agda-spa/Lattice.agda

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