Details depend on the proof system. But yes, in general you expect that the number of possible proofs goes up exponentially with their length.
But any single given proof can be checked in polynomial time.
That's almost the definition of NP: NP stands for 'nondeterministic polynomial time'. Which means if you can nondeterministically 'guess' a solution (or someone hands it to you etc), you can verify it in polynomial time.
But any single given proof can be checked in polynomial time.
That's almost the definition of NP: NP stands for 'nondeterministic polynomial time'. Which means if you can nondeterministically 'guess' a solution (or someone hands it to you etc), you can verify it in polynomial time.