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

{-# OPTIONS --cubical-compatible --safe #-}

open import Algebra

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

open AbelianGroup G
open import Function.Base using (_$_)
open import Relation.Binary.Reasoning.Setoid setoid

------------------------------------------------------------------------
-- Publicly re-export group properties

open import Algebra.Properties.Group group public

------------------------------------------------------------------------
-- Properties of abelian groups

⁻¹-anti-homo‿- :  x y  (x - y) ⁻¹  y - x
⁻¹-anti-homo‿- = ⁻¹-anti-homo-//

xyx⁻¹≈y :  x y  x  y  x ⁻¹  y
xyx⁻¹≈y x y = begin
  x  y  x ⁻¹    ≈⟨ ∙-congʳ $ comm _ _ 
  y  x  x ⁻¹    ≈⟨ assoc _ _ _ 
  y  (x  x ⁻¹)  ≈⟨ ∙-congˡ $ inverseʳ _ 
  y  ε           ≈⟨ identityʳ _ 
  y               

⁻¹-∙-comm :  x y  x ⁻¹  y ⁻¹  (x  y) ⁻¹
⁻¹-∙-comm x y = begin
  x ⁻¹  y ⁻¹ ≈⟨ ⁻¹-anti-homo-∙ y x 
  (y  x) ⁻¹  ≈⟨ ⁻¹-cong $ comm y x 
  (x  y) ⁻¹