model.tex 20.2 KB
 Ralf Jung committed Jan 31, 2016 1 2 3 4 \section{Model and semantics} The semantics closely follows the ideas laid out in~\cite{catlogic}.  Ralf Jung committed Mar 12, 2016 5 \subsection{Generic model of base logic}  Ralf Jung committed Jan 31, 2016 6   Ralf Jung committed Mar 12, 2016 7 The base logic including equality, later, always, and a notion of ownership is defined on $\UPred(\monoid)$ for any CMRA $\monoid$.  Ralf Jung committed Jan 31, 2016 8   Ralf Jung committed Mar 12, 2016 9 10 11 \typedsection{Interpretation of base assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \to \UPred(\monoid)} Remember that $\UPred(\monoid)$ is isomorphic to $\monoid \monra \SProp$. We are thus going to define the assertions as mapping CMRA elements to sets of step-indices.  Ralf Jung committed Jan 31, 2016 12   Ralf Jung committed Mar 12, 2016 13 We introduce an additional logical connective $\ownM\melt$, which will later be used to encode all of $\knowInv\iname\prop$, $\ownGGhost\melt$ and $\ownPhys\state$.  Ralf Jung committed Jan 31, 2016 14 15  \begin{align*}  Ralf Jung committed Mar 12, 2016 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  \Sem{\vctx \proves t =_\type u : \Prop}_\gamma &\eqdef \Lam \any. \setComp{n}{\Sem{\vctx \proves t : \type}_\gamma \nequiv{n} \Sem{\vctx \proves u : \type}_\gamma} \\ \Sem{\vctx \proves \FALSE : \Prop}_\gamma &\eqdef \Lam \any. \emptyset \\ \Sem{\vctx \proves \TRUE : \Prop}_\gamma &\eqdef \Lam \any. \mathbb{N} \\ \Sem{\vctx \proves \prop \land \propB : \Prop}_\gamma &\eqdef \Lam \melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt) \cap \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt) \\ \Sem{\vctx \proves \prop \lor \propB : \Prop}_\gamma &\eqdef \Lam \melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt) \cup \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt) \\ \Sem{\vctx \proves \prop \Ra \propB : \Prop}_\gamma &\eqdef \Lam \melt. \setComp{n}{\begin{aligned} \All m, \meltB.& m \leq n \land \melt \mincl \meltB \land \meltB \in \mval_m \Ra {} \\ & m \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt) \Ra {}\\& m \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt)\end{aligned}}\\ \Sem{\vctx \proves \All x : \type. \prop : \Prop}_\gamma &\eqdef \Lam \melt. \setComp{n}{ \All v \in \Sem{\type}. n \in \Sem{\vctx, x : \type \proves \prop : \Prop}_{\gamma[x \mapsto v]}(\melt) } \\ \Sem{\vctx \proves \Exists x : \type. \prop : \Prop}_\gamma &\eqdef \Lam \melt. \setComp{n}{ \Exists v \in \Sem{\type}. n \in \Sem{\vctx, x : \type \proves \prop : \Prop}_{\gamma[x \mapsto v]}(\melt) } \\ ~\\ \Sem{\vctx \proves \always{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \Sem{\vctx \proves \prop : \Prop}_\gamma(\mcore\melt) \\ \Sem{\vctx \proves \later{\prop} : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{n = 0 \lor n-1 \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\melt)}\\ \Sem{\vctx \proves \prop * \propB : \Prop}_\gamma &\eqdef \Lam\melt. \setComp{n}{\begin{aligned}\Exists \meltB_1, \meltB_2. &\melt \nequiv{n} \meltB_1 \mtimes \meltB_2 \land {}\\& n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\meltB_1) \land n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\meltB_2)\end{aligned}} \\ \Sem{\vctx \proves \prop \wand \propB : \Prop}_\gamma &\eqdef \Lam \melt. \setComp{n}{\begin{aligned} \All m, \meltB.& m \leq n \land \melt\mtimes\meltB \in \mval_m \Ra {} \\ & m \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\meltB) \Ra {}\\& m \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\melt\mtimes\meltB)\end{aligned}} \\ \Sem{\vctx \proves \ownM{\melt} : \Prop}_\gamma &\eqdef \Lam\meltB. \setComp{n}{\melt \mincl[n] \meltB} \\ \Sem{\vctx \proves \mval(\melt) : \Prop}_\gamma &\eqdef \Lam\any. \setComp{n}{\melt \in \mval_n} \\  Ralf Jung committed Jan 31, 2016 43 44 \end{align*}  Ralf Jung committed Mar 12, 2016 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 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\rtimes \rs''\\[1em] % % % \UPred(\textdom{Res}) &\eqdef& % \{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid % \All (k,\rs) \in p. % \All j\leq k. % \All \rs' \geq \rs. % (j,\rs')\in p \,\}\\[0.5em] % \restr{p}{k} &\eqdef& % \{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em] % p \nequiv{n} q & \eqdef & \restr{p}{n} = \restr{q}{n}\\[1em] % % % \textdom{PreProp} & \cong & % \latert\big( \textdom{World} \monra \UPred(\textdom{Res}) % \big)\\[0.5em] % % % \textdom{World} & \eqdef & % \mathbb{N} \fpfn \textdom{PreProp}\\[0.5em] % % % w \nequiv{n} w' & \eqdef & % n = 0 \lor % \bigl(\dom(w) = \dom(w') \land \All i\in\dom(w). w(i) \nequiv{n} w'(i)\bigr) % \\[0.5em] % % % w \leq w' & \eqdef & % \dom(w) \subseteq \dom(w') \land \All i \in \dom(w). w(i) = w'(i) % \\[0.5em] % % % \textdom{Prop} & \eqdef & \textdom{World} \monra \UPred(\textdom{Res}) % \end{array} %$ % For $p,q\in\UPred(\textdom{Res})$ with $p \nequiv{n} q$ defined % as above, $\UPred(\textdom{Res})$ is a % c.o.f.e. % $\textdom{Prop}$ is a c.o.f.e., which exists by America and Rutten's theorem~\cite{America-Rutten:JCSS89}. % We do not need to consider how the object is constructed. % We only need the isomorphism, given by maps % \begin{align*} % \wIso &: \latert \bigl(World \monra \UPred(\textdom{Res})\bigr) \to \textdom{PreProp} \\ % \wIso^{-1} &: \textdom{PreProp} \to \latert \bigl(World \monra \UPred(\textdom{Res})\bigr) % \end{align*} % which are inverses to each other. % Note: this is an isomorphism in $\cal U$, i.e., $\wIso$ and % $\wIso^{-1}$ are both non-expansive. % $\textdom{World}$ is a c.o.f.e.\ with the family of equivalence % relations defined as shown above. % \subsection{Semantic structures: types and environments} % For a set $X$, write $\Delta X$ for the discrete c.o.f.e.\ with $x \nequiv{n} % x'$ iff $n = 0$ or $x = x'$ % $% \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} % \Sem{\textsort{Unit}} &\eqdef& \Delta \{ \star \} \\ % \Sem{\textsort{InvName}} &\eqdef& \Delta \mathbb{N} \\ % \Sem{\textsort{InvMask}} &\eqdef& \Delta \pset{\mathbb{N}} \\ % \Sem{\textsort{Monoid}} &\eqdef& \Delta |\monoid| % \end{array} % \qquad\qquad % \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} % \Sem{\textsort{Val}} &\eqdef& \Delta \textdom{Val} \\ % \Sem{\textsort{Exp}} &\eqdef& \Delta \textdom{Exp} \\ % \Sem{\textsort{Ectx}} &\eqdef& \Delta \textdom{Ectx} \\ % \Sem{\textsort{State}} &\eqdef& \Delta \textdom{State} \\ % \end{array} % \qquad\qquad % \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} % \Sem{\sort \times \sort'} &\eqdef& \Sem{\sort} \times \Sem{\sort} \\ % \Sem{\sort \to \sort'} &\eqdef& \Sem{\sort} \to \Sem{\sort} \\ % \Sem{\Prop} &\eqdef& \textdom{Prop} \\ % \end{array} %$ % The balance of our signature $\Sig$ is interpreted as follows. % For each base type $\type$ not covered by the preceding table, we pick an object $X_\type$ in $\cal U$ and define % $% \Sem{\type} \eqdef X_\type %$ % For each function symbol $\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn$, we pick an arrow $\Sem{\sigfn} : \Sem{\type_1} \times \dots \times \Sem{\type_n} \to \Sem{\type_{n+1}}$ in $\cal U$. % An environment $\vctx$ is interpreted as the set of % maps $\rho$, with $\dom(\rho) = \dom(\vctx)$ and % $\rho(x)\in\Sem{\vctx(x)}$, % and % $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land % \All x\in\dom(\rho). \rho(x) \nequiv{n} \rho'(x)\bigr)$. % \ralf{Re-check all the following definitions with the Coq development.} % %\typedsection{Validity}{valid : \pset{\textdom{Prop}} \in Sets} % % % %\begin{align*} % %valid(p) &\iff \All n \in \mathbb{N}. \All \rs \in \textdom{Res}. \All W \in \textdom{World}. (n, \rs) \in p(W) % %\end{align*} % \typedsection{Later modality}{\later : \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \later p &\eqdef \Lam W. \{\, (n + 1, r) \mid (n, r) \in p(W) \,\} \cup \{\, (0, r) \mid r \in \textdom{Res} \,\} % \end{align*}  Ralf Jung committed Jan 31, 2016 168 % \begin{lem}  Ralf Jung committed Mar 12, 2016 169 170 171 172 173 % $\later{}$ is well-defined: $\later {p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\later{}$ itself is a \emph{contractive} map. % \end{lem} % \typedsection{Always modality}{\always{} : \textdom{Prop} \to \textdom{Prop} \in {\cal U}}  Ralf Jung committed Jan 31, 2016 174 % \begin{align*}  Ralf Jung committed Mar 12, 2016 175 % \always{p} \eqdef \Lam W. \{\, (n, r) \mid (n, \munit) \in p(W) \,\}  Ralf Jung committed Jan 31, 2016 176 % \end{align*}  Ralf Jung committed Mar 12, 2016 177 178 % \begin{lem} % $\always{}$ is well-defined: $\always{p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\always{}$ itself is a non-expansive map.  Ralf Jung committed Jan 31, 2016 179 180 % \end{lem}  Ralf Jung committed Mar 12, 2016 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 % % PDS: p \Rightarrow q not defined. % %\begin{lem}\label{lem:always-impl-valid} % %\begin{align*} % %&\forall p, q \in \textdom{Prop}.~\\ % %&\qquad % % (\forall n \in \mathbb{N}.~\forall \rs \in \textdom{Res}.~\forall W \in \textdom{World}.~(n, \rs) \in p(W) \Rightarrow (n, \rs) \in q(W)) \Leftrightarrow~valid(\always{(p \Rightarrow q)}) % %\end{align*} % %\end{lem} % \typedsection{Invariant definition}{inv : \Delta(\mathbb{N}) \times \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \mathit{inv}(\iota, p) &\eqdef \Lam W. \{\, (n, r) \mid \iota\in\dom(W) \land W(\iota) \nequiv{n+1}_{\textdom{PreProp}} \wIso(p) \,\} % \end{align*} % \begin{lem} % $\mathit{inv}$ is well-defined: $\mathit{inv}(\iota, p)$ is a valid proposition (this amounts to showing non-expansiveness), and $\mathit{inv}$ itself is a non-expansive map. % \end{lem}  Ralf Jung committed Jan 31, 2016 197   Ralf Jung committed Mar 12, 2016 198 199 200 201 202 203 % \typedsection{World satisfaction}{\wsat{-}{-}{-}{-} : % \textdom{State} \times % \pset{\mathbb{N}} \times % \textdom{Res} \times % \textdom{World} \to \psetdown{\mathbb{N}} \in {\cal U}} % \ralf{Make this Dave-compatible: Explicitly compose all the things in $s$}  Ralf Jung committed Feb 01, 2016 204 % \begin{align*}  Ralf Jung committed Mar 12, 2016 205 % \wsat{\state}{\mask}{\rs}{W} &=  Ralf Jung committed Feb 01, 2016 206 % \begin{aligned}[t]  Ralf Jung committed Mar 12, 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 % \{\, n + 1 \in \mathbb{N} \mid &\Exists \rsB:\mathbb{N} \fpfn \textdom{Res}. (\rs \rtimes \rsB).\pres = \state \land{}\\ % &\quad \All \iota \in \dom(W). \iota \in \dom(W) \leftrightarrow \iota \in \dom(\rsB) \land {}\\ % &\quad\quad \iota \in \mask \ra (n, \rsB(\iota)) \in \wIso^{-1}(W(\iota))(W) \,\} \cup \{ 0 \} % \end{aligned} % \end{align*} % \begin{lem}\label{lem:fullsat-nonexpansive} % $\wsat{-}{-}{-}{-}$ is well-defined: It maps into $\psetdown{\mathbb{N}}$. (There is no need for it to be a non-expansive map, it doesn't itself live in $\cal U$.) % \end{lem} % \begin{lem}\label{lem:fullsat-weaken-mask} % \begin{align*} % \MoveEqLeft % \All \state \in \Delta(\textdom{State}). % \All \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}). % \All \rs, \rsB \in \Delta(\textdom{Res}). % \All W \in \textdom{World}. \\& % \mask_1 \subseteq \mask_2 \implies (\wsat{\state}{\mask_2}{\rs}{W}) \subseteq (\wsat{\state}{\mask_1}{\rs}{W}) % \end{align*} % \end{lem} % \begin{lem}\label{lem:nequal_ext_world} % \begin{align*} % & % \All n \in \mathbb{N}. % \All W_1, W_1', W_2 \in \textdom{World}. % W_1 \nequiv{n} W_2 \land W_1 \leq W_1' \implies \Exists W_2' \in \textdom{World}. W_1' \nequiv{n} W_2' \land W_2 \leq W_2' % \end{align*} % \end{lem} % \typedsection{Timeless}{\textit{timeless} : \textdom{Prop} \to \textdom{Prop}} % \begin{align*} % \textit{timeless}(p) \eqdef % \begin{aligned}[t] % \Lam W. % \{\, (n, r) &\mid \All W' \geq W. \All k \leq n. \All r' \in \textdom{Res}. \\  Ralf Jung committed Feb 01, 2016 243 % &\qquad  Ralf Jung committed Mar 12, 2016 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 % k > 0 \land (k - 1, r') \in p(W') \implies (k, r') \in p(W') \,\} % \end{aligned} % \end{align*} % \begin{lem} % \textit{timeless} is well-defined: \textit{timeless}(p) is a valid proposition, and \textit{timeless} itself is a non-expansive map. % \end{lem} % % PDS: \Ra undefined. % %\begin{lem} % %\begin{align*} % %& % % \All p \in \textdom{Prop}. % % \All \mask \in \pset{\mathbb{N}}. % %valid(\textit{timeless}(p) \Ra (\later p \vs[\mask][\mask] p)) % %\end{align*} % %\end{lem} % \typedsection{View-shift}{\mathit{vs} : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \mathit{vs}_{\mask_1}^{\mask_2}(q) &= \Lam W. % \begin{aligned}[t] % \{\, (n, \rs) &\mid \All W_F \geq W. \All \rs_F, \mask_F, \state. \All k \leq n.\\ % &\qquad % k \in (\wsat{\state}{\mask_1 \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \land k > 0 \land \mask_F \sep (\mask_1 \cup \mask_2) \implies{} \\  Ralf Jung committed Feb 01, 2016 269 % &\qquad  Ralf Jung committed Mar 12, 2016 270 % \Exists W' \geq W_F. \Exists \rs'. k \in (\wsat{\state}{\mask_2 \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(W')  Ralf Jung committed Feb 01, 2016 271 272 273 % \,\} % \end{aligned} % \end{align*}  Ralf Jung committed Mar 12, 2016 274 275 276 % \begin{lem} % $\mathit{vs}$ is well-defined: $\mathit{vs}_{\mask_1}^{\mask_2}(q)$ is a valid proposition, and $\mathit{vs}$ is a non-expansive map. % \end{lem}  Ralf Jung committed Jan 31, 2016 277 278   Ralf Jung committed Mar 12, 2016 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 % %\begin{lem}\label{lem:prim_view_shift_trans} % %\begin{align*} % %\MoveEqLeft % % \All \mask_1, \mask_2, \mask_3 \in \Delta(\pset{\mathbb{N}}). % % \All p, q \in \textdom{Prop}. \All W \in \textdom{World}. % % \All n \in \mathbb{N}.\\ % %& % % \mask_2 \subseteq \mask_1 \cup \mask_3 \land % % \bigl(\All W' \geq W. \All r \in \textdom{Res}. \All k \leq n. (k, r) \in p(W') \implies (k, r) \in vs_{\mask_2}^{\mask_3}(q)(W')\bigr) \\ % %&\qquad % % {}\implies \All r \in \textdom{Res}. (n, r) \in vs_{\mask_1}^{\mask_2}(p)(W) \implies (n, r) \in vs_{\mask_1}^{\mask_3}(q)(W) % %\end{align*} % %\end{lem} % % PDS: E_1 ==>> E_2 undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall \mask_1, \mask_2, \mask_3 \in \Delta(\pset{\mathbb{N}}).~ % % \forall p_1, p_2, p_3 \in \textdom{Prop}.~\\ % %&\qquad % % \mask_2 \subseteq \mask_1 \cup \mask_3 \Rightarrow % % valid(((p_1 \vs[\mask_1][\mask_2] p_2) \land (p_2 \vs[\mask_2][\mask_3] p_3)) \Rightarrow (p_1 \vs[\mask_1][\mask_3] p_3)) % %\end{align*} % %\end{lem} % %\begin{lem} % %\begin{align*} % %\MoveEqLeft % % \All \iota \in \mathbb{N}. % % \All p \in \textdom{Prop}. % % \All W \in \textdom{World}. % % \All \rs \in \textdom{Res}. % % \All n \in \mathbb{N}. \\ % %& % % (n, \rs) \in inv(\iota, p)(W) \implies (n, \rs) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W) % %\end{align*} % %\end{lem} % % PDS: * undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall \iota \in \mathbb{N}.~ % % \forall p \in \textdom{Prop}.~ % % \forall W \in \textdom{World}.~ % % \forall \rs \in \textdom{Res}.~ % % \forall n \in \mathbb{N}.~\\ % %&\qquad % % (n, \rs) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \rs) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W) % %\end{align*} % %\end{lem} % % \begin{lem} % % \begin{align*} % % & % % \forall \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}).~ % % valid(\bot \vs[\mask_1][\mask_2] \bot) % % \end{align*} % % \end{lem} % % PDS: E_1 ==>> E_2 undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall p, q \in \textdom{Prop}.~ % % \forall \mask \in \pset{\mathbb{N}}.~ % %valid(\always{(p \Rightarrow q)} \Rightarrow (p \vs[\mask][\mask] q)) % %\end{align*} % %\end{lem} % % PDS: E # E' and E_1 ==>> E_2 undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall p_1, p_2, p_3 \in \textdom{Prop}.~ % % \forall \mask_1, \mask_2, \mask \in \pset{\mathbb{N}}.~ % %valid(\mask \sep \mask_1 \Ra \mask \sep \mask_2 \Ra (p_1 \vs[\mask_1][\mask_2] p_2) \Rightarrow (p_1 * p_3 \vs[\mask_1 \cup \mask][\mask_2 \cup \mask] p_2 * p_3)) % %\end{align*} % %\end{lem} % \typedsection{Weakest precondition}{\mathit{wp} : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \to \textdom{Prop}) \to \textdom{Prop} \in {\cal U}} % % \begin{align*} % % \mathit{wp}_\mask(\expr, q) &\eqdef \Lam W. % % \begin{aligned}[t] % % \{\, (n, \rs) &\mid \All W_F \geq W; k \leq n; \rs_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\wsat{\state}{\mask \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \implies{}\\ % % &\qquad % % (\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \rs'. \\ % % &\qquad\qquad % % k \in (\wsat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(\expr)(W'))~\land \\ % % &\qquad % % (\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \rs'. \\ % % &\qquad\qquad % % k - 1 \in (\wsat{\state'}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k-1, \rs') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\ % % &\qquad % % (\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \rs', \rs_1', \rs_2'. \\ % % &\qquad\qquad % % k - 1 \in (\wsat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land \rs' = \rs_1' \rtimes \rs_2'~\land \\ % % &\qquad\qquad % % (k-1, \rs_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land % % (k-1, \rs_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W')) % % \,\} % % \end{aligned} % % \end{align*} % \begin{lem} % $\mathit{wp}$ is well-defined: $\mathit{wp}_{\mask}(\expr, q)$ is a valid proposition, and $\mathit{wp}$ is a non-expansive map. Besides, the dependency on the recursive occurrence is contractive, so $\mathit{wp}$ has a fixed-point. % \end{lem}  Ralf Jung committed Jan 31, 2016 387   Ralf Jung committed Mar 12, 2016 388 389 390 391 % \begin{lem} % $\mathit{wp}$ on values and non-mask-changing $\mathit{vs}$ agree: % $\mathit{wp}_\mask(\val, q) = \mathit{vs}_{\mask}^{\mask}(q \: \val)$ % \end{lem}  Ralf Jung committed Jan 31, 2016 392   Ralf Jung committed Mar 12, 2016 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 % \begin{align*} % \Sem{\vctx \proves x : \sort}_\gamma &= \gamma(x) \\ % \Sem{\vctx \proves \sigfn(\term_1, \dots, \term_n) : \type_{n+1}}_\gamma &= \Sem{\sigfn}(\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \dots, \Sem{\vctx \proves \term_n : \type_n}_\gamma) \ \WHEN \sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn \\ % \Sem{\vctx \proves \Lam x. \term : \sort \to \sort'}_\gamma &= % \Lam v : \Sem{\sort}. \Sem{\vctx, x : \sort \proves \term : \sort'}_{\gamma[x \mapsto v]} \\ % \Sem{\vctx \proves \term~\termB : \sort'}_\gamma &= % \Sem{\vctx \proves \term : \sort \to \sort'}_\gamma(\Sem{\vctx \proves \termB : \sort}_\gamma) \\ % \Sem{\vctx \proves \unitval : \unitsort}_\gamma &= \star \\ % \Sem{\vctx \proves (\term_1, \term_2) : \sort_1 \times \sort_2}_\gamma &= (\Sem{\vctx \proves \term_1 : \sort_1}_\gamma, \Sem{\vctx \proves \term_2 : \sort_2}_\gamma) \\ % \Sem{\vctx \proves \pi_i~\term : \sort_1}_\gamma &= \pi_i(\Sem{\vctx \proves \term : \sort_1 \times \sort_2}_\gamma) % \end{align*} % % % \begin{align*} % \Sem{\vctx \proves \mzero : \textsort{Monoid}}_\gamma &= \mzero \\ % \Sem{\vctx \proves \munit : \textsort{Monoid}}_\gamma &= \munit \\ % \Sem{\vctx \proves \melt \mtimes \meltB : \textsort{Monoid}}_\gamma &= % \Sem{\vctx \proves \melt : \textsort{Monoid}}_\gamma \mtimes \Sem{\vctx \proves \meltB : \textsort{Monoid}}_\gamma % \end{align*} % % % \Sem{\vctx \proves \MU x. \pred : \sort \to \Prop}_\gamma &\eqdef % \mathit{fix}(\Lam v : \Sem{\sort \to \Prop}. \Sem{\vctx, x : \sort \to \Prop \proves \pred : \sort \to \Prop}_{\gamma[x \mapsto v]}) \\ % \Sem{\vctx \proves \knowInv{\iname}{\prop} : \Prop}_\gamma &= % inv(\Sem{\vctx \proves \iname : \textsort{InvName}}_\gamma, \Sem{\vctx \proves \prop : \Prop}_\gamma) \\ % \Sem{\vctx \proves \ownGGhost{\melt} : \Prop}_\gamma &= % \Lam W. \{\, (n, \rs) \mid \rs.\ghostRes \geq \Sem{\vctx \proves \melt : \textsort{Monoid}}_\gamma \,\} \\ % \Sem{\vctx \proves \ownPhys{\state} : \Prop}_\gamma &= % \Lam W. \{\, (n, \rs) \mid \rs.\pres = \Sem{\vctx \proves \state : \textsort{State}}_\gamma \,\} % % % \begin{align*} % \Sem{\vctx \proves \pvsA{\prop}{\mask_1}{\mask_2} : \Prop}_\gamma &= % \textdom{vs}^{\Sem{\vctx \proves \mask_2 : \textsort{InvMask}}_\gamma}_{\Sem{\vctx \proves \mask_1 : \textsort{InvMask}}_\gamma}(\Sem{\vctx \proves \prop : \Prop}_\gamma) \\ % \Sem{\vctx \proves \dynA{\expr}{\pred}{\mask} : \Prop}_\gamma &= % \textdom{wp}_{\Sem{\vctx \proves \mask : \textsort{InvMask}}_\gamma}(\Sem{\vctx \proves \expr : \textsort{Exp}}_\gamma, \Sem{\vctx \proves \pred : \textsort{Val} \to \Prop}_\gamma) \\ % \Sem{\vctx \proves \wtt{\timeless{\prop}}{\Prop}}_\gamma &= % \textdom{timeless}(\Sem{\vctx \proves \prop : \Prop}_\gamma) % \end{align*}  Ralf Jung committed Jan 31, 2016 433   Ralf Jung committed Mar 12, 2016 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 % \typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : 2 \in \mathit{Sets}} % % \Sem{\vctx \mid \pfctx \proves \propB} \eqdef % \begin{aligned}[t] % \MoveEqLeft % \forall n \in \mathbb{N}.\; % \forall W \in \textdom{World}.\; % \forall \rs \in \textdom{Res}.\; % \forall \gamma \in \Sem{\vctx},\; % \\& % \bigl(\All \propB \in \pfctx. (n, \rs) \in \Sem{\vctx \proves \propB : \Prop}_\gamma(W)\bigr) % \implies (n, \rs) \in \Sem{\vctx \proves \prop : \Prop}_\gamma(W) % \end{aligned} %  Ralf Jung committed Jan 31, 2016 449 450 451 452 453  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: