We introduce refutationally complete superposition calculi for intentional and extensional clausal $\lambda$-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 $\lambda$-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 higher-order logic.Comment: arXiv admin note: text overlap with arXiv:2102.00453

@article{paperbot1327,
  title = {Superposition for Lambda-Free Higher-Order Logic},
  author = {Alexander Bentkamp and Jasmin Blanchette and Simon Cruanes and Uwe Waldmann},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 17, Issue 2},
  year = {2021},
  doi = {10.23638/lmcs-17(2:1)2021}
}