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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/reals/CReals1".
19 (* $Id: CReals1.v,v 1.4 2004/04/23 10:01:04 lcf Exp $ *)
34 Section More_Cauchy_Props.
38 *** More properties of Cauchy sequences
39 We will now define some special Cauchy sequences and prove some
40 more useful properties about them.
42 The sequence defined by $x_n=\frac2{n+1}$#x(n)=2/(n+1)#.
45 inline cic:/CoRN/reals/CReals1/twice_inv_seq_Lim.con.
47 inline cic:/CoRN/reals/CReals1/twice_inv_seq.con.
50 Next, we prove that the sequence of absolute values of a Cauchy
51 sequence is also Cauchy.
54 inline cic:/CoRN/reals/CReals1/Cauchy_Lim_abs.con.
56 inline cic:/CoRN/reals/CReals1/Cauchy_abs.con.
58 inline cic:/CoRN/reals/CReals1/Lim_abs.con.
61 End More_Cauchy_Props.
69 We will now examine (although without formalizing it) the concept
70 of subsequence and some of its properties.
72 %\begin{convention}% Let [seq1,seq2:nat->IR].
75 In order for [seq1] to be a subsequence of [seq2], there must be an
76 increasing function [f] growing to infinity such that
77 [forall (n :nat), (seq1 n) [=] (seq2 (f n))]. We assume [f] to be such a
80 Finally, for some of our results it is important to assume that
84 inline cic:/CoRN/reals/CReals1/seq1.var.
86 inline cic:/CoRN/reals/CReals1/seq2.var.
88 inline cic:/CoRN/reals/CReals1/f.var.
90 inline cic:/CoRN/reals/CReals1/monF.var.
92 inline cic:/CoRN/reals/CReals1/crescF.var.
94 inline cic:/CoRN/reals/CReals1/subseq.var.
96 inline cic:/CoRN/reals/CReals1/mon_seq2.var.
98 inline cic:/CoRN/reals/CReals1/unbnd_f.con.
102 inline cic:/CoRN/reals/CReals1/mon_F'.con.
106 inline cic:/CoRN/reals/CReals1/conv_subseq_imp_conv_seq.con.
108 inline cic:/CoRN/reals/CReals1/Cprop2_seq_imp_Cprop2_subseq.con.
110 inline cic:/CoRN/reals/CReals1/conv_seq_imp_conv_subseq.con.
112 inline cic:/CoRN/reals/CReals1/Cprop2_subseq_imp_Cprop2_seq.con.
119 Section Cauchy_Subsequences.
122 inline cic:/CoRN/reals/CReals1/seq1.var.
124 inline cic:/CoRN/reals/CReals1/seq2.var.
126 inline cic:/CoRN/reals/CReals1/f.var.
128 inline cic:/CoRN/reals/CReals1/monF.var.
130 inline cic:/CoRN/reals/CReals1/crescF.var.
132 inline cic:/CoRN/reals/CReals1/subseq.var.
134 inline cic:/CoRN/reals/CReals1/mon_seq2.var.
136 inline cic:/CoRN/reals/CReals1/Lim_seq_eq_Lim_subseq.con.
138 inline cic:/CoRN/reals/CReals1/Lim_subseq_eq_Lim_seq.con.
141 End Cauchy_Subsequences.
145 Section Properties_of_Exponentiation.
148 (*#* *** More properties of Exponentiation
150 Finally, we prove that [x[^]n] grows to infinity if [x [>] One].
153 inline cic:/CoRN/reals/CReals1/power_big'.con.
155 inline cic:/CoRN/reals/CReals1/power_big.con.
157 inline cic:/CoRN/reals/CReals1/qi_yields_zero.con.
159 inline cic:/CoRN/reals/CReals1/qi_lim_zero.con.
162 End Properties_of_Exponentiation.
165 (*#* *** [IR] has characteristic zero *)
167 inline cic:/CoRN/reals/CReals1/char0_IR.con.
169 inline cic:/CoRN/reals/CReals1/poly_apzero_IR.con.
171 inline cic:/CoRN/reals/CReals1/poly_IR_extensional.con.