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.<ua2,ua6,ux2,uy2,ux1,uy1,uX1,uY1,uZ1,uX2,uY2,uZ2> = PolynomialRing(GF(2),12,order='invlex')
I = R.ideal([
  mynumerator((uy1^2+ux1*uy1)-(ux1^3+ua2*ux1^2+ua6))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1^2))
, mynumerator((uy2^2+ux2*uy2)-(ux2^3+ua2*ux2^2+ua6))
, mynumerator((ux2)-(uX2/uZ2))
, mynumerator((uy2)-(uY2/uZ2^2))
, mynumerator((ua2)-(1))
])

J = I + R.ideal([0
, uX1-uX2
, uY1-uY2
, uZ1-uZ2
])

ua2 = fastfrac(ua2)
ua6 = fastfrac(ua6)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)
uX2 = fastfrac(uX2)
uY2 = fastfrac(uY2)
uZ2 = fastfrac(uZ2)


uA = ((uX1*uZ2))
uB = ((uX2*uZ1))
uC = ((uA^2))
uD = ((uB^2))
uE = ((uA+uB))
uF = ((uC+uD))
uG = ((uY1*uZ2^2))
uH = ((uY2*uZ1^2))
uI = ((uG+uH))
uJ = ((uI*uE))
uZ3 = ((uF*uZ1*uZ2))
uX3 = ((uA*(uH+uD)+uB*(uC+uG)))
uY3 = (((uA*uJ+uF*uG)*uF+(uJ+uZ3)*uX3))

ux3 = ((((uy1+uy2)/(ux1+ux2))^2+((uy1+uy2)/(ux1+ux2))+ux1+ux2+ua2)).reduce()
uy3 = ((((uy1+uy2)/(ux1+ux2))^3+(ux2+ua2+fastfrac(1))*((uy1+uy2)/(ux1+ux2))+ux1+ux2+ua2+uy1)).reduce()

print(isidentity((uy3^2+ux3*uy3)-(ux3^3+ua2*ux3^2+ua6)))
print(isidentity((ux3)-(uX3/uZ3)))
print(isidentity((uy3)-(uY3/uZ3^2)))

unified = True
uX4 = uX3
uY4 = uY3
uZ4 = uZ3
ux4 = (((ux1+uy1/ux1)^2+(ux1+uy1/ux1)+ua2)).reduce()
uy4 = (((ux1+uy1/ux1)^3+(ux1+ua2+fastfrac(1))*(ux1+uy1/ux1)+ua2+uy1)).reduce()
if unified: unified = isdoublingidentity((uy4^2+ux4*uy4)-(ux4^3+ua2*ux4^2+ua6))
if unified: unified = isdoublingidentity((ux4)-(uX4/uZ4))
if unified: unified = isdoublingidentity((uy4)-(uY4/uZ4^2))
if unified: print("Unified")