(** ** Memory invariants *)
(** A stack location is private if it is not the image of a valid location and we have full rights on it. *)
Definition loc_private (F: meminj) (m m': mem) (sp: block) (ofs: Z) : Prop :=
Mem.perm m' sp ofs Cur Freeable /\
(forall b delta, F b = Some(sp, delta) -> ~Mem.perm m b (ofs - delta) Max Nonempty).
(** Likewise, for a range of locations. *)
Definition range_private (F: meminj) (m m': mem) (sp: block) (lo hi: Z) : Prop :=
forall ofs, lo <= ofs < hi -> loc_private F m m' sp ofs.

Lemma range_private_invariant:
forall F m m' sp lo hi F1 m1 m1',
range_private F m m' sp lo hi ->
(forall b delta ofs,
F1 b = Some(sp, delta) ->
Mem.perm m1 b ofs Max Nonempty ->
lo <= ofs + delta < hi ->
F b = Some(sp, delta) /\ Mem.perm m b ofs Max Nonempty) ->
(forall ofs, Mem.perm m' sp ofs Cur Freeable -> Mem.perm m1' sp ofs Cur Freeable) ->
range_private F1 m1 m1' sp lo hi.