mathlib VS swipl-devel

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)

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.

SWIProlog[1] has so far been my go to due to the extensive support system it has. However, I've been meaning to explore higher order logic a bit and Ciao[2] caught my attention there, with sugar for function-like notation and higher order programming including "lambda" style predicate expressions .... and it compiles down to executable. The function notation in this context is along the same lines as Mozart/Oz and can be convenient. Not explore the higher order aspects much though.

[1]: https://www.swi-prolog.org/

[2]: https://en.wikipedia.org/wiki/Ciao_(programming_language)

?- version(). Welcome to SWI-Prolog (threaded, 64 bits, version 9.0.0)SWI-Prolog comes with ABSOLUTELY NO WARRANTY. This is free software. Please run ?- license. for legal details.For online help and background, visit https://www.swi-prolog.org For built-in help, use ?- help(Topic). or ?- apropos(Word). true. ?- del(a, L, [1,2,3]). L = [a, 1, 2, 3] ; L = [1, a, 2, 3] ; L = [1, 2, a, 3] ; L = [1, 2, 3, a] ; false.

$ swipl Welcome to SWI-Prolog (threaded, 64 bits, version 8.5.10) SWI-Prolog comes with ABSOLUTELY NO WARRANTY. This is free software. Please run ?- license. for legal details. For online help and background, visit https://www.swi-prolog.org For built-in help, use ?- help(Topic). or ?- apropos(Word). ?- [fm2gp_primes]. true. ?- time( setup_call_cleanup(open('prolog-primes.txt', write, Out), with_output_to(Out, primes(500_000)), close(Out)) ). % 8,766,852 inferences, 1.055 CPU in 1.198 seconds (88% CPU, 8311018 Lips) Out = (0x600000648100).

The utilities in dgc/bacics.pl that you linked yourself are not too advanced, too quickly. Understanding those is exactly what you need in order to be able to write useful grammars for two reasons. They show how to approach many common issues with DCGs; and you know what building blocks you have at your disposal. I feel you discarded those too fast and strongly suggest you try to revisit them.

- And last but not least... the ability to convert authorization logic into SQL [4]. Which is done by having the language return constraints over any unbound (free) variables.

To me this is what makes logic programming exciting for authorization. It gives you this small kernel of declarative programming, and gives you a ton of freedom to build on top.

[1] https://www.swi-prolog.org/

(Btw. I'm using SWI Prolog.)

Welcome to SWI-Prolog (threaded, 64 bits, version 8.0.2) SWI-Prolog comes with ABSOLUTELY NO WARRANTY. This is free software. Please run ?- license. for legal details. For online help and background, visit http://www.swi-prolog.org For built-in help, use ?- help(Topic). or ?- apropos(Word). ?- use_module(library(lists)). true. ?- clumped([a,a,a,b,b,c], Rs). ERROR: Undefined procedure: clumped/2 (DWIM could not correct goal) ?-

Many library predicates do the argument reordering to take advantage of this special case argument indexing as explained in the answer by u/mycl. For example library(apply) in SWI-Prolog. is full of those.