On the Conservativity of Leibniz Equality

We embed a rst order theory with equality in the Pure Type System L that is a subsystem of the well-known type system PRED2. The embedding is based on the Curry-Howard isomorphism, i.e. ? ? ? > and 8 coincide with ! and. Formulas of the form t1 s = t2 are treated as Leibniz equalities. That is, t1 s = t2 is identiied with the second order formula 8P : P (t1)? ? ? >P(t2), which contains only ? ? ? >'s and 8's and can hence be embedded straightforwardly. We give a syntactic proof for the equivalence between derivability in the logic and inhabitance in L. The idea of the proof is to introduce extra reduction steps, that reduce those proofterms that do not correspond to derivations in the logic to ones that do correspond to derivations in the logic.