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

Lemma cache_lookup_term (p: cbv) (ext: cache): wf p -> (forall t: term, assoc_cache t ext <> None -> assoc_cache t (cache_right p) = None) -> cache_lookup (ext ++ cache_right p) (proj_left p) = @term_from_value _ _ _ (proj_right p).