[Add] Consequences of identity for monoids (#2692)

* Adds reasonig combinator for semigroup

* Adds reasonig combinator for semigroup

* Adds reasonig combinator for semigroup

* Adds reasonig combinator for semigroup

* Add some more missing reasoning combinators

* add module Extends

* rename SemiGroup to Semigroup

* fix-whitespace

* Update CHANGELOG.md

Co-authored-by: jamesmckinna <[email protected]>

* New names

* improve syntx

* fix-whitespace

* Name changes

* Names change

* Space

* Proof of assoc with PUshes and Pulles

* Proof of assoc with PUshes and Pulles

* white space

* Update src/Algebra/Properties/Semigroup/Reasoning.agda

Co-authored-by: jamesmckinna <[email protected]>

* Reasoning to Semigroup and explicit variables

* Update CHANGELOG.md

Co-authored-by: jamesmckinna <[email protected]>

* Add reasonining combinators for monoids

* chore: move imports around in Algebra.Definitions

merge

* Syntax modification

* rebase

* update semigroup

* One line proof

* move importation

* fix-whitespace

* Add reasonining combinators for monoids

* chore: move imports around in Algebra.Definitions

merge

* Syntax modification

* rebase

merge

* move importation

* fix-whitespace

* explicit varaibles

* used of semigroup propreties

* update

* Update CHANGELOG.md

Co-authored-by: jamesmckinna <[email protected]>

* merge and proof refactoring

* Changelog update

* Update src/Algebra/Properties/Monoid.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Monoid.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Monoid.agda

* rename id to ε

---------

Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: Jacques Carette <[email protected]>

Author: 3 people

CHANGELOG.md

@@ -186,6 +186,10 @@ New modules
 Additions to existing modules
 -----------------------------
 
+* In `Algebra.Properties.Monoid` adding consequences for identity for monoids
+
+* In `Algebra.Properties.Semigroup` adding consequences for associativity for semigroups
+
 * In `Algebra.Consequences.Base`:
   ```agda
   module Congruence (_≈_ : Rel A ℓ) (cong : Congruent₂ _≈_ _∙_) (refl : Reflexive _≈_)

src/Algebra/Properties/Monoid.agda

@@ -0,0 +1,67 @@
+------------------------------------------------------------------------
+-- 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)
+