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,ud2,ux1,uy1,ux2,uy2> = PolynomialRing(GF(2),6,order='invlex')
I = R.ideal([
  mynumerator((ud1*(ux1+uy1)+ud2*(ux1^2+uy1^2))-((ux1+ux1^2)*(uy1+uy1^2)))
, mynumerator((ud1*(ux2+uy2)+ud2*(ux2^2+uy2^2))-((ux2+ux2^2)*(uy2+uy2^2)))
])

J = I + R.ideal([0
, ux1-ux2
, uy1-uy2
])

ud1 = fastfrac(ud1)
ud2 = fastfrac(ud2)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)

ux3 = (((ud1*(ux1+ux2)+ud2*(ux1+uy1)*(ux2+uy2)+(ux1+ux1^2)*(ux2*(uy1+uy2+fastfrac(1))+uy1*uy2))/(ud1+(ux1+ux1^2)*(ux2+uy2)))).reduce()
uy3 = (((ud1*(uy1+uy2)+ud2*(ux1+uy1)*(ux2+uy2)+(uy1+uy1^2)*(uy2*(ux1+ux2+fastfrac(1))+ux1*ux2))/(ud1+(uy1+uy1^2)*(ux2+uy2)))).reduce()
ux4 = (((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()
uy4 = (((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()
a0 = fastfrac((fastfrac(1)))
a1 = fastfrac((fastfrac(1)))
a2 = fastfrac((ud1^2+ud2))
a3 = fastfrac((fastfrac(0)))
a4 = fastfrac((fastfrac(0)))
a6 = fastfrac((ud1^4*(ud1^4+ud1^2+ud2^2)))
wu1 = ((ud1*(ud1^2+ud1+ud2)*(ux1+uy1)/(ux1*uy1+ud1*(ux1+uy1)))).reduce().sreduce()
wv1 = ((ud1*(ud1^2+ud1+ud2)*(ux1/(ux1*uy1+ud1*(ux1+uy1))+ud1+fastfrac(1)))).reduce().sreduce()
print isidentity(a0*(wv1^2)+a1*(wu1*wv1)+a3*wv1-(((wu1+a2)*wu1+a4)*wu1+a6))
print isidentity(ux1-(ud1*(wu1+ud1^2+ud1+ud2)/(wu1+wv1+(ud1^2+ud1)*(ud1^2+ud1+ud2))))
print isidentity(uy1-(ud1*(wu1+ud1^2+ud1+ud2)/(wv1+(ud1^2+ud1)*(ud1^2+ud1+ud2))))
wu2 = ((ud1*(ud1^2+ud1+ud2)*(ux2+uy2)/(ux2*uy2+ud1*(ux2+uy2)))).reduce().sreduce()
wv2 = ((ud1*(ud1^2+ud1+ud2)*(ux2/(ux2*uy2+ud1*(ux2+uy2))+ud1+fastfrac(1)))).reduce().sreduce()
wu3 = ((ud1*(ud1^2+ud1+ud2)*(ux3+uy3)/(ux3*uy3+ud1*(ux3+uy3)))).reduce().sreduce()
wv3 = ((ud1*(ud1^2+ud1+ud2)*(ux3/(ux3*uy3+ud1*(ux3+uy3))+ud1+fastfrac(1)))).reduce().sreduce()
wu4 = ((ud1*(ud1^2+ud1+ud2)*(ux4+uy4)/(ux4*uy4+ud1*(ux4+uy4)))).reduce().sreduce()
wv4 = ((ud1*(ud1^2+ud1+ud2)*(ux4/(ux4*uy4+ud1*(ux4+uy4))+ud1+fastfrac(1)))).reduce().sreduce()
slope = ((wv2-wv1)/(wu2-wu1)).reduce().sreduce()
print isidentity(a0*slope^2+a1*slope-a2-wu1-wu2-wu3)
print isidentity(slope*(wu1-wu3)-wv1-a1*wu3-a3-wv3)
slope = ((fastfrac(3)*wu1^2+fastfrac(2)*a2*wu1+a4-a1*wv1)/(fastfrac(2)*a0*wv1+a1*wu1+a3)).reduce().sreduce()
print isidentity(a0*slope^2+a1*slope-a2-wu1-wu1-wu4)
print isidentity(slope*(wu1-wu4)-wv1-a1*wu4-a3-wv4)