\newcommand{paren}{1}{(#1)}
\newcommand{ang}{1}{\langle#1\rangle}
\newcommand{suchthat}{0}{\mid}
\newcommand{eqdef}{0}{\font{Euclid}{\bf{::=}}}
\newcommand{bar}{1}{\overline{#1}}
\newcommand{and}{0}{\wedge}
\newcommand{conj}{0}{\And}
\newcommand{smand}{0}{\And}
\newcommand{AND}{0}{\bigwedge}
\newcommand{Land}{0}{\AND}
\newcommand{lgand}{0}{\AND}
\newcommand{Or}{0}{\vee}
\newcommand{disj}{0}{\Or}
\newcommand{smor}{0}{\Or}
\newcommand{OR}{0}{\bigvee}
\newcommand{Lor}{0}{\OR}
\newcommand{lgor}{0}{\OR}
\newcommand{implies}{0}{\rightarrow}
\newcommand{iff}{0}{\leftrightarrow}
\newcommand{bicond}{0}{\leftrightarrow}
\newcommand{equivalent}{0}{\Leftrightarrow}
\newcommand{set}{1}{\lbrace{#1}\rbrace}
\newcommand{card}{1}{\abs{#1}}
\newcommand{union}{0}{\cup}
\newcommand{lgunion}{0}{\bigcup}
\newcommand{intersect}{0}{\cap}
\newcommand{lgintersect}{0}{\bigcap}
\newcommand{cross}{0}{\times}
\newcommand{compose}{0}{\circ}
\newcommand{composition}{0}{\circ}
\newcommand{power}{0}{\cP}
\newcommand{range}{1}{\font{times new roman}{range}(#1)}
\newcommand{domain}{1}{\font{times new roman}{domain}(#1)}
\newcommand{emptystring}{0}{\lambda}
\newcommand{naturals}{0}{\bbn}}
\newcommand{integers}{0}{\bbz}
\newcommand{rationals}{0}{\bbq}
\newcommand{reals}{0}{\bbr}
\newcommand{complexes}{0}{\bbc}
\newcommand{abs}{1}{\mid#1\mid}
\newcommand{floor}{1}{\lfloor#1\rfloor}
\newcommand{ceil}{1}{\lceil#1\rceil}
%\newcommand{divides}{0}{\mathbin{|}}
\newcommand{divides}{0}{\mid}
\newcommand{sspace}{0}{\cS}
\newcommand{pr}{1}{\font{times new roman}{Pr}\lbrace#1\rbrace}
\newcommand{prob}{1}{\pr{#1}}
\newcommand{prsub}{2}{\font{times new roman}{Pr}_{#2}\lbrace{#1}\rbrace}
\newcommand{prcond}{2}{\font{times new roman}{Pr}\lbrace{#1}\,\mid\,#2\rbrace}
\newcommand{Ex}{0}{\font{times new roman}{E}}
\newcommand{Var}{0}{\font{times new roman}{Var}}
\newcommand{Cov}{0}{\font{times new roman}{Cov}}
\newcommand{expect}{1}{\Ex[#1]}
\newcommand{expectsq}{1}{{\Ex}^2[#1]}
\newcommand{expcond}{2}{\expect{#1\mid#2}}
\newcommand{variance}{1}{\Var[#1]}
\newcommand{varsq}{1}{{\Var}^2[#1]}
\newcommand{covar}{1}{\Cov[#1]}
\newcommand{covariance}{2}{\Cov[#1,#2}}