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[helm.git] / matita / contribs / CoRN-Decl / reals / Cauchy_CReals.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/reals/Cauchy_CReals".
+
+include "CoRN.ma".
+
+(* $Id: Cauchy_CReals.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
+
+include "algebra/Cauchy_COF.ma".
+
+include "reals/CReals.ma".
+
+(* UNEXPORTED
+Section R_CReals
+*)
+
+(*#* * The Real Number Structure
+
+We will now apply our Cauchy sequence construction to an archimedean ordered field in order to obtain a model of the real numbers.
+
+** Injection of [Q]
+
+We start by showing how to inject the rational numbers in the field of Cauchy sequences; this embedding preserves the algebraic operations.
+
+%\begin{convention}% Let [F] be an ordered field.
+%\end{convention}%
+*)
+
+alias id "F" = "cic:/CoRN/reals/Cauchy_CReals/R_CReals/F.var".
+
+(* NOTATION
+Notation "'R_COrdField''" := (R_COrdField F).
+*)
+
+inline "cic:/CoRN/reals/Cauchy_CReals/inject_Q.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_eq.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_plus.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_min.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_lt.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_ap.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_eq.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_less.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_le.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_leEq.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_AbsSmall.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_One.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_nring'.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_nring.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_mult.con".
+
+(* UNEXPORTED
+Opaque R_COrdField.
+*)
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_div_three.con".
+
+(* UNEXPORTED
+Transparent R_COrdField.
+*)
+
+inline "cic:/CoRN/reals/Cauchy_CReals/ing_n.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/conv_modulus.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/R_CReals/T.con" "R_CReals__".
+
+(*#* We now assume our original field is archimedean and prove that the
+resulting one is, too.
+*)
+
+alias id "F_is_archemaedian" = "cic:/CoRN/reals/Cauchy_CReals/R_CReals/F_is_archemaedian.var".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/R_is_archemaedian.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/reals/Cauchy_CReals/R_CReals/PT.con" "R_CReals__".
+
+(* end hide *)
+
+inline "cic:/CoRN/reals/Cauchy_CReals/modulus_property.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/modulus_property_2.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R_2.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/CS_seq_diagonal.con".
+
+(*#* ** Cauchy Completeness
+We can also define a limit operator.
+*)
+
+inline "cic:/CoRN/reals/Cauchy_CReals/Q_dense_in_R.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/LimR_CauchySeq.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/R_is_complete.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/R_is_CReals.con".
+
+inline "cic:/CoRN/reals/Cauchy_CReals/R_as_CReals.con".
+
+(* UNEXPORTED
+End R_CReals
+*)
+