%%%%%%%%% New environments and commands %%%%%%%%%%%

% proof environments

\newenvironment{proof}%
{\begin{trivlist}\item[]{\bf Proof.  }}%
{  {\everymath{}$\Box$}\end{trivlist}}
\newenvironment{proofof}[1]%
{\begin{trivlist}\item[]{\bf Proof of #1.  }}%
{  {\everymath{}$\Box$}\end{trivlist}}

% math tabbing environment

\newenvironment{mtabbing}
{\begin{tabbing}\hspace*{\leftmargini}\=\+\kill$}%
{$\end{tabbing}}
\newcommand{\TS}{{}$\=\+${}}% for use with mtabbing environment
\newcommand{\NL}{{}$\\${}}% for use with mtabbing environment
\newcommand{\TSN}{{}$\=${}}% for use with mtabbing environment (no \+)
\newcommand{\NT}{{}$\>${}}% for use with mtabbing environment

% for centered mtabbing's, within displaymath's

\newcommand{\mtab}[1]{\mbox{\kern-2.3ex\vbox{
\topsep 0pt\begin{mtabbing}#1\end{mtabbing}\kern-1ex}}}

% centered tabbing environment

\newenvironment{ctabbing}
{\begin{center}
\begin{minipage}{10in}
\begin{tabbing}}%
{\end{tabbing}
\end{minipage}
\end{center}}

% alltt environment in which text has size \small
% manually indent each line with \hspace{\leftmargini}
% (would be nice to automate this)

\newenvironment{myalltt}%
{\small\alltt}%
{\endalltt}

% theorem-like environments

\theoremstyle{break}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{openproblem}[theorem]{Open Problem}
\newtheorem{example}[theorem]{Example}
\newtheorem{counterexample}[theorem]{Counterexample}
{\theorembodyfont{\normalfont}
\newtheorem{exercise}[theorem]{Exercise}}
{\theorembodyfont{\normalfont}
\newtheorem{definition}[theorem]{Definition}}
\newcommand{\theoremnumber}[1]{\setcounter{theorem}{#1-1}}

\newcommand{\clearemptydoublepage}{\newpage{\pagestyle{empty}\cleardoublepage}}

\newcommand{\abr}{\allowbreak}
\newcommand{\gbr}{\penalty1000 } % Allow break grudgingly
\newcommand{\hquad}{\;\;} % Half quad in math mode

\newcommand{\oper}[1]{\!\!\!& #1 &\!\!\!}% For use with eqnarray
\newcommand{\moper}[1]{\!\!& #1 &\!\!}% Medium size operator
\newcommand{\boper}[1]{\!& #1 &\!}% Big size operator

% proof justification in display

\newcommand{\by}[1]{\qquad{\textrm{(#1)}}}
\newcommand{\bylemma}[1]{\by{Lemma \ref{#1}}}
\newcommand{\bytheorem}[1]{\by{Theorem \ref{#1}}}
\newcommand{\bycorollary}[1]{\by{Corollary \ref{#1}}}
\newcommand{\byproposition}[1]{\by{Proposition \ref{#1}}}

% functions and restriction

\newcommand{\fun}{\mathbin{\to}}
\newcommand{\restr}{\mathord{\mid}}

% sets

\newcommand{\setof}[2]{\{ \, #1 \mid #2 \, \}}
\newcommand{\singset}[1]{\{ #1 \}}
\newcommand{\powset}[1]{{\cal P} #1}
\newcommand{\sub}{\subseteq}% Subset

% spacing in math

\newcommand{\eqtxt}[1]{\;\abr\mbox{#1}\abr\;}
\newcommand{\eqtxtl}[1]{\;\abr\mbox{#1}\abr}
\newcommand{\eqtxtr}[1]{\abr\mbox{#1}\abr\;}
\newcommand{\eqtxtn}[1]{\abr\mbox{#1}\abr}

% definition by cases

\newcommand{\casesdef}[4]{
\left\{ \begin{array}{ll}
#1 & #2 \\
#3 & #4
\end{array} \right.}

% macros for producing ascii symbols in typewriter font

\newcommand{\tildesym}{\symbol{'176}}       % produces ~
\newcommand{\underscoresym}{\symbol{'137}}  % produces _
\newcommand{\primesym}{\symbol{'047}}       % produces '
\newcommand{\quotesym}{\symbol{'042}}       % produces "
\newcommand{\uparrowsym}{\symbol{'136}}     % produces ^