137 lines
5.3 KiB
Agda
137 lines
5.3 KiB
Agda
module Lattice where
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open import Equivalence
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import Chain
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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 Data.Product using (_×_; Σ; _,_)
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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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IsDecidable : ∀ {a} {A : Set a} (R : A → A → Set a) → Set a
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IsDecidable {a} {A} R = ∀ (a₁ a₂ : A) → Dec (R a₁ a₂)
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record IsSemilattice {a} (A : Set a)
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(_≈_ : A → A → Set a)
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(_⊔_ : A → A → A) : Set a where
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_≼_ : A → A → Set a
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a ≼ b = Σ A (λ c → (a ⊔ c) ≈ b)
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_≺_ : A → A → Set a
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a ≺ b = (a ≼ b) × (¬ a ≈ b)
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field
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≈-equiv : IsEquivalence A _≈_
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≈-⊔-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≈ a₂ → a₃ ≈ a₄ → (a₁ ⊔ a₃) ≈ (a₂ ⊔ a₄)
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⊔-assoc : (x y z : A) → ((x ⊔ y) ⊔ z) ≈ (x ⊔ (y ⊔ z))
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⊔-comm : (x y : A) → (x ⊔ y) ≈ (y ⊔ x)
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⊔-idemp : (x : A) → (x ⊔ x) ≈ x
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open IsEquivalence ≈-equiv public
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≼-refl : ∀ (a : A) → a ≼ a
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≼-refl a = (a , ⊔-idemp a)
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≼-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≼ a₃ → a₂ ≼ a₄
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≼-cong a₁≈a₂ a₃≈a₄ (c₁ , a₁⊔c₁≈a₃) = (c₁ , ≈-trans (≈-⊔-cong (≈-sym a₁≈a₂) ≈-refl) (≈-trans a₁⊔c₁≈a₃ a₃≈a₄))
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≺-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≺ a₃ → a₂ ≺ a₄
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≺-cong a₁≈a₂ a₃≈a₄ (a₁≼a₃ , a₁̷≈a₃) =
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( ≼-cong a₁≈a₂ a₃≈a₄ a₁≼a₃
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, λ a₂≈a₄ → a₁̷≈a₃ (≈-trans a₁≈a₂ (≈-trans a₂≈a₄ (≈-sym a₃≈a₄)))
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)
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record IsLattice {a} (A : Set a)
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(_≈_ : A → A → Set a)
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(_⊔_ : A → A → A)
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(_⊓_ : A → A → A) : Set a where
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field
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joinSemilattice : IsSemilattice A _≈_ _⊔_
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meetSemilattice : IsSemilattice A _≈_ _⊓_
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absorb-⊔-⊓ : (x y : A) → (x ⊔ (x ⊓ y)) ≈ x
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absorb-⊓-⊔ : (x y : A) → (x ⊓ (x ⊔ y)) ≈ x
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open IsSemilattice joinSemilattice public
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open IsSemilattice meetSemilattice public using () renaming
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( ⊔-assoc to ⊓-assoc
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; ⊔-comm to ⊓-comm
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; ⊔-idemp to ⊓-idemp
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; ≈-⊔-cong to ≈-⊓-cong
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; _≼_ to _≽_
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; _≺_ to _≻_
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; ≼-refl to ≽-refl
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)
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FixedHeight : ∀ (h : ℕ) → Set a
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FixedHeight h = Chain.Height (_≈_) ≈-equiv _≺_ ≺-cong h
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record IsFiniteHeightLattice {a} (A : Set a)
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(h : ℕ)
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(_≈_ : A → A → Set a)
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(_⊔_ : A → A → A)
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(_⊓_ : A → A → A) : Set (lsuc a) where
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field
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isLattice : IsLattice A _≈_ _⊔_ _⊓_
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open IsLattice isLattice public
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field
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fixedHeight : FixedHeight h
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module _ {a b} {A : Set a} {B : Set b}
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(_≼₁_ : A → A → Set a) (_≼₂_ : B → B → Set b) where
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Monotonic : (A → B) → Set (a ⊔ℓ b)
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Monotonic f = ∀ {a₁ a₂ : A} → a₁ ≼₁ a₂ → f a₁ ≼₂ f a₂
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module ChainMapping {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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{_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}
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(slA : IsSemilattice A _≈₁_ _⊔₁_) (slB : IsSemilattice B _≈₂_ _⊔₂_) where
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open IsSemilattice slA renaming (_≼_ to _≼₁_; _≺_ to _≺₁_; ≈-equiv to ≈₁-equiv; ≺-cong to ≺₁-cong)
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open IsSemilattice slB renaming (_≼_ to _≼₂_; _≺_ to _≺₂_; ≈-equiv to ≈₂-equiv; ≺-cong to ≺₂-cong)
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open Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong using () renaming (Chain to Chain₁; step to step₁; done to done₁)
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open Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong using () renaming (Chain to Chain₂; step to step₂; done to done₂)
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Chain-map : ∀ (f : A → B) →
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Monotonic _≼₁_ _≼₂_ f →
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Injective _≈₁_ _≈₂_ f →
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f Preserves _≈₁_ ⟶ _≈₂_ →
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∀ {a₁ a₂ : A} {n : ℕ} → Chain₁ a₁ a₂ n → Chain₂ (f a₁) (f a₂) n
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Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ (done₁ a₁≈a₂) =
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done₂ (Preservesᶠ a₁≈a₂)
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Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ (step₁ (a₁≼₁a , a₁̷≈₁a) a≈₁a' a'a₂) =
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let fa₁≺₂fa = (Monotonicᶠ a₁≼₁a , λ fa₁≈₂fa → a₁̷≈₁a (Injectiveᶠ fa₁≈₂fa))
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fa≈fa' = Preservesᶠ a≈₁a'
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in step₂ fa₁≺₂fa fa≈fa' (Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ a'a₂)
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record Semilattice {a} (A : Set a) : Set (lsuc a) where
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field
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_≈_ : A → A → Set a
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_⊔_ : A → A → A
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isSemilattice : IsSemilattice A _≈_ _⊔_
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open IsSemilattice isSemilattice public
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record Lattice {a} (A : Set a) : Set (lsuc a) where
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field
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_≈_ : A → A → Set a
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_⊔_ : A → A → A
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_⊓_ : A → A → A
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isLattice : IsLattice A _≈_ _⊔_ _⊓_
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open IsLattice isLattice public
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