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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Require Export DistributedReferenceCounting.machine4.machine.
Require Export DistributedReferenceCounting.machine4.cardinal.
Require Export DistributedReferenceCounting.machine4.comm.
Unset Standard Proposition Elimination Names.
Section ROOTED.
Variable s0 : Site.
Definition rooted_fun (s1 s2 : Site) (m : Message) :=
match m with
| copy => if eq_site_dec s1 s0 then 1%Z else 0%Z
| dec => if eq_site_dec s2 s0 then 1%Z else 0%Z
| inc_dec s3 =>
if eq_site_dec s3 s0
then
if eq_site_dec s2 owner then 1%Z else 0%Z
else 0%Z
end.
Definition rooted (s1 s2 : Site) (q : queue Message) :=
reduce Message (rooted_fun s1 s2) q.
Definition sigma_rooted (bm : Bag_of_message) :=
sigma2_table Site LS LS (queue Message) rooted bm.
End ROOTED.
Section ROOTED1.

Lemma sigma_rooted_collect_message :
forall (m : Message) (s0 s1 s2 : Site) (b : Bag_of_message),
first Message (b s1 s2) = value Message m ->
s0 <> owner ->
sigma_rooted s0 (Collect_message Message b s1 s2) =
(sigma_rooted s0 b - rooted_fun s0 s1 s2 m)%Z.