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Formalised Cut Admissibility for Display Logic
 In Proc. TPHOLS'02, LNCS 2410, 131147
, 2002
"... We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machinechecked proof of cutadmissibility for RA. Unlike other "implementations ", we explicitly formalise the structural induction in Isabelle /HOL and believ ..."
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We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machinechecked proof of cutadmissibility for RA. Unlike other "implementations ", we explicitly formalise the structural induction in Isabelle /HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.
A New Machinechecked Proof of Strong Normalisation for Display Logic
 Electronic Notes in Theoretical Computer Science
, 2002
"... We use a deep embedding of the display calculus for relation algebras #RA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for #RA which does not use measures on the size of derivations. Our formalisation generalises easily ..."
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Cited by 6 (2 self)
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We use a deep embedding of the display calculus for relation algebras #RA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for #RA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCKlogic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework.
Abstract A New Machinechecked Proof of Strong Normalisation for Display Logic
"... We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for δRA which does not use measures on the size of derivations. Our formalisation generalises easily ..."
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We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for δRA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCKlogic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework. 1
A New Machinechecked Proof of Strong Normalisation for Display Logic
"... 1 Introduction Sequent calculi provide a rigorous basis for metatheoretic studies of logics. The central theorem is cutelimination which states that detours through lemmata can be avoided, and it can be used to show many important logical properties like consistency, interpolation, and Beth defina ..."
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1 Introduction Sequent calculi provide a rigorous basis for metatheoretic studies of logics. The central theorem is cutelimination which states that detours through lemmata can be avoided, and it can be used to show many important logical properties like consistency, interpolation, and Beth definability. Cutfree sequent calculi are also useful for automated deduction [14], nonclassical extensions of logic programming [22], and studying deep connections between cut elimination, lambda calculi and functional programming. Sequent calculi, and their extensions, therefore play an important role in theoretical computer science.
Transformation Rules for Z
"... Z is a formal specification language combining typed set theory, predicate calculus, and a schema calculus. This paper describes an extension of Z that allows transformation and reasoning rules to be written in a Zlike notation. This gives a highlevel, declarative, way of specifying transformation ..."
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Z is a formal specification language combining typed set theory, predicate calculus, and a schema calculus. This paper describes an extension of Z that allows transformation and reasoning rules to be written in a Zlike notation. This gives a highlevel, declarative, way of specifying transformations of Z terms, which makes it easier to build new Z manipulation tools. We describe the syntax and semantics of these rules, plus some example reasoning engines that use sets of rules to manipulate Z terms. The utility of these rules is demonstrated by discussing two sets of rules. One set defines expansion of Z schema expressions. The other set is used by the ZLive animator to preprocess Z expressions into a form more suitable for animation. 1
Tools and Techniques for Formalising Structural Proof Theory
, 2009
"... Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the l ..."
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Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3c, G3s, G3LC and G4ip. Invertibility is important in another respect; that of proofsearch. Should all rules in a calculus be invertible, then terminating rootfirst proof search gives a decision procedure
Tool support for reasoning in display calculi
"... Abstract. We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calcu ..."
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Abstract. We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of metatools, that allows us to adapt the tool for a wide variety of user defined calculi. 1
Verification
"... Traversal strategies are at the heart of transformational programming with rewritingbased frameworks such as Stratego/XT or Tom and specific approaches for generic functional programming such as Strafunski or “Scrap your boilerplate”. Such traversal strategies are distinctively based on onelayer ..."
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Traversal strategies are at the heart of transformational programming with rewritingbased frameworks such as Stratego/XT or Tom and specific approaches for generic functional programming such as Strafunski or “Scrap your boilerplate”. Such traversal strategies are distinctively based on onelayer traversal primitives from which traversal schemes are derived by recursive closure. We describe a mechanized, formal model of such strategies. The model covers two different semantics of strategies, strategic programming laws, termination conditions for strategy combinators as well as properties related to the success/failure behavior of strategies. The model has been mechanized in Isabelle/HOL.