------------------------------------------------------------------------
-- The Agda standard library
--
-- Equational reasoning for monoids
-- (Utilities for identity and cancellation reasoning, extending semigroup reasoning)
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
open import Algebra.Bundles using (Monoid)
open import Function using (_∘_)
module Algebra.Properties.Monoid {o ℓ} (M : Monoid o ℓ) where
open Monoid M
using (Carrier; _∙_; _≈_; setoid; isMagma; semigroup; ε; sym; identityˡ
; identityʳ ; ∙-cong; refl; assoc; ∙-congˡ; ∙-congʳ; trans)
open import Relation.Binary.Reasoning.Setoid setoid
open import Algebra.Properties.Semigroup semigroup public
private
variable
a b c d : Carrier
ε-unique : ∀ a → (∀ b → b ∙ a ≈ b) → a ≈ ε
ε-unique a b∙a≈b = trans (sym (identityˡ a)) (b∙a≈b ε)
ε-comm : ∀ a → a ∙ ε ≈ ε ∙ a
ε-comm a = trans (identityʳ a) (sym (identityˡ a))
module _ (a≈ε : a ≈ ε) where
elimʳ : ∀ b → b ∙ a ≈ b
elimʳ = trans (∙-congˡ a≈ε) ∘ identityʳ
elimˡ : ∀ b → a ∙ b ≈ b
elimˡ = trans (∙-congʳ a≈ε) ∘ identityˡ
introʳ : ∀ b → b ≈ b ∙ a
introʳ = sym ∘ elimʳ
introˡ : ∀ b → b ≈ a ∙ b
introˡ = sym ∘ elimˡ
introᶜ : ∀ c → b ∙ c ≈ b ∙ (a ∙ c)
introᶜ c = trans (∙-congˡ (sym (identityˡ c))) (∙-congˡ (∙-congʳ (sym a≈ε)))
module _ (inv : a ∙ c ≈ ε) where
cancelʳ : ∀ b → (b ∙ a) ∙ c ≈ b
cancelʳ b = trans (uv≈w⇒xu∙v≈xw inv b) (identityʳ b)
cancelˡ : ∀ b → a ∙ (c ∙ b) ≈ b
cancelˡ b = trans (uv≈w⇒u∙vx≈wx inv b) (identityˡ b)
insertˡ : ∀ b → b ≈ a ∙ (c ∙ b)
insertˡ = sym ∘ cancelˡ
insertʳ : ∀ b → b ≈ (b ∙ a) ∙ c
insertʳ = sym ∘ cancelʳ
cancelᶜ : ∀ b d → (b ∙ a) ∙ (c ∙ d) ≈ b ∙ d
cancelᶜ b d = trans (uv≈w⇒xu∙vy≈x∙wy inv b d) (∙-congˡ (identityˡ d))
insertᶜ : ∀ b d → b ∙ d ≈ (b ∙ a) ∙ (c ∙ d)
insertᶜ = λ b d → sym (cancelᶜ b d)
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