## A Structural Proof of Cut Elimination and Its Representation in a Logical Framework (1994)

Citations: | 17 - 4 self |

### BibTeX

@TECHREPORT{Pfenning94astructural,

author = {Frank Pfenning},

title = {A Structural Proof of Cut Elimination and Its Representation in a Logical Framework},

institution = {},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present new proofs of cut elimination for intuitionistic and classical sequent calculi. In both cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and concise representations in LF, which are given in full detail. This work was supported by NSF Grant CCR-9303383 The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of NSF or the U.S. government. Keywords: Logic, Cut Elimination, Logical Framework Contents 1 Introduction 1 2 Intuitionistic Sequent Calculus 2 3 Proof Terms for the Sequent Calculus 8 4 Representing Sequent Derivations in LF 10 5 Admissibility of Cut 13 6 Extension to Classical Logic 18 7 Conclusion 24 A Detailed Admissibility Proofs for Cut 26 A.1 Intuitionistic Calculus : : : : : : : : : : : : : : : : : : :...

### Citations

762 |
Introduction to Metamathematics
- Kleene
(Show Context)
Citation Context ...ee nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. Starting point for the proofs is Kleene's sequent system G 3 =-=[Kle52]-=-, which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other se... |

716 | A framework for defining logics
- Harper, Honsell, et al.
- 1993
(Show Context)
Citation Context ...f parameters in sequent derivations. In this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF =-=[HHP93]-=- as implemented in the Elf system [Pfe91]. Multi-sets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as var... |

669 | Light linear logic
- Girard
- 1998
(Show Context)
Citation Context ...entation [Fel89] in Prolog, for example, uses lists of hypotheses which is advantageous for search but makes a formal meta-theory prohibitively complex. Frameworks based on sequent calculi such as LU =-=[Gir93]-=- or Forum [Mil94] allow direct encodings, but they lack a notation for the proof terms that are required to describe cut elimination. In this section we develop a formulation of the sequent calculus f... |

597 |
Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
Citation Context ... : : : 56 B.2 Classical Calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 Acknowledgments 62 Referencess1 INTRODUCTION 1 1 Introduction Gentzen's sequent calculi =-=[Gen35]-=- for intuitionistic and classical logic have been the central tool in many proof-theoretical investigations and applications of logic in computer science such as logic programming (e.g. [MNPS91]) or a... |

385 | Uniform proofs as a foundation for logic programming
- Miller, Nadathur, et al.
- 1991
(Show Context)
Citation Context ...alculi [Gen35] for intuitionistic and classical logic have been the central tool in many proof-theoretical investigations and applications of logic in computer science such as logic programming (e.g. =-=[MNPS91]-=-) or automated theorem proving (e.g. [Wal90]). The central property of sequent calculi is cut elimination (Gentzen's Hauptsatz) which yields consistency of the logic as a corollary. The algorithm for ... |

290 | Computational interpretation of linear logic
- Abramsky
- 1990
(Show Context)
Citation Context ...milar to the way that normalization for natural deduction may be viewed as functional computation. For the case of linear logic, this point was made by Girard [Gir87] and later elaborated by Abramsky =-=[Abr93]-=-; see also [Gal93] for a tutorial introduction. Many proofs of cut elimination have been given in the literature yet, to our knowledge, none of them have been formalized even though this is clearly po... |

177 | Logic programming in the LF logical framework
- Pfenning
- 1991
(Show Context)
Citation Context ...his paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system =-=[Pfe91]-=-. Multi-sets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus natural... |

132 |
Higher-order critical pairs
- Nipkow
- 1991
(Show Context)
Citation Context ...zation and cut elimination (see, for example, [Zuc74]) in the same framework. Another direction is to study cut elimination in a formulation as a higher-order rewrite system along the lines of Nipkow =-=[Nip91]-=-, but using dependent types. We first note that our system of rules is terminating (note that we cannot permute adjacent cuts!). Assuming the completeness of a critical pair criterion for the dependen... |

86 | A multiple-conclusion meta-logic
- Miller
- 1994
(Show Context)
Citation Context ...in Prolog, for example, uses lists of hypotheses which is advantageous for search but makes a formal meta-theory prohibitively complex. Frameworks based on sequent calculi such as LU [Gir93] or Forum =-=[Mil94]-=- allow direct encodings, but they lack a notation for the proof terms that are required to describe cut elimination. In this section we develop a formulation of the sequent calculus for intuitionistic... |

69 | A typed pattern calculus
- Breazu-Tannen, Kesner, et al.
- 1993
(Show Context)
Citation Context ...toward the representation of the rules in LF. There are at least three distinct roles that proof terms may play for a sequent calculus, an issue recognized by Gallier [Gal93] and Breazu-Tannen et al. =-=[BTKP93]-=-. The most immediate perhaps is to annotate sequent derivations with -terms that represent the natural deductions they correspond to. The second is to think of proof terms as expressions in a programm... |

60 | Constructive logics. Part I: A tutorial on proof systems and typed λ-calculi
- Gallier
- 1991
(Show Context)
Citation Context ...hat normalization for natural deduction may be viewed as functional computation. For the case of linear logic, this point was made by Girard [Gir87] and later elaborated by Abramsky [Abr93]; see also =-=[Gal93]-=- for a tutorial introduction. Many proofs of cut elimination have been given in the literature yet, to our knowledge, none of them have been formalized even though this is clearly possible (see, for e... |

50 | Specifying and Implementing Theorem Provers in a Higher-Order Logic Programming Language
- Felty
- 1989
(Show Context)
Citation Context ...Finding an elegant encoding of sequents and sequent derivations in a logical framework is therefore the first critical issue in an implementation of a proof of cut elimination. Felty's representation =-=[Fel89]-=- in Prolog, for example, uses lists of hypotheses which is advantageous for search but makes a formal meta-theory prohibitively complex. Frameworks based on sequent calculi such as LU [Gir93] or Forum... |

48 |
The correspondence between cut-elimination and normalization
- Zucker
- 1974
(Show Context)
Citation Context ...he (meta-meta-)theoretical analysis of schema-checker itself is not yet complete. In other future work we plan to reexamine the connection between normalization and cut elimination (see, for example, =-=[Zuc74]-=-) in the same framework. Another direction is to study cut elimination in a formulation as a higher-order rewrite system along the lines of Nipkow [Nip91], but using dependent types. We first note tha... |

46 | The Coq proof assistant user's guide, Rapport Techniques 154 - Dowek, Felty, et al. - 1993 |

44 | Natural semantics and some of its metatheory in Elf - Michaylov, Pfenning - 1991 |

24 |
Automated Proof Search in Non-Classical Logics: Efficient Matrix Proof Methods for Modal and Intuitionistic Logics
- Wallen
- 1987
(Show Context)
Citation Context ...al logic have been the central tool in many proof-theoretical investigations and applications of logic in computer science such as logic programming (e.g. [MNPS91]) or automated theorem proving (e.g. =-=[Wal90]-=-). The central property of sequent calculi is cut elimination (Gentzen's Hauptsatz) which yields consistency of the logic as a corollary. The algorithm for cut elimination may be interpreted computati... |

7 | A theory and its metatheory in FS0
- Matthews
- 1994
(Show Context)
Citation Context ...ntroduction. Many proofs of cut elimination have been given in the literature yet, to our knowledge, none of them have been formalized even though this is clearly possible (see, for example, Matthews =-=[Mat94]-=- pencil-and-paper analysis of cut elimination for the (; :) fragment of classical propositional logic in FS 0 ). They are difficult to mechanize for a number of reasons which in combination are quite ... |

7 |
Verifying the meta-theory of deductive systems. Thesis Proposal
- Rohwedder
- 1994
(Show Context)
Citation Context ...ge which, we hope, others will take up. In future work we plan to verify mechanically that the given signatures indeed implement proofs. The prototype implementation of the schema-checker sketched in =-=[Roh94]-=- currently accepts them, but the (meta-meta-)theoretical analysis of schema-checker itself is not yet complete. In other future work we plan to reexamine the connection between normalization and cut e... |