Abstract We present the spinal atomic $$\lambda $$ λ -calculus, a typed $$\lambda $$ λ -calculus with explicit sharing and atomic duplication that achieves spinal full laziness: duplicating only the direct paths between a binder and bound variables is enough for beta reduction to proceed. We show this calculus is the result of a Curry–Howard style interpretation of a deep-inference proof system, and prove that it has natural properties with respect to the $$\lambda $$ λ -calculus: confluence and preservation of strong normalisation.

An explicit-sharing lambda-calculus is presented, based on a Curry-Howard-style interpretation of the deep inference proof formalism. Duplication of subterms during reduction proceeds `atomically', i.e. on individual constructors, similar to optimal graph reduction in the style of Lamping. The calculus preserves strong normalisation with respect to the lambda-calculus, and achieves fully lazy sharing.

In usual proof systems, like the sequent calculus, only a very limited way of combining proofs is available through the tree structure. We present in this paper a logic-independent proof calculus, where proofs can be freely composed by connectives, and prove its basic properties. The main advantage of this proof calculus is that it allows to avoid certain types of syntactic bureaucracy inherent to all usual proof systems, in particular the sequent calculus. Proofs in this system closely reflect their atomic flow, which traces the behaviour of atoms through structural rules. The general definition is illustrated by the standard deep-inference system for propositional logic, for which there are known rewriting techniques that achieve cut elimination based only on the information in atomic flows.

Abstract We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.

The strong normalization theorem for second-order classical natural deduction is proved. The method used is an adaptation of the one of reducibility candidates introduced in a thesis by J.Y. Girard (Univ. Paris 7, 1972) for second-order intuitionistic natural deduction. The extension to the classical case requires, in particular, a simplification of the notion of reducibility candidates.>

Les arbres sont les ordres partiels qui vérifient l'axiome supplémental ∀x∀y∀z(y ≤x ∧z ≤ x → y ≤ z ∨ z ≤ y), i.e. l'ensemble des minorants de chaque élément est totalement ordonné. Le principal résultat de cet article concerne l'instabilité des arbres: nous prouvons (§2) qu'aucune théorie d'arbre n'a la propriété d'indépendance, ce qui généralise un theoreme de Poizat [3] sur les ordres totaux. II s'en suit au moyen d'un resultat d'interprétation [4] qu'aucune théorie de structure arborescente n'a la propriété d'indépendance; en particulier ceci vaut pour les arbres colorés. Le §3 est consacré aux arbres stables. II est prouvé qu'un arbre est stable ssi il est superstable ssi il est de hauteur bornée. Sous l'hypothèse de stabilité, les arbres premiers, minimaux, et strictement minimaux sont caractérisés en termes d'al-gébricité.