Consistency proof of a feasible arithmetic inside a bounded arithmetic

This paper presents proof that the weakest theory of bounded arithmetic of Buss's hierarchy is capable of proving the consistency of a system based on a feasible arithmetic of Cook and Urquhart from which induction has been removed. This result apparently contradicts the result of Buss and Ignjatovi\'c, who stated that it is not possible to prove such a result. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system we considered is strictly equational, which is a property on which our proof relies. Our proof relies on the big-step semantics of the terms that appear in the theory of Cook and Urquhart. In our work, we first prove that, in the system under consideration, if an equation is proved and either its left or right hand side is computed, then there is a corresponding computation for its right or left hand side, respectively. By carefully computing the bound of the size of the computation, the proof inside a bounded arithmetic of this theorem is obtained, from which the consistency of the system is readily proven.