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 higher-order patterns, a notion investigated for the simply-typed -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 anti-unification algorithm we have in mind is that of proof generalization. 1 Introduction Higher-order logic with an embedded simply-typed - 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 pre-unification algorithm for the simply-typed -calculus [12] which has shown it...

Logic programming provides a uniform framework in which all aspects of explanation-based generalization and learning may be defined and carried out, but first-order Horn logic is not well suited to application domains such as theorem proving or program synthesis where functions and predicates are the objects of computation. We explore the use of a higher-order representation language and extend EBG to a higher-order logic programming language. Variables may now range over functions and predicates, which leads to an expansion of the space of possible generalizations. We address this problem by extending the logic with the modal ⊔ ⊓ operator (indicating necessary truth) which leads to the language λ ⊔ ⊓ Prolog. We develop a meta-interpreter realizing EBG for λ ⊔ ⊓ Prolog and give some examples in an expanded version of this extended abstract which is available as a technical report [2]. 1