helm/software/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
  20 COrdFields
  21 *)
  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 inline cic:/CoRN/algebra/COrdFields2/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
 196 Transparent_algebra
 197 *)
 198
 199 inline cic:/CoRN/algebra/COrdFields2/nexp_resp_pos.con.
 200
 201 (* INCLUDE
 202 Opaque_algebra
 203 *)
 204
 205 inline cic:/CoRN/algebra/COrdFields2/mult_resp_nonneg.con.
 206
 207 (* INCLUDE
 208 Transparent_algebra
 209 *)
 210
 211 inline cic:/CoRN/algebra/COrdFields2/nexp_resp_nonneg.con.
 212
 213 inline cic:/CoRN/algebra/COrdFields2/power_resp_leEq.con.
 214
 215 inline cic:/CoRN/algebra/COrdFields2/nexp_resp_less.con.
 216
 217 inline cic:/CoRN/algebra/COrdFields2/power_cancel_leEq.con.
 218
 219 inline cic:/CoRN/algebra/COrdFields2/power_cancel_less.con.
 220
 221 inline cic:/CoRN/algebra/COrdFields2/nat_less_bin_nexp.con.
 222
 223 inline cic:/CoRN/algebra/COrdFields2/Sum_resp_leEq.con.
 224
 225 inline cic:/CoRN/algebra/COrdFields2/Sumx_resp_leEq.con.
 226
 227 inline cic:/CoRN/algebra/COrdFields2/Sum2_resp_leEq.con.
 228
 229 inline cic:/CoRN/algebra/COrdFields2/approach_zero.con.
 230
 231 inline cic:/CoRN/algebra/COrdFields2/approach_zero_weak.con.
 232
 233 (* UNEXPORTED
 234 End misc.
 235 *)
 236
 237 inline cic:/CoRN/algebra/COrdFields2/equal_less_leEq.con.
 238
 239 (* UNEXPORTED
 240 End Properties_of_leEq.
 241 *)
 242
 243 (*#**********************************)
 244
 245 (* UNEXPORTED
 246 Section PosP_properties.
 247 *)
 248
 249 (*#**********************************)
 250
 251 (*#*
 252 ** Properties of positive numbers
 253 %\begin{convention}% Let [R] be an ordered field.
 254 %\end{convention}%
 255 *)
 256
 257 inline cic:/CoRN/algebra/COrdFields2/R.var.
 258
 259 (* begin hide *)
 260
 261 (* end hide *)
 262
 263 inline cic:/CoRN/algebra/COrdFields2/mult_pos_imp.con.
 264
 265 inline cic:/CoRN/algebra/COrdFields2/plus_resp_pos_nonneg.con.
 266
 267 inline cic:/CoRN/algebra/COrdFields2/plus_resp_nonneg_pos.con.
 268
 269 inline cic:/CoRN/algebra/COrdFields2/pos_square.con.
 270
 271 inline cic:/CoRN/algebra/COrdFields2/mult_cancel_pos_rht.con.
 272
 273 inline cic:/CoRN/algebra/COrdFields2/mult_cancel_pos_lft.con.
 274
 275 inline cic:/CoRN/algebra/COrdFields2/pos_wd.con.
 276
 277 inline cic:/CoRN/algebra/COrdFields2/even_power_pos.con.
 278
 279 inline cic:/CoRN/algebra/COrdFields2/odd_power_cancel_pos.con.
 280
 281 inline cic:/CoRN/algebra/COrdFields2/plus_resp_pos.con.
 282
 283 inline cic:/CoRN/algebra/COrdFields2/pos_nring_S.con.
 284
 285 inline cic:/CoRN/algebra/COrdFields2/square_eq_pos.con.
 286
 287 inline cic:/CoRN/algebra/COrdFields2/square_eq_neg.con.
 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 cic:/CoRN/algebra/COrdFields2/one_div_succ.con.
 304
 305 (* UNEXPORTED
 306 Implicit Arguments one_div_succ [R].
 307 *)
 308
 309 (* UNEXPORTED
 310 Section One_div_succ_properties.
 311 *)
 312
 313 inline cic:/CoRN/algebra/COrdFields2/R.var.
 314
 315 inline cic:/CoRN/algebra/COrdFields2/one_div_succ_resp_leEq.con.
 316
 317 inline cic:/CoRN/algebra/COrdFields2/one_div_succ_pos.con.
 318
 319 inline cic:/CoRN/algebra/COrdFields2/one_div_succ_resp_less.con.
 320
 321 (* UNEXPORTED
 322 End One_div_succ_properties.
 323 *)
 324
 325 (*#*
 326 ** Properties of [Half]
 327 *)
 328
 329 inline cic:/CoRN/algebra/COrdFields2/Half.con.
 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 inline cic:/CoRN/algebra/COrdFields2/R.var.
 346
 347 inline cic:/CoRN/algebra/COrdFields2/half_1.con.
 348
 349 (* UNEXPORTED
 350 Hint Resolve half_1: algebra.
 351 *)
 352
 353 inline cic:/CoRN/algebra/COrdFields2/pos_half.con.
 354
 355 inline cic:/CoRN/algebra/COrdFields2/half_1'.con.
 356
 357 inline cic:/CoRN/algebra/COrdFields2/half_2.con.
 358
 359 inline cic:/CoRN/algebra/COrdFields2/half_lt1.con.
 360
 361 inline cic:/CoRN/algebra/COrdFields2/half_3.con.
 362
 363 (* UNEXPORTED
 364 End Half_properties.
 365 *)
 366
 367 (* UNEXPORTED
 368 Hint Resolve half_1 half_1' half_2: algebra.
 369 *)
 370