1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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 *********************)
21 (* file : bridges_gives_our.v *)
23 (* version : 1.50 - 09/05/2001 *)
25 (* version : 1.00 - 09/03/2001 *)
27 (* author : Milad Niqui *)
29 (* language : coq7.0bet26feb *)
31 (* dependency : least_upper_bound_principle *)
33 (* description : Bridges' proof of Cauchy completeness in TCS-219 *)
35 include "reals/Bridges_LUB.ma".
37 (* This lemma comes from lemmas.v of Martijn Oostdijk *)
39 inline procedural "cic:/CoRN/reals/Bridges_iso/le_witness_informative.con" as lemma.
42 Section bridges_axioms_imply_ours
45 alias id "OF" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/OF.var".
47 alias id "lubp" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/lubp.var".
49 alias id "is_Archimedes" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/is_Archimedes.var".
51 inline procedural "cic:/CoRN/reals/Bridges_iso/is_Archimedes'.con" as lemma.
57 inline procedural "cic:/CoRN/reals/Bridges_iso/leEq_geEq.con" as lemma.
59 inline procedural "cic:/CoRN/reals/Bridges_iso/glbp.con" as theorem.
65 alias id "P" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/P.var".
67 inline procedural "cic:/CoRN/reals/Bridges_iso/inequality1.con" as lemma.
69 inline procedural "cic:/CoRN/reals/Bridges_iso/inequality2.con" as lemma.
71 inline procedural "cic:/CoRN/reals/Bridges_iso/inequality3.con" as lemma.
73 inline procedural "cic:/CoRN/reals/Bridges_iso/inequality4.con" as lemma.
75 inline procedural "cic:/CoRN/reals/Bridges_iso/Hum.con" as definition.
77 inline procedural "cic:/CoRN/reals/Bridges_iso/bound_tk1.con" as lemma.
79 inline procedural "cic:/CoRN/reals/Bridges_iso/bound_tk2.con" as lemma.
81 inline procedural "cic:/CoRN/reals/Bridges_iso/trick.con" as lemma.
83 inline procedural "cic:/CoRN/reals/Bridges_iso/trick'.con" as lemma.
85 inline procedural "cic:/CoRN/reals/Bridges_iso/up_bound_for_n_element.con" as theorem.
87 inline procedural "cic:/CoRN/reals/Bridges_iso/low_bound_for_n_element.con" as lemma.
89 inline procedural "cic:/CoRN/reals/Bridges_iso/saghf.con" as definition.
91 inline procedural "cic:/CoRN/reals/Bridges_iso/Psaghf.con" as lemma.
93 inline procedural "cic:/CoRN/reals/Bridges_iso/kaf.con" as definition.
95 inline procedural "cic:/CoRN/reals/Bridges_iso/Pkaf.con" as lemma.
97 alias id "is_finite_P" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/is_finite_P.var".
99 inline procedural "cic:/CoRN/reals/Bridges_iso/card.con" as definition.
101 inline procedural "cic:/CoRN/reals/Bridges_iso/Pcard1.con" as lemma.
103 inline procedural "cic:/CoRN/reals/Bridges_iso/seq.con" as definition.
105 inline procedural "cic:/CoRN/reals/Bridges_iso/Pseq1.con" as definition.
107 inline procedural "cic:/CoRN/reals/Bridges_iso/Pseq1_unfolded.con" as lemma.
109 inline procedural "cic:/CoRN/reals/Bridges_iso/indeks.con" as definition.
111 inline procedural "cic:/CoRN/reals/Bridges_iso/Pindeks.con" as lemma.
113 alias id "is_onto_seq_P" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/is_onto_seq_P.var".
115 inline procedural "cic:/CoRN/reals/Bridges_iso/P_is_inhabited.con" as lemma.
118 Lemma bounded_quantifier:(N:nat;phi,psi:nat->Prop)
119 ((m:nat)(le m N)->(phi m)\/(psi m))->
120 ((m:nat)(le m N)->(phi m))\/(Ex [j:nat](le j N)/\(psi j)).
124 Cut (phi O)\/(psi O).
130 Rewrite <- (le_n_O_eq m H2).
146 Apply le_trans with m:=N.
155 Case (le_lt_eq_dec m (S N)).
159 Apply (lt_n_Sm_le m N).
179 Apply le_trans with m:=N.
186 inline procedural "cic:/CoRN/reals/Bridges_iso/bounded_quantifier_informative.con" as lemma.
188 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_lemma1a.con" as lemma.
190 alias id "P_is_strongly_extensional" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/P_is_strongly_extensional.var".
192 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_lemma1b.con" as lemma.
198 (*#**********************************)
200 (*#**********************************)
202 (*#**********************************)
204 (*#**********************************)
207 Section Every_Cauchy_Sequence_is_bounded
210 inline procedural "cic:/CoRN/reals/Bridges_iso/seq2set.con" as definition.
212 alias id "g" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/Every_Cauchy_Sequence_is_bounded/g.var".
214 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/Every_Cauchy_Sequence_is_bounded/g_.con" "bridges_axioms_imply_ours__proofs_in_TCS__Every_Cauchy_Sequence_is_bounded__" as definition.
216 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/Every_Cauchy_Sequence_is_bounded/pg.con" "bridges_axioms_imply_ours__proofs_in_TCS__Every_Cauchy_Sequence_is_bounded__" as definition.
218 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/Every_Cauchy_Sequence_is_bounded/P.con" "bridges_axioms_imply_ours__proofs_in_TCS__Every_Cauchy_Sequence_is_bounded__" as definition.
220 inline procedural "cic:/CoRN/reals/Bridges_iso/fin_is_fin.con" as lemma.
222 inline procedural "cic:/CoRN/reals/Bridges_iso/card_fin.con" as lemma.
224 inline procedural "cic:/CoRN/reals/Bridges_iso/finite_seq.con" as lemma.
226 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_lemma2a.con" as lemma.
228 inline procedural "cic:/CoRN/reals/Bridges_iso/sup.con" as definition.
230 inline procedural "cic:/CoRN/reals/Bridges_iso/Psup.con" as definition.
232 inline procedural "cic:/CoRN/reals/Bridges_iso/Psup_proj1.con" as lemma.
234 inline procedural "cic:/CoRN/reals/Bridges_iso/Psup_unfolded1.con" as lemma.
236 inline procedural "cic:/CoRN/reals/Bridges_iso/Psup_unfolded2.con" as lemma.
238 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_lemma2b.con" as lemma.
240 inline procedural "cic:/CoRN/reals/Bridges_iso/inf.con" as definition.
242 inline procedural "cic:/CoRN/reals/Bridges_iso/Pinf.con" as definition.
244 inline procedural "cic:/CoRN/reals/Bridges_iso/Pinf_proj1.con" as lemma.
246 inline procedural "cic:/CoRN/reals/Bridges_iso/Pinf_unfolded1.con" as lemma.
248 inline procedural "cic:/CoRN/reals/Bridges_iso/Pinf_unfolded2.con" as lemma.
250 (* I tried very much not to mention this lemma! *)
252 inline procedural "cic:/CoRN/reals/Bridges_iso/sup_leEq.con" as lemma.
254 inline procedural "cic:/CoRN/reals/Bridges_iso/inf_geEq.con" as lemma.
256 inline procedural "cic:/CoRN/reals/Bridges_iso/tail_is_Cauchy.con" as lemma.
258 inline procedural "cic:/CoRN/reals/Bridges_iso/tail_seq.con" as definition.
261 End Every_Cauchy_Sequence_is_bounded
264 alias id "g" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/g.var".
266 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/g_.con" "bridges_axioms_imply_ours__proofs_in_TCS__" as definition.
268 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/pg.con" "bridges_axioms_imply_ours__proofs_in_TCS__" as definition.
270 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/sup_tail.con" "bridges_axioms_imply_ours__proofs_in_TCS__" as definition.
272 inline procedural "cic:/CoRN/reals/Bridges_iso/sup_tail_leEq.con" as lemma.
274 inline procedural "cic:/CoRN/reals/Bridges_iso/sup_tail_is_Cauchy.con" as lemma.
276 inline procedural "cic:/CoRN/reals/Bridges_iso/sup_tail_as_Cauchy.con" as definition.
278 inline procedural "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/L.con" "bridges_axioms_imply_ours__proofs_in_TCS__" as definition.
280 inline procedural "cic:/CoRN/reals/Bridges_iso/sup_tail_decrease.con" as lemma.
282 inline procedural "cic:/CoRN/reals/Bridges_iso/L_less_sup_n.con" as lemma.
284 inline procedural "cic:/CoRN/reals/Bridges_iso/Psup_unfolded2_informative.con" as lemma.
286 inline procedural "cic:/CoRN/reals/Bridges_iso/Pinf_unfolded2_informative.con" as lemma.
288 inline procedural "cic:/CoRN/reals/Bridges_iso/convergent_subseq.con" as lemma.
290 (* very elegant proof almost as short as text version! *)
292 inline procedural "cic:/CoRN/reals/Bridges_iso/lubp_gives_Cauchy.con" as theorem.
298 inline procedural "cic:/CoRN/reals/Bridges_iso/Bridges_R_is_CReals.con" as definition.
300 inline procedural "cic:/CoRN/reals/Bridges_iso/Bridges_R_as_CReals.con" as definition.
303 End bridges_axioms_imply_ours
308 (*#* remove printing Q *)