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 include "CoRN.ma".
  20
  21 (* $Id: CPseudoMSpaces.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
  22
  23 include "reals/Intervals.ma".
  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 alias id "A" = "cic:/CoRN/metrics/CPseudoMSpaces/Relations/A.var".
  39
  40 alias id "d" = "cic:/CoRN/metrics/CPseudoMSpaces/Relations/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 coercion cic:/matita/CoRN-Decl/metrics/CPseudoMSpaces/cms_crr.con 0 (* compounds *).
  91
  92 (* UNEXPORTED
  93 End Definition_PsMS0
  94 *)
  95
  96 (* UNEXPORTED
  97 Implicit Arguments cms_d [c].
  98 *)
  99
 100 (* NOTATION
 101 Infix "[-d]" := cms_d (at level 68, left associativity).
 102 *)
 103
 104 (* UNEXPORTED
 105 Section PsMS_axioms
 106 *)
 107
 108 (*#* **Pseudo Metric Space axioms
 109 %\begin{convention}%
 110 Let [A] be a pseudo metric space.
 111 %\end{convention}%
 112 *)
 113
 114 alias id "A" = "cic:/CoRN/metrics/CPseudoMSpaces/PsMS_axioms/A.var".
 115
 116 inline "cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace_is_CPsMetricSpace.con".
 117
 118 inline "cic:/CoRN/metrics/CPseudoMSpaces/d_com.con".
 119
 120 inline "cic:/CoRN/metrics/CPseudoMSpaces/d_nneg.con".
 121
 122 inline "cic:/CoRN/metrics/CPseudoMSpaces/d_pos_imp_ap.con".
 123
 124 inline "cic:/CoRN/metrics/CPseudoMSpaces/d_tri_ineq.con".
 125
 126 (* UNEXPORTED
 127 End PsMS_axioms
 128 *)
 129
 130 (* UNEXPORTED
 131 Section PsMS_basics
 132 *)
 133
 134 (*#* **Pseudo Metric Space basics
 135 %\begin{convention}%
 136 Let [Y] be a pseudo metric space.
 137 %\end{convention}%
 138 *)
 139
 140 alias id "Y" = "cic:/CoRN/metrics/CPseudoMSpaces/PsMS_basics/Y.var".
 141
 142 inline "cic:/CoRN/metrics/CPseudoMSpaces/rev_tri_ineq.con".
 143
 144 (*#*
 145 Instead of taking [pos_imp_ap] as axiom,
 146 we could as well have taken [diag_zero].
 147 *)
 148
 149 inline "cic:/CoRN/metrics/CPseudoMSpaces/diag_zero_imp_pos_imp_ap.con".
 150
 151 inline "cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap_imp_diag_zero.con".
 152
 153 inline "cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace_diag_zero.con".
 154
 155 (* UNEXPORTED
 156 End PsMS_basics
 157 *)
 158
 159 (* UNEXPORTED
 160 Section Zerof
 161 *)
 162
 163 (*#* **Zero function
 164 *)
 165
 166 (*#*
 167 Every setoid forms with the binary function that always returns zero,
 168 a pseudo metric space.
 169 *)
 170
 171 inline "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun.con".
 172
 173 inline "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_strext.con".
 174
 175 inline "cic:/CoRN/metrics/CPseudoMSpaces/Zero_fun.con".
 176
 177 inline "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_com.con".
 178
 179 inline "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_nneg.con".
 180
 181 inline "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_pos_imp_ap.con".
 182
 183 inline "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_tri_ineq.con".
 184
 185 inline "cic:/CoRN/metrics/CPseudoMSpaces/zf_is_CPsMetricSpace.con".
 186
 187 inline "cic:/CoRN/metrics/CPseudoMSpaces/zf_as_CPsMetricSpace.con".
 188
 189 (* UNEXPORTED
 190 End Zerof
 191 *)
 192