Definition sepcon {worlds: Type} {R: Relation worlds} {J: Join worlds} (X: Ensemble worlds) (Y: Ensemble worlds): Ensemble worlds :=
  fun m => exists m0 m1 m2, m0 <= m /\ join m1 m2 m0 /\ X m1 /\ Y m2.Definition wand {worlds: Type} {R: Relation worlds} {J: Join worlds} (X: Ensemble worlds) (Y: Ensemble worlds): Ensemble worlds :=
  fun m => forall m0 m1 m2, m <= m0 -> join m0 m1 m2 -> X m1 -> Y m2.Definition emp {worlds: Type} {R: Relation worlds} {J: Join worlds}: Ensemble worlds := increasing'.

(* Specifications: *)
Lemma sepcon_closed {worlds: Type} {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds}: forall (X: Ensemble worlds) (Y: Ensemble worlds), upwards_closed_Kdenote X -> upwards_closed_Kdenote Y -> upwards_closed_Kdenote (sepcon X Y).
Lemma wand_closed {worlds: Type} {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds}: forall (X: Ensemble worlds) (Y: Ensemble worlds), upwards_closed_Kdenote X -> upwards_closed_Kdenote Y -> upwards_closed_Kdenote (wand X Y).
Lemma emp_closed {worlds: Type} {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds}: upwards_closed_Kdenote emp.