# Re: [isabelle] natural number arithmetic normalisation

```It would certainly be nice to get rid of these rewrites, but the task is
daunting.

Tobias

Am 14/11/2011 07:23, schrieb Brian Huffman:
> On Mon, Nov 14, 2011 at 12:34 AM, Michael Norrish
> <Michael.Norrish at nicta.com.au> wrote:
>> The following term arose inside a side-condition that the simplifier was attempting to discharge:
>>
>>  (2::nat) ^ (2 * (2 * (2 * (2 * (2 * 1)))))
>>
>> The simp tactic being used included field_simps as a rewrite.
>>
>> The result was an apparent "hang" as Isabelle attempted to calculate 2 ^ 32 in unary arithmetic.
>>
>> You can see the behaviour by doing
>>
>>  lemma "(2::nat) ^ (2 * (2 * (2 * (2 * (2 * 1))))) = X"
>
> This is a very interesting puzzle, especially since, as you say,
> field_simps doesn't even mention Suc!
>
> After looking at the simp trace to see which rules were involved I
> realized that you can get the same blowup using "simp only" with a
> small set of rules, none of which are in field_simps, and all of which
> are in the default simpset:
>
> lemma "(2::nat) ^ (2 * (2 * (2 * (2 * (2 * 1))))) = X"
> apply (simp only: One_nat_def mult_Suc_right mult_0_right add_2_eq_Suc)
>
> Yet simply writing "apply simp" on the same goal reduces everything to
> just a numeral.
>
> The weirdness involves these rewrite rules:
>
> lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
> lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
>
> These rules originate quite a while ago:
> http://isabelle.in.tum.de/repos/isabelle/rev/9d6514fcd584
>
> Now, what happens if we simplify a term like "2 + 0" or "0 + 2", where
> more than one possible simp rule can apply? It turns out that the
> simplifier will rewrite "2 + 0" to "2" (using the additive zero law),
> but in the other order, "0 + 2" rewrites to "Suc (Suc 0)" (using rule
> add_2_eq_Suc'). So the presence of the add_commute rule really makes a
> difference here:
>
> lemma "(2::nat) ^ (2 * (2 * (2 * (2 * (2 * 1))))) = X"
>
>
>> It seems to me that this is yet more evidence that using 1 = Suc 0 as a rewrite is a bad idea.
>
> I agree. I think that a good guideline for the Isabelle simpset should
> be that no simp rule should ever insert a Suc into a subgoal that
>
> We have discussed removing "1 = Suc 0" as a simp rule on the dev
> mailing list before:
> https://mailmanbroy.informatik.tu-muenchen.de/pipermail/isabelle-dev/2009-February/000484.html
>
> My conclusion back then was that the only reason we have "1 = Suc 0"
> [simp] is historical, since "1" used to be a mere abbreviation for
> "Suc 0". It would be nice to finally get rid of it (along with