Consider the “letter game” patterned after an article by Don Gernes from Ponderosa High School. We have three undefined terms, the letters M, I, and U, and one definition, “x means any string of I’s and U’s.” Finally, there are four axioms for this letter game.

Axioms 1. If a string of letters ends in I, you may add a U at the end. This is the addition rule.
2. If you have Mx, then you may add x to get Mxx, by what we call the doubling rule.
3. If three I’s occur—that is, III—then you may replace that string by U. We call this the substitution rule.
4. If UU occurs, you drop it. This is the deletion rule.
Given: MI

Prove: MUIU; this is a theorem.