helm/software/matita/contribs/procedural/CoRN/ftc/WeakIVT.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: WeakIVT.v,v 1.9 2004/04/23 10:01:01 lcf Exp $ *)
  20
  21 (*#* printing ** %\ensuremath\times% #&times;# *)
  22
  23 (* begin hide *)
  24
  25 (* NOTATION
  26 Infix "**" := prodT (at level 20).
  27 *)
  28
  29 (* end hide *)
  30
  31 include "ftc/Continuity.ma".
  32
  33 (*#* *IVT for Partial Functions
  34
  35 In general, we cannot prove the classically valid Intermediate Value
  36 Theorem for arbitrary partial functions, which states that in any
  37 interval [[a,b]], for any value [z] between [f(a)] and [f(b)]
  38 there exists $x\in[a,b]$#x&isin;[a,b]# such that [f(x) [=] z].
  39
  40 However, as is usually the case, there are some good aproximation results.  We
  41 will prove them here.
  42 *)
  43
  44 (* UNEXPORTED
  45 Section Lemma1
  46 *)
  47
  48 (* UNEXPORTED
  49 cic:/CoRN/ftc/WeakIVT/Lemma1/a.var
  50 *)
  51
  52 (* UNEXPORTED
  53 cic:/CoRN/ftc/WeakIVT/Lemma1/b.var
  54 *)
  55
  56 (* UNEXPORTED
  57 cic:/CoRN/ftc/WeakIVT/Lemma1/Hab.var
  58 *)
  59
  60 (* begin hide *)
  61
  62 inline procedural "cic:/CoRN/ftc/WeakIVT/Lemma1/I.con" "Lemma1__" as definition.
  63
  64 (* end hide *)
  65
  66 (* UNEXPORTED
  67 cic:/CoRN/ftc/WeakIVT/Lemma1/F.var
  68 *)
  69
  70 (* UNEXPORTED
  71 cic:/CoRN/ftc/WeakIVT/Lemma1/contF.var
  72 *)
  73
  74 (*#* **First Lemmas
  75
  76 %\begin{convention}% Let [a, b : IR] and [Hab : a [<=] b] and denote by [I]
  77 the interval [[a,b]].  Let [F] be a continuous function on [I].
  78 %\end{convention}%
  79
  80 We begin by proving that, if [f(a) [<] f(b)], then for every [y] in
  81 [[f(a),f(b)]] there is an $x\in[a,b]$#x&isin;[a,b]# such that [f(x)] is close
  82 enough to [z].
  83 *)
  84
  85 inline procedural "cic:/CoRN/ftc/WeakIVT/Weak_IVT_ap_lft.con" as lemma.
  86
  87 (* UNEXPORTED
  88 End Lemma1
  89 *)
  90
  91 (* UNEXPORTED
  92 Section Lemma2
  93 *)
  94
  95 (* UNEXPORTED
  96 cic:/CoRN/ftc/WeakIVT/Lemma2/a.var
  97 *)
  98
  99 (* UNEXPORTED
 100 cic:/CoRN/ftc/WeakIVT/Lemma2/b.var
 101 *)
 102
 103 (* UNEXPORTED
 104 cic:/CoRN/ftc/WeakIVT/Lemma2/Hab.var
 105 *)
 106
 107 (* begin hide *)
 108
 109 inline procedural "cic:/CoRN/ftc/WeakIVT/Lemma2/I.con" "Lemma2__" as definition.
 110
 111 (* end hide *)
 112
 113 (* UNEXPORTED
 114 cic:/CoRN/ftc/WeakIVT/Lemma2/F.var
 115 *)
 116
 117 (* UNEXPORTED
 118 cic:/CoRN/ftc/WeakIVT/Lemma2/contF.var
 119 *)
 120
 121 (*#*
 122 If [f(b) [<] f(a)], a similar result holds:
 123 *)
 124
 125 inline procedural "cic:/CoRN/ftc/WeakIVT/Weak_IVT_ap_rht.con" as lemma.
 126
 127 (* UNEXPORTED
 128 End Lemma2
 129 *)
 130
 131 (* UNEXPORTED
 132 Section IVT
 133 *)
 134
 135 (*#* **The IVT
 136
 137 We will now assume that [a [<] b] and that [F] is not only
 138 continuous, but also strictly increasing in [I].  Under
 139 these assumptions, we can build two sequences of values which
 140 converge to [x0] such that [f(x0) [=] z].
 141 *)
 142
 143 (* UNEXPORTED
 144 cic:/CoRN/ftc/WeakIVT/IVT/a.var
 145 *)
 146
 147 (* UNEXPORTED
 148 cic:/CoRN/ftc/WeakIVT/IVT/b.var
 149 *)
 150
 151 (* UNEXPORTED
 152 cic:/CoRN/ftc/WeakIVT/IVT/Hab'.var
 153 *)
 154
 155 (* UNEXPORTED
 156 cic:/CoRN/ftc/WeakIVT/IVT/Hab.var
 157 *)
 158
 159 (* begin hide *)
 160
 161 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/I.con" "IVT__" as definition.
 162
 163 (* end hide *)
 164
 165 (* UNEXPORTED
 166 cic:/CoRN/ftc/WeakIVT/IVT/F.var
 167 *)
 168
 169 (* UNEXPORTED
 170 cic:/CoRN/ftc/WeakIVT/IVT/contF.var
 171 *)
 172
 173 (* begin hide *)
 174
 175 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/incF.con" "IVT__" as definition.
 176
 177 (* end hide *)
 178
 179 (* begin show *)
 180
 181 (* UNEXPORTED
 182 cic:/CoRN/ftc/WeakIVT/IVT/incrF.var
 183 *)
 184
 185 (* end show *)
 186
 187 (* begin hide *)
 188
 189 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/Ha.con" "IVT__" as definition.
 190
 191 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/Hb.con" "IVT__" as definition.
 192
 193 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/HFab'.con" "IVT__" as definition.
 194
 195 (* end hide *)
 196
 197 (* begin show *)
 198
 199 (* UNEXPORTED
 200 cic:/CoRN/ftc/WeakIVT/IVT/z.var
 201 *)
 202
 203 (* UNEXPORTED
 204 cic:/CoRN/ftc/WeakIVT/IVT/Haz.var
 205 *)
 206
 207 (* UNEXPORTED
 208 cic:/CoRN/ftc/WeakIVT/IVT/Hzb.var
 209 *)
 210
 211 (* end show *)
 212
 213 (*#* Given any two points [x [<] y] in [[a,b]] such that [x [<=] z [<=] y],
 214 we can find [x' [<] y'] such that $|x'-y'|=\frac23|x-y|$#|x'-y'|=2/3|x-y|#
 215 and [x' [<=] z [<=] y'].
 216 *)
 217
 218 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT_seq_lemma.con" as lemma.
 219
 220 (* end hide *)
 221
 222 (*#*
 223 We now iterate this construction.
 224 *)
 225
 226 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT_aux_seq_type.ind".
 227
 228 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT_iter.con" as definition.
 229
 230 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT_seq.con" as definition.
 231
 232 (*#*
 233 We now define the sequences built from this iteration, starting with [a] and [b].
 234 *)
 235
 236 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq.con" as definition.
 237
 238 inline procedural "cic:/CoRN/ftc/WeakIVT/b_seq.con" as definition.
 239
 240 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_I.con" as definition.
 241
 242 inline procedural "cic:/CoRN/ftc/WeakIVT/b_seq_I.con" as definition.
 243
 244 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_less_b_seq.con" as lemma.
 245
 246 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_leEq_z.con" as lemma.
 247
 248 inline procedural "cic:/CoRN/ftc/WeakIVT/z_leEq_b_seq.con" as lemma.
 249
 250 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_mon.con" as lemma.
 251
 252 inline procedural "cic:/CoRN/ftc/WeakIVT/b_seq_mon.con" as lemma.
 253
 254 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_b_seq_dist_n.con" as lemma.
 255
 256 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_b_seq_dist.con" as lemma.
 257
 258 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_Cauchy.con" as lemma.
 259
 260 inline procedural "cic:/CoRN/ftc/WeakIVT/b_seq_Cauchy.con" as lemma.
 261
 262 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/xa.con" "IVT__" as definition.
 263
 264 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT/xb.con" "IVT__" as definition.
 265
 266 inline procedural "cic:/CoRN/ftc/WeakIVT/a_seq_b_seq_lim.con" as lemma.
 267
 268 inline procedural "cic:/CoRN/ftc/WeakIVT/xa_in_interval.con" as lemma.
 269
 270 inline procedural "cic:/CoRN/ftc/WeakIVT/IVT_I.con" as lemma.
 271
 272 (* UNEXPORTED
 273 End IVT
 274 *)
 275