The Coq mailing list

[Altenkirch Thorsten <psztxa AT exmail.nottingham.ac.uk>]

>
>
>
>Higher-order datatypes are a bit more difficult, and I admit I never
>worked it out. I believe it's possible though.
>
>>> The fact that the implementation is behind the theoretical capabilities
>>> is just that, an implementation problem.
>> The eliminators are like axioms in set theory and it is worthwhile to
>> translate any extension by reducing them to the axiom instead of adding
>> new ones even though they may be semantically justified.
>That's fair, but I don't see why eliminators have a more powerful
>ontological status than sized-types.

It is maybe not the ontological status but they are a hell easier to write
down!

Do I believe in a foundational system which consists of pages with rules
full of greek (and other alphabets) and super sub and whatever scripts?

Actually I also don't like the schemes for datatypes either, hence I would
like to reduce everything to fixed collection of type formers (we have
done a fair bit of this using containers).

So my target type theory is 0,1,2,Pi,Sigma,Id,U_n with eliminators. Ok we
will need to consider extensions for induction recursion and HITs but they
should also presented by a finite collection of constants with eliminators.

I know there are some gaps in between what we like to do and what we know
it is safe to do but I think we should be able to narrow this gap.

Thorsten

>They require work to be derived as
>well, and the correctness proofs for them are quite similar to that for
>sized types. This seems like a matter of preference. Obviously I'm not
>neutral in this matter though.
>
>Best,
>
>Cody
>
>
>
>> Thorsten
>>
>>
>>> Best,
>>>
>>> Cody
>>>
>>>
>>>> V.
>>>>
>>>>
>>>> On Jan 6, 2014, at 6:27 PM, Cody Roux 
>>>> <cody.roux AT andrew.cmu.edu>
>>>>wrote:
>>>>
>>>>> The thing about Coq is that it's very disturbing not to have a
>>>>> Set-theoretic interpretation where Prop is the set of Booleans {True,
>>>>> False}. Agda of course doesn't have these qualms, but the fact that
>>>>> this
>>>>> issue appears in Coq without use of anything fancy (impredicativity,
>>>>> etc) makes me still a bit worried. In particular there should be a
>>>>> Set-theoretic model for Agda where Empty -> Box and Box -are- equal:
>>>>> with a sufficiently clever interpretation of inductives, I believe I
>>>>> can
>>>>> get [[Empty -> Box ]] = [[Box]] = { {} } (where {} is just the empty
>>>>> set). Can someone with more Agda knowledge try to type the original
>>>>> problematic term?
>>>>>
>>>>>
>>>>> Hypothesis Heq : (False -> False) = True.
>>>>>
>>>>> Fixpoint contradiction (u : True) : False :=
>>>>> contradiction (
>>>>> match Heq in (_ = T) return T with
>>>>> | eq_refl => fun f:False => match f with end
>>>>> end
>>>>> ).
>>>>>
>>>>>
>>>>>
>>>>> On 01/06/2014 06:12 PM, Altenkirch Thorsten wrote:
>>>>>> Hi Maxime,
>>>>>>
>>>>>> your postulate seems correct since certainly Empty → Box and Box are
>>>>>> isomorphic and hence equal as a consequence of univalence.
>>>>>>
>>>>>> However, I think the definition of loop looks suspicious in the
>>>>>> presence
>>>>>> of proof-relevant equality: if we replace rewrite by an explicit use
>>>>>> of
>>>>>> subst, loop isn't structural recursive anymore. Hence my diagnosis
>>>>>> would
>>>>>> be that rewrite is simply incompatible with univalence.
>>>>>>
>>>>>> Cheers,
>>>>>> Thorsten
>>>>>>
>>>>>> On 06/01/2014 20:42, "Maxime Dénès" 
>>>>>> <mail AT maximedenes.fr>
>>>>>>  wrote:
>>>>>>
>>>>>>> Bingo, Agda seems to have the same problem:
>>>>>>>
>>>>>>> module Termination where
>>>>>>>
>>>>>>> open import Relation.Binary.Core
>>>>>>>
>>>>>>> data Empty : Set where
>>>>>>>
>>>>>>> data Box : Set where
>>>>>>> wrap : (Empty → Box) → Box
>>>>>>>
>>>>>>> postulate
>>>>>>> iso : (Empty → Box) ≡ Box
>>>>>>>
>>>>>>> loop : Box -> Empty
>>>>>>> loop (wrap x) rewrite iso = loop x
>>>>>>>
>>>>>>> gift : Empty → Box
>>>>>>> gift ()
>>>>>>>
>>>>>>> bug : Empty
>>>>>>> bug = loop (wrap gift)
>>>>>>>
>>>>>>> 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.
>>>>>>>
>>>>>>> Maxime.
>>>>>>>
>>>>>>> On 01/06/2014 01:15 PM, Maxime Dénès wrote:
>>>>>>>> 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.
>>>>>>>>
>>>>>>>> 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.
>>>>>>>>
>>>>>>>> I am also curious as to whether Agda is impacted. Let's try it :)
>>>>>>>>
>>>>>>>> Maxime.
>>>>>>>>
>>>>>>>> On 01/06/2014 12:59 PM, Altenkirch Thorsten wrote:
>>>>>>>>> Which bug was this?
>>>>>>>>>
>>>>>>>>> I only saw the one which allowed you to prove
>>>>>>>>>anti-extensionality?
>>>>>>>>> But
>>>>>>>>> this wasn't related to termination, or?
>>>>>>>>>
>>>>>>>>> Thorsten
>>>>>>>>>
>>>>>>>>> On 06/01/2014 16:54, "Cody Roux" 
>>>>>>>>> <cody.roux AT andrew.cmu.edu>
>>>>>>>>>wrote:
>>>>>>>>>
>>>>>>>>>> Nice summary!
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On 01/06/2014 08:49 AM, Altenkirch Thorsten wrote:
>>>>>>>>>>> Agda enforces termination via a termination checker which is
>>>>>>>>>>>more
>>>>>>>>>>> flexible but I think less principled than Coq's approach.
>>>>>>>>>> That's a bit scary given that there was an inconsistency found
>>>>>>>>>>in
>>>>>>>>>> the Coq termination checker just a couple of weeks ago :)
>>>>>>>>>>
>>>>>>>>>> BTW, has anyone tried reproducing the bug in Agda?
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Cody
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>>>>>>>>>
>>>>>>>>>
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