Idris2 VS Agda

Have you looked into Idris2 at all. While looking into these theorum provers, it always felt like they had an impedance mismatch with normal programming.

Idris2 portends to a general purpose language that also has a more advanced type system for the theorum proving.

https://github.com/idris-lang/Idris2

The original paper also does plain STLC first in section 2, and then adds dependent types in section 3. (And finally it adds the naturals in section 4.)

In the Idris2 github repository, Guillaume Allais goes a step further and shows a well-named version. There the types of terms and values are indexed by the list of names in the environment and the compiler checks that the manipulation of deBruin levels and indices is correct:

https://github.com/idris-lang/Idris2/blob/main/libs/papers/L...

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.

Here's the full tab list: - https://tjpalmer.github.io/languish/ - https://blog.python.org/2022/10/python-3110-is-now-available.html - https://devblogs.microsoft.com/python/python-311-faster-cpython-team/ - https://github.com/tc39/proposals/blob/main/finished-proposals.md - https://devblogs.microsoft.com/typescript/ten-years-of-typescript/ - https://devblogs.microsoft.com/typescript/announcing-typescript-4-6/#cfa-destructured-discriminated-unions - https://devblogs.microsoft.com/typescript/announcing-typescript-4-9/#the-satisfies-operator - https://devblogs.microsoft.com/typescript/announcing-typescript-4-7/#go-to-source-definition - https://devblogs.microsoft.com/typescript/announcing-typescript-4-8/#build-watch-incremental-improvements - https://openjdk.org/projects/jdk/18/ - https://openjdk.org/projects/jdk/19/ - https://blog.jetbrains.com/clion/2022/07/july-2022-iso-cpp/ - https://en.wikipedia.org/wiki/C%2B%2B23 - https://en.cppreference.com/w/cpp/23 - https://www.open-std.org/jtc1/sc22/wg21/docs/papers/2021/p2128r6.pdf - https://devblogs.microsoft.com/dotnet/announcing-dotnet-7/ - https://devblogs.microsoft.com/dotnet/welcome-to-csharp-11/ - https://devblogs.microsoft.com/dotnet/announcing-fsharp-7/ - https://learn.microsoft.com/en-us/dotnet/core/deploying/native-aot/ - https://go.dev/blog/go1.19 - https://go.dev/blog/go1.18 - https://thephd.dev/c23-is-coming-here-is-what-is-on-the-menu - https://thephd.dev/c23-is-coming-here-is-what-is-on-the-menu#n3017---embed - https://thephd.dev/c23-is-coming-here-is-what-is-on-the-menu#n3006--n3007---type-inference-for-object-definitions - https://www.php.net/archive/2022.php#2022-12-08-1 - https://wiki.php.net/rfc/dnf_types - https://blog.rust-lang.org/ - https://blog.rust-lang.org/2022/01/13/Rust-1.58.0.html#captured-identifiers-in-format-strings - https://blog.rust-lang.org/2022/02/24/Rust-1.59.0.html#inline-assembly - https://blog.rust-lang.org/2022/05/19/Rust-1.61.0.html#more-capabilities-for-const-fn - https://blog.rust-lang.org/2022/08/11/Rust-1.63.0.html#scoped-threads - https://blog.rust-lang.org/2022/11/03/Rust-1.65.0.html#generic-associated-types-gats - https://blog.jetbrains.com/kotlin/2022/06/kotlin-1-7-0-released/ - https://stat.ethz.ch/pipermail/r-announce/2022/000683.html - https://dart.dev/guides/whats-new - https://medium.com/dartlang/dart-2-18-f4b3101f146c - https://medium.com/dartlang/the-road-to-dart-3-afdd580fbefa - https://www.swift.org/blog/swift-5.6-released/ - https://www.swift.org/blog/swift-5.7-released/ - https://www.swift.org/blog/swift-language-updates-from-wwdc22/ - https://www.ruby-lang.org/en/news/2022/12/25/ruby-3-2-0-released/ - https://www.lua.org/news.html - https://www.scala-lang.org/blog/2022/09/05/scala-3.2.0-released.html - https://tjpalmer.github.io/languish/#y=mean&weights=issues%3D1%26pulls%3D0%26stars%3D1%26soQuestions%3D1&names=solidity%2Chaskell%2Cjulia%2Celixir%2Cclojure%2Cperl%2Cgroovy%2Cocaml%2Cgdscript%2Ccmake%2Cnix%2Cvisual+basic+.net - https://blog.soliditylang.org/ - https://downloads.haskell.org/~ghc/9.4.1/docs/users_guide/9.4.1-notes.html - https://julialang.org/blog/2022/08/julia-1.8-highlights/ - https://discourse.julialang.org/t/julia-v1-9-0-beta2-is-fast/92290 - https://elixir-lang.org/blog/2022/09/01/elixir-v1-14-0-released/ - https://elixir-lang.org/blog/2022/10/05/my-future-with-elixir-set-theoretic-types/ - https://clojure.org/news/2022/03/22/clojure-1-11-0 - https://godotengine.org/en/news/default/1 - https://ocaml.org/news/ocaml-5.0 - https://tjpalmer.github.io/languish/#y=mean&weights=issues%3D1%26pulls%3D0%26stars%3D1%26soQuestions%3D1&names=gdscript%2Czig%2Cpascal%2Cfortran%2Cnim%2Cf%23%2Ccommon+lisp%2Cwebassembly%2Ccrystal%2Ccython%2Cvala%2Cerlang%2Chaxe%2Cv%2Cd - https://ziglang.org/download/0.10.0/release-notes.html - https://ziglang.org/news/goodbye-cpp/ - https://nim-lang.org/blog.html - https://nim-lang.org/blog/2022/12/21/version-20-rc.html - https://www.erlang.org/news/157 - https://github.com/WebAssembly/proposals/commits/main - https://github.com/crystal-lang/crystal/releases - https://dlang.org/changelog/2.099.0.html - https://dlang.org/changelog/2.100.0.html - https://dlang.org/changelog/2.101.0.html - https://github.com/odin-lang/Odin/releases - https://gleam.run/news/ - https://gleam.run/news/gleam-v0.22-released/ - https://gleam.run/news/gleam-v0.24-released/ - https://github.com/idris-lang/Idris2/blob/102d7ebc18a9e881021ed4b05186cccda5274cbe/CHANGELOG.md - https://github.com/diku-dk/futhark/blob/master/CHANGELOG.md#02111 - https://grain-lang.org/blog/2022/06/06/new-release-grain-v0.5-durum/ - https://rescript-lang.org/blog/release-10-0-0 - https://www.roc-lang.org/ - https://simon.peytonjones.org/assets/pdfs/haskell-exchange-22.pdf - https://vale.dev/ - https://www.val-lang.dev/

Idris2 has a great syntax for this, see e.g. node018:

I asked because it just baffles me any time I see a dependently typed language using unary numbers. I think to myself, "are these people even educated? Do they know about number systems?" I mean, cavemen were the last group using unary number system as their mainstay, and that was during the Paleolithic. Then they have issues open like this when they discuss optimizing those damn unaries. But this shouldn't even have been a problem for anyone even remotely related to programming! It's giving a terrible impression of dependently-typed languages. Getting rid of those should be the first step of popularizing them.

addOne : Int -> EitherT String IO Int addOne x = pure $ x + 1 add : Int -> Int -> EitherT String IO Int add x y = pure $ x + y main : IO () main = do exportFn `{add} the (IO ()) $ exportFn `{addOne} -- `the (IO ())` is needed due to issue https://github.com/idris-lang/Idris2/issues/2851

The idea of world state is an attraction of the state that side effects change. I got the idea from Idris internals: https://github.com/idris-lang/Idris2/blob/main/libs/prelude/PrimIO.idr

This was recently deemed inappropriate:

"Bye bye Set"

"Set and Prop are removed as keywords"

https://github.com/agda/agda/pull/4629

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.

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 well-known but is very cool.

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.

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?

That's absolutely untrue. From the horse's mouth:

Agda has the so-called 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/mixfix-operators.html - https://github.com/agda/agda/blob/master/examples/Introduction/Operators.agda - https://github.com/agda/agda-stdlib/blob/master/src/Data/Product/Base.agda

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.

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.