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
  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.<uminustwo,ui,ua,u2d,ud,ux1,uy1,uX1,uY1,uZ1> = PolynomialRing(QQ,10,order='invlex')
I = R.ideal([
  mynumerator((ua*ux1^3+uy1^3+1)-(ud*ux1*uy1))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1))
, mynumerator((ui^2)-(-1))
, mynumerator((uminustwo)-(-2))
, mynumerator((u2d)-(2*ud))
])

uminustwo = fastfrac(uminustwo)
ui = fastfrac(ui)
ua = fastfrac(ua)
u2d = fastfrac(u2d)
ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uiZ = ((ui*uZ1))
uA = (((uY1-uiZ)*(uY1+uiZ)))
uB = ((uY1*uZ1))
uC = (((uA-uB)*(uY1+uZ1)))
uD = (((uA+uB)*(uZ1-uY1)))
uE = ((fastfrac(3)*uC-u2d*uX1*uB))
uX3 = ((uminustwo*uX1*uD))
uY3 = (((uD-uE)*uZ1))
uZ3 = (((uD+uE)*uY1))

ux3 = (((ux1-uy1^3*ux1)/(ua*uy1*ux1^3-uy1))).reduce()
uy3 = (((uy1^3-ua*ux1^3)/(ua*uy1*ux1^3-uy1))).reduce()

print isidentity((ua*ux3^3+uy3^3+fastfrac(1))-(ud*ux3*uy3))
print isidentity((ux3)-(uX3/uZ3))
print isidentity((uy3)-(uY3/uZ3))