helm/software/matita/contribs/CoRN-Decl/metrics/CPseudoMSpaces.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/metrics/CPseudoMSpaces".
  18
  19 (* $Id: CPseudoMSpaces.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
  20
  21 (* INCLUDE
  22 Intervals
  23 *)
  24
  25 (*#* *Metric Spaces
  26 *)
  27
  28 (* UNEXPORTED
  29 Section Relations.
  30 *)
  31
  32 (*#* **Relations necessary for Pseudo Metric Spaces and Metric Spaces
  33 %\begin{convention}%
  34 Let [A : CSetoid], [d : (CSetoid_bin_fun A A IR)].
  35 %\end{convention}%
  36 *)
  37
  38 inline cic:/CoRN/metrics/CPseudoMSpaces/A.var.
  39
  40 inline cic:/CoRN/metrics/CPseudoMSpaces/d.var.
  41
  42 (* UNEXPORTED
  43 Set Implicit Arguments.
  44 *)
  45
  46 (* UNEXPORTED
  47 Unset Strict Implicit.
  48 *)
  49
  50 inline cic:/CoRN/metrics/CPseudoMSpaces/com.con.
  51
  52 inline cic:/CoRN/metrics/CPseudoMSpaces/nneg.con.
  53
  54 inline cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap.con.
  55
  56 inline cic:/CoRN/metrics/CPseudoMSpaces/tri_ineq.con.
  57
  58 (* UNEXPORTED
  59 Set Strict Implicit.
  60 *)
  61
  62 (* UNEXPORTED
  63 Unset Implicit Arguments.
  64 *)
  65
  66 inline cic:/CoRN/metrics/CPseudoMSpaces/diag_zero.con.
  67
  68 inline cic:/CoRN/metrics/CPseudoMSpaces/apdiag_imp_grzero.con.
  69
  70 (* UNEXPORTED
  71 End Relations.
  72 *)
  73
  74 (* UNEXPORTED
  75 Section Definition_PsMS0.
  76 *)
  77
  78 (*#* **Definition of Pseudo Metric Space
  79 *)
  80
  81 (*#*
  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.
  84 *)
  85
  86 inline cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace.ind.
  87
  88 inline cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace.ind.
  89
  90 (* UNEXPORTED
  91 End Definition_PsMS0.
  92 *)
  93
  94 (* UNEXPORTED
  95 Implicit Arguments cms_d [c].
  96 *)
  97
  98 (* UNEXPORTED
  99 Section PsMS_axioms.
 100 *)
 101
 102 (*#* **Pseudo Metric Space axioms
 103 %\begin{convention}%
 104 Let [A] be a pseudo metric space.
 105 %\end{convention}%
 106 *)
 107
 108 inline cic:/CoRN/metrics/CPseudoMSpaces/A.var.
 109
 110 inline cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace_is_CPsMetricSpace.con.
 111
 112 inline cic:/CoRN/metrics/CPseudoMSpaces/d_com.con.
 113
 114 inline cic:/CoRN/metrics/CPseudoMSpaces/d_nneg.con.
 115
 116 inline cic:/CoRN/metrics/CPseudoMSpaces/d_pos_imp_ap.con.
 117
 118 inline cic:/CoRN/metrics/CPseudoMSpaces/d_tri_ineq.con.
 119
 120 (* UNEXPORTED
 121 End PsMS_axioms.
 122 *)
 123
 124 (* UNEXPORTED
 125 Section PsMS_basics.
 126 *)
 127
 128 (*#* **Pseudo Metric Space basics
 129 %\begin{convention}%
 130 Let [Y] be a pseudo metric space.
 131 %\end{convention}%
 132 *)
 133
 134 inline cic:/CoRN/metrics/CPseudoMSpaces/Y.var.
 135
 136 inline cic:/CoRN/metrics/CPseudoMSpaces/rev_tri_ineq.con.
 137
 138 (*#*
 139 Instead of taking [pos_imp_ap] as axiom,
 140 we could as well have taken [diag_zero].
 141 *)
 142
 143 inline cic:/CoRN/metrics/CPseudoMSpaces/diag_zero_imp_pos_imp_ap.con.
 144
 145 inline cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap_imp_diag_zero.con.
 146
 147 inline cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace_diag_zero.con.
 148
 149 (* UNEXPORTED
 150 End PsMS_basics.
 151 *)
 152
 153 (* UNEXPORTED
 154 Section Zerof.
 155 *)
 156
 157 (*#* **Zero function
 158 *)
 159
 160 (*#*
 161 Every setoid forms with the binary function that always returns zero,
 162 a pseudo metric space.
 163 *)
 164
 165 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun.con.
 166
 167 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_strext.con.
 168
 169 inline cic:/CoRN/metrics/CPseudoMSpaces/Zero_fun.con.
 170
 171 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_com.con.
 172
 173 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_nneg.con.
 174
 175 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_pos_imp_ap.con.
 176
 177 inline cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_tri_ineq.con.
 178
 179 inline cic:/CoRN/metrics/CPseudoMSpaces/zf_is_CPsMetricSpace.con.
 180
 181 inline cic:/CoRN/metrics/CPseudoMSpaces/zf_as_CPsMetricSpace.con.
 182
 183 (* UNEXPORTED
 184 End Zerof.
 185 *)
 186