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
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
|
(************************************************************************)
(* 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 SeqSeries.
Require Import Rtrigo_def.
Require Import Omega.
Local Open Scope R_scope.
Definition A1 (x:R) (N:nat) : R :=
sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) N.
Definition B1 (x:R) (N:nat) : R :=
sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
N.
Definition C1 (x y:R) (N:nat) : R :=
sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) N.
Definition Reste1 (x y:R) (N:nat) : R :=
sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
y ^ (2 * (N - l))) (pred (N - k))) (pred N).
Definition Reste2 (x y:R) (N:nat) : R :=
sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
x ^ (2 * S (l + k) + 1) *
((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
y ^ (2 * (N - l) + 1)) (pred (N - k))) (
pred N).
Definition Reste (x y:R) (N:nat) : R := Reste2 x y N - Reste1 x y (S N).
(* Here is the main result that will be used to prove that (cos (x+y))=(cos x)(cos y)-(sin x)(sin y) *)
Theorem cos_plus_form :
forall (x y:R) (n:nat),
(0 < n)%nat ->
A1 x (S n) * A1 y (S n) - B1 x n * B1 y n + Reste x y n = C1 x y (S n).
intros.
unfold A1, B1.
rewrite
(cauchy_finite (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k))
(fun k:nat => (-1) ^ k / INR (fact (2 * k)) * y ^ (2 * k)) (
S n)).
rewrite
(cauchy_finite
(fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
(fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * y ^ (2 * k + 1)) n H)
.
unfold Reste.
replace
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
x ^ (2 * S (l + k)) *
((-1) ^ (S n - l) / INR (fact (2 * (S n - l))) *
y ^ (2 * (S n - l)))) (pred (S n - k))) (
pred (S n))) with (Reste1 x y (S n)).
replace
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
x ^ (2 * S (l + k) + 1) *
((-1) ^ (n - l) / INR (fact (2 * (n - l) + 1)) *
y ^ (2 * (n - l) + 1))) (pred (n - k))) (
pred n)) with (Reste2 x y n).
replace
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun p:nat =>
(-1) ^ p / INR (fact (2 * p)) * x ^ (2 * p) *
((-1) ^ (k - p) / INR (fact (2 * (k - p))) * y ^ (2 * (k - p))))
k) (S n)) with
(sum_f_R0
(fun k:nat =>
(-1) ^ k / INR (fact (2 * k)) *
sum_f_R0
(fun l:nat => C (2 * k) (2 * l) * x ^ (2 * l) * y ^ (2 * (k - l))) k)
(S n)).
pose
(sin_nnn :=
fun n:nat =>
match n with
| O => 0
| S p =>
(-1) ^ S p / INR (fact (2 * S p)) *
sum_f_R0
(fun l:nat =>
C (2 * S p) (S (2 * l)) * x ^ S (2 * l) * y ^ S (2 * (p - l))) p
end).
ring_simplify.
unfold Rminus.
replace
(* (- old ring compat *)
(-
sum_f_R0
(fun k:nat =>
sum_f_R0
(fun p:nat =>
(-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) *
((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
y ^ (2 * (k - p) + 1))) k) n) with (sum_f_R0 sin_nnn (S n)).
rewrite <- sum_plus.
unfold C1.
apply sum_eq; intros.
induction i as [| i Hreci].
simpl.
unfold C; simpl.
field; discrR.
unfold sin_nnn.
rewrite <- Rmult_plus_distr_l.
apply Rmult_eq_compat_l.
rewrite binomial.
pose (Wn := fun i0:nat => C (2 * S i) i0 * x ^ i0 * y ^ (2 * S i - i0)).
replace
(sum_f_R0
(fun l:nat => C (2 * S i) (2 * l) * x ^ (2 * l) * y ^ (2 * (S i - l)))
(S i)) with (sum_f_R0 (fun l:nat => Wn (2 * l)%nat) (S i)).
replace
(sum_f_R0
(fun l:nat =>
C (2 * S i) (S (2 * l)) * x ^ S (2 * l) * y ^ S (2 * (i - l))) i) with
(sum_f_R0 (fun l:nat => Wn (S (2 * l))) i).
apply sum_decomposition.
apply sum_eq; intros.
unfold Wn.
apply Rmult_eq_compat_l.
replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))).
reflexivity.
omega.
apply sum_eq; intros.
unfold Wn.
apply Rmult_eq_compat_l.
replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat.
reflexivity.
omega.
replace
(-
sum_f_R0
(fun k:nat =>
sum_f_R0
(fun p:nat =>
(-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) *
((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
y ^ (2 * (k - p) + 1))) k) n) with
(-1 *
sum_f_R0
(fun k:nat =>
sum_f_R0
(fun p:nat =>
(-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) *
((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
y ^ (2 * (k - p) + 1))) k) n);[idtac|ring].
rewrite scal_sum.
rewrite decomp_sum.
replace (sin_nnn 0%nat) with 0.
rewrite Rplus_0_l.
change (pred (S n)) with n.
(* replace (pred (S n)) with n; [ idtac | reflexivity ]. *)
apply sum_eq; intros.
rewrite Rmult_comm.
unfold sin_nnn.
rewrite scal_sum.
rewrite scal_sum.
apply sum_eq; intros.
unfold Rdiv.
(*repeat rewrite Rmult_assoc.*)
(* rewrite (Rmult_comm (/ INR (fact (2 * S i)))). *)
repeat rewrite <- Rmult_assoc.
rewrite <- (Rmult_comm (/ INR (fact (2 * S i)))).
repeat rewrite <- Rmult_assoc.
replace (/ INR (fact (2 * S i)) * C (2 * S i) (S (2 * i0))) with
(/ INR (fact (2 * i0 + 1)) * / INR (fact (2 * (i - i0) + 1))).
replace (S (2 * i0)) with (2 * i0 + 1)%nat; [ idtac | ring ].
replace (S (2 * (i - i0))) with (2 * (i - i0) + 1)%nat; [ idtac | ring ].
replace ((-1) ^ S i) with (-1 * (-1) ^ i0 * (-1) ^ (i - i0)).
ring.
simpl.
pattern i at 2; replace i with (i0 + (i - i0))%nat.
rewrite pow_add.
ring.
symmetry ; apply le_plus_minus; assumption.
unfold C.
unfold Rdiv; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
rewrite Rinv_mult_distr.
replace (S (2 * i0)) with (2 * i0 + 1)%nat;
[ apply Rmult_eq_compat_l | ring ].
replace (2 * S i - (2 * i0 + 1))%nat with (2 * (i - i0) + 1)%nat.
reflexivity.
omega.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
reflexivity.
apply lt_O_Sn.
(* ring. *)
apply sum_eq; intros.
rewrite scal_sum.
apply sum_eq; intros.
unfold Rdiv.
repeat rewrite <- Rmult_assoc.
rewrite <- (Rmult_comm (/ INR (fact (2 * i)))).
repeat rewrite <- Rmult_assoc.
replace (/ INR (fact (2 * i)) * C (2 * i) (2 * i0)) with
(/ INR (fact (2 * i0)) * / INR (fact (2 * (i - i0)))).
replace ((-1) ^ i) with ((-1) ^ i0 * (-1) ^ (i - i0)).
ring.
pattern i at 2; replace i with (i0 + (i - i0))%nat.
rewrite pow_add.
ring.
symmetry ; apply le_plus_minus; assumption.
unfold C.
unfold Rdiv; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
rewrite Rinv_mult_distr.
replace (2 * i - 2 * i0)%nat with (2 * (i - i0))%nat.
reflexivity.
omega.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
unfold Reste2; apply sum_eq; intros.
apply sum_eq; intros.
unfold Rdiv; ring.
unfold Reste1; apply sum_eq; intros.
apply sum_eq; intros.
unfold Rdiv; ring.
apply lt_O_Sn.
Qed.
Lemma pow_sqr : forall (x:R) (i:nat), x ^ (2 * i) = (x * x) ^ i.
intros.
assert (H := pow_Rsqr x i).
unfold Rsqr in H; exact H.
Qed.
Lemma A1_cvg : forall x:R, Un_cv (A1 x) (cos x).
intro.
unfold cos; destruct (exist_cos (Rsqr x)) as (x0,p).
unfold cos_in, cos_n, infinite_sum, R_dist in p.
unfold Un_cv, R_dist; intros.
destruct (p eps H) as (x1,H0).
exists x1; intros.
unfold A1.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) n) with
(sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * (x * x) ^ i) n).
apply H0; assumption.
apply sum_eq.
intros.
replace ((x * x) ^ i) with (x ^ (2 * i)).
reflexivity.
apply pow_sqr.
Qed.
Lemma C1_cvg : forall x y:R, Un_cv (C1 x y) (cos (x + y)).
intros.
unfold cos.
destruct (exist_cos (Rsqr (x + y))) as (x0,p).
unfold cos_in, cos_n, infinite_sum, R_dist in p.
unfold Un_cv, R_dist; intros.
destruct (p eps H) as (x1,H0).
exists x1; intros.
unfold C1.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) n)
with
(sum_f_R0
(fun i:nat => (-1) ^ i / INR (fact (2 * i)) * ((x + y) * (x + y)) ^ i) n).
apply H0; assumption.
apply sum_eq.
intros.
replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)).
reflexivity.
apply pow_sqr.
Qed.
Lemma B1_cvg : forall x:R, Un_cv (B1 x) (sin x).
intro.
case (Req_dec x 0); intro.
rewrite H.
rewrite sin_0.
unfold B1.
unfold Un_cv; unfold R_dist; intros; exists 0%nat; intros.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k + 1))
n) with 0.
unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
induction n as [| n Hrecn].
simpl; ring.
rewrite tech5; rewrite <- Hrecn.
simpl; ring.
unfold ge; apply le_O_n.
unfold sin. destruct (exist_sin (Rsqr x)) as (x0,p).
unfold sin_in, sin_n, infinite_sum, R_dist in p.
unfold Un_cv, R_dist; intros.
cut (0 < eps / Rabs x);
[ intro
| unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ].
destruct (p (eps / Rabs x) H1) as (x1,H2).
exists x1; intros.
unfold B1.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
n) with
(x *
sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n).
replace
(x *
sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n -
x * x0) with
(x *
(sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n -
x0)); [ idtac | ring ].
rewrite Rabs_mult.
apply Rmult_lt_reg_l with (/ Rabs x).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
rewrite <- Rmult_assoc, <- Rinv_l_sym, Rmult_1_l, <- (Rmult_comm eps). apply H2;
assumption.
apply Rabs_no_R0; assumption.
rewrite scal_sum.
apply sum_eq.
intros.
rewrite pow_add.
rewrite pow_sqr.
simpl.
ring.
Qed.
|
