Abstract. In this paper we define an inductive set that is bijective with the ff-equated lambda-terms. Unlike de-Bruijn indices, however, our inductive definition includes names and reasoning about this definition is very similar to informal reasoning on paper. For this we provide a structural induction principle that requires to prove the lambda-case for fresh binders only. The main technical novelty of this work is that it is compatible with the axiom-of-choice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser and strongnormalisation. Keywords. Lambda-calculus, nominal logic, structural induction, theoremassistants.

In this paper we present a strongly normalising cut-elimination procedure for classical logic. This procedure adapts Gentzen's standard cut-reductions, but is less restrictive than previous strongly normalising cut-elimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cut-rules to pass over other cut-rules. In order to adapt the notion of symmetric reducibility candidates for proving the strong normalisation property, we introduce a novel term assignment for sequent proofs of classical logic and formalise cut-reductions as term rewriting rules.

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

X is an untyped continuation-style formal language with a typed subset which provides a Curry-Howard isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λ-calculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LK-proof with only atomic cuts. In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cut-elimination method. The system CERES already proved efficient in handling very large proofs.

& the advisers At the heart of the connections between Proof Theory and Type Theory, the Curry-Howard correspondence provides proof-terms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of proof-theoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proof-search, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proof-terms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proof-terms for natural deduction (λcalculus) and sequent calculus, and investigate normalisation and cut-elimination, with call-by-name and call-by-value semantics. In particular, it introduces proofterm calculi for multiplicative natural deduction and for the depth-bounded sequent calculus G4. The former gives rise to the calculus λlxr with explicit substitutions,

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skeleton of a proof with only atomic cuts. In this paper we present a systematic experiment with the implementation of CERES on a proof of reasonable size and complexity. It turns out that the proof with cuts can be transformed into two mathematically different proofs of the theorem. In particular, the application of positive and negative hyperresolution yield different mathematical arguments. As an unexpected side-effect the derived clauses of the resolution refutation proved particularly interesting as they can be considered as meaningful universal lemmas. Though the proof under investigation is intuitively simple, the experiment demonstrates that new (and relevant) mathematical information on proofs can be obtained by computational methods. It can be considered as a first step in the development of an experimental culture of computer-aided proof analysis in mathematics.

Abstract. We study the π-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have a representation in π.

Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term-rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the first authors PhD. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.