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

Lemma term_size_proj_left_bound : forall i s p' c1 t c2 v,
let p := cbv_update i s p' c1 t c2 v in
let S := max_active_size p in
wf p ->
forall p',
InCBV p' p -> term_size (proj_left p') <= activation_bound prog S + S + cache_size c1.