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.<ud1,uee,ud2,uff,ue,uf,ux3,uy3,ux2,uy2,uW1,uZ1,uW2,uZ2,uW3,uZ3> = PolynomialRing(GF(2),16,order='invlex')
ux_2 = (uy2)
uy_2 = (ux2)
ux1 = ((ud1*(ux3+ux_2)+ud2*(ux3+uy3)*(ux_2+uy_2)+(ux3+ux3^2)*(ux_2*(uy3+uy_2+1)+uy3*uy_2))/(ud1+(ux3+ux3^2)*(ux_2+uy_2)))
uy1 = ((ud1*(uy3+uy_2)+ud2*(ux3+uy3)*(ux_2+uy_2)+(uy3+uy3^2)*(uy_2*(ux3+ux_2+1)+ux3*ux_2))/(ud1+(uy3+uy3^2)*(ux_2+uy_2)))
I = R.ideal([
  mynumerator((ud1*(ux1+uy1)+ud2*(ux1^2+uy1^2))-((ux1+ux1^2)*(uy1+uy1^2)))
, mynumerator((ux1+uy1)-(uW1/uZ1))
, mynumerator((ud1*(ux2+uy2)+ud2*(ux2^2+uy2^2))-((ux2+ux2^2)*(uy2+uy2^2)))
, mynumerator((ux2+uy2)-(uW2/uZ2))
, mynumerator((ud1*(ux3+uy3)+ud2*(ux3^2+uy3^2))-((ux3+ux3^2)*(uy3+uy3^2)))
, mynumerator((ux3+uy3)-(uW3/uZ3))
, mynumerator((ud1)-(ud2))
, mynumerator((ue^4)-(ud1))
, mynumerator((uf^4)-(ud2/ud1+1))
, mynumerator((uee)-(ue*ue))
, mynumerator((uff)-(uf*uf))
])

ud1 = fastfrac(ud1)
uee = fastfrac(uee)
ud2 = fastfrac(ud2)
uff = fastfrac(uff)
ue = fastfrac(ue)
uf = fastfrac(uf)
ux3 = fastfrac(ux3)
uy3 = fastfrac(uy3)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)
uW1 = fastfrac(uW1)
uZ1 = fastfrac(uZ1)
uW2 = fastfrac(uW2)
uZ2 = fastfrac(uZ2)
uW3 = fastfrac(uW3)
uZ3 = fastfrac(uZ3)
ux_2 = fastfrac(ux_2)
uy_2 = fastfrac(uy_2)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)


uC = ((uW2*(uZ2+uW2)))
uD = ((uW3*(uZ3+uW3)))
uW4 = ((uC^2))
uZ4 = ((uW4+((ue*uZ2+uf*uW2)^2)^2))
uE = ((uZ2*uZ3))
uF = ((uW2*uW3))
uV = ((uC*uD))
uU = ((uV+(uee*uE+uff*uF)^2))
uW5 = ((uV*uZ1+uU*uW1))
uZ5 = ((uU*uZ1))

ux4 = (((ud1*(ux2+ux2)+ud2*(ux2+uy2)*(ux2+uy2)+(ux2+ux2^2)*(ux2*(uy2+uy2+fastfrac(1))+uy2*uy2))/(ud1+(ux2+ux2^2)*(ux2+uy2)))).reduce()
ux5 = (((ud1*(ux3+ux2)+ud2*(ux3+uy3)*(ux2+uy2)+(ux3+ux3^2)*(ux2*(uy3+uy2+fastfrac(1))+uy3*uy2))/(ud1+(ux3+ux3^2)*(ux2+uy2)))).reduce()
uy4 = (((ud1*(uy2+uy2)+ud2*(ux2+uy2)*(ux2+uy2)+(uy2+uy2^2)*(uy2*(ux2+ux2+fastfrac(1))+ux2*ux2))/(ud1+(uy2+uy2^2)*(ux2+uy2)))).reduce()
uy5 = (((ud1*(uy3+uy2)+ud2*(ux3+uy3)*(ux2+uy2)+(uy3+uy3^2)*(uy2*(ux3+ux2+fastfrac(1))+ux3*ux2))/(ud1+(uy3+uy3^2)*(ux2+uy2)))).reduce()

print isidentity((ud1*(ux5+uy5)+ud2*(ux5^2+uy5^2))-((ux5+ux5^2)*(uy5+uy5^2))) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ud1*(ux5+uy5)+ud2*(ux5^2+uy5^2))-((ux5+ux5^2)*(uy5+uy5^2))))
print isidentity((ux5+uy5)-(uW5/uZ5)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ux5+uy5)-(uW5/uZ5)))
print isidentity((ud1*(ux5+uy5)+ud2*(ux5^2+uy5^2))-((ux5+ux5^2)*(uy5+uy5^2))) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ud1*(ux5+uy5)+ud2*(ux5^2+uy5^2))-((ux5+ux5^2)*(uy5+uy5^2))))
print isidentity((ux5+uy5)-(uW5/uZ5)) or isidentity(uy1*uy1*uy2*uy3*ux1*ux1*ux2*ux3*((ux5+uy5)-(uW5/uZ5)))