Definition cbv_cache_entry (proof_tree : cbv) := (proj_left proof_tree, proj_right proof_tree).

Lemma size_judgement : forall i s p' c1 t c2 v, let p := cbv_update i s p' c1 t c2 v in let A := length (activations p) in let S := max_active_size p in let C := maxl (map (fun tv : term * value => value_size (snd tv)) c1) in wf p -> max_judgement_size p <= activation_bound prog S + S + cache_size c1 + (1 + max S C) * (1 + (A * activation_bound prog S + A) + max_arity * (A * activation_bound prog S + A)).