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=601baed778a190b580982b588ebe49ba3f762b30;hp=8fca8e19f901f5504114f8d3556daed47b996fea;hpb=7a8f91f8aa2d6ba24bf6b3093866f759ee16e690;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 8fca8e19f..18f5f6ebb 100644 --- a/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunSeries.ma +++ b/helm/software/matita/contribs/CoRN-Decl/ftc/MoreFunSeries.ma @@ -16,7 +16,7 @@ set "baseuri" "cic:/matita/CoRN-Decl/ftc/MoreFunSeries". -include "CoRN_notation.ma". +include "CoRN.ma". (* $Id: MoreFunSeries.v,v 1.4 2004/04/23 10:00:59 lcf Exp $ *) @@ -27,7 +27,7 @@ include "ftc/MoreFunctions.ma". (*#* printing FSeries_Sum %\ensuremath{\sum_{\infty}}% #∑'# *) (* UNEXPORTED -Section Definitions. +Section Definitions *) (*#* *More on Sequences and Series @@ -52,15 +52,15 @@ 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". @@ -83,26 +83,26 @@ inline "cic:/CoRN/ftc/MoreFunSeries/Cauchy_fun_seq2_seq_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 *) @@ -111,11 +111,11 @@ 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. +End More_Definitions *) (* UNEXPORTED -Section Irrelevance_of_Proofs. +Section Irrelevance_of_Proofs *) (*#* ***Basic Properties @@ -123,25 +123,25 @@ 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 *) @@ -152,7 +152,7 @@ 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. +End Irrelevance_of_Proofs *) (* UNEXPORTED @@ -160,42 +160,42 @@ 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/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 *) @@ -214,7 +214,7 @@ 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. +End More_Properties *) (* UNEXPORTED @@ -222,7 +222,7 @@ Hint Resolve Cauchy_cont_Lim_IR: continuous. *) (* UNEXPORTED -Section Algebraic_Properties. +Section Algebraic_Properties *) (*#* ***Algebraic Properties @@ -230,15 +230,15 @@ 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". @@ -248,19 +248,19 @@ 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/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 *) @@ -271,32 +271,32 @@ 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. +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 *) @@ -317,11 +317,11 @@ 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. +End More_Algebraic_Properties *) (* UNEXPORTED -Section Other. +Section Other *) (*#* ***Miscellaneous @@ -340,11 +340,11 @@ Another interesting fact: if a sequence of constant functions converges then it inline "cic:/CoRN/ftc/MoreFunSeries/fun_const_Lim.con". (* UNEXPORTED -End Other. +End Other *) (* UNEXPORTED -Section Series_Definitions. +Section Series_Definitions *) (*#* **Series @@ -354,9 +354,9 @@ 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". @@ -364,7 +364,7 @@ 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 *) @@ -374,7 +374,7 @@ inline "cic:/CoRN/ftc/MoreFunSeries/fun_series_inc_IR.con". (* begin hide *) -inline "cic:/CoRN/ftc/MoreFunSeries/h.con". +inline "cic:/CoRN/ftc/MoreFunSeries/Series_Definitions/h.con" "Series_Definitions__". (* end hide *) @@ -385,7 +385,7 @@ inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum.con". inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_char.con". (* UNEXPORTED -End Series_Definitions. +End Series_Definitions *) (* UNEXPORTED @@ -393,12 +393,12 @@ 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. @@ -407,20 +407,20 @@ Absolute convergence still exists. 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". @@ -431,7 +431,7 @@ inline "cic:/CoRN/ftc/MoreFunSeries/convergent_imp_Continuous.con". inline "cic:/CoRN/ftc/MoreFunSeries/Continuous_FSeries_Sum.con". (* UNEXPORTED -End Convergence_Results. +End Convergence_Results *) (* UNEXPORTED @@ -443,7 +443,7 @@ Hint Resolve convergent_imp_Continuous Continuous_FSeries_Sum: continuous. *) (* UNEXPORTED -Section Operations. +Section Operations *) (*#* **Algebraic Operations @@ -451,7 +451,7 @@ 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". @@ -461,17 +461,17 @@ 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/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". (* 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 *) @@ -494,22 +494,22 @@ inline "cic:/CoRN/ftc/MoreFunSeries/FSeries_Sum_minus.con". %\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.con". (* UNEXPORTED -End Operations. +End Operations *) (* UNEXPORTED -Section Convergence_Criteria. +Section Convergence_Criteria *) (*#* ***Convergence Criteria @@ -518,11 +518,11 @@ 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". @@ -533,11 +533,11 @@ 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. +End Convergence_Criteria *) (* UNEXPORTED -Section Insert_Series. +Section Insert_Series *) (*#* ***Translation @@ -549,11 +549,11 @@ $f_i$#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". @@ -566,6 +566,6 @@ inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_conv.con". inline "cic:/CoRN/ftc/MoreFunSeries/insert_series_sum.con". (* UNEXPORTED -End Insert_Series. +End Insert_Series *)