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Simplify Expression.__mul__(), Expression.__truediv__()
[linpy.git] / pypol / linexprs.py
diff --git a/pypol/linexprs.py b/pypol/linexprs.py
index ed68493..b74628b 100644 (file)
--- a/pypol/linexprs.py
+++ b/pypol/linexprs.py
@@ -3,14 +3,14 @@ import functools
 import numbers
 import re
 
 import numbers
 import re
 
-from collections import OrderedDict
+from collections import OrderedDict, defaultdict, Mapping
 from fractions import Fraction, gcd
 
 
 __all__ = [
     'Expression',
 from fractions import Fraction, gcd
 
 
 __all__ = [
     'Expression',
-    'Symbol', 'symbols', 'symbolname', 'symbolnames',
-    'Constant',
+    'Symbol', 'Dummy', 'symbols',
+    'Rational',
 ]
 
 
 ]
 
 
@@ -20,7 +20,7 @@ def _polymorphic(func):
         if isinstance(right, Expression):
             return func(left, right)
         elif isinstance(right, numbers.Rational):
         if isinstance(right, Expression):
             return func(left, right)
         elif isinstance(right, numbers.Rational):
-            right = Constant(right)
+            right = Rational(right)
             return func(left, right)
         return NotImplemented
     return wrapper
             return func(left, right)
         return NotImplemented
     return wrapper
@@ -31,68 +31,56 @@ class Expression:
     This class implements linear expressions.
     """
 
     This class implements linear expressions.
     """
 
-    __slots__ = (
-        '_coefficients',
-        '_constant',
-        '_symbols',
-        '_dimension',
-        '_hash',
-    )
-
     def __new__(cls, coefficients=None, constant=0):
         if isinstance(coefficients, str):
     def __new__(cls, coefficients=None, constant=0):
         if isinstance(coefficients, str):
-            if constant:
+            if constant != 0:
                 raise TypeError('too many arguments')
                 raise TypeError('too many arguments')
-            return cls.fromstring(coefficients)
-        if isinstance(coefficients, dict):
-            coefficients = coefficients.items()
+            return Expression.fromstring(coefficients)
         if coefficients is None:
         if coefficients is None:
-            return Constant(constant)
-        coefficients = [(symbol, coefficient)
+            return Rational(constant)
+        if isinstance(coefficients, Mapping):
+            coefficients = coefficients.items()
+        for symbol, coefficient in coefficients:
+            if not isinstance(symbol, Symbol):
+                raise TypeError('symbols must be Symbol instances')
+            if not isinstance(coefficient, numbers.Rational):
+                raise TypeError('coefficients must be rational numbers')
+        coefficients = [(symbol, Fraction(coefficient))
             for symbol, coefficient in coefficients if coefficient != 0]
             for symbol, coefficient in coefficients if coefficient != 0]
+        if not isinstance(constant, numbers.Rational):
+            raise TypeError('constant must be a rational number')
+        constant = Fraction(constant)
         if len(coefficients) == 0:
         if len(coefficients) == 0:
-            return Constant(constant)
-        elif len(coefficients) == 1 and constant == 0:
+            return Rational(constant)
+        if len(coefficients) == 1 and constant == 0:
             symbol, coefficient = coefficients[0]
             if coefficient == 1:
             symbol, coefficient = coefficients[0]
             if coefficient == 1:
-                return Symbol(symbol)
+                return symbol
         self = object().__new__(cls)
         self = object().__new__(cls)
-        self._coefficients = {}
-        for symbol, coefficient in coefficients:
-            symbol = symbolname(symbol)
-            if isinstance(coefficient, Constant):
-                coefficient = coefficient.constant
-            if not isinstance(coefficient, numbers.Rational):
-                raise TypeError('coefficients must be rational numbers '
-                    'or Constant instances')
-            self._coefficients[symbol] = coefficient
-        self._coefficients = OrderedDict(sorted(self._coefficients.items()))
-        if isinstance(constant, Constant):
-            constant = constant.constant
-        if not isinstance(constant, numbers.Rational):
-            raise TypeError('constant must be a rational number '
-                'or a Constant instance')
+        self._coefficients = OrderedDict(sorted(coefficients,
+            key=lambda item: item[0].sortkey()))
         self._constant = constant
         self._symbols = tuple(self._coefficients)
         self._dimension = len(self._symbols)
         self._constant = constant
         self._symbols = tuple(self._coefficients)
         self._dimension = len(self._symbols)
-        self._hash = hash((tuple(self._coefficients.items()), self._constant))
         return self
 
     def coefficient(self, symbol):
         return self
 
     def coefficient(self, symbol):
-        symbol = symbolname(symbol)
+        if not isinstance(symbol, Symbol):
+            raise TypeError('symbol must be a Symbol instance')
         try:
         try:
-            return self._coefficients[symbol]
+            return Rational(self._coefficients[symbol])
         except KeyError:
         except KeyError:
-            return 0
+            return Rational(0)
 
     __getitem__ = coefficient
 
     def coefficients(self):
 
     __getitem__ = coefficient
 
     def coefficients(self):
-        yield from self._coefficients.items()
+        for symbol, coefficient in self._coefficients.items():
+            yield symbol, Rational(coefficient)
 
     @property
     def constant(self):
 
     @property
     def constant(self):
-        return self._constant
+        return Rational(self._constant)
 
     @property
     def symbols(self):
 
     @property
     def symbols(self):
@@ -103,7 +91,7 @@ class Expression:
         return self._dimension
 
     def __hash__(self):
         return self._dimension
 
     def __hash__(self):
-        return self._hash
+        return hash((tuple(self._coefficients.items()), self._constant))
 
     def isconstant(self):
         return False
 
     def isconstant(self):
         return False
@@ -112,9 +100,9 @@ class Expression:
         return False
 
     def values(self):
         return False
 
     def values(self):
-        for symbol in self.symbols:
-            yield self.coefficient(symbol)
-        yield self.constant
+        for coefficient in self._coefficients.values():
+            yield Rational(coefficient)
+        yield Rational(self._constant)
 
     def __bool__(self):
         return True
 
     def __bool__(self):
         return True
@@ -127,68 +115,44 @@ class Expression:
 
     @_polymorphic
     def __add__(self, other):
 
     @_polymorphic
     def __add__(self, other):
-        coefficients = dict(self.coefficients())
-        for symbol, coefficient in other.coefficients():
-            if symbol in coefficients:
-                coefficients[symbol] += coefficient
-            else:
-                coefficients[symbol] = coefficient
-        constant = self.constant + other.constant
+        coefficients = defaultdict(Fraction, self._coefficients)
+        for symbol, coefficient in other._coefficients.items():
+            coefficients[symbol] += coefficient
+        constant = self._constant + other._constant
         return Expression(coefficients, constant)
 
     __radd__ = __add__
 
     @_polymorphic
     def __sub__(self, other):
         return Expression(coefficients, constant)
 
     __radd__ = __add__
 
     @_polymorphic
     def __sub__(self, other):
-        coefficients = dict(self.coefficients())
-        for symbol, coefficient in other.coefficients():
-            if symbol in coefficients:
-                coefficients[symbol] -= coefficient
-            else:
-                coefficients[symbol] = -coefficient
-        constant = self.constant - other.constant
+        coefficients = defaultdict(Fraction, self._coefficients)
+        for symbol, coefficient in other._coefficients.items():
+            coefficients[symbol] -= coefficient
+        constant = self._constant - other._constant
         return Expression(coefficients, constant)
 
     def __rsub__(self, other):
         return -(self - other)
 
         return Expression(coefficients, constant)
 
     def __rsub__(self, other):
         return -(self - other)
 
-    @_polymorphic
     def __mul__(self, other):
     def __mul__(self, other):
-        if other.isconstant():
-            coefficients = dict(self.coefficients())
+        if isinstance(other, numbers.Rational):
+            coefficients = dict(self._coefficients)
             for symbol in coefficients:
             for symbol in coefficients:
-                coefficients[symbol] *= other.constant
-            constant = self.constant * other.constant
+                coefficients[symbol] *= other
+            constant = self._constant * other
             return Expression(coefficients, constant)
             return Expression(coefficients, constant)
-        if isinstance(other, Expression) and not self.isconstant():
-            raise ValueError('non-linear expression: '
-                    '{} * {}'.format(self._parenstr(), other._parenstr()))
         return NotImplemented
 
     __rmul__ = __mul__
 
         return NotImplemented
 
     __rmul__ = __mul__
 
-    @_polymorphic
     def __truediv__(self, other):
     def __truediv__(self, other):
-        if other.isconstant():
-            coefficients = dict(self.coefficients())
+        if isinstance(other, numbers.Rational):
+            coefficients = dict(self._coefficients)
             for symbol in coefficients:
             for symbol in coefficients:
-                coefficients[symbol] = \
-                        Fraction(coefficients[symbol], other.constant)
-            constant = Fraction(self.constant, other.constant)
+                coefficients[symbol] /= other
+            constant = self._constant / other
+            # import pdb; pdb.set_trace()
             return Expression(coefficients, constant)
             return Expression(coefficients, constant)
-        if isinstance(other, Expression):
-            raise ValueError('non-linear expression: '
-                '{} / {}'.format(self._parenstr(), other._parenstr()))
-        return NotImplemented
-
-    def __rtruediv__(self, other):
-        if isinstance(other, self):
-            if self.isconstant():
-                constant = Fraction(other, self.constant)
-                return Expression(constant=constant)
-            else:
-                raise ValueError('non-linear expression: '
-                        '{} / {}'.format(other._parenstr(), self._parenstr()))
         return NotImplemented
 
     @_polymorphic
         return NotImplemented
 
     @_polymorphic
@@ -196,8 +160,8 @@ class Expression:
         # "normal" equality
         # see http://docs.sympy.org/dev/tutorial/gotchas.html#equals-signs
         return isinstance(other, Expression) and \
         # "normal" equality
         # see http://docs.sympy.org/dev/tutorial/gotchas.html#equals-signs
         return isinstance(other, Expression) and \
-                self._coefficients == other._coefficients and \
-                self.constant == other.constant
+            self._coefficients == other._coefficients and \
+            self._constant == other._constant
 
     @_polymorphic
     def __le__(self, other):
 
     @_polymorphic
     def __le__(self, other):
@@ -219,11 +183,30 @@ class Expression:
         from .polyhedra import Gt
         return Gt(self, other)
 
         from .polyhedra import Gt
         return Gt(self, other)
 
-    def _toint(self):
+    def scaleint(self):
         lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
             [value.denominator for value in self.values()])
         return self * lcm
 
         lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
             [value.denominator for value in self.values()])
         return self * lcm
 
+    def subs(self, symbol, expression=None):
+        if expression is None:
+            if isinstance(symbol, Mapping):
+                symbol = symbol.items()
+            substitutions = symbol
+        else:
+            substitutions = [(symbol, expression)]
+        result = self
+        for symbol, expression in substitutions:
+            if not isinstance(symbol, Symbol):
+                raise TypeError('symbols must be Symbol instances')
+            coefficients = [(othersymbol, coefficient)
+                for othersymbol, coefficient in result._coefficients.items()
+                if othersymbol != symbol]
+            coefficient = result._coefficients.get(symbol, 0)
+            constant = result._constant
+            result = Expression(coefficients, constant) + coefficient*expression
+        return result
+
     @classmethod
     def _fromast(cls, node):
         if isinstance(node, ast.Module) and len(node.body) == 1:
     @classmethod
     def _fromast(cls, node):
         if isinstance(node, ast.Module) and len(node.body) == 1:
@@ -233,7 +216,7 @@ class Expression:
         elif isinstance(node, ast.Name):
             return Symbol(node.id)
         elif isinstance(node, ast.Num):
         elif isinstance(node, ast.Name):
             return Symbol(node.id)
         elif isinstance(node, ast.Num):
-            return Constant(node.n)
+            return Rational(node.n)
         elif isinstance(node, ast.UnaryOp) and isinstance(node.op, ast.USub):
             return -cls._fromast(node.operand)
         elif isinstance(node, ast.BinOp):
         elif isinstance(node, ast.UnaryOp) and isinstance(node.op, ast.USub):
             return -cls._fromast(node.operand)
         elif isinstance(node, ast.BinOp):
@@ -254,47 +237,58 @@ class Expression:
     @classmethod
     def fromstring(cls, string):
         # add implicit multiplication operators, e.g. '5x' -> '5*x'
     @classmethod
     def fromstring(cls, string):
         # add implicit multiplication operators, e.g. '5x' -> '5*x'
-        string = cls._RE_NUM_VAR.sub(r'\1*\2', string)
+        string = Expression._RE_NUM_VAR.sub(r'\1*\2', string)
         tree = ast.parse(string, 'eval')
         return cls._fromast(tree)
 
         tree = ast.parse(string, 'eval')
         return cls._fromast(tree)
 
-    def __str__(self):
+    def __repr__(self):
         string = ''
         string = ''
-        i = 0
-        for symbol in self.symbols:
-            coefficient = self.coefficient(symbol)
+        for i, (symbol, coefficient) in enumerate(self.coefficients()):
             if coefficient == 1:
             if coefficient == 1:
-                if i == 0:
-                    string += symbol
-                else:
-                    string += ' + {}'.format(symbol)
+                if i != 0:
+                    string += ' + '
             elif coefficient == -1:
             elif coefficient == -1:
-                if i == 0:
-                    string += '-{}'.format(symbol)
-                else:
-                    string += ' - {}'.format(symbol)
+                string += '-' if i == 0 else ' - '
+            elif i == 0:
+                string += '{}*'.format(coefficient)
+            elif coefficient > 0:
+                string += ' + {}*'.format(coefficient)
             else:
             else:
-                if i == 0:
-                    string += '{}*{}'.format(coefficient, symbol)
-                elif coefficient > 0:
-                    string += ' + {}*{}'.format(coefficient, symbol)
-                else:
-                    assert coefficient < 0
-                    coefficient *= -1
-                    string += ' - {}*{}'.format(coefficient, symbol)
-            i += 1
+                string += ' - {}*'.format(-coefficient)
+            string += '{}'.format(symbol)
         constant = self.constant
         constant = self.constant
-        if constant != 0 and i == 0:
+        if len(string) == 0:
             string += '{}'.format(constant)
         elif constant > 0:
             string += ' + {}'.format(constant)
         elif constant < 0:
             string += '{}'.format(constant)
         elif constant > 0:
             string += ' + {}'.format(constant)
         elif constant < 0:
-            constant *= -1
-            string += ' - {}'.format(constant)
-        if string == '':
-            string = '0'
+            string += ' - {}'.format(-constant)
         return string
 
         return string
 
+    def _repr_latex_(self):
+        string = ''
+        for i, (symbol, coefficient) in enumerate(self.coefficients()):
+            if coefficient == 1:
+                if i != 0:
+                    string += ' + '
+            elif coefficient == -1:
+                string += '-' if i == 0 else ' - '
+            elif i == 0:
+                string += '{}'.format(coefficient._repr_latex_().strip('$'))
+            elif coefficient > 0:
+                string += ' + {}'.format(coefficient._repr_latex_().strip('$'))
+            elif coefficient < 0:
+                string += ' - {}'.format((-coefficient)._repr_latex_().strip('$'))
+            string += '{}'.format(symbol._repr_latex_().strip('$'))
+        constant = self.constant
+        if len(string) == 0:
+            string += '{}'.format(constant._repr_latex_().strip('$'))
+        elif constant > 0:
+            string += ' + {}'.format(constant._repr_latex_().strip('$'))
+        elif constant < 0:
+            string += ' - {}'.format((-constant)._repr_latex_().strip('$'))
+        return '${}$'.format(string)
+
     def _parenstr(self, always=False):
         string = str(self)
         if not always and (self.isconstant() or self.issymbol()):
     def _parenstr(self, always=False):
         string = str(self)
         if not always and (self.isconstant() or self.issymbol()):
@@ -302,30 +296,27 @@ class Expression:
         else:
             return '({})'.format(string)
 
         else:
             return '({})'.format(string)
 
-    def __repr__(self):
-        return '{}({!r})'.format(self.__class__.__name__, str(self))
-
     @classmethod
     def fromsympy(cls, expr):
         import sympy
     @classmethod
     def fromsympy(cls, expr):
         import sympy
-        coefficients = {}
+        coefficients = []
         constant = 0
         for symbol, coefficient in expr.as_coefficients_dict().items():
             coefficient = Fraction(coefficient.p, coefficient.q)
             if symbol == sympy.S.One:
                 constant = coefficient
             elif isinstance(symbol, sympy.Symbol):
         constant = 0
         for symbol, coefficient in expr.as_coefficients_dict().items():
             coefficient = Fraction(coefficient.p, coefficient.q)
             if symbol == sympy.S.One:
                 constant = coefficient
             elif isinstance(symbol, sympy.Symbol):
-                symbol = symbol.name
-                coefficients[symbol] = coefficient
+                symbol = Symbol(symbol.name)
+                coefficients.append((symbol, coefficient))
             else:
                 raise ValueError('non-linear expression: {!r}'.format(expr))
             else:
                 raise ValueError('non-linear expression: {!r}'.format(expr))
-        return cls(coefficients, constant)
+        return Expression(coefficients, constant)
 
     def tosympy(self):
         import sympy
         expr = 0
         for symbol, coefficient in self.coefficients():
 
     def tosympy(self):
         import sympy
         expr = 0
         for symbol, coefficient in self.coefficients():
-            term = coefficient * sympy.Symbol(symbol)
+            term = coefficient * sympy.Symbol(symbol.name)
             expr += term
         expr += self.constant
         return expr
             expr += term
         expr += self.constant
         return expr
@@ -333,16 +324,15 @@ class Expression:
 
 class Symbol(Expression):
 
 
 class Symbol(Expression):
 
-    __slots__ = (
-        '_name',
-        '_hash',
-    )
-
     def __new__(cls, name):
     def __new__(cls, name):
-        name = symbolname(name)
+        if not isinstance(name, str):
+            raise TypeError('name must be a string')
         self = object().__new__(cls)
         self = object().__new__(cls)
-        self._name = name
-        self._hash = hash(self._name)
+        self._name = name.strip()
+        self._coefficients = {self: Fraction(1)}
+        self._constant = Fraction(0)
+        self._symbols = (self,)
+        self._dimension = 1
         return self
 
     @property
         return self
 
     @property
@@ -350,35 +340,20 @@ class Symbol(Expression):
         return self._name
 
     def __hash__(self):
         return self._name
 
     def __hash__(self):
-        return self._hash
+        return hash(self.sortkey())
 
 
-    def coefficient(self, symbol):
-        symbol = symbolname(symbol)
-        if symbol == self.name:
-            return 1
-        else:
-            return 0
-
-    def coefficients(self):
-        yield self.name, 1
-
-    @property
-    def constant(self):
-        return 0
-
-    @property
-    def symbols(self):
+    def sortkey(self):
         return self.name,
 
         return self.name,
 
-    @property
-    def dimension(self):
-        return 1
-
     def issymbol(self):
         return True
 
     def __eq__(self, other):
     def issymbol(self):
         return True
 
     def __eq__(self, other):
-        return isinstance(other, Symbol) and self.name == other.name
+        return not isinstance(other, Dummy) and isinstance(other, Symbol) \
+            and self.name == other.name
+
+    def asdummy(self):
+        return Dummy(self.name)
 
     @classmethod
     def _fromast(cls, node):
 
     @classmethod
     def _fromast(cls, node):
@@ -391,7 +366,10 @@ class Symbol(Expression):
         raise SyntaxError('invalid syntax')
 
     def __repr__(self):
         raise SyntaxError('invalid syntax')
 
     def __repr__(self):
-        return '{}({!r})'.format(self.__class__.__name__, self._name)
+        return self.name
+
+    def _repr_latex_(self):
+        return '${}$'.format(self.name)
 
     @classmethod
     def fromsympy(cls, expr):
 
     @classmethod
     def fromsympy(cls, expr):
@@ -402,90 +380,96 @@ class Symbol(Expression):
             raise TypeError('expr must be a sympy.Symbol instance')
 
 
             raise TypeError('expr must be a sympy.Symbol instance')
 
 
-def symbols(names):
-    if isinstance(names, str):
-        names = names.replace(',', ' ').split()
-    return (Symbol(name) for name in names)
+class Dummy(Symbol):
 
 
-def symbolname(symbol):
-    if isinstance(symbol, str):
-        return symbol.strip()
-    elif isinstance(symbol, Symbol):
-        return symbol.name
-    else:
-        raise TypeError('symbol must be a string or a Symbol instance')
+    _count = 0
 
 
-def symbolnames(symbols):
-    if isinstance(symbols, str):
-        return symbols.replace(',', ' ').split()
-    return (symbolname(symbol) for symbol in symbols)
+    def __new__(cls, name=None):
+        if name is None:
+            name = 'Dummy_{}'.format(Dummy._count)
+        self = object().__new__(cls)
+        self._index = Dummy._count
+        self._name = name.strip()
+        self._coefficients = {self: Fraction(1)}
+        self._constant = Fraction(0)
+        self._symbols = (self,)
+        self._dimension = 1
+        Dummy._count += 1
+        return self
 
 
+    def __hash__(self):
+        return hash(self.sortkey())
 
 
-class Constant(Expression):
+    def sortkey(self):
+        return self._name, self._index
 
 
-    __slots__ = (
-        '_constant',
-        '_hash',
-    )
+    def __eq__(self, other):
+        return isinstance(other, Dummy) and self._index == other._index
 
 
-    def __new__(cls, numerator=0, denominator=None):
-        self = object().__new__(cls)
-        if denominator is None and isinstance(numerator, Constant):
-            self._constant = numerator.constant
-        else:
-            self._constant = Fraction(numerator, denominator)
-        self._hash = hash(self._constant)
-        return self
+    def __repr__(self):
+        return '_{}'.format(self.name)
 
 
-    def __hash__(self):
-        return self._hash
+    def _repr_latex_(self):
+        return '${}_{{{}}}$'.format(self.name, self._index)
 
 
-    def coefficient(self, symbol):
-        symbol = symbolname(symbol)
-        return 0
 
 
-    def coefficients(self):
-        yield from []
+def symbols(names):
+    if isinstance(names, str):
+        names = names.replace(',', ' ').split()
+    return tuple(Symbol(name) for name in names)
 
 
-    @property
-    def symbols(self):
-        return ()
+
+class Rational(Expression, Fraction):
+
+    def __new__(cls, numerator=0, denominator=None):
+        self = Fraction.__new__(cls, numerator, denominator)
+        self._coefficients = {}
+        self._constant = Fraction(self)
+        self._symbols = ()
+        self._dimension = 0
+        return self
+
+    def __hash__(self):
+        return Fraction.__hash__(self)
 
     @property
 
     @property
-    def dimension(self):
-        return 0
+    def constant(self):
+        return self
 
     def isconstant(self):
         return True
 
 
     def isconstant(self):
         return True
 
-    @_polymorphic
-    def __eq__(self, other):
-        return isinstance(other, Constant) and self.constant == other.constant
-
     def __bool__(self):
     def __bool__(self):
-        return self.constant != 0
+        return Fraction.__bool__(self)
 
     @classmethod
     def fromstring(cls, string):
 
     @classmethod
     def fromstring(cls, string):
-        if isinstance(string, str):
-            return Constant(Fraction(string))
-        else:
+        if not isinstance(string, str):
             raise TypeError('string must be a string instance')
             raise TypeError('string must be a string instance')
+        return Rational(string)
 
     def __repr__(self):
 
     def __repr__(self):
-        if self.constant.denominator == 1:
-            return '{}({!r})'.format(self.__class__.__name__,
-                self.constant.numerator)
+        if self.denominator == 1:
+            return '{!r}'.format(self.numerator)
+        else:
+            return '{!r}/{!r}'.format(self.numerator, self.denominator)
+
+    def _repr_latex_(self):
+        if self.denominator == 1:
+            return '${}$'.format(self.numerator)
+        elif self.numerator < 0:
+            return '$-\\frac{{{}}}{{{}}}$'.format(-self.numerator,
+                self.denominator)
         else:
         else:
-            return '{}({!r}, {!r})'.format(self.__class__.__name__,
-                self.constant.numerator, self.constant.denominator)
+            return '$\\frac{{{}}}{{{}}}$'.format(self.numerator,
+                self.denominator)
 
     @classmethod
     def fromsympy(cls, expr):
         import sympy
         if isinstance(expr, sympy.Rational):
 
     @classmethod
     def fromsympy(cls, expr):
         import sympy
         if isinstance(expr, sympy.Rational):
-            return cls(expr.p, expr.q)
+            return Rational(expr.p, expr.q)
         elif isinstance(expr, numbers.Rational):
         elif isinstance(expr, numbers.Rational):
-            return cls(expr)
+            return Rational(expr)
         else:
             raise TypeError('expr must be a sympy.Rational instance')
         else:
             raise TypeError('expr must be a sympy.Rational instance')
A polyhedral library based on ISL