X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FCoRN-Decl%2Freals%2FSeries.ma;h=40315e1179aa41364aec8a9ea0f5f45ebaae0d26;hb=a5bfa65b18a876fca982270f673c686a7d124f65;hp=50dd1005f84c5ef4610de0cce83e97d95867a0aa;hpb=f104e234238586ac846881feb30e1b56a509cfd3;p=helm.git diff --git a/matita/contribs/CoRN-Decl/reals/Series.ma b/matita/contribs/CoRN-Decl/reals/Series.ma index 50dd1005f..40315e117 100644 --- a/matita/contribs/CoRN-Decl/reals/Series.ma +++ b/matita/contribs/CoRN-Decl/reals/Series.ma @@ -31,7 +31,7 @@ include "reals/CSumsReals.ma". include "reals/NRootIR.ma". (* UNEXPORTED -Section Definitions. +Section Definitions *) (*#* *Series of Real Numbers @@ -45,7 +45,7 @@ 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". @@ -92,25 +92,25 @@ 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". (* UNEXPORTED -End More_Definitions. +End More_Definitions *) (* UNEXPORTED -Section Power_Series. +Section Power_Series *) (*#* **Power Series @@ -129,11 +129,11 @@ 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". @@ -144,11 +144,11 @@ inline "cic:/CoRN/reals/Series/power_series_conv.con". inline "cic:/CoRN/reals/Series/power_series_sum.con". (* UNEXPORTED -End Power_Series. +End Power_Series *) (* UNEXPORTED -Section Operations. +Section Operations *) (*#* **Operations @@ -168,13 +168,13 @@ 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". @@ -190,23 +190,23 @@ 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/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. *) @@ -215,11 +215,11 @@ inline "cic:/CoRN/reals/Series/conv_series_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 @@ -230,13 +230,13 @@ 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_equal.var". +alias id "aew_equal" = "cic:/CoRN/reals/Series/Almost_Everywhere/aew_equal.var". inline "cic:/CoRN/reals/Series/aew_Cauchy.con". @@ -245,29 +245,29 @@ inline "cic:/CoRN/reals/Series/aew_Cauchy2.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". (* UNEXPORTED -End Cauchy_Almost_Everywhere. +End Cauchy_Almost_Everywhere *) (* UNEXPORTED -Section Convergence_Criteria. +Section Convergence_Criteria *) (*#* **Convergence Criteria @@ -276,7 +276,7 @@ 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. @@ -299,25 +299,25 @@ inline "cic:/CoRN/reals/Series/tail_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_div.con". (* UNEXPORTED -End More_CC. +End More_CC *) (* UNEXPORTED -Section Alternate_Series. +Section Alternate_Series *) (*#* **Alternate Series @@ -326,36 +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". (* UNEXPORTED -End Alternate_Series. +End Alternate_Series *) (* UNEXPORTED -Section Important_Numbers. +Section Important_Numbers *) (*#* **Important Numbers @@ -377,6 +377,6 @@ inline "cic:/CoRN/reals/Series/pi_series_conv.con". inline "cic:/CoRN/reals/Series/pi.con". (* UNEXPORTED -End Important_Numbers. +End Important_Numbers *)