{-# OPTIONS --without-K --safe #-}
open import Algebra
module Algebra.Properties.Group {g₁ g₂} (G : Group g₁ g₂) where
open Group G
open import Algebra.FunctionProperties _≈_
open import Relation.Binary.Reasoning.Setoid setoid
open import Function
open import Data.Product
ε⁻¹≈ε : ε ⁻¹ ≈ ε
ε⁻¹≈ε = begin
ε ⁻¹ ≈⟨ sym $ identityʳ (ε ⁻¹) ⟩
ε ⁻¹ ∙ ε ≈⟨ inverseˡ ε ⟩
ε ∎
∙-cancelˡ : LeftCancellative _∙_
∙-cancelˡ x {y} {z} eq = begin
y ≈⟨ sym $ identityˡ y ⟩
ε ∙ y ≈⟨ sym $ ∙-congʳ $ inverseˡ x ⟩
(x ⁻¹ ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩
x ⁻¹ ∙ (x ∙ y) ≈⟨ ∙-congˡ eq ⟩
x ⁻¹ ∙ (x ∙ z) ≈⟨ sym $ assoc (x ⁻¹) x z ⟩
(x ⁻¹ ∙ x) ∙ z ≈⟨ ∙-congʳ $ inverseˡ x ⟩
ε ∙ z ≈⟨ identityˡ z ⟩
z ∎
∙-cancelʳ : RightCancellative _∙_
∙-cancelʳ {x} y z eq = begin
y ≈⟨ sym $ identityʳ y ⟩
y ∙ ε ≈⟨ sym $ ∙-congˡ $ inverseʳ x ⟩
y ∙ (x ∙ x ⁻¹) ≈⟨ sym $ assoc y x (x ⁻¹) ⟩
(y ∙ x) ∙ x ⁻¹ ≈⟨ ∙-congʳ eq ⟩
(z ∙ x) ∙ x ⁻¹ ≈⟨ assoc z x (x ⁻¹) ⟩
z ∙ (x ∙ x ⁻¹) ≈⟨ ∙-congˡ $ inverseʳ x ⟩
z ∙ ε ≈⟨ identityʳ z ⟩
z ∎
∙-cancel : Cancellative _∙_
∙-cancel = ∙-cancelˡ , ∙-cancelʳ
⁻¹-involutive : ∀ x → x ⁻¹ ⁻¹ ≈ x
⁻¹-involutive x = begin
x ⁻¹ ⁻¹ ≈⟨ sym $ identityʳ _ ⟩
x ⁻¹ ⁻¹ ∙ ε ≈⟨ ∙-congˡ $ sym (inverseˡ _) ⟩
x ⁻¹ ⁻¹ ∙ (x ⁻¹ ∙ x) ≈⟨ sym $ assoc _ _ _ ⟩
x ⁻¹ ⁻¹ ∙ x ⁻¹ ∙ x ≈⟨ ∙-congʳ $ inverseˡ _ ⟩
ε ∙ x ≈⟨ identityˡ _ ⟩
x ∎
private
left-helper : ∀ x y → x ≈ (x ∙ y) ∙ y ⁻¹
left-helper x y = begin
x ≈⟨ sym (identityʳ x) ⟩
x ∙ ε ≈⟨ ∙-congˡ $ sym (inverseʳ y) ⟩
x ∙ (y ∙ y ⁻¹) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩
(x ∙ y) ∙ y ⁻¹ ∎
right-helper : ∀ x y → y ≈ x ⁻¹ ∙ (x ∙ y)
right-helper x y = begin
y ≈⟨ sym (identityˡ y) ⟩
ε ∙ y ≈⟨ ∙-congʳ $ sym (inverseˡ x) ⟩
(x ⁻¹ ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩
x ⁻¹ ∙ (x ∙ y) ∎
identityˡ-unique : ∀ x y → x ∙ y ≈ y → x ≈ ε
identityˡ-unique x y eq = begin
x ≈⟨ left-helper x y ⟩
(x ∙ y) ∙ y ⁻¹ ≈⟨ ∙-congʳ eq ⟩
y ∙ y ⁻¹ ≈⟨ inverseʳ y ⟩
ε ∎
identityʳ-unique : ∀ x y → x ∙ y ≈ x → y ≈ ε
identityʳ-unique x y eq = begin
y ≈⟨ right-helper x y ⟩
x ⁻¹ ∙ (x ∙ y) ≈⟨ refl ⟨ ∙-cong ⟩ eq ⟩
x ⁻¹ ∙ x ≈⟨ inverseˡ x ⟩
ε ∎
identity-unique : ∀ {x} → Identity x _∙_ → x ≈ ε
identity-unique {x} id = identityˡ-unique x x (proj₂ id x)
inverseˡ-unique : ∀ x y → x ∙ y ≈ ε → x ≈ y ⁻¹
inverseˡ-unique x y eq = begin
x ≈⟨ left-helper x y ⟩
(x ∙ y) ∙ y ⁻¹ ≈⟨ ∙-congʳ eq ⟩
ε ∙ y ⁻¹ ≈⟨ identityˡ (y ⁻¹) ⟩
y ⁻¹ ∎
inverseʳ-unique : ∀ x y → x ∙ y ≈ ε → y ≈ x ⁻¹
inverseʳ-unique x y eq = begin
y ≈⟨ sym (⁻¹-involutive y) ⟩
y ⁻¹ ⁻¹ ≈⟨ ⁻¹-cong (sym (inverseˡ-unique x y eq)) ⟩
x ⁻¹ ∎
left-identity-unique = identityˡ-unique
{-# WARNING_ON_USAGE left-identity-unique
"Warning: left-identity-unique was deprecated in v1.1.
Please use identityˡ-unique instead."
#-}
right-identity-unique = identityʳ-unique
{-# WARNING_ON_USAGE right-identity-unique
"Warning: right-identity-unique was deprecated in v1.1.
Please use identityʳ-unique instead."
#-}
left-inverse-unique = inverseˡ-unique
{-# WARNING_ON_USAGE left-inverse-unique
"Warning: left-inverse-unique was deprecated in v1.1.
Please use inverseˡ-unique instead."
#-}
right-inverse-unique = inverseʳ-unique
{-# WARNING_ON_USAGE right-inverse-unique
"Warning: right-inverse-unique was deprecated in v1.1.
Please use inverseʳ-unique instead."
#-}