Uncurry for n-ary functions

Question

I have a type level numbers

data Z   deriving Typeable
data S n deriving Typeable

and n-ary functions (code from fixed-vector package)

Answer 1

You can do this without any type classes by constructing a datatype which can represent the type Nat on the data level:

data Nat = Z | S Nat

type family   Fn (n :: Nat) a b
type instance Fn Z     a b = b
type instance Fn (S n) a b = a -> Fn n a b

type family   Add (n :: Nat) (m :: Nat) :: Nat
type instance Add Z          m = m
type instance Add (S n)      m = S (Add n m)

newtype Fun n a b = Fun { unFun :: Fn n a b }

data SNat (n :: Nat) where 
  SZ :: SNat Z
  SS :: SNat n -> SNat (S n)

uncurryN :: forall n m a b . SNat n -> Fun (Add n m) a b -> Fun n a (Fun m a b) 
uncurryN SZ f = Fun f
uncurryN (SS (n :: SNat n')) g = Fun (\x -> unFun (uncurryN n (Fun (unFun g x)) :: Fun n' a (Fun m a b)))

If you don't like explicitly mentioning the n parameter, thats ok since you can always go back and forth between a function which takes an parameter as a type class and which takes a parameter as data:

class SingI (a :: k) where
  type Sing :: k -> * 
  sing :: Sing a

instance SingI Z where 
  type Sing = SNat
  sing = SZ

instance SingI n => SingI (S n) where
  type Sing = SNat
  sing = SS sing 

toNatSing :: (SNat n -> t) -> (SingI n => t)
toNatSing f = f sing 

fromNatSing :: (SingI n => t) -> (SNat n -> t)
fromNatSing f SZ = f 
fromNatSing f (SS n) = fromNatSing f n 

uncurryN' :: SingI n => Fun (Add n m) a b -> Fun n a (Fun m a b) 
uncurryN' = toNatSing uncurryN