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(* 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.             *)
(*                                                                    *)
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(* 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                                                     *)
(** this file contains some cardinal functions, which are message dependent *)
Require Export reduce.
Require Export DistributedReferenceCounting.machine4.machine.
Section COUNT_MESSAGE.
Definition dec_predicate (m : Message) :=
match m with
| dec => true
| _ => false
end.
Definition inc_predicate (m : Message) :=
match m with
| inc_dec _ => true
| _ => false
end.
(* the next one only recognises inc_dec messages related to s0 *)
Variable s0 : Site.
Definition site_inc_predicate (m : Message) :=
match m with
| inc_dec s => if eq_site_dec s s0 then true else false
| _ => false
end.
Definition copy_predicate (m : Message) :=
match m with
| copy => true
| _ => false
end.
Definition dec_count (m : Message) :=
match m with
| dec => 1%Z
| _ => 0%Z
end.
Definition inc_count (m : Message) := 0%Z.
Definition copy_count (m : Message) :=
match m with
| copy => 1%Z
| _ => 0%Z
end.
Definition cardinal_count :=
fun_sum Message dec_count (fun_sum Message inc_count copy_count).
Definition cardinal := reduce Message cardinal_count.

Lemma cardinal_first_out :
forall (q : queue Message) (m : Message),
first Message q = value Message m ->
cardinal (first_out Message q) = (cardinal q - cardinal_count m)%Z.