from dedekint import REAL
# mpq for rational numbers
from gmpy2 import mpq

# Theorem 3.1 (4)
# Die Menge {q ∈ Q | q < x} ist eine entscheidbare Menge rationaler Zahlen (Dedekind’scher Schnitt)

def sqrt_two(q):
    if q <= 0 or q * q < 2: return True
    return False


x = REAL(sqrt_two)

print("1.41422 is smaller:", x.smaller(mpq(141422, 100000)))
print("1.41421 is smaller:", x.smaller(mpq(141421, 100000)))

k = 0
n = 1
z = 0

for k in range(100):
    if x.smaller(mpq(z, n)):
        z = z + 1
    else:
        print(z, "/", n)
        z = (z - 1) * 10
        n = n * 10