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. ... @@ -90,10 +90,9 @@ Completeness is necessary to take fixed-points. \begin{thm}[Banach's fixed-point] \begin{thm}[Banach's fixed-point] \label{thm:banach} \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 $f(\fixp_T f) = \fixp_T f$. 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} \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}] \begin{thm}[America and Rutten~\cite{America-Rutten:JCSS89,birkedal:metric-space}] \label{thm:america_rutten} \label{thm:america_rutten} Let $1$ be the discrete COFE on the unit type: $1 \eqdef \Delta \{ () \}$. 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 $\textdom{ResF} ... @@ -36,7 +36,7 @@ Furthermore, since the$\iFunc_i$are locally contractive, so is$\textdom{ResF} Now we can write down the recursive domain equation: Now we can write down the recursive domain equation: $\iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp))$ $\iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp))$ 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: We do not need to consider how the object $\iPreProp$ is constructed, we only need the isomorphism, given by: \begin{align*} \begin{align*} \Res &\eqdef \textdom{ResF}(\iPreProp, \iPreProp) \\ \Res &\eqdef \textdom{ResF}(\iPreProp, \iPreProp) \\ ... ...
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