(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU Lesser General Public License for more details.                *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: Ordered.v  Definition of ordered list and some properties  Definition: ordered Laurent.Thery@inria.fr (2003) **********************************************************************)
Require Export List.
From Huffman Require Export Permutation.
From Huffman Require Export Aux.
Section ordered.
Variable A : Type.
Variable order : A -> A -> Prop.
Variable order_trans : forall a b c : A, order a b -> order b c -> order a c.
(* Ordered list with respect to an order *)
Inductive ordered : list A -> Prop :=
| ordered_nil : ordered nil
| ordered_one : forall a : A, ordered (a :: nil)
| ordered_cons :
forall (a b : A) (l : list A),
order a b -> ordered (b :: l) -> ordered (a :: b :: l).
Hint Constructors ordered : core.

(* Specifications: *)
(* Inversion theorem *)
Theorem ordered_inv_order :
forall (a b : A) (l : list A), ordered (a :: b :: l) -> order a b.