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.<ud1,ud2,ue,uf,ux1,uy1,uW1,uZ1> = PolynomialRing(GF(2),8,order='invlex')
I = R.ideal([
  mynumerator((ud1*(ux1+uy1)+ud2*(ux1^2+uy1^2))-((ux1+ux1^2)*(uy1+uy1^2)))
, mynumerator((ux1+uy1)-(uW1/uZ1))
, mynumerator((ue^4)-(ud1))
, mynumerator((uf^4)-(ud2/ud1+1))
])

ud1 = fastfrac(ud1)
ud2 = fastfrac(ud2)
ue = fastfrac(ue)
uf = fastfrac(uf)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uW1 = fastfrac(uW1)
uZ1 = fastfrac(uZ1)


uC = ((uW1*(uZ1+uW1)))
uW3 = ((uC^2))
uZ3 = ((uW3+((ue*uZ1+uf*uW1)^2)^2))

ux3 = (((ud1*(ux1+ux1)+ud2*(ux1+uy1)*(ux1+uy1)+(ux1+ux1^2)*(ux1*(uy1+uy1+fastfrac(1))+uy1*uy1))/(ud1+(ux1+ux1^2)*(ux1+uy1)))).reduce()
uy3 = (((ud1*(uy1+uy1)+ud2*(ux1+uy1)*(ux1+uy1)+(uy1+uy1^2)*(uy1*(ux1+ux1+fastfrac(1))+ux1*ux1))/(ud1+(uy1+uy1^2)*(ux1+uy1)))).reduce()

print(isidentity((ud1*(ux3+uy3)+ud2*(ux3^2+uy3^2))-((ux3+ux3^2)*(uy3+uy3^2))))
print(isidentity((ux3+uy3)-(uW3/uZ3)))