Commit a072e355 authored by Ralf Jung's avatar Ralf Jung

docs: define RAs

parent 610698ec
......@@ -52,7 +52,31 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th
\subsection{RA}
\ralf{Copy this from the paper, when that one is more polished.}
\begin{defn}
A \emph{resource algebra} (RA) is a tuple \\
$(\monoid, \mval \subseteq \monoid, \mcore{-}:
\monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid)$ satisfying
\begin{align*}
\All \melt, \meltB, \meltC.& (\melt \mtimes \meltB) \mtimes \meltC = \melt \mtimes (\meltB \mtimes \meltC) \tagH{ra-assoc} \\
\All \melt, \meltB.& \melt \mtimes \meltB = \meltB \mtimes \melt \tagH{ra-comm} \\
\All \melt.& \mcore\melt \mtimes \melt = \melt \tagH{ra-core-id} \\
\All \melt.& \mcore{\mcore\melt} = \mcore\melt \tagH{ra-core-idem} \\
\All \melt, \meltB.& \melt \mincl \meltB \Ra \mcore\melt \mincl \mcore\meltB \tagH{ra-core-mono} \\
\All \melt, \meltB.& (\melt \mtimes \meltB) \in \mval \Ra \melt \in \mval \tagH{ra-valid-op} \\
\text{where}\qquad %\qquad\\
\melt \mincl \meltB \eqdef{}& \Exists \meltC. \meltB = \melt \mtimes \meltC \tagH{ra-incl}
\end{align*}
\end{defn}
\begin{defn}
It is possible to do a \emph{frame-preserving update} from $\melt \in \monoid$ to $\meltsB \subseteq \monoid$, written $\melt \mupd \meltsB$, if
\[ \All \melt_f. \melt \mtimes \melt_f \in \mval \Ra \Exists \meltB \in \meltsB. \meltB \mtimes \melt_f \in \mval \]
We further define $\melt \mupd \meltB \eqdef \melt \mupd \set\meltB$.
\end{defn}
\ralf{Copy the explanation from the paper, when that one is more polished.}
\subsection{CMRA}
......
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