matita/contribs/CoRN-Decl/reals/CReals1.ma

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  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/reals/CReals1".
  18
  19 include "CoRN.ma".
  20
  21 (* $Id: CReals1.v,v 1.4 2004/04/23 10:01:04 lcf Exp $ *)
  22
  23 include "reals/Max_AbsIR.ma".
  24
  25 include "algebra/Expon.ma".
  26
  27 include "algebra/CPoly_ApZero.ma".
  28
  29 (* UNEXPORTED
  30 Section More_Cauchy_Props
  31 *)
  32
  33 (*#* **Miscellaneous
  34 *** More properties of Cauchy sequences
  35 We will now define some special Cauchy sequences and prove some
  36 more useful properties about them.
  37
  38 The sequence defined by $x_n=\frac2{n+1}$#x(n)=2/(n+1)#.
  39 *)
  40
  41 inline "cic:/CoRN/reals/CReals1/twice_inv_seq_Lim.con".
  42
  43 inline "cic:/CoRN/reals/CReals1/twice_inv_seq.con".
  44
  45 (*#*
  46 Next, we prove that the sequence of absolute values of a Cauchy
  47 sequence is also Cauchy.
  48 *)
  49
  50 inline "cic:/CoRN/reals/CReals1/Cauchy_Lim_abs.con".
  51
  52 inline "cic:/CoRN/reals/CReals1/Cauchy_abs.con".
  53
  54 inline "cic:/CoRN/reals/CReals1/Lim_abs.con".
  55
  56 (* UNEXPORTED
  57 End More_Cauchy_Props
  58 *)
  59
  60 (* UNEXPORTED
  61 Section Subsequences
  62 *)
  63
  64 (*#* *** Subsequences
  65 We will now examine (although without formalizing it) the concept
  66 of subsequence and some of its properties.
  67
  68 %\begin{convention}% Let [seq1,seq2:nat->IR].
  69 %\end{convention}%
  70
  71 In order for [seq1] to be a subsequence of [seq2], there must be an
  72 increasing function [f] growing to infinity such that
  73 [forall (n :nat), (seq1 n) [=] (seq2 (f n))].  We assume [f] to be such a
  74 function.
  75
  76 Finally, for some of our results it is important to assume that
  77 [seq2] is monotonous.
  78 *)
  79
  80 alias id "seq1" = "cic:/CoRN/reals/CReals1/Subsequences/seq1.var".
  81
  82 alias id "seq2" = "cic:/CoRN/reals/CReals1/Subsequences/seq2.var".
  83
  84 alias id "f" = "cic:/CoRN/reals/CReals1/Subsequences/f.var".
  85
  86 alias id "monF" = "cic:/CoRN/reals/CReals1/Subsequences/monF.var".
  87
  88 alias id "crescF" = "cic:/CoRN/reals/CReals1/Subsequences/crescF.var".
  89
  90 alias id "subseq" = "cic:/CoRN/reals/CReals1/Subsequences/subseq.var".
  91
  92 alias id "mon_seq2" = "cic:/CoRN/reals/CReals1/Subsequences/mon_seq2.var".
  93
  94 inline "cic:/CoRN/reals/CReals1/unbnd_f.con".
  95
  96 (* begin hide *)
  97
  98 inline "cic:/CoRN/reals/CReals1/Subsequences/mon_F'.con" "Subsequences__".
  99
 100 (* end hide *)
 101
 102 inline "cic:/CoRN/reals/CReals1/conv_subseq_imp_conv_seq.con".
 103
 104 inline "cic:/CoRN/reals/CReals1/Cprop2_seq_imp_Cprop2_subseq.con".
 105
 106 inline "cic:/CoRN/reals/CReals1/conv_seq_imp_conv_subseq.con".
 107
 108 inline "cic:/CoRN/reals/CReals1/Cprop2_subseq_imp_Cprop2_seq.con".
 109
 110 (* UNEXPORTED
 111 End Subsequences
 112 *)
 113
 114 (* UNEXPORTED
 115 Section Cauchy_Subsequences
 116 *)
 117
 118 alias id "seq1" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq1.var".
 119
 120 alias id "seq2" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq2.var".
 121
 122 alias id "f" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/f.var".
 123
 124 alias id "monF" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/monF.var".
 125
 126 alias id "crescF" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/crescF.var".
 127
 128 alias id "subseq" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/subseq.var".
 129
 130 alias id "mon_seq2" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/mon_seq2.var".
 131
 132 inline "cic:/CoRN/reals/CReals1/Lim_seq_eq_Lim_subseq.con".
 133
 134 inline "cic:/CoRN/reals/CReals1/Lim_subseq_eq_Lim_seq.con".
 135
 136 (* UNEXPORTED
 137 End Cauchy_Subsequences
 138 *)
 139
 140 (* UNEXPORTED
 141 Section Properties_of_Exponentiation
 142 *)
 143
 144 (*#* *** More properties of Exponentiation
 145
 146 Finally, we prove that [x[^]n] grows to infinity if [x [>] One].
 147 *)
 148
 149 inline "cic:/CoRN/reals/CReals1/power_big'.con".
 150
 151 inline "cic:/CoRN/reals/CReals1/power_big.con".
 152
 153 inline "cic:/CoRN/reals/CReals1/qi_yields_zero.con".
 154
 155 inline "cic:/CoRN/reals/CReals1/qi_lim_zero.con".
 156
 157 (* UNEXPORTED
 158 End Properties_of_Exponentiation
 159 *)
 160
 161 (*#* *** [IR] has characteristic zero *)
 162
 163 inline "cic:/CoRN/reals/CReals1/char0_IR.con".
 164
 165 inline "cic:/CoRN/reals/CReals1/poly_apzero_IR.con".
 166
 167 inline "cic:/CoRN/reals/CReals1/poly_IR_extensional.con".
 168