OK, thanks. I stand corrected. But in that case, I'm with Thorsten: 2-level seems easier. On Monday, February 11, 2019 at 3:04:58 AM UTC-5, Валерий Исаев wrote: > > Hi Matt, > > I should point out that when I say "a type theory" I mean an ordinary > dependently typed language, without any strict equalities or 2-levelness. > Any such theory has only (dependent) types and terms and usual structural > rules. > Any "coherence problem" is solved by simply adding an infinite tower of > terms. For example, in MLTT, if f : A -> B -> C, you can define function > swap(f) = \x y -> f y x : B -> A -> C. Then swap \circ swap = id > definitionally. Of course, you cannot have such a definitional equality in > Q(MLTT), but instead you have a propositional equality p : Id(swap \circ > swap, id) and also an infinite tower of terms, which assure that p is > coherent. Any rule that holds definitionally in MLTT will be true in > Q(MLTT) propositionally. > > Regards, > Valery Isaev > > > пн, 11 февр. 2019 г. в 10:01, Matt Oliveri > >: > >> I think you're right. From discussions about autophagy, it seems like no >> one knows how to match judgmental equality using equality types, unless >> that equality type family is propositionally truncated in some way. >> >> Consequently, my guess is that Valery's Q transformation actually yields >> something rather like a 2-level system. >> >> On Saturday, February 9, 2019 at 7:30:07 AM UTC-5, Thorsten Altenkirch >> wrote: >>> >>> Hi, >>> >>> >>> >>> what we need is a strict equality on all types. If we would state the >>> laws of type theory just using the equality type we would also need to add >>> coherence laws. Since I would include the laws for substitution (never >>> understood why substitution is different from application) this would >>> include the laws for infinity categories and this would make even basic >>> type theory certainly much more complicated if not unusable. Instead one >>> introduces a 2-level system with strict equality on one level and weak >>> equality on another. For historic and pragmatic reasons this is combined >>> with the computational aspects of type theory which is expressed as >>> judgemental equality. However, there are reasons to separate these >>> concerns, e.g. to work with higher dimensional constructions in type theory >>> such as semi-simplicial types it is helpful to work with hypothetical >>> strict equalities (see our paper ( >>> http://www.cs.nott.ac.uk/~psztxa/publ/csl16.pdf). >>> >>> >>> >>> I do think that the computational behaviour of type theory is important >>> too. However, this can be expressed by demandic a form of computational >>> adequacy, that is for every term there is a strictly equal normal form. It >>> is not necessary that strict equality in general is decidable (indeed >>> different applications of type theory may demand different decision >>> procedures). >>> >>> >>> >>> Thorsten >>> >> -- 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. For more options, visit https://groups.google.com/d/optout.