--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/reals/Bridges_iso".
+
+include "CoRN.ma".
+
+(* begin hide *)
+
+(* file : bridges_gives_our.v *)
+
+(* version : 1.50 - 09/05/2001 *)
+
+(* version : 1.00 - 09/03/2001 *)
+
+(* author : Milad Niqui *)
+
+(* language : coq7.0bet26feb *)
+
+(* dependency : least_upper_bound_principle *)
+
+(* description : Bridges' proof of Cauchy completeness in TCS-219 *)
+
+include "reals/Bridges_LUB.ma".
+
+(* This lemma comes from lemmas.v of Martijn Oostdijk *)
+
+inline "cic:/CoRN/reals/Bridges_iso/le_witness_informative.con".
+
+(* UNEXPORTED
+Section bridges_axioms_imply_ours
+*)
+
+alias id "OF" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/OF.var".
+
+alias id "lubp" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/lubp.var".
+
+alias id "is_Archimedes" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/is_Archimedes.var".
+
+inline "cic:/CoRN/reals/Bridges_iso/is_Archimedes'.con".
+
+(* UNEXPORTED
+Section proofs_in_TCS
+*)
+
+inline "cic:/CoRN/reals/Bridges_iso/leEq_geEq.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/glbp.con".
+
+(* UNEXPORTED
+Section supremum
+*)
+
+alias id "P" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/P.var".
+
+inline "cic:/CoRN/reals/Bridges_iso/inequality1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/inequality2.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/inequality3.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/inequality4.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Hum.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/bound_tk1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/bound_tk2.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/trick.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/trick'.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/up_bound_for_n_element.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/low_bound_for_n_element.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/saghf.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Psaghf.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/kaf.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pkaf.con".
+
+alias id "is_finite_P" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/is_finite_P.var".
+
+inline "cic:/CoRN/reals/Bridges_iso/card.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pcard1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/seq.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pseq1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pseq1_unfolded.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/indeks.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pindeks.con".
+
+alias id "is_onto_seq_P" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/is_onto_seq_P.var".
+
+inline "cic:/CoRN/reals/Bridges_iso/P_is_inhabited.con".
+
+(*
+Lemma bounded_quantifier:(N:nat;phi,psi:nat->Prop)
+ ((m:nat)(le m N)->(phi m)\/(psi m))->
+ ((m:nat)(le m N)->(phi m))\/(Ex [j:nat](le j N)/\(psi j)).
+Proof.
+ Intros.
+ Induction N.
+ Cut (phi O)\/(psi O).
+ Intro.
+ Case H0.
+ Intros.
+ Left.
+ Intros.
+ Rewrite <- (le_n_O_eq m H2).
+ Assumption.
+ Intro.
+ Right.
+ Exists O.
+ Split.
+ Constructor.
+ Assumption.
+ Apply H.
+ Constructor.*)
+
+(* n=(S n0) *)
+
+(* Case HrecN.
+ Intros.
+ Apply H.
+ Apply le_trans with m:=N.
+ Assumption.
+ Apply le_n_Sn.
+ Intro.
+ Case (H (S N)).
+ Apply le_n.
+ Intros.
+ Left.
+ Intros.
+ Case (le_lt_eq_dec m (S N)).
+ Assumption.
+ Intros.
+ Apply H0.
+ Apply (lt_n_Sm_le m N).
+ Assumption.
+ Intro.
+ Rewrite e.
+ Assumption.
+ Intro.
+ Right.
+ Exists (S N).
+ Split.
+ Apply le_n.
+ Assumption.
+ Intro.
+ Right.
+ Case H0.
+ Intro j.
+ Intros.
+ Exists j.
+ Elim H1.
+ Intros.
+ Split.
+ Apply le_trans with m:=N.
+ Assumption.
+ Apply le_n_Sn.
+ Assumption.
+Qed.
+*)
+
+inline "cic:/CoRN/reals/Bridges_iso/bounded_quantifier_informative.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_lemma1a.con".
+
+alias id "P_is_strongly_extensional" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/supremum/P_is_strongly_extensional.var".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_lemma1b.con".
+
+(* UNEXPORTED
+End supremum
+*)
+
+(*#**********************************)
+
+(*#**********************************)
+
+(*#**********************************)
+
+(*#**********************************)
+
+(* UNEXPORTED
+Section Every_Cauchy_Sequence_is_bounded
+*)
+
+inline "cic:/CoRN/reals/Bridges_iso/seq2set.con".
+
+alias id "g" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/Every_Cauchy_Sequence_is_bounded/g.var".
+
+inline "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__".
+
+inline "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__".
+
+inline "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__".
+
+inline "cic:/CoRN/reals/Bridges_iso/fin_is_fin.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/card_fin.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/finite_seq.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_lemma2a.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/sup.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Psup.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Psup_proj1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Psup_unfolded1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Psup_unfolded2.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_lemma2b.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/inf.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pinf.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pinf_proj1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pinf_unfolded1.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pinf_unfolded2.con".
+
+(* I tried very much not to mention this lemma! *)
+
+inline "cic:/CoRN/reals/Bridges_iso/sup_leEq.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/inf_geEq.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/tail_is_Cauchy.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/tail_seq.con".
+
+(* UNEXPORTED
+End Every_Cauchy_Sequence_is_bounded
+*)
+
+alias id "g" = "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/g.var".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/g_.con" "bridges_axioms_imply_ours__proofs_in_TCS__".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/pg.con" "bridges_axioms_imply_ours__proofs_in_TCS__".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/sup_tail.con" "bridges_axioms_imply_ours__proofs_in_TCS__".
+
+inline "cic:/CoRN/reals/Bridges_iso/sup_tail_leEq.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/sup_tail_is_Cauchy.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/sup_tail_as_Cauchy.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/bridges_axioms_imply_ours/proofs_in_TCS/L.con" "bridges_axioms_imply_ours__proofs_in_TCS__".
+
+inline "cic:/CoRN/reals/Bridges_iso/sup_tail_decrease.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/L_less_sup_n.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Psup_unfolded2_informative.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Pinf_unfolded2_informative.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/convergent_subseq.con".
+
+(* very elegant proof almost as short as text version! *)
+
+inline "cic:/CoRN/reals/Bridges_iso/lubp_gives_Cauchy.con".
+
+(* UNEXPORTED
+End proofs_in_TCS
+*)
+
+inline "cic:/CoRN/reals/Bridges_iso/Bridges_R_is_CReals.con".
+
+inline "cic:/CoRN/reals/Bridges_iso/Bridges_R_as_CReals.con".
+
+(* UNEXPORTED
+End bridges_axioms_imply_ours
+*)
+
+(* end hide *)
+
+(*#* remove printing Q *)
+
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