Q:

Oracle complexity of a problem in the Counting Hierarchy

In "On The Complexity of Numerical Analysis" (SIAM J. Comp. Vol. 38, 2009), Allender et al. introduce the problem of PosSLP and show that its complexity lies in the counting hierarchy, and more precisely in
$P^{\mathit{PP}^{\mathit{PP}^{\mathit{PP}}}}$.
I have a problem, call it $X$, that I have shown can be solved in
$\mathit{NP}^{\mathit{PosSLP}}$. Can I correctly conclude that $X$ lies in 
$\mathit{NP}^{\mathit{PP}^{\mathit{PP}^{\mathit{PP}}}}$? 

A:

Yes. Each time your $\sf NP$ machine want to query the $\sf PosSLP$ oracle, simply simulate the polynomial time oracle Turing machine underlying the inclusion $\sf PosSLP \subseteq {P}^{{PP}^{PP^{PP}}}$, passing its oracle queries to the $\sf {PP}^{PP^{PP}}$ oracle.

