S. Soloviev and Z. Luo. Coercion completion and conservativity in coercive subtyping. To be published in Annals of Pure and Applied Logic, 2001. Appendix A The following gives the rules of the logical framework LF.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... found in our recent work on mathematical language [LC98b] Coercions also provide an elegant analysis of certain phenomena of lexical semantics which have been problematic for other approaches [LC98a] The metatheory of this approach to subtyping has been studied in other papers [Luo97] JLS98] SL98a] SL98b] and the approach was recently extended to cover dependent coercions [LS99a] discussed in section 2.4.4) A significant problem for this approach is the issue of coherence checking. The coherence property guarantees that where several coercions are applicable at in some derivation, ....

S. Soloviev and Z. Luo. Coercion completion and conservativity in coercive subtyping. Talk given at TYPES'98, 1998.

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S. Soloviev and Z. Luo. Coercion completion and conservativity in coercive subtyping. To be published in Annals of Pure and Applied Logic, 2001. Appendix A The following gives the rules of the logical framework LF.

....parameterised inductive types and prove its coherence and the admissibility of substitution and weak transitivity in the coercive subtyping framework. 1 Introduction Coercive subtyping represents a general approach to subtyping and inheritance in type theory (see, for example, Luo97,Luo99,SL02] In particular, it provides a framework in which subtyping, inheritance, and abbreviation can be understood in dependent type theories. This paper investigates the issue of transitivity in coercive subtyping. A problem with transitivity In the presentation of coercive subtyping in [Luo99] the ....

....for details of some of these development and applications of coercive subtyping. Some important meta theoretic aspects of coercive subtyping have been studied. In particular, the results on conservativity and on transitivity elimination for subkinding have been proved in [JLS98,SL02] The conservativity result says, intuitively, that every judgement that is derivable in the theory with coercive Note that the harmful dependency is some form of dependency between parameters. Forbidding such dependency does not mean that all of the dependent types are excluded. See more ....

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S. Soloviev and Z. Luo. Coercion completion and conservativity in coercive subtyping. Annals of Pure and Applied Logic, 2002. Appendix A: The following gives the rules of the logical framework LF.

....and substitution. In particular, we consider suitable subtyping rules for types and types and prove its coherence and the admissibility of substitution and transitivity rules at the type level in the coercive subtyping framework. 1 Introduction Coercive subtyping, as studied in [Luo97,Luo99,SL01] represents a novel general approach to subtyping and inheritance in type theory. In particular, it provides a framework in which subtyping, inheritance, and abbreviation can be understood in dependent type theories where types are understood as consisting of canonical objects. In this paper, we ....

....for details of some of these development and applications of coercive subtyping. Some important meta theoretic aspects of coercive subtyping have been studied. In particular, the results on conservativity and on transitivity elimination for subkinding have been proved in [JLS98,SL01] The conservativity result says, intuitively, that every judgement that is derivable in the theory with coercive subtyping and that does not contain coercive applications is derivable in the 2 original type theory. Furthermore, for every derivation in the theory with coercive subtyping, one ....

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S. Soloviev and Z. Luo. Coercion completion and conservativity in coercive subtyping. To be published in Annals of Pure and Applied Logic, 2001. Appendix A The following gives the rules of the logical framework LF.

....for these is given at the end of C R. A standard example is the family of vectors: the type name contains the length of the vector, so an empty vector has type V ec(zero) and so on. 2.3. Coercive Subtyping LF is extended with a notion of Coercive Subtyping in [25] with additional studies in [20, 38, 28]. Coercive Subtyping is viewed as an abbreviational mechanism of the meta language (LF) and not part of a particular object type theory. The machinery of subtyping is expressed as a fundamental part of LF and object type theories can make use of it by virtue of their definition in LF. Expressing ....

....one can always insert coercions correctly ( coercion completion ) to obtain a derivation in the original type theory. A related result is conservativity: every judgement that is derivable in the extended type theory and that does not use coercions, can be derived in the original type theory [38]. Elimination of transitivity in sub kinding has also been established; although elimination of transitivity in sub typing has not yet been proved. The correct notion of reduction in LF is typed reduction. Notice that a term f(x) with an implicit coercion is not a fij redex because it cannot ....

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Soloviev, S. and Z. Luo: 2000, `Coercion completion and conservativity in coercive subtyping'. Annals of Pure and Applied Logic (to appear). impl-universes.tex; 26/09/2000; 14:27; p.28

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