The Coq mailing list

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

On 27/01/2016 15:36, "Jonathan Leivent" 
<jonikelee AT gmail.com>
 wrote:

>
>
>On 01/27/2016 03:39 AM, Thorsten Altenkirch wrote:
>> Indeed you need functional extensionality which isn¹t provable in coq.
>> Imho this is a bug, but one which isn¹t easy to fix.
>
>Why would it be considered a bug?  It is always possible to add the
>functional extensionality axiom to Coq when needed.

This destroys the computational behaviour of Type Theory.

Type Theory is not an axiomatic theory you should be able to prove things
by constructing an inhabitant of a type.

> And, without it, 
>one can express concepts such as that two functions of the same type
>where one sorts lists in O(N^2) time and the other in O(NlogN) time are
>not identical.

You cannot express the difference in Coq anyway. If you want to reason
about function codes instead of functions you can do this even in the
presence of ext.

>
>Maybe this just indicates a difference in Coq usage between software
>developers and type theorists.  Although Curry-Howard says we should
>correspond ;)

Yes, this is true Coq is being abused in various ways. I think Type Theory
is actually quite a good system both for Mathematics and for Programming.
No reason not to use it.

Thorsten





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