In general, if you’re trying to prove something, you can think of the various techniques and tools you have much like chess moves or Hanabi lines. Just like in turn-based games, you’ll find there are dead ends, e.g. trying to apply so-and-so theorem to reduce the problem to proving X, but X turns out to be false. Or there will be lots of paths that look more promising, but you can’t see far enough into the future to completely evaluate them all, and then you have to use heuristics and intuition to prioritize between approaches.