By V. Michele Abrusci (auth.), Marta Cialdea Mayer, Fiora Pirri (eds.)

This publication constitutes the refereed complaints of the foreign convention on automatic Reasoning with Analytic Tableaux and comparable equipment, TABLEAUX 2003, held in Rome, Italy in September 2003.

The 20 revised complete papers provided have been rigorously reviewed and chosen for inclusion within the publication. All present concerns surrounding the mechanization of logical reasoning with tableaux and related tools are addressed within the context of a huge number of good judgment calculi.

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**Sample text**

Another property can be stated: given a proof Π of a sequent Γ [ϕ]Λ , any sequent Δ [ψ]Ω in Π is such that Λ ≤ Ω. 1 Given a proof Π of a sequent Γ that Λ ≤ Λ . [ϕ]Λ , any term [ψ]Λ appearing in Π is such This property suggests an interesting manipulation of the localizations appearing in a proof. 1 asserts that any localization Ω in Π can be written as Ω1 ·Ω2 with Λ ≤ Ω2 . One would 26 Olivier Brunet then want to replace this localization by Ω1 ·Λ or even Ω1 ·Λ for some Λ ≤ Λ. Such manipulation is in fact necessary if one wants to have the cut-elimination property for , since if one has a proof of Π of Γ [ϕ]Λ , then for Λ ≤ Λ, the following provides a proof of Γ [ϕ]Λ : Λ ≤ Λ Π Γ [ϕ]Λ Γ [ϕ]Λ [ϕ]Λ [ϕ]Λ Axiom Cut In that situation, eliminating the cut implies that one has a way to transform Π into a proof of Γ [ϕ]Λ .

K , Πk Δk , Σk , (μk − λk )q is derivable in GLBn . Hence Q is derivable in GLBn by (nC) as required. ✷ We now check that applications of (gencut) on atomic cut formulae can be eliminated in GLBn . Proposition 9. If Q = Γ1 , λ1 q Δ1 , λ1 q| . . |Γk , λk q Δk , λk q is derivable in GLBn for q atomic and λ1 , . . , λk ≥ 0, then Q = Γ1 Δ1 | . . |Γk Δk is derivable in GLBn . Proof. We follow the proof of Proposition 6 and just check the extra case where the last step in the proof of Q is an application of (nC).

We now show that applications of (cut) on atomic 7 formulae are eliminable. Proposition 5. If G|Γ, λq Δ and G |Π λq, Σ are derivable in GL for q atomic and λ ≥ 0, then G|G |Γ, Π Δ, Σ is derivable in GL. Proof. We prove the more general result that: If Q = Γ1 , λ1 q Δ1 | . . |Γk , λk q Δk and Qi = Gi |Πi Σi , μi q for i = 1, . . , k are derivable in GL for q atomic and μi ≥ λi ≥ 0 for i = 1, . . , k, then Q = G1 | . . |Gk |Γ1 , Π1 Δ1 , Σ1 , (μ1 − λ1 )q| . . |Γk , Πk Δk , Σk , (μk − λk )q is derivable in GL.