(******************************************  Definitions of spacer and withspacer  Comment: spacer only has offset_zero property but does not have local_facts and isptr property.  ******************************************)
(* Arguments spacer sh be ed / _ . *)
Definition withspacer sh (be: Z) (ed: Z) P (p: val): mpred :=
if Z.eq_dec (ed - be) 0
then P p
else P p * spacer sh be ed p.

Lemma at_offset_preserve_local_facts: forall P pos, (forall p, P p |-- !!(isptr p)) -> (forall p, at_offset P pos p |-- !!(isptr p)).