Is it possible to write this function in Haskell?
I'm learning dependent types: In Haskell I have defined the canonical type
data Vec ∷ Type → Nat → Type where
Nil ∷ Vec a Z
(:-) ∷ a → Vec a n → Vec a (S n)
and implemented most of the functions from Data.List however I don't know how to write, if possible at all, functions like
delete ∷ Eq a ⇒ a → Vec a n → Vec a (??)
since the length of the result is not known. I have found it in Agda and it's implemented this way
delete : {A : Set}{n : Nat}(x : A)(xs : Vec A (suc n)) → x ∈ xs → Vec A n
delete .x (x ∷ xs) hd = xs
delete {A}{zero } _ ._ (tl ())
delete {A}{suc _} y (x ∷ xs) (tl p) = x ∷ delete y xs p
If I understand correctly delete it's defined with the constrain of x being an element of xs , in that case you just remove x and subtract 1 from the length. Can I write something like this in Haskell?
The problem is that you need a dependent quantifier which Haskell currently lacks. Ie the (x : A)(xs : Vec A (suc n)) → ... part is not directly expressible. You can probably cook up something using singletons, but it'll be really ugly and complicated.
I would just define
delete ∷ Eq a ⇒ a → Vec a (S n) → Maybe (Vec a n)
and be fine with it. I'd also change the order of arguments to Vec to make it possible to provide Applicative , Traversable and other instances.
Actually, no. In order to define delete you just need to provide an index at which to delete:
{-# LANGUAGE GADTs, DataKinds #-}
data Nat = Z | S Nat
data Index n where
IZ :: Index n
IS :: Index n -> Index (S n)
data Vec n a where
Nil :: Vec Z a
(:-) :: a -> Vec n a -> Vec (S n) a
delete :: Index n -> Vec (S n) a -> Vec n a
delete IZ (x :- xs) = xs
delete (IS n) (x :- (y :- xs)) = x :- delete n (y :- xs)
Note that x ∈ xs is nothing more than a richly typed index.
Here is a solution with singletons:
{-# LANGUAGE GADTs, DataKinds, PolyKinds, KindSignatures, UndecidableInstances, TypeFamilies, RankNTypes, TypeOperators #-}
infixr 5 :-
data Nat = Z | S Nat
data family Sing (x :: a)
data instance Sing (b :: Bool) where
STrue :: Sing True
SFalse :: Sing False
data instance Sing (n :: Nat) where
SZ :: Sing Z
SS :: Sing n -> Sing (S n)
type family (:==) (x :: a) (y :: a) :: Bool
class SEq a where
(===) :: forall (x :: a) (y :: a). Sing x -> Sing y -> Sing (x :== y)
type instance Z :== Z = True
type instance S n :== Z = False
type instance Z :== S m = False
type instance S n :== S m = n :== m
instance SEq Nat where
SZ === SZ = STrue
SS n === SZ = SFalse
SZ === SS m = SFalse
SS n === SS m = n === m
data Vec xs a where
Nil :: Vec '[] a
(:-) :: Sing x -> Vec xs a -> Vec (x ': xs) a
type family If b x y where
If True x y = x
If False x y = y
type family Delete x xs where
Delete x '[] = '[]
Delete x (y ': xs) = If (x :== y) xs (y ': Delete x xs)
delete :: forall (x :: a) xs. SEq a => Sing x -> Vec xs a -> Vec (Delete x xs) a
delete x Nil = Nil
delete x (y :- xs) = case x === y of
STrue -> xs
SFalse -> y :- delete x xs
test :: Vec '[S Z, S (S (S Z)), Z] Nat
test = delete (SS (SS SZ)) (SS SZ :- SS (SS (SS SZ)) :- SS (SS SZ) :- SZ :- Nil)
Here we index Vec s by lists of their elements and store singletons as elements of vectors. We also define SEq which is a type class that contains a method that receives two singletons and returns either a proof of equality of values they promote or their inequality. Next we define a type family Delete that is like usual delete for lists, but at the type level. Finally in the actual delete we pattern match on x === y and thus reveal whether x is equal to y or not, which makes the type family compute.