Atomic polymorphism

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.

η-conversions of IPC implemented in atomic F

“…Next we will observe that the strategy to prove strong normalization for F at presented in [7] also works to prove strong normalization for F ∧ at . By the Curry-Howard isomorphism also known as "formulas-as-types paradigm", F ∧ at can be presented in the (operational) λ-calculus style.…”

“…The atomic polymorphic calculus F at [3,7] 1 is the restriction of Jean-Yves Girard's system F [9] to atomic universal instantiations. The restriction occurs only in the derivations (terms) allowed, not in the formulas (types) permitted.…”

“…Thus, any deduction in IPC can be performed into F at -a predicative system with no bad connectives, with no commuting conversions, with a simple strong normalization proof [7,4] and whose normal proofs enjoy the subformula property [3].…”

“…It is an easy exercise to adapt the proof of strong normalization presented in [7] (for F at ) to F ∧ at . We sketch such a proof in the next section.…”

“…We sketch such a proof in the next section. It was shown in [7] that the β-conversions of IPC could be implemented in F at through βη-conversions and, as an application of strong normalization for F at considering βη-conversions, the strong normalization for full IPC considering β-conversions was derived. What about the η-conversions?…”