Lexicographic Path Induction

  • Jeffrey Sarnat
  • Carsten Schürmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)


Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers, who often prefer alternatives such as proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a consistency proof of Martin-Löf’s intuitionistic theory of inductive definitions. The consistency of Heyting arithmetic follows directly, and weak normalization for Gödel’s T follows indirectly; both have been formalized in a prototypical extension of Twelf.


Atomic Formula Predicate Symbol Sequent Calculus Proof Tree Proof Rule 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jeffrey Sarnat
    • 1
  • Carsten Schürmann
    • 2
  1. 1.Yale UniversityUSA
  2. 2.IT University of CopenhagenDenmark

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