summaryrefslogtreecommitdiff
path: root/theories/Reals/Rderiv.v
blob: ba42bad9f6255353386b623dc27d09c783cbf924 (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
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
(************************************************************************)
(*  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: Rderiv.v 10710 2008-03-23 09:24:09Z herbelin $ i*)

(*********************************************************)
(**     Definition of the derivative,continuity          *)
(*                                                       *)
(*********************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import Rlimit.
Require Import Fourier.
Require Import Classical_Prop.
Require Import Classical_Pred_Type.
Require Import Omega.
Open Local Scope R_scope.

(*********)
Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.

(*********)
Definition continue_in (f:R -> R) (D:R -> Prop) (x0:R) : Prop :=
  limit1_in f (D_x D x0) (f x0) x0.

(*********)
Definition D_in (f d:R -> R) (D:R -> Prop) (x0:R) : Prop :=
  limit1_in (fun x:R => (f x - f x0) / (x - x0)) (D_x D x0) (d x0) x0.

(*********)
Lemma cont_deriv :
  forall (f d:R -> R) (D:R -> Prop) (x0:R),
    D_in f d D x0 -> continue_in f D x0.
Proof.
  unfold continue_in in |- *; unfold D_in in |- *; unfold limit1_in in |- *;
    unfold limit_in in |- *; unfold Rdiv in |- *; simpl in |- *; 
      intros; elim (H eps H0); clear H; intros; elim H; 
        clear H; intros; elim (Req_dec (d x0) 0); intro.
  split with (Rmin 1 x); split.
  elim (Rmin_Rgt 1 x 0); intros a b; apply (b (conj Rlt_0_1 H)).
  intros; elim H3; clear H3; intros;
    generalize (let (H1, H2) := Rmin_Rgt 1 x (R_dist x1 x0) in H1);
      unfold Rgt in |- *; intro; elim (H5 H4); clear H5; 
        intros; generalize (H1 x1 (conj H3 H6)); clear H1; 
          intro; unfold D_x in H3; elim H3; intros.
  rewrite H2 in H1; unfold R_dist in |- *; unfold R_dist in H1;
    cut (Rabs (f x1 - f x0) < eps * Rabs (x1 - x0)).
  intro; unfold R_dist in H5;
    generalize (Rmult_lt_compat_l eps (Rabs (x1 - x0)) 1 H0 H5);
      rewrite Rmult_1_r; intro; apply Rlt_trans with (r2 := eps * Rabs (x1 - x0));
        assumption.
  rewrite (Rminus_0_r ((f x1 - f x0) * / (x1 - x0))) in H1;
    rewrite Rabs_mult in H1; cut (x1 - x0 <> 0).
  intro; rewrite (Rabs_Rinv (x1 - x0) H9) in H1;
    generalize
      (Rmult_lt_compat_l (Rabs (x1 - x0)) (Rabs (f x1 - f x0) * / Rabs (x1 - x0))
        eps (Rabs_pos_lt (x1 - x0) H9) H1); intro; rewrite Rmult_comm in H10;
      rewrite Rmult_assoc in H10; rewrite Rinv_l in H10.
  rewrite Rmult_1_r in H10; rewrite Rmult_comm; assumption.
  apply Rabs_no_R0; auto.
  apply Rminus_eq_contra; auto.
(**)
  split with (Rmin (Rmin (/ 2) x) (eps * / Rabs (2 * d x0))); split.
  cut (Rmin (/ 2) x > 0).
  cut (eps * / Rabs (2 * d x0) > 0).
  intros; elim (Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) 0);
    intros a b; apply (b (conj H4 H3)).
  apply Rmult_gt_0_compat; auto.
  unfold Rgt in |- *; apply Rinv_0_lt_compat; apply Rabs_pos_lt;
    apply Rmult_integral_contrapositive; split.
  discrR.
  assumption.
  elim (Rmin_Rgt (/ 2) x 0); intros a b; cut (0 < 2).
  intro; generalize (Rinv_0_lt_compat 2 H3); intro; fold (/ 2 > 0) in H4;
    apply (b (conj H4 H)).
  fourier.
  intros; elim H3; clear H3; intros;
    generalize
      (let (H1, H2) :=
        Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) (R_dist x1 x0) in
        H1); unfold Rgt in |- *; intro; elim (H5 H4); clear H5; 
      intros; generalize (let (H1, H2) := Rmin_Rgt (/ 2) x (R_dist x1 x0) in H1);
        unfold Rgt in |- *; intro; elim (H7 H5); clear H7; 
          intros; clear H4 H5; generalize (H1 x1 (conj H3 H8)); 
            clear H1; intro; unfold D_x in H3; elim H3; intros;
              generalize (sym_not_eq H5); clear H5; intro H5;
                generalize (Rminus_eq_contra x1 x0 H5); intro; generalize H1;
                  pattern (d x0) at 1 in |- *;
                    rewrite <- (let (H1, H2) := Rmult_ne (d x0) in H2);
                      rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist in |- *;
                        unfold Rminus at 1 in |- *; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0)));
                          rewrite (Rmult_comm (/ (x1 - x0) * (x1 - x0)) (d x0));
                            rewrite <- (Ropp_mult_distr_l_reverse (d x0) (/ (x1 - x0) * (x1 - x0)));
                              rewrite (Rmult_comm (- d x0) (/ (x1 - x0) * (x1 - x0)));
                                rewrite (Rmult_assoc (/ (x1 - x0)) (x1 - x0) (- d x0));
                                  rewrite <-
                                    (Rmult_plus_distr_l (/ (x1 - x0)) (f x1 - f x0) ((x1 - x0) * - d x0))
                                    ; rewrite (Rabs_mult (/ (x1 - x0)) (f x1 - f x0 + (x1 - x0) * - d x0));
                                      clear H1; intro;
                                        generalize
                                          (Rmult_lt_compat_l (Rabs (x1 - x0))
                                            (Rabs (/ (x1 - x0)) * Rabs (f x1 - f x0 + (x1 - x0) * - d x0)) eps
                                            (Rabs_pos_lt (x1 - x0) H9) H1);
                                          rewrite <-
                                            (Rmult_assoc (Rabs (x1 - x0)) (Rabs (/ (x1 - x0)))
                                              (Rabs (f x1 - f x0 + (x1 - x0) * - d x0)));
                                            rewrite (Rabs_Rinv (x1 - x0) H9);
                                              rewrite (Rinv_r (Rabs (x1 - x0)) (Rabs_no_R0 (x1 - x0) H9));
                                                rewrite
                                                  (let (H1, H2) := Rmult_ne (Rabs (f x1 - f x0 + (x1 - x0) * - d x0)) in H2)
                                                  ; generalize (Rabs_triang_inv (f x1 - f x0) ((x1 - x0) * d x0)); 
                                                    intro; rewrite (Rmult_comm (x1 - x0) (- d x0));
                                                      rewrite (Ropp_mult_distr_l_reverse (d x0) (x1 - x0));
                                                        fold (f x1 - f x0 - d x0 * (x1 - x0)) in |- *;
                                                          rewrite (Rmult_comm (x1 - x0) (d x0)) in H10; clear H1; 
                                                            intro;
                                                              generalize
                                                                (Rle_lt_trans (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0)))
                                                                  (Rabs (f x1 - f x0 - d x0 * (x1 - x0))) (Rabs (x1 - x0) * eps) H10 H1);
                                                                clear H1; intro;
                                                                  generalize
                                                                    (Rplus_lt_compat_l (Rabs (d x0 * (x1 - x0)))
                                                                      (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0))) (
                                                                        Rabs (x1 - x0) * eps) H1); unfold Rminus at 2 in |- *;
                                                                    rewrite (Rplus_comm (Rabs (f x1 - f x0)) (- Rabs (d x0 * (x1 - x0))));
                                                                      rewrite <-
                                                                        (Rplus_assoc (Rabs (d x0 * (x1 - x0))) (- Rabs (d x0 * (x1 - x0)))
                                                                          (Rabs (f x1 - f x0))); rewrite (Rplus_opp_r (Rabs (d x0 * (x1 - x0))));
                                                                        rewrite (let (H1, H2) := Rplus_ne (Rabs (f x1 - f x0)) in H2); 
                                                                          clear H1; intro; cut (Rabs (d x0 * (x1 - x0)) + Rabs (x1 - x0) * eps < eps).
  intro;
    apply
      (Rlt_trans (Rabs (f x1 - f x0))
        (Rabs (d x0 * (x1 - x0)) + Rabs (x1 - x0) * eps) eps H1 H11). 
  clear H1 H5 H3 H10; generalize (Rabs_pos_lt (d x0) H2); intro;
    unfold Rgt in H0;
      generalize (Rmult_lt_compat_l eps (R_dist x1 x0) (/ 2) H0 H7); 
        clear H7; intro;
          generalize
            (Rmult_lt_compat_l (Rabs (d x0)) (R_dist x1 x0) (
              eps * / Rabs (2 * d x0)) H1 H6); clear H6; intro;
            rewrite (Rmult_comm eps (R_dist x1 x0)) in H3; unfold R_dist in H3, H5;
              rewrite <- (Rabs_mult (d x0) (x1 - x0)) in H5;
                rewrite (Rabs_mult 2 (d x0)) in H5; cut (Rabs 2 <> 0).
  intro; fold (Rabs (d x0) > 0) in H1;
    rewrite
      (Rinv_mult_distr (Rabs 2) (Rabs (d x0)) H6
        (Rlt_dichotomy_converse (Rabs (d x0)) 0 (or_intror (Rabs (d x0) < 0) H1)))
      in H5;
      rewrite (Rmult_comm (Rabs (d x0)) (eps * (/ Rabs 2 * / Rabs (d x0)))) in H5;
        rewrite <- (Rmult_assoc eps (/ Rabs 2) (/ Rabs (d x0))) in H5;
          rewrite (Rmult_assoc (eps * / Rabs 2) (/ Rabs (d x0)) (Rabs (d x0))) in H5;
            rewrite
              (Rinv_l (Rabs (d x0))
                (Rlt_dichotomy_converse (Rabs (d x0)) 0 (or_intror (Rabs (d x0) < 0) H1)))
              in H5; rewrite (let (H1, H2) := Rmult_ne (eps * / Rabs 2) in H1) in H5;
                cut (Rabs 2 = 2).
  intro; rewrite H7 in H5;
    generalize
      (Rplus_lt_compat (Rabs (d x0 * (x1 - x0))) (eps * / 2)
        (Rabs (x1 - x0) * eps) (eps * / 2) H5 H3); intro; 
      rewrite eps2 in H10; assumption.
  unfold Rabs in |- *; case (Rcase_abs 2); auto.
  intro; cut (0 < 2).
  intro; generalize (Rlt_asym 0 2 H7); intro; elimtype False; auto.
  fourier.
  apply Rabs_no_R0.
  discrR.
Qed.


(*********)
Lemma Dconst :
  forall (D:R -> Prop) (y x0:R), D_in (fun x:R => y) (fun x:R => 0) D x0.
Proof.
  unfold D_in in |- *; intros; unfold limit1_in in |- *;
    unfold limit_in in |- *; unfold Rdiv in |- *; intros; 
      simpl in |- *; split with eps; split; auto.
  intros; rewrite (Rminus_diag_eq y y (refl_equal y)); rewrite Rmult_0_l;
    unfold R_dist in |- *; rewrite (Rminus_diag_eq 0 0 (refl_equal 0));
      unfold Rabs in |- *; case (Rcase_abs 0); intro.
  absurd (0 < 0); auto.
  red in |- *; intro; apply (Rlt_irrefl 0 H1).
  unfold Rgt in H0; assumption.
Qed.

(*********)
Lemma Dx :
  forall (D:R -> Prop) (x0:R), D_in (fun x:R => x) (fun x:R => 1) D x0.
Proof.
  unfold D_in in |- *; unfold Rdiv in |- *; intros; unfold limit1_in in |- *;
    unfold limit_in in |- *; intros; simpl in |- *; split with eps; 
      split; auto.
  intros; elim H0; clear H0; intros; unfold D_x in H0; elim H0; intros;
    rewrite (Rinv_r (x - x0) (Rminus_eq_contra x x0 (sym_not_eq H3)));
      unfold R_dist in |- *; rewrite (Rminus_diag_eq 1 1 (refl_equal 1));
        unfold Rabs in |- *; case (Rcase_abs 0); intro.
  absurd (0 < 0); auto.
  red in |- *; intro; apply (Rlt_irrefl 0 r).
  unfold Rgt in H; assumption.
Qed. 

(*********)
Lemma Dadd :
  forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
    D_in f df D x0 ->
    D_in g dg D x0 ->
    D_in (fun x:R => f x + g x) (fun x:R => df x + dg x) D x0.
Proof.
  unfold D_in in |- *; intros;
    generalize
      (limit_plus (fun x:R => (f x - f x0) * / (x - x0))
        (fun x:R => (g x - g x0) * / (x - x0)) (D_x D x0) (
          df x0) (dg x0) x0 H H0); clear H H0; unfold limit1_in in |- *;
      unfold limit_in in |- *; simpl in |- *; intros; elim (H eps H0); 
        clear H; intros; elim H; clear H; intros; split with x; 
          split; auto; intros; generalize (H1 x1 H2); clear H1; 
            intro; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1;
              rewrite (Rmult_comm (g x1 - g x0) (/ (x1 - x0))) in H1;
                rewrite <- (Rmult_plus_distr_l (/ (x1 - x0)) (f x1 - f x0) (g x1 - g x0))
                  in H1;
                  rewrite (Rmult_comm (/ (x1 - x0)) (f x1 - f x0 + (g x1 - g x0))) in H1;
                    cut (f x1 - f x0 + (g x1 - g x0) = f x1 + g x1 - (f x0 + g x0)).
  intro; rewrite H3 in H1; assumption.
  ring.
Qed.

(*********)
Lemma Dmult :
  forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
    D_in f df D x0 ->
    D_in g dg D x0 ->
    D_in (fun x:R => f x * g x) (fun x:R => df x * g x + f x * dg x) D x0.
Proof.
  intros; unfold D_in in |- *; generalize H H0; intros; unfold D_in in H, H0;
    generalize (cont_deriv f df D x0 H1); unfold continue_in in |- *; 
      intro;
        generalize
          (limit_mul (fun x:R => (g x - g x0) * / (x - x0)) (
            fun x:R => f x) (D_x D x0) (dg x0) (f x0) x0 H0 H3); 
          intro; cut (limit1_in (fun x:R => g x0) (D_x D x0) (g x0) x0).
  intro;
    generalize
      (limit_mul (fun x:R => (f x - f x0) * / (x - x0)) (
        fun _:R => g x0) (D_x D x0) (df x0) (g x0) x0 H H5);
      clear H H0 H1 H2 H3 H5; intro;
        generalize
          (limit_plus (fun x:R => (f x - f x0) * / (x - x0) * g x0)
            (fun x:R => (g x - g x0) * / (x - x0) * f x) (
              D_x D x0) (df x0 * g x0) (dg x0 * f x0) x0 H H4); 
          clear H4 H; intro; unfold limit1_in in H; unfold limit_in in H; 
            simpl in H; unfold limit1_in in |- *; unfold limit_in in |- *; 
              simpl in |- *; intros; elim (H eps H0); clear H; intros; 
                elim H; clear H; intros; split with x; split; auto; 
                  intros; generalize (H1 x1 H2); clear H1; intro;
                    rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1;
                      rewrite (Rmult_comm (g x1 - g x0) (/ (x1 - x0))) in H1;
                        rewrite (Rmult_assoc (/ (x1 - x0)) (f x1 - f x0) (g x0)) in H1;
                          rewrite (Rmult_assoc (/ (x1 - x0)) (g x1 - g x0) (f x1)) in H1;
                            rewrite <-
                              (Rmult_plus_distr_l (/ (x1 - x0)) ((f x1 - f x0) * g x0)
                                ((g x1 - g x0) * f x1)) in H1;
                              rewrite
                                (Rmult_comm (/ (x1 - x0)) ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1))
                                in H1; rewrite (Rmult_comm (dg x0) (f x0)) in H1;
                                  cut
                                    ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1 = f x1 * g x1 - f x0 * g x0).
  intro; rewrite H3 in H1; assumption.
  ring.
  unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros;
    split with eps; split; auto; intros; elim (R_dist_refl (g x0) (g x0));
      intros a b; rewrite (b (refl_equal (g x0))); unfold Rgt in H; 
        assumption.
Qed.

(*********)
Lemma Dmult_const :
  forall (D:R -> Prop) (f df:R -> R) (x0 a:R),
    D_in f df D x0 -> D_in (fun x:R => a * f x) (fun x:R => a * df x) D x0.
Proof.
  intros;
    generalize (Dmult D (fun _:R => 0) df (fun _:R => a) f x0 (Dconst D a x0) H);
      unfold D_in in |- *; intros; rewrite (Rmult_0_l (f x0)) in H0;
        rewrite (let (H1, H2) := Rplus_ne (a * df x0) in H2) in H0; 
          assumption.
Qed.

(*********)
Lemma Dopp :
  forall (D:R -> Prop) (f df:R -> R) (x0:R),
    D_in f df D x0 -> D_in (fun x:R => - f x) (fun x:R => - df x) D x0.
Proof.
  intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in in |- *;
    unfold limit1_in in |- *; unfold limit_in in |- *; 
      intros; generalize (H0 eps H1); clear H0; intro; elim H0; 
        clear H0; intros; elim H0; clear H0; simpl in |- *; 
          intros; split with x; split; auto.
  intros; generalize (H2 x1 H3); clear H2; intro;
    rewrite Ropp_mult_distr_l_reverse in H2;
      rewrite Ropp_mult_distr_l_reverse in H2;
        rewrite Ropp_mult_distr_l_reverse in H2;
          rewrite (let (H1, H2) := Rmult_ne (f x1) in H2) in H2;
            rewrite (let (H1, H2) := Rmult_ne (f x0) in H2) in H2;
              rewrite (let (H1, H2) := Rmult_ne (df x0) in H2) in H2; 
                assumption.
Qed.

(*********)
Lemma Dminus :
  forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
    D_in f df D x0 ->
    D_in g dg D x0 ->
    D_in (fun x:R => f x - g x) (fun x:R => df x - dg x) D x0.
Proof.
  unfold Rminus in |- *; intros; generalize (Dopp D g dg x0 H0); intro;
    apply (Dadd D df (fun x:R => - dg x) f (fun x:R => - g x) x0); 
      assumption. 
Qed.

(*********)
Lemma Dx_pow_n :
  forall (n:nat) (D:R -> Prop) (x0:R),
    D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0.
Proof.
  simple induction n; intros.
  simpl in |- *; rewrite Rmult_0_l; apply Dconst.
  intros; cut (n0 = (S n0 - 1)%nat);
    [ intro a; rewrite <- a; clear a | simpl in |- *; apply minus_n_O ].
  generalize
    (Dmult D (fun _:R => 1) (fun x:R => INR n0 * x ^ (n0 - 1)) (
      fun x:R => x) (fun x:R => x ^ n0) x0 (Dx D x0) (
        H D x0)); unfold D_in in |- *; unfold limit1_in in |- *;
    unfold limit_in in |- *; simpl in |- *; intros; elim (H0 eps H1); 
      clear H0; intros; elim H0; clear H0; intros; split with x; 
        split; auto.
  intros; generalize (H2 x1 H3); clear H2 H3; intro;
    rewrite (let (H1, H2) := Rmult_ne (x0 ^ n0) in H2) in H2;
      rewrite (tech_pow_Rmult x1 n0) in H2; rewrite (tech_pow_Rmult x0 n0) in H2;
        rewrite (Rmult_comm (INR n0) (x0 ^ (n0 - 1))) in H2;
          rewrite <- (Rmult_assoc x0 (x0 ^ (n0 - 1)) (INR n0)) in H2;
            rewrite (tech_pow_Rmult x0 (n0 - 1)) in H2; elim (classic (n0 = 0%nat));
              intro cond.
  rewrite cond in H2; rewrite cond; simpl in H2; simpl in |- *;
    cut (1 + x0 * 1 * 0 = 1 * 1);
      [ intro A; rewrite A in H2; assumption | ring ].
  cut (n0 <> 0%nat -> S (n0 - 1) = n0); [ intro | omega ];
    rewrite (H3 cond) in H2; rewrite (Rmult_comm (x0 ^ n0) (INR n0)) in H2;
      rewrite (tech_pow_Rplus x0 n0 n0) in H2; assumption.
Qed.

(*********)
Lemma Dcomp :
  forall (Df Dg:R -> Prop) (df dg f g:R -> R) (x0:R),
    D_in f df Df x0 ->
    D_in g dg Dg (f x0) ->
    D_in (fun x:R => g (f x)) (fun x:R => df x * dg (f x)) (Dgf Df Dg f) x0.
Proof.
  intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in in |- *;
    unfold Rdiv in |- *; intros;
      generalize
        (limit_comp f (fun x:R => (g x - g (f x0)) * / (x - f x0)) (
          D_x Df x0) (D_x Dg (f x0)) (f x0) (dg (f x0)) x0); 
        intro; generalize (cont_deriv f df Df x0 H); intro; 
          unfold continue_in in H4; generalize (H3 H4 H2); clear H3; 
            intro;
              generalize
                (limit_mul (fun x:R => (g (f x) - g (f x0)) * / (f x - f x0))
                  (fun x:R => (f x - f x0) * / (x - x0))
                  (Dgf (D_x Df x0) (D_x Dg (f x0)) f) (dg (f x0)) (
                    df x0) x0 H3); intro;
                cut
                  (limit1_in (fun x:R => (f x - f x0) * / (x - x0))
                    (Dgf (D_x Df x0) (D_x Dg (f x0)) f) (df x0) x0).
  intro; generalize (H5 H6); clear H5; intro;
    generalize
      (limit_mul (fun x:R => (f x - f x0) * / (x - x0)) (
        fun x:R => dg (f x0)) (D_x Df x0) (df x0) (dg (f x0)) x0 H1
      (limit_free (fun x:R => dg (f x0)) (D_x Df x0) x0 x0)); 
      intro; unfold limit1_in in |- *; unfold limit_in in |- *; 
        simpl in |- *; unfold limit1_in in H5, H7; unfold limit_in in H5, H7;
          simpl in H5, H7; intros; elim (H5 eps H8); elim (H7 eps H8); 
            clear H5 H7; intros; elim H5; elim H7; clear H5 H7; 
              intros; split with (Rmin x x1); split.
  elim (Rmin_Rgt x x1 0); intros a b; apply (b (conj H9 H5)); clear a b.
  intros; elim H11; clear H11; intros; elim (Rmin_Rgt x x1 (R_dist x2 x0));
    intros a b; clear b; unfold Rgt in a; elim (a H12); 
      clear H5 a; intros; unfold D_x, Dgf in H11, H7, H10; 
        clear H12; elim (classic (f x2 = f x0)); intro.
  elim H11; clear H11; intros; elim H11; clear H11; intros;
    generalize (H10 x2 (conj (conj H11 H14) H5)); intro;
      rewrite (Rminus_diag_eq (f x2) (f x0) H12) in H16;
        rewrite (Rmult_0_l (/ (x2 - x0))) in H16;
          rewrite (Rmult_0_l (dg (f x0))) in H16; rewrite H12;
            rewrite (Rminus_diag_eq (g (f x0)) (g (f x0)) (refl_equal (g (f x0))));
              rewrite (Rmult_0_l (/ (x2 - x0))); assumption.
  clear H10 H5; elim H11; clear H11; intros; elim H5; clear H5; intros;
    cut
      (((Df x2 /\ x0 <> x2) /\ Dg (f x2) /\ f x0 <> f x2) /\ R_dist x2 x0 < x1);
      auto; intro; generalize (H7 x2 H14); intro;
        generalize (Rminus_eq_contra (f x2) (f x0) H12); intro;
          rewrite
            (Rmult_assoc (g (f x2) - g (f x0)) (/ (f x2 - f x0))
              ((f x2 - f x0) * / (x2 - x0))) in H15;
            rewrite <- (Rmult_assoc (/ (f x2 - f x0)) (f x2 - f x0) (/ (x2 - x0)))
              in H15; rewrite (Rinv_l (f x2 - f x0) H16) in H15;
                rewrite (let (H1, H2) := Rmult_ne (/ (x2 - x0)) in H2) in H15;
                  rewrite (Rmult_comm (df x0) (dg (f x0))); assumption.
  clear H5 H3 H4 H2; unfold limit1_in in |- *; unfold limit_in in |- *;
    simpl in |- *; unfold limit1_in in H1; unfold limit_in in H1; 
      simpl in H1; intros; elim (H1 eps H2); clear H1; intros; 
        elim H1; clear H1; intros; split with x; split; auto; 
          intros; unfold D_x, Dgf in H4, H3; elim H4; clear H4; 
            intros; elim H4; clear H4; intros; exact (H3 x1 (conj H4 H5)).
Qed.   

(*********)
Lemma D_pow_n :
  forall (n:nat) (D:R -> Prop) (x0:R) (expr dexpr:R -> R),
    D_in expr dexpr D x0 ->
    D_in (fun x:R => expr x ^ n)
    (fun x:R => INR n * expr x ^ (n - 1) * dexpr x) (
      Dgf D D expr) x0.
Proof.
  intros n D x0 expr dexpr H;
    generalize
      (Dcomp D D dexpr (fun x:R => INR n * x ^ (n - 1)) expr (
        fun x:R => x ^ n) x0 H (Dx_pow_n n D (expr x0))); 
      intro; unfold D_in in |- *; unfold limit1_in in |- *;
        unfold limit_in in |- *; simpl in |- *; intros; unfold D_in in H0;
          unfold limit1_in in H0; unfold limit_in in H0; simpl in H0; 
            elim (H0 eps H1); clear H0; intros; elim H0; clear H0; 
              intros; split with x; split; intros; auto.
  cut
    (dexpr x0 * (INR n * expr x0 ^ (n - 1)) =
      INR n * expr x0 ^ (n - 1) * dexpr x0);
    [ intro Rew; rewrite <- Rew; exact (H2 x1 H3) | ring ].
Qed.