Prove the foldr-implies lemma
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -11,6 +11,7 @@ open import Data.Empty using (⊥-elim)
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open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
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open import Data.Nat using (suc)
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open import Data.Product using (_×_; proj₁; proj₂; _,_)
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open import Data.Sum using (inj₁; inj₂)
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open import Data.List using (List; _∷_; []; foldr; foldl; cartesianProduct; cartesianProductWith)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Data.List.Relation.Unary.Any as Any using ()
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@ -19,7 +20,7 @@ open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Data.Unit using (⊤)
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open import Function using (_∘_; flip)
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open import Utils using (Pairwise; _⇒_)
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open import Utils using (Pairwise; _⇒_; _∨_)
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import Lattice.FiniteValueMap
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open IsFiniteHeightLattice isFiniteHeightLatticeˡ
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@ -62,6 +63,11 @@ module WithProg (prog : Program) where
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; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
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; ⊔-idemp to ⊔ᵛ-idemp
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)
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open Lattice.FiniteValueMap.IterProdIsomorphism _≟ˢ_ isLatticeˡ
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using ()
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renaming
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( Provenance-union to Provenance-unionᵐ
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)
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open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ
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using ()
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renaming
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@ -236,7 +242,11 @@ module WithProg (prog : Program) where
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module WithInterpretation (latticeInterpretationˡ : LatticeInterpretation isLatticeˡ) where
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open LatticeInterpretation latticeInterpretationˡ
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using ()
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renaming (⟦_⟧ to ⟦_⟧ˡ; ⟦⟧-respects-≈ to ⟦⟧ˡ-respects-≈ˡ)
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renaming
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( ⟦_⟧ to ⟦_⟧ˡ
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; ⟦⟧-respects-≈ to ⟦⟧ˡ-respects-≈ˡ
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; ⟦⟧-⊔-∨ to ⟦⟧ˡ-⊔ˡ-∨
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)
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⟦_⟧ᵛ : VariableValues → Env → Set
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⟦_⟧ᵛ vs ρ = ∀ {k l} → (k , l) ∈ᵛ vs → ∀ {v} → (k , v) Language.∈ ρ → ⟦ l ⟧ˡ v
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@ -251,9 +261,21 @@ module WithProg (prog : Program) where
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in
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⟦⟧ˡ-respects-≈ˡ (≈ˡ-sym l≈l') v ⟦l'⟧v
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⟦⟧ᵛ-⊔ᵛ-∨ : ∀ {vs₁ vs₂ : VariableValues} → (⟦ vs₁ ⟧ᵛ ∨ ⟦ vs₂ ⟧ᵛ) ⇒ ⟦ vs₁ ⊔ᵛ vs₂ ⟧ᵛ
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⟦⟧ᵛ-⊔ᵛ-∨ {vs₁} {vs₂} ρ ⟦vs₁⟧ρ∨⟦vs₂⟧ρ {k} {l} k,l∈vs₁₂ {v} k,v∈ρ
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with ((l₁ , l₂) , (refl , (k,l₁∈vs₁ , k,l₂∈vs₂)))
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← Provenance-unionᵐ vs₁ vs₂ k,l∈vs₁₂
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with ⟦vs₁⟧ρ∨⟦vs₂⟧ρ
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... | inj₁ ⟦vs₁⟧ρ = ⟦⟧ˡ-⊔ˡ-∨ {l₁} {l₂} v (inj₁ (⟦vs₁⟧ρ k,l₁∈vs₁ k,v∈ρ))
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... | inj₂ ⟦vs₂⟧ρ = ⟦⟧ˡ-⊔ˡ-∨ {l₁} {l₂} v (inj₂ (⟦vs₂⟧ρ k,l₂∈vs₂ k,v∈ρ))
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⟦⟧ᵛ-foldr : ∀ {vs : VariableValues} {vss : List VariableValues} {ρ : Env} →
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⟦ vs ⟧ᵛ ρ → vs ∈ˡ vss → ⟦ foldr _⊔ᵛ_ ⊥ᵛ vss ⟧ᵛ ρ
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⟦⟧ᵛ-foldr = {!!}
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⟦⟧ᵛ-foldr {vs} {vs ∷ vss'} {ρ = ρ} ⟦vs⟧ρ (Any.here refl) =
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⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ (inj₁ ⟦vs⟧ρ)
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⟦⟧ᵛ-foldr {vs} {vs' ∷ vss'} {ρ = ρ} ⟦vs⟧ρ (Any.there vs∈vss') =
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⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs'} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ
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(inj₂ (⟦⟧ᵛ-foldr ⟦vs⟧ρ vs∈vss'))
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InterpretationValid : Set
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InterpretationValid = ∀ {vs ρ e v} → ρ , e ⇒ᵉ v → ⟦ vs ⟧ᵛ ρ → ⟦ eval e vs ⟧ˡ v
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@ -153,6 +153,26 @@ module IterProdIsomorphism where
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fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
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in (v'' , (v'≈v'' , there k',v''∈fm'₁))
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FromBothMaps : ∀ (k : A) (v : B) {ks : List A} (fm₁ fm₂ : FiniteMap ks) → Set
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FromBothMaps k v fm₁ fm₂ =
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Σ (B × B)
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(λ (v₁ , v₂) → ( (v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
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Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} →
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(k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → FromBothMaps k v fm₁ fm₂
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Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
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with Expr-Provenance-≡ ((` m₁) ∪ (` m₂)) k,v∈fm₁fm₂
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... | in₁ (single k,v∈m₁) k∉km₂
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with k∈km₁ ← (forget k,v∈m₁)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₂ k∈km₁)
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... | in₂ k∉km₁ (single k,v∈m₂)
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with k∈km₂ ← (forget k,v∈m₂)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₁ k∈km₂)
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... | bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) =
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((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
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private
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first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
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Σ B (λ v → (k , v) ∈ᵐ fm)
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@ -204,26 +224,6 @@ module IterProdIsomorphism where
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k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
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k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'
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FromBothMaps : ∀ (k : A) (v : B) {ks : List A} (fm₁ fm₂ : FiniteMap ks) → Set
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FromBothMaps k v fm₁ fm₂ =
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Σ (B × B)
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(λ (v₁ , v₂) → ( (v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
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Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} →
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(k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → FromBothMaps k v fm₁ fm₂
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Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
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with Expr-Provenance-≡ ((` m₁) ∪ (` m₂)) k,v∈fm₁fm₂
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... | in₁ (single k,v∈m₁) k∉km₂
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with k∈km₁ ← (forget k,v∈m₁)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₂ k∈km₁)
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... | in₂ k∉km₁ (single k,v∈m₂)
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with k∈km₂ ← (forget k,v∈m₂)
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rewrite trans ks₁≡ks (sym ks₂≡ks) =
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⊥-elim (k∉km₁ k∈km₂)
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... | bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) =
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((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
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pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
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pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
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pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) =
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