@@ -75,6 +75,7 @@ The function space $\ofe \nfn \cofeB$ is a COFE if $\cofeB$ is a COFE (\ie the d

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@@ -75,6 +75,7 @@ The function space $\ofe \nfn \cofeB$ is a COFE if $\cofeB$ is a COFE (\ie the d

Completeness is necessary to take fixed-points.

Completeness is necessary to take fixed-points.

For once, every \emph{contractive function}$f : \ofe\to\cofeB$ where $\cofeB$ is a COFE and inhabited has a \emph{unique} fixed-point $\fix(f)$ such that $\fix(f)= f(\fix(f))$.

For once, every \emph{contractive function}$f : \ofe\to\cofeB$ where $\cofeB$ 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.

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.