<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; 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 (£f 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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