loudandclear

I don't think the debate is whether over or under was the intention of the inventors. The debate is whether over or under is "better", and what the patent says is pretty irrelevant.

Since a problem in P is can be solved in polynomial time, then a solution to it can be verified in polynomial time. If it can be verified in polynomial time, then it is in NP. It's this simple.

However, it has nothing to do with "non-polynomial" time solutions, as you originally stated. NP and non-polynomial doesn't have a correspondence by their definitions.

"P problems are in fact contained within the set of NP problems i.e. non-polynomial time solutions exist for P problems. However, all problems in NP may or may not be in P."

Yes, P is in contained in NP. However, the reason is wrong. It is contained in NP because any problem in P is verifiable in polytime (i.e. by actually solving it).