```------------------------------------------------------------------------
-- The Agda standard library
--
-- Some basic properties of Rings
------------------------------------------------------------------------

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

open import Algebra using (Ring)

module Algebra.Properties.Ring {r₁ r₂} (R : Ring r₁ r₂) where

open Ring R

import Algebra.Properties.AbelianGroup as AbelianGroupProperties
open import Function.Base using (_\$_)
open import Relation.Binary.Reasoning.Setoid setoid
open import Algebra.Definitions _≈_
open import Data.Product

------------------------------------------------------------------------
-- Export properties of abelian groups

open AbelianGroupProperties +-abelianGroup public
renaming
( ε⁻¹≈ε            to -0#≈0#
; ∙-cancelˡ        to +-cancelˡ
; ∙-cancelʳ        to +-cancelʳ
; ∙-cancel         to +-cancel
; ⁻¹-involutive    to -‿involutive
; ⁻¹-injective     to -‿injective
; ⁻¹-anti-homo-∙   to -‿anti-homo-+
; identityˡ-unique to +-identityˡ-unique
; identityʳ-unique to +-identityʳ-unique
; identity-unique  to +-identity-unique
; inverseˡ-unique  to +-inverseˡ-unique
; inverseʳ-unique  to +-inverseʳ-unique
; ⁻¹-∙-comm        to -‿+-comm
)

------------------------------------------------------------------------
-- Properties of -_

-‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y
-‿distribˡ-* x y = sym \$ begin
- x * y                        ≈⟨ sym \$ +-identityʳ _ ⟩
- x * y + 0#                   ≈⟨ +-congˡ \$ sym (-‿inverseʳ _) ⟩
- x * y + (x * y + - (x * y))  ≈⟨ sym \$ +-assoc _ _ _  ⟩
- x * y + x * y + - (x * y)    ≈⟨ +-congʳ \$ sym (distribʳ _ _ _) ⟩
(- x + x) * y + - (x * y)      ≈⟨ +-congʳ \$ *-congʳ \$ -‿inverseˡ _ ⟩
0# * y + - (x * y)             ≈⟨ +-congʳ \$ zeroˡ _ ⟩
0# + - (x * y)                 ≈⟨ +-identityˡ _ ⟩
- (x * y)                      ∎

-‿distribʳ-* : ∀ x y → - (x * y) ≈ x * - y
-‿distribʳ-* x y = sym \$ begin
x * - y                        ≈⟨ sym \$ +-identityˡ _ ⟩
0# + x * - y                   ≈⟨ +-congʳ \$ sym (-‿inverseˡ _) ⟩
- (x * y) + x * y + x * - y    ≈⟨ +-assoc _ _ _  ⟩
- (x * y) + (x * y + x * - y)  ≈⟨ +-congˡ \$ sym (distribˡ _ _ _)  ⟩
- (x * y) + x * (y + - y)      ≈⟨ +-congˡ \$ *-congˡ \$ -‿inverseʳ _ ⟩
- (x * y) + x * 0#             ≈⟨ +-congˡ \$ zeroʳ _ ⟩
- (x * y) + 0#                 ≈⟨ +-identityʳ _ ⟩
- (x * y)                      ∎

-1*x≈-x : ∀ x → - 1# * x ≈ - x
-1*x≈-x x = begin
- 1# * x    ≈⟨ sym (-‿distribˡ-* 1# x ) ⟩
- (1# * x)  ≈⟨ -‿cong ( *-identityˡ x ) ⟩
- x         ∎

x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
x+x≈x⇒x≈0 x eq = +-identityˡ-unique x x eq

x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z
x[y-z]≈xy-xz x y z = begin
x * (y - z)      ≈⟨ distribˡ x y (- z) ⟩
x * y + x * - z  ≈⟨ +-congˡ (sym (-‿distribʳ-* x z)) ⟩
x * y - x * z    ∎

[y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)
[y-z]x≈yx-zx x y z = begin
(y - z) * x      ≈⟨ distribʳ x y (- z) ⟩
y * x + - z * x  ≈⟨ +-congˡ (sym (-‿distribˡ-* z x)) ⟩
y * x - z * x    ∎
```