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.<uc,ud,ur,us,ux3,uy3,ux2,uy2,uY1,uZ1,uY2,uZ2,uY3,uZ3> = PolynomialRing(QQ,14,order='invlex')
ux_2 = (-ux2)
uy_2 = (uy2)
ux1 = ((ux3*uy_2+uy3*ux_2)/(uc*(1+ud*ux3*ux_2*uy3*uy_2)))
uy1 = ((uy3*uy_2-ux3*ux_2)/(uc*(1-ud*ux3*ux_2*uy3*uy_2)))
I = R.ideal([
  mynumerator((ux1^2+uy1^2)-(uc^2*(1+ud*ux1^2*uy1^2)))
, mynumerator((ur*uy1)-(uY1/uZ1))
, mynumerator((ux2^2+uy2^2)-(uc^2*(1+ud*ux2^2*uy2^2)))
, mynumerator((ur*uy2)-(uY2/uZ2))
, mynumerator((ux3^2+uy3^2)-(uc^2*(1+ud*ux3^2*uy3^2)))
, mynumerator((ur*uy3)-(uY3/uZ3))
, mynumerator((uc)-(1))
, mynumerator((ud)-(ur^2))
, mynumerator((us)-((1+ur)/(1-ur)))
, mynumerator((uZ1)-(1))
])

uc = fastfrac(uc)
ud = fastfrac(ud)
ur = fastfrac(ur)
us = fastfrac(us)
ux3 = fastfrac(ux3)
uy3 = fastfrac(uy3)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)
uY2 = fastfrac(uY2)
uZ2 = fastfrac(uZ2)
uY3 = fastfrac(uY3)
uZ3 = fastfrac(uZ3)
ux_2 = fastfrac(ux_2)
uy_2 = fastfrac(uy_2)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)


uYY2 = ((uY2^2))
uZZ2 = ((ur*uZ2^2))
uA = ((uZZ2-uYY2))
uB = ((uZZ2+uYY2))
uYY3 = ((uY3^2))
uZZ3 = ((ur*uZ3^2))
uV2 = ((us*uA^2))
uW2 = ((uB^2))
uY4 = ((uW2-uV2))
uZ4 = ((uW2+uV2))
uV = ((us*uA*(uZZ3-uYY3)))
uW = ((uB*(uZZ3+uYY3)))
uY5 = ((ur*(uW-uV)))
uZ5 = ((uY1*(uW+uV)))

ux4 = (((ux2*uy2+uy2*ux2)/(uc*(fastfrac(1)+ud*ux2*ux2*uy2*uy2)))).reduce()
ux5 = (((ux3*uy2+uy3*ux2)/(uc*(fastfrac(1)+ud*ux3*ux2*uy3*uy2)))).reduce()
uy4 = (((uy2*uy2-ux2*ux2)/(uc*(fastfrac(1)-ud*ux2*ux2*uy2*uy2)))).reduce()
uy5 = (((uy3*uy2-ux3*ux2)/(uc*(fastfrac(1)-ud*ux3*ux2*uy3*uy2)))).reduce()

print(isidentity((ux5^2+uy5^2)-(uc^2*(fastfrac(1)+ud*ux5^2*uy5^2))) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ux5^2+uy5^2)-(uc^2*(fastfrac(1)+ud*ux5^2*uy5^2)))))
print(isidentity((ur*uy5)-(uY5/uZ5)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ur*uy5)-(uY5/uZ5))))
print(isidentity((ux5^2+uy5^2)-(uc^2*(fastfrac(1)+ud*ux5^2*uy5^2))) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ux5^2+uy5^2)-(uc^2*(fastfrac(1)+ud*ux5^2*uy5^2)))))
print(isidentity((ur*uy5)-(uY5/uZ5)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ur*uy5)-(uY5/uZ5))))