Section LIST_FOLD.
Variables A B: Type.
Variable f: A -> B -> B.
(** 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.
End LIST_FOLD.
(** Properties of list membership. *)
(** [list_disjoint l1 l2] holds iff [l1] and [l2] have no elements in common. *)
Definition list_disjoint (A: Type) (l1 l2: list A) : Prop :=
forall (x y: A), In x l1 -> In y l2 -> x <> y.

Lemma list_disjoint_notin:
forall (A: Type) (l1 l2: list A) (a: A),
list_disjoint l1 l2 -> In a l1 -> ~(In a l2).