## Eta Expansions in System F (1996)

Venue: | LIENS-DMI, Ecole Normale Superieure |

Citations: | 6 - 0 self |

### BibTeX

@TECHREPORT{Ghani96etaexpansions,

author = {Neil Ghani},

title = {Eta Expansions in System F},

institution = {LIENS-DMI, Ecole Normale Superieure},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

The use of expansionary j-rewrite rules in various typed -calculi has become increasingly common in recent years as their advantages over contractive j-rewrite rules have become apparent. Not only does one obtain the decidability of fij-equality, but rewrite relations based on expansions give a natural interpretation of long fij-normal forms, generalise more easily to other type constructors, retain key properties when combined with other rewrite relations, and are supported by a categorical theory of reduction. This paper extends the initial results concerning the simply typed -calculus to System F, that is, we prove strong normalisation and confluence for a rewrite relation consisting of traditional fi-reductions and j-expansions satisfying certain restrictions. Further, we characterise the second order long fij-normal forms as precisely the normal forms of the restricted rewrite relation. These results are an important step towards showing that j-expansions are compatible with the m...

### Citations

132 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ...quation 2 demonstrates --- for a detailed discussion the reader should consult [3]. These deficiencies in j-contractions have recently led several authors [1, 2, 6, 12] to reconsider the old proposal =-=[11, 13, 14]-=- that j-conversion be interpreted as an expansion f ) x T :fx if f : T!T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such ... |

43 |
OnMintsâ€™ reductions for ccc-Calculus
- Akama
- 1993
(Show Context)
Citation Context ...r with j-contractions confluence is lost as equation 2 demonstrates --- for a detailed discussion the reader should consult [3]. These deficiencies in j-contractions have recently led several authors =-=[1, 2, 6, 12]-=- to reconsider the old proposal [11, 13, 14] that j-conversion be interpreted as an expansion f ) x T :fx if f : T!T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In th... |

37 | The Virtues of Eta-expansion
- Jay, Ghani
- 1995
(Show Context)
Citation Context ...ned with rewrite rules arising from either algebraic rewrite systems or from other type constructors. For instance, the presence of the unit type with j-rewrite rule t ) leads to a loss of confluence =-=[12]-=-. Specifically if f is a variable of type 1!1 then the following divergence cannot be completed. x 1 :s( x 1 :fx ) f (2) The combination of type theories with algebraic rewrite relations is another ar... |

28 | Beta-eta equality for coproducts - Ghani - 1995 |

25 | Adjoint Rewriting
- Ghani
- 1995
(Show Context)
Citation Context ...ormal forms of this restricted rewrite 2 relation are exactly Huet's long fij-normal forms [11, 14]. Most pleasingly of all, these properties tend to be maintained if one adds other type constructors =-=[7]-=-, or algebraic rewrite rules [3]. In addition to these practical arguments, the category-theoretic analysis of reduction [7] provides another argument in favour of interpreting j as an expansion. In t... |

17 |
R'esolution d"equations dans des langages d'ordre 1; 2
- Huet
- 1976
(Show Context)
Citation Context ...ertaining to the use of j-expansions in System F. First, we fulfill our earlier promise of showing that a term is a )F - normal form iff it is a long fij-normal form. A term is a long fij-normal form =-=[11]-=- iff it is a fi-normal form and all subterms are either of base type, introduction terms, occur negatively or are actually types. A term is an internal long fijnormal form iff it is a fi-normal form a... |

16 | Combining first order algebraic rewriting systems, recursion and extensional lambda calculi
- Cosmo, Kesner
- 1994
(Show Context)
Citation Context ... 1 and rewrite rule fx ). This relation is confluent, but when taken together with j-contractions confluence is lost as equation 2 demonstrates --- for a detailed discussion the reader should consult =-=[3]-=-. These deficiencies in j-contractions have recently led several authors [1, 4, 6, 12] to reconsider the old proposal [11, 13, 14] that j-conversion be interpreted as an expansion f ) x : T:fx if f : ... |

15 |
Teorija categorii i teoria dokazatelstv.I. Aktualnye problemy logiki i metodologii nauky
- Mints
- 1979
(Show Context)
Citation Context ...quation 2 demonstrates --- for a detailed discussion the reader should consult [3]. These deficiencies in j-contractions have recently led several authors [1, 2, 6, 12] to reconsider the old proposal =-=[11, 13, 14]-=- that j-conversion be interpreted as an expansion f ) x T :fx if f : T!T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such ... |

12 | Expanding extensional polymorphism
- Cosmo, Piperno
- 1995
(Show Context)
Citation Context ...is requires several alterations to the traditional definition a reducibility candidate to cope with the presence of expansions. This reducibility candidate method differs from the modular approach of =-=[5]-=- which investigates rewriting in System F but does not consider the second order j-rewrite rule. The reader is encouraged to consult [4] where a similar approach to rewriting in System F to that of th... |

8 |
Simulating expansions without expansions
- Cosmo, Kesner
- 1994
(Show Context)
Citation Context ...r with j-contractions confluence is lost as equation 2 demonstrates --- for a detailed discussion the reader should consult [3]. These deficiencies in j-contractions have recently led several authors =-=[1, 2, 6, 12]-=- to reconsider the old proposal [11, 13, 14] that j-conversion be interpreted as an expansion f ) x T :fx if f : T!T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In th... |

7 |
Some -calculi with categorical sums and products
- Dougherty
- 1993
(Show Context)
Citation Context ...r with j-contractions confluence is lost as equation 2 demonstrates --- for a detailed discussion the reader should consult [3]. These deficiencies in j-contractions have recently led several authors =-=[1, 2, 6, 12]-=- to reconsider the old proposal [11, 13, 14] that j-conversion be interpreted as an expansion f ) x T :fx if f : T!T 0 and x 6 2FV(f) and the resulting rewrite relation has been shown confluent. In th... |

6 |
The system F of variable types - 15 years later
- Girard
- 1986
(Show Context)
Citation Context ...r may consult the above references. This paper extends the initial results in a different direction by investigating the use of expansionary j-rewrite rules in a polymorphic -calculus called System F =-=[10, 9]-=-. This calculus was introduced by Girard over twenty years ago and may be thought of as the simply typed -calculus enriched with type variables and a mechanism for forming \Pi-types by universally qua... |

4 |
Rewriting with polymorphic extensional - calculus
- Cosmo, Kesner
- 1995
(Show Context)
Citation Context ...ucibility candidate method differs from the modular approach of [5] which investigates rewriting in System F but does not consider the second order j-rewrite rule. The reader is encouraged to consult =-=[4]-=- where a similar approach to rewriting in System F to that of this paper is taken. In conclusion, there are many reasons for using j-expansions in type theory, e.g. the ease with which they generalise... |

3 |
Combining algebraic rewriting, extensional -calculi and fixpoints
- Cosmo, Kesner
- 1995
(Show Context)
Citation Context ... 1 and rewrite rule fx ). This relation is confluent, but when taken together with j-contractions confluence is lost as equation 2 demonstrates --- for a detailed discussion the reader should consult =-=[3]-=-. These deficiencies in j-contractions have recently led several authors [1, 2, 6, 12] to reconsider the old proposal [11, 13, 14] that j-conversion be interpreted as an expansion f ) x T :fx if f : T... |