matita/contribs/CoRN-Decl/metrics/ContFunctions.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/ContFunctions".
  18
  19 include "CoRN.ma".
  20
  21 (* $Id: ContFunctions.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
  22
  23 include "metrics/CPseudoMSpaces.ma".
  24
  25 (* UNEXPORTED
  26 Section Continuous_functions
  27 *)
  28
  29 (*#* **Continuous functions, uniformly continuous functions and Lipschitz functions
  30 %\begin{convention}%
  31 Let [A] and [B] be pseudo metric spaces.
  32 %\end{convention}%
  33 *)
  34
  35 alias id "A" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/A.var".
  36
  37 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/B.var".
  38
  39 (*#*
  40 We will look at some notions of continuous functions.
  41 *)
  42
  43 inline "cic:/CoRN/metrics/ContFunctions/continuous.con".
  44
  45 inline "cic:/CoRN/metrics/ContFunctions/continuous'.con".
  46
  47 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous.con".
  48
  49 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous'.con".
  50
  51 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous''.con".
  52
  53 inline "cic:/CoRN/metrics/ContFunctions/lipschitz.con".
  54
  55 inline "cic:/CoRN/metrics/ContFunctions/lipschitz'.con".
  56
  57 (* UNEXPORTED
  58 End Continuous_functions
  59 *)
  60
  61 (* UNEXPORTED
  62 Implicit Arguments continuous [A B].
  63 *)
  64
  65 (* UNEXPORTED
  66 Implicit Arguments uni_continuous [A B].
  67 *)
  68
  69 (* UNEXPORTED
  70 Implicit Arguments lipschitz [A B].
  71 *)
  72
  73 (* UNEXPORTED
  74 Implicit Arguments continuous' [A B].
  75 *)
  76
  77 (* UNEXPORTED
  78 Implicit Arguments uni_continuous' [A B].
  79 *)
  80
  81 (* UNEXPORTED
  82 Implicit Arguments uni_continuous'' [A B].
  83 *)
  84
  85 (* UNEXPORTED
  86 Implicit Arguments lipschitz' [A B].
  87 *)
  88
  89 (* UNEXPORTED
  90 Section Lemmas
  91 *)
  92
  93 (* begin hide *)
  94
  95 inline "cic:/CoRN/metrics/ContFunctions/nexp_power.con".
  96
  97 (* end hide *)
  98
  99 inline "cic:/CoRN/metrics/ContFunctions/continuous_imp_continuous'.con".
 100
 101 inline "cic:/CoRN/metrics/ContFunctions/continuous'_imp_continuous.con".
 102
 103 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_uni_continuous'.con".
 104
 105 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous.con".
 106
 107 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous''.con".
 108
 109 inline "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_lipschitz'.con".
 110
 111 inline "cic:/CoRN/metrics/ContFunctions/lipschitz'_imp_lipschitz.con".
 112
 113 (*#*
 114 Every uniformly continuous function is continuous and
 115 every Lipschitz function is uniformly continuous.
 116 *)
 117
 118 inline "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_continuous.con".
 119
 120 inline "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_uni_continuous.con".
 121
 122 (* UNEXPORTED
 123 End Lemmas
 124 *)
 125
 126 (* UNEXPORTED
 127 Section Identity
 128 *)
 129
 130 (*#* **Identity
 131 *)
 132
 133 (*#*
 134 The identity function is Lipschitz.
 135 Hence it is uniformly continuous and continuous.
 136 *)
 137
 138 inline "cic:/CoRN/metrics/ContFunctions/id_is_lipschitz.con".
 139
 140 inline "cic:/CoRN/metrics/ContFunctions/id_is_uni_continuous.con".
 141
 142 inline "cic:/CoRN/metrics/ContFunctions/id_is_continuous.con".
 143
 144 (* UNEXPORTED
 145 End Identity
 146 *)
 147
 148 (* UNEXPORTED
 149 Section Constant
 150 *)
 151
 152 (*#* **Constant functions
 153 %\begin{convention}%
 154 Let [B] and [X] be pseudo metric spaces.
 155 %\end{convention}%
 156 *)
 157
 158 (*#*
 159 Any constant function is Lipschitz.
 160 Hence it is uniformly continuous and continuous.
 161 *)
 162
 163 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Constant/B.var".
 164
 165 alias id "X" = "cic:/CoRN/metrics/ContFunctions/Constant/X.var".
 166
 167 inline "cic:/CoRN/metrics/ContFunctions/const_fun_is_lipschitz.con".
 168
 169 inline "cic:/CoRN/metrics/ContFunctions/const_fun_is_uni_continuous.con".
 170
 171 inline "cic:/CoRN/metrics/ContFunctions/const_fun_is_continuous.con".
 172
 173 (* UNEXPORTED
 174 End Constant
 175 *)
 176
 177 (* UNEXPORTED
 178 Section Composition
 179 *)
 180
 181 (*#* **Composition
 182 %\begin{convention}%
 183 Let [B],[C] and [X] be pseudo metric spaces.
 184 Let [f : (CSetoid_fun X B)] and
 185 [g : (CSetoid_fun  B C)].
 186 %\end{convention}%
 187 *)
 188
 189 (*#*
 190 The composition of two Lipschitz/uniformly continous/continuous functions is
 191 again Lipschitz/uniformly continuous/continuous.
 192 *)
 193
 194 alias id "X" = "cic:/CoRN/metrics/ContFunctions/Composition/X.var".
 195
 196 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Composition/B.var".
 197
 198 alias id "f" = "cic:/CoRN/metrics/ContFunctions/Composition/f.var".
 199
 200 alias id "C" = "cic:/CoRN/metrics/ContFunctions/Composition/C.var".
 201
 202 alias id "g" = "cic:/CoRN/metrics/ContFunctions/Composition/g.var".
 203
 204 inline "cic:/CoRN/metrics/ContFunctions/comp_resp_lipschitz.con".
 205
 206 inline "cic:/CoRN/metrics/ContFunctions/comp_resp_uni_continuous.con".
 207
 208 inline "cic:/CoRN/metrics/ContFunctions/comp_resp_continuous.con".
 209
 210 (* UNEXPORTED
 211 End Composition
 212 *)
 213
 214 (* UNEXPORTED
 215 Section Limit
 216 *)
 217
 218 (*#* **Limit
 219 *)
 220
 221 inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit.con".
 222
 223 (* UNEXPORTED
 224 Implicit Arguments MSseqLimit [X].
 225 *)
 226
 227 inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit'.con".
 228
 229 (* UNEXPORTED
 230 Implicit Arguments MSseqLimit' [X].
 231 *)
 232
 233 inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit_imp_MSseqLimit'.con".
 234
 235 inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit'_imp_MSseqLimit.con".
 236
 237 inline "cic:/CoRN/metrics/ContFunctions/seqcontinuous'.con".
 238
 239 (* UNEXPORTED
 240 Implicit Arguments seqcontinuous' [A B].
 241 *)
 242
 243 inline "cic:/CoRN/metrics/ContFunctions/continuous'_imp_seqcontinuous'.con".
 244
 245 (* UNEXPORTED
 246 End Limit
 247 *)
 248