[Agda] A different implementation of --without-K

Martin Escardo m.escardo at cs.bham.ac.uk
Sun Dec 1 21:58:03 CET 2013

On 29/11/13 19:01, Altenkirch Thorsten wrote:
> Shouldn't the semantics simply be that you can only prove things you can
> prove using only J but not K?

I like this specification. But is it complete?

Let me propose a possible elaboration of this specification.

In MLTT, you don't have pattern matching: everything has to be defined
in terms of predefined combinators, which include J but not K.

We can imagine pattern matching as syntax sugar for definitions with

So an ideal Agda compiler would translate Agda into MLTT combinators
(perhaps for MLTT with various extensions for inductive definitions).

In this case, the correctness criterion is simply that Streicher's
combinator K is not needed for this (ideal) translation.

Actually, it would be very good if Agda did work like the idealized
Agda discussed above, by compilation to combinators. This would allow
for proof terms that can be exported to other proof assistants, for
instance, e.g. for checking that an Agda proof is a proof, without
having to rely on Agda to be sure.


> Thorsten
> On 29/11/2013 17:36, "Andreas Abel" <andreas.abel at ifi.lmu.de> wrote:
>> On 29.11.2013 18:11, Harley D. Eades III wrote:
>>> On Nov 29, 2013, at 5:00 AM, Nils Anders Danielsson <nad at cse.gu.se>
>>> wrote:
>>>> On 2013-11-28 23:34, Andreas Abel wrote:
>>>>> Yes, some of the fail test cases seem only to be there to document
>>>>> what *is* rejected.  I do not know what purpose this serves, since I
>>>>> would expect only stuff there that *must be* rejected.
>>>> The point is that the semantics of --without-K should not be changed by
>>>> accident. These examples "must" be rejected until we make a conscious
>>>> decision to change the semantics.
>>> Where is the semantics of --without-K formally written down?
>> There is not much written.  The documentation of the --without-K
>> variants is in the respective release notes.
>> Cheers,
>> Andreas
>> --
>> 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
>> http://www2.tcs.ifi.lmu.de/~abel/
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