Constant names Empty and Universe, pretty printing

[linpy.git] / pypol / linear.py

diff --git a/pypol/linear.py b/pypol/linear.py
index 6550e92..3cd7633 100644 (file)
--- a/pypol/linear.py
+++ b/pypol/linear.py
@@ -1,19 +1,48 @@
-

 import functools
 import numbers
 
 from fractions import Fraction, gcd
 
 import functools
 import numbers
 
 from fractions import Fraction, gcd
 

+from pypol import isl
+from pypol.isl import libisl
+

 
 __all__ = [
 
 __all__ = [

-    'Expression',
-    'constant', 'symbol', 'symbols',
+    'Expression', 'Constant', 'Symbol', 'symbols',

     'eq', 'le', 'lt', 'ge', 'gt',
     'Polyhedron',
     'eq', 'le', 'lt', 'ge', 'gt',
     'Polyhedron',

-    'empty', 'universe'
+    'Empty', 'Universe'

 ]
 
 
 ]
 
 

+def _polymorphic_method(func):
+    @functools.wraps(func)
+    def wrapper(a, b):
+        if isinstance(b, Expression):
+            return func(a, b)
+        if isinstance(b, numbers.Rational):
+            b = Constant(b)
+            return func(a, b)
+        return NotImplemented
+    return wrapper
+
+def _polymorphic_operator(func):
+    # A polymorphic operator should call a polymorphic method, hence we just
+    # have to test the left operand.
+    @functools.wraps(func)
+    def wrapper(a, b):
+        if isinstance(a, numbers.Rational):
+            a = Constant(a)
+            return func(a, b)
+        elif isinstance(a, Expression):
+            return func(a, b)
+        raise TypeError('arguments must be linear expressions')
+    return wrapper
+
+
+_main_ctx = isl.Context()
+
+

 class Expression:
     """
     This class implements linear expressions.
 class Expression:
     """
     This class implements linear expressions.


@@ -24,37 +53,56 @@ class Expression:
             if constant:
                 raise TypeError('too many arguments')
             return cls.fromstring(coefficients)
             if constant:
                 raise TypeError('too many arguments')
             return cls.fromstring(coefficients)

-        self = super().__new__(cls)
-        self._coefficients = {}

         if isinstance(coefficients, dict):
             coefficients = coefficients.items()
         if isinstance(coefficients, dict):
             coefficients = coefficients.items()

-        if coefficients is not None:
-            for symbol, coefficient in coefficients:
-                if isinstance(symbol, Expression) and symbol.issymbol():
-                    symbol = str(symbol)
-                elif not isinstance(symbol, str):
-                    raise TypeError('symbols must be strings')
-                if not isinstance(coefficient, numbers.Rational):
-                    raise TypeError('coefficients must be rational numbers')
-                if coefficient != 0:
-                    self._coefficients[symbol] = coefficient
+        if coefficients is None:
+            return Constant(constant)
+        coefficients = [(symbol, coefficient)
+                for symbol, coefficient in coefficients if coefficient != 0]
+        if len(coefficients) == 0:
+            return Constant(constant)
+        elif len(coefficients) == 1 and constant == 0:
+            symbol, coefficient = coefficients[0]
+            if coefficient == 1:
+                return Symbol(symbol)
+        self = object().__new__(cls)
+        self._coefficients = {}
+        for symbol, coefficient in coefficients:
+            if isinstance(symbol, Symbol):
+                symbol = str(symbol)
+            elif not isinstance(symbol, str):
+                raise TypeError('symbols must be strings or Symbol instances')
+            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
+        if isinstance(constant, Constant):
+            constant = constant.constant

         if not isinstance(constant, numbers.Rational):
         if not isinstance(constant, numbers.Rational):

-            raise TypeError('constant must be a rational number')
+            raise TypeError('constant must be a rational number or a Constant instance')

         self._constant = constant
         self._constant = constant

+        self._symbols = tuple(sorted(self._coefficients))
+        self._dimension = len(self._symbols)

         return self
 
         return self
 

+    @classmethod
+    def fromstring(cls, string):
+        raise NotImplementedError
+
+    @property

     def symbols(self):
     def symbols(self):

-        yield from sorted(self._coefficients)
+        return self._symbols

 
     @property
     def dimension(self):
 
     @property
     def dimension(self):

-        return len(list(self.symbols()))
+        return self._dimension

 
     def coefficient(self, symbol):
 
     def coefficient(self, symbol):

-        if isinstance(symbol, Expression) and symbol.issymbol():
+        if isinstance(symbol, Symbol):

             symbol = str(symbol)
         elif not isinstance(symbol, str):
             symbol = str(symbol)
         elif not isinstance(symbol, str):

-            raise TypeError('symbol must be a string')
+            raise TypeError('symbol must be a string or a Symbol instance')

         try:
             return self._coefficients[symbol]
         except KeyError:
         try:
             return self._coefficients[symbol]
         except KeyError:


@@ -63,7 +111,7 @@ class Expression:
     __getitem__ = coefficient
 
     def coefficients(self):
     __getitem__ = coefficient
 
     def coefficients(self):

-        for symbol in self.symbols():
+        for symbol in self.symbols:

             yield symbol, self.coefficient(symbol)
 
     @property
             yield symbol, self.coefficient(symbol)
 
     @property


@@ -71,24 +119,22 @@ class Expression:
         return self._constant
 
     def isconstant(self):
         return self._constant
 
     def isconstant(self):

-        return len(self._coefficients) == 0
+        return False

 
     def values(self):
 
     def values(self):

-        for symbol in self.symbols():
+        for symbol in self.symbols:

             yield self.coefficient(symbol)
         yield self.constant
 
             yield self.coefficient(symbol)
         yield self.constant
 

+    @property

     def symbol(self):
     def symbol(self):

-        if not self.issymbol():
-            raise ValueError('not a symbol: {}'.format(self))
-        for symbol in self.symbols():
-            return symbol
+        raise ValueError('not a symbol: {}'.format(self))

 
     def issymbol(self):
 
     def issymbol(self):

-        return len(self._coefficients) == 1 and self._constant == 0
+        return False

 
     def __bool__(self):
 
     def __bool__(self):

-        return (not self.isconstant()) or bool(self.constant)
+        return True

 
     def __pos__(self):
         return self
 
     def __pos__(self):
         return self


@@ -96,18 +142,7 @@ class Expression:
     def __neg__(self):
         return self * -1
 
     def __neg__(self):
         return self * -1
 

-    def _polymorphic(func):
-        @functools.wraps(func)
-        def wrapper(self, other):
-            if isinstance(other, Expression):
-                return func(self, other)
-            if isinstance(other, numbers.Rational):
-                other = Expression(constant=other)
-                return func(self, other)
-            return NotImplemented
-        return wrapper
-
-    @_polymorphic
+    @_polymorphic_method

     def __add__(self, other):
         coefficients = dict(self.coefficients())
         for symbol, coefficient in other.coefficients():
     def __add__(self, other):
         coefficients = dict(self.coefficients())
         for symbol, coefficient in other.coefficients():


@@ -120,7 +155,7 @@ class Expression:
 
     __radd__ = __add__
 
 
     __radd__ = __add__
 

-    @_polymorphic
+    @_polymorphic_method

     def __sub__(self, other):
         coefficients = dict(self.coefficients())
         for symbol, coefficient in other.coefficients():
     def __sub__(self, other):
         coefficients = dict(self.coefficients())
         for symbol, coefficient in other.coefficients():


@@ -131,9 +166,10 @@ class Expression:
         constant = self.constant - other.constant
         return Expression(coefficients, constant)
 
         constant = self.constant - other.constant
         return Expression(coefficients, constant)
 

-    __rsub__ = __sub__
+    def __rsub__(self, other):
+        return -(self - other)

 
 

-    @_polymorphic
+    @_polymorphic_method

     def __mul__(self, other):
         if other.isconstant():
             coefficients = dict(self.coefficients())
     def __mul__(self, other):
         if other.isconstant():
             coefficients = dict(self.coefficients())


@@ -148,7 +184,7 @@ class Expression:
 
     __rmul__ = __mul__
 
 
     __rmul__ = __mul__
 

-    @_polymorphic
+    @_polymorphic_method

     def __truediv__(self, other):
         if other.isconstant():
             coefficients = dict(self.coefficients())
     def __truediv__(self, other):
         if other.isconstant():
             coefficients = dict(self.coefficients())


@@ -163,7 +199,7 @@ class Expression:
         return NotImplemented
 
     def __rtruediv__(self, other):
         return NotImplemented
 
     def __rtruediv__(self, other):

-        if isinstance(other, Rational):
+        if isinstance(other, self):

             if self.isconstant():
                 constant = Fraction(other, self.constant)
                 return Expression(constant=constant)
             if self.isconstant():
                 constant = Fraction(other, self.constant)
                 return Expression(constant=constant)


@@ -174,10 +210,9 @@ class Expression:
 
     def __str__(self):
         string = ''
 
     def __str__(self):
         string = ''

-        symbols = sorted(self.symbols())

         i = 0
         i = 0

-        for symbol in symbols:
-            coefficient = self[symbol]
+        for symbol in self.symbols:
+            coefficient = self.coefficient(symbol)

             if coefficient == 1:
                 if i == 0:
                     string += symbol
             if coefficient == 1:
                 if i == 0:
                     string += symbol


@@ -226,11 +261,7 @@ class Expression:
         string += '}}, {!r})'.format(self.constant)
         return string
 
         string += '}}, {!r})'.format(self.constant)
         return string
 

-    @classmethod
-    def fromstring(cls, string):
-        raise NotImplementedError
-
-    @_polymorphic
+    @_polymorphic_method

     def __eq__(self, other):
         # "normal" equality
         # see http://docs.sympy.org/dev/tutorial/gotchas.html#equals-signs
     def __eq__(self, other):
         # "normal" equality
         # see http://docs.sympy.org/dev/tutorial/gotchas.html#equals-signs


@@ -239,79 +270,112 @@ class Expression:
                 self.constant == other.constant
 
     def __hash__(self):
                 self.constant == other.constant
 
     def __hash__(self):

-        return hash((self._coefficients, self._constant))
+        return hash((tuple(sorted(self._coefficients.items())), self._constant))

 
 

-    def _canonify(self):
+    def _toint(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
 

-    @_polymorphic
+    @_polymorphic_method

     def _eq(self, other):
     def _eq(self, other):

-        return Polyhedron(equalities=[(self - other)._canonify()])
+        return Polyhedron(equalities=[(self - other)._toint()])

 
 

-    @_polymorphic
+    @_polymorphic_method

     def __le__(self, other):
     def __le__(self, other):

-        return Polyhedron(inequalities=[(self - other)._canonify()])
+        return Polyhedron(inequalities=[(other - self)._toint()])

 
 

-    @_polymorphic
+    @_polymorphic_method

     def __lt__(self, other):
     def __lt__(self, other):

-        return Polyhedron(inequalities=[(self - other)._canonify() + 1])
+        return Polyhedron(inequalities=[(other - self)._toint() - 1])

 
 

-    @_polymorphic
+    @_polymorphic_method

     def __ge__(self, other):
     def __ge__(self, other):

-        return Polyhedron(inequalities=[(other - self)._canonify()])
+        return Polyhedron(inequalities=[(self - other)._toint()])

 
 

-    @_polymorphic
+    @_polymorphic_method

     def __gt__(self, other):
     def __gt__(self, other):

-        return Polyhedron(inequalities=[(other - self)._canonify() + 1])
+        return Polyhedron(inequalities=[(self - other)._toint() - 1])

 
 
 
 

-def constant(numerator=0, denominator=None):
-    return Expression(constant=Fraction(numerator, denominator))
+class Constant(Expression):

 
 

-def symbol(name):
-    if not isinstance(name, str):
-        raise TypeError('name must be a string')
-    return Expression(coefficients={name: 1})
+    def __new__(cls, numerator=0, denominator=None):
+        self = object().__new__(cls)
+        if denominator is None:
+            if isinstance(numerator, numbers.Rational):
+                self._constant = numerator
+            elif isinstance(numerator, Constant):
+                self._constant = numerator.constant
+            else:
+                raise TypeError('constant must be a rational number or a Constant instance')
+        else:
+            self._constant = Fraction(numerator, denominator)
+        self._coefficients = {}
+        self._symbols = ()
+        self._dimension = 0
+        return self
+
+    def isconstant(self):
+        return True
+
+    def __bool__(self):
+        return bool(self.constant)
+
+    def __repr__(self):
+        return '{}({!r})'.format(self.__class__.__name__, self._constant)
+
+
+class Symbol(Expression):
+
+    def __new__(cls, name):
+        if isinstance(name, Symbol):
+            name = name.symbol
+        elif not isinstance(name, str):
+            raise TypeError('name must be a string or a Symbol instance')
+        self = object().__new__(cls)
+        self._coefficients = {name: 1}
+        self._constant = 0
+        self._symbols = tuple(name)
+        self._symbol = name
+        self._dimension = 1
+        return self
+
+    @property
+    def symbol(self):
+        return self._symbol
+
+    def issymbol(self):
+        return True
+
+    def __repr__(self):
+        return '{}({!r})'.format(self.__class__.__name__, self._symbol)

 
 def symbols(names):
     if isinstance(names, str):
         names = names.replace(',', ' ').split()
 
 def symbols(names):
     if isinstance(names, str):
         names = names.replace(',', ' ').split()

-    return (symbol(name) for name in names)
+    return (Symbol(name) for name in names)

 
 
 
 

-def _operator(func):
-    @functools.wraps(func)
-    def wrapper(a, b):
-        if isinstance(a, numbers.Rational):
-            a = constant(a)
-        if isinstance(b, numbers.Rational):
-            b = constant(b)
-        if isinstance(a, Expression) and isinstance(b, Expression):
-            return func(a, b)
-        raise TypeError('arguments must be linear expressions')
-    return wrapper
-
-@_operator
+@_polymorphic_operator

 def eq(a, b):
 def eq(a, b):

-    return a._eq(b)
+    return a.__eq__(b)

 
 

-@_operator
+@_polymorphic_operator

 def le(a, b):
 def le(a, b):

-    return a <= b
+    return a.__le__(b)

 
 

-@_operator
+@_polymorphic_operator

 def lt(a, b):
 def lt(a, b):

-    return a < b
+    return a.__lt__(b)

 
 

-@_operator
+@_polymorphic_operator

 def ge(a, b):
 def ge(a, b):

-    return a >= b
+    return a.__ge__(b)

 
 

-@_operator
+@_polymorphic_operator

 def gt(a, b):
 def gt(a, b):

-    return a > b
+    return a.__gt__(b)

 
 
 class Polyhedron:
 
 
 class Polyhedron:


@@ -333,6 +397,7 @@ class Polyhedron:
                         raise TypeError('non-integer constraint: '
                                 '{} == 0'.format(constraint))
                 self._equalities.append(constraint)
                         raise TypeError('non-integer constraint: '
                                 '{} == 0'.format(constraint))
                 self._equalities.append(constraint)

+        self._equalities = tuple(self._equalities)

         self._inequalities = []
         if inequalities is not None:
             for constraint in inequalities:
         self._inequalities = []
         if inequalities is not None:
             for constraint in inequalities:


@@ -341,59 +406,82 @@ class Polyhedron:
                         raise TypeError('non-integer constraint: '
                                 '{} <= 0'.format(constraint))
                 self._inequalities.append(constraint)
                         raise TypeError('non-integer constraint: '
                                 '{} <= 0'.format(constraint))
                 self._inequalities.append(constraint)

+        self._inequalities = tuple(self._inequalities)
+        self._constraints = self._equalities + self._inequalities
+        self._symbols = set()
+        for constraint in self._constraints:
+            self.symbols.update(constraint.symbols)
+        self._symbols = tuple(sorted(self._symbols))

         return self
 
         return self
 

+    @classmethod
+    def fromstring(cls, string):
+        raise NotImplementedError
+

     @property
     def equalities(self):
     @property
     def equalities(self):

-        yield from self._equalities
+        return self._equalities

 
     @property
     def inequalities(self):
 
     @property
     def inequalities(self):

-        yield from self._inequalities
+        return self._inequalities

 
 

+    @property

     def constraints(self):
     def constraints(self):

-        yield from self.equalities
-        yield from self.inequalities
+        return self._constraints

 
 

+    @property

     def symbols(self):
     def symbols(self):

-        s = set()
-        for constraint in self.constraints():
-            s.update(constraint.symbols)
-        yield from sorted(s)
+        return self._symbols

 
     @property
     def dimension(self):
 
     @property
     def dimension(self):

-        return len(self.symbols())
+        return len(self.symbols)

 
     def __bool__(self):
 
     def __bool__(self):

-        # return false if the polyhedron is empty, true otherwise
-        raise NotImplementedError
+        return not self.is_empty()

 
     def __contains__(self, value):
         # is the value in the polyhedron?
         raise NotImplementedError
 
     def __eq__(self, other):
 
     def __contains__(self, value):
         # is the value in the polyhedron?
         raise NotImplementedError
 
     def __eq__(self, other):

-        raise NotImplementedError
+        # works correctly when symbols is not passed
+        # should be equal if values are the same even if symbols are different
+        bset = self._toisl()
+        other = other._toisl()
+        return bool(libisl.isl_basic_set_plain_is_equal(bset, other))

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

-        return self == empty
+        bset = self._toisl()
+        return bool(libisl.isl_basic_set_is_empty(bset))

 
     def isuniverse(self):
 
     def isuniverse(self):

-        return self == universe
+        bset = self._toisl()
+        return bool(libisl.isl_basic_set_is_universe(bset))

 
     def isdisjoint(self, other):
         # return true if the polyhedron has no elements in common with other
 
     def isdisjoint(self, other):
         # return true if the polyhedron has no elements in common with other

-        raise NotImplementedError
+        #symbols = self._symbolunion(other)
+        bset = self._toisl()
+        other = other._toisl()
+        return bool(libisl.isl_set_is_disjoint(bset, other))

 
     def issubset(self, other):
 
     def issubset(self, other):

-        raise NotImplementedError
+        # check if self(bset) is a subset of other
+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        return bool(libisl.isl_set_is_strict_subset(other, bset))

 
     def __le__(self, other):
         return self.issubset(other)
 
     def __lt__(self, other):
 
     def __le__(self, other):
         return self.issubset(other)
 
     def __lt__(self, other):

-        raise NotImplementedError
+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        return bool(libisl.isl_set_is_strict_subset(other, bset))

 
     def issuperset(self, other):
         # test whether every element in other is in the polyhedron
 
     def issuperset(self, other):
         # test whether every element in other is in the polyhedron


@@ -403,6 +491,10 @@ class Polyhedron:
         return self.issuperset(other)
 
     def __gt__(self, other):
         return self.issuperset(other)
 
     def __gt__(self, other):

+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        bool(libisl.isl_set_is_strict_subset(other, bset))

         raise NotImplementedError
 
     def union(self, *others):
         raise NotImplementedError
 
     def union(self, *others):


@@ -428,10 +520,14 @@ class Polyhedron:
     def __and__(self, other):
         return self.intersection(other)
 
     def __and__(self, other):
         return self.intersection(other)
 

-    def difference(self, *others):
-        # return a new polyhedron with elements in the polyhedron that are not
-        # in the others
-        raise NotImplementedError
+    def difference(self, other):
+        # return a new polyhedron with elements in the polyhedron that are not in the other
+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        difference = libisl.isl_set_subtract(bset, other)
+        return difference
+

 
     def __sub__(self, other):
         return self.difference(other)
 
     def __sub__(self, other):
         return self.difference(other)


@@ -441,20 +537,81 @@ class Polyhedron:
         for constraint in self.equalities:
             constraints.append('{} == 0'.format(constraint))
         for constraint in self.inequalities:
         for constraint in self.equalities:
             constraints.append('{} == 0'.format(constraint))
         for constraint in self.inequalities:

-            constraints.append('{} <= 0'.format(constraint))
+            constraints.append('{} >= 0'.format(constraint))

         return '{{{}}}'.format(', '.join(constraints))
 
     def __repr__(self):
         return '{{{}}}'.format(', '.join(constraints))
 
     def __repr__(self):

-        equalities = list(self.equalities)
-        inequalities = list(self.inequalities)
-        return '{}(equalities={!r}, inequalities={!r})' \
-                ''.format(self.__class__.__name__, equalities, inequalities)
+        if self.isempty():
+            return 'Empty'
+        elif self.isuniverse():
+            return 'Universe'
+        else:
+            equalities = list(self.equalities)
+            inequalities = list(self.inequalities)
+            return '{}(equalities={!r}, inequalities={!r})' \
+                    ''.format(self.__class__.__name__, equalities, inequalities)
+
+    def _symbolunion(self, *others):
+        symbols = set(self.symbols)
+        for other in others:
+            symbols.update(other.symbols)
+        return sorted(symbols)
+
+    def _toisl(self, symbols=None):
+        if symbols is None:
+            symbols = self.symbols
+        num_coefficients = len(symbols)
+        space = libisl.isl_space_set_alloc(_main_ctx, 0, num_coefficients)
+        bset = libisl.isl_basic_set_universe(libisl.isl_space_copy(space))
+        ls = libisl.isl_local_space_from_space(space)
+        #if there are equalities/inequalities, take each constant and coefficient and add as a constraint to the basic set
+        for eq in self.equalities:
+            ceq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(ls))
+            coeff_eq = dict(eq.coefficients())
+            if eq.constant:
+                value = str(eq.constant).encode()
+                val = libisl.isl_val_read_from_str(_main_ctx, value)
+                ceq = libisl.isl_constraint_set_constant_val(ceq, val)
+            for eq in coeff_eq:
+                number = str(coeff_eq.get(eq)).encode()
+                num = libisl.isl_val_read_from_str(_main_ctx, number)
+                iden = symbols.index(eq)
+                ceq = libisl.isl_constraint_set_coefficient_val(ceq, libisl.isl_dim_set, iden, num)  #use 3 for type isl_dim_set
+            bset = libisl.isl_basic_set_add_constraint(bset, ceq)
+        for ineq in self.inequalities:
+            cin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(ls))
+            coeff_in = dict(ineq.coefficients())
+            if ineq.constant:
+                value = str(ineq.constant).encode()
+                val = libisl.isl_val_read_from_str(_main_ctx, value)
+                cin = libisl.isl_constraint_set_constant_val(cin, val)
+            for ineq in coeff_in:
+                number = str(coeff_in.get(ineq)).encode()
+                num = libisl.isl_val_read_from_str(_main_ctx, number)
+                iden = symbols.index(ineq)
+                cin = libisl.isl_constraint_set_coefficient_val(cin, libisl.isl_dim_set, iden, num)  #use 3 for type isl_dim_set
+            bset = libisl.isl_basic_set_add_constraint(bset, cin)
+        bset = isl.BasicSet(bset)
+        return bset

 
     @classmethod
 
     @classmethod

-    def fromstring(cls, string):
+    def _fromisl(cls, bset):

         raise NotImplementedError
         raise NotImplementedError

-
-
-empty = le(1, 0)
-
-universe = Polyhedron()
+        equalities = ...
+        inequalities = ...
+        return cls(equalities, inequalities)
+        '''takes basic set  in isl form and puts back into python version of polyhedron
+        isl example code gives isl form as:
+            "{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}")
+            our printer is giving form as:
+            { [i0, i1] : 2i1 >= -2 - i0 } '''
+
+Empty = eq(0,1)
+Universe = Polyhedron()
+
+if __name__ == '__main__':
+    ex1 = Expression(coefficients={'a': 6, 'b': 6}, constant= 3) #this is the expression that does not work (even without adding values)
+    ex2 = Expression(coefficients={'x': 4, 'y': 2}, constant= 3)
+    p = Polyhedron(equalities=[ex2])
+    p2 = Polyhedron(equalities=[ex2])
+    print(p._toisl()) # checking is values works for toisl