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Dependently Typed Functional Programs and their Proofs
, 1999
"... Research in dependent type theories [ML71a] has, in the past, concentrated on its use in the presentation of theorems and theoremproving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs ..."
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Cited by 70 (13 self)
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Research in dependent type theories [ML71a] has, in the past, concentrated on its use in the presentation of theorems and theoremproving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs may readily be specified and established. In particular, it develops technology for programming with dependent inductive families of datatypes and proving those programs correct. It demonstrates the considerable advantage to be gained by indexing data structures with pertinent characteristic information whose soundness is ensured by typechecking, rather than human effort. Type theory traditionally presents safe and terminating computation on inductive datatypes by means of elimination rules which serve as induction principles and, via their associated reduction behaviour, recursion operators [Dyb91]. In the programming language arena, these appear somewhat cumbersome and give rise to unappealing code, complicated by the inevitable interaction between case analysis on dependent types and equational reasoning on their indices which must appear explicitly in the terms. Thierry Coquand’s proposal [Coq92] to equip type theory directly with the kind of
Proof Representation in Type Theory: State of the Art
, 1996
"... In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to repres ..."
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Cited by 5 (0 self)
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In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called placesholders. We present a theoretical approach to placeholders in type theory. In this approach placeholders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities.
Pure Type Systems with Explicit Substitution
, 2000
"... We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is ..."
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Cited by 5 (0 self)
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We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is established. A more complicated extension is proposed for which subject reduction does hold in general.
A Calculus of Lambda Calculus Contexts
 Journal of Automated Reasoning
, 2001
"... The calculus c serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole lling, by a mechanism of delayed substitution. The context calculus c is given in the form of an extension of th ..."
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Cited by 3 (0 self)
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The calculus c serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole lling, by a mechanism of delayed substitution. The context calculus c is given in the form of an extension of the lambda calculus. Many notions of context can be represented within the framework; a particular variation can be obtained by the choice of a pretyping, which we illustrate by three examples. 1.
Proof Representation in Type Theory: State of the Art
, 1996
"... In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to repres ..."
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Cited by 1 (0 self)
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In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called placesholders. We present a theoretical approach to placeholders in type theory. In this approach placeholders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities. Representaci'on de pruebas en la teor'ia de tipos: Estado del arte Resumen En el marco de la l...
Higher Order Unification via ...Style of Explicit Substitution
"... A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach ..."
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A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.
A Leftlinear Variant of λσ
, 1997
"... In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcal ..."
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In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcalculus enjoys the same general characteristics as oe, i.e. a simple and finitary firstorder presentation, confluent on expressions with metavariables of terms and weakly normalizing on typed expressions. Moreover, L does not have the nonleftlinear surjective pairing rule of oe which raises technical problems in some frameworks.