open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; cong) open import Relation.Binary.Definitions using (Decidable) open import Relation.Binary.Core using (Rel) open import Relation.Nullary using (Dec; yes; no) open import Agda.Primitive using (Level; _⊔_) 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.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.Empty using (⊥) keys : List (A × B) → List A keys [] = [] keys ((k , v) ∷ xs) = k ∷ keys xs 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) Map : Set (a ⊔ b) Map = Σ (List (A × B)) (λ l → Unique (keys l)) 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 x'≢x : ¬ x' ≡ x x'≢x x'≡x = x∉xs (here (sym x'≡x)) help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (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 = MemProp._∈_ p m 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) absurd : ∀ {a} {A : Set a} → ⊥ → A absurd () private module ImplRelation (_≈_ : B → B → Set b) where subset : 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)) private module ImplInsert (f : B → B → B) where _∈k_ : A → List (A × B) → Set a _∈k_ k m = MemProp._∈_ k (keys m) 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 merge : List (A × B) → List (A × B) → List (A × B) merge m₁ m₂ = foldr insert m₂ m₁ 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) 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))) ∈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 ImplInsert f renaming ( insert to insert-impl ; merge to merge-impl ) insert : A → B → Map → Map insert k v (kvs , uks) = (insert-impl k v kvs , insert-preserves-unique k v kvs uks) merge : Map → Map → Map merge (kvs₁ , _) (kvs₂ , uks₂) = (merge-impl kvs₁ kvs₂ , merge-preserves-unique kvs₁ kvs₂ uks₂) module _ (_≈_ : B → B → Set b) where open ImplRelation _≈_ renaming (subset to subset-impl) subset : Map → Map → Set (a ⊔ b) subset (kvs₁ , _) (kvs₂ , _) = subset-impl kvs₁ kvs₂ lift : Map → Map → Set (a ⊔ b) lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁