Ëì - Calculus and Böhm ’ S Theorem

The-calculus is an extension of the-calculus that has been introduced by M Parigot to give an algorithmic content to classical proofs. We show that B ¨ ohm's theorem fails in this calculus. §1. Introduction. The-calculus (its typed and untyped versions) has been introduced by M. Parigot in [6]. Its typed version is an extension of the typed-calculus intended to give an algorithmic content to classical proofs. The main computational rules are (the usual one of the-calculus) and. This new rule corresponds (cf. [9]), in the typed version, to the elimination of a logical cut related to the classical rule : If Γ, α : ¬A M : ⊥ , then Γ α.M : A. Two other rules and (that look like, for the-variables, the-rule) also are introduced. In [6], Parigot proved that the (untyped)-calculus with the rules , , and satisfies the Church Rosser property. He also proved ([8]) that every typed term is strongly normalizing. This paper is concerned with B ¨ ohm's theorem. This theorem, in the-calculus, says that if two normal closed terms are computationally equivalent (i.e., when applied to any sequence of arguments the first one is solvable iff the second one also is solvable), then they are-equivalent. We thus also have to consider the-rule. However in the-calculus, the-reduction has not the Church-Rosser's property, because of the following critical pair.   α.M In order to be able to state an equivalent form of B ¨ ohm's theorem we have to restore the confluence. We thus consider another reduction (called the-reduction) which is an-expansion followed by a-reduction : α.M → x.α.M [x/ * α] stands for α.M → exp x.(α.M x) → x.α.M [x/ * α]. This reduction, which corresponds exactly to the reduction defined by Prawitz ([9] Chap. ˜ III, § 1, Theorem I), has also been considered by Parigot ([6]) but only in the typed version. It is proved in [10] that the-reduction satisfies the Church-Rosser's property. In the presence of , Parigot's-reduction is no longer needed. Indeed, any-reduction may be simulated by a-reduction immediately followed by a-reduction.