intros T mon_T t suc_t.
apply is_monotone_intro.
intros t0 suc_t0 i Pa t1 suc_1.
inversion_clear mon_T.
apply H with t0.
apply succs_trans with t; assumption.
assumption.
assumption.
Qed.

Lemma is_monotone_tree_is_monotone :
forall t : Tree, Is_Monotone_Tree t -> Is_Monotone t.