logic.tex 29.4 KB
 Ralf Jung committed Jan 31, 2016 1 2 3 4  \section{Parameters to the logic} \begin{itemize}  Ralf Jung committed Feb 01, 2016 5 6 7 8 9 10 11 12 % \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*} % \fork{\expr} &\notin \textdom{Val} \\ % \fork{\expr_1} = \fork{\expr_2} &\implies \expr_1 = \expr_2 % \end{align*}  Ralf Jung committed Jan 31, 2016 13 14 15 16 \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 17 18 19 20 21 22  [\; ][ \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 = [\;] \\  Ralf Jung committed Feb 01, 2016 23 % \ectx[\expr] = \fork{\expr'} &\implies \ectx = [\;]  Ralf Jung committed Jan 31, 2016 24 25 26 27 28 29 30 31 32  \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 33 % \textlog{stuck}(\expr) &\iff \All \ectx, \expr'. \expr = \ectx[\expr'] \implies  Ralf Jung committed Jan 31, 2016 34 35 36  \lnot \textlog{reducible}(\expr') \end{align*} and the following hold  Ralf Jung committed Feb 01, 2016 37 38 39 40 41 42 43 44 % \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*}  Ralf Jung committed Jan 31, 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  \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']]}}  Ralf Jung committed Feb 01, 2016 81 82 83 84 85 % \and % \infer % {} % {\cfg{\state}{\tpool [i \mapsto \ectx[\fork{\expr}]]} \step % \cfg{\state}{\tpool [i \mapsto \ectx[\textsf{fRet}]] [j \mapsto \expr]}}  Ralf Jung committed Jan 31, 2016 86 87 88 89 90 91 92 93 \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 94 A \emph{signature} $\Sig = (\SigType, \SigFn)$ comprises a set  Ralf Jung committed Jan 31, 2016 95 96 97 98 99 100 101 102 103 104 105 106 107 $\SigType \supseteq \{ \textsort{Val}, \textsort{Exp}, \textsort{Ectx}, \textsort{State}, \textsort{Monoid}, \textsort{InvName}, \textsort{InvMask}, \Prop \}$ 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 108 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 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 \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 133 134 135  \MU \var. \pred \mid \Exists \var:\sort. \prop \mid \All \var:\sort. \prop \mid  Ralf Jung committed Jan 31, 2016 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 \\& \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 152 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 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183  \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 184 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 185 186 187 188 189 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 190 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\sort}}  Ralf Jung committed Jan 31, 2016 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 \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 262 263  \vctx, \var:\sort\to\Prop \proves \wtt{\pred}{\sort\to\Prop} \and \text{$\var$ is guarded in $\pred$}  Ralf Jung committed Jan 31, 2016 264  }{  Ralf Jung committed Jan 31, 2016 265  \vctx \proves \wtt{\MU \var. \pred}{\sort\to\Prop}  Ralf Jung committed Jan 31, 2016 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327  } \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 328 % 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 329   Ralf Jung committed Jan 31, 2016 330 This is entirely standard.  Ralf Jung committed Jan 31, 2016 331 332  Soundness follows from the theorem that ${\cal U}(\any, \textdom{Prop})  Ralf Jung committed Jan 31, 2016 333 : {\cal U}^{\textrm{op}} \to \textrm{Poset}$ is a hyperdoctrine.  Ralf Jung committed Jan 31, 2016 334 335 336 337 338 339 340 341 342 343  \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 344 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 345 346 347  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 348 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 349 350 351  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 352 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 353 354 355  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 356 \infer[$\vee$E]  Ralf Jung committed Jan 31, 2016 357 358 359 360 361  {\pfctx \proves \prop \vee \propB \\ \pfctx, \prop \proves \propC \\ \pfctx, \propB \proves \propC} {\pfctx \proves \propC} \and  Ralf Jung committed Jan 31, 2016 362 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 363 364 365  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 366 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 367 368 369  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 370 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 371 372 373  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 374 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 375 376 377  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 378 \infer[$\forall_1$I]  Ralf Jung committed Jan 31, 2016 379 380 381  {\pfctx, x : \sort \proves \prop} {\pfctx \proves \forall x: \sort.\; \prop} \and  Ralf Jung committed Jan 31, 2016 382 \infer[$\forall_1$E]  Ralf Jung committed Jan 31, 2016 383 384 385 386  {\pfctx \proves \forall X \in \sort.\; \prop \\ \pfctx \proves \term: \sort} {\pfctx \proves \prop[\term/X]} \and  Ralf Jung committed Jan 31, 2016 387 \infer[$\exists_1$E]  Ralf Jung committed Jan 31, 2016 388 389 390 391  {\pfctx \proves \exists X\in \sort.\; \prop \\ \pfctx, X : \sort, \prop \proves \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 392 \infer[$\exists_1$I]  Ralf Jung committed Jan 31, 2016 393 394 395 396  {\pfctx \proves \prop[\term/X] \\ \pfctx \proves \term: \sort} {\pfctx \proves \exists X: \sort. \prop} \and  Ralf Jung committed Jan 31, 2016 397 \infer[$\forall_2$I]  Ralf Jung committed Jan 31, 2016 398 399  {\pfctx, \var: \Pred(\sort) \proves \prop} {\pfctx \proves \forall \var\in \Pred(\sort).\; \prop}  Ralf Jung committed Jan 31, 2016 400 \and  Ralf Jung committed Jan 31, 2016 401 \infer[$\forall_2$E]  Ralf Jung committed Jan 31, 2016 402  {\pfctx \proves \forall \var. \prop \\  Ralf Jung committed Jan 31, 2016 403  \pfctx \proves \propB: \Prop}  Ralf Jung committed Jan 31, 2016 404  {\pfctx \proves \prop[\propB/\var]}  Ralf Jung committed Jan 31, 2016 405 \and  Ralf Jung committed Jan 31, 2016 406 \infer[$\exists_2$E]  Ralf Jung committed Jan 31, 2016 407 408  {\pfctx \proves \exists \var \in \Pred(\sort).\prop \\ \pfctx, \var : \Pred(\sort), \prop \proves \propB}  Ralf Jung committed Jan 31, 2016 409 410  {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 411 \infer[$\exists_2$I]  Ralf Jung committed Jan 31, 2016 412  {\pfctx \proves \prop[\propB/\var] \\  Ralf Jung committed Jan 31, 2016 413  \pfctx \proves \propB: \Prop}  Ralf Jung committed Jan 31, 2016 414  {\pfctx \proves \exists \var. \prop}  Ralf Jung committed Jan 31, 2016 415 \and  Ralf Jung committed Jan 31, 2016 416 \inferB[Elem]  Ralf Jung committed Jan 31, 2016 417 418 419  {\pfctx \proves \term \in (X \in \sort). \prop} {\pfctx \proves \prop[\term/X]} \and  Ralf Jung committed Jan 31, 2016 420 \inferB[Elem-$\mu$]  Ralf Jung committed Jan 31, 2016 421 422  {\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 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 \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']}  Ralf Jung committed Feb 01, 2016 570 571 572 573 % \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]}  Ralf Jung committed Jan 31, 2016 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 \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 793 794 795 796 797  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: