From: Valery Isaev <valery.isaev@gmail.com> To: Jon Sterling <jon@jonmsterling.com> Cc: "'Martin Escardo' via Homotopy Type Theory" <HomotopyTypeTheory@googlegroups.com> Subject: Re: [HoTT] New theorem prover Arend is released Date: Thu, 8 Aug 2019 18:22:52 +0300 Message-ID: <CAA520fuU-BEcdg6mrAkTxmqpDhG9pHpD-1a=X7ZU-n8q57dKsg@mail.gmail.com> (raw) In-Reply-To: <d0fd1d18-136b-41fa-b721-f64b9c487376@www.fastmail.com> [-- Attachment #1: Type: text/plain, Size: 9272 bytes --] чт, 8 авг. 2019 г. в 18:11, Jon Sterling <jon@jonmsterling.com>: > Hi Valery (and Bas), > > Thanks very much for the reference! Can you say a bit more about the > interpretation in model categories? I have the following specific questions: > > - By "Cartesian model category" do you refer to the pushout-product axiom? > I apologize for my ignorance, model categories are not my area of expertise. > > Yes. > - In most models of HoTT that I'm familiar with, including simplicial sets > and several versions of cubical sets, the interval is not fibrant. What > interpretation do have in mind for I? (btw, sorry if this has been > discussed before!). > You can take any contractible fibrant object such that the map 1+1 -> I is a cofibration. Such an object always exists (just factor the map 1+1 -> 1 as a cofibration followed by trivial fibration), so the only additional assumption is that the exponential (-)^I exists and the pushout-product axiom holds. > > Thanks, > Jon > > > On Thu, Aug 8, 2019, at 10:45 AM, Valery Isaev wrote: > > 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 > > <id,left> : 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 <b.a.w.spitters@gmail.com>: > > > 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 <jon@jonmsterling.com> > 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 > > > > > <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>. > > > > > > > > > > -- > > > > > 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 > <mailto:HomotopyTypeTheory%2Bunsubscribe@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 <mailto: > HomotopyTypeTheory%2Bunsubscribe@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 <mailto: > HomotopyTypeTheory%2Bunsubscribe@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. > > 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/CAA520ft6xBR1fJz4N0c5NvB%2BpWD%2B14RPCu5g32cxv%2BYdbEmd0g%40mail.gmail.com > < > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAA520ft6xBR1fJz4N0c5NvB%2BpWD%2B14RPCu5g32cxv%2BYdbEmd0g%40mail.gmail.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/d0fd1d18-136b-41fa-b721-f64b9c487376%40www.fastmail.com > . > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. 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next prev parent reply indexThread overview:20+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-08-06 22:16 Валерий Исаев 2019-08-07 15:01 ` Andrej Bauer 2019-08-07 22:13 ` Nicolai Kraus 2019-08-08 9:55 ` Valery Isaev 2019-08-10 9:47 ` Michael Shulman 2019-08-10 12:30 ` Valery Isaev 2019-08-10 12:37 ` Valery Isaev 2019-08-08 12:20 ` Jon Sterling 2019-08-08 12:29 ` Bas Spitters 2019-08-08 14:44 ` Valery Isaev 2019-08-08 15:11 ` Jon Sterling2019-08-08 15:22 ` Valery Isaev [this message]2019-08-10 9:42 ` Michael Shulman 2019-08-10 12:24 ` Valery Isaev 2019-08-10 23:37 ` Michael Shulman 2019-08-11 10:46 ` Valery Isaev 2019-08-11 12:39 ` Michael Shulman 2019-08-11 16:55 ` Michael Shulman 2019-08-12 14:44 ` Daniel R. Grayson 2019-08-12 17:32 ` Michael Shulman

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