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).
Record SM_wd (mu:SM_Injection):Prop := {
disjoint_extern_local_Src: forall b, locBlocksSrc mu b = false \/ extBlocksSrc mu b = false;
disjoint_extern_local_Tgt: forall b, locBlocksTgt mu b = false \/ extBlocksTgt mu b = false;
local_DomRng: forall b1 b2 z, local_of mu b1 = Some(b2,z) ->
(locBlocksSrc mu b1 = true /\ locBlocksTgt mu b2 = true);
extern_DomRng: forall b1 b2 z, extern_of mu b1 = Some(b2,z) ->
(extBlocksSrc mu b1 = true /\ extBlocksTgt mu b2 = true);
pubSrcAx: forall b1, pubBlocksSrc mu b1 = true ->
exists b2 z, local_of mu b1 = Some(b2,z) /\
pubBlocksTgt mu b2 = true;
frgnSrcAx: forall b1, frgnBlocksSrc mu b1 = true ->
exists b2 z, extern_of mu b1 = Some(b2,z) /\
frgnBlocksTgt mu b2 = true;
pubBlocksLocalTgt: forall b, pubBlocksTgt mu b = true ->
locBlocksTgt mu b = true;
frgnBlocksExternTgt: forall b, frgnBlocksTgt mu b = true ->
extBlocksTgt mu b = true
}.

Lemma extern_foreignunknown: forall mu (WD: SM_wd mu),
extern_of mu = join (foreign_of mu) (unknown_of mu).