let parent_states = if parent == T.nil then auto.Ata.top_states else get cache tree parent in
let fc_states = if fc == T.nil then auto.Ata.bottom_states else get cache tree fc in
let ns_states = if ns == T.nil then auto.Ata.bottom_states else get cache tree ns in
let parent_states = if parent == T.nil then auto.Ata.top_states else get cache tree parent in
let fc_states = if fc == T.nil then auto.Ata.bottom_states else get cache tree fc in
let ns_states = if ns == T.nil then auto.Ata.bottom_states else get cache tree ns in