<div dir="ltr">Actually, if I am not mistaken, this is not a problem after all — because one has to also add a corresponding extra clause for eqU in the recursive definition of El, which one adds as<div><br></div><div>ap El (eqU p) == p</div>
<div><br></div><div>and from this, one can prove that El is an equivalence on paths. (Thorsten and I have been discussing this in a little more detail off-list; I think he will be posting a longer follow-up email later.)</div>
<div><br></div><div>–p.</div><div><br></div><div><br></div></div><div class="gmail_extra"><br><br><div class="gmail_quote">On Fri, May 23, 2014 at 5:27 PM, Guillaume Brunerie <span dir="ltr"><<a href="mailto:guillaume.brunerie@gmail.com" target="_blank">guillaume.brunerie@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><p dir="ltr">Hi Thorsten,</p>
<p dir="ltr">I don't know about the elimination rule, but I just wanted to point out that you won't get a univalent universe in this way because eqU refl and refl won't be identified, and similarly for concat (eqU p) (eqU q) and eqU (concat p q).<br>
And the obvious way to fix requires something like defining semi-simplicial types internally...</p>
<p dir="ltr">Guillaume</p>
<div class="gmail_quote">Le 23 mai 2014 15:23, "Altenkirch Thorsten" <<a href="mailto:psztxa@exmail.nottingham.ac.uk" target="_blank">psztxa@exmail.nottingham.ac.uk</a>> a écrit :<br type="attribution"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div><div class="h5">
<div style="word-wrap:break-word;color:rgb(0,0,0);font-size:14px;font-family:Calibri,sans-serif"><div>I would like to combine higher inductive definitions (I.e. have path constructors) with induction-recursion. One application would be to define a closed universe which is univalent. However, I cannot see any reasonable way to define an eliminator.</div>
<div><br></div><div>Ok, I start with a simple universe(using Agda) - an inductive recursive definition</div><div><br></div><div><div>data U : Set </div><div>El : U -> Set</div><div><br></div><div>data U where</div><div>
nat : U</div><div> pi : (a : U)(b : El a -> U) -> U</div><div><br></div><div>El nat = Nat</div><div>El (pi a b) = (x : El a) -> El (b x)</div></div><div><br></div><div>Now I can define an eliminator for the universe which allows me to define dependent functions by recursion over type codes:</div>
<div><br></div><div><div>ElimU : (X : U -> Set) </div><div> -> (X nat)</div><div> -> ((a : U) -> X a -> (b : El a -> U) -> ((x : El a) -> X (b x)) -> X (pi a b))</div><div> -> (a : U) -> X a</div>
<div>ElimU X n p nat = n</div><div>ElimU X n p (pi a b) = p a (ElimU X n p a) b (λ x -> ElimU X n p (b x))</div><div> </div></div><div>However, I also would like to add a path constructor, which identifies codes if the have the same semantics:</div>
<div><br></div><div><div>postulate </div><div> eqU : forall {a b} -> El a == El b -> a == b</div></div><div><br></div><div>But I don't see a good way to modify the Eliminator. It seems that this corresponds to a condition on the eliminator as a whole.</div>
<div><br></div><div>An alternative is to quotient the universe afterwards. However, the problem is that in this case I cannot lift the pi constructor to the quotiented universe – the usual problem when quotienting infinitary constructors. This can usually be overcome by defining the path constructors mutually..</div>
<div><br></div><div>Any ideas? Maybe the whole thing doesn't make sense semantically?</div><div><br></div><div>Thorsten</div>
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