logic.tex 30.3 KB
 Ralf Jung committed Jan 31, 2016 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 % CONVENTION: % Use \Ra/Lra for the logic and \implies/\iff for the metalogic. % This short (for now) note lays out a \emph{generic} separation logic which % manages sharing through invariants and ownership through (partial commutative) % monoids. The logic is generic in that the actual language it applies to is % taken as a parameter, giving in particular the atomic (per-thread) reduction % relation. Over this, we layer concurrency (by giving a semantics to \kw{fork} % and lifting to thread pools). The generic logic provides numerous logical % connectives and the semantics of Hoare triples and view shifts, together with a % large portion of the proof theory---including, in particular, the structural % rules for Hoare logic. Ultimately, these are proved sound relative to some % simple assumptions about the language. It should be possible, moreover, to give % a generic adequacy proof for Hoare triples as applied to the lifted thread-pool % semantics. \section{Parameters to the logic} \begin{itemize} \item A set \textdom{Exp} of \emph{expressions} (metavariable $\expr$) with a subset \textdom{Val} of values ($\val$). We assume that if $\expr$ is an expression then so is $\fork{\expr}$. We moreover assume a value \textsf{fRet} (giving the intended return value of a fork), and we assume that \begin{align*}  Ralf Jung committed Jan 31, 2016 25 26  \fork{\expr} &\notin \textdom{Val} \\ \fork{\expr_1} = \fork{\expr_2} &\implies \expr_1 = \expr_2  Ralf Jung committed Jan 31, 2016 27 28 29 30 31  \end{align*} \item A set $\textdom{Ectx}$ of \emph{evaluation contexts} ($\ectx$) that includes the empty context $[\; ]$, a plugging operation $\ectx[\expr]$ that produces an expression, and context composition $\circ$ satisfying the following axioms: \begin{align*}  Ralf Jung committed Jan 31, 2016 32 33 34 35 36 37 38  [\; ][ \expr ] &= \expr \\ \ectx_1[\ectx_2[\expr]] &= (\ectx_1 \circ \ectx_2) [\expr] \\ \ectx_1[\expr] = \ectx_2[\expr] &\implies \ectx_1 = \ectx_2 \\ \ectx[\expr_1] = \ectx[\expr_2] &\implies \expr_1 = \expr_2 \\ \ectx_1 \circ \ectx_2 = [\; ] &\implies \ectx_1 = \ectx_2 = [\; ] \\ \ectx[\expr] \in \textdom{Val} &\implies \ectx = [\;] \\ \ectx[\expr] = \fork{\expr'} &\implies \ectx = [\;]  Ralf Jung committed Jan 31, 2016 39 40 41 42 43 44 45 46 47  \end{align*} \item A set \textdom{State} of shared machine states (\eg heaps), metavariable $\state$. \item An \emph{atomic stepping relation} $(- \step -) \subseteq (\textdom{State} \times \textdom{Exp}) \times (\textdom{State} \times \textdom{Exp})$ and notions of an expression to be \emph{reducible} or \emph{stuck}, such that \begin{align*} \textlog{reducible}(\expr) &\iff \Exists \state, \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \\  Ralf Jung committed Jan 31, 2016 48 % \textlog{stuck}(\expr) &\iff \All \ectx, \expr'. \expr = \ectx[\expr'] \implies  Ralf Jung committed Jan 31, 2016 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  \lnot \textlog{reducible}(\expr') \end{align*} and the following hold \begin{align*} &\textlog{stuck}(\fork{\expr})& \\ &\textlog{stuck}(\val)&\\ &\ectx[\expr] = \ectx'[\expr'] \implies \textlog{reducible}(\expr') \implies \expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(step-by-value)} \\ &\ectx[\expr] = \ectx'[\fork{\expr'}] \implies \expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(fork-by-value)} \\ \end{align*} \item A predicate \textlog{atomic} on expressions satisfying \begin{align*} &\textlog{atomic}(\expr) \implies \textlog{reducible}(\expr) &\\ &\textlog{atomic}(\expr) \implies \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \expr_2 \in \textdom{Val} &\mbox{(atomic-step)} \end{align*} \item A commutative monoid with zero, $M$. That is, a set $\mcar{M}$ with two distinguished elements $\mzero$ (zero, undefined) and $\munit$ (one, unit) and an operation $\mtimes$ (times, combine) such that \begin{align*} \melt \mtimes \meltB &= \meltB \mtimes \melt \\ \munit \mtimes \melt &= \melt \\ (\melt \mtimes \meltB) \mtimes \meltC &= \melt \mtimes (\meltB \mtimes \meltC) \\ \mzero \mtimes \melt &= \mzero \\ \mzero &\neq \munit \end{align*} Let $\mcarp{M} \eqdef |\monoid| \setminus \{\mzero\}$. \item Arbitrary additional types and terms. \end{itemize} \section{The concurrent language} \paragraph{Machine syntax} $\tpool \in \textdom{ThreadPool} \eqdef \mathbb{N} \fpfn \textdom{Exp}$ \judgment{Machine reduction} {\cfg{\state}{\tpool} \step \cfg{\state'}{\tpool'}} \begin{mathpar} \infer {\cfg{\state}{\expr} \step \cfg{\state'}{\expr'}} {\cfg{\state}{\tpool [i \mapsto \ectx[\expr]]} \step \cfg{\state'}{\tpool [i \mapsto \ectx[\expr']]}} \and \infer {} {\cfg{\state}{\tpool [i \mapsto \ectx[\fork{\expr}]]} \step \cfg{\state}{\tpool [i \mapsto \ectx[\textsf{fRet}]] [j \mapsto \expr]}} \end{mathpar} \section{Syntax} \subsection{Grammar}\label{sec:grammar} \paragraph{Signatures.} We use a signature to account syntactically for the logic's parameters.  Ralf Jung committed Jan 31, 2016 109 A \emph{signature} $\Sig = (\SigType, \SigFn)$ comprises a set  Ralf Jung committed Jan 31, 2016 110 111 112 113 114 115 116 117 118 119 120 121 122 $\SigType \supseteq \{ \textsort{Val}, \textsort{Exp}, \textsort{Ectx}, \textsort{State}, \textsort{Monoid}, \textsort{InvName}, \textsort{InvMask}, \Prop \}$ of base types (or base \emph{sorts}) and a set $\SigFn$ of typed function symbols. This means that each function symbol has an associated \emph{arity} comprising a natural number $n$ and an ordered list of $n+1$ base types. We write $\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn$ to express that $\sigfn$ is a function symbol with the indicated arity. \dave{Say something not-too-shabby about adequacy: We don't spell out what it means.} \paragraph{Syntax.}  Ralf Jung committed Jan 31, 2016 123 Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\textdom{Var}$ of variables (ranged over by metavariables $x$, $y$, $z$):  Ralf Jung committed Jan 31, 2016 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 \newcommand{\unitterm}{()}% \newcommand{\unitsort}{1}% \unit is bold. \begin{align*} \term, \prop, \pred ::={}& x \mid \sigfn(\term_1, \dots, \term_n) \mid \unitterm \mid (\term, \term) \mid \pi_i\; \term \mid \Lam x.\term \mid \term\;\term \mid \mzero \mid \munit \mid \term \mtimes \term \mid \\& \FALSE \mid \TRUE \mid \term =_\sort \term \mid \prop \Ra \prop \mid \prop \land \prop \mid \prop \lor \prop \mid \prop * \prop \mid \prop \wand \prop \mid \\&  Ralf Jung committed Jan 31, 2016 148 149 150  \MU \var. \pred \mid \Exists \var:\sort. \prop \mid \All \var:\sort. \prop \mid  Ralf Jung committed Jan 31, 2016 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 \\& \knowInv{\term}{\prop} \mid \ownGGhost{\term} \mid \ownPhys{\term} \mid \always\prop \mid {\later\prop} \mid \pvsA{\prop}{\term}{\term} \mid \dynA{\term}{\pred}{\term} \mid \timeless{\prop} \0.4em] \sort ::={}& \type \mid \unitsort \mid \sort \times \sort \mid \sort \to \sort \end{align*}  Ralf Jung committed Jan 31, 2016 167 Recursive predicates must be \emph{guarded}: in \MU \var. \pred, the variable \var can only appear under the later \later modality.  Ralf Jung committed Jan 31, 2016 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  \paragraph{Metavariable conventions.} We introduce additional metavariables ranging over terms and generally let the choice of metavariable indicate the term's sort: \[ \begin{array}{r|l} \text{metavariable} & \text{sort} \\\hline \term, \termB & \text{arbitrary} \\ \val, \valB & \textsort{Val} \\ \expr & \textsort{Exp} \\ \ectx & \textsort{Ectx} \\ \state & \textsort{State} \\ \end{array} \qquad\qquad \begin{array}{r|l} \text{metavariable} & \text{sort} \\\hline \iname & \textsort{InvName} \\ \mask & \textsort{InvMask} \\ \melt, \meltB & \textsort{Monoid} \\ \prop, \propB, \propC & \Prop \\ \pred, \predB, \predC & \sort\to\Prop \text{ (when \sort is clear from context)} \\ \end{array} \paragraph{Variable conventions.} We often abuse notation, using the preceding \emph{term} metavariables to range over (bound) \emph{variables}. We omit type annotations in binders, when the type is clear from context. \subsection{Types}\label{sec:types} Iris terms are simply-typed.  Ralf Jung committed Jan 31, 2016 199 The judgment $\vctx \proves_\Sig \wtt{\term}{\sort}$ expresses that, in signature $\Sig$ and variable context $\vctx$, the term $\term$ has sort $\sort$.  Ralf Jung committed Jan 31, 2016 200 201 202 203 204 In giving the rules for this judgment, we omit the signature (which does not change). A variable context, $\vctx = x_1:\sort_1, \dots, x_n:\sort_n$, declares a list of variables and their sorts. In writing $\vctx, x:\sort$, we presuppose that $x$ is not already declared in $\vctx$.  Ralf Jung committed Jan 31, 2016 205 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\sort}}  Ralf Jung committed Jan 31, 2016 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 \begin{mathparpagebreakable} %%% variables and function symbols \axiom{x : \sort \proves \wtt{x}{\sort}} \and \infer{\vctx \proves \wtt{\term}{\sort}} {\vctx, x:\sort' \proves \wtt{\term}{\sort}} \and \infer{\vctx, x:\sort', y:\sort' \proves \wtt{\term}{\sort}} {\vctx, x:\sort' \proves \wtt{\term[x/y]}{\sort}} \and \infer{\vctx_1, x:\sort', y:\sort'', \vctx_2 \proves \wtt{\term}{\sort}} {\vctx_1, x:\sort'', y:\sort', \vctx_2 \proves \wtt{\term[y/x,x/y]}{\sort}} \and \infer{ \vctx \proves \wtt{\term_1}{\type_1} \and \cdots \and \vctx \proves \wtt{\term_n}{\type_n} \and \sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn }{ \vctx \proves \wtt {\sigfn(\term_1, \dots, \term_n)} {\type_{n+1}} } %%% products \and \axiom{\vctx \proves \wtt{\unitterm}{\unitsort}} \and \infer{\vctx \proves \wtt{\term}{\sort_1} \and \vctx \proves \wtt{\termB}{\sort_2}} {\vctx \proves \wtt{(\term,\termB)}{\sort_1 \times \sort_2}} \and \infer{\vctx \proves \wtt{\term}{\sort_1 \times \sort_2} \and i \in \{1, 2\}} {\vctx \proves \wtt{\pi_i\,\term}{\sort_i}} %%% functions \and \infer{\vctx, x:\sort \proves \wtt{\term}{\sort'}} {\vctx \proves \wtt{\Lam x. \term}{\sort \to \sort'}} \and \infer {\vctx \proves \wtt{\term}{\sort \to \sort'} \and \wtt{\termB}{\sort}} {\vctx \proves \wtt{\term\;\termB}{\sort'}} %%% monoids \and \axiom{\vctx \proves \wtt{\mzero}{\textsort{Monoid}}} \and \axiom{\vctx \proves \wtt{\munit}{\textsort{Monoid}}} \and \infer{\vctx \proves \wtt{\melt}{\textsort{Monoid}} \and \vctx \proves \wtt{\meltB}{\textsort{Monoid}}} {\vctx \proves \wtt{\melt \mtimes \meltB}{\textsort{Monoid}}} %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and \infer{\vctx \proves \wtt{\term}{\sort} \and \vctx \proves \wtt{\termB}{\sort}} {\vctx \proves \wtt{\term =_\sort \termB}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}} {\vctx \proves \wtt{\prop \Ra \propB}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}} {\vctx \proves \wtt{\prop \land \propB}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}} {\vctx \proves \wtt{\prop \lor \propB}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}} {\vctx \proves \wtt{\prop * \propB}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\propB}{\Prop}} {\vctx \proves \wtt{\prop \wand \propB}{\Prop}} \and \infer{  Ralf Jung committed Jan 31, 2016 277 278  \vctx, \var:\sort\to\Prop \proves \wtt{\pred}{\sort\to\Prop} \and \text{$\var$ is guarded in $\pred$}  Ralf Jung committed Jan 31, 2016 279  }{  Ralf Jung committed Jan 31, 2016 280  \vctx \proves \wtt{\MU \var. \pred}{\sort\to\Prop}  Ralf Jung committed Jan 31, 2016 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  } \and \infer{\vctx, x:\sort \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\sort. \prop}{\Prop}} \and \infer{\vctx, x:\sort \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\sort. \prop}{\Prop}} \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\iname}{\textsort{InvName}} }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and \infer{\vctx \proves \wtt{\melt}{\textsort{Monoid}}} {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and \infer{\vctx \proves \wtt{\state}{\textsort{State}}} {\vctx \proves \wtt{\ownPhys{\state}}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\always\prop}{\Prop}} \and \infer{\vctx \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\later\prop}{\Prop}} \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\mask}{\textsort{InvMask}} \and \vctx \proves \wtt{\mask'}{\textsort{InvMask}} }{ \vctx \proves \wtt{\pvsA{\prop}{\mask}{\mask'}}{\Prop} } \and \infer{ \vctx \proves \wtt{\expr}{\textsort{Exp}} \and \vctx \proves \wtt{\pred}{\textsort{Val} \to \Prop} \and \vctx \proves \wtt{\mask}{\textsort{InvMask}} }{ \vctx \proves \wtt{\dynA{\expr}{\pred}{\mask}}{\Prop} } \and \infer{ \vctx \proves \wtt{\prop}{\Prop} }{ \vctx \proves \wtt{\timeless{\prop}}{\Prop} } \end{mathparpagebreakable} \section{Base logic} The judgment $\vctx \mid \pfctx \proves \prop$ says that with free variables $\vctx$, proposition $\prop$ holds whenever all assumptions $\pfctx$ hold. We implicitly assume that an arbitrary variable context, $\vctx$, is added to every constituent of the rules. Axioms $\prop \Ra \propB$ stand for judgments $\vctx \mid \cdot \proves \prop \Ra \propB$ with no assumptions. (Bi-implications are analogous.) % \subsubsection{Judgments} % % Proof rules implicitly assume well-sortedness.  Ralf Jung committed Jan 31, 2016 343 % e\subsection{Laws of intuitionistic higher-order logic with guarded recursion over a simply-typed lambda calculus}\label{sec:HOL}  Ralf Jung committed Jan 31, 2016 344   Ralf Jung committed Jan 31, 2016 345 This is entirely standard.  Ralf Jung committed Jan 31, 2016 346 347  Soundness follows from the theorem that ${\cal U}(\any, \textdom{Prop})  Ralf Jung committed Jan 31, 2016 348 : {\cal U}^{\textrm{op}} \to \textrm{Poset}$ is a hyperdoctrine.  Ralf Jung committed Jan 31, 2016 349 350 351 352 353 354 355 356 357 358  \begin{mathpar} \inferH{Asm} {\prop \in \pfctx} {\pfctx \proves \prop} \and \inferH{Eq} {\pfctx \proves \prop(\term) \\ \pfctx \proves \term = \term'} {\pfctx \proves \prop(\term')} \and  Ralf Jung committed Jan 31, 2016 359 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 360 361 362  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 363 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 364 365 366  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 367 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 368 369 370  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 371 \infer[$\vee$E]  Ralf Jung committed Jan 31, 2016 372 373 374 375 376  {\pfctx \proves \prop \vee \propB \\ \pfctx, \prop \proves \propC \\ \pfctx, \propB \proves \propC} {\pfctx \proves \propC} \and  Ralf Jung committed Jan 31, 2016 377 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 378 379 380  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 381 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 382 383 384  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 385 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 386 387 388  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 389 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 390 391 392  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 393 \infer[$\forall_1$I]  Ralf Jung committed Jan 31, 2016 394 395 396  {\pfctx, x : \sort \proves \prop} {\pfctx \proves \forall x: \sort.\; \prop} \and  Ralf Jung committed Jan 31, 2016 397 \infer[$\forall_1$E]  Ralf Jung committed Jan 31, 2016 398 399 400 401  {\pfctx \proves \forall X \in \sort.\; \prop \\ \pfctx \proves \term: \sort} {\pfctx \proves \prop[\term/X]} \and  Ralf Jung committed Jan 31, 2016 402 \infer[$\exists_1$E]  Ralf Jung committed Jan 31, 2016 403 404 405 406  {\pfctx \proves \exists X\in \sort.\; \prop \\ \pfctx, X : \sort, \prop \proves \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 407 \infer[$\exists_1$I]  Ralf Jung committed Jan 31, 2016 408 409 410 411  {\pfctx \proves \prop[\term/X] \\ \pfctx \proves \term: \sort} {\pfctx \proves \exists X: \sort. \prop} \and  Ralf Jung committed Jan 31, 2016 412 \infer[$\forall_2$I]  Ralf Jung committed Jan 31, 2016 413 414  {\pfctx, \var: \Pred(\sort) \proves \prop} {\pfctx \proves \forall \var\in \Pred(\sort).\; \prop}  Ralf Jung committed Jan 31, 2016 415 \and  Ralf Jung committed Jan 31, 2016 416 \infer[$\forall_2$E]  Ralf Jung committed Jan 31, 2016 417  {\pfctx \proves \forall \var. \prop \\  Ralf Jung committed Jan 31, 2016 418  \pfctx \proves \propB: \Prop}  Ralf Jung committed Jan 31, 2016 419  {\pfctx \proves \prop[\propB/\var]}  Ralf Jung committed Jan 31, 2016 420 \and  Ralf Jung committed Jan 31, 2016 421 \infer[$\exists_2$E]  Ralf Jung committed Jan 31, 2016 422 423  {\pfctx \proves \exists \var \in \Pred(\sort).\prop \\ \pfctx, \var : \Pred(\sort), \prop \proves \propB}  Ralf Jung committed Jan 31, 2016 424 425  {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 426 \infer[$\exists_2$I]  Ralf Jung committed Jan 31, 2016 427  {\pfctx \proves \prop[\propB/\var] \\  Ralf Jung committed Jan 31, 2016 428  \pfctx \proves \propB: \Prop}  Ralf Jung committed Jan 31, 2016 429  {\pfctx \proves \exists \var. \prop}  Ralf Jung committed Jan 31, 2016 430 \and  Ralf Jung committed Jan 31, 2016 431 \inferB[Elem]  Ralf Jung committed Jan 31, 2016 432 433 434  {\pfctx \proves \term \in (X \in \sort). \prop} {\pfctx \proves \prop[\term/X]} \and  Ralf Jung committed Jan 31, 2016 435 \inferB[Elem-$\mu$]  Ralf Jung committed Jan 31, 2016 436 437  {\pfctx \proves \term \in (\mu\var \in \Pred(\sort). \pred)} {\pfctx \proves \term \in \pred[\mu\var \in \Pred(\sort). \pred/\var]}  Ralf Jung committed Jan 31, 2016 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 \end{mathpar} \subsection{Axioms from the logic of (affine) bunched implications} \begin{mathpar} \begin{array}{rMcMl} \prop * \propB &\Lra& \propB * \prop \\ (\prop * \propB) * \propC &\Lra& \prop * (\propB * \propC) \\ \prop * \propB &\Ra& \prop \end{array} \and \begin{array}{rMcMl} (\prop \vee \propB) * \propC &\Lra& (\prop * \propC) \vee (\propB * \propC) \\ (\prop \wedge \propB) * \propC &\Ra& (\prop * \propC) \wedge (\propB * \propC) \\ (\Exists x. \prop) * \propB &\Lra& \Exists x. (\prop * \propB) \\ (\All x. \prop) * \propB &\Ra& \All x. (\prop * \propB) \end{array} \and \infer {\pfctx, \prop_1 \proves \propB_1 \and \pfctx, \prop_2 \proves \propB_2} {\pfctx, \prop_1 * \prop_2 \proves \propB_1 * \propB_2} \and \infer {\pfctx, \prop * \propB \proves \propC} {\pfctx, \prop \proves \propB \wand \propC} \and \infer {\pfctx, \prop \proves \propB \wand \propC} {\pfctx, \prop * \propB \proves \propC} \end{mathpar} \subsection{Laws for ghosts and physical resources} \begin{mathpar} \begin{array}{rMcMl} \ownGGhost{\melt} * \ownGGhost{\meltB} &\Lra& \ownGGhost{\melt \mtimes \meltB} \\ \TRUE &\Ra& \ownGGhost{\munit}\\ \ownGGhost{\mzero} &\Ra& \FALSE\\ \multicolumn{3}{c}{\timeless{\ownGGhost{\melt}}} \end{array} \and \begin{array}{c} \ownPhys{\state} * \ownPhys{\state'} \Ra \FALSE \\ \timeless{\ownPhys{\state}} \end{array} \end{mathpar} \subsection{Laws for the later modality}\label{sec:later} \begin{mathpar} \inferH{Mono} {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and \inferhref{L{\"o}b}{Loeb} {\pfctx, \later{\prop} \proves \prop} {\pfctx \proves \prop} \and \begin{array}[b]{rMcMl} \later{\always{\prop}} &\Lra& \always{\later{\prop}} \\ \later{(\prop \wedge \propB)} &\Lra& \later{\prop} \wedge \later{\propB} \\ \later{(\prop \vee \propB)} &\Lra& \later{\prop} \vee \later{\propB} \\ \end{array} \and \begin{array}[b]{rMcMl} \later{\All x.\prop} &\Lra& \All x. \later\prop \\ \later{\Exists x.\prop} &\Lra& \Exists x. \later\prop \\ \later{(\prop * \propB)} &\Lra& \later\prop * \later\propB \end{array} \end{mathpar} \subsection{Laws for the always modality}\label{sec:always} \begin{mathpar} \axiomH{Necessity} {\always{\prop} \Ra \prop} \and \inferhref{$\always$I}{AlwaysIntro} {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and \begin{array}[b]{rMcMl} \always(\term =_\sort \termB) &\Lra& \term=_\sort \termB \\ \always{\prop} * \propB &\Lra& \always{\prop} \land \propB \\ \always{(\prop \Ra \propB)} &\Ra& \always{\prop} \Ra \always{\propB} \\ \end{array} \and \begin{array}[b]{rMcMl} \always{(\prop \land \propB)} &\Lra& \always{\prop} \land \always{\propB} \\ \always{(\prop \lor \propB)} &\Lra& \always{\prop} \lor \always{\propB} \\ \always{\All x. \prop} &\Lra& \All x. \always{\prop} \\ \always{\Exists x. \prop} &\Lra& \Exists x. \always{\prop} \\ \end{array} \end{mathpar} Note that $\always$ binds more tightly than $*$, $\land$, $\lor$, and $\Ra$. \section{Program logic}\label{sec:proglog} Hoare triples and view shifts are syntactic sugar for weakest (liberal) preconditions and primitive view shifts, respectively: \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask] \eqdef \always{(\prop \Ra \dynA{\expr}{\lambda\Ret\val.\propB}{\mask})} \qquad\qquad \begin{aligned} \prop \vs[\mask_1][\mask_2] \propB &\eqdef \always{(\prop \Ra \pvsA{\propB}{\mask_1}{\mask_2})} \\ \prop \vsE[\mask_1][\mask_2] \propB &\eqdef \prop \vs[\mask_1][\mask_2] \propB \land \propB \vs[\mask2][\mask_1] \prop \end{aligned} We write just one mask for a view shift when $\mask_1 = \mask_2$. The convention for omitted masks is generous: An omitted $\mask$ is $\top$ for Hoare triples and $\emptyset$ for view shifts. % PDS: We're repeating ourselves. We gave Γ conventions and we're about to give Θ conventions. Also, the scope of "Below" is unclear. % Below, we implicitly assume the same context for all judgements which don't have an explicit context at \emph{all} pre-conditions \emph{and} the conclusion. Henceforward, we implicitly assume a proof context, $\pfctx$, is added to every constituent of the rules. Generally, this is an arbitrary proof context. We write $\provesalways$ to denote judgments that can only be extended with a boxed proof context. \ralf{Give the actual base rules from the Coq development instead} \subsection{Hoare triples} \begin{mathpar} \inferH{Ret} {} {\hoare{\TRUE}{\valB}{\Ret\val. \val = \valB}[\mask]} \and \inferH{Bind} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \\ \All \val. \hoare{\propB}{K[\val]}{\Ret\valB.\propC}[\mask]} {\hoare{\prop}{K[\expr]}{\Ret\valB.\propC}[\mask]} \and \inferH{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{Frame} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask]} {\hoare{\prop * \propC}{\expr}{\Ret\val. \propB * \propC}[\mask \uplus \mask']} \and \inferH{AFrame} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \and \text{$\expr$ not a value} } {\hoare{\prop * \later\propC}{\expr}{\Ret\val. \propB * \propC}[\mask \uplus \mask']} \and \inferH{Fork} {\hoare{\prop}{\expr}{\Ret\any. \TRUE}[\top]} {\hoare{\later\prop * \later\propB}{\fork{\expr}}{\Ret\val. \val = \textsf{fRet} \land \propB}[\mask]} \and \inferH{ACsq} {\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} } {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask \uplus \mask']} \end{mathpar} \subsection{View shifts} \begin{mathpar} \inferH{NewInv} {\infinite(\mask)} {\later{\prop} \vs[\mask] \exists \iname\in\mask.\; \knowInv{\iname}{\prop}} \and \inferH{FpUpd} {\melt \mupd \meltsB} {\ownGGhost{\melt} \vs \exists \meltB \in \meltsB.\; \ownGGhost{\meltB}} \and \inferH{VSTrans} {\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 \inferH{VSImp} {\always{(\prop \Ra \propB)}} {\prop \vs[\emptyset] \propB} \and \inferH{VSFrame} {\prop \vs[\mask_1][\mask_2] \propB} {\prop * \propC \vs[\mask_1 \uplus \mask'][\mask_2 \uplus \mask'] \propB * \propC} \and \inferH{VSTimeless} {\timeless{\prop}} {\later \prop \vs \prop} \and \axiomH{InvOpen} {\knowInv{\iname}{\prop} \proves \TRUE \vs[\{ \iname \} ][\emptyset] \later \prop} \and \axiomH{InvClose} {\knowInv{\iname}{\prop} \proves \later \prop \vs[\emptyset][\{ \iname \} ] \TRUE } \end{mathpar} \vspace{5pt} Note that $\timeless{\prop}$ means that $\prop$ does not depend on the step index. Furthermore, $$\melt \mupd \meltsB \eqdef \always{\All \melt_f. \melt \sep \melt_f \Ra \Exists \meltB \in \meltsB. \meltB \sep \melt_f}$$ \subsection{Derived rules} \paragraph{Derived structural rules.} The following are easily derived by unfolding the sugar for Hoare triples and view shifts. \begin{mathpar} \inferHB{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{VSDisj} {\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 \inferHB{Exist} {\All \var. \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} {\hoare{\Exists \var. \prop}{\expr}{\Ret\val.\propB}[\mask]} \and \inferHB{VSExist} {\All \var. (\prop \vs[\mask_1][\mask_2] \propB)} {(\Exists \var. \prop) \vs[\mask_1][\mask_2] \propB} \and \inferHB{BoxOut} {\always\propB \provesalways \hoare{\prop}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \land \always{\propB}}{\expr}{\Ret\val.\propC}[\mask]} \and \inferHB{VSBoxOut} {\always\propB \provesalways \prop \vs[\mask_1][\mask_2] \propC} {\prop \land \always{\propB} \vs[\mask_1][\mask_2] \propC} \and \inferH{False} {} {\hoare{\FALSE}{\expr}{\Ret \val. \prop}[\mask]} \and \inferH{VSFalse} {} {\FALSE \vs[\mask_1][\mask_2] \prop } \end{mathpar} The proofs all follow the same pattern, so we only show two of them in detail. \begin{proof}[Proof of \ruleref{Exist}] After unfolding the syntactic sugar for Hoare triples and removing the boxes from premise and conclusion, our goal becomes $(\Exists \var. \prop(\var)) \Ra \dynA{\expr}{\Lam\val. \propB}{\mask}$ (remember that $\var$ is free in $\prop$) and the premise reads $\All \var. \prop(\var) \Ra \dynA{\expr}{\Lam\val. \propB}{\mask}.$ Let $\var$ be given and assume $\prop(\var)$. To show $\dynA{\expr}{\Lam\val. \propB}{\mask}$, apply the premise to $\var$ and $\prop(\var)$. For the other direction, assume $\hoare{\Exists \var. \prop(\var)}{\expr}{\Ret\val. \propB}[\mask]$ and let $\var$ be given. We have to show $\hoare{\prop(\var)}{\expr}{\Ret\val. \propB}[\mask]$. This trivially follows from \ruleref{Csq} with $\prop(\var) \Ra \Exists \var. \prop(\var)$. \end{proof} \begin{proof}[Proof of \ruleref{BoxOut}] After unfolding the syntactic sugar for Hoare triples, our goal becomes \label{eq:boxin:goal} \always\pfctx \proves \always\bigl(\prop\land\always \propB \Ra \dynA{\expr}{\Lam\val. \propC}{\mask}\bigr) while our premise reads \label{eq:boxin:as} \always\pfctx, \always\propB \proves \always(\prop \Ra \dynA{\expr}{\Lam\val. \propC}{\mask}) By the introduction rules for $\always$ and implication, it suffices to show $(\always\pfctx), \prop,\always \propB \proves \dynA{\expr}{\Lam\val. \propC}{\mask}$ By modus ponens and \ruleref{Necessity}, it suffices to show~\eqref{eq:boxin:as}, which is exactly our assumption. For the other direction, assume~\eqref{eq:boxin:goal}. We have to show~\eqref{eq:boxin:as}. By \ruleref{AlwaysIntro} and implication introduction, it suffices to show $(\always\pfctx), \prop,\always \propB \proves \dynA{\expr}{\Lam\val. \propC}{\mask}$ which easily follows from~\eqref{eq:boxin:goal}. \end{proof} \paragraph{Derived rules for invariants.} Invariants can be opened around atomic expressions and view shifts. \begin{mathpar} \inferH{Inv} {\hoare{\later{\propC} * \prop } {\expr} {\Ret\val. \later{\propC} * \propB }[\mask] \and \physatomic{\expr} } {\knowInv{\iname}{\propC} \proves \hoare{\prop} {\expr} {\Ret\val. \propB}[\mask \uplus \{ \iname \}] } \and \inferH{VSInv} {\later{\prop} * \propB \vs[\mask_1][\mask_2] \later{\prop} * \propC} {\knowInv{\iname}{\prop} \proves \propB \vs[\mask_1 \uplus \{ \iname \}][\mask_2 \uplus \{ \iname \}] \propC} \end{mathpar} \begin{proof}[Proof of \ruleref{Inv}] Use \ruleref{ACsq} with $\mask_1 \eqdef \mask \cup \{\iname\}$, $\mask_2 \eqdef \mask$. The view shifts are obtained by \ruleref{InvOpen} and \ruleref{InvClose} with framing of $\mask$ and $\prop$ or $\propB$, respectively. \end{proof} \begin{proof}[Proof of \ruleref{VSInv}] Analogous to the proof of \ruleref{Inv}, using \ruleref{VSTrans} instead of \ruleref{ACsq}. \end{proof} \subsubsection{Unsound rules} Some rule suggestions (or rather, wishes) keep coming up, which are unsound. We collect them here. \begin{mathpar} \infer {P \vs Q} {\later P \vs \later Q} \and \infer {\later(P \vs Q)} {\later P \vs \later Q} \end{mathpar} Of course, the second rule implies the first, so let's focus on that. Since implications work under $\later$, from $\later P$ we can get $\later \pvs{Q}$. If we now try to prove $\pvs{\later Q}$, we will be unable to establish world satisfaction in the new world: We have no choice but to use $\later \pvs{Q}$ at one step index below what we are operating on (because we have it under a $\later$). We can easily get world satisfaction for that lower step-index (by downwards-closedness of step-indexed predicates). We can, however, not make much use of the world satisfaction that we get out, becaase it is one step-index too low. \subsection{Adequacy} The adequacy statement reads as follows: \begin{align*} &\All \mask, \expr, \val, \pred, i, \state, \state', \tpool'. \\&( \proves \hoare{\ownPhys\state}{\expr}{x.\; \pred(x)}[\mask]) \implies \\&\cfg{\state}{[i \mapsto \expr]} \step^\ast \cfg{\state'}{[i \mapsto \val] \uplus \tpool'} \implies \\&\pred(\val) \end{align*} where $\pred$ can mention neither resources nor invariants. \subsection{Axiom lifting}\label{sec:lifting} The following lemmas help in proving axioms for a particular language. The first applies to expressions with side-effects, and the second to side-effect-free expressions. \dave{Update the others, and the example, wrt the new treatment of $\predB$.} \begin{align*} &\All \expr, \state, \pred, \prop, \propB, \mask. \\ &\textlog{reducible}(e) \implies \\ &(\All \expr', \state'. \cfg{\state}{\expr} \step \cfg{\state'}{\expr'} \implies \pred(\expr', \state')) \implies \\ &{} \proves \bigl( (\All \expr', \state'. \pred (\expr', \state') \Ra \hoare{\prop}{\expr'}{\Ret\val. \propB}[\mask]) \Ra \hoare{ \later \prop * \ownPhys{\state} }{\expr}{\Ret\val. \propB}[\mask] \bigr) \\ \quad\\ &\All \expr, \pred, \prop, \propB, \mask. \\ &\textlog{reducible}(e) \implies \\ &(\All \state, \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \state_2 = \state \land \pred(\expr_2)) \implies \\ &{} \proves \bigl( (\All \expr'. \pred(\expr') \Ra \hoare{\prop}{\expr'}{\Ret\val. \propB}[\mask]) \Ra \hoare{\later\prop}{\expr}{\Ret\val. \propB}[\mask] \bigr) \end{align*} Note that $\pred$ is a meta-logic predicate---it does not depend on any world or resources being owned. The following specializations cover all cases of a heap-manipulating lambda calculus like $F_{\mu!}$. \begin{align*} &\All \expr, \expr', \prop, \propB, \mask. \\ &\textlog{reducible}(e) \implies \\ &(\All \state, \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \state_2 = \state \land \expr_2 = \expr') \implies \\ &{} \proves (\hoare{\prop}{\expr'}{\Ret\val. \propB}[\mask] \Ra \hoare{\later\prop}{\expr}{\Ret\val. \propB}[\mask] ) \\ \quad \\ &\All \expr, \state, \pred, \mask. \\ &\textlog{atomic}(e) \implies \\ &\bigl(\All \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \pred(\expr_2, \state_2)\bigr) \implies \\ &{} \proves (\hoare{ \ownPhys{\state} }{\expr}{\Ret\val. \Exists\state'. \ownPhys{\state'} \land \pred(\val, \state') }[\mask] ) \end{align*} The first is restricted to deterministic pure reductions, like $\beta$-reduction. The second is suited to proving triples for (possibly non-deterministic) atomic expressions; for example, with $\expr \eqdef \;!\ell$ (dereferencing $\ell$) and $\state \eqdef h \mtimes \ell \mapsto \valB$ and $\pred(\val, \state') \eqdef \state' = (h \mtimes \ell \mapsto \valB) \land \val = \valB$, one obtains the axiom $\All h, \ell, \valB. \hoare{\ownPhys{h \mtimes \ell \mapsto \valB}}{!\ell}{\Ret\val. \val = \valB \land \ownPhys{h \mtimes \ell \mapsto \valB} }$. %Axioms for CAS-like operations can be obtained by first deriving rules for the two possible cases, and then using the disjunction rule.  Ralf Jung committed Jan 31, 2016 808 809 810 811 812  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: