Haskell function application and currying

Question

I am always interested in learning new languages, a fact that keeps me on my toes and makes me (I believe) a better programmer. My attempts at conquering Haskell come and go - t

Answer 1

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

This is true. You concluded from that that

(f . g) x y = f (g x y)

must also be true, but that is not the case. In fact, the following is true:

(f . g) x y = f (g x) y

which is not the same.

Why is this true? Well (f . g) x y is the same as ((f . g) x) y and since we know that (f . g) x = f (g x) we can reduce that to (f (g x)) y, which is again the same as f (g x) y.

So (concatMap . ins) 1 [[2,3]] is equivalent to concatMap (ins 1) [[2,3]]. There is no magic going on here.

Another way to approach this is via the types:

. has the type (b -> c) -> (a -> b) -> a -> c, concatMap has the type (x -> [y]) -> [x] -> [y], ins has the type t -> [t] -> [[t]]. So if we use concatMap as the b -> c argument and ins as the a -> b argument, then a becomes t, b becomes [t] -> [[t]] and c becomes [[t]] -> [[t]] (with x = [t] and y = [t]).

So the type of concatMap . ins is t -> [[t]] -> [[t]], which means a function taking a whatever and a list of lists (of whatevers) and returning a list of lists (of the same type).