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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 "CoRN.ma".
18
19 (* begin hide *)
20
21 (* file        : least_upper_bound_principle.v                     *)
22
23 (* version     : 1.50 - 03/05/2001                                 *)
24
25 (* version     : 1.00 - 27/02/2001                                 *)
26
27 (* author      : Milad Niqui                                       *)
28
29 (* language    : coq 7.0beta26feb                                  *)
30
31 (* dependency  : iso_CReals.v , Expon.v                            *)
32
33 (* description : proof of the Bridges' least upper bound principle *)
34
35 include "reals/iso_CReals.ma".
36
37 include "algebra/Expon.ma".
38
39 (* UNEXPORTED
40 Section LUBP
41 *)
42
43 (* UNEXPORTED
44 cic:/CoRN/reals/Bridges_LUB/LUBP/R1.var
45 *)
46
47 (* SUBSECTION ON GENRAL DEFINITIONS *)
48
49 (* UNEXPORTED
50 Section lub_definitions
51 *)
52
53 (* UNEXPORTED
54 cic:/CoRN/reals/Bridges_LUB/LUBP/lub_definitions/OF.var
55 *)
56
57 (* UNEXPORTED
58 cic:/CoRN/reals/Bridges_LUB/LUBP/lub_definitions/SS.var
59 *)
60
61 inline procedural "cic:/CoRN/reals/Bridges_LUB/member.con" as definition.
62
63 inline procedural "cic:/CoRN/reals/Bridges_LUB/Pmember.con" as definition.
64
65 inline procedural "cic:/CoRN/reals/Bridges_LUB/is_upper_bound.con" as definition.
66
67 inline procedural "cic:/CoRN/reals/Bridges_LUB/l_u_b.con" as definition.
68
69 inline procedural "cic:/CoRN/reals/Bridges_LUB/supremum.con" as definition.
70
71 inline procedural "cic:/CoRN/reals/Bridges_LUB/Psupremum.con" as definition.
72
73 (* the following definitions are not used in *)
74
75 (* this file but later we will need them     *)
76
77 inline procedural "cic:/CoRN/reals/Bridges_LUB/is_lower_bound.con" as definition.
78
79 inline procedural "cic:/CoRN/reals/Bridges_LUB/g_l_b.con" as definition.
80
81 inline procedural "cic:/CoRN/reals/Bridges_LUB/infimum.con" as definition.
82
83 inline procedural "cic:/CoRN/reals/Bridges_LUB/Pinfimum.con" as definition.
84
85 (* UNEXPORTED
86 End lub_definitions
87 *)
88
89 (* MAIN SECTION *)
90
91 (* UNEXPORTED
92 Section upper_bound_sequence
93 *)
94
95 (* UNEXPORTED
96 cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/A.var
97 *)
98
99 (* UNEXPORTED
100 cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/is_inhabitted.var
101 *)
102
103 (* UNEXPORTED
104 cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/bounded_above.var
105 *)
106
107 (* UNEXPORTED
108 cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/located.var
109 *)
110
111 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/s.con" "LUBP__upper_bound_sequence__" as definition.
112
113 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/Ps.con" "LUBP__upper_bound_sequence__" as definition.
114
115 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/b0.con" "LUBP__upper_bound_sequence__" as definition.
116
117 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/Pb0.con" "LUBP__upper_bound_sequence__" as definition.
118
119 inline procedural "cic:/CoRN/reals/Bridges_LUB/b0_is_upper_bound.con" as lemma.
120
121 inline procedural "cic:/CoRN/reals/Bridges_LUB/s_inhabits_A.con" as lemma.
122
123 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/dstart_l.con" "LUBP__upper_bound_sequence__" as definition.
124
125 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/dstart_r.con" "LUBP__upper_bound_sequence__" as definition.
126
127 inline procedural "cic:/CoRN/reals/Bridges_LUB/dl_less_dr.con" as lemma.
128
129 inline procedural "cic:/CoRN/reals/Bridges_LUB/shrink23d.con" as lemma.
130
131 inline procedural "cic:/CoRN/reals/Bridges_LUB/shrink13d.con" as lemma.
132
133 inline procedural "cic:/CoRN/reals/Bridges_LUB/shrink24d.con" as lemma.
134
135 inline procedural "cic:/CoRN/reals/Bridges_LUB/Real_Interval.con" as definition.
136
137 inline procedural "cic:/CoRN/reals/Bridges_LUB/dcotrans_analyze.con" as definition.
138
139 inline procedural "cic:/CoRN/reals/Bridges_LUB/dcotrans_analyze_strong.con" as lemma.
140
141 (* NOTATION
142 Notation "( p , q )" := (pairT p q).
143 *)
144
145 inline procedural "cic:/CoRN/reals/Bridges_LUB/dif_cotrans.con" as definition.
146
147 inline procedural "cic:/CoRN/reals/Bridges_LUB/dif_cotrans_strong.con" as lemma.
148
149 inline procedural "cic:/CoRN/reals/Bridges_LUB/dIntrvl.con" as definition.
150
151 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/U.con" "LUBP__upper_bound_sequence__" as definition.
152
153 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/V.con" "LUBP__upper_bound_sequence__" as definition.
154
155 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/W.con" "LUBP__upper_bound_sequence__" as definition.
156
157 inline procedural "cic:/CoRN/reals/Bridges_LUB/delta_dIntrvl.con" as lemma.
158
159 inline procedural "cic:/CoRN/reals/Bridges_LUB/Length_dIntrvl.con" as lemma.
160
161 inline procedural "cic:/CoRN/reals/Bridges_LUB/dIntrvl_inside_l_n.con" as lemma.
162
163 inline procedural "cic:/CoRN/reals/Bridges_LUB/dIntrvl_inside_r_n.con" as lemma.
164
165 inline procedural "cic:/CoRN/reals/Bridges_LUB/V_increase.con" as lemma.
166
167 inline procedural "cic:/CoRN/reals/Bridges_LUB/W_decrease.con" as lemma.
168
169 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_m_n_V.con" as lemma.
170
171 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_m_n_W.con" as lemma.
172
173 (*  These lemma are *very* similar to those in  *)
174
175 (*  Cauchy_rationals_approach_reals.v           *)
176
177 inline procedural "cic:/CoRN/reals/Bridges_LUB/a_familiar_simple_inequality.con" as lemma.
178
179 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_conversion_rate2.con" as lemma.
180
181 inline procedural "cic:/CoRN/reals/Bridges_LUB/CS_seq_U.con" as lemma.
182
183 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_as_CauchySeq.con" as definition.
184
185 inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/B.con" "LUBP__upper_bound_sequence__" as definition.
186
187 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_minus_V.con" as lemma.
188
189 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_minus_W.con" as lemma.
190
191 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_V_upper.con" as lemma.
192
193 inline procedural "cic:/CoRN/reals/Bridges_LUB/U_W_lower.con" as lemma.
194
195 inline procedural "cic:/CoRN/reals/Bridges_LUB/AbsSmall_U_V.con" as lemma.
196
197 inline procedural "cic:/CoRN/reals/Bridges_LUB/AbsSmall_U_W.con" as lemma.
198
199 (* Two properties of exponentiation in COrdFields *)
200
201 inline procedural "cic:/CoRN/reals/Bridges_LUB/nexp_resp_great_One.con" as lemma.
202
203 inline procedural "cic:/CoRN/reals/Bridges_LUB/very_weak_binomial.con" as lemma.
204
205 (* A consequence of Archimedean property -         *)
206
207 (* the every basis of definition of e=lim(1+1/n)^n *)
208
209 inline procedural "cic:/CoRN/reals/Bridges_LUB/nexp_resp_Two.con" as lemma.
210
211 inline procedural "cic:/CoRN/reals/Bridges_LUB/twisted_archimedean.con" as lemma.
212
213 inline procedural "cic:/CoRN/reals/Bridges_LUB/B_limit_V.con" as lemma.
214
215 inline procedural "cic:/CoRN/reals/Bridges_LUB/B_limit_W.con" as lemma.
216
217 inline procedural "cic:/CoRN/reals/Bridges_LUB/W_n_is_upper.con" as lemma.
218
219 inline procedural "cic:/CoRN/reals/Bridges_LUB/A_bounds_V_n.con" as lemma.
220
221 inline procedural "cic:/CoRN/reals/Bridges_LUB/cauchy_gives_lub.con" as theorem.
222
223 (* UNEXPORTED
224 End upper_bound_sequence
225 *)
226
227 (* UNEXPORTED
228 End LUBP
229 *)
230
231 (* end hide *)
232