## Cut Elimination in Deduction Modulo by Abstract Completion

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Venue: | in "FoSSaCS 2007, 24/03/2007, Braga/Portugal |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Burel_cutelimination,

author = {Guillaume Burel and Claude Kirchner},

title = {Cut Elimination in Deduction Modulo by Abstract Completion},

booktitle = {in "FoSSaCS 2007, 24/03/2007, Braga/Portugal},

year = {},

pages = {00115556}

}

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### Abstract

### Citations

1015 |
T.: Term Rewriting and All That
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...ion modulo. We use standard definitions for terms, predicates, propositions (with connectors ¬, ⇒, ∧, ∨ and quantifiers ∀, ∃), substitutions, term rewrite rules and term rewriting, as can be found in =-=[2,19]-=-. The set of terms built from a signature ΣCut Elimination in Deduction Modulo by Abstract Completion 5 and a set of variables V is denoted by T (Σ, V ), the replacement of a variable x by a term t i... |

605 | Untersuchungen über das logische Schliessen - Gentzen - 1934 |

274 | Orderings for term-rewriting systems
- Dershowitz
- 1982
(Show Context)
Citation Context ...cedence > : for all formulæ P, Q, if P is greater than Q for the subformula relation, then Cut(P ) > Cut(Q), and for all other inference rules r of Fig. 1, Cut(P ) > r. We order proofs using the RPO (=-=Dershowitz, 1982-=-) based on this precedence. Since the precedence is well-founded, so is the RPO (Dershowitz, 1982). We restrict this ordering to proofs which have the same sequent as conclusion, modulo weakening. Bec... |

167 | Logic for Computer Science: Foundations of Automatic Theorem Proving - Gallier - 1987 |

86 | Theorem Proving Modulo - DOWEK, HARDIN, et al. - 1998 |

51 | Proof normalization modulo, in "J - DOWEK, WERNER |

33 | Proofs and Types, volume 7 of Cambridge Tracts in Theoret Computer Science - Girard, Lafont, et al. - 1989 |

31 | HOL-λσ an intentional first-order expression of higher-order logic - Dowek, Hardin, et al. - 2001 |

23 | Autarkic computations in formal proofs
- Barendregt, Barendsen
- 2002
(Show Context)
Citation Context ...ntage from this complementarity. Deduction2 Guillaume Burel and Claude Kirchner modulo is at the heart of proof assistants and proof search methods, either implicitly or explicitly (see for instance =-=[24,3,16,5]-=-) and getting a deep understanding of its logical behavior is of prime interest either for theoretical or practical purposes. In deduction modulo, computations are modeled by a congruence relation bet... |

17 | Abstract canonical inference
- Bonacina, Dershowitz
- 2006
(Show Context)
Citation Context ...e system. To fully solve this question, including unlimited use of quantifiers, we propose here a quite different approach based on the notion of abstract canonical system and inference introduced in =-=[12,4]-=-. This abstract framework is based on a proof ordering whose goal is to apprehend the notion of proof quality from which the notions of canonicity, completeness and redundancy follow up. It is shown t... |

15 | Term rewriting and all that. Cambridge university press. ISBN - Baader, Nipkow - 1998 |

12 | Abstract saturation-based inference - Dershowitz, Kirchner - 2003 |

9 | Confluence as a cut elimination property - Dowek - 2003 |

7 | Bonacina and Nachum Dershowitz. Abstract canonical inference - Paola - 2007 |

7 | Abstract canonical presentations - Dershowitz, Kirchner |

7 | What is a theory - Dowek - 2002 |

5 | Completion is an instance of abstract canonical system inference
- Burel, Kirchner
- 2006
(Show Context)
Citation Context ...ons of canonicity, completeness and redundancy follow up. It is shown to be adapted to existing completion procedures such as ground completion [10] and standard (a.k.a. Knuth-Bendix [23]) completion =-=[6]-=-. 4 In the case of deduction modulo, the intuitive notion of subformula must of course take into account the equivalence relation.Cut Elimination in Deduction Modulo by Abstract Completion 3 To prese... |

4 |
Tamed: A tableau method for deduction modulo
- Bonichon
- 2004
(Show Context)
Citation Context ... modulo is at the heart of proof assistants and proof search methods, either implicitly or explicitly (see for instance Peterson and Stickel, 1981; Barendregt and Barendsen, 2002; Dowek et al., 2003; =-=Bonichon, 2004-=-) and getting a deep understanding of its logical behavior is of prime interest either for theoretical or practical purposes. In deduction modulo, computations are modeled by a congruence relation bet... |

4 |
Non-normalisation de la théorie de Zermelo, manuscript
- Crabbé
- 1974
(Show Context)
Citation Context ...odulo, the Hauptsatz, i.e. the fact that cuts are not needed to build proofs, is no longer true as one can see from an example derived from Crabbé’s proof of the non-normalization of Zermelo’s theory =-=[8]-=- (see for instance [16]). But we know that the admissibility of the cut rule is fundamental for at least two related reasons: first, if a system admits the cut rule, then the formulæ needed to build a... |

1 |
On generalized theorems for normalization of proofs
- Aiguier, Boin, et al.
- 2005
(Show Context)
Citation Context ...The last case corresponds in fact to an η-expansion.) It is proved in [17] that such an order condition provides normalization in the quantifier-free case. This result was also independently found by =-=[1]-=-, with the same kind of ordering over proofs. 4.2 The Completion Procedure As we wrote in Sect. 2.3, we want to define a completing deduction mechanism by adding to a presentation A a presentation C(A... |