Mostly, yes. Oil is an input to near everything and, so, inflation tends to go toghether with its price.
today
As I see, two factors can change real oil prices:
1 - A recession, where people stop buying lots of goods and concentrate their spending on oil. That would make oil prices go up.
2 - Food inflation, where people stop buying other goods to concentrate their spending on food. That would make oil prices to go down.

And 2 seems to be dominating 1 by now.

But forget about all that. Current oil prices are due to the weak Dollar (and a weak Dollar is partialy due to peak oil, but that is another topic). Have I said that gasoline prices are falling as a rock at Brazil?

I'm sorry, I couldn't disagree more.

1. IF there's a recession, that'll mean less, not more oil demand, thus leading to lower prices.

2. Food inflation does not impact oil inflation unless there is a recession, in which case, oil goes down.

The dollar has stabilized at 1,42 in euros since last month, when oil hit 82. The current 5$ rise is therefore not because of the descending dollar. And oil is also rising in euros, so this is no "excuse" at all.

The underlying fundamental is shrinking supply (non-OPEC), shrinking stocks (USA) and growing demand (Chindia). Demand has been higher than supply all year long.

"1. IF there's a recession, that'll mean less, not more oil demand, thus leading to lower prices."

Ah, but you discount the effect of panic and hoarding (personal and national) on prices.

People don't 'hoard' during recessions. They bemoan their woes to congress and the Democrats give them $5000 baby bonds.

I am talking about inflation adjusted prices.

On a recession, every price go down (deflation), but oil moves less than average. On a food inflation, every price go up, but oil, again, moves less than average.