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Constructions, Inductive Types and Strong Normalization
, 1993
"... This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic ..."
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Cited by 35 (3 self)
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This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notion of model, CCstructures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to nonalgebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a nontrivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...
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 frame ..."
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Cited by 27 (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...
An Implementation of LF with Coercive Subtyping & Universes
 Journal of Automated Reasoning
"... . We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of MartinLof's logical framework, with explicitly typed abstractions. We outline the system of LF with its extensions of inductive types and coercions. ..."
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Cited by 20 (12 self)
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. We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of MartinLof's logical framework, with explicitly typed abstractions. We outline the system of LF with its extensions of inductive types and coercions. Plastic is the first implementation of this extended system; we discuss motivations and basic architecture, and give examples of its use. LF is used to specify type theories. The theory UTT includes a hierarchy of universes which is specified in Tarski style. We outline the theory of these universes and explain how they are implemented in Plastic. Of particular interest is the relationship between universes and inductive types, and the relationship between universes and coercive subtyping. We claim that the combination of Tarskistyle universes together with coercive subtyping provides an ideal formulation of universes which is both semantically clear and practical to use. Keywords: type theory, un...
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...
The Open Calculus of Constructions: An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving
"... The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equatio ..."
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Cited by 6 (0 self)
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The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higherorder style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a firstorder semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.
Inheritance of Proofs
, 1996
"... The CurryHoward isomorphism, a fundamental property shared by many type theories, establishes a direct correspondence between programs and proofs. This suggests that the same structuring principles that ease programming be used to simplify proving as well. To exploit objectoriented structuring me ..."
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Cited by 4 (0 self)
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The CurryHoward isomorphism, a fundamental property shared by many type theories, establishes a direct correspondence between programs and proofs. This suggests that the same structuring principles that ease programming be used to simplify proving as well. To exploit objectoriented structuring mechanisms for verification, we extend the objectmodel of Pierce and Turner, based on the higher order typed calculus F ! , with a proof component. By enriching the (functional) signature of objects with a specification, the methods and their correctness proofs are packed together in the objects. The uniform treatment of methods and proofs gives rise in a natural way to objectoriented proving principles  including inheritance of proofs, late binding of proofs, and encapsulation of proofs  as analogues to objectoriented programming principles. We have used Lego, a typetheoretic proof checker, to explore the feasibility of this approach. In particular, we have verified a small hier...
Implementation Techniques for Inductive Types in Plastic
 Types for Proofs and Programs, volume 1956 of LNCS
, 2000
"... . In the context of Plastic, a proof assistant for a variant of MartinLof's Logical Framework LF with explicitly typed abstractions, we outline the technique used for implementing inductive types from their declarations. This form of inductive types gives rise to a problem of nonlinear p ..."
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Cited by 4 (2 self)
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. In the context of Plastic, a proof assistant for a variant of MartinLof's Logical Framework LF with explicitly typed abstractions, we outline the technique used for implementing inductive types from their declarations. This form of inductive types gives rise to a problem of nonlinear pattern matching; we propose this match can be ignored in welltyped terms, and outline a proof of this. The paper then explains how the inductive types are realised inside the reduction mechanisms of Plastic, and briefly considers optimisations for inductive types. Key words: type theory, inductive types, LF, implementation. 1 Introduction This paper considers implementation techniques for a particular approach to inductive types in constructive type theory. The inductive types considered are those given in Chapter 9 of [15], in which Luo presents a variant of MartinLof's Logical Framework LF which has explicitly typed abstractions, and a schema for inductive types within this LF which is...
Recursive Models of General Inductive Types
 Fundam. Inf
, 1993
"... We give an interpretation of MartinLof's type theory (with universes) extended with generalized inductive types. The model is an extension of the recursive model given by Beeson. By restricting our attention to PER model, we show that the strictness of positivity condition in the definition of ..."
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We give an interpretation of MartinLof's type theory (with universes) extended with generalized inductive types. The model is an extension of the recursive model given by Beeson. By restricting our attention to PER model, we show that the strictness of positivity condition in the definition of generalized inductive types can be dropped. It therefore gives an interpretation of general inductive types in MartinLof's type theory. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Science, Manchester University, are available by anonymous ftp from m1.cs.man.ac.uk (130.88.13.4) in the directory /pub/TR. The files are stored as PostScript, in compressed form, with the report number as filename. Alternative...
Understanding Inductive Types in Constructions
, 1993
"... In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different ..."
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In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different style is wrong.