Definition 3.  PROG\((\phi)\) is true with respect to interval \(I\) and world \(w\) if and only if \(\forall w' \in IE(w)\) there exists an interval \(I' \supseteq I\) such that \(\phi\) is true with respect to \(I'\) and \(w'\). In addition \(I\) should not be the final subinterval of \(I'\). \(IE(w)\) is the set of inertia worlds for \(w\).