X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Fftc%2FMoreFunSeries.ma;h=18f5f6ebb4d1fd5377950d8ee7b845c4c9e9bd73;hb=434258767bd3307ea05d9eab48892a6fff73888d;hp=0aae72baaf58770db90e4e755819290d45dddd97;hpb=3199f2a8428b5d19a3a803c1b03d9f90e4120f74;p=helm.git diff --git a/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunSeries.ma b/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunSeries.ma index 0aae72baa..18f5f6ebb 100644 --- a/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunSeries.ma +++ b/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunSeries.ma @@ -16,20 +16,18 @@ 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 -FunctSeries -*) +include "ftc/FunctSeries.ma". -(* INCLUDE -MoreFunctions -*) +include "ftc/MoreFunctions.ma". (*#* printing FSeries_Sum %\ensuremath{\sum_{\infty}}% #∑'# *) (* UNEXPORTED -Section Definitions. +Section Definitions *) (*#* *More on Sequences and Series @@ -54,70 +52,70 @@ Some of the definitions do not make sense in this more general setting but the ones which do we simply adapt in the usual way. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/F.var. +alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/F.var". -inline cic:/CoRN/ftc/MoreFunSeries/contf.var. +alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/Definitions/contf.var". -inline cic:/CoRN/ftc/MoreFunSeries/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/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/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. +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/conv_Cauchy_fun_seq'_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_seq2_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_seq2_seq_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_real_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_real_IR.con". (* UNEXPORTED -End Definitions. +End Definitions *) (* UNEXPORTED -Section More_Definitions. +Section More_Definitions *) (*#* Limit is defined and works in the same way as before. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/contf.var. +alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/More_Definitions/contf.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/conv.var. +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_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_Lim_char.con. +inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_Lim_char.con". (* UNEXPORTED -End More_Definitions. +End More_Definitions *) (* UNEXPORTED -Section Irrelevance_of_Proofs. +Section Irrelevance_of_Proofs *) (*#* ***Basic Properties @@ -125,36 +123,36 @@ Section Irrelevance_of_Proofs. Proofs are irrelevant as before---they just have to be present. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/f.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/contf.var. +alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contf.var". -inline cic:/CoRN/ftc/MoreFunSeries/contf0.var. +alias id "contf0" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contf0.var". (* end show *) -inline cic:/CoRN/ftc/MoreFunSeries/F.var. +alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/F.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/contF.var. +alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/Irrelevance_of_Proofs/contF.var". -inline cic:/CoRN/ftc/MoreFunSeries/contF0.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/conv_fun_seq'_wd_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq2_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. +inline "cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq_wd_IR.con". (* UNEXPORTED -End Irrelevance_of_Proofs. +End Irrelevance_of_Proofs *) (* UNEXPORTED @@ -162,61 +160,61 @@ Opaque Cauchy_fun_seq_Lim_IR. *) (* UNEXPORTED -Section More_Properties. +Section More_Properties *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/g.var. +alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/g.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/contf.var. +alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contf.var". -inline cic:/CoRN/ftc/MoreFunSeries/contf0.var. +alias id "contf0" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contf0.var". -inline cic:/CoRN/ftc/MoreFunSeries/contg.var. +alias id "contg" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contg.var". -inline cic:/CoRN/ftc/MoreFunSeries/contg0.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. +inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_conv_fun_seq'_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/F.var. +alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/F.var". -inline cic:/CoRN/ftc/MoreFunSeries/G.var. +alias id "G" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/G.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/contF.var. +alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contF.var". -inline cic:/CoRN/ftc/MoreFunSeries/contF0.var. +alias id "contF0" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contF0.var". -inline cic:/CoRN/ftc/MoreFunSeries/contG.var. +alias id "contG" = "cic:/CoRN/ftc/MoreFunSeries/More_Properties/contG.var". -inline cic:/CoRN/ftc/MoreFunSeries/contG0.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'_wdl_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wdr_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'_wdl'_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/conv_fun_seq'_wdr'_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_cont_Lim_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/Cauchy_conv_fun_seq_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_conv_fun_seq_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/conv_Cauchy_fun_seq_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/conv_Cauchy_fun_seq_IR.con". (* UNEXPORTED -End More_Properties. +End More_Properties *) (* UNEXPORTED @@ -224,7 +222,7 @@ Hint Resolve Cauchy_cont_Lim_IR: continuous. *) (* UNEXPORTED -Section Algebraic_Properties. +Section Algebraic_Properties *) (*#* ***Algebraic Properties @@ -232,98 +230,98 @@ Section Algebraic_Properties. Algebraic operations still work well. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/g.var. +alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/g.var". -inline cic:/CoRN/ftc/MoreFunSeries/contf.var. +alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contf.var". -inline cic:/CoRN/ftc/MoreFunSeries/contg.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/FLim_unique_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq_wd_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_Lim_seq_const_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_const_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_const_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/F.var. +alias id "F" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/F.var". -inline cic:/CoRN/ftc/MoreFunSeries/G.var. +alias id "G" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/G.var". -inline cic:/CoRN/ftc/MoreFunSeries/contF.var. +alias id "contF" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contF.var". -inline cic:/CoRN/ftc/MoreFunSeries/contG.var. +alias id "contG" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/contG.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/convF.var. +alias id "convF" = "cic:/CoRN/ftc/MoreFunSeries/Algebraic_Properties/convF.var". -inline cic:/CoRN/ftc/MoreFunSeries/convG.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_plus'_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_minus'_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_minus'_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_mult'_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_Lim_seq_mult'_IR.con". (* UNEXPORTED -End Algebraic_Properties. +End Algebraic_Properties *) (* UNEXPORTED -Section More_Algebraic_Properties. +Section More_Algebraic_Properties *) (*#* If we work with the limit function things fit in just as well. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/g.var. +alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/g.var". -inline cic:/CoRN/ftc/MoreFunSeries/contf.var. +alias id "contf" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/contf.var". -inline cic:/CoRN/ftc/MoreFunSeries/contg.var. +alias id "contg" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/contg.var". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/Hf.var. +alias id "Hf" = "cic:/CoRN/ftc/MoreFunSeries/More_Algebraic_Properties/Hf.var". -inline cic:/CoRN/ftc/MoreFunSeries/Hg.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_Lim_seq_plus_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_plus.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_Lim_seq_inv_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_inv.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_Lim_seq_minus_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_minus.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_Lim_seq_mult_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_mult.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_Cauchy_prop_mult.con". (* UNEXPORTED -End More_Algebraic_Properties. +End More_Algebraic_Properties *) (* UNEXPORTED -Section Other. +Section Other *) (*#* ***Miscellaneous @@ -331,22 +329,22 @@ Section Other. 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/seq_to_funseq.con". -inline cic:/CoRN/ftc/MoreFunSeries/funseq_conv.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. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_const_Lim.con". (* UNEXPORTED -End Other. +End Other *) (* UNEXPORTED -Section Series_Definitions. +Section Series_Definitions *) (*#* **Series @@ -356,38 +354,38 @@ We now consider series of functions defined in arbitrary intervals. Convergence is defined as expected---through convergence in every compact interval. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.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_convergent_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_series_conv_imp_conv_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_conv_imp_conv_IR.con". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/H.var. +alias id "H" = "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/H.var". (* end show *) -inline cic:/CoRN/ftc/MoreFunSeries/fun_series_inc_IR.con. +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/h.con. +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_strext_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum.con. +inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_char.con. +inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_char.con". (* UNEXPORTED -End Series_Definitions. +End Series_Definitions *) (* UNEXPORTED @@ -395,45 +393,45 @@ Implicit Arguments FSeries_Sum [J f]. *) (* UNEXPORTED -Section More_Series_Definitions. +Section More_Series_Definitions *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/More_Series_Definitions/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.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. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_abs_convergent_IR.con". (* UNEXPORTED -End More_Series_Definitions. +End More_Series_Definitions *) (* UNEXPORTED -Section Convergence_Results. +Section Convergence_Results *) (*#* As before, any series converges to its sum. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Results/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.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/FSeries_conv.con". -inline cic:/CoRN/ftc/MoreFunSeries/convergent_imp_inc.con. +inline "cic:/CoRN/ftc/MoreFunSeries/convergent_imp_inc.con". -inline cic:/CoRN/ftc/MoreFunSeries/convergent_imp_Continuous.con. +inline "cic:/CoRN/ftc/MoreFunSeries/convergent_imp_Continuous.con". -inline cic:/CoRN/ftc/MoreFunSeries/Continuous_FSeries_Sum.con. +inline "cic:/CoRN/ftc/MoreFunSeries/Continuous_FSeries_Sum.con". (* UNEXPORTED -End Convergence_Results. +End Convergence_Results *) (* UNEXPORTED @@ -445,7 +443,7 @@ Hint Resolve convergent_imp_Continuous Continuous_FSeries_Sum: continuous. *) (* UNEXPORTED -Section Operations. +Section Operations *) (*#* **Algebraic Operations @@ -453,65 +451,65 @@ Section Operations. Convergence is well defined and preserved by operations. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +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/conv_fun_const_series_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_const_series_Sum_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/conv_zero_fun_series_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_zero_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_zero_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Operations/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/g.var. +alias id "g" = "cic:/CoRN/ftc/MoreFunSeries/Operations/g.var". -inline cic:/CoRN/ftc/MoreFunSeries/fun_series_convergent_wd_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_convergent_wd_IR.con". (* begin show *) -inline cic:/CoRN/ftc/MoreFunSeries/convF.var. +alias id "convF" = "cic:/CoRN/ftc/MoreFunSeries/Operations/convF.var". -inline cic:/CoRN/ftc/MoreFunSeries/convG.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_wd'.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_plus_conv.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_plus.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_inv_conv.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_inv.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus_conv.con. +inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus_conv.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus.con. +inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus.con". (*#* %\begin{convention}% Let [c:IR] and [H:PartIR] be continuous in [J]. %\end{convention}% *) -inline cic:/CoRN/ftc/MoreFunSeries/c.var. +alias id "c" = "cic:/CoRN/ftc/MoreFunSeries/Operations/c.var". -inline cic:/CoRN/ftc/MoreFunSeries/H.var. +alias id "H" = "cic:/CoRN/ftc/MoreFunSeries/Operations/H.var". -inline cic:/CoRN/ftc/MoreFunSeries/contH.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_conv.con". -inline cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_scal.con. +inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_scal.con". (* UNEXPORTED -End Operations. +End Operations *) (* UNEXPORTED -Section Convergence_Criteria. +Section Convergence_Criteria *) (*#* ***Convergence Criteria @@ -520,26 +518,26 @@ The most important tests for convergence of series still apply: the comparison test (in both versions) and the ratio test. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Criteria/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Convergence_Criteria/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/contF.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_str_comparison_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_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/abs_imp_conv_IR.con". -inline cic:/CoRN/ftc/MoreFunSeries/fun_ratio_test_conv_IR.con. +inline "cic:/CoRN/ftc/MoreFunSeries/fun_ratio_test_conv_IR.con". (* UNEXPORTED -End Convergence_Criteria. +End Convergence_Criteria *) (* UNEXPORTED -Section Insert_Series. +Section Insert_Series *) (*#* ***Translation @@ -551,23 +549,23 @@ $f_i$#fi# and inserting the null function in the first position. This does not affect convergence or the sum of the series. *) -inline cic:/CoRN/ftc/MoreFunSeries/J.var. +alias id "J" = "cic:/CoRN/ftc/MoreFunSeries/Insert_Series/J.var". -inline cic:/CoRN/ftc/MoreFunSeries/f.var. +alias id "f" = "cic:/CoRN/ftc/MoreFunSeries/Insert_Series/f.var". -inline cic:/CoRN/ftc/MoreFunSeries/convF.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.con". -inline cic:/CoRN/ftc/MoreFunSeries/insert_series_cont.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_sum_char.con". -inline cic:/CoRN/ftc/MoreFunSeries/insert_series_conv.con. +inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_conv.con". -inline cic:/CoRN/ftc/MoreFunSeries/insert_series_sum.con. +inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_sum.con". (* UNEXPORTED -End Insert_Series. +End Insert_Series *)