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___,, * 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 Rbase Rfunctions SeqSeries Rtrigo_fun Max.
Local Open Scope R_scope.
(********************************)
(** * Definition of exponential *)
(********************************)
Definition exp_in (x l:R) : Prop :=
infinite_sum (fun i:nat => / INR (fact i) * x ^ i) l.
Lemma exp_cof_no_R0 : forall n:nat, / INR (fact n) <> 0.
Proof.
intro.
apply Rinv_neq_0_compat.
apply INR_fact_neq_0.
Qed.
Lemma exist_exp : forall x:R, { l:R | exp_in x l }.
Proof.
intro;
generalize
(Alembert_C3 (fun n:nat => / INR (fact n)) x exp_cof_no_R0 Alembert_exp).
unfold Pser, exp_in.
trivial.
Defined.
Definition exp (x:R) : R := proj1_sig (exist_exp x).
Lemma pow_i : forall i:nat, (0 < i)%nat -> 0 ^ i = 0.
Proof.
intros; apply pow_ne_zero.
red; intro; rewrite H0 in H; elim (lt_irrefl _ H).
Qed.
Lemma exist_exp0 : { l:R | exp_in 0 l }.
Proof.
exists 1.
unfold exp_in; unfold infinite_sum; intros.
exists 0%nat.
intros; replace (sum_f_R0 (fun i:nat => / INR (fact i) * 0 ^ i) n) with 1.
unfold R_dist; replace (1 - 1) with 0;
[ rewrite Rabs_R0; assumption | ring ].
induction n as [| n Hrecn].
simpl; rewrite Rinv_1; ring.
rewrite tech5.
rewrite <- Hrecn.
simpl.
ring.
unfold ge; apply le_O_n.
Defined.
(* Value of [exp 0] *)
Lemma exp_0 : exp 0 = 1.
Proof.
cut (exp_in 0 (exp 0)).
cut (exp_in 0 1).
unfold exp_in; intros; eapply uniqueness_sum.
apply H0.
apply H.
exact (proj2_sig exist_exp0).
exact (proj2_sig (exist_exp 0)).
Qed.
(*****************************************)
(** * Definition of hyperbolic functions *)
(*****************************************)
Definition cosh (x:R) : R := (exp x + exp (- x)) / 2.
Definition sinh (x:R) : R := (exp x - exp (- x)) / 2.
Definition tanh (x:R) : R := sinh x / cosh x.
Lemma cosh_0 : cosh 0 = 1.
Proof.
unfold cosh; rewrite Ropp_0; rewrite exp_0.
unfold Rdiv; rewrite <- Rinv_r_sym; [ reflexivity | discrR ].
Qed.
Lemma sinh_0 : sinh 0 = 0.
Proof.
unfold sinh; rewrite Ropp_0; rewrite exp_0.
unfold Rminus, Rdiv; rewrite Rplus_opp_r; apply Rmult_0_l.
Qed.
Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)).
Lemma simpl_cos_n :
forall n:nat, cos_n (S n) / cos_n n = - / INR (2 * S n * (2 * n + 1)).
Proof.
intro; unfold cos_n; replace (S n) with (n + 1)%nat; [ idtac | ring ].
rewrite pow_add; unfold Rdiv; rewrite Rinv_mult_distr.
rewrite Rinv_involutive.
replace
((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1))) *
(/ (-1) ^ n * INR (fact (2 * n)))) with
((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1))) * INR (fact (2 * n)) *
(-1) ^ 1); [ idtac | ring ].
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; unfold pow; rewrite Rmult_1_r.
replace (2 * (n + 1))%nat with (S (S (2 * n))); [ idtac | ring ].
do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
repeat rewrite Rinv_mult_distr; try (apply not_O_INR; discriminate).
rewrite <- (Rmult_comm (-1)).
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
replace (S (2 * n)) with (2 * n + 1)%nat; [ idtac | ring ].
rewrite mult_INR; rewrite Rinv_mult_distr.
ring.
apply not_O_INR; discriminate.
replace (2 * n + 1)%nat with (S (2 * n));
[ apply not_O_INR; discriminate | ring ].
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 pow_nonzero; discrR.
apply INR_fact_neq_0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat; apply INR_fact_neq_0.
Qed.
Lemma archimed_cor1 :
forall eps:R, 0 < eps -> exists N : nat, / INR N < eps /\ (0 < N)%nat.
Proof.
intros; cut (/ eps < IZR (up (/ eps))).
intro; cut (0 <= up (/ eps))%Z.
intro; assert (H2 := IZN _ H1); elim H2; intros; exists (max x 1).
split.
cut (0 < IZR (Z.of_nat x)).
intro; rewrite INR_IZR_INZ; apply Rle_lt_trans with (/ IZR (Z.of_nat x)).
apply Rmult_le_reg_l with (IZR (Z.of_nat x)).
assumption.
rewrite <- Rinv_r_sym;
[ idtac | red; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ].
apply Rmult_le_reg_l with (IZR (Z.of_nat (max x 1))).
apply Rlt_le_trans with (IZR (Z.of_nat x)).
assumption.
repeat rewrite <- INR_IZR_INZ; apply le_INR; apply le_max_l.
rewrite Rmult_1_r; rewrite (Rmult_comm (IZR (Z.of_nat (max x 1))));
rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; repeat rewrite <- INR_IZR_INZ; apply le_INR;
apply le_max_l.
rewrite <- INR_IZR_INZ; apply not_O_INR.
red; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat;
[ intro | apply lt_O_Sn ]; assert (H8 := lt_le_trans _ _ _ H7 H6);
rewrite H5 in H8; elim (lt_irrefl _ H8).
pattern eps at 1; rewrite <- Rinv_involutive.
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat; [ apply Rinv_0_lt_compat; assumption | assumption ].
rewrite H3 in H0; assumption.
red; intro; rewrite H5 in H; elim (Rlt_irrefl _ H).
apply Rlt_trans with (/ eps).
apply Rinv_0_lt_compat; assumption.
rewrite H3 in H0; assumption.
apply lt_le_trans with 1%nat; [ apply lt_O_Sn | apply le_max_r ].
apply le_IZR; replace (IZR 0) with 0; [ idtac | reflexivity ]; left;
apply Rlt_trans with (/ eps);
[ apply Rinv_0_lt_compat; assumption | assumption ].
assert (H0 := archimed (/ eps)).
elim H0; intros; assumption.
Qed.
Lemma Alembert_cos : Un_cv (fun n:nat => Rabs (cos_n (S n) / cos_n n)) 0.
Proof.
unfold Un_cv; intros.
assert (H0 := archimed_cor1 eps H).
elim H0; intros; exists x.
intros; rewrite simpl_cos_n; unfold R_dist; unfold Rminus;
rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu;
rewrite Rabs_Ropp; rewrite Rabs_right.
rewrite mult_INR; rewrite Rinv_mult_distr.
cut (/ INR (2 * S n) < 1).
intro; cut (/ INR (2 * n + 1) < eps).
intro; rewrite <- (Rmult_1_l eps).
apply Rmult_gt_0_lt_compat; try assumption.
change (0 < / INR (2 * n + 1)); apply Rinv_0_lt_compat;
apply lt_INR_0.
replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].
apply Rlt_0_1.
cut (x < 2 * n + 1)%nat.
intro; assert (H5 := lt_INR _ _ H4).
apply Rlt_trans with (/ INR x).
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat.
apply lt_INR_0.
elim H1; intros; assumption.
apply lt_INR_0; replace (2 * n + 1)%nat with (S (2 * n));
[ apply lt_O_Sn | ring ].
assumption.
elim H1; intros; assumption.
apply lt_le_trans with (S n).
unfold ge in H2; apply le_lt_n_Sm; assumption.
replace (2 * n + 1)%nat with (S (2 * n)); [ idtac | ring ].
apply le_n_S; apply le_n_2n.
apply Rmult_lt_reg_l with (INR (2 * S n)).
apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n))).
apply lt_O_Sn.
replace (S n) with (n + 1)%nat; [ idtac | ring ].
ring.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ].
replace (2 * S n)%nat with (S (S (2 * n))).
apply lt_n_S; apply lt_O_Sn.
replace (S n) with (n + 1)%nat; [ ring | ring ].
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
replace (2 * n + 1)%nat with (S (2 * n));
[ apply not_O_INR; discriminate | ring ].
apply Rle_ge; left; apply Rinv_0_lt_compat.
apply lt_INR_0.
replace (2 * S n * (2 * n + 1))%nat with (S (S (4 * (n * n) + 6 * n))).
apply lt_O_Sn.
apply INR_eq.
repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR;
rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
replace (INR 0) with 0; [ ring | reflexivity ].
Qed.
Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0.
Proof.
intro; unfold cos_n; unfold Rdiv; apply prod_neq_R0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat.
apply INR_fact_neq_0.
Qed.
(**********)
Definition cos_in (x l:R) : Prop :=
infinite_sum (fun i:nat => cos_n i * x ^ i) l.
(**********)
Lemma exist_cos : forall x:R, { l:R | cos_in x l }.
Proof.
intro; generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos).
unfold Pser, cos_in; trivial.
Qed.
(** Definition of cosinus *)
Definition cos (x:R) : R := let (a,_) := exist_cos (Rsqr x) in a.
Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).
Lemma simpl_sin_n :
forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)).
Proof.
intro; unfold sin_n; replace (S n) with (n + 1)%nat; [ idtac | ring ].
rewrite pow_add; unfold Rdiv; rewrite Rinv_mult_distr.
rewrite Rinv_involutive.
replace
((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1) + 1)) *
(/ (-1) ^ n * INR (fact (2 * n + 1)))) with
((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1) + 1)) *
INR (fact (2 * n + 1)) * (-1) ^ 1); [ idtac | ring ].
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; unfold pow; rewrite Rmult_1_r;
replace (2 * (n + 1) + 1)%nat with (S (S (2 * n + 1))).
do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
repeat rewrite Rinv_mult_distr.
rewrite <- (Rmult_comm (-1)); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; replace (S (2 * n + 1)) with (2 * (n + 1))%nat.
repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr.
ring.
apply not_O_INR; discriminate.
replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ].
apply not_O_INR; discriminate.
apply prod_neq_R0.
apply not_O_INR; discriminate.
replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ].
apply not_O_INR; discriminate.
replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ].
rewrite mult_plus_distr_l; cut (forall n:nat, S n = (n + 1)%nat).
intros; rewrite (H (2 * n + 1)%nat).
ring.
intros; ring.
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
cut (forall n:nat, S (S n) = (n + 2)%nat);
[ intros; rewrite (H (2 * n + 1)%nat); ring | intros; ring ].
apply pow_nonzero; discrR.
apply INR_fact_neq_0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat; apply INR_fact_neq_0.
Qed.
Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0.
Proof.
unfold Un_cv; intros; assert (H0 := archimed_cor1 eps H).
elim H0; intros; exists x.
intros; rewrite simpl_sin_n; unfold R_dist; unfold Rminus;
rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu;
rewrite Rabs_Ropp; rewrite Rabs_right.
rewrite mult_INR; rewrite Rinv_mult_distr.
cut (/ INR (2 * S n) < 1).
intro; cut (/ INR (2 * S n + 1) < eps).
intro; rewrite <- (Rmult_1_l eps); rewrite (Rmult_comm (/ INR (2 * S n + 1)));
apply Rmult_gt_0_lt_compat; try assumption.
change (0 < / INR (2 * S n + 1)); apply Rinv_0_lt_compat;
apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n));
[ apply lt_O_Sn | ring ].
apply Rlt_0_1.
cut (x < 2 * S n + 1)%nat.
intro; assert (H5 := lt_INR _ _ H4); apply Rlt_trans with (/ INR x).
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat.
apply lt_INR_0; elim H1; intros; assumption.
apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n));
[ apply lt_O_Sn | ring ].
assumption.
elim H1; intros; assumption.
apply lt_le_trans with (S n).
unfold ge in H2; apply le_lt_n_Sm; assumption.
replace (2 * S n + 1)%nat with (S (2 * S n)); [ idtac | ring ].
apply le_S; apply le_n_2n.
apply Rmult_lt_reg_l with (INR (2 * S n)).
apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n)));
[ apply lt_O_Sn | replace (S n) with (n + 1)%nat; [ idtac | ring ]; ring ].
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ].
replace (2 * S n)%nat with (S (S (2 * n))).
apply lt_n_S; apply lt_O_Sn.
replace (S n) with (n + 1)%nat; [ ring | ring ].
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
left; change (0 < / INR ((2 * S n + 1) * (2 * S n)));
apply Rinv_0_lt_compat.
apply lt_INR_0.
replace ((2 * S n + 1) * (2 * S n))%nat with
(S (S (S (S (S (S (4 * (n * n) + 10 * n))))))).
apply lt_O_Sn.
apply INR_eq; repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR;
rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
replace (INR 0) with 0; [ ring | reflexivity ].
Qed.
Lemma sin_no_R0 : forall n:nat, sin_n n <> 0.
Proof.
intro; unfold sin_n; unfold Rdiv; apply prod_neq_R0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat; apply INR_fact_neq_0.
Qed.
(**********)
Definition sin_in (x l:R) : Prop :=
infinite_sum (fun i:nat => sin_n i * x ^ i) l.
(**********)
Lemma exist_sin : forall x:R, { l:R | sin_in x l }.
Proof.
intro; generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin).
unfold Pser, sin_n; trivial.
Defined.
(***********************)
(* Definition of sinus *)
Definition sin (x:R) : R := let (a,_) := exist_sin (Rsqr x) in x * a.
(*********************************************)
(** * Properties *)
(*********************************************)
Lemma cos_sym : forall x:R, cos x = cos (- x).
Proof.
intros; unfold cos; replace (Rsqr (- x)) with (Rsqr x).
reflexivity.
apply Rsqr_neg.
Qed.
Lemma sin_antisym : forall x:R, sin (- x) = - sin x.
Proof.
intro; unfold sin; replace (Rsqr (- x)) with (Rsqr x);
[ idtac | apply Rsqr_neg ].
case (exist_sin (Rsqr x)); intros; ring.
Qed.
Lemma sin_0 : sin 0 = 0.
Proof.
unfold sin; case (exist_sin (Rsqr 0)).
intros; ring.
Qed.
Lemma exist_cos0 : { l:R | cos_in 0 l }.
Proof.
exists 1.
unfold cos_in; unfold infinite_sum; intros; exists 0%nat.
intros.
unfold R_dist.
induction n as [| n Hrecn].
unfold cos_n; simpl.
unfold Rdiv; rewrite Rinv_1.
do 2 rewrite Rmult_1_r.
unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
rewrite tech5.
replace (cos_n (S n) * 0 ^ S n) with 0.
rewrite Rplus_0_r.
apply Hrecn; unfold ge; apply le_O_n.
simpl; ring.
Defined.
(* Value of [cos 0] *)
Lemma cos_0 : cos 0 = 1.
Proof.
cut (cos_in 0 (cos 0)).
cut (cos_in 0 1).
unfold cos_in; intros; eapply uniqueness_sum.
apply H0.
apply H.
exact (proj2_sig exist_cos0).
assert (H := proj2_sig (exist_cos (Rsqr 0))); unfold cos;
pattern 0 at 1; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ].
Qed.
|
