### The λ-Calculus and the Unity of Structural Proof Theory

José Espírito Santo
2009 Theory of Computing Systems
In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cutelimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato's calculus. It is a calculus with modus ponens and primitive substitution; it is also a "coercion calculus", in the
more » ... e of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a "multiary" calculus, because "applicative terms" may exhibit a list of several arguments. But the combination of "multiarity" and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions. Two of the most important systems of formal deduction are sequent calculus and natural deduction, both introduced in Gentzen's seminal paper [15] . When they were introduced, the two systems seemed to differ substantially. Natural deduction manipulated formulas, tried to model informal reasoning, and had an implicit management of structural rules. Sequent calculus manipulated "sequents" (formal instances of the provability relation), tried to model a symmetry * The idea is that, if t, u i and v are mapped by ( ) * to M , N i and P , respectively, then t(u 1 :: · · · u m :: (x)v) is mapped to M N 1 · · · N m /x P .