Forked from
Iris / Iris
8071 commits behind the upstream repository.
logic.tex 29.34 KiB
\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
\begin{equation}\label{eq:boxin:goal}
\always\pfctx \proves \always\bigl(\prop\land\always \propB \Ra \dynA{\expr}{\Lam\val. \propC}{\mask}\bigr)
\end{equation}
while our premise reads
\begin{equation}\label{eq:boxin:as}
\always\pfctx, \always\propB \proves \always(\prop \Ra \dynA{\expr}{\Lam\val. \propC}{\mask})
\end{equation}
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: