Title: Well-Quasi-Orders
Author: Christian Sternagel
Submission date: 2012-04-13
Abstract: Based on Isabelle/HOL's type class for preorders, we introduce a type class for well-quasi-orders (wqo) which is characterized by the absence of "bad" sequences (our proofs are along the lines of the proof of Nash-Williams, from which we also borrow terminology). Our main results are instantiations for the product type, the list type, and a type of finite trees, which (almost) directly follow from our proofs of (1) Dickson's Lemma, (2) Higman's Lemma, and (3) Kruskal's Tree Theorem. More concretely:
  • If the sets A and B are wqo then their Cartesian product is wqo.
  • If the set A is wqo then the set of finite lists over A is wqo.
  • If the set A is wqo then the set of finite trees over A is wqo.
The research was funded by the Austrian Science Fund (FWF): J3202.
Change history: [2012-06-11]: Added Kruskal's Tree Theorem.
[2012-12-19]: New variant of Kruskal's tree theorem for terms (as opposed to variadic terms, i.e., trees), plus finite version of the tree theorem as corollary.
[2013-05-16]: Simplified construction of minimal bad sequences.
[2014-07-09]: Simplified proofs of Higman's lemma and Kruskal's tree theorem, based on homogeneous sequences.
[2016-01-03]: An alternative proof of Higman's lemma by open induction.
[2017-06-08]: Proved (classical) equivalence to inductive definition of almost-full relations according to the ITP 2012 paper "Stop When You Are Almost-Full" by Vytiniotis, Coquand, and Wahlstedt.
  author  = {Christian Sternagel},
  title   = {Well-Quasi-Orders},
  journal = {Archive of Formal Proofs},
  month   = apr,
  year    = 2012,
  note    = {\url{https://isa-afp.org/entries/Well_Quasi_Orders.html},
            Formal proof development},
  ISSN    = {2150-914x},
License: BSD License
Depends on: Abstract-Rewriting, Open_Induction
Used by: Decreasing-Diagrams-II, Myhill-Nerode, Polynomials, Saturation_Framework, Saturation_Framework_Extensions