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_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
    print('omega0_regular', omega0_regular)
    omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
    #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
    print('omega8', omega8, 'second_patch(omega8)', second_patch(omega8))
    omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
    print('omega8_regular 1', omega8_regular)
    omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
    print('omega8_regular 2', omega8_regular)
    u = (C.x)^(-1)
    v = (C.y)/(C.x)^(g+1)
    omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
    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_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
    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