\section{Language} 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 \begin{itemize} \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} $(-,- \step -,-,-) \subseteq \textdom{Expr} \times \textdom{State} \times \textdom{Expr} \times \textdom{State} \times (\textdom{Expr} \uplus \set{\bot})$ We will write $\expr_1, \state_1 \step \expr_2, \state_2$ for $\expr_1, \state_1 \step \expr_2, \state_2, \bot$. \\ 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. \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_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 \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. \end{itemize} \begin{defn} An expression $\expr$ and state $\state$ are \emph{reducible} (written $\red(\expr, \state)$) if $\Exists \expr_2, \state_2, \expr_f. \expr,\state \step \expr_2,\state_2,\expr_f$ \end{defn} \begin{defn}[Context] A function $\lctx : \textdom{Expr} \to \textdom{Expr}$ is a \emph{context} if the following conditions are satisfied: \begin{enumerate}[itemsep=0pt] \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 $\lctx$:\\ $\All \expr_1, \state_1, \expr_2, \state_2, \expr_f. \expr_1, \state_1 \step \expr_2,\state_2,\expr_f \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr_f$ \item Reductions stay below $\lctx$ until there is a value in the hole:\\ $\All \expr_1', \state_1, \expr_2, \state_2, \expr_f. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr_f \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr_f$ \end{enumerate} \end{defn} \subsection{Concurrent language} For any language $\Lang$, we define the corresponding thread-pool semantics. \paragraph{Machine syntax} $\tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Exp}^n$ \judgment{Machine reduction} {\cfg{\tpool}{\state} \step \cfg{\tpool'}{\state'}} \begin{mathpar} \infer {\expr_1, \state_1 \step \expr_2, \state_2, \expr_f \and \expr_f \neq \bot} {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_f]}{\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'}} \end{mathpar} \clearpage \section{Logic} To instantiate Iris, you need to define the following parameters: \begin{itemize} \item A language $\Lang$ \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 \end{itemize} \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: $\SigType \supseteq \{ \textlog{Val}, \textlog{Expr}, \textlog{State}, \textlog{M}, \textlog{InvName}, \textlog{InvMask}, \Prop \}$ 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 $\SigType$). 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. 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} \paragraph{Syntax.} Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\textdom{Var}$ of variables (ranged over by metavariables $x$, $y$, $z$): \begin{align*} \type \bnfdef{}& \sigtype \mid 1 \mid \type \times \type \mid \type \to \type \0.4em] \term, \prop, \pred \bnfdef{}& \var \mid \sigfn(\term_1, \dots, \term_n) \mid () \mid (\term, \term) \mid \pi_i\; \term \mid \Lam \var:\type.\term \mid \term(\term) \mid \munit \mid \mcore\term \mid \term \mtimes \term \mid \\& \FALSE \mid \TRUE \mid \term =_\type \term \mid \prop \Ra \prop \mid \prop \land \prop \mid \prop \lor \prop \mid \prop * \prop \mid \prop \wand \prop \mid \\& \MU \var:\type. \pred \mid \Exists \var:\type. \prop \mid \All \var:\type. \prop \mid \\& \knowInv{\term}{\prop} \mid \ownGGhost{\term} \mid \mval(\term) \mid \ownPhys{\term} \mid \always\prop \mid {\later\prop} \mid \pvs[\term][\term] \prop\mid \wpre{\term}[\term]{\Ret\var.\term} \end{align*} Recursive predicates must be \emph{guarded}: in \MU \var. \pred, the variable \var can only appear under the later \later modality. Note that \always and \later bind more tightly than *, \wand, \land, \lor, and \Ra. We will write \pvs[\term] \prop for \pvs[\term][\term] \prop. 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. 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 \paragraph{Metavariable conventions.} We introduce additional metavariables ranging over terms and generally let the choice of metavariable indicate the term's type: $\begin{array}{r|l} \text{metavariable} & \text{type} \\\hline \term, \termB & \text{arbitrary} \\ \val, \valB & \textlog{Val} \\ \expr & \textlog{Expr} \\ \state & \textlog{State} \\ \end{array} \qquad\qquad \begin{array}{r|l} \text{metavariable} & \text{type} \\\hline \iname & \textlog{InvName} \\ \mask & \textlog{InvMask} \\ \melt, \meltB & \textlog{M} \\ \prop, \propB, \propC & \Prop \\ \pred, \predB, \predC & \type\to\Prop \text{ (when \type is clear from context)} \\ \end{array}$ \paragraph{Variable conventions.} 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. \subsection{Types}\label{sec:types} Iris terms are simply-typed. The judgment $\vctx \proves \wtt{\term}{\type}$ expresses that, in variable context $\vctx$, the term $\term$ has type $\type$. 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}} \begin{mathparpagebreakable} %%% variables and function symbols \axiom{x : \type \proves \wtt{x}{\type}} \and \infer{\vctx \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term}{\type}} \and \infer{\vctx, x:\type', y:\type' \proves \wtt{\term}{\type}} {\vctx, x:\type' \proves \wtt{\term[x/y]}{\type}} \and \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}} \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 \axiom{\vctx \proves \wtt{()}{1}} \and \infer{\vctx \proves \wtt{\term}{\type_1} \and \vctx \proves \wtt{\termB}{\type_2}} {\vctx \proves \wtt{(\term,\termB)}{\type_1 \times \type_2}} \and \infer{\vctx \proves \wtt{\term}{\type_1 \times \type_2} \and i \in \{1, 2\}} {\vctx \proves \wtt{\pi_i\,\term}{\type_i}} %%% functions \and \infer{\vctx, x:\type \proves \wtt{\term}{\type'}} {\vctx \proves \wtt{\Lam x. \term}{\type \to \type'}} \and \infer {\vctx \proves \wtt{\term}{\type \to \type'} \and \wtt{\termB}{\type}} {\vctx \proves \wtt{\term(\termB)}{\type'}} %%% monoids \and \infer{}{\vctx \proves \wtt\munit{\textlog{M}}} \and \infer{\vctx \proves \wtt\melt{\textlog{M}}}{\vctx \proves \wtt{\mcore\melt}{\textlog{M}}} \and \infer{\vctx \proves \wtt{\melt}{\textlog{M}} \and \vctx \proves \wtt{\meltB}{\textlog{M}}} {\vctx \proves \wtt{\melt \mtimes \meltB}{\textlog{M}}} %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and \infer{\vctx \proves \wtt{\term}{\type} \and \vctx \proves \wtt{\termB}{\type}} {\vctx \proves \wtt{\term =_\type \termB}{\Prop}} \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{ \vctx, \var:\type \proves \wtt{\term}{\type} \and \text{$\var$ is guarded in $\term$} }{ \vctx \proves \wtt{\MU \var:\type. \term}{\type} } \and \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\type. \prop}{\Prop}} \and \infer{\vctx, x:\type \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\type. \prop}{\Prop}} \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\iname}{\textlog{InvName}} }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and \infer{\vctx \proves \wtt{\melt}{\textlog{M}}} {\vctx \proves \wtt{\ownGGhost{\melt}}{\Prop}} \and \infer{\vctx \proves \wtt{\melt}{\textlog{M}}} {\vctx \proves \wtt{\mval(\melt)}{\Prop}} \and \infer{\vctx \proves \wtt{\state}{\textlog{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}{\textlog{InvMask}} \and \vctx \proves \wtt{\mask'}{\textlog{InvMask}} }{ \vctx \proves \wtt{\pvs[\mask][\mask'] \prop}{\Prop} } \and \infer{ \vctx \proves \wtt{\expr}{\textlog{Expr}} \and \vctx,\var:\textlog{Val} \proves \wtt{\term}{\Prop} \and \vctx \proves \wtt{\mask}{\textlog{InvMask}} }{ \vctx \proves \wtt{\wpre{\expr}[\mask]{\Ret\var.\term}}{\Prop} } \end{mathparpagebreakable} \subsection{Proof rules} 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. 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} \paragraph{Laws of intuitionistic higher-order logic with equality.} This is entirely standard. \begin{mathparpagebreakable} \infer[Asm] {\prop \in \pfctx} {\pfctx \proves \prop} \and \infer[Eq] {\pfctx \proves \prop \\ \pfctx \proves \term =_\type \term'} {\pfctx \proves \prop[\term'/\term]} \and \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 \infer[$\wedge$I] {\pfctx \proves \prop \\ \pfctx \proves \propB} {\pfctx \proves \prop \wedge \propB} \and \infer[$\wedge$EL] {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \prop} \and \infer[$\wedge$ER] {\pfctx \proves \prop \wedge \propB} {\pfctx \proves \propB} \and \infer[$\vee$IL] {\pfctx \proves \prop } {\pfctx \proves \prop \vee \propB} \and \infer[$\vee$IR] {\pfctx \proves \propB} {\pfctx \proves \prop \vee \propB} \and \infer[$\vee$E] {\pfctx \proves \prop \vee \propB \\ \pfctx, \prop \proves \propC \\ \pfctx, \propB \proves \propC} {\pfctx \proves \propC} \and \infer[$\Ra$I] {\pfctx, \prop \proves \propB} {\pfctx \proves \prop \Ra \propB} \and \infer[$\Ra$E] {\pfctx \proves \prop \Ra \propB \\ \pfctx \proves \prop} {\pfctx \proves \propB} \and \infer[$\forall$I] { \vctx,\var : \type\mid\pfctx \proves \prop} {\vctx\mid\pfctx \proves \forall \var: \type.\; \prop} \and \infer[$\forall$E] {\vctx\mid\pfctx \proves \forall \var :\type.\; \prop \\ \vctx \proves \wtt\term\type} {\vctx\mid\pfctx \proves \prop[\term/\var]} \and \infer[$\exists$I] {\vctx\mid\pfctx \proves \prop[\term/\var] \\ \vctx \proves \wtt\term\type} {\vctx\mid\pfctx \proves \exists \var: \type. \prop} \and \infer[$\exists$E] {\vctx\mid\pfctx \proves \exists \var: \type.\; \prop \\ \vctx,\var : \type\mid\pfctx , \prop \proves \propB} {\vctx\mid\pfctx \proves \propB} \and \infer[$\lambda$] {} {\pfctx \proves (\Lam\var: \type. \prop)(\term) =_{\type\to\type'} \prop[\term/\var]} \and \infer[$\mu$] {} {\pfctx \proves \mu\var: \type. \prop =_{\type} \prop[\mu\var: \type. \prop/\var]} \end{mathparpagebreakable} \paragraph{Laws of (affine) bunched implications.} \begin{mathpar} \begin{array}{rMcMl} \TRUE * \prop &\provesIff& \prop \\ \prop * \propB &\provesIff& \propB * \prop \\ (\prop * \propB) * \propC &\provesIff& \prop * (\propB * \propC) \end{array} \and \infer[$*$-mono] {\prop_1 \proves \propB_1 \and \prop_2 \proves \propB_2} {\prop_1 * \prop_2 \proves \propB_1 * \propB_2} \and \inferB[$\wand$I-E] {\prop * \propB \proves \propC} {\prop \proves \propB \wand \propC} \end{mathpar} \paragraph{Laws for ghosts and physical resources.} \begin{mathpar} \begin{array}{rMcMl} \ownGGhost{\melt} * \ownGGhost{\meltB} &\provesIff& \ownGGhost{\melt \mtimes \meltB} \\ \ownGGhost{\melt} &\provesIff& \mval(\melt) \\ \TRUE &\proves& \ownGGhost{\munit} \end{array} \and \and \begin{array}{c} \ownPhys{\state} * \ownPhys{\state'} \proves \FALSE \end{array} \end{mathpar} \paragraph{Laws for the later modality.} \begin{mathpar} \infer[$\later$-mono] {\pfctx \proves \prop} {\pfctx \proves \later{\prop}} \and \infer[L{\"o}b] {} {(\later\prop\Ra\prop) \proves \prop} \and \infer[$\later$-$\exists$] {\text{$\type$ is inhabited}} {\later{\Exists x:\type.\prop} \proves \Exists x:\type. \later\prop} \\\\ \begin{array}[c]{rMcMl} \later{(\prop \wedge \propB)} &\provesIff& \later{\prop} \wedge \later{\propB} \\ \later{(\prop \vee \propB)} &\provesIff& \later{\prop} \vee \later{\propB} \\ \end{array} \and \begin{array}[c]{rMcMl} \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 \end{array} \end{mathpar} \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 {\text{$\melt$ is a discrete COFE element}} {\timeless{\mval(\melt)}} \infer{} {\timeless{\ownPhys\state}} \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} \paragraph{Laws for the always modality.} \begin{mathpar} \infer[$\always$I] {\always{\pfctx} \proves \prop} {\always{\pfctx} \proves \always{\prop}} \and \infer[$\always$E]{} {\always{\prop} \proves \prop} \and \begin{array}[c]{rMcMl} \always{(\prop * \propB)} &\proves& \always{(\prop \land \propB)} \\ \always{\prop} * \propB &\proves& \always{\prop} \land \propB \\ \always{\later\prop} &\provesIff& \later\always{\prop} \\ \end{array} \and \begin{array}[c]{rMcMl} \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} \\ \end{array} \and { \term =_\type \term' \proves \always \term =_\type \term'} \and { \knowInv\iname\prop \proves \always \knowInv\iname\prop} \and { \ownGGhost{\mcore\melt} \proves \always \ownGGhost{\mcore\melt}} \end{mathpar} \paragraph{Laws of primitive view shifts.} \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} \inferH{pvs-allocI} {\text{$\mask$ is infinite}} {\later\prop \proves \pvs[\mask] \Exists \iname \in \mask. \knowInv\iname\prop} \inferH{pvs-openI} {}{\knowInv\iname\prop \proves \pvs[\set\iname][\emptyset] \later\prop} \inferH{pvs-closeI} {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE} \inferH{pvs-update} {\melt \mupd \meltsB} {\ownGGhost\melt \proves \pvs[\mask] \Exists\meltB\in\meltsB. \ownGGhost\meltB} \end{mathpar} \paragraph{Laws of weakest preconditions.} \begin{mathpar} \infer[wp-value] {}{\prop[\val/\var] \proves \wpre{\val}[\mask]{\Ret\var.\prop}} \infer[wp-mono] {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB} {\wpre\expr[\mask_1]{\Ret\var.\prop} \proves \wpre\expr[\mask_2]{\Ret\var.\propB}} \infer[pvs-wp] {}{\pvs[\mask] \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}} \infer[wp-pvs] {}{\wpre\expr[\mask]{\Ret\var.\pvs[\mask] \prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}} \infer[wp-atomic] {\mask_2 \subseteq \mask_1 \and \physatomic{\expr}} {\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}} \infer[wp-frame] {}{\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}} \infer[wp-frame-step] {\toval(\expr) = \bot} {\later\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}} \infer[wp-bind] {\text{$\lctx$ is a context}} {\wpre\expr[\mask]{\Ret\var. \wpre{\lctx(\ofval(\var))}[\mask]{\Ret\varB.\prop}} \proves \wpre{\lctx(\expr)}[\mask]{\Ret\varB.\prop}} \end{mathpar} \paragraph{Lifting of operational semantics.}~ \begin{mathpar} \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_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}} } \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_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}} \end{mathpar} 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). \subsection{Adequacy} The adequacy statement concerning functional correctness reads as follows: \begin{align*} &\All \mask, \expr, \val, \pred, \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'}{[\val] \dplus \tpool'} \Ra \\&\pred(\val) \end{align*} where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants. 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. % 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. %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: