[mmarx]

> Specifically: when you have a less-than relationship "<", then !(a<b) && !(b<a) implies that a and b are equal (a==b). And if a==b and b==c then it must be the case that a==c, or the requirements of the ordering predicate are not met.

¬(a<b) ∧ ¬(b<a) → a=b is not, in fact, a requirement on <. Rather, the point is that behind the scenes, any two elements satisfying ¬(a<b) ∧ ¬(b<a) are treated as equivalent by these containers.

To see the difference, consider the (somewhat counter-intuitive) behavior of NaNs: for any two NaNs m, n, we have ¬(m<n) and ¬(n<m), yet also m≠n. If that implication were an actual requirement, then the usual ordering < on floats would not be a suitable ordering predicate. What happens, though, is that the containers will implicitly treat NaNs as equivalent, i.e., the notion of equivalence the container uses for the elements depends only on < and might not coincide with the usual ==.