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.<ub2,ua,ub4,ub,ux1,uy1,uX1,uZ1> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
  mynumerator((uy1^2)-(ux1^3+ua*ux1+ub))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((ub2)-(2*ub))
, mynumerator((ub4)-(4*ub))
])

ub2 = fastfrac(ub2)
ua = fastfrac(ua)
ub4 = fastfrac(ub4)
ub = fastfrac(ub)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uZ1 = fastfrac(uZ1)


uXX = ((uX1^2))
uZZ = ((uZ1^2))
uA = ((fastfrac(2)*((uX1+uZ1)^2-uXX-uZZ)))
uaZZ = ((ua*uZZ))
uX3 = (((uXX-uaZZ)^2-ub2*uA*uZZ))
uZ3 = ((uA*(uXX+uaZZ)+ub4*uZZ^2))

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

print(isidentity((uy3^2)-(ux3^3+ua*ux3+ub)))
print(isidentity((ux3)-(uX3/uZ3)))