helm/software/matita/contribs/CoRN-Procedural/reals/IVT.mma

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   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 alias id "a" = "cic:/CoRN/reals/IVT/Nested_Intervals/a.var".
  41
  42 alias id "b" = "cic:/CoRN/reals/IVT/Nested_Intervals/b.var".
  43
  44 alias id "a_mon" = "cic:/CoRN/reals/IVT/Nested_Intervals/a_mon.var".
  45
  46 alias id "b_mon" = "cic:/CoRN/reals/IVT/Nested_Intervals/b_mon.var".
  47
  48 alias id "a_b" = "cic:/CoRN/reals/IVT/Nested_Intervals/a_b.var".
  49
  50 alias id "b_a" = "cic:/CoRN/reals/IVT/Nested_Intervals/b_a.var".
  51
  52 inline procedural "cic:/CoRN/reals/IVT/a_mon'.con".
  53
  54 inline procedural "cic:/CoRN/reals/IVT/b_mon'.con".
  55
  56 inline procedural "cic:/CoRN/reals/IVT/a_b'.con".
  57
  58 inline procedural "cic:/CoRN/reals/IVT/intervals_cauchy.con".
  59
  60 (* begin hide *)
  61
  62 inline procedural "cic:/CoRN/reals/IVT/Nested_Intervals/a'.con" "Nested_Intervals__".
  63
  64 (* end hide *)
  65
  66 inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_limit.con".
  67
  68 (*#* %\begin{convention}% Let [f] be a continuous real function.
  69 %\end{convention}%
  70 *)
  71
  72 alias id "f" = "cic:/CoRN/reals/IVT/Nested_Intervals/f.var".
  73
  74 alias id "f_contin" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_contin.var".
  75
  76 inline procedural "cic:/CoRN/reals/IVT/f_contin_pos.con".
  77
  78 inline procedural "cic:/CoRN/reals/IVT/f_contin_neg.con".
  79
  80 (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *)
  81
  82 alias id "f_a" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_a.var".
  83
  84 alias id "f_b" = "cic:/CoRN/reals/IVT/Nested_Intervals/f_b.var".
  85
  86 inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_zero.con".
  87
  88 (* UNEXPORTED
  89 End Nested_Intervals
  90 *)
  91
  92 (* UNEXPORTED
  93 Section Bisection
  94 *)
  95
  96 (*#* ** Bissections *)
  97
  98 alias id "f" = "cic:/CoRN/reals/IVT/Bisection/f.var".
  99
 100 alias id "f_apzero_interval" = "cic:/CoRN/reals/IVT/Bisection/f_apzero_interval.var".
 101
 102 alias id "a" = "cic:/CoRN/reals/IVT/Bisection/a.var".
 103
 104 alias id "b" = "cic:/CoRN/reals/IVT/Bisection/b.var".
 105
 106 alias id "a_b" = "cic:/CoRN/reals/IVT/Bisection/a_b.var".
 107
 108 alias id "f_a" = "cic:/CoRN/reals/IVT/Bisection/f_a.var".
 109
 110 alias id "f_b" = "cic:/CoRN/reals/IVT/Bisection/f_b.var".
 111
 112 (*#*
 113 %\begin{convention}% Let [Small] denote [Two[/]ThreeNZ], [lft] be [(Two[*]a[+]b) [/]ThreeNZ] and [rht] be [(a[+]Two[*]b) [/]ThreeNZ].
 114 %\end{convention}%
 115 *)
 116
 117 (* begin hide *)
 118
 119 inline procedural "cic:/CoRN/reals/IVT/Bisection/Small.con" "Bisection__".
 120
 121 inline procedural "cic:/CoRN/reals/IVT/Bisection/lft.con" "Bisection__".
 122
 123 inline procedural "cic:/CoRN/reals/IVT/Bisection/rht.con" "Bisection__".
 124
 125 (* end hide *)
 126
 127 inline procedural "cic:/CoRN/reals/IVT/a_lft.con".
 128
 129 inline procedural "cic:/CoRN/reals/IVT/rht_b.con".
 130
 131 inline procedural "cic:/CoRN/reals/IVT/lft_rht.con".
 132
 133 inline procedural "cic:/CoRN/reals/IVT/smaller_lft.con".
 134
 135 inline procedural "cic:/CoRN/reals/IVT/smaller_rht.con".
 136
 137 (* UNEXPORTED
 138 Hint Resolve smaller_lft smaller_rht: algebra.
 139 *)
 140
 141 inline procedural "cic:/CoRN/reals/IVT/Cbisect'.con".
 142
 143 (* UNEXPORTED
 144 End Bisection
 145 *)
 146
 147 (* UNEXPORTED
 148 Section Bisect_Interval
 149 *)
 150
 151 alias id "f" = "cic:/CoRN/reals/IVT/Bisect_Interval/f.var".
 152
 153 alias id "C_f_apzero_interval" = "cic:/CoRN/reals/IVT/Bisect_Interval/C_f_apzero_interval.var".
 154
 155 (* begin hide *)
 156
 157 inline procedural "cic:/CoRN/reals/IVT/Bisect_Interval/Small.con" "Bisect_Interval__".
 158
 159 (* end hide *)
 160
 161 inline procedural "cic:/CoRN/reals/IVT/bisect_interval.ind".
 162
 163 inline procedural "cic:/CoRN/reals/IVT/Cbisect_exists.con".
 164
 165 inline procedural "cic:/CoRN/reals/IVT/bisect.con".
 166
 167 inline procedural "cic:/CoRN/reals/IVT/bisect_prop.con".
 168
 169 (* UNEXPORTED
 170 End Bisect_Interval
 171 *)
 172
 173 (* UNEXPORTED
 174 Section IVT_Op
 175 *)
 176
 177 (*#* ** IVT for operations
 178 Same conventions as before.
 179 *)
 180
 181 alias id "f" = "cic:/CoRN/reals/IVT/IVT_Op/f.var".
 182
 183 alias id "f_contin" = "cic:/CoRN/reals/IVT/IVT_Op/f_contin.var".
 184
 185 alias id "f_apzero_interval" = "cic:/CoRN/reals/IVT/IVT_Op/f_apzero_interval.var".
 186
 187 alias id "a" = "cic:/CoRN/reals/IVT/IVT_Op/a.var".
 188
 189 alias id "b" = "cic:/CoRN/reals/IVT/IVT_Op/b.var".
 190
 191 alias id "a_b" = "cic:/CoRN/reals/IVT/IVT_Op/a_b.var".
 192
 193 alias id "f_a" = "cic:/CoRN/reals/IVT/IVT_Op/f_a.var".
 194
 195 alias id "f_b" = "cic:/CoRN/reals/IVT/IVT_Op/f_b.var".
 196
 197 (* begin hide *)
 198
 199 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/Small.con" "IVT_Op__".
 200
 201 (* end hide *)
 202
 203 inline procedural "cic:/CoRN/reals/IVT/interval_sequence.con".
 204
 205 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/a_.con" "IVT_Op__".
 206
 207 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/b_.con" "IVT_Op__".
 208
 209 inline procedural "cic:/CoRN/reals/IVT/intervals_smaller.con".
 210
 211 inline procedural "cic:/CoRN/reals/IVT/intervals_small''.con".
 212
 213 inline procedural "cic:/CoRN/reals/IVT/intervals_small'.con".
 214
 215 inline procedural "cic:/CoRN/reals/IVT/intervals_small.con".
 216
 217 inline procedural "cic:/CoRN/reals/IVT/Civt_op.con".
 218
 219 (* UNEXPORTED
 220 End IVT_Op
 221 *)
 222
 223 (* UNEXPORTED
 224 Section IVT_Poly
 225 *)
 226
 227 (*#* ** IVT for polynomials *)
 228
 229 inline procedural "cic:/CoRN/reals/IVT/Civt_poly.con".
 230
 231 (* UNEXPORTED
 232 End IVT_Poly
 233 *)
 234