helm/software/matita/contribs/CoRN-Procedural/reals/CReals1.mma

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   4 (*      ||A||       A project by Andrea Asperti                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "CoRN.ma".
  18
  19 (* $Id: CReals1.v,v 1.4 2004/04/23 10:01:04 lcf Exp $ *)
  20
  21 include "reals/Max_AbsIR.ma".
  22
  23 include "algebra/Expon.ma".
  24
  25 include "algebra/CPoly_ApZero.ma".
  26
  27 (* UNEXPORTED
  28 Section More_Cauchy_Props
  29 *)
  30
  31 (*#* **Miscellaneous
  32 *** More properties of Cauchy sequences
  33 We will now define some special Cauchy sequences and prove some
  34 more useful properties about them.
  35
  36 The sequence defined by $x_n=\frac2{n+1}$#x(n)=2/(n+1)#.
  37 *)
  38
  39 inline procedural "cic:/CoRN/reals/CReals1/twice_inv_seq_Lim.con".
  40
  41 inline procedural "cic:/CoRN/reals/CReals1/twice_inv_seq.con".
  42
  43 (*#*
  44 Next, we prove that the sequence of absolute values of a Cauchy
  45 sequence is also Cauchy.
  46 *)
  47
  48 inline procedural "cic:/CoRN/reals/CReals1/Cauchy_Lim_abs.con".
  49
  50 inline procedural "cic:/CoRN/reals/CReals1/Cauchy_abs.con".
  51
  52 inline procedural "cic:/CoRN/reals/CReals1/Lim_abs.con".
  53
  54 (* UNEXPORTED
  55 End More_Cauchy_Props
  56 *)
  57
  58 (* UNEXPORTED
  59 Section Subsequences
  60 *)
  61
  62 (*#* *** Subsequences
  63 We will now examine (although without formalizing it) the concept
  64 of subsequence and some of its properties.
  65
  66 %\begin{convention}% Let [seq1,seq2:nat->IR].
  67 %\end{convention}%
  68
  69 In order for [seq1] to be a subsequence of [seq2], there must be an
  70 increasing function [f] growing to infinity such that
  71 [forall (n :nat), (seq1 n) [=] (seq2 (f n))].  We assume [f] to be such a
  72 function.
  73
  74 Finally, for some of our results it is important to assume that
  75 [seq2] is monotonous.
  76 *)
  77
  78 alias id "seq1" = "cic:/CoRN/reals/CReals1/Subsequences/seq1.var".
  79
  80 alias id "seq2" = "cic:/CoRN/reals/CReals1/Subsequences/seq2.var".
  81
  82 alias id "f" = "cic:/CoRN/reals/CReals1/Subsequences/f.var".
  83
  84 alias id "monF" = "cic:/CoRN/reals/CReals1/Subsequences/monF.var".
  85
  86 alias id "crescF" = "cic:/CoRN/reals/CReals1/Subsequences/crescF.var".
  87
  88 alias id "subseq" = "cic:/CoRN/reals/CReals1/Subsequences/subseq.var".
  89
  90 alias id "mon_seq2" = "cic:/CoRN/reals/CReals1/Subsequences/mon_seq2.var".
  91
  92 inline procedural "cic:/CoRN/reals/CReals1/unbnd_f.con".
  93
  94 (* begin hide *)
  95
  96 inline procedural "cic:/CoRN/reals/CReals1/Subsequences/mon_F'.con" "Subsequences__".
  97
  98 (* end hide *)
  99
 100 inline procedural "cic:/CoRN/reals/CReals1/conv_subseq_imp_conv_seq.con".
 101
 102 inline procedural "cic:/CoRN/reals/CReals1/Cprop2_seq_imp_Cprop2_subseq.con".
 103
 104 inline procedural "cic:/CoRN/reals/CReals1/conv_seq_imp_conv_subseq.con".
 105
 106 inline procedural "cic:/CoRN/reals/CReals1/Cprop2_subseq_imp_Cprop2_seq.con".
 107
 108 (* UNEXPORTED
 109 End Subsequences
 110 *)
 111
 112 (* UNEXPORTED
 113 Section Cauchy_Subsequences
 114 *)
 115
 116 alias id "seq1" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq1.var".
 117
 118 alias id "seq2" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq2.var".
 119
 120 alias id "f" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/f.var".
 121
 122 alias id "monF" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/monF.var".
 123
 124 alias id "crescF" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/crescF.var".
 125
 126 alias id "subseq" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/subseq.var".
 127
 128 alias id "mon_seq2" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/mon_seq2.var".
 129
 130 inline procedural "cic:/CoRN/reals/CReals1/Lim_seq_eq_Lim_subseq.con".
 131
 132 inline procedural "cic:/CoRN/reals/CReals1/Lim_subseq_eq_Lim_seq.con".
 133
 134 (* UNEXPORTED
 135 End Cauchy_Subsequences
 136 *)
 137
 138 (* UNEXPORTED
 139 Section Properties_of_Exponentiation
 140 *)
 141
 142 (*#* *** More properties of Exponentiation
 143
 144 Finally, we prove that [x[^]n] grows to infinity if [x [>] One].
 145 *)
 146
 147 inline procedural "cic:/CoRN/reals/CReals1/power_big'.con".
 148
 149 inline procedural "cic:/CoRN/reals/CReals1/power_big.con".
 150
 151 inline procedural "cic:/CoRN/reals/CReals1/qi_yields_zero.con".
 152
 153 inline procedural "cic:/CoRN/reals/CReals1/qi_lim_zero.con".
 154
 155 (* UNEXPORTED
 156 End Properties_of_Exponentiation
 157 *)
 158
 159 (*#* *** [IR] has characteristic zero *)
 160
 161 inline procedural "cic:/CoRN/reals/CReals1/char0_IR.con".
 162
 163 inline procedural "cic:/CoRN/reals/CReals1/poly_apzero_IR.con".
 164
 165 inline procedural "cic:/CoRN/reals/CReals1/poly_IR_extensional.con".
 166