(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Require Export DistributedReferenceCounting.machine3.alternate.
Require Export DistributedReferenceCounting.machine3.invariant0.
Require Export DistributedReferenceCounting.machine3.invariant1.
Require Export DistributedReferenceCounting.machine3.invariant2.
Require Export DistributedReferenceCounting.machine3.invariant3.
Require Export DistributedReferenceCounting.machine3.invariant4.
Unset Standard Proposition Elimination Names.
(* Changes - proof of  1  for false_rt_D_queue. - proof of  1  for  legal_alternate. - owner_st_positive and its proof. *)
Section INVARIANT5.
(* In this file, we establish the following inequality Int + Sum copy + Sum dec - Sum inc_dec >= 0  *)
(* The message queue between a site and the owner has a particular structure.  Messages dec and inc_dec are alternated. *)

(* Specifications: *)
Lemma legal_alternate :
forall (s : Site) (c : Config),
legal c ->
alternate (bm c s owner) /\ (rt c s = false -> D_queue (bm c s owner)).
Lemma count_inc_and_dec_alt_queue :
forall q : queue Message,
alternate q ->
(reduce Message dec_count q + 1 >= reduce Message new_inc_count q)%Z.
Lemma count_inc_and_dec_D_queue :
forall q : queue Message,
alternate q ->
D_queue q ->
(reduce Message dec_count q >= reduce Message new_inc_count q)%Z.
Lemma copy_count_is_positive :
forall q : queue Message, (reduce Message copy_count q >= 0)%Z.
Lemma dec_count_is_positive :
forall q : queue Message, (reduce Message dec_count q >= 0)%Z.
Lemma invariant5 :
forall c : Config,
legal c ->
forall s : Site,
(Int (rt c s) + reduce Message dec_count (bm c s owner) +
reduce Message copy_count (bm c owner s) -
reduce Message new_inc_count (bm c s owner) >= 0)%Z.