## 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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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 ...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 |