X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FCoRN-Decl%2Freals%2FSeries.ma;h=40315e1179aa41364aec8a9ea0f5f45ebaae0d26;hb=1470ff47df1349333c6b721a1c162cc7dfc6806f;hp=ed331dbd1ff2d7d1234a9048665ec928aa86fe70;hpb=3199f2a8428b5d19a3a803c1b03d9f90e4120f74;p=helm.git diff --git a/helm/software/matita/contribs/CoRN-Decl/reals/Series.ma b/helm/software/matita/contribs/CoRN-Decl/reals/Series.ma index ed331dbd1..40315e117 100644 --- a/helm/software/matita/contribs/CoRN-Decl/reals/Series.ma +++ b/helm/software/matita/contribs/CoRN-Decl/reals/Series.ma @@ -16,6 +16,8 @@ set "baseuri" "cic:/matita/CoRN-Decl/reals/Series". +include "CoRN.ma". + (* $Id: Series.v,v 1.6 2004/04/23 10:01:05 lcf Exp $ *) (*#* printing seq_part_sum %\ensuremath{\sum^n}% #∑n# *) @@ -24,16 +26,12 @@ set "baseuri" "cic:/matita/CoRN-Decl/reals/Series". (*#* printing pi %\ensuremath{\pi}% #π *) -(* INCLUDE -CSumsReals -*) +include "reals/CSumsReals.ma". -(* INCLUDE -NRootIR -*) +include "reals/NRootIR.ma". (* UNEXPORTED -Section Definitions. +Section Definitions *) (*#* *Series of Real Numbers @@ -47,9 +45,9 @@ To each such sequence we can assign a sequence of partial sums. %\end{convention}% *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/Definitions/x.var". -inline cic:/CoRN/reals/Series/seq_part_sum.con. +inline "cic:/CoRN/reals/Series/seq_part_sum.con". (*#* For subsequent purposes it will be very useful to be able to write the @@ -57,15 +55,15 @@ difference between two arbitrary elements of the sequence of partial sums as a sum of elements of the original sequence. *) -inline cic:/CoRN/reals/Series/seq_part_sum_n.con. +inline "cic:/CoRN/reals/Series/seq_part_sum_n.con". (*#* A series is convergent iff its sequence of partial Sums is a Cauchy sequence. To each convergent series we can assign a Sum. *) -inline cic:/CoRN/reals/Series/convergent.con. +inline "cic:/CoRN/reals/Series/convergent.con". -inline cic:/CoRN/reals/Series/series_sum.con. +inline "cic:/CoRN/reals/Series/series_sum.con". (*#* Divergence can be characterized in a positive way, which will sometimes be useful. We thus define divergence of sequences and series and prove the @@ -73,46 +71,46 @@ obvious fact that no sequence can be both convergent and divergent, whether considered either as a sequence or as a series. *) -inline cic:/CoRN/reals/Series/divergent_seq.con. +inline "cic:/CoRN/reals/Series/divergent_seq.con". -inline cic:/CoRN/reals/Series/divergent.con. +inline "cic:/CoRN/reals/Series/divergent.con". -inline cic:/CoRN/reals/Series/conv_imp_not_div.con. +inline "cic:/CoRN/reals/Series/conv_imp_not_div.con". -inline cic:/CoRN/reals/Series/div_imp_not_conv.con. +inline "cic:/CoRN/reals/Series/div_imp_not_conv.con". -inline cic:/CoRN/reals/Series/convergent_imp_not_divergent.con. +inline "cic:/CoRN/reals/Series/convergent_imp_not_divergent.con". -inline cic:/CoRN/reals/Series/divergent_imp_not_convergent.con. +inline "cic:/CoRN/reals/Series/divergent_imp_not_convergent.con". (*#* Finally we have the well known fact that every convergent series converges to zero as a sequence. *) -inline cic:/CoRN/reals/Series/series_seq_Lim.con. +inline "cic:/CoRN/reals/Series/series_seq_Lim.con". -inline cic:/CoRN/reals/Series/series_seq_Lim'.con. +inline "cic:/CoRN/reals/Series/series_seq_Lim'.con". (* UNEXPORTED -End Definitions. +End Definitions *) (* UNEXPORTED -Section More_Definitions. +Section More_Definitions *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/More_Definitions/x.var". (*#* We also define absolute convergence. *) -inline cic:/CoRN/reals/Series/abs_convergent.con. +inline "cic:/CoRN/reals/Series/abs_convergent.con". (* UNEXPORTED -End More_Definitions. +End More_Definitions *) (* UNEXPORTED -Section Power_Series. +Section Power_Series *) (*#* **Power Series @@ -120,7 +118,7 @@ Section Power_Series. Power series are an important special case. *) -inline cic:/CoRN/reals/Series/power_series.con. +inline "cic:/CoRN/reals/Series/power_series.con". (*#* Specially important is the case when [c] is a positive real number @@ -131,26 +129,26 @@ we can also compute its sum. %\end{convention}% *) -inline cic:/CoRN/reals/Series/c.var. +alias id "c" = "cic:/CoRN/reals/Series/Power_Series/c.var". -inline cic:/CoRN/reals/Series/H0c.var. +alias id "H0c" = "cic:/CoRN/reals/Series/Power_Series/H0c.var". -inline cic:/CoRN/reals/Series/Hc1.var. +alias id "Hc1" = "cic:/CoRN/reals/Series/Power_Series/Hc1.var". -inline cic:/CoRN/reals/Series/c_exp_Lim.con. +inline "cic:/CoRN/reals/Series/c_exp_Lim.con". -inline cic:/CoRN/reals/Series/power_series_Lim1.con. +inline "cic:/CoRN/reals/Series/power_series_Lim1.con". -inline cic:/CoRN/reals/Series/power_series_conv.con. +inline "cic:/CoRN/reals/Series/power_series_conv.con". -inline cic:/CoRN/reals/Series/power_series_sum.con. +inline "cic:/CoRN/reals/Series/power_series_sum.con". (* UNEXPORTED -End Power_Series. +End Power_Series *) (* UNEXPORTED -Section Operations. +Section Operations *) (*#* **Operations @@ -159,9 +157,9 @@ Some operations with series preserve convergence. We start by defining the series that is zero everywhere. *) -inline cic:/CoRN/reals/Series/conv_zero_series.con. +inline "cic:/CoRN/reals/Series/conv_zero_series.con". -inline cic:/CoRN/reals/Series/series_sum_zero.con. +inline "cic:/CoRN/reals/Series/series_sum_zero.con". (*#* Next we consider extensionality, as well as the sum and difference of two convergent series. @@ -170,58 +168,58 @@ of two convergent series. %\end{convention}% *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/Operations/x.var". -inline cic:/CoRN/reals/Series/y.var. +alias id "y" = "cic:/CoRN/reals/Series/Operations/y.var". -inline cic:/CoRN/reals/Series/convX.var. +alias id "convX" = "cic:/CoRN/reals/Series/Operations/convX.var". -inline cic:/CoRN/reals/Series/convY.var. +alias id "convY" = "cic:/CoRN/reals/Series/Operations/convY.var". -inline cic:/CoRN/reals/Series/convergent_wd.con. +inline "cic:/CoRN/reals/Series/convergent_wd.con". -inline cic:/CoRN/reals/Series/series_sum_wd.con. +inline "cic:/CoRN/reals/Series/series_sum_wd.con". -inline cic:/CoRN/reals/Series/conv_series_plus.con. +inline "cic:/CoRN/reals/Series/conv_series_plus.con". -inline cic:/CoRN/reals/Series/series_sum_plus.con. +inline "cic:/CoRN/reals/Series/series_sum_plus.con". -inline cic:/CoRN/reals/Series/conv_series_minus.con. +inline "cic:/CoRN/reals/Series/conv_series_minus.con". -inline cic:/CoRN/reals/Series/series_sum_minus.con. +inline "cic:/CoRN/reals/Series/series_sum_minus.con". (*#* Multiplication by a scalar [c] is also permitted. *) -inline cic:/CoRN/reals/Series/c.var. +alias id "c" = "cic:/CoRN/reals/Series/Operations/c.var". -inline cic:/CoRN/reals/Series/conv_series_mult_scal.con. +inline "cic:/CoRN/reals/Series/conv_series_mult_scal.con". -inline cic:/CoRN/reals/Series/series_sum_mult_scal.con. +inline "cic:/CoRN/reals/Series/series_sum_mult_scal.con". (* UNEXPORTED -End Operations. +End Operations *) (* UNEXPORTED -Section More_Operations. +Section More_Operations *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/More_Operations/x.var". -inline cic:/CoRN/reals/Series/convX.var. +alias id "convX" = "cic:/CoRN/reals/Series/More_Operations/convX.var". (*#* As a corollary, we get the series of the inverses. *) -inline cic:/CoRN/reals/Series/conv_series_inv.con. +inline "cic:/CoRN/reals/Series/conv_series_inv.con". -inline cic:/CoRN/reals/Series/series_sum_inv.con. +inline "cic:/CoRN/reals/Series/series_sum_inv.con". (* UNEXPORTED -End More_Operations. +End More_Operations *) (* UNEXPORTED -Section Almost_Everywhere. +Section Almost_Everywhere *) (*#* ** Almost Everywhere @@ -232,44 +230,44 @@ and derive an important corollary for series. Let [x,y : nat->IR] be equal after some natural number. *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/Almost_Everywhere/x.var". -inline cic:/CoRN/reals/Series/y.var. +alias id "y" = "cic:/CoRN/reals/Series/Almost_Everywhere/y.var". -inline cic:/CoRN/reals/Series/aew_eq.con. +inline "cic:/CoRN/reals/Series/aew_eq.con". -inline cic:/CoRN/reals/Series/aew_equal.var. +alias id "aew_equal" = "cic:/CoRN/reals/Series/Almost_Everywhere/aew_equal.var". -inline cic:/CoRN/reals/Series/aew_Cauchy.con. +inline "cic:/CoRN/reals/Series/aew_Cauchy.con". -inline cic:/CoRN/reals/Series/aew_Cauchy2.con. +inline "cic:/CoRN/reals/Series/aew_Cauchy2.con". -inline cic:/CoRN/reals/Series/aew_series_conv.con. +inline "cic:/CoRN/reals/Series/aew_series_conv.con". (* UNEXPORTED -End Almost_Everywhere. +End Almost_Everywhere *) (* UNEXPORTED -Section Cauchy_Almost_Everywhere. +Section Cauchy_Almost_Everywhere *) (*#* Suppose furthermore that [x,y] are Cauchy sequences. *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/Cauchy_Almost_Everywhere/x.var". -inline cic:/CoRN/reals/Series/y.var. +alias id "y" = "cic:/CoRN/reals/Series/Cauchy_Almost_Everywhere/y.var". -inline cic:/CoRN/reals/Series/aew_equal.var. +alias id "aew_equal" = "cic:/CoRN/reals/Series/Cauchy_Almost_Everywhere/aew_equal.var". -inline cic:/CoRN/reals/Series/aew_Lim.con. +inline "cic:/CoRN/reals/Series/aew_Lim.con". (* UNEXPORTED -End Cauchy_Almost_Everywhere. +End Cauchy_Almost_Everywhere *) (* UNEXPORTED -Section Convergence_Criteria. +Section Convergence_Criteria *) (*#* **Convergence Criteria @@ -278,48 +276,48 @@ Section Convergence_Criteria. %\end{convention}% *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/Convergence_Criteria/x.var". (*#* We include the comparison test for series, both in a strong and in a less general (but simpler) form. *) -inline cic:/CoRN/reals/Series/str_comparison.con. +inline "cic:/CoRN/reals/Series/str_comparison.con". -inline cic:/CoRN/reals/Series/comparison.con. +inline "cic:/CoRN/reals/Series/comparison.con". (*#* As a corollary, we get that every absolutely convergent series converges. *) -inline cic:/CoRN/reals/Series/abs_imp_conv.con. +inline "cic:/CoRN/reals/Series/abs_imp_conv.con". (*#* Next we have the ratio test, both as a positive and negative result. *) -inline cic:/CoRN/reals/Series/divergent_crit.con. +inline "cic:/CoRN/reals/Series/divergent_crit.con". -inline cic:/CoRN/reals/Series/tail_series.con. +inline "cic:/CoRN/reals/Series/tail_series.con". -inline cic:/CoRN/reals/Series/join_series.con. +inline "cic:/CoRN/reals/Series/join_series.con". (* UNEXPORTED -End Convergence_Criteria. +End Convergence_Criteria *) (* UNEXPORTED -Section More_CC. +Section More_CC *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/More_CC/x.var". -inline cic:/CoRN/reals/Series/ratio_test_conv.con. +inline "cic:/CoRN/reals/Series/ratio_test_conv.con". -inline cic:/CoRN/reals/Series/ratio_test_div.con. +inline "cic:/CoRN/reals/Series/ratio_test_div.con". (* UNEXPORTED -End More_CC. +End More_CC *) (* UNEXPORTED -Section Alternate_Series. +Section Alternate_Series *) (*#* **Alternate Series @@ -328,36 +326,36 @@ Alternate series are a special case. Suppose that [x] is nonnegative and decreasing convergent to 0. *) -inline cic:/CoRN/reals/Series/x.var. +alias id "x" = "cic:/CoRN/reals/Series/Alternate_Series/x.var". -inline cic:/CoRN/reals/Series/pos_x.var. +alias id "pos_x" = "cic:/CoRN/reals/Series/Alternate_Series/pos_x.var". -inline cic:/CoRN/reals/Series/Lim_x.var. +alias id "Lim_x" = "cic:/CoRN/reals/Series/Alternate_Series/Lim_x.var". -inline cic:/CoRN/reals/Series/mon_x.var. +alias id "mon_x" = "cic:/CoRN/reals/Series/Alternate_Series/mon_x.var". (* begin hide *) -inline cic:/CoRN/reals/Series/y.con. +inline "cic:/CoRN/reals/Series/Alternate_Series/y.con" "Alternate_Series__". -inline cic:/CoRN/reals/Series/alternate_lemma1.con. +inline "cic:/CoRN/reals/Series/Alternate_Series/alternate_lemma1.con" "Alternate_Series__". -inline cic:/CoRN/reals/Series/alternate_lemma2.con. +inline "cic:/CoRN/reals/Series/Alternate_Series/alternate_lemma2.con" "Alternate_Series__". -inline cic:/CoRN/reals/Series/alternate_lemma3.con. +inline "cic:/CoRN/reals/Series/Alternate_Series/alternate_lemma3.con" "Alternate_Series__". -inline cic:/CoRN/reals/Series/alternate_lemma4.con. +inline "cic:/CoRN/reals/Series/Alternate_Series/alternate_lemma4.con" "Alternate_Series__". (* end hide *) -inline cic:/CoRN/reals/Series/alternate_series_conv.con. +inline "cic:/CoRN/reals/Series/alternate_series_conv.con". (* UNEXPORTED -End Alternate_Series. +End Alternate_Series *) (* UNEXPORTED -Section Important_Numbers. +Section Important_Numbers *) (*#* **Important Numbers @@ -366,19 +364,19 @@ We end this chapter by defining two important numbers in mathematics: [pi] and $e$#e#, both as sums of convergent series. *) -inline cic:/CoRN/reals/Series/e_series.con. +inline "cic:/CoRN/reals/Series/e_series.con". -inline cic:/CoRN/reals/Series/e_series_conv.con. +inline "cic:/CoRN/reals/Series/e_series_conv.con". -inline cic:/CoRN/reals/Series/E.con. +inline "cic:/CoRN/reals/Series/E.con". -inline cic:/CoRN/reals/Series/pi_series.con. +inline "cic:/CoRN/reals/Series/pi_series.con". -inline cic:/CoRN/reals/Series/pi_series_conv.con. +inline "cic:/CoRN/reals/Series/pi_series_conv.con". -inline cic:/CoRN/reals/Series/pi.con. +inline "cic:/CoRN/reals/Series/pi.con". (* UNEXPORTED -End Important_Numbers. +End Important_Numbers *)