## Barendregt’s variable convention in rule inductions (2007)

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Venue: | In Proc. of the 21th International Conference on Automated Deduction (CADE), volume 4603 of LNAI |

Citations: | 21 - 8 self |

### BibTeX

@INPROCEEDINGS{Urban07barendregt’svariable,

author = {Christian Urban and Stefan Berghofer and Michael Norrish},

title = {Barendregt’s variable convention in rule inductions},

booktitle = {In Proc. of the 21th International Conference on Automated Deduction (CADE), volume 4603 of LNAI},

year = {2007},

pages = {35--50},

publisher = {Springer}

}

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### Abstract

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 "pencil-and-paper " 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]

### Citations

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Citation Context ...en extensively used in formalisations: for example in our formalisations of the CR and SN properties in the λ-calculus, in a formalisation by Bengtson and Parrow for several proofs in the pi-calculus =-=[3]-=-, in a formalisation of Crary’s chapter on logical relation [4], and in various formalised proofs on structural operational semantics. 2 Nominal Logic Before proceeding, we briefly introduce some impo... |

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Citation Context ...ion principles once and for all. Proofs using the vc-compatible principles then do not need to perform any explicit renaming steps. Somewhat similar to our approach is the work of Pollack and McKinna =-=[6]-=-. Starting from the standard induction principle that is associated with an inductive definition, we derived an induction principle that allows emulation of Barendregt’s variable convention. Pollack a... |

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Citation Context ... induction relying on the substitution lemma, and the lemma fresh-atm, which states that x # y is the same as x �= y when y is an atom. 6 Related Work Apart from our own preliminary work in this area =-=[11]-=-, we believe the prettiest formal proof of the weakening lemma to be that in Pitts [9]. This proof uses the equivariance property of the typing relation, and includes a renaming step using permutation... |

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Citation Context ...lisations of the CR and SN properties in the λ-calculus, in a formalisation by Bengtson and Parrow for several proofs in the pi-calculus [3], in a formalisation of Crary’s chapter on logical relation =-=[4]-=-, and in various formalised proofs on structural operational semantics. 2 Nominal Logic Before proceeding, we briefly introduce some important notions from nominal logic [9,12]. In particular, we will... |

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Citation Context ...s performing the informal proof. Importantly, this new principle can be derived from the original inductive definition of the typing relation in a mechanical way. This method extends our earlier work =-=[11,7]-=-, where we constructed our new induction principles by hand. By formally deriving principles that avoid the need to rename bound variables, we advance the state-of-the-art in mechanical theorem-provin... |