[Agda] without-K problem

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Wed Jun 20 12:32:06 CEST 2012

On 20/06/2012 08:39, "Andreas Abel" <andreas.abel at ifi.lmu.de> wrote:

>On 18.06.12 9:18 PM, Guillaume Brunerie wrote:
>> Hi all,
>> For what it’s worth, I cannot reproduce the bug using the following
>> definition of the identity types:
>> data _≡_ {A : Set} : A → A → Set where
>>    refl : {x : A} → x ≡ x
>> But I can reproduce it using the following definition (which is the one
>> used in the standard library):
>> data _≡_ {A : Set} (x : A) : A → Set where
>>    refl : x ≡ x
>Yes, it seems --without-K should also forbid data definitions where a
>parameter variable ("x") is free in a index position.
>Otherwise, index variables can be "hidden" in parameter positions and
>are not spotted by --without-K when it checks whether a split is legal.

Yes, and this is what it says in the specification of without-K
(I checked this with Nisse yesterday).

Hence this seems to be a bug in the implementation only.


>But then, of course, --without-K is no longer a local check and
>--without-K modules may only import --without-K modules.
>> 2012/6/18 Altenkirch Thorsten <psztxa at exmail.nottingham.ac.uk
>> <mailto:psztxa at exmail.nottingham.ac.uk>>
>>     Hi Nisse,
>>     I noticed that a weak version of K is provable even though the
>>     without-K flag is set:
>>     {-# OPTIONS --without-K #-}
>>     module K-bug where
>>     open import Relation.Binary.PropositionalEquality
>>     weakK : {A : Set}{a b : A}(p q : a ≡ b)(α β : p ≡ q) → α ≡ β
>>     weakK refl .refl refl refl = refl
>>     This would imply that all types have dimension of at most 3. I don't
>>     think it is provable with J.
>>     A simpler term which is provable but shouldn't was found by my
>>     student Nicolai:
>>     weak2 : {A : Set} {a : A} (α : refl {x = a} ≡ refl) → α ≡ refl
>>     weak2 refl = refl
>>     It seems to me that these patterns satisfy the specification of
>>     without-K (I.e. the condition is too weak).
>>     Do you see a fix?
>>     Cheers,
>>     Thorsten
>Andreas Abel  <><      Du bist der geliebte Mensch.
>Theoretical Computer Science, University of Munich
>Oettingenstr. 67, D-80538 Munich, GERMANY
>andreas.abel at ifi.lmu.de

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