(* Nats smaller than n *)
Fixpoint natsLess n : list nat :=
match n with
0 => []
| S n => n :: natsLess n
end.
(* Sum *)
Fixpoint sumn (A:list nat) :=
match A with
[] => 0
| a::A => a + sumn A
end.
Hint Rewrite sumn_app : list.

Lemma length_concat X (A : list (list X)) :
length (concat A) = sumn (map (@length _) A).