diff --git a/Map.agda b/Map.agda index 24f2577..164d139 100644 --- a/Map.agda +++ b/Map.agda @@ -8,14 +8,14 @@ module Map {a b : Level} (A : Set a) (B : Set b) (≡-dec-A : Decidable (_≡_ {a} {A})) where +import Data.List.Membership.Propositional as MemProp + open import Relation.Nullary using (¬_) open import Data.Nat using (ℕ) open import Data.List using (List; []; _∷_; _++_) -open import Data.List.Membership.Propositional using () 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.Product using (_×_; _,_; Σ; proj₁ ; proj₂) -open import Data.Unit using (⊤) open import Data.Empty using (⊥) Map : Set (a ⊔ b) @@ -32,7 +32,7 @@ data Unique {c} {C : Set c} : List C → Set c where → Unique xs → Unique (x ∷ xs) -Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ Data.List.Membership.Propositional._∈_ x xs → Unique xs → Unique (xs ++ (x ∷ [])) +Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ MemProp._∈_ x xs → Unique xs → Unique (xs ++ (x ∷ [])) Unique-append {c} {C} {x} {[]} _ _ = push [] empty Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs') where @@ -43,12 +43,8 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x' help {[]} _ = x'≢x ∷ [] help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es - _∈_ : (A × B) → List (A × B) → Set (a ⊔ b) -_∈_ p m = Data.List.Membership.Propositional._∈_ p m - -_∈k_ : A → List (A × B) → Set a -_∈k_ k m = Data.List.Membership.Propositional._∈_ k (keys m) +_∈_ p m = MemProp._∈_ p 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)) @@ -63,57 +59,64 @@ 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 → 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 -... | no _ = x ∷ insert f k v xs - -merge : (B → B → B) → List (A × B) → List (A × B) → List (A × B) -merge f m₁ m₂ = foldr (insert f) m₂ m₁ - 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) +private module Impl (f : B → B → B) where + _∈k_ : A → List (A × B) → Set a + _∈k_ k m = MemProp._∈_ k (keys m) -insert-keys-∉ : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert f k v l) -insert-keys-∉ f k v [] _ = refl -insert-keys-∉ f k v ((k' , v') ∷ xs) k∉kl with (≡-dec-A k k') -... | yes k≡k' = absurd (k∉kl (here k≡k')) -... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ f k v xs (λ k∈kxs → k∉kl (there k∈kxs))) + insert : A → B → List (A × B) → List (A × B) + insert k v [] = (k , v) ∷ [] + insert k v (x@(k' , v') ∷ xs) with ≡-dec-A k k' + ... | yes _ = (k' , f v v') ∷ xs + ... | no _ = x ∷ insert k v xs -∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈k l) -∈k-dec k [] = no (λ ()) -∈k-dec k ((k' , v) ∷ xs) with (≡-dec-A k k') -... | yes k≡k' = yes (here k≡k') -... | no k≢k' with (∈k-dec k xs) -... | yes k∈kxs = yes (there k∈kxs) -... | no k∉kxs = no witness - where - witness : ¬ k ∈k ((k' , v) ∷ xs) - witness (here k≡k') = k≢k' k≡k' - witness (there k∈kxs) = k∉kxs k∈kxs + merge : List (A × B) → List (A × B) → List (A × B) + merge m₁ m₂ = foldr insert m₂ m₁ -insert-preserves-unique : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert f k v l)) -insert-preserves-unique f k v l u with (∈k-dec k l) -... | yes k∈kl rewrite insert-keys-∈ f k v l k∈kl = u -... | no k∉kl rewrite sym (insert-keys-∉ f k v l k∉kl) = Unique-append k∉kl u + insert-keys-∈ : ∀ (k : A) (v : B) (l : List (A × B)) → k ∈k l → keys l ≡ keys (insert k v l) + insert-keys-∈ 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-∈ k v ((k' , _) ∷ xs) (there k∈kxs) with (≡-dec-A k k') + ... | yes _ = refl + ... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ k v xs k∈kxs) -merge-preserves-unique : ∀ (f : B → B → B) (l₁ l₂ : List (A × B)) → Unique (keys l₂) → Unique (keys (merge f l₁ l₂)) -merge-preserves-unique f [] l₂ u₂ = u₂ -merge-preserves-unique f ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-unique f k₁ v₁ (merge f xs₁ l₂) (merge-preserves-unique f xs₁ l₂ u₂) + insert-keys-∉ : ∀ (k : A) (v : B) (l : List (A × B)) → ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert k v l) + insert-keys-∉ k v [] _ = refl + insert-keys-∉ k v ((k' , v') ∷ xs) k∉kl with (≡-dec-A k k') + ... | yes k≡k' = absurd (k∉kl (here k≡k')) + ... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ k v xs (λ k∈kxs → k∉kl (there 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' + ∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈k l) + ∈k-dec k [] = no (λ ()) + ∈k-dec k ((k' , v) ∷ xs) with (≡-dec-A k k') + ... | yes k≡k' = yes (here k≡k') + ... | no k≢k' with (∈k-dec k xs) + ... | yes k∈kxs = yes (there k∈kxs) + ... | no k∉kxs = no witness + where + witness : ¬ k ∈k ((k' , v) ∷ xs) + witness (here k≡k') = k≢k' k≡k' + witness (there k∈kxs) = k∉kxs k∈kxs + + insert-preserves-unique : ∀ (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert k v l)) + insert-preserves-unique k v l u with (∈k-dec k l) + ... | yes k∈kl rewrite insert-keys-∈ k v l k∈kl = u + ... | no k∉kl rewrite sym (insert-keys-∉ k v l k∉kl) = Unique-append k∉kl u + + merge-preserves-unique : ∀ (l₁ l₂ : List (A × B)) → Unique (keys l₂) → Unique (keys (merge l₁ l₂)) + 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) + +Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → MemProp._∈_ (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 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) + +module _ (f : B → B → B) where + open Impl f public using (insert; merge)