Cut Elimination, Substitution and Normalisation

  • Roy DyckhoffEmail author
Part of the Outstanding Contributions to Logic book series (OCTR, volume 7)


We present a proof (of the main parts of which there is a formal version, checked with the Isabelle proof assistant) that, for a G3-style calculus covering all of intuitionistic zero-order logic, with an associated term calculus, and with a particular strongly normalising and confluent system of cut-reduction rules, every reduction step has, as its natural deduction translation, a sequence of zero or more reduction steps (detour reductions, permutation reductions or simplifications). This complements and (we believe) clarifies earlier work by (e.g.) Zucker and Pottinger on a question raised in 1971 by Kreisel.


Intuitionistic logic Minimal logic Sequent calculus Natural deduction Cut-elimination Substitution Normalisation 



Thanks are due to Jan von Plato, Peter Chapman, Jacob Howe, Stéphane Graham-Lengrand and Christian Urban for helpful comments and (to the last of these) for a copy of (Urban 2014) prior to its publication, albeit many years after its 2001 presentation in Rio. Chapman’s work (Dyckhoff and Chapman 2009) (incorporating also ideas by Urban) was invaluable in checking the correctness of all the lemmata about substitution. The work was motivated by requirements for some not yet published work (joint with James Caldwell) supported by EPSRC grant EP/F031114/1.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Computer ScienceSt Andrews UniversityScotlandUK

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