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Coercive Subtyping in Type Theory
 Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258
, 1996
"... We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof'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; ..."
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Cited by 26 (14 self)
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We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof'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 MartinLof's type theory [ML84, NPS90...
Coercion Completion and Conservativity in Coercive Subtyping
 ANNALS OF PURE AND APPLIED LOGIC
, 2000
"... Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory suc ..."
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Cited by 14 (9 self)
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Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory such as MartinLof's type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.
Dependent Coercions
, 1999
"... A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. A dependent coercion introduces a subtyping relation between a ty ..."
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Cited by 8 (5 self)
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A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. A dependent coercion introduces a subtyping relation between a type 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 metatheory, 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 extend the framework of coercive subtyping to introduce a notion of dependent coer...
Some Algorithmic and ProofTheoretical Aspects of Coercive Subtyping
 In Proceedings of TYPES'96, Lecture Notes in Computer Science
, 1996
"... . 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 prooftheoretic and computational properties. 1 Introduction Coercive subtyping, as first introduced in [Luo96], offers a conc ..."
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Cited by 6 (0 self)
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. 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 prooftheoretic 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...
Coercive Subtyping for the Calculus of Constructions (extended abstract)
"... We present a coercive subtyping system for the calculus of constructions. The proposed system λC co ≤ is obtained essentially by adding coercions and ηconversion to λC≤[10], which is a subtyping extension to the calculus of constructions without coercions. Following [17, 18], the coercive subtyping ..."
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Cited by 3 (1 self)
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We present a coercive subtyping system for the calculus of constructions. The proposed system λC co ≤ is obtained essentially by adding coercions and ηconversion to λC≤[10], which is a subtyping extension to the calculus of constructions without coercions. Following [17, 18], the coercive subtyping c: A ≤ B is understood as a special case of typing in arrow type c: A → B such that the term c behaves like an identity function. We prove that, with respect to this semantic interpretation, the proposed coercive subtyping system is sound and complete, and that this completeness leads to transitivity elimination (transitivity rule is admissible). In and CCβη, this fact implies that λC co ≤ has confluence, subject reduction and strong normalization. We propose a formalization of coercion inference problem and present a sound and complete coercion inference algorithm. addition, we establish the equivalence between λC co
Proof Reutilization in MartinLöf's Logical Framework Extended with Record Types and Subtyping
, 2000
"... The extension of MartinLöf's theory of types with record types and subtyping has elsewhere been presented. We give a concise description of that theory and motivate its use for the formalization of systems of algebras. We also give a short account of a proof checker that has been implemented on mac ..."
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Cited by 1 (0 self)
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The extension of MartinLöf's theory of types with record types and subtyping has elsewhere been presented. We give a concise description of that theory and motivate its use for the formalization of systems of algebras. We also give a short account of a proof checker that has been implemented on machine. The logical heart of the checker is constituted by the procedures for the mechanical verification of the forms of judgement of a particular formulation of the extension. The case study that we put forward in this work has been developed and mechanically verified using the implemented system. We illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions.
Dependent Type System with Subtyping  Type Level Transitivity Elimination
, 1996
"... Dependent type systems have been the basis of many proof development enviroments. In [AC96], a system P is proposed as a subtyping extension of the dependent type system P[Bar92] (also called \Pi[Dow95]). P has nice metatheoretic properties including subject reduction and decidability, but transit ..."
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Dependent type systems have been the basis of many proof development enviroments. In [AC96], a system P is proposed as a subtyping extension of the dependent type system P[Bar92] (also called \Pi[Dow95]). P has nice metatheoretic properties including subject reduction and decidability, but transitivity elimination is restricted to the fi 2 normalized types. In this report, we propose a system \Pi , which is equivalent to P , but it has type level transitivity eliminatioin property. This feature distinguishes our approach from the existing subtyping systems with reduction relations in types. e.g. P[AC96], F ! [SP94], F ! [Com94], where transitivity elimination only holds for normalized types. Metatheoretic properties including subject reduction and decidability are established. The system is shown to be equivalent with P in typing, kinding and context formation. The type checking algorithm is more clear and efficient than P . The technique is suitable for future extensions and...