Move proof of least element into FiniteHeightLattice

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Analysis/Forward.agda

@@ -21,7 +21,7 @@ open IsFiniteHeightLattice isFiniteHeightLatticeˡ
     using () renaming (isLattice to isLatticeˡ)
 
 module WithProg (prog : Program) where
-    open import Analysis.Forward.Lattices L prog hiding (≈ᵛ-Decidable) -- to disambiguate instance resolution
+    open import Analysis.Forward.Lattices L prog hiding (≈ᵛ-Decidable; ≈ᵐ-Decidable) -- to disambiguate instance resolution
     open import Analysis.Forward.Evaluation L prog
     open Program prog
 
@@ -66,7 +66,7 @@ module WithProg (prog : Program) where
                            (joinAll-Mono {sv₁} {sv₂} sv₁≼sv₂)
 
         -- The fixed point of the 'analyze' function is our final goal.
-        open import Fixedpoint ≈ᵐ-Decidable isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
+        open import Fixedpoint analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
             using ()
             renaming (aᶠ to result; aᶠ≈faᶠ to result≈analyze-result)
             public

Fixedpoint.agda

@@ -5,8 +5,8 @@ module Fixedpoint {a} {A : Set a}
                   {h : ℕ}
                   {_≈_ : A → A → Set a}
                   {_⊔_ : A → A → A} {_⊓_ : A → A → A}
-                  (≈-Decidable : IsDecidable _≈_)
-                  (flA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_)
+                  {{ ≈-Decidable : IsDecidable _≈_ }}
+                  {{flA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_}}
                   (f : A → A)
                   (Monotonicᶠ : Monotonic (IsFiniteHeightLattice._≼_ flA)
                                           (IsFiniteHeightLattice._≼_ flA) f) where
@@ -28,24 +28,9 @@ private
         using ()
         renaming
             ( ⊥ to ⊥ᴬ
-            ; longestChain to longestChainᴬ
             ; bounded to boundedᴬ
             )
 
-
-    ⊥ᴬ≼ : ∀ (a : A) → ⊥ᴬ ≼ a
-    ⊥ᴬ≼ a with ≈-dec a ⊥ᴬ
-    ...   | yes a≈⊥ᴬ = ≼-cong a≈⊥ᴬ ≈-refl (≼-refl a)
-    ...   | no a̷≈⊥ᴬ with ≈-dec ⊥ᴬ (a ⊓ ⊥ᴬ)
-    ...         | yes ⊥ᴬ≈a⊓⊥ᴬ = ≈-trans (⊔-comm ⊥ᴬ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥ᴬ≈a⊓⊥ᴬ) (absorb-⊔-⊓ a ⊥ᴬ))
-    ...         | no ⊥ᴬ̷≈a⊓⊥ᴬ = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ (ChainA.step x≺⊥ᴬ ≈-refl longestChainᴬ))
-                    where
-                        ⊥ᴬ⊓a̷≈⊥ᴬ : ¬ (⊥ᴬ ⊓ a) ≈ ⊥ᴬ
-                        ⊥ᴬ⊓a̷≈⊥ᴬ = λ ⊥ᴬ⊓a≈⊥ᴬ → ⊥ᴬ̷≈a⊓⊥ᴬ (≈-trans (≈-sym ⊥ᴬ⊓a≈⊥ᴬ) (⊓-comm _ _))
-
-                        x≺⊥ᴬ : (⊥ᴬ ⊓ a) ≺ ⊥ᴬ
-                        x≺⊥ᴬ = (≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ᴬ ⊔ (⊥ᴬ ⊓ a)}) (absorb-⊔-⊓ ⊥ᴬ a)) , ⊥ᴬ⊓a̷≈⊥ᴬ)
-
     -- using 'g', for gas, here helps make sure the function terminates.
     -- since A forms a fixed-height lattice, we must find a solution after
     -- 'h' steps at most. Gas is set up such that as soon as it runs
@@ -65,7 +50,7 @@ private
                     c' rewrite +-comm 1 hᶜ = ChainA.concat c (ChainA.step a₂≺fa₂ ≈-refl (ChainA.done (≈-refl {f a₂})))
 
     fix : Σ A (λ a → a ≈ f a)
-    fix = doStep (suc h) 0 ⊥ᴬ ⊥ᴬ (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))
+    fix = doStep (suc h) 0 ⊥ᴬ ⊥ᴬ (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥≼ (f ⊥ᴬ))
 
 aᶠ : A
 aᶠ = proj₁ fix
@@ -85,4 +70,4 @@ private
     ...   | no  _ = stepPreservesLess g' _ _ _ b b≈fb (≼-cong ≈-refl (≈-sym b≈fb) (Monotonicᶠ a₂≼b)) _ _ _
 
 aᶠ≼ : ∀ (a : A) → a ≈ f a → aᶠ ≼ a
-aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥ᴬ≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))
+aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥≼ (f ⊥ᴬ))

Isomorphism.agda

@@ -68,6 +68,7 @@ module TransportFiniteHeight
         renaming (⊥ to ⊥₁; ⊤ to ⊤₁; bounded to bounded₁; longestChain to c)
 
     instance
+        fixedHeight : IsLattice.FixedHeight lB height
         fixedHeight = record
             { ⊥ = f ⊥₁
             ; ⊤ = f ⊤₁

Lattice.agda

@@ -4,7 +4,8 @@ open import Equivalence
 import Chain
 
 open import Relation.Binary.Core using (_Preserves_⟶_ )
-open import Relation.Nullary using (Dec; ¬_)
+open import Relation.Nullary using (Dec; ¬_; yes; no)
+open import Data.Empty using (⊥-elim)
 open import Data.Nat as Nat using (ℕ)
 open import Data.Product using (_×_; Σ; _,_)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
@@ -236,6 +237,28 @@ record IsFiniteHeightLattice {a} (A : Set a)
     field
         {{fixedHeight}} : FixedHeight h
 
+    -- If the equality is decidable, we can prove that the top and bottom
+    -- elements of the chain are least and greatest elements of the lattice
+    module _ {{≈-Decidable : IsDecidable _≈_}} where
+        open IsDecidable ≈-Decidable using () renaming (R-dec to ≈-dec)
+
+        module MyChain = Chain _≈_ ≈-equiv _≺_ ≺-cong
+        open MyChain.Height fixedHeight using (⊥; ⊤) public
+        open MyChain.Height fixedHeight using (bounded; longestChain)
+
+        ⊥≼ : ∀ (a : A) → ⊥ ≼ a
+        ⊥≼ a with ≈-dec a ⊥
+        ...   | yes a≈⊥ = ≼-cong a≈⊥ ≈-refl (≼-refl a)
+        ...   | no a̷≈⊥ with ≈-dec ⊥ (a ⊓ ⊥)
+        ...         | yes ⊥≈a⊓⊥ = ≈-trans (⊔-comm ⊥ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥≈a⊓⊥) (absorb-⊔-⊓ a ⊥))
+        ...         | no ⊥ᴬ̷≈a⊓⊥ = ⊥-elim (MyChain.Bounded-suc-n bounded (MyChain.step x≺⊥ ≈-refl longestChain))
+                        where
+                            ⊥⊓a̷≈⊥ : ¬ (⊥ ⊓ a) ≈ ⊥
+                            ⊥⊓a̷≈⊥ = λ ⊥⊓a≈⊥ → ⊥ᴬ̷≈a⊓⊥ (≈-trans (≈-sym ⊥⊓a≈⊥) (⊓-comm _ _))
+
+                            x≺⊥ : (⊥ ⊓ a) ≺ ⊥
+                            x≺⊥ = (≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ ⊔ (⊥ ⊓ a)}) (absorb-⊔-⊓ ⊥ a)) , ⊥⊓a̷≈⊥)
+
 module ChainMapping {a b} {A : Set a} {B : Set b}
     {_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b}
     {_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}

Lattice/Builder.agda

@@ -148,6 +148,10 @@ record IsPartialSemilattice {a} {A : Set a}
                 with a₁ ⊔? x | refl⇒a₁⊔?x=nothing refl | x-absorbʳ a₁
         ...       | nothing | refl | ()
 
+
+PartialAbsorb : ∀ {a} {A : Set a} (_≈_ : A → A → Set a) (_⊗₁_ : A → A → Maybe A) (_⊗₂_ : A → A → Maybe A) → Set a
+PartialAbsorb {a} {A} _≈_ _⊗₁_ _⊗₂_ = ∀ (x y : A) →  maybe (λ x⊗₂y → lift-≈ _≈_ (x ⊗₁ x⊗₂y) (just x)) (Trivial _) (x ⊗₂ y)
+
 record IsPartialLattice {a} {A : Set a}
     (_≈_ : A → A → Set a)
     (_⊔?_ : A → A → Maybe A)
@@ -157,8 +161,8 @@ record IsPartialLattice {a} {A : Set a}
         {{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)
+        absorb-⊔-⊓ : PartialAbsorb _≈_ _⊔?_ _⊓?_
+        absorb-⊓-⊔ : PartialAbsorb _≈_ _⊓?_ _⊔?_
 
     open IsPartialSemilattice partialJoinSemilattice
         renaming (HasAbsorbingElement to HasGreatestElement)
@@ -213,6 +217,7 @@ record PartialLattice {a} (A : Set a) : Set (lsuc a) where
     least-⊔-identˡ le x = ≈?-trans (⊔-comm (HasLeastElement.x le) x) (least-⊔-identʳ le x)
 
 record PartialLatticeType (a : Level) : Set (lsuc a) where
+    constructor mkPartialLatticeType
     field
         EltType : Set a
         {{partialLattice}} : PartialLattice EltType
@@ -508,6 +513,24 @@ eval _⊔?_ (e₁ ∨ e₂) = (eval _⊔?_ e₁) >>= (λ v₁ →  (eval _⊔?_
 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)
 
+PartialAbsorb-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) →
+                    ∀ (_⊕₁_ : A → A → Maybe A) (_⊕₂_ : A → A → Maybe A)
+                      (_⊗₁_ : B → B → Maybe B) (_⊗₂_ : B → B → Maybe B) →
+                      PartiallySubHomo f _⊕₁_ _⊗₁_ →
+                      PartiallySubHomo f _⊕₂_ _⊗₂_ →
+                      PartialAbsorb _≈ᵃ_ _⊕₁_ _⊕₂_ →
+                      ∀ (x y : A) →  maybe (λ x⊗₂y → lift-≈ _≈ᵇ_ (f x ⊗₁ x⊗₂y) (just (f x))) (Trivial _) (f x ⊗₂ f y)
+PartialAbsorb-map f _≈ᵃ_ _≈ᵇ_ ≈ᵃ⇒≈ᵇ _⊕₁_ _⊕₂_ _⊗₁_ _⊗₂_ psh₁ psh₂ absorbᵃ x y
+    with x ⊕₂ y | f x ⊗₂ f y | psh₂ x y | absorbᵃ x y
+...   | nothing | nothing | refl | _ = mkTrivial
+...   | just x⊕₂y | just fx⊗₂fy | refl | x⊕₁xy≈?x
+        with x ⊕₁ x⊕₂y | f x ⊗₁ (f x⊕₂y) | psh₁ x x⊕₂y | x⊕₁xy≈?x
+...       |  just x⊕₁xy | just fx⊗₁fx⊕₂y | refl | ≈-just x⊕₁xy≈x = ≈-just (≈ᵃ⇒≈ᵇ _ _ x⊕₁xy≈x)
+
+
 -- 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.
@@ -609,6 +632,30 @@ lvMeet-comm = lvCombine-comm PartialLatticeType._⊓?_ PartialLatticeType.⊓-co
 lvMeet-idemp : ∀ {a} (L : List (PartialLatticeType a)) → PartialIdemp (eqL L) (lvMeet L)
 lvMeet-idemp = lvCombine-idemp PartialLatticeType._⊓?_ PartialLatticeType.⊓-idemp
 
+lvCombine-absorb : ∀ {a} (f₁ f₂ : CombineForPLT a) →
+                   (∀ plt →  PartialAbsorb (PartialLatticeType._≈_ plt) (f₁ plt) (f₂ plt)) →
+                   ∀ (L : List (PartialLatticeType a)) →
+                   PartialAbsorb (eqL L) (lvCombine f₁ L) (lvCombine f₂ L)
+lvCombine-absorb f₁ f₂ absorb-f₁-f₂ [] ()
+lvCombine-absorb f₁ f₂ absorb-f₁-f₂ (plt ∷ plts) (inj₁ v₁) (inj₁ v₂)
+    = PartialAbsorb-map inj₁ _ _ (λ _ _ x → x) (f₁ plt) (f₂ plt)
+                                 (lvCombine f₁ (plt ∷ plts)) (lvCombine f₂ (plt ∷ plts))
+                                 (λ _ _ → refl) (λ _ _ → refl)
+                                 (absorb-f₁-f₂ plt) v₁ v₂
+lvCombine-absorb f₁ f₂ absorb-f₁-f₂ (plt ∷ plts) (inj₂ lv₁) (inj₁ v₂) = mkTrivial
+lvCombine-absorb f₁ f₂ absorb-f₁-f₂ (plt ∷ plts) (inj₁ v₁) (inj₂ lv₂) = mkTrivial
+lvCombine-absorb f₁ f₂ absorb-f₁-f₂ (plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂)
+    = PartialAbsorb-map inj₂ _ _ (λ _ _ x → x) (lvCombine f₁ plts) (lvCombine f₂ plts)
+                                 (lvCombine f₁ (plt ∷ plts)) (lvCombine f₂ (plt ∷ plts))
+                                 (λ _ _ → refl) (λ _ _ → refl)
+                                 (lvCombine-absorb f₁ f₂  absorb-f₁-f₂ plts) lv₁ lv₂
+
+absorb-lvJoin-lvMeet : ∀ {a} (L : List (PartialLatticeType a)) → PartialAbsorb (eqL L) (lvJoin L) (lvMeet L)
+absorb-lvJoin-lvMeet = lvCombine-absorb PartialLatticeType._⊔?_ PartialLatticeType._⊓?_ PartialLatticeType.absorb-⊔-⊓
+
+absorb-lvMeet-lvJoin : ∀ {a} (L : List (PartialLatticeType a)) → PartialAbsorb (eqL L) (lvMeet L) (lvJoin L)
+absorb-lvMeet-lvJoin = lvCombine-absorb PartialLatticeType._⊓?_ PartialLatticeType._⊔?_ PartialLatticeType.absorb-⊓-⊔
+
 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
@@ -946,6 +993,73 @@ pathMeet'-idemp {Ls = add-via-greatest l ls} (inj₁ lv)
 ...   | just lv⊔lv | ≈-just lv⊔lv≈lv = ≈-just lv⊔lv≈lv
 pathMeet'-idemp {Ls = add-via-greatest l ls} (inj₂ p) = lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (pathMeet'-idemp {Ls = ls} p)
 
+absorb-pathJoin'-pathMeet' : ∀ {a} {Ls : Layers a} → PartialAbsorb (eqPath' Ls) (pathJoin' Ls) (pathMeet' Ls)
+absorb-pathJoin'-pathMeet' {Ls = single l} lv₁ lv₂ = absorb-lvJoin-lvMeet (toList l) lv₁ lv₂
+absorb-pathJoin'-pathMeet' {Ls = Ls@(add-via-least l ls)} (inj₁ lv₁) (inj₁ lv₂)
+    = PartialAbsorb-map inj₁ _ _ (λ _ _ x → x) (lvJoin (toList l)) (lvMeet (toList l))
+                                 (pathJoin' Ls) (pathMeet' Ls)
+                                 (λ _ _ → refl) (λ _ _ → refl)
+                                 (absorb-lvJoin-lvMeet (toList l)) lv₁ lv₂
+absorb-pathJoin'-pathMeet' {Ls = add-via-least l {{least}} ls} (inj₂ p₁) (inj₁ lv₂)
+    = pathJoin'-idemp {Ls = add-via-least l {{least}} ls} (inj₂ p₁)
+absorb-pathJoin'-pathMeet' {Ls = add-via-least l ls} (inj₁ lv₁) (inj₂ p₂)
+    = ≈-just (eqLv-refl l lv₁)
+absorb-pathJoin'-pathMeet' {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₂ p₁) (inj₂ p₂)
+    with pathMeet' ls p₁ p₂ | absorb-pathJoin'-pathMeet' {Ls = ls} p₁ p₂
+...   | nothing | _ = mkTrivial
+...   | just p₁⊓p₂ | p₁⊔p₁p₂≈?p₁
+        with pathJoin' ls p₁ p₁⊓p₂ | p₁⊔p₁p₂≈?p₁
+...       | just _ | ≈-just p₁⊔p₁p₂≈p₁ = ≈-just p₁⊔p₁p₂≈p₁
+absorb-pathJoin'-pathMeet' {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂)
+    with lvMeet (toList l) lv₁ lv₂ | absorb-lvJoin-lvMeet (toList l) lv₁ lv₂
+...   | nothing | _ = ≈-just (eqLv-refl l lv₁)
+...   | just lv₁⊓lv₂ | lv₁⊔lv₁lv₂≈?lv₁
+        with lvJoin (toList l) lv₁ lv₁⊓lv₂ | lv₁⊔lv₁lv₂≈?lv₁
+...       | just _ | ≈-just lv₁⊔lv₁lv₂≈lv₁ = ≈-just lv₁⊔lv₁lv₂≈lv₁
+absorb-pathJoin'-pathMeet' {Ls = add-via-greatest l ls {{greatest}} } (inj₂ p₁) (inj₁ lv₂)
+    = pathJoin'-idemp {Ls = add-via-greatest l ls {{greatest}}} (inj₂ p₁)
+absorb-pathJoin'-pathMeet' {Ls = add-via-greatest l ls} (inj₁ lv₁) (inj₂ p₂)
+    = ≈-just (eqLv-refl l lv₁)
+absorb-pathJoin'-pathMeet' {Ls = Ls@(add-via-greatest l ls)} (inj₂ p₁) (inj₂ p₂)
+    = PartialAbsorb-map inj₂ _ _ (λ _ _ x → x) (pathJoin' ls) (pathMeet' ls)
+                                 (pathJoin' Ls) (pathMeet' Ls)
+                                 (λ _ _ → refl) (λ _ _ → refl)
+                                 (absorb-pathJoin'-pathMeet' {Ls = ls}) p₁ p₂
+
+absorb-pathMeet'-pathJoin' : ∀ {a} {Ls : Layers a} → PartialAbsorb (eqPath' Ls) (pathMeet' Ls) (pathJoin' Ls)
+absorb-pathMeet'-pathJoin' {Ls = single l} lv₁ lv₂ = absorb-lvMeet-lvJoin (toList l) lv₁ lv₂
+absorb-pathMeet'-pathJoin' {Ls = Ls@(add-via-least l ls)} (inj₁ lv₁) (inj₁ lv₂)
+    = PartialAbsorb-map inj₁ _ _ (λ _ _ x → x) (lvMeet (toList l)) (lvJoin (toList l))
+                                 (pathMeet' Ls) (pathJoin' Ls)
+                                 (λ _ _ → refl) (λ _ _ → refl)
+                                 (absorb-lvMeet-lvJoin (toList l)) lv₁ lv₂
+absorb-pathMeet'-pathJoin' {Ls = add-via-least l ls} (inj₂ p₁) (inj₁ lv₂)
+    = ≈-just (eqPath'-refl ls p₁)
+absorb-pathMeet'-pathJoin' {Ls = add-via-least l {{least}} ls} (inj₁ lv₁) (inj₂ p₂)
+    = pathMeet'-idemp {Ls = add-via-least l {{least}} ls} (inj₁ lv₁)
+absorb-pathMeet'-pathJoin' {Ls = add-via-least l@(plt ∷⁺ []) {{MkLayerLeast {{hasLeast = hasLeast}}}} ls} (inj₂ p₁) (inj₂ p₂)
+    with pathJoin' ls p₁ p₂ | absorb-pathMeet'-pathJoin' {Ls = ls} p₁ p₂
+...   | nothing | _ = ≈-just (eqPath'-refl ls p₁)
+...   | just p₁⊓p₂ | p₁⊔p₁p₂≈?p₁
+        with pathMeet' ls p₁ p₁⊓p₂ | p₁⊔p₁p₂≈?p₁
+...       | just _ | ≈-just p₁⊔p₁p₂≈p₁ = ≈-just p₁⊔p₁p₂≈p₁
+absorb-pathMeet'-pathJoin' {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₁ lv₂)
+    with lvJoin (toList l) lv₁ lv₂ | absorb-lvMeet-lvJoin (toList l) lv₁ lv₂
+...   | nothing | _ = mkTrivial
+...   | just lv₁⊓lv₂ | lv₁⊔lv₁lv₂≈?lv₁
+        with lvMeet (toList l) lv₁ lv₁⊓lv₂ | lv₁⊔lv₁lv₂≈?lv₁
+...       | just _ | ≈-just lv₁⊔lv₁lv₂≈lv₁ = ≈-just lv₁⊔lv₁lv₂≈lv₁
+absorb-pathMeet'-pathJoin' {Ls = add-via-greatest l ls} (inj₂ p₁) (inj₁ lv₂)
+    = ≈-just (eqPath'-refl ls p₁)
+absorb-pathMeet'-pathJoin' {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁) (inj₂ p₂)
+    = pathMeet'-idemp {Ls = add-via-greatest l ls {{greatest}}} (inj₁ lv₁)
+absorb-pathMeet'-pathJoin' {Ls = Ls@(add-via-greatest l ls)} (inj₂ p₁) (inj₂ p₂)
+    = PartialAbsorb-map inj₂ _ _ (λ _ _ x → x) (pathMeet' ls) (pathJoin' ls)
+                                 (pathMeet' Ls) (pathJoin' Ls)
+                                 (λ _ _ → refl) (λ _ _ → refl)
+                                 (absorb-pathMeet'-pathJoin' {Ls = ls}) p₁ p₂
+
+
 record _≈_ {a} {Ls : Layers a} (p₁ p₂ : Path Ls) : Set a where
     constructor mk-≈
     field pathEq : eqPath' Ls (Path.path p₁) (Path.path p₂)
@@ -1026,6 +1140,59 @@ instance
             ; ≈-⊔-cong = ≈-⊓̇-cong {Ls = Ls}
             }
 
+⊔̇-⊓̇-absorb : ∀ {a} {Ls : Layers a} → PartialAbsorb (_≈_ {Ls = Ls}) (_⊔̇_ {Ls = Ls}) (_⊓̇_ {Ls = Ls})
+⊔̇-⊓̇-absorb {Ls = Ls} (MkPath p₁) (MkPath p₂)
+    = PartialAbsorb-map MkPath _ _ (λ _ _ → mk-≈) (pathJoin' Ls) (pathMeet' Ls)
+                                   (_⊔̇_ {Ls = Ls}) (_⊓̇_ {Ls = Ls})
+                                   (λ _ _ → refl) (λ _ _ → refl)
+                                   (absorb-pathJoin'-pathMeet' {Ls = Ls}) p₁ p₂
+
+⊓̇-⊔̇-absorb : ∀ {a} {Ls : Layers a} → PartialAbsorb (_≈_ {Ls = Ls}) (_⊓̇_ {Ls = Ls}) (_⊔̇_ {Ls = Ls})
+⊓̇-⊔̇-absorb {Ls = Ls} (MkPath p₁) (MkPath p₂)
+    = PartialAbsorb-map MkPath _ _ (λ _ _ → mk-≈) (pathMeet' Ls) (pathJoin' Ls)
+                                   (_⊓̇_ {Ls = Ls}) (_⊔̇_ {Ls = Ls})
+                                   (λ _ _ → refl) (λ _ _ → refl)
+                                   (absorb-pathMeet'-pathJoin' {Ls = Ls}) p₁ p₂
+
+instance
+    Path-IsPartialLattice : ∀ {a} {Ls : Layers a} → IsPartialLattice (_≈_ {Ls = Ls}) (_⊔̇_ {Ls = Ls}) (_⊓̇_ {Ls = Ls})
+    Path-IsPartialLattice {Ls = Ls} =
+        record
+            { absorb-⊔-⊓ = ⊔̇-⊓̇-absorb {Ls = Ls}
+            ; absorb-⊓-⊔ = ⊓̇-⊔̇-absorb {Ls = Ls}
+            }
+
+instance
+    -- IsLattice-IsPartialLattice : ∀ {a} {A : Set a}
+    --     {_≈_ : A → A → Set a} {_⊔_ : A → A → A} {_⊓_ : A → A → A}
+    --     {{lA : IsLattice A _≈_ _⊔_ _⊓_}} → IsPartialLattice _≈_ _⊔_ _⊓_
+    -- IsLattice-IsPartialLattice = {!!}
+
+    Lattice-PartialLattice : ∀ {a} {A : Set a}
+        {{lA : Lattice A }} →  PartialLattice A
+    Lattice-PartialLattice = {!!}
+
+    Lattice-Least : ∀ {a} {A : Set a}
+        {{lA : Lattice A }} → PartialLattice.HasLeastElement (Lattice-PartialLattice {{lA = lA}})
+    Lattice-Least = {!!}
+
+open import Lattice.Unit
+
+ThreeElements : Set
+ThreeElements = Path (add-via-least ((mkPartialLatticeType ⊤) ∷⁺ []) (add-via-least ((mkPartialLatticeType ⊤) ∷⁺ []) (single ((mkPartialLatticeType ⊤) ∷⁺ []))))
+
+e₁ : ThreeElements
+e₁ = MkPath (inj₁ (inj₁ tt))
+
+e₂ : ThreeElements
+e₂ = MkPath (inj₂ (inj₁ (inj₁ tt)))
+
+e₃ : ThreeElements
+e₃ = MkPath (inj₂ (inj₂ (inj₁ tt)))
+
+ex1 : e₁ ⊔̇ e₂ ≡ just e₁
+ex1 = refl
+
 -- 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)