On Sun, Feb 11, 2001 at 01:37:28PM +1300, Brian Boutel wrote:
Let me restate my question more carefully:
Can you demonstrate a revised hierarchy without Eq? What would happen to Ord and the numeric classes with default class method definitions that use (==) either explicitly or in pattern matching against numeric literals? Both Integral and RealFrac do this to compare or test the value of signum.
I've been working on writing up my preferred hierarchy, but the short answer is that classes that are currently derived from Ord often do require Eq as superclasses. In the specific cases: I think possibly divMod and quotRem should be split into separate classes. It seems to me that divMod is the more fundamental pair: it satisfies the identity mod (a+b) b === mod a b div (a+b) b === 1 + div a b in addition to (div a b)*b + mod a b === a. This identity is not enough to specify divMod competely; another reasonable choice for Integers would be to round to the nearest integer. But this is enough to make it useful for many applications. quotRem is also useful (although it only satisfies the second of these), and does require the ordering (and ==) to define sensibly, so I would make it a method of a subclass of Ord (and hence Eq). So I would tend to put these into two separate classes: class (Ord a, Num a) => Real a class (Num a) => Integral a where div, mod :: a -> a -> a divMod :: a -> a -> (a,a) class (Integral a, Real a) => RealIntegral a where quot, rem :: a -> a -> a quotRem :: a -> a -> (a,a) I haven't thought about the operations in RealFrac and their semantics enough to say much sensible, but probably they will again require Ord as a superclass. In general, I think a good approach is to think carefully about the semantics of a class and its operations, and to declare exactly the superclasses that are necessary to define the semantics. Note that sometimes there are no additional operations. For instance, declaring a class to be an instance of Real a should mean that the ordering (from Ord) and the numeric structure (from Num) are compatible. Note also that we cannot require Eq to state laws (the '===' above); consider the laws required for the Monad class to convince yourself. Best, Dylan Thurston