\usepackage{amssymb,amsmath}
\newcommand{\N}{\mathbb{N}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\gmdl}[1]{\mathcal{#1}}


\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\st}{ \; | \; } % x such that ...
\newcommand{\res}{\mathord{\upharpoonright}}
\newcommand{\concat}{\mathord{\hat{\;}}}


%Jeremy's macros
\newcommand{\limplies}{\rightarrow}
%\newcommand{\lor}{\vee}
%\newcommand{\land}{\wedge} 
\newcommand{\liff}{\leftrightarrow}
%\newcommand{\lnot}{\neg}
\newcommand{\ph}{\varphi}
\newcommand{\ex}{\exists}
\newcommand{\al}{\forall}