X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FCoRN-Decl%2Fftc%2FFunctSeries.ma;fp=matita%2Fcontribs%2FCoRN-Decl%2Fftc%2FFunctSeries.ma;h=917cbe2bcbde4bfa3b5c5f172a2fe589491da743;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/CoRN-Decl/ftc/FunctSeries.ma b/matita/contribs/CoRN-Decl/ftc/FunctSeries.ma new file mode 100644 index 000000000..917cbe2bc --- /dev/null +++ b/matita/contribs/CoRN-Decl/ftc/FunctSeries.ma @@ -0,0 +1,366 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/CoRN-Decl/ftc/FunctSeries". + +include "CoRN.ma". + +(* $Id: FunctSeries.v,v 1.6 2004/04/23 10:00:58 lcf Exp $ *) + +include "ftc/FunctSequence.ma". + +include "reals/Series.ma". + +(*#* printing fun_seq_part_sum %\ensuremath{\sum^n}% #∑n# *) + +(*#* printing Fun_Series_Sum %\ensuremath{\sum_0^{\infty}}% #∑0# *) + +(* UNEXPORTED +Section Definitions +*) + +(*#* *Series of Functions + +We now turn our attention to series of functions. Like it was already +the case for sequences, we will mainly rewrite the results we proved +for series of real numbers in a different way. + +%\begin{convention}% Throughout this section: + - [a] and [b] will be real numbers and the interval [[a,b]] will be denoted +by [I]; + - [f, g] and [h] will denote sequences of continuous functions; + - [F, G] and [H] will denote continuous functions. + +%\end{convention}% + +** Definitions + +As before, we will consider only sequences of continuous functions +defined in a compact interval. For this, partial sums are defined and +convergence is simply the convergence of the sequence of partial sums. +*) + +alias id "a" = "cic:/CoRN/ftc/FunctSeries/Definitions/a.var". + +alias id "b" = "cic:/CoRN/ftc/FunctSeries/Definitions/b.var". + +alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Definitions/Hab.var". + +(* begin hide *) + +inline "cic:/CoRN/ftc/FunctSeries/Definitions/I.con" "Definitions__". + +(* end hide *) + +alias id "f" = "cic:/CoRN/ftc/FunctSeries/Definitions/f.var". + +inline "cic:/CoRN/ftc/FunctSeries/fun_seq_part_sum.con". + +inline "cic:/CoRN/ftc/FunctSeries/fun_seq_part_sum_cont.con". + +inline "cic:/CoRN/ftc/FunctSeries/fun_series_convergent.con". + +(*#* +For what comes up next we need to know that the convergence of a +series of functions implies pointwise convergence of the corresponding +real number series. +*) + +inline "cic:/CoRN/ftc/FunctSeries/fun_series_conv_imp_conv.con". + +(*#* We then define the sum of the series as being the pointwise sum of +the corresponding series. +*) + +(* begin show *) + +alias id "H" = "cic:/CoRN/ftc/FunctSeries/Definitions/H.var". + +(* end show *) + +(* begin hide *) + +inline "cic:/CoRN/ftc/FunctSeries/Definitions/contf.con" "Definitions__". + +inline "cic:/CoRN/ftc/FunctSeries/Definitions/incf.con" "Definitions__". + +(* end hide *) + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_strext.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum.con". + +(* UNEXPORTED +End Definitions +*) + +(* UNEXPORTED +Implicit Arguments Fun_Series_Sum [a b Hab f]. +*) + +(* UNEXPORTED +Hint Resolve fun_seq_part_sum_cont: continuous. +*) + +(* UNEXPORTED +Section More_Definitions +*) + +alias id "a" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/a.var". + +alias id "b" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/b.var". + +alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/Hab.var". + +alias id "f" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/f.var". + +(*#* A series can also be absolutely convergent. *) + +inline "cic:/CoRN/ftc/FunctSeries/fun_series_abs_convergent.con". + +(* UNEXPORTED +End More_Definitions +*) + +(* UNEXPORTED +Section Operations +*) + +(* **Algebraic Properties + +All of these are analogous to the properties for series of real numbers, so we won't comment much about them. +*) + +alias id "a" = "cic:/CoRN/ftc/FunctSeries/Operations/a.var". + +alias id "b" = "cic:/CoRN/ftc/FunctSeries/Operations/b.var". + +alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Operations/Hab.var". + +(* begin hide *) + +inline "cic:/CoRN/ftc/FunctSeries/Operations/I.con" "Operations__". + +(* end hide *) + +inline "cic:/CoRN/ftc/FunctSeries/fun_seq_part_sum_n.con". + +inline "cic:/CoRN/ftc/FunctSeries/conv_fun_const_series.con". + +inline "cic:/CoRN/ftc/FunctSeries/fun_const_series_sum.con". + +inline "cic:/CoRN/ftc/FunctSeries/conv_zero_fun_series.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_zero.con". + +(* begin show *) + +alias id "f" = "cic:/CoRN/ftc/FunctSeries/Operations/f.var". + +alias id "g" = "cic:/CoRN/ftc/FunctSeries/Operations/g.var". + +(* end show *) + +inline "cic:/CoRN/ftc/FunctSeries/fun_series_convergent_wd.con". + +(* begin show *) + +alias id "convF" = "cic:/CoRN/ftc/FunctSeries/Operations/convF.var". + +alias id "convG" = "cic:/CoRN/ftc/FunctSeries/Operations/convG.var". + +(* end show *) + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_wd'.con". + +inline "cic:/CoRN/ftc/FunctSeries/conv_fun_series_plus.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_plus.con". + +inline "cic:/CoRN/ftc/FunctSeries/conv_fun_series_minus.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_min.con". + +(*#* +%\begin{convention}% Let [c:IR]. +%\end{convention}% +*) + +alias id "c" = "cic:/CoRN/ftc/FunctSeries/Operations/c.var". + +alias id "H" = "cic:/CoRN/ftc/FunctSeries/Operations/H.var". + +alias id "contH" = "cic:/CoRN/ftc/FunctSeries/Operations/contH.var". + +inline "cic:/CoRN/ftc/FunctSeries/conv_fun_series_scal.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_scal.con". + +(* UNEXPORTED +End Operations +*) + +(* UNEXPORTED +Section More_Operations +*) + +alias id "a" = "cic:/CoRN/ftc/FunctSeries/More_Operations/a.var". + +alias id "b" = "cic:/CoRN/ftc/FunctSeries/More_Operations/b.var". + +alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/More_Operations/Hab.var". + +(* begin hide *) + +inline "cic:/CoRN/ftc/FunctSeries/More_Operations/I.con" "More_Operations__". + +(* end hide *) + +alias id "f" = "cic:/CoRN/ftc/FunctSeries/More_Operations/f.var". + +alias id "convF" = "cic:/CoRN/ftc/FunctSeries/More_Operations/convF.var". + +inline "cic:/CoRN/ftc/FunctSeries/conv_fun_series_inv.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_inv.con". + +(* UNEXPORTED +End More_Operations +*) + +(* UNEXPORTED +Section Other_Results +*) + +alias id "a" = "cic:/CoRN/ftc/FunctSeries/Other_Results/a.var". + +alias id "b" = "cic:/CoRN/ftc/FunctSeries/Other_Results/b.var". + +alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Other_Results/Hab.var". + +alias id "f" = "cic:/CoRN/ftc/FunctSeries/Other_Results/f.var". + +alias id "convF" = "cic:/CoRN/ftc/FunctSeries/Other_Results/convF.var". + +(*#* +The following relate the sum series with the limit of the sequence of +partial sums; as a corollary we get the continuity of the sum of the +series. +*) + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_char'.con". + +inline "cic:/CoRN/ftc/FunctSeries/fun_series_conv.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_cont.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_char.con". + +inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_as_Lim.con". + +(* UNEXPORTED +End Other_Results +*) + +(* UNEXPORTED +Hint Resolve Fun_Series_Sum_cont: continuous. +*) + +(* UNEXPORTED +Section Convergence_Criteria +*) + +(*#* **Convergence Criteria + +Most of the convergence criteria for series of real numbers carry over to series of real-valued functions, so again we just present them without comments. +*) + +alias id "a" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/a.var". + +alias id "b" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/b.var". + +alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/Hab.var". + +(* begin hide *) + +inline "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/I.con" "Convergence_Criteria__". + +(* end hide *) + +alias id "f" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/f.var". + +alias id "contF" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/contF.var". + +(* UNEXPORTED +Opaque Frestr. +*) + +(* UNEXPORTED +Transparent Frestr. +*) + +(* UNEXPORTED +Opaque FAbs. +*) + +(* UNEXPORTED +Transparent FAbs. +*) + +(* UNEXPORTED +Opaque fun_seq_part_sum. +*) + +(* UNEXPORTED +Opaque FAbs. +*) + +(* UNEXPORTED +Transparent FAbs. +*) + +(* UNEXPORTED +Opaque FAbs. +*) + +(* UNEXPORTED +Transparent fun_seq_part_sum. +*) + +(* UNEXPORTED +Opaque fun_seq_part_sum. +*) + +inline "cic:/CoRN/ftc/FunctSeries/fun_str_comparison.con". + +(* UNEXPORTED +Transparent FAbs. +*) + +inline "cic:/CoRN/ftc/FunctSeries/fun_comparison.con". + +inline "cic:/CoRN/ftc/FunctSeries/abs_imp_conv.con". + +(* UNEXPORTED +Opaque FAbs. +*) + +inline "cic:/CoRN/ftc/FunctSeries/fun_ratio_test_conv.con". + +(* UNEXPORTED +End Convergence_Criteria +*) +