def test_var():
    var("a")
    assert a  == Symbol("a")

    var("b bb cc zz _x")
    assert b  == Symbol("b")
    assert bb == Symbol("bb")
    assert cc == Symbol("cc")
    assert zz == Symbol("zz")
    assert _x == Symbol("_x")

    v = var(['d','e','fg'])
    assert d  == Symbol('d')
    assert e  == Symbol('e')
    assert fg == Symbol('fg')

    # check return value
    assert v  == (d, e, fg)

    # see if var() really injects into global namespace
    raises(NameError, "z1")
    make_z1()
    assert z1 == Symbol("z1")

    raises(NameError, "z2")
    make_z2()
    assert z2 == Symbol("z2")

def test_Integer_new():
    """
    Test for Integer constructor
    """
    _test_rational_new(Integer)

    raises(ValueError, 'Integer("10.5")')

def test_args():
    p = Permutation([(0, 3, 1, 2), (4, 5)])
    assert p._cyclic_form is None
    assert Permutation(p) == p
    assert p.cyclic_form == [[0, 3, 1, 2], [4, 5]]
    assert p._array_form == [3, 2, 0, 1, 5, 4]
    p = Permutation((0, 3, 1, 2))
    assert p._cyclic_form is None
    assert p._array_form == [0, 3, 1, 2]
    assert Permutation([0]) == Permutation((0, ))
    assert Permutation([[0], [1]]) == Permutation(((0, ), (1, ))) == \
        Permutation(((0, ), [1]))
    assert Permutation([[1, 2]]) == Permutation([0, 2, 1])
    assert Permutation([[1], [4, 2]]) == Permutation([0, 1, 4, 3, 2])
    assert Permutation([[1], [4, 2]], size=1) == Permutation([0, 1, 4, 3, 2])
    assert Permutation(
        [[1], [4, 2]], size=6) == Permutation([0, 1, 4, 3, 2, 5])
    assert Permutation([], size=3) == Permutation([0, 1, 2])
    assert Permutation(3).list(5) == [0, 1, 2, 3, 4]
    assert Permutation(3).list(-1) == []
    assert Permutation(5)(1, 2).list(-1) == [0, 2, 1]
    assert Permutation(5)(1, 2).list() == [0, 2, 1, 3, 4, 5]
    raises(TypeError, lambda: Permutation([1, 2], [0]))
           # enclosing brackets needed
    raises(ValueError, lambda: Permutation([[1, 2], 0]))
           # enclosing brackets needed on 0
    raises(ValueError, lambda: Permutation([1, 1, 0]))
    raises(ValueError, lambda: Permutation([[1], [1, 2]]))
    raises(ValueError, lambda: Permutation([4, 5], size=10))  # where are 0-3?
    # but this is ok because cycles imply that only those listed moved
    assert Permutation(4, 5) == Permutation([0, 1, 2, 3, 5, 4])

def test_product_basic():
    H, T = 'H', 'T'
    unit_line = Interval(0, 1)
    d6 = FiniteSet(1, 2, 3, 4, 5, 6)
    d4 = FiniteSet(1, 2, 3, 4)
    coin = FiniteSet(H, T)

    square = unit_line * unit_line

    assert (0, 0) in square
    assert 0 not in square
    assert (H, T) in coin ** 2
    assert (.5, .5, .5) in square * unit_line
    assert (H, 3, 3) in coin * d6* d6
    HH, TT = sympify(H), sympify(T)
    assert set(coin**2) == set(((HH, HH), (HH, TT), (TT, HH), (TT, TT)))

    assert (d4*d4).is_subset(d6*d6)

    assert square.complement(Interval(-oo, oo)*Interval(-oo, oo)) == Union(
        (Interval(-oo, 0, True, True) +
         Interval(1, oo, True, True))*Interval(-oo, oo),
         Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
                  Interval(1, oo, True, True)))

    assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
    assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
    assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)

    assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square)  # segment in square

    assert len(coin*coin*coin) == 8
    assert len(S.EmptySet*S.EmptySet) == 0
    assert len(S.EmptySet*coin) == 0
    raises(TypeError, lambda: len(coin*Interval(0, 2)))

def test_Routine_argument_order():
    a, x, y, z = symbols('a x y z')
    expr = (x + y)*z
    raises(CodeGenArgumentListError, lambda: make_routine("test", expr,
           argument_sequence=[z, x]))
    raises(CodeGenArgumentListError, lambda: make_routine("test", Eq(a,
           expr), argument_sequence=[z, x, y]))
    r = make_routine('test', Eq(a, expr), argument_sequence=[z, x, a, y])
    assert [ arg.name for arg in r.arguments ] == [z, x, a, y]
    assert [ type(arg) for arg in r.arguments ] == [
        InputArgument, InputArgument, OutputArgument, InputArgument  ]
    r = make_routine('test', Eq(z, expr), argument_sequence=[z, x, y])
    assert [ type(arg) for arg in r.arguments ] == [
        InOutArgument, InputArgument, InputArgument ]

    from sympy.tensor import IndexedBase, Idx
    A, B = map(IndexedBase, ['A', 'B'])
    m = symbols('m', integer=True)
    i = Idx('i', m)
    r = make_routine('test', Eq(A[i], B[i]), argument_sequence=[B, A, m])
    assert [ arg.name for arg in r.arguments ] == [B.label, A.label, m]

    expr = Integral(x*y*z, (x, 1, 2), (y, 1, 3))
    r = make_routine('test', Eq(a, expr), argument_sequence=[z, x, a, y])
    assert [ arg.name for arg in r.arguments ] == [z, x, a, y]

def test_intersection():
    # iterable
    i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False)
    assert i.is_iterable
    assert set(i) == {S(2), S(3)}

    # challenging intervals
    x = Symbol('x', real=True)
    i = Intersection(Interval(0, 3), Interval(x, 6))
    assert (5 in i) is False
    raises(TypeError, lambda: 2 in i)

    # Singleton special cases
    assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet
    assert Intersection(Interval(-oo, oo), Interval(-oo, x)) == Interval(-oo, x)

    # Products
    line = Interval(0, 5)
    i = Intersection(line**2, line**3, evaluate=False)
    assert (2, 2) not in i
    assert (2, 2, 2) not in i
    raises(ValueError, lambda: list(i))

    assert Intersection(Intersection(S.Integers, S.Naturals, evaluate=False),
                        S.Reals, evaluate=False) == \
            Intersection(S.Integers, S.Naturals, S.Reals, evaluate=False)

    assert Intersection(S.Complexes, FiniteSet(S.ComplexInfinity)) == S.EmptySet

def test_is_subset():
    assert Interval(0, 1).is_subset(Interval(0, 2)) is True
    assert Interval(0, 3).is_subset(Interval(0, 2)) is False

    assert FiniteSet(1, 2).is_subset(FiniteSet(1, 2, 3, 4))
    assert FiniteSet(4, 5).is_subset(FiniteSet(1, 2, 3, 4)) is False
    assert FiniteSet(1).is_subset(Interval(0, 2))
    assert FiniteSet(1, 2).is_subset(Interval(0, 2, True, True)) is False
    assert (Interval(1, 2) + FiniteSet(3)).is_subset(
        (Interval(0, 2, False, True) + FiniteSet(2, 3)))

    assert Interval(3, 4).is_subset(Union(Interval(0, 1), Interval(2, 5))) is True
    assert Interval(3, 6).is_subset(Union(Interval(0, 1), Interval(2, 5))) is False

    assert FiniteSet(1, 2, 3, 4).is_subset(Interval(0, 5)) is True
    assert S.EmptySet.is_subset(FiniteSet(1, 2, 3)) is True

    assert Interval(0, 1).is_subset(S.EmptySet) is False
    assert S.EmptySet.is_subset(S.EmptySet) is True

    raises(ValueError, lambda: S.EmptySet.is_subset(1))

    # tests for the issubset alias
    assert FiniteSet(1, 2, 3, 4).issubset(Interval(0, 5)) is True
    assert S.EmptySet.issubset(FiniteSet(1, 2, 3)) is True

def test_egyptian_fraction():
    def test_equality(r, alg="Greedy"):
        return r == Add(*[Rational(1, i) for i in egyptian_fraction(r, alg)])

    r = random_complex_number(a=0, c=1, b=0, d=0, rational=True)
    assert test_equality(r)

    assert egyptian_fraction(Rational(4, 17)) == [5, 29, 1233, 3039345]
    assert egyptian_fraction(Rational(7, 13), "Greedy") == [2, 26]
    assert egyptian_fraction(Rational(23, 101), "Greedy") == \
        [5, 37, 1438, 2985448, 40108045937720]
    assert egyptian_fraction(Rational(18, 23), "Takenouchi") == \
        [2, 6, 12, 35, 276, 2415]
    assert egyptian_fraction(Rational(5, 6), "Graham Jewett") == \
        [6, 7, 8, 9, 10, 42, 43, 44, 45, 56, 57, 58, 72, 73, 90, 1806, 1807,
         1808, 1892, 1893, 1980, 3192, 3193, 3306, 5256, 3263442, 3263443,
         3267056, 3581556, 10192056, 10650056950806]
    assert egyptian_fraction(Rational(5, 6), "Golomb") == [2, 6, 12, 20, 30]
    assert egyptian_fraction(Rational(5, 121), "Golomb") == [25, 1225, 3577, 7081, 11737]
    raises(ValueError, lambda: egyptian_fraction(Rational(-4, 9)))
    assert egyptian_fraction(Rational(8, 3), "Golomb") == [1, 2, 3, 4, 5, 6, 7,
                                                           14, 574, 2788, 6460,
                                                           11590, 33062, 113820]
    assert egyptian_fraction(Rational(355, 113)) == [1, 2, 3, 4, 5, 6, 7, 8, 9,
                                                     10, 11, 12, 27, 744, 893588,
                                                     1251493536607,
                                                     20361068938197002344405230]

def test_solve_poly_system():
    assert solve_poly_system([x-1], x) == [(S.One,)]

    assert solve_poly_system([y - x, y - x - 1], x, y) == None

    assert solve_poly_system([y - x**2, y + x**2], x, y) == [(S.Zero, S.Zero)]

    assert solve_poly_system([2*x - 3, 3*y/2 - 2*x, z - 5*y], x, y, z) == \
        [(Rational(3, 2), Integer(2), Integer(10))]

    assert solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) == \
       [(0, 0), (2, -sqrt(2)), (2, sqrt(2))]

    assert solve_poly_system([y - x**2, y + x**2 + 1], x, y) == \
           [(I*sqrt(S.Half), -S.Half), (-I*sqrt(S.Half), -S.Half)]

    f_1 = x**2 + y + z - 1
    f_2 = x + y**2 + z - 1
    f_3 = x + y + z**2 - 1

    a, b = -sqrt(2) - 1, sqrt(2) - 1

    assert solve_poly_system([f_1, f_2, f_3], x, y, z) == \
        [(a, a, a), (0, 0, 1), (0, 1, 0), (b, b, b), (1, 0, 0)]

    solution = [(1, -1), (1, 1)]

    assert solve_poly_system([Poly(x**2 - y**2), Poly(x - 1)]) == solution
    assert solve_poly_system([x**2 - y**2, x - 1], x, y) == solution
    assert solve_poly_system([x**2 - y**2, x - 1]) == solution

    assert solve_poly_system([x + x*y - 3, y + x*y - 4], x, y) == [(-3, -2), (1, 2)]

    raises(NotImplementedError, "solve_poly_system([x**3-y**3], x, y)")
    raises(PolynomialError, "solve_poly_system([1/x], x)")

def test_math_lambda():
    mpmath.mp.dps = 50
    sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
    f = lambdify(x, sin(x), "math")
    prec = 1e-15
    assert -prec < f(0.2) - sin02 < prec
    raises(TypeError, lambda: f(x))

def test_mpmath_lambda():
    mpmath.mp.dps = 50
    sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
    f = lambdify(x, sin(x), "mpmath")
    prec = 1e-49  # mpmath precision is around 50 decimal places
    assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
    raises(TypeError, lambda: f(x))

def test_lambda():
    x = Symbol('x')
    assert sympify('lambda : 1') == Lambda((), 1)
    assert sympify('lambda x: 2*x') == Lambda(x, 2*x)
    assert sympify('lambda x, y: 2*x+y') == Lambda([x, y], 2*x+y)

    raises(SympifyError, "_sympify('lambda : 1')")

def test_Max():
    from sympy.abc import x, y, z
    n = Symbol('n', negative=True)
    n_ = Symbol('n_', negative=True)
    nn = Symbol('nn', nonnegative=True)
    nn_ = Symbol('nn_', nonnegative=True)
    p = Symbol('p', positive=True)
    p_ = Symbol('p_', positive=True)
    np = Symbol('np', nonpositive=True)
    np_ = Symbol('np_', nonpositive=True)

    assert Max(5, 4) == 5

    # lists

    raises(ValueError, lambda: Max())
    assert Max(x, y) == Max(y, x)
    assert Max(x, y, z) == Max(z, y, x)
    assert Max(x, Max(y, z)) == Max(z, y, x)
    assert Max(x, Min(y, oo)) == Max(x, y)
    assert Max(n, -oo, n_,  p, 2) == Max(p, 2)
    assert Max(n, -oo, n_,  p) == p
    assert Max(2, x, p, n, -oo, S.NegativeInfinity, n_,  p, 2) == Max(2, x, p)
    assert Max(0, x, 1, y) == Max(1, x, y)
    assert Max(x, x + 1, x - 1) == 1 + x
    assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
    assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
    assert Max(cos(x), sin(x)).subs(x, 1) == sin(1)
    assert Max(cos(x), sin(x)).subs(x, S(1)/2) == cos(S(1)/2)
    raises(ValueError, lambda: Max(cos(x), sin(x)).subs(x, I))
    raises(ValueError, lambda: Max(I))
    raises(ValueError, lambda: Max(I, x))
    raises(ValueError, lambda: Max(S.ComplexInfinity, 1))

def test_julia_piecewise():
    expr = Piecewise((x, x < 1), (x**2, True))
    assert julia_code(expr) == "((x < 1) ? (x) : (x.^2))"
    assert julia_code(expr, assign_to="r") == (
        "r = ((x < 1) ? (x) : (x.^2))")
    assert julia_code(expr, assign_to="r", inline=False) == (
        "if (x < 1)\n"
        "    r = x\n"
        "else\n"
        "    r = x.^2\n"
        "end")
    expr = Piecewise((x**2, x < 1), (x**3, x < 2), (x**4, x < 3), (x**5, True))
    expected = ("((x < 1) ? (x.^2) :\n"
                "(x < 2) ? (x.^3) :\n"
                "(x < 3) ? (x.^4) : (x.^5))")
    assert julia_code(expr) == expected
    assert julia_code(expr, assign_to="r") == "r = " + expected
    assert julia_code(expr, assign_to="r", inline=False) == (
        "if (x < 1)\n"
        "    r = x.^2\n"
        "elseif (x < 2)\n"
        "    r = x.^3\n"
        "elseif (x < 3)\n"
        "    r = x.^4\n"
        "else\n"
        "    r = x.^5\n"
        "end")
    # Check that Piecewise without a True (default) condition error
    expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0))
    raises(ValueError, lambda: julia_code(expr))

def test_erf2():

    assert erf2(0, 0) == S.Zero
    assert erf2(x, x) == S.Zero
    assert erf2(nan, 0) == nan

    assert erf2(-oo,  y) ==  erf(y) + 1
    assert erf2( oo,  y) ==  erf(y) - 1
    assert erf2(  x, oo) ==  1 - erf(x)
    assert erf2(  x,-oo) == -1 - erf(x)
    assert erf2(x, erf2inv(x, y)) == y

    assert erf2(-x, -y) == -erf2(x,y)
    assert erf2(-x,  y) == erf(y) + erf(x)
    assert erf2( x, -y) == -erf(y) - erf(x)
    assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels)
    assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc)
    assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper)
    assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg)
    assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
    assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint)

    assert erf2(I, 0).is_real is False
    assert erf2(0, 0).is_real is True

    assert expand_func(erf(x) + erf2(x, y)) == erf(y)

    assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))

    assert erf2(x, y).rewrite('erf')  == erf(y) - erf(x)
    assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
    assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y))

    raises(ArgumentIndexError, lambda: erfi(x).fdiff(3))

def test_erf():
    assert erf(nan) == nan

    assert erf(oo) == 1
    assert erf(-oo) == -1

    assert erf(0) == 0

    assert erf(I*oo) == oo*I
    assert erf(-I*oo) == -oo*I

    assert erf(-2) == -erf(2)
    assert erf(-x*y) == -erf(x*y)
    assert erf(-x - y) == -erf(x + y)

    assert erf(I).is_real == False
    assert erf(0).is_real == True

    assert conjugate(erf(z)) == erf(conjugate(z))

    assert erf(x).as_leading_term(x) == x
    assert erf(1/x).as_leading_term(x) == erf(1/x)

    assert erf(z).rewrite('uppergamma') == sqrt(z**2)*erf(sqrt(z**2))/z

    assert limit(exp(x)*exp(x**2)*(erf(x+1/exp(x))-erf(x)), x, oo) == 2/sqrt(pi)
    assert limit((1-erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi)
    assert limit((1-erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1
    assert limit(((1-erf(x))*exp(x**2)*sqrt(pi)*x-1)*2*x**2, x, oo) == -1

    raises(ArgumentIndexError, 'erf(x).fdiff(2)')

def test_evalf_trig_zero_detection():
    a = sin(160*pi, evaluate=False)
    t = a.evalf(maxn=100)
    assert abs(t) < 1e-100
    assert t._prec < 2
    assert a.evalf(chop=True) == 0
    raises(PrecisionExhausted, lambda: a.evalf(strict=True))

def test_elgamal():
    dk = elgamal_private_key(5)
    ek = elgamal_public_key(dk)
    P = ek[0]
    assert P - 1 == decipher_elgamal(encipher_elgamal(P - 1, ek), dk)
    raises(ValueError, lambda: encipher_elgamal(P, dk))
    raises(ValueError, lambda: encipher_elgamal(-1, dk))

def test_interval_arguments():
    assert Interval(0, oo) == Interval(0, oo, False, True)
    assert Interval(0, oo).right_open is true
    assert Interval(-oo, 0) == Interval(-oo, 0, True, False)
    assert Interval(-oo, 0).left_open is true
    assert Interval(oo, -oo) == S.EmptySet
    assert Interval(oo, oo) == S.EmptySet
    assert Interval(-oo, -oo) == S.EmptySet

    assert isinstance(Interval(1, 1), FiniteSet)
    e = Sum(x, (x, 1, 3))
    assert isinstance(Interval(e, e), FiniteSet)

    assert Interval(1, 0) == S.EmptySet
    assert Interval(1, 1).measure == 0

    assert Interval(1, 1, False, True) == S.EmptySet
    assert Interval(1, 1, True, False) == S.EmptySet
    assert Interval(1, 1, True, True) == S.EmptySet


    assert isinstance(Interval(0, Symbol('a')), Interval)
    assert Interval(Symbol('a', real=True, positive=True), 0) == S.EmptySet
    raises(ValueError, lambda: Interval(0, S.ImaginaryUnit))
    raises(ValueError, lambda: Interval(0, Symbol('z', real=False)))

    raises(NotImplementedError, lambda: Interval(0, 1, And(x, y)))
    raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y)))
    raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y)))

def test_dh_shared_key():
    prk = dh_private_key(digit = 100)
    p, _, ga = dh_public_key(prk)
    b = randrange(2, p)
    sk = dh_shared_key((p, _, ga), b)
    assert sk == pow(ga, b, p)
    raises(ValueError, lambda: dh_shared_key((1031, 14, 565), 2000))