The following lemma and proof are from Takeuti's Proof Theory (2nd edition, pp. 126-127), I've highlighted the problematic part in blue:
How does Takeuti get this inequality? If the proof $x$ is not reducible then $r(x) = x$ and also $|x| = |r(x)|$, but to apply the induction hypothesis we need $|r(x)| < |x|$.
It would also be very beneficial to me if someone could explain the role that $u(x)$ is supposed to play in this proof.