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 ≡-dec-A ⊤-isLattice
open UnitMap
    using (Map; Expr; ⟦_⟧)
    renaming
        ( Expr-Provenance to Expr-Provenanceᵐ
        )
open UnitMap
    using
        ( _⊆_; _≈_; ≈-equiv; _⊔_; _⊓_; _∪_ ; _∩_ ; `_; empty; forget
        ; isUnionSemilattice; isIntersectSemilattice; isLattice; lattice
        ; Provenance
        ; ⊔-preserves-∈k₁
        ; ⊔-preserves-∈k₂
        )
    renaming (_∈k_ to _∈_) public
open Provenance 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

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

Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈ ⟦ e ⟧ → Provenance k tt e
Expr-Provenance k e k∈e = let (tt , (prov , _)) = Expr-Provenanceᵐ k e k∈e in prov