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 (*#***********************************************************************)
21 (* v * The Coq Proof Assistant / The Coq Development Team *)
23 (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
25 (* \VV/ **************************************************************)
27 (* // * This file is distributed under the terms of the *)
29 (* * GNU Lesser General Public License Version 2.1 *)
31 (*#***********************************************************************)
33 (*i $Id: AltSeries.v,v 1.12.2.1 2004/07/16 19:31:10 herbelin Exp $ i*)
35 include "Reals/Rbase.ma".
37 include "Reals/Rfunctions.ma".
39 include "Reals/Rseries.ma".
41 include "Reals/SeqProp.ma".
43 include "Reals/PartSum.ma".
45 include "Arith/Max.ma".
48 Open Local Scope R_scope.
53 inline procedural "cic:/Coq/Reals/AltSeries/tg_alt.con" as definition.
55 inline procedural "cic:/Coq/Reals/AltSeries/positivity_seq.con" as definition.
57 inline procedural "cic:/Coq/Reals/AltSeries/CV_ALT_step0.con" as lemma.
59 inline procedural "cic:/Coq/Reals/AltSeries/CV_ALT_step1.con" as lemma.
63 inline procedural "cic:/Coq/Reals/AltSeries/CV_ALT_step2.con" as lemma.
65 (* A more general inequality *)
67 inline procedural "cic:/Coq/Reals/AltSeries/CV_ALT_step3.con" as lemma.
71 inline procedural "cic:/Coq/Reals/AltSeries/CV_ALT_step4.con" as lemma.
73 (* This lemma gives an interesting result about alternated series *)
75 inline procedural "cic:/Coq/Reals/AltSeries/CV_ALT.con" as lemma.
77 (*#***********************************************)
79 (* Convergence of alternated series *)
83 (* Applications: PI, cos, sin *)
85 (*#***********************************************)
87 inline procedural "cic:/Coq/Reals/AltSeries/alternated_series.con" as theorem.
89 inline procedural "cic:/Coq/Reals/AltSeries/alternated_series_ineq.con" as theorem.
91 (*#***********************************)
93 (* Application : construction of PI *)
95 (*#***********************************)
97 inline procedural "cic:/Coq/Reals/AltSeries/PI_tg.con" as definition.
99 inline procedural "cic:/Coq/Reals/AltSeries/PI_tg_pos.con" as lemma.
101 inline procedural "cic:/Coq/Reals/AltSeries/PI_tg_decreasing.con" as lemma.
103 inline procedural "cic:/Coq/Reals/AltSeries/PI_tg_cv.con" as lemma.
105 inline procedural "cic:/Coq/Reals/AltSeries/exist_PI.con" as lemma.
107 (* Now, PI is defined *)
109 inline procedural "cic:/Coq/Reals/AltSeries/PI.con" as definition.
111 (* We can get an approximation of PI with the following inequality *)
113 inline procedural "cic:/Coq/Reals/AltSeries/PI_ineq.con" as lemma.
115 inline procedural "cic:/Coq/Reals/AltSeries/PI_RGT_0.con" as lemma.