Austria-Japan Workshop on Symbolic Computation and Software Verification

We show that continuation passing style translations (CPS-translations) can be used to prove strong cut-elimination of sequent cal-culi. One of the simplest methods to prove strong normalizability (SN) of calculi isto give a reduction preserving translation from the system to another for whichSN is already proved. For a very easy example, SN of the Church-style simplytyped λ-calculus is easily proved by reducing to SN of the Curry-style via theforgetful function λx : A.m = λ.|M |, which preserves one-step reduction relationand typability. For calculi with control operators (or type systems corresponding classicalnatural deduction), such as Parigot’s λμ-calculus, CPS-translations have beenused to prove SN, since it translates programs with control operators to pro-grams without them [2, 12, 6, 3] (these proofs, in fact, contain errors due to thesame problem, and we give a correction for them in [10, 8]). Furthermore, in [4],de Groote introduced a CPS-translation which maps a λ-calculus λ withconjunction and disjunction to λ, and he proved SN of λ by reducingit to the well-known result, SN of λ. In his proof, since the CPS-translationcollapses the permutative conversion for disjunction, we must separately showSN of permutative conversion. In [8], a modified CPS-translation, continuationand garbage passing translation(CGPS-translation), for λ is given, and SNof the system is proved. Since the CGPS-translation preserves one or more stepsreduction including permutative conversion, the proof is simpler than [3]. In this talk, we prove SN of a local-step cut-elimination procedure of anintuitionistic sequent calculus by a CGPS-translation, which is the result of [11].The proof of SN consists of two parts: (1) proving SN of a subsystem LJp of thesequent calculus by a CGPS-translation, (2) reducing SN of the sequent calculusto SN of the LJp. For (2), we adopt the method of [1], which was introducedfor systems with explicit substitutions. We can see that the sequent calculusis isomorphic to a natural deduction with general elimination rules [13] andexplicit substitutions. In this correspondence, the subsystem LJp correspondsto the natural deduction with general elimination rules, for which SN has beenalready proved by Joachimski and Matthes [9] by inductive characterization ofSN terms.