Although the idea is straightforward, the proof requires some messy details and it's not clear whether it can be generalised to other systems in the spirit of PV omega . Roughly, our proof goes as follows.
Some work has to be done to interpret the recursor.
Preliminary investigation has shown that using a different category the same argument can be carried out for a higher-order version of Bellantoni's system BC which unlike his system BC omega provides safe composition at all types and no primitive functions other than those of BC . A draft paper on this subject is available here .