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.<ua,ud,ux1,uy1,uT1,uX1,uY1,uZ1> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
  mynumerator((ua*ux1^2+uy1^2)-(1+ud*ux1^2*uy1^2))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1))
, mynumerator((ux1*uy1)-(uT1/uZ1))
])

ua = fastfrac(ua)
ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uT1 = fastfrac(uT1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uYY = ((uY1^2))
uaXX = ((ua*uX1^2))
uAp = ((uYY+uaXX))
uB = ((fastfrac(2)*(fastfrac(2)*uZ1^2-uAp)))
uxB = ((uaXX*uB))
uyB = ((uYY*uB))
uAA = ((uAp*(uYY-uaXX)))
uF = ((uAA-uyB))
uG = ((uAA+uxB))
uxE = ((uX1*(uyB+uAA)))
uyH = ((uY1*(uxB-uAA)))
uzF = ((uZ1*uF))
uzG = ((uZ1*uG))
uX3 = ((uxE*uzF))
uY3 = ((uyH*uzG))
uZ3 = ((uzF*uzG))
uT3 = ((uxE*uyH))

ux2 = (((ux1*uy1+uy1*ux1)/(fastfrac(1)+ud*ux1*ux1*uy1*uy1))).reduce()
uy2 = (((uy1*uy1-ua*ux1*ux1)/(fastfrac(1)-ud*ux1*ux1*uy1*uy1))).reduce()
ux3 = (((ux1*uy2+uy1*ux2)/(fastfrac(1)+ud*ux1*ux2*uy1*uy2))).reduce()
uy3 = (((uy1*uy2-ua*ux1*ux2)/(fastfrac(1)-ud*ux1*ux2*uy1*uy2))).reduce()

print(isidentity((ua*ux3^2+uy3^2)-(fastfrac(1)+ud*ux3^2*uy3^2)))
print(isidentity((ux3)-(uX3/uZ3)))
print(isidentity((uy3)-(uY3/uZ3)))
print(isidentity((ux3*uy3)-(uT3/uZ3)))