Definition bobby_eight_poly x := length (functions_of_prog prog) * Nat.pow x max_arity.

Theorem A_T_bound i s p c1 f lt c2 v: let proof_tree := cbv_update i s p c1 (fapply f lt) c2 v in PPO_prog -> wf proof_tree -> A_T proof_tree <= bobby_eight_poly (max_active_size proof_tree).