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45
A module calculus for Pure Type Systems
, 1996
"... Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verif ..."
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Cited by 23 (3 self)
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Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verification of distincts parts of large proofs. Unfortunately, few theoretical basis are given for these features. In this paper we propose an extension of Pure Type Systems with a module calculus adapted from SMLlike module systems for programming languages. Our module calculus gives a theoretical framework addressing the need for these features. We show that our module extension is conservative, and that type inference in the module extension of a given PTS is decidable under some hypotheses over the considered PTS.
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 22 (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.
A Proof of Strong Normalization For the Theory of Constructions Using a KripkeLike Interpretation
 In Workshop on Logical FrameworksPreliminary Proceedings
, 1990
"... . We give a proof that all terms that typecheck in the theory of constructions are strongly normalizing (under fireduction). The main novelty of this proof is that it uses a "Kripkelike" interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequence ..."
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Cited by 20 (0 self)
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. We give a proof that all terms that typecheck in the theory of constructions are strongly normalizing (under fireduction). The main novelty of this proof is that it uses a "Kripkelike" interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequences of strong normalization, consistency and decidability of typechecking. We also show that our proof yields another proof of strong normalization for LF (under fireduction), using the reducibility method. 1 Introduction We give a proof that all terms that typecheck in the theory of constructions are strongly normalizing (under fireduction). The main novelty of this proof is that it uses a "Kripkelike " interpretation of the types and kinds, and that it does not use infinite contexts. The idea used for avoiding infinite contexts comes from Coquand's thesis [Coq85]. Our proof yields as a corollary another proof of strong normalization (under fireduction) of wellformed terms of LF . In f...
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 16 (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
A short and flexible proof of Strong Normalization for the Calculus of Constructions
, 1994
"... this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through ..."
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Cited by 15 (0 self)
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this paper can still go through (with slightly more technical effort) in case one can distinguish cases according to whether a specific subterm is a type or kind in a fixed context. The other property of type systems that is really actually required for the constructions in this paper to go through is a slight strengthening of the Stripping property (also called Generation). This property says, for example, that if \Gamma ` v:T:M : U has a derivation D, then one can find a subderivation of
A Tutorial on Recursive Types in Coq
, 1996
"... Contents 1 Recursive Types and Case Analysis 2 1.1 The predecessor function : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 The empty type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.3 The singleton type : : : : : : : : : : : : : : : ..."
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Cited by 14 (1 self)
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Contents 1 Recursive Types and Case Analysis 2 1.1 The predecessor function : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 The empty type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.3 The singleton type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Families of Recursive Types : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.5 The propositional equality type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2 Discrimination of introduction rules 7 3 Injectivity of introduction rules 8 4 Case Analysis and Propositional Equality 9 5 Positive Recursive Types 12 5.1 Mutually Dependent Declarations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 5.2 Impredicative Recursive Types : : : : : : : : : : : : : : : : : :
Sets in Types, Types in Sets
 Proceedings of TACS'97
, 1997
"... . We present two mutual encodings, respectively of the Calculus of Inductive Constructions in ZermeloFraenkel set theory and the opposite way. More precisely, we actually construct two families of encodings, relating the number of universes in the type theory with the number of inaccessible cardina ..."
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Cited by 12 (1 self)
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. We present two mutual encodings, respectively of the Calculus of Inductive Constructions in ZermeloFraenkel set theory and the opposite way. More precisely, we actually construct two families of encodings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r.t. expressive power and thus are essentially equivalent. Both encodings are quite elementary: type theory is interpreted in set theory through a generalization of Coquand 's simple proofirrelevance interpretation. Set theory is encoded in type theory using a variant of Aczel's encoding; we have formally checked this last part using the Coq proof assistant. 1 Introduction This work is an attempt towards better understanding of the expressiveness of powerful type theories. We here investigate the Calculus of Inductive Constructions (CIC); this formalism is, with some variants, the one implemen...
Developing (Meta)Theory of lambdacalculus in the Theory of Contexts
 Proc. MERLIN’01, TR 2001/26, Dept. of Math. and Comp. Sci., Univ. of Leicester
, 2001
"... . We present a case study on the formal development of a non trivial (meta)theory in the Theory of Contexts using the Coq proof assistant. The methodology underlying the Theory of Contexts for reasoning on systems presented in HOAS is based on an axiomatic syntactic standpoint. We feel that one ..."
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Cited by 12 (2 self)
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. We present a case study on the formal development of a non trivial (meta)theory in the Theory of Contexts using the Coq proof assistant. The methodology underlying the Theory of Contexts for reasoning on systems presented in HOAS is based on an axiomatic syntactic standpoint. We feel that one of the main advantages of this approach, is that it requires a very low logical overhead. The object logic we focus on is the lazy, callbyname #calculus (#cbn ), both untyped and simply typed. We will see that the formal, fully detailed development of the theory of #cbn in the Theory of Contexts introduces a small, sustainable overhead with respect to the proofs "on the paper". Moreover, this will allow for comparison with similar case studies developed in other approaches to the metatheoretical reasoning in higherorder abstract syntax. Keywords: higherorder abstract syntax, induction, logical frameworks.
The not so simple proofirrelevant model of CC
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... It is wellknown that the Calculus of Constructions (CC) bears a simple settheoretical model in which proofterms are mapped onto a single object—a property which is known as proofirrelevance. In this paper, we show that when going into the (generally omitted) technical details, this naive model r ..."
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Cited by 10 (1 self)
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It is wellknown that the Calculus of Constructions (CC) bears a simple settheoretical model in which proofterms are mapped onto a single object—a property which is known as proofirrelevance. In this paper, we show that when going into the (generally omitted) technical details, this naive model raises several unexpected difficulties related to the interpretation of the impredicative level, especially for the soundness property which is surprisingly difficult to be given a correct proof in this simple framework. We propose a way to tackle these difficulties, thus giving a (more) detailed elementary consistency proof of CC without going back to a translation to Fω. We also discuss some possible alternatives and possible extensions of our construction.
The structure of nuprl’s type theory
, 1997
"... on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) ..."
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Cited by 9 (3 self)
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on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html)