From 404749206c0e83e8bed5f866245b432268c876d9 Mon Sep 17 00:00:00 2001 From: Ralf Jung Date: Sun, 10 Dec 2017 14:11:16 +0100 Subject: [PATCH] docs: banach: make f^k part of the theorem, nut just a remark; extend ref to america-rutten --- docs/algebra.tex | 3 +-- docs/program-logic.tex | 2 +- 2 files changed, 2 insertions(+), 3 deletions(-) diff --git a/docs/algebra.tex b/docs/algebra.tex index dc5dfad2..dc8bdead 100644 --- a/docs/algebra.tex +++ b/docs/algebra.tex @@ -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: $1 \eqdef \Delta \{ () \}$. diff --git a/docs/program-logic.tex b/docs/program-logic.tex index de5055c1..7e7f2e00 100644 --- a/docs/program-logic.tex +++ b/docs/program-logic.tex @@ -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) \\ -- 2.26.2