Commit cd3a1805 authored by Ralf Jung's avatar Ralf Jung

docs: notation for nonexp functions

parent d7cae999
Pipeline #299 passed with stage
......@@ -25,7 +25,7 @@
\end{defn}
\begin{defn}
A function $f : \cofe \to \cofeB$ between two COFEs is \emph{non-expansive} if
A function $f : \cofe \to \cofeB$ between two COFEs is \emph{non-expansive} (written $f : \cofe \nfn \cofeB$) if
\[\All n, x \in \cofe, y \in \cofe. x \nequiv{n} y \Ra f(x) \nequiv{n} f(y) \]
It is \emph{contractive} if
\[ \All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x) \]
......@@ -49,8 +49,9 @@ Note that $\COFEs$ is cartesian closed.
\subsection{CMRA}
\begin{defn}
A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \mcore{-}: \monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid, (\mdiv) : \monoid \times \monoid \to \monoid)$ satisfying
A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \mcore{-}: \monoid \nfn \monoid, (\mtimes) : \monoid \times \monoid \nfn \monoid, (\mdiv) : \monoid \times \monoid \nfn \monoid)$ satisfying
\begin{align*}
\All n, \melt, \meltB.& \melt \nequiv{n} \meltB \land \melt\in\mval_n \Ra \meltB\in\mval_n \tagH{cmra-valid-nonexp} \\
\All n, m.& n \geq m \Ra V_n \subseteq V_m \tagH{cmra-valid-mono} \\
\All \melt, \meltB, \meltC.& (\melt \mtimes \meltB) \mtimes \meltC = \melt \mtimes (\meltB \mtimes \meltC) \tagH{cmra-assoc} \\
\All \melt, \meltB.& \melt \mtimes \meltB = \meltB \mtimes \melt \tagH{cmra-comm} \\
......@@ -143,7 +144,7 @@ Note that every RA is a discrete CMRA, by picking the discrete COFE for the equi
Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE structure, as well as the step-index of $\mval$.
\begin{defn}
A function $f : \monoid_1 \to \monoid_2$ 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} (written $f : \monoid_1 \monra \monoid_2$) if it satisfies the following conditions:
\begin{enumerate}[itemsep=0pt]
\item $f$ is non-expansive
\item $f$ preserves validity: \\
......
......@@ -52,6 +52,7 @@
\newcommand{\Ra}{\Rightarrow}
\newcommand{\Lra}{\Leftrightarrow}
\newcommand\monra{\xrightarrow{\kern-0.25ex\textrm{mon}\kern-0.25ex}}
\newcommand\nfn{\xrightarrow{\kern-0.15ex\textrm{n}\kern-0.05ex}}
\newcommand{\eqdef}{\triangleq}
\newcommand{\bnfdef}{\vcentcolon\vcentcolon=}
......
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