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.<ud1,ud2,ux2,uy2,ux1,uy1,uX1,uY1,uZ1,uX2,uY2,uZ2> = PolynomialRing(GF(2),12,order='invlex')
I = R.ideal([
  mynumerator((ud1*(ux1+uy1)+ud2*(ux1^2+uy1^2))-((ux1+ux1^2)*(uy1+uy1^2)))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1))
, mynumerator((ud1*(ux2+uy2)+ud2*(ux2^2+uy2^2))-((ux2+ux2^2)*(uy2+uy2^2)))
, mynumerator((ux2)-(uX2/uZ2))
, mynumerator((uy2)-(uY2/uZ2))
])

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

ud1 = fastfrac(ud1)
ud2 = fastfrac(ud2)
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)


uW1 = ((uX1+uY1))
uW2 = ((uX2+uY2))
uA = ((uX1*(uX1+uZ1)))
uB = ((uY1*(uY1+uZ1)))
uC = ((uZ1*uZ2))
uD = ((uW2*uZ2))
uE = ((ud1*uC^2))
uH = (((ud1*uZ2+ud2*uW2)*uW1*uC))
uI = ((ud1*uC*uZ1))
uU = ((uE+uA*uD))
uV = ((uE+uB*uD))
uS = ((uU*uV))
uX3 = ((uS*uY1+(uH+uX2*(uI+uA*(uY2+uZ2)))*uV*uZ1))
uY3 = ((uS*uX1+(uH+uY2*(uI+uB*(uX2+uZ2)))*uU*uZ1))
uZ3 = ((uS*uZ1))

ux3 = (((ud1*(ux1+ux2)+ud2*(ux1+uy1)*(ux2+uy2)+(ux1+ux1^2)*(ux2*(uy1+uy2+fastfrac(1))+uy1*uy2))/(ud1+(ux1+ux1^2)*(ux2+uy2)))).reduce()
uy3 = (((ud1*(uy1+uy2)+ud2*(ux1+uy1)*(ux2+uy2)+(uy1+uy1^2)*(uy2*(ux1+ux2+fastfrac(1))+ux1*ux2))/(ud1+(uy1+uy1^2)*(ux2+uy2)))).reduce()

print(isidentity((ud1*(ux3+uy3)+ud2*(ux3^2+uy3^2))-((ux3+ux3^2)*(uy3+uy3^2))))
print(isidentity((ux3)-(uX3/uZ3)))
print(isidentity((uy3)-(uY3/uZ3)))

unified = True
uX4 = uX3
uY4 = uY3
uZ4 = uZ3
ux4 = (((ud1*(ux1+ux1)+ud2*(ux1+uy1)*(ux1+uy1)+(ux1+ux1^2)*(ux1*(uy1+uy1+fastfrac(1))+uy1*uy1))/(ud1+(ux1+ux1^2)*(ux1+uy1)))).reduce()
uy4 = (((ud1*(uy1+uy1)+ud2*(ux1+uy1)*(ux1+uy1)+(uy1+uy1^2)*(uy1*(ux1+ux1+fastfrac(1))+ux1*ux1))/(ud1+(uy1+uy1^2)*(ux1+uy1)))).reduce()
if unified: unified = isdoublingidentity((ud1*(ux4+uy4)+ud2*(ux4^2+uy4^2))-((ux4+ux4^2)*(uy4+uy4^2)))
if unified: unified = isdoublingidentity((ux4)-(uX4/uZ4))
if unified: unified = isdoublingidentity((uy4)-(uY4/uZ4))
if unified: print("Unified")