$\lambda\rho$-calculus II

Abstract

In [4], the author introduced the system $\lambda\rho$-calculus and stated without proof that the strong normalization theorem holds. Here we introduce a lemma (Lemma 4.10) and use it to prove the strong normalization theorem. While a typed $\lambda$-term itself is a derivation of the natural deduction for intuitionistic implicational logic (cf. [2]), a typed $\lambda\rho$-term itself is a derivation of the natural deduction for classical implicational logic. Our system is simpler than the implicational fragment of Parigot's $\lambda\mu$-calculus (cf. [5]).