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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.             *)
(*                                                                    *)
(* 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 Bool.
Require Export fifo.
Unset Standard Proposition Elimination Names.
Section REDUCE.
Variable E : Set.
Variable f : E -> Z.
Fixpoint reduce (q : queue E) : Z :=
match q with
| empty => 0%Z
| input d q' => (reduce q' + f d)%Z
end.
Hypothesis positive_function : forall a : E, (f a > 0)%Z.

Lemma sum_positive :
forall a b : Z, (a >= 0)%Z -> (b > 0)%Z -> (a + b >= 0)%Z.