Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1), 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of reusability. Specic forms of subtyping connected to reusability include record subtyping [10] and constructor subtyping [7] but these forms of subtyping are too specialised to provide a general solution to the problem of reusability. Implicit coercions and coercive subtyping These paradigms [3, 5, 28, 35] provide a very powerful mechanism that subsumes a variety of previous and seemingly unrelated proposals in the area of subtyping. In a nutshell, a coercion from A to B is a function f : A B, declared in a coercion context, and which allows to view a as an element of B and a shorthand for f a ....

....coherence to the detriment of conceptual clarity; computational interpretation of coercions: in most existing works, coercions are not given a computational meaning through rewrite rules. Instead, the meaning of coercive subtyping is given in a logical framework with equality judgements, as in [28], or by a translation of the extended language (with coercions) into the original language (without coercions) as in [3, 35] We prefer to follow the standard approach (for proof development systems) where type systems are dened on their own and expressions are given a computational ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9:105130, February 1999.

....(cof crr (crl crr IR) x y) Instead of what seems to be just one function symbol [ the term actually contains 33 function symbols. This shows that the terms in our framework are relatively heavy . But note that most of this big term is inferable coercions. In Luo s coercive subtyping [13] these parts of the term are actually elided, not just suppressed in printing. This may be an important optimization for large scale formal mathematics. 8 Conclusion We presented a framework for writing algebraic expressions in the Coq proof assistant. The features of Coq that made our approach ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1), 1999.

....enhance proof reuse in type systems. Specic forms of subtyping connected to reusability include record subtyping [11] and constructor subtyping [8] but these forms of subtyping are too specialised to provide a general solution to the problem of reusability. Implicit coercions Implicit coercions [3, 5, 28, 36] provide a very powerful mechanism that subsumes a variety of previous and seemingly unrelated proposals in the area of subtyping. In a nutshell, a coercion from A to B is a function f : A B, declared in a coercion context, and which allows to view a as an element of B and a shorthand for f a ....

....coherence to the detriment of conceptual clarity; computational interpretation of coercions: in most existing works, coercions are not given a computational meaning through rewrite rules. Instead, the meaning of coercive subtyping is given in a logical framework with equality judgements, as in [28], or by a translation of the extended language with Type Isomorphisms and Proof Reuse in Dependent Type Theory 3 coercions into the original language without coercions, as in [3, 36] We prefer to follow the usual approach where expressions are given a computational interpretation independently of ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9:105130, February 1999.

....used, rather than trying to check it globally over all possible coercions whether they are used or not; this technique we call dynamic coherence checking . 1 Introduction In recent papers, Luo defines Coercive Subtyping as an abbreviational mechanism in a logical framework [Luo97, JLS98, Luo99] Other work has introduced a coercion mechanism in to a particular type theory, for example: Bailey in Lego s ECC [Bai98] and Saibi in Coq s CIC [Sai99] But, it is argued that the most natural interpretation of Coercive Subtyping is as a general mechanism independent of particular type ....

....making heavy use of inductive types; some earlier approaches cannot be adapted to these circumstances. A 2 coercions based explanation of universes in the Tarski style (as used in UTT) is a good example of the benefits of coercive subtyping in the presentation of a type theory (introduced in [Luo99] and discussed in [CL00b] compared to Luo s earlier version in ECC that used Russell style universes and a built in cumulativity relation [Luo94] the new presentation is much clearer. See section 5 of [Luo99] for a more complete discussion of approaches to subtyping. Forms of coercive ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....of Aspinall and Compagnoni s P with late binding. Their calculus is a signicant improvement over P and allows to formalize the examples of [29] However, declarative subtyping, even combined with late binding, is not appropriate for the inductive approach to formalization; implicit coercions [5, 24] allow to view a term a of type A as a term of type B whenever there is a previously agreed upon function, called coercion, from A to B. This approach, which leads to extremely powerful type systems, is implemented in several proof development systems, including Coq and Lego, and has proved useful ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9:105130, February 1999.

....when structures are not flat , but are constructed from substructures, as is desirable in practice. Peter Aczel [1] suggested a notion of coercive subtyping for type theory. Both Coq and Lego now support ad hoc, but very useful, notions of coercive subtyping [20, 2] Zhaohui Luo and colleagues [15, 16] have studied coercive subtyping foundationally, as an abbreviational mechanism which introduces definitional equalities at a logical framework level. This approach is more expressive in some ways than the implementations, but in other respects does not yet equal them. Also, we do not claim that ....

....Sigma Types Since records are constructed like nested Sigma types, it is useful to ask how subtyping propagates through Sigma. Suppose L is a subtype of L # by coercion c : L)L # , and A : L # )Type . c can be lifted to a coercion c # : #LA#c)#L # A , showing that #LA#c is a subtype of #L # A [15]. L c L # A : L # )Type #LA#c c # #L # A c # is defined by c # (#l, a#) #c(l) a# . The composition, A#c , is needed for the subtype to be well formed, as A expects an L # object, not an L object. This idiom will appear again in Sect. 5.4. 5 The illusion is not complete, as Inner ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1), 1999.

....Calculus of Constructions with subtyping: A = fi C Gamma C D D = fi B Gamma A; B; C; D : s Gamma A B where Gamma A; B; C; D : s denotes that A; B; C; D are well formed. In his study on coercive subtyping for UTT (a system more expressive than the Calculus of Constructions) Luo [Luo97] deals with type conversion by the following rule: A = C Gamma C c 0 D D = B Gamma A; B; C; D : c = c 0 Gamma A c B where c; c 0 are coercions, is the type conversion defined in his system and Gamma A; B; C; D : s states that A; B; C; D are well formed. A significant ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(13), 1997.

.... and to enforce a subsumption rule a : A A B a : B This basic mechanism of subtyping is powerful enough to capture a variety of concepts in computer science, see e.g. 9] and its use is spreading both in functional programming languages, see e.g. 23, 28, 29] and in proof assistants, see e.g. [7, 22, 30]. Constructor subtyping is a basic form of subtyping, suggested in [12] and developed in [5] in which an inductive type is viewed as a subtype of another inductive type if has more constructors than . As such, constructor subtyping captures in a type theoretic context the ubiquitous use ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 199x. To appear.

.... and to enforce a subsumption rule a : A A B a : B This basic mechanism of subtyping is powerful enough to capture a variety of concepts in computer science, see e.g. 9] and its use is spreading both in functional programming languages, see e.g. 25, 30, 31] and in proof assistants, see e.g. [7, 24, 32]. Constructor subtyping is a basic form of subtyping, suggested in [12] and developed in [5] in which an inductive type oe is viewed as a subtype of another inductive type if has more constructors than oe. As such, constructor subtyping captures in a type theoretic context the ubiquitous use ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 199x. To appear.

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105130, 1999.

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105-130, 1999.

....certain 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 ....

....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 transitivity rule A c 0 is included. Intuitively, it says that the composition of two coercions is also the coercion corresponding to transitivity. In [LL01] it has been proved that the transitivity rule is admissible for certain subtyping rules, such as those for types and ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105 130, 1999.

....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 ....

....problem In this section, we give a brief introduction to coercive subtyping, explain the notion of coherence and its importance, and lay down the necessary formal details to be used in the following sections. 2. 1 Coercive subtyping The basic idea of coercive subtyping, as studied in e.g. Luo99] is that A is a subtype of B if there is a (unique) coercion c from A to B, and therefore, any object of type A may be regarded as object of type B via c, where c is a functional operation from A to B in the type theory. A coercion plays the role of abbreviation. More precisely, if c is a ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105-130, 1999.

....components of Martin Lof s intensional type theory can be introduced using this LF. We use the form of LF from Luo s monograph [12] and subsequent publications, featuring type annotations on all abstraction forms, a mechanism for introducing inductive types, and an extension for coercive subtyping [13]. Being a meta language, LF is a deliberately restricted language. To use it for specific applications, the components of an object language must be introduced as new constants and computation rules, typically by defining suitable inductive types. For example, the theory UTT (related to Lego s ....

....of type A. Representing such a family by means of an internal Pi type, as done in systems like Lego, is inappropriate and leads to possible misunderstandings in use of type theory. In later work, Luo proposed an extension of LF which provides an abbreviational mechanism of coercive subtyping [13]. Coercive subtyping allows an application of a functional operation f : x : B)C to an object y : A to be wellformed if a coercion c from type A to type B is known (written A c B) Thus f(y) is a well typed abbreviation of f(c(y) Placing coercive subtyping in the framework allows any theory ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....only be introduced between two objects of a type, or between two types. 5 In the following, we give several examples of introducing type constructors and their associated operators as constants. We omit El to write A for El(A) in the examples. 5 This conforms to the restriction considered in [Luo99], where LF is used to specify type theories. 10 Example 3.1 Examples of parametric constants include those for introducing the type of natural numbers: N : Type 0 : N s : x:N) N R : C: x:N)Type; c:C[0] f : x:N; y:C[x] C[s[x] z:N) C[z] The corresponding computation rules are: R[C; c; f; ....

....the meta theoretic results of the object type theories such as UTT. 20 6 Conclusions We have presented and studied PAL , a logical framework based on parameterisation and definitions rather than lambda abstraction. Further extensions of meta features such as coercive subtyping (see eg, [Luo99]) may be considered. PAL is developed partly as an underlying framework for implementing proof development systems. We expect that the simplicity gained would benefit implementation as well as the user (eg, it is expected that the use of de Bruijn indices would be simplified, and the ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....for inductive families, coercive subtyping, and universes. There are several motivations for implementing this variant of LF (henceforth LF; Martin Lof s LF will be named explicitly) including: A tool for research on subtyping. Recent work on Coercive Subtyping by Luo, Soloviev, and Jones [16, 13] analyses subtyping as a mechanism for abbreviation that is built directly in to the (meta level) LF. Object type theories specified in LF (such as UTT) then automatically inherit subtyping. Plastic is currently being used in case studies of coercion use, and to experiment with algorithms for ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....relation is obtained by overloading object terms (e.g. terms) It is not clear (if possible) how the traditional approach to subtyping can be applied to type theory with inductive types in accordance with the view that types consist of canonical objects. The framework on coercive subtyping [Luo97, Luo99] takes a new approach to subtyping and inheritance taking coercions seriously and directly at the proof theoretic level and providing a coherent view on how subtyping and inheritance can be studied in a type theory with inductive data types. The basic idea of coercive subtyping in this context ....

....requires an object of K; then the argument k 0 in the context f stands for its image of the coercion, c(k 0 ) Therefore, one can use f(k 0 ) as an abbreviation of f(c(k 0 ) The above simple idea, when formulated in a typed logical framework [Luo94] becomes very powerful. In our early work [Luo97, Luo99], we have developed the framework that covers subtyping relations represented by the following kinds of coercions: ffl Simple coercions: representing subtyping between two types. For example, coercions between basic inductive types: the type of even numbers E is a subtype of the type of natural ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

.... rule is: E Pi [A; B; C; f; A; B; g] f [g] C[ A; B; g] We can define the application operator app : A:Type; B: x:A)Type; F : Pi[A; B] x:A) B[x] as follows: app[A; B; F; a] df E Pi [A; B; G: Pi[A; B] B[a] g: x:A)B[x] g[a] F ] 3 This confirms to the restriction considered in [Luo99], where LF is used to specify type theories. 7 Note that parametric abstraction is used in the above definition. We can show the expected equalities hold. For example, we can show that the usual fi equality app[A; B; A; B; g] a] g[a] Furthermore, for propositional equality = Pi[A;B] eg, ....

....in implementations requires further research and a real development of a proof system. 6 Conclusion and discussions We have presented and studied PAL , a logical framework simpler than those based on lambda abstraction. Further extensions of meta features such as coercive subtyping (see eg, [Luo99]) may be considered. Among the related work, a comparison with Cartmell s notion contextual categories [Car78, Car86] may lead to further insight. Acknowledgements I am grateful to Paul Callaghan, Conor McBride, Bengt Nordstrom, Randy Pollack, and an LFM 00 referee, who have read drafts of this ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....type theory in Lego) Inversion and pattern matching (recent work by Conor McBride [21] Simple tacticals. 3 Work closely related to Lego ffl Development of expressive type theory: Work on ECC and UTT (see eg, 15] Typed operational semantics [7, 8] Coercive subtyping [16, 29] (cf, implementation of coercions in Lego [2] and in Coq [27] Extensionality and related issues [9] ffl Application examples: Formalisation of Pure Type Systems [3] and verification of proof checker [24, 22, 25] Verification of functional programs [4] imperative programs [11, 28] and ....

Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....and a family of types in that an object of the type is mapped into one of the types in the family. We present the formal framework, discuss its meta theory, and consider applications such as its use in functional programming with dependent types. 1 Introduction Coercive subtyping, as studied in [Luo97, Luo99, JLS98], represents a new 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 ....

....K; then the argument k 0 in the context f stands for its image of the coercion, c(k 0 ) Therefore, one can use f(k 0 ) as an abbreviation of f(c(k 0 ) Power of the framework The above simple idea, when formulated in a typed logical framework [Luo94] becomes very powerful. In our early work [Luo97, Luo99], we have developed the framework that covers subtyping relations represented by the following kinds of coercions: ffl Simple coercions: representing subtyping between two types. For example, coercions between basic inductive types: Even is a subtype of Nat. ffl Parameterised coercions: ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

....in the interactive process of using MV, on the other. Another contribution of this paper in studying the conceptual structure of MV is the treatment of the inheritance relation between conceptual categories based on coercive subtyping a general theory of subtyping and inheritance in type theory [Luo97, Luo98, SL98]. It is interesting to see that coercive subtyping, a theory developed in a rather different context, can be successfully applied to understanding the linguistic structure of MV. In Section 2, we shall briefly elaborate the background of this work, giving a general discussion of informal ....

....between mathematical concepts. Many linguistic conventions, in particular, abbreviational conventions, are based on them. In this section, we extend our study of conceptual categories to investigate such relationships between categories. In particular, we shall study how coercive subtyping [Luo97, Luo98], a new theory of subtyping and inheritance in type theory, can be applied in our typetheoretic analysis of mathematical language. The benefits are to provide adequate abbreviational mechanisms, to understand the inheritance relationships between mathematical concepts, and to deal with contrastive ....

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 1998. To appear.

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1), 1999.

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Zhaohui Luo. Coercive subtyping. Journal of Logic and Computation, 9(1):105--130, 1999.

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Z. Luo. Coercive subtyping. Journal of Logic and Computation, 9(1), 1999.

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