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:

\(\ds \)

\(=\)

\(\ds \)

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \)

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

Therefore:

$proposition_n$