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\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}

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% TODO: These need syncing with Coq
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% \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}.

% An STS invariant asserts authoritative ownership of an STS's current state and that state's interpretation:
% \begin{align*}
%   \STSInv(\STSS, \pred, \gname) \eqdef{}& \Exists s \in \STSS. \ownGhost{\gname}{(s, \STSS, \emptyset):\STSMon{\STSS}} * \pred(s) \\
%   \STS(\STSS, \pred, \gname, \iname) \eqdef{}& \knowInv{\iname}{\STSInv(\STSS, \pred, \gname)}
% \end{align*}

% We can specialize \ruleref{NewInv}, \ruleref{InvOpen}, and \ruleref{InvClose} to STS invariants:
% \begin{mathpar}
%  \inferH{NewSts}
%   {\infinite(\mask)}
%   {\later\pred(s) \vs[\mask] \Exists \iname \in \mask, \gname.   \STS(\STSS, \pred, \gname, \iname) * \ownGhost{\gname}{(s, \STST \setminus \STSL(s)) : \STSMon{\STSS}}}
%  \and
%  \axiomH{StsOpen}
%   {  \STS(\STSS, \pred, \gname, \iname) \vdash \ownGhost{\gname}{(s_0, T) : \STSMon{\STSS}} \vsE[\{\iname\}][\emptyset] \Exists s\in \upclose(\{s_0\}, T). \later\pred(s) * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T):\STSMon{\STSS}}}
%  \and
%  \axiomH{StsClose}
%   {  \STS(\STSS, \pred, \gname, \iname), (s, T) \ststrans (s', T')  \proves \later\pred(s') * \ownGhost{\gname}{(s, S, T):\STSMon{\STSS}} \vs[\emptyset][\{\iname\}] \ownGhost{\gname}{(s', T') : \STSMon{\STSS}} }
% \end{mathpar}
% \begin{proof}
% \ruleref{NewSts} uses \ruleref{NewGhost} to allocate $\ownGhost{\gname}{(s, \upclose(s, T), T) : \STSMon{\STSS}}$ where $T \eqdef \STST \setminus \STSL(s)$, and \ruleref{NewInv}.

% \ruleref{StsOpen} just uses \ruleref{InvOpen} and \ruleref{InvClose} on $\iname$, and the monoid equality $(s, \upclose(\{s_0\}, T), T) = (s, \STSS, \emptyset) \mtimes (\munit, \upclose(\{s_0\}, T), T)$.

% \ruleref{StsClose} applies \ruleref{StsStep} and \ruleref{InvClose}.
% \end{proof}
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% Using these view shifts, we can prove STS variants of the invariant rules \ruleref{Inv} and \ruleref{VSInv}~(compare the former to CaReSL's island update rule~\cite{caresl}):
% \begin{mathpar}
%  \inferH{Sts}
%   {\All s \in \upclose(\{s_0\}, T). \hoare{\later\pred(s) * P}{\expr}{\Ret \val. \Exists s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * Q}[\mask]
%    \and \physatomic{\expr}}
%   {  \STS(\STSS, \pred, \gname, \iname) \vdash \hoare{\ownGhost{\gname}{(s_0, T):\STSMon{\STSS}} * P}{\expr}{\Ret \val. \Exists s', T'. \ownGhost{\gname}{(s', T'):\STSMon{\STSS}} * Q}[\mask \uplus \{\iname\}]}
%  \and
%  \inferH{VSSts}
%   {\forall s \in \upclose(\{s_0\}, T).\; \later\pred(s) * P \vs[\mask_1][\mask_2] \exists s', T'.\; (s, T) \ststrans (s', T') * \later\pred(s') * Q}
%   {  \STS(\STSS, \pred, \gname, \iname) \vdash \ownGhost{\gname}{(s_0, T):\STSMon{\STSS}} * P \vs[\mask_1 \uplus \{\iname\}][\mask_2 \uplus \{\iname\}] \Exists s', T'. \ownGhost{\gname}{(s', T'):\STSMon{\STSS}} * Q}
% \end{mathpar}

% \begin{proof}[Proof of \ruleref{Sts}]\label{pf:sts}
%  We have to show
%  \[\hoare{\ownGhost{\gname}{(s_0, T):\STSMon{\STSS}} * P}{\expr}{\Ret \val. \Exists s', T'. \ownGhost{\gname}{(s', T'):\STSMon{\STSS}} * Q}[\mask \uplus \{\iname\}]\]
%  where $\val$, $s'$, $T'$ are free in $Q$.
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%  First, by \ruleref{ACsq} with \ruleref{StsOpen} and \ruleref{StsClose} (after moving $(s, T) \ststrans (s', T')$ into the view shift using \ruleref{VSBoxOut}), it suffices to show
%  \[\hoareV{\Exists s\in \upclose(\{s_0\}, T). \later\pred(s) * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)} * P}{\expr}{\Ret \val. \Exists s, T, S, s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * \ownGhost{\gname}{(s, S, T):\STSMon{\STSS}} * Q(\val, s', T')}[\mask]\]
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%  Now, use \ruleref{Exist} to move the $s$ from the precondition into the context and use \ruleref{Csq} to (i)~fix the $s$ and $T$ in the postcondition to be the same as in the precondition, and (ii)~fix $S \eqdef \upclose(\{s_0\}, T)$.
%  It remains to show:
%  \[\hoareV{s\in \upclose(\{s_0\}, T) * \later\pred(s) * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)} * P}{\expr}{\Ret \val. \Exists s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)} * Q(\val, s', T')}[\mask]\]
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%  Finally, use \ruleref{BoxOut} to move $s\in \upclose(\{s_0\}, T)$ into the context, and \ruleref{Frame} on $\ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)}$:
%  \[s\in \upclose(\{s_0\}, T) \vdash \hoare{\later\pred(s) * P}{\expr}{\Ret \val. \Exists s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * Q(\val, s', T')}[\mask]\]
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%  This holds by our premise.
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% \end{proof}
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% % \begin{proof}[Proof of \ruleref{VSSts}]
% % This is similar to above, so we only give the proof in short notation:

% % \hproof{%
% % 	Context: $\knowInv\iname{\STSInv(\STSS, \pred, \gname)}$ \\
% % 	\pline[\mask_1 \uplus \{\iname\}]{
% % 		\ownGhost\gname{(s_0, T)} * P
% % 	} \\
% % 	\pline[\mask_1]{%
% % 		\Exists s. \later\pred(s) * \ownGhost\gname{(s, S, T)} * P
% % 	} \qquad by \ruleref{StsOpen} \\
% % 	Context: $s \in S \eqdef \upclose(\{s_0\}, T)$ \\
% % 	\pline[\mask_2]{%
% % 		 \Exists s', T'. \later\pred(s') * Q(s', T') * \ownGhost\gname{(s, S, T)}
% % 	} \qquad by premiss \\
% % 	Context: $(s, T) \ststrans (s', T')$ \\
% % 	\pline[\mask_2 \uplus \{\iname\}]{
% % 		\ownGhost\gname{(s', T')} * Q(s', T')
% % 	} \qquad by \ruleref{StsClose}
% % }
% % \end{proof}

% \subsection{Authoritative monoids with interpretation}\label{sec:authinterp}

% Building on \Sref{sec:auth}, after constructing the monoid $\auth{M}$ for a cancellative monoid $M$, we can tie an interpretation, $\pred : \mcarp{M} \to \Prop$, to the authoritative element of $M$, recovering reasoning that is close to the sharing rule in~\cite{krishnaswami+:icfp12}.

% Let $\pred_\bot$ be the extension of $\pred$ to $\mcar{M}$ with $\pred_\bot(\mzero) = \FALSE$.
% Now define
% \begin{align*}
%   \AuthInv(M, \pred, \gname) \eqdef{}& \exists \melt \in \mcar{M}.\; \ownGhost{\gname}{\authfull \melt:\auth{M}} * \pred_\bot(\melt) \\
%   \Auth(M, \pred, \gname, \iname) \eqdef{}& M~\textlog{cancellative} \land \knowInv{\iname}{\AuthInv(M, \pred, \gname)}
% \end{align*}

% The frame-preserving updates for $\auth{M}$ gives rise to the following view shifts:
% \begin{mathpar}
%  \inferH{NewAuth}
%   {\infinite(\mask) \and M~\textlog{cancellative}}
%   {\later\pred_\bot(a) \vs[\mask] \exists \iname \in \mask, \gname.\; \Auth(M, \pred, \gname, \iname) * \ownGhost{\gname}{\authfrag a : \auth{M}}}
%  \and
%  \axiomH{AuthOpen}
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%   {\Auth(M, \pred, \gname, \iname) \vdash \ownGhost{\gname}{\authfrag \melt : \auth{M}} \vsE[\{\iname\}][\emptyset] \exists \melt_\f.\; \later\pred_\bot(\melt \mtimes \melt_\f) * \ownGhost{\gname}{\authfull \melt \mtimes \melt_\f, \authfrag a:\auth{M}}}
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%  \and
%  \axiomH{AuthClose}
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%   {\Auth(M, \pred, \gname, \iname) \vdash \later\pred_\bot(\meltB \mtimes \melt_\f) * \ownGhost{\gname}{\authfull a \mtimes \melt_\f, \authfrag a:\auth{M}} \vs[\emptyset][\{\iname\}] \ownGhost{\gname}{\authfrag \meltB : \auth{M}} }
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% \end{mathpar}

% These view shifts in turn can be used to prove variants of the invariant rules:
% \begin{mathpar}
%  \inferH{Auth}
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%   {\forall \melt_\f.\; \hoare{\later\pred_\bot(a \mtimes \melt_\f) * P}{\expr}{\Ret\val. \exists \meltB.\; \later\pred_\bot(\meltB\mtimes \melt_\f) * Q}[\mask]
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%    \and \physatomic{\expr}}
%   {\Auth(M, \pred, \gname, \iname) \vdash \hoare{\ownGhost{\gname}{\authfrag a:\auth{M}} * P}{\expr}{\Ret\val. \exists \meltB.\; \ownGhost{\gname}{\authfrag \meltB:\auth{M}} * Q}[\mask \uplus \{\iname\}]}
%  \and
%  \inferH{VSAuth}
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%   {\forall \melt_\f.\; \later\pred_\bot(a \mtimes \melt_\f) * P \vs[\mask_1][\mask_2] \exists \meltB.\; \later\pred_\bot(\meltB \mtimes \melt_\f) * Q(\meltB)}
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%   {\Auth(M, \pred, \gname, \iname) \vdash
%    \ownGhost{\gname}{\authfrag a:\auth{M}} * P \vs[\mask_1 \uplus \{\iname\}][\mask_2 \uplus \{\iname\}]
%    \exists \meltB.\; \ownGhost{\gname}{\authfrag \meltB:\auth{M}} * Q(\meltB)}
% \end{mathpar}


% \subsection{Ghost heap}
% \label{sec:ghostheap}%
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% FIXME use the finmap provided by the global ghost ownership, instead of adding our own
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% We define a simple ghost heap with fractional permissions.
% Some modules require a few ghost names per module instance to properly manage ghost state, but would like to expose to clients a single logical name (avoiding clutter).
% In such cases we use these ghost heaps.

% We seek to implement the following interface:
% \newcommand{\GRefspecmaps}{\textsf{GMapsTo}}%
% \begin{align*}
%  \exists& {\fgmapsto[]} : \textsort{Val} \times \mathbb{Q}_{>} \times \textsort{Val} \ra \textsort{Prop}.\;\\
%   & \All x, q, v. x \fgmapsto[q] v \Ra x \fgmapsto[q] v \land q \in (0, 1] \\
%   &\forall x, q_1, q_2, v, w.\; x \fgmapsto[q_1] v * x \fgmapsto[q_2] w \Leftrightarrow x \fgmapsto[q_1 + q_2] v * v = w\\
%   & \forall v.\; \TRUE \vs[\emptyset] \exists x.\; x \fgmapsto[1] v \\
%   & \forall x, v, w.\; x \fgmapsto[1] v \vs[\emptyset] x \fgmapsto[1] w
% \end{align*}
% We write $x \fgmapsto v$ for $\exists q.\; x \fgmapsto[q] v$ and $x \gmapsto v$ for $x \fgmapsto[1] v$.
% Note that $x \fgmapsto v$ is duplicable but cannot be boxed (as it depends on resources); \ie we have $x \fgmapsto v \Lra x \fgmapsto v * x \fgmapsto v$ but not $x \fgmapsto v \Ra \always x \fgmapsto v$.

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% To implement this interface, allocate an instance $\gname_G$ of $\FHeap(\Val)$ and define
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% \[
% 	x \fgmapsto[q] v \eqdef
% 	  \begin{cases}
%     	\ownGhost{\gname_G}{x \mapsto (q, v)} & \text{if $q \in (0, 1]$} \\
%     	\FALSE & \text{otherwise}
%     \end{cases}
% \]
% The view shifts in the specification follow immediately from \ruleref{GhostUpd} and the frame-preserving updates in~\Sref{sec:fheapm}.
% The first implication is immediate from the definition.
% The second implication follows by case distinction on $q_1 + q_2 \in (0, 1]$.
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