ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <iostream>
  24 #include <sstream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "indexed.h"
  29 #include "idx.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "power.h"
  34 #include "relational.h"
  35 #include "symmetry.h"
  36 #include "operators.h"
  37 #include "lst.h"
  38 #include "archive.h"
  39 #include "symbol.h"
  40 #include "utils.h"
  41 #include "integral.h"
  42 #include "matrix.h"
  43 #include "inifcns.h"
  44
  45 namespace GiNaC {
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
  48   print_func<print_context>(&indexed::do_print).
  49   print_func<print_latex>(&indexed::do_print_latex).
  50   print_func<print_tree>(&indexed::do_print_tree))
  51
  52 //////////
  53 // default constructor
  54 //////////
  55
  56 indexed::indexed() : symtree(not_symmetric())
  57 {
  58         tinfo_key = &indexed::tinfo_static;
  59 }
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
  66 {
  67         tinfo_key = &indexed::tinfo_static;
  68         validate();
  69 }
  70
  71 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
  72 {
  73         tinfo_key = &indexed::tinfo_static;
  74         validate();
  75 }
  76
  77 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
  78 {
  79         tinfo_key = &indexed::tinfo_static;
  80         validate();
  81 }
  82
  83 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
  84 {
  85         tinfo_key = &indexed::tinfo_static;
  86         validate();
  87 }
  88
  89 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(not_symmetric())
  90 {
  91         tinfo_key = &indexed::tinfo_static;
  92         validate();
  93 }
  94
  95 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  96 {
  97         tinfo_key = &indexed::tinfo_static;
  98         validate();
  99 }
 100
 101 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 102 {
 103         tinfo_key = &indexed::tinfo_static;
 104         validate();
 105 }
 106
 107 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
 108 {
 109         tinfo_key = &indexed::tinfo_static;
 110         validate();
 111 }
 112
 113 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
 114 {
 115         seq.insert(seq.end(), v.begin(), v.end());
 116         tinfo_key = &indexed::tinfo_static;
 117         validate();
 118 }
 119
 120 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 121 {
 122         seq.insert(seq.end(), v.begin(), v.end());
 123         tinfo_key = &indexed::tinfo_static;
 124         validate();
 125 }
 126
 127 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 128 {
 129         tinfo_key = &indexed::tinfo_static;
 130 }
 131
 132 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 133 {
 134         tinfo_key = &indexed::tinfo_static;
 135 }
 136
 137 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
 138 {
 139         tinfo_key = &indexed::tinfo_static;
 140 }
 141
 142 //////////
 143 // archiving
 144 //////////
 145
 146 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 147 {
 148         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 149                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 150                 unsigned symm = 0;
 151                 n.find_unsigned("symmetry", symm);
 152                 switch (symm) {
 153                         case 1:
 154                                 symtree = sy_symm();
 155                                 break;
 156                         case 2:
 157                                 symtree = sy_anti();
 158                                 break;
 159                         default:
 160                                 symtree = not_symmetric();
 161                                 break;
 162                 }
 163                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 164         }
 165 }
 166
 167 void indexed::archive(archive_node &n) const
 168 {
 169         inherited::archive(n);
 170         n.add_ex("symmetry", symtree);
 171 }
 172
 173 DEFAULT_UNARCHIVE(indexed)
 174
 175 //////////
 176 // functions overriding virtual functions from base classes
 177 //////////
 178
 179 void indexed::printindices(const print_context & c, unsigned level) const
 180 {
 181         if (seq.size() > 1) {
 182
 183                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 184
 185                 if (is_a<print_latex>(c)) {
 186
 187                         // TeX output: group by variance
 188                         bool first = true;
 189                         bool covariant = true;
 190
 191                         while (it != itend) {
 192                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 193                                 if (first || cur_covariant != covariant) { // Variance changed
 194                                         // The empty {} prevents indices from ending up on top of each other
 195                                         if (!first)
 196                                                 c.s << "}{}";
 197                                         covariant = cur_covariant;
 198                                         if (covariant)
 199                                                 c.s << "_{";
 200                                         else
 201                                                 c.s << "^{";
 202                                 }
 203                                 it->print(c, level);
 204                                 c.s << " ";
 205                                 first = false;
 206                                 it++;
 207                         }
 208                         c.s << "}";
 209
 210                 } else {
 211
 212                         // Ordinary output
 213                         while (it != itend) {
 214                                 it->print(c, level);
 215                                 it++;
 216                         }
 217                 }
 218         }
 219 }
 220
 221 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
 222 {
 223         if (precedence() <= level)
 224                 c.s << openbrace << '(';
 225         c.s << openbrace;
 226         seq[0].print(c, precedence());
 227         c.s << closebrace;
 228         printindices(c, level);
 229         if (precedence() <= level)
 230                 c.s << ')' << closebrace;
 231 }
 232
 233 void indexed::do_print(const print_context & c, unsigned level) const
 234 {
 235         print_indexed(c, "", "", level);
 236 }
 237
 238 void indexed::do_print_latex(const print_latex & c, unsigned level) const
 239 {
 240         print_indexed(c, "{", "}", level);
 241 }
 242
 243 void indexed::do_print_tree(const print_tree & c, unsigned level) const
 244 {
 245         c.s << std::string(level, ' ') << class_name() << " @" << this
 246             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 247             << ", " << seq.size()-1 << " indices"
 248             << ", symmetry=" << symtree << std::endl;
 249         seq[0].print(c, level + c.delta_indent);
 250         printindices(c, level + c.delta_indent);
 251 }
 252
 253 bool indexed::info(unsigned inf) const
 254 {
 255         if (inf == info_flags::indexed) return true;
 256         if (inf == info_flags::has_indices) return seq.size() > 1;
 257         return inherited::info(inf);
 258 }
 259
 260 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 261         bool operator() (const ex & e, unsigned inf) const {
 262                 return !(ex_to<idx>(e).get_value().info(inf));
 263         }
 264 };
 265
 266 bool indexed::all_index_values_are(unsigned inf) const
 267 {
 268         // No indices? Then no property can be fulfilled
 269         if (seq.size() < 2)
 270                 return false;
 271
 272         // Check all indices
 273         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 274 }
 275
 276 int indexed::compare_same_type(const basic & other) const
 277 {
 278         GINAC_ASSERT(is_a<indexed>(other));
 279         return inherited::compare_same_type(other);
 280 }
 281
 282 ex indexed::eval(int level) const
 283 {
 284         // First evaluate children, then we will end up here again
 285         if (level > 1)
 286                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 287
 288         const ex &base = seq[0];
 289
 290         // If the base object is 0, the whole object is 0
 291         if (base.is_zero())
 292                 return _ex0;
 293
 294         // If the base object is a product, pull out the numeric factor
 295         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 296                 exvector v(seq);
 297                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 298                 v[0] = seq[0] / f;
 299                 return f * thiscontainer(v);
 300         }
 301
 302         if(this->tinfo()==&indexed::tinfo_static && seq.size()==1)
 303                 return base;
 304
 305         // Canonicalize indices according to the symmetry properties
 306         if (seq.size() > 2) {
 307                 exvector v = seq;
 308                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 309                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 310                 if (sig != std::numeric_limits<int>::max()) {
 311                         // Something has changed while sorting indices, more evaluations later
 312                         if (sig == 0)
 313                                 return _ex0;
 314                         return ex(sig) * thiscontainer(v);
 315                 }
 316         }
 317
 318         // Let the class of the base object perform additional evaluations
 319         return ex_to<basic>(base).eval_indexed(*this);
 320 }
 321
 322 ex indexed::real_part() const
 323 {
 324         if(op(0).info(info_flags::real))
 325                 return *this;
 326         return real_part_function(*this).hold();
 327 }
 328
 329 ex indexed::imag_part() const
 330 {
 331         if(op(0).info(info_flags::real))
 332                 return 0;
 333         return imag_part_function(*this).hold();
 334 }
 335
 336 ex indexed::thiscontainer(const exvector & v) const
 337 {
 338         return indexed(ex_to<symmetry>(symtree), v);
 339 }
 340
 341 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
 342 {
 343         return indexed(ex_to<symmetry>(symtree), vp);
 344 }
 345
 346 unsigned indexed::return_type() const
 347 {
 348         if(is_a<matrix>(op(0)))
 349                 return return_types::commutative;
 350         else
 351                 return op(0).return_type();
 352 }
 353
 354 ex indexed::expand(unsigned options) const
 355 {
 356         GINAC_ASSERT(seq.size() > 0);
 357
 358         if (options & expand_options::expand_indexed) {
 359                 ex newbase = seq[0].expand(options);
 360                 if (is_exactly_a<add>(newbase)) {
 361                         ex sum = _ex0;
 362                         for (size_t i=0; i<newbase.nops(); i++) {
 363                                 exvector s = seq;
 364                                 s[0] = newbase.op(i);
 365                                 sum += thiscontainer(s).expand(options);
 366                         }
 367                         return sum;
 368                 }
 369                 if (!are_ex_trivially_equal(newbase, seq[0])) {
 370                         exvector s = seq;
 371                         s[0] = newbase;
 372                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
 373                 }
 374         }
 375         return inherited::expand(options);
 376 }
 377
 378 //////////
 379 // virtual functions which can be overridden by derived classes
 380 //////////
 381
 382 // none
 383
 384 //////////
 385 // non-virtual functions in this class
 386 //////////
 387
 388 /** Check whether all indices are of class idx and validate the symmetry
 389  *  tree. This function is used internally to make sure that all constructed
 390  *  indexed objects really carry indices and not some other classes. */
 391 void indexed::validate() const
 392 {
 393         GINAC_ASSERT(seq.size() > 0);
 394         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 395         while (it != itend) {
 396                 if (!is_a<idx>(*it))
 397                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 398                 it++;
 399         }
 400
 401         if (!symtree.is_zero()) {
 402                 if (!is_exactly_a<symmetry>(symtree))
 403                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 404                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 405         }
 406 }
 407
 408 /** Implementation of ex::diff() for an indexed object always returns 0.
 409  *
 410  *  @see ex::diff */
 411 ex indexed::derivative(const symbol & s) const
 412 {
 413         return _ex0;
 414 }
 415
 416 //////////
 417 // global functions
 418 //////////
 419
 420 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 421         bool operator() (const ex &lh, const ex &rh) const
 422         {
 423                 if (lh.is_equal(rh))
 424                         return true;
 425                 else
 426                         try {
 427                                 // Replacing the dimension might cause an error (e.g. with
 428                                 // index classes that only work in a fixed number of dimensions)
 429                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 430                         } catch (...) {
 431                                 return false;
 432                         }
 433         }
 434 };
 435
 436 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 437 static bool indices_consistent(const exvector & v1, const exvector & v2)
 438 {
 439         // Number of indices must be the same
 440         if (v1.size() != v2.size())
 441                 return false;
 442
 443         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 444 }
 445
 446 exvector indexed::get_indices() const
 447 {
 448         GINAC_ASSERT(seq.size() >= 1);
 449         return exvector(seq.begin() + 1, seq.end());
 450 }
 451
 452 exvector indexed::get_dummy_indices() const
 453 {
 454         exvector free_indices, dummy_indices;
 455         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 456         return dummy_indices;
 457 }
 458
 459 exvector indexed::get_dummy_indices(const indexed & other) const
 460 {
 461         exvector indices = get_free_indices();
 462         exvector other_indices = other.get_free_indices();
 463         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 464         exvector dummy_indices;
 465         find_dummy_indices(indices, dummy_indices);
 466         return dummy_indices;
 467 }
 468
 469 bool indexed::has_dummy_index_for(const ex & i) const
 470 {
 471         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 472         while (it != itend) {
 473                 if (is_dummy_pair(*it, i))
 474                         return true;
 475                 it++;
 476         }
 477         return false;
 478 }
 479
 480 exvector indexed::get_free_indices() const
 481 {
 482         exvector free_indices, dummy_indices;
 483         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 484         return free_indices;
 485 }
 486
 487 exvector add::get_free_indices() const
 488 {
 489         exvector free_indices;
 490         for (size_t i=0; i<nops(); i++) {
 491                 if (i == 0)
 492                         free_indices = op(i).get_free_indices();
 493                 else {
 494                         exvector free_indices_of_term = op(i).get_free_indices();
 495                         if (!indices_consistent(free_indices, free_indices_of_term))
 496                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 497                 }
 498         }
 499         return free_indices;
 500 }
 501
 502 exvector mul::get_free_indices() const
 503 {
 504         // Concatenate free indices of all factors
 505         exvector un;
 506         for (size_t i=0; i<nops(); i++) {
 507                 exvector free_indices_of_factor = op(i).get_free_indices();
 508                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 509         }
 510
 511         // And remove the dummy indices
 512         exvector free_indices, dummy_indices;
 513         find_free_and_dummy(un, free_indices, dummy_indices);
 514         return free_indices;
 515 }
 516
 517 exvector ncmul::get_free_indices() const
 518 {
 519         // Concatenate free indices of all factors
 520         exvector un;
 521         for (size_t i=0; i<nops(); i++) {
 522                 exvector free_indices_of_factor = op(i).get_free_indices();
 523                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 524         }
 525
 526         // And remove the dummy indices
 527         exvector free_indices, dummy_indices;
 528         find_free_and_dummy(un, free_indices, dummy_indices);
 529         return free_indices;
 530 }
 531
 532 struct is_summation_idx : public std::unary_function<ex, bool> {
 533         bool operator()(const ex & e)
 534         {
 535                 return is_dummy_pair(e, e);
 536         }
 537 };
 538
 539 exvector integral::get_free_indices() const
 540 {
 541         if (a.get_free_indices().size() || b.get_free_indices().size())
 542                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
 543         return f.get_free_indices();
 544 }
 545
 546 template<class T> size_t number_of_type(const exvector&v)
 547 {
 548         size_t number = 0;
 549         for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
 550                 if(is_exactly_a<T>(*i))
 551                         ++number;
 552         return number;
 553 }
 554
 555 /** Rename dummy indices in an expression.
 556  *
 557  *  @param e Expression to work on
 558  *  @param local_dummy_indices The set of dummy indices that appear in the
 559  *    expression "e"
 560  *  @param global_dummy_indices The set of dummy indices that have appeared
 561  *    before and which we would like to use in "e", too. This gets updated
 562  *    by the function */
 563 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 564 {
 565         size_t global_size = number_of_type<T>(global_dummy_indices),
 566                local_size = number_of_type<T>(local_dummy_indices);
 567
 568         // Any local dummy indices at all?
 569         if (local_size == 0)
 570                 return e;
 571
 572         if (global_size < local_size) {
 573
 574                 // More local indices than we encountered before, add the new ones
 575                 // to the global set
 576                 size_t old_global_size = global_size;
 577                 int remaining = local_size - global_size;
 578                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 579                 while (it != itend && remaining > 0) {
 580                         if (is_exactly_a<T>(*it) && find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(idx_is_equal_ignore_dim(), *it)) == global_dummy_indices.end()) {
 581                                 global_dummy_indices.push_back(*it);
 582                                 global_size++;
 583                                 remaining--;
 584                         }
 585                         it++;
 586                 }
 587
 588                 // If this is the first set of local indices, do nothing
 589                 if (old_global_size == 0)
 590                         return e;
 591         }
 592         GINAC_ASSERT(local_size <= global_size);
 593
 594         // Construct vectors of index symbols
 595         exvector local_syms, global_syms;
 596         local_syms.reserve(local_size);
 597         global_syms.reserve(local_size);
 598         for (size_t i=0; local_syms.size()!=local_size; i++)
 599                 if(is_exactly_a<T>(local_dummy_indices[i]))
 600                         local_syms.push_back(local_dummy_indices[i].op(0));
 601         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 602         for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
 603                 if(is_exactly_a<T>(global_dummy_indices[i]))
 604                         global_syms.push_back(global_dummy_indices[i].op(0));
 605         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 606
 607         // Remove common indices
 608         exvector local_uniq, global_uniq;
 609         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 610         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 611
 612         // Replace remaining non-common local index symbols by global ones
 613         if (local_uniq.empty())
 614                 return e;
 615         else {
 616                 while (global_uniq.size() > local_uniq.size())
 617                         global_uniq.pop_back();
 618                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 619         }
 620 }
 621
 622 /** Given a set of indices, extract those of class varidx. */
 623 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 624 {
 625         exvector::const_iterator it1, itend;
 626         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 627                 if (is_exactly_a<varidx>(*it1))
 628                         variant_indices.push_back(*it1);
 629         }
 630 }
 631
 632 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 633  *  variance.
 634  *
 635  *  @param e Object to work on
 636  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 637  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 638  *  @return true if 'e' was changed */
 639 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 640 {
 641         bool something_changed = false;
 642
 643         // Find dummy symbols that occur twice in the same indexed object.
 644         exvector local_var_dummies;
 645         local_var_dummies.reserve(e.nops()/2);
 646         for (size_t i=1; i<e.nops(); ++i) {
 647                 if (!is_a<varidx>(e.op(i)))
 648                         continue;
 649                 for (size_t j=i+1; j<e.nops(); ++j) {
 650                         if (is_dummy_pair(e.op(i), e.op(j))) {
 651                                 local_var_dummies.push_back(e.op(i));
 652                                 for (exvector::iterator k = variant_dummy_indices.begin();
 653                                                 k!=variant_dummy_indices.end(); ++k) {
 654                                         if (e.op(i).op(0) == k->op(0)) {
 655                                                 variant_dummy_indices.erase(k);
 656                                                 break;
 657                                         }
 658                                 }
 659                                 break;
 660                         }
 661                 }
 662         }
 663
 664         // In the case where a dummy symbol occurs twice in the same indexed object
 665         // we try all posibilities of raising/lowering and keep the least one in
 666         // the sense of ex_is_less.
 667         ex optimal_e = e;
 668         size_t numpossibs = 1 << local_var_dummies.size();
 669         for (size_t i=0; i<numpossibs; ++i) {
 670                 ex try_e = e;
 671                 for (size_t j=0; j<local_var_dummies.size(); ++j) {
 672                         exmap m;
 673                         if (1<<j & i) {
 674                                 ex curr_idx = local_var_dummies[j];
 675                                 ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
 676                                 m[curr_idx] = curr_toggle;
 677                                 m[curr_toggle] = curr_idx;
 678                         }
 679                         try_e = e.subs(m, subs_options::no_pattern);
 680                 }
 681                 if(ex_is_less()(try_e, optimal_e))
 682                 {       optimal_e = try_e;
 683                         something_changed = true;
 684                 }
 685         }
 686         e = optimal_e;
 687
 688         if (!is_a<indexed>(e))
 689                 return true;
 690
 691         exvector seq = ex_to<indexed>(e).seq;
 692
 693         // If a dummy index is encountered for the first time in the
 694         // product, pull it up, otherwise, pull it down
 695         for (exvector::iterator it2 = seq.begin()+1, it2end = seq.end();
 696                         it2 != it2end; ++it2) {
 697                 if (!is_exactly_a<varidx>(*it2))
 698                         continue;
 699
 700                 exvector::iterator vit, vitend;
 701                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 702                         if (it2->op(0).is_equal(vit->op(0))) {
 703                                 if (ex_to<varidx>(*it2).is_covariant()) {
 704                                         /*
 705                                          * N.B. we don't want to use
 706                                          *
 707                                          *  e = e.subs(lst(
 708                                          *  *it2 == ex_to<varidx>(*it2).toggle_variance(),
 709                                          *  ex_to<varidx>(*it2).toggle_variance() == *it2
 710                                          *  ), subs_options::no_pattern);
 711                                          *
 712                                          * since this can trigger non-trivial repositioning of indices,
 713                                          * e.g. due to non-trivial symmetry properties of e, thus
 714                                          * invalidating iterators
 715                                          */
 716                                         *it2 = ex_to<varidx>(*it2).toggle_variance();
 717                                         something_changed = true;
 718                                 }
 719                                 moved_indices.push_back(*vit);
 720                                 variant_dummy_indices.erase(vit);
 721                                 goto next_index;
 722                         }
 723                 }
 724
 725                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 726                         if (it2->op(0).is_equal(vit->op(0))) {
 727                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 728                                         *it2 = ex_to<varidx>(*it2).toggle_variance();
 729                                         something_changed = true;
 730                                 }
 731                                 goto next_index;
 732                         }
 733                 }
 734
 735 next_index: ;
 736         }
 737
 738         if (something_changed)
 739                 e = ex_to<indexed>(e).thiscontainer(seq);
 740
 741         return something_changed;
 742 }
 743
 744 /* Ordering that only compares the base expressions of indexed objects. */
 745 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 746         bool operator() (const ex &lh, const ex &rh) const
 747         {
 748                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 749         }
 750 };
 751
 752 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
 753  * It returns an exvector of factors from the supplied product */
 754 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
 755 {
 756         // Remember whether the product was commutative or noncommutative
 757         // (because we chop it into factors and need to reassemble later)
 758         non_commutative = is_exactly_a<ncmul>(e);
 759
 760         // Collect factors in an exvector, store squares twice
 761         v.reserve(e.nops() * 2);
 762
 763         if (is_exactly_a<power>(e)) {
 764                 // We only get called for simple squares, split a^2 -> a*a
 765                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 766                 v.push_back(e.op(0));
 767                 v.push_back(e.op(0));
 768         } else {
 769                 for (size_t i=0; i<e.nops(); i++) {
 770                         ex f = e.op(i);
 771                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 772                                 v.push_back(f.op(0));
 773                                 v.push_back(f.op(0));
 774                         } else if (is_exactly_a<ncmul>(f)) {
 775                                 // Noncommutative factor found, split it as well
 776                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 777                                 for (size_t j=0; j<f.nops(); j++)
 778                                         v.push_back(f.op(j));
 779                         } else
 780                                 v.push_back(f);
 781                 }
 782         }
 783 }
 784
 785 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
 786 {       exvector dummy_syms;
 787         dummy_syms.reserve(r.nops());
 788         for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 789                         if(is_exactly_a<T>(*it))
 790                                 dummy_syms.push_back(it->op(0));
 791         if(dummy_syms.size() < 2)
 792                 return r;
 793         ex q=symmetrize(r, dummy_syms);
 794         return q;
 795 }
 796
 797 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
 798 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
 799
 800 /** Simplify product of indexed expressions (commutative, noncommutative and
 801  *  simple squares), return list of free indices. */
 802 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 803 {
 804         // Collect factors in an exvector
 805         exvector v;
 806
 807         // Remember whether the product was commutative or noncommutative
 808         // (because we chop it into factors and need to reassemble later)
 809         bool non_commutative;
 810         product_to_exvector(e, v, non_commutative);
 811
 812         // Perform contractions
 813         bool something_changed = false;
 814         GINAC_ASSERT(v.size() > 1);
 815         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 816         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 817
 818 try_again:
 819                 if (!is_a<indexed>(*it1))
 820                         continue;
 821
 822                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 823
 824                 // Indexed factor found, get free indices and look for contraction
 825                 // candidates
 826                 exvector free1, dummy1;
 827                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 828
 829                 exvector::iterator it2;
 830                 for (it2 = it1 + 1; it2 != itend; it2++) {
 831
 832                         if (!is_a<indexed>(*it2))
 833                                 continue;
 834
 835                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 836
 837                         // Find free indices of second factor and merge them with free
 838                         // indices of first factor
 839                         exvector un;
 840                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 841                         un.insert(un.end(), free1.begin(), free1.end());
 842
 843                         // Check whether the two factors share dummy indices
 844                         exvector free, dummy;
 845                         find_free_and_dummy(un, free, dummy);
 846                         size_t num_dummies = dummy.size();
 847                         if (num_dummies == 0)
 848                                 continue;
 849
 850                         // At least one dummy index, is it a defined scalar product?
 851                         bool contracted = false;
 852                         if (free.empty() && it1->nops()==2 && it2->nops()==2) {
 853
 854                                 ex dim = minimal_dim(
 855                                         ex_to<idx>(it1->op(1)).get_dim(),
 856                                         ex_to<idx>(it2->op(1)).get_dim()
 857                                 );
 858
 859                                 // User-defined scalar product?
 860                                 if (sp.is_defined(*it1, *it2, dim)) {
 861
 862                                         // Yes, substitute it
 863                                         *it1 = sp.evaluate(*it1, *it2, dim);
 864                                         *it2 = _ex1;
 865                                         goto contraction_done;
 866                                 }
 867                         }
 868
 869                         // Try to contract the first one with the second one
 870                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 871                         if (!contracted) {
 872
 873                                 // That didn't work; maybe the second object knows how to
 874                                 // contract itself with the first one
 875                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 876                         }
 877                         if (contracted) {
 878 contraction_done:
 879                                 if (first_noncommutative || second_noncommutative
 880                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 881                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 882                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 883
 884                                         // One of the factors became a sum or product:
 885                                         // re-expand expression and run again
 886                                         // Non-commutative products are always re-expanded to give
 887                                         // eval_ncmul() the chance to re-order and canonicalize
 888                                         // the product
 889                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 890                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 891                                 }
 892
 893                                 // Both objects may have new indices now or they might
 894                                 // even not be indexed objects any more, so we have to
 895                                 // start over
 896                                 something_changed = true;
 897                                 goto try_again;
 898                         }
 899                 }
 900         }
 901
 902         // Find free indices (concatenate them all and call find_free_and_dummy())
 903         // and all dummy indices that appear
 904         exvector un, individual_dummy_indices;
 905         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 906                 exvector free_indices_of_factor;
 907                 if (is_a<indexed>(*it1)) {
 908                         exvector dummy_indices_of_factor;
 909                         find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
 910                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 911                 } else
 912                         free_indices_of_factor = it1->get_free_indices();
 913                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 914         }
 915         exvector local_dummy_indices;
 916         find_free_and_dummy(un, free_indices, local_dummy_indices);
 917         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 918
 919         // Filter out the dummy indices with variance
 920         exvector variant_dummy_indices;
 921         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 922
 923         // Any indices with variance present at all?
 924         if (!variant_dummy_indices.empty()) {
 925
 926                 // Yes, bring the product into a canonical order that only depends on
 927                 // the base expressions of indexed objects
 928                 if (!non_commutative)
 929                         std::sort(v.begin(), v.end(), ex_base_is_less());
 930
 931                 exvector moved_indices;
 932
 933                 // Iterate over all indexed objects in the product
 934                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 935                         if (!is_a<indexed>(*it1))
 936                                 continue;
 937
 938                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 939                                 something_changed = true;
 940                 }
 941         }
 942
 943         ex r;
 944         if (something_changed)
 945                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 946         else
 947                 r = e;
 948
 949         // The result should be symmetric with respect to exchange of dummy
 950         // indices, so if the symmetrization vanishes, the whole expression is
 951         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 952         ex q = idx_symmetrization<idx>(r, local_dummy_indices);
 953         if (q.is_zero()) {
 954                 free_indices.clear();
 955                 return _ex0;
 956         }
 957         q = idx_symmetrization<varidx>(q, local_dummy_indices);
 958         if (q.is_zero()) {
 959                 free_indices.clear();
 960                 return _ex0;
 961         }
 962         q = idx_symmetrization<spinidx>(q, local_dummy_indices);
 963         if (q.is_zero()) {
 964                 free_indices.clear();
 965                 return _ex0;
 966         }
 967
 968         // Dummy index renaming
 969         r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
 970         r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
 971         r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
 972
 973         // Product of indexed object with a scalar?
 974         if (is_exactly_a<mul>(r) && r.nops() == 2
 975          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 976                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 977         else
 978                 return r;
 979 }
 980
 981 /** This structure stores the original and symmetrized versions of terms
 982  *  obtained during the simplification of sums. */
 983 class terminfo {
 984 public:
 985         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 986
 987         ex orig; /**< original term */
 988         ex symm; /**< symmtrized term */
 989 };
 990
 991 class terminfo_is_less {
 992 public:
 993         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 994         {
 995                 return (ti1.symm.compare(ti2.symm) < 0);
 996         }
 997 };
 998
 999 /** This structure stores the individual symmetrized terms obtained during
1000  *  the simplification of sums. */
1001 class symminfo {
1002 public:
1003         symminfo() : num(0) {}
1004
1005         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
1006         {
1007                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
1008                         coeff = symmterm_.op(symmterm_.nops()-1);
1009                         symmterm = symmterm_ / coeff;
1010                 } else {
1011                         coeff = 1;
1012                         symmterm = symmterm_;
1013                 }
1014         }
1015
1016         ex symmterm;  /**< symmetrized term */
1017         ex coeff;     /**< coefficient of symmetrized term */
1018         ex orig;      /**< original term */
1019         size_t num; /**< how many symmetrized terms resulted from the original term */
1020 };
1021
1022 class symminfo_is_less_by_symmterm {
1023 public:
1024         bool operator() (const symminfo & si1, const symminfo & si2) const
1025         {
1026                 return (si1.symmterm.compare(si2.symmterm) < 0);
1027         }
1028 };
1029
1030 class symminfo_is_less_by_orig {
1031 public:
1032         bool operator() (const symminfo & si1, const symminfo & si2) const
1033         {
1034                 return (si1.orig.compare(si2.orig) < 0);
1035         }
1036 };
1037
1038 bool hasindex(const ex &x, const ex &sym)
1039 {
1040         if(is_a<idx>(x) && x.op(0)==sym)
1041                 return true;
1042         else
1043                 for(size_t i=0; i<x.nops(); ++i)
1044                         if(hasindex(x.op(i), sym))
1045                                 return true;
1046         return false;
1047 }
1048
1049 /** Simplify indexed expression, return list of free indices. */
1050 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1051 {
1052         // Expand the expression
1053         ex e_expanded = e.expand();
1054
1055         // Simplification of single indexed object: just find the free indices
1056         // and perform dummy index renaming/repositioning
1057         if (is_a<indexed>(e_expanded)) {
1058
1059                 // Find the dummy indices
1060                 const indexed &i = ex_to<indexed>(e_expanded);
1061                 exvector local_dummy_indices;
1062                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1063
1064                 // Filter out the dummy indices with variance
1065                 exvector variant_dummy_indices;
1066                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1067
1068                 // Any indices with variance present at all?
1069                 if (!variant_dummy_indices.empty()) {
1070
1071                         // Yes, reposition them
1072                         exvector moved_indices;
1073                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1074                 }
1075
1076                 // Rename the dummy indices
1077                 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1078                 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1079                 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1080                 return e_expanded;
1081         }
1082
1083         // Simplification of sum = sum of simplifications, check consistency of
1084         // free indices in each term
1085         if (is_exactly_a<add>(e_expanded)) {
1086                 bool first = true;
1087                 ex sum;
1088                 free_indices.clear();
1089
1090                 for (size_t i=0; i<e_expanded.nops(); i++) {
1091                         exvector free_indices_of_term;
1092                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1093                         if (!term.is_zero()) {
1094                                 if (first) {
1095                                         free_indices = free_indices_of_term;
1096                                         sum = term;
1097                                         first = false;
1098                                 } else {
1099                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
1100                                                 std::ostringstream s;
1101                                                 s << "simplify_indexed: inconsistent indices in sum: ";
1102                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1103                                                 throw (std::runtime_error(s.str()));
1104                                         }
1105                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1106                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1107                                         else
1108                                                 sum += term;
1109                                 }
1110                         }
1111                 }
1112
1113                 // If the sum turns out to be zero, we are finished
1114                 if (sum.is_zero()) {
1115                         free_indices.clear();
1116                         return sum;
1117                 }
1118
1119                 // More than one term and more than one dummy index?
1120                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1121                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1122                         return sum;
1123
1124                 // Chop the sum into terms and symmetrize each one over the dummy
1125                 // indices
1126                 std::vector<terminfo> terms;
1127                 for (size_t i=0; i<sum.nops(); i++) {
1128                         const ex & term = sum.op(i);
1129                         exvector dummy_indices_of_term;
1130                         dummy_indices_of_term.reserve(dummy_indices.size());
1131                         for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1132                                 if(hasindex(term,i->op(0)))
1133                                         dummy_indices_of_term.push_back(*i);
1134                         ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1135                         term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1136                         term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1137                         if (term_symm.is_zero())
1138                                 continue;
1139                         terms.push_back(terminfo(term, term_symm));
1140                 }
1141
1142                 // Sort by symmetrized terms
1143                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1144
1145                 // Combine equal symmetrized terms
1146                 std::vector<terminfo> terms_pass2;
1147                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1148                         size_t num = 1;
1149                         std::vector<terminfo>::const_iterator j = i + 1;
1150                         while (j != terms.end() && j->symm == i->symm) {
1151                                 num++;
1152                                 j++;
1153                         }
1154                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1155                         i = j;
1156                 }
1157
1158                 // If there is only one term left, we are finished
1159                 if (terms_pass2.size() == 1)
1160                         return terms_pass2[0].orig;
1161
1162                 // Chop the symmetrized terms into subterms
1163                 std::vector<symminfo> sy;
1164                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1165                         if (is_exactly_a<add>(i->symm)) {
1166                                 size_t num = i->symm.nops();
1167                                 for (size_t j=0; j<num; j++)
1168                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1169                         } else
1170                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1171                 }
1172
1173                 // Sort by symmetrized subterms
1174                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1175
1176                 // Combine equal symmetrized subterms
1177                 std::vector<symminfo> sy_pass2;
1178                 exvector result;
1179                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1180
1181                         // Combine equal terms
1182                         std::vector<symminfo>::const_iterator j = i + 1;
1183                         if (j != sy.end() && j->symmterm == i->symmterm) {
1184
1185                                 // More than one term, collect the coefficients
1186                                 ex coeff = i->coeff;
1187                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1188                                         coeff += j->coeff;
1189                                         j++;
1190                                 }
1191
1192                                 // Add combined term to result
1193                                 if (!coeff.is_zero())
1194                                         result.push_back(coeff * i->symmterm);
1195
1196                         } else {
1197
1198                                 // Single term, store for second pass
1199                                 sy_pass2.push_back(*i);
1200                         }
1201
1202                         i = j;
1203                 }
1204
1205                 // Were there any remaining terms that didn't get combined?
1206                 if (sy_pass2.size() > 0) {
1207
1208                         // Yes, sort by their original terms
1209                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1210
1211                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1212
1213                                 // How many symmetrized terms of this original term are left?
1214                                 size_t num = 1;
1215                                 std::vector<symminfo>::const_iterator j = i + 1;
1216                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1217                                         num++;
1218                                         j++;
1219                                 }
1220
1221                                 if (num == i->num) {
1222
1223                                         // All terms left, then add the original term to the result
1224                                         result.push_back(i->orig);
1225
1226                                 } else {
1227
1228                                         // Some terms were combined with others, add up the remaining symmetrized terms
1229                                         std::vector<symminfo>::const_iterator k;
1230                                         for (k=i; k!=j; k++)
1231                                                 result.push_back(k->coeff * k->symmterm);
1232                                 }
1233
1234                                 i = j;
1235                         }
1236                 }
1237
1238                 // Add all resulting terms
1239                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1240                 if (sum_symm.is_zero())
1241                         free_indices.clear();
1242                 return sum_symm;
1243         }
1244
1245         // Simplification of products
1246         if (is_exactly_a<mul>(e_expanded)
1247          || is_exactly_a<ncmul>(e_expanded)
1248          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1249                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1250
1251         // Cannot do anything
1252         free_indices.clear();
1253         return e_expanded;
1254 }
1255
1256 /** Simplify/canonicalize expression containing indexed objects. This
1257  *  performs contraction of dummy indices where possible and checks whether
1258  *  the free indices in sums are consistent.
1259  *
1260  *  @param options Simplification options (currently unused)
1261  *  @return simplified expression */
1262 ex ex::simplify_indexed(unsigned options) const
1263 {
1264         exvector free_indices, dummy_indices;
1265         scalar_products sp;
1266         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1267 }
1268
1269 /** Simplify/canonicalize expression containing indexed objects. This
1270  *  performs contraction of dummy indices where possible, checks whether
1271  *  the free indices in sums are consistent, and automatically replaces
1272  *  scalar products by known values if desired.
1273  *
1274  *  @param sp Scalar products to be replaced automatically
1275  *  @param options Simplification options (currently unused)
1276  *  @return simplified expression */
1277 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1278 {
1279         exvector free_indices, dummy_indices;
1280         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1281 }
1282
1283 /** Symmetrize expression over its free indices. */
1284 ex ex::symmetrize() const
1285 {
1286         return GiNaC::symmetrize(*this, get_free_indices());
1287 }
1288
1289 /** Antisymmetrize expression over its free indices. */
1290 ex ex::antisymmetrize() const
1291 {
1292         return GiNaC::antisymmetrize(*this, get_free_indices());
1293 }
1294
1295 /** Symmetrize expression by cyclic permutation over its free indices. */
1296 ex ex::symmetrize_cyclic() const
1297 {
1298         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1299 }
1300
1301 //////////
1302 // helper classes
1303 //////////
1304
1305 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1306 {
1307         // If indexed, extract base objects
1308         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1309         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1310
1311         // Enforce canonical order in pair
1312         if (s1.compare(s2) > 0) {
1313                 v1 = s2;
1314                 v2 = s1;
1315         } else {
1316                 v1 = s1;
1317                 v2 = s2;
1318         }
1319 }
1320
1321 bool spmapkey::operator==(const spmapkey &other) const
1322 {
1323         if (!v1.is_equal(other.v1))
1324                 return false;
1325         if (!v2.is_equal(other.v2))
1326                 return false;
1327         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1328                 return true;
1329         else
1330                 return dim.is_equal(other.dim);
1331 }
1332
1333 bool spmapkey::operator<(const spmapkey &other) const
1334 {
1335         int cmp = v1.compare(other.v1);
1336         if (cmp)
1337                 return cmp < 0;
1338         cmp = v2.compare(other.v2);
1339         if (cmp)
1340                 return cmp < 0;
1341
1342         // Objects are equal, now check dimensions
1343         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1344                 return false;
1345         else
1346                 return dim.compare(other.dim) < 0;
1347 }
1348
1349 void spmapkey::debugprint() const
1350 {
1351         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1352 }
1353
1354 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1355 {
1356         spm[spmapkey(v1, v2)] = sp;
1357 }
1358
1359 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1360 {
1361         spm[spmapkey(v1, v2, dim)] = sp;
1362 }
1363
1364 void scalar_products::add_vectors(const lst & l, const ex & dim)
1365 {
1366         // Add all possible pairs of products
1367         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1368                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1369                         add(*it1, *it2, *it1 * *it2);
1370 }
1371
1372 void scalar_products::clear()
1373 {
1374         spm.clear();
1375 }
1376
1377 /** Check whether scalar product pair is defined. */
1378 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1379 {
1380         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1381 }
1382
1383 /** Return value of defined scalar product pair. */
1384 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1385 {
1386         return spm.find(spmapkey(v1, v2, dim))->second;
1387 }
1388
1389 void scalar_products::debugprint() const
1390 {
1391         std::cerr << "map size=" << spm.size() << std::endl;
1392         spmap::const_iterator i = spm.begin(), end = spm.end();
1393         while (i != end) {
1394                 const spmapkey & k = i->first;
1395                 std::cerr << "item key=";
1396                 k.debugprint();
1397                 std::cerr << ", value=" << i->second << std::endl;
1398                 ++i;
1399         }
1400 }
1401
1402 exvector get_all_dummy_indices_safely(const ex & e)
1403 {
1404         if (is_a<indexed>(e))
1405                 return ex_to<indexed>(e).get_dummy_indices();
1406         else if (is_a<power>(e) && e.op(1)==2) {
1407                 return e.op(0).get_free_indices();
1408         }
1409         else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1410                 exvector dummies;
1411                 exvector free_indices;
1412                 for (int i=0; i<e.nops(); ++i) {
1413                         exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1414                         dummies.insert(dummies.end(), dummies_of_factor.begin(),
1415                                 dummies_of_factor.end());
1416                         exvector free_of_factor = e.op(i).get_free_indices();
1417                         free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1418                                 free_of_factor.end());
1419                 }
1420                 exvector free_out, dummy_out;
1421                 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1422                         dummy_out);
1423                 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1424                 return dummies;
1425         }
1426         else if(is_a<add>(e)) {
1427                 exvector result;
1428                 for(int i=0; i<e.nops(); ++i) {
1429                         exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1430                         sort(dummies_of_term.begin(), dummies_of_term.end());
1431                         exvector new_vec;
1432                         set_union(result.begin(), result.end(), dummies_of_term.begin(),
1433                                 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1434                                 ex_is_less());
1435                         result.swap(new_vec);
1436                 }
1437                 return result;
1438         }
1439         return exvector();
1440 }
1441
1442 /** Returns all dummy indices from the exvector */
1443 exvector get_all_dummy_indices(const ex & e)
1444 {
1445         exvector p;
1446         bool nc;
1447         product_to_exvector(e, p, nc);
1448         exvector::const_iterator ip = p.begin(), ipend = p.end();
1449         exvector v, v1;
1450         while (ip != ipend) {
1451                 if (is_a<indexed>(*ip)) {
1452                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1453                         v.insert(v.end(), v1.begin(), v1.end());
1454                         exvector::const_iterator ip1 = ip+1;
1455                         while (ip1 != ipend) {
1456                                 if (is_a<indexed>(*ip1)) {
1457                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1458                                         v.insert(v.end(), v1.begin(), v1.end());
1459                                 }
1460                                 ++ip1;
1461                         }
1462                 }
1463                 ++ip;
1464         }
1465         return v;
1466 }
1467
1468 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1469 {
1470         exvector common_indices;
1471         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1472         if (common_indices.empty()) {
1473                 return lst(lst(), lst());
1474         } else {
1475                 exvector new_indices, old_indices;
1476                 old_indices.reserve(2*common_indices.size());
1477                 new_indices.reserve(2*common_indices.size());
1478                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1479                 while (ip != ipend) {
1480                         ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1481                         ex newidx;
1482                         if(is_exactly_a<spinidx>(*ip))
1483                                 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1484                                                 ex_to<spinidx>(*ip).is_covariant(),
1485                                                 ex_to<spinidx>(*ip).is_dotted()))
1486                                         -> setflag(status_flags::dynallocated);
1487                         else if (is_exactly_a<varidx>(*ip))
1488                                 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1489                                                 ex_to<varidx>(*ip).is_covariant()))
1490                                         -> setflag(status_flags::dynallocated);
1491                         else
1492                                 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1493                                         -> setflag(status_flags::dynallocated);
1494                         old_indices.push_back(*ip);
1495                         new_indices.push_back(newidx);
1496                         if(is_a<varidx>(*ip)) {
1497                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1498                                 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1499                         }
1500                         ++ip;
1501                 }
1502                 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1503         }
1504 }
1505
1506 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1507 {
1508         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1509         return (indices_subs.op(0).nops()>0 ? b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming) : b);
1510 }
1511
1512 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1513 {
1514         exvector va = get_all_dummy_indices_safely(a);
1515         if (va.size() > 0) {
1516                 exvector vb = get_all_dummy_indices_safely(b);
1517                 if (vb.size() > 0) {
1518                         sort(va.begin(), va.end(), ex_is_less());
1519                         sort(vb.begin(), vb.end(), ex_is_less());
1520                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1521                         if (indices_subs.op(0).nops() > 0)
1522                                 return b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming);
1523                 }
1524         }
1525         return b;
1526 }
1527
1528 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1529 {
1530         if (va.size() > 0) {
1531                 exvector vb = get_all_dummy_indices_safely(b);
1532                 if (vb.size() > 0) {
1533                         sort(vb.begin(), vb.end(), ex_is_less());
1534                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1535                         if (indices_subs.op(0).nops() > 0) {
1536                                 if (modify_va) {
1537                                         for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1538                                                 va.push_back(*i);
1539                                         exvector uncommon_indices;
1540                                         set_difference(vb.begin(), vb.end(), indices_subs.op(0).begin(), indices_subs.op(0).end(), std::back_insert_iterator<exvector>(uncommon_indices), ex_is_less());
1541                                         exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1542                                         while (ip != ipend) {
1543                                                 va.push_back(*ip);
1544                                                 ++ip;
1545                                         }
1546                                         sort(va.begin(), va.end(), ex_is_less());
1547                                 }
1548                                 return b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming);
1549                         }
1550                 }
1551         }
1552         return b;
1553 }
1554
1555 ex expand_dummy_sum(const ex & e, bool subs_idx)
1556 {
1557         ex e_expanded = e.expand();
1558         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1559         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1560                 return e_expanded.map(fcn);
1561         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1562                 exvector v;
1563                 if (is_a<indexed>(e_expanded))
1564                         v = ex_to<indexed>(e_expanded).get_dummy_indices();
1565                 else
1566                         v = get_all_dummy_indices(e_expanded);
1567                 ex result = e_expanded;
1568                 for(exvector::const_iterator it=v.begin(); it!=v.end(); ++it) {
1569                         ex nu = *it;
1570                         if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1571                                 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1572                                 ex en = 0;
1573                                 for (int i=0; i < idim; i++) {
1574                                         if (subs_idx && is_a<varidx>(nu)) {
1575                                                 ex other = ex_to<varidx>(nu).toggle_variance();
1576                                                 en += result.subs(lst(
1577                                                         nu == idx(i, idim),
1578                                                         other == idx(i, idim)
1579                                                 ));
1580                                         } else {
1581                                                 en += result.subs( nu.op(0) == i );
1582                                         }
1583                                 }
1584                                 result = en;
1585                         }
1586                 }
1587                 return result;
1588         } else {
1589                 return e;
1590         }
1591 }
1592
1593 } // namespace GiNaC