Hint Resolve closed_wrt_lift4  closed_wrtl_lift4 : closed.

Lemma closed_wrt_lift4C: forall {A1 A2 A3 A4}{B} S (f: A1 -> A2 -> A3 -> A4 -> B) P1 P2 P3 P4,
closed_wrt_vars S P1 ->
closed_wrt_vars S P2 ->
closed_wrt_vars S P3 ->
closed_wrt_vars S P4 ->
closed_wrt_vars S (@liftx (Tarrow A1 (Tarrow A2 (Tarrow A3 (Tarrow A4 (LiftEnviron B))))) f P1 P2 P3 P4).