def test_diff():
    x, y = symbols('x, y')
    assert Rational(1, 3).diff(x) is S.Zero
    assert I.diff(x) is S.Zero
    assert pi.diff(x) is S.Zero
    assert x.diff(x, 0) == x
    assert (x**2).diff(x, 2, x) == 0
    assert (x**2).diff(x, y, 0) == 2*x
    assert (x**2).diff(x, y) == 0
    raises(ValueError, lambda: x.diff(1, x))

    a = Symbol("a")
    b = Symbol("b")
    c = Symbol("c")
    p = Rational(5)
    e = a*b + b**p
    assert e.diff(a) == b
    assert e.diff(b) == a + 5*b**4
    assert e.diff(b).diff(a) == Rational(1)
    e = a*(b + c)
    assert e.diff(a) == b + c
    assert e.diff(b) == a
    assert e.diff(b).diff(a) == Rational(1)
    e = c**p
    assert e.diff(c, 6) == Rational(0)
    assert e.diff(c, 5) == Rational(120)
    e = c**Rational(2)
    assert e.diff(c) == 2*c
    e = a*b*c
    assert e.diff(c) == a*b

def reduce_rational_inequalities_wrap(condition, var):
    if condition.is_Relational:
        return _reduce_inequalities([[condition]], var, relational=False)
    if isinstance(condition, Or):
        return Union(*[_reduce_inequalities([[arg]], var, relational=False)
            for arg in condition.args])
    if isinstance(condition, And):
        intervals = [_reduce_inequalities([[arg]], var, relational=False)
            for arg in condition.args]
        I = intervals[0]
        for i in intervals:
            I = I.intersect(i)
        return I

def reduce_rational_inequalities_wrap(condition, var):
    if condition.is_Relational:
        return _reduce_inequalities([[condition]], var, relational=False)
    if condition.__class__ is Or:
        return _reduce_inequalities([list(condition.args)],
                var, relational=False)
    if condition.__class__ is And:
        intervals = [_reduce_inequalities([[arg]], var, relational=False)
            for arg in condition.args]
        I = intervals[0]
        for i in intervals:
            I = I.intersect(i)
        return I

def test_issue_8919():
    c = symbols('c:5')
    x = symbols("x")
    f1 = Piecewise((c[1], x < 1), (c[2], True))
    f2 = Piecewise((c[3], x < S(1)/3), (c[4], True))
    assert integrate(f1*f2, (x, 0, 2)
        ) == c[1]*c[3]/3 + 2*c[1]*c[4]/3 + c[2]*c[4]
    f1 = Piecewise((0, x < 1), (2, True))
    f2 = Piecewise((3, x < 2), (0, True))
    assert integrate(f1*f2, (x, 0, 3)) == 6

    y = symbols("y", positive=True)
    a, b, c, x, z = symbols("a,b,c,x,z", real=True)
    I = Integral(Piecewise(
        (0, (x >= y) | (x < 0) | (b > c)),
        (a, True)), (x, 0, z))
    ans = I.doit()
    assert ans == Piecewise((0, b > c), (a*Min(y, z) - a*Min(0, z), True))
    for cond in (True, False):
         for yy in range(1, 3):
             for zz in range(-yy, 0, yy):
                 reps = [(b > c, cond), (y, yy), (z, zz)]
                 assert ans.subs(reps) == I.subs(reps).doit()

def test_issue_2387_bug():
    from sympy import I, Expr
    assert abs(Expr._from_mpmath(I._to_mpmath(15), 15) - I) < 1.0e-15