+  In other words, the following expressions should have the same result:
+
+    ``x == y`` and ``not x != y``
+
+    ``x < y`` and ``not x >= y``
+
+    ``x > y`` and ``not x <= y``

I think the second and third items here go too far: sets don't obey these rules, for example.  Not all uses of comparisons need to force a total ordering.

OTOH, you leave out a more fundamental relation, namely that `x < y` and `y > x` should ordinarily give the same result, as should `x <= y` and `y >= x`.