helm/software/matita/contribs/procedural/CoRN/algebra/COrdFields2.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 include "algebra/COrdFields.ma".
  20
  21 (*#* printing one_div_succ %\ensuremath{\frac1{\cdot+1}}% *)
  22
  23 (*#* printing Half %\ensuremath{\frac12}% #&frac12;# *)
  24
  25 (*#**********************************)
  26
  27 (* UNEXPORTED
  28 Section Properties_of_leEq
  29 *)
  30
  31 (*#**********************************)
  32
  33 (*#*
  34 ** Properties of [[<=]]
  35 %\begin{convention}% Let [R] be an ordered field.
  36 %\end{convention}%
  37 *)
  38
  39 (* UNEXPORTED
  40 cic:/CoRN/algebra/COrdFields2/Properties_of_leEq/R.var
  41 *)
  42
  43 (* UNEXPORTED
  44 Section addition
  45 *)
  46
  47 (*#*
  48 *** Addition and subtraction%\label{section:leEq-plus-minus}%
  49 *)
  50
  51 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq.con" as lemma.
  52
  53 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq_lft.con" as lemma.
  54
  55 inline procedural "cic:/CoRN/algebra/COrdFields2/minus_resp_leEq.con" as lemma.
  56
  57 inline procedural "cic:/CoRN/algebra/COrdFields2/inv_resp_leEq.con" as lemma.
  58
  59 (* UNEXPORTED
  60 Transparent cg_minus.
  61 *)
  62
  63 inline procedural "cic:/CoRN/algebra/COrdFields2/minus_resp_leEq_rht.con" as lemma.
  64
  65 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq_both.con" as lemma.
  66
  67 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_less_leEq.con" as lemma.
  68
  69 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq_less.con" as lemma.
  70
  71 inline procedural "cic:/CoRN/algebra/COrdFields2/minus_resp_less_leEq.con" as lemma.
  72
  73 inline procedural "cic:/CoRN/algebra/COrdFields2/minus_resp_leEq_both.con" as lemma.
  74
  75 (*#* Cancellation properties
  76 *)
  77
  78 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_cancel_leEq_rht.con" as lemma.
  79
  80 inline procedural "cic:/CoRN/algebra/COrdFields2/inv_cancel_leEq.con" as lemma.
  81
  82 (*#* Laws for shifting
  83 *)
  84
  85 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_plus_leEq.con" as lemma.
  86
  87 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_plus.con" as lemma.
  88
  89 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_plus_leEq'.con" as lemma.
  90
  91 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_plus'.con" as lemma.
  92
  93 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_rht.con" as lemma.
  94
  95 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_lft.con" as lemma.
  96
  97 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_minus_leEq.con" as lemma.
  98
  99 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_minus.con" as lemma.
 100
 101 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_minus'.con" as lemma.
 102
 103 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_zero_leEq_minus.con" as lemma.
 104
 105 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_zero_leEq_minus'.con" as lemma.
 106
 107 (* UNEXPORTED
 108 End addition
 109 *)
 110
 111 (* UNEXPORTED
 112 Section multiplication
 113 *)
 114
 115 (*#*
 116 *** Multiplication and division
 117
 118 Multiplication and division respect [[<=]]
 119 *)
 120
 121 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_resp_leEq_rht.con" as lemma.
 122
 123 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_resp_leEq_lft.con" as lemma.
 124
 125 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_resp_leEq_both.con" as lemma.
 126
 127 inline procedural "cic:/CoRN/algebra/COrdFields2/recip_resp_leEq.con" as lemma.
 128
 129 inline procedural "cic:/CoRN/algebra/COrdFields2/div_resp_leEq.con" as lemma.
 130
 131 (* UNEXPORTED
 132 Hint Resolve recip_resp_leEq: algebra.
 133 *)
 134
 135 (*#* Cancellation properties
 136 *)
 137
 138 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_cancel_leEq.con" as lemma.
 139
 140 (*#* Laws for shifting
 141 *)
 142
 143 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_mult_leEq.con" as lemma.
 144
 145 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_mult_leEq'.con" as lemma.
 146
 147 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_mult'.con" as lemma.
 148
 149 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_div_leEq.con" as lemma.
 150
 151 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_div_leEq'.con" as lemma.
 152
 153 inline procedural "cic:/CoRN/algebra/COrdFields2/shift_leEq_div.con" as lemma.
 154
 155 (* UNEXPORTED
 156 Hint Resolve shift_leEq_div: algebra.
 157 *)
 158
 159 inline procedural "cic:/CoRN/algebra/COrdFields2/eps_div_leEq_eps.con" as lemma.
 160
 161 inline procedural "cic:/CoRN/algebra/COrdFields2/nonneg_div_two.con" as lemma.
 162
 163 inline procedural "cic:/CoRN/algebra/COrdFields2/nonneg_div_two'.con" as lemma.
 164
 165 inline procedural "cic:/CoRN/algebra/COrdFields2/nonneg_div_three.con" as lemma.
 166
 167 inline procedural "cic:/CoRN/algebra/COrdFields2/nonneg_div_three'.con" as lemma.
 168
 169 inline procedural "cic:/CoRN/algebra/COrdFields2/nonneg_div_four.con" as lemma.
 170
 171 inline procedural "cic:/CoRN/algebra/COrdFields2/nonneg_div_four'.con" as lemma.
 172
 173 (* UNEXPORTED
 174 End multiplication
 175 *)
 176
 177 (* UNEXPORTED
 178 Section misc
 179 *)
 180
 181 (*#*
 182 *** Miscellaneous Properties
 183 *)
 184
 185 inline procedural "cic:/CoRN/algebra/COrdFields2/sqr_nonneg.con" as lemma.
 186
 187 inline procedural "cic:/CoRN/algebra/COrdFields2/nring_nonneg.con" as lemma.
 188
 189 inline procedural "cic:/CoRN/algebra/COrdFields2/suc_leEq_dub.con" as lemma.
 190
 191 inline procedural "cic:/CoRN/algebra/COrdFields2/leEq_nring.con" as lemma.
 192
 193 inline procedural "cic:/CoRN/algebra/COrdFields2/cc_abs_aid.con" as lemma.
 194
 195 include "tactics/Transparent_algebra.ma".
 196
 197 inline procedural "cic:/CoRN/algebra/COrdFields2/nexp_resp_pos.con" as lemma.
 198
 199 include "tactics/Opaque_algebra.ma".
 200
 201 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_resp_nonneg.con" as lemma.
 202
 203 include "tactics/Transparent_algebra.ma".
 204
 205 inline procedural "cic:/CoRN/algebra/COrdFields2/nexp_resp_nonneg.con" as lemma.
 206
 207 inline procedural "cic:/CoRN/algebra/COrdFields2/power_resp_leEq.con" as lemma.
 208
 209 inline procedural "cic:/CoRN/algebra/COrdFields2/nexp_resp_less.con" as lemma.
 210
 211 inline procedural "cic:/CoRN/algebra/COrdFields2/power_cancel_leEq.con" as lemma.
 212
 213 inline procedural "cic:/CoRN/algebra/COrdFields2/power_cancel_less.con" as lemma.
 214
 215 inline procedural "cic:/CoRN/algebra/COrdFields2/nat_less_bin_nexp.con" as lemma.
 216
 217 inline procedural "cic:/CoRN/algebra/COrdFields2/Sum_resp_leEq.con" as lemma.
 218
 219 inline procedural "cic:/CoRN/algebra/COrdFields2/Sumx_resp_leEq.con" as lemma.
 220
 221 inline procedural "cic:/CoRN/algebra/COrdFields2/Sum2_resp_leEq.con" as lemma.
 222
 223 inline procedural "cic:/CoRN/algebra/COrdFields2/approach_zero.con" as lemma.
 224
 225 inline procedural "cic:/CoRN/algebra/COrdFields2/approach_zero_weak.con" as lemma.
 226
 227 (* UNEXPORTED
 228 End misc
 229 *)
 230
 231 inline procedural "cic:/CoRN/algebra/COrdFields2/equal_less_leEq.con" as lemma.
 232
 233 (* UNEXPORTED
 234 End Properties_of_leEq
 235 *)
 236
 237 (*#**********************************)
 238
 239 (* UNEXPORTED
 240 Section PosP_properties
 241 *)
 242
 243 (*#**********************************)
 244
 245 (*#*
 246 ** Properties of positive numbers
 247 %\begin{convention}% Let [R] be an ordered field.
 248 %\end{convention}%
 249 *)
 250
 251 (* UNEXPORTED
 252 cic:/CoRN/algebra/COrdFields2/PosP_properties/R.var
 253 *)
 254
 255 (* begin hide *)
 256
 257 (* NOTATION
 258 Notation ZeroR := (Zero:R).
 259 *)
 260
 261 (* NOTATION
 262 Notation OneR := (One:R).
 263 *)
 264
 265 (* end hide *)
 266
 267 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_pos_imp.con" as lemma.
 268
 269 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_pos_nonneg.con" as lemma.
 270
 271 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_nonneg_pos.con" as lemma.
 272
 273 inline procedural "cic:/CoRN/algebra/COrdFields2/pos_square.con" as lemma.
 274
 275 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_cancel_pos_rht.con" as lemma.
 276
 277 inline procedural "cic:/CoRN/algebra/COrdFields2/mult_cancel_pos_lft.con" as lemma.
 278
 279 inline procedural "cic:/CoRN/algebra/COrdFields2/pos_wd.con" as lemma.
 280
 281 inline procedural "cic:/CoRN/algebra/COrdFields2/even_power_pos.con" as lemma.
 282
 283 inline procedural "cic:/CoRN/algebra/COrdFields2/odd_power_cancel_pos.con" as lemma.
 284
 285 inline procedural "cic:/CoRN/algebra/COrdFields2/plus_resp_pos.con" as lemma.
 286
 287 inline procedural "cic:/CoRN/algebra/COrdFields2/pos_nring_S.con" as lemma.
 288
 289 inline procedural "cic:/CoRN/algebra/COrdFields2/square_eq_pos.con" as lemma.
 290
 291 inline procedural "cic:/CoRN/algebra/COrdFields2/square_eq_neg.con" as lemma.
 292
 293 (* UNEXPORTED
 294 End PosP_properties
 295 *)
 296
 297 (* UNEXPORTED
 298 Hint Resolve mult_resp_nonneg.
 299 *)
 300
 301 (*#*
 302 ** Properties of one over successor
 303 %\begin{convention}% Let [R] be an ordered field.
 304 %\end{convention}%
 305 *)
 306
 307 inline procedural "cic:/CoRN/algebra/COrdFields2/one_div_succ.con" as definition.
 308
 309 (* UNEXPORTED
 310 Implicit Arguments one_div_succ [R].
 311 *)
 312
 313 (* UNEXPORTED
 314 Section One_div_succ_properties
 315 *)
 316
 317 (* UNEXPORTED
 318 cic:/CoRN/algebra/COrdFields2/One_div_succ_properties/R.var
 319 *)
 320
 321 inline procedural "cic:/CoRN/algebra/COrdFields2/one_div_succ_resp_leEq.con" as lemma.
 322
 323 inline procedural "cic:/CoRN/algebra/COrdFields2/one_div_succ_pos.con" as lemma.
 324
 325 inline procedural "cic:/CoRN/algebra/COrdFields2/one_div_succ_resp_less.con" as lemma.
 326
 327 (* UNEXPORTED
 328 End One_div_succ_properties
 329 *)
 330
 331 (*#*
 332 ** Properties of [Half]
 333 *)
 334
 335 inline procedural "cic:/CoRN/algebra/COrdFields2/Half.con" as definition.
 336
 337 (* UNEXPORTED
 338 Implicit Arguments Half [R].
 339 *)
 340
 341 (* UNEXPORTED
 342 Section Half_properties
 343 *)
 344
 345 (*#*
 346 %\begin{convention}%
 347 Let [R] be an ordered field.
 348 %\end{convention}%
 349 *)
 350
 351 (* UNEXPORTED
 352 cic:/CoRN/algebra/COrdFields2/Half_properties/R.var
 353 *)
 354
 355 inline procedural "cic:/CoRN/algebra/COrdFields2/half_1.con" as lemma.
 356
 357 (* UNEXPORTED
 358 Hint Resolve half_1: algebra.
 359 *)
 360
 361 inline procedural "cic:/CoRN/algebra/COrdFields2/pos_half.con" as lemma.
 362
 363 inline procedural "cic:/CoRN/algebra/COrdFields2/half_1'.con" as lemma.
 364
 365 inline procedural "cic:/CoRN/algebra/COrdFields2/half_2.con" as lemma.
 366
 367 inline procedural "cic:/CoRN/algebra/COrdFields2/half_lt1.con" as lemma.
 368
 369 inline procedural "cic:/CoRN/algebra/COrdFields2/half_3.con" as lemma.
 370
 371 (* UNEXPORTED
 372 End Half_properties
 373 *)
 374
 375 (* UNEXPORTED
 376 Hint Resolve half_1 half_1' half_2: algebra.
 377 *)
 378