helm/software/matita/contribs/procedural/CoRN/algebra/CGroups.mma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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 (* $Id: CGroups.v,v 1.9 2004/04/23 10:00:52 lcf Exp $ *)
  20
  21 (*#* printing [-] %\ensuremath-% #&minus;# *)
  22
  23 (*#* printing [--] %\ensuremath-% #&minus;# *)
  24
  25 (*#* printing {-} %\ensuremath-% #&minus;# *)
  26
  27 (*#* printing {--} %\ensuremath-% #&minus;# *)
  28
  29 include "algebra/CMonoids.ma".
  30
  31 (* Begin_SpecReals *)
  32
  33 (*#*
  34 * Groups
  35 ** Definition of the notion of Group
  36 *)
  37
  38 inline procedural "cic:/CoRN/algebra/CGroups/is_inverse.con" as definition.
  39
  40 (* UNEXPORTED
  41 Implicit Arguments is_inverse [S].
  42 *)
  43
  44 inline procedural "cic:/CoRN/algebra/CGroups/is_CGroup.con" as definition.
  45
  46 inline procedural "cic:/CoRN/algebra/CGroups/CGroup.ind".
  47
  48 (* COERCION
  49 cic:/matita/CoRN-Procedural/algebra/CGroups/cg_crr.con
  50 *)
  51
  52 (* End_SpecReals *)
  53
  54 (* Begin_SpecReals *)
  55
  56 (* UNEXPORTED
  57 Implicit Arguments cg_inv [c].
  58 *)
  59
  60 (* NOTATION
  61 Notation "[--] x" := (cg_inv x) (at level 2, right associativity).
  62 *)
  63
  64 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus.con" as definition.
  65
  66 (*#*
  67 %\begin{nameconvention}%
  68 In the names of lemmas, we will denote [[--] ] with [inv],
  69 and [ [-] ] with [minus].
  70 %\end{nameconvention}%
  71 *)
  72
  73 (* UNEXPORTED
  74 Implicit Arguments cg_minus [G].
  75 *)
  76
  77 (* NOTATION
  78 Infix "[-]" := cg_minus (at level 50, left associativity).
  79 *)
  80
  81 (* End_SpecReals *)
  82
  83 (*#*
  84 ** Group axioms
  85 %\begin{convention}% Let [G] be a group.
  86 %\end{convention}%
  87 *)
  88
  89 (* UNEXPORTED
  90 Section CGroup_axioms
  91 *)
  92
  93 (* UNEXPORTED
  94 cic:/CoRN/algebra/CGroups/CGroup_axioms/G.var
  95 *)
  96
  97 inline procedural "cic:/CoRN/algebra/CGroups/cg_inverse.con" as lemma.
  98
  99 (* UNEXPORTED
 100 End CGroup_axioms
 101 *)
 102
 103 (*#*
 104 ** Group basics
 105 General properties of groups.
 106 %\begin{convention}% Let [G] be a group.
 107 %\end{convention}%
 108 *)
 109
 110 (* UNEXPORTED
 111 Section CGroup_basics
 112 *)
 113
 114 (* UNEXPORTED
 115 cic:/CoRN/algebra/CGroups/CGroup_basics/G.var
 116 *)
 117
 118 inline procedural "cic:/CoRN/algebra/CGroups/cg_rht_inv_unfolded.con" as lemma.
 119
 120 inline procedural "cic:/CoRN/algebra/CGroups/cg_lft_inv_unfolded.con" as lemma.
 121
 122 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_correct.con" as lemma.
 123
 124 (* UNEXPORTED
 125 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded cg_minus_correct:
 126   algebra.
 127 *)
 128
 129 inline procedural "cic:/CoRN/algebra/CGroups/cg_inverse'.con" as lemma.
 130
 131 (* Hints for Auto *)
 132
 133 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_unfolded.con" as lemma.
 134
 135 (* UNEXPORTED
 136 Hint Resolve cg_minus_unfolded: algebra.
 137 *)
 138
 139 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_wd.con" as lemma.
 140
 141 (* UNEXPORTED
 142 Hint Resolve cg_minus_wd: algebra_c.
 143 *)
 144
 145 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_strext.con" as lemma.
 146
 147 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_is_csetoid_bin_op.con" as definition.
 148
 149 inline procedural "cic:/CoRN/algebra/CGroups/grp_inv_assoc.con" as lemma.
 150
 151 (* UNEXPORTED
 152 Hint Resolve grp_inv_assoc: algebra.
 153 *)
 154
 155 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_unique.con" as lemma.
 156
 157 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_inv.con" as lemma.
 158
 159 (* UNEXPORTED
 160 Hint Resolve cg_inv_inv: algebra.
 161 *)
 162
 163 inline procedural "cic:/CoRN/algebra/CGroups/cg_cancel_lft.con" as lemma.
 164
 165 inline procedural "cic:/CoRN/algebra/CGroups/cg_cancel_rht.con" as lemma.
 166
 167 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_unique'.con" as lemma.
 168
 169 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_unique_2.con" as lemma.
 170
 171 inline procedural "cic:/CoRN/algebra/CGroups/cg_zero_inv.con" as lemma.
 172
 173 (* UNEXPORTED
 174 Hint Resolve cg_zero_inv: algebra.
 175 *)
 176
 177 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_zero.con" as lemma.
 178
 179 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_op.con" as lemma.
 180
 181 (*#*
 182 Useful for interactive proof development.
 183 *)
 184
 185 inline procedural "cic:/CoRN/algebra/CGroups/x_minus_x.con" as lemma.
 186
 187 (*#*
 188 ** Sub-groups
 189 %\begin{convention}% Let [P] be a predicate on [G] containing
 190 [Zero] and closed under [[+]] and [[--] ].
 191 %\end{convention}%
 192 *)
 193
 194 (* UNEXPORTED
 195 Section SubCGroups
 196 *)
 197
 198 (* UNEXPORTED
 199 cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/P.var
 200 *)
 201
 202 (* UNEXPORTED
 203 cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/Punit.var
 204 *)
 205
 206 (* UNEXPORTED
 207 cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/op_pres_P.var
 208 *)
 209
 210 (* UNEXPORTED
 211 cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/inv_pres_P.var
 212 *)
 213
 214 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/subcrr.con" "CGroup_basics__SubCGroups__" as definition.
 215
 216 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/subinv.con" "CGroup_basics__SubCGroups__" as definition.
 217
 218 inline procedural "cic:/CoRN/algebra/CGroups/isgrp_scrr.con" as lemma.
 219
 220 inline procedural "cic:/CoRN/algebra/CGroups/Build_SubCGroup.con" as definition.
 221
 222 (* UNEXPORTED
 223 End SubCGroups
 224 *)
 225
 226 (* UNEXPORTED
 227 End CGroup_basics
 228 *)
 229
 230 (* UNEXPORTED
 231 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded: algebra.
 232 *)
 233
 234 (* UNEXPORTED
 235 Hint Resolve cg_inv_inv cg_minus_correct cg_zero_inv cg_inv_zero: algebra.
 236 *)
 237
 238 (* UNEXPORTED
 239 Hint Resolve cg_minus_unfolded grp_inv_assoc cg_inv_op: algebra.
 240 *)
 241
 242 (* UNEXPORTED
 243 Hint Resolve cg_minus_wd: algebra_c.
 244 *)
 245
 246 (*#*
 247 ** Associative properties of groups
 248 %\begin{convention}% Let [G] be a group.
 249 %\end{convention}%
 250 *)
 251
 252 (* UNEXPORTED
 253 Section Assoc_properties
 254 *)
 255
 256 (* UNEXPORTED
 257 cic:/CoRN/algebra/CGroups/Assoc_properties/G.var
 258 *)
 259
 260 inline procedural "cic:/CoRN/algebra/CGroups/assoc_2.con" as lemma.
 261
 262 inline procedural "cic:/CoRN/algebra/CGroups/zero_minus.con" as lemma.
 263
 264 inline procedural "cic:/CoRN/algebra/CGroups/cg_cancel_mixed.con" as lemma.
 265
 266 inline procedural "cic:/CoRN/algebra/CGroups/plus_resp_eq.con" as lemma.
 267
 268 (* UNEXPORTED
 269 End Assoc_properties
 270 *)
 271
 272 (* UNEXPORTED
 273 Hint Resolve assoc_2 minus_plus zero_minus cg_cancel_mixed plus_resp_eq:
 274   algebra.
 275 *)
 276
 277 (*#*
 278 ** Apartness in Constructive Groups
 279 Specific properties of apartness.
 280 %\begin{convention}% Let [G] be a group.
 281 %\end{convention}%
 282 *)
 283
 284 (* UNEXPORTED
 285 Section cgroups_apartness
 286 *)
 287
 288 (* UNEXPORTED
 289 cic:/CoRN/algebra/CGroups/cgroups_apartness/G.var
 290 *)
 291
 292 inline procedural "cic:/CoRN/algebra/CGroups/cg_add_ap_zero.con" as lemma.
 293
 294 inline procedural "cic:/CoRN/algebra/CGroups/op_rht_resp_ap.con" as lemma.
 295
 296 inline procedural "cic:/CoRN/algebra/CGroups/cg_ap_cancel_rht.con" as lemma.
 297
 298 inline procedural "cic:/CoRN/algebra/CGroups/plus_cancel_ap_rht.con" as lemma.
 299
 300 inline procedural "cic:/CoRN/algebra/CGroups/minus_ap_zero.con" as lemma.
 301
 302 inline procedural "cic:/CoRN/algebra/CGroups/zero_minus_apart.con" as lemma.
 303
 304 inline procedural "cic:/CoRN/algebra/CGroups/inv_resp_ap_zero.con" as lemma.
 305
 306 inline procedural "cic:/CoRN/algebra/CGroups/inv_resp_ap.con" as lemma.
 307
 308 inline procedural "cic:/CoRN/algebra/CGroups/minus_resp_ap_rht.con" as lemma.
 309
 310 inline procedural "cic:/CoRN/algebra/CGroups/minus_resp_ap_lft.con" as lemma.
 311
 312 inline procedural "cic:/CoRN/algebra/CGroups/minus_cancel_ap_rht.con" as lemma.
 313
 314 (* UNEXPORTED
 315 End cgroups_apartness
 316 *)
 317
 318 (* UNEXPORTED
 319 Hint Resolve op_rht_resp_ap: algebra.
 320 *)
 321
 322 (* UNEXPORTED
 323 Hint Resolve minus_ap_zero zero_minus_apart inv_resp_ap_zero: algebra.
 324 *)
 325
 326 (* UNEXPORTED
 327 Section CGroup_Ops
 328 *)
 329
 330 (*#*
 331 ** Functional operations
 332
 333 As before, we lift our group operations to the function space of the group.
 334
 335 %\begin{convention}%
 336 Let [G] be a group and [F,F':(PartFunct G)] with domains given respectively
 337 by [P] and [Q].
 338 %\end{convention}%
 339 *)
 340
 341 (* UNEXPORTED
 342 cic:/CoRN/algebra/CGroups/CGroup_Ops/G.var
 343 *)
 344
 345 (* UNEXPORTED
 346 cic:/CoRN/algebra/CGroups/CGroup_Ops/F.var
 347 *)
 348
 349 (* UNEXPORTED
 350 cic:/CoRN/algebra/CGroups/CGroup_Ops/F'.var
 351 *)
 352
 353 (* begin hide *)
 354
 355 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_Ops/P.con" "CGroup_Ops__" as definition.
 356
 357 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_Ops/Q.con" "CGroup_Ops__" as definition.
 358
 359 (* end hide *)
 360
 361 (* UNEXPORTED
 362 Section Part_Function_Inv
 363 *)
 364
 365 inline procedural "cic:/CoRN/algebra/CGroups/part_function_inv_strext.con" as lemma.
 366
 367 inline procedural "cic:/CoRN/algebra/CGroups/Finv.con" as definition.
 368
 369 (* UNEXPORTED
 370 End Part_Function_Inv
 371 *)
 372
 373 (* UNEXPORTED
 374 Section Part_Function_Minus
 375 *)
 376
 377 inline procedural "cic:/CoRN/algebra/CGroups/part_function_minus_strext.con" as lemma.
 378
 379 inline procedural "cic:/CoRN/algebra/CGroups/Fminus.con" as definition.
 380
 381 (* UNEXPORTED
 382 End Part_Function_Minus
 383 *)
 384
 385 (*#*
 386 %\begin{convention}% Let [R:G->CProp].
 387 %\end{convention}%
 388 *)
 389
 390 (* UNEXPORTED
 391 cic:/CoRN/algebra/CGroups/CGroup_Ops/R.var
 392 *)
 393
 394 inline procedural "cic:/CoRN/algebra/CGroups/included_FInv.con" as lemma.
 395
 396 inline procedural "cic:/CoRN/algebra/CGroups/included_FInv'.con" as lemma.
 397
 398 inline procedural "cic:/CoRN/algebra/CGroups/included_FMinus.con" as lemma.
 399
 400 inline procedural "cic:/CoRN/algebra/CGroups/included_FMinus'.con" as lemma.
 401
 402 inline procedural "cic:/CoRN/algebra/CGroups/included_FMinus''.con" as lemma.
 403
 404 (* UNEXPORTED
 405 End CGroup_Ops
 406 *)
 407
 408 (* UNEXPORTED
 409 Implicit Arguments Finv [G].
 410 *)
 411
 412 (* NOTATION
 413 Notation "{--} x" := (Finv x) (at level 2, right associativity).
 414 *)
 415
 416 (* UNEXPORTED
 417 Implicit Arguments Fminus [G].
 418 *)
 419
 420 (* NOTATION
 421 Infix "{-}" := Fminus (at level 50, left associativity).
 422 *)
 423
 424 (* UNEXPORTED
 425 Hint Resolve included_FInv included_FMinus : included.
 426 *)
 427
 428 (* UNEXPORTED
 429 Hint Immediate included_FInv' included_FMinus' included_FMinus'' : included.
 430 *)
 431