(* 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: SubstPred.v  Definition of the substitution in a tree with respect to a cover  Definition: subst_pred  Laurent.Thery@inria.fr (2003) **********************************************************************)
Section SubstPred.
Variable A : Type.
(* Take two covers an substitute the elements of one by the element of the other *)
Inductive subst_pred :
list (btree A) -> list (btree A) -> btree A -> btree A -> Prop :=
| subst_pred_id :
forall (t1 t2 : btree A) (l1 l2 : list (btree A)),
subst_pred (t1 :: nil) (t2 :: nil) t1 t2
| subst_pred_node :
forall (t1 t2 t3 t4 : btree A) (l1 l2 l3 l4 l5 l6 : list (btree A)),
subst_pred l1 l2 t1 t2 ->
subst_pred l3 l4 t3 t4 ->
subst_pred (l1 ++ l3) (l2 ++ l4) (node t1 t3) (node t2 t4).
Hint Resolve subst_pred_id subst_pred_node : core.

(* The first cover of the substitution is an ordered cover *)
Theorem subst_pred_ordered_cover_l :
forall (t1 t2 : btree A) (l1 l2 : list (btree A)),
subst_pred l1 l2 t1 t2 -> ordered_cover l1 t1.