lean
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lean | CoqGym | |
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4 | 2 | |
1,915 | 369 | |
- | 2.2% | |
0.0 | 3.6 | |
over 3 years ago | 10 months ago | |
C++ | Coq | |
Apache License 2.0 | GNU Lesser General Public License v3.0 only |
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lean
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Paper from 2021 claims P=NP with poorly specified algorithm for maximum clique using dynamical systems theory
Apparently, it even still segfaulted in 2018 https://github.com/leanprover/lean/issues/1958. I don't expect my tools to segfault.
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Low-level format file of mathlib
Does anyone happen to have the mathlib library in the [low-level format](https://github.com/leanprover/lean/blob/master/doc/export_format.md)? I've been trying to run lean --export to obtain it, but I keep getting various errors.
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Mathematics: our overlooked ability
I have spent a good deal of time trying to formalize elementary mathematics and computer science textbooks in the Lean Theorem Prover, and in trying to extend and improve Lean to make the process easier. I have been able to translate entire chapters of several textbooks into Lean in a natural way, with every line of Lean seemingly isomorphic to the informal presentation. However, once in a while I will hit a statement or proof step that may seem simple to me but that requires a major refactor, or adding new features to Lean itself, or just seems like a brick wall. My brain is able to perform massive refactorings of mathematical knowledge and abstractions, synthesize the equivalent of tens of thousands of lines of tricky and performance-critical software, and maybe even expand the logic I am effectively operating in, all in the blink of an eye.
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How do I get back into math?
However, mathlib makes some weird design choices. For example, (semi)groups are defined using multiplicative notation -- and then immediately followed by an entire section giving the exact same definitions using additive notation! The claimed reason for this is that the more abstract approach is inconvenient for automation. I did it in Coq using the abstract approach, and indeed, noticed that doing so broke automation, which I then worked around in various ways. But it's just weird to me as a mathematician to have additive and multiplicative groups be different objects, so I wouldn't want to do it the Lean way come hell or high water. The Lean approach causes practical difficulties as well: you have to prove every theorem about groups twice. In some cases (e.g. product groups), you have to prove every theorem FOUR times. Ugh.
CoqGym
- Lean4 helped Terence Tao discover a small bug in his recent paper
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Discussion Thread
This has been an active area of research for a few years. See for example https://arxiv.org/abs/1905.09381. It's still immature as a field, and most results are essentially "we got basic stuff down but it hasn't gotten powerful enough to prove anything truly challenging" but it definitely exists and is being developed.
What are some alternatives?
Agda - Agda is a dependently typed programming language / interactive theorem prover.
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.
FStar - A Proof-oriented Programming Language
trepplein - Lean type-checker written in Scala.
Coq-HoTT - A Coq library for Homotopy Type Theory
mathlib - Lean 3's obsolete mathematical components library: please use mathlib4
arend-lib
lean-chat
ttlite - A SuperCompiler for Martin-Löf's Type Theory
symmetric_project
daisy-nfsd - DaisyNFS is an NFS server verified using Dafny and Perennial.
cedar-spec - Definitional implementation of Cedar language and utilities for DRT