Convert to Pointfree Style?

Question

Learn You a Haskell discusses "Making a Monad" with the following Prob type:

import Data.Ratio

newtype Prob a = Prob { getProb :: [(a,Rational)] } deriving Show  

Prob represents an a type, and then a Rational representing the probability of this a being used.

Let's look at a Prob instance:

*Main> Prob [('a', 1%2), ('b', 1%2)]
Prob {getProb = [('a',1 % 2),('b',1 % 2)]}

LYAH poses an exercise to figure out how to turn thisSituation, of type Prob(Prob Char) into Prob Char:

thisSituation :: Prob (Prob Char)
thisSituation = Prob
  [( Prob [('a', 1%2),('b',1%2)], 1%4)
  ,( Prob [('c', 1%2),('d',1%2)], 3%4)
  ]

Here's what I came up with:

flatten :: Prob (Prob a) -> Prob a
flatten pp = Prob $ convert $ getProb pp

convert :: [(Prob a, Rational)] -> [(a, Rational)]
convert xs = concat $ map f xs

f :: (Prob a, Rational) -> [(a, Rational)]
f (p, r) = map (mult r) (getProb p)

mult :: Rational -> (a, Rational) -> (a, Rational)
mult r (x, y) = (x, r*y)

I tried point-free as so:

flatten :: Prob (Prob a) -> Prob a
flatten = Prob $ convert $ getProb 

But got this error:

*Main> :l MakingMonad.hs
[1 of 1] Compiling Main             ( MakingMonad.hs, interpreted )

MakingMonad.hs:37:11:
    Couldn't match expected type `Prob (Prob a) -> Prob a'
                with actual type `Prob a0'
    In the expression: Prob $ convert $ getProb
    In an equation for `flatten': flatten = Prob $ convert $ getProb

MakingMonad.hs:37:28:
    Couldn't match expected type `[(Prob a0, Rational)]'
                with actual type `Prob a1 -> [(a1, Rational)]'
    In the second argument of `($)', namely `getProb'
    In the second argument of `($)', namely `convert $ getProb'
    In the expression: Prob $ convert $ getProb
Failed, modules loaded: none.

Can I make flatten point-free? If so, please show me how. If not, please explain why.

Answer 1

When you use $ in flatten, you get code that looks like

flatten = Prob $ convert $ getProb
==> Prob (convert (getProb))

Which is not what you want.

You want Prob . convert . getProb