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 *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/metrics/CPseudoMSpaces".
19 (* $Id: CPseudoMSpaces.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
32 (*#* **Relations necessary for Pseudo Metric Spaces and Metric Spaces
34 Let [A : CSetoid], [d : (CSetoid_bin_fun A A IR)].
38 inline cic:/CoRN/metrics/CPseudoMSpaces/A.var.
40 inline cic:/CoRN/metrics/CPseudoMSpaces/d.var.
43 Set Implicit Arguments.
47 Unset Strict Implicit.
50 inline cic:/CoRN/metrics/CPseudoMSpaces/com.con.
52 inline cic:/CoRN/metrics/CPseudoMSpaces/nneg.con.
54 inline cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap.con.
56 inline cic:/CoRN/metrics/CPseudoMSpaces/tri_ineq.con.
63 Unset Implicit Arguments.
66 inline cic:/CoRN/metrics/CPseudoMSpaces/diag_zero.con.
68 inline cic:/CoRN/metrics/CPseudoMSpaces/apdiag_imp_grzero.con.
75 Section Definition_PsMS0.
78 (*#* **Definition of Pseudo Metric Space
82 A pseudo metric space consists of a setoid and a %''pseudo metric''% #"pseudo metric"#, also called
83 %''distance''% #"distance"#, a binairy function that fulfils certain properties.
86 inline cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace.ind.
88 inline cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace.ind.
95 Implicit Arguments cms_d [c].
102 (*#* **Pseudo Metric Space axioms
104 Let [A] be a pseudo metric space.
108 inline cic:/CoRN/metrics/CPseudoMSpaces/A.var.
110 inline cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace_is_CPsMetricSpace.con.
112 inline cic:/CoRN/metrics/CPseudoMSpaces/d_com.con.
114 inline cic:/CoRN/metrics/CPseudoMSpaces/d_nneg.con.
116 inline cic:/CoRN/metrics/CPseudoMSpaces/d_pos_imp_ap.con.
118 inline cic:/CoRN/metrics/CPseudoMSpaces/d_tri_ineq.con.
128 (*#* **Pseudo Metric Space basics
130 Let [Y] be a pseudo metric space.
134 inline cic:/CoRN/metrics/CPseudoMSpaces/Y.var.
136 inline cic:/CoRN/metrics/CPseudoMSpaces/rev_tri_ineq.con.
139 Instead of taking [pos_imp_ap] as axiom,
140 we could as well have taken [diag_zero].
143 inline cic:/CoRN/metrics/CPseudoMSpaces/diag_zero_imp_pos_imp_ap.con.
145 inline cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap_imp_diag_zero.con.
147 inline cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace_diag_zero.con.
161 Every setoid forms with the binary function that always returns zero,
162 a pseudo metric space.
165 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun.con.
167 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_strext.con.
169 inline cic:/CoRN/metrics/CPseudoMSpaces/Zero_fun.con.
171 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_com.con.
173 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_nneg.con.
175 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_pos_imp_ap.con.
177 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_tri_ineq.con.
179 inline cic:/CoRN/metrics/CPseudoMSpaces/zf_is_CPsMetricSpace.con.
181 inline cic:/CoRN/metrics/CPseudoMSpaces/zf_as_CPsMetricSpace.con.