def test_piecewise_solve():
    abs2 = Piecewise((-x, x <= 0), (x, x > 0))
    f = abs2.subs(x, x - 2)
    assert solve(f, x) == [2]
    assert solve(f - 1,x) == [1, 3]

    f = Piecewise(((x - 2)**2, x >= 0), (1, True))
    assert solve(f, x) == [2]

    g = Piecewise(((x - 5)**5, x >= 4), (f, True))
    assert solve(g, x) == [2, 5]

    g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
    assert solve(g, x) == [2, 5]

    g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
    assert solve(g, x) == [5]

    g = Piecewise(((x - 5)**5, x >= 2), (f, True))
    assert solve(g, x) == [5]

    g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
    assert solve(g, x) == [5]

    assert solve(Piecewise((x, x < 0), (x, -1))) == [0]

def test_piecewise_interval():
    p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
    assert p1.subs(x, -0.5) == 0
    assert p1.subs(x, 0.5) == 0.5
    assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
    assert integrate(
        p1, x) == Piecewise((x**2/2, Interval(0, 1).contains(x)), (0, True))

def test_piecewise_simplify():
    p = Piecewise(((x**2 + 1)/x**2, Eq(x*(1 + x) - x**2, 0)),
                  ((-1)**x*(-1), True))
    assert p.simplify() == \
        Piecewise((zoo, Eq(x, 0)), ((-1)**(x + 1), True))
    # simplify when there are Eq in conditions
    assert Piecewise(
        (a, And(Eq(a, 0), Eq(a + b, 0))), (1, True)).simplify(
        ) == Piecewise(
        (0, And(Eq(a, 0), Eq(b, 0))), (1, True))
    assert Piecewise((2*x*factorial(a)/(factorial(y)*factorial(-y + a)),
        Eq(y, 0) & Eq(-y + a, 0)), (2*factorial(a)/(factorial(y)*factorial(-y
        + a)), Eq(y, 0) & Eq(-y + a, 1)), (0, True)).simplify(
        ) == Piecewise(
            (2*x, And(Eq(a, 0), Eq(y, 0))),
            (2, And(Eq(a, 1), Eq(y, 0))),
            (0, True))
    args = (2, And(Eq(x, 2), Ge(y ,0))), (x, True)
    assert Piecewise(*args).simplify() == Piecewise(*args)
    args = (1, Eq(x, 0)), (sin(x)/x, True)
    assert Piecewise(*args).simplify() == Piecewise(*args)
    assert Piecewise((2 + y, And(Eq(x, 2), Eq(y, 0))), (x, True)
        ).simplify() == x
    # check that x or f(x) are recognized as being Symbol-like for lhs
    args = Tuple((1, Eq(x, 0)), (sin(x) + 1 + x, True))
    ans = x + sin(x) + 1
    f = Function('f')
    assert Piecewise(*args).simplify() == ans
    assert Piecewise(*args.subs(x, f(x))).simplify() == ans.subs(x, f(x))

def test_issue_4313():
    u = Piecewise((0, x <= 0), (1, x >= a), (x/a, True))
    e = (u - u.subs(x, y))**2/(x - y)**2
    M = Max(0, a)
    assert integrate(e,  x).expand() == Piecewise(
        (Piecewise(
            (0, x <= 0),
            (-y**2/(a**2*x - a**2*y) + x/a**2 - 2*y*log(-y)/a**2 +
                2*y*log(x - y)/a**2 - y/a**2, x <= M),
            (-y**2/(-a**2*y + a**2*M) + 1/(-y + M) -
                1/(x - y) - 2*y*log(-y)/a**2 + 2*y*log(-y +
                M)/a**2 - y/a**2 + M/a**2, True)),
        ((a <= y) & (y <= 0)) | ((y <= 0) & (y > -oo))),
        (Piecewise(
            (-1/(x - y), x <= 0),
            (-a**2/(a**2*x - a**2*y) + 2*a*y/(a**2*x - a**2*y) -
                y**2/(a**2*x - a**2*y) + 2*log(-y)/a - 2*log(x - y)/a +
                2/a + x/a**2 - 2*y*log(-y)/a**2 + 2*y*log(x - y)/a**2 -
                y/a**2, x <= M),
            (-a**2/(-a**2*y + a**2*M) + 2*a*y/(-a**2*y +
                a**2*M) - y**2/(-a**2*y + a**2*M) +
                2*log(-y)/a - 2*log(-y + M)/a + 2/a -
                2*y*log(-y)/a**2 + 2*y*log(-y + M)/a**2 -
                y/a**2 + M/a**2, True)),
        a <= y),
        (Piecewise(
            (-y**2/(a**2*x - a**2*y), x <= 0),
            (x/a**2 + y/a**2, x <= M),
            (a**2/(-a**2*y + a**2*M) -
                a**2/(a**2*x - a**2*y) - 2*a*y/(-a**2*y + a**2*M) +
                2*a*y/(a**2*x - a**2*y) + y**2/(-a**2*y + a**2*M) -
                y**2/(a**2*x - a**2*y) + y/a**2 + M/a**2, True)),
        True))

def test_Function_subs():
    from sympy.abc import x, y

    f, g, h, i = symbols("f g h i", cls=Function)
    p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
    assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
    assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)

def test_issue_12587():
    # sort holes into intervals
    p = Piecewise((1, x > 4), (2, Not((x <= 3) & (x > -1))), (3, True))
    assert p.integrate((x, -5, 5)) == 23
    p = Piecewise((1, x > 1), (2, x < y), (3, True))
    lim = x, -3, 3
    ans = p.integrate(lim)
    for i in range(-1, 3):
        assert ans.subs(y, i) == p.subs(y, i).integrate(lim)

def test_conjugate_transpose():
    A, B = symbols("A B", commutative=False)
    p = Piecewise((A*B**2, x > 0), (A**2*B, True))
    assert p.adjoint() == \
        Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
    assert p.conjugate() == \
        Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
    assert p.transpose() == \
        Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))

def test_issue_6900():
    t0, t1, T, t = symbols('t0, t1 T t')
    f = Piecewise((0, t < t0), (x, And(t0 <= t, t < t1)), (0, t >= t1))
    assert f.integrate(t) == Piecewise(
        (0, t <= t0),
        (t*x - t0*x, t <= Max(t0, t1)),
        (-t0*x + x*Max(t0, t1), True))
    assert f.integrate((t, t0, T)) == Piecewise(
        (-t0*x + x*Min(T, Max(t0, t1)), T > t0),
        (0, True))

def test_piecewise_integrate2():
    from itertools import permutations
    lim = Tuple(x, c, d)
    p = Piecewise((1, x < a), (2, x > b), (3, True))
    q = p.integrate(lim)
    assert q == Piecewise(
        (-c + 2*d - 2*Min(d, Max(a, c)) + Min(d, Max(a, b, c)), c < d),
        (-2*c + d + 2*Min(c, Max(a, d)) - Min(c, Max(a, b, d)), True))
    for v in permutations((1, 2, 3, 4)):
        r = dict(zip((a, b, c, d), v))
        assert p.subs(r).integrate(lim.subs(r)) == q.subs(r)

def test_unevaluated_integrals():
    f = Function('f')
    p = Piecewise((1, Eq(f(x) - 1, 0)), (2, x - 10 < 0), (0, True))
    assert p.integrate(x) == Integral(p, x)
    assert p.integrate((x, 0, 5)) == Integral(p, (x, 0, 5))
    # test it by replacing f(x) with x%2 which will not
    # affect the answer: the integrand is essentially 2 over
    # the domain of integration
    assert Integral(p, (x, 0, 5)).subs(f(x), x%2).n() == 10

    # this is a test of using _solve_inequality when
    # solve_univariate_inequality fails
    assert p.integrate(y) == Piecewise(
        (y, Eq(f(x), 1) | ((x < 10) & Eq(f(x), 1))),
        (2*y, (x >= -oo) & (x < 10)), (0, True))

def test_piecewise_as_leading_term():
    p1 = Piecewise((1/x, x>1), (0, True))
    p2 = Piecewise((x, x>1), (0, True))
    p3 = Piecewise((1/x, x>1), (x, True))
    p4 = Piecewise((x, x>1), (1/x, True))
    assert p1.as_leading_term(x) == 0
    assert p2.as_leading_term(x) == 0
    assert p3.as_leading_term(x) == x
    assert p4.as_leading_term(x) == 1/x

def test_piecewise_simplify():
    p = Piecewise(((x**2 + 1)/x**2, Eq(x*(1 + x) - x**2, 0)),
                  ((-1)**x*(-1), True))
    assert p.simplify() == \
        Piecewise((zoo, Eq(x, 0)), ((-1)**(x + 1), True))
    # simplify when there are Eq in conditions
    assert Piecewise(
        (a, And(Eq(a, 0), Eq(a + b, 0))), (1, True)).simplify(
        ) == Piecewise(
        (0, And(Eq(a, 0), Eq(b, 0))), (1, True))
    assert Piecewise((2*x*factorial(a)/(factorial(y)*factorial(-y + a)),
        Eq(y, 0) & Eq(-y + a, 0)), (2*factorial(a)/(factorial(y)*factorial(-y
        + a)), Eq(y, 0) & Eq(-y + a, 1)), (0, True)).simplify(
        ) == Piecewise(
            (2*x, And(Eq(a, 0), Eq(y, 0))),
            (2, And(Eq(a, 1), Eq(y, 0))),
            (0, True))

def test_piecewise_fold_piecewise_in_cond():
    p1 = Piecewise((cos(x), x < 0), (0, True))
    p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
    p3 = piecewise_fold(p2)
    assert p2.subs(x, -pi / 2) == 0.0
    assert p2.subs(x, 1) == 0.0
    assert p2.subs(x, -pi / 4) == 1.0
    p4 = Piecewise((0, Eq(p1, 0)), (1, True))
    assert piecewise_fold(p4) == Piecewise((0, Or(And(Eq(cos(x), 0), x < 0), Not(x < 0))), (1, True))

    r1 = 1 < Piecewise((1, x < 1), (3, True))
    assert piecewise_fold(r1) == Not(x < 1)

    p5 = Piecewise((1, x < 0), (3, True))
    p6 = Piecewise((1, x < 1), (3, True))
    p7 = piecewise_fold(Piecewise((1, p5 < p6), (0, True)))
    assert Piecewise((1, And(Not(x < 1), x < 0)), (0, True))

def test_piecewise_integrate1ca():
    y = symbols('y', real=True)
    g = Piecewise(
        (1 - x, Interval(0, 1).contains(x)),
        (1 + x, Interval(-1, 0).contains(x)),
        (0, True)
        )
    gy1 = g.integrate((x, y, 1))
    g1y = g.integrate((x, 1, y))

    assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
    assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
    assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
    assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
    # XXX Make test pass without simplify
    assert g.integrate((x, 2, 1)) == gy1.subs(y, 2).simplify()
    assert g.integrate((x, 1, 2)) == g1y.subs(y, 2).simplify()

    assert piecewise_fold(gy1.rewrite(Piecewise)) == \
        Piecewise(
            (1, y <= -1),
            (-y**2/2 - y + S(1)/2, y <= 0),
            (y**2/2 - y + S(1)/2, y < 1),
            (0, True))
    assert piecewise_fold(g1y.rewrite(Piecewise)) == \
        Piecewise(
            (-1, y <= -1),
            (y**2/2 + y - S(1)/2, y <= 0),
            (-y**2/2 + y - S(1)/2, y < 1),
            (0, True))

    # g1y and gy1 should simplify if the condition that y < 1
    # is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
    # XXX Make test pass without simplify
    assert gy1.simplify() == Piecewise(
        (
            -Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
            Min(1, Max(0, y))**2 + S(1)/2, y < 1),
        (0, True)
        )
    assert g1y.simplify() == Piecewise(
        (
            Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
            Min(1, Max(0, y))**2 - S(1)/2, y < 1),
        (0, True))

def test_issue_10137():
    a = Symbol('a', real=True, finite=True)
    b = Symbol('b', real=True, finite=True)
    x = Symbol('x', real=True, finite=True)
    y = Symbol('y', real=True, finite=True)
    p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
    p1 = Piecewise((0, Or(a > x, b < x)), (1, True))
    assert integrate(p0, (x, y, oo)) == integrate(p1, (x, y, oo))
    p3 = Piecewise((1, And(0 < x, x < a)), (0, True))
    p4 = Piecewise((1, And(a > x, x > 0)), (0, True))
    ip3 = integrate(p3, x)
    assert ip3 == Piecewise(
        (0, x <= 0),
        (x, x <= Max(0, a)),
        (Max(0, a), True))
    ip4 = integrate(p4, x)
    assert ip4 == ip3
    assert p3.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
    assert p4.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2

def test_piecewise_complex():
    p1 = Piecewise((2, x < 0), (1, 0 <= x))
    p2 = Piecewise((2*I, x < 0), (I, 0 <= x))
    p3 = Piecewise((I*x, x > 1), (1 + I, True))
    p4 = Piecewise((-I*conjugate(x), x > 1), (1 - I, True))

    assert conjugate(p1) == p1
    assert conjugate(p2) == piecewise_fold(-p2)
    assert conjugate(p3) == p4

    assert p1.is_imaginary is False
    assert p1.is_real is True
    assert p2.is_imaginary is True
    assert p2.is_real is False
    assert p3.is_imaginary is None
    assert p3.is_real is None

    assert p1.as_real_imag() == (p1, 0)
    assert p2.as_real_imag() == (0, -I*p2)

def test_piecewise_integrate1cb():
    y = symbols('y', real=True)
    g = Piecewise(
        (0, Or(x <= -1, x >= 1)),
        (1 - x, x > 0),
        (1 + x, True)
        )
    gy1 = g.integrate((x, y, 1))
    g1y = g.integrate((x, 1, y))

    assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
    assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
    assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
    assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
    assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
    assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)

    assert piecewise_fold(gy1.rewrite(Piecewise)) == \
        Piecewise(
            (1, y <= -1),
            (-y**2/2 - y + S(1)/2, y <= 0),
            (y**2/2 - y + S(1)/2, y < 1),
            (0, True))
    assert piecewise_fold(g1y.rewrite(Piecewise)) == \
        Piecewise(
            (-1, y <= -1),
            (y**2/2 + y - S(1)/2, y <= 0),
            (-y**2/2 + y - S(1)/2, y < 1),
            (0, True))

    # g1y and gy1 should simplify if the condition that y < 1
    # is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
    assert gy1 == Piecewise(
        (
            -Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
            Min(1, Max(0, y))**2 + S(1)/2, y < 1),
        (0, True)
        )
    assert g1y == Piecewise(
        (
            Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
            Min(1, Max(0, y))**2 - S(1)/2, y < 1),
        (0, True))

    def point_cflexure(self):
        """
        Returns a Set of point(s) with zero bending moment and
        where bending moment curve of the beam object changes
        its sign from negative to positive or vice versa.
        Examples
        ========
        There is is 10 meter long overhanging beam. There are
        two simple supports below the beam. One at the start
        and another one at a distance of 6 meters from the start.
        Point loads of magnitude 10KN and 20KN are applied at
        2 meters and 4 meters from start respectively. A Uniformly
        distribute load of magnitude of magnitude 3KN/m is also
        applied on top starting from 6 meters away from starting
        point till end.
        Using the sign convention of upward forces and clockwise moment
        being positive.
        >>> from sympy.physics.continuum_mechanics.beam import Beam
        >>> from sympy import symbols
        >>> E, I = symbols('E, I')
        >>> b = Beam(10, E, I)
        >>> b.apply_load(-4, 0, -1)
        >>> b.apply_load(-46, 6, -1)
        >>> b.apply_load(10, 2, -1)
        >>> b.apply_load(20, 4, -1)
        >>> b.apply_load(3, 6, 0)
        >>> b.point_cflexure()
        [10/3]
        """
        from sympy import solve, Piecewise

        # To restrict the range within length of the Beam
        moment_curve = Piecewise((float("nan"), self.variable<=0),
                (self.bending_moment(), self.variable<self.length),
                (float("nan"), True))

        points = solve(moment_curve.rewrite(Piecewise), self.variable,
                        domain=S.Reals)
        return points

def test_issue_8458():
    x, y = symbols('x y')
    # Original issue
    p1 = Piecewise((0, Eq(x, 0)), (sin(x), True))
    assert p1.simplify() == sin(x)
    # Slightly larger variant
    p2 = Piecewise((x, Eq(x, 0)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
    assert p2.simplify() == sin(x)
    # Test for problem highlighted during review
    p3 = Piecewise((x+1, Eq(x, -1)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
    assert p3.simplify() == Piecewise((0, Eq(x, -1)), (sin(x), True))

def test_has_piecewise():
    f = (x * y + 3 / y) ** (3 + 2)
    g = Function("g")
    h = Function("h")
    p = Piecewise((g, x < -1), (1, x <= 1), (f, True))

    assert p.has(x)
    assert p.has(y)
    assert not p.has(z)
    assert p.has(1)
    assert p.has(3)
    assert not p.has(4)
    assert p.has(f)
    assert p.has(g)
    assert not p.has(h)