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+\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}}