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https://github.com/Smaug123/agdaproofs
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Fix up the structure of Actions and SymmetricGroups (#67)
This commit is contained in:
Patrick Stevens
GitHub
26
Groups/Actions/Definition.agda
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26
Groups/Actions/Definition.agda
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{-# 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.Groups2
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open import Sets.EquivalenceRelations
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module Groups.Actions.Definition where
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record GroupAction {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·_ : A → A → A} {B : Set n} (G : Group S _·_) (X : Setoid {n} {p} B) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Group G
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open Setoid S renaming (_∼_ to _∼G_)
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open Setoid X renaming (_∼_ to _∼X_)
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field
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action : A → B → B
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actionWellDefined1 : {g h : A} → {x : B} → (g ∼G h) → action g x ∼X action h x
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actionWellDefined2 : {g : A} → {x y : B} → (x ∼X y) → action g x ∼X action g y
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identityAction : {x : B} → action 0G x ∼X x
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associativeAction : {x : B} → {g h : A} → action (g · h) x ∼X action g (action h x)
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25
Groups/Actions/Orbit.agda
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25
Groups/Actions/Orbit.agda
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{-# 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.Actions.Definition
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open import Groups.Groups2
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open import Sets.EquivalenceRelations
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module Groups.Actions.Orbit where
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data Orbit {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) (x : B) : Set (a ⊔ b ⊔ c ⊔ d) where
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orbitElt : (g : A) → Orbit action x
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orbitSetoid : {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) (x : B) → Setoid (Orbit action x)
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Setoid._∼_ (orbitSetoid {T = T} action x) (orbitElt a) (orbitElt b) = Setoid._∼_ T (GroupAction.action action a x) (GroupAction.action action b x)
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Equivalence.reflexive (Setoid.eq (orbitSetoid {T = T} action x)) {orbitElt g} = Equivalence.reflexive (Setoid.eq T)
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Equivalence.symmetric (Setoid.eq (orbitSetoid {T = T} action x)) {orbitElt g} {orbitElt h} = Equivalence.symmetric (Setoid.eq T)
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Equivalence.transitive (Setoid.eq (orbitSetoid {T = T} action x)) {orbitElt g} {orbitElt h} {orbitElt i} = Equivalence.transitive (Setoid.eq T)
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116
Groups/Actions/OrbitStabiliser.agda
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116
Groups/Actions/OrbitStabiliser.agda
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{-# 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.Actions.Definition
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open import Groups.Actions.Stabiliser
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open import Groups.Groups2
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open import Sets.EquivalenceRelations
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module Groups.Actions.OrbitStabiliser where
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orbitStabiliserEquivRel1 : {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) (x : B) → Rel A
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orbitStabiliserEquivRel1 {T = T} action x g1 g2 = Setoid._∼_ T (GroupAction.action action g1 x) (GroupAction.action action g2 x)
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orbitStabiliserEquivRel2 : {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) (x : B) → Rel A
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orbitStabiliserEquivRel2 {T = T} {_+_ = _+_} {G = G} action x g1 g2 = Setoid._∼_ T (GroupAction.action action ((Group.inverse G g2) + g1) x) x
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osEquivRel1Equiv : {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) (x : B) → Equivalence (orbitStabiliserEquivRel1 action x)
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Equivalence.reflexive (osEquivRel1Equiv {T = T} action x) {a} = Equivalence.reflexive (Setoid.eq T)
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Equivalence.symmetric (osEquivRel1Equiv {T = T} action x) {a} {b} = Equivalence.symmetric (Setoid.eq T)
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Equivalence.transitive (osEquivRel1Equiv {T = T} action x) {a} {b} {c} = Equivalence.transitive (Setoid.eq T)
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osEquivRel2Equiv : {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) (x : B) → Equivalence (orbitStabiliserEquivRel2 action x)
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Equivalence.reflexive (osEquivRel2Equiv {T = T} {G = G} action x) = transitive (GroupAction.actionWellDefined1 action (Group.invLeft G)) (GroupAction.identityAction action)
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where
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open Equivalence (Setoid.eq T)
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Equivalence.symmetric (osEquivRel2Equiv {S = S} {T = T} {_+_ = _+_} {G = G} gAction x) {b} {c} b=c = need
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where
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open Equivalence (Setoid.eq T)
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open Equivalence (Setoid.eq S) renaming (symmetric to symmetricS ; transitive to transitiveS ; reflexive to reflexiveS)
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open Setoid T
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open GroupAction gAction
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open Group G
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have : action ((inverse c) + b) x ∼ x
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have = b=c
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p : action (inverse (inverse b + c)) x ∼ x
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p = transitive (actionWellDefined1 (transitiveS (invContravariant G) (Group.+WellDefined G reflexiveS (invInv G)))) have
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q : action ((inverse b) + c) (action (inverse (inverse b + c)) x) ∼ action ((inverse b) + c) x
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q = actionWellDefined2 p
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r : action (((inverse b) + c) + (inverse (inverse b + c))) x ∼ action ((inverse b) + c) x
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r = transitive associativeAction q
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s : action identity x ∼ action ((inverse b) + c) x
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s = transitive (actionWellDefined1 (symmetricS invRight)) r
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need : action ((inverse b) + c) x ∼ x
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need = symmetric (transitive (symmetric identityAction) s)
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Equivalence.transitive (osEquivRel2Equiv {S = S} {T = T} {_+_ = _+_} {G = G} gAction x) {a} {b} {c} a=b b=c = need
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where
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open Equivalence (Setoid.eq T)
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open Equivalence (Setoid.eq S) renaming (symmetric to symmetricS ; reflexive to reflexiveS ; transitive to transitiveS)
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open Setoid T
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open GroupAction gAction
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open Group G
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have1 : action ((inverse c) + b) x ∼ x
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have1 = b=c
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have2 : action ((inverse b) + a) x ∼ x
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have2 = a=b
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p : action ((inverse c) + b) (action (inverse b + a) x) ∼ x
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p = transitive (actionWellDefined2 have2) have1
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q : action ((inverse c + b) + (inverse b + a)) x ∼ x
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q = transitive associativeAction p
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need : action ((inverse c) + a) x ∼ x
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need = transitive (actionWellDefined1 (transitiveS (+WellDefined reflexiveS (transitiveS (symmetricS identLeft) (+WellDefined (symmetricS invRight) reflexiveS))) (fourWayAssoc G))) q
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osEquivRelsEqual' : {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) (x : B) {g h : A} → (orbitStabiliserEquivRel2 action x) g h → (orbitStabiliserEquivRel1 action x) g h
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osEquivRelsEqual' {S = S} {T = T} {_+_ = _+_} {G = G} action x {g} {h} g=h = need
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where
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open Setoid T
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open Group G
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open Equivalence eq
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open Equivalence (Setoid.eq S) renaming (symmetric to symmetricS ; transitive to transitiveS)
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have : (GroupAction.action action ((inverse h) + g) x) ∼ x
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have = g=h
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p : (GroupAction.action action (inverse h) (GroupAction.action action g x)) ∼ x
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p = transitive (symmetric (GroupAction.associativeAction action)) have
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q : (GroupAction.action action h (GroupAction.action action (inverse h) (GroupAction.action action g x))) ∼ GroupAction.action action h x
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q = GroupAction.actionWellDefined2 action p
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r : (GroupAction.action action (h + inverse h) (GroupAction.action action g x)) ∼ GroupAction.action action h x
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r = transitive (GroupAction.associativeAction action) q
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s : (GroupAction.action action identity (GroupAction.action action g x)) ∼ GroupAction.action action h x
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s = transitive (GroupAction.actionWellDefined1 action (symmetricS (Group.invRight G))) r
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need : (GroupAction.action action g x) ∼ (GroupAction.action action h x)
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need = transitive (symmetric (GroupAction.identityAction action)) s
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osEquivRelsEqual : {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) (x : B) {g h : A} → (orbitStabiliserEquivRel1 action x) g h → (orbitStabiliserEquivRel2 action x) g h
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osEquivRelsEqual {T = T} {_+_ = _+_} {G = G} action x {g} {h} g=h = need
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where
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open Setoid T
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open Group G
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open Equivalence eq
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have : (GroupAction.action action g x) ∼ (GroupAction.action action h x)
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have = g=h
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p : GroupAction.action action (inverse h) (GroupAction.action action g x) ∼ GroupAction.action action (inverse h) (GroupAction.action action h x)
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p = GroupAction.actionWellDefined2 action have
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need : (GroupAction.action action ((inverse h) + g) x) ∼ x
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need = transitive (GroupAction.associativeAction action) (transitive p (transitive (symmetric (GroupAction.associativeAction action)) (transitive (GroupAction.actionWellDefined1 action (Group.invLeft G)) (GroupAction.identityAction action))))
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orbitStabiliserBijection : {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) (x : B) → A → ((Stabiliser action x) && Orbit action x)
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orbitStabiliserBijection action x g = stab {!GroupAction.action action ? ?!} {!!} ,, orbitElt g
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where
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osBijWellDefined : {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) (x : B) → {r s : A} → (Setoid._∼_ S r s) → Setoid._∼_ (directSumSetoid (stabiliserSetoid action x) (orbitSetoid action x)) (orbitStabiliserBijection action x r) (orbitStabiliserBijection action x s)
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_&&_.fst (osBijWellDefined action x r~s) = {!!}
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_&&_.snd (osBijWellDefined action x r~s) = GroupAction.actionWellDefined1 action r~s
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orbitStabiliserTheorem : {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) (x : B) → SetoidBijection S (directSumSetoid (stabiliserSetoid action x) (orbitSetoid action x)) (orbitStabiliserBijection action x)
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SetoidInjection.+WellDefined (SetoidBijection.inj (orbitStabiliserTheorem {S = S} action x)) = osBijWellDefined action x
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SetoidInjection.injective (SetoidBijection.inj (orbitStabiliserTheorem {S = S} action x)) {g} {h} (fst ,, gx=hx) = {!!}
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SetoidSurjection.+WellDefined (SetoidBijection.surj (orbitStabiliserTheorem action x)) = osBijWellDefined action x
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SetoidSurjection.surjective (SetoidBijection.surj (orbitStabiliserTheorem action x)) {stab g gx=x ,, orbitElt h} = h , ({!!} ,, {!!})
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62
Groups/Actions/Stabiliser.agda
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62
Groups/Actions/Stabiliser.agda
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{-# 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.Subgroups.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Actions.Definition
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open import Groups.Groups2
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open import Sets.EquivalenceRelations
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open import Groups.Actions.Definition
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module Groups.Actions.Stabiliser where
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data Stabiliser {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) (x : B) : Set (a ⊔ b ⊔ c ⊔ d) where
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stab : (g : A) → Setoid._∼_ T (GroupAction.action action g x) x → Stabiliser action x
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stabiliserSetoid : {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) (x : B) → Setoid (Stabiliser action x)
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Setoid._∼_ (stabiliserSetoid {S = S} action x) (stab g gx=x) (stab h hx=x) = Setoid._∼_ S g h
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Equivalence.reflexive (Setoid.eq (stabiliserSetoid {S = S} action x)) {stab g _} = Equivalence.reflexive (Setoid.eq S)
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Equivalence.symmetric (Setoid.eq (stabiliserSetoid {S = S} action x)) {stab g _} {stab h _} = Equivalence.symmetric (Setoid.eq S)
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Equivalence.transitive (Setoid.eq (stabiliserSetoid {S = S} action x)) {stab g _} {stab h _} {stab i _} = Equivalence.transitive (Setoid.eq S)
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stabiliserGroupOp : {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) {x : B} → Stabiliser action x → Stabiliser action x → Stabiliser action x
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stabiliserGroupOp {T = T} {_+_ = _+_} action (stab p px=x) (stab q qx=x) = stab (p + q) (transitive (GroupAction.associativeAction action) (transitive (GroupAction.actionWellDefined2 action qx=x) px=x))
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where
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open Setoid T
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open Equivalence eq
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stabiliserGroup : {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) {x : B} → Group (stabiliserSetoid action x) (stabiliserGroupOp action)
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Group.+WellDefined (stabiliserGroup {T = T} {G = G} action {x}) {stab m mx=x} {stab n nx=x} {stab r rx=x} {stab s sx=x} m=r n=s = Group.+WellDefined G m=r n=s
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where
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open Setoid T
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open Equivalence eq
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Group.0G (stabiliserGroup {G = G} action) = stab (Group.0G G) (GroupAction.identityAction action)
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Group.inverse (stabiliserGroup {T = T} {_+_ = _+_} {G = G} action {x}) (stab g gx=x) = stab (Group.inverse G g) (transitive {_} {GroupAction.action action ((inverse g) + g) x} (symmetric (transitive (GroupAction.associativeAction action) (GroupAction.actionWellDefined2 action gx=x))) (transitive (GroupAction.actionWellDefined1 action invLeft) (GroupAction.identityAction action)))
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where
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open Group G
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open Setoid T
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open Equivalence eq
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Group.+Associative (stabiliserGroup {G = G} action) {stab m mx=x} {stab n nx=x} {stab o ox=x} = Group.+Associative G
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Group.identRight (stabiliserGroup {G = G} action) {stab m mx=x} = Group.identRight G
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Group.identLeft (stabiliserGroup {G = G} action) {stab m mx=x }= Group.identLeft G
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Group.invLeft (stabiliserGroup {G = G} action) {stab m mx=x} = Group.invLeft G
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Group.invRight (stabiliserGroup {G = G} action) {stab m mx=x} = Group.invRight G
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stabiliserInjection : {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) {x : B} → Stabiliser action x → A
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stabiliserInjection action (stab g gx=x) = g
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stabiliserInjectionIsHom : {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) {x : B} → GroupHom (stabiliserGroup action {x}) G (stabiliserInjection action {x})
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GroupHom.groupHom (stabiliserInjectionIsHom {S = S} action) {stab g gx=x} {stab h hx=x} = Equivalence.reflexive (Setoid.eq S)
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GroupHom.wellDefined (stabiliserInjectionIsHom action) {stab g gx=x} {stab h hx=x} g=h = g=h
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stabiliserIsSubgroup : {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) {x : B} → Subgroup G (stabiliserGroup action) (stabiliserInjectionIsHom action {x})
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SetoidInjection.wellDefined (Subgroup.fInj (stabiliserIsSubgroup action)) {stab g gx=x} {stab h hx=x} g=h = g=h
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SetoidInjection.injective (Subgroup.fInj (stabiliserIsSubgroup action)) {stab g gx=x} {stab h hx=x} g=h = g=h
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