1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Compare.
Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import PartSum.
Require Import Binomial.
Require Import Omega.
Local Open Scope R_scope.
(** TT Ak; 0<=k<=N *)
Fixpoint prod_f_R0 (f:nat -> R) (N:nat) : R :=
match N with
| O => f O
| S p => prod_f_R0 f p * f (S p)
end.
Notation prod_f_SO := (fun An N => prod_f_R0 (fun n => An (S n)) N).
(**********)
Lemma prod_SO_split :
forall (An:nat -> R) (n k:nat),
(k < n)%nat ->
prod_f_R0 An n =
prod_f_R0 An k * prod_f_R0 (fun l:nat => An (k +1+l)%nat) (n - k -1).
Proof.
intros; induction n as [| n Hrecn].
absurd (k < 0)%nat; omega.
cut (k = n \/ (k < n)%nat);[intro; elim H0; intro|omega].
replace (S n - k - 1)%nat with O; [rewrite H1; simpl|omega].
replace (n+1+0)%nat with (S n); ring.
replace (S n - k-1)%nat with (S (n - k-1));[idtac|omega].
simpl; replace (k + S (n - k))%nat with (S n).
replace (k + 1 + S (n - k - 1))%nat with (S n).
rewrite Hrecn; [ ring | assumption ].
omega.
omega.
Qed.
(**********)
Lemma prod_SO_pos :
forall (An:nat -> R) (N:nat),
(forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_R0 An N.
Proof.
intros; induction N as [| N HrecN].
simpl; apply H; trivial.
simpl; apply Rmult_le_pos.
apply HrecN; intros; apply H; apply le_trans with N;
[ assumption | apply le_n_Sn ].
apply H; apply le_n.
Qed.
(**********)
Lemma prod_SO_Rle :
forall (An Bn:nat -> R) (N:nat),
(forall n:nat, (n <= N)%nat -> 0 <= An n <= Bn n) ->
prod_f_R0 An N <= prod_f_R0 Bn N.
Proof.
intros; induction N as [| N HrecN].
elim H with O; trivial.
simpl; apply Rle_trans with (prod_f_R0 An N * Bn (S N)).
apply Rmult_le_compat_l.
apply prod_SO_pos; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros;
assumption.
elim (H (S N) (le_n (S N))); intros; assumption.
do 2 rewrite <- (Rmult_comm (Bn (S N))); apply Rmult_le_compat_l.
elim (H (S N) (le_n (S N))); intros.
apply Rle_trans with (An (S N)); assumption.
apply HrecN; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros;
split; assumption.
Qed.
(** Application to factorial *)
Lemma fact_prodSO :
forall n:nat, INR (fact n) = prod_f_R0 (fun k:nat =>
(match (eq_nat_dec k 0) with
| left _ => 1%R
| right _ => INR k
end)) n.
Proof.
intro; induction n as [| n Hrecn].
reflexivity.
simpl; rewrite <- Hrecn.
case n; auto with real.
intros; repeat rewrite plus_INR;rewrite mult_INR;ring.
Qed.
Lemma le_n_2n : forall n:nat, (n <= 2 * n)%nat.
Proof.
simple induction n.
replace (2 * 0)%nat with 0%nat; [ apply le_n | ring ].
intros; replace (2 * S n0)%nat with (S (S (2 * n0))).
apply le_n_S; apply le_S; assumption.
replace (S (S (2 * n0))) with (2 * n0 + 2)%nat; [ idtac | ring ].
replace (S n0) with (n0 + 1)%nat; [ idtac | ring ].
ring.
Qed.
(** We prove that (N!)^2<=(2N-k)!*k! forall k in [|O;2N|] *)
Lemma RfactN_fact2N_factk :
forall N k:nat,
(k <= 2 * N)%nat ->
Rsqr (INR (fact N)) <= INR (fact (2 * N - k)) * INR (fact k).
Proof.
assert (forall (n:nat), 0 <= (if eq_nat_dec n 0 then 1 else INR n)).
intros; case (eq_nat_dec n 0); auto with real.
assert (forall (n:nat), (0 < n)%nat ->
(if eq_nat_dec n 0 then 1 else INR n) = INR n).
intros n; case (eq_nat_dec n 0); auto with real.
intros; absurd (0 < n)%nat; omega.
intros; unfold Rsqr; repeat rewrite fact_prodSO.
cut ((k=N)%nat \/ (k < N)%nat \/ (N < k)%nat).
intro H2; elim H2; intro H3.
rewrite H3; replace (2*N-N)%nat with N;[right; ring|omega].
case H3; intro; clear H2 H3.
rewrite (prod_SO_split (fun l:nat => if eq_nat_dec l 0 then 1 else INR l) (2 * N - k) N).
rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply prod_SO_pos; intros; auto.
replace (2 * N - k - N-1)%nat with (N - k-1)%nat.
rewrite Rmult_comm; rewrite (prod_SO_split
(fun l:nat => if eq_nat_dec l 0 then 1 else INR l) N k).
apply Rmult_le_compat_l.
apply prod_SO_pos; intros; auto.
apply prod_SO_Rle; intros; split; auto.
rewrite H0.
rewrite H0.
apply le_INR; omega.
omega.
omega.
assumption.
omega.
omega.
rewrite <- (Rmult_comm (prod_f_R0 (fun l:nat =>
if eq_nat_dec l 0 then 1 else INR l) k));
rewrite (prod_SO_split (fun l:nat =>
if eq_nat_dec l 0 then 1 else INR l) k N).
rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply prod_SO_pos; intros; auto.
rewrite Rmult_comm;
rewrite (prod_SO_split (fun l:nat =>
if eq_nat_dec l 0 then 1 else INR l) N (2 * N - k)).
apply Rmult_le_compat_l.
apply prod_SO_pos; intros; auto.
replace (N - (2 * N - k)-1)%nat with (k - N-1)%nat.
apply prod_SO_Rle; intros; split; auto.
rewrite H0.
rewrite H0.
apply le_INR; omega.
omega.
omega.
omega.
omega.
assumption.
omega.
Qed.
(**********)
Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n).
Proof.
intro; apply lt_INR_0; apply neq_O_lt; red; intro;
elim (fact_neq_0 n); symmetry ; assumption.
Qed.
(** We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *)
Lemma C_maj : forall N k:nat, (k <= 2 * N)%nat -> C (2 * N) k <= C (2 * N) N.
Proof.
intros; unfold C; unfold Rdiv; apply Rmult_le_compat_l.
apply pos_INR.
replace (2 * N - N)%nat with N.
apply Rmult_le_reg_l with (INR (fact N) * INR (fact N)).
apply Rmult_lt_0_compat; apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
rewrite Rmult_comm;
apply Rmult_le_reg_l with (INR (fact k) * INR (fact (2 * N - k))).
apply Rmult_lt_0_compat; apply INR_fact_lt_0.
rewrite Rmult_1_r; rewrite <- mult_INR; rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; rewrite mult_INR; rewrite (Rmult_comm (INR (fact k)));
replace (INR (fact N) * INR (fact N)) with (Rsqr (INR (fact N))).
apply RfactN_fact2N_factk.
assumption.
reflexivity.
rewrite mult_INR; apply prod_neq_R0; apply INR_fact_neq_0.
apply prod_neq_R0; apply INR_fact_neq_0.
omega.
Qed.
|
