Commit 82aee390 authored by Ralf Jung's avatar Ralf Jung

docs: describe the part of the model that works for any UPred

parent 7c354ddb
......@@ -2,6 +2,7 @@
\subsection{COFE}
This definition varies slightly from the original one in~\cite{catlogic}.
\begin{defn}[Chain]
Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N} \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$.
\end{defn}
......@@ -94,7 +95,8 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th
\All n, \melt, \meltB_1, \meltB_2.& \omit\rlap{$\melt \in \mval_n \land \melt \nequiv{n} \meltB_1 \mtimes \meltB_2 \Ra {}$} \\
&\Exists \meltC_1, \meltC_2. \melt = \meltC_1 \mtimes \meltC_2 \land \meltC_1 \nequiv{n} \meltB_1 \land \meltC_2 \nequiv{n} \meltB_2 \tagH{cmra-extend} \\
\text{where}\qquad\qquad\\
\melt \mincl \meltB \eqdef{}& \Exists \meltC. \meltB = \melt \mtimes \meltC \tagH{cmra-incl}
\melt \mincl \meltB \eqdef{}& \Exists \meltC. \meltB = \melt \mtimes \meltC \tagH{cmra-incl}\\
\melt \mincl[n] \meltB \eqdef{}& \Exists \meltC. \meltB \nequiv{n} \melt \mtimes \meltC \tagH{cmra-inclN}
\end{align*}
\end{defn}
......
......@@ -25,7 +25,7 @@ where $\mProp$ is the set of meta-level propositions, \eg Coq's \texttt{Prop}.
$\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$.
One way to understand this definition is to re-write it a little.
We start by defining the COFE of \emph{step-indexed propositions}:
We start by defining the COFE of \emph{step-indexed propositions}: For every step-index, we proposition either holds or does not hold.
\begin{align*}
\SProp \eqdef{}& \psetdown{\mathbb{N}} \\
\eqdef{}& \setComp{\prop \in \pset{\mathbb{N}}}{ \All n, m. n \geq m \Ra n \in \prop \Ra m \in \prop } \\
......@@ -149,6 +149,7 @@ We obtain the following frame-preserving updates:
{\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}}
\end{mathpar}
%TODO: These need syncing with Coq
% \subsection{Exclusive monoid}
% Given a set $X$, we define a monoid such that at most one $x \in X$ can be owned.
......@@ -373,8 +374,6 @@ We obtain the following frame-preserving updates:
% \subsection{STS with tokens monoid}
% \label{sec:stsmon}
% \ralf{This needs syncing with the Coq development.}
% Given a state-transition system~(STS) $(\STSS, \ra)$, a set of tokens $\STSS$, and a labeling $\STSL: \STSS \ra \mathcal{P}(\STST)$ of \emph{protocol-owned} tokens for each state, we construct a monoid modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens.
% The construction follows the idea of STSs as described in CaReSL \cite{caresl}.
......
......@@ -337,6 +337,7 @@ We can now derive the following rules for this derived form of the invariant ass
{\knowInv\namesp\prop \proves \propB \vs[\mask] \propC}
\end{mathpar}
% TODO: These need syncing with Coq
% \subsection{STSs with interpretation}\label{sec:stsinterp}
% Building on \Sref{sec:stsmon}, after constructing the monoid $\STSMon{\STSS}$ for a particular STS, we can use an invariant to tie an interpretation, $\pred : \STSS \to \Prop$, to the STS's current state, recovering CaReSL-style reasoning~\cite{caresl}.
......
......@@ -221,7 +221,7 @@
\newcommand*{\knowInv}[2]{\boxedassert{#2}[#1]}
\newcommand*{\ownGhost}[2]{\boxedassert[densely dashed]{#2}[#1]}
\newcommand*{\ownGGhost}[1]{\boxedassert[densely dashed]{#1}}
\newcommand{\ownM}[1]{\textlog{Own}(#1)}
\newcommand{\ownPhys}[1]{\textlog{Phy}(#1)}
%% View Shifts
......
......@@ -33,8 +33,8 @@
\endgroup\clearpage\begingroup
\input{logic}
\endgroup\clearpage\begingroup
%\input{model}
%\endgroup\clearpage\begingroup
\input{model}
\endgroup\clearpage\begingroup
\input{derived}
\endgroup\clearpage\begingroup
\printbibliography
......
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