On Thu, Jan 01, 2015 at 08:58:35AM -0500, oleg@okmij.org wrote:
Tom Ellis wrote:
Is there a type with an Eq instance for which on Ord instance could not also be provided (i.e. I'm interested in Clark's "converse")?
Of course. One of the very many examples is Complex. One can say we can compare complex numbers, by their absolute value. That is true; however we end up in a position where compare x y returns EQ but x==y returns False.
I don't understand your objection. `Complex a` isomorphic to `(a, a)` which can be totally ordered. (The complex field cannot be given a total order *compatible with the field structure*, but that's not relevant to my question.)
In general, Ord represents total order. There are many structures on which total order cannot be established. Take nodes in the tree, the Tree data type. One can compare nodes by taking a parent to be less than any of its children. But then two brothers are not comparable. Searching for "partial orders" and pre-orders gives many more examples.
I think you may be missing the intent of my question. I am merely asking whether there is a type which supports a (law abiding) Eq instance that cannot support a (law abiding) Ord instance. Whether that Ord instance is compatible with additional structures on that type is beside the point. Tom