helm/software/matita/contribs/procedural/Coq/Reals/Rfunctions.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 "Coq.ma".
  18
  19 (*#***********************************************************************)
  20
  21 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
  22
  23 (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
  24
  25 (*   \VV/  **************************************************************)
  26
  27 (*    //   *      This file is distributed under the terms of the       *)
  28
  29 (*         *       GNU Lesser General Public License Version 2.1        *)
  30
  31 (*#***********************************************************************)
  32
  33 (*i $Id: Rfunctions.v,v 1.31.2.1 2004/07/16 19:31:12 herbelin Exp $ i*)
  34
  35 (*i Some properties about pow and sum have been made with John Harrison i*)
  36
  37 (*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*)
  38
  39 (*#*******************************************************)
  40
  41 (*#*          Definition of the sum functions            *)
  42
  43 (*                                                      *)
  44
  45 (*#*******************************************************)
  46
  47 include "Reals/Rbase.ma".
  48
  49 include "Reals/R_Ifp.ma".
  50
  51 include "Reals/Rbasic_fun.ma".
  52
  53 include "Reals/R_sqr.ma".
  54
  55 include "Reals/SplitAbsolu.ma".
  56
  57 include "Reals/SplitRmult.ma".
  58
  59 include "Reals/ArithProp.ma".
  60
  61 include "ZArith/Zpower.ma".
  62
  63 (* UNEXPORTED
  64 Open Local Scope nat_scope.
  65 *)
  66
  67 (* UNEXPORTED
  68 Open Local Scope R_scope.
  69 *)
  70
  71 (*#******************************)
  72
  73 (*#*  Lemmas about factorial    *)
  74
  75 (*#******************************)
  76
  77 (*#********)
  78
  79 inline procedural "cic:/Coq/Reals/Rfunctions/INR_fact_neq_0.con" as lemma.
  80
  81 (*#********)
  82
  83 inline procedural "cic:/Coq/Reals/Rfunctions/fact_simpl.con" as lemma.
  84
  85 (*#********)
  86
  87 inline procedural "cic:/Coq/Reals/Rfunctions/simpl_fact.con" as lemma.
  88
  89 (*#******************************)
  90
  91 (*          Power              *)
  92
  93 (*#******************************)
  94
  95 (*#********)
  96
  97 inline procedural "cic:/Coq/Reals/Rfunctions/pow.con" as definition.
  98
  99 (* NOTATION
 100 Infix "^" := pow : R_scope.
 101 *)
 102
 103 inline procedural "cic:/Coq/Reals/Rfunctions/pow_O.con" as lemma.
 104
 105 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1.con" as lemma.
 106
 107 inline procedural "cic:/Coq/Reals/Rfunctions/pow_add.con" as lemma.
 108
 109 inline procedural "cic:/Coq/Reals/Rfunctions/pow_nonzero.con" as lemma.
 110
 111 (* UNEXPORTED
 112 Hint Resolve pow_O pow_1 pow_add pow_nonzero: real.
 113 *)
 114
 115 inline procedural "cic:/Coq/Reals/Rfunctions/pow_RN_plus.con" as lemma.
 116
 117 inline procedural "cic:/Coq/Reals/Rfunctions/pow_lt.con" as lemma.
 118
 119 (* UNEXPORTED
 120 Hint Resolve pow_lt: real.
 121 *)
 122
 123 inline procedural "cic:/Coq/Reals/Rfunctions/Rlt_pow_R1.con" as lemma.
 124
 125 (* UNEXPORTED
 126 Hint Resolve Rlt_pow_R1: real.
 127 *)
 128
 129 inline procedural "cic:/Coq/Reals/Rfunctions/Rlt_pow.con" as lemma.
 130
 131 (* UNEXPORTED
 132 Hint Resolve Rlt_pow: real.
 133 *)
 134
 135 (*#********)
 136
 137 inline procedural "cic:/Coq/Reals/Rfunctions/tech_pow_Rmult.con" as lemma.
 138
 139 (*#********)
 140
 141 inline procedural "cic:/Coq/Reals/Rfunctions/tech_pow_Rplus.con" as lemma.
 142
 143 inline procedural "cic:/Coq/Reals/Rfunctions/poly.con" as lemma.
 144
 145 inline procedural "cic:/Coq/Reals/Rfunctions/Power_monotonic.con" as lemma.
 146
 147 inline procedural "cic:/Coq/Reals/Rfunctions/RPow_abs.con" as lemma.
 148
 149 inline procedural "cic:/Coq/Reals/Rfunctions/Pow_x_infinity.con" as lemma.
 150
 151 inline procedural "cic:/Coq/Reals/Rfunctions/pow_ne_zero.con" as lemma.
 152
 153 inline procedural "cic:/Coq/Reals/Rfunctions/Rinv_pow.con" as lemma.
 154
 155 inline procedural "cic:/Coq/Reals/Rfunctions/pow_lt_1_zero.con" as lemma.
 156
 157 inline procedural "cic:/Coq/Reals/Rfunctions/pow_R1.con" as lemma.
 158
 159 inline procedural "cic:/Coq/Reals/Rfunctions/pow_Rsqr.con" as lemma.
 160
 161 inline procedural "cic:/Coq/Reals/Rfunctions/pow_le.con" as lemma.
 162
 163 (*#*********)
 164
 165 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1_even.con" as lemma.
 166
 167 (*#*********)
 168
 169 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1_odd.con" as lemma.
 170
 171 (*#*********)
 172
 173 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1_abs.con" as lemma.
 174
 175 inline procedural "cic:/Coq/Reals/Rfunctions/pow_mult.con" as lemma.
 176
 177 inline procedural "cic:/Coq/Reals/Rfunctions/pow_incr.con" as lemma.
 178
 179 inline procedural "cic:/Coq/Reals/Rfunctions/pow_R1_Rle.con" as lemma.
 180
 181 inline procedural "cic:/Coq/Reals/Rfunctions/Rle_pow.con" as lemma.
 182
 183 inline procedural "cic:/Coq/Reals/Rfunctions/pow1.con" as lemma.
 184
 185 inline procedural "cic:/Coq/Reals/Rfunctions/pow_Rabs.con" as lemma.
 186
 187 inline procedural "cic:/Coq/Reals/Rfunctions/pow_maj_Rabs.con" as lemma.
 188
 189 (*#******************************)
 190
 191 (*#*         PowerRZ            *)
 192
 193 (*#******************************)
 194
 195 (*i Due to L.Thery i*)
 196
 197 (* UNEXPORTED
 198 Ltac case_eq name :=
 199   generalize (refl_equal name); pattern name at -1 in |- *; case name.
 200 *)
 201
 202 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ.con" as definition.
 203
 204 (* NOTATION
 205 Infix Local "^Z" := powerRZ (at level 30, right associativity) : R_scope.
 206 *)
 207
 208 inline procedural "cic:/Coq/Reals/Rfunctions/Zpower_NR0.con" as lemma.
 209
 210 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_O.con" as lemma.
 211
 212 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_1.con" as lemma.
 213
 214 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_NOR.con" as lemma.
 215
 216 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_add.con" as lemma.
 217
 218 (* UNEXPORTED
 219 Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real.
 220 *)
 221
 222 inline procedural "cic:/Coq/Reals/Rfunctions/Zpower_nat_powerRZ.con" as lemma.
 223
 224 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_lt.con" as lemma.
 225
 226 (* UNEXPORTED
 227 Hint Resolve powerRZ_lt: real.
 228 *)
 229
 230 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_le.con" as lemma.
 231
 232 (* UNEXPORTED
 233 Hint Resolve powerRZ_le: real.
 234 *)
 235
 236 inline procedural "cic:/Coq/Reals/Rfunctions/Zpower_nat_powerRZ_absolu.con" as lemma.
 237
 238 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_R1.con" as lemma.
 239
 240 (*#******************************)
 241
 242 (* For easy interface          *)
 243
 244 (*#******************************)
 245
 246 (* decimal_exp r z is defined as r 10^z *)
 247
 248 inline procedural "cic:/Coq/Reals/Rfunctions/decimal_exp.con" as definition.
 249
 250 (*#******************************)
 251
 252 (*#* Sum of n first naturals    *)
 253
 254 (*#******************************)
 255
 256 (*#********)
 257
 258 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat_f_O.con" as definition.
 259
 260 (*#********)
 261
 262 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat_f.con" as definition.
 263
 264 (*#********)
 265
 266 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat_O.con" as definition.
 267
 268 (*#********)
 269
 270 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat.con" as definition.
 271
 272 (*#******************************)
 273
 274 (*#*            Sum             *)
 275
 276 (*#******************************)
 277
 278 (*#********)
 279
 280 inline procedural "cic:/Coq/Reals/Rfunctions/sum_f_R0.con" as definition.
 281
 282 (*#********)
 283
 284 inline procedural "cic:/Coq/Reals/Rfunctions/sum_f.con" as definition.
 285
 286 inline procedural "cic:/Coq/Reals/Rfunctions/GP_finite.con" as lemma.
 287
 288 inline procedural "cic:/Coq/Reals/Rfunctions/sum_f_R0_triangle.con" as lemma.
 289
 290 (*#******************************)
 291
 292 (*        Distance  in R       *)
 293
 294 (*#******************************)
 295
 296 (*#********)
 297
 298 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist.con" as definition.
 299
 300 (*#********)
 301
 302 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_pos.con" as lemma.
 303
 304 (*#********)
 305
 306 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_sym.con" as lemma.
 307
 308 (*#********)
 309
 310 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_refl.con" as lemma.
 311
 312 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_eq.con" as lemma.
 313
 314 (*#**********)
 315
 316 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_tri.con" as lemma.
 317
 318 (*#********)
 319
 320 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_plus.con" as lemma.
 321
 322 (*#******************************)
 323
 324 (*#*       Infinit Sum          *)
 325
 326 (*#******************************)
 327
 328 (*#********)
 329
 330 inline procedural "cic:/Coq/Reals/Rfunctions/infinit_sum.con" as definition.
 331