------------------------------------------------------------------------
-- The Agda standard library
--
-- Some derivable properties
------------------------------------------------------------------------

open import Algebra

module Algebra.Properties.AbelianGroup
         {g₁ g₂} (G : AbelianGroup g₁ g₂) where

import Algebra.Properties.Group as GP
open import Data.Product
open import Function
import Relation.Binary.EqReasoning as EqR

open AbelianGroup G
open EqR setoid

open GP group public

private
  lemma :  x y  x  y  x ⁻¹  y
  lemma x y = begin
    x  y  x ⁻¹    ≈⟨ comm _ _  ∙-cong  refl 
    y  x  x ⁻¹    ≈⟨ assoc _ _ _ 
    y  (x  x ⁻¹)  ≈⟨ refl  ∙-cong  proj₂ inverse _ 
    y  ε           ≈⟨ proj₂ identity _ 
    y               

⁻¹-∙-comm :  x y  x ⁻¹  y ⁻¹  (x  y) ⁻¹
⁻¹-∙-comm x y = begin
  x ⁻¹  y ⁻¹                         ≈⟨ comm _ _ 
  y ⁻¹  x ⁻¹                         ≈⟨ sym $ lem  ∙-cong  refl 
  x  (y  (x  y) ⁻¹  y ⁻¹)  x ⁻¹  ≈⟨ lemma _ _ 
  y  (x  y) ⁻¹  y ⁻¹               ≈⟨ lemma _ _ 
  (x  y) ⁻¹                          
  where
  lem = begin
    x  (y  (x  y) ⁻¹  y ⁻¹)  ≈⟨ sym $ assoc _ _ _ 
    x  (y  (x  y) ⁻¹)  y ⁻¹  ≈⟨ sym $ assoc _ _ _  ∙-cong  refl 
    x  y  (x  y) ⁻¹  y ⁻¹    ≈⟨ proj₂ inverse _  ∙-cong  refl 
    ε  y ⁻¹                     ≈⟨ proj₁ identity _ 
    y ⁻¹