Binomial.v

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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Require Import Rbase.
Require Import Rfunctions.
Require Import PartSum.
Local Open Scope R_scope.

Definition C (n p:nat) : R :=
  INR (fact n) / (INR (fact p) * INR (fact (n - p))).

Lemma pascal_step1 : forall n i:nat, (i <= n)%nat -> C n i = C n (n - i).forall n i : nat, (i <= n)%nat -> C n i = C n (n - i)
Proof.forall n i : nat, (i <= n)%nat -> C n i = C n (n - i)
  intros; unfold C; replace (n - (n - i))%nat with i.n, i:nat
H:(i <= n)%nat
INR (fact n) / (INR (fact i) * INR (fact (n - i))) =
INR (fact n) / (INR (fact (n - i)) * INR (fact i))
n, i:nat
H:(i <= n)%nat
i = (n - (n - i))%nat
  rewrite Rmult_comm.n, i:nat
H:(i <= n)%nat
INR (fact n) / (INR (fact (n - i)) * INR (fact i)) =
INR (fact n) / (INR (fact (n - i)) * INR (fact i))
n, i:nat
H:(i <= n)%nat
i = (n - (n - i))%nat
  reflexivity.n, i:nat
H:(i <= n)%nat
i = (n - (n - i))%nat
  apply plus_minus; rewrite plus_comm; apply le_plus_minus; assumption.
Qed.

Lemma pascal_step2 :
  forall n i:nat,
    (i <= n)%nat -> C (S n) i = INR (S n) / INR (S n - i) * C n i.forall n i : nat,
(i <= n)%nat ->
C (S n) i = INR (S n) / INR (S n - i) * C n i
Proof.forall n i : nat,
(i <= n)%nat ->
C (S n) i = INR (S n) / INR (S n - i) * C n i
  intros; unfold C; replace (S n - i)%nat with (S (n - i)).n, i:nat
H:(i <= n)%nat
INR (fact (S n)) /
(INR (fact i) * INR (fact (S (n - i)))) =
INR (S n) / INR (S (n - i)) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  cut (forall n:nat, fact (S n) = (S n * fact n)%nat).n, i:nat
H:(i <= n)%nat
(forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat) ->
INR (fact (S n)) /
(INR (fact i) * INR (fact (S (n - i)))) =
INR (S n) / INR (S (n - i)) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  intro; repeat rewrite H0.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S n * fact n) /
(INR (fact i) * INR (S (n - i) * fact (n - i))) =
INR (S n) / INR (S (n - i)) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  unfold Rdiv; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S n) * INR (fact n) *
(/ INR (fact i) *
 (/ INR (S (n - i)) * / INR (fact (n - i)))) =
INR (S n) * / INR (S (n - i)) *
(INR (fact n) *
 (/ INR (fact i) * / INR (fact (n - i))))
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  ring.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply not_O_INR; discriminate.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact i) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) * INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply prod_neq_R0.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (S (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply not_O_INR; discriminate.n, i:nat
H:(i <= n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact (n - i)) <> 0
n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i <= n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  intro; reflexivity.n, i:nat
H:(i <= n)%nat
S (n - i) = (S n - i)%nat
  apply minus_Sn_m; assumption.
Qed.

Lemma pascal_step3 :
  forall n i:nat, (i < n)%nat -> C n (S i) = INR (n - i) / INR (S i) * C n i.forall n i : nat,
(i < n)%nat ->
C n (S i) = INR (n - i) / INR (S i) * C n i
Proof.forall n i : nat,
(i < n)%nat ->
C n (S i) = INR (n - i) / INR (S i) * C n i
  intros; unfold C.n, i:nat
H:(i < n)%nat
INR (fact n) /
(INR (fact (S i)) * INR (fact (n - S i))) =
INR (n - i) / INR (S i) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
  cut (forall n:nat, fact (S n) = (S n * fact n)%nat).n, i:nat
H:(i < n)%nat
(forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat) ->
INR (fact n) /
(INR (fact (S i)) * INR (fact (n - S i))) =
INR (n - i) / INR (S i) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  intro.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
INR (fact n) /
(INR (fact (S i)) * INR (fact (n - S i))) =
INR (n - i) / INR (S i) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  cut ((n - i)%nat = S (n - S i)).n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i) ->
INR (fact n) /
(INR (fact (S i)) * INR (fact (n - S i))) =
INR (n - i) / INR (S i) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  intro.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact n) /
(INR (fact (S i)) * INR (fact (n - S i))) =
INR (n - i) / INR (S i) *
(INR (fact n) / (INR (fact i) * INR (fact (n - i))))
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  pattern (n - i)%nat at 2; rewrite H1.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact n) /
(INR (fact (S i)) * INR (fact (n - S i))) =
INR (n - i) / INR (S i) *
(INR (fact n) /
 (INR (fact i) * INR (fact (S (n - S i)))))
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  repeat rewrite H0; unfold Rdiv; repeat rewrite mult_INR;
    repeat rewrite Rinv_mult_distr.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact n) *
(/ INR (S i) * / INR (fact i) * / INR (fact (n - S i))) =
INR (n - i) * / INR (S i) *
(INR (fact n) *
 (/ INR (fact i) *
  (/ INR (S (n - S i)) * / INR (fact (n - S i)))))
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  rewrite <- H1; rewrite (Rmult_comm (/ INR (n - i)));
    repeat rewrite Rmult_assoc; rewrite (Rmult_comm (INR (n - i)));
      repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact n) *
(/ INR (S i) *
 (/ INR (fact i) * / INR (fact (n - S i)))) =
/ INR (S i) *
(INR (fact n) *
 (/ INR (fact i) * (/ INR (fact (n - S i)) * 1)))
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (n - i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  ring.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (n - i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply not_O_INR; apply minus_neq_O; assumption.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply not_O_INR; discriminate.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S (n - S i)) * INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply not_O_INR; discriminate.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (S i) * INR (fact i) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
H1:(n - i)%nat = S (n - S i)
INR (fact (n - S i)) <> 0
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply INR_fact_neq_0.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = S (n - S i)
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  rewrite minus_Sn_m.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(n - i)%nat = (S n - S i)%nat
n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(S i <= n)%nat
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  simpl; reflexivity.n, i:nat
H:(i < n)%nat
H0:forall n0 : nat,
fact (S n0) = (S n0 * fact n0)%nat
(S i <= n)%nat
n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  apply lt_le_S; assumption.n, i:nat
H:(i < n)%nat
forall n0 : nat, fact (S n0) = (S n0 * fact n0)%nat
  intro; reflexivity.
Qed.

  (**********)
Lemma pascal :
  forall n i:nat, (i < n)%nat -> C n i + C n (S i) = C (S n) (S i).forall n i : nat,
(i < n)%nat -> C n i + C n (S i) = C (S n) (S i)
Proof.forall n i : nat,
(i < n)%nat -> C n i + C n (S i) = C (S n) (S i)
  intros.n, i:nat
H:(i < n)%nat
C n i + C n (S i) = C (S n) (S i)
  rewrite pascal_step3; [ idtac | assumption ].n, i:nat
H:(i < n)%nat
C n i + INR (n - i) / INR (S i) * C n i =
C (S n) (S i)
  replace (C n i + INR (n - i) / INR (S i) * C n i) with
    (C n i * (1 + INR (n - i) / INR (S i))); [ idtac | ring ].n, i:nat
H:(i < n)%nat
C n i * (1 + INR (n - i) / INR (S i)) = C (S n) (S i)
  replace (1 + INR (n - i) / INR (S i)) with (INR (S n) / INR (S i)).n, i:nat
H:(i < n)%nat
C n i * (INR (S n) / INR (S i)) = C (S n) (S i)
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  rewrite pascal_step1.n, i:nat
H:(i < n)%nat
C n (n - i) * (INR (S n) / INR (S i)) = C (S n) (S i)
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  rewrite Rmult_comm; replace (S i) with (S n - (n - i))%nat.n, i:nat
H:(i < n)%nat
INR (S n) / INR (S n - (n - i)) * C n (n - i) =
C (S n) (S n - (n - i))
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  rewrite <- pascal_step2.n, i:nat
H:(i < n)%nat
C (S n) (n - i) = C (S n) (S n - (n - i))
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply pascal_step1.n, i:nat
H:(i < n)%nat
(n - i <= S n)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply le_trans with n.n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(n <= S n)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply le_minusni_n.n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
(n <= S n)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply lt_le_weak; assumption.n, i:nat
H:(i < n)%nat
(n <= S n)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply le_n_Sn.n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply le_minusni_n.n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply lt_le_weak; assumption.n, i:nat
H:(i < n)%nat
(S n - (n - i))%nat = S i
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  rewrite <- minus_Sn_m.n, i:nat
H:(i < n)%nat
S (n - (n - i)) = S i
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  cut ((n - (n - i))%nat = i).n, i:nat
H:(i < n)%nat
(n - (n - i))%nat = i -> S (n - (n - i)) = S i
n, i:nat
H:(i < n)%nat
(n - (n - i))%nat = i
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  intro; rewrite H0; reflexivity.n, i:nat
H:(i < n)%nat
(n - (n - i))%nat = i
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  symmetry ; apply plus_minus.n, i:nat
H:(i < n)%nat
n = (n - i + i)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  rewrite plus_comm; rewrite le_plus_minus_r.n, i:nat
H:(i < n)%nat
n = n
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  reflexivity.n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply lt_le_weak; assumption.n, i:nat
H:(i < n)%nat
(n - i <= n)%nat
n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply le_minusni_n; apply lt_le_weak; assumption.n, i:nat
H:(i < n)%nat
(i <= n)%nat
n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  apply lt_le_weak; assumption.n, i:nat
H:(i < n)%nat
INR (S n) / INR (S i) = 1 + INR (n - i) / INR (S i)
  unfold Rdiv.n, i:nat
H:(i < n)%nat
INR (S n) * / INR (S i) =
1 + INR (n - i) * / INR (S i)
  repeat rewrite S_INR.n, i:nat
H:(i < n)%nat
(INR n + 1) * / (INR i + 1) =
1 + INR (n - i) * / (INR i + 1)
  rewrite minus_INR.n, i:nat
H:(i < n)%nat
(INR n + 1) * / (INR i + 1) =
1 + (INR n - INR i) * / (INR i + 1)
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  cut (INR i + 1 <> 0).n, i:nat
H:(i < n)%nat
INR i + 1 <> 0 ->
(INR n + 1) * / (INR i + 1) =
1 + (INR n - INR i) * / (INR i + 1)
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  intro.n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR n + 1) * / (INR i + 1) =
1 + (INR n - INR i) * / (INR i + 1)
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  apply Rmult_eq_reg_l with (INR i + 1); [ idtac | assumption ].n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR i + 1) * ((INR n + 1) * / (INR i + 1)) =
(INR i + 1) * (1 + (INR n - INR i) * / (INR i + 1))
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  rewrite Rmult_plus_distr_l.n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR i + 1) * ((INR n + 1) * / (INR i + 1)) =
(INR i + 1) * 1 +
(INR i + 1) * ((INR n - INR i) * / (INR i + 1))
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  rewrite Rmult_1_r.n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR i + 1) * ((INR n + 1) * / (INR i + 1)) =
INR i + 1 +
(INR i + 1) * ((INR n - INR i) * / (INR i + 1))
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  do 2 rewrite (Rmult_comm (INR i + 1)).n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR n + 1) * / (INR i + 1) * (INR i + 1) =
INR i + 1 +
(INR n - INR i) * / (INR i + 1) * (INR i + 1)
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  repeat rewrite Rmult_assoc.n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR n + 1) * (/ (INR i + 1) * (INR i + 1)) =
INR i + 1 +
(INR n - INR i) * (/ (INR i + 1) * (INR i + 1))
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  rewrite <- Rinv_l_sym; [ idtac | assumption ].n, i:nat
H:(i < n)%nat
H0:INR i + 1 <> 0
(INR n + 1) * 1 = INR i + 1 + (INR n - INR i) * 1
n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  ring.n, i:nat
H:(i < n)%nat
INR i + 1 <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  rewrite <- S_INR.n, i:nat
H:(i < n)%nat
INR (S i) <> 0
n, i:nat
H:(i < n)%nat
(i <= n)%nat
  apply not_O_INR; discriminate.n, i:nat
H:(i < n)%nat
(i <= n)%nat
  apply lt_le_weak; assumption.
Qed.

  (*********************)
  (*********************)
Lemma binomial :
  forall (x y:R) (n:nat),
    (x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n.forall (x y : R) (n : nat),
(x + y) ^ n =
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
Proof.forall (x y : R) (n : nat),
(x + y) ^ n =
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
  intros; induction  n as [| n Hrecn].x, y:R
(x + y) ^ 0 =
sum_f_R0 (fun i : nat => C 0 i * x ^ i * y ^ (0 - i))
  0
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
(x + y) ^ S n =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n)
  unfold C; simpl; unfold Rdiv;
    repeat rewrite Rmult_1_r; rewrite Rinv_1; ring.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
(x + y) ^ S n =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n)
  pattern (S n) at 1; replace (S n) with (n + 1)%nat; [ idtac | ring ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
(x + y) ^ (n + 1) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n)
  rewrite pow_add; rewrite Hrecn.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) ^ 1 =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n)
  replace ((x + y) ^ 1) with (x + y); [ idtac | simpl; ring ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n)
  rewrite tech5.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i)) n +
C (S n) (S n) * x ^ S n * y ^ (S n - S n)
  cut (forall p:nat, C p p = 1).x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
(forall p : nat, C p p = 1) ->
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i)) n +
C (S n) (S n) * x ^ S n * y ^ (S n - S n)
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  cut (forall p:nat, C p 0 = 1).x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
(forall p : nat, C p 0 = 1) ->
(forall p : nat, C p p = 1) ->
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i)) n +
C (S n) (S n) * x ^ S n * y ^ (S n - S n)
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intros; rewrite H0; rewrite <- minus_n_n; rewrite Rmult_1_l.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n * y ^ 0
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl; reflexivity ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
sum_f_R0 (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  induction  n as [| n Hrecn0].x, y:R
Hrecn:(x + y) ^ 0 =
sum_f_R0
  (fun i : nat => C 0 i * x ^ i * y ^ (0 - i))
  0
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
sum_f_R0 (fun i : nat => C 0 i * x ^ i * y ^ (0 - i))
  0 * (x + y) =
sum_f_R0 (fun i : nat => C 1 i * x ^ i * y ^ (1 - i))
  0 + x ^ 1
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n) * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S (S n)) i * x ^ i * y ^ (S (S n) - i)) 
  (S n) + x ^ S (S n)
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  simpl; do 2 rewrite H; ring.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
sum_f_R0
  (fun i : nat => C (S n) i * x ^ i * y ^ (S n - i))
  (S n) * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S (S n)) i * x ^ i * y ^ (S (S n) - i)) 
  (S n) + x ^ S (S n)
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  (* N >= 1 *)
  set (N := S n).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * (x + y) =
sum_f_R0
  (fun i : nat => C (S N) i * x ^ i * y ^ (S N - i)) N +
x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite Rmult_plus_distr_l.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x +
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y =
sum_f_R0
  (fun i : nat => C (S N) i * x ^ i * y ^ (S N - i)) N +
x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (N - i)) N * x) with
    (sum_f_R0 (fun i:nat => C N i * x ^ S i * y ^ (N - i)) N).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N +
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y =
sum_f_R0
  (fun i : nat => C (S N) i * x ^ i * y ^ (S N - i)) N +
x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (N - i)) N * y) with
    (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (S N - i)) N).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0
  (fun i : nat => C (S N) i * x ^ i * y ^ (S N - i)) N +
x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite (decomp_sum (fun i:nat => C (S N) i * x ^ i * y ^ (S N - i)) N).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
C (S N) 0 * x ^ 0 * y ^ (S N - 0) +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i))
  (Init.Nat.pred N) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite H; replace (x ^ 0) with 1; [ idtac | reflexivity ].x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
1 * 1 * y ^ (S N - 0) +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i))
  (Init.Nat.pred N) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  do 2 rewrite Rmult_1_l.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ (S N - 0) +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i))
  (Init.Nat.pred N) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (S N - 0)%nat with (S N); [ idtac | reflexivity ].x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i))
  (Init.Nat.pred N) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  set (An := fun i:nat => C N i * x ^ S i * y ^ (N - i)).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i))
  (Init.Nat.pred N) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  set (Bn := fun i:nat => C N (S i) * x ^ S i * y ^ (N - i)).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i))
  (Init.Nat.pred N) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (pred N) with n.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N +
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n +
x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (sum_f_R0 (fun i:nat => C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n)
    with (sum_f_R0 (fun i:nat => An i + Bn i) n).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + sum_f_R0 (fun i : nat => An i + Bn i) n +
x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite plus_sum.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + (sum_f_R0 An n + sum_f_R0 Bn n) + x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (x ^ S N) with (An (S n)).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + (sum_f_R0 An n + sum_f_R0 Bn n) + An (S n)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite (Rplus_comm (sum_f_R0 An n)).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + (sum_f_R0 Bn n + sum_f_R0 An n) + An (S n)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  repeat rewrite Rplus_assoc.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + (sum_f_R0 Bn n + (sum_f_R0 An n + An (S n)))
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite <- tech5.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + (sum_f_R0 Bn n + sum_f_R0 An (S n))
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  fold N.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 An N +
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
y ^ S N + (sum_f_R0 Bn n + sum_f_R0 An N)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  set (Cn := fun i:nat => C N i * x ^ i * y ^ (S N - i)).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
sum_f_R0 An N + sum_f_R0 Cn N =
y ^ S N + (sum_f_R0 Bn n + sum_f_R0 An N)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  cut (forall i:nat, (i < N)%nat -> Cn (S i) = Bn i).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
(forall i : nat, (i < N)%nat -> Cn (S i) = Bn i) ->
sum_f_R0 An N + sum_f_R0 Cn N =
y ^ S N + (sum_f_R0 Bn n + sum_f_R0 An N)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intro; replace (sum_f_R0 Bn n) with (sum_f_R0 (fun i:nat => Cn (S i)) n).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 An N + sum_f_R0 Cn N =
y ^ S N +
(sum_f_R0 (fun i : nat => Cn (S i)) n + sum_f_R0 An N)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (y ^ S N) with (Cn 0%nat).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 An N + sum_f_R0 Cn N =
Cn 0%nat +
(sum_f_R0 (fun i : nat => Cn (S i)) n + sum_f_R0 An N)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
Cn 0%nat = y ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite <- Rplus_assoc; rewrite (decomp_sum Cn N).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 An N +
(Cn 0%nat +
 sum_f_R0 (fun i : nat => Cn (S i)) (Init.Nat.pred N)) =
Cn 0%nat + sum_f_R0 (fun i : nat => Cn (S i)) n +
sum_f_R0 An N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
Cn 0%nat = y ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (pred N) with n.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 An N +
(Cn 0%nat + sum_f_R0 (fun i : nat => Cn (S i)) n) =
Cn 0%nat + sum_f_R0 (fun i : nat => Cn (S i)) n +
sum_f_R0 An N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
Cn 0%nat = y ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  ring.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
Cn 0%nat = y ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold N; simpl; reflexivity.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
Cn 0%nat = y ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold N; apply lt_O_Sn.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
Cn 0%nat = y ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold Cn; rewrite H; simpl; ring.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
sum_f_R0 (fun i : nat => Cn (S i)) n = sum_f_R0 Bn n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  apply sum_eq.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
H1:forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
forall i : nat, (i <= n)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intros; apply H1.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
An:=fun i0 : nat => C N i0 * x ^ S i0 * y ^ (N - i0):nat -> R
Bn:=fun i0 : nat =>
C N (S i0) * x ^ S i0 * y ^ (N - i0):nat -> R
Cn:=fun i0 : nat => C N i0 * x ^ i0 * y ^ (S N - i0):nat -> R
H1:forall i0 : nat,
(i0 < N)%nat -> Cn (S i0) = Bn i0
i:nat
H2:(i <= n)%nat
(i < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold N; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ].x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
Cn:=fun i : nat => C N i * x ^ i * y ^ (S N - i):nat -> R
forall i : nat, (i < N)%nat -> Cn (S i) = Bn i
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  reflexivity.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
An (S n) = x ^ S N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold An; fold N; rewrite <- minus_n_n; rewrite H0;
    simpl; ring.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
sum_f_R0 (fun i : nat => An i + Bn i) n =
sum_f_R0
  (fun i : nat =>
   C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  apply sum_eq.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
forall i : nat,
(i <= n)%nat ->
An i + Bn i =
C (S N) (S i) * x ^ S i * y ^ (S N - S i)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intros; unfold An, Bn.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
An:=fun i0 : nat => C N i0 * x ^ S i0 * y ^ (N - i0):nat -> R
Bn:=fun i0 : nat =>
C N (S i0) * x ^ S i0 * y ^ (N - i0):nat -> R
i:nat
H1:(i <= n)%nat
C N i * x ^ S i * y ^ (N - i) +
C N (S i) * x ^ S i * y ^ (N - i) =
C (S N) (S i) * x ^ S i * y ^ (S N - S i)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  change (S N - S i)%nat with (N - i)%nat.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
An:=fun i0 : nat => C N i0 * x ^ S i0 * y ^ (N - i0):nat -> R
Bn:=fun i0 : nat =>
C N (S i0) * x ^ S i0 * y ^ (N - i0):nat -> R
i:nat
H1:(i <= n)%nat
C N i * x ^ S i * y ^ (N - i) +
C N (S i) * x ^ S i * y ^ (N - i) =
C (S N) (S i) * x ^ S i * y ^ (N - i)
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite <- pascal;
    [ ring
      | apply le_lt_trans with n; [ assumption | unfold N; apply lt_n_Sn ] ].x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
An:=fun i : nat => C N i * x ^ S i * y ^ (N - i):nat -> R
Bn:=fun i : nat => C N (S i) * x ^ S i * y ^ (N - i):nat -> R
n = Init.Nat.pred N
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold N; reflexivity.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
(0 < N)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  unfold N; apply lt_O_Sn.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ i * y ^ (S N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite <- (Rmult_comm y); rewrite scal_sum; apply sum_eq.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
forall i : nat,
(i <= N)%nat ->
C N i * x ^ i * y ^ (S N - i) =
C N i * x ^ i * y ^ (N - i) * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intros; replace (S N - i)%nat with (S (N - i)).x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
i:nat
H1:(i <= N)%nat
C N i * x ^ i * y ^ S (N - i) =
C N i * x ^ i * y ^ (N - i) * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
i:nat
H1:(i <= N)%nat
S (N - i) = (S N - i)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (S (N - i)) with (N - i + 1)%nat; [ idtac | ring ].x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
i:nat
H1:(i <= N)%nat
C N i * x ^ i * y ^ (N - i + 1) =
C N i * x ^ i * y ^ (N - i) * y
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
i:nat
H1:(i <= N)%nat
S (N - i) = (S N - i)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl; ring ];
    ring.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n ->
sum_f_R0
  (fun i0 : nat =>
   C n i0 * x ^ i0 * y ^ (n - i0)) n *
(x + y) =
sum_f_R0
  (fun i0 : nat =>
   C (S n) i0 * x ^ i0 * y ^ (S n - i0)) n +
x ^ S n
N:=S n:nat
i:nat
H1:(i <= N)%nat
S (N - i) = (S N - i)%nat
x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  apply minus_Sn_m; assumption.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
sum_f_R0
  (fun i : nat => C N i * x ^ S i * y ^ (N - i)) N =
sum_f_R0 (fun i : nat => C N i * x ^ i * y ^ (N - i))
  N * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite <- (Rmult_comm x); rewrite scal_sum; apply sum_eq.x, y:R
n:nat
Hrecn:(x + y) ^ S n =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) 
  (S n)
H:forall p : nat, C p 0 = 1
H0:forall p : nat, C p p = 1
Hrecn0:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n ->
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n * (x + y) =
sum_f_R0
  (fun i : nat =>
   C (S n) i * x ^ i * y ^ (S n - i)) n +
x ^ S n
N:=S n:nat
forall i : nat,
(i <= N)%nat ->
C N i * x ^ S i * y ^ (N - i) =
C N i * x ^ i * y ^ (N - i) * x
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intros; replace (S i) with (i + 1)%nat; [ idtac | ring ]; rewrite pow_add;
    replace (x ^ 1) with x; [ idtac | simpl; ring ];
      ring.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p 0 = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intro; unfold C.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
p:nat
INR (fact p) / (INR (fact 0) * INR (fact (p - 0))) = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (INR (fact 0)) with 1; [ idtac | reflexivity ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
p:nat
INR (fact p) / (1 * INR (fact (p - 0))) = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  replace (p - 0)%nat with p; [ idtac | apply minus_n_O ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
p:nat
INR (fact p) / (1 * INR (fact p)) = 1
x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  rewrite Rmult_1_l; unfold Rdiv; rewrite <- Rinv_r_sym;
    [ reflexivity | apply INR_fact_neq_0 ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
forall p : nat, C p p = 1
  intro; unfold C.x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
p:nat
INR (fact p) / (INR (fact p) * INR (fact (p - p))) = 1
  replace (p - p)%nat with 0%nat; [ idtac | apply minus_n_n ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
p:nat
INR (fact p) / (INR (fact p) * INR (fact 0)) = 1
  replace (INR (fact 0)) with 1; [ idtac | reflexivity ].x, y:R
n:nat
Hrecn:(x + y) ^ n =
sum_f_R0
  (fun i : nat => C n i * x ^ i * y ^ (n - i))
  n
p:nat
INR (fact p) / (INR (fact p) * 1) = 1
  rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym;
    [ reflexivity | apply INR_fact_neq_0 ].
Qed.