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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/CReals1".
+
+include "CoRN.ma".
+
+(* $Id: CReals1.v,v 1.4 2004/04/23 10:01:04 lcf Exp $ *)
+
+include "reals/Max_AbsIR.ma".
+
+include "algebra/Expon.ma".
+
+include "algebra/CPoly_ApZero.ma".
+
+(* UNEXPORTED
+Section More_Cauchy_Props
+*)
+
+(*#* **Miscellaneous
+*** More properties of Cauchy sequences
+We will now define some special Cauchy sequences and prove some 
+more useful properties about them.
+
+The sequence defined by $x_n=\frac2{n+1}$#x(n)=2/(n+1)#.
+*)
+
+inline "cic:/CoRN/reals/CReals1/twice_inv_seq_Lim.con".
+
+inline "cic:/CoRN/reals/CReals1/twice_inv_seq.con".
+
+(*#* 
+Next, we prove that the sequence of absolute values of a Cauchy 
+sequence is also Cauchy.
+*)
+
+inline "cic:/CoRN/reals/CReals1/Cauchy_Lim_abs.con".
+
+inline "cic:/CoRN/reals/CReals1/Cauchy_abs.con".
+
+inline "cic:/CoRN/reals/CReals1/Lim_abs.con".
+
+(* UNEXPORTED
+End More_Cauchy_Props
+*)
+
+(* UNEXPORTED
+Section Subsequences
+*)
+
+(*#* *** Subsequences
+We will now examine (although without formalizing it) the concept 
+of subsequence and some of its properties.
+
+%\begin{convention}% Let [seq1,seq2:nat->IR].
+%\end{convention}%
+
+In order for [seq1] to be a subsequence of [seq2], there must be an
+increasing function [f] growing to infinity such that
+[forall (n :nat), (seq1 n) [=] (seq2 (f n))].  We assume [f] to be such a
+function.
+
+Finally, for some of our results it is important to assume that 
+[seq2] is monotonous.
+*)
+
+alias id "seq1" = "cic:/CoRN/reals/CReals1/Subsequences/seq1.var".
+
+alias id "seq2" = "cic:/CoRN/reals/CReals1/Subsequences/seq2.var".
+
+alias id "f" = "cic:/CoRN/reals/CReals1/Subsequences/f.var".
+
+alias id "monF" = "cic:/CoRN/reals/CReals1/Subsequences/monF.var".
+
+alias id "crescF" = "cic:/CoRN/reals/CReals1/Subsequences/crescF.var".
+
+alias id "subseq" = "cic:/CoRN/reals/CReals1/Subsequences/subseq.var".
+
+alias id "mon_seq2" = "cic:/CoRN/reals/CReals1/Subsequences/mon_seq2.var".
+
+inline "cic:/CoRN/reals/CReals1/unbnd_f.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/reals/CReals1/Subsequences/mon_F'.con" "Subsequences__".
+
+(* end hide *)
+
+inline "cic:/CoRN/reals/CReals1/conv_subseq_imp_conv_seq.con".
+
+inline "cic:/CoRN/reals/CReals1/Cprop2_seq_imp_Cprop2_subseq.con".
+
+inline "cic:/CoRN/reals/CReals1/conv_seq_imp_conv_subseq.con".
+
+inline "cic:/CoRN/reals/CReals1/Cprop2_subseq_imp_Cprop2_seq.con".
+
+(* UNEXPORTED
+End Subsequences
+*)
+
+(* UNEXPORTED
+Section Cauchy_Subsequences
+*)
+
+alias id "seq1" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq1.var".
+
+alias id "seq2" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/seq2.var".
+
+alias id "f" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/f.var".
+
+alias id "monF" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/monF.var".
+
+alias id "crescF" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/crescF.var".
+
+alias id "subseq" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/subseq.var".
+
+alias id "mon_seq2" = "cic:/CoRN/reals/CReals1/Cauchy_Subsequences/mon_seq2.var".
+
+inline "cic:/CoRN/reals/CReals1/Lim_seq_eq_Lim_subseq.con".
+
+inline "cic:/CoRN/reals/CReals1/Lim_subseq_eq_Lim_seq.con".
+
+(* UNEXPORTED
+End Cauchy_Subsequences
+*)
+
+(* UNEXPORTED
+Section Properties_of_Exponentiation
+*)
+
+(*#* *** More properties of Exponentiation
+
+Finally, we prove that [x[^]n] grows to infinity if [x [>] One].
+*)
+
+inline "cic:/CoRN/reals/CReals1/power_big'.con".
+
+inline "cic:/CoRN/reals/CReals1/power_big.con".
+
+inline "cic:/CoRN/reals/CReals1/qi_yields_zero.con".
+
+inline "cic:/CoRN/reals/CReals1/qi_lim_zero.con".
+
+(* UNEXPORTED
+End Properties_of_Exponentiation
+*)
+
+(*#* *** [IR] has characteristic zero *)
+
+inline "cic:/CoRN/reals/CReals1/char0_IR.con".
+
+inline "cic:/CoRN/reals/CReals1/poly_apzero_IR.con".
+
+inline "cic:/CoRN/reals/CReals1/poly_IR_extensional.con".
+