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Refinement Types for Logical Frameworks
 Informal Proceedings of the Workshop on Types for Proofs and Programs
, 1993
"... We propose a refinement of the type theory underlying the LF logical framework by a form of subtypes and intersection types. This refinement preserves desirable features of LF, such as decidability of typechecking, and at the same time considerably simplifies the representations of many deductive s ..."
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Cited by 43 (9 self)
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We propose a refinement of the type theory underlying the LF logical framework by a form of subtypes and intersection types. This refinement preserves desirable features of LF, such as decidability of typechecking, and at the same time considerably simplifies the representations of many deductive systems. A subtheory can be applied directly to hereditary Harrop formulas which form the basis of Prolog and Isabelle. 1 Introduction Over the past two years we have carried out extensive experiments in the application of the LF Logical Framework [HHP93] to represent and implement deductive systems and their metatheory. Such systems arise naturally in the study of logic and the theory of programming languages. For example, we have formalized the operational semantics and type system of MiniML and implemented a proof of type preservation [MP91] and the correctness of a compiler to a variant of the Categorical Abstract Machine [HP92]. LF is based on a predicative type theory with dependent t...
Unification in a λCalculus with Intersection Types
"... We propose related algorithms for unification and constraint simplification in !& , a refinement of the simplytyped λcalculus with subtypes and bounded intersection types. !& is intended as the basis of a logical framework in order to achieve more succinct and declarative axiomatizations of deduct ..."
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Cited by 3 (1 self)
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We propose related algorithms for unification and constraint simplification in !& , a refinement of the simplytyped λcalculus with subtypes and bounded intersection types. !& is intended as the basis of a logical framework in order to achieve more succinct and declarative axiomatizations of deductive systems than possible with the simplytyped λcalculus. The unification and constraint simplification algorithms described here lay the groundwork for a mechanization of such frameworks as constraint logic programming languages and theorem provers.
Reduction and Unification in Lambda Calculi with a General Notion of Subtype
 J. of Automated Reasoning
, 1994
"... . Reduction, equality and unification are studied for a family of simply typed calculi with subtypes. The subtype relation is required to relate base types only to base types and to satisfy some ordertheoretic conditions. Constants are required to have a least type, i.e. "no overloading". We defin ..."
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Cited by 2 (1 self)
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. Reduction, equality and unification are studied for a family of simply typed calculi with subtypes. The subtype relation is required to relate base types only to base types and to satisfy some ordertheoretic conditions. Constants are required to have a least type, i.e. "no overloading". We define the usual fi and a subtypedependent jreduction. These are related to a typed equality relation and shown to be confluent in a certain sense. We present a generic algorithm for preunification modulo fijconversion and an arbitrary subtype relation. Furthermore it is shown that unification with respect to any subtype relation is universal. Key words. Simply typed calculi, subtypes, reduction, higherorder unification. 1 Introduction Subtypes have long been recognized as an important means of succinctly expressing inheritance relations. They are particularly valuable in the area of automated deduction because they can be built into the inference engine by means of ordersorted unificat...
HigherOrder OrderSorted Resolution
 FB Informatik, Universitat des Saarlandes
, 1994
"... The introduction of sorts to firstorder automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search space. It is therefore promising to treat sorts in higher order theorem proving as well. In this paper we present a generalizati ..."
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Cited by 1 (1 self)
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The introduction of sorts to firstorder automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search space. It is therefore promising to treat sorts in higher order theorem proving as well. In this paper we present a generalization of Huet's Constrained Resolution to an ordersorted type theory \SigmaT with term declarations. This system builds certain taxonomic axioms into the unification and conducts reasoning with them in a controlled way. We make this notion precise by giving a relativization operator that totally and faithfully encodes \SigmaT into simple type theory.
ΩMKRP: A Proof Development Environment
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] wi ..."
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Cited by 1 (0 self)
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In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] with the firstorder theorem prover mkrp [11]. An important finding was that although current automated theorem provers have evidently reached the power to solve nontrivial problems, they do not provide sufficient assistance for proving the theorems contained in such a textbook. On account of this, we believe that significantly more support for proof development can be provided by a system with the following two features:  The system must provide a comfortable humanoriented problemsolving environment. In particular, a human user should be able to specify the problem to be solved in a natural way and communicate on proof
Unification in a Sorted λCalculus with Term Declarations and Function Sorts
, 1994
"... The introduction of sorts to firstorder automated deduction... ..."