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

Lemma nb_judgements_bound i s p' c1 t c2 v : let p := cbv_update i s p' c1 t c2 v in nb_judgements p <= suml (map nb_judgements_sub (activations p)) + length (activations p) + nb_read p.