[0x53]

What does provably correct mean here? I think you mean that the code doesn’t have any memory corruption vulnerabilities. However, that is only one class of vulnerability, so more techniques then just relying on a memory safe language are required for secure software.

[anyfoo]

It means that the type system can prove certain properties for you. For example, in languages with dependent type systems like Agda, you can construct a sorted list type that the compiler will prove it sorted at all times, otherwise it won’t compile. Or a complex tree type that is always balanced. Or a set of only even numbers, and again it won’t compile otherwise…

(Sadly, if you go that far, it isn’t generally Turing complete anymore. Though in some cases that’s a good thing.)