%auto-ignore
\usepackage{theorem}
\usepackage[english]{babel}
\usepackage{a4}
\usepackage{epsfig}
\usepackage{amsfonts}
\usepackage{latexsym}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{definition}[theorem]{Definition}
\theorembodyfont{\rmfamily}
\newtheorem{remark}[theorem]{Remark}
\theorembodyfont{\rmfamily}
\newtheorem{example}[theorem]{Example}

\newenvironment{proof}[1][\it Proof.]{\begin{trivlist}\item[\hskip \labelsep {\bfseries #1}]}{$_\Box$\end{trivlist}}


\def\N{\ensuremath{\mathbb{N}}}
\def\Z{\ensuremath{\mathbb{Z}}}
\def\Q{\ensuremath{\mathbb{Q}}}
\def\R{\ensuremath{\mathbb{R}}}

\def\F{\ensuremath{\mathcal{F}}}
\def\G{\ensuremath{\mathcal{G}}}
\def\K{\ensuremath{\mathcal{K}}}

\def\k{\ensuremath{{\bf{k}}}}
\def\x{\ensuremath{{\bf{x}}}}
\def\t{\ensuremath{{\bf{t}}}}