# Strong Induction

Posted on 2017-04-06 by nbloomf

This page is part of a series on Arithmetic Made Difficult.

In this post we’ll take a break from defining programs to establish two important equivalent versions of the induction principle: first a version of induction that allows us to make a much “stronger” induction hypothesis, and then a nonconstructive result about $$\nats$$ that will make some proofs easier.

(Strong Induction.) Suppose we have a map $$f : \nats \rightarrow A$$ and a subset $$B \subseteq A$$ such that the following hold.

1. $$f(\zero) \in B$$.
2. If $$f(k) \in B$$ whenever $$\nleq(k,n)$$, then $$f(\next(n)) \in B$$.

Then $$f(n) \in B$$ for all $$n \in \nats$$.

Let $$B$$ be such a subset.

Define $T = \{ n \in \nats \mid \mathrm{if}\ \nleq(k,n)\ \mathrm{then}\ f(k) \in B \}.$ We will show that $$T = \nats$$ by induction. For the base case $$n = \zero$$, note that if $$\nleq(k,\zero)$$, then $$k = \zero$$, and we have $$f(\zero) \in B$$ by condition (1).

Next suppose $$n \in T$$, and suppose further that $$\nleq(k,\next(n))$$. Now either $$\nleq(k,n)$$ or $$k = \next(n)$$. If $$\nleq(k,n)$$, then since $$n \in T$$ we have $$f(k) \in B$$. If $$k = \next(n)$$, we have $$f(\next(n)) \in B$$ by condition (2). Thus $$\next(n) \in T$$ as needed.

Now suppose $$n \in \nats$$. Since $$\nleq(n,n)$$ and $$n \in T$$, we have $$f(n) \in B$$ as claimed.

As with ordinary induction, we will typically apply this theorem with $$A = \nats$$ and $$f$$ the identity.

Note that condition (2) in the Strong Induction theorem appears to be much stronger than the corresponding condition in ordinary induction. Here we are allowed to assume that a given statement is true for all integers up to some bound $$n$$, while plain induction makes that assumption only for $$n$$. However it turns out that these two results are equivalent; in fact it’s not too difficult to see that strong induction implies ordinary induction. Despite this, there are times when the stronger induction hypothesis makes for a simpler proof.

(Strong Induction on $$f$$.) Suppose we have a map $$f : A \rightarrow \nats$$ and a subset $$B \subseteq A$$ satisfying the following.

1. If $$f(a) = \zero$$, then $$a \in B$$.
2. If we have $$n \in \nats$$ such that if $$f(a) = k$$ with $$\nleq(k,n)$$ then $$a \in B$$, then if $$f(a) = \next(n)$$ then $$a \in B$$.

Then we have $$B = A$$.

Let $$B$$ be such a subset. We define $$T \subseteq \nats$$ by $T = \{n \in \nats \mid \forall a \in A, \mathrm{if}\ f(a) = n\ \mathrm{then}\ a \in B\}.$ We will show that $$T = \nats$$ by strong induction.

For the base case, suppose $$f(a) = \zero$$. By condition (1) we have $$a \in B$$. Thus $$\zero \in T$$.

For the inductive step, suppose we have $$n \in \nats$$ such that $$k \in T$$ whenever $$\nleq(k,n)$$. Now suppose we have $$a \in A$$ such that $$f(a) = \next(n)$$. By condition (2) we have $$a \in B$$ as needed.

So $$T = \nats$$, and thus if $$a \in A$$, we have $$f(a) = n$$ for some $$n \in \nats$$, and thus $$a \in B$$.

Again, when using this principle we will say we’re using “strong induction on $$f(a)$$”.

(Well-Ordering Property.) Let $$A$$ be a nonempty set and $$f : A \rightarrow \nats$$. Then there exists $$a \in A$$ such that $$\nleq(f(a),f(b))$$ for all $$b \in A$$.

Suppose to the contrary that some set $$A$$ and map $$f : A \rightarrow \nats$$ exist which do not have this property; that is, for all $$a \in A$$, there exists $$b \in A$$ such that $$\nleq(f(a),f(b))$$ is false.

Define $T = \{ n \in \nats \mid \exists a \in A, n = f(a) \},$ and let $$K = \nats \setminus T$$. Note that $$T \neq \emptyset$$. We will show that $$K = \nats$$ by strong induction.

For the base case, note that if $$f(a) = \zero$$, then $$\nleq(f(a),f(b))$$ is true for all $$b \in A$$. So we have $$\zero \notin T$$, and thus $$\zero \in K$$.

For the inductive step, suppose we have $$n \in \nats$$ such that $$k \in K$$ whenever $$\nleq(k,n)$$. Now suppose that $$\next(n) \notin K$$, that is, $$\next(n) \in T$$; say $$\next(n) = f(a)$$. Now suppose we have $$b \in A$$. If $$\nleq(f(a),f(b))$$ is false, then $$\nleq(f(b),f(a))$$ is true and $$f(b) \neq \next(n)$$. But now $$\nleq(f(b),n)$$, so that $$f(b) \in K$$, a contradiction. Thus we have $$\nleq(f(a),f(b)) = \btrue$$ for all $$b$$, a contradiction. So in fact $$\next(n) \in K$$.

By induction, then, we have $$K = \nats$$, so that $$T = \emptyset$$, a contradiction. So in fact no such $$A$$ and $$f$$ can exist.

Again, in typical applications we will have $$A = \nats$$ and $$f$$ the identity.