Prove commutativity and associativity of value joining
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -80,10 +80,7 @@ record IsPartialSemilattice {a} {A : Set a}
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renaming (≈-trans to ≈?-trans; ≈-sym to ≈?-sym; ≈-refl to ≈?-refl)
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≈-≼-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≼ a₃ → a₂ ≼ a₄
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≈-≼-cong {a₁} {a₂} {a₃} {a₄} a₁≈a₂ a₃≈a₄ a₁≼a₃
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with a₁ ⊔? a₃ | a₂ ⊔? a₄ | ≈-⊔-cong a₁≈a₂ a₃≈a₄ | a₁≼a₃
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... | just a₁⊔a₃ | just a₂⊔a₄ | ≈-just a₁⊔a₃≈a₂≈a₄ | ≈-just a₁⊔a₃≈a₃
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= ≈-just (≈-trans (≈-trans (≈-sym a₁⊔a₃≈a₂≈a₄) a₁⊔a₃≈a₃) a₃≈a₄)
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≈-≼-cong {a₁} {a₂} {a₃} {a₄} a₁≈a₂ a₃≈a₄ a₁≼a₃ = ≈?-trans (≈?-trans (≈?-sym (≈-⊔-cong a₁≈a₂ a₃≈a₄)) a₁≼a₃) (≈-just a₃≈a₄)
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-- curious: similar property (properties?) to partial commutative monoids (PCMs)
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-- from Iris.
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@ -242,10 +239,7 @@ CombineForPLT a = ∀ (plt : PartialLatticeType a) → PartialLatticeType.EltTyp
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lvCombine : ∀ {a} (f : CombineForPLT a) (l : List (PartialLatticeType a)) (lv₁ lv₂ : ListValue l) → Maybe (ListValue l)
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lvCombine _ [] _ _ = nothing
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lvCombine f (plt ∷ plts) (inj₁ v₁) (inj₁ v₂)
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with f plt v₁ v₂
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... | just v = just (inj₁ v)
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... | nothing = nothing
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lvCombine f (plt ∷ plts) (inj₁ v₁) (inj₁ v₂) = Maybe.map inj₁ (f plt v₁ v₂)
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lvCombine f (_ ∷ plts) (inj₂ lv₁) (inj₂ lv₂) = Maybe.map inj₂ (lvCombine f plts lv₁ lv₂)
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lvCombine _ (_ ∷ plts) (inj₁ _) (inj₂ _) = nothing
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lvCombine _ (_ ∷ plts) (inj₂ _) (inj₁ _) = nothing
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@ -326,29 +320,95 @@ eqPath'-refl (add-via-greatest l ls) (inj₂ p) = eqPath'-refl ls p
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-- Now that we can compare paths for "equality", we can state and
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-- prove theorems such as commutativity and idempotence.
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data Expr {a} (A : Set a) : Set a where
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`_ : A → Expr A
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_∨_ : Expr A → Expr A → Expr A
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map : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) → Expr A → Expr B
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map f (` x) = ` (f x)
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map f (e₁ ∨ e₂) = map f e₁ ∨ map f e₂
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eval : ∀ {a} {A : Set a} (_⊔?_ : A → A → Maybe A) (e : Expr A) → Maybe A
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eval _⊔?_ (` x) = just x
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eval _⊔?_ (e₁ ∨ e₂) = (eval _⊔?_ e₁) >>= (λ v₁ → (eval _⊔?_ e₂) >>= (v₁ ⊔?_))
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-- the two lvCombine homomorphism properties essentially say that, when all layer values
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-- are from the same type, the lvCombine preserves the structure of operations
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-- on these layer values.
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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)
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lvCombine-homo₁ plt plts f (` _) = refl
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lvCombine-homo₁ plt plts f (e₁ ∨ e₂)
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rewrite lvCombine-homo₁ plt plts f e₁ rewrite lvCombine-homo₁ plt plts f e₂
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with eval (f plt) e₁ | eval (f plt) e₂
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... | just x₁ | just x₂ = refl
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... | nothing | nothing = refl
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... | just x₁ | nothing = refl
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... | nothing | just x₂ = refl
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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)
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lvCombine-homo₂ plt plts f (` _) = refl
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lvCombine-homo₂ plt plts f (e₁ ∨ e₂)
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rewrite lvCombine-homo₂ plt plts f e₁ rewrite lvCombine-homo₂ plt plts f e₂
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with eval (lvCombine f plts) e₁ | eval (lvCombine f plts) e₂
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... | just x₁ | just x₂ = refl
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... | nothing | nothing = refl
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... | just x₁ | nothing = refl
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... | nothing | just x₂ = refl
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lvCombine-comm : ∀ {a} (f : CombineForPLT a) →
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(∀ plt → PartialComm (PartialLatticeType._≈_ plt) (f plt)) →
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∀ (L : List (PartialLatticeType a)) →
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PartialComm (eqL L) (lvCombine f L)
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lvCombine-comm f f-comm L@(plt ∷ plts) lv₁@(inj₁ v₁) (inj₁ v₂)
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with f plt v₁ v₂
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| f plt v₂ v₁
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| f-comm plt v₁ v₂
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... | _ | _ | ≈-just v₁v₂≈v₂v₁ = ≈-just v₁v₂≈v₂v₁
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... | _ | _ | ≈-nothing = ≈-nothing
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lvCombine-comm f f-comm (plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂)
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with lvCombine f plts lv₁ lv₂ | lvCombine f plts lv₂ lv₁ | lvCombine-comm f f-comm plts lv₁ lv₂
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... | _ | _ | ≈-just lv₁lv₂≈lv₂lv₁ = ≈-just lv₁lv₂≈lv₂lv₁
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... | _ | _ | ≈-nothing = ≈-nothing
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rewrite lvCombine-homo₁ plt plts f ((` v₁) ∨ (` v₂)) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (f-comm plt v₁ v₂)
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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)
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lvCombine-comm f f-comm (plt ∷ plts) (inj₁ v₁) (inj₂ lv₂) = ≈-nothing
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lvCombine-comm f f-comm (plt ∷ plts) (inj₂ lv₁) (inj₁ v₂) = ≈-nothing
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lvCombine-assoc : ∀ {a} (f : CombineForPLT a) →
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(∀ plt → PartialAssoc (PartialLatticeType._≈_ plt) (f plt)) →
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∀ (L : List (PartialLatticeType a)) →
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PartialAssoc (eqL L) (lvCombine f L)
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ v₁) (inj₁ v₂) (inj₁ v₃)
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rewrite lvCombine-homo₁ plt plts f (((` v₁) ∨ (` v₂)) ∨ (` v₃))
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rewrite lvCombine-homo₁ plt plts f ((` v₁) ∨ ((` v₂) ∨ (` v₃)))
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= lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (f-assoc plt v₁ v₂ v₃)
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ _) (inj₁ v₂) (inj₁ v₃)
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with f plt v₂ v₃
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... | just _ = ≈-nothing
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... | nothing = ≈-nothing
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ _) (inj₂ _) (inj₁ _) = ≈-nothing
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂) (inj₁ _)
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with lvCombine f plts lv₁ lv₂
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... | just _ = ≈-nothing
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... | nothing = ≈-nothing
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ v₁) (inj₁ v₂) (inj₂ _)
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with f plt v₁ v₂
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... | just _ = ≈-nothing
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... | nothing = ≈-nothing
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ _) (inj₁ _) (inj₂ _) = ≈-nothing
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₁ _) (inj₂ lv₂) (inj₂ lv₃)
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with lvCombine f plts lv₂ lv₃
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... | just _ = ≈-nothing
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... | nothing = ≈-nothing
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lvCombine-assoc f f-assoc L@(plt ∷ plts) (inj₂ lv₁) (inj₂ lv₂) (inj₂ lv₃)
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rewrite lvCombine-homo₂ plt plts f (((` lv₁) ∨ (` lv₂)) ∨ (` lv₃))
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rewrite lvCombine-homo₂ plt plts f ((` lv₁) ∨ ((` lv₂) ∨ (` lv₃)))
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= lift-≈-map inj₂ _ _ (λ _ _ x → x) _ _ (lvCombine-assoc f f-assoc plts lv₁ lv₂ lv₃)
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lvJoin-comm : ∀ {a} (L : List (PartialLatticeType a)) → PartialComm (eqL L) (lvJoin L)
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lvJoin-comm = lvCombine-comm PartialLatticeType._⊔?_ PartialLatticeType.⊔-comm
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lvJoin-assoc : ∀ {a} (L : List (PartialLatticeType a)) → PartialAssoc (eqL L) (lvJoin L)
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lvJoin-assoc = lvCombine-assoc PartialLatticeType._⊔?_ PartialLatticeType.⊔-assoc
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lvMeet-comm : ∀ {a} (L : List (PartialLatticeType a)) → PartialComm (eqL L) (lvMeet L)
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lvMeet-comm = lvCombine-comm PartialLatticeType._⊓?_ PartialLatticeType.⊓-comm
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lvMeet-assoc : ∀ {a} (L : List (PartialLatticeType a)) → PartialAssoc (eqL L) (lvMeet L)
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lvMeet-assoc = lvCombine-assoc PartialLatticeType._⊓?_ PartialLatticeType.⊓-assoc
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pathJoin'-comm : ∀ {a} {Ls : Layers a} → PartialComm (eqPath' Ls) (pathJoin' Ls)
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pathJoin'-comm {Ls = single l} lv₁ lv₂ = lvJoin-comm (toList l) lv₁ lv₂
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pathJoin'-comm {Ls = add-via-least l ls} (inj₁ lv₁) (inj₁ lv₂) = lift-≈-map inj₁ _ _ (λ _ _ x → x) _ _ (lvJoin-comm (toList l) lv₁ lv₂)
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