{-# OPTIONS --cubical-compatible --safe #-}
open import Algebra.Bundles using (Loop)
module Algebra.Properties.Loop {l₁ l₂} (L : Loop l₁ l₂) where
open Loop L
open import Algebra.Definitions _≈_
open import Algebra.Properties.Quasigroup quasigroup
open import Data.Product.Base using (proj₂)
open import Relation.Binary.Reasoning.Setoid setoid
x//x≈ε : ∀ x → x // x ≈ ε
x//x≈ε x = begin
x // x ≈⟨ //-congʳ (identityˡ x) ⟨
(ε ∙ x) // x ≈⟨ rightDividesʳ x ε ⟩
ε ∎
x\\x≈ε : ∀ x → x \\ x ≈ ε
x\\x≈ε x = begin
x \\ x ≈⟨ \\-congˡ (identityʳ x ) ⟨
x \\ (x ∙ ε) ≈⟨ leftDividesʳ x ε ⟩
ε ∎
ε\\x≈x : ∀ x → ε \\ x ≈ x
ε\\x≈x x = begin
ε \\ x ≈⟨ identityˡ (ε \\ x) ⟨
ε ∙ (ε \\ x) ≈⟨ leftDividesˡ ε x ⟩
x ∎
x//ε≈x : ∀ x → x // ε ≈ x
x//ε≈x x = begin
x // ε ≈⟨ identityʳ (x // ε) ⟨
(x // ε) ∙ ε ≈⟨ rightDividesˡ ε x ⟩
x ∎
identityˡ-unique : ∀ x y → x ∙ y ≈ y → x ≈ ε
identityˡ-unique x y eq = begin
x ≈⟨ x≈z//y x y y eq ⟩
y // y ≈⟨ x//x≈ε y ⟩
ε ∎
identityʳ-unique : ∀ x y → x ∙ y ≈ x → y ≈ ε
identityʳ-unique x y eq = begin
y ≈⟨ y≈x\\z x y x eq ⟩
x \\ x ≈⟨ x\\x≈ε x ⟩
ε ∎
identity-unique : ∀ {x} → Identity x _∙_ → x ≈ ε
identity-unique {x} id = identityˡ-unique x x (proj₂ id x)