Definition add := fold_right plus (0%nat).

(* Specifications: *)
Lemma add_same c l: (forall x, In x l -> x = c) -> add l = length l * c.
Lemma length_concat T (l: list (list T)): length (concat l) = add (map (@length _) l).
Lemma concat_map_nil T U (l: list T): concat (map (fun _ => nil) l) = @nil U.