According to the history of the treaty of Paris, the United States would be independent, but it would be restrainted to the east area of the appallachian mountains. That was in the beginning. That was the deal purposed by France.
Later, the American realized they could get a better agreement from London. They cut off with France and spain, and negotiated with Great britain. The news terms were: the United states would have all of the area east of the Mississipi river, south of canada and north of florida. They agreed the british merchants and loyalists to get their properties back.