# 1.8 Induction

## Induction Rules

Consider the following three fundamental principles for reasoning about nonnegative integers:

• the Induction Principle,
• the Strong Induction Principle,
• the Well Ordering Principle.

Indicate the principle behind each of the inference rules over natural numbers below:

1. $$\dfrac{P(0), \forall m. (\forall k \leq m. P(k)) \text{ IMPLIES } P(m+1)}{\forall n. P(n)}$$

This is a formulation of:

Exercise 1
Notice "$$(\forall k \leq m. P(k))$$"
2. $$\dfrac{P(b), \forall k \geq b. P(k) \text{ IMPLIES } P(k+1)}{\forall m \geq b. P(m)}$$

This is a formulation of:

Exercise 2
This is just the induction rule with the base case starting at $$b$$ instead of 0.
3. $$\dfrac{\exists n. P(n)}{\exists m. (P(m) \text{ AND } (\forall k. P(k) \text{ IMPLIES } k \geq m))}$$

This is a formulation of:

Exercise 3
If we let $$S$$ be the set $$\{ k \; | \;P(k) \}$$, then "$$\exists n. P(n)$$" tells us that $$S$$ is nonempty, and "$$P(m) \text{ AND } \forall k. P(k) \text{ IMPLIES } k \ge m)$$" says that $$m$$ is the smallest number in $$S$$.
4. $$\dfrac{P(0), \forall k > 0. P(k) \text{ IMPLIES } P(k+1)}{\forall n. P(n)}$$

This is a formulation of:

Exercise 4
The rule looks like ordinary induction, but in the antecedent, $$k$$ is strictly greater than 0, which leaves the possibility that $$P(0)$$ does not imply $$P(1)$$. Thus, the induction hypothesis $$P(n)$$ may not propagate from 0 to all other nonnegative numbers.
5. $$\dfrac{\forall m. (\forall k < m. P(k)) \text{ IMPLIES } P(m)}{\forall n. P(n)}$$

This is a formulation of:

Exercise 5
This rule looks identical to the rule for strong induction, as in Part 1, except with $$P(0)$$ missing. It is still a strong induction because the base case is provided: when $$m$$ is 0, the assumption of the implication is "vacuously" true, and the conclusion in this case would be precisely that $$P(0)$$ is true.