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 inject_separated_incr_fwd2: forall j0 j j' m10 m20 m1 m2, inject_incr j j' -> inject_separated j j' m1 m2 -> inject_incr j0 j -> mem_forward m10 m1 -> inject_separated j0 j m10 m20 -> mem_forward m20 m2 -> inject_separated j0 j' m10 m20.