# Re: [isabelle] ln on negative numbers

I don't object to give ln a known value for negative numbers.
But I also think you need to add a carrier set (like a O_on A f) to your
O-notation! Many definitions in Isabelle/HOL nowadays have it.
Btw: if you integrate filters into your O-notation this is not necessary
any more, as this is taken care of by the "at _ within _" filter.
- Johannes
Am Mittwoch, den 29.04.2015, 16:02 +0200 schrieb Manuel Eberl:
> Hallo,
>
> I am currently in the process of writing some automation for Landau
> symbols. One of the problems that I currently have is that I would like
> to rewrite things like
>
> O(Îx. (c*x) powr p)
>
> to
>
> O(Îx. c powr p * x powr p)
>
> However, for c < 0 this is simply not true (well, morally not true). If
> c < 0, and w.l.o.g. x > 0, we have
>
> c powr p = (c*x) powr p = exp (p * THE u. False)
>
>
> I therefore suggest to define the real logarithm as "ln x = (if x â 0
> then 0 else THE u. exp u = x)". The practical implications of this
> should be small, and it would allow stating a lot of lemmas involving
> "powr" in a much simpler form and allow transformations such as the one
> above.
>
> Cheers,
>
> Manuel
>

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