Whoops, I completely forgot about libsparkcrypto (https://github.com/Componolit/libsparkcrypto) which is a "formally verified implementation of several widely used cryptographic algorithms using the SPARK 2014 programming language and toolset. For the complete library proofs of the absence of run-time errors like type range violations, division by zero and numerical overflows are available. Some of its subprograms include proofs of partial correctness.".

Features:

* AES-128, AES-192, AES-256

* AES-CBC (all supported AES modes)

* SHA-1

* HMAC-SHA1

* SHA-256, SHA-384, SHA-512

* HMAC-SHA-256, HMAC-SHA-384, HMAC-SHA-512

* PRF-HMAC-SHA-256, PRF-HMAC-SHA-384, PRF-HMAC-SHA-512

* RIPEMD-160

* HMAC-RIPEMD-160

* ECDSA, ECGDSA

It is pretty awesome. :)

Looking at the examples you've provided, I'm pretty sure the Linux kernel's codebase dwarfs their combined codebases into insignificance. Since it is very hard to not have NP-hard asymptotic complexity when formally verifying a codebase as large and as demanding (in terms of performance, etc) as the Linux kernel, I would expect that if you tried to rewrite Linux in the languages you've mentioned and formally verify the rewrite, you will be able to keep the entire AWS cloud (globally) very busy for a very, very long time. On the other hand, if you were to do the rewrite in Rust and use some hypothetical formally verified Rust compiler to type check it (which is mathematically equivalent to such a formal verification in, say, F* limited to only verifying certain properties), it'll probably type check in minutes or even seconds on the laptop I'm currently writing this reply on.

For a less hypothetical example. RedoxOS has a codebase that dwarfs Sel4 (but it is still much smaller than Linux in turn), yet it type checks with much less compute cost than it takes to formally verify Sel4. Or heck, Redleaf actually formally verifies in a tiny fraction of the time it takes to formally verify Sel4, because the Rust type checker not only significantly reduces the amount of requirements that still needs to be formally verified but also reduces the complexity to verify those requirements to linear to the codebase size due to the restrictions that the Rust type system and module system puts on valid Rust programs (with a little bit of care when writing the code). Of course, until you have a certified Rust compiler there might still be miscompiles of Redleaf and until all the features of Rust used in the Linux kernel is formally specified, we cannot do the same with the Linux kernel as for Redleaf even if somehow all of Linux got rewritten in Rust. But in Rust at least it is conceivably possible given sufficient resources and sufficient time. I very much doubt it will ever be possible without help from a sufficiently expressive and practical type system.

Oh yea. Then there's another problem. Most (all?) of the examples you've mentioned have requirements that are well specified and pretty much set in stone. The requirements fro the Linux kernel... aren't quite as obliging.