Online Appendix to: Model Checking of Recursive Probabilistic Systems

PROOF. It suffices, by standard facts about probability measure, to prove the claim for cylinders C(w′) ⊆ ′, where w′ = w0, . . . wk. We use induction on k. The base case (k = 0) follows from Lemma 7. Namely, C( ) = ′, and ρ−1( ′) = \ ρ−1( ). Thus Pr (ρ−1( ′)) = 1 − Pr (ρ−1( )) = 1. For the induction step, suppose that the claim holds for the prefix w′ = w0w1 . . . wk. Let D[w′] = ρ−1(C(w′)). Define the event Ji,y ∈ F to be Ji,y = {t ∈ | ρ(t) = w0 . . . wi . . . ,and wi = y}. Note that by definition of conditional probability, Pr (D[w′wk+1]) = Pr (D[w′]) Pr (Jk+1,wk+1 | D[w′]). We want to show that Pr (D[w′wk+1]) = Pr ′(C(w′wk+1)). We distinguish three cases, based on what type of edge (wk, wk+1) is in HA , as in the proof of Lemma 7.