% CONVENTION: % Use \Ra/Lra for the logic and \implies/\iff for the metalogic. % This short (for now) note lays out a \emph{generic} separation logic which % manages sharing through invariants and ownership through (partial commutative) % monoids. The logic is generic in that the actual language it applies to is % taken as a parameter, giving in particular the atomic (per-thread) reduction % relation. Over this, we layer concurrency (by giving a semantics to \kw{fork} % and lifting to thread pools). The generic logic provides numerous logical % connectives and the semantics of Hoare triples and view shifts, together with a % large portion of the proof theory---including, in particular, the structural % rules for Hoare logic. Ultimately, these are proved sound relative to some % simple assumptions about the language. It should be possible, moreover, to give % a generic adequacy proof for Hoare triples as applied to the lifted thread-pool % semantics. \section{Parameters to the logic} \begin{itemize} \item A set \textdom{Exp} of \emph{expressions} (metavariable $\expr$) with a subset \textdom{Val} of values ($\val$). We assume that if $\expr$ is an expression then so is $\fork{\expr}$. We moreover assume a value \textsf{fRet} (giving the intended return value of a fork), and we assume that \begin{align*} \fork{\expr} &\notin \textdom{Val} \\ \fork{\expr_1} = \fork{\expr_2} &\implies \expr_1 = \expr_2 \end{align*} \item A set $\textdom{Ectx}$ of \emph{evaluation contexts} ($\ectx$) that includes the empty context $[\; ]$, a plugging operation $\ectx[\expr]$ that produces an expression, and context composition $\circ$ satisfying the following axioms: \begin{align*} [\; ][ \expr ] &= \expr \\ \ectx_1[\ectx_2[\expr]] &= (\ectx_1 \circ \ectx_2) [\expr] \\ \ectx_1[\expr] = \ectx_2[\expr] &\implies \ectx_1 = \ectx_2 \\ \ectx[\expr_1] = \ectx[\expr_2] &\implies \expr_1 = \expr_2 \\ \ectx_1 \circ \ectx_2 = [\; ] &\implies \ectx_1 = \ectx_2 = [\; ] \\ \ectx[\expr] \in \textdom{Val} &\implies \ectx = [\;] \\ \ectx[\expr] = \fork{\expr'} &\implies \ectx = [\;] \end{align*} \item A set \textdom{State} of shared machine states (\eg heaps), metavariable $\state$. \item An \emph{atomic stepping relation} $(- \step -) \subseteq (\textdom{State} \times \textdom{Exp}) \times (\textdom{State} \times \textdom{Exp})$ and notions of an expression to be \emph{reducible} or \emph{stuck}, such that \begin{align*} \textlog{reducible}(\expr) &\iff \Exists \state, \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \\ % \textlog{stuck}(\expr) &\iff \All \ectx, \expr'. \expr = \ectx[\expr'] \implies \lnot \textlog{reducible}(\expr') \end{align*} and the following hold \begin{align*} &\textlog{stuck}(\fork{\expr})& \\ &\textlog{stuck}(\val)&\\ &\ectx[\expr] = \ectx'[\expr'] \implies \textlog{reducible}(\expr') \implies \expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(step-by-value)} \\ &\ectx[\expr] = \ectx'[\fork{\expr'}] \implies \expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(fork-by-value)} \\ \end{align*} \item A predicate \textlog{atomic} on expressions satisfying \begin{align*} &\textlog{atomic}(\expr) \implies \textlog{reducible}(\expr) &\\ &\textlog{atomic}(\expr) \implies \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \expr_2 \in \textdom{Val} &\mbox{(atomic-step)} \end{align*} \item A commutative monoid with zero, $M$. That is, a set $\mcar{M}$ with two distinguished elements $\mzero$ (zero, undefined) and $\munit$ (one, unit) and an operation $\mtimes$ (times, combine) such that \begin{align*} \melt \mtimes \meltB &= \meltB \mtimes \melt \\ \munit \mtimes \melt &= \melt \\ (\melt \mtimes \meltB) \mtimes \meltC &= \melt \mtimes (\meltB \mtimes \meltC) \\ \mzero \mtimes \melt &= \mzero \\ \mzero &\neq \munit \end{align*} Let $\mcarp{M} \eqdef |\monoid| \setminus \{\mzero\}$. \item Arbitrary additional types and terms. \end{itemize} \section{The concurrent language} \paragraph{Machine syntax} $\tpool \in \textdom{ThreadPool} \eqdef \mathbb{N} \fpfn \textdom{Exp}$ \judgment{Machine reduction} {\cfg{\state}{\tpool} \step \cfg{\state'}{\tpool'}} \begin{mathpar} \infer {\cfg{\state}{\expr} \step \cfg{\state'}{\expr'}} {\cfg{\state}{\tpool [i \mapsto \ectx[\expr]]} \step \cfg{\state'}{\tpool [i \mapsto \ectx[\expr']]}} \and \infer {} {\cfg{\state}{\tpool [i \mapsto \ectx[\fork{\expr}]]} \step \cfg{\state}{\tpool [i \mapsto \ectx[\textsf{fRet}]] [j \mapsto \expr]}} \end{mathpar} \section{Syntax} \subsection{Grammar}\label{sec:grammar} \paragraph{Signatures.} We use a signature to account syntactically for the logic's parameters. A \emph{signature} $\Sig = (\SigType, \SigFn)$ comprises a set $\SigType \supseteq \{ \textsort{Val}, \textsort{Exp}, \textsort{Ectx}, \textsort{State}, \textsort{Monoid}, \textsort{InvName}, \textsort{InvMask}, \Prop \}$ of base types (or base \emph{sorts}) and a set $\SigFn$ of typed function symbols. This means that each function symbol has an associated \emph{arity} comprising a natural number $n$ and an ordered list of $n+1$ base types. 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. \dave{Say something not-too-shabby about adequacy: We don't spell out what it means.} \paragraph{Syntax.} Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\textdom{Var}$ of variables (ranged over by metavariables $x$, $y$, $z$): \newcommand{\unitterm}{()}% \newcommand{\unitsort}{1}% \unit is bold. \begin{align*} \term, \prop, \pred ::={}& x \mid \sigfn(\term_1, \dots, \term_n) \mid \unitterm \mid (\term, \term) \mid \pi_i\; \term \mid \Lam x.\term \mid \term\;\term \mid \mzero \mid \munit \mid \term \mtimes \term \mid \\& \FALSE \mid \TRUE \mid \term =_\sort \term \mid \prop \Ra \prop \mid \prop \land \prop \mid \prop \lor \prop \mid \prop * \prop \mid \prop \wand \prop \mid \\& \MU \var. \pred \mid \Exists \var:\sort. \prop \mid \All \var:\sort. \prop \mid \\& \knowInv{\term}{\prop} \mid \ownGGhost{\term} \mid \ownPhys{\term} \mid \always\prop \mid {\later\prop} \mid \pvsA{\prop}{\term}{\term} \mid \dynA{\term}{\pred}{\term} \mid \timeless{\prop} \0.4em] \sort ::={}& \type \mid \unitsort \mid \sort \times \sort \mid \sort \to \sort \end{align*} Recursive predicates must be \emph{guarded}: in \MU \var. \pred, the variable \var can only appear under the later \later modality. \paragraph{Metavariable conventions.} We introduce additional metavariables ranging over terms and generally let the choice of metavariable indicate the term's sort: \[ \begin{array}{r|l} \text{metavariable} & \text{sort} \\\hline \term, \termB & \text{arbitrary} \\ \val, \valB & \textsort{Val} \\ \expr & \textsort{Exp} \\ \ectx & \textsort{Ectx} \\ \state & \textsort{State} \\ \end{array} \qquad\qquad \begin{array}{r|l} \text{metavariable} & \text{sort} \\\hline \iname & \textsort{InvName} \\ \mask & \textsort{InvMask} \\ \melt, \meltB & \textsort{Monoid} \\ \prop, \propB, \propC & \Prop \\ \pred, \predB, \predC & \sort\to\Prop \text{ (when \sort is clear from context)} \\ \end{array} \paragraph{Variable conventions.} We often abuse notation, using the preceding \emph{term} metavariables to range over (bound) \emph{variables}. We omit type annotations in binders, when the type is clear from context. \subsection{Types}\label{sec:types} Iris terms are simply-typed. The judgment $\vctx \proves_\Sig \wtt{\term}{\sort}$ expresses that, in signature $\Sig$ and variable context $\vctx$, the term $\term$ has sort $\sort$. In giving the rules for this judgment, we omit the signature (which does not change). A variable context, $\vctx = x_1:\sort_1, \dots, x_n:\sort_n$, declares a list of variables and their sorts. In writing $\vctx, x:\sort$, we presuppose that $x$ is not already declared in $\vctx$. \judgment{Well-typed terms}{\vctx \proves_\Sig \wtt{\term}{\sort}} \begin{mathparpagebreakable} %%% variables and function symbols \axiom{x : \sort \proves \wtt{x}{\sort}} \and \infer{\vctx \proves \wtt{\term}{\sort}} {\vctx, x:\sort' \proves \wtt{\term}{\sort}} \and \infer{\vctx, x:\sort', y:\sort' \proves \wtt{\term}{\sort}} {\vctx, x:\sort' \proves \wtt{\term[x/y]}{\sort}} \and \infer{\vctx_1, x:\sort', y:\sort'', \vctx_2 \proves \wtt{\term}{\sort}} {\vctx_1, x:\sort'', y:\sort', \vctx_2 \proves \wtt{\term[y/x,x/y]}{\sort}} \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{\unitterm}{\unitsort}} \and \infer{\vctx \proves \wtt{\term}{\sort_1} \and \vctx \proves \wtt{\termB}{\sort_2}} {\vctx \proves \wtt{(\term,\termB)}{\sort_1 \times \sort_2}} \and \infer{\vctx \proves \wtt{\term}{\sort_1 \times \sort_2} \and i \in \{1, 2\}} {\vctx \proves \wtt{\pi_i\,\term}{\sort_i}} %%% functions \and \infer{\vctx, x:\sort \proves \wtt{\term}{\sort'}} {\vctx \proves \wtt{\Lam x. \term}{\sort \to \sort'}} \and \infer {\vctx \proves \wtt{\term}{\sort \to \sort'} \and \wtt{\termB}{\sort}} {\vctx \proves \wtt{\term\;\termB}{\sort'}} %%% monoids \and \axiom{\vctx \proves \wtt{\mzero}{\textsort{Monoid}}} \and \axiom{\vctx \proves \wtt{\munit}{\textsort{Monoid}}} \and \infer{\vctx \proves \wtt{\melt}{\textsort{Monoid}} \and \vctx \proves \wtt{\meltB}{\textsort{Monoid}}} {\vctx \proves \wtt{\melt \mtimes \meltB}{\textsort{Monoid}}} %%% props and predicates \\ \axiom{\vctx \proves \wtt{\FALSE}{\Prop}} \and \axiom{\vctx \proves \wtt{\TRUE}{\Prop}} \and \infer{\vctx \proves \wtt{\term}{\sort} \and \vctx \proves \wtt{\termB}{\sort}} {\vctx \proves \wtt{\term =_\sort \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:\sort\to\Prop \proves \wtt{\pred}{\sort\to\Prop} \and \text{$\var$ is guarded in $\pred$} }{ \vctx \proves \wtt{\MU \var. \pred}{\sort\to\Prop} } \and \infer{\vctx, x:\sort \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\Exists x:\sort. \prop}{\Prop}} \and \infer{\vctx, x:\sort \proves \wtt{\prop}{\Prop}} {\vctx \proves \wtt{\All x:\sort. \prop}{\Prop}} \and \infer{ \vctx \proves \wtt{\prop}{\Prop} \and \vctx \proves \wtt{\iname}{\textsort{InvName}} }{ \vctx \proves \wtt{\knowInv{\iname}{\prop}}{\Prop} } \and \infer{\vctx \proves \wtt{\melt}{\textsort{Monoid}}} {\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} } \and \infer{ \vctx \proves \wtt{\prop}{\Prop} }{ \vctx \proves \wtt{\timeless{\prop}}{\Prop} } \end{mathparpagebreakable} \section{Base logic} 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.) % \subsubsection{Judgments} % % Proof rules implicitly assume well-sortedness. % e\subsection{Laws of intuitionistic higher-order logic with guarded recursion over a simply-typed lambda calculus}\label{sec:HOL} This is entirely standard. Soundness follows from the theorem that ${\cal U}(\any, \textdom{Prop}) : {\cal U}^{\textrm{op}} \to \textrm{Poset}$ is a hyperdoctrine. \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 \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[$\vee$E] {\pfctx \proves \prop \vee \propB \\ \pfctx, \prop \proves \propC \\ \pfctx, \propB \proves \propC} {\pfctx \proves \propC} \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[$\Ra$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[$\forall_1$I] {\pfctx, x : \sort \proves \prop} {\pfctx \proves \forall x: \sort.\; \prop} \and \infer[$\forall_1$E] {\pfctx \proves \forall X \in \sort.\; \prop \\ \pfctx \proves \term: \sort} {\pfctx \proves \prop[\term/X]} \and \infer[$\exists_1$E] {\pfctx \proves \exists X\in \sort.\; \prop \\ \pfctx, X : \sort, \prop \proves \propB} {\pfctx \proves \propB} \and \infer[$\exists_1$I] {\pfctx \proves \prop[\term/X] \\ \pfctx \proves \term: \sort} {\pfctx \proves \exists X: \sort. \prop} \and \infer[$\forall_2$I] {\pfctx, \var: \Pred(\sort) \proves \prop} {\pfctx \proves \forall \var\in \Pred(\sort).\; \prop} \and \infer[$\forall_2$E] {\pfctx \proves \forall \var. \prop \\ \pfctx \proves \propB: \Prop} {\pfctx \proves \prop[\propB/\var]} \and \infer[$\exists_2$E] {\pfctx \proves \exists \var \in \Pred(\sort).\prop \\ \pfctx, \var : \Pred(\sort), \prop \proves \propB} {\pfctx \proves \propB} \and \infer[$\exists_2$I] {\pfctx \proves \prop[\propB/\var] \\ \pfctx \proves \propB: \Prop} {\pfctx \proves \exists \var. \prop} \and \inferB[Elem] {\pfctx \proves \term \in (X \in \sort). \prop} {\pfctx \proves \prop[\term/X]} \and \inferB[Elem-$\mu$] {\pfctx \proves \term \in (\mu\var \in \Pred(\sort). \pred)} {\pfctx \proves \term \in \pred[\mu\var \in \Pred(\sort). \pred/\var]} \end{mathpar} \subsection{Axioms from the logic of (affine) bunched implications} \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} \subsection{Laws for ghosts and physical resources} \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} \subsection{Laws for the later modality}\label{sec:later} \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} \subsection{Laws for the always modality}\label{sec:always} \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 $\Ra$. \section{Program logic}\label{sec:proglog} Hoare triples and view shifts are syntactic sugar for weakest (liberal) preconditions and primitive view shifts, respectively: \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask] \eqdef \always{(\prop \Ra \dynA{\expr}{\lambda\Ret\val.\propB}{\mask})} \qquad\qquad \begin{aligned} \prop \vs[\mask_1][\mask_2] \propB &\eqdef \always{(\prop \Ra \pvsA{\propB}{\mask_1}{\mask_2})} \\ \prop \vsE[\mask_1][\mask_2] \propB &\eqdef \prop \vs[\mask_1][\mask_2] \propB \land \propB \vs[\mask2][\mask_1] \prop \end{aligned} We write just one mask for a view shift when $\mask_1 = \mask_2$. The convention for omitted masks is generous: An omitted $\mask$ is $\top$ for Hoare triples and $\emptyset$ for view shifts. % PDS: We're repeating ourselves. We gave Γ conventions and we're about to give Θ conventions. Also, the scope of "Below" is unclear. % Below, we implicitly assume the same context for all judgements which don't have an explicit context at \emph{all} pre-conditions \emph{and} the conclusion. Henceforward, we implicitly assume a proof context, $\pfctx$, is added to every constituent of the rules. Generally, this is an arbitrary proof context. We write $\provesalways$ to denote judgments that can only be extended with a boxed proof context. \ralf{Give the actual base rules from the Coq development instead} \subsection{Hoare triples} \begin{mathpar} \inferH{Ret} {} {\hoare{\TRUE}{\valB}{\Ret\val. \val = \valB}[\mask]} \and \inferH{Bind} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \\ \All \val. \hoare{\propB}{K[\val]}{\Ret\valB.\propC}[\mask]} {\hoare{\prop}{K[\expr]}{\Ret\valB.\propC}[\mask]} \and \inferH{Csq} {\prop \vs \prop' \\ \hoare{\prop'}{\expr}{\Ret\val.\propB'}[\mask] \\ \All \val. \propB' \vs \propB} {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} \and \inferH{Frame} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask]} {\hoare{\prop * \propC}{\expr}{\Ret\val. \propB * \propC}[\mask \uplus \mask']} \and \inferH{AFrame} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \and \text{$\expr$ not a value} } {\hoare{\prop * \later\propC}{\expr}{\Ret\val. \propB * \propC}[\mask \uplus \mask']} \and \inferH{Fork} {\hoare{\prop}{\expr}{\Ret\any. \TRUE}[\top]} {\hoare{\later\prop * \later\propB}{\fork{\expr}}{\Ret\val. \val = \textsf{fRet} \land \propB}[\mask]} \and \inferH{ACsq} {\prop \vs[\mask \uplus \mask'][\mask] \prop' \\ \hoare{\prop'}{\expr}{\Ret\val.\propB'}[\mask] \\ \All\val. \propB' \vs[\mask][\mask \uplus \mask'] \propB \\ \physatomic{\expr} } {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask \uplus \mask']} \end{mathpar} \subsection{View shifts} \begin{mathpar} \inferH{NewInv} {\infinite(\mask)} {\later{\prop} \vs[\mask] \exists \iname\in\mask.\; \knowInv{\iname}{\prop}} \and \inferH{FpUpd} {\melt \mupd \meltsB} {\ownGGhost{\melt} \vs \exists \meltB \in \meltsB.\; \ownGGhost{\meltB}} \and \inferH{VSTrans} {\prop \vs[\mask_1][\mask_2] \propB \and \propB \vs[\mask_2][\mask_3] \propC \and \mask_2 \subseteq \mask_1 \cup \mask_3} {\prop \vs[\mask_1][\mask_3] \propC} \and \inferH{VSImp} {\always{(\prop \Ra \propB)}} {\prop \vs[\emptyset] \propB} \and \inferH{VSFrame} {\prop \vs[\mask_1][\mask_2] \propB} {\prop * \propC \vs[\mask_1 \uplus \mask'][\mask_2 \uplus \mask'] \propB * \propC} \and \inferH{VSTimeless} {\timeless{\prop}} {\later \prop \vs \prop} \and \axiomH{InvOpen} {\knowInv{\iname}{\prop} \proves \TRUE \vs[\{ \iname \} ][\emptyset] \later \prop} \and \axiomH{InvClose} {\knowInv{\iname}{\prop} \proves \later \prop \vs[\emptyset][\{ \iname \} ] \TRUE } \end{mathpar} \vspace{5pt} Note that $\timeless{\prop}$ means that $\prop$ does not depend on the step index. Furthermore, $$\melt \mupd \meltsB \eqdef \always{\All \melt_f. \melt \sep \melt_f \Ra \Exists \meltB \in \meltsB. \meltB \sep \melt_f}$$ \subsection{Derived rules} \paragraph{Derived structural rules.} The following are easily derived by unfolding the sugar for Hoare triples and view shifts. \begin{mathpar} \inferHB{Disj} {\hoare{\prop}{\expr}{\Ret\val.\propC}[\mask] \and \hoare{\propB}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \lor \propB}{\expr}{\Ret\val.\propC}[\mask]} \and \inferHB{VSDisj} {\prop \vs[\mask_1][\mask_2] \propC \and \propB \vs[\mask_1][\mask_2] \propC} {\prop \lor \propB \vs[\mask_1][\mask_2] \propC} \and \inferHB{Exist} {\All \var. \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} {\hoare{\Exists \var. \prop}{\expr}{\Ret\val.\propB}[\mask]} \and \inferHB{VSExist} {\All \var. (\prop \vs[\mask_1][\mask_2] \propB)} {(\Exists \var. \prop) \vs[\mask_1][\mask_2] \propB} \and \inferHB{BoxOut} {\always\propB \provesalways \hoare{\prop}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \land \always{\propB}}{\expr}{\Ret\val.\propC}[\mask]} \and \inferHB{VSBoxOut} {\always\propB \provesalways \prop \vs[\mask_1][\mask_2] \propC} {\prop \land \always{\propB} \vs[\mask_1][\mask_2] \propC} \and \inferH{False} {} {\hoare{\FALSE}{\expr}{\Ret \val. \prop}[\mask]} \and \inferH{VSFalse} {} {\FALSE \vs[\mask_1][\mask_2] \prop } \end{mathpar} The proofs all follow the same pattern, so we only show two of them in detail. \begin{proof}[Proof of \ruleref{Exist}] After unfolding the syntactic sugar for Hoare triples and removing the boxes from premise and conclusion, our goal becomes $(\Exists \var. \prop(\var)) \Ra \dynA{\expr}{\Lam\val. \propB}{\mask}$ (remember that $\var$ is free in $\prop$) and the premise reads $\All \var. \prop(\var) \Ra \dynA{\expr}{\Lam\val. \propB}{\mask}.$ Let $\var$ be given and assume $\prop(\var)$. To show $\dynA{\expr}{\Lam\val. \propB}{\mask}$, apply the premise to $\var$ and $\prop(\var)$. For the other direction, assume $\hoare{\Exists \var. \prop(\var)}{\expr}{\Ret\val. \propB}[\mask]$ and let $\var$ be given. We have to show $\hoare{\prop(\var)}{\expr}{\Ret\val. \propB}[\mask]$. This trivially follows from \ruleref{Csq} with $\prop(\var) \Ra \Exists \var. \prop(\var)$. \end{proof} \begin{proof}[Proof of \ruleref{BoxOut}] After unfolding the syntactic sugar for Hoare triples, our goal becomes $$\label{eq:boxin:goal} \always\pfctx \proves \always\bigl(\prop\land\always \propB \Ra \dynA{\expr}{\Lam\val. \propC}{\mask}\bigr)$$ while our premise reads $$\label{eq:boxin:as} \always\pfctx, \always\propB \proves \always(\prop \Ra \dynA{\expr}{\Lam\val. \propC}{\mask})$$ By the introduction rules for $\always$ and implication, it suffices to show $(\always\pfctx), \prop,\always \propB \proves \dynA{\expr}{\Lam\val. \propC}{\mask}$ By modus ponens and \ruleref{Necessity}, it suffices to show~\eqref{eq:boxin:as}, which is exactly our assumption. For the other direction, assume~\eqref{eq:boxin:goal}. We have to show~\eqref{eq:boxin:as}. By \ruleref{AlwaysIntro} and implication introduction, it suffices to show $(\always\pfctx), \prop,\always \propB \proves \dynA{\expr}{\Lam\val. \propC}{\mask}$ which easily follows from~\eqref{eq:boxin:goal}. \end{proof} \paragraph{Derived rules for invariants.} Invariants can be opened around atomic expressions and view shifts. \begin{mathpar} \inferH{Inv} {\hoare{\later{\propC} * \prop } {\expr} {\Ret\val. \later{\propC} * \propB }[\mask] \and \physatomic{\expr} } {\knowInv{\iname}{\propC} \proves \hoare{\prop} {\expr} {\Ret\val. \propB}[\mask \uplus \{ \iname \}] } \and \inferH{VSInv} {\later{\prop} * \propB \vs[\mask_1][\mask_2] \later{\prop} * \propC} {\knowInv{\iname}{\prop} \proves \propB \vs[\mask_1 \uplus \{ \iname \}][\mask_2 \uplus \{ \iname \}] \propC} \end{mathpar} \begin{proof}[Proof of \ruleref{Inv}] Use \ruleref{ACsq} with $\mask_1 \eqdef \mask \cup \{\iname\}$, $\mask_2 \eqdef \mask$. The view shifts are obtained by \ruleref{InvOpen} and \ruleref{InvClose} with framing of $\mask$ and $\prop$ or $\propB$, respectively. \end{proof} \begin{proof}[Proof of \ruleref{VSInv}] Analogous to the proof of \ruleref{Inv}, using \ruleref{VSTrans} instead of \ruleref{ACsq}. \end{proof} \subsubsection{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. \subsection{Adequacy} The adequacy statement reads as follows: \begin{align*} &\All \mask, \expr, \val, \pred, i, \state, \state', \tpool'. \\&( \proves \hoare{\ownPhys\state}{\expr}{x.\; \pred(x)}[\mask]) \implies \\&\cfg{\state}{[i \mapsto \expr]} \step^\ast \cfg{\state'}{[i \mapsto \val] \uplus \tpool'} \implies \\&\pred(\val) \end{align*} where $\pred$ can mention neither resources nor invariants. \subsection{Axiom lifting}\label{sec:lifting} The following lemmas help in proving axioms for a particular language. The first applies to expressions with side-effects, and the second to side-effect-free expressions. \dave{Update the others, and the example, wrt the new treatment of $\predB$.} \begin{align*} &\All \expr, \state, \pred, \prop, \propB, \mask. \\ &\textlog{reducible}(e) \implies \\ &(\All \expr', \state'. \cfg{\state}{\expr} \step \cfg{\state'}{\expr'} \implies \pred(\expr', \state')) \implies \\ &{} \proves \bigl( (\All \expr', \state'. \pred (\expr', \state') \Ra \hoare{\prop}{\expr'}{\Ret\val. \propB}[\mask]) \Ra \hoare{ \later \prop * \ownPhys{\state} }{\expr}{\Ret\val. \propB}[\mask] \bigr) \\ \quad\\ &\All \expr, \pred, \prop, \propB, \mask. \\ &\textlog{reducible}(e) \implies \\ &(\All \state, \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \state_2 = \state \land \pred(\expr_2)) \implies \\ &{} \proves \bigl( (\All \expr'. \pred(\expr') \Ra \hoare{\prop}{\expr'}{\Ret\val. \propB}[\mask]) \Ra \hoare{\later\prop}{\expr}{\Ret\val. \propB}[\mask] \bigr) \end{align*} Note that $\pred$ is a meta-logic predicate---it does not depend on any world or resources being owned. The following specializations cover all cases of a heap-manipulating lambda calculus like $F_{\mu!}$. \begin{align*} &\All \expr, \expr', \prop, \propB, \mask. \\ &\textlog{reducible}(e) \implies \\ &(\All \state, \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \state_2 = \state \land \expr_2 = \expr') \implies \\ &{} \proves (\hoare{\prop}{\expr'}{\Ret\val. \propB}[\mask] \Ra \hoare{\later\prop}{\expr}{\Ret\val. \propB}[\mask] ) \\ \quad \\ &\All \expr, \state, \pred, \mask. \\ &\textlog{atomic}(e) \implies \\ &\bigl(\All \expr_2, \state_2. \cfg{\state}{\expr} \step \cfg{\state_2}{\expr_2} \implies \pred(\expr_2, \state_2)\bigr) \implies \\ &{} \proves (\hoare{ \ownPhys{\state} }{\expr}{\Ret\val. \Exists\state'. \ownPhys{\state'} \land \pred(\val, \state') }[\mask] ) \end{align*} The first is restricted to deterministic pure reductions, like $\beta$-reduction. The second is suited to proving triples for (possibly non-deterministic) atomic expressions; for example, with $\expr \eqdef \;!\ell$ (dereferencing $\ell$) and $\state \eqdef h \mtimes \ell \mapsto \valB$ and $\pred(\val, \state') \eqdef \state' = (h \mtimes \ell \mapsto \valB) \land \val = \valB$, one obtains the axiom $\All h, \ell, \valB. \hoare{\ownPhys{h \mtimes \ell \mapsto \valB}}{!\ell}{\Ret\val. \val = \valB \land \ownPhys{h \mtimes \ell \mapsto \valB} }$. %Axioms for CAS-like operations can be obtained by first deriving rules for the two possible cases, and then using the disjunction rule. %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: