# [Haskell-cafe] function types as instances of Num

Dan Weston westondan at imageworks.com
Thu Oct 26 19:51:14 EDT 2006

```How about:

{-# OPTIONS -fglasgow-exts #-}

import Control.Arrow

type Alpha alpha = alpha -> (Integer,alpha)

test  = square . (lit 4)

lit :: Integer -> Alpha alpha
lit val stack    = (val, stack)

instance Eq (Alpha alpha) where
x == y = uncurry (==) . (fst . x &&& fst . y) \$ undefined

instance Show (Alpha alpha) where
show x = show . fst \$ x undefined

instance Num (Alpha alpha) where
fromInteger i = (\s -> (i,s))
(+)           = fBinary (+)
(-)           = fBinary (-)
(*)           = fBinary (*)
negate        = fUnary   negate
abs           = fUnary   abs
signum        = fUnary   signum

fUnary  op x   = (op         .  fst          &&&  snd       ) .  x
fBinary op x y = (uncurry op . (fst *** fst) &&& (snd . fst)) . (x &&& y)

Greg Buchholz wrote:
>     Let's say we've got a little stack language, where you compute
> things by transformations of stacks, using compositions of functions
> from stacks to stacks (represented here as nested tuples). (See also
> Chris Okasaki's "Techniques for Embedding Postfix Languages in Haskell"
>  www.eecs.harvard.edu/~nr/ cs252r/archive/chris-okasaki/hw02.ps )
>
>   For example, the simple program below calculates the square of 4...
>> {-# OPTIONS -fglasgow-exts #-}
>>
>> main = print \$ test ()
>> test  = square . (lit 4)
>>
>> lit :: Integer -> a -> (Integer,a)
>> lit val stack    = (val, stack)
>>
>> dup  (a, b)      = (a, (a, b))
>> mult (a, (b, c)) = (b*a, c)
>> square = mult . dup
>
> ...now let's say I find that using the function "lit" to annotation
> numeric literals ugly.  What I really want is something like...
>
>> test' = square . 4
>
> ...Seems simple enough, I'll just make an appropriate instance of Num
> and I'll be able to use fromInteger...
>
>> instance Eq   (a -> (Integer, a)) instance Show (a -> (Integer, a))
instance Num  (a -> (Integer, a)) where
>>     fromInteger = lit
>
> ...but now when I try it, GHC complains...
>
>     No instance for (Num (a -> (Integer, t)))
>       arising from the literal `4' at final.hs:15:17
>     Possible fix:
>       add an instance declaration for (Num (a -> (Integer, t)))
>     In the second argument of `(.)', namely `4'
>     In the expression: square . 4
>     In the definition of `test'': test' = square . 4
>
> ...so it seems that (a -> (Integer, t)) can't be unified with (a ->
> (Integer, a)), or somesuch.  Any thoughts on how to get this to work?
>
>
> Thanks,
>
> Greg Buchholz
>
>
> _______________________________________________