from collections import defaultdict

from sympy.core.core import C
from sympy.core.singleton import S
from sympy.core.operations import AssocOp
from sympy.core.cache import cacheit
from sympy.core.numbers import ilcm, igcd
from sympy.core.expr import Expr


def _addsort(args):
    # in-place sorting of args

    # Currently we sort things using hashes, as it is quite fast. A better
    # solution is not to sort things at all - but this needs some more
    # fixing.
    args.sort(key=hash)


def _unevaluated_Add(*args):
    """Return a well-formed unevaluated Add: Numbers are collected and
    put in slot 0 and args are sorted. Use this when args have changed
    but you still want to return an unevaluated Add.

    Examples
    ========

    >>> from sympy.core.add import _unevaluated_Add as uAdd
    >>> from sympy import S, Add
    >>> from sympy.abc import x, y
    >>> a = uAdd(*[S(1.0), x, S(2)])
    >>> a.args[0]
    3.00000000000000
    >>> a.args[1]
    x

    Beyond the Number being in slot 0, there is no other assurance of
    order for the arguments since they are hash sorted. So, for testing
    purposes, output produced by this in some other function can only
    be tested against the output of this function or as one of several
    options:

    >>> opts = (Add(x, y, evaluated=False), Add(y, x, evaluated=False))
    >>> a = uAdd(x, y)
    >>> assert a in opts and a == uAdd(x, y)

    """
    args = list(args)
    newargs = []
    co = S.Zero
    while args:
        a = args.pop()
        if a.is_Add:
            # this will keep nesting from building up
            # so that x + (x + 1) -> x + x + 1 (3 args)
            args.extend(a.args)
        elif a.is_Number:
            co += a
        else:
            newargs.append(a)
    _addsort(newargs)
    if co:
        newargs.insert(0, co)
    return Add._from_args(newargs)


class Add(Expr, AssocOp):

    __slots__ = []

    is_Add = True

    #identity = S.Zero
    # cyclic import, so defined in numbers.py

    @classmethod
    def flatten(cls, seq):
        """
        Takes the sequence "seq" of nested Adds and returns a flatten list.

        Returns: (commutative_part, noncommutative_part, order_symbols)

        Applies associativity, all terms are commutable with respect to
        addition.

        NB: the removal of 0 is already handled by AssocOp.__new__

        See also
        ========

        sympy.core.mul.Mul.flatten

        """
        rv = None
        if len(seq) == 2:
            a, b = seq
            if b.is_Rational:
                a, b = b, a
            if a.is_Rational:
                if b.is_Mul:
                    rv = [a, b], [], None
            if rv:
                if all(s.is_commutative for s in rv[0]):
                    return rv
                return [], rv[0], None

        terms = {}      # term -> coeff
                        # e.g. x**2 -> 5   for ... + 5*x**2 + ...

        coeff = S.Zero  # coefficient (Number or zoo) to always be in slot 0
                        # e.g. 3 + ...
        order_factors = []

        for o in seq:

            # O(x)
            if o.is_Order:
                for o1 in order_factors:
                    if o1.contains(o):
                        o = None
                        break
                if o is None:
                    continue
                order_factors = [o] + [
                    o1 for o1 in order_factors if not o.contains(o1)]
                continue

            # 3 or NaN
            elif o.is_Number:
                if (o is S.NaN or coeff is S.ComplexInfinity and
                        o.is_bounded is False):
                    # we know for sure the result will be nan
                    return [S.NaN], [], None
                if coeff.is_Number:
                    coeff += o
                    if coeff is S.NaN:
                        # we know for sure the result will be nan
                        return [S.NaN], [], None
                continue

            elif o is S.ComplexInfinity:
                if coeff.is_bounded is False:
                    # we know for sure the result will be nan
                    return [S.NaN], [], None
                coeff = S.ComplexInfinity
                continue

            # Add([...])
            elif o.is_Add:
                # NB: here we assume Add is always commutative
                seq.extend(o.args)  # TODO zerocopy?
                continue

            # Mul([...])
            elif o.is_Mul:
                c, s = o.as_coeff_Mul()

            # check for unevaluated Pow, e.g. 2**3 or 2**(-1/2)
            elif o.is_Pow:
                b, e = o.as_base_exp()
                if b.is_Number and (e.is_Integer or
                                   (e.is_Rational and e.is_negative)):
                    seq.append(b**e)
                    continue
                c, s = S.One, o

            else:
                # everything else
                c = S.One
                s = o

            # now we have:
            # o = c*s, where
            #
            # c is a Number
            # s is an expression with number factor extracted
            # let's collect terms with the same s, so e.g.
            # 2*x**2 + 3*x**2  ->  5*x**2
            if s in terms:
                terms[s] += c
            else:
                terms[s] = c

        # now let's construct new args:
        # [2*x**2, x**3, 7*x**4, pi, ...]
        newseq = []
        noncommutative = False
        for s, c in terms.items():
            # 0*s
            if c is S.Zero:
                continue
            # 1*s
            elif c is S.One:
                newseq.append(s)
            # c*s
            else:
                if s.is_Mul:
                    # Mul, already keeps its arguments in perfect order.
                    # so we can simply put c in slot0 and go the fast way.
                    cs = s._new_rawargs(*((c,) + s.args))
                    newseq.append(cs)
                elif s.is_Add:
                    # we just re-create the unevaluated Mul
                    newseq.append(Mul(c, s, evaluate=False))
                else:
                    # alternatively we have to call all Mul's machinery (slow)
                    newseq.append(Mul(c, s))

            noncommutative = noncommutative or not s.is_commutative

        # oo, -oo
        if coeff is S.Infinity:
            newseq = [f for f in newseq if not
                      (f.is_nonnegative or f.is_real and
                       (f.is_bounded or f.is_infinitesimal))]

        elif coeff is S.NegativeInfinity:
            newseq = [f for f in newseq if not
                      (f.is_nonpositive or f.is_real and
                       (f.is_bounded or f.is_infinitesimal))]

        if coeff is S.ComplexInfinity:
            # zoo might be
            #   unbounded_real + bounded_im
            #   bounded_real + unbounded_im
            #   unbounded_real + unbounded_im
            # addition of a bounded real or imaginary number won't be able to
            # change the zoo nature; if unbounded a NaN condition could result
            # if the unbounded symbol had sign opposite of the unbounded
            # portion of zoo, e.g., unbounded_real - unbounded_real.
            newseq = [c for c in newseq if not (c.is_bounded and
                                                c.is_real is not None)]

        # process O(x)
        if order_factors:
            newseq2 = []
            for t in newseq:
                for o in order_factors:
                    # x + O(x) -> O(x)
                    if o.contains(t):
                        t = None
                        break
                # x + O(x**2) -> x + O(x**2)
                if t is not None:
                    newseq2.append(t)
            newseq = newseq2 + order_factors
            # 1 + O(1) -> O(1)
            for o in order_factors:
                if o.contains(coeff):
                    coeff = S.Zero
                    break

        # order args canonically
        _addsort(newseq)

        # current code expects coeff to be first
        if coeff is not S.Zero:
            newseq.insert(0, coeff)

        # we are done
        if noncommutative:
            return [], newseq, None
        else:
            return newseq, [], None

    @classmethod
    def class_key(cls):
        """Nice order of classes"""
        return 3, 1, cls.__name__

    def as_coefficients_dict(a):
        """Return a dictionary mapping terms to their Rational coefficient.
        Since the dictionary is a defaultdict, inquiries about terms which
        were not present will return a coefficient of 0. If an expression is
        not an Add it is considered to have a single term.

        Examples
        ========

        >>> from sympy.abc import a, x
        >>> (3*x + a*x + 4).as_coefficients_dict()
        {1: 4, x: 3, a*x: 1}
        >>> _[a]
        0
        >>> (3*a*x).as_coefficients_dict()
        {a*x: 3}
        """

        d = defaultdict(list)
        for ai in a.args:
            c, m = ai.as_coeff_Mul()
            d[m].append(c)
        for k, v in d.iteritems():
            if len(v) == 1:
                d[k] = v[0]
            else:
                d[k] = Add(*v)
        di = defaultdict(int)
        di.update(d)
        return di

    @cacheit
    def as_coeff_add(self, *deps):
        """
        Returns a tuple (coeff, args) where self is treated as an Add and coeff
        is the Number term and args is a tuple of all other terms.

        Examples
        ========

        >>> from sympy.abc import x
        >>> (7 + 3*x).as_coeff_add()
        (7, (3*x,))
        >>> (7*x).as_coeff_add()
        (0, (7*x,))
        """
        if deps:
            l1 = []
            l2 = []
            for f in self.args:
                if f.has(*deps):
                    l2.append(f)
                else:
                    l1.append(f)
            return self._new_rawargs(*l1), tuple(l2)
        coeff, notrat = self.args[0].as_coeff_add()
        if coeff is not S.Zero:
            return coeff, notrat + self.args[1:]
        return S.Zero, self.args

    def as_coeff_Add(self):
        """Efficiently extract the coefficient of a summation. """
        coeff, args = self.args[0], self.args[1:]

        if coeff.is_Number:
            if len(args) == 1:
                return coeff, args[0]
            else:
                return coeff, self._new_rawargs(*args)
        else:
            return S.Zero, self

    # Note, we intentionally do not implement Add.as_coeff_mul().  Rather, we
    # let Expr.as_coeff_mul() just always return (S.One, self) for an Add.  See
    # issue 2425.

    def _eval_derivative(self, s):
        return Add(*[f.diff(s) for f in self.args])

    def _eval_nseries(self, x, n, logx):
        terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
        return Add(*terms)

    def _matches_simple(self, expr, repl_dict):
        # handle (w+3).matches('x+5') -> {w: x+2}
        coeff, terms = self.as_coeff_add()
        if len(terms) == 1:
            return terms[0].matches(expr - coeff, repl_dict)
        return

    def matches(self, expr, repl_dict={}, old=False):
        return AssocOp._matches_commutative(self, expr, repl_dict, old)

    @staticmethod
    def _combine_inverse(lhs, rhs):
        """
        Returns lhs - rhs, but treats arguments like symbols, so things like
        oo - oo return 0, instead of a nan.
        """
        from sympy import oo, I, expand_mul
        if lhs == oo and rhs == oo or lhs == oo*I and rhs == oo*I:
            return S.Zero
        return expand_mul(lhs - rhs)

    @cacheit
    def as_two_terms(self):
        """Return head and tail of self.

        This is the most efficient way to get the head and tail of an
        expression.

        - if you want only the head, use self.args[0];
        - if you want to process the arguments of the tail then use
          self.as_coef_add() which gives the head and a tuple containing
          the arguments of the tail when treated as an Add.
        - if you want the coefficient when self is treated as a Mul
          then use self.as_coeff_mul()[0]

        >>> from sympy.abc import x, y
        >>> (3*x*y).as_two_terms()
        (3, x*y)
        """
        if len(self.args) == 1:
            return S.Zero, self
        return self.args[0], self._new_rawargs(*self.args[1:])

    def as_numer_denom(self):

        # clear rational denominator
        content, expr = self.primitive()
        ncon, dcon = content.as_numer_denom()

        # collect numerators and denominators of the terms
        nd = defaultdict(list)
        for f in expr.args:
            ni, di = f.as_numer_denom()
            nd[di].append(ni)
        # put infinity in the numerator
        if S.Zero in nd:
            n = nd.pop(S.Zero)
            assert len(n) == 1
            n = n[0]
            nd[S.One].append(n/S.Zero)

        # check for quick exit
        if len(nd) == 1:
            d, n = nd.popitem()
            return Add(
                *[_keep_coeff(ncon, ni) for ni in n]), _keep_coeff(dcon, d)

        # sum up the terms having a common denominator
        for d, n in nd.iteritems():
            if len(n) == 1:
                nd[d] = n[0]
            else:
                nd[d] = Add(*n)

        # assemble single numerator and denominator
        denoms, numers = [list(i) for i in zip(*nd.iteritems())]
        n, d = Add(*[Mul(*(denoms[:i] + [numers[i]] + denoms[i + 1:]))
                   for i in xrange(len(numers))]), Mul(*denoms)

        return _keep_coeff(ncon, n), _keep_coeff(dcon, d)

    def _eval_is_polynomial(self, syms):
        return all(term._eval_is_polynomial(syms) for term in self.args)

    def _eval_is_rational_function(self, syms):
        return all(term._eval_is_rational_function(syms) for term in self.args)

    def _eval_is_algebraic_expr(self, syms):
        return all(term._eval_is_algebraic_expr(syms) for term in self.args)

    # assumption methods
    _eval_is_real = lambda self: self._eval_template_is_attr(
        'is_real', when_multiple=None)
    _eval_is_antihermitian = lambda self: self._eval_template_is_attr(
        'is_antihermitian', when_multiple=None)
    _eval_is_bounded = lambda self: self._eval_template_is_attr(
        'is_bounded', when_multiple=None)
    _eval_is_hermitian = lambda self: self._eval_template_is_attr(
        'is_hermitian', when_multiple=None)
    _eval_is_imaginary = lambda self: self._eval_template_is_attr(
        'is_imaginary', when_multiple=None)
    _eval_is_integer = lambda self: self._eval_template_is_attr(
        'is_integer', when_multiple=None)
    _eval_is_commutative = lambda self: self._eval_template_is_attr(
        'is_commutative')

    def _eval_is_odd(self):
        l = [f for f in self.args if not (f.is_even is True)]
        if not l:
            return False
        if l[0].is_odd:
            return self._new_rawargs(*l[1:]).is_even

    def _eval_is_irrational(self):
        for t in self.args:
            a = t.is_irrational
            if a:
                others = list(self.args)
                others.remove(t)
                if all(x.is_rational is True for x in others):
                    return True
                return None
            if a is None:
                return
        return False

    def _eval_is_positive(self):
        if self.is_number:
            return super(Add, self)._eval_is_positive()
        pos = nonneg = nonpos = unknown_sign = False
        unbounded = set()
        args = [a for a in self.args if not a.is_zero]
        if not args:
            return False
        for a in args:
            ispos = a.is_positive
            ubound = a.is_unbounded
            if ubound:
                unbounded.add(ispos)
                if len(unbounded) > 1:
                    return None
            if ispos:
                pos = True
                continue
            elif a.is_nonnegative:
                nonneg = True
                continue
            elif a.is_nonpositive:
                nonpos = True
                continue
            elif a.is_zero:
                continue

            if ubound is None:
                # sign is unknown; if we don't know the boundedness
                # we're done: we don't know. That is technically true,
                # but the only option is that we have something like
                # oo - oo which is NaN and it really doesn't matter
                # what sign we apply to that because it (when finally
                # computed) will trump any sign. So instead of returning
                # None, we pass.
                pass
            else:
                return None
            unknown_sign = True

        if unbounded:
            return unbounded.pop()
        elif unknown_sign:
            return None
        elif not nonpos and not nonneg and pos:
            return True
        elif not nonpos and pos:
            return True
        elif not pos and not nonneg:
            return False

    def _eval_is_negative(self):
        if self.is_number:
            return super(Add, self)._eval_is_negative()
        neg = nonpos = nonneg = unknown_sign = False
        unbounded = set()
        args = [a for a in self.args if not a.is_zero]
        if not args:
            return False
        for a in args:
            isneg = a.is_negative
            ubound = a.is_unbounded
            if ubound:
                unbounded.add(isneg)
                if len(unbounded) > 1:
                    return None
            if isneg:
                neg = True
                continue
            elif a.is_nonpositive:
                nonpos = True
                continue
            elif a.is_nonnegative:
                nonneg = True
                continue
            elif a.is_zero:
                continue

            if ubound is None:
                # sign is unknown; if we don't know the boundedness
                # we're done: we don't know. That is technically true,
                # but the only option is that we have something like
                # oo - oo which is NaN and it really doesn't matter
                # what sign we apply to that because it (when finally
                # computed) will trump any sign. So instead of returning
                # None, we pass.
                pass
            unknown_sign = True

        if unbounded:
            return unbounded.pop()
        elif unknown_sign:
            return None
        elif not nonneg and not nonpos and neg:
            return True
        elif not nonneg and neg:
            return True
        elif not neg and not nonpos:
            return False

    def _eval_subs(self, old, new):
        if not old.is_Add:
            return None

        coeff_self, terms_self = self.as_coeff_Add()
        coeff_old, terms_old = old.as_coeff_Add()

        if coeff_self.is_Rational and coeff_old.is_Rational:
            if terms_self == terms_old:   # (2 + a).subs( 3 + a, y) -> -1 + y
                return Add(new, coeff_self, -coeff_old)
            if terms_self == -terms_old:  # (2 + a).subs(-3 - a, y) -> -1 - y
                return Add(-new, coeff_self, coeff_old)

        if coeff_self.is_Rational and coeff_old.is_Rational \
                or coeff_self == coeff_old:
            args_old, args_self = Add.make_args(
                terms_old), Add.make_args(terms_self)
            if len(args_old) < len(args_self):  # (a+b+c).subs(b+c,x) -> a+x
                self_set = set(args_self)
                old_set = set(args_old)

                if old_set < self_set:
                    ret_set = self_set - old_set
                    return Add(new, coeff_self, -coeff_old,
                               *[s._subs(old, new) for s in ret_set])

                args_old = Add.make_args(
                    -terms_old)     # (a+b+c+d).subs(-b-c,x) -> a-x+d
                old_set = set(args_old)
                if old_set < self_set:
                    ret_set = self_set - old_set
                    return Add(-new, coeff_self, coeff_old,
                               *[s._subs(old, new) for s in ret_set])

    def removeO(self):
        args = [a for a in self.args if not a.is_Order]
        return self._new_rawargs(*args)

    def getO(self):
        args = [a for a in self.args if a.is_Order]
        if args:
            return self._new_rawargs(*args)

    @cacheit
    def extract_leading_order(self, *symbols):
        """
        Returns the leading term and it's order.

        Examples
        ========

        >>> from sympy.abc import x
        >>> (x + 1 + 1/x**5).extract_leading_order(x)
        ((x**(-5), O(x**(-5))),)
        >>> (1 + x).extract_leading_order(x)
        ((1, O(1)),)
        >>> (x + x**2).extract_leading_order(x)
        ((x, O(x)),)

        """
        lst = []
        seq = [(f, C.Order(f, *symbols)) for f in self.args]
        for ef, of in seq:
            for e, o in lst:
                if o.contains(of) and o != of:
                    of = None
                    break
            if of is None:
                continue
            new_lst = [(ef, of)]
            for e, o in lst:
                if of.contains(o) and o != of:
                    continue
                new_lst.append((e, o))
            lst = new_lst
        return tuple(lst)

    def as_real_imag(self, deep=True, **hints):
        """
        returns a tuple represeting a complex numbers

        Examples
        ========

        >>> from sympy import I
        >>> (7 + 9*I).as_real_imag()
        (7, 9)
        """
        sargs, terms = self.args, []
        re_part, im_part = [], []
        for term in sargs:
            re, im = term.as_real_imag(deep=deep)
            re_part.append(re)
            im_part.append(im)
        return (self.func(*re_part), self.func(*im_part))

    def _eval_as_leading_term(self, x):
        from sympy import expand_mul, factor_terms

        old = self

        self = expand_mul(self)
        if not self.is_Add:
            return self.as_leading_term(x)

        unbounded = [t for t in self.args if t.is_unbounded]
        if unbounded:
            return Add._from_args(unbounded)

        self = Add(*[t.as_leading_term(x) for t in self.args]).removeO()
        if not self:
            # simple leading term analysis gave us 0 but we have to send
            # back a term, so compute the leading term (via series)
            return old.compute_leading_term(x)
        elif not self.is_Add:
            return self
        else:
            plain = Add(*[s for s, _ in self.extract_leading_order(x)])
            rv = factor_terms(plain, fraction=False)
            rv_fraction = factor_terms(rv, fraction=True)
            # if it simplifies to an x-free expression, return that;
            # tests don't fail if we don't but it seems nicer to do this
            if x not in rv_fraction.free_symbols:
                return rv_fraction
            return rv

    def _eval_adjoint(self):
        return Add(*[t.adjoint() for t in self.args])

    def _eval_conjugate(self):
        return Add(*[t.conjugate() for t in self.args])

    def _eval_transpose(self):
        return Add(*[t.transpose() for t in self.args])

    def __neg__(self):
        return Add(*[-t for t in self.args])

    def _sage_(self):
        s = 0
        for x in self.args:
            s += x._sage_()
        return s

    def primitive(self):
        """
        Return ``(R, self/R)`` where ``R``` is the Rational GCD of ``self```.

        ``R`` is collected only from the leading coefficient of each term.

        Examples
        ========

        >>> from sympy.abc import x, y

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

        >>> (2*x/3 + 4*y/9).primitive()
        (2/9, 3*x + 2*y)

        >>> (2*x/3 + 4.2*y).primitive()
        (1/3, 2*x + 12.6*y)

        No subprocessing of term factors is performed:

        >>> ((2 + 2*x)*x + 2).primitive()
        (1, x*(2*x + 2) + 2)

        Recursive subprocessing can be done with the as_content_primitive()
        method:

        >>> ((2 + 2*x)*x + 2).as_content_primitive()
        (2, x*(x + 1) + 1)

        See also: primitive() function in polytools.py

        """

        terms = []
        inf = False
        for a in self.args:
            c, m = a.as_coeff_Mul()
            if not c.is_Rational:
                c = S.One
                m = a
            inf = inf or m is S.ComplexInfinity
            terms.append((c.p, c.q, m))

        if not inf:
            ngcd = reduce(igcd, [t[0] for t in terms], 0)
            dlcm = reduce(ilcm, [t[1] for t in terms], 1)
        else:
            ngcd = reduce(igcd, [t[0] for t in terms if t[1]], 0)
            dlcm = reduce(ilcm, [t[1] for t in terms if t[1]], 1)

        if ngcd == dlcm == 1:
            return S.One, self
        if not inf:
            for i, (p, q, term) in enumerate(terms):
                terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term)
        else:
            for i, (p, q, term) in enumerate(terms):
                if q:
                    terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term)
                else:
                    terms[i] = _keep_coeff(Rational(p, q), term)

        # we don't need a complete re-flattening since no new terms will join
        # so we just use the same sort as is used in Add.flatten. When the
        # coefficient changes, the ordering of terms may change, e.g.
        #     (3*x, 6*y) -> (2*y, x)
        #
        # We do need to make sure that term[0] stays in position 0, however.
        #
        if terms[0].is_Number or terms[0] is S.ComplexInfinity:
            c = terms.pop(0)
        else:
            c = None
        _addsort(terms)
        if c:
            terms.insert(0, c)
        return Rational(ngcd, dlcm), self._new_rawargs(*terms)

    def as_content_primitive(self, radical=False):
        """Return the tuple (R, self/R) where R is the positive Rational
        extracted from self. If radical is True (default is False) then
        common radicals will be removed and included as a factor of the
        primitive expression.

        Examples
        ========

        >>> from sympy import sqrt
        >>> (3 + 3*sqrt(2)).as_content_primitive()
        (3, 1 + sqrt(2))

        Radical content can also be factored out of the primitive:

        >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
        (2, sqrt(2)*(1 + 2*sqrt(5)))

        See docstring of Expr.as_content_primitive for more examples.
        """
        con, prim = Add(*[_keep_coeff(*a.as_content_primitive(
            radical=radical)) for a in self.args]).primitive()
        if radical and prim.is_Add:
            # look for common radicals that can be removed
            args = prim.args
            rads = []
            common_q = None
            for m in args:
                term_rads = defaultdict(list)
                for ai in Mul.make_args(m):
                    if ai.is_Pow:
                        b, e = ai.as_base_exp()
                        if e.is_Rational and b.is_Integer:
                            term_rads[e.q].append(abs(int(b))**e.p)
                if not term_rads:
                    break
                if common_q is None:
                    common_q = set(term_rads.keys())
                else:
                    common_q = common_q & set(term_rads.keys())
                    if not common_q:
                        break
                rads.append(term_rads)
            else:
                # process rads
                # keep only those in common_q
                for r in rads:
                    for q in r.keys():
                        if q not in common_q:
                            r.pop(q)
                    for q in r:
                        r[q] = prod(r[q])
                # find the gcd of bases for each q
                G = []
                for q in common_q:
                    g = reduce(igcd, [r[q] for r in rads], 0)
                    if g != 1:
                        G.append(g**Rational(1, q))
                if G:
                    G = Mul(*G)
                    args = [ai/G for ai in args]
                    prim = G*Add(*args)

        return con, prim

    @property
    def _sorted_args(self):
        from sympy.core.compatibility import default_sort_key
        return sorted(self.args, key=lambda w: default_sort_key(w))

from mul import Mul, _keep_coeff, prod
from sympy.core.numbers import Rational
