logic.tex 16.5 KB
Newer Older
Ralf Jung's avatar
Ralf Jung committed
1
\section{Language}
2

Ralf Jung's avatar
Ralf Jung committed
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
4
\begin{itemize}
Ralf Jung's avatar
Ralf Jung committed
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.
25 26
\end{itemize}

Ralf Jung's avatar
Ralf Jung committed
27 28 29
\subsection{The concurrent language}

For any language $\Lang$, we define the corresponding thread-pool semantics.
30 31 32

\paragraph{Machine syntax}
\[
Ralf Jung's avatar
Ralf Jung committed
33
	\tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n
34 35
\]

Ralf Jung's avatar
Ralf Jung committed
36 37
\judgment{Machine reduction} {\cfg{\tpool}{\state} \step
  \cfg{\tpool'}{\state'}}
38 39
\begin{mathpar}
\infer
Ralf Jung's avatar
Ralf Jung committed
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'}}
47 48 49
\end{mathpar}


Ralf Jung's avatar
Ralf Jung committed
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}
57

Ralf Jung's avatar
Ralf Jung committed
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:
61
\[
Ralf Jung's avatar
Ralf Jung committed
62
	\SigType \supseteq \{ \textsort{Val}, \textsort{Expr}, \textsort{State}, \textsort{M}, \textsort{InvName}, \textsort{InvMask}, \Prop \}
63
\]
Ralf Jung's avatar
Ralf Jung committed
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 $\SigType$).
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's avatar
Ralf Jung committed
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}
80 81

\paragraph{Syntax.}
Ralf Jung's avatar
Ralf Jung committed
82
Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\textdom{Var}$ of variables (ranged over by metavariables $x$, $y$, $z$):
83

84
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
85 86 87 88 89 90
  \type ::={}&
      \sigtype \mid
      \unitsort \mid
      \type \times \type \mid
      \type \to \type
\\[0.4em]
91 92 93
  \term, \prop, \pred ::={}&
      x \mid
      \sigfn(\term_1, \dots, \term_n) \mid
94
      \unitval \mid
95 96 97
      (\term, \term) \mid
      \pi_i\; \term \mid
      \Lam x.\term \mid
Ralf Jung's avatar
Ralf Jung committed
98
      \term(\term)  \mid
99 100 101 102 103 104
      \mzero \mid
      \munit \mid
      \term \mtimes \term \mid
\\&
    \FALSE \mid
    \TRUE \mid
Ralf Jung's avatar
Ralf Jung committed
105
    \term =_\type \term \mid
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's avatar
Ralf Jung committed
112
    \MU \var. \pred  \mid
Ralf Jung's avatar
Ralf Jung committed
113 114
    \Exists \var:\type. \prop \mid
    \All \var:\type. \prop \mid
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's avatar
Ralf Jung committed
122
    \dynA{\term}{\pred}{\term}
123
\end{align*}
Ralf Jung's avatar
Ralf Jung committed
124
Recursive predicates must be \emph{guarded}: in $\MU \var. \pred$, the variable $\var$ can only appear under the later $\later$ modality.
125 126

\paragraph{Metavariable conventions.}
Ralf Jung's avatar
Ralf Jung committed
127
We introduce additional metavariables ranging over terms and generally let the choice of metavariable indicate the term's type:
128 129
\[
\begin{array}{r|l}
Ralf Jung's avatar
Ralf Jung committed
130
 \text{metavariable} & \text{type} \\\hline
131 132 133 134 135 136 137
  \term, \termB & \text{arbitrary} \\
  \val, \valB & \textsort{Val} \\
  \expr & \textsort{Exp} \\
  \state & \textsort{State} \\
\end{array}
\qquad\qquad
\begin{array}{r|l}
Ralf Jung's avatar
Ralf Jung committed
138
 \text{metavariable} & \text{type} \\\hline
139 140
  \iname & \textsort{InvName} \\
  \mask & \textsort{InvMask} \\
Ralf Jung's avatar
Ralf Jung committed
141
  \melt, \meltB & \textsort{M} \\
142
  \prop, \propB, \propC & \Prop \\
Ralf Jung's avatar
Ralf Jung committed
143
  \pred, \predB, \predC & \type\to\Prop \text{ (when $\type$ is clear from context)} \\
144 145 146 147
\end{array}
\]

\paragraph{Variable conventions.}
148
We often abuse notation, using the preceding \emph{term} meta-variables to range over (bound) \emph{variables}.
149
We omit type annotations in binders, when the type is clear from context.
Ralf Jung's avatar
Ralf Jung committed
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.
151 152 153 154 155


\subsection{Types}\label{sec:types}

Iris terms are simply-typed.
Ralf Jung's avatar
Ralf Jung committed
156
The judgment $\vctx \proves \wtt{\term}{\type}$ expresses that, in variable context $\vctx$, the term $\term$ has type $\type$.
157

Ralf Jung's avatar
Ralf Jung committed
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 $\vctx$.
160

Ralf Jung's avatar
Ralf Jung committed
161
\judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}}
162 163
\begin{mathparpagebreakable}
%%% variables and function symbols
Ralf Jung's avatar
Ralf Jung committed
164
	\axiom{x : \type \proves \wtt{x}{\type}}
165
\and
Ralf Jung's avatar
Ralf Jung committed
166 167
	\infer{\vctx \proves \wtt{\term}{\type}}
		{\vctx, x:\type' \proves \wtt{\term}{\type}}
168
\and
Ralf Jung's avatar
Ralf Jung committed
169 170
	\infer{\vctx, x:\type', y:\type' \proves \wtt{\term}{\type}}
		{\vctx, x:\type' \proves \wtt{\term[x/y]}{\type}}
171
\and
Ralf Jung's avatar
Ralf Jung committed
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}}
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
185
	\axiom{\vctx \proves \wtt{\unitval}{\unitsort}}
186
\and
Ralf Jung's avatar
Ralf Jung committed
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}}
189
\and
Ralf Jung's avatar
Ralf Jung committed
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}}
192 193
%%% functions
\and
Ralf Jung's avatar
Ralf Jung committed
194 195
	\infer{\vctx, x:\type \proves \wtt{\term}{\type'}}
		{\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}}
196 197
\and
	\infer
Ralf Jung's avatar
Ralf Jung committed
198 199
	{\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}}
	{\vctx \proves \wtt{\term(\termB)}{\type'}}
200 201
%%% monoids
\and
Ralf Jung's avatar
Ralf Jung committed
202
	\infer{\vctx \proves \wtt{\term}{\textsort{M}}}{\vctx \proves \wtt{\munit(\term)}{\textsort{M}}}
203
\and
Ralf Jung's avatar
Ralf Jung committed
204 205
	\infer{\vctx \proves \wtt{\melt}{\textsort{M}} \and \vctx \proves \wtt{\meltB}{\textsort{M}}}
		{\vctx \proves \wtt{\melt \mtimes \meltB}{\textsort{M}}}
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's avatar
Ralf Jung committed
212 213
	\infer{\vctx \proves \wtt{\term}{\type} \and \vctx \proves \wtt{\termB}{\type}}
		{\vctx \proves \wtt{\term =_\type \termB}{\Prop}}
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's avatar
Ralf Jung committed
231
		\vctx, \var:\type\to\Prop \proves \wtt{\pred}{\type\to\Prop} \and
Ralf Jung's avatar
Ralf Jung committed
232
		\text{$\var$ is guarded in $\pred$}
233
	}{
Ralf Jung's avatar
Ralf Jung committed
234
		\vctx \proves \wtt{\MU \var. \pred}{\type\to\Prop}
235 236
	}
\and
Ralf Jung's avatar
Ralf Jung committed
237 238
	\infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}}
		{\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}}
239
\and
Ralf Jung's avatar
Ralf Jung committed
240 241
	\infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}}
		{\vctx \proves \wtt{\All x:\type. \prop}{\Prop}}
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's avatar
Ralf Jung committed
250
	\infer{\vctx \proves \wtt{\melt}{\textsort{M}}}
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's avatar
Ralf Jung committed
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.
283

Ralf Jung's avatar
Ralf Jung committed
284
\judgment{Timeless Propositions}{\timeless{P}}
285

Ralf Jung's avatar
Ralf Jung committed
286 287 288 289 290
\ralf{Define a judgment that defines them.}

\subsection{Base logic}

\ralf{Go on checking below.}
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's avatar
Ralf Jung committed
296
\paragraph{Laws of intuitionistic higher-order logic.}
297
This is entirely standard.
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
308
\infer[$\wedge$I]
309 310 311
  {\pfctx \proves \prop \\ \pfctx \proves \propB}
  {\pfctx \proves \prop \wedge \propB}
\and
312
\infer[$\wedge$EL]
313 314 315
  {\pfctx \proves \prop \wedge \propB}
  {\pfctx \proves \prop}
\and
316
\infer[$\wedge$ER]
317 318 319
  {\pfctx \proves \prop \wedge \propB}
  {\pfctx \proves \propB}
\and
320
\infer[$\vee$E]
321 322 323 324 325
  {\pfctx \proves \prop \vee \propB \\
   \pfctx, \prop \proves \propC \\
   \pfctx, \propB \proves \propC}
  {\pfctx \proves \propC}
\and
326
\infer[$\vee$IL]
327 328 329
  {\pfctx \proves \prop }
  {\pfctx \proves \prop \vee \propB}
\and
330
\infer[$\vee$IR]
331 332 333
  {\pfctx \proves \propB}
  {\pfctx \proves \prop \vee \propB}
\and
334
\infer[$\Ra$I]
335 336 337
  {\pfctx, \prop \proves \propB}
  {\pfctx \proves \prop \Ra \propB}
\and
338
\infer[$\Ra$E]
339 340 341
  {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop}
  {\pfctx \proves \propB}
\and
342
\infer[$\forall_1$I]
343 344 345
  {\pfctx, x : \sort \proves \prop}
  {\pfctx \proves \forall x: \sort.\; \prop}
\and
346
\infer[$\forall_1$E]
347 348 349 350
  {\pfctx \proves \forall X \in \sort.\; \prop \\
   \pfctx \proves \term: \sort}
  {\pfctx \proves \prop[\term/X]}
\and
351
\infer[$\exists_1$E]
352 353 354 355
  {\pfctx \proves \exists X\in \sort.\; \prop \\
   \pfctx, X : \sort, \prop \proves \propB}
  {\pfctx \proves \propB}
\and
356
\infer[$\exists_1$I]
357 358 359 360
  {\pfctx \proves \prop[\term/X] \\
   \pfctx \proves \term: \sort}
  {\pfctx \proves \exists X: \sort. \prop}
\and
361
\infer[$\forall_2$I]
Ralf Jung's avatar
Ralf Jung committed
362 363
  {\pfctx, \var: \Pred(\sort) \proves \prop}
  {\pfctx \proves \forall \var\in \Pred(\sort).\; \prop}
364
\and
365
\infer[$\forall_2$E]
Ralf Jung's avatar
Ralf Jung committed
366
  {\pfctx \proves \forall \var. \prop \\
367
   \pfctx \proves \propB: \Prop}
Ralf Jung's avatar
Ralf Jung committed
368
  {\pfctx \proves \prop[\propB/\var]}
369
\and
370
\infer[$\exists_2$E]
Ralf Jung's avatar
Ralf Jung committed
371 372
  {\pfctx \proves \exists \var \in \Pred(\sort).\prop \\
   \pfctx, \var : \Pred(\sort), \prop \proves \propB}
373 374
  {\pfctx \proves \propB}
\and
375
\infer[$\exists_2$I]
Ralf Jung's avatar
Ralf Jung committed
376
  {\pfctx \proves \prop[\propB/\var] \\
377
   \pfctx \proves \propB: \Prop}
Ralf Jung's avatar
Ralf Jung committed
378
  {\pfctx \proves \exists \var. \prop}
379
\and
380
\inferB[Elem]
381 382 383
  {\pfctx \proves \term \in (X \in \sort). \prop}
  {\pfctx \proves \prop[\term/X]}
\and
384
\inferB[Elem-$\mu$]
Ralf Jung's avatar
Ralf Jung committed
385 386
  {\pfctx \proves \term \in (\mu\var \in \Pred(\sort). \pred)}
  {\pfctx \proves \term \in \pred[\mu\var \in \Pred(\sort). \pred/\var]}
387 388
\end{mathpar}

Ralf Jung's avatar
Ralf Jung committed
389
\paragraph{Laws of (affine) bunched implications.}
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's avatar
Ralf Jung committed
420
\paragraph{Laws for ghosts and physical resources.}
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's avatar
Ralf Jung committed
436
\paragraph{Laws for the later modality.}
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's avatar
Ralf Jung committed
460
\paragraph{Laws for the always modality.}
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's avatar
Ralf Jung committed
485
\paragraph{Laws of primitive view shifts.}
486

Ralf Jung's avatar
Ralf Jung committed
487
\paragraph{Laws of weakest preconditions.}
488

489 490 491 492 493

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "iris"
%%% End: