X-Git-Url: https://svn.cri.ensmp.fr/git/linpy.git/blobdiff_plain/7afcb0a1ded9e9a331b131689d68c085f712143f..d585b06ccf67b2837519f4b48c6800dcdb924d9d:/linpy/domains.py?ds=inline

diff --git a/linpy/domains.py b/linpy/domains.py
index 238248b..7f4fa65 100644
--- a/linpy/domains.py
+++ b/linpy/domains.py
@@ -15,13 +15,6 @@
 # You should have received a copy of the GNU General Public License
 # along with LinPy.  If not, see <http://www.gnu.org/licenses/>.
 
-"""
-Polyhedral domains
-
-This module provides classes and functions to deal with polyhedral
-domains, i.e. unions of polyhedra.
-"""
-
 import ast
 import functools
 import re
@@ -30,21 +23,27 @@ import math
 from fractions import Fraction
 
 from . import islhelper
-from .islhelper import mainctx, libisl
-from .linexprs import LinExpr, Symbol, Rational
 from .geometry import GeometricObject, Point, Vector
+from .islhelper import libisl
+from .linexprs import LinExpr, Symbol
 
 
 __all__ = [
+    'And',
     'Domain',
-    'And', 'Or', 'Not',
+    'Not',
+    'Or',
 ]
 
 
 @functools.total_ordering
 class Domain(GeometricObject):
     """
-    This class represents polyhedral domains, i.e. unions of polyhedra.
+    A domain is a union of polyhedra. Unlike polyhedra, domains allow exact
+    computation of union, subtraction and complementary operations.
+
+    A domain with a unique polyhedron is automatically subclassed as a
+    Polyhedron instance.
     """
 
     __slots__ = (
@@ -55,7 +54,28 @@ class Domain(GeometricObject):
 
     def __new__(cls, *polyhedra):
         """
-        Create and return a new domain from a string or a list of polyhedra.
+        Return a domain from a sequence of polyhedra.
+
+        >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+        >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+        >>> dom = Domain(square1, square2)
+        >>> dom
+        Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y),
+           And(x <= 3, 1 <= x, y <= 3, 1 <= y))
+
+        It is also possible to build domains from polyhedra using arithmetic
+        operators Domain.__or__(), Domain.__invert__() or functions Or() and
+        Not(), using one of the following instructions:
+
+        >>> dom = square1 | square2
+        >>> dom = Or(square1, square2)
+
+        Alternatively, a domain can be built from a string:
+
+        >>> dom = Domain('0 <= x <= 2, 0 <= y <= 2; 1 <= x <= 3, 1 <= y <= 3')
+
+        Finally, a domain can be built from a GeometricObject instance, calling
+        the GeometricObject.asdomain() method.
         """
         from .polyhedra import Polyhedron
         if len(polyhedra) == 1:
@@ -66,7 +86,7 @@ class Domain(GeometricObject):
                 return argument.aspolyhedron()
             else:
                 raise TypeError('argument must be a string '
-                    'or a GeometricObject instance')
+                                'or a GeometricObject instance')
         else:
             for polyhedron in polyhedra:
                 if not isinstance(polyhedron, Polyhedron):
@@ -88,35 +108,28 @@ class Domain(GeometricObject):
     @property
     def polyhedra(self):
         """
-        The tuple of polyhedra which constitute the domain.
+        The tuple of polyhedra present in the domain.
         """
         return self._polyhedra
 
     @property
     def symbols(self):
         """
-        The tuple of symbols present in the domain equations.
+        The tuple of symbols present in the domain equations, sorted according
+        to Symbol.sortkey().
         """
         return self._symbols
 
     @property
     def dimension(self):
         """
-        The dimension of the domain, i.e. the number of symbols.
+        The dimension of the domain, i.e. the number of symbols present in it.
         """
         return self._dimension
 
-    def make_disjoint(self):
-        """
-        Return an equivalent domain, whose polyhedra are 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):
         """
-        Return True if the domain is empty.
+        Return True if the domain is empty, that is, equal to Empty.
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         empty = bool(libisl.isl_set_is_empty(islset))
@@ -131,7 +144,7 @@ class Domain(GeometricObject):
 
     def isuniverse(self):
         """
-        Return True if the domain is universal, i.e. with no constraint.
+        Return True if the domain is universal, that is, equal to Universe.
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         universe = bool(libisl.isl_set_plain_is_universe(islset))
@@ -149,20 +162,24 @@ class Domain(GeometricObject):
 
     def __eq__(self, other):
         """
-        Return True if the two domains are equal.
+        Return True if two domains 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
+        if isinstance(other, Domain):
+            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
+        return NotImplemented
 
     def isdisjoint(self, other):
         """
         Return True if two domains have a null intersection.
         """
+        if not isinstance(other, Domain):
+            raise TypeError('other must be a Domain instance')
         symbols = self._xsymbols([self, other])
         islset1 = self._toislset(self.polyhedra, symbols)
         islset2 = self._toislset(other.polyhedra, symbols)
@@ -175,29 +192,33 @@ class Domain(GeometricObject):
         """
         Report whether another domain contains the domain.
         """
-        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
+        return self < other
 
     def __le__(self, other):
-        return self.issubset(other)
+        if isinstance(other, Domain):
+            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
+        return NotImplemented
     __le__.__doc__ = issubset.__doc__
 
     def __lt__(self, other):
         """
         Report whether another domain is contained within the domain.
         """
-        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
+        if isinstance(other, Domain):
+            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
+        return NotImplemented
 
     def complement(self):
         """
@@ -211,10 +232,18 @@ class Domain(GeometricObject):
         return self.complement()
     __invert__.__doc__ = complement.__doc__
 
+    def make_disjoint(self):
+        """
+        Return an equivalent domain, whose polyhedra are disjoint.
+        """
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_make_disjoint(islset)
+        return self._fromislset(islset, self.symbols)
+
     def coalesce(self):
         """
         Simplify the representation of the domain by trying to combine pairs of
-        polyhedra into a single polyhedron.
+        polyhedra into a single polyhedron, and return the resulting domain.
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_coalesce(islset)
@@ -223,7 +252,7 @@ class Domain(GeometricObject):
     def detect_equalities(self):
         """
         Simplify the representation of the domain by detecting implicit
-        equalities.
+        equalities, and return the resulting domain.
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_detect_equalities(islset)
@@ -231,16 +260,14 @@ class Domain(GeometricObject):
 
     def remove_redundancies(self):
         """
-        Remove redundant constraints in the domain.
+        Remove redundant constraints in the domain, and return the resulting
+        domain.
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_remove_redundancies(islset)
         return self._fromislset(islset, self.symbols)
 
     def aspolyhedron(self):
-        """
-        Return the polyhedral hull of the domain.
-        """
         from .polyhedra import Polyhedron
         islset = self._toislset(self.polyhedra, self.symbols)
         islbset = libisl.isl_set_polyhedral_hull(islset)
@@ -251,25 +278,32 @@ class Domain(GeometricObject):
 
     def project(self, symbols):
         """
-        Project out the symbols given in arguments.
+        Project out the sequence of symbols given in arguments, and return the
+        resulting domain.
         """
+        symbols = list(symbols)
+        for symbol in symbols:
+            if not isinstance(symbol, Symbol):
+                raise TypeError('symbols must be Symbol instances')
         islset = self._toislset(self.polyhedra, self.symbols)
         n = 0
         for index, symbol in reversed(list(enumerate(self.symbols))):
             if symbol in symbols:
                 n += 1
             elif n > 0:
-                islset = libisl.isl_set_project_out(islset,
-                    libisl.isl_dim_set, index + 1, n)
+                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)
+            islset = libisl.isl_set_project_out(
+                islset, libisl.isl_dim_set, 0, n)
         symbols = [symbol for symbol in self.symbols if symbol not in symbols]
         return Domain._fromislset(islset, symbols)
 
     def sample(self):
         """
-        Return a sample of the domain.
+        Return a sample of the domain, as an integer instance of Point. If the
+        domain is empty, a ValueError exception is raised.
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islpoint = libisl.isl_set_sample_point(islset)
@@ -278,8 +312,8 @@ class Domain(GeometricObject):
             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 = 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)
@@ -287,54 +321,62 @@ class Domain(GeometricObject):
 
     def intersection(self, *others):
         """
-        Return the intersection of two domains as a new domain.
+        Return the intersection of two or more domains as a new domain. As an
+        alternative, function And() can be used.
         """
-        if len(others) == 0:
-            return self
-        symbols = self._xsymbols((self,) + others)
-        islset1 = self._toislset(self.polyhedra, symbols)
+        result = self
         for other in others:
-            islset2 = other._toislset(other.polyhedra, symbols)
-            islset1 = libisl.isl_set_intersect(islset1, islset2)
-        return self._fromislset(islset1, symbols)
+            result &= other
+        return result
 
     def __and__(self, other):
-        return self.intersection(other)
+        if isinstance(other, Domain):
+            symbols = self._xsymbols([self, other])
+            islset1 = self._toislset(self.polyhedra, symbols)
+            islset2 = other._toislset(other.polyhedra, symbols)
+            islset = libisl.isl_set_intersect(islset1, islset2)
+            return self._fromislset(islset, symbols)
+        return NotImplemented
     __and__.__doc__ = intersection.__doc__
 
     def union(self, *others):
         """
-        Return the union of two domains as a new domain.
+        Return the union of two or more domains as a new domain. As an
+        alternative, function Or() can be used.
         """
-        if len(others) == 0:
-            return self
-        symbols = self._xsymbols((self,) + others)
-        islset1 = self._toislset(self.polyhedra, symbols)
+        result = self
         for other in others:
-            islset2 = other._toislset(other.polyhedra, symbols)
-            islset1 = libisl.isl_set_union(islset1, islset2)
-        return self._fromislset(islset1, symbols)
+            result |= other
+        return result
 
     def __or__(self, other):
-        return self.union(other)
+        if isinstance(other, Domain):
+            symbols = self._xsymbols([self, other])
+            islset1 = self._toislset(self.polyhedra, symbols)
+            islset2 = other._toislset(other.polyhedra, symbols)
+            islset = libisl.isl_set_union(islset1, islset2)
+            return self._fromislset(islset, symbols)
+        return NotImplemented
     __or__.__doc__ = union.__doc__
 
     def __add__(self, other):
-        return self.union(other)
+        return self | other
     __add__.__doc__ = union.__doc__
 
     def difference(self, other):
         """
         Return the difference of two domains as a new domain.
         """
-        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)
+        return self - other
 
     def __sub__(self, other):
-        return self.difference(other)
+        if isinstance(other, Domain):
+            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)
+        return NotImplemented
     __sub__.__doc__ = difference.__doc__
 
     def lexmin(self):
@@ -353,42 +395,48 @@ class Domain(GeometricObject):
         islset = libisl.isl_set_lexmax(islset)
         return self._fromislset(islset, self.symbols)
 
-    _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
+    if islhelper.isl_version >= '0.13':
+        _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
+    else:
+        _RE_COORDINATE = None
 
     def vertices(self):
         """
-        Return the vertices of the domain.
+        Return the vertices of the domain, as a list of rational instances of
+        Point.
         """
-        from .polyhedra import Polyhedron
         if not self.isbounded():
             raise ValueError('domain must be bounded')
         islbset = self._toislbasicset(self.equalities, self.inequalities,
-            self.symbols)
-        vertices = libisl.isl_basic_set_compute_vertices(islbset);
+                                      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)
+            expression = libisl.isl_vertex_get_expr(vertex)
             coordinates = []
-            if islhelper.isl_version < '0.13':
-                constraints = islhelper.isl_basic_set_constraints(expr)
+            if self._RE_COORDINATE is None:
+                constraints = islhelper.isl_basic_set_constraints(expression)
                 for constraint in constraints:
-                    constant = libisl.isl_constraint_get_constant_val(constraint)
+                    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 = \
+                            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)
+                string = islhelper.isl_multi_aff_to_str(expression)
                 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)
+                    denominator = \
+                        1 if denominator is None else int(denominator)
                     coordinate = Fraction(numerator, denominator)
                     coordinates.append((symbol, coordinate))
             points.append(Point(coordinates))
@@ -396,24 +444,34 @@ class Domain(GeometricObject):
 
     def points(self):
         """
-        Return the points with integer coordinates contained in the domain.
+        Return the integer points of a bounded domain, as a list of integer
+        instances of Point. If the domain is not bounded, a ValueError
+        exception is raised.
         """
         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 = 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
 
+    def __contains__(self, point):
+        """
+        Return True if the point if contained within the domain.
+        """
+        for polyhedron in self.polyhedra:
+            if point in polyhedron:
+                return True
+        return False
+
     @classmethod
     def _polygon_inner_point(cls, points):
         symbols = points[0].symbols
@@ -467,7 +525,9 @@ class Domain(GeometricObject):
 
     def faces(self):
         """
-        Return the vertices of the domain, grouped by face.
+        Return the list of faces of a bounded domain. Each face is represented
+        by a list of vertices, in the form of rational instances of Point. If
+        the domain is not bounded, a ValueError exception is raised.
         """
         faces = []
         for polyhedron in self.polyhedra:
@@ -531,7 +591,11 @@ class Domain(GeometricObject):
 
     def plot(self, plot=None, **kwargs):
         """
-        Plot the domain using matplotlib.
+        Plot a 2D or 3D domain using matplotlib. Draw it to the current plot
+        object if present, otherwise create a new one. options are keyword
+        arguments passed to the matplotlib drawing functions, they can be used
+        to set the drawing color for example. Raise ValueError is the domain is
+        not 2D or 3D.
         """
         if not self.isbounded():
             raise ValueError('domain must be bounded')
@@ -540,24 +604,17 @@ class Domain(GeometricObject):
         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):
-        """
-        Return True if point if contained within the domain.
-        """
-        for polyhedron in self.polyhedra:
-            if point in polyhedron:
-                return True
-        return False
+            raise ValueError('domain must be 2 or 3-dimensional')
 
     def subs(self, symbol, expression=None):
         """
-        Subsitute symbol by expression in equations and return the resulting
-        domain.
+        Substitute the given symbol by an expression in the domain constraints.
+        To perform multiple substitutions at once, pass a sequence or a
+        dictionary of (old, new) pairs to subs. The syntax of this function is
+        similar to LinExpr.subs().
         """
         polyhedra = [polyhedron.subs(symbol, expression)
-            for polyhedron in self.polyhedra]
+                     for polyhedron in self.polyhedra]
         return Domain(*polyhedra)
 
     @classmethod
@@ -585,12 +642,12 @@ class Domain(GeometricObject):
     @classmethod
     def _toislset(cls, polyhedra, symbols):
         polyhedron = polyhedra[0]
-        islbset = polyhedron._toislbasicset(polyhedron.equalities,
-            polyhedron.inequalities, symbols)
+        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)
+            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
@@ -648,19 +705,20 @@ class Domain(GeometricObject):
     @classmethod
     def fromstring(cls, string):
         """
-        Convert a string into a domain.
+        Create a domain from a string. Raise SyntaxError if the string is not
+        properly formatted.
         """
-        # remove curly brackets
+        # Remove curly brackets.
         string = cls._RE_BRACES.sub(r'', string)
-        # replace '=' by '=='
+        # Replace '=' by '=='.
         string = cls._RE_EQ.sub(r'\1==\2', string)
-        # replace 'and', 'or', 'not'
+        # 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'
+        # Add implicit multiplication operators, e.g. '5x' -> '5*x'.
         string = cls._RE_NUM_VAR.sub(r'\1*\2', string)
-        # add parentheses to force precedence
+        # Add parentheses to force precedence.
         tokens = cls._RE_OPERATORS.split(string)
         for i, token in enumerate(tokens):
             if i % 2 == 0:
@@ -671,23 +729,14 @@ class Domain(GeometricObject):
         return cls._fromast(tree)
 
     def __repr__(self):
-        """
-        Return 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):
+    def fromsympy(cls, expression):
         """
-        Convert a SymPy expression into a domain.
+        Create a domain from a SymPy expression.
         """
         import sympy
         from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
@@ -697,38 +746,46 @@ class Domain(GeometricObject):
             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 LinExpr.fromsympy(expr)
-        raise ValueError('non-domain expression: {!r}'.format(expr))
+        if expression.func in funcmap:
+            args = [Domain.fromsympy(arg) for arg in expression.args]
+            return funcmap[expression.func](*args)
+        elif isinstance(expression, sympy.Expr):
+            return LinExpr.fromsympy(expression)
+        raise ValueError('non-domain expression: {!r}'.format(expression))
 
     def tosympy(self):
         """
-        Convert a domain into a SymPy expression.
+        Convert the domain to a SymPy expression.
         """
         import sympy
-        polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
+        polyhedra = [polyhedron.tosympy() for polyhedron in self.polyhedra]
         return sympy.Or(*polyhedra)
 
 
 def And(*domains):
+    """
+    Create the intersection domain of the domains given in arguments.
+    """
     if len(domains) == 0:
         from .polyhedra import Universe
         return Universe
     else:
         return domains[0].intersection(*domains[1:])
-And.__doc__ = Domain.intersection.__doc__
+
 
 def Or(*domains):
+    """
+    Create the union domain of the domains given in arguments.
+    """
     if len(domains) == 0:
         from .polyhedra import Empty
         return Empty
     else:
         return domains[0].union(*domains[1:])
-Or.__doc__ = Domain.union.__doc__
+
 
 def Not(domain):
+    """
+    Create the complementary domain of the domain given in argument.
+    """
     return ~domain
-Not.__doc__ = Domain.complement.__doc__