Definition list_length_induction :
  forall P : list A -> Type,
  (forall l1 : list A,
   (forall l2 : list A, length l2 < length l1 -> P l2) -> P l1) ->
  forall l : list A, P l.

Theorem length_app : forall l1 l2 : list A, length (l1 ++ l2) = length l1 + length l2.