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

TPS is a theorem proving system for first- and higher-order logic which runs in Common Lisp and can operate in automatic, semi-automatic, and interactive modes. As its logical language TPS uses the typed A-calculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be used to search for an expansion proof [10, 11] of a theorem, which represents in a nonredtmdant way the basic combinatorial information required to construct a proof of

. 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...

Introduction This is a brief update on the Tps automated theorem proving system for classical type theory, which was described in [3]. Manuals and information about obtaining Tps can be found at http://gtps.math.cmu.edu/tps.html. In Section 2 we discuss some examples of theorems which Tps can now prove automatically, and in Section 3 we discuss an example which illustrates one of the many challenges of theorem proving in higher-order logic. We rst provide a brief summary of the key features of Tps . Tps uses Church's type theory [8] (typed -calculus) as its logical language. Ws are displayed on the screen and in printed proofs in the notation of this system of symbolic logic. One can use Tps in automatic, semi-automatic, or interactive mode to construct proofs in natural deduction style, and a mixture of these modes of operation is most useful fo

We present a procedure for proving the validity of first-order formulas in the presence of decision procedures for an interpreted subset of the language. The procedure is designed to be practical: formulas can have large complex boolean structure, and include structure sharing in the form of let-expressions. The decision procedures are only required to decide the unsatisfiability of sets of literals. However, T-refuting substitutions are used whenever they can be computed; we show how this can be done for a theory of partial orders and equality. The procedure has been implemented as part of STeP, a tool for the formal verification of reactive systems. Although the procedure is incomplete, it eliminates the need for user interaction in the proof of many verification conditions.

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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...