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