def mynumerator(x):
  if parent(x) == R:
    return x
  return numerator(x)

class fastfrac:
  def __init__(self,top,bot=1):
    if parent(top) == ZZ or parent(top) == R:
      self.top = R(top)
      self.bot = R(bot)
    elif top.__class__ == fastfrac:
      self.top = top.top
      self.bot = top.bot * bot
    else:
      self.top = R(numerator(top))
      self.bot = R(denominator(top)) * bot
  def reduce(self):
    return fastfrac(self.top / self.bot)
  def sreduce(self):
    return fastfrac(I.reduce(self.top),I.reduce(self.bot))
  def iszero(self):
    return self.top in I and not (self.bot in I)
  def isdoublingzero(self):
    return self.top in J and not (self.bot in J)
  def __add__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top + self.bot * other,self.bot)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.bot + self.bot * other.top,self.bot * other.bot)
    return NotImplemented
  def __sub__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top - self.bot * other,self.bot)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.bot - self.bot * other.top,self.bot * other.bot)
    return NotImplemented
  def __neg__(self):
    return fastfrac(-self.top,self.bot)
  def __mul__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top * other,self.bot)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.top,self.bot * other.bot)
    return NotImplemented
  def __rmul__(self,other):
    return self.__mul__(other)
  def __div__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top,self.bot * other)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.bot,self.bot * other.top)
    return NotImplemented
  __truediv__ = __div__
  def __pow__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top ^ other,self.bot ^ other)
    return NotImplemented

def isidentity(x):
  return x.iszero()

def isdoublingidentity(x):
  return x.isdoublingzero()

R.<ua,uc1,ud1,us1,uD1,uS1,uZ1,uDZ1,uSC1,uC1> = PolynomialRing(QQ,10,order='invlex')
I = R.ideal([
  mynumerator((us1^2+uc1^2)-(1))
, mynumerator((ua*us1^2+ud1^2)-(1))
, mynumerator((us1)-(uS1/uZ1))
, mynumerator((uc1)-(uC1/uZ1))
, mynumerator((ud1)-(uD1/uZ1))
, mynumerator((uSC1)-(uS1*uC1))
, mynumerator((uDZ1)-(uD1*uZ1))
])

ua = fastfrac(ua)
uc1 = fastfrac(uc1)
ud1 = fastfrac(ud1)
us1 = fastfrac(us1)
uD1 = fastfrac(uD1)
uS1 = fastfrac(uS1)
uZ1 = fastfrac(uZ1)
uDZ1 = fastfrac(uDZ1)
uSC1 = fastfrac(uSC1)
uC1 = fastfrac(uC1)


ua0 = ((uS1))
ua1 = ((uC1))
ua2 = ((uD1))
ua3 = ((uZ1))
ul1 = ((ua3*ua1))
um = ((ul1^2))
ul2 = ((ua0*ua2))
un = ((ul2^2))
ul3 = ((fastfrac(2)*(ua1*ua2)^2))
ur3 = ((um+un))
ur0 = (((ul1+ul2)^2-ur3))
ur1 = ((um-un))
ur2 = ((ul3-ur1))
uS3 = ((ur0))
uC3 = ((ur1))
uD3 = ((ur2))
uZ3 = ((ur3))
uSC3 = ((uS3*uC3))
uDZ3 = ((uD3*uZ3))

uc3 = (((uc1*uc1-ud1*us1*us1*ud1)/(uc1^2+(ud1*us1)^2))).reduce()
ud3 = (((ud1*ud1-ua*us1*uc1*us1*uc1)/(uc1^2+(ud1*us1)^2))).reduce()
us3 = (((uc1*us1*ud1+ud1*us1*uc1)/(uc1^2+(ud1*us1)^2))).reduce()

print(isidentity((us3^2+uc3^2)-(fastfrac(1))))
print(isidentity((ua*us3^2+ud3^2)-(fastfrac(1))))
print(isidentity((us3)-(uS3/uZ3)))
print(isidentity((uc3)-(uC3/uZ3)))
print(isidentity((ud3)-(uD3/uZ3)))
print(isidentity((uSC3)-(uS3*uC3)))
print(isidentity((uDZ3)-(uD3*uZ3)))