Coq: Happy to See it as a Geek Toy

Daniel, I'm really happy you defined Coq a geek toy. Usually such pieces of software are more seen as academic toys, seeing it promoted as a geeky software is a joy to me.

I'm a developer of Matita, a concurrent system of Coq (yet less mature a of today). I'm happy to give some more info about Coq (and hence about Matita, which is based on the same logical foundation of Coq).

Coq is not really a programming language (even if the CoqArt book says so, to attract the developer community), it is mainly a proof assistant; i.e. a software in which you give definitions, state theorems and prove them. The added value of the system wrt plain pen and paper is that it grants that you won't be able to produce incorrect proofs (of course you are still free to prove useless theorems :-) ).

It is more expressive than predicate calculus, technically it is based on the Calculus of (Co)Inductive Constructions, but the most important differences are that you can give inductive definitions of predicates (or define recursive data structures) and the system will let you play with them freely automatically constructing recursors on these data types. You can even define infinite data structures (using co-induction) mimicking lazy types available in languages as Haskell.

Regarding the synergy of Coq with OCaml: yes, you can prove in Coq the correctness of OCaml program, but actually not more than you can do with programs written in other programming languages (see for example Why, Krakatoa, and Caduceus for some practical tools to prove properties about Java and C programs).

Yet the synergy with OCaml is even more than that: once you proved properties like the existence of, say, a prime number greater than a given prime number, you can ask Coq to generate for you (using a feature called extraction) an OCaml program which consumes as input a prime number and returns the next one. More generally you can extract programs from (suitable) proofs.

Some info on the Debian side: thanks to smimou Coq has been in Debian for a long time now and Why/Krakatoa/Caduceus are coming soon.

Happy hacking!