Higher-order unification is equational unification for βη-conversion. But it is not first-order equational unification, as substitution has to avoid capture. In this paper higher-order unification is reduced to first-order equational unification in a suitable theory: the λσ-calculus of explicit substitutions.

ions x 1 : : : xn are assumed to have arity one. For instance, x 1 x 2 :c(x 3 :x 3 ; x 2 (x 1 )) (assumed in normal form) has the following representation as a tree: x 1 x 2 c \Gamma \Gamma @ @ x 3 x 2 x 3 x 1 In what follows, we assume that F is finite. This is not a restriction as, for countably infinite alphabets, there is always another alphabet F 0 , which is finite, and an injective tree homomorphism h from T (F) into T (F) 0 such that h(T (F)) is recognizable by a finite tree automaton and the size of h(t) is linear with respect to the size of t. 1 However, for sake of clarity, we will keep the standard notations instead of using the encodings of F . 3.2 2-automata We will use a slight modification of tree automata. The main difference with the definitions of [13, 4] is the presence of special symbols 2 ø which should be interpreted as any term of type ø . This slight modification is necessary because, for instance, the set of all closed terms is not recognizable by ...

The system Coq is an environment for proof development based on the Calculus of Constructions extended by inductive definitions. Functional programs can be extracted from constructive proofs written in Coq. The extracted program and its corresponding proof are strongly related. The idea in this paper is to use this link to have another approach: to give a program and to generate automatically the proof from which it could be extracted. Moreover, we introduce a notion of annotated programs.

The smallest transitive relation ! on well-typed 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 well-founded in the type systems of the cube. Thus every term admits a j-long normal form. Introduction In this paper we prove that the smallest transitive relation ! on well-typed 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 well-founded 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 j-long form of a normal term in these type systems. In simply typed -calculus, to define the j-long form of a normal term we first define the j-long 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...

Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because first-order logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these auxiliaries to be present in any equivalence program.

We show that it is decidable whether a third-order matching problem in ! (an extension of the simply typed lambda calculus with type constructors) has a solution or not. We present an algorithm which, given such a problem, returns a solution for this problem if the problem has a solution and returns fail otherwise. We also show that it is undecidable whether a third-order matching problem in ! has a closed solution or not. 1 Introduction It is well-known that type theory is a good basis for the implementation of proof checkers. Although there are various ways to use type theory for proof checking, they all exploit the fact that type theory provides a uniform way to represent and manipulate proofs, formulas and data types. The man-machine interaction of proof checking can be considerably improved if some kind of matching algorithm can be implemented for the terms of the underlying type theory. For if one wants to prove OE(t) for a certain formula OE and term t, and one already has a pr...

We introduce the main concepts and problems in the theory of proof-search in type-theoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proof-search in type-theoretic languages; rather, we present some key ideas and problems, starting from well-motivated points of departure such as a definition of a type-theoretic language or the relationship between languages and proof-objects. The strong connections between different proof-search methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context.

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. We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higher-order logic can be adapted to provide increased automation in proof search in the Calculus of Constructions (CC). Bledsoe's procedure operates on an extension of first-order logic that allows existential quantification over set variables. The method finds maximal solutions for this special class of higher-order variables. This class of variables can also be identified in CC. The existence of a correspondence between higher-order logic and higher-order type theories such as CC is well-known. CC can be viewed as an extension of higher-order logic where the basic terms of the language, the simply-typed -terms, are replaced with terms containing dependent types. We adapt Bledsoe's procedure to the corresponding class of variables in CC and extend it to handle terms with dependent types. 1 Introduction Both higher-order logic and higher-order type theories serve as th...

We show that it is decidable whether a third-order matching problem in the polymorphic lambda calculus has a solution. The proof is constructive in the sense that an algorithm can be extracted from it that, given such a problem, returns a substitution if it has a solution and fail otherwise. 1 Introduction This paper is a contribution to the theory of (pattern) matching in higher order type theory. The starting point is the fact that third-order matching is decidable in the simply typed lambda calculus with constant types (see [5]). The question we would like to answer is: what happens if we extend this calculus with the type features that are characteristic for the Calculus of Constructions [2]: dependent types, type constructors and polymorphism. In [3], Dowek showed that in lambda calculi with dependent types third-order matching is undecidable. In contrast, we showed in [15] that the presence of type constructors is not sufficient to make third-order matching undecidable. In this ...