[isabelle-dev] HOL/Number_Theory/Primes

Tobias Nipkow nipkow at in.tum.de
Fri Nov 7 18:34:27 CET 2014

```Thanks for finding this out. The theorem is

"a dvd b ==> b mod a = 0"

This applies to any term "a mod b" and creates a subgoal "a dvd b". Normally,
that is not too bad. But if a and b are numerals, this leads to a loop with the
rewrite rule Divides.dvd_eq_mod_eq_0_numeral:

(numeral ?x dvd numeral ?y) = (numeral ?y mod numeral ?x = 0)

The enormous runtimes where due to this loop. It was not an infinite loop
because the simplifier has a depth limit.

Clearly, we cannot have such a loop. Either mod can use dvd or the other way
around, but not both.

Thanks for simp_trace_new/Lars Hupel, it made it easy to find out what was going
on. [It would be nice if the trace could also show when the depth limit is
exceeded, it does not seem to].

Tobias

On 07/11/2014 17:45, Dmitriy Traytel wrote:
> The culprit seems to be dvd_imp_mod_0 introduced as a simp rule in 773b378d9313.
>
> The following takes again only 2 seconds.
>
> declare dvd_imp_mod_0[simp del]
> lemma "prime(97::nat)" by simp
>
> Dmitriy
>
> On 07.11.2014 15:31, Tobias Nipkow wrote:
>> Very nice observations, thank you. I was obviously too hasty to remove the
>> test which exposed this time leak. Once this issue has been fixed, I will put
>> the "long" test back in, with a better comment.
>>
>> Tobias
>>
>> On 07/11/2014 15:27, Dmitriy Traytel wrote:
>>> This is in Isabelle2014. In 229765cc3414 I make the same measurements as Larry.
>>> So indeed (as the text above those lemmas suggests) there seems to be a
>>> regression with the simplifier setup.
>>>
>>> Dmitriy
>>>
>>> On 07.11.2014 15:10, Julian Brunner wrote:
>>>> The proof that 97 is prime only takes 1.3s on my machine (2.7 GHz i7),
>>>>
>>>> On Wed, Nov 5, 2014 at 8:37 PM, Florian Haftmann
>>>> <florian.haftmann at informatik.tu-muenchen.de> wrote:
>>>>>> This theory takes quite a while to load, and I have found out why:
>>>>>>
>>>>>> text{* A bit of regression testing: *}
>>>>>>
>>>>>> lemma "prime(97::nat)" by simp
>>>>>> lemma "prime(997::nat)" by eval
>>>>>>
>>>>>> The proof that 97 is prime takes 35 seconds on a very fast machine. Can we
>>>>>> get rid of this or at least substitute a smaller number?
>>>>> The question is whether this has really to be performed using simp.
>>>>>
>>>>> As an alternative, a suitable code equations could be proven using the
>>>>> primes_upto in Eratosthenes.thy, but I did never take any measurements
>>>>> at which threshold the additional data structures outperform brute-force
>>>>> calculation.
>>>>>
>>>>>          Florian
>>>>>
>>>>> --
>>>>>
>>>>> PGP available:
>>>>> http://home.informatik.tu-muenchen.de/haftmann/pgp/florian_haftmann_at_informatik_tu_muenchen_de
>>>>>
>>>>>
>>>>>
>>>>>
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