We introduce refutationally complete superposition calculi for intentional and extensional clausal λ-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the λ-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone towards complete, highly efficient automatic theorem provers for full higherorder logic.Key words and phrases: superposition calculus, clausal lambda-free higher-order logic, refutational completeness.Extended version of Bentkamp et al., "Superposition for lambda-free higher-order logic" [11].

E 2.3 is a theorem prover for many-sorted first-order logic with equality. We describe the basic logical and software architecture of the system, as well as core features of the implementation. We particularly discuss recently added features and extensions, including the extension to many-sorted logic, optional limited support for higher-order logic, and the integration of SAT techniques via PicoSAT. Minor additions include improved support for TPTP standard features, always-on internal proof objects, and lazy orphan removal. The paper also gives an overview of the performance of the system, and describes ongoing and future work.

Superposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved. We present a new extension of typed superposition that can perform structural induction. Several inductive goals can be attempted within a single saturation loop, by leveraging AVATAR [1]. Lemmas obtained by generalization or theory exploration can be introduced during search, used, and proved, all in the same search space. We describe an implementation and present some promising results.

Decades of work have gone into developing efficient proof calculi, data structures, algorithms, and heuristics for first-order automatic theorem proving. Higher-order provers lag behind in terms of efficiency. Instead of developing a new higher-order prover from the ground up, we propose to start with the state-of-the-art superposition-based prover E and gradually enrich it with higher-order features. We explain how to extend the prover's data structures, algorithms, and heuristics to λ-free higher-order logic, a formalism that supports partial application and applied variables. Our extension outperforms the traditional encoding and appears promising as a stepping stone towards full higher-order logic.

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SMT solvers have recently been extended with techniques for finding models of universally quantified formulas in some restricted fragments of firstorder logic. This paper introduces a translation that reduces axioms specifying a large class of recursive functions, including terminating functions, to universally quantified formulas for which these techniques are applicable. An evaluation confirms that the approach improves the performance of existing solvers on benchmarks from three sources. The translation is implemented as a preprocessor in the CVC4 solver and in a new higher-order model finder called Nunchaku.

Decades of work have gone into developing efficient proof calculi, data structures, algorithms, and heuristics for first-order automatic theorem proving. Higher-order provers lag behind in terms of efficiency. Instead of developing a new higher-order prover from the ground up, we propose to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features. We explain how to extend the prover’s data structures, algorithms, and heuristics to $$\lambda $$ λ -free higher-order logic, a formalism that supports partial application and applied variables. Our extension outperforms the traditional encoding and appears promising as a stepping stone toward full higher-order logic.

Superposition is among the most successful calculi for firstorder logic. Its extension to higher-order logic introduces new challenges such as infinitely branching inference rules, new possibilities such as reasoning about formulas, and the need to curb the explosion of specific higher-order rules. We describe techniques that address these issues and extensively evaluate their implementation in the Zipperposition theorem prover. Largely thanks to their use, Zipperposition won the higher-order division of the CASC-J10 competition.