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Atomic polymorphism

Published online by Cambridge University Press:  12 March 2014

Fernando Ferreira
Affiliation:
Faculdade de Ciências da Universidade de Lisboa, Departamento de Matemática, Campo Grande, ED. C6, 1749-016 Lisboa, Portugal, E-mail: fjferreira@fc.ul.pt
Gilda Ferreira
Affiliation:
Universidade Lusófona de Humanidades e Tecnologias, Departamento de Matemática, AV DO Campo Grande, 376, 1749-024 Lisboa, Portugal, E-mail: gildafer@cii.fc.ul.pt

Abstract

It has been known for six years that the restriction of Girard's polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the full intuitionistic propositional calculus translates into one or more βη-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to η-conversions.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2013

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