On Induction

Tags: logic

Published on Wednesday, February 1st, 2017

In this post, I will write about three kinds of inductions: weak induction, strong induction, and structural induction. I in fact dislike these terms because, as I will show below, they are equivalent. Personally, when I think of induction, I will default to the structural induction. Last semester I took a logic class, and they introduce these terms, and also “formula induction”, “proof induction”, “term induction”, “sequence induction”, etc. I was like: *sigh* THEY ARE THE SAME WHY DO YOU NEED DIFFERENT NAMES.

Anyway, for the sake of this article, we will do use these terms for clarity and show the following.

Strong Induction Doesn’t Need A Base Case

Weak Induction states: for any proposition $P$, $(P(0) \land \forall x. (P(x) \to P(x + 1))) \to \forall x. P(x)$. One must prove both $P(0)$ which is known as base case and $\forall x. (P(x) \to P(x + 1))$ which is known as inductive case to show that $P$ holds for every natural number. One common mistake is, sometimes we forget to prove the base case, so we will be able to (incorrectly) prove something clearly wrong like $\forall x. x > x + 1$. Let’s actually try this:

Proof. Let $x$ be a natural number. The induction hypothesis states $x > x + 1$. Our goal is to show that $x + 1 > x + 2$, but this is just adding 1 to both sides in the inequality from the induction hypothesis. Therefore, $\forall x. x > x + 1$.

This is incorrect because we have not proved the base case that $0 > 1$ which, indeed, is unprovable.

However, in strong induction, we can omit the base case! The traditional strong induction is: for any proposition $P$, $(P(0) \land \forall x. ((\forall y < x. P(y)) \to P(x))) \to \forall x. P(x)$. But as stated above, in fact just only $\forall x. ((\forall y < x. P(y)) \to P(x)) \to \forall x. P(x)$ is enough.

As an example, if you try to use strong induction to prove that $\forall x. x > x + 1$, we would have:

Let $x$ be a natural number. The induction hypothesis states: $\forall y < x. y > y + 1$ and we want to prove $\forall x. x > x + 1$. By letting $y = x - 1$, we obtain that $x - 1 > x - 1 + 1 = x$. By adding 1 both sides, we obtain $x > x + 1$. So we conclude that $\forall x. x > x + 1$...

Wait what!?! We are not supposed to be able to prove this!

Well, there is a subtle error in the above proof. Specifically, when we let $y = x - 1$. How can we be sure that $y$ is a natural number? The answer is we can’t, as when $x = 0$, $y = x - 1$ is not a natural number! The right way is to do a case analysis here: case $x = 0$ and case $x > 0$. When $x = 0$, the induction hypothesis states $\forall y < 0. y > y + 1$ which is a blank statement, and because we can’t prove that $0 > 1$, we can’t prove that $x > x + 1$ in case $x = 0$, so the proof fails!

To be formal, we will derive $(P(0) \land \forall x. (\forall y < x. P(y) \to P(x))) \to \forall x. P(x)$ from $\forall x. ((\forall y < x. P(y)) \to P(x)) \to \forall x. P(x)$ and vice versa. This shows that both variants of strong induction are equivalent.

Proof. $(\to)$ Suppose $(P(0) \land \forall x. ((\forall y < x. P(y)) \to P(x))) \to \forall x. P(x)$. Our goal is to show that $\forall x. ((\forall y < x. P(y)) \to P(x)) \to \forall x. P(x)$. By the deduction theorem, assume $\forall x. ((\forall y < x. P(y)) \to P(x))$, we then only need to prove $\forall x. P(x)$. By applying the hypothesis when $x = 0$, we obtain $(\forall y < 0. P(y)) \to P(0)$. However, $\forall y < 0. P(y)$ is vacuously true, so we obtain $P(0)$. With both $P(0)$ and $\forall x. ((\forall y < x. P(y)) \to P(x))$, we can use the first hypothesis to obtain $\forall x. P(x)$, finishing the proof.

$(\leftarrow)$ Suppose $(\forall x. ((\forall y < x. P(y)) \to P(x))) \to \forall x. P(x)$. Our goal is to show that $(P(0) \land \forall x. ((\forall y < x. P(y)) \to P(x))) \to \forall x. P(x)$. By the deduction theorem, assume $P(0)$ and $\forall x. ((\forall y < x. P(y)) \to P(x))$, we then only need to prove $\forall x. P(x)$. But with $\forall x. ((\forall y < x. P(y)) \to P(x))$, we can use the first hypothesis to obtain $\forall x. P(x)$, finishing the proof.

Equivalence of Weak Induction and Strong Induction

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Require Import Omega.

Definition NProp := nat -> Prop.
Theorem weak_eq_to_strong :
  (forall P : NProp, P 0 -> (forall n, P n -> P (S n)) -> forall n, P n) <->
  (forall P : NProp, (forall n, (forall m, (m < n) -> P m) -> P n) -> forall n, P n).
Proof.
  split.
  - intros ind_w P ind_H_s n.
    apply (ind_w (fun k => ((forall x, x <= k -> P x)))) with (n := n); intuition.
    + inversion H; intuition.
      apply ind_H_s; intuition.
      inversion H1.
  - intros; intuition.
    apply H; intuition.
    destruct n0; intuition.
Qed.

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