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https://github.com/Smaug123/agdaproofs
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39 lines
2.7 KiB
Agda
39 lines
2.7 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.Naturals
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open import Sets.FinSet
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open import Groups.Definition
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open import Groups.Lemmas
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open import Groups.Groups
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open import Groups.SymmetryGroups
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open import Groups.Groups2
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open import Groups.Actions
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module Groups.ActionIsSymmetry where
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actionPermutation : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} → (action : GroupAction G T) → (g : A) → SymmetryGroupElements T
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actionPermutation {B = B} {T = T} {_+_ = _+_} {G = G} action g = sym {f = λ x → (GroupAction.action action g x)} (record { inj = record { injective = inj ; wellDefined = GroupAction.actionWellDefined2 action } ; surj = record { surjective = surj ; wellDefined = GroupAction.actionWellDefined2 action } })
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where
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open Setoid T
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq T))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq T))
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open Transitive (Equivalence.transitiveEq (Setoid.eq T))
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open Group G
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inj : {x y : B} → (Setoid._∼_ T (GroupAction.action action g x) (GroupAction.action action g y)) → Setoid._∼_ T x y
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inj {x} {y} gx=gy = transitive (symmetric (GroupAction.identityAction action)) (transitive (transitive (symmetric (GroupAction.actionWellDefined1 action (invLeft {g}))) (transitive (transitive (GroupAction.associativeAction action) (transitive (GroupAction.actionWellDefined2 action gx=gy) (symmetric (GroupAction.associativeAction action)))) (GroupAction.actionWellDefined1 action (invLeft {g})))) (GroupAction.identityAction action))
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surj : {x : B} → Sg B (λ a → GroupAction.action action g a ∼ x)
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surj {x} = GroupAction.action action (inverse g) x , transitive (symmetric (GroupAction.associativeAction action)) (transitive (GroupAction.actionWellDefined1 action invRight) (GroupAction.identityAction action))
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actionPermutationMapIsHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) → GroupHom G (symmetricGroup T) (actionPermutation action)
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GroupHom.groupHom (actionPermutationMapIsHom {T = T} action) = ExtensionallyEqual.eq λ {z} → GroupAction.associativeAction action
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where
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open Setoid T
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq T))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq T))
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open Transitive (Equivalence.transitiveEq (Setoid.eq T))
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GroupHom.wellDefined (actionPermutationMapIsHom action) x=y = ExtensionallyEqual.eq λ {z} → GroupAction.actionWellDefined1 action x=y
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