--- /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/MoreFunSeries".
+
+include "CoRN.ma".
+
+(* $Id: MoreFunSeries.v,v 1.4 2004/04/23 10:00:59 lcf Exp $ *)
+
+include "ftc/FunctSeries.ma".
+
+include "ftc/MoreFunctions.ma".
+
+(*#* printing FSeries_Sum %\ensuremath{\sum_{\infty}}% #∑'<sub>∞</sub># *)
+
+(* UNEXPORTED
+Section Definitions
+*)
+
+(*#* *More on Sequences and Series
+
+We will now extend our convergence definitions and results for
+sequences and series of functions defined in compact intervals to
+arbitrary intervals.
+
+%\begin{convention}% Throughout this file, [J] will be an interval,
+[f, g] will be sequences of continuous (in [J]) functions and [F,G]
+will be continuous (in [J]) functions.
+%\end{convention}%
+
+**Sequences
+
+First we will consider the case of sequences.
+
+***Definitions
+
+Some of the definitions do not make sense in this more general setting
+(for instance, because the norm of a function is no longer defined),
+but the ones which do we simply adapt in the usual way.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/f.var".
+
+alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/F.var".
+
+alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/contf.var".
+
+alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/contF.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq2_IR.con".
+
+(*#*
+The equivalences between these definitions still hold.
+*)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_Cauchy_fun_seq'_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_seq2_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq2_seq_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_real_IR.con".
+
+(* UNEXPORTED
+End Definitions
+*)
+
+(* UNEXPORTED
+Section More_Definitions
+*)
+
+(*#*
+Limit is defined and works in the same way as before.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/f.var".
+
+alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/contf.var".
+
+(* begin show *)
+
+alias id "conv" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/conv.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_Lim_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_Lim_char.con".
+
+(* UNEXPORTED
+End More_Definitions
+*)
+
+(* UNEXPORTED
+Section Irrelevance_of_Proofs
+*)
+
+(*#* ***Basic Properties
+
+Proofs are irrelevant as before---they just have to be present.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/f.var".
+
+(* begin show *)
+
+alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contf.var".
+
+alias id "contf0" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contf0.var".
+
+(* end show *)
+
+alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/F.var".
+
+(* begin show *)
+
+alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contF.var".
+
+alias id "contF0" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contF0.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wd_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq2_wd_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq_wd_IR.con".
+
+(* UNEXPORTED
+End Irrelevance_of_Proofs
+*)
+
+(* UNEXPORTED
+Opaque Cauchy_fun_seq_Lim_IR.
+*)
+
+(* UNEXPORTED
+Section More_Properties
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/g.var".
+
+(* begin show *)
+
+alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contf.var".
+
+alias id "contf0" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contf0.var".
+
+alias id "contg" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contg.var".
+
+alias id "contg0" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contg0.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_conv_fun_seq'_IR.con".
+
+alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/G.var".
+
+(* begin show *)
+
+alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contF.var".
+
+alias id "contF0" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contF0.var".
+
+alias id "contG" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contG.var".
+
+alias id "contG0" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contG0.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wdl_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wdr_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wdl'_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wdr'_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_cont_Lim_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_conv_fun_seq_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_Cauchy_fun_seq_IR.con".
+
+(* UNEXPORTED
+End More_Properties
+*)
+
+(* UNEXPORTED
+Hint Resolve Cauchy_cont_Lim_IR: continuous.
+*)
+
+(* UNEXPORTED
+Section Algebraic_Properties
+*)
+
+(*#* ***Algebraic Properties
+
+Algebraic operations still work well.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/g.var".
+
+alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contf.var".
+
+alias id "contg" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contg.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FLim_unique_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_wd_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_const_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_const_IR.con".
+
+alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/G.var".
+
+alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contG.var".
+
+(* begin show *)
+
+alias id "convF" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/convF.var".
+
+alias id "convG" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/convG.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_plus'_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_minus'_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_mult'_IR.con".
+
+(* UNEXPORTED
+End Algebraic_Properties
+*)
+
+(* UNEXPORTED
+Section More_Algebraic_Properties
+*)
+
+(*#*
+If we work with the limit function things fit in just as well.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/g.var".
+
+alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/contf.var".
+
+alias id "contg" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/contg.var".
+
+(* begin show *)
+
+alias id "Hf" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/Hf.var".
+
+alias id "Hg" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/Hg.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_plus_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_plus.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_inv_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_inv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_minus_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_minus.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_mult_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_mult.con".
+
+(* UNEXPORTED
+End More_Algebraic_Properties
+*)
+
+(* UNEXPORTED
+Section Other
+*)
+
+(*#* ***Miscellaneous
+
+Finally, we define a mapping between sequences of real numbers and sequences of (constant) functions and prove that convergence is preserved.
+*)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/seq_to_funseq.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/funseq_conv.con".
+
+(*#*
+Another interesting fact: if a sequence of constant functions converges then it must converge to a constant function.
+*)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_const_Lim.con".
+
+(* UNEXPORTED
+End Other
+*)
+
+(* UNEXPORTED
+Section Series_Definitions
+*)
+
+(*#* **Series
+
+We now consider series of functions defined in arbitrary intervals.
+
+Convergence is defined as expected---through convergence in every compact interval.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/f.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_convergent_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_conv_imp_conv_IR.con".
+
+(* begin show *)
+
+alias id "H" = "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/H.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_inc_IR.con".
+
+(*#* Assume [h(x)] is the pointwise series of [f(x)] *)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/h.con" "Series_Definitions__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_strext_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_char.con".
+
+(* UNEXPORTED
+End Series_Definitions
+*)
+
+(* UNEXPORTED
+Implicit Arguments FSeries_Sum [J f].
+*)
+
+(* UNEXPORTED
+Section More_Series_Definitions
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Series_Definitions/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Series_Definitions/f.var".
+
+(*#*
+Absolute convergence still exists.
+*)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_abs_convergent_IR.con".
+
+(* UNEXPORTED
+End More_Series_Definitions
+*)
+
+(* UNEXPORTED
+Section Convergence_Results
+*)
+
+(*#*
+As before, any series converges to its sum.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Results/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Results/f.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_conv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/convergent_imp_inc.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/convergent_imp_Continuous.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/Continuous_FSeries_Sum.con".
+
+(* UNEXPORTED
+End Convergence_Results
+*)
+
+(* UNEXPORTED
+Hint Resolve convergent_imp_inc: included.
+*)
+
+(* UNEXPORTED
+Hint Resolve convergent_imp_Continuous Continuous_FSeries_Sum: continuous.
+*)
+
+(* UNEXPORTED
+Section Operations
+*)
+
+(*#* **Algebraic Operations
+
+Convergence is well defined and preserved by operations.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Operations/J.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_const_series_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_const_series_Sum_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/conv_zero_fun_series_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_zero_IR.con".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Operations/f.var".
+
+alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/Operations/g.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_convergent_wd_IR.con".
+
+(* begin show *)
+
+alias id "convF" = "cic:/CoRN/ftc/MoreFunSeries/Operations/convF.var".
+
+alias id "convG" = "cic:/CoRN/ftc/MoreFunSeries/Operations/convG.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_wd'.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_plus_conv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_plus.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_inv_conv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_inv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus_conv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus.con".
+
+(*#*
+%\begin{convention}% Let [c:IR] and [H:PartIR] be continuous in [J].
+%\end{convention}%
+*)
+
+alias id "c" = "cic:/CoRN/ftc/MoreFunSeries/Operations/c.var".
+
+alias id "H" = "cic:/CoRN/ftc/MoreFunSeries/Operations/H.var".
+
+alias id "contH" = "cic:/CoRN/ftc/MoreFunSeries/Operations/contH.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_scal_conv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_scal.con".
+
+(* UNEXPORTED
+End Operations
+*)
+
+(* UNEXPORTED
+Section Convergence_Criteria
+*)
+
+(*#* ***Convergence Criteria
+
+The most important tests for convergence of series still apply: the
+comparison test (in both versions) and the ratio test.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Criteria/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Criteria/f.var".
+
+alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Criteria/contF.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_str_comparison_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_comparison_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/abs_imp_conv_IR.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/fun_ratio_test_conv_IR.con".
+
+(* UNEXPORTED
+End Convergence_Criteria
+*)
+
+(* UNEXPORTED
+Section Insert_Series
+*)
+
+(*#* ***Translation
+
+When working in particular with power series and Taylor series, it is
+sometimes useful to ``shift'' all the terms in the series one position
+forward, that is, replacing each $f_{i+1}$#f<sub>i+1</sub># with
+$f_i$#f<sub>i</sub># and inserting the null function in the first
+position. This does not affect convergence or the sum of the series.
+*)
+
+alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Insert_Series/J.var".
+
+alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Insert_Series/f.var".
+
+alias id "convF" = "cic:/CoRN/ftc/MoreFunSeries/Insert_Series/convF.var".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/insert_series.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_cont.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_sum_char.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_conv.con".
+
+inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_sum.con".
+
+(* UNEXPORTED
+End Insert_Series
+*)
+
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