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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}
\newtheorem{example}{Example}


\title{Consistent n-Approximators ($\beta$)}
\author{}
\date{}

\begin{document}

\maketitle

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


\begin{definition}
    \AnNdaoO is {\em extra consistent} if for every $\lte$-\Prefixpoint $y$ of $o(x, \cdot)_2$, $x \lte y$.
\end{definition}



\begin{definition}
    "beta" stable revision
    \begin{align*}
        S(o)(x, y)_2 \define \minLattice_{\lte}(\fixpointsOf(o(x, \cdot)) \setminus ((y \downclosure) \setminus (x \upclosure)))
    \end{align*}
\end{definition}


\begin{theorem}
    For an extra consistent \Ndao $o$, regular stable revision is equivalent to beta stable revision
\end{theorem}

\begin{example}
    SHowing without ultra consistency
    \begin{align*}
        o(x, y) \define (\{ \bot \}, \{ \bot \})
    \end{align*}
    below properties don't hold
\end{example}

\begin{proposition}\label{prop:ultra-consistency}
    Works fo rdouble sided ordering
    Given \AnNdaoO $o: \LL^2 \rightarrow \powersetO(\L)^2$ that is \Monotone from $\lte_p^2$ to $<_p^2$,\ we have for any consistent pair $(x, y) \in \LL^2$,
    \begin{align*}
        o(x, y) \lte_p^2 (o(x, y)_2, o(x, y)_1)
    \end{align*}
\end{proposition}

\begin{lemma}
    This probably needs double sides :()
    Given an \Ndao $o$, if $y$ is a \Prefixpoint of $o(x, \cdot)_2$, then for some $y' \in o(x, \cdot)_2$, we have
        $x \lte y' \lte y$.
\end{lemma}
\begin{proof}
    begin
\end{proof}


% 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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