(*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 as_injD_None: forall mu b1, as_inj mu b1 = None -> extern_of mu b1 = None /\ local_of mu b1 = None.