User:Lord Farin/Proof Structures

Proof by Mathematical Induction (type 1 - 'Normal' Induction)
The proof proceeds by induction on $n$, the describe the integer $n$.

Basis for the Induction
The case $n = n_0$ is verified as follows:


 * Reasoning for $n_0$

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$ with $n \ge n_0$.

Assume to-be-proved property holds for $n$.

This is our induction hypothesis.

Induction Step
This is our induction step:

''Insert reasoning. Whenever the induction hypothesis is used, write down explicitly that this is done.''

When desired, use the reference induction hypothesis.

We conclude to-be-proved property holds for $n+1$.

The result follows by the Principle of Mathematical Induction.

Proof by Mathematical Induction (type 2 - Complete Induction)
The proof proceeds by induction on $n$, the describe the integer $n$.

Basis for the Induction
The case $n = n_0$ is verified as follows:


 * Reasoning for $n_0$

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$ with $n \ge n_0$.

Assume to-be-proved property holds for all $k$ such that $n_0 \le k \le n$.

This is our induction hypothesis.

Induction Step
This is our induction step:

''Insert reasoning. Whenever the induction hypothesis is used, write down explicitly that this is done.''

When desired, use the reference induction hypothesis.

We conclude to-be-proved property holds for $n+1$.

The result follows by the Second Principle of Mathematical Induction.

Notes to (Induction) Proofs
The above are only guidelines. In most cases, a different formulation is natural. Just use this as a template, so as to structure all induction proofs uniformly. Some proofs require induction to multiple variables. I haven't encountered such proofs on PW; when I do, I will put up a proof template. For further proof templates, cf. User:Prime.mover/Proof Structures.