ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2006 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
  27 #include "indexed.h"
  28 #include "idx.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "symmetry.h"
  35 #include "operators.h"
  36 #include "lst.h"
  37 #include "archive.h"
  38 #include "symbol.h"
  39 #include "utils.h"
  40 #include "integral.h"
  41 #include "matrix.h"
  42
  43 namespace GiNaC {
  44
  45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
  46   print_func<print_context>(&indexed::do_print).
  47   print_func<print_latex>(&indexed::do_print_latex).
  48   print_func<print_tree>(&indexed::do_print_tree))
  49
  50 //////////
  51 // default constructor
  52 //////////
  53
  54 indexed::indexed() : symtree(not_symmetric())
  55 {
  56         tinfo_key = &indexed::tinfo_static;
  57 }
  58
  59 //////////
  60 // other constructors
  61 //////////
  62
  63 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
  64 {
  65         tinfo_key = &indexed::tinfo_static;
  66         validate();
  67 }
  68
  69 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
  70 {
  71         tinfo_key = &indexed::tinfo_static;
  72         validate();
  73 }
  74
  75 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
  76 {
  77         tinfo_key = &indexed::tinfo_static;
  78         validate();
  79 }
  80
  81 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
  82 {
  83         tinfo_key = &indexed::tinfo_static;
  84         validate();
  85 }
  86
  87 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())
  88 {
  89         tinfo_key = &indexed::tinfo_static;
  90         validate();
  91 }
  92
  93 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  94 {
  95         tinfo_key = &indexed::tinfo_static;
  96         validate();
  97 }
  98
  99 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 100 {
 101         tinfo_key = &indexed::tinfo_static;
 102         validate();
 103 }
 104
 105 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)
 106 {
 107         tinfo_key = &indexed::tinfo_static;
 108         validate();
 109 }
 110
 111 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
 112 {
 113         seq.insert(seq.end(), v.begin(), v.end());
 114         tinfo_key = &indexed::tinfo_static;
 115         validate();
 116 }
 117
 118 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 119 {
 120         seq.insert(seq.end(), v.begin(), v.end());
 121         tinfo_key = &indexed::tinfo_static;
 122         validate();
 123 }
 124
 125 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 126 {
 127         tinfo_key = &indexed::tinfo_static;
 128 }
 129
 130 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 131 {
 132         tinfo_key = &indexed::tinfo_static;
 133 }
 134
 135 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
 136 {
 137         tinfo_key = &indexed::tinfo_static;
 138 }
 139
 140 //////////
 141 // archiving
 142 //////////
 143
 144 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 145 {
 146         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 147                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 148                 unsigned symm = 0;
 149                 n.find_unsigned("symmetry", symm);
 150                 switch (symm) {
 151                         case 1:
 152                                 symtree = sy_symm();
 153                                 break;
 154                         case 2:
 155                                 symtree = sy_anti();
 156                                 break;
 157                         default:
 158                                 symtree = not_symmetric();
 159                                 break;
 160                 }
 161                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 162         }
 163 }
 164
 165 void indexed::archive(archive_node &n) const
 166 {
 167         inherited::archive(n);
 168         n.add_ex("symmetry", symtree);
 169 }
 170
 171 DEFAULT_UNARCHIVE(indexed)
 172
 173 //////////
 174 // functions overriding virtual functions from base classes
 175 //////////
 176
 177 void indexed::printindices(const print_context & c, unsigned level) const
 178 {
 179         if (seq.size() > 1) {
 180
 181                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 182
 183                 if (is_a<print_latex>(c)) {
 184
 185                         // TeX output: group by variance
 186                         bool first = true;
 187                         bool covariant = true;
 188
 189                         while (it != itend) {
 190                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 191                                 if (first || cur_covariant != covariant) { // Variance changed
 192                                         // The empty {} prevents indices from ending up on top of each other
 193                                         if (!first)
 194                                                 c.s << "}{}";
 195                                         covariant = cur_covariant;
 196                                         if (covariant)
 197                                                 c.s << "_{";
 198                                         else
 199                                                 c.s << "^{";
 200                                 }
 201                                 it->print(c, level);
 202                                 c.s << " ";
 203                                 first = false;
 204                                 it++;
 205                         }
 206                         c.s << "}";
 207
 208                 } else {
 209
 210                         // Ordinary output
 211                         while (it != itend) {
 212                                 it->print(c, level);
 213                                 it++;
 214                         }
 215                 }
 216         }
 217 }
 218
 219 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
 220 {
 221         if (precedence() <= level)
 222                 c.s << openbrace << '(';
 223         c.s << openbrace;
 224         seq[0].print(c, precedence());
 225         c.s << closebrace;
 226         printindices(c, level);
 227         if (precedence() <= level)
 228                 c.s << ')' << closebrace;
 229 }
 230
 231 void indexed::do_print(const print_context & c, unsigned level) const
 232 {
 233         print_indexed(c, "", "", level);
 234 }
 235
 236 void indexed::do_print_latex(const print_latex & c, unsigned level) const
 237 {
 238         print_indexed(c, "{", "}", level);
 239 }
 240
 241 void indexed::do_print_tree(const print_tree & c, unsigned level) const
 242 {
 243         c.s << std::string(level, ' ') << class_name() << " @" << this
 244             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 245             << ", " << seq.size()-1 << " indices"
 246             << ", symmetry=" << symtree << std::endl;
 247         seq[0].print(c, level + c.delta_indent);
 248         printindices(c, level + c.delta_indent);
 249 }
 250
 251 bool indexed::info(unsigned inf) const
 252 {
 253         if (inf == info_flags::indexed) return true;
 254         if (inf == info_flags::has_indices) return seq.size() > 1;
 255         return inherited::info(inf);
 256 }
 257
 258 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 259         bool operator() (const ex & e, unsigned inf) const {
 260                 return !(ex_to<idx>(e).get_value().info(inf));
 261         }
 262 };
 263
 264 bool indexed::all_index_values_are(unsigned inf) const
 265 {
 266         // No indices? Then no property can be fulfilled
 267         if (seq.size() < 2)
 268                 return false;
 269
 270         // Check all indices
 271         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 272 }
 273
 274 int indexed::compare_same_type(const basic & other) const
 275 {
 276         GINAC_ASSERT(is_a<indexed>(other));
 277         return inherited::compare_same_type(other);
 278 }
 279
 280 ex indexed::eval(int level) const
 281 {
 282         // First evaluate children, then we will end up here again
 283         if (level > 1)
 284                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 285
 286         const ex &base = seq[0];
 287
 288         // If the base object is 0, the whole object is 0
 289         if (base.is_zero())
 290                 return _ex0;
 291
 292         // If the base object is a product, pull out the numeric factor
 293         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 294                 exvector v(seq);
 295                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 296                 v[0] = seq[0] / f;
 297                 return f * thiscontainer(v);
 298         }
 299
 300         if(this->tinfo()==&indexed::tinfo_static && seq.size()==1)
 301                 return base;
 302
 303         // Canonicalize indices according to the symmetry properties
 304         if (seq.size() > 2) {
 305                 exvector v = seq;
 306                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 307                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 308                 if (sig != INT_MAX) {
 309                         // Something has changed while sorting indices, more evaluations later
 310                         if (sig == 0)
 311                                 return _ex0;
 312                         return ex(sig) * thiscontainer(v);
 313                 }
 314         }
 315
 316         // Let the class of the base object perform additional evaluations
 317         return ex_to<basic>(base).eval_indexed(*this);
 318 }
 319
 320 ex indexed::thiscontainer(const exvector & v) const
 321 {
 322         return indexed(ex_to<symmetry>(symtree), v);
 323 }
 324
 325 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
 326 {
 327         return indexed(ex_to<symmetry>(symtree), vp);
 328 }
 329
 330 unsigned indexed::return_type() const
 331 {
 332         if(is_a<matrix>(op(0)))
 333                 return return_types::commutative;
 334         else
 335                 return op(0).return_type();
 336 }
 337
 338 ex indexed::expand(unsigned options) const
 339 {
 340         GINAC_ASSERT(seq.size() > 0);
 341
 342         if (options & expand_options::expand_indexed) {
 343                 ex newbase = seq[0].expand(options);
 344                 if (is_exactly_a<add>(newbase)) {
 345                         ex sum = _ex0;
 346                         for (size_t i=0; i<newbase.nops(); i++) {
 347                                 exvector s = seq;
 348                                 s[0] = newbase.op(i);
 349                                 sum += thiscontainer(s).expand(options);
 350                         }
 351                         return sum;
 352                 }
 353                 if (!are_ex_trivially_equal(newbase, seq[0])) {
 354                         exvector s = seq;
 355                         s[0] = newbase;
 356                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
 357                 }
 358         }
 359         return inherited::expand(options);
 360 }
 361
 362 //////////
 363 // virtual functions which can be overridden by derived classes
 364 //////////
 365
 366 // none
 367
 368 //////////
 369 // non-virtual functions in this class
 370 //////////
 371
 372 /** Check whether all indices are of class idx and validate the symmetry
 373  *  tree. This function is used internally to make sure that all constructed
 374  *  indexed objects really carry indices and not some other classes. */
 375 void indexed::validate() const
 376 {
 377         GINAC_ASSERT(seq.size() > 0);
 378         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 379         while (it != itend) {
 380                 if (!is_a<idx>(*it))
 381                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 382                 it++;
 383         }
 384
 385         if (!symtree.is_zero()) {
 386                 if (!is_exactly_a<symmetry>(symtree))
 387                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 388                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 389         }
 390 }
 391
 392 /** Implementation of ex::diff() for an indexed object always returns 0.
 393  *
 394  *  @see ex::diff */
 395 ex indexed::derivative(const symbol & s) const
 396 {
 397         return _ex0;
 398 }
 399
 400 //////////
 401 // global functions
 402 //////////
 403
 404 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 405         bool operator() (const ex &lh, const ex &rh) const
 406         {
 407                 if (lh.is_equal(rh))
 408                         return true;
 409                 else
 410                         try {
 411                                 // Replacing the dimension might cause an error (e.g. with
 412                                 // index classes that only work in a fixed number of dimensions)
 413                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 414                         } catch (...) {
 415                                 return false;
 416                         }
 417         }
 418 };
 419
 420 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 421 static bool indices_consistent(const exvector & v1, const exvector & v2)
 422 {
 423         // Number of indices must be the same
 424         if (v1.size() != v2.size())
 425                 return false;
 426
 427         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 428 }
 429
 430 exvector indexed::get_indices() const
 431 {
 432         GINAC_ASSERT(seq.size() >= 1);
 433         return exvector(seq.begin() + 1, seq.end());
 434 }
 435
 436 exvector indexed::get_dummy_indices() const
 437 {
 438         exvector free_indices, dummy_indices;
 439         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 440         return dummy_indices;
 441 }
 442
 443 exvector indexed::get_dummy_indices(const indexed & other) const
 444 {
 445         exvector indices = get_free_indices();
 446         exvector other_indices = other.get_free_indices();
 447         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 448         exvector dummy_indices;
 449         find_dummy_indices(indices, dummy_indices);
 450         return dummy_indices;
 451 }
 452
 453 bool indexed::has_dummy_index_for(const ex & i) const
 454 {
 455         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 456         while (it != itend) {
 457                 if (is_dummy_pair(*it, i))
 458                         return true;
 459                 it++;
 460         }
 461         return false;
 462 }
 463
 464 exvector indexed::get_free_indices() const
 465 {
 466         exvector free_indices, dummy_indices;
 467         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 468         return free_indices;
 469 }
 470
 471 exvector add::get_free_indices() const
 472 {
 473         exvector free_indices;
 474         for (size_t i=0; i<nops(); i++) {
 475                 if (i == 0)
 476                         free_indices = op(i).get_free_indices();
 477                 else {
 478                         exvector free_indices_of_term = op(i).get_free_indices();
 479                         if (!indices_consistent(free_indices, free_indices_of_term))
 480                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 481                 }
 482         }
 483         return free_indices;
 484 }
 485
 486 exvector mul::get_free_indices() const
 487 {
 488         // Concatenate free indices of all factors
 489         exvector un;
 490         for (size_t i=0; i<nops(); i++) {
 491                 exvector free_indices_of_factor = op(i).get_free_indices();
 492                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 493         }
 494
 495         // And remove the dummy indices
 496         exvector free_indices, dummy_indices;
 497         find_free_and_dummy(un, free_indices, dummy_indices);
 498         return free_indices;
 499 }
 500
 501 exvector ncmul::get_free_indices() const
 502 {
 503         // Concatenate free indices of all factors
 504         exvector un;
 505         for (size_t i=0; i<nops(); i++) {
 506                 exvector free_indices_of_factor = op(i).get_free_indices();
 507                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 508         }
 509
 510         // And remove the dummy indices
 511         exvector free_indices, dummy_indices;
 512         find_free_and_dummy(un, free_indices, dummy_indices);
 513         return free_indices;
 514 }
 515
 516 struct is_summation_idx : public std::unary_function<ex, bool> {
 517         bool operator()(const ex & e)
 518         {
 519                 return is_dummy_pair(e, e);
 520         }
 521 };
 522
 523 exvector integral::get_free_indices() const
 524 {
 525         if (a.get_free_indices().size() || b.get_free_indices().size())
 526                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
 527         return f.get_free_indices();
 528 }
 529
 530 template<class T> size_t number_of_type(const exvector&v)
 531 {
 532         size_t number = 0;
 533         for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
 534                 if(is_exactly_a<T>(*i))
 535                         ++number;
 536         return number;
 537 }
 538
 539 /** Rename dummy indices in an expression.
 540  *
 541  *  @param e Expression to work on
 542  *  @param local_dummy_indices The set of dummy indices that appear in the
 543  *    expression "e"
 544  *  @param global_dummy_indices The set of dummy indices that have appeared
 545  *    before and which we would like to use in "e", too. This gets updated
 546  *    by the function */
 547 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 548 {
 549         size_t global_size = number_of_type<T>(global_dummy_indices),
 550                local_size = number_of_type<T>(local_dummy_indices);
 551
 552         // Any local dummy indices at all?
 553         if (local_size == 0)
 554                 return e;
 555
 556         if (global_size < local_size) {
 557
 558                 // More local indices than we encountered before, add the new ones
 559                 // to the global set
 560                 size_t old_global_size = global_size;
 561                 int remaining = local_size - global_size;
 562                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 563                 while (it != itend && remaining > 0) {
 564                         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()) {
 565                                 global_dummy_indices.push_back(*it);
 566                                 global_size++;
 567                                 remaining--;
 568                         }
 569                         it++;
 570                 }
 571
 572                 // If this is the first set of local indices, do nothing
 573                 if (old_global_size == 0)
 574                         return e;
 575         }
 576         GINAC_ASSERT(local_size <= global_size);
 577
 578         // Construct vectors of index symbols
 579         exvector local_syms, global_syms;
 580         local_syms.reserve(local_size);
 581         global_syms.reserve(local_size);
 582         for (size_t i=0; local_syms.size()!=local_size; i++)
 583                 if(is_exactly_a<T>(local_dummy_indices[i]))
 584                         local_syms.push_back(local_dummy_indices[i].op(0));
 585         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 586         for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
 587                 if(is_exactly_a<T>(global_dummy_indices[i]))
 588                         global_syms.push_back(global_dummy_indices[i].op(0));
 589         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 590
 591         // Remove common indices
 592         exvector local_uniq, global_uniq;
 593         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 594         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 595
 596         // Replace remaining non-common local index symbols by global ones
 597         if (local_uniq.empty())
 598                 return e;
 599         else {
 600                 while (global_uniq.size() > local_uniq.size())
 601                         global_uniq.pop_back();
 602                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 603         }
 604 }
 605
 606 /** Given a set of indices, extract those of class varidx. */
 607 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 608 {
 609         exvector::const_iterator it1, itend;
 610         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 611                 if (is_exactly_a<varidx>(*it1))
 612                         variant_indices.push_back(*it1);
 613         }
 614 }
 615
 616 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 617  *  variance.
 618  *
 619  *  @param e Object to work on
 620  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 621  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 622  *  @return true if 'e' was changed */
 623 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 624 {
 625         bool something_changed = false;
 626
 627         // If a dummy index is encountered for the first time in the
 628         // product, pull it up, otherwise, pull it down
 629         exvector::const_iterator it2, it2start, it2end;
 630         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 631                 if (!is_exactly_a<varidx>(*it2))
 632                         continue;
 633
 634                 exvector::iterator vit, vitend;
 635                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 636                         if (it2->op(0).is_equal(vit->op(0))) {
 637                                 if (ex_to<varidx>(*it2).is_covariant()) {
 638                                         e = e.subs(lst(
 639                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 640                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 641                                         ), subs_options::no_pattern);
 642                                         something_changed = true;
 643                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 644                                         it2start = ex_to<indexed>(e).seq.begin();
 645                                         it2end = ex_to<indexed>(e).seq.end();
 646                                 }
 647                                 moved_indices.push_back(*vit);
 648                                 variant_dummy_indices.erase(vit);
 649                                 goto next_index;
 650                         }
 651                 }
 652
 653                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 654                         if (it2->op(0).is_equal(vit->op(0))) {
 655                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 656                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
 657                                         something_changed = true;
 658                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 659                                         it2start = ex_to<indexed>(e).seq.begin();
 660                                         it2end = ex_to<indexed>(e).seq.end();
 661                                 }
 662                                 goto next_index;
 663                         }
 664                 }
 665
 666 next_index: ;
 667         }
 668
 669         return something_changed;
 670 }
 671
 672 /* Ordering that only compares the base expressions of indexed objects. */
 673 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 674         bool operator() (const ex &lh, const ex &rh) const
 675         {
 676                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 677         }
 678 };
 679
 680 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
 681  * It returns an exvector of factors from the supplied product */
 682 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
 683 {
 684         // Remember whether the product was commutative or noncommutative
 685         // (because we chop it into factors and need to reassemble later)
 686         non_commutative = is_exactly_a<ncmul>(e);
 687
 688         // Collect factors in an exvector, store squares twice
 689         v.reserve(e.nops() * 2);
 690
 691         if (is_exactly_a<power>(e)) {
 692                 // We only get called for simple squares, split a^2 -> a*a
 693                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 694                 v.push_back(e.op(0));
 695                 v.push_back(e.op(0));
 696         } else {
 697                 for (size_t i=0; i<e.nops(); i++) {
 698                         ex f = e.op(i);
 699                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 700                                 v.push_back(f.op(0));
 701                                 v.push_back(f.op(0));
 702                         } else if (is_exactly_a<ncmul>(f)) {
 703                                 // Noncommutative factor found, split it as well
 704                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 705                                 for (size_t j=0; j<f.nops(); j++)
 706                                         v.push_back(f.op(j));
 707                         } else
 708                                 v.push_back(f);
 709                 }
 710         }
 711 }
 712
 713 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
 714 {       exvector dummy_syms;
 715         dummy_syms.reserve(r.nops());
 716         for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 717                         if(is_exactly_a<T>(*it))
 718                                 dummy_syms.push_back(it->op(0));
 719         if(dummy_syms.size() < 2)
 720                 return r;
 721         ex q=symmetrize(r, dummy_syms);
 722         return q;
 723 }
 724
 725 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
 726 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
 727
 728 /** Simplify product of indexed expressions (commutative, noncommutative and
 729  *  simple squares), return list of free indices. */
 730 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 731 {
 732         // Collect factors in an exvector
 733         exvector v;
 734
 735         // Remember whether the product was commutative or noncommutative
 736         // (because we chop it into factors and need to reassemble later)
 737         bool non_commutative;
 738         product_to_exvector(e, v, non_commutative);
 739
 740         // Perform contractions
 741         bool something_changed = false;
 742         GINAC_ASSERT(v.size() > 1);
 743         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 744         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 745
 746 try_again:
 747                 if (!is_a<indexed>(*it1))
 748                         continue;
 749
 750                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 751
 752                 // Indexed factor found, get free indices and look for contraction
 753                 // candidates
 754                 exvector free1, dummy1;
 755                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 756
 757                 exvector::iterator it2;
 758                 for (it2 = it1 + 1; it2 != itend; it2++) {
 759
 760                         if (!is_a<indexed>(*it2))
 761                                 continue;
 762
 763                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 764
 765                         // Find free indices of second factor and merge them with free
 766                         // indices of first factor
 767                         exvector un;
 768                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 769                         un.insert(un.end(), free1.begin(), free1.end());
 770
 771                         // Check whether the two factors share dummy indices
 772                         exvector free, dummy;
 773                         find_free_and_dummy(un, free, dummy);
 774                         size_t num_dummies = dummy.size();
 775                         if (num_dummies == 0)
 776                                 continue;
 777
 778                         // At least one dummy index, is it a defined scalar product?
 779                         bool contracted = false;
 780                         if (free.empty() && it1->nops()==2 && it2->nops()==2) {
 781
 782                                 ex dim = minimal_dim(
 783                                         ex_to<idx>(it1->op(1)).get_dim(),
 784                                         ex_to<idx>(it2->op(1)).get_dim()
 785                                 );
 786
 787                                 // User-defined scalar product?
 788                                 if (sp.is_defined(*it1, *it2, dim)) {
 789
 790                                         // Yes, substitute it
 791                                         *it1 = sp.evaluate(*it1, *it2, dim);
 792                                         *it2 = _ex1;
 793                                         goto contraction_done;
 794                                 }
 795                         }
 796
 797                         // Try to contract the first one with the second one
 798                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 799                         if (!contracted) {
 800
 801                                 // That didn't work; maybe the second object knows how to
 802                                 // contract itself with the first one
 803                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 804                         }
 805                         if (contracted) {
 806 contraction_done:
 807                                 if (first_noncommutative || second_noncommutative
 808                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 809                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 810                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 811
 812                                         // One of the factors became a sum or product:
 813                                         // re-expand expression and run again
 814                                         // Non-commutative products are always re-expanded to give
 815                                         // eval_ncmul() the chance to re-order and canonicalize
 816                                         // the product
 817                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 818                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 819                                 }
 820
 821                                 // Both objects may have new indices now or they might
 822                                 // even not be indexed objects any more, so we have to
 823                                 // start over
 824                                 something_changed = true;
 825                                 goto try_again;
 826                         }
 827                 }
 828         }
 829
 830         // Find free indices (concatenate them all and call find_free_and_dummy())
 831         // and all dummy indices that appear
 832         exvector un, individual_dummy_indices;
 833         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 834                 exvector free_indices_of_factor;
 835                 if (is_a<indexed>(*it1)) {
 836                         exvector dummy_indices_of_factor;
 837                         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);
 838                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 839                 } else
 840                         free_indices_of_factor = it1->get_free_indices();
 841                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 842         }
 843         exvector local_dummy_indices;
 844         find_free_and_dummy(un, free_indices, local_dummy_indices);
 845         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 846
 847         // Filter out the dummy indices with variance
 848         exvector variant_dummy_indices;
 849         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 850
 851         // Any indices with variance present at all?
 852         if (!variant_dummy_indices.empty()) {
 853
 854                 // Yes, bring the product into a canonical order that only depends on
 855                 // the base expressions of indexed objects
 856                 if (!non_commutative)
 857                         std::sort(v.begin(), v.end(), ex_base_is_less());
 858
 859                 exvector moved_indices;
 860
 861                 // Iterate over all indexed objects in the product
 862                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 863                         if (!is_a<indexed>(*it1))
 864                                 continue;
 865
 866                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 867                                 something_changed = true;
 868                 }
 869         }
 870
 871         ex r;
 872         if (something_changed)
 873                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 874         else
 875                 r = e;
 876
 877         // The result should be symmetric with respect to exchange of dummy
 878         // indices, so if the symmetrization vanishes, the whole expression is
 879         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 880         ex q = idx_symmetrization<idx>(r, local_dummy_indices);
 881         if (q.is_zero()) {
 882                 free_indices.clear();
 883                 return _ex0;
 884         }
 885         q = idx_symmetrization<varidx>(q, local_dummy_indices);
 886         if (q.is_zero()) {
 887                 free_indices.clear();
 888                 return _ex0;
 889         }
 890         q = idx_symmetrization<spinidx>(q, local_dummy_indices);
 891         if (q.is_zero()) {
 892                 free_indices.clear();
 893                 return _ex0;
 894         }
 895
 896         // Dummy index renaming
 897         r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
 898         r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
 899         r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
 900
 901         // Product of indexed object with a scalar?
 902         if (is_exactly_a<mul>(r) && r.nops() == 2
 903          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 904                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 905         else
 906                 return r;
 907 }
 908
 909 /** This structure stores the original and symmetrized versions of terms
 910  *  obtained during the simplification of sums. */
 911 class terminfo {
 912 public:
 913         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 914
 915         ex orig; /**< original term */
 916         ex symm; /**< symmtrized term */
 917 };
 918
 919 class terminfo_is_less {
 920 public:
 921         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 922         {
 923                 return (ti1.symm.compare(ti2.symm) < 0);
 924         }
 925 };
 926
 927 /** This structure stores the individual symmetrized terms obtained during
 928  *  the simplification of sums. */
 929 class symminfo {
 930 public:
 931         symminfo() : num(0) {}
 932
 933         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 934         {
 935                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 936                         coeff = symmterm_.op(symmterm_.nops()-1);
 937                         symmterm = symmterm_ / coeff;
 938                 } else {
 939                         coeff = 1;
 940                         symmterm = symmterm_;
 941                 }
 942         }
 943
 944         ex symmterm;  /**< symmetrized term */
 945         ex coeff;     /**< coefficient of symmetrized term */
 946         ex orig;      /**< original term */
 947         size_t num; /**< how many symmetrized terms resulted from the original term */
 948 };
 949
 950 class symminfo_is_less_by_symmterm {
 951 public:
 952         bool operator() (const symminfo & si1, const symminfo & si2) const
 953         {
 954                 return (si1.symmterm.compare(si2.symmterm) < 0);
 955         }
 956 };
 957
 958 class symminfo_is_less_by_orig {
 959 public:
 960         bool operator() (const symminfo & si1, const symminfo & si2) const
 961         {
 962                 return (si1.orig.compare(si2.orig) < 0);
 963         }
 964 };
 965
 966 bool hasindex(const ex &x, const ex &sym)
 967 {
 968         if(is_a<idx>(x) && x.op(0)==sym)
 969                 return true;
 970         else
 971                 for(size_t i=0; i<x.nops(); ++i)
 972                         if(hasindex(x.op(i), sym))
 973                                 return true;
 974         return false;
 975 }
 976
 977 /** Simplify indexed expression, return list of free indices. */
 978 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 979 {
 980         // Expand the expression
 981         ex e_expanded = e.expand();
 982
 983         // Simplification of single indexed object: just find the free indices
 984         // and perform dummy index renaming/repositioning
 985         if (is_a<indexed>(e_expanded)) {
 986
 987                 // Find the dummy indices
 988                 const indexed &i = ex_to<indexed>(e_expanded);
 989                 exvector local_dummy_indices;
 990                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 991
 992                 // Filter out the dummy indices with variance
 993                 exvector variant_dummy_indices;
 994                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 995
 996                 // Any indices with variance present at all?
 997                 if (!variant_dummy_indices.empty()) {
 998
 999                         // Yes, reposition them
1000                         exvector moved_indices;
1001                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1002                 }
1003
1004                 // Rename the dummy indices
1005                 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1006                 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1007                 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1008                 return e_expanded;
1009         }
1010
1011         // Simplification of sum = sum of simplifications, check consistency of
1012         // free indices in each term
1013         if (is_exactly_a<add>(e_expanded)) {
1014                 bool first = true;
1015                 ex sum;
1016                 free_indices.clear();
1017
1018                 for (size_t i=0; i<e_expanded.nops(); i++) {
1019                         exvector free_indices_of_term;
1020                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1021                         if (!term.is_zero()) {
1022                                 if (first) {
1023                                         free_indices = free_indices_of_term;
1024                                         sum = term;
1025                                         first = false;
1026                                 } else {
1027                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
1028                                                 std::ostringstream s;
1029                                                 s << "simplify_indexed: inconsistent indices in sum: ";
1030                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1031                                                 throw (std::runtime_error(s.str()));
1032                                         }
1033                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1034                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1035                                         else
1036                                                 sum += term;
1037                                 }
1038                         }
1039                 }
1040
1041                 // If the sum turns out to be zero, we are finished
1042                 if (sum.is_zero()) {
1043                         free_indices.clear();
1044                         return sum;
1045                 }
1046
1047                 // More than one term and more than one dummy index?
1048                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1049                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1050                         return sum;
1051
1052                 // Chop the sum into terms and symmetrize each one over the dummy
1053                 // indices
1054                 std::vector<terminfo> terms;
1055                 for (size_t i=0; i<sum.nops(); i++) {
1056                         const ex & term = sum.op(i);
1057                         exvector dummy_indices_of_term;
1058                         dummy_indices_of_term.reserve(dummy_indices.size());
1059                         for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1060                                 if(hasindex(term,i->op(0)))
1061                                         dummy_indices_of_term.push_back(*i);
1062                         ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1063                         term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1064                         term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1065                         if (term_symm.is_zero())
1066                                 continue;
1067                         terms.push_back(terminfo(term, term_symm));
1068                 }
1069
1070                 // Sort by symmetrized terms
1071                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1072
1073                 // Combine equal symmetrized terms
1074                 std::vector<terminfo> terms_pass2;
1075                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1076                         size_t num = 1;
1077                         std::vector<terminfo>::const_iterator j = i + 1;
1078                         while (j != terms.end() && j->symm == i->symm) {
1079                                 num++;
1080                                 j++;
1081                         }
1082                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1083                         i = j;
1084                 }
1085
1086                 // If there is only one term left, we are finished
1087                 if (terms_pass2.size() == 1)
1088                         return terms_pass2[0].orig;
1089
1090                 // Chop the symmetrized terms into subterms
1091                 std::vector<symminfo> sy;
1092                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1093                         if (is_exactly_a<add>(i->symm)) {
1094                                 size_t num = i->symm.nops();
1095                                 for (size_t j=0; j<num; j++)
1096                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1097                         } else
1098                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1099                 }
1100
1101                 // Sort by symmetrized subterms
1102                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1103
1104                 // Combine equal symmetrized subterms
1105                 std::vector<symminfo> sy_pass2;
1106                 exvector result;
1107                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1108
1109                         // Combine equal terms
1110                         std::vector<symminfo>::const_iterator j = i + 1;
1111                         if (j != sy.end() && j->symmterm == i->symmterm) {
1112
1113                                 // More than one term, collect the coefficients
1114                                 ex coeff = i->coeff;
1115                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1116                                         coeff += j->coeff;
1117                                         j++;
1118                                 }
1119
1120                                 // Add combined term to result
1121                                 if (!coeff.is_zero())
1122                                         result.push_back(coeff * i->symmterm);
1123
1124                         } else {
1125
1126                                 // Single term, store for second pass
1127                                 sy_pass2.push_back(*i);
1128                         }
1129
1130                         i = j;
1131                 }
1132
1133                 // Were there any remaining terms that didn't get combined?
1134                 if (sy_pass2.size() > 0) {
1135
1136                         // Yes, sort by their original terms
1137                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1138
1139                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1140
1141                                 // How many symmetrized terms of this original term are left?
1142                                 size_t num = 1;
1143                                 std::vector<symminfo>::const_iterator j = i + 1;
1144                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1145                                         num++;
1146                                         j++;
1147                                 }
1148
1149                                 if (num == i->num) {
1150
1151                                         // All terms left, then add the original term to the result
1152                                         result.push_back(i->orig);
1153
1154                                 } else {
1155
1156                                         // Some terms were combined with others, add up the remaining symmetrized terms
1157                                         std::vector<symminfo>::const_iterator k;
1158                                         for (k=i; k!=j; k++)
1159                                                 result.push_back(k->coeff * k->symmterm);
1160                                 }
1161
1162                                 i = j;
1163                         }
1164                 }
1165
1166                 // Add all resulting terms
1167                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1168                 if (sum_symm.is_zero())
1169                         free_indices.clear();
1170                 return sum_symm;
1171         }
1172
1173         // Simplification of products
1174         if (is_exactly_a<mul>(e_expanded)
1175          || is_exactly_a<ncmul>(e_expanded)
1176          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1177                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1178
1179         // Cannot do anything
1180         free_indices.clear();
1181         return e_expanded;
1182 }
1183
1184 /** Simplify/canonicalize expression containing indexed objects. This
1185  *  performs contraction of dummy indices where possible and checks whether
1186  *  the free indices in sums are consistent.
1187  *
1188  *  @param options Simplification options (currently unused)
1189  *  @return simplified expression */
1190 ex ex::simplify_indexed(unsigned options) const
1191 {
1192         exvector free_indices, dummy_indices;
1193         scalar_products sp;
1194         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1195 }
1196
1197 /** Simplify/canonicalize expression containing indexed objects. This
1198  *  performs contraction of dummy indices where possible, checks whether
1199  *  the free indices in sums are consistent, and automatically replaces
1200  *  scalar products by known values if desired.
1201  *
1202  *  @param sp Scalar products to be replaced automatically
1203  *  @param options Simplification options (currently unused)
1204  *  @return simplified expression */
1205 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1206 {
1207         exvector free_indices, dummy_indices;
1208         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1209 }
1210
1211 /** Symmetrize expression over its free indices. */
1212 ex ex::symmetrize() const
1213 {
1214         return GiNaC::symmetrize(*this, get_free_indices());
1215 }
1216
1217 /** Antisymmetrize expression over its free indices. */
1218 ex ex::antisymmetrize() const
1219 {
1220         return GiNaC::antisymmetrize(*this, get_free_indices());
1221 }
1222
1223 /** Symmetrize expression by cyclic permutation over its free indices. */
1224 ex ex::symmetrize_cyclic() const
1225 {
1226         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1227 }
1228
1229 //////////
1230 // helper classes
1231 //////////
1232
1233 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1234 {
1235         // If indexed, extract base objects
1236         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1237         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1238
1239         // Enforce canonical order in pair
1240         if (s1.compare(s2) > 0) {
1241                 v1 = s2;
1242                 v2 = s1;
1243         } else {
1244                 v1 = s1;
1245                 v2 = s2;
1246         }
1247 }
1248
1249 bool spmapkey::operator==(const spmapkey &other) const
1250 {
1251         if (!v1.is_equal(other.v1))
1252                 return false;
1253         if (!v2.is_equal(other.v2))
1254                 return false;
1255         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1256                 return true;
1257         else
1258                 return dim.is_equal(other.dim);
1259 }
1260
1261 bool spmapkey::operator<(const spmapkey &other) const
1262 {
1263         int cmp = v1.compare(other.v1);
1264         if (cmp)
1265                 return cmp < 0;
1266         cmp = v2.compare(other.v2);
1267         if (cmp)
1268                 return cmp < 0;
1269
1270         // Objects are equal, now check dimensions
1271         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1272                 return false;
1273         else
1274                 return dim.compare(other.dim) < 0;
1275 }
1276
1277 void spmapkey::debugprint() const
1278 {
1279         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1280 }
1281
1282 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1283 {
1284         spm[spmapkey(v1, v2)] = sp;
1285 }
1286
1287 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1288 {
1289         spm[spmapkey(v1, v2, dim)] = sp;
1290 }
1291
1292 void scalar_products::add_vectors(const lst & l, const ex & dim)
1293 {
1294         // Add all possible pairs of products
1295         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1296                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1297                         add(*it1, *it2, *it1 * *it2);
1298 }
1299
1300 void scalar_products::clear()
1301 {
1302         spm.clear();
1303 }
1304
1305 /** Check whether scalar product pair is defined. */
1306 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1307 {
1308         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1309 }
1310
1311 /** Return value of defined scalar product pair. */
1312 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1313 {
1314         return spm.find(spmapkey(v1, v2, dim))->second;
1315 }
1316
1317 void scalar_products::debugprint() const
1318 {
1319         std::cerr << "map size=" << spm.size() << std::endl;
1320         spmap::const_iterator i = spm.begin(), end = spm.end();
1321         while (i != end) {
1322                 const spmapkey & k = i->first;
1323                 std::cerr << "item key=";
1324                 k.debugprint();
1325                 std::cerr << ", value=" << i->second << std::endl;
1326                 ++i;
1327         }
1328 }
1329
1330 exvector get_all_dummy_indices_safely(const ex & e)
1331 {
1332         if (is_a<indexed>(e))
1333                 return ex_to<indexed>(e).get_dummy_indices();
1334         else if (is_a<power>(e) && e.op(1)==2) {
1335                 return e.op(0).get_free_indices();
1336         }
1337         else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1338                 exvector dummies;
1339                 exvector free_indices;
1340                 for (int i=0; i<e.nops(); ++i) {
1341                         exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1342                         dummies.insert(dummies.end(), dummies_of_factor.begin(),
1343                                 dummies_of_factor.end());
1344                         exvector free_of_factor = e.op(i).get_free_indices();
1345                         free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1346                                 free_of_factor.end());
1347                 }
1348                 exvector free_out, dummy_out;
1349                 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1350                         dummy_out);
1351                 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1352                 return dummies;
1353         }
1354         else if(is_a<add>(e)) {
1355                 exvector result;
1356                 for(int i=0; i<e.nops(); ++i) {
1357                         exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1358                         sort(dummies_of_term.begin(), dummies_of_term.end());
1359                         exvector new_vec;
1360                         set_union(result.begin(), result.end(), dummies_of_term.begin(),
1361                                 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1362                                 ex_is_less());
1363                         result.swap(new_vec);
1364                 }
1365                 return result;
1366         }
1367         return exvector();
1368 }
1369
1370 /** Returns all dummy indices from the exvector */
1371 exvector get_all_dummy_indices(const ex & e)
1372 {
1373         exvector p;
1374         bool nc;
1375         product_to_exvector(e, p, nc);
1376         exvector::const_iterator ip = p.begin(), ipend = p.end();
1377         exvector v, v1;
1378         while (ip != ipend) {
1379                 if (is_a<indexed>(*ip)) {
1380                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1381                         v.insert(v.end(), v1.begin(), v1.end());
1382                         exvector::const_iterator ip1 = ip+1;
1383                         while (ip1 != ipend) {
1384                                 if (is_a<indexed>(*ip1)) {
1385                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1386                                         v.insert(v.end(), v1.begin(), v1.end());
1387                                 }
1388                                 ++ip1;
1389                         }
1390                 }
1391                 ++ip;
1392         }
1393         return v;
1394 }
1395
1396 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1397 {
1398         exvector common_indices;
1399         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1400         if (common_indices.empty()) {
1401                 return lst(lst(), lst());
1402         } else {
1403                 exvector new_indices, old_indices;
1404                 old_indices.reserve(2*common_indices.size());
1405                 new_indices.reserve(2*common_indices.size());
1406                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1407                 while (ip != ipend) {
1408                         ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1409                         ex newidx;
1410                         if(is_exactly_a<spinidx>(*ip))
1411                                 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1412                                                 ex_to<spinidx>(*ip).is_covariant(),
1413                                                 ex_to<spinidx>(*ip).is_dotted()))
1414                                         -> setflag(status_flags::dynallocated);
1415                         else if (is_exactly_a<varidx>(*ip))
1416                                 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1417                                                 ex_to<varidx>(*ip).is_covariant()))
1418                                         -> setflag(status_flags::dynallocated);
1419                         else
1420                                 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1421                                         -> setflag(status_flags::dynallocated);
1422                         old_indices.push_back(*ip);
1423                         new_indices.push_back(newidx);
1424                         if(is_a<varidx>(*ip)) {
1425                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1426                                 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1427                         }
1428                         ++ip;
1429                 }
1430                 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1431         }
1432 }
1433
1434 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1435 {
1436         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1437         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);
1438 }
1439
1440 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1441 {
1442         exvector va = get_all_dummy_indices_safely(a);
1443         if (va.size() > 0) {
1444                 exvector vb = get_all_dummy_indices_safely(b);
1445                 if (vb.size() > 0) {
1446                         sort(va.begin(), va.end(), ex_is_less());
1447                         sort(vb.begin(), vb.end(), ex_is_less());
1448                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1449                         if (indices_subs.op(0).nops() > 0)
1450                                 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);
1451                 }
1452         }
1453         return b;
1454 }
1455
1456 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1457 {
1458         if (va.size() > 0) {
1459                 exvector vb = get_all_dummy_indices_safely(b);
1460                 if (vb.size() > 0) {
1461                         sort(vb.begin(), vb.end(), ex_is_less());
1462                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1463                         if (indices_subs.op(0).nops() > 0) {
1464                                 if (modify_va) {
1465                                         for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1466                                                 va.push_back(*i);
1467                                         exvector uncommon_indices;
1468                                         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());
1469                                         exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1470                                         while (ip != ipend) {
1471                                                 va.push_back(*ip);
1472                                                 ++ip;
1473                                         }
1474                                         sort(va.begin(), va.end(), ex_is_less());
1475                                 }
1476                                 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);
1477                         }
1478                 }
1479         }
1480         return b;
1481 }
1482
1483 ex expand_dummy_sum(const ex & e, bool subs_idx)
1484 {
1485         ex e_expanded = e.expand();
1486         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1487         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1488                 return e_expanded.map(fcn);
1489         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded)) {
1490                 exvector v = get_all_dummy_indices(e_expanded);
1491                 exvector::const_iterator it = v.begin(), itend = v.end();
1492                 while (it != itend) {
1493                         varidx nu = ex_to<varidx>(*it);
1494                         if (nu.is_dim_numeric()) {
1495                                 ex en = 0;
1496                                 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1497                                         if (is_a<varidx>(nu) && !subs_idx) {
1498                                                 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1499                                         } else {
1500                                                 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1501                                         }
1502                                 }
1503                                 return expand_dummy_sum(en, subs_idx);
1504                         }
1505                         ++it;
1506                 }
1507                 return e;
1508         } else if (is_a<indexed>(e_expanded)) {
1509                 exvector v = ex_to<indexed>(e_expanded).get_dummy_indices();
1510                 exvector::const_iterator it = v.begin(), itend = v.end();
1511                 while (it != itend) {
1512                         varidx nu = ex_to<varidx>(*it);
1513                         if (nu.is_dim_numeric()) {
1514                                 ex en = 0;
1515                                 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1516                                         if (is_a<varidx>(nu) && !subs_idx) {
1517                                                 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1518                                         } else {
1519                                                 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1520                                         }
1521                                 }
1522                                 return expand_dummy_sum(en, subs_idx);
1523                         }
1524                         ++it;
1525                 }
1526                 return e;
1527         } else {
1528                 return e;
1529         }
1530 }
1531
1532 } // namespace GiNaC