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

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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

In higher-order logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. Using the Knaster-Tarski Fixed Point Theorem [ 15 ] , constraints whose solutions require recursive de nitions can be solved as xed points of monotone set functions. In this paper, we consider an approach to higher-order theorem proving which intertwines conventional theorem proving in the form of mating search with generating and solving set constraints.

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...

tax of a higher-order logic is to introduce some kind of typing scheme. One approach types first-order individuals with #, sets of individuals with ###, sets of pairs of individuals with ####, sets of sets of individuals with #####, etc. Such a typing scheme does not provide types for function symbols. Since in some treatments of higher-order logic, functions can be represented by their graphs, i.e. certain kinds of sets of ordered pairs, this lack is not a serious restriction. Identifying functions up to their graphs does, of course, treat functions extensionally, something that might be 1 too strong in some applications. (A logic is extensional if whenever two predicates or two functions are equal on all their arguments, they themselves are equal.) A more general approach to typing is that used in the Simple Theory of Types (Church, 1940). Here again, the type # is used to denote the set of first-order individuals, and the type o is used to denote the sort of booleans, false

The sequent calculus is often criticized for requiring proofs to be laden with large volumes of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cut-free sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers revolt against the sequent calculus and replace it with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. In this paper, we propose taking, instead, an evolutionary approach to recover canonicity within the sequent calculus, an approach we illustrate for classical first-order logic. We use a multi-focused sequent system as our means of abstracting away the details from classical sequent proofs. We then show that, among the focused sequent proofs, the maximally multi-focused proofs, which make the foci as parallel as possible, are canonical. Moreover, such proofs are isomorphic to expansion tree proofs—a well known, simple, and parallel generalization of Herbrand disjunctions—for classical first-order logic. We thus provide a systematic method of recovering the essence of any sequent proof without abandoning the sequent calculus. 1