130 lines
3.6 KiB
Agda
130 lines
3.6 KiB
Agda
module Lattice.Unit where
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open import Data.Empty using (⊥-elim)
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open import Data.Product using (_,_)
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open import Data.Nat using (ℕ; _≤_; z≤n)
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open import Data.Unit using (⊤; tt) public
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open import Data.Unit.Properties using (_≟_; ≡-setoid)
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open import Relation.Binary using (Setoid)
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
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open import Relation.Nullary using (Dec; ¬_; yes; no)
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open import Equivalence
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open import Lattice
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import Chain
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open Setoid ≡-setoid using (refl; sym; trans)
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_≈_ : ⊤ → ⊤ → Set
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_≈_ = _≡_
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≈-equiv : IsEquivalence ⊤ _≈_
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≈-equiv = record
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{ ≈-refl = refl
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; ≈-sym = sym
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; ≈-trans = trans
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}
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≈-dec : IsDecidable _≈_
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≈-dec = _≟_
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_⊔_ : ⊤ → ⊤ → ⊤
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tt ⊔ tt = tt
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_⊓_ : ⊤ → ⊤ → ⊤
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tt ⊓ tt = tt
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≈-⊔-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ → (ab₁ ⊔ ab₃) ≈ (ab₂ ⊔ ab₄)
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≈-⊔-cong {tt} {tt} {tt} {tt} _ _ = Eq.refl
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⊔-assoc : (x y z : ⊤) → ((x ⊔ y) ⊔ z) ≈ (x ⊔ (y ⊔ z))
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⊔-assoc tt tt tt = Eq.refl
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⊔-comm : (x y : ⊤) → (x ⊔ y) ≈ (y ⊔ x)
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⊔-comm tt tt = Eq.refl
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⊔-idemp : (x : ⊤) → (x ⊔ x) ≈ x
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⊔-idemp tt = Eq.refl
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instance
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isJoinSemilattice : IsSemilattice ⊤ _≈_ _⊔_
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isJoinSemilattice = record
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{ ≈-equiv = ≈-equiv
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; ≈-⊔-cong = ≈-⊔-cong
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; ⊔-assoc = ⊔-assoc
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; ⊔-comm = ⊔-comm
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; ⊔-idemp = ⊔-idemp
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}
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≈-⊓-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ → (ab₁ ⊓ ab₃) ≈ (ab₂ ⊓ ab₄)
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≈-⊓-cong {tt} {tt} {tt} {tt} _ _ = Eq.refl
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⊓-assoc : (x y z : ⊤) → ((x ⊓ y) ⊓ z) ≈ (x ⊓ (y ⊓ z))
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⊓-assoc tt tt tt = Eq.refl
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⊓-comm : (x y : ⊤) → (x ⊓ y) ≈ (y ⊓ x)
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⊓-comm tt tt = Eq.refl
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⊓-idemp : (x : ⊤) → (x ⊓ x) ≈ x
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⊓-idemp tt = Eq.refl
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instance
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isMeetSemilattice : IsSemilattice ⊤ _≈_ _⊓_
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isMeetSemilattice = record
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{ ≈-equiv = ≈-equiv
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; ≈-⊔-cong = ≈-⊓-cong
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; ⊔-assoc = ⊓-assoc
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; ⊔-comm = ⊓-comm
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; ⊔-idemp = ⊓-idemp
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}
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instance
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isLattice : IsLattice ⊤ _≈_ _⊔_ _⊓_
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isLattice = record
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{ joinSemilattice = isJoinSemilattice
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; meetSemilattice = isMeetSemilattice
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; absorb-⊔-⊓ = λ { tt tt → Eq.refl }
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; absorb-⊓-⊔ = λ { tt tt → Eq.refl }
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}
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lattice : Lattice ⊤
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lattice = record
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{ _≈_ = _≈_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isLattice = isLattice
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}
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open Chain _≈_ ≈-equiv (IsLattice._≺_ isLattice) (IsLattice.≺-cong isLattice)
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private
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longestChain : Chain tt tt 0
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longestChain = done refl
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isLongest : ∀ {t₁ t₂ : ⊤} {n : ℕ} → Chain t₁ t₂ n → n ≤ 0
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isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl)
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isLongest (done _) = z≤n
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instance
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fixedHeight : IsLattice.FixedHeight isLattice 0
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fixedHeight = record
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{ ⊥ = tt
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; ⊤ = tt
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; longestChain = longestChain
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; bounded = isLongest
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}
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isFiniteHeightLattice : IsFiniteHeightLattice ⊤ 0 _≈_ _⊔_ _⊓_
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isFiniteHeightLattice = record
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{ isLattice = isLattice
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; fixedHeight = fixedHeight
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}
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finiteHeightLattice : FiniteHeightLattice ⊤
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finiteHeightLattice = record
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{ height = 0
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; _≈_ = _≈_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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; isFiniteHeightLattice = isFiniteHeightLattice
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}
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