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(* 02110-1301 USA                                                     *)
(***********************************************************************)
(*    Proof of Huffman algorithm: ISort.v                              *)
(*                                                                     *)
(*    Definition of sorting by insertion and its proof of correctness  *)
(*                                                                     *)
(*    Definitions: isort, insert                                       *)
(*                                                                     *)
(*                                    Laurent.Thery@inria.fr (2003)    *)
(***********************************************************************)
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).