TITLE: On the definition of regularity
QUESTION [13 upvotes]: 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.

REPLY [5 votes]: There are some comparison results in Chapter 5 of Bjork's `Analytic D-modules and Applications'. Also see Chapter 8. In particular, I think Thm. 8.7.3 combined with Thm. 5.6.5 (almost) gives (1) iff (3). Further, I think Prop. 5.6.22 gives the equivalence with (2). There are also results in there comparing Deligne's description.
I must admit though that I find Bjork quite notationally dense and am not particularly familiar with it, so I may be quite off with the references above. I am interested in this question also, so please comment/post if you find better references.