Abstract
Well-founded induction is very general. For example the so-called “principle of strong induction” is well-founded induction on the set of natural numbers with their usual d -ordering, while so-called “ transfinite induction” is well-founded induction when the ordering d is total. Proofs using well-founded induction may not proceed as smoothly as one would like when the theorem to be proved is not in the neat form (Vx: x E C: Q.x). We address this inconvenience in what follows by formulating well-founded induction for formulae of the shape (Vx: f.x E C: Q.x) where C is a well-founded set and f is any function (whose range may or may not be a subset of C). We will use the result to make a short formula-driven proof of the fundamental invariance theorem for loops, and indeed this is the immediate motivation of the work. Let (C, <) abbreviate “set C partially ordered by Q “9 and define x<y to be xgy~x#y for all x, y and any partial ordering d . “(C, G ) is well-founded” means that for each subset S of C the following holds:
Published Version
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