On Friday, 8 March 2019 15:13:14 UTC, dlicata wrote: > > nice! the negation trick is clever. > Yes, this is really nice. Martin, I don’t understand why this situation is any different than natural > numbers, though. If I have a closed term of type nat, I can normalize it, > but then I (externally) need to read through all of the successors to > figure out what number it is. (Or maybe I can only weak head normalize it, > in which case I need to interleave further weak head normalization after > every successor.) Is this an unbounded search? The parallel is to read > “hcom []” as successor and “|x,a|” as zero. > > Except that you would have different normal forms for the same thing, and this thing would be prefixed by an unbounded number of *printed* clock ticks. It is not right to say that unbounded search is needed, I agree. But it wouldn't be pleasant to have to see 10^6 ticks. I would rather wait for 10^6 units of time (which should be fast) and then see the answer than see a million hcomp's followed by the answer on my computer screen . :-) Fortunately, Anders says that the clock ticks are not printed in the version of cubical type theory implemented by Agda --cubical, which is nice! Martin > -Dan > > > On Mar 8, 2019, at 9:59 AM, Anders Mortberg > wrote: > > > > In fact, in Cubical Agda you will not get these hcomp's with empty > > systems. The reason is that because of the way hcomp works in Agda > > there is a very nice trick to implement the "generalized hcomp" > > operation of the paper that Dan linked to. I summarized the trick in: > > > > https://github.com/agda/agda/issues/3415 > > > > I added this to Agda some month ago and it should be possible to > > update Simon's canonicity proof to get a stronger result saying that > > the only elements of HITs in the empty context are point constructors > > (like in the AFH paper). For this to work you also have to impose a > > "validity" constraint (Def 12 in the ccctt paper Dan linked to) so > > that empty systems cannot result from substitutions. This is currently > > not done in Cubical Agda, but if you start with a term with only valid > > systems then you should never get an empty system. > > > > So the extraction of witnesses from existence statements should work > > as Martín said in Cubical Agda. > > > > -- > > Anders > > > > On Thu, Mar 7, 2019 at 6:23 PM Martín Hötzel Escardó > > > wrote: > >> > >> And this is a wildly speculative question. If we used Andrew Swan's > identity type derived from the cubical path type only, as in the abstract > library file > https://github.com/agda/cubical/blob/master/Cubical/Core/HoTT-UF.agda) > would we still get this phenomenon? Maybe not? What I mean is that we use > normal Agda, together with what is offered in that file and nothing else > (so that we are using HoTT book axiomatic type theory). Martin > >> > >> On Thursday, 7 March 2019 23:01:33 UTC, Martín Hötzel Escardó wrote: > >>> > >>> Oh, this is annoying, because it seems to mean that we would need > unbounded search (to drop all "hcom []"'s) until we can read the |x,a|, > which is against the spirit of, say, Martin-Loef type theories. Martin > >>> > >>> On Thursday, 7 March 2019 22:51:20 UTC, dlicata wrote: > >>>> > >>>> That would be true if the term you are normalizing is in the empty > interval context, and the cubical type theory has “empty system regularity” > (like https://www.cs.cmu.edu/~cangiuli/papers/ccctt.pdf). > >>>> > >>>> Otherwise, if you evaluate something in the empty interval context, > you might see something like > >>>> hcom [] (hcom [] (hcom [] (hcom [] (… |x,a| … )))) > >>>> with |x,a| in there somewhere. In HITs, Kan composition is treated > as a constructor of the type, and though there are no interesting lines to > compose in the empty interval context, the uninteresting compositions don’t > vanish in all flavors of cubical type theory. > >>>> > >>>>> On Mar 7, 2019, at 5:41 PM, Martín Hötzel Escardó < > escardo...@gmail.com> wrote: > >>>>> > >>>>> So I presume that when we ask cubical Agda to normalize a term of > type || Sigma (x:X), A x || we will get a term of the form |x,a| and so we > will see the x in normal form, where |-| is the map into the truncation, > right? Martin. > >>>>> > >>>>> On Thursday, 7 March 2019 21:52:12 UTC, Anders Mörtberg wrote: > >>>>> The existence property is proved for CCHM cubicaltt by Simon in: > >>>>> > >>>>> https://arxiv.org/abs/1607.04156 > >>>>> > >>>>> See corollary 5.2. This works a bit more generally than what Martín > said, in particular in any context with only dimension variables we can > compute a witness to an existence. So if in context G = i_1 : II, ..., i_n > : II (possibly empty) we have: > >>>>> > >>>>> G |- t : exists (x : X), A(x) > >>>>> > >>>>> then we can compute G |- u : X so that G |- B(u). > >>>>> > >>>>> -- > >>>>> Anders > >>>>> > >>>>> On Thursday, March 7, 2019 at 11:16:48 AM UTC-5, Martín Hötzel > Escardó wrote: > >>>>> I got confused now. :-) > >>>>> > >>>>> Seriously now, what you say seems related to the fact that from a > proof |- t : || X || in the empty context, you get |- x : X in cubical type > theory. This follows from Simon's canonicity result (at least for X=natural > numbers), and is like the so-called "existence property" in the internal > language of the free elementary topos. This says that from a proof |- > exists (x:X), A x in the empty context, you get |- x : X and |- A x. This > says that exists in the empty context behaves like Sigma. But only in the > empty context, because otherwise it behaves like "local existence" as in > Kripke-Joyal semantics. > >>>>> > >>>>> Martin > >>>>> > >>>>> On Thursday, 7 March 2019 14:10:56 UTC, dlicata wrote: > >>>>> Just in case anyone reading this thread later is confused about a > more beginner point than the ones Nicolai and Martin made, one possible > stumbling block here is that, if someone means “is inhabited” in an > external sense (there is a closed term of that type), then the answer is > yes (at least in some models): if ||A|| is inhabited then A is inhabited. > For example, in cubical models with canonicity, it is true that a closed > term of type ||A|| evaluates to a value that has as a subterm a closed term > of type A (the other values of ||A|| are some “formal compositions” of > values of ||A||, but there has to be an |a| in there at the base case). > This is consistent with what Martin and Nicolai said because “if A is > inhabited then B is inhabited” (in this external sense) doesn’t necessarily > mean there is a map A -> B internally. > >>>>> > >>>>> -Dan > >>>>> > >>>>>> On Mar 5, 2019, at 6:07 PM, Martín Hötzel Escardó < > escardo...@gmail.com> wrote: > >>>>>> > >>>>>> Or you can read the paper https://lmcs.episciences.org/3217/ > regarding what Nicolai said. > >>>>>> > >>>>>> Moreover, in the HoTT book, it is shown that if || X||->X holds for > all X, then univalence can't hold. (It is global choice, which can't be > invariant under equivalence.) > >>>>>> > >>>>>> The above paper shows that unrestricted ||X||->X it gives excluded > middle. > >>>>>> > >>>>>> However, for a lot of kinds of types one can show that ||X||->X > does hold. For example, if they have a constant endo-function. Moreover, > for any type X, the availability of ||X||->X is logically equivalent to the > availability of a constant map X->X (before we know whether X has a point > or not, in which case the availability of a constant endo-map is trivial). > >>>>>> > >>>>>> Martin > >>>>>> > >>>>>> On Tuesday, 5 March 2019 22:47:55 UTC, Nicolai Kraus wrote: > >>>>>> You can't have a function which, for all A, gives you ||A|| -> A. > See the exercises 3.11 and 3.12! > >>>>>> -- Nicolai > >>>>>> > >>>>>> On 05/03/19 22:31, Jean Joseph wrote: > >>>>>>> Hi, > >>>>>>> > >>>>>>> From the HoTT book, the truncation of any type A has two > constructors: > >>>>>>> > >>>>>>> 1) for any a : A, there is |a| : ||A|| > >>>>>>> 2) for any x,y : ||A||, x = y. > >>>>>>> > >>>>>>> I get that if A is inhabited, then ||A|| is inhabited by (1). But > is it true that, if ||A|| is inhabited, then A is inhabited? > >>>>>>> -- > >>>>>>> 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. > >>>>>> > >>>>>> > >>>>>> -- > >>>>>> 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. > >>>>> > >>>>> > >>>>> -- > >>>>> 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. > >>>> > >> -- > >> 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. > > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. 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