ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2016 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 "indexed.h"
  24 #include "idx.h"
  25 #include "add.h"
  26 #include "mul.h"
  27 #include "ncmul.h"
  28 #include "power.h"
  29 #include "relational.h"
  30 #include "symmetry.h"
  31 #include "operators.h"
  32 #include "lst.h"
  33 #include "archive.h"
  34 #include "symbol.h"
  35 #include "utils.h"
  36 #include "integral.h"
  37 #include "matrix.h"
  38 #include "inifcns.h"
  39
  40 #include <iostream>
  41 #include <limits>
  42 #include <sstream>
  43 #include <stdexcept>
  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 }
  59
  60 //////////
  61 // other constructors
  62 //////////
  63
  64 indexed::indexed(const ex & b) : inherited{b}, symtree(not_symmetric())
  65 {
  66         validate();
  67 }
  68
  69 indexed::indexed(const ex & b, const ex & i1) : inherited{b, i1}, symtree(not_symmetric())
  70 {
  71         validate();
  72 }
  73
  74 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited{b, i1, i2}, symtree(not_symmetric())
  75 {
  76         validate();
  77 }
  78
  79 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited{b, i1, i2, i3}, symtree(not_symmetric())
  80 {
  81         validate();
  82 }
  83
  84 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())
  85 {
  86         validate();
  87 }
  88
  89 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited{b, i1, i2}, symtree(symm)
  90 {
  91         validate();
  92 }
  93
  94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited{b, i1, i2, i3}, symtree(symm)
  95 {
  96         validate();
  97 }
  98
  99 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)
 100 {
 101         validate();
 102 }
 103
 104 indexed::indexed(const ex & b, const exvector & v) : inherited{b}, symtree(not_symmetric())
 105 {
 106         seq.insert(seq.end(), v.begin(), v.end());
 107         validate();
 108 }
 109
 110 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited{b}, symtree(symm)
 111 {
 112         seq.insert(seq.end(), v.begin(), v.end());
 113         validate();
 114 }
 115
 116 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 117 {
 118 }
 119
 120 indexed::indexed(const symmetry & symm, const exvector & v) : inherited(v), symtree(symm)
 121 {
 122 }
 123
 124 indexed::indexed(const symmetry & symm, exvector && v) : inherited(std::move(v)), symtree(symm)
 125 {
 126 }
 127
 128 //////////
 129 // archiving
 130 //////////
 131
 132 void indexed::read_archive(const archive_node &n, lst &sym_lst)
 133 {
 134         inherited::read_archive(n, sym_lst);
 135         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 136                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 137                 unsigned symm = 0;
 138                 n.find_unsigned("symmetry", symm);
 139                 switch (symm) {
 140                         case 1:
 141                                 symtree = sy_symm();
 142                                 break;
 143                         case 2:
 144                                 symtree = sy_anti();
 145                                 break;
 146                         default:
 147                                 symtree = not_symmetric();
 148                                 break;
 149                 }
 150                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 151         }
 152 }
 153 GINAC_BIND_UNARCHIVER(indexed);
 154
 155 void indexed::archive(archive_node &n) const
 156 {
 157         inherited::archive(n);
 158         n.add_ex("symmetry", symtree);
 159 }
 160
 161 //////////
 162 // functions overriding virtual functions from base classes
 163 //////////
 164
 165 void indexed::printindices(const print_context & c, unsigned level) const
 166 {
 167         if (seq.size() > 1) {
 168
 169                 auto it = seq.begin() + 1, itend = seq.end();
 170
 171                 if (is_a<print_latex>(c)) {
 172
 173                         // TeX output: group by variance
 174                         bool first = true;
 175                         bool covariant = true;
 176
 177                         while (it != itend) {
 178                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 179                                 if (first || cur_covariant != covariant) { // Variance changed
 180                                         // The empty {} prevents indices from ending up on top of each other
 181                                         if (!first)
 182                                                 c.s << "}{}";
 183                                         covariant = cur_covariant;
 184                                         if (covariant)
 185                                                 c.s << "_{";
 186                                         else
 187                                                 c.s << "^{";
 188                                 }
 189                                 it->print(c, level);
 190                                 c.s << " ";
 191                                 first = false;
 192                                 it++;
 193                         }
 194                         c.s << "}";
 195
 196                 } else {
 197
 198                         // Ordinary output
 199                         while (it != itend) {
 200                                 it->print(c, level);
 201                                 it++;
 202                         }
 203                 }
 204         }
 205 }
 206
 207 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
 208 {
 209         if (precedence() <= level)
 210                 c.s << openbrace << '(';
 211         c.s << openbrace;
 212         seq[0].print(c, precedence());
 213         c.s << closebrace;
 214         printindices(c, level);
 215         if (precedence() <= level)
 216                 c.s << ')' << closebrace;
 217 }
 218
 219 void indexed::do_print(const print_context & c, unsigned level) const
 220 {
 221         print_indexed(c, "", "", level);
 222 }
 223
 224 void indexed::do_print_latex(const print_latex & c, unsigned level) const
 225 {
 226         print_indexed(c, "{", "}", level);
 227 }
 228
 229 void indexed::do_print_tree(const print_tree & c, unsigned level) const
 230 {
 231         c.s << std::string(level, ' ') << class_name() << " @" << this
 232             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 233             << ", " << seq.size()-1 << " indices"
 234             << ", symmetry=" << symtree << std::endl;
 235         seq[0].print(c, level + c.delta_indent);
 236         printindices(c, level + c.delta_indent);
 237 }
 238
 239 bool indexed::info(unsigned inf) const
 240 {
 241         if (inf == info_flags::indexed) return true;
 242         if (inf == info_flags::has_indices) return seq.size() > 1;
 243         return inherited::info(inf);
 244 }
 245
 246 bool indexed::all_index_values_are(unsigned inf) const
 247 {
 248         // No indices? Then no property can be fulfilled
 249         if (seq.size() < 2)
 250                 return false;
 251
 252         // Check all indices
 253         return find_if(seq.begin() + 1, seq.end(),
 254                        [inf](const ex & e) { return !(ex_to<idx>(e).get_value().info(inf)); }) == seq.end();
 255 }
 256
 257 int indexed::compare_same_type(const basic & other) const
 258 {
 259         GINAC_ASSERT(is_a<indexed>(other));
 260         return inherited::compare_same_type(other);
 261 }
 262
 263 ex indexed::eval() const
 264 {
 265         const ex &base = seq[0];
 266
 267         // If the base object is 0, the whole object is 0
 268         if (base.is_zero())
 269                 return _ex0;
 270
 271         // If the base object is a product, pull out the numeric factor
 272         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 273                 exvector v(seq);
 274                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 275                 v[0] = seq[0] / f;
 276                 return f * thiscontainer(v);
 277         }
 278
 279         if((typeid(*this) == typeid(indexed)) && seq.size()==1)
 280                 return base;
 281
 282         // Canonicalize indices according to the symmetry properties
 283         if (seq.size() > 2) {
 284                 exvector v = seq;
 285                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 286                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 287                 if (sig != std::numeric_limits<int>::max()) {
 288                         // Something has changed while sorting indices, more evaluations later
 289                         if (sig == 0)
 290                                 return _ex0;
 291                         return ex(sig) * thiscontainer(v);
 292                 }
 293         }
 294
 295         // Let the class of the base object perform additional evaluations
 296         return ex_to<basic>(base).eval_indexed(*this);
 297 }
 298
 299 ex indexed::real_part() const
 300 {
 301         if(op(0).info(info_flags::real))
 302                 return *this;
 303         return real_part_function(*this).hold();
 304 }
 305
 306 ex indexed::imag_part() const
 307 {
 308         if(op(0).info(info_flags::real))
 309                 return 0;
 310         return imag_part_function(*this).hold();
 311 }
 312
 313 ex indexed::thiscontainer(const exvector & v) const
 314 {
 315         return indexed(ex_to<symmetry>(symtree), v);
 316 }
 317
 318 ex indexed::thiscontainer(exvector && v) const
 319 {
 320         return indexed(ex_to<symmetry>(symtree), std::move(v));
 321 }
 322
 323 unsigned indexed::return_type() const
 324 {
 325         if(is_a<matrix>(op(0)))
 326                 return return_types::commutative;
 327         else
 328                 return op(0).return_type();
 329 }
 330
 331 ex indexed::expand(unsigned options) const
 332 {
 333         GINAC_ASSERT(seq.size() > 0);
 334
 335         if (options & expand_options::expand_indexed) {
 336                 ex newbase = seq[0].expand(options);
 337                 if (is_exactly_a<add>(newbase)) {
 338                         ex sum = _ex0;
 339                         for (size_t i=0; i<newbase.nops(); i++) {
 340                                 exvector s = seq;
 341                                 s[0] = newbase.op(i);
 342                                 sum += thiscontainer(s).expand(options);
 343                         }
 344                         return sum;
 345                 }
 346                 if (!are_ex_trivially_equal(newbase, seq[0])) {
 347                         exvector s = seq;
 348                         s[0] = newbase;
 349                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
 350                 }
 351         }
 352         return inherited::expand(options);
 353 }
 354
 355 //////////
 356 // virtual functions which can be overridden by derived classes
 357 //////////
 358
 359 // none
 360
 361 //////////
 362 // non-virtual functions in this class
 363 //////////
 364
 365 /** Check whether all indices are of class idx and validate the symmetry
 366  *  tree. This function is used internally to make sure that all constructed
 367  *  indexed objects really carry indices and not some other classes. */
 368 void indexed::validate() const
 369 {
 370         GINAC_ASSERT(seq.size() > 0);
 371         auto it = seq.begin() + 1, itend = seq.end();
 372         while (it != itend) {
 373                 if (!is_a<idx>(*it))
 374                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 375                 it++;
 376         }
 377
 378         if (!symtree.is_zero()) {
 379                 if (!is_exactly_a<symmetry>(symtree))
 380                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 381                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 382         }
 383 }
 384
 385 /** Implementation of ex::diff() for an indexed object always returns 0.
 386  *
 387  *  @see ex::diff */
 388 ex indexed::derivative(const symbol & s) const
 389 {
 390         return _ex0;
 391 }
 392
 393 //////////
 394 // global functions
 395 //////////
 396
 397 struct idx_is_equal_ignore_dim {
 398         bool operator() (const ex &lh, const ex &rh) const
 399         {
 400                 if (lh.is_equal(rh))
 401                         return true;
 402                 else
 403                         try {
 404                                 // Replacing the dimension might cause an error (e.g. with
 405                                 // index classes that only work in a fixed number of dimensions)
 406                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 407                         } catch (...) {
 408                                 return false;
 409                         }
 410         }
 411 };
 412
 413 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 414 static bool indices_consistent(const exvector & v1, const exvector & v2)
 415 {
 416         // Number of indices must be the same
 417         if (v1.size() != v2.size())
 418                 return false;
 419
 420         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 421 }
 422
 423 exvector indexed::get_indices() const
 424 {
 425         GINAC_ASSERT(seq.size() >= 1);
 426         return exvector(seq.begin() + 1, seq.end());
 427 }
 428
 429 exvector indexed::get_dummy_indices() const
 430 {
 431         exvector free_indices, dummy_indices;
 432         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 433         return dummy_indices;
 434 }
 435
 436 exvector indexed::get_dummy_indices(const indexed & other) const
 437 {
 438         exvector indices = get_free_indices();
 439         exvector other_indices = other.get_free_indices();
 440         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 441         exvector dummy_indices;
 442         find_dummy_indices(indices, dummy_indices);
 443         return dummy_indices;
 444 }
 445
 446 bool indexed::has_dummy_index_for(const ex & i) const
 447 {
 448         auto it = seq.begin() + 1, itend = seq.end();
 449         while (it != itend) {
 450                 if (is_dummy_pair(*it, i))
 451                         return true;
 452                 it++;
 453         }
 454         return false;
 455 }
 456
 457 exvector indexed::get_free_indices() const
 458 {
 459         exvector free_indices, dummy_indices;
 460         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 461         return free_indices;
 462 }
 463
 464 exvector add::get_free_indices() const
 465 {
 466         exvector free_indices;
 467         for (size_t i=0; i<nops(); i++) {
 468                 if (i == 0)
 469                         free_indices = op(i).get_free_indices();
 470                 else {
 471                         exvector free_indices_of_term = op(i).get_free_indices();
 472                         if (!indices_consistent(free_indices, free_indices_of_term))
 473                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 474                 }
 475         }
 476         return free_indices;
 477 }
 478
 479 exvector mul::get_free_indices() const
 480 {
 481         // Concatenate free indices of all factors
 482         exvector un;
 483         for (size_t i=0; i<nops(); i++) {
 484                 exvector free_indices_of_factor = op(i).get_free_indices();
 485                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 486         }
 487
 488         // And remove the dummy indices
 489         exvector free_indices, dummy_indices;
 490         find_free_and_dummy(un, free_indices, dummy_indices);
 491         return free_indices;
 492 }
 493
 494 exvector ncmul::get_free_indices() const
 495 {
 496         // Concatenate free indices of all factors
 497         exvector un;
 498         for (size_t i=0; i<nops(); i++) {
 499                 exvector free_indices_of_factor = op(i).get_free_indices();
 500                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 501         }
 502
 503         // And remove the dummy indices
 504         exvector free_indices, dummy_indices;
 505         find_free_and_dummy(un, free_indices, dummy_indices);
 506         return free_indices;
 507 }
 508
 509 struct is_summation_idx {
 510         bool operator()(const ex & e)
 511         {
 512                 return is_dummy_pair(e, e);
 513         }
 514 };
 515
 516 exvector integral::get_free_indices() const
 517 {
 518         if (a.get_free_indices().size() || b.get_free_indices().size())
 519                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
 520         return f.get_free_indices();
 521 }
 522
 523 template<class T> size_t number_of_type(const exvector&v)
 524 {
 525         size_t number = 0;
 526         for (auto & it : v)
 527                 if (is_exactly_a<T>(it))
 528                         ++number;
 529         return number;
 530 }
 531
 532 /** Rename dummy indices in an expression.
 533  *
 534  *  @param e Expression to work on
 535  *  @param local_dummy_indices The set of dummy indices that appear in the
 536  *    expression "e"
 537  *  @param global_dummy_indices The set of dummy indices that have appeared
 538  *    before and which we would like to use in "e", too. This gets updated
 539  *    by the function */
 540 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 541 {
 542         size_t global_size = number_of_type<T>(global_dummy_indices),
 543                local_size = number_of_type<T>(local_dummy_indices);
 544
 545         // Any local dummy indices at all?
 546         if (local_size == 0)
 547                 return e;
 548
 549         if (global_size < local_size) {
 550
 551                 // More local indices than we encountered before, add the new ones
 552                 // to the global set
 553                 size_t old_global_size = global_size;
 554                 int remaining = local_size - global_size;
 555                 auto it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 556                 while (it != itend && remaining > 0) {
 557                         if (is_exactly_a<T>(*it) &&
 558                             find_if(global_dummy_indices.begin(), global_dummy_indices.end(),
 559                                     [it](const ex &lh) { return idx_is_equal_ignore_dim()(lh, *it); }) == global_dummy_indices.end()) {
 560                                 global_dummy_indices.push_back(*it);
 561                                 global_size++;
 562                                 remaining--;
 563                         }
 564                         it++;
 565                 }
 566
 567                 // If this is the first set of local indices, do nothing
 568                 if (old_global_size == 0)
 569                         return e;
 570         }
 571         GINAC_ASSERT(local_size <= global_size);
 572
 573         // Construct vectors of index symbols
 574         exvector local_syms, global_syms;
 575         local_syms.reserve(local_size);
 576         global_syms.reserve(local_size);
 577         for (size_t i=0; local_syms.size()!=local_size; i++)
 578                 if(is_exactly_a<T>(local_dummy_indices[i]))
 579                         local_syms.push_back(local_dummy_indices[i].op(0));
 580         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 581         for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
 582                 if(is_exactly_a<T>(global_dummy_indices[i]))
 583                         global_syms.push_back(global_dummy_indices[i].op(0));
 584         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 585
 586         // Remove common indices
 587         exvector local_uniq, global_uniq;
 588         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 589         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 590
 591         // Replace remaining non-common local index symbols by global ones
 592         if (local_uniq.empty())
 593                 return e;
 594         else {
 595                 while (global_uniq.size() > local_uniq.size())
 596                         global_uniq.pop_back();
 597                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 598         }
 599 }
 600
 601 /** Given a set of indices, extract those of class varidx. */
 602 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 603 {
 604         exvector::const_iterator it1, itend;
 605         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 606                 if (is_exactly_a<varidx>(*it1))
 607                         variant_indices.push_back(*it1);
 608         }
 609 }
 610
 611 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 612  *  variance.
 613  *
 614  *  @param e Object to work on
 615  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 616  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 617  *  @return true if 'e' was changed */
 618 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 619 {
 620         bool something_changed = false;
 621
 622         // Find dummy symbols that occur twice in the same indexed object.
 623         exvector local_var_dummies;
 624         local_var_dummies.reserve(e.nops()/2);
 625         for (size_t i=1; i<e.nops(); ++i) {
 626                 if (!is_a<varidx>(e.op(i)))
 627                         continue;
 628                 for (size_t j=i+1; j<e.nops(); ++j) {
 629                         if (is_dummy_pair(e.op(i), e.op(j))) {
 630                                 local_var_dummies.push_back(e.op(i));
 631                                 for (auto k = variant_dummy_indices.begin(); k!=variant_dummy_indices.end(); ++k) {
 632                                         if (e.op(i).op(0) == k->op(0)) {
 633                                                 variant_dummy_indices.erase(k);
 634                                                 break;
 635                                         }
 636                                 }
 637                                 break;
 638                         }
 639                 }
 640         }
 641
 642         // In the case where a dummy symbol occurs twice in the same indexed object
 643         // we try all possibilities of raising/lowering and keep the least one in
 644         // the sense of ex_is_less.
 645         ex optimal_e = e;
 646         size_t numpossibs = 1 << local_var_dummies.size();
 647         for (size_t i=0; i<numpossibs; ++i) {
 648                 ex try_e = e;
 649                 for (size_t j=0; j<local_var_dummies.size(); ++j) {
 650                         exmap m;
 651                         if (1<<j & i) {
 652                                 ex curr_idx = local_var_dummies[j];
 653                                 ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
 654                                 m[curr_idx] = curr_toggle;
 655                                 m[curr_toggle] = curr_idx;
 656                         }
 657                         try_e = e.subs(m, subs_options::no_pattern);
 658                 }
 659                 if(ex_is_less()(try_e, optimal_e))
 660                 {       optimal_e = try_e;
 661                         something_changed = true;
 662                 }
 663         }
 664         e = optimal_e;
 665
 666         if (!is_a<indexed>(e))
 667                 return true;
 668
 669         exvector seq = ex_to<indexed>(e).seq;
 670
 671         // If a dummy index is encountered for the first time in the
 672         // product, pull it up, otherwise, pull it down
 673         for (auto it2 = seq.begin()+1, it2end = seq.end(); it2 != it2end; ++it2) {
 674                 if (!is_exactly_a<varidx>(*it2))
 675                         continue;
 676
 677                 exvector::iterator vit, vitend;
 678                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 679                         if (it2->op(0).is_equal(vit->op(0))) {
 680                                 if (ex_to<varidx>(*it2).is_covariant()) {
 681                                         /*
 682                                          * N.B. we don't want to use
 683                                          *
 684                                          *  e = e.subs(lst{
 685                                          *  *it2 == ex_to<varidx>(*it2).toggle_variance(),
 686                                          *  ex_to<varidx>(*it2).toggle_variance() == *it2
 687                                          *  }, subs_options::no_pattern);
 688                                          *
 689                                          * since this can trigger non-trivial repositioning of indices,
 690                                          * e.g. due to non-trivial symmetry properties of e, thus
 691                                          * invalidating iterators
 692                                          */
 693                                         *it2 = ex_to<varidx>(*it2).toggle_variance();
 694                                         something_changed = true;
 695                                 }
 696                                 moved_indices.push_back(*vit);
 697                                 variant_dummy_indices.erase(vit);
 698                                 goto next_index;
 699                         }
 700                 }
 701
 702                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 703                         if (it2->op(0).is_equal(vit->op(0))) {
 704                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 705                                         *it2 = ex_to<varidx>(*it2).toggle_variance();
 706                                         something_changed = true;
 707                                 }
 708                                 goto next_index;
 709                         }
 710                 }
 711
 712 next_index: ;
 713         }
 714
 715         if (something_changed)
 716                 e = ex_to<indexed>(e).thiscontainer(seq);
 717
 718         return something_changed;
 719 }
 720
 721 /* Ordering that only compares the base expressions of indexed objects. */
 722 struct ex_base_is_less {
 723         bool operator() (const ex &lh, const ex &rh) const
 724         {
 725                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 726         }
 727 };
 728
 729 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
 730  * It returns an exvector of factors from the supplied product */
 731 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
 732 {
 733         // Remember whether the product was commutative or noncommutative
 734         // (because we chop it into factors and need to reassemble later)
 735         non_commutative = is_exactly_a<ncmul>(e);
 736
 737         // Collect factors in an exvector, store squares twice
 738         v.reserve(e.nops() * 2);
 739
 740         if (is_exactly_a<power>(e)) {
 741                 // We only get called for simple squares, split a^2 -> a*a
 742                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 743                 v.push_back(e.op(0));
 744                 v.push_back(e.op(0));
 745         } else {
 746                 for (size_t i=0; i<e.nops(); i++) {
 747                         ex f = e.op(i);
 748                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 749                                 v.push_back(f.op(0));
 750                                 v.push_back(f.op(0));
 751                         } else if (is_exactly_a<ncmul>(f)) {
 752                                 // Noncommutative factor found, split it as well
 753                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 754                                 for (size_t j=0; j<f.nops(); j++)
 755                                         v.push_back(f.op(j));
 756                         } else
 757                                 v.push_back(f);
 758                 }
 759         }
 760 }
 761
 762 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
 763 {       exvector dummy_syms;
 764         dummy_syms.reserve(r.nops());
 765         for (auto & it : local_dummy_indices)
 766                         if(is_exactly_a<T>(it))
 767                                 dummy_syms.push_back(it.op(0));
 768         if(dummy_syms.size() < 2)
 769                 return r;
 770         ex q=symmetrize(r, dummy_syms);
 771         return q;
 772 }
 773
 774 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
 775 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
 776
 777 /** Simplify product of indexed expressions (commutative, noncommutative and
 778  *  simple squares), return list of free indices. */
 779 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 780 {
 781         // Collect factors in an exvector
 782         exvector v;
 783
 784         // Remember whether the product was commutative or noncommutative
 785         // (because we chop it into factors and need to reassemble later)
 786         bool non_commutative;
 787         product_to_exvector(e, v, non_commutative);
 788
 789         // Perform contractions
 790         bool something_changed = false;
 791         bool has_nonsymmetric = false;
 792         GINAC_ASSERT(v.size() > 1);
 793         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 794         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 795
 796 try_again:
 797                 if (!is_a<indexed>(*it1))
 798                         continue;
 799
 800                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 801                 bool first_nonsymmetric = ex_to<symmetry>(ex_to<indexed>(*it1).get_symmetry()).has_nonsymmetric();
 802
 803                 // Indexed factor found, get free indices and look for contraction
 804                 // candidates
 805                 exvector free1, dummy1;
 806                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 807
 808                 exvector::iterator it2;
 809                 for (it2 = it1 + 1; it2 != itend; it2++) {
 810
 811                         if (!is_a<indexed>(*it2))
 812                                 continue;
 813
 814                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 815
 816                         // Find free indices of second factor and merge them with free
 817                         // indices of first factor
 818                         exvector un;
 819                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 820                         un.insert(un.end(), free1.begin(), free1.end());
 821
 822                         // Check whether the two factors share dummy indices
 823                         exvector free, dummy;
 824                         find_free_and_dummy(un, free, dummy);
 825                         size_t num_dummies = dummy.size();
 826                         if (num_dummies == 0)
 827                                 continue;
 828
 829                         // At least one dummy index, is it a defined scalar product?
 830                         bool contracted = false;
 831                         if (free.empty() && it1->nops()==2 && it2->nops()==2) {
 832
 833                                 ex dim = minimal_dim(
 834                                         ex_to<idx>(it1->op(1)).get_dim(),
 835                                         ex_to<idx>(it2->op(1)).get_dim()
 836                                 );
 837
 838                                 // User-defined scalar product?
 839                                 if (sp.is_defined(*it1, *it2, dim)) {
 840
 841                                         // Yes, substitute it
 842                                         *it1 = sp.evaluate(*it1, *it2, dim);
 843                                         *it2 = _ex1;
 844                                         goto contraction_done;
 845                                 }
 846                         }
 847
 848                         // Try to contract the first one with the second one
 849                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 850                         if (!contracted) {
 851
 852                                 // That didn't work; maybe the second object knows how to
 853                                 // contract itself with the first one
 854                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 855                         }
 856                         if (contracted) {
 857 contraction_done:
 858                                 if (first_noncommutative || second_noncommutative
 859                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 860                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 861                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 862
 863                                         // One of the factors became a sum or product:
 864                                         // re-expand expression and run again
 865                                         // Non-commutative products are always re-expanded to give
 866                                         // eval_ncmul() the chance to re-order and canonicalize
 867                                         // the product
 868                                         bool is_a_product = (is_exactly_a<mul>(*it1) || is_exactly_a<ncmul>(*it1)) &&
 869                                                             (is_exactly_a<mul>(*it2) || is_exactly_a<ncmul>(*it2));
 870                                         ex r = (non_commutative ? ex(ncmul(std::move(v))) : ex(mul(std::move(v))));
 871
 872                                         // If new expression is a product we can call this function again,
 873                                         // otherwise we need to pass argument to simplify_indexed() to be expanded
 874                                         if (is_a_product)
 875                                                 return simplify_indexed_product(r, free_indices, dummy_indices, sp);
 876                                         else
 877                                                 return simplify_indexed(r, free_indices, dummy_indices, sp);
 878                                 }
 879
 880                                 // Both objects may have new indices now or they might
 881                                 // even not be indexed objects any more, so we have to
 882                                 // start over
 883                                 something_changed = true;
 884                                 goto try_again;
 885                         }
 886                         else if (!has_nonsymmetric &&
 887                                         (first_nonsymmetric ||
 888                                          ex_to<symmetry>(ex_to<indexed>(*it2).get_symmetry()).has_nonsymmetric())) {
 889                                 has_nonsymmetric = true;
 890                         }
 891                 }
 892         }
 893
 894         // Find free indices (concatenate them all and call find_free_and_dummy())
 895         // and all dummy indices that appear
 896         exvector un, individual_dummy_indices;
 897         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 898                 exvector free_indices_of_factor;
 899                 if (is_a<indexed>(*it1)) {
 900                         exvector dummy_indices_of_factor;
 901                         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);
 902                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 903                 } else
 904                         free_indices_of_factor = it1->get_free_indices();
 905                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 906         }
 907         exvector local_dummy_indices;
 908         find_free_and_dummy(un, free_indices, local_dummy_indices);
 909         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 910
 911         // Filter out the dummy indices with variance
 912         exvector variant_dummy_indices;
 913         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 914
 915         // Any indices with variance present at all?
 916         if (!variant_dummy_indices.empty()) {
 917
 918                 // Yes, bring the product into a canonical order that only depends on
 919                 // the base expressions of indexed objects
 920                 if (!non_commutative)
 921                         std::sort(v.begin(), v.end(), ex_base_is_less());
 922
 923                 exvector moved_indices;
 924
 925                 // Iterate over all indexed objects in the product
 926                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 927                         if (!is_a<indexed>(*it1))
 928                                 continue;
 929
 930                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 931                                 something_changed = true;
 932                 }
 933         }
 934
 935         ex r;
 936         if (something_changed)
 937                 r = non_commutative ? ex(ncmul(std::move(v))) : ex(mul(std::move(v)));
 938         else
 939                 r = e;
 940
 941         // The result should be symmetric with respect to exchange of dummy
 942         // indices, so if the symmetrization vanishes, the whole expression is
 943         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 944         if (has_nonsymmetric) {
 945                 ex q = idx_symmetrization<idx>(r, local_dummy_indices);
 946                 if (q.is_zero()) {
 947                         free_indices.clear();
 948                         return _ex0;
 949                 }
 950                 q = idx_symmetrization<varidx>(q, local_dummy_indices);
 951                 if (q.is_zero()) {
 952                         free_indices.clear();
 953                         return _ex0;
 954                 }
 955                 q = idx_symmetrization<spinidx>(q, local_dummy_indices);
 956                 if (q.is_zero()) {
 957                         free_indices.clear();
 958                         return _ex0;
 959                 }
 960         }
 961
 962         // Dummy index renaming
 963         r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
 964         r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
 965         r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
 966
 967         // Product of indexed object with a scalar?
 968         if (is_exactly_a<mul>(r) && r.nops() == 2
 969          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 970                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 971         else
 972                 return r;
 973 }
 974
 975 /** This structure stores the original and symmetrized versions of terms
 976  *  obtained during the simplification of sums. */
 977 class terminfo {
 978 public:
 979         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 980
 981         ex orig; /**< original term */
 982         ex symm; /**< symmetrized term */
 983 };
 984
 985 class terminfo_is_less {
 986 public:
 987         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 988         {
 989                 return (ti1.symm.compare(ti2.symm) < 0);
 990         }
 991 };
 992
 993 /** This structure stores the individual symmetrized terms obtained during
 994  *  the simplification of sums. */
 995 class symminfo {
 996 public:
 997         symminfo() : num(0) {}
 998
 999         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
1000         {
1001                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
1002                         coeff = symmterm_.op(symmterm_.nops()-1);
1003                         symmterm = symmterm_ / coeff;
1004                 } else {
1005                         coeff = 1;
1006                         symmterm = symmterm_;
1007                 }
1008         }
1009
1010         ex symmterm;  /**< symmetrized term */
1011         ex coeff;     /**< coefficient of symmetrized term */
1012         ex orig;      /**< original term */
1013         size_t num; /**< how many symmetrized terms resulted from the original term */
1014 };
1015
1016 class symminfo_is_less_by_symmterm {
1017 public:
1018         bool operator() (const symminfo & si1, const symminfo & si2) const
1019         {
1020                 return (si1.symmterm.compare(si2.symmterm) < 0);
1021         }
1022 };
1023
1024 class symminfo_is_less_by_orig {
1025 public:
1026         bool operator() (const symminfo & si1, const symminfo & si2) const
1027         {
1028                 return (si1.orig.compare(si2.orig) < 0);
1029         }
1030 };
1031
1032 bool hasindex(const ex &x, const ex &sym)
1033 {
1034         if(is_a<idx>(x) && x.op(0)==sym)
1035                 return true;
1036         else
1037                 for(size_t i=0; i<x.nops(); ++i)
1038                         if(hasindex(x.op(i), sym))
1039                                 return true;
1040         return false;
1041 }
1042
1043 /** Simplify indexed expression, return list of free indices. */
1044 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1045 {
1046         // Expand the expression
1047         ex e_expanded = e.expand();
1048
1049         // Simplification of single indexed object: just find the free indices
1050         // and perform dummy index renaming/repositioning
1051         if (is_a<indexed>(e_expanded)) {
1052
1053                 // Find the dummy indices
1054                 const indexed &i = ex_to<indexed>(e_expanded);
1055                 exvector local_dummy_indices;
1056                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1057
1058                 // Filter out the dummy indices with variance
1059                 exvector variant_dummy_indices;
1060                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1061
1062                 // Any indices with variance present at all?
1063                 if (!variant_dummy_indices.empty()) {
1064
1065                         // Yes, reposition them
1066                         exvector moved_indices;
1067                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1068                 }
1069
1070                 // Rename the dummy indices
1071                 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1072                 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1073                 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1074                 return e_expanded;
1075         }
1076
1077         // Simplification of sum = sum of simplifications, check consistency of
1078         // free indices in each term
1079         if (is_exactly_a<add>(e_expanded)) {
1080                 bool first = true;
1081                 ex sum;
1082                 free_indices.clear();
1083
1084                 for (size_t i=0; i<e_expanded.nops(); i++) {
1085                         exvector free_indices_of_term;
1086                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1087                         if (!term.is_zero()) {
1088                                 if (first) {
1089                                         free_indices = free_indices_of_term;
1090                                         sum = term;
1091                                         first = false;
1092                                 } else {
1093                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
1094                                                 std::ostringstream s;
1095                                                 s << "simplify_indexed: inconsistent indices in sum: ";
1096                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1097                                                 throw (std::runtime_error(s.str()));
1098                                         }
1099                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1100                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1101                                         else
1102                                                 sum += term;
1103                                 }
1104                         }
1105                 }
1106
1107                 // If the sum turns out to be zero, we are finished
1108                 if (sum.is_zero()) {
1109                         free_indices.clear();
1110                         return sum;
1111                 }
1112
1113                 // More than one term and more than one dummy index?
1114                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1115                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1116                         return sum;
1117
1118                 // Chop the sum into terms and symmetrize each one over the dummy
1119                 // indices
1120                 std::vector<terminfo> terms;
1121                 for (size_t i=0; i<sum.nops(); i++) {
1122                         const ex & term = sum.op(i);
1123                         exvector dummy_indices_of_term;
1124                         dummy_indices_of_term.reserve(dummy_indices.size());
1125                         for (auto & i : dummy_indices)
1126                                 if (hasindex(term,i.op(0)))
1127                                         dummy_indices_of_term.push_back(i);
1128                         ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1129                         term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1130                         term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1131                         if (term_symm.is_zero())
1132                                 continue;
1133                         terms.push_back(terminfo(term, term_symm));
1134                 }
1135
1136                 // Sort by symmetrized terms
1137                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1138
1139                 // Combine equal symmetrized terms
1140                 std::vector<terminfo> terms_pass2;
1141                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1142                         size_t num = 1;
1143                         auto j = i + 1;
1144                         while (j != terms.end() && j->symm == i->symm) {
1145                                 num++;
1146                                 j++;
1147                         }
1148                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1149                         i = j;
1150                 }
1151
1152                 // If there is only one term left, we are finished
1153                 if (terms_pass2.size() == 1)
1154                         return terms_pass2[0].orig;
1155
1156                 // Chop the symmetrized terms into subterms
1157                 std::vector<symminfo> sy;
1158                 for (auto & i : terms_pass2) {
1159                         if (is_exactly_a<add>(i.symm)) {
1160                                 size_t num = i.symm.nops();
1161                                 for (size_t j=0; j<num; j++)
1162                                         sy.push_back(symminfo(i.symm.op(j), i.orig, num));
1163                         } else
1164                                 sy.push_back(symminfo(i.symm, i.orig, 1));
1165                 }
1166
1167                 // Sort by symmetrized subterms
1168                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1169
1170                 // Combine equal symmetrized subterms
1171                 std::vector<symminfo> sy_pass2;
1172                 exvector result;
1173                 for (auto i=sy.begin(); i!=sy.end(); ) {
1174
1175                         // Combine equal terms
1176                         auto j = i + 1;
1177                         if (j != sy.end() && j->symmterm == i->symmterm) {
1178
1179                                 // More than one term, collect the coefficients
1180                                 ex coeff = i->coeff;
1181                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1182                                         coeff += j->coeff;
1183                                         j++;
1184                                 }
1185
1186                                 // Add combined term to result
1187                                 if (!coeff.is_zero())
1188                                         result.push_back(coeff * i->symmterm);
1189
1190                         } else {
1191
1192                                 // Single term, store for second pass
1193                                 sy_pass2.push_back(*i);
1194                         }
1195
1196                         i = j;
1197                 }
1198
1199                 // Were there any remaining terms that didn't get combined?
1200                 if (sy_pass2.size() > 0) {
1201
1202                         // Yes, sort by their original terms
1203                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1204
1205                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1206
1207                                 // How many symmetrized terms of this original term are left?
1208                                 size_t num = 1;
1209                                 auto j = i + 1;
1210                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1211                                         num++;
1212                                         j++;
1213                                 }
1214
1215                                 if (num == i->num) {
1216
1217                                         // All terms left, then add the original term to the result
1218                                         result.push_back(i->orig);
1219
1220                                 } else {
1221
1222                                         // Some terms were combined with others, add up the remaining symmetrized terms
1223                                         std::vector<symminfo>::const_iterator k;
1224                                         for (k=i; k!=j; k++)
1225                                                 result.push_back(k->coeff * k->symmterm);
1226                                 }
1227
1228                                 i = j;
1229                         }
1230                 }
1231
1232                 // Add all resulting terms
1233                 ex sum_symm = dynallocate<add>(result);
1234                 if (sum_symm.is_zero())
1235                         free_indices.clear();
1236                 return sum_symm;
1237         }
1238
1239         // Simplification of products
1240         if (is_exactly_a<mul>(e_expanded)
1241          || is_exactly_a<ncmul>(e_expanded)
1242          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1243                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1244
1245         // Cannot do anything
1246         free_indices.clear();
1247         return e_expanded;
1248 }
1249
1250 /** Simplify/canonicalize expression containing indexed objects. This
1251  *  performs contraction of dummy indices where possible and checks whether
1252  *  the free indices in sums are consistent.
1253  *
1254  *  @param options Simplification options (currently unused)
1255  *  @return simplified expression */
1256 ex ex::simplify_indexed(unsigned options) const
1257 {
1258         exvector free_indices, dummy_indices;
1259         scalar_products sp;
1260         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1261 }
1262
1263 /** Simplify/canonicalize expression containing indexed objects. This
1264  *  performs contraction of dummy indices where possible, checks whether
1265  *  the free indices in sums are consistent, and automatically replaces
1266  *  scalar products by known values if desired.
1267  *
1268  *  @param sp Scalar products to be replaced automatically
1269  *  @param options Simplification options (currently unused)
1270  *  @return simplified expression */
1271 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1272 {
1273         exvector free_indices, dummy_indices;
1274         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1275 }
1276
1277 /** Symmetrize expression over its free indices. */
1278 ex ex::symmetrize() const
1279 {
1280         return GiNaC::symmetrize(*this, get_free_indices());
1281 }
1282
1283 /** Antisymmetrize expression over its free indices. */
1284 ex ex::antisymmetrize() const
1285 {
1286         return GiNaC::antisymmetrize(*this, get_free_indices());
1287 }
1288
1289 /** Symmetrize expression by cyclic permutation over its free indices. */
1290 ex ex::symmetrize_cyclic() const
1291 {
1292         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1293 }
1294
1295 //////////
1296 // helper classes
1297 //////////
1298
1299 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1300 {
1301         // If indexed, extract base objects
1302         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1303         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1304
1305         // Enforce canonical order in pair
1306         if (s1.compare(s2) > 0) {
1307                 v1 = s2;
1308                 v2 = s1;
1309         } else {
1310                 v1 = s1;
1311                 v2 = s2;
1312         }
1313 }
1314
1315 bool spmapkey::operator==(const spmapkey &other) const
1316 {
1317         if (!v1.is_equal(other.v1))
1318                 return false;
1319         if (!v2.is_equal(other.v2))
1320                 return false;
1321         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1322                 return true;
1323         else
1324                 return dim.is_equal(other.dim);
1325 }
1326
1327 bool spmapkey::operator<(const spmapkey &other) const
1328 {
1329         int cmp = v1.compare(other.v1);
1330         if (cmp)
1331                 return cmp < 0;
1332         cmp = v2.compare(other.v2);
1333         if (cmp)
1334                 return cmp < 0;
1335
1336         // Objects are equal, now check dimensions
1337         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1338                 return false;
1339         else
1340                 return dim.compare(other.dim) < 0;
1341 }
1342
1343 void spmapkey::debugprint() const
1344 {
1345         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1346 }
1347
1348 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1349 {
1350         spm[spmapkey(v1, v2)] = sp;
1351 }
1352
1353 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1354 {
1355         spm[spmapkey(v1, v2, dim)] = sp;
1356 }
1357
1358 void scalar_products::add_vectors(const lst & l, const ex & dim)
1359 {
1360         // Add all possible pairs of products
1361         for (auto & it1 : l)
1362                 for (auto & it2 : l)
1363                         add(it1, it2, it1 * it2);
1364 }
1365
1366 void scalar_products::clear()
1367 {
1368         spm.clear();
1369 }
1370
1371 /** Check whether scalar product pair is defined. */
1372 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1373 {
1374         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1375 }
1376
1377 /** Return value of defined scalar product pair. */
1378 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1379 {
1380         return spm.find(spmapkey(v1, v2, dim))->second;
1381 }
1382
1383 void scalar_products::debugprint() const
1384 {
1385         std::cerr << "map size=" << spm.size() << std::endl;
1386         for (auto & it : spm) {
1387                 const spmapkey & k = it.first;
1388                 std::cerr << "item key=";
1389                 k.debugprint();
1390                 std::cerr << ", value=" << it.second << std::endl;
1391         }
1392 }
1393
1394 exvector get_all_dummy_indices_safely(const ex & e)
1395 {
1396         if (is_a<indexed>(e))
1397                 return ex_to<indexed>(e).get_dummy_indices();
1398         else if (is_a<power>(e) && e.op(1)==2) {
1399                 return e.op(0).get_free_indices();
1400         }
1401         else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1402                 exvector dummies;
1403                 exvector free_indices;
1404                 for (std::size_t i = 0; i < e.nops(); ++i) {
1405                         exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1406                         dummies.insert(dummies.end(), dummies_of_factor.begin(),
1407                                 dummies_of_factor.end());
1408                         exvector free_of_factor = e.op(i).get_free_indices();
1409                         free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1410                                 free_of_factor.end());
1411                 }
1412                 exvector free_out, dummy_out;
1413                 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1414                         dummy_out);
1415                 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1416                 return dummies;
1417         }
1418         else if(is_a<add>(e)) {
1419                 exvector result;
1420                 for(std::size_t i = 0; i < e.nops(); ++i) {
1421                         exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1422                         sort(dummies_of_term.begin(), dummies_of_term.end());
1423                         exvector new_vec;
1424                         set_union(result.begin(), result.end(), dummies_of_term.begin(),
1425                                 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1426                                 ex_is_less());
1427                         result.swap(new_vec);
1428                 }
1429                 return result;
1430         }
1431         return exvector();
1432 }
1433
1434 /** Returns all dummy indices from the exvector */
1435 exvector get_all_dummy_indices(const ex & e)
1436 {
1437         exvector p;
1438         bool nc;
1439         product_to_exvector(e, p, nc);
1440         auto ip = p.begin(), ipend = p.end();
1441         exvector v, v1;
1442         while (ip != ipend) {
1443                 if (is_a<indexed>(*ip)) {
1444                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1445                         v.insert(v.end(), v1.begin(), v1.end());
1446                         auto ip1 = ip + 1;
1447                         while (ip1 != ipend) {
1448                                 if (is_a<indexed>(*ip1)) {
1449                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1450                                         v.insert(v.end(), v1.begin(), v1.end());
1451                                 }
1452                                 ++ip1;
1453                         }
1454                 }
1455                 ++ip;
1456         }
1457         return v;
1458 }
1459
1460 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1461 {
1462         exvector common_indices;
1463         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1464         if (common_indices.empty()) {
1465                 return lst{lst{}, lst{}};
1466         } else {
1467                 exvector new_indices, old_indices;
1468                 old_indices.reserve(2*common_indices.size());
1469                 new_indices.reserve(2*common_indices.size());
1470                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1471                 while (ip != ipend) {
1472                         ex newsym = dynallocate<symbol>();
1473                         ex newidx;
1474                         if(is_exactly_a<spinidx>(*ip))
1475                                 newidx = dynallocate<spinidx>(newsym, ex_to<spinidx>(*ip).get_dim(),
1476                                                               ex_to<spinidx>(*ip).is_covariant(),
1477                                                               ex_to<spinidx>(*ip).is_dotted());
1478                         else if (is_exactly_a<varidx>(*ip))
1479                                 newidx = dynallocate<varidx>(newsym, ex_to<varidx>(*ip).get_dim(),
1480                                                              ex_to<varidx>(*ip).is_covariant());
1481                         else
1482                                 newidx = dynallocate<idx>(newsym, ex_to<idx>(*ip).get_dim());
1483                         old_indices.push_back(*ip);
1484                         new_indices.push_back(newidx);
1485                         if(is_a<varidx>(*ip)) {
1486                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1487                                 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1488                         }
1489                         ++ip;
1490                 }
1491                 return lst{lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end())};
1492         }
1493 }
1494
1495 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1496 {
1497         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1498         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);
1499 }
1500
1501 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1502 {
1503         exvector va = get_all_dummy_indices_safely(a);
1504         if (va.size() > 0) {
1505                 exvector vb = get_all_dummy_indices_safely(b);
1506                 if (vb.size() > 0) {
1507                         sort(va.begin(), va.end(), ex_is_less());
1508                         sort(vb.begin(), vb.end(), ex_is_less());
1509                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1510                         if (indices_subs.op(0).nops() > 0)
1511                                 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);
1512                 }
1513         }
1514         return b;
1515 }
1516
1517 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1518 {
1519         if (va.size() > 0) {
1520                 exvector vb = get_all_dummy_indices_safely(b);
1521                 if (vb.size() > 0) {
1522                         sort(vb.begin(), vb.end(), ex_is_less());
1523                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1524                         if (indices_subs.op(0).nops() > 0) {
1525                                 if (modify_va) {
1526                                         for (auto & i : ex_to<lst>(indices_subs.op(1)))
1527                                                 va.push_back(i);
1528                                         exvector uncommon_indices;
1529                                         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());
1530                                         for (auto & ip : uncommon_indices)
1531                                                 va.push_back(ip);
1532                                         sort(va.begin(), va.end(), ex_is_less());
1533                                 }
1534                                 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);
1535                         }
1536                 }
1537         }
1538         return b;
1539 }
1540
1541 ex expand_dummy_sum(const ex & e, bool subs_idx)
1542 {
1543         ex e_expanded = e.expand();
1544         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1545         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1546                 return e_expanded.map(fcn);
1547         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1548                 exvector v;
1549                 if (is_a<indexed>(e_expanded))
1550                         v = ex_to<indexed>(e_expanded).get_dummy_indices();
1551                 else
1552                         v = get_all_dummy_indices(e_expanded);
1553                 ex result = e_expanded;
1554                 for (const auto & nu : v) {
1555                         if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1556                                 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1557                                 ex en = 0;
1558                                 for (int i=0; i < idim; i++) {
1559                                         if (subs_idx && is_a<varidx>(nu)) {
1560                                                 ex other = ex_to<varidx>(nu).toggle_variance();
1561                                                 en += result.subs(lst{
1562                                                         nu == idx(i, idim),
1563                                                         other == idx(i, idim)
1564                                                 });
1565                                         } else {
1566                                                 en += result.subs( nu.op(0) == i );
1567                                         }
1568                                 }
1569                                 result = en;
1570                         }
1571                 }
1572                 return result;
1573         } else {
1574                 return e;
1575         }
1576 }
1577
1578 } // namespace GiNaC