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A bit of cleanup (#69)

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Patrick Stevens
2019-11-13 18:35:41 +00:00
committed by GitHub
parent 4db82b1afc
commit 2ef7348d30
7 changed files with 64 additions and 56 deletions
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37
Groups/ActionIsSymmetry.agda
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@@ -8,31 +8,28 @@ open import Numbers.Naturals.Naturals
open import Sets.FinSet
open import Groups.Definition
open import Groups.Lemmas
open import Groups.Homomorphisms.Definition
open import Groups.Groups
open import Groups.SymmetricGroups.Definition
open import Groups.Groups2
open import Groups.Actions
open import Groups.Actions.Definition
open import Sets.EquivalenceRelations
module Groups.ActionIsSymmetry where
module Groups.ActionIsSymmetry {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 _+_} (gAction : GroupAction G T) where
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
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 } })
open Group G
open GroupAction gAction
actionPermutation : (g : A) → SymmetryGroupElements T
actionPermutation g = sym {f = λ x → action g x} (record { inj = record { injective = inj ; wellDefined = actionWellDefined2 } ; surj = record { surjective = surj ; wellDefined = actionWellDefined2 } })
where
open Setoid T
open Reflexive (Equivalence.reflexiveEq (Setoid.eq T))
open Symmetric (Equivalence.symmetricEq (Setoid.eq T))
open Transitive (Equivalence.transitiveEq (Setoid.eq T))
open Group G
inj : {x y : B} → (Setoid._∼_ T (GroupAction.action action g x) (GroupAction.action action g y)) → Setoid._∼_ T x y
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))
surj : {x : B} → Sg B (λ a → GroupAction.action action g a ∼ x)
surj {x} = GroupAction.action action (inverse g) x , transitive (symmetric (GroupAction.associativeAction action)) (transitive (GroupAction.actionWellDefined1 action invRight) (GroupAction.identityAction action))
open Equivalence eq
inj : {x y : B} → (Setoid._∼_ T (action g x) (action g y)) → Setoid._∼_ T x y
inj {x} {y} gx=gy = transitive (symmetric identityAction) (transitive (transitive (symmetric (actionWellDefined1 (invLeft {g}))) (transitive (transitive associativeAction (transitive (actionWellDefined2 gx=gy) (symmetric associativeAction))) (actionWellDefined1 (invLeft {g})))) identityAction)
surj : {x : B} → Sg B (λ a → action g a ∼ x)
surj {x} = action (inverse g) x , transitive (symmetric associativeAction) (transitive (actionWellDefined1 invRight) identityAction)
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)
GroupHom.groupHom (actionPermutationMapIsHom {T = T} action) = ExtensionallyEqual.eq λ {z} → GroupAction.associativeAction action
where
open Setoid T
open Reflexive (Equivalence.reflexiveEq (Setoid.eq T))
open Symmetric (Equivalence.symmetricEq (Setoid.eq T))
open Transitive (Equivalence.transitiveEq (Setoid.eq T))
GroupHom.wellDefined (actionPermutationMapIsHom action) x=y = ExtensionallyEqual.eq λ {z} → GroupAction.actionWellDefined1 action x=y
actionPermutationMapIsHom : GroupHom G (symmetricGroup T) actionPermutation
GroupHom.groupHom actionPermutationMapIsHom = associativeAction
GroupHom.wellDefined actionPermutationMapIsHom x=y = actionWellDefined1 x=y