From f01839f757fc9a67dfb982f1c64cc42961a605ea Mon Sep 17 00:00:00 2001 From: Ralf Jung Date: Wed, 12 Apr 2017 11:56:42 +0200 Subject: [PATCH] =?UTF-8?q?add=20Ale=C5=A1's=20proof=20that=20agree=20is?= =?UTF-8?q?=20not=20complete?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- theories/algebra/agree.v | 21 +++++++++++++++++++++ 1 file changed, 21 insertions(+) diff --git a/theories/algebra/agree.v b/theories/algebra/agree.v index 5f699ec7..13f87dbf 100644 --- a/theories/algebra/agree.v +++ b/theories/algebra/agree.v @@ -6,6 +6,27 @@ Local Arguments valid _ _ !_ /. Local Arguments op _ _ _ !_ /. Local Arguments pcore _ _ !_ /. +(** Define an agreement construction such that Agree A is discrete when A is discrete. + Notice that this construction is NOT complete. The fullowing is due to Aleš: + + +Proposition: Ag(T) is not necessarily complete. +Proof. + Let T be the set of binary streams (infinite sequences) with the usual + ultrametric, measuring how far they agree. + + Let Aₙ be the set of all binary strings of length n. Thus for Aₙ to be a + subset of T we have them continue as a stream of zeroes. + + Now Aₙ is a finite non-empty subset of T. Moreover {Aₙ} is a Cauchy sequence + in the defined (Hausdorff) metric. + + However the limit (if it were to exist as an element of Ag(T)) would have to + be the set of all binary streams, which is not exactly finite. + + Thus Ag(T) is not necessarily complete. +*) + Record agree (A : Type) : Type := Agree { agree_car : A; agree_with : list A; -- GitLab