OK, let me be more precise.  Obviously if the implementation in a class is:

class Foo:
    def __lt__(self, other):
        return random.random() < 0.5

Then we aren't going to rely on much.  

* If comparison of any two items in a list (under __lt__) is deterministic, is the resulting sort order deterministic? (Pretty sure this is a yes)
* If the pairwise comparisons are deterministic, is sorting idempotent?

This statement is certainly false:

* If two items are equal, and pairwise inequality is deterministic, exchanging the items does not affect the sorting of other items in the list.

On Sun, Jan 6, 2019 at 11:09 PM Tim Peters <tim.peters@gmail.com> wrote:
[David Mertz <david.mertz@gmail.com>]
> Thanks Tim for clarifying.  Is it even the case that sorts are STABLE in
> the face of non-total orderings under __lt__?  A couple quick examples
> don't refute that, but what I tried was not very thorough, nor did I
> think much about TimSort itself.

I'm not clear on what "stable" could mean in the absence of a total
ordering.  Not only does sort not assume __lt__ is a total ordering,
it doesn't assume it's transitive, or even deterministic.  We really
can't assume anything about potentially user-defined functions.

What sort does guarantee is that the result list is some permutation
of the input list, regardless of how insanely __lt__ may behave.  If
__lt__ sanely defines a deterministic total order, then "stable" and
"sorted" are guaranteed too, with their obvious meanings.

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