## Reflecting proofs in first-order logic with equality (2005)

Venue: | In Proceedings of EUROSPEECH’97 |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{Contejean05reflectingproofs,

author = {Evelyne Contejean and Pierre Corbineau},

title = {Reflecting proofs in first-order logic with equality},

booktitle = {In Proceedings of EUROSPEECH’97},

year = {2005},

pages = {7--22},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Our general goal is to provide better automation in interactive proof assistants such as Coq. We present an interpreter of proof traces in first-order multi-sorted logic with equality. Thanks to the reflection ability of Coq, this interpreter is both implemented and formally proved sound — with respect to a reflective interpretation of formulae as Coq properties — inside Coq’s type theory. Our generic framework allows to interpret proofs traces computed by any automated theorem prover, as long as they are precise enough: we illustrate that on traces produced by the CiME tool when solving unifiability problems by ordered completion. We discuss some benchmark results obtained on the TPTP library. The aim of this paper is twofold: first we want to validate a reflective approach for proofs in interactive proof assistants, and second show how to provide a better automation for such assistants. Both aspects can be achieved by using external provers designed to automatically solve some problems of interest: these provers can “feed ” the assistant with large proofs, and help to compare the direct and the reflective approaches, and they can also release the user from (parts of) the proof.

### Citations

138 |
Contraction-free sequent calculi for intuitionistic logic
- Dyckhoff
- 1992
(Show Context)
Citation Context ...s and their meaning in Figure 3. The judgement π : Γ ⊢ G means “π is a correct proof trace for the sequent Γ ⊢ G”. We adapted Roy Dychkoff’s contraction free sequent calculus for intuitionistic logic =-=[8]-=-, in order to allow multiple sorts and possibly empty domains. Even though the choice of this sequent calculus may seem exotic, the proof technique is quite generic and our proofs may be easily adapte... |

125 | Completion without failure
- Bachmair, Dershowitz, et al.
- 1989
(Show Context)
Citation Context ...e sorted predicate. The valid predicate states that an integer is a valid index for a given array. The function and predicate signatures corresponding to this theory will be: Fn = 1 : {∅, 1, 0}; 2 : {=-=[2; 1]-=-, 1, get}; 3 : {[2; 1; 1], 2, set} Pred = 1 : {[1; 1], <}; 2 : {[1; 1], ≥}; 3 : {[2], sorted}; 4 : {[1; 2], valid} Definition 1 (term, formula). Terms and formulae are recursively defined as follows: ... |

68 | A compiled implementation of strong reduction
- Gregoire, Leroy
(Show Context)
Citation Context ...actics, a cut-free reified proof and a cut-free proof with tactics. We used the current CVS versions of CiME3 and Coq, with the virtual machine turned on, which helps the Coq kernel reduce terms (see =-=[9]-=-). Coq has been run on each of the 4*230 generated proofs, again with a timeout of 600s. We have observed that the short proofs are always checked in less that 1 second, for reified proofs as well as ... |

59 |
On word problems in equational theories
- Hsiang, Rusinowitch
- 1987
(Show Context)
Citation Context ...interpretation functions (an interpretation is preserved by the addition of variables in the context). 3 Rewriting Traces for Ordered Completion 3.1 Completion Rules The purpose of ordered completion =-=[15, 13, 2]-=- is to build a convergent rewriting system from a set of equations in order to decide the word problem. Some provers such as CiME [4] or Waldmeister [12] use an enhanced version of ordered completion ... |

58 | Metatheory and reflection in theorem proving: A survey and critique,” SRI
- Harrison
- 1995
(Show Context)
Citation Context ... usually have a very fine grained proof notion, whereas automated theorem provers tend to do complex inferences such as term normalization and paramodulation in one single step. Reflection techniques =-=[10]-=- provide a good intermediate layer to turn traces missing a lot of implicit information into fully explicit proofs. They rely on the computation abilities of the proof assistant for trivial parts of p... |

16 | Nivelle. Automated proof construction in type theory using resolution
- Bezem, Hendriks, et al.
- 2002
(Show Context)
Citation Context ...ly explicit proofs. They rely on the computation abilities of the proof assistant for trivial parts of proofs, leaving the hard but interesting work of finding proofs to automated tools. Bezem et al. =-=[3]-=- use reflection techniques to handle the clausification part of a proof but the derivation of the empty clause is provided by an external tool. Our approach extends the reflection technique to the pro... |

11 |
de Bruijn. Lambda calculus with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem
- G
- 1972
(Show Context)
Citation Context ...sents free variables and their indices will refer to the slot they use in the sequent context (see below). The Bv constructor represents bound variables under quantifiers, using the deBruijn notation =-=[7]-=-: the indices are unary integers N , 0 standing for the variable bound by the innermost quantifier over the position of the variable, 1 for the next innermost, etc. The indices in the App and Atom con... |

7 | First-order reasoning in the calculus of inductive constructions - Corbineau - 2004 |

7 | The next WALDMEISTER loop
- Hillenbrand, Löchner
- 2002
(Show Context)
Citation Context ...Rules The purpose of ordered completion [15, 13, 2] is to build a convergent rewriting system from a set of equations in order to decide the word problem. Some provers such as CiME [4] or Waldmeister =-=[12]-=- use an enhanced version of ordered completion in order to solve unifiability problems instead of word problems. We adopt the classical presentation of the completion process as a sequence of applicat... |

5 | Proof reflection in Coq
- Hendriks
(Show Context)
Citation Context ...some automation may be introduced in Coq and release the user from a part of the proof. Previous works on reflection either aimed at proving meta-properties of proof trees in a very general framework =-=[11]-=- or at actually solving domain specific problems and at providing some automation for interactive provers [3, 6]. We claim that our work belongs to the second trend but without loss of generality sinc... |

3 |
Réflexion pour la réécriture dans le calcul des constructions inductives
- Alvarado
- 2002
(Show Context)
Citation Context ...verimag.imag.fr/AVERROES/sOther works integrate either reflection and/or rewriting inside Coq. Nguyen [14] explains how to produce term rewriting proofs, but does not use reflection, whereas Alvarado =-=[1]-=- provides a reflection framework dedicated to proofs of equality of terms. Both of them consider that the rewriting system is fixed a priori. Our approach is close to the work of Crégut [6] who also i... |

3 | Certifying term rewriting proof in elan
- Nguyen
- 2001
(Show Context)
Citation Context ...action TYPES and the French RNTL project AVERROES www.cs.chalmers.se/Cs/Research/Logic/Types/ www-verimag.imag.fr/AVERROES/sOther works integrate either reflection and/or rewriting inside Coq. Nguyen =-=[14]-=- explains how to produce term rewriting proofs, but does not use reflection, whereas Alvarado [1] provides a reflection framework dedicated to proofs of equality of terms. Both of them consider that t... |

1 |
Une procédure de décision réflexive pour l’arithmétique de Presburger en Coq
- Crégut
- 2001
(Show Context)
Citation Context ...s Alvarado [1] provides a reflection framework dedicated to proofs of equality of terms. Both of them consider that the rewriting system is fixed a priori. Our approach is close to the work of Crégut =-=[6]-=- who also interprets proof traces thanks to reflection, but for quantifier-free formulae in Peano’s Arithmetic. In our work, the rewriting system changes during the completion process, and in order to... |