I would consider a proof to be a "repeatable argument". One you could show to someone and expect them to be convinced by it. I think it is a defensible viewpoint that a proof in coq is not 100% convincing. If you think otherwise, then how can you reconcile the existence of falso (a coq verified proof of "false")?
Proof and belief I think are pretty strongly intertwined, but I'm not going to pretend to have a particularly rigorous philosophy on the matter. Similarly, when the proof of Fermat's last theorem was published, I don't know if I should consider that to be a proof because it is well beyond my comprehension. I have no reason to question it, but should I consider it a proof? I know that people smarter than me (e.g. Wiles) thought the original version of it was a proof, but it had a subtle error in it which required a fix. While I haven't looked at the proof and revision, I would be surprised if I could look at the two versions as labelled and tell which one is the correct version.
https://github.com/clarus/falso
Proof and belief I think are pretty strongly intertwined, but I'm not going to pretend to have a particularly rigorous philosophy on the matter. Similarly, when the proof of Fermat's last theorem was published, I don't know if I should consider that to be a proof because it is well beyond my comprehension. I have no reason to question it, but should I consider it a proof? I know that people smarter than me (e.g. Wiles) thought the original version of it was a proof, but it had a subtle error in it which required a fix. While I haven't looked at the proof and revision, I would be surprised if I could look at the two versions as labelled and tell which one is the correct version.