## Some Algorithmic and Proof-Theoretical Aspects of Coercive Subtyping (1996)

Venue: | In Proceedings of TYPES'96, Lecture Notes in Computer Science |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Jones96somealgorithmic,

author = {Alex Jones and Zhaohui Luo and sergei Soloviev},

title = {Some Algorithmic and Proof-Theoretical Aspects of Coercive Subtyping},

booktitle = {In Proceedings of TYPES'96, Lecture Notes in Computer Science},

year = {1996},

publisher = {Springer}

}

### OpenURL

### Abstract

. Coercive subtyping offers a conceptually simple but powerful framework to understand subtyping and subset relationships in type theory. In this paper we study some of its proof-theoretic and computational properties. 1 Introduction Coercive subtyping, as first introduced in [Luo96], offers a conceptually simple but powerful framework to understand subtyping and subset relationships in type theories with sophisticated type structures such as dependent types, inductive types, and type universes. A basic idea behind coercive subtyping is that subtyping provides a powerful mechanism for notational abbreviation in type theory. If A is a subtype of B given by a specified coercion function, an object of type A can be regarded as an object of type B, that is, its image via the coercion function, and hence objects of a subtype can be used as abbreviations for objects of a supertype. With coercive subtyping, this abbreviational mechanism is formally treated at the level of the logical framewo...

### Citations

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Citation Context ... definitionally equal to f(c(a)), where c is the coercion between K and K 0 . This simple extension of the logical framework (and hence of the specified type theories such as Martin-Lof's type theory =-=[NPS90]-=- and the type theory UTT [Luo94,Gog94]), provides a surprisingly powerful mechanism that facilitates useful ways of reasoning about subsets of objects, helps proof reuse and modularisation, and gives ... |

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Citation Context ...towards further study of, for example, more efficient reduction strategies. A lot of work has been presented already on subtyping. People such as Cardelli ([Car88,Car89]) and Aspinall and Compagnoni (=-=[AC96]-=-) present systems with subtyping that do not use coercions. Bailey [Bai96] and Saibi [Sai97] have already implemented coercive subtyping into current proof assistant programs as notational abbreviatio... |

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Citation Context ...ups as a subset of the type of monoids, then any proof to do with monoids can then be automatically applied to groups. Such a mechanism has been implemented by Bailey in LEGO [Bai96] and Saibi in Coq =-=[Sai97]-=- to support proof reuse and notational abbreviation. Similar mechanisms can be used for development of program specification and verification proofs (c.f., [Luo93]). In this paper, we study some of th... |

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Citation Context ...e reduction steps would not be admissible, but what we can do is the following: [x:K 1 ]([y:K 2 ]y)(x)sc [x:K 1 ]([y:K 2 ]y)(c(x))sfi [x:K 1 ]c(x)sj c 4.2 Type-checking In type-checkers such as LEGO (=-=[LP92]-=-) and Coq ([H + 96]) type-checking is based upon the methods described in [Hue89]. The main thrust here is placed on a conversion algorithm as it is the most tricky part of type-checking. Conversion i... |

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Citation Context ...ley in LEGO [Bai96] and Saibi in Coq [Sai97] to support proof reuse and notational abbreviation. Similar mechanisms can be used for development of program specification and verification proofs (c.f., =-=[Luo93]-=-). In this paper, we study some of the basic proof-theoretic and computational properties of coercive subtyping. The formal system we consider, as presented in Section 2, is essentially the same as th... |

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Citation Context ...nderstand subtyping and subset relationships in type theory. In this paper we study some of its proof-theoretic and computational properties. 1 Introduction Coercive subtyping, as first introduced in =-=[Luo96]-=-, offers a conceptually simple but powerful framework to understand subtyping and subset relationships in type theories with sophisticated type structures such as dependent types, inductive types, and... |

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Citation Context ...we define the type of groups as a subset of the type of monoids, then any proof to do with monoids can then be automatically applied to groups. Such a mechanism has been implemented by Bailey in LEGO =-=[Bai96]-=- and Saibi in Coq [Sai97] to support proof reuse and notational abbreviation. Similar mechanisms can be used for development of program specification and verification proofs (c.f., [Luo93]). In this p... |

7 | Representing Algebra in LEGO - Bailey |

2 | Typeful Programming, Lecture notes for the IFIP State of the Art Seminar on Formal Description of Programming Concepts, Rio de - Cardelli - 1989 |

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1 |
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Citation Context ...K ! c K 0 . We also restrict the basic coercions to be closed terms between closed types (as considered in [Luo96]). The presentation of a more general framework of coercive subtyping can be found in =-=[Luo97]-=-, which uses subkinding judgements with explicit coercion terms and also allows more general basic subtyping rules such as those for parameterised inductive types. In section 3 we prove several import... |