def de_rham_witt_lift_form0(omega):
    C = omega.curve
    omega_regular = omega.regular_form() #Present omega0 in the form P dx + Q dy
    #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
    return omega_regular.dx.teichmuller()*(C.x.teichmuller().diffn()) + omega_regular.dy.teichmuller()*(C.y.teichmuller().diffn())

def de_rham_witt_lift_form8(omega):
    C = omega.curve
    g = C.genus()
    omega_regular = second_patch(omega).regular_form()
    omega_regular = (second_patch(omega_regular.dx), second_patch(omega_regular.dy))
    u = (C.x)^(-1)
    v = (C.y)/(C.x)^(g+1)
    omega_lift = omega_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega_regular[1].teichmuller()*(v.teichmuller().diffn())
    return omega_lift
    
def de_rham_witt_lift(cech_class, prec = 50):
    C = cech_class.curve
    g = C.genus()
    omega0 = cech_class.omega0
    omega8 = cech_class.omega8
    fct = cech_class.f
    omega0_lift = de_rham_witt_lift_form0(omega0)
    omega8_lift = de_rham_witt_lift_form8(omega8)
    print('omega0_lift, omega8_lift', omega0_lift, omega8_lift)
    aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
    #return aux
    if aux.h1.function != 0:
        raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
    decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec) #decompose dV(smth) in aux as smth regular on U0 - smth regular on U8.
    aux_h2 = decom_aux_h2[0]
    aux_f = decom_aux_h2[2]
    aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
    result = superelliptic_drw_cech(omega0_lift - aux_omega0.verschiebung(), fct.teichmuller() + aux_h2.verschiebung() + aux_f.verschiebung())
    compare = omega8_lift-decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
    print("result.omega8 == compare", result.omega8 == compare)
    print("result.omega8 - compare", result.omega8 - compare)
    
    #print('test:', omega0_lift - omega8_lift - fct.teichmuller().diffn() == decom_aux_h2[0].verschiebung().diffn() - decom_aux_h2[1].verschiebung().diffn() + decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
    #print('test 1:', omega0_lift - decom_aux_h2[0].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - fct.teichmuller().diffn() == omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
    #A = omega0_lift - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung()
    #B = decom_aux_h2[0].verschiebung() + fct.teichmuller()
    #C = omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
    #print('test 2:', A - B.diffn() == C)
    #print('test 3:', result.omega0 == A, result.f == B, result.omega8 == C)
    #print(result.omega8, '\n \n', compare, '\n \n', aux_f, '\n \n')
    return result#.reduce()

def crystalline_cohomology_basis(self, prec = 50):
    result = []
    for a in self.de_rham_basis():
        result += [de_rham_witt_lift(a, prec = prec)]
    return result

superelliptic.crystalline_cohomology_basis = crystalline_cohomology_basis