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 in_flat_map_ex : forall (l : list B) (f : B -> list C) a, In a (flat_map f l) -> exists b : _, In b l /\ In a (f b).