148 lines
5.5 KiB
Plaintext
148 lines
5.5 KiB
Plaintext
module Lattice where
|
||
|
||
open import Equivalence
|
||
import Chain
|
||
|
||
open import Relation.Binary.Core using (_Preserves_⟶_ )
|
||
open import Relation.Nullary using (Dec; ¬_)
|
||
open import Data.Nat as Nat using (ℕ)
|
||
open import Data.Product using (_×_; Σ; _,_)
|
||
open import Data.Sum using (_⊎_; inj₁; inj₂)
|
||
open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)
|
||
open import Function.Definitions using (Injective)
|
||
|
||
IsDecidable : ∀ {a} {A : Set a} (R : A → A → Set a) → Set a
|
||
IsDecidable {a} {A} R = ∀ (a₁ a₂ : A) → Dec (R a₁ a₂)
|
||
|
||
record IsSemilattice {a} (A : Set a)
|
||
(_≈_ : A → A → Set a)
|
||
(_⊔_ : A → A → A) : Set a where
|
||
|
||
_≼_ : A → A → Set a
|
||
a ≼ b = (a ⊔ b) ≈ b
|
||
|
||
_≺_ : A → A → Set a
|
||
a ≺ b = (a ≼ b) × (¬ a ≈ b)
|
||
|
||
field
|
||
≈-equiv : IsEquivalence A _≈_
|
||
≈-⊔-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≈ a₂ → a₃ ≈ a₄ → (a₁ ⊔ a₃) ≈ (a₂ ⊔ a₄)
|
||
|
||
⊔-assoc : (x y z : A) → ((x ⊔ y) ⊔ z) ≈ (x ⊔ (y ⊔ z))
|
||
⊔-comm : (x y : A) → (x ⊔ y) ≈ (y ⊔ x)
|
||
⊔-idemp : (x : A) → (x ⊔ x) ≈ x
|
||
|
||
open IsEquivalence ≈-equiv public
|
||
|
||
open import Relation.Binary.Reasoning.Base.Single _≈_ ≈-refl ≈-trans
|
||
|
||
≼-refl : ∀ (a : A) → a ≼ a
|
||
≼-refl a = ⊔-idemp a
|
||
|
||
≼-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≼ a₃ → a₂ ≼ a₄
|
||
≼-cong {a₁} {a₂} {a₃} {a₄} a₁≈a₂ a₃≈a₄ a₁⊔a₃≈a₃ =
|
||
begin
|
||
a₂ ⊔ a₄
|
||
∼⟨ ≈-⊔-cong (≈-sym a₁≈a₂) (≈-sym a₃≈a₄) ⟩
|
||
a₁ ⊔ a₃
|
||
∼⟨ a₁⊔a₃≈a₃ ⟩
|
||
a₃
|
||
∼⟨ a₃≈a₄ ⟩
|
||
a₄
|
||
∎
|
||
|
||
≺-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≺ a₃ → a₂ ≺ a₄
|
||
≺-cong a₁≈a₂ a₃≈a₄ (a₁≼a₃ , a₁̷≈a₃) =
|
||
( ≼-cong a₁≈a₂ a₃≈a₄ a₁≼a₃
|
||
, λ a₂≈a₄ → a₁̷≈a₃ (≈-trans a₁≈a₂ (≈-trans a₂≈a₄ (≈-sym a₃≈a₄)))
|
||
)
|
||
|
||
record IsLattice {a} (A : Set a)
|
||
(_≈_ : A → A → Set a)
|
||
(_⊔_ : A → A → A)
|
||
(_⊓_ : A → A → A) : Set a where
|
||
|
||
field
|
||
joinSemilattice : IsSemilattice A _≈_ _⊔_
|
||
meetSemilattice : IsSemilattice A _≈_ _⊓_
|
||
|
||
absorb-⊔-⊓ : (x y : A) → (x ⊔ (x ⊓ y)) ≈ x
|
||
absorb-⊓-⊔ : (x y : A) → (x ⊓ (x ⊔ y)) ≈ x
|
||
|
||
open IsSemilattice joinSemilattice public
|
||
open IsSemilattice meetSemilattice public using () renaming
|
||
( ⊔-assoc to ⊓-assoc
|
||
; ⊔-comm to ⊓-comm
|
||
; ⊔-idemp to ⊓-idemp
|
||
; ≈-⊔-cong to ≈-⊓-cong
|
||
; _≼_ to _≽_
|
||
; _≺_ to _≻_
|
||
; ≼-refl to ≽-refl
|
||
)
|
||
|
||
FixedHeight : ∀ (h : ℕ) → Set a
|
||
FixedHeight h = Chain.Height (_≈_) ≈-equiv _≺_ ≺-cong h
|
||
|
||
record IsFiniteHeightLattice {a} (A : Set a)
|
||
(h : ℕ)
|
||
(_≈_ : A → A → Set a)
|
||
(_⊔_ : A → A → A)
|
||
(_⊓_ : A → A → A) : Set (lsuc a) where
|
||
|
||
field
|
||
isLattice : IsLattice A _≈_ _⊔_ _⊓_
|
||
|
||
open IsLattice isLattice public
|
||
|
||
field
|
||
fixedHeight : FixedHeight h
|
||
|
||
module _ {a b} {A : Set a} {B : Set b}
|
||
(_≼₁_ : A → A → Set a) (_≼₂_ : B → B → Set b) where
|
||
|
||
Monotonic : (A → B) → Set (a ⊔ℓ b)
|
||
Monotonic f = ∀ {a₁ a₂ : A} → a₁ ≼₁ a₂ → f a₁ ≼₂ f a₂
|
||
|
||
module ChainMapping {a b} {A : Set a} {B : Set b}
|
||
{_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b}
|
||
{_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}
|
||
(slA : IsSemilattice A _≈₁_ _⊔₁_) (slB : IsSemilattice B _≈₂_ _⊔₂_) where
|
||
|
||
open IsSemilattice slA renaming (_≼_ to _≼₁_; _≺_ to _≺₁_; ≈-equiv to ≈₁-equiv; ≺-cong to ≺₁-cong)
|
||
open IsSemilattice slB renaming (_≼_ to _≼₂_; _≺_ to _≺₂_; ≈-equiv to ≈₂-equiv; ≺-cong to ≺₂-cong)
|
||
|
||
open Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong using () renaming (Chain to Chain₁; step to step₁; done to done₁)
|
||
open Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong using () renaming (Chain to Chain₂; step to step₂; done to done₂)
|
||
|
||
Chain-map : ∀ (f : A → B) →
|
||
Monotonic _≼₁_ _≼₂_ f →
|
||
Injective _≈₁_ _≈₂_ f →
|
||
f Preserves _≈₁_ ⟶ _≈₂_ →
|
||
∀ {a₁ a₂ : A} {n : ℕ} → Chain₁ a₁ a₂ n → Chain₂ (f a₁) (f a₂) n
|
||
Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ (done₁ a₁≈a₂) =
|
||
done₂ (Preservesᶠ a₁≈a₂)
|
||
Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ (step₁ (a₁≼₁a , a₁̷≈₁a) a≈₁a' a'a₂) =
|
||
let fa₁≺₂fa = (Monotonicᶠ a₁≼₁a , λ fa₁≈₂fa → a₁̷≈₁a (Injectiveᶠ fa₁≈₂fa))
|
||
fa≈fa' = Preservesᶠ a≈₁a'
|
||
in step₂ fa₁≺₂fa fa≈fa' (Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ a'a₂)
|
||
|
||
record Semilattice {a} (A : Set a) : Set (lsuc a) where
|
||
field
|
||
_≈_ : A → A → Set a
|
||
_⊔_ : A → A → A
|
||
|
||
isSemilattice : IsSemilattice A _≈_ _⊔_
|
||
|
||
open IsSemilattice isSemilattice public
|
||
|
||
|
||
record Lattice {a} (A : Set a) : Set (lsuc a) where
|
||
field
|
||
_≈_ : A → A → Set a
|
||
_⊔_ : A → A → A
|
||
_⊓_ : A → A → A
|
||
|
||
isLattice : IsLattice A _≈_ _⊔_ _⊓_
|
||
|
||
open IsLattice isLattice public
|