\newcommand{\arrow}{\rightarrow}
\newcommand{\bairespace}{\omega^{\omega}}
\newcommand{\binseq}{2^{<\omega}}
\newcommand{\cantorspace}{2^{\omega}}
\newcommand{\claim}{\par\bigskip\noindent {\bf Claim: }}
\newcommand{\concat}{{\;\hat{ }\;}}
\newcommand{\Conj}{\mathop{\bigwedge\kern -6pt\bigwedge}}
\newcommand{\conj}{\mathop{\wedge}}
\newcommand{\continuum}{{\goth c}}
\newcommand{\cross}{\times}
\newcommand{\elemequiv}{\equiv}
\newcommand{\elemsub}{\preceq}
\newcommand{\eq}{\approx}
\newcommand{\example}{\par\bigskip\noindent {\bf Example: }}
\newcommand{\finsets}{[\omega]^{< \omega}}
\newcommand{\finsubsets}[1]{[#1]^{<\omega}}
\newcommand{\forces}{\mid\vdash}
\newcommand{\implies}{\rightarrow}
\newcommand{\infsets}{[\omega]^{\omega}}
\newcommand{\infsubsets}[1]{ [#1]^{\omega} }
\newcommand{\Intersect}{\bigcap}
\newcommand{\intersect}{\cap}
\newcommand{\isom}{\simeq}
\newcommand{\poset}{{\Bbb P}}
\newcommand{\problem}{\par\bigskip\noindent {\bf Problem: }}
\newcommand{\proof}{\par\noindent proof:\par}
\newcommand{\proves}{\vdash}
\newcommand{\rationals}{{\Bbb Q}}
\newcommand{\rmand}{{\mbox{ and }}}
\newcommand{\rmiff}{{\mbox{ iff }}}
\newcommand{\reals}{{\Bbb R}}
\newcommand{\satisfies}{\models}
\newcommand{\seq}{\omega^{< \omega}}
\newcommand{\setof}{\;\;|\;\;}
\newcommand{\superset}{\supset}
\newcommand{\Union}{\bigcup}
\newcommand{\union}{\cup}

%%%%%%%%%  counters

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\newcounter{probno}
\newcommand{\prob}{\addtocounter{probno}{1}
\par\smallskip\noindent\theprobno . }

%%%%%%%% AMS  symbols


%%%%%% if amssym.def and amssym.tex are not available
%%%%%% uncomment next 8 lines
%\newcommand{\res}{\uparrow}
%\newcommand{\propsub}{\subset}
%\newcommand{\qed}{\nopagebreak\par\noindent\nopagebreak$\Box$\par}
%\newcommand{\Bbb}[1]{{\bf #1}}
%\newcommand{\goth}[1]{{\bf #1}}
%\newcommand{\ulcorner}{\vdash}
%\newcommand{\urcorner}{\dashv}
%\endinput

\newcommand{\res}{\upharpoonright}
\newcommand{\propsub}{\subsetneqq}
\newcommand{\qed}{\nopagebreak\par\noindent\nopagebreak$\blacksquare$\par}
\input amssym.def
\input amssym