(*The join of all four components - useful for stating
  match_state conditions*)
Definition as_inj (mu: SM_Injection) : meminj :=
  join (extern_of mu) (local_of mu).

Lemma local_ofD_None: forall mu b1, local_of mu b1 = None -> pub_of mu b1 = None /\ priv_of mu b1 = None.