helm/software/matita/contribs/procedural/CoRN/reals/IVT.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: IVT.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
  20
  21 include "reals/CPoly_Contin.ma".
  22
  23 (* UNEXPORTED
  24 Section Nested_Intervals
  25 *)
  26
  27 (*#* * Intermediate Value Theorem
  28
  29 ** Nested intervals
  30
  31 %\begin{convention}% Let [a,b:nat->IR] be sequences such that:
  32 - [a] is increasing;
  33 - [b] is decreasing;
  34 - [forall (i:nat), (a i) [<] (b i)];
  35 - for every positive real number [eps], there is an [i] such that [(b i) [<] (a i) [+]eps].
  36
  37 %\end{convention}%
  38 *)
  39
  40 (* UNEXPORTED
  41 cic:/CoRN/reals/IVT/Nested_Intervals/a.var
  42 *)
  43
  44 (* UNEXPORTED
  45 cic:/CoRN/reals/IVT/Nested_Intervals/b.var
  46 *)
  47
  48 (* UNEXPORTED
  49 cic:/CoRN/reals/IVT/Nested_Intervals/a_mon.var
  50 *)
  51
  52 (* UNEXPORTED
  53 cic:/CoRN/reals/IVT/Nested_Intervals/b_mon.var
  54 *)
  55
  56 (* UNEXPORTED
  57 cic:/CoRN/reals/IVT/Nested_Intervals/a_b.var
  58 *)
  59
  60 (* UNEXPORTED
  61 cic:/CoRN/reals/IVT/Nested_Intervals/b_a.var
  62 *)
  63
  64 inline procedural "cic:/CoRN/reals/IVT/a_mon'.con" as lemma.
  65
  66 inline procedural "cic:/CoRN/reals/IVT/b_mon'.con" as lemma.
  67
  68 inline procedural "cic:/CoRN/reals/IVT/a_b'.con" as lemma.
  69
  70 inline procedural "cic:/CoRN/reals/IVT/intervals_cauchy.con" as lemma.
  71
  72 (* begin hide *)
  73
  74 inline procedural "cic:/CoRN/reals/IVT/Nested_Intervals/a'.con" "Nested_Intervals__" as definition.
  75
  76 (* end hide *)
  77
  78 inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_limit.con" as lemma.
  79
  80 (*#* %\begin{convention}% Let [f] be a continuous real function.
  81 %\end{convention}%
  82 *)
  83
  84 (* UNEXPORTED
  85 cic:/CoRN/reals/IVT/Nested_Intervals/f.var
  86 *)
  87
  88 (* UNEXPORTED
  89 cic:/CoRN/reals/IVT/Nested_Intervals/f_contin.var
  90 *)
  91
  92 inline procedural "cic:/CoRN/reals/IVT/f_contin_pos.con" as lemma.
  93
  94 inline procedural "cic:/CoRN/reals/IVT/f_contin_neg.con" as lemma.
  95
  96 (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *)
  97
  98 (* UNEXPORTED
  99 cic:/CoRN/reals/IVT/Nested_Intervals/f_a.var
 100 *)
 101
 102 (* UNEXPORTED
 103 cic:/CoRN/reals/IVT/Nested_Intervals/f_b.var
 104 *)
 105
 106 inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_zero.con" as lemma.
 107
 108 (* UNEXPORTED
 109 End Nested_Intervals
 110 *)
 111
 112 (* UNEXPORTED
 113 Section Bisection
 114 *)
 115
 116 (*#* ** Bissections *)
 117
 118 (* UNEXPORTED
 119 cic:/CoRN/reals/IVT/Bisection/f.var
 120 *)
 121
 122 (* UNEXPORTED
 123 cic:/CoRN/reals/IVT/Bisection/f_apzero_interval.var
 124 *)
 125
 126 (* UNEXPORTED
 127 cic:/CoRN/reals/IVT/Bisection/a.var
 128 *)
 129
 130 (* UNEXPORTED
 131 cic:/CoRN/reals/IVT/Bisection/b.var
 132 *)
 133
 134 (* UNEXPORTED
 135 cic:/CoRN/reals/IVT/Bisection/a_b.var
 136 *)
 137
 138 (* UNEXPORTED
 139 cic:/CoRN/reals/IVT/Bisection/f_a.var
 140 *)
 141
 142 (* UNEXPORTED
 143 cic:/CoRN/reals/IVT/Bisection/f_b.var
 144 *)
 145
 146 (*#*
 147 %\begin{convention}% Let [Small] denote [Two[/]ThreeNZ], [lft] be [(Two[*]a[+]b) [/]ThreeNZ] and [rht] be [(a[+]Two[*]b) [/]ThreeNZ].
 148 %\end{convention}%
 149 *)
 150
 151 (* begin hide *)
 152
 153 inline procedural "cic:/CoRN/reals/IVT/Bisection/Small.con" "Bisection__" as definition.
 154
 155 inline procedural "cic:/CoRN/reals/IVT/Bisection/lft.con" "Bisection__" as definition.
 156
 157 inline procedural "cic:/CoRN/reals/IVT/Bisection/rht.con" "Bisection__" as definition.
 158
 159 (* end hide *)
 160
 161 inline procedural "cic:/CoRN/reals/IVT/a_lft.con" as lemma.
 162
 163 inline procedural "cic:/CoRN/reals/IVT/rht_b.con" as lemma.
 164
 165 inline procedural "cic:/CoRN/reals/IVT/lft_rht.con" as lemma.
 166
 167 inline procedural "cic:/CoRN/reals/IVT/smaller_lft.con" as lemma.
 168
 169 inline procedural "cic:/CoRN/reals/IVT/smaller_rht.con" as lemma.
 170
 171 (* UNEXPORTED
 172 Hint Resolve smaller_lft smaller_rht: algebra.
 173 *)
 174
 175 inline procedural "cic:/CoRN/reals/IVT/Cbisect'.con" as lemma.
 176
 177 (* UNEXPORTED
 178 End Bisection
 179 *)
 180
 181 (* UNEXPORTED
 182 Section Bisect_Interval
 183 *)
 184
 185 (* UNEXPORTED
 186 cic:/CoRN/reals/IVT/Bisect_Interval/f.var
 187 *)
 188
 189 (* UNEXPORTED
 190 cic:/CoRN/reals/IVT/Bisect_Interval/C_f_apzero_interval.var
 191 *)
 192
 193 (* begin hide *)
 194
 195 inline procedural "cic:/CoRN/reals/IVT/Bisect_Interval/Small.con" "Bisect_Interval__" as definition.
 196
 197 (* end hide *)
 198
 199 inline procedural "cic:/CoRN/reals/IVT/bisect_interval.ind".
 200
 201 inline procedural "cic:/CoRN/reals/IVT/Cbisect_exists.con" as lemma.
 202
 203 inline procedural "cic:/CoRN/reals/IVT/bisect.con" as definition.
 204
 205 inline procedural "cic:/CoRN/reals/IVT/bisect_prop.con" as lemma.
 206
 207 (* UNEXPORTED
 208 End Bisect_Interval
 209 *)
 210
 211 (* UNEXPORTED
 212 Section IVT_Op
 213 *)
 214
 215 (*#* ** IVT for operations
 216 Same conventions as before.
 217 *)
 218
 219 (* UNEXPORTED
 220 cic:/CoRN/reals/IVT/IVT_Op/f.var
 221 *)
 222
 223 (* UNEXPORTED
 224 cic:/CoRN/reals/IVT/IVT_Op/f_contin.var
 225 *)
 226
 227 (* UNEXPORTED
 228 cic:/CoRN/reals/IVT/IVT_Op/f_apzero_interval.var
 229 *)
 230
 231 (* UNEXPORTED
 232 cic:/CoRN/reals/IVT/IVT_Op/a.var
 233 *)
 234
 235 (* UNEXPORTED
 236 cic:/CoRN/reals/IVT/IVT_Op/b.var
 237 *)
 238
 239 (* UNEXPORTED
 240 cic:/CoRN/reals/IVT/IVT_Op/a_b.var
 241 *)
 242
 243 (* UNEXPORTED
 244 cic:/CoRN/reals/IVT/IVT_Op/f_a.var
 245 *)
 246
 247 (* UNEXPORTED
 248 cic:/CoRN/reals/IVT/IVT_Op/f_b.var
 249 *)
 250
 251 (* begin hide *)
 252
 253 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/Small.con" "IVT_Op__" as definition.
 254
 255 (* end hide *)
 256
 257 inline procedural "cic:/CoRN/reals/IVT/interval_sequence.con" as definition.
 258
 259 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/a_.con" "IVT_Op__" as definition.
 260
 261 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/b_.con" "IVT_Op__" as definition.
 262
 263 inline procedural "cic:/CoRN/reals/IVT/intervals_smaller.con" as lemma.
 264
 265 inline procedural "cic:/CoRN/reals/IVT/intervals_small''.con" as lemma.
 266
 267 inline procedural "cic:/CoRN/reals/IVT/intervals_small'.con" as lemma.
 268
 269 inline procedural "cic:/CoRN/reals/IVT/intervals_small.con" as lemma.
 270
 271 inline procedural "cic:/CoRN/reals/IVT/Civt_op.con" as lemma.
 272
 273 (* UNEXPORTED
 274 End IVT_Op
 275 *)
 276
 277 (* UNEXPORTED
 278 Section IVT_Poly
 279 *)
 280
 281 (*#* ** IVT for polynomials *)
 282
 283 inline procedural "cic:/CoRN/reals/IVT/Civt_poly.con" as lemma.
 284
 285 (* UNEXPORTED
 286 End IVT_Poly
 287 *)
 288