On Mon, Jul 26, 2010 at 11:55 AM, John Lato
Hello,
I was wondering today, is this generally true?
instance (Monad m, Monoid a) => Monoid (m a) where mempty = return mempty mappend = liftM2 mappend
Yes.
I know it isn't a good idea to use this instance, but assuming that the instance head does what I mean, is it valid? Or more generally is it true for applicative functors as well? I think it works for a few tricky monads, but that's not any sort of proof. I don't even know how to express what would need to be proven here.
There are multiple potential monoids that you may be interested in here. There is the monoid formed by MonadPlus, there is the monoid formed by wrapping a monad (or applicative) around a monoid, which usually forms part of a right seminearring because of the left-distributive law, there are also potentially other monoids for particular monads. See the monad module in my monoids package: http://hackage.haskell.org/packages/archive/monoids/0.2.0.2/doc/html/Data-Mo...
Any resources for how I could develop a means to reason about this sort of property?
The types are not enough. What you need is the associativity of Kleisli arrow composition and the two identity laws. The three monad laws are precisely what you need to form this monoid. There are analogous laws for Applicative that serve the same purpose. -Edward Kmett