X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/82e100a9d5e7db532fda649849dc784148e55069..25ce908cffca380f930182a77c1e5a4491042a1c:/pypol/domains.py~

diff --git a/pypol/domains.py~ b/pypol/domains.py~
new file mode 100644
index 0000000..be9dd4b
--- /dev/null
+++ b/pypol/domains.py~
@@ -0,0 +1,695 @@
+import ast
+import functools
+import re
+import math
+
+from fractions import Fraction
+
+from . import islhelper
+from .islhelper import mainctx, libisl
+from .linexprs import Expression, Symbol, Rational
+from .geometry import GeometricObject, Point, Vector
+
+
+__all__ = [
+    'Domain',
+    'And', 'Or', 'Not',
+]
+
+
+@functools.total_ordering
+class Domain(GeometricObject):
+
+    __slots__ = (
+        '_polyhedra',
+        '_symbols',
+        '_dimension',
+    )
+
+    def __new__(cls, *polyhedra):
+        from .polyhedra import Polyhedron
+        if len(polyhedra) == 1:
+            argument = polyhedra[0]
+            if isinstance(argument, str):
+                return cls.fromstring(argument)
+            elif isinstance(argument, GeometricObject):
+                return argument.aspolyhedron()
+            else:
+                raise TypeError('argument must be a string '
+                    'or a GeometricObject instance')
+        else:
+            for polyhedron in polyhedra:
+                if not isinstance(polyhedron, Polyhedron):
+                    raise TypeError('arguments must be Polyhedron instances')
+            symbols = cls._xsymbols(polyhedra)
+            islset = cls._toislset(polyhedra, symbols)
+            return cls._fromislset(islset, symbols)
+
+    @classmethod
+    def _xsymbols(cls, iterator):
+        """
+        Return the ordered tuple of symbols present in iterator.
+        """
+        symbols = set()
+        for item in iterator:
+            symbols.update(item.symbols)
+        return tuple(sorted(symbols, key=Symbol.sortkey))
+
+    @property
+    def polyhedra(self):
+        return self._polyhedra
+
+    @property
+    def symbols(self):
+        return self._symbols
+
+    @property
+    def dimension(self):
+        return self._dimension
+
+    def disjoint(self):
+        """
+        Returns this set as disjoint.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_make_disjoint(mainctx, islset)
+        return self._fromislset(islset, self.symbols)
+
+    def isempty(self):
+        """
+        Returns true if this set is an Empty set.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        empty = bool(libisl.isl_set_is_empty(islset))
+        libisl.isl_set_free(islset)
+        return empty
+
+    def __bool__(self):
+        return not self.isempty()
+
+    def isuniverse(self):
+        """
+        Returns true if this set is the Universe set.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        universe = bool(libisl.isl_set_plain_is_universe(islset))
+        libisl.isl_set_free(islset)
+        return universe
+
+    def isbounded(self):
+        """
+        Returns true if this set is bounded.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        bounded = bool(libisl.isl_set_is_bounded(islset))
+        libisl.isl_set_free(islset)
+        return bounded
+
+    def __eq__(self, other):
+        """
+        Returns true if two sets are equal.
+        """
+        symbols = self._xsymbols([self, other])
+        islset1 = self._toislset(self.polyhedra, symbols)
+        islset2 = other._toislset(other.polyhedra, symbols)
+        equal = bool(libisl.isl_set_is_equal(islset1, islset2))
+        libisl.isl_set_free(islset1)
+        libisl.isl_set_free(islset2)
+        return equal
+
+    def isdisjoint(self, other):
+        """
+        Return True if two sets have a null intersection.
+        """
+        symbols = self._xsymbols([self, other])
+        islset1 = self._toislset(self.polyhedra, symbols)
+        islset2 = self._toislset(other.polyhedra, symbols)
+        equal = bool(libisl.isl_set_is_disjoint(islset1, islset2))
+        libisl.isl_set_free(islset1)
+        libisl.isl_set_free(islset2)
+        return equal
+
+    def issubset(self, other):
+        """
+        Report whether another set contains this set.
+        """
+        symbols = self._xsymbols([self, other])
+        islset1 = self._toislset(self.polyhedra, symbols)
+        islset2 = self._toislset(other.polyhedra, symbols)
+        equal = bool(libisl.isl_set_is_subset(islset1, islset2))
+        libisl.isl_set_free(islset1)
+        libisl.isl_set_free(islset2)
+        return equal
+
+    def __le__(self, other):
+        """
+        Returns true if this set is less than or equal to another set.
+        """
+        return self.issubset(other)
+
+    def __lt__(self, other):
+        """
+        Returns true if this set is less than another set.
+        """
+        symbols = self._xsymbols([self, other])
+        islset1 = self._toislset(self.polyhedra, symbols)
+        islset2 = self._toislset(other.polyhedra, symbols)
+        equal = bool(libisl.isl_set_is_strict_subset(islset1, islset2))
+        libisl.isl_set_free(islset1)
+        libisl.isl_set_free(islset2)
+        return equal
+
+    def complement(self):
+        """
+        Returns the complement of this set.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_complement(islset)
+        return self._fromislset(islset, self.symbols)
+
+    def __invert__(self):
+        """
+        Returns the complement of this set.
+        """
+        return self.complement()
+
+    def simplify(self):
+        """
+        Returns a set without redundant constraints.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_remove_redundancies(islset)
+        return self._fromislset(islset, self.symbols)
+
+    def aspolyhedron(self):
+        """
+        Returns polyhedral hull of set.
+        """
+        from .polyhedra import Polyhedron
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islbset = libisl.isl_set_polyhedral_hull(islset)
+        return Polyhedron._fromislbasicset(islbset, self.symbols)
+
+    def asdomain(self):
+        return self
+
+    def project(self, dims):
+        """
+        Return new set with given dimensions removed.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        n = 0
+        for index, symbol in reversed(list(enumerate(self.symbols))):
+            if symbol in dims:
+                n += 1
+            elif n > 0:
+                islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, index + 1, n)
+                n = 0
+        if n > 0:
+            islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, 0, n)
+        dims = [symbol for symbol in self.symbols if symbol not in dims]
+        return Domain._fromislset(islset, dims)
+
+    def sample(self):
+        """
+        Returns a single subset of the input.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islpoint = libisl.isl_set_sample_point(islset)
+        if bool(libisl.isl_point_is_void(islpoint)):
+            libisl.isl_point_free(islpoint)
+            raise ValueError('domain must be non-empty')
+        point = {}
+        for index, symbol in enumerate(self.symbols):
+            coordinate = libisl.isl_point_get_coordinate_val(islpoint,
+                libisl.isl_dim_set, index)
+            coordinate = islhelper.isl_val_to_int(coordinate)
+            point[symbol] = coordinate
+        libisl.isl_point_free(islpoint)
+        return point
+
+    def intersection(self, *others):
+        """
+         Return the intersection of two sets as a new set.
+        """
+        if len(others) == 0:
+            return self
+        symbols = self._xsymbols((self,) + others)
+        islset1 = self._toislset(self.polyhedra, symbols)
+        for other in others:
+            islset2 = other._toislset(other.polyhedra, symbols)
+            islset1 = libisl.isl_set_intersect(islset1, islset2)
+        return self._fromislset(islset1, symbols)
+
+    def __and__(self, other):
+        """
+         Return the intersection of two sets as a new set.
+        """
+        return self.intersection(other)
+
+    def union(self, *others):
+        """
+        Return the union of sets as a new set.
+        """
+        if len(others) == 0:
+            return self
+        symbols = self._xsymbols((self,) + others)
+        islset1 = self._toislset(self.polyhedra, symbols)
+        for other in others:
+            islset2 = other._toislset(other.polyhedra, symbols)
+            islset1 = libisl.isl_set_union(islset1, islset2)
+        return self._fromislset(islset1, symbols)
+
+    def __or__(self, other):
+        """
+        Return a new set with elements from both sets.
+        """
+        return self.union(other)
+
+    def __add__(self, other):
+        """
+        Return new set containing all elements in both sets.
+        """
+        return self.union(other)
+
+    def difference(self, other):
+        """
+        Return the difference of two sets as a new set.
+        """
+        symbols = self._xsymbols([self, other])
+        islset1 = self._toislset(self.polyhedra, symbols)
+        islset2 = other._toislset(other.polyhedra, symbols)
+        islset = libisl.isl_set_subtract(islset1, islset2)
+        return self._fromislset(islset, symbols)
+
+    def __sub__(self, other):
+        """
+        Return the difference of two sets as a new set.
+        """
+        return self.difference(other)
+
+    def lexmin(self):
+        """
+        Return a new set containing the lexicographic minimum of the elements in the set.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_lexmin(islset)
+        return self._fromislset(islset, self.symbols)
+
+    def lexmax(self):
+        """
+        Return a new set containing the lexicographic maximum of the elements in the set.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_lexmax(islset)
+        return self._fromislset(islset, self.symbols)
+
+    def num_parameters(self):
+        """
+        Return the total number of parameters, input, output or set dimensions.
+        """
+        islbset = self._toislbasicset(self.equalities, self.inequalities, self.symbols)
+        num = libisl.isl_basic_set_dim(islbset, libisl.isl_dim_set)
+        return num
+
+    def involves_dims(self, dims):
+        """
+        Returns true if set depends on given dimensions.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        dims = sorted(dims)
+        symbols = sorted(list(self.symbols))
+        n = 0
+        if len(dims)>0:
+            for dim in dims:
+                if dim in symbols:
+                    first = symbols.index(dims[0])
+                    n +=1
+                else:
+                    first = 0
+        else:
+            return False
+        value = bool(libisl.isl_set_involves_dims(islset, libisl.isl_dim_set, first, n))
+        libisl.isl_set_free(islset)
+        return value
+
+    _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
+
+    def vertices(self):
+        """
+        Return a list of vertices for this Polygon.
+        """
+        from .polyhedra import Polyhedron
+        islbset = self._toislbasicset(self.equalities, self.inequalities, self.symbols)
+        vertices = libisl.isl_basic_set_compute_vertices(islbset);
+        vertices = islhelper.isl_vertices_vertices(vertices)
+        points = []
+        for vertex in vertices:
+            expr = libisl.isl_vertex_get_expr(vertex)
+            coordinates = []
+            if islhelper.isl_version < '0.13':
+                constraints = islhelper.isl_basic_set_constraints(expr)
+                for constraint in constraints:
+                    constant = libisl.isl_constraint_get_constant_val(constraint)
+                    constant = islhelper.isl_val_to_int(constant)
+                    for index, symbol in enumerate(self.symbols):
+                        coefficient = libisl.isl_constraint_get_coefficient_val(constraint,
+                            libisl.isl_dim_set, index)
+                        coefficient = islhelper.isl_val_to_int(coefficient)
+                        if coefficient != 0:
+                            coordinate = -Fraction(constant, coefficient)
+                            coordinates.append((symbol, coordinate))
+            else:
+                string = islhelper.isl_multi_aff_to_str(expr)
+                matches = self._RE_COORDINATE.finditer(string)
+                for symbol, match in zip(self.symbols, matches):
+                    numerator = int(match.group('num'))
+                    denominator = match.group('den')
+                    denominator = 1 if denominator is None else int(denominator)
+                    coordinate = Fraction(numerator, denominator)
+                    coordinates.append((symbol, coordinate))
+            points.append(Point(coordinates))
+        return points
+
+    def points(self):
+        """
+        Returns the points contained in the set.
+        """
+        if not self.isbounded():
+            raise ValueError('domain must be bounded')
+        from .polyhedra import Universe, Eq
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islpoints = islhelper.isl_set_points(islset)
+        points = []
+        for islpoint in islpoints:
+            coordinates = {}
+            for index, symbol in enumerate(self.symbols):
+                coordinate = libisl.isl_point_get_coordinate_val(islpoint,
+                    libisl.isl_dim_set, index)
+                coordinate = islhelper.isl_val_to_int(coordinate)
+                coordinates[symbol] = coordinate
+            points.append(Point(coordinates))
+        return points
+
+    @classmethod
+    def _polygon_inner_point(cls, points):
+        symbols = points[0].symbols
+        coordinates = {symbol: 0 for symbol in symbols}
+        for point in points:
+            for symbol, coordinate in point.coordinates():
+                coordinates[symbol] += coordinate
+        for symbol in symbols:
+            coordinates[symbol] /= len(points)
+        return Point(coordinates)
+
+    @classmethod
+    def _sort_polygon_2d(cls, points):
+        if len(points) <= 3:
+            return points
+        o = cls._polygon_inner_point(points)
+        angles = {}
+        for m in points:
+            om = Vector(o, m)
+            dx, dy = (coordinate for symbol, coordinate in om.coordinates())
+            angle = math.atan2(dy, dx)
+            angles[m] = angle
+        return sorted(points, key=angles.get)
+
+    @classmethod
+    def _sort_polygon_3d(cls, points):
+        if len(points) <= 3:
+            return points
+        o = cls._polygon_inner_point(points)
+        a = points[0]
+        oa = Vector(o, a)
+        norm_oa = oa.norm()
+        for b in points[1:]:
+            ob = Vector(o, b)
+            u = oa.cross(ob)
+            if not u.isnull():
+                u = u.asunit()
+                break
+        else:
+            raise ValueError('degenerate polygon')
+        angles = {a: 0.}
+        for m in points[1:]:
+            om = Vector(o, m)
+            normprod = norm_oa * om.norm()
+            cosinus = max(oa.dot(om) / normprod, -1.)
+            sinus = u.dot(oa.cross(om)) / normprod
+            angle = math.acos(cosinus)
+            angle = math.copysign(angle, sinus)
+            angles[m] = angle
+        return sorted(points, key=angles.get)
+
+    def faces(self):
+        faces = []
+        for polyhedron in self.polyhedra:
+            vertices = polyhedron.vertices()
+            for constraint in polyhedron.constraints:
+                face = []
+                for vertex in vertices:
+                    if constraint.subs(vertex.coordinates()) == 0:
+                        face.append(vertex)
+                if len(face) >= 3:
+                    faces.append(face)
+        return faces
+
+    def _plot_2d(self, plot=None, **kwargs):
+        import matplotlib.pyplot as plt
+        from matplotlib.patches import Polygon
+        if plot is None:
+            fig = plt.figure()
+            plot = fig.add_subplot(1, 1, 1)
+        xmin, xmax = plot.get_xlim()
+        ymin, ymax = plot.get_ylim()
+        for polyhedron in self.polyhedra:
+            vertices = polyhedron._sort_polygon_2d(polyhedron.vertices())
+            xys = [tuple(vertex.values()) for vertex in vertices]
+            xs, ys = zip(*xys)
+            xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
+            ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
+            plot.add_patch(Polygon(xys, closed=True, **kwargs))
+        plot.set_xlim(xmin, xmax)
+        plot.set_ylim(ymin, ymax)
+        return plot
+
+    def _plot_3d(self, plot=None, **kwargs):
+        import matplotlib.pyplot as plt
+        from mpl_toolkits.mplot3d import Axes3D
+        from mpl_toolkits.mplot3d.art3d import Poly3DCollection
+        if plot is None:
+            fig = plt.figure()
+            axes = Axes3D(fig)
+        else:
+            axes = plot
+        xmin, xmax = axes.get_xlim()
+        ymin, ymax = axes.get_ylim()
+        zmin, zmax = axes.get_zlim()
+        poly_xyzs = []
+        for vertices in self.faces():
+            vertices = self._sort_polygon_3d(vertices)
+            vertices.append(vertices[0])
+            face_xyzs = [tuple(vertex.values()) for vertex in vertices]
+            xs, ys, zs = zip(*face_xyzs)
+            xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
+            ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
+            zmin, zmax = min(zmin, float(min(zs))), max(zmax, float(max(zs)))
+            poly_xyzs.append(face_xyzs)
+        collection = Poly3DCollection(poly_xyzs, **kwargs)
+        axes.add_collection3d(collection)
+        axes.set_xlim(xmin, xmax)
+        axes.set_ylim(ymin, ymax)
+        axes.set_zlim(zmin, zmax)
+        return axes
+
+    def plot(self, plot=None, **kwargs):
+        """
+        Display plot of this set.
+        """
+        if not self.isbounded():
+            raise ValueError('domain must be bounded')
+        elif self.dimension == 2:
+            return self._plot_2d(plot=plot, **kwargs)
+        elif self.dimension == 3:
+            return self._plot_3d(plot=plot, **kwargs)
+        else:
+            raise ValueError('polyhedron must be 2 or 3-dimensional')
+
+    def __contains__(self, point):
+        for polyhedron in self.polyhedra:
+            if point in polyhedron:
+                return True
+        return False
+
+    def subs(self, symbol, expression=None):
+        polyhedra = [polyhedron.subs(symbol, expression)
+            for polyhedron in self.polyhedra]
+        return Domain(*polyhedra)
+
+    @classmethod
+    def _fromislset(cls, islset, symbols):
+        from .polyhedra import Polyhedron
+        islset = libisl.isl_set_remove_divs(islset)
+        islbsets = islhelper.isl_set_basic_sets(islset)
+        libisl.isl_set_free(islset)
+        polyhedra = []
+        for islbset in islbsets:
+            polyhedron = Polyhedron._fromislbasicset(islbset, symbols)
+            polyhedra.append(polyhedron)
+        if len(polyhedra) == 0:
+            from .polyhedra import Empty
+            return Empty
+        elif len(polyhedra) == 1:
+            return polyhedra[0]
+        else:
+            self = object().__new__(Domain)
+            self._polyhedra = tuple(polyhedra)
+            self._symbols = cls._xsymbols(polyhedra)
+            self._dimension = len(self._symbols)
+            return self
+
+    @classmethod
+    def _toislset(cls, polyhedra, symbols):
+        polyhedron = polyhedra[0]
+        islbset = polyhedron._toislbasicset(polyhedron.equalities,
+            polyhedron.inequalities, symbols)
+        islset1 = libisl.isl_set_from_basic_set(islbset)
+        for polyhedron in polyhedra[1:]:
+            islbset = polyhedron._toislbasicset(polyhedron.equalities,
+                polyhedron.inequalities, symbols)
+            islset2 = libisl.isl_set_from_basic_set(islbset)
+            islset1 = libisl.isl_set_union(islset1, islset2)
+        return islset1
+
+    @classmethod
+    def _fromast(cls, node):
+        from .polyhedra import Polyhedron
+        if isinstance(node, ast.Module) and len(node.body) == 1:
+            return cls._fromast(node.body[0])
+        elif isinstance(node, ast.Expr):
+            return cls._fromast(node.value)
+        elif isinstance(node, ast.UnaryOp):
+            domain = cls._fromast(node.operand)
+            if isinstance(node.operand, ast.invert):
+                return Not(domain)
+        elif isinstance(node, ast.BinOp):
+            domain1 = cls._fromast(node.left)
+            domain2 = cls._fromast(node.right)
+            if isinstance(node.op, ast.BitAnd):
+                return And(domain1, domain2)
+            elif isinstance(node.op, ast.BitOr):
+                return Or(domain1, domain2)
+        elif isinstance(node, ast.Compare):
+            equalities = []
+            inequalities = []
+            left = Expression._fromast(node.left)
+            for i in range(len(node.ops)):
+                op = node.ops[i]
+                right = Expression._fromast(node.comparators[i])
+                if isinstance(op, ast.Lt):
+                    inequalities.append(right - left - 1)
+                elif isinstance(op, ast.LtE):
+                    inequalities.append(right - left)
+                elif isinstance(op, ast.Eq):
+                    equalities.append(left - right)
+                elif isinstance(op, ast.GtE):
+                    inequalities.append(left - right)
+                elif isinstance(op, ast.Gt):
+                    inequalities.append(left - right - 1)
+                else:
+                    break
+                left = right
+            else:
+                return Polyhedron(equalities, inequalities)
+        raise SyntaxError('invalid syntax')
+
+    _RE_BRACES = re.compile(r'^\{\s*|\s*\}$')
+    _RE_EQ = re.compile(r'([^<=>])=([^<=>])')
+    _RE_AND = re.compile(r'\band\b|,|&&|/\\|∧|∩')
+    _RE_OR = re.compile(r'\bor\b|;|\|\||\\/|∨|∪')
+    _RE_NOT = re.compile(r'\bnot\b|!|¬')
+    _RE_NUM_VAR = Expression._RE_NUM_VAR
+    _RE_OPERATORS = re.compile(r'(&|\||~)')
+
+    @classmethod
+    def fromstring(cls, string):
+        # remove curly brackets
+        string = cls._RE_BRACES.sub(r'', string)
+        # replace '=' by '=='
+        string = cls._RE_EQ.sub(r'\1==\2', string)
+        # replace 'and', 'or', 'not'
+        string = cls._RE_AND.sub(r' & ', string)
+        string = cls._RE_OR.sub(r' | ', string)
+        string = cls._RE_NOT.sub(r' ~', string)
+        # add implicit multiplication operators, e.g. '5x' -> '5*x'
+        string = cls._RE_NUM_VAR.sub(r'\1*\2', string)
+        # add parentheses to force precedence
+        tokens = cls._RE_OPERATORS.split(string)
+        for i, token in enumerate(tokens):
+            if i % 2 == 0:
+                token = '({})'.format(token)
+                tokens[i] = token
+        string = ''.join(tokens)
+        tree = ast.parse(string, 'eval')
+        return cls._fromast(tree)
+
+    def __repr__(self):
+        assert len(self.polyhedra) >= 2
+        strings = [repr(polyhedron) for polyhedron in self.polyhedra]
+        return 'Or({})'.format(', '.join(strings))
+
+    def _repr_latex_(self):
+        strings = []
+        for polyhedron in self.polyhedra:
+            strings.append('({})'.format(polyhedron._repr_latex_().strip('$')))
+        return '${}$'.format(' \\vee '.join(strings))
+
+    @classmethod
+    def fromsympy(cls, expr):
+        import sympy
+        from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
+        funcmap = {
+            sympy.And: And, sympy.Or: Or, sympy.Not: Not,
+            sympy.Lt: Lt, sympy.Le: Le,
+            sympy.Eq: Eq, sympy.Ne: Ne,
+            sympy.Ge: Ge, sympy.Gt: Gt,
+        }
+        if expr.func in funcmap:
+            args = [Domain.fromsympy(arg) for arg in expr.args]
+            return funcmap[expr.func](*args)
+        elif isinstance(expr, sympy.Expr):
+            return Expression.fromsympy(expr)
+        raise ValueError('non-domain expression: {!r}'.format(expr))
+
+    def tosympy(self):
+        import sympy
+        polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
+        return sympy.Or(*polyhedra)
+
+
+def And(*domains):
+    """
+    Return the intersection of two sets as a new set.
+    """
+    if len(domains) == 0:
+        from .polyhedra import Universe
+        return Universe
+    else:
+        return domains[0].intersection(*domains[1:])
+
+def Or(*domains):
+    """
+    Return the union of sets as a new set.
+    """
+    if len(domains) == 0:
+        from .polyhedra import Empty
+        return Empty
+    else:
+        return domains[0].union(*domains[1:])
+
+def Not(domain):
+    """
+    Returns the complement of this set.
+    """
+    return ~domain