Consistency proof of an arithmetic with substitution inside a bounded arithmetic

In this paper, we prove that S 2 can prove consistency of PV−, the system obtained from Cook and Urquhart’s PV [3] by removing induction. This apparently contradicts Buss and Ignjatović [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 BASICe-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 〈t, ρ〉 ↓ v where ρ is an evaluation of variables and v is the value of t, then there is a derivation of 〈u, ρ〉 ↓ v. By carefully computing the bound of the derivation and ρ, we get S 2 -proof of consistency of PV−.