[Haskell-beginners] Pointfree expeiments

[Stephen Tetley]

Hi Paul

Its Pointfree that's choosing `ap` - I've never got round to using it
myself but I assume it only re-writes using only "standard" functions
(Prelude + Base?). As there isn't a specific definition of Starling
available for the functional type it chooses Monadic ap. Nowadays it
could also choose (<*>) from Control.Applicative.

The monad instance for functions is the Reader monad without a newtype wrapper:

m x == (r1 -> x)

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

Subsituting you get:
ap       ::  (r1 -> a -> b) -> (r1 -> a) -> (r1 -> b)

If you want you can knocking out the bracket of the last part...

ap       ::  (r1 -> a -> b) -> (r1 -> a) -> r1 -> b

Same with liftM2

liftM2  :: (Monad m) => (a1 -> a2 -> r) -> m a1 -> m a2 -> m r

Nicer var names:

liftM2  :: (Monad m) => (a -> b -> ans) -> m a -> m b -> m ans

Substituting, m x == (r1 -> x)

liftM2  :: (a -> b -> c) -> (r1 -> a) -> (r1 -> b) -> (r1 -> ans)

This matches the Phoenix combinator also known as Big Phi. I think it
was in the combinator set used by David Turner in the implementation
of SASL (SASL's abstract machine used combinators as its machine
code).

phoenix :: (a -> b -> ans) -> (r1 -> a) -> (r1 -> b) -> r1 -> ans