Ivan Perez ivanperezdominguez at gmail.com
Fri Feb 10 21:27:31 CET 2012

```To understand how liftM2 achieves the cartesian product, I think
one way is to find liftM2's implementation and (>>=) implementation
as part of []'s instantiation of the Monad class.

You can find the first in Control.Monad, and the second in
the standard prelude.

Lists are monads, and as John (almost) said, liftM2 f x y is equivalent to
liftM2 f m1 m2 = do
x1 <- m1
x2 <- m2
return (f x1 x2)

Which is syntactic sugar (fancy Haskell) for

liftM2 f m1 m2 =
m1 >>= (\x1 -> m2 >>= (\x2 -> return (f x1 x2)))

In the prelude, you can find
m >>= k             = foldr ((++) . k) [] m

Fhe right-hand side of (>>=) here is roughly equivalent to
concat (map k m).

The last step, which I leave as an exercise to the reader (I always wanted
to say that), is use the right hand side of the definition of (>>=) for lists
in the right hand side of liftM2 when applied to (,) and two lists.

You can see the type of the function (,) (yes, comma is a function!)
by executing, in ghci:

:type (,)

Cheers,
Ivan.

On 9 February 2012 19:23, John Meacham <john at repetae.net> wrote:
> A good first step would be understanding how the other entry works:
>
> cartProd :: [a] -> [b] -> [(a,b)]
> cartProd xs ys = do
>        x <- xs
>        y <- ys
>        return (x,y)
>
> It is about halfway between the two choices.
>
>    John
>
> On Thu, Feb 9, 2012 at 9:37 AM, readams <richard.adams at lvvwd.com> wrote:
>> Nice explanation.  However, at
>> http://stackoverflow.com/questions/4119730/cartesian-product it was pointed
>> out that this
>>
>> cartProd :: [a] -> [b] -> [(a, b)]
>> cartProd = liftM2 (,)
>>
>> is equivalent to the cartesian product produced using a list comprehension:
>>
>> cartProd xs ys = [(x,y) | x <- xs, y <- ys]
>>
>> I do not see how your method of explanation can be used to explain this
>> equivalence?  Nevertheless, can you help me to understand how liftM2 (,)
>> achieves the cartesian product?  For example,
>>
>> [(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)]
>>
>> Thank you!
>>
>> --
>> View this message in context: http://haskell.1045720.n5.nabble.com/Cannot-understand-liftM2-tp3085649p5470185.html
>> Sent from the Haskell - Haskell-Cafe mailing list archive at Nabble.com.
>>
>> _______________________________________________