Termination of Permutative Conversions in Intuitionistic Gentzen Calculi

It is shown that permutative conversions terminate for the cut-free intuitionistic Gentzen (i.e. sequent) calculus; this proves a conjecture by Dyckhoo and Pinto. The main technical tool is a term notation for derivations in Gentzen calculi. These terms may be seen as-terms with explicit substitution, where the latter corresponds to the left introduction rules. Prawitz 7] deenes a map F transforming derivations in the intuitionistic Gentzen (or sequent) calculus LJ into natural deductions in NJ. Moreover he (essentially) proved surjectivity of F by constructing an inverse map G from NJ back to LJ. Howard 3] identiied a class of cut-free sequent derivations { to be called-normal below { corresponding to normal derivation terms (this is stated without proof in 3, Theorem 2]) 1. A number of authors made use of`permutative conversions' to clarify this situation. Zucker 9] showed for the negative fragment of LJ c (i.e. LJ with cut) that two derivations have the same value under F ii they can be transformed into each other by means of permutative conversions (more precisely: permutative conversions in the sense of Kleene 4] and in addition permutations with the cut rule). Pottinger 6] extended Zucker's results to formulas with _ and 9. Mints 5] and independently Dyckhoo and Pinto 1] prove a similar result for cut free calculi, where in Dyckhoo and Pinto 1] the orientation of permutative conversions is taken into account and connuence is proven. Here we prove termination of some versions of the permutative conversion rules; a weaker form of one of our results was conjectured in 1]. For simplicity we restrict attention to the negative fragment of intuitionistic logic (i.e. of minimal logic, since no symbol ? for falsity is present); however, the arguments below can be extended to the full language with _; 9 (thanks to Matthias HH olzl who has checked this). I would like to thank Roy Dyckhoo for making 1] available to me and patiently explaining its results. Moreover, Roy Dyckhoo and Luis Pinto provided useful comments on an earlier draft. The present paper would not have been possible without this interaction. Thanks are also due to JJ org Hudelmaier, Grigori Mints, Anne Troelstra and three anonymous referees for further helpful comments. 1 Derivations as sequent terms Cut-free derivations in the negative fragment of LJ are denoted by sequent terms; the diierence to-terms and hence to derivation terms in natural deduction is that a …