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The Theory of LEGO  A Proof Checker for the Extended Calculus of Constructions
, 1994
"... LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO ..."
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Cited by 68 (10 self)
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LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the metatheory of LEGO's type systems leading to a machinechecked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...
Some lambda calculus and type theory formalized
 Journal of Automated Reasoning
, 1999
"... Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention ..."
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Cited by 52 (7 self)
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Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.
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
The Implementation of ALF  a Proof Editor based on MartinLöf's Monomorphic Type Theory with Explicit Substitution
, 1995
"... This thesis describes the implementation of ALF, which is an interactive proof editor based on MartinLöf's type theory with explicit substitutions. ALF is a general purpose proof assistant, in which different logics can be represented. Proof objects are manipulated directly, by the usual editing op ..."
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Cited by 43 (0 self)
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This thesis describes the implementation of ALF, which is an interactive proof editor based on MartinLöf's type theory with explicit substitutions. ALF is a general purpose proof assistant, in which different logics can be represented. Proof objects are manipulated directly, by the usual editing operations. A partial proof is represented as an incomplete proof object, i.e., a proof object containing placeholders. A modular type/proof checking algorithm for complete proof objects is presented, and it is proved sound and complete assuming some basic meta theory properties of the substitution calculus. The algorithm is extended to handle incomplete objects in such a way that the type checking problem is reduced to a unication problem, i.e., the problem of finding instantiations to the placeholders in the object. Placeholders are represented together with their expected type and local context. We show that checking the correctness of instantiations can be localised, which means that it is e...
Pure type systems formalized
 Proceedings of the International Conference on Typed Lambda Calculi and Applications
, 1993
"... ..."
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.
Closure Under AlphaConversion
 In The Informal Proceeding of the 1993 Workshop on Types for Proofs and Programs
, 1993
"... this paper appears in Types for Proofs and Programs: International Workshop TYPES'93, Nijmegen, May 1993, Selected Papers, LNCS 806. abstraction, compute a type for its body in an extended context; to compute a type for an application, compute types for its left and right components, and check that ..."
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Cited by 23 (3 self)
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this paper appears in Types for Proofs and Programs: International Workshop TYPES'93, Nijmegen, May 1993, Selected Papers, LNCS 806. abstraction, compute a type for its body in an extended context; to compute a type for an application, compute types for its left and right components, and check that they match appropriately. Lets use the algorithm to compute a type for a = [x:ø ][x:oe]x. FAILURE: no rule applies because x 2 Dom (x:ø )
A Verified Typechecker
 PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON TYPED LAMBDA CALCULI AND APPLICATIONS, VOLUME 902 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1995
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Pure Type Systems in Rewriting Logic
 In Proc. of LFM’99: Workshop on Logical Frameworks and MetaLanguages
, 1999
"... . The logical and operational aspects of rewriting logic as a logical framework are illustrated in detail by representing pure type systems as object logics. More precisely, we apply membership equational logic, the equational sublogic of rewriting logic, to specify pure type systems as they can be ..."
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Cited by 10 (2 self)
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. The logical and operational aspects of rewriting logic as a logical framework are illustrated in detail by representing pure type systems as object logics. More precisely, we apply membership equational logic, the equational sublogic of rewriting logic, to specify pure type systems as they can be found in the literature and also a new variant of pure type systems with explicit names that solves the problems with closure under conversion in a very satisfactory way. Furthermore, we use rewriting logic itself to give a formal operational description of type checking, that directly serves as an ecient type checking algorithm. The work reported here is part of a more ambitious project concerned with the development in Maude [7] of a proof assistant for OCC, the open calculus of constructions, an equational extension of the calculus of constructions. 1 Introduction This paper is a detailed case study on the ease and naturalness with which a family of higherorder formal systems, namely...
Normalisation and Equivalence in Proof Theory and Type Theory
, 2006
"... & the advisers At the heart of the connections between Proof Theory and Type Theory, the CurryHoward correspondence provides proofterms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the d ..."
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Cited by 9 (2 self)
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& the advisers At the heart of the connections between Proof Theory and Type Theory, the CurryHoward correspondence provides proofterms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of prooftheoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proofsearch, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proofterms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proofterms for natural deduction (λcalculus) and sequent calculus, and investigate normalisation and cutelimination, with callbyname and callbyvalue semantics. In particular, it introduces proofterm calculi for multiplicative natural deduction and for the depthbounded sequent calculus G4. The former gives rise to the calculus λlxr with explicit substitutions,