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 Ralf Jung committed Jan 31, 2016 1 2 \section{Model and semantics}  Ralf Jung committed Mar 07, 2016 3 4 \ralf{What also needs to be done here: Define uPred and its later function; define black later; define the resource CMRA}  Ralf Jung committed Jan 31, 2016 5 6 7 8 9 10 11 12 13 14 15 The semantics closely follows the ideas laid out in~\cite{catlogic}. We just repeat some of the most important definitions here. An \emph{ordered family of equivalence relations} (o.f.e.\@) is a pair $(X,(\nequiv{n})_{n\in\mathbb{N}})$, with $X$ a set, and each $\nequiv{n}$ an equivalence relation over $X$ satisfying \begin{itemize} \item $\All x,x'. x \nequiv{0} x',$ \item $\All x,x',n. x \nequiv{n+1} x' \implies x \nequiv{n} x',$ \item $\All x,x'. (\All n. x\nequiv{n} x') \implies x = x'.$ \end{itemize}  Ralf Jung committed Feb 02, 2016 16 \a  Ralf Jung committed Jan 31, 2016 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 43 44 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 Let $(X,(\nequivset{n}{X})_{n\in\mathbb{N}})$ and $(Y,(\nequivset{n}{Y})_{n\in\mathbb{N}})$ be o.f.e.'s. A function $f: X\to Y$ is \emph{non-expansive} if, for all $x$, $x'$ and $n$, $x \nequivset{n}{X} x' \implies fx \nequivset{n}{Y} f x'.$ Let $(X,(\nequiv{n})_{n\in\mathbb{N}})$ be an o.f.e. A sequence $(x_i)_{i\in\mathbb{N}}$ of elements in $X$ is a \emph{chain} (aka \emph{Cauchy sequence}) if $\All k. \Exists n. \All i,j\geq n. x_i \nequiv{k} x_j.$ A \emph{limit} of a chain $(x_i)_{i\in\mathbb{N}}$ is an element $x\in X$ such that $\All n. \Exists k. \All i\geq k. x_i \nequiv{n} x.$ An o.f.e.\ $(X,(\nequiv{n})_{n\in\mathbb{N}})$ is \emph{complete} if all chains have a limit. A complete o.f.e.\ is called a c.o.f.e.\ (pronounced coffee''). When the family of equivalence relations is clear from context we simply write $X$ for a c.o.f.e.\ $(X,(\nequiv{n})_{n\in\mathbb{N}})$. Let $\cal U$ be the category of c.o.f.e.'s and nonexpansive maps. Products and function spaces are defined as follows. For c.o.f.e.'s $(X,(\nequivset{n}{X})_{n\in\mathbb{N}})$ and $(Y,(\nequivset{n}{Y})_{n\in\mathbb{N}})$, their product is $(X\times Y, (\nequiv{n})_{n\in\mathbb{N}}),$ where $(x,y) \nequiv{n} (x',y') \iff x \nequiv{n} x' \land y \nequiv{n} y'.$ The function space is $(\{\, f : X\to Y \mid f \text{ is non-expansive}\,\}, (\nequiv{n})_{n\in\mathbb{N}}),$ where $f \nequiv{n} g \iff \All x. f(x) \nequiv{n} g(x).$ For a c.o.f.e.\ $(X,(\nequiv{n}_{n\in\mathbb{N}}))$, $\latert (X,(\nequiv{n}_{n\in\mathbb{N}}))$ is the c.o.f.e.\@ $(X,(\nequivB{n}_{n\in\mathbb{N}}))$, where $x \nequivB{n} x' \iff \begin{cases} \top &\IF n=0 \\ x \nequiv{n-1} x' &\IF n>0 \end{cases}$ (Sidenote: $\latert$ extends to a functor on $\cal U$ by the identity action on morphisms). \subsection{Semantic structures: propositions} \ralf{This needs to be synced with the Coq development again.} $\begin{array}[t]{rcl} % \protStatus &::=& \enabled \ALT \disabled \\[0.4em] \textdom{Res} &\eqdef&  Ralf Jung committed Jan 31, 2016 87 \{\, \rs = (\pres, \ghostRes) \mid  Ralf Jung committed Jan 31, 2016 88 \pres \in \textdom{State} \uplus \{\munit\} \land \ghostRes \in \mcarp{\monoid} \,\} \\[0.5em]  Ralf Jung committed Jan 31, 2016 89 (\pres, \ghostRes) \rtimes  Ralf Jung committed Jan 31, 2016 90 91 92 93 94 95 96 (\pres', \ghostRes') &\eqdef& \begin{cases} (\pres, \ghostRes \mtimes \ghostRes') & \mbox{if \pres' = \munit and \ghostRes \mtimes \ghostRes' \neq \mzero} \\ (\pres', \ghostRes \mtimes \ghostRes') & \mbox{if \pres = \munit and \ghostRes \mtimes \ghostRes' \neq \mzero} \end{cases} \\[0.5em] %  Ralf Jung committed Jan 31, 2016 97 \rs \leq \rs' & \eqdef &  Ralf Jung committed Jan 31, 2016 98 \Exists \rs''. \rs' = \rs \rtimes \rs''\\[1em]  Ralf Jung committed Jan 31, 2016 99 100 101 % \UPred(\textdom{Res}) &\eqdef& \{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid  Ralf Jung committed Jan 31, 2016 102 \All (k,\rs) \in p.  Ralf Jung committed Jan 31, 2016 103 \All j\leq k.  Ralf Jung committed Jan 31, 2016 104 105 \All \rs' \geq \rs. (j,\rs')\in p \,\}\\[0.5em]  Ralf Jung committed Jan 31, 2016 106 \restr{p}{k} &\eqdef&  Ralf Jung committed Jan 31, 2016 107 \{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em]  Ralf Jung committed Jan 31, 2016 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 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 $x = x'$ $\begin{array}[t]{@{}l@{\ }c@{\ }l@{}}  Ralf Jung committed Jan 31, 2016 154 155 156 157 \Sem{\textsort{Unit}} &\eqdef& \Delta \{ \star \} \\ \Sem{\textsort{InvName}} &\eqdef& \Delta \mathbb{N} \\ \Sem{\textsort{InvMask}} &\eqdef& \Delta \pset{\mathbb{N}} \\ \Sem{\textsort{Monoid}} &\eqdef& \Delta |\monoid|  Ralf Jung committed Jan 31, 2016 158 159 160 \end{array} \qquad\qquad \begin{array}[t]{@{}l@{\ }c@{\ }l@{}}  Ralf Jung committed Jan 31, 2016 161 162 163 164 \Sem{\textsort{Val}} &\eqdef& \Delta \textdom{Val} \\ \Sem{\textsort{Exp}} &\eqdef& \Delta \textdom{Exp} \\ \Sem{\textsort{Ectx}} &\eqdef& \Delta \textdom{Ectx} \\ \Sem{\textsort{State}} &\eqdef& \Delta \textdom{State} \\  Ralf Jung committed Jan 31, 2016 165 166 167 \end{array} \qquad\qquad \begin{array}[t]{@{}l@{\ }c@{\ }l@{}}  Ralf Jung committed Jan 31, 2016 168 169 170 \Sem{\sort \times \sort'} &\eqdef& \Sem{\sort} \times \Sem{\sort} \\ \Sem{\sort \to \sort'} &\eqdef& \Sem{\sort} \to \Sem{\sort} \\ \Sem{\Prop} &\eqdef& \textdom{Prop} \\  Ralf Jung committed Jan 31, 2016 171 172 173 \end{array}$  Ralf Jung committed Jan 31, 2016 174 The balance of our signature $\Sig$ is interpreted as follows.  Ralf Jung committed Jan 31, 2016 175 176 For each base type $\type$ not covered by the preceding table, we pick an object $X_\type$ in $\cal U$ and define $ Ralf Jung committed Jan 31, 2016 177 \Sem{\type} \eqdef X_\type  Ralf Jung committed Jan 31, 2016 178 $  Ralf Jung committed Jan 31, 2016 179 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 $\cal U$.  Ralf Jung committed Jan 31, 2016 180 181 182  An environment $\vctx$ is interpreted as the set of maps $\rho$, with $\dom(\rho) = \dom(\vctx)$ and  Ralf Jung committed Jan 31, 2016 183 $\rho(x)\in\Sem{\vctx(x)}$,  Ralf Jung committed Jan 31, 2016 184 185 186 187 188 189 190 191 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*}  Ralf Jung committed Jan 31, 2016 192 %valid(p) &\iff \All n \in \mathbb{N}. \All \rs \in \textdom{Res}. \All W \in \textdom{World}. (n, \rs) \in p(W)  Ralf Jung committed Jan 31, 2016 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 %\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*} \begin{lem} $\later{}$ 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}} \begin{align*} \always{p} \eqdef \Lam W. \{\, (n, r) \mid (n, \munit) \in p(W) \,\} \end{align*} \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. \end{lem} % PDS: p \Rightarrow q not defined. %\begin{lem}\label{lem:always-impl-valid} %\begin{align*} %&\forall p, q \in \textdom{Prop}.~\\ %&\qquad  Ralf Jung committed Jan 31, 2016 218 % (\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)})  Ralf Jung committed Jan 31, 2016 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 %\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} \typedsection{World satisfaction}{\fullSat{-}{-}{-}{-} : \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 $s$} \begin{align*}  Ralf Jung committed Jan 31, 2016 237  \fullSat{\state}{\mask}{\rs}{W} &=  Ralf Jung committed Jan 31, 2016 238  \begin{aligned}[t]  Ralf Jung committed Jan 31, 2016 239  \{\, n + 1 \in \mathbb{N} \mid &\Exists \rsB:\mathbb{N} \fpfn \textdom{Res}. (\rs \rtimes \rsB).\pres = \state \land{}\\  Ralf Jung committed Jan 31, 2016 240 241  &\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 \}  Ralf Jung committed Jan 31, 2016 242 243 244 245 246 247 248 249 250 251 252  \end{aligned} \end{align*} \begin{lem}\label{lem:fullsat-nonexpansive} $\fullSat{-}{-}{-}{-}$ 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}}).  Ralf Jung committed Jan 31, 2016 253  \All \rs, \rsB \in \Delta(\textdom{Res}).  Ralf Jung committed Jan 31, 2016 254  \All W \in \textdom{World}. \\&  Ralf Jung committed Jan 31, 2016 255  \mask_1 \subseteq \mask_2 \implies (\fullSat{\state}{\mask_2}{\rs}{W}) \subseteq (\fullSat{\state}{\mask_1}{\rs}{W})  Ralf Jung committed Jan 31, 2016 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  \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}. \\ &\qquad 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]  Ralf Jung committed Jan 31, 2016 298  \{\, (n, \rs) &\mid \All W_F \geq W. \All \rs_F, \mask_F, \state. \All k \leq n.\\  Ralf Jung committed Jan 31, 2016 299  &\qquad  Ralf Jung committed Jan 31, 2016 300  k \in (\fullSat{\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 Jan 31, 2016 301  &\qquad  Ralf Jung committed Jan 31, 2016 302  \Exists W' \geq W_F. \Exists \rs'. k \in (\fullSat{\state}{\mask_2 \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(W')  Ralf Jung committed Jan 31, 2016 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  \,\} \end{aligned} \end{align*} \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} %\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}.  Ralf Jung committed Jan 31, 2016 343 % \All \rs \in \textdom{Res}.  Ralf Jung committed Jan 31, 2016 344 345 % \All n \in \mathbb{N}. \\ %&  Ralf Jung committed Jan 31, 2016 346 % (n, \rs) \in inv(\iota, p)(W) \implies (n, \rs) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W)  Ralf Jung committed Jan 31, 2016 347 348 349 350 351 352 353 354 355 356 %\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}.~  Ralf Jung committed Jan 31, 2016 357 % \forall \rs \in \textdom{Res}.~  Ralf Jung committed Jan 31, 2016 358 359 % \forall n \in \mathbb{N}.~\\ %&\qquad  Ralf Jung committed Jan 31, 2016 360 % (n, \rs) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \rs) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W)  Ralf Jung committed Jan 31, 2016 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 %\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}}  Ralf Jung committed Feb 01, 2016 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 % \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 (\fullSat{\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 (\fullSat{\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 (\fullSat{\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 (\fullSat{\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*}  Ralf Jung committed Jan 31, 2016 416 417 418 419 420 421 422 423 424 \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} \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 425 \typedsection{Interpretation of terms}{\Sem{\vctx \proves \term : \sort} : \Sem{\vctx} \to \Sem{\sort} \in {\cal U}}  Ralf Jung committed Jan 31, 2016 426   Ralf Jung committed Jan 31, 2016 427 %A term $\vctx \proves \term : \sort$ is interpreted as a non-expansive map from $\Sem{\vctx}$ to $\Sem{\sort}$.  Ralf Jung committed Jan 31, 2016 428 429  \begin{align*}  Ralf Jung committed Jan 31, 2016 430 431 432 433 434 435  \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) \\  Ralf Jung committed Feb 02, 2016 436  \Sem{\vctx \proves \unitval : \unitsort}_\gamma &= \star \\  Ralf Jung committed Jan 31, 2016 437 438  \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)  Ralf Jung committed Jan 31, 2016 439 440 441 \end{align*} % \begin{align*}  Ralf Jung committed Jan 31, 2016 442 443 444 445  \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  Ralf Jung committed Jan 31, 2016 446 447 448 \end{align*} % \begin{align*}  Ralf Jung committed Jan 31, 2016 449 450 451 452 453 454 455 456 457  \Sem{\vctx \proves t =_\sort u : \Prop}_\gamma &= \Lam W. \{\, (n, r) \mid \Sem{\vctx \proves t : \sort}_\gamma \nequiv{n+1} \Sem{\vctx \proves u : \sort}_\gamma \,\} \\ \Sem{\vctx \proves \FALSE : \Prop}_\gamma &= \Lam W. \emptyset \\ \Sem{\vctx \proves \TRUE : \Prop}_\gamma &= \Lam W. \mathbb{N} \times \textdom{Res} \\ \Sem{\vctx \proves P \land Q : \Prop}_\gamma &= \Lam W. \Sem{\vctx \proves P : \Prop}_\gamma(W) \cap \Sem{\vctx \proves Q : \Prop}_\gamma(W) \\ \Sem{\vctx \proves P \lor Q : \Prop}_\gamma &= \Lam W. \Sem{\vctx \proves P : \Prop}_\gamma(W) \cup \Sem{\vctx \proves Q : \Prop}_\gamma(W) \\ \Sem{\vctx \proves P \Ra Q : \Prop}_\gamma &=  Ralf Jung committed Jan 31, 2016 458 459 460  \Lam W. \begin{aligned}[t] \{\, (n, r) &\mid \All n' \leq n. \All W' \geq W. \All r' \geq r. \\ &\qquad  Ralf Jung committed Jan 31, 2016 461  (n', r') \in \Sem{\vctx \proves P : \Prop}_\gamma(W')~ \\  Ralf Jung committed Jan 31, 2016 462  &\qquad  Ralf Jung committed Jan 31, 2016 463  \implies (n', r') \in \Sem{\vctx \proves Q : \Prop}_\gamma(W') \,\}  Ralf Jung committed Jan 31, 2016 464  \end{aligned} \\  Ralf Jung committed Jan 31, 2016 465 466 467 468  \Sem{\vctx \proves \All x : \sort. P : \Prop}_\gamma &= \Lam W. \{\, (n, r) \mid \All v \in \Sem{\sort}. (n, r) \in \Sem{\vctx, x : \sort \proves P : \Prop}_{\gamma[x \mapsto v]}(W) \,\} \\ \Sem{\vctx \proves \Exists x : \sort. P : \Prop}_\gamma &= \Lam W. \{\, (n, r) \mid \Exists v \in \Sem{\sort}. (n, r) \in \Sem{\vctx, x : \sort \proves P : \Prop}_{\gamma[x \mapsto v]}(W) \,\}  Ralf Jung committed Jan 31, 2016 469 470 471 \end{align*} % \begin{align*}  Ralf Jung committed Jan 31, 2016 472 473 474 475 476  \Sem{\vctx \proves \always{\prop} : \Prop}_\gamma &= \always{\Sem{\vctx \proves \prop : \Prop}_\gamma} \\ \Sem{\vctx \proves \later{\prop} : \Prop}_\gamma &= \later \Sem{\vctx \proves \prop : \Prop}_\gamma\\ \Sem{\vctx \proves \MU x. \pred : \sort \to \Prop}_\gamma &= \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 \prop * \propB : \Prop}_\gamma &=  Ralf Jung committed Jan 31, 2016 477 478 479  \begin{aligned}[t] \Lam W. \{\, (n, r) &\mid \Exists r_1, r_2. r = r_1 \bullet r_2 \land{} \\ &\qquad  Ralf Jung committed Jan 31, 2016 480  (n, r_1) \in \Sem{\vctx \proves \prop : \Prop}_\gamma \land{} \\  Ralf Jung committed Jan 31, 2016 481  &\qquad  Ralf Jung committed Jan 31, 2016 482  (n, r_2) \in \Sem{\vctx \proves \propB : \Prop}_\gamma \,\}  Ralf Jung committed Jan 31, 2016 483  \end{aligned} \\  Ralf Jung committed Jan 31, 2016 484  \Sem{\vctx \proves \prop \wand \propB : \Prop}_\gamma &=  Ralf Jung committed Jan 31, 2016 485 486 487  \begin{aligned}[t] \Lam W. \{\, (n, r) &\mid \All n' \leq n. \All W' \geq W. \All r'. \\ &\qquad  Ralf Jung committed Jan 31, 2016 488  (n', r') \in \Sem{\vctx \proves \prop : \Prop}_\gamma(W') \land r \sep r' \\  Ralf Jung committed Jan 31, 2016 489  &\qquad  Ralf Jung committed Jan 31, 2016 490  \implies (n', r \bullet r') \in \Sem{\vctx \proves \propB : \Prop}_\gamma(W')  Ralf Jung committed Jan 31, 2016 491 492  \} \end{aligned} \\  Ralf Jung committed Jan 31, 2016 493 494 495 496 497 498  \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 \,\}  Ralf Jung committed Jan 31, 2016 499 500 501 \end{align*} % \begin{align*}  Ralf Jung committed Jan 31, 2016 502 503 504 505 506 507  \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)  Ralf Jung committed Jan 31, 2016 508 509 510 511 512 513 514 515 516 517 \end{align*} \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}.\;  Ralf Jung committed Jan 31, 2016 518 \forall \rs \in \textdom{Res}.\;  Ralf Jung committed Jan 31, 2016 519 \forall \gamma \in \Sem{\vctx},\;  Ralf Jung committed Jan 31, 2016 520 \\&  Ralf Jung committed Jan 31, 2016 521 522 \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)  Ralf Jung committed Jan 31, 2016 523 524 \end{aligned}  Ralf Jung committed Jan 31, 2016 525 526 527 528 529  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: