### EXAMPLE-INDUCTION-SCHEME-UPWARDS

induction upwards
```Major Section:  INTRODUCTION-TO-THE-THEOREM-PROVER
```

See logic-knowledge-taken-for-granted-inductive-proof for an explanation of what we mean by the induction suggested by a recursive function or a term.

Induction Upwards: To `(p i max)` for all `i` and all `max`, prove each of the following:

```Base Case:
(implies (not (and (natp i)
(natp max)
(< i max)))
(p i max))
```

```Induction Step:
(implies (and  (natp i)
(natp max)
(< i max)
(p (+ i 1) max))
(p i max))
```
Note that the induction hypothesis is about an `i` that is bigger than the `i` in in the conclusion. In induction, as in recursion, the sense of one thing being ``smaller'' than another is determined by an arbitrary measure of all the variables, not the magnitude or extent of some particular variable.

A function that suggests this induction is shown below. ACL2 has to be told the measure, namely the difference between `max` and `i` (coerced to a natural number to insure that the measure is an ordinal).

```(defun count-up (i max)
(declare (xargs :measure (nfix (- max i))))
(if (and (natp i)
(natp max)
(< i max))
(cons i (count-up (+ 1 i) max))
nil)).
```