Version 1.0

[linpy.git] / linpy / domains.py

diff --git a/linpy/domains.py b/linpy/domains.py
index 21e78db..f26b2fc 100644 (file)
--- a/linpy/domains.py
+++ b/linpy/domains.py
@@ -24,7 +24,7 @@ from fractions import Fraction
 
 from . import islhelper
 from .islhelper import mainctx, libisl
 
 from . import islhelper
 from .islhelper import mainctx, libisl

-from .linexprs import Expression, Symbol, Rational
+from .linexprs import LinExpr, Symbol

 from .geometry import GeometricObject, Point, Vector
 
 
 from .geometry import GeometricObject, Point, Vector
 
 


@@ -36,6 +36,13 @@ __all__ = [
 
 @functools.total_ordering
 class Domain(GeometricObject):
 
 @functools.total_ordering
 class Domain(GeometricObject):

+    """
+    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__ = (
         '_polyhedra',
 
     __slots__ = (
         '_polyhedra',


@@ -44,6 +51,30 @@ class Domain(GeometricObject):
     )
 
     def __new__(cls, *polyhedra):
     )
 
     def __new__(cls, *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:
             argument = polyhedra[0]
         from .polyhedra import Polyhedron
         if len(polyhedra) == 1:
             argument = polyhedra[0]


@@ -74,27 +105,29 @@ class Domain(GeometricObject):
 
     @property
     def polyhedra(self):
 
     @property
     def polyhedra(self):

+        """
+        The tuple of polyhedra present in the domain.
+        """

         return self._polyhedra
 
     @property
     def symbols(self):
         return self._polyhedra
 
     @property
     def symbols(self):

+        """
+        The tuple of symbols present in the domain equations, sorted according
+        to Symbol.sortkey().
+        """

         return self._symbols
 
     @property
     def dimension(self):
         return self._symbols
 
     @property
     def dimension(self):

-        return self._dimension
-
-    def disjoint(self):

         """
         """

-        Returns this set as disjoint.
+        The dimension of the domain, i.e. the number of symbols present in it.

         """
         """

-        islset = self._toislset(self.polyhedra, self.symbols)
-        islset = libisl.isl_set_make_disjoint(mainctx, islset)
-        return self._fromislset(islset, self.symbols)
+        return self._dimension

 
     def isempty(self):
         """
 
     def isempty(self):
         """

-        Returns true if this set is an Empty set.
+        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))
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         empty = bool(libisl.isl_set_is_empty(islset))


@@ -102,11 +135,14 @@ class Domain(GeometricObject):
         return empty
 
     def __bool__(self):
         return empty
 
     def __bool__(self):

+        """
+        Return True if the domain is non-empty.
+        """

         return not self.isempty()
 
     def isuniverse(self):
         """
         return not self.isempty()
 
     def isuniverse(self):
         """

-        Returns true if this set is the Universe set.
+        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))
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         universe = bool(libisl.isl_set_plain_is_universe(islset))


@@ -115,7 +151,7 @@ class Domain(GeometricObject):
 
     def isbounded(self):
         """
 
     def isbounded(self):
         """

-        Returns true if this set is bounded.
+        Return True if the domain is bounded.

         """
         islset = self._toislset(self.polyhedra, self.symbols)
         bounded = bool(libisl.isl_set_is_bounded(islset))
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         bounded = bool(libisl.isl_set_is_bounded(islset))


@@ -124,20 +160,24 @@ class Domain(GeometricObject):
 
     def __eq__(self, other):
         """
 
     def __eq__(self, other):
         """

-        Returns true if two sets 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):
         """
 
     def isdisjoint(self, other):
         """

-        Return True if two sets have a null intersection.
+        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)
         symbols = self._xsymbols([self, other])
         islset1 = self._toislset(self.polyhedra, symbols)
         islset2 = self._toislset(other.polyhedra, symbols)


@@ -148,60 +188,84 @@ class Domain(GeometricObject):
 
     def issubset(self, other):
         """
 
     def issubset(self, other):
         """

-        Report whether another set contains this set.
+        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):
 
     def __le__(self, other):

-        """
-        Returns true if this set is less than or equal to another set.
-        """
-        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):
         """
 
     def __lt__(self, other):
         """

-        Returns true if this set is less than another set.
+        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):
         """
 
     def complement(self):
         """

-        Returns the complement of this set.
+        Return the complementary domain of the domain.

         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_complement(islset)
         return self._fromislset(islset, self.symbols)
 
     def __invert__(self):
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_complement(islset)
         return self._fromislset(islset, self.symbols)
 
     def __invert__(self):

+        return self.complement()
+    __invert__.__doc__ = complement.__doc__
+
+    def make_disjoint(self):

         """
         """

-        Returns the complement of this set.
+        Return an equivalent domain, whose polyhedra are disjoint.

         """
         """

-        return self.complement()
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_make_disjoint(islset)
+        return self._fromislset(islset, self.symbols)

 
 

-    def simplify(self):
+    def coalesce(self):

         """
         """

-        Returns a set without redundant constraints.
+        Simplify the representation of the domain by trying to combine pairs of
+        polyhedra into a single polyhedron, and return the resulting domain.

         """
         islset = self._toislset(self.polyhedra, self.symbols)
         """
         islset = self._toislset(self.polyhedra, self.symbols)

-        islset = libisl.isl_set_remove_redundancies(islset)
+        islset = libisl.isl_set_coalesce(islset)

         return self._fromislset(islset, self.symbols)
 
         return self._fromislset(islset, self.symbols)
 

-    def aspolyhedron(self):
+    def detect_equalities(self):
+        """
+        Simplify the representation of the domain by detecting implicit
+        equalities, and return the resulting domain.

         """
         """

-        Returns polyhedral hull of set.
+        islset = self._toislset(self.polyhedra, self.symbols)
+        islset = libisl.isl_set_detect_equalities(islset)
+        return self._fromislset(islset, self.symbols)
+
+    def remove_redundancies(self):
+        """
+        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):

         from .polyhedra import Polyhedron
         islset = self._toislset(self.polyhedra, self.symbols)
         islbset = libisl.isl_set_polyhedral_hull(islset)
         from .polyhedra import Polyhedron
         islset = self._toislset(self.polyhedra, self.symbols)
         islbset = libisl.isl_set_polyhedral_hull(islset)


@@ -210,26 +274,33 @@ class Domain(GeometricObject):
     def asdomain(self):
         return self
 
     def asdomain(self):
         return self
 

-    def project(self, dims):
+    def project(self, symbols):

         """
         """

-        Return new set with given dimensions removed.
+        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))):
         islset = self._toislset(self.polyhedra, self.symbols)
         n = 0
         for index, symbol in reversed(list(enumerate(self.symbols))):

-            if symbol in dims:
+            if symbol in symbols:

                 n += 1
             elif n > 0:
                 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)
                 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)
+        symbols = [symbol for symbol in self.symbols if symbol not in symbols]
+        return Domain._fromislset(islset, symbols)

 
     def sample(self):
         """
 
     def sample(self):
         """

-        Returns a single subset of the input.
+        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)
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islpoint = libisl.isl_set_sample_point(islset)


@@ -247,67 +318,67 @@ class Domain(GeometricObject):
 
     def intersection(self, *others):
         """
 
     def intersection(self, *others):
         """

-         Return the intersection of two sets as a new set.
+        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:
         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):
 
     def __and__(self, other):

-        """
-         Return the intersection of two sets as a new set.
-        """
-        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):
         """
 
     def union(self, *others):
         """

-        Return the union of sets as a new set.
+        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:
         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):
 
     def __or__(self, other):

-        """
-        Return a new set with elements from both sets.
-        """
-        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):
 
     def __add__(self, other):

-        """
-        Return new set containing all elements in both sets.
-        """
-        return self.union(other)
+        return self | other
+    __add__.__doc__ = union.__doc__

 
     def difference(self, other):
         """
 
     def difference(self, other):
         """

-        Return the difference of two sets as a new set.
+        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):
 
     def __sub__(self, other):

-        """
-        Return the difference of two sets as a new set.
-        """
-        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):
         """
 
     def lexmin(self):
         """

-        Return a new set containing the lexicographic minimum of the elements in the set.
+        Return the lexicographic minimum of the elements in the domain.

         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_lexmin(islset)
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_lexmin(islset)


@@ -315,52 +386,35 @@ class Domain(GeometricObject):
 
     def lexmax(self):
         """
 
     def lexmax(self):
         """

-        Return a new set containing the lexicographic maximum of the elements in the set.
+        Return the lexicographic maximum of the elements in the domain.

         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_lexmax(islset)
         return self._fromislset(islset, self.symbols)
 
         """
         islset = self._toislset(self.polyhedra, self.symbols)
         islset = libisl.isl_set_lexmax(islset)
         return self._fromislset(islset, self.symbols)
 

-
-    def involves_vars(self, vars):
-        """
-        Returns true if a set depends on given dimensions.
-        """
-        islset = self._toislset(self.polyhedra, self.symbols)
-        dims = sorted(vars)
-        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+))?')
+    if islhelper.isl_version >= '0.13':
+        _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
+    else:
+        _RE_COORDINATE = None

 
     def vertices(self):
         """
 
     def vertices(self):
         """

-        Return a list of vertices for this Polygon.
+        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')
         """
         from .polyhedra import Polyhedron
         if not self.isbounded():
             raise ValueError('domain must be bounded')

-        islbset = self._toislbasicset(self.equalities, self.inequalities, self.symbols)
+        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:
         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 = []
             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 = islhelper.isl_val_to_int(constant)
                 for constraint in constraints:
                     constant = libisl.isl_constraint_get_constant_val(constraint)
                     constant = islhelper.isl_val_to_int(constant)


@@ -372,7 +426,7 @@ class Domain(GeometricObject):
                             coordinate = -Fraction(constant, coefficient)
                             coordinates.append((symbol, coordinate))
             else:
                             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'))
                 matches = self._RE_COORDINATE.finditer(string)
                 for symbol, match in zip(self.symbols, matches):
                     numerator = int(match.group('num'))


@@ -385,7 +439,9 @@ class Domain(GeometricObject):
 
     def points(self):
         """
 
     def points(self):
         """

-        Returns the points contained in the set.
+        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')
         """
         if not self.isbounded():
             raise ValueError('domain must be bounded')


@@ -403,6 +459,15 @@ class Domain(GeometricObject):
             points.append(Point(coordinates))
         return points
 
             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
     @classmethod
     def _polygon_inner_point(cls, points):
         symbols = points[0].symbols


@@ -456,7 +521,9 @@ class Domain(GeometricObject):
 
     def faces(self):
         """
 
     def faces(self):
         """

-        Returns the vertices of the faces of a polyhedra.
+        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:
         """
         faces = []
         for polyhedron in self.polyhedra:


@@ -518,10 +585,13 @@ class Domain(GeometricObject):
         axes.set_zlim(zmin, zmax)
         return axes
 
         axes.set_zlim(zmin, zmax)
         return axes
 

-

     def plot(self, plot=None, **kwargs):
         """
     def plot(self, plot=None, **kwargs):
         """

-        Display plot of this set.
+        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')
         """
         if not self.isbounded():
             raise ValueError('domain must be bounded')


@@ -530,18 +600,14 @@ class Domain(GeometricObject):
         elif self.dimension == 3:
             return self._plot_3d(plot=plot, **kwargs)
         else:
         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
+            raise ValueError('domain must be 2 or 3-dimensional')

 
     def subs(self, symbol, expression=None):
         """
 
     def subs(self, symbol, expression=None):
         """

-        Subsitute the given value into an expression and return the resulting
-        expression.
+        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]
         """
         polyhedra = [polyhedron.subs(symbol, expression)
             for polyhedron in self.polyhedra]


@@ -603,10 +669,10 @@ class Domain(GeometricObject):
         elif isinstance(node, ast.Compare):
             equalities = []
             inequalities = []
         elif isinstance(node, ast.Compare):
             equalities = []
             inequalities = []

-            left = Expression._fromast(node.left)
+            left = LinExpr._fromast(node.left)

             for i in range(len(node.ops)):
                 op = node.ops[i]
             for i in range(len(node.ops)):
                 op = node.ops[i]

-                right = Expression._fromast(node.comparators[i])
+                right = LinExpr._fromast(node.comparators[i])

                 if isinstance(op, ast.Lt):
                     inequalities.append(right - left - 1)
                 elif isinstance(op, ast.LtE):
                 if isinstance(op, ast.Lt):
                     inequalities.append(right - left - 1)
                 elif isinstance(op, ast.LtE):


@@ -629,22 +695,26 @@ class Domain(GeometricObject):
     _RE_AND = re.compile(r'\band\b|,|&&|/\\|∧|∩')
     _RE_OR = re.compile(r'\bor\b|;|\|\||\\/|∨|∪')
     _RE_NOT = re.compile(r'\bnot\b|!|¬')
     _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_NUM_VAR = LinExpr._RE_NUM_VAR

     _RE_OPERATORS = re.compile(r'(&|\||~)')
 
     @classmethod
     def fromstring(cls, string):
     _RE_OPERATORS = re.compile(r'(&|\||~)')
 
     @classmethod
     def fromstring(cls, string):

-        # remove curly brackets
+        """
+        Create a domain from a string. Raise SyntaxError if the string is not
+        properly formatted.
+        """
+        # Remove curly brackets.

         string = cls._RE_BRACES.sub(r'', string)
         string = cls._RE_BRACES.sub(r'', string)

-        # replace '=' by '=='
+        # Replace '=' by '=='.

         string = cls._RE_EQ.sub(r'\1==\2', string)
         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)
         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)
         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:
         tokens = cls._RE_OPERATORS.split(string)
         for i, token in enumerate(tokens):
             if i % 2 == 0:


@@ -659,14 +729,11 @@ class Domain(GeometricObject):
         strings = [repr(polyhedron) for polyhedron in self.polyhedra]
         return 'Or({})'.format(', '.join(strings))
 
         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
     @classmethod

-    def fromsympy(cls, expr):
+    def fromsympy(cls, expression):
+        """
+        Create a domain from a SymPy expression.
+        """

         import sympy
         from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
         funcmap = {
         import sympy
         from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
         funcmap = {


@@ -675,14 +742,17 @@ class Domain(GeometricObject):
             sympy.Eq: Eq, sympy.Ne: Ne,
             sympy.Ge: Ge, sympy.Gt: Gt,
         }
             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))
+        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):
 
     def tosympy(self):

+        """
+        Convert the domain to a SymPy expression.
+        """

         import sympy
         polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
         return sympy.Or(*polyhedra)
         import sympy
         polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
         return sympy.Or(*polyhedra)


@@ -690,7 +760,7 @@ class Domain(GeometricObject):
 
 def And(*domains):
     """
 
 def And(*domains):
     """

-    Return the intersection of two sets as a new set.
+    Create the intersection domain of the domains given in arguments.

     """
     if len(domains) == 0:
         from .polyhedra import Universe
     """
     if len(domains) == 0:
         from .polyhedra import Universe


@@ -700,7 +770,7 @@ def And(*domains):
 
 def Or(*domains):
     """
 
 def Or(*domains):
     """

-    Return the union of sets as a new set.
+    Create the union domain of the domains given in arguments.

     """
     if len(domains) == 0:
         from .polyhedra import Empty
     """
     if len(domains) == 0:
         from .polyhedra import Empty


@@ -710,6 +780,6 @@ def Or(*domains):
 
 def Not(domain):
     """
 
 def Not(domain):
     """

-    Returns the complement of this set.
+    Create the complementary domain of the domain given in argument.

     """
     return ~domain
     """
     return ~domain