matita/contribs/CoRN-Decl/complex/CComplex.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/complex/CComplex".
  18
  19 (* $Id: CComplex.v,v 1.8 2004/04/23 10:00:55 lcf Exp $ *)
  20
  21 (*#* printing Re %\ensuremath{\Re}% #&real;# *)
  22
  23 (*#* printing Im %\ensuremath{\Im}% #&image;# *)
  24
  25 (*#* printing CC %\ensuremath{\mathbb C}% #<b>C</b># *)
  26
  27 (*#* printing II %\ensuremath{\imath}% #i# *)
  28
  29 (*#* printing [+I*] %\ensuremath{+\imath}% *)
  30
  31 (*#* printing AbsCC %\ensuremath{|\cdot|_{\mathbb C}}% *)
  32
  33 (*#* printing CCX %\ensuremath{\mathbb C[X]}% #<b>C</b>[X]# *)
  34
  35 (* INCLUDE
  36 NRootIR
  37 *)
  38
  39 (*#* * Complex Numbers
  40 ** Algebraic structure
  41 *)
  42
  43 (* UNEXPORTED
  44 Section Complex_Numbers.
  45 *)
  46
  47 inline cic:/CoRN/complex/CComplex/CC_set.ind.
  48
  49 inline cic:/CoRN/complex/CComplex/cc_ap.con.
  50
  51 inline cic:/CoRN/complex/CComplex/cc_eq.con.
  52
  53 inline cic:/CoRN/complex/CComplex/cc_is_CSetoid.con.
  54
  55 inline cic:/CoRN/complex/CComplex/cc_csetoid.con.
  56
  57 inline cic:/CoRN/complex/CComplex/cc_plus.con.
  58
  59 inline cic:/CoRN/complex/CComplex/cc_mult.con.
  60
  61 inline cic:/CoRN/complex/CComplex/cc_zero.con.
  62
  63 inline cic:/CoRN/complex/CComplex/cc_one.con.
  64
  65 inline cic:/CoRN/complex/CComplex/cc_i.con.
  66
  67 inline cic:/CoRN/complex/CComplex/cc_inv.con.
  68
  69 (* not needed anymore
  70 Lemma cc_plus_op_proof : (bin_op_wd cc_csetoid cc_plus).
  71 Unfold bin_op_wd. Unfold bin_fun_wd.
  72 Intros x1 x2 y1 y2. Elim x1. Elim x2. Elim y1. Elim y2.
  73 Simpl. Unfold cc_eq. Simpl. Intros.
  74 Elim H. Clear H. Intros. Elim H0. Clear H0. Intros.
  75 Split; Algebra.
  76 Qed.
  77
  78 Lemma cc_mult_op_proof : (bin_op_wd cc_csetoid cc_mult).
  79 Unfold bin_op_wd. Unfold bin_fun_wd.
  80 Intros x1 x2 y1 y2. Elim x1. Elim x2. Elim y1. Elim y2.
  81 Simpl. Unfold cc_eq. Simpl. Intros.
  82 Elim H. Clear H. Intros. Elim H0. Clear H0. Intros.
  83 Split; Algebra.
  84 Qed.
  85
  86 Lemma cc_inv_op_proof : (un_op_wd cc_csetoid cc_inv).
  87 Unfold un_op_wd. Unfold fun_wd.
  88 Intros x y. Elim x. Elim y.
  89 Simpl. Unfold cc_eq. Simpl. Intros.
  90 Elim H. Clear H. Intros.
  91 Split; Algebra.
  92 Qed.
  93 *)
  94
  95 inline cic:/CoRN/complex/CComplex/cc_inv_strext.con.
  96
  97 inline cic:/CoRN/complex/CComplex/cc_plus_strext.con.
  98
  99 inline cic:/CoRN/complex/CComplex/cc_mult_strext.con.
 100
 101 inline cic:/CoRN/complex/CComplex/cc_inv_op.con.
 102
 103 inline cic:/CoRN/complex/CComplex/cc_plus_op.con.
 104
 105 inline cic:/CoRN/complex/CComplex/cc_mult_op.con.
 106
 107 inline cic:/CoRN/complex/CComplex/cc_csg_associative.con.
 108
 109 inline cic:/CoRN/complex/CComplex/cc_cr_mult_associative.con.
 110
 111 inline cic:/CoRN/complex/CComplex/cc_csemi_grp.con.
 112
 113 inline cic:/CoRN/complex/CComplex/cc_cm_proof.con.
 114
 115 inline cic:/CoRN/complex/CComplex/cc_cmonoid.con.
 116
 117 inline cic:/CoRN/complex/CComplex/cc_cg_proof.con.
 118
 119 inline cic:/CoRN/complex/CComplex/cc_cr_dist.con.
 120
 121 inline cic:/CoRN/complex/CComplex/cc_cr_non_triv.con.
 122
 123 inline cic:/CoRN/complex/CComplex/cc_cgroup.con.
 124
 125 inline cic:/CoRN/complex/CComplex/cc_cabgroup.con.
 126
 127 inline cic:/CoRN/complex/CComplex/cc_cr_mult_mon.con.
 128
 129 inline cic:/CoRN/complex/CComplex/cc_mult_commutes.con.
 130
 131 inline cic:/CoRN/complex/CComplex/cc_isCRing.con.
 132
 133 inline cic:/CoRN/complex/CComplex/cc_cring.con.
 134
 135 inline cic:/CoRN/complex/CComplex/cc_ap_zero.con.
 136
 137 inline cic:/CoRN/complex/CComplex/cc_inv_aid.con.
 138
 139 (*#*
 140 If [x [~=] Zero] or [y [~=] Zero], then  [x [/] x[^]2 [+] y[^]2 [~=] Zero] or
 141 [[--]y[/]x[^]2[+]y[^]2 [~=] Zero].
 142 *)
 143
 144 inline cic:/CoRN/complex/CComplex/cc_inv_aid2.con.
 145
 146 (*
 147 REMARK KEPT FOR SENTIMENTAL REASONS...
 148
 149 This definition seems clever.  Even though we *cannot* construct an
 150 element of (NonZeros cc_cring) (a Set) by deciding which part of the
 151 input (Re or Im) is NonZero (a Prop), we manage to construct the
 152 actual function.
 153 *)
 154
 155 inline cic:/CoRN/complex/CComplex/cc_recip.con.
 156
 157 inline cic:/CoRN/complex/CComplex/cc_cfield_proof.con.
 158
 159 inline cic:/CoRN/complex/CComplex/cc_Recip_proof.con.
 160
 161 (* UNEXPORTED
 162 Opaque cc_recip.
 163 *)
 164
 165 (* UNEXPORTED
 166 Opaque cc_inv.
 167 *)
 168
 169 inline cic:/CoRN/complex/CComplex/cc_cfield.con.
 170
 171 inline cic:/CoRN/complex/CComplex/CC.con.
 172
 173 (*#*
 174 Maps from reals to complex and vice-versa are defined, as well as conjugate,
 175 absolute value and the imaginary unit [I] *)
 176
 177 inline cic:/CoRN/complex/CComplex/cc_set_CC.con.
 178
 179 inline cic:/CoRN/complex/CComplex/cc_IR.con.
 180
 181 inline cic:/CoRN/complex/CComplex/CC_conj.con.
 182
 183 (* old def
 184 Definition CC_conj' : CC->CC := [z:CC_set] (CC_set_rec [_:CC_set]CC_set [Re0,Im0:IR] (Build_CC_set Re0 [--]Im0) z).
 185 *)
 186
 187 inline cic:/CoRN/complex/CComplex/AbsCC.con.
 188
 189 inline cic:/CoRN/complex/CComplex/TwoCC_ap_zero.con.
 190
 191 (* UNEXPORTED
 192 End Complex_Numbers.
 193 *)
 194
 195 (* begin hide *)
 196
 197 (* end hide *)
 198
 199 inline cic:/CoRN/complex/CComplex/II.con.
 200
 201 (*#* ** Properties of [II] *)
 202
 203 (* UNEXPORTED
 204 Section I_properties.
 205 *)
 206
 207 inline cic:/CoRN/complex/CComplex/I_square.con.
 208
 209 (* UNEXPORTED
 210 Hint Resolve I_square: algebra.
 211 *)
 212
 213 inline cic:/CoRN/complex/CComplex/I_square'.con.
 214
 215 inline cic:/CoRN/complex/CComplex/I_recip_lft.con.
 216
 217 inline cic:/CoRN/complex/CComplex/I_recip_rht.con.
 218
 219 inline cic:/CoRN/complex/CComplex/mult_I.con.
 220
 221 inline cic:/CoRN/complex/CComplex/I_wd.con.
 222
 223 (*#* ** Properties of [Re] and [Im] *)
 224
 225 inline cic:/CoRN/complex/CComplex/calculate_norm.con.
 226
 227 inline cic:/CoRN/complex/CComplex/calculate_Re.con.
 228
 229 inline cic:/CoRN/complex/CComplex/calculate_Im.con.
 230
 231 inline cic:/CoRN/complex/CComplex/Re_wd.con.
 232
 233 inline cic:/CoRN/complex/CComplex/Im_wd.con.
 234
 235 inline cic:/CoRN/complex/CComplex/Re_resp_plus.con.
 236
 237 inline cic:/CoRN/complex/CComplex/Re_resp_inv.con.
 238
 239 inline cic:/CoRN/complex/CComplex/Im_resp_plus.con.
 240
 241 inline cic:/CoRN/complex/CComplex/Im_resp_inv.con.
 242
 243 inline cic:/CoRN/complex/CComplex/cc_calculate_square.con.
 244
 245 (* UNEXPORTED
 246 End I_properties.
 247 *)
 248
 249 (* UNEXPORTED
 250 Hint Resolve I_square I_square' I_recip_lft I_recip_rht mult_I calculate_norm
 251   cc_calculate_square: algebra.
 252 *)
 253
 254 (* UNEXPORTED
 255 Hint Resolve I_wd Re_wd Im_wd: algebra_c.
 256 *)
 257
 258 (*#* ** Properties of conjugation *)
 259
 260 (* UNEXPORTED
 261 Section Conj_properties.
 262 *)
 263
 264 inline cic:/CoRN/complex/CComplex/CC_conj_plus.con.
 265
 266 inline cic:/CoRN/complex/CComplex/CC_conj_mult.con.
 267
 268 (* UNEXPORTED
 269 Hint Resolve CC_conj_mult: algebra.
 270 *)
 271
 272 inline cic:/CoRN/complex/CComplex/CC_conj_strext.con.
 273
 274 inline cic:/CoRN/complex/CComplex/CC_conj_conj.con.
 275
 276 inline cic:/CoRN/complex/CComplex/CC_conj_zero.con.
 277
 278 inline cic:/CoRN/complex/CComplex/CC_conj_one.con.
 279
 280 (* UNEXPORTED
 281 Hint Resolve CC_conj_one: algebra.
 282 *)
 283
 284 inline cic:/CoRN/complex/CComplex/CC_conj_nexp.con.
 285
 286 (* UNEXPORTED
 287 End Conj_properties.
 288 *)
 289
 290 (* UNEXPORTED
 291 Hint Resolve CC_conj_plus CC_conj_mult CC_conj_nexp CC_conj_conj
 292   CC_conj_zero: algebra.
 293 *)
 294
 295 (*#* ** Properties of the real axis *)
 296
 297 (* UNEXPORTED
 298 Section cc_IR_properties.
 299 *)
 300
 301 inline cic:/CoRN/complex/CComplex/Re_cc_IR.con.
 302
 303 inline cic:/CoRN/complex/CComplex/Im_cc_IR.con.
 304
 305 inline cic:/CoRN/complex/CComplex/cc_IR_wd.con.
 306
 307 (* UNEXPORTED
 308 Hint Resolve cc_IR_wd: algebra_c.
 309 *)
 310
 311 inline cic:/CoRN/complex/CComplex/cc_IR_resp_ap.con.
 312
 313 inline cic:/CoRN/complex/CComplex/cc_IR_mult.con.
 314
 315 (* UNEXPORTED
 316 Hint Resolve cc_IR_mult: algebra.
 317 *)
 318
 319 inline cic:/CoRN/complex/CComplex/cc_IR_mult_lft.con.
 320
 321 inline cic:/CoRN/complex/CComplex/cc_IR_mult_rht.con.
 322
 323 inline cic:/CoRN/complex/CComplex/cc_IR_plus.con.
 324
 325 (* UNEXPORTED
 326 Hint Resolve cc_IR_plus: algebra.
 327 *)
 328
 329 inline cic:/CoRN/complex/CComplex/cc_IR_minus.con.
 330
 331 inline cic:/CoRN/complex/CComplex/cc_IR_zero.con.
 332
 333 (* UNEXPORTED
 334 Hint Resolve cc_IR_zero: algebra.
 335 *)
 336
 337 inline cic:/CoRN/complex/CComplex/cc_IR_one.con.
 338
 339 (* UNEXPORTED
 340 Hint Resolve cc_IR_one: algebra.
 341 *)
 342
 343 inline cic:/CoRN/complex/CComplex/cc_IR_nring.con.
 344
 345 inline cic:/CoRN/complex/CComplex/cc_IR_nexp.con.
 346
 347 (* UNEXPORTED
 348 End cc_IR_properties.
 349 *)
 350
 351 (* UNEXPORTED
 352 Hint Resolve Re_cc_IR Im_cc_IR: algebra.
 353 *)
 354
 355 (* UNEXPORTED
 356 Hint Resolve cc_IR_wd: algebra_c.
 357 *)
 358
 359 (* UNEXPORTED
 360 Hint Resolve cc_IR_mult cc_IR_nexp cc_IR_mult_lft cc_IR_mult_rht cc_IR_plus
 361   cc_IR_minus: algebra.
 362 *)
 363
 364 (* UNEXPORTED
 365 Hint Resolve cc_IR_nring cc_IR_zero: algebra.
 366 *)
 367
 368 (*#* ** [CC] has characteristic zero *)
 369
 370 (* INCLUDE
 371 Transparent_algebra
 372 *)
 373
 374 inline cic:/CoRN/complex/CComplex/char0_CC.con.
 375
 376 (* INCLUDE
 377 Opaque_algebra
 378 *)
 379
 380 inline cic:/CoRN/complex/CComplex/poly_apzero_CC.con.
 381
 382 inline cic:/CoRN/complex/CComplex/poly_CC_extensional.con.
 383