Commit 40474920 authored by Ralf Jung's avatar Ralf Jung

docs: banach: make f^k part of the theorem, nut just a remark; extend ref to america-rutten

parent 8a2a27eb
......@@ -90,10 +90,9 @@ Completeness is necessary to take fixed-points.
\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 $f(\fixp_T f) = \fixp_T f$.
Moreover, this theorem also holds if $f$ is just non-expansive, and $f^k$ is contractive for an arbitrary $k$.
\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: $1 \eqdef \Delta \{ () \}$.
......
......@@ -36,7 +36,7 @@ Furthermore, since the $\iFunc_i$ are locally contractive, so is $\textdom{ResF}
Now we can write down the recursive domain equation:
\[ \iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp)) \]
Here, $\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor, which exists by \thmref{thm:america_rutten}.
Here, $\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor, which exists and is unique up to isomorphism by \thmref{thm:america_rutten}.
We do not need to consider how the object $\iPreProp$ is constructed, we only need the isomorphism, given by:
\begin{align*}
\Res &\eqdef \textdom{ResF}(\iPreProp, \iPreProp) \\
......
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