lean4 VS mathlib

Recently replaced by Lean, though.

https://github.com/cedar-policy/cedar-spec

https://lean-lang.org

Nice read! Rust has pushed, and will continue to push, the limits of practical, bare metal, memory safe languages. And it's interesting to think about what's next, maybe eventually there will be some form of practical theorem proving "for the masses". Lean 4 looks great and has potential, but it's still mostly a language for mathematicians. There has been some research on AI constructed proofs, which could be the best of both worlds because then the type checker can verify that the AI generated code/proof is indeed correct. Tools like Kani are also a step forward in program correctness.

Yeah, I believe they said intend for it to be used as a general purpose programming language. I used it to complete Advent of Code last year.

There are some really interesting features for general purpose programming in there. For example: you can code updates to arrays in a functional style (change a value, get a new array back), but if the refcount is 1, it updates in place. This works for inductive types and structures, too. So I was able to efficiently use C-style arrays (O(1) update/lookup) while writing functional code. (paper: https://arxiv.org/abs/1908.05647 )

Another interesting feature is that the "do" blocks include mutable variables and for loops (with continue / break / return), that gets compiled down to monad operations. (paper: https://dl.acm.org/doi/10.1145/3547640 )

And I'm impressed that you can add to the syntax of the language, in the same way that the language is implemented, and then use that syntax in the next line of code. (paper: https://lmcs.episciences.org/9362/pdf ). There is an example in the source repository that adds and then uses a JSX-like syntax. (https://github.com/leanprover/lean4/blob/master/tests/playgr... )

My latest inspiration to use code to a) replace my video editor, b) learn the basics of EDM production and c) understand a few topics in higher maths. This might sound very strange given there are specialised tools for these jobs. There's iMovie / Adobe Premier for video, there's GarageBand and FL studio for music and old good pen and pencil for math proofs. But these tools have three big limitations. First they have a lot of idiosyncratic learning, you have to spend quite some time getting used to these tools and my experience is that this time is quite upsetting. In contrast, you only have to learn to code one, maybe spend a few hours getting used to the syntax of another language. I'm not sure if that's true for most people but it was true for me using the tools mentioned above and wanted a place to discuss and see other people ideas and experiments. The second issue is that all these custom-made tools, are not composing easily. I can't search for all math proofs that used a single theorem. I can't create a plugin for iMovie and apply it to all my videos. I can't pick easily pick a rhythm from the internet and build upon for fun. There's also the issue of costs and version control, all tools I'm using today are open source and my work is stored in my repositories. This way I can create branches and test my ideas and I'm also confident that I can work in these projects in years.

Dafny and Whiley are two examples with explicit verification support. Idris and other dependently typed languages should all be rich enough to express the required predicate but might not necessarily be able to accept a reasonable implementation as proof. Isabelle, Lean, Coq, and other theorem provers definitely can express the capability but aren't going to churn out much in the way of executable programs; they're more useful to guide an implementation in a more practical functional language but then the proof is separated from the implementation, and you could also use tools like TLA+.

https://dafny.org/

https://whiley.org/

https://www.idris-lang.org/

https://isabelle.in.tum.de/

https://leanprover.github.io/

https://coq.inria.fr/

http://lamport.azurewebsites.net/tla/tla.html

It's been on my mind lately as well. I was trying out `symbolics.jl` (a CAS written in Julia), and it turned out that it didn't support symbolic integration beyond simple linear functions or polynomials (at least back then, things have changed now it seems). Implementing a generic algorithm for finding integrals is hard, but I was expecting more from that CAS since this seems to be implemented in most other CASs. The thing is that every single CAS that covers general maths knowledge will have to implement the same algorithm, while it's hard to do it even once!

I feel like at least a large part of the functionality of a general purpose CAS can be written down once, and every CAS out there could benefit from it, similar to what the Language Server Protocol did for programming tools. They also had to rewrite the same tool for some language multiple times because there are lots of editors out there, and the LSP cut the time investment down a lot. They did have to invest a large amount of time to get LSP up and running, and it'll have to be maintained, but I think it's orders of magnitudes more efficient than having every tool developed and maintained for every single (programming language, editor) pair out there.

Main problem is like you said how to write down mathematical knowledge in a way that all CASs can understand it. I've been learning about Mathlib lately [0], which seems like a great starting point for this. It is as far as I know one of the first machine readable libraries of mathematical knowledge; it has a large community which has been pushing it continuously forward for years into research-level mathematics and covering the entire undergraduate maths curriculum and it's still accelerating. If some kind of protocol can be designed to read from libraries like this and turn it into CAS code, that would be a major step towards making the CAS ecosystem more sustainable I think.

It's not exactly what you were talking about, as in, this would allow multiple CASs to co-exist and benefit from each other, but I think that's better than having one massive CAS that has a monopoly. No software is perfect, but having a diverse set of choices that are open source would be more than enough to satisfy everyone.

(I have posted about this before on the Lean Zulip forum, it's open to everyone to read without an account [1])

[0] https://leanprover-community.github.io/

Kinda agree but Mathlib and its documentation makes for a big corpus to learn by example from. Not ideal but it helps.

https://github.com/leanprover-community/mathlib

https://github.com/leanprover-community/mathlib

https://1lab.dev/

You can watch the next generation, or participate, right now.

Still, there are many useful tools based on these ideas, used by programmers and mathematicians alike. What you describe sounds rather like Datalog (e.g. Soufflé Datalog), where you supply some rules and an initial fact, and the system repeatedly expands out the set of facts until nothing new can be derived. (This has to be finite, if you want to get anywhere.) In Prolog (e.g. SWI Prolog) you also supply a set of rules and facts, but instead of a fact as your starting point, you give a query containing some unknown variables, and the system tries to find an assignment of the variables that proves the query. And finally there is a rich array of theorem provers and proof assistants such as Agda, Coq, Lean, and Twelf, which can all be used to help check your reasoning or explore new ideas.

For the math that you mention, I would suggest looking at mathlib (https://github.com/leanprover-community/mathlib). I agree that the foundations of Coq are somewhat distanced from the foundations most mathematicians are trained in. Lean/mathlib might be a bit more familiar, not sure. That said, I don't see any obstacles to developing classical real analysis or linear algebra in Coq, once you've gotten used to writing proofs in it.

At this point I would go with Lean because of mathlib. Mathlib's goal is to formalize modern mathematics, so many of the theorems you would need for analysis should already be there for you.

I think about that every day. Lean's mathlib is a gigantic (with respect to this kind of project) code base and each function, each definition has a precise and rigorous natural language counterpart (in a maths book, somewhere).

It's been around since 2013, although there are LLM that interact with Lean to do automated theorem proving. Anyway, you can learn more about Lean here. I enjoyed their natural numbers game (which reminds, me I should finish the last two levels)