Given some countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn\monoid$ is equipped with a COFE and CMRA structure by lifting everything pointwise.
% If $\melt_f = \munit$, then the statement is trivial.
% If $\melt_f \neq \munit$, then if $\melt = \munit$, we must have $\meltB = \munit$ and we are done.
% Similar so for $\meltB = \munit$.
% So let $\melt \neq \munit \neq \meltB$ and $\melt_f \mtimes \melt = \melt_f \mtimes \meltB \neq \mzero$.
% It follows immediately that $\melt = \melt_f = \meltB$.
% \end{proof}
% \subsection{Finite Powerset Monoid}
...
...
@@ -137,35 +159,6 @@ $\oneshotm(-)$ is a locally non-expansive bifunctor from $\CMRAs$ to $\CMRAs$.
% The other direction works the same way.
% \end{proof}
% \subsection{Product monoid}
% \label{sec:prodm}
% Given a family $(M_i)_{i \in I}$ of monoids ($I$ countable), we construct a product monoid.
% Let $\prod_{i \in I} M_i$ be the monoid with carrier $\prod_{i \in I} \mcarp{M_i}$ and point-wise multiplication, non-zero when \emph{all} individual multiplications are non-zero.
% For $f \in \prod_{i \in I} \mcarp{M_i}$, we write $f[i \mapsto a]$ for the disjoint union $f \uplus [i \mapsto a]$.
% Frame-preserving updates on the $M_i$ lift to the product:
% By the definition of multiplication, this means that for all $i \in I$ we have $\melt_f(i) \mtimes \melt(i) = \melt_f(i) \mtimes \meltB(i) \neq \mzero_{M_i}$.
% As all base monoids are cancellative, we obtain $\forall i \in I.\; \melt(i) = \meltB(i)$ from which we immediately get $\melt = \meltB$.
% \end{proof}
% \subsection{Fractional monoid}
% \label{sec:fracm}
...
...
@@ -214,32 +207,6 @@ $\oneshotm(-)$ is a locally non-expansive bifunctor from $\CMRAs$ to $\CMRAs$.
% The first is trivial, the second follows from cancellativitiy of $M$.
% \end{proof}
% \subsection{Finite partial function monoid}
% \label{sec:fpfunm}
% Given a countable set $X$ and a monoid $M$, we construct a monoid representing finite partial functions from $X$ to (non-unit, non-zero elements of) $M$.
% \ralf{all outdated}
% Let ${X} \fpfn {M}$ be the product monoid $\prod_{x \in X} M$, as defined in \secref{sec:prodm} but restricting the carrier to functions $f$ where the set $\dom(f) \eqdef \{ x \mid f(x) \neq \munit_M \}$ is finite.
% This is well-defined as the set of these $f$ contains the unit and is closed under multiplication.
% (We identify finite partial functions from $X$ to $\mcarp{M}\setminus\{\munit_M\}$ and total functions from $X$ to $\mcarp{M}$ with finite $\munit_M$-support.)
% Assume some $g \sep f$. Since $\dom(f \mtimes g)$ is finite, there will be some undefined element $x \notin \dom(f \mtimes g)$. Let $f' \eqdef f[x \mapsto a]$. This is compatible with $g$, so we are done.
% \end{proof}
% We write $[x \mapsto a]$ for the function mapping $x$ to $a$ and everything else in $X$ to $\munit$.
@@ -205,38 +205,37 @@ We can derive some specialized forms of the lifting axioms for the operational s
\ralf{Add these.}
\subsection{Global functor and ghost ownership}
\ralf{Describe this.}
% \subsection{Global monoid}
Hereinafter we assume the global CMRA functor (served up as a parameter to Iris) is obtained from a family of functors $(F_i)_{i \in I}$ for some finite $I$ by picking
We don't care so much about what concretely $\textdom{GhName}$ is, as long as it is countable and infinite.
With $M_i \eqdef F_i(\iProp)$, we write $\ownGhost{\gname}{\melt : M_i}$ (or just $\ownGhost{\gname}{\melt}$ if $M_i$ is clear from the context) for $\ownGGhost{[i \mapsto[\gname\mapsto\melt]]}$.
In other words, $\ownGhost{\gname}{\melt : M_i}$ asserts that in the current state of monoid $M_i$, the ``ghost location'' $\gname$ is allocated and we own piece $\melt$.
% Hereinafter we assume the global monoid (served up as a parameter to Iris) is obtained from a family of monoids $(M_i)_{i \in I}$ by first applying the construction for finite partial functions to each~(\Sref{sec:fpfunm}), and then applying the product construction~(\Sref{sec:prodm}):
% We don't care so much about what concretely $\textdom{GhName}$ is, as long as it is countable and infinite.
% We write $\ownGhost{\gname}{\melt : M_i}$ (or just $\ownGhost{\gname}{\melt}$ if $M_i$ is clear from the context) for $\ownGGhost{[i \mapsto [\gname \mapsto \melt]]}$ when $\melt \in \mcarp {M_i}$, and for $\FALSE$ when $\melt = \mzero_{M_i}$.
% In other words, $\ownGhost{\gname}{\melt : M_i}$ asserts that in the current state of monoid $M_i$, the name $\gname$ is allocated and has at least value $\melt$.
% From~\ruleref{FpUpd} and the multiplications and frame-preserving updates in~\Sref{sec:prodm} and~\Sref{sec:fpfunm}, we have the following derived rules.
From~\ruleref{pvs-update}, \ruleref{vs-update} and the frame-preserving updates in~\Sref{sec:prodm} and~\Sref{sec:fpfnm}, we have the following derived rules.
\begin{mathparpagebreakable}
\inferH{NewGhostStrong}{\text{$G$ infinite}}
{\TRUE\vs\Exists\gname\in G. \ownGhost\gname{\melt : M_i}
