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If there is a proof that an NP-Hard problem which is not NP-Complete can be solved in P time, it does say that the verification time is polynomial too.
Why doesn't it then mean that all NP-Hard are NP (i.e. NPC)?
But, if there is something with exponential verification i.e. outside NP and since it can be reduced to an NPC problem in polynomial time which means it is also verifiable in polynomial time (As we assumed above). Isn't it? Which might say that all NP-hard is NP-complete...
I'm unable to grasp what is NP-hard but not NPC here? What is wrong in this reasoning?
Are there any problems which are outside Recursive Enumerable Languages in NP-Hard? Is that even possible?