Which is the type of (flip .)?

Question

I'm trying to understand why the type of:

(flip .) is: (a -> a1 -> b -> c) -> a -> b -> a1 -> c

First of all, the type of:

flip: is (a -> b -> c) -> b -> a -> c (.): is (b -> c) -> (a -> b) -> a -> c

I will rename the variables to be more clear in my explanation, so the types:

flip: is (ax -> bx -> cx) -> bx -> ax -> cx (.): is (by -> cy) -> (ay -> by) -> ay -> cy

Then I try substituing like this:

ax = (by -> cy) bx = (ay -> by) cx = ay -> cy

So the resulting type is: (ay -> by) (by -> cy) -> ay -> cy, which is different with the correct result.

Any help?

Thanks, Sebastián.

Answer 1

(flip .) is (.) flip, so:

• (.) :: (bx -> cx) -> (ax -> bx) -> ax -> cx
• flip :: (ay -> by -> cy) -> by -> ay -> cy
• In (.) flip,
  • bx is ay -> by -> cy
  • cx is by -> ay -> cy
  • so it’s all (ax -> (ay -> by -> cy)) -> ax -> (by -> ay -> cy),
    which is just (ax -> ay -> by -> cy) -> ax -> by -> ay -> cy,
    which matches up with (flip .) :: (a -> a1 -> b -> c) -> a -> b -> a1 -> c.