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Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 75 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
Combining HigherOrder and FirstOrder Computation Using ρcalculus: Towards a Semantics of ELAN
 In Frontiers of Combining Systems 2
, 1999
"... The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using ..."
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Cited by 20 (9 self)
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The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using an explicit substitution setting, called ρσ, that generalizes the λσcalculus. The basic properties of the nonexplicit and explicit substitution versions are presented. We then detail how to use the ρcalculus to give an operational semantics to the rewrite rules of the ELAN language. 1
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 ..."
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Cited by 19 (0 self)
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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 ..."
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Cited by 16 (2 self)
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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.
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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Cited by 15 (0 self)
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider
Comparing and Implementing Calculi of Explicit Substitutions with Eta Reduction
 Annals of Pure and Applied Logic
, 2005
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that e ..."
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Cited by 10 (8 self)
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The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of betacontraction.
Tradeoffs in the Intensional Representation of Lambda Terms
 Rewriting Techniques and Applications (RTA 2002), volume 2378 of LNCS
, 2002
"... Higherorder representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. ..."
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Cited by 10 (3 self)
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Higherorder representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. Refined lambda calculus notations have been proposed that can be used in realizing such implementations. There are, however, choices in the actual deployment of such notations whose practical consequences are not well understood. Towards addressing this lacuna, the impact of three specific ideas is examined: the de Bruijn representation of bound variables, the explicit encoding of substitutions in terms and the annotation of terms to indicate their independence on external abstractions. Qualitative assessments are complemented by experiments over actual computations using the lambdaProlog language.
A Typed Context Calculus
, 2001
"... This paper develops a typed calculus for contexts i.e., lambda terms with "holes". In addition to ordinary lambda terms, the calculus contains labeled holes, hole abstraction and context application for manipulating firstclass contexts. The primary operation for contexts is holefilling, which capt ..."
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Cited by 9 (0 self)
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This paper develops a typed calculus for contexts i.e., lambda terms with "holes". In addition to ordinary lambda terms, the calculus contains labeled holes, hole abstraction and context application for manipulating firstclass contexts. The primary operation for contexts is holefilling, which captures free variables. This operation conicts with substitution of the lambda calculus, and a straightforward mixture of the two results in an inconsistent system. We solve this problem by defining a type system that precisely specifies the variablecapturing nature of contexts and that keeps track of bound variable renaming. These mechanisms enable us to define a reduction system that properly integrates betareduction and holefilling. The resulting calculus is ChurchRosser and the type system has the subject reduction property. We believe that the context calculus will serve as a basis for developing a programming language with advanced features that call for manipulation of open terms.
The simply typed rewriting calculus
 WRLA 2004
, 2004
"... The rewriting calculus is a rule construction and application framework. As such it embeds in a uniform way term rewriting and lambdacalculus. Since rule application is an explicit object of the calculus, it allows us also to handle the set of results explicitly. We present a simply typed version o ..."
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Cited by 9 (5 self)
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The rewriting calculus is a rule construction and application framework. As such it embeds in a uniform way term rewriting and lambdacalculus. Since rule application is an explicit object of the calculus, it allows us also to handle the set of results explicitly. We present a simply typed version of the rewriting calculus. With a good choice of the type system, we show that the calculus is type preserving and terminating, i.e. verifies the subject reduction and strong normalization properties.