Given some set $T$ and an indexed family $({\nequiv{n}}\subseteq T \times T)_{n \in\mathbb{N}}$ of equivalence relations, a \emph{chain} is a function $c : \mathbb{N}\to T$ such that $\All n, m. n < m \Ra c (m)\nequiv{n} c (n+1)$.
\end{defn}
\begin{defn}
A COFE is a tuple $(T, (\nequiv{n})_{n \in\mathbb{N}}, c : (\mathbb{N}\toT)\to T)$ satisfying
A \emph{complete ordered family of equivalences} (COFE) is a tuple $(T, ({\nequiv{n}}\subseteq T \times T)_{n \in\mathbb{N}}, \lim : \chain(T)\to T)$ 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}\\
\All x, y.& x = y \Lra (\All n. x \nequiv{n} y) \tagH{cofe-limit}\\
\All n, X.&c(X) \nequiv{n}X(n+1) \tagH{cofe-compl}
\All n, c.&\lim(c) \nequiv{n}c(n+1) \tagH{cofe-compl}
\end{align*}
\end{defn}
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@@ -17,7 +21,7 @@
\subsection{CMRA}
\begin{defn}
A CMRA is a tuple $(\monoid, (\mval_n \subseteq\monoid)_{n \in\mathbb{N}}, \munit: \monoid\to\monoid, (\mtimes) : \monoid\times\monoid\to\monoid, (\mdiv) : \monoid\times\monoid\to\monoid)$ satisfying
A \emph{CMRA} is a tuple $(\monoid, (\mval_n \subseteq\monoid)_{n \in\mathbb{N}}, \munit: \monoid\to\monoid, (\mtimes) : \monoid\times\monoid\to\monoid, (\mdiv) : \monoid\times\monoid\to\monoid)$ satisfying
\begin{align*}
\All n, m.& n \geq m \Ra V_n \subseteq V_m \tagH{cmra-valid-mono}\\
