agda-spa/Lattice/Map.agda

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.Map {a b : Level} {A : Set a} {B : Set b}
    {_≈₂_ : B → B → Set b}
    {_⊔₂_ : B → B → B} {_⊓₂_ : B → B → B}
    (≡-dec-A : Decidable (_≡_ {a} {A}))
    (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where

open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)

open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Nat using (ℕ)
open import Data.List using (List; map; []; _∷_; _++_) renaming (foldr to foldrˡ)
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.Empty using (⊥; ⊥-elim)
open import Equivalence
open import Utils using (Unique; push; Unique-append; All¬-¬Any; All-x∈xs)
open import Data.String using () renaming (_++_ to _++ˢ_)
open import Showable using (Showable; show)

open IsLattice lB using () renaming
    ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans
    ; ≈-⊔-cong to ≈₂-⊔₂-cong; ≈-⊓-cong to ≈₂-⊓₂-cong
    ; ⊔-idemp to ⊔₂-idemp; ⊔-comm to ⊔₂-comm; ⊔-assoc to ⊔₂-assoc
    ; ⊓-idemp to ⊓₂-idemp; ⊓-comm to ⊓₂-comm; ⊓-assoc to ⊓₂-assoc
    ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
    ; _≼_ to _≼₂_
    )

private module ImplKeys where
    keys : List (A × B) → List A
    keys = map proj₁

-- See note on `forget` for why this is defined in global scope even though
-- it operates on lists.
∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈ˡ (ImplKeys.keys 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 ∈ˡ (ImplKeys.keys ((k' , v) ∷ xs))
                  witness (here k≡k') = k≢k' k≡k'
                  witness (there k∈kxs) = k∉kxs k∈kxs

private module _ where
    open MemProp using (_∈_)
    open ImplKeys

    unique-not-in : ∀ {k : A} {v : B} {l : List (A × B)}  →
                    ¬ (All (λ k' → ¬ k ≡ k') (keys l) × (k , v) ∈ l)
    unique-not-in {l = (k' , _) ∷ xs} (k≢k' ∷ _ , here k',≡x) =
        k≢k' (cong proj₁ k',≡x)
    unique-not-in {l = _ ∷ xs} (_ ∷ rest , there k,v'∈xs) =
        unique-not-in (rest , k,v'∈xs)

    ListAB-functional : ∀ {k : A} {v v' : B} {l : List (A × B)} →
                        Unique (keys l) → (k , v) ∈ l → (k , v') ∈ l → v ≡ v'
    ListAB-functional _ (here k,v≡x) (here k,v'≡x) =
        cong proj₂ (trans k,v≡x (sym k,v'≡x))
    ListAB-functional (push k≢xs _) (here k,v≡x) (there k,v'∈xs)
        rewrite sym k,v≡x = ⊥-elim (unique-not-in (k≢xs , k,v'∈xs))
    ListAB-functional (push k≢xs _) (there k,v∈xs) (here k,v'≡x)
        rewrite sym k,v'≡x = ⊥-elim (unique-not-in (k≢xs , k,v∈xs))
    ListAB-functional {l = _ ∷ xs } (push _ uxs) (there k,v∈xs) (there k,v'∈xs) =
        ListAB-functional uxs k,v∈xs k,v'∈xs

    k∈-dec : ∀ (k : A) (l : List A) → Dec (k ∈ l)
    k∈-dec k [] = no (λ ())
    k∈-dec k (x ∷ xs)
        with (≡-dec-A k x)
    ...   | yes refl = yes (here refl)
    ...   | no k≢x with (k∈-dec k xs)
    ...       | yes k∈xs = yes (there k∈xs)
    ...       | no k∉xs = no (λ { (here k≡x) → k≢x k≡x; (there k∈xs) → k∉xs k∈xs })

    ∈-cong : ∀ {c d} {C : Set c} {D : Set d} {c : C} {l : List C} →
             (f : C → D) → c ∈ l → f c ∈ map f l
    ∈-cong f (here c≡c') = here (cong f c≡c')
    ∈-cong f (there c∈xs) = there (∈-cong f c∈xs)

    locate-impl : ∀ {k : A} {l : List (A × B)} → k ∈ keys l → Σ B (λ v → (k , v) ∈ l)
    locate-impl {k} {(k' , v) ∷ xs} (here k≡k') rewrite k≡k' = (v , here refl)
    locate-impl {k} {(k' , v) ∷ xs} (there k∈kxs) = let (v , k,v∈xs) = locate-impl k∈kxs in (v , there k,v∈xs)

private module ImplRelation where
    open MemProp using (_∈_)

    subset : List (A × B) → List (A × B) → Set (a ⊔ℓ b)
    subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ →
                   Σ B (λ v' → v ≈₂ v' × ((k , v') ∈ m₂))

private module ImplInsert (f : B → B → B) where
    open import Data.List using (map)
    open MemProp using (_∈_)
    open ImplKeys

    private
        _∈k_ : A → List (A × B) → Set a
        _∈k_ k m = k ∈ (keys m)

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

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

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

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

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

    union-subset-keys : ∀ {l₁ l₂ : List (A × B)} →
                        All (λ k → k ∈k l₂) (keys l₁) →
                        keys l₂ ≡ keys (union l₁ l₂)
    union-subset-keys {[]} _ = refl
    union-subset-keys {(k , v) ∷ l₁'} (k∈kl₂ ∷ kl₁'⊆kl₂)
        rewrite union-subset-keys kl₁'⊆kl₂ =
        insert-keys-∈ k∈kl₂

    union-equal-keys : ∀ {l₁ l₂ : List (A × B)} →
                       keys l₁ ≡ keys l₂ → keys l₁ ≡ keys (union l₁ l₂)
    union-equal-keys {l₁} {l₂} kl₁≡kl₂
        with subst (λ l → All (λ k → k ∈ l) (keys l₁)) kl₁≡kl₂ (All-x∈xs (keys l₁))
    ...   | kl₁⊆kl₂ = trans kl₁≡kl₂ (union-subset-keys {l₁} {l₂} kl₁⊆kl₂)

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

    insert-fresh : ∀ {k : A} {v : B} {l : List (A × B)} →
                                 ¬ k ∈k l → (k , v) ∈ insert k v l
    insert-fresh {l = []} k∉kl = here refl
    insert-fresh {k} {l = (k' , v') ∷ xs} k∉kl
        with ≡-dec-A k k'
    ...   | yes k≡k' = ⊥-elim (k∉kl (here k≡k'))
    ...   | no _ = there (insert-fresh (λ k∈kxs → k∉kl (there k∈kxs)))

    insert-preserves-∉k : ∀ {k k' : A} {v' : B} {l : List (A × B)} →
                                  ¬ k ≡ k' → ¬ k ∈k l → ¬ k ∈k insert k' v' l
    insert-preserves-∉k {l = []} k≢k' k∉kl (here k≡k') = k≢k' k≡k'
    insert-preserves-∉k {l = []} k≢k' k∉kl (there ())
    insert-preserves-∉k {k} {k'} {v'} {(k'' , v'') ∷ xs} k≢k' k∉kl k∈kil
        with ≡-dec-A k k''
    ...   | yes k≡k'' = k∉kl (here k≡k'')
    ...   | no k≢k'' with ≡-dec-A k' k'' | k∈kil
    ...       | yes k'≡k'' | here k≡k'' = k≢k'' k≡k''
    ...       | yes k'≡k'' | there k∈kxs = k∉kl (there k∈kxs)
    ...       | no k'≢k''  | here k≡k'' = k∉kl (here k≡k'')
    ...       | no k'≢k''  | there k∈kxs = insert-preserves-∉k k≢k'
                                           (λ k∈kxs → k∉kl (there k∈kxs)) k∈kxs

    union-preserves-∉ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                        ¬ k ∈k l₁ → ¬ k ∈k l₂ → ¬ k ∈k union l₁ l₂
    union-preserves-∉ {l₁ = []} _ k∉kl₂ = k∉kl₂
    union-preserves-∉ {k} {(k' , v') ∷ xs₁} k∉kl₁ k∉kl₂
        with ≡-dec-A k k'
    ...   | yes k≡k' = ⊥-elim (k∉kl₁ (here k≡k'))
    ...   | no k≢k' = insert-preserves-∉k k≢k' (union-preserves-∉ (λ k∈kxs₁ → k∉kl₁ (there k∈kxs₁)) k∉kl₂)

    insert-preserves-∈k : ∀ {k k' : A} {v' : B} {l : List (A × B)} →
                          k ∈k l → k ∈k insert k' v' l
    insert-preserves-∈k {k} {k'} {v'} {(k'' , v'') ∷ xs} (here k≡k'')
        with (≡-dec-A k' k'')
    ...   | yes _ = here k≡k''
    ...   | no _ = here k≡k''
    insert-preserves-∈k {k} {k'} {v'} {(k'' , v'') ∷ xs} (there k∈kxs)
        with (≡-dec-A k' k'')
    ...   | yes _ = there k∈kxs
    ...   | no _ = there (insert-preserves-∈k k∈kxs)

    union-preserves-∈k₁ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                          k ∈k l₁ → k ∈k (union l₁ l₂)
    union-preserves-∈k₁ {k} {(k' , v') ∷ xs} {l₂} (here k≡k')
        with ∈k-dec k (union xs l₂)
    ...   | yes k∈kxsl₂ = insert-preserves-∈k k∈kxsl₂
    ...   | no k∉kxsl₂ rewrite k≡k' = ∈-cong proj₁ (insert-fresh k∉kxsl₂)
    union-preserves-∈k₁ {k} {(k' , v') ∷ xs} {l₂} (there k∈kxs) =
        insert-preserves-∈k (union-preserves-∈k₁ k∈kxs)

    union-preserves-∈k₂ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                          k ∈k l₂ → k ∈k (union l₁ l₂)
    union-preserves-∈k₂ {k} {[]} {l₂} k∈kl₂ = k∈kl₂
    union-preserves-∈k₂ {k} {(k' , v') ∷ xs} {l₂} k∈kl₂ =
        insert-preserves-∈k (union-preserves-∈k₂ {l₁ = xs} k∈kl₂)

    ∉-union-∉-either : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                       ¬ k ∈k union l₁ l₂ → ¬ k ∈k l₁ × ¬ k ∈k l₂
    ∉-union-∉-either {k} {l₁} {l₂} k∉l₁l₂
        with ∈k-dec k l₁
    ...   | yes k∈kl₁ = ⊥-elim (k∉l₁l₂ (union-preserves-∈k₁ k∈kl₁))
    ...   | no k∉kl₁ with ∈k-dec k l₂
    ...         | yes k∈kl₂ = ⊥-elim (k∉l₁l₂ (union-preserves-∈k₂ {l₁ = l₁} k∈kl₂))
    ...         | no k∉kl₂ = (k∉kl₁ , k∉kl₂)


    insert-preserves-∈ : ∀ {k k' : A} {v v' : B} {l : List (A × B)} →
                         ¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ insert k' v' l
    insert-preserves-∈ {k} {k'} {l = x ∷ xs} k≢k' (here k,v=x)
        rewrite sym k,v=x with ≡-dec-A k' k
    ...   | yes k'≡k = ⊥-elim (k≢k' (sym k'≡k))
    ...   | no _ = here refl
    insert-preserves-∈ {k} {k'} {l = (k'' , _) ∷ xs} k≢k' (there k,v∈xs)
        with ≡-dec-A k' k''
    ...   | yes _ = there k,v∈xs
    ...   | no _ = there (insert-preserves-∈ k≢k' k,v∈xs)

    union-preserves-∈₂ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
                         ¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ union l₁ l₂
    union-preserves-∈₂ {l₁ = []} _ k,v∈l₂ = k,v∈l₂
    union-preserves-∈₂ {l₁ = (k' , v') ∷ xs₁} k∉kl₁ k,v∈l₂ =
        let recursion = union-preserves-∈₂ (λ k∈xs₁ → k∉kl₁ (there k∈xs₁)) k,v∈l₂
        in insert-preserves-∈ (λ k≡k' → k∉kl₁ (here k≡k')) recursion

    union-preserves-∈₁ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
                         Unique (keys l₁) → (k , v) ∈ l₁ → ¬ k ∈k l₂ → (k , v) ∈ union l₁ l₂
    union-preserves-∈₁ {k} {v} {(k' , v') ∷ xs₁} (push k'≢xs₁ uxs₁) (there k,v∈xs₁) k∉kl₂ =
        insert-preserves-∈ k≢k' k,v∈mxs₁l
        where
            k,v∈mxs₁l = union-preserves-∈₁ uxs₁ k,v∈xs₁ k∉kl₂

            k≢k' : ¬ k ≡ k'
            k≢k' with ≡-dec-A k k'
            ...    | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v∈xs₁))
            ...    | no  k≢k' = k≢k'
    union-preserves-∈₁ {l₁ = (k' , v') ∷ xs₁} (push k'≢xs₁ uxs₁) (here k,v≡k',v') k∉kl₂
        rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
        insert-fresh (union-preserves-∉ (All¬-¬Any k'≢xs₁) k∉kl₂)

    insert-combines : ∀ {k : A} {v v' : B} {l : List (A × B)} →
                      Unique (keys l) → (k , v') ∈ l → (k , f v v') ∈ (insert k v l)
    insert-combines {l = (k' , v'') ∷ xs} _ (here k,v'≡k',v'')
        rewrite cong proj₁ k,v'≡k',v'' rewrite cong proj₂ k,v'≡k',v''
        with ≡-dec-A k' k'
    ...   | yes _ = here refl
    ...   | no k≢k' = ⊥-elim (k≢k' refl)
    insert-combines {k} {l = (k' , v'') ∷ xs} (push k'≢xs uxs) (there k,v'∈xs)
        with ≡-dec-A k k'
    ...   | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs (∈-cong proj₁ k,v'∈xs))
    ...   | no k≢k' = there (insert-combines uxs k,v'∈xs)

    union-combines : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
                     Unique (keys l₁) → Unique (keys l₂) →
                     (k , v₁) ∈ l₁ → (k , v₂) ∈ l₂ → (k , f v₁ v₂) ∈ union l₁ l₂
    union-combines {l₁ = (k' , v) ∷ xs₁} {l₂} (push k'≢xs₁ uxs₁) ul₂ (here k,v₁≡k',v) k,v₂∈l₂
        rewrite cong proj₁ (sym (k,v₁≡k',v)) rewrite cong proj₂ (sym (k,v₁≡k',v)) =
        insert-combines (union-preserves-Unique xs₁ l₂ ul₂) (union-preserves-∈₂ (All¬-¬Any k'≢xs₁) k,v₂∈l₂)
    union-combines {k} {l₁ = (k' , v) ∷ xs₁} (push k'≢xs₁ uxs₁) ul₂ (there k,v₁∈xs₁) k,v₂∈l₂ =
        insert-preserves-∈ k≢k' (union-combines uxs₁ ul₂ k,v₁∈xs₁ k,v₂∈l₂)
        where
            k≢k' : ¬ k ≡ k'
            k≢k' with ≡-dec-A k k'
            ...    | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs₁ (∈-cong proj₁ k,v₁∈xs₁))
            ...    | no  k≢k' = k≢k'

    update : A → B → List (A × B) → List (A × B)
    update k v [] = []
    update k v ((k' , v') ∷ xs) with ≡-dec-A k k'
    ...                            | yes _ = (k' , f v v') ∷ xs
    ...                            | no _ = (k' , v') ∷ update k v xs

    updates : List (A × B) → List (A × B) → List (A × B)
    updates l₁ l₂ = foldr update l₂ l₁

    restrict : List (A × B) → List (A × B) → List (A × B)
    restrict l [] = []
    restrict l ((k' , v') ∷ xs) with ∈k-dec k' l
    ...                           | yes _ = (k' , v') ∷ restrict l xs
    ...                           | no _ = restrict l xs

    intersect : List (A × B) → List (A × B) → List (A × B)
    intersect l₁ l₂ = restrict l₁ (updates l₁ l₂)

    update-keys : ∀ {k : A} {v : B} {l : List (A × B)} →
                  keys l ≡ keys (update k v l)
    update-keys {l = []} = refl
    update-keys {k} {v} {l = (k' , v') ∷ xs}
        with ≡-dec-A k k'
    ...   | yes _ = refl
    ...   | no _ rewrite update-keys {k} {v} {xs} = refl

    updates-keys : ∀ {l₁ l₂ : List (A × B)} →
                   keys l₂ ≡ keys (updates l₁ l₂)
    updates-keys {[]} = refl
    updates-keys {(k , v) ∷ xs} {l₂}
        rewrite updates-keys {xs} {l₂}
        rewrite update-keys {k} {v} {updates xs l₂} = refl

    update-preserves-Unique : ∀ {k : A} {v : B} {l : List (A × B)} →
                              Unique (keys l) → Unique (keys (update k v l ))
    update-preserves-Unique {k} {v} {l} u rewrite update-keys {k} {v} {l} = u

    updates-preserve-Unique : ∀ {l₁ l₂ : List (A × B)} →
                              Unique (keys l₂) → Unique (keys (updates l₁ l₂))
    updates-preserve-Unique {[]} u = u
    updates-preserve-Unique {(k , v) ∷ xs} u = update-preserves-Unique (updates-preserve-Unique {xs} u)

    restrict-preserves-k≢ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                            All (λ k' → ¬ k ≡ k') (keys l₂) → All (λ k' → ¬ k ≡ k') (keys (restrict l₁ l₂))
    restrict-preserves-k≢ {k} {l₁} {[]} k≢l₂ = k≢l₂
    restrict-preserves-k≢ {k} {l₁} {(k' , v') ∷ xs} (k≢k' ∷ k≢xs)
        with ∈k-dec k' l₁
    ...   | yes _ = k≢k' ∷ restrict-preserves-k≢ k≢xs
    ...   | no _ = restrict-preserves-k≢ k≢xs

    restrict-preserves-Unique : ∀ {l₁ l₂ : List (A × B)} →
                                Unique (keys l₂) → Unique (keys (restrict l₁ l₂))
    restrict-preserves-Unique {l₁} {[]} _ = Utils.empty
    restrict-preserves-Unique {l₁} {(k , v) ∷ xs} (push k≢xs uxs)
        with ∈k-dec k l₁
    ...   | yes _ = push (restrict-preserves-k≢ k≢xs) (restrict-preserves-Unique uxs)
    ...   | no _ = restrict-preserves-Unique uxs

    intersect-preserves-Unique : ∀ {l₁ l₂ : List (A × B)} →
                                 Unique (keys l₂) → Unique (keys (intersect l₁ l₂))
    intersect-preserves-Unique {l₁} u = restrict-preserves-Unique (updates-preserve-Unique {l₁} u)

    updates-preserve-∉₂ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                          ¬ k ∈k l₂ → ¬ k ∈k updates l₁ l₂
    updates-preserve-∉₂ {k} {l₁} {l₂} k∉kl₁ k∈kl₁l₂
        rewrite updates-keys {l₁} {l₂} = k∉kl₁ k∈kl₁l₂

    restrict-needs-both : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                          k ∈k restrict l₁ l₂ → (k ∈k l₁ × k ∈k l₂)
    restrict-needs-both {k} {l₁} {[]} ()
    restrict-needs-both {k} {l₁} {(k' , _) ∷ xs} k∈l₁l₂
        with ∈k-dec k' l₁ | k∈l₁l₂
    ...   | yes k'∈kl₁    | here k≡k'
            rewrite k≡k' =
            (k'∈kl₁ , here refl)
    ...   | yes _         | there k∈l₁xs =
            let (k∈kl₁ , k∈kxs) = restrict-needs-both k∈l₁xs
            in (k∈kl₁ , there k∈kxs)
    ...   | no k'∉kl₁     | k∈l₁xs =
            let (k∈kl₁ , k∈kxs) = restrict-needs-both k∈l₁xs
            in (k∈kl₁ , there k∈kxs)

    restrict-subset-keys : ∀ {l₁ l₂ : List (A × B)} →
                           All (λ k → k ∈k l₁) (keys l₂) →
                           keys l₂ ≡ keys (restrict l₁ l₂)
    restrict-subset-keys {l₁} {[]} _ = refl
    restrict-subset-keys {l₁} {(k , v) ∷ l₂'} (k∈kl₁ ∷ kl₂'⊆kl₁)
        with ∈k-dec k l₁
    ...   | no k∉kl₁ = ⊥-elim (k∉kl₁ k∈kl₁)
    ...   | yes _ rewrite restrict-subset-keys {l₁} {l₂'} kl₂'⊆kl₁ = refl

    restrict-equal-keys : ∀ {l₁ l₂ : List (A × B)} →
                          keys l₁ ≡ keys l₂ →
                          keys l₁ ≡ keys (restrict l₁ l₂)
    restrict-equal-keys {l₁} {l₂} kl₁≡kl₂
        with subst (λ l → All (λ k → k ∈ l) (keys l₂)) (sym kl₁≡kl₂) (All-x∈xs (keys l₂))
    ...   | kl₂⊆kl₁ = trans kl₁≡kl₂ (restrict-subset-keys {l₁} {l₂} kl₂⊆kl₁)

    intersect-equal-keys : ∀ {l₁ l₂ : List (A × B)} →
                           keys l₁ ≡ keys l₂ →
                           keys l₁ ≡ keys (intersect l₁ l₂)
    intersect-equal-keys {l₁} {l₂} kl₁≡kl₂
        rewrite restrict-equal-keys (trans kl₁≡kl₂ (updates-keys {l₁} {l₂}))
        rewrite updates-keys {l₁} {l₂} = refl

    restrict-preserves-∉₁ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                            ¬ k ∈k l₁ → ¬ k ∈k restrict l₁ l₂
    restrict-preserves-∉₁ {k} {l₁} {l₂} k∉kl₁ k∈kl₁l₂ =
        let (k∈kl₁ , _) = restrict-needs-both {l₂ = l₂} k∈kl₁l₂ in k∉kl₁ k∈kl₁

    restrict-preserves-∉₂ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                            ¬ k ∈k l₂ → ¬ k ∈k restrict l₁ l₂
    restrict-preserves-∉₂ {k} {l₁} {l₂} k∉kl₂ k∈kl₁l₂ =
        let (_ , k∈kl₂) = restrict-needs-both {l₂ = l₂} k∈kl₁l₂ in k∉kl₂ k∈kl₂

    intersect-preserves-∉₁ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                             ¬ k ∈k l₁ → ¬ k ∈k intersect l₁ l₂
    intersect-preserves-∉₁ {k} {l₁} {l₂} = restrict-preserves-∉₁ {k} {l₁} {updates l₁ l₂}

    intersect-preserves-∉₂ : ∀ {k : A} {l₁ l₂ : List (A × B)} →
                             ¬ k ∈k l₂ → ¬ k ∈k intersect l₁ l₂
    intersect-preserves-∉₂ {k} {l₁} {l₂} k∉l₂ = restrict-preserves-∉₂ (updates-preserve-∉₂ {l₁ = l₁} k∉l₂ )

    restrict-preserves-∈₂ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
                            k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ restrict l₁ l₂
    restrict-preserves-∈₂ {k} {v} {l₁} {(k' , v') ∷ xs} k∈kl₁ (here k,v≡k',v')
        rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v'
        with ∈k-dec k' l₁
    ...   | yes _ = here refl
    ...   | no k'∉kl₁ = ⊥-elim (k'∉kl₁ k∈kl₁)
    restrict-preserves-∈₂ {l₁ = l₁} {l₂ = (k' , v') ∷ xs} k∈kl₁ (there k,v∈xs)
        with ∈k-dec k' l₁
    ...   | yes _ = there (restrict-preserves-∈₂ k∈kl₁ k,v∈xs)
    ...   | no _ = restrict-preserves-∈₂ k∈kl₁ k,v∈xs

    update-preserves-∈ : ∀ {k k' : A} {v v' : B} {l : List (A × B)} →
                         ¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ update k' v' l
    update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') ∷ xs} k≢k' (here k,v≡k'',v'')
        rewrite cong proj₁ k,v≡k'',v'' rewrite cong proj₂ k,v≡k'',v''
        with ≡-dec-A k' k''
    ...   | yes k'≡k'' = ⊥-elim (k≢k' (sym k'≡k''))
    ...   | no _ = here refl
    update-preserves-∈ {k} {k'} {v} {v'} {(k'' , v'') ∷ xs} k≢k' (there k,v∈xs)
        with ≡-dec-A k' k''
    ...   | yes _ = there k,v∈xs
    ...   | no _ = there (update-preserves-∈ k≢k' k,v∈xs)

    updates-preserve-∈₂ : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
                         ¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ updates l₁ l₂
    updates-preserve-∈₂ {k} {v} {[]} {l₂} _ k,v∈l₂ = k,v∈l₂
    updates-preserve-∈₂ {k} {v} {(k' , v') ∷ xs} {l₂} k∉kl₁ k,v∈l₂ =
        update-preserves-∈ (λ k≡k' → k∉kl₁ (here k≡k')) (updates-preserve-∈₂ (λ k∈kxs → k∉kl₁ (there k∈kxs)) k,v∈l₂)

    update-combines : ∀ {k : A} {v v' : B} {l : List (A × B)} →
                      Unique (keys l) → (k , v) ∈ l → (k , f v' v) ∈ update k v' l
    update-combines {k} {v} {v'} {(k' , v'') ∷ xs} _ (here k,v=k',v'')
        rewrite cong proj₁ k,v=k',v'' rewrite cong proj₂ k,v=k',v''
        with ≡-dec-A k' k'
    ...   | yes _ = here refl
    ...   | no k'≢k' = ⊥-elim (k'≢k' refl)
    update-combines {k} {v} {v'} {(k' , v'') ∷ xs} (push k'≢xs uxs) (there k,v∈xs)
        with ≡-dec-A k k'
    ...   | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs (∈-cong proj₁ k,v∈xs))
    ...   | no _ = there (update-combines uxs k,v∈xs)

    updates-combine : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
                      Unique (keys l₁) → Unique (keys l₂) →
                      (k , v₁) ∈ l₁ → (k , v₂) ∈ l₂ → (k , f v₁ v₂) ∈ updates l₁ l₂
    updates-combine {k} {v₁} {v₂} {(k' , v') ∷ xs} {l₂} (push k'≢xs uxs₁) u₂ (here k,v₁≡k',v') k,v₂∈l₂
        rewrite cong proj₁ k,v₁≡k',v' rewrite cong proj₂ k,v₁≡k',v' =
        update-combines (updates-preserve-Unique {l₁ = xs} u₂) (updates-preserve-∈₂ (All¬-¬Any k'≢xs) k,v₂∈l₂)
    updates-combine {k} {v₁} {v₂} {(k' , v') ∷ xs} {l₂} (push k'≢xs uxs₁) u₂ (there k,v₁∈xs) k,v₂∈l₂ =
        update-preserves-∈ k≢k' (updates-combine uxs₁ u₂ k,v₁∈xs k,v₂∈l₂)
        where
            k≢k' : ¬ k ≡ k'
            k≢k' with ≡-dec-A k k'
            ...    | yes k≡k' rewrite k≡k' = ⊥-elim (All¬-¬Any k'≢xs (∈-cong proj₁ k,v₁∈xs))
            ...    | no  k≢k' = k≢k'

    intersect-combines : ∀ {k : A} {v₁ v₂ : B} {l₁ l₂ : List (A × B)} →
                      Unique (keys l₁) → Unique (keys l₂) →
                      (k , v₁) ∈ l₁ → (k , v₂) ∈ l₂ → (k , f v₁ v₂) ∈ intersect l₁ l₂
    intersect-combines u₁ u₂ k,v₁∈l₁ k,v₂∈l₂ =
        restrict-preserves-∈₂ (∈-cong proj₁ k,v₁∈l₁) (updates-combine u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)

Map : Set (a ⊔ℓ b)
Map = Σ (List (A × B)) (λ l → Unique (ImplKeys.keys l))

instance
    showable : {{ showableA : Showable A }} {{ showableB : Showable B }} →
               Showable Map
    showable = record
        { show = λ (kvs , _) →
            "{" ++ˢ foldrˡ (λ (x , y) rest → show x ++ˢ " ↦ " ++ˢ show y ++ˢ ", " ++ˢ rest) "" kvs ++ˢ "}"
        }

empty : Map
empty = ([] , Utils.empty)

keys : Map → List A
keys (kvs , _) = ImplKeys.keys kvs

_∈_ : (A × B) → Map → Set (a ⊔ℓ b)
_∈_ p (kvs , _) = MemProp._∈_ p kvs

_∈k_ : A → Map → Set a
_∈k_ k m = MemProp._∈_ k (keys m)

locate : ∀ {k : A} {m : Map} → k ∈k m → Σ B (λ v → (k , v) ∈ m)
locate k∈km = locate-impl k∈km

-- `forget` and `∈k-dec` are defined this way because ∈ for maps uses
-- projection, so the full map can't be guessed. On the other hand, list can
-- be guessed.
forget : ∀ {k : A} {v : B} {l : List (A × B)} → (k , v) ∈ˡ l → k ∈ˡ (ImplKeys.keys l)
forget = ∈-cong proj₁

Map-functional : ∀ {k : A} {v v' : B} {m : Map} → (k , v) ∈ m → (k , v') ∈ m → v ≡ v'
Map-functional {m = (l , ul)} k,v∈m k,v'∈m = ListAB-functional ul k,v∈m k,v'∈m

open ImplRelation using () renaming (subset to subset-impl) public

_⊆_ : Map → Map → Set (a ⊔ℓ b)
_⊆_ (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂

⊆-refl : (m : Map) →  m ⊆ m
⊆-refl _ k v k,v∈m = (v , (≈₂-refl , k,v∈m))

⊆-trans : (m₁ m₂ m₃ : Map) →  m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃
⊆-trans _ _ _ m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ =
    let
        (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
        (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂
    in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))