@@ -23,6 +23,8 @@ This definition varies slightly from the original one in~\cite{catlogic}.
...
@@ -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
An element $x \in\cofe$ of a COFE is called \emph{discrete} if
\[\All y \in\cofe. x \nequiv{0} y \Ra x = y\]
\[\All y \in\cofe. x \nequiv{0} y \Ra x = y\]
A COFE $A$ is called \emph{discrete} if all its elements are discrete.
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}
\end{defn}
\begin{defn}
\begin{defn}
...
@@ -31,6 +33,7 @@ This definition varies slightly from the original one in~\cite{catlogic}.
...
@@ -31,6 +33,7 @@ This definition varies slightly from the original one in~\cite{catlogic}.
It is \emph{contractive} if
It is \emph{contractive} if
\[\All n, x \in\cofe, y \in\cofe. (\All m < n. x \nequiv{m} y)\Ra f(x)\nequiv{n} f(x)\]
\[\All n, x \in\cofe, y \in\cofe. (\All m < n. x \nequiv{m} y)\Ra f(x)\nequiv{n} f(x)\]
\end{defn}
\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}
\begin{defn}
The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows.
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
...
@@ -33,7 +33,7 @@ We start by defining the COFE of \emph{step-indexed propositions}: For every ste
\end{align*}
\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.
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*}
\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)}
\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*}
\end{align*}
The reason we chose the first definition is that it is easier to work with in Coq.
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 $\CMRAs$.
...
@@ -77,35 +77,35 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
\subsection{Agreement}
\subsection{Agreement}
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
\newcommand{\agc}{\mathrm{c}}% the "c" field of an agreement element
\newcommand{\aginjc}{\mathrm{c}}% the "c" field of an agreement element
\newcommand{\agV}{\mathrm{V}}% the "V" field of an agreement element
\newcommand{\aginjV}{\mathrm{V}}% the "V" field of an agreement element
\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\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.\agV\Lra m \in\meltB.\agV) \land (\All m \leq n. m \in\melt.\agV\Ra\melt.\agc(m) \nequiv{m}\meltB.\agc(m)) \\
\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.\agV\land\All m \leq n. \melt.\agc(n) \nequiv{m}\melt.\agc(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\\
\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*}
\end{align*}
$\agm(-)$ is a locally non-expansive functor 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.
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 $\agm(\cofe)$.
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 $\cofe$?
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 $\cofe$.
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:
We define an injection $\aginj$ into $\agm(\cofe)$ as follows:
\[\ag(x)\eqdef\record{\mathrm c \eqdef\Lam\any. x, \mathrm V \eqdef\mathbb{N}}\]
\[\aginj(x)\eqdef\record{\mathrm c \eqdef\Lam\any. x, \mathrm V \eqdef\mathbb{N}}\]
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
% 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
% \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.
% $\always{}$ is well-defined: $\always{p}$ is a valid proposition (this amounts to showing non-expansiveness), and $\always{}$ itself is a non-expansive map.
% $\mathit{inv}$ 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.
% $\wsat{-}{-}{-}{-}$ 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 $\cal U$.)
% $\mathit{wp}$ 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.
\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{}\\&
% \Lam v : \Sem{\sort}. \Sem{\vctx, x : \sort \proves \term : \sort'}_{\gamma[x \mapsto v]} \\
\Exists\rsB. k \in\prop(\rsB) \land k \in\wsat\state{\mask_2 \cup\mask_\f}{\rsB\mtimes\rs_\f}
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 a function $\Sem{\sigfn} : \Sem{\type_1}\times\dots\times\Sem{\type_n}\nfn\Sem{\type_{n+1}}$.
\typedsection{Interpretation of non-propositional terms}{\Sem{\vctx\proves\term : \type} : \Sem{\vctx}\nfn\Sem{\type}}
\begin{align*}
\Sem{\vctx\proves x : \type}_\gamma&\eqdef\gamma(x) \\
