aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Reals/Cos_rel.v
blob: ba108e95ec6e265a2ff06e443749e1e52f913b5f (plain)
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
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)
 
(*i $Id$ i*)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Open Local 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 in |- *.
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 in |- *.
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).
ring.
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)).
set
 (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).
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 (sum_f_R0 sin_nnn (S n)).
rewrite <- sum_plus.
unfold C1 in |- *.
apply sum_eq; intros.
induction  i as [| i Hreci].
simpl in |- *.
rewrite Rplus_0_l.
replace (C 0 0) with 1.
unfold Rdiv in |- *; rewrite Rinv_1.
ring.
unfold C in |- *.
rewrite <- minus_n_n.
simpl in |- *.
unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rinv_1; ring.
unfold sin_nnn in |- *.
rewrite <- Rmult_plus_distr_l.
apply Rmult_eq_compat_l.
rewrite binomial.
set (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).
rewrite Rplus_comm.
apply sum_decomposition.
apply sum_eq; intros.
unfold Wn in |- *.
apply Rmult_eq_compat_l.
replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))).
reflexivity.
apply INR_eq.
rewrite S_INR; rewrite mult_INR.
repeat rewrite minus_INR.
rewrite mult_INR; repeat rewrite S_INR.
rewrite mult_INR; repeat rewrite S_INR; ring.
replace (2 * S i)%nat with (S (S (2 * i))).
apply le_n_S.
apply le_trans with (2 * i)%nat.
apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
apply le_n_Sn.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR;
 ring.
assumption.
apply sum_eq; intros.
unfold Wn in |- *.
apply Rmult_eq_compat_l.
replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat.
reflexivity.
apply INR_eq.
rewrite mult_INR.
repeat rewrite minus_INR.
rewrite mult_INR; repeat rewrite S_INR.
rewrite mult_INR; repeat rewrite S_INR; ring.
apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
assumption.
rewrite <- (Ropp_involutive (sum_f_R0 sin_nnn (S n))).
apply Ropp_eq_compat.
replace (- sum_f_R0 sin_nnn (S n)) with (-1 * sum_f_R0 sin_nnn (S n));
 [ idtac | ring ].
rewrite scal_sum.
rewrite decomp_sum.
replace (sin_nnn 0%nat) with 0.
rewrite Rmult_0_l; rewrite Rplus_0_l.
replace (pred (S n)) with n; [ idtac | reflexivity ].
apply sum_eq; intros.
rewrite Rmult_comm.
unfold sin_nnn in |- *.
rewrite scal_sum.
rewrite scal_sum.
apply sum_eq; intros.
unfold Rdiv in |- *.
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 in |- *.
pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat.
rewrite pow_add.
ring.
symmetry  in |- *; apply le_plus_minus; assumption.
unfold C in |- *.
unfold Rdiv in |- *; 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.
apply INR_eq.
rewrite plus_INR; rewrite mult_INR; repeat rewrite minus_INR.
rewrite plus_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; ring.
replace (2 * i0 + 1)%nat with (S (2 * i0)).
replace (2 * S i)%nat with (S (S (2 * i))).
apply le_n_S.
apply le_trans with (2 * i)%nat.
apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
apply le_n_Sn.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR;
 ring.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
 repeat rewrite S_INR; ring.
assumption.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
reflexivity.
apply lt_O_Sn.
apply sum_eq; intros.
rewrite scal_sum.
apply sum_eq; intros.
unfold Rdiv in |- *.
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 in |- *; replace i with (i0 + (i - i0))%nat.
rewrite pow_add.
ring.
symmetry  in |- *; apply le_plus_minus; assumption.
unfold C in |- *.
unfold Rdiv in |- *; 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.
apply INR_eq.
rewrite mult_INR; repeat rewrite minus_INR.
do 2 rewrite mult_INR; repeat rewrite S_INR; ring.
apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
assumption.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
unfold Reste2 in |- *; apply sum_eq; intros.
apply sum_eq; intros.
unfold Rdiv in |- *; ring. 
unfold Reste1 in |- *; apply sum_eq; intros.
apply sum_eq; intros.
unfold Rdiv in |- *; 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. 
assert (H := exist_cos (x * x)). 
elim H; intros. 
assert (p_i := p). 
unfold cos_in in p. 
unfold cos_n, infinit_sum in p. 
unfold R_dist in p. 
cut (cos x = x0). 
intro. 
rewrite H0. 
unfold Un_cv in |- *; unfold R_dist in |- *; intros. 
elim (p eps H1); intros. 
exists x1; intros. 
unfold A1 in |- *. 
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 H2; assumption. 
apply sum_eq. 
intros. 
replace ((x * x) ^ i) with (x ^ (2 * i)). 
reflexivity. 
apply pow_sqr. 
unfold cos in |- *. 
case (exist_cos (Rsqr x)). 
unfold Rsqr in |- *; intros. 
unfold cos_in in p_i. 
unfold cos_in in c. 
apply uniqueness_sum with (fun i:nat => cos_n i * (x * x) ^ i); assumption. 
Qed. 
 
Lemma C1_cvg : forall x y:R, Un_cv (C1 x y) (cos (x + y)). 
intros. 
assert (H := exist_cos ((x + y) * (x + y))). 
elim H; intros. 
assert (p_i := p). 
unfold cos_in in p. 
unfold cos_n, infinit_sum in p. 
unfold R_dist in p. 
cut (cos (x + y) = x0). 
intro. 
rewrite H0. 
unfold Un_cv in |- *; unfold R_dist in |- *; intros. 
elim (p eps H1); intros. 
exists x1; intros. 
unfold C1 in |- *. 
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 H2; assumption. 
apply sum_eq. 
intros. 
replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)). 
reflexivity. 
apply pow_sqr. 
unfold cos in |- *. 
case (exist_cos (Rsqr (x + y))). 
unfold Rsqr in |- *; intros. 
unfold cos_in in p_i. 
unfold cos_in in c. 
apply uniqueness_sum with (fun i:nat => cos_n i * ((x + y) * (x + y)) ^ i);
 assumption. 
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 in |- *. 
unfold Un_cv in |- *; unfold R_dist in |- *; 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 in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. 
induction  n as [| n Hrecn]. 
simpl in |- *; ring. 
rewrite tech5; rewrite <- Hrecn. 
simpl in |- *; ring. 
unfold ge in |- *; apply le_O_n. 
assert (H0 := exist_sin (x * x)). 
elim H0; intros. 
assert (p_i := p). 
unfold sin_in in p. 
unfold sin_n, infinit_sum in p. 
unfold R_dist in p. 
cut (sin x = x * x0). 
intro. 
rewrite H1. 
unfold Un_cv in |- *; unfold R_dist in |- *; intros. 
cut (0 < eps / Rabs x);
 [ intro
 | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ]. 
elim (p (eps / Rabs x) H3); intros. 
exists x1; intros. 
unfold B1 in |- *. 
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. 
rewrite <- Rinv_l_sym. 
rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H4; apply H4;
 assumption. 
apply Rabs_no_R0; assumption. 
rewrite scal_sum. 
apply sum_eq. 
intros. 
rewrite pow_add. 
rewrite pow_sqr. 
simpl in |- *. 
ring. 
unfold sin in |- *. 
case (exist_sin (Rsqr x)). 
unfold Rsqr in |- *; intros. 
unfold sin_in in p_i. 
unfold sin_in in s. 
assert
 (H1 := uniqueness_sum (fun i:nat => sin_n i * (x * x) ^ i) x0 x1 p_i s). 
rewrite H1; reflexivity. 
Qed.