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-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem*{remark}{Remark}
-
-\DeclareMathOperator{\E}{\mathbb{E}}
-\let\P\relax
-\DeclareMathOperator{\P}{\mathbb{P}}
-\newcommand{\ex}[1]{\E\left[#1\right]}
-\newcommand{\prob}[1]{\P\left[#1\right]}
-\newcommand{\inprod}[1]{\left\langle #1 \right\rangle}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\N}{\mathbb{N}}
-\newcommand{\eps}{\varepsilon}
-\newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}