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

Lemma cache_path_proj_left_not_in_init t lp c1 c2 : cache_path c1 c2 lp = true -> andl (map wf lp) -> In t (map proj_left (revflatten (map activations_cache_order lp))) -> ~ In t (map fst c1).