Proof with auto with real.
intros.
rewrite H.
rewrite monoid_expec_bind_leaf_plus.
unfold monoid_expec.
rewrite expec_Leaf.
reflexivity.
Qed.
Hint Resolve ne_tree.In_head.

Lemma monoid_expec_plus (A B: Set)
(f: MonoidMonadTrans.M m ne_tree_monad.ext A)
(g: A -> MonoidMonadTrans.M m ne_tree_monad.ext B) :
forall (gc: forall x y, ne_tree.In x f -> ne_tree.In y f ->
monoid_expec (g (snd x)) = monoid_expec (g (snd y))),
monoid_expec (f >>= g) = monoid_expec f + monoid_expec (g (snd (ne_tree.head f))).