[Agda] termination proofs

[Jacques Carette]

"natural" is hard to define.  I guess I would first pursue the ideas 
mentioned in this MO question
http://mathoverflow.net/questions/69229/proof-strength-of-calculus-of-inductive-constructions

and then look through the FOM archives 
(http://www.cs.nyu.edu/mailman/listinfo/fom/) for posts by Harvey 
Friedman.  He's been spending the last umpteenth years proving that 
simple-sounding combinatorial statements require ever crazier cardinals 
to show that they define total functions.   By making the combinatorial 
statements quite 'simple', he hopes that these will also be 'natural'.

Jacques

On 12-09-24 03:55 PM, Altenkirch Thorsten wrote:
> Nice idea. However, I think that termination of Goodstein sequences is
> provable in Agda.
>
> We are far beyond Arithmetic. I am sure Anton can give a better answer but
> I'd suspect that inductive-recursive definitions give you Pi_1^2-CA. Are
> there natural functions which are not provable total in such a theory?
>
> Thorsten
>
> On 24/09/2012 20:42, "Jacques Carette" <carette at mcmaster.ca> wrote:
>
>> On 12-09-24 03:30 PM, Altenkirch Thorsten wrote:
>>> Re: Does there exist an algorithmic map in  Nat -> Nat  for which there
>>> is not any Agda program with a termination proof accepted by Agda ?
>>> [...]
>>> Another question: Is there such a program which we can actually write
>>> down?
>>>
>> Wouldn't a program based on Goodstein sequences [1] be quite easy to
>> write, but with quite a difficult termination proof, at least as far as
>> Agda is concerned?
>>
>> If that's not fast enough, then one can go up the hierarchy [2] (without
>> the need to go all Goedelian) of fast growing functions to Friedman's
>> TREE function [3] for something quite frightening (although still
>> 'predicative' in a certain sense).
>>
>> Jacques
>>
>> [1] http://en.wikipedia.org/wiki/Goodstein%27s_theorem
>> [2] http://en.wikipedia.org/wiki/Fast-growing_hierarchy
>> [3]
>> http://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem#Friedman.27s_finite_
>> form
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