def test_diopcoverage():
eq = (2*x + y + 1)**2
assert diop_solve(eq) == {(t_0, -2*t_0 - 1)}
eq = 2*x**2 + 6*x*y + 12*x + 4*y**2 + 18*y + 18
assert diop_solve(eq) == {(t_0, -t_0 - 3), (2*t_0 - 3, -t_0)}
assert diop_quadratic(x + y**2 - 3) == {(-t**2 + 3, -t)}
assert diop_quadratic(x + y) is None # wrong type
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
assert diop_linear(x**2 - 1) is None # wrong type
assert base_solution_linear(0, 1, 2, t=None) == (0, 0)
ans = (3*t - 1, -2*t + 1)
assert base_solution_linear(4, 8, 12, t) == ans
assert base_solution_linear(4, 8, 12, t=None) == tuple(_.subs({t: 0}) for _ in ans)
assert cornacchia(1, 1, 20) is None
assert cornacchia(1, 1, 5) == {(1, 2)}
assert cornacchia(1, 2, 17) == {(3, 2)}
assert cornacchia(2, 3, 31) == set()
pytest.raises(ValueError, lambda: reconstruct(4, 20, 1))
assert gaussian_reduce(4, 1, 3) == (1, 1)
eq = -w**2 - x**2 - y**2 + z**2
assert (diop_general_pythagorean(eq) == diop_general_pythagorean(-eq) ==
(m1**2 + m2**2 - m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2))
assert check_param(Integer(3) + x/3, Integer(4) + x/2,
Integer(2), x) == (None, None)
assert check_param(Rational(3, 2), Integer(4) + x,
Integer(2), x) == (None, None)
assert check_param(Integer(4) + x, Rational(3, 2),
Integer(2), x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) is True
assert _odd(0) == (not _even(0)) is False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
pytest.raises(TypeError, lambda: _remove_gcd((2, 4, 6)))
assert sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11) == (11, 1, 5)
# it's ok if these pass some day when the solvers are implemented
pytest.raises(NotImplementedError, lambda: diophantine(x**2 + y**2 + x*y + 2*y*z - 12))
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
# issue sympy/sympy#11026
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**3 - 2))
assert transformation_to_DN(x + y) is None # wrong type
assert find_DN(x + y) is None # wrong type
assert diop_ternary_quadratic(x + y) is None # wrong type
assert transformation_to_normal(x + y) is None # wrong type
assert parametrize_ternary_quadratic(x + y) is None # wrong type
assert diop_general_pythagorean(x + y) is None # wrong type
assert diop_general_sum_of_squares(x + y) is None # wrong type
assert diop_general_sum_of_even_powers(x + y) is None # wrong type
def test_quadratic_elliptical_case():
# Elliptical case: B**2 - 4AC < 0
# Two test cases highlighted require lot of memory due to quadratic_congruence() method.
# This above method should be replaced by Pernici's square_mod() method when his PR gets merged.
# assert diop_solve(42*x**2 + 8*x*y + 15*y**2 + 23*x + 17*y - 4915) == {(-11, -1)}
assert diop_solve(4 * x**2 + 3 * y**2 + 5 * x - 11 * y + 12) == set()
assert diop_solve(x**2 + y**2 + 2 * x + 2 * y + 2) == {(-1, -1)}
# assert diop_solve(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) == {(-15, 6)}
assert diop_solve(10*x**2 + 12*x*y + 12*y**2 - 34) == \
{(-1, -1), (-1, 2), (1, -2), (1, 1)}
def test_quadratic_elliptical_case():
# Elliptical case: B**2 - 4AC < 0
# Two test cases highlighted require lot of memory due to quadratic_congruence() method.
# This above method should be replaced by Pernici's square_mod() method when his PR gets merged.
# assert diop_solve(42*x**2 + 8*x*y + 15*y**2 + 23*x + 17*y - 4915) == {(-11, -1)}
assert diop_solve(4*x**2 + 3*y**2 + 5*x - 11*y + 12) == set()
assert diop_solve(x**2 + y**2 + 2*x + 2*y + 2) == {(-1, -1)}
# assert diop_solve(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) == {(-15, 6)}
assert diop_solve(10*x**2 + 12*x*y + 12*y**2 - 34) == \
{(1, -2), (-1, -1), (1, 1), (-1, 2)}
def test_diop_ternary_quadratic():
# Commented out test cases should be uncommented after
# the bug with factor_list() gets merged.
assert check_solutions(2*x**2 + z**2 + y**2 - 4*x*y)
assert check_solutions(x**2 - y**2 - z**2 - x*y - y*z)
assert check_solutions(3*x**2 - x*y - y*z - x*z)
assert check_solutions(x**2 - y*z - x*z)
# assert check_solutions(5*x**2 - 3*x*y - x*z)
assert check_solutions(4*x**2 - 5*y**2 - x*z)
assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z)
assert check_solutions(8*x**2 - 12*y*z)
assert check_solutions(45*x**2 - 7*y**2 - 8*x*y - z**2)
assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y - 8*x*y)
assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 17*y*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 16*y*z + 12*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z)
assert check_solutions(x*y - 7*y*z + 13*x*z)
assert diop_ternary_quadratic_normal(x**2 + y**2 + z**2) == (None, None, None)
assert diop_ternary_quadratic_normal(x**2 + y**2) is None
pytest.raises(ValueError,
lambda: _diop_ternary_quadratic_normal((x, y, z),
{x*y: 1, x**2: 2,
y**2: 3, z**2: 0}))
eq = -2*x*y - 6*x*z + 7*y**2 - 3*y*z + 4*z**2
assert diop_ternary_quadratic(eq) == (7, 2, 0)
assert diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2) == (1, 0, 2)
assert diop_ternary_quadratic(x*y + 2*y*z) == (-2, 0, n1)
eq = -5*x*y - 8*x*z - 3*y*z + 8*z**2
assert parametrize_ternary_quadratic(eq) == (64*p**2 - 24*p*q, -64*p*q + 64*q**2, 40*p*q)
# this cannot be tested with diophantine because it will
# factor into a product
assert diop_solve(x*y + 2*y*z) == (-4*p*q, -2*n1*p**2 + 2*p**2, 2*p*q)
def test_diop_sum_of_even_powers():
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123
ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used
assert len(ans) == 14
eq = x**4 + y**4 + z**4 - 2673
assert diop_solve(eq) == {(3, 6, 6), (2, 4, 7)}
assert diop_general_sum_of_even_powers(eq, 2) == {(3, 6, 6), (2, 4, 7)}
pytest.raises(NotImplementedError, lambda: diop_general_sum_of_even_powers(-eq, 2))
neg = symbols('neg', negative=True)
eq = x**4 + y**4 + neg**4 - 2673
assert diop_general_sum_of_even_powers(eq) == {(-3, 6, 6)}
assert diophantine(x**4 + y**4 + 2) == set()
assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set()
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3*x*y + 34*x - 12*y + 1) == {(-133, -11), (5, -57)}
assert diop_solve(6*x*y + 2*x + 3*y + 1) == set()
assert diop_solve(-13*x*y + 2*x - 4*y - 54) == {(27, 0)}
assert diop_solve(-27*x*y - 30*x - 12*y - 54) == {(-14, -1)}
assert diop_solve(2*x*y + 5*x + 56*y + 7) == {(-161, -3),
(-47, -6), (-35, -12), (-29, -69),
(-27, 64), (-21, 7), (-9, 1),
(105, -2)}
assert diop_solve(6*x*y + 9*x + 2*y + 3) == set()
assert diop_solve(x*y + x + y + 1) == {(-1, t), (t, -1)}
assert diophantine(48*x*y)
def test_diop_sum_of_even_powers():
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123
ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used
assert len(ans) == 14
eq = x**4 + y**4 + z**4 - 2673
assert diop_solve(eq) == {(3, 6, 6), (2, 4, 7)}
assert diop_general_sum_of_even_powers(eq, 2) == {(3, 6, 6), (2, 4, 7)}
pytest.raises(NotImplementedError,
lambda: diop_general_sum_of_even_powers(-eq, 2))
neg = symbols('neg', negative=True)
eq = x**4 + y**4 + neg**4 - 2673
assert diop_general_sum_of_even_powers(eq) == {(-3, 6, 6)}
assert diophantine(x**4 + y**4 + 2) == set()
assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set()
def test_diop_ternary_quadratic():
# Commented out test cases should be uncommented after
# the bug with factor_list() gets merged.
assert check_solutions(2 * x**2 + z**2 + y**2 - 4 * x * y)
assert check_solutions(x**2 - y**2 - z**2 - x * y - y * z)
assert check_solutions(3 * x**2 - x * y - y * z - x * z)
assert check_solutions(x**2 - y * z - x * z)
# assert check_solutions(5*x**2 - 3*x*y - x*z)
assert check_solutions(4 * x**2 - 5 * y**2 - x * z)
assert check_solutions(3 * x**2 + 2 * y**2 - z**2 - 2 * x * y + 5 * y * z -
7 * y * z)
assert check_solutions(8 * x**2 - 12 * y * z)
assert check_solutions(45 * x**2 - 7 * y**2 - 8 * x * y - z**2)
assert check_solutions(x**2 - 49 * y**2 - z**2 + 13 * z * y - 8 * x * y)
assert check_solutions(90 * x**2 + 3 * y**2 + 5 * x * y + 2 * z * y +
5 * x * z)
assert check_solutions(x**2 + 3 * y**2 + z**2 - x * y - 17 * y * z)
assert check_solutions(x**2 + 3 * y**2 + z**2 - x * y - 16 * y * z +
12 * x * z)
assert check_solutions(x**2 + 3 * y**2 + z**2 - 13 * x * y - 16 * y * z +
12 * x * z)
assert check_solutions(x * y - 7 * y * z + 13 * x * z)
assert diop_ternary_quadratic_normal(x**2 + y**2 + z**2) == (None, None,
None)
assert diop_ternary_quadratic_normal(x**2 + y**2) is None
pytest.raises(
ValueError, lambda: _diop_ternary_quadratic_normal((x, y, z), {
x * y: 1,
x**2: 2,
y**2: 3,
z**2: 0
}))
eq = -2 * x * y - 6 * x * z + 7 * y**2 - 3 * y * z + 4 * z**2
assert diop_ternary_quadratic(eq) == (7, 2, 0)
assert diop_ternary_quadratic_normal(4 * x**2 + 5 * y**2 - z**2) == (1, 0,
2)
assert diop_ternary_quadratic(x * y + 2 * y * z) == (-2, 0, n1)
eq = -5 * x * y - 8 * x * z - 3 * y * z + 8 * z**2
assert parametrize_ternary_quadratic(eq) == (64 * p**2 - 24 * p * q,
-64 * p * q + 64 * q**2,
40 * p * q)
# this cannot be tested with diophantine because it will
# factor into a product
assert diop_solve(x * y + 2 * y * z) == (-4 * p * q,
-2 * n1 * p**2 + 2 * p**2,
2 * p * q)
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3 * x * y + 34 * x - 12 * y + 1) == {(-133, -11),
(5, -57)}
assert diop_solve(6 * x * y + 2 * x + 3 * y + 1) == set()
assert diop_solve(-13 * x * y + 2 * x - 4 * y - 54) == {(27, 0)}
assert diop_solve(-27 * x * y - 30 * x - 12 * y - 54) == {(-14, -1)}
assert diop_solve(2 * x * y + 5 * x + 56 * y + 7) == {
(-161, -3), (-47, -6), (-35, -12), (-29, -69), (-27, 64), (-21, 7),
(-9, 1), (105, -2)
}
assert diop_solve(6 * x * y + 9 * x + 2 * y + 3) == set()
assert diop_solve(x * y + x + y + 1) == {(-1, t), (t, -1)}
assert diophantine(48 * x * y)
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3*x*y + 34*x - 12*y + 1) == \
{(-Integer(133), -Integer(11)), (Integer(5), -Integer(57))}
assert diop_solve(6*x*y + 2*x + 3*y + 1) == set()
assert diop_solve(-13*x*y + 2*x - 4*y - 54) == {(Integer(27), Integer(0))}
assert diop_solve(-27*x*y - 30*x - 12*y - 54) == {(-Integer(14), -Integer(1))}
assert diop_solve(2*x*y + 5*x + 56*y + 7) == {(-Integer(161), -Integer(3)),
(-Integer(47), -Integer(6)), (-Integer(35), -Integer(12)), (-Integer(29), -Integer(69)),
(-Integer(27), Integer(64)), (-Integer(21), Integer(7)), (-Integer(9), Integer(1)),
(Integer(105), -Integer(2))}
assert diop_solve(6*x*y + 9*x + 2*y + 3) == set()
assert diop_solve(x*y + x + y + 1) == {(-Integer(1), t), (t, -Integer(1))}
assert diophantine(48*x*y)
def test_diopcoverage():
eq = (2 * x + y + 1)**2
assert diop_solve(eq) == {(t_0, -2 * t_0 - 1)}
eq = 2 * x**2 + 6 * x * y + 12 * x + 4 * y**2 + 18 * y + 18
assert diop_solve(eq) == {(t_0, -t_0 - 3), (2 * t_0 - 3, -t_0)}
assert diop_quadratic(x + y**2 - 3) == {(-t**2 + 3, -t)}
assert diop_quadratic(x + y) is None # wrong type
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
assert diop_linear(x**2 - 1) is None # wrong type
assert base_solution_linear(0, 1, 2, t=None) == (0, 0)
ans = (3 * t - 1, -2 * t + 1)
assert base_solution_linear(4, 8, 12, t) == ans
assert base_solution_linear(4, 8, 12,
t=None) == tuple(_.subs({t: 0}) for _ in ans)
assert cornacchia(1, 1, 20) is None
assert cornacchia(1, 1, 5) == {(2, 1)}
assert cornacchia(1, 2, 17) == {(3, 2)}
assert cornacchia(2, 3, 31) == set()
assert cornacchia(1, 4, 52) == {(4, 3)}
pytest.raises(ValueError, lambda: reconstruct(4, 20, 1))
assert gaussian_reduce(4, 1, 3) == (1, 1)
eq = -w**2 - x**2 - y**2 + z**2
assert (diop_general_pythagorean(eq) == diop_general_pythagorean(-eq) ==
(m1**2 + m2**2 - m3**2, 2 * m1 * m3, 2 * m2 * m3,
m1**2 + m2**2 + m3**2))
assert check_param(Integer(3) + x / 3,
Integer(4) + x / 2, Integer(2), x) == (None, None)
assert check_param(Rational(3, 2),
Integer(4) + x, Integer(2), x) == (None, None)
assert check_param(Integer(4) + x, Rational(3, 2), Integer(2),
x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) is True
assert _odd(0) == (not _even(0)) is False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
assert sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11) == (11, 1, 5)
# it's ok if these pass some day when the solvers are implemented
pytest.raises(NotImplementedError,
lambda: diophantine(x**2 + y**2 + x * y + 2 * y * z - 12))
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
# issue sympy/sympy#11026
pytest.raises(NotImplementedError, lambda: diophantine(x**3 + y**3 - 2))
assert transformation_to_DN(x + y) is None # wrong type
assert find_DN(x + y) is None # wrong type
assert diop_ternary_quadratic(x + y) is None # wrong type
assert transformation_to_normal(x + y) is None # wrong type
assert parametrize_ternary_quadratic(x + y) is None # wrong type
assert diop_general_pythagorean(x + y) is None # wrong type
assert diop_general_sum_of_squares(x + y) is None # wrong type
assert diop_general_sum_of_even_powers(x + y) is None # wrong type
assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \
{(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)}
def test_univariate():
assert diop_solve((x - 1) * (x - 2)**2) == {(1, ), (2, )}
assert diop_solve((x - 1) * (x - 2)) == {(1, ), (2, )}
assert diop_solve(2 * a**2 + a - 1) == {(-1, )}
def test_univariate():
assert diop_solve((x - 1)*(x - 2)**2) == {(Integer(1),), (Integer(2),)}
assert diop_solve((x - 1)*(x - 2)) == {(Integer(1),), (Integer(2),)}
def test_linear():
assert diop_solve(x) == (0,)
assert diop_solve(1*x) == (0,)
assert diop_solve(3*x) == (0,)
assert diop_solve(x + 1) == (-1,)
assert diop_solve(2*x + 1) == (None,)
assert diop_solve(2*x + 4) == (-2,)
assert diop_solve(y + x) == (t_0, -t_0)
assert diop_solve(y + x + 0) == (t_0, -t_0)
assert diop_solve(y + x - 0) == (t_0, -t_0)
assert diop_solve(0*x - y - 5) == (-5,)
assert diop_solve(3*y + 2*x - 5) == (3*t_0 - 5, -2*t_0 + 5)
assert diop_solve(2*x - 3*y - 5) == (3*t_0 - 5, 2*t_0 - 5)
assert diop_solve(-2*x - 3*y - 5) == (3*t_0 + 5, -2*t_0 - 5)
assert diop_solve(7*x + 5*y) == (5*t_0, -7*t_0)
assert diop_solve(2*x + 4*y) == (2*t_0, -t_0)
assert diop_solve(4*x + 6*y - 4) == (3*t_0 - 2, -2*t_0 + 2)
assert diop_solve(4*x + 6*y - 3) == (None, None)
assert diop_solve(0*x + 3*y - 4*z + 5) == (4*t_0 + 5, 3*t_0 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5) == (t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5, None) == (0, 5, 5)
assert diop_solve(4*x + 2*y + 8*z - 5) == (None, None, None)
assert diop_solve(5*x + 7*y - 2*z - 6) == (t_0, -3*t_0 + 2*t_1 + 6, -8*t_0 + 7*t_1 + 18)
assert diop_solve(3*x - 6*y + 12*z - 9) == (2*t_0 + 3, t_0 + 2*t_1, t_1)
assert diop_solve(6*w + 9*x + 20*y - z) == (t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 20*t_2)
# to ignore constant factors, use diophantine
pytest.raises(TypeError, lambda: diop_solve(x/2))
def test_univariate():
assert diop_solve((x - 1)*(x - 2)**2) == {(1,), (2,)}
assert diop_solve((x - 1)*(x - 2)) == {(1,), (2,)}
assert diop_solve(2*a**2 + a - 1) == {(-1,)}
def test_linear():
assert diop_solve(x) == (0, )
assert diop_solve(1 * x) == (0, )
assert diop_solve(3 * x) == (0, )
assert diop_solve(x + 1) == (-1, )
assert diop_solve(2 * x + 1) == (None, )
assert diop_solve(2 * x + 4) == (-2, )
assert diop_solve(y + x) == (t_0, -t_0)
assert diop_solve(y + x + 0) == (t_0, -t_0)
assert diop_solve(y + x - 0) == (t_0, -t_0)
assert diop_solve(0 * x - y - 5) == (-5, )
assert diop_solve(3 * y + 2 * x - 5) == (3 * t_0 - 5, -2 * t_0 + 5)
assert diop_solve(2 * x - 3 * y - 5) == (3 * t_0 - 5, 2 * t_0 - 5)
assert diop_solve(-2 * x - 3 * y - 5) == (3 * t_0 + 5, -2 * t_0 - 5)
assert diop_solve(7 * x + 5 * y) == (5 * t_0, -7 * t_0)
assert diop_solve(2 * x + 4 * y) == (2 * t_0, -t_0)
assert diop_solve(4 * x + 6 * y - 4) == (3 * t_0 - 2, -2 * t_0 + 2)
assert diop_solve(4 * x + 6 * y - 3) == (None, None)
assert diop_solve(0 * x + 3 * y - 4 * z + 5) == (4 * t_0 + 5, 3 * t_0 + 5)
assert diop_solve(4 * x + 3 * y - 4 * z + 5) == (t_0,
8 * t_0 + 4 * t_1 + 5,
7 * t_0 + 3 * t_1 + 5)
assert diop_solve(4 * x + 3 * y - 4 * z + 5, None) == (0, 5, 5)
assert diop_solve(4 * x + 2 * y + 8 * z - 5) == (None, None, None)
assert diop_solve(5 * x + 7 * y - 2 * z - 6) == (t_0,
-3 * t_0 + 2 * t_1 + 6,
-8 * t_0 + 7 * t_1 + 18)
assert diop_solve(3 * x - 6 * y + 12 * z - 9) == (2 * t_0 + 3,
t_0 + 2 * t_1, t_1)
assert diop_solve(6 * w + 9 * x + 20 * y -
z) == (t_0, t_1, t_1 + t_2,
6 * t_0 + 29 * t_1 + 20 * t_2)
# to ignore constant factors, use diophantine
pytest.raises(TypeError, lambda: diop_solve(x / 2))
def test_linear():
assert diop_solve(x) == (0,)
assert diop_solve(1*x) == (0,)
assert diop_solve(3*x) == (0,)
assert diop_solve(x + 1) == (-1,)
assert diop_solve(2*x + 1) == (None,)
assert diop_solve(2*x + 4) == (-2,)
assert diop_solve(y + x) == (t_0, -t_0)
assert diop_solve(y + x + 0) == (t_0, -t_0)
assert diop_solve(y + x - 0) == (t_0, -t_0)
assert diop_solve(0*x - y - 5) == (-5,)
assert diop_solve(3*y + 2*x - 5) == (3*t_0 - 5, -2*t_0 + 5)
assert diop_solve(2*x - 3*y - 5) == (-3*t_0 - 5, -2*t_0 - 5)
assert diop_solve(-2*x - 3*y - 5) == (-3*t_0 + 5, 2*t_0 - 5)
assert diop_solve(7*x + 5*y) == (5*t_0, -7*t_0)
assert diop_solve(2*x + 4*y) == (2*t_0, -t_0)
assert diop_solve(4*x + 6*y - 4) == (3*t_0 - 2, -2*t_0 + 2)
assert diop_solve(4*x + 6*y - 3) == (None, None)
assert diop_solve(0*x + 3*y - 4*z + 5) == (-4*t_0 + 5, -3*t_0 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5) == (t_0, -4*t_1 + 5, t_0 - 3*t_1 + 5)
assert diop_solve(4*x + 2*y + 8*z - 5) == (None, None, None)
assert diop_solve(5*x + 7*y - 2*z - 6) == (t_0, -7*t_0 - 2*t_1 + 6, -22*t_0 - 7*t_1 + 18)
assert diop_solve(3*x - 6*y + 12*z - 9) == (2*t_0 + 3, t_0 + 2*t_1, t_1)
assert diop_solve(6*w + 9*x + 20*y - z) == (t_0, t_1, -t_1 - t_2, 6*t_0 - 11*t_1 - 20*t_2)
