) (f x)) x
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ans {x} with SetoidSurjection.surjective (SetoidBijection.surj fBij) {f x}
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ans {x} | a , b = SetoidInjection.injective (SetoidBijection.inj fBij) b
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symmetricGroupInvIsRight : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → {x : SymmetryGroupElements S} → Setoid._∼_ (symmetricSetoid S) (symmetricGroupOp x (symmetricGroupInv S x)) (sym setoidIdIsBijective)
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symmetricGroupInvIsRight {A = A} S {sym {f = f} fBij} = ExtensionallyEqual.eq ans
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symmetricGroupInvIsRight {A = A} S {sym {f = f} fBij} = ans
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where
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ans : {x : A} → Setoid._∼_ S (f (SetoidInvertible.inverse (setoidBijectiveImpliesInvertible fBij) x)) x
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ans {x} with SetoidSurjection.surjective (SetoidBijection.surj fBij) {x}
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ans {x} | a , b = b
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symmetricGroup : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → Group (symmetricSetoid S) (symmetricGroupOp {A = A})
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Group.+WellDefined (symmetricGroup A) {sym {m} bijM} {sym {n} bijN} {sym {x} bijX} {sym {y} bijY} (ExtensionallyEqual.eq m~x) (ExtensionallyEqual.eq n~y) = ExtensionallyEqual.eq (transitive m~x (SetoidBijection.wellDefined bijX n~y))
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Group.+WellDefined (symmetricGroup A) {sym {m} bijM} {sym {n} bijN} {sym {x} bijX} {sym {y} bijY} m~x n~y = transitive m~x (SetoidBijection.wellDefined bijX n~y)
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where
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open Equivalence (Setoid.eq A)
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Group.0G (symmetricGroup A) = sym setoidIdIsBijective
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Group.inverse (symmetricGroup S) = symmetricGroupInv S
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Group.+Associative (symmetricGroup A) {sym {f} bijF} {sym {g} bijG} {sym {h} bijH} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
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Group.identRight (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
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Group.identLeft (symmetricGroup A) {sym {f} bijF} = ExtensionallyEqual.eq λ {x} → Equivalence.reflexive (Setoid.eq A)
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Group.+Associative (symmetricGroup A) {sym {f} bijF} {sym {g} bijG} {sym {h} bijH} = Equivalence.reflexive (Setoid.eq A)
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Group.identRight (symmetricGroup A) {sym {f} bijF} = Equivalence.reflexive (Setoid.eq A)
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Group.identLeft (symmetricGroup A) {sym {f} bijF} = Equivalence.reflexive (Setoid.eq A)
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Group.invLeft (symmetricGroup S) {x} = symmetricGroupInvIsLeft S {x}
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Group.invRight (symmetricGroup S) {x} = symmetricGroupInvIsRight S {x}
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