Module Semantics.
Definition andp {worlds: Type} (X: Ensemble worlds) (Y: Ensemble worlds): Ensemble worlds :=
fun m => X m /\ Y m.
Definition orp {worlds: Type} (X: Ensemble worlds) (Y: Ensemble worlds): Ensemble worlds :=
fun m => X m \/ Y m.
Definition falsep {worlds: Type}: Ensemble worlds := fun m => False.

Lemma andp_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 (andp X Y).