\melt\nequiv{n}\meltB\eqdef{}& (\All x \in\melt. \Exists y \in\meltB. x \nequiv{n} y) \land (\All y \in\meltB. \Exists x \in\melt. x \nequiv{n} y) \\

\textnormal{where }&\melt\sim\meltB\eqdef{}\All n. \melt\nequiv{n}\meltB\\

~\\

% \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\

\melt\nequiv{n}\meltB\eqdef{}& (\All m \leq n. m \in\melt.V \Lra m \in\meltB.V) \land (\All m \leq n. m \in\melt.V \Ra\melt.c(m) \nequiv{m}\meltB.c(m)) \\

\mval_n \eqdef{}&\setComp{\melt\in\agm(\cofe)}{ n \in\melt.V \land\All m \leq n. \melt.c(n) \nequiv{m}\melt.c(m) }\\

\mval_n \eqdef{}&\setComp{\melt\in\agm(\cofe)}{\All x, y \in\melt. x \nequiv{n} y }\\

\mcore\melt\eqdef{}&\melt\\

\melt\mtimes\meltB\eqdef{}&\left(\melt.c, \setComp{n}{n \in\melt.V \land n \in\meltB.V \land\melt\nequiv{n}\meltB}\right)

\melt\mtimes\meltB\eqdef{}&\melt\cup\meltB

\end{align*}

%Note that the carrier $\agm(\cofe)$ is a \emph{record} consisting of the two fields $c$ and $V$.

$\agm(-)$ is a locally non-expansive functor from $\OFEs$ to $\CMRAs$.

You can think of the $c$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in V$ 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 $V$ 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 $V =\nat$ 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:

\[\aginj(x)\eqdef\record{\mathrm c \eqdef\Lam\any. x, \mathrm V \eqdef\nat}\]

We define a non-expansive injection $\aginj$ into $\agm(\cofe)$ as follows:

\[\aginj(x)\eqdef\set{x}\]

There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: