Say we want to factor a large number.
Factoring it is in NP.
Verifying it is definitely in P.
However, how do we know that there is no other factors besides the one we found?
So it seems that verifying it isn't in NP either.
Now I am getting confused with NP definition here. I know that NP includes all P.
But then there is this side definition that NP are problems with polynomial verifier.
Is there a proof that all NP problems have P verifier?
If the answer is true, yes all NP problems and hence all NP complete problems have P verifier, then please tell me the proof.