  We prove that for a homologically smooth and proper DG algebra over a field
of characteristic 0, the Hodge-to-de Rham spectral sequence degenerates. This
has been conjectured by M. Kontsevich and Y. Soibelman arXiv:math/0606241 and
proved in arXiv:math/0611623 under a technical assumption. In this paper, the
assumption is removed, and the argument is considerably simplified (in
particular, it no longer uses Dold-Kan equivalence and simplicial methods). We
also analyse the degeneration of the conjugate spectral sequence in positive
caracteristic constructed in arXiv:1509.08784.
