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44
Calculating ChurchRosser Proofs in Kleene Algebra
 Relational Methods in Computer Science, 6th International Conference, volume 2561 of LNCS
, 2002
"... We prove ChurchRosser theorems for nonsymmetric transitive relations, quasiorderings and equations in Kleene algebra. Proofs are simple, rigorous and general, using solely algebraic properties of the regular operations. They are fixed pointbased, inductionfree and often amenable to automata. The ..."
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Cited by 9 (4 self)
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We prove ChurchRosser theorems for nonsymmetric transitive relations, quasiorderings and equations in Kleene algebra. Proofs are simple, rigorous and general, using solely algebraic properties of the regular operations. They are fixed pointbased, inductionfree and often amenable to automata. They are mere calculations as opposed to deduction and in particular suited to automation. In the ChurchRosser proofs for the calculus, the term and algebra part are cleanly separated. In all our considerations, Kleene algebra is an excellent means of abstraction.
Mechanizing the Metatheory of LF
, 2008
"... LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF’s judgments. Although detailed informal proofs of these properties hav ..."
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Cited by 9 (6 self)
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LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF’s judgments. Although detailed informal proofs of these properties have been published, they have not been formally verified in a theorem prover. We have formalized these properties within Isabelle/HOL using the Nominal Datatype Package, closely following a recent article by Harper and Pfenning. In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others. Besides its intrinsic interest, our formalization provides a foundation for studying the adequacy of LF encodings, the correctness of Twelfstyle metatheoretic reasoning, and the metatheory of extensions to LF.
The Mechanisation of BarendregtStyle Equational Proofs (the Residual Perspective)
, 2001
"... We show how to mechanise equational proofs about higherorder languages by using the primitive proof principles of firstorder abstract syntax over onesorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for ..."
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Cited by 7 (4 self)
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We show how to mechanise equational proofs about higherorder languages by using the primitive proof principles of firstorder abstract syntax over onesorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for the λcalculus: the residual theory of β is renamingfree upto an initiality condition akin to the socalled Barendregt Variable Convention. We use our results to give a new diagrambased proof of the development part of the strong finite development property for the λcalculus. The proof has the same equational implications (e.g., confluence) as the proof of the full property but without the need to prove SN. We account for two other uses of the proof method, as presented elsewhere. One has been mechanised in full in Isabelle/HOL.
A General Mathematics of Names
 Information and Computation
, 2007
"... We introduce FMG (FraenkelMostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and namecarrying sy ..."
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Cited by 7 (4 self)
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We introduce FMG (FraenkelMostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and namecarrying syntax — have a relation generalising to all sets and not only sets of syntax trees. We also give syntaxfree accounts of Barendregt representatives, scope extrusion, and other phenomena associated to αequivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.
A Theory of Hygienic Macros
"... Abstract. Hygienic macro systems, such as Scheme’s, automatically rename variables to prevent unintentional variable capture—in short, they “just work. ” Yet hygiene has never been formally presented as a specification rather than an algorithm. According to folklore, the definition of hygienic macro ..."
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Abstract. Hygienic macro systems, such as Scheme’s, automatically rename variables to prevent unintentional variable capture—in short, they “just work. ” Yet hygiene has never been formally presented as a specification rather than an algorithm. According to folklore, the definition of hygienic macro expansion hinges on the preservation of alphaequivalence. But the only known notion of alphaequivalence for programs with macros depends on the results of macro expansion! We break this circularity by introducing explicit binding specifications into the syntax of macro definitions, permitting a definition of alphaequivalence independent of expansion. We define a semantics for a firstorder subset of Schemelike macros and prove hygiene as a consequence of confluence. The subject of macro hygiene is not at all decided, and more research is needed to precisely state what hygiene formally means and [precisely which] assurances it provides. —Oleg Kiselyov [1]
Proof pearl: de bruijn terms really do work
 In TPHOLs, volume 4732 of LNCS
, 2007
"... Abstract. Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λcalculus (à laHuet,Nipkowand Shankar) is isomorphic to an αquotiented λcalculus. In order to establish the link, we introduce an “indexcarrying ” abstraction mechanism over de Bruijn t ..."
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Cited by 6 (1 self)
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Abstract. Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λcalculus (à laHuet,Nipkowand Shankar) is isomorphic to an αquotiented λcalculus. In order to establish the link, we introduce an “indexcarrying ” abstraction mechanism over de Bruijn terms, and consider it alongside a simplified substitution mechanism. Relating the new notions to those of the αquotiented and the proper de Bruijn formalisms draws on techniques from the theory of nominal sets. 1
A Comparison of Formalizations of the MetaTheory of a Language with Variable Bindings in Isabelle
 Supplemental Proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics
, 2001
"... Abstract. Theorem provers can be used to reason formally about programming languages and there are various general methods for the formalization of variable binding operators. Hence there are choices for the style of formalization of such languages, even within a single theorem prover. The choice of ..."
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Cited by 5 (2 self)
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Abstract. Theorem provers can be used to reason formally about programming languages and there are various general methods for the formalization of variable binding operators. Hence there are choices for the style of formalization of such languages, even within a single theorem prover. The choice of formalization can affect how easy or difficult it is to do automated reasoning. The aim of this paper is to compare and contrast three formalizations (termed de Bruijn, weak HOAS and full HOAS) of a typical functional programming language. Our contribution is a detailed report on our formalizations, a survey of related work, and a final comparative summary, in which we mention a novel approach to a hybrid de Bruijn/HOAS syntax. 1
A Framework for the Formalisation of Pi Calculus Type Systems in Isabelle/HOL
 in Isabelle/HOL, Proc. TPHOLs01
, 2000
"... . We present a formalisation, in the theorem proving system Isabelle/HOL, of a linear type system for the pi calculus, including a proof of runtime safety of typed processes. The use of a uniform encoding of pi calculus syntax in a meta language, the development of a general theory of type envir ..."
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Cited by 4 (0 self)
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. We present a formalisation, in the theorem proving system Isabelle/HOL, of a linear type system for the pi calculus, including a proof of runtime safety of typed processes. The use of a uniform encoding of pi calculus syntax in a meta language, the development of a general theory of type environments, and the structured formalisation of the main proofs, facilitate the adaptation of the Isabelle theories and proof scripts to variations on the language and other type systems. Keywords: Types; pi calculus; automatic theorem proving; semantics. 1
Nominal renaming sets
"... Abstract. Nominal techniques are based on the idea of sets with a finitelysupported atomspermutation action. We consider the idea of nominal renaming sets, which are sets with a finitelysupported atomsrenaming action; renamings can identify atoms, permutations cannot. We show that nominal renaming ..."
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Cited by 4 (2 self)
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Abstract. Nominal techniques are based on the idea of sets with a finitelysupported atomspermutation action. We consider the idea of nominal renaming sets, which are sets with a finitelysupported atomsrenaming action; renamings can identify atoms, permutations cannot. We show that nominal renaming sets exhibit many of the useful qualities found in (permutative) nominal sets; an elementary setsbased presentation, inductive datatypes of syntax up to binding, cartesian closure, and being a topos. Unlike is the case for nominal sets, the notion of namesabstraction coincides with functional abstraction. Thus we obtain a concrete presentation of sheaves on