We define $\mathbf P$ as the set of problems solvable in polynomial time. We define $\mathbf{NP}$ as the set of problems with a verifier $ \in \mathbf P$.
Is there a name for problems whose verifiers are $\in \mathbf {NP}$ (e.g., $\mathbf{N(NP)}$)? I can't see this being a very useful complexity class, but, for example we have that $\mathbf{NP} \neq \mathbf{N(NP)} \implies \mathbf{P} \neq \mathbf{NP}$, so it might be an area of research for that reason alone.