\providecommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\providecommand{\ceil}[1]{\left\lceil#1\right\rceil}
\def\sp#1#2{\langle #1, #2 \rangle}
+\def\spv#1#2{\langle\vec #1,\vec #2\rangle}
\newtheorem{theorem}{Theorem}
\newaliascnt{example}{theorem}