The importance of this question doesn’t involve whether or not there is an “option” in the first case or what you can or can’t do in the second. What matters is whether, hypothetically, you would always obey such a proof or would potentially disobey one. The hypothetical here matters independently of what actually happens because the hypothetical commitments of an agent can potentially be used in cases like this to prove things about it via Lob’s theorem. Another type of in which how an agent would hypothetically behave can have real influence on its actual circumstances is Newcomb’s problem. See this post.
The importance of this question doesn’t involve whether or not there is an “option” in the first case or what you can or can’t do in the second. What matters is whether, hypothetically, you would always obey such a proof or would potentially disobey one. The hypothetical here matters independently of what actually happens because the hypothetical commitments of an agent can potentially be used in cases like this to prove things about it via Lob’s theorem.
Another type of in which how an agent would hypothetically behave can have real influence on its actual circumstances is Newcomb’s problem. See this post.