------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between magma-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.Core
open import Algebra.Bundles
open import Algebra.Morphism.Structures
open import Relation.Binary.Core

module Algebra.Morphism.MagmaMonomorphism
  {a b ℓ₁ ℓ₂} {M₁ : RawMagma a ℓ₁} {M₂ : RawMagma b ℓ₂} {⟦_⟧}
  (isMagmaMonomorphism : IsMagmaMonomorphism M₁ M₂ ⟦_⟧)
  where

open IsMagmaMonomorphism isMagmaMonomorphism
open RawMagma M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_)
open RawMagma M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_)

open import Algebra.Structures
open import Algebra.Definitions
open import Data.Product.Base using (map)
open import Data.Sum.Base using (inj₁; inj₂)
import Relation.Binary.Reasoning.Setoid as SetoidReasoning
import Relation.Binary.Morphism.RelMonomorphism isRelMonomorphism as RelMorphism

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

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

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

  cong : Congruent₂ _≈₁_ _∙_
  cong {x} {y} {u} {v} x≈y u≈v = injective (begin
     x  u       ≈⟨  homo x u 
     x    u   ≈⟨  ◦-cong (⟦⟧-cong x≈y) (⟦⟧-cong u≈v) 
     y    v   ≈⟨ homo y v 
     y  v       )

  assoc : Associative _≈₂_ _◦_  Associative _≈₁_ _∙_
  assoc assoc x y z = injective (begin
     (x  y)  z           ≈⟨  homo (x  y) z 
     x  y    z         ≈⟨  ◦-cong (homo x y) refl 
    ( x    y )   z   ≈⟨  assoc  x   y   z  
     x   ( y    z )  ≈⟨ ◦-cong refl (homo y z) 
     x    y  z         ≈⟨ homo x (y  z) 
     x  (y  z)           )

  comm : Commutative _≈₂_ _◦_  Commutative _≈₁_ _∙_
  comm comm x y = injective (begin
     x  y       ≈⟨  homo x y 
     x    y   ≈⟨  comm  x   y  
     y    x   ≈⟨ homo y x 
     y  x       )

  idem : Idempotent _≈₂_ _◦_  Idempotent _≈₁_ _∙_
  idem idem x = injective (begin
     x  x      ≈⟨ homo x x 
     x    x  ≈⟨ idem  x  
     x          )

  sel : Selective _≈₂_ _◦_  Selective _≈₁_ _∙_
  sel sel x y with sel  x   y 
  ... | inj₁ x◦y≈x = inj₁ (injective (begin
     x  y       ≈⟨ homo x y 
     x    y   ≈⟨ x◦y≈x 
     x           ))
  ... | inj₂ x◦y≈y = inj₂ (injective (begin
     x  y       ≈⟨ homo x y 
     x    y   ≈⟨ x◦y≈y 
     y           ))

  cancelˡ : LeftCancellative _≈₂_ _◦_  LeftCancellative _≈₁_ _∙_
  cancelˡ cancelˡ x y z x∙y≈x∙z = injective (cancelˡ  x   y   z  (begin
     x    y   ≈⟨ homo x y 
     x  y       ≈⟨  ⟦⟧-cong x∙y≈x∙z 
     x  z       ≈⟨  homo x z 
     x    z   ))

  cancelʳ : RightCancellative _≈₂_ _◦_  RightCancellative _≈₁_ _∙_
  cancelʳ cancelʳ x y z y∙x≈z∙x = injective (cancelʳ  x   y   z  (begin
     y    x   ≈⟨ homo y x 
     y  x       ≈⟨  ⟦⟧-cong y∙x≈z∙x 
     z  x       ≈⟨  homo z x 
     z    x   ))

  cancel : Cancellative _≈₂_ _◦_  Cancellative _≈₁_ _∙_
  cancel = map cancelˡ cancelʳ

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

isMagma : IsMagma _≈₂_ _◦_  IsMagma _≈₁_ _∙_
isMagma isMagma = record
  { isEquivalence = RelMorphism.isEquivalence M.isEquivalence
  ; ∙-cong        = cong isMagma
  } where module M = IsMagma isMagma

isSemigroup : IsSemigroup _≈₂_ _◦_  IsSemigroup _≈₁_ _∙_
isSemigroup isSemigroup = record
  { isMagma = isMagma S.isMagma
  ; assoc   = assoc   S.isMagma S.assoc
  } where module S = IsSemigroup isSemigroup

isBand : IsBand _≈₂_ _◦_  IsBand _≈₁_ _∙_
isBand isBand = record
  { isSemigroup = isSemigroup B.isSemigroup
  ; idem        = idem        B.isMagma B.idem
  } where module B = IsBand isBand

isSelectiveMagma : IsSelectiveMagma _≈₂_ _◦_  IsSelectiveMagma _≈₁_ _∙_
isSelectiveMagma isSelMagma = record
  { isMagma = isMagma S.isMagma
  ; sel     = sel     S.isMagma S.sel
  } where module S = IsSelectiveMagma isSelMagma