open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
open import Relation.Binary.Definitions using (Decidable)
open import Relation.Binary.Core using (Rel)
open import Relation.Nullary using (Dec; yes; no)
open import Agda.Primitive using (Level; _⊔_)

module Map {a b : Level} (A : Set a) (B : Set b)
    (≡-dec-A : Decidable (_≡_ {a} {A}))
    where

open import Data.Nat using (ℕ)
open import Data.String using (String; _++_)
open import Data.List using (List; []; _∷_)
open import Data.List.Membership.Propositional using ()
open import Data.Product using (_×_; _,_; Σ)
open import Data.Unit using (⊤)
open import Data.Empty using (⊥)

Map : Set (a ⊔ b)
Map = List (A × B)

insert : (B → B → B) → A → B → Map → Map
insert f k v [] = (k , v) ∷ []
insert f k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
...                             | yes _ = (k , f v v') ∷ xs
...                             | no _ = x ∷ insert f k v xs

foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> Map -> C
foldr f b [] = b
foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)

_∈_ : (A × B) → Map → Set (a ⊔ b)
_∈_ p m = Data.List.Membership.Propositional._∈_ p m

subset : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
subset _≈_ m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))

lift : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
    where
        _⊆_ : Map → Map → Set (a ⊔ b)
        _⊆_ = subset _≈_