[pron]

> but the CompCert compiler is a highly non-trivial example of a program verified using dependent types, and which would not really be possible using automated methods.

First of all, it wasn't really verified using dependent types; it was verified in Coq. Coq's theory is, indeed, based on dependent types, but Coq is used more like a traditional proof assistant (HOL or TLA+) than a dependently typed programming language like Agda or Idris.

Second, CompCert, while certainly non-trivial, is also about 1/5 the size of jQuery, which means it's at least one to two orders-of-magnitude away from the average business application.

> You can similarly formalize the kind of reasoning that a model checker (or any other automated method) does and then use it in your proof assistant.

Yes. In the few cases where deductive proofs are used in industry, they're often used to tie together results from model checking.