Commit 94042bb7 by Ralf Jung

### docs: UPred

parent 3d79ec6c
Pipeline #328 passed with stage
 ... ... @@ -34,12 +34,20 @@ \begin{defn} The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows. \end{defn} Note that $\COFEs$ is cartesian closed. Note that $\COFEs$ is cartesian closed: \begin{defn} Given two COFEs $\cofe$ and $\cofeB$, the set of non-expansive functions $\set{f : \cofe \nfn \cofeB}$ is itself a COFE with \begin{align*} f \nequiv{n} g \eqdef{}& \All x \in \cofe. f(x) \nequiv{n} g(x) \end{align*} \end{defn} \begin{defn} A (bi)functor $F : \COFEs \to \COFEs$ is called \emph{locally non-expansive} if its action $F_1$ on arrows is itself a non-expansive map. Similarly, $F$ is called \emph{locally contractive} if $F_1$ is a contractive map. \end{defn} The function space $(-) \nfn (-)$ is a locally non-expansive bifunctor. Note that the composition of non-expansive (bi)functors is non-expansive, and the composition of a non-expansive and a contractive (bi)functor is contractive. \subsection{RA} ... ...
 ... ... @@ -8,7 +8,35 @@ Given a COFE $\cofe$, we define $\latert\cofe$ as follows: \latert\cofe \eqdef{}& \latertinj(\cofe) \\ \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y \end{align*} $\latert(-)$ is a locally \emph{contractive} bifunctor from $\COFEs$ to $\COFEs$. $\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$. \subsection{Uniform Predicates} 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}{ \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)) \end{inbox} } \end{align*} where $\mProp$ is the set of meta-level propositions, \eg Coq's \texttt{Prop}. $\UPred(-)$ 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}: \begin{align*} \SProp \eqdef{}& \psetdown{\mathbb{N}} \\ \prop \nequiv{n} \propB \eqdef{}& \All m \leq n. m \in \prop \Lra m \in \propB \end{align*} where $\psetdown{N}$ denotes the set of \emph{down-closed} sets of natural numbers: If $n$ is in the set, then all smaller numbers also have to be in there. 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 \\ \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. \clearpage \section{CMRA constructions} ... ... @@ -44,6 +72,7 @@ We obtain the following frame-preserving updates: {\melt \mupd \meltsB} {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}} \end{mathpar} $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$. \subsection{Agreement} ... ... @@ -59,7 +88,7 @@ Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows: \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 }) \end{align*} $\agm(-)$ is a locally non-expansive bifunctor from $\COFEs$ to $\CMRAs$. $\agm(-)$ 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. 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 $\agm(\cofe)$. ... ... @@ -84,7 +113,7 @@ There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can The purpose of the one-shot CMRA is to lazily initialize the state of a ghost location. Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows: \begin{align*} \oneshot(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\ \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*} ... ... @@ -107,7 +136,7 @@ The step-indexed equivalence is inductively defined as follows: \axiom{\bot \nequiv{n} \bot} \end{mathpar} $\oneshotm(-)$ is a locally non-expansive bifunctor from $\CMRAs$ to $\CMRAs$. $\oneshotm(-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$. We obtain the following frame-preserving updates: \begin{mathpar} ... ...
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