Ltac grab_indexes_SEP ns :=
  rewrite (grab_indexes_SEP ns);
    unfold grab_indexes; simpl grab_calc;
   unfold grab_indexes', insert;
   repeat simpl_nat_of_P; cbv beta iota;
   unfold app_alt; fold @app_alt.(* The simpl_nat_of_P tactic is a complete hack,
  needed for compatibility between Coq 8.3/8.4,
  because the name of the thing to unfold varies
  between the two versions *)
Ltac simpl_nat_of_P :=
match goal with |- context [nat_of_P ?n] =>
  match n with xI _ => idtac | xO _ => idtac | xH => idtac | _ => fail end;
  let N := fresh "N" in
  set (N:= nat_of_P n);
  compute in N;
  unfold N; clear N
end.

Lemma lower_andp: forall P Q rho, @andp (environ->mpred) _ P Q rho = andp (P rho) (Q rho).