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Focusing in linear metalogic
 In Proceedings of IJCAR: International Joint Conference on Automated Reasoning, volume 5195 of LNAI
, 2008
"... nigam at lix.polytechnique.fr dale.miller at inria.fr Abstract. It is well known how to use an intuitionistic metalogic to specify natural deduction systems. It is also possible to use linear logic as a metalogic for the specification of a variety of sequent calculus proof systems. Here, we show t ..."
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nigam at lix.polytechnique.fr dale.miller at inria.fr Abstract. It is well known how to use an intuitionistic metalogic to specify natural deduction systems. It is also possible to use linear logic as a metalogic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and nonnormal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics. 1
Validity concepts in prooftheoretic semantics
 ProofTheoretic Semantics. Special issue of Synthese
"... Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in det ..."
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Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in detail various notions of prooftheoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart. 1. Introduction: Prooftheoretic
A Cube of Proof Systems for the Intuitionistic Predicate mu,nuLogic
 Dept. of Informatics, Univ. of Oslo
, 1997
"... This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight pr ..."
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This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight prooftheoretically interesting naturaldeduction calculi for this logic and propose a classification of these into a cube on the basis of the embeddibility relationships between these. 1 Introduction ¯,logics, i.e. logics with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers, have turned out to be a useful formalism in a number of computer science areas. The classical 1storder predicate ¯,logic can been used as a logic of (nondeterministic) imperative programs and as a database query language. It is also one of the relation description languages studied in descriptive complexity theory (finite model theory) (for a survey on this hi...
Tool Support for Logics of Programs
 Mathematical Methods in Program Development: Summer School Marktoberdorf 1996, NATO ASI Series F
, 1996
"... Proof tools must be well designed if they... ..."
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2004).On the notion of assumption in logical systems
 In R
"... When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either ass ..."
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When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either assumptions or axioms, formulas of the sort (ii) are conclusions of (proper) inference rules. Axioms may be viewed as conclusions of (improper) inference rules, viz. inference rules without premisses. In what follows I refer to conclusions of proper or improper inference rules as assertions. 1 In natural deduction systems, inference rules deal both with assumptions and assertions, as the assumptions on which the conclusion of an inference rule depends, are not necessarily given by the collection of all assumptions on which the premisses depend, in case the rule permits the discharging of assumptions. For example, the rule of implication introduction
Using LJF as a Framework for Proof Systems
, 2009
"... In this work we show how to use the focused intuitionistic logic system LJF as a framework for encoding several different intuitionistic and classical proof systems. The proof systems are encoded in a strong level of adequacy, namely the level of (open) derivations. Furthermore we show how to prove ..."
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In this work we show how to use the focused intuitionistic logic system LJF as a framework for encoding several different intuitionistic and classical proof systems. The proof systems are encoded in a strong level of adequacy, namely the level of (open) derivations. Furthermore we show how to prove relative completeness between the different systems. By relative completeness we mean that the systems prove the same formulas. The proofs of relative completeness exploit the encodings to give, in most cases, fairly simple proofs. This work is heavily based on the recent work by Nigam and Miller, which uses the focused linear logic system LLF to encode the same proof systems as we do. Our work shows that the features of linear logic are not needed for the full adequacy result, and furthermore we show that even though encoding in LLF is more generic and streamlined, the encoding in LJF sometimes gives simpler, more natural encodings and easier proofs.
Focusing in linear metalogic: Extended report. Available from http://hal.inria.fr/inria00281631
, 2008
"... Abstract. It is well known how to use an intuitionistic metalogic to specify natural deduction systems. It is also possible to use linear logic as a metalogic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for suc ..."
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Cited by 4 (4 self)
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Abstract. It is well known how to use an intuitionistic metalogic to specify natural deduction systems. It is also possible to use linear logic as a metalogic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and nonnormal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics. 1
Structured induction proofs in Isabelle/Isar
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM 2006), LNAI
, 2006
"... Isabelle/Isar is a generic framework for humanreadable formal proof documents, based on higherorder natural deduction. The Isar proof language provides general principles that may be instantiated to particular objectlogics and applications. We discuss specific Isar language elements that support ..."
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Cited by 4 (1 self)
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Isabelle/Isar is a generic framework for humanreadable formal proof documents, based on higherorder natural deduction. The Isar proof language provides general principles that may be instantiated to particular objectlogics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out wellstructured and readable. Our techniques can be applied to nonstandard variants of induction as well, such as coinduction and nominal induction. This demonstrates that Isar provides a viable platform for building domainspecific tools that support fullyformal mathematical proof composition.