(** This is exactly [List.fold_left] from Coq's standard library,
  with [f] taking arguments in a different order. *)

Fixpoint list_fold_left (accu: B) (l: list A) : B :=
  match l with nil => accu | x :: l' => list_fold_left (f x accu) l' end.(** This is exactly [List.fold_right] from Coq's standard library,
  except that it runs in constant stack space. *)

Definition list_fold_right (l: list A) (base: B) : B :=
  list_fold_left base (List.rev' l).Remark list_fold_left_app:
  forall l1 l2 accu,
  list_fold_left accu (l1 ++ l2) = list_fold_left (list_fold_left accu l1) l2.

Lemma list_in_insert: forall (A: Type) (x: A) (l1 l2: list A) (y: A), In x (l1 ++ l2) -> In x (l1 ++ y :: l2).