--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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}% #∑<sup>n</sup># *)
+
+(*#* printing Fun_Series_Sum %\ensuremath{\sum_0^{\infty}}% #∑<sub>0</sub><sup>∞</sup># *)
+
+(* 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
+*)
+
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