## Extensional rewriting with sums (2007)

Venue: | In TLCA |

Citations: | 5 - 3 self |

### BibTeX

@INPROCEEDINGS{Lindley07extensionalrewriting,

author = {Sam Lindley},

title = {Extensional rewriting with sums},

booktitle = {In TLCA},

year = {2007},

pages = {255--271}

}

### OpenURL

### Abstract

Abstract. Inspired by recent work on normalisation by evaluation for sums, we propose a normalising and confluent extensional rewriting theory for the simply-typed λ-calculus extended with sum types. As a corollary of confluence we obtain decidability for the extensional equational theory of simply-typed λ-calculus extended with sum types. Unlike previous decidability results, which rely on advanced rewriting techniques or advanced category theory, we only use standard techniques. 1

### Citations

369 |
Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems
- Huet
- 1980
(Show Context)
Citation Context ... the axioms of their equational theory into a reduction relation and an equivalence relation, and use Huet’s technique for proving confluence of the reduction relation modulo the equivalence relation =-=[7]-=-. Balat et al use an equivalence for defining normal forms and implementing normalisation by evaluation with sums. Their equivalence is the least congruence satisfying the move-case4 and redundant-gua... |

360 |
Proofs and Types
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ..., x1.n, x2.n) = n, x1, x2 /∈ fv(n) The local η axiom for sums +.η is a special case of +.η † in which n is just x. The move-case axiom is a generalisation of the usual commuting conversions for λ →×+ =-=[6,11]-=-. As well as allowing cases to move across elimination frames (F1[ ]), move-case also allows them to be moved across neutral frames (F2[ ]), lambda frames (F3[ ]) and continuation frames (F4[ ]). (Fra... |

127 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ..., x1.n, x2.n) = n, x1, x2 /∈ fv(n) The local η axiom for sums +.η is a special case of +.η † in which n is just x. The move-case axiom is a generalisation of the usual commuting conversions for λ →×+ =-=[6,11]-=-. As well as allowing cases to move across elimination frames (F1[ ]), move-case also allows them to be moved across neutral frames (F2[ ]), lambda frames (F3[ ]) and continuation frames (F4[ ]). (Fra... |

39 | Normalization by evaluation for typed lambda calculus with coproducts
- Altenkirch, Dybjer, et al.
- 2001
(Show Context)
Citation Context ...cult in the presence of η-rules. Existing rewriting theories, with the exception of Ghani’s [5], are either incomplete with respect to the equational theory or non-confluent. Quoting Altenkirch et al =-=[1]-=-, Ghani’s work involves ‘intricate rewriting techniques whose details are daunting’. Our aim is to introduce a straightforward rewriting theory using standard techniques. The essential reason why the ... |

31 | Extensional normalisation and type-directed partial evaluation for typed lambda calculus with sums - Balat, Cosmo, et al. - 2004 |

26 | The lambda calculus: its syntax and semantics. Number 103 - Barendregt - 1985 |

26 | fij-equality for coproducts
- Ghani
- 1995
(Show Context)
Citation Context ...mply-typed λ-calculus, in the absence of η-rules. However, adding sum types to the rewriting theory is difficult in the presence of η-rules. Existing rewriting theories, with the exception of Ghani’s =-=[5]-=-, are either incomplete with respect to the equational theory or non-confluent. Quoting Altenkirch et al [1], Ghani’s work involves ‘intricate rewriting techniques whose details are daunting’. Our aim... |

15 | Reducibility and >>-lifting for computation types
- Lindley, Stark
- 2005
(Show Context)
Citation Context ...theory that generates the equational theory. Sect. 4 gives a reducibility proof of strong normalisation for a fragment of the rewriting theory following the approach of Lindley and Stark [8, Chaper 3]=-=[9]-=-. Sect. 5 uses strong normalisation results for fragments of the rewriting theory to prove weak normalisation and confluence modulo ∼ for the full rewriting theory, and hence decidability for the equa... |

9 | Normalisation by evaluation in the compilation of typed functional programming languages - Lindley - 2005 |

8 | Normalization by evaluation
- Berger, Eberl, et al.
- 1998
(Show Context)
Citation Context ...oblem, all suggested by the work of Altenkirch et al [1] and Balat et al [2] on normalisation by evaluation for the simply-typed λ-calculus extended with sums. The goal of normalisation by evaluation =-=[4]-=- is to find a unique normal form with respect to the equational theory. In contrast, we shall be interested in normal forms with respect to a rewriting theory. In fact, the case ordering problem also ... |

6 | A terminating and confluent linear lambda calculus
- Ohta, Hasegawa
- 2006
(Show Context)
Citation Context ...ons can be applied indefinitely as m appears as a subterm of n. m = δ(p, x.n, x.δ(p, x.n, x.n)) −→move-case4 −→move-case4 δ(p, x.δ(p, x.n, x.n), x.δ(p, x.n, x.n)) δ(p, x.m, x.m) = n Ohta and Hasegawa =-=[10]-=- face a similar problem for a linear lambda calculus. Their solution is to separate the axioms of their equational theory into a reduction relation and an equivalence relation, and use Huet’s techniqu... |