Kiddie Talk - The Diamond Lemma and its applications

Proof. The proof is by induction on N , and is obviously true for N = 0 (won position) or N = 1 (only one vertex has negative evaluation, so there’s only one legal move which is known to be winning). Suppose then that for every k < N , if a pair (G, ē) admits a win in k moves then every legal sequence of moves from (G, ē) leads to a win in k moves. Assume that (G, e) admits a win in N moves starting with the move at the vertex vi, executed by the player A. Take any other legal move at vj executed by the player D, we need to show that after this move, (G, ej) admits a win in N − 1 moves. Note that what we can describe our claim in a very figurative way: let Γ be the directed graph whose vertices are evaluations on G (i.e. ordered real n-uples (e1, . . . , en)) and where e and e ′ are linked by an edge going from e to e′ if there is a move at some vertex vi that brings e to e ′. Then we’re saying that Γ is a poset with the ’diamond condition’: