Lexicographic Path Induction

13 years 11 months ago

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Abstract. 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-L¨of’s intuitionistic theory of inductive deﬁnitions. The consistency of Heyting arithmetic follows directly, and weak normalization for G¨odel’s T follows indirectly; both have ...

Jeffrey Sarnat, Carsten Schürmann

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