derived.tex 19.8 KB
 Ralf Jung committed Mar 08, 2016 1 \section{Derived proof rules and other constructions}  Ralf Jung committed Mar 07, 2016 2 3 4  \subsection{Base logic}  Ralf Jung committed Mar 08, 2016 5 6 7 8 9 10 We collect here some important and frequently used derived proof rules. \begin{mathparpagebreakable} \infer{} {\prop \Ra \propB \proves \prop \wand \propB} \infer{}  Ralf Jung committed Mar 08, 2016 11  {\prop * \Exists\var.\propB \provesIff \Exists\var. \prop * \propB}  Ralf Jung committed Mar 08, 2016 12 13 14 15 16  \infer{} {\prop * \Exists\var.\propB \proves \Exists\var. \prop * \propB} \infer{}  Ralf Jung committed Mar 08, 2016 17  {\always(\prop*\propB) \provesIff \always\prop * \always\propB}  Ralf Jung committed Mar 08, 2016 18 19 20 21 22 23 24 25  \infer{} {\always(\prop \Ra \propB) \proves \always\prop \Ra \always\propB} \infer{} {\always(\prop \wand \propB) \proves \always\prop \wand \always\propB} \infer{}  Ralf Jung committed Mar 08, 2016 26  {\always(\prop \wand \propB) \provesIff \always(\prop \Ra \propB)}  Ralf Jung committed Mar 08, 2016 27 28 29 30 31 32 33 34 35 36 37  \infer{} {\later(\prop \Ra \propB) \proves \later\prop \Ra \later\propB} \infer{} {\later(\prop \wand \propB) \proves \later\prop \wand \later\propB} \infer {\pfctx, \later\prop \proves \prop} {\pfctx \proves \prop} \end{mathparpagebreakable}  Ralf Jung committed Mar 07, 2016 38   Ralf Jung committed Mar 07, 2016 39 40 41 42 43 44 \paragraph{Persistent assertions.} \begin{defn} An assertion $\prop$ is \emph{persistent} if $\prop \proves \always\prop$. \end{defn} Of course, $\always\prop$ is persistent for any $\prop$.  Ralf Jung committed Mar 08, 2016 45 Furthermore, by the proof rules given above, $t = t'$ as well as $\ownGGhost{\mcore\melt}$ and $\knowInv\iname\prop$ are persistent.  Ralf Jung committed Mar 07, 2016 46 47 48 49 Persistence is preserved by conjunction, disjunction, separating conjunction as well as universal and existential quantification. In our proofs, we will implicitly add and remove $\always$ from persistent assertions as necessary, and generally treat them like normal, non-linear assumptions.  Ralf Jung committed Mar 08, 2016 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 \paragraph{Timeless assertions.} We can show that the following additional closure properties hold for timeless assertions: \begin{mathparpagebreakable} \infer {\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop \land \propB}} \infer {\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop \lor \propB}} \infer {\vctx \proves \timeless{\prop} \and \vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop * \propB}} \infer {\vctx \proves \timeless{\prop}} {\vctx \proves \timeless{\always\prop}} \end{mathparpagebreakable}  Ralf Jung committed Mar 07, 2016 73 74 \subsection{Program logic}  Ralf Jung committed Mar 06, 2016 75 Hoare triples and view shifts are syntactic sugar for weakest (liberal) preconditions and primitive view shifts, respectively:  Ralf Jung committed Mar 07, 2016 76  Ralf Jung committed Mar 08, 2016 77 \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask] \eqdef \always{(\prop \Ra \wpre{\expr}[\mask]{\lambda\Ret\val.\propB})}  Ralf Jung committed Mar 07, 2016 78 79 80 81 82 83 \qquad\qquad \begin{aligned} \prop \vs[\mask_1][\mask_2] \propB &\eqdef \always{(\prop \Ra \pvs[\mask_1][\mask_2] {\propB})} \\ \prop \vsE[\mask_1][\mask_2] \propB &\eqdef \prop \vs[\mask_1][\mask_2] \propB \land \propB \vs[\mask2][\mask_1] \prop \end{aligned}  Ralf Jung committed Mar 06, 2016 84 We write just one mask for a view shift when $\mask_1 = \mask_2$.  Ralf Jung committed Mar 07, 2016 85 86 Clearly, all of these assertions are persistent. The convention for omitted masks is similar to the base logic:  Ralf Jung committed Mar 06, 2016 87 88 89 An omitted $\mask$ is $\top$ for Hoare triples and $\emptyset$ for view shifts.  Ralf Jung committed Mar 08, 2016 90 \paragraph{View shifts.}  Ralf Jung committed Mar 07, 2016 91 The following rules can be derived for view shifts.  Ralf Jung committed Mar 06, 2016 92   Ralf Jung committed Mar 07, 2016 93 94 \begin{mathparpagebreakable} \inferH{vs-update}  Ralf Jung committed Mar 06, 2016 95 96 97  {\melt \mupd \meltsB} {\ownGGhost{\melt} \vs \exists \meltB \in \meltsB.\; \ownGGhost{\meltB}} \and  Ralf Jung committed Mar 07, 2016 98 \inferH{vs-trans}  Ralf Jung committed Mar 06, 2016 99 100 101  {\prop \vs[\mask_1][\mask_2] \propB \and \propB \vs[\mask_2][\mask_3] \propC \and \mask_2 \subseteq \mask_1 \cup \mask_3} {\prop \vs[\mask_1][\mask_3] \propC} \and  Ralf Jung committed Mar 07, 2016 102 \inferH{vs-imp}  Ralf Jung committed Mar 06, 2016 103 104 105  {\always{(\prop \Ra \propB)}} {\prop \vs[\emptyset] \propB} \and  Ralf Jung committed Mar 07, 2016 106 \inferH{vs-mask-frame}  Ralf Jung committed Mar 06, 2016 107  {\prop \vs[\mask_1][\mask_2] \propB}  Ralf Jung committed Mar 07, 2016 108  {\prop \vs[\mask_1 \uplus \mask'][\mask_2 \uplus \mask'] \propB}  Ralf Jung committed Mar 06, 2016 109 \and  Ralf Jung committed Mar 07, 2016 110 111 112 113 114 \inferH{vs-frame} {\prop \vs[\mask_1][\mask_2] \propB} {\prop * \propC \vs[\mask_1][\mask_2] \propB * \propC} \and \inferH{vs-timeless}  Ralf Jung committed Mar 06, 2016 115 116 117  {\timeless{\prop}} {\later \prop \vs \prop} \and  Ralf Jung committed Mar 07, 2016 118 119 120 121 122 \inferH{vs-allocI} {\infinite(\mask)} {\later{\prop} \vs[\mask] \exists \iname\in\mask.\; \knowInv{\iname}{\prop}} \and \axiomH{vs-openI}  Ralf Jung committed Mar 06, 2016 123 124  {\knowInv{\iname}{\prop} \proves \TRUE \vs[\{ \iname \} ][\emptyset] \later \prop} \and  Ralf Jung committed Mar 07, 2016 125 \axiomH{vs-closeI}  Ralf Jung committed Mar 06, 2016 126 127  {\knowInv{\iname}{\prop} \proves \later \prop \vs[\emptyset][\{ \iname \} ] \TRUE }  Ralf Jung committed Mar 07, 2016 128 \inferHB{vs-disj}  Ralf Jung committed Mar 06, 2016 129 130 131  {\prop \vs[\mask_1][\mask_2] \propC \and \propB \vs[\mask_1][\mask_2] \propC} {\prop \lor \propB \vs[\mask_1][\mask_2] \propC} \and  Ralf Jung committed Mar 07, 2016 132 \inferHB{vs-exist}  Ralf Jung committed Mar 06, 2016 133 134 135  {\All \var. (\prop \vs[\mask_1][\mask_2] \propB)} {(\Exists \var. \prop) \vs[\mask_1][\mask_2] \propB} \and  Ralf Jung committed Mar 07, 2016 136 \inferHB{vs-box}  Ralf Jung committed Mar 07, 2016 137  {\always\propB \proves \prop \vs[\mask_1][\mask_2] \propC}  Ralf Jung committed Mar 06, 2016 138 139  {\prop \land \always{\propB} \vs[\mask_1][\mask_2] \propC} \and  Ralf Jung committed Mar 07, 2016 140 \inferH{vs-false}  Ralf Jung committed Mar 06, 2016 141 142  {} {\FALSE \vs[\mask_1][\mask_2] \prop }  Ralf Jung committed Mar 07, 2016 143 \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 144 145   Ralf Jung committed Mar 07, 2016 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 \paragraph{Hoare triples.} The following rules can be derived for Hoare triples. \begin{mathparpagebreakable} \inferH{Ht-ret} {} {\hoare{\TRUE}{\valB}{\Ret\val. \val = \valB}[\mask]} \and \inferH{Ht-bind} {\text{$\lctx$ is a context} \and \hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \\ \All \val. \hoare{\propB}{\lctx(\val)}{\Ret\valB.\propC}[\mask]} {\hoare{\prop}{\lctx(\expr)}{\Ret\valB.\propC}[\mask]} \and \inferH{Ht-csq} {\prop \vs \prop' \\ \hoare{\prop'}{\expr}{\Ret\val.\propB'}[\mask] \\ \All \val. \propB' \vs \propB} {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} \and \inferH{Ht-mask-weaken} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask]} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask \uplus \mask']} \\\\ \inferH{Ht-frame} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask]} {\hoare{\prop * \propC}{\expr}{\Ret\val. \propB * \propC}[\mask]} \and \inferH{Ht-frame-step} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \and \toval(\expr) = \bot} {\hoare{\prop * \later\propC}{\expr}{\Ret\val. \propB * \propC}[\mask]} \and \inferH{Ht-atomic} {\prop \vs[\mask \uplus \mask'][\mask] \prop' \\ \hoare{\prop'}{\expr}{\Ret\val.\propB'}[\mask] \\ \All\val. \propB' \vs[\mask][\mask \uplus \mask'] \propB \\ \physatomic{\expr}  Ralf Jung committed Mar 06, 2016 182  }  Ralf Jung committed Mar 07, 2016 183  {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask \uplus \mask']}  Ralf Jung committed Mar 06, 2016 184 \and  Ralf Jung committed Mar 07, 2016 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 \inferHB{Ht-disj} {\hoare{\prop}{\expr}{\Ret\val.\propC}[\mask] \and \hoare{\propB}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \lor \propB}{\expr}{\Ret\val.\propC}[\mask]} \and \inferHB{Ht-exist} {\All \var. \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} {\hoare{\Exists \var. \prop}{\expr}{\Ret\val.\propB}[\mask]} \and \inferHB{Ht-box} {\always\propB \proves \hoare{\prop}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \land \always{\propB}}{\expr}{\Ret\val.\propC}[\mask]} \and \inferH{Ht-false} {} {\hoare{\FALSE}{\expr}{\Ret \val. \prop}[\mask]} \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 201   Ralf Jung committed Mar 08, 2016 202 203 204 205 206 \paragraph{Lifting of operational semantics.} We can derive some specialized forms of the lifting axioms for the operational semantics, as well as some forms that involve view shifts and Hoare triples. \ralf{Add these.}  Ralf Jung committed Mar 10, 2016 207 \subsection{Global functor and ghost ownership}  Ralf Jung committed Mar 08, 2016 208 \ralf{Describe this.}  Ralf Jung committed Mar 07, 2016 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  % \subsection{Global monoid} % 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}): % $M \eqdef \prod_{i \in I} \textdom{GhName} \fpfn M_i$ % 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. % \begin{mathpar} % \axiomH{NewGhost}{ % \TRUE \vs \Exists\gname. \ownGhost\gname{\melt : M_i} % } % \and % \inferH{GhostUpd} % {\melt \mupd_{M_i} B} % {\ownGhost\gname{\melt : M_i} \vs \Exists \meltB\in B. \ownGhost\gname{\meltB : M_i}} % \and % \axiomH{GhostEq} % {\ownGhost\gname{\melt : M_i} * \ownGhost\gname{\meltB : M_i} \Lra \ownGhost\gname{\melt\mtimes\meltB : M_i}} % \axiomH{GhostUnit} % {\TRUE \Ra \ownGhost{\gname}{\munit : M_i}} % \axiomH{GhostZero} % {\ownGhost\gname{\mzero : M_i} \Ra \FALSE} % \axiomH{GhostTimeless} % {\timeless{\ownGhost\gname{\melt : M_i}}} % \end{mathpar}  Ralf Jung committed Mar 08, 2016 241 \subsection{Invariant identifier namespaces}  Ralf Jung committed Mar 11, 2016 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256  Let $\namesp \ni \textlog{InvNamesp} \eqdef \textlog{list}(\textlog{InvName})$ be the type of \emph{namespaces} for invariant names. Notice that there is an injection $\textlog{namesp\_inj}: \textlog{InvNamesp} \ra \textlog{InvName}$. Whenever needed (in particular, for masks at view shifts and Hoare triples), we coerce $\namesp$ to its suffix-closure: $\namecl\namesp \eqdef \setComp{\iname}{\Exists \namesp'. \iname = \textlog{namesp\_inj}(\namesp' \dplus \namesp)}$ We use the notation $\namesp.\iname$ for the namespace $[\iname] \dplus \namesp$. We will overload the usual Iris notation for invariant assertions in the following: $\knowInv\namesp\prop \eqdef \Exists \iname \in \namecl\namesp. \knowInv\iname{\prop}$ We define the inclusion relation on namespaces as $\namesp_1 \sqsubseteq \namesp_2 \Lra \Exists \namesp_3. \namesp_2 = \namesp_3 \dplus \namesp_1$, \ie $\namesp_1$ is a suffix of $\namesp_2$. We have that $\namesp_1 \sqsubseteq \namesp_2 \Ra \namecl\namesp_2 \subseteq \namecl\namesp_1$. Similarly, we define $\namesp_1 \sep \namesp_2 \eqdef \Exists \namesp_1', \namesp_2'. \namesp_1' \sqsubseteq \namesp_1 \land \namesp_2' \sqsubseteq \namesp_2 \land |\namesp_1'| = |\namesp_2'| \land \namesp_1' \neq \namesp_2'$, \ie there exists a distinguishing suffix. We have that $\namesp_1 \sep \namesp_2 \Ra \namecl\namesp_2 \sep \namecl\namesp_1$, and furthermore $\iname_1 \neq \iname_2 \Ra \namesp.\iname_1 \sep \namesp.\iname_2$. \ralf{Give derived rules for invariants.}  Ralf Jung committed Mar 08, 2016 257   Ralf Jung committed Mar 07, 2016 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 % \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}  Ralf Jung committed Jan 31, 2016 287   Ralf Jung committed Mar 07, 2016 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 % 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 $Q$.  Ralf Jung committed Jan 31, 2016 304   Ralf Jung committed Mar 07, 2016 305 306 % 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 % $\hoareV{\Exists s\in \upclose(\{s_0\}, T). \later\pred(s) * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)} * P}{\expr}{\Ret \val. \Exists s, T, S, s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * \ownGhost{\gname}{(s, S, T):\STSMon{\STSS}} * Q(\val, s', T')}[\mask]$  Ralf Jung committed Jan 31, 2016 307   Ralf Jung committed Mar 07, 2016 308 309 310 % 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: % $\hoareV{s\in \upclose(\{s_0\}, T) * \later\pred(s) * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)} * P}{\expr}{\Ret \val. \Exists s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * \ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)} * Q(\val, s', T')}[\mask]$  Ralf Jung committed Jan 31, 2016 311   Ralf Jung committed Mar 07, 2016 312 313 % Finally, use \ruleref{BoxOut} to move $s\in \upclose(\{s_0\}, T)$ into the context, and \ruleref{Frame} on $\ownGhost{\gname}{(s, \upclose(\{s_0\}, T), T)}$: % $s\in \upclose(\{s_0\}, T) \vdash \hoare{\later\pred(s) * P}{\expr}{\Ret \val. \Exists s', T'. (s, T) \ststrans (s', T') * \later\pred(s') * Q(\val, s', T')}[\mask]$  Ralf Jung committed Jan 31, 2016 314   Ralf Jung committed Mar 07, 2016 315 % This holds by our premise.  Ralf Jung committed Jan 31, 2016 316 % \end{proof}  Ralf Jung committed Jan 31, 2016 317   Ralf Jung committed Mar 07, 2016 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 % % \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} % {\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}}} % \and % \axiomH{AuthClose} % {\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}} } % \end{mathpar} % These view shifts in turn can be used to prove variants of the invariant rules: % \begin{mathpar} % \inferH{Auth} % {\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] % \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} % {\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)} % {\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}%  Ralf Jung committed Mar 10, 2016 381 % FIXME use the finmap provided by the global ghost ownership, instead of adding our own  Ralf Jung committed Mar 07, 2016 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 % 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 $x \fgmapsto v \Ra \always x \fgmapsto v$. % To implement this interface, allocate an instance $\gname_G$ of $\FHeap(\textdom{Val})$ and define % $% x \fgmapsto[q] v \eqdef % \begin{cases} % \ownGhost{\gname_G}{x \mapsto (q, v)} & \text{if q \in (0, 1]} \\ % \FALSE & \text{otherwise} % \end{cases} %$ % 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 $q_1 + q_2 \in (0, 1]$.  Ralf Jung committed Jan 31, 2016 409   Ralf Jung committed Jan 31, 2016 410 411 412 413 414  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: