Cut-elimination for a logic with definitions and induction
Theoretical Computer Science - Special issue on proof-search in type-theoretic languages
Proof Normalization for a First-Order Formulation of Higher-Order Logic
TPHOLs '97 Proceedings of the 10th International Conference on Theorem Proving in Higher Order Logics
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CADE-12 Proceedings of the 12th International Conference on Automated Deduction
Journal of Automated Reasoning
HOL-λσ: an intentional first-order expression of higher-order logic
Mathematical Structures in Computer Science
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
Cut Elimination in Deduction Modulo by Abstract Completion
LFCS '07 Proceedings of the international symposium on Logical Foundations of Computer Science
Focusing and polarization in linear, intuitionistic, and classical logics
Theoretical Computer Science
Embedding pure type systems in the lambda-pi-calculus modulo
TLCA'07 Proceedings of the 8th international conference on Typed lambda calculi and applications
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RTA'07 Proceedings of the 18th international conference on Term rewriting and applications
Truth values algebras and proof normalization
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
Semantic cut elimination in the intuitionistic sequent calculus
TLCA'05 Proceedings of the 7th international conference on Typed Lambda Calculi and Applications
Rewriting Computation and Proof
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CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
Unbounded proof-length speed-up in deduction modulo
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
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Superdeduction and deduction modulo are two methods designed to ease the use of first-order theories in predicate logic. Superdeduction modulo combines both in a single framework. Although soundness is ensured, using superdeduction modulo to extend deduction with awkward theories can jeopardize cut-elimination or completeness w .r .t predicate logic. In this paper our aim is to design criteria for theories which will ensure these properties. We revisit the superdeduction paradigm by comparing it with the focusing approach. In particular we prove a focalization theorem for cut-free superdeduction modulo: we show that permutations of inference rules can transform any cut-free proof in deduction modulo into a cut-free proof in superdeduction modulo and conversely, provided that some hypotheses on the synchrony of reasoning axioms are verified. It implies that cut-elimination for deduction modulo and for superdeduction modulo are equivalent. Since several criteria have already been proposed for theories that do not break cut-elimination of the corresponding deduction modulo system, these criteria also imply cut-elimination of the superdeduction modulo system, provided our synchrony hypotheses hold.