(*
Lemma gvar_globals_only:
  forall i v rho, gvar i v rho -> gvar i v (globals_only rho).
Proof.
unfold gvar; intros.
unfold Map.get in *.
destruct (ve_of rho i) as [[? ?]|] eqn:?; try contradiction.
unfold globals_only.
simpl. auto.
Qed.
Hint Resolve gvar_globals_only.
*)

Ltac safe_auto_with_closed :=
   (* won't instantiate evars by accident *)
 match goal with |- ?A =>
          solve [first [has_evar A | auto 50 with closed]]
 end.

Lemma closed_wrt_lift3: forall {A1 A2 A3}{B} S (f: A1 -> A2 -> A3 -> B) P1 P2 P3, closed_wrt_vars S P1 -> closed_wrt_vars S P2 -> closed_wrt_vars S P3 -> closed_wrt_vars S (lift3 f P1 P2 P3).