I was reviewing some code and came across the following gem, which I'd wager is a copy-paste of pointfree output:
(I thought the following would more appropriate than the usual foo/bar for this particular question :P)
import Control.Monad (liftM2)
data Battleship = Battleship { x :: Int
, y :: Int
} deriving Show
placeBattleship :: Int -> Int -> Battleship
placeBattleship x' y' = Battleship { x = x', y = y' }
coordinates :: Battleship -> (Int, Int)
coordinates = liftM2 (,) x y
Would someone be kind enough to explain the steps needed to simplify from:
(i) coordinates b = (x b, y b)
to:
(ii) coordinates = liftM2 (,) x y?
In particular, I'm a bit confused as to the use of liftM2 as I wasn't even aware that a monad was lurking in the background.
I know that (i) can also be represented as: coordinates s = (,) (x s) (y s) but I'm not sure where/how to proceed.
P.S. The following is why I suspect it's from pointfree (output is from GHCI and :pl is aliased to pointfree):
λ: :pl coordinates s = (x s, y s)
coordinates = liftM2 (,) x y
This takes advantage of the Monad instance for (->) r, also called the "reader monad". This is the monad of functions from a specific type to a. (Take a look here for motivation on why it exists in the first place.)
To see how it works for various functions, replace m with (r -> in m a. For example, if we just do liftM, we get:
liftM :: (a -> b) -> (m a -> m b)
liftM :: (a -> b) -> ((r -> a) -> (r -> b))
:: (a -> b) -> (r -> a) -> (r -> b) -- simplify parentheses
...which is just function composition. Neat.
We can do the same thing for liftM2:
liftM2 :: (a -> b -> c) -> m a -> m b -> m c
liftM2 :: (a -> b -> c) -> (r -> a) -> (r -> b) -> (r -> c)
So what we see is a way to compose two one-argument functions with a two-argument function. It's a way of generalizing normal function composition to more than one argument. The idea is that we create a function that takes a single r by passing that through both of the one-argument functions, getting two arguments to pass into the two-argument function. So if we have f :: (r -> a), g :: (r -> b) and h :: (a -> b -> c), we produce:
\ r -> h (f r) (h r)
Now, how does this apply to your code? (,) is the two-argument function, and x and y are one-argument functions of the type Battleship -> Int (because that's how field accessors work). With this in mind:
liftM2 (,) x y = \ r -> (,) (x r) (y r)
= \ r -> (x r, y r)
Once you've internalized the idea of multiple function composition like this, point-free code like this becomes quite a bit more readable—no need to use the pointfree tool! In this case, I think the non-pointfree version is still better, but the pointfree one isn't terrible itself.