------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomomorphism between right semimodules
------------------------------------------------------------------------

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

open import Algebra.Module.Bundles.Raw using (RawRightSemimodule)
open import Algebra.Module.Morphism.Structures using (IsRightSemimoduleMonomorphism)

module Algebra.Module.Morphism.RightSemimoduleMonomorphism
  {r a b ℓ₁ ℓ₂} {R : Set r} {M : RawRightSemimodule R a ℓ₁} {N : RawRightSemimodule R b ℓ₂} {⟦_⟧}
  (isRightSemimoduleMonomorphism : IsRightSemimoduleMonomorphism M N ⟦_⟧)
  where

open IsRightSemimoduleMonomorphism isRightSemimoduleMonomorphism
private
  module M = RawRightSemimodule M
  module N = RawRightSemimodule N

open import Algebra.Bundles using (Semiring)
open import Algebra.Core using (Op₂)
import Algebra.Module.Definitions.Right as RightDefs
open import Algebra.Module.Structures  using (IsRightSemimodule)
open import Algebra.Structures using (IsMagma; IsSemiring)
open import Function.Base using (flip; _$_)
open import Level using (Level)
open import Relation.Binary.Core using (Rel)
import Relation.Binary.Reasoning.Setoid as SetoidReasoning

private
  variable
    ℓr : Level
    _≈_ : Rel R ℓr
    _+_ _*_ : Op₂ R
    0# 1# : R

------------------------------------------------------------------------
-- Re-exports

open import Algebra.Morphism.MonoidMonomorphism
  +ᴹ-isMonoidMonomorphism public
    using ()
    renaming
      ( cong to +ᴹ-cong
      ; assoc to +ᴹ-assoc
      ; comm to +ᴹ-comm
      ; identityˡ to +ᴹ-identityˡ
      ; identityʳ to +ᴹ-identityʳ
      ; identity to +ᴹ-identity
      ; isMagma to +ᴹ-isMagma
      ; isSemigroup to +ᴹ-isSemigroup
      ; isMonoid to +ᴹ-isMonoid
      ; isCommutativeMonoid to +ᴹ-isCommutativeMonoid
      )

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

module _ (+ᴹ-isMagma : IsMagma N._≈ᴹ_ N._+ᴹ_) where

  open IsMagma +ᴹ-isMagma
    using (setoid)
    renaming (∙-cong to +ᴹ-cong′)
  open SetoidReasoning setoid

  module MDefs = RightDefs R M._≈ᴹ_
  module NDefs = RightDefs R N._≈ᴹ_

  *ᵣ-cong : NDefs.Congruent _≈_ N._*ᵣ_ → MDefs.Congruent _≈_ M._*ᵣ_
  *ᵣ-cong *ᵣ-cong {x} {y} {u} {v} x≈ᴹy u≈v = injective $ begin
    ⟦ x M.*ᵣ u ⟧ ≈⟨ *ᵣ-homo u x ⟩
    ⟦ x ⟧ N.*ᵣ u ≈⟨ *ᵣ-cong (⟦⟧-cong x≈ᴹy) u≈v ⟩
    ⟦ y ⟧ N.*ᵣ v ≈˘⟨ *ᵣ-homo v y ⟩
    ⟦ y M.*ᵣ v ⟧ ∎

  *ᵣ-zeroʳ : NDefs.RightZero 0# N.0ᴹ N._*ᵣ_ → MDefs.RightZero 0# M.0ᴹ M._*ᵣ_
  *ᵣ-zeroʳ {0# = 0#} *ᵣ-zeroʳ x = injective $ begin
    ⟦ x M.*ᵣ 0# ⟧ ≈⟨ *ᵣ-homo 0# x ⟩
    ⟦ x ⟧ N.*ᵣ 0# ≈⟨ *ᵣ-zeroʳ ⟦ x ⟧ ⟩
    N.0ᴹ          ≈˘⟨ 0ᴹ-homo ⟩
    ⟦ M.0ᴹ ⟧      ∎

  *ᵣ-distribˡ : N._*ᵣ_ NDefs.DistributesOverˡ _+_ ⟶ N._+ᴹ_ → M._*ᵣ_ MDefs.DistributesOverˡ _+_ ⟶ M._+ᴹ_
  *ᵣ-distribˡ {_+_ = _+_} *ᵣ-distribˡ x m n = injective $ begin
    ⟦ x M.*ᵣ (m + n) ⟧             ≈⟨ *ᵣ-homo (m + n) x ⟩
    ⟦ x ⟧ N.*ᵣ (m + n)             ≈⟨ *ᵣ-distribˡ ⟦ x ⟧ m n ⟩
    ⟦ x ⟧ N.*ᵣ m N.+ᴹ ⟦ x ⟧ N.*ᵣ n ≈˘⟨ +ᴹ-cong′ (*ᵣ-homo m x) (*ᵣ-homo n x) ⟩
    ⟦ x M.*ᵣ m ⟧ N.+ᴹ ⟦ x M.*ᵣ n ⟧ ≈˘⟨ +ᴹ-homo (x M.*ᵣ m) (x M.*ᵣ n) ⟩
    ⟦ x M.*ᵣ m M.+ᴹ x M.*ᵣ n ⟧     ∎

  *ᵣ-identityʳ : NDefs.RightIdentity 1# N._*ᵣ_ → MDefs.RightIdentity 1# M._*ᵣ_
  *ᵣ-identityʳ {1# = 1#} *ᵣ-identityʳ m = injective $ begin
    ⟦ m M.*ᵣ 1# ⟧ ≈⟨ *ᵣ-homo 1# m ⟩
    ⟦ m ⟧ N.*ᵣ 1# ≈⟨ *ᵣ-identityʳ ⟦ m ⟧ ⟩
    ⟦ m ⟧         ∎

  *ᵣ-assoc : NDefs.RightCongruent N._*ᵣ_ → NDefs.Associative _*_ N._*ᵣ_ → MDefs.Associative _*_ M._*ᵣ_
  *ᵣ-assoc {_*_ = _*_} *ᵣ-congʳ *ᵣ-assoc m x y = injective $ begin
    ⟦ m M.*ᵣ x M.*ᵣ y ⟧ ≈⟨ *ᵣ-homo y (m M.*ᵣ x) ⟩
    ⟦ m M.*ᵣ x ⟧ N.*ᵣ y ≈⟨ *ᵣ-congʳ (*ᵣ-homo x m) ⟩
    ⟦ m ⟧ N.*ᵣ x N.*ᵣ y ≈⟨ *ᵣ-assoc ⟦ m ⟧ x y ⟩
    ⟦ m ⟧ N.*ᵣ (x * y)  ≈˘⟨ *ᵣ-homo (x * y) m ⟩
    ⟦ m M.*ᵣ (x * y) ⟧  ∎

  *ᵣ-zeroˡ : NDefs.RightCongruent N._*ᵣ_ → NDefs.LeftZero N.0ᴹ N._*ᵣ_ → MDefs.LeftZero M.0ᴹ M._*ᵣ_
  *ᵣ-zeroˡ *ᵣ-congʳ *ᵣ-zeroˡ x = injective $ begin
    ⟦ M.0ᴹ M.*ᵣ x ⟧ ≈⟨ *ᵣ-homo x M.0ᴹ ⟩
    ⟦ M.0ᴹ ⟧ N.*ᵣ x ≈⟨ *ᵣ-congʳ 0ᴹ-homo ⟩
    N.0ᴹ N.*ᵣ x     ≈⟨ *ᵣ-zeroˡ x ⟩
    N.0ᴹ            ≈˘⟨ 0ᴹ-homo ⟩
    ⟦ M.0ᴹ ⟧        ∎

  *ᵣ-distribʳ : NDefs.RightCongruent N._*ᵣ_ → N._*ᵣ_ NDefs.DistributesOverʳ N._+ᴹ_ → M._*ᵣ_ MDefs.DistributesOverʳ M._+ᴹ_
  *ᵣ-distribʳ *ᵣ-congʳ *ᵣ-distribʳ x m n = injective $ begin
    ⟦ (m M.+ᴹ n) M.*ᵣ x ⟧          ≈⟨ *ᵣ-homo x (m M.+ᴹ n) ⟩
    ⟦ m M.+ᴹ n ⟧ N.*ᵣ x            ≈⟨ *ᵣ-congʳ (+ᴹ-homo m n) ⟩
    (⟦ m ⟧ N.+ᴹ ⟦ n ⟧) N.*ᵣ x      ≈⟨ *ᵣ-distribʳ x ⟦ m ⟧ ⟦ n ⟧ ⟩
    ⟦ m ⟧ N.*ᵣ x N.+ᴹ ⟦ n ⟧ N.*ᵣ x ≈˘⟨ +ᴹ-cong′ (*ᵣ-homo x m) (*ᵣ-homo x n) ⟩
    ⟦ m M.*ᵣ x ⟧ N.+ᴹ ⟦ n M.*ᵣ x ⟧ ≈˘⟨ +ᴹ-homo (m M.*ᵣ x) (n M.*ᵣ x) ⟩
    ⟦ m M.*ᵣ x M.+ᴹ n M.*ᵣ x ⟧     ∎

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

module _ (R-isSemiring : IsSemiring _≈_ _+_ _*_ 0# 1#) where

  private
    R-semiring : Semiring _ _
    R-semiring = record { isSemiring = R-isSemiring }

  isRightSemimodule
    : IsRightSemimodule R-semiring N._≈ᴹ_ N._+ᴹ_ N.0ᴹ N._*ᵣ_
    → IsRightSemimodule R-semiring M._≈ᴹ_ M._+ᴹ_ M.0ᴹ M._*ᵣ_
  isRightSemimodule isRightSemimodule = record
    { +ᴹ-isCommutativeMonoid = +ᴹ-isCommutativeMonoid NN.+ᴹ-isCommutativeMonoid
    ; isPrerightSemimodule = record
      { *ᵣ-cong = *ᵣ-cong NN.+ᴹ-isMagma NN.*ᵣ-cong
      ; *ᵣ-zeroʳ = *ᵣ-zeroʳ NN.+ᴹ-isMagma NN.*ᵣ-zeroʳ
      ; *ᵣ-distribˡ = *ᵣ-distribˡ NN.+ᴹ-isMagma NN.*ᵣ-distribˡ
      ; *ᵣ-identityʳ = *ᵣ-identityʳ NN.+ᴹ-isMagma NN.*ᵣ-identityʳ
      ; *ᵣ-assoc = *ᵣ-assoc NN.+ᴹ-isMagma NN.*ᵣ-congʳ NN.*ᵣ-assoc
      ; *ᵣ-zeroˡ = *ᵣ-zeroˡ NN.+ᴹ-isMagma NN.*ᵣ-congʳ NN.*ᵣ-zeroˡ
      ; *ᵣ-distribʳ = *ᵣ-distribʳ NN.+ᴹ-isMagma NN.*ᵣ-congʳ NN.*ᵣ-distribʳ
      }
    } where module NN = IsRightSemimodule isRightSemimodule