logic.tex 23.6 KB
 Ralf Jung committed Mar 06, 2016 1 \section{Language}  Ralf Jung committed Jan 31, 2016 2   Ralf Jung committed Mar 07, 2016 3 A \emph{language} $\Lang$ consists of a set \textdom{Expr} of \emph{expressions} (metavariable $\expr$), a set \textdom{Val} of \emph{values} (metavariable $\val$), and a set \textdom{State} of \emph{states} (metvariable $\state$) such that  Ralf Jung committed Jan 31, 2016 4 \begin{itemize}  Ralf Jung committed Mar 06, 2016 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 \item There exist functions $\ofval : \textdom{Val} \to \textdom{Expr}$ and $\toval : \textdom{Expr} \pfn \textdom{val}$ (notice the latter is partial), such that \begin{mathpar} {\All \expr, \val. \toval(\expr) = \val \Ra \ofval(\val) = \expr} \and {\All\val. \toval(\ofval(\val)) = \val} \end{mathpar} \item There exists a \emph{primitive reduction relation} $(-,- \step -,-,-) \subseteq \textdom{Expr} \times \textdom{State} \times \textdom{Expr} \times \textdom{State} \times (\textdom{Expr} \uplus \set{()})$ We will write $\expr_1, \state_1 \step \expr_2, \state_2$ for $\expr_1, \state_1 \step \expr_2, \state_2, ()$. \\ A reduction $\expr_1, \state_1 \step \expr_2, \state_2, \expr'$ indicates that, when $\expr_1$ reduces to $\expr$, a \emph{new thread} $\expr'$ is forked off. \item All values are stuck: $\expr, \_ \step \_, \_, \_ \Ra \toval(\expr) = \bot$ \item There is a predicate defining \emph{atomic} expressions satisfying \let\oldcr\cr \begin{mathpar} {\All\expr. \atomic(\expr) \Ra \toval(\expr) = \bot} \and {{ \begin{inbox} \All\expr_1, \state_1, \expr_2, \state_2, \expr'. \atomic(\expr_1) \land \expr_1, \state_1 \step \expr_2, \state_2, \expr' \Ra {}\\\qquad\qquad\qquad\quad~~ \Exists \val_2. \toval(\expr_2) = \val_2 \end{inbox} }} \end{mathpar} In other words, atomic expression \emph{reduce in one step to a value}. It does not matter whether they fork off an arbitrary expression.  Ralf Jung committed Jan 31, 2016 25 26 \end{itemize}  Ralf Jung committed Mar 07, 2016 27 \begin{defn}[Context]  Ralf Jung committed Mar 07, 2016 28 29 30 31 32 33 34 35 36  A function $\lctx : \textdom{Expr} \to \textdom{Expr}$ is a \emph{context} if the following conditions are satisfied: \begin{enumerate} \item $\lctx$ does not turn non-values into values:\\ $\All\expr. \toval(\expr) = \bot \Ra \toval(\lctx(\expr)) = \bot$ \item One can perform reductions below $\lctx$:\\ $\All \expr_1, \state_1, \expr_2, \state_2, \expr'. \expr_1, \state_1 \step \expr_2,\state_2,\expr' \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr'$ \item Reductions stay below $\lctx$ until there is a value in the hole:\\ $\All \expr_1', \state_1, \expr_2, \state_2, \expr'. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr' \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr'$ \end{enumerate}  Ralf Jung committed Mar 07, 2016 37 38 \end{defn}  Ralf Jung committed Mar 06, 2016 39 40 41 \subsection{The concurrent language} For any language $\Lang$, we define the corresponding thread-pool semantics.  Ralf Jung committed Jan 31, 2016 42 43 44  \paragraph{Machine syntax} $ Ralf Jung committed Mar 06, 2016 45  \tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n  Ralf Jung committed Jan 31, 2016 46 47 $  Ralf Jung committed Mar 06, 2016 48 49 \judgment{Machine reduction} {\cfg{\tpool}{\state} \step \cfg{\tpool'}{\state'}}  Ralf Jung committed Jan 31, 2016 50 51 \begin{mathpar} \infer  Ralf Jung committed Mar 06, 2016 52 53 54 55 56 57 58  {\expr_1, \state_1 \step \expr_2, \state_2, \expr' \and \expr' \neq ()} {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr']}{\state'}} \and\infer {\expr_1, \state_1 \step \expr_2, \state_2} {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool'}{\state'}}  Ralf Jung committed Jan 31, 2016 59 60 \end{mathpar}  Ralf Jung committed Mar 07, 2016 61 \clearpage  Ralf Jung committed Mar 06, 2016 62 63 64 65 66 67 \section{The logic} To instantiate Iris, you need to define the following parameters: \begin{itemize} \item A language $\Lang$ \item A locally contractive functor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state  Ralf Jung committed Mar 07, 2016 68  \ralf{$\iFunc$ also needs to have a single-unit.}  Ralf Jung committed Mar 06, 2016 69 \end{itemize}  Ralf Jung committed Jan 31, 2016 70   Ralf Jung committed Mar 06, 2016 71 72 73 \noindent As usual for higher-order logics, you can furthermore pick a \emph{signature} $\Sig = (\SigType, \SigFn, \SigAx)$ to add more types, symbols and axioms to the language. You have to make sure that $\SigType$ includes the base types:  Ralf Jung committed Jan 31, 2016 74 $ Ralf Jung committed Mar 08, 2016 75  \SigType \supseteq \{ \textlog{Val}, \textlog{Expr}, \textlog{State}, \textlog{M}, \textlog{InvName}, \textlog{InvMask}, \Prop \}  Ralf Jung committed Jan 31, 2016 76 $  Ralf Jung committed Mar 06, 2016 77 78 79 Elements of $\SigType$ are ranged over by $\sigtype$. Each function symbol in $\SigFn$ has an associated \emph{arity} comprising a natural number $n$ and an ordered list of $n+1$ types $\type$ (the grammar of $\type$ is defined below, and depends only on $\SigType$).  Ralf Jung committed Jan 31, 2016 80 81 82 83 84 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.  Ralf Jung committed Mar 06, 2016 85 86 87 88 89 90  Furthermore, $\SigAx$ is a set of \emph{axioms}, that is, terms $\term$ of type $\Prop$. Again, the grammar of terms and their typing rules are defined below, and depends only on $\SigType$ and $\SigFn$, not on $\SigAx$. Elements of $\SigAx$ are ranged over by $\sigax$. \subsection{Grammar}\label{sec:grammar}  Ralf Jung committed Jan 31, 2016 91 92  \paragraph{Syntax.}  Ralf Jung committed Jan 31, 2016 93 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 Feb 02, 2016 94   Ralf Jung committed Jan 31, 2016 95 \begin{align*}  Ralf Jung committed Mar 08, 2016 96  \type \bnfdef{}&  Ralf Jung committed Mar 06, 2016 97  \sigtype \mid  Ralf Jung committed Mar 08, 2016 98  1 \mid  Ralf Jung committed Mar 06, 2016 99 100 101  \type \times \type \mid \type \to \type \0.4em]  Ralf Jung committed Mar 08, 2016 102  \term, \prop, \pred \bnfdef{}&  Ralf Jung committed Mar 06, 2016 103  \var \mid  Ralf Jung committed Jan 31, 2016 104  \sigfn(\term_1, \dots, \term_n) \mid  Ralf Jung committed Mar 08, 2016 105  () \mid  Ralf Jung committed Jan 31, 2016 106 107  (\term, \term) \mid \pi_i\; \term \mid  Ralf Jung committed Mar 06, 2016 108  \Lam \var:\type.\term \mid  Ralf Jung committed Mar 06, 2016 109  \term(\term) \mid  Ralf Jung committed Jan 31, 2016 110 111 112 113 114  \munit \mid \term \mtimes \term \mid \\& \FALSE \mid \TRUE \mid  Ralf Jung committed Mar 06, 2016 115  \term =_\type \term \mid  Ralf Jung committed Jan 31, 2016 116 117 118 119 120 121  \prop \Ra \prop \mid \prop \land \prop \mid \prop \lor \prop \mid \prop * \prop \mid \prop \wand \prop \mid \\&  Ralf Jung committed Mar 06, 2016 122  \MU \var:\type. \pred \mid  Ralf Jung committed Mar 06, 2016 123 124  \Exists \var:\type. \prop \mid \All \var:\type. \prop \mid  Ralf Jung committed Jan 31, 2016 125 126 127 128 129 130 \\& \knowInv{\term}{\prop} \mid \ownGGhost{\term} \mid \ownPhys{\term} \mid \always\prop \mid {\later\prop} \mid  Ralf Jung committed Mar 07, 2016 131  \pvs[\term][\term] \prop\mid  Ralf Jung committed Mar 07, 2016 132  \wpre{\term}{\Ret\var.\term}[\term]  Ralf Jung committed Jan 31, 2016 133 \end{align*}  Ralf Jung committed Jan 31, 2016 134 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 135   Ralf Jung committed Mar 06, 2016 136 Note that \always and \later bind more tightly than *, \wand, \land, \lor, and \Ra.  Ralf Jung committed Mar 07, 2016 137 We will write \pvs[\term] \prop for \pvs[\term][\term] \prop.  Ralf Jung committed Mar 07, 2016 138 139 If we omit the mask, then it is \top for weakest precondition \wpre\expr{\Ret\var.\prop} and \emptyset for primitive view shifts \pvs \prop.  Ralf Jung committed Mar 06, 2016 140   Ralf Jung committed Jan 31, 2016 141 \paragraph{Metavariable conventions.}  Ralf Jung committed Mar 06, 2016 142 We introduce additional metavariables ranging over terms and generally let the choice of metavariable indicate the term's type:  Ralf Jung committed Jan 31, 2016 143 144 \[ \begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 145  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Jan 31, 2016 146  \term, \termB & \text{arbitrary} \\  Ralf Jung committed Mar 08, 2016 147 148 149  \val, \valB & \textlog{Val} \\ \expr & \textlog{Expr} \\ \state & \textlog{State} \\  Ralf Jung committed Jan 31, 2016 150 151 152 \end{array} \qquad\qquad \begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 153  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Mar 08, 2016 154 155 156  \iname & \textlog{InvName} \\ \mask & \textlog{InvMask} \\ \melt, \meltB & \textlog{M} \\  Ralf Jung committed Jan 31, 2016 157  \prop, \propB, \propC & \Prop \\  Ralf Jung committed Mar 06, 2016 158  \pred, \predB, \predC & \type\to\Prop \text{ (when \type is clear from context)} \\  Ralf Jung committed Jan 31, 2016 159 160 161 162 \end{array} \paragraph{Variable conventions.}  Ralf Jung committed Feb 02, 2016 163 We often abuse notation, using the preceding \emph{term} meta-variables to range over (bound) \emph{variables}.  Ralf Jung committed Jan 31, 2016 164 We omit type annotations in binders, when the type is clear from context.  Ralf Jung committed Mar 06, 2016 165 We assume that, if a term occurs multiple times in a rule, its free variables are exactly those binders which are available at every occurrence.  Ralf Jung committed Jan 31, 2016 166 167 168 169 170  \subsection{Types}\label{sec:types} Iris terms are simply-typed.  Ralf Jung committed Mar 06, 2016 171 The judgment $\vctx \proves \wtt{\term}{\type}$ expresses that, in variable context $\vctx$, the term $\term$ has type $\type$.  Ralf Jung committed Jan 31, 2016 172   Ralf Jung committed Mar 06, 2016 173 174 A variable context, $\vctx = x_1:\type_1, \dots, x_n:\type_n$, declares a list of variables and their types. In writing $\vctx, x:\type$, we presuppose that $x$ is not already declared in $\vctx$.  Ralf Jung committed Jan 31, 2016 175   Ralf Jung committed Mar 06, 2016 176 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 177 178 \begin{mathparpagebreakable} %%% variables and function symbols  Ralf Jung committed Mar 06, 2016 179  \axiom{x : \type \proves \wtt{x}{\type}}  Ralf Jung committed Jan 31, 2016 180 \and  Ralf Jung committed Mar 06, 2016 181 182  \infer{\vctx \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 183 \and  Ralf Jung committed Mar 06, 2016 184 185  \infer{\vctx, x:\type', y:\type' \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term[x/y]}{\type}}  Ralf Jung committed Jan 31, 2016 186 \and  Ralf Jung committed Mar 06, 2016 187 188  \infer{\vctx_1, x:\type', y:\type'', \vctx_2 \proves \wtt{\term}{\type}} {\vctx_1, x:\type'', y:\type', \vctx_2 \proves \wtt{\term[y/x,x/y]}{\type}}  Ralf Jung committed Jan 31, 2016 189 190 191 192 193 194 195 196 197 198 199 \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  Ralf Jung committed Mar 08, 2016 200  \axiom{\vctx \proves \wtt{()}{1}}  Ralf Jung committed Jan 31, 2016 201 \and  Ralf Jung committed Mar 06, 2016 202 203  \infer{\vctx \proves \wtt{\term}{\type_1} \and \vctx \proves \wtt{\termB}{\type_2}} {\vctx \proves \wtt{(\term,\termB)}{\type_1 \times \type_2}}  Ralf Jung committed Jan 31, 2016 204 \and  Ralf Jung committed Mar 06, 2016 205 206  \infer{\vctx \proves \wtt{\term}{\type_1 \times \type_2} \and i \in \{1, 2\}} {\vctx \proves \wtt{\pi_i\,\term}{\type_i}}  Ralf Jung committed Jan 31, 2016 207 208 %%% functions \and  Ralf Jung committed Mar 06, 2016 209 210  \infer{\vctx, x:\type \proves \wtt{\term}{\type'}} {\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}  Ralf Jung committed Jan 31, 2016 211 212 \and \infer  Ralf Jung committed Mar 06, 2016 213 214  {\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}} {\vctx \proves \wtt{\term(\termB)}{\type'}}  Ralf Jung committed Jan 31, 2016 215 216 %%% monoids \and  Ralf Jung committed Mar 08, 2016 217  \infer{}{\vctx \proves \wtt{\munit}{\textlog{M} \to \textlog{M}}}  Ralf Jung committed Jan 31, 2016 218 \and  Ralf Jung committed Mar 08, 2016 219 220  \infer{\vctx \proves \wtt{\melt}{\textlog{M}} \and \vctx \proves \wtt{\meltB}{\textlog{M}}} {\vctx \proves \wtt{\melt \mtimes \meltB}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 221 222 223 224 225 226 %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and  Ralf Jung committed Mar 06, 2016 227 228  \infer{\vctx \proves \wtt{\term}{\type} \and \vctx \proves \wtt{\termB}{\type}} {\vctx \proves \wtt{\term =_\type \termB}{\Prop}}  Ralf Jung committed Jan 31, 2016 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 \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 Mar 06, 2016 246 247  \vctx, \var:\type \proves \wtt{\term}{\type} \and \text{$\var$ is guarded in $\term$}  Ralf Jung committed Jan 31, 2016 248  }{  Ralf Jung committed Mar 06, 2016 249  \vctx \proves \wtt{\MU \var:\type. \term}{\type}  Ralf Jung committed Jan 31, 2016 250 251  } \and  Ralf Jung committed Mar 06, 2016 252 253  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 254 \and  Ralf Jung committed Mar 06, 2016 255 256  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 257 258 259 \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and  Ralf Jung committed Mar 08, 2016 260  \vctx \proves \wtt{\iname}{\textlog{InvName}}  Ralf Jung committed Jan 31, 2016 261 262 263 264  }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and  Ralf Jung committed Mar 08, 2016 265  \infer{\vctx \proves \wtt{\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 266 267  {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and  Ralf Jung committed Mar 08, 2016 268  \infer{\vctx \proves \wtt{\state}{\textlog{State}}}  Ralf Jung committed Jan 31, 2016 269 270 271 272 273 274 275 276 277 278  {\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  Ralf Jung committed Mar 08, 2016 279 280  \vctx \proves \wtt{\mask}{\textlog{InvMask}} \and \vctx \proves \wtt{\mask'}{\textlog{InvMask}}  Ralf Jung committed Jan 31, 2016 281  }{  Ralf Jung committed Mar 07, 2016 282  \vctx \proves \wtt{\pvs[\mask][\mask'] \prop}{\Prop}  Ralf Jung committed Jan 31, 2016 283 284 285  } \and \infer{  Ralf Jung committed Mar 08, 2016 286 287 288  \vctx \proves \wtt{\expr}{\textlog{Expr}} \and \vctx,\var:\textlog{Val} \proves \wtt{\term}{\Prop} \and \vctx \proves \wtt{\mask}{\textlog{InvMask}}  Ralf Jung committed Jan 31, 2016 289  }{  Ralf Jung committed Mar 07, 2016 290  \vctx \proves \wtt{\wpre{\expr}{\Ret\var.\term}[\mask]}{\Prop}  Ralf Jung committed Jan 31, 2016 291 292 293  } \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 294 \subsection{Timeless propositions}  Ralf Jung committed Mar 06, 2016 295 296 297  Some propositions are \emph{timeless}, which intuitively means that step-indexing does not affect them. This is a \emph{meta-level} assertions about propositions, defined by the following judgment.  Ralf Jung committed Jan 31, 2016 298   Ralf Jung committed Mar 06, 2016 299 \judgment{Timeless Propositions}{\timeless{P}}  Ralf Jung committed Jan 31, 2016 300   Ralf Jung committed Mar 06, 2016 301 302 \ralf{Define a judgment that defines them.}  Ralf Jung committed Mar 06, 2016 303 \subsection{Proof rules}  Ralf Jung committed Mar 06, 2016 304   Ralf Jung committed Jan 31, 2016 305 306 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.  Ralf Jung committed Mar 07, 2016 307 Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent.  Ralf Jung committed Jan 31, 2016 308 309 310 Axioms $\prop \Ra \propB$ stand for judgments $\vctx \mid \cdot \proves \prop \Ra \propB$ with no assumptions. (Bi-implications are analogous.)  Ralf Jung committed Mar 06, 2016 311 \judgment{}{\vctx \mid \pfctx \proves \prop}  Ralf Jung committed Mar 06, 2016 312 \paragraph{Laws of intuitionistic higher-order logic.}  Ralf Jung committed Jan 31, 2016 313 This is entirely standard.  Ralf Jung committed Mar 06, 2016 314 315 \begin{mathparpagebreakable} \infer[Asm]  Ralf Jung committed Jan 31, 2016 316 317 318  {\prop \in \pfctx} {\pfctx \proves \prop} \and  Ralf Jung committed Mar 06, 2016 319 \infer[Eq]  Ralf Jung committed Mar 07, 2016 320 321  {\pfctx \proves \prop \\ \pfctx \proves \term =_\type \term'} {\pfctx \proves \prop[\term'/\term]}  Ralf Jung committed Jan 31, 2016 322 \and  Ralf Jung committed Mar 06, 2016 323 324 325 326 327 328 329 330 331 332 333 334 \infer[Refl] {} {\pfctx \proves \term =_\type \term} \and \infer[$\bot$E] {\pfctx \proves \FALSE} {\pfctx \proves \prop} \and \infer[$\top$I] {} {\pfctx \proves \TRUE} \and  Ralf Jung committed Jan 31, 2016 335 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 336 337 338  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 339 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 340 341 342  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 343 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 344 345 346  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 347 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 348 349 350  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 351 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 352 353 354  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Mar 06, 2016 355 356 357 358 359 360 \infer[$\vee$E] {\pfctx \proves \prop \vee \propB \\ \pfctx, \prop \proves \propC \\ \pfctx, \propB \proves \propC} {\pfctx \proves \propC} \and  Ralf Jung committed Jan 31, 2016 361 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 362 363 364  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 365 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 366 367 368  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Mar 06, 2016 369 370 371 \infer[$\forall$I] { \vctx,\var : \type\mid\pfctx \proves \prop} {\vctx\mid\pfctx \proves \forall \var: \type.\; \prop}  Ralf Jung committed Jan 31, 2016 372 \and  Ralf Jung committed Mar 06, 2016 373 374 375 376 \infer[$\forall$E] {\vctx\mid\pfctx \proves \forall \var :\type.\; \prop \\ \vctx \proves \wtt\term\type} {\vctx\mid\pfctx \proves \prop[\term/\var]}  Ralf Jung committed Jan 31, 2016 377 \and  Ralf Jung committed Mar 06, 2016 378 379 380 381 \infer[$\exists$I] {\vctx\mid\pfctx \proves \prop[\term/\var] \\ \vctx \proves \wtt\term\type} {\vctx\mid\pfctx \proves \exists \var: \type. \prop}  Ralf Jung committed Jan 31, 2016 382 \and  Ralf Jung committed Mar 06, 2016 383 384 385 386 \infer[$\exists$E] {\vctx\mid\pfctx \proves \exists \var: \type.\; \prop \\ \vctx,\var : \type\mid\pfctx , \prop \proves \propB} {\vctx\mid\pfctx \proves \propB}  Ralf Jung committed Jan 31, 2016 387 \and  Ralf Jung committed Mar 06, 2016 388 389 390 \infer[$\lambda$] {} {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}  Ralf Jung committed Jan 31, 2016 391 \and  Ralf Jung committed Mar 06, 2016 392 393 394 395 \infer[$\mu$] {} {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]} \end{mathparpagebreakable}  Ralf Jung committed Jan 31, 2016 396   Ralf Jung committed Mar 06, 2016 397 \paragraph{Laws of (affine) bunched implications.}  Ralf Jung committed Jan 31, 2016 398 399 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 06, 2016 400  \TRUE * \prop &\Lra& \prop \\  Ralf Jung committed Jan 31, 2016 401  \prop * \propB &\Lra& \propB * \prop \\  Ralf Jung committed Mar 06, 2016 402  (\prop * \propB) * \propC &\Lra& \prop * (\propB * \propC)  Ralf Jung committed Jan 31, 2016 403 404 \end{array} \and  Ralf Jung committed Mar 06, 2016 405 \infer[$*$-mono]  Ralf Jung committed Mar 06, 2016 406 407 408  {\prop_1 \proves \propB_1 \and \prop_2 \proves \propB_2} {\prop_1 * \prop_2 \proves \propB_1 * \propB_2}  Ralf Jung committed Jan 31, 2016 409 \and  Ralf Jung committed Mar 06, 2016 410 \inferB[$\wand$I-E]  Ralf Jung committed Mar 06, 2016 411 412  {\prop * \propB \proves \propC} {\prop \proves \propB \wand \propC}  Ralf Jung committed Jan 31, 2016 413 414 \end{mathpar}  Ralf Jung committed Mar 06, 2016 415 \paragraph{Laws for ghosts and physical resources.}  Ralf Jung committed Jan 31, 2016 416 417 418 \begin{mathpar} \begin{array}{rMcMl} \ownGGhost{\melt} * \ownGGhost{\meltB} &\Lra& \ownGGhost{\melt \mtimes \meltB} \\  Ralf Jung committed Mar 06, 2016 419 420 %\TRUE &\Ra& \ownGGhost{\munit}\\ \ownGGhost{\melt} &\Ra& \melt \in \mval % * \ownGGhost{\melt}  Ralf Jung committed Jan 31, 2016 421 422 423 \end{array} \and \begin{array}{c}  Ralf Jung committed Mar 06, 2016 424 \ownPhys{\state} * \ownPhys{\state'} \Ra \FALSE  Ralf Jung committed Jan 31, 2016 425 426 427 \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 428 \paragraph{Laws for the later modality.}  Ralf Jung committed Jan 31, 2016 429 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 430 \infer[$\later$-mono]  Ralf Jung committed Jan 31, 2016 431 432 433  {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and  Ralf Jung committed Mar 06, 2016 434 435 436 \infer[L{\"o}b] {} {(\later\prop\Ra\prop) \proves \prop}  Ralf Jung committed Jan 31, 2016 437 \and  Ralf Jung committed Mar 06, 2016 438 439 440 441 442 \infer[$\later$-$\exists$] {\text{$\type$ is inhabited}} {\later{\Exists x:\type.\prop} \proves \Exists x:\type. \later\prop} \\\\ \begin{array}[c]{rMcMl}  Ralf Jung committed Jan 31, 2016 443 444 445 446  \later{(\prop \wedge \propB)} &\Lra& \later{\prop} \wedge \later{\propB} \\ \later{(\prop \vee \propB)} &\Lra& \later{\prop} \vee \later{\propB} \\ \end{array} \and  Ralf Jung committed Mar 06, 2016 447 \begin{array}[c]{rMcMl}  Ralf Jung committed Jan 31, 2016 448  \later{\All x.\prop} &\Lra& \All x. \later\prop \\  Ralf Jung committed Mar 06, 2016 449  \Exists x. \later\prop &\Ra& \later{\Exists x.\prop} \\  Ralf Jung committed Jan 31, 2016 450 451 452 453  \later{(\prop * \propB)} &\Lra& \later\prop * \later\propB \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 454 \paragraph{Laws for the always modality.}  Ralf Jung committed Jan 31, 2016 455 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 456 \infer[$\always$I]  Ralf Jung committed Jan 31, 2016 457 458 459  {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and  Ralf Jung committed Mar 06, 2016 460 461 462 463 464 465 466 \infer[$\always$E]{} {\always{\prop} \Ra \prop} \and \begin{array}[c]{rMcMl} \always{(\prop * \propB)} &\Ra& \always{(\prop \land \propB)} \\ \always{\prop} * \propB &\Ra& \always{\prop} \land \propB \\ \always{\later\prop} &\Lra& \later\always{\prop} \\  Ralf Jung committed Jan 31, 2016 467 468 \end{array} \and  Ralf Jung committed Mar 06, 2016 469 \begin{array}[c]{rMcMl}  Ralf Jung committed Jan 31, 2016 470 471 472 473 474  \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}  Ralf Jung committed Mar 07, 2016 475 476 477 478 479 480 \and { \term =_\type \term' \Ra \always \term =_\type \term'} \and { \knowInv\iname\prop \Ra \always \knowInv\iname\prop} \and { \ownGGhost{\munit(\melt)} \Ra \always \ownGGhost{\munit(\melt)}}  Ralf Jung committed Jan 31, 2016 481 482 \end{mathpar}  Ralf Jung committed Mar 06, 2016 483 \paragraph{Laws of primitive view shifts.}  Ralf Jung committed Mar 07, 2016 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 \begin{mathpar} \infer[pvs-intro] {}{\prop \proves \pvs[\mask] \prop} \infer[pvs-mono] {\prop \proves \propB} {\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1][\mask_2] \propB} \infer[pvs-timeless] {\timeless\prop} {\later\prop \proves \pvs[\mask] \prop} \infer[pvs-trans] {\mask_2 \subseteq \mask_1 \cup \mask_3} {\pvs[\mask_1][\mask_2] \pvs[\mask_2][\mask_3] \prop \proves \pvs[\mask_1][\mask_3] \prop} \infer[pvs-mask-frame] {}{\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1 \uplus \mask_f][\mask_2 \uplus \mask_f] \prop} \infer[pvs-frame] {}{\propB * \pvs[\mask_1][\mask_2]\prop \proves \pvs[\mask_1][\mask_2] \propB * \prop} \infer[pvs-allocI] {\text{$\mask$ is infinite}} {\later\prop \proves \pvs[\mask] \Exists \iname \in \mask. \knowInv\iname\prop} \infer[pvs-openI] {}{\knowInv\iname\prop \proves \pvs[\set\iname][\emptyset] \later\prop} \infer[pvs-closeI] {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE} \infer[pvs-update] {\melt \mupd \meltsB} {\ownGGhost\melt \proves \pvs[\mask] \Exists\meltB\in\meltsB. \ownGGhost\meltB} \end{mathpar}  Ralf Jung committed Jan 31, 2016 520   Ralf Jung committed Mar 06, 2016 521 \paragraph{Laws of weakest preconditions.}  Ralf Jung committed Mar 07, 2016 522 523 524 525 526 \begin{mathpar} \infer[wp-value] {}{\prop[\val/\var] \proves \wpre{\val}{\Ret\var.\prop}[\mask]} \infer[wp-mono]  Ralf Jung committed Mar 08, 2016 527 {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB}  Ralf Jung committed Mar 07, 2016 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 {\wpre\expr{\Ret\var.\prop}[\mask_1] \proves \wpre\expr{\Ret\var.\propB}[\mask_2]} \infer[pvs-wp] {}{\pvs[\mask] \wpre\expr{\Ret\var.\prop}[\mask] \proves \wpre\expr{\Ret\var.\prop}[\mask]} \infer[wp-pvs] {}{\wpre\expr{\Ret\var.\pvs[\mask] \prop}[\mask] \proves \wpre\expr{\Ret\var.\prop}[\mask]} \infer[wp-atomic] {\mask_2 \subseteq \mask_1 \and \physatomic{\expr}} {\pvs[\mask_1][\mask_2] \wpre\expr{\Ret\var. \pvs[\mask_2][\mask_1]\prop}[\mask_2] \proves \wpre\expr{\Ret\var.\prop}[\mask_1]} \infer[wp-frame] {}{\propB * \wpre\expr{\Ret\var.\prop}[\mask] \proves \wpre\expr{\Ret\var.\propB*\prop}[\mask]} \infer[wp-frame-step] {\toval(\expr) = \bot} {\later\propB * \wpre\expr{\Ret\var.\prop}[\mask] \proves \wpre\expr{\Ret\var.\propB*\prop}[\mask]} \infer[wp-bind] {\text{$\lctx$ is a context}} {\wpre\expr{\Ret\var. \wpre{\lctx(\ofval(\var))}{\Ret\varB.\prop}[\mask]}[\mask] \proves \wpre{\lctx(\expr)}{\Ret\varB.\prop}[\mask]} \end{mathpar}  Ralf Jung committed Jan 31, 2016 552   Ralf Jung committed Mar 07, 2016 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 \subsection{Lifting of operational semantics}\label{sec:lifting} ~\\\ralf{Add this.} % 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. \subsection{Adequacy} The adequacy statement reads as follows: \begin{align*}  Ralf Jung committed Mar 07, 2016 593  &\All \mask, \expr, \val, \pred, \state, \melt, \state', \tpool'.  Ralf Jung committed Mar 07, 2016 594 595 596 597  \\&(\All n. \melt \in \mval_n) \Ra \\&( \ownPhys\state * \ownGGhost\melt \proves \wpre{\expr}{x.\; \pred(x)}[\mask]) \Ra \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{[\val] \dplus \tpool'} \Ra  Ralf Jung committed Mar 07, 2016 598 599  \\&\pred(\val) \end{align*}  Ralf Jung committed Mar 07, 2016 600 where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants.  Ralf Jung committed Mar 07, 2016 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  % RJ: If we want this section back, we should port it to primitive view shifts and prove it in Coq. % \subsection{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.  Ralf Jung committed Jan 31, 2016 627 628 629 630 %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: