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
  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,ud2,ud,ux1,uy1,uX1,uY1,uZ1> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
  mynumerator((ua*ux1^2+uy1^2)-(1+ud*ux1^2*uy1^2))
, mynumerator((ux1)-(uZ1/uX1))
, mynumerator((uy1)-(uZ1/uY1))
, mynumerator((ud2)-(2*ud))
])

ua = fastfrac(ua)
ud2 = fastfrac(ud2)
ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uA = ((uX1^2))
uB = ((uY1^2))
uU = ((ua*uB))
uC = ((uA+uU))
uD = ((uA-uU))
uE = (((uX1+uY1)^2-uA-uB))
uX3 = ((uC*uD))
uY3 = ((uE*(uC-ud2*uZ1^2)))
uZ3 = ((uD*uE))

ux3 = (((ux1*uy1+uy1*ux1)/(fastfrac(1)+ud*ux1*ux1*uy1*uy1))).reduce()
uy3 = (((uy1*uy1-ua*ux1*ux1)/(fastfrac(1)-ud*ux1*ux1*uy1*uy1))).reduce()

print isidentity((ua*ux3^2+uy3^2)-(fastfrac(1)+ud*ux3^2*uy3^2))
print isidentity((ux3)-(uZ3/uX3))
print isidentity((uy3)-(uZ3/uY3))