I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is more "tech support" than math.
But: after reading that Georges Gonthier and collaborators had formalized a proof of the Feit-Thompson theorem in Coq, I thought it would be fun to try to verify it on my own computer.
So I installed Coq (8.4pl2), OCaml, etc, on my Ubuntu box, and then went here and downloaded feit_thompson.tar.gz (v2.0). I unpacked it and ran make. It ran various invocations of coqc for about two hours, and then finished with no error reported. A bunch of .vo files were produced, including one for theories/stripped_odd_order_theorem.v, which even the casual observer can see contains what appears to be a statement of the Feit-Thompson theorem.
Is that it? Does that mean that Coq agrees with the correctness of the proof? Or are there more steps?
I was particularly curious because I saw that Makefile.coq has a validate target (which isn't run by just doing make). So I ran make -f Makefile.coq validate. This started a coqchk command which ran for about five minutes and then failed with:
User error: Signature components for label fun_of_fin do not match
make: *** [validate] Error 1
Was I not supposed to do that? Or did I just find a flaw in their proof? 🙂
Again, I freely admit that I do not really know what I'm doing. In particular, it's not clear to me whether compilation of a Coq file actually verifies the theorems and proofs within it, or merely prepares it for later validation. If an expert answers "Go away until you understand Coq" I'll meekly accept it. But it does seem like it might be helpful for even a newbie to be able to understand how to properly run Coq on a proof created by someone else, and I couldn't find documentation that clearly explained it.
Best Answer
The error you get is a real one, but is not in the proof of the odd order theorem. It is in Coq. Let me be more clear: a bug in the kernel of Coq was making the .vo files (the files coqchk checks) incomplete. Some universe constraints coming from module sub typing were forgotten. coqchk correctly spots it, and actually revealed the bug I fixed a while ago. A patchlevel release of Coq including the fix is already on its way.
The bugreport: https://coq.inria.fr/bugs/show_bug.cgi?id=3243
The commit that fixed the bug: https://github.com/coq/coq/commit/a07deb4eac1d5f886159784ef5d8d006892be547