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

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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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  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 alias id "F" = "cic:/CoRN/reals/Cauchy_CReals/R_CReals/F.var".
  42
  43 (* NOTATION
  44 Notation "'R_COrdField''" := (R_COrdField F).
  45 *)
  46
  47 inline procedural "cic:/CoRN/reals/Cauchy_CReals/inject_Q.con".
  48
  49 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_eq.con".
  50
  51 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_plus.con".
  52
  53 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_min.con".
  54
  55 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_lt.con".
  56
  57 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_ap.con".
  58
  59 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_eq.con".
  60
  61 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_less.con".
  62
  63 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_le.con".
  64
  65 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_leEq.con".
  66
  67 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_AbsSmall.con".
  68
  69 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_One.con".
  70
  71 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_nring'.con".
  72
  73 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_nring.con".
  74
  75 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_mult.con".
  76
  77 (* UNEXPORTED
  78 Opaque R_COrdField.
  79 *)
  80
  81 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_div_three.con".
  82
  83 (* UNEXPORTED
  84 Transparent R_COrdField.
  85 *)
  86
  87 inline procedural "cic:/CoRN/reals/Cauchy_CReals/ing_n.con".
  88
  89 inline procedural "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R.con".
  90
  91 inline procedural "cic:/CoRN/reals/Cauchy_CReals/conv_modulus.con".
  92
  93 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_CReals/T.con" "R_CReals__".
  94
  95 (*#* We now assume our original field is archimedean and prove that the
  96 resulting one is, too.
  97 *)
  98
  99 alias id "F_is_archemaedian" = "cic:/CoRN/reals/Cauchy_CReals/R_CReals/F_is_archemaedian.var".
 100
 101 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_is_archemaedian.con".
 102
 103 (* begin hide *)
 104
 105 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_CReals/PT.con" "R_CReals__".
 106
 107 (* end hide *)
 108
 109 inline procedural "cic:/CoRN/reals/Cauchy_CReals/modulus_property.con".
 110
 111 inline procedural "cic:/CoRN/reals/Cauchy_CReals/modulus_property_2.con".
 112
 113 inline procedural "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R_2.con".
 114
 115 inline procedural "cic:/CoRN/reals/Cauchy_CReals/CS_seq_diagonal.con".
 116
 117 (*#* ** Cauchy Completeness
 118 We can also define a limit operator.
 119 *)
 120
 121 inline procedural "cic:/CoRN/reals/Cauchy_CReals/Q_dense_in_R.con".
 122
 123 inline procedural "cic:/CoRN/reals/Cauchy_CReals/LimR_CauchySeq.con".
 124
 125 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_is_complete.con".
 126
 127 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_is_CReals.con".
 128
 129 inline procedural "cic:/CoRN/reals/Cauchy_CReals/R_as_CReals.con".
 130
 131 (* UNEXPORTED
 132 End R_CReals
 133 *)
 134