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

   1 (**************************************************************************)
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   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 include "algebra/CGroups.ma".
  20
  21 (* UNEXPORTED
  22 Section Abelian_Groups
  23 *)
  24
  25 (*#*
  26 * Abelian Groups
  27 Now we introduce commutativity and add some results.
  28 *)
  29
  30 inline procedural "cic:/CoRN/algebra/CAbGroups/is_CAbGroup.con" as definition.
  31
  32 inline procedural "cic:/CoRN/algebra/CAbGroups/CAbGroup.ind".
  33
  34 (* COERCION
  35 cic:/matita/CoRN-Procedural/algebra/CAbGroups/cag_crr.con
  36 *)
  37
  38 (* UNEXPORTED
  39 Section AbGroup_Axioms
  40 *)
  41
  42 (* UNEXPORTED
  43 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/AbGroup_Axioms/G.var
  44 *)
  45
  46 (*#*
  47 %\begin{convention}% Let [G] be an Abelian Group.
  48 %\end{convention}%
  49 *)
  50
  51 inline procedural "cic:/CoRN/algebra/CAbGroups/CAbGroup_is_CAbGroup.con" as lemma.
  52
  53 inline procedural "cic:/CoRN/algebra/CAbGroups/cag_commutes.con" as lemma.
  54
  55 inline procedural "cic:/CoRN/algebra/CAbGroups/cag_commutes_unfolded.con" as lemma.
  56
  57 (* UNEXPORTED
  58 End AbGroup_Axioms
  59 *)
  60
  61 (* UNEXPORTED
  62 Section SubCAbGroups
  63 *)
  64
  65 (*#*
  66 ** Subgroups of an Abelian Group
  67 *)
  68
  69 (* UNEXPORTED
  70 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/G.var
  71 *)
  72
  73 (* UNEXPORTED
  74 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/P.var
  75 *)
  76
  77 (* UNEXPORTED
  78 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/Punit.var
  79 *)
  80
  81 (* UNEXPORTED
  82 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/op_pres_P.var
  83 *)
  84
  85 (* UNEXPORTED
  86 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/inv_pres_P.var
  87 *)
  88
  89 (*#*
  90 %\begin{convention}%
  91 Let [G] be an Abelian Group and [P] be a ([CProp]-valued) predicate on [G]
  92 that contains [Zero] and is closed under [[+]] and [[--]].
  93 %\end{convention}%
  94 *)
  95
  96 inline procedural "cic:/CoRN/algebra/CAbGroups/Abelian_Groups/SubCAbGroups/subcrr.con" "Abelian_Groups__SubCAbGroups__" as definition.
  97
  98 inline procedural "cic:/CoRN/algebra/CAbGroups/isabgrp_scrr.con" as lemma.
  99
 100 inline procedural "cic:/CoRN/algebra/CAbGroups/Build_SubCAbGroup.con" as definition.
 101
 102 (* UNEXPORTED
 103 End SubCAbGroups
 104 *)
 105
 106 (* UNEXPORTED
 107 Section Various
 108 *)
 109
 110 (*#*
 111 ** Basic properties of Abelian groups
 112 *)
 113
 114 (* UNEXPORTED
 115 Hint Resolve cag_commutes_unfolded: algebra.
 116 *)
 117
 118 (* UNEXPORTED
 119 cic:/CoRN/algebra/CAbGroups/Abelian_Groups/Various/G.var
 120 *)
 121
 122 (*#*
 123 %\begin{convention}% Let [G] be an Abelian Group.
 124 %\end{convention}%
 125 *)
 126
 127 inline procedural "cic:/CoRN/algebra/CAbGroups/cag_op_inv.con" as lemma.
 128
 129 (* UNEXPORTED
 130 Hint Resolve cag_op_inv: algebra.
 131 *)
 132
 133 inline procedural "cic:/CoRN/algebra/CAbGroups/assoc_1.con" as lemma.
 134
 135 inline procedural "cic:/CoRN/algebra/CAbGroups/minus_plus.con" as lemma.
 136
 137 inline procedural "cic:/CoRN/algebra/CAbGroups/op_lft_resp_ap.con" as lemma.
 138
 139 inline procedural "cic:/CoRN/algebra/CAbGroups/cag_ap_cancel_lft.con" as lemma.
 140
 141 inline procedural "cic:/CoRN/algebra/CAbGroups/plus_cancel_ap_lft.con" as lemma.
 142
 143 (* UNEXPORTED
 144 End Various
 145 *)
 146
 147 (* UNEXPORTED
 148 End Abelian_Groups
 149 *)
 150
 151 (* UNEXPORTED
 152 Hint Resolve cag_commutes_unfolded: algebra.
 153 *)
 154
 155 (* UNEXPORTED
 156 Hint Resolve cag_op_inv assoc_1 zero_minus minus_plus op_lft_resp_ap: algebra.
 157 *)
 158
 159 (* UNEXPORTED
 160 Section Nice_Char
 161 *)
 162
 163 (*#*
 164 ** Building an abelian group
 165
 166 In order to actually define concrete abelian groups,
 167 it is not in general practical to construct first a semigroup,
 168 then a monoid, then a group and finally an abelian group.  The
 169 presence of commutativity, for example, makes many of the monoid
 170 proofs trivial.  In this section, we provide a constructor that
 171 will allow us to go directly from a setoid to an abelian group.
 172
 173 We start from a setoid S with an element [unit], a
 174 commutative and associative binary operation [plus] which
 175 is strongly extensional in its first argument and admits [unit]
 176 as a left unit, and a unary setoid
 177 function [inv] which inverts elements respective to [plus].
 178 *)
 179
 180 (* UNEXPORTED
 181 cic:/CoRN/algebra/CAbGroups/Nice_Char/S.var
 182 *)
 183
 184 (* UNEXPORTED
 185 cic:/CoRN/algebra/CAbGroups/Nice_Char/unit.var
 186 *)
 187
 188 (* UNEXPORTED
 189 cic:/CoRN/algebra/CAbGroups/Nice_Char/plus.var
 190 *)
 191
 192 (*#*
 193 %\begin{convention}%
 194 Let [S] be a Setoid and [unit:S], [plus:S->S->S] and [inv] a unary
 195 setoid operation on [S].
 196 Assume that [plus] is commutative, associative and `left-strongly-extensional
 197 ([(plus x z) [#] (plus y z) -> x [#] y]), that [unit] is a left-unit
 198 for [plus] and [(inv x)] is a right-inverse of [x] w.r.t.%\% [plus].
 199 %\end{convention}%
 200 *)
 201
 202 (* UNEXPORTED
 203 cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_lext.var
 204 *)
 205
 206 (* UNEXPORTED
 207 cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_lunit.var
 208 *)
 209
 210 (* UNEXPORTED
 211 cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_comm.var
 212 *)
 213
 214 (* UNEXPORTED
 215 cic:/CoRN/algebra/CAbGroups/Nice_Char/plus_assoc.var
 216 *)
 217
 218 (* UNEXPORTED
 219 cic:/CoRN/algebra/CAbGroups/Nice_Char/inv.var
 220 *)
 221
 222 (* UNEXPORTED
 223 cic:/CoRN/algebra/CAbGroups/Nice_Char/inv_inv.var
 224 *)
 225
 226 inline procedural "cic:/CoRN/algebra/CAbGroups/plus_rext.con" as lemma.
 227
 228 inline procedural "cic:/CoRN/algebra/CAbGroups/plus_runit.con" as lemma.
 229
 230 inline procedural "cic:/CoRN/algebra/CAbGroups/plus_is_fun.con" as lemma.
 231
 232 inline procedural "cic:/CoRN/algebra/CAbGroups/inv_inv'.con" as lemma.
 233
 234 inline procedural "cic:/CoRN/algebra/CAbGroups/plus_fun.con" as definition.
 235
 236 inline procedural "cic:/CoRN/algebra/CAbGroups/Build_CSemiGroup'.con" as definition.
 237
 238 inline procedural "cic:/CoRN/algebra/CAbGroups/Build_CMonoid'.con" as definition.
 239
 240 inline procedural "cic:/CoRN/algebra/CAbGroups/Build_CGroup'.con" as definition.
 241
 242 inline procedural "cic:/CoRN/algebra/CAbGroups/Build_CAbGroup'.con" as definition.
 243
 244 (* UNEXPORTED
 245 End Nice_Char
 246 *)
 247
 248 (*#* ** Iteration
 249
 250 For reflection the following is needed; hopefully it is also useful.
 251 *)
 252
 253 (* UNEXPORTED
 254 Section Group_Extras
 255 *)
 256
 257 (* UNEXPORTED
 258 cic:/CoRN/algebra/CAbGroups/Group_Extras/G.var
 259 *)
 260
 261 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult.con" as definition.
 262
 263 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_wd.con" as lemma.
 264
 265 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_one.con" as lemma.
 266
 267 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_Zero.con" as lemma.
 268
 269 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_plus.con" as lemma.
 270
 271 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_mult.con" as lemma.
 272
 273 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_inv.con" as lemma.
 274
 275 inline procedural "cic:/CoRN/algebra/CAbGroups/nmult_plus'.con" as lemma.
 276
 277 (* UNEXPORTED
 278 Hint Resolve nmult_wd nmult_Zero nmult_inv nmult_plus nmult_plus': algebra.
 279 *)
 280
 281 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult.con" as definition.
 282
 283 (*
 284 Lemma Zeq_imp_nat_eq : forall m n:nat, m = n -> m = n.
 285 auto.
 286 intro m; induction m.
 287 intro n; induction n; auto.
 288
 289 intro; induction n.
 290 intro. inversion H.
 291 intros.
 292 rewrite (IHm n).
 293 auto.
 294 repeat rewrite inj_S in H.
 295 auto with zarith.
 296 Qed.
 297 *)
 298
 299 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_char.con" as lemma.
 300
 301 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_wd.con" as lemma.
 302
 303 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_one.con" as lemma.
 304
 305 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_min_one.con" as lemma.
 306
 307 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_zero.con" as lemma.
 308
 309 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_Zero.con" as lemma.
 310
 311 (* UNEXPORTED
 312 Hint Resolve zmult_zero: algebra.
 313 *)
 314
 315 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_plus.con" as lemma.
 316
 317 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_mult.con" as lemma.
 318
 319 inline procedural "cic:/CoRN/algebra/CAbGroups/zmult_plus'.con" as lemma.
 320
 321 (* UNEXPORTED
 322 End Group_Extras
 323 *)
 324
 325 (* UNEXPORTED
 326 Hint Resolve nmult_wd nmult_one nmult_Zero nmult_plus nmult_inv nmult_mult
 327   nmult_plus' zmult_wd zmult_one zmult_min_one zmult_zero zmult_Zero
 328   zmult_plus zmult_mult zmult_plus': algebra.
 329 *)
 330
 331 (* UNEXPORTED
 332 Implicit Arguments nmult [G].
 333 *)
 334
 335 (* UNEXPORTED
 336 Implicit Arguments zmult [G].
 337 *)
 338