[Haskell-cafe] Curiose types

[Andrew Coppin]

As I lay in bed last night, a curios fact occurred to me. (Yes, I don't 
get out very much...)


Consider the map function:

  map :: (a -> b) -> [a] -> [b]

There are two ways you can think about this function. First, you can see 
it as meaning

  map :: (a -> b) -> ([a] -> [b])

Which is beautifully symmetric. Alternatively, you can think about how 
you actually use it:

  map :: ((a -> b) -> [a]) -> [b]

Now this got me thinking: is (->) associative? x_x


Well now, let's consider a general 3-argument function:

  foo :: a -> b -> c -> d

According to the rules, this is of course equivilent to

  foo :: a -> (b -> (c -> d))

If you understand what currying is and how it works, this makes perfect 
sense. Now let's bracket that the other way:

  foo :: ((a -> b) -> c) -> d

Well, while that type *does* make sense, it's clearly a completely 
*different* type!


So, clearly, (->) is not associative. Now I'm left wondering why you can 
bracket the type for map in two different ways, but not in general. Hmm...