Definition judgsize (p:cbv) := term_size (proj_left p) + value_size (proj_right p).

Lemma active_size_bound qic qif cs: forall i sub p c f lv d v, forall pi, valid_QI qic qif cs -> let t := (fapply f lv) in let proof_tree := cbv_update i sub p c t d v in let l:= (map (fun x=> mcs * (term_size t)) lv) in wf proof_tree -> In pi (activations proof_tree) -> cache_bounded qic qif c -> judgsize pi <= (max_arity + 1) * (qif f l) + 1.