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Move the implementation details into a private module

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2023-07-24 23:12:04 -07:00
parent 232bd65809
commit c2bc1c5421
1 changed files with 51 additions and 48 deletions
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Map.agda
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@ -8,14 +8,14 @@ module Map {a b : Level} (A : Set a) (B : Set b)
(≡-dec-A : Decidable (_≡_ {a} {A})) (≡-dec-A : Decidable (_≡_ {a} {A}))
where where
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; []; _∷_; _++_)
open import Data.List.Membership.Propositional using ()
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.Unit using (⊤)
open import Data.Empty using (⊥) open import Data.Empty using (⊥)
Map : Set (a ⊔ b) Map : Set (a ⊔ b)
@ -32,7 +32,7 @@ data Unique {c} {C : Set c} : List C → Set c where
→ Unique xs → Unique xs
→ Unique (x ∷ 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} {[]} _ _ = 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') 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 where
@ -43,12 +43,8 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') = push (help x'
help {[]} _ = x'≢x ∷ [] help {[]} _ = x'≢x ∷ []
help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
_∈_ : (A × B) → List (A × B) → Set (a ⊔ b) _∈_ : (A × B) → List (A × B) → Set (a ⊔ b)
_∈_ p m = Data.List.Membership.Propositional._∈_ p m _∈_ p m = MemProp._∈_ p m
_∈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 : ∀ (_≈_ : 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)) 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 [] = b
foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs) 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 : ∀ {a} {A : Set a} → ⊥ → A
absurd () 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) private module Impl (f : B → B → B) where
insert-keys-∈ f k v ((k' , v') ∷ xs) (here k≡k') with (≡-dec-A k k') _∈k_ : A → List (A × B) → Set a
... | yes _ = refl _∈k_ k m = MemProp._∈_ k (keys m)
... | 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)
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 : A → B → List (A × B) → List (A × B)
insert-keys-∉ f k v [] _ = refl insert k v [] = (k , v) ∷ []
insert-keys-∉ f k v ((k' , v') ∷ xs) k∉kl with (≡-dec-A k k') insert k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
... | yes k≡k' = absurd (k∉kl (here k≡k')) ... | yes _ = (k' , f v v') ∷ xs
... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ f k v xs (λ k∈kxs → k∉kl (there k∈kxs))) ... | no _ = x ∷ insert k v xs
∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈k l) merge : List (A × B) → List (A × B) → List (A × B)
∈k-dec k [] = no (λ ()) merge m₁ m₂ = foldr insert m₂ m₁
∈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 : ∀ (f : B → B → B) (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert f k v l)) insert-keys-∈ : ∀ (k : A) (v : B) (l : List (A × B)) → k ∈k l → keys l ≡ keys (insert k v l)
insert-preserves-unique f k v l u with (∈k-dec k l) insert-keys-∈ k v ((k' , v') ∷ xs) (here k≡k') with (≡-dec-A k k')
... | yes k∈kl rewrite insert-keys-∈ f k v l k∈kl = u ... | yes _ = refl
... | no k∉kl rewrite sym (insert-keys-∉ f k v l k∉kl) = Unique-append k∉kl u ... | 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)) insert-keys-∉ : ∀ (k : A) (v : B) (l : List (A × B)) → ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert k v l)
merge-preserves-unique f [] l₂ u₂ = u₂ insert-keys-∉ k v [] _ = refl
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 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' _ _ (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)) Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
where where
unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys 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 ((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) 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)