Prove that AxB is a finite height semilattice
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -16,7 +16,7 @@ module _ where
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data Chain : A → A → ℕ → Set a where
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done : ∀ {a a' : A} → a ≈ a' → Chain a a' 0
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step : ∀ {a₁ a₂ a₂' a₃ : A} {n : ℕ} → a₁ R a₂ → a₂ ≈ a₂' → Chain a₂ a₃ n → Chain a₁ a₃ (suc n)
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step : ∀ {a₁ a₂ a₂' a₃ : A} {n : ℕ} → a₁ R a₂ → a₂ ≈ a₂' → Chain a₂' a₃ n → Chain a₁ a₃ (suc n)
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Chain-≈-cong₁ : ∀ {a₁ a₁' a₂} {n : ℕ} → a₁ ≈ a₁' → Chain a₁ a₂ n → Chain a₁' a₂ n
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Chain-≈-cong₁ a₁≈a₁' (done a₁≈a₂) = done (≈-trans (≈-sym a₁≈a₁') a₁≈a₂)
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51
Lattice.agda
51
Lattice.agda
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@ -4,17 +4,22 @@ open import Equivalence
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import Chain
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import Data.Nat.Properties as NatProps
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym)
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; subst)
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open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; step-≡; _∎)
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open import Relation.Binary.Definitions
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open import Relation.Binary.Core using (_Preserves_⟶_ )
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open import Relation.Nullary using (Dec; ¬_)
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open import Data.Nat as Nat using (ℕ; _≤_; _+_)
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open import Relation.Nullary using (Dec; ¬_; yes; no)
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open import Data.Nat as Nat using (ℕ; _≤_; _+_; suc)
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open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)
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open import Function.Definitions using (Injective)
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open import Data.Empty using (⊥)
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record IsDecidable {a} (A : Set a) (R : A → A → Set a) : Set a where
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absurd : ∀ {a} {A : Set a} → ⊥ → A
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absurd ()
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record IsDecidable {a} {A : Set a} (R : A → A → Set a) : Set a where
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field
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R-dec : ∀ (a₁ a₂ : A) → Dec (R a₁ a₂)
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@ -356,20 +361,32 @@ module IsLatticeInstances where
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module IsFiniteHeightLatticeInstances where
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module ForProd {a} {A B : Set a}
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(_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
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(decA : IsDecidable _≈₁_) (decB : IsDecidable _≈₂_)
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(_⊔₁_ : A → A → A) (_⊓₁_ : A → A → A)
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(_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
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(h₁ h₂ : ℕ)
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(lA : IsFiniteHeightLattice A h₁ _≈₁_ _⊔₁_ _⊓₁_) (lB : IsFiniteHeightLattice B h₂ _≈₂_ _⊔₂_ _⊓₂_) where
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open NatProps
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module ProdLattice = IsLatticeInstances.ForProd _≈₁_ _≈₂_ _⊔₁_ _⊓₁_ _⊔₂_ _⊓₂_ (IsFiniteHeightLattice.isLattice lA) (IsFiniteHeightLattice.isLattice lB)
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open ProdLattice using (_⊔_; _⊓_; _≈_) public
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open IsLattice ProdLattice.ProdIsLattice using (_≼_; _≺_; ≺-cong; ≈-equiv)
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open IsFiniteHeightLattice lA using () renaming (⊔-idemp to ⊔₁-idemp; _≼_ to _≼₁_; ≈-refl to ≈₁-refl)
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open IsFiniteHeightLattice lB using () renaming (⊔-idemp to ⊔₂-idemp; _≼_ to _≼₂_; ≈-refl to ≈₂-refl)
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open IsFiniteHeightLattice lA using () renaming (⊔-idemp to ⊔₁-idemp; _≼_ to _≼₁_; ≈-equiv to ≈₁-equiv; ≈-refl to ≈₁-refl; ≈-trans to ≈₁-trans; ≈-sym to ≈₁-sym; _≺_ to _≺₁_; ≺-cong to ≺₁-cong)
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open IsFiniteHeightLattice lB using () renaming (⊔-idemp to ⊔₂-idemp; _≼_ to _≼₂_; ≈-equiv to ≈₂-equiv; ≈-refl to ≈₂-refl; ≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym; _≺_ to _≺₂_; ≺-cong to ≺₂-cong)
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open IsDecidable decA using () renaming (R-dec to ≈₁-dec)
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open IsDecidable decB using () renaming (R-dec to ≈₂-dec)
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module ChainMapping₁ = ChainMapping (IsFiniteHeightLattice.joinSemilattice lA) (IsLattice.joinSemilattice ProdLattice.ProdIsLattice)
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module ChainMapping₂ = ChainMapping (IsFiniteHeightLattice.joinSemilattice lB) (IsLattice.joinSemilattice ProdLattice.ProdIsLattice)
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module ChainA = Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong
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module ChainB = Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong
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module ProdChain = Chain _≈_ ≈-equiv _≺_ ≺-cong
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open ChainA using () renaming (Chain to Chain₁; done to done₁; step to step₁; Chain-≈-cong₁ to Chain₁-≈-cong₁)
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open ChainB using () renaming (Chain to Chain₂; done to done₂; step to step₂; Chain-≈-cong₁ to Chain₂-≈-cong₁)
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open ProdChain using (Chain; concat; done; step)
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private
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a,∙-Monotonic : ∀ (a : A) → Monotonic _≼₂_ _≼_ (λ b → (a , b))
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a,∙-Monotonic a {b₁} {b₂} (b , b₁⊔b≈b₂) = ((a , b) , (⊔₁-idemp a , b₁⊔b≈b₂))
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@ -395,15 +412,33 @@ module IsFiniteHeightLatticeInstances where
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bmax : B
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bmax = proj₂ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lB)))
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unzip : ∀ {a₁ a₂ : A} {b₁ b₂ : B} {n : ℕ} → Chain (a₁ , b₁) (a₂ , b₂) n → Σ (ℕ × ℕ) (λ (n₁ , n₂) → ((Chain₁ a₁ a₂ n₁ × Chain₂ b₁ b₂ n₂) × (n ≤ n₁ + n₂)))
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unzip (done (a₁≈a₂ , b₁≈b₂)) = ((0 , 0) , ((done₁ a₁≈a₂ , done₂ b₁≈b₂) , ≤-refl))
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unzip {a₁} {a₂} {b₁} {b₂} {n} (step {(a₁ , b₁)} {(a , b)} (((d₁ , d₂) , (a₁⊔d₁≈a , b₁⊔d₂≈b)) , a₁b₁̷≈ab) (a≈a' , b≈b') a'b'a₂b₂)
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with ≈₁-dec a₁ a | ≈₂-dec b₁ b | unzip a'b'a₂b₂
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... | yes a₁≈a | yes b₁≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) = absurd (a₁b₁̷≈ab (a₁≈a , b₁≈b))
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... | no a₁̷≈a | yes b₁≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
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((suc n₁ , n₂) , ((step₁ ((d₁ , a₁⊔d₁≈a) , a₁̷≈a) a≈a' c₁ , Chain₂-≈-cong₁ (≈₂-sym (≈₂-trans b₁≈b b≈b')) c₂), +-monoʳ-≤ 1 (n≤n₁+n₂)))
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... | yes a₁≈a | no b₁̷≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
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((n₁ , suc n₂) , ( (Chain₁-≈-cong₁ (≈₁-sym (≈₁-trans a₁≈a a≈a')) c₁ , step₂ ((d₂ , b₁⊔d₂≈b) , b₁̷≈b) b≈b' c₂)
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, subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂)
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))
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... | no a₁̷≈a | no b₁̷≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
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((suc n₁ , suc n₂) , ( (step₁ ((d₁ , a₁⊔d₁≈a) , a₁̷≈a) a≈a' c₁ , step₂ ((d₂ , b₁⊔d₂≈b) , b₁̷≈b) b≈b' c₂)
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, ≤-stepsˡ 1 (subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂))
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))
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ProdIsFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_
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ProdIsFiniteHeightLattice = record
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{ isLattice = ProdLattice.ProdIsLattice
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; fixedHeight =
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( ( ((amin , bmin) , (amax , bmax))
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, Chain.concat _≈_ ≈-equiv _≺_ ≺-cong
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, concat
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(ChainMapping₁.Chain-map (λ a → (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ (IsFiniteHeightLattice.fixedHeight lA))))
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(ChainMapping₂.Chain-map (λ b → (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ (IsFiniteHeightLattice.fixedHeight lB))))
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)
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, _
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, λ a₁b₁a₂b₂ → let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
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in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ (IsFiniteHeightLattice.fixedHeight lA) a₁a₂)
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(proj₂ (IsFiniteHeightLattice.fixedHeight lB) b₁b₂))
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)
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}
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