Only valid for simply-typed languages, so not (yet) contradicting Jon's claim: admissibility of substitution is important in the development of Howe's method for proving that applicative bisimilarity is a congruence. Essentially, the reason is that it provides a simpler induction principle. This is implicit in my recent work with Peio Borthelle and Ambroise Lafont on a categorical framework for Howe's method (e.g., [1]), used as motivation for "A unified treatment of structural definitions on syntax..." [2], and explicitly used (and extended to operations other than substitution) in ongoing work on a generalisation of [1]. More generally, I suspect that it is useful in programming language theory, where people tend to work with extrinsic presentations. [1] https://hal.archives-ouvertes.fr/hal-02966439v6 [2] https://hal.archives-ouvertes.fr/hal-03633933 Le ven. 18 nov. 2022 à 03:35, Michael Shulman a écrit : > As far as the mathematical study of type theories and their models goes, > that may be true. But I believe that when talking about the way type > theories are used in practice, either on paper or in a proof assistant, > there is still a difference. > > Suppose I am teaching a calculus class, and I define f(x) = x^2 + 1 and I > want to evaluate f(3). I don't write > > f(3) = (x^2+1)[3/x] = (x^2)[3/x] + 1[3/x] = 3^2 + 1 = 9 + 1 = 10. > > Instead, I jump right to f(3) = 3^2+1, because substitution is an > operation that happens immediately in my head, not a computational step > analogous to 3^2 = 9. Similarly, the user of a proof assistant never types > or sees substitution as part of the syntax; it is an operation *on* syntax > that happens behind the scenes. > > Yes, this is a property of a particular *presentation* of a free structure > rather than a property of the structure itself, but that doesn't > intrinsically make it unimportant. Groups and group presentations are both > important. Maybe you want to say that "a type theory" refers to the free > structure rather than its presentation, but choosing to use words in that > way doesn't by itself make "presentations of type theories" (or whatever > you call them) less important. Maybe you want to define an "admissible > rule" to be a property that holds in the syntax but fails in some actual > models; but that doesn't make "rules that hold in all models and can be > made to hold in a particular presentation of the free model without being > given explicitly as generating operations/equalities" (or whatever you call > them) less important. > > I do agree that Andrej's formulation sounds backwards. In practice (in my > experience) one doesn't write the rules down first and then try to figure > out what kind of substitution is admissible. Instead one decides what the > substitution rule should be, and *then* (hopefully) works out a way of > presenting the syntax to make that substitution rule admissible. This is > particularly tricky for modal type theories, where the categorically > "obvious" rules do not admit substitution, and leads to the introduction of > split contexts as used in the BFP paper. I have trouble imagining how I > could have written that paper if I had been forced to write explicit > substitutions everywhere. Thorsten and Jon, do you maintain that all the > work that's gone into figuring out ways to present modal type theories with > "admissible substitution" is meaningless? > > On Thu, Nov 17, 2022 at 5:37 AM Jon Sterling wrote: > >> Indeed, I echo Thorsten's comment — to put it another way, even being >> able to tell whether these rules are derivable or only admissible is like >> knowing what an angel's favorite TV show is (in other words, a form of >> knowledge that cannot be applied toward anything by human beings). At least >> for structural type theory, there is nothing worth saying that cannot be >> phrased in a way that does not depend on whether structural rules are >> admissible or derivable. It may be that admissiblity of structural rules >> starts to play a role in substructural type theory, however, but this is >> not my area of expertise. >> >> It is revealing that nobody has proposed a notion of **model** of type >> theory in which the admissible structural rules do not hold; this would be >> the necessary form taken by any evidence for the thesis that it is >> important for structural rules to not be derivable. Absent such a notion of >> model and evidence that it is at all compelling/useful, we would have to >> conclude that worrying about admissibility vs. derivability of structural >> rules in the official presentation of type theory is fundementally >> misguided. >> >> On 16 Nov 2022, at 4:52, 'Thorsten Altenkirch' via Homotopy Type Theory >> wrote: >> >> That depends on what presentation of Type Theory you are using. Your >> remarks apply to the extrinsic approach from the last millennium. More >> recent presentation of Type Theory built in substitution and weakening and >> use an intrinsic approach which avoids talking about preterms you don’t >> really care about. >> >> >> >> https://dl.acm.org/doi/10.1145/2837614.2837638 >> >> >> >> Cheers, >> >> Thorsten >> >> >> >> *From:* homotopytypetheory@googlegroups.com < >> homotopytypetheory@googlegroups.com> on behalf of andrej.bauer@andrej.com >> >> *Date:* Tuesday, 15 November 2022 at 22:39 >> *To:* Homotopy Type Theory >> *Subject:* Re: [HoTT] Question about the formal rules of cohesive >> homotopy type theory >> >> > Does this also include the structural rules of type theory such as the >> substitution and weakening rules? >> >> I would just like to point out that substutition and weakening typically >> are not part of the rules. They are shown to be admissible. In this spirit, >> the question should have been: what is the precise version of substitution >> and weakening (which is a special case of substitution) that is admissible >> in cohesive type theory? >> >> With kind regards, >> >> Andrej >> >> -- >> 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/D66F4584-A005-4F69-8E75-E976E0FF9957%40andrej.com >> . >> >> This message and any attachment are intended solely for the addressee >> and may contain confidential information. If you have received this >> message in error, please contact the sender and delete the email and >> attachment. >> >> Any views or opinions expressed by the author of this email do not >> necessarily reflect the views of the University of Nottingham. Email >> communications with the University of Nottingham may be monitored >> where permitted by law. >> >> >> >> >> -- >> 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/PAXPR06MB786979CA94519BCC98EDD32FCD079%40PAXPR06MB7869.eurprd06.prod.outlook.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/41C2FBD7-7C3B-4D6D-A444-13FA43EDD1CF%40jonmsterling.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/CADYavpxcTpvy6%2BBS%2B-5yjOjVFkdXFHdmCX0U3Qre2J6t8Lfh_g%40mail.gmail.com > > . > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. 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