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 > > > . > > > > > > -- > > > You received this message because you are subscribed to the Google > > > Groups "Homotopy Type Theory" group. > > > To unsubscribe from this group and stop receiving emails from it, send > > > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > > > To view this discussion on the web visit > > > > https://groups.google.com/d/msgid/HomotopyTypeTheory/9d23061c-4b7a-4d69-9c22-f28261ad3b33%40googlegroups.com > < > https://groups.google.com/d/msgid/HomotopyTypeTheory/9d23061c-4b7a-4d69-9c22-f28261ad3b33%40googlegroups.com?utm_medium=email&utm_source=footer > >. > > > > -- > > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/06e24c98-7409-4e75-88ee-a6e1bb891e1e%40www.fastmail.com > . > > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuSLoX8gKy54NQM6SNoi43wVA0A1Ad59qKs6prULkh6zBw%40mail.gmail.com > . > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. 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