% \iffalse meta-comment
%
% Copyright (C) 2016-2026 by Pierre Senellart
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
%   http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2005/12/01 or later.
%
% This work has the LPPL maintenance status `maintained'.
%
% The Current Maintainer of this work is Pierre Senellart
% <pierre@senellart.com> and a version control system for this work
% is available at http://github.com/PierreSenellart/apxproof
%
% This work consists of the files apxproof.dtx and apxproof.ins
% and the derived file apxproof.sty.
%
% \fi
%
% \iffalse
%<package>\NeedsTeXFormat{LaTeX2e}[2005/12/01]
%<package>\ProvidesPackage{apxproof}
%<package>  [2026/05/14 v1.4.0 Automatic proofs in appendix]
%
%<*driver>
\documentclass{ltxdoc}
\usepackage{apxproof}
\usepackage{hypdoc}
\usepackage{textcomp}
\usepackage[TS1,T1]{fontenc}
\usepackage{lmodern}
\usepackage{microtype}
\newtheorem{example}{Example}
\newtheoremrep{foobar}{Foobar}
\newtheoremrep*{foobarstar}{Foobarstar}
\EnableCrossrefs
\CodelineIndex
\RecordChanges
\begin{document}
  \DocInput{apxproof.dtx}
\end{document}
%</driver>
% \fi
%
% \CheckSum{1275}
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%
% \changes{v1.0.4}{2017/03/02}{Show options commented on in margin and index}
%\iffalse
% Taken from xkeyval.dtx
%\fi
%\makeatletter
%\def\DescribeOption#1{\leavevmode\@bsphack
%              \marginpar{\raggedleft\PrintDescribeOption{#1}}%
%              \SpecialOptionIndex{#1}\@esphack\ignorespaces}
%\def\PrintDescribeOption#1{\strut\emph{option}\\\MacroFont #1\ }
%\def\SpecialOptionIndex#1{\@bsphack
%    \index{#1\actualchar{\protect\ttfamily#1}
%           (option)\encapchar usage}%
%    \index{options:\levelchar#1\actualchar{\protect\ttfamily#1}\encapchar
%           usage}\@esphack}
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%  \@for\@tempa:=#1\do{%
%    \\\strut\MacroFont\@tempa\SpecialOptionIndex\@tempa
%  }}\@esphack\ignorespaces}
%\makeatother
%
% \changes{v1.0.0}{2016/10/31}{Initial released version}
% \changes{v1.0.1}{2016/11/07}{Prevent empty bibliography environment;
% fix typos}
%
% \GetFileInfo{apxproof.sty}
%
% \makeatletter\c@IndexColumns=2\makeatother
% \DoNotIndex{
%  \newcommand,\newenvironment,\end,\begin,\edef,\if,\else,\fi,\def,
%  \begingroup,\endgroup,\csname,\let,\noexpand,\protect,\expandafter,
%  \ifthenelse,\equal,\endcsname,\@empty,\@ifnotempty,\immediate,
%  \addtocounter,\newcounter,\newtoggle,\global,\ifdefined,\iftoggle,
%  \ifx,\jobname,\makeatletter,\makeatother,\newwrite,\NewEnviron,
%  \patchcmd,\relax,\renewcommand,\renewenvironment,\RequirePackage,
%  \roman,\space,\undefined,\unexpanded
% }
%
% \title{The \textsf{apxproof} package}
%
% \author{Pierre Senellart \\ \texttt{pierre@senellart.com} \\
% \url{http://github.com/PierreSenellart/apxproof}}
% \date{\filedate \quad \fileversion}
%
% \maketitle
%
% \begin{abstract}
% This package makes it easier to write articles where proofs and other material
% are deferred to the appendix. The appendix material is written in the \LaTeX{}
% code along with the main text which it naturally complements, and it is
% automatically deferred. The package can automatically send proofs to the
% appendix, can repeat in the appendix the theorem environments stated in the
% main text, can section the appendix automatically based on the sectioning of
% the main text, and supports a separate bibliography for the appendix material.
% \end{abstract}
%
% \section{Usage}
% The \textsf{apxproof} package is intended to simplify the writing of articles where some of
% the content needs to be deferred to an appendix. This is in particular
% useful for the submission of scientific articles to conferences or
% journals that limit the number of pages in the main text but allow an
% extra appendix, where proofs of theorems and other material can be added.
%
% \subsection{Basics}
% To use \textsf{apxproof}, first load it in the header of your document:
% \begin{quote}|\usepackage{apxproof}|\end{quote}
% On its own, this does not do anything and should not change the
% appearance of your document. To add an appendix with some material from
% your document, use the |toappendix| environment:
% \begin{quote}
% |\begin{toappendix}|\DescribeEnv{toappendix}\\
% \hspace*{1em}\dots\\
% |\end{toappendix}|
% \end{quote}
% The content will appear at the end of your document, in an
% automatically generated section that refers to the current section in the main
% text.
% \begin{example}
% Throughout this documentation, all examples produce content
% deferred to the appendix, at the very end of this document.
% \begin{verbatim}
% \begin{toappendix}
% This content is in the appendix.
% \end{toappendix}
% \end{verbatim}
% \begin{toappendix}
% This content is in the appendix.
% \end{toappendix}
% \end{example}
% \changes{v1.0.3}{2017/01/10}{Note on entire sections in appendix}
% When the content to put in appendix is an entire section, make sure
% that \verb|\section| is the very first command that appears within the
% \verb|toappendix| environment. It will disable the automatic production
% of a section heading.
% \subsection{Repeated Theorems and Proofs}
% In some scientific papers that include proofs, it is common to defer
% proofs to the appendix. This can easily be achieved using the
% |appendixproof| environment:
% \begin{quote}
% |\begin{appendixproof}|\DescribeEnv{appendixproof}\\
% \hspace*{1em}\dots\\
% |\end{appendixproof}|
% \end{quote}
% This behaves like the |toappendix| environment, except that
% a proof environment is generated.
% \begin{example}We now send a proof to the appendix:
% \begin{verbatim}
% \begin{appendixproof}
% This proof is in the appendix.
% \end{appendixproof}
% \end{verbatim}
% \begin{appendixproof}
% This proof is in the appendix.
% \end{appendixproof}
% \end{example}
%
% When deferring proofs to the appendix, an annoying problem is that the
% statement of the theorem remains in the main text; it is hard to read a
% proof that is far away from the statement it proves. \textsf{apxproof}
% offers two ways to address this. With
% |\newtheoremrep|\DescribeMacro{\newtheoremrep}, the statement
% of the theorem is \emph{repeated} in the appendix, immediately above
% the deferred proof. With the starred variant
% |\newtheoremrep*|\DescribeMacro{\newtheoremrep*}, the
% theorem is \emph{not} repeated, but the proof is still automatically
% deferred to the appendix; this is convenient when the proofs are short
% enough or close enough to the main text that a restatement would be
% redundant, especially when combined with the
% |\appendixproofname|\DescribeMacro{\appendixproofname}{} hook (see
% Section~\ref{sec:customization}) to title proofs with the type and
% number of the theorem they refer to (e.g., ``Proof of Theorem~1''). In
% summary, three behaviors are available for a theorem environment:
% \begin{description}
% \item[\textbackslash|newtheorem|] (\LaTeX's standard command):
%   the theorem appears in the main text only; the |proof| environment is
%   left unchanged by \textsf{apxproof}.
% \item[\textbackslash|newtheoremrep|]: the theorem appears in the main
%   text \emph{and} is restated in the appendix above its proof; the
%   |proof| that follows in the source is automatically deferred to the
%   appendix.
% \item[\textbackslash|newtheoremrep*|]: the theorem appears in the
%   main text only; the |proof| that follows is automatically deferred to
%   the appendix, without restating the theorem there.
% \end{description}
% In all cases, theorems can share counters and a single document can
% mix the three behaviors freely.
%
% To use the repeated-theorem feature, you can define a new
% \emph{repeated theorem} environment using the |\newtheoremrep| command:
% \changes{v1.0.5}{2017/05/31}{Ability to specify a sectioning counter in newtheoremrep}
% \begin{quote}
% |\newtheoremrep|\marg{name}\oarg{counter}\marg{title}\oarg{countersec}\DescribeMacro{\newtheoremrep}
% \end{quote}
% or, equivalently, its starred form to obtain a \emph{proof-deferred}
% (but not repeated) theorem:
% \begin{quote}
% |\newtheoremrep*|\marg{name}\oarg{counter}\marg{title}\oarg{countersec}\DescribeMacro{\newtheoremrep*}
% \end{quote}
% Usage of either form is exactly the same as that of AMS \LaTeX{}'s
% |\newtheorem| macro:
% \begin{itemize}
% \item \meta{name} (e.g., |theorem|) is the name of an environment that is created for this kind of
% theorem;
% \item \meta{counter} (e.g., |definition|) is an optional counter describing
% from which kind of environment the numbering of these environments should be
% inherited;
% \item \meta{title}
% (e.g., |Theorem|) is
% the title that will be used to display this theorem environment;
% \item \meta{countersec} (e.g., |section|) is an optional counter of a
% sectioning command indicating that counters for this theorem should
% be prefixed by this counter (and reset at each occurrence of the
% sectioning command).
% \end{itemize}
% \meta{counter} and \meta{countersec} should not be used together.
% What
% differs from |\newtheorem| is that, when the following is written:
% \begin{quote}
% |\newtheoremrep{foobar}{Foobar}|
% \end{quote}
% then \emph{two} environments are defined: the \verb|foobar|
% environment, which behaves as if |\newtheorem| had been used, and the
% |foobarrep| environment, which results in the statement of this
% environment being repeated in the appendix.
% The starred form |\newtheoremrep*{foobarstar}{Foobarstar}| similarly
% defines two environments |foobarstar| and |foobarstarrep|, except that
% |\begin{foobarstarrep}| does not produce any restatement of the
% theorem in the appendix; only the following |proof| is deferred.
%
% One interesting feature of \textsf{apxproof} is that in most
% situations, there is no need to use the |appendixproof| environment.
% Indeed, the |proof| \DescribeEnv{proof} environment is redefined by \textsf{apxproof} to
% automatically put the proof either in the main text (if it follows a
% regular theorem) or in the appendix (if it follows a repeated
% theorem).
% \begin{example}
% Assume we have first defined a repeated theorem environment |foobar| as
% above.
% We can now use this theorem environment, first for a regular theorem in
% the main text, then for a theorem repeated in the main text and in the
% appendix:
% \begin{quote}
% \begin{verbatim}
% \begin{foobar}
% This foobar is a regular one, in the main text.
% \end{foobar}
% \begin{proof}
% This is the proof of the regular foobar.
% \end{proof}
% \end{verbatim}
% \end{quote}
% We obtain:
% \begin{foobar}
% This foobar is a regular one, in the main text.
% \end{foobar}
% \begin{proof}
% This is the proof of the regular foobar.
% \end{proof}
% Now, if we use a repeated theorem:
% \begin{quote}
% \begin{verbatim}
% \begin{foobarrep}
% This foobar is repeated in the appendix.
% \end{foobarrep}
% \begin{proof}
% This is the proof of the repeated foobar.
% \end{proof}
% \end{verbatim}
% \end{quote}
% We now obtain:
% \begin{foobarrep}
% This foobar is repeated in the appendix.
% \end{foobarrep}
% \begin{proof}
% This is the proof of the repeated foobar.
% \end{proof}
% Note that, since |hyperref| is loaded, there are hyperlinks
% created between the statements of the theorems in the main text and in
% the appendix.
% \end{example}
% \begin{example}
% In contrast, the starred form of |\newtheoremrep| produces a theorem
% whose proof is deferred to the appendix \emph{without} restating the
% theorem there. Assuming the declaration
% |\newtheoremrep*{foobarstar}{Foobarstar}| at the top of the document,
% writing:
% \begin{quote}
% \begin{verbatim}
% \begin{foobarstarrep}
% This foobar will not be repeated in the appendix,
% but the proof below will be deferred there.
% \end{foobarstarrep}
% \begin{proof}
% This is the proof of the (non-repeated) foobar.
% \end{proof}
% \end{verbatim}
% \end{quote}
% yields:
% \begin{foobarstarrep}
% This foobar will not be repeated in the appendix,
% but the proof below will be deferred there.
% \end{foobarstarrep}
% \begin{proof}
% This is the proof of the (non-repeated) foobar.
% \end{proof}
% In this mode, the title of the deferred proof in the appendix is
% controlled by the |\appendixproofname| hook documented in
% Section~\ref{sec:customization}; by default, this hook is empty, so
% proofs are titled simply ``Proof.'' as usual. The forward hyperlink
% from the theorem number in the main text (when \textsf{hyperref} is
% loaded) points to the location of the proof in the appendix.
% \end{example}
%
% When the proof is deferred to the appendix, it is common practice to
% add a proof sketch in the main text. \textsf{apxproof} defines a simple
% |proofsketch| environment for this purpose:
% \changes{v1.0.5}{2017/05/31}{Fix compilation of proofsketch environment in inline mode}
% \begin{quote}
% |\begin{proofsketch}|\DescribeEnv{proofsketch}\\
% \hspace*{1em}\dots\\
% |\end{proofsketch}|
% \end{quote}
% The proof sketch is typeset similarly to a proof, but is always in the
% main text. Similarly, an |inlineproof| \DescribeEnv{inlineproof}
% environment is provided so as to
% be able to have both a proof in the appendix (using the regular |proof|
% environment, or alternatively the |appendixproof| environment) and a
% different proof in the main text (using the |inlineproof| environment).
% \begin{example} Here are simple examples of proof sketches and inline
% proofs:
% \begin{verbatim}
% \begin{proofsketch}
% This is a proof sketch.
% \end{proofsketch}
% \end{verbatim}
% \begin{proofsketch}
% This is a proof sketch.
% \end{proofsketch}
% \begin{verbatim}
% \begin{inlineproof}
% This is an inline proof.
% \end{inlineproof}
% \end{verbatim}
% \begin{inlineproof}
% This is an inline proof.
% \end{inlineproof}
% \end{example}
%
% \subsection{Bibliography}
% By default, \textsf{apxproof} automatically adds a bibliography in the appendix
% with only the references cited in the appendix material. This allows for a
% clean separation of references used solely in the main text, and those
% used in the appendix.
% \begin{example}
% Assume we have citations both in the main text and in the appendix.
% \begin{verbatim}
% This is a citation in the main text~\cite{lamport86}.
% \begin{toappendix}
% This is a citation in the appendix~\cite{proofsAreHard}.
% \end{toappendix}
% \end{verbatim}
% This is a citation in the main text~\cite{lamport86}.
% \begin{toappendix}
% This is a citation in the appendix~\cite{proofsAreHard}.
% \end{toappendix}
% \end{example}
% The bibliography in the appendix can use
% a different style and heading than the bibliography in the main text (and, by
% default, it does). See
% Section~\ref{sec:customization} for how to configure the appearance of
% that bibliography.
%
% \DescribeOption{bibliography}
% In order to use a single appendix for the main text and the
% bibliography, one can specify the
% value |common| to the |bibliography| option when loading the package.
%
% \DescribeOption{bibengine}
% By default, \textsf{apxproof} relies on the \textsf{bibunits} package
% to typeset a separate bibliography for the appendix material; this is
% the |bibengine=bibtex| mode (the default), suitable for documents
% that use \textsf{bibtex} (with or without \textsf{natbib}) for
% citations and references. Documents using \textsf{biblatex} can
% instead set |bibengine=biblatex|, in which case \textsf{apxproof}
% wraps the appendix material in a \textsf{biblatex} |refsection|
% environment and emits the appendix bibliography with
% |\printbibliography|. In this mode the user is responsible for
% loading \textsf{biblatex} and declaring bibliography resources with
% |\addbibresource| in the preamble; resources are shared between the
% main document and the appendix |refsection|, but citations and the
% printed bibliography are scoped independently. The |bibengine| option
% only has an effect when |bibliography=separate|.
% (By default this option is set to |separate|.)
% \subsection{Mode}
% \DescribeOption{appendix}
% An optional \meta{mode} can be specified when loading the package:
% \begin{quote}|\usepackage[appendix=|\meta{mode}|]{apxproof}|\end{quote}
% \meta{mode} can take one of the following four values:
% \begin{description}
% \item[|append|] This is the default.
% Appendix material gathered by \textsf{apxproof} is appended to
% the main text.
% \item[|inline|] In this mode, \textsf{apxproof} simply inlines the
% content along with the main text.
% \item[|strip|] This mode functions similarly to |append| except that
% the appendix is not appended at the end of the document. All appendix
% material is therefore removed.
% \item[|chapterend|] In this mode, appendix material for each chapter
% is gathered separately and emitted at the end of that chapter
% (i.e., just before the next |\chapter|, |\part|, |\frontmatter|,
% |\mainmatter|, or |\backmatter| command, or at the end of the
% document), using the |subappendices| environment from the
% \textsf{appendix} package. Only |bibliography=common| is supported
% in this mode. See Section~\ref{sec:customization} for the
% |\chapterappendixprelim| and |\flushchapterappendix| customization
% macros.
% \end{description}
%
% \subsection{Customization}
% \label{sec:customization}
% \textsf{apxproof} provides a few macros that can be redefined (using
% |\renewcommand|) to customize the appearance of the appendix:
% \begin{description}
% \item[\textbackslash|mainbodyrepeatedtheorem|]\DescribeMacro{\mainbodyrepeatedtheorem}
% is a macro that is executed at the beginning of the body of every
% repeated theorem. This can be used to notify the reader that the
% theorem is repeated in appendix in some way, e.g., with a margin note.
% \item[\textbackslash|appendixsectionformat|\marg{number}\marg{title}]\DescribeMacro{\appendixsectionformat} is a macro
% that indicates how to format the section titles in the Appendix, given
% the number and title of the section in the main text. By
% default, they appear as ``Proofs for Section~\meta{number}
% (\meta{title})''.
% \item[\textbackslash|appendixproofname|\marg{type}\marg{ref}]\DescribeMacro{\appendixproofname}
% is a macro that determines the title of a proof deferred to the
% appendix when it follows a |\newtheoremrep| or |\newtheoremrep*|
% environment. By default, this macro is empty, which means proofs are
% titled ``Proof.'' as usual. When redefined to a non-empty value, the
% expansion is used as the title of the appendix proof; \meta{type} is
% the title of the theorem environment (e.g., |Theorem|) and \meta{ref}
% is a reference to the theorem statement in the main text. To title
% all such proofs ``Proof of \meta{type}~\meta{number}'', one would
% typically write:
% \begin{quote}
% \begin{verbatim}
% \renewcommand{\appendixproofname}[2]{Proof of #1~#2}
% \end{verbatim}
% \end{quote}
% This hook applies whether the theorem is restated in the appendix
% (|\newtheoremrep|) or not (|\newtheoremrep*|). It has no effect on
% manually written |appendixproof| environments or on proofs given an
% explicit optional title.
% \item[\textbackslash|appendixrefname|]\DescribeMacro{\appendixrefname} contains the heading that is displayed before
% the bibliography. By default, this is ``References for the
% Appendix''. (Note that this command is also defined and used by the
% |memoir| document class.)
% \item[\textbackslash|appendixbibliographystyle|]\DescribeMacro{\appendixbibliographystyle} contains the |.bst| bibliography
% style that is used in the bibliography in appendix. By default, this is
% |alpha|.
% \item[\textbackslash|appendixbibliographyprelim|]\DescribeMacro{\appendixbibliographyprelim} contains arbitrary code that is executed
% just before the production of the bibliography in appendix, which can
% be used to configure the way it is displayed.
% \item[\textbackslash|appendixprelim|]\DescribeMacro{\appendixprelim} contains arbitrary code that is executed
% just before the production of the appendix, which can
% be used to configure the way it is displayed. By default, this command
% contains |\clearpage\onecolumn| (the appendix is typeset on a new page
% in single-column mode) but redefining this option allows changing this
% behavior. This macro has no effect in |chapterend| mode; use
% |\chapterappendixprelim| instead.
% \item[\textbackslash|chapterappendixprelim|]\DescribeMacro{\chapterappendixprelim}
% (|chapterend| mode only) contains arbitrary code that is executed
% just before each chapter's appendix block. By default, this command
% contains |\clearpage| (each chapter appendix starts on a new page).
% This macro has no effect in |append| or |strip| mode; use
% |\appendixprelim| instead.
% \item[\textbackslash|flushchapterappendix|]\DescribeMacro{\flushchapterappendix}
% (|chapterend| mode only) manually emits the current chapter's
% accumulated appendix material at the point of invocation, and starts
% a fresh per-chapter buffer. The package automatically inserts an
% equivalent flush before each |\chapter|, |\part|, |\frontmatter|,
% |\mainmatter|, |\backmatter|, and at the end of the document, so
% this macro is only useful in unusual document structures where an
% appendix needs to be emitted at a location that is not a standard
% structural boundary.
% \end{description}
% \DescribeOption{repeqn}
% Another customization capability concerns \emph{numbered equations} that are present
% within repeated theorems. An optional |repeqn| option can be specified
% when loading the package, which controls whether
% equation numbers should be as in the main text (by setting this option
% to |same|, the default) or independently
% numbered (by setting this option to |independent|). In the latter case,
% whenever a referenceable counter is set with |\label{|\meta{counter}|}|,
% |\ref{|\meta{counter}|}| references the counter in the main text, while
% |\ref{|\meta{counter}|-apx}| references the counter in the appendix
% (except in |inline| mode, where both have the same effect).
%
% \DescribeOption{forwardlinking}
% Another customization option concerns hyperlinking.
% Usually, when \textsf{hyperref} is loaded, |foobarrep|
% environments in the main text have their number link to their repetition in the
% appendix. To suppress this behavior and have |foobarrep| environments
% treated as if \textsf{hyperref} were not loaded, one can specify the
% value |no| to the |forwardlinking| option when loading the package. (By
% default this option is set to |yes|.)
%
% \subsection{Advanced Features}
% We now describe a few advanced macros and environments, the usage of
% which is limited to special cases:
% \begin{description}
% \item[|nestedproof|]\DescribeEnv{nestedproof} is an environment that
% can be used within a |proof| environment deferred in the appendix; this
% is required because, for technical reasons, no |proof| environment can
% be nested within a deferred |proof| environment.
% \item[\textbackslash|noproofinappendix|]\DescribeMacro{\noproofinappendix}
% can be used inside repeated theorems that are not followed by a
% |proof| or |appendixproof| environment; the point is to ensure that a further
% |proof| environment cannot be mistakenly understood as a proof of the repeated
% theorem. It should not be needed in most situations as
% \textsf{apxproof} tries figuring out when a proof follows a repeated
% theorem automatically, but may occasionally be needed in complex
% scenarios.
% \item[\textbackslash|nosectionappendix|]\DescribeMacro{\nosectionappendix}
% is to be used inside a section that \emph{does} contain appendix material, but
% for which a section in the appendix should not be created. This should
% be rarely needed. When this command is present, appendix material is
% appended to the end of the previously created section.
% \end{description}
% \subsection{Grouping Appendix Material Under a Single Section}
% \changes{v1.4.0}{2026/05/14}{Documented workarounds for grouping
%   per-section appendices under a single wrapping section. See \#56.}
% \label{sec:appendix-grouping}
% By default, \textsf{apxproof} produces one section in the appendix for
% each section of the main text that contains appendix material (with
% the format controlled by |\appendixsectionformat|). If you would
% rather have a single section in the appendix (e.g., titled
% ``Proofs'') with the per-section entries appearing as
% \emph{subsections} of that single section, two recipes are available.
%
% The first is fully explicit: disable the automatic per-section
% heading with |\nosectionappendix|, and write the desired structure
% directly inside |toappendix| environments.
% \begin{quote}
% \begin{verbatim}
% \begin{toappendix}
%   \section{Proofs}
% \end{toappendix}
%
% \section{First}\label{sec:first}
% \nosectionappendix
% \begin{toappendix}
%   \subsection{Proofs of Section~\ref{sec:first}}
% \end{toappendix}
%
% % ... theorems and proofs of section ``First'' ...
%
% \section{Second}\label{sec:second}
% \nosectionappendix
% \begin{toappendix}
%   \subsection{Proofs of Section~\ref{sec:second}}
% \end{toappendix}
%
% % ... theorems and proofs of section ``Second'' ...
% \end{verbatim}
% \end{quote}
% The reader of the source can then see exactly what will be produced
% in the appendix, at the cost of some verbosity.
%
% The second recipe is more concise. The section level used by
% \textsf{apxproof} for the auto-generated appendix headings is swapped
% from |\section| to |\subsection| for the duration of an explicit
% |\begingroup|/|\endgroup| group:
% \begin{quote}
% \begin{verbatim}
% \begin{toappendix}
%   \section{Proofs}
%   \begingroup\makeatletter
%     \let\axp@oldsection\subsection
%   \makeatother
% \end{toappendix}
%
% \section{First}\label{sec:first}
% % ... theorems and proofs of section ``First'' ...
%
% \section{Second}\label{sec:second}
% % ... theorems and proofs of section ``Second'' ...
%
% \begin{toappendix}
%   \endgroup
% \end{toappendix}
% \end{verbatim}
% \end{quote}
% The wrapping |\section{Proofs}| is written once, and every subsequent
% auto-generated appendix section heading is demoted to a |\subsection|
% for the scope of the group. The trade-off is that this recipe relies
% on an internal macro (|\axp@oldsection|) and on a manually balanced
% |\begingroup|/|\endgroup| pair across two separate |toappendix|
% environments.
% \section{Supported Document Classes}
% Because \textsf{apxproof} modifies sectioning commands, bibliographies,
% and proofs, it may not work straight away with arbitrary document classes.
% It has currently been tested with and is supported for the following
% document classes:
% \begin{itemize}
% \item \LaTeX{} standard document classes (e.g., |article.cls|)
% \item \href{https://www.ctan.org/pkg/koma-script}{KOMA-Script}
% (e.g., |scrartcl.cls|, |scrbook.cls|)
% \item \href{https://ctan.org/pkg/memoir}{|memoir.cls|}
% \item \href{https://www.acm.org/publications/proceedings-template}{ACM
% SIG Proceedings} (e.g., |sig-alternate.cls|, |acmart.cls|)
% \item \href{https://www.springer.com/computer/lncs/lncs+authors}{
% Springer's Lecture Notes in Computer Science} (e.g.,
% |llncs.cls|)
% \item
% \href{https://www.dagstuhl.de/en/publications/lipics}{Schlo\ss{}
% Dagstuhl's Leibniz International Proceedings in Informatics} (e.g.,
% |lipics.cls|, |lipcs-v2016.cls|)
% \end{itemize}
% Other classes may work out of the box. Adding support for specific
% classes is possible and can be requested from the
% author of this package.
% \section{Known Issues and Limitations}
% We report here some issues we are currently aware of:
% \begin{itemize}
% \item When using \textsf{hyperref}, the appendix in the bibliography is
% not hyperlinked. This is to avoid possible issues with multiply defined
% bibliography entries.
% \item |appendixproof|, |proof|, |toappendix| environments cannot be nested. This is a limitation of
% the \textsf{fancyvrb} package that \textsf{apxproof} relies on. Note
% the existence of the |nestedproof| environment for nested proofs.
% \item \textsf{apxproof} poorly interacts with Sync\TeX: identifying
% which source line has produced which box does not work for appendix
% content managed by \textsf{apxproof} or repeated theorems. No obvious
% fix is known, though this issue will be investigated in the long term.
% \item By default, when |bibliography=separate|, \textsf{apxproof}
% uses the \textsf{bibunits} package to generate a second bibliography.
% Packages incompatible with \textsf{bibunits} (notably
% \textsf{biblatex}) will therefore not work in that default
% configuration. For \textsf{biblatex} users, set |bibengine=biblatex|
% (in addition to |bibliography=separate|) to use \textsf{biblatex}'s
% own |refsection| machinery instead. The |bibliography=common| mode
% remains a third option that works with most bibliography packages.
% \end{itemize}
%
% Issues not listed here should be reported to the author.
%
% \section{License}
% Copyright \textcopyright{} 2016--2026 by Pierre Senellart.
%
% This work may be distributed and/or modified under the conditions of the
% \LaTeX{} Project Public License, either version 1.3 of this license or (at
% your option) any later version. The latest version of this license is in
% \url{http://www.latex-project.org/lppl.txt} and version 1.3 or later is part of
% all distributions of \LaTeX{} version 2005/12/01 or later.
%
% \section{Contact}
% \begin{itemize}
% \item \url{https://github.com/PierreSenellart/apxproof}
% \item
% Pierre Senellart
% \href{mailto:pierre@senellart.com}{<pierre@senellart.com>}
% \end{itemize}
% Bug reports and feature requests should
% preferably be submitted through the \emph{Issues} feature of GitHub.
%
% \section{Acknowledgments}
% Thanks to Antoine Amarilli for feedback and proofreading. Thanks to K.~D.
% Bauer for the implementation of the forward-linking mechanism, and
% for various bugfixes. Thanks to  Leonid Kostrykin for an initial
% implementation of the |forwardlinking| option.
%
% \StopEventually{
%   \PrintChanges
%   \PrintIndex
% }
%
% \section{Implementation}
% We now describe the entire code of the package, in a literate programming
% fashion. Throughout the package, we use the |axp@| prefix to identify
% local macros and environment names, which are not meant to be used by the final
% user.
% \subsection{Dependencies}
% We first load a few package dependencies:
% \begin{itemize}
% \item \textsf{environ} to easily define the repeated theorem
% environments.
%    \begin{macrocode}
\RequirePackage{environ}
%    \end{macrocode}
% \item \textsf{etoolbox} to define simple toggles.
%    \begin{macrocode}
\RequirePackage{etoolbox}
%    \end{macrocode}
% \item \textsf{fancyvrb} for the bulk of the work of exporting appendix
% material in an auxiliary file.
%    \begin{macrocode}
\RequirePackage{fancyvrb}
%    \end{macrocode}
% \item \textsf{ifthen} for easier comparison of character strings.
%    \begin{macrocode}
\RequirePackage{ifthen}
%    \end{macrocode}
% \item \textsf{kvoptions} to manage options passed to the package.
%    \begin{macrocode}
\RequirePackage{kvoptions}
%    \end{macrocode}
% \item \textsf{catchfile} to be able to check the content of files
% |\input| within appendix content.
%    \begin{macrocode}
\RequirePackage{catchfile}
%    \end{macrocode}
% \item \textsf{amsthm} for its |\newteorem| macro.
% Some document classes (e.g., \textsf{lipics})
% preload \textsf{amsthm}: this is fine, |\RequirePackage{amsthm}|
% will simply have no effect. On the other hand, some other document
% classes (e.g., \textsf{llncs} or \textsf{sig-alternate}) define a
% |proof| environment that conflicts with \textsf{amsthm}, so we have to
% undefine this environment before loading \textsf{amsthm}. In that case,
% we reestablish the existing proof environments, in case they had been
% customized (e.g., \textsf{sig-alternate})
% \changes{v1.0.4}{2017/03/08}{Re-establish custom proof environments}
%    \begin{macrocode}
\@ifpackageloaded{amsthm}{
  }{
    \let\apx@oldamsthmproof\proof
    \let\apx@oldamsthmendproof\endproof
    \let\proof\undefined
    \let\endproof\undefined
  }
\RequirePackage{amsthm}
\ifdefined\apx@oldamsthmproof
  \let\proof\apx@oldamsthmproof
  \let\endproof\apx@oldamsthmendproof
\fi
%    \end{macrocode}
% \end{itemize}
% \subsection{Option Processing}
% Many names throughout the package use an arobase (|@|) to avoid name
% conflict with user-defined names. To simplify the compilation of the
% documentation, we simply make it a regular character in all the rest.
%    \begin{macrocode}
\makeatletter
%    \end{macrocode}
% We setup the processing of options using \textsf{keyval} facilities.
%    \begin{macrocode}
\SetupKeyvalOptions{
  family=axp,
  prefix=axp@
}
%    \end{macrocode}
% We declare the following options:
% \begin{itemize}
%   \item |appendix|, with a default value of |append| (other possible
%     values: |strip|, |inline|);
%   \item |bibliography|, with a default value of |separate| (other
%     possible value: |common|);
%   \item |repeqn|, with a default value of |same| (other possible value:
%   |independent|).
% \end{itemize}
% \begin{macro}{\axp@appendix}
%    \begin{macrocode}
\DeclareStringOption[append]{appendix}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@bibliography}
% \changes{v1.0.4}{2017/03/02}{\texttt{bibliography} option}
%    \begin{macrocode}
\DeclareStringOption[separate]{bibliography}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@bibengine}
% \changes{v1.4.0}{2026/05/14}{\texttt{bibengine} option to select
%   the engine used for separate bibliographies: \textsf{bibunits}
%   (default) or \textsf{biblatex}. See \#29.}
%    \begin{macrocode}
\DeclareStringOption[bibtex]{bibengine}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@repeqn}
% \changes{v1.1.0}{2018/07/20}{\texttt{repeqn} option}
%    \begin{macrocode}
\DeclareStringOption[same]{repeqn}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@forwardlinking}
% \changes{v1.2.3}{2021/07/12}{\texttt{forwardlinking} option}
%    \begin{macrocode}
\DeclareStringOption[yes]{forwardlinking}
%    \end{macrocode}
% \end{macro}
%    \begin{macrocode}
\ProcessLocalKeyvalOptions*
%    \end{macrocode}
% We check that the value of the options are valid, and add a
% message to the compilation log.
%    \begin{macrocode}
\ifthenelse{\equal{\axp@appendix}{append}}{
  \message{apxproof: Appendix material appended to the document}
}{\ifthenelse{\equal{\axp@appendix}{strip}}{
  \message{apxproof: Appendix material stripped}
}{\ifthenelse{\equal{\axp@appendix}{inline}}{
  \message{apxproof: Appendix material inlined within the document}
}{\ifthenelse{\equal{\axp@appendix}{chapterend}}{
  \message{apxproof: Appendix material appended at the end of each chapter}
}{
  \errmessage{Error: unsupported option appendix=\axp@appendix\ for
  package apxproof}
}}}}
\ifthenelse{\equal{\axp@bibengine}{bibtex}}{
}{\ifthenelse{\equal{\axp@bibengine}{biblatex}}{
}{
  \errmessage{Error: unsupported option bibengine=\axp@bibengine\ for
  package apxproof}
}}
\ifthenelse{\equal{\axp@bibliography}{separate}}{
%    \end{macrocode}
% In |bibengine=bibtex| mode (the default), the external
% \textsf{bibunits} package is used to add a second bibliography for
% the appendix material. In |bibengine=biblatex| mode, no extra
% package is loaded here: \textsf{biblatex} is expected to have been
% loaded by the user, and its |refsection| environment is used at
% appendix typeset time to scope citations.
% \changes{v1.2.0}{2019/04/17}{Do not load \textsf{bibunits} if
% \texttt{bibliography} is set to \texttt{common}}
%    \begin{macrocode}
  \ifthenelse{\equal{\axp@bibengine}{bibtex}}{
    \RequirePackage{bibunits}
    \message{apxproof: Separate bibliography for appendix material (bibunits)}
  }{
    \message{apxproof: Separate bibliography for appendix material (biblatex)}
  }
}{\ifthenelse{\equal{\axp@bibliography}{common}}{
  \message{apxproof: Common bibliography for appendix and main text}
}{
  \errmessage{Error: unsupported option bibliography=\axp@bibliography\ for
  package apxproof}
}}
\ifthenelse{\equal{\axp@appendix}{chapterend}
            \AND\equal{\axp@bibliography}{separate}}{
  \PackageError{apxproof}{%
    bibliography=separate is not supported with appendix=chapterend}{%
    Use bibliography=common instead.}
}{}
\ifthenelse{\equal{\axp@repeqn}{same}}{
  \message{apxproof: Repeated equations keep the same numbering}
}{\ifthenelse{\equal{\axp@repeqn}{independent}}{
  \message{apxproof: Repeated equations are independently numbered}
}{
  \errmessage{Error: unsupported option repeqn=\axp@repeqn\ for
  package apxproof}
}}
%    \end{macrocode}
% \begin{macro}{\axp@forward@suppress}
%    \begin{macrocode}
\newbool{axp@forward@suppress}
\ifthenelse{\equal{\axp@forwardlinking}{yes}}{
}{\ifthenelse{\equal{\axp@forwardlinking}{no}}{
  \message{apxproof: Disable forward linking}
  \global\booltrue{axp@forward@suppress}%
}{
  \errmessage{Error: unsupported option forwardlinking=\axp@repeqn\ for
  package apxproof}
}}
%    \end{macrocode}
% \end{macro}
% \subsection{Macros Common to All Compilation Modes}
% \begin{macro}{\axp@newtheoremrep@definetheorem}
% \changes{v1.2.0}{2019/10/01}{Restore predefined theorem counters}
% \changes{v1.2.6}{2025/08/04}{Undefine counter printing command to
% avoid warning}
% Common to all compilation modes, we define
% |\axp@newtheoremrep@definetheorem|.
% When called with first argument |foobar|, we first undefine the existing
% |foobar| environment (and its counter) if it has already been defined (e.g., by the
% document class), then invoke |\axp@newtheorem| for the regular version
% of the theorem |foobar|, saving and restoring any existing theorem counter unless the
% |\newtheoremdep| redefines the base counter.
%    \begin{macrocode}
\def\axp@newtheoremrep@definetheorem#1#2#3#4{%
  \expandafter\let\csname #1\endcsname\undefined
  \expandafter\let\csname the#1\endcsname\undefined
  \ifcsname c@#1\endcsname
    \expandafter\expandafter\expandafter\let\expandafter\expandafter
      \csname c@axp@#1\endcsname\csname c@#1\endcsname
    \expandafter\let\csname c@#1\endcsname\undefined
  \fi
  \axp@newtheorem{#1}{#2}{#3}{#4}%
  \ifcsname c@axp@#1\endcsname
    \ifx\relax#2\relax
      \expandafter\expandafter\expandafter\let\expandafter\expandafter
        \csname c@#1\endcsname\csname c@axp@#1\endcsname
    \else
    \fi
  \fi
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@newtheorem}
% \changes{v1.0.6}{2018/05/10}{Introduce intermediary command for theorem
% macro}
% \begin{macro}{\@axp@newtheorem}
% \begin{macro}{\@@axp@newtheorem}
% We introduce an intermediate |\axp@newtheorem| command to define a new
% theorem, differently depending on whether there is a section counter or
% not. This will be useful, in particular to allow changing this
% definition depending on the document class. This command uses two
% intermediary commands, |\@axp@newtheorem| and |\@@axp@newtheorem|, for
% the non-starred and starred versions.
%    \begin{macrocode}
\def\axp@newtheorem{\@ifstar\@@axp@newtheorem\@axp@newtheorem}
\def\@axp@newtheorem#1#2#3#4{%
  \ifx\relax#4\relax
    \newtheorem{#1}[#2]{#3}%
  \else
    \newtheorem{#1}{#3}[#4]%
  \fi
}
\def\@@axp@newtheorem#1#2{%
  \newtheorem*{#1}{#2}%
}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{\newtheoremrep}
% \begin{macro}{\axp@newtheoremreptmp}
% We define the high-level |\newtheoremrep| to have the same syntax as
% \textsf{amsthm}'s |\newtheorem|, plus an optional |*| variant that
% defers the proof to the appendix without restating the theorem.
% For this purpose, we need a little
% trick to deal with the second and fourth optional arguments, which is what
% |\@oparg| and |\axp@newtheoremreptmp| are
% used for. |\axp@newtheoremrep| is defined differently
% depending on the compilation mode.
% \changes{v1.4.0}{2026/05/14}{Added starred form of \cs{newtheoremrep}
% for proof-deferred theorems that are not restated in the appendix}
%    \begin{macrocode}
\newcommand\newtheoremrep{%
  \@ifstar{\axp@newtheoremrep@star}{\axp@newtheoremrep@nostar}%
}
\newcommand\axp@newtheoremrep@star[1]{%
  \global\csdef{axp@proofof@#1}{}%
  \@oparg{\axp@newtheoremreptmp{#1}}[]%
}
\newcommand\axp@newtheoremrep@nostar[1]{%
  \global\csundef{axp@proofof@#1}%
  \@oparg{\axp@newtheoremreptmp{#1}}[]%
}
\def\axp@newtheoremreptmp#1[#2]#3{%
  \@oparg{\axp@newtheoremrep{#1}[#2]{#3}}[]%
}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{environment}{proofsketch}
% Simple |proofsketch| environment.
% \changes{v1.0.3}{2016/12/16}{Ignore spaces after beginning of Proof sketch}
%    \begin{macrocode}
  \newenvironment{proofsketch}{\begin{axp@oldproof}[Proof sketch]}{\end{axp@oldproof}}
%    \end{macrocode}
% \end{environment}
% \begin{macro}{\mainbodyrepeatedtheorem}
% \changes{v1.2.0}{2019/09/21}{Configurable repeated theorem command}
% \begin{macro}{\appendixrefname}
% \changes{v1.2.1}{2020/10/09}{Fix compatibility with
% \texttt{memoir} document class}
% \begin{macro}{\appendixbibliographystyle}
% \begin{macro}{\appendixbibliographyprelim}
% \begin{macro}{\appendixprelim}
% \changes{v1.0.4}{2017/03/03}{Configurable appendix style}
% \begin{macro}{\appendixsectionformat}
% \changes{v1.1.0}{2018/07/17}{Fix missing space in default
% \cs{appendixsectionformat}}
% \begin{macro}{\appendixproofname}
% \changes{v1.4.0}{2026/05/14}{Added \cs{appendixproofname} hook for
% customizing the title of a proof deferred to the appendix}
% We provide sensible defaults for these user-customizable macros.
% Even though they are not all useful in all modes, we define them for
% all modes so that a |\renewcommand| works in all cases.
%    \begin{macrocode}
\newcommand{\mainbodyrepeatedtheorem}{}
\providecommand{\appendixrefname}{References for the Appendix}
\newcommand{\appendixbibliographystyle}{alpha}
\newcommand{\appendixbibliographyprelim}{}
\newcommand{\appendixprelim}{\clearpage\onecolumn}
\newcommand{\appendixsectionformat}[2]{Proofs for Section~#1\ (#2)}
\newcommand{\appendixproofname}[2]{}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{environment}{axp@oldproof}
%   We save the definition of the existing |proof| environment.
%    \begin{macrocode}
  \let\axp@oldproof\proof
  \let\endaxp@oldproof\endproof
%    \end{macrocode}
% \end{environment}
% We define a utility macro that will be used to properly set the
% |\label| command (and its \textsf{amsmath} counterpart,
% |\label@in@display|) for equations within repeated theorems, depending on the
% compilation mode.
% \begin{macro}{\axp@redefinelabels}
% \changes{v1.1.0}{2018/07/04}{Fix \cs{label} not being disabled
% in \textsf{amsmath} environments, where \cs{label@in@display} is
% used instead (K. D. Bauer)}
% \changes{v1.2.0}{2019/02/15}{Fix extra spacing erroneously
% introduced within the \texttt{\textbackslash axp@redefinelabels}
% macro}
%    \begin{macrocode}
  \newcommand{\axp@redefinelabels}{%
    \providecommand\label@in@display{}%
    \ifthenelse{\equal{\axp@appendix}{inline}}{%
      \let\axp@oldlabel\label
      \let\axp@oldlabel@in@display\label@in@display
      \renewcommand\label[1]{%
        \axp@oldlabel{##1}%
        \axp@oldlabel{##1-apx}%
      }%
      \renewcommand\label@in@display[1]{%
        \axp@oldlabel@in@display{##1}%
        \axp@oldlabel{##1-apx}%
      }%
    }{%
      \let\axp@oldlabel\label
      \let\axp@oldlabel@in@display\label@in@display
      \renewcommand\label[1]{\axp@oldlabel{##1-apx}}%
      \renewcommand\label@in@display[1]{\axp@oldlabel@in@display{##1-apx}}%
    }%
  }
%    \end{macrocode}
% \end{macro}
% \subsubsection{Class-Specific Behavior}
% Finally, some class-specific behavior common to all compilation modes.
% \paragraph{\textsf{llncs} and other Springer document classes}
% \changes{v1.0.6}{2018/05/10}{Better support of Springer document
% classes}
%     \begin{macrocode}
   \ifdefined\spnewtheorem
%    \end{macrocode}
% \begin{macro}{\@axp@newtheorem}
% \begin{macro}{\@@axp@newtheorem}
% It is necessary to use |\spnewtheorem| instead of |\newtheorem| in
% Springer document classes to obtain standard formatting.
%    \begin{macrocode}
\def\@axp@newtheorem#1#2#3#4{%
  \ifx\relax#4\relax
    \ifx\relax#2\relax
      \spnewtheorem{#1}{#3}{\bfseries}{\itshape}%
    \else
      \spnewtheorem{#1}[#2]{#3}{\bfseries}{\itshape}%
    \fi
  \else
    \spnewtheorem{#1}{#3}[#4]{\bfseries}{\itshape}%
  \fi
}
\def\@@axp@newtheorem#1#2{%
  \spnewtheorem*{#1}{#2}{\upshape\bfseries}{\itshape}%
}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{macro}{proofsketch}
% \changes{v1.2.0}{2019/02/15}{Fix proof sketches in inline
% compilation mode for Springer document classes}
% We redefine the |proofsketch| environment, which is used
% differently in the base class.
%    \begin{macrocode}
\renewenvironment{proofsketch}{\begin{axp@oldproof}[sketch]}{\end{axp@oldproof}}
%    \end{macrocode}
% \end{macro}
% We have to redefine the macro |\@thmcountersep| for proper sectioned
% counters.
%    \begin{macrocode}
\def\@thmcountersep{.}
  \fi
%    \end{macrocode}
% \paragraph{\textsf{acmart}}
% \changes{v1.2.6}{2025/09/29}{Fix compilation issues in recent versions of \textsf{acmart}}
% Some versions of \textsf{acmart} define a |\ACM@origsection| macro and
% verify that |\section| has the same content. We need to update
% |\ACM@origsection| accordingly.
%    \begin{macrocode}
\ifdefined\ACM@origsection
  \AtBeginDocument{\let\ACM@origsection\section}
\fi
%    \end{macrocode}
% \subsection{Inline Compilation Mode}
%    \begin{macrocode}
\ifthenelse{\equal{\axp@appendix}{inline}}{
%    \end{macrocode}
% \begin{macro}{\axp@newtheoremrep}
% In inline mode, |\axp@newtheoremrep| uses
% |\axp@newtheoremrep@definetheorem| to define the regular theorem
% environment and creates a repeated theorem
% environment that behaves exactly as the regular theorem environment,
% while calling |\axp@redefinelabels| to make sure that |-axp| variants
% of equation counters are defined.
% \changes{v1.2.0}{2019/02/15}{Fix formatting of theorems without
% notes in some document classes in inline mode}
%    \begin{macrocode}
  \def\axp@newtheoremrep#1[#2]#3[#4]{%
    \axp@newtheoremrep@definetheorem{#1}{#2}{#3}{#4}%
    \NewEnviron{#1rep}[1][]{%
      \ifx\relax##1\relax
        \begin{#1}\axp@redefinelabels\BODY\end{#1}%
      \else
        \begin{#1}[##1]\axp@redefinelabels\BODY\end{#1}%
      \fi
    }
  }
%    \end{macrocode}
% \end{macro}
% \begin{environment}{inlineproof}
% \begin{environment}{nestedproof}
% \begin{environment}{appendixproof}
% In inline mode, these environments behave like the regular |proof|
% environment.
%    \begin{macrocode}
  \let\inlineproof\proof
  \let\endinlineproof\endproof
  \let\nestedproof\proof
  \let\endnestedproof\endproof
  \let\appendixproof\proof
  \let\endappendixproof\endproof
%    \end{macrocode}
% \end{environment}
% \end{environment}
% \end{environment}
% \begin{environment}{toappendix}
% \begin{macro}{\noproofinappendix}
% \begin{macro}{\nosectionappendix}
% In inline mode, this environment and these macros are no-ops.
%    \begin{macrocode}
  \newenvironment{toappendix}{}{}
  \let\noproofinappendix\relax
  \let\nosectionappendix\relax
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{environment}
%    \begin{macrocode}
}
%    \end{macrocode}
% \subsection{Append, Strip, or Chapter End Compilation Modes}
%    \begin{macrocode}
{
%    \end{macrocode}
% We now deal with the case where \textsf{apxproof} really does something
% useful: either append the appendix material to the document, or strip
% it entirely.
% \bigskip
%    \subsubsection{Auxiliary File for the Appendix}
% \begin{macro}{\axp@proofsfile}
% We open a new auxiliary file, with extension |.axp|, where the appendix
% material will be dumped.
% \changes{v1.1.0}{2018/07/11}{Initialization deferred to \cs{AtBeginDocument}
% for compatibility with \cs{dump}ed precompiled preambles (K. D.
% Bauer)}
%    \begin{macrocode}
  \AtBeginDocument{
    \newwrite\axp@proofsfile
    \ifthenelse{\equal{\axp@appendix}{chapterend}}{%
      \axp@openchapterfile
    }{%
      \immediate\openout\axp@proofsfile=\jobname.axp
    }%
  }
%    \end{macrocode}
% \end{macro}
% \begin{environment}{proof}
% \begin{macro}{\section}
% At the beginning of this file, we make |@| a regular character (since
% it will be used in several places for internal names) and reestablish the original definition of
% the |proof| environment and the |\section| macro.
%    \begin{macrocode}
  \newcommand{\axp@writeproofsfilepreamble}{%
    \immediate\write\axp@proofsfile{%
      \noexpand\makeatletter
      \noexpand\let\noexpand\proof\noexpand\axp@oldproof
      \noexpand\let\noexpand\endproof\noexpand\endaxp@oldproof
      \noexpand\let\noexpand\claimproof\noexpand\axp@oldclaimproof
      \noexpand\let\noexpand\endclaimproof\noexpand\endaxp@oldclaimproof
      \noexpand\let\noexpand\section\noexpand\axp@oldsection
    }%
  }
  \AtBeginDocument{
    \ifthenelse{\equal{\axp@appendix}{chapterend}}{}{%
      \axp@writeproofsfilepreamble
    }%
  }
%    \end{macrocode}
% \end{macro}
% \end{environment}
% \begin{macro}{\axp@unactivateeightbit}
% \changes{v1.1.0}{2019/01/28}{Fix compilation of non-ASCII
% characters with \texttt{\textbackslash usepackage[utf8]\{inputenc\}}}
% We need an auxiliary macro to disable active characters that have the
% high bit set when writing to the |.axp| file. See
% \url{https://tex.stackexchange.com/a/145361/166858}
%    \begin{macrocode}
  \def\axp@unactivateeightbit{%
    \count@=128%
    \loop
      \catcode\count@=12%
      \ifnum\count@<255%
      \advance\count@\@ne
    \repeat}
%    \end{macrocode}
% \end{macro}
% \begin{environment}{axp@VerbatimOut}
% \begin{macro}{\FVB@axp@VerbatimOut}
% \begin{macro}{\FVE@axp@VerbatimOut}
% \changes{v1.1.0}{2018/10/02}{Make \textsf{apxproof} compatible with
% independent use of \textsf{fancyvrb}}
% Using the functionalities of the \textsf{fancyvrb} package, we define a
% custom verbatim environment |axp@VerbatimOut| that writes every line
% to the |\axp@proofsfile|. We also use the previous macro to disable active
% characters with the eighth bit set, and we make sure the catcode of |@|
% is reset for every verbatim environment, in case it is used by the user (e.g.,
% as in the \textsf{xypic} package). Finally, as an additional
% precaution, we reset |\FV@CatCodesHook| that is for example set by the
% |commandchars| or |commentchar| option of |\fvset|.
% \changes{v1.2.1}{2020/01/01}{Fix compatibility with
% \textsf{xypic} package}
% \changes{v1.2.1}{2020/10/15}{Fix compatibility with other uses of \textsf{fancyvrb} that set
% \texttt{\textbackslash FV@CatCodesHook}}
%    \begin{macrocode}
  \DefineVerbatimEnvironment{axp@VerbatimOut}{axp@VerbatimOut}{}
  \def\FVB@axp@VerbatimOut{%
    \@bsphack
    \begingroup
      \axp@unactivateeightbit
      \FV@DefineWhiteSpace
      \def\FV@Space{\space}%
      \FV@DefineTabOut
      \def\FV@ProcessLine{\immediate\write\axp@proofsfile}%
      \let\FV@FontScanPrep\relax
      \let\@noligs\relax
      \def\FV@CatCodesHook{}%
      \FV@Scan}
  \def\FVE@axp@VerbatimOut{%
      \immediate\write\axp@proofsfile{\noexpand\makeatletter}%
      \endgroup\@esphack}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{environment}
% \begin{macro}{\axp@markchapterhasappendix}
% No-op by default; overridden in |chapterend| mode to set a flag
% indicating that the current chapter has appendix content.
%    \begin{macrocode}
  \newcommand{\axp@markchapterhasappendix}{}
%    \end{macrocode}
% \end{macro}
% \begin{environment}{toappendix}
% The entire content of this environment is put in appendix,
% starting a new appendix section beforehand if needed.
%    \begin{macrocode}
  \newenvironment{toappendix}
    {\axp@markchapterhasappendix\axp@writesection\axp@VerbatimOut}
    {\endaxp@VerbatimOut}
%    \end{macrocode}
% \end{environment}
%    \subsubsection{Definition of New Theorems}
% \begin{macro}{axp@seenreptheorem}
% Used to indicate whether a repeated theorem was just typeset, without
% its proof.
%    \begin{macrocode}
  \newtoggle{axp@seenreptheorem}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{axp@rpcounter}
% Sequentially incremented for every repeated theorem, used to create labels.
%    \begin{macrocode}
  \newcounter{axp@rpcounter}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{axp@equation}
% \begin{macro}{axp@equationx}
% Used to save the value of the |equation| counter, when |repeqn| is set
% to |same|.
%    \begin{macrocode}
  \newcounter{axp@equation}
  \newcounter{axp@equationx}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{macro}{axp@newtheoremrep}
% With first argument |foobar|, we use
% |\axp@newtheoremrep@definetheorem| to define the regular version of the
% theorem |foobar|. We then patch |\begin{foobar}| so as not to
% expect a proof in the appendix and define an internal theorem |axp@foobarrp| that
% will be used in the appendix to restate the existing theorem.
% In the starred mode, the theorem is not restated in the appendix,
% so we do not define the |axp@foobarrp| environment.
%    \begin{macrocode}
  \def\axp@newtheoremrep#1[#2]#3[#4]{%
    \axp@newtheoremrep@definetheorem{#1}{#2}{#3}{#4}%
    \expandafter\pretocmd\csname #1\endcsname{\noproofinappendix}{}{}%
    \ifcsdef{axp@proofof@#1}{}{%
      \axp@newtheorem*{axp@#1rp}{#3}%
    }%
    \axp@forward@setup{#1}{#2}{#3}{#4}%
%    \end{macrocode}
% We then define a |foobarrep| environment that increments the
% |axp@rpcounter| and typeset the regular |foobar| theorem with a label
% derived from the counter, along with a possible custom command to
% identify repeated theorems. We distinguish the case when the theorem
% argument has a note and when it does not.
% \changes{v1.0.6}{2018/05/10}{Better handling of note-free theorems in
% document classes that treat theorems differently when they have an
% empty note}
% We save the equation counter
% before typesetting the theorem environment, to reset it to the same
% value in the repeated environment when |repeqn| is set to |same|.
% \changes{v1.2.2}{2020/11/25}{Fix handling of optional arguments of
% repeated theorems containing optional arguments}
%    \begin{macrocode}
    \NewEnviron{#1rep}[1][]{%
      \ifthenelse{\equal{\axp@repeqn}{same}}{%
        \setcounter{axp@equation}{\value{equation}}%
      }{}%
      \addtocounter{axp@rpcounter}{1}%
      \ifx\relax##1\relax
        \axp@with@forward{#1}{\begin{#1}}\label{axp@r\roman{axp@rpcounter}}%
      \else
        \axp@with@forward{#1}{\begin{#1}[{##1}]}\label{axp@r\roman{axp@rpcounter}}%
      \fi
      \mainbodyrepeatedtheorem
      \BODY\end{#1}%
%    \end{macrocode}
% We set the |axp@seenreptheorem| toggle to indicate that we are looking
% for the proof of the theorem, then store in a macro the content of the
% theorem's body.
%    \begin{macrocode}
      \global\toggletrue{axp@seenreptheorem}%
      \global\expandafter\let\csname rplet\roman{axp@rpcounter}%
                             \endcsname
      \BODY
%    \end{macrocode}
% Possibly after starting a new appendix section if needed, we typeset a
% repeated version of the theorem using the |axp@foobarrp| environment
% and a reference to the previously defined label.
% We use |\axp@redefinelabels| in this environment to avoid multiply
% defined labels. We have to deal in a careful way with theorem notes: we
% want to use a theorem note to display the number of the repeated
% theorem, but theorem notes are usually typeset in a much
% different way (different font, parentheses) than theorem
% headings. In the case of the Springer document
% classes, we use the  |\theopargself| macro to disable parentheses. For
% other document classes, we need to manually patch the |\thmhead|
% command at the right time. We also specially cover the case of
% the ACM document class where |\@acmplainnotefont| is used instead
% of |\thm@notefont|.
% \changes{v1.0.2}{2016/12/13}{Fix missing space between repeated theorem
% counter and theorem note}
% \changes{v1.0.2}{2016/12/13}{Fix display of repeated theorem counter in
% some document classes}
% \changes{v1.0.6}{2018/05/10}{Fix incorrect use of \texttt{\textbackslash noexpand}
% in optional argument of macro environment}
% \changes{v1.2.0}{2019/10/07}{Fix display of theorem notes}
%    \begin{macrocode}
      \axp@markchapterhasappendix
      \axp@writesection%
      \immediate\write\axp@proofsfile{\noexpand\makeatletter}%
      \ifthenelse{\equal{\axp@repeqn}{same}}{%
        \immediate\write\axp@proofsfile{%
          \noexpand\setcounter{axp@equationx}{\value{equation}}%
          \noexpand\setcounter{equation}{\theaxp@equation}%
        }%
      }{}%
      \ifbool{axp@forward@suppress}{%
        \global\def\axp@refstar{\ref*}
      }{%
        \global\def\axp@refstar{\ref}
      }
%    \end{macrocode}
% We write the auto-title state to the |.axp| file so that the following
% deferred proof can title itself via |\appendixproofname|. In starred
% mode, we also emit a forward-link target into the appendix so that
% hyperlinks from the inline theorem number always land somewhere
% defined, even if the user does not provide a proof.
% \changes{v1.4.0}{2026/05/14}{Write auto-title state for
% \cs{appendixproofname} hook}
%    \begin{macrocode}
      \immediate\write\axp@proofsfile{%
        \noexpand\gdef\noexpand\axp@nextproofof@type{#3}%
        \noexpand\gdef\noexpand\axp@nextproofof@label{axp@r\roman{axp@rpcounter}}%
      }%
      \ifcsdef{axp@proofof@#1}{%
        \immediate\write\axp@proofsfile{%
          \noexpand\axp@forward@target{axp@fw@r\roman{axp@rpcounter}}{}%
        }%
      }{%
%    \end{macrocode}
% In the default (unstarred) mode, we emit the repeated version of the
% theorem to the appendix; the forward-link target sits inside the
% restated theorem. In starred mode, we skip this block entirely.
%    \begin{macrocode}
        \immediate\write\axp@proofsfile{{%
          \ifdefined\theopargself
            \noexpand\theopargself
          \else
            \noexpand\pretocmd{\noexpand\@begintheorem}{%
              \noexpand\patchcmd{\noexpand\thmhead}{\noexpand\@acmplainnotefont}{}{}{}%
              \noexpand\patchcmd{\noexpand\thmhead}{\noexpand\the\noexpand\thm@notefont}{}{}{}%
              \noexpand\patchcmd{\noexpand\thmhead}{(}{}{}{}%
              \noexpand\patchcmd{\noexpand\thmhead}{)}{}{}{}%
            }{}{}
          \fi
          \noexpand\begin{axp@#1rp}
            [%
              \noexpand\axp@refstar{axp@r\roman{axp@rpcounter}}%
              \@ifnotempty{##1}{%
                \ifdefined\theopargself
                \else
                  \ifdefined\@acmplainnotefont
                    \noexpand\@acmplainnotefont
                  \else
                    \noexpand\ifdefined\noexpand\thm@notefont
                      \noexpand\the\noexpand\thm@notefont
                    \noexpand\fi
                  \fi
                \fi
                {} (\unexpanded{{##1}})%
              }%
            ]%
            \noexpand\axp@forward@target{axp@fw@r\roman{axp@rpcounter}}{}%
            \noexpand\axp@redefinelabels
            \expandafter\noexpand\csname rplet\roman{axp@rpcounter}%
                                 \endcsname
          \noexpand\end{axp@#1rp}
        }}%
      }%
      \ifthenelse{\equal{\axp@repeqn}{same}}{%
        \immediate\write\axp@proofsfile{%
          \noexpand\setcounter{equation}{\value{axp@equationx}}%
        }%
      }{}%
    }%
  }
%    \end{macrocode}
% \end{macro}
%
%    \subsubsection{Forward-Linking Mechanism}
% \changes{v1.1.0}{2018/07/02}{Added forward-link mechanism (K. D. Bauer)}
%
% When hyperref is loaded, |foobarrep| environments
% in the main text have their number link to their
% repetition in the appendix.
%
%

%
% \begin{macro}{\axp@with@forward}
% In order to make the number of the |foobarrep| theorem a link
% to its repeated version, we temporarily redefine the
% |\thefoobar| command, or, if we inherited the counter
% from a |bazbar| environment, the |\thebazbar| command.
% This seems to be the only robust way to make the number
% a |\hyperlink|, without adding extensive dependence
% on internals of |amsthm|, the builtin |\newtheorem|
% and possibly document-class specific definitions.
%
% In order to allow users to redefine |\thefoobar| without
% breaking this feature, we redefine |\thefoobar| only
% for the duration of the |\begin{foobar}| form,
% resetting it to the old value as soon as possible.
%
% Redefining |\thefoobar| has the side effect of changing |\newlabel| entries
% in the |.aux| file, so we need to to be able to disable
% addition of the hyperlink, which is why we use an
% intermediate |\axp@forward@link|\marg{target}\marg{text} macro,
% We also redefine |\theHfoobar| which is used by |hyperref| but not
% defined if |hyperref| was loaded after |\newtheoremrep| was used
% and |\protect| it to output it verbatim into the |.aux| file.
%
% These hyperlinks are of course disabled in the |strip| compilation
% mode.
%    \begin{macrocode}
   \newcommand{\axp@with@forward}[2]{%
     \ifthenelse{\equal{\axp@appendix}{strip}}{#2}{
       \global\booltrue{axp@forward}%
       \ifcsundef{axp@old@the\csname axp@cn@#1\endcsname}{%
         \csletcs{axp@old@the\csname axp@cn@#1\endcsname}{the\csname axp@cn@#1\endcsname}%
         \csletcs{theH\csname axp@cn@#1\endcsname}{the\csname axp@cn@#1\endcsname}%
         \csdef{the\csname axp@cn@#1\endcsname}{%
           \protect\axp@forward@link{axp@fw@r\roman{axp@rpcounter}}%
             {\csname axp@old@the\csname axp@cn@#1\endcsname\endcsname}%
         }%
       }{}%
       #2%
       \ifcsdef{axp@old@the\csname axp@cn@#1\endcsname}{%
         \csletcs{the\csname axp@cn@#1\endcsname}{axp@old@the\csname axp@cn@#1\endcsname}%
       }{}%
       \global\boolfalse{axp@forward}
     }}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\axp@forward@link}
% \begin{macro}{axp@forward}
% Dummy macro, for handling the unwanted change of
% the |\newlabel| entry in the |.aux| file caused by
% changing the definition of |\thefoobar|. We also use |\texorpdfstring|
% to detect whether the reference appears within a PDF bookmark, in which
% case we skip the test altogether.
% \changes{v1.2.5}{2025/07/25}{Remove forward linking command from PDF bookmarks}
%
%    \begin{macrocode}
  \newbool{axp@forward}
  \newcommand{\axp@forward@link}[2]{%
    \ifdefined\texorpdfstring\else\newcommand\texorpdfstring[2]{#2}\fi
    \texorpdfstring{%
      \ifboolexpr{ bool {axp@forward} and not bool {axp@forward@suppress} }{%
        \ifcsdef{hyperlink}{%
          \hyperlink{#1}{#2}%
        }{%
          #2%
        }%
      }{%
        #2%
      }%
    }{%
      #2%
    }%
  }%
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\axp@forward@target}
% Provides the needed |\hypertarget|.
% Intended to be written to the |.axp| file.
%    \begin{macrocode}
  \newcommand{\axp@forward@target}[2]{%
    \ifcsname hypertarget\endcsname
      \hypertarget{#1}{#2}%
    \else
      #2%
    \fi
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\axp@forward@setup}
% \changes{v1.2.6}{2025/08/04}{Support for \texttt{aliascnt} counters}
% In order to support counter inheritance with the first
% optional argument of |\newtheoremrep|,
% we need access to the name of the counter.
% For compliance with the behavior of |\@axp@newtheorem|,
% the first optional argument (|#2|) is ignored
% if the second optional argument (|#4|) is given.
% We also check whether |aliascnt| was used to define this counter;
% if so, we use the original name as the counter name to handle
% the way such aliases work.
%    \begin{macrocode}
  \newcommand{\axp@forward@setup}[4]{%
    \csedef{axp@cn@#1}{\ifblank{#4}{\ifblank{#2}{#1}{\ifcsdef{AC@cnt@#1}{#1}{#2}}}{#1}}%
  }
%    \end{macrocode}
% \end{macro}
%
%
%    \subsubsection{Proof Environments}
% \begin{macro}{\noproofinappendix}
%   Utility macro that toggles |axp@seenreptheorem| to false.
%    \begin{macrocode}
  \newcommand\noproofinappendix{%
    \global\togglefalse{axp@seenreptheorem}%
  }
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@apxproofstart}
% \changes{v1.4.0}{2026/05/14}{Helper that consults \cs{appendixproofname}
% at appendix typeset time to auto-title deferred proofs}
% Helper invoked from the |.axp| file at appendix typeset time to open
% a deferred proof. Reads the |\axp@nextproofof@*| state (set by the
% preceding |theoremrep|) and consults |\appendixproofname| to decide
% whether to pass an optional title argument.
%    \begin{macrocode}
  \gdef\axp@nextproofof@type{}
  \gdef\axp@nextproofof@label{}
  \def\axp@apxproofstart#1{%
    \protected@xdef\axp@apxproofstart@title{%
      \appendixproofname{\axp@nextproofof@type}%
                        {\noexpand\ref{\axp@nextproofof@label}}%
    }%
    \ifx\axp@apxproofstart@title\@empty
      \begin{#1}%
    \else
      \expandafter\axp@apxproofstart@withtitle
        \expandafter{\axp@apxproofstart@title}{#1}%
    \fi
  }
  \def\axp@apxproofstart@withtitle#1#2{\begin{#2}[#1]}
%    \end{macrocode}
% \end{macro}
% \begin{environment}{appendixproof}
% \changes{v1.2.4}{2021/11/03}{Support optional arguments in proofs}
% We dump the content of this in appendix, within an original |proof|
% environment, possibly after creating a new appendix section.
% We support
% optional arguments in proofs, but we need to be careful not to gobble
% any newline character, that would make it impossible to process the
% first line with \textsf{fancyvrb}.
% \changes{v1.4.0}{2026/05/14}{Route deferred proofs through
% \cs{axp@apxproofstart} to support \cs{appendixproofname}}
% When the proof is auto-routed from a |theoremrep| (the toggle
% |axp@seenreptheorem| is true at this point and no explicit optional
% argument was given), we delegate the proof opening to
% |\axp@apxproofstart| so that |\appendixproofname| is consulted at
% appendix typeset time. A user-supplied optional argument or a manual
% |\appendixproof| call (toggle false) bypasses this and uses
% the standard ``Proof.'' heading.
%    \begin{macrocode}
  \def\appendixproof{\catcode`\^^M=\active\@ifnextchar[{\catcode`\^^M=5\@@appendixproof[axp@oldproof]}{\catcode`\^^M=5\@appendixproof[axp@oldproof]}}
  \def\@appendixproof[#1]%
    {%
      \axp@markchapterhasappendix
      \axp@writesection
      \iftoggle{axp@seenreptheorem}{%
        \immediate\write\axp@proofsfile{%
          \noexpand\makeatletter\noexpand\axp@apxproofstart{#1}\noexpand\makeatother%
        }%
      }{%
        \immediate\write\axp@proofsfile{%
          \noexpand\makeatletter\noexpand\begin{#1}\noexpand\makeatother%
        }%
      }%
      \axp@VerbatimOut
    }
  \def\@@appendixproof[#1][#2]%
    {%
      \axp@markchapterhasappendix
      \axp@writesection
      \immediate\write\axp@proofsfile{%
        \noexpand\makeatletter\noexpand\begin{#1}[\unexpanded{#2}]\noexpand\makeatother%
      }%
      \axp@VerbatimOut
    }
  \def\endappendixproof
    {%
      \endaxp@VerbatimOut
      \immediate\write\axp@proofsfile{%
        \noexpand\end{axp@oldproof}%
      }%
      \noproofinappendix
    }
%    \end{macrocode}
% \end{environment}
% \begin{environment}{proof}
% This environment either puts the proof in appendix, if we are after a repeated
% theorem without its proof, or inlines it otherwise. We support
% optional arguments in proofs, but we need to be careful not to gobble
% any newline character, that would make it impossible to process the
% first line with \textsf{fancyvrb}.
%    \begin{macrocode}
  \def\proof{\catcode`\^^M=\active\ltx@ifnextchar@nospace[{\catcode`\^^M=5\axp@@proof}{\catcode`\^^M=5\axp@proof}}
  \def\axp@proof
    {%
      \iftoggle{axp@seenreptheorem}{%
        \appendixproof
      }{%
        \axp@oldproof
      }%
    }
  \def\axp@@proof[#1]%
    {%
      \iftoggle{axp@seenreptheorem}{%
        \appendixproof[#1]
      }{%
        \axp@oldproof[#1]
      }%
    }
  \def\endproof
    {%
      \iftoggle{axp@seenreptheorem}{%
        \endappendixproof
      }{%
        \endaxp@oldproof
      }%
    }
%    \end{macrocode}
% \end{environment}
% \begin{environment}{claimproof}
% \changes{v1.2.4}{2021/11/03}{Support for \texttt{claimproof} environment
% from \textsf{lipics}}
% \changes{v1.2.5}{2024/01/26}{Support for locally defined \texttt{claimproof} environment}
% If the |claimproof| environment exists (\textsf{lipics} document
% class or user-defined), we also redefine it, in the same way.
%    \begin{macrocode}
  \AtBeginDocument{
    \ifdefined\claimproof
    \let\axp@oldclaimproof\claimproof
    \let\endaxp@oldclaimproof\endclaimproof
    \def\claimproof{\catcode`\^^M=\active\ltx@ifnextchar@nospace[{\catcode`\^^M=5\axp@@claimproof}{\catcode`\^^M=5\axp@claimproof}}
    \def\appendixclaimproof{\catcode`\^^M=\active\@ifnextchar[{\catcode`\^^M=5\@@appendixproof[axp@oldclaimproof]}{\catcode`\^^M=5\expandafter\@appendixproof[axp@oldclaimproof]}}
    \def\axp@claimproof
      {%
        \iftoggle{axp@seenreptheorem}{%
          \appendixclaimproof
        }{%
          \axp@oldclaimproof
        }%
      }
    \def\axp@@claimproof[#1]%
      {%
        \iftoggle{axp@seenreptheorem}{%
          \appendixclaimproof[#1]%
        }{%
          \axp@oldclaimproof[#1]%
        }%
      }
    \def\endclaimproof
      {%
        \iftoggle{axp@seenreptheorem}{%
          \endappendixclaimproof
        }{%
          \endaxp@oldclaimproof
        }%
      }
    \def\endappendixclaimproof
    {%
      \endaxp@VerbatimOut
      \immediate\write\axp@proofsfile{%
      \noexpand\end{axp@oldclaimproof}%
      }%
      \noproofinappendix
    }
    \fi
  }
%    \end{macrocode}
% \end{environment}

% \begin{environment}{inlineproof}
% \begin{environment}{nestedproof}
% These two environments are synonyms for the original |proof|
% environment.
%    \begin{macrocode}
  \let\inlineproof\axp@oldproof
  \let\endinlineproof\endaxp@oldproof
  \let\nestedproof\axp@oldproof
  \let\endnestedproof\endaxp@oldproof
%    \end{macrocode}
% \end{environment}
% \end{environment}
%    \subsubsection{Section Management}
% \begin{macro}{axp@seccounter}
% Sequentially incremented for every section, used to create labels.
%    \begin{macrocode}
  \newcounter{axp@seccounter}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@sectitle}
% Saves the title of the last encountered section.
%    \begin{macrocode}
  \def\axp@sectitle{}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@section}
% \changes{v1.2.2}{2021/06/23}{Detect a section within an included
% file to avoid produced useless sections}
% \changes{v1.3.0}{2026/04/09}{Skip blank lines before detecting
% a section command}
% \changes{v1.3.0}{2026/04/09}{Detect a section within an
% \cs{include}'d file}
% \begin{macro}{\axp@@sectionskippar}
% \begin{macro}{\axp@@sectiontestinput}
% \begin{macro}{\axp@@sectiontestinclude}
% \begin{macro}{\axp@@sectionfinishinclude}
% \begin{macro}{\axp@@sectiontestsection}
% This command |\axp@section| behaves similarly to |\axp@oldsection|,
% except that it first tests whether a |\section| follows, and if so,
% does not produce anything. This is useful to avoid producing empty
% sections in the appendix. Using the \textsf{catchfile} package, we
% also check whether a |\section| is within an |\input| or |\include|
% that immediately follows. We also skip any |\par| tokens (produced by
% blank lines) before performing the check.
%    \begin{macrocode}
  \def\axp@section#1{%
    \@ifnextchar\par
      {\axp@@sectionskippar{#1}}%
      {\@ifnextchar\input
        {\axp@@sectiontestinput{#1}}%
        {\@ifnextchar\include
          {\axp@@sectiontestinclude{#1}}%
          {\axp@@sectiontestsection{#1}}}}%
  }
  \def\axp@@sectionskippar#1\par{\axp@section{#1}}
  \def\axp@@sectiontestinput#1\input#2{%
    \CatchFileDef{\axp@tmp}{#2}{}%
    \def\axp@tmptmp{\axp@@sectiontestsection{#1}}%
    \expandafter\axp@tmptmp\axp@tmp%
  }
  \def\axp@@sectiontestinclude#1\include#2{%
    \set@curr@file{#2}%
    \edef\axp@tmp{\@strip@tex@ext\@curr@file.tex}%
    \expandafter\CatchFileDef\expandafter\axp@tmp\expandafter{\axp@tmp}{}%
    \def\axp@tmptmp{\axp@@sectionfinishinclude{#1}{#2}}%
    \expandafter\axp@tmptmp\axp@tmp\@nil
  }
  \long\def\axp@@sectionfinishinclude#1#2#3#4\@nil{%
    \ifx#3\section
      \makeatother
    \else
      \axp@oldsection{#1}\makeatother
    \fi
    \include{#2}%
  }
  \def\axp@@sectiontestsection#1{\@ifnextchar\section{\makeatother}{\axp@oldsection{#1}\makeatother}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{\axp@oldsection}
% \begin{macro}{\section}
% \changes{v1.1.0}{2018/07/25}{Rewrote definition of \cs{section}
% to enable optional argument. See \#23. (K. D. Bauer)}
% \changes{v1.1.0}{2018/07/25}{Fix handling of fragile macros within
% section headings. See \#22.}
% \begin{macro}{\@section}
% \begin{macro}{\@@section}
% We redefine the |\section| command to create a label based on
% |axp@seccounter| and to store its title in |\axp@sectitle|.
% In order to support starred and unstarred versions, as well
% as the optional short-title argument, the intermediate macros
% \cs{@section} and \cs{@@section} are needed.
%    \begin{macrocode}
  \let\axp@oldsection\section
  \def\section{\@ifstar\@section\@@section}
  \newcommand{\@section}[2][\relax]{\axp@@@section{*}{#1}{#2}}%
  \newcommand{\@@section}[2][\relax]{\axp@@@section{}{#1}{#2}}%
  \newcommand{\axp@@@section}[3]{%
    \global\def\axp@sectitle{#3}%
    \ifx\relax#2\relax
      \axp@oldsection#1{#3}%
    \else
      \axp@oldsection#1[{#2}]{#3}%
    \fi
    \addtocounter{axp@seccounter}{1}%
    \label{axp@s\roman{axp@seccounter}}%
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{\nosectionappendix}
% We remove the current section title, to indicate no section should be
% created in the appendix.
%    \begin{macrocode}
  \newcommand{\nosectionappendix}{
    \global\def\axp@sectitle{}%
  }
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@writesection}
% If |\axp@sectitle| is not empty, we create
% a new section in the appendix, referring to the main text section.
%
% Here, we wrap |\ref{axp@si}| into |\axp@protectref@i|, in order
% to protect the label name from wrongly being converted to uppercase,
% e.g., in \textsf{fancyhdr} with |\pagestyle{fancy}|.
%
% This macro is defined both in the |.aux| file (in order to ensure availability
% when typesetting the |\tableofcontents|), and immediately before typesetting
% the appendix section (to ensure availability in the |\section| command).
%
% \changes{v1.0.6}{2018/05/10}{Fix extraneous space after section number
% in appendix titles}
% \changes{v1.1.0}{2018/07/09}{Make \cs{axp@tmp} wrapper more robust.
% Resolves issues from use of section title in \textsf{fancyhdr},
% and in \cs{tableofcontents} (K. D. Bauer).}
%    \begin{macrocode}
  \newcommand\axp@writesection{%
    \ifx\axp@sectitle\@empty
    \else
      \edef\axp@tmp{%
        \noexpand\global\noexpand\def
        \expandonce{\csname axp@protectref@\roman{axp@seccounter}\endcsname}{%
          \noexpand\ref{axp@s\roman{axp@seccounter}}%
        }%
      }%
      \immediate\write\@auxout{\expandonce\axp@tmp}
      \immediate\write\axp@proofsfile{%
        \expandonce\axp@tmp^^J%
        \noexpand\axp@section{%
          \noexpand\appendixsectionformat{%
            \protect
            \expandonce{\csname axp@protectref@\roman{axp@seccounter}\endcsname}%
          }{\expandonce\axp@sectitle}%
        }%
      }%
      \nosectionappendix
    \fi
  }
%    \end{macrocode}
% \end{macro}
% Finally, in a somewhat ad hoc manner, we disable the whole section management
% for \cs{tableofcontents}, which may be typeset using a section heading,
% but for which automatic section management does not make sense.
%
% \begin{macro}{\axp@oldtableofcontents}
% \begin{macro}{\tableofcontents}
% \changes{v1.1.0}{2018/07/09}{Disable section management for table
% of contents}
%    \begin{macrocode}
  \let\axp@oldtableofcontents\tableofcontents
  \def\tableofcontents{{\let\section\axp@oldsection\axp@oldtableofcontents}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%    \subsubsection{Append Compilation Mode}
%    \begin{macrocode}
  \ifthenelse{\equal{\axp@appendix}{append}}{
%    \end{macrocode}
% \begin{macro}{\axp@oldbibliography}
% \begin{macro}{\bibliography}
% In |bibengine=bibtex| mode, when the |bibliography| option is set to
% |separate|, we need to set the appendix bibliography source to be
% the same as that of the main text, thanks to \textsf{bibunits}'s
% |\defaultbibliography| macro. In |bibengine=biblatex| mode, no such
% rebinding is needed: \textsf{biblatex} resources declared with
% |\addbibresource| are visible to all |refsection|s globally.
% \changes{v1.4.0}{2026/05/14}{Only rebind for bibunits engine.
%   See \#29.}
%    \begin{macrocode}
    \ifthenelse{\equal{\axp@bibliography}{separate}
                \AND\equal{\axp@bibengine}{bibtex}}{
      \let\axp@oldbibliography\bibliography
      \renewcommand\bibliography[1]{%
        \defaultbibliography{#1}%
        \axp@oldbibliography{#1}%
      }
    }{}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% After the end of the main text, we add the appendix (after the command
% |\appendixprelim| is issued)
% within a |bibunit| environment so as to typeset
% a separate bibliography for the appendix (unless the |bibliography|
% option is set to |common|). This is done using |\pretocmd| on
% |\@enddocumenthook| instead of |\AtEndDocument| because we want the
% code to be run before any code in the |\@enddocumenthook| that has
% been set in the document class, as in the \textsf{amsart} document
% class.
% \changes{v1.2.2}{2021/03/23}{Compatibility with AMS document classes: do not
% use \texttt{\textbackslash AtEndDocument}}
% There is an extra test to
% ensure an empty bibliography environment is not produced.
% The name of the bibliography is changed to |\appendixrefname|; in most
% document classes, it is called |\refname| but it is occasionally
% (|scrartcl|, |scrreprt|) called |\bibname|. An ad-hoc test is added
% to fix a conflict with the \textsf{natbib} package which redefines
% |bibcite| at the end of the document.
% \changes{v1.0.6}{2018/05/10}{Deal with document classes where the
% bibliography is called \texttt{\textbackslash bibname}}
% \changes{v1.2.1}{2020/02/27}{Ad hoc fix for \textsf{natbib}
% package conflict}
% \changes{v1.2.3}{2021/08/03}{More robust redefinition of
% \texttt{thebibliography} environment, for compatibility with
% \textsf{tocbibind}}
%    \begin{macrocode}
    \pretocmd{\@enddocumenthook}{%
      \ifdefined\NAT@testdef
        \renewcommand\bibcite[2]{%
          \global\@namedef{b@#1\@extra@binfo}{#2}%
        }
      \fi
      \appendixprelim
      \appendix
      \ifthenelse{\equal{\axp@bibliography}{separate}}{
        \ifthenelse{\equal{\axp@bibengine}{bibtex}}{
          \begin{bibunit}[\appendixbibliographystyle]
        }{
          \begin{refsection}
        }
      }{}
        \immediate\closeout\axp@proofsfile
        \input{\jobname.axp}
      \ifthenelse{\equal{\axp@bibliography}{separate}}{
        \ifthenelse{\equal{\axp@bibengine}{bibtex}}{
          \ifdefined\refname
            \renewcommand{\refname}{\appendixrefname}
          \else\ifdefined\bibname
            \renewcommand{\bibname}{\appendixrefname}
          \fi\fi
          \let\axp@oldthebibliography\thebibliography
          \let\endaxp@oldthebibliography\endthebibliography
          \renewenvironment{thebibliography}[1]{%
            \def\axp@tmp{#1}%
            \ifx\axp@tmp\empty
              \gdef\axp@noappendixbibliography1\relax
            \else
              \begin{axp@oldthebibliography}{#1}%
            \fi
          }{%
            \ifdefined\axp@noappendixbibliography\relax\else\end{axp@oldthebibliography}%
          \fi}
          \appendixbibliographyprelim
          \putbib
          \end{bibunit}
        }{
          \appendixbibliographyprelim
          \printbibliography[title=\appendixrefname]
          \end{refsection}
        }
        \ifdefined\NAT@testdef
          \let\bibcite\NAT@testdef
        \fi
      }{}
    }{}{}
%    \end{macrocode}
%    \begin{macrocode}
  }{}
%    \end{macrocode}
%    \subsubsection{Chapter End Compilation Mode}
% \changes{v1.3.0}{2026/04/14}{Added \texttt{chapterend} appendix mode}
% \changes{v1.4.0}{2026/04/29}{Flush \texttt{chapterend} appendices at
%   \cs{part}, \cs{frontmatter}, \cs{mainmatter}, and \cs{backmatter}
%   boundaries so the last chapter of a part/matter is not emitted under
%   the next one}
% \changes{v1.4.0}{2026/04/29}{Public \cs{flushchapterappendix} macro
%   for manually emitting the current chapter's accumulated appendix in
%   \texttt{chapterend} mode}
% In |chapterend| mode, appendix material for each chapter is gathered
% in a per-chapter auxiliary file and emitted immediately after that
% chapter's content (and bibliography). This requires the
% \textsf{appendix} package for its |subappendices| environment.
%    \begin{macrocode}
  \ifthenelse{\equal{\axp@appendix}{chapterend}}{
%    \end{macrocode}
%    \begin{macrocode}
    \RequirePackage{appendix}
%    \end{macrocode}
% \begin{macro}{\chapterappendixprelim}
% \begin{macro}{\axp@beginchapterappendix}
% \begin{macro}{\axp@endchapterappendix}
% User-customizable hooks surrounding each chapter's appendix block.
% By default they open and close a |subappendices| environment.
%    \begin{macrocode}
    \newcommand{\chapterappendixprelim}{\clearpage}
    \newcommand{\axp@beginchapterappendix}{%
      \chapterappendixprelim
      \begin{subappendices}%
    }
    \newcommand{\axp@endchapterappendix}{%
      \end{subappendices}%
    }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{axp@chaptercounter}
% \begin{macro}{axp@chapterhasappendix}
% \begin{macro}{\axp@currentaxpfile}
% Counter for unique per-chapter file names, a boolean tracking whether
% the current chapter has any appendix content, and the current file name.
%    \begin{macrocode}
    \newcounter{axp@chaptercounter}
    \newbool{axp@chapterhasappendix}
    \def\axp@currentaxpfile{}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{\axp@markchapterhasappendix}
% Override the no-op to set the boolean when content is written.
%    \begin{macrocode}
    \renewcommand{\axp@markchapterhasappendix}{%
      \global\booltrue{axp@chapterhasappendix}%
    }
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@openchapterfile}
% Open a fresh auxiliary file for the current chapter.
% We write |\begingroup| first so that all the preamble |\let|
% assignments (and any catcode changes from |\makeatletter|) are
% scoped to this chapter's appendix and do not leak into the rest of
% the document when the file is |\input|'d. We also reset the
% section-title tracker and the repeated-theorem toggle so that state
% from the previous chapter cannot bleed into the new file.
%    \begin{macrocode}
    \newcommand{\axp@openchapterfile}{%
      \edef\axp@currentaxpfile{\jobname-\arabic{axp@chaptercounter}.axp}%
      \immediate\openout\axp@proofsfile=\axp@currentaxpfile
      \immediate\write\axp@proofsfile{\noexpand\begingroup}%
      \axp@writeproofsfilepreamble
      \global\boolfalse{axp@chapterhasappendix}%
      \global\def\axp@sectitle{}%
      \global\togglefalse{axp@seenreptheorem}%
    }
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@flushchapterappendix}
% Close the current chapter's auxiliary file (appending |\endgroup|
% to close the group opened in |\axp@openchapterfile|) and, if it
% contains any content, emit it inside a |subappendices| block.
%    \begin{macrocode}
    \newcommand{\axp@flushchapterappendix}{%
      \immediate\write\axp@proofsfile{\noexpand\endgroup}%
      \immediate\closeout\axp@proofsfile
      \ifbool{axp@chapterhasappendix}{%
        \axp@beginchapterappendix
        \input{\axp@currentaxpfile}%
        \axp@endchapterappendix
      }{}%
    }
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\axp@chapterboundary}
% \begin{macro}{\flushchapterappendix}
% |\axp@chapterboundary| flushes the previous chapter's auxiliary file,
% increments the counter, and opens the next one. It is used as a hook
% for |\chapter|, |\part|, and the |book|-class matter-switching
% commands |\frontmatter|, |\mainmatter|, |\backmatter|, so that
% per-chapter appendices are emitted before any of these structural
% transitions (otherwise the appendix of the last chapter of a
% part/matter would end up under the next part/matter).
% |\flushchapterappendix| is a public alias for the same operation,
% which can be invoked manually wherever an appendix flush boundary is
% needed (e.g., before a bibliography that does not produce its own
% chapter heading).
%    \begin{macrocode}
    \newcommand{\axp@chapterboundary}{%
      \axp@flushchapterappendix
      \stepcounter{axp@chaptercounter}%
      \axp@openchapterfile
    }
    \let\flushchapterappendix\axp@chapterboundary
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{macro}{\axp@hookboundary}
% Helper to install |\axp@chapterboundary| as a |\pretocmd| hook on a
% structural command (|\chapter|, |\part|, |\frontmatter|, etc.),
% issuing a package warning if the hook fails.
%    \begin{macrocode}
    \newcommand{\axp@hookboundary}[1]{%
      \pretocmd{#1}{\axp@chapterboundary}{}{%
        \PackageWarning{apxproof}{%
          Could not hook into \string#1;^^J%
          chapter appendices may not be emitted correctly}}%
    }
%    \end{macrocode}
% \end{macro}
% At the start of each |\chapter|, |\part|, or matter-switching
% command, run the boundary hook. At |\end{document}|, flush the
% last chapter's file.
%    \begin{macrocode}
    \AtBeginDocument{%
      \ifdef{\chapter}{%
        \axp@hookboundary{\chapter}%
      }{%
        \PackageWarning{apxproof}{%
          appendix=chapterend used but \string\chapter\space
          is not defined}%
      }%
      \ifdef{\part}{\axp@hookboundary{\part}}{}%
      \ifdef{\frontmatter}{\axp@hookboundary{\frontmatter}}{}%
      \ifdef{\mainmatter}{\axp@hookboundary{\mainmatter}}{}%
      \ifdef{\backmatter}{\axp@hookboundary{\backmatter}}{}%
    }
    \pretocmd{\@enddocumenthook}{%
      \axp@flushchapterappendix
    }{}{}
%    \end{macrocode}
%    \begin{macrocode}
  }{}
%    \end{macrocode}
% \subsubsection{Class-Specific Behavior}
% We conclude with some class-specific behavior.
% \paragraph{ACM Document Classes (old versions, till 2017)}
% \changes{v1.0.4}{2017/03/08}{More faithful theorem style for ACM templates}
% \changes{v1.0.4}{2017/03/08}{More robust coherent styling of proof sketches}
%    \begin{macrocode}
  \ifdefined\@acmtitlebox
%    \end{macrocode}
% We first redefine the |proofsketch| environment, which is used
% differently in the base class.
%    \begin{macrocode}
  \renewenvironment{proofsketch}{\begin{axp@oldproof}[sketch]}{\end{axp@oldproof}}
%    \end{macrocode}
% We adjust the styling of theorems for the needs of \textsf{apxproof}.
%    \begin{macrocode}
  \newtheoremstyle{mystyle}
    {6pt}
    {6pt}
    {\itshape}
    {10pt}
    {\scshape}
    {.}
    {.5em}
    {}
  \theoremstyle{mystyle}
%    \end{macrocode}
% \begin{macro}{\thebibliography}
% \begin{macro}{\refname}
% \begin{macro}{\appendixrefname}
% The section title of the bibliography is in uppercase in these document
% classes. In addition, the |\thebibliography| macro hard-codes twice
% the section title, so we un-hardcode it so that it can be modified
% in the appendix.
%    \begin{macrocode}
    \patchcmd{\thebibliography}{References}{\protect\refname}{}{}
    \patchcmd{\thebibliography}{References}{\protect\refname}{}{}
    \newcommand{\refname}{REFERENCES}
    \renewcommand{\appendixrefname}{REFERENCES FOR THE APPENDIX}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
%    \begin{macrocode}
  \fi
%    \end{macrocode}
% \changes{v1.0.6}{2018/05/10}{Support of new ACM document class
% (\texttt{acmart.cls})}
% \paragraph{\textsf{lipcs}}
%    \begin{macrocode}
   \ifdefined\lipics@opterrshort
%    \end{macrocode}
% \begin{macro}{\appendixbibliographyprelim}
% \changes{v1.0.3}{2017/02/06}{Support for lipics-v2016}
% The default bibliography in the \textsf{lipics} document class
% formatting is not compatible with the
% \textsf{alpha} bibliography style. We fix this here.
%    \begin{macrocode}
    \renewcommand{\appendixbibliographyprelim}{%
      \global\let\@oldbiblabel\@biblabel
      \def\@biblabel{\hspace*{-2em}\small\@oldbiblabel}%
    }
%    \end{macrocode}
% \end{macro}
%    \begin{macrocode}
  \fi
}
%    \end{macrocode}
% \Finale
% \bibliographystyle{plain}
% \bibliography{apxproof}