matita/contribs/CoRN-Decl/reals/IVT.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/reals/IVT".
  18
  19 (* $Id: IVT.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
  20
  21 (* INCLUDE
  22 CPoly_Contin
  23 *)
  24
  25 (* UNEXPORTED
  26 Section Nested_Intervals.
  27 *)
  28
  29 (*#* * Intermediate Value Theorem
  30
  31 ** Nested intervals
  32
  33 %\begin{convention}% Let [a,b:nat->IR] be sequences such that:
  34 - [a] is increasing;
  35 - [b] is decreasing;
  36 - [forall (i:nat), (a i) [<] (b i)];
  37 - for every positive real number [eps], there is an [i] such that [(b i) [<] (a i) [+]eps].
  38
  39 %\end{convention}%
  40 *)
  41
  42 inline cic:/CoRN/reals/IVT/a.var.
  43
  44 inline cic:/CoRN/reals/IVT/b.var.
  45
  46 inline cic:/CoRN/reals/IVT/a_mon.var.
  47
  48 inline cic:/CoRN/reals/IVT/b_mon.var.
  49
  50 inline cic:/CoRN/reals/IVT/a_b.var.
  51
  52 inline cic:/CoRN/reals/IVT/b_a.var.
  53
  54 inline cic:/CoRN/reals/IVT/a_mon'.con.
  55
  56 inline cic:/CoRN/reals/IVT/b_mon'.con.
  57
  58 inline cic:/CoRN/reals/IVT/a_b'.con.
  59
  60 inline cic:/CoRN/reals/IVT/intervals_cauchy.con.
  61
  62 (* begin hide *)
  63
  64 inline cic:/CoRN/reals/IVT/a'.con.
  65
  66 (* end hide *)
  67
  68 inline cic:/CoRN/reals/IVT/Cnested_intervals_limit.con.
  69
  70 (*#* %\begin{convention}% Let [f] be a continuous real function.
  71 %\end{convention}%
  72 *)
  73
  74 inline cic:/CoRN/reals/IVT/f.var.
  75
  76 inline cic:/CoRN/reals/IVT/f_contin.var.
  77
  78 inline cic:/CoRN/reals/IVT/f_contin_pos.con.
  79
  80 inline cic:/CoRN/reals/IVT/f_contin_neg.con.
  81
  82 (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *)
  83
  84 inline cic:/CoRN/reals/IVT/f_a.var.
  85
  86 inline cic:/CoRN/reals/IVT/f_b.var.
  87
  88 inline cic:/CoRN/reals/IVT/Cnested_intervals_zero.con.
  89
  90 (* UNEXPORTED
  91 End Nested_Intervals.
  92 *)
  93
  94 (* UNEXPORTED
  95 Section Bisection.
  96 *)
  97
  98 (*#* ** Bissections *)
  99
 100 inline cic:/CoRN/reals/IVT/f.var.
 101
 102 inline cic:/CoRN/reals/IVT/f_apzero_interval.var.
 103
 104 inline cic:/CoRN/reals/IVT/a.var.
 105
 106 inline cic:/CoRN/reals/IVT/b.var.
 107
 108 inline cic:/CoRN/reals/IVT/a_b.var.
 109
 110 inline cic:/CoRN/reals/IVT/f_a.var.
 111
 112 inline cic:/CoRN/reals/IVT/f_b.var.
 113
 114 (*#*
 115 %\begin{convention}% Let [Small] denote [Two[/]ThreeNZ], [lft] be [(Two[*]a[+]b) [/]ThreeNZ] and [rht] be [(a[+]Two[*]b) [/]ThreeNZ].
 116 %\end{convention}%
 117 *)
 118
 119 (* begin hide *)
 120
 121 inline cic:/CoRN/reals/IVT/Small.con.
 122
 123 inline cic:/CoRN/reals/IVT/lft.con.
 124
 125 inline cic:/CoRN/reals/IVT/rht.con.
 126
 127 (* end hide *)
 128
 129 inline cic:/CoRN/reals/IVT/a_lft.con.
 130
 131 inline cic:/CoRN/reals/IVT/rht_b.con.
 132
 133 inline cic:/CoRN/reals/IVT/lft_rht.con.
 134
 135 inline cic:/CoRN/reals/IVT/smaller_lft.con.
 136
 137 inline cic:/CoRN/reals/IVT/smaller_rht.con.
 138
 139 (* UNEXPORTED
 140 Hint Resolve smaller_lft smaller_rht: algebra.
 141 *)
 142
 143 inline cic:/CoRN/reals/IVT/Cbisect'.con.
 144
 145 (* UNEXPORTED
 146 End Bisection.
 147 *)
 148
 149 (* UNEXPORTED
 150 Section Bisect_Interval.
 151 *)
 152
 153 inline cic:/CoRN/reals/IVT/f.var.
 154
 155 inline cic:/CoRN/reals/IVT/C_f_apzero_interval.var.
 156
 157 (* begin hide *)
 158
 159 inline cic:/CoRN/reals/IVT/Small.con.
 160
 161 (* end hide *)
 162
 163 inline cic:/CoRN/reals/IVT/bisect_interval.ind.
 164
 165 inline cic:/CoRN/reals/IVT/Cbisect_exists.con.
 166
 167 inline cic:/CoRN/reals/IVT/bisect.con.
 168
 169 inline cic:/CoRN/reals/IVT/bisect_prop.con.
 170
 171 (* UNEXPORTED
 172 End Bisect_Interval.
 173 *)
 174
 175 (* UNEXPORTED
 176 Section IVT_Op.
 177 *)
 178
 179 (*#* ** IVT for operations
 180 Same conventions as before.
 181 *)
 182
 183 inline cic:/CoRN/reals/IVT/f.var.
 184
 185 inline cic:/CoRN/reals/IVT/f_contin.var.
 186
 187 inline cic:/CoRN/reals/IVT/f_apzero_interval.var.
 188
 189 inline cic:/CoRN/reals/IVT/a.var.
 190
 191 inline cic:/CoRN/reals/IVT/b.var.
 192
 193 inline cic:/CoRN/reals/IVT/a_b.var.
 194
 195 inline cic:/CoRN/reals/IVT/f_a.var.
 196
 197 inline cic:/CoRN/reals/IVT/f_b.var.
 198
 199 (* begin hide *)
 200
 201 inline cic:/CoRN/reals/IVT/Small.con.
 202
 203 (* end hide *)
 204
 205 inline cic:/CoRN/reals/IVT/interval_sequence.con.
 206
 207 inline cic:/CoRN/reals/IVT/a_.con.
 208
 209 inline cic:/CoRN/reals/IVT/b_.con.
 210
 211 inline cic:/CoRN/reals/IVT/intervals_smaller.con.
 212
 213 inline cic:/CoRN/reals/IVT/intervals_small''.con.
 214
 215 inline cic:/CoRN/reals/IVT/intervals_small'.con.
 216
 217 inline cic:/CoRN/reals/IVT/intervals_small.con.
 218
 219 inline cic:/CoRN/reals/IVT/Civt_op.con.
 220
 221 (* UNEXPORTED
 222 End IVT_Op.
 223 *)
 224
 225 (* UNEXPORTED
 226 Section IVT_Poly.
 227 *)
 228
 229 (*#* ** IVT for polynomials *)
 230
 231 inline cic:/CoRN/reals/IVT/Civt_poly.con.
 232
 233 (* UNEXPORTED
 234 End IVT_Poly.
 235 *)
 236