\newcommand{\fixpointsOf}{\textbf{\textit{fix}}}
\newcommand{\prefixpointsOf}{\textbf{\textit{prefix}}}
\newcommand{\postfixpointsOf}{\textbf{\textit{postfix}}}
\newcommand{\LL}{\mathcal{L}}
\newcommand{\lte}{\preceq}
\newcommand{\ltePrecision}{\preceq_p^2}
\newcommand{\lteSub}[1]{\lteNoLink_{#1}}
\newcommand{\gteSub}[1]{\gteNoLink_{#1}}
\newcommand{\gte}{\succeq}
\newcommand{\lteSetR}{\preccurlyeq}
\newcommand{\lteSetRPrecision}{\preccurlyeq_p^2}
\newcommand{\lteSetL}{\curlyeqprec}
\newcommand{\lteSetLPrecision}{\curlyeqprec_p^2}
\newcommand{\lteSetLR}{\mathbin{\stackMath\topinset{\preccurlyeq}{\curlyeqprec}{}{}}}
\newcommand{\lteSetLRPrecision}{\mathbin{\stackMath\topinset{\preccurlyeq}{\curlyeqprec}{}{}_p^2}}
\newcommand{\image}[1]{[#1]}
\newcommand{\define}{\coloneqq}
\newcommand{\union}{\mathbin{\cup}}
\newcommand{\unionBig}{\bigcup}
\newcommand{\intersect}{\mathbin{\cap}}
\newcommand{\intersectBig}{\bigcap}
\newcommand{\symmetricdifference}{\mathbin{\Delta}}
\newcommand{\glb}{\bigwedge}
\newcommand{\lub}{\bigvee}
\newcommand{\glbBinary}{\land}
\newcommand{\lubBinary}{\lor}
\newcommand{\powerset}{\wp}
\newcommand{\powersetO}{\wp^o}
\newcommand{\upclosure}{\uparrow}
\newcommand{\upprecision}{\uparrow^2_p}
\newcommand{\downclosure}{\downarrow}
\newcommand{\partialApp}{\cdot}
\newcommand{\lfp}{\textbf{lfp}}
\newcommand{\latticeinterval}[2]{[#1, #2]}

\newcommand{\Sdet}{S}
\newcommand{\Sdetbeta}{S^{\beta}}
\let\restrictionWithSpaces\restriction
\renewcommand{\restriction}{{\restrictionWithSpaces}}

\newcommand{\quotient}[2]{\sfrac{#1}{#2}}

\newcommand{\antisymmequiv}[1]{\quotient{#1}{\upprecision}}

\newcommand{\LLc}{\LLNoLink^{c}}

\newcommand{\projL}{\Pi_{1}}
\newcommand{\projR}{\Pi_{2}}