------------------------------------------------------------------------
-- 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 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
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