------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between ring-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.Bundles using (RawRing)
open import Algebra.Morphism.Structures using (IsRingMonomorphism)
module Algebra.Morphism.RingMonomorphism
{a b ℓ₁ ℓ₂} {R₁ : RawRing a ℓ₁} {R₂ : RawRing b ℓ₂} {⟦_⟧}
(isRingMonomorphism : IsRingMonomorphism R₁ R₂ ⟦_⟧)
where
open IsRingMonomorphism isRingMonomorphism
open RawRing R₁ renaming (Carrier to A; _≈_ to _≈₁_)
open RawRing R₂ renaming
( Carrier to B; _≈_ to _≈₂_; _+_ to _⊕_
; _*_ to _⊛_; 1# to 1#₂; 0# to 0#₂; -_ to ⊝_)
import Algebra.Morphism.GroupMonomorphism as GroupMonomorphism
import Algebra.Morphism.MonoidMonomorphism as MonoidMonomorphism
open import Relation.Binary.Core using (Rel)
open import Algebra.Definitions
using (_DistributesOverˡ_; _DistributesOverʳ_ ; _DistributesOver_
; LeftZero; RightZero; Zero)
open import Algebra.Structures
using (IsRing; IsCommutativeRing; IsGroup; IsMagma)
open import Data.Product.Base using (proj₁; proj₂; _,_)
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
------------------------------------------------------------------------
-- Re-export all properties of group and monoid monomorphisms
open GroupMonomorphism +-isGroupMonomorphism public
renaming
( assoc to +-assoc
; comm to +-comm
; cong to +-cong
; idem to +-idem
; sel to +-sel
; ⁻¹-cong to neg-cong
; identity to +-identity; identityˡ to +-identityˡ; identityʳ to +-identityʳ
; cancel to +-cancel; cancelˡ to +-cancelˡ; cancelʳ to +-cancelʳ
; zero to +-zero; zeroˡ to +-zeroˡ; zeroʳ to +-zeroʳ
; isMagma to +-isMagma
; isSemigroup to +-isSemigroup
; isMonoid to +-isMonoid
; isSelectiveMagma to +-isSelectiveMagma
; isBand to +-isBand
; isCommutativeMonoid to +-isCommutativeMonoid
)
open MonoidMonomorphism *-isMonoidMonomorphism public
renaming
( assoc to *-assoc
; comm to *-comm
; cong to *-cong
; idem to *-idem
; sel to *-sel
; identity to *-identity; identityˡ to *-identityˡ; identityʳ to *-identityʳ
; cancel to *-cancel; cancelˡ to *-cancelˡ; cancelʳ to *-cancelʳ
; zero to *-zero; zeroˡ to *-zeroˡ; zeroʳ to *-zeroʳ
; isMagma to *-isMagma
; isSemigroup to *-isSemigroup
; isMonoid to *-isMonoid
; isSelectiveMagma to *-isSelectiveMagma
; isBand to *-isBand
; isCommutativeMonoid to *-isCommutativeMonoid
)
------------------------------------------------------------------------
-- Properties
module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
(*-isMagma : IsMagma _≈₂_ _⊛_) where
open IsGroup +-isGroup hiding (setoid; refl; sym)
open IsMagma *-isMagma renaming (∙-cong to ◦-cong)
open ≈-Reasoning setoid
distribˡ : _DistributesOverˡ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverˡ_ _≈₁_ _*_ _+_
distribˡ distribˡ x y z = injective (begin
⟦ x * (y + z) ⟧ ≈⟨ *-homo x (y + z) ⟩
⟦ x ⟧ ⊛ ⟦ y + z ⟧ ≈⟨ ◦-cong refl (+-homo y z) ⟩
⟦ x ⟧ ⊛ (⟦ y ⟧ ⊕ ⟦ z ⟧) ≈⟨ distribˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
⟦ x ⟧ ⊛ ⟦ y ⟧ ⊕ ⟦ x ⟧ ⊛ ⟦ z ⟧ ≈⟨ ∙-cong (*-homo x y) (*-homo x z) ⟨
⟦ x * y ⟧ ⊕ ⟦ x * z ⟧ ≈⟨ +-homo (x * y) (x * z) ⟨
⟦ x * y + x * z ⟧ ∎)
distribʳ : _DistributesOverʳ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverʳ_ _≈₁_ _*_ _+_
distribʳ distribˡ x y z = injective (begin
⟦ (y + z) * x ⟧ ≈⟨ *-homo (y + z) x ⟩
⟦ y + z ⟧ ⊛ ⟦ x ⟧ ≈⟨ ◦-cong (+-homo y z) refl ⟩
(⟦ y ⟧ ⊕ ⟦ z ⟧) ⊛ ⟦ x ⟧ ≈⟨ distribˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
⟦ y ⟧ ⊛ ⟦ x ⟧ ⊕ ⟦ z ⟧ ⊛ ⟦ x ⟧ ≈⟨ ∙-cong (*-homo y x) (*-homo z x) ⟨
⟦ y * x ⟧ ⊕ ⟦ z * x ⟧ ≈⟨ +-homo (y * x) (z * x) ⟨
⟦ y * x + z * x ⟧ ∎)
distrib : _DistributesOver_ _≈₂_ _⊛_ _⊕_ → _DistributesOver_ _≈₁_ _*_ _+_
distrib distrib = distribˡ (proj₁ distrib) , distribʳ (proj₂ distrib)
zeroˡ : LeftZero _≈₂_ 0#₂ _⊛_ → LeftZero _≈₁_ 0# _*_
zeroˡ zeroˡ x = injective (begin
⟦ 0# * x ⟧ ≈⟨ *-homo 0# x ⟩
⟦ 0# ⟧ ⊛ ⟦ x ⟧ ≈⟨ ◦-cong 0#-homo refl ⟩
0#₂ ⊛ ⟦ x ⟧ ≈⟨ zeroˡ ⟦ x ⟧ ⟩
0#₂ ≈⟨ 0#-homo ⟨
⟦ 0# ⟧ ∎)
zeroʳ : RightZero _≈₂_ 0#₂ _⊛_ → RightZero _≈₁_ 0# _*_
zeroʳ zeroʳ x = injective (begin
⟦ x * 0# ⟧ ≈⟨ *-homo x 0# ⟩
⟦ x ⟧ ⊛ ⟦ 0# ⟧ ≈⟨ ◦-cong refl 0#-homo ⟩
⟦ x ⟧ ⊛ 0#₂ ≈⟨ zeroʳ ⟦ x ⟧ ⟩
0#₂ ≈⟨ 0#-homo ⟨
⟦ 0# ⟧ ∎)
zero : Zero _≈₂_ 0#₂ _⊛_ → Zero _≈₁_ 0# _*_
zero zero = zeroˡ (proj₁ zero) , zeroʳ (proj₂ zero)
neg-distribˡ-* : (∀ x y → (⊝ (x ⊛ y)) ≈₂ ((⊝ x) ⊛ y)) → (∀ x y → (- (x * y)) ≈₁ ((- x) * y))
neg-distribˡ-* neg-distribˡ-* x y = injective (begin
⟦ - (x * y) ⟧ ≈⟨ -‿homo (x * y) ⟩
⊝ ⟦ x * y ⟧ ≈⟨ ⁻¹-cong (*-homo x y) ⟩
⊝ (⟦ x ⟧ ⊛ ⟦ y ⟧) ≈⟨ neg-distribˡ-* ⟦ x ⟧ ⟦ y ⟧ ⟩
⊝ ⟦ x ⟧ ⊛ ⟦ y ⟧ ≈⟨ ◦-cong (sym (-‿homo x)) refl ⟩
⟦ - x ⟧ ⊛ ⟦ y ⟧ ≈⟨ sym (*-homo (- x) y) ⟩
⟦ - x * y ⟧ ∎)
neg-distribʳ-* : (∀ x y → (⊝ (x ⊛ y)) ≈₂ (x ⊛ (⊝ y))) → (∀ x y → (- (x * y)) ≈₁ (x * (- y)))
neg-distribʳ-* neg-distribʳ-* x y = injective (begin
⟦ - (x * y) ⟧ ≈⟨ -‿homo (x * y) ⟩
⊝ ⟦ x * y ⟧ ≈⟨ ⁻¹-cong (*-homo x y) ⟩
⊝ (⟦ x ⟧ ⊛ ⟦ y ⟧) ≈⟨ neg-distribʳ-* ⟦ x ⟧ ⟦ y ⟧ ⟩
⟦ x ⟧ ⊛ ⊝ ⟦ y ⟧ ≈⟨ ◦-cong refl (sym (-‿homo y)) ⟩
⟦ x ⟧ ⊛ ⟦ - y ⟧ ≈⟨ sym (*-homo x (- y)) ⟩
⟦ x * - y ⟧ ∎)
isRing : IsRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ → IsRing _≈₁_ _+_ _*_ -_ 0# 1#
isRing isRing = record
{ +-isAbelianGroup = isAbelianGroup R.+-isAbelianGroup
; *-cong = *-cong R.*-isMagma
; *-assoc = *-assoc R.*-isMagma R.*-assoc
; *-identity = *-identity R.*-isMagma R.*-identity
; distrib = distrib R.+-isGroup R.*-isMagma R.distrib
} where module R = IsRing isRing
isCommutativeRing : IsCommutativeRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ →
IsCommutativeRing _≈₁_ _+_ _*_ -_ 0# 1#
isCommutativeRing isCommRing = record
{ isRing = isRing C.isRing
; *-comm = *-comm C.*-isMagma C.*-comm
} where module C = IsCommutativeRing isCommRing
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