Commit 7cb94e3e by Ralf Jung

### docs: complete model description

parent 82aee390
Pipeline #331 passed with stage
 ... ... @@ -23,6 +23,8 @@ This definition varies slightly from the original one in~\cite{catlogic}. An element $x \in \cofe$ of a COFE is called \emph{discrete} if $\All y \in \cofe. x \nequiv{0} y \Ra x = y$ A COFE $A$ is called \emph{discrete} if all its elements are discrete. For a set $X$, we write $\Delta X$ for the discrete COFE with $x \nequiv{n} x' \eqdef x = x'$ \end{defn} \begin{defn} ... ... @@ -31,6 +33,7 @@ This definition varies slightly from the original one in~\cite{catlogic}. It is \emph{contractive} if $\All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x)$ \end{defn} The reason that contractive functions are interesting is that for every contractive $f : \cofe \to \cofe$ with $\cofe$ inhabited, there exists a fixed-point $\fix(f)$ such that $\fix(f) = f(\fix(f))$. \begin{defn} The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows. ... ...
 ... ... @@ -33,7 +33,7 @@ We start by defining the COFE of \emph{step-indexed propositions}: For every ste \end{align*} Now we can rewrite $\UPred(\monoid)$ as monotone step-indexed predicates over $\monoid$, where the definition of a monotone'' function here is a little funny. \begin{align*} \UPred(\monoid) \approx{}& \monoid \monra \SProp \\ \UPred(\monoid) \cong{}& \monoid \monra \SProp \\ \eqdef{}& \setComp{\pred: \monoid \nfn \SProp}{\All n, m, x, y. n \in \pred(x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra m \in \pred(y)} \end{align*} The reason we chose the first definition is that it is easier to work with in Coq. ... ... @@ -77,35 +77,35 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$. \subsection{Agreement} Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows: \newcommand{\agc}{\mathrm{c}} % the "c" field of an agreement element \newcommand{\agV}{\mathrm{V}} % the "V" field of an agreement element \newcommand{\aginjc}{\mathrm{c}} % the "c" field of an agreement element \newcommand{\aginjV}{\mathrm{V}} % the "V" field of an agreement element \begin{align*} \agm(\cofe) \eqdef{}& \record{\agc : \mathbb{N} \to \cofe , \agV : \SProp} \\ \agm(\cofe) \eqdef{}& \record{\aginjc : \mathbb{N} \to \cofe , \aginjV : \SProp} \\ & \text{quotiented by} \\ \melt \equiv \meltB \eqdef{}& \melt.\agV = \meltB.\agV \land \All n. n \in \melt.\agV \Ra \melt.\agc(n) \nequiv{n} \meltB.\agc(n) \\ \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\agV \Lra m \in \meltB.\agV) \land (\All m \leq n. m \in \melt.\agV \Ra \melt.\agc(m) \nequiv{m} \meltB.\agc(m)) \\ \mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\agV \land \All m \leq n. \melt.\agc(n) \nequiv{m} \melt.\agc(m) } \\ \melt \equiv \meltB \eqdef{}& \melt.\aginjV = \meltB.\aginjV \land \All n. n \in \melt.\aginjV \Ra \melt.\aginjc(n) \nequiv{n} \meltB.\aginjc(n) \\ \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\aginjV \Lra m \in \meltB.\aginjV) \land (\All m \leq n. m \in \melt.\aginjV \Ra \melt.\aginjc(m) \nequiv{m} \meltB.\aginjc(m)) \\ \mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\aginjV \land \All m \leq n. \melt.\aginjc(n) \nequiv{m} \melt.\aginjc(m) } \\ \mcore\melt \eqdef{}& \melt \\ \melt \mtimes \meltB \eqdef{}& (\melt.\agc, \setComp{n}{n \in \melt.\agV \land n \in \meltB.\agV \land \melt \nequiv{n} \meltB }) \melt \mtimes \meltB \eqdef{}& (\melt.\aginjc, \setComp{n}{n \in \melt.\aginjV \land n \in \meltB.\aginjV \land \melt \nequiv{n} \meltB }) \end{align*} $\agm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$. You can think of the $\agc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \agV$ steps. You can think of the $\aginjc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \aginjV$ steps. The reason we store a chain, rather than a single element, is that $\agm(\cofe)$ needs to be a COFE itself, so we need to be able to give a limit for every chain of $\agm(\cofe)$. However, given such a chain, we cannot constructively define its limit: Clearly, the $\agV$ of the limit is the limit of the $\agV$ of the chain. However, given such a chain, we cannot constructively define its limit: Clearly, the $\aginjV$ of the limit is the limit of the $\aginjV$ of the chain. But what to pick for the actual data, for the element of $\cofe$? Only if $\agV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\agV$ is smaller, the chain cancels'', \ie stops converging as we reach indices $n \notin \agV$. Only if $\aginjV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\aginjV$ is smaller, the chain cancels'', \ie stops converging as we reach indices $n \notin \aginjV$. To mitigate this, we apply the usual construction to close a set; we go from elements of $\cofe$ to chains of $\cofe$. We define an injection $\ag$ into $\agm(\cofe)$ as follows: $\ag(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}}$ We define an injection $\aginj$ into $\agm(\cofe)$ as follows: $\aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}}$ There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar} \axiomH{ag-val}{\ag(x) \in \mval_n} \axiomH{ag-val}{\aginj(x) \in \mval_n} \axiomH{ag-dup}{\ag(x) = \ag(x)\mtimes\ag(x)} \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)} \axiomH{ag-agree}{\ag(x) \mtimes \ag(y) \in \mval_n \Ra x \nequiv{n} y} \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y} \end{mathpar} \subsection{One-shot} ... ... @@ -115,17 +115,17 @@ Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows: \begin{align*} \oneshotm(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\ \mval_n \eqdef{}& \set{\ospending, \munit} \cup \setComp{\osshot(\melt)}{\melt \in \mval_n} \end{align*} \begin{align*} \mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\ \mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot \end{align*} \begin{align*} \\%\end{align*} %\begin{align*} \osshot(\melt) \mtimes \osshot(\meltB) \eqdef{}& \osshot(\melt \mtimes \meltB) \\ \munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\ \munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt) \end{align*} \end{align*}% The remaining cases of composition go to $\bot$. \begin{align*} \mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\ \mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot \end{align*} The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \axiom{\ospending \nequiv{n} \ospending} ... ... @@ -149,34 +149,38 @@ We obtain the following frame-preserving updates: {\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}} \end{mathpar} %TODO: These need syncing with Coq % \subsection{Exclusive monoid} \subsection{Exclusive CMRA} % Given a set $X$, we define a monoid such that at most one $x \in X$ can be owned. % Let $\exm{X}$ be the monoid with carrier $X \uplus \{ \munit \}$ and multiplication % $% \melt \cdot \meltB \;\eqdef\; % \begin{cases} % \melt & \mbox{if } \meltB = \munit \\ % \meltB & \mbox{if } \melt = \munit % \end{cases} %$ Given a cofe $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned: \begin{align*} \exm(\cofe) \eqdef{}& \exinj(\cofe) + \munit + \bot \\ \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} \\ \munit \mtimes \exinj(x) \eqdef{}& \exinj(x) \mtimes \munit \eqdef \exinj(x) \end{align*} The remaining cases of composition go to $\bot$. \begin{align*} \mcore{\exinj(x)} \eqdef{}& \munit & \mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot \end{align*} The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)} % The frame-preserving update % \begin{mathpar} % \inferH{ExUpd} % {x \in X} % {x \mupd \melt} % \end{mathpar} % is easily shown, as the only possible frame for $x$ is $\munit$. \axiom{\munit \nequiv{n} \munit} % Exclusive monoids are cancellative. % \begin{proof}[Proof of cancellativity] % If $\melt_f = \munit$, then the statement is trivial. % If $\melt_f \neq \munit$, then we must have $\melt = \meltB = \munit$, as otherwise one of the two products would be $\mzero$. % \end{proof} \axiom{\bot \nequiv{n} \bot} \end{mathpar} $\exm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$. We obtain the following frame-preserving update: \begin{mathpar} \inferH{ex-update}{} {\exinj(x) \mupd \exinj(y)} \end{mathpar} %TODO: These need syncing with Coq % \subsection{Finite Powerset Monoid} % Given an infinite set $X$, we define a monoid $\textmon{PowFin}$ with carrier $\mathcal{P}^{\textrm{fin}}(X)$ as follows: ... ...
 ... ... @@ -86,13 +86,15 @@ \newcommand{\rs}{r} \newcommand{\rsB}{s} \newcommand{\rss}{R} \newcommand{\pres}{\pi} \newcommand{\wld}{w} \newcommand{\ghostRes}{g} %% Various pieces of syntax \newcommand{\wsat}[4]{#1 \models_{#2} #3; #4} \newcommand{\wsat}[3]{#1 \models_{#2} #3} \newcommand{\wsatpre}{\textdom{pre-wsat}} \newcommand{\wtt}[2]{#1 : #2} % well-typed term ... ... @@ -114,6 +116,7 @@ \newcommand{\UPred}{\textdom{UPred}} \newcommand{\mProp}{\textdom{Prop}} % meta-level prop \newcommand{\iProp}{\textdom{iProp}} \newcommand{\iPreProp}{\textdom{iPreProp}} \newcommand{\Wld}{\textdom{Wld}} \newcommand{\Res}{\textdom{Res}} ... ... @@ -121,6 +124,7 @@ \newcommand{\cofeB}{U} \newcommand{\COFEs}{\mathcal{U}} % category of COFEs \newcommand{\iFunc}{\Sigma} \newcommand{\fix}{\textdom{fix}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % CMRA (RESOURCE ALGEBRA) SYMBOLS & NOTATION & IDENTIFIERS ... ... @@ -136,6 +140,8 @@ \newcommand{\melts}{A} \newcommand{\meltsB}{B} \newcommand{\f}{\mathrm{f}} % for "frame" \newcommand{\mcar}[1]{|#1|} \newcommand{\mcarp}[1]{\mcar{#1}^{+}} \newcommand{\munit}{\varepsilon} ... ... @@ -321,13 +327,14 @@ % Agreement \newcommand{\agm}{\ensuremath{\textmon{Ag}}} \newcommand{\ag}{\textlog{ag}} \newcommand{\aginj}{\textlog{ag}} % Fraction \newcommand{\fracm}{\ensuremath{\textmon{Frac}}} % Exclusive \newcommand{\exm}{\ensuremath{\textmon{Ex}}} \newcommand{\exinj}{\textlog{ex}} % Auth \newcommand{\authm}{\textmon{Auth}} ... ...
 ... ... @@ -124,7 +124,7 @@ Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\t \prop * \prop \mid \prop \wand \prop \mid \\& \MU \var:\type. \pred \mid \MU \var:\type. \term \mid \Exists \var:\type. \prop \mid \All \var:\type. \prop \mid \\& ... ... @@ -136,7 +136,7 @@ Iris syntax is built up from a signature$\Sig$and a countably infinite set$\t \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. Recursive predicates must be \emph{guarded}: in $\MU \var. \term$, 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 $\pvs[\term][\term] \prop$. ... ...
 ... ... @@ -3,10 +3,11 @@ The semantics closely follows the ideas laid out in~\cite{catlogic}. \subsection{Generic model of base logic} \label{sec:upred-logic} The base logic including equality, later, always, and a notion of ownership is defined on $\UPred(\monoid)$ for any CMRA $\monoid$. \typedsection{Interpretation of base assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \to \UPred(\monoid)} \typedsection{Interpretation of base assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \nfn \UPred(\monoid)} Remember that $\UPred(\monoid)$ is isomorphic to $\monoid \monra \SProp$. We are thus going to define the assertions as mapping CMRA elements to sets of step-indices. ... ... @@ -42,410 +43,165 @@ We introduce an additional logical connective $\ownM\melt$, which will later be \Sem{\vctx \proves \mval(\melt) : \Prop}_\gamma &\eqdef \Lam\any. \setComp{n}{\melt \in \mval_n} \\ \end{align*} For every definition, we have to show all the side-conditions: The maps have to be non-expansive and monotone. %\subsection{Iris model} % \subsection{Semantic structures: propositions} % \ralf{This needs to be synced with the Coq development again.} \subsection{Iris model} % $% \begin{array}[t]{rcl} % % \protStatus &::=& \enabled \ALT \disabled \\[0.4em] % \textdom{Res} &\eqdef& % \{\, \rs = (\pres, \ghostRes) \mid % \pres \in \textdom{State} \uplus \{\munit\} \land \ghostRes \in \mcarp{\monoid} \,\} \\[0.5em] % (\pres, \ghostRes) \rtimes % (\pres', \ghostRes') &\eqdef& % \begin{cases} % (\pres, \ghostRes \mtimes \ghostRes') & \mbox{if \pres' = \munit and \ghostRes \mtimes \ghostRes' \neq \mzero} \\ % (\pres', \ghostRes \mtimes \ghostRes') & \mbox{if \pres = \munit and \ghostRes \mtimes \ghostRes' \neq \mzero} % \end{cases} % \\[0.5em] % % % \rs \leq \rs' & \eqdef & % \Exists \rs''. \rs' = \rs \rtimes \rs''\\[1em] % % % \UPred(\textdom{Res}) &\eqdef& % \{\, p \subseteq \mathbb{N} \times \textdom{Res} \mid % \All (k,\rs) \in p. % \All j\leq k. % \All \rs' \geq \rs. % (j,\rs')\in p \,\}\\[0.5em] % \restr{p}{k} &\eqdef& % \{\, (j, \rs) \in p \mid j < k \,\}\\[0.5em] % p \nequiv{n} q & \eqdef & \restr{p}{n} = \restr{q}{n}\\[1em] % % % \textdom{PreProp} & \cong & % \latert\big( \textdom{World} \monra \UPred(\textdom{Res}) % \big)\\[0.5em] % % % \textdom{World} & \eqdef & % \mathbb{N} \fpfn \textdom{PreProp}\\[0.5em] % % % w \nequiv{n} w' & \eqdef & % n = 0 \lor % \bigl(\dom(w) = \dom(w') \land \All i\in\dom(w). w(i) \nequiv{n} w'(i)\bigr) % \\[0.5em] % % % w \leq w' & \eqdef & % \dom(w) \subseteq \dom(w') \land \All i \in \dom(w). w(i) = w'(i) % \\[0.5em] % % % \textdom{Prop} & \eqdef & \textdom{World} \monra \UPred(\textdom{Res}) % \end{array} %$ % For $p,q\in\UPred(\textdom{Res})$ with $p \nequiv{n} q$ defined % as above, $\UPred(\textdom{Res})$ is a % c.o.f.e. % $\textdom{Prop}$ is a c.o.f.e., which exists by America and Rutten's theorem~\cite{America-Rutten:JCSS89}. % We do not need to consider how the object is constructed. % We only need the isomorphism, given by maps % \begin{align*} % \wIso &: \latert \bigl(World \monra \UPred(\textdom{Res})\bigr) \to \textdom{PreProp} \\ % \wIso^{-1} &: \textdom{PreProp} \to \latert \bigl(World \monra \UPred(\textdom{Res})\bigr) % \end{align*} % which are inverses to each other. % Note: this is an isomorphism in $\cal U$, i.e., $\wIso$ and % $\wIso^{-1}$ are both non-expansive. % $\textdom{World}$ is a c.o.f.e.\ with the family of equivalence % relations defined as shown above. % \subsection{Semantic structures: types and environments} % For a set $X$, write $\Delta X$ for the discrete c.o.f.e.\ with $x \nequiv{n} % x'$ iff $n = 0$ or $x = x'$ % $% \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} % \Sem{\textsort{Unit}} &\eqdef& \Delta \{ \star \} \\ % \Sem{\textsort{InvName}} &\eqdef& \Delta \mathbb{N} \\ % \Sem{\textsort{InvMask}} &\eqdef& \Delta \pset{\mathbb{N}} \\ % \Sem{\textsort{Monoid}} &\eqdef& \Delta |\monoid| % \end{array} % \qquad\qquad % \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} % \Sem{\textsort{Val}} &\eqdef& \Delta \textdom{Val} \\ % \Sem{\textsort{Exp}} &\eqdef& \Delta \textdom{Exp} \\ % \Sem{\textsort{Ectx}} &\eqdef& \Delta \textdom{Ectx} \\ % \Sem{\textsort{State}} &\eqdef& \Delta \textdom{State} \\ % \end{array} % \qquad\qquad % \begin{array}[t]{@{}l@{\ }c@{\ }l@{}} % \Sem{\sort \times \sort'} &\eqdef& \Sem{\sort} \times \Sem{\sort} \\ % \Sem{\sort \to \sort'} &\eqdef& \Sem{\sort} \to \Sem{\sort} \\ % \Sem{\Prop} &\eqdef& \textdom{Prop} \\ % \end{array} %$ % The balance of our signature $\Sig$ is interpreted as follows. % For each base type $\type$ not covered by the preceding table, we pick an object $X_\type$ in $\cal U$ and define % $% \Sem{\type} \eqdef X_\type %$ % For each function symbol $\sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn$, we pick an arrow $\Sem{\sigfn} : \Sem{\type_1} \times \dots \times \Sem{\type_n} \to \Sem{\type_{n+1}}$ in $\cal U$. % An environment $\vctx$ is interpreted as the set of % maps $\rho$, with $\dom(\rho) = \dom(\vctx)$ and % $\rho(x)\in\Sem{\vctx(x)}$, % and % $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land % \All x\in\dom(\rho). \rho(x) \nequiv{n} \rho'(x)\bigr)$. % \ralf{Re-check all the following definitions with the Coq development.} % %\typedsection{Validity}{valid : \pset{\textdom{Prop}} \in Sets} % % % %\begin{align*} % %valid(p) &\iff \All n \in \mathbb{N}. \All \rs \in \textdom{Res}. \All W \in \textdom{World}. (n, \rs) \in p(W) % %\end{align*} % \typedsection{Later modality}{\later : \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \later p &\eqdef \Lam W. \{\, (n + 1, r) \mid (n, r) \in p(W) \,\} \cup \{\, (0, r) \mid r \in \textdom{Res} \,\} % \end{align*} % \begin{lem} % $\later{}$ is well-defined: $\later {p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\later{}$ itself is a \emph{contractive} map. % \end{lem} % \typedsection{Always modality}{\always{} : \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \always{p} \eqdef \Lam W. \{\, (n, r) \mid (n, \munit) \in p(W) \,\} % \end{align*} % \begin{lem} % $\always{}$ is well-defined: $\always{p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\always{}$ itself is a non-expansive map. % \end{lem} % % PDS: p \Rightarrow q not defined. % %\begin{lem}\label{lem:always-impl-valid} % %\begin{align*} % %&\forall p, q \in \textdom{Prop}.~\\ % %&\qquad % % (\forall n \in \mathbb{N}.~\forall \rs \in \textdom{Res}.~\forall W \in \textdom{World}.~(n, \rs) \in p(W) \Rightarrow (n, \rs) \in q(W)) \Leftrightarrow~valid(\always{(p \Rightarrow q)}) % %\end{align*} % %\end{lem} % \typedsection{Invariant definition}{inv : \Delta(\mathbb{N}) \times \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \mathit{inv}(\iota, p) &\eqdef \Lam W. \{\, (n, r) \mid \iota\in\dom(W) \land W(\iota) \nequiv{n+1}_{\textdom{PreProp}} \wIso(p) \,\} % \end{align*} % \begin{lem} % $\mathit{inv}$ is well-defined: $\mathit{inv}(\iota, p)$ is a valid proposition (this amounts to showing non-expansiveness), and $\mathit{inv}$ itself is a non-expansive map. % \end{lem} % \typedsection{World satisfaction}{\wsat{-}{-}{-}{-} : % \textdom{State} \times % \pset{\mathbb{N}} \times % \textdom{Res} \times % \textdom{World} \to \psetdown{\mathbb{N}} \in {\cal U}} % \ralf{Make this Dave-compatible: Explicitly compose all the things in $s$} % \begin{align*} % \wsat{\state}{\mask}{\rs}{W} &= % \begin{aligned}[t] % \{\, n + 1 \in \mathbb{N} \mid &\Exists \rsB:\mathbb{N} \fpfn \textdom{Res}. (\rs \rtimes \rsB).\pres = \state \land{}\\ % &\quad \All \iota \in \dom(W). \iota \in \dom(W) \leftrightarrow \iota \in \dom(\rsB) \land {}\\ % &\quad\quad \iota \in \mask \ra (n, \rsB(\iota)) \in \wIso^{-1}(W(\iota))(W) \,\} \cup \{ 0 \} % \end{aligned} % \end{align*} % \begin{lem}\label{lem:fullsat-nonexpansive} % $\wsat{-}{-}{-}{-}$ is well-defined: It maps into $\psetdown{\mathbb{N}}$. (There is no need for it to be a non-expansive map, it doesn't itself live in $\cal U$.) % \end{lem} % \begin{lem}\label{lem:fullsat-weaken-mask} % \begin{align*} % \MoveEqLeft % \All \state \in \Delta(\textdom{State}). % \All \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}). % \All \rs, \rsB \in \Delta(\textdom{Res}). % \All W \in \textdom{World}. \\& % \mask_1 \subseteq \mask_2 \implies (\wsat{\state}{\mask_2}{\rs}{W}) \subseteq (\wsat{\state}{\mask_1}{\rs}{W}) % \end{align*} % \end{lem} % \begin{lem}\label{lem:nequal_ext_world} % \begin{align*} % & % \All n \in \mathbb{N}. % \All W_1, W_1', W_2 \in \textdom{World}. % W_1 \nequiv{n} W_2 \land W_1 \leq W_1' \implies \Exists W_2' \in \textdom{World}. W_1' \nequiv{n} W_2' \land W_2 \leq W_2' % \end{align*} % \end{lem} % \typedsection{Timeless}{\textit{timeless} : \textdom{Prop} \to \textdom{Prop}} % \begin{align*} % \textit{timeless}(p) \eqdef % \begin{aligned}[t] % \Lam W. % \{\, (n, r) &\mid \All W' \geq W. \All k \leq n. \All r' \in \textdom{Res}. \\ % &\qquad % k > 0 \land (k - 1, r') \in p(W') \implies (k, r') \in p(W') \,\} % \end{aligned} % \end{align*} % \begin{lem} % \textit{timeless} is well-defined: \textit{timeless}(p) is a valid proposition, and \textit{timeless} itself is a non-expansive map. % \end{lem} % % PDS: \Ra undefined. % %\begin{lem} % %\begin{align*} % %& % % \All p \in \textdom{Prop}. % % \All \mask \in \pset{\mathbb{N}}. % %valid(\textit{timeless}(p) \Ra (\later p \vs[\mask][\mask] p)) % %\end{align*} % %\end{lem} % \typedsection{View-shift}{\mathit{vs} : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \textdom{Prop} \to \textdom{Prop} \in {\cal U}} % \begin{align*} % \mathit{vs}_{\mask_1}^{\mask_2}(q) &= \Lam W. % \begin{aligned}[t] % \{\, (n, \rs) &\mid \All W_F \geq W. \All \rs_F, \mask_F, \state. \All k \leq n.\\ % &\qquad % k \in (\wsat{\state}{\mask_1 \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \land k > 0 \land \mask_F \sep (\mask_1 \cup \mask_2) \implies{} \\ % &\qquad % \Exists W' \geq W_F. \Exists \rs'. k \in (\wsat{\state}{\mask_2 \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(W') % \,\} % \end{aligned} % \end{align*} % \begin{lem} % $\mathit{vs}$ is well-defined: $\mathit{vs}_{\mask_1}^{\mask_2}(q)$ is a valid proposition, and $\mathit{vs}$ is a non-expansive map. % \end{lem} % %\begin{lem}\label{lem:prim_view_shift_trans} % %\begin{align*} % %\MoveEqLeft % % \All \mask_1, \mask_2, \mask_3 \in \Delta(\pset{\mathbb{N}}). % % \All p, q \in \textdom{Prop}. \All W \in \textdom{World}. % % \All n \in \mathbb{N}.\\ % %& % % \mask_2 \subseteq \mask_1 \cup \mask_3 \land % % \bigl(\All W' \geq W. \All r \in \textdom{Res}. \All k \leq n. (k, r) \in p(W') \implies (k, r) \in vs_{\mask_2}^{\mask_3}(q)(W')\bigr) \\ % %&\qquad % % {}\implies \All r \in \textdom{Res}. (n, r) \in vs_{\mask_1}^{\mask_2}(p)(W) \implies (n, r) \in vs_{\mask_1}^{\mask_3}(q)(W) % %\end{align*} % %\end{lem} % % PDS: E_1 ==>> E_2 undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall \mask_1, \mask_2, \mask_3 \in \Delta(\pset{\mathbb{N}}).~ % % \forall p_1, p_2, p_3 \in \textdom{Prop}.~\\ % %&\qquad % % \mask_2 \subseteq \mask_1 \cup \mask_3 \Rightarrow % % valid(((p_1 \vs[\mask_1][\mask_2] p_2) \land (p_2 \vs[\mask_2][\mask_3] p_3)) \Rightarrow (p_1 \vs[\mask_1][\mask_3] p_3)) % %\end{align*} % %\end{lem} % %\begin{lem} % %\begin{align*} % %\MoveEqLeft % % \All \iota \in \mathbb{N}. % % \All p \in \textdom{Prop}. % % \All W \in \textdom{World}. % % \All \rs \in \textdom{Res}. % % \All n \in \mathbb{N}. \\ % %& % % (n, \rs) \in inv(\iota, p)(W) \implies (n, \rs) \in vs_{\{ \iota \}}^{\emptyset}(\later p)(W) % %\end{align*} % %\end{lem} % % PDS: * undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall \iota \in \mathbb{N}.~ % % \forall p \in \textdom{Prop}.~ % % \forall W \in \textdom{World}.~ % % \forall \rs \in \textdom{Res}.~ % % \forall n \in \mathbb{N}.~\\ % %&\qquad % % (n, \rs) \in (inv(\iota, p) * \later p)(W) \Rightarrow (n, \rs) \in vs^{\{ \iota \}}_{\emptyset}(\top)(W) % %\end{align*} % %\end{lem} % % \begin{lem} % % \begin{align*} % % & % % \forall \mask_1, \mask_2 \in \Delta(\pset{\mathbb{N}}).~ % % valid(\bot \vs[\mask_1][\mask_2] \bot) % % \end{align*} % % \end{lem} % % PDS: E_1 ==>> E_2 undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall p, q \in \textdom{Prop}.~ % % \forall \mask \in \pset{\mathbb{N}}.~ % %valid(\always{(p \Rightarrow q)} \Rightarrow (p \vs[\mask][\mask] q)) % %\end{align*} % %\end{lem} \paragraph{Semantic domain of assertions.} The first complicated task in building a model of full Iris is defining the semantic model of $\Prop$. We start by defining the functor that assembles the CMRAs we need to the global resource CMRA: \begin{align*} \textdom{ResF}(\cofe) \eqdef{}& \record{\wld: \agm(\latert \cofe), \pres: \exm(\textdom{State}), \ghostRes: F(\cofe)} \end{align*} where $F$ is the user-chosen bifunctor from $\COFEs$ to $\CMRAs$. $\textdom{ResF}(\cofe)$ is a CMRA by lifting the individual CMRAs pointwise. Furthermore, if $F$ is locally contractive, then so is $\textdom{ResF}(-)$. Now we can write down the recursive domain equation: $\iPreProp \cong \UPred(\textdom{ResF}(\iPreProp))$ $\iPreProp$ is a COFE, which exists by America and Rutten's theorem~\cite{America-Rutten:JCSS89}. We do not need to consider how the object is constructed. We only need the isomorphism, given by \begin{align*} \Res &\eqdef \textdom{ResF}(\iPreProp) \\ \iProp &\eqdef \UPred(\Res) \\ \wIso &: \iProp \nfn \iPreProp \\ \wIso^{-1} &: \iPreProp \nfn \iProp \end{align*} % % PDS: E # E' and E_1 ==>> E_2 undefined. % %\begin{lem} % %\begin{align*} % %& % % \forall p_1, p_2, p_3 \in \textdom{Prop}.~ % % \forall \mask_1, \mask_2, \mask \in \pset{\mathbb{N}}.~ % %valid(\mask \sep \mask_1 \Ra \mask \sep \mask_2 \Ra (p_1 \vs[\mask_1][\mask_2] p_2) \Rightarrow (p_1 * p_3 \vs[\mask_1 \cup \mask][\mask_2 \cup \mask] p_2 * p_3)) % %\end{align*} % %\end{lem} We then pick $\iProp$ as the interpretation of $\Prop$: $\Sem{\Prop} \eqdef \iProp$ % \typedsection{Weakest precondition}{\mathit{wp} : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \to \textdom{Prop}) \to \textdom{Prop} \in {\cal U}} % % \begin{align*} % % \mathit{wp}_\mask(\expr, q) &\eqdef \Lam W. % % \begin{aligned}[t] % % \{\, (n, \rs) &\mid \All W_F \geq W; k \leq n; \rs_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\wsat{\state}{\mask \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \implies{}\\ % % &\qquad % % (\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \rs'. \\ % % &\qquad\qquad % % k \in (\wsat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(\expr)(W'))~\land \\ % % &\qquad % % (\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \rs'. \\ % % &\qquad\qquad % % k - 1 \in (\wsat{\state'}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k-1, \rs') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\ % % &\qquad % % (\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \rs', \rs_1', \rs_2'. \\ % % &\qquad\qquad % % k - 1 \in (\wsat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land \rs' = \rs_1' \rtimes \rs_2'~\land \\ % % &\qquad\qquad % % (k-1, \rs_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land % % (k-1, \rs_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W')) % % \,\} % % \end{aligned} % % \end{align*} % \begin{lem} % $\mathit{wp}$ is well-defined: $\mathit{wp}_{\mask}(\expr, q)$ is a valid proposition, and $\mathit{wp}$ is a non-expansive map. Besides, the dependency on the recursive occurrence is contractive, so $\mathit{wp}$ has a fixed-point. % \end{lem} \paragraph{Interpretation of assertions.} $\iProp$ is a $\UPred$, and hence the definitions from \Sref{sec:upred-logic} apply. We only have to define the missing connectives, the most interesting bits being are primitive view shifts and weakest preconditions. % \begin{lem} % $\mathit{wp}$ on values and non-mask-changing $\mathit{vs}$ agree: % $\mathit{wp}_\mask(\val, q) = \mathit{vs}_{\mask}^{\mask}(q \: \val)$ % \end{lem} \typedsection{World satisfaction}{\wsat{-}{-}{-} : \Delta\textdom{State} \times \Delta\pset{\mathbb{N}} \times \textdom{Res} \nfn \SProp } \begin{align*} \wsatpre(n, \mask, \state, \rss, \rs) & \eqdef \begin{inbox}[t] \rs \in \mval_{n+1} \land \rs.\pres = \exinj(\sigma) \land \dom(\rss) \subseteq \mask \cap \dom( \rs.\wld) \land {}\\ \All\iname \in \mask, \prop. \rs.\wld(\iname) \nequiv{n+1} \aginj(\latertinj(\wIso(\prop))) \Ra n \in \prop(\rss(\iname)) \end{inbox}\\ \wsat{\state}{\mask}{\rs} &\eqdef \set{0}\cup\setComp{n+1}{\Exists \rss : \mathbb{N} \fpfn \textdom{Res}. \wsatpre(n, \mask, \state, \rss, \rs \mtimes \prod_\iname \rss(\iname))} \end{align*} % \begin{align*} % \Sem{\vctx \proves x : \sort}_\gamma &= \gamma(x) \\ % \Sem{\vctx \proves \sigfn(\term_1, \dots, \term_n) : \type_{n+1}}_\gamma &= \Sem{\sigfn}(\Sem{\vctx \proves \term_1 : \type_1}_\gamma, \dots, \Sem{\vctx \proves \term_n : \type_n}_\gamma) \ \WHEN \sigfn : \type_1, \dots, \type_n \to \type_{n+1} \in \SigFn \\ % \Sem{\vctx \proves \Lam x. \term : \sort \to \sort'}_\gamma &= % \Lam v : \Sem{\sort}. \Sem{\vctx, x : \sort \proves \term : \sort'}_{\gamma[x \mapsto v]} \\ % \Sem{\vctx \proves \term~\termB : \sort'}_\gamma &= % \Sem{\vctx \proves \term : \sort \to \sort'}_\gamma(\Sem{\vctx \proves \termB : \sort}_\gamma) \\ % \Sem{\vctx \proves \unitval : \unitsort}_\gamma &= \star \\ % \Sem{\vctx \proves (\term_1, \term_2) : \sort_1 \times \sort_2}_\gamma &= (\Sem{\vctx \proves \term_1 : \sort_1}_\gamma, \Sem{\vctx \proves \term_2 : \sort_2}_\gamma) \\ % \Sem{\vctx \proves \pi_i~\term : \sort_1}_\gamma &= \pi_i(\Sem{\vctx \proves \term : \sort_1 \times \sort_2}_\gamma) % \end{align*} % % % \begin{align*} % \Sem{\vctx \proves \mzero : \textsort{Monoid}}_\gamma &= \mzero \\ % \Sem{\vctx \proves \munit : \textsort{Monoid}}_\gamma &= \munit \\ % \Sem{\vctx \proves \melt \mtimes \meltB : \textsort{Monoid}}_\gamma &= % \Sem{\vctx \proves \melt : \textsort{Monoid}}_\gamma \mtimes \Sem{\vctx \proves \meltB : \textsort{Monoid}}_\gamma % \end{align*} % % % \Sem{\vctx \proves \MU x. \pred : \sort \to \Prop}_\gamma &\eqdef % \mathit{fix}(\Lam v : \Sem{\sort \to \Prop}. \Sem{\vctx, x : \sort \to \Prop \proves \pred : \sort \to \Prop}_{\gamma[x \mapsto v]}) \\ \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 {}\\& \Exists \rsB. k \in \prop(\rsB) \land k \in \wsat\state{\mask_2 \cup \mask_\f}{\rsB \mtimes \rs_\f} \end{aligned}} \end{align*} \typedsection{Weakest precondition}{\mathit{wp} : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \nfn \iProp) \nfn \iProp} % \Sem{\vctx \proves \knowInv{\iname}{\prop} : \Prop}_\gamma &= % inv(\Sem{\vctx \proves \iname : \textsort{InvName}}_\gamma, \Sem{\vctx \proves \prop : \Prop}_\gamma) \\ % \Sem{\vctx \proves \ownGGhost{\melt} : \Prop}_\gamma &= % \Lam W. \{\, (n, \rs) \mid \rs.\ghostRes \geq \Sem{\vctx \proves \melt : \textsort{Monoid}}_\gamma \,\} \\ % \Sem{\vctx \proves \ownPhys{\state} : \Prop}_\gamma &= % \Lam W. \{\, (n, \rs) \mid \rs.\pres = \Sem{\vctx \proves \state : \textsort{State}}_\gamma \,\} $\textdom{wp}$ 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\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 {}&\\ &\qquad\qquad (\expr_\f = \bot \lor m \in \textdom{wp}(\top, \expr_\f, \Lam\any.\Lam\any.\mathbb{N})(\rsB_2)) \end{aligned}} \\ \textdom{wp}_\mask(\expr, \pred)