## Termination Proofs for Higher-order Rewrite Systems (1994)

Venue: | IN 1ST INTERNATIONAL WORKSHOP ON HIGHER-ORDER ALGEBRA, LOGIC AND TERM REWRITING |

Citations: | 14 - 0 self |

### BibTeX

@INPROCEEDINGS{Pol94terminationproofs,

author = {Jaco van de Pol},

title = {Termination Proofs for Higher-order Rewrite Systems},

booktitle = {IN 1ST INTERNATIONAL WORKSHOP ON HIGHER-ORDER ALGEBRA, LOGIC AND TERM REWRITING},

year = {1994},

pages = {305--325},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [12]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a wellfounded monotone algebra", described in [8, 19]. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be strictly monotonic in this ordering. This choice generates a model for the HRS. If the choice can be made in such a way that for each rule the interpretation of the left hand side is greater than the interpretation of the right hand side, then the HRS is terminating. At the end of the paper some applications of this technique are given, which show that this technique is natural and can easily be applied.

### Citations

1164 |
The Lambda Calculus, Its Syntax and Semantics
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- 1981
(Show Context)
Citation Context ...lated (Lemma 13). If s = mn: 1. By induction hypothesis (1) both [[m]] ff and [[n]] ff are monotonic. Then by Definition 11, also [[s]] ff = [[m]] ff [[n]] ff is monotonic. 2. By induction hypothesis =-=(2)-=- [[m]] ff mon [[m]] fi . By induction hypothesis (1) [[n]] ff is monotonic, so we have, with Definition 11, that [[m]] ff [[n]] ff mons[[m]] fi [[n]] ff . We also get from the induction hypotheses (1,... |

187 |
Isabelle: The next 700 theorem provers
- Paulson
- 1990
(Show Context)
Citation Context ...echnique is natural and can easily be applied. 1 Introduction In the field of automated proof verification one sees a development towards higher-order concepts. In the generic theorem prover Isabelle =-=[15]-=-, typed lambda calculus is used as the syntax for the formulae. In other systems, as Coq [14], typed lambda calculus is even used for the logic, using the Curry-Howard isomorphism which links formulae... |

173 | Inductive definitions in the system coq - rules and properties
- Paulin-Mohring
- 1993
(Show Context)
Citation Context ... verification one sees a development towards higher-order concepts. In the generic theorem prover Isabelle [15], typed lambda calculus is used as the syntax for the formulae. In other systems, as Coq =-=[14]-=-, typed lambda calculus is even used for the logic, using the Curry-Howard isomorphism which links formulae to types and proofs to terms. This development is mirrored in the research on Term Rewriting... |

133 |
Higher-order critical pairs
- Nipkow
- 1991
(Show Context)
Citation Context ...Utrecht University Heidelberglaan 8, 3584 CS Utrecht, The Netherlands email: jaco@phil.ruu.nl Abstract. This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in =-=[12]. Thi-=-s formalism combines the computational aspects of term rewriting and simply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique "Te... |

104 |
The syntax and semantics of CRL
- Groote, Ponse
- 1995
(Show Context)
Citation Context ... ? [[r]]. In the next section the applicability of this proof method is shown. 6 Applications 6.1 Process Algebra The first application comes from Process Algebra, or better an extension of it: ��=-=CRL [7]. We-=- only concentrate on the fragment of Process Algebra with choice (+), sequential composition ( \Delta ) and deadlock (ffi) and the data dependent choice (\Sigma) from ��CRL. The Process Algebra pa... |

100 | Equations and rewrite rules: a survey
- Huet, Oppen
- 1980
(Show Context)
Citation Context ...ply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a wellfounded monotone algebra", des=-=cribed in [8, 19]-=-. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be str... |

59 |
Combining algebra and higher-order types
- Breazu-Tannen
- 1988
(Show Context)
Citation Context ... which is very similar to CRSs in essence, but rather different in presentation. A precise comparison is given in [17]. A more general setting is given in [18]. Quite other approaches can be found in =-=[4, 9]-=-. Two important issues concerning rewrite systems are termination and confluence. For results about local confluence of HRSs and confluence of orthogonal HRSs the reader is referred to [12] and [13] r... |

38 | Orthogonal higher-order rewrite systems are confluent - Nipkow - 1993 |

30 |
Proofs of strong normalization
- Gandy
- 1980
(Show Context)
Citation Context ...n of terms can be extended to the interpretation of higher-order terms. The orderings and the notion of strictness can also be generalised. The techniques to achieve this are similar to those used in =-=[5, 6]-=-. Moreover, the result that termination proofs can be given with a well-founded monotone algebra in [19] carries over to HRSs with simple conditions on the well-founded ordering. With this technique s... |

28 |
A Computation Model for Executable Higher-Order Algebraic Specification Languages
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- 1991
(Show Context)
Citation Context ... which is very similar to CRSs in essence, but rather different in presentation. A precise comparison is given in [17]. A more general setting is given in [18]. Quite other approaches can be found in =-=[4, 9]-=-. Two important issues concerning rewrite systems are termination and confluence. For results about local confluence of HRSs and confluence of orthogonal HRSs the reader is referred to [12] and [13] r... |

23 |
Combinatory Reduction Systems, volume 127 of Mathematical centre tracts. Mathematisch Centrum
- Klop
- 1980
(Show Context)
Citation Context ... to terms. This development is mirrored in the research on Term Rewriting Systems (TRS). There are different formalisms dealing with the combination of term rewriting and an abstraction mechanism. In =-=[11]-=- the concept of Combinatory Reduction Systems (CRS) was introduced. These systems essentially are TRSs with bound variables. In [12, 13] the formalism of Higher-order Rewrite Systems (HRS) is describe... |

19 |
Surjective Pairing and Strong Normalization: Two Themes
- Vrijer
- 1987
(Show Context)
Citation Context ...n of terms can be extended to the interpretation of higher-order terms. The orderings and the notion of strictness can also be generalised. The techniques to achieve this are similar to those used in =-=[5, 6]-=-. Moreover, the result that termination proofs can be given with a well-founded monotone algebra in [19] carries over to HRSs with simple conditions on the well-founded ordering. With this technique s... |

19 | Comparing Combinatory Reduction Systems and Higher-order Rewrite Systems
- Oostrom, Raamsdonk
(Show Context)
Citation Context ...riables. In [12, 13] the formalism of Higher-order Rewrite Systems (HRS) is described, which is very similar to CRSs in essence, but rather different in presentation. A precise comparison is given in =-=[17]-=-. A more general setting is given in [18]. Quite other approaches can be found in [4, 9]. Two important issues concerning rewrite systems are termination and confluence. For results about local conflu... |

19 |
The Clausal Theory of Types, volume 21 of Cambridge Tracts
- Wolfram
- 1993
(Show Context)
Citation Context ...her-order Rewrite Systems (HRS) is described, which is very similar to CRSs in essence, but rather different in presentation. A precise comparison is given in [17]. A more general setting is given in =-=[18]-=-. Quite other approaches can be found in [4, 9]. Two important issues concerning rewrite systems are termination and confluence. For results about local confluence of HRSs and confluence of orthogonal... |

17 | Verifying process algebra proofs in type theory - Sellink - 1993 |

13 | Termination of term rewriting by interpretation
- ZANTEMA
- 1993
(Show Context)
Citation Context ...ply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a wellfounded monotone algebra", des=-=cribed in [8, 19]-=-. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be str... |

10 |
Term rewriting analysis in process algebra
- Akkerman, Baeten
- 1990
(Show Context)
Citation Context ...ential composition ( \Delta ) and deadlock (ffi) and the data dependent choice (\Sigma) from ��CRL. The Process Algebra part can be formulated in a first order Term Rewriting System (see for insta=-=nce [1]). T-=-he rules for the Sumoperator require higher-order rewrite rules to deal with the bound variables. This reformulation of ��CRL can be found in [16, p. 33]. There are two base types: fProc; Datag. F... |

3 | Perpetual reductions and strong normalization in orthogonal term rewriting systems
- Khasidashvili
- 1993
(Show Context)
Citation Context ...or termination of regular CRSs is given. With this condition, stated in terms of redexes and descendants, a termination proof for CRSs remains a syntactical matter. Other work on this line is done in =-=[10]-=-. We also refer to [9] where a recursion scheme for higher-order rules is given that guarantees termination. Termination of first-order Term Rewriting is already an undecidable problem. But as the ter... |

1 | The APIC-series 31 - In |