Haskell中类型的推理

Question

Chapter 16 of "Haskell Programming from First Principles" on page 995 has an exercise to manually work out how (fmap. fmap) typechecks. It suggests substituting the type of each fmap for the function types in the type of the composition operator:

T1 (.):: (b -> c) -> (a -> b) -> a -> c

T2 fmap:: Functor f => (m -> n) -> fm -> fn

T3 fmap:: Functor g => (x -> y) -> gx -> gy

By (attempting to) substitute T2 and T3 into T1, I arrived at the following:

T4: ((m -> n) -> fm -> fn) -> ((x -> y) -> gx -> gy) -> a -> c

Further, it suggests checking the type of (fmap. fmap) to see what the end type should look like.

T5: (fmap. fmap):: (Functor f1, Functor f2) => (a -> b) -> f1 (f2 a) -> f1 (f2 b)

I'm having trouble understanding what I should be doing here. Could any knowledgeable haskellers help get me started, or maybe provide examples of similar exercises that show how to work out types by hand?

Answer 1

We proceed step by careful step:

--- fmap . fmap  =  (.) fmap fmap
--- Functor f, g, ... => ..... 

(.)      :: (   b     ->      c    ) -> (a ->  b ) -> a ->     c
    fmap ::  (d -> e) -> f d -> f e
             --------    ----------
(.) fmap ::                             (a ->d->e) -> a -> f d -> f e
                                             ----          ----------
-- then,

(.) fmap      :: (   a     ->  d  ->  e ) -> a   -> f   d   -> f   e
         fmap ::  (b -> c) -> g b -> g c
                  --------    ---    ---
(.) fmap fmap ::                          (b->c) -> f (g b) -> f (g c)
                                          ------      -----      -----

It is important to consistently rename all the type variables on each separate use of a type, to avoid conflation.

We use the fact that the arrows associate on the right,

 A -> B -> C   ~   A -> (B -> C)

and the type inference rule is

   f   :: A -> B
     x :: C
   --------------
   f x ::      B    ,  A ~ C

(f:: A -> B) (x:: C):: B under the equivalence / unification of types A ~ C and all that it entails.