The problem of Oracle-SAT is given below:
Given oracle query access to some machine, $U$ that has $2^N$ inputs, determine if there is an input such that the machine accepts.
This is very similar to the regular SAT problem. The only alteration in SAT is that we are given the structure of the circuit; i.e. the CNF formula.
Oracle-SAT is clearly intractable, as we would need to examine all $2^N$ inputs. And SAT is thought to also be intractable, as $P \neq NP$.
Is it possible for Oracle-SAT to reduce to SAT? I am using the notion of Cook-reduction here. If we had an oracle that solved SAT, could we also solve Oracle-SAT in Polynomial time?
If it possible for Oracle-SAT to reduce to SAT, then this would imply $P \neq NP$, right?