Agda
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Agda  opentyperep  

27  1  
2,248  4  
0.8%    
9.6  0.0  
3 days ago  over 7 years ago  
Haskell  Haskell  
MIT License  BSD 3clause "New" or "Revised" License 
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Agda

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?
 Doom emacs and agdamode

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.

Haskell is the greatest programming language of all time ... the rational adult in a room full of children ... When I program in Haskell, I am in utopia. I am in a different world than 99.9% of what I see posted on Reddit.
"Haskal is a mess, that's why I use a language that's written with Haskal."

The comment with the most upvotes decides what language I write my finals in this year will be.
Do it in Agda.

Wren is a small, fast, classbased concurrent scripting language
Agda is fun[1]. And there's also Idris[2]  more programming language, less proof assistant.
[1] https://wiki.portal.chalmers.se/agda/pmwiki.php https://github.com/agda/agda
opentyperep
What are some alternatives?
lean  Lean 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 semiinteractive development of machinechecked proofs.
HoleyMonoid  Automatically exported from code.google.com/p/monoidcont
zig  Generalpurpose programming language and toolchain for maintaining robust, optimal, and reusable software.
distributive  Dual Traversable
agdavim  Agda interaction in vim
fgl  A Functional Graph Library for Haskell
miso  :ramen: A tasty Haskell frontend framework
eliminators  Dependently typed elimination functions using singletons
agda.nvim  NeoVim plugin for interacting with Agda written in Lua
helf  Haskell implementation of the Edinburgh Logical Framework
permutation  git import of patrick perry permutations lib from darcs