helm/software/matita/contribs/CoRN-Procedural/metrics/ContFunctions.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: ContFunctions.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
  20
  21 include "metrics/CPseudoMSpaces.ma".
  22
  23 (* UNEXPORTED
  24 Section Continuous_functions
  25 *)
  26
  27 (*#* **Continuous functions, uniformly continuous functions and Lipschitz functions
  28 %\begin{convention}%
  29 Let [A] and [B] be pseudo metric spaces.
  30 %\end{convention}%
  31 *)
  32
  33 alias id "A" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/A.var".
  34
  35 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/B.var".
  36
  37 (*#*
  38 We will look at some notions of continuous functions.
  39 *)
  40
  41 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous.con".
  42
  43 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous'.con".
  44
  45 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous.con".
  46
  47 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous'.con".
  48
  49 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous''.con".
  50
  51 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz.con".
  52
  53 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz'.con".
  54
  55 (* UNEXPORTED
  56 End Continuous_functions
  57 *)
  58
  59 (* UNEXPORTED
  60 Implicit Arguments continuous [A B].
  61 *)
  62
  63 (* UNEXPORTED
  64 Implicit Arguments uni_continuous [A B].
  65 *)
  66
  67 (* UNEXPORTED
  68 Implicit Arguments lipschitz [A B].
  69 *)
  70
  71 (* UNEXPORTED
  72 Implicit Arguments continuous' [A B].
  73 *)
  74
  75 (* UNEXPORTED
  76 Implicit Arguments uni_continuous' [A B].
  77 *)
  78
  79 (* UNEXPORTED
  80 Implicit Arguments uni_continuous'' [A B].
  81 *)
  82
  83 (* UNEXPORTED
  84 Implicit Arguments lipschitz' [A B].
  85 *)
  86
  87 (* UNEXPORTED
  88 Section Lemmas
  89 *)
  90
  91 (* begin hide *)
  92
  93 inline procedural "cic:/CoRN/metrics/ContFunctions/nexp_power.con".
  94
  95 (* end hide *)
  96
  97 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous_imp_continuous'.con".
  98
  99 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous'_imp_continuous.con".
 100
 101 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_uni_continuous'.con".
 102
 103 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous.con".
 104
 105 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous''.con".
 106
 107 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_lipschitz'.con".
 108
 109 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz'_imp_lipschitz.con".
 110
 111 (*#*
 112 Every uniformly continuous function is continuous and
 113 every Lipschitz function is uniformly continuous.
 114 *)
 115
 116 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_continuous.con".
 117
 118 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_uni_continuous.con".
 119
 120 (* UNEXPORTED
 121 End Lemmas
 122 *)
 123
 124 (* UNEXPORTED
 125 Section Identity
 126 *)
 127
 128 (*#* **Identity
 129 *)
 130
 131 (*#*
 132 The identity function is Lipschitz.
 133 Hence it is uniformly continuous and continuous.
 134 *)
 135
 136 inline procedural "cic:/CoRN/metrics/ContFunctions/id_is_lipschitz.con".
 137
 138 inline procedural "cic:/CoRN/metrics/ContFunctions/id_is_uni_continuous.con".
 139
 140 inline procedural "cic:/CoRN/metrics/ContFunctions/id_is_continuous.con".
 141
 142 (* UNEXPORTED
 143 End Identity
 144 *)
 145
 146 (* UNEXPORTED
 147 Section Constant
 148 *)
 149
 150 (*#* **Constant functions
 151 %\begin{convention}%
 152 Let [B] and [X] be pseudo metric spaces.
 153 %\end{convention}%
 154 *)
 155
 156 (*#*
 157 Any constant function is Lipschitz.
 158 Hence it is uniformly continuous and continuous.
 159 *)
 160
 161 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Constant/B.var".
 162
 163 alias id "X" = "cic:/CoRN/metrics/ContFunctions/Constant/X.var".
 164
 165 inline procedural "cic:/CoRN/metrics/ContFunctions/const_fun_is_lipschitz.con".
 166
 167 inline procedural "cic:/CoRN/metrics/ContFunctions/const_fun_is_uni_continuous.con".
 168
 169 inline procedural "cic:/CoRN/metrics/ContFunctions/const_fun_is_continuous.con".
 170
 171 (* UNEXPORTED
 172 End Constant
 173 *)
 174
 175 (* UNEXPORTED
 176 Section Composition
 177 *)
 178
 179 (*#* **Composition
 180 %\begin{convention}%
 181 Let [B],[C] and [X] be pseudo metric spaces.
 182 Let [f : (CSetoid_fun X B)] and
 183 [g : (CSetoid_fun  B C)].
 184 %\end{convention}%
 185 *)
 186
 187 (*#*
 188 The composition of two Lipschitz/uniformly continous/continuous functions is
 189 again Lipschitz/uniformly continuous/continuous.
 190 *)
 191
 192 alias id "X" = "cic:/CoRN/metrics/ContFunctions/Composition/X.var".
 193
 194 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Composition/B.var".
 195
 196 alias id "f" = "cic:/CoRN/metrics/ContFunctions/Composition/f.var".
 197
 198 alias id "C" = "cic:/CoRN/metrics/ContFunctions/Composition/C.var".
 199
 200 alias id "g" = "cic:/CoRN/metrics/ContFunctions/Composition/g.var".
 201
 202 inline procedural "cic:/CoRN/metrics/ContFunctions/comp_resp_lipschitz.con".
 203
 204 inline procedural "cic:/CoRN/metrics/ContFunctions/comp_resp_uni_continuous.con".
 205
 206 inline procedural "cic:/CoRN/metrics/ContFunctions/comp_resp_continuous.con".
 207
 208 (* UNEXPORTED
 209 End Composition
 210 *)
 211
 212 (* UNEXPORTED
 213 Section Limit
 214 *)
 215
 216 (*#* **Limit
 217 *)
 218
 219 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit.con".
 220
 221 (* UNEXPORTED
 222 Implicit Arguments MSseqLimit [X].
 223 *)
 224
 225 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit'.con".
 226
 227 (* UNEXPORTED
 228 Implicit Arguments MSseqLimit' [X].
 229 *)
 230
 231 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit_imp_MSseqLimit'.con".
 232
 233 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit'_imp_MSseqLimit.con".
 234
 235 inline procedural "cic:/CoRN/metrics/ContFunctions/seqcontinuous'.con".
 236
 237 (* UNEXPORTED
 238 Implicit Arguments seqcontinuous' [A B].
 239 *)
 240
 241 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous'_imp_seqcontinuous'.con".
 242
 243 (* UNEXPORTED
 244 End Limit
 245 *)
 246