\newcommand{\nop}[1]{}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{observation}{Observation}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{example}{Example}
\newtheorem{claim}{Claim}

\DeclareMathOperator{\op}{op}
\DeclareMathOperator{\args}{arg}
\DeclareMathOperator{\cost}{cost}
\DeclareMathOperator{\lleft}{left}
\DeclareMathOperator{\rright}{right}
\DeclareMathOperator{\argmax}{argmax}
\DeclareMathOperator{\argmin}{argmin}

\newcommand{\Paragraph}[1]{~\vspace*{-0.8\baselineskip}\\{\bf #1}}
\newcommand\mathbftt[1]{\textnormal{\ttfamily\bfseries #1}}

\newcommand\note[1]{\marginpar{\color{red}\tiny #1}}