Section ISortExample.
Variable A : Type.
Variable order : A -> A -> Prop.
Variable order_fun : A -> A -> bool.
Hypothesis order_fun_true : forall a b : A, order_fun a b = true -> order a b.
Hypothesis order_fun_false : forall a b : A, order_fun a b = false -> order b a.
(* Insert an element *)
Fixpoint insert (a : A) (l : list A) {struct l} : list A :=
match l with
| nil => a :: nil
| b :: l1 =>
match order_fun a b with
| true => a :: l
| false => b :: insert a l1
end
end.

(* Inserting preserves ordering *)
Theorem insert_ordered :
forall l : list A,
ordered order l -> forall a : A, ordered order (insert a l).