\input diagrams
\font\tenbb=msbm10
\font\sevenbb=msbm7
\font\fivebb=msbm5

\newfam\bbfam \def\bb{\fam\bbfam}
\textfont\bbfam=\tenbb
\scriptfont\bbfam=\sevenbb
\scriptscriptfont\bbfam=\fivebb

\font\tengoth=eufm10
\font\sevengoth=eufm7
\font\fivegoth=eufm5

\newfam\gothfam \def\goth{\fam\gothfam}
\textfont\gothfam=\tengoth
\scriptfont\gothfam=\sevengoth
\scriptscriptfont\gothfam=\fivegoth

\font\tensans=cmss10
\font\sevensans=cmss7
\font\fivesans=cmss5

\newfam\sansfam \def\sans{\fam\sansfam}
\textfont\sansfam=\tensans
\scriptfont\sansfam=\sevensans
\scriptscriptfont\sansfam=\fivesans

\font\sc=cmcsc10

\font\bigcm=cmb10 scaled \magstep2

% general macros
\def\assign#1#2{\line{{\bigcm #1 Assignment \##2}\hfil Jason Reed}
\vskip 1mm \hrule}
\def\spassign#1#2{\line{{\bigcm #1 #2}\hfil Jason Reed}
\vskip 1mm \hrule}
\def\spsec#1{\line{\bigcm #1 \hfil} \vskip 1mm \hrule\vskip 5mm}
\def\problem#1{\vskip 3mm\item{\tt #1}}
\def\sub#1{\itemitem{#1)} }
\def\psub#1 {\vskip 3mm\itemitem{#1)}\par\noindent}
\def\itemitemitem{\par\indent\indent\hangindent3\parindent\textindent}
\let\iii=\itemitemitem
\def\manual{\vskip 1.8in}
\def\littlemanual{\vskip .9in}
% Dammit, conflicts with LaTeX
%\def\figure#1{$$\epsfbox{figure.#1}$$\centerline{\sl Figure #1.}\vskip 3mm}
\def\part#1 {\par\noindent$\bullet$ #1\par\noindent}
%          ^ this space is necessary!

% macros mostly used for 15-212 homework
% but they might come in handy whenever I need to write up proofs...
\def\proc#1{\par\noindent{\sc #1}}
\def\lit#1{\hbox{\tt #1}}
\def\rep#1{{\overline{#1}}}
\def\gsto{\Longrightarrow}
\def\evto{\hookrightarrow}
\def\evaluation#1{\vskip 1mm\halign{$##$\quad\strut&$##$\hfil\quad&$##$\hfil\quad&\hfill##\cr#1}}
\def\qed{\rightline{\vrule width .5em height .5em depth 0em}} % after a $$...$$
\def\cqed{\hskip2mm{\vrule width .5em height .5em depth 0em}} % at the end of a |P.
\def\putbefore#1{\llap{#1\enspace}\ignorespaces}
\def\shout#1{\par\noindent\putbefore{$\triangleright$}{\sc #1}}
\def\case#1{\par\noindent\putbefore{$\circ$}{\sc Case #1:}}
\def\floor#1{{\left\lfloor{#1}\right\rfloor}}
\def\N{{\bb N}}
\def\Z{{\bb Z}}
\def\Q{{\bb Q}}
\def\R{{\bb R}}
\def\C{{\bb C}}
\def\MLneg{{\char"7E}}

% macros for either linear algebra or physics
\def\vect#1{{\vec{\bf #1}}}
\def\ihat{{\hat{\hbox{\bf \char16}}}}
\def\jhat{{\hat{\hbox{\bf \char17}}}}
\def\khat{{\hat{\hbox{\bf k}}}}

% macros for physics
% I really do loathe physics.
\def\degrees{^\circ}
\def\dir#1{{{\vect #1}\over{|\vect #1|}}}
\def\m{{\rm m}}
\def\km{{\rm km}}
\def\s{{\rm s}}
\def\Newt{{\rm N}}
\def\kg{{\rm kg}}
\def\ms{\,^{\rm m}\kern -.5 ex/\kern -.3ex_{\rm s}\kern .2ex}
\def\mss{\,^{\rm m}\kern -.5 ex/\kern -.3ex_{{\rm s}^2}\kern .2ex}
\def\perp{{\scriptscriptstyle \bot}}
\def\parallel{{\scriptscriptstyle \|}}
\def\normal{{\cal N}}
\def\J{{\rm J}}
\def\W{{\rm W}}
\def\kW{{\rm kW}}
\def\u#1{\,{\rm #1}}

% macros for 21-241 Matrix algebra
\def\rref{\mathop{\rm rref}}
\def\rank{\mathop{\rm rank}}
\def\image{\mathop{\rm image}}
\def\im{\mathop{\rm im}}
\def\span{\mathop{\rm span}}
\def\tr{\mathop{\rm tr}}
%...and plain TeX already gives us \ker. Yay!
\def\frac#1#2{{#1/ #2}} % for importing stuff from Mathematica
\def\column{{\vrule height 4ex}}
\def\coldots{{\vbox to 4ex{\vfil\hbox{$\cdots$}\vfil}}}
\def\proj{\mathop{\rm proj}\nolimits}
\def\coord#1#2{\bigl[{#1}\bigr]_{\cal{#2}}}

% For 21-260 Diff Eq
\def\L#1{{\cal L}\left\{#1\right\}}

% For 21-235 Math Studies
\def\disclaim{\vskip 3mm \par\noindent{\bf Notation caveat}: I use $\subseteq$ for 
{\sl is a subset of}, and $\subset$ for {\sl is a strict
subset of}.}
\def\Rng{\mathop{\rm Rng}}
\def\Fix{\mathop{\rm Fix}}
\def\F{{\bb F}}
\def\P{{\bb P}}
\def\Pcross{{\bb P}^\times}
\def\adjust{\big|}
\def\from{\bigg|}
\def\st{\mathop|}
\def\del{\partial}

% For 21-800 Recursion Theory
\def\marksize{.75em}
\def\markspace{.125em}
\def\mark#1{\vcenter{\vbox{\hrule\hbox{\vrule
\vbox to \marksize{\vfil \hbox to \marksize
{\hfil$\scriptstyle #1$\hfil}\vfil}\vrule}\hrule}}}

\def\bigmark#1{\vcenter{\vbox{\hrule\hbox{\vrule
\vbox {\vskip\markspace \hbox
{\hskip\markspace $\scriptstyle #1$\hskip\markspace}\vskip\markspace}\vrule}\hrule}}}

\def\iff{\Leftrightarrow}
\def\imp{\Rightarrow}

% For 80-413 Category Theory

\def\id{{\rm id}}
\def\op{{\rm op}}
\def\Rel{{\bf Rel}}
\def\Sets{{\bf Sets}}
\def\Monoids{{\bf Monoids}}
\def\Posets{{\bf Posets}}
\def\Cat{{\bf Cat}}
\def\Mon{{\bf Mon}}
\def\Grp{{\bf Grp}}
\def\Grph{{\bf Grph}}
\def\Ab{{\bf Ab}}
\def\dash{\hbox{---}}
\def\hom{{\rm hom}}
\def\nat{{\rm nat}}
\newarrow{Bi}>---{>>}
\newarrow{Cong}=====
\newarrow{Iso}<--->
\def\pair#1#2{{\langle #1 , #2 \rangle}}
\def\invlim{{\lim\limits_{\longleftarrow}}}
\def\dirlim{{\lim\limits_{\longrightarrow}}}

% For 80-820 Categorical Logic

\def\mod{{\rm mod}}
\def\T{{\bb T}}
\def\1{{\bf 1}}
\def\Graph{{\bf Graph}}
\def\y{{\bf y}}

% For 21-229 Set Theory

\def\dom{\mathop{\rm dom}}
\def\rng{\mathop{\rm rng}}
\def\Seq{{\rm Seq}}
\def\ps{{\cal P}}
\def\an{{\aleph_0}}
\def\on{{\rm On}}
\def\cf{\mathop{\rm cf}}

\catcode`@=11
\def\bdots{\mathinner{\mkern1mu\raise\p@\vbox{\kern7\p@\hbox{.}}\mkern2mu
    \raise3\p@\hbox{.}\mkern2mu\raise5\p@\hbox{.}\mkern1mu}}
\catcode`@=12

% For 21-651 Topology

\def\O{{\cal O}}

% For 15-312 Programming Languages

\def\prov{\vdash}
\def\rule#1:{\vskip 3mm\noindent [{\bf#1}]\quad}
\def\erule#1:{\vskip 3mm\noindent (E)[{\bf#1}]\quad}
\def\crule#1:{\vskip 3mm\noindent (C)[{\bf#1}]\quad}
\def\multib{\vskip 3mm\noindent}
\def\multi#1:{[{\bf#1}]}
\def\multie{\quad}
\def\sep{\vskip 3mm \hrule \vskip 3mm}
\def\fbox{\vbox{\hrule\hbox{\vrule
\vbox to .6em{\vfil \hbox to .6em {\hfil}\vfil}\vrule}\hrule}}
\def\bif{\mathop{\bf if}}
\def\bthen{\mathop{\bf then}}
\def\belse{\mathop{\bf else}}
\def\bapply{\mathop{\bf apply}}
\def\bfi{\mathop{\bf fi}}
\def\bbind{\mathop{\bf bind}}
\def\bto{\mathop{\bf to}}
\def\bin{\mathop{\bf in}}
\def\bbool{\mathop{\bf bool}}
\def\btrue{\mathop{\bf true}}
\def\bfalse{\mathop{\bf false}}
\def\bint{\mathop{\bf int}}
\def\bfun{\mathop{\bf fun}}
\def\bis{\mathop{\bf is}}
\def\bend{\mathop{\bf end}}
\def\bone{\mathop{\bf 1}}
\def\btwo{\mathop{\bf 2}}
\def\binl{\mathop{\bf inl}\nolimits}
\def\binr{\mathop{\bf inr}\nolimits}
\def\bcase{\mathop{\bf case}}
\def\bof{\mathop{\bf of}}
\def\bfold{\mathop{\bf fold}}
\def\bunfold{\mathop{\bf unfold}}
\def\bref{\mathop{\bf ref}}
\def\btry{\mathop{\bf try}}
\def\bow{\mathop{\bf ow}}
\def\bfail{\mathop{\bf fail}}
\def\braise{\mathop{\bf raise}}
\def\bexnEq{\mathop{\bf exnEq}}
\def\bletexn{\mathop{\bf letexn}}
\def\bexn{\mathop{\bf exn}}
\def\bstack{\mathop{\bf stack}}
\def\value{\mathop{\sans value}}
\def\ok{\mathop{\sans ok}}
\def\frame{\mathop{\sans frame}}
\def\stack{\mathop{\sans stack}}
\def\hstack{\mathop{\sans hstack}}
\def\empt{\mathop{\langle\rangle}}

\catcode`@=11
\def\sendl#1{\vbox{\m@th\ialign{##\crcr
      $\hfil\displaystyle{#1}\hfil$\crcr
      \noalign{\kern\p@\nointerlineskip}
      \hbox to 2in{\leftarrowfill}\crcr}}}
\def\sendr#1{\vbox{\m@th\ialign{##\crcr
      $\hfil\displaystyle{#1}\hfil$\crcr
      \noalign{\kern\p@\nointerlineskip}
      \hbox to 2in{\rightarrowfill}\crcr}}}
\catcode`@=12 % at signs are no longer letters

% Dammit, these conflict with LaTeX
%\def\[[{[\![}
%\def\]]{]\!]}

% LaTeX stuf...
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{fact}[theorem]{Fact}
\newenvironment{proof}[1][Proof]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{definition}[1][Definition]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{example}[1][Example]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{remark}[1][Remark]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}