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37
Pure Type Systems with Definitions
, 1993
"... In this paper, an extension of Pure Type Systems (PTS's) with definitions is presented. We prove this extension preserves many of the properties of PTS's. The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation. ..."
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In this paper, an extension of Pure Type Systems (PTS's) with definitions is presented. We prove this extension preserves many of the properties of PTS's. The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation.
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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. 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...
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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The Implicit Calculus of Constructions  Extending Pure Type Systems with an Intersection Type Binder and Subtyping
 Proc. of 5th Int. Conf. on Typed Lambda Calculi and Applications, TLCA'01, Krakow
, 2001
"... In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Currystyle variant of the Calculus of Constructions that we extend by adding an intersection type binder called the implicit dependent product. Unlike the usual approach of Type Assignment Systems ..."
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In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Currystyle variant of the Calculus of Constructions that we extend by adding an intersection type binder called the implicit dependent product. Unlike the usual approach of Type Assignment Systems, the implicit product can be used at every place in the universe hierarchy. We study syntactical properties of this calculus such as the subject reduction property, and we show that the implicit product induces a rich subtyping relation over the type system in a natural way. We also illustrate the specicities of this calculus by revisitting the impredicative encodings of the Calculus of Constructions, and we show that their translation into the implicit calculus helps to reect the computational meaning of the underlying terms in a more accurate way.
From formal proofs to mathematical proofs: A safe, incremental way for building in firstorder decision procedures
 In TCS 2008: 5th IFIP International Conference on Theoretical Computer Science
, 2008
"... (CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary firstorder decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated firstorder decision proc ..."
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(CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary firstorder decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated firstorder decision procedures that can be taken from the shelves provided they deliver a proof certificate. The soundness of the whole system becomes an incremental property following from the soundness of the certificate checkers and that of the kernel. A detailed example shows that the resulting style of proofs becomes closer to that of the working mathematician. 1
Modularity of Strong Normalization in the Algebraicλcube
, 1996
"... In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating and the ..."
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Cited by 8 (2 self)
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In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating and the higherorder rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraicλcube. We also prove that local confluence is a modular property of all the systems in the algebraicλcube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.
Comparing cubes of typed and type assignment systems
 Annals of Pure and Applied Logic
, 1997
"... We study the cube of type assignment systems, as introduced in [13], and confront it with Barendregt’s typed λcube [4]. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address th ..."
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We study the cube of type assignment systems, as introduced in [13], and confront it with Barendregt’s typed λcube [4]. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for the systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one.
On the Definition of the Etalong Normal Form in Type Systems of the Cube
 Informal Proceedings of the Workshop on Types for Proofs and Programs
, 1993
"... The smallest transitive relation ! on welltyped normal terms such that if t is a strict subterm of u then t ! u and if T is the normal form of the type of t and the term t is not a sort then T ! t is wellfounded in the type systems of the cube. Thus every term admits a jlong normal form. Introdu ..."
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The smallest transitive relation ! on welltyped normal terms such that if t is a strict subterm of u then t ! u and if T is the normal form of the type of t and the term t is not a sort then T ! t is wellfounded in the type systems of the cube. Thus every term admits a jlong normal form. Introduction In this paper we prove that the smallest transitive relation ! on welltyped normal terms such that ffl if t is a strict subterm of u then t ! u, ffl if T is the normal form of the type of t and the term t is not a sort then T ! t is wellfounded in the type systems of the cube [1]. This result is proved using the notion of marked terms introduced by de Vrijer [6]. A motivation for this theorem is to define the jlong form of a normal term in these type systems. In simply typed calculus, to define the jlong form of a normal term we first define the jlong form of a variable x of type P 1 ! ::: ! P n ! P (P atomic) as the term [y 1 : P 1 ]:::[y n : P n ](x y 0 1 ::: y 0 n ) w...
A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping
"... We introduce a new model based on coherence spaces for interpreting large impredicative type systems such as the Extended Calculus of Constructions (ECC). Moreover, we show that this model is wellsuited for interpreting intersection types and subtyping too, and we illustrate this by interpreting a ..."
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We introduce a new model based on coherence spaces for interpreting large impredicative type systems such as the Extended Calculus of Constructions (ECC). Moreover, we show that this model is wellsuited for interpreting intersection types and subtyping too, and we illustrate this by interpreting a variant of ECC with an additional intersection type binder. Furthermore, we propose a general method for interpreting the impredicative level in a nonsyntactical way, by allowing the model to be parametrized by an arbitrarily large coherence space in order to interpret inhabitants of impredicative types. As an application, we show that uncountable types such as the type of real numbers or ZermeloFrnkel sets can safely be axiomatized on the impredicative level of, say, ECC, without harm for consistency. 1
Typechecking in Pure Type Systems
 Informal proceedings of Logical Frameworks'92
, 1992
"... Introduction This work is motivated by two related problems. The first is to find reasonable algorithms for typechecking Pure Type Systems [Bar91] (PTS); the second is a technical question about PTS, the so called Expansion Postponment property (EP), which is tantalizingly simple but remains unsolv ..."
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Cited by 5 (2 self)
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Introduction This work is motivated by two related problems. The first is to find reasonable algorithms for typechecking Pure Type Systems [Bar91] (PTS); the second is a technical question about PTS, the so called Expansion Postponment property (EP), which is tantalizingly simple but remains unsolved. There are several implementations of formal systems that are either PTS or closely related to PTS. For example, LEGO [LP92] implements the Pure Calculus of Constructions [CH88] (PCC), the Extended Calculus of Constructions [Luo90] and the Edinburgh Logical Framework [HHP87]. ELF [Pfe89] implements LF; CONSTRUCTOR [Hel91] implements arbitrary PTS with finite set of sorts. Are these implementations actually correct? It is not difficult to find a natural efficient algorithm that is provably sound (Section 3), but completeness is more difficult. In fact Jutting has shown typechecking is decidable for all normalizing PTS with a finite set of sorts [vBJ92], but his algorithm, which com