Structural subtyping for inductive types with functorial equality rulesâ
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Structural subtyping for inductive types with functorial equality rulesâ
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Under consideration of Math. Struct. in Comp. Science
Structural subtyping for inductive types with functorial equality rules∗ Zhaohui Luo and Robin Adams Department of Computer Science Royal Holloway, University of London Egham, Surrey TW20 0EX, U.K. Received February 27, 2006 and Revised December 21, 2006
Abstract Subtyping for inductive types in dependent type theories is studied in the framework of coercive subtyping. General structural subtyping rules for parameterised inductive types are formulated based on the notion of inductive schemata. Certain extensional equality rules play an important role in proving some of the crucial properties of the type system with these subtyping rules. In particular, it is shown that the structural subtyping rules are coherent and that transitivity is admissible in the presence of the functorial rules of computational equality.
1. Introduction Coercive subtyping [Luo97, Luo99] is a general framework for subtyping and abbreviation in dependent type theories. In particular, it incorporates subtyping for inductive types. In this paper, structural subtyping for inductive types is studied in the framework of coercive subtyping. Inductive types in dependent type theories have a rich structure and include many interesting types such as the types of natural numbers, lists, trees, ordinals, dependent pairs, etc. In general, inductive types can be considered as generated by inductive schemata [Dyb91, PM93, Luo94] and this gives us a clear guide as to how to consider natural subtyping rules for the inductive types. The idea of deriving the general structural subtyping rules from the inductive schemata was first considered in [Luo97] and the subtyping rules for typical inductive types were given in [Luo99]. However, from the PhD work of Y. Luo [Luo05], it has become clear that, in an intensional type theory, the structural subtyping rules for some of the inductive types are not compatible with the notion of transitivity (sometimes called ‘strong transitivity’) as given by the following rule in coercive subtyping:
∗
This work is partially supported by the following research grants: UK EPSRC grant GR/R84092, Leverhulme Trust grant F/07 537/AA and EU TYPES grant 510996.
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