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_map_inv : forall (b : B) (l : list A), In b (map f l) -> exists a : A, In a l /\ b = f a.