Agda
lean
Agda  lean  

27  4  
2,421  1,915  
1.2%    
9.8  0.0  
1 day ago  almost 4 years ago  
Haskell  C++  
MIT License  Apache License 2.0 
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Agda

Types versus sets (and what about categories?)
This was recently deemed inappropriate:
"Bye bye Set"
"Set and Prop are removed as keywords"
https://github.com/agda/agda/pull/4629

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.

What can Category Theory do?
Haskell and Agda are probably the most obvious examples. Ocaml too, but it is much older, so its type system is not as categorical. There is also Idris, which is not as wellknown but is very cool.

What are the current hot topics in type theory and static analysis?
Most of the proof assistants out there: Lean, Coq, Dafny, Isabelle, F*, Idris 2, and Agda. And the main concepts are dependent types, Homotopy Type Theory AKA HoTT, and Category Theory. Warning: HoTT and Category Theory are really dense, you're going to really need to research them.

Amendmend proposal: Changed syntax for Or patterns
Does this come with plans to separately unify the body with each of the contexts induced by matching on each of the respective patterns (similar to what’s discussed here), or will it behave like the _ pattern and use only the most general context?

Functional Programming and Maths <> How can a code monkey learn Agda?
That's absolutely untrue. From the horse's mouth:
 Doom emacs and agdamode

FP language idea  would this is possible to infer and type check?
Agda has the socalled mixfix operators (which are powerful enough to cover pre/in/postfix cases with an arbitrary number of arguments), check that out:  https://agda.readthedocs.io/en/v2.6.1/language/mixfixoperators.html  https://github.com/agda/agda/blob/master/examples/Introduction/Operators.agda  https://github.com/agda/agdastdlib/blob/master/src/Data/Product/Base.agda

Best Programming Language for Computational Proof
Coq, Agda, Lean, Isabelle, and probably some others which are not coming to my mind at the moment, but those would be considered the major ones.

Do you use Idris or Coq, and why?
Funny that you say this, because there are some obvious long standing open feature requests with looking up the type of the term under cursor — № 4295 and № 516. I am not blaming anyone in particular — this is the way it is. I wish I could find time to rewrite the proof search engine (how hard can it be), but I am already buried under a pile of other commitments and a good chunk of overwhelming sadness.
lean

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.

Lowlevel format file of mathlib
Does anyone happen to have the mathlib library in the [lowlevel 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.

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 performancecritical software, and maybe even expand the logic I am effectively operating in, all in the blink of an eye.

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.
What are some alternatives?
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.
FStar  A Prooforiented Programming Language
opentyperep  Open type representations and dynamic types
CoqHoTT  A Coq library for Homotopy Type Theory
HoleyMonoid  Automatically exported from code.google.com/p/monoidcont
CoqGym  A Learning Environment for Theorem Proving with the Coq proof assistant
lean4  Lean 4 programming language and theorem prover
arendlib
distributive  Dual Traversable
ttlite  A SuperCompiler for MartinLöf's Type Theory
agdavim  Agda interaction in vim
daisynfsd  DaisyNFS is an NFS server verified using Dafny and Perennial.