[Haskell-cafe] Type classes

[Johannes Gerer]

Dear Haskellers,

While trying to understand the interconnection and hierarchy behind
the important typeclasses, I stumbled upon the following question that
I am not able to answer:

There seems to be a hierachy of the type classes, in the sense, that
the more powerfull a class is, the more fleixblility you have for
combining them to complex programs. (Functor -> Applicative ->
Arrow[Choice,Plus,Apply,..] -> Monad). It was nice to read in the
Typeclassopedia, that ArrowApply and Monad are equivalent, which is
shown by deriving two instances from each other:

instance Monad m => ArrowApply (Kleisli m)
instance ArrowApply a => Monad (a anyType)

The logic seems to be, that if I can derive from every instance of
class A an instance of class B, then A is more powerfull than B and
(in general) it is easier to be of class B than of class A (e.g. more
types can be made Applicatives, than Monads)

So far, I think I can follow. But what really hit me was the Cokleisli
type. Using it and the logic from above, I can show that ANY type
class is more (or equally) powerfull than the Monad:

instance AnyClass m => Monad (Cokleilsi m anyType)

I know this makes no sense, but where is the fallacy? Why even bother
with the above derivation, if any type class can be made into a monad?

I can see, that the Monad instance from above does not really
transform the type "a", but instead simply fix its first argument. But
then on the other hand, the ArrowApply Instance does transform the "m"
type (in a way similar to Cokleisli). If attention needs to be paid to
the details, then what are they and why did they not matter above?

Thanks,

Johannes