By Robert S. Boyer

Not like so much texts on common sense and arithmetic, this ebook is set how you can end up theorems instead of facts of particular effects. We supply our solutions to such questions as: - whilst may still induction be used? - How does one invent a suitable induction argument? - whilst may still a definition be accelerated?

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The first term is just a variable, the second is the application of the 0-ary function symbol TRUE to no arguments (and hence denotes a constant), and the third is the application of the dyadic function symbol P to the term (ADDI X) and the variable Y. To talk about terms, it is convenient to use so-called "metavari ables" that are understood by the reader to stand for certain vari ables, function symbols, or terms. We will use only lowercase type written words as metavariables, and we will make clear what type of syntactic object the symbol is to denote.

I. DEFINITIONS / 47 precise, we assume that: There exists a set D such that each function symbol f mentioned as a function symbol in any axiom denotes a function whose domain is Dn and whose range is a subset of D, where n is the number of arguments of f. If G is a function whose domain is a subset of D n , for some n , and whose range is a subset of D, then the extension of G is the function on Dn to D that is defined to be ( G XI . . Xn ) if

D. E. SUMMARY The purpose of this chapter was to provide an introduction to our function-based theory and to indicate how we prove theorems in the theory. As noted, all our proof techniques have b e e n implemented in an automatic theorem-proving program. In fact, the last section was written, in its entirety, by our automatic theorem-prover in response to three user commands supplying the definitions of FLATTEN and MC. FLATTEN and the statement of the theorem to be proved. This book is about such questions as how function definitions are analyzed by our theorem-proving system to establish their admissibility, how the system discovers that (LISTP (FLATTEN X) ) is a theorem w h e n presented with the definition of FLATTEN, why the system chooses the inductions it does, and why some functions are expanded and others are not.