agda-spa/Lattice/Builder.agda

module Lattice.Builder where

open import Lattice
open import Equivalence
open import Data.Maybe as Maybe using (Maybe; just; nothing; _>>=_; maybe)
open import Data.Maybe.Properties using (just-injective)
open import Data.Unit using (⊤; tt)
open import Data.List.NonEmpty using (List⁺; tail; toList) renaming (_∷_ to _∷⁺_)
open import Data.List using (List; _∷_; [])
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Data.Product using (Σ; _,_)
open import Data.Empty using (⊥; ⊥-elim)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)

maybe-injection : ∀ {a} {A : Set a} (x : A) → just x ≡ nothing → ⊥
maybe-injection x ()

data lift-≈ {a} {A : Set a} (_≈_ : A → A → Set a) : Maybe A → Maybe A → Set a where
    ≈-just : ∀ {a₁ a₂ : A} →  a₁ ≈ a₂ →  lift-≈ _≈_ (just a₁) (just a₂)
    ≈-nothing : lift-≈ _≈_ nothing nothing

lift-≈-map : ∀ {a b} {A : Set a} {B : Set b} (f : A → B)
               (_≈ᵃ_ : A → A → Set a) (_≈ᵇ_ : B → B → Set b) →
               (∀ a₁ a₂ → a₁ ≈ᵃ a₂ → f a₁ ≈ᵇ f a₂) →
               ∀ m₁ m₂ → lift-≈ _≈ᵃ_ m₁ m₂ → lift-≈ _≈ᵇ_ (Maybe.map f m₁) (Maybe.map f m₂)
lift-≈-map f _≈ᵃ_ _≈ᵇ_ ≈ᵃ→≈ᵇ (just a₁) (just a₂) (≈-just a₁≈a₂) = ≈-just (≈ᵃ→≈ᵇ a₁ a₂ a₁≈a₂)
lift-≈-map f _≈ᵃ_ _≈ᵇ_ ≈ᵃ→≈ᵇ nothing nothing ≈-nothing = ≈-nothing

instance
    lift-≈-Equivalence : ∀ {a} {A : Set a} {_≈_ : A → A → Set a} {{isEquivalence : IsEquivalence A _≈_}} → IsEquivalence (Maybe A) (lift-≈ _≈_)
    lift-≈-Equivalence {{isEquivalence}} = record
        { ≈-trans =
            (λ { (≈-just a₁≈a₂) (≈-just a₂≈a₃) → ≈-just (IsEquivalence.≈-trans isEquivalence a₁≈a₂ a₂≈a₃)
               ; ≈-nothing ≈-nothing → ≈-nothing
               })
        ; ≈-sym =
            (λ { (≈-just a₁≈a₂) → ≈-just (IsEquivalence.≈-sym isEquivalence a₁≈a₂)
               ; ≈-nothing → ≈-nothing
               })
        ; ≈-refl = λ { {just a} → ≈-just (IsEquivalence.≈-refl isEquivalence)
                     ; {nothing} → ≈-nothing
                     }
        }

PartialCong : ∀ {a} {A : Set a} (_≈_ : A → A → Set a) (_⊗_ : A → A → Maybe A) → Set a
PartialCong {a} {A} _≈_ _⊗_ = ∀ {a₁ a₂ a₃ a₄} → a₁ ≈ a₂ → a₃ ≈ a₄ → lift-≈ _≈_ (a₁ ⊗ a₃) (a₂ ⊗ a₄)

PartialAssoc : ∀ {a} {A : Set a} (_≈_ : A → A → Set a) (_⊗_ : A → A → Maybe A) → Set a
PartialAssoc {a} {A} _≈_ _⊗_ = ∀ (x y z : A) →  lift-≈ _≈_ ((x ⊗ y) >>= (_⊗ z)) ((y ⊗ z) >>= (x ⊗_))

PartialComm : ∀ {a} {A : Set a} (_≈_ : A → A → Set a) (_⊗_ : A → A → Maybe A) → Set a
PartialComm {a} {A} _≈_ _⊗_ = ∀ (x y : A) →  lift-≈ _≈_ (x ⊗ y) (y ⊗ x)

PartialIdemp : ∀ {a} {A : Set a} (_≈_ : A → A → Set a) (_⊗_ : A → A → Maybe A) → Set a
PartialIdemp {a} {A} _≈_ _⊗_ = ∀ (x : A) →  lift-≈ _≈_ (x ⊗ x) (just x)

data Trivial a : Set a where
    instance
        mkTrivial : Trivial a

data Empty a : Set a where

record IsPartialSemilattice {a} {A : Set a}
    (_≈_ : A → A → Set a)
    (_⊔?_ : A → A → Maybe A) : Set a where

    _≈?_ : Maybe A → Maybe A → Set a
    _≈?_ = lift-≈ _≈_

    _≼_ : A → A → Set a
    _≼_ x y = (x ⊔? y) ≈? (just y)

    field
        ≈-equiv : IsEquivalence A _≈_
        ≈-⊔-cong : PartialCong _≈_ _⊔?_

        ⊔-assoc : PartialAssoc _≈_ _⊔?_
        ⊔-comm : PartialComm _≈_ _⊔?_
        ⊔-idemp : PartialIdemp _≈_ _⊔?_

    open IsEquivalence ≈-equiv public
    open IsEquivalence (lift-≈-Equivalence {{≈-equiv}}) public using ()
        renaming (≈-trans to ≈?-trans; ≈-sym to ≈?-sym; ≈-refl to ≈?-refl)

    ≈-≼-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≈ a₂ → a₃ ≈ a₄ →  a₁ ≼ a₃ → a₂ ≼ a₄
    ≈-≼-cong {a₁} {a₂} {a₃} {a₄} a₁≈a₂ a₃≈a₄ a₁≼a₃ = ≈?-trans (≈?-trans (≈?-sym (≈-⊔-cong a₁≈a₂ a₃≈a₄)) a₁≼a₃) (≈-just a₃≈a₄)

    -- curious: similar property (properties?) to partial commutative monoids (PCMs)
    -- from Iris.
    ≼-partialˡ : ∀ {a a₁ a₂} → a₁ ≼ a₂ →  (a ⊔? a₁) ≡ nothing → (a ⊔? a₂) ≡ nothing
    ≼-partialˡ {a} {a₁} {a₂} a₁≼a₂ a⊔a₁≡nothing
        with a₁ ⊔? a₂ | a₁≼a₂ | a ⊔? a₁ | a⊔a₁≡nothing | ⊔-assoc a a₁ a₂
    ...   | just a₁⊔a₂ | ≈-just a₁⊔a₂≈a₂ | nothing | refl | aa₁⊔a₂≈a⊔a₁a₂
            with a ⊔? a₁⊔a₂ | aa₁⊔a₂≈a⊔a₁a₂ | a ⊔? a₂ | ≈-⊔-cong (≈-refl {a = a}) a₁⊔a₂≈a₂
    ...       | nothing | ≈-nothing | nothing | ≈-nothing = refl

    ≼-partialʳ : ∀ {a₁ a₂ a} → a₁ ≼ a₂ →  (a₁ ⊔? a) ≡ nothing → (a₂ ⊔? a) ≡ nothing
    ≼-partialʳ {a₁} {a₂} {a} a₁≼a₂ a₁⊔a≡nothing
        with a₁ ⊔? a | a ⊔? a₁ | a₁⊔a≡nothing | ⊔-comm a₁ a | ≼-partialˡ {a} {a₁} {a₂} a₁≼a₂
    ...   | nothing | nothing | refl | ≈-nothing | refl⇒a⊔a₂=nothing
            with a₂ ⊔? a | a ⊔? a₂ | ⊔-comm a₂ a | refl⇒a⊔a₂=nothing refl
    ...       | nothing | nothing | ≈-nothing | refl = refl

    ≼-refl : ∀ (x : A) →  x ≼ x
    ≼-refl x = ⊔-idemp x

    ≼-refl' : ∀ {a₁ a₂ : A} → a₁ ≈ a₂ → a₁ ≼ a₂
    ≼-refl' {a₁} {a₂} a₁≈a₂ = ≈-≼-cong ≈-refl a₁≈a₂ (≼-refl a₁)

    x⊔?x≼x : ∀ (x : A) →  maybe (λ x⊔x →  x⊔x ≼ x) (Empty a) (x ⊔? x)
    x⊔?x≼x x
        with x ⊔? x | ⊔-idemp x
    ...   | just x⊔x | ≈-just x⊔x≈x = ≈-≼-cong (≈-sym x⊔x≈x) ≈-refl (≼-refl x)

    x≼x⊔?y : ∀ (x y : A) →  maybe (λ x⊔y →  x ≼ x⊔y) (Trivial a) (x ⊔? y)
    x≼x⊔?y x y
        with x ⊔? y in x⊔?y= | x ⊔? x | ⊔-idemp x | ⊔-assoc x x y | x⊔?x≼x x
    ...   | nothing  | _ | _ | _ | _ = mkTrivial
    ...   | just x⊔y | just x⊔x | ≈-just x⊔x≈x | ⊔-assoc-xxy | x⊔x≼x
            with x⊔x ⊔? y in xx⊔?y= | x ⊔? x⊔y
    ...       | nothing | nothing =
                ⊥-elim (maybe-injection _ (trans (sym x⊔?y=) (≼-partialʳ x⊔x≼x xx⊔?y=)))
    ...       | just xx⊔y | just x⊔xy rewrite (sym xx⊔?y=) rewrite (sym x⊔?y=) =
                ≈?-trans (≈?-sym ⊔-assoc-xxy) (≈-⊔-cong x⊔x≈x (≈-refl {a = y}))

    y≼x⊔?y : ∀ (x y : A) →  maybe (λ x⊔y →  y ≼ x⊔y) (Trivial a) (x ⊔? y)
    y≼x⊔?y x y
        with x ⊔? y | y ⊔? x | ⊔-comm y x | x≼x⊔?y y x
    ...   | nothing | nothing | ≈-nothing | _ = mkTrivial
    ...   | just x⊔y | just y⊔x | ≈-just y⊔x≈x⊔y | y⊔yx≈yx
          = ≈?-trans (≈?-sym (≈-⊔-cong (≈-refl {a = y}) y⊔x≈x⊔y))
                     (≈?-trans y⊔yx≈yx (≈-just y⊔x≈x⊔y))

    record HasAbsorbingElement : Set a where
        field
            x : A
            x-absorbˡ : (a : A) → (x ⊔? a) ≈? just x

        x-absorbʳ : (a : A) → (a ⊔? x) ≈? just x
        x-absorbʳ a = ≈?-trans (⊔-comm a x) (x-absorbˡ a)

        not-partial : ∀ (a₁ a₂ : A) →  Σ A (λ a₃ →  (a₁ ⊔? a₂) ≡ just a₃)
        not-partial a₁ a₂
            with a₁ ⊔? a₂ | ≼-partialˡ {a = a₁} (x-absorbʳ a₂)
        ...   | just a₁⊔a₂ | _ = (a₁⊔a₂ , refl)
        ...   | nothing | refl⇒a₁⊔?x=nothing
                with a₁ ⊔? x | refl⇒a₁⊔?x=nothing refl | x-absorbʳ a₁
        ...       | nothing | refl | ()

record IsPartialLattice {a} {A : Set a}
    (_≈_ : A → A → Set a)
    (_⊔?_ : A → A → Maybe A)
    (_⊓?_ : A → A → Maybe A) : Set a where

    field
        {{partialJoinSemilattice}} : IsPartialSemilattice _≈_ _⊔?_
        {{partialMeetSemilattice}} : IsPartialSemilattice _≈_ _⊓?_

        absorb-⊔-⊓ : (x y : A) →  maybe (λ x⊓y → lift-≈ _≈_ (x ⊔? x⊓y) (just x)) (Trivial _) (x ⊓? y)
        absorb-⊓-⊔ : (x y : A) →  maybe (λ x⊔y → lift-≈ _≈_ (x ⊓? x⊔y) (just x)) (Trivial _) (x ⊔? y)

    open IsPartialSemilattice partialJoinSemilattice
        renaming (HasAbsorbingElement to HasGreatestElement)
        public
    open IsPartialSemilattice partialMeetSemilattice using ()
        renaming
            ( HasAbsorbingElement to HasLeastElement
            ; ≈-⊔-cong to ≈-⊓-cong
            ; ⊔-assoc to ⊓-assoc
            ; ⊔-comm to ⊓-comm
            ; ⊔-idemp to ⊓-idemp
            ; _≼_ to _≽_
            ; ≈-≼-cong to ≈-≽-cong
            ; ≼-partialˡ to ≽-partialˡ
            ; ≼-partialʳ to ≽-partialʳ
            ; ≼-refl to ≽-refl
            ; ≼-refl' to ≽-refl'
            ; x⊔?x≼x to x⊓?x≽x
            ; x≼x⊔?y to x≽x⊓?y
            ; y≼x⊔?y to y≽x⊓?y
            )
        public

record PartialLattice {a} (A : Set a) : Set (lsuc a) where
    field
        _≈_ : A → A → Set a
        _⊔?_ : A → A → Maybe A
        _⊓?_ : A → A → Maybe A

        {{isPartialLattice}} : IsPartialLattice _≈_ _⊔?_ _⊓?_

    open IsPartialLattice isPartialLattice public

    greatest-⊓-identʳ : ∀ (le : HasGreatestElement) (x : A) →
        (x ⊓? HasGreatestElement.x le) ≈? (just x)
    greatest-⊓-identʳ le x
        with x ⊔? (HasGreatestElement.x le)  | HasGreatestElement.x-absorbʳ le x | absorb-⊓-⊔ x (HasGreatestElement.x le) 
    ...   | just x⊔greatest | ≈-just x⊔greatest≈greatest | x⊓xgreatest≈?x = ≈?-trans (≈?-sym (≈-⊓-cong (≈-refl {a = x}) x⊔greatest≈greatest)) x⊓xgreatest≈?x

    greatest-⊓-identˡ : ∀ (le : HasGreatestElement) (x : A) →
        (HasGreatestElement.x le ⊓? x) ≈? (just x)
    greatest-⊓-identˡ le x = ≈?-trans (⊓-comm (HasGreatestElement.x le) x) (greatest-⊓-identʳ le x)

    least-⊔-identʳ : ∀ (le : HasLeastElement) (x : A) →
        (x ⊔? HasLeastElement.x le) ≈? (just x)
    least-⊔-identʳ le x
        with x ⊓? (HasLeastElement.x le)  | HasLeastElement.x-absorbʳ le x | absorb-⊔-⊓ x (HasLeastElement.x le) 
    ...   | just x⊓least | ≈-just x⊓least≈least | x⊔xleast≈?x = ≈?-trans (≈?-sym (≈-⊔-cong (≈-refl {a = x}) x⊓least≈least)) x⊔xleast≈?x

    least-⊔-identˡ : ∀ (le : HasLeastElement) (x : A) →
        (HasLeastElement.x le ⊔? x) ≈? (just x)
    least-⊔-identˡ le x = ≈?-trans (⊔-comm (HasLeastElement.x le) x) (least-⊔-identʳ le x)

record PartialLatticeType (a : Level) : Set (lsuc a) where
    field
        EltType : Set a
        {{partialLattice}} : PartialLattice EltType

    open PartialLattice partialLattice public

Layer : (a : Level) → Set (lsuc a)
Layer a = List⁺ (PartialLatticeType a)

data LayerLeast {a} : Layer a → Set (lsuc a) where
    instance
        MkLayerLeast : {T : Set a}
                       {{ partialLattice : PartialLattice T }}
                       {{ hasLeast : PartialLattice.HasLeastElement partialLattice }} →
                       LayerLeast (record { EltType = T; partialLattice = partialLattice } ∷⁺ [])

data LayerGreatest {a} : Layer a → Set (lsuc a) where
    instance
        MkLayerGreatest : {T : Set a}
                       {{ partialLattice : PartialLattice T }}
                       {{ hasGreatest : PartialLattice.HasGreatestElement partialLattice }} →
                       LayerGreatest (record { EltType = T; partialLattice = partialLattice } ∷⁺ [])

interleaved mutual
    data Layers a : Set (lsuc a)
    head : ∀ {a} → Layers a → Layer a

    data Layers where
        single : Layer a → Layers a
        add-via-least : (l : Layer a) {{ least : LayerLeast l }} (ls : Layers a) → Layers a
        add-via-greatest : (l : Layer a) (ls : Layers a) {{ greatest : LayerGreatest (head ls) }} → Layers a

    head (single l) = l
    head (add-via-greatest l _) = l
    head (add-via-least l _) = l

ListValue : ∀ {a} → List (PartialLatticeType a) → Set a
ListValue [] = Empty _
ListValue (plt ∷ plts) = PartialLatticeType.EltType plt ⊎ ListValue plts

-- Define LayerValue and Path' as functions to avoid increasing the universe level.
-- Path' in particular needs to be at the same level as the other lattices
-- in the builder to ensure composability.

LayerValue : ∀ {a} →  Layer a → Set a
LayerValue l⁺ = ListValue (toList l⁺)

Path' : ∀ {a} → Layers a → Set a
Path' (single l) = LayerValue l
Path' (add-via-least l ls) = LayerValue l ⊎ Path' ls
Path' (add-via-greatest l ls) = LayerValue l ⊎ Path' ls

record Path {a} (Ls : Layers a) : Set a where
    constructor MkPath
    field path : Path' Ls

valueAtHead : ∀ {a} (ls : Layers a) (lv : LayerValue (head ls)) → Path' ls
valueAtHead (single _) lv = lv
valueAtHead (add-via-least _ _) lv = inj₁ lv
valueAtHead (add-via-greatest _ _) lv = inj₁ lv

CombineForPLT : ∀ a → Set (lsuc a)
CombineForPLT a = ∀ (plt : PartialLatticeType a) → PartialLatticeType.EltType plt → PartialLatticeType.EltType plt → Maybe (PartialLatticeType.EltType plt)

lvCombine : ∀ {a} (f : CombineForPLT a) (l : List (PartialLatticeType a)) (lv₁ lv₂ : ListValue l) →  Maybe (ListValue l)
lvCombine _ [] _ _ = nothing
lvCombine f (plt ∷ plts) (inj₁ v₁) (inj₁ v₂) = Maybe.map inj₁ (f plt v₁ v₂)
lvCombine f (_ ∷ plts) (inj₂ lv₁) (inj₂ lv₂) = Maybe.map inj₂ (lvCombine f plts lv₁ lv₂)
lvCombine _ (_ ∷ plts) (inj₁ _) (inj₂ _) = nothing
lvCombine _ (_ ∷ plts) (inj₂ _) (inj₁ _) = nothing

lvJoin : ∀ {a} (l : List (PartialLatticeType a)) (lv₁ lv₂ : ListValue l) →  Maybe (ListValue l)
lvJoin = lvCombine PartialLatticeType._⊔?_

lvMeet : ∀ {a} (l : List (PartialLatticeType a)) (lv₁ lv₂ : ListValue l) →  Maybe (ListValue l)
lvMeet = lvCombine PartialLatticeType._⊓?_

pathJoin' : ∀ {a} (Ls : Layers a) (p₁ p₂ : Path' Ls) →  Maybe (Path' Ls)
pathJoin' (single l) lv₁ lv₂ = lvJoin (toList l) lv₁ lv₂
pathJoin' (add-via-least l ls) (inj₁ lv₁) (inj₁ lv₂) = Maybe.map inj₁ (lvJoin (toList l) lv₁ lv₂)
pathJoin' (add-via-least l ls) (inj₁ lv₁) (inj₂ p₂) = just (inj₁ lv₁)
pathJoin' (add-via-least l ls) (inj₂ p₁) (inj₁ lv₂) = just (inj₁ lv₂)
pathJoin' (add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls) (inj₂ p₁) (inj₂ p₂)
    with pathJoin' ls p₁ p₂
...   | just p = just (inj₂ p)
...   | nothing = just (inj₁ (inj₁ (IsPartialSemilattice.HasAbsorbingElement.x hasLeast)))
pathJoin' (add-via-greatest l ls) (inj₁ lv₁) (inj₁ lv₂) = Maybe.map inj₁ (lvJoin (toList l) lv₁ lv₂)
pathJoin' (add-via-greatest l ls) (inj₁ lv₁) (inj₂ p₂) = just (inj₁ lv₁)
pathJoin' (add-via-greatest l ls) (inj₂ p₁) (inj₁ lv₂) = just (inj₁ lv₂)
pathJoin' (add-via-greatest l ls {{greatest}}) (inj₂ p₁) (inj₂ p₂) = Maybe.map inj₂ (pathJoin' ls p₁ p₂) -- not our job to provide the least element here. If there was a "greatest" there, recursion would've found it (?)

greatestPath : ∀ {a} (Ls : Layers a) (greatest : LayerGreatest (head Ls)) → Path' Ls
greatestPath Ls greatest
    with head Ls | greatest | valueAtHead Ls
...   | _ | MkLayerGreatest {{hasGreatest = hasGreatest}} | vah
      = vah (inj₁ (IsPartialSemilattice.HasAbsorbingElement.x hasGreatest))

pathMeet' : ∀ {a} (Ls : Layers a) (p₁ p₂ : Path' Ls) →  Maybe (Path' Ls)
pathMeet' (single l) lv₁ lv₂ = lvMeet (toList l) lv₁ lv₂
pathMeet' (add-via-least l ls) (inj₁ lv₁) (inj₁ lv₂) = Maybe.map inj₁ (lvMeet (toList l) lv₁ lv₂)
pathMeet' (add-via-least l ls) (inj₁ lv₁) (inj₂ p₂) = just (inj₂ p₂)
pathMeet' (add-via-least l ls) (inj₂ p₁) (inj₁ lv₂) = just (inj₂ p₁)
pathMeet' (add-via-least l ls) (inj₂ p₁) (inj₂ p₂) = Maybe.map inj₂ (pathMeet' ls p₁ p₂)
pathMeet' (add-via-greatest l ls {{greatest}}) (inj₁ lv₁) (inj₁ lv₂)
    with lvMeet (toList l) lv₁ lv₂
...   | just lv = just (inj₁ lv)
...   | nothing = just (inj₂ (greatestPath ls greatest))
pathMeet' (add-via-greatest l ls) (inj₁ lv₁) (inj₂ p₂) = just (inj₂ p₂)
pathMeet' (add-via-greatest l ls) (inj₂ p₁) (inj₁ lv₂) = just (inj₂ p₁)
pathMeet' (add-via-greatest l ls) (inj₂ p₁) (inj₂ p₂) = Maybe.map inj₂ (pathMeet' ls p₁ p₂)

eqL : ∀ {a} (L : List (PartialLatticeType a)) →  ListValue L → ListValue L → Set a
eqL [] ()
eqL (plt ∷ plts) (inj₁ v₁ ) (inj₁ v₂) = PartialLattice._≈_ (PartialLatticeType.partialLattice plt) v₁ v₂
eqL (plt ∷ plts) (inj₂ v₁ ) (inj₂ v₂) = eqL plts v₁ v₂
eqL (plt ∷ plts) (inj₁ _) (inj₂ _) = Empty _
eqL (plt ∷ plts) (inj₂ _) (inj₁ _) = Empty _

eqL-trans : ∀ {a} (L : List (PartialLatticeType a)) {lv₁ lv₂ lv₃ : ListValue L} →  eqL L lv₁ lv₂ → eqL L lv₂ lv₃ → eqL L lv₁ lv₃
eqL-trans [] {()}
eqL-trans (plt ∷ plts) {inj₁ v₁} {inj₁ v₂} {inj₁ v₃} v₁≈v₂ v₂≈v₃ = PartialLatticeType.≈-trans plt v₁≈v₂ v₂≈v₃
eqL-trans (plt ∷ plts) {inj₂ lv₁} {inj₂ lv₂} {inj₂ lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqL-trans plts lv₁≈lv₂ lv₂≈lv₃

eqL-sym : ∀ {a} (L : List (PartialLatticeType a)) {lv₁ lv₂ : ListValue L} →  eqL L lv₁ lv₂ → eqL L lv₂ lv₁
eqL-sym [] {()}
eqL-sym (plt ∷ plts) {inj₁ v₁} {inj₁ v₂} v₁≈v₂ = PartialLatticeType.≈-sym plt v₁≈v₂
eqL-sym (plt ∷ plts) {inj₂ lv₁} {inj₂ lv₂} lv₁≈lv₂ = eqL-sym plts lv₁≈lv₂

eqL-refl : ∀ {a} (L : List (PartialLatticeType a)) (lv : ListValue L) →  eqL L lv lv
eqL-refl [] ()
eqL-refl (plt ∷ plts) (inj₁ v) = IsEquivalence.≈-refl (PartialLatticeType.≈-equiv plt)
eqL-refl (plt ∷ plts) (inj₂ v) = eqL-refl plts v

instance
    eqL-Equivalence : ∀ {a} {L : List (PartialLatticeType a)} → IsEquivalence (ListValue L) (eqL L)
    eqL-Equivalence {L = L} = record
        { ≈-trans = eqL-trans L
        ; ≈-sym = eqL-sym L
        ; ≈-refl = λ {lv} → eqL-refl L lv
        }

eqL-lvCombine-cong : ∀ {a} (f : CombineForPLT a) →
                     (∀ plt →  PartialCong (PartialLatticeType._≈_ plt) (f plt)) →
                     ∀ (L : List (PartialLatticeType a)) →
                     PartialCong (eqL L) (lvCombine f L)
eqL-lvCombine-cong f f-Cong [] {()}
eqL-lvCombine-cong f f-Cong (plt ∷ plts) {inj₁ v₁} {inj₁ v₂} {inj₁ v₃} {inj₁ v₄} v₁≈v₂ v₃≈v₄
    with f plt v₁ v₃ | f plt v₂ v₄ | f-Cong plt v₁≈v₂ v₃≈v₄
...   | just _ | just _ | ≈-just v₁⊗v₃≈v₂⊗v₄ = ≈-just v₁⊗v₃≈v₂⊗v₄
...   | nothing | nothing | ≈-nothing = ≈-nothing
eqL-lvCombine-cong f f-Cong (plt ∷ plts) {inj₂ lv₁} {inj₂ lv₂} {inj₁ v₃} {inj₁ v₄} _ _ = ≈-nothing
eqL-lvCombine-cong f f-Cong (plt ∷ plts) {inj₁ v₁} {inj₁ v₂} {inj₂ lv₃} {inj₂ lv₄} _ _ = ≈-nothing
eqL-lvCombine-cong f f-Cong (plt ∷ plts) {inj₂ lv₁} {inj₂ lv₂} {inj₂ lv₃} {inj₂ lv₄} lv₁≈lv₂ lv₃≈lv₄
    with lvCombine f plts lv₁ lv₃ | lvCombine f plts lv₂ lv₄ | eqL-lvCombine-cong f f-Cong plts {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
...   | just _ | just _ | ≈-just lv₁⊗v₃≈v₂⊗v₄ = ≈-just lv₁⊗v₃≈v₂⊗v₄
...   | nothing | nothing | ≈-nothing = ≈-nothing

eqLv : ∀ {a} (L : Layer a) →  LayerValue L → LayerValue L → Set a
eqLv L = eqL (toList L)

eqLv-trans : ∀ {a} (L : Layer a) →  {lv₁ lv₂ lv₃ : LayerValue L} → eqLv L lv₁ lv₂ → eqLv L lv₂ lv₃ → eqLv L lv₁ lv₃
eqLv-trans L {lv₁} {lv₂} {lv₃} = eqL-trans (toList L) {lv₁} {lv₂} {lv₃}

eqLv-sym : ∀ {a} (L : Layer a) →  {lv₁ lv₂ : LayerValue L} → eqLv L lv₁ lv₂ → eqLv L lv₂ lv₁
eqLv-sym L {lv₁} {lv₂} = eqL-sym (toList L) {lv₁} {lv₂}

eqLv-refl : ∀ {a} (L : Layer a) →  (lv : LayerValue L) → eqLv L lv lv
eqLv-refl L = eqL-refl (toList L)

instance
    eqLv-Equivalence : ∀ {a} {L : Layer a} → IsEquivalence (LayerValue L) (eqLv L)
    eqLv-Equivalence {L = L} = record
        { ≈-trans = λ {lv₁} {lv₂} {lv₃} → eqLv-trans L {lv₁} {lv₂} {lv₃}
        ; ≈-sym = λ {lv₁} {lv₂} → eqLv-sym L {lv₁} {lv₂}
        ; ≈-refl = λ {lv} → eqLv-refl L lv
        }

eqLv-lvCombine-cong : ∀ {a} (f : CombineForPLT a) →
                      (∀ plt →  PartialCong (PartialLatticeType._≈_ plt) (f plt)) →
                      ∀ (L : Layer a) →
                      PartialCong (eqLv L) (lvCombine f (toList L))
eqLv-lvCombine-cong f f-Cong L {lv₁} {lv₂} {lv₃} {lv₄} = eqL-lvCombine-cong f f-Cong (toList L) {lv₁} {lv₂} {lv₃} {lv₄}

eqLv-lvJoin-cong : ∀ {a} (L : Layer a) → PartialCong (eqLv L) (lvJoin (toList L))
eqLv-lvJoin-cong = eqLv-lvCombine-cong PartialLatticeType._⊔?_ PartialLatticeType.≈-⊔-cong

eqLv-lvMeet-cong : ∀ {a} (L : Layer a) → PartialCong (eqLv L) (lvMeet (toList L))
eqLv-lvMeet-cong = eqLv-lvCombine-cong PartialLatticeType._⊓?_ PartialLatticeType.≈-⊓-cong

eqPath' : ∀ {a} (Ls : Layers a) → Path' Ls → Path' Ls → Set a
eqPath' (single l) lv₁ lv₂ = eqLv l lv₁ lv₂
eqPath' (add-via-least l ls) (inj₁ lv₁) (inj₁ lv₂) = eqLv l lv₁ lv₂
eqPath' (add-via-least l ls) (inj₂ p₁) (inj₂ p₂) = eqPath' ls p₁ p₂
eqPath' (add-via-least l ls) (inj₁ lv₁) (inj₂ p₂) = Empty _
eqPath' (add-via-least l ls) (inj₂ p₁) (inj₁ lv₂) = Empty _
eqPath' (add-via-greatest l ls) (inj₁ lv₁) (inj₁ lv₂) = eqLv l lv₁ lv₂
eqPath' (add-via-greatest l ls) (inj₂ p₁) (inj₂ p₂) = eqPath' ls p₁ p₂
eqPath' (add-via-greatest l ls) (inj₁ lv₁) (inj₂ p₂) = Empty _
eqPath' (add-via-greatest l ls) (inj₂ p₁) (inj₁ lv₂) = Empty _

eqPath'-trans : ∀ {a} (Ls : Layers a) →  {p₁ p₂ p₃ : Path' Ls} →  eqPath' Ls p₁ p₂ → eqPath' Ls p₂ p₃ → eqPath' Ls p₁ p₃
eqPath'-trans (single l) {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqLv-trans l {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃
eqPath'-trans (add-via-least l ls) {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqLv-trans l {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃
eqPath'-trans (add-via-least l ls) {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} p₁≈p₂ p₂≈p₃ = eqPath'-trans ls p₁≈p₂ p₂≈p₃
eqPath'-trans (add-via-greatest l ls) {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} lv₁≈lv₂ lv₂≈lv₃ = eqLv-trans l {lv₁} {lv₂} {lv₃} lv₁≈lv₂ lv₂≈lv₃
eqPath'-trans (add-via-greatest l ls) {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} p₁≈p₂ p₂≈p₃ = eqPath'-trans ls p₁≈p₂ p₂≈p₃

eqPath'-sym : ∀ {a} (Ls : Layers a) →  {p₁ p₂ : Path' Ls} →  eqPath' Ls p₁ p₂ → eqPath' Ls p₂ p₁
eqPath'-sym (single l) {lv₁} {lv₂} lv₁≈lv₂ = eqLv-sym l {lv₁} {lv₂} lv₁≈lv₂
eqPath'-sym (add-via-least l ls) {inj₁ lv₁} {inj₁ lv₂}lv₁≈lv₂ = eqLv-sym l {lv₁} {lv₂} lv₁≈lv₂
eqPath'-sym (add-via-least l ls) {inj₂ p₁} {inj₂ p₂} p₁≈p₂ = eqPath'-sym ls p₁≈p₂
eqPath'-sym (add-via-greatest l ls) {inj₁ lv₁} {inj₁ lv₂}lv₁≈lv₂ = eqLv-sym l {lv₁} {lv₂} lv₁≈lv₂
eqPath'-sym (add-via-greatest l ls) {inj₂ p₁} {inj₂ p₂} p₁≈p₂ = eqPath'-sym ls p₁≈p₂

eqPath'-refl : ∀ {a} (Ls : Layers a) →  (p : Path' Ls) →  eqPath' Ls p p
eqPath'-refl (single l) lv = eqLv-refl l lv
eqPath'-refl (add-via-least l ls) (inj₁ lv) = eqLv-refl l lv
eqPath'-refl (add-via-least l ls) (inj₂ p) = eqPath'-refl ls p
eqPath'-refl (add-via-greatest l ls) (inj₁ lv) = eqLv-refl l lv
eqPath'-refl (add-via-greatest l ls) (inj₂ p) = eqPath'-refl ls p

instance
    eqPath'-Equivalence : ∀ {a} {Ls : Layers a} → IsEquivalence (Path' Ls) (eqPath' Ls)
    eqPath'-Equivalence {Ls = Ls} = record
        { ≈-trans = eqPath'-trans Ls
        ; ≈-sym = eqPath'-sym Ls
        ; ≈-refl = λ {x} → eqPath'-refl Ls x
        }

eqPath'-pathJoin'-cong : ∀ {a} {Ls : Layers a} {a₁ a₂ a₃ a₄} →  eqPath' Ls a₁ a₂ →  eqPath' Ls a₃ a₄ →  lift-≈ (eqPath' Ls) (pathJoin' Ls a₁ a₃) (pathJoin' Ls a₂ a₄)
eqPath'-pathJoin'-cong {Ls = single l} {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄ = eqLv-lvJoin-cong l {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
eqPath'-pathJoin'-cong {Ls = add-via-least l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} {inj₁ lv₄} lv₁≈lv₂ lv₃≈lv₄
    with lvJoin (toList l) lv₁ lv₃ | lvJoin (toList l) lv₂ lv₄ | eqLv-lvJoin-cong l {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
...   | just _ | just _ | ≈-just lv₁⊔lv₃≈lv₂⊔lv₄ = ≈-just lv₁⊔lv₃≈lv₂⊔lv₄
...   | nothing | nothing | ≈-nothing = ≈-nothing
eqPath'-pathJoin'-cong {Ls = add-via-least l ls} {inj₂ p₁} {inj₂ p₂} {inj₁ lv₃} {inj₁ lv₄} p₁≈p₂ lv₃≈lv₄ = ≈-just lv₃≈lv₄
eqPath'-pathJoin'-cong {Ls = add-via-least l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₂ p₃} {inj₂ p₄} lv₁≈lv₂ p₃≈p₄ = ≈-just lv₁≈lv₂
eqPath'-pathJoin'-cong {Ls = Ls@(add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls)} {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} {inj₂ p₄} p₁≈p₂ p₃≈p₄
    with pathJoin' ls p₁ p₃ | pathJoin' ls p₂ p₄ | eqPath'-pathJoin'-cong {Ls = ls} {p₁} {p₂} {p₃} {p₄} p₁≈p₂ p₃≈p₄
...   | just p₁⊔p₃ | just p₂⊔p₄ | ≈-just p₁⊔p₃≈p₂⊔p₄ = ≈-just p₁⊔p₃≈p₂⊔p₄
...   | nothing | nothing | ≈-nothing = ≈-just (eqPath'-refl Ls (inj₁ (inj₁ (PartialLatticeType.HasLeastElement.x {_} {plt} hasLeast))))
eqPath'-pathJoin'-cong {Ls = add-via-greatest l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} {inj₁ lv₄} lv₁≈lv₂ lv₃≈lv₄
    with lvJoin (toList l) lv₁ lv₃ | lvJoin (toList l) lv₂ lv₄ | eqLv-lvJoin-cong l {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
...   | just _ | just _ | ≈-just lv₁⊔lv₃≈lv₂⊔lv₄ = ≈-just lv₁⊔lv₃≈lv₂⊔lv₄
...   | nothing | nothing | ≈-nothing = ≈-nothing
eqPath'-pathJoin'-cong {Ls = add-via-greatest l ls} {inj₂ p₁} {inj₂ p₂} {inj₁ lv₃} {inj₁ lv₄} p₁≈p₂ lv₃≈lv₄ = ≈-just lv₃≈lv₄
eqPath'-pathJoin'-cong {Ls = add-via-greatest l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₂ p₃} {inj₂ p₄} lv₁≈lv₂ p₃≈p₄ = ≈-just lv₁≈lv₂
eqPath'-pathJoin'-cong {Ls = add-via-greatest l ls {{greatest}}} {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} {inj₂ p₄} p₁≈p₂ p₃≈p₄
    with pathJoin' ls p₁ p₃ | pathJoin' ls p₂ p₄ | eqPath'-pathJoin'-cong {Ls = ls} {p₁} {p₂} {p₃} {p₄} p₁≈p₂ p₃≈p₄
...   | just p₁⊔p₃ | just p₂⊔p₄ | ≈-just p₁⊔p₃≈p₂⊔p₄ = ≈-just p₁⊔p₃≈p₂⊔p₄
...   | nothing | nothing | ≈-nothing = ≈-nothing

eqPath'-pathMeet'-cong : ∀ {a} {Ls : Layers a} {a₁ a₂ a₃ a₄} →  eqPath' Ls a₁ a₂ →  eqPath' Ls a₃ a₄ →  lift-≈ (eqPath' Ls) (pathMeet' Ls a₁ a₃) (pathMeet' Ls a₂ a₄)
eqPath'-pathMeet'-cong {Ls = single l} {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄ = eqLv-lvMeet-cong l {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
eqPath'-pathMeet'-cong {Ls = add-via-least l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} {inj₁ lv₄} lv₁≈lv₂ lv₃≈lv₄
    with lvMeet (toList l) lv₁ lv₃ | lvMeet (toList l) lv₂ lv₄ | eqLv-lvMeet-cong l {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
...   | just _ | just _ | ≈-just lv₁⊓lv₃≈lv₂⊓lv₄ = ≈-just lv₁⊓lv₃≈lv₂⊓lv₄
...   | nothing | nothing | ≈-nothing = ≈-nothing
eqPath'-pathMeet'-cong {Ls = add-via-least l ls} {inj₂ p₁} {inj₂ p₂} {inj₁ lv₃} {inj₁ lv₄} p₁≈p₂ lv₃≈lv₄ = ≈-just p₁≈p₂
eqPath'-pathMeet'-cong {Ls = add-via-least l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₂ p₃} {inj₂ p₄} lv₁≈lv₂ p₃≈p₄ = ≈-just p₃≈p₄
eqPath'-pathMeet'-cong {Ls = add-via-least l {{least}}  ls} {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} {inj₂ p₄} p₁≈p₂ p₃≈p₄
    with pathMeet' ls p₁ p₃ | pathMeet' ls p₂ p₄ | eqPath'-pathMeet'-cong {Ls = ls} {p₁} {p₂} {p₃} {p₄} p₁≈p₂ p₃≈p₄
...   | just p₁⊓p₃ | just p₂⊓p₄ | ≈-just p₁⊓p₃≈p₂⊓p₄ = ≈-just p₁⊓p₃≈p₂⊓p₄
...   | nothing | nothing | ≈-nothing = ≈-nothing
eqPath'-pathMeet'-cong {Ls = add-via-greatest l ls {{greatest}}} {inj₁ lv₁} {inj₁ lv₂} {inj₁ lv₃} {inj₁ lv₄} lv₁≈lv₂ lv₃≈lv₄
    with lvMeet (toList l) lv₁ lv₃ | lvMeet (toList l) lv₂ lv₄ | eqLv-lvMeet-cong l {lv₁} {lv₂} {lv₃} {lv₄} lv₁≈lv₂ lv₃≈lv₄
...   | just _ | just _ | ≈-just lv₁⊓lv₃≈lv₂⊓lv₄ = ≈-just lv₁⊓lv₃≈lv₂⊓lv₄
...   | nothing | nothing | ≈-nothing = ≈-just (eqPath'-refl ls (greatestPath ls greatest))
eqPath'-pathMeet'-cong {Ls = add-via-greatest l ls} {inj₂ p₁} {inj₂ p₂} {inj₁ lv₃} {inj₁ lv₄} p₁≈p₂ lv₃≈lv₄ = ≈-just p₁≈p₂
eqPath'-pathMeet'-cong {Ls = add-via-greatest l ls} {inj₁ lv₁} {inj₁ lv₂} {inj₂ p₃} {inj₂ p₄} lv₁≈lv₂ p₃≈p₄ = ≈-just p₃≈p₄
eqPath'-pathMeet'-cong {Ls = add-via-greatest l ls {{greatest}}} {inj₂ p₁} {inj₂ p₂} {inj₂ p₃} {inj₂ p₄} p₁≈p₂ p₃≈p₄
    with pathMeet' ls p₁ p₃ | pathMeet' ls p₂ p₄ | eqPath'-pathMeet'-cong {Ls = ls} {p₁} {p₂} {p₃} {p₄} p₁≈p₂ p₃≈p₄
...   | just p₁⊓p₃ | just p₂⊓p₄ | ≈-just p₁⊓p₃≈p₂⊓p₄ = ≈-just p₁⊓p₃≈p₂⊓p₄
...   | nothing | nothing | ≈-nothing = ≈-nothing

-- Now that we can compare paths for "equality", we can state and
-- prove theorems such as commutativity and idempotence.

data Expr {a} (A : Set a) : Set a where
    `_ : A →  Expr A
    _∨_ : Expr A → Expr A → Expr A

map : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) → Expr A → Expr B
map f (` x) = ` (f x)
map f (e₁ ∨ e₂) = map f e₁ ∨ map f e₂

eval : ∀ {a} {A : Set a} (_⊔?_ : A → A → Maybe A) (e : Expr A) → Maybe A
eval _⊔?_ (` x) = just x
eval _⊔?_ (e₁ ∨ e₂) = (eval _⊔?_ e₁) >>= (λ v₁ →  (eval _⊔?_ e₂) >>= (v₁ ⊔?_))

-- a function is "partially sub-homomorphic" for some subset of B (image of f) if
-- for (f A), it preserves the structure of operations _⊕ on A in B.
PartiallySubHomo : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (_⊕_ : A → A → Maybe A) (_⊗_ : B → B → Maybe B) → Set (a ⊔ℓ b)
PartiallySubHomo {A = A} f _⊕_ _⊗_ = ∀ (a₁ a₂ : A) →  (f a₁ ⊗ f a₂) ≡ Maybe.map f (a₁ ⊕ a₂)

-- this evaluation property, which depends on PartiallySubHomo, effectively makes
-- it easier to talk about compound operations and the preservation of their
-- structure when _⊗_ is PartiallySubHomo.
--
-- i.e., if (f a) ⊗ (f b) = Maybe.map f (a ⊕ b), then we can make similar claims about f (a ⊕ b ⊕ c)
eval-subHomo : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (_⊕_ : A → A → Maybe A) (_⊗_ : B → B → Maybe B) (e : Expr A) →
               PartiallySubHomo f _⊕_ _⊗_ →
               eval _⊗_ (map f e) ≡ Maybe.map f (eval _⊕_ e)
eval-subHomo f _⊕_ _⊗_ (` _) psh = refl
eval-subHomo f _⊕_ _⊗_ (e₁ ∨ e₂) psh
    rewrite eval-subHomo f _⊕_ _⊗_ e₁ psh rewrite eval-subHomo f _⊕_ _⊗_ e₂ psh
    with eval _⊕_ e₁ | eval _⊕_ e₂
...   | just x₁ | just x₂ = psh x₁ x₂
...   | nothing | nothing = refl
...   | just x₁ | nothing = refl
...   | nothing | just x₂ = refl

lvCombine-homo₁ : ∀ {a} plt plts (f : CombineForPLT a) (e : Expr (PartialLatticeType.EltType plt)) → eval (lvCombine f (plt ∷ plts)) (map inj₁ e) ≡ Maybe.map inj₁ (eval (f plt) e)
lvCombine-homo₁ plt plts f e = eval-subHomo inj₁ (f plt) (lvCombine f (plt ∷ plts)) e (λ a₁ a₂ → refl)

lvCombine-homo₂ : ∀ {a} plt plts (f : CombineForPLT a) (e : Expr (ListValue plts)) → eval (lvCombine f (plt ∷ plts)) (map inj₂ e) ≡ Maybe.map inj₂ (eval (lvCombine f plts) e)
lvCombine-homo₂ plt plts f e = eval-subHomo inj₂ (lvCombine f plts) (lvCombine f (plt ∷ plts)) e (λ a₁ a₂ → refl)

pathJoin'-homo-least₁ : ∀ {a} (l : Layer a) (ls : Layers a) least (e : Expr (LayerValue l)) → eval (pathJoin' (add-via-least l {{least}} ls)) (map inj₁ e) ≡ Maybe.map inj₁ (eval (lvJoin (toList l)) e)
pathJoin'-homo-least₁ l ls least e = eval-subHomo inj₁ (lvJoin (toList l)) (pathJoin' (add-via-least l {{least}} ls)) e (λ a₁ a₂ → refl)

pathJoin'-homo-greatest₁ : ∀ {a} (l : Layer a) (ls : Layers a) greatest (e : Expr (LayerValue l)) → eval (pathJoin' (add-via-greatest l ls {{greatest}})) (map inj₁ e) ≡ Maybe.map inj₁ (eval (lvJoin (toList l)) e)
pathJoin'-homo-greatest₁ l ls greatest e = eval-subHomo inj₁ (lvJoin (toList l)) (pathJoin' (add-via-greatest l ls {{greatest}})) e (λ a₁ a₂ → refl)

pathJoin'-homo-greatest₂ : ∀ {a} (l : Layer a) (ls : Layers a) greatest (e : Expr (Path' ls)) → eval (pathJoin' (add-via-greatest l ls {{greatest}})) (map inj₂ e) ≡ Maybe.map inj₂ (eval (pathJoin' ls) e)
pathJoin'-homo-greatest₂ l ls greatest e = eval-subHomo inj₂ (pathJoin' ls) (pathJoin' (add-via-greatest l ls {{greatest}})) e (λ a₁ a₂ → refl)

pathMeet'-homo-least₁ : ∀ {a} (l : Layer a) (ls : Layers a) least (e : Expr (LayerValue l)) → eval (pathMeet' (add-via-least l {{least}} ls)) (map inj₁ e) ≡ Maybe.map inj₁ (eval (lvMeet (toList l)) e)
pathMeet'-homo-least₁ l ls least e = eval-subHomo inj₁ (lvMeet (toList l)) (pathMeet' (add-via-least l {{least}} ls)) e (λ a₁ a₂ → refl)

lvCombine-assoc : ∀ {a} (f : CombineForPLT a) →
                  (∀ plt →  PartialAssoc (PartialLatticeType._≈_ plt) (f plt)) →
                  ∀ (L : List (PartialLatticeType a)) →
                  PartialAssoc (eqL L) (lvCombine f L)
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ v₁) (inj₁ v₂) (inj₁ v₃)
    rewrite lvCombine-homo₁ plt plts f (((` v₁) ∨ (` v₂)) ∨ (` v₃))
    rewrite lvCombine-homo₁ plt plts f ((` v₁) ∨ ((` v₂) ∨ (` v₃)))
    = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (f-assoc plt v₁ v₂ v₃)
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ _) (inj₁ v₂) (inj₁ v₃)
    with f plt v₂ v₃
...   | just _ = ≈-nothing
...   | nothing = ≈-nothing
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ _) (inj₂ _) (inj₁ _) = ≈-nothing
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂) (inj₁ _)
    with lvCombine f plts lv₁ lv₂
...   | just _ = ≈-nothing
...   | nothing = ≈-nothing
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ v₁) (inj₁ v₂) (inj₂ _)
    with f plt v₁ v₂
...   | just _ = ≈-nothing
...   | nothing = ≈-nothing
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ _) (inj₁ _) (inj₂ _) = ≈-nothing
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ _) (inj₂ lv₂) (inj₂ lv₃)
    with lvCombine f plts lv₂ lv₃
...   | just _ = ≈-nothing
...   | nothing = ≈-nothing
lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂) (inj₂ lv₃)
    rewrite lvCombine-homo₂ plt plts f (((` lv₁) ∨ (` lv₂)) ∨ (` lv₃))
    rewrite lvCombine-homo₂ plt plts f ((` lv₁) ∨ ((` lv₂) ∨ (` lv₃)))
    = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (lvCombine-assoc f f-assoc plts lv₁ lv₂ lv₃)

lvCombine-comm : ∀ {a} (f : CombineForPLT a) →
                 (∀ plt →  PartialComm (PartialLatticeType._≈_ plt) (f plt)) →
                 ∀ (L : List (PartialLatticeType a)) →
                 PartialComm (eqL L) (lvCombine f L)
lvCombine-comm f f-comm L@(plt ∷ plts) lv₁@(inj₁ v₁) (inj₁ v₂)
    rewrite lvCombine-homo₁ plt plts f ((` v₁) ∨ (` v₂)) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (f-comm plt v₁ v₂)
lvCombine-comm f f-comm (plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂) = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (lvCombine-comm f f-comm plts lv₁ lv₂)
lvCombine-comm f f-comm (plt ∷ plts) (inj₁ v₁) (inj₂ lv₂) = ≈-nothing
lvCombine-comm f f-comm (plt ∷ plts) (inj₂ lv₁) (inj₁ v₂) = ≈-nothing

lvCombine-idemp : ∀ {a} (f : CombineForPLT a) →
                  (∀ plt →  PartialIdemp (PartialLatticeType._≈_ plt) (f plt)) →
                  ∀ (L : List (PartialLatticeType a)) →
                  PartialIdemp (eqL L) (lvCombine f L)
lvCombine-idemp f f-idemp (plt ∷ plts) (inj₁ v) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (f-idemp plt v)
lvCombine-idemp f f-idemp (plt ∷ plts) (inj₂ lv) = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (lvCombine-idemp f f-idemp plts lv)

lvJoin-assoc : ∀ {a} (L : List (PartialLatticeType a)) → PartialAssoc (eqL L) (lvJoin L)
lvJoin-assoc = lvCombine-assoc PartialLatticeType._⊔?_ PartialLatticeType.⊔-assoc

lvJoin-comm : ∀ {a} (L : List (PartialLatticeType a)) → PartialComm (eqL L) (lvJoin L)
lvJoin-comm = lvCombine-comm PartialLatticeType._⊔?_ PartialLatticeType.⊔-comm

lvJoin-idemp : ∀ {a} (L : List (PartialLatticeType a)) → PartialIdemp (eqL L) (lvJoin L)
lvJoin-idemp = lvCombine-idemp PartialLatticeType._⊔?_ PartialLatticeType.⊔-idemp

lvMeet-assoc : ∀ {a} (L : List (PartialLatticeType a)) → PartialAssoc (eqL L) (lvMeet L)
lvMeet-assoc = lvCombine-assoc PartialLatticeType._⊓?_ PartialLatticeType.⊓-assoc

lvMeet-comm : ∀ {a} (L : List (PartialLatticeType a)) → PartialComm (eqL L) (lvMeet L)
lvMeet-comm = lvCombine-comm PartialLatticeType._⊓?_ PartialLatticeType.⊓-comm

lvMeet-idemp : ∀ {a} (L : List (PartialLatticeType a)) → PartialIdemp (eqL L) (lvMeet L)
lvMeet-idemp = lvCombine-idemp PartialLatticeType._⊓?_ PartialLatticeType.⊓-idemp

lvJoin-greatest-total : ∀ {a} {L : Layer a} →  (lv₁ lv₂ : LayerValue L) →  LayerGreatest L →  lvJoin (toList L) lv₁ lv₂ ≡ nothing → ⊥
lvJoin-greatest-total {L = plt ∷⁺ []} (inj₁ v₁) (inj₁ v₂) (MkLayerGreatest {{hasGreatest = hasGreatest}}) v₁⊔v₂≡nothing
    with IsPartialLattice.HasGreatestElement.not-partial (PartialLatticeType.isPartialLattice plt) hasGreatest v₁ v₂
...   | (v₁v₂ , v₁⊔v₂≡justv₁v₂)
         with trans (cong (Maybe.map inj₁) (sym v₁⊔v₂≡justv₁v₂)) v₁⊔v₂≡nothing
...        | ()

pathJoin'-greatest-total : ∀ {a} {Ls : Layers a} →  (p₁ p₂ : Path' Ls) →  LayerGreatest (head Ls) → pathJoin' Ls p₁ p₂ ≡ nothing → ⊥
pathJoin'-greatest-total {Ls = single L} lv₁ lv₂ layerGreatest lv₁⊔lv₂≡nothing = lvJoin-greatest-total {L = L} lv₁ lv₂ layerGreatest lv₁⊔lv₂≡nothing
pathJoin'-greatest-total {Ls = add-via-greatest L _} (inj₁ lv₁) (inj₁ lv₂) layerGreatest _
    with nothing <- lvJoin (toList L) lv₁ lv₂ in lv₁⊔?lv₂=? = lvJoin-greatest-total {L = L} lv₁ lv₂ layerGreatest lv₁⊔?lv₂=?
pathJoin'-greatest-total {Ls = add-via-least L _} (inj₁ lv₁) (inj₁ lv₂) layerGreatest _
    with nothing <- lvJoin (toList L) lv₁ lv₂ in lv₁⊔?lv₂=? = lvJoin-greatest-total {L = L} lv₁ lv₂ layerGreatest lv₁⊔?lv₂=?
pathJoin'-greatest-total {Ls = add-via-least (plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} Ls'} (inj₂ p₁) (inj₂ p₂) (MkLayerGreatest {{hasGreatest = hasGreatest}}) inj₂p₁⊔inj₂p₂≡nothing
    with pathJoin' Ls' p₁ p₂ | inj₂p₁⊔inj₂p₂≡nothing
...   | nothing | ()
...   | just _ | ()
pathJoin'-greatest-total {Ls = add-via-greatest L Ls {{layerGreatest}}} (inj₂ p₁) (inj₂ p₂) _ inj₂p₁⊔inj₂p₂≡nothing
    with pathJoin' Ls p₁ p₂ in p₁⊔p₂=? | inj₂p₁⊔inj₂p₂≡nothing
...   | just p₁⊔p₂ | ()
...   | nothing | refl = pathJoin'-greatest-total {Ls = Ls} p₁ p₂ layerGreatest p₁⊔p₂=?

pathJoin'-comm : ∀ {a} {Ls : Layers a} → PartialComm (eqPath' Ls) (pathJoin' Ls)
pathJoin'-comm {Ls = single l} lv₁ lv₂ = lvJoin-comm (toList l) lv₁ lv₂
pathJoin'-comm {Ls = add-via-least l ls} (inj₁ lv₁) (inj₁ lv₂) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvJoin-comm (toList l) lv₁ lv₂)
pathJoin'-comm {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₂ p₁) (inj₂ p₂)
    with pathJoin' ls p₁ p₂ | pathJoin' ls p₂ p₁ | pathJoin'-comm {Ls = ls} p₁ p₂
...   | just p | just p' | ≈-just p≈p' = ≈-just p≈p'
...   | nothing | nothing | ≈-nothing = ≈-just (eqLv-refl l (inj₁ (IsPartialSemilattice.HasAbsorbingElement.x hasLeast)))
pathJoin'-comm {Ls = add-via-least l ls} (inj₁ lv₁) (inj₂ p₂) = ≈-just (eqLv-refl l lv₁)
pathJoin'-comm {Ls = add-via-least l ls} (inj₂ p₁) (inj₁ lv₂) = ≈-just (eqLv-refl l lv₂)
pathJoin'-comm {Ls = add-via-greatest l ls} (inj₁ lv₁) (inj₁ lv₂) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvJoin-comm (toList l) lv₁ lv₂)
pathJoin'-comm {Ls = add-via-greatest l ls} (inj₂ p₁) (inj₂ p₂) = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (pathJoin'-comm {Ls = ls} p₁ p₂)
pathJoin'-comm {Ls = add-via-greatest l ls} (inj₁ lv₁) (inj₂ p₂) = ≈-just (eqLv-refl l lv₁)
pathJoin'-comm {Ls = add-via-greatest l ls} (inj₂ p₁) (inj₁ lv₂) = ≈-just (eqLv-refl l lv₂)

lvCombine-related : ∀ {a} (f : CombineForPLT a) →
                      (∀ plt {a₁} {a₂} {a} →
                         PartialLatticeType._≈?_ plt (f plt a₁ a₂) (just a₂) → 
                         (f plt a₁ a) ≡ nothing → (f plt a₂ a) ≡ nothing) →
                      (∀ plt a₁ a₂ →
                         maybe (λ a₁⊔a₂ →  PartialLatticeType._≈?_ plt (f plt a₂ a₁⊔a₂) (just a₁⊔a₂))
                               (Trivial a) (f plt a₁ a₂)) →
                    ∀ (L : List (PartialLatticeType a))
                      (lv₁ lv₂ lv₃ : ListValue L) →  (lv₁⊔lv₂ : ListValue L) → lvCombine f L lv₁ lv₂ ≡ just lv₁⊔lv₂ → lvCombine f L lv₂ lv₃ ≡ nothing → lvCombine f L lv₁⊔lv₂ lv₃ ≡ nothing
lvCombine-related f f-partial f-Mono [] ()
lvCombine-related f f-partial f-Mono (plt ∷ plts) (inj₁ v₁) (inj₁ v₂) lv₃ lv₁⊗lv₂ lv₁⊗?lv₂≡justlv₁⊗lv₂ lv₂⊗?lv₃≡nothing
    with f plt v₁ v₂ | lv₁⊗?lv₂≡justlv₁⊗lv₂ | lv₃ | f-Mono plt v₁ v₂
...   | just _ | refl | inj₂ _ | _ = refl
...   | just v₁⊗v₂ | refl | inj₁ v₃ | v₂≼v₁⊗v₂
        with f plt v₂ v₃ | lv₂⊗?lv₃≡nothing | f-partial plt {v₂} {v₁⊗v₂} {v₃} v₂≼v₁⊗v₂ 
...       | nothing | refl | refl⇒v₁v₂⊗?v₃≡nothing = cong (Maybe.map inj₁) (refl⇒v₁v₂⊗?v₃≡nothing refl)
lvCombine-related f f-partial f-Mono (plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂) lv₃ lv₁⊔lv₂ lv₁⊗?lv₂≡justlv₁⊗lv₂ lv₂⊗?lv₃≡nothing
    with lvCombine f plts lv₁ lv₂ in lv₁⊗?lv₂≡? | lv₁⊗?lv₂≡justlv₁⊗lv₂ | lv₃
...   | just lv₁⊔lv₂ | refl | inj₁ v₃ = refl
...   | just lv₁⊔lv₂ | refl | inj₂ lv₃'
        with lvCombine f plts lv₂ lv₃' in lv₂⊗?lv₃'≡? | lv₂⊗?lv₃≡nothing
...        | nothing | refl = cong (Maybe.map inj₂) (lvCombine-related f f-partial f-Mono plts lv₁ lv₂ lv₃' lv₁⊔lv₂ lv₁⊗?lv₂≡? lv₂⊗?lv₃'≡?)

lvJoin-related : ∀ {a} (L : List (PartialLatticeType a))
                   (lv₁ lv₂ lv₃ : ListValue L) →  (lv₁⊔lv₂ : ListValue L) → lvJoin L lv₁ lv₂ ≡ just lv₁⊔lv₂ → lvJoin L lv₂ lv₃ ≡ nothing → lvJoin L lv₁⊔lv₂ lv₃ ≡ nothing
lvJoin-related = lvCombine-related PartialLatticeType._⊔?_ PartialLatticeType.≼-partialʳ PartialLatticeType.y≼x⊔?y

lvMeet-related : ∀ {a} (L : List (PartialLatticeType a))
                   (lv₁ lv₂ lv₃ : ListValue L) →  (lv₁⊔lv₂ : ListValue L) → lvMeet L lv₁ lv₂ ≡ just lv₁⊔lv₂ → lvMeet L lv₂ lv₃ ≡ nothing → lvMeet L lv₁⊔lv₂ lv₃ ≡ nothing
lvMeet-related = lvCombine-related PartialLatticeType._⊓?_ PartialLatticeType.≽-partialʳ PartialLatticeType.y≽x⊓?y

-- crucially for the well-behavedness of path joins etc., divergences (e.g.,
-- "couldn't find path join") don't propagate far if they happen near
-- the end of a path. As a result, it should not be possible to have two
-- "deep" paths that produce `nothing`. The *-shallow-* lemmas state that.

pathJoin'-shallow-least : ∀ {a} (l : Layer a) (ls : Layers a) least (p₁ p₂ : Path' ls) → pathJoin' (add-via-least l {{least}} ls) (inj₂ p₁) (inj₂ p₂) ≡ nothing → ⊥
pathJoin'-shallow-least l@(plt ∷⁺ []) ls (MkLayerLeast {{hasLeast = hasLeast}}) p₁ p₂ inj₂p₁⊔inj₂p₂≡nothing
    with pathJoin' ls p₁ p₂ | inj₂p₁⊔inj₂p₂≡nothing
...   | just _ | ()
...   | nothing | ()

pathJoin'-shallow-greatest : ∀ {a} (l : Layer a) (ls : Layers a) greatest (p₁ p₂ : Path' ls) → pathJoin' (add-via-greatest l ls {{greatest}}) (inj₂ p₁) (inj₂ p₂) ≡ nothing → ⊥
pathJoin'-shallow-greatest l ls greatest p₁ p₂ inj₂p₁⊔inj₂p₂≡nothing
    with pathJoin' ls p₁ p₂ | pathJoin'-greatest-total {Ls = ls} p₁ p₂ greatest | inj₂p₁⊔inj₂p₂≡nothing
...   | just p₁⊔p₂ | _ | ()
...   | nothing | refl⇒⊥ | _ = ⊥-elim (refl⇒⊥ refl)

pathJoin'-related : ∀ {a} {Ls : Layers a} (p₁ p₂ p₃ : Path' Ls) → (p₁⊔p₂ : Path' Ls) → pathJoin' Ls p₁ p₂ ≡ just p₁⊔p₂ → pathJoin' Ls p₂ p₃ ≡ nothing → pathJoin' Ls p₁⊔p₂ p₃ ≡ nothing
pathJoin'-related {Ls = single l} = lvJoin-related (toList l)
pathJoin'-related {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃) _ lv₁⊔?lv₂=justlv₁⊔lv₂ lv₂⊔?lv₃=nothing
    with lvJoin (toList l) lv₁ lv₂ | lv₁⊔?lv₂=justlv₁⊔lv₂ | lvJoin (toList l) lv₂ lv₃ | lv₂⊔?lv₃=nothing | lvJoin-related (toList l) lv₁ lv₂ lv₃
...   | just lv₁⊔lv₂ | refl | nothing | refl | refl⇒refl⇒lv₁lv₂⊔lv₃=nothing = cong (Maybe.map inj₁) (refl⇒refl⇒lv₁lv₂⊔lv₃=nothing lv₁⊔lv₂ refl refl)
pathJoin'-related {Ls = add-via-least l {{least}} ls} (inj₂ p₁) (inj₁ lv₂) (inj₁ lv₃) _ refl lv₂⊔?lv₃=nothing rewrite lv₂⊔?lv₃=nothing = refl
pathJoin'-related {Ls = add-via-least l {{least}} ls} _ (inj₂ p₂) (inj₂ p₃) _ _ injp₂⊔?injp₃=nothing = ⊥-elim (pathJoin'-shallow-least l ls least p₂ p₃ injp₂⊔?injp₃=nothing)
pathJoin'-related {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃) _ lv₁⊔?lv₂=justlv₁⊔lv₂ lv₂⊔?lv₃=nothing
    with lvJoin (toList l) lv₁ lv₂ | lv₁⊔?lv₂=justlv₁⊔lv₂ | lvJoin (toList l) lv₂ lv₃ | lv₂⊔?lv₃=nothing | lvJoin-related (toList l) lv₁ lv₂ lv₃
...   | just lv₁⊔lv₂ | refl | nothing | refl | refl⇒refl⇒lv₁lv₂⊔lv₃=nothing = cong (Maybe.map inj₁) (refl⇒refl⇒lv₁lv₂⊔lv₃=nothing lv₁⊔lv₂ refl refl)
pathJoin'-related {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁) (inj₁ lv₂) (inj₁ lv₃) _ refl lv₂⊔?lv₃=nothing rewrite lv₂⊔?lv₃=nothing = refl
pathJoin'-related {Ls = add-via-greatest l ls {{greatest}}} _ (inj₂ p₂) (inj₂ p₃) _ _ injp₂⊔?injp₃=nothing
    with pathJoin' ls p₂ p₃ in p₂⊔?p₃=? | injp₂⊔?injp₃=nothing
...   | nothing | refl = ⊥-elim (pathJoin'-greatest-total {Ls = ls} p₂ p₃ greatest p₂⊔?p₃=?)

pathJoin'-related' : ∀ {a} {Ls : Layers a} (p₁ p₂ p₃ : Path' Ls) → (p₂⊔p₃ : Path' Ls) → pathJoin' Ls p₂ p₃ ≡ just p₂⊔p₃ → pathJoin' Ls p₁ p₂ ≡ nothing → pathJoin' Ls p₁ p₂⊔p₃ ≡ nothing
pathJoin'-related' {Ls = Ls} p₁ p₂ p₃ p₂⊔p₃ p₂⊔?p₃=justp₂⊔p₃ p₁⊔?p₂=nothing
    rewrite p₂⊔?p₃=justp₂⊔p₃
    rewrite p₁⊔?p₂=nothing
    with pathJoin' Ls p₂ p₃ in p₂⊔?p₃=justp₂⊔p₃' | pathJoin' Ls p₃ p₂ in p₃⊔p₂=? | pathJoin'-comm {Ls = Ls} p₂ p₃
      | pathJoin' Ls p₂ p₁ in p₂⊔p₁=? | pathJoin' Ls p₁ p₂ | pathJoin'-comm {Ls = Ls} p₂ p₁
...   | just p₂⊔p₃' | just p₃⊔p₂ | ≈-just p₂⊔p₃≈p₃⊔p₂
      | nothing | nothing | ≈-nothing
      rewrite just-injective p₂⊔?p₃=justp₂⊔p₃
      with pathJoin' Ls p₃⊔p₂ p₁ | pathJoin' Ls p₁ p₃⊔p₂ | pathJoin' Ls p₁ p₂⊔p₃
        | pathJoin'-comm {Ls = Ls} p₃⊔p₂ p₁
        | pathJoin'-related {Ls = Ls} p₃ p₂ p₁ p₃⊔p₂ p₃⊔p₂=? p₂⊔p₁=? 
        | eqPath'-pathJoin'-cong {Ls = Ls} (eqPath'-refl Ls p₁) p₂⊔p₃≈p₃⊔p₂
...     | nothing | nothing | nothing | ≈-nothing | refl | ≈-nothing = refl


-- fact: pathJoin' Ls p₃ p₂ ≡ just p₃⊔p₂

pathJoin'-assoc : ∀ {a} {Ls : Layers a} → PartialAssoc (eqPath' Ls) (pathJoin' Ls)
pathJoin'-assoc {Ls = single l} lv₁ lv₂ lv₃ = lvJoin-assoc (toList l) lv₁ lv₂ lv₃
pathJoin'-assoc {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
    rewrite pathJoin'-homo-least₁ l ls least (((` lv₁) ∨ (` lv₂)) ∨ (` lv₃))
    rewrite pathJoin'-homo-least₁ l ls least ((` lv₁) ∨ ((` lv₂) ∨ (` lv₃)))
    = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvJoin-assoc (toList l) lv₁ lv₂ lv₃)
pathJoin'-assoc {Ls = add-via-least l {{least}} ls} (inj₂ p₁) (inj₁ lv₂) (inj₁ lv₃)
    with lvJoin (toList l) lv₂ lv₃
...   | m@(just lv₂⊔lv₃) = ≈-just (eqLv-refl l lv₂⊔lv₃)
...   | nothing = ≈-nothing
pathJoin'-assoc {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₂ p₂) (inj₁ lv₃)
    = lift-≈-map inj₁ _ _ (λ _ _ x → x)
                          (lvJoin (toList l) lv₁ lv₃)
                          (lvJoin (toList l) lv₁ lv₃)
                          (IsEquivalence.≈-refl (lift-≈-Equivalence {{eqLv-Equivalence {L = l}}}))
pathJoin'-assoc {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₂ p₁) (inj₂ p₂) (inj₁ lv₃@(inj₁ v₃))
    with pathJoin' ls p₁ p₂
...   | just _ = ≈-just (eqLv-refl l lv₃)
...   | nothing = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _
                  (lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (PartialLatticeType.least-⊔-identˡ plt hasLeast v₃))
pathJoin'-assoc {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₁ lv₂) (inj₂ p₃)
    with lvJoin (toList l) lv₁ lv₂
...   | m@(just lv₁⊔lv₂) = ≈-just (eqLv-refl l lv₁⊔lv₂)
...   | nothing = ≈-nothing
pathJoin'-assoc {Ls = add-via-least l {{least}} ls} (inj₂ p₁) (inj₁ lv₂) (inj₂ p₃)
    = ≈-just (eqLv-refl l lv₂)
pathJoin'-assoc {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₁ lv₁@(inj₁ v₁)) (inj₂ p₂) (inj₂ p₃)
    with pathJoin' ls p₂ p₃
...   | just _ = ≈-just (eqLv-refl l lv₁)
...   | nothing = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _
                  (lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (IsEquivalence.≈-sym (lift-≈-Equivalence {{PartialLatticeType.≈-equiv plt}}) (PartialLatticeType.least-⊔-identʳ plt hasLeast v₁)))
pathJoin'-assoc {Ls = Ls@(add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls)} (inj₂ p₁) (inj₂ p₂) (inj₂ p₃)
    with pathJoin' ls p₁ p₂ in p₁⊔?p₂=? | pathJoin' ls p₂ p₃ in p₂⊔?p₃=?
      | pathJoin'-assoc {Ls = ls} p₁ p₂ p₃
      | pathJoin'-related {Ls = ls} p₁ p₂ p₃
      | pathJoin'-related' {Ls = ls} p₁ p₂ p₃
...   | nothing | just p₂⊔p₃ | _ | _ | refl⇒refl⇒p₁⊔p₂p₃=nothing
        rewrite refl⇒refl⇒p₁⊔p₂p₃=nothing p₂⊔p₃ refl refl
        = ≈-just (eqPath'-refl Ls (inj₁ (inj₁ (PartialLatticeType.HasLeastElement.x {_} {plt} hasLeast))))
...   | just p₁⊔p₂ | nothing | _ | refl⇒refl⇒p₁p₂⊔p₃=nothing | _
        rewrite refl⇒refl⇒p₁p₂⊔p₃=nothing p₁⊔p₂ refl refl
        = ≈-just (eqPath'-refl Ls (inj₁ (inj₁ (PartialLatticeType.HasLeastElement.x {_} {plt} hasLeast))))
...   | nothing | nothing | _ | _ | _ = ≈-just (eqPath'-refl Ls (inj₁ (inj₁ (PartialLatticeType.HasLeastElement.x {_} {plt} hasLeast))))
...   | just p₁⊔p₂ | just p₂⊔p₃ | p₁⊔p₂p₃≈?p₁p₂⊔p₃ | _ | _
        with pathJoin' ls p₁⊔p₂ p₃ | pathJoin' ls p₁ p₂⊔p₃ | p₁⊔p₂p₃≈?p₁p₂⊔p₃
...       | just p₁⊔p₂p₃ | just p₁p₂⊔p₃ | ≈-just p₁⊔p₂p₃≈p₁p₂⊔p₃ = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (≈-just p₁⊔p₂p₃≈p₁p₂⊔p₃)
...       | nothing | nothing | ≈-nothing = ≈-just (eqPath'-refl Ls (inj₁ (inj₁ (PartialLatticeType.HasLeastElement.x {_} {plt} hasLeast))))
pathJoin'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
    rewrite pathJoin'-homo-greatest₁ l ls greatest (((` lv₁) ∨ (` lv₂)) ∨ (` lv₃))
    rewrite pathJoin'-homo-greatest₁ l ls greatest ((` lv₁) ∨ ((` lv₂) ∨ (` lv₃)))
    = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvJoin-assoc (toList l) lv₁ lv₂ lv₃)
pathJoin'-assoc {Ls = add-via-greatest l ls} (inj₂ p₁) (inj₁ lv₂) (inj₁ lv₃)
    with lvJoin (toList l) lv₂ lv₃
...   | m@(just lv₂⊔lv₃) = ≈-just (eqLv-refl l lv₂⊔lv₃)
...   | nothing = ≈-nothing
pathJoin'-assoc {Ls = add-via-greatest l ls} (inj₁ lv₁) (inj₂ p₂) (inj₁ lv₃)
    = lift-≈-map inj₁ _ _ (λ _ _ x → x)
                          (lvJoin (toList l) lv₁ lv₃)
                          (lvJoin (toList l) lv₁ lv₃)
                          (IsEquivalence.≈-refl (lift-≈-Equivalence {{eqLv-Equivalence {L = l}}}))
pathJoin'-assoc {Ls = add-via-greatest l ls {{greatest = greatest}}} (inj₂ p₁) (inj₂ p₂) (inj₁ lv₃)
    with pathJoin' ls p₁ p₂ in p₁⊔?p₂=?
...   | just p₁⊔p₂ = ≈-just (eqLv-refl l lv₃)
...   | nothing = ⊥-elim (pathJoin'-greatest-total {Ls = ls} p₁ p₂ greatest p₁⊔?p₂=?)
pathJoin'-assoc {Ls = add-via-greatest l ls} (inj₁ lv₁) (inj₁ lv₂) (inj₂ p₃)
    with lvJoin (toList l) lv₁ lv₂
...   | m@(just lv₁⊔lv₂) = ≈-just (eqLv-refl l lv₁⊔lv₂)
...   | nothing = ≈-nothing
pathJoin'-assoc {Ls = add-via-greatest l ls} (inj₂ p₁) (inj₁ lv₂) (inj₂ p₃)
    = ≈-just (eqLv-refl l lv₂)
pathJoin'-assoc {Ls = add-via-greatest l ls {{greatest = greatest}}} (inj₁ lv₁) (inj₂ p₂) (inj₂ p₃)
    with pathJoin' ls p₂ p₃ in p₂⊔?p₃=?
...   | just p₂⊔p₃ = ≈-just (eqLv-refl l lv₁)
...   | nothing = ⊥-elim (pathJoin'-greatest-total {Ls = ls} p₂ p₃ greatest p₂⊔?p₃=?)
pathJoin'-assoc {Ls = add-via-greatest l ls {{greatest = greatest}}} (inj₂ p₁) (inj₂ p₂) (inj₂ p₃)
    with pathJoin' ls p₁ p₂ in p₁⊔?p₂=? | pathJoin' ls p₂ p₃ in p₂⊔?p₃=? | pathJoin'-assoc {Ls = ls} p₁ p₂ p₃
...   | just p₁⊔p₂ | just p₂⊔p₃ | p₁⊔p₂p₃≈p₁p₂⊔p₃ = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ p₁⊔p₂p₃≈p₁p₂⊔p₃
...   | nothing | _ | _ = ⊥-elim (pathJoin'-greatest-total {Ls = ls} p₁ p₂ greatest p₁⊔?p₂=?)
...   | _ | nothing | _ = ⊥-elim (pathJoin'-greatest-total {Ls = ls} p₂ p₃ greatest p₂⊔?p₃=?)

pathMeet'-comm : ∀ {a} {Ls : Layers a} → PartialComm (eqPath' Ls) (pathMeet' Ls)
pathMeet'-comm {Ls = single l} lv₁ lv₂ = lvMeet-comm (toList l) lv₁ lv₂
pathMeet'-comm {Ls = add-via-least l ls} (inj₁ lv₁) (inj₁ lv₂) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvMeet-comm (toList l) lv₁ lv₂)
pathMeet'-comm {Ls = add-via-least l ls} (inj₂ p₁) (inj₂ p₂) = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (pathMeet'-comm {Ls = ls} p₁ p₂)
pathMeet'-comm {Ls = add-via-least l ls} (inj₁ lv₁) (inj₂ p₂) = ≈-just (eqPath'-refl ls p₂)
pathMeet'-comm {Ls = add-via-least l ls} (inj₂ p₁) (inj₁ lv₂) = ≈-just (eqPath'-refl ls p₁)
pathMeet'-comm {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂)
    with lvMeet (toList l) lv₁ lv₂ | lvMeet (toList l) lv₂ lv₁ | lvMeet-comm (toList l) lv₁ lv₂
...   | just lv | just lv' | ≈-just lv≈lv' = ≈-just lv≈lv'
...   | nothing | nothing | ≈-nothing
        with ls | greatest | valueAtHead ls
-- begin dumb: don't know why, but abstracting on `head ls` leads to an ill-formed case
...       | single l' | MkLayerGreatest {{hasGreatest = hasGreatest}} | vah
          = ≈-just (eqLv-refl l' (vah (inj₁ (IsPartialSemilattice.HasAbsorbingElement.x hasGreatest))))
...       | add-via-least l' {{least = least}} ls' | MkLayerGreatest {{hasGreatest = hasGreatest}} | vah
          = ≈-just (eqPath'-refl (add-via-least l' {{least}} ls') (vah (inj₁ (IsPartialSemilattice.HasAbsorbingElement.x hasGreatest))))
...       | add-via-greatest l' ls' {{greatest = greatest'}} | MkLayerGreatest {{hasGreatest = hasGreatest}} | vah
          = ≈-just (eqPath'-refl (add-via-greatest l' ls' {{greatest'}}) (vah (inj₁ (IsPartialSemilattice.HasAbsorbingElement.x hasGreatest))))
-- end dumb
pathMeet'-comm {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁) (inj₂ p₂) = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (pathMeet'-comm {Ls = ls} p₁ p₂)
pathMeet'-comm {Ls = add-via-greatest l ls} (inj₁ lv₁) (inj₂ p₂) = ≈-just (eqPath'-refl ls p₂)
pathMeet'-comm {Ls = add-via-greatest l ls} (inj₂ p₁) (inj₁ lv₂) = ≈-just (eqPath'-refl ls p₁)

greatestPath-pathMeet'-identˡ : ∀ {a} (Ls : Layers a) (greatest : LayerGreatest (head Ls))
                                   (p : Path' Ls) →
                                   let pᵍ = greatestPath Ls greatest
                                   in lift-≈ (eqPath' Ls) (pathMeet' Ls pᵍ p) (just p)
greatestPath-pathMeet'-identˡ (single (plt ∷⁺ []) ) (MkLayerGreatest {{hasGreatest = hasGreatest}}) (inj₁ v)
    = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (PartialLatticeType.greatest-⊓-identˡ plt hasGreatest v)
greatestPath-pathMeet'-identˡ (add-via-least (plt ∷⁺ []) ls) (MkLayerGreatest {{hasGreatest = hasGreatest}}) (inj₁ (inj₁ v))
    = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _
        (lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (PartialLatticeType.greatest-⊓-identˡ plt hasGreatest v))
greatestPath-pathMeet'-identˡ (add-via-least l ls) (MkLayerGreatest {{hasGreatest = hasGreatest}}) (inj₂ p) = ≈-just (eqPath'-refl ls p)
greatestPath-pathMeet'-identˡ {a = a} (add-via-greatest (plt ∷⁺ []) ls) (MkLayerGreatest {{hasGreatest = hasGreatest}}) (inj₁ (inj₁ v))
    with PartialLatticeType._⊓?_ plt (PartialLatticeType.HasGreatestElement.x {a = a} {plt} hasGreatest) v | PartialLatticeType.greatest-⊓-identˡ plt hasGreatest v
...   | just vᵍ⊔v | ≈-just vᵍ⊔v≈v = ≈-just vᵍ⊔v≈v
greatestPath-pathMeet'-identˡ (add-via-greatest l ls) (MkLayerGreatest {{hasGreatest = hasGreatest}}) (inj₂ p) = ≈-just (eqPath'-refl ls p)

greatestPath-pathMeet'-identʳ : ∀ {a} (Ls : Layers a) (greatest : LayerGreatest (head Ls))
                                   (p : Path' Ls) →
                                   let pᵍ = greatestPath Ls greatest
                                   in lift-≈ (eqPath' Ls) (pathMeet' Ls p pᵍ) (just p)
greatestPath-pathMeet'-identʳ Ls greatest p
    = IsEquivalence.≈-trans (lift-≈-Equivalence {{eqPath'-Equivalence {Ls = Ls}}})
        (pathMeet'-comm {Ls = Ls} p (greatestPath Ls greatest))
        (greatestPath-pathMeet'-identˡ Ls greatest p)

pathMeet'-assoc : ∀ {a} {Ls : Layers a} → PartialAssoc (eqPath' Ls) (pathMeet' Ls)
pathMeet'-assoc {Ls = single l} lv₁ lv₂ lv₃ = lvMeet-assoc (toList l) lv₁ lv₂ lv₃
pathMeet'-assoc {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
    rewrite pathMeet'-homo-least₁ l ls least (((` lv₁) ∨ (` lv₂)) ∨ (` lv₃))
    rewrite pathMeet'-homo-least₁ l ls least ((` lv₁) ∨ ((` lv₂) ∨ (` lv₃)))
    = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvMeet-assoc (toList l) lv₁ lv₂ lv₃)
pathMeet'-assoc {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₂ p₁) (inj₁ (inj₁ v₂)) (inj₁ (inj₁ v₃))
    with PartialLatticeType._⊓?_ plt v₂ v₃ | IsPartialLattice.HasLeastElement.not-partial (PartialLatticeType.isPartialLattice plt) hasLeast v₂ v₃
...   | just v₂⊓l₃ | (_ , refl) = ≈-just (eqPath'-refl ls p₁)
pathMeet'-assoc {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₂ p₂) (inj₁ lv₃)
    = ≈-just (eqPath'-refl ls p₂)
pathMeet'-assoc {Ls = add-via-least l {{least}} ls} (inj₂ p₁) (inj₂ p₂) (inj₁ lv₃)
    with pathMeet' ls p₁ p₂
...   | just p₁⊓p₂ = ≈-just (eqPath'-refl ls p₁⊓p₂)
...   | nothing = ≈-nothing
pathMeet'-assoc {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₁ (inj₁ v₁)) (inj₁ (inj₁ v₂)) (inj₂ p₃)
    with PartialLatticeType._⊓?_ plt v₁ v₂ | IsPartialLattice.HasLeastElement.not-partial (PartialLatticeType.isPartialLattice plt) hasLeast v₁ v₂
...   | just _ | (_ , refl) = ≈-just (eqPath'-refl ls p₃)
pathMeet'-assoc {Ls = add-via-least l {{least}} ls} (inj₂ p₁) (inj₁ lv₂) (inj₂ p₃)
    with pathMeet' ls p₁ p₃
...   | just p₁⊓p₃ = ≈-just (eqPath'-refl ls p₁⊓p₃)
...   | nothing = ≈-nothing
pathMeet'-assoc {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₂ p₂) (inj₂ p₃)
    with pathMeet' ls p₂ p₃
...   | just p₂⊓p₃ = ≈-just (eqPath'-refl ls p₂⊓p₃)
...   | nothing = ≈-nothing
pathMeet'-assoc {Ls = add-via-least l ls} (inj₂ p₁) (inj₂ p₂) (inj₂ p₃)
    with pathMeet' ls p₁ p₂ in p₁⊓?p₂=? | pathMeet' ls p₂ p₃ in p₂⊓?p₃=? | pathMeet'-assoc {Ls = ls} p₁ p₂ p₃
...   | just p₁⊓p₂ | just p₂⊓p₃ | p₁⊓p₂p₃≈p₁p₂⊓p₃ = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ p₁⊓p₂p₃≈p₁p₂⊓p₃
...   | nothing | nothing | _ = ≈-nothing
...   | nothing | just p₂⊓p₃ | p₁⊓p₂p₃≈p₁p₂⊓p₃ with nothing ← pathMeet' ls p₁ p₂⊓p₃ = ≈-nothing
...   | just p₁⊓p₂ | nothing | p₁⊓p₂p₃≈p₁p₂⊓p₃ with nothing ← pathMeet' ls p₁⊓p₂ p₃ = ≈-nothing
-- begin dumb: due to annoying nested with-abstractions, we have several written-out cases here
-- for the same inj₁/inj₁/inj₁ combination.
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
    with lvMeet (toList l) lv₁ lv₂ | lvMeet (toList l) lv₂ lv₃ | lvMeet-assoc (toList l) lv₁ lv₂ lv₃
...   | just lv₁⊓lv₂ | just lv₂⊓lv₃ | p₁⊓?p₂p₃≈p₁p₂⊓?p₃
        with lvMeet (toList l) lv₁⊓lv₂ lv₃ | lvMeet (toList l) lv₁ lv₂⊓lv₃ | p₁⊓?p₂p₃≈p₁p₂⊓?p₃
...       | just _ | just _ | ≈-just p₁⊓p₂p₃≈p₁p₂⊓p₃  = ≈-just p₁⊓p₂p₃≈p₁p₂⊓p₃
...       | nothing | nothing | ≈-nothing = ≈-just (eqPath'-refl ls (greatestPath ls greatest))
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
      | nothing | nothing | _ = ≈-just (eqPath'-refl ls (greatestPath ls greatest))
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
      | just lv₁⊓lv₂ | nothing | p₁⊓?p₂p₃≈p₁p₂⊓?p₃
        with nothing <- lvMeet (toList l) lv₁⊓lv₂ lv₃ = ≈-just (eqPath'-refl ls (greatestPath ls greatest))
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₁ lv₃)
      | nothing | just lv₂⊓lv₃ | p₁⊓?p₂p₃≈p₁p₂⊓?p₃
        with nothing <- lvMeet (toList l) lv₁ lv₂⊓lv₃ = ≈-just (eqPath'-refl ls (greatestPath ls greatest))
-- end dumb
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁) (inj₁ lv₂) (inj₁ lv₃)
    with lvMeet (toList l) lv₂ lv₃
...   | just _ = ≈-just (eqPath'-refl ls p₁)
...   | nothing = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (IsEquivalence.≈-sym (lift-≈-Equivalence {{eqPath'-Equivalence {Ls = ls}}}) (greatestPath-pathMeet'-identʳ ls greatest p₁))
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₂ p₂) (inj₁ lv₃)
    = ≈-just (eqPath'-refl ls p₂)
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁) (inj₂ p₂) (inj₁ lv₃)
    with pathMeet' ls p₁ p₂
...   | just p₁⊓p₂ = ≈-just (eqPath'-refl ls p₁⊓p₂)
...   | nothing = ≈-nothing
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂) (inj₂ p₃)
    with lvMeet (toList l) lv₁ lv₂
...   | just _ = ≈-just (eqPath'-refl ls p₃)
...   | nothing = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (greatestPath-pathMeet'-identˡ ls greatest p₃)
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁) (inj₁ lv₂) (inj₂ p₃)
    with pathMeet' ls p₁ p₃
...   | just p₁⊓p₃ = ≈-just (eqPath'-refl ls p₁⊓p₃)
...   | nothing = ≈-nothing
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₂ p₂) (inj₂ p₃)
    with pathMeet' ls p₂ p₃
...   | just p₂⊓p₃ = ≈-just (eqPath'-refl ls p₂⊓p₃)
...   | nothing = ≈-nothing
pathMeet'-assoc {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁) (inj₂ p₂) (inj₂ p₃)
    with pathMeet' ls p₁ p₂ in p₁⊓?p₂=? | pathMeet' ls p₂ p₃ in p₂⊓?p₃=? | pathMeet'-assoc {Ls = ls} p₁ p₂ p₃
...   | just p₁⊓p₂ | just p₂⊓p₃ | p₁⊓p₂p₃≈p₁p₂⊓p₃ = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ p₁⊓p₂p₃≈p₁p₂⊓p₃
...   | nothing | nothing | _ = ≈-nothing
...   | nothing | just p₂⊓p₃ | p₁⊓p₂p₃≈p₁p₂⊓p₃ with nothing ← pathMeet' ls p₁ p₂⊓p₃ = ≈-nothing
...   | just p₁⊓p₂ | nothing | p₁⊓p₂p₃≈p₁p₂⊓p₃ with nothing ← pathMeet' ls p₁⊓p₂ p₃ = ≈-nothing

record _≈_ {a} {Ls : Layers a} (p₁ p₂ : Path Ls) : Set a where
    field pathEq : eqPath' Ls (Path.path p₁) (Path.path p₂)

_⊔̇_ : ∀ {a} {Ls : Layers a} (p₁ p₂ : Path Ls) → Maybe (Path Ls)
_⊔̇_ {a} {ls} p₁ p₂ = Maybe.map MkPath (pathJoin' ls (Path.path p₁) (Path.path p₂))

_⊓̇_ : ∀ {a} {Ls : Layers a} (p₁ p₂ : Path Ls) → Maybe (Path Ls)
_⊓̇_ {a} {ls} p₁ p₂ = Maybe.map MkPath (pathMeet' ls (Path.path p₁) (Path.path p₂))

-- data ListValue {a : Level} : List (PartialLatticeType a) → Set (lsuc a) where
--     here : ∀ {plt : PartialLatticeType a} {pltl : List (PartialLatticeType a)}
--              (v : PartialLatticeType.EltType plt) → ListValue (plt ∷ pltl)
--     there : ∀ {plt : PartialLatticeType a} {pltl : List (PartialLatticeType a)} →
--              ListValue pltl → ListValue (plt ∷ pltl)