def test_string_ordinals():
assert str(omega) == 'w'
assert str(Ordinal(OmegaPower(5, 3), OmegaPower(3,
2))) == 'w**5*3 + w**3*2'
assert str(Ordinal(OmegaPower(5, 3), OmegaPower(0, 5))) == 'w**5*3 + 5'
assert str(Ordinal(OmegaPower(1, 3), OmegaPower(0, 5))) == 'w*3 + 5'
assert str(Ordinal(OmegaPower(omega + 1, 1),
OmegaPower(3, 2))) == 'w**(w + 1) + w**3*2'
def test_addition_with_ordinals():
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + Ordinal(OmegaPower(3, 3)) == \
Ordinal(OmegaPower(5, 3), OmegaPower(3, 5))
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + Ordinal(OmegaPower(4, 2)) == \
Ordinal(OmegaPower(5, 3), OmegaPower(4, 2))
assert Ordinal(OmegaPower(omega, 2), OmegaPower(3, 2)) + Ordinal(OmegaPower(4, 2)) == \
Ordinal(OmegaPower(omega, 2), OmegaPower(4, 2))
def test_exponentiation():
w = omega
assert w**2 == w * w
assert w**3 == w * w * w
assert w**(w + 1) == Ordinal(OmegaPower(omega + 1, 1))
assert (w**w) * (w**w) == w**(w * 2)
def test_multiplication_with_integers():
w = omega
assert 3 * w == w
assert w * 9 == Ordinal(OmegaPower(1, 9))
def test_comparison():
assert Ordinal(OmegaPower(5, 3)) > Ordinal(OmegaPower(4, 3),
OmegaPower(2, 1))
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) < Ordinal(
OmegaPower(5, 4))
assert Ordinal(OmegaPower(5, 4)) < Ordinal(OmegaPower(5, 5),
OmegaPower(4, 1))
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) == \
Ordinal(OmegaPower(5, 3), OmegaPower(3, 2))
assert not Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) == Ordinal(
OmegaPower(5, 3))
assert Ordinal(OmegaPower(omega, 3)) > Ordinal(OmegaPower(5, 3))
def test_addition_with_integers():
assert 3 + Ordinal(OmegaPower(5, 3)) == Ordinal(OmegaPower(5, 3))
assert Ordinal(OmegaPower(5, 3)) + 3 == Ordinal(OmegaPower(5, 3),
OmegaPower(0, 3))
assert Ordinal(OmegaPower(5, 3), OmegaPower(0, 2))+3 == \
Ordinal(OmegaPower(5, 3), OmegaPower(0, 5))
def test_is_successort():
w = Ordinal(OmegaPower(5, 1))
assert not w.is_successor_ordinal
