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
\begin{align*}
\monoid\eqdef{}&\recordComp{c : \mathbb{N}\to T , V : \pset{\mathbb{N}}}{\All n, m. n \geq m \Ra n \in V \Ra m \in V }\\
\monoid\eqdef{}&\recordComp{\agc : \mathbb{N}\to T , \agV : \pset{\mathbb{N}}}{\All n, m. n \geq m \Ra n \in\agV\Ra m \in\agV}\\
&\text{quotiented by}\\
\melt\equiv\meltB\eqdef{}&\melt.V = \meltB.V \land\All n. n \in\melt.V \Ra\melt.c(n) \nequiv{n}\meltB.c(n) \\
\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\monoid}{ n \in\melt.V \land\All m \leq n. \melt.c(n) \nequiv{m}\melt.c(m) }\\
\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) }\\
\mcore\melt\eqdef{}&\melt\\
\melt\mtimes\meltB\eqdef{}& (\melt.c, \setComp{n}{n \in\melt.V \land n \in\meltB.V_2\land\melt\nequiv{n}\meltB})
\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 $\CMRAs$.
The reason we store a \emph{chain}$c$ of elements of $T$, 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.
\ralf{Figure out why exactly this is not possible without adding an explicit chain here.}
We define an injection $\ag$ into $\agm(\cofe)$ as follows:
\[\ag(x)\eqdef\record{\mathrm c \eqdef\Lam\any. x, \mathrm V \eqdef\mathbb{N}}\]
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
