LeftSemimoduleMonomorphism.agda

------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between left semimodules
------------------------------------------------------------------------

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

open import Algebra.Module.Bundles.Raw using (RawLeftSemimodule)
open import Algebra.Module.Morphism.Structures using (IsLeftSemimoduleMonomorphism)

module Algebra.Module.Morphism.LeftSemimoduleMonomorphism
  {r a b ℓ₁ ℓ₂} {R : Set r} {M₁ : RawLeftSemimodule R a ℓ₁} {M₂ : RawLeftSemimodule R b ℓ₂} {⟦_⟧}
  (isLeftSemimoduleMonomorphism : IsLeftSemimoduleMonomorphism M₁ M₂ ⟦_⟧)
  where

open IsLeftSemimoduleMonomorphism M₁ M₂ isLeftSemimoduleMonomorphism
private
  module M = RawLeftSemimodule M₁
  module N = RawLeftSemimodule M₂

open import Algebra.Bundles using (Semiring)
open import Algebra.Core using (Op₂)
import Algebra.Module.Definitions.Left as LeftDefs
open import Algebra.Module.Structures using (IsLeftSemimodule)
open import Algebra.Structures using (IsSemiring; IsMagma)
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-export most properties of monoid monomorphisms

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 = LeftDefs R M._≈ᴹ_
  module NDefs = LeftDefs R N._≈ᴹ_

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

  *ₗ-zeroˡ : NDefs.LeftZero 0# N.0ᴹ N._*ₗ_ → MDefs.LeftZero 0# M.0ᴹ M._*ₗ_
  *ₗ-zeroˡ {0# = 0#} *ₗ-zeroˡ x = injective $ begin
    ⟦ 0# M.*ₗ x ⟧ ≈⟨ *ₗ-homo 0# x ⟩
    0# N.*ₗ ⟦ x ⟧ ≈⟨ *ₗ-zeroˡ ⟦ x ⟧ ⟩
    N.0ᴹ          ≈˘⟨ 0ᴹ-homo ⟩
    ⟦ M.0ᴹ ⟧      ∎

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

  *ₗ-identityˡ : NDefs.LeftIdentity 1# N._*ₗ_ → MDefs.LeftIdentity 1# M._*ₗ_
  *ₗ-identityˡ {1# = 1#} *ₗ-identityˡ m = injective $ begin
    ⟦ 1# M.*ₗ m ⟧ ≈⟨ *ₗ-homo 1# m ⟩
    1# N.*ₗ ⟦ m ⟧ ≈⟨ *ₗ-identityˡ ⟦ m ⟧ ⟩
    ⟦ m ⟧         ∎

  *ₗ-assoc : NDefs.LeftCongruent N._*ₗ_ → NDefs.Associative _*_ N._*ₗ_ → MDefs.Associative _*_ M._*ₗ_
  *ₗ-assoc {_*_ = _*_} *ₗ-congˡ *ₗ-assoc x y m = injective $ begin
    ⟦ (x * y) M.*ₗ m ⟧  ≈⟨ *ₗ-homo (x * y) m ⟩
    (x * y) N.*ₗ ⟦ m ⟧  ≈⟨ *ₗ-assoc x y ⟦ m ⟧ ⟩
    x N.*ₗ y N.*ₗ ⟦ m ⟧ ≈˘⟨ *ₗ-congˡ (*ₗ-homo y m) ⟩
    x N.*ₗ ⟦ y M.*ₗ m ⟧ ≈˘⟨ *ₗ-homo x (y M.*ₗ m) ⟩
    ⟦ x M.*ₗ y M.*ₗ m ⟧ ∎

  *ₗ-zeroʳ : NDefs.LeftCongruent N._*ₗ_ → NDefs.RightZero N.0ᴹ N._*ₗ_ → MDefs.RightZero M.0ᴹ M._*ₗ_
  *ₗ-zeroʳ *ₗ-congˡ *ₗ-zeroʳ x = injective $ begin
    ⟦ x M.*ₗ M.0ᴹ ⟧ ≈⟨ *ₗ-homo x M.0ᴹ ⟩
    x N.*ₗ ⟦ M.0ᴹ ⟧ ≈⟨ *ₗ-congˡ 0ᴹ-homo ⟩
    x N.*ₗ N.0ᴹ     ≈⟨ *ₗ-zeroʳ x ⟩
    N.0ᴹ            ≈˘⟨ 0ᴹ-homo ⟩
    ⟦ M.0ᴹ ⟧        ∎

  *ₗ-distribˡ : NDefs.LeftCongruent N._*ₗ_ → N._*ₗ_ NDefs.DistributesOverˡ N._+ᴹ_ → M._*ₗ_ MDefs.DistributesOverˡ M._+ᴹ_
  *ₗ-distribˡ *ₗ-congˡ *ₗ-distribˡ x m n = injective $ begin
    ⟦ x M.*ₗ (m M.+ᴹ n) ⟧          ≈⟨ *ₗ-homo x (m M.+ᴹ n) ⟩
    x N.*ₗ ⟦ m M.+ᴹ n ⟧            ≈⟨ *ₗ-congˡ (+ᴹ-homo m n) ⟩
    x N.*ₗ (⟦ m ⟧ N.+ᴹ ⟦ n ⟧)      ≈⟨ *ₗ-distribˡ x ⟦ m ⟧ ⟦ n ⟧ ⟩
    x N.*ₗ ⟦ m ⟧ N.+ᴹ x N.*ₗ ⟦ n ⟧ ≈˘⟨ +ᴹ-cong′ (*ₗ-homo x m) (*ₗ-homo x n) ⟩
    ⟦ x M.*ₗ m ⟧ N.+ᴹ ⟦ x M.*ₗ n ⟧ ≈˘⟨ +ᴹ-homo (x M.*ₗ m) (x M.*ₗ n) ⟩
    ⟦ x M.*ₗ m M.+ᴹ x M.*ₗ n ⟧     ∎

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

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

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

  isLeftSemimodule
    : IsLeftSemimodule R-semiring N._≈ᴹ_ N._+ᴹ_ N.0ᴹ N._*ₗ_
    → IsLeftSemimodule R-semiring M._≈ᴹ_ M._+ᴹ_ M.0ᴹ M._*ₗ_
  isLeftSemimodule isLeftSemimodule = record
    { +ᴹ-isCommutativeMonoid = +ᴹ-isCommutativeMonoid NN.+ᴹ-isCommutativeMonoid
    ; isPreleftSemimodule = record
      { *ₗ-cong = *ₗ-cong NN.+ᴹ-isMagma NN.*ₗ-cong
      ; *ₗ-zeroˡ = *ₗ-zeroˡ NN.+ᴹ-isMagma NN.*ₗ-zeroˡ
      ; *ₗ-distribʳ = *ₗ-distribʳ NN.+ᴹ-isMagma NN.*ₗ-distribʳ
      ; *ₗ-identityˡ = *ₗ-identityˡ NN.+ᴹ-isMagma NN.*ₗ-identityˡ
      ; *ₗ-assoc = *ₗ-assoc NN.+ᴹ-isMagma NN.*ₗ-congˡ NN.*ₗ-assoc
      ; *ₗ-zeroʳ = *ₗ-zeroʳ NN.+ᴹ-isMagma NN.*ₗ-congˡ NN.*ₗ-zeroʳ
      ; *ₗ-distribˡ = *ₗ-distribˡ NN.+ᴹ-isMagma NN.*ₗ-congˡ NN.*ₗ-distribˡ
      }
    } where module NN = IsLeftSemimodule isLeftSemimodule