the price of oil does not cause inflation. The amount of money in circulation is inflation. If that has not increased inordinately then as oil goes up other things must go down commensurately. That is not currently the case. Inflation is strong. Much of it is exported to China as China uses its dollars to buy US securities and prices inside China go up. Here we have exploding housing prices and, of course, oil. These manifestations are dismissed as not being part of the "core" inflation rate.Because these prices are rising so strongly and the so-called core rate is NOT declining, inflation is strongly indicated.
After your first three sentences your analysis falls apart.
Problem is, the strongly increasing prices are only in a few select regions. Where I'm at, prices (real estate) are in some cases declining.