intros P step.
cut (forall (n : nat) (t : Tree), height_tree t <= n -> P t).
intro claim.
intro t.
apply claim with (height_tree t).
trivial.
intros n; elim n; clear n.
intros t; elim t; clear t.
intros a succs u0.
elimtype False.
inversion_clear u0.
intros n ih t.
elim t; clear t.
intros a succs u0.
apply step; clear step.
intros t in_t.
apply ih; clear ih P.
apply le_S_n.
apply le_trans with (S (height_forest succs)).
apply le_n_S.
apply height_in_le; assumption.
assumption.
Qed.

(* Specifications: *)
Lemma My_Tree_rec :
forall P : Tree -> Set,
(forall (a : A) (succs : Forest),
(forall t : Tree, In_Forest t succs -> P t) -> P (node a succs)) ->
forall t : Tree, P t.