Section Sublist.
Inductive sublist (A : Type) : (list A) -> (list A) -> Prop :=
| sublist_refl : forall l, sublist l l
| sublist_skip : forall l1 h t, sublist l1 t -> sublist l1 (h :: t)
| sublist_cons : forall h t1 t2, sublist t1 t2 -> sublist (h :: t1) (h :: t2).
Hint Constructors sublist.

(* Specifications: *)
Lemma sublist_nil A (l : list A) : sublist [] l.
Lemma sublist_incl A (l1 l2 : list A) : sublist l1 l2 -> incl l1 l2.
Lemma sublist_app_skip A (l1 l2 l3 : list A) : sublist l1 l3 -> sublist l1 (l2 ++ l3).
Lemma sublist_app_left A (l1 l2 l3 : list A) : sublist l1 l3 -> sublist (l2 ++ l1) (l2 ++ l3).
Lemma sublist_app_compat A (l1 l2 l3 l4 : list A) :
sublist l1 l3 ->
sublist l2 l4 ->
sublist (l1 ++ l2) (l3 ++ l4).
Lemma sublist_flatmap_in_ext A B (f : A -> list B) g l:
(forall x : A , In x l -> sublist (f x) (g x)) ->
sublist (flat_map f l) (flat_map g l).
Lemma NoDup_sublist A (l1 l2 : list A) : sublist l1 l2 -> NoDup l2 -> NoDup l1.
End Sublist.