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A Lambda Proof of the P-W Theorem

Sachio Hirokawa, Yuichi Komori and Misao Nagayama
The Journal of Symbolic Logic
Vol. 65, No. 4 (Dec., 2000), pp. 1841-1849
DOI: 10.2307/2695080
Stable URL: http://www.jstor.org/stable/2695080
Page Count: 9
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A Lambda Proof of the P-W Theorem
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Abstract

The logical system P-W is an implicational non-commutative intuitionistic logic defined 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 λ x.x. Here the HRML λ-terms correspond, via the Curry-Howard isomorphism, to the P-W proofs in natural deduction style.

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