Commit 861ee760 by Robbert Krebbers

### Docs: put Banach and America/Rutten in a theorem environment.

parent c73f0c68
 ... ... @@ -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 $\COFEs$). \end{thm} \subsection{RA} ... ...
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