def test_cartes():
    assert list(cartes([1, 2], [3, 4, 5])) == \
        [(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)]
    assert list(cartes()) == [()]
    assert list(cartes('a')) == [('a',)]
    assert list(cartes('a', repeat=2)) == [('a', 'a')]
    assert list(cartes(range(2))) == [(0,), (1,)]

def test_roots_quadratic():
    assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
    assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [-Rational(3, 2), 0]
    assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
    assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]

    f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)

    assert roots_quadratic(Poly(f, x)) == \
        [-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2),
         -e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)]

    # check for simplification
    f = Poly(y*x**2 - 2*x - 2*y, x)
    assert roots_quadratic(f) == \
        [-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
    f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
    assert roots_quadratic(f) == \
        [y**2/2 - sqrt(y**4)/2 + 1, y**2/2 + sqrt(y**4)/2 + 1]

    f = Poly(sqrt(2)*x**2 - 1, x)
    r = roots_quadratic(f)
    assert r == _nsort(r)

    # issue 8255
    f = Poly(-24*x**2 - 180*x + 264)
    assert [w.n(2) for w in f.all_roots(radicals=True)] == \
           [w.n(2) for w in f.all_roots(radicals=False)]
    for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)):
        f = Poly(_a*x**2 + _b*x + _c)
        roots = roots_quadratic(f)
        assert roots == _nsort(roots)

def test_roots_binomial():
    assert roots_binomial(Poly(5*x, x)) == [0]
    assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
    assert roots_binomial(Poly(5*x + 2, x)) == [-Rational(2, 5)]

    A = 10**Rational(3, 4)/10

    assert roots_binomial(Poly(5*x**4 + 2, x)) == \
        [-A - A*I, -A + A*I, A - A*I, A + A*I]

    a1 = Symbol('a1', nonnegative=True)
    b1 = Symbol('b1', nonnegative=True)

    r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
    r1 = roots_binomial(Poly(a1*x**2 + b1, x))

    assert powsimp(r0[0]) == powsimp(r1[0])
    assert powsimp(r0[1]) == powsimp(r1[1])
    for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
        if a == b and a != 1:  # a == b == 1 is sufficient
            continue
        p = Poly(a*x**n + s*b)
        ans = roots_binomial(p)
        assert ans == _nsort(ans)

    # issue 8813
    assert roots(Poly(2*x**3 - 16*y**3, x)) == {
        2*y*(-S(1)/2 - sqrt(3)*I/2): 1,
        2*y: 1,
        2*y*(-S(1)/2 + sqrt(3)*I/2): 1}

 def _contains(self, other):
     from sympy.matrices import Matrix
     from sympy.solvers.solveset import solveset, linsolve
     from sympy.utilities.iterables import iterable, cartes
     L = self.lamda
     if self._is_multivariate():
         if not iterable(L.expr):
             if iterable(other):
                 return S.false
             return other.as_numer_denom() in self.func(
                 Lambda(L.variables, L.expr.as_numer_denom()), self.base_set)
         if len(L.expr) != len(self.lamda.variables):
             raise NotImplementedError(filldedent('''
 Dimensions of input and output of Lambda are different.'''))
         eqs = [expr - val for val, expr in zip(other, L.expr)]
         variables = L.variables
         free = set(variables)
         if all(i.is_number for i in list(Matrix(eqs).jacobian(variables))):
             solns = list(linsolve([e - val for e, val in
             zip(L.expr, other)], variables))
         else:
             syms = [e.free_symbols & free for e in eqs]
             solns = {}
             for i, (e, s, v) in enumerate(zip(eqs, syms, other)):
                 if not s:
                     if e != v:
                         return S.false
                     solns[vars[i]] = [v]
                     continue
                 elif len(s) == 1:
                     sy = s.pop()
                     sol = solveset(e, sy)
                     if sol is S.EmptySet:
                         return S.false
                     elif isinstance(sol, FiniteSet):
                         solns[sy] = list(sol)
                     else:
                         raise NotImplementedError
                 else:
                     raise NotImplementedError
             solns = cartes(*[solns[s] for s in variables])
     else:
         # assume scalar -> scalar mapping
         solnsSet = solveset(L.expr - other, L.variables[0])
         if solnsSet.is_FiniteSet:
             solns = list(solnsSet)
         else:
             raise NotImplementedError(filldedent('''
             Determining whether an ImageSet contains %s has not
             been implemented.''' % func_name(other)))
     for soln in solns:
         try:
             if soln in self.base_set:
                 return S.true
         except TypeError:
             return self.base_set.contains(soln.evalf())
     return S.false

def test_fuzzy_group():
    from sympy.utilities.iterables import cartes
    v = [T, F, U]
    for i in cartes(*[v]*3):
        assert _fuzzy_group(i) is (
            None if None in i else (True if all(j for j in i) else False))
        assert _fuzzy_group(i, quick_exit=True) is (
            None if (i.count(False) > 1) else (None if None in i else (
            True if all(j for j in i) else False)))

def test_piecewise_integrate3_inequality_conditions():
    from sympy.utilities.iterables import cartes
    lim = (x, 0, 5)
    # set below includes two pts below range, 2 pts in range,
    # 2 pts above range, and the boundaries
    N = (-2, -1, 0, 1, 2, 5, 6, 7)

    p = Piecewise((1, x > a), (2, x > b), (0, True))
    ans = p.integrate(lim)
    for i, j in cartes(N, repeat=2):
        reps = dict(zip((a, b), (i, j)))
        assert ans.subs(reps) == p.subs(reps).integrate(lim)
    assert ans.subs(a, 4).subs(b, 1) == 0 + 2*3 + 1

    p = Piecewise((1, x > a), (2, x < b), (0, True))
    ans = p.integrate(lim)
    for i, j in cartes(N, repeat=2):
        reps = dict(zip((a, b), (i, j)))
        assert ans.subs(reps) == p.subs(reps).integrate(lim)

def piecewise_fold(expr):
    """
    Takes an expression containing a piecewise function and returns the
    expression in piecewise form. In addition, any ITE conditions are
    rewritten in negation normal form and simplified.

    Examples
    ========

    >>> from sympy import Piecewise, piecewise_fold, sympify as S
    >>> from sympy.abc import x
    >>> p = Piecewise((x, x < 1), (1, S(1) <= x))
    >>> piecewise_fold(x*p)
    Piecewise((x**2, x < 1), (x, 1 <= x))

    See Also
    ========

    Piecewise
    """
    if not isinstance(expr, Basic) or not expr.has(Piecewise):
        return expr

    new_args = []
    if isinstance(expr, (ExprCondPair, Piecewise)):
        for e, c in expr.args:
            if not isinstance(e, Piecewise):
                e = piecewise_fold(e)
            # we don't keep Piecewise in condition because
            # it has to be checked to see that it's complete
            # and we convert it to ITE at that time
            assert not c.has(Piecewise)  # pragma: no cover
            if isinstance(c, ITE):
                c = c.to_nnf()
                c = simplify_logic(c, form='cnf')
            if isinstance(e, Piecewise):
                new_args.extend([(piecewise_fold(ei), And(ci, c))
                    for ei, ci in e.args])
            else:
                new_args.append((e, c))
    else:
        from sympy.utilities.iterables import cartes
        folded = list(map(piecewise_fold, expr.args))
        for ec in cartes(*[
                (i.args if isinstance(i, Piecewise) else
                 [(i, true)]) for i in folded]):
            e, c = zip(*ec)
            new_args.append((expr.func(*e), And(*c)))

    return Piecewise(*new_args)

def piecewise_fold(expr):
    """
    Takes an expression containing a piecewise function and returns the
    expression in piecewise form.

    Examples
    ========

    >>> from sympy import Piecewise, piecewise_fold, sympify as S
    >>> from sympy.abc import x
    >>> p = Piecewise((x, x < 1), (1, S(1) <= x))
    >>> piecewise_fold(x*p)
    Piecewise((x**2, x < 1), (x, 1 <= x))

    See Also
    ========

    Piecewise
    """
    if not isinstance(expr, Basic) or not expr.has(Piecewise):
        return expr

    new_args = []
    if isinstance(expr, (ExprCondPair, Piecewise)):
        for e, c in expr.args:
            if not isinstance(e, Piecewise):
                e = piecewise_fold(e)
            if isinstance(e, Piecewise):
                new_args.extend([(piecewise_fold(ei), And(ci, c))
                    for ei, ci in e.args])
            else:
                new_args.append((e, c))
    else:
        from sympy.utilities.iterables import cartes
        folded = list(map(piecewise_fold, expr.args))
        for ec in cartes(*[
                (i.args if isinstance(i, Piecewise) else
                 [(i, S.true)]) for i in folded]):
            e, c = zip(*ec)
            new_args.append((expr.func(*e), And(*c)))

    return Piecewise(*new_args)

def test_issue1447():
    # using list(...) so py.test can recalculate values
    from sympy import sign
    tests = list(cartes([cot, tan],
                        [-pi/2, 0, pi/2, pi, 3*pi/2],
                        ['-', '+']))
    results = (0, 0, -oo, oo, 0, 0, -oo, oo, 0, 0,
               oo, -oo, 0, 0, oo, -oo, 0, 0, oo, -oo)
    assert len(tests) == len(results)
    for i, (args, res) in enumerate(zip(tests, results)):
        f, l, d= args
        eq=f(x)
        try:
            assert limit(eq, x, l, dir=d) == res
        except AssertionError:
            if 0: # change to 1 if you want to see the failing tests
                print
                print i, res, eq, l, d, limit(eq, x, l, dir=d)
            else:
                assert None

def test_issue2084():
    # using list(...) so py.test can recalculate values
    tests = list(cartes([x, -x],
                        [-1, 1],
                        [2, 3, Rational(1, 2), Rational(2, 3)],
                        ['-', '+']))
    results = (oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3), oo,
               0, 0, 0, 0, 0, 0, 0, 0,
               oo, oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3),
               0, 0, 0, 0, 0, 0, 0, 0)
    assert len(tests) == len(results)
    for i, (args, res) in enumerate(zip(tests, results)):
        y, s, e, d = args
        eq=y**(s*e)
        try:
            assert limit(eq, x, 0, dir=d) == res
        except AssertionError:
            if 0: # change to 1 if you want to see the failing tests
                print
                print i, res, eq, d, limit(eq, x, 0, dir=d)
            else:
                assert None

def test_roots_binomial():
    assert roots_binomial(Poly(5 * x, x)) == [0]
    assert roots_binomial(Poly(5 * x ** 4, x)) == [0, 0, 0, 0]
    assert roots_binomial(Poly(5 * x + 2, x)) == [-Rational(2, 5)]

    A = 10 ** Rational(3, 4) / 10

    assert roots_binomial(Poly(5 * x ** 4 + 2, x)) == [-A - A * I, -A + A * I, A - A * I, A + A * I]

    a1 = Symbol("a1", nonnegative=True)
    b1 = Symbol("b1", nonnegative=True)

    r0 = roots_quadratic(Poly(a1 * x ** 2 + b1, x))
    r1 = roots_binomial(Poly(a1 * x ** 2 + b1, x))

    assert powsimp(r0[0]) == powsimp(r1[0])
    assert powsimp(r0[1]) == powsimp(r1[1])
    for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
        if a == b and a != 1:  # a == b == 1 is sufficient
            continue
        p = Poly(a * x ** n + s * b)
        roots = roots_binomial(p)
        assert roots == _nsort(roots)

def test_cartes():
    assert list(cartes([1, 2], [3, 4, 5])) == \
           [[1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5]]
    assert list(cartes()) == [[]]

#.........这里部分代码省略.........
    assert Range(-oo, 1, 1)[-1] is S.Zero
    assert Range(oo, 1, -1)[-1] == 2
    assert Range(0, -oo, -2)[-1] == -oo
    assert Range(1, 10, 1)[-1] == 9

    it = iter(Range(-oo, 0, 2))
    raises(ValueError, lambda: next(it))

    assert empty.intersect(S.Integers) == empty
    assert Range(-1, 10, 1).intersect(S.Integers) == Range(-1, 10, 1)
    assert Range(-1, 10, 1).intersect(S.Naturals) == Range(1, 10, 1)

    # test slicing
    assert Range(1, 10, 1)[5] == 6
    assert Range(1, 12, 2)[5] == 11
    assert Range(1, 10, 1)[-1] == 9
    assert Range(1, 10, 3)[-1] == 7
    raises(ValueError, lambda: Range(oo,0,-1)[1:3:0])
    raises(ValueError, lambda: Range(oo,0,-1)[:1])
    raises(ValueError, lambda: Range(1, oo)[-2])
    raises(ValueError, lambda: Range(-oo, 1)[2])
    raises(IndexError, lambda: Range(10)[-20])
    raises(IndexError, lambda: Range(10)[20])
    raises(ValueError, lambda: Range(2, -oo, -2)[2:2:0])
    assert Range(2, -oo, -2)[2:2:2] == empty
    assert Range(2, -oo, -2)[:2:2] == Range(2, -2, -4)
    raises(ValueError, lambda: Range(-oo, 4, 2)[:2:2])
    assert Range(-oo, 4, 2)[::-2] == Range(2, -oo, -4)
    raises(ValueError, lambda: Range(-oo, 4, 2)[::2])
    assert Range(oo, 2, -2)[::] == Range(oo, 2, -2)
    assert Range(-oo, 4, 2)[:-2:-2] == Range(2, 0, -4)
    assert Range(-oo, 4, 2)[:-2:2] == Range(-oo, 0, 4)
    raises(ValueError, lambda: Range(-oo, 4, 2)[:0:-2])
    raises(ValueError, lambda: Range(-oo, 4, 2)[:2:-2])
    assert Range(-oo, 4, 2)[-2::-2] == Range(0, -oo, -4)
    raises(ValueError, lambda: Range(-oo, 4, 2)[-2:0:-2])
    raises(ValueError, lambda: Range(-oo, 4, 2)[0::2])
    assert Range(oo, 2, -2)[0::] == Range(oo, 2, -2)
    raises(ValueError, lambda: Range(-oo, 4, 2)[0:-2:2])
    assert Range(oo, 2, -2)[0:-2:] == Range(oo, 6, -2)
    raises(ValueError, lambda: Range(oo, 2, -2)[0:2:])
    raises(ValueError, lambda: Range(-oo, 4, 2)[2::-1])
    assert Range(-oo, 4, 2)[-2::2] == Range(0, 4, 4)
    assert Range(oo, 0, -2)[-10:0:2] == empty
    raises(ValueError, lambda: Range(oo, 0, -2)[-10:10:2])
    raises(ValueError, lambda: Range(oo, 0, -2)[0::-2])
    assert Range(oo, 0, -2)[0:-4:-2] == empty
    assert Range(oo, 0, -2)[:0:2] == empty
    raises(ValueError, lambda: Range(oo, 0, -2)[:1:-1])

    # test empty Range
    assert empty.reversed == empty
    assert 0 not in empty
    assert list(empty) == []
    assert len(empty) == 0
    assert empty.size is S.Zero
    assert empty.intersect(FiniteSet(0)) is S.EmptySet
    assert bool(empty) is False
    raises(IndexError, lambda: empty[0])
    assert empty[:0] == empty
    raises(NotImplementedError, lambda: empty.inf)
    raises(NotImplementedError, lambda: empty.sup)

    AB = [None] + list(range(12))
    for R in [
            Range(1, 10),
            Range(1, 10, 2),
        ]:
        r = list(R)
        for a, b, c in cartes(AB, AB, [-3, -1, None, 1, 3]):
            for reverse in range(2):
                r = list(reversed(r))
                R = R.reversed
                result = list(R[a:b:c])
                ans = r[a:b:c]
                txt = ('\n%s[%s:%s:%s] = %s -> %s' % (
                R, a, b, c, result, ans))
                check = ans == result
                assert check, txt

    assert Range(1, 10, 1).boundary == Range(1, 10, 1)

    for r in (Range(1, 10, 2), Range(1, oo, 2)):
        rev = r.reversed
        assert r.inf == rev.inf and r.sup == rev.sup
        assert r.step == -rev.step

    # Make sure to use range in Python 3 and xrange in Python 2 (regardless of
    # compatibility imports above)
    if PY3:
        builtin_range = range
    else:
        builtin_range = xrange

    assert Range(builtin_range(10)) == Range(10)
    assert Range(builtin_range(1, 10)) == Range(1, 10)
    assert Range(builtin_range(1, 10, 2)) == Range(1, 10, 2)
    if PY3:
        assert Range(builtin_range(1000000000000)) == \
            Range(1000000000000)

def piecewise_fold(expr):
    """
    Takes an expression containing a piecewise function and returns the
    expression in piecewise form. In addition, any ITE conditions are
    rewritten in negation normal form and simplified.

    Examples
    ========

    >>> from sympy import Piecewise, piecewise_fold, sympify as S
    >>> from sympy.abc import x
    >>> p = Piecewise((x, x < 1), (1, S(1) <= x))
    >>> piecewise_fold(x*p)
    Piecewise((x**2, x < 1), (x, True))

    See Also
    ========

    Piecewise
    """
    if not isinstance(expr, Basic) or not expr.has(Piecewise):
        return expr

    new_args = []
    if isinstance(expr, (ExprCondPair, Piecewise)):
        for e, c in expr.args:
            if not isinstance(e, Piecewise):
                e = piecewise_fold(e)
            # we don't keep Piecewise in condition because
            # it has to be checked to see that it's complete
            # and we convert it to ITE at that time
            assert not c.has(Piecewise)  # pragma: no cover
            if isinstance(c, ITE):
                c = c.to_nnf()
                c = simplify_logic(c, form='cnf')
            if isinstance(e, Piecewise):
                new_args.extend([(piecewise_fold(ei), And(ci, c))
                    for ei, ci in e.args])
            else:
                new_args.append((e, c))
    else:
        from sympy.utilities.iterables import cartes, sift, common_prefix
        # Given
        #     P1 = Piecewise((e11, c1), (e12, c2), A)
        #     P2 = Piecewise((e21, c1), (e22, c2), B)
        #     ...
        # the folding of f(P1, P2) is trivially
        # Piecewise(
        #   (f(e11, e21), c1),
        #   (f(e12, e22), c2),
        #   (f(Piecewise(A), Piecewise(B)), True))
        # Certain objects end up rewriting themselves as thus, so
        # we do that grouping before the more generic folding.
        # The following applies this idea when f = Add or f = Mul
        # (and the expression is commutative).
        if expr.is_Add or expr.is_Mul and expr.is_commutative:
            p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
            pc = sift(p, lambda x: x.args[0].cond)
            for c in pc:
                if len(pc[c]) > 1:
                    pargs = [list(i.args) for i in pc[c]]
                    # the first one is the same; there may be more
                    com = common_prefix(*[
                        [i.cond for i in j] for j in pargs])
                    n = len(com)
                    collected = []
                    for i in range(n):
                        collected.append((
                            expr.func(*[ai[i].expr for ai in pargs]),
                            com[i]))
                    remains = []
                    for a in pargs:
                        if n == len(a):  # no more args
                            continue
                        if a[n].cond == True:  # no longer Piecewise
                            remains.append(a[n].expr)
                        else:  # restore the remaining Piecewise
                            remains.append(
                                Piecewise(*a[n:], evaluate=False))
                    if remains:
                        collected.append((expr.func(*remains), True))
                    args.append(Piecewise(*collected, evaluate=False))
                    continue
                args.extend(pc[c])
        else:
            args = expr.args
        # fold
        folded = list(map(piecewise_fold, args))
        for ec in cartes(*[
                (i.args if isinstance(i, Piecewise) else
                 [(i, true)]) for i in folded]):
            e, c = zip(*ec)
            new_args.append((expr.func(*e), And(*c)))

    return Piecewise(*new_args)

def test_cartes():
    assert list(cartes([1, 2], [3, 4, 5])) == \
           [(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)]
    assert list(cartes()) == [()]

def test_arguments():
    """Functions accepting `Point` objects in `geometry`
    should also accept tuples and lists and
    automatically convert them to points."""

    singles2d = ((1,2), [1,2], Point(1,2))
    singles2d2 = ((1,3), [1,3], Point(1,3))
    doubles2d = cartes(singles2d, singles2d2)
    p2d = Point2D(1,2)
    singles3d = ((1,2,3), [1,2,3], Point(1,2,3))
    doubles3d = subsets(singles3d, 2)
    p3d = Point3D(1,2,3)
    singles4d = ((1,2,3,4), [1,2,3,4], Point(1,2,3,4))
    doubles4d = subsets(singles4d, 2)
    p4d = Point(1,2,3,4)

    # test 2D
    test_single = ['distance', 'is_scalar_multiple', 'taxicab_distance', 'midpoint', 'intersection', 'dot', 'equals', '__add__', '__sub__']
    test_double = ['is_concyclic', 'is_collinear']
    for p in singles2d:
        Point2D(p)
    for func in test_single:
        for p in singles2d:
            getattr(p2d, func)(p)
    for func in test_double:
        for p in doubles2d:
            getattr(p2d, func)(*p)

    # test 3D
    test_double = ['is_collinear']
    for p in singles3d:
        Point3D(p)
    for func in test_single:
        for p in singles3d:
            getattr(p3d, func)(p)
    for func in test_double:
        for p in doubles2d:
            getattr(p3d, func)(*p)

    # test 4D
    test_double = ['is_collinear']
    for p in singles4d:
        Point(p)
    for func in test_single:
        for p in singles4d:
            getattr(p4d, func)(p)
    for func in test_double:
        for p in doubles4d:
            getattr(p4d, func)(*p)

    # test evaluate=False for ops
    x = Symbol('x')
    a = Point(0, 1)
    assert a + (0.1, x) == Point(0.1, 1 + x)
    a = Point(0, 1)
    assert a/10.0 == Point(0.0, 0.1)
    a = Point(0, 1)
    assert a*10.0 == Point(0.0, 10.0)

    # test evaluate=False when changing dimensions
    u = Point(.1, .2, evaluate=False)
    u4 = Point(u, dim=4, on_morph='ignore')
    assert u4.args == (.1, .2, 0, 0)
    assert all(i.is_Float for i in u4.args[:2])
    # and even when *not* changing dimensions
    assert all(i.is_Float for i in Point(u).args)

    # never raise error if creating an origin
    assert Point(dim=3, on_morph='error')

def test_cartes():
    assert list(cartes([1, 2], [3, 4, 5])) == [(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)]
    assert list(cartes()) == [()]
    assert list(cartes("a")) == [("a",)]
    assert list(cartes("a", repeat=2)) == [("a", "a")]
    assert list(cartes(range(2))) == [(0,), (1,)]

#.........这里部分代码省略.........

        >>> type(_[0])
        <class 'sympy.core.function.UndefinedFunction'>

    """
    result = []

    if isinstance(names, string_types):
        marker = 0
        literals = ['\,', '\:', '\ ']
        for i in range(len(literals)):
            lit = literals.pop(0)
            if lit in names:
                while chr(marker) in names:
                    marker += 1
                lit_char = chr(marker)
                marker += 1
                names = names.replace(lit, lit_char)
                literals.append((lit_char, lit[1:]))
        def literal(s):
            if literals:
                for c, l in literals:
                    s = s.replace(c, l)
            return s

        names = names.strip()
        as_seq = names.endswith(',')
        if as_seq:
            names = names[:-1].rstrip()
        if not names:
            raise ValueError('no symbols given')

        # split on commas
        names = [n.strip() for n in names.split(',')]
        if not all(n for n in names):
            raise ValueError('missing symbol between commas')
        # split on spaces
        for i in range(len(names) - 1, -1, -1):
            names[i: i + 1] = names[i].split()

        cls = args.pop('cls', Symbol)
        seq = args.pop('seq', as_seq)

        for name in names:
            if not name:
                raise ValueError('missing symbol')

            if ':' not in name:
                symbol = cls(literal(name), **args)
                result.append(symbol)
                continue

            split = _range.split(name)
            # remove 1 layer of bounding parentheses around ranges
            for i in range(len(split) - 1):
                if i and ':' in split[i] and split[i] != ':' and \
                        split[i - 1].endswith('(') and \
                        split[i + 1].startswith(')'):
                    split[i - 1] = split[i - 1][:-1]
                    split[i + 1] = split[i + 1][1:]
            for i, s in enumerate(split):
                if ':' in s:
                    if s[-1].endswith(':'):
                        raise ValueError('missing end range')
                    a, b = s.split(':')
                    if b[-1] in string.digits:
                        a = 0 if not a else int(a)
                        b = int(b)
                        split[i] = [str(c) for c in range(a, b)]
                    else:
                        a = a or 'a'
                        split[i] = [string.ascii_letters[c] for c in range(
                            string.ascii_letters.index(a),
                            string.ascii_letters.index(b) + 1)]  # inclusive
                    if not split[i]:
                        break
                else:
                    split[i] = [s]
            else:
                seq = True
                if len(split) == 1:
                    names = split[0]
                else:
                    names = [''.join(s) for s in cartes(*split)]
                if literals:
                    result.extend([cls(literal(s), **args) for s in names])
                else:
                    result.extend([cls(s, **args) for s in names])

        if not seq and len(result) <= 1:
            if not result:
                return ()
            return result[0]

        return tuple(result)
    else:
        for name in names:
            result.append(symbols(name, **args))

        return type(names)(result)

def test_torf():
    from sympy.utilities.iterables import cartes
    v = [T, F, U]
    for i in cartes(*[v]*3):
        assert _torf(i) is (True if all(j for j in i) else
                            (False if all(j is False for j in i) else None))