I want to prove that a problem $P_1$ is NP-complete. Let say that I want to do a reduction from SAT problem.
If the instance of problem $P_1$ depends on $M$ and $N$, can I specify the sturcture of the instance of the SAT problem?
More precisely, can I say that, for example, the instance of SAT problem is composed of $M\times N +1$ clauses and every clause has $M+N$ literals ?
Based on this structure of the instance of SAT problem, I construct an instance of $P_1$. Is this proof correct?