helm/software/matita/contribs/CoRN-Decl/reals/Cauchy_CReals.ma

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   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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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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  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/reals/Cauchy_CReals".
  18
  19 include "CoRN.ma".
  20
  21 (* $Id: Cauchy_CReals.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
  22
  23 include "algebra/Cauchy_COF.ma".
  24
  25 include "reals/CReals.ma".
  26
  27 (* UNEXPORTED
  28 Section R_CReals.
  29 *)
  30
  31 (*#* * The Real Number Structure
  32
  33 We will now apply our Cauchy sequence construction to an archimedean ordered field in order to obtain a model of the real numbers.
  34
  35 ** Injection of [Q]
  36
  37 We start by showing how to inject the rational numbers in the field of Cauchy sequences; this embedding preserves the algebraic operations.
  38
  39 %\begin{convention}% Let [F] be an ordered field.
  40 %\end{convention}%
  41 *)
  42
  43 inline "cic:/CoRN/reals/Cauchy_CReals/F.var".
  44
  45 inline "cic:/CoRN/reals/Cauchy_CReals/inject_Q.con".
  46
  47 inline "cic:/CoRN/reals/Cauchy_CReals/ing_eq.con".
  48
  49 inline "cic:/CoRN/reals/Cauchy_CReals/ing_plus.con".
  50
  51 inline "cic:/CoRN/reals/Cauchy_CReals/ing_min.con".
  52
  53 inline "cic:/CoRN/reals/Cauchy_CReals/ing_lt.con".
  54
  55 inline "cic:/CoRN/reals/Cauchy_CReals/ing_ap.con".
  56
  57 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_eq.con".
  58
  59 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_less.con".
  60
  61 inline "cic:/CoRN/reals/Cauchy_CReals/ing_le.con".
  62
  63 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_leEq.con".
  64
  65 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_AbsSmall.con".
  66
  67 inline "cic:/CoRN/reals/Cauchy_CReals/ing_One.con".
  68
  69 inline "cic:/CoRN/reals/Cauchy_CReals/ing_nring'.con".
  70
  71 inline "cic:/CoRN/reals/Cauchy_CReals/ing_nring.con".
  72
  73 inline "cic:/CoRN/reals/Cauchy_CReals/ing_mult.con".
  74
  75 (* UNEXPORTED
  76 Opaque R_COrdField.
  77 *)
  78
  79 inline "cic:/CoRN/reals/Cauchy_CReals/ing_div_three.con".
  80
  81 (* UNEXPORTED
  82 Transparent R_COrdField.
  83 *)
  84
  85 inline "cic:/CoRN/reals/Cauchy_CReals/ing_n.con".
  86
  87 inline "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R.con".
  88
  89 inline "cic:/CoRN/reals/Cauchy_CReals/conv_modulus.con".
  90
  91 inline "cic:/CoRN/reals/Cauchy_CReals/T.con".
  92
  93 (*#* We now assume our original field is archimedean and prove that the
  94 resulting one is, too.
  95 *)
  96
  97 inline "cic:/CoRN/reals/Cauchy_CReals/F_is_archemaedian.var".
  98
  99 inline "cic:/CoRN/reals/Cauchy_CReals/R_is_archemaedian.con".
 100
 101 (* begin hide *)
 102
 103 inline "cic:/CoRN/reals/Cauchy_CReals/PT.con".
 104
 105 (* end hide *)
 106
 107 inline "cic:/CoRN/reals/Cauchy_CReals/modulus_property.con".
 108
 109 inline "cic:/CoRN/reals/Cauchy_CReals/modulus_property_2.con".
 110
 111 inline "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R_2.con".
 112
 113 inline "cic:/CoRN/reals/Cauchy_CReals/CS_seq_diagonal.con".
 114
 115 (*#* ** Cauchy Completeness
 116 We can also define a limit operator.
 117 *)
 118
 119 inline "cic:/CoRN/reals/Cauchy_CReals/Q_dense_in_R.con".
 120
 121 inline "cic:/CoRN/reals/Cauchy_CReals/LimR_CauchySeq.con".
 122
 123 inline "cic:/CoRN/reals/Cauchy_CReals/R_is_complete.con".
 124
 125 inline "cic:/CoRN/reals/Cauchy_CReals/R_is_CReals.con".
 126
 127 inline "cic:/CoRN/reals/Cauchy_CReals/R_as_CReals.con".
 128
 129 (* UNEXPORTED
 130 End R_CReals.
 131 *)
 132