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,ux2,uy2> = PolynomialRing(QQ,6,order='invlex')
I = R.ideal([
  mynumerator((ua*ux1^3+uy1^3+1)-(ud*ux1*uy1))
, mynumerator((ua*ux2^3+uy2^3+1)-(ud*ux2*uy2))
])

J = I + R.ideal([0
, ux1-ux2
, uy1-uy2
])

ua = fastfrac(ua)
ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)

ux3 = (((ux1-uy1^2*ux2*uy2)/(ua*ux1*uy1*ux2^2-uy2))).reduce()
uy3 = (((uy1*uy2^2-ua*ux1^2*ux2)/(ua*ux1*uy1*ux2^2-uy2))).reduce()
ux4 = (((ux1-uy1^3*ux1)/(ua*uy1*ux1^3-uy1))).reduce()
uy4 = (((uy1^3-ua*ux1^3)/(ua*uy1*ux1^3-uy1))).reduce()
a0 = fastfrac((fastfrac(1)))
a1 = fastfrac((-fastfrac(3)*(ud/fastfrac(3))/(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))))
a2 = fastfrac((-fastfrac(9)*(ud/fastfrac(3))*(ud/fastfrac(3))/((ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3))))))
a3 = fastfrac((-fastfrac(9)/((ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3))))))
a4 = fastfrac((-fastfrac(27)*(ud/fastfrac(3))/((ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3))))))
a6 = fastfrac((-fastfrac(27)/((ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3))))))
wu1 = (((-fastfrac(3)/(ua-ud*ud*ud/fastfrac(27)))*ux1/(ud*ux1/fastfrac(3)-(-uy1)+fastfrac(1)))).reduce().sreduce()
wv1 = (((-fastfrac(9)/((ua-ud*ud*ud/fastfrac(27))*(ua-ud*ud*ud/fastfrac(27))))*(-uy1)/(ud*ux1/fastfrac(3)-(-uy1)+fastfrac(1)))).reduce().sreduce()
print(isidentity(a0*(wv1^2)+a1*(wu1*wv1)+a3*wv1-(((wu1+a2)*wu1+a4)*wu1+a6)))
print(isidentity(ux1-(fastfrac(3)*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*wu1/((ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*wv1-fastfrac(3)*(ud/fastfrac(3))*wu1*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))-fastfrac(9)))))
print(isidentity(uy1-(-(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*wv1/((ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))*wv1-fastfrac(3)*(ud/fastfrac(3))*wu1*(ua-(ud/fastfrac(3))*(ud/fastfrac(3))*(ud/fastfrac(3)))-fastfrac(9)))))
wu2 = (((-fastfrac(3)/(ua-ud*ud*ud/fastfrac(27)))*ux2/(ud*ux2/fastfrac(3)-(-uy2)+fastfrac(1)))).reduce().sreduce()
wv2 = (((-fastfrac(9)/((ua-ud*ud*ud/fastfrac(27))*(ua-ud*ud*ud/fastfrac(27))))*(-uy2)/(ud*ux2/fastfrac(3)-(-uy2)+fastfrac(1)))).reduce().sreduce()
wu3 = (((-fastfrac(3)/(ua-ud*ud*ud/fastfrac(27)))*ux3/(ud*ux3/fastfrac(3)-(-uy3)+fastfrac(1)))).reduce().sreduce()
wv3 = (((-fastfrac(9)/((ua-ud*ud*ud/fastfrac(27))*(ua-ud*ud*ud/fastfrac(27))))*(-uy3)/(ud*ux3/fastfrac(3)-(-uy3)+fastfrac(1)))).reduce().sreduce()
wu4 = (((-fastfrac(3)/(ua-ud*ud*ud/fastfrac(27)))*ux4/(ud*ux4/fastfrac(3)-(-uy4)+fastfrac(1)))).reduce().sreduce()
wv4 = (((-fastfrac(9)/((ua-ud*ud*ud/fastfrac(27))*(ua-ud*ud*ud/fastfrac(27))))*(-uy4)/(ud*ux4/fastfrac(3)-(-uy4)+fastfrac(1)))).reduce().sreduce()
slope = ((wv2-wv1)/(wu2-wu1)).reduce().sreduce()
print(isidentity(a0*slope^2+a1*slope-a2-wu1-wu2-wu3))
print(isidentity(slope*(wu1-wu3)-wv1-a1*wu3-a3-wv3))
slope = ((fastfrac(3)*wu1^2+fastfrac(2)*a2*wu1+a4-a1*wv1)/(fastfrac(2)*a0*wv1+a1*wu1+a3)).reduce().sreduce()
print(isidentity(a0*slope^2+a1*slope-a2-wu1-wu1-wu4))
print(isidentity(slope*(wu1-wu4)-wv1-a1*wu4-a3-wv4))