open import Lattice
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Relation.Binary.Definitions using (Decidable)
open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)

module Lattice.MapSet {a : Level} (A : Set a) (≡-dec-A : Decidable (_≡_ {a} {A})) where

open import Data.List using (List; map)
open import Data.Product using (proj₁)
open import Function using (_∘_)

open import Lattice.Unit using (⊤; tt) renaming (_≈_ to _≈₂_; _⊔_ to _⊔₂_; _⊓_ to _⊓₂_; isLattice to ⊤-isLattice)
import Lattice.Map

private module UnitMap = Lattice.Map A ⊤ _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A ⊤-isLattice
open UnitMap using (Map)
open UnitMap using
    ( _⊆_; _≈_; ≈-equiv; _⊔_; _⊓_; empty
    ; isUnionSemilattice; isIntersectSemilattice; isLattice; lattice
    ) public

MapSet : Set a
MapSet = Map

to-List : MapSet → List A
to-List = map proj₁ ∘ proj₁

insert : A → MapSet → MapSet
insert k = UnitMap.insert k tt