There are many combinatorial properties that are shared by all constructions using the trees of strategies approach, some of which are relevant only to constructions of a fixed level. We gather some of these properties together in this chapter. We have attempted to isolate many combinatorial lemmas in this chapter, so that the proofs presented in other chapters depend only on the statements of these lemmas together with some special properties of the construction.

The lemmas of the current chapter are true of all assignments of requirements and derivatives to trees that satisfy the conditions of Chapter 2. These lemmas deal, among other things, with existence of initial and principal derivatives, an analysis of the ability to take both switching and nonswitching extensions of nodes, an analysis of the link creation process, and an analysis of the situations in which we can find free derivatives of a node. We begin, in Section 8.1, with some lemmas about the path generating and outcome functions λ and out. In Section 8.2, we demonstrate the ability to take both switching and nonswitching extensions. We prove some technical lemmas about links in Section 8.3, and apply these in Section 8.4 to conclude that certain nodes will be free at various stages of the construction. More general theorems are proved in Section 8.5. The remaining sections deal with lemmas that are level-specific, i.e., depend on the specific starting tree, or are more technical in nature.

Many of the lemmas deal with relationships between successive trees Tk and Tk+1. We will state these lemmas in full generality, but frequently present only the proof for k = 0, noting that the proofs relativize.