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Prove the foldr-implies lemma

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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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40
Lattice/FiniteValueMap.agda
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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
... | in₁ (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)
... | in₂ 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
... | in₁ (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)
... | in₂ 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 , _) =