Operator section for applicative with <$> and <*>

Question

Consider functions of type a -> b -> c, and the applicative values a1, a2 :: (Applicative f) => f a.

I wish to construct a function which may be applied to functions of type a -> b -> c to obtain values of type Applicative f :: f c. I can do this in the following way:

g :: (Applicative f) => (a -> b -> c) -> f c
g = \f -> f <$> a1 <*> a2

(The explicit lambda is deliberate, as I am considering the construction of this function at any level, not just the top level).

If I try to write g in point-free style:

g = (<$> a1 <*> a2)

I get the following compile error:

The operator `<$>' [infixl 4] of a section
    must have lower precedence than that of the operand,
      namely `<*>' [infixl 4]
    in the section: `<$> gen1 <*> gen2'

I could write this point-free implementation:

g = flip (flip liftA2 a1) a2

but I feel that this is less readable, and it is simpler to refactor the infix function-based implementation to, e.g., add another argument, than changing the above to use liftA3.

One can write a chain of compositions:

g = (<*> a2) . (<$> a1)

This achieves point-free style and it is simple to add arguments - but they get prepended on the left instead of appended on the right, so you lose the correspondence with the function type (a -> b -> c). Furthermore, with more arguments you end up with a much longer expression than just using a lambda as in the first implementation.

So, is there are nice, terse way to write the section I desire, or am I stuck with a lambda?

Answer 1

<*> operates on the result of <$>, so:

g = (<*> a2) . (<$> a1)

Answer 2

I'm not really convinced pointfree can be made better than using an explicit argument here, but a couple more ideas:

• You can use flip infix:

  g = liftA2 `flip` a1 `flip` a2
  

• You can use >>> from Control.Category, which is . flipped:

  g = (<$> a1) >>> (<*> a2)