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  "body": "\n## Statement\n\nLet $(X, \\le)$ be a [[well-quasi-ordered|NH25G6GC07]] set, and let $X^*$ be the set of finite words over $X$, the [[free monoid|CV013KHVJG]] on $X$.\nFor words $u = a_1 \\cdots a_m$ and $v = b_1 \\cdots b_n$ in $X^*$, write $u \\le^* v$ when there are indices $1 \\le i_1 \u003c i_2 \u003c \\cdots \u003c i_m \\le n$ with $a_k \\le b_{i_k}$ for $k = 1, \\ldots, m$.\nThen $(X^*, \\le^*)$ is well-quasi-ordered.\n\n## Remarks\n\nThe usual proof is the minimal bad sequence argument of Nash-Williams.\nAssume a bad sequence exists, and choose one in which each word is as short as possible given the earlier ones.\nRemove the first letter of each word; since $X$ is well-quasi-ordered, an infinite subsequence has non-decreasing first letters.\nThe shortened words of this subsequence, put after the beginning of the minimal sequence, give a bad sequence that contradicts minimality.\n\nExample: take $X = \\set{a, b}$ with equality as the order.\nThen $u \\le^* v$ means that $u$ is a [[subsequence|0F1PR74BRX]] of $v$, a scattered subword.\nSo in every infinite list of words over two letters, some word is a scattered subword of a later one.\n\nA consequence concerns any language $L \\subseteq X^*$ over a finite alphabet.\nThe set of words having a scattered subword in $L$ is regular, since it is generated by its finitely many minimal elements.\n\n## Explanations\n\n### L1 human\n\nWrite down an endless list of words in any finite alphabet.\nHigman's lemma says that some word in your list can be found inside a later word by deleting letters.\nMore generally the letters may come from any well-quasi-ordered set, and a letter may be replaced by a larger one.\n\n### L2 human\n\nThe lemma is the first step from simple well-quasi-orders, such as finite sets and $\\N$, to orders on structured objects.\nKruskal's tree theorem extends it from words to finite trees, and its proof uses Higman's lemma on the sequences of subtrees.\nIn computer science it gives termination and decidability results for well-structured transition systems such as lossy channel systems.\n",
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