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)

-- | Type family for n-ary functions.
type family   Fn n a b
type instance Fn Z     a b = b
type instance Fn (S n) a b = a -> Fn n a b

-- | Newtype wrapper which is used to make 'Fn' injective. It's also a
--   reader monad.
newtype Fun n a b = Fun { unFun :: Fn n a b }

I need function like

uncurryN :: Fun (n + k) a b -> Fun n a (Fun k a b)

I read several articles about type level computations, but all about type safe list concatenation.

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

Answer 2

This required a bit of care in unwrapping/rewrapping the Fun newtype. I also exploited the DataKinds extension.

{-# LANGUAGE DataKinds, KindSignatures, TypeFamilies, 
    MultiParamTypeClasses, ScopedTypeVariables, FlexibleInstances #-}
{-# OPTIONS -Wall #-}

-- | Type-level naturals.
data Nat = Z | S Nat

-- | Type family for n-ary functions.
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

-- | Addition.
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 wrapper which is used to make 'Fn' injective.
newtype Fun n a b = Fun { unFun :: Fn n a b }

class UncurryN (n :: Nat) (m :: Nat) a b where
    uncurryN :: Fun (Add n m) a b -> Fun n a (Fun m a b)

instance UncurryN Z m a b where
    uncurryN g = Fun g

instance UncurryN n m a b => UncurryN (S n) m a b where
    uncurryN g = Fun (\x -> unFun (uncurryN (Fun (unFun g x)) :: Fun n a (Fun m a b)))

{- An expanded equivalent with more signatures:

instance UncurryN n m a b => UncurryN (S n) m a b where
    uncurryN g = let f :: a -> Fn n a (Fun m a b)
                     f x = let h :: Fun (Add n m) a b
                               h = Fun ((unFun g :: Fn (Add (S n) m) a b) x)
                           in unFun (uncurryN h :: Fun n a (Fun m a b))
                     in Fun f
-}