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13
Proofassistants using Dependent Type Systems
, 2001
"... this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs ..."
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Cited by 48 (4 self)
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this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs
Subset coercions in Coq
 In Selected papers from the International Workshop on Types for Proofs and Programs (TYPES’06
, 2006
"... Abstract. We propose a new language for writing programs with dependent types which can be elaborated into partial Coq terms. This language permits to establish a phase distinction between writing and proving algorithms in the Coq environment. Concretely, this means allowing to write algorithms as e ..."
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Cited by 39 (2 self)
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Abstract. We propose a new language for writing programs with dependent types which can be elaborated into partial Coq terms. This language permits to establish a phase distinction between writing and proving algorithms in the Coq environment. Concretely, this means allowing to write algorithms as easily as in a practical functional programming language whilst giving them as rich a specification as desired and proving that the code meets the specification using the whole Coq proof apparatus. This is achieved by extending conversion to an equivalence which relates types and subsets based on them, a technique originating from the “Predicate subtyping ” feature of PVS and following mathematical convention. The typing judgements can be translated to the Calculus of (Co)Inductive Constructions (Cic) by means of an interpretation which inserts coercions at the appropriate places. These coercions can contain existential variables representing the propositional parts of the final term, corresponding to proof obligations (or PVS typechecking conditions). A prototype implementation of this process is integrated with the Coq environment. 1
Un Calcul De Constructions Infinies Et Son Application A La Verification De Systemes Communicants
, 1996
"... m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to ..."
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Cited by 17 (0 self)
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m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to Luc Boug'e, who accepted to be my oficial supervisor, and to the chair of the jury, Michel Cosnard, who opened to me the doors of the LIP. During these last three years in Lyon I met many wonderful people, who then become wonderful friends. Miguel, Nuria, Veronique, Patricia, Philippe, Pia, Rodrigo, Salvador, Sophie : : : with you I have shared the happiness and sadness of everyday life, those little things which make us to remember someone forever. I also would like to thank the people from "Tango de Soie", for all those funny nights at the Caf'e Moulin Joly. Thanks too to the Uruguayan research community in Computer Science (specially to Cristina Cornes and Alberto Pardo) w
CoInductive Types in Coq: An Experiment with the Alternating Bit Protocol
, 1995
"... We describe an experience concerning the implementation and use of coinductive types in the proof editor Coq. Coinductive types are recursive types which, opposite to inductive ones, may be inhabited by infinite objects. In order to illustrate their use in Coq, we describe an axiomatisation of ..."
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Cited by 12 (1 self)
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We describe an experience concerning the implementation and use of coinductive types in the proof editor Coq. Coinductive types are recursive types which, opposite to inductive ones, may be inhabited by infinite objects. In order to illustrate their use in Coq, we describe an axiomatisation of a calculus of broadcasting systems where recursive processes are represented using infinite objects. This calculus is used for developing a verification proof of the alternating bit protocol. Keywords: Program Verification, Type Theory, CoInductive Types, Communicating Processes R'esum'e Dans cet article nous d'ecrivons une exp'erience concernant l'implantation et l'utilisation de types coinductifs dans l'environnement de preuves Coq. Les types coinductifs sont des types recursifs qui, `a la diff'erence des types inductifs, peuvent etre habit'es par des objets infinis. Pour illustrer leur utilisation dans Coq nous d'ecrivons comment axiomatiser un calcul de processus qui communiq...
PROGRAMing finger trees in Coq
 In ACM SIGPLAN International Conference on Functional Programming. Association for Computing Machinery
, 2007
"... Finger Trees (Hinze and Paterson 2006) are a general purpose persistent data structure with good performance. Their genericity permits developing a wealth of structures like ordered sequences or interval trees on top of a single implementation. However, the type systems used by current functional la ..."
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Cited by 12 (0 self)
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Finger Trees (Hinze and Paterson 2006) are a general purpose persistent data structure with good performance. Their genericity permits developing a wealth of structures like ordered sequences or interval trees on top of a single implementation. However, the type systems used by current functional languages do not guarantee the coherent parameterization and specialization of Finger Trees, let alone the correctness of their implementation.We present a certified implementation of Finger Trees solving these problems using the PROGRAM extension of COQ. We not only implement the structure but also prove its invariants along the way, which permit building certified structures on top of Finger Trees in an elegant way. Categories and Subject Descriptors D.2.4 [Software/Program
Verifying programs in the Calculus of Inductive Constructions
, 1997
"... . This paper deals with a particular approach to the verification of functional programs. A specification of a program can be represented by a logical formula [Con86, NPS90]. In a constructive framework, developing a program then corresponds to proving this formula. Given a specification and a progr ..."
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Cited by 6 (0 self)
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. This paper deals with a particular approach to the verification of functional programs. A specification of a program can be represented by a logical formula [Con86, NPS90]. In a constructive framework, developing a program then corresponds to proving this formula. Given a specification and a program, we focus on reconstructing a proof of the specification whose algorithmic contents corresponds to the given program. The best we can hope is to generate proof obligations on atomic parts of the program corresponding to logical properties to be verified. First, this paper studies a weak extraction of a program from a proof that keeps track of intermediate specifications. From such a program, we prove the determinism of retrieving proof obligations. Then, heuristic methods are proposed for retrieving the proof from a natural program containing only partial annotations. Finally, the implementation of this method as a tactic of the Coq proof assistant is presented. 1. Introduction A large p...
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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Cited by 2 (0 self)
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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 ...
System ST  Toward A Type System for Extraction AND Proof of Programs
, 2001
"... We introduce a new type system called \System ST" (ST stands for SubTyping), based on subtyping, and prove the basic property of the system. We show the extraordinary expressive power of the system which leads us to think that it could be a good candidate for doing both proof and extraction of pr ..."
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Cited by 1 (1 self)
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We introduce a new type system called \System ST" (ST stands for SubTyping), based on subtyping, and prove the basic property of the system. We show the extraordinary expressive power of the system which leads us to think that it could be a good candidate for doing both proof and extraction of programs.
System ST, βreduction and completeness
 In Logic In Computer Science
, 2003
"... We prove that system ST (introduced in a previous work) enjoys subject reduction and is complete for realizability semantics. As far as the author knows, this is the only type system enjoying the second property. System ST is a very expressive type system, whose principle is to use two kinds of form ..."
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We prove that system ST (introduced in a previous work) enjoys subject reduction and is complete for realizability semantics. As far as the author knows, this is the only type system enjoying the second property. System ST is a very expressive type system, whose principle is to use two kinds of formulae: types (formulae with algorithmic content) and propositions (formulae without algorithmic content). The fact that subtyping is used to build propositions and that propositions can be used in types trough a special implication gives its great expressive power to the system: all the operators you can imagine are definable (union, intersection, singleton,...).
Axiomatisations, Proofs, and Formal Specifications of Algorithms: Commented Case Studies In the Coq Proof Assistant
"... this paper is but a tiny initial fragment of the theory of categories. However, it is quite promising, in that the power of dependent types and inductive types (or at least \Sigmatypes) is put to full use; note in particular the dependent equality between morphisms of possibly nonconvertible types ..."
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this paper is but a tiny initial fragment of the theory of categories. However, it is quite promising, in that the power of dependent types and inductive types (or at least \Sigmatypes) is put to full use; note in particular the dependent equality between morphisms of possibly nonconvertible types.