{-# OPTIONS --safe --warning=error --without-K #-}

open import Setoids.Setoids
open import Groups.Definition
open import Groups.Lemmas
open import Groups.Homomorphisms.Definition
open import Groups.QuotientGroup.Definition
open import Groups.Homomorphisms.Lemmas
open import Groups.Actions.Definition
open import Sets.EquivalenceRelations

module Groups.SymmetricGroups.Lemmas where

trivialAction : {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) → GroupAction G X
trivialAction G X = record { action = λ _ x → x ; actionWellDefined1 = λ _ → reflexive ; actionWellDefined2 = λ wd1 → wd1 ; identityAction = reflexive ; associativeAction = reflexive }
  where
    open Setoid X renaming (eq to setoidEq)
    open Equivalence (Setoid.eq X)

leftRegularAction : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) → GroupAction G S
GroupAction.action (leftRegularAction {_·_ = _·_} G) g h = g · h
  where
    open Group G
GroupAction.actionWellDefined1 (leftRegularAction {S = S} G) eq1 = +WellDefined eq1 reflexive
  where
    open Group G
    open Setoid S renaming (eq to setoidEq)
    open Equivalence setoidEq
GroupAction.actionWellDefined2 (leftRegularAction {S = S} G) {g} {x} {y} eq1 = +WellDefined reflexive eq1
  where
    open Group G
    open Setoid S
    open Equivalence eq
GroupAction.identityAction (leftRegularAction G) = identLeft
  where
    open Group G
GroupAction.associativeAction (leftRegularAction {S = S} G) = symmetric +Associative
  where
    open Group G
    open Setoid S
    open Equivalence eq

conjugationAction : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → GroupAction G S
conjugationAction {S = S} {_·_ = _·_} G = record { action = λ g h → (g · h) · (inverse g) ; actionWellDefined1 = λ gh → +WellDefined (+WellDefined gh reflexive) (inverseWellDefined G gh) ; actionWellDefined2 = λ x~y → +WellDefined (+WellDefined reflexive x~y) reflexive ; identityAction = transitive (+WellDefined reflexive (invIdent G)) (transitive identRight identLeft)  ; associativeAction = λ {x} {g} {h} → transitive (+WellDefined reflexive (invContravariant G)) (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (symmetric (+Associative)) (+WellDefined reflexive +Associative))) reflexive)) }
  where
    open Group G
    open Setoid S
    open Equivalence eq

conjugationNormalSubgroupAction : {m n o p : _} {A : Set m} {B : Set o} {S : Setoid {m} {n} A} {T : Setoid {o} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} → (G : Group S _·A_) → (H : Group T _·B_) → {underF : A → B} (f : GroupHom G H underF) → GroupAction G (quotientGroupSetoid G f)
GroupAction.action (conjugationNormalSubgroupAction {_·A_ = _·A_} G H {f} fHom) a b = a ·A (b ·A (Group.inverse G a))
GroupAction.actionWellDefined1 (conjugationNormalSubgroupAction {S = S} {T = T} {_·A_ = _·A_} G H {f} fHom) {g} {h} {x} g~h = ans
  where
    open Group G
    open Setoid T
    open Equivalence eq
    ans : f ((g ·A (x ·A (inverse g))) ·A inverse (h ·A (x ·A inverse h))) ∼ Group.0G H
    ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+WellDefined G g~h (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (inverseWellDefined G g~h))))) (imageOfIdentityIsIdentity fHom)
GroupAction.actionWellDefined2 (conjugationNormalSubgroupAction {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G H {f} fHom) {g} {x} {y} x~y = ans
  where
    open Group G
    open Setoid T
    open Equivalence (Setoid.eq S)
    open Equivalence (Setoid.eq T) renaming (transitive to transitiveH ; symmetric to symmetricH ; reflexive to reflexiveH)
    input : f (x ·A inverse y) ∼ Group.0G H
    input = x~y
    p1 : Setoid._∼_ S ((g ·A (x ·A inverse g)) ·A inverse (g ·A (y ·A inverse g))) ((g ·A (x ·A inverse g)) ·A (inverse (y ·A (inverse g)) ·A inverse g))
    p1 = Group.+WellDefined G reflexive (invContravariant G)
    p2 : Setoid._∼_ S ((g ·A (x ·A inverse g)) ·A (inverse (y ·A (inverse g)) ·A inverse g)) ((g ·A (x ·A inverse g)) ·A ((inverse (inverse g) ·A inverse y) ·A inverse g))
    p2 = Group.+WellDefined G reflexive (Group.+WellDefined G (invContravariant G) reflexive)
    p3 : Setoid._∼_ S ((g ·A (x ·A inverse g)) ·A ((inverse (inverse g) ·A inverse y) ·A inverse g)) (g ·A (((x ·A inverse g) ·A (inverse (inverse g) ·A inverse y)) ·A inverse g))
    p3 = symmetric (transitive (+WellDefined reflexive (symmetric +Associative)) +Associative)
    p4 : Setoid._∼_ S (g ·A (((x ·A inverse g) ·A (inverse (inverse g) ·A inverse y)) ·A inverse g)) (g ·A ((x ·A ((inverse g ·A inverse (inverse g)) ·A inverse y)) ·A inverse g))
    p4 = Group.+WellDefined G reflexive (Group.+WellDefined G (symmetric (transitive (+WellDefined reflexive (symmetric +Associative)) +Associative)) reflexive)
    p5 : Setoid._∼_ S (g ·A ((x ·A ((inverse g ·A inverse (inverse g)) ·A inverse y)) ·A inverse g)) (g ·A ((x ·A (0G ·A inverse y)) ·A inverse g))
    p5 = Group.+WellDefined G reflexive (Group.+WellDefined G (Group.+WellDefined G reflexive (Group.+WellDefined G invRight reflexive)) reflexive)
    p6 : Setoid._∼_ S  (g ·A ((x ·A (0G ·A inverse y)) ·A inverse g)) (g ·A ((x ·A inverse y) ·A inverse g))
    p6 = Group.+WellDefined G reflexive (Group.+WellDefined G (Group.+WellDefined G reflexive identLeft) reflexive)
    intermediate : Setoid._∼_ S ((g ·A (x ·A inverse g)) ·A inverse (g ·A (y ·A inverse g))) (g ·A ((x ·A inverse y) ·A inverse g))
    intermediate = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 p6))))
    p7 : f ((g ·A (x ·A inverse g)) ·A inverse (g ·A (y ·A inverse g))) ∼ f (g ·A ((x ·A inverse y) ·A inverse g))
    p7 = GroupHom.wellDefined fHom intermediate
    p8 : f (g ·A ((x ·A inverse y) ·A inverse g)) ∼ (f g) ·B (f ((x ·A inverse y) ·A inverse g))
    p8 = GroupHom.groupHom fHom
    p9 : (f g) ·B (f ((x ·A inverse y) ·A inverse g)) ∼ (f g) ·B (f (x ·A inverse y) ·B f (inverse g))
    p9 = Group.+WellDefined H reflexiveH (GroupHom.groupHom fHom)
    p10 : (f g) ·B (f (x ·A inverse y) ·B f (inverse g)) ∼ (f g) ·B (Group.0G H ·B f (inverse g))
    p10 = Group.+WellDefined H reflexiveH (Group.+WellDefined H input reflexiveH)
    p11 : (f g) ·B (Group.0G H ·B f (inverse g)) ∼ (f g) ·B (f (inverse g))
    p11 = Group.+WellDefined H reflexiveH (Group.identLeft H)
    p12 : (f g) ·B (f (inverse g)) ∼ f (g ·A (inverse g))
    p12 = symmetricH (GroupHom.groupHom fHom)
    intermediate2 : f ((g ·A (x ·A inverse g)) ·A inverse (g ·A (y ·A inverse g))) ∼ (f (g ·A (inverse g)))
    intermediate2 = transitiveH p7 (transitiveH p8 (transitiveH p9 (transitiveH p10 (transitiveH p11 p12))))
    ans : f ((g ·A (x ·A inverse g)) ·A inverse (g ·A (y ·A inverse g))) ∼ Group.0G H
    ans = transitiveH intermediate2 (transitiveH (GroupHom.wellDefined fHom invRight) (imageOfIdentityIsIdentity fHom))
GroupAction.identityAction (conjugationNormalSubgroupAction {S = S} {T = T} {_·A_ = _·A_} G H {f} fHom) {x} = ans
  where
    open Group G
    open Setoid S
    open Setoid T renaming (_∼_ to _∼T_)
    open Equivalence (Setoid.eq T)
    i : Setoid._∼_ S (x ·A inverse 0G) x
    i = Equivalence.transitive (Setoid.eq S) (+WellDefined (Equivalence.reflexive (Setoid.eq S)) (invIdent G)) identRight
    h : 0G ·A (x ·A inverse 0G) ∼ x
    h = Equivalence.transitive (Setoid.eq S) identLeft i
    g : ((0G ·A (x ·A inverse 0G)) ·A inverse x) ∼ 0G
    g = transferToRight'' G h
    ans : f ((0G ·A (x ·A inverse 0G)) ·A Group.inverse G x) ∼T Group.0G H
    ans = transitive (GroupHom.wellDefined fHom g) (imageOfIdentityIsIdentity fHom)
GroupAction.associativeAction (conjugationNormalSubgroupAction {S = S} {T = T} {_·A_ = _·A_} G H {f} fHom) {x} {g} {h} = ans
  where
    open Group G
    open Setoid T renaming (_∼_ to _∼T_)
    open Setoid S renaming (_∼_ to _∼S_)
    open Equivalence (Setoid.eq T) renaming (transitive to transitiveH)
    open Equivalence (Setoid.eq S) renaming (transitive to transitiveG ; symmetric to symmetricG ; reflexive to reflexiveG)
    ans : f (((g ·A h) ·A (x ·A inverse (g ·A h))) ·A inverse ((g ·A ((h ·A (x ·A inverse h)) ·A inverse g)))) ∼T Group.0G H
    ans = transitiveH (GroupHom.wellDefined fHom (transferToRight'' G (transitiveG (symmetricG +Associative) (Group.+WellDefined G reflexiveG (transitiveG (+WellDefined reflexiveG (transitiveG (+WellDefined reflexiveG (invContravariant G)) +Associative)) +Associative))))) (imageOfIdentityIsIdentity fHom)