A lambda proof of the P-W theorem

Abstract

AbstractThe logical system P-W is an implicational non-commutative intuitionistic logic denned by axiom schemes B − (b → c) → (a → b) → a → c. B′ = (a → b) → (b → c) → a → c. I - a → a with the rules of modus ponens and substitution. The P-W problem is a problem asking whether α - β holds if α → β and β → α are both provable in P-W. The answer is affirmative. The first to prove this was E. P. Martin by a semantical method.In this paper, we give the first proof of Martin's theorem based on the theory of simply typed λ-calculus. This proof is obtained as a corollary to the main theorem of this paper, shown without using Martin's Theorem, that any closed hereditary right-maximal linear (HRML) λ-term of type α → α is βη-reducible to λxx. Here the HRML λ-terms correspond, via the Curry-Howard isomorphism, to the P-W proofs in natural deduction style.