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

open import LogicalFormulae
open import Setoids.Setoids
open import Groups.Definition
open import Sets.EquivalenceRelations

module Groups.Groups where

reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
  where
    open Group G renaming (inverse to _^-1)
    open Setoid S
    open Equivalence eq
    p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
    p2 : (r · (s · t)) · u ∼ ((r · s) · t) · u
    p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
    p1 = Group.+Associative G
    p2 = Group.+WellDefined G (Group.+Associative G) reflexive
    p3 = symmetric (Group.+Associative G)

fourWay+Associative' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
  where
    open Group G
    open Setoid S
    open Equivalence eq

fourWay+Associative'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
  where
    open Group G
    open Setoid S
    open Equivalence eq