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
-FunctSequence
-*)
+include "ftc/FunctSequence.ma".
-(* INCLUDE
-Series
-*)
+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.
+Section Definitions
*)
(*#* *Series of Functions
convergence is simply the convergence of the sequence of partial sums.
*)
-inline cic:/CoRN/ftc/FunctSeries/a.var.
+alias id "a" = "cic:/CoRN/ftc/FunctSeries/Definitions/a.var".
-inline cic:/CoRN/ftc/FunctSeries/b.var.
+alias id "b" = "cic:/CoRN/ftc/FunctSeries/Definitions/b.var".
-inline cic:/CoRN/ftc/FunctSeries/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Definitions/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/FunctSeries/I.con.
+inline "cic:/CoRN/ftc/FunctSeries/Definitions/I.con" "Definitions__".
(* end hide *)
-inline cic:/CoRN/ftc/FunctSeries/f.var.
+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.con".
-inline cic:/CoRN/ftc/FunctSeries/fun_seq_part_sum_cont.con.
+inline "cic:/CoRN/ftc/FunctSeries/fun_seq_part_sum_cont.con".
-inline cic:/CoRN/ftc/FunctSeries/fun_series_convergent.con.
+inline "cic:/CoRN/ftc/FunctSeries/fun_series_convergent.con".
(*#*
For what comes up next we need to know that the convergence of a
real number series.
*)
-inline cic:/CoRN/ftc/FunctSeries/fun_series_conv_imp_conv.con.
+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 *)
-inline cic:/CoRN/ftc/FunctSeries/H.var.
+alias id "H" = "cic:/CoRN/ftc/FunctSeries/Definitions/H.var".
(* end show *)
(* begin hide *)
-inline cic:/CoRN/ftc/FunctSeries/contf.con.
+inline "cic:/CoRN/ftc/FunctSeries/Definitions/contf.con" "Definitions__".
-inline cic:/CoRN/ftc/FunctSeries/incf.con.
+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_strext.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum.con.
+inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum.con".
(* UNEXPORTED
-End Definitions.
+End Definitions
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section More_Definitions.
+Section More_Definitions
*)
-inline cic:/CoRN/ftc/FunctSeries/a.var.
+alias id "a" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/a.var".
-inline cic:/CoRN/ftc/FunctSeries/b.var.
+alias id "b" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/b.var".
-inline cic:/CoRN/ftc/FunctSeries/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/More_Definitions/Hab.var".
-inline cic:/CoRN/ftc/FunctSeries/f.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.
+inline "cic:/CoRN/ftc/FunctSeries/fun_series_abs_convergent.con".
(* UNEXPORTED
-End More_Definitions.
+End More_Definitions
*)
(* UNEXPORTED
-Section Operations.
+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.
*)
-inline cic:/CoRN/ftc/FunctSeries/a.var.
+alias id "a" = "cic:/CoRN/ftc/FunctSeries/Operations/a.var".
-inline cic:/CoRN/ftc/FunctSeries/b.var.
+alias id "b" = "cic:/CoRN/ftc/FunctSeries/Operations/b.var".
-inline cic:/CoRN/ftc/FunctSeries/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Operations/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/FunctSeries/I.con.
+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/fun_seq_part_sum_n.con".
-inline cic:/CoRN/ftc/FunctSeries/conv_fun_const_series.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/fun_const_series_sum.con".
-inline cic:/CoRN/ftc/FunctSeries/conv_zero_fun_series.con.
+inline "cic:/CoRN/ftc/FunctSeries/conv_zero_fun_series.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_zero.con.
+inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_zero.con".
(* begin show *)
-inline cic:/CoRN/ftc/FunctSeries/f.var.
+alias id "f" = "cic:/CoRN/ftc/FunctSeries/Operations/f.var".
-inline cic:/CoRN/ftc/FunctSeries/g.var.
+alias id "g" = "cic:/CoRN/ftc/FunctSeries/Operations/g.var".
(* end show *)
-inline cic:/CoRN/ftc/FunctSeries/fun_series_convergent_wd.con.
+inline "cic:/CoRN/ftc/FunctSeries/fun_series_convergent_wd.con".
(* begin show *)
-inline cic:/CoRN/ftc/FunctSeries/convF.var.
+alias id "convF" = "cic:/CoRN/ftc/FunctSeries/Operations/convF.var".
-inline cic:/CoRN/ftc/FunctSeries/convG.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/Fun_Series_Sum_wd'.con".
-inline cic:/CoRN/ftc/FunctSeries/conv_fun_series_plus.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/Fun_Series_Sum_plus.con".
-inline cic:/CoRN/ftc/FunctSeries/conv_fun_series_minus.con.
+inline "cic:/CoRN/ftc/FunctSeries/conv_fun_series_minus.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_min.con.
+inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_min.con".
(*#*
%\begin{convention}% Let [c:IR].
%\end{convention}%
*)
-inline cic:/CoRN/ftc/FunctSeries/c.var.
+alias id "c" = "cic:/CoRN/ftc/FunctSeries/Operations/c.var".
-inline cic:/CoRN/ftc/FunctSeries/H.var.
+alias id "H" = "cic:/CoRN/ftc/FunctSeries/Operations/H.var".
-inline cic:/CoRN/ftc/FunctSeries/contH.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/conv_fun_series_scal.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_scal.con.
+inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_scal.con".
(* UNEXPORTED
-End Operations.
+End Operations
*)
(* UNEXPORTED
-Section More_Operations.
+Section More_Operations
*)
-inline cic:/CoRN/ftc/FunctSeries/a.var.
+alias id "a" = "cic:/CoRN/ftc/FunctSeries/More_Operations/a.var".
-inline cic:/CoRN/ftc/FunctSeries/b.var.
+alias id "b" = "cic:/CoRN/ftc/FunctSeries/More_Operations/b.var".
-inline cic:/CoRN/ftc/FunctSeries/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/More_Operations/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/FunctSeries/I.con.
+inline "cic:/CoRN/ftc/FunctSeries/More_Operations/I.con" "More_Operations__".
(* end hide *)
-inline cic:/CoRN/ftc/FunctSeries/f.var.
+alias id "f" = "cic:/CoRN/ftc/FunctSeries/More_Operations/f.var".
-inline cic:/CoRN/ftc/FunctSeries/convF.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/conv_fun_series_inv.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_inv.con.
+inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_inv.con".
(* UNEXPORTED
-End More_Operations.
+End More_Operations
*)
(* UNEXPORTED
-Section Other_Results.
+Section Other_Results
*)
-inline cic:/CoRN/ftc/FunctSeries/a.var.
+alias id "a" = "cic:/CoRN/ftc/FunctSeries/Other_Results/a.var".
-inline cic:/CoRN/ftc/FunctSeries/b.var.
+alias id "b" = "cic:/CoRN/ftc/FunctSeries/Other_Results/b.var".
-inline cic:/CoRN/ftc/FunctSeries/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Other_Results/Hab.var".
-inline cic:/CoRN/ftc/FunctSeries/f.var.
+alias id "f" = "cic:/CoRN/ftc/FunctSeries/Other_Results/f.var".
-inline cic:/CoRN/ftc/FunctSeries/convF.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
series.
*)
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_char'.con.
+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_conv.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_cont.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_char.con".
-inline cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_as_Lim.con.
+inline "cic:/CoRN/ftc/FunctSeries/Fun_Series_Sum_as_Lim.con".
(* UNEXPORTED
-End Other_Results.
+End Other_Results
*)
(* UNEXPORTED
*)
(* UNEXPORTED
-Section Convergence_Criteria.
+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.
*)
-inline cic:/CoRN/ftc/FunctSeries/a.var.
+alias id "a" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/a.var".
-inline cic:/CoRN/ftc/FunctSeries/b.var.
+alias id "b" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/b.var".
-inline cic:/CoRN/ftc/FunctSeries/Hab.var.
+alias id "Hab" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/Hab.var".
(* begin hide *)
-inline cic:/CoRN/ftc/FunctSeries/I.con.
+inline "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/I.con" "Convergence_Criteria__".
(* end hide *)
-inline cic:/CoRN/ftc/FunctSeries/f.var.
+alias id "f" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/f.var".
-inline cic:/CoRN/ftc/FunctSeries/contF.var.
+alias id "contF" = "cic:/CoRN/ftc/FunctSeries/Convergence_Criteria/contF.var".
(* UNEXPORTED
Opaque Frestr.
Opaque fun_seq_part_sum.
*)
-inline cic:/CoRN/ftc/FunctSeries/fun_str_comparison.con.
+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/fun_comparison.con".
-inline cic:/CoRN/ftc/FunctSeries/abs_imp_conv.con.
+inline "cic:/CoRN/ftc/FunctSeries/abs_imp_conv.con".
(* UNEXPORTED
Opaque FAbs.
*)
-inline cic:/CoRN/ftc/FunctSeries/fun_ratio_test_conv.con.
+inline "cic:/CoRN/ftc/FunctSeries/fun_ratio_test_conv.con".
(* UNEXPORTED
-End Convergence_Criteria.
+End Convergence_Criteria
*)