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21
Nominal techniques in Isabelle/HOL
 Proceedings of the 20th International Conference on Automated Deduction (CADE20
, 2005
"... Abstract. In this paper we define an inductive set that is bijective with the ffequated lambdaterms. Unlike deBruijn indices, however, our inductive definition includes names and reasoning about this definition is very similar to informal reasoning on paper. For this we provide a structural induc ..."
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Cited by 87 (14 self)
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Abstract. In this paper we define an inductive set that is bijective with the ffequated lambdaterms. Unlike deBruijn indices, however, our inductive definition includes names and reasoning about this definition is very similar to informal reasoning on paper. For this we provide a structural induction principle that requires to prove the lambdacase for fresh binders only. The main technical novelty of this work is that it is compatible with the axiomofchoice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for ChurchRosser and strongnormalisation. Keywords. Lambdacalculus, nominal logic, structural induction, theoremassistants.
Barendregt’s variable convention in rule inductions
 In Proc. of the 21th International Conference on Automated Deduction (CADE), volume 4603 of LNAI
, 2007
"... Abstract. Inductive definitions and rule inductions are two fundamental reasoning tools in logic and computer science. When inductive definitions involve binders, then Barendregt's variable convention is nearly always employed (explicitly or implicitly) in order to obtain simple proofs. Using t ..."
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Cited by 21 (8 self)
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Abstract. Inductive definitions and rule inductions are two fundamental reasoning tools in logic and computer science. When inductive definitions involve binders, then Barendregt's variable convention is nearly always employed (explicitly or implicitly) in order to obtain simple proofs. Using this convention, one does not consider truly arbitrary bound names, as required by the rule induction principle, but rather bound names about which various freshness assumptions are made. Unfortunately, neither Barendregt nor others give a formal justification for the variable convention, which makes it hard to formalise such proofs. In this paper we identify conditions an inductive definition has to satisfy so that a form of the variable convention can be built into the rule induction principle. In practice this means we come quite close to the informal reasoning of &quot;pencilandpaper &quot; proofs, while remaining completely formal. Our conditions also reveal circumstances in which Barendregt's variable convention is not applicable, and can even lead to faulty reasoning. 1 Introduction In informal proofs about languages that feature bound variables, one often assumes (explicitly or implicitly) a rather convenient convention about those bound variables. Barendregt's statement of the convention is: Variable Convention: If M1; : : : ; Mn occur in a certain mathematical context (e.g. definition, proof), then in these terms all bound variables are chosen to be different from the free variables. [2, Page 26]
A Definitional TwoLevel Approach to Reasoning with HigherOrder Abstract Syntax
 Journal of Automated Reasoning
, 2010
"... Abstract. Combining higherorder abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co ..."
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Cited by 16 (4 self)
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Abstract. Combining higherorder abstract syntax and (co)induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use it in a multilevel reasoning fashion, similar in spirit to other metalogics such as Linc and Twelf. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of nonstratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuationmachine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly
A HeadtoHead Comparison of de Bruijn Indices and Names
 IN PROC. INT. WORKSHOP ON LOGICAL FRAMEWORKS AND METALANGUAGES: THEORY AND PRACTICE
, 2006
"... Often debates about pros and cons of various techniques for formalising lambdacalculi rely on subjective arguments, such as de Bruijn indices are hard to read for humans or nominal approaches come close to the style of reasoning employed in informal proofs. In this paper we will compare four formal ..."
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Cited by 13 (1 self)
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Often debates about pros and cons of various techniques for formalising lambdacalculi rely on subjective arguments, such as de Bruijn indices are hard to read for humans or nominal approaches come close to the style of reasoning employed in informal proofs. In this paper we will compare four formalisations based on de Bruijn indices and on names from the nominal logic work, thus providing some hard facts about the pros and cons of these two formalisation techniques. We conclude that the relative merits of the different approaches, as usual, depend on what task one has at hand and which goals one pursues with a formalisation.
Structural Recursion with Locally Scoped Names
"... This paper introduces a new recursion principle for inductively defined data modulo αequivalence of bound names that makes use of Oderskystyle local names when recursing over bound names. It is formulated in simply typed λcalculus extended with names that can be restricted to a lexical scope, tes ..."
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Cited by 8 (2 self)
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This paper introduces a new recursion principle for inductively defined data modulo αequivalence of bound names that makes use of Oderskystyle local names when recursing over bound names. It is formulated in simply typed λcalculus extended with names that can be restricted to a lexical scope, tested for equality, explicitly swapped and abstracted. The new recursion principle is motivated by the nominal sets notion of “αstructural recursion”, whose use of names and associated freshness sideconditions in recursive definitions formalizes common practice with binders. The new calculus has a simple interpretation in nominal sets equipped with name restriction operations. It is shown to adequately represent αstructural recursion while avoiding the need to verify freshness sideconditions in definitions and computations. The paper is a revised and expanded version of (Pitts, 2010). 1
Simple nominal type theory
"... Abstract. Nominal logic is an extension of firstorder logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standa ..."
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Cited by 8 (1 self)
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Abstract. Nominal logic is an extension of firstorder logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standard type theories for propositional, first or higherorder logic. This has proven difficult, largely because of complex interactions between nominal logic’s nameabstraction operation and ordinary functional abstraction. This difficulty already arises in the case of propositional logic and simple type theory. In this paper we show how this difficulty can be overcome, and present a simple nominal type theory which enjoys properties such as type soundness and strong normalization, and which can be soundly interpreted using existing nominal set models of nominal logic. We also sketch how recursion combinators for languages with binding structure can be provided. This is an important first step towards understanding the constructive content of nominal logic and incorporating it into existing logics and type theories. 1
Nominal System T
, 2010
"... This paper introduces a new recursion principle for inductive data modulo ..."
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Cited by 6 (1 self)
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This paper introduces a new recursion principle for inductive data modulo
Revisiting cutelimination: One difficult proof is really a proof
 RTA 2008
, 2008
"... Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. ..."
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Cited by 5 (3 self)
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Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the first authors PhD. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.
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.