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,urecipd,ud,ux1,uy1,uX1,uY1,uZ1> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
  mynumerator((ua*ux1^3+uy1^3+1)-(ud*ux1*uy1))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1))
, mynumerator((ud*urecipd)-(1))
])

ua = fastfrac(ua)
urecipd = fastfrac(urecipd)
ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uU = ((ua*uX1*uX1^2))
uV = ((uY1*uY1^2))
uW = ((uZ1*uZ1^2))
uA = (((uU-uV)^2))
uB = (((uU-uW)^2))
uC = (((uV-uW)^2))
uD = ((uA+uC))
uE = ((uA+uB))
uX3 = ((urecipd*(uU+uV+uW)*(uB+uD)))
uY3 = ((fastfrac(2)*uU*uC-uV*(uC-uE)))
uZ3 = ((fastfrac(2)*uV*uB-uU*(uB-uD)))

ux2 = (((ux1-uy1^3*ux1)/(ua*uy1*ux1^3-uy1))).reduce()
uy2 = (((uy1^3-ua*ux1^3)/(ua*uy1*ux1^3-uy1))).reduce()
ux3 = (((ux1-uy1^2*ux2*uy2)/(ua*ux1*uy1*ux2^2-uy2))).reduce()
uy3 = (((uy1*uy2^2-ua*ux1^2*ux2)/(ua*ux1*uy1*ux2^2-uy2))).reduce()

print(isidentity((ua*ux3^3+uy3^3+fastfrac(1))-(ud*ux3*uy3)))
print(isidentity((ux3)-(uX3/uZ3)))
print(isidentity((uy3)-(uY3/uZ3)))