Generate function of given arity in Haskell using type numbers

Question

Assume I have encoded the natural numbers in Haskell types, and that I have a way of adding and subtracting from them:

data Zero
data Succ n
-- ...

Answer 1

OK. Yes. Definitely, by threading a numeric type around the recursive instances.

First, some boilerplate:

{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE MultiParamTypeClasses  #-}
{-# LANGUAGE EmptyDataDecls         #-}
{-# LANGUAGE FlexibleInstances      #-}
{-# LANGUAGE FlexibleContexts       #-}
{-# LANGUAGE ScopedTypeVariables    #-}

Your nats:

data Zero
data Succ n

A recursive builder for the variadic functions, now with an n argument:

class BuildList n a r | r -> a where
    build' :: n -> [a] -> a -> r

A base case: stop when we get to Zero:

instance BuildList Zero a [a] where
    build' _ l x = reverse $ x:l

Otherwise, decrement by one and recurse:

instance BuildList n a r => BuildList (Succ n) a (a->r) where
    build' (_ :: Succ n) l x y = build' (undefined :: n) (x:l) y

Now, we only want to loop 3 times, so write that down:

build :: BuildList (Succ (Succ Zero)) a r => a -> r
build x = build' (undefined :: Succ (Succ Zero)) [] x

Done.

Testing:

> build "one" "two" "three" :: [[Char]]
["one","two","three"]

Any less or more are errors:

*Main> build "one" "two" "three" "four" :: [[Char]]

<interactive>:1:1:
    No instance for (BuildList Zero [Char] ([Char] -> [[Char]]))

*Main> build "one" "two" :: [[Char]]

<interactive>:1:1:
    No instance for (BuildList (Succ Zero) [Char] [[Char]])

Answer 2

Inlining Martijn's code gives a very simple solution:

zero xs = xs
suc n xs x = n (xs ++ [x])
buildList n = n []

Answer 3

Oh, my... FlexibleContexts??? NoMonomorphismRestriction??? Come on, guys, isn't it exactly what TypeFamilies are for?

{-# LANGUAGE TypeFamilies #-}
data Zero = Zero
newtype Succ n = Succ n
zero = Zero
one = Succ zero
two = Succ one
three = Succ two
class BuildList n where
    type BL n
    buildListPrefix :: n -> ([String] -> [String]) -> BL n
instance BuildList Zero where
    type BL Zero = [String]
    buildListPrefix Zero h = h []
instance BuildList n => BuildList (Succ n) where
    type BL (Succ n) = String -> BL n
    buildListPrefix (Succ n) h s = buildListPrefix n (h . (s:))
buildList:: BuildList n => n -> BL n
buildList n = buildListPrefix n id

Answer 4

I see your functional-dependency multiparameter empty-datatype flexibly scoped type variables and raise you a Haskell 98 version! It uses HoleyMonoid which is available on hackage:

{-# LANGUAGE NoMonomorphismRestriction #-}

import Prelude hiding (id, (.))
import Control.Category
import Data.HoleyMonoid

suc n = later (:[]) . n

zero  = id
one   = suc zero
two   = suc one
three = suc two

buildList = run

Testing (feel free to omit any type signatures):

> run three "one" "two" "three"
["one","two","three"]