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The Rho Cube
 In Proc. of FOSSACS, volume 2030 of LNCS
, 2001
"... www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. The rewriting calculus, or Rho Calculus (ρCal), is a simple calculus that uniformly integrates abstraction on patterns and nondeterminism. Therefore, it fully integrates rewriting and λcalculus. The original presentation of the calculus was unty ..."
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Cited by 31 (16 self)
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www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. The rewriting calculus, or Rho Calculus (ρCal), is a simple calculus that uniformly integrates abstraction on patterns and nondeterminism. Therefore, it fully integrates rewriting and λcalculus. The original presentation of the calculus was untyped. In this paper we present a uniform way to decorate the terms of the calculus with types. This gives raise to a new presentation à la Church, together with nine (8+1) type systems which can be placed in a ρcube that extends the λcube of Barendregt. Due to the matching capabilities of the calculus, the type systems use only one abstraction mechanism and therefore gives an original answer to the identification of the standard “λ ” and “Π” abstractors. As a consequence, this brings matching and rewriting as the first class concepts of the Rhoversions of the Logical Framework (LF) of Harper
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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Cited by 7 (3 self)
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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.
Comparing Cubes
"... We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subje ..."
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Cited by 5 (3 self)
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We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject reduction and strong normalization. We study the relationship between the two cubes, which leads to some unexpected results in the eld of systems with dependent types.
The ChurchRosser Property for Pure Type Systems with βηreduction
, 1992
"... this paper is to give a proof of the ChurchRosser property (or confluence) with respect to ## reduction for type theories with labelled lambda abstraction. This property is interesting in general since many other useful properties of a system depends on it and it is crucial when trying to use such ..."
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Cited by 2 (0 self)
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this paper is to give a proof of the ChurchRosser property (or confluence) with respect to ## reduction for type theories with labelled lambda abstraction. This property is interesting in general since many other useful properties of a system depends on it and it is crucial when trying to use such systems as framework for defining logics. The usual way of encoding logics in these frameworks implies that proof checking corresponds to type checking, and since we want proof checking to be decidable we also need decidability of the frameworks. Strong Normalization and the Church Rosser property give us an algorithm to decide if two wellformed types are equal, simply compute the two terms to normal forms and see if they are identical. This together with Subject Reduction (typing is preserved by reduction) and Uniqness of Types (the type of a term is unique modulo equality on types) give a quite simple algorithm for type checking, not involving higher order unification or backtracking. Subject Reduction alone is also an important property in general, this ensures a certain kind of soundness of a type system, reduction or computation does not change the typing of a term. The standard way of establishing these properties when only # reduction is present is to use the fact that the ChurchRosser property is true for the preset of terms (labeled lambda terms in general) and from this again one can establish Subject Reduction. Though Uniqness of Types is not true for arbitrary type systems the standard proof strategy also use the ChurchRosser property. When it comes to Strong Normalization the ChurchRosser property is usually used in the proofs. With labelled systems one often translate them into unlabelled ones in a way that preserve reduction. Since the unlabelled systems a...