------------------------------------------------------------------------
-- Some derivable properties
------------------------------------------------------------------------

open import Algebra

module Algebra.Props.AbelianGroup (g : AbelianGroup) where

open AbelianGroup g
import Relation.Binary.EqReasoning as EqR; open EqR setoid
open import Data.Function
open import Data.Product

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

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