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