> What specific program properties have you tried proving that have required hand proofs you don't think are that difficult?

I don't do much proving by hand; I tend to sketch out possible encodings by hand, but work on actual proofs on a computer. Incidentally I find that such exploratory work is where CoqIDE, ProofGeneral, etc. shine; as opposed to Agda or Idris which seem to work best for "writing up" and "filling in the holes" of proofs we're already confident in.

Most non-difficult things I've formally proven been toy examples. I did enjoy formalising a particular AI approach, which was pretty simple to prove properties about once I figured out a nice encoding https://github.com/warbo/powerplay (for theorems see https://github.com/Warbo/powerplay/blob/master/SimpleTests.v ). Though that's more of a toy theory, rather than a huge programming effort like seL4, CompCert, etc.

> The core issue is that once you try to capture a program property in one part of your program, you have to start capturing it everywhere and it's very difficult to contain the complexity. Simple looking properties can require very deep knowledge and expertise of writing proofs.

Very true. There are essentially two ways to do it:

- Use standard encodings (e.g. lists and peano naturals), then build up a mountain of lemmas about them, and plug those together in different combinations each time we want to prove a property (e.g. the elements are sorted).

- Use "correct by construction" encodings (e.g. a list of differences) which guarantee the properties we want, then build up a mountain of helper functions which act on those encodings and preserve the properties (e.g. `mapDiffs`, `foldDiffs`, `filterDiffs`, etc.)

I've seen a couple of potential improvements for this proposed; though I'm not sure whether they scale:

- Adam Chlipala likes to build up a custom proof tactic (in Certified Programming with Dependent Types this is called `crush`). Whenever that tactic doesn't work, he proves the statement manually, then alters the tactic's search algorithm so it can find that proof, then replaces the manual proof by a call to the improved tactic.

- Conor McBride proposed "ornaments" as a way to "add on" property-guaranteeing information to standard encodings, e.g. treating normal lists as lists-of-differences to preserve sortedness.

> OCaml etc. are only likely to add feature that give a good amount of power for a small amount of expertise whereas with Coq, Isabelle etc. they don't need to hold back on what level of expertise is expected from users (e.g. many users are in research publishing journal papers from the expertise required of them). Dependent types give a massive amount of power for a massive rise in the expertise expected from the user.

I don't think dependent types themselves are particularly difficult to grok, although they can certainly be used for all sorts of elaborate shenanigans. I don't think that's any different from programming languages themselves though, e.g. some languages like Basic and Scheme are routinely taught to children, despite there presumably being some monstrously fiendish projects out there which use those languages. I've also encountered some bewilderingly complex spreadsheets in my time, but I wouldn't say that spreadsheets themselves require difficult expertise.

I certainly find the few, simple rules of dependent types much easier to understand than the mess that is type level programming in Haskell (data kinds, singletons, associated types, open/closed type families, etc.). I still don't fully understand that stuff, but I do know that:

- Data kinds in Haskell are just normal values in dependently typed languages

- (Closed) type families in Haskell are just normal functions in dependently typed languages

- Type classes and instances in Haskell are just normal record types and values in dependently typed languages

- Constraints in Haskell are just function arguments in dependently typed languages (this also generalises constraint solving to implicit arguments)

- Associated types in Haskell are just normal record fields in dependently typed languages

Most of these features are just a "shadow language" ( https://gbracha.blogspot.com/2014/09/a-domain-of-shadows.htm... ), which are completely unnecessary when types are first-class, and especially when dependent types are allowed.

Again, just because something is available doesn't mean that we need to use it. The only pressure to do so comes from existing libraries. As another example, many Haskell libraries, including the Prelude (built-ins) and "do notation" make use of monads (via the `Monad` type class). Yet we're free to ignore all of that and never touch monads at all; we could spend our entire programming career in the language, blisfully unaware that such a thing even exists. The language does require that we define a `main` value as our program's entry point (just like Java and C), and that value has a type like `IO ()`; hence we're forced to use `IO`. Yet we can use `IO` without ever knowing or caring that it just-so-happens to implement the `Monad` type class: we can do everything we want via functions like `runIO :: IO (IO a) -> IO a` instead of `join :: (Monad m) => m (m a) -> m a`, and `sequenceIO :: [IO a] -> IO a` instead of `sequence :: (Monad m) => [m a] -> m a`, etc. in the same way that many programmers spend their entire careers making heavy use of lists (perhaps even using a LISt Processing language!) without having to know or care that `List` just-so-happens to be a monad.