Huffman.SameSumLeaves
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(* GNU Lesser General Public License for more details. *)
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(* 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: SameSumLeaves.v
Definition of the equality of sum leaves
Definition: same_sum_leaves
Initial author: Laurent.Thery@inria.fr (2003)
From Huffman Require Export Cover.
From Huffman Require Export WeightTree.
Section SameSumLeaves.
Variable A : Type.
Variable f : A -> nat.
(* the sum leaves are the same upto permutation *)
Definition same_sum_leaves (l1 l2 : list (btree A)) : Prop :=
exists l3 : list (btree A),
(exists l4 : list (btree A),
permutation l1 l3 /\
permutation l2 l4 /\ map (sum_leaves f) l3 = map (sum_leaves f) l4).
(* if the sum leaves are the same, the list are of same length *)
Theorem same_sum_leaves_length :
forall l1 l2 : list (btree A),
same_sum_leaves l1 l2 -> length l1 = length l2.
Proof using.
intros l1 l2 (l3, (l4, (H0, (H1, H2)))).
rewrite (permutation_length _ _ _ H0).
rewrite (permutation_length _ _ _ H1).
repeat rewrite <- (map_length (sum_leaves f)); auto.
apply f_equal with (f := length (A:=nat)); auto.
Qed.
End SameSumLeaves.
Arguments same_sum_leaves [A].