Bug in expand_dummy_sum is fixed.
[ginac.git] / ginac / indexed.cpp
1 /** @file indexed.cpp
2  *
3  *  Implementation of GiNaC's indexed expressions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
21  */
22
23 #include <iostream>
24 #include <sstream>
25 #include <stdexcept>
26
27 #include "indexed.h"
28 #include "idx.h"
29 #include "add.h"
30 #include "mul.h"
31 #include "ncmul.h"
32 #include "power.h"
33 #include "relational.h"
34 #include "symmetry.h"
35 #include "operators.h"
36 #include "lst.h"
37 #include "archive.h"
38 #include "symbol.h"
39 #include "utils.h"
40 #include "integral.h"
41 #include "matrix.h"
42 #include "inifcns.h"
43
44 namespace GiNaC {
45
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
47   print_func<print_context>(&indexed::do_print).
48   print_func<print_latex>(&indexed::do_print_latex).
49   print_func<print_tree>(&indexed::do_print_tree))
50
51 //////////
52 // default constructor
53 //////////
54
55 indexed::indexed() : symtree(not_symmetric())
56 {
57         tinfo_key = &indexed::tinfo_static;
58 }
59
60 //////////
61 // other constructors
62 //////////
63
64 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
65 {
66         tinfo_key = &indexed::tinfo_static;
67         validate();
68 }
69
70 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
71 {
72         tinfo_key = &indexed::tinfo_static;
73         validate();
74 }
75
76 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
77 {
78         tinfo_key = &indexed::tinfo_static;
79         validate();
80 }
81
82 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
83 {
84         tinfo_key = &indexed::tinfo_static;
85         validate();
86 }
87
88 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())
89 {
90         tinfo_key = &indexed::tinfo_static;
91         validate();
92 }
93
94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
95 {
96         tinfo_key = &indexed::tinfo_static;
97         validate();
98 }
99
100 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
101 {
102         tinfo_key = &indexed::tinfo_static;
103         validate();
104 }
105
106 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)
107 {
108         tinfo_key = &indexed::tinfo_static;
109         validate();
110 }
111
112 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
113 {
114         seq.insert(seq.end(), v.begin(), v.end());
115         tinfo_key = &indexed::tinfo_static;
116         validate();
117 }
118
119 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
120 {
121         seq.insert(seq.end(), v.begin(), v.end());
122         tinfo_key = &indexed::tinfo_static;
123         validate();
124 }
125
126 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
127 {
128         tinfo_key = &indexed::tinfo_static;
129 }
130
131 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
132 {
133         tinfo_key = &indexed::tinfo_static;
134 }
135
136 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
137 {
138         tinfo_key = &indexed::tinfo_static;
139 }
140
141 //////////
142 // archiving
143 //////////
144
145 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
146 {
147         if (!n.find_ex("symmetry", symtree, sym_lst)) {
148                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
149                 unsigned symm = 0;
150                 n.find_unsigned("symmetry", symm);
151                 switch (symm) {
152                         case 1:
153                                 symtree = sy_symm();
154                                 break;
155                         case 2:
156                                 symtree = sy_anti();
157                                 break;
158                         default:
159                                 symtree = not_symmetric();
160                                 break;
161                 }
162                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
163         }
164 }
165
166 void indexed::archive(archive_node &n) const
167 {
168         inherited::archive(n);
169         n.add_ex("symmetry", symtree);
170 }
171
172 DEFAULT_UNARCHIVE(indexed)
173
174 //////////
175 // functions overriding virtual functions from base classes
176 //////////
177
178 void indexed::printindices(const print_context & c, unsigned level) const
179 {
180         if (seq.size() > 1) {
181
182                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
183
184                 if (is_a<print_latex>(c)) {
185
186                         // TeX output: group by variance
187                         bool first = true;
188                         bool covariant = true;
189
190                         while (it != itend) {
191                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
192                                 if (first || cur_covariant != covariant) { // Variance changed
193                                         // The empty {} prevents indices from ending up on top of each other
194                                         if (!first)
195                                                 c.s << "}{}";
196                                         covariant = cur_covariant;
197                                         if (covariant)
198                                                 c.s << "_{";
199                                         else
200                                                 c.s << "^{";
201                                 }
202                                 it->print(c, level);
203                                 c.s << " ";
204                                 first = false;
205                                 it++;
206                         }
207                         c.s << "}";
208
209                 } else {
210
211                         // Ordinary output
212                         while (it != itend) {
213                                 it->print(c, level);
214                                 it++;
215                         }
216                 }
217         }
218 }
219
220 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
221 {
222         if (precedence() <= level)
223                 c.s << openbrace << '(';
224         c.s << openbrace;
225         seq[0].print(c, precedence());
226         c.s << closebrace;
227         printindices(c, level);
228         if (precedence() <= level)
229                 c.s << ')' << closebrace;
230 }
231
232 void indexed::do_print(const print_context & c, unsigned level) const
233 {
234         print_indexed(c, "", "", level);
235 }
236
237 void indexed::do_print_latex(const print_latex & c, unsigned level) const
238 {
239         print_indexed(c, "{", "}", level);
240 }
241
242 void indexed::do_print_tree(const print_tree & c, unsigned level) const
243 {
244         c.s << std::string(level, ' ') << class_name() << " @" << this
245             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
246             << ", " << seq.size()-1 << " indices"
247             << ", symmetry=" << symtree << std::endl;
248         seq[0].print(c, level + c.delta_indent);
249         printindices(c, level + c.delta_indent);
250 }
251
252 bool indexed::info(unsigned inf) const
253 {
254         if (inf == info_flags::indexed) return true;
255         if (inf == info_flags::has_indices) return seq.size() > 1;
256         return inherited::info(inf);
257 }
258
259 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
260         bool operator() (const ex & e, unsigned inf) const {
261                 return !(ex_to<idx>(e).get_value().info(inf));
262         }
263 };
264
265 bool indexed::all_index_values_are(unsigned inf) const
266 {
267         // No indices? Then no property can be fulfilled
268         if (seq.size() < 2)
269                 return false;
270
271         // Check all indices
272         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
273 }
274
275 int indexed::compare_same_type(const basic & other) const
276 {
277         GINAC_ASSERT(is_a<indexed>(other));
278         return inherited::compare_same_type(other);
279 }
280
281 ex indexed::eval(int level) const
282 {
283         // First evaluate children, then we will end up here again
284         if (level > 1)
285                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
286
287         const ex &base = seq[0];
288
289         // If the base object is 0, the whole object is 0
290         if (base.is_zero())
291                 return _ex0;
292
293         // If the base object is a product, pull out the numeric factor
294         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
295                 exvector v(seq);
296                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
297                 v[0] = seq[0] / f;
298                 return f * thiscontainer(v);
299         }
300
301         if(this->tinfo()==&indexed::tinfo_static && seq.size()==1)
302                 return base;
303
304         // Canonicalize indices according to the symmetry properties
305         if (seq.size() > 2) {
306                 exvector v = seq;
307                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
308                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
309                 if (sig != INT_MAX) {
310                         // Something has changed while sorting indices, more evaluations later
311                         if (sig == 0)
312                                 return _ex0;
313                         return ex(sig) * thiscontainer(v);
314                 }
315         }
316
317         // Let the class of the base object perform additional evaluations
318         return ex_to<basic>(base).eval_indexed(*this);
319 }
320
321 ex indexed::real_part() const
322 {
323         if(op(0).info(info_flags::real))
324                 return *this;
325         return real_part_function(*this).hold();
326 }
327
328 ex indexed::imag_part() const
329 {
330         if(op(0).info(info_flags::real))
331                 return 0;
332         return imag_part_function(*this).hold();
333 }
334
335 ex indexed::thiscontainer(const exvector & v) const
336 {
337         return indexed(ex_to<symmetry>(symtree), v);
338 }
339
340 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
341 {
342         return indexed(ex_to<symmetry>(symtree), vp);
343 }
344
345 unsigned indexed::return_type() const
346 {
347         if(is_a<matrix>(op(0)))
348                 return return_types::commutative;
349         else
350                 return op(0).return_type();
351 }
352
353 ex indexed::expand(unsigned options) const
354 {
355         GINAC_ASSERT(seq.size() > 0);
356
357         if (options & expand_options::expand_indexed) {
358                 ex newbase = seq[0].expand(options);
359                 if (is_exactly_a<add>(newbase)) {
360                         ex sum = _ex0;
361                         for (size_t i=0; i<newbase.nops(); i++) {
362                                 exvector s = seq;
363                                 s[0] = newbase.op(i);
364                                 sum += thiscontainer(s).expand(options);
365                         }
366                         return sum;
367                 }
368                 if (!are_ex_trivially_equal(newbase, seq[0])) {
369                         exvector s = seq;
370                         s[0] = newbase;
371                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
372                 }
373         }
374         return inherited::expand(options);
375 }
376
377 //////////
378 // virtual functions which can be overridden by derived classes
379 //////////
380
381 // none
382
383 //////////
384 // non-virtual functions in this class
385 //////////
386
387 /** Check whether all indices are of class idx and validate the symmetry
388  *  tree. This function is used internally to make sure that all constructed
389  *  indexed objects really carry indices and not some other classes. */
390 void indexed::validate() const
391 {
392         GINAC_ASSERT(seq.size() > 0);
393         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
394         while (it != itend) {
395                 if (!is_a<idx>(*it))
396                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
397                 it++;
398         }
399
400         if (!symtree.is_zero()) {
401                 if (!is_exactly_a<symmetry>(symtree))
402                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
403                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
404         }
405 }
406
407 /** Implementation of ex::diff() for an indexed object always returns 0.
408  *
409  *  @see ex::diff */
410 ex indexed::derivative(const symbol & s) const
411 {
412         return _ex0;
413 }
414
415 //////////
416 // global functions
417 //////////
418
419 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
420         bool operator() (const ex &lh, const ex &rh) const
421         {
422                 if (lh.is_equal(rh))
423                         return true;
424                 else
425                         try {
426                                 // Replacing the dimension might cause an error (e.g. with
427                                 // index classes that only work in a fixed number of dimensions)
428                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
429                         } catch (...) {
430                                 return false;
431                         }
432         }
433 };
434
435 /** Check whether two sorted index vectors are consistent (i.e. equal). */
436 static bool indices_consistent(const exvector & v1, const exvector & v2)
437 {
438         // Number of indices must be the same
439         if (v1.size() != v2.size())
440                 return false;
441
442         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
443 }
444
445 exvector indexed::get_indices() const
446 {
447         GINAC_ASSERT(seq.size() >= 1);
448         return exvector(seq.begin() + 1, seq.end());
449 }
450
451 exvector indexed::get_dummy_indices() const
452 {
453         exvector free_indices, dummy_indices;
454         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
455         return dummy_indices;
456 }
457
458 exvector indexed::get_dummy_indices(const indexed & other) const
459 {
460         exvector indices = get_free_indices();
461         exvector other_indices = other.get_free_indices();
462         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
463         exvector dummy_indices;
464         find_dummy_indices(indices, dummy_indices);
465         return dummy_indices;
466 }
467
468 bool indexed::has_dummy_index_for(const ex & i) const
469 {
470         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
471         while (it != itend) {
472                 if (is_dummy_pair(*it, i))
473                         return true;
474                 it++;
475         }
476         return false;
477 }
478
479 exvector indexed::get_free_indices() const
480 {
481         exvector free_indices, dummy_indices;
482         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
483         return free_indices;
484 }
485
486 exvector add::get_free_indices() const
487 {
488         exvector free_indices;
489         for (size_t i=0; i<nops(); i++) {
490                 if (i == 0)
491                         free_indices = op(i).get_free_indices();
492                 else {
493                         exvector free_indices_of_term = op(i).get_free_indices();
494                         if (!indices_consistent(free_indices, free_indices_of_term))
495                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
496                 }
497         }
498         return free_indices;
499 }
500
501 exvector mul::get_free_indices() const
502 {
503         // Concatenate free indices of all factors
504         exvector un;
505         for (size_t i=0; i<nops(); i++) {
506                 exvector free_indices_of_factor = op(i).get_free_indices();
507                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
508         }
509
510         // And remove the dummy indices
511         exvector free_indices, dummy_indices;
512         find_free_and_dummy(un, free_indices, dummy_indices);
513         return free_indices;
514 }
515
516 exvector ncmul::get_free_indices() const
517 {
518         // Concatenate free indices of all factors
519         exvector un;
520         for (size_t i=0; i<nops(); i++) {
521                 exvector free_indices_of_factor = op(i).get_free_indices();
522                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
523         }
524
525         // And remove the dummy indices
526         exvector free_indices, dummy_indices;
527         find_free_and_dummy(un, free_indices, dummy_indices);
528         return free_indices;
529 }
530
531 struct is_summation_idx : public std::unary_function<ex, bool> {
532         bool operator()(const ex & e)
533         {
534                 return is_dummy_pair(e, e);
535         }
536 };
537
538 exvector integral::get_free_indices() const
539 {
540         if (a.get_free_indices().size() || b.get_free_indices().size())
541                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
542         return f.get_free_indices();
543 }
544
545 template<class T> size_t number_of_type(const exvector&v)
546 {
547         size_t number = 0;
548         for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
549                 if(is_exactly_a<T>(*i))
550                         ++number;
551         return number;
552 }
553
554 /** Rename dummy indices in an expression.
555  *
556  *  @param e Expression to work on
557  *  @param local_dummy_indices The set of dummy indices that appear in the
558  *    expression "e"
559  *  @param global_dummy_indices The set of dummy indices that have appeared
560  *    before and which we would like to use in "e", too. This gets updated
561  *    by the function */
562 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
563 {
564         size_t global_size = number_of_type<T>(global_dummy_indices),
565                local_size = number_of_type<T>(local_dummy_indices);
566
567         // Any local dummy indices at all?
568         if (local_size == 0)
569                 return e;
570
571         if (global_size < local_size) {
572
573                 // More local indices than we encountered before, add the new ones
574                 // to the global set
575                 size_t old_global_size = global_size;
576                 int remaining = local_size - global_size;
577                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
578                 while (it != itend && remaining > 0) {
579                         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()) {
580                                 global_dummy_indices.push_back(*it);
581                                 global_size++;
582                                 remaining--;
583                         }
584                         it++;
585                 }
586
587                 // If this is the first set of local indices, do nothing
588                 if (old_global_size == 0)
589                         return e;
590         }
591         GINAC_ASSERT(local_size <= global_size);
592
593         // Construct vectors of index symbols
594         exvector local_syms, global_syms;
595         local_syms.reserve(local_size);
596         global_syms.reserve(local_size);
597         for (size_t i=0; local_syms.size()!=local_size; i++)
598                 if(is_exactly_a<T>(local_dummy_indices[i]))
599                         local_syms.push_back(local_dummy_indices[i].op(0));
600         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
601         for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
602                 if(is_exactly_a<T>(global_dummy_indices[i]))
603                         global_syms.push_back(global_dummy_indices[i].op(0));
604         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
605
606         // Remove common indices
607         exvector local_uniq, global_uniq;
608         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
609         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
610
611         // Replace remaining non-common local index symbols by global ones
612         if (local_uniq.empty())
613                 return e;
614         else {
615                 while (global_uniq.size() > local_uniq.size())
616                         global_uniq.pop_back();
617                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
618         }
619 }
620
621 /** Given a set of indices, extract those of class varidx. */
622 static void find_variant_indices(const exvector & v, exvector & variant_indices)
623 {
624         exvector::const_iterator it1, itend;
625         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
626                 if (is_exactly_a<varidx>(*it1))
627                         variant_indices.push_back(*it1);
628         }
629 }
630
631 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
632  *  variance.
633  *
634  *  @param e Object to work on
635  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
636  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
637  *  @return true if 'e' was changed */
638 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
639 {
640         bool something_changed = false;
641
642         // If a dummy index is encountered for the first time in the
643         // product, pull it up, otherwise, pull it down
644         exvector::const_iterator it2, it2start, it2end;
645         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
646                 if (!is_exactly_a<varidx>(*it2))
647                         continue;
648
649                 exvector::iterator vit, vitend;
650                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
651                         if (it2->op(0).is_equal(vit->op(0))) {
652                                 if (ex_to<varidx>(*it2).is_covariant()) {
653                                         e = e.subs(lst(
654                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
655                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
656                                         ), subs_options::no_pattern);
657                                         something_changed = true;
658                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
659                                         it2start = ex_to<indexed>(e).seq.begin();
660                                         it2end = ex_to<indexed>(e).seq.end();
661                                 }
662                                 moved_indices.push_back(*vit);
663                                 variant_dummy_indices.erase(vit);
664                                 goto next_index;
665                         }
666                 }
667
668                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
669                         if (it2->op(0).is_equal(vit->op(0))) {
670                                 if (ex_to<varidx>(*it2).is_contravariant()) {
671                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
672                                         something_changed = true;
673                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
674                                         it2start = ex_to<indexed>(e).seq.begin();
675                                         it2end = ex_to<indexed>(e).seq.end();
676                                 }
677                                 goto next_index;
678                         }
679                 }
680
681 next_index: ;
682         }
683
684         return something_changed;
685 }
686
687 /* Ordering that only compares the base expressions of indexed objects. */
688 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
689         bool operator() (const ex &lh, const ex &rh) const
690         {
691                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
692         }
693 };
694
695 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum() 
696  * It returns an exvector of factors from the supplied product */
697 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
698 {
699         // Remember whether the product was commutative or noncommutative
700         // (because we chop it into factors and need to reassemble later)
701         non_commutative = is_exactly_a<ncmul>(e);
702
703         // Collect factors in an exvector, store squares twice
704         v.reserve(e.nops() * 2);
705
706         if (is_exactly_a<power>(e)) {
707                 // We only get called for simple squares, split a^2 -> a*a
708                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
709                 v.push_back(e.op(0));
710                 v.push_back(e.op(0));
711         } else {
712                 for (size_t i=0; i<e.nops(); i++) {
713                         ex f = e.op(i);
714                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
715                                 v.push_back(f.op(0));
716                                 v.push_back(f.op(0));
717                         } else if (is_exactly_a<ncmul>(f)) {
718                                 // Noncommutative factor found, split it as well
719                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
720                                 for (size_t j=0; j<f.nops(); j++)
721                                         v.push_back(f.op(j));
722                         } else
723                                 v.push_back(f);
724                 }
725         }
726 }
727
728 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
729 {       exvector dummy_syms;
730         dummy_syms.reserve(r.nops());
731         for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
732                         if(is_exactly_a<T>(*it))
733                                 dummy_syms.push_back(it->op(0));
734         if(dummy_syms.size() < 2)
735                 return r;
736         ex q=symmetrize(r, dummy_syms);
737         return q;
738 }
739
740 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
741 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
742
743 /** Simplify product of indexed expressions (commutative, noncommutative and
744  *  simple squares), return list of free indices. */
745 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
746 {
747         // Collect factors in an exvector
748         exvector v;
749
750         // Remember whether the product was commutative or noncommutative
751         // (because we chop it into factors and need to reassemble later)
752         bool non_commutative;
753         product_to_exvector(e, v, non_commutative);
754
755         // Perform contractions
756         bool something_changed = false;
757         GINAC_ASSERT(v.size() > 1);
758         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
759         for (it1 = v.begin(); it1 != next_to_last; it1++) {
760
761 try_again:
762                 if (!is_a<indexed>(*it1))
763                         continue;
764
765                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
766
767                 // Indexed factor found, get free indices and look for contraction
768                 // candidates
769                 exvector free1, dummy1;
770                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
771
772                 exvector::iterator it2;
773                 for (it2 = it1 + 1; it2 != itend; it2++) {
774
775                         if (!is_a<indexed>(*it2))
776                                 continue;
777
778                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
779
780                         // Find free indices of second factor and merge them with free
781                         // indices of first factor
782                         exvector un;
783                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
784                         un.insert(un.end(), free1.begin(), free1.end());
785
786                         // Check whether the two factors share dummy indices
787                         exvector free, dummy;
788                         find_free_and_dummy(un, free, dummy);
789                         size_t num_dummies = dummy.size();
790                         if (num_dummies == 0)
791                                 continue;
792
793                         // At least one dummy index, is it a defined scalar product?
794                         bool contracted = false;
795                         if (free.empty() && it1->nops()==2 && it2->nops()==2) {
796
797                                 ex dim = minimal_dim(
798                                         ex_to<idx>(it1->op(1)).get_dim(),
799                                         ex_to<idx>(it2->op(1)).get_dim()
800                                 );
801
802                                 // User-defined scalar product?
803                                 if (sp.is_defined(*it1, *it2, dim)) {
804
805                                         // Yes, substitute it
806                                         *it1 = sp.evaluate(*it1, *it2, dim);
807                                         *it2 = _ex1;
808                                         goto contraction_done;
809                                 }
810                         }
811
812                         // Try to contract the first one with the second one
813                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
814                         if (!contracted) {
815
816                                 // That didn't work; maybe the second object knows how to
817                                 // contract itself with the first one
818                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
819                         }
820                         if (contracted) {
821 contraction_done:
822                                 if (first_noncommutative || second_noncommutative
823                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
824                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
825                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
826
827                                         // One of the factors became a sum or product:
828                                         // re-expand expression and run again
829                                         // Non-commutative products are always re-expanded to give
830                                         // eval_ncmul() the chance to re-order and canonicalize
831                                         // the product
832                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
833                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
834                                 }
835
836                                 // Both objects may have new indices now or they might
837                                 // even not be indexed objects any more, so we have to
838                                 // start over
839                                 something_changed = true;
840                                 goto try_again;
841                         }
842                 }
843         }
844
845         // Find free indices (concatenate them all and call find_free_and_dummy())
846         // and all dummy indices that appear
847         exvector un, individual_dummy_indices;
848         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
849                 exvector free_indices_of_factor;
850                 if (is_a<indexed>(*it1)) {
851                         exvector dummy_indices_of_factor;
852                         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);
853                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
854                 } else
855                         free_indices_of_factor = it1->get_free_indices();
856                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
857         }
858         exvector local_dummy_indices;
859         find_free_and_dummy(un, free_indices, local_dummy_indices);
860         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
861
862         // Filter out the dummy indices with variance
863         exvector variant_dummy_indices;
864         find_variant_indices(local_dummy_indices, variant_dummy_indices);
865
866         // Any indices with variance present at all?
867         if (!variant_dummy_indices.empty()) {
868
869                 // Yes, bring the product into a canonical order that only depends on
870                 // the base expressions of indexed objects
871                 if (!non_commutative)
872                         std::sort(v.begin(), v.end(), ex_base_is_less());
873
874                 exvector moved_indices;
875
876                 // Iterate over all indexed objects in the product
877                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
878                         if (!is_a<indexed>(*it1))
879                                 continue;
880
881                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
882                                 something_changed = true;
883                 }
884         }
885
886         ex r;
887         if (something_changed)
888                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
889         else
890                 r = e;
891
892         // The result should be symmetric with respect to exchange of dummy
893         // indices, so if the symmetrization vanishes, the whole expression is
894         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
895         ex q = idx_symmetrization<idx>(r, local_dummy_indices);
896         if (q.is_zero()) {
897                 free_indices.clear();
898                 return _ex0;
899         }
900         q = idx_symmetrization<varidx>(q, local_dummy_indices);
901         if (q.is_zero()) {
902                 free_indices.clear();
903                 return _ex0;
904         }
905         q = idx_symmetrization<spinidx>(q, local_dummy_indices);
906         if (q.is_zero()) {
907                 free_indices.clear();
908                 return _ex0;
909         }
910
911         // Dummy index renaming
912         r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
913         r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
914         r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
915
916         // Product of indexed object with a scalar?
917         if (is_exactly_a<mul>(r) && r.nops() == 2
918          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
919                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
920         else
921                 return r;
922 }
923
924 /** This structure stores the original and symmetrized versions of terms
925  *  obtained during the simplification of sums. */
926 class terminfo {
927 public:
928         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
929
930         ex orig; /**< original term */
931         ex symm; /**< symmtrized term */
932 };
933
934 class terminfo_is_less {
935 public:
936         bool operator() (const terminfo & ti1, const terminfo & ti2) const
937         {
938                 return (ti1.symm.compare(ti2.symm) < 0);
939         }
940 };
941
942 /** This structure stores the individual symmetrized terms obtained during
943  *  the simplification of sums. */
944 class symminfo {
945 public:
946         symminfo() : num(0) {}
947
948         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
949         {
950                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
951                         coeff = symmterm_.op(symmterm_.nops()-1);
952                         symmterm = symmterm_ / coeff;
953                 } else {
954                         coeff = 1;
955                         symmterm = symmterm_;
956                 }
957         }
958
959         ex symmterm;  /**< symmetrized term */
960         ex coeff;     /**< coefficient of symmetrized term */
961         ex orig;      /**< original term */
962         size_t num; /**< how many symmetrized terms resulted from the original term */
963 };
964
965 class symminfo_is_less_by_symmterm {
966 public:
967         bool operator() (const symminfo & si1, const symminfo & si2) const
968         {
969                 return (si1.symmterm.compare(si2.symmterm) < 0);
970         }
971 };
972
973 class symminfo_is_less_by_orig {
974 public:
975         bool operator() (const symminfo & si1, const symminfo & si2) const
976         {
977                 return (si1.orig.compare(si2.orig) < 0);
978         }
979 };
980
981 bool hasindex(const ex &x, const ex &sym)
982 {       
983         if(is_a<idx>(x) && x.op(0)==sym)
984                 return true;
985         else
986                 for(size_t i=0; i<x.nops(); ++i)
987                         if(hasindex(x.op(i), sym))
988                                 return true;
989         return false;
990 }
991
992 /** Simplify indexed expression, return list of free indices. */
993 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
994 {
995         // Expand the expression
996         ex e_expanded = e.expand();
997
998         // Simplification of single indexed object: just find the free indices
999         // and perform dummy index renaming/repositioning
1000         if (is_a<indexed>(e_expanded)) {
1001
1002                 // Find the dummy indices
1003                 const indexed &i = ex_to<indexed>(e_expanded);
1004                 exvector local_dummy_indices;
1005                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1006
1007                 // Filter out the dummy indices with variance
1008                 exvector variant_dummy_indices;
1009                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1010
1011                 // Any indices with variance present at all?
1012                 if (!variant_dummy_indices.empty()) {
1013
1014                         // Yes, reposition them
1015                         exvector moved_indices;
1016                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1017                 }
1018
1019                 // Rename the dummy indices
1020                 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1021                 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1022                 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1023                 return e_expanded;
1024         }
1025
1026         // Simplification of sum = sum of simplifications, check consistency of
1027         // free indices in each term
1028         if (is_exactly_a<add>(e_expanded)) {
1029                 bool first = true;
1030                 ex sum;
1031                 free_indices.clear();
1032
1033                 for (size_t i=0; i<e_expanded.nops(); i++) {
1034                         exvector free_indices_of_term;
1035                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1036                         if (!term.is_zero()) {
1037                                 if (first) {
1038                                         free_indices = free_indices_of_term;
1039                                         sum = term;
1040                                         first = false;
1041                                 } else {
1042                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
1043                                                 std::ostringstream s;
1044                                                 s << "simplify_indexed: inconsistent indices in sum: ";
1045                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1046                                                 throw (std::runtime_error(s.str()));
1047                                         }
1048                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1049                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1050                                         else
1051                                                 sum += term;
1052                                 }
1053                         }
1054                 }
1055
1056                 // If the sum turns out to be zero, we are finished
1057                 if (sum.is_zero()) {
1058                         free_indices.clear();
1059                         return sum;
1060                 }
1061
1062                 // More than one term and more than one dummy index?
1063                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1064                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1065                         return sum;
1066
1067                 // Chop the sum into terms and symmetrize each one over the dummy
1068                 // indices
1069                 std::vector<terminfo> terms;
1070                 for (size_t i=0; i<sum.nops(); i++) {
1071                         const ex & term = sum.op(i);
1072                         exvector dummy_indices_of_term;
1073                         dummy_indices_of_term.reserve(dummy_indices.size());
1074                         for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1075                                 if(hasindex(term,i->op(0)))
1076                                         dummy_indices_of_term.push_back(*i);
1077                         ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1078                         term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1079                         term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1080                         if (term_symm.is_zero())
1081                                 continue;
1082                         terms.push_back(terminfo(term, term_symm));
1083                 }
1084
1085                 // Sort by symmetrized terms
1086                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1087
1088                 // Combine equal symmetrized terms
1089                 std::vector<terminfo> terms_pass2;
1090                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1091                         size_t num = 1;
1092                         std::vector<terminfo>::const_iterator j = i + 1;
1093                         while (j != terms.end() && j->symm == i->symm) {
1094                                 num++;
1095                                 j++;
1096                         }
1097                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1098                         i = j;
1099                 }
1100
1101                 // If there is only one term left, we are finished
1102                 if (terms_pass2.size() == 1)
1103                         return terms_pass2[0].orig;
1104
1105                 // Chop the symmetrized terms into subterms
1106                 std::vector<symminfo> sy;
1107                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1108                         if (is_exactly_a<add>(i->symm)) {
1109                                 size_t num = i->symm.nops();
1110                                 for (size_t j=0; j<num; j++)
1111                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1112                         } else
1113                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1114                 }
1115
1116                 // Sort by symmetrized subterms
1117                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1118
1119                 // Combine equal symmetrized subterms
1120                 std::vector<symminfo> sy_pass2;
1121                 exvector result;
1122                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1123
1124                         // Combine equal terms
1125                         std::vector<symminfo>::const_iterator j = i + 1;
1126                         if (j != sy.end() && j->symmterm == i->symmterm) {
1127
1128                                 // More than one term, collect the coefficients
1129                                 ex coeff = i->coeff;
1130                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1131                                         coeff += j->coeff;
1132                                         j++;
1133                                 }
1134
1135                                 // Add combined term to result
1136                                 if (!coeff.is_zero())
1137                                         result.push_back(coeff * i->symmterm);
1138
1139                         } else {
1140
1141                                 // Single term, store for second pass
1142                                 sy_pass2.push_back(*i);
1143                         }
1144
1145                         i = j;
1146                 }
1147
1148                 // Were there any remaining terms that didn't get combined?
1149                 if (sy_pass2.size() > 0) {
1150
1151                         // Yes, sort by their original terms
1152                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1153
1154                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1155
1156                                 // How many symmetrized terms of this original term are left?
1157                                 size_t num = 1;
1158                                 std::vector<symminfo>::const_iterator j = i + 1;
1159                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1160                                         num++;
1161                                         j++;
1162                                 }
1163
1164                                 if (num == i->num) {
1165
1166                                         // All terms left, then add the original term to the result
1167                                         result.push_back(i->orig);
1168
1169                                 } else {
1170
1171                                         // Some terms were combined with others, add up the remaining symmetrized terms
1172                                         std::vector<symminfo>::const_iterator k;
1173                                         for (k=i; k!=j; k++)
1174                                                 result.push_back(k->coeff * k->symmterm);
1175                                 }
1176
1177                                 i = j;
1178                         }
1179                 }
1180
1181                 // Add all resulting terms
1182                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1183                 if (sum_symm.is_zero())
1184                         free_indices.clear();
1185                 return sum_symm;
1186         }
1187
1188         // Simplification of products
1189         if (is_exactly_a<mul>(e_expanded)
1190          || is_exactly_a<ncmul>(e_expanded)
1191          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1192                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1193
1194         // Cannot do anything
1195         free_indices.clear();
1196         return e_expanded;
1197 }
1198
1199 /** Simplify/canonicalize expression containing indexed objects. This
1200  *  performs contraction of dummy indices where possible and checks whether
1201  *  the free indices in sums are consistent.
1202  *
1203  *  @param options Simplification options (currently unused)
1204  *  @return simplified expression */
1205 ex ex::simplify_indexed(unsigned options) const
1206 {
1207         exvector free_indices, dummy_indices;
1208         scalar_products sp;
1209         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1210 }
1211
1212 /** Simplify/canonicalize expression containing indexed objects. This
1213  *  performs contraction of dummy indices where possible, checks whether
1214  *  the free indices in sums are consistent, and automatically replaces
1215  *  scalar products by known values if desired.
1216  *
1217  *  @param sp Scalar products to be replaced automatically
1218  *  @param options Simplification options (currently unused)
1219  *  @return simplified expression */
1220 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1221 {
1222         exvector free_indices, dummy_indices;
1223         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1224 }
1225
1226 /** Symmetrize expression over its free indices. */
1227 ex ex::symmetrize() const
1228 {
1229         return GiNaC::symmetrize(*this, get_free_indices());
1230 }
1231
1232 /** Antisymmetrize expression over its free indices. */
1233 ex ex::antisymmetrize() const
1234 {
1235         return GiNaC::antisymmetrize(*this, get_free_indices());
1236 }
1237
1238 /** Symmetrize expression by cyclic permutation over its free indices. */
1239 ex ex::symmetrize_cyclic() const
1240 {
1241         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1242 }
1243
1244 //////////
1245 // helper classes
1246 //////////
1247
1248 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1249 {
1250         // If indexed, extract base objects
1251         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1252         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1253
1254         // Enforce canonical order in pair
1255         if (s1.compare(s2) > 0) {
1256                 v1 = s2;
1257                 v2 = s1;
1258         } else {
1259                 v1 = s1;
1260                 v2 = s2;
1261         }
1262 }
1263
1264 bool spmapkey::operator==(const spmapkey &other) const
1265 {
1266         if (!v1.is_equal(other.v1))
1267                 return false;
1268         if (!v2.is_equal(other.v2))
1269                 return false;
1270         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1271                 return true;
1272         else
1273                 return dim.is_equal(other.dim);
1274 }
1275
1276 bool spmapkey::operator<(const spmapkey &other) const
1277 {
1278         int cmp = v1.compare(other.v1);
1279         if (cmp)
1280                 return cmp < 0;
1281         cmp = v2.compare(other.v2);
1282         if (cmp)
1283                 return cmp < 0;
1284
1285         // Objects are equal, now check dimensions
1286         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1287                 return false;
1288         else
1289                 return dim.compare(other.dim) < 0;
1290 }
1291
1292 void spmapkey::debugprint() const
1293 {
1294         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1295 }
1296
1297 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1298 {
1299         spm[spmapkey(v1, v2)] = sp;
1300 }
1301
1302 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1303 {
1304         spm[spmapkey(v1, v2, dim)] = sp;
1305 }
1306
1307 void scalar_products::add_vectors(const lst & l, const ex & dim)
1308 {
1309         // Add all possible pairs of products
1310         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1311                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1312                         add(*it1, *it2, *it1 * *it2);
1313 }
1314
1315 void scalar_products::clear()
1316 {
1317         spm.clear();
1318 }
1319
1320 /** Check whether scalar product pair is defined. */
1321 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1322 {
1323         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1324 }
1325
1326 /** Return value of defined scalar product pair. */
1327 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1328 {
1329         return spm.find(spmapkey(v1, v2, dim))->second;
1330 }
1331
1332 void scalar_products::debugprint() const
1333 {
1334         std::cerr << "map size=" << spm.size() << std::endl;
1335         spmap::const_iterator i = spm.begin(), end = spm.end();
1336         while (i != end) {
1337                 const spmapkey & k = i->first;
1338                 std::cerr << "item key=";
1339                 k.debugprint();
1340                 std::cerr << ", value=" << i->second << std::endl;
1341                 ++i;
1342         }
1343 }
1344
1345 exvector get_all_dummy_indices_safely(const ex & e)
1346 {
1347         if (is_a<indexed>(e))
1348                 return ex_to<indexed>(e).get_dummy_indices();
1349         else if (is_a<power>(e) && e.op(1)==2) {
1350                 return e.op(0).get_free_indices();
1351         }       
1352         else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1353                 exvector dummies;
1354                 exvector free_indices;
1355                 for (int i=0; i<e.nops(); ++i) {
1356                         exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1357                         dummies.insert(dummies.end(), dummies_of_factor.begin(),
1358                                 dummies_of_factor.end());
1359                         exvector free_of_factor = e.op(i).get_free_indices();
1360                         free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1361                                 free_of_factor.end());
1362                 }
1363                 exvector free_out, dummy_out;
1364                 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1365                         dummy_out);
1366                 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1367                 return dummies;
1368         }
1369         else if(is_a<add>(e)) {
1370                 exvector result;
1371                 for(int i=0; i<e.nops(); ++i) {
1372                         exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1373                         sort(dummies_of_term.begin(), dummies_of_term.end());
1374                         exvector new_vec;
1375                         set_union(result.begin(), result.end(), dummies_of_term.begin(),
1376                                 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1377                                 ex_is_less());
1378                         result.swap(new_vec);
1379                 }
1380                 return result;
1381         }
1382         return exvector();
1383 }
1384
1385 /** Returns all dummy indices from the exvector */
1386 exvector get_all_dummy_indices(const ex & e)
1387 {
1388         exvector p;
1389         bool nc;
1390         product_to_exvector(e, p, nc);
1391         exvector::const_iterator ip = p.begin(), ipend = p.end();
1392         exvector v, v1;
1393         while (ip != ipend) {
1394                 if (is_a<indexed>(*ip)) {
1395                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1396                         v.insert(v.end(), v1.begin(), v1.end());
1397                         exvector::const_iterator ip1 = ip+1;
1398                         while (ip1 != ipend) {
1399                                 if (is_a<indexed>(*ip1)) {
1400                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1401                                         v.insert(v.end(), v1.begin(), v1.end());
1402                                 }
1403                                 ++ip1;
1404                         }
1405                 }
1406                 ++ip;
1407         }
1408         return v;
1409 }
1410
1411 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1412 {
1413         exvector common_indices;
1414         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1415         if (common_indices.empty()) {
1416                 return lst(lst(), lst());
1417         } else {
1418                 exvector new_indices, old_indices;
1419                 old_indices.reserve(2*common_indices.size());
1420                 new_indices.reserve(2*common_indices.size());
1421                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1422                 while (ip != ipend) {
1423                         ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1424                         ex newidx;
1425                         if(is_exactly_a<spinidx>(*ip))
1426                                 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1427                                                 ex_to<spinidx>(*ip).is_covariant(),
1428                                                 ex_to<spinidx>(*ip).is_dotted()))
1429                                         -> setflag(status_flags::dynallocated);
1430                         else if (is_exactly_a<varidx>(*ip))
1431                                 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1432                                                 ex_to<varidx>(*ip).is_covariant()))
1433                                         -> setflag(status_flags::dynallocated);
1434                         else
1435                                 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1436                                         -> setflag(status_flags::dynallocated);
1437                         old_indices.push_back(*ip);
1438                         new_indices.push_back(newidx);
1439                         if(is_a<varidx>(*ip)) {
1440                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1441                                 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1442                         }
1443                         ++ip;
1444                 }
1445                 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1446         }
1447 }
1448
1449 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1450 {
1451         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1452         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);
1453 }
1454
1455 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1456 {
1457         exvector va = get_all_dummy_indices_safely(a);
1458         if (va.size() > 0) {
1459                 exvector vb = get_all_dummy_indices_safely(b);
1460                 if (vb.size() > 0) {
1461                         sort(va.begin(), va.end(), ex_is_less());
1462                         sort(vb.begin(), vb.end(), ex_is_less());
1463                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1464                         if (indices_subs.op(0).nops() > 0)
1465                                 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);
1466                 }
1467         }
1468         return b;
1469 }
1470
1471 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1472 {
1473         if (va.size() > 0) {
1474                 exvector vb = get_all_dummy_indices_safely(b);
1475                 if (vb.size() > 0) {
1476                         sort(vb.begin(), vb.end(), ex_is_less());
1477                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1478                         if (indices_subs.op(0).nops() > 0) {
1479                                 if (modify_va) {
1480                                         for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1481                                                 va.push_back(*i);
1482                                         exvector uncommon_indices;
1483                                         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());
1484                                         exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1485                                         while (ip != ipend) {
1486                                                 va.push_back(*ip);
1487                                                 ++ip;
1488                                         }
1489                                         sort(va.begin(), va.end(), ex_is_less());
1490                                 }
1491                                 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);
1492                         }
1493                 }
1494         }
1495         return b;
1496 }
1497
1498 ex expand_dummy_sum(const ex & e, bool subs_idx)
1499 {
1500         ex e_expanded = e.expand();
1501         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1502         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1503                 return e_expanded.map(fcn);
1504         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1505                 exvector v;
1506                 if (is_a<indexed>(e_expanded))
1507                         v = ex_to<indexed>(e_expanded).get_dummy_indices();
1508                 else
1509                         v = get_all_dummy_indices(e_expanded);
1510                 ex result = e_expanded;
1511                 for(exvector::const_iterator it=v.begin(); it!=v.end(); ++it) {
1512                         ex nu = *it;
1513                         if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1514                                 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1515                                 ex en = 0;
1516                                 for (int i=0; i < idim; i++) {
1517                                         if (subs_idx && is_a<varidx>(nu)) {
1518                                                 ex other = ex_to<varidx>(nu).toggle_variance();
1519                                                 en += result.subs(lst(
1520                                                         nu == idx(i, idim),
1521                                                         other == idx(i, idim)
1522                                                 ));
1523                                         } else {
1524                                                 en += result.subs( nu.op(0) == i );
1525                                         }
1526                                 }
1527                                 result = en;
1528                         }
1529                 }
1530                 return result;
1531         } else {
1532                 return e;
1533         }
1534 }
1535
1536 } // namespace GiNaC