you can still verify arbitrarily long running programs - there are instances of such software, such as sel4 (https://sel4.systems/) and certikos (https://flint.cs.yale.edu/certikos/), you simply model them as finite programs that run on an infinite stream of events.
> finite programs that run on an infinite stream of events
This requires coinduction, right? (That's my understanding of the formal representation of infinite streams.) If so, that does limit your options, since most of the proof assistants don't handle coinductive data, as I understand it.