[Haskell-cafe] Why does `flip` cause function type so different ?

[Daniel Fischer]

Am Freitag 19 März 2010 04:34:53 schrieb zaxis:
> >let f x xs = [x:xs,xs]
> >
> > :t  f
>
> f :: a -> [a] -> [[a]]
>
> >:t  (>>=) .f
>
> (>>=) .f :: a -> ([[a]] -> [a] -> b) -> [a] -> b
>
> > :t (flip (>>=) .f)
>
> (flip (>>=) .f) :: a -> [[a]] -> [[a]]
>
> Why is the type of `(>>=) .f` and `flip (>>=) .f` so different ?
>

Because the types of (>>=) and flip (>>=) are different.

(>>=) :: Monad m => m a -> (a -> m b) -> m b
flip (>>=) :: Monad m => (a -> m b) -> m a -> m b

Now, for whatever f, for ((>>=) . f) to be well-typed, f must have the type

f :: s -> m t

for some Monad m. (It must be a function because it's an argument of 
composition (.), and since

((>>=) . f) x === (>>=) (f x) === (f x >>=),

f's return type must be monadic.) Then (f x >>=) takes a function 
g :: t -> mu, so that (f x >>= g) is a value of type m u.

The given f has type a -> [a] -> [[a]] === a -> ([a] -> [[a]]), so
m t === [a] -> [[a]] === ((->) [a]) [[a]],
i.e. 
m === ((->) a)
and
t === [[a]]

(f x >>=) then takes a function g of type (t -> m b), expanded

([[a]] -> ([a] -> b)), and then f x >>= g is a value of type m b, expanded

[a] -> b.

This works because for any type w, ((->) w) is indeed a Monad, the Reader 
Monad, to be specific.

On the other hand, for (flip (>>=) . f) to be well typed, f must be a 
function returning something of type (t -> m u) for some Monad m, so f's 
type must be s -> (t -> m u).

The given f has type

a -> [a] -> [[a]],

so s === a, t === [a], m === [] and u === [a] === t.

Then (flip (>>=) . f) x, which is (>>= f x), takes a further argument of 
type m t === [[a]] and then returns a value of type m u === m t === [[a]]