1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: TrigMon.v,v 1.9 2004/04/23 10:01:08 lcf Exp $ *)
21 include "transc/Pi.ma".
28 Sign properties: cosine is positive in
29 $(-\frac{\pi}2,\frac{\pi}2)$#(-π/2,π/2)#, sine in
30 $(0,\pi)$#(0,π)# and tangent in $(0,\frac{\pi}2)$#0,π/2)#.
33 inline procedural "cic:/CoRN/transc/TrigMon/Cos_pos.con" as lemma.
35 inline procedural "cic:/CoRN/transc/TrigMon/Sin_pos.con" as lemma.
37 inline procedural "cic:/CoRN/transc/TrigMon/Tan_pos.con" as lemma.
39 inline procedural "cic:/CoRN/transc/TrigMon/Cos_nonneg.con" as lemma.
41 inline procedural "cic:/CoRN/transc/TrigMon/Sin_nonneg.con" as lemma.
45 inline procedural "cic:/CoRN/transc/TrigMon/Abs_Sin_less_One.con" as lemma.
47 inline procedural "cic:/CoRN/transc/TrigMon/Abs_Cos_less_One.con" as lemma.
50 Sine is (strictly) increasing in [[ [--]Pi[/]Two,Pi[/]Two]]; cosine
51 is (strictly) decreasing in [[Zero,Pi]].
54 inline procedural "cic:/CoRN/transc/TrigMon/Sin_resp_leEq.con" as lemma.
56 inline procedural "cic:/CoRN/transc/TrigMon/Cos_resp_leEq.con" as lemma.
60 inline procedural "cic:/CoRN/transc/TrigMon/Cos_resp_less_aux.con" as lemma.
62 inline procedural "cic:/CoRN/transc/TrigMon/Cos_resp_less_aux'.con" as lemma.
66 inline procedural "cic:/CoRN/transc/TrigMon/Cos_resp_less.con" as lemma.
68 inline procedural "cic:/CoRN/transc/TrigMon/Sin_resp_less.con" as lemma.
74 (*#* **Derivative of Tangent
76 Finally, two formulas for the derivative of the tangent function and
77 monotonicity properties.
80 inline procedural "cic:/CoRN/transc/TrigMon/bnd_Cos.con" as lemma.
86 inline procedural "cic:/CoRN/transc/TrigMon/Derivative_Tan_1.con" as lemma.
88 inline procedural "cic:/CoRN/transc/TrigMon/Derivative_Tan_2.con" as lemma.
90 inline procedural "cic:/CoRN/transc/TrigMon/Tan_resp_less.con" as lemma.
92 inline procedural "cic:/CoRN/transc/TrigMon/Tan_resp_leEq.con" as lemma.
99 Hint Resolve Derivative_Tan_1 Derivative_Tan_2: derivate.