Inductive Definitions with Decidable Atomic Formulas

We introduce a type theory for innnitely branching trees, called the theory of free algebras. In this type theory we deene an exten-sional equality based on decidable atomic formulas only. We show, that equality axioms, which add full extensionality to the theory, yield a conservative extension of the (intensional) type theory for formulas having types of level 1. Types like nat ! nat and well-founded trees with branching over the natural numbers (Kleene's O) have this property. We can therefore extract constructive proofs and programs from classical proofs of 2-sentences with this restriction on the types.