
\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}
