Move the product instances into its own file
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -9,29 +9,3 @@ record IsEquivalence {a} (A : Set a) (_≈_ : A → A → Set a) : Set a where
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≈-refl : {a : A} → a ≈ a
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≈-refl : {a : A} → a ≈ a
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≈-sym : {a b : A} → a ≈ b → b ≈ a
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≈-sym : {a b : A} → a ≈ b → b ≈ a
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≈-trans : {a b c : A} → a ≈ b → b ≈ c → a ≈ c
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≈-trans : {a b c : A} → a ≈ b → b ≈ c → a ≈ c
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module IsEquivalenceInstances 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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(eA : IsEquivalence A _≈₁_) (eB : IsEquivalence B _≈₂_) where
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infix 4 _≈_
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_≈_ : A × B → A × B → Set a
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(a₁ , b₁) ≈ (a₂ , b₂) = (a₁ ≈₁ a₂) × (b₁ ≈₂ b₂)
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ProdEquivalence : IsEquivalence (A × B) _≈_
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ProdEquivalence = record
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{ ≈-refl = λ {p} →
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( IsEquivalence.≈-refl eA
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, IsEquivalence.≈-refl eB
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)
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; ≈-sym = λ {p₁} {p₂} (a₁≈a₂ , b₁≈b₂) →
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( IsEquivalence.≈-sym eA a₁≈a₂
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, IsEquivalence.≈-sym eB b₁≈b₂
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)
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; ≈-trans = λ {p₁} {p₂} {p₃} (a₁≈a₂ , b₁≈b₂) (a₂≈a₃ , b₂≈b₃) →
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( IsEquivalence.≈-trans eA a₁≈a₂ a₂≈a₃
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, IsEquivalence.≈-trans eB b₁≈b₂ b₂≈b₃
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)
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}
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150
Lattice.agda
150
Lattice.agda
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@ -6,7 +6,6 @@ import Chain
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import Data.Nat.Properties as NatProps
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import Data.Nat.Properties as NatProps
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; subst)
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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 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.Binary.Core using (_Preserves_⟶_ )
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open import Relation.Nullary using (Dec; ¬_; yes; no)
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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.Nat as Nat using (ℕ; _≤_; _+_; suc)
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@ -142,152 +141,3 @@ record Lattice {a} (A : Set a) : Set (lsuc a) where
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isLattice : IsLattice A _≈_ _⊔_ _⊓_
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isLattice : IsLattice A _≈_ _⊔_ _⊓_
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open IsLattice isLattice public
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open IsLattice isLattice public
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module IsSemilatticeInstances 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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(_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
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(sA : IsSemilattice A _≈₁_ _⊔₁_) (sB : IsSemilattice B _≈₂_ _⊔₂_) where
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open Eq
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open Data.Product
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module ProdEquiv = IsEquivalenceInstances.ForProd _≈₁_ _≈₂_ (IsSemilattice.≈-equiv sA) (IsSemilattice.≈-equiv sB)
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open ProdEquiv using (_≈_) public
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infixl 20 _⊔_
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_⊔_ : A × B → A × B → A × B
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(a₁ , b₁) ⊔ (a₂ , b₂) = (a₁ ⊔₁ a₂ , b₁ ⊔₂ b₂)
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ProdIsSemilattice : IsSemilattice (A × B) _≈_ _⊔_
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ProdIsSemilattice = record
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{ ≈-equiv = ProdEquiv.ProdEquivalence
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; ≈-⊔-cong = λ (a₁≈a₂ , b₁≈b₂) (a₃≈a₄ , b₃≈b₄) →
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( IsSemilattice.≈-⊔-cong sA a₁≈a₂ a₃≈a₄
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, IsSemilattice.≈-⊔-cong sB b₁≈b₂ b₃≈b₄
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)
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; ⊔-assoc = λ (a₁ , b₁) (a₂ , b₂) (a₃ , b₃) →
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( IsSemilattice.⊔-assoc sA a₁ a₂ a₃
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, IsSemilattice.⊔-assoc sB b₁ b₂ b₃
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)
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; ⊔-comm = λ (a₁ , b₁) (a₂ , b₂) →
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( IsSemilattice.⊔-comm sA a₁ a₂
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, IsSemilattice.⊔-comm sB b₁ b₂
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)
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; ⊔-idemp = λ (a , b) →
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( IsSemilattice.⊔-idemp sA a
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, IsSemilattice.⊔-idemp sB b
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)
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}
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module IsLatticeInstances 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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(_⊔₁_ : A → A → A) (_⊓₁_ : A → A → A)
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(_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
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(lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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module ProdJoin = IsSemilatticeInstances.ForProd _≈₁_ _≈₂_ _⊔₁_ _⊔₂_ (IsLattice.joinSemilattice lA) (IsLattice.joinSemilattice lB)
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open ProdJoin using (_⊔_; _≈_) public
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module ProdMeet = IsSemilatticeInstances.ForProd _≈₁_ _≈₂_ _⊓₁_ _⊓₂_ (IsLattice.meetSemilattice lA) (IsLattice.meetSemilattice lB)
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open ProdMeet using () renaming (_⊔_ to _⊓_) public
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ProdIsLattice : IsLattice (A × B) _≈_ _⊔_ _⊓_
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ProdIsLattice = record
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{ joinSemilattice = ProdJoin.ProdIsSemilattice
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; meetSemilattice = ProdMeet.ProdIsSemilattice
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; absorb-⊔-⊓ = λ (a₁ , b₁) (a₂ , b₂) →
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( IsLattice.absorb-⊔-⊓ lA a₁ a₂
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, IsLattice.absorb-⊔-⊓ lB b₁ b₂
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)
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; absorb-⊓-⊔ = λ (a₁ , b₁) (a₂ , b₂) →
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( IsLattice.absorb-⊓-⊔ lA a₁ a₂
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, IsLattice.absorb-⊓-⊔ lB b₁ b₂
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)
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}
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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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(≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : 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_; ≈-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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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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a,∙-Preserves-≈₂ : ∀ (a : A) → (λ b → (a , b)) Preserves _≈₂_ ⟶ _≈_
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a,∙-Preserves-≈₂ a {b₁} {b₂} b₁≈b₂ = (≈₁-refl , b₁≈b₂)
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∙,b-Monotonic : ∀ (b : B) → Monotonic _≼₁_ _≼_ (λ a → (a , b))
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∙,b-Monotonic b {a₁} {a₂} (a , a₁⊔a≈a₂) = ((a , b) , (a₁⊔a≈a₂ , ⊔₂-idemp b))
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∙,b-Preserves-≈₁ : ∀ (b : B) → (λ a → (a , b)) Preserves _≈₁_ ⟶ _≈_
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∙,b-Preserves-≈₁ b {a₁} {a₂} a₁≈a₂ = (a₁≈a₂ , ≈₂-refl)
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amin : A
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amin = proj₁ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lA)))
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amax : A
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amax = proj₂ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lA)))
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bmin : B
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bmin = proj₁ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lB)))
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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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, 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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, 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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168
Lattice/Prod.agda
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168
Lattice/Prod.agda
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@ -0,0 +1,168 @@
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open import Lattice
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module Lattice.Prod {a} {A B : Set a}
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(_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
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(_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
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(_⊓₁_ : A → A → A) (_⊓₂_ : B → B → B)
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(lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
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open import Data.Nat using (ℕ; _≤_; _+_; suc)
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open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
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open import Equivalence
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open import Relation.Binary.Core using (_Preserves_⟶_ )
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open import Relation.Binary.Definitions
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open import Relation.Binary.PropositionalEquality using (sym; subst)
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open import Relation.Nullary using (¬_; yes; no)
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import Chain
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open IsLattice lA using () renaming
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( ≈-equiv to ≈₁-equiv; ≈-refl to ≈₁-refl; ≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans
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; joinSemilattice to joinSemilattice₁
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; meetSemilattice to meetSemilattice₁
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; FixedHeight to FixedHeight₁
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; ⊔-idemp to ⊔₁-idemp
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; _≼_ to _≼₁_; _≺_ to _≺₁_
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; ≺-cong to ≺₁-cong
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)
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open IsLattice lB using () renaming
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( ≈-equiv to ≈₂-equiv; ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans
|
||||||
|
; joinSemilattice to joinSemilattice₂
|
||||||
|
; meetSemilattice to meetSemilattice₂
|
||||||
|
; FixedHeight to FixedHeight₂
|
||||||
|
; ⊔-idemp to ⊔₂-idemp
|
||||||
|
; _≼_ to _≼₂_; _≺_ to _≺₂_
|
||||||
|
; ≺-cong to ≺₂-cong
|
||||||
|
)
|
||||||
|
|
||||||
|
_≈_ : A × B → A × B → Set a
|
||||||
|
(a₁ , b₁) ≈ (a₂ , b₂) = (a₁ ≈₁ a₂) × (b₁ ≈₂ b₂)
|
||||||
|
|
||||||
|
≈-equiv : IsEquivalence (A × B) _≈_
|
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|
≈-equiv = record
|
||||||
|
{ ≈-refl = λ {p} → (≈₁-refl , ≈₂-refl)
|
||||||
|
; ≈-sym = λ {p₁} {p₂} (a₁≈a₂ , b₁≈b₂) → (≈₁-sym a₁≈a₂ , ≈₂-sym b₁≈b₂)
|
||||||
|
; ≈-trans = λ {p₁} {p₂} {p₃} (a₁≈a₂ , b₁≈b₂) (a₂≈a₃ , b₂≈b₃) →
|
||||||
|
( ≈₁-trans a₁≈a₂ a₂≈a₃ , ≈₂-trans b₁≈b₂ b₂≈b₃ )
|
||||||
|
}
|
||||||
|
|
||||||
|
_⊔_ : A × B → A × B → A × B
|
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|
(a₁ , b₁) ⊔ (a₂ , b₂) = (a₁ ⊔₁ a₂ , b₁ ⊔₂ b₂)
|
||||||
|
|
||||||
|
_⊓_ : A × B → A × B → A × B
|
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|
(a₁ , b₁) ⊓ (a₂ , b₂) = (a₁ ⊓₁ a₂ , b₁ ⊓₂ b₂)
|
||||||
|
|
||||||
|
private module ProdIsSemilattice (f₁ : A → A → A) (f₂ : B → B → B) (sA : IsSemilattice A _≈₁_ f₁) (sB : IsSemilattice B _≈₂_ f₂) where
|
||||||
|
isSemilattice : IsSemilattice (A × B) _≈_ (λ (a₁ , b₁) (a₂ , b₂) → (f₁ a₁ a₂ , f₂ b₁ b₂))
|
||||||
|
isSemilattice = record
|
||||||
|
{ ≈-equiv = ≈-equiv
|
||||||
|
; ≈-⊔-cong = λ (a₁≈a₂ , b₁≈b₂) (a₃≈a₄ , b₃≈b₄) →
|
||||||
|
( IsSemilattice.≈-⊔-cong sA a₁≈a₂ a₃≈a₄
|
||||||
|
, IsSemilattice.≈-⊔-cong sB b₁≈b₂ b₃≈b₄
|
||||||
|
)
|
||||||
|
; ⊔-assoc = λ (a₁ , b₁) (a₂ , b₂) (a₃ , b₃) →
|
||||||
|
( IsSemilattice.⊔-assoc sA a₁ a₂ a₃
|
||||||
|
, IsSemilattice.⊔-assoc sB b₁ b₂ b₃
|
||||||
|
)
|
||||||
|
; ⊔-comm = λ (a₁ , b₁) (a₂ , b₂) →
|
||||||
|
( IsSemilattice.⊔-comm sA a₁ a₂
|
||||||
|
, IsSemilattice.⊔-comm sB b₁ b₂
|
||||||
|
)
|
||||||
|
; ⊔-idemp = λ (a , b) →
|
||||||
|
( IsSemilattice.⊔-idemp sA a
|
||||||
|
, IsSemilattice.⊔-idemp sB b
|
||||||
|
)
|
||||||
|
}
|
||||||
|
|
||||||
|
isJoinSemilattice : IsSemilattice (A × B) _≈_ _⊔_
|
||||||
|
isJoinSemilattice = ProdIsSemilattice.isSemilattice _⊔₁_ _⊔₂_ joinSemilattice₁ joinSemilattice₂
|
||||||
|
|
||||||
|
isMeetSemilattice : IsSemilattice (A × B) _≈_ _⊓_
|
||||||
|
isMeetSemilattice = ProdIsSemilattice.isSemilattice _⊓₁_ _⊓₂_ meetSemilattice₁ meetSemilattice₂
|
||||||
|
|
||||||
|
isLattice : IsLattice (A × B) _≈_ _⊔_ _⊓_
|
||||||
|
isLattice = record
|
||||||
|
{ joinSemilattice = isJoinSemilattice
|
||||||
|
; meetSemilattice = isMeetSemilattice
|
||||||
|
; absorb-⊔-⊓ = λ (a₁ , b₁) (a₂ , b₂) →
|
||||||
|
( IsLattice.absorb-⊔-⊓ lA a₁ a₂
|
||||||
|
, IsLattice.absorb-⊔-⊓ lB b₁ b₂
|
||||||
|
)
|
||||||
|
; absorb-⊓-⊔ = λ (a₁ , b₁) (a₂ , b₂) →
|
||||||
|
( IsLattice.absorb-⊓-⊔ lA a₁ a₂
|
||||||
|
, IsLattice.absorb-⊓-⊔ lB b₁ b₂
|
||||||
|
)
|
||||||
|
}
|
||||||
|
|
||||||
|
module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
|
||||||
|
(h₁ h₂ : ℕ)
|
||||||
|
(fhA : FixedHeight₁ h₁) (fhB : FixedHeight₂ h₂) where
|
||||||
|
|
||||||
|
open import Data.Nat.Properties
|
||||||
|
open IsLattice isLattice using (_≼_; _≺_; ≺-cong)
|
||||||
|
|
||||||
|
module ChainMapping₁ = ChainMapping joinSemilattice₁ isJoinSemilattice
|
||||||
|
module ChainMapping₂ = ChainMapping joinSemilattice₂ isJoinSemilattice
|
||||||
|
|
||||||
|
module ChainA = Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong
|
||||||
|
module ChainB = Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong
|
||||||
|
module ProdChain = Chain _≈_ ≈-equiv _≺_ ≺-cong
|
||||||
|
|
||||||
|
open ChainA using () renaming (Chain to Chain₁; done to done₁; step to step₁; Chain-≈-cong₁ to Chain₁-≈-cong₁)
|
||||||
|
open ChainB using () renaming (Chain to Chain₂; done to done₂; step to step₂; Chain-≈-cong₁ to Chain₂-≈-cong₁)
|
||||||
|
open ProdChain using (Chain; concat; done; step)
|
||||||
|
|
||||||
|
private
|
||||||
|
a,∙-Monotonic : ∀ (a : A) → Monotonic _≼₂_ _≼_ (λ b → (a , b))
|
||||||
|
a,∙-Monotonic a {b₁} {b₂} (b , b₁⊔b≈b₂) = ((a , b) , (⊔₁-idemp a , b₁⊔b≈b₂))
|
||||||
|
|
||||||
|
a,∙-Preserves-≈₂ : ∀ (a : A) → (λ b → (a , b)) Preserves _≈₂_ ⟶ _≈_
|
||||||
|
a,∙-Preserves-≈₂ a {b₁} {b₂} b₁≈b₂ = (≈₁-refl , b₁≈b₂)
|
||||||
|
|
||||||
|
∙,b-Monotonic : ∀ (b : B) → Monotonic _≼₁_ _≼_ (λ a → (a , b))
|
||||||
|
∙,b-Monotonic b {a₁} {a₂} (a , a₁⊔a≈a₂) = ((a , b) , (a₁⊔a≈a₂ , ⊔₂-idemp b))
|
||||||
|
|
||||||
|
∙,b-Preserves-≈₁ : ∀ (b : B) → (λ a → (a , b)) Preserves _≈₁_ ⟶ _≈_
|
||||||
|
∙,b-Preserves-≈₁ b {a₁} {a₂} a₁≈a₂ = (a₁≈a₂ , ≈₂-refl)
|
||||||
|
|
||||||
|
amin : A
|
||||||
|
amin = proj₁ (proj₁ (proj₁ fhA))
|
||||||
|
|
||||||
|
amax : A
|
||||||
|
amax = proj₂ (proj₁ (proj₁ fhA))
|
||||||
|
|
||||||
|
bmin : B
|
||||||
|
bmin = proj₁ (proj₁ (proj₁ fhB))
|
||||||
|
|
||||||
|
bmax : B
|
||||||
|
bmax = proj₂ (proj₁ (proj₁ fhB))
|
||||||
|
|
||||||
|
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)))
|
||||||
|
unzip (done (a₁≈a₂ , b₁≈b₂)) = ((0 , 0) , ((done₁ a₁≈a₂ , done₂ b₁≈b₂) , ≤-refl))
|
||||||
|
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)
|
||||||
|
with ≈₁-dec a₁ a | ≈₂-dec b₁ b | unzip a'b'a₂b₂
|
||||||
|
... | yes a₁≈a | yes b₁≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) = absurd (a₁b₁̷≈ab (a₁≈a , b₁≈b))
|
||||||
|
... | no a₁̷≈a | yes b₁≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
|
||||||
|
((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)))
|
||||||
|
... | yes a₁≈a | no b₁̷≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
|
||||||
|
((n₁ , suc n₂) , ( (Chain₁-≈-cong₁ (≈₁-sym (≈₁-trans a₁≈a a≈a')) c₁ , step₂ ((d₂ , b₁⊔d₂≈b) , b₁̷≈b) b≈b' c₂)
|
||||||
|
, subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂)
|
||||||
|
))
|
||||||
|
... | no a₁̷≈a | no b₁̷≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
|
||||||
|
((suc n₁ , suc n₂) , ( (step₁ ((d₁ , a₁⊔d₁≈a) , a₁̷≈a) a≈a' c₁ , step₂ ((d₂ , b₁⊔d₂≈b) , b₁̷≈b) b≈b' c₂)
|
||||||
|
, ≤-stepsˡ 1 (subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂))
|
||||||
|
))
|
||||||
|
|
||||||
|
isFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_
|
||||||
|
isFiniteHeightLattice = record
|
||||||
|
{ isLattice = isLattice
|
||||||
|
; fixedHeight =
|
||||||
|
( ( ((amin , bmin) , (amax , bmax))
|
||||||
|
, concat
|
||||||
|
(ChainMapping₁.Chain-map (λ a → (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ fhA)))
|
||||||
|
(ChainMapping₂.Chain-map (λ b → (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ fhB)))
|
||||||
|
)
|
||||||
|
, λ a₁b₁a₂b₂ → let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
|
||||||
|
in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ fhA a₁a₂) (proj₂ fhB b₁b₂))
|
||||||
|
)
|
||||||
|
}
|
||||||
Loading…
Reference in New Issue
Block a user