3. Notes

A. Techniques that were applied to make the proof of the theorem easier to understand:

(a) Use a minimal counterexample in place of an induction, especially if more than one inductive parameter is involved.

(b) Use of ``without loss of generality'' (wlog) arguments when possible. A single letter can thereby be used in place of two (such as $D$ and $P$). This approach is an excellent way to analyze a problem in the first place. (The ``re-choosing'' of ${\bf s}$ in the second paragraph is another wlog statement in disguise.)

(c) Split a problem into simpler subproblems, for example by expressing $(P ^ *)_{\not\geq {\bf s}}$ as the union of two wpo sets.

(d) Take technical lemmas useful for the proof and if possible fashion them into more general principles (here, the Observations) that are interesting enough to stand on their own.

(e) Use a suggestive notation, such as $(P ^ *)_{\geq q}$, etc., that is almost self-explanatory, and choose letters to suggest kinds of objects (elements of $P$ versus sequences).

(f) Use an abbreviation, e.g., ``wpo'', for a long phrase if the phrase occurs many times. (It is better not to abbreviate a phrase that occurs only several times. In general, do not use abbreviations in statements of theorems if the abbreviations are special to your writeup.)

B. Generalization: The theory is often studied in the more general form of well quasi-ordered sets; the results are similar. (A quasi order $\leq$ is a relation that is reflexive and transitive but not necessarily antisymmetric, so that $a \leq b$ and $b \leq a$ is allowed for $a \neq b$.)