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Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack&a ..."
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Cited by 25 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
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.
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 22 (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 22 (1 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 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 c ..."
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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...
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 19 (2 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...
A minimalist twolevel foundation for constructive mathematics
, 2008
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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Cited by 19 (7 self)
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms” paradigm and acts as a programming language.
Common nouns as types
 Logical Aspects of Computational Linguistics (LACL’2012). LNCS 7351
, 2012
"... Abstract. When modern type theories are employed for formal semantics, common nouns (CNs) are interpreted as types, not as predicates. Although this brings about some technical advantages, it is worthwhile to ask: what is special about CNs that merits them to be interpreted as types? We discuss the ..."
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Abstract. When modern type theories are employed for formal semantics, common nouns (CNs) are interpreted as types, not as predicates. Although this brings about some technical advantages, it is worthwhile to ask: what is special about CNs that merits them to be interpreted as types? We discuss the observation made by Geach that, unlike other lexical categories, CNs have criteria of identity, a component of meaning that makes it legitimate to compare, count and quantify. This is closely related to the notion of set (type) in constructive mathematics, where a set (type) is not given solely by specifying its objects, but together with an equality between its objects, and explains and justifies to some extent why types are used to interpret CNs in modern type theories. It is shown that, in order to faithfully interpret modified CNs as Σtypes so that the associated criteria of identity can be captured correctly, it is important to assume proof irrelevance in type theory. We shall also briefly discuss a proposal to interpret mass noun phrases as types in a uniform approach to the semantics of CNs. 1
The metaobject facility typed
 In SAC
, 2006
"... The Object Managment Group’s MetaObject Facility (MOF) [9] is a semiformal approach to writing models and metamodels (models of models). The MOF was developed to enable systematic model/metamodel interchange and integration. The approach is problematic, unless metamodels are correctly specified: an ..."
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The Object Managment Group’s MetaObject Facility (MOF) [9] is a semiformal approach to writing models and metamodels (models of models). The MOF was developed to enable systematic model/metamodel interchange and integration. The approach is problematic, unless metamodels are correctly specified: an error in a metamodel specification will propagate throughout instantiating models and final model implementations. An important open question is how to develop provably correct metamodels. This paper outlines a solution to the question, in which the MOF metamodelling approach is formalized within constructive type theory.