Abstract We present a sound and complete unification procedure for deterministic higher-order patterns, a class of simply-typed lambda terms introduced by Yokoyama et al. which comes with a deterministic matching problem. Our unification procedure can be seen as a special case of full higher-order unification where flex-flex pairs can be solved in a most general way. Moreover, our method generalizes Libal and Miller’s recent functions-as-constructors higher-order unification (FCU) by dropping their global restriction on variable arguments, thereby losing the property that every solvable problem has a most general unifier. In fact, minimal complete sets of unifiers of deterministic higher-order patterns may be infinite, so decidability of the unification problem remains an open question.

Abstract We lift the computability path order and its extensions from plain higher-order rewriting to higher-order rewriting on $$\beta \eta $$ β η -normal forms where matching modulo $$\beta \eta $$ β η is employed. The resulting order NCPO is shown to be useful on practical examples. In particular, it can handle systems where its cousin NHORPO fails even when it is used together with the powerful transformation technique of neutralization. We also argue that automating NCPO efficiently is straightforward using SAT/SMT solvers whereas this cannot be said about the transformation technique of neutralization. Our prototype implementation supports automatic termination proof search for NCPO and is also the first one to automate NHORPO with neutralization.

We revisit completion modulo equational theories for left-linear term rewrite systems where unification modulo the theory is avoided and the normal rewrite relation can be used in order to decide validity questions. To that end, we give a new correctness proof for finite runs and establish a simulation result between the two inference systems known from the literature. Given a concrete reduction order, novel canonicity results show that the resulting complete systems are unique up to the representation of their rules' right-hand sides. Furthermore, we show how left-linear AC completion can be simulated by general AC completion. In particular, this result allows us to switch from the former to the latter at any point during a completion process.

Abstract Dependent type theory gives an expressive type system facilitating succinct formalizations of mathematical concepts. In practice, it is mainly used for interactive theorem proving with intensional type theories, with PVS being a notable exception. In this paper, we present native rules for automated reasoning in a dependently-typed version (DHOL) of classical higher-order logic (HOL). DHOL has an extensional type theory with an undecidable type checking problem which contains theorem proving. We implemented the inference rules as well as an automatic type checking mode in Lash, a fork of Satallax, the leading tableaux-based prover for HOL. Our method is sound and complete with respect to provability in DHOL. Completeness is guaranteed by the incorporation of a sound and complete translation from DHOL to HOL recently proposed by Rothgang et al. While this translation can already be used as a preprocessing step to any HOL prover, to achieve better performance, our system directly works in DHOL. Moreover, experimental results show that the DHOL version of Lash can outperform all major HOL provers executed on the translation.

Abstract We revisit AC completion for left-linear term rewrite systems where AC unification is avoided and the normal rewrite relation can be used in order to decide validity questions. To that end, we give a new correctness proof for finite runs and establish a simulation result between the two inference systems known from the literature. Furthermore, we show how left-linear AC completion can be simulated by general AC completion. In particular, this result allows us to switch from the former to the latter at any point during a completion process. Finally, we present experimental results for our implementation of left-linear AC completion in the tool .