def test_expand_relational():
    n = symbols('n', negative=True)
    p, q = symbols('p q', positive=True)
    r = ((n + q*(-n/q + 1))/(q*(-n/q + 1)) < 0)
    assert r is not S.false
    assert r.expand() is S.false
    assert (q > 0).expand() is S.true

def test_relational_simplification_numerically():
    def test_simplification_numerically_function(original, simplified):
        symb = original.free_symbols
        n = len(symb)
        valuelist = list(set(list(combinations(list(range(-(n-1), n))*n, n))))
        for values in valuelist:
            sublist = dict(zip(symb, values))
            originalvalue = original.subs(sublist)
            simplifiedvalue = simplified.subs(sublist)
            assert originalvalue == simplifiedvalue, "Original: {}\nand"\
                " simplified: {}\ndo not evaluate to the same value for {}"\
                "".format(original, simplified, sublist)

    w, x, y, z = symbols('w x y z', real=True)
    d, e = symbols('d e', real=False)

    expressions = (And(Eq(x, y), x >= y, w < y, y >= z, z < y),
                   And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
                   Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
                   And(x >= y, Eq(y, x)),
                   Or(And(Eq(x, y), x >= y, w < y, Or(y >= z, z < y)),
                      And(Eq(x, y), x >= 1, 2 < y, y >= -1, z < y)),
                   (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)),
                   )

    for expression in expressions:
        test_simplification_numerically_function(expression,
                                                 expression.simplify())

def test_rewrite():
    x, y, z = symbols('x y z')
    a, b = symbols('a b')
    f1 = sin(x) + cos(x)
    assert f1.rewrite(cos,exp) == exp(I*x)/2 + sin(x) + exp(-I*x)/2
    assert f1.rewrite([cos],sin) == sin(x) + sin(x + pi/2, evaluate=False)
    f2 = sin(x) + cos(y)/gamma(z)
    assert f2.rewrite(sin,exp) == -I*(exp(I*x) - exp(-I*x))/2 + cos(y)/gamma(z)

def test_as_Boolean():
    nz = symbols('nz', nonzero=True)
    assert all(as_Boolean(i) is S.true for i in (True, S.true, 1, nz))
    z = symbols('z', zero=True)
    assert all(as_Boolean(i) is S.false for i in (False, S.false, 0, z))
    assert all(as_Boolean(i) == i for i in (x, x < 0))
    for i in (2, S(2), x + 1, []):
        raises(TypeError, lambda: as_Boolean(i))

def test_relational_simplification():
    w, x, y, z = symbols('w x y z', real=True)
    d, e = symbols('d e', real=False)
    # Test all combinations or sign and order
    assert Or(x >= y, x < y).simplify() == S.true
    assert Or(x >= y, y > x).simplify() == S.true
    assert Or(x >= y, -x > -y).simplify() == S.true
    assert Or(x >= y, -y < -x).simplify() == S.true
    assert Or(-x <= -y, x < y).simplify() == S.true
    assert Or(-x <= -y, -x > -y).simplify() == S.true
    assert Or(-x <= -y, y > x).simplify() == S.true
    assert Or(-x <= -y, -y < -x).simplify() == S.true
    assert Or(y <= x, x < y).simplify() == S.true
    assert Or(y <= x, y > x).simplify() == S.true
    assert Or(y <= x, -x > -y).simplify() == S.true
    assert Or(y <= x, -y < -x).simplify() == S.true
    assert Or(-y >= -x, x < y).simplify() == S.true
    assert Or(-y >= -x, y > x).simplify() == S.true
    assert Or(-y >= -x, -x > -y).simplify() == S.true
    assert Or(-y >= -x, -y < -x).simplify() == S.true

    assert Or(x < y, x >= y).simplify() == S.true
    assert Or(y > x, x >= y).simplify() == S.true
    assert Or(-x > -y, x >= y).simplify() == S.true
    assert Or(-y < -x, x >= y).simplify() == S.true
    assert Or(x < y, -x <= -y).simplify() == S.true
    assert Or(-x > -y, -x <= -y).simplify() == S.true
    assert Or(y > x, -x <= -y).simplify() == S.true
    assert Or(-y < -x, -x <= -y).simplify() == S.true
    assert Or(x < y, y <= x).simplify() == S.true
    assert Or(y > x, y <= x).simplify() == S.true
    assert Or(-x > -y, y <= x).simplify() == S.true
    assert Or(-y < -x, y <= x).simplify() == S.true
    assert Or(x < y, -y >= -x).simplify() == S.true
    assert Or(y > x, -y >= -x).simplify() == S.true
    assert Or(-x > -y, -y >= -x).simplify() == S.true
    assert Or(-y < -x, -y >= -x).simplify() == S.true

    # Some other tests
    assert Or(x >= y, w < z, x <= y).simplify() == S.true
    assert And(x >= y, x < y).simplify() == S.false
    assert Or(x >= y, Eq(y, x)).simplify() == (x >= y)
    assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y)
    assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \
        Or(x >= y, y > Min(w, z))
    assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \
        And(Eq(x, y), y > Max(w, z))
    assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
        (Eq(x, y) | (x >= 1) | (y > Min(2, z)))
    assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
        (Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
    assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \
        (Eq(x, y) & Eq(d, e) & (d >= e))
    assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0))
    assert Xor(x >= y, x <= y).simplify() == Ne(x, y)

def test_replace_map():
    F, G = symbols('F, G', cls=Function)
    K = OperationsOnlyMatrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1) \
                                                                              : G(1)}), (G(2), {F(2): G(2)})])
    M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
    N = M.replace(F, G, True)
    assert N == K

def test_rewrite():
    x, y, z = symbols('x y z')
    a, b = symbols('a b')
    f1 = sin(x) + cos(x)
    assert f1.rewrite(cos,exp) == exp(I*x)/2 + sin(x) + exp(-I*x)/2
    assert f1.rewrite([cos],sin) == sin(x) + sin(x + pi/2, evaluate=False)
    f2 = sin(x) + cos(y)/gamma(z)
    assert f2.rewrite(sin,exp) == -I*(exp(I*x) - exp(-I*x))/2 + cos(y)/gamma(z)
    assert Max(a, b).rewrite(Piecewise) == Piecewise((a, a >= b), (b, True))
    assert Max(x, y, z).rewrite(Piecewise) == Piecewise((x, (x >= y) & (x >= z)), (y, y >= z), (z, True))
    assert Max(x, y, a, b).rewrite(Piecewise) == Piecewise((a, (a >= b) & (a >= x) & (a >= y)),
        (b, (b >= x) & (b >= y)), (x, x >= y), (y, True))
    assert Min(a, b).rewrite(Piecewise) == Piecewise((a, a <= b), (b, True))
    assert Min(x, y, z).rewrite(Piecewise) == Piecewise((x, (x <= y) & (x <= z)), (y, y <= z), (z, True))
    assert Min(x,  y, a, b).rewrite(Piecewise) ==  Piecewise((a, (a <= b) & (a <= x) & (a <= y)),
        (b, (b <= x) & (b <= y)), (x, x <= y), (y, True))

def test_preorder_traversal():
    expr = Basic(b21, b3)
    assert list(preorder_traversal(expr)) == [expr, b21, b2, b1, b1, b3, b2, b1]
    assert list(preorder_traversal(("abc", ("d", "ef")))) == [("abc", ("d", "ef")), "abc", ("d", "ef"), "d", "ef"]

    result = []
    pt = preorder_traversal(expr)
    for i in pt:
        result.append(i)
        if i == b2:
            pt.skip()
    assert result == [expr, b21, b2, b1, b3, b2]

    w, x, y, z = symbols("w:z")
    expr = z + w * (x + y)
    assert list(preorder_traversal([expr], key=default_sort_key)) == [
        [w * (x + y) + z],
        w * (x + y) + z,
        z,
        w * (x + y),
        w,
        x + y,
        x,
        y,
    ]

def test_issue_14700():
    A, B, C, D, E, F, G, H = symbols('A B C D E F G H')
    q = ((B & D & H & ~F) | (B & H & ~C & ~D) | (B & H & ~C & ~F) |
         (B & H & ~D & ~G) | (B & H & ~F & ~G) | (C & G & ~B & ~D) |
         (C & G & ~D & ~H) | (C & G & ~F & ~H) | (D & F & H & ~B) |
         (D & F & ~G & ~H) | (B & D & F & ~C & ~H) | (D & E & F & ~B & ~C) |
         (D & F & ~A & ~B & ~C) | (D & F & ~A & ~C & ~H) |
         (A & B & D & F & ~E & ~H))
    soldnf = ((B & D & H & ~F) | (D & F & H & ~B) | (B & H & ~C & ~D) |
              (B & H & ~D & ~G) | (C & G & ~B & ~D) | (C & G & ~D & ~H) |
              (C & G & ~F & ~H) | (D & F & ~G & ~H) | (D & E & F & ~C & ~H) |
              (D & F & ~A & ~C & ~H) | (A & B & D & F & ~E & ~H))
    solcnf = ((B | C | D) & (B | D | G) & (C | D | H) & (C | F | H) &
              (D | G | H) & (F | G | H) & (B | F | ~D | ~H) &
              (~B | ~D | ~F | ~H) & (D | ~B | ~C | ~G | ~H) &
              (A | H | ~C | ~D | ~F | ~G) & (H | ~C | ~D | ~E | ~F | ~G) &
              (B | E | H | ~A | ~D | ~F | ~G))
    assert simplify_logic(q, "dnf") == soldnf
    assert simplify_logic(q, "cnf") == solcnf

    minterms = [[0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1],
                [0, 0, 1, 1], [1, 0, 1, 1]]
    dontcares = [[1, 0, 0, 0], [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1]]
    assert SOPform([w, x, y, z], minterms) == (x & ~w) | (y & z & ~x)
    # Should not be more complicated with don't cares
    assert SOPform([w, x, y, z], minterms, dontcares) == \
        (x & ~w) | (y & z & ~x)

 def execute(self, grp):
     x = symbols('x')
     x_val = self.x.get_data(Int32).values[0]
     fun = self.fun.get_data(Text).values[0]
     expr = parse_expr(fun)
     res = expr.subs(x, x_val)
     self.out.put_data(Int32(int(res)))

def test_multivariate_bool_as_set():
    x, y = symbols("x,y")

    assert And(x >= 0, y >= 0).as_set() == Interval(0, oo) * Interval(0, oo)
    assert Or(x >= 0, y >= 0).as_set() == S.Reals * S.Reals - Interval(-oo, 0, True, True) * Interval(
        -oo, 0, True, True
    )

    def as_real_imag(self, deep=True, **hints):
        from sympy.core.symbol import symbols
        from sympy.polys.polytools import poly
        from sympy.core.function import expand_multinomial
        if self.exp.is_Integer:
            exp = self.exp
            re, im = self.base.as_real_imag(deep=deep)
            if re.func == C.re or im.func == C.im:
                return self, S.Zero
            a, b = symbols('a b', cls=Dummy)
            if exp >= 0:
                if re.is_Number and im.is_Number:
                    # We can be more efficient in this case
                    expr = expand_multinomial(self.base**exp)
                    return expr.as_real_imag()

                expr = poly((a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp
            else:
                mag = re**2 + im**2
                re, im = re/mag, -im/mag
                if re.is_Number and im.is_Number:
                    # We can be more efficient in this case
                    expr = expand_multinomial((re + im*S.ImaginaryUnit)**-exp)
                    return expr.as_real_imag()

                expr = poly((a + b)**-exp)

            # Terms with even b powers will be real
            r = [i for i in expr.terms() if not i[0][1] % 2]
            re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
            # Terms with odd b powers will be imaginary
            r = [i for i in expr.terms() if i[0][1] % 4 == 1]
            im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
            r = [i for i in expr.terms() if i[0][1] % 4 == 3]
            im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])

            return (re_part.subs({a: re, b: S.ImaginaryUnit*im}),
            im_part1.subs({a: re, b: im}) + im_part3.subs({a: re, b: -im}))

        elif self.exp.is_Rational:
            # NOTE: This is not totally correct since for x**(p/q) with
            #       x being imaginary there are actually q roots, but
            #       only a single one is returned from here.
            re, im = self.base.as_real_imag(deep=deep)
            if re.func == C.re or im.func == C.im:
                return self, S.Zero
            r = Pow(Pow(re, 2) + Pow(im, 2), S.Half)
            t = C.atan2(im, re)

            rp, tp = Pow(r, self.exp), t*self.exp

            return (rp*C.cos(tp), rp*C.sin(tp))
        else:

            if deep:
                hints['complex'] = False
                return (C.re(self.expand(deep, **hints)),
                        C.im(self.expand(deep, **hints)))
            else:
                return (C.re(self), C.im(self))

def test_literal_evalf_is_number_is_zero_is_comparable():
    from sympy.integrals.integrals import Integral
    from sympy.core.symbol import symbols
    from sympy.core.function import Function
    from sympy.functions.elementary.trigonometric import cos, sin
    x = symbols('x')
    f = Function('f')

    # issue 5033
    assert f.is_number is False
    # issue 6646
    assert f(1).is_number is False
    i = Integral(0, (x, x, x))
    # expressions that are symbolically 0 can be difficult to prove
    # so in case there is some easy way to know if something is 0
    # it should appear in the is_zero property for that object;
    # if is_zero is true evalf should always be able to compute that
    # zero
    assert i.n() == 0
    assert i.is_zero
    assert i.is_number is False
    assert i.evalf(2, strict=False) == 0

    # issue 10268
    n = sin(1)**2 + cos(1)**2 - 1
    assert n.is_comparable is False
    assert n.n(2).is_comparable is False
    assert n.n(2).n(2).is_comparable

def test_log():
    R, x = ring('x', QQ)
    p = 1 + x
    p1 = rs_log(p, x, 4)/x**2
    assert p1 == S(1)/3*x - S(1)/2 + x**(-1)
    p = 1 + x +2*x**2/3
    p1 = rs_log(p, x, 9)
    assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \
      7*x**4/36 - x**3/3 + x**2/6 + x
    p2 = rs_series_inversion(p, x, 9)
    p3 = rs_log(p2, x, 9)
    assert p3 == -p1

    R, x, y = ring('x, y', QQ)
    p = 1 + x + 2*y*x**2
    p1 = rs_log(p, x, 6)
    assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y -
                  x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x)

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', EX)
    assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \
        - EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a))
    assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \
        EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \
        EX(1/a)*x + EX(log(a))

    p = x + x**2 + 3
    assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + S(19281291595)/9920232)

def test_is_cnf():
    x, y, z = symbols("x,y,z")
    assert is_cnf(x) is True
    assert is_cnf(x | y | z) is True
    assert is_cnf(x & y & z) is True
    assert is_cnf((x | y) & z) is True
    assert is_cnf((x & y) | z) is False

def test_literal_evalf_is_number_is_zero_is_comparable():
    from sympy.integrals.integrals import Integral
    from sympy.core.symbol import symbols
    from sympy.core.function import Function
    from sympy.functions.elementary.trigonometric import cos, sin
    x = symbols('x')
    f = Function('f')

    # the following should not be changed without a lot of dicussion
    # `foo.is_number` should be equivalent to `not foo.free_symbols`
    # it should not attempt anything fancy; see is_zero, is_constant
    # and equals for more rigorous tests.
    assert f(1).is_number is True
    i = Integral(0, (x, x, x))
    # expressions that are symbolically 0 can be difficult to prove
    # so in case there is some easy way to know if something is 0
    # it should appear in the is_zero property for that object;
    # if is_zero is true evalf should always be able to compute that
    # zero
    assert i.n() == 0
    assert i.is_zero
    assert i.is_number is False
    assert i.evalf(2, strict=False) == 0

    # issue 10268
    n = sin(1)**2 + cos(1)**2 - 1
    assert n.is_comparable is False
    assert n.n(2).is_comparable is False
    assert n.n(2).n(2).is_comparable

def test_is_dnf():
    x, y, z = symbols('x,y,z')
    assert is_dnf(x) is True
    assert is_dnf(x | y | z) is True
    assert is_dnf(x & y & z) is True
    assert is_dnf((x & y) | z) is True
    assert is_dnf((x | y) & z) is False

def test_sin():
    R, x, y = ring('x, y', QQ)
    assert rs_sin(x, x, 9)/x**5 == \
        -S(1)/5040*x**2 + S(1)/120 - S(1)/6*x**(-2) + x**(-4)
    assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \
        x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \
        x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \
        x**3*y**3/6 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
    assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
        sin(a)*x**2/2 + cos(a)*x + sin(a)
    assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
        cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
        cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)

    R, x, y = ring('x, y', EX)
    assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
        EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
    assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
        EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
        EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
        EX(cos(a))*x + EX(sin(a))

def test_cos():
    R, x, y = ring('x, y', QQ)
    assert rs_cos(x, x, 9)/x**5 == \
        S(1)/40320*x**3 - S(1)/720*x + S(1)/24*x**(-1) - S(1)/2*x**(-3) + x**(-5)
    assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \
        x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \
        x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \
        x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
    assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
        cos(a)*x**2/2 - sin(a)*x + cos(a)
    assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
        sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
        sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)

    R, x, y = ring('x, y', EX)
    assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
        EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
    assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
        EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
        EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
        EX(sin(a))*x + EX(cos(a))

def test_exp():
    R, x = ring('x', QQ)
    p = x + x**4
    for h in [10, 30]:
        q = rs_series_inversion(1 + p, x, h) - 1
        p1 = rs_exp(q, x, h)
        q1 = rs_log(p1, x, h)
        assert q1 == q
    p1 = rs_exp(p, x, 30)
    assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
    prec = 21
    p = rs_log(1 + x, x, prec)
    p1 = rs_exp(p, x, prec)
    assert p1 == x + 1

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[exp(a), a])
    assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
        exp(a)*x**2/2 + exp(a)*x + exp(a)
    assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
            exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
            exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)

    R, x, y = ring('x, y', EX)
    assert rs_exp(x + a, x, 5) ==  EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
        EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
    assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
        EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
        EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
        EX(exp(a))*x + EX(exp(a))