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48
Nominal techniques in Isabelle/HOL
 Proceedings of the 20th International Conference on Automated Deduction (CADE20
, 2005
"... Abstract. In this paper we define an inductive set that is bijective with the ffequated lambdaterms. Unlike deBruijn 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 induc ..."
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Cited by 80 (12 self)
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Abstract. In this paper we define an inductive set that is bijective with the ffequated lambdaterms. Unlike deBruijn 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 lambdacase for fresh binders only. The main technical novelty of this work is that it is compatible with the axiomofchoice (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 ChurchRosser and strongnormalisation. Keywords. Lambdacalculus, nominal logic, structural induction, theoremassistants.
Strong Normalisation of CutElimination in Classical Logic
, 2000
"... In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos ..."
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Cited by 35 (4 self)
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In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cutrules to pass over other cutrules. 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 cutreductions as term rewriting rules.
Orderenriched categorical models of the classical sequent calculus
 LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS
, 2003
"... It is wellknown 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 wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contra ..."
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Cited by 21 (2 self)
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It is wellknown 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 wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cutreduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish nondeterministic choices of cutelimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.
Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward 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 ..."
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Cited by 16 (16 self)
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X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward 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 λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
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Cited by 13 (0 self)
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Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
From X to π; representing the classical sequent calculus
"... Abstract. We study the πcalculus, enriched with pairing and nonblocking 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 (cutelimination) and assignable types are preserved. ..."
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Cited by 12 (12 self)
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Abstract. We study the πcalculus, enriched with pairing and nonblocking 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 (cutelimination) and assignable types are preserved. Since X enjoys the CurryHoward isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have a representation in π.
Proof Transformation by CERES
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM) 2006, VOLUME 4108 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... Cutelimination 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 cutelimination method CERES (cutelimination by resolution) works by constructing a set o ..."
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Cited by 9 (8 self)
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Cutelimination 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 cutelimination method CERES (cutelimination 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 LKproof 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 cutelimination method. The system CERES already proved efficient in handling very large proofs.
Categorical Proof Theory of Classical Propositional Calculus
, 2005
"... We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite wellbehaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. ..."
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Cited by 9 (1 self)
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We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite wellbehaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour.
Normalisation and Equivalence in Proof Theory and Type Theory
, 2006
"... & the advisers At the heart of the connections between Proof Theory and Type Theory, the CurryHoward correspondence provides proofterms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the d ..."
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Cited by 9 (2 self)
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& the advisers At the heart of the connections between Proof Theory and Type Theory, the CurryHoward correspondence provides proofterms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of prooftheoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proofsearch, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proofterms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proofterms for natural deduction (λcalculus) and sequent calculus, and investigate normalisation and cutelimination, with callbyname and callbyvalue semantics. In particular, it introduces proofterm calculi for multiplicative natural deduction and for the depthbounded sequent calculus G4. The former gives rise to the calculus λlxr with explicit substitutions,
CutElimination: Experiments with CERES
, 2005
"... Cutelimination 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 cutelimination method CERES (cutelimination by resolution) works by constructing a set of ..."
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Cited by 9 (8 self)
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Cutelimination 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 cutelimination method CERES (cutelimination 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 sideeffect 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 computeraided proof analysis in mathematics.