Petri Nets

Abstract

Exercise 1 (Dickson’s Lemma). A quasi-order (A,≤) is a set A endowed with a reflexive and transitive ordering relation ≤. A well quasi order (wqo) is a quasi order (A,≤) s.t., for any infinite sequence a0a1 · · · in Aω, there exist indices i < j with ai ≤ aj . 1. Let (A,≤) be a wqo and B ⊆ A. Show that (B,≤) is a wqo. 2. Show that (N ] {ω},≤) is a wqo. 3. Let (A,≤) be a wqo. Show that any infinite sequence a0a1 · · · in Aω embeds an infinite increasing subsequence ai0 ≤ ai1 ≤ ai2 ≤ · · · with i0 < i1 < i2 < · · · .