System Description: LEO - A Higher-Order Theorem Prover

Many (mathematical) problems, such as Cantor’s theorem, can be expressed very elegantly in higher-order logic, but lead to an exhaustive and un-intuitive formulation when coded in first-order logic. Thus, despite the difficulty of higher-order automated theorem proving, which has to deal with problems like the undecidability of higher-order unification (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of first-order theorem provers, but instead can be solved easily by an higher-order theorem prover (HOATP) like Leo. This is due to the expressiveness of higher-order Logic and, in the special case of Leo, due to an appropriate handling of the extensionality principles (functional extensionality and extensionality on truth values). Leo uses a higher-order Logic based upon Church’s simply typed λ-calculus, so that the comprehension axioms are implicitly handled by αβη-equality. Leo employs a higher-order resolution calculus ERES (see [3] in this volume for details), where the search for empty clauses and higher-order pre-unification [6] are interleaved: the unifiability preconditions of the resolution and factoring rules are residuated as special negative equality literals that are treated by special unification rules. In contrast to other HOATP’s (such as Tps [1]) extensionality principles are build in into Leo’s unification, and hence do not have to be axiomatized in order to achieve Henkin completeness.