Definition expr_closed_wrt_lvars (S: ident -> Prop) (e: expr) : Prop :=
  forall (cs: compspecs) rho ve',
     (forall i, S i \/ Map.get (ve_of rho) i = Map.get ve' i) ->
     eval_expr e rho = eval_expr e (mkEnviron (ge_of rho) ve' (te_of rho)).Definition lvalue_closed_wrt_lvars (S: ident -> Prop) (e: expr) : Prop :=
  forall (cs: compspecs) rho ve',
     (forall i, S i \/ Map.get (ve_of rho) i = Map.get ve' i) ->
     eval_lvalue e rho = eval_lvalue e (mkEnviron (ge_of rho) ve'  (te_of rho)).

Lemma closed_wrt_allp: forall A S P, (forall x: A, closed_wrt_vars S (P x)) -> closed_wrt_vars S (allp P).