X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/98936866ae400d45b7b74f7ba0d04c66ace0424f..dc449ca80b432de202188a4300ef990abeb968a1:/pypol/domains.py~

diff --git a/pypol/domains.py~ b/pypol/domains.py~
deleted file mode 100644
index be9dd4b..0000000
--- a/pypol/domains.py~
+++ /dev/null
@@ -1,695 +0,0 @@
-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