What about something like data AddMult a b = AddMult a b class Monoid a where operation :: a -> a -> a identity :: a instance (Monoid a, Monoid b) => Monoid (AddMult a b) where operation (AddMult a1 m1) (AddMult a2 m2) = AddMult (operation a1 a2) (operation m1 m2) identity = AddMult identity identity class Commutative a where -- Nothing, this is a programmer proof obligation class Monoid a => Group a where inverse :: a -> a class (Commutative a, Group a) => AbelianGroup a where class (AbelianGroup a, AbelianGroup b) => Field a b where instance AbelianGroup a => Field a a where George Pollard wrote:
Is there a good way of doing this? My running example is Monoid:
class Monoid a where operation :: a -> a -> a identity :: a
With the obvious examples on Num:
instance (Num a) => Monoid a where operation = (+) identity = 1
instance (Num a) => Monoid a where operation = (*) identity = 0
Of course, this won't work. I could introduce a newtype wrapper:
newtype (Num a) => MulNum a = MulNum a newtype (Num a) => AddNum a = AddNum a
instance (Num a) => Monoid (MulNum a) where operation (MulNum x) (MulNum y) = MulNum (x * y) identity = MulNum 1
instance (Num a) => Monoid (AddNum a) where ... -- etc
However, when it comes to defining (e.g.) a Field class you have two Abelian groups over the same type, which won't work straight off:
class Field a where ... instance (AbelianGroup a, AbelianGroup a) => Field a where ...
Could try using the newtypes again:
instance (AbelianGroup (x a), AbelianGroup (y a) => Field a where ...
... but this requires undecidable instances. I'm not even sure if it will do what I want. (For one thing it would also require an indication of which group distributes over the other, and this may restore decidability.)
I'm beginning to think that the best way to do things would be to drop the newtype wrappers and include instead an additional parameter of a type-level Nat to allow multiple definitions per type. Is this a good way to do things?
Has anyone else done something similar? I've taken a look at the Numeric Prelude but it seems to be doing things a bit differently. (e.g. there aren't constraints on Ring that require Monoid, etc)
- George