The original SUBSET-SUM problem is "given a set of integers, is there a non-empty subset whose sum is zero?"
If we look at the inverse problem: "given a finite set of integers, does every non-empty subset have a nonzero sum?"
I understand how the verifier and certificate are used to determine SUBSET-SUM to be in NP.
I know it is unknown is NP = coNP.
But am I right when I say that the inverse problem has a verifier and certificate? The certificate being "every non-empty subset" and the verifier verifies every single subset. This will not be a polynomial verifier so it won't be in NP, but am I correct to assume that this is indeed a verifier and certificate?