Re: [isabelle] primrec of datatype containing fset

Personally, I would suggest something like this:

lemma single_le_sumI:
  assumes "âbâA. a â (f b :: nat)" "finite A"
  shows   "a â (âbâA. f b)"
proof -
  from assms obtain b where "b â A" "a â f b" by blast
  note this(2)
  also from âb â Aâ and âfinite Aâ
    have "f b â (âbâA. f b)" by (intro member_le_sum) auto
  finally show ?thesis .
function count :: "val â nat" where
  "count (VNat n) = 1" |
  "count (VRel A) = (â(x,y)âfset A. count x + count y)"
  by pat_completeness auto
termination by (relation "measure size") (force simp: less_Suc_eq_le
intro: single_le_sumI)+

This reuses the summation function for sets, and I think that will be
much nicer to work with than a fold. The termination proof is still a
bit messy, unfortunately, and I don't know what can be done about it in
general, but at least similar cases should also work with this "one-liner".


On 2017-07-09 12:56, Siek, Jeremy wrote:
> Hi Andrei,
> Good catch!
> But now Iâm a bit unsatisfiedâ the function version of count required a fair bit of work and
> looks like itâs rather specific, so I may have to repeat this amount of work for every
> other recursive function on datatype val.
> I wonder if the termination argument involving fsum could be adapted to ffold?
> Cheers,
> Jeremy
> On Jul 9, 2017, at 7:19 AM, Andrei Popescu <A.Popescu at<mailto:A.Popescu at>> wrote:
> Hi Jeremy and Lars,
> Special care is needed when recursing through permutative structures like sets and multisets. For example, the primrec version discussed here is _not_ equal to the 'fun' version of 'count', since it recursively collapses the numbers coming from subtrees that happen to have the same count. The 'fun' version is the correct one.
> All the best,
> Andrei
> ________________________________
> From: cl-isabelle-users-bounces at<mailto:cl-isabelle-users-bounces at> <cl-isabelle-users-bounces at<mailto:cl-isabelle-users-bounces at>> on behalf of Lars Hupel <hupel at<mailto:hupel at>>
> Sent: 09 July 2017 11:28
> To: Siek, Jeremy
> Cc: cl-isabelle-users at<mailto:cl-isabelle-users at>
> Subject: Re: [isabelle] primrec of datatype containing fset
>> Thank you for your very thorough answer! It looks like option 1 will
> suite
>> my needs.
> On second thought, I realised that your use of "ffold" is merely to
> compute the sum of some things in a set. Luckily, we can make "function"
> cope with that. See attachment for a possible solution (also a version with
> "primrec" that doesn't use "ffold").
> Cheers
> Lars
> __________________________________________
> Jeremy G. Siek    <jsiek at<mailto:jsiek at>>
> Associate Professor
> School of Informatics and Computing
> Indiana University Bloomington

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