Local Open Scope logic_base.
Local Open Scope syntax.
Local Open Scope kripke_model.
Module Semantics.
Definition boxp {worlds: Type} {R: KM.Relation worlds} (X: Ensemble worlds): Ensemble worlds :=
fun m => forall n, KM.Krelation m n -> X n.

Lemma boxp_closed
{worlds: Type}
{R1: KI.Relation worlds}
{po_R1: PreOrder KI.Krelation}
{R2: KM.Relation worlds}
{ukmM: UpwardsClosedOrderedKripkeModel worlds}:
forall (X: Ensemble worlds),
upwards_closed_Kdenote X ->
upwards_closed_Kdenote (boxp X).