(* 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 MiscRsa.
(*********************************************************************** ********************************************************************** **********************************************************************  ITERATED SUMS *)
Fixpoint sum_nm (n : nat) : nat -> (nat -> nat) -> nat :=
fun (m : nat) (f : nat -> nat) =>
match n with
| O => f m
| S n' => f m + sum_nm n' (S m) f
end.
auto.
Qed.
intros m n f; generalize m; clear m; elim n; simpl in |- *; intros.
rewrite (plus_comm m); auto.
rewrite H; repeat (rewrite (plus_comm m); simpl in |- *); auto with arith.
Qed.
(*********************************************************************** ********************************************************************** **********************************************************************  BINOMIAL *)
Fixpoint binomial (a : nat) : nat -> nat :=
fun b : nat =>
match a, b with
| _, O => 1
| O, S b' => 0
| S a', S b' => binomial a' (S b') + binomial a' b'
end.

Lemma binomial_def1 : forall n : nat, binomial n 0 = 1.