What Wei Dai said. The first change you propose breaks A’s ability to obtain step 2 by simple reasoning about itself. Originally I came up with a different problem: systematically search all proofs up to a large length, and if we find a proof that we output some number, then output that number. That problem is way more crazy than mine, I don’t know what the answer is, it might actually depend on the proof checker’s implementation.
What Wei Dai said. The first change you propose breaks A’s ability to obtain step 2 by simple reasoning about itself. Originally I came up with a different problem: systematically search all proofs up to a large length, and if we find a proof that we output some number, then output that number. That problem is way more crazy than mine, I don’t know what the answer is, it might actually depend on the proof checker’s implementation.