  We provide a simple abstract formalism of integration by parts under which we
obtain some regularization lemmas. These lemmas apply to any sequence of random
variables $(F_n)$ which are smooth and non-degenerated in some sense and enable
one to upgrade the distance of convergence from smooth Wasserstein distances to
total variation in a quantitative way. This is a well studied topic and one can
consult for instance Bally and Caramellino [Electron. J. Probab. 2014],
Bogachev, Kosov and Zelenov [Trans. Amer. Math. Soc. 2018], Hu, Lu and Nualart
[J. Funct. Anal. 2014], Nourdin and Poly [Stoch. Proc. Appl. 2013] and the
references therein for an overview of this issue. Each of the aforementioned
references share the fact that some non-degeneracy is required along the whole
sequence. We provide here the first result removing this costly assumption as
we require only non-degeneracy at the limit. The price to pay is to control the
smooth Wasserstein distance between the Malliavin matrix of the sequence and
the Malliavin matrix of the limit which is particularly easy in the context of
Gaussian limit as their Malliavin matrix is deterministic. We then recover, in
a slightly weaker form, the main findings of Nourdin, Peccati and Swan [J.
Funct. Anal. 2014]. Another application concerns the approximation of the
semi-group of a diffusion process by the Euler scheme in a quantitative way and
under the H\"ormander condition.
