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(* 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.
simple induction n; simpl in |- *; auto.
intros m; case m; simpl in |- *; auto.
intros H'; inversion H'; auto.
intros n0 H' m; case m; simpl in |- *; auto.
intros H'0; inversion H'0; auto.
intros n1 H'0.
rewrite H'; auto with arith.
rewrite H'; auto with arith.
Qed.

Theorem exp_Pascal :
forall a b n : nat,
power (a + b) n =
sum_nm n 0 (fun k : nat => binomial n k * (power a (n - k) * power b k)).