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 val_lessdef_inject_compose: forall v1 v2 (LD12 : Val.lessdef v1 v2) j v3 (InjV23 : val_inject j v2 v3), val_inject j v1 v3.