matita/contribs/CoRN-Decl/algebra/COrdFields2.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/algebra/COrdFields2".
  18
  19 include "CoRN.ma".
  20
  21 include "algebra/COrdFields.ma".
  22
  23 (*#* printing one_div_succ %\ensuremath{\frac1{\cdot+1}}% *)
  24
  25 (*#* printing Half %\ensuremath{\frac12}% #&frac12;# *)
  26
  27 (*#**********************************)
  28
  29 (* UNEXPORTED
  30 Section Properties_of_leEq
  31 *)
  32
  33 (*#**********************************)
  34
  35 (*#*
  36 ** Properties of [[<=]]
  37 %\begin{convention}% Let [R] be an ordered field.
  38 %\end{convention}%
  39 *)
  40
  41 alias id "R" = "cic:/CoRN/algebra/COrdFields2/Properties_of_leEq/R.var".
  42
  43 (* UNEXPORTED
  44 Section addition
  45 *)
  46
  47 (*#*
  48 *** Addition and subtraction%\label{section:leEq-plus-minus}%
  49 *)
  50
  51 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq.con".
  52
  53 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq_lft.con".
  54
  55 inline "cic:/CoRN/algebra/COrdFields2/minus_resp_leEq.con".
  56
  57 inline "cic:/CoRN/algebra/COrdFields2/inv_resp_leEq.con".
  58
  59 (* UNEXPORTED
  60 Transparent cg_minus.
  61 *)
  62
  63 inline "cic:/CoRN/algebra/COrdFields2/minus_resp_leEq_rht.con".
  64
  65 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq_both.con".
  66
  67 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_less_leEq.con".
  68
  69 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_leEq_less.con".
  70
  71 inline "cic:/CoRN/algebra/COrdFields2/minus_resp_less_leEq.con".
  72
  73 inline "cic:/CoRN/algebra/COrdFields2/minus_resp_leEq_both.con".
  74
  75 (*#* Cancellation properties
  76 *)
  77
  78 inline "cic:/CoRN/algebra/COrdFields2/plus_cancel_leEq_rht.con".
  79
  80 inline "cic:/CoRN/algebra/COrdFields2/inv_cancel_leEq.con".
  81
  82 (*#* Laws for shifting
  83 *)
  84
  85 inline "cic:/CoRN/algebra/COrdFields2/shift_plus_leEq.con".
  86
  87 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_plus.con".
  88
  89 inline "cic:/CoRN/algebra/COrdFields2/shift_plus_leEq'.con".
  90
  91 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_plus'.con".
  92
  93 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_rht.con".
  94
  95 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_lft.con".
  96
  97 inline "cic:/CoRN/algebra/COrdFields2/shift_minus_leEq.con".
  98
  99 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_minus.con".
 100
 101 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_minus'.con".
 102
 103 inline "cic:/CoRN/algebra/COrdFields2/shift_zero_leEq_minus.con".
 104
 105 inline "cic:/CoRN/algebra/COrdFields2/shift_zero_leEq_minus'.con".
 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 "cic:/CoRN/algebra/COrdFields2/mult_resp_leEq_rht.con".
 122
 123 inline "cic:/CoRN/algebra/COrdFields2/mult_resp_leEq_lft.con".
 124
 125 inline "cic:/CoRN/algebra/COrdFields2/mult_resp_leEq_both.con".
 126
 127 inline "cic:/CoRN/algebra/COrdFields2/recip_resp_leEq.con".
 128
 129 inline "cic:/CoRN/algebra/COrdFields2/div_resp_leEq.con".
 130
 131 (* UNEXPORTED
 132 Hint Resolve recip_resp_leEq: algebra.
 133 *)
 134
 135 (*#* Cancellation properties
 136 *)
 137
 138 inline "cic:/CoRN/algebra/COrdFields2/mult_cancel_leEq.con".
 139
 140 (*#* Laws for shifting
 141 *)
 142
 143 inline "cic:/CoRN/algebra/COrdFields2/shift_mult_leEq.con".
 144
 145 inline "cic:/CoRN/algebra/COrdFields2/shift_mult_leEq'.con".
 146
 147 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_mult'.con".
 148
 149 inline "cic:/CoRN/algebra/COrdFields2/shift_div_leEq.con".
 150
 151 inline "cic:/CoRN/algebra/COrdFields2/shift_div_leEq'.con".
 152
 153 inline "cic:/CoRN/algebra/COrdFields2/shift_leEq_div.con".
 154
 155 (* UNEXPORTED
 156 Hint Resolve shift_leEq_div: algebra.
 157 *)
 158
 159 inline "cic:/CoRN/algebra/COrdFields2/eps_div_leEq_eps.con".
 160
 161 inline "cic:/CoRN/algebra/COrdFields2/nonneg_div_two.con".
 162
 163 inline "cic:/CoRN/algebra/COrdFields2/nonneg_div_two'.con".
 164
 165 inline "cic:/CoRN/algebra/COrdFields2/nonneg_div_three.con".
 166
 167 inline "cic:/CoRN/algebra/COrdFields2/nonneg_div_three'.con".
 168
 169 inline "cic:/CoRN/algebra/COrdFields2/nonneg_div_four.con".
 170
 171 inline "cic:/CoRN/algebra/COrdFields2/nonneg_div_four'.con".
 172
 173 (* UNEXPORTED
 174 End multiplication
 175 *)
 176
 177 (* UNEXPORTED
 178 Section misc
 179 *)
 180
 181 (*#*
 182 *** Miscellaneous Properties
 183 *)
 184
 185 inline "cic:/CoRN/algebra/COrdFields2/sqr_nonneg.con".
 186
 187 inline "cic:/CoRN/algebra/COrdFields2/nring_nonneg.con".
 188
 189 inline "cic:/CoRN/algebra/COrdFields2/suc_leEq_dub.con".
 190
 191 inline "cic:/CoRN/algebra/COrdFields2/leEq_nring.con".
 192
 193 inline "cic:/CoRN/algebra/COrdFields2/cc_abs_aid.con".
 194
 195 include "tactics/Transparent_algebra.ma".
 196
 197 inline "cic:/CoRN/algebra/COrdFields2/nexp_resp_pos.con".
 198
 199 include "tactics/Opaque_algebra.ma".
 200
 201 inline "cic:/CoRN/algebra/COrdFields2/mult_resp_nonneg.con".
 202
 203 include "tactics/Transparent_algebra.ma".
 204
 205 inline "cic:/CoRN/algebra/COrdFields2/nexp_resp_nonneg.con".
 206
 207 inline "cic:/CoRN/algebra/COrdFields2/power_resp_leEq.con".
 208
 209 inline "cic:/CoRN/algebra/COrdFields2/nexp_resp_less.con".
 210
 211 inline "cic:/CoRN/algebra/COrdFields2/power_cancel_leEq.con".
 212
 213 inline "cic:/CoRN/algebra/COrdFields2/power_cancel_less.con".
 214
 215 inline "cic:/CoRN/algebra/COrdFields2/nat_less_bin_nexp.con".
 216
 217 inline "cic:/CoRN/algebra/COrdFields2/Sum_resp_leEq.con".
 218
 219 inline "cic:/CoRN/algebra/COrdFields2/Sumx_resp_leEq.con".
 220
 221 inline "cic:/CoRN/algebra/COrdFields2/Sum2_resp_leEq.con".
 222
 223 inline "cic:/CoRN/algebra/COrdFields2/approach_zero.con".
 224
 225 inline "cic:/CoRN/algebra/COrdFields2/approach_zero_weak.con".
 226
 227 (* UNEXPORTED
 228 End misc
 229 *)
 230
 231 inline "cic:/CoRN/algebra/COrdFields2/equal_less_leEq.con".
 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 alias id "R" = "cic:/CoRN/algebra/COrdFields2/PosP_properties/R.var".
 252
 253 (* begin hide *)
 254
 255 (* NOTATION
 256 Notation ZeroR := (Zero:R).
 257 *)
 258
 259 (* NOTATION
 260 Notation OneR := (One:R).
 261 *)
 262
 263 (* end hide *)
 264
 265 inline "cic:/CoRN/algebra/COrdFields2/mult_pos_imp.con".
 266
 267 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_pos_nonneg.con".
 268
 269 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_nonneg_pos.con".
 270
 271 inline "cic:/CoRN/algebra/COrdFields2/pos_square.con".
 272
 273 inline "cic:/CoRN/algebra/COrdFields2/mult_cancel_pos_rht.con".
 274
 275 inline "cic:/CoRN/algebra/COrdFields2/mult_cancel_pos_lft.con".
 276
 277 inline "cic:/CoRN/algebra/COrdFields2/pos_wd.con".
 278
 279 inline "cic:/CoRN/algebra/COrdFields2/even_power_pos.con".
 280
 281 inline "cic:/CoRN/algebra/COrdFields2/odd_power_cancel_pos.con".
 282
 283 inline "cic:/CoRN/algebra/COrdFields2/plus_resp_pos.con".
 284
 285 inline "cic:/CoRN/algebra/COrdFields2/pos_nring_S.con".
 286
 287 inline "cic:/CoRN/algebra/COrdFields2/square_eq_pos.con".
 288
 289 inline "cic:/CoRN/algebra/COrdFields2/square_eq_neg.con".
 290
 291 (* UNEXPORTED
 292 End PosP_properties
 293 *)
 294
 295 (* UNEXPORTED
 296 Hint Resolve mult_resp_nonneg.
 297 *)
 298
 299 (*#*
 300 ** Properties of one over successor
 301 %\begin{convention}% Let [R] be an ordered field.
 302 %\end{convention}%
 303 *)
 304
 305 inline "cic:/CoRN/algebra/COrdFields2/one_div_succ.con".
 306
 307 (* UNEXPORTED
 308 Implicit Arguments one_div_succ [R].
 309 *)
 310
 311 (* UNEXPORTED
 312 Section One_div_succ_properties
 313 *)
 314
 315 alias id "R" = "cic:/CoRN/algebra/COrdFields2/One_div_succ_properties/R.var".
 316
 317 inline "cic:/CoRN/algebra/COrdFields2/one_div_succ_resp_leEq.con".
 318
 319 inline "cic:/CoRN/algebra/COrdFields2/one_div_succ_pos.con".
 320
 321 inline "cic:/CoRN/algebra/COrdFields2/one_div_succ_resp_less.con".
 322
 323 (* UNEXPORTED
 324 End One_div_succ_properties
 325 *)
 326
 327 (*#*
 328 ** Properties of [Half]
 329 *)
 330
 331 inline "cic:/CoRN/algebra/COrdFields2/Half.con".
 332
 333 (* UNEXPORTED
 334 Implicit Arguments Half [R].
 335 *)
 336
 337 (* UNEXPORTED
 338 Section Half_properties
 339 *)
 340
 341 (*#*
 342 %\begin{convention}%
 343 Let [R] be an ordered field.
 344 %\end{convention}%
 345 *)
 346
 347 alias id "R" = "cic:/CoRN/algebra/COrdFields2/Half_properties/R.var".
 348
 349 inline "cic:/CoRN/algebra/COrdFields2/half_1.con".
 350
 351 (* UNEXPORTED
 352 Hint Resolve half_1: algebra.
 353 *)
 354
 355 inline "cic:/CoRN/algebra/COrdFields2/pos_half.con".
 356
 357 inline "cic:/CoRN/algebra/COrdFields2/half_1'.con".
 358
 359 inline "cic:/CoRN/algebra/COrdFields2/half_2.con".
 360
 361 inline "cic:/CoRN/algebra/COrdFields2/half_lt1.con".
 362
 363 inline "cic:/CoRN/algebra/COrdFields2/half_3.con".
 364
 365 (* UNEXPORTED
 366 End Half_properties
 367 *)
 368
 369 (* UNEXPORTED
 370 Hint Resolve half_1 half_1' half_2: algebra.
 371 *)
 372