44 lines
1.3 KiB
TeX
44 lines
1.3 KiB
TeX
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\begin{definition}
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\AnNdaoO is {\em extra consistent} if for every $\lte$-\Prefixpoint $y$ of $o(x, \cdot)_2$, $x \lte y$.
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\end{definition}
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\begin{definition}
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"beta" stable revision
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\begin{align*}
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S(o)(x, y)_2 \define \minLattice_{\lte}(\fixpointsOf(o(x, \cdot)) \setminus ((y \downclosure) \setminus (x \upclosure)))
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\end{align*}
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\end{definition}
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\begin{theorem}
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For an extra consistent \Ndao $o$, regular stable revision is equivalent to beta stable revision
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\end{theorem}
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\begin{example}
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SHowing without ultra consistency
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\begin{align*}
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o(x, y) \define (\{ \bot \}, \{ \bot \})
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\end{align*}
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below properties don't hold
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\end{example}
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\begin{proposition}\label{prop:ultra-consistency}
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Works fo rdouble sided ordering
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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 $(x, y) \in \LL^2$,
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\begin{align*}
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o(x, y) \lte_p^2 (o(x, y)_2, o(x, y)_1)
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\end{align*}
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\end{proposition}
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\begin{lemma}
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This probably needs double sides :()
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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
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$x \lte y' \lte y$.
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\end{lemma}
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\begin{proof}
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begin
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\end{proof}
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