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 *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/transc/TrigMon".
21 (* $Id: TrigMon.v,v 1.9 2004/04/23 10:01:08 lcf Exp $ *)
23 include "transc/Pi.ma".
30 Sign properties: cosine is positive in
31 $(-\frac{\pi}2,\frac{\pi}2)$#(-π/2,π/2)#, sine in
32 $(0,\pi)$#(0,π)# and tangent in $(0,\frac{\pi}2)$#0,π/2)#.
35 inline "cic:/CoRN/transc/TrigMon/Cos_pos.con".
37 inline "cic:/CoRN/transc/TrigMon/Sin_pos.con".
39 inline "cic:/CoRN/transc/TrigMon/Tan_pos.con".
41 inline "cic:/CoRN/transc/TrigMon/Cos_nonneg.con".
43 inline "cic:/CoRN/transc/TrigMon/Sin_nonneg.con".
47 inline "cic:/CoRN/transc/TrigMon/Abs_Sin_less_One.con".
49 inline "cic:/CoRN/transc/TrigMon/Abs_Cos_less_One.con".
52 Sine is (strictly) increasing in [[ [--]Pi[/]Two,Pi[/]Two]]; cosine
53 is (strictly) decreasing in [[Zero,Pi]].
56 inline "cic:/CoRN/transc/TrigMon/Sin_resp_leEq.con".
58 inline "cic:/CoRN/transc/TrigMon/Cos_resp_leEq.con".
62 inline "cic:/CoRN/transc/TrigMon/Cos_resp_less_aux.con".
64 inline "cic:/CoRN/transc/TrigMon/Cos_resp_less_aux'.con".
68 inline "cic:/CoRN/transc/TrigMon/Cos_resp_less.con".
70 inline "cic:/CoRN/transc/TrigMon/Sin_resp_less.con".
76 (*#* **Derivative of Tangent
78 Finally, two formulas for the derivative of the tangent function and
79 monotonicity properties.
82 inline "cic:/CoRN/transc/TrigMon/bnd_Cos.con".
88 inline "cic:/CoRN/transc/TrigMon/Derivative_Tan_1.con".
90 inline "cic:/CoRN/transc/TrigMon/Derivative_Tan_2.con".
92 inline "cic:/CoRN/transc/TrigMon/Tan_resp_less.con".
94 inline "cic:/CoRN/transc/TrigMon/Tan_resp_leEq.con".
101 Hint Resolve Derivative_Tan_1 Derivative_Tan_2: derivate.