logic.tex 23.8 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 06, 2016 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  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 08, 2016 27 28 29 30 31 \begin{defn} An expression $\expr$ and state $\state$ are \emph{reducible} (written $\red(\expr, \state)$) if $\Exists \expr_2, \state_2, \expr'. \expr,\state \step \expr_2,\state_2,\expr'$ \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 38 39 40 41  \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 42 43 \end{defn}  Ralf Jung committed Mar 06, 2016 44 45 46 \subsection{The concurrent language} 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 06, 2016 57 58 59 60 61 62 63  {\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 64 65 \end{mathpar}  Ralf Jung committed Mar 07, 2016 66 \clearpage  Ralf Jung committed Mar 06, 2016 67 68 69 70 71 \section{The logic} 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 130 131 132 133 134 135 \\& \knowInv{\term}{\prop} \mid \ownGGhost{\term} \mid \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 Feb 02, 2016 173 We often abuse notation, using the preceding \emph{term} meta-variables to range over (bound) \emph{variables}.  Ralf Jung committed Jan 31, 2016 174 We omit type annotations in binders, when the type is clear from context.  Ralf Jung committed Mar 06, 2016 175 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 176 177 178 179 180  \subsection{Types}\label{sec:types} Iris terms are simply-typed.  Ralf Jung committed Mar 06, 2016 181 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 182   Ralf Jung committed Mar 06, 2016 183 184 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 185   Ralf Jung committed Mar 06, 2016 186 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 187 188 \begin{mathparpagebreakable} %%% variables and function symbols  Ralf Jung committed Mar 06, 2016 189  \axiom{x : \type \proves \wtt{x}{\type}}  Ralf Jung committed Jan 31, 2016 190 \and  Ralf Jung committed Mar 06, 2016 191 192  \infer{\vctx \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 193 \and  Ralf Jung committed Mar 06, 2016 194 195  \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 196 \and  Ralf Jung committed Mar 06, 2016 197 198  \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 199 200 201 202 203 204 205 206 207 208 209 \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 210  \axiom{\vctx \proves \wtt{()}{1}}  Ralf Jung committed Jan 31, 2016 211 \and  Ralf Jung committed Mar 06, 2016 212 213  \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 214 \and  Ralf Jung committed Mar 06, 2016 215 216  \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 217 218 %%% functions \and  Ralf Jung committed Mar 06, 2016 219 220  \infer{\vctx, x:\type \proves \wtt{\term}{\type'}} {\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}  Ralf Jung committed Jan 31, 2016 221 222 \and \infer  Ralf Jung committed Mar 06, 2016 223 224  {\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}} {\vctx \proves \wtt{\term(\termB)}{\type'}}  Ralf Jung committed Jan 31, 2016 225 %%% monoids  Ralf Jung committed Mar 08, 2016 226 227 \and \infer{}{\vctx \proves \wtt\munit{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 228 \and  Ralf Jung committed Mar 08, 2016 229  \infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 230 \and  Ralf Jung committed Mar 08, 2016 231 232  \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 233 234 235 236 237 238 %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and  Ralf Jung committed Mar 06, 2016 239 240  \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 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 \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 258 259  \vctx, \var:\type \proves \wtt{\term}{\type} \and \text{$\var$ is guarded in $\term$}  Ralf Jung committed Jan 31, 2016 260  }{  Ralf Jung committed Mar 06, 2016 261  \vctx \proves \wtt{\MU \var:\type. \term}{\type}  Ralf Jung committed Jan 31, 2016 262 263  } \and  Ralf Jung committed Mar 06, 2016 264 265  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 266 \and  Ralf Jung committed Mar 06, 2016 267 268  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 269 270 271 \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and  Ralf Jung committed Mar 08, 2016 272  \vctx \proves \wtt{\iname}{\textlog{InvName}}  Ralf Jung committed Jan 31, 2016 273 274 275 276  }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and  Ralf Jung committed Mar 08, 2016 277  \infer{\vctx \proves \wtt{\melt}{\textlog{M}}}  Ralf Jung committed Jan 31, 2016 278 279  {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and  Ralf Jung committed Mar 08, 2016 280  \infer{\vctx \proves \wtt{\state}{\textlog{State}}}  Ralf Jung committed Jan 31, 2016 281 282 283 284 285 286 287 288 289 290  {\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 291 292  \vctx \proves \wtt{\mask}{\textlog{InvMask}} \and \vctx \proves \wtt{\mask'}{\textlog{InvMask}}  Ralf Jung committed Jan 31, 2016 293  }{  Ralf Jung committed Mar 07, 2016 294  \vctx \proves \wtt{\pvs[\mask][\mask'] \prop}{\Prop}  Ralf Jung committed Jan 31, 2016 295 296 297  } \and \infer{  Ralf Jung committed Mar 08, 2016 298 299 300  \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 301  }{  Ralf Jung committed Mar 08, 2016 302  \vctx \proves \wtt{\wpre{\expr}[\mask]{\Ret\var.\term}}{\Prop}  Ralf Jung committed Jan 31, 2016 303 304 305  } \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 306 \subsection{Proof rules}  Ralf Jung committed Mar 06, 2016 307   Ralf Jung committed Jan 31, 2016 308 309 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 310 Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent.  Ralf Jung committed Mar 08, 2016 311 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 312   Ralf Jung committed Mar 06, 2016 313 \judgment{}{\vctx \mid \pfctx \proves \prop}  Ralf Jung committed Mar 08, 2016 314 \paragraph{Laws of intuitionistic higher-order logic with equality.}  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 08, 2016 402 403 404  \TRUE * \prop &\provesIff& \prop \\ \prop * \propB &\provesIff& \propB * \prop \\ (\prop * \propB) * \propC &\provesIff& \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 \begin{mathpar} \begin{array}{rMcMl}  Ralf Jung committed Mar 08, 2016 420 421 422 \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 423 424 \end{array} \and  Ralf Jung committed Mar 08, 2016 425 \and  Ralf Jung committed Jan 31, 2016 426 \begin{array}{c}  Ralf Jung committed Mar 08, 2016 427 \ownPhys{\state} * \ownPhys{\state'} \proves \FALSE  Ralf Jung committed Jan 31, 2016 428 429 430 \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 431 \paragraph{Laws for the later modality.}  Ralf Jung committed Jan 31, 2016 432 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 433 \infer[$\later$-mono]  Ralf Jung committed Jan 31, 2016 434 435 436  {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and  Ralf Jung committed Mar 06, 2016 437 438 439 \infer[L{\"o}b] {} {(\later\prop\Ra\prop) \proves \prop}  Ralf Jung committed Jan 31, 2016 440 \and  Ralf Jung committed Mar 06, 2016 441 442 443 444 445 \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 446 447  \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 448 449 \end{array} \and  Ralf Jung committed Mar 06, 2016 450 \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 451 452 453  \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 454 455 456 \end{array} \end{mathpar}  Ralf Jung committed Mar 08, 2016 457 458 459 460 461 462 463 464 465 466 \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}} \infer{}  Ralf Jung committed Mar 08, 2016 467 {\timeless{\ownPhys\state}}  Ralf Jung committed Mar 08, 2016 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486  \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 487 \paragraph{Laws for the always modality.}  Ralf Jung committed Jan 31, 2016 488 \begin{mathpar}  Ralf Jung committed Mar 06, 2016 489 \infer[$\always$I]  Ralf Jung committed Jan 31, 2016 490 491 492  {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and  Ralf Jung committed Mar 06, 2016 493 \infer[$\always$E]{}  Ralf Jung committed Mar 08, 2016 494  {\always{\prop} \proves \prop}  Ralf Jung committed Mar 06, 2016 495 496 \and \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 497 498 499  \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 500 501 \end{array} \and  Ralf Jung committed Mar 06, 2016 502 \begin{array}[c]{rMcMl}  Ralf Jung committed Mar 08, 2016 503 504 505 506  \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 507 \end{array}  Ralf Jung committed Mar 07, 2016 508 \and  Ralf Jung committed Mar 08, 2016 509 { \term =_\type \term' \proves \always \term =_\type \term'}  Ralf Jung committed Mar 07, 2016 510 \and  Ralf Jung committed Mar 08, 2016 511 { \knowInv\iname\prop \proves \always \knowInv\iname\prop}  Ralf Jung committed Mar 07, 2016 512 \and  Ralf Jung committed Mar 08, 2016 513 { \ownGGhost{\mcore\melt} \proves \always \ownGGhost{\mcore\melt}}  Ralf Jung committed Jan 31, 2016 514 515 \end{mathpar}  Ralf Jung committed Mar 06, 2016 516 \paragraph{Laws of primitive view shifts.}  Ralf Jung committed Mar 07, 2016 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 \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 553   Ralf Jung committed Mar 06, 2016 554 \paragraph{Laws of weakest preconditions.}  Ralf Jung committed Mar 07, 2016 555 556 \begin{mathpar} \infer[wp-value]  Ralf Jung committed Mar 08, 2016 557 {}{\prop[\val/\var] \proves \wpre{\val}[\mask]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 558 559  \infer[wp-mono]  Ralf Jung committed Mar 08, 2016 560 {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB}  Ralf Jung committed Mar 08, 2016 561 {\wpre\expr[\mask_1]{\Ret\var.\prop} \proves \wpre\expr[\mask_2]{\Ret\var.\propB}}  Ralf Jung committed Mar 07, 2016 562 563  \infer[pvs-wp]  Ralf Jung committed Mar 08, 2016 564 {}{\pvs[\mask] \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 565 566  \infer[wp-pvs]  Ralf Jung committed Mar 08, 2016 567 {}{\wpre\expr[\mask]{\Ret\var.\pvs[\mask] \prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}}  Ralf Jung committed Mar 07, 2016 568 569 570  \infer[wp-atomic] {\mask_2 \subseteq \mask_1 \and \physatomic{\expr}}  Ralf Jung committed Mar 08, 2016 571 572 {\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 573 574  \infer[wp-frame]  Ralf Jung committed Mar 08, 2016 575 {}{\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}}  Ralf Jung committed Mar 07, 2016 576 577 578  \infer[wp-frame-step] {\toval(\expr) = \bot}  Ralf Jung committed Mar 08, 2016 579 {\later\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}}  Ralf Jung committed Mar 07, 2016 580 581 582  \infer[wp-bind] {\text{$\lctx$ is a context}}  Ralf Jung committed Mar 08, 2016 583 {\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 584 \end{mathpar}  Ralf Jung committed Jan 31, 2016 585   Ralf Jung committed Mar 07, 2016 586 \subsection{Lifting of operational semantics}\label{sec:lifting}  Ralf Jung committed Mar 08, 2016 587 588 589 590 591 592 593  \begin{mathparpagebreakable} \infer[wp-lift-step] {\mask_2 \subseteq \mask_1 \and \toval(\expr_1) = \bot \and \red(\expr_1, \state_1) \and \All \expr_2, \state_2, \expr'. \expr_1,\state_1 \step \expr_2,\state_2,\expr' \Ra \pred(\expr_2,\state_2,\expr')}  Ralf Jung committed Mar 08, 2016 594  {\pvs[\mask_1][\mask_2] \later\ownPhys{\state_1} * \later\All \expr_2, \state_2, \expr'. \pred(\expr_2, \state_2, \expr') \land \ownPhys{\state_2} \wand \pvs[\mask_2][\mask_1] \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr'}[\top]{\Ret\var.\TRUE} {}\\\proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}  Ralf Jung committed Mar 08, 2016 595 596 597 598 599  \infer[wp-lift-pure-step] {\toval(\expr_1) = \bot \and \All \state_1. \red(\expr_1, \state_1) \and \All \state_1, \expr_2, \state_2, \expr'. \expr_1,\state_1 \step \expr_2,\state_2,\expr' \Ra \state_1 = \state_2 \land \pred(\expr_2,\expr')}  Ralf Jung committed Mar 08, 2016 600  {\later\All \expr_2, \expr'. \pred(\expr_2, \expr') \wand \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr'}[\top]{\Ret\var.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}}  Ralf Jung committed Mar 08, 2016 601 602 \end{mathparpagebreakable}  Ralf Jung committed Mar 08, 2016 603 Here we define $\wpre{\expr'}[\mask]{\Ret\var.\prop} \eqdef \TRUE$ if $\expr' = \bot$ (remember that our stepping relation can, but does not have to, define a forked-off expression).  Ralf Jung committed Mar 07, 2016 604 605 606  \subsection{Adequacy}  Ralf Jung committed Mar 08, 2016 607 The adequacy statement concerning functional correctness reads as follows:  Ralf Jung committed Mar 07, 2016 608 \begin{align*}  Ralf Jung committed Mar 07, 2016 609  &\All \mask, \expr, \val, \pred, \state, \melt, \state', \tpool'.  Ralf Jung committed Mar 07, 2016 610  \\&(\All n. \melt \in \mval_n) \Ra  Ralf Jung committed Mar 08, 2016 611  \\&( \ownPhys\state * \ownGGhost\melt \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra  Ralf Jung committed Mar 07, 2016 612 613  \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{[\val] \dplus \tpool'} \Ra  Ralf Jung committed Mar 07, 2016 614 615  \\&\pred(\val) \end{align*}  Ralf Jung committed Mar 07, 2016 616 where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants.  Ralf Jung committed Mar 07, 2016 617   Ralf Jung committed Mar 08, 2016 618 619 620 621 622 623 624 625 626 627 628 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 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  % 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 654 655 656 657 %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: