@book{SR--2005--01,
author ={Claus-Peter Wirth},
title ={{$\lim$$+$}, {$\delta^+$}, and Non-Permutability of
{$\beta$}-Steps},
publisher ={{SEKI Publications}},
series ={{SEKI-Report SR--2005--01 (ISSN 1437--4447)}},
address ={Saarland Univ.},
year ={2006},
note ={\rev\,\ed\ of \Jul\,30, 2006},
url={www.ags.uni-sb.de/~cp/p/nonpermut},
abstract ={Using a human-oriented formal example proof of the
({$\lim$$+$}) theorem,
{i.e.\ } that the sum of limits is the limit of the sum
which is of value for reference on its own,
we exhibit a non-permutability of {$\beta$}-steps and {$\delta^+$}-steps
(according to Smullyan's classification), which is not visible with
non-liberalized {$\delta$}-rules
and not serious with further liberalized {$\delta$}-rules,
such as the {$\delta^{+^+}$}-rule.
Besides a careful presentation of the search for a proof of
({$\lim$$+$}) with several pedagogical intentions,
the main subject is to explain why the order of
{$\beta$}-steps plays such a practically important role in some calculi.}}