helm/software/matita/contribs/CoRN-Procedural/metrics/CPseudoMSpaces.mma

   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 include "CoRN.ma".
  18
  19 (* $Id: CPseudoMSpaces.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
  20
  21 include "reals/Intervals.ma".
  22
  23 (*#* *Metric Spaces
  24 *)
  25
  26 (* UNEXPORTED
  27 Section Relations
  28 *)
  29
  30 (*#* **Relations necessary for Pseudo Metric Spaces and Metric Spaces
  31 %\begin{convention}%
  32 Let [A : CSetoid], [d : (CSetoid_bin_fun A A IR)].
  33 %\end{convention}%
  34 *)
  35
  36 alias id "A" = "cic:/CoRN/metrics/CPseudoMSpaces/Relations/A.var".
  37
  38 alias id "d" = "cic:/CoRN/metrics/CPseudoMSpaces/Relations/d.var".
  39
  40 (* UNEXPORTED
  41 Set Implicit Arguments.
  42 *)
  43
  44 (* UNEXPORTED
  45 Unset Strict Implicit.
  46 *)
  47
  48 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/com.con" as definition.
  49
  50 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/nneg.con" as definition.
  51
  52 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap.con" as definition.
  53
  54 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/tri_ineq.con" as definition.
  55
  56 (* UNEXPORTED
  57 Set Strict Implicit.
  58 *)
  59
  60 (* UNEXPORTED
  61 Unset Implicit Arguments.
  62 *)
  63
  64 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/diag_zero.con" as definition.
  65
  66 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/apdiag_imp_grzero.con" as definition.
  67
  68 (* UNEXPORTED
  69 End Relations
  70 *)
  71
  72 (* UNEXPORTED
  73 Section Definition_PsMS0
  74 *)
  75
  76 (*#* **Definition of Pseudo Metric Space
  77 *)
  78
  79 (*#*
  80 A pseudo metric space consists of a setoid and a %''pseudo metric''% #"pseudo metric"#, also called
  81 %''distance''% #"distance"#, a binairy function that fulfils certain properties.
  82 *)
  83
  84 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace.ind".
  85
  86 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace.ind".
  87
  88 (* COERCION
  89 cic:/matita/CoRN-Procedural/metrics/CPseudoMSpaces/cms_crr.con
  90 *)
  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 procedural "cic:/CoRN/metrics/CPseudoMSpaces/CPsMetricSpace_is_CPsMetricSpace.con" as lemma.
 117
 118 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_com.con" as lemma.
 119
 120 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_nneg.con" as lemma.
 121
 122 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_pos_imp_ap.con" as lemma.
 123
 124 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/d_tri_ineq.con" as lemma.
 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 procedural "cic:/CoRN/metrics/CPseudoMSpaces/rev_tri_ineq.con" as lemma.
 143
 144 (*#*
 145 Instead of taking [pos_imp_ap] as axiom,
 146 we could as well have taken [diag_zero].
 147 *)
 148
 149 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/diag_zero_imp_pos_imp_ap.con" as lemma.
 150
 151 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/pos_imp_ap_imp_diag_zero.con" as lemma.
 152
 153 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/is_CPsMetricSpace_diag_zero.con" as lemma.
 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 procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun.con" as definition.
 172
 173 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_strext.con" as lemma.
 174
 175 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/Zero_fun.con" as definition.
 176
 177 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_com.con" as lemma.
 178
 179 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_nneg.con" as lemma.
 180
 181 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_pos_imp_ap.con" as lemma.
 182
 183 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zero_fun_tri_ineq.con" as lemma.
 184
 185 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zf_is_CPsMetricSpace.con" as definition.
 186
 187 inline procedural "cic:/CoRN/metrics/CPseudoMSpaces/zf_as_CPsMetricSpace.con" as definition.
 188
 189 (* UNEXPORTED
 190 End Zerof
 191 *)
 192