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Inductive Definitions in the System Coq Rules and Properties
, 1992
"... In the pure Calculus of Constructions, it is possible to represent data structures and predicates using higherorder quantification. However, this representation is not satisfactory, from the point of view of both the efficiency of the underlying programs and the power of the logical system. For ..."
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Cited by 191 (2 self)
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In the pure Calculus of Constructions, it is possible to represent data structures and predicates using higherorder quantification. However, this representation is not satisfactory, from the point of view of both the efficiency of the underlying programs and the power of the logical system. For these reasons, the calculus was extended with a primitive notion of inductive definitions [8]. This paper describes the rules for inductive definitions in the system Coq. They are general enough to be seen as one formulation of adding inductive definitions to a typed lambdacalculus. We prove strong normalization for a subsystem of Coq corresponding to the pure Calculus of Constructions plus Inductive Definitions with only weak nondependent eliminations.
TypeTheoretic Methodology For Practical Programming Languages
 DEPARTMENT OF COMPUTER SCIENCE, CORNELL UNIVERSITY
, 1998
"... The significance of type theory to the theory of programming languages has long been recognized. Advances in programming languages have often derived from understanding that stems from type theory. However, these applications of type theory to practical programming languages have been indirect; the ..."
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Cited by 30 (3 self)
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The significance of type theory to the theory of programming languages has long been recognized. Advances in programming languages have often derived from understanding that stems from type theory. However, these applications of type theory to practical programming languages have been indirect; the differences between practical languages and type theory have prevented direct connections between the two. This dissertation presents systematic techniques directly relating practical programming languages to type theory. These techniques allow programming languages to be interpreted in the rich mathematical domain of type theory. Such interpretations lead to semantics that are at once denotational and operational, combining the advantages of each, and they also lay the foundation for formal verification of computer programs in type theory. Previous type theories either have not provided adequate expressiveness to interpret practical languages, or have provided such expressiveness at the expense of essential features of the type theory. In particular, no previous type theory has supported a notion of partial functions (needed to interpret recursion in practical languages), and a notion of total functions and objects (needed to reason about data values), and an intrinsic notion of equality (needed for most interesting results). This dissertation presents the first type theory incorporating all three, and discusses issues arising in the design of that type theory. This type theory is used as the target of a typetheoretic semantics for a expressive programming calculus. This calculus may serve as an internal language for a variety of functional programming languages. The semantics is stated as a syntaxdirected embedding of the programming calculus into type theory. A critical point arising in both the type theory and the typetheoretic semantics is the issue of admissibility. Admissibility governs what types it is legal to form recursive functions over. To build a useful type theory for partial functions it is necessary to have a wide class of admissible types. In particular, it is necessary for all the types arising in the typetheoretic semantics to be admissible. In this dissertation I present a class of admissible types that is considerably wider than any previously known class.
Nested General Recursion and Partiality in Type Theory
 Theorem Proving in Higher Order Logics: 14th International Conference, TPHOLs 2001, volume 2152 of Lecture Notes in Computer Science
, 2000
"... We extend Bove's technique for formalising simple general recursive algorithms in constructive type theory to nested recursive algorithms. The method consists in defining an inductive specialpurpose accessibility predicate, that characterises the inputs on which the algorithm terminates. As a ..."
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We extend Bove's technique for formalising simple general recursive algorithms in constructive type theory to nested recursive algorithms. The method consists in defining an inductive specialpurpose accessibility predicate, that characterises the inputs on which the algorithm terminates. As a result, the typetheoretic version of the algorithm can be defined by structural recursion on the proof that the input values satisfy this predicate. This technique results in definitions in which the computational and logical parts are clearly separated; hence, the typetheoretic version of the algorithm is given by its purely functional content, similarly to the corresponding program in a functional programming language. In the case of nested recursion, the special predicate and the typetheoretic algorithm must be defined simultaneously, because they depend on each other. This kind of definitions is not allowed in ordinary type theory, but it is provided in type theories extended wit...
Computational Complexity and Induction for Partial Computable Functions in Type Theory
 In Preprint
, 1999
"... An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in ..."
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Cited by 12 (7 self)
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An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory. This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computationa...
Monadic Type Systems: Pure Type Systems for Impure Settings (Preliminary Report)
 In Proceedings of the Second HOOTS Workshop
, 1997
"... Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameteriz ..."
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Cited by 9 (2 self)
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Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameterized higherorder type structure of pure type systems and the monadic term and type structure used to capture computational effects in the theory of computational monads. We demonstrate that monadic type systems nicely characterize previous work and suggest how they can support several new theoretical and practical applications. A technical foundation for monadic type systems is laid by recasting and scaling up the main results from pure type systems (confluence, subject reduction, strong normalisation for particular classes of systems, etc.) and from operational presentations of computational monads (notions of operational equivalence based on applicative similarity, coinduction proof techni...
Admissibility of Fixpoint Induction over Partial Types
 Automated deduction  CADE15. Lect. Notes in Comp. Sci
, 1998
"... Partial types allow the reasoning about partial functions in type theory. The partial functions of main interest are recursively computed functions, which are commonly assigned types using fixpoint induction. However, fixpoint induction is valid only on admissible types. Previous work has shown many ..."
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Cited by 6 (2 self)
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Partial types allow the reasoning about partial functions in type theory. The partial functions of main interest are recursively computed functions, which are commonly assigned types using fixpoint induction. However, fixpoint induction is valid only on admissible types. Previous work has shown many types to be admissible, but has not shown any dependent products to be admissible. Disallowing recursion on dependent product types substantially reduces the expressiveness of the logic; for example, it prevents much reasoning about modules, objects and algebras. In this paper I present two new tools, predicateadmissibility and monotonicity, for showing types to be admissible. These tools show a wide class of types to be admissible; in particular, they show many dependent products to be admissible. This alleviates difficulties in applying partial types to theorem proving in practice. I also present a general least upper bound theorem for fixed points with regard to a computational approxim...
Hybrid PartialTotal Type Theory
, 1995
"... In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be memb ..."
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Cited by 5 (0 self)
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In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ¸ on computations, and a fixed point induction principle. The resulting theory is thus intended as a generalpurpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...
Safe proof checking in type theory with Y
 IN CSL
, 1999
"... We present an extension of type theory with a fixed point combinator Y. We are particularly interested in using this Y for doing unbounded proof search in the proof system. Therefore we treat in some detail a typed λcalculus for higher order predicate logic with inductive types (a reasonable subsy ..."
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Cited by 3 (0 self)
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We present an extension of type theory with a fixed point combinator Y. We are particularly interested in using this Y for doing unbounded proof search in the proof system. Therefore we treat in some detail a typed λcalculus for higher order predicate logic with inductive types (a reasonable subsystem of the theory implemented in [Dowek e.a. 1991]) and show how bounded proof search can be done in this system, and how unbounded proof search can be done if we add Y. Of course, proof search can also be implemented (as a tactic) in the meta language. This may give faster results, but asks from the user to be able to program the implementation. In our approach the user works completely in the proof system itself. We also provide the meta theory of type theory with Y that allows to use the fixed point combinator in a safe way. Most importantly, we prove a kind of conservativity result, showing that, if we can generate a proof term M of formula ' in the extended system, and M does not contain Y, then M is already a proof of ' in the original system.
The Coq Proof Assistant  Reference Manual Version 6.1
, 1997
"... : Coq is a proof assistant based on a higherorder logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.enslyon.fr:/pub/LIP/COQ/V6.1 Keywords: Coq, Proof Assistant, Formal Proofs, Calculus of Inductives Constru ..."
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: Coq is a proof assistant based on a higherorder logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.enslyon.fr:/pub/LIP/COQ/V6.1 Keywords: Coq, Proof Assistant, Formal Proofs, Calculus of Inductives Constructions (R'esum'e : tsvp) This research was partly supported by ESPRIT Basic Research Action "Types" and by the GDR "Programmation " cofinanced by MREPRC and CNRS. Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11  T'el'ecopie : (33 1) 39 63 53 30 Manuel de r'ef'erence du syst`eme Coq version V6.1 R'esum'e : Coq est un syst`eme permettant le d'eveloppement et la v'erification de preuves formelles dans une logique d'ordre sup'erieure incluant un riche langage de d'efinitions de fonctions. Ce document constitue le manuel de r'ef'erence de la version V6.1 qui est distribu 'ee par ftp ...
CC+: An extension of the Calculus of Constructions with fixpoints
, 1993
"... We follow an original idea suggested by Constable and Smith [6, 7] providing a way for reasoning about non terminating computations in a typed framework. A former study has been worked out within NuPrl by Smith [21]. We investigate how these ideas can be developed within the Calculus of Construct ..."
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We follow an original idea suggested by Constable and Smith [6, 7] providing a way for reasoning about non terminating computations in a typed framework. A former study has been worked out within NuPrl by Smith [21]. We investigate how these ideas can be developed within the Calculus of Constructions (CC). The adaptation provides an conservative extension, denoted CC+. Strong normalisation for fireductions is preserved. We recover the alternate "recursive" coding for integers introduced in AF2 by Parigot [12, 13]. Thus, the computational behaviour for terms coding integers is improved. Moreover, as expected, all partial recursive functions are now definable. Relationships with primitive coding through "Church" integers within the pure Calculus is studied, giving some insights into logical expressiveness issue. All these results easily generalize to all the usual data structures.