By Robert S. Boyer

Not like such a lot 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: - whilst should still induction be used? - How does one invent a suitable induction argument? - whilst may still a definition be improved?

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Because we will have many kinds of shells, we will need a function, called the "recognizer" of the class that returns T on objects of the class and F otherwise. , the predecessor function is the "ac cessor" corresponding to the shell for numbers). Finally, we posit that any object in the class can be generated with a finite number of "con7 One way to make sure that T is not a n u m b e r or to escape from asking what is the successor of T is to employ a typed syntax. Indeed, Aubin [2] and Cartwright [14] have implemented theorem-provers for recursive functions that use typed syntax.

Our principle of definition is: To define f of xif . . , xnto be body (usually written Definition ( f xt . . x n ) = b o d y " ) , where: (a) f is a new function symbol of n arguments; (b) Xi, . . , x n are distinct variables; (c) body is a term and mentions no symbol as a variable other than xlf . . , x n ; and (d) there is a well-founded relation denoted by a function sym bol r and a function symbol m of n arguments, such that for each occurrence of a subterm of the form ( f y ^ . y n ) in body and the f-free terms t i , .

T h e base case and the induction steps produced by this application of the induction principle are those exhibited in Chapter II. 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 , .