η-conversions of IPC implemented in atomic F

Abstract

It is known that the β-conversions of the full intuitionistic propositional calculus (IPC) translate into βη-conversions of the atomic polymorphic calculus F at . Since F at enjoys the property of strong normalization for βη-conversions, an alternative proof of strong normalization for IPC considering β-conversions can be derived. In the present paper we improve the previous result by analyzing the translation of the η-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus F ∧ at ). In fact, from the strong normalization of F ∧ at we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard (β and η) conversions.

A Refined Interpretation of Intuitionistic Logic by Means of Atomic Polymorphism

“…Being ∧ a "good" connective there is no obstacle in considering it as primitive in F at . On the contrary, recent studies in the canonical translation [5,4] show that the implementation of the η-conversions into atomic F or the validity of the Rasiowa-Harrop disjunction property in F at require conjunction to be primitive in the system. To help the comparison with the alternative translation of the previous sections which considers ∧ as primitive in F at we take the canonical translation with primitive conjunction in the target system.…”

“…As an alternative to the canonical embedding of IPC into F at [2,3,4], in this section we introduce another translation (•) • : IPC → F at . The alternative translation comprises the Russell-Prawitz translation of formulas and a translation of proof-terms (which induces a translation of derivations).…”

“…The canonical embedding is by now described in several publications but always in natural deduction style. See for instance [3,2] or [5,4]. In the former references conjunction is interpreted in F at by the Russell-Prawitz's translation of formulas while in the latter references conjunction is a primitive symbol of F at .…”

“…The constructive contents of the proof of this lemma is the proof transformation that underlies instantiation overflow as captured in [5,4]. Def.…”

“…((A ⊃ X) ∧ (B ⊃ X)) ⊃ X, and on instantiation overflow -the possibility of deriving in F at the instantiation of the above universal formulas by any (not necessarily atomic) formula 1 . This embedding, which we call the canonical embedding, works at the levels of provability and proof reduction, with βη-conversions being preserved by the translation [3,4], and commutative conversions being mapped to βηequalities [2].…”