Commit f01839f7 authored by Ralf Jung's avatar Ralf Jung

add Aleš's proof that agree is not complete

parent 1c1ae879
...@@ -6,6 +6,27 @@ Local Arguments valid _ _ !_ /. ...@@ -6,6 +6,27 @@ Local Arguments valid _ _ !_ /.
Local Arguments op _ _ _ !_ /. Local Arguments op _ _ _ !_ /.
Local Arguments pcore _ _ !_ /. 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 { Record agree (A : Type) : Type := Agree {
agree_car : A; agree_car : A;
agree_with : list A; agree_with : list A;
......
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