------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between monoid-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 (RawMonoid)
open import Algebra.Morphism.Structures using (IsMonoidMonomorphism)
module Algebra.Morphism.MonoidMonomorphism
{a b ℓ₁ ℓ₂} {M₁ : RawMonoid a ℓ₁} {M₂ : RawMonoid b ℓ₂} {⟦_⟧}
(isMonoidMonomorphism : IsMonoidMonomorphism M₁ M₂ ⟦_⟧)
where
open IsMonoidMonomorphism isMonoidMonomorphism
open RawMonoid M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; ε to ε₁)
open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; ε to ε₂)
open import Algebra.Definitions
using (Identity; Zero; LeftIdentity; RightIdentity; LeftZero; RightZero)
open import Algebra.Structures
using (IsMagma; IsMonoid; IsCommutativeMonoid)
open import Data.Product.Base using (map)
open import Relation.Binary.Core using (Rel)
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
------------------------------------------------------------------------
-- Re-export all properties of magma monomorphisms
open import Algebra.Morphism.MagmaMonomorphism
isMagmaMonomorphism public
------------------------------------------------------------------------
-- Properties
module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)
open ≈-Reasoning setoid
identityˡ : LeftIdentity _≈₂_ ε₂ _◦_ → LeftIdentity _≈₁_ ε₁ _∙_
identityˡ idˡ x = injective (begin
⟦ ε₁ ∙ x ⟧ ≈⟨ homo ε₁ x ⟩
⟦ ε₁ ⟧ ◦ ⟦ x ⟧ ≈⟨ ◦-cong ε-homo refl ⟩
ε₂ ◦ ⟦ x ⟧ ≈⟨ idˡ ⟦ x ⟧ ⟩
⟦ x ⟧ ∎)
identityʳ : RightIdentity _≈₂_ ε₂ _◦_ → RightIdentity _≈₁_ ε₁ _∙_
identityʳ idʳ x = injective (begin
⟦ x ∙ ε₁ ⟧ ≈⟨ homo x ε₁ ⟩
⟦ x ⟧ ◦ ⟦ ε₁ ⟧ ≈⟨ ◦-cong refl ε-homo ⟩
⟦ x ⟧ ◦ ε₂ ≈⟨ idʳ ⟦ x ⟧ ⟩
⟦ x ⟧ ∎)
identity : Identity _≈₂_ ε₂ _◦_ → Identity _≈₁_ ε₁ _∙_
identity = map identityˡ identityʳ
zeroˡ : LeftZero _≈₂_ ε₂ _◦_ → LeftZero _≈₁_ ε₁ _∙_
zeroˡ zeˡ x = injective (begin
⟦ ε₁ ∙ x ⟧ ≈⟨ homo ε₁ x ⟩
⟦ ε₁ ⟧ ◦ ⟦ x ⟧ ≈⟨ ◦-cong ε-homo refl ⟩
ε₂ ◦ ⟦ x ⟧ ≈⟨ zeˡ ⟦ x ⟧ ⟩
ε₂ ≈⟨ ε-homo ⟨
⟦ ε₁ ⟧ ∎)
zeroʳ : RightZero _≈₂_ ε₂ _◦_ → RightZero _≈₁_ ε₁ _∙_
zeroʳ zeʳ x = injective (begin
⟦ x ∙ ε₁ ⟧ ≈⟨ homo x ε₁ ⟩
⟦ x ⟧ ◦ ⟦ ε₁ ⟧ ≈⟨ ◦-cong refl ε-homo ⟩
⟦ x ⟧ ◦ ε₂ ≈⟨ zeʳ ⟦ x ⟧ ⟩
ε₂ ≈⟨ ε-homo ⟨
⟦ ε₁ ⟧ ∎)
zero : Zero _≈₂_ ε₂ _◦_ → Zero _≈₁_ ε₁ _∙_
zero = map zeroˡ zeroʳ
------------------------------------------------------------------------
-- Structures
isMonoid : IsMonoid _≈₂_ _◦_ ε₂ → IsMonoid _≈₁_ _∙_ ε₁
isMonoid isMonoid = record
{ isSemigroup = isSemigroup M.isSemigroup
; identity = identity M.isMagma M.identity
} where module M = IsMonoid isMonoid
isCommutativeMonoid : IsCommutativeMonoid _≈₂_ _◦_ ε₂ →
IsCommutativeMonoid _≈₁_ _∙_ ε₁
isCommutativeMonoid isCommMonoid = record
{ isMonoid = isMonoid C.isMonoid
; comm = comm C.isMagma C.comm
} where module C = IsCommutativeMonoid isCommMonoid
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4