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Start moving away from purely list-based maps.

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The eventual goal is to make a map be a list and a proof
that all the keys are unique.

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2023-07-24 20:38:34 -07:00
parent f2e72b54ce
commit c50195942d
1 changed files with 34 additions and 19 deletions
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53
Map.agda
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@ -22,45 +22,60 @@ open import Data.Empty using (⊥)
Map : Set (a ⊔ b)
Map = List (A × B)
data Unique : List (A × B) → Set (a ⊔ b) where
empty : Unique []
push : forall {k : A} {v : B} {xs : List (A × B)}
→ All (λ (k' , _) → ¬ k ≡ k') xs
→ Unique xs
→ Unique ((k , v) ∷ xs)
keys : List (A × B) → List A
keys [] = []
keys ((k , v) ∷ xs) = k ∷ keys xs
_∈_ : (A × B) → Map → Set (a ⊔ b)
data Unique {c} {C : Set c} : List C → Set c where
empty : Unique []
push : forall {x : C} {xs : List C}
→ All (λ x' → ¬ x ≡ x') xs
→ Unique xs
→ Unique (x ∷ xs)
_∈_ : (A × B) → List (A × B) → Set (a ⊔ b)
_∈_ p m = Data.List.Membership.Propositional._∈_ p m
subset : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
_∈k_ : A → List (A × B) → Set a
_∈k_ k m = Data.List.Membership.Propositional._∈_ k (keys m)
subset : ∀ (_≈_ : B → B → Set b) → List (A × B) → List (A × B) → Set (a ⊔ b)
subset _≈_ m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
lift : ∀ (_≈_ : B → B → Set b) → Map → Map → Set (a ⊔ b)
lift : ∀ (_≈_ : B → B → Set b) → List (A × B) → List (A × B) → Set (a ⊔ b)
lift _≈_ m₁ m₂ = (m₁ ⊆ m₂) × (m₂ ⊆ m₁)
where
_⊆_ : Map → Map → Set (a ⊔ b)
_⊆_ : List (A × B) → List (A × B) → Set (a ⊔ b)
_⊆_ = subset _≈_
foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> Map -> C
foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> List (A × B) -> C
foldr f b [] = b
foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
insert : (B → B → B) → A → B → Map → Map
insert : (B → B → B) → A → B → List (A × B) → List (A × B)
insert f k v [] = (k , v) ∷ []
insert f k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
... | yes _ = (k , f v v') ∷ xs
... | yes _ = (k' , f v v') ∷ xs
... | no _ = x ∷ insert f k v xs
merge : (B → B → B) → Map → Map → Map
merge : (B → B → B) → List (A × B) → List (A × B) → List (A × B)
merge f m₁ m₂ = foldr (insert f) m₂ m₁
Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique ((k , v) ∷ xs) → Data.List.Membership.Propositional._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
absurd : ∀ {a} {A : Set a} → ⊥ → A
absurd ()
insert-keys-∈ : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → k ∈k l → keys l ≡ keys (insert f k v l)
insert-keys-∈ f k v ((k' , v') ∷ xs) (here k≡k') with (≡-dec-A k k')
... | yes _ = refl
... | no k≢k' = absurd (k≢k' k≡k')
insert-keys-∈ f k v ((k' , _) ∷ xs) (there k∈kxs) with (≡-dec-A k k')
... | yes _ = refl
... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ f k v xs k∈kxs)
Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → Data.List.Membership.Propositional._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
where
absurd : ∀ {a} {A : Set a} → ⊥ → A
absurd ()
unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ (k' , _) → ¬ k ≡ k') xs × (k , v') ∈ xs)
unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)