In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules.

(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its restriction to any curve is regular

(2) Mebkhout defines the irregularity complexes of a complex of D-modules along an hypersurface. The complex is then regular if its irregularity complexes are 0 along any hypersurface.

(3) Kashiwara defines a D-module as regular if it admits a good filtration $F_*M$ such that $Ann(Gr^F M)$ is a radical ideal of $Gr^F D_X = \pi_*O_{T^*X}$.

I think there are other definitions (in Deligne for example)

Where can I find proofs that all these definitions are equivalent? Thanks.

In the literature on D-modules, there are many definitions of regularity of holonomic D-modules.

(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its restriction to any curve is regular

(2) Mebkhout defines the irregularity complexes of a complex of D-modules along an hypersurface. The complex is then regular if its irregularity complexes are 0 along any hypersurface.

(3) Kashiwara defines a D-module as regular if it admits a good filtration $F_*M$ such that $\operatorname{Ann}(Gr^F M)$ is a radical ideal of $Gr^F D_X = \pi_*O_{T^*X}$.

I think there are other definitions (in Deligne for example)

Where can I find proofs that all these definitions are equivalent? Thanks.