\newtheorem{theorem}{Theorem}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{openprob}[theorem]{Open Problem}
\newtheorem{remk}[theorem]{Remark}
\newtheorem{apdxlemma}{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{problem}{Problem}

\newenvironment{remark}{\begin{remk}

\begin{normalfont}}{\end{normalfont}
\end{remk}}
\newtheorem{sublemma}[theorem]{Sublemma}


%%%%%%%%%%%%%%%%%%%%%%% text macros
\newcommand{\etal}{{\it et~al.\ }}
\newcommand{\ie} {{\it i.e.,\ }}
\newcommand{\eg} {{\it e.g.,\ }}
\newcommand{\cf}{{\it cf.,\ }}

%%%%%%%%%%%%%%%%%%%%%%% general useful macros
\newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\N}{{\mathbb{N}}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\poly}{{\mathrm{poly}}}
\newcommand{\polylog}{{\mathrm{polylog}}}
\newcommand{\loglog}{{\mathop{\mathrm{loglog}}}}
\newcommand{\zo}{\{0,1\}}
\newcommand{\suchthat}{{\;\; : \;\;}}
\newcommand{\pr}[1]{\Pr\left[#1\right]}
\newcommand{\deffont}{\em}
\newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\Var}{\mathop{\mathrm Var}\displaylimits}
\newcommand{\eps}{\varepsilon}
\newcommand{\one}{\mathbf{1}}

%%%%%%%%%%%%%%%%%%%%%%%%%% linear algebra
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\spn}{\operatorname{span}}

%%%%%%%%%%%%%%%%%%% macros particular to this course

\newcommand{\mathprob}[1]{\mathsf{#1}}
\newcommand{\SAT}{\mathprob{SAT}}
\newcommand{\yes}{{\sc yes}}
\newcommand{\no}{{\sc no}}

\newcommand{\class}[1]{\mathbf{#1}}
\newcommand{\TIME}{\class{DTIME}}
\newcommand{\NTIME}{\class{NTIME}}
\renewcommand{\P}{\class{P}}
\newcommand{\NP}{\class{NP}}
\newcommand{\coNP}{\class{coNP}}
\newcommand{\BPP}{\class{BPP}}
\newcommand{\BQP}{\class{BQP}}
\newcommand{\RP}{\class{RP}}
\newcommand{\coRP}{\class{coRP}}
\newcommand{\ZPP}{\class{ZPP}}
\newcommand{\MA}{\class{MA}}
\newcommand{\AM}{\class{AM}}
\newcommand{\Sigmacc}[1]{\class{\Sigma^p_{#1}}}
\newcommand{\Picc}[1]{\class{\Pi^p_{#1}}}
\newcommand{\SZK}{\class{SZK}}
\newcommand{\PP}{\class{PP}}
\newcommand{\sharpP}{\class{\#P}}
\newcommand{\IP}{\class{IP}}
\newcommand{\Ecc}{\class{E}}
\newcommand{\EXP}{\class{EXP}}
\newcommand{\NEXP}{\class{NEXP}}

\newcommand{\SPACE}{\class{SPACE}}
\newcommand{\NSPACE}{\class{NSPACE}}
\renewcommand{\L}{\class{L}}
\newcommand{\NL}{\class{NL}}
\newcommand{\coNL}{\class{coNL}}
\newcommand{\BPL}{\class{BPL}}
\newcommand{\RL}{\class{RL}}
\newcommand{\PSPACE}{\class{PSPACE}}

\newcommand{\Ppoly}{\class{P_{/poly}}}
\newcommand{\AC}{\class{AC}}
\newcommand{\NC}{\class{NC}}
\newcommand{\ACC}{\class{ACC}}

\newcommand{\PCP}{\class{PCP}}

\newcommand{\negl}{{\mathrm{neg}}}
\newcommand{\sgn}{\operatorname{sgn}}

\newcommand{\Enc}{\operatorname{Enc}}
\newcommand{\Dec}{\operatorname{Dec}}

%%%%%%Gates%%%%%%%%
\newcommand{\THR}{\mathrm{THR}}
\newcommand{\MAJ}{\mathrm{MAJ}}
\newcommand{\AND}{\mathrm{AND}}
\newcommand{\OR}{\mathrm{OR}}
\newcommand{\XOR}{\mathrm{XOR}}

%%%%%%%Functions%%%%%
\newcommand{\EQ}{\mathrm{EQ}}
\newcommand{\GT}{\mathrm{GT}}

