fricas
mathlib
fricas | mathlib | |
---|---|---|
8 | 36 | |
287 | 1,643 | |
1.0% | 1.2% | |
9.3 | 8.8 | |
4 days ago | 18 days ago | |
Clojure | Lean | |
BSD 3-clause "New" or "Revised" License | Apache License 2.0 |
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fricas
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Integral Calculator
But it's integration functionalities are less advanced and comprehensive than those of Fricas. Interestingly, the latter is, like Maxima, implemented using Lisp and stems from a ancient software lineage. Both systems are free and open-source.
Fricas home page: http://fricas.github.io
Some independent integration benchmarks, comparing multiple computer algebra systems: https://www.12000.org/my_notes/CAS_integration_tests/index.h...
- FriCAS – an advanced computer algebra system
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Strategies for doing symbolic integration algorithmically
Even partial implementations of the Risch algorithm can be pretty daunting but you might look at a couple heuristics that handle the easier bits, like Manel Bronstien's Poor Man's Integrator https://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html which doesn't need as many algebraic tools, but does need gcd, factor, and solve. I think FriCAS might use this https://github.com/fricas/fricas
- A Mature Library For Symbolic Computation?
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[2021 Day 6] [Fricas] Solution via finding a recurrence and solving it
Fricas home page: https://fricas.github.io
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Is Haskell a good language for CAS/numerical analysis?
I used to use Maxima back in the day, which is embedded in Lisp. With a quick googling I found FriCAS https://github.com/fricas/fricas , which aims to be "world class" AND its libraries are built in a strongly-typed DSL called Spad.
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"FriCAS algebra library, the largest and most advanced free general purpose computer algebra system" (as of September 2007)
BTW this is not a Clojure project. It contains .boot files that look like this and GitHub thinks they're Clojure. Trying to edit the .gitattributes through a PR.
mathlib
- An Easy-Sounding Problem Yields Numbers Too Big for Our Universe
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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 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/
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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/leanprover-community/mathlib
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It's not mathematics that you need to contribute to (2010)
https://github.com/leanprover-community/mathlib
https://1lab.dev/
You can watch the next generation, or participate, right now.
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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.
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Will Computers Redefine the Roots of Math?
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.
- Did studying proof based math topics e.g. analysis make you a better programmer?
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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.
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[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).
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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?
axiom - The dynamic infrastructure framework for everybody! Distribute the workload of many different scanning tools with ease, including nmap, ffuf, masscan, nuclei, meg and many more!
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 semi-interactive development of machine-checked proofs.
axiom - Axiom is a free, open source computer algebra system
Coq-Equations - A function definition package for Coq
Axiom - An FFmpeg GUI for Windows
mathquill - Easily type math in your webapp
Symbolics.jl - Symbolic programming for the next generation of numerical software
polynomial-algebra - polynomial-algebra Haskell library
cadabra2 - A field-theory motivated approach to computer algebra.
lean-liquid - 💧 Liquid Tensor Experiment
casadi - CasADi is a symbolic framework for numeric optimization implementing automatic differentiation in forward and reverse modes on sparse matrix-valued computational graphs. It supports self-contained C-code generation and interfaces state-of-the-art codes such as SUNDIALS, IPOPT etc. It can be used from C++, Python or Matlab/Octave.
natural_number_game - Building the natural numbers in Lean 3. The original natural number game, now frozen. See README for Lean 4 information.