101 lines
4.6 KiB
Agda
101 lines
4.6 KiB
Agda
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open import Equivalence
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module Homomorphism {a b} (A : Set a) (B : Set b)
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(_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set b)
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(≈₂-equiv : IsEquivalence B _≈₂_)
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(f : A → B) where
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open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)
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open import Function.Definitions using (Surjective)
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open import Relation.Binary.Core using (_Preserves_⟶_ )
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open import Data.Product using (_,_)
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open import Lattice
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open IsEquivalence ≈₂-equiv using () renaming (≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym; ≈-refl to ≈₂-refl)
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open import Relation.Binary.Reasoning.Base.Single _≈₂_ ≈₂-refl ≈₂-trans
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infixl 20 _∙₂_
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_∙₂_ = ≈₂-trans
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record SemilatticeHomomorphism (_⊔₁_ : A → A → A)
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(_⊔₂_ : B → B → B) : Set (a ⊔ℓ b) where
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field
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f-preserves-≈ : f Preserves _≈₁_ ⟶ _≈₂_
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f-⊔-distr : ∀ (a₁ a₂ : A) → f (a₁ ⊔₁ a₂) ≈₂ ((f a₁) ⊔₂ (f a₂))
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module _ (_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
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(sh : SemilatticeHomomorphism _⊔₁_ _⊔₂_)
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(≈₂-⊔₂-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≈₂ a₂ → a₃ ≈₂ a₄ → (a₁ ⊔₂ a₃) ≈₂ (a₂ ⊔₂ a₄))
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(surF : Surjective _≈₁_ _≈₂_ f) where
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open SemilatticeHomomorphism sh
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transportSemilattice : IsSemilattice A _≈₁_ _⊔₁_ → IsSemilattice B _≈₂_ _⊔₂_
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transportSemilattice sA = record
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{ ≈-equiv = ≈₂-equiv
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; ≈-⊔-cong = ≈₂-⊔₂-cong
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; ⊔-assoc = λ b₁ b₂ b₃ →
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let (a₁ , fa₁≈b₁) = surF b₁
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(a₂ , fa₂≈b₂) = surF b₂
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(a₃ , fa₃≈b₃) = surF b₃
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in
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begin
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(b₁ ⊔₂ b₂) ⊔₂ b₃
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∼⟨ ≈₂-⊔₂-cong (≈₂-⊔₂-cong (≈₂-sym fa₁≈b₁) (≈₂-sym fa₂≈b₂)) (≈₂-sym fa₃≈b₃) ⟩
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(f a₁ ⊔₂ f a₂) ⊔₂ f a₃
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∼⟨ ≈₂-⊔₂-cong (≈₂-sym (f-⊔-distr a₁ a₂)) ≈₂-refl ⟩
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f (a₁ ⊔₁ a₂) ⊔₂ f a₃
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∼⟨ ≈₂-sym (f-⊔-distr (a₁ ⊔₁ a₂) a₃) ⟩
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f ((a₁ ⊔₁ a₂) ⊔₁ a₃)
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∼⟨ f-preserves-≈ (IsSemilattice.⊔-assoc sA a₁ a₂ a₃) ⟩
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f (a₁ ⊔₁ (a₂ ⊔₁ a₃))
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∼⟨ f-⊔-distr a₁ (a₂ ⊔₁ a₃) ⟩
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f a₁ ⊔₂ f (a₂ ⊔₁ a₃)
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∼⟨ ≈₂-⊔₂-cong ≈₂-refl (f-⊔-distr a₂ a₃) ⟩
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f a₁ ⊔₂ (f a₂ ⊔₂ f a₃)
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∼⟨ ≈₂-⊔₂-cong fa₁≈b₁ (≈₂-⊔₂-cong fa₂≈b₂ fa₃≈b₃) ⟩
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b₁ ⊔₂ (b₂ ⊔₂ b₃)
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∎
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; ⊔-comm = λ b₁ b₂ →
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let (a₁ , fa₁≈b₁) = surF b₁
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(a₂ , fa₂≈b₂) = surF b₂
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in
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begin
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b₁ ⊔₂ b₂
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∼⟨ ≈₂-⊔₂-cong (≈₂-sym fa₁≈b₁) (≈₂-sym fa₂≈b₂) ⟩
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f a₁ ⊔₂ f a₂
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∼⟨ ≈₂-sym (f-⊔-distr a₁ a₂) ⟩
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f (a₁ ⊔₁ a₂)
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∼⟨ f-preserves-≈ (IsSemilattice.⊔-comm sA a₁ a₂) ⟩
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f (a₂ ⊔₁ a₁)
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∼⟨ f-⊔-distr a₂ a₁ ⟩
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f a₂ ⊔₂ f a₁
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∼⟨ ≈₂-⊔₂-cong fa₂≈b₂ fa₁≈b₁ ⟩
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b₂ ⊔₂ b₁
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∎
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; ⊔-idemp = λ b →
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let (a , fa≈b) = surF b
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in
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begin
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b ⊔₂ b
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∼⟨ ≈₂-⊔₂-cong (≈₂-sym fa≈b) (≈₂-sym fa≈b) ⟩
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f a ⊔₂ f a
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∼⟨ ≈₂-sym (f-⊔-distr a a) ⟩
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f (a ⊔₁ a)
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∼⟨ f-preserves-≈ (IsSemilattice.⊔-idemp sA a) ⟩
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f a
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∼⟨ fa≈b ⟩
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b
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∎
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}
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record LatticeHomomorphism (_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
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(_⊓₁_ : A → A → A) (_⊓₂_ : B → B → B) : Set (a ⊔ℓ b) where
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field
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⊔-homomorphism : SemilatticeHomomorphism _⊔₁_ _⊔₂_
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⊓-homomorphism : SemilatticeHomomorphism _⊓₁_ _⊓₂_
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open SemilatticeHomomorphism ⊔-homomorphism using (f-preserves-≈)
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open SemilatticeHomomorphism ⊓-homomorphism renaming (f-⊔-distr to f-⊓-distr)
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