The Church-Rosser Property for -reduction in Typed -Calculi

In this paper we investigate the Church-Rosser property (CR) for Pure Type Systems with-reduction. For Pure Type Systems with only-reduction, CR on well typed terms follows immediately from CR on the so called 'pseudoterms' and subject reduction. For-reduction, CR on the set of pseudoterms is just false, as was shown by Nederpelt 1973]. Here we prove that CR (for) on the well-typed terms of a xed type holds, which is the maximum we can expect in view of Neder-pelts counterexample. The proof is given for a large class of Pure Type systems that contains e.g. LF (for which CR for was proved by Salvesen 1989] and Coquand 1991]), F, F! and the Calculus of Constructions. In the proof, one key lemma (a very weak form of CR for on pseudoterms) takes a central position. It is remarkable that in the proof of this key lemma the counterexample to CR for is essentially used.