-- Because iterated products currently require both A and B to be of the same
-- universe, and the FiniteMap is written in a universe-polymorphic way,
-- specialize the FiniteMap module with Set-typed types only.

open import Lattice
open import Equivalence
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_)
open import Function.Definitions using (Inverseˡ; Inverseʳ)

module Lattice.FiniteValueMap (A : Set) (B : Set)
    (_≈₂_ : B → B → Set)
    (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
    (≈-dec-A : Decidable (_≡_ {_} {A}))
    (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where

open import Data.List using (List; length; []; _∷_; map)
open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
open import Data.Product using (Σ; proj₁; proj₂; _×_)
open import Data.Empty using (⊥-elim)
open import Utils using (Unique; push; empty; All¬-¬Any)
open import Data.Product using (_,_)
open import Data.List.Properties using (∷-injectiveʳ)
open import Data.List.Relation.Unary.All using (All)
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Relation.Nullary using (¬_)

open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB using (subset-impl; locate; forget; _∈_; Map-functional; Expr-Provenance; _∩_; _∪_; `_; in₁; in₂; bothᵘ; single; ⊔-combines)
open import Lattice.FiniteMap A B _≈₂_ _⊔₂_ _⊓₂_ ≈-dec-A lB public

module IterProdIsomorphism where
    open import Data.Unit using (⊤; tt)
    open import Lattice.Unit using () renaming (_≈_ to _≈ᵘ_; _⊔_ to _⊔ᵘ_; _⊓_ to _⊓ᵘ_; ≈-dec to ≈ᵘ-dec; isLattice to isLatticeᵘ; ≈-equiv to ≈ᵘ-equiv)
    open import Lattice.IterProd _≈₂_ _≈ᵘ_ _⊔₂_ _⊔ᵘ_ _⊓₂_ _⊓ᵘ_ lB isLatticeᵘ as IP using (IterProd)
    open IsLattice lB using () renaming (≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym)

    from : ∀ {ks : List A} → FiniteMap ks → IterProd (length ks)
    from {[]} (([] , _) , _) = tt
    from {k ∷ ks'} (((k' , v) ∷ fm' , push _ uks') , refl) = (v , from ((fm' , uks'), refl))

    to : ∀ {ks : List A} → Unique ks → IterProd (length ks) → FiniteMap ks
    to {[]} _ ⊤ = (([] , empty) , refl)
    to {k ∷ ks'} (push k≢ks' uks') (v , rest) =
        let
            ((fm' , ufm') , fm'≡ks') = to uks' rest

            -- This would be easier if we pattern matched on the equiality proof
            -- to get refl, but that makes it harder to reason about 'to' when
            -- the arguments are not known to be refl.
            k≢fm' = subst (λ ks → All (λ k' → ¬ k ≡ k') ks) (sym fm'≡ks') k≢ks'
            kvs≡ks = cong (k ∷_) fm'≡ks'
        in
            (((k , v) ∷ fm' , push k≢fm' ufm') , kvs≡ks)


    private
        _≈ᵐ_ : ∀ {ks : List A} → FiniteMap ks → FiniteMap ks → Set
        _≈ᵐ_ {ks} = _≈_ ks

        _⊔ᵐ_ : ∀ {ks : List A} → FiniteMap ks → FiniteMap ks → FiniteMap ks
        _⊔ᵐ_ {ks} = _⊔_ ks

        _⊆ᵐ_ : ∀ {ks₁ ks₂ : List A} → FiniteMap ks₁ → FiniteMap ks₂ → Set
        _⊆ᵐ_ fm₁ fm₂ = subset-impl (proj₁ (proj₁ fm₁)) (proj₁ (proj₁ fm₂))

        _≈ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → Set
        _≈ⁱᵖ_ {ks} = IP._≈_ (length ks)

        _⊔ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
        _⊔ⁱᵖ_ {ks} = IP._⊔_ (length ks)

        _∈ᵐ_ : ∀ {ks : List A} → A × B → FiniteMap ks → Set
        _∈ᵐ_ {ks} k,v fm = k,v ∈ proj₁ fm

    from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
                      Inverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- from (to x) = x
    from-to-inverseˡ {[]} _ _ = IsEquivalence.≈-refl (IP.≈-equiv 0)
    from-to-inverseˡ {k ∷ ks'} (push k≢ks' uks') (v , rest)
        with ((fm' , ufm') , refl) ← to uks' rest in p rewrite sym p =
            (IsLattice.≈-refl lB , from-to-inverseˡ {ks'} uks' rest)
            -- the rewrite here is needed because the IH is in terms of `to uks' rest`,
            -- but we end up with the 'unpacked' form (fm', ...). So, put it back
            -- in the 'packed' form after we've performed enough inspection
            -- to know we take the cons branch of `to`.

    -- The map has its own uniqueness proof, but the call to 'to' needs a standalone
    -- uniqueness proof too. Work with both proofs as needed to thread things through.
    from-to-inverseʳ : ∀ {ks : List A} (uks : Unique ks) →
                       Inverseʳ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- to (from x) = x
    from-to-inverseʳ {[]} _ (([] , empty) , kvs≡ks) rewrite kvs≡ks = ((λ k v ()) , (λ k v ()))
    from-to-inverseʳ {k ∷ ks'} uks@(push k≢ks'₁ uks'₁) fm₁@(m₁@((k , v) ∷ fm'₁ , push k≢ks'₂ uks'₂) , refl)
        with to uks'₁ (from ((fm'₁ , uks'₂) , refl)) | from-to-inverseʳ {ks'} uks'₁ ((fm'₁ , uks'₂) , refl)
    ...   | ((fm'₂ , ufm'₂) , _) | (fm'₂⊆fm'₁ , fm'₁⊆fm'₂) = (m₂⊆m₁ , m₁⊆m₂)
                where
                    kvs₁ = (k , v) ∷ fm'₁
                    kvs₂ = (k , v) ∷ fm'₂

                    m₁⊆m₂ : subset-impl kvs₁ kvs₂
                    m₁⊆m₂ k' v' (here refl) = (v' , (IsLattice.≈-refl lB , here refl))
                    m₁⊆m₂ k' v' (there k',v'∈fm'₁) =
                        let (v'' , (v'≈v'' , k',v''∈fm'₂)) = fm'₁⊆fm'₂ k' v' k',v'∈fm'₁
                        in (v'' , (v'≈v'' , there k',v''∈fm'₂))

                    m₂⊆m₁ : subset-impl kvs₂ kvs₁
                    m₂⊆m₁ k' v' (here refl) = (v' , (IsLattice.≈-refl lB , here refl))
                    m₂⊆m₁ k' v' (there k',v'∈fm'₂) =
                        let (v'' , (v'≈v'' , k',v''∈fm'₁)) = fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
                        in (v'' , (v'≈v'' , there k',v''∈fm'₁))

    private
        first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → Σ B (λ v → (k , v) ∈ proj₁ fm)
        first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)

        from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
        from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = refl

        -- We need pop because reasoning about two distinct 'refl' pattern
        -- matches is giving us unification errors. So, stash the 'refl' pattern
        -- matching into a helper functions, and write solutions in terms
        -- of that.
        pop : ∀ {k : A} {ks : List A} → FiniteMap (k ∷ ks) → FiniteMap ks
        pop (((_ ∷ fm') , push _ ufm') , refl) = ((fm' , ufm') , refl)

        pop-≈ :  ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → fm₁ ≈ᵐ fm₂ → pop fm₁ ≈ᵐ pop fm₂
        pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) = (narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
            where
                narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂  → pop fm₁ ⊆ᵐ fm₂
                narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁ = kvs₁⊆kvs₂ k' v' (there k',v'∈fm'₁)

                narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂  → fm₁ ⊆ᵐ pop fm₂
                narrow₂ {fm₁} {fm₂ = (_ ∷ fm'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
                    with kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
                ...   | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) = ⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈fm'₁))
                ...   | (v'' , (v'≈v'' , there k',v'∈fm'₂)) = (v'' , (v'≈v'' , k',v'∈fm'₂))

                narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂  → pop fm₁ ⊆ᵐ pop fm₂
                narrow {fm₁} {fm₂} x = narrow₂ {pop fm₁} (narrow₁ {fm₂ = fm₂} x)

        k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) → (k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
        k,v∈pop⇒k,v∈ {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k',v∈fm =
            ((λ { refl → All¬-¬Any k≢ks (forget {m = (fm' , uks')} k',v∈fm) }), there k',v∈fm)

        k,v∈⇒k,v∈pop : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) → ¬ k ≡ k' → (k' , v) ∈ᵐ fm → (k' , v) ∈ᵐ pop fm
        k,v∈⇒k,v∈pop {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
        k,v∈⇒k,v∈pop {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'

        Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} → (k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → Σ (B × B) (λ (v₁ , v₂) → ((v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
        Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
            with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
        ...   | (_ , (in₁ (single k,v∈m₁) k∉km₂ , _)) with k∈km₁ ← (forget {m = m₁} k,v∈m₁) rewrite trans ks₁≡ks (sym ks₂≡ks) = ⊥-elim (k∉km₂ k∈km₁)
        ...   | (_ , (in₂ k∉km₁ (single k,v∈m₂) , _)) with k∈km₂ ← (forget {m = m₂} k,v∈m₂) rewrite trans ks₁≡ks (sym ks₂≡ks) = ⊥-elim (k∉km₁ k∈km₂)
        ...   | (v₁⊔v₂ , (bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) , k,v₁⊔v₂∈m₁m₂))
                rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂ = ((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))

        pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
        pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) = (pfm₁fm₂⊆pfm₁pfm₂ , pfm₁pfm₂⊆pfm₁fm₂)
            where
                -- pfm₁fm₂⊆pfm₁pfm₂ = {!!}
                pfm₁fm₂⊆pfm₁pfm₂ : pop (fm₁ ⊔ᵐ fm₂) ⊆ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
                pfm₁fm₂⊆pfm₁pfm₂ k' v' k',v'∈pfm₁fm₂
                    with (k≢k' , k',v'∈fm₁fm₂) ← k,v∈pop⇒k,v∈ (fm₁ ⊔ᵐ fm₂) k',v'∈pfm₁fm₂
                    with ((v₁ , v₂) , (refl , (k,v₁∈fm₁ , k,v₂∈fm₂))) ← Provenance-union fm₁ fm₂ k',v'∈fm₁fm₂
                    with k',v₁∈pfm₁ ← k,v∈⇒k,v∈pop fm₁ k≢k' k,v₁∈fm₁
                    with k',v₂∈pfm₂ ← k,v∈⇒k,v∈pop fm₂ k≢k' k,v₂∈fm₂
                    = (v₁ ⊔₂ v₂ , (IsLattice.≈-refl lB , ⊔-combines {m₁ = proj₁ (pop fm₁)} {m₂ = proj₁ (pop fm₂)} k',v₁∈pfm₁ k',v₂∈pfm₂))

                pfm₁pfm₂⊆pfm₁fm₂ : (pop fm₁ ⊔ᵐ pop fm₂) ⊆ᵐ pop (fm₁ ⊔ᵐ fm₂)
                pfm₁pfm₂⊆pfm₁fm₂ k' v' k',v'∈pfm₁pfm₂
                    with ((v₁ , v₂) , (refl , (k,v₁∈pfm₁ , k,v₂∈pfm₂))) ← Provenance-union (pop fm₁) (pop fm₂) k',v'∈pfm₁pfm₂
                    with (k≢k' , k',v₁∈fm₁) ← k,v∈pop⇒k,v∈ fm₁ k,v₁∈pfm₁
                    with (_    , k',v₂∈fm₂) ← k,v∈pop⇒k,v∈ fm₂ k,v₂∈pfm₂
                    = (v₁ ⊔₂ v₂ , (IsLattice.≈-refl lB , k,v∈⇒k,v∈pop (fm₁ ⊔ᵐ fm₂) k≢k' (⊔-combines {m₁ = m₁} {m₂ = m₂} k',v₁∈fm₁ k',v₂∈fm₂)))

        from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₂ (from fm) ≡ from (pop fm)
        from-rest (((_ ∷ fm') , push _ ufm') , refl) = refl

    from-preserves-≈ : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → fm₁ ≈ᵐ fm₂ → (_≈ⁱᵖ_ {ks}) (from fm₁) (from fm₂)
    from-preserves-≈ {[]} (([] , _) , _) (([] , _) , _) _ = IsEquivalence.≈-refl ≈ᵘ-equiv
    from-preserves-≈ {k ∷ ks'} fm₁@(m₁ , _) fm₂@(m₂ , _) fm₁≈fm₂@(kvs₁⊆kvs₂ , kvs₂⊆kvs₁)
        with first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value fm₁ | from-first-value fm₂
    ...   | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl
            with kvs₁⊆kvs₂ _ _ k,v₁∈fm₁
    ...       | (v₁' , (v₁≈v₁' , k,v₁'∈fm₂))
                rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₁'∈fm₂
                rewrite from-rest fm₁ rewrite from-rest fm₂
                = (v₁≈v₁' , from-preserves-≈ (pop fm₁) (pop fm₂) (pop-≈ fm₁ fm₂ fm₁≈fm₂))

    to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) (ip₁ ip₂ : IterProd (length ks)) → _≈ⁱᵖ_ {ks} ip₁ ip₂ → to uks ip₁ ≈ᵐ to uks ip₂
    to-preserves-≈ {[]} empty tt tt _ = ((λ k v ()), (λ k v ()))
    to-preserves-≈ {k ∷ ks'} uks@(push k≢ks' uks') ip₁@(v₁ , rest₁) ip₂@(v₂ , rest₂) (v₁≈v₂ , rest₁≈rest₂) = (fm₁⊆fm₂ , fm₂⊆fm₁)
        where
            fm₁⊆fm₂ : to uks ip₁ ⊆ᵐ to uks ip₂
            fm₁⊆fm₂ k v k,v∈kvs₁
                with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
                with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
                with k,v∈kvs₁
            ...   | here refl = (v₂ , (v₁≈v₂ , here refl))
            ...   | there k,v∈fm'₁ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₁)) = proj₁ (to-preserves-≈ uks' rest₁ rest₂ rest₁≈rest₂) k v k,v∈fm'₁ in (v' , (v≈v' , there k,v'∈kvs₁))

            fm₂⊆fm₁ : to uks ip₂ ⊆ᵐ to uks ip₁
            fm₂⊆fm₁ k v k,v∈kvs₂
                with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
                with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
                with k,v∈kvs₂
            ...   | here refl = (v₁ , (IsLattice.≈-sym lB v₁≈v₂ , here refl))
            ...   | there k,v∈fm'₂ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₂)) = proj₂ (to-preserves-≈ uks' rest₁ rest₂ rest₁≈rest₂) k v k,v∈fm'₂ in (v' , (v≈v' , there k,v'∈kvs₂))

    from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → _≈ⁱᵖ_ {ks} (from (fm₁ ⊔ᵐ fm₂)) (_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
    from-⊔-distr {[]} fm₁ fm₂ = IsEquivalence.≈-refl ≈ᵘ-equiv
    from-⊔-distr {k ∷ ks} fm₁@(m₁ , _) fm₂@(m₂ , _)
        with first-key-in-map (fm₁ ⊔ᵐ fm₂) | first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value (fm₁ ⊔ᵐ fm₂) | from-first-value fm₁ | from-first-value fm₂
    ...   | (v , k,v∈fm₁fm₂) | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl | refl
            with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
    ...       | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget {m = m₂} k,v₂∈fm₂))
    ...       | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget {m = m₁} k,v₁∈fm₁))
    ...       | (v₁⊔v₂ , (bothᵘ {v₁'} {v₂'} (single k,v₁'∈m₁) (single k,v₂'∈m₂) , k,v₁⊔v₂∈m₁m₂))
                rewrite Map-functional {m = m₁} k,v₁∈fm₁ k,v₁'∈m₁
                rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₂'∈m₂
                rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂
                rewrite from-rest (fm₁ ⊔ᵐ fm₂) rewrite from-rest fm₁ rewrite from-rest fm₂
                = ( IsLattice.≈-refl lB
                  , IsEquivalence.≈-trans (IP.≈-equiv (length ks)) (from-preserves-≈ (pop (fm₁ ⊔ᵐ fm₂)) (pop fm₁ ⊔ᵐ pop fm₂) (pop-⊔-distr fm₁ fm₂)) ((from-⊔-distr (pop fm₁) (pop fm₂)))
                  )


    -- Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) (k : A) (v : B) → (k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → Σ (B × B) (λ (v₁ , v₂) → ((v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
    
    to-⊔-distr : ∀ {ks : List A} (uks : Unique ks) → (ip₁ ip₂ : IterProd (length ks)) → to uks (_⊔ⁱᵖ_ {ks} ip₁ ip₂) ≈ᵐ (to uks ip₁ ⊔ᵐ to uks ip₂)
    to-⊔-distr {[]} empty tt tt = ((λ k v ()), (λ k v ()))
    to-⊔-distr {ks@(k ∷ ks')} uks@(push k≢ks' uks') ip₁@(v₁ , rest₁) ip₂@(v₂ , rest₂) = (fm⊆fm₁fm₂ , fm₁fm₂⊆fm)
        where
            fm₁ = to uks ip₁
            fm₁' = to uks' rest₁
            fm₂ = to uks ip₂
            fm₂' = to uks' rest₂
            fm = to uks (_⊔ⁱᵖ_ {k ∷ ks'} ip₁ ip₂)

            fm⊆fm₁fm₂ : fm ⊆ᵐ (fm₁ ⊔ᵐ fm₂)
            fm⊆fm₁fm₂ k v (here refl) =
                (v₁ ⊔₂ v₂
                , (IsLattice.≈-refl lB
                  , ⊔-combines {k} {v₁} {v₂} {proj₁ fm₁} {proj₁ fm₂} (here refl) (here refl)
                  )
                )
            fm⊆fm₁fm₂ k' v (there k',v∈fm')
                with (fm'⊆fm'₁fm'₂ , _) ← to-⊔-distr uks' rest₁ rest₂
                with (v' , (v₁⊔v₂≈v' , k',v'∈fm'₁fm'₂)) ← fm'⊆fm'₁fm'₂ k' v k',v∈fm'
                with (_ , (refl , (v₁∈fm'₁ , v₂∈fm'₂))) ← Provenance-union fm₁' fm₂' k',v'∈fm'₁fm'₂ =
                (v' , (v₁⊔v₂≈v' , ⊔-combines {m₁ = proj₁ fm₁} {m₂ = proj₁ fm₂} (there v₁∈fm'₁) (there v₂∈fm'₂)))

            fm₁fm₂⊆fm : (fm₁ ⊔ᵐ fm₂) ⊆ᵐ fm
            fm₁fm₂⊆fm k' v k',v∈fm₁fm₂
                with (_ , fm'₁fm'₂⊆fm') ← to-⊔-distr uks' rest₁ rest₂
                with (_ , (refl , (v₁∈fm₁ , v₂∈fm₂))) ← Provenance-union fm₁ fm₂ k',v∈fm₁fm₂
                with v₁∈fm₁ | v₂∈fm₂
            ...   | here refl | here refl = (v , (IsLattice.≈-refl lB , here refl))
            ...   | here refl | there k',v₂∈fm₂' = ⊥-elim (All¬-¬Any k≢ks' (subst (λ list → k' ∈ˡ list) (proj₂ fm₂') (forget {m = proj₁ fm₂'} k',v₂∈fm₂')))
            ...   | there k',v₁∈fm₁' | here refl = ⊥-elim (All¬-¬Any k≢ks' (subst (λ list → k' ∈ˡ list) (proj₂ fm₁') (forget {m = proj₁ fm₁'} k',v₁∈fm₁')))
            ...   | there k',v₁∈fm₁' | there k',v₂∈fm₂' =
                        let
                            k',v₁v₂∈fm₁'fm₂' = ⊔-combines {m₁ = proj₁ fm₁'} {m₂ = proj₁ fm₂'} k',v₁∈fm₁' k',v₂∈fm₂'
                            (v' , (v₁⊔v₂≈v' , v'∈fm')) = fm'₁fm'₂⊆fm' _ _ k',v₁v₂∈fm₁'fm₂'
                        in
                            (v' , (v₁⊔v₂≈v' , there v'∈fm'))