Definition join (j k:meminj):meminj := fun b =>
match j b with
Some (b1, delta) => Some (b1,delta)
| None => k b
end.
Definition disjoint (j k:meminj):Prop :=
forall b, j b = None \/ k b = None.
Definition shared_of (mu: SM_Injection) : meminj :=
join (foreign_of mu) (pub_of mu).

Lemma foreign_in_shared: forall mu,
inject_incr (foreign_of mu) (shared_of mu).