Structural Induction - Well-ordering

Well-ordering

Just as standard mathematical induction is equivalent to the well-ordering principle, structural induction is also equivalent to a well-ordering principle. If the set of all structures of a certain kind admits a well-founded partial order, then every nonempty subset must have a minimal element. (This is the definition of "well-founded".) The significance of the lemma in this context is that it allows us to deduce that if there are any counterexamples to the theorem we want to prove, then there must be a minimal counterexample. If we can show the existence of the minimal counterexample implies an even smaller counterexample, we have a contradiction (since the minimal counterexample isn't minimal) and so the set of counterexamples must be empty.

As an example of this type of argument, consider the set of all binary trees. We will show that the number of leaves in a full binary tree is one more than the number of interior nodes. Suppose there is a counterexample; then there must exist one with the minimal possible number of interior nodes. This counterexample, C, has n interior nodes and l leaves, where n+1 ≠ l. Moreover, C must be nontrivial, because the trivial tree has n = 0 and l = 1 and is therefore not a counterexample. C therefore has at least one leaf whose parent node is an interior node. Delete this leaf and its parent from the tree, promoting the leaf's sibling node to the position formerly occupied by its parent. This reduces both n and l by 1, so the new tree also has n+1 ≠ l and is therefore a smaller counterexample. But by hypothesis, C was already the smallest counterexample; therefore, the supposition that there were any counterexamples to begin with must have been false. The partial ordering implied by 'smaller' here is the one that says that S < T whenever S has fewer nodes than T.