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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 54 (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 50 (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
Pure type systems formalized
 Proceedings of the International Conference on Typed Lambda Calculi and Applications
, 1993
"... ..."
Typechecking Injective Pure Type Systems
, 1993
"... Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(\GammajM ) and sort(\GammajM ), where \Gamma is a pseudocontext and M is a pseudoterm, and such that for every sort s, \Gamma ` M : A \Gamma ` A : s ) elmt(\Gamm ..."
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Cited by 3 (1 self)
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Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(\GammajM ) and sort(\GammajM ), where \Gamma is a pseudocontext and M is a pseudoterm, and such that for every sort s, \Gamma ` M : A \Gamma ` A : s ) elmt(\GammajM ) = s \Gamma ` M : s ) sort(\GammajM ) = s By eliminating the problematic clause in the (abstraction) rule in favor of constraints over elmt(:j:) and sort(:j:), we provide a sound and complete typechecking algorithm for injective Pure Type Systems. In addition, we prove Expansion Postponement for a variant of injective Pure Type Systems where the problematic clause in the (abstraction) rule is replaced in favor of constraints over elmt(:j:) and sort(:j:). 1
A new implementation of Automath
 Journal of Automated Reasoning
"... Abstract. This paper presents aut, a modern Automath checker. It is a straightforward reimplementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT68 and AUTQE dialects of Automath. This progr ..."
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Cited by 3 (0 self)
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Abstract. This paper presents aut, a modern Automath checker. It is a straightforward reimplementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT68 and AUTQE dialects of Automath. This program was written to restore a damaged version of Jutting’s translation of Landau’s Grundlagen. Some notable features: − It is fast. On a 1GHz machine it will check the full Jutting formalization (736K of nonwhitespace Automath source) in 0.6 seconds. − Its implementation of λterms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. − The program can compile an Automath text into one big ‘Automath single line’ style λterm. It outputs such a term using de Bruijn indices. (These λterms cannot be checked by modern systems like Coq or Agda, because the λtyped λcalculi of de Bruijn are different from the Πtyped λcalculi of modern type theory.) The source of aut is freely available on the Web at the address
Under consideration for publication in J. Functional Programming 1 Pure Type System conversion is always typable
, 2011
"... Pure Type Systems are usually described in two different ways, one that uses an external notion of computation like betareduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described ..."
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Pure Type Systems are usually described in two different ways, one that uses an external notion of computation like betareduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step towards this equivalence has been made by Adams for a particular class of Pure Type Systems (PTS) called functional. Then, his result has been relaxed to all semifull PTSs in previous work. In this paper, we finally give a positive answer to the general question, and prove that equivalence holds for any Pure Type System. 1