137 lines
6.8 KiB
Agda
137 lines
6.8 KiB
Agda
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; cong)
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open import Relation.Binary.Definitions using (Decidable)
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open import Relation.Binary.Core using (Rel)
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open import Relation.Nullary using (Dec; yes; no)
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open import Agda.Primitive using (Level; _⊔_)
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module Map {a b : Level} (A : Set a) (B : Set b)
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(≡-dec-A : Decidable (_≡_ {a} {A}))
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where
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import Data.List.Membership.Propositional as MemProp
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open import Relation.Nullary using (¬_)
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open import Data.Nat using (ℕ)
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open import Data.List using (List; []; _∷_; _++_)
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open import Data.List.Relation.Unary.All using (All; []; _∷_)
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open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
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open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
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open import Data.Empty using (⊥)
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keys : List (A × B) → List A
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keys [] = []
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keys ((k , v) ∷ xs) = k ∷ keys xs
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data Unique {c} {C : Set c} : List C → Set c where
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empty : Unique []
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push : forall {x : C} {xs : List C}
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→ All (λ x' → ¬ x ≡ x') xs
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→ Unique xs
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→ Unique (x ∷ xs)
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Map : Set (a ⊔ b)
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Map = Σ (List (A × B)) (λ l → Unique (keys l))
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Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} → ¬ MemProp._∈_ x xs → Unique xs → Unique (xs ++ (x ∷ []))
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Unique-append {c} {C} {x} {[]} _ _ = push [] empty
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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')
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where
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x'≢x : ¬ x' ≡ x
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x'≢x x'≡x = x∉xs (here (sym x'≡x))
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help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
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help {[]} _ = x'≢x ∷ []
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help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
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_∈_ : (A × B) → List (A × B) → Set (a ⊔ b)
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_∈_ p m = MemProp._∈_ p m
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foldr : ∀ {c} {C : Set c} → (A → B → C → C) -> C -> List (A × B) -> C
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foldr f b [] = b
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foldr f b ((k , v) ∷ xs) = f k v (foldr f b xs)
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absurd : ∀ {a} {A : Set a} → ⊥ → A
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absurd ()
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private module ImplRelation (_≈_ : B → B → Set b) where
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subset : List (A × B) → List (A × B) → Set (a ⊔ b)
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subset m₁ m₂ = ∀ (k : A) (v : B) → (k , v) ∈ m₁ → Σ B (λ v' → v ≈ v' × ((k , v') ∈ m₂))
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private module ImplInsert (f : B → B → B) where
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_∈k_ : A → List (A × B) → Set a
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_∈k_ k m = MemProp._∈_ k (keys m)
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insert : A → B → List (A × B) → List (A × B)
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insert k v [] = (k , v) ∷ []
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insert k v (x@(k' , v') ∷ xs) with ≡-dec-A k k'
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... | yes _ = (k' , f v v') ∷ xs
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... | no _ = x ∷ insert k v xs
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merge : List (A × B) → List (A × B) → List (A × B)
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merge m₁ m₂ = foldr insert m₂ m₁
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insert-keys-∈ : ∀ (k : A) (v : B) (l : List (A × B)) → k ∈k l → keys l ≡ keys (insert k v l)
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insert-keys-∈ k v ((k' , v') ∷ xs) (here k≡k') with (≡-dec-A k k')
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... | yes _ = refl
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... | no k≢k' = absurd (k≢k' k≡k')
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insert-keys-∈ k v ((k' , _) ∷ xs) (there k∈kxs) with (≡-dec-A k k')
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... | yes _ = refl
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... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∈ k v xs k∈kxs)
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insert-keys-∉ : ∀ (k : A) (v : B) (l : List (A × B)) → ¬ (k ∈k l) → (keys l ++ (k ∷ [])) ≡ keys (insert k v l)
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insert-keys-∉ k v [] _ = refl
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insert-keys-∉ k v ((k' , v') ∷ xs) k∉kl with (≡-dec-A k k')
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... | yes k≡k' = absurd (k∉kl (here k≡k'))
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... | no _ = cong (λ xs' → k' ∷ xs') (insert-keys-∉ k v xs (λ k∈kxs → k∉kl (there k∈kxs)))
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∈k-dec : ∀ (k : A) (l : List (A × B)) → Dec (k ∈k l)
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∈k-dec k [] = no (λ ())
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∈k-dec k ((k' , v) ∷ xs) with (≡-dec-A k k')
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... | yes k≡k' = yes (here k≡k')
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... | no k≢k' with (∈k-dec k xs)
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... | yes k∈kxs = yes (there k∈kxs)
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... | no k∉kxs = no witness
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where
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witness : ¬ k ∈k ((k' , v) ∷ xs)
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witness (here k≡k') = k≢k' k≡k'
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witness (there k∈kxs) = k∉kxs k∈kxs
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insert-preserves-unique : ∀ (k : A) (v : B) (l : List (A × B)) → Unique (keys l) → Unique (keys (insert k v l))
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insert-preserves-unique k v l u with (∈k-dec k l)
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... | yes k∈kl rewrite insert-keys-∈ k v l k∈kl = u
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... | no k∉kl rewrite sym (insert-keys-∉ k v l k∉kl) = Unique-append k∉kl u
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merge-preserves-unique : ∀ (l₁ l₂ : List (A × B)) → Unique (keys l₂) → Unique (keys (merge l₁ l₂))
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merge-preserves-unique [] l₂ u₂ = u₂
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merge-preserves-unique ((k₁ , v₁) ∷ xs₁) l₂ u₂ = insert-preserves-unique k₁ v₁ (merge xs₁ l₂) (merge-preserves-unique xs₁ l₂ u₂)
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-- Map-functional : ∀ (k : A) (v v' : B) (xs : List (A × B)) → Unique (keys ((k , v) ∷ xs)) → MemProp._∈_ (k , v') ((k , v) ∷ xs) → v ≡ v'
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-- Map-functional k v v' _ _ (here k,v'≡k,v) = sym (cong proj₂ k,v'≡k,v)
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-- Map-functional k v v' xs (push k≢ _) (there k,v'∈xs) = absurd (unique-not-in xs v' (k≢ , k,v'∈xs))
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-- where
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-- unique-not-in : ∀ (xs : List (A × B)) (v' : B) → ¬ (All (λ k' → ¬ k ≡ k') (keys xs) × (k , v') ∈ xs)
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-- unique-not-in ((k' , _) ∷ xs) v' (k≢k' ∷ _ , here k',≡x) = k≢k' (cong proj₁ k',≡x)
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-- unique-not-in (_ ∷ xs) v' (_ ∷ rest , there k,v'∈xs) = unique-not-in xs v' (rest , k,v'∈xs)
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module _ (f : B → B → B) where
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open ImplInsert f renaming
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( insert to insert-impl
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; merge to merge-impl
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)
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insert : A → B → Map → Map
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insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-unique k v kvs uks)
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merge : Map → Map → Map
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merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-unique kvs₁ kvs₂ uks₂)
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module _ (_≈_ : B → B → Set b) where
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open ImplRelation _≈_ renaming (subset to subset-impl)
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subset : Map → Map → Set (a ⊔ b)
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subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂
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lift : Map → Map → Set (a ⊔ b)
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lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
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