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