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  "body": "\n## Statement\n\nFor finite [[rooted trees|0C9J6W2A8A]] $T$ and $T'$, write $u \\wedge v$ for the nearest common ancestor of vertices $u, v$ of $T$.\nSay that $T$ embeds in $T'$ if some injective map $f$ from the vertices of $T$ to those of $T'$ satisfies $f(u \\wedge v) = f(u) \\wedge f(v)$ for all $u, v$.\nThen the finite rooted trees are well-quasi-ordered by embedding.\nThat is, for every infinite sequence $T_1, T_2, T_3, \\dots$ of finite rooted trees, there are indices $i \u003c j$ such that $T_i$ embeds in $T_j$.\n\n## Remarks\n\nThe labelled form also holds.\nLet the vertices carry labels from a well-quasi-ordered set $(X, \\le)$, and ask also that each label of $T$ is at most the label of its image.\nThen the trees are still well-quasi-ordered.\nJoseph Kruskal proved the theorem in 1960, and Crispin Nash-Williams gave a short proof by the minimal bad sequence argument.\nThat argument takes a bad sequence whose trees are as small as possible at each step.\nIt then applies Higman's lemma to the sets of subtrees at the roots to reach a contradiction.\nHarvey Friedman showed that a finite form of the theorem, although true, is not provable in the subsystem $\\mathrm{ATR}_0$ of second-order arithmetic.\nThe [[Robertson–Seymour theorem|WP3GDNWJJW]] on graph minors builds on its methods.\n\n## Explanations\n\n### L1 human\n\nWrite down an endless list of finite trees.\nThe theorem says that, no matter how you choose them, some tree in the list will be contained, in a branch-preserving way, in a later tree.\nYou cannot keep producing trees that avoid all earlier ones.\n\n### L2 human\n\nThe theorem is a basic example of a well-quasi-order on structured objects, after Higman's lemma for finite words.\nIt is used in term rewriting to prove that certain orderings, such as recursive path orderings, are well-founded.\nIts finite form grows so fast that it escapes strong formal systems, which makes it a standard example in proof theory.\n",
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