How to create a polyvariadic haskell function?

function-parameter, haskell, polyvariadic, variadic-functions

Solution

The key points of `printf` is the ability to either return a String or a function. Copied from http://www.haskell.org/ghc/docs/6.12.2/html/libraries/base-4.2.0.1/src/Text-Printf.html,

printf :: (PrintfType r) => String -> r
printf fmts = spr fmts []

class PrintfType t where
    spr :: String -> [UPrintf] -> t

instance (IsChar c) => PrintfType [c] where
    spr fmts args = map fromChar (uprintf fmts (reverse args))

instance (PrintfArg a, PrintfType r) => PrintfType (a -> r) where
    spr fmts args = \a -> spr fmts (toUPrintf a : args)

and the basic structure we can extract out is

variadicFunction :: VariadicReturnClass r => RequiredArgs -> r
variadicFunction reqArgs = variadicImpl reqArgs mempty

class VariadicReturnClass r where
   variadicImpl :: RequiredArgs -> AccumulatingType -> r

instance VariadicReturnClass ActualReturnType where
   variadicImpl reqArgs acc = constructActualResult reqArgs acc

instance (ArgClass a, VariadicReturnClass r) => VariadicReturnClass (a -> r) where
   variadicImpl reqArgs acc = \a -> variadicImpl reqArgs (specialize a `mappend` acc)

For instance:

class SumRes r where 
    sumOf :: Integer -> r

instance SumRes Integer where
    sumOf = id

instance (Integral a, SumRes r) => SumRes (a -> r) where
    sumOf x = sumOf . (x +) . toInteger

then we could use

*Main> sumOf 1 :: Integer
1
*Main> sumOf 1 4 7 10 :: Integer
22
*Main> sumOf 1 4 7 10 0 0  :: Integer
22
*Main> sumOf 1 4 7 10 2 5 8 22 :: Integer
59

Problem

I need a function which takes an arbitrary number of arguments (All of the same type), does something with them and afterwards gives a result back. A list of arguments is impracticable in my specific case. As I looked through the haskell libs, I saw that the function `printf` (from module `Text.Printf`) uses a similar trick. Unfortunately, I couldn't understand that magic by looking at the source. Can somebody explain how to achieve this, or at least some webpage/paper/whatever where I could find a good description for this? Motivation: The reason I need this is really quite simple. For school (computer science class), we are required to write a module that is able to "record" a mathematical expression, express it as a string (Via writing an instance of Num/Real/etc for an own datatype), and perform various operations on it. This datatype contains a special constructor for a variable, which may be replaced by a value or whatever by a specified function. One of the goals is to write a function, which takes such an expression with some number of variables (pairs of type `(Char,Rational)`) and calculates the result of the expression. We should look at how to express the goal of the function best. (My idea: The function returns another function which takes exactly as many arguments as vars that are defined in the function - seems to be impossible).

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