User:Prime.mover/Proof Structures/Proof by Complete Induction

Proof by Complete Induction
The proof proceeds by strong induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $proposition_n$

$\map P 0$ is the case:
 * $proposition_0$

Thus $\map P 0$ is seen to hold.

Basis for the Induction
$\map P 1$ is the case:
 * $proposition_1$

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that if $\map P j$ is true, for all $j$ such that $0 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.

This is the induction hypothesis:
 * $proposition_k$

from which it is to be shown that:
 * $proposition_{k + 1}$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.

Therefore:
 * $proposition_n$