.

betastable/report.out

@@ -1,3 +1,5 @@
 \BOOKMARK [1][-]{section.1}{\376\377\000P\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i\000e\000s}{}% 1
 \BOOKMARK [1][-]{section.2}{\376\377\000L\000a\000t\000t\000i\000c\000e\000\040\000T\000h\000e\000o\000r\000y}{}% 2
-\BOOKMARK [1][-]{section.3}{\376\377\000D\000e\000t\000e\000r\000n\000m\000i\000n\000i\000s\000t\000i\000c\000\040\000A\000F\000T}{}% 3
+\BOOKMARK [1][-]{section.3}{\376\377\000D\000e\000t\000e\000r\000m\000i\000n\000i\000s\000t\000i\000c\000\040\000A\000F\000T}{}% 3
+\BOOKMARK [1][-]{section.4}{\376\377\000S\000k\000e\000t\000c\000h\000\040\000N\000o\000v\000\040\0002\0000}{}% 4
+\BOOKMARK [1][-]{section.5}{\376\377\000S\000k\000e\000t\000c\000h\000\040\000J\000a\000n\000\040\0002\0000}{}% 5

betastable/report.tex

@@ -45,50 +45,11 @@ pdfpagemode=FullScreen,
 \section{Preliminaries}
 \input{sections/preliminaries.tex}
 
+\section{Sketch Nov 20}
+\input{sections/sketchnov20.tex}
 
-\begin{definition}
+\section{Sketch Jan 20}
-    \AnNdaoO is {\em extra consistent} if for every $\lte$-\Prefixpoint $y$ of $o(x, \cdot)_2$, $x \lte y$.
+\input{sections/sketchjan20.tex}
-\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*}

betastable/sections/preliminaries.tex

@@ -16,7 +16,7 @@ With abuse to notation, when given a set of functions $F$, we write $\bigcup \{
 
 
 
-\section{Deternministic AFT}
+\section{Deterministic AFT}
 
 The original definition of approximators from Denecker et al.\ \cite{denecker2000approximations}, which has been relaxed following Heyninck et al.\ \cite{nondet2}.
 

betastable/sections/sketchjan20.tex

@@ -0,0 +1,6 @@
+
+\begin{definitionOf}{chain}
+    Given a \Poset $\langle \LL, \lte \rangle$, a {\em chain} is a (possibly empty) set $C \subseteq \LL$ that is totally ordered, that is, 
+    for all $x, y \in C$ either $x \lte y$ or $y \lte x$.
+    
+\end{definitionOf}

betastable/sections/sketchnov20.tex

@@ -0,0 +1,43 @@
+
+\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}

glossary.tex

@@ -73,3 +73,5 @@
 
 \newcommand{\minLattice}{\bm{min}}
 \newcommand{\maxLattice}{\bm{max}}
+
+\newcommand{\Chain}{\definitionLink{chain}{chain}}