Add support for Texinfo-5.0.
[ginac.git] / ginac / indexed.cpp
1 /** @file indexed.cpp
2  *
3  *  Implementation of GiNaC's indexed expressions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2011 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, bool discardable) : inherited(v, discardable), symtree(symm)
121 {
122 }
123
124 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), 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                 exvector::const_iterator 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 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
247         bool operator() (const ex & e, unsigned inf) const {
248                 return !(ex_to<idx>(e).get_value().info(inf));
249         }
250 };
251
252 bool indexed::all_index_values_are(unsigned inf) const
253 {
254         // No indices? Then no property can be fulfilled
255         if (seq.size() < 2)
256                 return false;
257
258         // Check all indices
259         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
260 }
261
262 int indexed::compare_same_type(const basic & other) const
263 {
264         GINAC_ASSERT(is_a<indexed>(other));
265         return inherited::compare_same_type(other);
266 }
267
268 ex indexed::eval(int level) const
269 {
270         // First evaluate children, then we will end up here again
271         if (level > 1)
272                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
273
274         const ex &base = seq[0];
275
276         // If the base object is 0, the whole object is 0
277         if (base.is_zero())
278                 return _ex0;
279
280         // If the base object is a product, pull out the numeric factor
281         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
282                 exvector v(seq);
283                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
284                 v[0] = seq[0] / f;
285                 return f * thiscontainer(v);
286         }
287
288         if((typeid(*this) == typeid(indexed)) && seq.size()==1)
289                 return base;
290
291         // Canonicalize indices according to the symmetry properties
292         if (seq.size() > 2) {
293                 exvector v = seq;
294                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
295                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
296                 if (sig != std::numeric_limits<int>::max()) {
297                         // Something has changed while sorting indices, more evaluations later
298                         if (sig == 0)
299                                 return _ex0;
300                         return ex(sig) * thiscontainer(v);
301                 }
302         }
303
304         // Let the class of the base object perform additional evaluations
305         return ex_to<basic>(base).eval_indexed(*this);
306 }
307
308 ex indexed::real_part() const
309 {
310         if(op(0).info(info_flags::real))
311                 return *this;
312         return real_part_function(*this).hold();
313 }
314
315 ex indexed::imag_part() const
316 {
317         if(op(0).info(info_flags::real))
318                 return 0;
319         return imag_part_function(*this).hold();
320 }
321
322 ex indexed::thiscontainer(const exvector & v) const
323 {
324         return indexed(ex_to<symmetry>(symtree), v);
325 }
326
327 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
328 {
329         return indexed(ex_to<symmetry>(symtree), vp);
330 }
331
332 unsigned indexed::return_type() const
333 {
334         if(is_a<matrix>(op(0)))
335                 return return_types::commutative;
336         else
337                 return op(0).return_type();
338 }
339
340 ex indexed::expand(unsigned options) const
341 {
342         GINAC_ASSERT(seq.size() > 0);
343
344         if (options & expand_options::expand_indexed) {
345                 ex newbase = seq[0].expand(options);
346                 if (is_exactly_a<add>(newbase)) {
347                         ex sum = _ex0;
348                         for (size_t i=0; i<newbase.nops(); i++) {
349                                 exvector s = seq;
350                                 s[0] = newbase.op(i);
351                                 sum += thiscontainer(s).expand(options);
352                         }
353                         return sum;
354                 }
355                 if (!are_ex_trivially_equal(newbase, seq[0])) {
356                         exvector s = seq;
357                         s[0] = newbase;
358                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
359                 }
360         }
361         return inherited::expand(options);
362 }
363
364 //////////
365 // virtual functions which can be overridden by derived classes
366 //////////
367
368 // none
369
370 //////////
371 // non-virtual functions in this class
372 //////////
373
374 /** Check whether all indices are of class idx and validate the symmetry
375  *  tree. This function is used internally to make sure that all constructed
376  *  indexed objects really carry indices and not some other classes. */
377 void indexed::validate() const
378 {
379         GINAC_ASSERT(seq.size() > 0);
380         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
381         while (it != itend) {
382                 if (!is_a<idx>(*it))
383                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
384                 it++;
385         }
386
387         if (!symtree.is_zero()) {
388                 if (!is_exactly_a<symmetry>(symtree))
389                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
390                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
391         }
392 }
393
394 /** Implementation of ex::diff() for an indexed object always returns 0.
395  *
396  *  @see ex::diff */
397 ex indexed::derivative(const symbol & s) const
398 {
399         return _ex0;
400 }
401
402 //////////
403 // global functions
404 //////////
405
406 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
407         bool operator() (const ex &lh, const ex &rh) const
408         {
409                 if (lh.is_equal(rh))
410                         return true;
411                 else
412                         try {
413                                 // Replacing the dimension might cause an error (e.g. with
414                                 // index classes that only work in a fixed number of dimensions)
415                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
416                         } catch (...) {
417                                 return false;
418                         }
419         }
420 };
421
422 /** Check whether two sorted index vectors are consistent (i.e. equal). */
423 static bool indices_consistent(const exvector & v1, const exvector & v2)
424 {
425         // Number of indices must be the same
426         if (v1.size() != v2.size())
427                 return false;
428
429         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
430 }
431
432 exvector indexed::get_indices() const
433 {
434         GINAC_ASSERT(seq.size() >= 1);
435         return exvector(seq.begin() + 1, seq.end());
436 }
437
438 exvector indexed::get_dummy_indices() const
439 {
440         exvector free_indices, dummy_indices;
441         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
442         return dummy_indices;
443 }
444
445 exvector indexed::get_dummy_indices(const indexed & other) const
446 {
447         exvector indices = get_free_indices();
448         exvector other_indices = other.get_free_indices();
449         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
450         exvector dummy_indices;
451         find_dummy_indices(indices, dummy_indices);
452         return dummy_indices;
453 }
454
455 bool indexed::has_dummy_index_for(const ex & i) const
456 {
457         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
458         while (it != itend) {
459                 if (is_dummy_pair(*it, i))
460                         return true;
461                 it++;
462         }
463         return false;
464 }
465
466 exvector indexed::get_free_indices() const
467 {
468         exvector free_indices, dummy_indices;
469         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
470         return free_indices;
471 }
472
473 exvector add::get_free_indices() const
474 {
475         exvector free_indices;
476         for (size_t i=0; i<nops(); i++) {
477                 if (i == 0)
478                         free_indices = op(i).get_free_indices();
479                 else {
480                         exvector free_indices_of_term = op(i).get_free_indices();
481                         if (!indices_consistent(free_indices, free_indices_of_term))
482                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
483                 }
484         }
485         return free_indices;
486 }
487
488 exvector mul::get_free_indices() const
489 {
490         // Concatenate free indices of all factors
491         exvector un;
492         for (size_t i=0; i<nops(); i++) {
493                 exvector free_indices_of_factor = op(i).get_free_indices();
494                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
495         }
496
497         // And remove the dummy indices
498         exvector free_indices, dummy_indices;
499         find_free_and_dummy(un, free_indices, dummy_indices);
500         return free_indices;
501 }
502
503 exvector ncmul::get_free_indices() const
504 {
505         // Concatenate free indices of all factors
506         exvector un;
507         for (size_t i=0; i<nops(); i++) {
508                 exvector free_indices_of_factor = op(i).get_free_indices();
509                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
510         }
511
512         // And remove the dummy indices
513         exvector free_indices, dummy_indices;
514         find_free_and_dummy(un, free_indices, dummy_indices);
515         return free_indices;
516 }
517
518 struct is_summation_idx : public std::unary_function<ex, bool> {
519         bool operator()(const ex & e)
520         {
521                 return is_dummy_pair(e, e);
522         }
523 };
524
525 exvector integral::get_free_indices() const
526 {
527         if (a.get_free_indices().size() || b.get_free_indices().size())
528                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
529         return f.get_free_indices();
530 }
531
532 template<class T> size_t number_of_type(const exvector&v)
533 {
534         size_t number = 0;
535         for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
536                 if(is_exactly_a<T>(*i))
537                         ++number;
538         return number;
539 }
540
541 /** Rename dummy indices in an expression.
542  *
543  *  @param e Expression to work on
544  *  @param local_dummy_indices The set of dummy indices that appear in the
545  *    expression "e"
546  *  @param global_dummy_indices The set of dummy indices that have appeared
547  *    before and which we would like to use in "e", too. This gets updated
548  *    by the function */
549 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
550 {
551         size_t global_size = number_of_type<T>(global_dummy_indices),
552                local_size = number_of_type<T>(local_dummy_indices);
553
554         // Any local dummy indices at all?
555         if (local_size == 0)
556                 return e;
557
558         if (global_size < local_size) {
559
560                 // More local indices than we encountered before, add the new ones
561                 // to the global set
562                 size_t old_global_size = global_size;
563                 int remaining = local_size - global_size;
564                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
565                 while (it != itend && remaining > 0) {
566                         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()) {
567                                 global_dummy_indices.push_back(*it);
568                                 global_size++;
569                                 remaining--;
570                         }
571                         it++;
572                 }
573
574                 // If this is the first set of local indices, do nothing
575                 if (old_global_size == 0)
576                         return e;
577         }
578         GINAC_ASSERT(local_size <= global_size);
579
580         // Construct vectors of index symbols
581         exvector local_syms, global_syms;
582         local_syms.reserve(local_size);
583         global_syms.reserve(local_size);
584         for (size_t i=0; local_syms.size()!=local_size; i++)
585                 if(is_exactly_a<T>(local_dummy_indices[i]))
586                         local_syms.push_back(local_dummy_indices[i].op(0));
587         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
588         for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
589                 if(is_exactly_a<T>(global_dummy_indices[i]))
590                         global_syms.push_back(global_dummy_indices[i].op(0));
591         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
592
593         // Remove common indices
594         exvector local_uniq, global_uniq;
595         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
596         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
597
598         // Replace remaining non-common local index symbols by global ones
599         if (local_uniq.empty())
600                 return e;
601         else {
602                 while (global_uniq.size() > local_uniq.size())
603                         global_uniq.pop_back();
604                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
605         }
606 }
607
608 /** Given a set of indices, extract those of class varidx. */
609 static void find_variant_indices(const exvector & v, exvector & variant_indices)
610 {
611         exvector::const_iterator it1, itend;
612         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
613                 if (is_exactly_a<varidx>(*it1))
614                         variant_indices.push_back(*it1);
615         }
616 }
617
618 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
619  *  variance.
620  *
621  *  @param e Object to work on
622  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
623  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
624  *  @return true if 'e' was changed */
625 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
626 {
627         bool something_changed = false;
628
629         // Find dummy symbols that occur twice in the same indexed object.
630         exvector local_var_dummies;
631         local_var_dummies.reserve(e.nops()/2);
632         for (size_t i=1; i<e.nops(); ++i) {
633                 if (!is_a<varidx>(e.op(i)))
634                         continue;
635                 for (size_t j=i+1; j<e.nops(); ++j) {
636                         if (is_dummy_pair(e.op(i), e.op(j))) {
637                                 local_var_dummies.push_back(e.op(i));
638                                 for (exvector::iterator k = variant_dummy_indices.begin();
639                                                 k!=variant_dummy_indices.end(); ++k) {
640                                         if (e.op(i).op(0) == k->op(0)) {
641                                                 variant_dummy_indices.erase(k);
642                                                 break;
643                                         }
644                                 }
645                                 break;
646                         }
647                 }
648         }
649
650         // In the case where a dummy symbol occurs twice in the same indexed object
651         // we try all posibilities of raising/lowering and keep the least one in
652         // the sense of ex_is_less.
653         ex optimal_e = e;
654         size_t numpossibs = 1 << local_var_dummies.size();
655         for (size_t i=0; i<numpossibs; ++i) {
656                 ex try_e = e;
657                 for (size_t j=0; j<local_var_dummies.size(); ++j) {
658                         exmap m;
659                         if (1<<j & i) {
660                                 ex curr_idx = local_var_dummies[j];
661                                 ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
662                                 m[curr_idx] = curr_toggle;
663                                 m[curr_toggle] = curr_idx;
664                         }
665                         try_e = e.subs(m, subs_options::no_pattern);
666                 }
667                 if(ex_is_less()(try_e, optimal_e))
668                 {       optimal_e = try_e;
669                         something_changed = true;
670                 }
671         }
672         e = optimal_e;
673
674         if (!is_a<indexed>(e))
675                 return true;
676
677         exvector seq = ex_to<indexed>(e).seq;
678
679         // If a dummy index is encountered for the first time in the
680         // product, pull it up, otherwise, pull it down
681         for (exvector::iterator it2 = seq.begin()+1, it2end = seq.end();
682                         it2 != it2end; ++it2) {
683                 if (!is_exactly_a<varidx>(*it2))
684                         continue;
685
686                 exvector::iterator vit, vitend;
687                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
688                         if (it2->op(0).is_equal(vit->op(0))) {
689                                 if (ex_to<varidx>(*it2).is_covariant()) {
690                                         /*
691                                          * N.B. we don't want to use
692                                          *
693                                          *  e = e.subs(lst(
694                                          *  *it2 == ex_to<varidx>(*it2).toggle_variance(),
695                                          *  ex_to<varidx>(*it2).toggle_variance() == *it2
696                                          *  ), subs_options::no_pattern);
697                                          *
698                                          * since this can trigger non-trivial repositioning of indices,
699                                          * e.g. due to non-trivial symmetry properties of e, thus
700                                          * invalidating iterators
701                                          */
702                                         *it2 = ex_to<varidx>(*it2).toggle_variance();
703                                         something_changed = true;
704                                 }
705                                 moved_indices.push_back(*vit);
706                                 variant_dummy_indices.erase(vit);
707                                 goto next_index;
708                         }
709                 }
710
711                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
712                         if (it2->op(0).is_equal(vit->op(0))) {
713                                 if (ex_to<varidx>(*it2).is_contravariant()) {
714                                         *it2 = ex_to<varidx>(*it2).toggle_variance();
715                                         something_changed = true;
716                                 }
717                                 goto next_index;
718                         }
719                 }
720
721 next_index: ;
722         }
723
724         if (something_changed)
725                 e = ex_to<indexed>(e).thiscontainer(seq);
726
727         return something_changed;
728 }
729
730 /* Ordering that only compares the base expressions of indexed objects. */
731 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
732         bool operator() (const ex &lh, const ex &rh) const
733         {
734                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
735         }
736 };
737
738 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum() 
739  * It returns an exvector of factors from the supplied product */
740 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
741 {
742         // Remember whether the product was commutative or noncommutative
743         // (because we chop it into factors and need to reassemble later)
744         non_commutative = is_exactly_a<ncmul>(e);
745
746         // Collect factors in an exvector, store squares twice
747         v.reserve(e.nops() * 2);
748
749         if (is_exactly_a<power>(e)) {
750                 // We only get called for simple squares, split a^2 -> a*a
751                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
752                 v.push_back(e.op(0));
753                 v.push_back(e.op(0));
754         } else {
755                 for (size_t i=0; i<e.nops(); i++) {
756                         ex f = e.op(i);
757                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
758                                 v.push_back(f.op(0));
759                                 v.push_back(f.op(0));
760                         } else if (is_exactly_a<ncmul>(f)) {
761                                 // Noncommutative factor found, split it as well
762                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
763                                 for (size_t j=0; j<f.nops(); j++)
764                                         v.push_back(f.op(j));
765                         } else
766                                 v.push_back(f);
767                 }
768         }
769 }
770
771 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
772 {       exvector dummy_syms;
773         dummy_syms.reserve(r.nops());
774         for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
775                         if(is_exactly_a<T>(*it))
776                                 dummy_syms.push_back(it->op(0));
777         if(dummy_syms.size() < 2)
778                 return r;
779         ex q=symmetrize(r, dummy_syms);
780         return q;
781 }
782
783 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
784 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
785
786 /** Simplify product of indexed expressions (commutative, noncommutative and
787  *  simple squares), return list of free indices. */
788 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
789 {
790         // Collect factors in an exvector
791         exvector v;
792
793         // Remember whether the product was commutative or noncommutative
794         // (because we chop it into factors and need to reassemble later)
795         bool non_commutative;
796         product_to_exvector(e, v, non_commutative);
797
798         // Perform contractions
799         bool something_changed = false;
800         bool has_nonsymmetric = false;
801         GINAC_ASSERT(v.size() > 1);
802         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
803         for (it1 = v.begin(); it1 != next_to_last; it1++) {
804
805 try_again:
806                 if (!is_a<indexed>(*it1))
807                         continue;
808
809                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
810                 bool first_nonsymmetric = ex_to<symmetry>(ex_to<indexed>(*it1).get_symmetry()).has_nonsymmetric();
811
812                 // Indexed factor found, get free indices and look for contraction
813                 // candidates
814                 exvector free1, dummy1;
815                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
816
817                 exvector::iterator it2;
818                 for (it2 = it1 + 1; it2 != itend; it2++) {
819
820                         if (!is_a<indexed>(*it2))
821                                 continue;
822
823                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
824
825                         // Find free indices of second factor and merge them with free
826                         // indices of first factor
827                         exvector un;
828                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
829                         un.insert(un.end(), free1.begin(), free1.end());
830
831                         // Check whether the two factors share dummy indices
832                         exvector free, dummy;
833                         find_free_and_dummy(un, free, dummy);
834                         size_t num_dummies = dummy.size();
835                         if (num_dummies == 0)
836                                 continue;
837
838                         // At least one dummy index, is it a defined scalar product?
839                         bool contracted = false;
840                         if (free.empty() && it1->nops()==2 && it2->nops()==2) {
841
842                                 ex dim = minimal_dim(
843                                         ex_to<idx>(it1->op(1)).get_dim(),
844                                         ex_to<idx>(it2->op(1)).get_dim()
845                                 );
846
847                                 // User-defined scalar product?
848                                 if (sp.is_defined(*it1, *it2, dim)) {
849
850                                         // Yes, substitute it
851                                         *it1 = sp.evaluate(*it1, *it2, dim);
852                                         *it2 = _ex1;
853                                         goto contraction_done;
854                                 }
855                         }
856
857                         // Try to contract the first one with the second one
858                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
859                         if (!contracted) {
860
861                                 // That didn't work; maybe the second object knows how to
862                                 // contract itself with the first one
863                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
864                         }
865                         if (contracted) {
866 contraction_done:
867                                 if (first_noncommutative || second_noncommutative
868                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
869                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
870                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
871
872                                         // One of the factors became a sum or product:
873                                         // re-expand expression and run again
874                                         // Non-commutative products are always re-expanded to give
875                                         // eval_ncmul() the chance to re-order and canonicalize
876                                         // the product
877                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
878                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
879                                 }
880
881                                 // Both objects may have new indices now or they might
882                                 // even not be indexed objects any more, so we have to
883                                 // start over
884                                 something_changed = true;
885                                 goto try_again;
886                         }
887                         else if (!has_nonsymmetric &&
888                                         (first_nonsymmetric ||
889                                          ex_to<symmetry>(ex_to<indexed>(*it2).get_symmetry()).has_nonsymmetric())) {
890                                 has_nonsymmetric = true;
891                         }
892                 }
893         }
894
895         // Find free indices (concatenate them all and call find_free_and_dummy())
896         // and all dummy indices that appear
897         exvector un, individual_dummy_indices;
898         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
899                 exvector free_indices_of_factor;
900                 if (is_a<indexed>(*it1)) {
901                         exvector dummy_indices_of_factor;
902                         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);
903                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
904                 } else
905                         free_indices_of_factor = it1->get_free_indices();
906                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
907         }
908         exvector local_dummy_indices;
909         find_free_and_dummy(un, free_indices, local_dummy_indices);
910         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
911
912         // Filter out the dummy indices with variance
913         exvector variant_dummy_indices;
914         find_variant_indices(local_dummy_indices, variant_dummy_indices);
915
916         // Any indices with variance present at all?
917         if (!variant_dummy_indices.empty()) {
918
919                 // Yes, bring the product into a canonical order that only depends on
920                 // the base expressions of indexed objects
921                 if (!non_commutative)
922                         std::sort(v.begin(), v.end(), ex_base_is_less());
923
924                 exvector moved_indices;
925
926                 // Iterate over all indexed objects in the product
927                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
928                         if (!is_a<indexed>(*it1))
929                                 continue;
930
931                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
932                                 something_changed = true;
933                 }
934         }
935
936         ex r;
937         if (something_changed)
938                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
939         else
940                 r = e;
941
942         // The result should be symmetric with respect to exchange of dummy
943         // indices, so if the symmetrization vanishes, the whole expression is
944         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
945         if (has_nonsymmetric) {
946                 ex q = idx_symmetrization<idx>(r, local_dummy_indices);
947                 if (q.is_zero()) {
948                         free_indices.clear();
949                         return _ex0;
950                 }
951                 q = idx_symmetrization<varidx>(q, local_dummy_indices);
952                 if (q.is_zero()) {
953                         free_indices.clear();
954                         return _ex0;
955                 }
956                 q = idx_symmetrization<spinidx>(q, local_dummy_indices);
957                 if (q.is_zero()) {
958                         free_indices.clear();
959                         return _ex0;
960                 }
961         }
962
963         // Dummy index renaming
964         r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
965         r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
966         r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
967
968         // Product of indexed object with a scalar?
969         if (is_exactly_a<mul>(r) && r.nops() == 2
970          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
971                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
972         else
973                 return r;
974 }
975
976 /** This structure stores the original and symmetrized versions of terms
977  *  obtained during the simplification of sums. */
978 class terminfo {
979 public:
980         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
981
982         ex orig; /**< original term */
983         ex symm; /**< symmtrized term */
984 };
985
986 class terminfo_is_less {
987 public:
988         bool operator() (const terminfo & ti1, const terminfo & ti2) const
989         {
990                 return (ti1.symm.compare(ti2.symm) < 0);
991         }
992 };
993
994 /** This structure stores the individual symmetrized terms obtained during
995  *  the simplification of sums. */
996 class symminfo {
997 public:
998         symminfo() : num(0) {}
999
1000         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
1001         {
1002                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
1003                         coeff = symmterm_.op(symmterm_.nops()-1);
1004                         symmterm = symmterm_ / coeff;
1005                 } else {
1006                         coeff = 1;
1007                         symmterm = symmterm_;
1008                 }
1009         }
1010
1011         ex symmterm;  /**< symmetrized term */
1012         ex coeff;     /**< coefficient of symmetrized term */
1013         ex orig;      /**< original term */
1014         size_t num; /**< how many symmetrized terms resulted from the original term */
1015 };
1016
1017 class symminfo_is_less_by_symmterm {
1018 public:
1019         bool operator() (const symminfo & si1, const symminfo & si2) const
1020         {
1021                 return (si1.symmterm.compare(si2.symmterm) < 0);
1022         }
1023 };
1024
1025 class symminfo_is_less_by_orig {
1026 public:
1027         bool operator() (const symminfo & si1, const symminfo & si2) const
1028         {
1029                 return (si1.orig.compare(si2.orig) < 0);
1030         }
1031 };
1032
1033 bool hasindex(const ex &x, const ex &sym)
1034 {       
1035         if(is_a<idx>(x) && x.op(0)==sym)
1036                 return true;
1037         else
1038                 for(size_t i=0; i<x.nops(); ++i)
1039                         if(hasindex(x.op(i), sym))
1040                                 return true;
1041         return false;
1042 }
1043
1044 /** Simplify indexed expression, return list of free indices. */
1045 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1046 {
1047         // Expand the expression
1048         ex e_expanded = e.expand();
1049
1050         // Simplification of single indexed object: just find the free indices
1051         // and perform dummy index renaming/repositioning
1052         if (is_a<indexed>(e_expanded)) {
1053
1054                 // Find the dummy indices
1055                 const indexed &i = ex_to<indexed>(e_expanded);
1056                 exvector local_dummy_indices;
1057                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1058
1059                 // Filter out the dummy indices with variance
1060                 exvector variant_dummy_indices;
1061                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1062
1063                 // Any indices with variance present at all?
1064                 if (!variant_dummy_indices.empty()) {
1065
1066                         // Yes, reposition them
1067                         exvector moved_indices;
1068                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1069                 }
1070
1071                 // Rename the dummy indices
1072                 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1073                 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1074                 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1075                 return e_expanded;
1076         }
1077
1078         // Simplification of sum = sum of simplifications, check consistency of
1079         // free indices in each term
1080         if (is_exactly_a<add>(e_expanded)) {
1081                 bool first = true;
1082                 ex sum;
1083                 free_indices.clear();
1084
1085                 for (size_t i=0; i<e_expanded.nops(); i++) {
1086                         exvector free_indices_of_term;
1087                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1088                         if (!term.is_zero()) {
1089                                 if (first) {
1090                                         free_indices = free_indices_of_term;
1091                                         sum = term;
1092                                         first = false;
1093                                 } else {
1094                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
1095                                                 std::ostringstream s;
1096                                                 s << "simplify_indexed: inconsistent indices in sum: ";
1097                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1098                                                 throw (std::runtime_error(s.str()));
1099                                         }
1100                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1101                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1102                                         else
1103                                                 sum += term;
1104                                 }
1105                         }
1106                 }
1107
1108                 // If the sum turns out to be zero, we are finished
1109                 if (sum.is_zero()) {
1110                         free_indices.clear();
1111                         return sum;
1112                 }
1113
1114                 // More than one term and more than one dummy index?
1115                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1116                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1117                         return sum;
1118
1119                 // Chop the sum into terms and symmetrize each one over the dummy
1120                 // indices
1121                 std::vector<terminfo> terms;
1122                 for (size_t i=0; i<sum.nops(); i++) {
1123                         const ex & term = sum.op(i);
1124                         exvector dummy_indices_of_term;
1125                         dummy_indices_of_term.reserve(dummy_indices.size());
1126                         for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1127                                 if(hasindex(term,i->op(0)))
1128                                         dummy_indices_of_term.push_back(*i);
1129                         ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1130                         term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1131                         term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1132                         if (term_symm.is_zero())
1133                                 continue;
1134                         terms.push_back(terminfo(term, term_symm));
1135                 }
1136
1137                 // Sort by symmetrized terms
1138                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1139
1140                 // Combine equal symmetrized terms
1141                 std::vector<terminfo> terms_pass2;
1142                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1143                         size_t num = 1;
1144                         std::vector<terminfo>::const_iterator j = i + 1;
1145                         while (j != terms.end() && j->symm == i->symm) {
1146                                 num++;
1147                                 j++;
1148                         }
1149                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1150                         i = j;
1151                 }
1152
1153                 // If there is only one term left, we are finished
1154                 if (terms_pass2.size() == 1)
1155                         return terms_pass2[0].orig;
1156
1157                 // Chop the symmetrized terms into subterms
1158                 std::vector<symminfo> sy;
1159                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1160                         if (is_exactly_a<add>(i->symm)) {
1161                                 size_t num = i->symm.nops();
1162                                 for (size_t j=0; j<num; j++)
1163                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1164                         } else
1165                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1166                 }
1167
1168                 // Sort by symmetrized subterms
1169                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1170
1171                 // Combine equal symmetrized subterms
1172                 std::vector<symminfo> sy_pass2;
1173                 exvector result;
1174                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1175
1176                         // Combine equal terms
1177                         std::vector<symminfo>::const_iterator j = i + 1;
1178                         if (j != sy.end() && j->symmterm == i->symmterm) {
1179
1180                                 // More than one term, collect the coefficients
1181                                 ex coeff = i->coeff;
1182                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1183                                         coeff += j->coeff;
1184                                         j++;
1185                                 }
1186
1187                                 // Add combined term to result
1188                                 if (!coeff.is_zero())
1189                                         result.push_back(coeff * i->symmterm);
1190
1191                         } else {
1192
1193                                 // Single term, store for second pass
1194                                 sy_pass2.push_back(*i);
1195                         }
1196
1197                         i = j;
1198                 }
1199
1200                 // Were there any remaining terms that didn't get combined?
1201                 if (sy_pass2.size() > 0) {
1202
1203                         // Yes, sort by their original terms
1204                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1205
1206                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1207
1208                                 // How many symmetrized terms of this original term are left?
1209                                 size_t num = 1;
1210                                 std::vector<symminfo>::const_iterator j = i + 1;
1211                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1212                                         num++;
1213                                         j++;
1214                                 }
1215
1216                                 if (num == i->num) {
1217
1218                                         // All terms left, then add the original term to the result
1219                                         result.push_back(i->orig);
1220
1221                                 } else {
1222
1223                                         // Some terms were combined with others, add up the remaining symmetrized terms
1224                                         std::vector<symminfo>::const_iterator k;
1225                                         for (k=i; k!=j; k++)
1226                                                 result.push_back(k->coeff * k->symmterm);
1227                                 }
1228
1229                                 i = j;
1230                         }
1231                 }
1232
1233                 // Add all resulting terms
1234                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1235                 if (sum_symm.is_zero())
1236                         free_indices.clear();
1237                 return sum_symm;
1238         }
1239
1240         // Simplification of products
1241         if (is_exactly_a<mul>(e_expanded)
1242          || is_exactly_a<ncmul>(e_expanded)
1243          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1244                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1245
1246         // Cannot do anything
1247         free_indices.clear();
1248         return e_expanded;
1249 }
1250
1251 /** Simplify/canonicalize expression containing indexed objects. This
1252  *  performs contraction of dummy indices where possible and checks whether
1253  *  the free indices in sums are consistent.
1254  *
1255  *  @param options Simplification options (currently unused)
1256  *  @return simplified expression */
1257 ex ex::simplify_indexed(unsigned options) const
1258 {
1259         exvector free_indices, dummy_indices;
1260         scalar_products sp;
1261         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1262 }
1263
1264 /** Simplify/canonicalize expression containing indexed objects. This
1265  *  performs contraction of dummy indices where possible, checks whether
1266  *  the free indices in sums are consistent, and automatically replaces
1267  *  scalar products by known values if desired.
1268  *
1269  *  @param sp Scalar products to be replaced automatically
1270  *  @param options Simplification options (currently unused)
1271  *  @return simplified expression */
1272 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1273 {
1274         exvector free_indices, dummy_indices;
1275         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1276 }
1277
1278 /** Symmetrize expression over its free indices. */
1279 ex ex::symmetrize() const
1280 {
1281         return GiNaC::symmetrize(*this, get_free_indices());
1282 }
1283
1284 /** Antisymmetrize expression over its free indices. */
1285 ex ex::antisymmetrize() const
1286 {
1287         return GiNaC::antisymmetrize(*this, get_free_indices());
1288 }
1289
1290 /** Symmetrize expression by cyclic permutation over its free indices. */
1291 ex ex::symmetrize_cyclic() const
1292 {
1293         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1294 }
1295
1296 //////////
1297 // helper classes
1298 //////////
1299
1300 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1301 {
1302         // If indexed, extract base objects
1303         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1304         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1305
1306         // Enforce canonical order in pair
1307         if (s1.compare(s2) > 0) {
1308                 v1 = s2;
1309                 v2 = s1;
1310         } else {
1311                 v1 = s1;
1312                 v2 = s2;
1313         }
1314 }
1315
1316 bool spmapkey::operator==(const spmapkey &other) const
1317 {
1318         if (!v1.is_equal(other.v1))
1319                 return false;
1320         if (!v2.is_equal(other.v2))
1321                 return false;
1322         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1323                 return true;
1324         else
1325                 return dim.is_equal(other.dim);
1326 }
1327
1328 bool spmapkey::operator<(const spmapkey &other) const
1329 {
1330         int cmp = v1.compare(other.v1);
1331         if (cmp)
1332                 return cmp < 0;
1333         cmp = v2.compare(other.v2);
1334         if (cmp)
1335                 return cmp < 0;
1336
1337         // Objects are equal, now check dimensions
1338         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1339                 return false;
1340         else
1341                 return dim.compare(other.dim) < 0;
1342 }
1343
1344 void spmapkey::debugprint() const
1345 {
1346         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1347 }
1348
1349 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1350 {
1351         spm[spmapkey(v1, v2)] = sp;
1352 }
1353
1354 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1355 {
1356         spm[spmapkey(v1, v2, dim)] = sp;
1357 }
1358
1359 void scalar_products::add_vectors(const lst & l, const ex & dim)
1360 {
1361         // Add all possible pairs of products
1362         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1363                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1364                         add(*it1, *it2, *it1 * *it2);
1365 }
1366
1367 void scalar_products::clear()
1368 {
1369         spm.clear();
1370 }
1371
1372 /** Check whether scalar product pair is defined. */
1373 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1374 {
1375         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1376 }
1377
1378 /** Return value of defined scalar product pair. */
1379 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1380 {
1381         return spm.find(spmapkey(v1, v2, dim))->second;
1382 }
1383
1384 void scalar_products::debugprint() const
1385 {
1386         std::cerr << "map size=" << spm.size() << std::endl;
1387         spmap::const_iterator i = spm.begin(), end = spm.end();
1388         while (i != end) {
1389                 const spmapkey & k = i->first;
1390                 std::cerr << "item key=";
1391                 k.debugprint();
1392                 std::cerr << ", value=" << i->second << std::endl;
1393                 ++i;
1394         }
1395 }
1396
1397 exvector get_all_dummy_indices_safely(const ex & e)
1398 {
1399         if (is_a<indexed>(e))
1400                 return ex_to<indexed>(e).get_dummy_indices();
1401         else if (is_a<power>(e) && e.op(1)==2) {
1402                 return e.op(0).get_free_indices();
1403         }       
1404         else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1405                 exvector dummies;
1406                 exvector free_indices;
1407                 for (std::size_t i = 0; i < e.nops(); ++i) {
1408                         exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1409                         dummies.insert(dummies.end(), dummies_of_factor.begin(),
1410                                 dummies_of_factor.end());
1411                         exvector free_of_factor = e.op(i).get_free_indices();
1412                         free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1413                                 free_of_factor.end());
1414                 }
1415                 exvector free_out, dummy_out;
1416                 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1417                         dummy_out);
1418                 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1419                 return dummies;
1420         }
1421         else if(is_a<add>(e)) {
1422                 exvector result;
1423                 for(std::size_t i = 0; i < e.nops(); ++i) {
1424                         exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1425                         sort(dummies_of_term.begin(), dummies_of_term.end());
1426                         exvector new_vec;
1427                         set_union(result.begin(), result.end(), dummies_of_term.begin(),
1428                                 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1429                                 ex_is_less());
1430                         result.swap(new_vec);
1431                 }
1432                 return result;
1433         }
1434         return exvector();
1435 }
1436
1437 /** Returns all dummy indices from the exvector */
1438 exvector get_all_dummy_indices(const ex & e)
1439 {
1440         exvector p;
1441         bool nc;
1442         product_to_exvector(e, p, nc);
1443         exvector::const_iterator ip = p.begin(), ipend = p.end();
1444         exvector v, v1;
1445         while (ip != ipend) {
1446                 if (is_a<indexed>(*ip)) {
1447                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1448                         v.insert(v.end(), v1.begin(), v1.end());
1449                         exvector::const_iterator ip1 = ip+1;
1450                         while (ip1 != ipend) {
1451                                 if (is_a<indexed>(*ip1)) {
1452                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1453                                         v.insert(v.end(), v1.begin(), v1.end());
1454                                 }
1455                                 ++ip1;
1456                         }
1457                 }
1458                 ++ip;
1459         }
1460         return v;
1461 }
1462
1463 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1464 {
1465         exvector common_indices;
1466         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1467         if (common_indices.empty()) {
1468                 return lst(lst(), lst());
1469         } else {
1470                 exvector new_indices, old_indices;
1471                 old_indices.reserve(2*common_indices.size());
1472                 new_indices.reserve(2*common_indices.size());
1473                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1474                 while (ip != ipend) {
1475                         ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1476                         ex newidx;
1477                         if(is_exactly_a<spinidx>(*ip))
1478                                 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1479                                                 ex_to<spinidx>(*ip).is_covariant(),
1480                                                 ex_to<spinidx>(*ip).is_dotted()))
1481                                         -> setflag(status_flags::dynallocated);
1482                         else if (is_exactly_a<varidx>(*ip))
1483                                 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1484                                                 ex_to<varidx>(*ip).is_covariant()))
1485                                         -> setflag(status_flags::dynallocated);
1486                         else
1487                                 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1488                                         -> setflag(status_flags::dynallocated);
1489                         old_indices.push_back(*ip);
1490                         new_indices.push_back(newidx);
1491                         if(is_a<varidx>(*ip)) {
1492                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1493                                 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1494                         }
1495                         ++ip;
1496                 }
1497                 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1498         }
1499 }
1500
1501 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1502 {
1503         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1504         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);
1505 }
1506
1507 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1508 {
1509         exvector va = get_all_dummy_indices_safely(a);
1510         if (va.size() > 0) {
1511                 exvector vb = get_all_dummy_indices_safely(b);
1512                 if (vb.size() > 0) {
1513                         sort(va.begin(), va.end(), ex_is_less());
1514                         sort(vb.begin(), vb.end(), ex_is_less());
1515                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1516                         if (indices_subs.op(0).nops() > 0)
1517                                 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);
1518                 }
1519         }
1520         return b;
1521 }
1522
1523 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1524 {
1525         if (va.size() > 0) {
1526                 exvector vb = get_all_dummy_indices_safely(b);
1527                 if (vb.size() > 0) {
1528                         sort(vb.begin(), vb.end(), ex_is_less());
1529                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1530                         if (indices_subs.op(0).nops() > 0) {
1531                                 if (modify_va) {
1532                                         for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1533                                                 va.push_back(*i);
1534                                         exvector uncommon_indices;
1535                                         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());
1536                                         exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1537                                         while (ip != ipend) {
1538                                                 va.push_back(*ip);
1539                                                 ++ip;
1540                                         }
1541                                         sort(va.begin(), va.end(), ex_is_less());
1542                                 }
1543                                 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);
1544                         }
1545                 }
1546         }
1547         return b;
1548 }
1549
1550 ex expand_dummy_sum(const ex & e, bool subs_idx)
1551 {
1552         ex e_expanded = e.expand();
1553         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1554         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1555                 return e_expanded.map(fcn);
1556         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1557                 exvector v;
1558                 if (is_a<indexed>(e_expanded))
1559                         v = ex_to<indexed>(e_expanded).get_dummy_indices();
1560                 else
1561                         v = get_all_dummy_indices(e_expanded);
1562                 ex result = e_expanded;
1563                 for(exvector::const_iterator it=v.begin(); it!=v.end(); ++it) {
1564                         ex nu = *it;
1565                         if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1566                                 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1567                                 ex en = 0;
1568                                 for (int i=0; i < idim; i++) {
1569                                         if (subs_idx && is_a<varidx>(nu)) {
1570                                                 ex other = ex_to<varidx>(nu).toggle_variance();
1571                                                 en += result.subs(lst(
1572                                                         nu == idx(i, idim),
1573                                                         other == idx(i, idim)
1574                                                 ));
1575                                         } else {
1576                                                 en += result.subs( nu.op(0) == i );
1577                                         }
1578                                 }
1579                                 result = en;
1580                         }
1581                 }
1582                 return result;
1583         } else {
1584                 return e;
1585         }
1586 }
1587
1588 } // namespace GiNaC