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Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal L ..."
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Cited by 161 (15 self)
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This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
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 this c ..."
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Cited by 20 (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 "pencilandpaper " 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 formal treatment of the Barendregt Variable Convention in rule inductions
 In MERLIN ’05: Proceedings of the 3rd ACM SIGPLAN workshop on Mechanized
, 2005
"... Barendregt’s variable convention simplifies many informal proofs in the λcalculus by allowing the consideration of only those bound variables that have been suitably chosen. Barendregt does not give a formal justification for the variable convention, which makes it hard to formalise such informal p ..."
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Cited by 11 (4 self)
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Barendregt’s variable convention simplifies many informal proofs in the λcalculus by allowing the consideration of only those bound variables that have been suitably chosen. Barendregt does not give a formal justification for the variable convention, which makes it hard to formalise such informal proofs. In this paper we show how a form of the variable convention can be built into the reasoning principles for rule inductions. We give two examples explaining our technique.
A Formalisation Of Weak Normalisation (With Respect To Permutations) Of Sequent Calculus Proofs
, 1999
"... rule). This is also the case for NJ and LJ as defined in this formalisation. This is due to the particular nature of the logics in question, and does not necessarily generalise to other logics. In particular, a formalisation of linear logic would not work in this fashion, and a more complex variable ..."
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Cited by 3 (0 self)
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rule). This is also the case for NJ and LJ as defined in this formalisation. This is due to the particular nature of the logics in question, and does not necessarily generalise to other logics. In particular, a formalisation of linear logic would not work in this fashion, and a more complex variablereferencing mechanism would be required. See Section 6 for a further discussion of this problem. Other operations, such as substitutions (sub in Table 2) and weakening, require lift and drop operations as defined in [27] to ensure the correctness of the de Bruijn indexing.
Strong Induction Principles in the Locally Nameless Representation of Binders (Preliminary Notes)
"... Abstract. When using the locally nameless representation for binders, proofs by rule induction over an inductively defined relation traditionally involve a weak and strong version of this relation, and a proof that both versions derive the same judgements. In these notes we demonstrate with examples ..."
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Cited by 3 (1 self)
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Abstract. When using the locally nameless representation for binders, proofs by rule induction over an inductively defined relation traditionally involve a weak and strong version of this relation, and a proof that both versions derive the same judgements. In these notes we demonstrate with examples that it is often sufficient to define just the weak version, using the infrastructure provided by the nominal Isabelle package to automatically derive (in a uniform way) a strong induction principle for this weak version. The derived strong induction principle offers a similar convenience in induction proofs as the traditional approach using weak and strong versions of the definition. From our experience, we conjecture that our technique can be used in many rule and structural induction proofs. 1
Formalising formulasastypesasobjects
 Types for Proofs and Programs
, 2000
"... Abstract. We describe a formalisation of the CurryHowardLawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current ..."
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Cited by 2 (0 self)
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Abstract. We describe a formalisation of the CurryHowardLawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current assumption list Γ and the conclusion A and organise numerous useful lemmas about proof trees categorically. We prove categorical properties about proof trees up to (syntactic) identity as well as up to βηconvertibility. We prove that our notion of proof trees is equivalent in an appropriate sense to more traditional representations of lambda terms. The formalisation is carried out in the proof assistant ALF for MartinLöf type theory. 1
Formal SOSProofs in the LambdaCalculus
 SOS 2007
, 2007
"... We describe in this paper formalisations for the properties of weakening, typesubstitutivity, subjectreduction and terminationof the usual bigstep evaluation relation. Our language is the lambdacalculus whose simplicity allows us to give theoremprover code for the formal proofs. The formalisati ..."
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Cited by 1 (0 self)
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We describe in this paper formalisations for the properties of weakening, typesubstitutivity, subjectreduction and terminationof the usual bigstep evaluation relation. Our language is the lambdacalculus whose simplicity allows us to give theoremprover code for the formal proofs. The formalisations are done in the theorem prover Isabelle/HOL using the nominal datatypepackage. The point of these formalisations is to be as close as possible to the "pencilandpaper" proofs for these properties, but of course be completely rigorous. We describe where the nominal datatype package is of great help with such formalisationsand where one has to invest additional effort in order to obtain formal proofs.
A First Order Theory of Names and Binding
, 2001
"... This talk describes an approach to formalising certain notions that are common in the practice of representing and reasoning about syntax involving variable binding. We concentrate on the use of explicitly named bound variables, rather than the use of nameless terms, explicit substitutions, or highe ..."
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This talk describes an approach to formalising certain notions that are common in the practice of representing and reasoning about syntax involving variable binding. We concentrate on the use of explicitly named bound variables, rather than the use of nameless terms, explicit substitutions, or higher order abstract syntax. We introduce Nominal Logic, a version of firstorder manysorted logic that gives a mathematical status to the taxonomic distinction often made between free and bound names. Nominal Logic contains primitives for renaming via nameswapping and for freshness of names. Its axioms express key properties of these primitives derived from the FMsets model of syntax introduced by Gabbay and Pitts (2001). The main point of the talk is to indicate that nameswapping has much nicer properties than renaming and to emphasise the usefulness, for the practice of operational semantics, of making explicit the equivariance property of assertions about syntaxnamely that their validity is invariant under swapping bindable names. 1 Aim To formalise some informal but familiar practices used for representing and reasoning about syntax involving names and namebinding (e.g. explicitly named bound variables): " rename bound variables to be distinct from each other and from names of free variables in the current context" (the "BVC"); "choose a fresh name"; "by induction on the structure of the syntax, modulo alphaequivalence". Slide 1 Some existing approaches to being more formal about this 0 : Abstractions are pairs. Be naive, but careful, with conventional syntax trees.