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.

Lemma join_None_rightneutral: forall j, join j (fun b => None) = j.
Lemma join_None_leftneutral: forall j, join (fun b => None) j = j.