(**************************************************************************) (* ___ *) (* ||M|| *) (* ||A|| A project by Andrea Asperti *) (* ||T|| *) (* ||I|| Developers: *) (* ||T|| The HELM team. *) (* ||A|| http://helm.cs.unibo.it *) (* \ / *) (* \ / This file is distributed under the terms of the *) (* v GNU General Public License Version 2 *) (* *) (**************************************************************************) (* This file was automatically generated: do not edit *********************) include "CoRN.ma". (* begin hide *) (* file : least_upper_bound_principle.v *) (* version : 1.50 - 03/05/2001 *) (* version : 1.00 - 27/02/2001 *) (* author : Milad Niqui *) (* language : coq 7.0beta26feb *) (* dependency : iso_CReals.v , Expon.v *) (* description : proof of the Bridges' least upper bound principle *) include "reals/iso_CReals.ma". include "algebra/Expon.ma". (* UNEXPORTED Section LUBP *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/R1.var *) (* SUBSECTION ON GENRAL DEFINITIONS *) (* UNEXPORTED Section lub_definitions *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/lub_definitions/OF.var *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/lub_definitions/SS.var *) inline procedural "cic:/CoRN/reals/Bridges_LUB/member.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/Pmember.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/is_upper_bound.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/l_u_b.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/supremum.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/Psupremum.con" as definition. (* the following definitions are not used in *) (* this file but later we will need them *) inline procedural "cic:/CoRN/reals/Bridges_LUB/is_lower_bound.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/g_l_b.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/infimum.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/Pinfimum.con" as definition. (* UNEXPORTED End lub_definitions *) (* MAIN SECTION *) (* UNEXPORTED Section upper_bound_sequence *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/A.var *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/is_inhabitted.var *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/bounded_above.var *) (* UNEXPORTED cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/located.var *) inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/s.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/Ps.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/b0.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/Pb0.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/b0_is_upper_bound.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/s_inhabits_A.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/dstart_l.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/dstart_r.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/dl_less_dr.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/shrink23d.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/shrink13d.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/shrink24d.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/Real_Interval.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/dcotrans_analyze.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/dcotrans_analyze_strong.con" as lemma. (* NOTATION Notation "( p , q )" := (pairT p q). *) inline procedural "cic:/CoRN/reals/Bridges_LUB/dif_cotrans.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/dif_cotrans_strong.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/dIntrvl.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/U.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/V.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/W.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/delta_dIntrvl.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/Length_dIntrvl.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/dIntrvl_inside_l_n.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/dIntrvl_inside_r_n.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/V_increase.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/W_decrease.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_m_n_V.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_m_n_W.con" as lemma. (* These lemma are *very* similar to those in *) (* Cauchy_rationals_approach_reals.v *) inline procedural "cic:/CoRN/reals/Bridges_LUB/a_familiar_simple_inequality.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_conversion_rate2.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/CS_seq_U.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_as_CauchySeq.con" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/LUBP/upper_bound_sequence/B.con" "LUBP__upper_bound_sequence__" as definition. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_minus_V.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_minus_W.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_V_upper.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/U_W_lower.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/AbsSmall_U_V.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/AbsSmall_U_W.con" as lemma. (* Two properties of exponentiation in COrdFields *) inline procedural "cic:/CoRN/reals/Bridges_LUB/nexp_resp_great_One.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/very_weak_binomial.con" as lemma. (* A consequence of Archimedean property - *) (* the every basis of definition of e=lim(1+1/n)^n *) inline procedural "cic:/CoRN/reals/Bridges_LUB/nexp_resp_Two.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/twisted_archimedean.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/B_limit_V.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/B_limit_W.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/W_n_is_upper.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/A_bounds_V_n.con" as lemma. inline procedural "cic:/CoRN/reals/Bridges_LUB/cauchy_gives_lub.con" as theorem. (* UNEXPORTED End upper_bound_sequence *) (* UNEXPORTED End LUBP *) (* end hide *)