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,uc,ux1,uy1,uR1,uXX1,uX1,uZZ1,uY1,uZ1> = PolynomialRing(QQ,10,order='invlex')
I = R.ideal([
  mynumerator((uy1^2)-(ux1^4+2*ua*ux1^2+1))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZZ1))
, mynumerator((uXX1)-(uX1^2))
, mynumerator((uZZ1)-(uZ1^2))
, mynumerator((uR1)-(2*uX1*uZ1))
, mynumerator((ua^2+uc^2)-(1))
])

ua = fastfrac(ua)
uc = fastfrac(uc)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uR1 = fastfrac(uR1)
uXX1 = fastfrac(uXX1)
uX1 = fastfrac(uX1)
uZZ1 = fastfrac(uZZ1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uA = ((uXX1^2))
uB = ((uZZ1^2))
uC = ((uA+uB))
uD = ((fastfrac(2)*((uXX1+uZZ1)^2-uC)))
uE = ((uA-uB))
uF = ((fastfrac(2)*uA))
uG = ((fastfrac(2)*uB))
uJ = ((ua*uD+fastfrac(2)*uC))
uK = ((uJ+uE))
uL = ((uJ-uE))
uM = ((uC*uE))
uN = ((uG*uK))
uP = ((uF*uL))
uX3 = ((uX1*(uM-uN)))
uY3 = ((uY1*((uM+uN)*(uP-uM)+(uD*uE)^2)))
uZ3 = ((uZ1*(uP+uM)))
uXX3 = ((uX3^2))
uZZ3 = ((uZ3^2))
uR3 = (((uX3+uZ3)^2-uXX3-uZZ3))

ux2 = (((ux1*uy1+uy1*ux1)/(fastfrac(1)-(ux1*ux1)^2))).reduce()
uy2 = ((((fastfrac(1)+(ux1*ux1)^2)*(uy1*uy1+fastfrac(2)*ua*ux1*ux1)+fastfrac(2)*ux1*ux1*(ux1^2+ux1^2))/(fastfrac(1)-(ux1*ux1)^2)^2)).reduce()
ux3 = (((ux1*uy2+uy1*ux2)/(fastfrac(1)-(ux1*ux2)^2))).reduce()
uy3 = ((((fastfrac(1)+(ux1*ux2)^2)*(uy1*uy2+fastfrac(2)*ua*ux1*ux2)+fastfrac(2)*ux1*ux2*(ux1^2+ux2^2))/(fastfrac(1)-(ux1*ux2)^2)^2)).reduce()

print isidentity((uy3^2)-(ux3^4+fastfrac(2)*ua*ux3^2+fastfrac(1)))
print isidentity((ux3)-(uX3/uZ3))
print isidentity((uy3)-(uY3/uZZ3))
print isidentity((uXX3)-(uX3^2))
print isidentity((uZZ3)-(uZ3^2))
print isidentity((uR3)-(fastfrac(2)*uX3*uZ3))