aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Reals/Ranalysis1.v
blob: f24df53a7f011f7a6cab8f85af622e2bd37dbf85 (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
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
(************************************************************************)
(*  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 Export Rlimit.
Require Export Rderiv.
Local Open Scope R_scope.
Implicit Type f : R -> R.

(****************************************************)
(** *         Basic operations on functions         *)
(****************************************************)
Definition plus_fct f1 f2 (x:R) : R := f1 x + f2 x.
Definition opp_fct f (x:R) : R := - f x.
Definition mult_fct f1 f2 (x:R) : R := f1 x * f2 x.
Definition mult_real_fct (a:R) f (x:R) : R := a * f x.
Definition minus_fct f1 f2 (x:R) : R := f1 x - f2 x.
Definition div_fct f1 f2 (x:R) : R := f1 x / f2 x.
Definition div_real_fct (a:R) f (x:R) : R := a / f x.
Definition comp f1 f2 (x:R) : R := f1 (f2 x).
Definition inv_fct f (x:R) : R := / f x.

Delimit Scope Rfun_scope with F.

Arguments plus_fct (f1 f2)%F x%R.
Arguments mult_fct (f1 f2)%F x%R.
Arguments minus_fct (f1 f2)%F x%R.
Arguments div_fct (f1 f2)%F x%R.
Arguments inv_fct f%F x%R.
Arguments opp_fct f%F x%R.
Arguments mult_real_fct a%R f%F x%R.
Arguments div_real_fct a%R f%F x%R.
Arguments comp (f1 f2)%F x%R.

Infix "+" := plus_fct : Rfun_scope.
Notation "- x" := (opp_fct x) : Rfun_scope.
Infix "*" := mult_fct : Rfun_scope.
Infix "-" := minus_fct : Rfun_scope.
Infix "/" := div_fct : Rfun_scope.
Local Notation "f1 'o' f2" := (comp f1 f2)
  (at level 20, right associativity) : Rfun_scope.
Notation "/ x" := (inv_fct x) : Rfun_scope.

Definition fct_cte (a x:R) : R := a.
Definition id (x:R) := x.

(****************************************************)
(** *          Variations of functions              *)
(****************************************************)
Definition increasing f : Prop := forall x y:R, x <= y -> f x <= f y.
Definition decreasing f : Prop := forall x y:R, x <= y -> f y <= f x.
Definition strict_increasing f : Prop := forall x y:R, x < y -> f x < f y.
Definition strict_decreasing f : Prop := forall x y:R, x < y -> f y < f x.
Definition constant f : Prop := forall x y:R, f x = f y.

(**********)
Definition no_cond (x:R) : Prop := True.

(**********)
Definition constant_D_eq f (D:R -> Prop) (c:R) : Prop :=
  forall x:R, D x -> f x = c.

(***************************************************)
(** *    Definition of continuity as a limit       *)
(***************************************************)

(**********)
Definition continuity_pt f (x0:R) : Prop := continue_in f no_cond x0.
Definition continuity f : Prop := forall x:R, continuity_pt f x.

Arguments continuity_pt f%F x0%R.
Arguments continuity f%F.

Lemma continuity_pt_locally_ext :
  forall f g a x, 0 < a -> (forall y, R_dist y x < a -> f y = g y) ->
  continuity_pt f x -> continuity_pt g x.
intros f g a x a0 q cf eps ep.
destruct (cf eps ep) as [a' [a'p Pa']].
exists (Rmin a a'); split.
 unfold Rmin; destruct (Rle_dec a a').
  assumption.
 assumption.
intros y cy; rewrite <- !q.
  apply Pa'.
  split;[| apply Rlt_le_trans with (Rmin a a');[ | apply Rmin_r]];tauto.
 rewrite R_dist_eq; assumption.   
apply Rlt_le_trans with (Rmin a a');[ | apply Rmin_l]; tauto.
Qed.


(**********)
Lemma continuity_pt_plus :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0.
Proof.
  unfold continuity_pt, plus_fct; unfold continue_in; intros;
    apply limit_plus; assumption.
Qed.

Lemma continuity_pt_opp :
  forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0.
Proof.
  unfold continuity_pt, opp_fct; unfold continue_in; intros;
    apply limit_Ropp; assumption.
Qed.

Lemma continuity_pt_minus :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0.
Proof.
  unfold continuity_pt, minus_fct; unfold continue_in; intros;
    apply limit_minus; assumption.
Qed.

Lemma continuity_pt_mult :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0.
Proof.
  unfold continuity_pt, mult_fct; unfold continue_in; intros;
    apply limit_mul; assumption.
Qed.

Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0.
Proof.
  unfold constant, continuity_pt; unfold continue_in;
    unfold limit1_in; unfold limit_in;
      intros; exists 1; split;
        [ apply Rlt_0_1
          | intros; generalize (H x x0); intro; rewrite H2; simpl;
            rewrite R_dist_eq; assumption ].
Qed.

Lemma continuity_pt_scal :
  forall f (a x0:R),
    continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0.
Proof.
  unfold continuity_pt, mult_real_fct; unfold continue_in;
    intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0).
  unfold limit1_in; unfold limit_in; intros; exists 1; split.
  apply Rlt_0_1.
  intros; rewrite R_dist_eq; assumption.
  assumption.
Qed.

Lemma continuity_pt_inv :
  forall f (x0:R), continuity_pt f x0 -> f x0 <> 0 -> continuity_pt (/ f) x0.
Proof.
  intros.
  replace (/ f)%F with (fun x:R => / f x).
  unfold continuity_pt; unfold continue_in; intros;
    apply limit_inv; assumption.
  unfold inv_fct; reflexivity.
Qed.

Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F.
Proof.
  intros; reflexivity.
Qed.

Lemma continuity_pt_div :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 ->
    continuity_pt f2 x0 -> f2 x0 <> 0 -> continuity_pt (f1 / f2) x0.
Proof.
  intros; rewrite (div_eq_inv f1 f2); apply continuity_pt_mult;
    [ assumption | apply continuity_pt_inv; assumption ].
Qed.

Lemma continuity_pt_comp :
  forall f1 f2 (x:R),
    continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x.
Proof.
  unfold continuity_pt; unfold continue_in; intros;
    unfold comp.
  cut
    (limit1_in (fun x0:R => f2 (f1 x0))
      (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) (
        f2 (f1 x)) x ->
      limit1_in (fun x0:R => f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x).
  intro; apply H1.
  eapply limit_comp.
  apply H.
  apply H0.
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.
  assert (H3 := H1 eps H2).
  elim H3; intros.
  exists x0.
  split.
  elim H4; intros; assumption.
  intros; case (Req_dec (f1 x) (f1 x1)); intro.
  rewrite H6; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
    assumption.
  elim H4; intros; apply H8.
  split.
  unfold Dgf, D_x, no_cond.
  split.
  split.
  trivial.
  elim H5; unfold D_x, no_cond; intros.
  elim H9; intros; assumption.
  split.
  trivial.
  assumption.
  elim H5; intros; assumption.
Qed.

(**********)
Lemma continuity_plus :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2).
Proof.
  unfold continuity; intros;
    apply (continuity_pt_plus f1 f2 x (H x) (H0 x)).
Qed.

Lemma continuity_opp : forall f, continuity f -> continuity (- f).
Proof.
  unfold continuity; intros; apply (continuity_pt_opp f x (H x)).
Qed.

Lemma continuity_minus :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2).
Proof.
  unfold continuity; intros;
    apply (continuity_pt_minus f1 f2 x (H x) (H0 x)).
Qed.

Lemma continuity_mult :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2).
Proof.
  unfold continuity; intros;
    apply (continuity_pt_mult f1 f2 x (H x) (H0 x)).
Qed.

Lemma continuity_const : forall f, constant f -> continuity f.
Proof.
  unfold continuity; intros; apply (continuity_pt_const f x H).
Qed.

Lemma continuity_scal :
  forall f (a:R), continuity f -> continuity (mult_real_fct a f).
Proof.
  unfold continuity; intros; apply (continuity_pt_scal f a x (H x)).
Qed.

Lemma continuity_inv :
  forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f).
Proof.
  unfold continuity; intros; apply (continuity_pt_inv f x (H x) (H0 x)).
Qed.

Lemma continuity_div :
  forall f1 f2,
    continuity f1 ->
    continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2).
Proof.
  unfold continuity; intros;
    apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)).
Qed.

Lemma continuity_comp :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1).
Proof.
  unfold continuity; intros.
  apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))).
Qed.


(*****************************************************)
(** * Derivative's definition using Landau's kernel  *)
(*****************************************************)

Definition derivable_pt_lim f (x l:R) : Prop :=
  forall eps:R,
    0 < eps ->
    exists delta : posreal,
      (forall h:R,
        h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - l) < eps).

Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l.

Definition derivable_pt f (x:R) := { l:R | derivable_pt_abs f x l }.
Definition derivable f := forall x:R, derivable_pt f x.

Definition derive_pt f (x:R) (pr:derivable_pt f x) := proj1_sig pr.
Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x).

Arguments derivable_pt_lim f%F x%R l.
Arguments derivable_pt_abs f%F (x l)%R.
Arguments derivable_pt f%F x%R.
Arguments derivable f%F.
Arguments derive_pt f%F x%R pr.
Arguments derive f%F pr x.

Definition antiderivative f (g:R -> R) (a b:R) : Prop :=
  (forall x:R,
    a <= x <= b ->  exists pr : derivable_pt g x, f x = derive_pt g x pr) /\
  a <= b.
(**************************************)
(** * Class of differential functions *)
(**************************************)
Record Differential : Type := mkDifferential
  {d1 :> R -> R; cond_diff : derivable d1}.

Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.

(**********)
Lemma uniqueness_step1 :
  forall f (x l1 l2:R),
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l1 0 ->
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l2 0 ->
    l1 = l2.
Proof.
  intros;
    apply
      (single_limit (fun h:R => (f (x + h) - f x) / h) (
        fun h:R => h <> 0) l1 l2 0); try assumption.
  unfold adhDa; intros; exists (alp / 2).
  split.
  unfold Rdiv; apply prod_neq_R0.
  red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).
  apply Rinv_neq_0_compat; discrR.
  unfold R_dist; unfold Rminus; rewrite Ropp_0;
    rewrite Rplus_0_r; unfold Rdiv; rewrite Rabs_mult.
  replace (Rabs (/ 2)) with (/ 2).
  replace (Rabs alp) with alp.
  apply Rmult_lt_reg_l with 2.
  prove_sup0.
  rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
    [ idtac | discrR ]; rewrite Rmult_1_r; rewrite double;
      pattern alp at 1; replace alp with (alp + 0);
        [ idtac | ring ]; apply Rplus_lt_compat_l; assumption.
  symmetry ; apply Rabs_right; left; assumption.
  symmetry ; apply Rabs_right; left; change (0 < / 2);
    apply Rinv_0_lt_compat; prove_sup0.
Qed.

Lemma uniqueness_step2 :
  forall f (x l:R),
    derivable_pt_lim f x l ->
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0.
Proof.
  unfold derivable_pt_lim; intros; unfold limit1_in;
    unfold limit_in; intros.
  assert (H1 := H eps H0).
  elim H1; intros.
  exists (pos x0).
  split.
  apply (cond_pos x0).
  simpl; unfold R_dist; intros.
  elim H3; intros.
  apply H2;
    [ assumption
      | unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5;
        assumption ].
Qed.

Lemma uniqueness_step3 :
  forall f (x l:R),
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 ->
    derivable_pt_lim f x l.
Proof.
  unfold limit1_in, derivable_pt_lim; unfold limit_in;
    unfold dist; simpl; intros.
  elim (H eps H0).
  intros; elim H1; intros.
  exists (mkposreal x0 H2).
  simpl; intros; unfold R_dist in H3; apply (H3 h).
  split;
    [ assumption
      | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; assumption ].
Qed.

Lemma uniqueness_limite :
  forall f (x l1 l2:R),
    derivable_pt_lim f x l1 -> derivable_pt_lim f x l2 -> l1 = l2.
Proof.
  intros.
  assert (H1 := uniqueness_step2 _ _ _ H).
  assert (H2 := uniqueness_step2 _ _ _ H0).
  assert (H3 := uniqueness_step1 _ _ _ _ H1 H2).
  assumption.
Qed.

Lemma derive_pt_eq :
  forall f (x l:R) (pr:derivable_pt f x),
    derive_pt f x pr = l <-> derivable_pt_lim f x l.
Proof.
  intros; split.
  intro; assert (H1 := proj2_sig pr); unfold derive_pt in H; rewrite H in H1;
    assumption.
  intro; assert (H1 := proj2_sig pr); unfold derivable_pt_abs in H1.
  assert (H2 := uniqueness_limite _ _ _ _ H H1).
  unfold derive_pt; unfold derivable_pt_abs.
  symmetry ; assumption.
Qed.

(**********)
Lemma derive_pt_eq_0 :
  forall f (x l:R) (pr:derivable_pt f x),
    derivable_pt_lim f x l -> derive_pt f x pr = l.
Proof.
  intros; elim (derive_pt_eq f x l pr); intros.
  apply (H1 H).
Qed.

(**********)
Lemma derive_pt_eq_1 :
  forall f (x l:R) (pr:derivable_pt f x),
    derive_pt f x pr = l -> derivable_pt_lim f x l.
Proof.
  intros; elim (derive_pt_eq f x l pr); intros.
  apply (H0 H).
Qed.


(**********************************************************************)
(** * Equivalence of this definition with the one using limit concept *)
(**********************************************************************)
Lemma derive_pt_D_in :
  forall f (df:R -> R) (x:R) (pr:derivable_pt f x),
    D_in f df no_cond x <-> derive_pt f x pr = df x.
Proof.
  intros; split.
  unfold D_in; unfold limit1_in; unfold limit_in;
    simpl; unfold R_dist; intros.
  apply derive_pt_eq_0.
  unfold derivable_pt_lim.
  intros; elim (H eps H0); intros alpha H1; elim H1; intros;
    exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
      intro; cut (x + h - x = h);
        [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
          [ intro; generalize (H6 H8); rewrite H7; intro; assumption
            | split;
              [ unfold D_x; split;
                [ unfold no_cond; trivial
                  | apply Rminus_not_eq_right; rewrite H7; assumption ]
                | rewrite H7; assumption ] ]
          | ring ].
  intro.
  assert (H0 := derive_pt_eq_1 f x (df x) pr H).
  unfold D_in; unfold limit1_in; unfold limit_in;
    unfold dist; simpl; unfold R_dist;
      intros.
  elim (H0 eps H1); intros alpha H2; exists (pos alpha); split.
  apply (cond_pos alpha).
  intros; elim H3; intros; unfold D_x in H4; elim H4; intros; cut (x0 - x <> 0).
  intro; generalize (H2 (x0 - x) H8 H5); replace (x + (x0 - x)) with x0.
  intro; assumption.
  ring.
  auto with real.
Qed.

Lemma derivable_pt_lim_D_in :
  forall f (df:R -> R) (x:R),
    D_in f df no_cond x <-> derivable_pt_lim f x (df x).
Proof.
  intros; split.
  unfold D_in; unfold limit1_in; unfold limit_in;
    simpl; unfold R_dist; intros.
  unfold derivable_pt_lim.
  intros; elim (H eps H0); intros alpha H1; elim H1; intros;
    exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
      intro; cut (x + h - x = h);
        [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
          [ intro; generalize (H6 H8); rewrite H7; intro; assumption
            | split;
              [ unfold D_x; split;
                [ unfold no_cond; trivial
                  | apply Rminus_not_eq_right; rewrite H7; assumption ]
                | rewrite H7; assumption ] ]
          | ring ].
  intro.
  unfold derivable_pt_lim in H.
  unfold D_in; unfold limit1_in; unfold limit_in;
    unfold dist; simpl; unfold R_dist;
      intros.
  elim (H eps H0); intros alpha H2; exists (pos alpha); split.
  apply (cond_pos alpha).
  intros.
  elim H1; intros; unfold D_x in H3; elim H3; intros; cut (x0 - x <> 0).
  intro; generalize (H2 (x0 - x) H7 H4); replace (x + (x0 - x)) with x0.
  intro; assumption.
  ring.
  auto with real.
Qed.

(* Extensionally equal functions have the same derivative. *)

Lemma derivable_pt_lim_ext : forall f g x l, (forall z, f z = g z) -> 
  derivable_pt_lim f x l -> derivable_pt_lim g x l.
intros f g x l fg df e ep; destruct (df e ep) as [d pd]; exists d; intros h;
rewrite <- !fg; apply pd.
Qed.

(* extensionally equal functions have the same derivative, locally. *)

Lemma derivable_pt_lim_locally_ext : forall f g x a b l, 
  a < x < b ->
  (forall z, a < z < b -> f z = g z) ->
  derivable_pt_lim f x l -> derivable_pt_lim g x l.
intros f g x a b l axb fg df e ep.
destruct (df e ep) as [d pd].
assert (d'h : 0 < Rmin d (Rmin (b - x) (x - a))).
 apply Rmin_pos;[apply cond_pos | apply Rmin_pos; apply Rlt_Rminus; tauto].
exists (mkposreal _ d'h); simpl; intros h hn0 cmp.
rewrite <- !fg;[ |assumption | ].
  apply pd;[assumption |].
 apply Rlt_le_trans with (1 := cmp), Rmin_l.
assert (-h < x - a).
 apply Rle_lt_trans with (1 := Rle_abs _).
 rewrite Rabs_Ropp; apply Rlt_le_trans with (1 := cmp).
 rewrite Rmin_assoc; apply Rmin_r.
assert (h < b - x).
 apply Rle_lt_trans with (1 := Rle_abs _).
 apply Rlt_le_trans with (1 := cmp).
 rewrite Rmin_comm, <- Rmin_assoc; apply Rmin_l.
split.
 apply (Rplus_lt_reg_l (- h)).
 replace ((-h) + (x + h)) with x by ring.
 apply (Rplus_lt_reg_r (- a)).
 replace (((-h) + a) + - a) with (-h) by ring.
 assumption.
apply (Rplus_lt_reg_r (- x)).
replace (x + h + - x) with h by ring.
assumption.
Qed.


(***********************************)
(** * derivability -> continuity   *)
(***********************************)
(**********)
Lemma derivable_derive :
  forall f (x:R) (pr:derivable_pt f x),  exists l : R, derive_pt f x pr = l.
Proof.
  intros; exists (proj1_sig pr).
  unfold derive_pt; reflexivity.
Qed.

Theorem derivable_continuous_pt :
  forall f (x:R), derivable_pt f x -> continuity_pt f x.
Proof.
  intros f x X.
  generalize (derivable_derive f x X); intro.
  elim H; intros l H1.
  cut (l = fct_cte l x).
  intro.
  rewrite H0 in H1.
  generalize (derive_pt_D_in f (fct_cte l) x); intro.
  elim (H2 X); intros.
  generalize (H4 H1); intro.
  unfold continuity_pt.
  apply (cont_deriv f (fct_cte l) no_cond x H5).
  unfold fct_cte; reflexivity.
Qed.

Theorem derivable_continuous : forall f, derivable f -> continuity f.
Proof.
  unfold derivable, continuity; intros f X x.
  apply (derivable_continuous_pt f x (X x)).
Qed.

(****************************************************************)
(** *                    Main rules                             *)
(****************************************************************)

Lemma derivable_pt_lim_plus :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 + f2) x (l1 + l2).
  intros.
  apply uniqueness_step3.
  assert (H1 := uniqueness_step2 _ _ _ H).
  assert (H2 := uniqueness_step2 _ _ _ H0).
  unfold plus_fct.
  cut
    (forall h:R,
      (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
      (f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h).
  intro.
  generalize
    (limit_plus (fun h':R => (f1 (x + h') - f1 x) / h')
      (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.
  elim (H4 eps H5); intros.
  exists x0.
  elim H6; intros.
  split.
  assumption.
  intros; rewrite H3; apply H8; assumption.
  intro; unfold Rdiv; ring.
Qed.

Lemma derivable_pt_lim_opp :
  forall f (x l:R), derivable_pt_lim f x l -> derivable_pt_lim (- f) x (- l).
Proof.
  intros.
  apply uniqueness_step3.
  assert (H1 := uniqueness_step2 _ _ _ H).
  unfold opp_fct.
  cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)).
  intro.
  generalize
    (limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1).
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.
  elim (H2 eps H3); intros.
  exists x0.
  elim H4; intros.
  split.
  assumption.
  intros; rewrite H0; apply H6; assumption.
  intro; unfold Rdiv; ring.
Qed.

Lemma derivable_pt_lim_minus :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 - f2) x (l1 - l2).
Proof.
  intros.
  apply uniqueness_step3.
  assert (H1 := uniqueness_step2 _ _ _ H).
  assert (H2 := uniqueness_step2 _ _ _ H0).
  unfold minus_fct.
  cut
    (forall h:R,
      (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
      (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h).
  intro.
  generalize
    (limit_minus (fun h':R => (f1 (x + h') - f1 x) / h')
      (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.
  elim (H4 eps H5); intros.
  exists x0.
  elim H6; intros.
  split.
  assumption.
  intros; rewrite <- H3; apply H8; assumption.
  intro; unfold Rdiv; ring.
Qed.

Lemma derivable_pt_lim_mult :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 ->
    derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2).
Proof.
  intros.
  assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x).
  elim H1; intros.
  assert (H4 := H3 H).
  assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) x).
  elim H5; intros.
  assert (H8 := H7 H0).
  clear H1 H2 H3 H5 H6 H7.
  assert
    (H1 :=
      derivable_pt_lim_D_in (f1 * f2)%F (fun y:R => l1 * f2 x + f1 x * l2) x).
  elim H1; intros.
  clear H1 H3.
  apply H2.
  unfold mult_fct.
  apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption.
Qed.

Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0.
Proof.
  intros; unfold fct_cte, derivable_pt_lim.
  intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus;
    rewrite Rplus_opp_r; unfold Rdiv; rewrite Rmult_0_l;
      rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
Qed.

Lemma derivable_pt_lim_scal :
  forall f (a x l:R),
    derivable_pt_lim f x l -> derivable_pt_lim (mult_real_fct a f) x (a * l).
Proof.
  intros.
  assert (H0 := derivable_pt_lim_const a x).
  replace (mult_real_fct a f) with (fct_cte a * f)%F.
  replace (a * l) with (0 * f x + a * l); [ idtac | ring ].
  apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption.
  unfold mult_real_fct, mult_fct, fct_cte; reflexivity.
Qed.

Lemma derivable_pt_lim_div_scal :
  forall f x l a, derivable_pt_lim f x l ->
     derivable_pt_lim (fun y => f y / a) x (l / a).
intros f x l a df;
  apply (derivable_pt_lim_ext (fun y => /a * f y)).
 intros z; rewrite Rmult_comm; reflexivity.
unfold Rdiv; rewrite Rmult_comm; apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derivable_pt_lim_scal_right :
  forall f x l a, derivable_pt_lim f x l ->
     derivable_pt_lim (fun y => f y * a) x (l * a).
intros f x l a df;
  apply (derivable_pt_lim_ext (fun y => a * f y)).
 intros z; rewrite Rmult_comm; reflexivity.
unfold Rdiv; rewrite Rmult_comm; apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1.
Proof.
  intro; unfold derivable_pt_lim.
  intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2;
    unfold id; replace ((x + h - x) / h - 1) with 0.
  rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h).
  apply Rabs_pos.
  assumption.
  unfold Rminus; rewrite Rplus_assoc; rewrite (Rplus_comm x);
    rewrite Rplus_assoc.
  rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv;
    rewrite <- Rinv_r_sym.
  symmetry ; apply Rplus_opp_r.
  assumption.
Qed.

Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x).
Proof.
  intro; unfold derivable_pt_lim.
  unfold Rsqr; intros eps Heps; exists (mkposreal eps Heps);
    intros h H1 H2; replace (((x + h) * (x + h) - x * x) / h - 2 * x) with h.
  assumption.
  replace ((x + h) * (x + h) - x * x) with (2 * x * h + h * h);
  [ idtac | ring ].
  unfold Rdiv; rewrite Rmult_plus_distr_r.
  repeat rewrite Rmult_assoc.
  repeat rewrite <- Rinv_r_sym; [ idtac | assumption ].
  ring.
Qed.

Lemma derivable_pt_lim_comp :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 (f1 x) l2 -> derivable_pt_lim (f2 o f1) x (l2 * l1).
Proof.
  intros; assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x).
  elim H1; intros.
  assert (H4 := H3 H).
  assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) (f1 x)).
  elim H5; intros.
  assert (H8 := H7 H0).
  clear H1 H2 H3 H5 H6 H7.
  assert (H1 := derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x).
  elim H1; intros.
  clear H1 H3; apply H2.
  unfold comp;
    cut
      (D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1)
        (Dgf no_cond no_cond f1) x ->
        D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) no_cond x).
  intro; apply H1.
  rewrite Rmult_comm;
    apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x);
      assumption.
  unfold Dgf, D_in, no_cond; unfold limit1_in;
    unfold limit_in; unfold dist; simpl;
      unfold R_dist; intros.
  elim (H1 eps H3); intros.
  exists x0; intros; split.
  elim H5; intros; assumption.
  intros; elim H5; intros; apply H9; split.
  unfold D_x; split.
  split; trivial.
  elim H6; intros; unfold D_x in H10; elim H10; intros; assumption.
  elim H6; intros; assumption.
Qed.

Lemma derivable_pt_plus :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x.
Proof.
  unfold derivable_pt; intros f1 f2 x X X0.
  elim X; intros.
  elim X0; intros.
  exists (x0 + x1).
  apply derivable_pt_lim_plus; assumption.
Qed.

Lemma derivable_pt_opp :
  forall f (x:R), derivable_pt f x -> derivable_pt (- f) x.
Proof.
  unfold derivable_pt; intros f x X.
  elim X; intros.
  exists (- x0).
  apply derivable_pt_lim_opp; assumption.
Qed.

Lemma derivable_pt_minus :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x.
Proof.
  unfold derivable_pt; intros f1 f2 x X X0.
  elim X; intros.
  elim X0; intros.
  exists (x0 - x1).
  apply derivable_pt_lim_minus; assumption.
Qed.

Lemma derivable_pt_mult :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x.
Proof.
  unfold derivable_pt; intros f1 f2 x X X0.
  elim X; intros.
  elim X0; intros.
  exists (x0 * f2 x + f1 x * x1).
  apply derivable_pt_lim_mult; assumption.
Qed.

Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x.
Proof.
  intros; unfold derivable_pt.
  exists 0.
  apply derivable_pt_lim_const.
Qed.

Lemma derivable_pt_scal :
  forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x.
Proof.
  unfold derivable_pt; intros f1 a x X.
  elim X; intros.
  exists (a * x0).
  apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derivable_pt_id : forall x:R, derivable_pt id x.
Proof.
  unfold derivable_pt; intro.
  exists 1.
  apply derivable_pt_lim_id.
Qed.

Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x.
Proof.
  unfold derivable_pt; intro; exists (2 * x).
  apply derivable_pt_lim_Rsqr.
Qed.

Lemma derivable_pt_comp :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x.
Proof.
  unfold derivable_pt; intros f1 f2 x X X0.
  elim X; intros.
  elim X0; intros.
  exists (x1 * x0).
  apply derivable_pt_lim_comp; assumption.
Qed.

Lemma derivable_plus :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2).
Proof.
  unfold derivable; intros f1 f2 X X0 x.
  apply (derivable_pt_plus _ _ x (X _) (X0 _)).
Qed.

Lemma derivable_opp : forall f, derivable f -> derivable (- f).
Proof.
  unfold derivable; intros f X x.
  apply (derivable_pt_opp _ x (X _)).
Qed.

Lemma derivable_minus :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2).
Proof.
  unfold derivable; intros f1 f2 X X0 x.
  apply (derivable_pt_minus _ _ x (X _) (X0 _)).
Qed.

Lemma derivable_mult :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2).
Proof.
  unfold derivable; intros f1 f2 X X0 x.
  apply (derivable_pt_mult _ _ x (X _) (X0 _)).
Qed.

Lemma derivable_const : forall a:R, derivable (fct_cte a).
Proof.
  unfold derivable; intros.
  apply derivable_pt_const.
Qed.

Lemma derivable_scal :
  forall f (a:R), derivable f -> derivable (mult_real_fct a f).
Proof.
  unfold derivable; intros f a X x.
  apply (derivable_pt_scal _ a x (X _)).
Qed.

Lemma derivable_id : derivable id.
Proof.
  unfold derivable; intro; apply derivable_pt_id.
Qed.

Lemma derivable_Rsqr : derivable Rsqr.
Proof.
  unfold derivable; intro; apply derivable_pt_Rsqr.
Qed.

Lemma derivable_comp :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1).
Proof.
  unfold derivable; intros f1 f2 X X0 x.
  apply (derivable_pt_comp _ _ x (X _) (X0 _)).
Qed.

Lemma derive_pt_plus :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
    derive_pt (f1 + f2) x (derivable_pt_plus _ _ _ pr1 pr2) =
    derive_pt f1 x pr1 + derive_pt f2 x pr2.
Proof.
  intros.
  assert (H := derivable_derive f1 x pr1).
  assert (H0 := derivable_derive f2 x pr2).
  assert
    (H1 := derivable_derive (f1 + f2)%F x (derivable_pt_plus _ _ _ pr1 pr2)).
  elim H; clear H; intros l1 H.
  elim H0; clear H0; intros l2 H0.
  elim H1; clear H1; intros l H1.
  rewrite H; rewrite H0; apply derive_pt_eq_0.
  assert (H3 := proj2_sig pr1).
  unfold derive_pt in H; rewrite H in H3.
  assert (H4 := proj2_sig pr2).
  unfold derive_pt in H0; rewrite H0 in H4.
  apply derivable_pt_lim_plus; assumption.
Qed.

Lemma derive_pt_opp :
  forall f (x:R) (pr1:derivable_pt f x),
    derive_pt (- f) x (derivable_pt_opp _ _ pr1) = - derive_pt f x pr1.
Proof.
  intros.
  assert (H := derivable_derive f x pr1).
  assert (H0 := derivable_derive (- f)%F x (derivable_pt_opp _ _ pr1)).
  elim H; clear H; intros l1 H.
  elim H0; clear H0; intros l2 H0.
  rewrite H; apply derive_pt_eq_0.
  assert (H3 := proj2_sig pr1).
  unfold derive_pt in H; rewrite H in H3.
  apply derivable_pt_lim_opp; assumption.
Qed.

Lemma derive_pt_minus :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
    derive_pt (f1 - f2) x (derivable_pt_minus _ _ _ pr1 pr2) =
    derive_pt f1 x pr1 - derive_pt f2 x pr2.
Proof.
  intros.
  assert (H := derivable_derive f1 x pr1).
  assert (H0 := derivable_derive f2 x pr2).
  assert
    (H1 := derivable_derive (f1 - f2)%F x (derivable_pt_minus _ _ _ pr1 pr2)).
  elim H; clear H; intros l1 H.
  elim H0; clear H0; intros l2 H0.
  elim H1; clear H1; intros l H1.
  rewrite H; rewrite H0; apply derive_pt_eq_0.
  assert (H3 := proj2_sig pr1).
  unfold derive_pt in H; rewrite H in H3.
  assert (H4 := proj2_sig pr2).
  unfold derive_pt in H0; rewrite H0 in H4.
  apply derivable_pt_lim_minus; assumption.
Qed.

Lemma derive_pt_mult :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
    derive_pt (f1 * f2) x (derivable_pt_mult _ _ _ pr1 pr2) =
    derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2.
Proof.
  intros.
  assert (H := derivable_derive f1 x pr1).
  assert (H0 := derivable_derive f2 x pr2).
  assert
    (H1 := derivable_derive (f1 * f2)%F x (derivable_pt_mult _ _ _ pr1 pr2)).
  elim H; clear H; intros l1 H.
  elim H0; clear H0; intros l2 H0.
  elim H1; clear H1; intros l H1.
  rewrite H; rewrite H0; apply derive_pt_eq_0.
  assert (H3 := proj2_sig pr1).
  unfold derive_pt in H; rewrite H in H3.
  assert (H4 := proj2_sig pr2).
  unfold derive_pt in H0; rewrite H0 in H4.
  apply derivable_pt_lim_mult; assumption.
Qed.

Lemma derive_pt_const :
  forall a x:R, derive_pt (fct_cte a) x (derivable_pt_const a x) = 0.
Proof.
  intros.
  apply derive_pt_eq_0.
  apply derivable_pt_lim_const.
Qed.

Lemma derive_pt_scal :
  forall f (a x:R) (pr:derivable_pt f x),
    derive_pt (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr) =
    a * derive_pt f x pr.
Proof.
  intros.
  assert (H := derivable_derive f x pr).
  assert
    (H0 := derivable_derive (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr)).
  elim H; clear H; intros l1 H.
  elim H0; clear H0; intros l2 H0.
  rewrite H; apply derive_pt_eq_0.
  assert (H3 := proj2_sig pr).
  unfold derive_pt in H; rewrite H in H3.
  apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derive_pt_id : forall x:R, derive_pt id x (derivable_pt_id _) = 1.
Proof.
  intros.
  apply derive_pt_eq_0.
  apply derivable_pt_lim_id.
Qed.

Lemma derive_pt_Rsqr :
  forall x:R, derive_pt Rsqr x (derivable_pt_Rsqr _) = 2 * x.
Proof.
  intros.
  apply derive_pt_eq_0.
  apply derivable_pt_lim_Rsqr.
Qed.

Lemma derive_pt_comp :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 (f1 x)),
    derive_pt (f2 o f1) x (derivable_pt_comp _ _ _ pr1 pr2) =
    derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1.
Proof.
  intros.
  assert (H := derivable_derive f1 x pr1).
  assert (H0 := derivable_derive f2 (f1 x) pr2).
  assert
    (H1 := derivable_derive (f2 o f1)%F x (derivable_pt_comp _ _ _ pr1 pr2)).
  elim H; clear H; intros l1 H.
  elim H0; clear H0; intros l2 H0.
  elim H1; clear H1; intros l H1.
  rewrite H; rewrite H0; apply derive_pt_eq_0.
  assert (H3 := proj2_sig pr1).
  unfold derive_pt in H; rewrite H in H3.
  assert (H4 := proj2_sig pr2).
  unfold derive_pt in H0; rewrite H0 in H4.
  apply derivable_pt_lim_comp; assumption.
Qed.

(* Pow *)
Definition pow_fct (n:nat) (y:R) : R := y ^ n.

Lemma derivable_pt_lim_pow_pos :
  forall (x:R) (n:nat),
    (0 < n)%nat -> derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).
Proof.
  intros.
  induction  n as [| n Hrecn].
  elim (lt_irrefl _ H).
  cut (n = 0%nat \/ (0 < n)%nat).
  intro; elim H0; intro.
  rewrite H1; simpl.
  replace (fun y:R => y * 1) with (id * fct_cte 1)%F.
  replace (1 * 1) with (1 * fct_cte 1 x + id x * 0).
  apply derivable_pt_lim_mult.
  apply derivable_pt_lim_id.
  apply derivable_pt_lim_const.
  unfold fct_cte, id; ring.
  reflexivity.
  replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n).
  replace (pred (S n)) with n; [ idtac | reflexivity ].
  replace (fun y:R => y * y ^ n) with (id * (fun y:R => y ^ n))%F.
  set (f := fun y:R => y ^ n).
  replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)).
  apply derivable_pt_lim_mult.
  apply derivable_pt_lim_id.
  unfold f; apply Hrecn; assumption.
  unfold f.
  pattern n at 1 5; replace n with (S (pred n)).
  unfold id; rewrite S_INR; simpl.
  ring.
  symmetry ; apply S_pred with 0%nat; assumption.
  unfold mult_fct, id; reflexivity.
  reflexivity.
  inversion H.
  left; reflexivity.
  right.
  apply lt_le_trans with 1%nat.
  apply lt_O_Sn.
  assumption.
Qed.

Lemma derivable_pt_lim_pow :
  forall (x:R) (n:nat),
    derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).
Proof.
  intros.
  induction  n as [| n Hrecn].
  simpl.
  rewrite Rmult_0_l.
  replace (fun _:R => 1) with (fct_cte 1);
  [ apply derivable_pt_lim_const | reflexivity ].
  apply derivable_pt_lim_pow_pos.
  apply lt_O_Sn.
Qed.

Lemma derivable_pt_pow :
  forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x.
Proof.
  intros; unfold derivable_pt.
  exists (INR n * x ^ pred n).
  apply derivable_pt_lim_pow.
Qed.

Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n).
Proof.
  intro; unfold derivable; intro; apply derivable_pt_pow.
Qed.

Lemma derive_pt_pow :
  forall (n:nat) (x:R),
    derive_pt (fun y:R => y ^ n) x (derivable_pt_pow n x) = INR n * x ^ pred n.
Proof.
  intros; apply derive_pt_eq_0.
  apply derivable_pt_lim_pow.
Qed.

Lemma pr_nu :
  forall f (x:R) (pr1 pr2:derivable_pt f x),
    derive_pt f x pr1 = derive_pt f x pr2.
Proof.
  intros f x (x0,H0) (x1,H1).
  apply (uniqueness_limite f x x0 x1 H0 H1).
Qed.


(************************************************************)
(** *           Local extremum's condition                  *)
(************************************************************)

Theorem deriv_maximum :
  forall f (a b c:R) (pr:derivable_pt f c),
    a < c ->
    c < b ->
    (forall x:R, a < x -> x < b -> f x <= f c) -> derive_pt f c pr = 0.
Proof.
  intros; case (Rtotal_order 0 (derive_pt f c pr)); intro.
  assert (H3 := derivable_derive f c pr).
  elim H3; intros l H4; rewrite H4 in H2.
  assert (H5 := derive_pt_eq_1 f c l pr H4).
  cut (0 < l / 2);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
        [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
  elim (H5 (l / 2) H6); intros delta H7.
  cut (0 < (b - c) / 2).
  intro; cut (Rmin (delta / 2) ((b - c) / 2) <> 0).
  intro; cut (Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta).
  intro.
  assert (H11 := H7 (Rmin (delta / 2) ((b - c) / 2)) H9 H10).
  cut (0 < Rmin (delta / 2) ((b - c) / 2)).
  intro; cut (a < c + Rmin (delta / 2) ((b - c) / 2)).
  intro; cut (c + Rmin (delta / 2) ((b - c) / 2) < b).
  intro; assert (H15 := H1 (c + Rmin (delta / 2) ((b - c) / 2)) H13 H14).
  cut
    ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
      Rmin (delta / 2) ((b - c) / 2) <= 0).
  intro; cut (- l < 0).
  intro; unfold Rminus in H11.
  cut
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2) + - l < 0).
  intro;
    cut
      (Rabs
        ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
          Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2).
  unfold Rabs;
    case
    (Rcase_abs
      ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
        Rmin (delta / 2) ((b + - c) / 2) + - l)) as [Hlt|Hge].
  replace
  (-
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2) + - l)) with
  (l +
    -
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2))).
  intro;
    generalize
      (Rplus_lt_compat_l (- l)
        (l +
          -
          ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
            Rmin (delta / 2) ((b + - c) / 2))) (l / 2) H19);
      repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
        rewrite Rplus_0_l; replace (- l + l / 2) with (- (l / 2)).
  intro;
    generalize
      (Ropp_lt_gt_contravar
        (-
          ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
            Rmin (delta / 2) ((b + - c) / 2))) (- (l / 2)) H20);
      repeat rewrite Ropp_involutive; intro;
        generalize
          (Rlt_trans 0 (l / 2)
            ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
              Rmin (delta / 2) ((b + - c) / 2)) H6 H21); intro;
          elim
            (Rlt_irrefl 0
              (Rlt_le_trans 0
                ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
                  Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)).
  pattern l at 2; rewrite double_var.
  ring.
  ring.
  intro.
  assert
    (H20 :=
      Rge_le
      ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
        Rmin (delta / 2) ((b + - c) / 2) + - l) 0 Hge).
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H20 H18)).
  assumption.
  rewrite <- Ropp_0;
    replace
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2) + - l) with
    (-
      (l +
        -
        ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
          Rmin (delta / 2) ((b + - c) / 2)))).
  apply Ropp_gt_lt_contravar;
    change
      (0 <
        l +
        -
        ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
          Rmin (delta / 2) ((b + - c) / 2))); apply Rplus_lt_le_0_compat;
      [ assumption
        | rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ].
  unfold Rminus; ring.
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
  replace
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
    Rmin (delta / 2) ((b - c) / 2)) with
  (-
    ((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
      Rmin (delta / 2) ((b - c) / 2))).
  rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge;
    unfold Rdiv; apply Rmult_le_pos;
      [ generalize
        (Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2)))
          (f (c + Rmin (delta * / 2) ((b - c) * / 2))) (
            f c) H15); rewrite Rplus_opp_r; intro; assumption
        | left; apply Rinv_0_lt_compat; assumption ].
  unfold Rdiv.
  rewrite <- Ropp_mult_distr_l_reverse.
  repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))).
  apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)).
  repeat rewrite <- Rmult_assoc.
  rewrite <- Rinv_r_sym.
  repeat rewrite Rmult_1_l.
  ring.
  red; intro.
  unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).
  red; intro.
  unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).
  assert (H14 := Rmin_r (delta / 2) ((b - c) / 2)).
  assert
    (H15 :=
      Rplus_le_compat_l c (Rmin (delta / 2) ((b - c) / 2)) ((b - c) / 2) H14).
  apply Rle_lt_trans with (c + (b - c) / 2).
  assumption.
  apply Rmult_lt_reg_l with 2.
  prove_sup0.
  replace (2 * (c + (b - c) / 2)) with (c + b).
  replace (2 * b) with (b + b).
  apply Rplus_lt_compat_r; assumption.
  ring.
  unfold Rdiv; rewrite Rmult_plus_distr_l.
  repeat rewrite (Rmult_comm 2).
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
  rewrite Rmult_1_r.
  ring.
  discrR.
  apply Rlt_trans with c.
  assumption.
  pattern c at 1; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l;
    assumption.
  cut (0 < delta / 2).
  intro;
    apply
      (Rmin_stable_in_posreal (mkposreal (delta / 2) H12)
        (mkposreal ((b - c) / 2) H8)).
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
  unfold Rabs; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))) as [Hlt|Hge].
  cut (0 < delta / 2).
  intro.
  generalize
    (Rmin_stable_in_posreal (mkposreal (delta / 2) H10)
      (mkposreal ((b - c) / 2) H8)); simpl; intro;
    elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 Hlt)).
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
  apply Rle_lt_trans with (delta / 2).
  apply Rmin_l.
  unfold Rdiv; apply Rmult_lt_reg_l with 2.
  prove_sup0.
  rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
  rewrite Rmult_1_l.
  replace (2 * delta) with (delta + delta).
  pattern delta at 2; rewrite <- (Rplus_0_r delta);
    apply Rplus_lt_compat_l.
  rewrite Rplus_0_r; apply (cond_pos delta).
  symmetry ; apply double.
  discrR.
  cut (0 < delta / 2).
  intro;
    generalize
      (Rmin_stable_in_posreal (mkposreal (delta / 2) H9)
        (mkposreal ((b - c) / 2) H8)); simpl;
      intro; red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
  unfold Rdiv; apply Rmult_lt_0_compat.
  generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro;
    assumption.
  apply Rinv_0_lt_compat; prove_sup0.
  elim H2; intro.
  symmetry ; assumption.
  generalize (derivable_derive f c pr); intro; elim H4; intros l H5.
  rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro;
    cut (0 < - (l / 2)).
  intro; elim (H6 (- (l / 2)) H7); intros delta H9.
  cut (0 < (c - a) / 2).
  intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) < 0).
  intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) <> 0).
  intro; cut (Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta).
  intro; generalize (H9 (Rmax (- (delta / 2)) ((a - c) / 2)) H11 H12); intro;
    cut (a < c + Rmax (- (delta / 2)) ((a - c) / 2)).
  cut (c + Rmax (- (delta / 2)) ((a - c) / 2) < b).
  intros; generalize (H1 (c + Rmax (- (delta / 2)) ((a - c) / 2)) H15 H14);
    intro;
      cut
        (0 <=
          (f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
          Rmax (- (delta / 2)) ((a - c) / 2)).
  intro; cut (0 < - l).
  intro; unfold Rminus in H13;
    cut
      (0 <
        (f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
        Rmax (- (delta / 2)) ((a + - c) / 2) + - l).
  intro;
    cut
      (Rabs
        ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
          Rmax (- (delta / 2)) ((a + - c) / 2) + - l) <
        - (l / 2)).
  unfold Rabs;
    case
    (Rcase_abs
      ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
        Rmax (- (delta / 2)) ((a + - c) / 2) + - l)) as [Hlt|Hge].
  elim
      (Rlt_irrefl 0
        (Rlt_trans 0
          ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
            Rmax (- (delta / 2)) ((a + - c) / 2) + - l) 0 H19 Hlt)).
  intros;
    generalize
      (Rplus_lt_compat_r l
        ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
          Rmax (- (delta / 2)) ((a + - c) / 2) + - l) (
            - (l / 2)) H20); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
      rewrite Rplus_0_r; replace (- (l / 2) + l) with (l / 2).
  cut (l / 2 < 0).
  intros;
    generalize
      (Rlt_trans
        ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
          Rmax (- (delta / 2)) ((a + - c) / 2)) (l / 2) 0 H22 H21);
      intro;
        elim
          (Rlt_irrefl 0
            (Rle_lt_trans 0
              ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
                Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)).
  rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0;
    apply Ropp_lt_gt_contravar; assumption.
  pattern l at 3; rewrite double_var.
  ring.
  assumption.
  apply Rplus_le_lt_0_compat; assumption.
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
  unfold Rdiv;
    replace
    ((f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
      / Rmax (- (delta * / 2)) ((a - c) * / 2)) with
    (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
      / - Rmax (- (delta * / 2)) ((a - c) * / 2)).
  apply Rmult_le_pos.
  generalize
    (Rplus_le_compat_l (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)))
      (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) (
        f c) H16); rewrite Rplus_opp_l;
    replace (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)) with
    (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c).
  intro; assumption.
  ring.
  left; apply Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar;
    assumption.
  unfold Rdiv.
  rewrite <- Ropp_inv_permute.
  rewrite Rmult_opp_opp.
  reflexivity.
  unfold Rdiv in H11; assumption.
  generalize (Rplus_lt_compat_l c (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H10);
    rewrite Rplus_0_r; intro; apply Rlt_trans with c;
      assumption.
  generalize (RmaxLess2 (- (delta / 2)) ((a - c) / 2)); intro;
    generalize
      (Rplus_le_compat_l c ((a - c) / 2) (Rmax (- (delta / 2)) ((a - c) / 2)) H14);
      intro; apply Rlt_le_trans with (c + (a - c) / 2).
  apply Rmult_lt_reg_l with 2.
  prove_sup0.
  replace (2 * (c + (a - c) / 2)) with (a + c).
  rewrite double.
  apply Rplus_lt_compat_l; assumption.
  field; discrR.
  assumption.
  unfold Rabs; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))) as [Hlt|Hge].
  generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro;
    generalize
      (Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2))
        H12); rewrite Ropp_involutive; intro;
      generalize (Rge_le (delta / 2) (- Rmax (- (delta / 2)) ((a - c) / 2)) H13);
        intro; apply Rle_lt_trans with (delta / 2).
  assumption.
  apply Rmult_lt_reg_l with 2.
  prove_sup0.
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
    rewrite <- Rinv_r_sym.
  rewrite Rmult_1_l; rewrite double.
  pattern delta at 2; rewrite <- (Rplus_0_r delta);
    apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta).
  discrR.
  cut (- (delta / 2) < 0).
  cut ((a - c) / 2 < 0).
  intros;
    generalize
      (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13)
        (mknegreal ((a - c) / 2) H12)); simpl;
      intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 Hge);
        intro;
          elim
            (Rlt_irrefl 0
              (Rle_lt_trans 0 (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H15 H14)).
  rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
    apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).
  assumption.
  unfold Rdiv.
  rewrite <- Ropp_mult_distr_l_reverse.
  rewrite (Ropp_minus_distr a c).
  reflexivity.
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv;
    apply Rmult_lt_0_compat;
      [ apply (cond_pos delta)
        | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
  red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
  cut ((a - c) / 2 < 0).
  intro; cut (- (delta / 2) < 0).
  intro;
    apply
      (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11)
        (mknegreal ((a - c) / 2) H10)).
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv;
    apply Rmult_lt_0_compat;
      [ apply (cond_pos delta)
        | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
  rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
    apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).
  assumption.
  unfold Rdiv.
  rewrite <- Ropp_mult_distr_l_reverse.
  rewrite (Ropp_minus_distr a c).
  reflexivity.
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro;
      assumption
      | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
  replace (- (l / 2)) with (- l / 2).
  unfold Rdiv; apply Rmult_lt_0_compat.
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
  assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ].
  unfold Rdiv; apply Ropp_mult_distr_l_reverse.
Qed.

Theorem deriv_minimum :
  forall f (a b c:R) (pr:derivable_pt f c),
    a < c ->
    c < b ->
    (forall x:R, a < x -> x < b -> f c <= f x) -> derive_pt f c pr = 0.
Proof.
  intros.
  rewrite <- (Ropp_involutive (derive_pt f c pr)).
  apply Ropp_eq_0_compat.
  rewrite <- (derive_pt_opp f c pr).
  cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c).
  intro.
  apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2).
  intros; unfold opp_fct; apply Ropp_ge_le_contravar; apply Rle_ge.
  apply (H1 x H2 H3).
Qed.

Theorem deriv_constant2 :
  forall f (a b c:R) (pr:derivable_pt f c),
    a < c ->
    c < b -> (forall x:R, a < x -> x < b -> f x = f c) -> derive_pt f c pr = 0.
Proof.
  intros.
  eapply deriv_maximum with a b; try assumption.
  intros; right; apply (H1 x H2 H3).
Qed.

(**********)
Lemma nonneg_derivative_0 :
  forall f (pr:derivable f),
    increasing f -> forall x:R, 0 <= derive_pt f x (pr x).
Proof.
  intros; unfold increasing in H.
  assert (H0 := derivable_derive f x (pr x)).
  elim H0; intros l H1.
  rewrite H1; case (Rtotal_order 0 l); intro.
  left; assumption.
  elim H2; intro.
  right; assumption.
  assert (H4 := derive_pt_eq_1 f x l (pr x) H1).
  cut (0 < - (l / 2)).
  intro; elim (H4 (- (l / 2)) H5); intros delta H6.
  cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta).
  intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11);
    cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)).
  intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l).
  intro; unfold Rabs;
    case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)) as [Hlt|Hge].
  elim
      (Rlt_irrefl 0
        (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 H12 Hlt)).
  intros;
    generalize
      (Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l)
        (- (l / 2)) H13); unfold Rminus;
      replace (- (l / 2) + l) with (l / 2).
  rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro;
    generalize
      (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) (l / 2) H9 H14);
      intro; cut (l / 2 < 0).
  intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (l / 2) 0 H15 H16)).
  rewrite <- Ropp_0 in H5;
    generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5);
      repeat rewrite Ropp_involutive; intro; assumption.
  pattern l at 3; rewrite double_var.
  ring.
  unfold Rminus; apply Rplus_le_le_0_compat.
  unfold Rdiv; apply Rmult_le_pos.
  cut (x <= x + delta * / 2).
  intro; generalize (H x (x + delta * / 2) H12); intro;
    generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13);
      rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
    left; assumption.
  left; apply Rinv_0_lt_compat; assumption.
  left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
  unfold Rdiv; apply Rmult_le_pos.
  cut (x <= x + delta * / 2).
  intro; generalize (H x (x + delta * / 2) H9); intro;
    generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12);
      rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
    left; assumption.
  left; apply Rinv_0_lt_compat; assumption.
  split.
  unfold Rdiv; apply prod_neq_R0.
  generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H7;
    elim (Rlt_irrefl 0 H7).
  apply Rinv_neq_0_compat; discrR.
  split.
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
  replace (Rabs (delta / 2)) with (delta / 2).
  unfold Rdiv; apply Rmult_lt_reg_l with 2.
  prove_sup0.
  rewrite (Rmult_comm 2).
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
  rewrite Rmult_1_r.
  rewrite double.
  pattern (pos delta) at 1; rewrite <- Rplus_0_r.
  apply Rplus_lt_compat_l; apply (cond_pos delta).
  symmetry ; apply Rabs_right.
  left; change (0 < delta / 2); unfold Rdiv;
    apply Rmult_lt_0_compat;
      [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
  unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse;
    apply Rmult_lt_0_compat.
  apply Rplus_lt_reg_l with l.
  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption.
  apply Rinv_0_lt_compat; prove_sup0.
Qed.