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

Lemma whole_cache_path_consistent lp : forall c1 c2, cache_path c1 c2 lp = true -> andl (map wf lp) -> exists C, cache_beq c2 (C ++ c1) = true /\ (forall t, assoc_cache t C <> None -> assoc_cache t c1 = None).