pypol/domains.py

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