Cut-elimination for a logic with definitions and induction
Theoretical Computer Science - Special issue on proof-search in type-theoretic languages
Basic proof theory (2nd ed.)
Inductive Definitions in the system Coq - Rules and Properties
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
LICS '95 Proceedings of the 10th Annual IEEE Symposium on Logic in Computer Science
Combining Generic Judgments with Recursive Definitions
LICS '08 Proceedings of the 2008 23rd Annual IEEE Symposium on Logic in Computer Science
LICS '08 Proceedings of the 2008 23rd Annual IEEE Symposium on Logic in Computer Science
Strong normalisation for a gentzen-like cut-elimination procedure
TLCA'01 Proceedings of the 5th international conference on Typed lambda calculi and applications
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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.