@@ -39,7 +39,7 @@ In order to solve the recursive domain equation in \Sref{sec:model} it is also e

...

@@ -39,7 +39,7 @@ In order to solve the recursive domain equation in \Sref{sec:model} it is also e

\end{defn}

\end{defn}

Intuitively, applying a non-expansive function to some data will not suddenly introduce differences between seemingly equal data.

Intuitively, applying a non-expansive function to some data will not suddenly introduce differences between seemingly equal data.

Elements that cannot be distinguished by programs within $n$ steps remain indistinguishable after applying $f$.

Elements that cannot be distinguished by programs within $n$ steps remain indistinguishable after applying $f$.

The reason that contractive functions are interesting is that for every contractive $f : \cofe\to\cofe$ with $\cofe$ inhabited, there exists a fixed-point $\fix(f)$ such that $\fix(f)= f(\fix(f))$.

The reason that contractive functions are interesting is that for every contractive $f : \cofe\to\cofe$ with $\cofe$ inhabited, there exists a \emph{unique}\footnote{Uniqueness is not proven in Coq.}fixed-point $\fix(f)$ such that $\fix(f)= f(\fix(f))$.

\begin{defn}

\begin{defn}

The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows.

The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows.

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@@ -59,6 +59,7 @@ Note that $\COFEs$ is cartesian closed. In particular:

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@@ -59,6 +59,7 @@ Note that $\COFEs$ is cartesian closed. In particular:

\end{defn}

\end{defn}

The function space $(-)\nfn(-)$ is a locally non-expansive bifunctor.

The function space $(-)\nfn(-)$ is a locally non-expansive bifunctor.

Note that the composition of non-expansive (bi)functors is non-expansive, and the composition of a non-expansive and a contractive (bi)functor is contractive.

Note that the composition of non-expansive (bi)functors is non-expansive, and the composition of a non-expansive and a contractive (bi)functor is contractive.

The reason contractive (bi)functors are interesting is that by America and Rutten's theorem~\cite{America-Rutten:JCSS89,birkedal:metric-space}, they have a unique\footnote{Uniqueness is not proven in Coq.} fixed-point.