X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FCoRN-Decl%2Freals%2FCauchy_CReals.ma;fp=matita%2Fcontribs%2FCoRN-Decl%2Freals%2FCauchy_CReals.ma;h=2fb5a06c35994ad4f2987c0b68d43dc2883d621e;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/CoRN-Decl/reals/Cauchy_CReals.ma b/matita/contribs/CoRN-Decl/reals/Cauchy_CReals.ma new file mode 100644 index 000000000..2fb5a06c3 --- /dev/null +++ b/matita/contribs/CoRN-Decl/reals/Cauchy_CReals.ma @@ -0,0 +1,136 @@ +(**************************************************************************) +(* ___ *) +(* ||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 +*) +