  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.
