## Revisiting cut-elimination: One difficult proof is really a proof (2008)

Venue: | RTA 2008 |

Citations: | 5 - 3 self |

### BibTeX

@MISC{Urban08revisitingcut-elimination:,

author = {Christian Urban and Bozhi Zhu},

title = {Revisiting cut-elimination: One difficult proof is really a proof },

year = {2008}

}

### OpenURL

### Abstract

Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term-rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination 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.

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Citation Context ...he present paper we describe a formalisation of an informal 20-page proof given by the first author. This proof claims to establish a strong normalisation result of cut-elimination in classical logic =-=[14, 17]-=-. However, this formalisation, too, uncovers a number of errors in the informal proof, including one that required to restate two central lemmas. One of the main applications of cut-elimination proced... |

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(Show Context)
Citation Context ... the cut-formula is introduced. To specify this operation, we annotated terms to sequent proofs whose inference rules are inspired by Kleene’s sequent calculus G3a [7] and the sequent calculus G3c of =-=[13]-=-. These terms encode the structure of a proof and are defined as: �� � ��� ����� �� Axiom � ��������� ���� � Cut � ���������� ����� �� And-R � ��� � � ������ �� And-L� �� � �� �� � �� � � ������ �� Or... |

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Citation Context ...e lemuridæ system [4] and the typed version ofRevisiting Cut-Elimination: One Difficult Proof Is Really a Proof 411 the X -calculus [20]) or adapted the same proof-technique to other rewrite systems =-=[21]-=-, it seems prudent to reconsider whether the original informal proof is actually a proof. The Nominal Datatype Package [19] provides an infrastructure for reasoning conveniently about datatypes with a... |

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Citation Context ...s quite difficult and since a number of researchers have built their results directly on the strongnormalisation property (for example the lemuridæ system [4] and the typed version of the � -calculus =-=[20]-=-) or adapted the same proof-technique to other rewrite systems [21], it seems prudent to reconsider whether the original informal proof is actually a proof. The Nominal Datatype Package provides an in... |

23 |
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(Show Context)
Citation Context ...cut-formula is introduced. To specify this operation, we used terms to annotate sequent proofs, whose inference rules are inspired by Kleene’s sequent calculus G3a [7] and the sequent calculus G3c of =-=[13]-=-. These terms encode the structure of a proof and are defined as: M,N ::= Ax(x, a) Axiom | Cut(〈a〉M,(x)N) Cut | AndR(〈a〉M,〈b〉N,c) And-R | And i L ((x)M,y) And-Li (i =1, 2) | Or i R(〈a〉M,b) Or-Ri (i =1... |

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Citation Context ...uly arbitrary bound variables, as required by the induction principles, but rather bound variables about which various freshness assumptions are made. Such a reasoning step is in general unsound (see =-=[16]-=- for an example). In informal “pencil-and-paper” proofs such problems are usually ignored. While this is harmless in easy proofs of simple properties, in difficult ones ignoring such problems carries ... |

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Citation Context ... a strong structural induction principle (the strong one has the variable convention already built in [19]), and provides a recursion combinator for defining functions over the structure of the terms =-=[15]-=-. With this combinator, it is easy to define the capture-avoiding renaming functions � �� �� �� and � �� �� ��, although these definitions require that several proof-obligations are discharged by the ... |

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Citation Context ...uthor for the strong-normalisation property is quite difficult and since a number of researchers have built their results directly on the strongnormalisation property (for example the lemuridæ system =-=[4]-=- and the typed version of the � -calculus [20]) or adapted the same proof-technique to other rewrite systems [21], it seems prudent to reconsider whether the original informal proof is actually a proo... |

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Citation Context ...r results directly on the strongnormalisation property (for example the lemuridæ system [4] and the typed version of the � -calculus [20]) or adapted the same proof-technique to other rewrite systems =-=[21]-=-, it seems prudent to reconsider whether the original informal proof is actually a proof. The Nominal Datatype Package provides an infrastructure for reasoning conveniently about datatypes with a buil... |

4 |
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(Show Context)
Citation Context ...ntax—another existing technique for dealing with binders, seems not yet streamlined enough to deal conveniently with logical relation arguments on the scale that are used in the informal proofs above =-=[12]-=-. The formalisation of a weakly normalising cut-elimination procedure done by Pfenning using higher-order abstract syntax in Twelf does not seem to scale to our strong normalisation proof, as it is im... |

3 | Mechanizing the Metatheory of LF - Berhofer, Cheney, et al. - 2010 |