## Coercive Subtyping in Type Theory (1996)

Venue: | Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258 |

Citations: | 25 - 14 self |

### BibTeX

@INPROCEEDINGS{Luo96coercivesubtyping,

author = {Zhaohui Luo},

title = {Coercive Subtyping in Type Theory},

booktitle = {Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258},

year = {1996},

pages = {draft.}

}

### Years of Citing Articles

### OpenURL

### Abstract

We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-Lof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive definition rules for the definitional equality. It is shown that this provides a conceptually simple and uniform framework to understand subtyping and coercion relations in type theories with sophisticated type structures such as inductive types and universes. The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computerassisted formal reasoning. 1 Introduction A type in type theory is often intuitively thought of as a set. For example, types in Martin-Lof's type theory [ML84, NPS90...

### Citations

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Citation Context ... typed version of Martin-Lof's logical framework (see Chapter 19 of [NPS90] for a presentation of the latter). We should also point out that LF is different from the Edinburgh Logical Framework (ELF) =-=[HHP87]-=-. The presentation of LF and discussions on how it should be used in specifying type theories can be found in Chapter 9 of [Luo94]. The inference rules of LF are given in Figure 1, which include gener... |

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Citation Context ... Fax: (+44)191-3742560, Email: zhaohui.luo@durham.ac.uk Abstract We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-Lof's type theory =-=[NPS90]-=- and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion a... |

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Citation Context ...haohui.luo@durham.ac.uk Abstract We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-Lof's type theory [NPS90] and the type theory UTT =-=[Luo94]-=-. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that t... |

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Citation Context ...ctive data types (and inductive families, not covered above), and predicative type universes. UTT has nice meta-theoretic properties such as Church-Rosser, Subject Reduction, and Strong Normalisation =-=[Gog94]-=-. Implemented in the Lego proof development system, UTT has been applied to program specification and verification (eg, [Bur93, BM92]), data refinement [Luo93], and formalisation of mathematics [Pol94... |

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Citation Context ...s Lego and solves the above problem. 4.4 Interpreting practical forms of coercion Anthony Bailey has recently implemented various interesting and useful coercion mechanisms in Lego [Bai96]. (Also see =-=[Sai96]-=- for a related development in the Coq system.) Bailey has given interesting examples to illustrate the use of implicit coercion to make proof development easier (more readable etc.). We briefly discus... |

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Citation Context ...for principal typing, this gives a proof-theoretic (and direct meaningtheoretic) treatment of subtyping as implicit coercions, as compared with its possible model-theoretic semantic counterpart (cf., =-=[BCGS91]-=-). ffl Subtyping is treated as an extension of the underlying logical framework---the meta-language in which type theories are formulated. Making essential use of a typed logical framework LF [Luo94] ... |

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Citation Context ...ject Reduction, and Strong Normalisation [Gog94]. Implemented in the Lego proof development system, UTT has been applied to program specification and verification (eg, [Bur93, BM92]), data refinement =-=[Luo93]-=-, and formalisation of mathematics [Pol94]. 2.3 Definitional equality and computational equality We use LF seriously as a meta-level language (see Section 9.1.2 of [Luo94] for a discussion). Along the... |

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Citation Context ... development system, it is natural to assume such extensions of the basic subtyping relation, unless requested otherwise by the user (e.g., by giving a specific coercion between two \Sigma-types). In =-=[Luo92]-=- we have suggested how subtyping can in general be extended to inductive types on the basis of the inductive schemata. Here, we extend the idea to coercive subtyping, which gives a systematic extensio... |

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Citation Context ...ct algebraic theories (Galois theory), Bailey has implemented various forms of coercions in the Lego system [LP92], which are very useful in practical large-scale development of mathematical theories =-=[Bai96]-=-. In this paper, we present an equational formulation of coercive subtyping, a novel approach to introducing user-specified implicit coercions into dependent type theories that can be formulated in a ... |

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Citation Context ... believe that coercive subtyping provides easier and more powerful reasoning mechanisms for reasoning about subsets of objects as well as for reusing proven results in developed formal theories (cf., =-=[Acz94]-=-). In the following section, we briefly introduce the logical framework and explain how to use it as a metalanguage to specify type theories. In Section 3, the basic ideas of coercive subtyping are fu... |

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Citation Context ...s between specifications [Luo93] as coercions), development of mathematical theories in proof development (with theory morphisms [Luo91] as coercions [Bai96]), and library structuring for proof reuse =-=[Luo95]-=-. Subtyping is in general a subtle issue partly because, in the presence of (arbitrary) subtyping, a judgement of the form k :: K is a synthetic judgement in the sense of Martin-Lof [ML94]; that is, t... |

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