Local Open Scope logic_base.
Local Open Scope syntax.
Local Open Scope kripke_model.
Module Semantics.
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).