[Coq-Club] [Agda] Re: [HoTT] newbie questions about homotopy
theory & advantage of UF/Coq
Conor McBride
conor at strictlypositive.org
Tue Jan 7 18:25:27 CET 2014
Hi Maxime
On 7 Jan 2014, at 16:56, Maxime D=A8=A6n=A8=A8s <mail at maximedenes.fr> =
wrote:
> Still, I don't understand why replacing the pattern matching in my
> example by a call to subst makes the termination check fail.
A call to subst destroys the "proper subterm" status, whereas
pattern matching naively preserves it.
>=20
> Maxime.
>> {-# OPTIONS --without-K #-}
>>=20
>> module BadWithoutK where
>>=20
>> data Zero : Set where
>>=20
>> data WOne : Set where
>> wrap : (Zero -> WOne) -> WOne
>>=20
>> data _<->_ (X : Set) : Set -> Set where
>> Refl : X <-> X
>>=20
>> postulate
>> myIso : WOne <-> (Zero -> WOne)
>>=20
>> moo : WOne -> Zero
>> noo : (X : Set) -> (WOne <-> X) -> X -> Zero
>>=20
>> moo (wrap f) =3D noo (Zero -> WOne) myIso f
>> noo .WOne Refl x =3D moo x
moo's input is bigger than the third argument in its noo call
noo's third input is the same as the argument in its moo call
Size-change termination, how are ye?
When you do the translation to eliminators, you're obliged to
show how recursive calls correspond to invocations of the
inductive hypothesis: here that's just not going to happen.
Transporting f across myIso does indeed give an element of
WOne (your Box), but that does not make a nullary inductive
hypothesis any easier to invoke.
I'd quite like to see us defining a type theory in which the
only checking is type checking, then using it as a target for
elaboration. Eliminators are one candidate, but there are
others. Sized types are certainly another strong candidate.
I'm also interested to see whether there are "locally clocked"
explanations for termination, as there seem to be for
productivity.
Interesting times
Conor
>>=20
>> bad : Zero
>> bad =3D moo (wrap \ ())
>>=20
>>=20
>>=20
>>> T.
>>>=20
>>>> I am also not sure how specific the problem is to univalence. As =
Cody
>>>> said, I would expect to find some set-theoretical models where the
>>>> equality holds. So being able to prove the negation is disturbing.
>>>>=20
>>>> Maxime.
>>>>=20
>>>> On 01/06/2014 07:23 PM, Abhishek Anand wrote:
>>>>> It is known that UIP is inconsistent with univalence.
>>>>> Does this typecheck even after disabling UIP(=3D=3DK axiom?) ?
>>>>> I might be wrong, but I think that UIP is baked into the =
typechecker
>>>>> of Agda.
>>>>> But, UIP can be disabled somehow?
>>>>>=20
>>>>>=20
>>>>> -- Abhishek
>>>>> http://www.cs.cornell.edu/~aa755/ =
<http://www.cs.cornell.edu/%7Eaa755/>
>>>>>=20
>>>>>=20
>>>>> On Mon, Jan 6, 2014 at 3:42 PM, Maxime D=A8=A6n=A8=A8s =
<mail at maximedenes.fr
>>>>> <mailto:mail at maximedenes.fr>> wrote:
>>>>>=20
>>>>> Bingo, Agda seems to have the same problem:
>>>>>=20
>>>>> module Termination where
>>>>>=20
>>>>> open import Relation.Binary.Core
>>>>>=20
>>>>> data Empty : Set where
>>>>>=20
>>>>> data Box : Set where
>>>>> wrap : (Empty =A1=FA Box) =A1=FA Box
>>>>>=20
>>>>> postulate
>>>>> iso : (Empty =A1=FA Box) =A1=D4 Box
>>>>>=20
>>>>> loop : Box -> Empty
>>>>> loop (wrap x) rewrite iso =3D loop x
>>>>>=20
>>>>> gift : Empty =A1=FA Box
>>>>> gift ()
>>>>>=20
>>>>> bug : Empty
>>>>> bug =3D loop (wrap gift)
>>>>>=20
>>>>> However, I may be missing something due to my ignorance of Agda.
>>>>> It may be well known that the axiom I introduced is =
inconsistent.
>>>>> Forgive me if it is the case.
>>>>>=20
>>>>> Maxime.
>>>>>=20
>>>>>=20
>>>>> On 01/06/2014 01:15 PM, Maxime D=A8=A6n=A8=A8s wrote:
>>>>>=20
>>>>> The anti-extensionality bug is indeed related to =
termination.
>>>>> More precisely, it is the subterm relation used by the guard
>>>>> checker which is not defined quite the right way on =
dependent
>>>>> pattern matching.
>>>>>=20
>>>>> It is not too hard to fix (we have a patch), but doing so
>>>>> without ruling out any interesting legitimate example =
(dealing
>>>>> with recursion on dependently typed data structures) is more
>>>>> challenging.
>>>>>=20
>>>>> I am also curious as to whether Agda is impacted. Let's try =
it
>>>>> :)
>>>>>=20
>>>>> Maxime.
>>>>>=20
>>>>> On 01/06/2014 12:59 PM, Altenkirch Thorsten wrote:
>>>>>=20
>>>>> Which bug was this?
>>>>>=20
>>>>> I only saw the one which allowed you to prove
>>>>> anti-extensionality? But
>>>>> this wasn't related to termination, or?
>>>>>=20
>>>>> Thorsten
>>>>>=20
>>>>> On 06/01/2014 16:54, "Cody Roux" =
<cody.roux at andrew.cmu.edu
>>>>> <mailto:cody.roux at andrew.cmu.edu>> wrote:
>>>>>=20
>>>>> Nice summary!
>>>>>=20
>>>>>=20
>>>>> On 01/06/2014 08:49 AM, Altenkirch Thorsten wrote:
>>>>>=20
>>>>> Agda enforces termination via a termination
>>>>> checker which is more
>>>>> flexible but I think less principled than Coq's
>>>>> approach.
>>>>>=20
>>>>> That's a bit scary given that there was an
>>>>> inconsistency found in
>>>>> the Coq termination checker just a couple of weeks =
ago
>>>>> :)
>>>>>=20
>>>>> BTW, has anyone tried reproducing the bug in Agda?
>>>>>=20
>>>>>=20
>>>>> Cody
>>>>>=20
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