A Rewrite System for Strongly Normalizable Terms

In a 2012 paper, Richard Statman exhibited an inference system, based on second order monadic logic and non-terminating rewrite rules, that exactly types all strongly normalizable lambdaterms. In this paper, we show that this system can be simplified to first-order minimal logic with rewrite rules, along the Deduction modulo lines. We show that our rewrite system is terminating and that the conversion rule respects weak versions of invertibility of the arrow and of quantifiers. This requires additional care, in particular in the treatment of the latter. Then we study proof reduction, and show that every typable proof term is strongly normalizable and vice-versa. 1998 ACM Subject Classification F.4.1 Mathematical Logic, Dummy classification – please refer to http://www.acm.org/about/class/ccs98-html

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