[Haskell-cafe] monoids and monads
John Lato
jwlato at gmail.com
Tue Jul 27 05:47:26 EDT 2010
On Tue, Jul 27, 2010 at 8:32 AM, Henning Thielemann
<schlepptop at henning-thielemann.de> wrote:
> John Lato schrieb:
>> Hello,
>>
>> I was wondering today, is this generally true?
>>
>> instance (Monad m, Monoid a) => Monoid (m a) where
>> mempty = return mempty
>> mappend = liftM2 mappend
>>
>> 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.
>
> I translate 'valid' and 'true' to "Is 'm a' a Monoid, given that 'm' is
> a Monad and 'a' is a Monoid?" If this is the question then we have to
> show the Monoid laws for (m a), namely
Thanks very much, this is what I was unable to express properly (hence
the informal description).
> left identity: forall x. mappend mempty x = x
> right identity: forall x. mappend x mempty = x
> associativity:
> forall x y z.
> (x `mappend` y) `mappend` z = x `mappend` (y `mappend` z)
So the task now is to prove that these laws hold, given the instance I
defined above, the monoid laws for 'a', and the monad laws for 'm'.
Or alternatively using applicative laws for 'm' for the more general
case. I'll see how I get on from here then.
Thanks,
John
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