# [isabelle-dev] Reorganising Analysis

Tobias Nipkow nipkow at in.tum.de
Tue Nov 5 13:55:52 CET 2019

```I stumbled across this oddity - the important concept of winding numbers being
defined in the middle of Cauchy_Integral_Theorem - myself this morning just
looking at the toc. Thus I am all in favour of splitting off Winding Numbers
from Cauchy_Integral_Theorem: the latter contains 1000 lines of material
following the definition up to subsection "Cauchy's integral formula, again for
a convex enclosing set". Later there is some more material:
subsection ‹More winding number properties›

Wenda, would you mind performing this split that you suggested?
Not sure about the names: Maybe Winding_Numbers and Winding_Numbers_Applied? Any
better ideas?

Tobias

On 05/11/2019 01:54, Wenda Li wrote:
>
>> On 4 Nov 2019, at 12:48, Manuel Eberl <eberlm at in.tum.de
>> <mailto:eberlm at in.tum.de>> wrote:
>>
>> I don't really have an idea of what material you really need for winding
>> numbers. I recall that Wenda had some problems with them (about what happens
>> when there's a pole on the path) – perhaps it would make sense to give that
>> theory an overhaul altogether?
>
> Our definition of winding numbers is fine to me, as it follows classic
> analytical textbook definition which usually does not allow the target point on
> the path. Unless we want to further generalise the definition in a topological
> or algebraic direction (which does not seem too necessary at the moment), this
> point-on-the-path issue is probably what we have to live with.
>
> My two cents about our winding numbers is the organisation: winding numbers are
> (kind of secretly) defined in Cauchy_Integral_Theorem.th
> <http://cauchy_integral_theorem.th>y rather than in Winding_Numbers.thy
> <https://isabelle.in.tum.de/repos/isabelle/file/c85efa2be619/src/HOL/Analysis/Winding_Numbers.thy>,
> which only contains advanced facts related to winding numbers.  Maybe it is
> worth having two theories about winding numbers — a basic one for the integral
> theorem and an advanced one that is on the top of Jordan_Curve & Riemann_Mapping.
>
> Wenda
>
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