By Robert S. Boyer

In contrast to so much texts on common sense and arithmetic, this e-book is ready find out how to end up theorems instead of facts of particular effects. We supply our solutions to such questions as: - while should still induction be used? - How does one invent a suitable induction argument? - whilst may still a definition be multiplied?

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**Example text**

P/Si,hl) Note in particular that we have to prove k + 1 things (the "base case" and k "induction steps"). Each induction step distinguishes a given case with one of the q/s and provides hi inductive instances of the conjecture being proved. We now illustrate an application of the induction principle. FLATTEN have b e e n introduced as in Chapter II. T h e first induction performed in the proof of the FLATTEN. MC . FLATTEN theorem of Chapter II is obtained by the following instance of our induction principle, p is the term ( EQUAL ( MC .

We introduced a principle of definition which allows the introduc tion of recursive functions. We introduced the concept of a lexicographic relation. We used the principle of definition to introduce the usual "less than" function, assumed it was well-founded, and defined the mea sure function COUNT that computes the size of an object. IV The Correctness of a Tautology-Checker Before we describe how we prove theorems in the theory just pre sented, it is important that the reader be familiar with the theory itself.

X n ) , Z l ) a n d « X I , . . , X n ) , Z 2 ) t o F 0 . L e t F l ' and Gl ■ be the ex tensions of F l and G l . Both F l and Gl are defined upon all members of Dn RM-smaller than (XI, . . , Xn) because both are partially correct. F l and Gl have the same values on all members of Dn RM-smaller than (XI, . . , Xn) because (XI, . . , Xn) is RM-minimal. Therefore, by the lemma, b o d y [ F l ' ] = body[Gl f ] . But Z l = ( F l XI . . Xn) = b o d y [ F l ' ] = body[Gl f ] = ( Gl XI . .