Definition impp {worlds: Type} {R: Relation worlds} (X: Ensemble worlds) (Y: Ensemble worlds): Ensemble worlds :=
  fun m => forall n, m <= n -> X n -> Y n.

Lemma impp_closed {worlds: Type} {R: Relation worlds} {po_R: PreOrder Krelation}: forall (X: Ensemble worlds) (Y: Ensemble worlds), upwards_closed_Kdenote X -> upwards_closed_Kdenote Y -> upwards_closed_Kdenote (impp X Y).