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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
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).
Proof.
intros; unfold C; replace (n - (n - i))%nat with i.
rewrite Rmult_comm.
reflexivity.
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.
Proof.
intros; unfold C; replace (S n - i)%nat with (S (n - i)).
cut (forall n:nat, fact (S n) = (S n * fact n)%nat).
intro; repeat rewrite H0.
unfold Rdiv; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr.
ring.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply prod_neq_R0.
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
intro; reflexivity.
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.
Proof.
intros; unfold C.
cut (forall n:nat, fact (S n) = (S n * fact n)%nat).
intro.
cut ((n - i)%nat = S (n - S i)).
intro.
pattern (n - i)%nat at 2; rewrite H1.
repeat rewrite H0; unfold Rdiv; repeat rewrite mult_INR;
repeat rewrite Rinv_mult_distr.
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.
ring.
apply not_O_INR; apply minus_neq_O; assumption.
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
apply INR_fact_neq_0.
rewrite minus_Sn_m.
simpl; reflexivity.
apply lt_le_S; assumption.
intro; reflexivity.
Qed.
(**********)
Lemma pascal :
forall n i:nat, (i < n)%nat -> C n i + C n (S i) = C (S n) (S i).
Proof.
intros.
rewrite pascal_step3; [ idtac | assumption ].
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 ].
replace (1 + INR (n - i) / INR (S i)) with (INR (S n) / INR (S i)).
rewrite pascal_step1.
rewrite Rmult_comm; replace (S i) with (S n - (n - i))%nat.
rewrite <- pascal_step2.
apply pascal_step1.
apply le_trans with n.
apply le_minusni_n.
apply lt_le_weak; assumption.
apply le_n_Sn.
apply le_minusni_n.
apply lt_le_weak; assumption.
rewrite <- minus_Sn_m.
cut ((n - (n - i))%nat = i).
intro; rewrite H0; reflexivity.
symmetry ; apply plus_minus.
rewrite plus_comm; rewrite le_plus_minus_r.
reflexivity.
apply lt_le_weak; assumption.
apply le_minusni_n; apply lt_le_weak; assumption.
apply lt_le_weak; assumption.
unfold Rdiv.
repeat rewrite S_INR.
rewrite minus_INR.
cut (INR i + 1 <> 0).
intro.
apply Rmult_eq_reg_l with (INR i + 1); [ idtac | assumption ].
rewrite Rmult_plus_distr_l.
rewrite Rmult_1_r.
do 2 rewrite (Rmult_comm (INR i + 1)).
repeat rewrite Rmult_assoc.
rewrite <- Rinv_l_sym; [ idtac | assumption ].
ring.
rewrite <- S_INR.
apply not_O_INR; discriminate.
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.
Proof.
intros; induction n as [| n Hrecn].
unfold C; simpl; unfold Rdiv;
repeat rewrite Rmult_1_r; rewrite Rinv_1; ring.
pattern (S n) at 1; replace (S n) with (n + 1)%nat; [ idtac | ring ].
rewrite pow_add; rewrite Hrecn.
replace ((x + y) ^ 1) with (x + y); [ idtac | simpl; ring ].
rewrite tech5.
cut (forall p:nat, C p p = 1).
cut (forall p:nat, C p 0 = 1).
intros; rewrite H0; rewrite <- minus_n_n; rewrite Rmult_1_l.
replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl; reflexivity ].
induction n as [| n Hrecn0].
simpl; do 2 rewrite H; ring.
(* N >= 1 *)
set (N := S n).
rewrite Rmult_plus_distr_l.
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).
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).
rewrite (decomp_sum (fun i:nat => C (S N) i * x ^ i * y ^ (S N - i)) N).
rewrite H; replace (x ^ 0) with 1; [ idtac | reflexivity ].
do 2 rewrite Rmult_1_l.
replace (S N - 0)%nat with (S N); [ idtac | reflexivity ].
set (An := fun i:nat => C N i * x ^ S i * y ^ (N - i)).
set (Bn := fun i:nat => C N (S i) * x ^ S i * y ^ (N - i)).
replace (pred N) with n.
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).
rewrite plus_sum.
replace (x ^ S N) with (An (S n)).
rewrite (Rplus_comm (sum_f_R0 An n)).
repeat rewrite Rplus_assoc.
rewrite <- tech5.
fold N.
set (Cn := fun i:nat => C N i * x ^ i * y ^ (S N - i)).
cut (forall i:nat, (i < N)%nat -> Cn (S i) = Bn i).
intro; replace (sum_f_R0 Bn n) with (sum_f_R0 (fun i:nat => Cn (S i)) n).
replace (y ^ S N) with (Cn 0%nat).
rewrite <- Rplus_assoc; rewrite (decomp_sum Cn N).
replace (pred N) with n.
ring.
unfold N; simpl; reflexivity.
unfold N; apply lt_O_Sn.
unfold Cn; rewrite H; simpl; ring.
apply sum_eq.
intros; apply H1.
unfold N; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ].
intros; unfold Bn, Cn.
replace (S N - S i)%nat with (N - i)%nat; reflexivity.
unfold An; fold N; rewrite <- minus_n_n; rewrite H0;
simpl; ring.
apply sum_eq.
intros; unfold An, Bn; replace (S N - S i)%nat with (N - i)%nat;
[ idtac | reflexivity ].
rewrite <- pascal;
[ ring
| apply le_lt_trans with n; [ assumption | unfold N; apply lt_n_Sn ] ].
unfold N; reflexivity.
unfold N; apply lt_O_Sn.
rewrite <- (Rmult_comm y); rewrite scal_sum; apply sum_eq.
intros; replace (S N - i)%nat with (S (N - i)).
replace (S (N - i)) with (N - i + 1)%nat; [ idtac | ring ].
rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl; ring ];
ring.
apply minus_Sn_m; assumption.
rewrite <- (Rmult_comm x); rewrite scal_sum; apply sum_eq.
intros; replace (S i) with (i + 1)%nat; [ idtac | ring ]; rewrite pow_add;
replace (x ^ 1) with x; [ idtac | simpl; ring ];
ring.
intro; unfold C.
replace (INR (fact 0)) with 1; [ idtac | reflexivity ].
replace (p - 0)%nat with p; [ idtac | apply minus_n_O ].
rewrite Rmult_1_l; unfold Rdiv; rewrite <- Rinv_r_sym;
[ reflexivity | apply INR_fact_neq_0 ].
intro; unfold C.
replace (p - p)%nat with 0%nat; [ idtac | apply minus_n_n ].
replace (INR (fact 0)) with 1; [ idtac | reflexivity ].
rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym;
[ reflexivity | apply INR_fact_neq_0 ].
Qed.
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