Domino Induction

Theorem 2.11 (Domino Theorem). Let $S_1, S_2, S_3, \dots$ be a linear
sequence of dominos such that:
- $S_1$ falls, and
- for every $k$, if $S_k$ falls then $S_{k+1}$ also falls.

Then every $S_n$ falls.

This is the practical meaning of the Peano induction axiom (P5). The dominos
are the natural numbers; each $S_k$ falling means "the statement $P(k)$ is true".
The first domino falling is the base case. The rule "if $S_k$ falls then
$S_{k+1}$ falls" is the inductive step.

To prove $\forall n \in \mathbb{N}. P(n)$, you show:
1. $P(0)$ (first domino falls)
2. $\forall n. P(n) \Rightarrow P(s(n))$ (if one falls, the next does)

Then by P5, every $P(n)$ holds. ∎