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If we have a string rewriting system within the alphabet $\{X,Y\}^*$ and the rule $XY\to YX$. How can we prove by induction that on every string input the system terminates?

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It's your exercise, so I'll let you answer it, but here is a hint. Consider the sequence of words visited by repeatedly applying the rewrite rule and its relationship to lexicographic order. Work through a few examples and see if you can form a conjecture.

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