## Eta-Expansions in Dependent Type Theory - The Calculus of Constructions (1997)

Venue: | Proceedings of the Third International Conference on Typed Lambda Calculus and Applications (TLCA'97 |

Citations: | 13 - 0 self |

### BibTeX

@INPROCEEDINGS{Ghani97eta-expansionsin,

author = {Neil Ghani},

title = {Eta-Expansions in Dependent Type Theory - The Calculus of Constructions},

booktitle = {Proceedings of the Third International Conference on Typed Lambda Calculus and Applications (TLCA'97},

year = {1997},

pages = {164--180},

publisher = {Springer-Verlag LNCS}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Although the use of expansionary j-rewrite has become increasingly common in recent years, one area where j-contractions have until now remained the only possibility is in the more powerful type theories of the -cube. This paper rectifies this situation by applying j-expansions to the Calculus of Constructions --- we discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long fij-normal form and deduce the decidability of fij-equality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed -calculus requires j-conversion, whose interpretation as a rewrite rule has traditionally been as a contraction x : T:fx ) f where x 6 2 FV(t). When combined with the usual fi-reduction, the resulting rewrite relation is strongly normalising and confluent, and thus reduction to normal form provides a decision procedure for the associated equational theory. However j-contractions beh...

### Citations

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Citation Context ...ractions which may be over types which are only fij-equal. This defines an equivalence relation and confluence is first proved only modulo type equivalence. There is a considerable body of literature =-=[2, 15]-=- concerning rewriting modulo equational theories and we now present those definitions and results required to generalise lemma 2.1 to rewriting modulo equivalence relations. In the following R and S a... |

125 |
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Citation Context ...mple shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see [5] for further details. Recently several authors [1, 4, 6, 14] have accepted the old proposal =-=[13, 16, 17]-=- that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as x:t ) y:(x:t)y ) y:t[y=x] j x:t tu ) (x:tx)... |

42 |
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Citation Context ...s confluent. However, as the above counterexample shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see [5] for further details. Recently several authors =-=[1, 4, 6, 14]-=- have accepted the old proposal [13, 16, 17] that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as... |

36 | Eta-expansions in F
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- 1997
(Show Context)
Citation Context ...ms of C as axiomatized in Table 1 are used to define a rewrite relation in two stages. Firstly the purely expansionary j-rewrite relation, denoted ) jF , is presented in Table 2. As in previous works =-=[11, 10]-=-, the condition that terms applied to other terms may not be expanded is enforced by simultaneously defining a further subrelation ) jI of ) jF which is guaranteed not to include top-level expansions.... |

31 | Completion for rewriting modulo a congruence
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Citation Context ...ractions which may be over types which are only fij-equal. This defines an equivalence relation and confluence is first proved only modulo type equivalence. There is a considerable body of literature =-=[2, 15]-=- concerning rewriting modulo equational theories and we now present those definitions and results required to generalise lemma 2.1 to rewriting modulo equivalence relations. In the following R and S a... |

26 | equality for coproducts
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Citation Context ...y the use of expansions is supported by categorical models of reduction [11] which has led to new expansionary j-rewrite rules for the coproduct and the tensor and !-type constructors of linear logic =-=[12, 11]-=-. This paper extends the initial results in a different direction by generalising the techniques previously developed to the Calculus of Constructions (C). The major difference between C and other the... |

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Citation Context ...all, these properties are maintained if one adds other type constructors, base types and associated rewrite rules [5, 3]. Finally the use of expansions is supported by categorical models of reduction =-=[11]-=- which has led to new expansionary j-rewrite rules for the coproduct and the tensor and !-type constructors of linear logic [12, 11]. This paper extends the initial results in a different direction by... |

24 |
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Citation Context ...rms, we use a typed version of fij-equality and so avoid the inconsistencies inherent in the untyped version. What's more, typed and untyped fij-equality coincide on well-typed terms of the same type =-=[9]-=-. Geuvers' paper also proves that typed fij-equality is decidable by orienting the j-equation as a contraction and showing that the resulting rewrite relation, in conjunction with the usual fi-redexes... |

17 |
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Citation Context ...mple shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see [5] for further details. Recently several authors [1, 4, 6, 14] have accepted the old proposal =-=[13, 16, 17]-=- that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as x:t ) y:(x:t)y ) y:t[y=x] j x:t tu ) (x:tx)... |

14 |
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Citation Context ...mple shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see [5] for further details. Recently several authors [1, 4, 6, 14] have accepted the old proposal =-=[13, 16, 17]-=- that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as x:t ) y:(x:t)y ) y:t[y=x] j x:t tu ) (x:tx)... |

9 | On the power of simple diagrams
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Citation Context ...ural mathematical theory of this important class of terms. Perhaps most pleasingly of all, these properties are maintained if one adds other type constructors, base types and associated rewrite rules =-=[5, 3]-=-. Finally the use of expansions is supported by categorical models of reduction [11] which has led to new expansionary j-rewrite rules for the coproduct and the tensor and !-type constructors of linea... |

8 |
Simulating expansions without expansions
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(Show Context)
Citation Context ...s confluent. However, as the above counterexample shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see [5] for further details. Recently several authors =-=[1, 4, 6, 14]-=- have accepted the old proposal [13, 16, 17] that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as... |

7 |
Some -calculi with categorical sums and products
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(Show Context)
Citation Context ...s confluent. However, as the above counterexample shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see [5] for further details. Recently several authors =-=[1, 4, 6, 14]-=- have accepted the old proposal [13, 16, 17] that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as... |

7 |
Relative termination. Dissertation, Fakultät für Mathematik und Informatik, Universität
- Geser
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(Show Context)
Citation Context ...st individual rewrites but also sequences of rewrites. Consequently there is a lot of research into finding sufficient conditions to guarantee that R and S commute and we present one such result from =-=[3, 8]-=-. A relation R is said to strongly locally commute over S if and only if SLC(R;S) iff R \Gamma1 ; S ` S ; (R + ) \Gamma1 Lemma 2.1 If SLC(R;S) and R is strongly normalising, then R and S is commute. A... |

3 |
Combining algebraic rewriting, extensional -calculi and fixpoints
- Cosmo, Kesner
- 1995
(Show Context)
Citation Context ...: 1!1 ands: 1 and with rewrite rule fx ), then ) is confluent. However, as the above counterexample shows shows, the combination of ) with the contractive j-rewrite rule fails to be confluent --- see =-=[5]-=- for further details. Recently several authors [1, 4, 6, 14] have accepted the old proposal [13, 16, 17] that j-conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation ha... |

3 |
On the Definition of the j-long Normal Form in Type Systems of the Cube
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(Show Context)
Citation Context ...ecidable by orienting the j-equation as a contraction and showing that the resulting rewrite relation, in conjunction with the usual fi-redexes, is strongly normalising and confluent. A related paper =-=[7]-=- (which we draw upon) also uses strong normalisation and confluence of fi and j-contractions to prove the existence of fij-long normal forms for C. After collecting some defintions in section 2, prese... |