Re: [HoTT] New theorem prover Arend is released

[Nicolai Kraus <nicolai.kraus@gmail.com>]

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Exciting and impressive! Many of Arend's features sound very useful!
I find the universes most interesting, but I don't understand how exactly
they work. From your site, I learned:
* A universe in Arend has two parameters, for size and for homotopy level.
* Universes are cumulative in both parameters.

Is type-checking still decidable? Say, we have this judgment:
(1)  A : \3-Type 1
We have
(2)  A : \4-Type 1
by cumulativity, but it shouldn't be decidable whether we also have
(3)  A : \2-Type 1.
In book-HoTT, only (1) would type-check, since A would be a pair of a type
and a proof of its homotopy level (i.e. you can only get (2,3) by changing
the proof). But your site says these proofs don't need to be carried
around. Are they still somewhere hidden in the background?



On Wed, Aug 7, 2019 at 12:16 AM Валерий Исаев <valery.isaev@gmail.com>
wrote:

> Arend is a new theorem prover that have been developed at JetBrains
> <https://www.jetbrains.com/> for quite some time. We are proud to
> announce that the first version of the language was released! To learn more
> about Arend, visit our site <https://arend-lang.github.io/>.
>
> Arend is based on a version of homotopy type theory that includes some of
> the cubical features. In particular, it has native higher inductive types,
> including higher inductive-inductive types. It also has other features
> which are necessary for a theorem prover such as universe polymorphism and
> class system. We believe that a theorem prover should be convenient to use.
> That is why we also developed a plugin for IntelliJ IDEA
> <https://www.jetbrains.com/idea/> that turns it into a full-fledged IDE
> for the Arend language. It implements many standard features such as syntax
> highlighting, completion, auto import, and auto formatting. It also has
> some language-specific features such as incremental typechecking and
> various refactoring tools.
>
> To learn more about Arend, you can check out the documentation
> <https://arend-lang.github.io/documentation>. You can also learn a lot
> from studying the standard library
> <https://github.com/JetBrains/arend-lib>. It implements some basic
> algebra, including localization of rings, and homotopy theory, including
> joins, modalities, and localization of types.
>
> Frequently asked questions (that nobody asked):
>
>    - Why do we need another theorem prover? We believe that a theorem
>    prover should be convenient to use. This means that it should have an IDE
>    comparable to that of mainstream programming languages. That is why we
>    implemented IntelliJ Arend
>    <https://arend-lang.github.io/about/intellij-features>. This also
>    means that the underlying theory should be powerful and expressive. That is
>    why Arend is based on homotopy type theory and has features such as an
>    impredicative type of propositions and a powerful class system.
>    - Does Arend have tactics? Not yet, but we are working on it.
>    - Does Arend have the canonicity property, i.e. does it evaluate
>    closed expressions to their canonical forms? No, but it computes more terms
>    than ordinary homotopy type theory, which makes it more convenient in many
>    aspects.
>
> If you want to know about language updates, you can follow us on twitter
> <https://twitter.com/ArendLang>. Questions, suggestions, and comments are
> welcome at google groups
> <https://groups.google.com/forum/#!forum/arend-lang>.
>
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