Fixpoint closed_eval_expr (j: ident) (e: expr) : bool :=
 match e with
 | Econst_int i ty => true
 | Econst_long i ty => true
 | Econst_float f ty => true
 | Econst_single f ty => true
 | Etempvar id ty => negb (eqb_ident j id)
 | Eaddrof a ty => closed_eval_lvalue j a
 | Eunop op a ty =>  closed_eval_expr j a
 | Ebinop op a1 a2 ty =>  andb (closed_eval_expr j a1) (closed_eval_expr j a2)
 | Ecast a ty => closed_eval_expr j a
 | Evar id ty => true
 | Ederef a ty => closed_eval_expr j a
 | Efield a i ty => closed_eval_lvalue j a
 | Esizeof _ _ => true
 | Ealignof _ _ => true
 end

 with closed_eval_lvalue (j: ident) (e: expr) : bool :=
 match e with
 | Evar id ty => true
 | Ederef a ty => closed_eval_expr j a
 | Efield a i ty => closed_eval_lvalue j a
 | _  => false
 end.

Lemma closed_wrt_PROPx: forall S P Q, closed_wrt_vars S Q -> closed_wrt_vars S (PROPx P Q).