*To*: cl-isabelle-users at lists.cam.ac.uk*Subject*: Re: [isabelle] Infinity - infinity = infinity*From*: Johannes Hölzl <hoelzl at in.tum.de>*Date*: Fri, 02 Dec 2016 17:14:45 +0100*In-reply-to*: <2956700a-dc8e-f201-a811-c5c3e3c3a87a@in.tum.de>*Organization*: TU München*References*: <6592A55E-9169-4BBE-8D8E-5A2F8E7718C4@inria.fr> <2956700a-dc8e-f201-a811-c5c3e3c3a87a@in.tum.de>

Another idea from Tobias (and I think also Andreas) is to add a special simpproc which does case-distinction on enat/ereal/ennreal and calls linarith. I would assume the simpproc is quite slow but can be disable by default and just be activated by the user. - Johannes Am Freitag, den 02.12.2016, 16:57 +0100 schrieb Tobias Nipkow: > 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 weÂ > have not inplemented it, and it would be some work, but not > infeasible). InÂ > their system, "â - â = â". Unless we know that your proposed > modification stillÂ > 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": > > > > Â1. âa b. a + b - a = b > > Â2. âa b c. a - b - c = a - (b + c) > > > > and we lack the third property required by > > "comm_semiring_1_cancel": > > > > Â3. 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:*Tobias Nipkow

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

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

- Previous by Date: Re: [isabelle] Infinity - infinity = infinity
- Next by Date: Re: [isabelle] Infinity - infinity = infinity
- Previous by Thread: Re: [isabelle] Infinity - infinity = infinity
- Next by Thread: Re: [isabelle] Infinity - infinity = infinity
- Cl-isabelle-users December 2016 archives indexes sorted by: [ thread ] [ subject ] [ author ] [ date ]
- Cl-isabelle-users list archive Table of Contents
- More information about the Cl-isabelle-users mailing list