Haskell currying to remove argument variable at the end

Question

I am a newbie trying to learn haskell, I tried to search for something similar in other forums but couldn't find a similar question.

addPoly :: (Num a)=>[[a]]->[a]
addPoly  x = map sum $ transpose x

runs fine

but when I remove x at the end it errors out

addPoly :: (Num a)=>[[a]]->[a]
addPoly  = map sum $ transpose 

errors saying:

Couldn't match expected type `[[Integer]] -> [Integer]'
            with actual type `[Integer]'
In the expression: map sum $ transpose
In an equation for `addPoly': addPoly = map sum $ transpose

Couldn't match expected type `[[Integer]]'
            with actual type `[[a0]] -> [[a0]]'
In the second argument of `($)', namely `transpose'
In the expression: map sum $ transpose
In an equation for `addPoly': addPoly = map sum $ transpose

Couldn't figure out what I am missing here.

Disclaimer: this is not a homework problem

Answer 1

$ is defined in Haskell as

f $ x = f x
infixr 0 $

So if you expand the first snippet of your code,

map sum $ transpose x

becomes

map sum (transpose x)

which will work.

But the second snippet

map sum $ transpose 

becomes

map sum transpose

and when you call that with x, you get

map sum transpose x

which actually map's sum over transpose (and calls the result with argument x, which also doesn't make sense, and causes the error message you get, as map will return a List, not a function), and not over transpose x.

You need to use the . function for this, instead of $, which is defined as

(.) f g = \x -> f (g x)

If you do that, your code

map sum . transpose

becomes

\x -> map sum (transpose x)

and when you call this from on some parameter x, it simply becomes

map sum (transpose x)

which is the (correct) code we started with.

Let me know if something isn't clear.

Answer 2

The proper code is:

addPoly :: (Num a)=>[[a]]->[a]
addPoly  = map sum . transpose 

How to arrive at it? Remember the following two rules:

f $ x = f x
f. g $ x == (f.g) x == f (g x) == f $ g x

Thus,

addPoly  x = map sum $ transpose x

is rewritten as

addPoly  x = map sum $ transpose $ x

and then each $ but the last are replaced by ..

addPoly  x = map sum . transpose $ x

Now, since you have only one $ and the argument is only on the right of $ you can switch to point-free style