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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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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/Cauchy_CReals".
21 (* $Id: Cauchy_CReals.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
23 include "algebra/Cauchy_COF.ma".
25 include "reals/CReals.ma".
31 (*#* * The Real Number Structure
33 We will now apply our Cauchy sequence construction to an archimedean ordered field in order to obtain a model of the real numbers.
37 We start by showing how to inject the rational numbers in the field of Cauchy sequences; this embedding preserves the algebraic operations.
39 %\begin{convention}% Let [F] be an ordered field.
43 alias id "F" = "cic:/CoRN/reals/Cauchy_CReals/R_CReals/F.var".
46 Notation "'R_COrdField''" := (R_COrdField F).
49 inline "cic:/CoRN/reals/Cauchy_CReals/inject_Q.con".
51 inline "cic:/CoRN/reals/Cauchy_CReals/ing_eq.con".
53 inline "cic:/CoRN/reals/Cauchy_CReals/ing_plus.con".
55 inline "cic:/CoRN/reals/Cauchy_CReals/ing_min.con".
57 inline "cic:/CoRN/reals/Cauchy_CReals/ing_lt.con".
59 inline "cic:/CoRN/reals/Cauchy_CReals/ing_ap.con".
61 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_eq.con".
63 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_less.con".
65 inline "cic:/CoRN/reals/Cauchy_CReals/ing_le.con".
67 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_leEq.con".
69 inline "cic:/CoRN/reals/Cauchy_CReals/ing_cancel_AbsSmall.con".
71 inline "cic:/CoRN/reals/Cauchy_CReals/ing_One.con".
73 inline "cic:/CoRN/reals/Cauchy_CReals/ing_nring'.con".
75 inline "cic:/CoRN/reals/Cauchy_CReals/ing_nring.con".
77 inline "cic:/CoRN/reals/Cauchy_CReals/ing_mult.con".
83 inline "cic:/CoRN/reals/Cauchy_CReals/ing_div_three.con".
86 Transparent R_COrdField.
89 inline "cic:/CoRN/reals/Cauchy_CReals/ing_n.con".
91 inline "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R.con".
93 inline "cic:/CoRN/reals/Cauchy_CReals/conv_modulus.con".
95 inline "cic:/CoRN/reals/Cauchy_CReals/R_CReals/T.con" "R_CReals__".
97 (*#* We now assume our original field is archimedean and prove that the
98 resulting one is, too.
101 alias id "F_is_archemaedian" = "cic:/CoRN/reals/Cauchy_CReals/R_CReals/F_is_archemaedian.var".
103 inline "cic:/CoRN/reals/Cauchy_CReals/R_is_archemaedian.con".
107 inline "cic:/CoRN/reals/Cauchy_CReals/R_CReals/PT.con" "R_CReals__".
111 inline "cic:/CoRN/reals/Cauchy_CReals/modulus_property.con".
113 inline "cic:/CoRN/reals/Cauchy_CReals/modulus_property_2.con".
115 inline "cic:/CoRN/reals/Cauchy_CReals/expand_Q_R_2.con".
117 inline "cic:/CoRN/reals/Cauchy_CReals/CS_seq_diagonal.con".
119 (*#* ** Cauchy Completeness
120 We can also define a limit operator.
123 inline "cic:/CoRN/reals/Cauchy_CReals/Q_dense_in_R.con".
125 inline "cic:/CoRN/reals/Cauchy_CReals/LimR_CauchySeq.con".
127 inline "cic:/CoRN/reals/Cauchy_CReals/R_is_complete.con".
129 inline "cic:/CoRN/reals/Cauchy_CReals/R_is_CReals.con".
131 inline "cic:/CoRN/reals/Cauchy_CReals/R_as_CReals.con".