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.

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