Reachability and Büchi Games - Rich Model Toolkit

Safety Games
Given a safety game (G,F) with G = (S,S 0 ,E), i.e.,
φ S = {ρ ∈ S ω | Occ(ρ) ⊆ F},
consider the reachability game (G,S \F), i.e.,
φ R = {ρ ∈ S ω | Occ(ρ)∩(S \F) ≠ ∅}.
Then, S ω \φ R = {ρ ∈ S ω | Occ(ρ)∩(S \F) = ∅}
= {ρ ∈ S ω | Occ(ρ) ⊆ F}.
Player 0 has a safety objective in (G,F).
Player 1 has a reachability objective in (G,F).
So, W 0 in the safety game (G,F) corresponds to W 1 in the
reachability game (G,S \F).