Comment by Jweb_Guru

I'm talking about the proof length, not the length of the proposition. In CiC, you can have proofs like (for example) "eq_refl : eq <some short term that takes a very long time to compute> <some other short term that takes a very long time to compute>" (where both terms can be guaranteed to terminate before you actually execute them, so this isn't semidecidable or anything, just really slow). In other words, there exist short correct proofs that are not polynomial time to verify (where "verify" is essentially the same as type checking). So "find a proof of length n for this proposition" is not a problem in NP. You can't just say "ah but proofs that long aren't meaningful in our universe" when the proof can be short.