On the Correspondence Between Proofs and -Terms

The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors →, ×, +, ∀, ∃, and ⊥ (falsity) (with or without η-like rules). ∗This research was partially supported by ONR Grant NOOO14-88-K-0593.

[1]  D. Prawitz.  Natural Deduction: A Proof-Theoretical Study , 1965 .

[2]  William W. Tait,et al.  Intensional interpretations of functionals of finite type I , 1967, Journal of Symbolic Logic.

[3]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[4]  D. Prawitz.  Ideas and Results in Proof Theory , 1971 .

[5]  J. Girard.  Une Extension De ĽInterpretation De Gödel a ĽAnalyse, Et Son Application a ĽElimination Des Coupures Dans ĽAnalyse Et La Theorie Des Types , 1971 .

[6]  Soren Stenlund.  Combinators, Lambda-Terms and Proof Theory , 1972 .

[7]  J. Y. Girard,et al.  Interpretation fonctionelle et elimination des coupures dans l'aritmetique d'ordre superieur , 1972 .

[8]  Per Martin-Löf,et al.  An intuitionistic theory of types , 1972 .

[9]  W. Tait.  A realizability interpretation of the theory of species , 1975 .

[10]  Dirk van Dalen,et al.  Logic and structure , 1980 .

[11]  George Koletsos,et al.  Church-Rosser theorem for typed functional systems , 1985, Journal of Symbolic Logic.

[12]  J. Gallier.  On Girard's "Candidats de Reductibilité" , 1989 .

[13]  Jean-Yves Girard,et al.  Geometry of Interaction 1: Interpretation of System F , 1989 .

[14]  J. Krivine.  Lambda-calcul : types et modèles , 1990 .

[15]  Jean Gallier,et al.  Constructive Logics Part I: A Tutorial on Proof Systems and Typed gamma-Calculi , 1993, Theor. Comput. Sci..

[16]  Yuri Gurevich,et al.  Logic in Computer Science , 1993, Current Trends in Theoretical Computer Science.