Python rings.ring函数代码示例

def test_PolyElement_tail_degree():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R(0).tail_degree() == -oo
    assert R(1).tail_degree() == 0
    assert (x + 1).tail_degree() == 0
    assert (2*y**3 + x**3*z).tail_degree() == 0
    assert (x*y**3 + x**3*z).tail_degree() == 1
    assert (x**5*y**3 + x**3*z).tail_degree() == 3

    assert R(0).tail_degree(x) == -oo
    assert R(1).tail_degree(x) == 0
    assert (x + 1).tail_degree(x) == 0
    assert (2*y**3 + x**3*z).tail_degree(x) == 0
    assert (x*y**3 + x**3*z).tail_degree(x) == 1
    assert (7*x**5*y**3 + x**3*z).tail_degree(x) == 3

    assert R(0).tail_degree(y) == -oo
    assert R(1).tail_degree(y) == 0
    assert (x + 1).tail_degree(y) == 0
    assert (2*y**3 + x**3*z).tail_degree(y) == 0
    assert (x*y**3 + x**3*z).tail_degree(y) == 0
    assert (7*x**5*y**3 + x**3*z).tail_degree(y) == 0

    assert R(0).tail_degree(z) == -oo
    assert R(1).tail_degree(z) == 0
    assert (x + 1).tail_degree(z) == 0
    assert (2*y**3 + x**3*z).tail_degree(z) == 0
    assert (x*y**3 + x**3*z).tail_degree(z) == 0
    assert (7*x**5*y**3 + x**3*z).tail_degree(z) == 0

    R, = ring("", ZZ)
    assert R(0).tail_degree() == -oo
    assert R(1).tail_degree() == 0

def test_PolyElement_from_expr():
    x, y, z = symbols("x,y,z")
    R, X, Y, Z = ring((x, y, z), ZZ)

    f = R.from_expr(1)
    assert f == 1 and isinstance(f, R.dtype)

    f = R.from_expr(x)
    assert f == X and isinstance(f, R.dtype)

    f = R.from_expr(x*y*z)
    assert f == X*Y*Z and isinstance(f, R.dtype)

    f = R.from_expr(x*y*z + x*y + x)
    assert f == X*Y*Z + X*Y + X and isinstance(f, R.dtype)

    f = R.from_expr(x**3*y*z + x**2*y**7 + 1)
    assert f == X**3*Y*Z + X**2*Y**7 + 1 and isinstance(f, R.dtype)

    raises(ValueError, lambda: R.from_expr(1/x))
    raises(ValueError, lambda: R.from_expr(2**x))
    raises(ValueError, lambda: R.from_expr(7*x + sqrt(2)))

    R, = ring("", ZZ)
    f = R.from_expr(1)
    assert f == 1 and isinstance(f, R.dtype)

def test_tan():
    R, x, y = ring('x, y', QQ)
    assert rs_tan(x, x, 9)/x**5 == \
        S(17)/315*x**2 + S(2)/15 + S(1)/3*x**(-2) + x**(-4)
    assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
        4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
        x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[tan(a), a])
    assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**S(3)/3 +
        2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + S(1)/3)*x**3 + \
        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
    assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
        (tan(a)**4 + S(4)/3*tan(a)**2 + S(1)/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)

    R, x, y = ring('x, y', EX)
    assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
        2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
        EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
    assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
        2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
        EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
        EX(tan(a)**2 + 1)*x + EX(tan(a))

    p = x + x**2 + 5
    assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
        668083460499)

def test_PolyElement_cancel():
    R, x, y = ring("x,y", ZZ)

    f = 2*x**3 + 4*x**2 + 2*x
    g = 3*x**2 + 3*x
    F = 2*x + 2
    G = 3

    assert f.cancel(g) == (F, G)

    assert (-f).cancel(g) == (-F, G)
    assert f.cancel(-g) == (-F, G)

    R, x, y = ring("x,y", QQ)

    f = QQ(1,2)*x**3 + x**2 + QQ(1,2)*x
    g = QQ(1,3)*x**2 + QQ(1,3)*x
    F = 3*x + 3
    G = 2

    assert f.cancel(g) == (F, G)

    assert (-f).cancel(g) == (-F, G)
    assert f.cancel(-g) == (-F, G)

    Fx, x = field("x", ZZ)
    Rt, t = ring("t", Fx)

    f = (-x**2 - 4)/4*t
    g = t**2 + (x**2 + 2)/2

    assert f.cancel(g) == ((-x**2 - 4)*t, 4*t**2 + 2*x**2 + 4)

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 _series_inversion1(p, x, prec):
    """
    univariate series inversion ``1/p`` modulo ``O(x**prec)``

    The Newton method is used.

    Examples
    ========

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.rings import ring
    >>> from sympy.polys.ring_series import _series_inversion1
    >>> R, x = ring('x', QQ)
    >>> p = x + 1
    >>> _series_inversion1(p, x, 4)
    -x**3 + x**2 - x + 1

    """
    ring = p.ring
    zm = ring.zero_monom
    if zm not in p:
        raise ValueError('no constant term in series')
    if _has_constant_term(p - p[zm], x):
        raise ValueError('p cannot contain a constant term depending on parameters')
    if p[zm] != ring(1):
        # TODO add check that it is a unit
        p1 = ring(1)/p[zm]
    else:
        p1 = ring(1)
    for precx in _giant_steps(prec):
        tmp = p1.square()
        tmp = rs_mul(tmp, p, x, precx)
        p1 = 2*p1 - tmp
    return p1

def test_dup_cyclotomic_p():
    R, x = ring("x", ZZ)

    assert R.dup_cyclotomic_p(x - 1) is True
    assert R.dup_cyclotomic_p(x + 1) is True
    assert R.dup_cyclotomic_p(x**2 + x + 1) is True
    assert R.dup_cyclotomic_p(x**2 + 1) is True
    assert R.dup_cyclotomic_p(x**4 + x**3 + x**2 + x + 1) is True
    assert R.dup_cyclotomic_p(x**2 - x + 1) is True
    assert R.dup_cyclotomic_p(x**6 + x**5 + x**4 + x**3 + x**2 + x + 1) is True
    assert R.dup_cyclotomic_p(x**4 + 1) is True
    assert R.dup_cyclotomic_p(x**6 + x**3 + 1) is True

    assert R.dup_cyclotomic_p(0) is False
    assert R.dup_cyclotomic_p(1) is False
    assert R.dup_cyclotomic_p(x) is False
    assert R.dup_cyclotomic_p(x + 2) is False
    assert R.dup_cyclotomic_p(3*x + 1) is False
    assert R.dup_cyclotomic_p(x**2 - 1) is False

    f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1
    assert R.dup_cyclotomic_p(f) is False

    g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1
    assert R.dup_cyclotomic_p(g) is True

    R, x = ring("x", QQ)
    assert R.dup_cyclotomic_p(x**2 + x + 1) is True
    assert R.dup_cyclotomic_p(QQ(1,2)*x**2 + x + 1) is False

    R, x = ring("x", ZZ["y"])
    assert R.dup_cyclotomic_p(x**2 + x + 1) is False

def test_sin():
    R, x, y = ring('x, y', QQ)
    assert rs_sin(x, x, 9) == \
        x - x**3/6 + x**5/120 - x**7/5040
    assert rs_sin(x*y + x**2*y**3, x, 9) == 1/12*x**8*y**11 - \
        1/720*x**8*y**9 + 1/12*x**7*y**9 - 1/5040*x**7*y**7 - 1/6*x**6*y**9 \
        + 1/24*x**6*y**7 - 1/2*x**5*y**7 + 1/120*x**5*y**5 - 1/2*x**4*y**5 \
        - 1/6*x**3*y**3 + 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) == \
        1/40320*x**8 - 1/720*x**6 + 1/24*x**4 - 1/2*x**2 + 1
    assert rs_cos(x*y + x**2*y**3, x, 9) == 1/24*x**8*y**12 - \
        1/48*x**8*y**10 + 1/40320*x**8*y**8 + 1/6*x**7*y**10 - \
        1/120*x**7*y**8 + 1/4*x**6*y**8 - 1/720*x**6*y**6 + 1/6*x**5*y**6 \
        - 1/2*x**4*y**6 + 1/24*x**4*y**4 - x**3*y**4 - 1/2*x**2*y**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_tan():
    R, x, y = ring('x, y', QQ)
    assert rs_tan(x, x, 9) == \
        x + x**3/3 + 2*x**5/15 + 17*x**7/315
    assert rs_tan(x*y + x**2*y**3, x, 9) == 4/3*x**8*y**11 + 17/45*x**8*y**9 + \
        4/3*x**7*y**9 + 17/315*x**7*y**7 + 1/3*x**6*y**9 + 2/3*x**6*y**7 + \
        x**5*y**7 + 2/15*x**5*y**5 + x**4*y**5 + 1/3*x**3*y**3 + x**2*y**3 + x*y

    # Constant term in series
    a = symbols('a')
    R, x, y = ring('x, y', QQ[tan(a), a])
    assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 + \
        2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + 1/3)*x**3 + \
        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
    assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
        (tan(a)**4 + 4/3*tan(a)**2 + 1/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)

    R, x, y = ring('x, y', EX)
    assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 + \
        2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
        EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
    assert rs_tan(x + x**2*y + a, x, 4) ==  EX(2*tan(a)**3 + \
        2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
        EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
        EX(tan(a)**2 + 1)*x + EX(tan(a))

    p = x + x**2 + 5
    assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + 67701870330562640/ \
        668083460499)

def test_dmp_primitive():
    R, x,y = ring("x,y", ZZ)

    assert R.dmp_primitive(0) == (0, 0)
    assert R.dmp_primitive(1) == (1, 1)

    f, g, F = 3*y**2 + 2*y + 1, 1, 0

    for i in range(0, 5):
        g *= f
        F += x**i*g

    assert R.dmp_primitive(F) == (f.drop(x), F / f)

    R, x,y,z = ring("x,y,z", ZZ)

    cont, f = R.dmp_primitive(f_4)
    assert cont == 1 and f == f_4
    cont, f = R.dmp_primitive(f_5)
    assert cont == 1 and f == f_5

    R, x,y,z,t = ring("x,y,z,t", ZZ)

    cont, f = R.dmp_primitive(f_6)
    assert cont == 1 and f == f_6

def test_dmp_discriminant():
    R, x = ring("x", ZZ)

    assert R.dmp_discriminant(0) == 0

    R, x, y = ring("x,y", ZZ)

    assert R.dmp_discriminant(0) == 0
    assert R.dmp_discriminant(y) == 0

    assert R.dmp_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664
    assert R.dmp_discriminant(5*x**5 + x**3 + 2) == 31252160
    assert R.dmp_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0
    assert R.dmp_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112

    assert R.dmp_discriminant(x**2*y + 2*y) == (-8*y**2).drop(x)
    assert R.dmp_discriminant(x*y**2 + 2*x) == 1

    R, x, y, z = ring("x,y,z", ZZ)
    assert R.dmp_discriminant(x*y + z) == 1

    R, x, y, z, u = ring("x,y,z,u", ZZ)
    assert R.dmp_discriminant(x**2*y + x*z + u) == (-4*y*u + z**2).drop(x)

    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
    assert R.dmp_discriminant(x**3*y + x**2*z + x*u + v) == \
        (-27*y**2*v**2 + 18*y*z*u*v - 4*y*u**3 - 4*z**3*v + z**2*u**2).drop(x)

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))

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_PolyElement___sub__():
    Rt, t = ring("t", ZZ)
    Ruv, u,v = ring("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Ruv)

    assert dict(x - 3*y) == {(1, 0, 0): 1, (0, 1, 0): -3}

    assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u}
    assert dict(-u + x*y) == dict(x*y - u) == {(1, 1, 0): 1, (0, 0, 0): -u}
    assert dict(-u + x*y + z) == dict(x*y + z - u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): -u}

    assert dict(-u*x + x) == dict(x - u*x) == {(1, 0, 0): -u + 1}
    assert dict(-u*x + x*y) == dict(x*y - u*x) == {(1, 1, 0): 1, (1, 0, 0): -u}
    assert dict(-u*x + x*y + z) == dict(x*y + z - u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): -u}

    raises(TypeError, lambda: t - x)
    raises(TypeError, lambda: x - t)
    raises(TypeError, lambda: t - u)
    raises(TypeError, lambda: u - t)

    Fuv, u,v = field("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Fuv)

    assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u}

    Rxyz, x,y,z = ring("x,y,z", EX)

    assert dict(-EX(pi) + x*y*z) == dict(x*y*z - EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): -EX(pi)}