TPS: A theorem-proving system for classical type theory

Abstract

This is a description of TPS, a theorem proving system for classical type theory (Church's typed λ-calculus). TPS has been designed to be a general research tool for manipulating wffs of first-and higher-order logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems which TPS can prove completely automatically are given to illustrate certain aspects of TPS's behavior and problems of theorem proving in higher-order logic. 7

Automation of Higher-Order Logic

“…He also showed how to build a resolution procedure for ETT on pre-unification instead of unification by making flex-flex equations into "constraints" on resolution (Huet, 1972(Huet, , 1973b. The earliest theorem provers for various supersets of ETT-TPS (Andrews et al, 1996), Isabelle (Paulson, 1989), and λProlog (Nadathur and Miller, 1988;Miller and Nadathur, 2012)-all implemented rather directly Huet's search procedure for pre-unifiers.…”

Higher-order aspects and context in SUMO

“…More recently extensionality and equality reasoning in HOL has been studied [43,44,45,46]. The TPS system [47,48], which is based on a higher-order mating calculus, is a pioneering ATP system for HOL.…”

“…TPS is a pioneering higher-order theorem proving system [47,48]. It can be used to prove theorems of HOL automatically, interactively, or semi-automatically.…”

Nitpick: A Counterexample Generator for Isabelle/HOL Based on the Relational Model Finder Kodkod

“…In a preliminary run, it disproved 293 out of 2729 formulas (mostly theorems), compared with 214 for Refute. Much to our surprise, Nitpick exhibited counterexamples for five formulas that had been proved by TPS [1] or LEO-II [2], revealing two bugs in the former and one bug in the latter. In exchange, LEO-II exposed one bug in Nitpick.…”

“…Nitpick is integrated with the TPTP benchmark suite [13] and exposed three bugs in the higher-order provers TPS [1] and LEO-II [2].…”