Definition map_fst (A B C: Set) (f: A -> B) (p: A * C): B * C := (f (fst p), snd p).

Lemma monoid_expec_Node_map (A B: Set) (f: A -> MonoidMonadTrans.M m ne_tree_monad.ext B) (l: ne_list.L A): monoid_expec (ne_tree.Node (ne_list.map f l)) = monoid_expec_sum f l / INR (length l).