WAT: opCmp and opEquals woes

[Fool via Digitalmars-d]

On Sunday, 27 July 2014 at 16:39:01 UTC, Ola Fosheim Grøstad 
wrote:
> On Saturday, 26 July 2014 at 16:43:06 UTC, Fool wrote:
>>> NaN < x is false
>>> NaN > x is false
>>
>> ...which means that < as it is usually defined on floating 
>> point numbers does not define a strict weak ordering.
>
> Are you sure?

One can define a strict weak ordering using different (but 
equivalent) sets of axioms.

We have

NOT (0.0 < NaN) AND NOT (NaN < 0.0)   [0.0 and NaN are 
incomparable]

AND

NOT (NaN < 1.0) AND NOT (1.0 < NaN)   [NaN and 1.0 are 
incomparable]

However, it does NOT hold

NOT (0.0 < 1.0) AND NOT (1.0 < 0.0)   [0.0 and 1.0 are 
incomparable]

Thus we do not have transitivity of incomparability:

"For all x, y, and z, if x is incomparable with y, and y is 
incomparable with z, then x is incomparable with z." [1]

[1] 
https://en.wikipedia.org/wiki/Weak_ordering#Strict_weak_orderings