From a7be766ab690dc5b5b3cbaf974a5687736530ccc Mon Sep 17 00:00:00 2001
From: Ralf Jung <jung@mpi-sws.org>
Date: Tue, 8 Mar 2016 16:28:53 +0100
Subject: [PATCH] add: discrete CMRAs
---
docs/algebra.tex | 27 ++++++++++++++++++---------
docs/iris.sty | 1 +
2 files changed, 19 insertions(+), 9 deletions(-)
diff --git a/docs/algebra.tex b/docs/algebra.tex
index 63cb6f818..a83e99cdb 100644
--- a/docs/algebra.tex
+++ b/docs/algebra.tex
@@ -48,7 +48,7 @@ Note that $\COFEs$ is cartesian closed.
\subsection{CMRA}
\begin{defn}
- A \emph{CMRA} is a tuple $(\monoid, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \mcore{-}: \monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid, (\mdiv) : \monoid \times \monoid \to \monoid)$ satisfying
+ A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}}, \mcore{-}: \monoid \to \monoid, (\mtimes) : \monoid \times \monoid \to \monoid, (\mdiv) : \monoid \times \monoid \to \monoid)$ satisfying
\begin{align*}
\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} \\
@@ -65,8 +65,6 @@ Note that $\COFEs$ is cartesian closed.
\end{align*}
\end{defn}
-Note that every RA is a CMRA, by picking the discrete COFE for the equivalence relation.
-
\ralf{Copy the rest of the explanation from the paper, when that one is more polished.}
\paragraph{The division operator $\mdiv$.}
@@ -115,10 +113,10 @@ This operation is needed to prove that $\later$ commutes with existential quanti
\begin{defn}
An element $\munit$ of a CMRA $\monoid$ is called the \emph{unit} of $\monoid$ if it satisfies the following conditions:
\begin{enumerate}[itemsep=0pt]
- \item $\munit$ is valid: \\ $\All n, \munit \in \mval_n$
- \item $\munit$ is left-identity of the operation: \\
+ \item $\munit$ is valid: \\ $\All n. \munit \in \mval_n$
+ \item $\munit$ is a left-identity of the operation: \\
$\All \melt \in M. \munit \mtimes \melt = \melt$
- \item $\munit$ is discrete
+ \item $\munit$ is a discrete element
\end{enumerate}
\end{defn}
@@ -130,16 +128,27 @@ This operation is needed to prove that $\later$ commutes with existential quanti
\end{defn}
Note that for RAs, this and the RA-based definition of a frame-preserving update coincide.
-\ralf{Describe discrete CMRAs, and how they correspond to RAs.}
+\begin{defn}
+ A CMRA $\monoid$ is \emph{discrete} if it satisfies the following conditions:
+ \begin{enumerate}[itemsep=0pt]
+ \item $\monoid$ is a discrete COFE
+ \item $\val$ ignores the step-index: \\
+ $\All \melt \in \monoid. \melt \in \mval_0 \Ra \All n, \melt \in \mval_n$
+ \item $f$ preserves CMRA inclusion:\\
+ $\All \melt, \meltB. \melt \mincl \meltB \Ra f(\melt) \mincl f(\meltB)$
+ \end{enumerate}
+\end{defn}
+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 $\mval$.
\begin{defn}
A function $f : M \to N$ between two CMRAs is \emph{monotone} if it satisfies the following conditions:
\begin{enumerate}[itemsep=0pt]
\item $f$ is non-expansive
\item $f$ preserves validity: \\
- $\All n, x \in M. x \in \mval_n \Ra f(x) \in \mval_n$
+ $\All n, \melt \in M. \melt \in \mval_n \Ra f(\melt) \in \mval_n$
\item $f$ preserves CMRA inclusion:\\
- $\All x, y. x \mincl y \Ra f(x) \mincl f(y)$
+ $\All \melt, \meltB. \melt \mincl \meltB \Ra f(\melt) \mincl f(\meltB)$
\end{enumerate}
\end{defn}
diff --git a/docs/iris.sty b/docs/iris.sty
index ea1ef7760..25f12226b 100644
--- a/docs/iris.sty
+++ b/docs/iris.sty
@@ -70,6 +70,7 @@
\newcommand{\pset}[1]{\wp(#1)} % Powerset
\newcommand{\psetdown}[1]{\wp^\downarrow(#1)}
+\newcommand{\finpset}[1]{\wp^\mathrm{fin}(#1)}
\newcommand{\Func}{F} % functor
--
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