logic.tex 24.4 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  A function $\lctx : \textdom{Expr} \to \textdom{Expr}$ is a \emph{context} if the following conditions are satisfied:  Ralf Jung committed Mar 08, 2016 29  \begin{enumerate}[itemsep=0pt]  Ralf Jung committed Mar 07, 2016 30 31 32 33 34 35 36  \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 \section{The logic} To instantiate Iris, you need to define the following parameters: \begin{itemize} \item A language $\Lang$  Ralf Jung committed Mar 08, 2016 67 \item A locally contractive functor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state, such that for all COFEs $A$, the CMRA $\iFunc(A)$ has a unit  Ralf Jung committed Mar 06, 2016 68 \end{itemize}  Ralf Jung committed Jan 31, 2016 69   Ralf Jung committed Mar 06, 2016 70 71 72 \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 73 $ Ralf Jung committed Mar 08, 2016 74  \SigType \supseteq \{ \textlog{Val}, \textlog{Expr}, \textlog{State}, \textlog{M}, \textlog{InvName}, \textlog{InvMask}, \Prop \}  Ralf Jung committed Jan 31, 2016 75 $  Ralf Jung committed Mar 06, 2016 76 77 78 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 79 80 81 82 83 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 84 85 86 87 88 89  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 90 91  \paragraph{Syntax.}  Ralf Jung committed Jan 31, 2016 92 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 93   Ralf Jung committed Jan 31, 2016 94 \begin{align*}  Ralf Jung committed Mar 08, 2016 95  \type \bnfdef{}&  Ralf Jung committed Mar 06, 2016 96  \sigtype \mid  Ralf Jung committed Mar 08, 2016 97  1 \mid  Ralf Jung committed Mar 06, 2016 98 99 100  \type \times \type \mid \type \to \type \0.4em]  Ralf Jung committed Mar 08, 2016 101  \term, \prop, \pred \bnfdef{}&  Ralf Jung committed Mar 06, 2016 102  \var \mid  Ralf Jung committed Jan 31, 2016 103  \sigfn(\term_1, \dots, \term_n) \mid  Ralf Jung committed Mar 08, 2016 104  () \mid  Ralf Jung committed Jan 31, 2016 105 106  (\term, \term) \mid \pi_i\; \term \mid  Ralf Jung committed Mar 06, 2016 107  \Lam \var:\type.\term \mid  Ralf Jung committed Mar 06, 2016 108  \term(\term) \mid  Ralf Jung committed Mar 08, 2016 109  \munit \mid  Ralf Jung committed Mar 08, 2016 110  \mcore\term \mid  Ralf Jung committed Jan 31, 2016 111 112 113 114  \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 08, 2016 140 141 142 143 144 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 as follows: \[ \vctx \proves \timeless{\prop} \eqdef \vctx\mid\later\prop \proves \prop \lor \later\FALSE  Ralf Jung committed Mar 06, 2016 145   Ralf Jung committed Jan 31, 2016 146 \paragraph{Metavariable conventions.}  Ralf Jung committed Mar 06, 2016 147 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 148 149 $\begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 150  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Jan 31, 2016 151  \term, \termB & \text{arbitrary} \\  Ralf Jung committed Mar 08, 2016 152 153 154  \val, \valB & \textlog{Val} \\ \expr & \textlog{Expr} \\ \state & \textlog{State} \\  Ralf Jung committed Jan 31, 2016 155 156 157 \end{array} \qquad\qquad \begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 158  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Mar 08, 2016 159 160 161  \iname & \textlog{InvName} \\ \mask & \textlog{InvMask} \\ \melt, \meltB & \textlog{M} \\  Ralf Jung committed Jan 31, 2016 162  \prop, \propB, \propC & \Prop \\  Ralf Jung committed Mar 06, 2016 163  \pred, \predB, \predC & \type\to\Prop \text{ (when \type is clear from context)} \\  Ralf Jung committed Jan 31, 2016 164 165 166 167 \end{array}$ \paragraph{Variable conventions.}  Ralf Jung committed Feb 02, 2016 168 We often abuse notation, using the preceding \emph{term} meta-variables to range over (bound) \emph{variables}.  Ralf Jung committed Jan 31, 2016 169 We omit type annotations in binders, when the type is clear from context.  Ralf Jung committed Mar 06, 2016 170 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 171 172 173 174 175  \subsection{Types}\label{sec:types} Iris terms are simply-typed.  Ralf Jung committed Mar 06, 2016 176 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 177   Ralf Jung committed Mar 06, 2016 178 179 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 180   Ralf Jung committed Mar 06, 2016 181 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 182 183 \begin{mathparpagebreakable} %%% variables and function symbols  Ralf Jung committed Mar 06, 2016 184  \axiom{x : \type \proves \wtt{x}{\type}}  Ralf Jung committed Jan 31, 2016 185 \and  Ralf Jung committed Mar 06, 2016 186 187  \infer{\vctx \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 188 \and  Ralf Jung committed Mar 06, 2016 189 190  \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 191 \and  Ralf Jung committed Mar 06, 2016 192 193  \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 194 195 196 197 198 199 200 201 202 203 204 \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 205  \axiom{\vctx \proves \wtt{()}{1}}  Ralf Jung committed Jan 31, 2016 206 \and  Ralf Jung committed Mar 06, 2016 207 208  \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 209 \and  Ralf Jung committed Mar 06, 2016 210 211  \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 212 213 %%% functions \and  Ralf Jung committed Mar 06, 2016 214 215  \infer{\vctx, x:\type \proves \wtt{\term}{\type'}} {\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}  Ralf Jung committed Jan 31, 2016 216 217 \and \infer  Ralf Jung committed Mar 06, 2016 218 219  {\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}} {\vctx \proves \wtt{\term(\termB)}{\type'}}  Ralf Jung committed Jan 31, 2016 220 %%% monoids  Ralf Jung committed Mar 08, 2016 221 222 \and \infer{}{\vctx \proves \wtt\munit{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 223 \and  Ralf Jung committed Mar 08, 2016 224  \infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 225 \and  Ralf Jung committed Mar 08, 2016 226 227  \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 228 229 230 231 232 233 %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and  Ralf Jung committed Mar 06, 2016 234 235  \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 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 \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 253 254  \vctx, \var:\type \proves \wtt{\term}{\type} \and \text{$\var$ is guarded in $\term$}  Ralf Jung committed Jan 31, 2016 255  }{  Ralf Jung committed Mar 06, 2016 256  \vctx \proves \wtt{\MU \var:\type. \term}{\type}  Ralf Jung committed Jan 31, 2016 257 258  } \and  Ralf Jung committed Mar 06, 2016 259 260  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 261 \and  Ralf Jung committed Mar 06, 2016 262 263  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 264 265 266 \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and  Ralf Jung committed Mar 08, 2016 267  \vctx \proves \wtt{\iname}{\textlog{InvName}}  Ralf Jung committed Jan 31, 2016 268 269 270 271  }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and  Ralf Jung committed Mar 08, 2016 272  \infer{\vctx \proves \wtt{\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 273 274  {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and  Ralf Jung committed Mar 08, 2016 275  \infer{\vctx \proves \wtt{\state}{\textlog{State}}}  Ralf Jung committed Jan 31, 2016 276 277 278 279 280 281 282 283 284 285  {\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 286 287  \vctx \proves \wtt{\mask}{\textlog{InvMask}} \and \vctx \proves \wtt{\mask'}{\textlog{InvMask}}  Ralf Jung committed Jan 31, 2016 288  }{  Ralf Jung committed Mar 07, 2016 289  \vctx \proves \wtt{\pvs[\mask][\mask'] \prop}{\Prop}  Ralf Jung committed Jan 31, 2016 290 291 292  } \and \infer{  Ralf Jung committed Mar 08, 2016 293 294 295  \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 296  }{  Ralf Jung committed Mar 07, 2016 297  \vctx \proves \wtt{\wpre{\expr}{\Ret\var.\term}[\mask]}{\Prop}  Ralf Jung committed Jan 31, 2016 298 299 300  } \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 301 \subsection{Proof rules}  Ralf Jung committed Mar 06, 2016 302   Ralf Jung committed Jan 31, 2016 303 304 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 305 Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent.  Ralf Jung committed Mar 08, 2016 306 Axioms $\vctx \mid \prop \provesIff \propB$ indicate that both $\vctx \mid \prop \proves \propB$ and $\vctx \mid \propB \proves \prop$ can be derived.  Ralf Jung committed Jan 31, 2016 307   Ralf Jung committed Mar 06, 2016 308 \judgment{}{\vctx \mid \pfctx \proves \prop}  Ralf Jung committed Mar 08, 2016 309 \paragraph{Laws of intuitionistic higher-order logic with equality.}  Ralf Jung committed Jan 31, 2016 310 This is entirely standard.  Ralf Jung committed Mar 06, 2016 311 312 \begin{mathparpagebreakable} \infer[Asm]  Ralf Jung committed Jan 31, 2016 313 314 315  {\prop \in \pfctx} {\pfctx \proves \prop} \and  Ralf Jung committed Mar 06, 2016 316 \infer[Eq]  Ralf Jung committed Mar 07, 2016 317 318  {\pfctx \proves \prop \\ \pfctx \proves \term =_\type \term'} {\pfctx \proves \prop[\term'/\term]}  Ralf Jung committed Jan 31, 2016 319 \and  Ralf Jung committed Mar 06, 2016 320 321 322 323 324 325 326 327 328 329 330 331 \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 332 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 333 334 335  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 336 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 337 338 339  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 340 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 341 342 343  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 344 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 345 346 347  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 348 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 349 350 351  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Mar 06, 2016 352 353 354 355 356 357 \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 358 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 359 360 361  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 362 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 363 364 365  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Mar 06, 2016 366 367 368 \infer[$\forall$I] { \vctx,\var : \type\mid\pfctx \proves \prop} {\vctx\mid\pfctx \proves \forall \var: \type.\; \prop}  Ralf Jung committed Jan 31, 2016 369 \and  Ralf Jung committed Mar 06, 2016 370 371 372 373 \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 374 \and  Ralf Jung committed Mar 06, 2016 375 376 377 378 \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 379 \and  Ralf Jung committed Mar 06, 2016 380 381 382 383 \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 384 \and  Ralf Jung committed Mar 06, 2016 385 386 387 \infer[$\lambda$] {} {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}  Ralf Jung committed Jan 31, 2016 388 \and  Ralf Jung committed Mar 06, 2016 389 390 391 392 \infer[$\mu$] {} {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]} \end{mathparpagebreakable}  Ralf Jung committed Jan 31, 2016 393   Ralf Jung committed Mar 06, 2016 394 \paragraph{Laws of (affine) bunched implications.}  Ralf Jung committed Jan 31, 2016 395 396 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 08, 2016 397 398 399  \TRUE * \prop &\provesIff& \prop \\ \prop * \propB &\provesIff& \propB * \prop \\ (\prop * \propB) * \propC &\provesIff& \prop * (\propB * \propC)  Ralf Jung committed Jan 31, 2016 400 401 \end{array} \and  Ralf Jung committed Mar 06, 2016 402 \infer[$*$-mono]  Ralf Jung committed Mar 06, 2016 403 404 405  {\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 406 \and  Ralf Jung committed Mar 06, 2016 407 \inferB[$\wand$I-E]  Ralf Jung committed Mar 06, 2016 408 409  {\prop * \propB \proves \propC} {\prop \proves \propB \wand \propC}  Ralf Jung committed Jan 31, 2016 410 411 \end{mathpar}  Ralf Jung committed Mar 06, 2016 412 \paragraph{Laws for ghosts and physical resources.}  Ralf Jung committed Jan 31, 2016 413 414 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 08, 2016 415 416 417 \ownGGhost{\melt} * \ownGGhost{\meltB} &\provesIff& \ownGGhost{\melt \mtimes \meltB} \\ \ownGGhost{\melt} &\provesIff& \melt \in \mval \\ \TRUE &\proves& \ownGGhost{\munit}  Ralf Jung committed Jan 31, 2016 418 419 \end{array} \and  Ralf Jung committed Mar 08, 2016 420 \and  Ralf Jung committed Jan 31, 2016 421 \begin{array}{c}  Ralf Jung committed Mar 08, 2016 422 \ownPhys{\state} * \ownPhys{\state'} \proves \FALSE  Ralf Jung committed Jan 31, 2016 423 424 425 \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 426 \paragraph{Laws for the later modality.}  Ralf Jung committed Jan 31, 2016 427 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 428 \infer[$\later$-mono]  Ralf Jung committed Jan 31, 2016 429 430 431  {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and  Ralf Jung committed Mar 06, 2016 432 433 434 \infer[L{\"o}b] {} {(\later\prop\Ra\prop) \proves \prop}  Ralf Jung committed Jan 31, 2016 435 \and  Ralf Jung committed Mar 06, 2016 436 437 438 439 440 \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 Mar 08, 2016 441 442  \later{(\prop \wedge \propB)} &\provesIff& \later{\prop} \wedge \later{\propB} \\ \later{(\prop \vee \propB)} &\provesIff& \later{\prop} \vee \later{\propB} \\  Ralf Jung committed Jan 31, 2016 443 444 \end{array} \and  Ralf Jung committed Mar 06, 2016 445 \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 446 447 448  \later{\All x.\prop} &\provesIff& \All x. \later\prop \\ \Exists x. \later\prop &\proves& \later{\Exists x.\prop} \\ \later{(\prop * \propB)} &\provesIff& \later\prop * \later\propB  Ralf Jung committed Jan 31, 2016 449 450 451 \end{array} \end{mathpar}  Ralf Jung committed Mar 08, 2016 452 453 454 455 456 457 458 459 460 461 \begin{mathpar} \infer {\text{$\term$ or $\term'$ is a discrete COFE element}} {\timeless{\term =_\type \term'}} \infer {\text{$\melt$ is a discrete COFE element}} {\timeless{\ownGGhost\melt}} \infer{}  Ralf Jung committed Mar 08, 2016 462 {\timeless{\ownPhys\state}}  Ralf Jung committed Mar 08, 2016 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481  \infer {\vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop \Ra \propB}} \infer {\vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop \wand \propB}} \infer {\vctx,\var:\type \proves \timeless{\prop}} {\vctx \proves \timeless{\All\var:\type.\prop}} \infer {\vctx,\var:\type \proves \timeless{\prop}} {\vctx \proves \timeless{\Exists\var:\type.\prop}} \end{mathpar}  Ralf Jung committed Mar 06, 2016 482 \paragraph{Laws for the always modality.}  Ralf Jung committed Jan 31, 2016 483 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 484 \infer[$\always$I]  Ralf Jung committed Jan 31, 2016 485 486 487  {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and  Ralf Jung committed Mar 06, 2016 488 \infer[$\always$E]{}  Ralf Jung committed Mar 08, 2016 489  {\always{\prop} \proves \prop}  Ralf Jung committed Mar 06, 2016 490 491 \and \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 492 493 494  \always{(\prop * \propB)} &\proves& \always{(\prop \land \propB)} \\ \always{\prop} * \propB &\proves& \always{\prop} \land \propB \\ \always{\later\prop} &\provesIff& \later\always{\prop} \\  Ralf Jung committed Jan 31, 2016 495 496 \end{array} \and  Ralf Jung committed Mar 06, 2016 497 \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 498 499 500 501  \always{(\prop \land \propB)} &\provesIff& \always{\prop} \land \always{\propB} \\ \always{(\prop \lor \propB)} &\provesIff& \always{\prop} \lor \always{\propB} \\ \always{\All x. \prop} &\provesIff& \All x. \always{\prop} \\ \always{\Exists x. \prop} &\provesIff& \Exists x. \always{\prop} \\  Ralf Jung committed Jan 31, 2016 502 \end{array}  Ralf Jung committed Mar 07, 2016 503 \and  Ralf Jung committed Mar 08, 2016 504 { \term =_\type \term' \proves \always \term =_\type \term'}  Ralf Jung committed Mar 07, 2016 505 \and  Ralf Jung committed Mar 08, 2016 506 { \knowInv\iname\prop \proves \always \knowInv\iname\prop}  Ralf Jung committed Mar 07, 2016 507 \and  Ralf Jung committed Mar 08, 2016 508 { \ownGGhost{\mcore\melt} \proves \always \ownGGhost{\mcore\melt}}  Ralf Jung committed Jan 31, 2016 509 510 \end{mathpar}  Ralf Jung committed Mar 06, 2016 511 \paragraph{Laws of primitive view shifts.}  Ralf Jung committed Mar 07, 2016 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 \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 548   Ralf Jung committed Mar 06, 2016 549 \paragraph{Laws of weakest preconditions.}  Ralf Jung committed Mar 07, 2016 550 551 552 553 554 \begin{mathpar} \infer[wp-value] {}{\prop[\val/\var] \proves \wpre{\val}{\Ret\var.\prop}[\mask]} \infer[wp-mono]  Ralf Jung committed Mar 08, 2016 555 {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB}  Ralf Jung committed Mar 07, 2016 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 {\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 580   Ralf Jung committed Mar 07, 2016 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 \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 621  &\All \mask, \expr, \val, \pred, \state, \melt, \state', \tpool'.  Ralf Jung committed Mar 07, 2016 622 623 624 625  \\&(\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 626 627  \\&\pred(\val) \end{align*}  Ralf Jung committed Mar 07, 2016 628 where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants.  Ralf Jung committed Mar 07, 2016 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  % 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 655 656 657 658 %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: