### Use \nat macro.

 ... ... @@ -6,11 +6,11 @@ The model of Iris lives in the category of \emph{Complete Ordered Families of Eq This definition varies slightly from the original one in~\cite{catlogic}. \begin{defn}[Chain] Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N} \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$. Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \nat}$ of equivalence relations, a \emph{chain} is a function $c : \nat \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$. \end{defn} \begin{defn} A \emph{complete ordered family of equivalences} (COFE) is a tuple $(\cofe, ({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \mathbb{N}}, \lim : \chain(\cofe) \to \cofe)$ satisfying A \emph{complete ordered family of equivalences} (COFE) is a tuple $(\cofe, ({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \nat}, \lim : \chain(\cofe) \to \cofe)$ satisfying \begin{align*} \All n. (\nequiv{n}) ~& \text{is an equivalence relation} \tagH{cofe-equiv} \\ \All n, m.& n \geq m \Ra (\nequiv{n}) \subseteq (\nequiv{m}) \tagH{cofe-mono} \\ ... ... @@ -115,7 +115,7 @@ Since Iris ensures that the global ghost state is valid, this means that we can \subsection{CMRA} \begin{defn} A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \mathbb{N}},\\ \mcore{{-}}: \monoid \nfn \maybe\monoid, (\mtimes) : \monoid \times \monoid \nfn \monoid)$ satisfying: A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \nat},\\ \mcore{{-}}: \monoid \nfn \maybe\monoid, (\mtimes) : \monoid \times \monoid \nfn \monoid)$ satisfying: \begin{align*} \All n, \melt, \meltB.& \melt \nequiv{n} \meltB \land \melt\in\mval_n \Ra \meltB\in\mval_n \tagH{cmra-valid-ne} \\ \All n, m.& n \geq m \Ra \mval_n \subseteq \mval_m \tagH{cmra-valid-mono} \\ ... ... @@ -136,7 +136,7 @@ Since Iris ensures that the global ghost state is valid, this means that we can This is a natural generalization of RAs over COFEs. All operations have to be non-expansive, and the validity predicate $\mval$ can now also depend on the step-index. We define the plain $\mval$ as the limit'' of the $\mval_n$: $\mval \eqdef \bigcap_{n \in \mathbb{N}} \mval_n$ $\mval \eqdef \bigcap_{n \in \nat} \mval_n$ \paragraph{The extension axiom (\ruleref{cmra-extend}).} Notice that the existential quantification in this axiom is \emph{constructive}, \ie it is a sigma type in Coq. ... ...
 ... ... @@ -16,7 +16,7 @@ $\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$. Given a CMRA $\monoid$, we define the COFE $\UPred(\monoid)$ of \emph{uniform predicates} over $\monoid$ as follows: \begin{align*} \UPred(\monoid) \eqdef{} \setComp{\pred: \mathbb{N} \times \monoid \to \mProp}{ \UPred(\monoid) \eqdef{} \setComp{\pred: \nat \times \monoid \to \mProp}{ \begin{inbox}[c] (\All n, x, y. \pred(n, x) \land x \nequiv{n} y \Ra \pred(n, y)) \land {}\\ (\All n, m, x, y. \pred(n, x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra \pred(m, y)) ... ... @@ -29,8 +29,8 @@ $\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$. One way to understand this definition is to re-write it a little. We start by defining the COFE of \emph{step-indexed propositions}: For every step-index, the proposition either holds or does not hold. \begin{align*} \SProp \eqdef{}& \psetdown{\mathbb{N}} \\ \eqdef{}& \setComp{X \in \pset{\mathbb{N}}}{ \All n, m. n \geq m \Ra n \in X \Ra m \in X } \\ \SProp \eqdef{}& \psetdown{\nat} \\ \eqdef{}& \setComp{X \in \pset{\nat}}{ \All n, m. n \geq m \Ra n \in X \Ra m \in X } \\ X \nequiv{n} Y \eqdef{}& \All m \leq n. m \in X \Lra m \in Y \end{align*} Notice that this notion of $\SProp$ is already hidden in the validity predicate $\mval_n$ of a CMRA: ... ... @@ -114,7 +114,7 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$. Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows: \begin{align*} \agm(\cofe) \eqdef{}& \set{(c, V) \in (\mathbb{N} \to \cofe) \times \SProp}/\ {\sim} \$-0.2em] \agm(\cofe) \eqdef{}& \set{(c, V) \in (\nat \to \cofe) \times \SProp}/\ {\sim} \\[-0.2em] \textnormal{where }& \melt \sim \meltB \eqdef{} \melt.V = \meltB.V \land \All n. n \in \melt.V \Ra \melt.c(n) \nequiv{n} \meltB.c(n) \\ % \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\ ... ... @@ -131,11 +131,11 @@ You can think of the c as a \emph{chain} of elements of \cofe that has to co 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 V of the limit is the limit of the V of the chain. But what to pick for the actual data, for the element of \cofe? Only if V = \mathbb{N} we have a chain of \cofe that we can take a limit of; if the V is smaller, the chain cancels'', \ie stops converging as we reach indices n \notin V. Only if V = \nat we have a chain of \cofe that we can take a limit of; if the V is smaller, the chain cancels'', \ie stops converging as we reach indices n \notin V. 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 \aginj into \agm(\cofe) as follows: \[ \aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}}$ $\aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \nat}$ There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar} \axiomH{ag-val}{\aginj(x) \in \mval_n} ... ...
 ... ... @@ -156,7 +156,7 @@ To instantiate the DC logic (base logic with dynamic composeable resources), the From this, we construct the bifunctor defining the overall resources as follows: \begin{align*} \textdom{ResF}(\cofe^\op, \cofe) \eqdef{}& \prod_{i \in \mathcal I} \mathbb{N} \fpfn \iFunc_i(\cofe^\op, \cofe) \textdom{ResF}(\cofe^\op, \cofe) \eqdef{}& \prod_{i \in \mathcal I} \nat \fpfn \iFunc_i(\cofe^\op, \cofe) \end{align*} (We will motivate both the use of a product and the finite partial function below.) $\textdom{ResF}(\cofe^\op, \cofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions). ... ...
 ... ... @@ -15,9 +15,9 @@ To this end, we use tokens that manage which invariants are currently enabled. We assume to have the following four CMRAs available: \begin{align*} \textmon{State} \eqdef{}& \authm(\exm(\State)) \\ \textmon{Inv} \eqdef{}& \authm(\mathbb N \fpfn \agm(\latert \iPreProp)) \\ \textmon{En} \eqdef{}& \pset{\mathbb N} \\ \textmon{Dis} \eqdef{}& \finpset{\mathbb N} \textmon{Inv} \eqdef{}& \authm(\nat \fpfn \agm(\latert \iPreProp)) \\ \textmon{En} \eqdef{}& \pset{\nat} \\ \textmon{Dis} \eqdef{}& \finpset{\nat} \end{align*} The last two are the tokens used for managing invariants, $\textmon{Inv}$ is the monoid used to manage the invariants themselves. Finally, $\textmon{State}$ is used to provide the program with a view of the physical state of the machine. ... ... @@ -28,7 +28,7 @@ Furthermore, we assume that instances named $\gname_{\textmon{State}}$, $\gname_ \paragraph{World Satisfaction.} We can now define the assertion$W(\emph{world satisfaction}) which ensures that the enabled invariants are actually maintained: \begin{align*} W \eqdef{}& \Exists I : \mathbb N \fpfn \Prop. W \eqdef{}& \Exists I : \nat \fpfn \Prop. \begin{array}{@{} l} \ownGhost{\gname_{\textmon{Inv}}}{\authfull \mapsingletonComp {\iname} ... ... @@ -47,7 +47,7 @@ The following assertion states that an invariant with name\iname$exists and m Next, we define \emph{view updates}, which are essentially the same as the resource updates of the base logic ($\Sref{sec:base-logic}$), except that they also have access to world satisfaction and can enable and disable invariants: $\pvs[\mask_1][\mask_2] \prop \eqdef W * \ownGhost{\gname_{\textmon{En}}}{\mask_1} \wand \upd\diamond (W * \ownGhost{\gname_{\textmon{En}}}{\mask_2} * \prop)$ Here,$\mask_1$and$\mask_2$are the \emph{masks} of the view update, defining which invariants have to be (at least!) available before and after the update. We use$\top$as symbol for the largest possible mask,$\mathbb N$, and$\bot$for the smallest possible mask$\emptyset$. We use$\top$as symbol for the largest possible mask,$\nat$, and$\bot$for the smallest possible mask$\emptyset$. We will write$\pvs[\mask] \prop$for$\pvs[\mask][\mask]\prop$. % View updates satisfy the following basic proof rules: ... ... @@ -369,14 +369,14 @@ Furthermore, we will often know that namespaces are \emph{disjoint} just by look The namespaces$\namesp.\texttt{iris}$and$\namesp.\texttt{gps}$are disjoint no matter the choice of$\namesp$. As a result, there is often no need to track disjointness of namespaces, we just have to pick the namespaces that we allocate our invariants in accordingly. Formally speaking, let$\namesp \in \textlog{InvNamesp} \eqdef \List(\mathbb N)$be the type of \emph{invariant namespaces}. Formally speaking, let$\namesp \in \textlog{InvNamesp} \eqdef \List(\nat)$be the type of \emph{invariant namespaces}. We use the notation$\namesp.\iname$for the namespace$[\iname] \dplus \namesp$. (In other words, the list is backwards''. This is because cons-ing to the list, like the dot does above, is easier to deal with in Coq than appending at the end.) The elements of a namespaces are \emph{structured invariant names} (think: Java fully qualified class name). They, too, are lists of$\mathbb N$, the same type as namespaces. In order to connect this up to the definitions of \Sref{sec:invariants}, we need a way to map structued invariant names to$\mathbb N$, the type of plain'' invariant names. Any injective mapping$\textlog{namesp\_inj}$will do; and such a mapping has to exist because$\List(\mathbb N)$is countable. They, too, are lists of$\nat$, the same type as namespaces. In order to connect this up to the definitions of \Sref{sec:invariants}, we need a way to map structued invariant names to$\nat$, the type of plain'' invariant names. Any injective mapping$\textlog{namesp\_inj}$will do; and such a mapping has to exist because$\List(\nat)$is countable. Whenever needed, we (usually implicitly) coerce$\namesp\$ to its encoded suffix-closure, \ie to the set of encoded structured invariant names contained in the namespace: $\namecl\namesp \eqdef \setComp{\iname}{\Exists \namesp'. \iname = \textlog{namesp\_inj}(\namesp' \dplus \namesp)}$ We will overload the notation for invariant assertions for using namespaces instead of names: ... ...