[Haskell-cafe] Confusion on the third monad law when using lambda abstractions

[David Menendez]

On Wed, Jun 17, 2009 at 9:08 PM, Jon Strait<jstrait at moonloop.net> wrote:
> I use and love Haskell, but I just have this nagging concern, that maybe
> someone can help me reason about.  If I'm missing something completely
> obvious here and making the wrong assumptions, please be gentle.  :)
>
> I'm reading the third (bind associativity) law for monads in this form:
>
> m >>= (\x -> k x >>= h)  =  (m >>= k) >>= h
>
> Now considering the definition of liftM2:
>
> liftM2 f m1 m2 = m1 >>= (\x1 -> m2 >>= (\x2 -> return (f x1 x2)))
>
> Isn't this liftM2  definition in the same form as the LHS of the third law
> equation, with (\x2 -> return (f x1 x2)) being the h function?

The usual convention here is that m, k, and h are variables bound
outside the scope of the law. In other words, the law only applies to
expressions which can be rewritten to have the form

let m = ...
    k = ...
    h = ...
in m >>= (\x -> k x >>= h)

In the case of liftM2, you'd have to rewrite it as,

liftM2 f m1 m2 = m >>= \x -> k x >>= h
    where
    m = m1
    k = \x -> m2 >>= \y -> return (x,y)
    h = \(x,y) -> return (f x y)

which is awkward, but works perfectly fine with the third law.

-- 
Dave Menendez <dave at zednenem.com>
<http://www.eyrie.org/~zednenem/>