# Re: [isabelle] Infinity - infinity = infinity

```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
>>
>>
>

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