A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle

The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results. This thesis develops a sorted higher-order logic ΣHOL suitable for automatic theorem proving applications. ΣHOL is based on a sorted λ-calculus ΣΛ → , which is obtained by extending Church’s simply typed λ-calculus by a higher-order sort concept including term declarations and functional base sorts. The term declaration mechanism studied here is powerful enough to allow convenient formalization of a large body of mathematics, since it offers natural primitives for domains and codomains of functions, and allows to treat function restriction. Furthermore, it subsumes most other mechanisms for the declaration of sort information known from the literature, and can thus serve as a general framework for the study of sorted higher-order logics. For instance, the term declaration mechanism of ΣHOL subsumes the subsorting mechanism as a derived notion, and hence justifies our special form of subsort inference. We present sets of transformations for sorted higher-order unification and pre-unification, and prove the nondeterministic completeness of the algorithm induced by these transformations. Th