TITLE: Proof assistant, Cura te ipsum
QUESTION [19 upvotes]: By a  bona fide  bug  in a proof assistant I mean a   software   flaw   which is  serious enough to create a possibility of  "proving"  something which is  actually  false.  This is not a purely academic problem https://cstheory.stackexchange.com/questions/37299/has-a-proof-checker-bug-ever-invalidated-a-major-proof.   We mathematicians do not have a real reason to care about it for now, but I am convinced that this problem has a good potential to grow to scary  proportions once  a large scale program of formalization of mathematics is attempted. 
Modern mathematics is more  hierarchical   then common software, and in a very nontrivial way. Because of this, a cleanup after fixing a bug may become rather painful.
I am curious  about  methods to avoid this which are based on mathematics (as opposed to some sort of  management).  One method  I know about is verifying   the code of a proof assistant  using this same proof assistant.  Strictly speaking, this is theoretically impossible due to the second Goedel theorem,  but the way around it is  to make the system stronger  (for example,  by adding a new axiom).    There is a paper by J. Harrison  about how it may be done for HOL light,  J. Harrison,  "Towards self-verification of HOL Light",  Automated Reasoning, 2006 - Springer.  More recent works in this direction are   Myreen,   Owens,  Kumar, "Steps Towards Verified Implementations of HOL Light" , and   Anand,  Rahli, "Towards a Formally Verified Proof Assistant", ITP 2014: Interactive Theorem Proving. 
Question 1.  How far did it go?  (All the above papers have the word ``towards'' in the title.)
Question 2.  What  can  be said about  formal verification of a proof assistant which may be of interest to   mathematicians?   (For example, are there nontrivial alternatives to,  or variations of,  self-verification?) 
Remark. The question whether this  or similar  strategy  actually   makes  a proof assistant  perfectly bug-free (in the above sense)  would be more appropriate on Computer Science SE, but I would not mind if anyone touches this topic.

REPLY [10 votes]: See this dissertation by Ramana Kumar (Cambridge 2015).
http://www.sigplan.org/Awards/Dissertation/2017_kumar.pdf

I present a proof of consistency of higher-order logic (HOL), in
  particular for the entire inference system implemented by the kernel
  of the HOL Light theorem prover [24]. The main lemma is a proof of
  soundness against a new specification of the semantics of HOL. This
  formalisation extends work by Harrison [23] towards self-verification
  of HOL Light. Using the proof-grounded compilation technique, I show
  how to produce a concrete implementation of a proof checker for HOL
  based on the verified inference system. The result is a theorem prover
  with very strong guarantees of correctness, and, as I will sketch, the
  rare potential to verify its own concrete implementation in machine
  code.

On a different note regarding HOL-varieties, there is also HOL Zero.
http://proof-technologies.com/holzero/