QED.sty - right-justified end-of-proof mark
Paul Taylor
1994
The package itself and this manual in
DVI,
PDF,
compressed PostScript and
A5 PS booklet format.
1 Introduction
Euclid established a style in his Elements that, for better or
worse, has come down to us as the definitive one for presenting
mathematical proofs. A Theorem or Problem (that is,
something to be proved or constructed, respectively)
is first stated, with the heading "Theorem" or "Problem".
After the proof or construction has been given, the original statement
is re-iterated, followed by the letters "QED" (or "QEI" for a Problem).
QED stood for either quod erat demonstrandum or quod est
demonstatum, which respectively mean "which was to be shown" and
"which has been shown".
Modern custom is to use a little square instead of the letters QED.
It is also customary to put this square on the far right of the page,
which is where this macro package comes in.
It is surprisingly difficult to persuade T_{E}X to do this. The obvious
solution,
and so $2 n = p + q$ as required. \hfill $\square$
works 80% of the time, but fails if the preceding text happens to
fill the line, in which case the square appears at the left of
the next line.
This happens because, when T_{E}X breaks the line at \hfill, it
discards this stretchable space (called "glue") [T_{E}Xbook, page
97]. The reason for this behaviour is that, when a word occurs at the
beginning or end of a line, we like to have it printed hard against the
margin, so the space that would otherwise occur between words is lost
when the line is broken there.
T_{E}X and L^{A}T_{E}X have been in standard use in the mathematical
sciences for two decades now, so you would expect this problem to
have been solved, and solved many times. However, on examining
the documents and macro files that have come my way, I find that they
either employ the naïve solution above, which doesn't work,
or they use code that is mine.
With the help of Chris Thompson of the Cambridge University Computing
Service, I implemented the following solution for my Ph.D. thesis in
1986. As well as keeping the square on the right (though possibly on
the next line), it also stops it from being left over (as a "widow")
on the next page.
\def\qed {{% 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 % \nobreak means lines not pages
\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 following \hfill
\hfill % push \square to right
$\square$% % the end-of-proof mark
%
% vertical
\par}} % build paragraph
If you already have a macro called \qed, \endproof or
\pushright which is laid out in a similar way to this, then it
is almost certainly my code.
Please use this macro package in place of this fragment of my code.
Besides the basic case of the little square at the end of a piece
of text, it also works with displays ($$ or the
displaymath environment) and has hooks for compatibility with
other situations.
2 QED as a macro
The basic use of the package is as follows:
\documentclass{article}
\title{A Proof of Goldbach's Conjecture}
\author{Paul Taylor}
\usepackage[open-square,define-standard-theorems]{QED}
\usepackage{amssymb}
\begin{document}
\maketitle
\begin{Theorem}
Every even number may be expressed as a sum of two primes.
\end{Theorem}
\Proof Trivial. $2n = p+q$. \qed
\end{document}
In principle, the \qed macro goes logically where you want the
little square to appear in the text.
But this still doesn't work (and nor can it)
because of the way T_{E}X works.
Briefly, the \qed macro alters the way in which the paragraph
is formatted, but commands like \end{itemize} close the paragraph
themselves, leaving \qed to the next paragraph, and therefore
the next line.
The rule is that \qed has to be put inside the innermost
environment instead, for example
\begin{itemize}
\item This case is easy too.
\item In the final case we have
$$ 2n = p + q \qed $$
\end{itemize}
If you still have to put \qed outside the environment, it may be
appropriate to add
\vskip-\baselineskip
before \qed, unless of course this would over-print. There is,
however, still a danger that T_{E}X will break the page here
(which seems to be unavoidable if the environment ends with a
\penalty).
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 T_{E}X to put the square on the next page.
3 Options
The package may be customised using the following options in the
usepackage declaration:
- black-square
- Use [black] for \qed.
- open-square
- Use [open] for \qed.
- define-standard-theorems
- Define Answer, Axiom, Assumption,
Conjecture, Construction, Convention, Corollary, Counterexample,
Definition, Example, Exercise, Fact, Lemma, NB, Notation, Note,
Observation, Problem, Proposition, PS, Remark, Question, Theorem,
Thesis, Warning and their plurals.
- roman-theorems
- Print the theorem statements in roman (upright).
- italic-theorems
- Print the theorem statements in italic
(default).
- theorems-as-commands
- Use commands like \Theorem
to introduce a theorem statement, instead of wrapping it in
a \begin{Theorem} ... \end{Theorem} environment.
- auto-QED
- Make \end{Theorem} insert
a \qed automatically.
- explicit-QED
- Suppress automatic \qed in
\end{Theorem} (default).
- number-theorems
- (Default).
- dont-number-theorems
Beware that the [open] and [black] symbols come from the
AMS symbols fonts, not the standard T_{E}X ones. This is why you need
to \usepackage{amssymb}.
Besides the open-square and black-square options,
you can also change what \qed prints by redefining, for example,
\def\qedsymbol{\bigcirc}
to get a big circle. The macros \QED, \QEI and
\QEF are also provided for classically minded mathematicians,
and produce these three-letter acronyms (I and F stand for inveniendum
and faciendum - to be found or done).
The \Proof macro, of course, produces the word "Proof",
in bold, as a new un-indented paragraph. You can change this by
redefining, for example,
\def \TheWordProof {\kern\parindent\textsc{D\'emonstration}: }
to print it in SMALL CAPITALS with indentation and a colon.
4 Proof environments
Given that a proof is enclosed in the pair \Proof ... \qed,
you may like to use a L^{A}T_{E}X environment instead:
\begin{Theorem}
Every even number is the sum of two primes.
\end{Theorem}
\begin{Proof}
Easy.
\end{Proof}
The problem with this is that \end{Proof} comes after
any $$ or \end{itemize} that might finish the proof,
which is too late for \qed to do its stuff.
In this case, you have to insert \qed yourself within $$
or \end{itemize} as in Section 2. It will then
print in the correct place. QED.sty will not repeat the square
when \end{Proof} is executed.
In order to achieve this effect, and also to allow for nested Proof
environments (for example the Lemmas leading up to a Theorem),
QED.sty has to maintain an internal switch saying whether to
print the square or not. This switch is turned on by \Proof or
\begin{Proof} and off when \qed is executed.
Nested Proof environments save the outer value of the switch.
As a result of this,
\qed which doesn't follow \Proof will not print
the square!
Each properly nested \begin{Proof} ... \end{Proof} pair
gets a single square at the end, on the far right hand side.
5 Theorem styles and environments
You may feel that it is logical to put the Proof environment
within the Theorem environment. QED.sty supports
this. In the case where the Proof is absent (because it is
obvious or because the main Theorem is stated after its proof), you
can arrange for the square to be generated by \end{Theorem},
by selecting the auto-QED option in \usepackage.
I, on the other hand, regard the words Theorem etc.
as headings, and prefer not to wrap my enunciations in L^{A}T_{E}X
environments at all. I just write
\Theorem $2n = p+q$.
\Proof Obvious. \qed
This behaviour is declared with the theorems-as-commands option
in \usepackage.
This also suppresses italic typesetting of the statement,
because there is no \end{Theorem} to turn it off.
You can also suppress italicisation or numbering with the
\usepackage options
roman-theorems and dont-number-theorems
If you want to use Mittelbach's theorem.sty package, you have
to load that first, then QED.sty.
This is because of the way in which the two packages interact,
in particular regarding the internal macro \@thm.
The options theorems-as-commands,
roman-theorems and dont-number-theorems
will not work in this case.
6 Control of page breaking
This is not really concerned with theorems and proofs, but this problem
seems to arise in the same context.
Whilst T_{E}X is extremely good at breaking paragraphs into lines
nicely, it often makes a mess of page breaking. This is because the
program was written in the early 1980s, when computers had kilobytes
rather than megabytes of memory, so T_{E}X generates pages of DVI as
soon as it can.
Its mechanisms for breaking both lines and pages involve \penalty
items in its horizontal and vertical lists. The L^{A}T_{E}X commands
\nobreak and \goodbreak generate T_{E}X \penalty
commands, which are supposed to control (line and) page breaking.
In my experience, these don't work.
The macros \goalbreak and \allowlines fix this problem
by testing how full the page already is, and generating a
\newpage if necessary.
- \goalbreak{50pt} or \goalbreak{2cm} ensure that
this much space is available on the current page.
- \allowlines{5} measures the space in lines instead
(using \baselineskip).
You may want to use this just before a short Theorem statement in order
to keep it on the same page as its proof. (In fact, QED.sty
uses a small \penalty to encourage this anyway.)
The macro \closeupaline is useful after \Proof when
displayed material follows immediately:
\Proof\closeupaline
$$ 2n = p + q $$
7 Modifying other L^{A}T_{E}X environments
This section is for T_{E}Xperts.
The \qed macro will only work in those environments which
provide access to the right hand side of the page. In other words,
wherever you think "\hfill\square" ought to work then
\qed will. (Of course it is just because that naïve method
doesn't work that this macro file is needed at all.) The macro was
designed to solve the problem in the case of "unrestricted horizontal
mode" (i.e. at the end of a paragraph); it also works in $$
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.
Unfortunately some environments (we shall use picture as an
example) do not allow this at all. In this case, the (re)designers
of the environment may wish to handle the placing of the little square
for themselves.
Each individual environment (both the standard ones and those defined
by the user) needs to co-operate in order to make \qed work.
It is not the business of this package to rewrite L^{A}T_{E}X in this way,
but the following mechanism may be inserted into other environments.
- Put \WillHandleQED in the \begin part of the
environment definition. This notifies QED.sty that
the environment is going to ...
- put \HandleQED anywhere in the \end part of the
environment definition (even inside dollars, braces or another
environment), where you would like the square to be printed if
\qed is used.
For example:
\newenvironment{qedpicture}%
{\begin{picture}\WillHandleQED}%
{\put(0,0){\HandleQED}\end{picture}}
defines a version of the picture environment in which the
\qed command causes a square to be printed at the origin
of picture's coordinate system - wherever \qed
occurs within the body.
The width of the saved \qed symbol is available for testing as
the dimension \wd\QEDbox.
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
Also, \UnHandleQED restores normal processing; this may be
useful for the minipage environment, which has its own notion of
"right hand side" of the page.
8 Implementation
Underlying \qed with its switchable behaviour and configurable
symbol is a macro
\pushright{text}
that always prints its argument, on the far right.
The argument to \pushright is in text mode, so
\qed invokes it with {$\qedsymbol$}.
It does the following case analysis:
- outer horizontal text mode (in a paragraph) uses
\parag@pushright{#1}
- inner horizontal (text or maths) mode (\hbox) uses
\hbox@pushright{#1}
- outer maths mode (display) uses T_{E}X's \eqno mechanism
but has to suppress a L^{A}T_{E}X error check.
- outer vertical mode (between paragraphs on the current page)
uses
\page@pushright{#1};
- inner vertical mode (\vbox) uses \vbox@pushright{#1};
The interested reader can follow the comments in the file QED.sty
itself to see how these cases are handled.
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.
Ideally 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.
The reason why this isn't possible is that T_{E}X forbids the use of
\lastbox and similar commands (which can be used to take
\hbox and \vbox lists apart) on the current page (outer
vmode). It does this because its "galley" implementation (shipping
complete pages to the DVI file as soon as possible) might output the
item before we have a chance to remove it from the page.
In principle there is a more sophisticated method, which could be used
in inner vertical mode (i.e. inside T_{E}X's \vbox,
\vadjust or \insert or L^{A}T_{E}X's \footnote,
\parbox, minipage, figure or table).
In these circumstances \lastbox is allowed.
We would have to strip and restore glue and 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.
This method could even used to implement the Proof environment
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:
- footnotes, floats, marginpars, labels and index items may be lost;
- overflow of T_{E}X's main memory,
especially if \end{Proof} is missing;
- I don't know how page-breaking would be handled.
I don't think this is worth the trouble. The package has, since
January 1994, included an error message
QED.sty could be improved in this case: please ask
that appears when \qed is used in inner vertical mode, but
nobody has contacted me as a result of it.
9 Availability, compatibility, copyright and (no) warranty
QED.sty is Copyright © 1993-2003 Paul Taylor.
The package may be obtained from the Comprehensive T_{E}X Archive Network at
www.ctan.org
or one of its numerous mirror sites. Within the directory structure
there, the file is called
macros/generic/proofs/taylor/QED.sty
but on your own computer you may call it QED.sty,
QED.tex, qed.sty or qed.tex.
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, his
present or former employers 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.
This is
www.PaulTaylor.EU/proofs/QEDdoc.html
and it was derived from proofs/QEDdoc.tex
which was last modified on 9 June 2007.