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

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