(* Skip the first elements of a list *)
Fixpoint skip_n (l : list A) (n : nat) {struct n} : 
 list A :=
  match n with
  | O => l
  | S n1 => match l with
            | nil => nil
            | a :: l1 => skip_n l1 n1
            end
  end.

Theorem exist_first_max : forall l : list nat, l <> nil -> exists a : nat, (exists l1 : list nat, (exists l2 : list nat, l = l1 ++ a :: l2 /\ (forall n1, In n1 l1 -> n1 < a) /\ (forall n1, In n1 l2 -> n1 <= a))).