That is a really good question. I suppose you could reduce it further by saying that you want the proof of "A or B". Assuming both true, it suffices to either get a proof for A or for B (of course, this may not be true in general).
Regardless, this is a really good counter-example that will force me to think some more about it. Thanks!
> I suppose you could reduce it further by saying that you want the proof of "A or B". Assuming both true, it suffices to either get a proof for A or for B
Yes, absolutely :) I thought about this framing too, but figured the one I gave above might be more immediately convincing.
Regardless, this is a really good counter-example that will force me to think some more about it. Thanks!