helm/software/matita/contribs/CoRN-Decl/algebra/CGroups.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/CGroups".
  18
  19 include "CoRN_notation.ma".
  20
  21 (* $Id: CGroups.v,v 1.9 2004/04/23 10:00:52 lcf Exp $ *)
  22
  23 (*#* printing [-] %\ensuremath-% #&minus;# *)
  24
  25 (*#* printing [--] %\ensuremath-% #&minus;# *)
  26
  27 (*#* printing {-} %\ensuremath-% #&minus;# *)
  28
  29 (*#* printing {--} %\ensuremath-% #&minus;# *)
  30
  31 include "algebra/CMonoids.ma".
  32
  33 (* Begin_SpecReals *)
  34
  35 (*#*
  36 * Groups
  37 ** Definition of the notion of Group
  38 *)
  39
  40 inline "cic:/CoRN/algebra/CGroups/is_inverse.con".
  41
  42 (* UNEXPORTED
  43 Implicit Arguments is_inverse [S].
  44 *)
  45
  46 inline "cic:/CoRN/algebra/CGroups/is_CGroup.con".
  47
  48 inline "cic:/CoRN/algebra/CGroups/CGroup.ind".
  49
  50 coercion "cic:/matita/CoRN-Decl/algebra/CGroups/cg_crr.con" 0 (* compounds *).
  51
  52 (* End_SpecReals *)
  53
  54 (* Begin_SpecReals *)
  55
  56 (* UNEXPORTED
  57 Implicit Arguments cg_inv [c].
  58 *)
  59
  60 inline "cic:/CoRN/algebra/CGroups/cg_minus.con".
  61
  62 (*#*
  63 %\begin{nameconvention}%
  64 In the names of lemmas, we will denote [[--] ] with [inv],
  65 and [ [-] ] with [minus].
  66 %\end{nameconvention}%
  67 *)
  68
  69 (* UNEXPORTED
  70 Implicit Arguments cg_minus [G].
  71 *)
  72
  73 (* End_SpecReals *)
  74
  75 (*#*
  76 ** Group axioms
  77 %\begin{convention}% Let [G] be a group.
  78 %\end{convention}%
  79 *)
  80
  81 (* UNEXPORTED
  82 Section CGroup_axioms.
  83 *)
  84
  85 inline "cic:/CoRN/algebra/CGroups/G.var".
  86
  87 inline "cic:/CoRN/algebra/CGroups/cg_inverse.con".
  88
  89 (* UNEXPORTED
  90 End CGroup_axioms.
  91 *)
  92
  93 (*#*
  94 ** Group basics
  95 General properties of groups.
  96 %\begin{convention}% Let [G] be a group.
  97 %\end{convention}%
  98 *)
  99
 100 (* UNEXPORTED
 101 Section CGroup_basics.
 102 *)
 103
 104 inline "cic:/CoRN/algebra/CGroups/G.var".
 105
 106 inline "cic:/CoRN/algebra/CGroups/cg_rht_inv_unfolded.con".
 107
 108 inline "cic:/CoRN/algebra/CGroups/cg_lft_inv_unfolded.con".
 109
 110 inline "cic:/CoRN/algebra/CGroups/cg_minus_correct.con".
 111
 112 (* UNEXPORTED
 113 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded cg_minus_correct:
 114   algebra.
 115 *)
 116
 117 inline "cic:/CoRN/algebra/CGroups/cg_inverse'.con".
 118
 119 (* Hints for Auto *)
 120
 121 inline "cic:/CoRN/algebra/CGroups/cg_minus_unfolded.con".
 122
 123 (* UNEXPORTED
 124 Hint Resolve cg_minus_unfolded: algebra.
 125 *)
 126
 127 inline "cic:/CoRN/algebra/CGroups/cg_minus_wd.con".
 128
 129 (* UNEXPORTED
 130 Hint Resolve cg_minus_wd: algebra_c.
 131 *)
 132
 133 inline "cic:/CoRN/algebra/CGroups/cg_minus_strext.con".
 134
 135 inline "cic:/CoRN/algebra/CGroups/cg_minus_is_csetoid_bin_op.con".
 136
 137 inline "cic:/CoRN/algebra/CGroups/grp_inv_assoc.con".
 138
 139 (* UNEXPORTED
 140 Hint Resolve grp_inv_assoc: algebra.
 141 *)
 142
 143 inline "cic:/CoRN/algebra/CGroups/cg_inv_unique.con".
 144
 145 inline "cic:/CoRN/algebra/CGroups/cg_inv_inv.con".
 146
 147 (* UNEXPORTED
 148 Hint Resolve cg_inv_inv: algebra.
 149 *)
 150
 151 inline "cic:/CoRN/algebra/CGroups/cg_cancel_lft.con".
 152
 153 inline "cic:/CoRN/algebra/CGroups/cg_cancel_rht.con".
 154
 155 inline "cic:/CoRN/algebra/CGroups/cg_inv_unique'.con".
 156
 157 inline "cic:/CoRN/algebra/CGroups/cg_inv_unique_2.con".
 158
 159 inline "cic:/CoRN/algebra/CGroups/cg_zero_inv.con".
 160
 161 (* UNEXPORTED
 162 Hint Resolve cg_zero_inv: algebra.
 163 *)
 164
 165 inline "cic:/CoRN/algebra/CGroups/cg_inv_zero.con".
 166
 167 inline "cic:/CoRN/algebra/CGroups/cg_inv_op.con".
 168
 169 (*#*
 170 Useful for interactive proof development.
 171 *)
 172
 173 inline "cic:/CoRN/algebra/CGroups/x_minus_x.con".
 174
 175 (*#*
 176 ** Sub-groups
 177 %\begin{convention}% Let [P] be a predicate on [G] containing
 178 [Zero] and closed under [[+]] and [[--] ].
 179 %\end{convention}%
 180 *)
 181
 182 (* UNEXPORTED
 183 Section SubCGroups.
 184 *)
 185
 186 inline "cic:/CoRN/algebra/CGroups/P.var".
 187
 188 inline "cic:/CoRN/algebra/CGroups/Punit.var".
 189
 190 inline "cic:/CoRN/algebra/CGroups/op_pres_P.var".
 191
 192 inline "cic:/CoRN/algebra/CGroups/inv_pres_P.var".
 193
 194 inline "cic:/CoRN/algebra/CGroups/subcrr.con".
 195
 196 inline "cic:/CoRN/algebra/CGroups/subinv.con".
 197
 198 inline "cic:/CoRN/algebra/CGroups/isgrp_scrr.con".
 199
 200 inline "cic:/CoRN/algebra/CGroups/Build_SubCGroup.con".
 201
 202 (* UNEXPORTED
 203 End SubCGroups.
 204 *)
 205
 206 (* UNEXPORTED
 207 End CGroup_basics.
 208 *)
 209
 210 (* UNEXPORTED
 211 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded: algebra.
 212 *)
 213
 214 (* UNEXPORTED
 215 Hint Resolve cg_inv_inv cg_minus_correct cg_zero_inv cg_inv_zero: algebra.
 216 *)
 217
 218 (* UNEXPORTED
 219 Hint Resolve cg_minus_unfolded grp_inv_assoc cg_inv_op: algebra.
 220 *)
 221
 222 (* UNEXPORTED
 223 Hint Resolve cg_minus_wd: algebra_c.
 224 *)
 225
 226 (*#*
 227 ** Associative properties of groups
 228 %\begin{convention}% Let [G] be a group.
 229 %\end{convention}%
 230 *)
 231
 232 (* UNEXPORTED
 233 Section Assoc_properties.
 234 *)
 235
 236 inline "cic:/CoRN/algebra/CGroups/G.var".
 237
 238 inline "cic:/CoRN/algebra/CGroups/assoc_2.con".
 239
 240 inline "cic:/CoRN/algebra/CGroups/zero_minus.con".
 241
 242 inline "cic:/CoRN/algebra/CGroups/cg_cancel_mixed.con".
 243
 244 inline "cic:/CoRN/algebra/CGroups/plus_resp_eq.con".
 245
 246 (* UNEXPORTED
 247 End Assoc_properties.
 248 *)
 249
 250 (* UNEXPORTED
 251 Hint Resolve assoc_2 minus_plus zero_minus cg_cancel_mixed plus_resp_eq:
 252   algebra.
 253 *)
 254
 255 (*#*
 256 ** Apartness in Constructive Groups
 257 Specific properties of apartness.
 258 %\begin{convention}% Let [G] be a group.
 259 %\end{convention}%
 260 *)
 261
 262 (* UNEXPORTED
 263 Section cgroups_apartness.
 264 *)
 265
 266 inline "cic:/CoRN/algebra/CGroups/G.var".
 267
 268 inline "cic:/CoRN/algebra/CGroups/cg_add_ap_zero.con".
 269
 270 inline "cic:/CoRN/algebra/CGroups/op_rht_resp_ap.con".
 271
 272 inline "cic:/CoRN/algebra/CGroups/cg_ap_cancel_rht.con".
 273
 274 inline "cic:/CoRN/algebra/CGroups/plus_cancel_ap_rht.con".
 275
 276 inline "cic:/CoRN/algebra/CGroups/minus_ap_zero.con".
 277
 278 inline "cic:/CoRN/algebra/CGroups/zero_minus_apart.con".
 279
 280 inline "cic:/CoRN/algebra/CGroups/inv_resp_ap_zero.con".
 281
 282 inline "cic:/CoRN/algebra/CGroups/inv_resp_ap.con".
 283
 284 inline "cic:/CoRN/algebra/CGroups/minus_resp_ap_rht.con".
 285
 286 inline "cic:/CoRN/algebra/CGroups/minus_resp_ap_lft.con".
 287
 288 inline "cic:/CoRN/algebra/CGroups/minus_cancel_ap_rht.con".
 289
 290 (* UNEXPORTED
 291 End cgroups_apartness.
 292 *)
 293
 294 (* UNEXPORTED
 295 Hint Resolve op_rht_resp_ap: algebra.
 296 *)
 297
 298 (* UNEXPORTED
 299 Hint Resolve minus_ap_zero zero_minus_apart inv_resp_ap_zero: algebra.
 300 *)
 301
 302 (* UNEXPORTED
 303 Section CGroup_Ops.
 304 *)
 305
 306 (*#*
 307 ** Functional operations
 308
 309 As before, we lift our group operations to the function space of the group.
 310
 311 %\begin{convention}%
 312 Let [G] be a group and [F,F':(PartFunct G)] with domains given respectively
 313 by [P] and [Q].
 314 %\end{convention}%
 315 *)
 316
 317 inline "cic:/CoRN/algebra/CGroups/G.var".
 318
 319 inline "cic:/CoRN/algebra/CGroups/F.var".
 320
 321 inline "cic:/CoRN/algebra/CGroups/F'.var".
 322
 323 (* begin hide *)
 324
 325 inline "cic:/CoRN/algebra/CGroups/P.con".
 326
 327 inline "cic:/CoRN/algebra/CGroups/Q.con".
 328
 329 (* end hide *)
 330
 331 (* UNEXPORTED
 332 Section Part_Function_Inv.
 333 *)
 334
 335 inline "cic:/CoRN/algebra/CGroups/part_function_inv_strext.con".
 336
 337 inline "cic:/CoRN/algebra/CGroups/Finv.con".
 338
 339 (* UNEXPORTED
 340 End Part_Function_Inv.
 341 *)
 342
 343 (* UNEXPORTED
 344 Section Part_Function_Minus.
 345 *)
 346
 347 inline "cic:/CoRN/algebra/CGroups/part_function_minus_strext.con".
 348
 349 inline "cic:/CoRN/algebra/CGroups/Fminus.con".
 350
 351 (* UNEXPORTED
 352 End Part_Function_Minus.
 353 *)
 354
 355 (*#*
 356 %\begin{convention}% Let [R:G->CProp].
 357 %\end{convention}%
 358 *)
 359
 360 inline "cic:/CoRN/algebra/CGroups/R.var".
 361
 362 inline "cic:/CoRN/algebra/CGroups/included_FInv.con".
 363
 364 inline "cic:/CoRN/algebra/CGroups/included_FInv'.con".
 365
 366 inline "cic:/CoRN/algebra/CGroups/included_FMinus.con".
 367
 368 inline "cic:/CoRN/algebra/CGroups/included_FMinus'.con".
 369
 370 inline "cic:/CoRN/algebra/CGroups/included_FMinus''.con".
 371
 372 (* UNEXPORTED
 373 End CGroup_Ops.
 374 *)
 375
 376 (* UNEXPORTED
 377 Implicit Arguments Finv [G].
 378 *)
 379
 380 (* UNEXPORTED
 381 Implicit Arguments Fminus [G].
 382 *)
 383
 384 (* UNEXPORTED
 385 Hint Resolve included_FInv included_FMinus : included.
 386 *)
 387
 388 (* UNEXPORTED
 389 Hint Immediate included_FInv' included_FMinus' included_FMinus'' : included.
 390 *)
 391