(* 
 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_le_l : forall a b c d l1 l2 l3 l4 l5 l6, length l1 = length l4 -> length l2 = length l5 -> length l3 = length l6 -> sum_leaves f c <= sum_leaves f d -> a <= b -> prod2list (l1 ++ a :: l2 ++ b :: l3) (l4 ++ d :: l5 ++ c :: l6) <= prod2list (l1 ++ a :: l2 ++ b :: l3) (l4 ++ c :: l5 ++ d :: l6).