Yes, Arend implements the theory described in this document. Semantically,
the additional constructions of this theory correspond to the assumption
that the model has a fibrant object I such that maps : X -> X
\times I have the left lifting property with respect to fibrations, and the
path object functor is given by (-)^I. So, the usual interpretation in
model categories (and other similar models) of HoTT extends to an
interpretation of this theory if the model category is a Cartesian model
category.
Regards,
Valery Isaev
чт, 8 авг. 2019 г. в 15:29, Bas Spitters :
> I imagine it could be related to earlier discussions, but Valery will
> correct me:
> https://groups.google.com/forum/#!topic/homotopytypetheory/N8jw_5h2Qjs
> https://valis.github.io/doc.pdf
>
> On Thu, Aug 8, 2019 at 2:20 PM Jon Sterling wrote:
> >
> > Dear Valery,
> >
> > Arend looks really impressive, especially the IDE features! I look
> forward to trying it. I like the little screen demos on the website.
> >
> > We have been curious for some time if someone could begin to explain
> what type theory Arend implements --- I am not necessarily looking for
> something super precise, but it would be great to have a high-level gloss
> that would help experts in the semantics of HoTT understand where Arend's
> type theory lies. For instance, I can see that Arend uses an interval, but
> this interval seems to work a bit differently from the interval in some
> other type theories. Is there any note or document that explains some of
> the mathematics behind Arend?
> >
> > Nice work! And I look forward to hearing and reading more.
> >
> > Best,
> > Jon
> >
> >
> > On Tue, Aug 6, 2019, at 6:16 PM, Валерий Исаев wrote:
> > > Arend is a new theorem prover that have been developed at JetBrains
> > > 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 .
> > >
> > > 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 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
> > > . You can also learn a lot
> > > from studying the standard library
> > > . 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
> > > . 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 . Questions, suggestions, and
> > > comments are welcome at google groups
> > > .
> > >
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