Source code for sympy.core.basic

"""Base class for all the objects in SymPy"""
from __future__ import print_function, division

from .assumptions import BasicMeta, ManagedProperties
from .cache import cacheit
from .sympify import _sympify, sympify, SympifyError
from .compatibility import (iterable, Iterator, ordered,
    string_types, with_metaclass, zip_longest, range)
from .singleton import S

from inspect import getmro

class Basic(with_metaclass(ManagedProperties)):
    Base class for all objects in SymPy.


    1) Always use ``.args``, when accessing parameters of some instance:

        >>> from sympy import cot
        >>> from import x, y

        >>> cot(x).args

        >>> cot(x).args[0]

        >>> (x*y).args
        (x, y)

        >>> (x*y).args[1]

    2) Never use internal methods or variables (the ones prefixed with ``_``):

        >>> cot(x)._args    # do not use this, use cot(x).args instead

    __slots__ = ['_mhash',              # hash value
                 '_args',               # arguments

    # To be overridden with True in the appropriate subclasses
    is_number = False
    is_Atom = False
    is_Symbol = False
    is_Dummy = False
    is_Wild = False
    is_Function = False
    is_Add = False
    is_Mul = False
    is_Pow = False
    is_Number = False
    is_Float = False
    is_Rational = False
    is_Integer = False
    is_NumberSymbol = False
    is_Order = False
    is_Derivative = False
    is_Piecewise = False
    is_Poly = False
    is_AlgebraicNumber = False
    is_Relational = False
    is_Equality = False
    is_Boolean = False
    is_Not = False
    is_Matrix = False
    is_Vector = False
    is_Point = False

    def __new__(cls, *args):
        obj = object.__new__(cls)
        obj._assumptions = cls.default_assumptions
        obj._mhash = None  # will be set by __hash__ method.

        obj._args = args  # all items in args must be Basic objects
        return obj

    def copy(self):
        return self.func(*self.args)

    def __reduce_ex__(self, proto):
        """ Pickling support."""
        return type(self), self.__getnewargs__(), self.__getstate__()

    def __getnewargs__(self):
        return self.args

    def __getstate__(self):
        return {}

    def __setstate__(self, state):
        for k, v in state.items():
            setattr(self, k, v)

    def __hash__(self):
        # hash cannot be cached using cache_it because infinite recurrence
        # occurs as hash is needed for setting cache dictionary keys
        h = self._mhash
        if h is None:
            h = hash((type(self).__name__,) + self._hashable_content())
            self._mhash = h
        return h

    def _hashable_content(self):
        """Return a tuple of information about self that can be used to
        compute the hash. If a class defines additional attributes,
        like ``name`` in Symbol, then this method should be updated
        accordingly to return such relevant attributes.

        Defining more than _hashable_content is necessary if __eq__ has
        been defined by a class. See note about this in Basic.__eq__."""
        return self._args

    def assumptions0(self):
        Return object `type` assumptions.

        For example:

          Symbol('x', real=True)
          Symbol('x', integer=True)

        are different objects. In other words, besides Python type (Symbol in
        this case), the initial assumptions are also forming their typeinfo.


        >>> from sympy import Symbol
        >>> from import x
        >>> x.assumptions0
        {'commutative': True}
        >>> x = Symbol("x", positive=True)
        >>> x.assumptions0
        {'commutative': True, 'complex': True, 'hermitian': True,
        'imaginary': False, 'negative': False, 'nonnegative': True,
        'nonpositive': False, 'nonzero': True, 'positive': True, 'real': True,
        'zero': False}

        return {}

    def compare(self, other):
        Return -1, 0, 1 if the object is smaller, equal, or greater than other.

        Not in the mathematical sense. If the object is of a different type
        from the "other" then their classes are ordered according to
        the sorted_classes list.


        >>> from import x, y

        # all redefinitions of __cmp__ method should start with the
        # following lines:
        if self is other:
            return 0
        n1 = self.__class__
        n2 = other.__class__
        c = (n1 > n2) - (n1 < n2)
        if c:
            return c
        st = self._hashable_content()
        ot = other._hashable_content()
        c = (len(st) > len(ot)) - (len(st) < len(ot))
        if c:
            return c
        for l, r in zip(st, ot):
            l = Basic(*l) if isinstance(l, frozenset) else l
            r = Basic(*r) if isinstance(r, frozenset) else r
            if isinstance(l, Basic):
                c =
                c = (l > r) - (l < r)
            if c:
                return c
        return 0

    def _compare_pretty(a, b):
        from sympy.series.order import Order
        if isinstance(a, Order) and not isinstance(b, Order):
            return 1
        if not isinstance(a, Order) and isinstance(b, Order):
            return -1

        if a.is_Rational and b.is_Rational:
            l = a.p * b.q
            r = b.p * a.q
            return (l > r) - (l < r)
            from sympy.core.symbol import Wild
            p1, p2, p3 = Wild("p1"), Wild("p2"), Wild("p3")
            r_a = a.match(p1 * p2**p3)
            if r_a and p3 in r_a:
                a3 = r_a[p3]
                r_b = b.match(p1 * p2**p3)
                if r_b and p3 in r_b:
                    b3 = r_b[p3]
                    c =, b3)
                    if c != 0:
                        return c

        return, b)

    def fromiter(cls, args, **assumptions):
        Create a new object from an iterable.

        This is a convenience function that allows one to create objects from
        any iterable, without having to convert to a list or tuple first.


        >>> from sympy import Tuple
        >>> Tuple.fromiter(i for i in range(5))
        (0, 1, 2, 3, 4)

        return cls(*tuple(args), **assumptions)

    def class_key(cls):
        """Nice order of classes. """
        return 5, 0, cls.__name__

    def sort_key(self, order=None):
        Return a sort key.


        >>> from sympy.core import S, I

        >>> sorted([S(1)/2, I, -I], key=lambda x: x.sort_key())
        [1/2, -I, I]

        >>> S("[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]")
        [x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)]
        >>> sorted(_, key=lambda x: x.sort_key())
        [x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2]


        # XXX: remove this when issue 5169 is fixed
        def inner_key(arg):
            if isinstance(arg, Basic):
                return arg.sort_key(order)
                return arg

        args = self._sorted_args
        args = len(args), tuple([inner_key(arg) for arg in args])
        return self.class_key(), args, S.One.sort_key(), S.One

    def __eq__(self, other):
        """Return a boolean indicating whether a == b on the basis of
        their symbolic trees.

        This is the same as == 0 but faster.


        If a class that overrides __eq__() needs to retain the
        implementation of __hash__() from a parent class, the
        interpreter must be told this explicitly by setting __hash__ =
        <ParentClass>.__hash__. Otherwise the inheritance of __hash__()
        will be blocked, just as if __hash__ had been explicitly set to


        from sympy import Pow
        if self is other:
            return True

        from .function import AppliedUndef, UndefinedFunction as UndefFunc

        if isinstance(self, UndefFunc) and isinstance(other, UndefFunc):
            if self.class_key() == other.class_key():
                return True
                return False
        if type(self) is not type(other):
            # issue 6100 a**1.0 == a like a**2.0 == a**2
            if isinstance(self, Pow) and self.exp == 1:
                return self.base == other
            if isinstance(other, Pow) and other.exp == 1:
                return self == other.base
                other = _sympify(other)
            except SympifyError:
                return False    # sympy != other

            if isinstance(self, AppliedUndef) and isinstance(other,
                if self.class_key() != other.class_key():
                    return False
            elif type(self) is not type(other):
                return False

        return self._hashable_content() == other._hashable_content()

    def __ne__(self, other):
        """a != b  -> Compare two symbolic trees and see whether they are different

           this is the same as:

    != 0

           but faster
        return not self.__eq__(other)

    def dummy_eq(self, other, symbol=None):
        Compare two expressions and handle dummy symbols.


        >>> from sympy import Dummy
        >>> from import x, y

        >>> u = Dummy('u')

        >>> (u**2 + 1).dummy_eq(x**2 + 1)
        >>> (u**2 + 1) == (x**2 + 1)

        >>> (u**2 + y).dummy_eq(x**2 + y, x)
        >>> (u**2 + y).dummy_eq(x**2 + y, y)

        dummy_symbols = [s for s in self.free_symbols if s.is_Dummy]

        if not dummy_symbols:
            return self == other
        elif len(dummy_symbols) == 1:
            dummy = dummy_symbols.pop()
            raise ValueError(
                "only one dummy symbol allowed on the left-hand side")

        if symbol is None:
            symbols = other.free_symbols

            if not symbols:
                return self == other
            elif len(symbols) == 1:
                symbol = symbols.pop()
                raise ValueError("specify a symbol in which expressions should be compared")

        tmp = dummy.__class__()

        return self.subs(dummy, tmp) == other.subs(symbol, tmp)

    # Note, we always use the default ordering (lex) in __str__ and __repr__,
    # regardless of the global setting.  See issue 5487.
    def __repr__(self):
        from sympy.printing import sstr
        return sstr(self, order=None)

    def __str__(self):
        from sympy.printing import sstr
        return sstr(self, order=None)

    def atoms(self, *types):
        """Returns the atoms that form the current object.

           By default, only objects that are truly atomic and can't
           be divided into smaller pieces are returned: symbols, numbers,
           and number symbols like I and pi. It is possible to request
           atoms of any type, however, as demonstrated below.


           >>> from sympy import I, pi, sin
           >>> from import x, y
           >>> (1 + x + 2*sin(y + I*pi)).atoms()
           set([1, 2, I, pi, x, y])

           If one or more types are given, the results will contain only
           those types of atoms.


           >>> from sympy import Number, NumberSymbol, Symbol
           >>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
           set([x, y])

           >>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
           set([1, 2])

           >>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
           set([1, 2, pi])

           >>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
           set([1, 2, I, pi])

           Note that I (imaginary unit) and zoo (complex infinity) are special
           types of number symbols and are not part of the NumberSymbol class.

           The type can be given implicitly, too:

           >>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
           set([x, y])

           Be careful to check your assumptions when using the implicit option
           since ``S(1).is_Integer = True`` but ``type(S(1))`` is ``One``, a special type
           of sympy atom, while ``type(S(2))`` is type ``Integer`` and will find all
           integers in an expression:

           >>> from sympy import S
           >>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))

           >>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
           set([1, 2])

           Finally, arguments to atoms() can select more than atomic atoms: any
           sympy type (loaded in core/ can be listed as an argument
           and those types of "atoms" as found in scanning the arguments of the
           expression recursively:

           >>> from sympy import Function, Mul
           >>> from sympy.core.function import AppliedUndef
           >>> f = Function('f')
           >>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
           set([f(x), sin(y + I*pi)])
           >>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)

           >>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
           set([I*pi, 2*sin(y + I*pi)])

        if types:
            types = tuple(
                [t if isinstance(t, type) else type(t) for t in types])
            types = (Atom,)
        result = set()
        for expr in preorder_traversal(self):
            if isinstance(expr, types):
        return result

    def free_symbols(self):
        """Return from the atoms of self those which are free symbols.

        For most expressions, all symbols are free symbols. For some classes
        this is not true. e.g. Integrals use Symbols for the dummy variables
        which are bound variables, so Integral has a method to return all
        symbols except those. Derivative keeps track of symbols with respect
        to which it will perform a derivative; those are
        bound variables, too, so it has its own free_symbols method.

        Any other method that uses bound variables should implement a
        free_symbols method."""
        return set().union(*[a.free_symbols for a in self.args])

    def canonical_variables(self):
        """Return a dictionary mapping any variable defined in
        ``self.variables`` as underscore-suffixed numbers
        corresponding to their position in ``self.variables``. Enough
        underscores are added to ensure that there will be no clash with
        existing free symbols.


        >>> from sympy import Lambda
        >>> from import x
        >>> Lambda(x, 2*x).canonical_variables
        {x: 0_}
        from sympy import Symbol
        if not hasattr(self, 'variables'):
            return {}
        u = "_"
        while any( for s in self.free_symbols):
            u += "_"
        name = '%%i%s' % u
        V = self.variables
        return dict(list(zip(V, [Symbol(name % i, **v.assumptions0)
            for i, v in enumerate(V)])))

    def rcall(self, *args):
        """Apply on the argument recursively through the expression tree.

        This method is used to simulate a common abuse of notation for
        operators. For instance in SymPy the the following will not work:

        ``(x+Lambda(y, 2*y))(z) == x+2*z``,

        however you can use

        >>> from sympy import Lambda
        >>> from import x, y, z
        >>> (x + Lambda(y, 2*y)).rcall(z)
        x + 2*z
        return Basic._recursive_call(self, args)

    def _recursive_call(expr_to_call, on_args):
        from sympy import Symbol
        def the_call_method_is_overridden(expr):
            for cls in getmro(type(expr)):
                if '__call__' in cls.__dict__:
                    return cls != Basic

        if callable(expr_to_call) and the_call_method_is_overridden(expr_to_call):
            if isinstance(expr_to_call, Symbol):  # XXX When you call a Symbol it is
                return expr_to_call               # transformed into an UndefFunction
                return expr_to_call(*on_args)
        elif expr_to_call.args:
            args = [Basic._recursive_call(
                sub, on_args) for sub in expr_to_call.args]
            return type(expr_to_call)(*args)
            return expr_to_call

    def is_hypergeometric(self, k):
        from sympy.simplify import hypersimp
        return hypersimp(self, k) is not None

    def is_comparable(self):
        """Return True if self can be computed to a real number
        (or already is a real number) with precision, else False.


        >>> from sympy import exp_polar, pi, I
        >>> (I*exp_polar(I*pi/2)).is_comparable
        >>> (I*exp_polar(I*pi*2)).is_comparable

        A False result does not mean that `self` cannot be rewritten
        into a form that would be comparable. For example, the
        difference computed below is zero but without simplification
        it does not evaluate to a zero with precision:

        >>> e = 2**pi*(1 + 2**pi)
        >>> dif = e - e.expand()
        >>> dif.is_comparable
        >>> dif.n(2)._prec

        is_real = self.is_real
        if is_real is False:
            return False
        is_number = self.is_number
        if is_number is False:
            return False
        n, i = [p.evalf(2) if not p.is_Number else p
            for p in self.as_real_imag()]
        if not i.is_Number or not n.is_Number:
            return False
        if i:
            # if _prec = 1 we can't decide and if not,
            # the answer is False because numbers with
            # imaginary parts can't be compared
            # so return False
            return False
            return n._prec != 1

    def func(self):
        The top-level function in an expression.

        The following should hold for all objects::

            >> x == x.func(*x.args)


        >>> from import x
        >>> a = 2*x
        >>> a.func
        <class 'sympy.core.mul.Mul'>
        >>> a.args
        (2, x)
        >>> a.func(*a.args)
        >>> a == a.func(*a.args)

        return self.__class__

    def args(self):
        """Returns a tuple of arguments of 'self'.


        >>> from sympy import cot
        >>> from import x, y

        >>> cot(x).args

        >>> cot(x).args[0]

        >>> (x*y).args
        (x, y)

        >>> (x*y).args[1]


        Never use self._args, always use self.args.
        Only use _args in __new__ when creating a new function.
        Don't override .args() from Basic (so that it's easy to
        change the interface in the future if needed).
        return self._args

    def _sorted_args(self):
        The same as ``args``.  Derived classes which don't fix an
        order on their arguments should override this method to
        produce the sorted representation.
        return self.args

    def as_poly(self, *gens, **args):
        """Converts ``self`` to a polynomial or returns ``None``.

           >>> from sympy import sin
           >>> from import x, y

           >>> print((x**2 + x*y).as_poly())
           Poly(x**2 + x*y, x, y, domain='ZZ')

           >>> print((x**2 + x*y).as_poly(x, y))
           Poly(x**2 + x*y, x, y, domain='ZZ')

           >>> print((x**2 + sin(y)).as_poly(x, y))

        from sympy.polys import Poly, PolynomialError

            poly = Poly(self, *gens, **args)

            if not poly.is_Poly:
                return None
                return poly
        except PolynomialError:
            return None

    def as_content_primitive(self, radical=False, clear=True):
        """A stub to allow Basic args (like Tuple) to be skipped when computing
        the content and primitive components of an expression.

        See docstring of Expr.as_content_primitive
        return S.One, self

    def subs(self, *args, **kwargs):
        Substitutes old for new in an expression after sympifying args.

        `args` is either:
          - two arguments, e.g. foo.subs(old, new)
          - one iterable argument, e.g. foo.subs(iterable). The iterable may be
             o an iterable container with (old, new) pairs. In this case the
               replacements are processed in the order given with successive
               patterns possibly affecting replacements already made.
             o a dict or set whose key/value items correspond to old/new pairs.
               In this case the old/new pairs will be sorted by op count and in
               case of a tie, by number of args and the default_sort_key. The
               resulting sorted list is then processed as an iterable container
               (see previous).

        If the keyword ``simultaneous`` is True, the subexpressions will not be
        evaluated until all the substitutions have been made.


        >>> from sympy import pi, exp, limit, oo
        >>> from import x, y
        >>> (1 + x*y).subs(x, pi)
        pi*y + 1
        >>> (1 + x*y).subs({x:pi, y:2})
        1 + 2*pi
        >>> (1 + x*y).subs([(x, pi), (y, 2)])
        1 + 2*pi
        >>> reps = [(y, x**2), (x, 2)]
        >>> (x + y).subs(reps)
        >>> (x + y).subs(reversed(reps))
        x**2 + 2

        >>> (x**2 + x**4).subs(x**2, y)
        y**2 + y

        To replace only the x**2 but not the x**4, use xreplace:

        >>> (x**2 + x**4).xreplace({x**2: y})
        x**4 + y

        To delay evaluation until all substitutions have been made,
        set the keyword ``simultaneous`` to True:

        >>> (x/y).subs([(x, 0), (y, 0)])
        >>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)

        This has the added feature of not allowing subsequent substitutions
        to affect those already made:

        >>> ((x + y)/y).subs({x + y: y, y: x + y})
        >>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
        y/(x + y)

        In order to obtain a canonical result, unordered iterables are
        sorted by count_op length, number of arguments and by the
        default_sort_key to break any ties. All other iterables are left

        >>> from sympy import sqrt, sin, cos
        >>> from import a, b, c, d, e

        >>> A = (sqrt(sin(2*x)), a)
        >>> B = (sin(2*x), b)
        >>> C = (cos(2*x), c)
        >>> D = (x, d)
        >>> E = (exp(x), e)

        >>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)

        >>> expr.subs(dict([A, B, C, D, E]))
        a*c*sin(d*e) + b

        The resulting expression represents a literal replacement of the
        old arguments with the new arguments. This may not reflect the
        limiting behavior of the expression:

        >>> (x**3 - 3*x).subs({x: oo})

        >>> limit(x**3 - 3*x, x, oo)

        If the substitution will be followed by numerical
        evaluation, it is better to pass the substitution to
        evalf as

        >>> (1/x).evalf(subs={x: 3.0}, n=21)

        rather than

        >>> (1/x).subs({x: 3.0}).evalf(21)

        as the former will ensure that the desired level of precision is

        See Also
        replace: replacement capable of doing wildcard-like matching,
                 parsing of match, and conditional replacements
        xreplace: exact node replacement in expr tree; also capable of
                  using matching rules
        evalf: calculates the given formula to a desired level of precision

        from sympy.core.containers import Dict
        from sympy.utilities import default_sort_key
        from sympy import Dummy, Symbol

        unordered = False
        if len(args) == 1:
            sequence = args[0]
            if isinstance(sequence, set):
                unordered = True
            elif isinstance(sequence, (Dict, dict)):
                unordered = True
                sequence = sequence.items()
            elif not iterable(sequence):
                from sympy.utilities.misc import filldedent
                raise ValueError(filldedent("""
                   When a single argument is passed to subs
                   it should be a dictionary of old: new pairs or an iterable
                   of (old, new) tuples."""))
        elif len(args) == 2:
            sequence = [args]
            raise ValueError("subs accepts either 1 or 2 arguments")

        sequence = list(sequence)
        for i in range(len(sequence)):
            o, n = sequence[i]
            so, sn = sympify(o), sympify(n)
            if not isinstance(so, Basic):
                if type(o) is str:
                    so = Symbol(o)
            sequence[i] = (so, sn)
            if _aresame(so, sn):
                sequence[i] = None
        sequence = list(filter(None, sequence))

        if unordered:
            sequence = dict(sequence)
            if not all(k.is_Atom for k in sequence):
                d = {}
                for o, n in sequence.items():
                        ops = o.count_ops(), len(o.args)
                    except TypeError:
                        ops = (0, 0)
                    d.setdefault(ops, []).append((o, n))
                newseq = []
                for k in sorted(d.keys(), reverse=True):
                        sorted([v[0] for v in d[k]], key=default_sort_key))
                sequence = [(k, sequence[k]) for k in newseq]
                del newseq, d
                sequence = sorted([(k, v) for (k, v) in sequence.items()],

        if kwargs.pop('simultaneous', False):  # XXX should this be the default for dict subs?
            reps = {}
            rv = self
            kwargs['hack2'] = True
            m = Dummy()
            for old, new in sequence:
                d = Dummy(commutative=new.is_commutative)
                # using d*m so Subs will be used on dummy variables
                # in things like Derivative(f(x, y), x) in which x
                # is both free and bound
                rv = rv._subs(old, d*m, **kwargs)
                if not isinstance(rv, Basic):
                reps[d] = new
            reps[m] = S.One  # get rid of m
            return rv.xreplace(reps)
            rv = self
            for old, new in sequence:
                rv = rv._subs(old, new, **kwargs)
                if not isinstance(rv, Basic):
            return rv

    def _subs(self, old, new, **hints):
        """Substitutes an expression old -> new.

        If self is not equal to old then _eval_subs is called.
        If _eval_subs doesn't want to make any special replacement
        then a None is received which indicates that the fallback
        should be applied wherein a search for replacements is made
        amongst the arguments of self.

        >>> from sympy import Add
        >>> from import x, y, z


        Add's _eval_subs knows how to target x + y in the following
        so it makes the change:

            >>> (x + y + z).subs(x + y, 1)
            z + 1

        Add's _eval_subs doesn't need to know how to find x + y in
        the following:

            >>> Add._eval_subs(z*(x + y) + 3, x + y, 1) is None

        The returned None will cause the fallback routine to traverse the args and
        pass the z*(x + y) arg to Mul where the change will take place and the
        substitution will succeed:

            >>> (z*(x + y) + 3).subs(x + y, 1)
            z + 3

        ** Developers Notes **

        An _eval_subs routine for a class should be written if:

            1) any arguments are not instances of Basic (e.g. bool, tuple);

            2) some arguments should not be targeted (as in integration

            3) if there is something other than a literal replacement
               that should be attempted (as in Piecewise where the condition
               may be updated without doing a replacement).

        If it is overridden, here are some special cases that might arise:

            1) If it turns out that no special change was made and all
               the original sub-arguments should be checked for
               replacements then None should be returned.

            2) If it is necessary to do substitutions on a portion of
               the expression then _subs should be called. _subs will
               handle the case of any sub-expression being equal to old
               (which usually would not be the case) while its fallback
               will handle the recursion into the sub-arguments. For
               example, after Add's _eval_subs removes some matching terms
               it must process the remaining terms so it calls _subs
               on each of the un-matched terms and then adds them
               onto the terms previously obtained.

           3) If the initial expression should remain unchanged then
              the original expression should be returned. (Whenever an
              expression is returned, modified or not, no further
              substitution of old -> new is attempted.) Sum's _eval_subs
              routine uses this strategy when a substitution is attempted
              on any of its summation variables.

        def fallback(self, old, new):
            Try to replace old with new in any of self's arguments.
            hit = False
            args = list(self.args)
            for i, arg in enumerate(args):
                if not hasattr(arg, '_eval_subs'):
                arg = arg._subs(old, new, **hints)
                if not _aresame(arg, args[i]):
                    hit = True
                    args[i] = arg
            if hit:
                rv = self.func(*args)
                hack2 = hints.get('hack2', False)
                if hack2 and self.is_Mul and not rv.is_Mul:  # 2-arg hack
                    coeff = S.One
                    nonnumber = []
                    for i in args:
                        if i.is_Number:
                            coeff *= i
                    nonnumber = self.func(*nonnumber)
                    if coeff is S.One:
                        return nonnumber
                        return self.func(coeff, nonnumber, evaluate=False)
                return rv
            return self

        if _aresame(self, old):
            return new

        rv = self._eval_subs(old, new)
        if rv is None:
            rv = fallback(self, old, new)
        return rv

    def _eval_subs(self, old, new):
        """Override this stub if you want to do anything more than
        attempt a replacement of old with new in the arguments of self.

        See also: _subs
        return None

    def xreplace(self, rule):
        Replace occurrences of objects within the expression.

        rule : dict-like
            Expresses a replacement rule

        xreplace : the result of the replacement


        >>> from sympy import symbols, pi, exp
        >>> x, y, z = symbols('x y z')
        >>> (1 + x*y).xreplace({x: pi})
        pi*y + 1
        >>> (1 + x*y).xreplace({x: pi, y: 2})
        1 + 2*pi

        Replacements occur only if an entire node in the expression tree is

        >>> (x*y + z).xreplace({x*y: pi})
        z + pi
        >>> (x*y*z).xreplace({x*y: pi})
        >>> (2*x).xreplace({2*x: y, x: z})
        >>> (2*2*x).xreplace({2*x: y, x: z})
        >>> (x + y + 2).xreplace({x + y: 2})
        x + y + 2
        >>> (x + 2 + exp(x + 2)).xreplace({x + 2: y})
        x + exp(y) + 2

        xreplace doesn't differentiate between free and bound symbols. In the
        following, subs(x, y) would not change x since it is a bound symbol,
        but xreplace does:

        >>> from sympy import Integral
        >>> Integral(x, (x, 1, 2*x)).xreplace({x: y})
        Integral(y, (y, 1, 2*y))

        Trying to replace x with an expression raises an error:

        >>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) # doctest: +SKIP
        ValueError: Invalid limits given: ((2*y, 1, 4*y),)

        See Also
        replace: replacement capable of doing wildcard-like matching,
                 parsing of match, and conditional replacements
        subs: substitution of subexpressions as defined by the objects

        value, _ = self._xreplace(rule)
        return value

    def _xreplace(self, rule):
        Helper for xreplace. Tracks whether a replacement actually occurred.
        if self in rule:
            return rule[self], True
        elif rule:
            args = []
            changed = False
            for a in self.args:
                    a_xr = a._xreplace(rule)
                    changed |= a_xr[1]
                except AttributeError:
            args = tuple(args)
            if changed:
                return self.func(*args), True
        return self, False

    def has(self, *patterns):
        Test whether any subexpression matches any of the patterns.


        >>> from sympy import sin
        >>> from import x, y, z
        >>> (x**2 + sin(x*y)).has(z)
        >>> (x**2 + sin(x*y)).has(x, y, z)
        >>> x.has(x)

        Note that ``expr.has(*patterns)`` is exactly equivalent to
        ``any(expr.has(p) for p in patterns)``. In particular, ``False`` is
        returned when the list of patterns is empty.

        >>> x.has()

        return any(self._has(pattern) for pattern in patterns)

    def _has(self, pattern):
        """Helper for .has()"""
        from sympy.core.function import UndefinedFunction, Function
        if isinstance(pattern, UndefinedFunction):
            return any(f.func == pattern or f == pattern
            for f in self.atoms(Function, UndefinedFunction))

        pattern = sympify(pattern)
        if isinstance(pattern, BasicMeta):
            return any(isinstance(arg, pattern)
            for arg in preorder_traversal(self))

            match = pattern._has_matcher()
            return any(match(arg) for arg in preorder_traversal(self))
        except AttributeError:
            return any(arg == pattern for arg in preorder_traversal(self))

    def _has_matcher(self):
        """Helper for .has()"""
        return self.__eq__

    def replace(self, query, value, map=False, simultaneous=True, exact=False):
        Replace matching subexpressions of ``self`` with ``value``.

        If ``map = True`` then also return the mapping {old: new} where ``old``
        was a sub-expression found with query and ``new`` is the replacement
        value for it. If the expression itself doesn't match the query, then
        the returned value will be ``self.xreplace(map)`` otherwise it should
        be ``self.subs(ordered(map.items()))``.

        Traverses an expression tree and performs replacement of matching
        subexpressions from the bottom to the top of the tree. The default
        approach is to do the replacement in a simultaneous fashion so
        changes made are targeted only once. If this is not desired or causes
        problems, ``simultaneous`` can be set to False. In addition, if an
        expression containing more than one Wild symbol is being used to match
        subexpressions and  the ``exact`` flag is True, then the match will only
        succeed if non-zero values are received for each Wild that appears in
        the match pattern.

        The list of possible combinations of queries and replacement values
        is listed below:


        Initial setup

            >>> from sympy import log, sin, cos, tan, Wild, Mul, Add
            >>> from import x, y
            >>> f = log(sin(x)) + tan(sin(x**2))

        1.1. type -> type
            obj.replace(type, newtype)

            When object of type ``type`` is found, replace it with the
            result of passing its argument(s) to ``newtype``.

            >>> f.replace(sin, cos)
            log(cos(x)) + tan(cos(x**2))
            >>> sin(x).replace(sin, cos, map=True)
            (cos(x), {sin(x): cos(x)})
            >>> (x*y).replace(Mul, Add)
            x + y

        1.2. type -> func
            obj.replace(type, func)

            When object of type ``type`` is found, apply ``func`` to its
            argument(s). ``func`` must be written to handle the number
            of arguments of ``type``.

            >>> f.replace(sin, lambda arg: sin(2*arg))
            log(sin(2*x)) + tan(sin(2*x**2))
            >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))

        2.1. pattern -> expr
            obj.replace(pattern(wild), expr(wild))

            Replace subexpressions matching ``pattern`` with the expression
            written in terms of the Wild symbols in ``pattern``.

            >>> a = Wild('a')
            >>> f.replace(sin(a), tan(a))
            log(tan(x)) + tan(tan(x**2))
            >>> f.replace(sin(a), tan(a/2))
            log(tan(x/2)) + tan(tan(x**2/2))
            >>> f.replace(sin(a), a)
            log(x) + tan(x**2)
            >>> (x*y).replace(a*x, a)

            When the default value of False is used with patterns that have
            more than one Wild symbol, non-intuitive results may be obtained:

            >>> b = Wild('b')
            >>> (2*x).replace(a*x + b, b - a)

            For this reason, the ``exact`` option can be used to make the
            replacement only when the match gives non-zero values for all
            Wild symbols:

            >>> (2*x + y).replace(a*x + b, b - a, exact=True)
            y - 2
            >>> (2*x).replace(a*x + b, b - a, exact=True)

        2.2. pattern -> func
            obj.replace(pattern(wild), lambda wild: expr(wild))

            All behavior is the same as in 2.1 but now a function in terms of
            pattern variables is used rather than an expression:

            >>> f.replace(sin(a), lambda a: sin(2*a))
            log(sin(2*x)) + tan(sin(2*x**2))

        3.1. func -> func
            obj.replace(filter, func)

            Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
            is True.

            >>> g = 2*sin(x**3)
            >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)

        The expression itself is also targeted by the query but is done in
        such a fashion that changes are not made twice.

            >>> e = x*(x*y + 1)
            >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
            2*x*(2*x*y + 1)

        See Also
        subs: substitution of subexpressions as defined by the objects
        xreplace: exact node replacement in expr tree; also capable of
                  using matching rules

        from sympy.core.symbol import Dummy
        from sympy.simplify.simplify import bottom_up

            query = sympify(query)
        except SympifyError:
            value = sympify(value)
        except SympifyError:
        if isinstance(query, type):
            _query = lambda expr: isinstance(expr, query)

            if isinstance(value, type):
                _value = lambda expr, result: value(*expr.args)
            elif callable(value):
                _value = lambda expr, result: value(*expr.args)
                raise TypeError(
                    "given a type, replace() expects another "
                    "type or a callable")
        elif isinstance(query, Basic):
            _query = lambda expr: expr.match(query)

            # XXX remove the exact flag and make multi-symbol
            # patterns use exact=True semantics; to do this the query must
            # be tested to find out how many Wild symbols are present.
            # See
            # ?fromgroups=#!topic/sympy/zPzo5FtRiqI
            # for a method of inspecting a function to know how many
            # parameters it has.
            if isinstance(value, Basic):
                if exact:
                    _value = lambda expr, result: (value.subs(result)
                        if all(val for val in result.values()) else expr)
                    _value = lambda expr, result: value.subs(result)
            elif callable(value):
                # match dictionary keys get the trailing underscore stripped
                # from them and are then passed as keywords to the callable;
                # if ``exact`` is True, only accept match if there are no null
                # values amongst those matched.
                if exact:
                    _value = lambda expr, result: (value(**dict([(
                        str(key)[:-1], val) for key, val in result.items()]))
                        if all(val for val in result.values()) else expr)
                    _value = lambda expr, result: value(**dict([(
                        str(key)[:-1], val) for key, val in result.items()]))
                raise TypeError(
                    "given an expression, replace() expects "
                    "another expression or a callable")
        elif callable(query):
            _query = query

            if callable(value):
                _value = lambda expr, result: value(expr)
                raise TypeError(
                    "given a callable, replace() expects "
                    "another callable")
            raise TypeError(
                "first argument to replace() must be a "
                "type, an expression or a callable")

        mapping = {}  # changes that took place
        mask = []  # the dummies that were used as change placeholders

        def rec_replace(expr):
            result = _query(expr)
            if result or result == {}:
                new = _value(expr, result)
                if new is not None and new != expr:
                    mapping[expr] = new
                    if simultaneous:
                        # don't let this expression be changed during rebuilding
                        com = getattr(new, 'is_commutative', True)
                        if com is None:
                            com = True
                        d = Dummy(commutative=com)
                        mask.append((d, new))
                        expr = d
                        expr = new
            return expr

        rv = bottom_up(self, rec_replace, atoms=True)

        # restore original expressions for Dummy symbols
        if simultaneous:
            mask = list(reversed(mask))
            for o, n in mask:
                r = {o: n}
                rv = rv.xreplace(r)

        if not map:
            return rv
            if simultaneous:
                # restore subexpressions in mapping
                for o, n in mask:
                    r = {o: n}
                    mapping = dict([(k.xreplace(r), v.xreplace(r))
                        for k, v in mapping.items()])
            return rv, mapping

    def find(self, query, group=False):
        """Find all subexpressions matching a query. """
        query = _make_find_query(query)
        results = list(filter(query, preorder_traversal(self)))

        if not group:
            return set(results)
            groups = {}

            for result in results:
                if result in groups:
                    groups[result] += 1
                    groups[result] = 1

            return groups

    def count(self, query):
        """Count the number of matching subexpressions. """
        query = _make_find_query(query)
        return sum(bool(query(sub)) for sub in preorder_traversal(self))

    def matches(self, expr, repl_dict={}, old=False):
        Helper method for match() that looks for a match between Wild symbols
        in self and expressions in expr.


        >>> from sympy import symbols, Wild, Basic
        >>> a, b, c = symbols('a b c')
        >>> x = Wild('x')
        >>> Basic(a + x, x).matches(Basic(a + b, c)) is None
        >>> Basic(a + x, x).matches(Basic(a + b + c, b + c))
        {x_: b + c}
        expr = sympify(expr)
        if not isinstance(expr, self.__class__):
            return None

        if self == expr:
            return repl_dict

        if len(self.args) != len(expr.args):
            return None

        d = repl_dict.copy()
        for arg, other_arg in zip(self.args, expr.args):
            if arg == other_arg:
            d = arg.xreplace(d).matches(other_arg, d, old=old)
            if d is None:
                return None
        return d

    def match(self, pattern, old=False):
        Pattern matching.

        Wild symbols match all.

        Return ``None`` when expression (self) does not match
        with pattern. Otherwise return a dictionary such that::

          pattern.xreplace(self.match(pattern)) == self


        >>> from sympy import Wild
        >>> from import x, y
        >>> p = Wild("p")
        >>> q = Wild("q")
        >>> r = Wild("r")
        >>> e = (x+y)**(x+y)
        >>> e.match(p**p)
        {p_: x + y}
        >>> e.match(p**q)
        {p_: x + y, q_: x + y}
        >>> e = (2*x)**2
        >>> e.match(p*q**r)
        {p_: 4, q_: x, r_: 2}
        >>> (p*q**r).xreplace(e.match(p*q**r))

        The ``old`` flag will give the old-style pattern matching where
        expressions and patterns are essentially solved to give the
        match. Both of the following give None unless ``old=True``:

        >>> (x - 2).match(p - x, old=True)
        {p_: 2*x - 2}
        >>> (2/x).match(p*x, old=True)
        {p_: 2/x**2}

        pattern = sympify(pattern)
        return pattern.matches(self, old=old)

    def count_ops(self, visual=None):
        """wrapper for count_ops that returns the operation count."""
        from sympy import count_ops
        return count_ops(self, visual)

    def doit(self, **hints):
        """Evaluate objects that are not evaluated by default like limits,
           integrals, sums and products. All objects of this kind will be
           evaluated recursively, unless some species were excluded via 'hints'
           or unless the 'deep' hint was set to 'False'.

           >>> from sympy import Integral
           >>> from import x

           >>> 2*Integral(x, x)
           2*Integral(x, x)

           >>> (2*Integral(x, x)).doit()

           >>> (2*Integral(x, x)).doit(deep=False)
           2*Integral(x, x)

        if hints.get('deep', True):
            terms = [term.doit(**hints) if isinstance(term, Basic) else term
                                         for term in self.args]
            return self.func(*terms)
            return self

    def _eval_rewrite(self, pattern, rule, **hints):
        if self.is_Atom:
            if hasattr(self, rule):
                return getattr(self, rule)()
            return self

        if hints.get('deep', True):
            args = [a._eval_rewrite(pattern, rule, **hints)
                        if isinstance(a, Basic) else a
                        for a in self.args]
            args = self.args

        if pattern is None or isinstance(self.func, pattern):
            if hasattr(self, rule):
                rewritten = getattr(self, rule)(*args)
                if rewritten is not None:
                    return rewritten
        return self.func(*args)

    def rewrite(self, *args, **hints):
        """ Rewrite functions in terms of other functions.

        Rewrites expression containing applications of functions
        of one kind in terms of functions of different kind. For
        example you can rewrite trigonometric functions as complex
        exponentials or combinatorial functions as gamma function.

        As a pattern this function accepts a list of functions to
        to rewrite (instances of DefinedFunction class). As rule
        you can use string or a destination function instance (in
        this case rewrite() will use the str() function).

        There is also the possibility to pass hints on how to rewrite
        the given expressions. For now there is only one such hint
        defined called 'deep'. When 'deep' is set to False it will
        forbid functions to rewrite their contents.


        >>> from sympy import sin, exp
        >>> from import x

        Unspecified pattern:

        >>> sin(x).rewrite(exp)
        -I*(exp(I*x) - exp(-I*x))/2

        Pattern as a single function:

        >>> sin(x).rewrite(sin, exp)
        -I*(exp(I*x) - exp(-I*x))/2

        Pattern as a list of functions:

        >>> sin(x).rewrite([sin, ], exp)
        -I*(exp(I*x) - exp(-I*x))/2

        if not args:
            return self
            pattern = args[:-1]
            if isinstance(args[-1], string_types):
                rule = '_eval_rewrite_as_' + args[-1]
                rule = '_eval_rewrite_as_' + args[-1].__name__

            if not pattern:
                return self._eval_rewrite(None, rule, **hints)
                if iterable(pattern[0]):
                    pattern = pattern[0]

                pattern = [p.__class__ for p in pattern if self.has(p)]

                if pattern:
                    return self._eval_rewrite(tuple(pattern), rule, **hints)
                    return self

class Atom(Basic):
    A parent class for atomic things. An atom is an expression with no subexpressions.


    Symbol, Number, Rational, Integer, ...
    But not: Add, Mul, Pow, ...

    is_Atom = True

    __slots__ = []

    def matches(self, expr, repl_dict={}, old=False):
        if self == expr:
            return repl_dict

    def xreplace(self, rule, hack2=False):
        return rule.get(self, self)

    def doit(self, **hints):
        return self

    def class_key(cls):
        return 2, 0, cls.__name__

    def sort_key(self, order=None):
        return self.class_key(), (1, (str(self),)), S.One.sort_key(), S.One

    def _eval_simplify(self, ratio, measure):
        return self

    def _sorted_args(self):
        # this is here as a safeguard against accidentally using _sorted_args
        # on Atoms -- they cannot be rebuilt as atom.func(*atom._sorted_args)
        # since there are no args. So the calling routine should be checking
        # to see that this property is not called for Atoms.
        raise AttributeError('Atoms have no args. It might be necessary'
        ' to make a check for Atoms in the calling code.')

def _aresame(a, b):
    """Return True if a and b are structurally the same, else False.


    To SymPy, 2.0 == 2:

    >>> from sympy import S
    >>> 2.0 == S(2)

    Since a simple 'same or not' result is sometimes useful, this routine was
    written to provide that query:

    >>> from sympy.core.basic import _aresame
    >>> _aresame(S(2.0), S(2))

    from .function import AppliedUndef, UndefinedFunction as UndefFunc
    for i, j in zip_longest(preorder_traversal(a), preorder_traversal(b)):
        if i != j or type(i) != type(j):
            if ((isinstance(i, UndefFunc) and isinstance(j, UndefFunc)) or
                (isinstance(i, AppliedUndef) and isinstance(j, AppliedUndef))):
                if i.class_key() != j.class_key():
                    return False
                return False
        return True

def _atomic(e):
    """Return atom-like quantities as far as substitution is
    concerned: Derivatives, Functions and Symbols. Don't
    return any 'atoms' that are inside such quantities unless
    they also appear outside, too.


    >>> from sympy import Derivative, Function, cos
    >>> from import x, y
    >>> from sympy.core.basic import _atomic
    >>> f = Function('f')
    >>> _atomic(x + y)
    set([x, y])
    >>> _atomic(x + f(y))
    set([x, f(y)])
    >>> _atomic(Derivative(f(x), x) + cos(x) + y)
    set([y, cos(x), Derivative(f(x), x)])

    from sympy import Derivative, Function, Symbol
    pot = preorder_traversal(e)
    seen = set()
        free = e.free_symbols
    except AttributeError:
        return set([e])
    atoms = set()
    for p in pot:
        if p in seen:
        if isinstance(p, Symbol) and p in free:
        elif isinstance(p, (Derivative, Function)):
    return atoms

class preorder_traversal(Iterator):
    Do a pre-order traversal of a tree.

    This iterator recursively yields nodes that it has visited in a pre-order
    fashion. That is, it yields the current node then descends through the
    tree breadth-first to yield all of a node's children's pre-order

    For an expression, the order of the traversal depends on the order of
    .args, which in many cases can be arbitrary.

    node : sympy expression
        The expression to traverse.
    keys : (default None) sort key(s)
        The key(s) used to sort args of Basic objects. When None, args of Basic
        objects are processed in arbitrary order. If key is defined, it will
        be passed along to ordered() as the only key(s) to use to sort the
        arguments; if ``key`` is simply True then the default keys of ordered
        will be used.

    subtree : sympy expression
        All of the subtrees in the tree.


    >>> from sympy import symbols
    >>> from sympy.core.basic import preorder_traversal
    >>> x, y, z = symbols('x y z')

    The nodes are returned in the order that they are encountered unless key
    is given; simply passing key=True will guarantee that the traversal is

    >>> list(preorder_traversal((x + y)*z, keys=None)) # doctest: +SKIP
    [z*(x + y), z, x + y, y, x]
    >>> list(preorder_traversal((x + y)*z, keys=True))
    [z*(x + y), z, x + y, x, y]

    def __init__(self, node, keys=None):
        self._skip_flag = False
        self._pt = self._preorder_traversal(node, keys)

    def _preorder_traversal(self, node, keys):
        yield node
        if self._skip_flag:
            self._skip_flag = False
        if isinstance(node, Basic):
            if not keys and hasattr(node, '_argset'):
                # LatticeOp keeps args as a set. We should use this if we
                # don't care about the order, to prevent unnecessary sorting.
                args = node._argset
                args = node.args
            if keys:
                if keys != True:
                    args = ordered(args, keys, default=False)
                    args = ordered(args)
            for arg in args:
                for subtree in self._preorder_traversal(arg, keys):
                    yield subtree
        elif iterable(node):
            for item in node:
                for subtree in self._preorder_traversal(item, keys):
                    yield subtree

    def skip(self):
        Skip yielding current node's (last yielded node's) subtrees.


        >>> from sympy.core import symbols
        >>> from sympy.core.basic import preorder_traversal
        >>> x, y, z = symbols('x y z')
        >>> pt = preorder_traversal((x+y*z)*z)
        >>> for i in pt:
        ...     print(i)
        ...     if i == x+y*z:
        ...             pt.skip()
        z*(x + y*z)
        x + y*z
        self._skip_flag = True

    def __next__(self):
        return next(self._pt)

    def __iter__(self):
        return self

def _make_find_query(query):
    """Convert the argument of Basic.find() into a callable"""
        query = sympify(query)
    except SympifyError:
    if isinstance(query, type):
        return lambda expr: isinstance(expr, query)
    elif isinstance(query, Basic):
        return lambda expr: expr.match(query) is not None
    return query