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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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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/Bridges_LUB".
23 (* file : least_upper_bound_principle.v *)
25 (* version : 1.50 - 03/05/2001 *)
27 (* version : 1.00 - 27/02/2001 *)
29 (* author : Milad Niqui *)
31 (* language : coq 7.0beta26feb *)
33 (* dependency : iso_CReals.v , Expon.v *)
35 (* description : proof of the Bridges' least upper bound principle *)
37 include "reals/iso_CReals.ma".
39 include "algebra/Expon.ma".
45 alias id "R1" = "cic:/CoRN/reals/Bridges_LUB/LUBP/R1.var".
47 (* SUBSECTION ON GENRAL DEFINITIONS *)
50 Section lub_definitions
53 alias id "OF" = "cic:/CoRN/reals/Bridges_LUB/LUBP/lub_definitions/OF.var".
55 alias id "SS" = "cic:/CoRN/reals/Bridges_LUB/LUBP/lub_definitions/SS.var".
57 inline "cic:/CoRN/reals/Bridges_LUB/member.con".
59 inline "cic:/CoRN/reals/Bridges_LUB/Pmember.con".
61 inline "cic:/CoRN/reals/Bridges_LUB/is_upper_bound.con".
63 inline "cic:/CoRN/reals/Bridges_LUB/l_u_b.con".
65 inline "cic:/CoRN/reals/Bridges_LUB/supremum.con".
67 inline "cic:/CoRN/reals/Bridges_LUB/Psupremum.con".
69 (* the following definitions are not used in *)
71 (* this file but later we will need them *)
73 inline "cic:/CoRN/reals/Bridges_LUB/is_lower_bound.con".
75 inline "cic:/CoRN/reals/Bridges_LUB/g_l_b.con".
77 inline "cic:/CoRN/reals/Bridges_LUB/infimum.con".
79 inline "cic:/CoRN/reals/Bridges_LUB/Pinfimum.con".
88 Section upper_bound_sequence
91 alias id "A" = "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/A.var".
93 alias id "is_inhabitted" = "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/is_inhabitted.var".
95 alias id "bounded_above" = "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/bounded_above.var".
97 alias id "located" = "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/located.var".
99 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/s.con" "LUBP__upper_bound_sequence__".
101 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/Ps.con" "LUBP__upper_bound_sequence__".
103 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/b0.con" "LUBP__upper_bound_sequence__".
105 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/Pb0.con" "LUBP__upper_bound_sequence__".
107 inline "cic:/CoRN/reals/Bridges_LUB/b0_is_upper_bound.con".
109 inline "cic:/CoRN/reals/Bridges_LUB/s_inhabits_A.con".
111 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/dstart_l.con" "LUBP__upper_bound_sequence__".
113 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/dstart_r.con" "LUBP__upper_bound_sequence__".
115 inline "cic:/CoRN/reals/Bridges_LUB/dl_less_dr.con".
117 inline "cic:/CoRN/reals/Bridges_LUB/shrink23d.con".
119 inline "cic:/CoRN/reals/Bridges_LUB/shrink13d.con".
121 inline "cic:/CoRN/reals/Bridges_LUB/shrink24d.con".
123 inline "cic:/CoRN/reals/Bridges_LUB/Real_Interval.con".
125 inline "cic:/CoRN/reals/Bridges_LUB/dcotrans_analyze.con".
127 inline "cic:/CoRN/reals/Bridges_LUB/dcotrans_analyze_strong.con".
130 Notation "( p , q )" := (pairT p q).
133 inline "cic:/CoRN/reals/Bridges_LUB/dif_cotrans.con".
135 inline "cic:/CoRN/reals/Bridges_LUB/dif_cotrans_strong.con".
137 inline "cic:/CoRN/reals/Bridges_LUB/dIntrvl.con".
139 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/U.con" "LUBP__upper_bound_sequence__".
141 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/V.con" "LUBP__upper_bound_sequence__".
143 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/W.con" "LUBP__upper_bound_sequence__".
145 inline "cic:/CoRN/reals/Bridges_LUB/delta_dIntrvl.con".
147 inline "cic:/CoRN/reals/Bridges_LUB/Length_dIntrvl.con".
149 inline "cic:/CoRN/reals/Bridges_LUB/dIntrvl_inside_l_n.con".
151 inline "cic:/CoRN/reals/Bridges_LUB/dIntrvl_inside_r_n.con".
153 inline "cic:/CoRN/reals/Bridges_LUB/V_increase.con".
155 inline "cic:/CoRN/reals/Bridges_LUB/W_decrease.con".
157 inline "cic:/CoRN/reals/Bridges_LUB/U_m_n_V.con".
159 inline "cic:/CoRN/reals/Bridges_LUB/U_m_n_W.con".
161 (* These lemma are *very* similar to those in *)
163 (* Cauchy_rationals_approach_reals.v *)
165 inline "cic:/CoRN/reals/Bridges_LUB/a_familiar_simple_inequality.con".
167 inline "cic:/CoRN/reals/Bridges_LUB/U_conversion_rate2.con".
169 inline "cic:/CoRN/reals/Bridges_LUB/CS_seq_U.con".
171 inline "cic:/CoRN/reals/Bridges_LUB/U_as_CauchySeq.con".
173 inline "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/B.con" "LUBP__upper_bound_sequence__".
175 inline "cic:/CoRN/reals/Bridges_LUB/U_minus_V.con".
177 inline "cic:/CoRN/reals/Bridges_LUB/U_minus_W.con".
179 inline "cic:/CoRN/reals/Bridges_LUB/U_V_upper.con".
181 inline "cic:/CoRN/reals/Bridges_LUB/U_W_lower.con".
183 inline "cic:/CoRN/reals/Bridges_LUB/AbsSmall_U_V.con".
185 inline "cic:/CoRN/reals/Bridges_LUB/AbsSmall_U_W.con".
187 (* Two properties of exponentiation in COrdFields *)
189 inline "cic:/CoRN/reals/Bridges_LUB/nexp_resp_great_One.con".
191 inline "cic:/CoRN/reals/Bridges_LUB/very_weak_binomial.con".
193 (* A consequence of Archimedean property - *)
195 (* the every basis of definition of e=lim(1+1/n)^n *)
197 inline "cic:/CoRN/reals/Bridges_LUB/nexp_resp_Two.con".
199 inline "cic:/CoRN/reals/Bridges_LUB/twisted_archimedean.con".
201 inline "cic:/CoRN/reals/Bridges_LUB/B_limit_V.con".
203 inline "cic:/CoRN/reals/Bridges_LUB/B_limit_W.con".
205 inline "cic:/CoRN/reals/Bridges_LUB/W_n_is_upper.con".
207 inline "cic:/CoRN/reals/Bridges_LUB/A_bounds_V_n.con".
209 inline "cic:/CoRN/reals/Bridges_LUB/cauchy_gives_lub.con".
212 End upper_bound_sequence