Agda
eliminators
Agda  eliminators  

27    
2,421  27  
1.2%    
9.8  5.0  
3 days ago  about 2 months ago  
Haskell  Haskell  
MIT License  BSD 3clause "New" or "Revised" License 
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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.
eliminators
We haven't tracked posts mentioning eliminators yet.
Tracking mentions began in Dec 2020.
What are some alternatives?
lean  Lean Theorem Prover
hoq  A language based on homotopy type theory with an interval
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.
opentyperep  Open type representations and dynamic types
helf  Haskell implementation of the Edinburgh Logical Framework
HoleyMonoid  Automatically exported from code.google.com/p/monoidcont
agdasnippets  Library and tool to render the snippets in literate Agda files to hyperlinked HTML, leaving the rest of the text untouched.
lean4  Lean 4 programming language and theorem prover
Sit  Prototypical type checker for Type Theory with Sized Natural Numbers
distributive  Dual Traversable
nonemptycontainers  Efficient nonempty variants of containers data types, with full API