From 738d8bcc90f41dc05434609e4d35d0976a7c39dc Mon Sep 17 00:00:00 2001 From: Robbert Krebbers Date: Sun, 21 Aug 2016 23:32:12 +0200 Subject: [PATCH] Prove uniqueness of Banach's fixpoint. --- algebra/cofe.v | 11 +++++++++++ docs/algebra.tex | 2 +- 2 files changed, 12 insertions(+), 1 deletion(-) diff --git a/algebra/cofe.v b/algebra/cofe.v index 9742a091..d2f120f5 100644 --- a/algebra/cofe.v +++ b/algebra/cofe.v @@ -188,12 +188,23 @@ Definition fixpoint_eq : @fixpoint = @fixpoint_def := proj2_sig fixpoint_aux. Section fixpoint. Context {A : cofeT} {Inhabited A} (f : A → A) {!Contractive f}. + Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). Proof. apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. Qed. + + Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. + Proof. + rewrite !equiv_dist=> Hx n. + rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //=. + induction n as [|n IH]; simpl in *. + - rewrite Hx; eauto using contractive_0. + - rewrite Hx. apply (contractive_S _), IH. + Qed. + Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. Proof. diff --git a/docs/algebra.tex b/docs/algebra.tex index 494fb54c..e9b3208e 100644 --- a/docs/algebra.tex +++ b/docs/algebra.tex @@ -39,7 +39,7 @@ In order to solve the recursive domain equation in \Sref{sec:model} it is also e \end{defn} 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 $f$. -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 $\fix(f) = f(\fix(f))$. +The reason that contractive functions are interesting is that for every contractive $f : \cofe \to \cofe$ with $\cofe$ inhabited, there exists a \emph{unique} fixed-point $\fix(f)$ such that $\fix(f) = f(\fix(f))$. \begin{defn} The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows. -- GitLab