From 8dbf76724a7fe1f1e25b8f463e31b635ed79a5b5 Mon Sep 17 00:00:00 2001
From: Ralf Jung
Date: Thu, 23 Nov 2017 20:59:38 +0100
Subject: [PATCH] Appendix: CMRA morphisms are homomorphisms
---
CHANGELOG.md | 2 +-
docs/algebra.tex | 10 ++++++----
2 files changed, 7 insertions(+), 5 deletions(-)
diff --git a/CHANGELOG.md b/CHANGELOG.md
index a3305d69..de7d68ab 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -8,7 +8,7 @@ Coq development, but not every API-breaking change is listed. Changes marked
Changes in and extensions of the theory:
* [#] Add new modality: ■ ("plainly").
-* [#] Camera morphisms have to be homomorphisms, not just monotone functions.
+* Camera morphisms have to be homomorphisms, not just monotone functions.
* Add a proof that `f` has a fixed point if `f^k` is contractive.
* Constructions for least and greatest fixed points over monotone predicates
(defined in the logic of Iris using impredicative quantification).
diff --git a/docs/algebra.tex b/docs/algebra.tex
index 972778a8..9cfed25e 100644
--- a/docs/algebra.tex
+++ b/docs/algebra.tex
@@ -213,14 +213,16 @@ Note that for RAs, this and the RA-based definition of a frame-preserving update
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 $f : \monoid_1 \to \monoid_2$ between two CMRAs is \emph{monotone} (written $f : \monoid_1 \monra \monoid_2$) if it satisfies the following conditions:
+\begin{defn}[CMRA homomorphism]
+ A function $f : \monoid_1 \to \monoid_2$ between two CMRAs is \emph{a CMRA homomorphism} if it satisfies the following conditions:
\begin{enumerate}[itemsep=0pt]
\item $f$ is non-expansive
+ \item $f$ commutes with composition:\\
+ $\All \melt_1 \in \monoid_1, \melt_2 \in \monoid_1. f(\melt_1) \mtimes f(\melt_2) = f(\melt_1 \mtimes \melt_2)$
+ \item $f$ commutes with the core:\\
+ $\All \melt \in \monoid_1. \mcore{f(\melt)} = f(\mcore{\melt})$
\item $f$ preserves validity: \\
$\All n, \melt \in \monoid_1. \melt \in \mval_n \Ra f(\melt) \in \mval_n$
- \item $f$ preserves CMRA inclusion:\\
- $\All \melt \in \monoid_1, \meltB \in \monoid_1. \melt \mincl \meltB \Ra f(\melt) \mincl f(\meltB)$
\end{enumerate}
\end{defn}
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