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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 (* This file was automatically generated: do not edit *********************)
16
17 include "Coq.ma".
18
19 (*#**********************************************************************)
20
21 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
22
23 (* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
24
25 (*   \VV/  *************************************************************)
26
27 (*    //   *      This file is distributed under the terms of the      *)
28
29 (*         *       GNU Lesser General Public License Version 2.1       *)
30
31 (*#**********************************************************************)
32
33 (*i $Id: Rtopology.v,v 1.19 2003/12/24 10:27:07 barras Exp $ i*)
34
35 include "Reals/Rbase.ma".
36
37 include "Reals/Rfunctions.ma".
38
39 include "Reals/Ranalysis1.ma".
40
41 include "Reals/RList.ma".
42
43 include "Logic/Classical_Prop.ma".
44
45 include "Logic/Classical_Pred_Type.ma".
46
47 (* UNEXPORTED
48 Open Local Scope R_scope.
49 *)
50
51 inline procedural "cic:/Coq/Reals/Rtopology/included.con" as definition.
52
53 inline procedural "cic:/Coq/Reals/Rtopology/disc.con" as definition.
54
55 inline procedural "cic:/Coq/Reals/Rtopology/neighbourhood.con" as definition.
56
57 inline procedural "cic:/Coq/Reals/Rtopology/open_set.con" as definition.
58
59 inline procedural "cic:/Coq/Reals/Rtopology/complementary.con" as definition.
60
61 inline procedural "cic:/Coq/Reals/Rtopology/closed_set.con" as definition.
62
63 inline procedural "cic:/Coq/Reals/Rtopology/intersection_domain.con" as definition.
64
65 inline procedural "cic:/Coq/Reals/Rtopology/union_domain.con" as definition.
66
67 inline procedural "cic:/Coq/Reals/Rtopology/interior.con" as definition.
68
69 inline procedural "cic:/Coq/Reals/Rtopology/interior_P1.con" as lemma.
70
71 inline procedural "cic:/Coq/Reals/Rtopology/interior_P2.con" as lemma.
72
73 inline procedural "cic:/Coq/Reals/Rtopology/point_adherent.con" as definition.
74
75 inline procedural "cic:/Coq/Reals/Rtopology/adherence.con" as definition.
76
77 inline procedural "cic:/Coq/Reals/Rtopology/adherence_P1.con" as lemma.
78
79 inline procedural "cic:/Coq/Reals/Rtopology/included_trans.con" as lemma.
80
81 inline procedural "cic:/Coq/Reals/Rtopology/interior_P3.con" as lemma.
82
83 inline procedural "cic:/Coq/Reals/Rtopology/complementary_P1.con" as lemma.
84
85 inline procedural "cic:/Coq/Reals/Rtopology/adherence_P2.con" as lemma.
86
87 inline procedural "cic:/Coq/Reals/Rtopology/adherence_P3.con" as lemma.
88
89 inline procedural "cic:/Coq/Reals/Rtopology/eq_Dom.con" as definition.
90
91 (* NOTATION
92 Infix "=_D" := eq_Dom (at level 70, no associativity).
93 *)
94
95 inline procedural "cic:/Coq/Reals/Rtopology/open_set_P1.con" as lemma.
96
97 inline procedural "cic:/Coq/Reals/Rtopology/closed_set_P1.con" as lemma.
98
99 inline procedural "cic:/Coq/Reals/Rtopology/neighbourhood_P1.con" as lemma.
100
101 inline procedural "cic:/Coq/Reals/Rtopology/open_set_P2.con" as lemma.
102
103 inline procedural "cic:/Coq/Reals/Rtopology/open_set_P3.con" as lemma.
104
105 inline procedural "cic:/Coq/Reals/Rtopology/open_set_P4.con" as lemma.
106
107 inline procedural "cic:/Coq/Reals/Rtopology/open_set_P5.con" as lemma.
108
109 inline procedural "cic:/Coq/Reals/Rtopology/disc_P1.con" as lemma.
110
111 inline procedural "cic:/Coq/Reals/Rtopology/continuity_P1.con" as lemma.
112
113 inline procedural "cic:/Coq/Reals/Rtopology/image_rec.con" as definition.
114
115 (*#*********)
116
117 inline procedural "cic:/Coq/Reals/Rtopology/continuity_P2.con" as lemma.
118
119 (*#*********)
120
121 inline procedural "cic:/Coq/Reals/Rtopology/continuity_P3.con" as lemma.
122
123 (*#*********)
124
125 inline procedural "cic:/Coq/Reals/Rtopology/Rsepare.con" as theorem.
126
127 inline procedural "cic:/Coq/Reals/Rtopology/family.ind".
128
129 inline procedural "cic:/Coq/Reals/Rtopology/family_open_set.con" as definition.
130
131 inline procedural "cic:/Coq/Reals/Rtopology/domain_finite.con" as definition.
132
133 inline procedural "cic:/Coq/Reals/Rtopology/family_finite.con" as definition.
134
135 inline procedural "cic:/Coq/Reals/Rtopology/covering.con" as definition.
136
137 inline procedural "cic:/Coq/Reals/Rtopology/covering_open_set.con" as definition.
138
139 inline procedural "cic:/Coq/Reals/Rtopology/covering_finite.con" as definition.
140
141 inline procedural "cic:/Coq/Reals/Rtopology/restriction_family.con" as lemma.
142
143 inline procedural "cic:/Coq/Reals/Rtopology/subfamily.con" as definition.
144
145 inline procedural "cic:/Coq/Reals/Rtopology/compact.con" as definition.
146
147 (*#*********)
148
149 inline procedural "cic:/Coq/Reals/Rtopology/family_P1.con" as lemma.
150
151 inline procedural "cic:/Coq/Reals/Rtopology/bounded.con" as definition.
152
153 inline procedural "cic:/Coq/Reals/Rtopology/open_set_P6.con" as lemma.
154
155 (*#*********)
156
157 inline procedural "cic:/Coq/Reals/Rtopology/compact_P1.con" as lemma.
158
159 (*#*********)
160
161 inline procedural "cic:/Coq/Reals/Rtopology/compact_P2.con" as lemma.
162
163 (*#*********)
164
165 inline procedural "cic:/Coq/Reals/Rtopology/compact_EMP.con" as lemma.
166
167 inline procedural "cic:/Coq/Reals/Rtopology/compact_eqDom.con" as lemma.
168
169 (* Borel-Lebesgue's lemma *)
170
171 inline procedural "cic:/Coq/Reals/Rtopology/compact_P3.con" as lemma.
172
173 inline procedural "cic:/Coq/Reals/Rtopology/compact_P4.con" as lemma.
174
175 (*#*********)
176
177 inline procedural "cic:/Coq/Reals/Rtopology/compact_P5.con" as lemma.
178
179 (*#*********)
180
181 inline procedural "cic:/Coq/Reals/Rtopology/compact_carac.con" as lemma.
182
183 inline procedural "cic:/Coq/Reals/Rtopology/image_dir.con" as definition.
184
185 (*#*********)
186
187 inline procedural "cic:/Coq/Reals/Rtopology/continuity_compact.con" as lemma.
188
189 inline procedural "cic:/Coq/Reals/Rtopology/Rlt_Rminus.con" as lemma.
190
191 inline procedural "cic:/Coq/Reals/Rtopology/prolongement_C0.con" as lemma.
192
193 (*#*********)
194
195 inline procedural "cic:/Coq/Reals/Rtopology/continuity_ab_maj.con" as lemma.
196
197 (*#*********)
198
199 inline procedural "cic:/Coq/Reals/Rtopology/continuity_ab_min.con" as lemma.
200
201 (*#*******************************************************)
202
203 (*         Proof of Bolzano-Weierstrass theorem         *)
204
205 (*#*******************************************************)
206
207 inline procedural "cic:/Coq/Reals/Rtopology/ValAdh.con" as definition.
208
209 inline procedural "cic:/Coq/Reals/Rtopology/intersection_family.con" as definition.
210
211 inline procedural "cic:/Coq/Reals/Rtopology/ValAdh_un_exists.con" as lemma.
212
213 inline procedural "cic:/Coq/Reals/Rtopology/ValAdh_un.con" as definition.
214
215 inline procedural "cic:/Coq/Reals/Rtopology/ValAdh_un_prop.con" as lemma.
216
217 inline procedural "cic:/Coq/Reals/Rtopology/adherence_P4.con" as lemma.
218
219 inline procedural "cic:/Coq/Reals/Rtopology/family_closed_set.con" as definition.
220
221 inline procedural "cic:/Coq/Reals/Rtopology/intersection_vide_in.con" as definition.
222
223 inline procedural "cic:/Coq/Reals/Rtopology/intersection_vide_finite_in.con" as definition.
224
225 (*#*********)
226
227 inline procedural "cic:/Coq/Reals/Rtopology/compact_P6.con" as lemma.
228
229 inline procedural "cic:/Coq/Reals/Rtopology/Bolzano_Weierstrass.con" as theorem.
230
231 (*#*******************************************************)
232
233 (*               Proof of Heine's theorem               *)
234
235 (*#*******************************************************)
236
237 inline procedural "cic:/Coq/Reals/Rtopology/uniform_continuity.con" as definition.
238
239 inline procedural "cic:/Coq/Reals/Rtopology/is_lub_u.con" as lemma.
240
241 inline procedural "cic:/Coq/Reals/Rtopology/domain_P1.con" as lemma.
242
243 inline procedural "cic:/Coq/Reals/Rtopology/Heine.con" as theorem.
244