  In this paper we introduce a system AID (Alogtime Inductive Definitions) of
bounded arithmetic. The main feature of AID is to allow a form of inductive
definitions, which was extracted from Buss' propositional consistency proof of
Frege systems F. We show that AID proves the soundness of F, and conversely any
\Sigma^b_0-theorem in AID yields boolean sentences of which F has polysize
proofs. Further we define \Sigma^b_1-faithful interpretations between AID +
\Sigma b^_0 - CA and a quantified theory QALV of an equational system ALV in P.
Clote. Hence ALV also proves the soundness of F.
