helm/software/matita/contribs/procedural/CoRN/reals/CReals1.mma

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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" as lemma.
  40
  41 inline procedural "cic:/CoRN/reals/CReals1/twice_inv_seq.con" as definition.
  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" as lemma.
  49
  50 inline procedural "cic:/CoRN/reals/CReals1/Cauchy_abs.con" as lemma.
  51
  52 inline procedural "cic:/CoRN/reals/CReals1/Lim_abs.con" as lemma.
  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 (* UNEXPORTED
  79 cic:/CoRN/reals/CReals1/Subsequences/seq1.var
  80 *)
  81
  82 (* UNEXPORTED
  83 cic:/CoRN/reals/CReals1/Subsequences/seq2.var
  84 *)
  85
  86 (* UNEXPORTED
  87 cic:/CoRN/reals/CReals1/Subsequences/f.var
  88 *)
  89
  90 (* UNEXPORTED
  91 cic:/CoRN/reals/CReals1/Subsequences/monF.var
  92 *)
  93
  94 (* UNEXPORTED
  95 cic:/CoRN/reals/CReals1/Subsequences/crescF.var
  96 *)
  97
  98 (* UNEXPORTED
  99 cic:/CoRN/reals/CReals1/Subsequences/subseq.var
 100 *)
 101
 102 (* UNEXPORTED
 103 cic:/CoRN/reals/CReals1/Subsequences/mon_seq2.var
 104 *)
 105
 106 inline procedural "cic:/CoRN/reals/CReals1/unbnd_f.con" as lemma.
 107
 108 (* begin hide *)
 109
 110 inline procedural "cic:/CoRN/reals/CReals1/Subsequences/mon_F'.con" "Subsequences__" as definition.
 111
 112 (* end hide *)
 113
 114 inline procedural "cic:/CoRN/reals/CReals1/conv_subseq_imp_conv_seq.con" as lemma.
 115
 116 inline procedural "cic:/CoRN/reals/CReals1/Cprop2_seq_imp_Cprop2_subseq.con" as lemma.
 117
 118 inline procedural "cic:/CoRN/reals/CReals1/conv_seq_imp_conv_subseq.con" as lemma.
 119
 120 inline procedural "cic:/CoRN/reals/CReals1/Cprop2_subseq_imp_Cprop2_seq.con" as lemma.
 121
 122 (* UNEXPORTED
 123 End Subsequences
 124 *)
 125
 126 (* UNEXPORTED
 127 Section Cauchy_Subsequences
 128 *)
 129
 130 (* UNEXPORTED
 131 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq1.var
 132 *)
 133
 134 (* UNEXPORTED
 135 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq2.var
 136 *)
 137
 138 (* UNEXPORTED
 139 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/f.var
 140 *)
 141
 142 (* UNEXPORTED
 143 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/monF.var
 144 *)
 145
 146 (* UNEXPORTED
 147 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/crescF.var
 148 *)
 149
 150 (* UNEXPORTED
 151 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/subseq.var
 152 *)
 153
 154 (* UNEXPORTED
 155 cic:/CoRN/reals/CReals1/Cauchy_Subsequences/mon_seq2.var
 156 *)
 157
 158 inline procedural "cic:/CoRN/reals/CReals1/Lim_seq_eq_Lim_subseq.con" as lemma.
 159
 160 inline procedural "cic:/CoRN/reals/CReals1/Lim_subseq_eq_Lim_seq.con" as lemma.
 161
 162 (* UNEXPORTED
 163 End Cauchy_Subsequences
 164 *)
 165
 166 (* UNEXPORTED
 167 Section Properties_of_Exponentiation
 168 *)
 169
 170 (*#* *** More properties of Exponentiation
 171
 172 Finally, we prove that [x[^]n] grows to infinity if [x [>] One].
 173 *)
 174
 175 inline procedural "cic:/CoRN/reals/CReals1/power_big'.con" as lemma.
 176
 177 inline procedural "cic:/CoRN/reals/CReals1/power_big.con" as lemma.
 178
 179 inline procedural "cic:/CoRN/reals/CReals1/qi_yields_zero.con" as lemma.
 180
 181 inline procedural "cic:/CoRN/reals/CReals1/qi_lim_zero.con" as lemma.
 182
 183 (* UNEXPORTED
 184 End Properties_of_Exponentiation
 185 *)
 186
 187 (*#* *** [IR] has characteristic zero *)
 188
 189 inline procedural "cic:/CoRN/reals/CReals1/char0_IR.con" as lemma.
 190
 191 inline procedural "cic:/CoRN/reals/CReals1/poly_apzero_IR.con" as lemma.
 192
 193 inline procedural "cic:/CoRN/reals/CReals1/poly_IR_extensional.con" as lemma.
 194