## Type-checking Injective Pure Type Systems (1993)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Barthe93type-checkinginjective,

author = {Gilles Barthe},

title = {Type-checking Injective Pure Type Systems},

year = {1993}

}

### OpenURL

### Abstract

Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(\GammajM ) and sort(\GammajM ), where \Gamma is a pseudo-context and M is a pseudo-term, and such that for every sort s, \Gamma ` M : A \Gamma ` A : s ) elmt(\GammajM ) = s \Gamma ` M : s ) sort(\GammajM ) = s By eliminating the problematic clause in the (abstraction) rule in favor of constraints over elmt(:j:) and sort(:j:), we provide a sound and complete type-checking algorithm for injective Pure Type Systems. In addition, we prove Expansion Postponement for a variant of injective Pure Type Systems where the problematic clause in the (abstraction) rule is replaced in favor of constraints over elmt(:j:) and sort(:j:). 1

### Citations

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Citation Context ...ction) rule is replaced in favour of constraints over elmt(:j:) and sort(:j:). 1 Introduction Pure Type Systems provide an elegant and general framework for the definition and study of typed -calculi =-=[2, 7, 9, 10, 19]-=-. One central issue in the theory of Pure Type Systems is the problem of type-checking. Given a Pure Type System S, type-checking consists in deciding whether a judgment \Gamma ` M : A is derivable ac... |

109 | On the type structure of Standard ML - Harper, Mitchell - 1993 |

88 | Logics and Type Systems
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Citation Context ...ction) rule is replaced in favour of constraints over elmt(:j:) and sort(:j:). 1 Introduction Pure Type Systems provide an elegant and general framework for the definition and study of typed -calculi =-=[2, 7, 9, 10, 19]-=-. One central issue in the theory of Pure Type Systems is the problem of type-checking. Given a Pure Type System S, type-checking consists in deciding whether a judgment \Gamma ` M : A is derivable ac... |

69 | The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions
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Citation Context ... it is possible to find reasonable, sound and complete, algorithms for type-checking. Over the last years, several authors have proposed such algorithms for some specific classes of Pure Type Systems =-=[3, 6, 13, 16, 17, 18]-=-. However, the situation is in our view unsatisfactory because these algorithms either impose strong restrictions on the Pure Type Systems or make use of a complex derivability relation, see Section 6... |

47 | A short and proof of strong normalization for the Calculus of Constructions - Geuvers - 1995 |

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Citation Context ...ithm is sound and complete for injective Pure Type Systems, a class of Pure Type Systems that includes many of the systems occurring in the literature, in particular the systems of Barendregt's -cube =-=[1, 2]-=-. The idea is to define for every pseudo-context \Gamma and pseudo-term M two sorts elmt(\GammajM ) and sort(\GammajM ), which may be computed easily and without invoking conversion or substitution, a... |

29 | Checking Algorithms for Pure Type Systems
- Jutting, S, et al.
- 1993
(Show Context)
Citation Context ... it is possible to find reasonable, sound and complete, algorithms for type-checking. Over the last years, several authors have proposed such algorithms for some specific classes of Pure Type Systems =-=[3, 6, 13, 16, 17, 18]-=-. However, the situation is in our view unsatisfactory because these algorithms either impose strong restrictions on the Pure Type Systems or make use of a complex derivability relation, see Section 6... |

29 |
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Citation Context ...ction) rule is replaced in favour of constraints over elmt(:j:) and sort(:j:). 1 Introduction Pure Type Systems provide an elegant and general framework for the definition and study of typed -calculi =-=[2, 7, 9, 10, 19]-=-. One central issue in the theory of Pure Type Systems is the problem of type-checking. Given a Pure Type System S, type-checking consists in deciding whether a judgment \Gamma ` M : A is derivable ac... |

29 |
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23 |
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Citation Context ... whether a judgment \Gamma ` M : A is derivable according to the rules of Pure Type Systems. Although type-checking is undecidable in general [8], most systems of interest have decidable typechecking =-=[5]-=-. For such systems, the question remains whether it is possible to find reasonable, sound and complete, algorithms for type-checking. Over the last years, several authors have proposed such algorithms... |

22 |
A short and proof of Strong Normalisation for the Calculus of Constructions
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Citation Context |

15 | Normalisation in lambda calculus and its relation to type inference - Severi - 1996 |

10 | A-translation and looping combinators in pure type systems
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Citation Context ...n a Pure Type System S, type-checking consists in deciding whether a judgment \Gamma ` M : A is derivable according to the rules of Pure Type Systems. Although type-checking is undecidable in general =-=[8]-=-, most systems of interest have decidable typechecking [5]. For such systems, the question remains whether it is possible to find reasonable, sound and complete, algorithms for type-checking. Over the... |

6 |
A typechecker for bijective pure type systems
- Poll
- 1993
(Show Context)
Citation Context ... it is possible to find reasonable, sound and complete, algorithms for type-checking. Over the last years, several authors have proposed such algorithms for some specific classes of Pure Type Systems =-=[3, 6, 13, 16, 17, 18]-=-. However, the situation is in our view unsatisfactory because these algorithms either impose strong restrictions on the Pure Type Systems or make use of a complex derivability relation, see Section 6... |

6 | Typing in Pure Type Systems - Jutting, S - 1993 |

5 | Typechecking in pure type systems
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(Show Context)
Citation Context ...ing `essentially' the same set of derivable judgements, one may want to distribute the (conversion) rule over the remaining rules of Pure Type Systems. The algorithm below, which is due to R. Pollack =-=[15, 16]-=-, is the result of performing such a distribution---there is a certain irony in coining the next definition as Pollack's algorithm, since the algorithm is called worse in [16]. Definition 13 (Pollack'... |

4 | The semi-full closure of Pure Type Systems
- Barthe
- 1998
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4 |
Type inference for pure type systems
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2 | An induction principle for Pure Type Systems
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- 1998
(Show Context)
Citation Context ...e in the (abstraction) rule of Pure Type Systems and obtain a sound and complete algorithm for type-checking. Besides, we show that the same idea also applies to the problem of Expansion Postponement =-=[4, 14, 16]-=-. Contents The remaining of the paper is organized as follows: in Section 2, we provide a brief overview of Pure Type Systems. In Section 3, we present two motivating open problems, namely the complet... |

1 | Theoretical pearl: Expansion postponement for normalizing pure type systems - Poll - 1998 |

1 | The semi-full closure of Pure Type Systems. Pages 316325 of - Barthe - 1998 |

1 | A veried type-checker. Pages 365380 of - Pollack - 1995 |