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

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   3 (*      ||M||                                                             *)
   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 (* $Id: Cauchy_CReals.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
  20
  21 (* INCLUDE
  22 Cauchy_COF
  23 *)
  24
  25 (* INCLUDE
  26 CReals
  27 *)
  28
  29 (* UNEXPORTED
  30 Section R_CReals.
  31 *)
  32
  33 (*#* * The Real Number Structure
  34
  35 We will now apply our Cauchy sequence construction to an archimedean ordered field in order to obtain a model of the real numbers.
  36
  37 ** Injection of [Q]
  38
  39 We start by showing how to inject the rational numbers in the field of Cauchy sequences; this embedding preserves the algebraic operations.
  40
  41 %\begin{convention}% Let [F] be an ordered field.
  42 %\end{convention}%
  43 *)
  44
  45 inline cic:/CoRN/reals/Cauchy_CReals/F.var.
  46
  47 inline cic:/CoRN/reals/Cauchy_CReals/inject_Q.con.
  48
  49 inline cic:/CoRN/reals/Cauchy_CReals/ing_eq.con.
  50
  51 inline cic:/CoRN/reals/Cauchy_CReals/ing_plus.con.
  52
  53 inline cic:/CoRN/reals/Cauchy_CReals/ing_min.con.
  54
  55 inline cic:/CoRN/reals/Cauchy_CReals/ing_lt.con.
  56
  57 inline cic:/CoRN/reals/Cauchy_CReals/ing_ap.con.
  58
  59 inline cic:/CoRN/reals/Cauchy_CReals/ing_cancel_eq.con.
  60
  61 inline cic:/CoRN/reals/Cauchy_CReals/ing_cancel_less.con.
  62
  63 inline cic:/CoRN/reals/Cauchy_CReals/ing_le.con.
  64
  65 inline cic:/CoRN/reals/Cauchy_CReals/ing_cancel_leEq.con.
  66
  67 inline cic:/CoRN/reals/Cauchy_CReals/ing_cancel_AbsSmall.con.
  68
  69 inline cic:/CoRN/reals/Cauchy_CReals/ing_One.con.
  70
  71 inline cic:/CoRN/reals/Cauchy_CReals/ing_nring'.con.
  72
  73 inline cic:/CoRN/reals/Cauchy_CReals/ing_nring.con.
  74
  75 inline cic:/CoRN/reals/Cauchy_CReals/ing_mult.con.
  76
  77 (* UNEXPORTED
  78 Opaque R_COrdField.
  79 *)
  80
  81 inline cic:/CoRN/reals/Cauchy_CReals/ing_div_three.con.
  82
  83 (* UNEXPORTED
  84 Transparent R_COrdField.
  85 *)
  86
  87 inline cic:/CoRN/reals/Cauchy_CReals/ing_n.con.
  88
  89 inline cic:/CoRN/reals/Cauchy_CReals/expand_Q_R.con.
  90
  91 inline cic:/CoRN/reals/Cauchy_CReals/conv_modulus.con.
  92
  93 inline cic:/CoRN/reals/Cauchy_CReals/T.con.
  94
  95 (*#* We now assume our original field is archimedean and prove that the
  96 resulting one is, too.
  97 *)
  98
  99 inline cic:/CoRN/reals/Cauchy_CReals/F_is_archemaedian.var.
 100
 101 inline cic:/CoRN/reals/Cauchy_CReals/R_is_archemaedian.con.
 102
 103 (* begin hide *)
 104
 105 inline cic:/CoRN/reals/Cauchy_CReals/PT.con.
 106
 107 (* end hide *)
 108
 109 inline cic:/CoRN/reals/Cauchy_CReals/modulus_property.con.
 110
 111 inline cic:/CoRN/reals/Cauchy_CReals/modulus_property_2.con.
 112
 113 inline cic:/CoRN/reals/Cauchy_CReals/expand_Q_R_2.con.
 114
 115 inline 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 cic:/CoRN/reals/Cauchy_CReals/Q_dense_in_R.con.
 122
 123 inline cic:/CoRN/reals/Cauchy_CReals/LimR_CauchySeq.con.
 124
 125 inline cic:/CoRN/reals/Cauchy_CReals/R_is_complete.con.
 126
 127 inline cic:/CoRN/reals/Cauchy_CReals/R_is_CReals.con.
 128
 129 inline cic:/CoRN/reals/Cauchy_CReals/R_as_CReals.con.
 130
 131 (* UNEXPORTED
 132 End R_CReals.
 133 *)
 134