\newcommand{\nb}[1]{{\color{violet} NB: #1}}
\newcommand{\todo}[1]{{\color{red} TODO: #1}}
\newcommand{\anupam}[1]{{\color{orange} Anupam: #1}}
\newcommand{\patrick}[1]{{\color{blue} Patrick: #1}}

%\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{lemma}[theorem]{Lemma}
%
%
%\theoremstyle{definition}
%\newtheorem{definition}[theorem]{Definition}

\newcommand{\lhs}{\mathit{LHS}}
\newcommand{\rhs}{\mathit{RHS}}

\newcommand{\Word}{\mathbb{W}}
\newcommand{\Nat}{\mathbb{N}}
\newcommand{\arith}{B_2}
\newcommand{\basic}{\mathit{BASIC}}

\newcommand{\ind}{\mathit{IND}}
\newcommand{\pind}{\mathit{PIND}}
\newcommand{\lind}{\mathit{LIND}}
\newcommand{\cind}[1]{#1\text{-}\ind}
\newcommand{\cpind}[1]{#1\text{-}\pind}
\newcommand{\clind}[1]{#1\text{-}\lind}
\renewcommand{\min}{\mathit{MIN}}
\newcommand{\cmin}[1]{#1\text{-}\min}

\newcommand{\prefix}{\preccurlyeq}

\newcommand{\charfn}[2]{\chi^{#1}_{#2}}
\newcommand{\wit}[2]{\textsc{wit}^{#1}_{#2}}
\newcommand{\Wit}[2]{\mathit{Wit}^{#1}_{#2}}
\newcommand{\dfn}{:=}
\newcommand{\seqar}{\rightarrow}
\newcommand{\proves}{\vdash}

\renewcommand{\epsilon}{\varepsilon}
\newcommand{\pred}{\mathsf{p}}
\renewcommand{\succ}[1]{\mathsf{s}_{#1}}
\newcommand{\hlf}[1]{\lfloor \frac{#1}{2}\rfloor}
\newcommand{\cond}{C}
\newcommand{\smsh}{\#}
\newcommand{\pair}[3]{\langle  #1;#2 , #3 \rangle}
\newcommand{\eq}{\textsc{eq}}
\newcommand{\leqfn}{\textsc{leq}}
\newcommand{\bit}{\textsc{bit}}
\newcommand{\andfn}{\textsc{and}}
\newcommand{\orfn}{\textsc{or}}
\newcommand{\notfn}{\textsc{not}}
\newcommand{\equivfn}{\textsc{equiv}}
\newcommand{\shorten}{\textsc{shorten}}


\newcommand{\zerobit}{\textsc{0bit}}
\newcommand{\onebit}{\textsc{1bit}}
\newcommand{\pref}{\textsc{pref}}
\newcommand{\addtosequence}{\textsc{addtosequ}}

\newcommand{\safe}{{N_0}}
\newcommand{\normal}{{N_1}}

\newcommand{\pv}{\mathit{PV}}
\newcommand{\pvbci}[1]{\pv^{#1}_{\mathrm{BC}}}
\newcommand{\mubci}[1]{\mu\mathrm{BC}^{#1}}
\newcommand{\mubc}{\mu\mathrm{BC}}
\newcommand{\bc}{\mathrm{BC}}

\newcommand{\sigp}[1]{\Sigma^p_{#1}}
\newcommand{\pip}[1]{\Pi^p_{#1}}
\newcommand{\fphi}[1]{\Box^p_{#1}}
\newcommand{\fph}{\Box^p}
	\newcommand{\ph}{\mathbf{PH}}
	\newcommand{\pspace}{\mathbf{PSPACE}}
	\newcommand{\fpspace}{\mathbf{FPSPACE}}
	\newcommand{\ptime}{\mathbf{P}}
	\newcommand{\fptime}{\mathbf{FP}}
	\newcommand{\nc}{\mathbf{NC}}
	\newcommand{\ac}{\mathbf{AC}}
	\newcommand{\exptime}{\mathbf{EXP}}
	\newcommand{\np}{\mathbf{NP}}
	\newcommand{\conp}{\mathbf{coNP}}
	\newcommand{\poly}{\mathrm{poly}}


	

\newcommand{\cnot}{\neg}
\newcommand{\cimp}{\supset}
\newcommand{\cor}{\vee}
\newcommand{\cand}{\wedge}
\newcommand{\ciff}{\equiv}

\newcommand{\size}[1]{|#1|}                        %% length of a word
%\newcommand{\mode}{\; \underline{\mbox{mod}}\;}   %% mod 2^{|x|}
\newcommand{\mode}{\ \underline{\mathrm{mod}}\ }   %% mod 2^{|x|}

%%%%% RULES %%%%%%
\newcommand{\lefrul}[1]{#1\text{-}\mathit{l}}
	\newcommand{\rigrul}[1]{#1\text{-}\mathit{r}}
	\newcommand{\id}{\it{id}}
	\newcommand{\cut}{\it{cut}}
	\newcommand{\wk}{\mathit{wk}}
	\newcommand{\cntr}{\mathit{cntr}}
	\newcommand{\rais}{\mathit{raise}}
 
	\newcommand{\LK}{\mathit{LK}}
% theories
	\newcommand{\theory}{\mathcal T}
	\newcommand{\system}{\mathcal S}
	
	%induction
%	\newcommand{\ind}{\mathit{PIND}}
%	\newcommand{\pind}{\mathit{PIND}}
	\newcommand{\cax}[2]{#1\text{-}#2}

	\newcommand{\sigone}{\Sigma^{\word^+}_1 }
	\newcommand{\sigzer}{\Sigma^{\word^+}_0}
	\newcommand{\bharith}{\mathcal A^1_2}
	%\newcommand{\arith}{I\sigone}
	
	
