proofs
mathlib
proofs  mathlib  

5  36  
286  1,660  
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17 days ago  9 days ago  
Coq  Lean  
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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!
mathlib
 An EasySounding Problem Yields Numbers Too Big for Our Universe

Towards a new SymPy: part 2 – Polynomials
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 researchlevel 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 coexist 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://leanprovercommunity.github.io/

Lean 4.0.0, first official lean4 release
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/leanprovercommunity/mathlib

It's not mathematics that you need to contribute to (2010)
https://github.com/leanprovercommunity/mathlib
https://1lab.dev/
You can watch the next generation, or participate, right now.

If given a list of properties/definitions and relationship between them, could a machine come up with (mostly senseless, but) true implications?
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.

Will Computers Redefine the Roots of Math?
For the math that you mention, I would suggest looking at mathlib (https://github.com/leanprovercommunity/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.
 Did studying proof based math topics e.g. analysis make you a better programmer?

Which proof assistant is the best to formalize real analysis/probability/statistics?
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.

[R] Large Language Models trained on code reason better, even on benchmarks that have nothing to do with code
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).

Is there a paid service where someone can explain a paper to me like I am 15?
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)
What are some alternatives?
CompCert  The CompCert formallyverified C compiler
coq  Coq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semiinteractive development of machinechecked proofs.
masterthesis
CoqEquations  A function definition package for Coq
hacspec  Please see https://github.com/hacspec/hax
mathquill  Easily type math in your webapp
aneris  Program logic for developing and verifying distributed systems
fricas  Official repository of the FriCAS computer algebra system
ccctalk  Correct Code by Construction talk's code
polynomialalgebra  polynomialalgebra Haskell library
coqsimpleio  IO for Gallina
leanliquid  💧 Liquid Tensor Experiment