A preorder is a well-quasi-ordering if every infinite sequence has indices with . An infinite sequence with no such pair is a bad sequence. This formulation handles combinatorial embedding relations that need not be antisymmetric on syntactic codes.
The Rado order has domain and iff either and , or . It is a well-quasi-ordering: either one row recurs infinitely in a sequence, or the first coordinates are unbounded and yield a cross-row comparison. Its infinite rows form an infinite antichain under Hoare domination preorder, showing that arbitrary-subset lifting need not preserve well-quasi-ordering.
If is a well-quasi-ordering, its finite words form a well-quasi-ordering under subsequence embedding with coordinatewise increase of letters. A minimal bad sequence proof removes the last letter of selected words whose last letters form a nondecreasing subsequence, then contradicts minimality. The result implies finite-subset lifting of a well-quasi-order under Hoare domination preorder.
A bad sequence in a preorder is an infinite sequence with no indices satisfying . A preorder is a well-quasi-ordering precisely when no such sequence exists.
A minimal bad sequence chooses each successive object of least possible natural-number size among choices that admit an infinite bad continuation of the already fixed prefix. Replacing its next object by a strictly smaller one cannot leave a bad sequence. This contradiction principle proves Higman lemma and the labelled version of Kruskal's tree theorem.
Given a size bound with finitely many possible objects at each position, put finite bad sequences in a tree ordered by extension. The tree is finitely branching. Arbitrarily long bad sequences would give an infinite bad sequence by König infinity lemma. This compactness argument converts an infinite well-quasi-ordering theorem into a uniform finite length bound.
Finite rooted trees are well-quasi-ordered by rooted-tree homeomorphic embedding. Consequently every infinite sequence contains an earlier tree embedding into a later one. Size-controlled finite forms follow by applying König infinity lemma to the finitely branching tree of bad prefixes.
Finite rooted trees labelled in a well-quasi-ordering are a well-quasi-ordering under label-monotone tree embedding. A minimal bad sequence argument makes the collection of proper rooted subtrees a well-quasi-ordering; Higman lemma then compares their child lists while the root labels are compared in the label order.
For every natural parameter , there is a finite length such that every sequence of finite rooted trees satisfying has an earlier tree homeomorphically embedding into a later tree. It is a true finite-combinatorial principle unprovable in Peano arithmetic and in arithmetical transfinite recursion theory. Each fixed-parameter instance can still be verified by a sufficiently large finite search.
