logic.tex 16.5 KB
 Ralf Jung committed Mar 06, 2016 1 \section{Language}  Ralf Jung committed Jan 31, 2016 2   Ralf Jung committed Mar 06, 2016 3 A \emph{language} $\Lang$ consists of a set \textdom{Exp} 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 06, 2016 27 28 29 \subsection{The concurrent language} For any language $\Lang$, we define the corresponding thread-pool semantics.  Ralf Jung committed Jan 31, 2016 30 31 32  \paragraph{Machine syntax} $ Ralf Jung committed Mar 06, 2016 33  \tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n  Ralf Jung committed Jan 31, 2016 34 35 $  Ralf Jung committed Mar 06, 2016 36 37 \judgment{Machine reduction} {\cfg{\tpool}{\state} \step \cfg{\tpool'}{\state'}}  Ralf Jung committed Jan 31, 2016 38 39 \begin{mathpar} \infer  Ralf Jung committed Mar 06, 2016 40 41 42 43 44 45 46  {\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 47 48 49 \end{mathpar}  Ralf Jung committed Mar 06, 2016 50 51 52 53 54 55 56 \section{The logic} To instantiate Iris, you need to define the following parameters: \begin{itemize} \item A language $\Lang$ \item A locally contractive functor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state \end{itemize}  Ralf Jung committed Jan 31, 2016 57   Ralf Jung committed Mar 06, 2016 58 59 60 \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 61 $ Ralf Jung committed Mar 06, 2016 62  \SigType \supseteq \{ \textsort{Val}, \textsort{Expr}, \textsort{State}, \textsort{M}, \textsort{InvName}, \textsort{InvMask}, \Prop \}  Ralf Jung committed Jan 31, 2016 63 $  Ralf Jung committed Mar 06, 2016 64 65 66 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 67 68 69 70 71 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 72 73 74 75 76 77 78 79  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 $\sigax$. \section{Syntax} \subsection{Grammar}\label{sec:grammar}  Ralf Jung committed Jan 31, 2016 80 81  \paragraph{Syntax.}  Ralf Jung committed Jan 31, 2016 82 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 83   Ralf Jung committed Jan 31, 2016 84 \begin{align*}  Ralf Jung committed Mar 06, 2016 85 86 87 88 89 90  \type ::={}& \sigtype \mid \unitsort \mid \type \times \type \mid \type \to \type \0.4em]  Ralf Jung committed Jan 31, 2016 91 92 93  \term, \prop, \pred ::={}& x \mid \sigfn(\term_1, \dots, \term_n) \mid  Ralf Jung committed Feb 02, 2016 94  \unitval \mid  Ralf Jung committed Jan 31, 2016 95 96 97  (\term, \term) \mid \pi_i\; \term \mid \Lam x.\term \mid  Ralf Jung committed Mar 06, 2016 98  \term(\term) \mid  Ralf Jung committed Jan 31, 2016 99 100 101 102 103 104  \mzero \mid \munit \mid \term \mtimes \term \mid \\& \FALSE \mid \TRUE \mid  Ralf Jung committed Mar 06, 2016 105  \term =_\type \term \mid  Ralf Jung committed Jan 31, 2016 106 107 108 109 110 111  \prop \Ra \prop \mid \prop \land \prop \mid \prop \lor \prop \mid \prop * \prop \mid \prop \wand \prop \mid \\&  Ralf Jung committed Jan 31, 2016 112  \MU \var. \pred \mid  Ralf Jung committed Mar 06, 2016 113 114  \Exists \var:\type. \prop \mid \All \var:\type. \prop \mid  Ralf Jung committed Jan 31, 2016 115 116 117 118 119 120 121 \\& \knowInv{\term}{\prop} \mid \ownGGhost{\term} \mid \ownPhys{\term} \mid \always\prop \mid {\later\prop} \mid \pvsA{\prop}{\term}{\term} \mid  Ralf Jung committed Mar 06, 2016 122  \dynA{\term}{\pred}{\term}  Ralf Jung committed Jan 31, 2016 123 \end{align*}  Ralf Jung committed Jan 31, 2016 124 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 125 126  \paragraph{Metavariable conventions.}  Ralf Jung committed Mar 06, 2016 127 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 128 129 \[ \begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 130  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Jan 31, 2016 131 132 133 134 135 136 137  \term, \termB & \text{arbitrary} \\ \val, \valB & \textsort{Val} \\ \expr & \textsort{Exp} \\ \state & \textsort{State} \\ \end{array} \qquad\qquad \begin{array}{r|l}  Ralf Jung committed Mar 06, 2016 138  \text{metavariable} & \text{type} \\\hline  Ralf Jung committed Jan 31, 2016 139 140  \iname & \textsort{InvName} \\ \mask & \textsort{InvMask} \\  Ralf Jung committed Mar 06, 2016 141  \melt, \meltB & \textsort{M} \\  Ralf Jung committed Jan 31, 2016 142  \prop, \propB, \propC & \Prop \\  Ralf Jung committed Mar 06, 2016 143  \pred, \predB, \predC & \type\to\Prop \text{ (when \type is clear from context)} \\  Ralf Jung committed Jan 31, 2016 144 145 146 147 \end{array} \paragraph{Variable conventions.}  Ralf Jung committed Feb 02, 2016 148 We often abuse notation, using the preceding \emph{term} meta-variables to range over (bound) \emph{variables}.  Ralf Jung committed Jan 31, 2016 149 We omit type annotations in binders, when the type is clear from context.  Ralf Jung committed Mar 06, 2016 150 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 151 152 153 154 155  \subsection{Types}\label{sec:types} Iris terms are simply-typed.  Ralf Jung committed Mar 06, 2016 156 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 157   Ralf Jung committed Mar 06, 2016 158 159 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 160   Ralf Jung committed Mar 06, 2016 161 \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 162 163 \begin{mathparpagebreakable} %%% variables and function symbols  Ralf Jung committed Mar 06, 2016 164  \axiom{x : \type \proves \wtt{x}{\type}}  Ralf Jung committed Jan 31, 2016 165 \and  Ralf Jung committed Mar 06, 2016 166 167  \infer{\vctx \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term}{\type}}  Ralf Jung committed Jan 31, 2016 168 \and  Ralf Jung committed Mar 06, 2016 169 170  \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 171 \and  Ralf Jung committed Mar 06, 2016 172 173  \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 174 175 176 177 178 179 180 181 182 183 184 \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 Feb 02, 2016 185  \axiom{\vctx \proves \wtt{\unitval}{\unitsort}}  Ralf Jung committed Jan 31, 2016 186 \and  Ralf Jung committed Mar 06, 2016 187 188  \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 189 \and  Ralf Jung committed Mar 06, 2016 190 191  \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 192 193 %%% functions \and  Ralf Jung committed Mar 06, 2016 194 195  \infer{\vctx, x:\type \proves \wtt{\term}{\type'}} {\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}  Ralf Jung committed Jan 31, 2016 196 197 \and \infer  Ralf Jung committed Mar 06, 2016 198 199  {\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}} {\vctx \proves \wtt{\term(\termB)}{\type'}}  Ralf Jung committed Jan 31, 2016 200 201 %%% monoids \and  Ralf Jung committed Mar 06, 2016 202  \infer{\vctx \proves \wtt{\term}{\textsort{M}}}{\vctx \proves \wtt{\munit(\term)}{\textsort{M}}}  Ralf Jung committed Jan 31, 2016 203 \and  Ralf Jung committed Mar 06, 2016 204 205  \infer{\vctx \proves \wtt{\melt}{\textsort{M}} \and \vctx \proves \wtt{\meltB}{\textsort{M}}} {\vctx \proves \wtt{\melt \mtimes \meltB}{\textsort{M}}}  Ralf Jung committed Jan 31, 2016 206 207 208 209 210 211 %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and  Ralf Jung committed Mar 06, 2016 212 213  \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 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 \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 231  \vctx, \var:\type\to\Prop \proves \wtt{\pred}{\type\to\Prop} \and  Ralf Jung committed Jan 31, 2016 232  \text{$\var$ is guarded in $\pred$}  Ralf Jung committed Jan 31, 2016 233  }{  Ralf Jung committed Mar 06, 2016 234  \vctx \proves \wtt{\MU \var. \pred}{\type\to\Prop}  Ralf Jung committed Jan 31, 2016 235 236  } \and  Ralf Jung committed Mar 06, 2016 237 238  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 239 \and  Ralf Jung committed Mar 06, 2016 240 241  \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}  Ralf Jung committed Jan 31, 2016 242 243 244 245 246 247 248 249 \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\iname}{\textsort{InvName}} }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and  Ralf Jung committed Mar 06, 2016 250  \infer{\vctx \proves \wtt{\melt}{\textsort{M}}}  Ralf Jung committed Jan 31, 2016 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278  {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and \infer{\vctx \proves \wtt{\state}{\textsort{State}}} {\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 \vctx \proves \wtt{\mask}{\textsort{InvMask}} \and \vctx \proves \wtt{\mask'}{\textsort{InvMask}} }{ \vctx \proves \wtt{\pvsA{\prop}{\mask}{\mask'}}{\Prop} } \and \infer{ \vctx \proves \wtt{\expr}{\textsort{Exp}} \and \vctx \proves \wtt{\pred}{\textsort{Val} \to \Prop} \and \vctx \proves \wtt{\mask}{\textsort{InvMask}} }{ \vctx \proves \wtt{\dynA{\expr}{\pred}{\mask}}{\Prop} } \end{mathparpagebreakable}  Ralf Jung committed Mar 06, 2016 279 280 281 282 \subsection{Timeless Propositions} 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 283   Ralf Jung committed Mar 06, 2016 284 \judgment{Timeless Propositions}{\timeless{P}}  Ralf Jung committed Jan 31, 2016 285   Ralf Jung committed Mar 06, 2016 286 287 288 289 290 \ralf{Define a judgment that defines them.} \subsection{Base logic} \ralf{Go on checking below.}  Ralf Jung committed Jan 31, 2016 291 292 293 294 295 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. 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 296 \paragraph{Laws of intuitionistic higher-order logic.}  Ralf Jung committed Jan 31, 2016 297 This is entirely standard.  Ralf Jung committed Jan 31, 2016 298 299 300 301 302 303 304 305 306 307  \begin{mathpar} \inferH{Asm} {\prop \in \pfctx} {\pfctx \proves \prop} \and \inferH{Eq} {\pfctx \proves \prop(\term) \\ \pfctx \proves \term = \term'} {\pfctx \proves \prop(\term')} \and  Ralf Jung committed Jan 31, 2016 308 \infer[$\wedge$I]  Ralf Jung committed Jan 31, 2016 309 310 311  {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and  Ralf Jung committed Jan 31, 2016 312 \infer[$\wedge$EL]  Ralf Jung committed Jan 31, 2016 313 314 315  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and  Ralf Jung committed Jan 31, 2016 316 \infer[$\wedge$ER]  Ralf Jung committed Jan 31, 2016 317 318 319  {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 320 \infer[$\vee$E]  Ralf Jung committed Jan 31, 2016 321 322 323 324 325  {\pfctx \proves \prop \vee \propB \\ \pfctx, \prop \proves \propC \\ \pfctx, \propB \proves \propC} {\pfctx \proves \propC} \and  Ralf Jung committed Jan 31, 2016 326 \infer[$\vee$IL]  Ralf Jung committed Jan 31, 2016 327 328 329  {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 330 \infer[$\vee$IR]  Ralf Jung committed Jan 31, 2016 331 332 333  {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and  Ralf Jung committed Jan 31, 2016 334 \infer[$\Ra$I]  Ralf Jung committed Jan 31, 2016 335 336 337  {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and  Ralf Jung committed Jan 31, 2016 338 \infer[$\Ra$E]  Ralf Jung committed Jan 31, 2016 339 340 341  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 342 \infer[$\forall_1$I]  Ralf Jung committed Jan 31, 2016 343 344 345  {\pfctx, x : \sort \proves \prop} {\pfctx \proves \forall x: \sort.\; \prop} \and  Ralf Jung committed Jan 31, 2016 346 \infer[$\forall_1$E]  Ralf Jung committed Jan 31, 2016 347 348 349 350  {\pfctx \proves \forall X \in \sort.\; \prop \\ \pfctx \proves \term: \sort} {\pfctx \proves \prop[\term/X]} \and  Ralf Jung committed Jan 31, 2016 351 \infer[$\exists_1$E]  Ralf Jung committed Jan 31, 2016 352 353 354 355  {\pfctx \proves \exists X\in \sort.\; \prop \\ \pfctx, X : \sort, \prop \proves \propB} {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 356 \infer[$\exists_1$I]  Ralf Jung committed Jan 31, 2016 357 358 359 360  {\pfctx \proves \prop[\term/X] \\ \pfctx \proves \term: \sort} {\pfctx \proves \exists X: \sort. \prop} \and  Ralf Jung committed Jan 31, 2016 361 \infer[$\forall_2$I]  Ralf Jung committed Jan 31, 2016 362 363  {\pfctx, \var: \Pred(\sort) \proves \prop} {\pfctx \proves \forall \var\in \Pred(\sort).\; \prop}  Ralf Jung committed Jan 31, 2016 364 \and  Ralf Jung committed Jan 31, 2016 365 \infer[$\forall_2$E]  Ralf Jung committed Jan 31, 2016 366  {\pfctx \proves \forall \var. \prop \\  Ralf Jung committed Jan 31, 2016 367  \pfctx \proves \propB: \Prop}  Ralf Jung committed Jan 31, 2016 368  {\pfctx \proves \prop[\propB/\var]}  Ralf Jung committed Jan 31, 2016 369 \and  Ralf Jung committed Jan 31, 2016 370 \infer[$\exists_2$E]  Ralf Jung committed Jan 31, 2016 371 372  {\pfctx \proves \exists \var \in \Pred(\sort).\prop \\ \pfctx, \var : \Pred(\sort), \prop \proves \propB}  Ralf Jung committed Jan 31, 2016 373 374  {\pfctx \proves \propB} \and  Ralf Jung committed Jan 31, 2016 375 \infer[$\exists_2$I]  Ralf Jung committed Jan 31, 2016 376  {\pfctx \proves \prop[\propB/\var] \\  Ralf Jung committed Jan 31, 2016 377  \pfctx \proves \propB: \Prop}  Ralf Jung committed Jan 31, 2016 378  {\pfctx \proves \exists \var. \prop}  Ralf Jung committed Jan 31, 2016 379 \and  Ralf Jung committed Jan 31, 2016 380 \inferB[Elem]  Ralf Jung committed Jan 31, 2016 381 382 383  {\pfctx \proves \term \in (X \in \sort). \prop} {\pfctx \proves \prop[\term/X]} \and  Ralf Jung committed Jan 31, 2016 384 \inferB[Elem-$\mu$]  Ralf Jung committed Jan 31, 2016 385 386  {\pfctx \proves \term \in (\mu\var \in \Pred(\sort). \pred)} {\pfctx \proves \term \in \pred[\mu\var \in \Pred(\sort). \pred/\var]}  Ralf Jung committed Jan 31, 2016 387 388 \end{mathpar}  Ralf Jung committed Mar 06, 2016 389 \paragraph{Laws of (affine) bunched implications.}  Ralf Jung committed Jan 31, 2016 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 \begin{mathpar} \begin{array}{rMcMl} \prop * \propB &\Lra& \propB * \prop \\ (\prop * \propB) * \propC &\Lra& \prop * (\propB * \propC) \\ \prop * \propB &\Ra& \prop \end{array} \and \begin{array}{rMcMl} (\prop \vee \propB) * \propC &\Lra& (\prop * \propC) \vee (\propB * \propC) \\ (\prop \wedge \propB) * \propC &\Ra& (\prop * \propC) \wedge (\propB * \propC) \\ (\Exists x. \prop) * \propB &\Lra& \Exists x. (\prop * \propB) \\ (\All x. \prop) * \propB &\Ra& \All x. (\prop * \propB) \end{array} \and \infer {\pfctx, \prop_1 \proves \propB_1 \and \pfctx, \prop_2 \proves \propB_2} {\pfctx, \prop_1 * \prop_2 \proves \propB_1 * \propB_2} \and \infer {\pfctx, \prop * \propB \proves \propC} {\pfctx, \prop \proves \propB \wand \propC} \and \infer {\pfctx, \prop \proves \propB \wand \propC} {\pfctx, \prop * \propB \proves \propC} \end{mathpar}  Ralf Jung committed Mar 06, 2016 420 \paragraph{Laws for ghosts and physical resources.}  Ralf Jung committed Jan 31, 2016 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435  \begin{mathpar} \begin{array}{rMcMl} \ownGGhost{\melt} * \ownGGhost{\meltB} &\Lra& \ownGGhost{\melt \mtimes \meltB} \\ \TRUE &\Ra& \ownGGhost{\munit}\\ \ownGGhost{\mzero} &\Ra& \FALSE\\ \multicolumn{3}{c}{\timeless{\ownGGhost{\melt}}} \end{array} \and \begin{array}{c} \ownPhys{\state} * \ownPhys{\state'} \Ra \FALSE \\ \timeless{\ownPhys{\state}} \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 436 \paragraph{Laws for the later modality.}  Ralf Jung committed Jan 31, 2016 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459  \begin{mathpar} \inferH{Mono} {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and \inferhref{L{\"o}b}{Loeb} {\pfctx, \later{\prop} \proves \prop} {\pfctx \proves \prop} \and \begin{array}[b]{rMcMl} \later{\always{\prop}} &\Lra& \always{\later{\prop}} \\ \later{(\prop \wedge \propB)} &\Lra& \later{\prop} \wedge \later{\propB} \\ \later{(\prop \vee \propB)} &\Lra& \later{\prop} \vee \later{\propB} \\ \end{array} \and \begin{array}[b]{rMcMl} \later{\All x.\prop} &\Lra& \All x. \later\prop \\ \later{\Exists x.\prop} &\Lra& \Exists x. \later\prop \\ \later{(\prop * \propB)} &\Lra& \later\prop * \later\propB \end{array} \end{mathpar}  Ralf Jung committed Mar 06, 2016 460 \paragraph{Laws for the always modality.}  Ralf Jung committed Jan 31, 2016 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484  \begin{mathpar} \axiomH{Necessity} {\always{\prop} \Ra \prop} \and \inferhref{$\always$I}{AlwaysIntro} {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and \begin{array}[b]{rMcMl} \always(\term =_\sort \termB) &\Lra& \term=_\sort \termB \\ \always{\prop} * \propB &\Lra& \always{\prop} \land \propB \\ \always{(\prop \Ra \propB)} &\Ra& \always{\prop} \Ra \always{\propB} \\ \end{array} \and \begin{array}[b]{rMcMl} \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} \end{mathpar} Note that $\always$ binds more tightly than $*$, $\land$, $\lor$, and $\Ra$.  Ralf Jung committed Mar 06, 2016 485 \paragraph{Laws of primitive view shifts.}  Ralf Jung committed Jan 31, 2016 486   Ralf Jung committed Mar 06, 2016 487 \paragraph{Laws of weakest preconditions.}  Ralf Jung committed Jan 31, 2016 488   Ralf Jung committed Jan 31, 2016 489 490 491 492 493  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: