*To*: Michael Norrish <Michael.Norrish at nicta.com.au>*Subject*: Re: [isabelle] natural number arithmetic normalisation*From*: Brian Huffman <brianh at cs.pdx.edu>*Date*: Mon, 14 Nov 2011 07:23:40 +0100*Cc*: Isabelle Isabelle Users Mailing List <cl-isabelle-users at lists.cam.ac.uk>*In-reply-to*: <4EC0541C.1090406@nicta.com.au>*References*: <4EC0541C.1090406@nicta.com.au>*Sender*: huffman.brian.c at gmail.com

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" > apply (simp add: field_simps) 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" apply (simp add: add_commute) (* blows up with Suc *) > 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 didn't already contain one. 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 add_2_eq_Suc and friends). - Brian

**Follow-Ups**:**Re: [isabelle] natural number arithmetic normalisation***From:*Tobias Nipkow

**References**:**[isabelle] natural number arithmetic normalisation***From:*Michael Norrish

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