(* 
 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.

Theorem prod2list_app : forall l1 l2 l3 l4, length l1 = length l2 -> prod2list (l1 ++ l3) (l2 ++ l4) = prod2list l1 l2 + prod2list l3 l4.