Translating I 0+exp proofs into weaker systems

The purpose of this paper is to explore the relationship between I∆0+exp and its weaker subtheories. We give a method of translating certain classes of I∆0+exp proofs into weaker systems of arithmetic such as Buss’ systems S2. We show if IEi(exp) ` A with a proof P of expindrank(P ) ≤ n + 1 where all (∀ ≤: right) or (∃ ≤: left) have bounding terms not containing function symbols then Si 2 ⊇ IEi,2 ` An. Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) ( I∆0(exp). Here I∆0(exp) is a conservative extension of I∆0+exp obtained by adding to I∆0 a symbol for 2x to the language as well as defining axioms for it.