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Five axioms of alphaconversion
 Ninth international Conference on Theorem Proving in Higher Order Logics TPHOL
, 1996
"... Abstract. We present five axioms of namecarrying lambdaterms identified up to alphaconversion—that is, up to renaming of bound variables. We assume constructors for constants, variables, application and lambdaabstraction. Other constants represent a function Fv that returns the set of free variab ..."
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Abstract. We present five axioms of namecarrying lambdaterms identified up to alphaconversion—that is, up to renaming of bound variables. We assume constructors for constants, variables, application and lambdaabstraction. Other constants represent a function Fv that returns the set of free variables in a term and a function that substitutes a term for a variable free in another term. Our axioms are (1) equations relating Fv and each constructor, (2) equations relating substitution and each constructor, (3) alphaconversion itself, (4) unique existence of functions on lambdaterms defined by structural iteration, and (5) construction of lambdaabstractions given certain functions from variables to terms. By building a model from de Bruijn’s nameless lambdaterms, we show that our five axioms are a conservative extension of HOL. Theorems provable from the axioms include distinctness, injectivity and an exhaustion principle for the constructors, principles of structural induction and primitive recursion on lambdaterms, Hindley and Seldin’s substitution lemmas and
Structuring Metatheory on Inductive Definitions
, 2000
"... We examine a problem for machine supported metatheory. There are statements true about a theory that are true of some (but only some) extensions; however standard theorystructuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and s ..."
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Cited by 7 (0 self)
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We examine a problem for machine supported metatheory. There are statements true about a theory that are true of some (but only some) extensions; however standard theorystructuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and show how a statement about a theory can explicitly formalize the closure conditions extensions should satisfy for it to remain true. We show how metatheories based on inductive denitions allow theories and general metatheorems to be organized this way, and report on a case study using the theory FS0 . 1 Introduction Hierarchical theory structuring plays an important role in the application of theorem provers to nontrivial problems, and many systems provide support for it. For example HOL [6], Isabelle [13] and their predecessor LCF [7] support simple theory hierarchies. In these systems a theory is a specication of a language, using types and typed constants, and a collection of rules...
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... ..."
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.
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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Cited by 6 (3 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 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.
Metatheory in the HigherOrder Logic Framework Isabelle
, 1996
"... Isabelle [Pau94] is a generic theorem proving environment. It is written in ML, and is part of the LCF [GMW79] family of tacticbased theorem provers. The core system uses a metalogic consisting of the implicational/universal fragment of an intuitionistic higherorder logic with equality. Within thi ..."
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Cited by 1 (1 self)
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Isabelle [Pau94] is a generic theorem proving environment. It is written in ML, and is part of the LCF [GMW79] family of tacticbased theorem provers. The core system uses a metalogic consisting of the implicational/universal fragment of an intuitionistic higherorder logic with equality. Within this system, a large number object logics can be represented. A general method of encoding a sequent calculus as an object logic in a form suitable for proving properties of the calculus is presented. Some suggestions about general use of Isabelle in this area are made.
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.
Machinechecked Cutelimination for Display Logic
, 2006
"... Belnap’s Display Logic is a generalised sequentstyle framework which is able to capture many different logics in one uniform setting. Its main attractions are twofold: a “display property ” which allows every formula to be introduced as the whole of the left or right side of a sequent, and a single ..."
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Belnap’s Display Logic is a generalised sequentstyle framework which is able to capture many different logics in one uniform setting. Its main attractions are twofold: a “display property ” which allows every formula to be introduced as the whole of the left or right side of a sequent, and a single cutelimination theorem which works for all “proper ” display calculi. Belnap’s original proof of this theorem is short and succinct, but it has been criticised by some as being too informal. The only other published proof turns out to contain an error. Modern interactive proof assistants like Nqthm, HOL, Isabelle, Coq and PVS invariably trace their origins to the simplytyped lambdacalculus of Church (although Mizar is an exception). Such tools have matured to the point where deep theorems in logic and mathematics like Gödel’s Incompleteness Theorem and the Four Colour Theorem can be expressed and proved by expert users using these systems.
1. What Should Proof Tools Do For Us?................................ 1
, 1996
"... cryptographic protocols Summary. Proof tools must be well designed if they are to be more effective than pen and paper. Isabelle supports a range of formalisms, two of which are described (higherorder logic and set theory). Isabelle’s representation of logic is influenced by logic programming: its ..."
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cryptographic protocols Summary. Proof tools must be well designed if they are to be more effective than pen and paper. Isabelle supports a range of formalisms, two of which are described (higherorder logic and set theory). Isabelle’s representation of logic is influenced by logic programming: its “logical variables ” can be used to implement stepwise refinement. Its automatic proof procedures are based on search primitives that are directly available to users. While emphasizing basic concepts, the article also discusses applications such as an approach to the
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