User:Prime.mover/Proof Structures/Proof by General Induction
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Proof by General Induction
The proof proceeds by general induction.
For all $x \in M$, let $\map P x$ be the proposition:
- $\text {proposition}_x$
Basis for the Induction
$\map P \O$ is the case:
- $\text {proposition}_\O$
Thus $\map P \O$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P x$ is true, where $x \in M$, then it logically follows that $\map P {\map g x}$ is true.
So this is the induction hypothesis:
- $\text {proposition}_x$
from which it is to be shown that:
- $\text {proposition}_{\map g x}$
Induction Step
This is the induction step:
\(\ds \) | \(=\) | \(\ds \) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(=\) | \(\ds \) |
So $\map P x \implies \map P {\map g x}$ and the result follows by the Principle of General Induction.
Therefore:
- $\forall x \in M: \text {proposition}_x$