open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Relation.Binary.Definitions using (Decidable)
open import Relation.Binary.Core using (Rel)
open import Relation.Nullary using (Dec; yes; no; Reflects; ofʸ; ofⁿ)
open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)

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

import Data.List.Membership.Propositional as MemProp

open import Relation.Nullary using (¬_)
open import Data.Nat using (ℕ)
open import Data.List using (List; map; []; _∷_; _++_)
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 (⊥)

keys : List (A × B) → List A
keys = map proj₁

data Unique {c} {C : Set c} : List C → Set c where
    empty : Unique []
    push : ∀ {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} →
                ¬ MemProp._∈_ 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

All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs

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

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

    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 = absurd (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 = absurd (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 × B)) → Dec (k ∈ 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 ∈ keys ((k' , v) ∷ xs)
                      witness (here k≡k') = k≢k' k≡k'
                      witness (there k∈kxs) = k∉kxs k∈kxs

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

private module ImplRelation (_≈_ : B → B → Set b) 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 ≈ v' × ((k , v') ∈ m₂))

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

    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' = absurd (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' = absurd (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-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' = absurd (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' = absurd (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 = absurd (k∉l₁l₂ (union-preserves-∈k₁ k∈kl₁))
    ...   | no k∉kl₁ with ∈k-dec k l₂
    ...         | yes k∈kl₂ = absurd (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 = absurd (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' = absurd (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' = absurd (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' = absurd (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' = absurd (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} {[]} _ = 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-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 = absurd (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'' = absurd (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' = absurd (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' = absurd (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' = absurd (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 (keys l))

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

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

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

data Expr : Set (a ⊔ℓ b) where
    `_ : Map → Expr
    _∪_ : Expr → Expr → Expr
    _∩_ : Expr → Expr → Expr

module _ (f : B → B → B) where
    open ImplInsert f renaming
        ( insert to insert-impl
        ; union to union-impl
        ; intersect to intersect-impl
        )

    union : Map → Map → Map
    union (kvs₁ , _) (kvs₂ , uks₂) = (union-impl kvs₁ kvs₂ , union-preserves-Unique kvs₁ kvs₂ uks₂)

    intersect : Map → Map → Map
    intersect (kvs₁ , _) (kvs₂ , uks₂) = (intersect-impl kvs₁ kvs₂ , intersect-preserves-Unique {kvs₁} {kvs₂} uks₂)

module _ (fUnion : B → B → B) (fIntersect : B → B → B) where
    open ImplInsert fUnion using
        ( union-combines
        ; union-preserves-∈₁
        ; union-preserves-∈₂
        ; union-preserves-∉
        )

    open ImplInsert fIntersect using
        ( restrict-needs-both
        ; updates
        ; intersect-preserves-∉₁
        ; intersect-preserves-∉₂
        ; intersect-combines
        )

    ⟦_⟧ : Expr -> Map
    ⟦ ` m ⟧ = m
    ⟦ e₁ ∪ e₂ ⟧ = union fUnion ⟦ e₁ ⟧ ⟦ e₂ ⟧
    ⟦ e₁ ∩ e₂ ⟧ = intersect fIntersect ⟦ e₁ ⟧ ⟦ e₂ ⟧

    data Provenance (k : A) : B → Expr → Set (a ⊔ℓ b) where
        single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Provenance k v (` m)
        in₁ : ∀ {v : B} {e₁ e₂ : Expr} → Provenance k v e₁ → ¬ k ∈k ⟦ e₂ ⟧ → Provenance k v (e₁ ∪ e₂)
        in₂ : ∀ {v : B} {e₁ e₂ : Expr} → ¬ k ∈k ⟦ e₁ ⟧ → Provenance k v e₂ → Provenance k v (e₁ ∪ e₂)
        bothᵘ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (fUnion v₁ v₂) (e₁ ∪ e₂)
        bothⁱ : ∀ {v₁ v₂ : B} {e₁ e₂ : Expr} → Provenance k v₁ e₁ → Provenance k v₂ e₂ → Provenance k (fIntersect v₁ v₂) (e₁ ∩ e₂)

    Expr-Provenance : ∀ (k : A) (e : Expr) → k ∈k ⟦ e ⟧ → Σ B (λ v → (Provenance k v e × (k , v) ∈ ⟦ e ⟧))
    Expr-Provenance k (` m) k∈km = let (v , k,v∈m) = locate k∈km in (v , (single k,v∈m , k,v∈m))
    Expr-Provenance k (e₁ ∪ e₂) k∈ke₁e₂
        with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
    ...   | yes k∈ke₁ | yes k∈ke₂ =
            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
                (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
            in  (fUnion v₁ v₂ , (bothᵘ g₁ g₂ , union-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
    ...   | yes k∈ke₁ | no k∉ke₂ =
            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
            in (v₁ , (in₁ g₁ k∉ke₂ , union-preserves-∈₁ (proj₂ ⟦ e₁ ⟧) k,v₁∈e₁ k∉ke₂))
    ...   | no k∉ke₁ | yes k∈ke₂ =
            let (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
            in (v₂ , (in₂ k∉ke₁ g₂ , union-preserves-∈₂ k∉ke₁ k,v₂∈e₂))
    ...   | no k∉ke₁ | no k∉ke₂  = absurd (union-preserves-∉ k∉ke₁ k∉ke₂ k∈ke₁e₂)
    Expr-Provenance k (e₁ ∩ e₂) k∈ke₁e₂
        with ∈k-dec k (proj₁ ⟦ e₁ ⟧) | ∈k-dec k (proj₁ ⟦ e₂ ⟧)
    ...   | yes k∈ke₁ | yes k∈ke₂ =
            let (v₁ , (g₁ , k,v₁∈e₁)) = Expr-Provenance k e₁ k∈ke₁
                (v₂ , (g₂ , k,v₂∈e₂)) = Expr-Provenance k e₂ k∈ke₂
            in (fIntersect v₁ v₂ , (bothⁱ g₁ g₂ , intersect-combines (proj₂ ⟦ e₁ ⟧) (proj₂ ⟦ e₂ ⟧) k,v₁∈e₁ k,v₂∈e₂))
    ...   | yes k∈ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)
    ...   | no k∉ke₁ | yes k∈ke₂ = absurd (intersect-preserves-∉₁ {l₂ = proj₁ ⟦ e₂ ⟧} k∉ke₁ k∈ke₁e₂)
    ...   | no k∉ke₁ | no k∉ke₂ = absurd (intersect-preserves-∉₂ {l₁ = proj₁ ⟦ e₁ ⟧} k∉ke₂ k∈ke₁e₂)


module _ (_≈_ : B → B → Set b) where
    open ImplRelation _≈_ renaming (subset to subset-impl)

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

    lift : Map → Map → Set (a ⊔ℓ b)
    lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁

    private
        data SubsetInfo (m₁ m₂ : Map) : Set (a ⊔ℓ b) where
            extra : (k : A) → k ∈k m₁ → ¬ k ∈k m₂ → SubsetInfo m₁ m₂
            mismatch : (k : A) (v₁ v₂ : B) → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → ¬ v₁ ≈ v₂ → SubsetInfo m₁ m₂
            fine : subset m₁ m₂ → SubsetInfo m₁ m₂

        SubsetInfo-to-dec : ∀ {m₁ m₂ : Map} → SubsetInfo m₁ m₂ → Dec (subset m₁ m₂)
        SubsetInfo-to-dec (extra k k∈km₁ k∉km₂) =
            let (v , k,v∈m₁) = locate k∈km₁
            in no (λ m₁⊆m₂ →
                   let (v' , (_ , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
                   in k∉km₂ (∈-cong proj₁ k,v'∈m₂))
        SubsetInfo-to-dec {m₁} {m₂} (mismatch k v₁ v₂ k,v₁∈m₁ k,v₂∈m₂ v₁̷≈v₂) =
            no (λ m₁⊆m₂ →
                let (v' , (v₁≈v' , k,v'∈m₂)) = m₁⊆m₂ k v₁ k,v₁∈m₁
                in v₁̷≈v₂ (subst (λ v'' → v₁ ≈ v'') (Map-functional {k} {v'} {v₂} {m₂} k,v'∈m₂ k,v₂∈m₂) v₁≈v')) -- for some reason, can't just use subst...
        SubsetInfo-to-dec (fine m₁⊆m₂) = yes m₁⊆m₂


    module _ (≈-dec : ∀ (b₁ b₂ : B) → Dec (b₁ ≈ b₂)) where
        private
            compute-SubsetInfo : ∀ m₁ m₂ → SubsetInfo m₁ m₂
            compute-SubsetInfo ([] , _) m₂ = fine (λ k v ())
            compute-SubsetInfo m₁@((k , v) ∷ xs₁ , push k≢xs₁ uxs₁) m₂@(l₂ , u₂)
                with compute-SubsetInfo (xs₁ , uxs₁) m₂
            ...   | extra k' k'∈kxs₁ k'∉km₂ = extra k' (there k'∈kxs₁) k'∉km₂
            ...   | mismatch k' v₁ v₂ k',v₁∈xs₁ k',v₂∈m₂ v₁̷≈v₂ =
                    mismatch k' v₁ v₂ (there k',v₁∈xs₁) k',v₂∈m₂ v₁̷≈v₂
            ...   | fine xs₁⊆m₂ with ∈k-dec k l₂
            ...       | no k∉km₂ = extra k (here refl) k∉km₂
            ...       | yes k∈km₂ with locate k∈km₂
            ...           | (v' , k,v'∈m₂) with ≈-dec v v'
            ...               | no v̷≈v' = mismatch k v v' (here refl) (k,v'∈m₂) v̷≈v'
            ...               | yes v≈v' = fine m₁⊆m₂
                                    where
                                        m₁⊆m₂ : subset m₁ m₂
                                        m₁⊆m₂ k' v'' (here k,v≡k',v'')
                                            rewrite cong proj₁ k,v≡k',v''
                                            rewrite cong proj₂ k,v≡k',v'' =
                                            (v' , (v≈v' , k,v'∈m₂))
                                        m₁⊆m₂ k' v'' (there k,v≡k',v'') =
                                            xs₁⊆m₂ k' v'' k,v≡k',v''

        subset-dec : ∀ m₁ m₂ → Dec (subset m₁ m₂)
        subset-dec m₁ m₂ = SubsetInfo-to-dec (compute-SubsetInfo m₁ m₂)

        lift-dec : ∀ m₁ m₂ → Dec (lift m₁ m₂)
        lift-dec m₁ m₂
            with subset-dec m₁ m₂ | subset-dec m₂ m₁
        ...   | yes m₁⊆m₂ | yes m₂⊆m₁ = yes (m₁⊆m₂ , m₂⊆m₁)
        ...   | _         | no  m₂̷⊆m₁ = no (λ (_ , m₂⊆m₁) → m₂̷⊆m₁ m₂⊆m₁)
        ...   | no  m₁̷⊆m₂ | _         = no (λ (m₁⊆m₂ , _) → m₁̷⊆m₂ m₁⊆m₂)

    -- The Provenance type requires both union and intersection functions,
    -- but sometimes here we're working with one operation only. Just use the
    -- union/intersection function for both -- it doesn't matter, since we don't
    -- use the dual operations in these proofs.

    module _ (f : B → B → B)
             (≈-f-cong : ∀ {b₁ b₂ b₃ b₄} → b₁ ≈ b₂ → b₃ ≈ b₄ → f b₁ b₃ ≈ f b₂ b₄) where
        private module I = ImplInsert f

        union-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → lift m₁ m₂ → lift m₃ m₄ → lift (union f m₁ m₃) (union f m₂ m₄)
        union-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) =
            ( union-subset m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃)
            , union-subset m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄)
            )
            where
                ≈-∉-cong : ∀ {m₁ m₂ : Map} {k : A} → lift m₁ m₂ → ¬ k ∈k m₁ → ¬ k ∈k m₂
                ≈-∉-cong (m₁⊆m₂ , m₂⊆m₁) k∉km₁ k∈km₂ =
                    let (v₂ , k,v₂∈m₂) = locate k∈km₂
                        (_ , (_ , k,v₁∈m₁)) = m₂⊆m₁ _ v₂ k,v₂∈m₂
                    in k∉km₁ (∈-cong proj₁ k,v₁∈m₁)

                union-subset : ∀ (m₁ m₂ m₃ m₄ : Map) → lift m₁ m₂ → lift m₃ m₄ → subset (union f m₁ m₃) (union f m₂ m₄)
                union-subset m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃
                    with Expr-Provenance f f k ((` m₁) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃)
                ...   | (_ , (bothᵘ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃))
                        rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ =
                        let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
                            (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
                        in (f v₂ v₄ , (≈-f-cong v₁≈v₂ v₃≈v₄ , I.union-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄))
                ...   | (_ , (in₁ (single {v₁} v₁∈m₁) k∉km₃ , v₁∈m₁m₃))
                        rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₁∈m₁m₃ =
                        let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
                            k∉km₄ = ≈-∉-cong {m₃} {m₄} m₃≈m₄ k∉km₃
                        in (v₂ , (v₁≈v₂ , I.union-preserves-∈₁ (proj₂ m₂) k,v₂∈m₂ k∉km₄))
                ...   | (_ , (in₂ k∉km₁ (single {v₃} v₃∈m₃) , v₃∈m₁m₃))
                        rewrite Map-functional {m = union f m₁ m₃} k,v∈m₁m₃ v₃∈m₁m₃ =
                        let (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
                            k∉km₂ = ≈-∉-cong {m₁} {m₂} m₁≈m₂ k∉km₁
                        in (v₄ , (v₃≈v₄ , I.union-preserves-∈₂ k∉km₂ k,v₄∈m₄))

        intersect-cong : ∀ {m₁ m₂ m₃ m₄ : Map} → lift m₁ m₂ → lift m₃ m₄ → lift (intersect f m₁ m₃) (intersect f m₂ m₄)
        intersect-cong {m₁} {m₂} {m₃} {m₄} (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃) =
            ( intersect-subset m₁ m₂ m₃ m₄ (m₁⊆m₂ , m₂⊆m₁) (m₃⊆m₄ , m₄⊆m₃)
            , intersect-subset m₂ m₁ m₄ m₃ (m₂⊆m₁ , m₁⊆m₂) (m₄⊆m₃ , m₃⊆m₄)
            )
            where
                intersect-subset : ∀ (m₁ m₂ m₃ m₄ : Map) → lift m₁ m₂ → lift m₃ m₄ → subset (intersect f m₁ m₃) (intersect f m₂ m₄)
                intersect-subset m₁ m₂ m₃ m₄ m₁≈m₂ m₃≈m₄ k v k,v∈m₁m₃
                    with Expr-Provenance f f k ((` m₁) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁m₃)
                ...   | (_ , (bothⁱ (single {v₁} v₁∈m₁) (single {v₃} v₃∈m₃) , v₁v₃∈m₁m₃))
                        rewrite Map-functional {m = intersect f m₁ m₃} k,v∈m₁m₃ v₁v₃∈m₁m₃ =
                        let (v₂ , (v₁≈v₂ , k,v₂∈m₂)) = proj₁ m₁≈m₂ k v₁ v₁∈m₁
                            (v₄ , (v₃≈v₄ , k,v₄∈m₄)) = proj₁ m₃≈m₄ k v₃ v₃∈m₃
                        in (f v₂ v₄ , (≈-f-cong v₁≈v₂ v₃≈v₄ , I.intersect-combines (proj₂ m₂) (proj₂ m₄) k,v₂∈m₂ k,v₄∈m₄))

    module _ (≈-refl : ∀ {b : B} → b ≈ b)
             (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁)
             (f : B → B → B) where
        private module I = ImplInsert f

        module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
            union-idemp : ∀ (m : Map) → lift (union f m m) m
            union-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
                where
                    mm-m-subset : subset (union f m m) m
                    mm-m-subset k v k,v∈mm
                        with Expr-Provenance f f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
                    ...   | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
                            rewrite Map-functional {m = m} v'∈m v''∈m
                            rewrite Map-functional {m = union f m m} k,v∈mm v'v''∈mm =
                            (v'' , (f-idemp v'' , v''∈m))
                    ...   | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m))
                    ...   | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m))

                    m-mm-subset : subset m (union f m m)
                    m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m))

        module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
            union-comm : ∀ (m₁ m₂ : Map) → lift (union f m₁ m₂) (union f m₂ m₁)
            union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
                where
                    union-comm-subset : ∀ (m₁ m₂ : Map) → subset (union f m₁ m₂) (union f m₂ m₁)
                    union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
                        with Expr-Provenance f f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
                    ...   | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
                            (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
                    ...   | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
                            (v₁ , (≈-refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁))
                    ...   | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
                            (v₂ , (≈-refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))

        module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃))  where
            union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
            union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
                where
                    union-assoc₁ : subset (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
                    union-assoc₁ k v k,v∈m₁₂m₃
                        with Expr-Provenance f f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
                    ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
                            let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
                            in (v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
                    ...   | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃))
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ =
                            (v₂ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
                    ...   | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃))
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ =
                            (f v₂ v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁  (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
                    ...   | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃))
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ =
                            (v₁ , (≈-refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃)))
                    ...   | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃))
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ =
                            (f v₁ v₃ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
                    ...   | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃)
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ =
                            (f v₁ v₂ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
                    ...   | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
                            (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))

                    union-assoc₂ : subset (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃)
                    union-assoc₂ k v k,v∈m₁m₂₃
                        with Expr-Provenance f f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
                    ...   | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
                            (v₃ , (≈-refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))
                    ...   | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ =
                            (v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃))
                    ...   | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ =
                            (f v₂ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃))
                    ...   | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ =
                            let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃
                            in (v₁ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃))
                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ =
                            (f v₁ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃))
                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ =
                            (f v₁ v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃))
                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
                            (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))

        module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
            intersect-idemp : ∀ (m : Map) → lift (intersect f m m) m
            intersect-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
                where
                    mm-m-subset : subset (intersect f m m) m
                    mm-m-subset k v k,v∈mm
                        with Expr-Provenance f f k ((` m) ∩ (` m)) (∈-cong proj₁ k,v∈mm)
                    ...   | (_ , (bothⁱ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
                            rewrite Map-functional {m = m} v'∈m v''∈m
                            rewrite Map-functional {m = intersect f m m} k,v∈mm v'v''∈mm =
                            (v'' , (f-idemp v'' , v''∈m))

                    m-mm-subset : subset m (intersect f m m)
                    m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.intersect-combines u u k,v∈m k,v∈m))

        module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
            intersect-comm : ∀ (m₁ m₂ : Map) → lift (intersect f m₁ m₂) (intersect f m₂ m₁)
            intersect-comm m₁ m₂ = (intersect-comm-subset m₁ m₂ , intersect-comm-subset m₂ m₁)
                where
                    intersect-comm-subset : ∀ (m₁ m₂ : Map) → subset (intersect f m₁ m₂) (intersect f m₂ m₁)
                    intersect-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
                        with Expr-Provenance f f k ((` m₁) ∩ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
                    ...   | (_ , (bothⁱ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
                            rewrite Map-functional {m = intersect f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
                            (f v₂ v₁ , (f-comm v₁ v₂ , I.intersect-combines u₂ u₁ v₂∈m₂ v₁∈m₁))

        module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃))  where
            intersect-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃))
            intersect-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (intersect-assoc₁ , intersect-assoc₂)
                where
                    intersect-assoc₁ : subset (intersect f (intersect f m₁ m₂) m₃) (intersect f m₁ (intersect f m₂ m₃))
                    intersect-assoc₁ k v k,v∈m₁₂m₃
                        with Expr-Provenance f f k (((` m₁) ∩ (` m₂)) ∩ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
                    ...   | (_ , (bothⁱ (bothⁱ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
                            rewrite Map-functional {m = intersect f (intersect f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
                            (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.intersect-combines u₁ (I.intersect-preserves-Unique {l₂} {l₃} u₃) v₁∈e₁ (I.intersect-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))

                    intersect-assoc₂ : subset (intersect f m₁ (intersect f m₂ m₃)) (intersect f (intersect f m₁ m₂) m₃)
                    intersect-assoc₂ k v k,v∈m₁m₂₃
                        with Expr-Provenance f f k ((` m₁) ∩ ((` m₂) ∩ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
                    ...   | (_ , (bothⁱ (single {v₁} v₁∈e₁) (bothⁱ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
                            rewrite Map-functional {m = intersect f m₁ (intersect f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
                            (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.intersect-combines (I.intersect-preserves-Unique {l₁} {l₂} u₂) u₃ (I.intersect-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))

    module _ (≈-refl : ∀ {b : B} → b ≈ b)
             (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁)
             (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
             (⊔₂-idemp :  ∀ (b : B) → (b ⊔₂ b) ≈ b)
             (⊓₂-idemp :  ∀ (b : B) → (b ⊓₂ b) ≈ b)
             (⊔₂-⊓₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊔₂ (b₁ ⊓₂ b₂)) ≈ b₁)
             (⊓₂-⊔₂-absorb : ∀ (b₁ b₂ : B) → (b₁ ⊓₂ (b₁ ⊔₂ b₂)) ≈ b₁)
             where
        private module I⊔ = ImplInsert _⊔₂_
        private module I⊓ = ImplInsert _⊓₂_

        private
            _⊔_ = union _⊔₂_
            _⊓_ = intersect _⊓₂_

        intersect-union-absorb : ∀ (m₁ m₂ : Map) →  lift (m₁ ⊓ (m₁ ⊔ m₂)) m₁
        intersect-union-absorb m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (intersect-union-absorb₁ , intersect-union-absorb₂)
            where
                intersect-union-absorb₁ : subset (m₁ ⊓ (m₁ ⊔ m₂)) m₁
                intersect-union-absorb₁ k v k,v∈m₁m₁₂
                    with Expr-Provenance _ _ k ((` m₁) ∩ ((` m₁) ∪ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂)
                ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
                                    (bothᵘ (single {v₁'} k,v₁'∈m₁)
                                           (single {v₂} v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
                        rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
                        rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
                        (v₁' , (⊓₂-⊔₂-absorb v₁' v₂ , k,v₁'∈m₁))
                ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
                                    (in₁ (single {v₁'} k,v₁'∈m₁) _) , v₁v₁'∈m₁m₁₂))
                        rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
                        rewrite Map-functional {m = m₁ ⊓ (m₁ ⊔ m₂)} k,v∈m₁m₁₂ v₁v₁'∈m₁m₁₂ =
                        (v₁' , (⊓₂-idemp v₁' , k,v₁'∈m₁))
                ...   | (_ , (bothⁱ (single {v₁} k,v₁∈m₁)
                                    (in₂ k∉m₁ _ ) , _)) = absurd (k∉m₁ (∈-cong proj₁ k,v₁∈m₁))

                intersect-union-absorb₂ : subset m₁ (m₁ ⊓ (m₁ ⊔ m₂))
                intersect-union-absorb₂ k v k,v∈m₁
                    with ∈k-dec k l₂
                ...   | yes k∈km₂ =
                        let (v₂ , k,v₂∈m₂) = locate k∈km₂
                        in (v ⊓₂ (v ⊔₂ v₂) , (≈-sym (⊓₂-⊔₂-absorb v v₂) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
                ...   | no k∉km₂ = (v ⊓₂ v , (≈-sym (⊓₂-idemp v) , I⊓.intersect-combines u₁ (I⊔.union-preserves-Unique l₁ l₂ u₂) k,v∈m₁ (I⊔.union-preserves-∈₁ u₁ k,v∈m₁ k∉km₂)))

        union-intersect-absorb : ∀ (m₁ m₂ : Map) →  lift (m₁ ⊔ (m₁ ⊓ m₂)) m₁
        union-intersect-absorb m₁@(l₁ , u₁) m₂@(l₂ , u₂) = (union-intersect-absorb₁ , union-intersect-absorb₂)
            where
                union-intersect-absorb₁ : subset (m₁ ⊔ (m₁ ⊓ m₂)) m₁
                union-intersect-absorb₁ k v k,v∈m₁m₁₂
                    with Expr-Provenance _ _ k ((` m₁) ∪ ((` m₁) ∩ (` m₂))) (∈-cong proj₁ k,v∈m₁m₁₂)
                ...   | (_ , (bothᵘ (single {v₁} k,v₁∈m₁)
                                    (bothⁱ (single {v₁'} k,v₁'∈m₁)
                                           (single {v₂} k,v₂∈m₂)) , v₁v₁'v₂∈m₁m₁₂))
                        rewrite Map-functional {m = m₁} k,v₁∈m₁ k,v₁'∈m₁
                        rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ v₁v₁'v₂∈m₁m₁₂ =
                        (v₁' , (⊔₂-⊓₂-absorb v₁' v₂ , k,v₁'∈m₁))
                ...   | (_ , (in₁ (single {v₁} k,v₁∈m₁) k∉km₁₂ , k,v₁∈m₁m₁₂))
                        rewrite Map-functional {m = m₁ ⊔ (m₁ ⊓ m₂)} k,v∈m₁m₁₂ k,v₁∈m₁m₁₂ =
                        (v₁ , (≈-refl , k,v₁∈m₁))
                ...   | (_ , (in₂ k∉km₁ (bothⁱ (single {v₁'} k,v₁'∈m₁)
                                              (single {v₂} k,v₂∈m₂)) , _)) =
                        absurd (k∉km₁ (∈-cong proj₁ k,v₁'∈m₁))

                union-intersect-absorb₂ : subset m₁ (m₁ ⊔ (m₁ ⊓ m₂))
                union-intersect-absorb₂ k v k,v∈m₁
                    with ∈k-dec k l₂
                ...   | yes k∈km₂ =
                        let (v₂ , k,v₂∈m₂) = locate k∈km₂
                        in (v ⊔₂ (v ⊓₂ v₂) , (≈-sym (⊔₂-⊓₂-absorb v v₂) , I⊔.union-combines u₁ (I⊓.intersect-preserves-Unique {l₁} {l₂} u₂) k,v∈m₁ (I⊓.intersect-combines u₁ u₂ k,v∈m₁ k,v₂∈m₂)))
                ...   | no k∉km₂ = (v , (≈-refl , I⊔.union-preserves-∈₁ u₁ k,v∈m₁ (I⊓.intersect-preserves-∉₂ {k} {l₁} {l₂} k∉km₂)))