We prove the strong normalization theorem for the natural deduction system for the constructive arithmetic TRDB (the system with Definition by Transfinite Recursion and Bar induction), which was introduced by Yasugi and Hayashi. We also establish the consistency of this system, applying the strong normalization theorem.
"Strong Normalization Theorem for a Constructive Arithmetic with Definition by Transfinite Recursion and Bar Induction." Notre Dame J. Formal Logic 38 (3) 350 - 373, Summer 1997. https://doi.org/10.1305/ndjfl/1039700743