From: Danielle Bolan <n02702451@hawkmail.newpaltz.edu>
Date: Fri, 18 Jul 2014 15:27:24 +0000 (+0200)
Subject: Added some docs
X-Git-Tag: 1.0~96
X-Git-Url: https://svn.cri.ensmp.fr/git/linpy.git/commitdiff_plain/25ce908cffca380f930182a77c1e5a4491042a1c?ds=inline

Added some docs
---

diff --git a/doc/domain.rst b/doc/domain.rst
new file mode 100644
index 0000000..217acba
--- /dev/null
+++ b/doc/domain.rst
@@ -0,0 +1,167 @@
+Domains Module
+==============
+
+.. py:class :: Domain
+
+    .. py:method:: polyhedra(self)
+    
+        Return .
+        
+Domain Properties
+-----------------
+    .. py:method:: symbols(self)
+    
+        Returns a list of the symbols used in a set.
+
+    .. py:method:: dimension(self)
+    
+        Returns the number of variables in a set.
+
+    .. py:method:: disjoint(self)
+    
+        Returns a set as disjoint.
+        
+    .. py:method:: num_parameters(self)    
+        
+        Returns the total number of parameters, input, output or set dimensions.
+        
+    .. py:method:: involves_dims(self, dims)
+    
+       Returns true if set depends on given dimensions. 
+        
+Unary Properties
+----------------
+    .. py:method:: isempty(self)
+    
+        Return true is set is an Empty set.
+        
+    .. py:method:: isuniverse(self)
+    
+        Return true if set is the Universe set.
+                
+    .. py:method:: isbounded(self)
+    
+        Return true if set is bounded
+
+    .. py:method:: disjoint(self)
+    
+        Returns this set as a disjoint set.
+
+Binary Properties
+-----------------
+
+    .. py:method:: isdisjoint(self, other)
+    
+        Return true if the intersection of two sets results in an Empty set.
+        
+    .. py:method:: issubset(self, other)
+    
+        Returns true if one set contains the other set.
+
+    .. py:method:: __eq__(self, other)
+    
+        Return true if self == other.  
+        
+    .. py:method:: __lt__(self, other)
+    
+        Return true if self < other.  
+        
+    .. py:method:: __le__(self, other)
+    
+        Return true if self <= other.  
+         
+    .. py:method:: __gt__(self, other)
+    
+        Return true if self > other.  
+                
+    .. py:method:: __ge__(self, other)
+    
+        Return true if self >= other.  
+        
+
+Unary Operations
+----------------
+
+    .. py:method:: complement(self)
+    
+        Return the complement of a set.       
+        
+    .. py:method:: simplify(self)
+
+        Removes redundant constraints from a set.        
+
+    .. py:method:: project(self, dims)
+    
+        Return a new set with the given dimensions removed.
+
+    .. py:method:: aspolyhedron(self)
+    
+        Return polyhedral hull of a set.        
+        
+    .. py:method:: asdomain(self)
+    
+        Return 
+
+    .. py:method:: sample(self)
+    
+        Return a single sample subset of a set.
+
+Binary Operations
+-----------------
+
+    .. py:method:: intersection(self)
+    
+        Return the intersection of two sets as a new set.         
+
+    .. py:method:: union(self)
+    
+        Return the union of two sets as a new set.
+
+    .. py:method:: __and__(self, other)
+    
+        Return the union of two sets as a new set.        
+        
+    .. py:method:: __or__(self, other)
+    
+        Return the intersection of two sets as a new set.         
+        
+    .. py:method:: __add__(self, other)
+    
+        Return the sum of two sets. 
+
+    .. py:method:: difference(self, other)
+    
+        Return the difference of two sets.         
+
+Lexiographic Operations
+-----------------------
+
+    .. py:method:: lexmin(self)
+    
+        Return a new set containing the lexicographic minimum of the elements in the set. 
+        
+    .. py:method:: lexmax(self)
+    
+        Return a new set containing the lexicographic maximum of the elements in the set.       
+        
+Plot Properties
+---------------
+
+    .. py:method:: points(self)
+    
+        Return a list of the points contained in a set.
+
+    .. py:method:: vertices(self)
+    
+        Return a list of the verticies of this set.
+        
+    .. py:method:: faces(self)
+    
+        Return a list of the vertices for each face of a set.
+        
+    .. py:method:: plot(self, plot=None, **kwargs)
+    
+        Return a plot of the given set.       
+        
+          
+        
diff --git a/doc/domain.rst~ b/doc/domain.rst~
new file mode 100644
index 0000000..217acba
--- /dev/null
+++ b/doc/domain.rst~
@@ -0,0 +1,167 @@
+Domains Module
+==============
+
+.. py:class :: Domain
+
+    .. py:method:: polyhedra(self)
+    
+        Return .
+        
+Domain Properties
+-----------------
+    .. py:method:: symbols(self)
+    
+        Returns a list of the symbols used in a set.
+
+    .. py:method:: dimension(self)
+    
+        Returns the number of variables in a set.
+
+    .. py:method:: disjoint(self)
+    
+        Returns a set as disjoint.
+        
+    .. py:method:: num_parameters(self)    
+        
+        Returns the total number of parameters, input, output or set dimensions.
+        
+    .. py:method:: involves_dims(self, dims)
+    
+       Returns true if set depends on given dimensions. 
+        
+Unary Properties
+----------------
+    .. py:method:: isempty(self)
+    
+        Return true is set is an Empty set.
+        
+    .. py:method:: isuniverse(self)
+    
+        Return true if set is the Universe set.
+                
+    .. py:method:: isbounded(self)
+    
+        Return true if set is bounded
+
+    .. py:method:: disjoint(self)
+    
+        Returns this set as a disjoint set.
+
+Binary Properties
+-----------------
+
+    .. py:method:: isdisjoint(self, other)
+    
+        Return true if the intersection of two sets results in an Empty set.
+        
+    .. py:method:: issubset(self, other)
+    
+        Returns true if one set contains the other set.
+
+    .. py:method:: __eq__(self, other)
+    
+        Return true if self == other.  
+        
+    .. py:method:: __lt__(self, other)
+    
+        Return true if self < other.  
+        
+    .. py:method:: __le__(self, other)
+    
+        Return true if self <= other.  
+         
+    .. py:method:: __gt__(self, other)
+    
+        Return true if self > other.  
+                
+    .. py:method:: __ge__(self, other)
+    
+        Return true if self >= other.  
+        
+
+Unary Operations
+----------------
+
+    .. py:method:: complement(self)
+    
+        Return the complement of a set.       
+        
+    .. py:method:: simplify(self)
+
+        Removes redundant constraints from a set.        
+
+    .. py:method:: project(self, dims)
+    
+        Return a new set with the given dimensions removed.
+
+    .. py:method:: aspolyhedron(self)
+    
+        Return polyhedral hull of a set.        
+        
+    .. py:method:: asdomain(self)
+    
+        Return 
+
+    .. py:method:: sample(self)
+    
+        Return a single sample subset of a set.
+
+Binary Operations
+-----------------
+
+    .. py:method:: intersection(self)
+    
+        Return the intersection of two sets as a new set.         
+
+    .. py:method:: union(self)
+    
+        Return the union of two sets as a new set.
+
+    .. py:method:: __and__(self, other)
+    
+        Return the union of two sets as a new set.        
+        
+    .. py:method:: __or__(self, other)
+    
+        Return the intersection of two sets as a new set.         
+        
+    .. py:method:: __add__(self, other)
+    
+        Return the sum of two sets. 
+
+    .. py:method:: difference(self, other)
+    
+        Return the difference of two sets.         
+
+Lexiographic Operations
+-----------------------
+
+    .. py:method:: lexmin(self)
+    
+        Return a new set containing the lexicographic minimum of the elements in the set. 
+        
+    .. py:method:: lexmax(self)
+    
+        Return a new set containing the lexicographic maximum of the elements in the set.       
+        
+Plot Properties
+---------------
+
+    .. py:method:: points(self)
+    
+        Return a list of the points contained in a set.
+
+    .. py:method:: vertices(self)
+    
+        Return a list of the verticies of this set.
+        
+    .. py:method:: faces(self)
+    
+        Return a list of the vertices for each face of a set.
+        
+    .. py:method:: plot(self, plot=None, **kwargs)
+    
+        Return a plot of the given set.       
+        
+          
+        
diff --git a/doc/examples.rst b/doc/examples.rst
new file mode 100644
index 0000000..e69de29
diff --git a/doc/geometry.rst b/doc/geometry.rst
new file mode 100644
index 0000000..224fb3a
--- /dev/null
+++ b/doc/geometry.rst
@@ -0,0 +1,38 @@
+Geometry Module
+===============
+
+The geometry module is used to obtain information about the points and vertices of a ployhedra.
+
+.. py:class:: Points
+
+This class represents points in space.
+
+    .. py:method:: isorigin(self)
+    
+        Return true is given point is the origin.
+        
+    .. py:method:: 
+        
+.. py:class:: Vector
+
+This class represents displacements in space.        
+
+    .. py:method:: angle(self, other)
+    
+        Retrieve the angle required to rotate the vector into the vector passed
+        in argument. The result is an angle in radians, ranging between -pi and
+        pi.
+
+    .. py:method:: cross(self, other)
+    
+        Calculate the cross product of two Vector3D structures. 
+
+    .. py:method:: dot(self, other)
+    
+        Calculate the dot product of two vectors. 
+        
+    .. py:method:: __trudiv__(self, other)
+    
+       Divide the vector by the specified scalar and returns the result as a
+       vector.         
+                
diff --git a/doc/geometry.rst~ b/doc/geometry.rst~
new file mode 100644
index 0000000..d96cd6f
--- /dev/null
+++ b/doc/geometry.rst~
@@ -0,0 +1,39 @@
+
+Geometry Module
+===============
+
+The geometry module is used to obtain information about the points and vertices of a ployhedra.
+
+.. py:class:: Points
+
+This class represents points in space.
+
+    .. py:method:: isorigin(self)
+    
+        Return true is given point is the origin.
+        
+    .. py:method:: 
+        
+.. py:class:: Vector
+
+This class represents displacements in space.        
+
+    .. py:method:: angle(self, other)
+    
+        Retrieve the angle required to rotate the vector into the vector passed
+        in argument. The result is an angle in radians, ranging between -pi and
+        pi.
+
+    .. py:method:: cross(self, other)
+    
+        Calculate the cross product of two Vector3D structures. 
+
+    .. py:method:: dot(self, other)
+    
+        Calculate the dot product of two vectors. 
+        
+    .. py:method:: __trudiv__(self, other)
+    
+       Divide the vector by the specified scalar and returns the result as a
+       vector.         
+                
diff --git a/doc/index.rst b/doc/index.rst
index 0c5738e..12f6f4e 100644
--- a/doc/index.rst
+++ b/doc/index.rst
@@ -6,19 +6,20 @@
 Welcome to pypol's documentation!
 =================================
 
-Since Pythagoras, we know that :math:`c = \sqrt{a^2 + b^2}`.
+Pypol is a Python library for symbolic mathematics.
+If you are new to Pypol, start with the Tutorial.
+
+This is the central page for all of SymPy's documentation.
+
 
 Contents:
 
 .. toctree::
    :maxdepth: 2
 
+   install.rst
+   examples.rst
+   modules.rst
 
 
-Indices and tables
-==================
-
-* :ref:`genindex`
-* :ref:`modindex`
-* :ref:`search`
 
diff --git a/doc/index.rst~ b/doc/index.rst~
new file mode 100644
index 0000000..cbe9460
--- /dev/null
+++ b/doc/index.rst~
@@ -0,0 +1,25 @@
+.. pypol documentation master file, created by
+   sphinx-quickstart on Wed Jun 25 20:34:21 2014.
+   You can adapt this file completely to your liking, but it should at least
+   contain the root `toctree` directive.
+
+Welcome to pypol's documentation!
+=================================
+
+Pypol is a Python library for symbolic mathematics.
+If you are new to Pypol, start with the Tutorial.
+
+This is the central page for all of SymPy's documentation.
+
+
+Contents:
+
+.. toctree::
+   :maxdepth: 2
+
+   install.rst
+   example.rst
+   modules.rst
+
+
+
diff --git a/doc/install.rst b/doc/install.rst
new file mode 100644
index 0000000..35cd056
--- /dev/null
+++ b/doc/install.rst
@@ -0,0 +1,19 @@
+.. _installation:
+
+Installation
+------------
+
+Dependencies
+============
+
+Users will first need to install Integer Set Library (isl). The source files of isl are available as a tarball or a git repository. Both
+are available `here`_ .  
+
+Source
+======
+
+Git
+===
+
+
+.. _here: http://freshmeat.net/projects/isl/
diff --git a/doc/install.rst~ b/doc/install.rst~
new file mode 100644
index 0000000..8e0224a
--- /dev/null
+++ b/doc/install.rst~
@@ -0,0 +1,14 @@
+.. _installation:
+
+Installation
+------------
+
+Dependencies
+============
+
+Users will first need to install Integer Set Library (isl). The source files of isl are available as a tarball or a git repository. Both
+are available `here`_ .  
+
+
+
+.. _here: http://freshmeat.net/projects/isl/
diff --git a/doc/linexpr.rst b/doc/linexpr.rst
new file mode 100644
index 0000000..3771d5f
--- /dev/null
+++ b/doc/linexpr.rst
@@ -0,0 +1,45 @@
+Linear Expression Module
+========================
+
+
+This class implements linear expressions.
+
+    .. py:method:: coefficient(self, symbol)
+        
+        Return the coefficient value of the given symbol.
+        
+    .. py:method:: coefficients(self)
+        
+        Return a list of the coefficients of an expression
+        
+    .. py:method:: constant(self)
+    
+        Return the constant value of an expression.
+        
+    .. py:method:: symbols(self)
+    
+        Return a list of symbols in an expression.                            
+    
+    .. py:method:: dimension(self)
+    
+        Return the number of vriables in an expression.
+        
+    .. py:method:: __sub__(self, other)
+    
+        Return the difference between two expressions.
+               
+    .. py:method:: subs(self, symbol, expression=None)
+    
+        Subsitute the given value into an expression and return the resulting expression.
+
+    .. py:method:: fromsympy(self)
+    
+        Convert sympy object to an expression.
+        
+    .. py:method:: tosympy(self)
+    
+        Return an expression as a sympy object.    
+          
+.. py:class:: Dummy(Symbol)
+
+This class returns a dummy symbol to ensure that each no variables are repeated in an expression   
diff --git a/doc/linexpr.rst~ b/doc/linexpr.rst~
new file mode 100644
index 0000000..6a017bf
--- /dev/null
+++ b/doc/linexpr.rst~
@@ -0,0 +1,54 @@
+Linear Expression Module
+========================
+
+
+This class implements linear expressions.
+
+    .. py:method:: coefficient(self, symbol)
+        
+        Return the coefficient value of the given symbol.
+        
+    .. py:method:: coefficients(self)
+        
+        Return a list of the coefficients of an expression
+        
+    .. py:method:: constant(self)
+    
+        Return the constant value of an expression.
+        
+    .. py:method:: symbols(self)
+    
+        Return a list of symbols in an expression.                            
+    
+    .. py:method:: dimension(self)
+    
+        Return the number of vriables in an expression.
+        
+    .. py:method:: __sub__(self, other)
+    
+        Return the difference between two expressions.
+               
+    .. py:method:: subs(self, symbol, expression=None)
+    
+        Subsitute the given value into an expression and return the resulting expression.
+
+    .. py:method:: fromsympy(self)
+    
+        Convert sympy object to an expression.
+        
+    .. py:method:: tosympy(self)
+    
+        Return an expression as a sympy object.    
+          
+.. py:class:: Dummy(Symbol)
+
+This class returns a dummy symbol to ensure that each no variables are repeated in an expression 
+
+
+
+
+
+
+
+     
+        
diff --git a/doc/modules.rst b/doc/modules.rst
new file mode 100644
index 0000000..2ecf371
--- /dev/null
+++ b/doc/modules.rst
@@ -0,0 +1,15 @@
+.. module-docs:
+
+Pypol Module Reference
+======================
+
+There are four main Pypol modules:
+
+.. toctree::
+   :maxdepth: 2
+
+   polyhedra.rst
+   domain.rst
+   linexpr.rst
+   geometry.rst
+   
diff --git a/doc/modules.rst~ b/doc/modules.rst~
new file mode 100644
index 0000000..2ecf371
--- /dev/null
+++ b/doc/modules.rst~
@@ -0,0 +1,15 @@
+.. module-docs:
+
+Pypol Module Reference
+======================
+
+There are four main Pypol modules:
+
+.. toctree::
+   :maxdepth: 2
+
+   polyhedra.rst
+   domain.rst
+   linexpr.rst
+   geometry.rst
+   
diff --git a/doc/polyhedra.rst b/doc/polyhedra.rst
new file mode 100644
index 0000000..fd09b90
--- /dev/null
+++ b/doc/polyhedra.rst
@@ -0,0 +1,61 @@
+Polyhedra Module
+================
+
+.. py:class:: Polyhedron
+
+Polyhedra Properties
+--------------------
+
+    .. py:method:: equalities(self)
+    
+        Return a list of the equalities in a set.
+        
+    .. py:method:: inequalities(self)
+    
+        Return a list of the inequalities in a set.
+
+    .. py:method:: constraints(self)
+    
+        Return ta list of the constraints of a set.
+
+Unary Operations
+----------------
+    .. py:method:: disjoint(self)
+    
+        Returns this set as a disjoint set.
+               
+    .. py:method:: isuniverse(self)
+    
+        Return true if this set is the Universe set.    
+
+    .. py:method:: aspolyhedron(self)
+    
+        Return polyhedral hull of this set.
+
+    .. py:method:: subs(self, symbol, expression=None)
+    
+        Subsitute expression into given set and returns the result.      
+        
+    .. py:method:: Lt(left, right)
+    
+        Assert first set is less than the second set.     
+        
+    .. py:method:: Le(left, right)
+    
+        Assert first set is less than or equal to the second set.
+        
+    .. py:method:: Eq(left, right)
+    
+        Assert first set is equal to the second set.         
+        
+    .. py:method:: Ne(left, right)
+    
+        Assert first set is not equal to the second set.          
+        
+    .. py:method:: Gt(left, right)
+    
+        Assert first set is greater than the second set.        
+
+    .. py:method:: Ge(left, right)
+    
+        Assert first set is greater than or equal to the second set. 
diff --git a/doc/polyhedra.rst~ b/doc/polyhedra.rst~
new file mode 100644
index 0000000..fd09b90
--- /dev/null
+++ b/doc/polyhedra.rst~
@@ -0,0 +1,61 @@
+Polyhedra Module
+================
+
+.. py:class:: Polyhedron
+
+Polyhedra Properties
+--------------------
+
+    .. py:method:: equalities(self)
+    
+        Return a list of the equalities in a set.
+        
+    .. py:method:: inequalities(self)
+    
+        Return a list of the inequalities in a set.
+
+    .. py:method:: constraints(self)
+    
+        Return ta list of the constraints of a set.
+
+Unary Operations
+----------------
+    .. py:method:: disjoint(self)
+    
+        Returns this set as a disjoint set.
+               
+    .. py:method:: isuniverse(self)
+    
+        Return true if this set is the Universe set.    
+
+    .. py:method:: aspolyhedron(self)
+    
+        Return polyhedral hull of this set.
+
+    .. py:method:: subs(self, symbol, expression=None)
+    
+        Subsitute expression into given set and returns the result.      
+        
+    .. py:method:: Lt(left, right)
+    
+        Assert first set is less than the second set.     
+        
+    .. py:method:: Le(left, right)
+    
+        Assert first set is less than or equal to the second set.
+        
+    .. py:method:: Eq(left, right)
+    
+        Assert first set is equal to the second set.         
+        
+    .. py:method:: Ne(left, right)
+    
+        Assert first set is not equal to the second set.          
+        
+    .. py:method:: Gt(left, right)
+    
+        Assert first set is greater than the second set.        
+
+    .. py:method:: Ge(left, right)
+    
+        Assert first set is greater than or equal to the second set. 
diff --git a/pypol/domains.py b/pypol/domains.py
index be9dd4b..1c400bd 100644
--- a/pypol/domains.py
+++ b/pypol/domains.py
@@ -340,6 +340,8 @@ class Domain(GeometricObject):
         Return a list of vertices for this Polygon.
         """
         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);
         vertices = islhelper.isl_vertices_vertices(vertices)
@@ -390,7 +392,7 @@ class Domain(GeometricObject):
                 coordinates[symbol] = coordinate
             points.append(Point(coordinates))
         return points
-
+    
     @classmethod
     def _polygon_inner_point(cls, points):
         symbols = points[0].symbols
@@ -443,6 +445,9 @@ class Domain(GeometricObject):
         return sorted(points, key=angles.get)
 
     def faces(self):
+    """
+    Returns the vertices of the faces of a polyhedra.
+    """
         faces = []
         for polyhedron in self.polyhedra:
             vertices = polyhedron.vertices()
@@ -503,6 +508,7 @@ class Domain(GeometricObject):
         axes.set_zlim(zmin, zmax)
         return axes
 
+
     def plot(self, plot=None, **kwargs):
         """
         Display plot of this set.
@@ -523,6 +529,9 @@ class Domain(GeometricObject):
         return False
 
     def subs(self, symbol, expression=None):
+    """
+    Subsitute the given value into an expression and return the resulting expression.
+    """
         polyhedra = [polyhedron.subs(symbol, expression)
             for polyhedron in self.polyhedra]
         return Domain(*polyhedra)
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
diff --git a/pypol/polyhedra.py b/pypol/polyhedra.py
index ccb1a8c..fe143d4 100644
--- a/pypol/polyhedra.py
+++ b/pypol/polyhedra.py
@@ -56,14 +56,23 @@ class Polyhedron(Domain):
 
     @property
     def equalities(self):
+    """
+    Return a list of the equalities in a set.
+    """
         return self._equalities
 
     @property
     def inequalities(self):
+    """
+    Return a list of the inequalities in a set.
+    """
         return self._inequalities
 
     @property
     def constraints(self):
+    """
+    Return ta list of the constraints of a set.
+    """
         return self._constraints
 
     @property
@@ -72,13 +81,13 @@ class Polyhedron(Domain):
 
     def disjoint(self):
         """
-        Return this set as disjoint.
+        Return a set as disjoint.
         """
         return self
 
     def isuniverse(self):
         """
-        Return true if this set is the Universe set.
+        Return true if a set is the Universe set.
         """
         islbset = self._toislbasicset(self.equalities, self.inequalities,
             self.symbols)
@@ -88,7 +97,7 @@ class Polyhedron(Domain):
 
     def aspolyhedron(self):
         """
-        Return polyhedral hull of this set.
+        Return polyhedral hull of a set.
         """
         return self
 
@@ -106,6 +115,9 @@ class Polyhedron(Domain):
         return True
 
     def subs(self, symbol, expression=None):
+    """
+    Subsitute the given value into an expression and return the resulting expression.
+    """
         equalities = [equality.subs(symbol, expression)
             for equality in self.equalities]
         inequalities = [inequality.subs(symbol, expression)
@@ -139,10 +151,6 @@ class Polyhedron(Domain):
 
     @classmethod
     def _fromislbasicset(cls, islbset, symbols):
-        if libisl.isl_basic_set_is_empty(islbset):
-            return Empty
-        if libisl.isl_basic_set_is_universe(islbset):
-            return Universe
         islconstraints = islhelper.isl_basic_set_constraints(islbset)
         equalities = []
         inequalities = []
@@ -223,6 +231,7 @@ class Polyhedron(Domain):
         else:
             return 'And({})'.format(', '.join(strings))
 
+
     def _repr_latex_(self):
         strings = []
         for equality in self.equalities:
@@ -233,12 +242,18 @@ class Polyhedron(Domain):
 
     @classmethod
     def fromsympy(cls, expr):
+    """
+    Convert a sympy object to an expression.
+    """
         domain = Domain.fromsympy(expr)
         if not isinstance(domain, Polyhedron):
             raise ValueError('non-polyhedral expression: {!r}'.format(expr))
         return domain
 
     def tosympy(self):
+    """
+    Return an expression as a sympy object. 
+    """
         import sympy
         constraints = []
         for equality in self.equalities:
@@ -247,7 +262,6 @@ class Polyhedron(Domain):
             constraints.append(sympy.Ge(inequality.tosympy(), 0))
         return sympy.And(*constraints)
 
-
 class EmptyType(Polyhedron):
 
     __slots__ = Polyhedron.__slots__
@@ -318,41 +332,42 @@ def _polymorphic(func):
 @_polymorphic
 def Lt(left, right):
     """
-    Return true if the first set is less than the second.
+    Assert first set is less than the second set.
     """
     return Polyhedron([], [right - left - 1])
 
 @_polymorphic
 def Le(left, right):
     """
-    Return true the first set is less than or equal to the second.
+    Assert first set is less than or equal to the second set.
     """
     return Polyhedron([], [right - left])
 
 @_polymorphic
 def Eq(left, right):
     """
-    Return true if the sets are equal.
+    Assert first set is equal to the second set.
     """
     return Polyhedron([left - right], [])
 
 @_polymorphic
 def Ne(left, right):
     """
-    Return true if the sets are NOT equal.
+    Assert first set is not equal to the second set.
     """
     return ~Eq(left, right)
 
 @_polymorphic
 def Gt(left, right):
     """
-    Return true if the first set is greater than the second set.
+    Assert first set is greater than the second set.
     """
     return Polyhedron([], [left - right - 1])
 
 @_polymorphic
 def Ge(left, right):
     """
-    Return true if the first set is greater than or equal the second set.
+    Assert first set is greater than or equal to the second set.
     """
     return Polyhedron([], [left - right])
+
diff --git a/pypol/polyhedra.py~ b/pypol/polyhedra.py~
new file mode 100644
index 0000000..ccb1a8c
--- /dev/null
+++ b/pypol/polyhedra.py~
@@ -0,0 +1,358 @@
+import functools
+import math
+import numbers
+
+from . import islhelper
+
+from .islhelper import mainctx, libisl
+from .geometry import GeometricObject, Point
+from .linexprs import Expression, Rational
+from .domains import Domain
+
+
+__all__ = [
+    'Polyhedron',
+    'Lt', 'Le', 'Eq', 'Ne', 'Ge', 'Gt',
+    'Empty', 'Universe',
+]
+
+
+class Polyhedron(Domain):
+
+    __slots__ = (
+        '_equalities',
+        '_inequalities',
+        '_constraints',
+        '_symbols',
+        '_dimension',
+    )
+
+    def __new__(cls, equalities=None, inequalities=None):
+        if isinstance(equalities, str):
+            if inequalities is not None:
+                raise TypeError('too many arguments')
+            return cls.fromstring(equalities)
+        elif isinstance(equalities, GeometricObject):
+            if inequalities is not None:
+                raise TypeError('too many arguments')
+            return equalities.aspolyhedron()
+        if equalities is None:
+            equalities = []
+        else:
+            for i, equality in enumerate(equalities):
+                if not isinstance(equality, Expression):
+                    raise TypeError('equalities must be linear expressions')
+                equalities[i] = equality.scaleint()
+        if inequalities is None:
+            inequalities = []
+        else:
+            for i, inequality in enumerate(inequalities):
+                if not isinstance(inequality, Expression):
+                    raise TypeError('inequalities must be linear expressions')
+                inequalities[i] = inequality.scaleint()
+        symbols = cls._xsymbols(equalities + inequalities)
+        islbset = cls._toislbasicset(equalities, inequalities, symbols)
+        return cls._fromislbasicset(islbset, symbols)
+
+    @property
+    def equalities(self):
+        return self._equalities
+
+    @property
+    def inequalities(self):
+        return self._inequalities
+
+    @property
+    def constraints(self):
+        return self._constraints
+
+    @property
+    def polyhedra(self):
+        return self,
+
+    def disjoint(self):
+        """
+        Return this set as disjoint.
+        """
+        return self
+
+    def isuniverse(self):
+        """
+        Return true if this set is the Universe set.
+        """
+        islbset = self._toislbasicset(self.equalities, self.inequalities,
+            self.symbols)
+        universe = bool(libisl.isl_basic_set_is_universe(islbset))
+        libisl.isl_basic_set_free(islbset)
+        return universe
+
+    def aspolyhedron(self):
+        """
+        Return polyhedral hull of this set.
+        """
+        return self
+
+    def __contains__(self, point):
+        if not isinstance(point, Point):
+            raise TypeError('point must be a Point instance')
+        if self.symbols != point.symbols:
+            raise ValueError('arguments must belong to the same space')
+        for equality in self.equalities:
+            if equality.subs(point.coordinates()) != 0:
+                return False
+        for inequality in self.inequalities:
+            if inequality.subs(point.coordinates()) < 0:
+                return False
+        return True
+
+    def subs(self, symbol, expression=None):
+        equalities = [equality.subs(symbol, expression)
+            for equality in self.equalities]
+        inequalities = [inequality.subs(symbol, expression)
+            for inequality in self.inequalities]
+        return Polyhedron(equalities, inequalities)
+
+    def _asinequalities(self):
+        inequalities = list(self.equalities)
+        inequalities.extend([-expression for expression in self.equalities])
+        inequalities.extend(self.inequalities)
+        return inequalities
+
+    def widen(self, other):
+        if not isinstance(other, Polyhedron):
+            raise ValueError('argument must be a Polyhedron instance')
+        inequalities1 = self._asinequalities()
+        inequalities2 = other._asinequalities()
+        inequalities = []
+        for inequality1 in inequalities1:
+            if other <= Polyhedron(inequalities=[inequality1]):
+                inequalities.append(inequality1)
+        for inequality2 in inequalities2:
+            for i in range(len(inequalities1)):
+                inequalities3 = inequalities1[:i] + inequalities[i + 1:]
+                inequalities3.append(inequality2)
+                polyhedron3 = Polyhedron(inequalities=inequalities3)
+                if self == polyhedron3:
+                    inequalities.append(inequality2)
+                    break
+        return Polyhedron(inequalities=inequalities)
+
+    @classmethod
+    def _fromislbasicset(cls, islbset, symbols):
+        if libisl.isl_basic_set_is_empty(islbset):
+            return Empty
+        if libisl.isl_basic_set_is_universe(islbset):
+            return Universe
+        islconstraints = islhelper.isl_basic_set_constraints(islbset)
+        equalities = []
+        inequalities = []
+        for islconstraint in islconstraints:
+            constant = libisl.isl_constraint_get_constant_val(islconstraint)
+            constant = islhelper.isl_val_to_int(constant)
+            coefficients = {}
+            for index, symbol in enumerate(symbols):
+                coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint,
+                    libisl.isl_dim_set, index)
+                coefficient = islhelper.isl_val_to_int(coefficient)
+                if coefficient != 0:
+                    coefficients[symbol] = coefficient
+            expression = Expression(coefficients, constant)
+            if libisl.isl_constraint_is_equality(islconstraint):
+                equalities.append(expression)
+            else:
+                inequalities.append(expression)
+        libisl.isl_basic_set_free(islbset)
+        self = object().__new__(Polyhedron)
+        self._equalities = tuple(equalities)
+        self._inequalities = tuple(inequalities)
+        self._constraints = tuple(equalities + inequalities)
+        self._symbols = cls._xsymbols(self._constraints)
+        self._dimension = len(self._symbols)
+        return self
+
+    @classmethod
+    def _toislbasicset(cls, equalities, inequalities, symbols):
+        dimension = len(symbols)
+        indices = {symbol: index for index, symbol in enumerate(symbols)}
+        islsp = libisl.isl_space_set_alloc(mainctx, 0, dimension)
+        islbset = libisl.isl_basic_set_universe(libisl.isl_space_copy(islsp))
+        islls = libisl.isl_local_space_from_space(islsp)
+        for equality in equalities:
+            isleq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(islls))
+            for symbol, coefficient in equality.coefficients():
+                islval = str(coefficient).encode()
+                islval = libisl.isl_val_read_from_str(mainctx, islval)
+                index = indices[symbol]
+                isleq = libisl.isl_constraint_set_coefficient_val(isleq,
+                    libisl.isl_dim_set, index, islval)
+            if equality.constant != 0:
+                islval = str(equality.constant).encode()
+                islval = libisl.isl_val_read_from_str(mainctx, islval)
+                isleq = libisl.isl_constraint_set_constant_val(isleq, islval)
+            islbset = libisl.isl_basic_set_add_constraint(islbset, isleq)
+        for inequality in inequalities:
+            islin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(islls))
+            for symbol, coefficient in inequality.coefficients():
+                islval = str(coefficient).encode()
+                islval = libisl.isl_val_read_from_str(mainctx, islval)
+                index = indices[symbol]
+                islin = libisl.isl_constraint_set_coefficient_val(islin,
+                    libisl.isl_dim_set, index, islval)
+            if inequality.constant != 0:
+                islval = str(inequality.constant).encode()
+                islval = libisl.isl_val_read_from_str(mainctx, islval)
+                islin = libisl.isl_constraint_set_constant_val(islin, islval)
+            islbset = libisl.isl_basic_set_add_constraint(islbset, islin)
+        return islbset
+
+    @classmethod
+    def fromstring(cls, string):
+        domain = Domain.fromstring(string)
+        if not isinstance(domain, Polyhedron):
+            raise ValueError('non-polyhedral expression: {!r}'.format(string))
+        return domain
+
+    def __repr__(self):
+        strings = []
+        for equality in self.equalities:
+            strings.append('Eq({}, 0)'.format(equality))
+        for inequality in self.inequalities:
+            strings.append('Ge({}, 0)'.format(inequality))
+        if len(strings) == 1:
+            return strings[0]
+        else:
+            return 'And({})'.format(', '.join(strings))
+
+    def _repr_latex_(self):
+        strings = []
+        for equality in self.equalities:
+            strings.append('{} = 0'.format(equality._repr_latex_().strip('$')))
+        for inequality in self.inequalities:
+            strings.append('{} \\ge 0'.format(inequality._repr_latex_().strip('$')))
+        return '$${}$$'.format(' \\wedge '.join(strings))
+
+    @classmethod
+    def fromsympy(cls, expr):
+        domain = Domain.fromsympy(expr)
+        if not isinstance(domain, Polyhedron):
+            raise ValueError('non-polyhedral expression: {!r}'.format(expr))
+        return domain
+
+    def tosympy(self):
+        import sympy
+        constraints = []
+        for equality in self.equalities:
+            constraints.append(sympy.Eq(equality.tosympy(), 0))
+        for inequality in self.inequalities:
+            constraints.append(sympy.Ge(inequality.tosympy(), 0))
+        return sympy.And(*constraints)
+
+
+class EmptyType(Polyhedron):
+
+    __slots__ = Polyhedron.__slots__
+
+    def __new__(cls):
+        self = object().__new__(cls)
+        self._equalities = (Rational(1),)
+        self._inequalities = ()
+        self._constraints = self._equalities
+        self._symbols = ()
+        self._dimension = 0
+        return self
+
+    def widen(self, other):
+        if not isinstance(other, Polyhedron):
+            raise ValueError('argument must be a Polyhedron instance')
+        return other
+
+    def __repr__(self):
+        return 'Empty'
+
+    def _repr_latex_(self):
+        return '$$\\emptyset$$'
+
+Empty = EmptyType()
+
+
+class UniverseType(Polyhedron):
+
+    __slots__ = Polyhedron.__slots__
+
+    def __new__(cls):
+        self = object().__new__(cls)
+        self._equalities = ()
+        self._inequalities = ()
+        self._constraints = ()
+        self._symbols = ()
+        self._dimension = ()
+        return self
+
+    def __repr__(self):
+        return 'Universe'
+
+    def _repr_latex_(self):
+        return '$$\\Omega$$'
+
+Universe = UniverseType()
+
+
+def _polymorphic(func):
+    @functools.wraps(func)
+    def wrapper(left, right):
+        if not isinstance(left, Expression):
+            if isinstance(left, numbers.Rational):
+                left = Rational(left)
+            else:
+                raise TypeError('left must be a a rational number '
+                    'or a linear expression')
+        if not isinstance(right, Expression):
+            if isinstance(right, numbers.Rational):
+                right = Rational(right)
+            else:
+                raise TypeError('right must be a a rational number '
+                    'or a linear expression')
+        return func(left, right)
+    return wrapper
+
+@_polymorphic
+def Lt(left, right):
+    """
+    Return true if the first set is less than the second.
+    """
+    return Polyhedron([], [right - left - 1])
+
+@_polymorphic
+def Le(left, right):
+    """
+    Return true the first set is less than or equal to the second.
+    """
+    return Polyhedron([], [right - left])
+
+@_polymorphic
+def Eq(left, right):
+    """
+    Return true if the sets are equal.
+    """
+    return Polyhedron([left - right], [])
+
+@_polymorphic
+def Ne(left, right):
+    """
+    Return true if the sets are NOT equal.
+    """
+    return ~Eq(left, right)
+
+@_polymorphic
+def Gt(left, right):
+    """
+    Return true if the first set is greater than the second set.
+    """
+    return Polyhedron([], [left - right - 1])
+
+@_polymorphic
+def Ge(left, right):
+    """
+    Return true if the first set is greater than or equal the second set.
+    """
+    return Polyhedron([], [left - right])