helm/software/matita/contribs/CoRN-Procedural/algebra/CGroups.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 (* $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 alias id "G" = "cic:/CoRN/algebra/CGroups/CGroup_axioms/G.var".
  94
  95 inline procedural "cic:/CoRN/algebra/CGroups/cg_inverse.con" as lemma.
  96
  97 (* UNEXPORTED
  98 End CGroup_axioms
  99 *)
 100
 101 (*#*
 102 ** Group basics
 103 General properties of groups.
 104 %\begin{convention}% Let [G] be a group.
 105 %\end{convention}%
 106 *)
 107
 108 (* UNEXPORTED
 109 Section CGroup_basics
 110 *)
 111
 112 alias id "G" = "cic:/CoRN/algebra/CGroups/CGroup_basics/G.var".
 113
 114 inline procedural "cic:/CoRN/algebra/CGroups/cg_rht_inv_unfolded.con" as lemma.
 115
 116 inline procedural "cic:/CoRN/algebra/CGroups/cg_lft_inv_unfolded.con" as lemma.
 117
 118 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_correct.con" as lemma.
 119
 120 (* UNEXPORTED
 121 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded cg_minus_correct:
 122   algebra.
 123 *)
 124
 125 inline procedural "cic:/CoRN/algebra/CGroups/cg_inverse'.con" as lemma.
 126
 127 (* Hints for Auto *)
 128
 129 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_unfolded.con" as lemma.
 130
 131 (* UNEXPORTED
 132 Hint Resolve cg_minus_unfolded: algebra.
 133 *)
 134
 135 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_wd.con" as lemma.
 136
 137 (* UNEXPORTED
 138 Hint Resolve cg_minus_wd: algebra_c.
 139 *)
 140
 141 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_strext.con" as lemma.
 142
 143 inline procedural "cic:/CoRN/algebra/CGroups/cg_minus_is_csetoid_bin_op.con" as definition.
 144
 145 inline procedural "cic:/CoRN/algebra/CGroups/grp_inv_assoc.con" as lemma.
 146
 147 (* UNEXPORTED
 148 Hint Resolve grp_inv_assoc: algebra.
 149 *)
 150
 151 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_unique.con" as lemma.
 152
 153 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_inv.con" as lemma.
 154
 155 (* UNEXPORTED
 156 Hint Resolve cg_inv_inv: algebra.
 157 *)
 158
 159 inline procedural "cic:/CoRN/algebra/CGroups/cg_cancel_lft.con" as lemma.
 160
 161 inline procedural "cic:/CoRN/algebra/CGroups/cg_cancel_rht.con" as lemma.
 162
 163 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_unique'.con" as lemma.
 164
 165 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_unique_2.con" as lemma.
 166
 167 inline procedural "cic:/CoRN/algebra/CGroups/cg_zero_inv.con" as lemma.
 168
 169 (* UNEXPORTED
 170 Hint Resolve cg_zero_inv: algebra.
 171 *)
 172
 173 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_zero.con" as lemma.
 174
 175 inline procedural "cic:/CoRN/algebra/CGroups/cg_inv_op.con" as lemma.
 176
 177 (*#*
 178 Useful for interactive proof development.
 179 *)
 180
 181 inline procedural "cic:/CoRN/algebra/CGroups/x_minus_x.con" as lemma.
 182
 183 (*#*
 184 ** Sub-groups
 185 %\begin{convention}% Let [P] be a predicate on [G] containing
 186 [Zero] and closed under [[+]] and [[--] ].
 187 %\end{convention}%
 188 *)
 189
 190 (* UNEXPORTED
 191 Section SubCGroups
 192 *)
 193
 194 alias id "P" = "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/P.var".
 195
 196 alias id "Punit" = "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/Punit.var".
 197
 198 alias id "op_pres_P" = "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/op_pres_P.var".
 199
 200 alias id "inv_pres_P" = "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/inv_pres_P.var".
 201
 202 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/subcrr.con" "CGroup_basics__SubCGroups__" as definition.
 203
 204 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_basics/SubCGroups/subinv.con" "CGroup_basics__SubCGroups__" as definition.
 205
 206 inline procedural "cic:/CoRN/algebra/CGroups/isgrp_scrr.con" as lemma.
 207
 208 inline procedural "cic:/CoRN/algebra/CGroups/Build_SubCGroup.con" as definition.
 209
 210 (* UNEXPORTED
 211 End SubCGroups
 212 *)
 213
 214 (* UNEXPORTED
 215 End CGroup_basics
 216 *)
 217
 218 (* UNEXPORTED
 219 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded: algebra.
 220 *)
 221
 222 (* UNEXPORTED
 223 Hint Resolve cg_inv_inv cg_minus_correct cg_zero_inv cg_inv_zero: algebra.
 224 *)
 225
 226 (* UNEXPORTED
 227 Hint Resolve cg_minus_unfolded grp_inv_assoc cg_inv_op: algebra.
 228 *)
 229
 230 (* UNEXPORTED
 231 Hint Resolve cg_minus_wd: algebra_c.
 232 *)
 233
 234 (*#*
 235 ** Associative properties of groups
 236 %\begin{convention}% Let [G] be a group.
 237 %\end{convention}%
 238 *)
 239
 240 (* UNEXPORTED
 241 Section Assoc_properties
 242 *)
 243
 244 alias id "G" = "cic:/CoRN/algebra/CGroups/Assoc_properties/G.var".
 245
 246 inline procedural "cic:/CoRN/algebra/CGroups/assoc_2.con" as lemma.
 247
 248 inline procedural "cic:/CoRN/algebra/CGroups/zero_minus.con" as lemma.
 249
 250 inline procedural "cic:/CoRN/algebra/CGroups/cg_cancel_mixed.con" as lemma.
 251
 252 inline procedural "cic:/CoRN/algebra/CGroups/plus_resp_eq.con" as lemma.
 253
 254 (* UNEXPORTED
 255 End Assoc_properties
 256 *)
 257
 258 (* UNEXPORTED
 259 Hint Resolve assoc_2 minus_plus zero_minus cg_cancel_mixed plus_resp_eq:
 260   algebra.
 261 *)
 262
 263 (*#*
 264 ** Apartness in Constructive Groups
 265 Specific properties of apartness.
 266 %\begin{convention}% Let [G] be a group.
 267 %\end{convention}%
 268 *)
 269
 270 (* UNEXPORTED
 271 Section cgroups_apartness
 272 *)
 273
 274 alias id "G" = "cic:/CoRN/algebra/CGroups/cgroups_apartness/G.var".
 275
 276 inline procedural "cic:/CoRN/algebra/CGroups/cg_add_ap_zero.con" as lemma.
 277
 278 inline procedural "cic:/CoRN/algebra/CGroups/op_rht_resp_ap.con" as lemma.
 279
 280 inline procedural "cic:/CoRN/algebra/CGroups/cg_ap_cancel_rht.con" as lemma.
 281
 282 inline procedural "cic:/CoRN/algebra/CGroups/plus_cancel_ap_rht.con" as lemma.
 283
 284 inline procedural "cic:/CoRN/algebra/CGroups/minus_ap_zero.con" as lemma.
 285
 286 inline procedural "cic:/CoRN/algebra/CGroups/zero_minus_apart.con" as lemma.
 287
 288 inline procedural "cic:/CoRN/algebra/CGroups/inv_resp_ap_zero.con" as lemma.
 289
 290 inline procedural "cic:/CoRN/algebra/CGroups/inv_resp_ap.con" as lemma.
 291
 292 inline procedural "cic:/CoRN/algebra/CGroups/minus_resp_ap_rht.con" as lemma.
 293
 294 inline procedural "cic:/CoRN/algebra/CGroups/minus_resp_ap_lft.con" as lemma.
 295
 296 inline procedural "cic:/CoRN/algebra/CGroups/minus_cancel_ap_rht.con" as lemma.
 297
 298 (* UNEXPORTED
 299 End cgroups_apartness
 300 *)
 301
 302 (* UNEXPORTED
 303 Hint Resolve op_rht_resp_ap: algebra.
 304 *)
 305
 306 (* UNEXPORTED
 307 Hint Resolve minus_ap_zero zero_minus_apart inv_resp_ap_zero: algebra.
 308 *)
 309
 310 (* UNEXPORTED
 311 Section CGroup_Ops
 312 *)
 313
 314 (*#*
 315 ** Functional operations
 316
 317 As before, we lift our group operations to the function space of the group.
 318
 319 %\begin{convention}%
 320 Let [G] be a group and [F,F':(PartFunct G)] with domains given respectively
 321 by [P] and [Q].
 322 %\end{convention}%
 323 *)
 324
 325 alias id "G" = "cic:/CoRN/algebra/CGroups/CGroup_Ops/G.var".
 326
 327 alias id "F" = "cic:/CoRN/algebra/CGroups/CGroup_Ops/F.var".
 328
 329 alias id "F'" = "cic:/CoRN/algebra/CGroups/CGroup_Ops/F'.var".
 330
 331 (* begin hide *)
 332
 333 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_Ops/P.con" "CGroup_Ops__" as definition.
 334
 335 inline procedural "cic:/CoRN/algebra/CGroups/CGroup_Ops/Q.con" "CGroup_Ops__" as definition.
 336
 337 (* end hide *)
 338
 339 (* UNEXPORTED
 340 Section Part_Function_Inv
 341 *)
 342
 343 inline procedural "cic:/CoRN/algebra/CGroups/part_function_inv_strext.con" as lemma.
 344
 345 inline procedural "cic:/CoRN/algebra/CGroups/Finv.con" as definition.
 346
 347 (* UNEXPORTED
 348 End Part_Function_Inv
 349 *)
 350
 351 (* UNEXPORTED
 352 Section Part_Function_Minus
 353 *)
 354
 355 inline procedural "cic:/CoRN/algebra/CGroups/part_function_minus_strext.con" as lemma.
 356
 357 inline procedural "cic:/CoRN/algebra/CGroups/Fminus.con" as definition.
 358
 359 (* UNEXPORTED
 360 End Part_Function_Minus
 361 *)
 362
 363 (*#*
 364 %\begin{convention}% Let [R:G->CProp].
 365 %\end{convention}%
 366 *)
 367
 368 alias id "R" = "cic:/CoRN/algebra/CGroups/CGroup_Ops/R.var".
 369
 370 inline procedural "cic:/CoRN/algebra/CGroups/included_FInv.con" as lemma.
 371
 372 inline procedural "cic:/CoRN/algebra/CGroups/included_FInv'.con" as lemma.
 373
 374 inline procedural "cic:/CoRN/algebra/CGroups/included_FMinus.con" as lemma.
 375
 376 inline procedural "cic:/CoRN/algebra/CGroups/included_FMinus'.con" as lemma.
 377
 378 inline procedural "cic:/CoRN/algebra/CGroups/included_FMinus''.con" as lemma.
 379
 380 (* UNEXPORTED
 381 End CGroup_Ops
 382 *)
 383
 384 (* UNEXPORTED
 385 Implicit Arguments Finv [G].
 386 *)
 387
 388 (* NOTATION
 389 Notation "{--} x" := (Finv x) (at level 2, right associativity).
 390 *)
 391
 392 (* UNEXPORTED
 393 Implicit Arguments Fminus [G].
 394 *)
 395
 396 (* NOTATION
 397 Infix "{-}" := Fminus (at level 50, left associativity).
 398 *)
 399
 400 (* UNEXPORTED
 401 Hint Resolve included_FInv included_FMinus : included.
 402 *)
 403
 404 (* UNEXPORTED
 405 Hint Immediate included_FInv' included_FMinus' included_FMinus'' : included.
 406 *)
 407