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1.1
date	95.07.12.22.05.45;	author WB;	state Exp;
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@%%%-*- compile-command: "notangle plain.tw > plain.sty"              -*-
%%%=====================================================================
%%% @@Noweb-file{
%%%   author   = "W\l{}odek Bzyl",
%%%   version  = "$Id: plain.tw,v 4.12 1995/07/10 21:32:25 web Exp $",
%%%   date     = "25 April 1995",
%%%   time     = "22:05:47 MET",
%%%   filename = "plain.tw",
%%%   checksum = "45702 2586 10926 101053",
%%%   address  = "Instytut Matematyki,
%%%               Uniwersytet Gda\'nski
%%%               Wita Stwosza 57
%%%               80--952 Gda\'nsk, Poland",
%%%   email    = "matwb@@halina.univ.gda.pl (Internet)",
%%%   abstract = "",
%%% }
%%%=====================================================================

\title{PLAIN.TW}{W\l{}odek Bzyl\hfil matwb@@halina.univ.gda.pl}

\printcontents

@@
\chapter{Introduction}

{\it However, we will not include a verbatim description, because some
parts of that file are too boring, and because the actual macros have
been ``optimized'' with respect to memory space and running time.}
\hfill {\sl D.E.~Knuth about\/ [[plain.tex]] format.}
\bigskip
\bigskip
\noindent
What follows is devoted to the details of the plain \TeX\ format.
This file serves two purposes:
\medskip

\item{(1)} As a documentation of the [[plain.tex]] format.
Weaving and texing this document should produce a handy reference.

\item{(2)}
The division of this web source into `chunks' should ease
creation of other formats tailored to particular applications.
Chunks could be easily modified, removed, added, or replaced.

\medskip
\noindent The change file mechanism
is not needed in case of \TeX\ language. 
Change files are used to incorporate system 
dependent code into source file, but \TeX\ code is already 
system independent. \TeX\ code could be only `format dependent' 
and here change files could be used.
 Another feature of format file is that it evolves with time, yet some
intermediate versions are used for preparation of books, articles etc.
All these versions and configurations must be kept well organized,
otherwise you are lost.
The Revision Control System is the tool that assists with these tasks.
 With the RCS it is possible, with small overhead,
 to preserve \stress{all revisions}
 which evolved from given text document,
 merge changes made by others,
 compare different versions,
 keep log of changes.

This document consists mainly of excerpts from the
{\sl \TeX Book}, but it is organised around the macros as they
appear in the [[plain.tex]] rather than around the
topics as in a user manual. Therefore this document
is not a \stress{user manual}, although many definitions
are contained here.
@@
\chapter{The layout of the format}

<<*>>=
<<Establish standard category code values>>
\catcode`@@ = 11
<<Define commonly used constants>>
<<Provide programming constructs>>
<<Allocate registers>>
<<Assign initial values to parameters>>
<<Set up text fonts>>
<<Set up math fonts>>
<<Provide macros for text formatting>>
<<Provide macros for math formatting>>
<<Prepare page for output>>
<<Read hyphenation patterns>>
<<Initialize the layout>>
\catcode`@@ = 12
<<Identify the format>>
@@
There are 256 characters that \TeX\ might encounter at each step, in
a file or in a line of text typed directly on your terminal. These
256~characters are classified into 16 categories numbered 0 to 15:
\begindisplay \def\\{\hfill}
\hfil\hidewidth\it Category\hidewidth&\it \qquad Meaning\hidewidth&%
\it Default\hfil\cr
\noalign{\smallskip}
\\0&Escape character&[[\]]\cr
\\1&Beginning of group&[[{]]\cr
\\2&End of group&[[}]]\cr
\\3&Math shift&[[$]]\cr
\\4&Alignment tab&[[&]]\cr
\\5&End of line&\<return>\cr
\\6&Parameter&[[#]]\cr
\\7&Superscript&[[^]]\cr
\\8&Subscript&[[_]]\cr
\\9&Ignored character&\<null>\cr
10&Space&\ttt{\char"20}\cr
11&Letter&[[A]], \dots, [[Z]] and [[a]], \dots, [[z]]\cr
12&Other character&none of the above or below\cr
13&Active character&[[~]]\cr
14&Comment character&[[%]]\cr
15&Invalid character&\<delete>\cr
\enddisplay

When [[INITEX]] begins, category 12 (other) has been assigned to all~256
possible characters, except that the 52~letters
[[A...Z]] and [[a...z]] are
category~11 (letter), and assignments equivalent to the
following have been made:
\begindisplay \def\\{\hfill}
[[\catcode `\\ = 0]]&\cr
[[\catcode `\^^M = 5]]&\cr
[[\catcode `\^^@@ = 9]]&\cr
[[\catcode `\  = 10]]&\cr
[[\catcode `\% = 14]]&\cr
[[\catcode `\^^? = 15]]&\cr
\enddisplay
Thus `[[\]]' is already an escape character,
`[[\char"20]]' is a space,
and `[[%]]' is available for comments on the first line of the file;
ASCII \<null> is ignored, ASCII \<return> is an end-of-line character,
and ASCII \<delete> is invalid.
\medskip
Furthermore \<tab> is given category space, \<formfeed>
becomes an active character that will detect runaways on files that
have been divided into ``file pages'' by \<formfeed>
characters. Finally the control sequence [[\active]] is defined to
yield the constant~13.

To re-catcode these special characters---not counting ASCII
\begindisplay \def\\{\hfill}
\\\<null>&       [[^^@@]]\cr
\\\<tab>&        [[^^I]]\cr
\\\<linefeed>&   [[^^J]]\cr
\\\<formfeed>&   [[^^L]]\cr
\\\<return>&     [[^^M]]\cr
\\\<delete>&     [[^^?]]\cr
\enddisplay
---use the control sequence [[\dospecials]]
that lists all the characters whose catcodes should
probably be changed to~12 (other) when copying things verbatim.
Each symbol in the list is preceded by [[\do]], which can be defined
if you want to do something to every item in the list.

<<Establish standard category code values>>=
\catcode`\{=1
\catcode`\}=2
\catcode`\$=3
\catcode`\&=4
\catcode`\#=6
\catcode`\^=7 \catcode`\^^K=7 % uparrow is for superscripts
\catcode`\_=8 \catcode`\^^A=8 % downarrow are for subscripts
\catcode`\^^I=10

\chardef\active=13 \catcode`\~=\active % tilde is active
\catcode`\^^L=\active \outer\def^^L{\par} % ascii form-feed is "\outer\par"

\def\dospecials{\do\ \do\\\do\{\do\}\do\$\do\&%
  \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~}
@@
To make the plain macros more efficient in time and space, several
constant values are declared as control sequences.
\stress{If they were changed, anything could happen.}
So be careful!

<<Define commonly used constants>>=
\chardef\@@ne=1
\chardef\tw@@=2
\chardef\thr@@@@=3
\chardef\sixt@@@@n=16
\chardef\@@cclv=255
\mathchardef\@@cclvi=256
\mathchardef\@@m=1000
\mathchardef\@@M=10000
\mathchardef\@@MM=20000
@@
\section{Text fonts}

<<Set up text fonts>>=
<<Provide support for font scaling>>
<<Define text fonts>>
<<Encode special characters, and characters not available on the keyboard>>
<<Provide support for accented characters>>
<<Assign uppercase and lowercase code values>>
<<Assign space factor codes>>
@@
Fonts assigned to [[\preloaded]] are not part of the format,
but they are preloaded so that other format packages can use them.
For example, if another set of macros says [[\font\ninerm=cmr9]],
\TeX\ will not have to reload the font metric information for [[cmr9]].

<<Define text fonts>>=
\font\tenrm=cmr10 % roman text
\font\preloaded=cmr9
\font\preloaded=cmr8
\font\sevenrm=cmr7
\font\preloaded=cmr6
\font\fiverm=cmr5

\font\preloaded=cmss10 % sans serif
\font\preloaded=cmssq8

\font\preloaded=cmssi10 % sans serif italic
\font\preloaded=cmssqi8

\font\tenbf=cmbx10 % boldface extended
\font\preloaded=cmbx9
\font\preloaded=cmbx8
\font\sevenbf=cmbx7
\font\preloaded=cmbx6
\font\fivebf=cmbx5

\font\tentt=cmtt10 % typewriter
\font\preloaded=cmtt9
\font\preloaded=cmtt8

\font\preloaded=cmsltt10 % slanted typewriter

\font\tensl=cmsl10 % slanted roman
\font\preloaded=cmsl9
\font\preloaded=cmsl8

\font\tenit=cmti10 % text italic
\font\preloaded=cmti9
\font\preloaded=cmti8
\font\preloaded=cmti7

\font\preloaded=cmu10 % unslanted text italic

\font\preloaded=cmcsc10 % caps and small caps

\font\preloaded=cmssbx10 % sans serif bold extended

\font\preloaded=cmdunh10 % Dunhill style

\font\preloaded=cmr7 scaled \magstep4 % for titles
\font\preloaded=cmtt10 scaled \magstep2
\font\preloaded=cmssbx10 scaled \magstep2

\font\preloaded=manfnt % METAFONT logo and dragon curve and special symbols
@@
Additional [[\preloaded]] fonts can be specified here.
(And those that were [[\preloaded]] above can be eliminated.)

<<Define text fonts>>=
\let\preloaded=\undefined % preloaded fonts must be declared anew later.
@@
<<Provide support for font scaling>>=
\def\magstephalf{1095 }
\def\magstep#1{\ifcase#1 \@@m\or 1200\or 1440\or 1728\or 2074\or 2488\fi\relax}
\def\magnification{\afterassignment\m@@g\count@@}
\def\m@@g{\mag\count@@
  \hsize6.5truein\vsize8.9truein\dimen\footins8truein}
@@
\subsection{Font encoding}

We usually think of text files as containing characters.  It doesn't
cause any problems most of the time when we use plain ASCII
characters---letters A--Z, a--z, the numerals 0--9 and some
of punctuation characters.  This illusion is broken down when we
start using characters that do not belong to this limited set, for
example, accented characters / mathematical symbols.  Then what we
see on screen may not match what we key in. What gets printed may
not match what we see on screen.  Moreover, what gets shown on screen
and what gets printed depends on what machine we are on and how the
fonts that we are using are set up.
In reality text files contain just numeric codes (in range 0--255)
stored in 8-bit bytes, and the mapping between `character' and numeric
code is quite arbitrary.  This is because there are very many more
characters than the 256 numeric codes possible with 8-bits.
Consequently, there will be a need for more than one possible mapping
or `font encoding', or in other words, there would not be a `standard'
encoding that suits all purposes.
\medskip
When a symbol is built up by forming a box, the [[\leavevmode]] macro
is called first; this starts a new paragraph, if \TeX\ is in vertical
mode, but does nothing if \TeX\ is in horizontal mode or math mode.
[[\chardef]] positions are taken from the fonts [[cmr10]] and [[cmsy10]].

<<Encode special characters, and characters not available on the keyboard>>=
\chardef\%=`\%
\chardef\&=`\&
\chardef\#=`\#
\chardef\$=`\$
\chardef\ss="19
\chardef\ae="1A
\chardef\oe="1B
\chardef\o="1C
\chardef\AE="1D
\chardef\OE="1E
\chardef\O="1F
\chardef\i="10 \chardef\j="11 % dotless letters
\def\aa{\accent23a}
\def\l{\char32l}
\def\L{\leavevmode\setbox0\hbox{L}\hbox to\wd0{\hss\char32L}}

\def\leavevmode{\unhbox\voidb@@x} % begins a paragraph, if necessary
\def\_{\leavevmode \kern.06em \vbox{\hrule width.3em}}
\def\AA{\leavevmode\setbox0\hbox{h}\dimen@@\ht0\advance\dimen@@-1ex%
  \rlap{\raise.67\dimen@@\hbox{\char'27}}A}

\def\mathhexbox#1#2#3{\leavevmode
  \hbox{$\m@@th \mathchar"#1#2#3$}}
\def\dag{\mathhexbox279}
\def\ddag{\mathhexbox27A}
\def\S{\mathhexbox278}
\def\P{\mathhexbox27B}
@@
The accent positions are taken from Computer Modern font family. We
are about to `hard-wire' CM accent encoding into the format. Different
encoding will be necessary if other styles of type are used.

Three alternative control-symbol accents are defined,
suitable for keyboards with extended character sets:
[[\let\^^_=\v]], [[\let\^^S=\u]], [[\let\^^D=\^]].

<<Provide support for accented characters>>=
\def\oalign#1{\leavevmode\vtop{\baselineskip\z@@skip \lineskip.25ex%
  \ialign{##\crcr#1\crcr}}} \def\o@@lign{\lineskiplimit\z@@ \oalign}
\def\ooalign{\lineskiplimit-\maxdimen \oalign} % chars over each other
\def\sh@@ft#1{\dimen\z@@.00#1ex\multiply\dimen\z@@\fontdimen1\font
  \kern-.0156\dimen\z@@} % compensate for slant in lowered accents
\def\d#1{{\o@@lign{\relax#1\crcr\hidewidth\sh@@ft{10}.\hidewidth}}}
\def\b#1{{\o@@lign{\relax#1\crcr\hidewidth\sh@@ft{29}%
    \vbox to.2ex{\hbox{\char22}\vss}\hidewidth}}}
\def\c#1{\setbox\z@@\hbox{#1}\ifdim\ht\z@@=1ex\accent24 #1%
  \else{\ooalign{\unhbox\z@@\crcr\hidewidth\char24\hidewidth}}\fi}
\def\copyright{{\ooalign{\hfil\raise.07ex\hbox{c}\hfil\crcr\mathhexbox20D}}}

\def\dots{\relax\ifmmode\ldots\else$\m@@th\ldots\,$\fi}
\def\TeX{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125emX}

\def\`#1{{\accent18 #1}}
\def\'#1{{\accent19 #1}}
\def\v#1{{\accent20 #1}} \let\^^_=\v
\def\u#1{{\accent21 #1}} \let\^^S=\u
\def\=#1{{\accent22 #1}}
\def\^#1{{\accent94 #1}} \let\^^D=\^
\def\.#1{{\accent95 #1}}
\def\H#1{{\accent"7D #1}}
\def\~#1{{\accent"7E #1}}
\def\"#1{{\accent"7F #1}}
\def\t#1{{\edef\next{\the\font}\the\textfont1\accent"7F\next#1}}
@@
[[INITEX]] sets [[\uccode]]$\,`x=`X$ and
[[\uccode]]$\,`X=`X$ for all letters $x$,
and [[\lccode]]$\,`x=`x$, [[\lccode]]$\,`X=`x$;
all other values are zero.

<<Assign uppercase and lowercase code values>>=
%%% Thats all for English language.
@@
Space factor code affects setting of interword glue.
The space factor is normally 1000, which means that the interword glue
should not be modified. If the space factor $f$ is different from
1000, the interword glue is computed as follows: Take the normal space
glue for the current font, and add the extra space if $f\ge2000$.
Then the stretch component is multiplied by $f/1000$, while the shrink
component is multiplied by $1000/f$.
(Look up the Appendix for the values of normal space, normal stretch,
normal shrink, and extra space for some of CM fonts.)

[[INITEX]] sets space factor codes:
[[\sfcode]]$\,x=1000$ for all $x$,
except that [[\sfcode]]$\,`X=999$ for uppercase letters.

The characters
`[[)]]', `[[']]', and `[[]]]'
does not change space factor.

<<Assign space factor codes>>=
\sfcode`\)=0 \sfcode`\'=0 \sfcode`\]=0
@@
\section{Math fonts}

<<Set up math fonts>>=
<<Define math fonts>>
<<Encode math accents>>
<<Establish spacing around mathematical objects>>
<<Assign math codes>>
<<Assign delimiter codes>>
<<Define font families>>
@@
As was said earlier, the font metric information about preloaded font
will be build into the format.  But, if another set of macros says
[[\font\fiftyfiverm = cmr9 at 55pt]], \TeX\ will \stress{have to
reload again} the font metric information for [[cmr9]].

<<Define math fonts>>=
\font\teni=cmmi10 % math italic
\font\preloaded=cmmi9
\font\preloaded=cmmi8
\font\seveni=cmmi7
\font\preloaded=cmmi6
\font\fivei=cmmi5

\font\tensy=cmsy10 % math symbols
\font\preloaded=cmsy9
\font\preloaded=cmsy8
\font\sevensy=cmsy7
\font\preloaded=cmsy6
\font\fivesy=cmsy5

\font\tenex=cmex10 % math extension

\font\preloaded=cmmib10 % bold math italic
\font\preloaded=cmbsy10 % bold math symbols
@@
\subsection{Mathematical spacing}

Spacing around mathematical object is measured in
[[mu]]---`math units.'
[[1mu]] is equal to $1/18\,$th part of [[\fontdimen 6]] of
the font in family 2.

[[\quad]] spacing does not change with the style of formula, nor does it
depend on the math font families that are being used.
But thin spaces, medium spaces, and thick spaces do get bigger and
smaller as the size of type gets bigger and smaller; this is because
they are defined in terms of \<muglue>.

According to these specifications, thin spaces in plain
\TeX\ do not stretch or shrink; medium spaces can stretch a little, and
they can shrink to zero; thick spaces can stretch a lot, but they never shrink.

\medskip

The following table gives the complete definition of muglue between
mathematical objects.  A formula is converted to a math list, and the
math list consists chiefly of ``atoms'' of eight basic types:
Ord~(ordinary), Op~(large operator), Bin~(binary operation),
Rel~(relation), Open~(opening), Close~(closing), Punct~(punctuation),
and Inner~(a delimited subformula).  Other kinds of atoms, which arise
from commands like [[\overline]] or [[\mathaccent]] or [[\vcenter]],
etc., are all treated as type~Ord; fractions are treated as
type~Inner. The following (non-symmetric) table is used to determine
the spacing between pairs of adjacent atoms:
\medbreak
$$\baselineskip0pt\lineskip0pt
\halign to\hsize
 {\strut\hbox to1.5\parindent{\it#\hfil}& % for the legend "Left atom"
  #\hfil\quad& % for the row labels
  #\hfil\tabskip 0pt plus 20pt& % for the rule at the left
  \hbox to 25pt{\tt\hss#\hss}& % for column 1
  \hbox to 25pt{\tt\hss#\hss}& % for column 2
  \hbox to 25pt{\tt\hss#\hss}& % for column 3
  \hbox to 25pt{\tt\hss#\hss}& % for column 4
  \hbox to 25pt{\tt\hss#\hss}& % for column 5
  \hbox to 25pt{\tt\hss#\hss}& % for column 6
  \hbox to 25pt{\tt\hss#\hss}& % for column 7
  \hbox to 25pt{\tt\hss#\hss}& % for column 8
  #\hfil\tabskip0pt\cr % for the rule at the right
\noalign{\vskip-6pt} % it just happens that there's extra white space
&&&&\multispan7\hss\it Right atom\hss\cr \noalign{\vskip3pt} &&&\rm
Ord&\rm Op&\rm Bin&\rm Rel&\rm Open&\rm Close&\rm Punct&\rm Inner\cr
\noalign{\vskip2pt} \omit&&\multispan{10}\leaders\hrule\hfil\cr
\omit\vbox to 2pt{}&&\vrule&&&&&&&&&\vrule\cr
&Ord&\vrule&0&1&(2)&(3)&0&0&0&(1)&\vrule\cr
&Op&\vrule&1&1&*&(3)&0&0&0&(1)&\vrule\cr
&Bin&\vrule&(2)&(2)&*&*&(2)&*&*&(2)&\vrule\cr
Left&Rel&\vrule&(3)&(3)&*&0&(3)&0&0&(3)&\vrule\cr
atom&Open&\vrule&0&0&*&0&0&0&0&0&\vrule\cr
&Close&\vrule&0&1&(2)&(3)&0&0&0&(1)&\vrule\cr
&Punct&\vrule&(1)&(1)&*&(1)&(1)&(1)&(1)&(1)&\vrule\cr
&Inner&\vrule&(1)&1&(2)&(3)&(1)&0&(1)&(1)&\vrule\cr \omit\vbox to
2pt{}&&\vrule&&&&&&&&&\vrule\cr
\omit&&\multispan{10}\leaders\hrule\hfil\cr}$$
\medbreak
Here [[0]], [[1]], [[2]], and [[3]] stand for
no space, thin space, medium space, and thick space,
respectively.
Thin space, medium space, and thin space are equal to
values of [[\thinmuskim]], [[\medmuskip]], [[\thickmuskip]]
parameters, respectively.
The table entry is parenthesized if the space is to be
inserted only in display and text styles, not in script and
scriptscript styles. For example, many of the entries in the Rel row
and the Rel column are `{\tt(3)}'; this means that thick spaces are
normally inserted before and after relational symbols like `=', but
not in subscripts. Some of the entries in the table are `{\tt*}'; such
cases never arise, because Bin atoms must be preceded and followed by
atoms compatible with the nature of binary operations.
The conversion of math lists to horizontal lists is done
whenever \TeX\ is about to leave math mode, and the inter-atomic
spacing is inserted at that time.

<<Establish spacing around mathematical objects>>=
\thinmuskip=3mu
\medmuskip=4mu plus 2mu minus 4mu
\thickmuskip=5mu plus 5mu
@@
For the positioning of accents over single character the width of
[[\skewchar]] is used.  For most of fonts the default value of
[[\skewchar]] is [[-1]]; but the math italic (family 1)
and math symbol fonts (family 2) have
special [[\skewchar]] values equal to [['177]] and [['60]],
respectively. These are characters
`{\teni\char'177\/}' and `{\tensy\char'60}'.

<<Encode math accents>>=
\skewchar\teni='177 \skewchar\seveni='177 \skewchar\fivei='177
\skewchar\tensy='60 \skewchar\sevensy='60 \skewchar\fivesy='60
@@
A math code is relevant only when the corresponding category code is
11 or~12. When processing in math mode characters of categories~11 and~12,
[[\char]] and [[\chardef]] characters are replaced by their math code.

If we denote 15-bit number by [["uvwz]], then math codes are assigned by
\medskip
[[\mathcode]] \<8-bit number> = [["uvwz]],\quad where
\medskip
\itemitem{[[u]]}  --- the class code (see below for the list)
\itemitem{[[v]]}  --- the font family number (see the font tables at the
                      end of this document)
\itemitem{[[wz]]} --- the position of the character in the font
$$\halign{\indent#\hfil&\quad#\hfil&\quad#\hfil&
\hskip4em#\hfil&\quad#\hfil&\quad#\hfil\cr
\it \kern-2pt Class&\it Meaning&\kern-2pt\it Example&
\it \kern-2pt Class&\it Meaning&\kern-2pt\it Example\cr
\noalign{\vskip2pt}
0&Ordinary&[[/]]&
4&Opening&[[(]]\cr
1&Large operator&[[\sum]]&
5&Closing&[[)]]\cr
2&Binary operation&[[+]]&
6&Punctuation&[[,]]\cr
3&Relation&[[=]]&
7&Variable family&[[x]]\cr
}$$

A [[\mathcode]] can also have the special value \hex{8000}, which
causes the character to behave as if it has catcode~13 (active).  This
feature makes [[']] apostrophe expand to [[\prime]].  The mathcode of
[[']] does not interfere with the use of [[']] in octal constants.
The mathcode of [["8000]] is also assigned to space and underscore.

[[INITEX]] sets up
[[\mathcode]]$\,x=x$ for $x=0..255$, except that
[[\mathcode]]$\,x=x+\hex{7000}$
for each of the ten digits $x={}$[[`0]] to [[`9]];
[[\mathcode]]$\,x=x+\hex{7100}$ for each of the 52~letters.
\TeX\ looks at the mathcode only when it is typesetting a character whose
catcode is 11~(letter) or 12~(other), or when it encounters a character that
is given explicitly as [[\char]]\<number>.

Class 7 is a special case that allows math symbols to change families.
It behaves exactly like class~0, except that the specified family is
replaced by the current value of an integer parameter called [[\fam]],
provided that [[\fam]] is a legal family number (i.e., if it lies between
0 and~15). \TeX\ automatically sets [[\fam=-1]] whenever math mode is entered;
therefore class~7 and class~0 are equivalent unless [[\fam]] has been
given a new value. Plain \TeX\ changes [[\fam]] to~0 when the user
types `[[\rm]]'; this makes it convenient to get roman letters in formulas,
since letters belong to class~7.
The control sequence [[\rm]] is an abbreviation for `[[\fam=0 \tenrm]]';
thus, [[\rm]] causes [[\fam]] to become zero, and it makes
[[\tenrm]] the ``current font.''  In horizontal mode, the [[\fam]]
value is irrelevant and the current font governs the typesetting of
letters; but in math mode, the current font is irrelevant and the
[[\fam]] value governs the letters. The current font affects math mode
only if control space ([[\ ]]) is used or if dimensions are given in
[[ex]] or [[em]] units; it also has an effect if an [[\hbox]] appears
inside a formula, since the contents of an hbox are typeset in
horizontal mode.

<<Assign math codes>>=
\mathcode`\^^@@="2201 % \cdot
\mathcode`\^^A="3223 % \downarrow
\mathcode`\^^B="010B % \alpha
\mathcode`\^^C="010C % \beta
\mathcode`\^^D="225E % \land
\mathcode`\^^E="023A % \lnot
\mathcode`\^^F="3232 % \in
\mathcode`\^^G="0119 % \pi
\mathcode`\^^H="0115 % \lambda
\mathcode`\^^I="010D % \gamma
\mathcode`\^^J="010E % \delta
\mathcode`\^^K="3222 % \uparrow
\mathcode`\^^L="2206 % \pm
\mathcode`\^^M="2208 % \oplus
\mathcode`\^^N="0231 % \infty
\mathcode`\^^O="0140 % \partial
\mathcode`\^^P="321A % \subset
\mathcode`\^^Q="321B % \supset
\mathcode`\^^R="225C % \cap
\mathcode`\^^S="225B % \cup
\mathcode`\^^T="0238 % \forall
\mathcode`\^^U="0239 % \exists
\mathcode`\^^V="220A % \otimes
\mathcode`\^^W="3224 % \leftrightarrow
\mathcode`\^^X="3220 % \leftarrow
\mathcode`\^^Y="3221 % \rightarrow
\mathcode`\^^Z="8000 % \ne
\mathcode`\^^[="2205 % \diamond
\mathcode`\^^\="3214 % \le
\mathcode`\^^]="3215 % \ge
\mathcode`\^^^="3211 % \equiv
\mathcode`\^^_="225F % \lor
\mathcode`\ ="8000 % \space
\mathcode`\!="5021
\mathcode`\'="8000 % ^\prime
\mathcode`\(="4028
\mathcode`\)="5029
\mathcode`\*="2203 % \ast
\mathcode`\+="202B
\mathcode`\,="613B
\mathcode`\-="2200
\mathcode`\.="013A
\mathcode`\/="013D
\mathcode`\:="303A
\mathcode`\;="603B
\mathcode`\<="313C
\mathcode`\=="303D
\mathcode`\>="313E
\mathcode`\?="503F
\mathcode`\[="405B
\mathcode`\\="026E % \backslash
\mathcode`\]="505D
\mathcode`\_="8000 % \_
\mathcode`\{="4266
\mathcode`\|="026A
\mathcode`\}="5267
\mathcode`\^^?="1273 % \smallint

@@
Delimiter codes are used after [[\left]] and [[\right]] commands,
when \TeX\ is looking for a delimiter.

If we denote 24-bit number by [["qrstuv]], then
delimiter codes are assigned by
\medskip
[[\delcode]] \<8-bit number> = [["qrstuv]],\quad where
\medskip
\itemitem{[[q]]}   --- the font family number of
\itemitem{[[rs]]}  --- the position of the the small variant of the delimiter
\itemitem{[[t]]}   --- the font family number of
\itemitem{[[uv]]}  --- the position of the the large variant of the delimiter
\medskip
%%% Assigning delimiters codes to characters completes coding.
%%% The full list consists of:
%%% [[\catcode]], [[\lccode]], [[\uccode]], [[\mathcode]], [[\delcode]].

[[INITEX]] sets all [[\delcode]] values to [[-1]],
which means that no characters are recognized as delimiters
in math formulas,
except [[\delcode]]$\,`.=0$, so that `[[.]]' stands for
``null delimiter''.
[[{]] and [[}]] should \stress{not get} delcodes;
otherwise parameter grouping fails!

<<Assign delimiter codes>>=
\delcode`\(="028300
\delcode`\)="029301
\delcode`\[="05B302
\delcode`\]="05D303
\delcode`\<="26830A
\delcode`\>="26930B
\delcode`\/="02F30E
\delcode`\|="26A30C
\delcode`\\="26E30F
@@
All characters that are typeset in math mode belong to one of sixteen
{\sl families} of fonts, numbered internally from 0 to~15. Each of
these families consists of three fonts: one for text size, one for
script size, and one for scriptscriptsize. The commands [[\textfont]],
[[\scriptfont]], and [[\scriptscriptfont]] are used to specify the
members of each family.  Since there are up to 256~characters per
font, and 3~fonts per family, and 16~families, \TeX\ can access up to
12,288 characters in any one formula (4096 in~each of the three
sizes).

The [[plain.tex]] format uses family~1 for math italic letters, family~2 for
ordinary math symbols, and family~3 for large symbols.
Text italic is put in family~4, slanted roman in family~5, bold roman
in family~6, and typewriter type in family~7.
A macro [[\newfam]] will assign symbolic names to families that aren't
already used.

[[INITEX]] initializes the mathcodes of all letters [[A]] to~[[Z]] and
[[a]] to~[[z]] so that they are symbols of class~7 and family~1; that's
why it is natural to use family~1 for math italics.  Similarly, the
digits [[0]] to~[[9]] are class~7 and family~0.  None of the other
families is treated in any special way by \TeX.

\TeX\ doesn't check to see if the families are sensibly organized.
The only constraint is that the fonts in
families 2 and~3 have special [[\fontdimen]] parameters, which govern
mathematical spacing (see Appendix).
In Computer Modern only [[cmsy]] and [[cmex]] have these
parameters, so their assignment to families 2 and~3 is almost
mandatory.

During the time that a math formula is being read,
\TeX\ remembers each symbol as being ``character position so-and-so in
family number such-and-such,'' but it does not take note of what fonts
are actually in the families until reaching the end of the formula.

<<Define font families>>=
\textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
\def\rm{\fam\z@@\tenrm}
\textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
\def\mit{\fam\@@ne} \def\oldstyle{\fam\@@ne\teni}
\textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
\def\cal{\fam\tw@@}
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\newfam\itfam \def\it{\fam\itfam\tenit} % \it is family 4
\textfont\itfam=\tenit
\newfam\slfam \def\sl{\fam\slfam\tensl} % \sl is family 5
\textfont\slfam=\tensl
\newfam\bffam \def\bf{\fam\bffam\tenbf} % \bf is family 6
\textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
\scriptscriptfont\bffam=\fivebf
\newfam\ttfam \def\tt{\fam\ttfam\tentt} % \tt is family 7
\textfont\ttfam=\tentt
@@
\section{Registers allocation}

Here are macros for the automatic allocation of [[\count]],
[[\box]], [[\dimen]], [[\skip]], [[\muskip]], and [[\toks]]
registers, as well as [[\read]] and [[\write]]
stream numbers, [[\fam]] codes, [[\language]] codes,
and [[\insert]] numbers.

The main use of these macros is for registers that are defined by one
macro and used by others, possibly at different nesting levels.

The following counters are reserved:
\begindisplay
  0--9&	 page numbering\cr
    10&	 count allocation\cr
    11&	 dimen allocation\cr
    12&	 skip allocation\cr
    13&	 muskip allocation\cr
    14&  box allocation\cr
    15&  toks allocation\cr
    16&  read file allocation\cr
    17&  write file allocation\cr
    18&  math family allocation\cr
    19&  language allocation\cr
    20&  insert allocation\cr
    21&  the most recently allocated number\cr
    22&  constant -1\cr
\enddisplay
New counters are allocated starting with 23, 24, etc.  Other registers are
allocated starting with 10.  This leaves 0 through 9 for the user to play
with safely, except that counts 0 to 9 are considered to be the page and
subpage numbers (since they are displayed during output). In this scheme,
[[\count]] 10 always contains the number of the highest-numbered counter that
has been allocated, [[\count 14]] the highest-numbered box, etc.
Inserts are given numbers 254, 253, etc., since they require a [[\count]],
[[\dimen]], [[\skip]], and [[\box]] all with the same number;
[[\count]] 20 contains the lowest-numbered insert that has been allocated.
[[\box255]] is reserved for [[\output]];
[[\count255]], [[\dimen255]], and [[\skip255]] can be used freely.

It is recommended that macro designers always use
global assignments with respect to registers numbered 1, 3, 5, 7, 9, and
always non-global assignments with respect to registers 0, 2, 4, 6, 8, 255.
This will prevent ``save stack buildup'' that might otherwise occur.

<<Allocate registers>>=
\count10=22 % allocates \count registers 23, 24, ...
\count11=9 % allocates \dimen registers 10, 11, ...
\count12=9 % allocates \skip registers 10, 11, ...
\count13=9 % allocates \muskip registers 10, 11, ...
\count14=9 % allocates \box registers 10, 11, ...
\count15=9 % allocates \toks registers 10, 11, ...
\count16=-1 % allocates input streams 0, 1, ...
\count17=-1 % allocates output streams 0, 1, ...
\count18=3 % allocates math families 4, 5, ...
\count19=0 % allocates \language codes 1, 2, ...
\count20=255 % allocates insertions 254, 253, ...
\countdef\insc@@unt=20 % the insertion counter
\countdef\allocationnumber=21 % the most recent allocation
\countdef\m@@ne=22 \m@@ne=-1 % a handy constant
\def\wlog{\immediate\write\m@@ne} % write on log file (only)
<<Allocate scratch registers>>
<<Provide user-level register allocation macros>>
<<Define implementation-level register allocation macros>>
<<Initialize register constants>>
@@
Here are abbreviations for the names of scratch registers
that don't need to be allocated.

<<Allocate scratch registers>>=
\countdef\count@@=255
\dimendef\dimen@@=0
\dimendef\dimen@@i=1 % global only
\dimendef\dimen@@ii=2
\skipdef\skip@@=0
\toksdef\toks@@=0
@@
Now, we define [[\newcount]], [[\newbox]], etc.
so that you can say [[\newcount\foo]]
and [[\foo]] will be defined (with [[\countdef]]) to be the next counter.
To find out which counter [[\foo]] is, you can look at [[\allocationnumber]].
Since there's no [[\boxdef]] command, [[\chardef]] is used to define
a [[\newbox]], [[\newinsert]], [[\newfam]], and so on.

<<Provide user-level register allocation macros>>=
\outer\def\newcount{\alloc@@0\count\countdef\insc@@unt}
\outer\def\newdimen{\alloc@@1\dimen\dimendef\insc@@unt}
\outer\def\newskip{\alloc@@2\skip\skipdef\insc@@unt}
\outer\def\newmuskip{\alloc@@3\muskip\muskipdef\@@cclvi}
\outer\def\newbox{\alloc@@4\box\chardef\insc@@unt}
\let\newtoks=\relax % we do this to allow plain.tex to be read in twice
\outer\def\newhelp#1#2{\newtoks#1#1\expandafter{\csname#2\endcsname}}
\outer\def\newtoks{\alloc@@5\toks\toksdef\@@cclvi}
\outer\def\newread{\alloc@@6\read\chardef\sixt@@@@n}
\outer\def\newwrite{\alloc@@7\write\chardef\sixt@@@@n}
\outer\def\newfam{\alloc@@8\fam\chardef\sixt@@@@n}
\outer\def\newlanguage{\alloc@@9\language\chardef\@@cclvi}
@@
<<Define implementation-level register allocation macros>>=
\def\alloc@@#1#2#3#4#5{\global\advance\count1#1by\@@ne
  \ch@@ck#1#4#2% make sure there's still room
  \allocationnumber=\count1#1%
  \global#3#5=\allocationnumber
  \wlog{\string#5=\string#2\the\allocationnumber}}
\outer\def\newinsert#1{\global\advance\insc@@unt by\m@@ne
  \ch@@ck0\insc@@unt\count
  \ch@@ck1\insc@@unt\dimen
  \ch@@ck2\insc@@unt\skip
  \ch@@ck4\insc@@unt\box
  \allocationnumber=\insc@@unt
  \global\chardef#1=\allocationnumber
  \wlog{\string#1=\string\insert\the\allocationnumber}}
\def\ch@@ck#1#2#3{\ifnum\count1#1<#2%
  \else\errmessage{No room for a new #3}\fi}
@@
We finish with the initalization of some constants.
<<Initialize register constants>>=
\newdimen\maxdimen \maxdimen=16383.99999pt % the largest legal <dimen>
\newskip\hideskip \hideskip=-1000pt plus 1fill % negative but can grow
\newskip\centering \centering=0pt plus 1000pt minus 1000pt
\newdimen\p@@ \p@@=1pt % this saves macro space and time
\newdimen\z@@ \z@@=0pt % can be used both for 0pt and 0
\newskip\z@@skip \z@@skip=0pt plus0pt minus0pt
\newbox\voidb@@x % permanently void box register
@@
\section{Parameters}

Let's turn now to \TeX's parameters, which the previous chapters have
introduced one at a time; it will be convenient to assemble them
all together.
\bigskip
An \<integer parameter> is one of the following tokens:
\begindisplay\belowdisplayskip=3pt plus 6pt \abovedisplayskip=3pt plus 1pt%
\openup.15pt
[[\pretolerance]]\qquad badness tolerance before hyphenation\cr
[[\tolerance]]\qquad badness tolerance after hyphenation\cr
[[\hbadness]]\qquad badness above which bad hboxes will be shown\cr
[[\vbadness]]\qquad badness above which bad vboxes will be shown\cr
[[\linepenalty]]\qquad amount added to badness of every line in a paragraph\cr
[[\hyphenpenalty]]\qquad penalty for line break after discretionary hyphen\cr
[[\exhyphenpenalty]]\qquad penalty for line break after explicit hyphen\cr
[[\binoppenalty]]\qquad penalty for line break after binary operation\cr
[[\relpenalty]]\qquad penalty for line break after math relation\cr
[[\clubpenalty]]\qquad penalty for creating a club line at bottom of page\cr
[[\widowpenalty]]\qquad penalty for creating a widow line at top of page\cr
[[\displaywidowpenalty]]\qquad ditto, before a display\cr
[[\brokenpenalty]]\qquad penalty for page break after a hyphenated line\cr
[[\predisplaypenalty]]\qquad penalty for page break just before a display\cr
[[\postdisplaypenalty]]\qquad penalty for page break just after a display\cr
[[\interlinepenalty]]\qquad additional penalty for page break between lines\cr
[[\floatingpenalty]]\qquad penalty for insertions that are split\cr
[[\outputpenalty]]\qquad penalty at the current page break\cr
[[\doublehyphendemerits]]\qquad demerits for consecutive broken lines\cr
[[\finalhyphendemerits]]\qquad demerits for a penultimate broken line\cr
[[\adjdemerits]]\qquad demerits for adjacent incompatible lines\cr
[[\looseness]]\qquad change to the number of lines in a paragraph\cr
[[\pausing]]\qquad positive if pausing after each line is read from a file\cr
[[\holdinginserts]]\qquad positive if insertions
   remain dormant in output box\cr
[[\tracingonline]]\qquad positive if showing diagnostic info on the terminal\cr
[[\tracingmacros]]\qquad positive if showing macros as they are expanded\cr
[[\tracingstats]]\qquad positive if showing statistics about memory usage\cr
[[\tracingparagraphs]]\qquad positive if showing line-break calculations\cr
[[\tracingpages]]\qquad positive if showing page-break calculations\cr
[[\tracingoutput]]\qquad positive if showing boxes that are shipped out\cr
[[\tracinglostchars]]\qquad positive if showing characters not in the font\cr
[[\tracingcommands]]\qquad positive if showing commands
   before they are executed\cr
[[\tracingrestores]]\qquad positive if showing deassignments when groups end\cr
[[\language]]\qquad the current set of hyphenation rules\cr
[[\uchyph]]\qquad positive if hyphenating words
   beginning with capital letters\cr
[[\lefthyphenmin]]\qquad smallest fragment at beginning of hyphenated word\cr
[[\righthyphenmin]]\qquad smallest fragment at end of hyphenated word\cr
[[\globaldefs]]\qquad nonzero if overriding [[\global]] specifications\cr
[[\defaulthyphenchar]]\qquad [[\hyphenchar]] value when a font is loaded\cr
[[\defaultskewchar]]\qquad [[\skewchar]] value when a font is loaded\cr
[[\escapechar]]\qquad escape character in the output of
   control sequence tokens\cr
[[\endlinechar]]\qquad character placed at the right end of an input line\cr
[[\newlinechar]]\qquad character that starts a new output line\cr
[[\maxdeadcycles]]\qquad upper bound on [[\deadcycles]]\cr
[[\hangafter]]\qquad hanging indentation changes after this many lines\cr
[[\fam]]\qquad the current family number\cr
[[\mag]]\qquad magnification ratio, times 1000\cr
[[\delimiterfactor]]\qquad ratio for variable delimiters, times 1000\cr
[[\time]]\qquad current time of day in minutes since midnight\cr
[[\day]]\qquad current day of the month\cr
[[\month]]\qquad current month of the year\cr
[[\year]]\qquad current year of our Lord\cr
[[\showboxbreadth]]\qquad maximum items per level when boxes are shown\cr
[[\showboxdepth]]\qquad maximum level when boxes are shown\cr
[[\errorcontextlines]]\qquad maximum extra context shown when errors occur\cr
\enddisplay
The first few of these parameters have values in units of ``badness'' and
``penalties'' that affect line breaking and page breaking. Then come
demerit-oriented parameters; demerits are essentially given in units of
``badness squared,'' so those parameters tend to have larger values.
By contrast, the next few parameters ([[\looseness]], [[\pausing]], etc.)\
generally have quite small values (either $-1$ or 0 or 1 or~2).
Miscellaneous parameters complete the set.
\bigskip
A \<dimen parameter> is one of the following:
\begindisplay\openup.15pt
[[\hfuzz]]\qquad maximum overrun before overfull hbox messages occur\cr
[[\vfuzz]]\qquad maximum overrun before overfull vbox messages occur\cr
[[\overfullrule]]\qquad width of rules appended to overfull boxes\cr
[[\emergencystretch]]\qquad reduces badnesses on final pass of line-breaking\cr
[[\hsize]]\qquad line width in horizontal mode\cr
[[\vsize]]\qquad page height in vertical mode\cr
[[\maxdepth]]\qquad maximum depth of boxes on main pages\cr
[[\splitmaxdepth]]\qquad maximum depth of boxes on split pages\cr
[[\boxmaxdepth]]\qquad maximum depth of boxes on explicit pages\cr
[[\lineskiplimit]]\qquad threshold where [[\baselineskip]] changes
   to [[\lineskip]]\cr
[[\delimitershortfall]]\qquad maximum space not covered by a delimiter\cr
[[\nulldelimiterspace]]\qquad width of a null delimiter\cr
[[\scriptspace]]\qquad extra space after subscript or superscript\cr
[[\mathsurround]]\qquad kerning before and after math in text\cr
[[\predisplaysize]]\qquad length of text preceding a display\cr
[[\displaywidth]]\qquad length of line for displayed equation\cr
[[\displayindent]]\qquad indentation of line for displayed equation\cr
[[\parindent]]\qquad width of\/ [[\indent]]\cr
[[\hangindent]]\qquad amount of hanging indentation\cr
[[\hoffset]]\qquad horizontal offset in [[\shipout]]\cr
[[\voffset]]\qquad vertical offset in [[\shipout]]\cr
\enddisplay

Before typesetting a delimiter \TeX\ determines the size $f$ of the
formula to be covered as twice the maximum of the height and the depth
of the formula. The size $d$ of the delimiter should be
$$
\min\big\{ f - \hbox{[[\delimitershortfall]]},\>
           \hbox{[[\delimiterfactor]]} \cdot {f \over 1000} \big\}
\leq d
$$
\medskip
And the possibilities for \<glue parameter> are:
\begindisplay\openup.15pt
[[\baselineskip]]\qquad desired glue between baselines\cr
[[\lineskip]]\qquad interline glue if\/ [[\baselineskip]] isn't feasible\cr
[[\parskip]]\qquad extra glue just above paragraphs\cr
[[\abovedisplayskip]]\qquad extra glue just above displays\cr
[[\abovedisplayshortskip]]\qquad ditto, following short lines\cr
[[\belowdisplayskip]]\qquad extra glue just below displays\cr
[[\belowdisplayshortskip]]\qquad ditto, following short lines\cr
[[\leftskip]]\qquad glue at left of justified lines\cr
[[\rightskip]]\qquad glue at right of justified lines\cr
[[\topskip]]\qquad glue at top of main pages\cr
[[\splittopskip]]\qquad glue at top of split pages\cr
[[\tabskip]]\qquad glue between aligned entries\cr
[[\spaceskip]]\qquad glue between words, if nonzero\cr
[[\xspaceskip]]\qquad glue between sentences, if nonzero\cr
[[\parfillskip]]\qquad additional [[\rightskip]] at end of paragraphs\cr
\enddisplay

To above parameters (except [[\parfillskip]]) are assigned values
appropriate for CM family typeset at 12$\,$pt baseline.

\medskip
Finally, there are three permissible \<muglue parameter> tokens:
\begindisplay\openup.15pt
[[\thinmuskip]]\qquad thin space in math formulas\cr
[[\medmuskip]]\qquad medium space in math formulas\cr
[[\thickmuskip]]\qquad thick space in math formulas\cr
\enddisplay
\bigskip
\TeX\ also has parameters that are token lists. Such parameters do not
enter into the definitions of \<number> and such things.
A \<token parameter> is any of:
\begindisplay\openup.15pt
[[\output]]\qquad the user's output routine\cr
[[\everypar]]\qquad tokens to insert when a paragraph begins\cr
[[\everymath]]\qquad tokens to insert when math in text begins\cr
[[\everydisplay]]\qquad tokens to insert when display math begins\cr
[[\everyhbox]]\qquad tokens to insert when an hbox begins\cr
[[\everyvbox]]\qquad tokens to insert when a vbox begins\cr
[[\everyjob]]\qquad tokens to insert when the job begins\cr
[[\everycr]]\qquad tokens
   to insert after every [[\cr]] or nonredundant [[\crcr]]\cr
[[\errhelp]]\qquad tokens that supplement an [[\errmessage]]\cr
\enddisplay
\bigskip
All of numeric parameters are listed below,
but the code is commented out if no special value needs to be set.
[[INITEX]] makes all parameters zero except where noted.

<<Assign initial values to parameters>>=
<<Assign values to integer parameters>>
<<Assign values to dimen parameters>>
<<Assign values to glue parameters>>
<<Assign values to special registers>>
@@
<<Assign values to integer parameters>>=
\pretolerance=100
\tolerance=200 % INITEX sets this to 10000
\hbadness=1000
\vbadness=1000
\linepenalty=10
\hyphenpenalty=50
\exhyphenpenalty=50
\binoppenalty=700
\relpenalty=500
\clubpenalty=150
\widowpenalty=150
\displaywidowpenalty=50
\brokenpenalty=100
\predisplaypenalty=10000
% \postdisplaypenalty=0
% \interlinepenalty=0
% \floatingpenalty=0, set during \insert
% \outputpenalty=0, set before TeX enters \output
\doublehyphendemerits=10000
\finalhyphendemerits=5000
\adjdemerits=10000
% \looseness=0, cleared by TeX after each paragraph
% \pausing=0
% \holdinginserts=0
% \tracingonline=0
% \tracingmacros=0
% \tracingstats=0
% \tracingparagraphs=0
% \tracingpages=0
% \tracingoutput=0
\tracinglostchars=1
% \tracingcommands=0
% \tracingrestores=0
% \language=0
\uchyph=1
% \lefthyphenmin=2 \righthyphenmin=3 set below
% \globaldefs=0
% \maxdeadcycles=25 % INITEX does this
% \hangafter=1 % INITEX does this, also TeX after each paragraph
% \fam=0
% \mag=1000 % INITEX does this
% \escapechar=`\\ % INITEX does this
\defaulthyphenchar=`\-
\defaultskewchar=-1
% \endlinechar=`\^^M % INITEX does this
\newlinechar=-1
\delimiterfactor=901
% \time=now % TeX does this at beginning of job
% \day=now % TeX does this at beginning of job
% \month=now % TeX does this at beginning of job
% \year=now % TeX does this at beginning of job
\showboxbreadth=5
\showboxdepth=3
\errorcontextlines=5
@@
<<Assign values to dimen parameters>>=
\hfuzz=0.1pt
\vfuzz=0.1pt
\overfullrule=5pt
\hsize=6.5in
\vsize=8.9in
\maxdepth=4pt
\splitmaxdepth=\maxdimen
\boxmaxdepth=\maxdimen
% \lineskiplimit=0pt, changed by \normalbaselines
\delimitershortfall=5pt
\nulldelimiterspace=1.2pt
\scriptspace=0.5pt
% \mathsurround=0pt
% \predisplaysize=0pt, set before TeX enters $$
% \displaywidth=0pt, set before TeX enters $$
% \displayindent=0pt, set before TeX enters $$
\parindent=20pt
% \hangindent=0pt, zeroed by TeX after each paragraph
% \hoffset=0pt
% \voffset=0pt
@@
<<Assign values to glue parameters>>=
% \baselineskip=0pt, changed by \normalbaselines
% \lineskip=0pt, changed by \normalbaselines
\parskip=0pt plus 1pt
\abovedisplayskip=12pt plus 3pt minus 9pt
\abovedisplayshortskip=0pt plus 3pt
\belowdisplayskip=12pt plus 3pt minus 9pt
\belowdisplayshortskip=7pt plus 3pt minus 4pt
% \leftskip=0pt
% \rightskip=0pt
\topskip=10pt
\splittopskip=10pt
% \tabskip=0pt
% \spaceskip=0pt
% \xspaceskip=0pt
\parfillskip=0pt plus 1fil
@@
We also define special registers that function like parameters:
<<Assign values to special registers>>=
\newskip\smallskipamount \smallskipamount=3pt plus 1pt minus 1pt
\newskip\medskipamount \medskipamount=6pt plus 2pt minus 2pt
\newskip\bigskipamount \bigskipamount=12pt plus 4pt minus 4pt
\newskip\normalbaselineskip \normalbaselineskip=12pt
\newskip\normallineskip \normallineskip=1pt
\newdimen\normallineskiplimit \normallineskiplimit=0pt
\newdimen\jot \jot=3pt
\newcount\interdisplaylinepenalty \interdisplaylinepenalty=100
\newcount\interfootnotelinepenalty \interfootnotelinepenalty=100
\def\normalbaselines{\lineskip\normallineskip
  \baselineskip\normalbaselineskip \lineskiplimit\normallineskiplimit}
@@
\section{Macros for text}

Here we introduce macros that are used for basic formatting
unrelated to mathematics.

<<Provide macros for text formatting>>=
<<Supply basic macros for text formatting>>
<<Define space macros>>
<<Supply various paragraph shapes>>
<<Define sectioning macros>>
<<Supply ragged setting>>
<<Supply `boxing' macros>>
<<Supply strut>>
<<Provide alignment macros>>
<<Establish spacing after punctuation characters>>
<<Supply various ways to fill space>>
<<Define [[\showhyphens]] macro>>
<<Completing the job>>
@@
[[\]]\<tab> and [[\]]\<return> are defined so that they
expand to [[\]]\<space>; this helps to prevent confusion, since all three
cases look identical when displayed on most computer terminals.

<<Supply basic macros for text formatting>>=
\def\^^M{\ } % control <return> = control <space>
\def\^^I{\ } % same for <tab>
@@
The control sequences [[\endgraf]] and [[\endline]] are made equivalent
to \TeX's primitive [[\par]] and~[[\cr]] operations, since it is often
useful to redefine the meanings of [[\par]] and~[[\cr]] themselves. Then
come the definitions of [[\space]] (a blank space), [[\empty]] (a~list
of no tokens), and [[\null]] (an~empty hbox).

[[\bgroup]] and [[\egroup]] are made to provide ``implicit'' grouping
characters that turn out to be especially useful in macro definitions.

<<Supply basic macros for text formatting>>=
\let\endgraf=\par \let\endline=\cr

\def\space{ }
\def\empty{}
\def\null{\hbox{}}

\let\bgroup={ \let\egroup=}
@@
The [[\obeylines]] macro says `[[\let^^M=\par]]'
instead of `[[\def^^M{\par}]]' because the [[\let]]
technique allows constructions such as
`[[\let\par=\cr]] [[\obeylines]] [[\halign||{...}]]'
in which [[\cr]]'s need not be given within the alignment.

<<Supply basic macros for text formatting>>=
{\catcode`\^^M=\active % these lines must end with %
  \gdef\obeylines{\catcode`\^^M\active \let^^M\par}%
  \global\let^^M\par} % this is in case ^^M appears in a \write
\def\obeyspaces{\catcode`\ \active}
{\obeyspaces\global\let =\space}
@@
The macros [[\lq]], [[\rq]], [[\lbrack]], and [[\rbrack]] are defined,
for people who have difficulty typing quotation marks and/or brackets.

<<Supply basic macros for text formatting>>=
\def\lq{`} \def\rq{'}
\def\lbrack{[} \def\rbrack{]}
@@
The macros [[\enskip]], [[\quad]], and [[\qquad]]
provide spaces that are legitimate breakpoints within a paragraph;
[[\enspace]], [[\thinspace]], and [[\negthinspace]] produce space that cannot
cause a break (although the space will disappear if it occurs just
next to certain kinds of breaks). All six of these spaces are relative
to the current font.

You can get horizontal space that
never disappears by saying `[[\hglue]]\<glue>'; this space is able to
stretch or shrink. Similarly, there's a vertical analog,
`[[\vglue]]\<glue>'.

The [[\nointerlineskip]] macro suppresses interline glue that would
ordinarily be inserted before the next box in vertical mode; this is a
``one shot'' macro, but [[\offinterlineskip]] is more drastic---it sets
things up so that future interline glue will be present, but zero.
There also are macros for potentially breakable vertical spaces:
[[\smallskip]], [[\medskip]], and [[\bigskip]].

<<Define space macros>>=
\def\thinspace{\kern .16667em }
\def\negthinspace{\kern-.16667em }
\def\enspace{\kern.5em }

\def\enskip{\hskip.5em\relax}
\def\quad{\hskip1em\relax}
\def\qquad{\hskip2em\relax}

\def\smallskip{\vskip\smallskipamount}
\def\medskip{\vskip\medskipamount}
\def\bigskip{\vskip\bigskipamount}

\def\nointerlineskip{\prevdepth-1000\p@@}
\def\offinterlineskip{\baselineskip-1000\p@@
  \lineskip\z@@ \lineskiplimit\maxdimen}

\def\topglue{\nointerlineskip\vglue-\topskip\vglue} % for top of page
\def\vglue{\afterassignment\vgl@@\skip@@=}
\def\vgl@@{\par \dimen@@\prevdepth \hrule height\z@@
  \nobreak\vskip\skip@@ \prevdepth\dimen@@}
\def\hglue{\afterassignment\hgl@@\skip@@=}
\def\hgl@@{\leavevmode \count@@\spacefactor \vrule width\z@@
  \nobreak\hskip\skip@@ \spacefactor\count@@}
@@
The following macros introduce penalty markers
that make breaking less, or more, desirable. The [[\break]], [[\nobreak]],
and [[\allowbreak]] macros are intended for use in any mode; the [[~]]~(tie)
and [[\slash]] (hyphen-like~`/') macros are intended for horizontal mode.
The others are intended only for vertical mode, i.e., between paragraphs,
so they begin with [[\par]].

<<Define space macros>>=
\def~{\penalty\@@M \ } % tie
\def\slash{/\penalty\exhyphenpenalty} % a `/' that acts like a `-'

\def\break{\penalty-\@@M}
\def\nobreak{\penalty \@@M}
\def\allowbreak{\penalty \z@@}

\def\filbreak{\par\vfil\penalty-200\vfilneg}
\def\goodbreak{\par\penalty-500 }
\def\eject{\par\break}
\def\supereject{\par\penalty-\@@MM}

\def\removelastskip{\ifdim\lastskip=\z@@\else\vskip-\lastskip\fi}
\def\smallbreak{\par\ifdim\lastskip<\smallskipamount
  \removelastskip\penalty-50\smallskip\fi}
\def\medbreak{\par\ifdim\lastskip<\medskipamount
  \removelastskip\penalty-100\medskip\fi}
\def\bigbreak{\par\ifdim\lastskip<\bigskipamount
  \removelastskip\penalty-200\bigskip\fi}
@@
[[\line]], [[\leftline]], [[\rightline]], and [[\centerline]]
produce boxes of the full line width, while [[\llap]] and [[\rlap]] make
boxes whose effective width is zero. The [[\underbar]] macro puts its
argument into an hbox with a straight line at a fixed distance under it.

[[\underbar]] uses math mode to do its job, although the
operation is essentially non-mathematical in nature. A few of the other
macros below use math mode in similar ways; thus, \TeX's mathematical
abilities prove to be useful even when no mathematical typesetting
is actually being done. A special control sequence [[\m@@th]] is
used to ``turn off'' [[\mathsurround]] when such constructions are
being performed.)

<<Supply `boxing' macros>>=
\def\line{\hbox to\hsize}
\def\leftline#1{\line{#1\hss}}
\def\rightline#1{\line{\hss#1}}
\def\centerline#1{\line{\hss#1\hss}}

\def\rlap#1{\hbox to\z@@{#1\hss}}
\def\llap#1{\hbox to\z@@{\hss#1}}

\def\m@@th{\mathsurround\z@@}
\def\underbar#1{$\setbox\z@@\hbox{#1}\dp\z@@\z@@
  \m@@th \underline{\box\z@@}$}
@@
A [[\strut]] is implemented here as a rule of width zero.
The `[[\relax]]' in this macro and in others below is necessary in case
|\strut| appears first in an alignment entry, because \TeX\ is in a
somewhat unpredictable mode at such times.

<<Supply strut>>=
\newbox\strutbox
\setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width\z@@}
\def\strut{\relax\ifmmode\copy\strutbox\else\unhcopy\strutbox\fi}
@@
The [[\ialign]] macro provides for alignments when it is necessary to
be sure that [[\tabskip]] is initially zero. The [[\hidewidth]] macro can be
used essentially as [[\hfill]] in alignment entries that are permitted to
``stick out'' of their column. There's also [[\multispan]], which permits
alignment entries to span one or more columns.

<<Provide alignment macros>>=
\def\hidewidth{\hskip\hideskip} % for alignment entries that can stick out
\def\ialign{\everycr{}\tabskip\z@@skip\halign} % initialized \halign
\newcount\mscount
\def\multispan#1{\omit \mscount#1\relax
  \loop\ifnum\mscount>\@@ne \sp@@n\repeat}
\def\sp@@n{\span\omit\advance\mscount\m@@ne}
@@
Now we get to the ``tabbing'' macros.
They keep track of the tab positions by
maintaining boxes full of empty boxes having the specified widths.

The macro [[\+]] has been declared `[[\outer]]' here, so that \TeX\ will
be better able to detect runaway arguments and definitions.
A non-[[\outer]] version, called [[\tabalign]], has also been provided
in case it is necessary to use [[\+]] in some ``inner'' place.
You can use [[\tabalign]] just like [[\+]], except after [[\settabs]].

<<Provide alignment macros>>=
\newif\ifus@@ \newif\if@@cr
\newbox\tabs \newbox\tabsyet \newbox\tabsdone

\def\cleartabs{\global\setbox\tabsyet\null \setbox\tabs\null}
\def\settabs{\setbox\tabs\null \futurelet\next\sett@@b}
\let\+=\relax % in case this file is being read in twice
\def\sett@@b{\ifx\next\+\def\nxt{\afterassignment\s@@tt@@b\let\nxt}%
  \else\let\nxt\s@@tcols\fi \let\next\relax \nxt}
\def\s@@tt@@b{\let\nxt\relax \us@@false\m@@ketabbox}
\def\tabalign{\us@@true\m@@ketabbox} % non-\outer version of \+
\outer\def\+{\tabalign}
\def\s@@tcols#1\columns{\count@@#1\dimen@@\hsize
  \loop\ifnum\count@@>\z@@ \@@nother \repeat}
\def\@@nother{\dimen@@ii\dimen@@ \divide\dimen@@ii\count@@
  \setbox\tabs\hbox{\hbox to\dimen@@ii{}\unhbox\tabs}%
  \advance\dimen@@-\dimen@@ii \advance\count@@\m@@ne}

\def\m@@ketabbox{\begingroup
  \global\setbox\tabsyet\copy\tabs
  \global\setbox\tabsdone\null
  \def\cr{\@@crtrue\crcr\egroup\egroup
    \ifus@@\unvbox\z@@\lastbox\fi\endgroup
    \setbox\tabs\hbox{\unhbox\tabsyet\unhbox\tabsdone}}%
  \setbox\z@@\vbox\bgroup\@@crfalse
    \ialign\bgroup&\t@@bbox##\t@@bb@@x\crcr}

\def\t@@bbox{\setbox\z@@\hbox\bgroup}
\def\t@@bb@@x{\if@@cr\egroup % now \box\z@@ holds the column
  \else\hss\egroup \global\setbox\tabsyet\hbox{\unhbox\tabsyet
      \global\setbox\@@ne\lastbox}% now \box\@@ne holds its size
    \ifvoid\@@ne\global\setbox\@@ne\hbox to\wd\z@@{}%
    \else\setbox\z@@\hbox to\wd\@@ne{\unhbox\z@@}\fi
    \global\setbox\tabsdone\hbox{\box\@@ne\unhbox\tabsdone}\fi
  \box\z@@}
@@
Paragraph shapes of a limited but important kind are provided by
[[\item]], [[\itemitem]], and [[\narrower]].
A macro [[\hang]] causes hanging indentation by the normal
amount of [[\parindent]], after the first line;
thus, the entire paragraph will be indented by the same amount
(unless it began with [[\noindent]]).
[[\textindent]][[{stuff}]] is like [[\indent]],
but it puts the `[[stuff]]' into the indentation,
flush right except for an en~space;
it also removes spaces that might
follow the right brace in `[[{stuff}]]'.

<<Supply various paragraph shapes>>=
\def\hang{\hangindent\parindent}
\def\textindent#1{\indent\llap{#1\enspace}\ignorespaces}
\def\item{\par\hang\textindent}
\def\itemitem{\par\indent \hangindent2\parindent \textindent}
\def\narrower{\advance\leftskip\parindent
  \advance\rightskip\parindent}
@@
The [[\beginsection]] macro is intended to mark the beginning of a
new major subdivision in a document; to use it, you say
`[[\beginsection]]\<section title>' followed by a blank line (or~[[\par]]).
The macro first emits glue and penalties, designed to start a new page
if the present page is nearly full; then it makes a [[\bigskip]] and
puts the section title flush left on a line by itself, in boldface type.
The section title is also displayed on the terminal.
After a [[\smallskip]], with page break prohibited, a [[\noindent]]
command is given; this suppresses indentation in the next paragraph,
i.e., in the first paragraph of the new section.
(However, the next ``paragraph'' will be empty if vertical mode
material immediately follows the [[\beginsection]] command.)

Special statements in a mathematical paper are often called theorems,
lemmas, definitions, axioms, postulates, remarks, corollaries, algorithms,
facts, conjectures, or some such things, and they generally are
given special typographic treatment. The [[\proclaim]] macro
puts the title of the proclamation in boldface, then sets the rest
of the paragraph in slanted type. The paragraph is followed by
something similar to [[\medbreak]], except that the amount of penalty
is different so that page breaks are discouraged:

<<Define sectioning macros>>=
\outer\def\beginsection#1\par{\vskip\z@@ plus.3\vsize\penalty-250
  \vskip\z@@ plus-.3\vsize\bigskip\vskip\parskip
  \message{#1}\leftline{\bf#1}\nobreak\smallskip\noindent}
\outer\def\proclaim #1. #2\par{\medbreak
  \noindent{\bf#1.\enspace}{\sl#2\par}%
  \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi}
@@
Ragged-right setting is initiated by restricting the spaces between
words to have a fixed width, and by putting variable space at the
right of each line.
You should not call [[\raggedright]] until your
text font has already been specified.

It is assumed that the ragged-right material will not be in a variety
of different sizes.  If this assumption is not valid, a different
approach should be used: [[\fontdimen]] parameters 3 and~4 of the
fonts you will be using should be set to zero, by saying, e.g.,
`[[\fontdimen3\tenrm=0pt]]'. These parameters specify the
stretchability and shrinkability of interword spaces. A special macro
[[\ttraggedright]] should be used for ragged-right setting in
[[typewriter type]], since the spaces between words are generally
bigger in that style. (Spaces are already unstretchable and
unshrinkable in font~|cmtt|.)

<<Supply ragged setting>>=
\def\raggedright
   {\rightskip\z@@ plus2em \spaceskip.3333em \xspaceskip.5em\relax}
\def\ttraggedright
   {\tt\rightskip\z@@ plus2em\relax} % for use with \tt only
@@
The [[\nonfrenchspacing]] macro is be used to change the default
sfcodes (set by [[INITEX]]) of punctuation marks.  This macro,
when called, affects spacing after punctuation marks.

<<Establish spacing after punctuation characters>>=
\def\frenchspacing{\sfcode`\.\@@m \sfcode`\?\@@m \sfcode`\!\@@m
  \sfcode`\:\@@m \sfcode`\;\@@m \sfcode`\,\@@m}
\def\nonfrenchspacing{\sfcode`\.3000\sfcode`\?3000\sfcode`\!3000%
  \sfcode`\:2000\sfcode`\;1500\sfcode`\,1250 }
@@
[[\rightarrowfill]] and [[\leftarrowfill]] macros use
[[\cleaders]] with a repeatable box consisting of the middle 10 units
of a minus sign, where one unit
is ${1\over18}\em$. The leaders are preceded and followed by $-$ and~$\to$;
there's enough backspacing to compensate for up to 5~units of extra space,
fore and aft, that |\cleaders| might leave blank. In this way a macro is
obtained such that
[[\hbox to 100pt{\rightarrowfill}]]
yields `\hbox to 100pt{\rightarrowfill}'.

The [[\overbrace]] and [[\underbrace]] macros of plain \TeX\ are
constructed by combining characters with rules. Font [[cmex10]]
contains four symbols $\bracelu\;\braceld\;\bracerd\;\braceru$, each
of which has depth zero and height equal to the thickness of a rule
that joins them properly.
Therefore it's easy to define [[\upbracefill]] and [[\downbracefill]]
macros so that you can obtain, e.g.,
\begindisplay
\vbox{\kern4pt
  \hbox to 100pt{\downbracefill}\hbox to 50pt{\upbracefill}
  }
\enddisplay
by saying `[[\hbox to 100pt{\downbracefill}\hbox to 50pt{\upbracefill}]]'
in vertical mode.

<<Supply various ways to fill space>>=
\def\hrulefill{\leaders\hrule\hfill}
\def\dotfill{\cleaders\hbox{$\m@@th \mkern1.5mu.\mkern1.5mu$}\hfill}
\def\rightarrowfill{$\m@@th\smash-\mkern-6mu%
  \cleaders\hbox{$\mkern-2mu\smash-\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill{$\m@@th\mathord\leftarrow\mkern-6mu%
  \cleaders\hbox{$\mkern-2mu\smash-\mkern-2mu$}\hfill
  \mkern-6mu\smash-$}
\mathchardef\braceld="37A \mathchardef\bracerd="37B
\mathchardef\bracelu="37C \mathchardef\braceru="37D
\def\downbracefill{$\m@@th \setbox\z@@\hbox{$\braceld$}%
  \braceld\leaders\vrule height\ht\z@@ depth\z@@\hfill\braceru
  \bracelu\leaders\vrule height\ht\z@@ depth\z@@\hfill\bracerd$}
\def\upbracefill{$\m@@th \setbox\z@@\hbox{$\braceld$}%
  \bracelu\leaders\vrule height\ht\z@@ depth\z@@\hfill\bracerd
  \braceld\leaders\vrule height\ht\z@@ depth\z@@\hfill\braceru$}
@@
If you want to see all of the hyphens that plain \TeX\ will find in some
random text, you can say `[[\showhyphens{]]\<random text>[[}]]' and the
results will appear on your terminal (and in the log file).
The [[\showhyphens]] macro creates an hbox that is intentionally
underfull.

<<Define [[\showhyphens]] macro>>=
\def\showhyphens#1{\setbox0\vbox{\parfillskip\z@@skip\hsize\maxdimen\tenrm
  \pretolerance\m@@ne\tolerance\m@@ne\hbadness0\showboxdepth0\ #1}}
@@
At the end of a \TeX\ manuscript it's usually best to finish
everything off by typing `[[\bye]]'. The `[[\vfill\supereject]]' gets
\TeX\ to flush out all remaining insertions, with blank space filling
the bottom of incomplete pages, and `[[\end]]' sends the computer into
its endgame routine.

<<Completing the job>>=
\outer\def\bye{\par\vfill\supereject\end}
@@
\section{Macros for math}

Most of this section consists of long listings of special symbols
together with their font locations.

<<Provide macros for math formatting>>=
<<Define math space macros>>
<<Define Greek letters>>
<<Define math symbols>>
<<Encode large operators>>
<<Encode binary operations>>
<<Encode relations>>
<<Supply vertical and diagonal dots>>
<<Supply variable-width math accents>>
<<Supply extensible delimiters>>
<<Provide access to delimiters of various sizes>>
<<Supply common math functions>>
<<Provide [[\cases]] and [[\matrix]] macros>>
<<Provide support to typeset displayed equations>>
@@
\TeX\ does automatic spacing of math formulas so that they look right,
and this is almost true. But occasionally you must give \TeX\ some
help. The number of possible math formulas is vast, and \TeX's spacing
rules are rather simple, so it is natural that exceptions should
arise.  The basic elements of space that \TeX\ puts into formulas are
called: thin spaces, medium spaces, thick spaces.
The normal space between words of a paragraph is approximately equal
to two thin spaces.
\bigskip
\noindent You can add your own spacing whenever you want to,
by using the control sequences
$$\halign{\indent#\hfil&\quad#\hfil\cr
|\,|&thin space \ (normally 1/6 of a quad);\cr
|\>|&medium space \ (normally 2/9 of a quad);\cr
|\;|&thick space \ (normally 5/18 of a quad);\cr
|\!|&negative thin space \ (normally $-1/6$ of a quad).\cr}$$

<<Define math space macros>>=
\def\,{\mskip\thinmuskip}
\def\>{\mskip\medmuskip}
\def\;{\mskip\thickmuskip}
\def\!{\mskip-\thinmuskip}
@@
The next job is to define Greek letters and other symbols of type Ord.
Uppercase Greek letters are assigned hexadecimal codes of the form \hex{7xxx},
so that they will change families when [[\fam]] changes.

[[\mathchardef]] defines a control sequence to be a synonym for a math
character (check [[\mathchar]] for the meaning of hex number).

<<Define Greek letters>>=
\mathchardef\alpha="010B
\mathchardef\beta="010C
\mathchardef\gamma="010D
\mathchardef\delta="010E
\mathchardef\epsilon="010F
\mathchardef\zeta="0110
\mathchardef\eta="0111
\mathchardef\theta="0112
\mathchardef\iota="0113
\mathchardef\kappa="0114
\mathchardef\lambda="0115
\mathchardef\mu="0116
\mathchardef\nu="0117
\mathchardef\xi="0118
\mathchardef\pi="0119
\mathchardef\rho="011A
\mathchardef\sigma="011B
\mathchardef\tau="011C
\mathchardef\upsilon="011D
\mathchardef\phi="011E
\mathchardef\chi="011F
\mathchardef\psi="0120
\mathchardef\omega="0121
\mathchardef\varepsilon="0122
\mathchardef\vartheta="0123
\mathchardef\varpi="0124
\mathchardef\varrho="0125
\mathchardef\varsigma="0126
\mathchardef\varphi="0127
\mathchardef\Gamma="7000
\mathchardef\Delta="7001
\mathchardef\Theta="7002
\mathchardef\Lambda="7003
\mathchardef\Xi="7004
\mathchardef\Pi="7005
\mathchardef\Sigma="7006
\mathchardef\Upsilon="7007
\mathchardef\Phi="7008
\mathchardef\Psi="7009
\mathchardef\Omega="700A
@@
<<Define math symbols>>=
\mathchardef\aleph="0240
\def\hbar{{\mathchar'26\mkern-9muh}}
\mathchardef\imath="017B
\mathchardef\jmath="017C
\mathchardef\ell="0160
\mathchardef\wp="017D
\mathchardef\Re="023C
\mathchardef\Im="023D
\mathchardef\partial="0140
\mathchardef\infty="0231
\mathchardef\prime="0230
\mathchardef\emptyset="023B
\mathchardef\nabla="0272
\def\surd{{\mathchar"1270}}
\mathchardef\top="023E
\mathchardef\bot="023F
\def\angle{{\vbox{\ialign{$\m@@th\scriptstyle##$\crcr
      \not\mathrel{\mkern14mu}\crcr
      \noalign{\nointerlineskip}
      \mkern2.5mu\leaders\hrule height.34pt\hfill\mkern2.5mu\crcr}}}}
\mathchardef\triangle="0234
\mathchardef\forall="0238
\mathchardef\exists="0239
\mathchardef\neg="023A \let\lnot=\neg
\mathchardef\flat="015B
\mathchardef\natural="015C
\mathchardef\sharp="015D
\mathchardef\clubsuit="027C
\mathchardef\diamondsuit="027D
\mathchardef\heartsuit="027E
\mathchardef\spadesuit="027F
@@
Integral signs get special treatment so that their limits won't be set
above and below.

<<Encode large operators>>=
\mathchardef\coprod="1360
\mathchardef\bigvee="1357
\mathchardef\bigwedge="1356
\mathchardef\biguplus="1355
\mathchardef\bigcap="1354
\mathchardef\bigcup="1353
\mathchardef\intop="1352 \def\int{\intop\nolimits}
\mathchardef\prod="1351
\mathchardef\sum="1350
\mathchardef\bigotimes="134E
\mathchardef\bigoplus="134C
\mathchardef\bigodot="134A
\mathchardef\ointop="1348 \def\oint{\ointop\nolimits}
\mathchardef\bigsqcup="1346
\mathchardef\smallint="1273
@@
<<Encode binary operations>>=
\mathchardef\triangleleft="212F
\mathchardef\triangleright="212E
\mathchardef\bigtriangleup="2234
\mathchardef\bigtriangledown="2235
\mathchardef\wedge="225E \let\land=\wedge
\mathchardef\vee="225F \let\lor=\vee
\mathchardef\cap="225C
\mathchardef\cup="225B
\mathchardef\ddagger="227A
\mathchardef\dagger="2279
\mathchardef\sqcap="2275
\mathchardef\sqcup="2274
\mathchardef\uplus="225D
\mathchardef\amalg="2271
\mathchardef\diamond="2205
\mathchardef\bullet="220F
\mathchardef\wr="226F
\mathchardef\div="2204
\mathchardef\odot="220C
\mathchardef\oslash="220B
\mathchardef\otimes="220A
\mathchardef\ominus="2209
\mathchardef\oplus="2208
\mathchardef\mp="2207
\mathchardef\pm="2206
\mathchardef\circ="220E
\mathchardef\bigcirc="220D
\mathchardef\setminus="226E % for set difference A\setminus B
\mathchardef\cdot="2201
\mathchardef\ast="2203
\mathchardef\times="2202
\mathchardef\star="213F
@@
Relations are also fairly straightforward, except for the ones that are
constructed from other characters. The [[\mapstochar]] is a character
`$\mapstochar\mskip5mu$' of width zero that is quite useless by itself,
but it combines with right arrows to make [[\mapsto]] `$\mapsto$' and
[[\longmapsto]] `$\longmapsto$'. Similarly, [[\not]] is a relation character of
width zero that puts a slash over the character that follows.  When two
relations are adjacent in a math formula, \TeX\ puts no space between them.

<<Encode relations>>=
\mathchardef\propto="322F
\mathchardef\sqsubseteq="3276
\mathchardef\sqsupseteq="3277
\mathchardef\parallel="326B
\mathchardef\mid="326A
\mathchardef\dashv="3261
\mathchardef\vdash="3260
\mathchardef\nearrow="3225
\mathchardef\searrow="3226
\mathchardef\nwarrow="322D
\mathchardef\swarrow="322E
\mathchardef\Leftrightarrow="322C
\mathchardef\Leftarrow="3228
\mathchardef\Rightarrow="3229
\def\neq{\not=} \let\ne=\neq
\mathchardef\leq="3214 \let\le=\leq
\mathchardef\geq="3215 \let\ge=\geq
\mathchardef\succ="321F
\mathchardef\prec="321E
\mathchardef\approx="3219
\mathchardef\succeq="3217
\mathchardef\preceq="3216
\mathchardef\supset="321B
\mathchardef\subset="321A
\mathchardef\supseteq="3213
\mathchardef\subseteq="3212
\mathchardef\in="3232
\mathchardef\ni="3233 \let\owns=\ni
\mathchardef\gg="321D
\mathchardef\ll="321C
\mathchardef\not="3236
\mathchardef\leftrightarrow="3224
\mathchardef\leftarrow="3220 \let\gets=\leftarrow
\mathchardef\rightarrow="3221 \let\to=\rightarrow
\mathchardef\mapstochar="3237 \def\mapsto{\mapstochar\rightarrow}
\mathchardef\sim="3218
\mathchardef\simeq="3227
\mathchardef\perp="323F
\mathchardef\equiv="3211
\mathchardef\asymp="3210
\mathchardef\smile="315E
\mathchardef\frown="315F
\mathchardef\leftharpoonup="3128
\mathchardef\leftharpoondown="3129
\mathchardef\rightharpoonup="312A
\mathchardef\rightharpoondown="312B

\def\joinrel{\mathrel{\mkern-3mu}}
\def\relbar{\mathrel{\smash-}} % \smash, because - has the same height as +
\def\Relbar{\mathrel=}
\mathchardef\lhook="312C \def\hookrightarrow{\lhook\joinrel\rightarrow}
\mathchardef\rhook="312D \def\hookleftarrow{\leftarrow\joinrel\rhook}
\def\bowtie{\mathrel\triangleright\joinrel\mathrel\triangleleft}
\def\models{\mathrel|\joinrel=}
\def\Longrightarrow{\Relbar\joinrel\Rightarrow}
\def\longrightarrow{\relbar\joinrel\rightarrow}
\def\longleftarrow{\leftarrow\joinrel\relbar}
\def\Longleftarrow{\Leftarrow\joinrel\Relbar}
\def\longmapsto{\mapstochar\longrightarrow}
\def\longleftrightarrow{\leftarrow\joinrel\rightarrow}
\def\Longleftrightarrow{\Leftarrow\joinrel\Rightarrow}
\def\iff{\;\Longleftrightarrow\;}
@@
After defining characters [[\ldotp]] and [[\cdotp]] that act as math
punctuation, [[\ldots]] and [[\cdots]] macros are defined that
give the proper spacing in most circumstances. Vertical and diagonal
dots ([[\vdots]] and [[\ddots]]) are also provided here:

<<Supply vertical and diagonal dots>>=
\mathchardef\ldotp="613A % ldot as a punctuation mark
\mathchardef\cdotp="6201 % cdot as a punctuation mark
\mathchardef\colon="603A % colon as a punctuation mark
\def\ldots{\mathinner{\ldotp\ldotp\ldotp}}
\def\cdots{\mathinner{\cdotp\cdotp\cdotp}}
\def\vdots{\vbox{\baselineskip4\p@@ \lineskiplimit\z@@
    \kern6\p@@\hbox{.}\hbox{.}\hbox{.}}}
\def\ddots{\mathinner{\mkern1mu\raise7\p@@\vbox{\kern7\p@@\hbox{.}}\mkern2mu
    \raise4\p@@\hbox{.}\mkern2mu\raise\p@@\hbox{.}\mkern1mu}}
@@
Most of the math accents are handled entirely by the [[\mathaccent]]
primitive, but a few of the variable-width ones are constructed the
hard way.

<<Supply variable-width math accents>>=
\def\acute{\mathaccent"7013 }
\def\grave{\mathaccent"7012 }
\def\ddot{\mathaccent"707F }
\def\tilde{\mathaccent"707E }
\def\bar{\mathaccent"7016 }
\def\breve{\mathaccent"7015 }
\def\check{\mathaccent"7014 }
\def\hat{\mathaccent"705E }
\def\vec{\mathaccent"017E }
\def\dot{\mathaccent"705F }
\def\widetilde{\mathaccent"0365 }
\def\widehat{\mathaccent"0362 }
\def\overrightarrow#1{\vbox{\m@@th\ialign{##\crcr
      \rightarrowfill\crcr\noalign{\kern-\p@@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}
\def\overleftarrow#1{\vbox{\m@@th\ialign{##\crcr
      \leftarrowfill\crcr\noalign{\kern-\p@@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}
\def\overbrace#1{\mathop{\vbox{\m@@th\ialign{##\crcr\noalign{\kern3\p@@}
      \downbracefill\crcr\noalign{\kern3\p@@\nointerlineskip}
      $\hfil\displaystyle{#1}\hfil$\crcr}}}\limits}
\def\underbrace#1{\mathop{\vtop{\m@@th\ialign{##\crcr
      $\hfil\displaystyle{#1}\hfil$\crcr\noalign{\kern3\p@@\nointerlineskip}
      \upbracefill\crcr\noalign{\kern3\p@@}}}}\limits}
\def\skew#1#2#3{{\muskip\z@@#1mu\divide\muskip\z@@\tw@@ \mkern\muskip\z@@
    #2{\mkern-\muskip\z@@{#3}\mkern\muskip\z@@}\mkern-\muskip\z@@}{}}
@@
Now we come to 24 delimiters that can change their size.
These are denoted explicitly by a \<27-bit number>.

If we denote 27-bit number by [["cqrstuv]], then
delimiter codes are assigned by
\medskip
[[\delimiter "cqrstuv]],\quad where
\medskip
\itemitem{[[c]]}   --- the class
\itemitem{[[q]]}   --- the font family number of
\itemitem{[[rs]]}  --- the position of the the small variant of the delimiter
\itemitem{[[t]]}   --- the font family number of
\itemitem{[[uv]]}  --- the position of the the large variant of the delimiter
\medskip
After [[\left]] and [[\right]] commands the class digit is ignored.
When \TeX\ is not looking for a delimiter the righmost three
digits [[tuv]] are ignored, and the remaining four [[cqrs]]
are treated as a [[\mathchar]].

<<Supply extensible delimiters>>=
\def\lmoustache{\delimiter"437A340 } % top from (, bottom from )
\def\rmoustache{\delimiter"537B341 } % top from ), bottom from (
\def\lgroup{\delimiter"462833A } % extensible ( with sharper tips
\def\rgroup{\delimiter"562933B } % extensible ) with sharper tips
\def\arrowvert{\delimiter"26A33C } % arrow without arrowheads
\def\Arrowvert{\delimiter"26B33D } % double arrow without arrowheads
\def\bracevert{\delimiter"77C33E } % the vertical bar that extends braces
\def\Vert{\delimiter"26B30D } \let\|=\Vert
\def\vert{\delimiter"26A30C }
\def\uparrow{\delimiter"3222378 }
\def\downarrow{\delimiter"3223379 }
\def\updownarrow{\delimiter"326C33F }
\def\Uparrow{\delimiter"322A37E }
\def\Downarrow{\delimiter"322B37F }
\def\Updownarrow{\delimiter"326D377 }
\def\backslash{\delimiter"26E30F } % for double coset G\backslash H
\def\rangle{\delimiter"526930B }
\def\langle{\delimiter"426830A }
\def\rbrace{\delimiter"5267309 } \let\}=\rbrace
\def\lbrace{\delimiter"4266308 } \let\{=\lbrace
\def\rceil{\delimiter"5265307 }
\def\lceil{\delimiter"4264306 }
\def\rfloor{\delimiter"5263305 }
\def\lfloor{\delimiter"4262304 }
@@
In the [[plain.tex]] format and in the Computer Modern math fonts
there is only one radical---the square root.  The meaning of
[[\radical]] is analogous to the [[\delimiter]] commands. Only the
class number is dropped.  Joining the radical character and the
horizontal rule is done by leeting the radical character have a large
depth, and the height which is equal to the rule thickness. The rule
is placed on the baseline and the radical character is placed
below. Then the whole is centered around math axis.

<<Supply extensible delimiters>>=
\def\choose{\atopwithdelims()}
\def\brack{\atopwithdelims[]}
\def\brace{\atopwithdelims\{\}}

\def\sqrt{\radical"270370 }
@@
These macros depend on actual sizes of delimiters.

<<Provide access to delimiters of various sizes>>=
\def\bigl{\mathopen\big}
\def\bigm{\mathrel\big}
\def\bigr{\mathclose\big}
\def\Bigl{\mathopen\Big}
\def\Bigm{\mathrel\Big}
\def\Bigr{\mathclose\Big}
\def\biggl{\mathopen\bigg}
\def\biggm{\mathrel\bigg}
\def\biggr{\mathclose\bigg}
\def\Biggl{\mathopen\Bigg}
\def\Biggm{\mathrel\Bigg}
\def\Biggr{\mathclose\Bigg}
\def\big#1{{\hbox{$\left#1\vbox to8.5\p@@{}\right.\n@@space$}}}
\def\Big#1{{\hbox{$\left#1\vbox to11.5\p@@{}\right.\n@@space$}}}
\def\bigg#1{{\hbox{$\left#1\vbox to14.5\p@@{}\right.\n@@space$}}}
\def\Bigg#1{{\hbox{$\left#1\vbox to17.5\p@@{}\right.\n@@space$}}}
\def\n@@space{\nulldelimiterspace\z@@ \m@@th}
@@
The [[\mathpalette]] operation constructs a formula in all four
styles; it is applied here in the implementation of [[\phantom]],
[[\smash]], [[\root]], and other operations. (Actually [[\phantom]]
and [[\smash]] are not perfect: They assume that the current style is
uncramped.)

These definitions illustrate how other built-up
symbol combinations could be defined to work in all four styles.

The control sequences [[\sp]] and [[\sb]]
are provided for people who can't easily type [[^]] and [[_]];
a ``discretionary times sign'' [[\*]] is defined.

<<Provide macros for math formatting>>=
\def\mathpalette#1#2{\mathchoice{#1\displaystyle{#2}}%
  {#1\textstyle{#2}}{#1\scriptstyle{#2}}{#1\scriptscriptstyle{#2}}}
\newbox\rootbox
\def\root#1\of{\setbox\rootbox\hbox{$\m@@th\scriptscriptstyle{#1}$}
  \mathpalette\r@@@@t}
\def\r@@@@t#1#2{\setbox\z@@\hbox{$\m@@th#1\sqrt{#2}$}
  \dimen@@\ht\z@@ \advance\dimen@@-\dp\z@@
  \mkern5mu\raise.6\dimen@@\copy\rootbox \mkern-10mu \box\z@@}
\newif\ifv@@ \newif\ifh@@
\def\vphantom{\v@@true\h@@false\ph@@nt}
\def\hphantom{\v@@false\h@@true\ph@@nt}
\def\phantom{\v@@true\h@@true\ph@@nt}
\def\ph@@nt{\ifmmode\def\next{\mathpalette\mathph@@nt}%
  \else\let\next\makeph@@nt\fi\next}
\def\makeph@@nt#1{\setbox\z@@\hbox{#1}\finph@@nt}
\def\mathph@@nt#1#2{\setbox\z@@\hbox{$\m@@th#1{#2}$}\finph@@nt}
\def\finph@@nt{\setbox\tw@@\null
  \ifv@@ \ht\tw@@\ht\z@@ \dp\tw@@\dp\z@@\fi
  \ifh@@ \wd\tw@@\wd\z@@\fi \box\tw@@}
\def\mathstrut{\vphantom(}
\def\smash{\relax % \relax, in case this comes first in \halign
  \ifmmode\def\next{\mathpalette\mathsm@@sh}\else\let\next\makesm@@sh
  \fi\next}
\def\makesm@@sh#1{\setbox\z@@\hbox{#1}\finsm@@sh}
\def\mathsm@@sh#1#2{\setbox\z@@\hbox{$\m@@th#1{#2}$}\finsm@@sh}
\def\finsm@@sh{\ht\z@@\z@@ \dp\z@@\z@@ \box\z@@}

\def\cong{\mathrel{\mathpalette\@@vereq\sim}} % congruence sign
\def\@@vereq#1#2{\lower.5\p@@\vbox{\lineskiplimit\maxdimen\lineskip-.5\p@@
    \ialign{$\m@@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}
\def\notin{\mathrel{\mathpalette\c@@ncel\in}}
\def\c@@ncel#1#2{\m@@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}
\def\rightleftharpoons{\mathrel{\mathpalette\rlh@@{}}}
\def\rlh@@#1{\vcenter{\m@@th\hbox{\ooalign{\raise2pt
          \hbox{$#1\rightharpoonup$}\crcr
        $#1\leftharpoondown$}}}}
\def\buildrel#1\over#2{\mathrel{\mathop{\kern\z@@#2}\limits^{#1}}}
\def\doteq{\buildrel\textstyle.\over=}

\let\sp=^ \let\sb=_
\def\*{\discretionary{\thinspace\the\textfont2\char2}{}{}}
@@
The 32 common functions whose names generally appear in roman letters.

<<Supply common math functions>>=
\def\log{\mathop{\rm log}\nolimits}
\def\lg{\mathop{\rm lg}\nolimits}
\def\ln{\mathop{\rm ln}\nolimits}
\def\lim{\mathop{\rm lim}}
\def\limsup{\mathop{\rm lim\,sup}}
\def\liminf{\mathop{\rm lim\,inf}}
\def\sin{\mathop{\rm sin}\nolimits}
\def\arcsin{\mathop{\rm arcsin}\nolimits}
\def\sinh{\mathop{\rm sinh}\nolimits}
\def\cos{\mathop{\rm cos}\nolimits}
\def\arccos{\mathop{\rm arccos}\nolimits}
\def\cosh{\mathop{\rm cosh}\nolimits}
\def\tan{\mathop{\rm tan}\nolimits}
\def\arctan{\mathop{\rm arctan}\nolimits}
\def\tanh{\mathop{\rm tanh}\nolimits}
\def\cot{\mathop{\rm cot}\nolimits}
\def\coth{\mathop{\rm coth}\nolimits}
\def\sec{\mathop{\rm sec}\nolimits}
\def\csc{\mathop{\rm csc}\nolimits}
\def\max{\mathop{\rm max}}
\def\min{\mathop{\rm min}}
\def\sup{\mathop{\rm sup}}
\def\inf{\mathop{\rm inf}}
\def\arg{\mathop{\rm arg}\nolimits}
\def\ker{\mathop{\rm ker}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\def\hom{\mathop{\rm hom}\nolimits}
\def\det{\mathop{\rm det}}
\def\exp{\mathop{\rm exp}\nolimits}
\def\Pr{\mathop{\rm Pr}}
\def\gcd{\mathop{\rm gcd}}
\def\deg{\mathop{\rm deg}\nolimits}

\def\bmod{\nonscript\mskip-\medmuskip\mkern5mu
  \mathbin{\rm mod}\penalty900\mkern5mu\nonscript\mskip-\medmuskip}
\def\pmod#1{\allowbreak\mkern18mu({\rm mod}\,\,#1)}
@@
The definition of [[\matrix]] goes to some pains to ensure that two
$n$-rowed matrices will have the same height and the same depth, unless
at least one of their rows is unusually big. The definition of
[[\bordermatrix]] is even more complicated, but it seems to work
reasonably well; it uses a constant [[\p@@renwd]] that represents the
width of a big extensible left parenthesis.

<<Provide [[\cases]] and [[\matrix]] macros>>=
\def\cases#1{\left\{\,\vcenter{\normalbaselines\m@@th
    \ialign{$##\hfil$&\quad##\hfil\crcr#1\crcr}}\right.}
\def\matrix#1{\null\,\vcenter{\normalbaselines\m@@th
    \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr
      \mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}
\def\pmatrix#1{\left(\matrix{#1}\right)}
\newdimen\p@@renwd
\setbox0=\hbox{\tenex B} \p@@renwd=\wd0 % width of the big left (
\def\bordermatrix#1{\begingroup \m@@th
  \setbox\z@@\vbox{\def\cr{\crcr\noalign{\kern2\p@@\global\let\cr\endline}}%
    \ialign{$##$\hfil\kern2\p@@\kern\p@@renwd&\thinspace\hfil$##$\hfil
      &&\quad\hfil$##$\hfil\crcr
      \omit\strut\hfil\crcr\noalign{\kern-\baselineskip}%
      #1\crcr\omit\strut\cr}}%
  \setbox\tw@@\vbox{\unvcopy\z@@\global\setbox\@@ne\lastbox}%
  \setbox\tw@@\hbox{\unhbox\@@ne\unskip\global\setbox\@@ne\lastbox}%
  \setbox\tw@@\hbox{$\kern\wd\@@ne\kern-\p@@renwd\left(\kern-\wd\@@ne
    \global\setbox\@@ne\vbox{\box\@@ne\kern2\p@@}%
    \vcenter{\kern-\ht\@@ne\unvbox\z@@\kern-\baselineskip}\,\right)$}%
  \null\;\vbox{\kern\ht\@@ne\box\tw@@}\endgroup}
@@
The value of [[\lineskiplimit]] is assumed to be [[\normallineskiplimit]]
plus the accumulated amount of ``opening up.'' Thus, the [[\vskip]]
instructions in [[\displ@@y]] will compensate for the fact that the
first baseline of an alignment is separated by an opened-up baselineskip
from the last line preceding the display.

<<Provide support to typeset displayed equations>>=
\def\openup{\afterassignment\@@penup\dimen@@=}
\def\@@penup{\advance\lineskip\dimen@@
  \advance\baselineskip\dimen@@
  \advance\lineskiplimit\dimen@@}
\def\eqalign#1{\null\,\vcenter{\openup\jot\m@@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\newif\ifdt@@p
\def\displ@@y{\global\dt@@ptrue\openup\jot\m@@th
  \everycr{\noalign{\ifdt@@p \global\dt@@pfalse \ifdim\prevdepth>-1000\p@@
      \vskip-\lineskiplimit \vskip\normallineskiplimit \fi
      \else \penalty\interdisplaylinepenalty \fi}}}
\def\@@lign{\tabskip\z@@skip\everycr{}} % restore inside \displ@@y
\def\displaylines#1{\displ@@y \tabskip\z@@skip
  \halign{\hbox to\displaywidth{$\@@lign\hfil\displaystyle##\hfil$}\crcr
    #1\crcr}}
\def\eqalignno#1{\displ@@y \tabskip\centering
  \halign to\displaywidth{\hfil$\@@lign\displaystyle{##}$\tabskip\z@@skip
    &$\@@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\llap{$\@@lign##$}\tabskip\z@@skip\crcr
    #1\crcr}}
\def\leqalignno#1{\displ@@y \tabskip\centering
  \halign to\displaywidth{\hfil$\@@lign\displaystyle{##}$\tabskip\z@@skip
    &$\@@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\kern-\displaywidth\rlap{$\@@lign##$}\tabskip\displaywidth\crcr
    #1\crcr}}
@@
Below we have an interesting set of
macros that convert [[f''']] into [[f^{\prime\prime\prime}]].

<<Supply common math functions>>=
{\catcode`\'=\active \gdef'{^\bgroup\prim@@s}}
\def\prim@@s{\prime\futurelet\next\pr@@m@@s}
\def\pr@@m@@s{\ifx'\next\let\nxt\pr@@@@@@s \else\ifx^\next\let\nxt\pr@@@@@@t
  \else\let\nxt\egroup\fi\fi \nxt}
\def\pr@@@@@@s#1{\prim@@s} \def\pr@@@@@@t#1#2{#2\egroup}
{\catcode`\^^Z=\active \gdef^^Z{\not=}} % ^^Z is like \ne in math
{\catcode`\_=\active \global\let_=\_} % _ in math is either subscript or \_
@@
\section{Macros for output}

<<Prepare page for output>>=
<<Supply headers and footers>>
<<Supply footnotes>>
<<Supply floating insertions>>
<<Supply raggedbottom setting>>
<<Set up the output routine>>
@@
The [[\makeheadline]] macro constructs a vbox of height and depth
zero. The magic constant $-22.5\pt$ is equal to
$$
\hbox{[[\topskip]]}
       -\hbox{\<height of strut>}-2\,\hbox{[[\baselineskip]]}
$$
i.e., $10\pt-8.5\pt-24\pt$
(assuming default values of [[\topskip]] and the height of the strut);
this places the reference point of the headline exactly $24\pt$ above the
reference point of the top line on the page, unless the headline or
the top line are excessively large.

The [[\advancepageno]] macro normally advances [[\pageno]]
by~$+1$; but if [[\pageno]] is negative (for roman numerals), the advance is
by~$-1$. The new value of [[\pageno]] will be appropriate for the next time
the output routine is called into action.

<<Supply headers and footers>>=
\countdef\pageno=0 \pageno=1 % first page is number 1
\newtoks\headline \headline={\hfil} % headline is normally blank
\newtoks\footline \footline={\hss\tenrm\folio\hss}
% footline is normally a centered page number in font \tenrm
%
\def\makeheadline{\vbox to\z@@{\vskip-22.5\p@@
  \line{\vbox to8.5\p@@{}\the\headline}\vss}\nointerlineskip}
\def\makefootline{\baselineskip24\p@@\line{\the\footline}}
%
\def\folio{\ifnum\pageno<\z@@ \romannumeral-\pageno \else\number\pageno \fi}
\def\nopagenumbers{\footline{\hfil}} % blank out the footline
\def\advancepageno{\ifnum\pageno<\z@@ \global\advance\pageno\m@@ne
  \else\global\advance\pageno\@@ne \fi} % increase |pageno|
@@
Ragged-bottom setting is achieved by inserting
infinite glue, which overpowers the stretchability of [[\topskip]].
This macros assume that [[\topskip = 10pt]]

<<Supply raggedbottom setting>>=
\newif\ifr@@ggedbottom
\def\raggedbottom{\topskip 10\p@@ plus60\p@@ \r@@ggedbottomtrue}
\def\normalbottom{\topskip 10\p@@ \r@@ggedbottomfalse} % undoes \raggedbottom
@@
There are 255 classes of insertions, [[\insert0]] to [[\insert254]], and
they are tied to other registers of the same number. For example,
[[\insert100]] is connected with [[\count100]], [[\dimen100]],
[[\skip100]], and [[\box100]].

\def\n{\thinspace$n$}
For our purposes let's consider a particular class of insertions
called class $n$; we will then be dealing with \TeX's primitive command
\begindisplay
[[\insert]]\n[[{]]\<vertical mode material>[[}]]
\enddisplay
which puts an insertion
item into a horizontal or vertical list. For this class of insertions
\begindisplay
[[\box]]\n\qquad is where the material appears when a page is output;\cr
[[\count]]\n\qquad is the magnification factor for page breaking;\cr
[[\dimen]]\n\qquad is the maximum insertion size per page;\cr
[[\skip]]\n\qquad is the extra space to allocate on a page.\cr
\enddisplay
For example, material inserted with [[\insert100]] will
eventually appear in [[\box100]].
@@
[[\footnote]] macro depends on the value of [[\bigskipamount]], and
[[\parindent]].
Because the value of [[\vsize]] ($8.9\in$)
is greater than [[\dimen\footins]] ($8\in$),
footnotes never fill up the whole page.

<<Supply footnotes>>=
\newinsert\footins
\def\footnote#1{\let\@@sf\empty % parameter #2 (the text) is read later
  \ifhmode\edef\@@sf{\spacefactor\the\spacefactor}\/\fi
  #1\@@sf\vfootnote{#1}}
\def\vfootnote#1{\insert\footins\bgroup
  \interlinepenalty\interfootnotelinepenalty
  \splittopskip\ht\strutbox % top baseline for broken footnotes
  \splitmaxdepth\dp\strutbox \floatingpenalty\@@MM
  \leftskip\z@@skip \rightskip\z@@skip \spaceskip\z@@skip \xspaceskip\z@@skip
  \textindent{#1}\footstrut\futurelet\next\fo@@t}
\def\fo@@t{\ifcat\bgroup\noexpand\next \let\next\f@@@@t
  \else\let\next\f@@t\fi \next}
\def\f@@@@t{\bgroup\aftergroup\@@foot\let\next}
\def\f@@t#1{#1\@@foot}
\def\@@foot{\strut\egroup}
\def\footstrut{\vbox to\splittopskip{}}

\skip\footins=\bigskipamount % space added when footnote is present
\count\footins=1000 % footnote magnification factor (1 to 1)
\dimen\footins=8in % maximum footnotes per page

\def\footnoterule{\kern-3\p@@
  \hrule width 2truein \kern 2.6\p@@} % the \hrule is .4pt high
@@
Here the constant [[12\p@@ = 12pt]] is hard coded into the format.

<<Supply floating insertions>>=
\newinsert\topins
\newif\ifp@@ge \newif\if@@mid
\def\topinsert{\@@midfalse\p@@gefalse\@@ins}
\def\midinsert{\@@midtrue\@@ins}
\def\pageinsert{\@@midfalse\p@@getrue\@@ins}
\skip\topins=\z@@skip % no space added when a topinsert is present
\count\topins=1000 % magnification factor (1 to 1)
\dimen\topins=\maxdimen % no limit per page
\def\@@ins{\par\begingroup\setbox\z@@\vbox\bgroup} % start a \vbox
\def\endinsert{\egroup % finish the \vbox
  \if@@mid \dimen@@\ht\z@@ \advance\dimen@@\dp\z@@ \advance\dimen@@12\p@@
    \advance\dimen@@\pagetotal \advance\dimen@@-\pageshrink
    \ifdim\dimen@@>\pagegoal\@@midfalse\p@@gefalse\fi\fi
  \if@@mid \bigskip\box\z@@\bigbreak
  \else\insert\topins{\penalty100 % floating insertion
    \splittopskip\z@@skip
    \splitmaxdepth\maxdimen \floatingpenalty\z@@
    \ifp@@ge \dimen@@\dp\z@@
    \vbox to\vsize{\unvbox\z@@\kern-\dimen@@}% depth is zero
    \else \box\z@@\nobreak\bigskip\fi}\fi\endgroup}
@@
The value of [[\boxmaxdepth]] is set to [[\maxdepth]] so that the vbox
will be constructed under the assumptions that \TeX's page builder has
used to set up [[\box255]].

The [[\pagecontents]] macro produces a vertical list for
everything that belongs on the main body of the page, namely the contents
of [[\box255]] together with illustrations (inserted at the top)
and footnotes (inserted at the bottom).
[[\topins]] and [[\footins]] are the insertion class numbers for
the two kinds of insertions used in plain \TeX; if more classes of
insertions are added, [[\pagecontents]] should be changed accordingly.
Notice that the boxes are unboxed so that the glue coming from
insertions can help out the glue on the main page. The [[\footnoterule]]
macro places a dividing line between the page and its
footnotes; it makes a net contribution of $0\pt$ to the height of
the vertical list.

The [[\dosupereject]] macro is designed to clear out any insertions
that have been held over, whether they are illustrations or footnotes
or both. The negative [[\kern]] here cancels out the natural space of
the [[\topskip]] glue that goes above the empty [[\line]]; that empty
line box prevents the [[\vfill]] from disappearing into a page break.
The vertical list that results from [[\dosupereject]] is placed on
\TeX's list of things to put out next, just after the straggling
insertions have been reconsidered. Hence another super-eject will
occur, and the process will continue until no insertions remain.

<<Set up the output routine>>=
\output{\plainoutput}
\def\plainoutput{\shipout\vbox{\makeheadline\pagebody\makefootline}%
  \advancepageno
  \ifnum\outputpenalty>-\@@MM \else\dosupereject\fi}

\def\pagebody{\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}}

\def\dosupereject{\ifnum\insertpenalties>\z@@ % something is being held over
  \line{}\kern-\topskip\nobreak\vfill\supereject\fi}

\def\pagecontents{\ifvoid\topins\else\unvbox\topins\fi
  \dimen@@=\dp\@@cclv \unvbox\@@cclv % open up \box255
  \ifvoid\footins\else % footnote info is present
    \vskip\skip\footins
    \footnoterule
    \unvbox\footins\fi
  \ifr@@ggedbottom \kern-\dimen@@ \vfil \fi}
@@
\section{Hyphenation}

A discretionary break consists of three sequences of characters
called the {\sl pre-break}, {\sl post-break}, and {\sl no-break\/}
texts. The idea is that if a line break occurs here, the pre-break text
will appear at the end of the current line and the post-break text will
occur at the beginning of the next line; but if no break occurs, the
no-break text will appear in the current line.
The discretionary are specified by writing
\begindisplay
|\discretionary{|\<pre-break text>|}{|\<post-break text>|}{|\<no-break text>|}|
\enddisplay
where the three texts consist entirely of characters, boxes, and kerns.
If a word contains discretionary breaks \TeX\ will not hyphenate it.

Hyphenation exceptions are specified with the statements like
\begindisplay
[[\hyphenation{gal-axy iso-peri-met-ric}]]
\enddisplay
which gives to \TeX\ locations where these words may be hyphenated.

The default values of [[\hyphenchar]] is [[`-]].

Sometimes the typewriter fonts are given
[[\hyphenchar]]\<font name>[[=-1]] which value inhibits hyphenation.

<<Read hyphenation patterns>>=
\lefthyphenmin=2 \righthyphenmin=3 % disallow x- or -xx breaks
\input hyphen
@@
\section{Initalization}

<<Initialize the layout>>=
\normalbaselines\rm % select roman font
\nonfrenchspacing % punctuation affects the spacing
@@
\section{Programming support}

<<Provide programming constructs>>=
<<Supply loops and conditionals>>
<<Define [[\tracingall]] macro>>
@@
The [[\loop...\repeat]] macro provides for iterative operations.
In this macro and several others, the control sequence `[[\next]]' is
given a temporary value that is not going to be needed later; thus,
[[\next]] acts like a ``scratch control sequence.''

The macro [[\newif]] to be used for definitions of new conditionals.
For example, [[\newif\iffoo]] creates [[\footrue]], [[\foofalse]] to
go with [[\iffoo]].

<<Supply loops and conditionals>>=
\def\loop#1\repeat{\def\body{#1}\iterate}
\def\iterate{\body \let\next\iterate \else\let\next\relax\fi \next}
\let\repeat=\fi % this makes \loop...\if...\repeat skippable

\outer\def\newif#1{\count@@\escapechar \escapechar\m@@ne
  \expandafter\expandafter\expandafter
   \edef\@@if#1{true}{\let\noexpand#1=\noexpand\iftrue}%
  \expandafter\expandafter\expandafter
   \edef\@@if#1{false}{\let\noexpand#1=\noexpand\iffalse}%
  \@@if#1{false}\escapechar\count@@} % the condition starts out false
\def\@@if#1#2{\csname\expandafter\if@@\string#1#2\endcsname}
{\uccode`1=`i \uccode`2=`f \uppercase{\gdef\if@@12{}}} % `if' is required
@@
<<Define [[\tracingall]] macro>>=
\def\tracingall{\tracingonline\@@ne\tracingcommands\tw@@\tracingstats\tw@@
  \tracingpages\@@ne\tracingoutput\@@ne\tracinglostchars\@@ne
  \tracingmacros\tw@@\tracingparagraphs\@@ne\tracingrestores\@@ne
  \showboxbreadth\maxdimen\showboxdepth\maxdimen\errorstopmode}
@@
The format name and version number are recorded in control sequences,
in order to help the people who might have to explain why something
doesn't work.

<<Identify the format>>=
\def\fmtname{plain+W}\def\fmtversion{3.14159+W}
@@
\chapter{Appendices}

\section{Efficiency and memory-space considerations}

One difficulty with large sets of macros is that they take up space.
It would be nice to preload every macro that every \TeX\ user has ever
dreamed up; but there might not be enough room,
because \TeX's memory capacity is finite. You might find it necessary
to hold back and to load only the macros that are really needed.
\medskip
How much memory space does a macro require?
\medskip
\noindent There are four kinds of memory involved: token memory,
name memory, string memory, and character memory. (If any of these
becomes too full, it will be necessary to increase what \TeX\ calls the
macro memory size, the hash size, the number of strings, and/or the
pool size, respectively. The token memory is
most important; a macro takes one cell of token memory for each token
in its definition, including the `|{|' and the `|}|'. For example,
the comparatively short definition
\medskip
[[\def\example#1\two{\four}]]
\medskip
\noindent takes five tokens: |#1|, [[\two]], |{|$_1$, [[\four]], and |}|$_2$.
Each control sequence also takes up one cell of name memory, one cell
of string memory, and as many cells of character memory as there are
characters in the name (seven in the case of |\example|). Character
memory is comparatively cheap; four characters, or in some cases five,
will fit in the same number of bits as a single cell of token memory,
inside the machine. Therefore you don't save much by choosing short
macro names.

\TeX\ will tell you how close you come to exceeding its current
memory capacity if you say [[\tracingstats=1]].
\TeX\ governs fourteen kinds of memory:
\begindisplay
|number of strings|\qquad(names of control sequences and files)\cr
|pool size|\qquad(the characters in such names)\cr
|main memory size|\qquad(boxes, glue, breakpoints, token lists,
        characters, etc.)\cr
|hash size|\qquad(control sequence names)\cr
|font memory|\qquad(font metric data)\cr
|exception dictionary|\qquad(hyphenation exceptions)\cr
|input stack size|\qquad(simultaneous input sources)\cr
|semantic nest size|\qquad(unfinished lists being constructed)\cr
|parameter stack size|\qquad(macro parameters)\cr
|buffer size|\qquad(characters in lines being read from files)\cr
|save size|\qquad(values to restore at group ends)\cr
|text input levels|\qquad(|\input| files and error insertions)\cr
|grouping levels|\qquad(unfinished groups)\cr
|pattern memory|\qquad(hyphenation pattern data)\cr
\enddisplay
The current amount of memory available will also be shown.

One obvious way to keep from loading too many macros is to keep
the macro files short and to |\input| only the ones that you need.

\section{Extensible delimiters}

\TeX\ builds large delimiters by using ``extensible'' characters,
which are specified by giving top, midle, bottom, and repeatable
characters in an \bold{extensible} command.
For example, the extensible left parentheses in [[cmex10]] are
defined by (see Figure~4)
\medskip
[[extensible oct"060": oct"060", 0, oct"100", oct"102";]]
\medskip
\noindent this says that character code [[oct"060"]] specifies
an extensible delimiter constructed from itself as the top piece,
from character number [[oct"100"]] as the bottom piece,
and from character number [[oct"102"]] as the piece which
should be repeated as often as necessary to reach a desired size.
In this particular example there is no middle piece,
but characters like curly braces have a middle piece as well.
A zero value in the top, middle, or bottom position means that
no character should be used in that part of the construction;
but a zero value in the final position means that character
number zero is the repeater. The width of an extensible
character is taken to be the width of the repeater.

Also several characters of various sizes can be linked together
in a series by means of a \bold{charlist} command.
For example (see Figure~4),
\medskip
[[charlist oct"000": oct"020": oct"022": oct"040";]]
\medskip
\noindent is used in the font [[cmex10]] to specify the left
parentheses that \TeX\ uses in displayed math formulas.
\TeX\ follows charlist to make variable-size delimiters
and variable-size accents, as well to link
[[\textstyle]] and the [[\displaystyle]] operators.


\section{Font dimensions}

The main information about font consists of the dimensions of the characters.
These numbers \TeX\ finds in the font metric files.
Except character dimensions, font metric files contain:
values for [[\fontdimen]]
parameters, italic correction of characters, ligature and kerning
programs for characters.
We change the fontdimen parameters with the (global) assignment:
\begindisplay
[[\fontdimen]]\<number>\<font>\<equals>\<dimen>,
\enddisplay
for example, the assignment
[[\fontdimen8\tenex = 0.6pt]]
changes width of the fraction bar from default $0.4\pt$ to $0.6\pt$.
\bigskip
\noindent The first seven fontdimen parameters have the following meaning:
\medskip
\item{1.}  the slant per point
\item{2.}  the interword space; that is used unless [[\spaceskip]]
           is specified
\item{3.}  interword stretch
\item{4.}  interword shrink
\item{5.}  the x-height
\item{6.}  the quad width (for the font in family 2,
           $1/18\,$th quad width is equal to $1\,$mu)
\item{7.}  the extra space; that value is added to the interword space
           used whenever [[\spacefactor]]${}\geq 2000$,
           unless [[\xspaceskip]] is specified.
\bigskip
\noindent For the font in family 2 attributes 8--19 specify positioning
of fractions, subscripts, superscripts.
\medskip
\noindent \stress{fraction numerator attributes: minimum shift up,
from the main baseline, of the baseline of the numerator of
a generalized fraction}
\medskip
\item{8.}  num1: for display style
\item{9.}  num2: for text style or smaller if a fraction
          bar is present
\item{10.} num3: for text style or smaller if no a fraction
          bar is present
\medskip
\noindent
\stress{fraction denominator attributes: minimum shift down, from the main
baseline, of the baseline of the denominator of a generalized fraction}
\medskip
\item{11.} denom1: for display style
\item{12.} denom2: for text style or smaller
\medskip
\noindent
\stress{superscript attributes: minimum shift up, from the main baseline,
of the baseline of the superscript}
\medskip
\item{13.} sup1: for display style
\item{14.} sup2: for text style or smaller, non-cramped
\item{15.} sup3: for text style or smaller, cramped
\medskip
\noindent
\stress{subscript attributes: minimum shift down, from the main baseline,
of the baseline of a subscript}
\medskip
\item{16.} sub1: when no superscript is present
\item{17.} sub2: when superscript is present
\medskip
\noindent
\stress{script adjustment attributes: for use only with non-glyph,
that is, composite objects}
\medskip
\item{18.} sup\_drop: maximum distance of superscript baseline below
          top of nucleus
\item{19.} sub\_drop: minimum distance of subscript baseline below
          bottom of nucleus
\medskip
\noindent
Delimiter span attributes: height plus depth of delimiter
enclosing a generalized fraction.
\medskip
\item{20.} delim1: in display style
\item{21.} delim2: in text style or smaller
\medskip
\noindent
The last parameter, the height of the math axis, specifies the height
above the baseline of the fraction bar, and the centre of large
delimiters and most operators and relations. This position is used in
vertical centering.
\medskip
\item{22.} axis\_height.
\bigbreak
\noindent For the font in family 3 attributes 9--12 determine
extra space added when limits are attached to operators.
The attribute 8 specifies thickness of the rule used for
overlines, underlines, radical extenders, and fraction bars.
From that dimension are derived `clearances' around
fraction bar.
The attribute 13 specifies extra space added above and below
attached limits.
\medskip
\item{8.}  default\_rule\_thickness
\item{9.}  big\_op\_spacing1
\item{10.} big\_op\_spacing2
\item{11.} big\_op\_spacing3
\item{12.} big\_op\_spacing4
\item{13.} big\_op\_spacing5
\medskip
\noindent We have:
big\_op\_spacing1(2) $\leq$ space between upper (lower)
limit and top (bottom) of large operator $\leq$ big\_op\_spacing3(4)
\bigskip
\begingroup
\def \loop #1\repeat%
    {\global\def\body{#1}\iterate}

\newcount \dno

\def \?#1%
    {\ifdim\the\fontdimen\dno#1<10pt \hphantom{0}\fi%
     \the\fontdimen\dno#1}

\ialign{\global\advance \dno by 1\relax#&
        \hfil\number\dno.#\quad&&
        \quad\tentt#\cr
\omit&\omit\tensl fontdimen&cmsy10&cmex10&cmtt10&cmr10&cmti10&cmbx10
                                                    \cr\noalign{\bigskip}
\loop&&
\?\tensy&
\?\tenex&
\?\tentt&
\?\tenrm&
\?\tenit&
\?\tenbf&\cr
\ifnum\dno<7\repeat\cr}

\advance \dno by -1
\vskip-\baselineskip

\ialign{\global\advance \dno by 1\relax#&
        \hfil\number\dno.#\quad&&
        \quad\tentt#\cr
\omit&\omit\hphantom{\tensl fontdimen}&\cr
\loop&&
\?\tensy&
\?\tenex&\cr
\ifnum\dno<13\repeat\cr}

\advance \dno by -1
\vskip-\baselineskip

\ialign{\global\advance \dno by 1\relax#&
        \hfil\number\dno.#\quad&&
        \quad\tentt#\cr
\omit&\omit\hphantom{\tensl fontdimen}&\cr
\loop&&
\?\tensy&\cr
\ifnum\dno<22\repeat\cr}
\endgroup
@@
\section{Font tables}

\noindent {\bf Figure 1.\enspace}[[cmr10]]---family 0
\beginchart\tenrm
\normalchart
\endchart

\medskip\vfill
\noindent {\bf Figure 2.\enspace}[[cmmi10]]---family 1
\beginchart\teni
\normalchart
\endchart

\medskip\vfill
\noindent {\bf Figure 3.\enspace}[[cmsy10]]---family 2
\beginchart\tensy
\normalchart
\endchart

\medskip\vfill
\noindent {\bf Figure 4.\enspace}[[cmex10]]---family 3
\beginchart\tenex
\normalchart
\endchart
@