\usepackage{fullpage}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{psfrag}
\usepackage{amsthm}

%============Formatting======================


\def\tb{\hspace*{1cm}}   %tab

%===========Theorems and such ========================

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{claim}[theorem]{Claim}


%===========Some math symbols=========================

\def\ZZ{\mathbb{Z}}        % set of integers
\def\RR{\mathbb{R}}        % set of real numbers
\def\iff{\Leftrightarrow}  % if and only if symbol

% Floors and ceilings.
\providecommand{\floor}[1]{\lfloor #1 \rfloor}  
\providecommand{\ceil}[1]{\lceil #1 \rceil}     
\providecommand{\Floor}[1]{\left\lfloor #1 \right\rfloor}
\providecommand{\Ceil}[1]{\left\lceil #1 \right\rceil}