helm/software/matita/contribs/CoRN-Procedural/reals/Q_in_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: Q_in_CReals.v,v 1.10 2004/04/23 10:01:05 lcf Exp $ *)
  20
  21 (*#* * On density of the image of [Q] in an arbitrary real number structure
  22 In this file we introduce the image of the concrete rational numbers
  23 (as defined earlier) in an arbitrary structure of type
  24 [CReals]. At the end of this file we assign to any real number two
  25 rational numbers for which the real number lies betwen image of them;
  26 in other words we will prove that the image of rational numbers in
  27 dense in any real number structure. *)
  28
  29 include "model/reals/Cauchy_IR.ma".
  30
  31 include "model/monoids/Nmonoid.ma".
  32
  33 include "model/rings/Zring.ma".
  34
  35 (* UNEXPORTED
  36 Section Rational_sequence_prelogue
  37 *)
  38
  39 (*#*
  40 %\begin{convention}% Let [R1] be a real number structure.
  41 %\end{convention}%
  42 *)
  43
  44 alias id "R1" = "cic:/CoRN/reals/Q_in_CReals/Rational_sequence_prelogue/R1.var".
  45
  46 (* We clone these proofs from CReals1.v just because there IR is an axiom *)
  47
  48 (* begin hide *)
  49
  50 inline procedural "cic:/CoRN/reals/Q_in_CReals/CReals_is_CReals.con".
  51
  52 inline procedural "cic:/CoRN/reals/Q_in_CReals/Lim_Cauchy.con".
  53
  54 inline procedural "cic:/CoRN/reals/Q_in_CReals/Archimedes.con".
  55
  56 inline procedural "cic:/CoRN/reals/Q_in_CReals/Archimedes'.con".
  57
  58 (*#**************************************)
  59
  60 (* COERCION
  61 cic:/Coq/NArith/BinPos/nat_of_P.con
  62 *)
  63
  64 (* end hide *)
  65
  66 (*#*
  67 ** Injection from [Q] to an arbitrary real number structure
  68  First we need to define the injection from [Q] to [R1]. Note that in [Cauchy_CReals] we defined [inject_Q] from an arbitray field [F] to [(R_COrdField F)] which was the set of Cauchy sequences of that field. But since [R1] is an %\emph{arbitrary}%#<i>arbitrary</i># real number structure we can not use [inject_Q].
  69
  70  To define the injection we need one elemntary lemma about the denominator:
  71 *)
  72
  73 inline procedural "cic:/CoRN/reals/Q_in_CReals/den_is_nonzero.con".
  74
  75 (*#* And we define the injection in the natural way, using [zring] and [nring]. We call this [inj_Q], in contrast with [inject_Q] defined in [Cauchy_CReals]. *)
  76
  77 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q.con".
  78
  79 (*#* Next we need some properties of [nring], on the setoid of natural numbers: *)
  80
  81 inline procedural "cic:/CoRN/reals/Q_in_CReals/nring_strext.con".
  82
  83 inline procedural "cic:/CoRN/reals/Q_in_CReals/nring_wd.con".
  84
  85 inline procedural "cic:/CoRN/reals/Q_in_CReals/nring_eq.con".
  86
  87 inline procedural "cic:/CoRN/reals/Q_in_CReals/nring_leEq.con".
  88
  89 (* begin hide *)
  90
  91 (* UNEXPORTED
  92 Unset Printing Coercions.
  93 *)
  94
  95 (* end hide *)
  96
  97 (*#* Similarly we prove some properties of [zring] on the ring of integers: *)
  98
  99 inline procedural "cic:/CoRN/reals/Q_in_CReals/zring_strext.con".
 100
 101 inline procedural "cic:/CoRN/reals/Q_in_CReals/zring_wd.con".
 102
 103 inline procedural "cic:/CoRN/reals/Q_in_CReals/zring_less.con".
 104
 105 (*#* Using the above lemmata we prove the basic properties of [inj_Q], i.e.%\% it is a setoid function and preserves the ring operations and oreder operation. *)
 106
 107 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_strext.con".
 108
 109 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_wd.con".
 110
 111 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_plus.con".
 112
 113 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_mult.con".
 114
 115 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_less.con".
 116
 117 inline procedural "cic:/CoRN/reals/Q_in_CReals/less_inj_Q.con".
 118
 119 inline procedural "cic:/CoRN/reals/Q_in_CReals/leEq_inj_Q.con".
 120
 121 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_leEq.con".
 122
 123 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_min.con".
 124
 125 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_minus.con".
 126
 127 (*#* Moreover, and as expected, the [AbsSmall] predicate is also
 128 preserved under the [inj_Q] *)
 129
 130 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_AbsSmall.con".
 131
 132 inline procedural "cic:/CoRN/reals/Q_in_CReals/AbsSmall_inj_Q.con".
 133
 134 (*#* ** Injection preserves Cauchy property
 135 We apply the above lemmata to obtain following theorem, which says
 136 that a Cauchy sequence of elemnts of [Q] will be Cauchy in [R1].
 137 *)
 138
 139 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_Cauchy.con".
 140
 141 (*#* Furthermore we prove that applying [nring] (which is adding the
 142 ring unit [n] times) is the same whether we do it in [Q] or in [R1]:
 143 *)
 144
 145 inline procedural "cic:/CoRN/reals/Q_in_CReals/inj_Q_nring.con".
 146
 147 (*#* ** Injection of [Q] is dense
 148 Finally we are able to prove the density of image of [Q] in [R1]. We
 149 state this fact in two different ways. Both of them have their
 150 specific use.
 151
 152 The first theorem states the fact that any real number can be bound by
 153 the image of two rational numbers. This is called [start_of_sequence]
 154 because it can be used as an starting point for the typical "interval
 155 trisection" argument, which is ubiquitous in constructive analysis.
 156 *)
 157
 158 inline procedural "cic:/CoRN/reals/Q_in_CReals/start_of_sequence.con".
 159
 160 (*#* The second version of the density of [Q] in [R1] states that given
 161 any positive real number, there is a rational number between it and
 162 zero. This lemma can be used to prove the more general fact that there
 163 is a rational number between any two real numbers. *)
 164
 165 inline procedural "cic:/CoRN/reals/Q_in_CReals/Q_dense_in_CReals.con".
 166
 167 (* UNEXPORTED
 168 End Rational_sequence_prelogue
 169 *)
 170