.

betastable/.vscode/settings.json

@@ -0,0 +1,3 @@
+{
+    "discord.enabled": true
+}

betastable/report.tex

@@ -12,6 +12,8 @@
 \usepackage{amssymb}
 \usepackage{mathtools}
 \usepackage{hyperref}
+\usepackage{xfrac}
+\usepackage{stackengine}
 \usepackage[block]{calculation}
 \hypersetup{
 colorlinks=true,
@@ -32,6 +34,7 @@ pdfpagemode=FullScreen,
 \newtheorem{proposition}{Proposition}
 \newtheorem{definition}{Definition}
 \newtheorem{example}{Example}
+\newtheorem{remark}{Remark}
 
 
 \title{Consistent n-Approximators ($\beta$)}

betastable/sections/sketchjan20.tex

@@ -2,5 +2,62 @@
 \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}
+
+\begin{definitionOf}{setordering}
+    Given a preorder, $\langle \LL, \lte \rangle$
+    We define a set ordering $\langle \powersetO(\LL), \lte \rangle$
+    s.t. for any $X, Y \subseteq \LL$
+    \begin{align*}
+        X \lte Y \textrm{ iff } (X \upclosure) \supseteq Y
+    \end{align*}
+\end{definitionOf}
+
+\begin{definitionOf}{chaincompleteposet}
+    We call \Poset $\langle \LL, \lte \rangle$ {\em chain-complete} if $\lub \LL$ is unique and in $\LL$ and for ever chain $C \subseteq \LL$
+    \begin{enumerate}
+        \item $\lub \LL$ is unique and in $\LL$, and
+        \item for every chain $C \subseteq \LL$, we have $\lub C \in \LL$
+    \end{enumerate}
+\end{definitionOf}
+
+\begin{definitionOf}{antisymmequiv}
+    Given a set $\powersetO(\LL)^2$, define $\antisymmequiv{\powersetO(\LL)^2}$ to be the set obtained by removing any elements $x \in \powersetO(\LL)^2$ s.t. $x \not= x \upprecision$.
+\end{definitionOf}
+
+\begin{remark}
+    The \Preorder $\langle \powersetO(\LL)^2, \lteSetLPrecision \rangle$ lacks \Antisymmetry, thus it is not a \Poset.
+    We can construct and equivalence relation $\equiv$ based on the \Antisymmetry condition:
+        $x \equiv x' \textrm{ iff } x \lteSetLPrecision x' \textrm{ and } x' \lteSetLPrecision x$.
+    Each equivalence class $[x]$ can be uniquely represented by $x \upprecision$. Thus, another way of viewing 
+    $\langle \antisymmequiv{\powersetO(\LL)^2}, \lteSetLPrecision \rangle$
+    is as the quotient under the above equivalence relation.
+\end{remark}
+
+\begin{lemma}
+    Given $\langle \antisymmequiv{\powersetO(\LL)^2}, \lteSetLPrecision \rangle$, and $x, y \in  \powersetO(\LL)^2$
+    \begin{enumerate}
+        \item $x \lteSetRPrecision y$. 
+        \item if $x \lteSetLPrecision y$ then $x \lteSetLRPrecision y$. 
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+    (1) We have $\top_{\ltePrecision} \in y$. (2) Follows from (1).
+\end{proof}
+
+
+\begin{propositionOf}{chaincompletepreserve}
+    Let $\langle \LL, \lte \rangle$ be a \CompleteLattice and $o: \LL^2 \rightarrow \powersetO(\LL)$ \AnNdao.
+    The structure $\langle \antisymmequiv{o\image{R}}, \lteSetL \rangle$ is a \ChainCompletePoset where $R$ is one of the following:
+    \begin{enumerate}
+        \item\label{chaincompletepreserve:enum:a} $\LL^2$,
+        \item\label{chaincompletepreserve:enum:b} $\LLc$
+        \item\label{chaincompletepreserve:enum:c} $\postfixpointsOf(o)$
+        \item\label{chaincompletepreserve:enum:d} $\prefixpointsOf(S(o))$
+    \end{enumerate}
+\end{propositionOf}
+\begin{proof}
+    This should be simpler, the incoming is chain complete poset so should inject one ?
+
+
+\end{proof}

glossary.tex

@@ -57,12 +57,22 @@
 \let\powersetONoLink\powersetO{}
 \renewcommand{\powersetO}{\definitionLink{powerset}{\powersetONoLink}}
 
+\newcommand{\Preorder}{\definitionLink{preorder}{preorder}}
+\newcommand{\Antisymmetry}{\definitionLink{antisymmetry}{antisymmetry}}
 \newcommand{\Poset}{\definitionLink{poset}{poset}}
+\newcommand{\ChainCompletePoset}{\definitionLink{chaincompleteposet}{chain-complete poset}}
+\newcommand{\ChainCompletePosets}{\definitionLink{chaincompleteposet}{chain-complete posets}}
+\newcommand{\AChainCompletePoset}{\definitionLink{chaincompleteposet}{a chain-complete poset}}
 \let\lteNoLink\lte{}
 \renewcommand{\lte}{\definitionLink{poset}{\lteNoLink}}
 \let\lteSubNoLink\lteSub{}
 \renewcommand{\lteSub}[1]{\definitionLink{poset}{\lteSubNoLink{#1}}}
 
+\let\gteNoLink\gte{}
+\renewcommand{\gte}{\definitionLink{poset}{\gteNoLink}}
+\let\gteSubNoLink\gteSub{}
+\renewcommand{\gteSub}[1]{\definitionLink{poset}{\gteSubNoLink{#1}}}
+
 
 \newcommand{\Ndao}{\definitionLink{ndao}{ndao}}
 \newcommand{\AnNdao}{an \definitionLink{ndao}{ndao}}

notation.tex

@@ -1,11 +1,25 @@
+
 \newcommand{\fixpointsOf}{\textbf{\textit{fix}}}
+\newcommand{\prefixpointsOf}{\textbf{\textit{prefix}}}
+\newcommand{\postfixpointsOf}{\textbf{\textit{postfix}}}
 \newcommand{\LL}{\mathcal{L}}
 \newcommand{\lte}{\preceq}
+\newcommand{\ltePrecision}{\preceq_p^2}
 \newcommand{\lteSub}[1]{\lteNoLink_{#1}}
+\newcommand{\gteSub}[1]{\gteNoLink_{#1}}
+\newcommand{\gte}{\succeq}
+\newcommand{\lteSetR}{\preccurlyeq}
+\newcommand{\lteSetRPrecision}{\preccurlyeq_p^2}
+\newcommand{\lteSetL}{\curlyeqprec}
+\newcommand{\lteSetLPrecision}{\curlyeqprec_p^2}
+\newcommand{\lteSetLR}{\mathbin{\stackMath\topinset{\preccurlyeq}{\curlyeqprec}{}{}}}
+\newcommand{\lteSetLRPrecision}{\mathbin{\stackMath\topinset{\preccurlyeq}{\curlyeqprec}{}{}_p^2}}
 \newcommand{\image}[1]{[#1]}
 \newcommand{\define}{\coloneqq}
 \newcommand{\union}{\mathbin{\cup}}
+\newcommand{\unionBig}{\bigcup}
 \newcommand{\intersect}{\mathbin{\cap}}
+\newcommand{\intersectBig}{\bigcap}
 \newcommand{\symmetricdifference}{\mathbin{\Delta}}
 \newcommand{\glb}{\bigwedge}
 \newcommand{\lub}{\bigvee}
@@ -14,6 +28,7 @@
 \newcommand{\powerset}{\wp}
 \newcommand{\powersetO}{\wp^o}
 \newcommand{\upclosure}{\uparrow}
+\newcommand{\upprecision}{\uparrow^2_p}
 \newcommand{\downclosure}{\downarrow}
 \newcommand{\partialApp}{\cdot}
 \newcommand{\lfp}{\textbf{lfp}}
@@ -23,3 +38,12 @@
 \newcommand{\Sdetbeta}{S^{\beta}}
 \let\restrictionWithSpaces\restriction
 \renewcommand{\restriction}{{\restrictionWithSpaces}}
+
+\newcommand{\quotient}[2]{\sfrac{#1}{#2}}
+
+\newcommand{\antisymmequiv}[1]{\quotient{#1}{\upprecision}}
+
+\newcommand{\LLc}{\LLNoLink^{c}}
+
+\newcommand{\projL}{\Pi_{1}}
+\newcommand{\projR}{\Pi_{2}}