kami
proofs
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Coq  Coq  
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kami
 Kami: A Platform for Hardware Specification and Verification

There's an ongoing effort to rewrite Principia Mathematica using Coq
There are ongoing research projects about that, you may want to have a look at Kôika (https://github.com/mitplv/koika), Kami (https://github.com/mitplv/kami), Lutsig (https://github.com/CakeML/hardware) and silveroak (https://github.com/projectoak/silveroak). Closer to HLS there is also Vericert (https://github.com/ymherklotz/vericert). There may be other research project I am unaware of, feel free to add them in a reply, I am interested in it.
proofs

A Taste of Coq and Correct Code by Construction
If you're already familiar with a functional programming language like Haskell or OCaml, you have the prerequisite knowledge to work through my Coq tutorial here: https://github.com/stepchowfun/proofs/tree/main/proofs/Tutor...
My goal with this tutorial was to introduce the core aspects of the language (dependent types, tactics, etc.) in a "straight to the point" kind of way for readers who are already motivated to learn it. If you've heard about proof assistants like Coq or Lean and you're fascinated by what they can do, and you just want the TL;DR of how they work, then this tutorial is written for you.
Any feedback is appreciated!

Thoughts on proof assistants?
Personally I treat Coq like an extension of my brain. Whenever I'm uncertain about something, I formalize it in Coq. I have a repository of proofs with GitHub Actions set up in such a way forbids me from pushing commits containing mathematical mistakes. I've formalized various aspects of category theory, type theory, domain theory, etc., and I've also verified a few programs, such as this sorting algorithm. Lately I've been experimenting with a few novel types of graphs, proving various properties about them with the aim of eventually developing a way to organize all of my data (files, notes, photos, passwords, etc.) in some kind of graph structure like that.

Formally Verifying Rust's Opaque Types
It's always a pleasant surprise to see people using Coq and other formal verification technology. We need more rigor in programming! If this article gave you a thirst for interactive theorem proving and you want to learn it from the ground up, I've recently written a Coq tutorial [1] which covers topics like programming with dependent types, writing proofs as data, and extracting verified code. That repository also contains a handy tactic called `eMagic` [1] (a variant of another useful tactic called `magic` which solve goals with existentials) which can automatically prove the theorem from the article.
[1] https://github.com/stepchowfun/proofs/tree/main/proofs/Tutor...
[2] https://github.com/stepchowfun/proofs/blob/56438c9752c414560...

A complete compiler and VM in 150 lines of code
For anyone who wants to learn Coq, I've just finished writing a tutorial [1] that is aimed at programmers (rather than, say, computer scientists). It covers topics like programming with dependent types, writing proofs as data, universes & other type theory stuff, and extracting verified code—with exercises. I hope people find it useful, and any feedback would be appreciated!
[1] https://github.com/stepchowfun/proofs/tree/main/proofs/Tutor...

New Coq tutorial
Hi all, Coq is a "proof assistant" that allows you to write both code and proofs in the same language (thanks to the Curry–Howard correspondence). Its uses range from pure math (e.g., the Feit–Thompson theorem was proven in Coq!) to reasoning about programming languages (e.g., proving the soundness of a type system) to writing verified code (e.g., this verified C compiler!). You can "extract" your code (without the proofs) to OCaml/Haskell/Scheme for running it in production. Coq is awesome, but it's known for having a steep learning curve (it's based on type theory, which is a foundational system of mathematics). It took me several years to become proficient in it. I wanted to help people pick it up faster than I did, so I wrote this introductory tutorial. Hope you find it useful!
What are some alternatives?
koika  A core language for rulebased hardware design 🦑
CompCert  The CompCert formallyverified C compiler
silveroak  Formal specification and verification of hardware, especially for security and privacy.
masterthesis
vericert  A formally verified highlevel synthesis tool based on CompCert and written in Coq.
hacspec  Please see https://github.com/hacspec/hax
ccctalk  Correct Code by Construction talk's code
coqsimpleio  IO for Gallina
parson  Yet another PEG parser combinator library and DSL
aneris  Program logic for developing and verifying distributed systems
dafny  Dafny is a verificationaware programming language
smlredprl  The People's Refinement Logic