Definition val_inject_opt (j: meminj) (v1 v2: option val) :=
  match v1, v2 with Some v1', Some v2' => val_inject j v1' v2'
  | None, None => True
  | _, _ => False
  end.

Lemma free_readonly: forall b lo hi m m' (M: Mem.free m b lo hi = Some m'), forall b, Mem.valid_block m b -> readonly m b m'.