def doprint(self, funcname, args, expr):
        """Returns the function definition code as a string."""
        from sympy import Dummy

        funcbody = []

        if not iterable(args):
            args = [args]

        argstrs, expr = self._preprocess(args, expr)

        # Generate argument unpacking and final argument list
        funcargs = []
        unpackings = []

        for argstr in argstrs:
            if iterable(argstr):
                funcargs.append(self._argrepr(Dummy()))
                unpackings.extend(self._print_unpacking(argstr, funcargs[-1]))
            else:
                funcargs.append(argstr)

        funcsig = 'def {}({}):'.format(funcname, ', '.join(funcargs))

        # Wrap input arguments before unpacking
        funcbody.extend(self._print_funcargwrapping(funcargs))

        funcbody.extend(unpackings)

        funcbody.append('return ({})'.format(self._exprrepr(expr)))

        funclines = [funcsig]
        funclines.extend('    ' + line for line in funcbody)

        return '\n'.join(funclines) + '\n'

def reduce_inequalities(inequalities, symbols=[]):
    """Reduce a system of inequalities with rational coefficients.

    Examples
    ========

    >>> from sympy import sympify as S, Symbol
    >>> from sympy.abc import x, y
    >>> from sympy.solvers.inequalities import reduce_inequalities

    >>> reduce_inequalities(0 <= x + 3, [])
    And(-3 <= x, x < oo)

    >>> reduce_inequalities(0 <= x + y*2 - 1, [x])
    x >= -2*y + 1
    """
    if not iterable(inequalities):
        inequalities = [inequalities]
    inequalities = [sympify(i) for i in inequalities]

    gens = set().union(*[i.free_symbols for i in inequalities])

    if not iterable(symbols):
        symbols = [symbols]
    symbols = (set(symbols) or gens) & gens
    if any(i.is_real is False for i in symbols):
        raise TypeError(
            filldedent(
                """
            inequalities cannot contain symbols that are not real."""
            )
        )

    # make vanilla symbol real
    recast = dict([(i, Dummy(i.name, real=True)) for i in gens if i.is_real is None])
    inequalities = [i.xreplace(recast) for i in inequalities]
    symbols = set([i.xreplace(recast) for i in symbols])

    # prefilter
    keep = []
    for i in inequalities:
        if isinstance(i, Relational):
            i = i.func(i.lhs.as_expr() - i.rhs.as_expr(), 0)
        elif i not in (True, False):
            i = Eq(i, 0)
        if i == True:
            continue
        elif i == False:
            return S.false
        if i.lhs.is_number:
            raise NotImplementedError("could not determine truth value of %s" % i)
        keep.append(i)
    inequalities = keep
    del keep

    # solve system
    rv = _reduce_inequalities(inequalities, symbols)

    # restore original symbols and return
    return rv.xreplace(dict([(v, k) for k, v in recast.items()]))

def test_iterable_ordered_iter():
    ordered = [list(), tuple(), Tuple(), Matrix([[]])]
    unordered = [set()]
    not_sympy_iterable = [{}, '']
    assert all(ordered_iter(i) for i in ordered)
    assert all(not ordered_iter(i) for i in unordered)
    assert all(iterable(i) for i in ordered + unordered)
    assert all(not iterable(i) for i in not_sympy_iterable)
    assert all(iterable(i, exclude=None) for i in not_sympy_iterable)

def test_iterable_is_sequence():
    ordered = [list(), tuple(), Tuple(), Matrix([[]])]
    unordered = [set()]
    not_sympy_iterable = [{}, '', u('')]
    assert all(is_sequence(i) for i in ordered)
    assert all(not is_sequence(i) for i in unordered)
    assert all(iterable(i) for i in ordered + unordered)
    assert all(not iterable(i) for i in not_sympy_iterable)
    assert all(iterable(i, exclude=None) for i in not_sympy_iterable)

def test_iterable():
    assert iterable(0) is False
    assert iterable(1) is False
    assert iterable(None) is False

    class Test1(NotIterable):
        pass

    assert iterable(Test1()) is False

    class Test2(NotIterable):
        _iterable = True

    assert iterable(Test2()) is True

    class Test3(object):
        pass

    assert iterable(Test3()) is False

    class Test4(object):
        _iterable = True

    assert iterable(Test4()) is True

    class Test5(object):
        def __iter__(self):
            yield 1

    assert iterable(Test5()) is True

    class Test6(Test5):
        _iterable = False

    assert iterable(Test6()) is False

def is_sequence(i, include=None):
    """
    Return a boolean indicating whether ``i`` is a sequence in the SymPy
    sense. If anything that fails the test below should be included as
    being a sequence for your application, set 'include' to that object's
    type; multiple types should be passed as a tuple of types.

    Note: although generators can generate a sequence, they often need special
    handling to make sure their elements are captured before the generator is
    exhausted, so these are not included by default in the definition of a
    sequence.

    See also: iterable

    Examples
    ========

    >>> from sympy.utilities.iterables import is_sequence
    >>> from types import GeneratorType
    >>> is_sequence([])
    True
    >>> is_sequence(set())
    False
    >>> is_sequence('abc')
    False
    >>> is_sequence('abc', include=str)
    True
    >>> generator = (c for c in 'abc')
    >>> is_sequence(generator)
    False
    >>> is_sequence(generator, include=(str, GeneratorType))
    True

    """
    return hasattr(i, "__getitem__") and iterable(i) or bool(include) and isinstance(i, include)

def fuzzy_and(*args):
    """Return True (all True), False (any False) or None.

    If `a` is an iterable it must have more than one element."""

    if (len(args) == 1 and iterable(args[0]) or
        len(args) > 2):
        if len(args) == 1:
            args = args[0]
        rv = True
        i = 0
        for ai in args:
            ai = fuzzy_bool(ai)
            if ai is False:
                return False
            if rv: # this will stop updating if a None is ever trapped
                rv = ai
            i += 1
        if i < 2:
            raise ValueError('iterables must have 2 or more elements')
        return rv

    a, b = [fuzzy_bool(i) for i in args]
    if a is True and b is True:
        return True
    elif a is False or b is False:
        return False

    def _find_opts(expr):

        if expr.is_Atom or expr.is_Order:
            return

        if iterable(expr):
            list(map(_find_opts, expr))
            return

        if expr in seen_subexp:
            return expr
        seen_subexp.add(expr)

        list(map(_find_opts, expr.args))

        if _coeff_isneg(expr):
            neg_expr = -expr
            if not neg_expr.is_Atom:
                opt_subs[expr] = Mul(S.NegativeOne, neg_expr, evaluate=False)
                seen_subexp.add(neg_expr)
                expr = neg_expr

        if expr.is_Mul:
            muls.add(expr)

        elif expr.is_Add:
            adds.add(expr)

        elif expr.is_Pow:
            if _coeff_isneg(expr.exp):
                opt_subs[expr] = Pow(Pow(expr.base, -expr.exp), S.NegativeOne,
                                     evaluate=False)

    def _find_repeated(expr):
        if not isinstance(expr, Basic):
            return

        if expr.is_Atom or expr.is_Order:
            return

        if iterable(expr):
            args = expr

        else:
            if expr in seen_subexp:
                for ign in ignore:
                    if ign in expr.free_symbols:
                        break
                else:
                    to_eliminate.add(expr)
                    return

            seen_subexp.add(expr)

            if expr in opt_subs:
                expr = opt_subs[expr]

            args = expr.args

        list(map(_find_repeated, args))

    def _find_opts(expr):

        if not isinstance(expr, Basic):
            return

        if expr.is_Atom or expr.is_Order:
            return

        if iterable(expr):
            list(map(_find_opts, expr))
            return

        if expr in seen_subexp:
            return expr
        seen_subexp.add(expr)

        list(map(_find_opts, expr.args))

        if _coeff_isneg(expr):
            neg_expr = -expr
            if not neg_expr.is_Atom:
                opt_subs[expr] = Mul(S.NegativeOne, neg_expr, evaluate=False)
                seen_subexp.add(neg_expr)
                expr = neg_expr

        if isinstance(expr, (Mul, MatMul)):
            muls.add(expr)

        elif isinstance(expr, (Add, MatAdd)):
            adds.add(expr)

        elif isinstance(expr, (Pow, MatPow)):
            if _coeff_isneg(expr.exp):
                opt_subs[expr] = Pow(Pow(expr.base, -expr.exp), S.NegativeOne,
                                     evaluate=False)

    def convert(self, elem, M=None):
        """
        Convert ``elem`` into the internal representation.

        This method is called implicitly whenever computations involve elements
        not in the internal representation.

        >>> from sympy.abc import x
        >>> from sympy import QQ
        >>> F = QQ.old_poly_ring(x).free_module(2)
        >>> F.convert([1, 0])
        [1, 0]
        """
        if isinstance(elem, FreeModuleElement):
            if elem.module is self:
                return elem
            if elem.module.rank != self.rank:
                raise CoercionFailed
            return FreeModuleElement(self,
                     tuple(self.ring.convert(x, elem.module.ring) for x in elem.data))
        elif iterable(elem):
            tpl = tuple(self.ring.convert(x) for x in elem)
            if len(tpl) != self.rank:
                raise CoercionFailed
            return FreeModuleElement(self, tpl)
        elif elem is 0:
            return FreeModuleElement(self, (self.ring.convert(0),)*self.rank)
        else:
            raise CoercionFailed

def _walsh_hadamard_transform(seq, inverse=False):
    """Utility function for the Walsh Hadamard Transform"""

    if not iterable(seq):
        raise TypeError("Expected a sequence of coefficients "
                        "for Walsh Hadamard Transform")

    a = [sympify(arg) for arg in seq]
    n = len(a)
    if n < 2:
        return a

    if n&(n - 1):
        n = 2**n.bit_length()

    a += [S.Zero]*(n - len(a))
    h = 2
    while h <= n:
        hf, ut = h // 2, n // h
        for i in range(0, n, h):
            for j in range(hf):
                u, v = a[i + j], a[i + j + hf]
                a[i + j], a[i + j + hf] = u + v, u - v
        h *= 2

    if inverse:
        a = [x/n for x in a]

    return a

    def _preprocess(self, args, expr):
        """Preprocess args, expr to replace arguments that do not map
        to valid Python identifiers.

        Returns string form of args, and updated expr.
        """
        from sympy import Dummy, Function, flatten, Derivative, ordered, Basic
        from sympy.matrices import DeferredVector

        # Args of type Dummy can cause name collisions with args
        # of type Symbol.  Force dummify of everything in this
        # situation.
        dummify = self._dummify or any(
            isinstance(arg, Dummy) for arg in flatten(args))

        argstrs = [None]*len(args)
        for arg, i in reversed(list(ordered(zip(args, range(len(args)))))):
            if iterable(arg):
                s, expr = self._preprocess(arg, expr)
            elif isinstance(arg, DeferredVector):
                s = str(arg)
            elif isinstance(arg, Basic) and arg.is_symbol:
                s = self._argrepr(arg)
                if dummify or not self._is_safe_ident(s):
                    dummy = Dummy()
                    s = self._argrepr(dummy)
                    expr = self._subexpr(expr, {arg: dummy})
            elif dummify or isinstance(arg, (Function, Derivative)):
                dummy = Dummy()
                s = self._argrepr(dummy)
                expr = self._subexpr(expr, {arg: dummy})
            else:
                s = str(arg)
            argstrs[i] = s
        return argstrs, expr

    def __new__(cls, *args, **kwargs):
        evaluate = kwargs.get('evaluate', global_evaluate[0])

        if iterable(args[0]):
            if isinstance(args[0], Point) and not evaluate:
                return args[0]
            args = args[0]

        # unpack the arguments into a friendly Tuple
        # if we were already a Point, we're doing an excess
        # iteration, but we'll worry about efficiency later
        coords = Tuple(*args)
        if any(a.is_number and im(a) for a in coords):
            raise ValueError('Imaginary coordinates not permitted.')

        # Turn any Floats into rationals and simplify
        # any expressions before we instantiate
        if evaluate:
            coords = coords.xreplace(dict(
                [(f, simplify(nsimplify(f, rational=True)))
                for f in coords.atoms(Float)]))
        if len(coords) == 2:
            return Point2D(coords, **kwargs)
        if len(coords) == 3:
            return Point3D(coords, **kwargs)

        return GeometryEntity.__new__(cls, *coords)

def lambdastr(args, expr, printer=None):
    """
    Returns a string that can be evaluated to a lambda function.

    >>> from sympy.abc import x, y, z
    >>> from sympy.utilities.lambdify import lambdastr
    >>> lambdastr(x, x**2)
    'lambda x: (x**2)'
    >>> lambdastr((x,y,z), [z,y,x])
    'lambda x,y,z: ([z, y, x])'

    """
    if printer is not None:
        if inspect.isfunction(printer):
            lambdarepr = printer
        else:
            if inspect.isclass(printer):
                lambdarepr = lambda expr: printer().doprint(expr)
            else:
                lambdarepr = lambda expr: printer.doprint(expr)
    else:
        #XXX: This has to be done here because of circular imports
        from sympy.printing.lambdarepr import lambdarepr

    # Transform everything to strings.
    from sympy.matrices import DeferredVector
    expr = lambdarepr(expr)
    if isinstance(args, str):
        pass
    elif iterable(args, exclude=DeferredVector):
        args = ",".join(str(a) for a in args)
    else:
        args = str(args)

    return "lambda %s: (%s)" % (args, expr)

 def sub_args(args, dummies_dict):
     if isinstance(args, str):
         return args
     elif isinstance(args, DeferredVector):
         return str(args)
     elif iterable(args):
         flatten = lambda *n: (e for a in n for e in
                               (flatten(*a) if iterable(a) else (a,)))
         dummies = flatten([sub_args(a, dummies_dict) for a in args])
         return ",".join(str(a) for a in dummies)
     else:
         if isinstance(args, Function):
             dummies = Dummy()
             dummies_dict.update({args : dummies})
             return str(dummies)
         else:
             return str(args)

 def flatten(arg):
     if isinstance(arg, Set):
         if arg.is_Intersection:
             return sum(map(flatten, arg.args), [])
         else:
             return [arg]
     if iterable(arg):  # and not isinstance(arg, Set) (implicit)
         return sum(map(flatten, arg), [])
     raise TypeError("Input must be Sets or iterables of Sets")

 def _flatten(arg):
     if isinstance(arg, SeqBase):
         if isinstance(arg, SeqMul):
             return sum(map(_flatten, arg.args), [])
         else:
             return [arg]
     elif iterable(arg):
         return sum(map(_flatten, arg), [])
     raise TypeError("Input must be Sequences or " " iterables of Sequences")

 def group_to_finiteset(group):
     """
     Converts ``group`` to a :class:``FiniteSet`` if it is an
     iterable.
     """
     if iterable(group):
         return FiniteSet(group)
     else:
         return group

def posify(eq):
    """Return eq (with generic symbols made positive) and a
    dictionary containing the mapping between the old and new
    symbols.

    Any symbol that has positive=None will be replaced with a positive dummy
    symbol having the same name. This replacement will allow more symbolic
    processing of expressions, especially those involving powers and
    logarithms.

    A dictionary that can be sent to subs to restore eq to its original
    symbols is also returned.

    >>> from sympy import posify, Symbol, log, solve
    >>> from sympy.abc import x
    >>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True))
    (_x + n + p, {_x: x})

    >>> eq = 1/x
    >>> log(eq).expand()
    log(1/x)
    >>> log(posify(eq)[0]).expand()
    -log(_x)
    >>> p, rep = posify(eq)
    >>> log(p).expand().subs(rep)
    -log(x)

    It is possible to apply the same transformations to an iterable
    of expressions:

    >>> eq = x**2 - 4
    >>> solve(eq, x)
    [-2, 2]
    >>> eq_x, reps = posify([eq, x]); eq_x
    [_x**2 - 4, _x]
    >>> solve(*eq_x)
    [2]
    """
    eq = sympify(eq)
    if iterable(eq):
        f = type(eq)
        eq = list(eq)
        syms = set()
        for e in eq:
            syms = syms.union(e.atoms(Symbol))
        reps = {}
        for s in syms:
            reps.update(dict((v, k) for k, v in posify(s)[1].items()))
        for i, e in enumerate(eq):
            eq[i] = e.subs(reps)
        return f(eq), dict([(r, s) for s, r in reps.items()])

    reps = dict([(s, Dummy(s.name, positive=True))
                 for s in eq.free_symbols if s.is_positive is None])
    eq = eq.subs(reps)
    return eq, dict([(r, s) for s, r in reps.items()])