def test_f5_reduce():
# katsura3 with lex
R, x, y, z = ring("x,y,z", QQ, lex)
F = [
(((0, 0, 0), 1), x + 2 * y + 2 * z - 1, 1),
(((0, 0, 0), 2), 6 * y ** 2 + 8 * y * z - 2 * y + 6 * z ** 2 - 2 * z, 2),
(
((0, 0, 0), 3),
QQ(10, 3) * y * z - QQ(1, 3) * y + 4 * z ** 2 - QQ(4, 3) * z,
3,
),
(((0, 0, 1), 2), y + 30 * z ** 3 - QQ(79, 7) * z ** 2 + QQ(3, 7) * z, 4),
(
((0, 0, 2), 2),
z ** 4 - QQ(10, 21) * z ** 3 + QQ(1, 84) * z ** 2 + QQ(1, 84) * z,
5,
),
]
cp = critical_pair(F[0], F[1], R)
s = s_poly(cp)
assert f5_reduce(s, F) == (((0, 2, 0), 1), R.zero, 1)
s = lbp(sig(Sign(s)[0], 100), Polyn(s), Num(s))
assert f5_reduce(s, F) == s
def test_f5_reduce():
# katsura3 with lex
F = [(((0, 0, 0), 1), [((1, 0, 0), QQ(1,1)), ((0, 1, 0), QQ(2,1)), ((0, 0, 1), QQ(2,1)), ((0, 0, 0), QQ(-1,1))], 1), (((0, 0, 0), 2), [((0, 2, 0), QQ(6,1)), ((0, 1, 1), QQ(8,1)), ((0, 1, 0), QQ(-2,1)), ((0, 0, 2), QQ(6,1)), ((0, 0, 1), QQ(-2,1))], 2), (((0, 0, 0), 3), [((0, 1, 1), QQ(10,3)), ((0, 1, 0), QQ(-1,3)), ((0, 0, 2), QQ(4,1)), ((0, 0, 1), QQ(-4,3))], 3), (((0, 0, 1), 2), [((0, 1, 0), QQ(1,1)), ((0, 0, 3), QQ(30,1)), ((0, 0, 2), QQ(-79,7)), ((0, 0, 1), QQ(3,7))], 4), (((0, 0, 2), 2), [((0, 0, 4), QQ(1,1)), ((0, 0, 3), QQ(-10,21)), ((0, 0, 2), QQ(1,84)), ((0, 0, 1), QQ(1,84))], 5)]
cp = critical_pair(F[0], F[1], 2, lex, QQ)
s = s_poly(cp, 2, lex, QQ)
assert f5_reduce(s, F, 2, lex, QQ) == (((0, 2, 0), 1), [], 1)
s = lbp(sig(Sign(s)[0], 100), Polyn(s), Num(s))
assert f5_reduce(s, F, 2, lex, QQ) == s