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}% #∑<sup>n</sup># *)
(*#* 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
%\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
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
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
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
%\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
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.
%\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
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
%\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
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
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
*)