Results 11  20
of
132
External Rewriting for Skeptical Proof Assistants
, 2002
"... This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associativecommutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associativecommutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a speci c and ecient environment and to check the computations later in a proof assistant.
Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus
, 1996
"... Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe calculus of explicit s ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe calculus of explicit substitutions, proposed by Abadi, Cardelli, Curien andL evy, is a firstorder rewriting system that implements substitution and renaming mechanism of calculus. However, oe does not preserve strong normalisation of calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation. 1. Explicit substitutions The calculus is a higherorder theor...
Substitution: A formal methods case study using monads and transformations
, 1994
"... The specification and derivation of substitution for the de Bruijn representation of  terms is used to illustrate programming with a functionsequence monad. The resulting program is improved by interactive program transformation methods into an efficient implementation that uses primitive machine ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
The specification and derivation of substitution for the de Bruijn representation of  terms is used to illustrate programming with a functionsequence monad. The resulting program is improved by interactive program transformation methods into an efficient implementation that uses primitive machine arithmetic. These transformations illustrate new techniques that assist the discovery of the arithmetic structure of the solution. Introduction Substitution is one of many problems in computer science that, once understood in one context, is understood in all contexts. Why, then, must a different substitution function be written for every abstract syntax implemented? This paper shows how to define substitution once and use the monadic structure of the definition to instantiate it on different abstract syntax structures. It also shows how to interactively derive an efficient implementation of substitution from this very abstract definition. The authors are supported in part by a grant fr...
Functional BackEnds within the LambdaSigma Calculus
, 1996
"... We define a weak calculus, oe w , as a subsystem of the full calculus with explicit substitutions oe * . We claim that oe w could be the archetypal output language of functional compilers, just as the calculus is their universal input language. Furthermore, oe * could be the adequate theory to e ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
We define a weak calculus, oe w , as a subsystem of the full calculus with explicit substitutions oe * . We claim that oe w could be the archetypal output language of functional compilers, just as the calculus is their universal input language. Furthermore, oe * could be the adequate theory to establish the correctness of simplified functional compilers. Here, we illustrate these claims by proving the correctness of four simplified compilers and runtime systems modeled as abstract machines. The four machines we prove are the Krivine machine, the SECD, the FAM and the CAM. Thereby, we give the first formal proofs of Cardelli's FAM and of its compiler.
The rewriting calculus  Part I
, 2001
"... The ρcalculus integrates in a uniform and simple setting firstorder rewriting, λcalculus and nondeterministic computations. Its abstraction mechanism is based on the rewrite rule formation and its main evaluation rule is based on matching modulo a theory T. In this first part, the calculus is mot ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
The ρcalculus integrates in a uniform and simple setting firstorder rewriting, λcalculus and nondeterministic computations. Its abstraction mechanism is based on the rewrite rule formation and its main evaluation rule is based on matching modulo a theory T. In this first part, the calculus is motivated and its syntax and evaluation rules for any theory T are presented. In the syntactic case, i.e. when T is the empty theory, we study its basic properties for the untyped case. We first show how it uniformly encodes λcalculus as well as firstorder rewriting derivations. Then we provide sufficient conditions for ensuring confluence of the calculus.
The LambdaCalculus with Multiplicities
, 1993
"... We introduce a refinement of the λcalculus, where the argument of a function is a bag of resources, that is a multiset of terms, whose multiplicities indicate how many copies of them are available. We show that this "λcalculus with multiplicities" has a natural functionality theory, simi ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
We introduce a refinement of the λcalculus, where the argument of a function is a bag of resources, that is a multiset of terms, whose multiplicities indicate how many copies of them are available. We show that this "λcalculus with multiplicities" has a natural functionality theory, similar to Coppo and Dezani's intersection type discipline. In our functionality theory the conjunction is managed in a "multiplicative" manner, according to Girard's terminology. We show that this provides an adequate interpretation of the calculus, by establishing that a term is convergent if and only if it has a nontrivial functional character.
Calculi of Generalised βReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1998
"... Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substit ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the  calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs, and show that welltyped terms are strongly normalizing and that other properties,...
The scalculus: its typed and its extended versions
, 1995
"... We present in this paper the simply typed version of the scalculus (cf. [KR95]) and prove the strong normalisation of the well typed terms. We also present an extension of the scalculus: the s ecalculus and prove its local con
uence on open terms and the weak normalisation of its corresponding ca ..."
Abstract

Cited by 16 (10 self)
 Add to MetaCart
(Show Context)
We present in this paper the simply typed version of the scalculus (cf. [KR95]) and prove the strong normalisation of the well typed terms. We also present an extension of the scalculus: the s ecalculus and prove its local con
uence on open terms and the weak normalisation of its corresponding calculus of substitutions s e. The strong normalisation of s e is still an open problem to challenge the rewriting community.
Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus
, 1995
"... Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, f ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, for instance, to represent incomplete proofs in type based proof systems. The oe calculus of explicit substitutions proposed by Abadi, Cardelli, Curien and L'evy gives an elegant way to deal with management of variable names and substitutions of calculus. However, oe does not preserve strong normalisation of calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i  calculus an we present the full proofs of its main properties. This is, as far as we know, the...