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,ux1,uy1,ux2,uy2> = PolynomialRing(QQ,5,order='invlex')
I = R.ideal([
  mynumerator((uy1^2)-(ux1^3+3*ua*(ux1+1)^2))
, mynumerator((uy2^2)-(ux2^3+3*ua*(ux2+1)^2))
])

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

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

ux3 = (((-ux1^3+(ux2-fastfrac(3)*ua)*ux1^2+(ux2^2+fastfrac(6)*ua*ux2)*ux1+(uy1^2-fastfrac(2)*uy2*uy1+(-ux2^3-fastfrac(3)*ua*ux2^2+uy2^2)))/(ux1^2-fastfrac(2)*ux2*ux1+ux2^2))).reduce()
uy3 = ((((-uy1+fastfrac(2)*uy2)*ux1^3+(-fastfrac(3)*ua*uy1+(-fastfrac(3)*uy2*ux2+fastfrac(3)*ua*uy2))*ux1^2+((fastfrac(3)*ux2^2+fastfrac(6)*ua*ux2)*uy1-fastfrac(6)*ua*uy2*ux2)*ux1+(uy1^3-fastfrac(3)*uy2*uy1^2+(-fastfrac(2)*ux2^3-fastfrac(3)*ua*ux2^2+fastfrac(3)*uy2^2)*uy1+(uy2*ux2^3+fastfrac(3)*ua*uy2*ux2^2-uy2^3)))/(-ux1^3+fastfrac(3)*ux2*ux1^2-fastfrac(3)*ux2^2*ux1+ux2^3))).reduce()
ux4 = ((fastfrac(9)/(fastfrac(4)*uy1^2)*ux1^4+fastfrac(9)/uy1^2*ua*ux1^3+(fastfrac(9)/uy1^2*ua^2+fastfrac(9)/uy1^2*ua)*ux1^2+(fastfrac(18)/uy1^2*ua^2-fastfrac(2))*ux1+(fastfrac(9)/uy1^2*ua^2-fastfrac(3)*ua))).reduce()
uy4 = ((-fastfrac(27)/(fastfrac(8)*uy1^3)*ux1^6-fastfrac(81)/(fastfrac(4)*uy1^3)*ua*ux1^5+(-fastfrac(81)/(fastfrac(2)*uy1^3)*ua^2-fastfrac(81)/(fastfrac(4)*uy1^3)*ua)*ux1^4+(-fastfrac(27)/uy1^3*ua^3-fastfrac(81)/uy1^3*ua^2+fastfrac(9)/(fastfrac(2)*uy1))*ux1^3+(-fastfrac(81)/uy1^3*ua^3-fastfrac(81)/(fastfrac(2)*uy1^3)*ua^2+fastfrac(27)/(fastfrac(2)*uy1)*ua)*ux1^2+(-fastfrac(81)/uy1^3*ua^3+fastfrac(9)/uy1*ua^2+fastfrac(9)/uy1*ua)*ux1+(-fastfrac(27)/uy1^3*ua^3+fastfrac(9)/uy1*ua^2-uy1))).reduce()
a0 = fastfrac((fastfrac(1)))
a1 = fastfrac((fastfrac(0)))
a2 = fastfrac((fastfrac(3)*ua))
a3 = fastfrac((fastfrac(0)))
a4 = fastfrac((fastfrac(6)*ua))
a6 = fastfrac((fastfrac(3)*ua))
wu1 = ((ux1)).reduce().sreduce()
wv1 = ((uy1)).reduce().sreduce()
print(isidentity(a0*(wv1^2)+a1*(wu1*wv1)+a3*wv1-(((wu1+a2)*wu1+a4)*wu1+a6)))
print(isidentity(ux1-(wu1)))
print(isidentity(uy1-(wv1)))
wu2 = ((ux2)).reduce().sreduce()
wv2 = ((uy2)).reduce().sreduce()
wu3 = ((ux3)).reduce().sreduce()
wv3 = ((uy3)).reduce().sreduce()
wu4 = ((ux4)).reduce().sreduce()
wv4 = ((uy4)).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))