Commit 0aaa35e7 by Ralf Jung

update iris.sty

parent 5cb9dfc4
 ... ... @@ -387,20 +387,20 @@ The construction follows the idea of STSs as described in CaReSL \cite{caresl}. We first lift the transition relation to $\STSS \times \wp(\STST)$ (implementing a \emph{law of token conservation}) and define a stepping relation for the \emph{frame} of a given token set: \begin{align*} (s, T) \stsstep (s', T') \eqdef{}& s \stsstep s' \land \STSL(s) \uplus T = \STSL(s') \uplus T' \\ s \stsfstep{T} s' \eqdef{}& \Exists T_1, T_2. T_1 \sep \STSL(s) \cup T \l+and (s, T_1) \stsstep (s', T_2) s \stsfstep{T} s' \eqdef{}& \Exists T_1, T_2. T_1 \disj \STSL(s) \cup T \l+and (s, T_1) \stsstep (s', T_2) \end{align*} We further define \emph{closed} sets of states (given a particular set of tokens) as well as the \emph{closure} of a set: \begin{align*} \STSclsd(S, T) \eqdef{}& \All s \in S. \STSL(s) \sep T \land \All s'. s \stsfstep{T} s' \Ra s' \in S \\ \STSclsd(S, T) \eqdef{}& \All s \in S. \STSL(s) \disj T \land \All s'. s \stsfstep{T} s' \Ra s' \in S \\ \upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' } \end{align*} The STS RA is defined as follows \begin{align*} \monoid \eqdef{}& \setComp{\STSauth((s, T) \in \STSS \times \wp(\STST))}{\STSL(s) \sep T} +{}\\& \setComp{\STSfrag((S, T) \in \wp(\STSS) \times \wp(\STST))}{\STSclsd(S, T) \land S \neq \emptyset} + \bot \\ \STSfrag(S_1, T_1) \mtimes \STSfrag(S_2, T_2) \eqdef{}& \STSfrag(S_1 \cap S_2, T_1 \cup T_2) \qquad\qquad\qquad \text{if $T_1 \sep T_2$ and $S_1 \cap S_2 \neq \emptyset$} \\ \STSfrag(S, T) \mtimes \STSauth(s, T') \eqdef{}& \STSauth(s, T') \mtimes \STSfrag(S, T) \eqdef \STSauth(s, T \cup T') \qquad \text{if $T \sep T'$ and $s \in S$} \\ \monoid \eqdef{}& \setComp{\STSauth((s, T) \in \STSS \times \wp(\STST))}{\STSL(s) \disj T} +{}\\& \setComp{\STSfrag((S, T) \in \wp(\STSS) \times \wp(\STST))}{\STSclsd(S, T) \land S \neq \emptyset} + \bot \\ \STSfrag(S_1, T_1) \mtimes \STSfrag(S_2, T_2) \eqdef{}& \STSfrag(S_1 \cap S_2, T_1 \cup T_2) \qquad\qquad\qquad \text{if $T_1 \disj T_2$ and $S_1 \cap S_2 \neq \emptyset$} \\ \STSfrag(S, T) \mtimes \STSauth(s, T') \eqdef{}& \STSauth(s, T') \mtimes \STSfrag(S, T) \eqdef \STSauth(s, T \cup T') \qquad \text{if $T \disj T'$ and $s \in S$} \\ \mcore{\STSfrag(S, T)} \eqdef{}& \STSfrag(\upclose(S, \emptyset), \emptyset) \\ \mcore{\STSauth(s, T)} \eqdef{}& \STSfrag(\upclose(\set{s}, \emptyset), \emptyset) \end{align*} ... ...
 ... ... @@ -319,8 +319,8 @@ We use the notation $\namesp.\iname$ for the namespace $[\iname] \dplus \namesp$ We define the inclusion relation on namespaces as $\namesp_1 \sqsubseteq \namesp_2 \Lra \Exists \namesp_3. \namesp_2 = \namesp_3 \dplus \namesp_1$, \ie $\namesp_1$ is a suffix of $\namesp_2$. We have that $\namesp_1 \sqsubseteq \namesp_2 \Ra \namecl{\namesp_2} \subseteq \namecl{\namesp_1}$. Similarly, we define $\namesp_1 \sep \namesp_2 \eqdef \Exists \namesp_1', \namesp_2'. \namesp_1' \sqsubseteq \namesp_1 \land \namesp_2' \sqsubseteq \namesp_2 \land |\namesp_1'| = |\namesp_2'| \land \namesp_1' \neq \namesp_2'$, \ie there exists a distinguishing suffix. We have that $\namesp_1 \sep \namesp_2 \Ra \namecl{\namesp_2} \sep \namecl{\namesp_1}$, and furthermore $\iname_1 \neq \iname_2 \Ra \namesp.\iname_1 \sep \namesp.\iname_2$. Similarly, we define $\namesp_1 \disj \namesp_2 \eqdef \Exists \namesp_1', \namesp_2'. \namesp_1' \sqsubseteq \namesp_1 \land \namesp_2' \sqsubseteq \namesp_2 \land |\namesp_1'| = |\namesp_2'| \land \namesp_1' \neq \namesp_2'$, \ie there exists a distinguishing suffix. We have that $\namesp_1 \disj \namesp_2 \Ra \namecl{\namesp_2} \disj \namecl{\namesp_1}$, and furthermore $\iname_1 \neq \iname_2 \Ra \namesp.\iname_1 \disj \namesp.\iname_2$. We will overload the usual Iris notation for invariant assertions in the following: $\knowInv\namesp\prop \eqdef \Exists \iname \in \namecl\namesp. \knowInv\iname{\prop}$ ... ...
 ... ... @@ -29,9 +29,8 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator*{\Sep}{\scalerel*{\ast}{\sum}} \newcommand{\bigast}{\Sep} \newcommand*{\sep}[1][]{\mathrel{\#_{#1}}} % bad name; it's a different "sep" \newcommand*{\disj}[1][]{\mathrel{\#_{#1}}} \newcommand\pord{\sqsubseteq} \newcommand\dplus{\mathbin{+\kern-1.0ex+}} \newcommand{\upclose}{\mathord{\uparrow}} \newcommand{\ALT}{\ |\ } ... ... @@ -44,10 +43,11 @@ \newcommand{\any}{{\rule[-.2ex]{1ex}{.4pt}}}% \newcommand{\judgment}[2]{\paragraph{#1}\hspace{\stretch{1}}\fbox{$#2$}} \newcommand{\judgment}[2][]{\paragraph{#1}\hspace{\stretch{1}}\fbox{$#2$}} \newcommand{\pfn}{\rightharpoonup} \newcommand\fpfn{\xrightharpoonup{\kern-0.25ex\textrm{fin}\kern-0.25ex}} \newcommand{\la}{\leftarrow} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\Lra}{\Leftrightarrow} ... ...
 ... ... @@ -50,7 +50,7 @@ For any language $\Lang$, we define the corresponding thread-pool semantics. \tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n \] \judgment{Machine reduction} {\cfg{\tpool}{\state} \step \judgment[Machine reduction]{\cfg{\tpool}{\state} \step \cfg{\tpool'}{\state'}} \begin{mathpar} \infer ... ... @@ -181,7 +181,7 @@ The judgment $\vctx \proves \wtt{\term}{\type}$ expresses that, in variable cont 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 $\vctx$. \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\type}} \judgment[Well-typed terms]{\vctx \proves_\Sig \wtt{\term}{\type}} \begin{mathparpagebreakable} %%% variables and function symbols \axiom{x : \type \proves \wtt{x}{\type}} ... ... @@ -312,7 +312,7 @@ We implicitly assume that an arbitrary variable context, $\vctx$, is added to ev Furthermore, an arbitrary \emph{boxed} assertion context $\always\pfctx$ may be added to every constituent. Axioms $\vctx \mid \prop \provesIff \propB$ indicate that both $\vctx \mid \prop \proves \propB$ and $\vctx \mid \propB \proves \prop$ can be derived. \judgment{}{\vctx \mid \pfctx \proves \prop} \judgment{\vctx \mid \pfctx \proves \prop} \paragraph{Laws of intuitionistic higher-order logic with equality.} This is entirely standard. \begin{mathparpagebreakable} ... ...
 ... ... @@ -95,7 +95,7 @@ We only have to define the missing connectives, the most interesting bits being \typedsection{Primitive view-shift}{\mathit{pvs}_{-}^{-}(-) : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \iProp \nfn \iProp} \begin{align*} \mathit{pvs}_{\mask_1}^{\mask_2}(\prop) &= \Lam \rs. \setComp{n}{\begin{aligned} \All \rs_\f, m, \mask_\f, \state.& 0 < m \leq n \land (\mask_1 \cup \mask_2) \sep \mask_\f \land k \in \wsat\state{\mask_1 \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\& \All \rs_\f, m, \mask_\f, \state.& 0 < m \leq n \land (\mask_1 \cup \mask_2) \disj \mask_\f \land k \in \wsat\state{\mask_1 \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\& \Exists \rsB. k \in \prop(\rsB) \land k \in \wsat\state{\mask_2 \cup \mask_\f}{\rsB \mtimes \rs_\f} \end{aligned}} \end{align*} ... ... @@ -105,7 +105,7 @@ We only have to define the missing connectives, the most interesting bits being $\textdom{wp}$ is defined as the fixed-point of a contractive function. \begin{align*} \textdom{pre-wp}(\textdom{wp})(\mask, \expr, \pred) &\eqdef \Lam\rs. \setComp{n}{\begin{aligned} \All &\rs_\f, m, \mask_\f, \state. 0 \leq m < n \land \mask \sep \mask_\f \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\ \All &\rs_\f, m, \mask_\f, \state. 0 \leq m < n \land \mask \disj \mask_\f \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\ &(\All\val. \toval(\expr) = \val \Ra \Exists \rsB. m+1 \in \prop(\rs') \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs' \mtimes \rs_\f}) \land {}\\ &(\toval(\expr) = \bot \land 0 < m \Ra \red(\expr, \state) \land \All \expr_2, \state_2, \expr_\f. \expr,\state \step \expr_2,\state_2,\expr_\f \Ra {}\\ &\qquad \Exists \rsB_1, \rsB_2. m \in \wsat\state{\mask \cup \mask_\f}{\rs' \mtimes \rs_\f} \land m \in \textdom{wp}(\mask, \expr_2, \pred)(\rsB_1) \land {}&\\ ... ...
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