------------------------------------------------------------------------
-- The Agda standard library
--
-- Morphisms between algebraic structures
------------------------------------------------------------------------

module Algebra.Morphism where

open import Relation.Binary
open import Algebra
open import Algebra.FunctionProperties
import Algebra.Properties.Group as GroupP
open import Function
open import Data.Product
open import Level
import Relation.Binary.EqReasoning as EqR

------------------------------------------------------------------------
-- Basic definitions

module Definitions {f t }
                   (From : Set f) (To : Set t) (_≈_ : Rel To ) where
  Morphism : Set _
  Morphism = From  To

  Homomorphic₀ : Morphism  From  To  Set _
  Homomorphic₀ ⟦_⟧   =     

  Homomorphic₁ : Morphism  Fun₁ From  Op₁ To  Set _
  Homomorphic₁ ⟦_⟧ ∙_ ∘_ =  x    x   (  x )

  Homomorphic₂ : Morphism  Fun₂ From  Op₂ To  Set _
  Homomorphic₂ ⟦_⟧ _∙_ _∘_ =
     x y   x  y   ( x    y )

------------------------------------------------------------------------
-- An example showing how a morphism type can be defined

-- Ring homomorphisms.

record _-Ring⟶_ {r₁ r₂ r₃ r₄}
                (From : Ring r₁ r₂) (To : Ring r₃ r₄) :
                Set (r₁  r₂  r₃  r₄) where
  private
    module F = Ring From
    module T = Ring To
  open Definitions F.Carrier T.Carrier T._≈_

  field
    ⟦_⟧     : Morphism
    ⟦⟧-cong : ⟦_⟧ Preserves F._≈_  T._≈_
    +-homo  : Homomorphic₂ ⟦_⟧ F._+_ T._+_
    *-homo  : Homomorphic₂ ⟦_⟧ F._*_ T._*_
    1-homo  : Homomorphic₀ ⟦_⟧ F.1#  T.1#

  open EqR T.setoid

  0-homo : Homomorphic₀ ⟦_⟧ F.0# T.0#
  0-homo =
    GroupP.left-identity-unique T.+-group  F.0#   F.0#  (begin
      T._+_  F.0#   F.0#  ≈⟨ T.sym (+-homo F.0# F.0#) 
       F._+_ F.0# F.0#      ≈⟨ ⟦⟧-cong (proj₁ F.+-identity F.0#) 
       F.0#                 )

  -‿homo : Homomorphic₁ ⟦_⟧ F.-_ T.-_
  -‿homo x =
    GroupP.left-inverse-unique T.+-group  F.-_ x   x  (begin
      T._+_  F.-_ x   x  ≈⟨ T.sym (+-homo (F.-_ x) x) 
       F._+_ (F.-_ x) x    ≈⟨ ⟦⟧-cong (proj₁ F.-‿inverse x) 
       F.0#                ≈⟨ 0-homo 
      T.0#                   )