Consistency proof of a feasible arithmetic inside a bounded arithmetic

In this paper, we prove that S 1 2 can prove consistency of PV , the system obtained from Cook and Urquhart's PV (3) by removing induction. This apparently contradicts Buss and Ignjatovic (2), since they prove that PV 6⊢ Con(PV ). However, what they actually prove is unprovability of consistency of the system which is obtained from PV by addition of propositional logic and BASIC e -axioms. On the other hand, our PV is strictly equational and our proof relies on it. Our proof relies on big-step semantics of terms of PV. We prove that if PV ⊢ t = u and there is a derivation of ht,�i ↓ v whereis an evaluation of variables and v is the value of t, then there is a derivation of hu,�i ↓ v. By carefully computing the bound of the derivation and �, we get S 1 2-proof of consistency of PV .