During my talk yesterday, somebody raised an important point when I discussed the Hoare rule for non-deterministic choice in the recursive semantics and claimed that the postcondition couldn’t be disjunctive: What if it was the negation of a conjunction?

I think I waved this away with a gesture towards the constructive Coq proof assistant, but that was in error (the Coq real number library actually posits the decidability of the reals).

In truth, propositions of the form , or even are excluded from the post-condition too. The question at hand is to identify the class of propositions such that and implies .

In our expanded VPHL paper, we explicitly limited the propositions we were considering to those of the form , for (specifically excluding negation and ), and were therefore able to prove the following lemmas (from which the Hoare rule I introduced at PPS followed easily):

Lemma 4.1 For any *non-disjunctive* assertion , implies that for any .

Lemma 4.2 For any *non-probabilistic* assertion , implies and for any .

But I’d be interested in the broader case: Has any work been done on showing what class of mathematical propositions are closed under the combination of distributions?