aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories7/Reals/Rtrigo.v
blob: d42b6e8ab0ba033661b61fff73e3d0890f672c2c (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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*i $Id$ i*)

Require Rbase.
Require Rfunctions.
Require SeqSeries.
Require Export Rtrigo_fun.
Require Export Rtrigo_def.
Require Export Rtrigo_alt.
Require Export Cos_rel.
Require Export Cos_plus.
Require ZArith_base.
Require Zcomplements.
Require Classical_Prop.
V7only [Import nat_scope. Import Z_scope. Import R_scope.].
Open Local Scope nat_scope.
Open Local Scope R_scope.

(** sin_PI2 is the only remaining axiom **)
Axiom sin_PI2 : ``(sin (PI/2))==1``.

(**********)
Lemma PI_neq0 : ~``PI==0``.
Red; Intro; Assert H0 := PI_RGT_0; Rewrite H in H0; Elim (Rlt_antirefl ? H0).
Qed.

(**********) 
Lemma cos_minus : (x,y:R) ``(cos (x-y))==(cos x)*(cos y)+(sin x)*(sin y)``.
Intros; Unfold Rminus; Rewrite cos_plus.
Rewrite <- cos_sym; Rewrite sin_antisym; Ring.
Qed.

(**********)
Lemma sin2_cos2 : (x:R) ``(Rsqr (sin x)) + (Rsqr (cos x))==1``.
Intro; Unfold Rsqr; Rewrite Rplus_sym; Rewrite <- (cos_minus x x); Unfold Rminus; Rewrite Rplus_Ropp_r; Apply cos_0.
Qed.

Lemma cos2 : (x:R) ``(Rsqr (cos x))==1-(Rsqr (sin x))``.
Intro x; Generalize (sin2_cos2 x); Intro H1; Rewrite <- H1; Unfold Rminus; Rewrite <- (Rplus_sym (Rsqr (cos x))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Symmetry; Apply Rplus_Or.
Qed.

(**********)
Lemma cos_PI2 : ``(cos (PI/2))==0``.
Apply Rsqr_eq_0; Rewrite cos2; Rewrite sin_PI2; Rewrite Rsqr_1; Unfold Rminus; Apply Rplus_Ropp_r.
Qed.

(**********)
Lemma cos_PI : ``(cos PI)==-1``.
Replace ``PI`` with ``PI/2+PI/2``.
Rewrite cos_plus.
Rewrite sin_PI2; Rewrite cos_PI2.
Ring.
Symmetry; Apply double_var.
Qed.

Lemma sin_PI : ``(sin PI)==0``.
Assert H := (sin2_cos2 PI).
Rewrite cos_PI in H.
Rewrite <- Rsqr_neg in H.
Rewrite Rsqr_1 in H.
Cut (Rsqr (sin PI))==R0.
Intro; Apply (Rsqr_eq_0 ? H0).
Apply r_Rplus_plus with R1.
Rewrite Rplus_Or; Rewrite Rplus_sym; Exact H.
Qed.

(**********)
Lemma neg_cos : (x:R) ``(cos (x+PI))==-(cos x)``.
Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring.
Qed.

(**********)
Lemma sin_cos : (x:R) ``(sin x)==-(cos (PI/2+x))``.
Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring.
Qed.

(**********)
Lemma sin_plus : (x,y:R) ``(sin (x+y))==(sin x)*(cos y)+(cos x)*(sin y)``.
Intros.
Rewrite (sin_cos ``x+y``).
Replace ``PI/2+(x+y)`` with ``(PI/2+x)+y``; [Rewrite cos_plus | Ring].
Rewrite (sin_cos ``PI/2+x``).
Replace ``PI/2+(PI/2+x)`` with ``x+PI``.
Rewrite neg_cos.
Replace (cos ``PI/2+x``) with ``-(sin x)``.
Ring.
Rewrite sin_cos; Rewrite Ropp_Ropp; Reflexivity.
Pattern 1 PI; Rewrite (double_var PI); Ring.
Qed.

Lemma sin_minus : (x,y:R) ``(sin (x-y))==(sin x)*(cos y)-(cos x)*(sin y)``.
Intros; Unfold Rminus; Rewrite sin_plus.
Rewrite <- cos_sym; Rewrite sin_antisym; Ring.
Qed.

(**********)
Definition tan [x:R] : R := ``(sin x)/(cos x)``.

Lemma tan_plus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x+y))==0`` -> ~``1-(tan x)*(tan y)==0`` -> ``(tan (x+y))==((tan x)+(tan y))/(1-(tan x)*(tan y))``.
Intros; Unfold tan; Rewrite sin_plus; Rewrite cos_plus; Unfold Rdiv; Replace ``((cos x)*(cos y)-(sin x)*(sin y))`` with ``((cos x)*(cos y))*(1-(sin x)*/(cos x)*((sin y)*/(cos y)))``.
Rewrite Rinv_Rmult.
Repeat Rewrite <- Rmult_assoc; Replace ``((sin x)*(cos y)+(cos x)*(sin y))*/((cos x)*(cos y))`` with ``((sin x)*/(cos x)+(sin y)*/(cos y))``.
Reflexivity.
Rewrite Rmult_Rplus_distrl; Rewrite Rinv_Rmult.
Repeat Rewrite Rmult_assoc; Repeat Rewrite (Rmult_sym ``(sin x)``); Repeat Rewrite <- Rmult_assoc.
Repeat Rewrite Rinv_r_simpl_m; [Reflexivity | Assumption | Assumption].
Assumption.
Assumption.
Apply prod_neq_R0; Assumption.
Assumption.
Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Apply Rplus_plus_r; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``(sin x)``); Rewrite (Rmult_sym ``(cos y)``); Rewrite <- Ropp_mul3; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym.
Rewrite Rmult_1l; Rewrite (Rmult_sym (sin x)); Rewrite <- Ropp_mul3; Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r; Rewrite (Rmult_sym ``/(cos y)``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym.
Apply Rmult_1r.
Assumption.
Assumption.
Qed.

(*******************************************************)
(* Some properties of cos, sin and tan                 *)
(*******************************************************)

Lemma sin2 : (x:R) ``(Rsqr (sin x))==1-(Rsqr (cos x))``.
Intro x; Generalize (cos2 x); Intro H1; Rewrite -> H1.
Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Symmetry; Apply Ropp_Ropp.
Qed.

Lemma sin_2a : (x:R) ``(sin (2*x))==2*(sin x)*(cos x)``.
Intro x; Rewrite double; Rewrite sin_plus.
Rewrite <- (Rmult_sym (sin x)); Symmetry; Rewrite Rmult_assoc; Apply double.
Qed.

Lemma cos_2a : (x:R) ``(cos (2*x))==(cos x)*(cos x)-(sin x)*(sin x)``.
Intro x; Rewrite double; Apply cos_plus.
Qed.

Lemma cos_2a_cos : (x:R) ``(cos (2*x))==2*(cos x)*(cos x)-1``.
Intro x; Rewrite double; Unfold Rminus; Rewrite Rmult_assoc; Rewrite cos_plus; Generalize (sin2_cos2 x); Rewrite double; Intro H1; Rewrite <- H1; SqRing.
Qed.

Lemma cos_2a_sin : (x:R) ``(cos (2*x))==1-2*(sin x)*(sin x)``.
Intro x; Rewrite Rmult_assoc; Unfold Rminus; Repeat Rewrite double.
Generalize (sin2_cos2 x); Intro H1; Rewrite <- H1; Rewrite cos_plus; SqRing.
Qed.

Lemma tan_2a : (x:R) ~``(cos x)==0`` -> ~``(cos (2*x))==0`` -> ~``1-(tan x)*(tan x)==0`` ->``(tan (2*x))==(2*(tan x))/(1-(tan x)*(tan x))``.
Repeat Rewrite double; Intros; Repeat Rewrite double; Rewrite double in H0; Apply tan_plus; Assumption.
Qed.

Lemma sin_neg : (x:R) ``(sin (-x))==-(sin x)``.
Apply sin_antisym.
Qed.

Lemma cos_neg : (x:R) ``(cos (-x))==(cos x)``.
Intro; Symmetry; Apply cos_sym.
Qed.

Lemma tan_0 : ``(tan 0)==0``.
Unfold tan; Rewrite -> sin_0; Rewrite -> cos_0.
Unfold Rdiv; Apply Rmult_Ol. 
Qed.

Lemma tan_neg : (x:R) ``(tan (-x))==-(tan x)``.
Intros x; Unfold tan; Rewrite sin_neg; Rewrite cos_neg; Unfold Rdiv.
Apply Ropp_mul1.
Qed.

Lemma tan_minus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x-y))==0`` -> ~``1+(tan x)*(tan y)==0`` -> ``(tan (x-y))==((tan x)-(tan y))/(1+(tan x)*(tan y))``.
Intros; Unfold Rminus; Rewrite tan_plus.
Rewrite tan_neg; Unfold Rminus; Rewrite <- Ropp_mul1; Rewrite Ropp_mul2; Reflexivity.
Assumption.
Rewrite cos_neg; Assumption.
Assumption.
Rewrite tan_neg; Unfold Rminus; Rewrite <- Ropp_mul1; Rewrite Ropp_mul2; Assumption.
Qed.

Lemma cos_3PI2 : ``(cos (3*(PI/2)))==0``.
Replace ``3*(PI/2)`` with ``PI+(PI/2)``.
Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI2; Ring.
Pattern 1 PI; Rewrite (double_var PI).
Ring.
Qed.

Lemma sin_2PI : ``(sin (2*PI))==0``.
Rewrite -> sin_2a; Rewrite -> sin_PI; Ring.
Qed.

Lemma cos_2PI : ``(cos (2*PI))==1``.
Rewrite -> cos_2a; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring.
Qed.

Lemma neg_sin : (x:R) ``(sin (x+PI))==-(sin x)``.
Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring.
Qed.

Lemma sin_PI_x : (x:R) ``(sin (PI-x))==(sin x)``.
Intro x; Rewrite -> sin_minus; Rewrite -> sin_PI; Rewrite -> cos_PI; Rewrite Rmult_Ol; Unfold Rminus; Rewrite Rplus_Ol; Rewrite Ropp_mul1; Rewrite Ropp_Ropp; Apply Rmult_1l.
Qed.

Lemma sin_period : (x:R)(k:nat) ``(sin (x+2*(INR k)*PI))==(sin x)``.
Intros x k; Induction k.
Cut ``x+2*(INR O)*PI==x``; [Intro; Rewrite H; Reflexivity | Ring].
Replace ``x+2*(INR (S k))*PI`` with ``(x+2*(INR k)*PI)+(2*PI)``; [Rewrite -> sin_plus; Rewrite -> sin_2PI; Rewrite -> cos_2PI; Ring; Apply Hreck | Rewrite -> S_INR; Ring].
Qed.

Lemma cos_period : (x:R)(k:nat) ``(cos (x+2*(INR k)*PI))==(cos x)``.
Intros x k; Induction k.
Cut ``x+2*(INR O)*PI==x``; [Intro; Rewrite H; Reflexivity | Ring].
Replace ``x+2*(INR (S k))*PI`` with ``(x+2*(INR k)*PI)+(2*PI)``; [Rewrite -> cos_plus; Rewrite -> sin_2PI; Rewrite -> cos_2PI; Ring; Apply Hreck | Rewrite -> S_INR; Ring].
Qed.

Lemma sin_shift : (x:R) ``(sin (PI/2-x))==(cos x)``.
Intro x; Rewrite -> sin_minus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring.
Qed.

Lemma cos_shift : (x:R) ``(cos (PI/2-x))==(sin x)``.
Intro x; Rewrite -> cos_minus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring.
Qed.

Lemma cos_sin : (x:R) ``(cos x)==(sin (PI/2+x))``.
Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring.
Qed.

Lemma PI2_RGT_0 : ``0<PI/2``.
Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup].
Qed. 

Lemma SIN_bound : (x:R) ``-1<=(sin x)<=1``.
Intro; Case (total_order_Rle ``-1`` (sin x)); Intro.
Case (total_order_Rle (sin x) ``1``); Intro.
Split; Assumption.
Cut ``1<(sin x)``.
Intro; Generalize (Rsqr_incrst_1 ``1`` (sin x) H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` (sin x) (Rlt_trans ``0`` ``1`` (sin x) Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite sin2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (cos x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (cos x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (cos x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (cos x)) ``0`` H3 H2)).
Auto with real.
Cut ``(sin x)< -1``.
Intro; Generalize (Rlt_Ropp (sin x) ``-1`` H); Rewrite Ropp_Ropp; Clear H; Intro; Generalize (Rsqr_incrst_1 ``1`` ``-(sin x)`` H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` ``-(sin x)`` (Rlt_trans ``0`` ``1`` ``-(sin x)`` Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite <- Rsqr_neg in H0; Rewrite sin2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (cos x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (cos x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (cos x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (cos x)) ``0`` H3 H2)).
Auto with real.
Qed.

Lemma COS_bound : (x:R) ``-1<=(cos x)<=1``.
Intro; Rewrite <- sin_shift; Apply SIN_bound.
Qed.

Lemma cos_sin_0 : (x:R) ~(``(cos x)==0``/\``(sin x)==0``).
Intro; Red; Intro; Elim H; Intros; Generalize (sin2_cos2 x); Intro; Rewrite H0 in H2; Rewrite H1 in H2; Repeat Rewrite Rsqr_O in H2; Rewrite Rplus_Or in H2; Generalize Rlt_R0_R1; Intro; Rewrite <- H2 in H3; Elim (Rlt_antirefl ``0`` H3).
Qed.
  
Lemma cos_sin_0_var : (x:R) ~``(cos x)==0``\/~``(sin x)==0``.
Intro; Apply not_and_or; Apply cos_sin_0.
Qed.

(*****************************************************************)
(* Using series definitions of cos and sin                       *)
(*****************************************************************)

Definition sin_lb [a:R] : R := (sin_approx a (3)).
Definition sin_ub [a:R] : R := (sin_approx a (4)).
Definition cos_lb [a:R] : R := (cos_approx a (3)).
Definition cos_ub [a:R] : R := (cos_approx a (4)).

Lemma sin_lb_gt_0 : (a:R) ``0<a``->``a<=PI/2``->``0<(sin_lb a)``.
Intros.
Unfold sin_lb; Unfold sin_approx; Unfold sin_term.
Pose Un := [i:nat]``(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))``.
Replace (sum_f_R0 [i:nat] ``(pow ( -1) i)*(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))`` (S (S (S O)))) with (sum_f_R0 [i:nat]``(pow (-1) i)*(Un i)`` (3)); [Idtac | Apply sum_eq; Intros; Unfold Un; Reflexivity].
Cut (n:nat)``(Un (S n))<(Un n)``.
Intro; Simpl.
Repeat Rewrite Rmult_1l; Repeat Rewrite Rmult_1r; Replace ``-1*(Un (S O))`` with ``-(Un (S O))``; [Idtac | Ring]; Replace ``-1* -1*(Un (S (S O)))`` with ``(Un (S (S O)))``; [Idtac | Ring]; Replace ``-1*( -1* -1)*(Un (S (S (S O))))`` with ``-(Un (S (S (S O))))``; [Idtac | Ring]; Replace ``(Un O)+ -(Un (S O))+(Un (S (S O)))+ -(Un (S (S (S O))))`` with ``((Un O)-(Un (S O)))+((Un (S (S O)))-(Un (S (S (S O)))))``; [Idtac | Ring].
Apply gt0_plus_gt0_is_gt0.
Unfold Rminus; Apply Rlt_anti_compatibility with (Un (S O)); Rewrite Rplus_Or; Rewrite (Rplus_sym (Un (S O))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H1.
Unfold Rminus; Apply Rlt_anti_compatibility with (Un (S (S (S O)))); Rewrite Rplus_Or; Rewrite (Rplus_sym (Un (S (S (S O))))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H1.
Intro; Unfold Un.
Cut (plus (mult (2) (S n)) (S O)) = (plus (plus (mult (2) n) (S O)) (2)).
Intro; Rewrite H1.
Rewrite pow_add; Unfold Rdiv; Rewrite Rmult_assoc; Apply Rlt_monotony.
Apply pow_lt; Assumption.
Rewrite <- H1; Apply Rlt_monotony_contra with (INR (fact (plus (mult (S (S O)) n) (S O)))).
Apply lt_INR_0; Apply neq_O_lt.
Assert H2 := (fact_neq_0 (plus (mult (2) n) (1))).
Red; Intro; Elim H2; Symmetry; Assumption.
Rewrite <- Rinv_r_sym.
Apply Rlt_monotony_contra with (INR (fact (plus (mult (S (S O)) (S n)) (S O)))).
Apply lt_INR_0; Apply neq_O_lt.
Assert H2 := (fact_neq_0 (plus (mult (2) (S n)) (1))).
Red; Intro; Elim H2; Symmetry; Assumption.
Rewrite (Rmult_sym (INR (fact (plus (mult (S (S O)) (S n)) (S O))))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym.
Do 2 Rewrite Rmult_1r; Apply Rle_lt_trans with ``(INR (fact (plus (mult (S (S O)) n) (S O))))*4``.
Apply Rle_monotony.
Replace R0 with (INR O); [Idtac | Reflexivity]; Apply le_INR; Apply le_O_n.
Simpl; Rewrite Rmult_1r; Replace ``4`` with ``(Rsqr 2)``; [Idtac | SqRing]; Replace ``a*a`` with (Rsqr a); [Idtac | Reflexivity]; Apply Rsqr_incr_1.
Apply Rle_trans with ``PI/2``; [Assumption | Unfold Rdiv; Apply Rle_monotony_contra with ``2``; [ Sup0 | Rewrite <- Rmult_assoc;  Rewrite Rinv_r_simpl_m; [Replace ``2*2`` with ``4``; [Apply PI_4 | Ring] | DiscrR]]].
Left; Assumption.
Left; Sup0.
Rewrite H1; Replace (plus (plus (mult (S (S O)) n) (S O)) (S (S O))) with (S (S (plus (mult (S (S O)) n) (S O)))).
Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR.
Repeat Rewrite <- Rmult_assoc.
Rewrite <- (Rmult_sym (INR (fact (plus (mult (S (S O)) n) (S O))))).
Rewrite Rmult_assoc.
Apply Rlt_monotony.
Apply lt_INR_0; Apply neq_O_lt.
Assert H2 := (fact_neq_0 (plus (mult (2) n) (1))).
Red; Intro; Elim H2; Symmetry; Assumption.
Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Pose x := (INR n); Unfold INR.
Replace ``(2*x+1+1+1)*(2*x+1+1)`` with ``4*x*x+10*x+6``; [Idtac | Ring].
Apply Rlt_anti_compatibility with ``-4``; Rewrite Rplus_Ropp_l; Replace ``-4+(4*x*x+10*x+6)`` with ``(4*x*x+10*x)+2``; [Idtac | Ring].
Apply ge0_plus_gt0_is_gt0.
Cut ``0<=x``.
Intro; Apply ge0_plus_ge0_is_ge0; Repeat Apply Rmult_le_pos; Assumption Orelse Left; Sup.
Unfold x; Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity].
Sup0.
Apply INR_eq; Do 2 Rewrite S_INR; Do 3 Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring.
Apply INR_fact_neq_0.
Apply INR_fact_neq_0.
Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring.
Qed.

Lemma SIN : (a:R) ``0<=a`` -> ``a<=PI`` -> ``(sin_lb a)<=(sin a)<=(sin_ub a)``.
Intros; Unfold sin_lb sin_ub; Apply (sin_bound a (S O) H H0).
Qed.

Lemma COS : (a:R) ``-PI/2<=a`` -> ``a<=PI/2`` -> ``(cos_lb a)<=(cos a)<=(cos_ub a)``.
Intros; Unfold cos_lb cos_ub; Apply (cos_bound a (S O) H H0).
Qed.

(**********)
Lemma _PI2_RLT_0 : ``-(PI/2)<0``.
Rewrite <- Ropp_O; Apply Rlt_Ropp1; Apply PI2_RGT_0.
Qed.

Lemma PI4_RLT_PI2 : ``PI/4<PI/2``.
Unfold Rdiv; Apply Rlt_monotony.
Apply PI_RGT_0.
Apply Rinv_lt.
Apply Rmult_lt_pos; Sup0.
Pattern 1 ``2``; Rewrite <- Rplus_Or.
Replace ``4`` with ``2+2``; [Apply Rlt_compatibility; Sup0 | Ring].
Qed.

Lemma PI2_Rlt_PI : ``PI/2<PI``.
Unfold Rdiv; Pattern 2 PI; Rewrite <- Rmult_1r.
Apply Rlt_monotony.
Apply PI_RGT_0.
Pattern 3 R1; Rewrite <- Rinv_R1; Apply Rinv_lt.
Rewrite Rmult_1l; Sup0.
Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1.
Qed.

(********************************************)
(* Increasing and decreasing of COS and SIN *)
(********************************************)
Theorem sin_gt_0 : (x:R) ``0<x`` -> ``x<PI`` -> ``0<(sin x)``.
Intros; Elim (SIN x (Rlt_le R0 x H) (Rlt_le x PI H0)); Intros H1 _; Case (total_order x ``PI/2``); Intro H2.
Apply Rlt_le_trans with (sin_lb x).
Apply sin_lb_gt_0; [Assumption | Left; Assumption].
Assumption.
Elim H2; Intro H3.
Rewrite H3; Rewrite sin_PI2; Apply Rlt_R0_R1.
Rewrite <- sin_PI_x; Generalize (Rgt_Ropp x ``PI/2`` H3); Intro H4; Generalize (Rlt_compatibility PI (Ropp x) (Ropp ``PI/2``) H4).
Replace ``PI+(-x)`` with ``PI-x``.
Replace ``PI+ -(PI/2)`` with ``PI/2``.
Intro H5; Generalize (Rlt_Ropp x PI H0); Intro H6; Change ``-PI < -x`` in H6; Generalize (Rlt_compatibility PI (Ropp PI) (Ropp x) H6).
Rewrite Rplus_Ropp_r.
Replace ``PI+ -x`` with ``PI-x``.
Intro H7; Elim (SIN ``PI-x`` (Rlt_le R0 ``PI-x`` H7) (Rlt_le ``PI-x`` PI (Rlt_trans ``PI-x`` ``PI/2`` ``PI`` H5 PI2_Rlt_PI))); Intros H8 _; Generalize (sin_lb_gt_0 ``PI-x`` H7 (Rlt_le ``PI-x`` ``PI/2`` H5)); Intro H9; Apply (Rlt_le_trans ``0`` ``(sin_lb (PI-x))`` ``(sin (PI-x))`` H9 H8).
Reflexivity.
Pattern 2 PI; Rewrite double_var; Ring.
Reflexivity.
Qed.

Theorem cos_gt_0 : (x:R) ``-(PI/2)<x`` -> ``x<PI/2`` -> ``0<(cos x)``.
Intros; Rewrite cos_sin; Generalize (Rlt_compatibility ``PI/2`` ``-(PI/2)`` x H).
Rewrite Rplus_Ropp_r; Intro H1; Generalize (Rlt_compatibility ``PI/2`` x ``PI/2`` H0); Rewrite <- double_var; Intro H2; Apply (sin_gt_0 ``PI/2+x`` H1 H2).
Qed.

Lemma sin_ge_0 : (x:R) ``0<=x`` -> ``x<=PI`` -> ``0<=(sin x)``.
Intros x H1 H2; Elim H1; Intro H3; [ Elim H2; Intro H4; [ Left ; Apply (sin_gt_0 x H3 H4) | Rewrite H4; Right; Symmetry; Apply sin_PI ] | Rewrite <- H3; Right; Symmetry; Apply sin_0].
Qed.

Lemma cos_ge_0 : (x:R) ``-(PI/2)<=x`` -> ``x<=PI/2`` -> ``0<=(cos x)``.
Intros x H1 H2; Elim H1; Intro H3; [ Elim H2; Intro H4; [ Left ; Apply (cos_gt_0 x H3 H4) | Rewrite H4; Right; Symmetry; Apply cos_PI2 ] | Rewrite <- H3; Rewrite cos_neg; Right; Symmetry; Apply cos_PI2 ].
Qed.

Lemma sin_le_0 : (x:R) ``PI<=x`` -> ``x<=2*PI`` -> ``(sin x)<=0``.
Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rle_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_ge_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring].
Qed.

Lemma cos_le_0 : (x:R) ``PI/2<=x``->``x<=3*(PI/2)``->``(cos x)<=0``.
Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rle_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``.
Rewrite cos_period; Apply cos_ge_0.
Replace ``-(PI/2)`` with ``-PI+(PI/2)``.
Unfold Rminus; Rewrite (Rplus_sym x); Apply Rle_compatibility; Assumption.
Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring.
Unfold Rminus; Rewrite Rplus_sym; Replace ``PI/2`` with ``(-PI)+3*(PI/2)``.
Apply Rle_compatibility; Assumption.
Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. 
Unfold INR; Ring.
Qed.

Lemma sin_lt_0 : (x:R) ``PI<x`` -> ``x<2*PI`` -> ``(sin x)<0``.
Intros x H1 H2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rlt_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_gt_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rlt_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rlt_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring].
Qed.

Lemma sin_lt_0_var : (x:R) ``-PI<x`` -> ``x<0`` -> ``(sin x)<0``.
Intros; Generalize (Rlt_compatibility ``2*PI`` ``-PI`` x H); Replace ``2*PI+(-PI)`` with ``PI``; [Intro H1; Rewrite Rplus_sym in H1; Generalize (Rlt_compatibility ``2*PI`` x ``0`` H0); Intro H2; Rewrite (Rplus_sym ``2*PI``) in H2; Rewrite <- (Rplus_sym R0) in H2; Rewrite Rplus_Ol in H2; Rewrite <- (sin_period x (1)); Unfold INR; Replace ``2*1*PI`` with ``2*PI``; [Apply (sin_lt_0 ``x+2*PI`` H1 H2) | Ring] | Ring].
Qed.

Lemma cos_lt_0 : (x:R) ``PI/2<x`` -> ``x<3*(PI/2)``-> ``(cos x)<0``.
Intros x H1 H2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rlt_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``.
Rewrite cos_period; Apply cos_gt_0.
Replace ``-(PI/2)`` with ``-PI+(PI/2)``.
Unfold Rminus; Rewrite (Rplus_sym x); Apply Rlt_compatibility; Assumption.
Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. 
Unfold Rminus; Rewrite Rplus_sym; Replace ``PI/2`` with ``(-PI)+3*(PI/2)``.
Apply Rlt_compatibility; Assumption.
Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. 
Unfold INR; Ring.
Qed.

Lemma tan_gt_0 : (x:R) ``0<x`` -> ``x<PI/2`` -> ``0<(tan x)``.
Intros x H1 H2; Unfold tan; Generalize _PI2_RLT_0; Generalize (Rlt_trans R0 x ``PI/2`` H1 H2); Intros; Generalize (Rlt_trans ``-(PI/2)`` R0 x H0 H1); Intro H5; Generalize (Rlt_trans x ``PI/2`` PI H2 PI2_Rlt_PI); Intro H7; Unfold Rdiv;  Apply Rmult_lt_pos.
Apply sin_gt_0; Assumption.
Apply Rlt_Rinv; Apply cos_gt_0; Assumption.
Qed.

Lemma tan_lt_0 : (x:R) ``-(PI/2)<x``->``x<0``->``(tan x)<0``.
Intros x H1 H2; Unfold tan; Generalize (cos_gt_0 x H1 (Rlt_trans x ``0`` ``PI/2`` H2 PI2_RGT_0)); Intro H3; Rewrite <- Ropp_O; Replace ``(sin x)/(cos x)`` with ``- ((-(sin x))/(cos x))``.
Rewrite <- sin_neg; Apply Rgt_Ropp; Change ``0<(sin (-x))/(cos x)``; Unfold Rdiv; Apply Rmult_lt_pos.
Apply sin_gt_0.
Rewrite <- Ropp_O; Apply Rgt_Ropp; Assumption.
Apply Rlt_trans with ``PI/2``.
Rewrite <- (Ropp_Ropp ``PI/2``); Apply Rgt_Ropp; Assumption.
Apply PI2_Rlt_PI.
Apply Rlt_Rinv; Assumption.
Unfold Rdiv; Ring.
Qed.

Lemma cos_ge_0_3PI2 : (x:R) ``3*(PI/2)<=x``->``x<=2*PI``->``0<=(cos x)``.
Intros; Rewrite <- cos_neg; Rewrite <- (cos_period ``-x`` (1)); Unfold INR; Replace ``-x+2*1*PI`` with ``2*PI-x``.
Generalize (Rle_Ropp x ``2*PI`` H0); Intro H1; Generalize (Rle_sym2 ``-(2*PI)`` ``-x`` H1); Clear H1; Intro H1; Generalize (Rle_compatibility ``2*PI`` ``-(2*PI)`` ``-x`` H1).
Rewrite Rplus_Ropp_r. 
Intro H2; Generalize (Rle_Ropp ``3*(PI/2)`` x H); Intro H3; Generalize (Rle_sym2 ``-x`` ``-(3*(PI/2))`` H3); Clear H3; Intro H3;  Generalize (Rle_compatibility ``2*PI`` ``-x`` ``-(3*(PI/2))`` H3).
Replace ``2*PI+ -(3*PI/2)`` with ``PI/2``.
Intro H4; Apply (cos_ge_0 ``2*PI-x`` (Rlt_le ``-(PI/2)`` ``2*PI-x`` (Rlt_le_trans ``-(PI/2)`` ``0`` ``2*PI-x`` _PI2_RLT_0 H2)) H4).
Rewrite double; Pattern 2 3 PI; Rewrite double_var; Ring.
Ring.
Qed.

Lemma form1 : (p,q:R) ``(cos p)+(cos q)==2*(cos ((p-q)/2))*(cos ((p+q)/2))``.
Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``.
Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``.
Rewrite cos_plus; Rewrite cos_minus; Ring.
Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring.
Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring.
Qed.

Lemma form2 : (p,q:R) ``(cos p)-(cos q)==-2*(sin ((p-q)/2))*(sin ((p+q)/2))``.
Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``.
Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``.
Rewrite cos_plus; Rewrite cos_minus; Ring.
Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring.
Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring.
Qed.

Lemma form3 : (p,q:R) ``(sin p)+(sin q)==2*(cos ((p-q)/2))*(sin ((p+q)/2))``.
Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``.
Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``.
Rewrite sin_plus; Rewrite sin_minus; Ring.
Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring.
Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring.
Qed.

Lemma form4 : (p,q:R) ``(sin p)-(sin q)==2*(cos ((p+q)/2))*(sin ((p-q)/2))``.
Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``.
Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``.
Rewrite sin_plus; Rewrite sin_minus; Ring.
Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring.
Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring.

Qed.

Lemma sin_increasing_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<(sin y)``->``x<y``.
Intros; Cut ``(sin ((x-y)/2))<0``.
Intro H4; Case (total_order ``(x-y)/2`` ``0``); Intro H5.
Assert Hyp : ``0<2``.
Sup0.
Generalize (Rlt_monotony ``2`` ``(x-y)/2`` ``0`` Hyp H5).
Unfold Rdiv.
Rewrite <- Rmult_assoc.
Rewrite Rinv_r_simpl_m.
Rewrite Rmult_Or.
Clear H5; Intro H5; Apply Rminus_lt; Assumption.
DiscrR.
Elim H5; Intro H6.
Rewrite H6 in H4; Rewrite sin_0 in H4; Elim (Rlt_antirefl ``0`` H4).
Change ``0<(x-y)/2`` in H6; Generalize (Rle_Ropp ``-(PI/2)`` y H1).
Rewrite Ropp_Ropp.
Intro H7; Generalize (Rle_sym2 ``-y`` ``PI/2`` H7); Clear H7; Intro H7; Generalize (Rplus_le x ``PI/2`` ``-y`` ``PI/2`` H0 H7).
Rewrite <- double_var.
Intro H8.
Assert Hyp : ``0<2``.
Sup0.
Generalize (Rle_monotony ``(Rinv 2)`` ``x-y`` PI (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H8).
Repeat Rewrite (Rmult_sym ``/2``). 
Intro H9; Generalize (sin_gt_0 ``(x-y)/2`` H6 (Rle_lt_trans ``(x-y)/2`` ``PI/2`` PI H9 PI2_Rlt_PI)); Intro H10; Elim (Rlt_antirefl ``(sin ((x-y)/2))`` (Rlt_trans ``(sin ((x-y)/2))`` ``0`` ``(sin ((x-y)/2))`` H4 H10)).
Generalize (Rlt_minus (sin x) (sin y) H3); Clear H3; Intro H3; Rewrite form4 in H3; Generalize (Rplus_le x ``PI/2`` y ``PI/2`` H0 H2).
Rewrite <- double_var.
Assert Hyp : ``0<2``.
Sup0.
Intro H4; Generalize (Rle_monotony ``(Rinv 2)`` ``x+y`` PI (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H4).
Repeat Rewrite (Rmult_sym ``/2``). 
Clear H4; Intro H4; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` y H H1); Replace ``-(PI/2)+(-(PI/2))`` with ``-PI``.
Intro H5; Generalize (Rle_monotony ``(Rinv 2)`` ``-PI`` ``x+y`` (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H5).
Replace ``/2*(x+y)`` with ``(x+y)/2``.
Replace ``/2*(-PI)`` with ``-(PI/2)``.
Clear H5; Intro H5; Elim H4; Intro H40.
Elim H5; Intro H50.
Generalize (cos_gt_0 ``(x+y)/2`` H50 H40); Intro H6; Generalize (Rlt_monotony ``2`` ``0`` ``(cos ((x+y)/2))`` Hyp H6).
Rewrite Rmult_Or. 
Clear H6; Intro H6; Case (case_Rabsolu ``(sin ((x-y)/2))``); Intro H7.
Assumption.
Generalize (Rle_sym2 ``0`` ``(sin ((x-y)/2))`` H7); Clear H7; Intro H7; Generalize (Rmult_le_pos ``2*(cos ((x+y)/2))`` ``(sin ((x-y)/2))`` (Rlt_le ``0`` ``2*(cos ((x+y)/2))`` H6) H7); Intro H8; Generalize (Rle_lt_trans ``0`` ``2*(cos ((x+y)/2))*(sin ((x-y)/2))`` ``0`` H8 H3); Intro H9; Elim (Rlt_antirefl ``0`` H9).
Rewrite <- H50 in H3; Rewrite cos_neg in H3; Rewrite cos_PI2 in H3; Rewrite Rmult_Or in H3; Rewrite Rmult_Ol in H3; Elim (Rlt_antirefl ``0`` H3).
Unfold Rdiv in H3.
Rewrite H40 in H3; Assert H50 := cos_PI2; Unfold Rdiv in H50; Rewrite H50 in H3; Rewrite Rmult_Or in H3; Rewrite Rmult_Ol in H3; Elim (Rlt_antirefl ``0`` H3).
Unfold Rdiv.
Rewrite <- Ropp_mul1.
Apply Rmult_sym.
Unfold Rdiv; Apply Rmult_sym.
Pattern 1 PI; Rewrite double_var.
Rewrite Ropp_distr1.
Reflexivity.
Qed.

Lemma sin_increasing_1 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``x<y``->``(sin x)<(sin y)``.
Intros; Generalize (Rlt_compatibility ``x`` ``x`` ``y`` H3); Intro H4; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` x H H); Replace ``-(PI/2)+ (-(PI/2))`` with ``-PI``.
Assert Hyp : ``0<2``.
Sup0.
Intro H5; Generalize (Rle_lt_trans ``-PI`` ``x+x`` ``x+y`` H5 H4); Intro H6; Generalize (Rlt_monotony ``(Rinv 2)`` ``-PI`` ``x+y`` (Rlt_Rinv ``2`` Hyp) H6); Replace ``/2*(-PI)`` with ``-(PI/2)``.
Replace ``/2*(x+y)`` with ``(x+y)/2``.
Clear H4 H5 H6; Intro H4; Generalize (Rlt_compatibility ``y`` ``x`` ``y`` H3); Intro H5; Rewrite Rplus_sym in H5; Generalize (Rplus_le y ``PI/2`` y ``PI/2`` H2 H2).
Rewrite <- double_var.
Intro H6; Generalize (Rlt_le_trans ``x+y`` ``y+y`` PI H5 H6); Intro H7; Generalize (Rlt_monotony ``(Rinv 2)``  ``x+y`` PI (Rlt_Rinv ``2`` Hyp) H7); Replace ``/2*PI`` with ``PI/2``.
Replace ``/2*(x+y)`` with ``(x+y)/2``.
Clear H5 H6 H7; Intro H5; Generalize (Rle_Ropp ``-(PI/2)`` y H1); Rewrite Ropp_Ropp; Clear H1; Intro H1; Generalize (Rle_sym2 ``-y`` ``PI/2`` H1); Clear H1; Intro H1; Generalize (Rle_Ropp y ``PI/2`` H2); Clear H2; Intro H2; Generalize (Rle_sym2 ``-(PI/2)`` ``-y`` H2); Clear H2; Intro H2; Generalize (Rlt_compatibility ``-y`` x y H3); Replace ``-y+x`` with ``x-y``.
Rewrite Rplus_Ropp_l.
Intro H6; Generalize (Rlt_monotony ``(Rinv 2)``  ``x-y`` ``0`` (Rlt_Rinv ``2`` Hyp) H6); Rewrite Rmult_Or; Replace ``/2*(x-y)`` with ``(x-y)/2``.
Clear H6; Intro H6; Generalize (Rplus_le  ``-(PI/2)`` x ``-(PI/2)`` ``-y`` H H2); Replace ``-(PI/2)+ (-(PI/2))`` with ``-PI``.
Replace `` x+ -y`` with ``x-y``.
Intro H7; Generalize (Rle_monotony ``(Rinv 2)`` ``-PI`` ``x-y`` (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H7); Replace ``/2*(-PI)`` with ``-(PI/2)``.
Replace ``/2*(x-y)`` with ``(x-y)/2``.
Clear H7; Intro H7; Clear H H0 H1 H2; Apply Rminus_lt; Rewrite form4; Generalize (cos_gt_0 ``(x+y)/2`` H4 H5); Intro H8; Generalize (Rmult_lt_pos ``2`` ``(cos ((x+y)/2))`` Hyp H8); Clear H8; Intro H8; Cut ``-PI< -(PI/2)``.
Intro H9; Generalize (sin_lt_0_var ``(x-y)/2`` (Rlt_le_trans ``-PI`` ``-(PI/2)`` ``(x-y)/2`` H9 H7) H6); Intro H10; Generalize (Rlt_anti_monotony ``(sin ((x-y)/2))`` ``0`` ``2*(cos ((x+y)/2))`` H10 H8); Intro H11; Rewrite Rmult_Or in H11; Rewrite Rmult_sym; Assumption.
Apply Rlt_Ropp; Apply PI2_Rlt_PI.
Unfold Rdiv; Apply Rmult_sym.
Unfold Rdiv; Rewrite <- Ropp_mul1; Apply Rmult_sym.
Reflexivity.
Pattern 1 PI; Rewrite double_var.
Rewrite Ropp_distr1.
Reflexivity.
Unfold Rdiv; Apply Rmult_sym.
Unfold Rminus; Apply Rplus_sym.
Unfold Rdiv; Apply Rmult_sym.
Unfold Rdiv; Apply Rmult_sym.
Unfold Rdiv; Apply Rmult_sym.
Unfold Rdiv.
Rewrite <- Ropp_mul1.
Apply Rmult_sym.
Pattern 1 PI; Rewrite double_var.
Rewrite Ropp_distr1.
Reflexivity.
Qed.

Lemma sin_decreasing_0 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``(sin x)<(sin y)`` -> ``y<x``.
Intros; Rewrite <- (sin_PI_x x) in H3; Rewrite <- (sin_PI_x y) in H3; Generalize (Rlt_Ropp ``(sin (PI-x))`` ``(sin (PI-y))`` H3); Repeat Rewrite <- sin_neg; Generalize (Rle_compatibility ``-PI`` x ``3*(PI/2)`` H); Generalize (Rle_compatibility ``-PI`` ``PI/2`` x H0); Generalize (Rle_compatibility ``-PI`` y ``3*(PI/2)`` H1); Generalize (Rle_compatibility ``-PI`` ``PI/2`` y H2); Replace ``-PI+x`` with ``x-PI``.
Replace ``-PI+PI/2`` with ``-(PI/2)``.
Replace ``-PI+y`` with ``y-PI``.
Replace ``-PI+3*(PI/2)`` with ``PI/2``.
Replace ``-(PI-x)`` with ``x-PI``.
Replace ``-(PI-y)`` with ``y-PI``.
Intros; Change ``(sin (y-PI))<(sin (x-PI))`` in H8; Apply Rlt_anti_compatibility with ``-PI``; Rewrite Rplus_sym; Replace ``y+ (-PI)`` with ``y-PI``.
Rewrite Rplus_sym; Replace ``x+ (-PI)`` with ``x-PI``.
Apply (sin_increasing_0 ``y-PI`` ``x-PI`` H4 H5 H6 H7 H8).
Reflexivity.
Reflexivity.
Unfold Rminus; Rewrite Ropp_distr1.
Rewrite Ropp_Ropp.
Apply Rplus_sym.
Unfold Rminus; Rewrite Ropp_distr1.
Rewrite Ropp_Ropp.
Apply Rplus_sym.
Pattern 2 PI; Rewrite double_var.
Rewrite Ropp_distr1.
Ring.
Unfold Rminus; Apply Rplus_sym.
Pattern 2 PI; Rewrite double_var.
Rewrite Ropp_distr1.
Ring.
Unfold Rminus; Apply Rplus_sym.
Qed.

Lemma sin_decreasing_1 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``x<y``  -> ``(sin y)<(sin x)``.
Intros; Rewrite <- (sin_PI_x x); Rewrite <- (sin_PI_x y); Generalize (Rle_compatibility ``-PI`` x ``3*(PI/2)`` H); Generalize (Rle_compatibility ``-PI`` ``PI/2`` x H0); Generalize (Rle_compatibility ``-PI`` y ``3*(PI/2)`` H1); Generalize (Rle_compatibility ``-PI`` ``PI/2`` y H2); Generalize (Rlt_compatibility ``-PI`` x y H3); Replace ``-PI+PI/2`` with ``-(PI/2)``.
Replace ``-PI+y`` with ``y-PI``.
Replace ``-PI+3*(PI/2)`` with ``PI/2``.
Replace ``-PI+x`` with ``x-PI``.
Intros; Apply Ropp_Rlt; Repeat Rewrite <- sin_neg; Replace ``-(PI-x)`` with ``x-PI``.
Replace ``-(PI-y)`` with ``y-PI``.
Apply (sin_increasing_1 ``x-PI`` ``y-PI`` H7 H8 H5 H6 H4).
Unfold Rminus; Rewrite Ropp_distr1.
Rewrite Ropp_Ropp.
Apply Rplus_sym.
Unfold Rminus; Rewrite Ropp_distr1.
Rewrite Ropp_Ropp.
Apply Rplus_sym.
Unfold Rminus; Apply Rplus_sym.
Pattern 2 PI; Rewrite double_var; Ring.
Unfold Rminus; Apply Rplus_sym.
Pattern 2 PI; Rewrite double_var; Ring.
Qed.

Lemma cos_increasing_0 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``(cos x)<(cos y)`` -> ``x<y``.
Intros x y H1 H2 H3 H4; Rewrite <- (cos_neg x); Rewrite <- (cos_neg y); Rewrite <- (cos_period ``-x`` (1)); Rewrite <- (cos_period ``-y`` (1)); Unfold INR; Replace ``-x+2*1*PI`` with ``PI/2-(x-3*(PI/2))``.
Replace ``-y+2*1*PI`` with ``PI/2-(y-3*(PI/2))``.
Repeat Rewrite cos_shift; Intro H5; Generalize (Rle_compatibility ``-3*(PI/2)`` PI x H1); Generalize (Rle_compatibility ``-3*(PI/2)`` x ``2*PI`` H2); Generalize (Rle_compatibility ``-3*(PI/2)`` PI y H3); Generalize (Rle_compatibility ``-3*(PI/2)`` y ``2*PI`` H4).
Replace ``-3*(PI/2)+y`` with ``y-3*(PI/2)``.
Replace ``-3*(PI/2)+x`` with ``x-3*(PI/2)``.
Replace ``-3*(PI/2)+2*PI`` with ``PI/2``.
Replace ``-3*PI/2+PI`` with ``-(PI/2)``.
Clear H1 H2 H3 H4; Intros H1 H2 H3 H4; Apply Rlt_anti_compatibility with ``-3*(PI/2)``; Replace ``-3*PI/2+x`` with ``x-3*(PI/2)``.
Replace ``-3*PI/2+y`` with ``y-3*(PI/2)``.
Apply (sin_increasing_0 ``x-3*(PI/2)`` ``y-3*(PI/2)`` H4 H3 H2 H1 H5).
Unfold Rminus.
Rewrite Ropp_mul1. 
Apply Rplus_sym. 
Unfold Rminus.
Rewrite Ropp_mul1. 
Apply Rplus_sym. 
Pattern 3 PI; Rewrite double_var.
Ring.
Rewrite double; Pattern 3 4 PI; Rewrite double_var.
Ring.
Unfold Rminus.
Rewrite Ropp_mul1. 
Apply Rplus_sym. 
Unfold Rminus.
Rewrite Ropp_mul1. 
Apply Rplus_sym. 
Rewrite Rmult_1r.
Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var.
Ring.
Rewrite Rmult_1r.
Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var.
Ring.
Qed.

Lemma cos_increasing_1 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``x<y`` -> ``(cos x)<(cos y)``.
Intros x y H1 H2 H3 H4 H5; Generalize (Rle_compatibility ``-3*(PI/2)`` PI x H1); Generalize (Rle_compatibility ``-3*(PI/2)`` x ``2*PI`` H2); Generalize (Rle_compatibility ``-3*(PI/2)`` PI y H3); Generalize (Rle_compatibility ``-3*(PI/2)`` y ``2*PI`` H4); Generalize (Rlt_compatibility ``-3*(PI/2)`` x y H5); Rewrite <- (cos_neg x); Rewrite <- (cos_neg y); Rewrite <- (cos_period ``-x`` (1)); Rewrite <- (cos_period ``-y`` (1)); Unfold INR; Replace ``-3*(PI/2)+x`` with ``x-3*(PI/2)``.
Replace ``-3*(PI/2)+y`` with ``y-3*(PI/2)``.
Replace ``-3*(PI/2)+PI`` with ``-(PI/2)``.
Replace ``-3*(PI/2)+2*PI`` with ``PI/2``.
Clear H1 H2 H3 H4 H5; Intros H1 H2 H3 H4 H5; Replace ``-x+2*1*PI`` with ``(PI/2)-(x-3*(PI/2))``.
Replace ``-y+2*1*PI`` with ``(PI/2)-(y-3*(PI/2))``.
Repeat Rewrite cos_shift; Apply (sin_increasing_1 ``x-3*(PI/2)`` ``y-3*(PI/2)`` H5 H4 H3 H2 H1).
Rewrite Rmult_1r.
Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var.
Ring.
Rewrite Rmult_1r.
Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var.
Ring.
Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var.
Ring.
Pattern 3 PI; Rewrite double_var; Ring.
Unfold Rminus.
Rewrite <- Ropp_mul1.
Apply Rplus_sym.
Unfold Rminus.
Rewrite <- Ropp_mul1.
Apply Rplus_sym.
Qed.

Lemma cos_decreasing_0 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``(cos x)<(cos y)``->``y<x``.
Intros; Generalize (Rlt_Ropp (cos x) (cos y) H3); Repeat Rewrite <- neg_cos; Intro H4; Change ``(cos (y+PI))<(cos (x+PI))`` in H4; Rewrite (Rplus_sym x) in H4; Rewrite (Rplus_sym y) in H4; Generalize (Rle_compatibility PI ``0`` x H); Generalize (Rle_compatibility PI x PI H0); Generalize (Rle_compatibility PI ``0`` y H1); Generalize (Rle_compatibility PI y PI H2); Rewrite Rplus_Or.
Rewrite <- double.
Clear H H0 H1 H2 H3; Intros; Apply Rlt_anti_compatibility with ``PI``; Apply (cos_increasing_0 ``PI+y`` ``PI+x`` H0 H H2 H1 H4).
Qed.

Lemma cos_decreasing_1 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``x<y``->``(cos y)<(cos x)``.
Intros; Apply Ropp_Rlt; Repeat Rewrite <- neg_cos; Rewrite (Rplus_sym x); Rewrite (Rplus_sym y); Generalize (Rle_compatibility PI ``0`` x H); Generalize (Rle_compatibility PI x PI H0); Generalize (Rle_compatibility PI ``0`` y H1); Generalize (Rle_compatibility PI y PI H2); Rewrite Rplus_Or.
Rewrite <- double.
Generalize (Rlt_compatibility PI x y H3); Clear H H0 H1 H2 H3; Intros; Apply (cos_increasing_1 ``PI+x`` ``PI+y`` H3 H2 H1 H0 H).
Qed.

Lemma tan_diff : (x,y:R) ~``(cos x)==0``->~``(cos y)==0``->``(tan x)-(tan y)==(sin (x-y))/((cos x)*(cos y))``.
Intros; Unfold tan;Rewrite sin_minus.
Unfold Rdiv. 
Unfold Rminus.
Rewrite Rmult_Rplus_distrl.
Rewrite Rinv_Rmult.
Repeat Rewrite (Rmult_sym (sin x)).
Repeat Rewrite Rmult_assoc.
Rewrite (Rmult_sym (cos y)).
Repeat Rewrite Rmult_assoc.
Rewrite <- Rinv_l_sym.
Rewrite Rmult_1r.
Rewrite (Rmult_sym (sin x)).
Apply Rplus_plus_r.
Rewrite <- Ropp_mul1.
Rewrite <- Ropp_mul3.
Rewrite (Rmult_sym ``/(cos x)``).
Repeat Rewrite Rmult_assoc.
Rewrite (Rmult_sym (cos x)).
Repeat Rewrite Rmult_assoc.
Rewrite <- Rinv_l_sym.
Rewrite Rmult_1r.
Reflexivity.
Assumption.
Assumption.
Assumption.
Assumption.
Qed.

Lemma tan_increasing_0 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``(tan x)<(tan y)``->``x<y``.
Intros; Generalize PI4_RLT_PI2; Intro H4; Generalize (Rlt_Ropp ``PI/4`` ``PI/2`` H4); Intro H5; Change ``-(PI/2)< -(PI/4)`` in H5; Generalize (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)); Intro HP1; Generalize (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)); Intro HP2; Generalize (not_sym ``0`` (cos x) (Rlt_not_eq ``0`` (cos x) (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)))); Intro H6; Generalize (not_sym ``0`` (cos y) (Rlt_not_eq ``0`` (cos y) (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)))); Intro H7; Generalize (tan_diff x y H6 H7); Intro H8; Generalize (Rlt_minus (tan x) (tan y) H3); Clear H3; Intro H3; Rewrite H8 in H3; Cut ``(sin (x-y))<0``.
Intro H9; Generalize (Rle_Ropp ``-(PI/4)`` y H1); Rewrite Ropp_Ropp; Intro H10; Generalize (Rle_sym2 ``-y`` ``PI/4`` H10); Clear H10; Intro H10; Generalize (Rle_Ropp y ``PI/4`` H2); Intro H11; Generalize (Rle_sym2 ``-(PI/4)`` ``-y`` H11); Clear H11; Intro H11; Generalize (Rplus_le ``-(PI/4)`` x ``-(PI/4)`` ``-y`` H H11); Generalize (Rplus_le x ``PI/4`` ``-y`` ``PI/4`` H0 H10); Replace ``x+ -y`` with ``x-y``.
Replace ``PI/4+PI/4`` with ``PI/2``.
Replace ``-(PI/4)+ -(PI/4)`` with ``-(PI/2)``.
Intros; Case (total_order ``0`` ``x-y``); Intro H14.
Generalize (sin_gt_0 ``x-y`` H14 (Rle_lt_trans ``x-y`` ``PI/2`` PI H12 PI2_Rlt_PI)); Intro H15; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(sin (x-y))`` ``0`` H15 H9)).
Elim H14; Intro H15.
Rewrite <- H15 in H9; Rewrite -> sin_0 in H9;  Elim (Rlt_antirefl ``0`` H9). 
Apply Rminus_lt; Assumption.
Pattern 1 PI; Rewrite double_var.
Unfold Rdiv.
Rewrite Rmult_Rplus_distrl.
Repeat Rewrite Rmult_assoc.
Rewrite <- Rinv_Rmult.
Rewrite Ropp_distr1.
Replace ``2*2`` with ``4``.
Reflexivity.
Ring.
DiscrR.
DiscrR.
Pattern 1 PI; Rewrite double_var.
Unfold Rdiv.
Rewrite Rmult_Rplus_distrl.
Repeat Rewrite Rmult_assoc.
Rewrite <- Rinv_Rmult.
Replace ``2*2`` with ``4``.
Reflexivity.
Ring.
DiscrR.
DiscrR.
Reflexivity.
Case (case_Rabsolu ``(sin (x-y))``); Intro H9.
Assumption.
Generalize (Rle_sym2 ``0`` ``(sin (x-y))`` H9); Clear H9; Intro H9; Generalize (Rlt_Rinv (cos x) HP1); Intro H10; Generalize (Rlt_Rinv (cos y) HP2); Intro H11; Generalize (Rmult_lt_pos (Rinv (cos x)) (Rinv (cos y)) H10 H11); Replace ``/(cos x)*/(cos y)`` with ``/((cos x)*(cos y))``.
Intro H12; Generalize (Rmult_le_pos ``(sin (x-y))`` ``/((cos x)*(cos y))`` H9 (Rlt_le ``0`` ``/((cos x)*(cos y))`` H12)); Intro H13; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``(sin (x-y))*/((cos x)*(cos y))`` ``0`` H13 H3)).
Rewrite Rinv_Rmult.
Reflexivity. 
Assumption.
Assumption.
Qed.

Lemma tan_increasing_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``x<y``->``(tan x)<(tan y)``.
Intros; Apply Rminus_lt; Generalize PI4_RLT_PI2; Intro H4; Generalize (Rlt_Ropp ``PI/4`` ``PI/2`` H4); Intro H5; Change ``-(PI/2)< -(PI/4)`` in H5; Generalize (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)); Intro HP1; Generalize (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)); Intro HP2; Generalize (not_sym ``0`` (cos x) (Rlt_not_eq ``0`` (cos x) (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)))); Intro H6; Generalize (not_sym ``0`` (cos y) (Rlt_not_eq ``0`` (cos y) (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)))); Intro H7; Rewrite (tan_diff x y H6 H7); Generalize (Rlt_Rinv (cos x) HP1); Intro H10; Generalize (Rlt_Rinv (cos y) HP2); Intro H11; Generalize (Rmult_lt_pos (Rinv (cos x)) (Rinv (cos y)) H10 H11); Replace ``/(cos x)*/(cos y)`` with ``/((cos x)*(cos y))``.
Clear H10 H11; Intro H8; Generalize (Rle_Ropp y ``PI/4`` H2); Intro H11; Generalize (Rle_sym2 ``-(PI/4)`` ``-y`` H11); Clear H11; Intro H11; Generalize (Rplus_le ``-(PI/4)`` x ``-(PI/4)`` ``-y`` H H11); Replace ``x+ -y`` with ``x-y``.
Replace ``-(PI/4)+ -(PI/4)`` with ``-(PI/2)``.
Clear H11; Intro H9; Generalize (Rlt_minus x y H3); Clear H3; Intro H3; Clear H H0 H1 H2 H4 H5 HP1 HP2; Generalize PI2_Rlt_PI; Intro H1; Generalize (Rlt_Ropp ``PI/2`` PI H1); Clear H1; Intro H1; Generalize (sin_lt_0_var ``x-y`` (Rlt_le_trans ``-PI`` ``-(PI/2)`` ``x-y`` H1 H9) H3); Intro H2; Generalize (Rlt_anti_monotony ``(sin (x-y))`` ``0`` ``/((cos x)*(cos y))`` H2 H8); Rewrite Rmult_Or; Intro H4; Assumption.
Pattern 1 PI; Rewrite double_var.
Unfold Rdiv.
Rewrite Rmult_Rplus_distrl.
Repeat Rewrite Rmult_assoc.
Rewrite <- Rinv_Rmult.
Replace ``2*2`` with ``4``.
Rewrite Ropp_distr1.
Reflexivity.
Ring.
DiscrR.
DiscrR.
Reflexivity.
Apply Rinv_Rmult; Assumption.
Qed.

Lemma sin_incr_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<=(sin y)``->``x<=y``.
Intros; Case (total_order (sin x) (sin y)); Intro H4; [Left; Apply (sin_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (sin_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (sin y) H8)]] | Elim (Rlt_antirefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5))]].
Qed.

Lemma sin_incr_1 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``x<=y``->``(sin x)<=(sin y)``.
Intros; Case (total_order x y); Intro H4; [Left; Apply (sin_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (sin x) (sin y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (sin_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]].
Qed.

Lemma sin_decr_0 : (x,y:R) ``x<=3*(PI/2)``->``PI/2<=x``->``y<=3*(PI/2)``->``PI/2<=y``-> ``(sin x)<=(sin y)`` -> ``y<=x``.
Intros; Case (total_order (sin x) (sin y)); Intro H4; [Left; Apply (sin_decreasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Generalize (sin_decreasing_1 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (sin y) H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5))]].
Qed.

Lemma sin_decr_1 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``x<=y``  -> ``(sin y)<=(sin x)``.
Intros; Case (total_order x y); Intro H4; [Left; Apply (sin_decreasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (sin x) (sin y)); Intro H6; [Generalize (sin_decreasing_0 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]].
Qed.

Lemma cos_incr_0 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``(cos x)<=(cos y)`` -> ``x<=y``.
Intros; Case (total_order (cos x) (cos y)); Intro H4; [Left; Apply (cos_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (cos_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (cos y) H8)]] | Elim (Rlt_antirefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5))]].
Qed.

Lemma cos_incr_1 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``x<=y`` -> ``(cos x)<=(cos y)``.
Intros; Case (total_order x y); Intro H4; [Left; Apply (cos_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (cos x) (cos y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (cos_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]].
Qed.

Lemma cos_decr_0 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``(cos x)<=(cos y)`` -> ``y<=x``.
Intros; Case (total_order (cos x) (cos y)); Intro H4; [Left; Apply (cos_decreasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Generalize (cos_decreasing_1 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (cos y) H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5))]].
Qed.

Lemma cos_decr_1 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``x<=y``->``(cos y)<=(cos x)``.
Intros; Case (total_order x y); Intro H4; [Left; Apply (cos_decreasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (cos x) (cos y)); Intro H6; [Generalize (cos_decreasing_0 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]].
Qed.

Lemma tan_incr_0 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``(tan x)<=(tan y)``->``x<=y``.
Intros; Case (total_order (tan x) (tan y)); Intro H4; [Left; Apply (tan_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (tan y) H8)]] | Elim (Rlt_antirefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5))]].
Qed.

Lemma tan_incr_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``x<=y``->``(tan x)<=(tan y)``.
Intros; Case (total_order x y); Intro H4; [Left; Apply (tan_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (tan x) (tan y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]].
Qed.

(**********)
Lemma sin_eq_0_1 : (x:R) (EXT k:Z | x==(Rmult (IZR k) PI)) -> (sin x)==R0.
Intros.
Elim H; Intros.
Apply (Zcase_sign x0).
Intro.
Rewrite H1 in H0.
Simpl in H0.
Rewrite H0; Rewrite Rmult_Ol; Apply sin_0.
Intro.
Cut `0<=x0`.
Intro.
Elim (IZN x0 H2); Intros.
Rewrite H3 in H0.
Rewrite <- INR_IZR_INZ in H0.
Rewrite H0.
Elim (even_odd_cor x1); Intros.
Elim H4; Intro.
Rewrite H5.
Rewrite mult_INR.
Simpl.
Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``).
Rewrite sin_period.
Apply sin_0.
Rewrite H5.
Rewrite S_INR; Rewrite mult_INR.
Simpl.
Rewrite Rmult_Rplus_distrl.
Rewrite Rmult_1l; Rewrite sin_plus.
Rewrite sin_PI.
Rewrite Rmult_Or.
Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``).
Rewrite sin_period.
Rewrite sin_0; Ring.
Apply le_IZR.
Left; Apply IZR_lt.
Assert H2 := Zgt_iff_lt.
Elim (H2 x0 `0`); Intros.
Apply H3; Assumption.
Intro.
Rewrite H0.
Replace ``(sin ((IZR x0)*PI))`` with ``-(sin (-(IZR x0)*PI))``.
Cut `0<=-x0`.
Intro.
Rewrite <- Ropp_Ropp_IZR.
Elim (IZN `-x0` H2); Intros.
Rewrite H3.
Rewrite <- INR_IZR_INZ.
Elim (even_odd_cor x1); Intros.
Elim H4; Intro.
Rewrite H5.
Rewrite mult_INR.
Simpl.
Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``).
Rewrite sin_period.
Rewrite sin_0; Ring.
Rewrite H5.
Rewrite S_INR; Rewrite mult_INR.
Simpl.
Rewrite Rmult_Rplus_distrl.
Rewrite Rmult_1l; Rewrite sin_plus.
Rewrite sin_PI.
Rewrite Rmult_Or.
Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``).
Rewrite sin_period.
Rewrite sin_0; Ring.
Apply le_IZR.
Apply Rle_anti_compatibility with ``(IZR x0)``.
Rewrite Rplus_Or.
Rewrite Ropp_Ropp_IZR.
Rewrite Rplus_Ropp_r.
Left; Replace R0 with (IZR `0`); [Apply IZR_lt | Reflexivity].
Assumption.
Rewrite <- sin_neg.
Rewrite Ropp_mul1.
Rewrite Ropp_Ropp.
Reflexivity.
Qed.

Lemma sin_eq_0_0 : (x:R) (sin x)==R0 -> (EXT k:Z | x==(Rmult (IZR k) PI)).
Intros.
Assert H0 := (euclidian_division x PI PI_neq0).
Elim H0; Intros q H1.
Elim H1; Intros r H2.
Exists q.
Cut r==R0.
Intro.
Elim H2; Intros H4 _; Rewrite H4; Rewrite H3.
Apply Rplus_Or.
Elim H2; Intros.
Rewrite H3 in H.
Rewrite sin_plus in H.
Cut ``(sin ((IZR q)*PI))==0``.
Intro.
Rewrite H5 in H.
Rewrite Rmult_Ol in H.
Rewrite Rplus_Ol in H.
Assert H6 := (without_div_Od ? ? H).
Elim H6; Intro.
Assert H8 := (sin2_cos2 ``(IZR q)*PI``).
Rewrite H5 in H8; Rewrite H7 in H8.
Rewrite Rsqr_O in H8.
Rewrite Rplus_Or in H8.
Elim R1_neq_R0; Symmetry; Assumption.
Cut r==R0\/``0<r<PI``.
Intro; Elim H8; Intro.
Assumption.
Elim H9; Intros.
Assert H12 := (sin_gt_0 ? H10 H11).
Rewrite H7 in H12; Elim (Rlt_antirefl ? H12).
Rewrite Rabsolu_right in H4.
Elim H4; Intros.
Case (total_order R0 r); Intro.
Right; Split; Assumption.
Elim H10; Intro.
Left; Symmetry; Assumption.
Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 H11)).
Apply Rle_sym1.
Left; Apply PI_RGT_0.
Apply sin_eq_0_1.
Exists q; Reflexivity.
Qed.

Lemma cos_eq_0_0 : (x:R) (cos x)==R0 -> (EXT k : Z | ``x==(IZR k)*PI+PI/2``). 
Intros x H; Rewrite -> cos_sin in H; Generalize (sin_eq_0_0 (Rplus (Rdiv PI (INR (2))) x) H); Intro H2; Elim H2; Intros x0 H3; Exists (Zminus x0 (inject_nat (S O))); Rewrite <- Z_R_minus; Ring; Rewrite Rmult_sym; Rewrite <- H3; Unfold INR.
Rewrite (double_var ``-PI``); Unfold Rdiv; Ring.
Qed.

Lemma  cos_eq_0_1 : (x:R) (EXT k : Z | ``x==(IZR k)*PI+PI/2``) -> ``(cos x)==0``.
Intros x H1; Rewrite cos_sin; Elim H1; Intros x0 H2; Rewrite H2; Replace ``PI/2+((IZR x0)*PI+PI/2)`` with ``(IZR x0)*PI+PI``.
Rewrite neg_sin; Rewrite <- Ropp_O.
Apply eq_Ropp; Apply sin_eq_0_1; Exists x0; Reflexivity.
Pattern 2 PI; Rewrite (double_var PI); Ring.
Qed.

Lemma sin_eq_O_2PI_0 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``(sin x)==0`` -> ``x==0``\/``x==PI``\/``x==2*PI``.
Intros; Generalize (sin_eq_0_0 x H1); Intro.
Elim H2; Intros k0 H3.
Case (total_order PI x); Intro.
Rewrite H3 in H4; Rewrite H3 in H0.
Right; Right.
Generalize (Rlt_monotony_r ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv ``PI`` PI_RGT_0) H4); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym.
Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``(IZR k0)*PI`` ``2*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H0); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym.
Repeat Rewrite Rmult_1r; Intro; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H5); Rewrite <- plus_IZR.
Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``.
Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``2`` H6); Rewrite <- plus_IZR.
Replace ``(IZR (NEG (xO xH)))+2`` with ``0``.
Intro; Cut ``-1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``.
Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H9); Intro.
Cut k0=`2`.
Intro; Rewrite H11 in H3; Rewrite H3; Simpl.
Reflexivity.
Rewrite <- (Zplus_inverse_l `2`) in H10; Generalize (Zsimpl_plus_l `-2` k0 `2` H10); Intro; Assumption.
Split.
Assumption.
Apply Rle_lt_trans with ``0``.
Assumption.
Apply Rlt_R0_R1.
Simpl; Ring.
Simpl; Ring.
Apply PI_neq0.
Apply PI_neq0.
Elim H4; Intro.
Right; Left.
Symmetry; Assumption.
Left.
Rewrite H3 in H5; Rewrite H3 in H; Generalize (Rlt_monotony_r ``/PI``  ``(IZR k0)*PI`` PI (Rlt_Rinv ``PI`` PI_RGT_0) H5); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym.
Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``0`` ``(IZR k0)*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym.
Rewrite Rmult_1r; Rewrite Rmult_Ol; Intro.
Cut ``-1 < (IZR (k0)) < 1``.
Intro; Generalize (one_IZR_lt1 k0 H8); Intro; Rewrite H9 in H3; Rewrite H3; Simpl; Apply Rmult_Ol.
Split.
Apply Rlt_le_trans with ``0``.
Rewrite <- Ropp_O; Apply Rgt_Ropp; Apply Rlt_R0_R1.
Assumption.
Assumption.
Apply PI_neq0.
Apply PI_neq0.
Qed.

Lemma sin_eq_O_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==0``\/``x==PI``\/``x==2*PI`` -> ``(sin x)==0``.
Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> sin_0; Reflexivity | Elim H4; Intro H5; [Rewrite H5; Rewrite -> sin_PI; Reflexivity | Rewrite H5; Rewrite -> sin_2PI; Reflexivity]].
Qed.

Lemma cos_eq_0_2PI_0 : (x:R) ``R0<=x`` -> ``x<=2*PI`` -> ``(cos x)==0`` -> ``x==(PI/2)``\/``x==3*(PI/2)``.
Intros; Case (total_order x ``3*(PI/2)``); Intro.
Rewrite cos_sin in H1.
Cut ``0<=PI/2+x``.
Cut ``PI/2+x<=2*PI``.
Intros; Generalize (sin_eq_O_2PI_0 ``PI/2+x`` H4 H3 H1); Intros.
Decompose [or] H5.
Generalize (Rle_compatibility ``PI/2`` ``0`` x H); Rewrite Rplus_Or; Rewrite H6; Intro.
Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``PI/2`` ``0`` PI2_RGT_0 H7)).
Left.
Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` PI H7).
Replace ``-(PI/2)+(PI/2+x)`` with x.
Replace ``-(PI/2)+PI`` with ``PI/2``.
Intro; Assumption.
Pattern 3 PI; Rewrite (double_var PI); Ring.
Ring.
Right.
Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` ``2*PI`` H7).
Replace ``-(PI/2)+(PI/2+x)`` with x.
Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``.
Intro; Assumption.
Rewrite double; Pattern 3 4 PI; Rewrite (double_var PI); Ring.
Ring.
Left; Replace ``2*PI`` with ``PI/2+3*(PI/2)``.
Apply Rlt_compatibility; Assumption.
Rewrite (double PI); Pattern 3 4 PI; Rewrite (double_var PI); Ring.
Apply ge0_plus_ge0_is_ge0.
Left; Unfold Rdiv; Apply Rmult_lt_pos.
Apply PI_RGT_0.
Apply Rlt_Rinv; Sup0. 
Assumption.
Elim H2; Intro.
Right; Assumption.
Generalize (cos_eq_0_0 x H1); Intro; Elim H4; Intros k0 H5.
Rewrite H5 in H3; Rewrite H5 in H0; Generalize (Rlt_compatibility ``-(PI/2)`` ``3*PI/2`` ``(IZR k0)*PI+PI/2`` H3); Generalize (Rle_compatibility ``-(PI/2)`` ``(IZR k0)*PI+PI/2`` ``2*PI`` H0).
Replace ``-(PI/2)+3*PI/2`` with PI.
Replace ``-(PI/2)+((IZR k0)*PI+PI/2)`` with ``(IZR k0)*PI``.
Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``.
Intros; Generalize (Rlt_monotony ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv PI PI_RGT_0) H7); Generalize (Rle_monotony ``/PI`` ``(IZR k0)*PI`` ``3*(PI/2)`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv PI PI_RGT_0)) H6).
Replace ``/PI*((IZR k0)*PI)`` with (IZR k0).
Replace ``/PI*(3*PI/2)`` with ``3*/2``.
Rewrite <- Rinv_l_sym.
Intros; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H9); Rewrite <- plus_IZR.
Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``.
Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``3*/2`` H8); Rewrite <- plus_IZR.
Replace ``(IZR (NEG (xO xH)))+2`` with ``0``.
Intro; Cut `` -1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``.
Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H12); Intro.
Cut k0=`2`.
Intro; Rewrite H14 in H8.
Assert Hyp : ``0<2``.
Sup0.
Generalize (Rle_monotony ``2`` ``(IZR (POS (xO xH)))`` ``3*/2`` (Rlt_le ``0`` ``2`` Hyp) H8); Simpl.
Replace ``2*2`` with ``4``.
Replace ``2*(3*/2)`` with ``3``.
Intro; Cut ``3<4``.
Intro; Elim (Rlt_antirefl ``3`` (Rlt_le_trans ``3`` ``4`` ``3`` H16 H15)).
Generalize (Rlt_compatibility ``3`` ``0`` ``1`` Rlt_R0_R1); Rewrite Rplus_Or.
Replace ``3+1`` with ``4``.
Intro; Assumption.
Ring.
Symmetry; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m.
DiscrR.
Ring.
Rewrite <- (Zplus_inverse_l `2`) in H13; Generalize (Zsimpl_plus_l `-2` k0 `2` H13); Intro; Assumption.
Split.
Assumption.
Apply Rle_lt_trans with ``(IZR (NEG (xO xH)))+3*/2``.
Assumption.
Simpl; Replace ``-2+3*/2`` with ``-(1*/2)``.
Apply Rlt_trans with ``0``.
Rewrite <- Ropp_O; Apply Rlt_Ropp.
Apply Rmult_lt_pos; [Apply Rlt_R0_R1 | Apply Rlt_Rinv; Sup0].
Apply Rlt_R0_R1.
Rewrite Rmult_1l; Apply r_Rmult_mult with ``2``.
Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym.
Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m.
Ring.
DiscrR.
DiscrR.
DiscrR.
Simpl; Ring.
Simpl; Ring.
Apply PI_neq0.
Unfold Rdiv; Pattern 1 ``3``; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym.
Rewrite Rmult_1l; Apply Rmult_sym.
Apply PI_neq0.
Symmetry; Rewrite (Rmult_sym ``/PI``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym.
Apply Rmult_1r.
Apply PI_neq0.
Rewrite double; Pattern 3 4 PI; Rewrite double_var; Ring.
Ring.
Pattern 1 PI; Rewrite double_var; Ring.
Qed.

Lemma  cos_eq_0_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==PI/2``\/``x==3*(PI/2)`` -> ``(cos x)==0``.
Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> cos_PI2; Reflexivity | Rewrite H4; Rewrite -> cos_3PI2; Reflexivity ].
Qed.