logic.tex 23.7 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 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 %%% monoids  Ralf Jung committed Mar 08, 2016 216 217 \and \infer{}{\vctx \proves \wtt\munit{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 218 \and  Ralf Jung committed Mar 08, 2016 219  \infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 220 \and  Ralf Jung committed Mar 08, 2016 221 222  \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 223 224 225 226 227 228 %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and  Ralf Jung committed Mar 06, 2016 229 230  \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 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 \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 248 249  \vctx, \var:\type \proves \wtt{\term}{\type} \and \text{$\var$ is guarded in $\term$}  Ralf Jung committed Jan 31, 2016 250  }{  Ralf Jung committed Mar 06, 2016 251  \vctx \proves \wtt{\MU \var:\type. \term}{\type}  Ralf Jung committed Jan 31, 2016 252 253  } \and  Ralf Jung committed Mar 06, 2016 254 255  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 256 \and  Ralf Jung committed Mar 06, 2016 257 258  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 259 260 261 \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and  Ralf Jung committed Mar 08, 2016 262  \vctx \proves \wtt{\iname}{\textlog{InvName}}  Ralf Jung committed Jan 31, 2016 263 264 265 266  }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and  Ralf Jung committed Mar 08, 2016 267  \infer{\vctx \proves \wtt{\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 268 269  {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and  Ralf Jung committed Mar 08, 2016 270  \infer{\vctx \proves \wtt{\state}{\textlog{State}}}  Ralf Jung committed Jan 31, 2016 271 272 273 274 275 276 277 278 279 280  {\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 281 282  \vctx \proves \wtt{\mask}{\textlog{InvMask}} \and \vctx \proves \wtt{\mask'}{\textlog{InvMask}}  Ralf Jung committed Jan 31, 2016 283  }{  Ralf Jung committed Mar 07, 2016 284  \vctx \proves \wtt{\pvs[\mask][\mask'] \prop}{\Prop}  Ralf Jung committed Jan 31, 2016 285 286 287  } \and \infer{  Ralf Jung committed Mar 08, 2016 288 289 290  \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 291  }{  Ralf Jung committed Mar 07, 2016 292  \vctx \proves \wtt{\wpre{\expr}{\Ret\var.\term}[\mask]}{\Prop}  Ralf Jung committed Jan 31, 2016 293 294 295  } \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 296 \subsection{Timeless propositions}  Ralf Jung committed Mar 06, 2016 297 298 299  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 300   Ralf Jung committed Mar 06, 2016 301 \judgment{Timeless Propositions}{\timeless{P}}  Ralf Jung committed Jan 31, 2016 302   Ralf Jung committed Mar 06, 2016 303 304 \ralf{Define a judgment that defines them.}  Ralf Jung committed Mar 06, 2016 305 \subsection{Proof rules}  Ralf Jung committed Mar 06, 2016 306   Ralf Jung committed Jan 31, 2016 307 308 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 309 Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent.  Ralf Jung committed Jan 31, 2016 310 311 312 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 313 \judgment{}{\vctx \mid \pfctx \proves \prop}  Ralf Jung committed Mar 06, 2016 314 \paragraph{Laws of intuitionistic higher-order logic.}  Ralf Jung committed Jan 31, 2016 315 This is entirely standard.  Ralf Jung committed Mar 06, 2016 316 317 \begin{mathparpagebreakable} \infer[Asm]  Ralf Jung committed Jan 31, 2016 318 319 320  {\prop \in \pfctx} {\pfctx \proves \prop} \and  Ralf Jung committed Mar 06, 2016 321 \infer[Eq]  Ralf Jung committed Mar 07, 2016 322 323  {\pfctx \proves \prop \\ \pfctx \proves \term =_\type \term'} {\pfctx \proves \prop[\term'/\term]}  Ralf Jung committed Jan 31, 2016 324 \and  Ralf Jung committed Mar 06, 2016 325 326 327 328 329 330 331 332 333 334 335 336 \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 337 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 338 339 340  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 341 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 342 343 344  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 345 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 346 347 348  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 349 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 350 351 352  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 353 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 354 355 356  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Mar 06, 2016 357 358 359 360 361 362 \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 363 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 364 365 366  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 367 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 368 369 370  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Mar 06, 2016 371 372 373 \infer[$\forall$I] { \vctx,\var : \type\mid\pfctx \proves \prop} {\vctx\mid\pfctx \proves \forall \var: \type.\; \prop}  Ralf Jung committed Jan 31, 2016 374 \and  Ralf Jung committed Mar 06, 2016 375 376 377 378 \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 379 \and  Ralf Jung committed Mar 06, 2016 380 381 382 383 \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 384 \and  Ralf Jung committed Mar 06, 2016 385 386 387 388 \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 389 \and  Ralf Jung committed Mar 06, 2016 390 391 392 \infer[$\lambda$] {} {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}  Ralf Jung committed Jan 31, 2016 393 \and  Ralf Jung committed Mar 06, 2016 394 395 396 397 \infer[$\mu$] {} {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]} \end{mathparpagebreakable}  Ralf Jung committed Jan 31, 2016 398   Ralf Jung committed Mar 06, 2016 399 \paragraph{Laws of (affine) bunched implications.}  Ralf Jung committed Jan 31, 2016 400 401 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 06, 2016 402  \TRUE * \prop &\Lra& \prop \\  Ralf Jung committed Jan 31, 2016 403  \prop * \propB &\Lra& \propB * \prop \\  Ralf Jung committed Mar 06, 2016 404  (\prop * \propB) * \propC &\Lra& \prop * (\propB * \propC)  Ralf Jung committed Jan 31, 2016 405 406 \end{array} \and  Ralf Jung committed Mar 06, 2016 407 \infer[$*$-mono]  Ralf Jung committed Mar 06, 2016 408 409 410  {\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 411 \and  Ralf Jung committed Mar 06, 2016 412 \inferB[$\wand$I-E]  Ralf Jung committed Mar 06, 2016 413 414  {\prop * \propB \proves \propC} {\prop \proves \propB \wand \propC}  Ralf Jung committed Jan 31, 2016 415 416 \end{mathpar}  Ralf Jung committed Mar 06, 2016 417 \paragraph{Laws for ghosts and physical resources.}  Ralf Jung committed Jan 31, 2016 418 419 420 \begin{mathpar} \begin{array}{rMcMl} \ownGGhost{\melt} * \ownGGhost{\meltB} &\Lra& \ownGGhost{\melt \mtimes \meltB} \\  Ralf Jung committed Mar 08, 2016 421 \TRUE &\Ra& \ownGGhost{\munit}\\  Ralf Jung committed Mar 06, 2016 422 \ownGGhost{\melt} &\Ra& \melt \in \mval % * \ownGGhost{\melt}  Ralf Jung committed Jan 31, 2016 423 424 425 \end{array} \and \begin{array}{c}  Ralf Jung committed Mar 06, 2016 426 \ownPhys{\state} * \ownPhys{\state'} \Ra \FALSE  Ralf Jung committed Jan 31, 2016 427 428 429 \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 430 \paragraph{Laws for the later modality.}  Ralf Jung committed Jan 31, 2016 431 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 432 \infer[$\later$-mono]  Ralf Jung committed Jan 31, 2016 433 434 435  {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and  Ralf Jung committed Mar 06, 2016 436 437 438 \infer[L{\"o}b] {} {(\later\prop\Ra\prop) \proves \prop}  Ralf Jung committed Jan 31, 2016 439 \and  Ralf Jung committed Mar 06, 2016 440 441 442 443 444 \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 445 446 447 448  \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 449 \begin{array}[c]{rMcMl}  Ralf Jung committed Jan 31, 2016 450  \later{\All x.\prop} &\Lra& \All x. \later\prop \\  Ralf Jung committed Mar 06, 2016 451  \Exists x. \later\prop &\Ra& \later{\Exists x.\prop} \\  Ralf Jung committed Jan 31, 2016 452 453 454 455  \later{(\prop * \propB)} &\Lra& \later\prop * \later\propB \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 456 \paragraph{Laws for the always modality.}  Ralf Jung committed Jan 31, 2016 457 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 458 \infer[$\always$I]  Ralf Jung committed Jan 31, 2016 459 460 461  {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and  Ralf Jung committed Mar 06, 2016 462 463 464 465 466 467 468 \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 469 470 \end{array} \and  Ralf Jung committed Mar 06, 2016 471 \begin{array}[c]{rMcMl}  Ralf Jung committed Jan 31, 2016 472 473 474 475 476  \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 477 478 479 480 481 \and { \term =_\type \term' \Ra \always \term =_\type \term'} \and { \knowInv\iname\prop \Ra \always \knowInv\iname\prop} \and  Ralf Jung committed Mar 08, 2016 482 { \ownGGhost{\mcore\melt} \Ra \always \ownGGhost{\mcore\melt}}  Ralf Jung committed Jan 31, 2016 483 484 \end{mathpar}  Ralf Jung committed Mar 06, 2016 485 \paragraph{Laws of primitive view shifts.}  Ralf Jung committed Mar 07, 2016 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 \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 522   Ralf Jung committed Mar 06, 2016 523 \paragraph{Laws of weakest preconditions.}  Ralf Jung committed Mar 07, 2016 524 525 526 527 528 \begin{mathpar} \infer[wp-value] {}{\prop[\val/\var] \proves \wpre{\val}{\Ret\var.\prop}[\mask]} \infer[wp-mono]  Ralf Jung committed Mar 08, 2016 529 {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB}  Ralf Jung committed Mar 07, 2016 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 {\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 554   Ralf Jung committed Mar 07, 2016 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 \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 595  &\All \mask, \expr, \val, \pred, \state, \melt, \state', \tpool'.  Ralf Jung committed Mar 07, 2016 596 597 598 599  \\&(\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 600 601  \\&\pred(\val) \end{align*}  Ralf Jung committed Mar 07, 2016 602 where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants.  Ralf Jung committed Mar 07, 2016 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  % 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 629 630 631 632 %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: