By Robert S. Boyer

In contrast to so much texts on common sense and arithmetic, this booklet is set the way to end up theorems instead of facts of particular effects. We supply our solutions to such questions as: - while may still induction be used? - How does one invent a suitable induction argument? - while should still a definition be elevated?

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We now prove that our induction principle is sound. Suppose we have in mind particular p , r , m, Xi, q t , hi, and s ifj satisfying condi tions (a) through (g) above, and suppose the base case and induction steps are theorems. Below is a set theoretic proof of p . PROOF. Without loss of generality we assume that the Xi are X I , Χ2, . . , Xn ; that r is R ; that m is M ; that Xn + 1 , X n + 2 , . . , Xz are all of the variables other than X I , X2, . . , Xn in p , the qi, and either component of any pair in any su; that p is ( P XI .

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.

A c n are new function symbols of one argu ment, wfn is a new function symbol of two arguments, and all the above function symbols are distinct; (b) each t r j is a term that mentions no symbol as a variable be- 38 / III. A PRECISE DEFINITION OF THE THEORY sides Xi and mentions no symbol as a function symbol besides I F , TRUE, FALSE, previously introduced shell recognizers, and r ; and (c) if no bottom object is supplied, the dVi are bottom objects of previously introduced shells, and for each i , (IMPLIES (EQUAL Xi dVi) t Γΐ) is a theorem; if a bottom object is supplied, each dVi is either ( b t m ) or a bottom object of some previously introduced shell, and for each i , (IMPLIES (AND (EQUAL Xi dVi) (r (btm))) is a theorem, means to extend the theory by doing the following (using T for ( r ( btm ) ) and F for all terms of the form ( EQUAL x ( btm ) ) if no bot tom object is supplied): (1) assume the following axioms: (OR (EQUAL (r X) T) (EQUAL (r X) F)), (r (const XI ...