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.<ud,ux2,uy2,ux1,uy1,uXX1,uXZ1,uY1,uYZ1,uZZ1,uXY1,uX1,uZ1,uYY1,uXX2,uXZ2,uY2,uYZ2,uZZ2,uXY2,uX2,uZ2,uYY2> = PolynomialRing(QQ,23,order='invlex')
I = R.ideal([
  mynumerator((ux1^3+uy1^3+1)-(3*ud*ux1*uy1))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1))
, mynumerator((uXX1)-(uX1*uX1))
, mynumerator((uYY1)-(uY1*uY1))
, mynumerator((uZZ1)-(uZ1*uZ1))
, mynumerator((uXY1)-(2*uX1*uY1))
, mynumerator((uXZ1)-(2*uX1*uZ1))
, mynumerator((uYZ1)-(2*uY1*uZ1))
, mynumerator((ux2^3+uy2^3+1)-(3*ud*ux2*uy2))
, mynumerator((ux2)-(uX2/uZ2))
, mynumerator((uy2)-(uY2/uZ2))
, mynumerator((uXX2)-(uX2*uX2))
, mynumerator((uYY2)-(uY2*uY2))
, mynumerator((uZZ2)-(uZ2*uZ2))
, mynumerator((uXY2)-(2*uX2*uY2))
, mynumerator((uXZ2)-(2*uX2*uZ2))
, mynumerator((uYZ2)-(2*uY2*uZ2))
])

J = I + R.ideal([0
, uXX1-uXX2
, uXY1-uXY2
, uXZ1-uXZ2
, uZZ1-uZZ2
, uYY1-uYY2
, uYZ1-uYZ2
, uX1-uX2
, uY1-uY2
, uZ1-uZ2
])

ud = fastfrac(ud)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uXX1 = fastfrac(uXX1)
uXZ1 = fastfrac(uXZ1)
uY1 = fastfrac(uY1)
uYZ1 = fastfrac(uYZ1)
uZZ1 = fastfrac(uZZ1)
uXY1 = fastfrac(uXY1)
uX1 = fastfrac(uX1)
uZ1 = fastfrac(uZ1)
uYY1 = fastfrac(uYY1)
uXX2 = fastfrac(uXX2)
uXZ2 = fastfrac(uXZ2)
uY2 = fastfrac(uY2)
uYZ2 = fastfrac(uYZ2)
uZZ2 = fastfrac(uZZ2)
uXY2 = fastfrac(uXY2)
uX2 = fastfrac(uX2)
uZ2 = fastfrac(uZ2)
uYY2 = fastfrac(uYY2)


uX3 = ((uYY1*uXZ2-uXZ1*uYY2))
uY3 = ((uXX1*uYZ2-uYZ1*uXX2))
uZ3 = ((uZZ1*uXY2-uXY1*uZZ2))
uXX3 = ((uX3^2))
uYY3 = ((uY3^2))
uZZ3 = ((uZ3^2))
uXY3 = (((uX3+uY3)^2-uXX3-uYY3))
uXZ3 = (((uX3+uZ3)^2-uXX3-uZZ3))
uYZ3 = (((uY3+uZ3)^2-uYY3-uZZ3))

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

print(isidentity((ux3^3+uy3^3+fastfrac(1))-(fastfrac(3)*ud*ux3*uy3)))
print(isidentity((ux3)-(uX3/uZ3)))
print(isidentity((uy3)-(uY3/uZ3)))
print(isidentity((uXX3)-(uX3*uX3)))
print(isidentity((uYY3)-(uY3*uY3)))
print(isidentity((uZZ3)-(uZ3*uZ3)))
print(isidentity((uXY3)-(fastfrac(2)*uX3*uY3)))
print(isidentity((uXZ3)-(fastfrac(2)*uX3*uZ3)))
print(isidentity((uYZ3)-(fastfrac(2)*uY3*uZ3)))

unified = True
uXX4 = uXX3
uXY4 = uXY3
uXZ4 = uXZ3
uZZ4 = uZZ3
uYY4 = uYY3
uYZ4 = uYZ3
uX4 = uX3
uY4 = uY3
uZ4 = uZ3
ux4 = ((uy1*(fastfrac(1)-ux1^3)/(ux1^3-uy1^3))).reduce()
uy4 = ((ux1*(uy1^3-fastfrac(1))/(ux1^3-uy1^3))).reduce()
if unified: unified = isdoublingidentity((ux4^3+uy4^3+fastfrac(1))-(fastfrac(3)*ud*ux4*uy4))
if unified: unified = isdoublingidentity((ux4)-(uX4/uZ4))
if unified: unified = isdoublingidentity((uy4)-(uY4/uZ4))
if unified: unified = isdoublingidentity((uXX4)-(uX4*uX4))
if unified: unified = isdoublingidentity((uYY4)-(uY4*uY4))
if unified: unified = isdoublingidentity((uZZ4)-(uZ4*uZ4))
if unified: unified = isdoublingidentity((uXY4)-(fastfrac(2)*uX4*uY4))
if unified: unified = isdoublingidentity((uXZ4)-(fastfrac(2)*uX4*uZ4))
if unified: unified = isdoublingidentity((uYZ4)-(fastfrac(2)*uY4*uZ4))
if unified: print("Unified")