consistent letter for COFEs

parent 22e1e914
 ... @@ -3,11 +3,11 @@ ... @@ -3,11 +3,11 @@ \subsection{COFE} \subsection{COFE} \begin{defn}[Chain] \begin{defn}[Chain] Given some set $T$ and an indexed family $({\nequiv{n}} \subseteq T \times T)_{n \in \mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N} \to T$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$. Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N} \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$. \end{defn} \end{defn} \begin{defn} \begin{defn} A \emph{complete ordered family of equivalences} (COFE) is a tuple $(T, ({\nequiv{n}} \subseteq T \times T)_{n \in \mathbb{N}}, \lim : \chain(T) \to T)$ satisfying A \emph{complete ordered family of equivalences} (COFE) is a tuple $(\cofe, ({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \mathbb{N}}, \lim : \chain(\cofe) \to \cofe)$ satisfying \begin{align*} \begin{align*} \All n. (\nequiv{n}) ~& \text{is an equivalence relation} \tagH{cofe-equiv} \\ \All n. (\nequiv{n}) ~& \text{is an equivalence relation} \tagH{cofe-equiv} \\ \All n, m.& n \geq m \Ra (\nequiv{n}) \subseteq (\nequiv{m}) \tagH{cofe-mono} \\ \All n, m.& n \geq m \Ra (\nequiv{n}) \subseteq (\nequiv{m}) \tagH{cofe-mono} \\ ... @@ -19,16 +19,16 @@ ... @@ -19,16 +19,16 @@ \ralf{Copy the explanation from the paper, when that one is more polished.} \ralf{Copy the explanation from the paper, when that one is more polished.} \begin{defn} \begin{defn} An element $x \in A$ of a COFE is called \emph{discrete} if An element $x \in \cofe$ of a COFE is called \emph{discrete} if $\All y \in A. x \nequiv{0} y \Ra x = y$ $\All y \in \cofe. x \nequiv{0} y \Ra x = y$ A COFE $A$ is called \emph{discrete} if all its elements are discrete. A COFE $A$ is called \emph{discrete} if all its elements are discrete. \end{defn} \end{defn} \begin{defn} \begin{defn} A function $f : A \to B$ between two COFEs is \emph{non-expansive} if A function $f : \cofe \to \cofeB$ between two COFEs is \emph{non-expansive} if $\All n, x \in A, y \in A. x \nequiv{n} y \Ra f(x) \nequiv{n} f(y)$ $\All n, x \in \cofe, y \in \cofe. x \nequiv{n} y \Ra f(x) \nequiv{n} f(y)$ It is \emph{contractive} if It is \emph{contractive} if $\All n, x \in A, y \in A. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x)$ $\All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x)$ \end{defn} \end{defn} \begin{defn} \begin{defn} ... @@ -135,20 +135,20 @@ Note that for RAs, this and the RA-based definition of a frame-preserving update ... @@ -135,20 +135,20 @@ Note that for RAs, this and the RA-based definition of a frame-preserving update \item $\val$ ignores the step-index: \\ \item $\val$ ignores the step-index: \\ $\All \melt \in \monoid. \melt \in \mval_0 \Ra \All n, \melt \in \mval_n$ $\All \melt \in \monoid. \melt \in \mval_0 \Ra \All n, \melt \in \mval_n$ \item $f$ preserves CMRA inclusion:\\ \item $f$ preserves CMRA inclusion:\\ $\All \melt, \meltB. \melt \mincl \meltB \Ra f(\melt) \mincl f(\meltB)$ $\All \melt \in \monoid, \meltB \in \monoid. \melt \leq \meltB \Ra f(\melt) \leq f(\meltB)$ \end{enumerate} \end{enumerate} \end{defn} \end{defn} Note that every RA is a discrete CMRA, by picking the discrete COFE for the equivalence relation. Note that every RA is a discrete CMRA, by picking the discrete COFE for the equivalence relation. Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE structure, as well as the step-index of $\mval$. Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE structure, as well as the step-index of $\mval$. \begin{defn} \begin{defn} A function $f : M \to N$ between two CMRAs is \emph{monotone} if it satisfies the following conditions: A function $f : \monoid_1 \to \monoid_2$ between two CMRAs is \emph{monotone} if it satisfies the following conditions: \begin{enumerate}[itemsep=0pt] \begin{enumerate}[itemsep=0pt] \item $f$ is non-expansive \item $f$ is non-expansive \item $f$ preserves validity: \\ \item $f$ preserves validity: \\ $\All n, \melt \in M. \melt \in \mval_n \Ra f(\melt) \in \mval_n$ $\All n, \melt \in \monoid_1. \melt \in \mval_n \Ra f(\melt) \in \mval_n$ \item $f$ preserves CMRA inclusion:\\ \item $f$ preserves CMRA inclusion:\\ $\All \melt, \meltB. \melt \mincl \meltB \Ra f(\melt) \mincl f(\meltB)$ $\All \melt \in \monoid_1, \meltB \in \monoid_1. \melt \leq \meltB \Ra f(\melt) \leq f(\meltB)$ \end{enumerate} \end{enumerate} \end{defn} \end{defn} ... ...
 ... @@ -111,6 +111,8 @@ ... @@ -111,6 +111,8 @@ \newcommand{\iProp}{\textdom{iProp}} \newcommand{\iProp}{\textdom{iProp}} \newcommand{\Wld}{\textdom{Wld}} \newcommand{\Wld}{\textdom{Wld}} \newcommand{\cofe}{T} \newcommand{\cofeB}{U} \newcommand{\COFEs}{\mathcal{U}} % category of COFEs \newcommand{\COFEs}{\mathcal{U}} % category of COFEs \newcommand{\iFunc}{\Sigma} \newcommand{\iFunc}{\Sigma} ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!