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Logic Programming in the LF Logical Framework
, 1991
"... this paper we describe Elf, a metalanguage intended for environments dealing with deductive systems represented in LF. While this paper is intended to include a full description of the Elf core language, we only state, but do not prove here the most important theorems regarding the basic building b ..."
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Cited by 175 (50 self)
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this paper we describe Elf, a metalanguage intended for environments dealing with deductive systems represented in LF. While this paper is intended to include a full description of the Elf core language, we only state, but do not prove here the most important theorems regarding the basic building blocks of Elf. These proofs are left to a future paper. A preliminary account of Elf can be found in [26]. The range of applications of Elf includes theorem proving and proof transformation in various logics, definition and execution of structured operational and natural semantics for programming languages, type checking and type inference, etc. The basic idea behind Elf is to unify logic definition (in the style of LF) with logic programming (in the style of Prolog, see [22, 24]). It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Hornclauses) an operational interpretation. An alternative approach to logic programming in LF has been developed independently by Pym [28]. Here are some of the salient characteristics of our unified approach to logic definition and metaprogramming. First of all, the Elf search process automatically constructs terms that can represent objectlogic proofs, and thus a program need not construct them explicitly. This is in contrast to logic programming languages where executing a logic program corresponds to theorem proving in a metalogic, but a metaproof is never constructed or used and it is solely the programmer's responsibility to construct objectlogic proofs where they are needed. Secondly, the partial correctness of many metaprograms with respect to a given logic can be expressed and proved by Elf itself (see the example in Section 5). This creates the possibilit...
Unification and AntiUnification in the Calculus of Constructions
 In Sixth Annual IEEE Symposium on Logic in Computer Science
, 1991
"... We present algorithms for unification and antiunification in the Calculus of Constructions, where occurrences of free variables (the variables subject to instantiation) are restricted to higherorder patterns, a notion investigated for the simplytyped calculus by Miller. Most general unifiers and ..."
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Cited by 61 (15 self)
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We present algorithms for unification and antiunification in the Calculus of Constructions, where occurrences of free variables (the variables subject to instantiation) are restricted to higherorder patterns, a notion investigated for the simplytyped calculus by Miller. Most general unifiers and least common antiinstances are shown to exist and are unique up to a simple equivalence. The unification algorithm is used for logic program execution and type and term reconstruction in the current implementation of Elf and has shown itself to be practical. The main application of the antiunification algorithm we have in mind is that of proof generalization. 1 Introduction Higherorder logic with an embedded simplytyped  calculus has been used as the basis for a number of theorem provers (for example [1, 19]) and the programming language Prolog [16]. Central to these systems is an implementation of Huet's preunification algorithm for the simplytyped calculus [12] which has shown it...
Extensions and Applications of Higherorder Unification
, 1990
"... ... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The ric ..."
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Cited by 25 (1 self)
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... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The rich structure of a typedcalculus,asopposedtotraditional,rst generalideaistouseacalculusasametalanguageforrepresentingvariousotherlanguages thelattercase,thealgorithmisincomplete,thoughstillquiteusefulinpractice. Thelastpartofthedissertationprovidesexamplesoftheusefulnessofthealgorithms.The algorithmrstfordependentproduct()types,andsecondforimplicitpolymorphism.In involvessignicantcomplicationsnotarisingHuet'scorrespondingalgorithmforthesimply orderabstractsyntaxtrees,allowsustoexpressrules,e.g.,programtransformationand typedcalculus,primarilybecauseitmustdealwithilltypedterms.Wethenextendthis Wecanthenuseunicationinthemetalanguagetomechanizeapplicationoftheserules.
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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Cited by 7 (0 self)
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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 normalization proof for MartinLöf's type theory
, 1996
"... The theory we will be concerned with in this paper is MartinLöf's polymorphic type theory with intensional equality and the universe of small sets. We will give a different proof of normalization for this theory. ..."
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The theory we will be concerned with in this paper is MartinLöf's polymorphic type theory with intensional equality and the universe of small sets. We will give a different proof of normalization for this theory.
Informal Proceedings Of The 1993 Workshop On Types For Proofs And Programs, Nijmegen
, 1993
"... Clauses : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 375 iii Foreword This document is the preliminary proceedings of the workshop of the Esprit Basic Research Project 6453 "Types for Proofs and Programs" held at the University of Nijmegen, the Netherlands, from ..."
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Clauses : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 375 iii Foreword This document is the preliminary proceedings of the workshop of the Esprit Basic Research Project 6453 "Types for Proofs and Programs" held at the University of Nijmegen, the Netherlands, from May 24th until May 28th 1993. The workshop was organised by Henk Barendregt and Herman Geuvers. Local arrangements were made by Marielle van der Zandt, Erik Barendsen, Herman Geuvers and Mark Ruys. These proceedings have been collected from L a T E X sources, using email. It contains 22 papers from the 35 talks that were presented at the workshop. Very useful support in solving the L a T E X puzzles was provided by Erik Barendsen. This document can be obtained by anonymous ftp from the University of Nijmegen: Type ftp ftp.cs.kun.nl anonymous (as login) [your email address] (as password) cd /pub/csi/CompMath/Types bin get NijmegenTypes.ps.Z bye iv Workshop Programme Types for Proofs an...