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------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between monoid-like structures
------------------------------------------------------------------------

-- See Data.Nat.Binary.Properties for examples of how this and similar
-- modules can be used to easily translate properties between types.

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

open import Algebra.Bundles
open import Algebra.Morphism.Structures
open import Relation.Binary.Core

module Algebra.Morphism.MonoidMonomorphism
  {a b ℓ₁ ℓ₂} {M₁ : RawMonoid a ℓ₁} {M₂ : RawMonoid b ℓ₂} {⟦_⟧}
  (isMonoidMonomorphism : IsMonoidMonomorphism M₁ M₂ ⟦_⟧)
  where

open IsMonoidMonomorphism isMonoidMonomorphism
open RawMonoid M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; ε to ε₁)
open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; ε to ε₂)

open import Algebra.Definitions
open import Algebra.Structures
open import Data.Product using (map)
import Relation.Binary.Reasoning.Setoid as SetoidReasoning

------------------------------------------------------------------------
-- Re-export all properties of magma monomorphisms

open import Algebra.Morphism.MagmaMonomorphism
  isMagmaMonomorphism public

------------------------------------------------------------------------
-- Properties

module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where

  open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)
  open SetoidReasoning setoid

  identityˡ : LeftIdentity _≈₂_ ε₂ _◦_  LeftIdentity _≈₁_ ε₁ _∙_
  identityˡ idˡ x = injective (begin
     ε₁  x       ≈⟨ homo ε₁ x 
     ε₁    x   ≈⟨ ◦-cong ε-homo refl 
    ε₂   x       ≈⟨ idˡ  x  
     x            )

  identityʳ : RightIdentity _≈₂_ ε₂ _◦_  RightIdentity _≈₁_ ε₁ _∙_
  identityʳ idʳ x = injective (begin
     x  ε₁       ≈⟨ homo x ε₁ 
     x    ε₁   ≈⟨ ◦-cong refl ε-homo 
     x   ε₂      ≈⟨ idʳ  x  
     x            )

  identity : Identity _≈₂_ ε₂ _◦_  Identity _≈₁_ ε₁ _∙_
  identity = map identityˡ identityʳ

  zeroˡ : LeftZero _≈₂_ ε₂ _◦_  LeftZero _≈₁_ ε₁ _∙_
  zeroˡ zeˡ x = injective (begin
     ε₁  x      ≈⟨  homo ε₁ x 
     ε₁    x  ≈⟨  ◦-cong ε-homo refl 
    ε₂     x    ≈⟨  zeˡ  x  
    ε₂             ≈˘⟨ ε-homo 
     ε₁          )

  zeroʳ : RightZero _≈₂_ ε₂ _◦_  RightZero _≈₁_ ε₁ _∙_
  zeroʳ zeʳ x = injective (begin
     x  ε₁      ≈⟨  homo x ε₁ 
     x    ε₁  ≈⟨  ◦-cong refl ε-homo 
     x   ε₂     ≈⟨  zeʳ  x  
    ε₂             ≈˘⟨ ε-homo 
     ε₁          )

  zero : Zero _≈₂_ ε₂ _◦_  Zero _≈₁_ ε₁ _∙_
  zero = map zeroˡ zeroʳ

------------------------------------------------------------------------
-- Structures

isMonoid : IsMonoid _≈₂_ _◦_ ε₂  IsMonoid _≈₁_ _∙_ ε₁
isMonoid isMonoid = record
  { isSemigroup = isSemigroup M.isSemigroup
  ; identity    = identity    M.isMagma M.identity
  } where module M = IsMonoid isMonoid

isCommutativeMonoid : IsCommutativeMonoid _≈₂_ _◦_ ε₂ 
                      IsCommutativeMonoid _≈₁_ _∙_ ε₁
isCommutativeMonoid isCommMonoid = record
  { isMonoid = isMonoid C.isMonoid
  ; comm     = comm     C.isMagma C.comm
  } where module C = IsCommutativeMonoid isCommMonoid