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)).