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.<ua2,ua6,ux3,uy3,ux2,uy2,uX1,uZ1,uX2,uZ2,uX3,uZ3> = PolynomialRing(GF(2),12,order='invlex')
ux_2 = (ux2)
uy_2 = (ux2+uy2)
ux1 = (((uy3+uy_2)/(ux3+ux_2))^2+((uy3+uy_2)/(ux3+ux_2))+ux3+ux_2+ua2)
uy1 = (((uy3+uy_2)/(ux3+ux_2))^3+(ux_2+ua2+1)*((uy3+uy_2)/(ux3+ux_2))+ux3+ux_2+ua2+uy3)
I = R.ideal([
  mynumerator((uy1^2+ux1*uy1)-(ux1^3+ua2*ux1^2+ua6))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy2^2+ux2*uy2)-(ux2^3+ua2*ux2^2+ua6))
, mynumerator((ux2)-(uX2/uZ2))
, mynumerator((uy3^2+ux3*uy3)-(ux3^3+ua2*ux3^2+ua6))
, mynumerator((ux3)-(uX3/uZ3))
])

ua2 = fastfrac(ua2)
ua6 = fastfrac(ua6)
ux3 = fastfrac(ux3)
uy3 = fastfrac(uy3)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)
uX1 = fastfrac(uX1)
uZ1 = fastfrac(uZ1)
uX2 = fastfrac(uX2)
uZ2 = fastfrac(uZ2)
uX3 = fastfrac(uX3)
uZ3 = fastfrac(uZ3)
ux_2 = fastfrac(ux_2)
uy_2 = fastfrac(uy_2)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)


uX4 = ((uX2^4+ua6*uZ2^4))
uZ4 = (((uX2*uZ2)^2))
uX5 = ((uZ1*(uX2^2*uX3^2+ua6*uZ2^2*uZ3^2)))
uZ5 = ((uX1*((uX2+uZ2)*(uX3+uZ3)-uX2*uX3-uZ2*uZ3)^2))

ux4 = (((ux2+uy2/ux2)^2+(ux2+uy2/ux2)+ua2)).reduce()
ux5 = ((((uy3+uy2)/(ux3+ux2))^2+((uy3+uy2)/(ux3+ux2))+ux3+ux2+ua2)).reduce()
uy4 = (((ux2+uy2/ux2)^3+(ux2+ua2+fastfrac(1))*(ux2+uy2/ux2)+ua2+uy2)).reduce()
uy5 = ((((uy3+uy2)/(ux3+ux2))^3+(ux2+ua2+fastfrac(1))*((uy3+uy2)/(ux3+ux2))+ux3+ux2+ua2+uy3)).reduce()

print(isidentity((uy5^2+ux5*uy5)-(ux5^3+ua2*ux5^2+ua6)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((uy5^2+ux5*uy5)-(ux5^3+ua2*ux5^2+ua6))))
print(isidentity((ux5)-(uX5/uZ5)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ux5)-(uX5/uZ5))))
print(isidentity((uy5^2+ux5*uy5)-(ux5^3+ua2*ux5^2+ua6)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((uy5^2+ux5*uy5)-(ux5^3+ua2*ux5^2+ua6))))
print(isidentity((ux5)-(uX5/uZ5)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ux5)-(uX5/uZ5))))