In the theory of crystal bases of quantized enveloping algebras due to Kashiwara, he has a so-called "grand-loop" argument, consisting of 14 induction statements. Each is a conventional induction, on the face of it, but the method of proof interleaves them, so that the $l$th case of one statement also features in the proof of the $l$th case of others. The proof takes just over 13 pages.
(Reference: pages 489-503 of "On crystal bases of the Q-analogue of universal enveloping algebras." Duke Math. J. 63 (1991), no. 2, 465–516.)