Add intermediate state for insertion proofs
This commit is contained in:
parent
6b51cd4050
commit
489b0532df
73
Map.agda
73
Map.agda
|
@ -12,15 +12,14 @@ import Data.List.Membership.Propositional as MemProp
|
||||||
|
|
||||||
open import Relation.Nullary using (¬_)
|
open import Relation.Nullary using (¬_)
|
||||||
open import Data.Nat using (ℕ)
|
open import Data.Nat using (ℕ)
|
||||||
open import Data.List using (List; []; _∷_; _++_)
|
open import Data.List using (List; map; []; _∷_; _++_)
|
||||||
open import Data.List.Relation.Unary.All using (All; []; _∷_)
|
open import Data.List.Relation.Unary.All using (All; []; _∷_)
|
||||||
open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
|
open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
|
||||||
open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
|
open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
|
||||||
open import Data.Empty using (⊥)
|
open import Data.Empty using (⊥)
|
||||||
|
|
||||||
keys : List (A × B) → List A
|
keys : List (A × B) → List A
|
||||||
keys [] = []
|
keys = map proj₁
|
||||||
keys ((k , v) ∷ xs) = k ∷ keys xs
|
|
||||||
|
|
||||||
data Unique {c} {C : Set c} : List C → Set c where
|
data Unique {c} {C : Set c} : List C → Set c where
|
||||||
empty : Unique []
|
empty : Unique []
|
||||||
|
@ -63,6 +62,7 @@ private module ImplRelation (_≈_ : B → B → Set b) where
|
||||||
subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
|
subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
|
||||||
|
|
||||||
private module ImplInsert (f : B → B → B) where
|
private module ImplInsert (f : B → B → B) where
|
||||||
|
open import Data.List using (map)
|
||||||
open MemProp using (_∈_)
|
open MemProp using (_∈_)
|
||||||
|
|
||||||
private
|
private
|
||||||
|
@ -117,6 +117,52 @@ private module ImplInsert (f : B → B → B) where
|
||||||
merge-preserves-Unique [] l₂ u₂ = u₂
|
merge-preserves-Unique [] l₂ u₂ = u₂
|
||||||
merge-preserves-Unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-Unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-Unique xs₁ l₂ u₂)
|
merge-preserves-Unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-Unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-Unique xs₁ l₂ u₂)
|
||||||
|
|
||||||
|
insert-preserves-other-keys : ∀ (k k' : A) (v v' : B) (l : List (A × B)) → ¬ k ≡ k' → (k , v) ∈ l → (k , v) ∈ insert k' v' l
|
||||||
|
insert-preserves-other-keys k k' v v' (x ∷ xs) k≢k' (here k,v=x) rewrite sym k,v=x with ≡-dec-A k' k
|
||||||
|
... | yes k'≡k = absurd (k≢k' (sym k'≡k))
|
||||||
|
... | no _ = here refl
|
||||||
|
insert-preserves-other-keys k k' v v' ((k'' , _) ∷ xs) k≢k' (there k,v∈xs) with ≡-dec-A k' k''
|
||||||
|
... | yes _ = there k,v∈xs
|
||||||
|
... | no _ = there (insert-preserves-other-keys k k' v v' xs k≢k' k,v∈xs)
|
||||||
|
|
||||||
|
merge-preserves-keys₁ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) → ¬ k ∈k l₁ → (k , v) ∈ l₂ → (k , v) ∈ merge l₁ l₂
|
||||||
|
merge-preserves-keys₁ k v [] l₂ _ k,v∈l₂ = k,v∈l₂
|
||||||
|
merge-preserves-keys₁ k v ((k' , v') ∷ xs₁) l₂ k∉kl₁ k,v∈l₂ =
|
||||||
|
let recursion = merge-preserves-keys₁ k v xs₁ l₂ (λ k∈xs₁ → k∉kl₁ (there k∈xs₁)) k,v∈l₂
|
||||||
|
in insert-preserves-other-keys k k' v v' _ (λ k≡k' → k∉kl₁ (here k≡k')) recursion
|
||||||
|
|
||||||
|
insert-preserves-other-key : ∀ (k : A) (v : B) (l : List (A × B)) → ¬ k ∈k l → (k , v) ∈ insert k v l
|
||||||
|
insert-preserves-other-key k v [] k∉kl = here refl
|
||||||
|
insert-preserves-other-key k v ((k' , v') ∷ xs) k∉kl with ≡-dec-A k k'
|
||||||
|
... | yes k≡k' = absurd (k∉kl (here k≡k'))
|
||||||
|
... | no _ = there (insert-preserves-other-key k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
|
||||||
|
|
||||||
|
|
||||||
|
∈-cong : ∀ {c d} {C : Set c} {D : Set d} {c : C} {l : List C} → (f : C → D) → c ∈ l → f c ∈ map f l
|
||||||
|
∈-cong f (here c≡c') = here (cong f c≡c')
|
||||||
|
∈-cong f (there c∈xs) = there (∈-cong f c∈xs)
|
||||||
|
|
||||||
|
-- prove that ¬ k ∈k m → (k , v) ∈ insert k v m
|
||||||
|
merge-preserves-keys₂ : ∀ (k : A) (v : B) (l₁ l₂ : List (A × B)) → Unique (keys l₁) → (k , v) ∈ l₁ → ¬ k ∈k l₂ → (k , v) ∈ merge l₁ l₂
|
||||||
|
merge-preserves-keys₂ k v ((k' , v') ∷ xs₁) l₂ (push k'≢xs₁ uxs₁) (here _) k∉kl₂ = {!!} -- hard!
|
||||||
|
-- where
|
||||||
|
-- rest : ∀ (l l' : List (A × B)) → All (λ k'' → ¬ k ≡ k'') (keys l) → ¬ k ∈k l' → ¬ k ∈k merge l l'
|
||||||
|
-- rest [] l' _ k∉kl' = k∉kl'
|
||||||
|
-- rest l [] (k≢l) _ = help
|
||||||
|
-- where
|
||||||
|
-- help : ∀ (l : List (A × B)) → All (λ k'' → ¬ k ≡ k'') (keys l) → ¬ k ∈k l
|
||||||
|
-- help [] _ ()
|
||||||
|
-- help ((k'' , _) ∷ xs) (k≢k'' ∷ k≢xs) (here k≡k'') = k≢k'' k≡k''
|
||||||
|
-- help ((k'' , _) ∷ xs) (k≢k'' ∷ k≢xs) (there k∈kxs) = help xs k≢xs k∈kxs
|
||||||
|
-- -- rest (x@(k'' , _) ∷ xs) l' (k≢k'' ∷ k≢xs) k∉kl' with (≡-dec-A k'' = (rest xs l' k≢xs k∉kl')
|
||||||
|
-- -- where
|
||||||
|
-- -- help : ¬ k ∈k (merge (x ∷ xs) l') -- insert x (merge xs l')
|
||||||
|
-- -- help (here k≡k'') = {!!}
|
||||||
|
-- -- help (there k∈) = {!!}
|
||||||
|
-- -- let nested = (rest xs l' k≢xs k∉kl')
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
Map : Set (a ⊔ b)
|
Map : Set (a ⊔ b)
|
||||||
Map = Σ (List (A × B)) (λ l → Unique (keys l))
|
Map = Σ (List (A × B)) (λ l → Unique (keys l))
|
||||||
|
|
||||||
|
@ -154,6 +200,27 @@ module _ (f : B → B → B) where
|
||||||
merge-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k merge m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
|
merge-provenance : ∀ (m₁ m₂ : Map) (k : A) → k ∈k merge m₁ m₂ → Σ (Provenance k m₁ m₂) MergeResult
|
||||||
merge-provenance = {!!}
|
merge-provenance = {!!}
|
||||||
|
|
||||||
|
-- ------------------------------------------------------------------------
|
||||||
|
--
|
||||||
|
-- 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 (stuck)
|
||||||
|
-- prove that ¬ k ∈k m → (k , v) ∈ insert k v m
|
||||||
|
--
|
||||||
|
-- ------------------------------------------------------------------------
|
||||||
|
--
|
||||||
|
-- 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
|
||||||
|
--
|
||||||
|
-- ------------------------------------------------------------------------
|
||||||
|
--
|
||||||
|
-- The following can probably be proven via keys.
|
||||||
|
--
|
||||||
|
-- prove that k ∉k m₁ → k ∉k m₂ → k ∉k merge m₁ m₂
|
||||||
|
|
||||||
module _ (_≈_ : B → B → Set b) where
|
module _ (_≈_ : B → B → Set b) where
|
||||||
open ImplRelation _≈_ renaming (subset to subset-impl)
|
open ImplRelation _≈_ renaming (subset to subset-impl)
|
||||||
|
|
||||||
|
|
Loading…
Reference in New Issue
Block a user