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_fst_inv : forall a (l : list (B * C)), In a (map (fst (B:=_)) l) -> exists c : _, In (a, c) l.