Like I haven't run Coq but I assume that in this particular instance, the answer is a rather trivial "yes"? You'd think about, creating this proof is a gigantic research slog and publishable result; verifying it is just looking at the notes you wrote down about what worked (this turing machine doesn't halt and here's the input and here's the repetitive sequence it generates, that turing machine always halts and here's the way we proved it...).

But taken more generally, not for this specific instance, your question is actually a Millenium Prize problem worth a million dollars.