*To*: Tobias Nipkow <nipkow at in.tum.de>*Subject*: Re: [isabelle] Infinity - infinity = infinity*From*: Lawrence Paulson <lp15 at cam.ac.uk>*Date*: Fri, 2 Dec 2016 16:09:47 +0000*Cc*: cl-isabelle-users at lists.cam.ac.uk*In-reply-to*: <2956700a-dc8e-f201-a811-c5c3e3c3a87a@in.tum.de>*References*: <6592A55E-9169-4BBE-8D8E-5A2F8E7718C4@inria.fr> <2956700a-dc8e-f201-a811-c5c3e3c3a87a@in.tum.de>

As a rule, people should use non-standard analysis rather than the extended naturals or reals. Although the former are more complicated, they preserve all the first order properties of their standard counterparts. In particular, the non-standard naturals are still a semiring. --lcp > On 2 Dec 2016, at 15:57, Tobias Nipkow <nipkow at in.tum.de> wrote: > > Jasmin, there is a reason why I would not do this: > > Aless Lasaruk and Thomas Sturm. > Effective Quantifier Elimination for Presburger Arithmetic with Infinity > > This paper shows that our current enat has quantifier elimination (although we have not inplemented it, and it would be some work, but not infeasible). In their system, "â - â = â". Unless we know that your proposed modification still has quantifier elimination, I would be reluctant to give up that strong property. > > Tobias > >> On 02/12/2016 16:01, Jasmin Blanchette wrote: >> Dear all, >> >> As noted before on this mailing list, automation for "enat" ("Library/Extended_Nat.thy") is quite poor. Often, the only way to proceed is to perform case distinctions on all "enat" and use auto on the emerging subgoals. >> >> My impression is that many type classes are not available because of the definition of subtraction. Because "â - â = â" (where "â" is the infinity symbol), we lack one of the two properties required by "cancel_comm_monoid_add": >> >> 1. âa b. a + b - a = b >> 2. âa b c. a - b - c = a - (b + c) >> >> and we lack the third property required by "comm_semiring_1_cancel": >> >> 3. âa b c. a * (b - c) = a * b - a * c >> >> Counterexample for 1: a = â, b = 0. >> Counterexample for 3: a = â, b = c = 1. >> >> These omissions affect further layers in the type class hierarchy -- e.g. we cannot use "ordered_cancel_comm_monoid_diff", even though some of its theorems (e.g. "add_diff_assoc2") turn out to hold. >> >> My proposal is to change the definition of subtraction so that "â - â = 0" and to instantiate the missing type classes. I believe this would make "enat" much less painful to use, and mathematically I'm not so convinced that "â - â = â" is such a great idea anyway. Indeed, I have recently implemented ordinals below Î_0 in Isabelle and was able to have much better automation than with "enat", and there we have Ï - Ï = 0. >> >> "enat" occurs in about 70 ".thy" files in Isabelle and the AFP, so this change (including the type class instantiations) seems quite manageable. We (= Mathias and I) would wait until after the 2016-1 release to avoid any interference. >> >> Any opinions for or against? >> >> Jasmin >> >> >

**Follow-Ups**:**Re: [isabelle] Infinity - infinity = infinity***From:*Andreas Lochbihler

**Re: [isabelle] Infinity - infinity = infinity***From:*Johannes Hölzl

**References**:**[isabelle] Infinity - infinity = infinity***From:*Jasmin Blanchette

**Re: [isabelle] Infinity - infinity = infinity***From:*Tobias Nipkow

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