Update plot examples
[linpy.git] / linpy / domains.py
1 # Copyright 2014 MINES ParisTech
2 #
3 # This file is part of LinPy.
4 #
5 # LinPy is free software: you can redistribute it and/or modify
6 # it under the terms of the GNU General Public License as published by
7 # the Free Software Foundation, either version 3 of the License, or
8 # (at your option) any later version.
9 #
10 # LinPy is distributed in the hope that it will be useful,
11 # but WITHOUT ANY WARRANTY; without even the implied warranty of
12 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
13 # GNU General Public License for more details.
14 #
15 # You should have received a copy of the GNU General Public License
16 # along with LinPy. If not, see <http://www.gnu.org/licenses/>.
17
18 import ast
19 import functools
20 import re
21 import math
22
23 from fractions import Fraction
24
25 from . import islhelper
26 from .islhelper import mainctx, libisl
27 from .linexprs import Expression, Symbol, Rational
28 from .geometry import GeometricObject, Point, Vector
29
30
31 __all__ = [
32 'Domain',
33 'And', 'Or', 'Not',
34 ]
35
36
37 @functools.total_ordering
38 class Domain(GeometricObject):
39
40 __slots__ = (
41 '_polyhedra',
42 '_symbols',
43 '_dimension',
44 )
45
46 def __new__(cls, *polyhedra):
47 from .polyhedra import Polyhedron
48 if len(polyhedra) == 1:
49 argument = polyhedra[0]
50 if isinstance(argument, str):
51 return cls.fromstring(argument)
52 elif isinstance(argument, GeometricObject):
53 return argument.aspolyhedron()
54 else:
55 raise TypeError('argument must be a string '
56 'or a GeometricObject instance')
57 else:
58 for polyhedron in polyhedra:
59 if not isinstance(polyhedron, Polyhedron):
60 raise TypeError('arguments must be Polyhedron instances')
61 symbols = cls._xsymbols(polyhedra)
62 islset = cls._toislset(polyhedra, symbols)
63 return cls._fromislset(islset, symbols)
64
65 @classmethod
66 def _xsymbols(cls, iterator):
67 """
68 Return the ordered tuple of symbols present in iterator.
69 """
70 symbols = set()
71 for item in iterator:
72 symbols.update(item.symbols)
73 return tuple(sorted(symbols, key=Symbol.sortkey))
74
75 @property
76 def polyhedra(self):
77 return self._polyhedra
78
79 @property
80 def symbols(self):
81 return self._symbols
82
83 @property
84 def dimension(self):
85 return self._dimension
86
87 def disjoint(self):
88 """
89 Returns this set as disjoint.
90 """
91 islset = self._toislset(self.polyhedra, self.symbols)
92 islset = libisl.isl_set_make_disjoint(mainctx, islset)
93 return self._fromislset(islset, self.symbols)
94
95 def isempty(self):
96 """
97 Returns true if this set is an Empty set.
98 """
99 islset = self._toislset(self.polyhedra, self.symbols)
100 empty = bool(libisl.isl_set_is_empty(islset))
101 libisl.isl_set_free(islset)
102 return empty
103
104 def __bool__(self):
105 return not self.isempty()
106
107 def isuniverse(self):
108 """
109 Returns true if this set is the Universe set.
110 """
111 islset = self._toislset(self.polyhedra, self.symbols)
112 universe = bool(libisl.isl_set_plain_is_universe(islset))
113 libisl.isl_set_free(islset)
114 return universe
115
116 def isbounded(self):
117 """
118 Returns true if this set is bounded.
119 """
120 islset = self._toislset(self.polyhedra, self.symbols)
121 bounded = bool(libisl.isl_set_is_bounded(islset))
122 libisl.isl_set_free(islset)
123 return bounded
124
125 def __eq__(self, other):
126 """
127 Returns true if two sets are equal.
128 """
129 symbols = self._xsymbols([self, other])
130 islset1 = self._toislset(self.polyhedra, symbols)
131 islset2 = other._toislset(other.polyhedra, symbols)
132 equal = bool(libisl.isl_set_is_equal(islset1, islset2))
133 libisl.isl_set_free(islset1)
134 libisl.isl_set_free(islset2)
135 return equal
136
137 def isdisjoint(self, other):
138 """
139 Return True if two sets have a null intersection.
140 """
141 symbols = self._xsymbols([self, other])
142 islset1 = self._toislset(self.polyhedra, symbols)
143 islset2 = self._toislset(other.polyhedra, symbols)
144 equal = bool(libisl.isl_set_is_disjoint(islset1, islset2))
145 libisl.isl_set_free(islset1)
146 libisl.isl_set_free(islset2)
147 return equal
148
149 def issubset(self, other):
150 """
151 Report whether another set contains this set.
152 """
153 symbols = self._xsymbols([self, other])
154 islset1 = self._toislset(self.polyhedra, symbols)
155 islset2 = self._toislset(other.polyhedra, symbols)
156 equal = bool(libisl.isl_set_is_subset(islset1, islset2))
157 libisl.isl_set_free(islset1)
158 libisl.isl_set_free(islset2)
159 return equal
160
161 def __le__(self, other):
162 """
163 Returns true if this set is less than or equal to another set.
164 """
165 return self.issubset(other)
166
167 def __lt__(self, other):
168 """
169 Returns true if this set is less than another set.
170 """
171 symbols = self._xsymbols([self, other])
172 islset1 = self._toislset(self.polyhedra, symbols)
173 islset2 = self._toislset(other.polyhedra, symbols)
174 equal = bool(libisl.isl_set_is_strict_subset(islset1, islset2))
175 libisl.isl_set_free(islset1)
176 libisl.isl_set_free(islset2)
177 return equal
178
179 def complement(self):
180 """
181 Returns the complement of this set.
182 """
183 islset = self._toislset(self.polyhedra, self.symbols)
184 islset = libisl.isl_set_complement(islset)
185 return self._fromislset(islset, self.symbols)
186
187 def __invert__(self):
188 """
189 Returns the complement of this set.
190 """
191 return self.complement()
192
193 def simplify(self):
194 """
195 Returns a set without redundant constraints.
196 """
197 islset = self._toislset(self.polyhedra, self.symbols)
198 islset = libisl.isl_set_remove_redundancies(islset)
199 return self._fromislset(islset, self.symbols)
200
201 def aspolyhedron(self):
202 """
203 Returns polyhedral hull of set.
204 """
205 from .polyhedra import Polyhedron
206 islset = self._toislset(self.polyhedra, self.symbols)
207 islbset = libisl.isl_set_polyhedral_hull(islset)
208 return Polyhedron._fromislbasicset(islbset, self.symbols)
209
210 def asdomain(self):
211 return self
212
213 def project(self, dims):
214 """
215 Return new set with given dimensions removed.
216 """
217 islset = self._toislset(self.polyhedra, self.symbols)
218 n = 0
219 for index, symbol in reversed(list(enumerate(self.symbols))):
220 if symbol in dims:
221 n += 1
222 elif n > 0:
223 islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, index + 1, n)
224 n = 0
225 if n > 0:
226 islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, 0, n)
227 dims = [symbol for symbol in self.symbols if symbol not in dims]
228 return Domain._fromislset(islset, dims)
229
230 def sample(self):
231 """
232 Returns a single subset of the input.
233 """
234 islset = self._toislset(self.polyhedra, self.symbols)
235 islpoint = libisl.isl_set_sample_point(islset)
236 if bool(libisl.isl_point_is_void(islpoint)):
237 libisl.isl_point_free(islpoint)
238 raise ValueError('domain must be non-empty')
239 point = {}
240 for index, symbol in enumerate(self.symbols):
241 coordinate = libisl.isl_point_get_coordinate_val(islpoint,
242 libisl.isl_dim_set, index)
243 coordinate = islhelper.isl_val_to_int(coordinate)
244 point[symbol] = coordinate
245 libisl.isl_point_free(islpoint)
246 return point
247
248 def intersection(self, *others):
249 """
250 Return the intersection of two sets as a new set.
251 """
252 if len(others) == 0:
253 return self
254 symbols = self._xsymbols((self,) + others)
255 islset1 = self._toislset(self.polyhedra, symbols)
256 for other in others:
257 islset2 = other._toislset(other.polyhedra, symbols)
258 islset1 = libisl.isl_set_intersect(islset1, islset2)
259 return self._fromislset(islset1, symbols)
260
261 def __and__(self, other):
262 """
263 Return the intersection of two sets as a new set.
264 """
265 return self.intersection(other)
266
267 def union(self, *others):
268 """
269 Return the union of sets as a new set.
270 """
271 if len(others) == 0:
272 return self
273 symbols = self._xsymbols((self,) + others)
274 islset1 = self._toislset(self.polyhedra, symbols)
275 for other in others:
276 islset2 = other._toislset(other.polyhedra, symbols)
277 islset1 = libisl.isl_set_union(islset1, islset2)
278 return self._fromislset(islset1, symbols)
279
280 def __or__(self, other):
281 """
282 Return a new set with elements from both sets.
283 """
284 return self.union(other)
285
286 def __add__(self, other):
287 """
288 Return new set containing all elements in both sets.
289 """
290 return self.union(other)
291
292 def difference(self, other):
293 """
294 Return the difference of two sets as a new set.
295 """
296 symbols = self._xsymbols([self, other])
297 islset1 = self._toislset(self.polyhedra, symbols)
298 islset2 = other._toislset(other.polyhedra, symbols)
299 islset = libisl.isl_set_subtract(islset1, islset2)
300 return self._fromislset(islset, symbols)
301
302 def __sub__(self, other):
303 """
304 Return the difference of two sets as a new set.
305 """
306 return self.difference(other)
307
308 def lexmin(self):
309 """
310 Return a new set containing the lexicographic minimum of the elements in the set.
311 """
312 islset = self._toislset(self.polyhedra, self.symbols)
313 islset = libisl.isl_set_lexmin(islset)
314 return self._fromislset(islset, self.symbols)
315
316 def lexmax(self):
317 """
318 Return a new set containing the lexicographic maximum of the elements in the set.
319 """
320 islset = self._toislset(self.polyhedra, self.symbols)
321 islset = libisl.isl_set_lexmax(islset)
322 return self._fromislset(islset, self.symbols)
323
324
325 def involves_vars(self, vars):
326 """
327 Returns true if a set depends on given dimensions.
328 """
329 islset = self._toislset(self.polyhedra, self.symbols)
330 dims = sorted(vars)
331 symbols = sorted(list(self.symbols))
332 n = 0
333 if len(dims)>0:
334 for dim in dims:
335 if dim in symbols:
336 first = symbols.index(dims[0])
337 n +=1
338 else:
339 first = 0
340 else:
341 return False
342 value = bool(libisl.isl_set_involves_dims(islset, libisl.isl_dim_set, first, n))
343 libisl.isl_set_free(islset)
344 return value
345
346 _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
347
348 def vertices(self):
349 """
350 Return a list of vertices for this Polygon.
351 """
352 from .polyhedra import Polyhedron
353 if not self.isbounded():
354 raise ValueError('domain must be bounded')
355 islbset = self._toislbasicset(self.equalities, self.inequalities, self.symbols)
356 vertices = libisl.isl_basic_set_compute_vertices(islbset);
357 vertices = islhelper.isl_vertices_vertices(vertices)
358 points = []
359 for vertex in vertices:
360 expr = libisl.isl_vertex_get_expr(vertex)
361 coordinates = []
362 if islhelper.isl_version < '0.13':
363 constraints = islhelper.isl_basic_set_constraints(expr)
364 for constraint in constraints:
365 constant = libisl.isl_constraint_get_constant_val(constraint)
366 constant = islhelper.isl_val_to_int(constant)
367 for index, symbol in enumerate(self.symbols):
368 coefficient = libisl.isl_constraint_get_coefficient_val(constraint,
369 libisl.isl_dim_set, index)
370 coefficient = islhelper.isl_val_to_int(coefficient)
371 if coefficient != 0:
372 coordinate = -Fraction(constant, coefficient)
373 coordinates.append((symbol, coordinate))
374 else:
375 string = islhelper.isl_multi_aff_to_str(expr)
376 matches = self._RE_COORDINATE.finditer(string)
377 for symbol, match in zip(self.symbols, matches):
378 numerator = int(match.group('num'))
379 denominator = match.group('den')
380 denominator = 1 if denominator is None else int(denominator)
381 coordinate = Fraction(numerator, denominator)
382 coordinates.append((symbol, coordinate))
383 points.append(Point(coordinates))
384 return points
385
386 def points(self):
387 """
388 Returns the points contained in the set.
389 """
390 if not self.isbounded():
391 raise ValueError('domain must be bounded')
392 from .polyhedra import Universe, Eq
393 islset = self._toislset(self.polyhedra, self.symbols)
394 islpoints = islhelper.isl_set_points(islset)
395 points = []
396 for islpoint in islpoints:
397 coordinates = {}
398 for index, symbol in enumerate(self.symbols):
399 coordinate = libisl.isl_point_get_coordinate_val(islpoint,
400 libisl.isl_dim_set, index)
401 coordinate = islhelper.isl_val_to_int(coordinate)
402 coordinates[symbol] = coordinate
403 points.append(Point(coordinates))
404 return points
405
406 @classmethod
407 def _polygon_inner_point(cls, points):
408 symbols = points[0].symbols
409 coordinates = {symbol: 0 for symbol in symbols}
410 for point in points:
411 for symbol, coordinate in point.coordinates():
412 coordinates[symbol] += coordinate
413 for symbol in symbols:
414 coordinates[symbol] /= len(points)
415 return Point(coordinates)
416
417 @classmethod
418 def _sort_polygon_2d(cls, points):
419 if len(points) <= 3:
420 return points
421 o = cls._polygon_inner_point(points)
422 angles = {}
423 for m in points:
424 om = Vector(o, m)
425 dx, dy = (coordinate for symbol, coordinate in om.coordinates())
426 angle = math.atan2(dy, dx)
427 angles[m] = angle
428 return sorted(points, key=angles.get)
429
430 @classmethod
431 def _sort_polygon_3d(cls, points):
432 if len(points) <= 3:
433 return points
434 o = cls._polygon_inner_point(points)
435 a = points[0]
436 oa = Vector(o, a)
437 norm_oa = oa.norm()
438 for b in points[1:]:
439 ob = Vector(o, b)
440 u = oa.cross(ob)
441 if not u.isnull():
442 u = u.asunit()
443 break
444 else:
445 raise ValueError('degenerate polygon')
446 angles = {a: 0.}
447 for m in points[1:]:
448 om = Vector(o, m)
449 normprod = norm_oa * om.norm()
450 cosinus = max(oa.dot(om) / normprod, -1.)
451 sinus = u.dot(oa.cross(om)) / normprod
452 angle = math.acos(cosinus)
453 angle = math.copysign(angle, sinus)
454 angles[m] = angle
455 return sorted(points, key=angles.get)
456
457 def faces(self):
458 """
459 Returns the vertices of the faces of a polyhedra.
460 """
461 faces = []
462 for polyhedron in self.polyhedra:
463 vertices = polyhedron.vertices()
464 for constraint in polyhedron.constraints:
465 face = []
466 for vertex in vertices:
467 if constraint.subs(vertex.coordinates()) == 0:
468 face.append(vertex)
469 if len(face) >= 3:
470 faces.append(face)
471 return faces
472
473 def _plot_2d(self, plot=None, **kwargs):
474 import matplotlib.pyplot as plt
475 from matplotlib.patches import Polygon
476 if plot is None:
477 fig = plt.figure()
478 plot = fig.add_subplot(1, 1, 1)
479 xmin, xmax = plot.get_xlim()
480 ymin, ymax = plot.get_ylim()
481 for polyhedron in self.polyhedra:
482 vertices = polyhedron._sort_polygon_2d(polyhedron.vertices())
483 xys = [tuple(vertex.values()) for vertex in vertices]
484 xs, ys = zip(*xys)
485 xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
486 ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
487 plot.add_patch(Polygon(xys, closed=True, **kwargs))
488 plot.set_xlim(xmin, xmax)
489 plot.set_ylim(ymin, ymax)
490 return plot
491
492 def _plot_3d(self, plot=None, **kwargs):
493 import matplotlib.pyplot as plt
494 from mpl_toolkits.mplot3d import Axes3D
495 from mpl_toolkits.mplot3d.art3d import Poly3DCollection
496 if plot is None:
497 fig = plt.figure()
498 axes = Axes3D(fig)
499 else:
500 axes = plot
501 xmin, xmax = axes.get_xlim()
502 ymin, ymax = axes.get_ylim()
503 zmin, zmax = axes.get_zlim()
504 poly_xyzs = []
505 for vertices in self.faces():
506 vertices = self._sort_polygon_3d(vertices)
507 vertices.append(vertices[0])
508 face_xyzs = [tuple(vertex.values()) for vertex in vertices]
509 xs, ys, zs = zip(*face_xyzs)
510 xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
511 ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
512 zmin, zmax = min(zmin, float(min(zs))), max(zmax, float(max(zs)))
513 poly_xyzs.append(face_xyzs)
514 collection = Poly3DCollection(poly_xyzs, **kwargs)
515 axes.add_collection3d(collection)
516 axes.set_xlim(xmin, xmax)
517 axes.set_ylim(ymin, ymax)
518 axes.set_zlim(zmin, zmax)
519 return axes
520
521
522 def plot(self, plot=None, **kwargs):
523 """
524 Display plot of this set.
525 """
526 if not self.isbounded():
527 raise ValueError('domain must be bounded')
528 elif self.dimension == 2:
529 return self._plot_2d(plot=plot, **kwargs)
530 elif self.dimension == 3:
531 return self._plot_3d(plot=plot, **kwargs)
532 else:
533 raise ValueError('polyhedron must be 2 or 3-dimensional')
534
535 def __contains__(self, point):
536 for polyhedron in self.polyhedra:
537 if point in polyhedron:
538 return True
539 return False
540
541 def subs(self, symbol, expression=None):
542 """
543 Subsitute the given value into an expression and return the resulting
544 expression.
545 """
546 polyhedra = [polyhedron.subs(symbol, expression)
547 for polyhedron in self.polyhedra]
548 return Domain(*polyhedra)
549
550 @classmethod
551 def _fromislset(cls, islset, symbols):
552 from .polyhedra import Polyhedron
553 islset = libisl.isl_set_remove_divs(islset)
554 islbsets = islhelper.isl_set_basic_sets(islset)
555 libisl.isl_set_free(islset)
556 polyhedra = []
557 for islbset in islbsets:
558 polyhedron = Polyhedron._fromislbasicset(islbset, symbols)
559 polyhedra.append(polyhedron)
560 if len(polyhedra) == 0:
561 from .polyhedra import Empty
562 return Empty
563 elif len(polyhedra) == 1:
564 return polyhedra[0]
565 else:
566 self = object().__new__(Domain)
567 self._polyhedra = tuple(polyhedra)
568 self._symbols = cls._xsymbols(polyhedra)
569 self._dimension = len(self._symbols)
570 return self
571
572 @classmethod
573 def _toislset(cls, polyhedra, symbols):
574 polyhedron = polyhedra[0]
575 islbset = polyhedron._toislbasicset(polyhedron.equalities,
576 polyhedron.inequalities, symbols)
577 islset1 = libisl.isl_set_from_basic_set(islbset)
578 for polyhedron in polyhedra[1:]:
579 islbset = polyhedron._toislbasicset(polyhedron.equalities,
580 polyhedron.inequalities, symbols)
581 islset2 = libisl.isl_set_from_basic_set(islbset)
582 islset1 = libisl.isl_set_union(islset1, islset2)
583 return islset1
584
585 @classmethod
586 def _fromast(cls, node):
587 from .polyhedra import Polyhedron
588 if isinstance(node, ast.Module) and len(node.body) == 1:
589 return cls._fromast(node.body[0])
590 elif isinstance(node, ast.Expr):
591 return cls._fromast(node.value)
592 elif isinstance(node, ast.UnaryOp):
593 domain = cls._fromast(node.operand)
594 if isinstance(node.operand, ast.invert):
595 return Not(domain)
596 elif isinstance(node, ast.BinOp):
597 domain1 = cls._fromast(node.left)
598 domain2 = cls._fromast(node.right)
599 if isinstance(node.op, ast.BitAnd):
600 return And(domain1, domain2)
601 elif isinstance(node.op, ast.BitOr):
602 return Or(domain1, domain2)
603 elif isinstance(node, ast.Compare):
604 equalities = []
605 inequalities = []
606 left = Expression._fromast(node.left)
607 for i in range(len(node.ops)):
608 op = node.ops[i]
609 right = Expression._fromast(node.comparators[i])
610 if isinstance(op, ast.Lt):
611 inequalities.append(right - left - 1)
612 elif isinstance(op, ast.LtE):
613 inequalities.append(right - left)
614 elif isinstance(op, ast.Eq):
615 equalities.append(left - right)
616 elif isinstance(op, ast.GtE):
617 inequalities.append(left - right)
618 elif isinstance(op, ast.Gt):
619 inequalities.append(left - right - 1)
620 else:
621 break
622 left = right
623 else:
624 return Polyhedron(equalities, inequalities)
625 raise SyntaxError('invalid syntax')
626
627 _RE_BRACES = re.compile(r'^\{\s*|\s*\}$')
628 _RE_EQ = re.compile(r'([^<=>])=([^<=>])')
629 _RE_AND = re.compile(r'\band\b|,|&&|/\\|∧|∩')
630 _RE_OR = re.compile(r'\bor\b|;|\|\||\\/|∨|∪')
631 _RE_NOT = re.compile(r'\bnot\b|!|¬')
632 _RE_NUM_VAR = Expression._RE_NUM_VAR
633 _RE_OPERATORS = re.compile(r'(&|\||~)')
634
635 @classmethod
636 def fromstring(cls, string):
637 # remove curly brackets
638 string = cls._RE_BRACES.sub(r'', string)
639 # replace '=' by '=='
640 string = cls._RE_EQ.sub(r'\1==\2', string)
641 # replace 'and', 'or', 'not'
642 string = cls._RE_AND.sub(r' & ', string)
643 string = cls._RE_OR.sub(r' | ', string)
644 string = cls._RE_NOT.sub(r' ~', string)
645 # add implicit multiplication operators, e.g. '5x' -> '5*x'
646 string = cls._RE_NUM_VAR.sub(r'\1*\2', string)
647 # add parentheses to force precedence
648 tokens = cls._RE_OPERATORS.split(string)
649 for i, token in enumerate(tokens):
650 if i % 2 == 0:
651 token = '({})'.format(token)
652 tokens[i] = token
653 string = ''.join(tokens)
654 tree = ast.parse(string, 'eval')
655 return cls._fromast(tree)
656
657 def __repr__(self):
658 assert len(self.polyhedra) >= 2
659 strings = [repr(polyhedron) for polyhedron in self.polyhedra]
660 return 'Or({})'.format(', '.join(strings))
661
662 def _repr_latex_(self):
663 strings = []
664 for polyhedron in self.polyhedra:
665 strings.append('({})'.format(polyhedron._repr_latex_().strip('$')))
666 return '${}$'.format(' \\vee '.join(strings))
667
668 @classmethod
669 def fromsympy(cls, expr):
670 import sympy
671 from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
672 funcmap = {
673 sympy.And: And, sympy.Or: Or, sympy.Not: Not,
674 sympy.Lt: Lt, sympy.Le: Le,
675 sympy.Eq: Eq, sympy.Ne: Ne,
676 sympy.Ge: Ge, sympy.Gt: Gt,
677 }
678 if expr.func in funcmap:
679 args = [Domain.fromsympy(arg) for arg in expr.args]
680 return funcmap[expr.func](*args)
681 elif isinstance(expr, sympy.Expr):
682 return Expression.fromsympy(expr)
683 raise ValueError('non-domain expression: {!r}'.format(expr))
684
685 def tosympy(self):
686 import sympy
687 polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
688 return sympy.Or(*polyhedra)
689
690
691 def And(*domains):
692 """
693 Return the intersection of two sets as a new set.
694 """
695 if len(domains) == 0:
696 from .polyhedra import Universe
697 return Universe
698 else:
699 return domains[0].intersection(*domains[1:])
700
701 def Or(*domains):
702 """
703 Return the union of sets as a new set.
704 """
705 if len(domains) == 0:
706 from .polyhedra import Empty
707 return Empty
708 else:
709 return domains[0].union(*domains[1:])
710
711 def Not(domain):
712 """
713 Returns the complement of this set.
714 """
715 return ~domain