Prove the foldr-implies lemma

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-05-09 18:37:50 -07:00
parent 80069e76e6
commit 794c04eee9
2 changed files with 45 additions and 23 deletions

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