open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong)
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; _⊔_)

module 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 ⊔ 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' : 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)

    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 k≢k' k∈kl =
        let (v , k,v∈l) = locate k∈kl
        in ∈-cong proj₁ (insert-preserves-∈ k≢k' k,v∈l)

    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'


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

_∈_ : (A × B) → Map → Set (a ⊔ 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 ⊔ b) where
    `_ : Map → Expr
    _∪_ : Expr → Expr → Expr

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

    insert : A → B → Map → Map
    insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-Unique uks)

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

    ⟦_⟧ : Expr -> Map
    ⟦ ` m ⟧ = m
    ⟦ e₁ ∪ e₂ ⟧ = union ⟦ e₁ ⟧ ⟦ e₂ ⟧

    data Provenance (k : A) : B → Expr → Set (a ⊔ 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 (f 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  (f 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)


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

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

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

module _ (f : B → B → B) where
    module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≡ f b₂ b₁)
             (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≡ f b₁ (f b₁ b₂)) 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 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 , ImplInsert.union-combines f 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 , ImplInsert.union-preserves-∈₂ f 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 , ImplInsert.union-preserves-∈₁ f u₂ v₂∈m₂ k∉km₁))

        union-assoc₁ : ∀ (m₁ m₂ m₃ : Map) → subset (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
        union-assoc₁ m₁ m₂ m₃ k v k,v∈m₁₂m₃
            with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
        ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃)) = {!!}
        ...   | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke3 , v₂∈m₁₂m₃)) = {!!}
        ...   | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃)) = {!!}
        ...   | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃)) = {!!}
        ...   | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃)) = {!!}
        ...   | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃) = {!!}
        ...   | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃)) = {!!}