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(* 02110-1301 USA                                                     *)
(********************************************************************** Proof of Huffman algorithm: Prod2List.v  Definition of a product between an integer list and a list of trees and its properties  Definition: prod2list  Laurent.Thery@inria.fr (2003) **********************************************************************)
From Huffman Require Export WeightTree.
From Huffman Require Export Ordered.
Section Prod2List.
Variable A : Type.
Variable f : A -> nat.
(* In the product the sum of the leaves is multiplied by the integer and added to the weight of the tree *)
Definition prod2list l1 l2 :=
fold_left plus
(map2 (fun a b => a * sum_leaves f b + weight_tree f b) l1 l2) 0.

(* Specifications: *)
(* The product of the appended list is the sum of the product *)
Theorem prod2list_app :
forall l1 l2 l3 l4,
length l1 = length l2 ->
prod2list (l1 ++ l3) (l2 ++ l4) = prod2list l1 l2 + prod2list l3 l4.