open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; cong)
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 Relation.Nullary using (¬_)
open import Data.Nat using (ℕ)
open import Data.List using (List; []; _∷_; _++_)
open import Data.List.Membership.Propositional using ()
open import Data.List.Relation.Unary.All using (All; []; _∷_)
open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
open import Data.Unit using (⊤)
open import Data.Empty using (⊥)

Map : Set (a ⊔ b)
Map = List (A × B)

keys : List (A × B) → List A
keys [] = []
keys ((k , v) ∷ xs) = k ∷ keys xs

data Unique {c} {C : Set c} : List C → Set c where
    empty : Unique []
    push : forall {x : C} {xs : List C}
        → All (λ x' → ¬ x ≡ x') xs
        → Unique xs
        → Unique (x ∷ xs)

Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ Data.List.Membership.Propositional._∈_ x xs → Unique xs →  Unique (xs ++ (x ∷ []))
Unique-append {c} {C} {x} {[]} _ _ = push [] empty
Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
    where
        x'≢x : ¬ x' ≡ x
        x'≢x x'≡x = x∉xs (here (sym x'≡x))

        help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
        help {[]} _ = x'≢x ∷ []
        help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es


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

_∈k_ : A → List (A × B) → Set a
_∈k_ k m = Data.List.Membership.Propositional._∈_ k (keys m)

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

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

insert : (B → B → B) → A → B → List (A × B) → List (A × B)
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

merge : (B → B → B) → List (A × B) → List (A × B) → List (A × B)
merge f m₁ m₂ = foldr (insert f) m₂ m₁

absurd : ∀ {a} {A : Set a} →  ⊥ → A
absurd ()

insert-keys-∈ : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → k ∈k l → keys l ≡ keys (insert f k v l)
insert-keys-∈ f k v ((k' , v') ∷ xs) (here k≡k') with (≡-dec-A k k')
...                                                | yes _ = refl
...                                                | no k≢k' = absurd (k≢k' k≡k')
insert-keys-∈ f k v ((k' , _) ∷ xs) (there k∈kxs) with (≡-dec-A k k')
...                                                | yes _ = refl
...                                                | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ f k v xs k∈kxs)

insert-keys-∉ : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert f k v l)
insert-keys-∉ f k v [] _ = refl
insert-keys-∉ f k v ((k' , v') ∷ xs) k∉kl with (≡-dec-A k k')
...                                         | yes k≡k' = absurd (k∉kl (here k≡k'))
...                                         | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ f k v xs (λ k∈kxs → k∉kl (there k∈kxs)))

∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈k l)
∈k-dec k [] = no (λ ())
∈k-dec k ((k' , v) ∷ xs) with (≡-dec-A k k')
...                        | yes k≡k' = yes (here k≡k')
...                        | no k≢k' with (∈k-dec k xs)
...                                    | yes k∈kxs = yes (there k∈kxs)
...                                    | no k∉kxs = no witness
                                           where
                                               witness : ¬ k ∈k ((k' , v) ∷ xs)
                                               witness (here k≡k') = k≢k' k≡k'
                                               witness (there k∈kxs) = k∉kxs k∈kxs

insert-preserves-unique : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert f k v l))
insert-preserves-unique f k v l u with (∈k-dec k l)
...                               | yes k∈kl rewrite insert-keys-∈ f k v l k∈kl = u
...                               | no k∉kl rewrite sym (insert-keys-∉ f k v l k∉kl) = Unique-append k∉kl u

merge-preserves-unique : ∀ (f : B → B → B) (l₁ l₂ : List (A × B)) → Unique (keys l₂) → Unique (keys (merge f l₁ l₂))
merge-preserves-unique f [] l₂ u₂ = u₂
merge-preserves-unique f ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-unique f k₁ v₁ (merge f xs₁ l₂) (merge-preserves-unique f xs₁ l₂ u₂)

Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → Data.List.Membership.Propositional._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
    where
        unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
        unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
        unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)