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.<ua4,ua,ux1,uy1,uX1,uZZ1,uY1,uZ1> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
  mynumerator((uy1^2)-(ux1^3+ua*ux1^2+16*ua*ux1))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZZ1))
, mynumerator((uZZ1)-(uZ1^2))
, mynumerator((ua4)-(4*ua))
])

ua4 = fastfrac(ua4)
ua = fastfrac(ua)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uZZ1 = fastfrac(uZZ1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uA = ((uX1^2))
uU = ((ua4*uZZ1))
uB = ((uA-fastfrac(4)*uU))
uC = ((uA*uU))
uZ3 = (((fastfrac(2)*uY1)^2))
uX3 = ((uB^2))
uY3 = ((uY1*uB*(fastfrac(2)*(uX3+fastfrac(32)*uC)+ua*(uZ3-uC))))
uZZ3 = ((uZ3^2))

ux3 = ((fastfrac(1)/(fastfrac(4)*uy1^2)*ux1^4-fastfrac(8)*ua/uy1^2*ux1^2+fastfrac(64)*ua^2/uy1^2)).reduce()
uy3 = ((fastfrac(1)/(fastfrac(8)*uy1^3)*ux1^6+((-ua^2+fastfrac(40)*ua)/(fastfrac(4)*uy1^3))*ux1^4+((ua*uy1^2+(fastfrac(16)*ua^3-fastfrac(640)*ua^2))/(fastfrac(4)*uy1^3))*ux1^2+((-fastfrac(4)*ua^2*uy1^2-fastfrac(512)*ua^3)/uy1^3))).reduce()

print isidentity((uy3^2)-(ux3^3+ua*ux3^2+fastfrac(16)*ua*ux3))
print isidentity((ux3)-(uX3/uZ3))
print isidentity((uy3)-(uY3/uZZ3))
print isidentity((uZZ3)-(uZ3^2))