QED.sty

\message{<Paul Taylor's right-justified end-of-proof mark, 24 June 1995>}
%%========================================================================
%************************************************************************%
%*                                                                      *%
%*      Maybe you already have a \qed macro to put a little square      *%
%*      on the right hand side at the end of a mathematical proof?      *%
%*                                                                      *%
%*                      IT'S PROBABLY MINE!                             *%
%*                                                                      *%
%*           Please compare yours with \parag@pushright below.          *%
%*                                                                      *%
%*      This new file is compatible, it fixes the bugs, has extra       *%
%*      features, and is maintained and portable.  Please use it!       *%
%*                                                                      *%
%*      NEW FEATURE \WillHandleQED and \HandleQED for environments      *%
%*      which do not give access to the right hand side of the page.    *%
%*                                                                      *%
%*      Now also available: \DefineStandardTheorems or                  *%
%*      \usepackage[define-standard-theorems]{QED}                      *%
%*                                                                      *%
%************************************************************************%
%========================================================================%

% COPYRIGHT (C) 1993-6 Paul Taylor, Department of Computer Science,
% Queen Mary & Westfield College, London E1 4NS.
% You may use this file for any purpose. You may copy it on condition that
% you do so in exactly the form in which it was supplied, make no charge
% for the copy and impose the same conditions on the recipient.
% This software is supplied ``as is'' without warranty, express or implied,
% including but not limited to merchantability and fitness for any particular
% purpose. No liability is accepted by the author, Queen Mary and Westfield
% College, Imperial College or any third party for any direct or consequential
% damages, however caused.
% In all cases you must keep several backup versions of your files in a place
% to which your machine cannot write, and you must check documents visually
% before submitting the final version to a publisher or expensive printer.

% WARNING: If you use this with Frank Mittlebach's "theorem.sty" you
% MUST load theorem first, then QED, then do the \newtheorem commands.

%       Authoritative version:  /pub/tex/contrib/pt/proofs/QED.sty
%       by anonymous FTP from:  ftp.dcs.qmw.ac.uk
%       also CTAN:              macros/generic/proofs/taylor/QED.sty
%       author/maintained by:   Dr Paul Taylor <pt@dcs.qmw.ac.uk>

% WARNING: This file replaces some of the functions of "Paul.sty"
% and MUST NOT be used with it as serious conflicts will arise.

%=========================================================================

%       USER INTERFACE AND EXAMPLES (see also OPTIONS below)

% LaTeX2e \usepackage options: see \DeclareOption commands below
% theorems-as-commands roman-theorems italic-theorems
% auto-QED explicit-QED number-theorems dont-number-theorems
% black-square open-square

%       \begin{Proof} ... \end{Proof}
%       or    \Proof  ... \endProof
%       or    \Proof  ... \endproof
%       or    \Proof  ... \qed          also \QED \QEF and \QEI

% WARNING: \qed which doesn't follow \Proof will not print the square.

% Each properly nested Proof/endProof pair gets a single
% \qedsymbol at the end, on the far right hand side.
% Unfortunately, this may still be in the wrong place
% if the proof ends with a display or itemisation.
% To solve this, \qed may be put *inside* the last substantive
% environment; the \qedsymbol will be put at that point
% (the visual end) and suppressed at the logical end of proof.

% Examples:
%       \begin{Proof}                   \begin{Proof}
%         ...                             ...
%         \begin{eqnarray}                \begin{itemize}
%          ... &=& ... \\                   \item ...
%          ..  &=& ... &\qed                \item ...   \qed
%         \end{eqnarray}                  \end{itemize}
%       \end{Proof}                     \end{Proof}
% also works with $$...$$ and \[...\] displays, including fleqn.sty.

% Unfortunately some environments, such as picture, do not allow this at all.
% See the section "centering environments" below for the mechanism for
% passing \qed to such an environment for special processing.

% In fact \qed will only work in those environments which provide access to
% the right hand side of the page. In other words, wherever you think
% "\hfill\square" ought to work then \qed will; of course it is just because
% that naive method doesn't work that this macro file is needed at all.
% The macro was designed to solve the problem in the case of "unrestricted
% horizontal mode" (ie at the end of a paragraph); it also works in $$
% displays by pretending to be an equation number. In most other cases it
% basically does "\hfill\square" or "\hfill$\square$" and hopes for the best.

% If you have to put \qed outside the environment, it may be appropriate to
% put \vskip-\baselineskip before it, unless of course this would over-print.
% There is, however, still a danger that TeX will break the page here.
% This seems to be unavoidable if the environment ends with a \penalty.

% The eqnarray example above does not in fact work: if you try it you'll
% get "! Paragraph ended before \llap was complete" with LaTeX in a mess.
% This is a very common place where you might want to put the end of proof
% mark, and I also consider that the fourth column should be available for
% "reasons" for equational steps.  The following substitute for \eqnarray
% allows this, whilst being compatible with automatic numbering. I have
% proposed it to Rainer Sch\"opf for inclusion in LaTeX. If you want to
% use it, please copy it to another file rather than uncommenting it here.
% Don't forget to use \makeatletter or put it in a documentstyle (.sty) file.
%
% \def\eqnarray{\stepcounter{equation}\let\@currentlabel\theequation
% \global\@eqnswtrue\m@th\global\@eqcnt\z@\tabskip\@centering\let\\\@eqncr
% $$\halign to\displaywidth\bgroup\@eqnsel\hskip\@centering$\displaystyle
% \tabskip\z@{##}$&\global\@eqcnt\@ne\hskip 2\arraycolsep \hfil${##}$\hfil
% &\global\@eqcnt\tw@\hskip2\arraycolsep$\displaystyle\tabskip\z@{##}$\hfil
% \tabskip\@centering&%
%       \global\@eqcnt\thr@@\hbox to\z@\bgroup\hss##%*** instead of "\llap{##}"
% \tabskip\z@\cr}%
% \def\@@eqncr{\let\@tempa\relax
% \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}%
%       \or\def\@tempa{&}\else\@eqnswfalse%*** instead of "\else\def\@tempa{&}"
% \fi\@tempa\if@eqnsw\@eqnnum\stepcounter{equation}\fi
%       \egroup%*** added, to match \bgroup in fourth cell of preamble
% \global\@eqnswtrue\global\@eqcnt\z@\cr}

% Why can't we do all of this with \endProof? Because by that stage
% the damage has been done: the environments, paragraphs or
% displays have already been closed and their \hbox'es added
% to the current vertical list, and cannot be removed for
% alteration if this is the current page.

% Even if you don't need to put \qed inside an inner environment
% to avoid a blank line, it's still a good idea to do so, because
% otherwise it's possible that intervening end-of-environment
% material may allow TeX to put the mark on the next page.

% This file may be called QED.sty, QED.tex, qed.sty or qed.tex.
% It may be loaded using \input, \usepackage or as a \documentstyle option,
% as it is careful to handle the \catcode of @ appropriately.

% If you don't use theorem.sty, this file instead provides options to use
% commands to begin theorems, instead of enclosing them in environments,
% and to suppress numbering and italicisation. A rudimentary version of
% \newtheorem for plain TeX is also implemented (always using commands
% rather than environments).

% At the end of the file some useful \newtheorem
% commands are listed but not executed.

% NB \Theorem\Proof\endProof\qed will print the second square, whereas
% \begin{Theorem}\end{Theorem}\begin{Proof}\end{Proof}\qed will not.

%=========================================================================
% Make @ letter, saving its old code to restore at the end of this file.
% Look for this towards the end of the file if you think something's missing!
% "plain.tex" defines \lq as a synonym for back quote (character constant);
% I use it to avoid corruption when sending TeX files by email via BITNET.
\edef\qedrestoreat{\noexpand\catcode\lq\noexpand\@=\the\catcode\lq\@}%%
\catcode\lq\@=11

% in LaTeX, \protect the user commands;
% in plain TeX, ensure that it is defined.
\ifx\protect\undefined\let\protect\relax\fi

%=========================================================================

%                               OPTIONS

% These are the user commands.

\def\qed{\protect\@qed{$\qedsymbol$}}% suppressible printing of \qedsymbol
% \pushright takes an argument, which it always prints
% it is (re)defined by \WillHandleQED and \UnHandleQED below (12.4.94)

% theoremata in Euclid needed proof:    "which   was  to be shown"
\def\QED{\protect\@qed{{\rm Q.E.D.}}}%  "quod   erat  demonstrandum"

% but problemata needed construction:   "which   was  to be found"
\def\QEI{\protect\@qed{{\rm Q.E.I.}}}%  "quod   erat  inveniendum"

% alternatively:                        "which   was  to be done/made"
\def\QEF{\protect\@qed{{\rm Q.E.F.}}}%  "quod   erat  faciendum"

% a LaTeX environment
\def\Proof{\protect\@Proof}\def\endProof{\protect\@endProof}%
\let\proof\Proof\let\endproof\endProof


% These are the user-settable parameters.

% The actual (maths) symbol used by \qed
\def\qedsymbol{\square}% PLEASE NOTE: this is in the AMS symbols font.
%\def\qedsymbol{\blacksquare}%  use one of these instead
%\def\qedsymbol{\hbox{\rm Q.E.D.}}%        if you prefer
%\def\qedsymbol{\dashv}% this is what the Journal of Symbolc Logic likes.

% This is the text used by \Proof.
% A new paragraph is begun without indentation.
% Change it to {\kern\parindent\sc D\'emonstration: } if you like.
\def\TheWordProof{\bf Proof\enskip}

% This is the font command used for the text of proofs
\def\ProofFont{}

% The command \AutoQEDtrue causes a \qed in \end{Theorem}, so you do
%       \begin{Theorem} ... \begin{Proof} ... \end{Proof} \end{Theorem}
% but if you omit the proof, the statement gets a little square at the end.
\newif\ifAutoQED\AutoQEDfalse

% The remaining options are redundant with theorem.sty
% so are not defined if this has been loaded.
\newif\ifNumberResults
\ifx\theorem@style\undefined
   %
   % How to print the name (#1), number (#2) and attribution (#3) of a theorem.
   \NumberResultstrue
   % If the number (#2) is absent, "\unskip" removes the
   % unwanted space "\ " from before it (16 Feb 1993).
   \def\TheoremHeader#1#2#3{\bf #1\ifNumberResults\ #2\unskip\fi#3}
   %
   % This is the font command used for the text of theorem enunciations
   % change to {\rm} or {\small} if you like.
   \def\TheoremFont{\it}
   %
   % The command \NumberResultsfalse will suppress theorem numbering.
   %
   % The command \TheoremsAsCommands will allow you to write
   % \Theorem instead of \begin{Theorem} ... \end{Theorem}.
   % In this case italicisation is suppressed by default.
   % To do this with theorem.sty would involve rewriting the substyles.
   %
   % WARNING: if you subsequently redefine \TheoremFont, and use the command
   % instead of the environment, the font will remain in force afterwards!
\fi

% The document style option "theorems-as-commands" will allow you to
% write \Theorem instead of \begin{Theorem} ... \end{Theorem}.
% Moved here from further down. 10 Jan 1994
\def\TheoremsAsCommands{%
  \def\TheoremFont{}% suppress italicised theorems.
  \def\begin@theorem##1{%
     \par\removelastskip\smallskip
     \save@set@qed % enable \qed
     \noindent{##1}% the name
     %\TheoremFont % italicise at your own risk!
   }%
  \def\@endtheorem{\ifAutoQED\qed\fi\restore@qed}%
}%

\expandafter\let\csname ds@theorems-as-commands\endcsname\TheoremsAsCommands

%=========================================================================
%  If you already have a macro called \qed, \endproof or \pushright
%  which is laid out in a similar way to the following two definitions
%  (but with \hbox@pushright incorporated) then it is almost certainly
%  my code, which has been copied anonymously via numerous people since
%  my Cambridge PhD thesis in August 1986. Belated acknowledgements to
%  Chris Thompson of the Cambridge University Computing Service for his
%  help with my early attempts at TeX, in particular this problem.

%  The complexity of the macro necessary to get a little box on the
%  right-hand-side at the end of a proof is amazing.  It really does
%  have to be this long!  Otherwise you're liable to get it at the
%  beginning of the next line, or even on the next page. 

\def\parag@pushright#1{{% set up
    \parfillskip=0pt            % so \par doesnt push \square to left
    \widowpenalty=10000         % so we dont break the page before \square
    \displaywidowpenalty=10000  % ditto
    \finalhyphendemerits=0      % TeXbook exercise 14.32
    %
    %                   horizontal
    %\leavevmode                % not needed since we now only use it in hmode
    \hbox@pushright             % this used to be incorporated
    #1%                         % the end-of-proof mark (or whatever)
    %
    %                     vertical
    \par}}%                     % build paragraph with the above parameters

\def\hbox@pushright{% horizontal
    \unskip                     % remove previous space or glue
    \nobreak                    % don't break lines
    \hfil                       % ragged right if we spill over
    \penalty50                  % discouragement to do so
    \hskip.2em                  % ensure some space
    \null                       % anchor the following \hfill
    \hfill                      % push \square to right
}%

% Ideally \endProof should do all this. We should be able to add the symbol
% to the last line after the paragraph, display or whatever other structure
% has been completed and we've returned to vertical mode. This isn't possible
% on the current page (outer vmode) because \lastbox doesn't work there.

% But \ifinner is true inside a \vbox, \vadjust or \insert (LaTeX \parbox,
% minipage, figure or table), and in these circumstances we can use \lastbox.
% We would have to strip & restore glue & penalties (\insert, \mark and \write
% are obstacles to this) to get access to the last line (box), which may
% still not have the correct baseline. We could check its natural width
% against the page width, but if we re-set it, centering and (right)
% justification might be disturbed. Is this worth the trouble?
\def\vbox@pushright#1{\expandafter\message %  (2 Jan 1994)
  {QED.sty could be improved in this case (line \the\inputlineno): please ask}%
  \page@pushright{#1}}%

% Using this method the Proof environment could be implemented by collecting
% the entire proof in its own \vbox, modifying that and \unvbox'ing it to the
% current page. This approach may lead to the following problems:
%   (i) footnotes, floats, marginpars, labels and index items may be lost;
%  (ii) overflow of TeX's main memory, especially if \end{Proof} is missing;
% (iii) I don't know how page-breaking would be handled.

%========================================================================

% Switch to enable/suppress \qedsymbol; since suppression may be done
% within a nested scope it must be global, so we do our own save/restore
% in a symbol which is local to the Proof environment.
% On first use, we check to see whether an obsolete version of Paul.sty
% is around, and complain loudly if it is, because they interfere seriously.

\newif\if@qed\@qedfalse
\def\save@set@qed{\check@pt@sty@v\let\saved@ifqed\if@qed\global\@qedtrue}%
\def\restore@qed{\global\let\if@qed\saved@ifqed}

% prefer proofs with statements if possible - hence \penalty700
% the only essential part of this is \save@set@qed
\def\@Proof{%
   \par\removelastskip\smallskip\penalty700
   \save@set@qed
   \noindent\ProofFont{\TheWordProof\enskip}%
}%

\def\@endProof{%
   \qed\restore@qed
   \penalty-700 \smallskip
}

\def\@qed#1{\check@pt@fm@thm
\if@qed                                 % have we already done \qed?
     \global\@qedfalse\pushright{#1}%  - no  - do it now, but not again
\else\ifhmode\ifinner\else\par\fi\fi%  - yes - just end paragraph, if any
\fi}

% \pushright{text} always prints its argument, on the far right, in text mode
\def\@pushright#1{%
  \ifvmode
       \ifinner\vbox@pushright{#1}% vertical mode (see comments above)
       \else   \page@pushright{#1}%
       \fi
  \else\ifmmode\maths@pushright{\hbox{#1}}% maths (force a text argument)
       \else\ifinner\hbox@pushright{#1}%  inside an \hbox
            \else\parag@pushright{#1}%    in a paragraph
  \fi  \fi  \fi
}% (22 Feb 1993) removed extra {} which would destroy our \] below

% In the \hbox and maths cases we are really at the mercy of the environment.
% Assuming that we have $\qed$ simply embedded in a paragraph, the horizontal
% processing by \hbox@pushright will at least make sure that the square isn't
% at the beginning of the line, but we can't stop it being on the next page.
% However it is more likely that the reason why we're in maths mode is that
% we're inside an eqnarray or similar, in which case \hfill will do its best
% to put the square on the right of the available space. (See \vadjust below.)
\def\maths@pushright#1{%
  \ifinner
     \hbox@pushright{#1}%
  \else
     \eqno#1%   use TeX's right equation number feature (\eqno) within $$.
     \def\]{$$\ignorespaces}%    suppress LaTeX's error checking (HACK!)
  \fi
}% (22 Feb 1993) removed extra {} which would destroy our \] above

% outer vertical mode (the current page)
% There's very little we can do in these circumstances!
% Penalties still allow the page to be broken.
% If the user has put negative glue in, leave it alone. (15 Feb 1994)
\def\page@pushright#1{% 18 Jan 1994
  %\unpenalty % ineffective
  \skip@\lastskip
  \ifdim\skip@>\z@
       \unskip    % remove \parskip, but only (15 Feb 1994) if it's positive
  \else\skip@\z@\relax
  \fi
  %\unpenalty % TeX won't let me do this
  \dimen@\baselineskip
  \advance\dimen@-\prevdepth            % save \prevdepth to make a strut
  \nobreak                              % don't break the page here
  \nointerlineskip
  \hbox to\hsize{%
    \setbox\z@\null
    \ifdim\dimen@>\z@\ht\z@\dimen@\fi   % simulate \baselineskip
    \box\z@
    \hfill
    #1}%
  \vskip\skip@                  % replace old \parskip
}%

% An idea that didn't work: see TeXbook p188. (16 Feb 1993)
% Use \predisplaysize to find out the width of the previous box,
% in the case where it has already been completed.
% But this can only be used inside displays ($$), and a display in
% vertical mode begins a paragraph first, so we end up with
% \predisplaysize=\parindent+2em whatever happened in the previous
% paragraph.

% Another idea: it may be worth trying \vadjust, if only to insert \nobreak.
% The problem with this is that if we are too deeply nested in horizontal
% or maths lists, the adjustment may not be able to find its way out to the
% enclosing vertical list. In the case of array-like environments, it will
% appear (assuming it manages to get out) after the current row of the array.

%===========================================================================
% We have to add to the theorem environments the commands to enable \qed.
% This is done by re-defining \@thm#1#2.
% Note that Mittelbach's theorem.sty re-defines \@thm and \@endtheorem, so
% that must be loaded before us, otherwise our changes will get over-written
% and \qed will never be enabled.
% Paul.sty didn't use this method of gaining control inside a theorem,
% whereas it did define \qed and \pushright, so here is the one place
% where we can check for it.

\ifx\nonqed@thm\undefined % otherwise double loading causes a loop 22 Feb 1993
   \let\nonqed@thm\@thm % save original
   \let\nonqed@endthm\@endtheorem
\fi

\def\@thm{\save@set@qed\nonqed@thm}%
\def\@endtheorem{\ifAutoQED\qed\fi\restore@qed\nonqed@endthm}%

%===========================================================================
% "CENTERING" ENVIRONMENTS - 
% those which don't give their contents access to the right hand side
% of the page, such as center, picture, ...  (12 April 1994)

% Each individual environment (both the standard ones and those defined
% by the user) needs to co-operate in order to make \qed and \pushright work.
% It is not the business of this package to rewrite LaTeX in this way,
% but the following mechanism may be inserted in to other environments.
%  - put \WillHandleQED in the \begin part of the environment definition;
%    then if a \qed occurs in the body of the environment it is stored
%    in \QEDbox until...
%  - \HandleQED anywhere in the \end part of the environment definition
%    (even inside dollars, braces or another environment) prints it.
%
% For example:
% \newenvironment{qedpicture}%
%    {\begin{picture}\WillHandleQED}%
%    {\put(0,0){\HandleQED}\end{picture}}
%
% To make this compatible with *not* loading QED.sty,
% add the following to macro files which use \WillHandleQED and \HandleQED:
%   \ifx\WillHandleQED\undefined
%      \let\WillHandleQED\relax
%      \let\HandleQED\relax
%      \let\UnHandleQED\relax % if you use it
%      \let\QEDbox\voidb@x    % must have \makeatletter here!
%   \fi
%
% \UnHandleQED restores normal processing; this may be useful for
% the minipage environment, which has its own notion of "right hand side"
% of the page.
%
% The width of the saved \qed is available for testing as \wd\QEDbox

\newbox\qed@box % the box in which to save \qed for special handling

\def\WillHandleQED{\relax
\ifx\HandleQED\nohandle@qed
   % no enclosing environment has declared \WillHandleQED
   \def\pushright{\global\setbox\qed@box\hbox}% \qed will be saved
   \let\QEDbox\qed@box % point at the right box
   \def\HandleQED{\unhbox\QEDbox}% print it
   \aftergroup\check@handle@qed  % check that this really gets done
\else
   % an enclosing environment already wants to catch \qed
   % \HandleQED doesn't print anything - let the saved \qed pass through
   \let\QEDbox\voidb@x % by pointing at an empty box
\fi}

\def\nohandle@qed{%
\errhelp{One of them is missing: see QED.sty.}%
\errmessage{This environment uses \string\WillHandleQED\space and
\string\HandleQED\space incorrectly}}

\def\check@handle@qed{\relax
\ifvoid\qed@box\else\expandafter\nohandle@qed\fi}

\def\UnHandleQED{%
\let\HandleQED\nohandle@qed
\let\QEDbox\voidb@x
\def\pushright{\protect\@pushright}}%

\UnHandleQED

%===========================================================================
% Check for obsolete versions and clashes of this and other macro files.
% Why do I do this? Because of those people who complain to me that
% something doesn't work but refuse to read the manual, and those
% who refuse to update their software, and then publish papers with
% commutative diagrams full of the bugs which I fixed several years ago.

\def\obsolete@pt@version#1{\errhelp={%
%     *****************************************************************
%     *            The master version is available by                 *
        Anonymous FTP ftp.dcs.qmw.ac.uk /pub/tex/contrib/pt/tex/#1}%*
%     *****************************************************************
\errmessage{You have an obsolete version of #1 - please get a new one}}%

% First check QED.sty
%\ifnum\year>1996 \obsolete@pt@version{QED.sty}\fi

% If theorem.sty hasn't been loaded, arrange to check it again in \@qed.
% And check (then) whether you've actually defined \square.
% changed 2 June 1993: don't define it now, as that would confuse amssymb.sty
\ifx\theorem@style\undefined
   \def\check@pt@fm@thm{\relax
     \ifx\square\undefined
       \gdef\square{\bigcirc
          \errhelp={Anonymous ftp e-math.ams.com /ams/amsfonts}%
          \errmessage{\string\square\space is an AMS symbol}%
          \global\let\square\bigcirc}%
     \fi
      \ifx\theorem@style\undefined
          \global\let\check@pt@fm@thm\relax
      \else\errhelp={The macros \@thm and \@endtheorem need to be re-defined.}%
          \errmessage{QED.sty must be loaded AFTER theorem.sty but before
           using \string\newtheorem}%
      \fi
      \global\let\check@pt@fm@thm\relax
      }%
\else
   \def\check@pt@fm@thm{%
      \ifx\square\undefined
         \def\square{\bigcirc
         \errhelp={Anonymous ftp e-math.ams.com /ams/amsfonts}%
         \errmessage{\string\square\space is an AMS symbol}%
         \global\let\square\bigcirc}%
      \fi
      \global\let\check@pt@fm@thm\relax
      }%
\fi

% Check for Paul.sty, which SERIOUSLY conflicts with this file.
% and also for version 2 or version 3 of commutative diagrams.
\def\check@pt@sty@v{%
   \relax
   \ifx\@@begintheorem\undefined
   \else\message{*** QED.sty and the old Paul.sty seriously conflict! ***}%
        \obsolete@pt@version{Paul.sty}%
   \fi
   \ifx\execHorizontalMap\undefined
        \gdef\obsolete@pt@version##1{}% it's no longer needed
   \else\global\let\di@gr@m\diagram
        \gdef\diagram{\obsolete@pt@version{diagrams.tex}\di@gr@m}%
   \fi
   \global\let\check@pt@sty@v\relax
}%

%===========================================================================

% Define some standard ``theorem'' types.

\def\DefineStandardTheorems{%
%
% First set up uniform numbering in LaTeX;
% don't use this in plain TeX because this usage of \newcommand is not
% supported and we have already done by hand what it would do in LaTeX,
% viz to define the macro \theResult and the count \c@Result.
\relax\ifx\@@thm\undefined
        \newtheorem{Result}{Result}[subsection]%
        \@addtoreset{Result}{section}%
        \ifx\chapter\undefined\else\@addtoreset{Result}{chapter}\fi
\fi
%
\let\theUnnumbered\relax\countdef\c@Unnumbered255 \def\p@Unnumbered{}%
\let\DefineStandardTheorems\relax
%
\newtheorem{Answer}[Result]{Answer}%
\newtheorem{Answers}[Result]{Answers}%
\newtheorem{Conjecture}[Result]{Conjecture}%
\newtheorem{Construction}[Result]{Construction}%
\newtheorem{Convention}[Unnumbered]{Convention}%
\newtheorem{Corollary}[Result]{Corollary}%
\newtheorem{Counterexample}[Result]{Counterexample}%
\newtheorem{Definition}[Result]{Definition}%
\newtheorem{Example}[Result]{Example}%
\newtheorem{Examples}[Result]{Examples}%
\newtheorem{Exercise}[Result]{Exercise}%
\newtheorem{Exercises}[Result]{Exercises}%
\newtheorem{Fact}[Result]{Fact}%
\newtheorem{Facts}[Result]{Facts}%
\newtheorem{Lemma}[Result]{Lemma}%
\newtheorem{NB}[Unnumbered]{NB}%
\newtheorem{Notation}[Result]{Notation}%
\newtheorem{Note}[Unnumbered]{Note}%
\newtheorem{Observation}[Result]{Observation}%
\newtheorem{Observations}[Result]{Observations}%
\newtheorem{Problem}[Result]{Problem}%
\newtheorem{Proposition}[Result]{Proposition}%
\newtheorem{PS}[Unnumbered]{PS}%
\newtheorem{Remark}[Result]{Remark}%
\newtheorem{Question}[Result]{Question}%
\newtheorem{Questions}[Result]{Questions}%
\newtheorem{Theorem}[Result]{Theorem}%
\newtheorem{Thesis}[Result]{Thesis}%
\newtheorem{Warning}[Result]{Warning}%
}

%===========================================================================
% Interaction with LaTeX2e (2 Jan 1994)

\ifx\newtheorem\undefined\else\ifx\ProvidesPackage\undefined
   \message{^^J^^J}%
   \message{****************************************************************}%
   \message{! ***** You are using an obsolete implementation of LaTeX ******}%
   \message{* A new version - called LaTeX2e - has been available from TeX *}%
   \message{* FTP archives such as ftp.tex.ac.uk, ftp.uni-stuttgart.de and *}%
   \message{* pip.shsu.edu since December 1993. It incorporates NFSS2 (the *}%
   \message{* Mittelbach-Schoepf New Font Selection Scheme) but is reverse *}%
   \message{* compatible with LaTeX versions 2.09 and is now the standard. *}%
   \message{****************************************************************}%
   \message{^^J^^J}%
\else
   \ProvidesPackage{QED}[1996/06/24 right-justified end-of-proof mark]%
   %\ProvidesFile{QED.sty}% I don't understand what this is for
   \DeclareOption{roman-theorems}{\def\TheoremFont{\rm}}%
   \DeclareOption{italic-theorems}{\def\TheoremFont{\it}}%
   \DeclareOption{theorems-as-commands}{\TheoremsAsCommands}%
   \DeclareOption{auto-QED}{\AutoQEDtrue}%
   \DeclareOption{explicit-QED}{\AutoQEDfalse}%
   \DeclareOption{number-theorems}{\NumberResultstrue}%
   \DeclareOption{dont-number-theorems}{\NumberResultsfalse}%
   \DeclareOption{black-square}{\def\qedsymbol{\blacksquare}}%
   \DeclareOption{open-square}{\def\qedsymbol{\square}}%
   \DeclareOption{define-standard-theorems}{\DefineStandardTheorems}%
   \ProcessOptions
\fi\fi

%===========================================================================
% If Mittelbach's theorem.sty is loaded, skip the rest of the file.
\ifx\theorem@style\undefined\else\qedrestoreat\expandafter\endinput\fi
%===========================================================================

% Allow the user to suppress italicisation and numbering.

% The following two macros are called by LaTeX to begin a theorem
% respectively with and without an optional (``attribution'') argument.
% It is clear from latex.tex that they are intended to be customised.
% The code common to the two has been taken out as \begin@theorem and
% \TheoremHeader. Of these, the former is re-defined by \TheoremsAsCommands.

\def\@opargbegintheorem#1#2#3{\begin@theorem{\TheoremHeader{#1}{#2}{ (#3)} }}

\def\@begintheorem#1#2{\begin@theorem{\TheoremHeader{#1}{#2}{} }}

\def\begin@theorem#1{%
  \trivlist\item[\hskip\labelsep {#1}]%         % LaTeX likes this method.
  \TheoremFont                                  % Italicise the enunciation
  \ifx\ProofFont\empty\def\ProofFont{\rm}\fi    % but not any nested proof.
  }%


%%% Here's what LaTeX (92/1/14) does:
%%%% RmS 91/08/14 Moved \it after \item to make it work with NFSS
%%\def\@begintheorem#1#2{\trivlist \item[\hskip \labelsep{\bf #1\ #2}]\it}
%%\def\@opargbegintheorem#1#2#3{\trivlist
%%      \item[\hskip \labelsep{\bf #1\ #2\ (#3)}]\it}
%%\def\@endtheorem{\endtrivlist}



%===========================================================================
% If we're running under LaTeX, skip the rest of the file.
\ifx\newtheorem\undefined\else\qedrestoreat\expandafter\endinput\fi
%===========================================================================

% This implements a rudimentary version of \newtheorem for plain TeX.
% Only the unattributed theorems and the
% declarations \newtheorem{Theorem}[Result]{Theorem} and
% \newtheorem{Convention}[Unnumbered]{Convention} are supported.
% The numbering is based on the counts \chapternumber, \sectionnumber
% and \subsectionnumber, which it is up to you to define with \newcount.
% It's also up to you to reset \c@Result at the beginning of each section.

% Form the number: (chapter).(section).(subsection).(#1)
% where any of the first three parts which is zero
% negative or undefined is omitted.
\def\ther@sult#1{%
\ifx\chapternumber\undefined\else
   \ifnum\chapternumber>0 \number\chapternumber.\fi\fi
\ifx\sectionnumber\undefined\else
   \ifnum\sectionnumber>0 \number\sectionnumber.\fi\fi
\ifx\subsectionnumber\undefined\else
   \ifnum\subsectionnumber>0 \number\subsectionnumber.\fi\fi
\expandafter\number\csname c@#1\endcsname}%

% plain version of LaTeX's \@thm
\def\plain@thm#1#2{% #1=counter, #2=text
  \save@set@qed
  \expandafter\advance\csname c@#1\endcsname1
  \@begintheorem{#2}{\csname the#1\endcsname}%
}%

% Rudimentary \newtheorem
\def\newtheorem#1[#2]#3{%   #1=command, #2=counter, #3=text
  %
  % \def\#1{\plain@thm{#2}{#3}}
  \expandafter\def\csname #1\endcsname{\plain@thm{#2}{#3}}%
  %
  % \def\end#1{\@endtheorem}
  \expandafter\def\csname end#1\endcsname{\@endtheorem}%
  %
  % if \c@#2 is not defined
  %    \newcount\c@#2
  %    \def\the#2{\ther@sult{#2}}
  \expandafter\ifx\csname c@#2\endcsname\relax
     \expandafter\expandafter\csname newcount\endcsname
        \csname c@#2\endcsname
     \expandafter\def\csname the#2\endcsname{\ther@sult{#2}}%
  \fi
}%

\TheoremsAsCommands

\qedrestoreat\endinput

%===========================================================================
% BUGS
% \item\qed\end{letterlist} puts the square above the item label.

% ! You can't use `\unskip' in vertical mode.
% from \qed in a paragraph on its own.