@@ -83,14 +83,23 @@ COFEs are \emph{complete OFEs}, which means that we can take limits of arbitrary
\end{defn}
The function space $\ofe\nfn\cofeB$ is a COFE if $\cofeB$ is a COFE (\ie the domain $\ofe$ can actually be just an OFE).
$\SProp$ as defined above is complete, \ie it is a COFE.
Completeness is necessary to take fixed-points.
For once, every contractive function $f : \cofe\to\cofe$ where $\cofe$ is a COFE and inhabited has a \emph{unique} fixed-point $\fix(f)$ such that $\fix(f)= f(\fix(f))$.
This also holds if $f^k$ is contractive for an arbitrary $k$.
Furthermore, by America and Rutten's theorem~\cite{America-Rutten:JCSS89,birkedal:metric-space}, every contractive (bi)functor from $\COFEs$ to $\COFEs$ has a unique\footnote{Uniqueness is not proven in Coq.} fixed-point.
$\SProp$ as defined above is complete, \ie it is a COFE.
\begin{thm}[Banach's fixed-point]
\label{thm:banach}
Given an inhabited COFE $\ofe$ and a contractive function $f : \ofe\to\ofe$, there exists a unique fixed-point $\fixp_T f$ such that $f(\fixp_T f)=\fixp_T f$.
\end{thm}
The above theorem also holds if $f^k$ is contractive for an arbitrary $k$.
\begin{thm}[America and Rutten~\cite{America-Rutten:JCSS89,birkedal:metric-space}]
\label{thm:america_rutten}
Let $1$ be the discrete COFE on the unit type: $1\eqdef\Delta\{()\}$.
Given a locally contractive bifunctor $G : \COFEs^{\textrm{op}}\times\COFEs\to\COFEs$, and provided that \(G(1, 1)\) is inhabited,
then there exists a unique\footnote{Uniqueness is not proven in Coq.} COFE $\ofe$ such that $G(\ofe^{\textrm{op}}, \ofe)\cong\ofe$ (\ie the two are isomorphic in $\COFEs$).
