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\input{../notation.tex}
\input{../glossary.tex}
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\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
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\title{Consistent n-Approximators ($\beta$)}
\author{}
\date{}

\begin{document}

\maketitle

\section{Preliminaries}
\input{sections/preliminaries.tex}

\section{Sketch Nov 20}
\input{sections/sketchnov20.tex}

\section{Sketch Jan 20}
\input{sections/sketchjan20.tex}

% iven an \gls{ndao} $o$, its {\em $\beta$-n-stable fixpoints} are fixpoints of the following    
%     \begin{align*}
%         B^o_{high}(x) &\define \{ a ~|~ a \in o({x}, a), \neg \exists a',
%         \\ &\hspace{1.5cm}(\boxed{x \preceq_{}}~ a' \prec_{} a) \land (a' \in o({x}, a))) \}\\
%         S(o)(x, y) &\define (C^{o}_{low}(y), B^{o}_{high}(x))
%     \end{align*}}

% \newcommand{\betastablefixpoint}{\hyperlink{glossary:betastablefixpoint}{$\beta$-stable fixpoint}}
% \newglossaryentry{betastablefixpoint}{
%     name={$\beta$-stable fixpoint},
%     description={
%         An \gls{interpretation} $(T, P)$ is a {\em $\alpha$-stable fixpoint} (or a $\beta$-stable fixpoint) if it is a \gls{fixpoint} of some $h \in H$ and for each $h' \in H$, none of the following hold
%         \begin{enumerate}[(i.)]
%             \item $\stablerevisionoperator(h')(T, P)_1 \prec_{} T$,
%             \item ($\alpha$-stable only)~$\stablerevisionoperator(h')(T, P)_2 \prec_{} P$, nor
%             \item ($\beta$-stable only) $\exists Z \in \L, T \preceq_{} (h'(T, Z)_2 = Z) \prec_{} P$
%         \end{enumerate}
% }}






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