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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: CPseudoMSpaces.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
21 include "reals/Intervals.ma".
30 (*#* **Relations necessary for Pseudo Metric Spaces and Metric Spaces
32 Let [A : CSetoid], [d : (CSetoid_bin_fun A A IR)].
37 cic:/CoRN/metrics/CPseudoMSpaces/Relations/A.var
41 cic:/CoRN/metrics/CPseudoMSpaces/Relations/d.var
45 Set Implicit Arguments.
49 Unset Strict Implicit.
52 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/com.con" as definition.
54 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/nneg.con" as definition.
56 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap.con" as definition.
58 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/tri_ineq.con" as definition.
65 Unset Implicit Arguments.
68 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/diag_zero.con" as definition.
70 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/apdiag_imp_grzero.con" as definition.
77 Section Definition_PsMS0
80 (*#* **Definition of Pseudo Metric Space
84 A pseudo metric space consists of a setoid and a %''pseudo metric''% #"pseudo metric"#, also called
85 %''distance''% #"distance"#, a binairy function that fulfils certain properties.
88 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace.ind".
90 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace.ind".
93 cic:/matita/CoRN-Procedural/metrics/CPseudoMSpaces/cms_crr.con
101 Implicit Arguments cms_d [c].
105 Infix "[-d]" := cms_d (at level 68, left associativity).
112 (*#* **Pseudo Metric Space axioms
114 Let [A] be a pseudo metric space.
119 cic:/CoRN/metrics/CPseudoMSpaces/PsMS_axioms/A.var
122 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace_is_CPsMetricSpace.con" as lemma.
124 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_com.con" as lemma.
126 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_nneg.con" as lemma.
128 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_pos_imp_ap.con" as lemma.
130 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_tri_ineq.con" as lemma.
140 (*#* **Pseudo Metric Space basics
142 Let [Y] be a pseudo metric space.
147 cic:/CoRN/metrics/CPseudoMSpaces/PsMS_basics/Y.var
150 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/rev_tri_ineq.con" as lemma.
153 Instead of taking [pos_imp_ap] as axiom,
154 we could as well have taken [diag_zero].
157 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/diag_zero_imp_pos_imp_ap.con" as lemma.
159 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap_imp_diag_zero.con" as lemma.
161 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace_diag_zero.con" as lemma.
175 Every setoid forms with the binary function that always returns zero,
176 a pseudo metric space.
179 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun.con" as definition.
181 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_strext.con" as lemma.
183 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/Zero_fun.con" as definition.
185 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_com.con" as lemma.
187 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_nneg.con" as lemma.
189 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_pos_imp_ap.con" as lemma.
191 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_tri_ineq.con" as lemma.
193 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zf_is_CPsMetricSpace.con" as definition.
195 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zf_as_CPsMetricSpace.con" as definition.