We make the following temporary definitions so to cheat CutAt and the likes that definitions for the keywords exist.
<..book / report / article cut points..>
\Configure{UndefinedSec}{likepart}
\Configure{UndefinedSec}{likechapter}
\Configure{UndefinedSec}{likesection}
\Configure{UndefinedSec}{likesubsection}
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<..ams art, proc, book..>
\Configure{UndefinedSec}{likepart}
\Configure{UndefinedSec}{likechapter}
\Configure{UndefinedSec}{likesection}
\Configure{UndefinedSec}{likesubsection}
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<..latex html cut points..>
\def\cut:gr#1{\lk:#1like<.par del.>%
\ifx \:temp\empty \expand:after{%
\expand:after{\expandafter\let\csname #1\endcsname=}%
\csname :#1\endcsname
\ct:gr{#1}%
\expand:after{\expandafter\let\csname :#1\endcsname=}%
\csname #1\endcsname
\expandafter\let\csname #1\endcsname=\:UnDef}%
\else
\expand:after{\ct:gr{#1}}%
\fi}
\def\ct:gr#1{%
\edef\:temp{%
\let\:csname \HP:file : #1\endcsname =\:csname #1\endcsname
\let\:csname \HP:file :Cut:#1\endcsname =\:csname Cut:#1\endcsname
\def\:csname #1\endcsname{%
\noexpand \@ifnextchar*{\noexpand\after:gobble
\:csname :like#1\endcsname}%
{\:csname \HP:file :#1-\endcsname}}%
\def\:csname \HP:file :#1-\endcsname####1{%
\noexpand\cond:cs{\HP:file}{####1}%
\:csname \HP:file : #1\endcsname{####1}}%
\def\:csname Cut:#1\endcsname####1{%
\noexpand\cond:cs{Cut:\HP:file}{####1}%
\noexpand\cond:cs{\HP:file :Cut:#1}{####1}}%
}\:temp }
\def\lk:#1like#2<.par del.>{\def\:temp{#1}}
\def\after:gobble#1#2{#1}
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