The provably terminating operations of the subsystem PETJ of explicit mathematics

In Spescha and Strahm [15], a system PET of explicit mathematics in the style of Feferman [7, 8] is introduced, and in Spescha and Strahm [16] the addition of the join principle to PET is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of PETJ are the polytime functions on binary words. However, although strongly conjectured, it remained open whether the same holds true for the corresponding theory PETJ with classical logic. This note supplements a proof of this conjecture.