constructions.tex 18.3 KB
 Ralf Jung committed Jan 31, 2016 1 % !TEX root = ./appendix.tex  Ralf Jung committed Mar 11, 2016 2 \section{COFE constructions}  Ralf Jung committed Jan 31, 2016 3   Ralf Jung committed Mar 11, 2016 4 5 6 7 8 9 10 11 12 13 \subsection{Next (type-level later)} Given a COFE $\cofe$, we define $\latert\cofe$ as follows: \begin{align*} \latert\cofe \eqdef{}& \latertinj(\cofe) \\ \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y \end{align*} $\latert(-)$ is a locally \emph{contractive} bifunctor from $\COFEs$ to $\COFEs$. \clearpage  Ralf Jung committed Feb 29, 2016 14 15 \section{CMRA constructions}  Ralf Jung committed Mar 11, 2016 16 17 18 \subsection{Product} \label{sec:prodm}  Ralf Jung committed Mar 11, 2016 19 Given a family $(M_i)_{i \in I}$ of CMRAs ($I$ finite), we construct a CMRA for the product $\prod_{i \in I} M_i$ by lifting everything pointwise.  Ralf Jung committed Mar 11, 2016 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  Frame-preserving updates on the $M_i$ lift to the product: \begin{mathpar} \inferH{prod-update} {\melt \mupd_{M_i} \meltsB} {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}} \end{mathpar} \subsection{Finite partial function} \label{sec:fpfnm} 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. We obtain the following frame-preserving updates: \begin{mathpar} \inferH{fpfn-alloc-strong} {\text{$G$ infinite} \and \melt \in \mval} {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in G}} \inferH{fpfn-alloc} {\melt \in \mval}  Ralf Jung committed Mar 11, 2016 41  {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in K}}  Ralf Jung committed Mar 11, 2016 42 43 44 45 46 47  \inferH{fpfn-update} {\melt \mupd \meltsB} {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}} \end{mathpar}  Ralf Jung committed Feb 29, 2016 48 49 \subsection{Agreement}  Ralf Jung committed Mar 09, 2016 50 Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:  Ralf Jung committed Mar 11, 2016 51 52 \newcommand{\agc}{\mathrm{c}} % the "c" field of an agreement element \newcommand{\agV}{\mathrm{V}} % the "V" field of an agreement element  Ralf Jung committed Mar 09, 2016 53 \begin{align*}  Ralf Jung committed Mar 11, 2016 54  \agm(\cofe) \eqdef{}& \recordComp{\agc : \mathbb{N} \to \cofe , \agV : \pset{\mathbb{N}}}{ \All n, m. n \geq m \Ra n \in \agV \Ra m \in \agV } \\  Ralf Jung committed Mar 09, 2016 55  & \text{quotiented by} \\  Ralf Jung committed Mar 11, 2016 56 57 58  \melt \equiv \meltB \eqdef{}& \melt.\agV = \meltB.\agV \land \All n. n \in \melt.\agV \Ra \melt.\agc(n) \nequiv{n} \meltB.\agc(n) \\ \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\agV \Lra m \in \meltB.\agV) \land (\All m \leq n. m \in \melt.\agV \Ra \melt.\agc(m) \nequiv{m} \meltB.\agc(m)) \\ \mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\agV \land \All m \leq n. \melt.\agc(n) \nequiv{m} \melt.\agc(m) } \\  Ralf Jung committed Mar 09, 2016 59  \mcore\melt \eqdef{}& \melt \\  Ralf Jung committed Mar 11, 2016 60  \melt \mtimes \meltB \eqdef{}& (\melt.\agc, \setComp{n}{n \in \melt.\agV \land n \in \meltB.\agV \land \melt \nequiv{n} \meltB })  Ralf Jung committed Mar 09, 2016 61 62 63 \end{align*} $\agm(-)$ is a locally non-expansive bifunctor from $\COFEs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 64 65 66 67 68 69 You can think of the $\agc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \agV$ steps. The reason we store a chain, rather than a single element, is that $\agm(\cofe)$ needs to be a COFE itself, so we need to be able to give a limit for every chain of $\agm(\cofe)$. However, given such a chain, we cannot constructively define its limit: Clearly, the $\agV$ of the limit is the limit of the $\agV$ of the chain. But what to pick for the actual data, for the element of $\cofe$? Only if $\agV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\agV$ is smaller, the chain cancels'', \ie stops converging as we reach indices $n \notin \agV$. To mitigate this, we apply the usual construction to close a set; we go from elements of $\cofe$ to chains of $\cofe$.  Ralf Jung committed Mar 09, 2016 70   Ralf Jung committed Mar 11, 2016 71 72 We define an injection $\ag$ into $\agm(\cofe)$ as follows: $\ag(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}}$  Ralf Jung committed Mar 09, 2016 73 74 There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar}  Ralf Jung committed Mar 11, 2016 75 76 77 78 79  \axiomH{ag-val}{\ag(x) \in \mval_n} \axiomH{ag-dup}{\ag(x) = \ag(x)\mtimes\ag(x)} \axiomH{ag-agree}{\ag(x) \mtimes \ag(y) \in \mval_n \Ra x \nequiv{n} y}  Ralf Jung committed Mar 09, 2016 80 81 \end{mathpar}  Ralf Jung committed Mar 11, 2016 82 83 84 85 86 \subsection{One-shot} The purpose of the one-shot CMRA is to lazily initialize the state of a ghost location. Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows: \begin{align*}  Ralf Jung committed Mar 11, 2016 87  \oneshot(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\  Ralf Jung committed Mar 11, 2016 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111  \mval_n \eqdef{}& \set{\ospending, \munit} \cup \setComp{\osshot(\melt)}{\melt \in \mval_n} \end{align*} \begin{align*} \mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\ \mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot \end{align*} \begin{align*} \osshot(\melt) \mtimes \osshot(\meltB) \eqdef{}& \osshot(\melt \mtimes \meltB) \\ \munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\ \munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt) \end{align*} The remaining cases of composition go to $\bot$. The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \axiom{\ospending \nequiv{n} \ospending} \infer{\melt \nequiv{n} \meltB}{\osshot(\melt) \nequiv{n} \osshot(\meltB)} \axiom{\munit \nequiv{n} \munit} \axiom{\bot \nequiv{n} \bot} \end{mathpar} $\oneshotm(-)$ is a locally non-expansive bifunctor from $\CMRAs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 112 113 114 115 116 117 118 119 120 121 We obtain the following frame-preserving updates: \begin{mathpar} \inferH{oneshot-shoot} {\melt \in \mval} {\ospending \mupd \osshot(\melt)} \inferH{oneshot-update} {\melt \mupd \meltsB} {\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}} \end{mathpar}  Ralf Jung committed Feb 29, 2016 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205  % \subsection{Exclusive monoid} % Given a set $X$, we define a monoid such that at most one $x \in X$ can be owned. % Let $\exm{X}$ be the monoid with carrier $X \uplus \{ \munit \}$ and multiplication % $% \melt \cdot \meltB \;\eqdef\; % \begin{cases} % \melt & \mbox{if } \meltB = \munit \\ % \meltB & \mbox{if } \melt = \munit % \end{cases} %$ % The frame-preserving update % \begin{mathpar} % \inferH{ExUpd} % {x \in X} % {x \mupd \melt} % \end{mathpar} % is easily shown, as the only possible frame for $x$ is $\munit$. % Exclusive monoids are cancellative. % \begin{proof}[Proof of cancellativity] % If $\melt_f = \munit$, then the statement is trivial. % If $\melt_f \neq \munit$, then we must have $\melt = \meltB = \munit$, as otherwise one of the two products would be $\mzero$. % \end{proof} % \subsection{Finite Powerset Monoid} % Given an infinite set $X$, we define a monoid $\textmon{PowFin}$ with carrier $\mathcal{P}^{\textrm{fin}}(X)$ as follows: % $% \melt \cdot \meltB \;\eqdef\; \melt \cup \meltB \quad \mbox{if } \melt \cap \meltB = \emptyset %$ % We obtain: % \begin{mathpar} % \inferH{PowFinUpd}{} % {\emptyset \mupd \{ \{x\} \mid x \in X \}} % \end{mathpar} % \begin{proof}[Proof of \ruleref{PowFinUpd}] % Assume some frame $\melt_f \sep \emptyset$. Since $\melt_f$ is finite and $X$ is infinite, there exists an $x \notin \melt_f$. % Pick that for the result. % \end{proof} % The powerset monoids is cancellative. % \begin{proof}[Proof of cancellativity] % Let $\melt_f \mtimes \melt = \melt_f \mtimes \meltB \neq \mzero$. % So we have $\melt_f \sep \melt$ and $\melt_f \sep \meltB$, and we have to show $\melt = \meltB$. % Assume $x \in \melt$. Hence $x \in \melt_f \mtimes \melt$ and thus $x \in \melt_f \mtimes \meltB$. % By disjointness, $x \notin \melt_f$ and hence $x \in meltB$. % The other direction works the same way. % \end{proof} % \subsection{Fractional monoid} % \label{sec:fracm} % Given a monoid $M$, we define a monoid representing fractional ownership of some piece $\melt \in M$. % The idea is to preserve all the frame-preserving update that $M$ could have, while additionally being able to do \emph{any} update if we own the full state (as determined by the fraction being $1$). % Let $\fracm{M}$ be the monoid with carrier $(((0, 1] \cap \mathbb{Q}) \times M) \uplus \{\munit\}$ and multiplication % \begin{align*} % (q, a) \mtimes (q', a') &\eqdef (q + q', a \mtimes a') \qquad \mbox{if $q+q'\le 1$} \\ % (q, a) \mtimes \munit &\eqdef (q,a) \\ % \munit \mtimes (q,a) &\eqdef (q,a). % \end{align*} % We get the following frame-preserving update. % \begin{mathpar} % \inferH{FracUpdFull} % {a, b \in M} % {(1, a) \mupd (1, b)} % \and\inferH{FracUpdLocal} % {a \mupd_M B} % {(q, a) \mupd \{q\} \times B} % \end{mathpar} % \begin{proof}[Proof of \ruleref{FracUpdFull}] % Assume some $f \sep (1, a)$. This can only be $f = \munit$, so showing $f \sep (1, b)$ is trivial. % \end{proof} % \begin{proof}[Proof of \ruleref{FracUpdLocal}] % Assume some $f \sep (q, a)$. If $f = \munit$, then $f \sep (q, b)$ is trivial for any $b \in B$. Just pick the one we obtain by choosing $\munit_M$ as the frame for $a$.  Ralf Jung committed Jan 31, 2016 206   Ralf Jung committed Feb 29, 2016 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 % In the interesting case, we have $f = (q_f, a_f)$. % Obtain $b$ such that $b \in B \land b \sep a_f$. % Then $(q, b) \sep f$, and we are done. % \end{proof} % $\fracm{M}$ is cancellative if $M$ is cancellative. % \begin{proof}[Proof of cancellativitiy] % If $\melt_f = \munit$, we are trivially done. % So let $\melt_f = (q_f, \melt_f')$. % If $\melt = \munit$, then $\meltB = \munit$ as otherwise the fractions could not match up. % Again, we are trivially done. % Similar so for $\meltB = \munit$. % So let $\melt = (q_a, \melt')$ and $\meltB = (q_b, \meltB')$. % We have $(q_f + q_a, \melt_f' \mtimes \melt') = (q_f + q_b, \melt_f' \mtimes \meltB')$. % We have to show $q_a = q_b$ and $\melt' = \meltB'$. % The first is trivial, the second follows from cancellativitiy of $M$. % \end{proof} % %\subsection{Disposable monoid} % % % %Given a monoid $M$, we construct a monoid where, having full ownership of an element $\melt$ of $M$, one can throw it away, transitioning to a dead element. % %Let \dispm{M} be the monoid with carrier $\mcarp{M} \uplus \{ \disposed \}$ and multiplication % %% The previous unit must remain the unit of the new monoid, as is is always duplicable and hence we could not transition to \disposed if it were not composable with \disposed % %\begin{align*} % % \melt \mtimes \meltB &\eqdef \melt \mtimes_M \meltB & \IF \melt \sep[M] \meltB \\ % % \disposed \mtimes \disposed &\eqdef \disposed \\ % % \munit_M \mtimes \disposed &\eqdef \disposed \mtimes \munit_M \eqdef \disposed % %\end{align*} % %The unit is the same as in $M$. % % % %The frame-preserving updates are % %\begin{mathpar} % % \inferH{DispUpd} % % {a \in \mcarp{M} \setminus \{\munit_M\} \and a \mupd_M B} % % {a \mupd B} % % \and % % \inferH{Dispose} % % {a \in \mcarp{M} \setminus \{\munit_M\} \and \All b \in \mcarp{M}. a \sep b \Ra b = \munit_M} % % {a \mupd \disposed} % %\end{mathpar} % % % %\begin{proof}[Proof of \ruleref{DispUpd}] % %Assume a frame $f$. If $f = \disposed$, then $a = \munit_M$, which is a contradiction. % %Thus $f \in \mcarp{M}$ and we can use $a \mupd_M B$. % %\end{proof} % % % %\begin{proof}[Proof of \ruleref{Dispose}] % %The second premiss says that $a$ has no non-trivial frame in $M$. To show the update, assume a frame $f$ in $\dispm{M}$. Like above, we get $f \in \mcarp{M}$, and thus $f = \munit_M$. But $\disposed \sep \munit_M$ is trivial, so we are done. % %\end{proof} % \subsection{Authoritative monoid}\label{sec:auth} % Given a monoid $M$, we construct a monoid modeling someone owning an \emph{authoritative} element $x$ of $M$, and others potentially owning fragments $\melt \le_M x$ of $x$. % (If $M$ is an exclusive monoid, the construction is very similar to a half-ownership monoid with two asymmetric halves.) % Let $\auth{M}$ be the monoid with carrier % $% \setComp{ (x, \melt) }{ x \in \mcarp{\exm{\mcarp{M}}} \land \melt \in \mcarp{M} \land (x = \munit_{\exm{\mcarp{M}}} \lor \melt \leq_M x) } %$ % and multiplication % $% (x, \melt) \mtimes (y, \meltB) \eqdef % (x \mtimes y, \melt \mtimes \meltB) \quad \mbox{if } x \sep y \land \melt \sep \meltB \land (x \mtimes y = \munit_{\exm{\mcarp{M}}} \lor \melt \mtimes \meltB \leq_M x \mtimes y) %$ % Note that $(\munit_{\exm{\mcarp{M}}}, \munit_M)$ is the unit and asserts no ownership whatsoever, but $(\munit_{M}, \munit_M)$ asserts that the authoritative element is $\munit_M$. % Let $x, \melt \in \mcarp M$. % We write $\authfull x$ for full ownership $(x, \munit_M):\auth{M}$ and $\authfrag \melt$ for fragmental ownership $(\munit_{\exm{\mcarp{M}}}, \melt)$ and $\authfull x , \authfrag \melt$ for combined ownership $(x, \melt)$. % If $x$ or $a$ is $\mzero_{M}$, then the sugar denotes $\mzero_{\auth{M}}$. % \ralf{This needs syncing with the Coq development.} % The frame-preserving update involves a rather unwieldy side-condition: % \begin{mathpar} % \inferH{AuthUpd}{ % \All\melt_f\in\mcar{\monoid}. \melt\sep\meltB \land \melt\mtimes\melt_f \le \meltB\mtimes\melt_f \Ra \melt'\mtimes\melt_f \le \melt'\mtimes\meltB \and % \melt' \sep \meltB % }{ % \authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt' % } % \end{mathpar} % We therefore derive two special cases. % \paragraph{Local frame-preserving updates.} % \newcommand\authupd{f}% % Following~\cite{scsl}, we say that $\authupd: \mcar{M} \ra \mcar{M}$ is \emph{local} if % $% \All a, b \in \mcar{M}. a \sep b \land \authupd(a) \neq \mzero \Ra \authupd(a \mtimes b) = \authupd(a) \mtimes b %$ % Then, % \begin{mathpar} % \inferH{AuthUpdLocal} % {\text{$\authupd$ local} \and \authupd(\melt)\sep\meltB} % {\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \authupd(\melt) \mtimes \meltB, \authfrag \authupd(\melt)} % \end{mathpar} % \paragraph{Frame-preserving updates on cancellative monoids.} % Frame-preserving updates are also possible if we assume $M$ cancellative: % \begin{mathpar} % \inferH{AuthUpdCancel} % {\text{$M$ cancellative} \and \melt'\sep\meltB} % {\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt'} % \end{mathpar} % \subsection{Fractional heap monoid} % \label{sec:fheapm} % By combining the fractional, finite partial function, and authoritative monoids, we construct two flavors of heaps with fractional permissions and mention their important frame-preserving updates. % Hereinafter, we assume the set $\textdom{Val}$ of values is countable. % Given a set $Y$, define $\FHeap(Y) \eqdef \textdom{Val} \fpfn \fracm(Y)$ representing a fractional heap with codomain $Y$. % From \S\S\ref{sec:fracm} and~\ref{sec:fpfunm} we obtain the following frame-preserving updates as well as the fact that $\FHeap(Y)$ is cancellative. % \begin{mathpar} % \axiomH{FHeapUpd}{h[x \mapsto (1, y)] \mupd h[x \mapsto (1, y')]} \and % \axiomH{FHeapAlloc}{h \mupd \{\, h[x \mapsto (1, y)] \mid x \in \textdom{Val} \,\}} % \end{mathpar} % We will write $qh$ with $h : \textsort{Val} \fpfn Y$ for the function in $\FHeap(Y)$ mapping every $x \in \dom(h)$ to $(q, h(x))$, and everything else to $\munit$. % Define $\AFHeap(Y) \eqdef \auth{\FHeap(Y)}$ representing an authoritative fractional heap with codomain $Y$. % We easily obtain the following frame-preserving updates. % \begin{mathpar} % \axiomH{AFHeapUpd}{ % (\authfull h[x \mapsto (1, y)], \authfrag [x \mapsto (1, y)]) \mupd (\authfull h[x \mapsto (1, y')], \authfrag [x \mapsto (1, y')]) % } % \and % \inferH{AFHeapAdd}{ % x \notin \dom(h) % }{ % \authfull h \mupd (\authfull h[x \mapsto (q, y)], \authfrag [x \mapsto (q, y)]) % } % \and % \axiomH{AFHeapRemove}{ % (\authfull h[x \mapsto (q, y)], \authfrag [x \mapsto (q, y)]) \mupd \authfull h % } % \end{mathpar} % \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}. % We first lift the transition relation to $\STSS \times \mathcal{P}(\STST)$ (implementing a \emph{law of token conservation}) and define upwards closure: % \begin{align*} % (s, T) \ra (s', T') \eqdef&\, s \ra s' \land \STSL(s) \uplus T = \STSL(s') \uplus T' \\ % \textsf{frame}(s, T) \eqdef&\, (s, \STST \setminus (\STSL(s) \uplus T)) \\ % \upclose(S, T) \eqdef&\, \setComp{ s' \in \STSS}{\exists s \in S.\; \textsf{frame}(s, T) \ststrans \textsf{frame}(s', T) } % \end{align*} % \noindent % We have % \begin{quote} % If $(s, T) \ra (s', T')$\\ % and $T_f \sep (T \uplus \STSL(s))$,\\ % then $\textsf{frame}(s, T_f) \ra \textsf{frame}(s', T_f)$. % \end{quote} % \begin{proof} % This follows directly by framing the tokens in $\STST \setminus (T_f \uplus T \uplus \STSL(s))$ around the given transition, which yields $(s, \STST \setminus (T_f \uplus \STSL{T}(s))) \ra (s', T' \uplus (\STST \setminus (T_f \uplus T \uplus \STSL{T}(s))))$. % This is exactly what we have to show, since we know $\STSL(s) \uplus T = \STSL(s') \uplus T'$. % \end{proof} % Let $\STSMon{\STSS}$ be the monoid with carrier % % \setComp{ (s, S, T) \in \exm{\STSS} \times \mathcal{P}(\STSS) \times \mathcal{P}(\STST) }{ \begin{aligned} &(s = \munit \lor s \in S) \land \upclose(S, T) = S \land{} \\& S \neq \emptyset \land \All s \in S. \STSL(s) \sep T \end{aligned} } % % and multiplication % % (s, S, T) \mtimes (s', S', T') \eqdef (s'' \eqdef s \mtimes_{\exm{\STSS}} s', S'' \eqdef S \cap S', T'' \eqdef T \cup T') \quad \text{if }\begin{aligned}[t] &(s = \munit \lor s' = \munit) \land T \sep T' \land{} \\& S'' \neq \emptyset \land (s'' \neq \munit \Ra s'' \in S'') \end{aligned} % % Some sugar makes it more convenient to assert being at least in a certain state and owning some tokens: $(s, T) : \STSMon{\STSS} \eqdef (\munit, \upclose(\{s\}, T), T) : \STSMon{\STSS}$, and % $s : \STSMon{\STSS} \eqdef (s, \emptyset) : \STSMon{\STSS}$. % We will need the following frame-preserving update. % \begin{mathpar} % \inferH{StsStep}{(s, T) \ststrans (s', T')} % {(s, S, T) \mupd (s', \upclose(\{s'\}, T'), T')} % \end{mathpar} % \begin{proof}[Proof of \ruleref{StsStep}] % Assume some upwards-closed $S_f, T_f$ (the frame cannot be authoritative) s.t.\ $s \in S_f$ and $T_f \sep (T \uplus \STSL(s))$. We have to show that this frame combines with our final monoid element, which is the case if $s' \in S_f$ and $T_f \sep T'$. % By upward-closedness, it suffices to show $\textsf{frame}(s, T_f) \ststrans \textsf{frame}(s', T_f)$. % This follows by induction on the path $(s, T) \ststrans (s', T')$, and using the lemma proven above for each step. % \end{proof}  Ralf Jung committed Jan 31, 2016 393   Ralf Jung committed Jan 31, 2016 394 395 396 397 398  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: