40 lines
1.3 KiB
Agda
40 lines
1.3 KiB
Agda
module Lattice where
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open import Relation.Binary.PropositionalEquality
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open import Relation.Binary.Definitions
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open import Data.Nat using (ℕ; _≤_)
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open import Data.Nat.Properties using (≤-refl; ≤-trans; ≤-antisym)
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open import Agda.Primitive using (lsuc)
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record Preorder {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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≼-refl : Reflexive (_≼_)
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≼-trans : Transitive (_≼_)
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≼-antisym : Antisymmetric (_≡_) (_≼_)
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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 → A
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⊔-assoc : (x : A) → (y : A) → (z : A) → x ⊔ (y ⊔ z) ≡ (x ⊔ y) ⊔ z
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⊔-comm : (x : A) → (y : A) → x ⊔ y ≡ y ⊔ x
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⊔-idemp : (x : A) → x ⊔ x ≡ x
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record Lattice {a} (A : Set a) : Set (lsuc a) where
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field
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joinSemilattice : Semilattice A
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meetSemilattice : Semilattice A
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_⊔_ = Semilattice._⊔_ joinSemilattice
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_⊓_ = Semilattice._⊔_ meetSemilattice
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field
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absorb-⊔-⊓ : (x : A) → (y : A) → x ⊔ (x ⊓ y) ≡ x
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absorb-⊓-⊔ : (x : A) → (y : A) → x ⊓ (x ⊔ y) ≡ x
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instance
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NatPreorder : Preorder ℕ
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NatPreorder = record { _≼_ = _≤_; ≼-refl = ≤-refl; ≼-trans = ≤-trans; ≼-antisym = ≤-antisym }
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