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

Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higher-order logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and those resulting from instantiations. An expansion tree is called an expansion tree proof (ET-proof) if it encodes a tautology, and, in the form not using skolem functions, an “imbedding ” relation among the critical variables be acyclic. The relative completeness result for expansion tree proofs not using skolem functions, i.e. if A is provable in higher-order logic then A has such an expansion tree proof, is based on Andrews ’ formulation of Takahashi’s proof of the cut-elimination theorem for higher-order logic. If the occurrences of skolem functions in instantiation terms are restricted appropriately, the use of skolem functions in place of critical variables is equivalent to the requirement that the imbedding relation is acyclic. This fact not only resolves the open question of what

An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause.

. This is description of TPS, a theorem-proving system for classical type theory (Church's typed #-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 that TPS can prove completely automatically are given to illustrate certain aspects of TPS's behavior and problems of theorem proving in higher-order logic. AMS Subject Classification: 03-04, 68T15, 03B35, 03B15, 03B10. Key words: higher-order logic, type theory, mating, connection, expansion proof, natural deduction. 1. Introduction TPS is a theorem-proving system for classical type theory ## (Church's typed #-calculus [20]) which has been under development at Carnegie Mellon University for a number years. This paper gives a general...

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This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed l-calculus). TPS can be used 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. CATEGORY: Demonstration 1. Introduction This presentation is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed 3 l-calculus [14]) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the same as the connection method developed independently by Bibel [13]. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural deduction proofs and expansion proofs. Natural deduction proofs are human-readable formal proofs. An example of such a proof which was produced aut...