## Termination and confluence of higher-order rewrite systems (2000)

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Venue: | In Proc. RTA ’00, volume 1833 of LNCS |

Citations: | 18 - 7 self |

### BibTeX

@INPROCEEDINGS{Blanqui00terminationand,

author = {Frédéric Blanqui},

title = {Termination and confluence of higher-order rewrite systems},

booktitle = {In Proc. RTA ’00, volume 1833 of LNCS},

year = {2000},

pages = {47--61},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract: In the last twenty years, several approaches to higher-order rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higher-order Rewrite Systems (HRSs) and Jouannaud and Okada’s higher-order algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higher-order pattern-matching mechanism, resulting in simply-typed CRSs. Then, we show how the termination criterion developed for IDTSs with first-order pattern-matching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higher-order pattern-matching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow’s higher-order critical pair analysis technique for proving local confluence can be applied to IDTSs. 1

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Citation Context ...er-order case the decidability result of Knuth and Bendix about local confluence of first-order term rewrite systems. At the same time, after the works of Breazu-Tannen [6], Breazu-Tannen and Gallier =-=[7]-=- and Okada [24] on the combination of Church’s simply-typed l-calculus 1with first-order term rewriting, Jouannaud and Okada introduced higher-order algebraic specification languages [11, 12] to prov... |

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Citation Context ..., other criteria are available. Van de Pol extended to the higher-order case the use of strictly monotone interpretations [28]. This approach is of course very powerful but it cannot be automated. In =-=[13]-=-, Jouannaud and Rubio defined an extension to the higher-order case of Dershowitz’ Recursive Path Ordering (HORPO) exploiting the notion of computable closure introduced in [5] by Jouannaud, Okada and... |

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Citation Context ...r and higher-order rewrite definitions. Later, together with the present author, they extended these languages with (strictly positive) inductive types, leading to Inductive Data Type Systems (IDTSs) =-=[5]-=-. This approach has also been adapted to richer type disciplines like Coquand and Huet’s Calculus of Constructions [2, 4], in order to extend the equality used in proof assistants based on the Curry-D... |

42 |
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Citation Context ...also be applied to HRSs. Second, we show how Nipkow’s higher-order critical pair analysis technique for proving local confluence can be applied to IDTSs. 1 Introduction In 1980, after a work by Aczel =-=[1]-=-, Klop introduced the Combinatory Rewrite Systems (CRSs) [15, 16], to generalize both first-order term rewriting and rewrite systems with bound variables like Church’s l-calculus. In 1991, after Mille... |

35 |
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Citation Context ... and Gallier [7] and Okada [24] on the combination of Church’s simply-typed l-calculus 1with first-order term rewriting, Jouannaud and Okada introduced higher-order algebraic specification languages =-=[11, 12]-=- to provide a computational model for typed functional languages extended with first-order and higher-order rewrite definitions. Later, together with the present author, they extended these languages ... |

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Citation Context ...trictly positive) inductive types, leading to Inductive Data Type Systems (IDTSs) [5]. This approach has also been adapted to richer type disciplines like Coquand and Huet’s Calculus of Constructions =-=[2, 4]-=-, in order to extend the equality used in proof assistants based on the Curry-De Bruijn-Howard isomorphism like Coq [10] or Lego [17]. Although CRSs and HRSs seem quite different, they have been preci... |

28 | Expression reduction systems
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Citation Context ...both systems has not been clearly stated yet. Other approaches have been proposed like Wolfram’s Higher-Order Term Rewriting Systems (HOTRSs) [33], Khasidashvili’s Expression Reduction Systems (ERSs) =-=[14]-=-, Takahashi’s Conditional Lambda-Calculus (CLC) [27], . . . (see [29]). To tame this proliferation, van Oostrom and van Raamsdonk introduced Higher-Order Rewriting Systems (HORSs) [29, 32] in which th... |

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Citation Context ...the decidability result of Knuth and Bendix about local confluence of first-order term rewrite systems. At the same time, after the works of Breazu-Tannen [6], Breazu-Tannen and Gallier [7] and Okada =-=[24]-=- on the combination of Church’s simply-typed l-calculus 1with first-order term rewriting, Jouannaud and Okada introduced higher-order algebraic specification languages [11, 12] to provide a computati... |

20 | Comparing combinatory reduction systems and higher-order rewrite systems
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Citation Context ... assistants based on the Curry-De Bruijn-Howard isomorphism like Coq [10] or Lego [17]. Although CRSs and HRSs seem quite different, they have been precisely compared by van Oostrom and van Raamsdonk =-=[31]-=-, and shown to have the same expressive power, CRSs using a more lazy evaluation strategy than HRSs. On the other hand, although IDTSs seem very close in spirit to CRSs, the relation between both syst... |

19 |
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Citation Context ...ysis technique can be applied to IDTSs. 2For proving the termination of a HRS, other criteria are available. Van de Pol extended to the higher-order case the use of strictly monotone interpretations =-=[28]-=-. This approach is of course very powerful but it cannot be automated. In [13], Jouannaud and Rubio defined an extension to the higher-order case of Dershowitz’ Recursive Path Ordering (HORPO) exploit... |

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Citation Context ...approaches have been proposed like Wolfram’s Higher-Order Term Rewriting Systems (HOTRSs) [33], Khasidashvili’s Expression Reduction Systems (ERSs) [14], Takahashi’s Conditional Lambda-Calculus (CLC) =-=[27]-=-, . . . (see [29]). To tame this proliferation, van Oostrom and van Raamsdonk introduced Higher-Order Rewriting Systems (HORSs) [29, 32] in which the matching procedure is a parameter called “substitu... |

17 |
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Citation Context ...s seem very close in spirit to CRSs, the relation between both systems has not been clearly stated yet. Other approaches have been proposed like Wolfram’s Higher-Order Term Rewriting Systems (HOTRSs) =-=[33]-=-, Khasidashvili’s Expression Reduction Systems (ERSs) [14], Takahashi’s Conditional Lambda-Calculus (CLC) [27], . . . (see [29]). To tame this proliferation, van Oostrom and van Raamsdonk introduced H... |

10 | Confluence of the lambda calculus with left linear algebraic rewriting - Müller - 1992 |

10 | Modularity of strong normalization in the algebraic-λ-cube
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Citation Context ...trictly positive) inductive types, leading to Inductive Data Type Systems (IDTSs) [5]. This approach has also been adapted to richer type disciplines like Coquand and Huet’s Calculus of Constructions =-=[2, 4]-=-, in order to extend the equality used in proof assistants based on the Curry-De Bruijn-Howard isomorphism like Coq [10] or Lego [17]. Although CRSs and HRSs seem quite different, they have been preci... |

9 |
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Citation Context ... Systems (ERSs) [14], Takahashi’s Conditional Lambda-Calculus (CLC) [27], . . . (see [29]). To tame this proliferation, van Oostrom and van Raamsdonk introduced Higher-Order Rewriting Systems (HORSs) =-=[29, 32]-=- in which the matching procedure is a parameter called “substitution calculus”. It appears that most of the known approaches can be obtained by using an appropriate substitution calculus. Van Oostrom ... |

6 | Modularity of strong normalization in the algebraic--cube - Barbanera, Fern'andez, et al. - 1997 |

5 | Development closed critical pairs
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Citation Context ...fluent. This is the case if it is weakly orthogonal, i.e. it is left-linear and all (higher-order) critical pairs are equal [29], or if it is left-linear and all critical pairs are development closed =-=[30]-=-. Now, one may wonder whether Nipkow’s result for local confluence of HRSs [18] may be applied to IDTSs. To this end, we need to interpret an IDTS as a HRS. This can be done in the following natural w... |

1 |
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Citation Context ... to richer type disciplines like Coquand and Huet’s Calculus of Constructions [2, 4], in order to extend the equality used in proof assistants based on the Curry-De Bruijn-Howard isomorphism like Coq =-=[10]-=- or Lego [17]. Although CRSs and HRSs seem quite different, they have been precisely compared by van Oostrom and van Raamsdonk [31], and shown to have the same expressive power, CRSs using a more lazy... |