## Shallow Dependency Pairs

### BibTeX

@MISC{Krauss_shallowdependency,

author = {Alexander Krauss},

title = {Shallow Dependency Pairs},

year = {}

}

### OpenURL

### Abstract

Abstract. We show how the dependency pair approach, commonly used to modularize termination proofs of rewrite systems, can be adapted to establish termination of recursive functions in a system like Isabelle/HOL or Coq. It turns out that all that is required are two simple lemmas about wellfoundedness. 1

### Citations

729 |
L.C.: Isabelle/Hol: a Proof Assistant for Higher-Order Logic
- Nipkow, Paulson
- 2004
(Show Context)
Citation Context ...raditional dependency pair approach in §3. In §4, we explain dependency pair proofs in the setting of Isabelle/HOL, and give some simple examples in §5 and §6. 2 Preliminaries We work in Isabelle/HOL =-=[12]-=-, but as we do not rely on any special features, the same ideas also apply in a system with different foundations, such as Coq. All theorems and proofs presented in this paper were mechanically checke... |

264 |
Computer-Aided Reasoning: An Approach
- Kaufmann, Monolios, et al.
- 2000
(Show Context)
Citation Context ... In rewriting, this simply corresponds to the rewriting of the right hand sides of the dependency pairs [8, Thm. 21]. The same idea has also been used in an extended termination checker [11] for ACL2 =-=[9]-=-, where it is called merging, but only justified metatheoretically. In our framework, we can give a formal justification of this step, using relation composition: Lemma 2 allows us to replace some cal... |

214 | Termination of term rewriting using dependency pairs
- Arts, Giesl
- 2000
(Show Context)
Citation Context ...re two simple lemmas about wellfoundedness. 1 Introduction Termination proofs are essential in theorem proving, as they are required to justify the definition of recursive functions. Dependency pairs =-=[1,8]-=- are currently one of the most successful approaches to prove termination of term rewrite systems (TRSs). They have been successfully adapted to other situations like functional programs [7]. So it se... |

55 | P.: The dependency pair framework: Combining techniques for automated termination proofs
- Giesl, Thiemann, et al.
- 2004
(Show Context)
Citation Context ...re two simple lemmas about wellfoundedness. 1 Introduction Termination proofs are essential in theorem proving, as they are required to justify the definition of recursive functions. Dependency pairs =-=[1,8]-=- are currently one of the most successful approaches to prove termination of term rewrite systems (TRSs). They have been successfully adapted to other situations like functional programs [7]. So it se... |

33 | Automated termination analysis for haskell: from term rewriting to programming languages
- Giesl, Swiderski, et al.
- 2006
(Show Context)
Citation Context ...y pairs [1,8] are currently one of the most successful approaches to prove termination of term rewrite systems (TRSs). They have been successfully adapted to other situations like functional programs =-=[7]-=-. So it seems natural to try to use such techniques in a theorem proving context. However, there are two main obstacles that make an adoption difficult: First, formalizing the underlying theory is a m... |

32 |
Reasoning about Terminating Functional Programs
- Slind
- 1999
(Show Context)
Citation Context ...) (Suc m) foo (Suc n) (Suc m) = foo n m When a recursive function is defined in Isabelle, the system automatically produces a proof obligation that corresponds to the termination of the function (cf. =-=[10,13]-=-). This proof obligation states the wellfoundedness of the function’s call relation (which relates each possible argument of the function with the resulting recursive calls). For the above definition,... |

26 | Termination analysis with calling context graphs
- Manolios, Vroon
- 2006
(Show Context)
Citation Context ...ly decreasing. In rewriting, this simply corresponds to the rewriting of the right hand sides of the dependency pairs [8, Thm. 21]. The same idea has also been used in an extended termination checker =-=[11]-=- for ACL2 [9], where it is called merging, but only justified metatheoretically. In our framework, we can give a formal justification of this step, using relation composition: Lemma 2 allows us to rep... |

23 |
CoLoR, a Coq library on rewriting and termination
- Blanqui, Delobel, et al.
- 2006
(Show Context)
Citation Context ... relation over, say, natural numbers is wellfounded”), where terms, signatures and reductions never occur on the object level. The first obstacle has been attacked by two recent formalization efforts =-=[2,5]-=-, where large parts of the underlying metatheory were formally verified. This approach allows the certification of TRS proofs in an interactive proof assistant, but it cannot be used to justify the de... |

18 | Certification of automated termination proofs
- Contejean, Courtieu, et al.
- 2007
(Show Context)
Citation Context ... relation over, say, natural numbers is wellfounded”), where terms, signatures and reductions never occur on the object level. The first obstacle has been attacked by two recent formalization efforts =-=[2,5]-=-, where large parts of the underlying metatheory were formally verified. This approach allows the certification of TRS proofs in an interactive proof assistant, but it cannot be used to justify the de... |

15 | Finding lexicographic orders for termination proofs in Isabelle/HOL
- Bulwahn, Krauss, et al.
- 2007
(Show Context)
Citation Context ...ller” element in a relation appears on the left. Consequently, the arguments from the left hand sides of the equations above must go to the right and vice-versa. In previous work (e.g. Bulwahn et al. =-=[3]-=-), this proof obligation was sometimes presented differently, asking for an embedding into another wellfounded relation:1 . wf ?R 2 . V m. ((Suc m, Suc m), (0 , Suc m)) ∈ ?R 3 . V n m. ((n, m), (Suc ... |

8 | Verifying mixed real-integer quantifier elimination
- Chaieb
- 2006
(Show Context)
Citation Context ...ous relation. But the increase might only be temporary. These functions are usually difficult to handle. The following example occurs in a reflected arithmetic decision procedure formalized by Chaieb =-=[4]-=-. It operates on a datatype representing numeric expressions: datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num | Sub num num | Mul int num | Floor num | CF int num num The fun... |