### document non-atomic invariants

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 %\section{Derived constructions} \section{Derived constructions} \subsection{Non-atomic invariants} Sometimes it is necessary to maintain invariants that we need to open non-atomically. Clearly, for this mechanism to be sound we need something that prevents us from opening the same invariant twice. Access to these \emph{non-atomic invariants} is thus guarded by tokens that take the role that masks play for normal'', atomic invariants. One way to think about them is as thread-local invariants''. For every thread, we have a set of \emph{tokens}. By giving up a token, you can obtain the invariant, and vice versa. Such invariants can only be opened by their respective thread, and as a consequence they can be kept open around any sequence of expressions (\ie there is no restriction to atomic expressions). To tie the threads and the tokens together, every thread is assigned a \emph{thread ID}. Note that these thread IDs are completely fictional, there is no operational aspect to them. In principle, the tokens could move between threads; that's not an issue at all. Concretely, this is the monoid structure we need: \begin{align*} \textdom{TId} \eqdef{}& \nat \\ \textmon{TTok} \eqdef{}& \textdom{TId} \fpfn \pset{\textdom{InvName}}\\ \textmon{TDis} \eqdef{}& \textdom{TId} \fpfn \finpset{\textdom{InvName}} \end{align*} For every thread, there is a set of tokens designating which invariants are \emph{enabled} (closed). This corresponds to the mask of normal'' invariants. We re-use the structure given by namespaces for non-atomic invariants. Furthermore, there is a \emph{finite} set of invariants that is \emph{disabled} (open). We assume a global instance $\Gttok$ of \textmon{TTok}, and an instance $\Gtdis$ of $\textmon{TDis}$. Then we can define some sugar for owning tokens: \begin{align*} \TTokE\tid\mask \eqdef{}& \ownGhost{\Gttok}{ \mapsingleton\tid\mask : \textmon{TTok} } \\ \TTok\tid \eqdef{}& \TTokE\tid\top \end{align*} Next, we define non-atomic invariants. To simplify this construction,we piggy-back into normal'' invariants. \begin{align*} \NaInv\tid\namesp\prop \eqdef{}& \Exists \iname\in\namesp. \knowInv\namesp{\prop * \ownGhost\Gtdis{\set{\iname}} \lor \TTokE\tid{\set{\iname}}} \end{align*} We easily obtain: \begin{mathpar} \axiom {\TRUE \vs[\bot] \Exists\tid. \TTok\tid} \axiom {\TTokE\tid{\mask_1 \uplus \mask_2} \Lra \TTokE\tid{\mask_1} * \TTokE\tid{\mask_2}} \axiom {\later\prop \vs[\namesp] \always\NaInv\tid\namesp\prop} \axiom {\NaInv\tid\namesp\prop \proves \Acc[\namesp]{\TTokE\tid\namesp}{\later\prop}} \end{mathpar} from which we can derive \begin{mathpar} \infer {\namesp \subseteq \mask} {\NaInv\tid\namesp\prop \proves \Acc[\namesp]{\TTokE\tid\mask}{\later\prop * \TTokE\tid{\mask \setminus \namesp}}} \end{mathpar} % TODO: These need syncing with Coq % \subsection{STSs with interpretation}\label{sec:stsinterp} ... ...
 ... ... @@ -426,5 +426,14 @@ %% Stored Propositions \newcommand{\mapstoprop}{\mathrel{\kern-0.5ex\tikz[baseline=(m)]{\node at (0,0) (m){}; \draw[line cap=round] (0,0.16) -- (0,-0.004);}\kern-1.5ex\Ra}} % Non-atomic invariants \newcommand*\Gttok{\gname_\textrm{TTok}} \newcommand*\Gtdis{\gname_\textrm{TDis}} \newcommand*\tid{t} \newcommand\NaInv{\textlog{NAInv}^{#1.#2}(#3)} \newcommand*\TTok{{[}\textrm{T}:#1{]}} \newcommand*\TTokE{{[}\textrm{T}:#1.#2{]}} \endinput
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