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Finally prove the provenance properties of merge.

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Danila Fedorin 2023-07-26 20:58:41 -07:00
parent 461732244a
commit 77b8b6fad9
1 changed files with 23 additions and 25 deletions
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Map.agda
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@ -135,9 +135,9 @@ private module ImplInsert (f : B → B → B) where
∈-cong f (here c≡c') = here (cong f c≡c')
∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
locate : ∀ (k : A) (l : List (A × B)) → k ∈k l → Σ B (λ v → (k , v) ∈ l)
locate k ((k' , v) ∷ xs) (here k≡k') rewrite k≡k' = (v , here refl)
locate k ((k' , v) ∷ xs) (there k∈kxs) = let (v , k,v∈xs) = locate k xs k∈kxs in (v , there k,v∈xs)
locate : ∀ {k : A} {l : List (A × B)} → k ∈k l → Σ B (λ v → (k , v) ∈ l)
locate {k} {(k' , v) ∷ xs} (here k≡k') rewrite k≡k' = (v , here refl)
locate {k} {(k' , v) ∷ xs} (there k∈kxs) = let (v , k,v∈xs) = locate k∈kxs in (v , there k,v∈xs)
insert-preserves-Unique : ∀ (k : A) (v : B) (l : List (A × B))
→ Unique (keys l) → Unique (keys (insert k v l))
@ -167,7 +167,7 @@ private module ImplInsert (f : B → B → B) where
insert-preserves-∈k-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
¬ k ≡ k' → k ∈k l → k ∈k insert k' v' l
insert-preserves-∈k-right k k' v' l k≢k' k∈kl =
let (v , k,v∈l) = locate k l k∈kl
let (v , k,v∈l) = locate k∈kl
in ∈-cong proj₁ (insert-preserves-∈-right k k' v v' l k≢k' k,v∈l)
insert-preserves-∉-right : ∀ (k k' : A) (v' : B) (l : List (A × B)) →
@ -284,28 +284,26 @@ module _ (f : B → B → B) where
MergeResult {k} {m₁} {m₂} (in₂ v₂ _ _) = (k , v₂) ∈ merge m₁ m₂
merge-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k merge m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
merge-provenance = {!!}
merge-provenance m₁@(l₁ , u₁) m₂@(l₂ , u₂) k k∈km₁m₂
with ∈k-dec k l₁ | ∈k-dec k l₂
... | yes k∈kl₁ | yes k∈kl₂ =
let
(v₁ , k,v₁∈l₁) = locate k∈kl₁
(v₂ , k,v₂∈l₂) = locate k∈kl₂
in
(both v₁ v₂ k,v₁∈l₁ k,v₂∈l₂ , merge-combines k v₁ v₂ l₁ l₂ u₁ u₂ k,v₁∈l₁ k,v₂∈l₂)
... | yes k∈kl₁ | no k∉kl₂ =
let
(v₁ , k,v₁∈l₁) = locate k∈kl₁
in
(in₁ v₁ k,v₁∈l₁ k∉kl₂ , merge-preserves-keys₂ k v₁ l₁ l₂ u₁ k,v₁∈l₁ k∉kl₂)
... | no k∉kl₁ | yes k∈kl₂ =
let
(v₂ , k,v₂∈l₂) = locate k∈kl₂
in
(in₂ v₂ k∉kl₁ k,v₂∈l₂ , merge-preserves-keys₁ k v₂ l₁ l₂ k∉kl₁ k,v₂∈l₂)
... | no k∉kl₁ | no k∉kl₂ = absurd (merge-preserves-∉ k l₁ l₂ k∉kl₁ k∉kl₂ k∈km₁m₂)
-- ------------------------------------------------------------------------
--
-- The following can be proven using plain properties of insert:
--
-- prove that ¬ k ∈k m₁ → (k , v) ∈ m₂ → (k , v) ∈ merge m₁ m₂ (done)
-- prove that k ≢ k' → (k , v) ∈ m → (k , v) ∈ insert k' v' m (done)
-- prove that (k , v) ∈ m₁ → ¬ k ∈k m₂ → (k , v) ∈ merge m₁ m₂ (done)
-- prove that ¬ k ∈k m → (k , v) ∈ insert k v m (done)
--
-- ------------------------------------------------------------------------
--
-- The following relies on uniqueness, since inserts stops after the first encounter.
--
-- prove that (k , v) ∈ m₁ → (k , v') ∈ m₂ → (k, f v v') ∈ merge m₁ m₂ (done)
--
-- ------------------------------------------------------------------------
--
-- The following can probably be proven via keys.
--
-- prove that k ∉k m₁ → k ∉k m₂ → k ∉k merge m₁ m₂ (done)
module _ (_≈_ : B → B → Set b) where
open ImplRelation _≈_ renaming (subset to subset-impl)