logic.tex 24 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 \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}  Ralf Jung committed Mar 08, 2016 8 9 \item There exists a \emph{primitive reduction relation} $(-,- \step -,-,-) \subseteq \textdom{Expr} \times \textdom{State} \times \textdom{Expr} \times \textdom{State} \times (\textdom{Expr} \uplus \set{\bot})$ We will write $\expr_1, \state_1 \step \expr_2, \state_2$ for $\expr_1, \state_1 \step \expr_2, \state_2, \bot$. \\  Ralf Jung committed Mar 12, 2016 10  A reduction $\expr_1, \state_1 \step \expr_2, \state_2, \expr_f$ indicates that, when $\expr_1$ reduces to $\expr$, a \emph{new thread} $\expr_f$ is forked off.  Ralf Jung committed Mar 06, 2016 11 12 13 14 15 16 17 18 \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}  Ralf Jung committed Mar 12, 2016 19 \All\expr_1, \state_1, \expr_2, \state_2, \expr_f. \atomic(\expr_1) \land \expr_1, \state_1 \step \expr_2, \state_2, \expr_f \Ra {}\\\qquad\qquad\qquad\quad~~ \Exists \val_2. \toval(\expr_2) = \val_2  Ralf Jung committed Mar 06, 2016 20 21 22 23 24  \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 08, 2016 27 28 \begin{defn} An expression $\expr$ and state $\state$ are \emph{reducible} (written $\red(\expr, \state)$) if  Ralf Jung committed Mar 12, 2016 29  $\Exists \expr_2, \state_2, \expr_f. \expr,\state \step \expr_2,\state_2,\expr_f$  Ralf Jung committed Mar 08, 2016 30 31 \end{defn}  Ralf Jung committed Mar 07, 2016 32 \begin{defn}[Context]  Ralf Jung committed Mar 07, 2016 33  A function $\lctx : \textdom{Expr} \to \textdom{Expr}$ is a \emph{context} if the following conditions are satisfied:  Ralf Jung committed Mar 08, 2016 34  \begin{enumerate}[itemsep=0pt]  Ralf Jung committed Mar 07, 2016 35 36 37  \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 $\lctx$:\\  Ralf Jung committed Mar 12, 2016 38  $\All \expr_1, \state_1, \expr_2, \state_2, \expr_f. \expr_1, \state_1 \step \expr_2,\state_2,\expr_f \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr_f$  Ralf Jung committed Mar 07, 2016 39  \item Reductions stay below $\lctx$ until there is a value in the hole:\\  Ralf Jung committed Mar 12, 2016 40  $\All \expr_1', \state_1, \expr_2, \state_2, \expr_f. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr_f \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr_f$  Ralf Jung committed Mar 07, 2016 41  \end{enumerate}  Ralf Jung committed Mar 07, 2016 42 43 \end{defn}  Ralf Jung committed Mar 11, 2016 44 \subsection{Concurrent language}  Ralf Jung committed Mar 06, 2016 45 46  For any language $\Lang$, we define the corresponding thread-pool semantics.  Ralf Jung committed Jan 31, 2016 47 48 49  \paragraph{Machine syntax} $ Ralf Jung committed Mar 06, 2016 50  \tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n  Ralf Jung committed Jan 31, 2016 51 52 $  Ralf Jung committed Mar 06, 2016 53 54 \judgment{Machine reduction} {\cfg{\tpool}{\state} \step \cfg{\tpool'}{\state'}}  Ralf Jung committed Jan 31, 2016 55 56 \begin{mathpar} \infer  Ralf Jung committed Mar 12, 2016 57  {\expr_1, \state_1 \step \expr_2, \state_2, \expr_f \and \expr_f \neq \bot}  Ralf Jung committed Mar 06, 2016 58  {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step  Ralf Jung committed Mar 12, 2016 59  \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_f]}{\state'}}  Ralf Jung committed Mar 06, 2016 60 61 62 63 \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 64 65 \end{mathpar}  Ralf Jung committed Mar 07, 2016 66 \clearpage  Ralf Jung committed Mar 11, 2016 67 \section{Logic}  Ralf Jung committed Mar 06, 2016 68 69 70 71  To instantiate Iris, you need to define the following parameters: \begin{itemize} \item A language $\Lang$  Ralf Jung committed Mar 09, 2016 72 \item A locally contractive bifunctor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state, such that for all COFEs $A$, the CMRA $\iFunc(A)$ has a unit  Ralf Jung committed Mar 06, 2016 73 \end{itemize}  Ralf Jung committed Jan 31, 2016 74   Ralf Jung committed Mar 06, 2016 75 76 77 \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 78 $ Ralf Jung committed Mar 08, 2016 79  \SigType \supseteq \{ \textlog{Val}, \textlog{Expr}, \textlog{State}, \textlog{M}, \textlog{InvName}, \textlog{InvMask}, \Prop \}  Ralf Jung committed Jan 31, 2016 80 $  Ralf Jung committed Mar 06, 2016 81 82 83 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 84 85 86 87 88 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 89 90 91 92 93 94  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 95 96  \paragraph{Syntax.}  Ralf Jung committed Jan 31, 2016 97 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 98   Ralf Jung committed Jan 31, 2016 99 \begin{align*}  Ralf Jung committed Mar 08, 2016 100  \type \bnfdef{}&  Ralf Jung committed Mar 06, 2016 101  \sigtype \mid  Ralf Jung committed Mar 08, 2016 102  1 \mid  Ralf Jung committed Mar 06, 2016 103 104 105  \type \times \type \mid \type \to \type \0.4em]  Ralf Jung committed Mar 08, 2016 106  \term, \prop, \pred \bnfdef{}&  Ralf Jung committed Mar 06, 2016 107  \var \mid  Ralf Jung committed Jan 31, 2016 108  \sigfn(\term_1, \dots, \term_n) \mid  Ralf Jung committed Mar 08, 2016 109  () \mid  Ralf Jung committed Jan 31, 2016 110 111  (\term, \term) \mid \pi_i\; \term \mid  Ralf Jung committed Mar 06, 2016 112  \Lam \var:\type.\term \mid  Ralf Jung committed Mar 06, 2016 113  \term(\term) \mid  Ralf Jung committed Mar 08, 2016 114  \munit \mid  Ralf Jung committed Mar 08, 2016 115  \mcore\term \mid  Ralf Jung committed Jan 31, 2016 116 117 118 119  \term \mtimes \term \mid \\& \FALSE \mid \TRUE \mid  Ralf Jung committed Mar 06, 2016 120  \term =_\type \term \mid  Ralf Jung committed Jan 31, 2016 121 122 123 124 125 126  \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 127  \MU \var:\type. \pred \mid  Ralf Jung committed Mar 06, 2016 128 129  \Exists \var:\type. \prop \mid \All \var:\type. \prop \mid  Ralf Jung committed Jan 31, 2016 130 131 \\& \knowInv{\term}{\prop} \mid  Ralf Jung committed Mar 11, 2016 132  \ownGGhost{\term} \mid \mval(\term) \mid  Ralf Jung committed Jan 31, 2016 133 134 135  \ownPhys{\term} \mid \always\prop \mid {\later\prop} \mid  Ralf Jung committed Mar 07, 2016 136  \pvs[\term][\term] \prop\mid  Ralf Jung committed Mar 08, 2016 137  \wpre{\term}[\term]{\Ret\var.\term}  Ralf Jung committed Jan 31, 2016 138 \end{align*}  Ralf Jung committed Jan 31, 2016 139 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 140   Ralf Jung committed Mar 06, 2016 141 Note that \always and \later bind more tightly than *, \wand, \land, \lor, and \Ra.  Ralf Jung committed Mar 07, 2016 142 We will write \pvs[\term] \prop for \pvs[\term][\term] \prop.  Ralf Jung committed Mar 07, 2016 143 144 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 145 146 147 148 149 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 150   Ralf Jung committed Jan 31, 2016 151 \paragraph{Metavariable conventions.}  Ralf Jung committed Mar 06, 2016 152 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 153 154 $\begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 155  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Jan 31, 2016 156  \term, \termB & \text{arbitrary} \\  Ralf Jung committed Mar 08, 2016 157 158 159  \val, \valB & \textlog{Val} \\ \expr & \textlog{Expr} \\ \state & \textlog{State} \\  Ralf Jung committed Jan 31, 2016 160 161 162 \end{array} \qquad\qquad \begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 163  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Mar 08, 2016 164 165 166  \iname & \textlog{InvName} \\ \mask & \textlog{InvMask} \\ \melt, \meltB & \textlog{M} \\  Ralf Jung committed Jan 31, 2016 167  \prop, \propB, \propC & \Prop \\  Ralf Jung committed Mar 06, 2016 168  \pred, \predB, \predC & \type\to\Prop \text{ (when \type is clear from context)} \\  Ralf Jung committed Jan 31, 2016 169 170 171 172 \end{array}$ \paragraph{Variable conventions.}  Ralf Jung committed Mar 06, 2016 173 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 174 175 176 177 178  \subsection{Types}\label{sec:types} Iris terms are simply-typed.  Ralf Jung committed Mar 06, 2016 179 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 180   Ralf Jung committed Mar 06, 2016 181 182 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 183   Ralf Jung committed Mar 06, 2016 184 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 185 186 \begin{mathparpagebreakable} %%% variables and function symbols  Ralf Jung committed Mar 06, 2016 187  \axiom{x : \type \proves \wtt{x}{\type}}  Ralf Jung committed Jan 31, 2016 188 \and  Ralf Jung committed Mar 06, 2016 189 190  \infer{\vctx \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 191 \and  Ralf Jung committed Mar 06, 2016 192 193  \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 194 \and  Ralf Jung committed Mar 06, 2016 195 196  \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 197 198 199 200 201 202 203 204 205 206 207 \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 208  \axiom{\vctx \proves \wtt{()}{1}}  Ralf Jung committed Jan 31, 2016 209 \and  Ralf Jung committed Mar 06, 2016 210 211  \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 212 \and  Ralf Jung committed Mar 06, 2016 213 214  \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 215 216 %%% functions \and  Ralf Jung committed Mar 06, 2016 217 218  \infer{\vctx, x:\type \proves \wtt{\term}{\type'}} {\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}  Ralf Jung committed Jan 31, 2016 219 220 \and \infer  Ralf Jung committed Mar 06, 2016 221 222  {\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}} {\vctx \proves \wtt{\term(\termB)}{\type'}}  Ralf Jung committed Jan 31, 2016 223 %%% monoids  Ralf Jung committed Mar 08, 2016 224 225 \and \infer{}{\vctx \proves \wtt\munit{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 226 \and  Ralf Jung committed Mar 08, 2016 227  \infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 228 \and  Ralf Jung committed Mar 08, 2016 229 230  \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 231 232 233 234 235 236 %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and  Ralf Jung committed Mar 06, 2016 237 238  \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 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 \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 256 257  \vctx, \var:\type \proves \wtt{\term}{\type} \and \text{$\var$ is guarded in $\term$}  Ralf Jung committed Jan 31, 2016 258  }{  Ralf Jung committed Mar 06, 2016 259  \vctx \proves \wtt{\MU \var:\type. \term}{\type}  Ralf Jung committed Jan 31, 2016 260 261  } \and  Ralf Jung committed Mar 06, 2016 262 263  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 264 \and  Ralf Jung committed Mar 06, 2016 265 266  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 267 268 269 \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and  Ralf Jung committed Mar 08, 2016 270  \vctx \proves \wtt{\iname}{\textlog{InvName}}  Ralf Jung committed Jan 31, 2016 271 272 273 274  }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and  Ralf Jung committed Mar 08, 2016 275  \infer{\vctx \proves \wtt{\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 276  {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}}  Ralf Jung committed Mar 11, 2016 277 278 279 \and \infer{\vctx \proves \wtt{\melt}{\textlog{M}}} {\vctx \proves \wtt{\mval(\melt)}{\Prop}}  Ralf Jung committed Jan 31, 2016 280 \and  Ralf Jung committed Mar 08, 2016 281  \infer{\vctx \proves \wtt{\state}{\textlog{State}}}  Ralf Jung committed Jan 31, 2016 282 283 284 285 286 287 288 289 290 291  {\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 292 293  \vctx \proves \wtt{\mask}{\textlog{InvMask}} \and \vctx \proves \wtt{\mask'}{\textlog{InvMask}}  Ralf Jung committed Jan 31, 2016 294  }{  Ralf Jung committed Mar 07, 2016 295  \vctx \proves \wtt{\pvs[\mask][\mask'] \prop}{\Prop}  Ralf Jung committed Jan 31, 2016 296 297 298  } \and \infer{  Ralf Jung committed Mar 08, 2016 299 300 301  \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 302  }{  Ralf Jung committed Mar 08, 2016 303  \vctx \proves \wtt{\wpre{\expr}[\mask]{\Ret\var.\term}}{\Prop}  Ralf Jung committed Jan 31, 2016 304 305 306  } \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 307 \subsection{Proof rules}  Ralf Jung committed Mar 06, 2016 308   Ralf Jung committed Jan 31, 2016 309 310 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 311 Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent.  Ralf Jung committed Mar 08, 2016 312 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 313   Ralf Jung committed Mar 06, 2016 314 \judgment{}{\vctx \mid \pfctx \proves \prop}  Ralf Jung committed Mar 08, 2016 315 \paragraph{Laws of intuitionistic higher-order logic with equality.}  Ralf Jung committed Jan 31, 2016 316 This is entirely standard.  Ralf Jung committed Mar 06, 2016 317 318 \begin{mathparpagebreakable} \infer[Asm]  Ralf Jung committed Jan 31, 2016 319 320 321  {\prop \in \pfctx} {\pfctx \proves \prop} \and  Ralf Jung committed Mar 06, 2016 322 \infer[Eq]  Ralf Jung committed Mar 07, 2016 323 324  {\pfctx \proves \prop \\ \pfctx \proves \term =_\type \term'} {\pfctx \proves \prop[\term'/\term]}  Ralf Jung committed Jan 31, 2016 325 \and  Ralf Jung committed Mar 06, 2016 326 327 328 329 330 331 332 333 334 335 336 337 \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 338 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 339 340 341  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 342 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 343 344 345  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 346 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 347 348 349  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 350 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 351 352 353  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 354 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 355 356 357  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Mar 06, 2016 358 359 360 361 362 363 \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 364 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 365 366 367  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 368 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 369 370 371  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Mar 06, 2016 372 373 374 \infer[$\forall$I] { \vctx,\var : \type\mid\pfctx \proves \prop} {\vctx\mid\pfctx \proves \forall \var: \type.\; \prop}  Ralf Jung committed Jan 31, 2016 375 \and  Ralf Jung committed Mar 06, 2016 376 377 378 379 \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 380 \and  Ralf Jung committed Mar 06, 2016 381 382 383 384 \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 385 \and  Ralf Jung committed Mar 06, 2016 386 387 388 389 \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 390 \and  Ralf Jung committed Mar 06, 2016 391 392 393 \infer[$\lambda$] {} {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]}  Ralf Jung committed Jan 31, 2016 394 \and  Ralf Jung committed Mar 06, 2016 395 396 397 398 \infer[$\mu$] {} {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]} \end{mathparpagebreakable}  Ralf Jung committed Jan 31, 2016 399   Ralf Jung committed Mar 06, 2016 400 \paragraph{Laws of (affine) bunched implications.}  Ralf Jung committed Jan 31, 2016 401 402 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 08, 2016 403 404 405  \TRUE * \prop &\provesIff& \prop \\ \prop * \propB &\provesIff& \propB * \prop \\ (\prop * \propB) * \propC &\provesIff& \prop * (\propB * \propC)  Ralf Jung committed Jan 31, 2016 406 407 \end{array} \and  Ralf Jung committed Mar 06, 2016 408 \infer[$*$-mono]  Ralf Jung committed Mar 06, 2016 409 410 411  {\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 412 \and  Ralf Jung committed Mar 06, 2016 413 \inferB[$\wand$I-E]  Ralf Jung committed Mar 06, 2016 414 415  {\prop * \propB \proves \propC} {\prop \proves \propB \wand \propC}  Ralf Jung committed Jan 31, 2016 416 417 \end{mathpar}  Ralf Jung committed Mar 06, 2016 418 \paragraph{Laws for ghosts and physical resources.}  Ralf Jung committed Jan 31, 2016 419 420 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 08, 2016 421 \ownGGhost{\melt} * \ownGGhost{\meltB} &\provesIff& \ownGGhost{\melt \mtimes \meltB} \\  Ralf Jung committed Mar 11, 2016 422 \ownGGhost{\melt} &\provesIff& \mval(\melt) \\  Ralf Jung committed Mar 08, 2016 423 \TRUE &\proves& \ownGGhost{\munit}  Ralf Jung committed Jan 31, 2016 424 425 \end{array} \and  Ralf Jung committed Mar 08, 2016 426 \and  Ralf Jung committed Jan 31, 2016 427 \begin{array}{c}  Ralf Jung committed Mar 08, 2016 428 \ownPhys{\state} * \ownPhys{\state'} \proves \FALSE  Ralf Jung committed Jan 31, 2016 429 430 431 \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 432 \paragraph{Laws for the later modality.}  Ralf Jung committed Jan 31, 2016 433 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 434 \infer[$\later$-mono]  Ralf Jung committed Jan 31, 2016 435 436 437  {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and  Ralf Jung committed Mar 06, 2016 438 439 440 \infer[L{\"o}b] {} {(\later\prop\Ra\prop) \proves \prop}  Ralf Jung committed Jan 31, 2016 441 \and  Ralf Jung committed Mar 06, 2016 442 443 444 445 446 \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 447 448  \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 449 450 \end{array} \and  Ralf Jung committed Mar 06, 2016 451 \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 452 453 454  \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 455 456 457 \end{array} \end{mathpar}  Ralf Jung committed Mar 08, 2016 458 459 460 461 462 463 464 465 466 \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}}  Ralf Jung committed Mar 11, 2016 467 468 469 470 \infer {\text{$\melt$ is a discrete COFE element}} {\timeless{\mval(\melt)}}  Ralf Jung committed Mar 08, 2016 471 \infer{}  Ralf Jung committed Mar 08, 2016 472 {\timeless{\ownPhys\state}}  Ralf Jung committed Mar 08, 2016 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491  \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 492 \paragraph{Laws for the always modality.}  Ralf Jung committed Jan 31, 2016 493 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 494 \infer[$\always$I]  Ralf Jung committed Jan 31, 2016 495 496 497  {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and  Ralf Jung committed Mar 06, 2016 498 \infer[$\always$E]{}  Ralf Jung committed Mar 08, 2016 499  {\always{\prop} \proves \prop}  Ralf Jung committed Mar 06, 2016 500 501 \and \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 502 503 504  \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 505 506 \end{array} \and  Ralf Jung committed Mar 06, 2016 507 \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 508 509 510 511  \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 512 \end{array}  Ralf Jung committed Mar 07, 2016 513 \and  Ralf Jung committed Mar 08, 2016 514 { \term =_\type \term' \proves \always \term =_\type \term'}  Ralf Jung committed Mar 07, 2016 515 \and  Ralf Jung committed Mar 08, 2016 516 { \knowInv\iname\prop \proves \always \knowInv\iname\prop}  Ralf Jung committed Mar 07, 2016 517 \and  Ralf Jung committed Mar 08, 2016 518 { \ownGGhost{\mcore\melt} \proves \always \ownGGhost{\mcore\melt}}  Ralf Jung committed Jan 31, 2016 519 520 \end{mathpar}  Ralf Jung committed Mar 06, 2016 521 \paragraph{Laws of primitive view shifts.}  Ralf Jung committed Mar 07, 2016 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 \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}  Ralf Jung committed Mar 11, 2016 544 \inferH{pvs-allocI}  Ralf Jung committed Mar 07, 2016 545 546 547 {\text{$\mask$ is infinite}} {\later\prop \proves \pvs[\mask] \Exists \iname \in \mask. \knowInv\iname\prop}  Ralf Jung committed Mar 11, 2016 548 \inferH{pvs-openI}  Ralf Jung committed Mar 07, 2016 549 550 {}{\knowInv\iname\prop \proves \pvs[\set\iname][\emptyset] \later\prop}  Ralf Jung committed Mar 11, 2016 551 \inferH{pvs-closeI}  Ralf Jung committed Mar 07, 2016 552 553 {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE}  Ralf Jung committed Mar 11, 2016 554 \inferH{pvs-update}  Ralf Jung committed Mar 07, 2016 555 556 557 {\melt \mupd \meltsB} {\ownGGhost\melt \proves \pvs[\mask] \Exists\meltB\in\meltsB. \ownGGhost\meltB} \end{mathpar}  Ralf Jung committed Jan 31, 2016 558   Ralf Jung committed Mar 06, 2016 559 \paragraph{Laws of weakest preconditions.}  Ralf Jung committed Mar 07, 2016 560 561 \begin{mathpar} \infer[wp-value]  Ralf Jung committed Mar 08, 2016 562 {}{\prop[\val/\var] \proves \wpre{\val}[\mask]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 563 564  \infer[wp-mono]  Ralf Jung committed Mar 08, 2016 565 {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB}  Ralf Jung committed Mar 08, 2016 566 {\wpre\expr[\mask_1]{\Ret\var.\prop} \proves \wpre\expr[\mask_2]{\Ret\var.\propB}}  Ralf Jung committed Mar 07, 2016 567 568  \infer[pvs-wp]  Ralf Jung committed Mar 08, 2016 569 {}{\pvs[\mask] \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 570 571  \infer[wp-pvs]  Ralf Jung committed Mar 08, 2016 572 {}{\wpre\expr[\mask]{\Ret\var.\pvs[\mask] \prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 573 574 575  \infer[wp-atomic] {\mask_2 \subseteq \mask_1 \and \physatomic{\expr}}  Ralf Jung committed Mar 08, 2016 576 577 {\pvs[\mask_1][\mask_2] \wpre\expr[\mask_2]{\Ret\var. \pvs[\mask_2][\mask_1]\prop} \proves \wpre\expr[\mask_1]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 578 579  \infer[wp-frame]  Ralf Jung committed Mar 08, 2016 580 {}{\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}}  Ralf Jung committed Mar 07, 2016 581 582 583  \infer[wp-frame-step] {\toval(\expr) = \bot}  Ralf Jung committed Mar 08, 2016 584 {\later\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}}  Ralf Jung committed Mar 07, 2016 585 586 587  \infer[wp-bind] {\text{$\lctx$ is a context}}  Ralf Jung committed Mar 08, 2016 588 {\wpre\expr[\mask]{\Ret\var. \wpre{\lctx(\ofval(\var))}[\mask]{\Ret\varB.\prop}} \proves \wpre{\lctx(\expr)}[\mask]{\Ret\varB.\prop}}  Ralf Jung committed Mar 07, 2016 589 \end{mathpar}  Ralf Jung committed Jan 31, 2016 590   Ralf Jung committed Mar 12, 2016 591 592 \paragraph{Lifting of operational semantics.}~ \begin{mathpar}  Ralf Jung committed Mar 08, 2016 593 594 595 596  \infer[wp-lift-step] {\mask_2 \subseteq \mask_1 \and \toval(\expr_1) = \bot \and \red(\expr_1, \state_1) \and  Ralf Jung committed Mar 12, 2016 597 598 599 600  \All \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \pred(\expr_2,\state_2,\expr_f)} { {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below... ~~\pvs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\All \expr_2, \state_2, \expr_f. \pred(\expr_2, \state_2, \expr_f) \land {}\\\qquad\qquad\qquad\qquad\qquad \ownPhys{\state_2} \wand \pvs[\mask_2][\mask_1] \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_f}[\top]{\Ret\any.\TRUE} {}\\\proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop} \end{inbox}} }  Ralf Jung committed Mar 08, 2016 601 602 603 604  \infer[wp-lift-pure-step] {\toval(\expr_1) = \bot \and \All \state_1. \red(\expr_1, \state_1) \and  Ralf Jung committed Mar 12, 2016 605 606  \All \state_1, \expr_2, \state_2, \expr_f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_f \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr_f)} {\later\All \expr_2, \expr_f. \pred(\expr_2, \expr_f) \Ra \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}  Ralf Jung committed Mar 12, 2016 607 \end{mathpar}  Ralf Jung committed Mar 08, 2016 608   Ralf Jung committed Mar 12, 2016 609 Here we define $\wpre{\expr_f}[\mask]{\Ret\var.\prop} \eqdef \TRUE$ if $\expr_f = \bot$ (remember that our stepping relation can, but does not have to, define a forked-off expression).  Ralf Jung committed Mar 07, 2016 610 611 612  \subsection{Adequacy}  Ralf Jung committed Mar 08, 2016 613 The adequacy statement concerning functional correctness reads as follows:  Ralf Jung committed Mar 07, 2016 614 \begin{align*}  Ralf Jung committed Mar 07, 2016 615  &\All \mask, \expr, \val, \pred, \state, \melt, \state', \tpool'.  Ralf Jung committed Mar 07, 2016 616  \\&(\All n. \melt \in \mval_n) \Ra  Ralf Jung committed Mar 08, 2016 617  \\&( \ownPhys\state * \ownGGhost\melt \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra  Ralf Jung committed Mar 07, 2016 618 619  \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{[\val] \dplus \tpool'} \Ra  Ralf Jung committed Mar 07, 2016 620 621  \\&\pred(\val) \end{align*}  Ralf Jung committed Mar 07, 2016 622 where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants.  Ralf Jung committed Mar 07, 2016 623   Ralf Jung committed Mar 08, 2016 624 625 626 627 628 629 630 631 632 633 634 Furthermore, the following adequacy statement shows that our weakest preconditions imply that the execution never gets \emph{stuck}: Every expression in the thread pool either is a value, or can reduce further. \begin{align*} &\All \mask, \expr, \state, \melt, \state', \tpool'. \\&(\All n. \melt \in \mval_n) \Ra \\&( \ownPhys\state * \ownGGhost\melt \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{\tpool'} \Ra \\&\All\expr'\in\tpool'. \toval(\expr) \neq \bot \lor \red(\expr, \state') \end{align*} Notice that this is stronger than saying that the thread pool can reduce; we actually assert that \emph{every} non-finished thread can take a step.  Ralf Jung committed Mar 07, 2016 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659  % 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 660 661 662 663 %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: