Preservation of equivalence of derivations under reduction of depth of formulas

Abstract

In this paper we consider derivations in the (&, ⊃)-fragment of the intuitionistic propositional calculus. As is known, replacement of any occurrence of a formula Φ[F] in a sequent S by an occurrence of the formula Φ[p], where p is a new propositional variable, with the simultaneous addition to the antecedent of the formula F ⊃ p or p ⊃ F depending on the sign of the occurrence of F in S, leaves the derivability unchanged. We give a proof of the fact that the natural extension of this transformation to derivations preserves the relation of equivalence of derivations, i.e., transformed derivations are equivalent if and only if the originals are equivalent. (Derivations are considered equivalent if certain of their normal forms coincide, or, what is the same, if their deductive terms coincide.) It is proved that by the iteration of this transformation, each derivation of an arbitrary sequent S can be transformed into a derivation of a sequent S′, depending only on S, whose succedent is a variable, and in the antecedent there occur only formulas of the form a,a & b, a ⊃ b,,(a ⊃ b) ⊃ c, a & b ⊃ c, a ⊃(b & c), wherea, b, c are variables. Here if S is balanced, then S′ is also balanced. (A sequent is called balanced if each variable occurs in it no more than twice.) The familiar correspondence between certain concepts of the theory of categories and concepts of the theory of proofs allows one to assert that there has been constructed a univalent functor, mapping a free Cartesian closed category into itself.