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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "CoRN.ma".
  18
  19 (* $Id: Cauchy_CReals.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
  20
  21 include "algebra/Cauchy_COF.ma".
  22
  23 include "reals/CReals.ma".
  24
  25 (* UNEXPORTED
  26 Section R_CReals
  27 *)
  28
  29 (*#* * The Real Number Structure
  30
  31 We will now apply our Cauchy sequence construction to an archimedean ordered field in order to obtain a model of the real numbers.
  32
  33 ** Injection of [Q]
  34
  35 We start by showing how to inject the rational numbers in the field of Cauchy sequences; this embedding preserves the algebraic operations.
  36
  37 %\begin{convention}% Let [F] be an ordered field.
  38 %\end{convention}%
  39 *)
  40
  41 (* UNEXPORTED
  42 cic:/CoRN/reals/Cauchy_CReals/R_CReals/F.var
  43 *)
  44
  45 (* NOTATION
  46 Notation "'R_COrdField''" := (R_COrdField F).
  47 *)
  48
  49 inline procedural "cic:/CoRN/reals/Cauchy_CReals/inject_Q.con" as definition.
  50
  51 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_eq.con" as lemma.
  52
  53 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_plus.con" as lemma.
  54
  55 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_min.con" as lemma.
  56
  57 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_lt.con" as lemma.
  58
  59 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_ap.con" as lemma.
  60
  61 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_eq.con" as lemma.
  62
  63 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_less.con" as lemma.
  64
  65 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_le.con" as lemma.
  66
  67 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_leEq.con" as lemma.
  68
  69 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_AbsSmall.con" as lemma.
  70
  71 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_One.con" as lemma.
  72
  73 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_nring'.con" as lemma.
  74
  75 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_nring.con" as lemma.
  76
  77 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_mult.con" as lemma.
  78
  79 (* UNEXPORTED
  80 Opaque R_COrdField.
  81 *)
  82
  83 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_div_three.con" as lemma.
  84
  85 (* UNEXPORTED
  86 Transparent R_COrdField.
  87 *)
  88
  89 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_n.con" as lemma.
  90
  91 inline procedural "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R.con" as theorem.
  92
  93 inline procedural "cic:/CoRN/reals/Cauchy_CReals/conv_modulus.con" as lemma.
  94
  95 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_CReals/T.con" "R_CReals__" as definition.
  96
  97 (*#* We now assume our original field is archimedean and prove that the
  98 resulting one is, too.
  99 *)
 100
 101 (* UNEXPORTED
 102 cic:/CoRN/reals/Cauchy_CReals/R_CReals/F_is_archemaedian.var
 103 *)
 104
 105 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_is_archemaedian.con" as theorem.
 106
 107 (* begin hide *)
 108
 109 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_CReals/PT.con" "R_CReals__" as definition.
 110
 111 (* end hide *)
 112
 113 inline procedural "cic:/CoRN/reals/Cauchy_CReals/modulus_property.con" as lemma.
 114
 115 inline procedural "cic:/CoRN/reals/Cauchy_CReals/modulus_property_2.con" as lemma.
 116
 117 inline procedural "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R_2.con" as lemma.
 118
 119 inline procedural "cic:/CoRN/reals/Cauchy_CReals/CS_seq_diagonal.con" as lemma.
 120
 121 (*#* ** Cauchy Completeness
 122 We can also define a limit operator.
 123 *)
 124
 125 inline procedural "cic:/CoRN/reals/Cauchy_CReals/Q_dense_in_R.con" as lemma.
 126
 127 inline procedural "cic:/CoRN/reals/Cauchy_CReals/LimR_CauchySeq.con" as definition.
 128
 129 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_is_complete.con" as theorem.
 130
 131 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_is_CReals.con" as definition.
 132
 133 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_as_CReals.con" as definition.
 134
 135 (* UNEXPORTED
 136 End R_CReals
 137 *)
 138